ba^ill^irg£:
TT»»»» t <»TT»T>»
.THE
STEREOMETRICON.
XKW srSTKM OF MEASURING
ALL liODILS
BY ONE AND THE SAME RULE.
Gi:m:ral application ofthi' prismoidal formula.
NOMEXCi.A rriiK .\x:)(;i;xEiL\r, feaii-ues of each of jiie
200 MODELS OX THP: BOARD.
THF, ai;e\s of sPin-:ri[t:.vL xitrvNiiLEs and i'olygons to any uadius :
OR WAMKTEU,
T A B L I<: S
of the Areas of Circles^ Segments^ Zoii's — sec index ^ table of S'^ccific gravities. •
QUEBEC
PRINTED BY G. DAUVEAU
188i
B2iXluXj^L I B O"
Neyjv system of determining the solid contents of a I
{Extract from the " Quebec Daily Mi
Mr. Baillairg6's lecture on Wednesday evening last
before the Literary and Historical Society of Quebec, pro-
ved once more how very interesting, even in a popular
sense, an otherurise dry and alistruse subject, may be-
come, wlien ably handled.
The lecturer showed the relationship of geometry to all
the industries of life. He traced its origin from remote
antiquity, its gradual developemeut uu to the preseut
time. He showed how it is the basis of all our public
works, and how we are indebted to it for all the construc-
tive arts ; its relationship to mechanics, hydraulics, jntics,
and all the physical sciences. The fairer portion of man-
kind, said Mr. B., have the keenest, most appreciative
perception of its advantages and beauties, as evidenced in
the ever- varying combinations so cunningly devised iu
their designs for needle tracery, laces and embroidery. He
showed its relationship to chemistry iu crystallization and
polarization ; to botany and zoology iu the laws of
morphology ; to theology, and so on. Iu treating of the
circle and other conic sections, he drew quite a poetical
comparisou between the engineer who traces out his curves
among the woods an«l waters of the earth, and the astro-
nomer who sweeps out his mi^rhty circuits amidst the
starry forests of the heavens. The parabola was fully
illustrated in its applit-ation to the throwing of projectiles
of war, also as evidenced in jets of water, the speaking
trumpet, the mirror and the reflector, which , in light-
houses, gathers the rays of light, as it were, into a bundle,
and sends them forth together on their errand of humanity.
In treating <.f the ellipse, this alm«»st magic curve which
is traced out in the heavens by every planet that revolves
about the sun, by every satellite about its primary, he
alluded to that most beautiful of all ovals — the face of
lo«'ely woman. He showed how the re-appearance of a
comet may now be predicted even to the very day it
heaves in sights, and though it has been absent for a ceu-
tury, and how in former ages, when these phenomeua
were unpredicted, they burst upon the world in unex-
pected moments, carrying terror everywhere and giving
rise to the utmost anxiety and consternation, as if the end
of all things were at hand ; in a word, Mr. Baillairg^
we^t over the whole field of geometry and mensuration,
both plane and spherical ; a difficult feat within the limits
of a single lecture ; and kept the audience, so to say, en-
tranced with interest for two whole hours, which the pre-
sident, Dr. Anderson, remarked : were to him as but one ;
and no doubt it must have been so to others, since Mr.
Wilkie, in seconding the vote of thanks proposed by Capt.
Ashe, alluded to the pleasure with which he had listened
to the lecture as if, he said, it were like poetry to him,
instead of the unpromising matter foreshadowed in the
title. Mr. Baillairg^ next explained in detail his stereo-
metrical tableau, which we hope to see soon introduced
into all the schools of this Dominion. He showed how con-
ducive it will be in abortening the time heretofore devoted
to the study of solids and even to
superficies, sphericul trigouometrj
perspective, drawing the developei
and iihadovvs, and the like. Mr.
tunity had been afforded him of (i
corroborated Mr. B.'s statement ii
savine: in time, where many abstr
ncrally require*! hours or days to s«
be, as Mr. Baillairge asserts, so g(
as has been certified by so many
over their «)wn signatures,) with t
mula and tableau, be performed ii
say nothing of the use the model
glance a knowledsre of their nome
an acquaintaueeship with their va
He showed how, tu the architect ji
and mechanic, the uiodels are sug
relative proportions of buililinga,
quays, cisterns and reservoirs, eai
and other vessels of capacity, cm
comprising railroad and other cuti
the shaft of the Greek and Ron
winey timber, savv-lo 's, the ciin
splayed opening of a door or wind
a wail, the vault or arched ceiling
billiard or the cannon ball, or. on
earth, sun and planets. Mr. Bail
received an «>rder for a tableau frc
cation of New-Brimswick, with t
iiw.o all the solioitls of that Froviu
writing tO' Mr. Baillairge, from
January last, to advise him of thi
patent for that coimtry, says that \
the President <iud secretary of th(
lization of education in France, Ii
teution, at their next general meet
of distinction conferred on him f
' invention and discovery are likelj
Mr. Giard, in writing to Mr. BhII
lion. Mr. Chauveau, Minister of I
" II se fera un devoir d'en recom
" toutes les maistms d'education c
From the Seminary and Laval U
writes : " Plus on 4tudie, plus on
" du cubage des corps, plus on
" one marvels) de sa simplicity, d
" sa grande gendralite." Rev.
" shall be delighted to see the ol
" superseded by a formula so simj
ton, or Tale College, United Stat
" bleau a most useful arrangemc
" riety and extent of the applicati
College I'Assomption '' will ado(
*^ tern as part of their course of 1
has written to the author that "
intents of a body of any shape, by one and the same rule.
hec Daily Mercury'' of 2,0th March, 1872.)
ilids and even to that of phiue and convex
iciil trigouometry. ppoinelric^al projection,
'ing the developement of surfaces, shades,
1 the like. Mr. Wilkie, so far us oppor-
ifforded him of proving the calculations,
B.'s statoineut in relation to the immense
'here many abstruse probleuis wliich ge-
lours or days to solve, can now (if the rule
irge asserts, so generally applicahlo, and,
&ed by so many persons in testimonials
^natures,) with the help of the new for-
I, be performed in as many mitmtes ; to
le use the models are iu imparting at a
<re of their noMiouclaturo or names, and
liip with their varied shapes ami figures,
to the architect and engineer, the boihler
e :nodels are suggestive of the*foniis and
ms of buildings, roofs, (lomes, piers and
lid reservoirs, cauldrons, vats, casks, tubs
s of capacity, earthworks of all kinds,
ad and other cuttings and enibankmeuts,
Greek and Roman column, square and
iW-lo's, tlie cimpin<; tent, the square i*r
of a door or window, uich or loophole in
IU' arched ceiling of a church or ball, the
iinon ball, or. on a larger scale, theuuxui,
anets. Mr. Baillajr^^e, we may add, has
for a tableau from the Minister of Edu-
runswick, with the view of introdiu-ing it
>ls of that Province ; .md Mr. V.-muier, in
^aillairge, from France, on the 10th of
advise him of the granting of his leiters
luntry, says that Messrs. Humbert & Noe,
i secretary of the society for the eenera-
,ion in France, have intimated their in-
lexteeneral meeting of having some mark
ferred on him for the benefit which his
covery are likely to confer on education,
ting to Mr. Bnillairge, on the part of the
9au, Minister of Public Instruction, says:
ivoir d^eu recommander I'adopticm dans
ons d'education et dans toutes les ecules.''
iry and Laval University. M-. Maingui
[1 ^tndie, plus on approftuidit cette formule
corps, plus on est enchante (the more
e sa simplicity, de sa clarte et surtout de
i^ralite." Rev. Mr. MuQuarries, B. A.
ted to see the old and tedious processes
a formula so simple and so exact." New-
ege. United States : " considers the ta-
seful arrangement for showing the va-
t oftbe applications of the formula." The
•tion " will adopt Mr. Ba'llairge's sys-
their coarse of instruction." Mr. Wilkie
B author that '* the rule is precise and
** simple, and will greatly shorten the processes of calcu-
lation. " Tlie tableau,'* siys thi« comnetent judare, " com-
** prising as it does a great variety of elementary models,
" will serve admirably to educate the eye, and must great-
" ly facilitate the study of solid mensuration." " Again,"
Pays Mr. Wilkie. " the Governujent would confer a boon
" on schools of the middle and higher class by affording
" access to so snggesfive a collection." There are others
who, irrespective of considerations as to the comparative
accuracy of the formula, or of its advantages, as applied to
mere mensuration, are awake to the fact that the models
are so much more suggestive to the pupil and the teacher
than their me e ret>resentatioH on a blackboard or on paper,
and who, in their written opini(m, have alluded especially
to this feature of the prof)osed system. M. .Toly President
oftbe Quebec Branch of the Montreal School of Arts and
Design, in a letter on the subjects to Mr. Wearer, the Pre-
sident of the Board, ami after having himself witnessed
its advantages on more than one occasion, says, in his
expressive style, " the difference is enormous." Professor
Tousaint, of the Normal School, Dufresne, of the Mont-
nuigny Academy, Boivin, of St. Hyacinthe, and many
others, are of ihe same opinion ; among them MM. R. S.
M. Bouchette, O'Farrell, Fletcher. St. Aubin, Ste<'-kel,
Juneau, Venner. Gallagher, Lafrance. and the late Brother
Anthony, &c., &c. Neither will it be forgotten that the
professors oftbe Laval University, after reading the enun-
ciation of Mr. B.'a formula, as given in his treatise of 1 866,
•'Xpresseil Themselves thus : " Un douteinvolontaire s'em-
p-ire d'abord de I'esprit, lorsqu'on lit le No. l.'>2l ; " mais
" un examen attentif dcs paragraphes suivants, dissipe
*' bientot ce doute et I'ou reste etonne ^ la vue d'une for-
" mule, si claire, siaisee a retenir et dontl'applicati<m est
" si g6nerale." Mr Fletcher, of the Crown Lan<ls De-
partment, says : " I have compared, iu the case of seve-
" ral solids, the results obtained by your mode of compu-
** tati<m with those resulting from the ordinary and more
" lengthy processes, and congratulate you sincerely on
•' your enuiiciation of a formula so brief and simple in ita
" character, and so precise and satisfactory in its results."
Mr. Baillairge also took occasion during his lecture to
allude, in other relations, to his treatise on geometry and
mensuration, in which he showed he has introduced many
important modifications in the usual mode of treating the
subject of plane and spherical geometry and trigonometry.
In conclusion, we must add that the Council of Public
Instruction, at its last meeting, appointed a Committee,
composed of the Lord Bishop of Quebec, and of Bi8hoT>s
Langevin and Labrecque, to report to the Council at its
n"xt general meeting in June, and who, it may be taken
for granted, after the many flattering testimonials in re-
lation to the ntir "^ and many advantages of the stereo-
metrical tableau .jr purposes of education, cannot but
recommend and direct its adoption iu all the schools of
the Dominiuu.
BAILLAIRGE'S ST
Honorary Member of the Society for the Genef
New system of measuring all bodies, segments, frustums i
Thirtee7i Medals of honor and Seventeen Diplomas from France, Italy,
United- Slates of Ameri
This is a Case 5 feet long, 3 feet wide and 5 inches deep, with a h
exhibiting- and affording free access to soni- 200 well-finished Hardwo
form, each of which being merely attached to the board, by means of v
Student or Professor.
Th" nse of tlio TivWoan
an«l Mccoinpiiiiyiiiij Tre.itise,
reduces the wliole scioiicis
suid art of Meiisinatiou from
tlie stiiily of a year to tluvt
of a day or two, and so sim-
plifies the study and teacliinuj
ofSidid GeOTiietry, the Xo-
iiieiichitnre of Geometrical
and other forms, the (.evelope-
meht of surfaces, iriometri-
c;il projection and perspect-
ive, plane and «'.nrved areas.
Spherical Geometry and
Trigonojnetry, and the men-
suration of surfaces and
solids, that tlie several
branches hereinbefore men-
tioned may now be taught
even in the most, elementary
schools, and iu convents,
where such study could not
even have been dreamed of
heretofore.
Each tableau is accompa-
nied by a Treatise explana-
tory of the mode of meas-
urement by the " Prismoidal
Eormula, and an exi)lana-
tion of the S(did, its nature,
shape, opposite bases, and
middle section, its lateral
surface developed, etc.
Agents wanted for the sale
of the Stereometricon in Ca-
nada, the United States, <&^^
Pour tvouverle volume il'
iiauce, ii. !s. Jw.O.,eic.
Pali^les, »^onter quatrc
iois la smtace an f "t> '
sixieiue vaitie »ie la li'i*!-
remVlouLnienrau.o.ule.
For the use of Architects, Engineers, Surveyors, Students and App
Mathematics, Universities, Colleges, Seminaries, Convents and other Ec
Measurers, Gaugers, Ship-builders, Contractors, Artisans and others in <
STEREOMETRiCON.
)RTnii: Generalization of Education in France, etc., etc.
ts, frustums and ungulas of these bodies, by one and the same rule.
^rance, Italy, Belgium, Russia, Canada, Japan, etc. {Patented in Canada, in the
ites of America, and in Eiuope.)
leep, with a hinged Glass Cover, under Lockand Key, so as to exclude dust while
hed Hardwood Models of every conceivable Elementary, Geometrical or other
y means of wire, peg or nail, can be removed and replaced at pleasure, by the
STATES and \\\ i^u l^ui rj.
ATS-UNIS
^t6 pour la
tiou eu
France, i? ■ ^4- ^- ^^
To Hiul tho 8"lul content of
any body.
■RULE : To tho sum of tha
paialel and areas, add
four tunes tho middle area
au.l multiply tho whole
by one siKth part ot the
heiffht or length ot the
body.
4 ^
Approved by tho Council o^
Public Instruction of the Province
of Quebec, and already adopted
and ordered by many Establish-
ments in Canada and elsewhere.
For information and testimonials
apply (free of charge) to
C. CAILLAIRGfi,
QIJKBEC,
CANADA.
Honorary ^femher of the Society for
the Generalization of Education
in France, etc., etc.
SUBSCRIBERS.
The Archbishop of Quebec, tho
Bishop of Kimouski, the Bishnp of
Kingston, the Bishop of St. Hya-
cintbc, the Dominion Board of
Works, the Schools of Art and De-
sign, the r.aval University, the
Seminary. Q., the Colleges at; Otta-
wa, Nicolet. Kimouski, Montma-
gny. St. Michel, etc.. i'Kcolo 'Sot-
male Laval. IfS Ecoles des Fr^res,
the Commercial Academy, the
Board of Land Surveyors, the De-
partemeiit of Education. New
Brunswick, the corporation of
Quebec, It. Hamilton, Ksq., F. N.
Martin, and (;. Koy, Civil Engi-
neers, etc. La Soci6t6 pour la vul-
garisation de I'Enseigiiement da
Peuple, France, F. Peachy, J. Le-
page, etc., Architects, 2f. Piton,
T. Maguire. J. Marcotte. builders,
the Council of Public Instruction,
Q.. the Jacques Carrier Normal
School. M. Piton Manitoba, the
('ollegps of Aylmer, L' Assomption,
Ste Anne de fa Pocaliere, St. Hya-
cinthe, the High School, Q.. the
Morin College. Q., the Lafrance
Academy, Q., Government Boards
of "Works. Q , the Ursulines Con- .
vent, the Convent of the Good
Shepherd. Grev Nuns, Soeurs de
J6sus-Marie, Q.. and M. S. W.
Townsend, Hamilton, Stc, &c., &c.
Etc., Etc., Etc.
mts and Apprentices, Customs and Excise Officers, Professors of Geometry and
and other Educational Establishments, Schools of Art and Design, Mechanics,
id others in Canada and elsewhere. i
BAILLAIRG
.■ Honorary Member OF TH
New system of measuring all be
Thirteen Medals of honor and Seventeen Diplfi
schools, and iu convents,
wherfi such study conM not
<sven have been dreamed of
heretofore.
Each tableau is accompa-
nied by a Treatise explana-
tory of the mode of meas-
urement by the '* Prisuioidal
Eonnula, and an explana-
tion of the s<did, its nature,
shape, opposite bases, and
middle section, its lateral
surface developed, etc.
Agents wanted for the sale
of the Stereometricon in Ca-
nada, the United States, d^c.
e lies
Pour trouver le volume il'un
corps quelcouque.
lKt?GLE:A lasom.neu-
PaUfeles, ajouter qua re
lois la «uTtace au ceutie
etmultiplierletoutV.-
sixienie paitie "e l.v bau
teuvouloimueurdusoiule.
LE STf°
ncil o^
Tovince
Mlopted
tablish-
ewhere.
moniala
This is a Case 5 feet long, 3 feet wide aivhile
exhibiting and affording free access to somti 2«ther
form, each of which being merely attached to t the
Student or Professor.
Th'» use of th(> 'I'abloaii
and accompanyins; Treatise,
reduces tlu; whole science
and art of Mensuration from
the study of a year to that
of a day or two, and so sim-
plifies the study and teachinuf
ofS(did Geometry, the No-
menclature of Geometrical
and .)ther forms, the <levelope-
meut of surfaces, geometri-
cal projection and perspect-
ive, plane and curved areas,
Spherical Geometry and
Trig(»nometry, and the men-
suration of surfaces and
solids, that tlie several
branches hereinbefore men-
tioned may now be taught
even iu the most elementary
Brevet6 am
- u
lil
Mftinl)re
VuljZNADA.
1
cietyfor
iucation
6 1^
bee, tho
ishop of
t. Hya-
ard of
and De-
Jty. the
r^atOtta-
rfontina-
^^>1b Nor-
Fr^res,
^^tfy. the
-*^theDe-
\ New
itiov. of
_ .. F. Tf.
lent da
', J. Le-
Piton,
KJuildeis,
"'iruction,
"Normal
jba, the
ption,
t. Hya-
]Q.. the
lafrance
^Pl Boards
%|es ('on-
le Good
lenrs de
^^^ Ai S. W.
=1-
For the use of Architects, Engineers, Survj and
Mathematics, Universities, Colleges, Seminarieinics,
Measurers, Gaugers, Ship- builders. Contractors
THE -y
STEREOMETRICON.
Originator : C. BAILLAIRGfi, M. S.
Hkmbkk of thk Sociktt fok thk GKXKiiAr.iZATiOM OK Educatio!! im Fraxcr, ahd
OF eKVKKAI. LkaUNKI) AND SCIKSIIFIC SoCIKirKH; CHKVAI.IKR OF THB
OuuEB OF St. Sauvkur i)k. Moxtk-Rkai.k, Italy ; ktc, kto.
FkLI,0W OF THK RoYAL SOCIKTY, CANADA.
I
MEASUREMENT OF ALL SOLIDS BY ONE AND THK 8AMK BULK.
UNIVERSAL APPLICATION OF THE PBISMOIDAL FORMULA.
THIKTKRN MeDAI.8 and BKVKKTKEN DlPr.OMAS AND LKrrKHB AWARDKD THK AUTHOU
FUou RtsaiA, Fhanck, Italy, Bklgiuh, Japan, ktc.
PROMOTER : THOMAS WHITTY,
PROFESSOR AT ST. DENIS ACADEMY, MONTREAL.
Comprises 200 Solids rppresentative of all conceivaMe elementary funiis, as of
the Cumpoueut parts uf Compound bodies.
Name aud descriptioti of each solid. What it is represautative or »aggegti7e of,
ur that of vvliich it forms a compoueut part.
Nature and name of opposite bases and of middle section, as of lateral faces
aud remainder of bounding Area, including every species of Piaae,
Spherical, Spheroidal, and Conoidal figures.
Division I, classes Ito X : pl-tvie faceti So'.ids and Solids of single curvature.
Division II, classes XI to XX ; Sidids of oouble curvature.
QUEBEC
PRINTED BY C. DARVEAU
' r .1 ' / * {
., ^.
J -y-
I ".a
\ '
IISTDEX
Tlie SforooTTiptrlcon : noTnpnclatnro and gnneral feature of each of the
200 solids on the board ; see tho diiigr.ira at the beginning of thU
pnniphli't , ' 5
Tlio AvoMs of Spherical Tri:nigles & Poly^joiis to .my radius or dia-
lucttT : a iiaper read before the lloyal Socit'ty of Canada in 1833. 55
On the general applieatlDn of tlie pii.^m )i'.lal formula : a paper read be-
fore the Royal Society of Canada in lQi2 61
TABLES
I. Siinarcs and Square Roots of numbers from 1 to IGOO 4
II. Circumferences and areas of circles of diameter ^'j to 150, advan-
ciujr by ^ 11
III. Circumferences and areas of circles of diameter ^'^^ to lOJ, advan-
cing ^'y iff ' -- 19
IV. Circumferences and areas of circles of diameter 1 to 50 feet, ad-
vancing by 1 inch or ^^ 25
V. Sides of Squares equal in area to a circle of diameter 1 to 100 ad-
vancing by i 29
VI. Lengths of circular arcs to diameter 1 diviled into 1000 equal
l)arts .- 31
VII Lengths of semi-elliptic arcs to transverse diameter 1 divided
into lOOO equal parts 33
VIII. Areas of tho segments of a circle to diameter I divided into
1000 equal parts 37
IX. Areas of the zanes of a circle to diameter 1 divided into 1000
equal parts - - 3D
X. Specific gravities or weights of bodies of all kinds : solid, fluid,
liquid and gazeous 22
.54
THE STEREOMETRICON
Orioinatob : C. BAILLAIEGfi, M .S.
.\ \i^'.
Member of the Society for the Greaeralisation of Education in France and of several learned
and scientific Societies : Chevalier af the Orderof St. Sauveur de Monte. Realo, Italy ;
Fellow of the Royal Society of Canada, etc., etc.. etc.
Measurement of all solids by one and the same rule.
Universal application of the prismoidal formula.
TLirteen Medals and seventeen Diplomas and letters awarded the author,
from Franc:;, Russia, Italy, Belgium, Japan, etc.
Pbomotbr: THOMAS WHITTY, professor at St. Denis Academy, Montreal, etc.
RULE : To the sum of the opposite and parallel end areas, add
four times the area of a section midway between and parallel to the
opposite bases ; multiply the whole by ^ part of the length or height
or diamett, of the solid, perpendicular to the bases ; the result will be
tfie solidity or volume, the capacity or contents of the body, figure or
vessel under consideration. ^S . ; ^
For application of the rule and examples of all kinds fully worked
out, see " Key to Stereometricon."
For areas of all kinds, plane, and of single and double curvature,
see also "Key to Stereometricon," with tables of areas of circles to
eighths, tenths and twelfths of an inch, or of any other unit of measure,
tables of segments and zones of a circle, etc., etc., at end of " Key."
— 4 —
The tabfean comprises 200 models, disposed in 10 horizontal and
20 vertical rows, series, families or classes. The solids may be indif-
ferently placed, and numbered from the right or left and from below
upwards or the contrary.
The solids are representative of all conceivable elementary forms and
figures, as of the component parts of all compound bodies.
DIVISION I.
Plane faced solids and solids of single curvature, or of which the
surfaces are capable of being developed in a plane.
CLASS I.
Prisms.
NoTK.— The author uses the term "trapeziu n" and not "trapezoid," as the termination
"oid" conveys the idea of a solid as paraboloid, hyperbol)id, conoid, prismoid, etc.
For the same reason he uses the French " trapeziform " instead of trapezoidal.
Name of solid, object of which it Nature and name of opposite bases
is representative or suggestive, or and middle section, lateral faces
of which it forms a component part, and remainder of bounding surface.
Reference to " Key to Stereome- Reference to page or paragraph of
tricon," for computation of contents " Key " for calculation of areas and
and of factors necessary thereto. of factors necessary thereto. ^- ?
§:
X — The cube or hexaedron — Each of its three pairs of opposite
one of the five platonic bo- and parallel faces or of its six faces
dies or bases and middle sections, per-
Representative of any other rec- feet and equal squares. For de-
tangular prism, of a building or veloped surface. See "Key to Ster.,'
block of buildings or of one of the page 131.
component parts thereof ; a brick or Representative of the floor, ceiling^
cut stone, a pt^Jestal, a die or dado ; wall oi partitions of a rectancrnlar
a pier or r\ury '^ox, chest, pack- room or apartment, or of the bases
age of m "cha^idise or parcel ; a and sides of the various objects
cistern, bin, at or other vessel of mentioned under the name of the
capacity; a pile of bricks, stones, solid.
lumber, books, etc., etc., etc. See " Key to Ster.," page 60.
" Key to Ster.," p. 61, par. (78).
fi— Aright isosceles triangular Its opposite and parallel bases
prism and middle section, equal right-
On end, a triangular block or angled isosceles triangles. Its
building; on its base, a ridge roof; sides or lateral faces rectangles.
on one of its sides, the roof of a pent- For areas, see " Key to Ster.," pages
house or lean-to. "Key to Ster. p. 6 1 . 19, 22 and 60. Sides suggestive of
those of objects alluded to.
8 —A right regular pentagonal Its opposite and parallel bases and
prism. middle section, regular and equal
On end, the base or component pentagons ; sides or lateral faces,
part of the shaft of a pentagonal pier rectangles.
or column; on one of its sides, a Areas suggestive of those of ob-
baker's, butcher's or other van ; an jects mentioned in adjoining co-
ambulance, etc. "Key," page 61. lumu. "Key," pages 35 and 19.
4— A right regular octagonal Its parallel and opposite bases and
prism. section, regular and equal oota-
Base or shaft of a column, a pier gons ; its sides or lateral faces, reo-
or post, a bead, baluster, hand-rail, tangles. "Key," pages 36, 19.
etc. "Key to Ster., " page 61.
5 — Oblique hexagonal prism Its parallel bases and section,
An inclined post or strut or the symmetrical ai.d equal hexa-
section of a stair-rail, a baluster on gons ; its sides, parallelograms,
a rake, etc. Mitred section of a rail " Key," pp. 26j 19 and 63. compute
or bead. "Key to Ster,," page 64. half of sym. hex. as a trapezium.
— 6 —
6— Oblique rectangular prism. Two of its three pairs of opposite
On end, an inclined strut or post, and paralled faces or bases and
etc ; on its parallelogram base, the sections, equal rectangles ; the
pier of a skew bridge, portion of a other bases and section, equal pa-
. mitred fillet, etc. rallelograms. " Key," page 63,
See "Key to Ster.," page 64.
7— Oblique prism or parallelo-
pipedon.
Section of mitred fillet on an in-
clined or oblique surface, etc.
Each of its three pairs of parallel
faces or bases and sections, equal
parallelograms.
8— A righ rectangular trapezi-
form prism, or a prism of
-which the base or section is
a rectangular trapezium,
On end, a pier or block of that
shape ; on its larger parallel face or
base, the partially flat roof ot" a
pent-house or lean-to ; the base of a
rectangular stack of chimneys on a
sloped roof or gable, a corbel, etc.
See "Key to Ster.," page 61.
Its opposite and parallel bases
and section ; on end, equal rec-
tangular trap'-ziums; its lateral
fcices, rectangles ; on either of its
parallel sides or faces: its bases,
rectangles ; its lateral fiices, rec-
tangles and trapeziunas See
"Key to Ster.," j figes 60 and 29.
May be treated indifferently as
a prism or prismoid.
Q — A right trapeziform prism.
On end, the splayed opening of
a door or window or loop-hole in a
wall; on broader base, a partially
flat roof; on its lesser parallel base,
a bin or through or other vessel of
capacity, section of a ditch excava-
tion or of a railroad embaukmeut on
level ground, a scow or pontoon.
On end, its bases and section,
trapeziums, and sides, rectan-
gles ; on either of its parallel faces,
its bases and section, rectangles ;
its sides, rectangles and trape-
ziums.
N. B. Its solid contents, like
those of Nos. 2 and 8, may be com-
puted either as prisms or prismoids.
10 A right or oblique polygo- Eule for solid contents : multiply
nal ooxnpound prism, deoom- one-third the sum of the three vert-
- 7 — • .
posable into right or oblique ical edges or depths of each of the
triangular prisms or frusta of component triangular prisms, or
prisms frusta of triangular prisms by the
An excavation or filling, etc. area of a section perpendicular to
A spoil bank or a borrowing pit. sides or horizontal, and add the
Each frustum or component part results. Page 67, rule II, "Key."
may be treated as a prismoid, one
of its sides being the base.
CLASS ir.
Prisms, Prasta and Ungulae of Prisms. ^ •.
11 — A right regular trian3;ular Its parallel bases and section,
prism. equal equilateral triangles ; its
On end, a triangular building, faces, rectangles. Compute as
pier or block ; on one of its sides, the prismoid with rectangular bases,
gable of a wall, the roof of a gabled the upper base then being an arris
house, etc. or line. -.,
"Key to Ster.," page 61. t; -^ ^
12 — Lateral "wedge or ung^la One of its parallel bases a regu-
of a right hexagonal prism, lar hexagon ; its middle base a
by a plane through edge of half hexagon or trapezium ; its
base, upper base a line; its lateral faces
Portion of a mitred bead or hand- a line, a rectangle, triangles and
rail, end of stair baluster under trapeziums ; its sloped face a
hand-rail, ridge roof of an octagonal symmetrical hexagon or 2
tower against a wall; base of a trapeziums, base to base.
chimney stack on a sloped roof or
gable.
— 8 —
13 — Lateral ungula of a right One of its opposite and parallel
hexagonal prism, by a plane bases, a regular hexagon ; the
■ through opposite angles of other, a point ; its middle section
the solid. a half hexagon or t-wo rectan-
Bas'e of a chimney stack, vase or gular trapeziums base to base ;
ornament on a sloped roof or gable, its lateral faces, trapeziums and
etc. triangles ; its plane of section, a
N. B. — This solid and the last, symmetrical hexagon, which,
are not prismoids according to the for area, regard as two equal tra-
definition thereof, page 163, par. peziums base to base, compute and
(206), " Key to Ster. ; " but the up- add.
per half, folded over and applied to See " Key to Ster.," page 29.
the lower half, evidently completes Or the symmetrical hexagon may
the prism, and hence the solidity is be decomposed into a rectangle and
exactly obtained by the prismoidal two equal triangles, for computa-
formula, as it is of a like frustum of tion of area.
a cylinder or of an uugula thereof
by a plane through edge of base.
14~Gentral -wedge or ungula One of its parallel bases, a hexa-
of a right hexagonal prism j gon ; the other, a line ; its middle
a prismoid. section, a symmetrical hexagon
A wedge, the ridge roof of a or t"Wo trapeziums, base to
tower, the base of a chimney stack, base ; its lateral faces, triangles
vase or ornament between two and trapeziums.
gables. See " Key to Ster.," page 29.
15 — An oblique trapeziform Treated as a prismoid : its oppo-
prismL. site and parallel bases, unequal
The partially flat roof to a dormer rectangles ; its lateral faces, tra-
window, the roof of a building abut- peziums.
ting against another roof, the splay- The factors of its middle section
ed opening of a basement window, arithmetical means between those
mitred portion of a batten or moul- of its opposite and parallel bases,
ding, section of a ditch excavation,
or of an embankment on a slope.
— 9 —
16 — An oblique triangular Treated as a prisraoid : one of
prism. its opposite and parallel bases, a.
The roof of a dor ler window or rectangle ; the other, a line ; its
of a wing to a house with a sloped lateml fuces, equal triangles and
loof, a mitred moulding or fillet, etc. parallelograms.
17 — Frustum of a right trian-
gular prism.
Eidge roof of a building against
a wall, a mitred moulding, etc.
As a prismoid : one of its parallel
bases, a rectanglr^ ; its opposite
base, a line ; its middle section, a
rectangle.
18 — Irregular frustum of an
oblique triangular prism.
Eidge roof of a building of irre-
gular plan abutting on the unequal-
ly sloped roof of another building,
etc.
Considered as a prismoid : one
base, a trapezium, the other, a
line ; its middle section, a trape-
zium , its ends, non - parallel
triangl3S ; its sides, trapeziums.
19 — A right prism on a mixti-
linear base.
On end, the unsplayed opening
of a door or window in a wall, etc.
Note, for area of segment of cir-
cle or ellipse, " Key," pages 33, 44,
61, 53, 57, tables II, III, IV, VIIL
<:\'
Parallel bases and section mix-
tilinear figures, decomposable
into a rectangle and the segment
or half of a circle or ellispis ; the
lateral face, a continuous rect-
angle.
Note. — The segment of a circle
or ellipse may be equal to, less or
greater than a semi-circle.
20— Regular firustrum of an As a prismoid : one mse, a
oblique triangular prism. rectangle ; the other, a line ;
A ridge roo^ mitred fillet, etc the middle section, a rectangle.
— 10 —
. r
CLASS III. ' ■
Pmsta of Prisms, Prismoids, Wedges.
21 — T h e dodecahedron, or The six pairs of parallel bases or
t"welve-sided solid, one of the twelve component faces of the solid,
five platonic bodies. equal and regular pentagons ;
Assemblage of twelve equal py- the middle section a regular
ramids with pentagonal bases, their decagon, the side of which is
apices or summits meeting in the equal to half the diagonal of the
centre of the solid or of the cir- pentagon, tor area of which see
cumscribed sphere. " Key to Ster.," page 36, rule II ;
The capital or intermediate sec- or compute one of the component
tion of a pentagonal shaft or column, pyramids and multiply by twelve.
a finial or other ornament. For developed surface, see " Key to
Ster," page 132.
522 —A rectangular -wedge, the On end : its opposite and parallel
head or heel broader than bases, a rectangle and a line ; its
the blade or edge. middle base or section, a rectangle.
The frustum of a triangular prism, On one of either of its other two
or may be treated as a prismoid, pairs of parallel bases ; one base, a
using either of its three pairs of trapezium, the other, a line ; the
parallel bases. middle section a trapezium ; side
An inclined plane, a low pent faces, a rectangle and triangles.
roof, an ordinary wedge, etc.
23 -A rectangular "wedge or Each of its three pairs of parallel
inclined plane the head or bases, a rectangle and a line ; its
heel of equal breadth with middle sections, rectangles, res-
the edge or blade. pectively equal to half the corres-
A right triangular prism, body ponding bases. May also be treated
of a dormer window or base of a as a triangular prism, with bases
chimney stack on a low or steep and section equal triangles.
roof, etc.
— 11 —
24 —An isosceles "wedge, the
edge or l»Ude liroader than
the heel. ' -
May also be considered, the frug-
tum of a triangular prism or pris-
inoid with three pairs of parallel
bases.
As a prismoid : one of its pairs of
parallel bases, a rectangle and a
line ; middle section, a rectangle ;
each other pair of parallel bases, a
trapezium and a line ; middle sec-
tion, a trapezium.
25— Frustum of a right rec-
tangular trapeziform prism,
or a prismoid.
A roof, partially flat, abutting
against a vertical wall at one end
and in rear, against a sloped roof
at the other, etc.
As a prismoid : its opposite and
parallel bases, rectangles ; the
longer side of the one corresponding
to the shorter side of the other ; its
middle section, a rectangle ; all
its lateral faces, trapeziums.
26— Irregular firustum of an ob-
lique trapeziform prism.
A roof between two others not
parallel, irregular section of a ditch
or embankment
As a prismoid : its opposite and
parallel bases and middle section,
trapeziums ; its lateral faces, tra-
peziums.
Factors of middle section arith-
metic means between those of the
bases.
27 — Frustum of a right isos-
celes trapeziform prism, a
prismoid.
On its larger base, a roof, section
of an embankment, etc.; on its
lesser base, a bin or vessel of ca-
pacity ; the capital of a pilaster, a
corbel ; on end, a splayed opening
in a wall.
As a prismoid : its opposite and
parallel bases and middle section,
rectangles ; lateral faces, trape-
ziums.
In all such solids, the half way
fectors need never be measured, as
they are always means between the
parallel bases of the trapezium faces.
28 — Frustum of an isosceles As a prismoid: one of its opposite
triangular prism, a prismoid. and parallel bases, a rectangle ; the
Ridge roof with ends unequally other, a line ; its middle section, a
sloped, mitred moulding, etc. rectangle. " Key," page 19.
— 12 —
28 — Frustum of a trapezlforzn As a prismoid : its opposite paral-
prism, a prismoid. lei bases and middle section reo-
A flat roof, etc. ; on its lesser tangles ; its lateral faces, trape-
parallel base, a bin or reservoir, a ziums. Factors of intermediate sec-
vehicle of capacitj, a scow, a pon- tion or middle base, arithmetic
toon ; on end or its parallel faces means between those of the end
vertical, the splayed opening of a bases,
window. "Key to Ster.," page 29.
80 — A prismoid on a mixtili- Its opposite and parallel bases
r.sar base. and middle section, mixtilinear
The roof of a building, circular figures ; the one a rectangle and
at one end or coved celling of a a semi-oirole ; the other two, reo-
room ; on its lesser base, a bathing tangles and semi-ellipses ; its
tub, etc. ; vertically, the splayed arched end developed, a sort of tra-
opening of a circular headed window pezium with curved bases ; its area
in a wall. equal to half sum of bases by mean
breadth or height.
CLASS IV.
Prismoids, etc.
31 — The ioosahedron, or twen- The ten pairs of parallel bases or
ty-sided solid; one of the twenty component faces of the solid
five platonio bodies. are equal equilateral triangles.
An assemblage of twenty equal Its middle section, a regular do-
pyramids on triangular bases, their decagon. Its middle section pa-
apices or summits meeting in a rallel to two opposite apices or to
common point, the centre of the the bases of any two opposite pen-
solid or of the circumscribed or tagonal pyraTnids of the solid, a
inscribed sphere. regular decagon, whose side is
-13 —
A finial or other ornament, etc. equal to half that of one of the
More expeditious to treat it for edges of the solid. For developed
solidity by computing one of its surface, see " Key to Ster," p. 133.
component pyramids, and multiply-
ing the result by twenty.
32 — A prismoid, both its bases,
lines. Irregular triangular
pyramid.
Dormer or gablet abutting on a
sloped roof. Component section of
Its opposite bases — considering
the solid as a prismoid resting on
one of its parallel edges — lines ;
its middle section a rectangle.
See "Key to Ster.," page 164,
No. 79. " Key " p. 165, par. (212). par, (208).
33— A prismoid on a trapezi- One ofitsparalledbaseSjatrape-
form base. zium ; the other, a line ; its middle
A cutting or embankment, etc. section, a trapezium.
34 — A railroad prismoid on a Its end sections or bases and middle
side slope. parallel section equal quadrila-
Section of a railroad cutting or em- terals, for area of which see " Key
bankment on ground, sloping late- to Ster.," page 30.
rally or in one direction only. This prismoid is a prism on an
irregular base, and may be so
4.,;. : , treated.
35— A railroad prismoid on a
grade and side slope, or on
ground sloping both lateral-
ly and longitudinally.
Its narrow base upwards, an em-
bankment ; the same downwards, a
cutting or excavation.
Its opposite and parallel end bases
and middle section, quadrilaterals,
the factors of the middle section
being all arithmetic means between
those of the corresponding end
areas. ■'*■.':■•■■---•;-:'■ '^^
38— A square or rectangular Its end bases and middle section
prismoidal stick of timber, squares or rectangles.
A squared log, a tapering post. Timber is usually measured by
— 14 —
the shaft of a chimney or high
tower, a reducer between rectangu-
lar conduits of unequal size, etc.
Note. — 25 per cent, of the whole
or true content is 33J per cent., or
one-third of the erroneous result.
multiplying its middle section into
its length. This gives an erroneous
result ; the more tapering the timber
is, the more so. If it tapered to a
point the error would be 25 per
cent., or one-quarter of the whole
in defect.
37— A prismoidal stick of -wa-
ney timber
A log of waney timber ; on end,
the shaft of a chimney, a high tower,
a tapering post
Its opposite bases and middle
section, symmetrical octagons,
for area of which see ' Key," p. 176,
par. (272), or squares or rectangles
with chamfered corners or angles.
38— A concavo-convex pris-
moid or curved viredge.
A corbel, spandrel, finial, etc. ; a
brake, a cam, etc. " Key to Ster.,"
par. (141).
Its opposite bases, a rectangle
and a line ; its middle section, a rec-
tangle ; its developed faces, trape-
ziums ; sides, mixtilinear tri-
angle.
39— A recto-concave prismoid,
or frustun of a curved wedge.
A corbel, spandrel, buttress, etc.
May be decomposed, as also No. 38,
into two sections for more exact
computation of solid contents.
Its opposite and parallel bases and
middle section, rectangles ; its de-
veloped faces trapeziums ; its late-
ral faces mixtilinear trapeziums
For areas see "Key," page 57.
40— Frustum of a rectangular
trapeziform prism, a prismoid
A flat roof in a rectangular cor-
ner ; on its lesser base, an angular
corbel, a sink, cistern, bin, etc. ^.,^
As a prismoid, its opposite and
parallel bases and middle section,
rectangles ; its lateral faces, tra-
peziums. ;
" Key," page 104, par. (141),
■4*^«-;^
; «. :.,! -\. .
- f j "y} ??'-»-j^^-
— 15 —
CLASS V. , '' .
Frismoids, etc.
41 — The ootahedron or eight- Its four pairs of parallel bases or
sided figure ; one of the five eight component faces, equilateral
platonie bodies. triangles ; its middle section, a
Assemblage of eight equal p3n:a- regular hexagon ; its middle sec.
mids on triangular bases, their apices tion through opposite apices and
meeting in a common point, the perpendicular to intervening arris
centre of the solid ; or two quadran- or edge a lozenge; through four
gular pyramids, base to base. apices, a square. For developed
surface see " Key to Ster," page 132.
42 — A prisinoid,one of its bases Its opposite and parallel bases, a
a square,the other an octagon square and an ootagon ; the mid.
Base or capital of a column, roof die section, a symmetrical oota-
of a square tower, a tower, pier, gon ; its lateral faces, triangles and
vessel of capacity, component sec- trape2uums. For area of symme-
tion of a steeple, etc. trical octagon, see "Key," par. (272)
43 — A prismoid, its opposite One of its opposite and parallel
bases, a square and a eircle. bases a square ; the other, a oir-
Base or capital of a column, roof of ole ; the middle section, a mixti.
a square tower, a tower, pier, vessel linear figure or a square "with
of capacity, a lighthouse, a section roimded corners,
of steeple or belfry, a reducer be- Its lateral surface capable of de-
tween a square and circular conduit, velopment into a plane trapezi-
form figure, one bfise circular, the
other polygonal.
44 — A prismoid, its bases one- Its opposite bases unequal
qual squares set diagonally, squares set diargonally to each
Kepresentative of the same ob- other; the middle section, a sym-
jects as solids, Nos. 42 and 43. metrical oOtMgon ; its lateral faces>
triangles.
-■■;., :,-:,■■■-"' ■ .-■ -16-
45 — A prisxnoid its bases a hex- One of its bases, a hexagon ;
agon and a rectangle. ' other a rectangle ; its middle sec-
Eepresentative of nearly the tion a symmetrical octagon ; its
same objects as the three last solids, lateral faces, rectangles and tri-
angles.
46 — The lateral frustum of a Its parallel bases and section,
rectangular prolate spindle, squares ; its lateral surface, mix-
Roof of a square tower, compo- tilinear fig-ures capable of de' 1-
nent part of a steeple, etc. opment into plane surfaces, i'or
area of these see " Key," page 57.
47 — A prismoid, its bases, an Its middle section, a mixtili-
ellipsis and a square. near figure or approximate oval.
A reducer between an elliptic Its lateral surface developed, a
and square conduit, a roof, etc. curved trapezium, one base
curved, the other polygonal. See
"Key to Ster., " page 166.
48 — A prismoid, its bases a Its middle base, a symmetrical
symmetrical hexagon and a octagon ; its bteral surface, trian-
line. gles. For symmetrical hexagon,
Eidge roof, coping or finial to a area equal to double that of half
post, panel ornament, etc. the figure, which is a trapezium.
49— A prismoid, its bases, a Its middle section or base, a
symmetrical hexagon and a symmetrical decagon ; its lateral
lozenge faces, triangles. Area of hexagon,
Flat roof, ornament, etc. ; on its double that of component trapezium.
lesser base, a fancy basket, a disk,etc.
50— A groined ceiling or the Its base and middle section,
half of a rectangular oblate squares ; its opposite base, a point ;
spindle. its lateral flEtces, mixtilinear fi-
A roof, panel ornament, etc. For gures.
more exact computation of contents, For areas of mixtilinear figures
decompose into two parts. Bee " Key to Ster.," page 57.
— 17 —
CLASS VI.
*iif
P3n:aimds and Frusta of Pyramids.
61— The tetrahedoD, or four-
sided figure ; one of the five
platonic hodies. A regular
triangular pyramid.
Apex roof of a triangular building,
finial or other ornament, the com-
ponent element of the icosahedron
and octahedron.
Its base and middle section,
equilateral triangles, the lesser
equal in area to one-quarter the
greater, its upper or opposite base,
a point ; its faces, triangles. For
development of surface see " Key
to Ster.," page 131. For area of
bases and faces, see page 36, rule II.
52— A regular square or rec-
tangular pyramid.
"-•' The spire of a steeple, a pinnacle,
roof of square tower, a bin, a vessel
of capacity, a finial or other orna-
ment, etc.
One of its parallel bases, a
square ; the other, a point ; its
middle section, a square, of which
the area is one quarter that of the
base. Lateral faces, isosceles tri-
angles.
63 —A pyramid, t"WO of its faces
perpendicular to base. The
ungula of a rectangular
^ prism on either of its bases.
An apex roof, section of cutting
or embankment, component portion
of other solids, a roof saddle.
Its base and middle section, tri-
angles ; apex, a point. Factors of
middle section half those of the base*
Affords a demonstration of the
theorem that in right-angled spheri-
cal triangles the sines of the sides
are as the sines of the angles.
64 — Frustum of a right trian-
gular pyramid.
Eoof, base or capital of a post or
column, base of a table-lamp or
vase, a vessel of capacity, component
section of other solids.
Its parallel bases and middle
section similar triangles ; lateral
faces, trapeziums. Factors of
section arithmetic means between
those of bases.
— 18 —
65 — Frustum of an oblique Its bases and middle parallel
triangular pyramid. section, similar triangles ; lateral
Flat roof of triangular building faces, trapeziums; factors of
abutting against a sloped or battered section, arithmetic means between
wall ; portion of a ditch excavation, those of the bases. For areas see
component portion of other solids. " Key to Ster.," pages 19, 22 and 29.
£6 — Frustum of a right rectan- Its opposite bases and middle
gular pyramid. section, squares or rectangles
Flat roof to tower ; reducer bet- whose factors or sides are each
ween conduits of varied size, com- equal to half the sum of the corres-
ponent portion of an obelisk, capital ponding sides of the bases, or
or base of a post or column, a bin, arithmetic means between them,
vat or other vessel of capacity, the For areas see " Key to Ster.," pages
body of a lantern, etc., etc. 19 and 29.
57 — A regular octangular or Its base and middle section,
octagonal p3n:amid. similar octagons ; lesser area
Koof of a tower, spire of a steeple, one-quarter of the greater ; its
finial or other ornament, a funnel, upper base or opposite one, an apex
strainer or filter, etc. or a point ; lateral faces, isosceles
triangles.
^ ■_^.-; -*Vl ',t ^-, . , "■■-■'''3,'
58 — The firustuxn of a regular Its opposite and parallel bases
octagonal pyramid. and middle section, regular octa-
On its broader base, a roof, tower, gons ; factora of section means to
pier, quay, component part of a those of the bases ; its lateral faces,
steeple, etc.; base of a column, lamp trapeziums. For expeditious
or vase, etc.; on its lesser base, a mode of arriving at area of octagon,
vat, bin, vase, or other vessel of see "Key to Ster.," page 176 or
capacity ; the body of a lantern, page 26, rule II. Developed surface
etc, etc. ^^ •' a regular polygonal sector or tra-
^ A; k rc a-i2- pezium.
■:-:;■._ _19 —
59 — Irregular and oblique Its base, a quadrilateral or
pyramid on a quadrilateral irregular trapezium ; its sum-
base, mit or apex, a point. Middle sec-
Apex roof of an irregularly tion similar to base and equal in
shaped building against a battered area to one-quarter that of base.
wall or roof, a roof saddle, etc.
60 — Frustum of a pyramid When decomposed for computa.
■with non parallel bases. tion of solid contents : bases and
Decomposable into the frustum section of frustum, similar trian-
of a pyramid with parallel bases, gles ; bases and section of compo-
and an irregular pyramid, by a nent pyramid or upper portion^
plane parallel to the ba:>e and similar quadrilaterals. This
passing through the nearest corner pyramid has its base in one of the
or point of the upper, or non lateral faces of the solid,
parallel base.
CLASS VII.
Cylinder, Frusta and Ungulae. /^^
61 — A right cylinder or infini- Its parallel bases and middle
tary prism. section, equal circles ; its lateral
A tower or circular apartment ; a surface developed in a plane, a
bin, vat, tub, bucket, pail, vase, rectangle ; its height, that of the
drinking vessel, cauldron or other cylinder ; its length, the circum-
vessel of capacity ; a road or other ference of the solid,
roller : the cylinder of a steam or Foi areas of circles calculated to
other engine ; a gasometer, the barrel eighths, tenths and twelfths of unity,
of a pump, etc., etc., etc. see tables II., III., IV. at end of
: > . "KeytoSter. "
62 — Frustum of lateral ungnla Its base, a circle ; its opposite
or "wedge of a right cylinder, base, a semi -circle or other seg-
May represent a cylindrical win- ment ; its middle section, a seg-
— 20
dow or opening in a sloped roof ment greater than a semi-cir-
abutting to a vertical wall or sur- cle ; its plane of section the seg-
face, the liquid in a closed cylindri- ment of an ellipsis ; its cylindrical
cal vessel held obliquely, base to surface decomposable by lines pa-
chimney or vase partly on a hori- rallel to bases into trapeziums-
zontal, partly on a gabled wall. For areas of segments, see table
VIII., " Key," pages 53, 38, 44.
63 — A rectangular circular
ring ;
The difference between two con-
centric cylinders, or a solid aunulus.
Horizontal section of a tower
wall, cross section of a brick, iron
or other conduit, section of a boiler,
vat, tub, or other vessel of capacity,
etc., etc.
Its bases and parallel section,
concentric annuli : its interior
and exterior surfaces continuous
rectangles. The area of annulus
equal to the difference of the inner
and outer circles, or to the breadth
of annulus into half the sum of its
circumferences. See " Key," p. 39.
64 — Central ungula or -wedge
of a right cylinder.
Kidge roof oi a tower, a wedge,
loop hole in a wali component
portion of compound solid, a finial
or other ornament, a strainer, etc.
Its base, a circle ; its opposite
base, a line ; its middle section, the
zone or a circle ; its sloped faces,
each a semi-ellipsis. Its cylindri-
cal surface decomposable into tra-
peziums by arcs parallel to base.
See tables II., III. IV., IX., of
" Key to Ster.,* also pages 38, 46, 53.
65 — Frustum of central wedge
or ungula of oylinder No 64
Flat roof of tower or other buil-
ding, base or capiial of rectangular
pillar, vessel of capacity, component
portion of compound polid, base of
chimney stack or vase between two
gables.
Its greater base a circle ; its
lesser base, the central zone of a
circle ; its intermediate base, the
zone'of a circle : its lateral faces
equal segment of equal ellipses.
Its cylindrical surface decomposable
into trapeziums parallel to bases.
See "Key to Ster.," page 51.
— 21 —
66 — Lateral ungula of rij jht Its base, a semi-circle ; its inter-
cylinder or recto-cylindrical mediate base ur middle section pa-
■wedge. rallel to base also a segment > its
Lunette or arched headway of a opposite base, a point ; its plane of
door or window, etc., in a sloped section or sloped face, a semi-ellip-
roof, component of a compound sis. Its curved surface developed
solid, the liquid in an inclined cy- an approximate parabola, tra-
lindrical vessel, base of a salient peziums. etc. See " Key," pages
chimney shaft over a roof, etc., etc. 38, 44, 51, tables II., III., IV., VIII.
67 — Frustum of lateral wedge Its parallel bases and middle
or ungula of a right cylinder, section, segments of a circle, less
Lunette to arched opening in a than, more than, and equal to
sloped roof or ceiling abutting on a half ; sloped face, the excentrio
vertical wall or surface ; liquid in zone of an ellipsis ; cylindrical
an inclined closed cylindrical ves- surface, trapezium parallel to base,
sel ; base of engaged column against For areas of segment, see " Key,'
a battered wall, etc. page 44, rule I., rule II., table VIII.;
for zone of ellipsis,see p. 53, art. (62).
68 — Irregular ungula o*" wedge 1st base, the segment of a cir-
of right cylinder cle greater than half; its op-
Lunette to a partially circular posite base, a line ; its middle sec-
opening in an inclined ceiling, etc. tion, an eccentric zone of a cir-
Compoueut portion of a compound cle ; one of its side faces, the seg-
solid. For areas, see " Key to Ster.," ment of an ellipsis ; the other
pages 44, 46, 53, articles (61) and plane face, an eccentric zone of
(62), tables VIII. and IX. an ellipsis.
69 — Concavo-convex prismoid One of its bases, the lune of a
or cylindro-cylindrical solid circle greater than a senai-cir-
or concave frustuni of a cle ; the other the lune of a cir-
wedge or ungula of right cle less than a semi circle ; the
cylinder . middle section, a lune equal or
Deposit of sediment in a cylin- thereabouts to a semi-circle. Its
drical sewer, section of additional side surfaces, convex and concave
— 22 —
excavation or filling, or difference approximate trapezsiums. Tot
between two lunettes. areas of lunes, see " Key," page 47.
70— •Frustum of an oblique When decomposed, its bases and
cylinder. section ellipses ; the base of ungu-
May be decomposed into an la, an ellipsis equal to each of those
oblique cylinder and the an^ula of the inclined cylinder ; its middle
of one by a plane parallel to base, section half an ellipsis. For uii-
and passing through nearest point gulae, see Nos. 72, 73, 75.
of other base.
, CLASS VIII.
ObHiue Cylinder, Frusta, Ungulae, Cjlindroids, etc.
71 — Oblique cylinder or infini- Its parallel bases and section
tary prism eqnal ellipses ; its lateral surface
Mitred section of conduit, hand capable ofdevelopment into a plane
rail, moulding ; inclined column, niixtilineal figure. See " Key to
post, strut or brace, etc. ; inclined Ster.," fig. n. page 57. For area of
cylindrical opening in a wall, etc. ellipsis, see page 51 of same.
72 — Obtuse frustirai or ungula One of its opposite bases, an
of oblique cylinder. ellipsis of sligh eccentricity j
Oblique lunette inclined upwards its opposite base, a point ; its mid-
or arched headway to a circular or die section, a semi-ellipsis equal
elliptical opening in a sloped roof to half of base ; its plane of section
or ceiling. Component mitred por- or lateral face, an ellipsis of
tion of hand-rail, bead molding, etc. greater eccentricity ; its lateral
cylindrical face developed, a figure
} * ^:^^; -'i like m page 57 of " Key. "
73 — Acute frustum or ungula Same as No. 72. For developed
of oblique cylinder. cylindrical surface, see fig. h. page
Kepresentative of same as No. 67 of " Key to Stereometricon."
72, 1) t inclined downwards.
— 23 —
For area ot ellii^^iis, " K^y to
Ser." pages 51 and 53.
74 Concave un^u^a or fVus-
t ! m of oblique cylinder.
Representative of same as No.
73, but in arch roof or ceiling in-
stead of sloped roof.
Same as No. 73, with curved
instead cf plane section. Its cylin-
drical surface developed similar to
fig. h, page 57 of " Key ; " its cur-
ved or concave section developed
an oval or fig. like a, p. 57, " Key."
75_Frustum, ungula or wedge Same as No. 72. For developed
of right cylinder. cylindrical surface, see fig. g; for
Base of chimney shaft on sloped ellipsis, fig. b. p. 57, "Key."
roof, or same as No. 72 not inclined.
76 — A cylindroid ; its bases, a
circle and an elipsis ; infini-
tary prismoid,
Base or capital of elliiitic column,
reducer or connecting l^nk between
a circular and an ul-U-'tic conduit;
a tub, vat or other v.?ss-^l of capa-
city ; a hat with elliplic or oval head
and a circular crown, etc.
Its middle section, an ellipsis of
which the conjugate or lesser dia-
meter or axis is an arithmetic mean
between those of the opposite bases.
For area of circle, see table II, III,
IV, and of ellipses, p. 51, " Key."
Lateral surface develofied, a plane
trapeziform fig ; its greater base,
convex; lesser, concave; its area»
equal to periphery of middle section
into mean height.
77— Cylindroid or infinitary
prismoid ; its bases, an elip-
sis and a circle.
Same as No. 76, or frustum of a
conic metallic vessel, which has
become fiattened or battered at one
end.
Its lateral surface developes into
a plane trapeziform figure, with
greater periphery convex ; and les-
ser concave. Area equal to peri-
phery of middle section into mean
height.
— 24 —
7?^— Cyllndrold ; its bases ellip- Factors of middle section, arith-
ses at right angles to each metic means between those of the
other. bases. Lateral surface developed, a
Capital or base of elliptic column, plane trapeziform figure of
connecting link btitween conduits ; area equal to periphery of middle
metallic envelope or tube flatten- section into mean height, page 51
ed at ends in opposite directions. of " Key."
79_Cylindroid or prismoid ; Middle section, a mixtilineal
its bases an ellipsis and a figure with factors, arithmetic
line. means between those of bases. For
Kidgeroofto elliptical building or area of middle section, page 57 of
tower ; a hut, camping tent, a strai- " Key." Lateral surface developed,
ner of filter ; a finial or other orna- a plane trapeziform figure ; its
ment. base, convex ; its opposite base, an-
,, gular. Area equal circumference of
middle section mean height.
80— A compound solid; a cy- For cylinder, see No. 61, class
Under and a cone. VII; for cone, see No. 81, class IX.
A tower or other building, a hut, The developed surface of a right
tent, or camp with conical roof ; a cone is the sector of a cercle.
hay rick, canister, finial ; reversed : For area, see " Key," page 42.
a cauldron, cistern, tub, filter, etc. . i-- v,
:;';•-■"■; -"'"'■ CLASS IX. /. : . ^''- ■,,■'■; '■"-'-■■''
Right and inclined Cone, Frusta, Ungnlae, etc.
81— A right cone or infinitary Its base, a circle ; its opposite
pyramid. ;,;,,. base, a point ; its middle section
Eoof of tower, spire, finial or a circle equal in area to one
other ornament, pile of shot or shells, quarter that of the base. Its lateral
cornet, filter or strainer, funnel, etc. surface developed, the sector of a
circle. For area of circle, see tables
II, III, IV, « Key to Ster."
— 25 —
82— Fru-^trm of a right cone, Its opposite and parallel bases
considertd as a prismoid and middle section, circles; its la-
A tower, quay, })ier, base or ca- teral surface developed, the sector
pital of a column, tiat roof of tower, of a circular ring, or a curved
comf)oiient portion of a spire, a trapezium. The diameter of mid-
salting tub, etc. , reversed : a butter die section an arithmetic mean be-
iirkin, a tub or vat in a brewery or tween those of the opposite bases,
distillery, etc., a drinking goblet, For area of bases and section see
bucket, pail, dish, basket, lamp " Key to Ster.," page 38, for lateral
shade ; a vessel of capacity, the plug surface, page 43. Tables of areas of
of a stop cock, etc., etc. circles to eighths, tentiis & twelfths,
11,111, IV.
83 — Inclined or oblique cone. Its base and middle section, sim-
Loop hole in a wall, the liquid ilar ellipses — the latter equal in
or fluid substaiiice in a conical ves- area to one quarter the former ; the
sel inclined to the horizon ; a finial upper base, an apex or point ; la-
or ornament adapted to a raking teral surface developed an irregu-
cornice or pediment, etc. lar sector, which, for computation
of area, divide into triangles.
84 — Frustum of inclined cone. Its opposite and parallel bases
Unequally splayed cireular open- and middle section, similar ellip-
ing in a wall ; a coal scuttle : re- ses ; its lateral surface developed
ducer or connecting link between portion of an eccentric annulus,
two conduits of different diameters art. 39, page 33, of "Key to Ster.,"
laid eccentrically etc. Diameters of middle section, arith-
metic me£His between those of bases*
85 —Flat or Iotw cone. Its base, a circle ; opposite base
Roof to tower or circular con- or apex, a point ; its middle sec-
struction ; cover of a box, basket, tion, a circle equal in area to one
cauldron, etc, ; finial or other orna- quarter that of base ; its lateral face
ment ; a Chinese hat, a pile of shot developed in a plane, the sector
or shells, a sun shade ; reversed : a of a circle.
— 2G —
spinninj? top, bottom of cauldron or For area of circle, see tables II,
reservoir, a funnel, stainer or filter, ill, IV, of "Key to Ster. ;" for
etc. sector, see page 42 of same.
86 - Frustum of a lovr or
^ sui "based cone.
Flat roof to a pavillion, tower,
etc. ; a hat, the cover of a vessel
of capacity ; an uu finished or trun-
cated i)ile of shot or sh<dls ; a lamp
shade ; a finial or other ornament ;
the bottom, base, top or other
component section of a compound
solid, us of No. 100 ; reversed : a
dish, pan, saucer, cauldron, cistern,
Its opposite bases and paralled
middle section or intermediate base,
circles ; diameter of middle section,
an arithmetic mean between those
of the opjtosite bases ; the lateral
area developed in a plane, the sector
of a circular annulus.
For areas of circles, see tables II.,
III., IV. of " Key to Ster.," sector,
page 43 of same.
87 — Paraliolic conic ungula by
a plane parallel to side of
cone.
Lunette to a circular headed
opening in a wall and sloped
ceilHng ; liquid in a closed conic
Vessel inclined to the horizon.
N.B.-For ratio of chord of middle
section or segment to that of base,
see " Key to Ster.," ^age 143, where
it is shown that the squares of the
chords are proportional to the
abscissae.
The base, the segment of a
circld ; the opposite base, a point ;
the middle section, the segment of
a circle ; the plane of section a
parabola. For areas of segment,
see " Key to Ster.," page 44 and
table VIII. ; for area of parabola
page 54 of same. The lateral surface
developed an approximate sector
of a circle. The height or versed
sine of middle section segment is
hatf that of base.
88 Frustum of parabolic
conic ngula by a plane
parallel to base of cone.
S]>lay(^d oi)ening or embrasure to
a si'ij;iui' lit- shaped window or loop
The parallel bases and miudle
section, segments of a circle ;
the lateral plane face or figure, the
zone of a parabola, for area of
which see " Key to Ster,," page 55,.
hole in a wall ; lunette to opening art, (66) ; the developed conical
— 27 —
in sloped ceiling terminating in a surface, an approximate sector
vertical surface ; liquid in a closed of a circular annulus or, more
vessel in the shape of the frustum correctly, a trapezium "with
of a cone, No. 82, when inclined curved concentric or parallel
from the vertical. bases, for area of which see note
For chord of middle segment, page 29, " Key to Ster.," For area
measure solid or compute by page of segment, table VIII, and page
143 of " Ster." 44 of same.
89— Frustum of a right elon- Like No. 82, its opposite and
gated cone. parallel bases and middle section
Shaft of Grecian column, tapered circles ; diameter of middle section
post, high tower or chimney shaft, equal to the half sum of those of
funnel, pipe reducer, speaking the bases ; the developed lateral
trumpet or horn, plug of a stopcock surface, the sector of a concen-
or tap, deep drinking goblet, or trie annulus.
other vessel of capacity large or For areas of circles to eighths,
small, shaft of a gun, component tenths and twelfths, see tables II.,
portion of many compound solids, III., IV., of " Key to Ster. ; " for
etc. that of sector, page 43 of same.
80 — A compound solid, com- For nature and areas of bases
posed of or decomposable into and middle section of ; he component
the frustum of a right cone frustum or a cone and of its lateral
and the segment or half of a surface, see Nos. 82 and 89.
sphere or spheroid. For areas of bases and middle
May represent a piece of ord- section of hemisphere or hemisphe-
nance, a deep conical vessel with roid or of the segment of either,
hemi-spherical, hemi-spheroidal or greater or less than a hemisphere,
segmental bottom or top to it. see tables II., III., IV. in " Key to
For hemi-sphere, hemi-spheroid, Ster."
or segments thereof, greater or less For diameter of middle section
than half, see classes 18, 19, 20. in hemisphere or in segment
For diameter of middle section thereof, see " Baillairge Geometry,"
in segment of spheroid, see " Key par. 539 or "Key to Ster.," par. 154,
to Ster.," pages 139 and 140, where where oa = y Co . oD., and oD^=
— 28 —
AB : CD : : \/Ao . oB : o M and
CD : AB : : y VoTojT: o M., or,
the rectangle under the required
radius nnd either axis of the
spheroid is equal to that under the
square root of the rectangle or
product of the abscissas of the first
axis and the other axis.
diam. AB minus versed sine oC;
or, the square of the half cord equals
the rectangle under the versed sine
and remainder of the diameter ; or,
may be obtained directly by mea-
suring the solid.
r
CLASS X.
4>:
Conic Frusta and Ungulae, etc,
91 -Conic wel|;e or central
ungula of a cone by planes
dravrn from opposite edges
of the base to meet in the
axis of the cone.
Ptidge roof to a tower, splayed
opening or embrasure to a long
naiTow vertical loop hole in a \tall ;
component section of com])Ound
solid of a cone and cylinder or
of cones having their bases or
apices in opposite directions.
The base, a circle ; the parallel
upper base, an arris or line ; the
middle section parallel to bases
the zone of a circle ; the lateral
plane faces equal segments of
equal ellipses, each greater
than half; the curved or conical
tai-es developed, equal curvilinear
triangles
For areas, see pages 38, 46, 53
and 57, and tables II , 111, IV., of
" Ster." For area of zone, see table
IX, of same.
92 Frustum of a conic -wedge
or of the central ungula of a
cone by a plane parallel to
base ; or, may be considered the
frustum of a right cone, laterally
and equally truncated on op-
posite sides.
Arched and splayed embrasure
in a wall, component portion of a
compound solid.
The base, a circle ; the opposite
and parallel base, a zone of a
circle ; the middle section, a zone ;
the lateral plane faces, equal seg-
ments of equal ellipses the
developped conical surfaces resol-
vable into trapeziform figures.
For area of tiapezium, page 29,
" Key to Ster."
— 29 —
93— Lateral elliptic nngula of
a cone, by a plane passing
through edge ot base.
Splayed embrasure to elliptic
opening in wall and ihrongh sloped
roof or Ceiling; etc.
Its base, a circle ; its upper or
opposite base, a point ; its middle
section parallel to base, the seg-
ment of a circle ; its plane face
an ellipsis ; its conical surface
developed a concavo - convex
figure like h, page 97 of " Key to
Ster."
94 — Lateral elliptic conic un-
gula, by a plane passing
"Within the base.
The liquid in an inclined conical
vessel, lunette head of opening in
sloped roof or ceiling ; base of struc-
ture rising from an inclined surface,
roof, pediment, etc.
For area of parabola see key to
Ster., page 54 ; for area of hyper-
bola, page 55, or iigure e, page 57 ;
for ellipsis, page 51 and 53.
The base, a segment ot a cir-
cle ; the upper base, a point ; the
middle section, a segment of a
circle ; the plane lateral face, the
segment of an ellipsis ; the de-
veloped conical surface as in No. 87
or 94. If the cutting plane be pa-
rallel to side of cone the face will
be a parabola ; if at an angle
greater than side of cone to base,
a hyperbola ; if less, an ellipsis.
95— Central ungula of cone or
conic "Wedge, by planes
through opposite edges of
upper or lesser base and
meeting in the axis of the
cone.
An embrasure, etc., etc.
The plane lateral faces, segments
of elHpses if cutting planes more
inclined to base than side of cone ;
if less, hyperbolas ; if equally, pa-
rabolas.
Bases and sections same as "No.
91 ; developed conical surface, a
concavo-convex triangle cora-
putible as per page 57 of "Key."
The lateral plane faces, equal
segments of equal ellipses*
equal parabolas or equal hy-
perbolas, as case may be. — See
No. 94.
— 30 —
96 Pru'^tUTi of conic "wedge, Its base, a circle ; other base and
No 85, by a plane par llel to middle section, zones of circles,
the base. "■ for areas of which see " Key to
An embrasure; a reducer or con- Stereometricon, table IX,
necting link between a rectangular :f-i
and circular conduit, etc.
97 — Concave ungnla of a cone
or a conical recto-concave
'wedge.
Lunette of circular headed open-
ing in wall, reaching through vault-
ed, groined or arched ceiling ; cone
scribed to cylindrical surface, or to
a shaft of elliptical section.
The base, the segment of a cir-
cle ; the other base, a point or
curved arris ; its intermediate base
or section, or its bases or sections
if divided for computation of cubi-
cal contents, segments of circles.
Its sides like No. 94.
98— Portion of firustum of right
cone, ty a plane through
both bases.
Splayed segment headed opening
in wall, liquid in closed tub lying
on its side ; base or capita] of half
column against sloped wall ; com-
ponent section of base or capital of
clustered, gothic or other column.
Its parallel end bases and mid-
dle section, segments of circles ;
its conical surface developed a
figure of trapezium form, having
parallel or concentric arcs of circles
for its bases ; its plane face, the
zone of an ellipsis or of a para-
bola or hyperbola according to
inclination of cutting plane.
99. — Lateral conic ungula or
wedge, by a plane through
edge of lesser base of frustum
Embrasure, liquid in inclined co-
nical vessel, section of conical elbow
or mitre, base of chimney stack to
sloped roof. May be treated also as
lying on its lateral plane face.
Its base, a circle ; opposite base,
a point ; intermediate section a
segment of a circle ; its plane
face an ellipsis, its conical surface
developed a concavo-convex figure
like g or h, page 97 of Ster. but with
concave base. Treat on circular base
as easier of computation.
— 31 —
100 — A compound solid com- All its areas to be nsed in com-
posed of, decomposable or resolv- putation of solid contents or capa-
able into two conic frusta and city are circles, and can be mea-
; a lo"w or flat cone. * y v^ sured to eighths, tenths or tweltths
May represent a covered dish, a of an inch or other unity, and the
basket or hamper, a vase, a finial or areas found by mere inspection in
other ornament, an urn, a cauldron tables II., III. and IV. at end of
on a stand, etc., etc. Baillarg^'s " Key to Ster. "
DIVISION 2.
Solids of double curvature, or of which the surfaces are not capable
of development in a plane.
'. CLASS XI. ■ '; - ■ -. J/f-s':,
Concave Cones, Frusta and Ungulae.
«
101 — Right concave cone or Its base and parallel sections,
spindle. circles ; its upper or opposite base.
Camping tent ; roof of tower, pa- an apex or point. Its lateral surface
villon, hut, etc. ; spire, funnel, not capable of development in a
' strainer, trumpet ; finial or other plane or into a sector of a circle as
ornament. is the case with a regular right cone.
May be decomposed into two or but may be readily and very ap-
more frusta by planes parallel to proximately computed by division
base, to admit of more accurate de- into continuous trapeziums by
termination of sohd contents. lines parallel to circumference of
base. See " Key to Ster.," page 96.
102 — Frustum of a right con- Its bases and parallel sections,
cave cone hetTxreen parallel circles. Intermediate diameters
planes. not, as in No. 82, arithmetical means
— 32— .
lllnstrativeof most of the objects between Lhose of the opposite or
mentioned in No. 82, which see. end bases, but must be measured or
For more accurate computation computed. Lateral area may be
of contents, divide into two sections conceived as made up of a series of
or more, according to greater or super or juxta-posed continuous
lesser curvature of the solid, and trapeziums.
treat each section as a separate - cv^^r
prismoid and add the results. ; .■ - - :: ^
103 — Inclined concave cone. Its base and section, approxi-
rinial, or ornament on a raking mate ellipses of slight excen-
cornice ; liquid in an inclined ves- tricity or ovoid figures ; its other
sel, etc., as for No. 101, may be base, a point.
decomposed by imaginary planes In developing the lateral surface
parallel to base into two or more into a series of continuous trape-
sections or slices, so that slant side ziums, the lines are not as in the
of each may be sensibly a straight right cone parallel to base or to
line. See p. 103, par. 139 " Key. " circumferences of parallel sections
but are drawn equidistant from the
,, • apex, thus leaving at the base a
' ^^. figure like h, page. 57 of "Key."
104 — Frustum of oblique con- Its bases and sections parallel
cave cone between parallel thereto, approximate ellipses or
planes. ^ ovoid figures. See remarks to
Kepresentative of same as No. 84. No. 102 .. * • ,*;
105 Flat or lo-w concave cone. Its bases, a circle and a point ;
Kepresentative of many of the section, a circle ; lateral area
objects mentioned in No. 85. ^ reducible to continuous trape-
ziums, par. 126, " Key to Ster."
106 — Frustum of flat or low Its bases and section, circles,
cone. for arejs of which see tables II.,
Eepresentative of objects under III. and IV. of ** Key to Ster.," to
nead of No. 86. eighths, tenths and twelfths of inch
or other unity.
— 33 —
107 — Ungula of concave cone
by a plane through outer
edge of base.
See No. 92, as to what it repre-
sents, etc.
See No. 92. Lateral surface
reducible to trapeziums and
triangles.
Base and sections, ovoid figures ;
areas, page 57 of Key.
108 — Ungula of concave cone Base" and section, segments of
by a plane cutting the base, circles ; upper base, a point.
See No. 93 as to what it repre- Lateral surface as No. 107-
sents, etc. ? *
109— Ungula of hoUovr cone by
a plane through edge of
lesser base of frustum.
See No. 99, base of chimney
stack to a sloped roof. » : < . '^
Base, a circle ; opposite base, a
point ; middle section, the seg-
ment of a circle ; lateral area,
trapeziums and triangles.
110— Frustum of (No. 109) un-
gula by a plane parallel to
base. ■••/■>-■■■";
See Nos. 98, 116, 126. ^V-
Base or capital of a column, or
base of chimney shaft, etc., on or
outside of sloped roof or gable.
Its base, a circle ; other base, a
segment of a circle ; its middle
section parallel to bases, also a
segment. For areas of segments
of circles, see " Key to Ster.," table
VIII., or rules, page 44 of same.
• ■ ^-^'•- ■ :: CLASS XIL (■■ C : :,:■;;/, ■
ParaTDoloid or Paracolic Conoid, Frusta and
^' *^ . ^ Ungnlae, etc.
Ill — Right paraboloid or para- Its base and middle section,
bolic conoid. circles ; its opposite base or apex,
Dome, hut, hive, roof, finial or a point ; its lateral surface resol-
other ornament, shade, globe, cover, vablu into a small circle at apex,
hood, cowl, etc. ; reversed : a filter, and continuous trapeziums. The
— 34 —
cauldron, or other vessel of capacity, squares of its intermediate diame-
thc bowl of a cup or drinking ters, proportional to abscissae. See
goblet, etc., etc. . . " Key to Ster.," page 96.
112 Frustum of right parabo- End and middle bases, circles ;
1 jid bet"ween parallel planes, squares of diameters proportional
liepreseuts mostly the same to abscissae. For areas of circles,
objects as the frustum of a cone, see " Key to Ster.," tables II., III.,
No. 82. and IV. ; - > VT ■
See page 142 " Key to Ster."
113 oblque paraboloid. Its base and middle section,
" Key to Ster.," page 142. similar ellipses ; its opposite base
Liquid in a parabolic vessel or other end, an apex or point. For
inclined to the horizon, metal in an areas of ellipses see " Key to Ster.,"
inclined crucible, finial or ornament page 51 ; for lateral area see No.
on an inclined or raking molding 103. ■'.';!
or pediment, etc. .^^^ : ;
114 Frustum of oblique para- Its bases and middle section,
boloid bet"ween parallel similar ellipses; for areas of
planes. which see " Key to Ster., page 51.
Represents same as frustum of For lateral area, see No. 103 or
inclined cone No. 84, "Key to reduce to trapeziums by lines
Ster.," page 142. from base to base.
115— Parabolic wedge or cen-
tral ungula of paraboloid.
See No. 91.
Lateral or paraboloidal surface
capable of approximate develop-
ment. See No. 91.
116 Portion of a paraboloidal Its lesser base, a circle ; opposite
f^ustiun, by a plane through base, the segment of a circle ;
its greater base and edge of middle section, also a segment,
other or opposite base. Its lateral plane face, the se£fment
— 35 —
See No. 98 as to what it repre- of an ellipsis. This face would be
sents. Also, base of chimney stack, a parabola if angle of face equalled
partly on a horizontal and partly that of side ; if greater, a hyperbola,
on an inclined base, or sloped roof,
etc. •
117 — Lateral ungula of parabo-
loid
Very similar to No. 92, as to
what it represents.
Its base, a circle ; opposite base,
a point ; middle section, the seg-
ment of a circle. Its plane face
an ellipsis.
118— Lateral ungula of parabo-
loid; elliptic, parabolic or
hyperbolic, according as pla-
ne of section cuts the base at
an angle less than, equal to,
or greater than that of the
side and base.
Its base, the segment of circle ;
its middle section, a segment ; its
upper or opposite base, a point ; its
plane face, the segment of an el-
lipsis, parabola or hjrperholai
according to angle of plane of sec-
tion.
119— Obtuse eliptic ungula of a
paraboloidjby a plane through
edge of lesser base of frus-
tum.
Base of chimney stack, etc., to
eloped roof ; base of vase, statue,
etc., on a pediment; a lunette,
scoop, etc. , ,
Its base, a circle ; middle section,
a segment ; other base, a point ;
its plane face, an ellipsis. For a-
reas of segments of circles, table
VIII of " Key to Ster." For area of
ellipsis, page 51 of same.
120— Frustum af a paraboloid
between non-parallele bases.
" Key to Ster.," page 145.
Lunette through a vertical wall
and inclined ceiling, etc. For com-
putation of solid contents decom-
Its factor areas, circles and a
segment ; its plane face, an ellip-
sis. For areas of segments of circles,
table VIII of "Key." Area of
circle, tables II, III and IV, of
same; ellipsis, page 51 of same;
— 36 —
pose into a frustum with parallel lateral area, page 95 ; solidity, page
bases, and an ungula by a plane 145 of same.
parallel to base, through nearest
point of upper base. ; . . ' ^^>v'J .
^ CLASS XIII.
Hperboloid or Hypertolic Conoid, Frusta and
Ungnlae, eto.
121— Right hyperboloid or hy-
perbolic conoid.
Page 146, "KeytoSter." Repre-
sentative of same as No. 111.
For intermediate diameter or that
of middle section, see "Key to
Ster.," page 147, 3rd line, or by
direct measurement. ■*
19.9. — Frustum of right hyper- Except for diameter of middle
boloid. 5^ i section, same as No. 112, or the
Representative of same, nearly diameter may be measured directly.
as Nos. 112 and 82. ^--^ .
123 — Oblique hyperboloid.
See " Key to Ster.," p. 146. Re-
presentative of same, as No. 113.
Same as No. 113, except for dia-
meter of middle section for which
see "Key to Ster.," page 147, line
3, or the diameter may be mea-
sured. :a - - •
124— Frustum of oblique hy- Same as No. 114, except for dia-
perboloid. meter of middle section for which
Representative of same, nearly see " Key to Ster.," page 147, line
as Nos. 84 and 114. 3, or may be had by measurement.
125 — Hjrperboloid wedge or Except for diameter of middle
central ungula. section, same as No. 91 or 95. For
Similar solid to No. 95 of a cone area of zone, see " Key to Ster.,"
and representative of same objects, page 46 or table IX of same.
37 —
1526— TTngulaof h3rperboloidby Its base, a circle ; middle sect-
a plane through edge of base ion, the segment of a circle ;
For solid content, treat as pris- other base, a point. Plane lateral
moid or by par. 185 of "Key." face, an ellipsis, its lateral surface
Solid similar to No. 93 of cone, of double curvature, as all such
or to No. 117 of paraboloid. figures are, not capable of [develop-
ment, but reducible as required.
127 — Frustum of hyperboloid Bases same as in No. 116. La-
■wedge. teral area developes into trapezi-
Similar to No. 116 of paraboloid, urns by lines parallel to bases. For
Base of chimney stack, etc., resting areas of circles, segments, zones,
partly on a sloped roof. see tables of " Key to Ster."
128 — Ungula of hyperboloid by Bases and section same as No.
a plane through base. 118 of paraboloid. See table VIII,
Similar to No. 118 of paraboloid, of " Key to Ster.," for areas of seg-
ments.
129--Prustum of hyperboloid Same as No. 92. For area of
-wedge, or of central ungula circles to eighths, tenths & twelfths
of hyperboloid. see tables II, III, and IV of " Key
Similar to No. 92 of cone. to Ster." For area of zone, see table
IX, of same. Lateral surface de-
composable into trapeziums.
130 — A compound solid : two
equal firasta of cone or co-
noid, base to base.
Illustrative of a keg or cask, barrel,
hogshead, etc., of any size or shape.
Treat one-half of solid as Nos.
92, 112, 122, and double the result.
See "Key to Ster.," fig. on page
155, for mode of measuring half-
way diameter, when the half solid
is not the frustum of a cone, but
that of a conoid or of an ellipsoid
or spheroid. When of a cone middle
diameter equal to arithmetic mean
of end diameters.
— 38 —
CLASS XIV.
Sundry Solids.
131 — Three axed spheroid.
See " Key to Ster.," page xxxix.
May for measurement be supposed
to lie or stand on either of its sides
or apices.
Kepresentative of a pebble, a bean,
spindle, torpedoe, a shell fish, a
flattened ellipsoid, etc., etc.
AU its sections, ellipses ; all its
parallel sections, similar ellipses.
For areas of ellipses, " Ster.," page
51. Lateral area, see general for-
mula, page 95, "Key to Ster." Or,
as with the spheroid, suppose the
surface divided as a melon is or
orange into ungulae, terminating
in apices or poles of the fig. -
132— An ovoid or solid of the
shape of an egg.
Divide into two or three sections
and treat separately as conoid,
segment of sphere or spheroid, and
frustum of conoid.
All parallel areas perpendicular
to longer or fixed axis, circles,
which find ready calculated for all
sized diameters to eighths, tenths
and twelfths of an inch, or other
unity of measure, tables II., III.,
and IV., of Key to Ster. For late-
ral area, see page 96 of same.
133^ Circular disc -with round-
ed edge.
Treat as a compound fiolid, to
wit : a flat or low cylinder, and
a ring semi-circular or seg-
mental in section. Add the
results.
For cylinder, see No. 61. For
ring compute area of section
thereof as semi-circle or seg-
ment, and multiply into circum-
ference. For area, mean circum-
ference of ring into circumference
of section.
134— T-wisted prism.
Portion of a circular stair rail, a
twisted pillar or column, spiral
oruameAt, etc.
Its bases and sections similar
and equal figures. The lateral
surface of each face can be deve-
loped in a plane, a trapezium o\^
rectangle.
— 39 —
135— A compound solid.
Two frusta of cones, their
lesser basses joined.
A windlass, spool, handle, shaft,
axle-tree, etc.
Treat half the solid as the frus-
tum of a coue, and double the
result, either for solid content or
area of figure.
136— A compound solid.
Two frusta of hollow cones
Joined by their lesser bases.
A windlass, spool, handle, shaft,
axle-tree, etc.
Treat one half the solid as frus-
tum of cone No. 102, and double
the result.
Lateral area resolvable into con-
tinuous trapeziums.
131— Compound solid.
Two frusta of concave cones
joined by their greater bases
A windlass, shaft, axle-tree, etc.
Treat half the solid, and double
the result. For areas of circles, see
tables II., III. and IV. of Ster.
138 - Compound solid.
The segment or half of an
elongated or prolate spindle,
No. 151, and the segment or
half of an oblate spindle. No.
141, or the segment of a sphere
or spheroid, classes XVII, and
XIX., a buoy, etc.
Sections perpendicular to axis,
circles ; Area resolvable into con-
tinuous trapeziums, a circle
and the sector of a circle. The
circle at apex of segment of sphere
or spheroid ; the sector at apex of
spindle. See page 55 of " Key to
Ster."
139— Compound solid like the
• last with hollow cone in-
stead of spindle.
A finial or other ornament, a
ciQ-de-lampe or pendant.
Sections perpendicular to axis,
circles. Lateral surface, conti-
nuous trapeziums, a circle, and
the sector of a circle at apex of
cone.
140 — Compound solid : the Bases and sections, circles.
- frustum of a sphere or sphe- Lateral surface resolvable into
— 40 —
roid and a hollow cone. continuous trapeziums. See
A Moorish dome, a minaret, general formula, page 95 of " Key
chimney of a coal oil lamp, a to Ster." .
decanter, a vase, a pitcher. . ,
CLASS XV.
Otlate or flattened Spindle, Frusta, Segments,
Sundry. '
141 — Oblate spindle, as t"wo Treat one half as segment of
equal segments of sphere or sphere or spheroid, and double the
spheroid base to base. result. See classes 17 and 19.
A quoit, etc. ^
142 — Semi-oblate spindle by a Treat its two halves together as
plane parallel to fixed axis, one segment of sphere or spheroid.
Floating caisson to entrance of See classes 17 and 19.
dock, etc.
143 Middle frustum of oblate
spindle.
Fixed caisson or coffer-dam.
Treat as prismoid. ' : ? ..
The bases and middle section
each a double segment of a
circle or ellipsis, or tvro seg-
ments thereof, base to base.
Table VIIL, "Key to Ster."
144__Lateral frustum of oblate The bases and section half-way
spindle, bet-ween planes pa- between them, double segments
rallel to fixed axis. of circles or ellipses, for areas
A flau-bottomed boat or other of which see table VIII., " Key to
sailing vetjsei or a caisson, etc. - Ster.," and page 53 of same.
— 41 —
145 — Lateral frustum of oblate
spindle truncated at one
end.
A flat-bottomed boat or other
sailing vessel.
Bases and middle section, double
segments, base to base, of
circles or ellipses truncated at
one end. For areas, see page 57
^ " Key to Ster."
146-'Lateralfirustuin of oblate Bases, double segments of
spindle truncated at both circles or ellipses truncated at
ends. both ends. Divide into trapeziums
' A flat-bottomed boat or pontoon, and compute areas by page 57
a scow, lighter, etc. " Key to Ster."
147 — Quarter of an oblate sphe-
roid, No. 181.
The arched ceiling, roof or vault
of the apsis of a church or half-
groined ceiling of a circular apart-
ment. On its lesser base, the head
of a shallow niche in a wall, etc.
Its base and middle section,
semi-circles, if treated on its
broader base ; if on its lesser face,
its base and middle section, semi-
ellipses. On whatever base it
stands, treat as if on broader base,
it being easier to compute circles
than ellipses.
148 — A compound body, a cone, Treat separately as cone No. 81,
and the segment of a sphere and as segment of sphere. No. 173,
or spheroid. or of spheroid No. 182.
A buoy, covered filter, etc
149— Elliptic ring, or may be
called an eccentric ring.
Treat as circular or cylindrical
ring, taking for bases, its least, its
greater, and its mean sections ; a: id
for length the mean of the inner
and outer circumferences.
Compute half of solid as the la-
teral frustum of a half-prolate spin-
dle or the frustum of an elongated
cone-. The solid may be conceived
to be formed of the middle frustum
of an elongated spindle bent till its
ends meet.
— 42 —
150 Compound solid : a cylin-
der and the segment of a
spare or speroid.
A mortar, a tower with domed
roof, a hall or room with groined
ceiling, a hut, hive, hood.
For area of sphere or spheroid,
see page 95 " Key to Ster.,"or page
105, 110, 124, Ex. 3. Areas of cir.
cles tables II., III. and IV. of same.
Half-way diameter in segment of
circle or sphere a mean proportio-
nal between abscissae of diameter.
CLASS XVI.
Prolate or Elongated Spindle, Frusta, Segments, etc.
151 — Prolate spindle. Its sections perpendicular to axis,
A shuttle, a torpedoe, a cigar, a circles. Decompose its lateral area
sheath, case, etc. into continuous trapeziums and
' ... ' »■ " a sector.
152 — Semi-prolate spindle by a For solidity, compute planes per-
plane through its greater or pendicular to fixed axis, as seg-
fixed axis. ments of circles, semi-circles,
A boat or saiHng vessel, a canoe, while the sections parallel thereto
etc. are not so readily computed.
153 — Semi-prolate spindle by a For greater accuracy, divide into
plane perpendicular to fixed a frustum and segment, compute
axis. and add cubical contents. Areas of
A hut, roof, filter or vessel of ca- bases, tables II., III. and IV, of
pacity, a minaret or finial. " Key to Ster. "
«
154 — Middle frustum of pro- See page 149 of " Key to Ster.,"
late spindle bet'ween planes and for lateral surface, page 95 of
perpendicular to fixed axis. same. See page 155 of same. Bases
A cask or keg, puncheon, hogs- and sections, circles, tables II., III.
head, etc. ; see page 155 " Key." and IV. of Key to Ster."
— 43 —
155 — Semi-middle frustum of Bases and middle section, semi-
prolate spindle. circles, see page 160 of " Key to
The liquid in a cask lying on its Ster. " Lateral surface decomposa-
side, a boat with truncated ends. . ble into trapeziums.
Compute as No. 154 and take half.
156 — Lateral frustum of pro-
late spindle by planes paral-
lel to fixed or longer axis
A flat-bottomed boat or other
sailing vessel.
Treat as prismoid, the greater
base, a double segment of a cir-
cle. The other base and section,
oval figures for areas of which
see page 57 of " Key to Ster."
157 — Eccentric frustum of a
prolate spindle by planes
perpendicular to fixed or
larger axis of solid.
The shaft of a Eoman column.
Compute each frustum from centre
and add the results.
Its bases and • sections, circles,
for areas of which to eighths, tenths
and twelfths of inch or other unit
of measure, see tables II., III. and
IV., " Key to Ster. "
Its lateral surface decomposable
into continuous trapeziums, or
nearly equal to length of side into
mean circumference.
158— Middle frustum of elon-
gated spindle by planes per-
pendicular to fixed or longer
axis. - :---'^''-
The shaft of a windlass, a drum
or pulley, a cigar, torpedoe, etc.
Its bases and sections, circles,
for areas of which see " Key to
Ster.," page 38, or tables II., III.
and IV. of same.
Lateral area equal nearly length
of curved side into mean of circum-
ferences.
159 — A curved halfspindle or Base and sections circles or
cone. ellipses of slight eccentricity.
A horn, powder flask, tusk or Lateral area decomposable into con-
tooth of an elephant, etc., a sup- tinuous trapeziums and sector
porting bracket from face of wall, at apex.
— 44— - :■ •:-■:-:-
180 — Frustum of a prolate spin- Base and sections parallel thereto,
die bet'ween non parallel circles, base of ungula a circle ;
. bases. middle base of ungula, a semi-cir-
Decompose into a frustum cle ; apex of ungula or opposite
■with parallel bases and an un- base, a point ; lateral surface, con-
gula by a plane through nearest tinuous trapeziums, and a fig.
point of one of the bases. like h, page 57 " Key to Ster."
CLASS XVII.
Sphere, Segments, Frusta and Ungulae, etc.
161 — The sphere.
A billiard or other playing ball,
the ball of a vane or steeple, sphe-
rical shot and shell, school spheres,
lamp globe or well, component part
of compound solid, etc. Solid con-
tent mav be had hy 'computing one
of the component ungulae and mul-
tiplying into number thereof.
The opposite bases, points ; the
middle section, a circle. The area
of surface admits of approximate
development into a series of equal
figures in the shape of the longitu-
dinal section of a prolate spindle,
or of double segments of a cir-
cle, base to base.
Surface equal to four great cir-
cles or to four times that of a great
circle.
162. — A hemisphere.
A dome, arched celling, globe,
shade, cover, hut.hive, etc. ; revers-
ed : a bowl, cauldron, copper, vase,
etc. r .
Contents more easily computable
as half of those of a whole sphere,
where there is no intermediate dia-
meter to calculate or measure.
Its base, a circle ; opposite base,
a point ; its middle section, a cir-
cle, the half diameter of which
equals the square root of the rec-
tangle under the versed and su-
versed sines or portions of the dia-
meter of the sphere. The lateral
area equal to two great circles of
the sphere.
— 45 —
163. — Segment of a sphere less
than a hemisphere.
Eepresentative of same objects
as No. 162, cover or bottom of a
boiler. Solid contents also equal to
one of the component ungulae into
the number thereof.
Base and section, circles ; other
base, a point ; radius of middle
section for area thereof, equal to
root of rectangle of parts into which
it divides the diameter of the sphere
of which the segment forms part.
For later\l area see " KeytoSter.,"
page 110, or General Formula,
page 95. j- •;>
164. — Segment of sphere, great- Its base and section circles ;
er than a hemisphere. other base a point ; radius of middle
Eepresentative of same as No. section the root of rectangle of
162, and of a Moorish or Turkish parts into which it divides diameter
or horse-shoe dome. of sphere. Lateral area, see " Key-
to Ster.," pagee 117 and 123.
165. — Middle frustum of a Bases, f'qaal circles ; middle
sphere. sections, a circle ; see tables of
Base, capital or middle section of areas of circles to eighths, tenths,
a colunm or post, a puncheon, hogs- and twelfths of an inch or other
head, -i usher, roller, lamp shade, unity of measure, II., III., and IV.
etc., etc. of " Key to Ster. "
166. — Lateral frustum of
sphere.
Base or capital of column, coved
ceihng, cauldron, dish, soup plate,
saucer, etc. Eadii of bases and sec-
tions proportional to square roots
of rectangles of portions into which
such radii or ordinates divide the
diameter of which the solid forms
a part.
Bases and section, circles; lateral
area resolvable into continuous
trapeziums ; or lateral area may
be had very nearly at one operation,
if the frustum be low or Hat and
that its lateral curvature be not
considerable.-
— 46 —
167. — Sherical "wedge or oen- Its base, a circle ; opposite base,
tral ungula of a sphere by a ridge, or axis, or line ; middle
planes from opposite edges section, the zone of a circle; its
of base of hemisphere to plane faces, circles; and lateral
meet in apex. area resolvable into trapeziums
Component portion of a com- and triangles
pound solid.
168. — Frustum of a spherical Base, a circle ; other base and
wedge or central ungula middle section, zones of circles.
bet"ween parallel planes. For areas of zones, see table IX.^
Component portion of compound " Key to Ster. "
solid.
169 — Spherical pyramid, ob-
tuse-angled and triangular.
Illustrative of the tri-obtuse-
angular spherical triangle, and of
the fact that the sum of the angles
of a spherical triangle, may reach
to six right angles, when each of the
component angles increases to 180°.
Base, a spherical triangle
having three obtuse angles \
apex or opposite base, a point ;
middle section, a similar tri-
obtuse angular spherical trian-
gle, and whose area is equal to
one-quarter that of base, its factors
being halves of those of base, and
i X i = i
170.— Frustum of sphere be-
tween non-parallel bases.
Elbow or connecting link between
two portions of a rail or bead ; base
of a vase or other ornament on a
raking cornice.
Decompose into frustum and un-
gula of a sphere by a plane parallel
to one of the bases and passing
through nearest point of other base
or more readily and exactly, com-
pute whole sphere, and deduct seg-
ment.
— 47 —
CLASS XVIII.
Spherical Ungulae, Sectors, Pyramids and Frusta.
171 — Qnarter-sphere or rectan-
gular ungula of a sphere.
Domed roof to a semi-circular
plan, vault of the apsis of a church,
head of a niche, " Key to Ster., "
page 117.
On its base : one base, a semi-
circle ; opposite base, a point ;
middle section, the segment of a
circle. On end : each of its oppo-
site bases, points ; its middle sec-
tion, the sector of a circle. Only
Compute as a whole sphere, and one area to compute, and easier and
divide by 4, or treat as an ungula. quickei than a segment.
See opposite par.
172. — Acute>angled spherical
ungula.
Component portion of the ball of
a vane or steeple ; natural section
of an orange, or of a ribbed melon,
section of a buoy, cauldron, etc.,
etc., elbow of two semi-cylindrical
mouldings, etc., at an obtuse angle.
Its opposite bases, points ; its
middle section, the sector of a
circle ; the spherical surface, the
component of a hollow metallic or
other sphere or spherical vessel, or
of the covering for a racket or other
playing ball, etc.
For spherical area see " Key to
Ster.," page 117. ..
173. — Obtuse- angled ungula of Opposite bases points ; middle
a sphere. sections, the sector of a circle >
Head of niche reaching into a its plane faces, semi-circles. Sphe-
sloped ceiling; elbow of two half- rical area, page 117 "Key to Ster."
beads at an acute angle, etc.
— 48 —
174 -Spherical sector or cone, Its base, a spherical segment ;
or, to avoid computing spherical the other base, a point ; middle
- areas, may be treated as a com- section, a spherical segment con-
pound body, a cone and the centric to the base and equal in area
segment of a sphere.. one quarter of base ; its height equal
A buoy, a finial or ornament, a to radius of sphere, its lateral face
top, etc., a covered filter. For areas developed, the sector of a circle.
of circles see tables II, III and IV, See " Key to Ster.," page 110.
of « Key to Ster."
175 — Frustum of a spherical Its bases and middle section pa-
sector bet"ween parallel rallel thereto, concentric and si-
spherical bases. milar segments of spheres of
Portion of a shell or bomb or corresponding radii. Its height
hollow sphere. To avoid comput- the length of slant side. Solidity
ing spherical areas, treat as frus- also equal to difference between
turn of cone, adding greater and whole and partial spherical sectors,
deducting lesser segment. . ,^ ,^,.
176 — Hexagonal spherical py- Its base, a regular six-sided
ramid. spherical polygon; its middle
Its base illustrative of a spherical section a figure similar to the last,
polygon, page 127 of *' Key." and equal in area to one-quarter
Component portion of a solid thereof; its opposite base, a pointj
sphere or ball j keystone of a vault, the centre of the sphere of which it
finial or other ornament ; decompo- forms part. For area of base, see
sable for computation into six equal " Key to Ster.," page 127. For area
triangular spherical pyramids, "Key of component spherical triangle of
to Ster.," page 129. See rule for base, see page 123 of same. Its
spherical areas at end of this pam- plane faces equal sectors cf a
phlet. ^.-:^:;,:J;: ^..•>r-:^?^^---r..>':- circle.
177 prustiun of hexagonal Its bases and middle section, si-
spherical pyramid between milar spherical polygons; factor
parallel bases. of middle section, as in cone, an
Keystone of vault. Component arithmetic mean between those of
— 49 —
portion of hollow sphere. Surfaces the bases. Its lateral faces, equal
illustrative of similar spherical po- frusta of equal sectors '^f a cir-
lygons. Height of solid equal slant cle, or ccncavo - convex trape-
height of side. ziums. See rule at end of this work.
178 — Half- quarter or one-
eighth of sphere or tri-rec-
tangular spherical pyramid
Termination or stop to chamfer
on angle of wall or pillar.
Compute whole sphere and di-
vide by eight.
Its base illustrative of the tri-
rectangular spherical triangle,
page 123 of "Key."
May compute for solid contents
as the half of an ungula where
only one area is required, that of a
sector of a circle. See rule at end
of this work.
179 — Acute equilateral trian-
gular spherical pyramid.
Its base illustrative of the equi-
lateral spherical triangle.
Base and middle section similar
equilateral spherical triangles,
for areas of which, see " Key to
Ster.," page 123, and rule at end
of this work.
180 — Frustum of triangular Bases and middle section, simi-
spherical pyramid. lar spherical triangles whose
Illustrative in its bases of simi- areas are as the squares of the cor-
lar spherical triangles. Keystone of responding radii ; or factors of
a vault to a triangular plan. middle section, arithmetic means
■ between those of the opposite bases.
'■■:':■--'/ - CLASS XIX.
Otlate Spheroid, Frusta and Segments.
181 — Oblate spheroid. Treated perpendicularly to its
Eepresentative, in a less exag- fixed axis, its opposite bases are
gerated ratio of its diameters or axes, considered points, as in the sphere,
of the Earth and planets which are a plane touching the sohd only in
— 50 —
jflattened at the poles or extremities a point ; its middle section, a
of fixed axis and protuberant at the circle. If considered parallel to its
eciuatoi'. An orange, lamp-shade, or fixed axis, its middle section, an
ellipsis. For spheroidal surface or
area, see N. 161.
globe, or bowl.
182 — Semi-oblate spheroid by Base, a circle ; opposite base, a
a plane perpendicular to its point ; middle section, a circle ;
fixed or lesser axis. for diameter of which, if not from
Elliptical celling, dome, cauldron, direct measurement, see " Key to
basin, dish, vase, shade, globe, etc. Ster.," page 139, line 10 and page
140, line 20.
183 — Semi-oblate spheroid by Equal in area and solid contents
a plane parallel to its fixed to No. 182 and of easier and quic-
or lesser axis ker computation, if considered
Dome or ceiling to an elliptic such, the factors being circles
plan; glass globe or shade, dish instead of ellipses. As it stands,
cover, hut, a trough, cauldron, etc. its base and middle section, simi-
lar ellipses. ^
184 — Segment of oblate sphe-
roid, greater than half by a
plane perpendicular to fixed
axis,
Turkish, Moorish or horse-shoe,
dome or ceiling ; a cauldron or cop-
per, etc.
Its base and middle section,
circles ; opposite base, point.
Spheroidal surface continuous
trapeziums and a circle at apex.
For areas of circles, see tables II.,
III. and IV. of " Key to Ster." For
factors of middle section, see No.
182.
185— Middle frustum or solid Opposite bases and middle sec-
zone of an oblate spheroid be- tion, circles ; for areas of circles
tween planes perpendicular to to eighths, tenths and twelfths of
fixed or shorter axis. an inch or other unity, see tables
Kepresentative of same as No. II., III. and IV. of " Key to Ster."
165. Spheroidal area, see page 95 of
same.
— 51 —
186 -Middle frustum or solid
zone of oblate spheroid by
planes parallel to fixed or
lesser axis of solid.
Its bases and middle section si-
milar ellipses, for areas of which
see page 51 of "Key to Ster/*
Spheroidal area, page 95 of same.
187— Segment of oblate sphe- Its base, an ellipsis ; opposite
roid less than half, by a base, a point ; middle section, an
plane parallel to its fixed or ellipsis similar to base. For fac-
lesser axis. tors of middle section, see No. 182.
Kepresentative of same as as No.
183. ' '-.:,■-:..
188— Lateral frustum of oblate Its opposite parallel bases and
spheroid by planes parallel middle section, ellipses, for areas
to fixed or shorter axis. of which see "Key to Ster." p. 61.
Coved ceiling of elliptic plan ; re- Its spheroidal surface decompos-
versed : a boat, a scow, a vessel of able into continuous trapeziums
capacity, etc. of variable height.
189 — Halt or segment of oblate Its base and middle section, si-
ST)her"id by a plane irclined milar ellipses ; its opposite base,
o axis of solid a point ; its spheroidal surface tra-
Liquid or fluid in a serai-sphe- peziums, with ellipsis at apex
roidal vessel inclined from the ver- and a curvilinear triangle at
tical. Finial on a pediment or base of shape similar to fig. h. page
sloped surface. 57 of " Key to Ster.," or lateral
' area may be divided and computed
^ * as triangles.
190 — Frustum of oblate sphe-
roid betxyeen non-parallel
bases.
Decompose into a frustum 'with
peu-allel bases, and an ungula by
a plane parallel to one base and
drawn through nearest point of
Bases and middle section of com-
ponent frustum with parallel bases,
ellipses ; base of ungula, an ellip-
sis ; middle section of ungula the
segment of an ellipsis ; its other
base, a point.
For factors of middle sections,
— 62 ^
other base, or compute whole sphe- see " Key to Ster.," page 139, line 10
roid and deduct stgmeuts. and j^age 140, line 20, where AB :
y, , CD:.\/Ao^B'.oMa7idGD: AB::
<>.-.-■. .■ r: . ■. '•■ VCo.oB: oM. .-,,:-■ iV/-'-^-- .
CLASS XX.
Prolate Spheroid, Frusta and Segments.
191 — Prolate spheroid Its middle section perpendicular
Representative of a lemon, melon, to tixed or longer axis, a circle;
cucumber, etc. ; a case, sheath, etc. its opposite end bases, points.
The work of computation expe- Spheroidal surface, continuous
dited by treating circles instead of trapezoids, or a series of double
elhpses ; that is, areas perpendicular segments base to base as the
instead of parallel to fixed axis. component ribs a of melon. May
treat as plane segment with length
of cord equal to semi-eUiptical sec-
tion.
192 — Semi-prolate spheroid by For solid contents and spheroidal
a plane parallel to fixed axis, surface, treat perpendicular to fixed
Vaulted ceiling to elliptic plan ; axis, where factors are circles or
reversed : a boat or other sailing semi-circles instead of ellipses.
vessel, a cauldron or vessel of ca- For areas of circles, see tables II.,
pacity, etc., etc. III. and IV. of " Key to Ster."
193 —Semi-prolate spheroid by Base, a circle ; other base, a
a plane perpendicular to point ; middle section, a circle.
fixed axis. For radius of middle section, see
A hive, hut, roof or dome to cir- formula given in No. 190, or at
cular tower or apartment ; reversed : page 139, line 10, page 140, line
a copper or boiler, 20 of " Key to Ster." Spheroidal
area, see Xo. 191.
— 53 —
194 — Segment of prolate sphe- Base and middle section, oiroles ;
roid greater than half, by a its other base, an apex or point,
plane perpendicular to fixed Its spheroidal surface resolvable
axis. into continuous trapeziums and
A hut, hive, dome, a cauldron or a circle at apex,
copper, etc.
__ — — .■ -^-" '■ •
195— Middle frustum or solid End bases, equal circles ; mid-
zone of prolate spheroid by die section, a circle. Unlike the
parallel planes perpendicu- middle frustum of a spindle, the
lar to fixed axis. solid contents of this solid are ob-
A cask, keg, barrel, puncheon, tained exactly by treating the whole
hogshead, etc., "Key." page 138. figure at once.
196 — Middle frustum or solid Opposite bases and middle sec-
zone of prolate spheroid by tiou, similar ellipses. Spheroidal
parallel planes oblique to surface, trapeziums of which take
axis. ^^ mean height.
A boss on raking strut, etc. i
197 — Lateral frustum or solid Bases and section, circles, for
zone of prolate spheroid by areas of which see tables II., III.
planes perpendicular to fixed and IV. " Key to Ster." For dia-
axis. meter of middle section, measure
Coved ceiling, base of column, solid or compute by formula of
etc. ; reversed: capital of column, page 139, line 10; page 140. line
dish, basin, bowl, tub, hamper or 20, where it is shown that the rect-
basket,, stew pan, cauldron or other angle under the required radius,
vess"..;! of capacity, etc., etc. and either axis of the spheroid, is
f equal to that under the square root
of the rectangle or product of the
abscissae of the first axis and the
".'"p.
^ other axis.
— 54 —
188— Lateral firustuxn or solid Its parallel bases and middle
zone of prolate spheroid by section, similar ellipses ; for areas
planes parallel to eaoh other, of which see " Key to Ster." page
and to longer or fixed axis. 51. Its lateral area resolvable into
Coved ceiling of elliptical plan, continuous trapeziums of vary-
etc. ; reversed : a flat-bottomed boat, ing height if parallel to bases, but
a sc^w, a dish, basket, etc., etc. of uniform height, if lines be drawn
from extremities of fixed axis.
199— Segment of prolate sphe- Its base and middle section, sl-
roid by a plane inclined to milar ellipses ; its other base, a
axis. point ; its spheroidal surface re-
Liquid in spheroidal vessel in- solvable by circles drawn from ex-
clined from the vertical, a scoop, tremity of fixed axis, into a circle,
scuttle, etc. trapeziums and a triangle.
200 — Frustum of prolate sphe- Decompose into frustum with
roid bet"ween non-parallel paraUel bases, and an ungula. Com-
planes. pute separately, and add ; or com-
The one, perpendicular to fixed pute whole segment due to frustuia
axis, the other oblique or inclined and deduct lesser segment.
thereto.
SPHERICAL TRIANGLES & POLYGONS
TO ANY EADIUS OE DIAMETEK.
Head before the mathematical, physical and chemical section of the
Royal Society of Canada, May 22nd 1883.
Last year I laid before this section of the Eoyal Society my pro-
posal to substitute in schools the prismoidal formula for all other known
formulae pertaining to the cubing of solid forms.
I then showed that on this sole condition, the computation of soli-
dities, even the most difficult by ordinary rules, as of the segments,
frusta and ungulae of Conoids and Spheroids, was susceptible of gene-
ralisation and of being taught in the most elementary institutions.
I then submitted that the advantage of the proposed system con-
sisted in this ; that while he who had gone through a course of mathe-
matics would, in three months thereafter or out of college, have complete-
ly forgotten or have inextricably mixed up in his mind the numeroua
and ever varying formulae for arriving at the contents of solids ; the
simple artisan, on the contrary, who at an elementary, school would have
been taught the universal formula, and who from the fact of having to
learn but one, could not forget it nor mix it up in his mind with any
others, could apply it always and everywhere during a life time without
the aid even of any book excepting may be, to save time, a table of the"
areas of circles or of other figures lengthy of computation.
— 56 —
What I then did for the measurement of solid forms, I now propose
to do for the mensuration of areas of spherical triangles and polygons
on a sphere of any radius ; I mean a simple and expeditious mode of
getting at the doubly curved area of any portion of the terrestrial
spheroid as of every sphere great or small : interior or exterior surface of
a dome for example or of one of its component parts, as well of the bot-
tom or roof of a gasometer, boiler, or of one of the constituent sections
thereof, descending even to the surface of the ball of a spire, a shell, a
cannon or a billard ball. ■ - ■
TO THIS END :
-■ [ -
The area of a sphere to diameter I. being =3.141,592,653,580,793+
Dividing by,', wc get that of the hemisphere =l,570,71)(),3-2(5,794,896,5
This divided by 4=area of tri-rectgrr sph. triangle =0,3'J:>,699,0dl,698,724,l ' •
-;-90=area of L" or of bi-rect. sph. tri. with sp. ex=lo =0,004,36:5,323,129,985,8
H-60= " of r or of « " '• " 1' =0,000,072,722,052,166,43
r-60= " of 1" or of " " « « 1" =0,000,001,212,034,202,77
-elO= " of0.1"orof " « « « 0.1" =0,0it0,000,121,203,420,277
-1-10=" of 0.01" or of" « « « 0.01" =0,000,000,012,120,342,027,7
-i-10= " of 0.001" or of " " " « O.OOr =0,000,000,001,212,034,202,77
Find the spherical excess, that is, the excess of the sum of the
three spherical angles over two right angles, or from the sum of the three
spherical angles deduct 180°. Multiply the remainder, that is, the '
spherical excess, by the tabular number herein above given : the degrees
by the number set opposite to 1°, the minutes by that corresponding to
1' and so on of the seconds and fractions of a second ; add these areas
and multiply their sum by the square of the diameter of the sphere of
the surface of which the given triangle forms part ; the result is the area
required.
EXAMPLE.
Let the spherical excess of a triangle described on the surface of a
sphere of which the diameter is an inch, a foot, or a mile, etc., be 3°—
4' — 2.235". What is the area ? ""' " '"':"' ^" '' '
Area of 1° = 0.004,363,323,129,985,8 X 3 = 0.013,089,969,389,955
«« 1' = 0.000,072,722,052,166,43 X 4 = 0.000,-290,888,208,664
« 1" = 0.000,001,212,034,202 X 2 =0.000,002,424,068,404
« 0.1" = 0. 000,000, I2l,20,i,420 X 2 = 0.000,000,242,406,840
« 0.01" = 0.000,000,0 12,120,342 X 3 =0.000,000,036,361,026
« 0.001" = 0.000,000,001,212,034 X 5 =0.000,000,006,060,170
Area required 0.013,383,566,495,059
— 57-
The answer is of course in square units or fractions of a sqnnre unit
of the same name with the diameter. That is, if the dianu'ter is an inch,
the area is the fraction of a square inch ; if a mile, the franction of a
square mile, and so on.
Remark. — If the decimals of seconds are neglected, then of course
the operation is simplified by the omission of the three last lines for
tenths, hundredths and thousandths of a second or of so many of them
as may be omitted. .
If the seconds are omitted, as would be the case in dealing with
any other triangle but one on the earth's surface, on account of its size ;
there will in such case remain only the two upper lines for degrees and
minutes, which will prove of ample accuracy when dealing with any
triangular space, compartment, or component section of a sphere of
the size of a dome, vaulted ceiling, gasometer, or large copper or boiler,
etc ; and in dealing with such spheres as a billiard or otlier playing
ball, a cannon ball or shell, the ball of a vane or steeple, or any boiler,
copper, etc., of ordinary size, it will generally sufdce to compute for
degrees only. Whence the following
RULE TO DEGREES ONLY.
Multiply the spherical excess in degrees by 0.004,363 and the
result by the square of the diameter for the required area. For greater
accuracy use— 0.004,363,323.
KULE TO DEGREES AND MINUTES.
Proceed as by last rule for degrees. Multiply the spherical excess
in minutes by 0.000,073, or for greater accuracy by 0.000,072,722. Add
the results, and multiply their sum by the square of the diameter for the
required area. ,.
EXAMPLE I.
Sum of angles 140° + 92° + 68° = 300 ; 300 — 180 = 120° sphe-
rical excess. Diameter =30. Answer area of 1° 0.004,363
Multiply by spherical excess . 120°
We get 0.523,560
This multiplied by square of diameter 30= 900
Eequired area = 471.194,000
. -.- V. ,\y- - _58— ''■::<_. -'r -'--.■
A result correct to units. If now greater accuracy be required, it is be
obtained by taking in more decimals ; thus,say area 1°= 0.004,363,323
120 •
0.523,598,760
900
471.238,884,000
EXAMPLE II.
The three angles each 120° their sum 360°, from which deducting
180° we get spherical excess = 180°. Diameter 20, of which the square
= 400.
Answer Area to 1°= 0.004,363.323
180
0.785,398,140
400
314.159,256,000
EXAMPLE III.
The sum of the three angles of a triangle traced on the surface of
the Terrestrial sphere exceeds by (1") one second, 180° ; what is the area
of the triangle, supposing the Earth to be a perfect sphere with a diame-
ter = 7,912 English miles, or, which is the same thing, that the diame-
ter of the Terrestrial spheroid or of its osculatory circle at the given
point on its surface be 7,912 miles.
Answer. Area of 1" to diameter 1. = 0.000,001,212,034,202
Square of diameter 62,598,744
75.871,818,730,242,288
Remark. — This unit 75.87 etc., as applied to the Terrestrial sphere,
becomes a tabular number, which may be used for computing the area
of any triangle on the earth's surface, as it evidently suffices to multiply
the area 75.87 etc., coiTesponding to one second (1") by the number of
seconds in the spherical excess, to arrive at the result ; and the result
may be had true to the tenth, thousandth, or millionth of a second, or of
any other fraction thereof by successively adding the same figures
— 59 —
75.87 etc., with the decimal point shifted to ihe left, mi^ olace for every
place of decimals in the given fraction of such second : the tenth of a
second gi\dng 7.587 etc., square miles, the 0.01" = .7587 of a square
mile, the 0.001" = .07587 etc., of a square mile, and so on ; while, by
shifting the decimal point to the right, we get successively 10" = 7o8.7
square miles, 100" = 7587. etc., square miles, or 1 =^ 75.87 X bU (num-
ber of seconds in a minute), 1°= 75.87 x 60 x 60 (number of seconds
in a degree).
RULE.
To compute the area of any sph'^rioal "olygon.
Divide the polygon into triauglis, cmupuic . a.h triangle separaliely
by the foregoing rules for triangles and add the results.
OR,
From the sum of all the interior angles of the polygon subtract as
many times two right angles as then- are siiles less two. This will give
the spherical excess. This into the tabular area for degrees, minutes,
seconds and fractions of a second, as the case may be, and the sum of
such areas into the square of the diameter of the sphere on which the
polygon is traced, will give the correct area of the proposed figure.
It may be remarked here that the area of a spherical lune or the
convex surface of a spherical uugula is equal to the tabular number into
twice the spherical excess, since it is evident that every such lune is
equivalent to two bi-rectangular spherical triangles of which the angle
at the apex, that is the inclination of the planes forming the ungula, is
the spherical excess.
Kemark. — The area found for any given spherical excess, on . a
sphere of given diameter, may be reduced to that, for the same spheri-
cal excess, on a sphere of any other diameter ; these areas being as the
squares of the respective diameters.
The area found for any given spherical excess on the earth's sur-
face, where the diameter of the osculatory circle is supposed to be 7912
miles, may be reduced to that for the same spherical excess where the
OBculatory circle is of different radius ; these areas being as the squares
of the respective radii or diameters.
y
fe. •»'
('i^i-:-^,'f- ■■
ON THE APPLICATION OF THE
PRISMOIDAL FORMULA
TO THE MEASUEEMENT OF ALL SOLIDS
By CHS. BAILLAIRG^, M. A.,
Member of the Society for the Generalization of Education in France, and of several learned
a.d sc.entiflc Societies, Cheval.er of the Order of St. Sauveur de Monte-Keaie Italv
• &c Kecepient of 13 medala of honor and IT diplomas and letters from Uuasia France
Italy, Belgium. Japan, &c. Member of the Koyal Society of Canada.
Kead before the mathematical section of the Society on Saturday the 28th of May.
' « Cette formule V=^(B + B' + 4M) (Says « the late Eevd, N
« Maingui of the Laval University) que Mr. BaiUarg^ travaille k
« vulgariser, a rimmense avantage de pouvoir remplacer toutes les
" autres formules de stereom^trie,"
The prismoidal formula reads thus: « To the sum of the opposite
and parallel end areas of a prismoid, add four times tJie middle
area and multiply the whole into one sixth the length, or height of the
solid." •■ •'
Tlie following letter from the Minister of Education, Eussia, may
be considered interesting in its bearings on the subject matter of this
communication
MINISTEKE DE L'INSTRUCTION PUBLIQUE.
Saint-Petersburg, le U f^vrier 1877.
No. 1823.
A M. BAILLAIEGfe,
Architecte h Quebec,
Monsieur,
Le comit^ scientifique du minist^re de I'lnstruction Putlique, (de
Eussie,) reconnaissant I'incontestable utilite de votre " Tableau Ster^o-
m^trique " pour I'enseignement de la g^om^trie en general de meme que
pour son application pratique k d'autres sciences, eprouve un plaisir tout
particulier k joindre aux suffrages des savants de I'Europe et de I'Am^-
rique sa complete approbation, en vous informant que le susdit tableau,
avec toutes ses applications, sera recommande aux dcoles primaires et
moyennes, pour en completer les cabinets et les collections mathema-
tiques, et inscrit dans les catalogues des ouvrages approuv^s par le
minist^re de I'lnstruction Publique. .. i^ .'
"^ Agr^ez, monsieur, I'assurance de ma haute consideration. .. . v., --
Le chef du d^partement au miniature de I'lnstruction Publique,
E. DE Bradkek.
— 63 —
The following extract from the Quebec Mercury, July 10, 1878
further corroborates its importance.
" It will be remembered that in February, 1877, Mr. Baillairg^ re-
ceived an official letter from the Minister of Public Instruction, of St.
Petersburg, Eussia, informing him that his new system of mensuration
had been adopted in all the primary and medium schools of that vast
empire. After a lapse of eighteen months, the system having been found
to work well, Mr. Baillairg^ has received an additional testimonial from
the same source informing him that the system is to be applied in all
the polytechnic shools of the Russian Empire."
Should the Royal Society of Canada prove instrumental in the
introduction of the new system throughout the remainder of the
civilized world. It will have shown that its creation by the Marquis of
Lome, the Govr. Gen. of Canada, has been in no way premature.
The definition of a prismoid as generally given is understood to
apply to a soUd having parallel end areas bounded by parallel sides.
This parallelism of the sides or edges of the opposite bases or end
areas does not imply, not does it exclude any proportionality between
such sides or edges. . ■ . ■ : r ^ _ ; , _ ^;;/5
Therefore is the frustum of a pyramid a prismoid, as also that of a
cone which is nothing but an infinitary pyramid, or one having for its
base a polygon of an infinite number of sides.
Now let two of the parallel edges of either base of the frustum
approach each other until they meet or merge in a single line or arris,
when we have the wedge which is therefore to all intents and purposes
a prismoid.
Further let this edge or arris become shorter and shorter until it
reduces to a point and then have we the pyramid which is again a pris-
moid, as is the cone. ;, ..^ , , , ,..., ; >„ ^v
It need hardly be said that the prism and cylinder are prismoids,
whose opposite edges are equal as well as parallel in the same way as
for the frusta of the pyramid and cone the opposite edges are propor-
tional while parallel.
— 64 —
Kow, nine tentlis or more of all the vessels of capacity, the world
over, and either on a large or reduced scale, have the shape of the frustum
of a cone or pyramid ; the latter as evidenced in bins, troughs and
cisterns of all sizes, in vehicles of capacity ; the former, in the brewers
vat, the salting tub, the butter firkin, the commom wooden pail, the
drinking goblet, the pan or pie dish, the wash tub — of whatever shape
its base — the milk pan and what not else ; again the lamp shade, the
shaft of a gun or mortar, the buoy, quai, pier, reservoir, tower, hay-rick,
hamper, basket and the like.
These are forms which in every-day life the otherwise untutored
hand and eye are called upon to estimate. Why then not teach a mode
of doing it which every one can leam, and not only learn but what is of
greater import, retain in mind or 'memory when mastered.
Why continue the old routine when, as here evidenced, it is so
much more simple and concise, so much quicker to apply the prismoidal
formula to all these forms, than resort to one more difficult of apprehen-
sion and which to carry or work out requires tenfold the time the other
does.
r Legendre's formula requires a geometric mean between the areas of
the opposite bases of the solid under consideration. This mean is far
less easily conceivable than the arithmetic one ; and to arrive at it the
end areas are to be multiplied into each other, and the square root ex-
tracted of their product ; a long and tedious operation, one known only
to the few, most difficult to retain, forgotten as soon as learnt and
therefore useless. , ,. - , , ., u
With the formula proposed ou the contrary, the operation is one
which the merest child can master, the mere mechanic or the artisan
remember all his life and readily apply ; for he has been taught at school
to compute areas, that of the circle as well as others, a figure which he
readily sees is resolvable into triangles by hues drawn from the centre
to equidistant points, or not, in the circumference, and the area thence
equal to the circumference — sum of the bases of the component triangles
— into half the radius, or height of the successive sectors which make
up the figure.
Now, of almost all the solide herein above alluded to, the opposite
— 65 —
bases and middle section are circles and the operation can be further
expedited by taking the areas ready made, to inches and even hnes or
less, from tables prepared for the purpose.
The labour then reduces to the mere arithmetic of adding the areas
80 found, that is the end areas and four times the middle area, and of
multiplying the sum thereof into one sixth the altitude, or depth ; that
is, to the simplest form of arithmetic taught in the most elementary
schools, to wit : addition and multiplication, with division added when
the cubical contents in feet, inches or other unit of capacity, are to be
reduced, as of inches into gallons and the like. 5. •'-.;• v.? Vj
I would have but one formula applicable to all bodies, and it will
of course be asked : why, for instance in the case of the cylinder, the
whole cone or pyramid, substitute the more complex for the simpler form
of computation. My reason for doing so has its untold importance to
thousands of the human race. Memory is not a gift to every one. I
have none of it myself or hardly any, and its absence only entails a
little reasoning as I am now to show. '" ::''
I have seen students, only three months out of college doubtful as
to which of the ordinary formulae to apply, to this pyramid or cone, the
conoid, the spheroid. In one — the first — the volume, is due to the base
and one third the height ; in the second, the base and one half the height •
in the other, the base and two thirds the height. Any mistake is fatal
to the result. ^ • ^ . : -/,: vi-^
But with the one and only one, the unique and universal formula
which I propose to substitute for every other, no error can obtain. Take
hold of the pyramid or cone : set down its upper or one end area or that
of its apex, equal nought (0) or zero, its other end area, whatever that
may be. Its middle area, you see at once is one quarter that of its base ;
for the middle or half way diameter is half that of the base, and the
areas of similar figures as the squares of their homologous or like di-
mensions. Now, ere you have put this down on paper ; ere you have
had time to do so, the reasoning process is going on within your mind
and in far less time than it takes me to relate it — that four times the
middle area plus the area of the base is equal to twice the base, and
that twice the base into one sixth the altitude is precisely the same thing
;,-■■;-■; , , —66— :.':-■,'
as once the base, that is, the base into one third the altitude, and so come
you back to the old or ordinary rule, the simpler of the two in this case,
and without the necessity of having this formula stored in your mind as
a separate process.
And so with the cylinder where you see at once that the area of
each base and of the middle section being all equal quantities, the sum
of these bases and of four times the middle section is the same thing as
six time the base, and again that six times the base into one sixth the
altitude is the old rule of the base into the altitude, without the ne-
cessity of remembering it as a separate and additional formula.
But the great advantage of this one universal rule, its beauty so to
say is further evidenced and more strikingly in the computation of the
more difficult solids, that is of those which are more difficult under the
old or ordinary rules. - ^ °'-
In the sphere, spheroid and conoids, the one area, that at the apex
or crown is always nought or nothing, as a plane there touches them
in one and only one point. The formula applied to the sphere and
spheroid therefore reduces to four times the middle area into one sixth
the altitude or diameter or axis perpendicular to the plane of section.
Now, let it be required to measure the liquid in a conoidal or
spheroidal vessel inclined to the horison or out of the vertical. This by
ordinary rules, becomes an operation of much time, trouble and anxiety,
as the size of the whole body or solid of which the portion or figure
under consideration forms a part, has to be made known, its factors en-
tering into the formula for the content required ; whereas by the pris-
moidal formula, no concern need be had as to the dimensions of the
entire body of which the figure submitted to computation is a segment.
That the rule applies to all such cases, is and has been abundantly
proven by myself (see my treatise of 1866) as applied to any segment
of a sphere or spheroid, to any ungula of such solids contained between
planes passing in any direction through the centre, to any frustum of
these bodies, — lateral or central — contained between parallel planes
inclined in any way to the axes ; to any parabolic or hyperbolic conoid,
right or inclined, as well to any parallel frustum of eituer.
-67 —
This proof has been substantiated by MM. Steckel of the Dept. of
Dominion Public Works, Deville a member of this society, and the late
Eevd. M. Maingui, professor of Mathematics at the Laval University, aS
well by the Revd. M. Billion, of the Seminary of St. Sulpice— Montreal j
by His Grace, bishop Langevin of Rimouski, and by many other ma-
thematicians fully adequate to the task.
M. Maingui says (page IX of his pamphlet and as already quoted
from the french version) : " This formula "" -r- 'Ms that
" which Mr. Baillairge is endeavouring to introduce ; it has the im-
" mense advantage of replacing all other stereometrical formulae."
This is the only formula which will allow of teaching stereometry
in all schools however elementary, and as has just been shown, the appli-
cation of it is the more simple, so to say, the more complex the body is,
since in the conoid and segment of spheroid, one of the factors at least is
zero, while two of them are zeros in the sphere and spheroid as in their
ungulae. ■ • ■' - ■ ■ ^ -.^ . . ■.■".■ . ^.
Thus while the student at college or from a University after having
devoted much time to the acquisition of a hundred rules for the cubing
\ of as many solids, has hopelessly forgotten them in after life, the com-
paratively illiterate artisan, tradesman, merchant, &c. who has never fre-
quented ought but a village school, will, having but one rule wherewith
to charge his memory, remember it all his life and be ever ready to
apply it? ' - • ■
In the case of spindles and the masurement of their middle frusta
— the representatives of casks of all varieties and sizes, — the prismoidal
foimula does not bring out the true content to within the tenth or
twentieth and up to the half or thereabout of one per cent ; notwith-
standing which, it is the only practical formula which can bring out
anything like a reliable result. The true formulae for casks never can
nor will they ever be applied ; they are too lengtly, too abstruse, and the
wine merchant will teU you that the nearest the guage rod can come to
within the truth, the guage rod founded on these formulae, is to within
from one to three and even four per cent. This stands to reason, as
when operating on the half cask — which is always done with all figures
having symmetrical and equal halves — the half way diameter between
— 68 —
the head and hung, the very element by which the cask varies ita capa-
city, enters as a factor into the occupation, while the guaging rod can
take no note of it. .1*.
It remains but to say that in the case of hoofi j,nd ungulae of cones
and cylinders, of conoids and of spheroids, when the bounding planes do
not pass through the centre, the prismoidal formula is still the best to
be employed in practice, and again brings out the volume to within one
half or so of one per cent. The true lules applicable to these ungulae can
never be remembered, nor are or will they ever be applied in practice.
Hather than that, the fudging or so called rule of thumb system, some
averaging of the dimensions is sure to be resorted to and a result arrived
at, where two or three to five per cent of error is considered near enough
while the proposed application of the prismoidal formula would reduce
the error to almost nothing. „ .^-
>5- ? Compound bodies must of course be treated separately or in parts.
Thus, a gun oi mortar, as made up of a cylinder or the frustum of a
cone and the segment or half of a sphere or spheroid ; a morish or tur-
kish dome, as the frustum of a spheroid surmounted by a hollow cone ;
a roofed tower, as a cone and cyhnder, a cone and frustum of a cone or
two conic frusta as the case may be and so of other compound forms.
Again when frusta between non parallel bases are to be treated, the
solid is to be divided by a plane parallel to one of its bases and passing
through the nearest edge or point of its opposite base, into a frustum
proper and an angula, subject to the percentage of error already noticed
in the volume of the angula ; while, by cubing the whole conoid on
segment of a spheroid of which the frustum forms a part, and then the
segment which is wanting to make up the whole, the true content can
be arrived it. \u--: ^:<'6 "'^ m-v- ^/':-:K^'j inVi,- -^Ji re nu ^.-:^h _
There are a class of solid forms where it would appear at first
sight that a departure from the prismoidal formula becomes necessary •
not so however as will presently be seen. I allude to the cubing of the
fragment of a shell for instance, or of the material forming the vaulting
of a dome as contained between its intrades and extrados. This is simply
arrived at, when the inner and outer faces are parallel or when the dome
or arch is of uniform thickness by applying the spherical, spheroidal or
— 69 —
cylindrical surfaces of the opposite bases, and the equally curved surface
of the middle section ; while, when the faces are not parrallel or the
thickness of varying dimensions, as well when the faces are everywhere
aquidistant, the volume may be had by cubing the outer and inner com-
ponent pyramids and taking the difference between them.
And in the making out of such sj)herical areas as may enter as
factors into any computation, a most concise and easy rule will be found
at page 35 of my " stereometricon " published in 1880; when any such
area can in a few minutes be made up the mere multiplication and
adilition of the elemental quantities given in the text, and any portion
of the earths snrfiue thus arrived at when the radius of the osculatory
circle for the given latitude is known.
With irregular forms, the figure can be sliced up and treated by the
formula, and those forms when small and still more complex, such as
carving, statuary, bronzes and the like, can be measured with minute
accuracy by the indirect process of the quantity of fluid of any kind dis-
placed, as of water when non obsorbent or of sand or sawdust etc., when
the contiary.
Again may the specific gravities of bodies be applied, or their weights
to making out their, volumes by sim[)le rule of (hree, or the reverse
process of weighing them by ratio when their volumes are ascertained.
Finally the quantities anil respective weights of the separate subs-
tances which enter into amalgams or alloys are obtainable as taught by
a comparison of their wei«;hts in air and water, th.it is of the amalgam
itself and of its unalloyed constituents.
The whole field of solid meusuration is thus gone over in these few
pages, instead of the volume required to contain the niiny separate and
varied formulae which the old process of com[)Utation gives rise to and
renders indispensable. The whole I say is gone over in as many minutes
as the oil process requires hours or even days.
'S-' I
TABLES
OF
I. Squares and Square Eoots of numbers from 1 to 1600.
II. Circumferences and areas of circles of diameter bV to 150
advancing by ^.
III. Circumferences and areas of circles of diameter iV to 100
' advancing by tV. •
IV. Circumferences and areas of circles of diameter 1 to
50 feet, advancing by 1 inch.
v. Sides of Squares equal in area to a circle of a diameter
1 to 100 advancing by a ^. Vv ,; .
VI. Lengths of circular arcs, to diameter 1 diviled into 1000
equal parts.
VII. Lengths of semi-elliptic arcs to transverse diameter
1 divided into 1000 equal parts.
VIII. Areas of the segments of a circle to diameter 1 divided
into 1000 equal parts.
IX. Areas of the zones of a circle to a diameter 1 divided
into 1000 equal parts.
X. Specific gravities or weights of bodies of all kinds solid,
fluid, liquid and gazeous.
TABl,E OF SQUARES, SQUARE ROOTS
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
1
1
1.0000000
61
3721
7.8102497
121
14641
11.0000000
2
4
1.4142136
62
3844
7.8740079
122
14834
11.0453610
:i
9
1.7320508
()3
3969
7.9372539
123
15129
11.0905365
4
16
2.0000000
64
4096
8.0000000
12^.
15376
11.1355287
5
25
2.2360680
65
4225
8.0622577
125
15625
11.1803399
6
36
2.4494897
66
4356
8.1240384
126
15876
11.2249722
7
49
2.6457513
67
4488
8.1853528
127
16129
11.2694277
8
61
2 w.'-'»->71
68
4624
82162113
128
16384
11.3137085
■J
f^l
;;.uuO(.ouo
69
4761
8.3066239
129
16641
11.3578167
10
100
3.16-J27:7
70
4900
8.3666003
130
KiOOO
11.4017543
11
121
3.3166248
71
5041
8.4261498
131
17161
11.4455231
12
1-14
3. 4641016
72
5184
8.4852814
132
17424
11.4891253
i:{
169
3.60.">5513
73
5329
8. .5440037
133
17689
11.5325626
4
196
3.7 416574
74
5476
8.6023253
134
17956
11.5758369
15
225
3.8-.'29'-'33
75
5625
8.6602540
135
18225
11.6189500
16
256
4 OUOOOOO
76
.5776
8.7177979
136
18496
11.6619038
17
2H9
4.1-J3105()
77
5929
8.7749644
137
18769
11.7046999
18
■ 324
4.2426407
78
6084
8.83i7609
138
19044
11.7473401
19
361
4.35.-'.')989
79
6241
8.8881944
139
19321
11.7898261
20
400
4.47-.'1360
80
6400
8.9442719
140
19600
11.8321596
21
441
4..".>25757
81
6561
9.0000000
141
19881
11.8743421
22
484
4.69041.')8
82
6724
9.0553851
142
20164
11.9163753
23
529
4.795>315
83
6889
9.1104336
143
20349
11.9582607
24
576
4>it8979.')
84
70.56
9.1651514
144
20736
12.0000000
25
625
5.0(100000
85
7225
9.2195445
145
21025
12.0415946
26
676
5.0990195
86
73yt)
9.2736185
146
21316
12.0830460
27
729
5.1961524
87
7569
.3273791
147
21(;09
12.1243557
28
784
5.-i9l5026
fiS
7744
.) 3.-'08315
148
21904
12.1655251
29
841
5.3r'51618
89
7921
9.4339811
149
22201
12.2065556
30
900
5.47722.->6
90
8100
9.4868330
150
22500
12.2474487
31
961
5.5677(544
31
8281
9.5393920
151
22801
12.2882057
32
1024
5.6.".68542
92
8464
9.5916634
152
23104
12.3288280
33
1089
5.744r.626
93
8649
9.6436.508
153
23409
12.3693169
34
11 ->6
5.S3()9519
94
8836
9.6953597
154
23716
12.4096736
:i5
1225
5.9160798
95
9025
9.7467943
155
24025
12.4498996
36
129«;
6.0()0()()00
96
92 It)
9.7979.590
156
24336
12.4899960
37
1369
6 0827 6-^5
97
9409
9.8488.578
157
24649
12.5299i;;i
38
1444
6.1644140
98
9604
9.8994949
158
24964
12.5698051
39
1521
6.2449980
99
9H01
9.949x744
159
2.5281
12.6095202
40
1600
6 3-J45553
100
10000
10.0000000
160
2.5()00
12.6491106
41
1681
6.4031242
101
10201
10.0498756
161
2.5921
12.688.5775
42
1764
6.4807407
102
10404
10.0995049
162
26244
12.727i»221
43
1849
6.5.')74385
103
10609
10.14-i8916
163
26.569
12.7671453
44
1936
6.6332496
104
10816
10 1980390
164
26896
12.8062485
45
2025
6.7082039
105
11025
10.2469508
165
27225
12.84.52326
46
2116
5.78-i3300
106
11236
10.2956301
166
27556
12.8840987
47
2209
6.8556546
107
11449
10.3440804
167
27889
12.9228480
48
2304
6.9282032
108
11664
10.3923048
168
28224
12.9614814
49
2401
7.0O()(MJ00
109
11881
10.4403065
169
28561
13.0000000
50
2500
7.07l0ti7S
110
12100
10.4880885
170
28900
13.0384048
51
2601
7.1414284
111
1232 1
10.53.56.538
171
29241
13.0766968
52
2704
7.2illO-J6
112
12544
10,5.«'300.52
172
29.5S4
13.1118770
53
2809
7.280 1U99
113
12769
10.()301458
173
29929
13.1529464
54
2916
7.34Mti9-.'
114
12996
10.6770783
174
:{()-.»76
13.1909000
55
3025
7.416U»8.->
115
13225
10.7238053
175
30625
13.2287566
5(i
3136
7.4-'3:;i4-^
116
13156
10.7703->96
176
30976
13.2664992
57
3249
7.549H344
117
136-9
l(t.Hl66538
177
31329
13 30 41 3^ <
58
3364
7.6157731
118
13924
10.8627805
178
31684
13.3116641
59
3481
7.681i457
119
14161
10.9087121
179
3J011
13.37'.'0>^-^2
60
3600
y7. 7459667
120
14400
10.9544512
180
32400
13,4164079
1
OF
NUMBERS FROM 1 TO 1600.
1
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
1 isi
327(;i
13.45:56240
-241
58081
15.5-241747
301
90001
17.349.3516
i 1^2
33124
13.4907370
-242
.58504
15.556:5492
■M)2
91204
17.3781472
1^:^
331r'9
13.5277493
'243
5<)049
15.58-^4573
;io:5
91809
17.40689.52
lrt4
33^56
13. .".64 6600
244
595:56
15.0-204994
304
92410
17.4:5.55958
1*')
34225
1:5.6014705
245
60025
15.(5524758
305
93025
17.4642492
isi;
345.-0
13.6:581817
246
6051(5
15.(584:5*71
:506
9:56:50
17.49285.57
lb7
34!)()9
13.6747943
247
OI1109
15.710-2:5:56
307
94249
17..5214155
LS8
:{5344
1:3.711:5092
248
61504
15.7480157
308
94,-(54
17.5499^288
lyy
35721
l:i.7477271
'249
62,)01
15.7797:338
309
9.5481
17.5783958
IIH)
3(;i00
13.7840488
250
02500
15.811:58*3
310
90100
17.60(581(59
191
364H1
1:5.8-202750
•251
63001
15 8429795
311
90721
17.6:551921
11(2
3t;-fj4
13.8504065
2.52
6:5504
15.8745079
312
97344
17.66:55217
li);}
:!7216
13.^9-244(10
•25:5
04009
15.90597:57
313
9~969
17.(5918060
li)4
376: 56
13.9-28:58*3
•254
(54510
15.9:57:5775
314
9859(5
17.7200451
IDa
:!S025
13.961-2400
255
05025
15.9(587194
315
99225
17.748-2:593
11)6
:'>84I6
14.0000000
•25(>
055:50
16.0000000
310
99850
17.7763888
I'JT
36.-S()9
14.0:556088
257
6«i(J49
16.031-J195
317
100489
17.80449:38
hi6
:59-J04
14.071-2473
258
6(5564
16.002:5784
318
1011-24
17.8:5-25545
i;)L>
:596Ul
14.1067:560
259
07081
16.0934709
319
101701
17.8605711
'iOO
40000
14.1421:350
200
67(500
10.1-245155
320
10-2400
17.88854:58
'^(ti
40401
14.1774409
•201
08121
16.1554944
321
lo:S041
17.91047-29
•2in
40H{i4
14.2126704
202
68044
16.1,-64141
:522
10:5084
17.944:5.584
w.i
41209
14.-2478068
203
69169
16.217-2747
323
104:5-29
17.97-22008
•J04
41616
14.'282s569
264
69696
10.2480708
324
104970
18.0000000
ViU.')
42025
14.:!178211
205
70225
16.27H.-200
■25
105(;-25
18.0277504
aot;
424:56
14.3527001
200
70756
ie.:5095004
.520
106270
18.05.54701
•J07
42849
14.:5874946
267
7P289
10.3401:546
327
1069-29
18.0831413
•iUH
43264
14.4-2-2-2051
2m
718-24
10.:5707055
328
107584
18.1107703
i>09
4:5681
l4.4.-.08:3-23
209
72:561
16.4012195
329
108241
18.138:5571
210
44100
14.491:5767
270
7-2900
16.431(57(57
:330
108900
18.1059021
L>11
44521
14.525H390
'271
73441
10.462(t770
331
1095(51
18.1934054
•Jl-i
44944
14..5602198
'272
73984
10.4924-2-25
3:52
110-224
18.-2-208672
•jia
45:!69
14.5945195
27:5
745-J9
16.5-J-27116
333
110*89
18.'248'2876
•iI4
45796
14.0287:588
274
•75076
l6.55-2iM54
334
111.5.56
18.-2750069
■>ir)
46225
14.0(528783
275
75(i-25
16.5831-240
335
12-225
18.30:500.52
. 2i6
46650
14.69(59385
276
76176
10.61:52477
:S30
112896
18.3:503028
■J 17
47089
14.7:50i»199
277
76729
10.0433170
3:57
li:55f>9
18.3.575598
t>ls
47524
14.7(548-2:51
•278
77284
lti.(5783:5-20
3:58
114-244
18:5847763
•21!)
47961
14.79804.-*0
279
77841
16.7U:!'293l
3:59
114921
18.41195'26
2-JO
484(»0
14.8:523970
280
78400
16.73:;2005
340
11.5(500
18.4:5908*9
•221
4.^841
14..-()()0(587
'281
78961
10.7(5:50.540
341
116281
18.46618.53
222
492-4
14. -'99(5(544
282
79524
10.792S."..5(i
:542
1 16964
18.49:52420
22:{
49729
14.9:531845
28:5
80089
16.*-2-ji;038
313
117(i49
18 .5-202592
224
50176
14.9(560295
284
80t ;.-.(>
16.8.V2-2995
344
118:5:50
18.5472:570
225
50625
15.0000000
285
Hl-2-25
1(5.881 94 :;o
345
1190-25
18.5741756
•22(5
51076
15.03:5-2904
286
81796
10.9115:545.
346
119716
18.6010752
•>27
5l5-i'.»
15.0(«)5192
287
8'2:561>
16 9410743
347
1-20409
18.6279:560
tss
519-4
15.0990(589
288
8-2044
10.97056-27
:548
121104
18.6547.581
22!»
5-2441
15.1:5-27400
289
8:5521
17.0000000
:549
121801
18.681,5417
2:w
52900
15.16.57509
290
84100
17.0-29:;-64
350
122500
18.708-2869
•->:5l
5:5361
15. 19^(1842
291
84(5^^ I
17.05*7-2--'l
351
12:3201
18.7:549940
1 2:!2
5:5824
1 5.-23 154()2
292
85-2()4
17.0-80075
352
12:5i»04
I8.761(i630
i '2:53
54289
15.264:5375
•29:5
85819
17.117'24-28
:553
124009
18.7882912
2:u
54750
15.-ii>70.'>H5
294
861:50
17.1464282
.354
1-2: .3 16
18.8148877
! 2:55
552-25
15.:5-'97097
295
h7025
17.175.-m;40
3.-.5
126()-25
18.83144:57
2m
55(596
15. :5. ■.•2-29 15
'HiM
87610
17.J01().505
;556
1267:56
18."^t;79(;-23
237
56169
]5.:594H043
"297
88209
17.2:5:;()879
:557
127449
18.89444:56
2:58
5<)().l4
15.427-2486
'298
88804
17.'2<52i;765
:558
1'28H54
1 8.9*208879
239
57121
15.4596248
'21.»9
89401
17.'>916165
:559
1288«1
18.947-29.53
240
57000
15.4919:534
300
90000
17.:5-205081
:500
1-29000
18.97:50660
TABLE OF SQUAllES, SQUARE ROOTS
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
361
l:;03v!l
19.0000000
4-21
177-241
20.51 --2-45
481
231:^.61
21.9318122 j
:',u'2
131lt44
19.0-26-2976
4-2-2
17.--0-4
20.54-J(5:5-6
4.82
2:5-2:1-24
21.9544984 j
363
131769
19.05-255.-9
42:5
1 7.-^11-2.-'
20.5()()96:5S
48:5
2:5:;2-9
21.977-2610
364
13-2496
19.0787.- !0
4-24
179776
20.. 591-2(503
4-4
2342.->6
22.0000000 1
3()5
13:!-2Jr)
19.10497;'.-2
4-25
l-i)i;-25
•>o.(;i.">5-28l 1
485
2:5..-j25
•22. 0^227 155 !
36()
133966
19.1311-265
4-26
].-!1476
20.6:597674
4S6
2:!()llt6
22 0454077
1 :5t)7
134689
19.157-2441
4-27
182:5-29
20.6(5:597H3
4.«^7
2:57 1(59
-2'2. 06-0765
368
13;") 1'24
19.1833-261
4-28
183184
-20.6.8.-: 1609 !
488
23;- 144
•22.0907-220
369
13t)I6l
19.-20937-27
4-29
1.-4011
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4S9
2:39121
•22.113:3444 i
370
13()900
19.-235:1^4 1
4:50
18l'.t(iO
'20.7:564414 i
490
240100
22.13594:56 1
371
1376U
19.-26i:!60:{
431
185761
20.7605:595 i
491
2410S1
•22. 158'! 98 i
37-2
138384
19.-287:50l5
432
l,866-24
20.7.-46097
492
24-20(54
•22.18107:30
37:5
1391-29
19.3i:!-2079
433
1-71-9
20>i"^0.-6."-20
493
24:5049
•22.20:360:53
374
139.-'76
19. 3:5907 9()
424
18h:;.")()
-20..^:!-26(ii)7
494
2440:56
•22. •2*26 1108
37r>
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I9.:i649167
435
1-9-2-25
20.8.')6(>5:!6
495
2450-25
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3ro
1 11376
19.:}907191
4:^6
190( )'.)()
•20..-84.M:5(t
496
246016
22.-2710575
377
14-21-9
19.41(51.878
4:57
19(1969
•jO.904.^450
497
247009
•J2. -29:54968
37 -i
14-2-81
19.14-2-2-2-21
43S
191 >M4
20.9-281195
498
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22.31.591:36
379
143()41
19.46792-2;',
4:;9
19-2721
20.9523-268
499
249001
22.:538:5079
380
144100
li>.4.-':!."..-'-7
440
19:5()(M)
•20.9765770
500
250000
22.3606798
3«l
145161
19.. M 9-2-21:!
441
1941.-1
21.0000000
501
251001
22.:!830'293
38i
14.59J4
19.54 48-J()3
44-2
195:!61
2 1.02: '.79(50
502
252004
22. 405:5;" 65
383
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19.570:!8.").-<
44:5
196249
21.0475().V2
503
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384
1474..6
19.59.59179 i
444
197i:;(5
21.071:5075
504
254016
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:;8.")
M^-2-25
19. 6-2 14 169
445
1980-25
21.09.' 0-231
505
252025
22.4722051
386
14-^9'.M)
19.646-.8-27
446
l9.-^91t;
21.11S7I21
506
256036
22.49414:58
3-^7
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l9.67-23l:i6
447
199809
21.14-2:5745
r,07
257049
•22.5166605
388
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I9.f)97''!56
448
20(1704
21.1(5(50 105
508
25^064
•22.5:38.8553
389
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l9.7-2:50>-29
449
•201601
2l.H9(5-201
509
259041
•22..5t;iO'2,c!:5
390
1 5-2 100
19.74-1177
45(t
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21.21:5-20:54
510
2()0100
•22.;-).S:51796
i 391
1.V2881
19.77:57199
451
20:5401
2t.'2:5(57()06
511
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19.79-9-99
452
204:504
21.-2(50-2916
512
2(52144
22.6-274170
393
151149
19.8-24-2-276
453
•20.V209
21. •28:579(57
51:5
2631(59
•22.649." 0:53
394
155-236
19.-194:::!2
454
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514
2(54196
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19.87461 »(i9
4.55
2(»70-25
21.: 5:507-290
515
2(55225
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516
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397
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457
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517
2(57289
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19.9499:573
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209761
21.4009:54()
518
2(58:524
•22.75i)6l:54
399
159-201
19.9719S44
459
210(5^1
2l.4-242-^.".3
519
269:561
•22.7H1-,715
400
IGOOOO
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460
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520
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401
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461
21-2521
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521
271411
'22 8'254244
40-2
1616(14
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462
21:54 1 1
2l.4941-'.".3
5-22
27-24.^4
'22.8473193
403
16-2409
■20.071-599
4t):!
2i4:!»;9
21.517431-^
523
27:55-.9
'22.86919:53
4(t4
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-20.099751-2
464
215^296
21.5 I0(;.")92
524
27157(5
'22.8910163 i
40.")
1610-25
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4(55
21(522')
2l.5(i:585.-'7
525
275625
•22.9128775 '
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16483()
'20.1494 117
4(56
2l71.Vi
2l.(!.-^70:53l
526
27(567(5
•22. 9:54 6'^; »9 '
407
165649
-20. 174-24 lit
467
•jl.'^nS,*
2l.6l01.-i2''
527
2777-29
•22.9.564.-106 '
408
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4(58
219021
2l.6:5:!:!07r
528
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22.97.^2506
409
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469
219961
2l.6.".t54(t78
529
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23 0000000
410
168 KfO
•20.-2181567
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220901 1
21.6791^:54
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2c<09(!0
2:5.0217-2-^9
411
16-9-21
•2(l.-273i:;49
471
221.^11
21.70-2.">:514
531
2809(51
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41i
1697 14
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47-2
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21.7-25.')(510
5:52
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413
170.'.69
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47:5
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2*^ 10-19
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414
17139(5
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474
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475
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5:}5
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2:5. 1:500(570 .
416
173056
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21.8174-242
:.'M
•287296
•23.15167:58
417
173'J89
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477
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2l..-'40:5'297 ;
5:57
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418
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478
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21.86:52111
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419
175561
•20.4694895
479
2-294 11
21.^^8()0().S) 1
5:59
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•2:5.2i(;;57:;5
4-20
176400
•20.49:59015
480
'230400
2l.90890r23 j
, 1
; 540
j
•291(500
•23.*2:57900l
: I
OF NUMBERS FUOM 1 TO 1600.
5
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
541
292081
23.2.594007
001
361201
24.51.5:5013
601
4:;«5921
25.7099203
ol-i
29:57 ()4
23.280.>^935
002
:>02404
24.5356883
002
43-244
25.729:1607
543
294849
23.:502:!0U4
00:5
:5o:)009
24..55605.83
003
4:19.509
25.74.>'7864
544
2959:5(5
23.323.-07t)
604
:J0481()
24.,5704115
064
440.-i90
25.7681975
545
297025
23.:i452:551
605
36(5025
24. .5907478
605
442225
25.78759:19
546
298116
23.3000429
600
3072:50
24. '5170(573
00(5
443555
25.80097.58
547
299209
23.3880311
007
308449
24.0:57:5700
607
4448:59
25.82(53431
548
:500:504
23.409:1998
608
:509()04
24.(5.57(5560
(508
44(5224
25.84.50900
549
301401
23.4307490
009
370^81
24.0779254
(509
447.501
25.. -650:543
550
."^02500
23.4520788
610
372100
24.0981781
670
44>900
25.884:1.582
551
3o;iooi
23.473:5892
611
37:5:121
24.7184142
671
450241
25.903(5(577
552
:!()4704
23.4940802
612
:!74544
24.7:580:W8
072
451.584
25.9229028
55:i
:!05SU9
23.51 59.')20
613
:57.5709
24.7588:)68
67:5
452929
25.94224:15
554
:;oo9io
23.,5:S72040
014
37(5990
24.77902:54
(574
454-J70
25.9015100
555
:508O25
23.5584:180
015
378225
24.79919:55
675
455025
25.9>07<52l
550
:509i:50
23., 5790.522
610
37945(5
24.819:5473
070
4.50970
20.0000000
557
:510249
2:i.600>474
017
:5f-O089
24.8:594847
077
458:129
20.01922:57
558
311:504
2:5.62202:5(5
618
381924
24.8590058
078
459084
26.01584:531
559
312481
23.6431. -^08
019
383161
24.8797106
079
401041
20.0.57(52.-4
5(iU
:5l:5(ioo
2:5.604:1191
020
:5844()0
24.8997992
680
402400
2(5.090^^096
501
314721
2:5.6854:580
021
385(541
24.919.-'710
6(^1
40:1701
26.07.597(57
50-J
315844
23.7005:592
622
386.-^84
24.9:599278
082
405124
2(5.1151297
5(>:i
310909
23.7270210
62:5
388129
24.9099679
08:5
4004.89
20.i:5426s7
.5(i4
313U9t)
23.7480-42
024
389370
24.97:;9920
084
4078.")6
2().15:5;5.>^:57
5()5
319225
23.7097280
625
390025
25.0000000
685
469225
26.17-.'.5047
5i;t)
320:5r)t)
23.7807545
ti2()
'581870
25.0199920
(58(5
470.590
26.191001T
507
• :V21489
23.8117018
027
;i93l29
25.0399681
087
471969
20.210(5848
508
:«2024
23.8:527500
628
:59i:!84
25.(1599282
088
47:!:54i
20.2-.:9754»
509
32:5704
23.8537209
029
395(541
25.0798724
689
474721
2(5.24.^8095
570
32 t9(H>
23.8740728
o:;o
390900
25.0998008
0'.)0
470100
•.().26T8511
571
:52(504 1
23.895000:5 '
031
398101
25.11971:54
691
477 481
20.28687^9
57:>
327184
23.910.5215
6:!2
399424
25.1:;90102
692
47.^''t)l
20.305-929
57;i
328:529
23.9:574184
0:13
40(I(M»
25.1591913
693
480249
20. 152489:12
574
:529470
23.9.582971
0:54
4019.50
25.179:1.500
094
481():!6
20.34:1^797
575
3:i0025
23.9791576
635
40:i225
25.19920(53
095
48:5025
20.:502852»
570
:531770
24.0(100000
0:50
404490
25.2190404
096
484410
20.;5818119
577
3:52929
24.020.^243
6:57
405709
25.2:5-8589
097
485809
20.4(.f07.570
578
:534084
24.041(i:500
6:58
407044
25.25-0019
098
487204
20.419()890
ry79
3:5-.241
24.0024188
639
408:521
25,278449:5
099
4.8.>^(501
20.43-0081
580
3:50400
24.0831891
040
409(i00
25.29.-i2213
7(10
490000
20.4.575131
5H1
3:575()1
24.10:59410
641
410881
25.3179778
701
491401
20.4704046
58v;
338724
24.1240702
642
4121(54
25.:5:577189
702
4t>2r04
2(5.4952^20
58:5
3:59889
24.14.5:5929
643
4i;il49
25.:5574447
703
494209
20.5141472
584
:54i(;56
24.1000919
644
414;:5()
25.:577I551
704
49.iOtO
20..-.:i2998:5
585
342225
24.18077:12
64;)
410025
25.:!9()-5{t2
705
497(t25
20.5518:161
59(i
34:5390
24.2074:5(i9
040
417310
25.4i(;.5:50l
706
4984:50
20. .570(5005
5?7
344509
24.2280829
647
418009
25.4:101947
707
499819
20.5894716
588
1545744
24.2487113
048
419904
25.4.55-441
708
501264
20.(5082094
589
34t)921
24.269:5222
649
421201
25:4754784
709
.502081
20.02705:59
590
318100
24.2899150
650
422500
25.49.'><t970
710
501100
20.6458-j.52
591
349281
24.3104916
651
42:5801
25.5147010
711
505.521
20.664.5.x:5:5
ri92
:550404
24.:53lO501
652
425104
25.5342907
712
506i»44
■J0.08:>:!281
59;{
:i51049
24:5515913
<;53
420409
25.5.5:58047
713
5081509
26.7020598
.594
352s:;!j
24:5721152
654
427710
25.. 57:542:57
714
5097iM5
2().72077.-<4
595
354025
24.:5926218
655
429025
25.5929078
715
51 1225
•' ..7:-5i?4H:l9
596
35.5210
24.4131112
050
4:5o:!:50
25.0.524909
710
5120.5-
Jl. 758 1 7(53
597
350409
24.43:5.58:14
057
4:51049
25.0320112
717
51 4089
26.77(58.557
598
357004
24.4.5403.><5
058
432901
25.(5515107
718
515.5-J4
26.7955220
599
:558801
24.47447(55
059
434281
25.0709953
719
51(5961
20.8141754
600
360000
24.4948974
600
435600
25.6904652
720
518400
26.8:528159' ■
TABLE OF SQUARES, SQUARE ROOTS
No.
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
7()2
763
761
765
7()()
767
768
769
770
771
772
773
//4
775
77(5
777
778
779
780
Square.
519841
521284
522729
524176
525625
527076
528529
529984
531441
532900
534361
535824
537-289
538756
54! .J5
541696
543169
544644
546121
547600
549081
550564
552049
553536
555025
566516
558U09
55<)504
561001
562500
564001
565504
567009
568516
570025
571536
573049
574564
576081
577600
579121
580644
582169
5>3(596
585225
585756
588-,'89
589824
591:561
58290(»
594441
595984
597529
591H)76
600625
(■)02176
603729
605284
606841
608400
Sqre. root.
26.8514442
•J6.8700.577
26.8886593
26.9072481
26.9258240
26.9443872
26.9629375
26.9814751
27.0000000
27.0185122
27.0370117
27.05549.-^5
27.07;iSt727
27.0924344
27.1108834
27.1293199
27.1477439
27.1661554
27.1845544
27.2029410
27.2213152
27.2396769
27.2580263
27.27(53634
27.2946^81
27.3130006
27.3313007
27.3495887
27.3678644
27.3861279
27.4043792
27.422<)184
27.4408455
27.4590604
27.4772633
27.49.")4542
27.5136330
27.5317998
27.5499546
27.5680975
27. 68622^4
27.6043475
27.6224546
27.6405499
27.6.58()334
27.6767050
27.6947()48
27.7128129
27.7308492
27.748f-(739
•*7.766S,<68
27.7849880
27.8020775
27.820^555
27.8388218
27.f<.")677()6
27.8747197
27.892()514
27.9105715
27.9284801
No.
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
82(i
827
828
829
830
831
832
833
8:!4
835
836
837
838
839
840
Square.
609961
611524
613089
61465C)
616225
617796
»il9369
620944
622521
624100
625681
627624
628849
630436
632025
633616
635209
636804
638401
640000
641601
643204
644809
64641()
648025
649635
651249
652864
654481
656100
657721
659344
660969
662596
664225
665856
667489
669124
670761
672400
674041
675()84
677329
678976
680625
682276
683929
685584
'687241
688900
69(»561
69-J224
693889
6955.56
697225
698,<96
700569
702244
703921
705600
Sqre. root.
27.9463772
27.9642629
27.9821372
28.0000000
2,-!. 0178515
28.0356915
28.0535203
28.0713377
28.0- 438
28. 10(, 1)386
28. 1247222
28.1424946
28.1602.557
28.17K»0.56
28.19.57444
28.2134720
28.2311884
28.2488938
28.2661881
28.2842712
28.30194:54
28.319<)045
28.:5:^72546
28.35489:«
28.:J7252:"
28.3901:391
28.4077454
28.425:'>408
28.4429253
28.4604989
28.4780617
28.49561:^7
28.5131549
28.5:^06852
28.5482048
28.5657137
28.58:52119
28.6006993
28.61817(i0
28.6:^56421
28.6530976
28.6705424
2H.6879766
28.7054002
28.7228130
28.7402157
28.7507677
28.7749.^91
28.792:5601
28.b097206
28.8270706
28.844410-J
2K.H617:594
28.8790.5S2
28.896:5666
28.9i:!6()46
28.9309523
28.9482297
28.9654967
28.98275:«
No.
841
H42
843
844
845
846
847
848
849
850
851
8.52
853
854
855
856
857
858
859
860
861
862
86:;
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
864
8.S5
88(5
887
888
889
890
891
892
89:?
894
895
896
897
898
899
900
Square.
Sqre. root.
707281
708964
710(549
712:536
714025
71.5716
717409
719104
720801
722500
724201
25'
904
/-,-/609
729316
731025
7:W7:56
731449
736164
7:57H81
739600
74l:i2l
74:1044
744769
74(5496
748225
749956
751(589
753424
755161
756900
758641
760:584
7(52129
763876
765625
7(57376
769129
770884
772641
774400
77(5161
777924
779(589
78145(5
78.3225
784996
786769
788544
790:^21
792100
79:5881
795664
797449
7992:56
801025
80281(5
804609
80(5404
808201
810000
29.0000000
29.0172:563
29.0344(523
29.0516781
29.0688837
29.0860791
29.1032644
29.1204:396
29.1376046
29.1547595
29.1719043
29.1890390
29.20(516:37
29.22:32784
29.240:3830
29.257 till
29.2745(523
29.291(5370
29.3087018
29.:5257566
29.3428015
29.3598365
29.:3764616
29.:?938769
29.4108823
29.4278779
29.4*448637
29.4618:597
29.47880.59
29.4957624
29.5127091
29.5296461
29.54(5.5734
29.56:34910
29.580.3989
29.5972972
29.61418.58
29.6310648
29.6479342
29.(5647939
29.6816442
29.(5984848
29.7153159
29.7:521:375
29.7488496
29.7657521
29.7825452
29.799:3289
29.81610:30
29.8328678
29.8496231
29.8663690
•>9.8S31056
29.8998:328
29.91(5.5506
29.93:32591
29.9499583
29.9666481
29.98:33687
30.0000000
■'.-
OF
NUMBERS FROM 1 TO 1600.
1
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
901
811801
30.0106621
961
92:5521
31.0000000
i
! 1021
1042441
31.9530906
yo-i
81:5004
30.0:5:5:5148
962
925444
31.0161248
1 1022
1044484
31.9t5f57:;47
U(j:i
815409
3(t.0499584
9ti3
927369
31.0:522413
10j:5
104()529
31 984:5712
904
817210
:50.0005928
964
929-96
31.04.S5494
1024
1048576
32.00000(»(»
905
b 19025
30.0-32179
965
931225
:!1. 0644491
1025
1050625
32.0156212
1 90(i
8208:56
30.099p:5::9
966
9:5:5156
31.0.-^05405
1026
1052676
32.0312:548
1 907
822049
:50. 11 (54407
967
9:55089
31.09662:56
1027
1054729
:52. 0408407
1 90d
824401
:50.i:5:5o:!83
968
9:57024
31.1l269-!4
1028
1050784
:52. 062 4:591
n 909
820-.'81
30.1490269
969
93-961
31.1287648
1029
I0.5f^841
:32. 07^029-
1) 910
828100
;50.10ti2O63
970
940900
31.14482:50
lo:!0
10(;0i)00
::2.09:i0l31
» 911
829921
:50. 1827705
971
942841
31.1608729
1031
1069! 01
:52. 1091877
' ::\'^
8:51744
:50. 199:5:577
972
944784
31.1709145
10:52
1065024
32.1247568
9l:J
8:5:5509
30.2158899
973
940729
:il. 1929479
103:5
1007089
:52. 1403 173
911
8::5:597
;50.2:524:329
974
948ti76
31.20.-'9731
10:14
1069156
32.155^704
915
8:57225
:50.24«9609
975
950625
31.2249900
10:55
1071225
32.1714159
91(j
1^:59050
:50.2051919
976
952576
31.2409987
lo:56
107:5296
::2. 18095:59
917
840889
30.2-20079
977
954529
31.250999'J
1037
1075:5(59
32.2024844
918
842724
:30.2985148
978
956484
31.2729915
10: !8
1077444
32.2180074
1 919
844501
:i0.3150!28
979
958441
31.2^89757
10:59
1079521
32.2:5:55229
9::^0
840400
30.:5:!15018
980
900400
31.:50I9517
1040
1081000
:52.2490310
9-21
848241
:50 3479818
981
9()2:561
31.: 1209 195
1041
108:5081
32.264531(5
9'2-J
850084
:50.:!044529
982
964:524
31.:5:;08792
1042
1085764
:52.2K)0248
9-S.i
851929
30.:!809151
98:5
966289'
:51.3.- -:508
1043
1087849
32.2955105
9-21
85:5770
30 3973683
984
968250
31.:50-7743
1044
10898:56
:52.:: 109888
9-25
855025
;50.4K58127
985
970225
31.;5847097
1045
1092025
32.:5264598
9-.'(5
857470
:50.43024.-'l
986
972196
31.4000:569
1046
1094116
:52.:54 192:53
9-J7
859:529
30.4406747
987
974169
31.4165561
1047
109(i209
:52.:557:5794
928
801184
30.40:50924
988
976144
31.4:524673
1048
1098:504
32.:57 28281
9-29
8():;041
:50.4795013
989
978121
31.44K!704
1049
1100401
:52.:5-rt2695
9:50
ri()4900
30.4959014
990
980100
31.461J6.'.4
1050
1102500
:52.4037O;55
9:51
800701
:50.5122926
991
982081
31 4901525
1051
1104001
:52.4 191:501
9:3-j
80>024
30.52S075't
992
9H4064
3l.49(io:!15 :
1052
1 106704
:52.4:545495
9:{:j
8704-9
:50.54:.0487
99:5
986049
31.5119025 •
1053
1108899
:52.4499615
9:51
872:5.56
30.56141:56
994
9880:56
31.5277*i55 i
1054
1110916
32.4(55:5(502
9:i5
874225
30.5777097
995
990025
31.543620*)
1055
111:5025
:52.4S07635
9:!C)
87009*;
30.5941171
996
992016
31.5.594577
10.')6
11151:56
32.4901536
9:57
?^77'.)09
30. () 104557
997
994009
31.575:500-^
10.-7
1117249
:!2. 51 15:564
9:58
879.->44
:5062O7Hr.6
998
990004
31.5911:580
10.58
1119:564
:52.52(;9119
9:59
881721
:50.64:51009
999
1998001
31.60.')90l:!
10.59
11214S1
32.5422802
940
88:5000
30.6594194
1000
1000000
31.6227700
lOiiO
1 12:5000
:52. 55764 12
941
885181
:50. 675723:5
1001
1000201
31.0:58.5840
1001
1 125721
:52. 5729949
942
887:504
:50.0920185
1002
1004004
31.()5l:!-:!0
100 J
1127H14
:52.588:3415
94:5
889249
30.708:5051
1003
100()009
31.1)701 752
106:;
1129969
32.60:56807
944
8911:56
:50.7245.-:50
1004
100-016
31.o-.".m;m)
10<)4
1 1:52090
:52.(;i90129
945
89:5025
:50. 7408523
1005
1010025
31.7017;!I9
1065
11:54225
:52.634:5:577
940
894910
:50 7571 1:50
1000
10100:56
:!i.7i:5o:!o
1060
11:50:556
:52. 6496554
917
896^08
;50.773:5051
1007
1014049
31 7:5:520:5:5
1067
11: '.8489
32.6049659
94rt
898704
30.7896080
1008
101(;064
31.7490157
1008
1140624
32.(;802693
949
900001
30.80581:56
1009
10180S1
31 7617003
10(59
1142761
:52. 0955654
950
902500
30.8220700
1010
1020IO(»
31.7-04972
1070
1144900
32.7108544
951
904401
:50. 8:582879
1011
10 JO 121
:!1.7962.'»i2
1071
1147041
:52. 7261:563
952
906:5(»4
:50.H544972
1012
1024144
:i!.811<.»4r4
1072
1149184
:52.74]4111
95:J
908209
:50.H706981
1013
1026169
31.>'27tS(;0'.t
1073
1151:J29
32.7566787
954
910116
:50.8HO-'904
1014
102HI96
3I.84:5:50(;6
1074
115:5476
:52. 7719:192
955
912025
:50.90:5074:5
1015
10:HI225
:5i.><.'i90iiif;
1075
1155625
:52.7871926
950
9l:!9:50
:50.9192477
1010
10:52256
:5 1.-7475 49
1076
1157776
32..«024:598
957
915819
:50. 9:554 166
1017
1031289
:;i.H«t()4:574
1077
1159929
32.8172782
958
917761
:50.95 15751
1018
10:56:524
31.9061123
1078
1162084
:32.8:529io:5
959
919081
:;0.9677251
1019
1o:sh:!61
:!1. 9217794 !
1079
1164241
:52. 848 1354
900
921600
30.9838668
1020
1040400
31.9374:588
1080
1166400
32.863:5535
8
TABLE OF SQUARES, SQUARE ROOTS
Nc.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
1081
116^s561
32.8785644
1141
1:501881
:53.77H6915
1201
1442401
:54. 6554469
1082
1170724
:52.r^9:57684
1142
1:504614
:53. 7934905
1202
1444804
34.6(59871(5
10r*;{
1172889
:52.90«965:J
1143
1:506449
33.8082830
1203
1447209
.34.6842904
1034
1175056
32.924155:5
1144
i:5087:;6
:5:5.82:«i69i
1204
1449616
:54. 6987031
1085
1177225
: 12.9:59:5:5^2
1145
; 31 1025
;53.n:57rt4H6
1205
1452025
34.7131099
1086
in9:59«i
:52. 9545141
1146
1313316
:53.8526218
1206
1454436
34.7275107
1087
1181569
:52.969(W:50
1147
1315609
:5:!.867:5884
1207
1456849
;54.74 19055
loss
118:5744
:52.9M8450
1148
1317904
:53. 882 1487
1208
1459264
34.7562944
108'J
1185921
3:!.00OU0()O
1149
1:520201
3:5.8969025
1209
14(51681
34.770(5773
lOUO
llbr-lOO
:5:5.O151480
1150
K5225()0
33.911(5499
1210
14(54100
34.7850543
1091
1190281
33.0:;02891
1151
i:524>^01
:53.92(5:!909
1211
1466521
34.7994253
1092
1192464
:53. 04 542:53
1152
1:527104
:53.94 11255
1212
1468944
;54. 81:57904
109:{
1191649
3:;.-< 1605505
1153
1329409
3:5.95585:57
1213
1471:569
:54.8281495
1094
]196s:56
3:5.0756708
1154
1:53 17 16
:53. 9705755
1214
1473796
34.8425028
1095
119:t(i25
:5:5.o;»(i7Hi2
1155
1:5:54 (»25
:53.9-5291(l
1215
1476225
34,8568.501
I09t)
1201216
:!3 105^907
1156
13:5(5:5:56
:i4. 0000000
121(5
1478656
:54,8711915
1097
120:;409
:53. 1209903
11.57
l:53-'t)49
:54.0 147027
1217
1481089
34.8855271
109S
1205601
:53.i:5()08:;o
1158
1340964
:5 4.029:5990
1218
148:5524
34.?^998,5(57
1099
1207-01
;53. 151 1689
1 159
134:'.281
:54.O44Of^90
1219
1485961
34.9141805
1100
1210000
:i3.1(;()2479
11(50
1345t)0()
34.0587727
1220
14884(»0
34.9281984
1101
1212201
:{3.181:52(»0
1161
1:547921
:54. 0734501
1221
1490841
34.9428984
1102
1214404
:5:r. 196:5-53
11 ()2
1:550244
:54. 0881211
1222
1493284
:54.9428104
1103
l21()t;09
:5:!.2 11 44:58
1163
13525(59
34.10278.-8
122:5
1495729
:54.95711(56
1104
l21.S-'i()
:;:5.2266955
1164
1:554896
34.1174442
1224
1498176
34.9714169
1105
1221025
:5:!.241540:!
1 165
1357225
:!4. 1:5209(5:5
1225
1500(525
.34.9857114
1106
I22:5.::ui
:5:5. 25657-3
1 1(56
1:55955(5
31.14(57422
1226
1503076
35.0000000
1107
1225449
:>:;.2716095 j
1167
1:561889
;54. 161:5817
1227
1505529
:55. 0142828
1108
1-J276t;4
3:;.2ri()():5:!9 |
11(58
1364224
:54. 17(50 l.-.O
.1 »28
1507984
35.0285598
1109
1229--81
:!3.:!(»l(i616 j
1169
1:566561
:54. 1906420
1229
1510441
35.0428:509
1110
12:52100
:53,31()(i(rj5
1170
1:5(58900
34.2052(527
12:50
1512900
:55. 0570963
1111
12:54:!21
:!:5.:{:5i()6ii()
1171
l::712il
:54.2--'98773
1231
1515:561
:55.07i:55.58
1112
12:5(5541
:53.:;(()()6io
1172
137:5584
:54. 2:544855
1232
1517824
:55.0>S5609()
Hi:}
12:58769
:5:5.;;6l()5i6
1173
l:;75929
31.2490875
12:53
1520289
:55 0998575
1114
124099()
:5:i.:57(5()3^5
1174
1:578276
34 26:5(;m:54
12:54
1522756
35.1140997
1115
12^:5225
:53.:59161.-.7
1175
13-^0(525
34.27827:50
12:55
152.5225
:35. 128:5:561
lilt;
I24545ii
3;5.40t;5,-'<i2
1176
i:!S2976
:54. 2928564
12:5(5
1527(59(5
.35.142.5568
1117
1247.is9
:!:;.42I5499
1177
1:5^5:529
34.3074:5:56
12:57
l5;;oi69
:55. 1.5(57917
1118
1249924
3:!.4:5fi5(»70
1178
1:587684
34.:5220046
12:58
15:12(541
:55.171(H()8
1119
1252161
:5:;. 151457:!
1179
1:590041
:54. 3:5(55(594
12:59
15:55121
:55. 18.52242
1120
1254400
:53. 46(540 11
ll.<0
1392400
34.:5511281
1240
15:57(500
:55. 1994:1 18
] 121
1256641
33.191:53-1
1181
1394761
34.3(55(5805
1241
1540081
:i5.2i:5(5:5:57
1122
12588-4
:'.3.49t;2(5^4
1182
1:597124
34.380226^
1242
1542564
35.2278299
112:5
1261129
3:;.5iilsii
1183
1:599489
34.:5947670
1243
1545049
:55. 2420204
1124
126::37t)
:5:5.52(;i(»:t-i
1184
140185(5
:54. 409:5011
1244
1. ■.475:56
:55.256150I
1125
1265625
:53.54 10196
1185
1404225
34.4238289
1245
1550025
:55. 2703812
1126
1267i^76
:5:5 55592:54
1186
1406596
34.4:58:5507
1246
1552516
:55. 2845575
1127
1270129
:5:!.5708-jO(;
1187
140^969
34.4528663
1247
1555009
:!5.2987252
1128
1272:5^4
:!3.5857n2
1188
1411:544
:54. 4(573759
1248
15.-)7509
:55.:5128H72
1129
1274t)4l
;5:;.600.-)952
1189
141:5721
31.4818793
1249
15500.'1
35.32704:55
ll:!0
I27t);)00
:53. 6154726
1190
1416100
34.490:5766
1250
1562500
35. :14 11941
ll:u
12791(11
:'.:5.6:5o:m;54
1191
1418481
:M. 5108(578
1251
15(55001
;55.:5553:59I
li:i2
1281424
3:1.6452077
1192
1420864
34.5253530
1252
1567504
:55.:569478l
11:5::
12s:!6-;»
:5:!.(;6oo().-)3
1193
1423249
34.5:598321
1253
1570009
:55.:58:56120
11:54
128595(i
:<3.6749165
1194
14256:56
34.5543051
1254
1572516
:55.:3977400
11:55
128H225
:5:!. 6-976 10
1195
1428025
34.5687720
12.55
15-5025
:55. 41 18624
11:56
1290496
:5:5. 7045991
1196
14:50416
34.58:52329
1256
15775:56
;55. 4259792
11:5;
12927<)9
:•.:;. 7 194:',06
1197
14:52809
34.5976S79
1257
1580049
:55.44OO903 j
11:58
1295044
:53,7:5l05.-.6
1198
1435204
34.6121366
125^
1. '.82564
:55.45419.-)8:
11:59
1297:521
:53. 7 190741
1199
1437601
34.62(55794
1259
158.-.081
:55. 4(582957 |
1140
1299600
:53.76:58860
1200
1440000
34.6410162
1
1260
1587600
35.4823900 i
1
OF NUMBERS FROM 1 TO 1609.
9
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
i-.'(;i
ir)901'»l
35.51056.18
1.321
1745041
3(5.3455637
1:5^1
19071(51
37.16180S4
i"it/j
l.V.t-J{)14
3... 5-24 6:593
13J2
17476-4
:5(5 :5.59:5179
i:!82
1909924
37.1752606
l"Jt):{
l;V.).Mt;()
:!5 .".3>7113
13-23
1759:5-29
:56.:57:?o(>70
l:5.S5
191-2(5.-9
37.1887079
l'i64
i.')y7»i9t)
35. .55-27777
13-24
17.5-2976
:5(5. 3-^6- 108
1:;h4
19154.56
37.2021505
1 •,'»;.")
leooM-jf)
35.5f)6,^3-5
13-25
1755625
:56.4005494
1:5.-5
191.-225
37.2155881
1 •,'()()
lti(l-J75ii
35..5>(»-'.t:'>7
13-26
175?S276
:;(5. 4 14-28-29
i:586
192099(5
37.2^290209
lv"i7
KKt.VJS')
:;5.. 59 19434
1327
17609-29
36.4-280112
i:!-7
192:!769
37.2^224489
1 vK).-^
lt)07tt-.'4
35(i(Kt.-76
1328
176:55-4
:;6.441 7:543
i:58,i
19-26544
37.-2558720
l"2«)i»
lt)l():!()l
35.(;-2:!(l-J6-2
1329
17(5624 1
;5(5. 4.5545-23
1:5.-9
19-29:521
37.269-2903
l-JTO
IHl-i'.XtO
35.6370.593
V.VM\
176-900
:5( 5. 4 69 1(550
1:590
19321 (JO
37.28^270:57
1-J71
i6ir)in
:;5.651(»-()9
13;! I
1771..61
:;6 4828727
1:591
19:541-81
37.29611-24
l-JTi.
ii;i7'.t-4
35.t)6510'.IO
13:!2
1774-224
:56. 49(55752
1:592
19376(54
37.3095162
1-J7:{
1 i)-20:,2[)
:;5.(;79 1-255
1333
177(5.S<9
:!(5.5 10-27-25
1:593
1940419
37.:«-29152
1-J74
1G-J:'.07()
;;5.69:!1366
1334
17795.56
:{6.52:59(547
1:594
1943-2:5(5
37.336:5094
i5>7r)
Hj-j.")*;".*.')
35.70714-21
1335
178-2-2-25
3(5.5:57(5548
1 :595
194(5025
37.:M9(i988
I'iHj
I(j-i8l7«)
35.7-2114-22
1336
1784>9ti
;J6.55i:5:5s-!
1:596
1948.-16
37.3630834
l-i7l
it;307-.'i»
35.7:>5i:;67
1337
17?7569
;5(5.5(5:.0106
1:597
1951(509
37.3764632
\-27f*
lH33->4
35.749 i -258
133S
1790244
:56.5786823
1:598
1954404
37.3898382
l--'7<)
1»)3.".8U
35.71131095
1339
179-2921
:5(5.592:!4'«9
i:!99
19.57-201
37.4032084
1-M)
1(;384(K»
35.7770876
1340
179. ,(500
:5( 5. 60(50 104
1400
19(50000
37.4165738
1-,'81
i()4(Hitil
35.7910(i03
l::41
17982-^1
:;6.(511t(56(58
1401
1962801
37.4299345
U&2
KM 35-24
35.80.50-276
1342
180(J964
;56.6:5:53I81
1402
19(55604
37.443-2904
r.v:5
I(i4<5(l8i»
35.8189^94
1343
180:5649
:{(S.(51(;9(544
140:5
19(58409
37.4566416
l-,'>4
l(M8()5t)
35..-^3-29157
1344
1806:;: 56
:5(5.6(50605(5
1404
1971216
37.4699880
Itid")
U)r>iv!->.")
35.846S'.)66
1345
1.-090-25
:5(5.ti74-2416
1405
19740-25
37.48:5:5296
l<)r>371>6
:'..5.r6081-21
1346
1811716
:i6.(;.«'787-J6
140(5
19768:J6
37.4966665
l-,'d7
I6r)63t;9
35.87478-2-2
i:i47
l'^144(»9
:?6.7014986
1407
1979649
37.5099987
l::iAS
1658944
35 8887169
1348
1817104
:56.7151I95
1408
1982464
37.523:5261
l-J-^9
I6(>ir)-ji
35.90J6461
1349
1-19-01
;;6. 7287:5.53
1409
1985-281
37.53(5(5487
l-2'.tO
l()t)4lU0
35.!>!(m699
1350
18-2-2500
:56. 7 4-255461
1410
19.-8100
:}7. 5499(567
l-ilH
ItihtiC.al
35.9301--8^
1351
1825-201
:{6.75.59519
1411
1990921
37.5(5:52799
1-J1I2
ii;()9-it;4
35.iM44015
1352
18-27904
:5(5. 76955-26
1412
199:!744
37.5765855
12'J-S
ltiVlH49
35.958309-2
1353
]8:}(i609
:56.78314.'<3
1413
1996569
37.58989-22
\-2[)i
lti744::ii
35.97-2-2115
13.54
]8:;:5316
3(5.79!57:51;0
1414
1999:596
37.(5031913
l-2[)b
16770-25
35.9861084
13." 5
18:<60-25
:56.f« 103-246
1415
200-2225
37.6164857
1-Ji»6
Iti79tu0
36.0000000
1356
18:587:56
36.'e2:;905:5
1416
200.5056
"7.6297754
lv!97
16^-i-.'09
36.01 388t;-2 1
1357
1841449
36.8374.-^09
1417
20078.89
37.6430604
l-^!)8
lt;8».-(t4
36.0-277671
1358
1844164
:56.8510515
1418
2010724
37.6563407
Iv'iCj
1687401
36.01161-26
1359
1846881
:;6.8646172
1419
20135(51
37.6696164
l:itiO
biyoodo
36.05551-28
13(;0
1849600
36.8781778
1420
2016400
37.6828874
1301
169-J601
36.069:!776
1361
I852:wi
36.8917:535
1421
2019-241
37.6961536
i:;o->
l*)95-.'04
36.0-:!-2:!7l
1362
18.5.5044
:?3.9052842
14-22
2022084
37.7094153
1303
1694809
3i;.09709l3
1363
1857769
36.9188-299
1423
20249-29
37,72-267-22
i:?04
1700416
;!6. 110910-2
1364
18(50496
:56. 9:5-2:57 06
14-24
20-27776
37.7:^59245
1 ;!().')
17030-25
36.1-247837
1365
18():i22.5
:56.9459064
1425
20:50625
37.7491722
i:5Ut)
1705636
36.1386-2-20
13()6
18H5956
3(5.9594:572
1426
20:53476
37.76-24152
1:W7
1708->49
36.1524.550
13()7
1866689
36.97-296:51
14-27
2036:5-29
37.7756535
i:kw
1710864
36.166-2-!-26
1368
18714-24
:56.9864840
14-28
20:59184
37.7888873
i:!oy
1713481
36.18010.50
1369
1874161
:i7.000(»000
;4'29
•2042041
37.8021163
1310
1716100
3(5.19392-21
1370
187(5900
37.01:55110
14:50
2044900
37.8153408
KUl
17187-21
36.-2077340 i
1371
1879641
37,0270172
1431
-2047761
37.8285606
1312
17-,M344
36.-2-2 15406 i
1372
188-2:5.-^4
37.0405184
1432
2050624
37.8417759
1313
17-23969
36. -23534 19 |
137:!
1885129
37.0540146
1133
2053489
37.8549864
1314
17-J«)596
36.'249i379
i:?74
18-7M7(5
37.0(575060
14:54
2056356
37.86819-24
131;")
17-29-2-!5
36.-2626-2rf7
1:575
1890(525
:}7. 0899924
1435
2059225
37.881:^9:58
1316
1731'^56
36.-2767 U3 |
1:576
1893.576
37.0944740
1436
2062096
37.8945906
1317.
17344.89
36.-2904946 ;
1:577
1K>6129
:i7. 1079506
1437
•2064959
37.90778-28
131H 1
17371-24
36.304-2697 :
1:578
l.-'98"'.-^4
:57. 1214-22!
14:58
2067844
37.9209704
1319
1739761
36.3l8o;;96 :
1:579
1901(541
:'>7.i:U8,89:!
14:59
2()/'j721
37.9341535
13'i0
174-2400
36.33l!:'U42
]:i8U
1904100
:57.14f:5.M2
1
1440
•207:5600
37.9473319
10
TABLE OP SQUARES, SQUARE ROOTS
I-: — —
No.
Square,
Sqre. root.
No.
Square.
Sqre. root.
No.
Square.
Sqre. root.
1441
207(>481
37.9605058
1495
22:55025
38 6652299
1548
2396304
39.3446311
1442
2079364
37.97:567.M
1496
22:58016
38.6781.59:5
1549
2:599401
:59.:557:5:573
1443
2082249
37.98689:58
1497
2-.M10U9
:58. 6910843
1550
2402500
:i9.:5700394
1444
2085136
38.()00()0(I0
149><
22110(14
:;,-'. 70 nto50
1551
•2405601
:i9.: 18-27373
1445
2088025
38.01315.")«)
1499
2247001
:5f-. 7169214
1552
2408704
39.39.54312
1446
2090915
:58.o.6:!()()7
15(10
225(1(100
:5-^.7298:!:!5
15.)3
-2411^09
159.408 1210
1447
209:1809
:5rt.03;t4..:;2
l.idl
2-J5;5O01
:58.7 4274i2
1554
-241491()
:59.4208067
1448
2096704
:58.(t.".25>t52
15()-J
•J.;. 6004
:58.7. 56447
1 555
•241c!0-25
39.43:54883
1449
2099f.01
:58. 0657:526
1.-.03
•rJ.)9(iO.!
3.-<.76-.i4:i;»
15.56
24211:56
39,4461658
1450
2102500
38.07 rtH6.M5
1 50 4
2v:()-^ol(i
::8.7.-l 1:5.-^9
1.5.57
2424-249
:59.458-:593
1451
2105401
38.09199:5".!
1. ")().')
22r..-)0-J5
:5.-'. 794:129 4
1558
24-27:5(;4
:59. 4715087
145-2
2108:i04
;58.10.'>117.-
l.'.OO
•J2ti.H>3i
:'.«.H).!2 !.'>.-
i 559
24:!(t4>l
:59.4841740
1453
2111209
:58. 11 8:5:171
1.07
22710 49
3-i.82(J0978
1560
24;5:i6UO
:59. 4968:153
1454
2114116
:58.1313.')19
l.-)ttS
•.•,'71064
:58.8:;297.57
1561
•24:51.721
:{9...Uit49-25
1455
2117025
3,-!. 144462-.:
i.-.on
-277081
:}8.-'4:.H4i>l
i:.62
•24:59844
39.5-221 4..7
1456
21199:i6
;58.1."):568l
15 1
•J-^-Ol(»0
;5>'.8.5>7184
1 .5(5:5
2442969
:59.5347948
1457
2122849
38.1 7066. t3
151 1
•j2f^;;l.:l
:>8.871.'"'^::4
15()4
244609(i
:59..5474:599
1458
2125764
38. 1^:57 ii62
1..12
2286144
:i-.88 4 44 4-'
1565
2449^2-25
39.5600809
1459
2128f.S4
38.1968.'.8.". 1
1513
2-.-'91()9
:5H ,-<i)7:soo()
i 156ii
24523.".6
:59.57-27179
1460
2131600
38.2099463
1..14
22'.tJl9ti
:iH.910l529
: 1.567
2 155489
:59.5S5;}508
1461
2i:{4521
38.2220297
1515
22'.t.".225
:5.>i. 92:50009
15()8
245H(;-24
:{9.5979797
146-i
21:57444
:{H.2:561085
1..16
•J-.;'.t-25()
3s.i):{.-,8l47
1569
•2461761
:59 610(j046
1463
214U:569
38.249 18J9 j
1517
2:5ol-i>9
:iS.<t4M)''4 1
1.570
24ii4;t00
39 62:5-2255
1464
214:}296
:W.2()2i.".-^9
151H
•.':5(»4:534
:5s.i>r,i.>i94
1571
•24(3H041
;5i).6.1584-24
1465
2146225
38.-J7531S-1
1519
2:5073til
3,S.'.t7 4:5.-05
1572
•2471184
;59.64,-^45..2
1466
2149156
:58.2'-8:5794
1.520
2310400
.'•KA)!*. 1774
1573
247 4:519
3:>.66l0640
1467
2152089
:58.:!0 14:560
1521
231:5441
:>9.ooooooo
I..74
2 477475
3i».67.16{i."'8
1468
2] 55024
:58.:n44H-^i
15J2
2316184
:!9.ol2-^l.-i4
1.57..
24^06-25
: 19. 6862- .96
1469
2157961
:58.:5275358
1..23
2319.529
:5:( (i-J. 6:526
1576
2 4~!;;776
:59. 1.98- 665 \
1470
2160900
:i8.:54(»5T90
1.524
2:522576
:5it.O.;,-il426
1577
24-(>929
:59.71145i»3 \
1471
216:5841
38.35:56178
1525
2:525()'J5
:5i».(i.',i24-:!
1..78
24.H)08J
:J9.7j4(4.^1
1472
2166784
38.3661)522
1526
2:;2>(;76
:!9.064ol99
1579
•249:5241
:59.7:!6,)3j9 '
1473
21()9729
:58.:5796H21
1527
2:5:! 1 7 -^9
::ii.()7';.-<i73
15s()
•2496400
:59.749j1:58
1474
2172676
38.:5927()76
1528
2:'.34781
3:t. 0-96406
15-1
2499561
39.7617907
1475
2175625
:J8.40,-)7287
1529
2:5:57r^41
39 102429(>
1.5H2
2502724
:59.774:56:!6
1476
2178576
:i8.41874.-)4
15:50
2340900
;'.9. 11.52144
1.5.S5
•25051-9
:59. 7669: 1-25
1477
2181529
:58.4317577
1531
2:54:5961
5tt. 12791 (51
15.-^4
250i)056
.•59.799497() .
1478
2184484
38.4447656
i.5:«
2:54702!
19. 1 1 ^716
I5S5
251-2-2^25
3:).>'l-2o5S5
1479
2187441
:58.4577691
1533
2:5500-^. '
.9. i.5;;54:59
1 ..S(,
■251.5:596
;59.'^i-.'4 11.55
1480
219(>400
:58. 4707681
l.-.:54
•j:5.-.:5l.".(i
!9 ((;6:!120
1.5-^7
•251-5(i9
:i.>..-'376646 '
1481
2193:361
38.48:57627
1.-):^)
2:55t;-.>J5
:'>9 179(»7()0
15M.S
•2521744
:59.84'.)7177
1482
2196:J24
38.4967530
i.-.:i6
2:!59-J96
9 191-^:5-">9
1 . .S9
•2521921
:}9.:5()^22()28
1483
2199289
38.5097:590
15:57
2:!C,2:5(i;»
;;'.».-.'04V.>15
15;m)
•25J-^I00
;59. -74-040
1484
2202256
38.5227206
15:58
2:5(^5444
:5'.). •217:54:51
1.591
•2.5:! 1^281
:i9>-'7:54l3
1485
2205225
38.5:556977
15:59
2:568; .21
:t9.2:5oo',»()5
1.5'.I2
•25:'.-i4(S4
:59.H9.»8747 [
1486
2208196
38.5486705
1540
2:571600
:59.^24-j.'<:5:J7
1.513
25:!7()49
:19.91-'4041 ir
1487
2211169
:58.56163S'J
1541
2:37 4 •■."'I
:59.-j.".5:)7-2-'
159 4
•254(K5t;
:59.9-24'.t295
1488
2214144
:58.57460:50
1542
2:57 7764
:5'.> •i6H:5(t7H
15;»5
25440-25
:i9.;»:i745l 1
1489
2217121
38.5875t;27
154:5
2:5-0-^19
:59.2-'io:'>87
1 .">9t)
•2.547-216
:i9.749.-'(>-<7 p
39.9624-^24
1490
2220100
:58.600r>l81
1544
2:n:59;!6
:!9.-i9376.">4
1.597
•2.550109
1491
2223081
38.61:^4691
1.-.45
2:5^'7()25
:;9.3()61SH)
1.598
-2.55:5ti04
:59.9749922
1492
2226004
:38.6264l.->8
1546
2:590116
:59.:5 19-2065
1599
-2.5.56-^41
:59.9-^749-'0
1493
2229049
38.6393582
1547
2:59:5209
:59 :53l9-.!08
ItioO
-2560000
40.0000000
1494
2232036
38.65->2962
•-i
tabtje: II. a.
AREAS OF CIRCLES, FROM ^ TO 150,
[Advancing by an Eighth.']
Diam.
Area.
Diam.
Area.
Diam.
Area.
Diam.
Area.
Diam.
Area.
bV
.00019
4.
12.5664
10.
78.54
16.
201.062
22.
380.134
,u
.00077
■K
13.364
■H
80.5157
■A
204.21(5
■A
384.465
Uf
•*8
14.1862
■A
82.5161
•M
207.394
ft
388.822
tV
.00307
15.0331
■A
84.5409
■A
210.597
393.203
i
.01-2t>7
■I 2
15.9043
16.8001
:M
86.59
88.6(543
'A
■A
2l3,r25
217.073
%
397.608
402.038
tV
.027H1
■¥
17.7205
■%
90.7628
■A
220.:;.')3
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406.493
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18.6655
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92.8858
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440.972
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41(5.477
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20.629
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97.2055
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230.33
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420.004
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21 6475
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99.4022
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233.705
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237.104
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429.135
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103.8091
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240.528
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433.731
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106. 1394
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471.436
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520.769
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151.201
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310.245
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525.837
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AREAS OF CIRCLES.
TABLE.— (Continued.)
Diam.
Area.
Diam.
Area.
1
Diam.
Area.
1
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Area.
Diam.
Area.
28.
61. "..7. ".4
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190.-.. 03
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2485.05
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1934.15
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57.
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1973.33
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2562.97
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201-2.89
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2619 36
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AREAS OF CIRCLES.
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6.593.54
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4778.37 85
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5657.84
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6629.57
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4824.43
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3.2
•78
5459.62
%
6411.86
h
7417.08
^ o
'A
3793.68
•>"
45'.t»i.36
•3:>
547«i.Ol
%
613262
H
71 61!. 21
3807.34
'%
4611.39
•%
5492.41
%
6450 4
li
74-5.:56
;}8-*1.02
■%
46-2i;.45
•4
550-^.84
%
6168.21
¥'
7.')04.r):)
-k
3834.73
■X
4641.53
.%
5525.3
%
6486 04
h
7523.75
14 AREAS OF CIRCLES.
T\JiLE~{Continued).— {Advancing by a Quarter nnd a half]
Diam.
Area.
1
Diiim.
Area.
Diam.
Areii.
10207. Oii
Diam.
Area.
11S82.32
1
Difim.
!
Area.
ii'^.
7.')1-J.9r(
105
h(>.'i0.03
Ii4.
123.
139.
15171. 71
■H
75i^.v4
■U
8;0(t.3-J
•%
1025 1.>-
■Va
11030.67
■}2
15284.0>!
7r).-^i.:)i
■h
8741.7
'h
10296.79
■h
1 1979.2
140.
153.t3.«4
7HU0.H-
-A
87f-3.18
■%
10341.8
■h
1 21127.66
v'o
1550.3.0^
■^>
7620.15
106.
8>24.75
115.
li)3;^ti.91
124.
12076.31
141.
1561 4. .',3
•^
imd.b
■%
8866.43
■M
l<i43J.12
■H
12125.05
■h
15725.47
•1
99.
7t)o."'.?*8
'%
8908.2
'%
10477.43
•\.
12173.9
142.
15>:^36.8
767f<.2;^
•%
r!"50.07
■%
10522. r;4
■h
I222V.S4
■'A
1594-:.52
76i>7.71
107.
8902.04
1116.
10..68.34
125.
12271.87
143.
16060.54
■%
7717.16
•Va
9U34.11
-H
10613.04
.1.,
I237tt.25
■^^
16173.15
7736.6:{
■h
9{t76.28
■%
106.-)0.64
12(). ~
12469.01
144.
l62(-6.05
77r)'o.i:{
■h
91;K53
1 Si
10705.44
■H
12568.17
•>2
16399..34
■H
777.').66
108.
OltiO.iU
1117.
10751.34
127.
12667.72
14.5.
16.M3.03
•%
7795.2
■H
9-J03.37
' -^
'0707.34
^2
12767.66
■}2
16627.il
"4
1(»0.
7f^l4.7ri
h
9245.ii2
: y^
)()r-4;i.43
128.
12H()7.99
146.
H;741.59
7f«:?4.;{rt
'%
9-J8.i.58
i ..^4
10-r0.ti2
•>2
129()8.71
•3-2
16856.44
7ri.")4.
100.
9331.34
118.
1 0935.0
129.
13069.84
147.
16971.71
•K
7i»9:{.:i2
■H
9374.10
■Va
109.S2.3
■}2
13171.35
.i.f 17087.36
.>o
79:i2.74
■h
9U7.14
■%
11028.78
130.
13273.26
148. 17203.4
-fi
7972.21
3/
•74
9460.19
■%
11075.37
■H
13371.55
K 17319.83
101.
8011.87
110.
9.>03.34
119.
11122.06
131.
13178.25
149. ": 17436.67
•^
8or.i.r)8
■H
9546.69
■Va
11168.83
•>^
l3.-)t^l.33
•H
175.53.89
•3
80;)1 39
A
9o89.'.'3
•>o
11215.71
132.
136«4.81
150.
17(571.5
•^
8131.3
■%
9633.37
■%
11262.69
v^
i.378.-.t)7
■'A
177c9.51
102.
8171.3
HI.
9676.91
120.
11309.7 6
133
13902.94
•K
8211.41
.1
9720.73
•%
1 135ti.03
•>2
13997.54
•S
825 1. (il
1
•/Si
9764.29
■h
11404.2
134.
14102.64
■^
8291.91
3>
•74
980.>.12
■%
11 451. .57
M
1420.-.0:
103.
8332.31
112.
98.52.(16
121.
11499.04
135. "
14313.01
•>^
6372.8 i
■H
9896.{)0
■Va
11546.61
■'A
1442(1.14
■K
8413.4
■h
9940.22
'%
11594.27
136.
14526.76
■^
8464.09
•^
99>'4.45
Si
11612.09
■H
14633.76
lOJ.
8494.89
113.
10028.77
122.
116^0.89
13V.
14741.17
•^
8.")35.7a
•H
100:3.2
H
11737.85
■H
14H4.'^.96
•>•>
8576.77
.h
10117.72
.^■i'
117H5.9:.
138.
149.57.16
•^
8617.^5
%
10162.34
.% Ile34.06
1
M 15065.73 1
1
To Compute the Area of a Diameter greaterthan any in the preceding Table.
Rule — Divide thp dimension by two, three, four, etc., if practicable to do so. until it is reduced
to a diameter to He found in ihe table.
Take the tabular aiea for the diameter, multiply it by the square of the diviser, and the product
will give the area required.
Example — What is the area for a diameter of 1050 ?
1050-4-7=150 ; tab. area, 150=17071 5, whichx 7 ^=865903.5. area re jMirei. ;-
To Compute the Area of an Integer and a Fraction not given in the Table.
^iTLS — Double, treble, or quadruple the dimension given, until the fraction is increased to a
whole number, or to one of thuse in the table, as i. i, etc., provided it is practicable tc do so.
T ike the area lor this niameter; and if it i.s double ot that for which the an'ais required, take one
fourth of It ; if treble, take one 9th. of it and if quadruple, takj one sixteenth of it, etc., etc.
Example — Required the area for a circle of 2. ^''^ inches.
3
Iff
X2=45, area for which = 15.0331, wliich -5-4= 3,708 ins.
TABLM II. l>.
ClfiCUMPEBENCES OP CIECLES, PHOM ^ TO 150.
[Advancing by an Eighth.}
Diaii
t
u
i
I.
■38
■M
■%
•>_
3/
.1^
■$
■'A
Ciicuin.
.04iK)y
.O'JSIT
,19»i3r.
.*.J9-.'7
.589
.7854
.9«i7r)
1.1781
1 :!744:>
l.i")708
1.7(5715
l.%:;5
•^.1;VJ85
■l.l\>i'd
•2.94.'..'.-)
:i. I 4 1(5
:<.:".:n;{
:{.y.'7
4.:{197
4.7 iv:
f). IO..I
5.4978
5.-905
6.-i8;;2
(5.(5759
7.(KJH(5
7.4(5i:{
7..-54
H.2t(i7
8.6:!94
9.(j:{-.'I
9.4-248
9.8175
i(».-l()2
l(t.()0«9
l(>.99:)6
I1.:5ws:{
11.781
I i. 17.17
Diam.
%
H
¥
I
1.^
3/
7''
>8
Ciicum.
Diam.
12.5(564
10.
1-2.959J
■H
1:<.:?518
•¥
l:V7445
■%
H.i:{7-i
■%
14.5-i99
■'4
14.9-2-26
■%
15.31.5:!
■%
15.708
11.
1(5 1007
■ M
l(i.4;»:!4
M
l(5..-;ft6l
■4
17. -2788
.y>
17.(5715
%
l8.0(il-2
%
If-f. 4.5(59
■%
18.849G
\-z.
19.-24-J:}
■%
I9.«:r>
%
•20.0^77
%
•20.4J04
•%
:2(l.8l:?l
•%
•21 -2058
■%
21.59."'5
•%
■21 99 1-2
13.
•22.:{8:{9
%
•22.77<5t5
>4
•23.1(593
%
•23 5(5^2
.%
•23.9.547
■%
24.3174
■%
21.7401
.%
•25.13-28
14.
•25.5255
%
•25.sn8^2
H
26.3109
•%
•2(5.7036
%
•27.0963
■%
•27.489
3i
•27.8817
■%
•28.-2; 41
15.
•28.6()71
%
29.0598
■H
•29.4.5-25
.%
•29..-^4.52
■%
30-2379
■t^
30.6306
■\
31.6233
•%
Circum. ' Diam.
31.416
31.8087
3-2. -20 14
32 59U
32.9f56.-5
33.3795
:{3.772i
34.1649
31.5.57(5
34.9503
35.343
36.1-284
36 5211
3(5.9138
37.3065
37 6992
38 0919
38.484(5
38.8773
:j9.-27
:!9.6(5-2'/
40.0554
40.4481
40.810-
41.-2335
41.(5262
42. (t 189
42.4 1 16
42.-043
43.197
43. .-.897
43.9824
44.3751
44.7678
45.1605
45.5532
45.9459
4(5.33i-«6
46.7313
47.124
47.51(57
47.9094
4-'.302l
4?^. (5948
49.0875
49 4H02
49.8729
16
17
18
19
-20
21
%
%
%
%
%
!^
H
>4
K
h
n
J?
Ys
H
H
H
%
%
Circmn.
50.265(5
50.6.583
51.051
51.4437
51.8364
52.-2-291
5-2.6;; 18
53.0145
53 4072
53.7999
54.19-26
54.5853
54.978
55.3707
55.7634
56.1561
56.54-8
56.9415
57.3342
57.7-269
58.1196
.58.51-23
58.905
59.2977
59.(5901
60.08:51
(50.4758
60.86-5
(51.-2612
61.(5.5:J9
62.0466
(52.4:593
62.8:?2
6:i.-2247
(53.6174
64.0101
64.4028
64.79.55
65. iH-^a
65.. 5809
65.97:56
66.:{(563
66.759
67.1517
(57 r '44
67.9:i;i
68. :;-298
68.72-25
Diam.
22
23
•24
-25
•26
K
Yt
%
%
'J
¥■
%
K
P
I
H
y^
Circum.
69.1152
69.5079
69.9066
70.-2933
70.686
71.0787
71.4714
71.8641
72.2568
.72.6495
73,04-22
73.4319
73.8-276
74.-2^203
74.613
75.0057
75.:,984
75.7911
76.1838
76.5765
76.9692
77-3619
77.7546
78.1473
78.9327
79.:V2o4
79.7181
80.1102
80.5035
80.8962
81. -2889
81.6816
8-2 0743
82.467
82.8597
83.^2524
83.6451
84.0:578
84.4305
84.8232
85.2159
85.6086
86.0013
86.394
86.7867
87.1794
87.5721
16
CniCUMFERENCES OF CIRCLES.
TABT.E.— (Conifinucrf.)
Diani.
Ciiciim.
Diam.
Ciicum.
Dmm.
Ciicum.
Diam.
1
Ciicum. Dia
im.
Ciicum.
•2'i.
ciT.JMUs
3.')
lt)'.».9.'it) :
42
l:! 1.947
49.
ir.{.938 56
175.93
•<«
«5;.:i;.7.)
^8
1 Id .WW \
%
132 :4
K
154.3:;!
'%
176.:{22
■k
n-^.VM-i
^4
lh).7ll
•>4
13j.7:i.5
■H
154.724
■Ya
176.715
Sy. U:>9
•%
lli.l:u
•%
13.5. 125
■%
155.117
H
177.108
'%
89..-):}5tJ
•%
lll.i.27
■%
1:;3.518
•K
1.55.509
'A
177.5
■%
8S>.9-283
•%
111.919
•%
133.911
■^B
155.902
H
177.893
•1^
90.3-il
•%
112.312
■il
134.303
■%
156. -295
K
178.286
'%
90.7137
■%
112.705
■%
134.696
■%
156.687
%
178.679
•29.
91.10t)4
36.
113.01W
43.
1; 15. 089
5(t.
I..7 08 57
179.071
'%
91.4991
■%
1)3.49
■%
135.4^1
■%
1.57.473
H
179.464
-¥
91.H91rt
■K
113.'-K{
H
l:!5.874
■H
1.57.865
H
179.8.57
%
9'-i.-JH4.">
%
m.276 ,
%
i.3i).j67
■H
1.58.258
H
1«0.249
■h
9-.'.h:7->
%
ll4.()ti< 1
M
136.66
■Yi
l.5-<.651
Ys
180.642
■H
9;i.0«9'.t
'A
ll.).(t.il 1
H
137.0.'.2
■%
159(144
%
181.035
i^
9:{.4r)-2H
X
11...454
%
137.415
■%
159.4:{6
yA
181.427
■%
9:i.8.)r)3
%
n...84t)
%
137.K}.-!
Vb
159.82:?
Yb
181.82
:iO.
94.248
37.
116 239
44-
13«.23
51.
16(t.222 58
1V2.213
•H
94.(1407
H
IK) 632
■%
138.623
•M
160.614
H
182.606
•H
95.o:{:{4
K
117.025
%
139 016
■Va
161.007
Ya
182.998
■%
95.4-J<)l
%
117.417 i
%
139.408
.%
161.4
H
I83.:i9l
'Vi
9:).8I88
■Vi
117.81
•K
I39.rt0l
■H
161.792
%
183.784
•1^
9t).-J115
%
1 1-.2U3
%
140.194
■%
l()2.1rt5
%
184.176
■4
9«.604->
h
nf<..-.95
%
141 •..■>H7
Va
162.578
Ya
184.569
o/^
9B.St9t)9
■h
Jl,-.986
■%
140.979
%
1(52.971
%
184.962
31.
97.:{896
;;8
119 301 '
45.
111.372
52.
lH:!.:i6:; 59
lr^5.:{51
•%
97.7823
h
119.77 1
%
141.765
■%
16:5.7.56
H
ln5.747
■H
98.17;>
■fi
120. I6t)
i
142.1.57
164.149
H
1-^6.14
H
98.5H77
%
!-i0...59
%
142.55
164. .54 1
%
l.--6.53:$
•K
98.9604
^
12().9.'.2
M
1 12.943
•%
1()4.931
>2^
186.925
H
99.3.-):{l
'A
121.314
%
14:{.33t;
■%
165.327
%
1.-7.318
■X
99.74r>H
%
1: 1.7:57
■^
143.728
■h
U)5.719
Ya
1H7.711
%
100.1:^85
■'A
12.; 13
■'A
144.121
%
16().112
Yb
188.1(»3
3-2.
100..5312
39
12... -.22
46.
141511
.53
1(i6.505 60
188.496
•%
100.9239
■Vb
l22.1)l.->
■%
141.906
■%
166.89!?
H
188.rt.-«9
:¥,
101.31t5(>
■Va
I2:!..i08
■H
145.299
■Va
167.29
H
189.2^1
10i.70.»3
■%
123.701
■%
145 692
%
167.6-'3
H
189.674
■%
102.102
■%
124.093
y^
14().084
-y^
168. 07()
H
190.007
■H
102.4947
■%
121.4S6
■%
14.1.477
■%
16H.468
%
190.46
•^
102.8S74
■%
124. r:9
■%
lit; -7
■%
l(i-r6I
Ya
l9(t.8.52
■ys
103.2801
■'A
12.-..271
■%
147.263
■%
169.254
%
191.245
•.V3.
103.073
40.
125.664
47.
147.655
54.
169.(546 61
191.53a
'%
104. 00«
H
126057
M
14.-.(tl8
■H
•70.039
H
192.03
%
104.4,58
■H
126.44;»
■%
148.441
■ Ya
170.4:12
Ya
li>2.423
104.8r)l
■%
126.842
■%
1 4.-'. 8:53
•%
170.-i25
%
192.816
-%
10.5.244
A
127.235
■%
149.226
■y^
171.217
%
193.208
•%
105.03H
4
127.627
%
149.619
■%
171.61
%
193.601
•¥
100.0'.'9
■%
12>.02
■%
l.O.oll
%
172.003
Ya
19:5.994
o ^
10H.422
%
!28.4r!
■%
150404
_.%
172.:i96
%
194.:{87
34.
106.814
41.
128.806
18.
150.797
55.
172.788 . 62.
194.779
■%
107.207
H
1-9.198
■M
151.19
■%
173.1^1
H
195.172
t
107.6
H
129.591
M
151.5H2
•H
173.573
H
195.565
107.5n)3
4
I29.W4
■K
151.975
4
173.966
H.
195.9.57
'%
108. 3H5
■H
130.376
-H
i:.2.3()8
•K
174.3.59
Y2
196. :15
'%
108.778
■^
130 769
■%
152.76
•^
174.7.52
Vb
196.743
'¥
109.171
4
131.1(2
■fi
I. -.3. 1.53
■H
17.5.144
Ya
197.1:55
-%
109.563
■%
131 5.-.4
■%
153 516
•%
175.537
Yb
197.528
CntCUMFERENCES OF GUCLES.
TABLE.—iContinued.)
17
Diaiii.
Circum.
63.
64.
■%
'■H
■h
■%
•74
65.
()6.
•7a
■%
'•%
•>4
■%
■h
CT.
6'i.
69.
•M
.%
■%
■H
■%
I
:|
■>^
•K
'%
•%
197.L»-J1
liK:il4
lys.Tue
i'.ty.Ui>9
iyy.-49-j
199.^rt4
2UU.-.>77
•-iOO.C)?
i>41.06-i
201.4;-);-.
•^Ul.C!4ri
202.241
2U2.6:i:{
20:5, 026
20:}.419
203..- U
204.204
204.597
2(M.9.''9
20.S.3O2
2IM.775
20t). Itid
•-:06.;-)6
•^06.y53
207.346
207.73f<
20ti.l31
20ei.r)24
20.-i.916
20i».309
209.; 02
210.095
2i0.4s7
2i0.p>,-i
211.273
21l.oti5
212.0.">8
212.451
212.843
213.-.>36
213.629
214.022
214.414
214.807
215.2
215.;')92
215.985
216 378
216.77
217.163
217.;->;')6
217.948
218.341
218.734
219.127
219.519
Dium.
71
73
/o
)8
%
'A
4
H
h
H
%
%
%
¥
f
A
%
I
Circum.
219.912
220.305
2-0.697
221.09
•^21.483
221. ",76
222.2t;8
222.r.t)l
223.051
2v:3.146
223.M39
224.232
224.624
32.) 017
225.41
225.803
226.195
226.5f<8
226.981
227.373
227.766
228.159
22,-^.:.51
228.944
229.337
229.73
230. 122
230.51;->
230.908
231.3
231.t;'.'3
232.0^6
232.478
23-.'.87 I
233.2()4
233.657
234.049
234. 443
234.«:!5
235.227
235.62
236.013
236.405
236.79s
237.191
23;.;-)84
237.. <76
238.369
23-<.762
239.154
239.547
239.94
240.332
240.725
241.118
241511
Diam.
80
81
82
83
Va
Va
X
I
Circum.
2)1 903
2 4-:. 296
242 6."'9
213.(181
243.474
243. .-67
244 259
•-'44.6.2
245.045
24;). 438
245.Ki
246.223
24ti.616
247.008
247.401
247.794
24r*.l.-6
24^^.579
24-.972
249.365
249.7;-)7
2;-.().15
2:)0.543
2;50.935
251.:!28
251.72!
2.52.113
2:)2.506
2.52. >99
2. 52. 2. '2
253. (i84
254.077
254.17
254.862
25.->.255
255 ti48
256.04
2.".t>.433
2.56.-26
257.219
2;-)7.61l
2;58.004
258.:!97
2i-)8.7f*9
2.59.1-2
259.575
2.-.9.967
260.36
260.753
261.146
261.53rt
261.931
262.3J4
262.716
2(i3.109
263.502
Diam.
84.
■K
•If
e6.
•H
Va
%
>8
89.
H
■A
■¥
■H
•H
■%
Circum.
90.
1
-.■■X
263.894
264. 2n7
264. 6n
26;) 073
2f.5.465
265.. -5.''
2(16. -.'51
2()6.(i43
■.;67.h:;6
267.429
267.821
26f«.2ll
2()8.6U7
26-.9it9
26!».:!92
269.785
27().I7H
270.:.7
2T(t.'.)63
2:1.3:6
271.7).-'
•J72.ni
272.-534
272.926
273.319
27:: 7 12
274 1(15
374.4'.i7
■.;74.89
275.2.-^3
2:."..()75
27t).()i;"'
•J76. Kil
27().-..3
277.24(1
277.629
27 .-'.032
278.424
•j7«..-^!7
279 21
279.602
279.995
280.388
280.781
281.173
281.5()6
281 .9:59
2e2.35l
282.744
2.-'3.137
2-^3. .529
283.922
284.315
284.708
285. 1
285.493
Diam.
91.
5Z
92.
93.
94
•%
78
■%
95.
■Va
90.
97.
^8
•%
■Va
■Vs
M
I
Circum.
H
I
285.886
286.278
286.671
287.064
287.456
287.849
288.242
288.634
289.027
289,42
289.813
290.205
290.. 598
290.991
281.383
281.776
292.169
292.;)62
292.954
293.347
293 74
294.132
294.;525
294.918
291.31
295.703
29().096
296.489
296.881
297.274
297.667
299.059
299.452
298.845
299.237
299.63
300.023
300.416
300.808
301.201
301.;-)94
301.986
302 379
302.772
3(13.164
303.5:57
303.95
304.343
304.735
305.128
305.:-)21
305.913
306.306
306.699
307.091
307.484
18
emCUMFERENCES OF CIRCLES.
TABLE.— {Continued.)
Diam.
l»H.
99.
loo.
101.
•I
loy.
•1^
103.
104.
•4
Circum.
307.877
30H.'i7
308.662
309.0:m
3U9.44H
30i),«4
310.233
310.626
3U.0lci
311.411
311.H04
312.196
3l2.r).'S9
312.9.-^2
313.375
313.767
314.16
314.945
315.731
316.516
317.302
31d.M87
318.«72
319.65H
320.443
321.229
322.014
322.799
323.5,-^5
324.37
32.=.. 156
325.941
326.726
327 .5 1 i
328.297
329.083
Diam.
105.
106.
Circum.
107
•4
108.
•Va
109.
110.
•k
111.
■Va
112.
•>4
113.
.%
1 /
321). ^68
330.(>.)3
331.439
332.224
333.01
33;;. 7 95
334. .58
3;i5.366
3:Ui. l.M
336.937
3:^7.722
33.-'.507
339.293
340 078
340.8ti4
341.649
342.434
343.22
314.005
344.791
345.576
34(i.3til
347.147
347.i(32
34rt.718
349.503
350-^88
.350.074
351.859
352.645
3,>3.43
Diam.
Ii4.
354.
:r)5.
355.
356.
357.
215
001
786
572
357
115.
•k
116.
117.
118.
119.
120.
121.
•>4
'-Va
■4
Va
'■Ya
Va
■Va
Va
■Va
%
■Va
122.
Circum.
Diam.
35.-'.142
123.
35«.9:8
■Va
35y.713
• k
3(;o.499
Va
361.284
124.
362.069
■Va
3ti2.ft55
.\^
36:<.64
■Va
364.426
125.
365.211
■'4
365.996
126.
36t>.782
■^
367.56/
127.
368.353
y?
369.138
128.
369.923
■%
370.70<>
129.
371.494
■y^
372.28
130.
373.065
•>2
373.N5
131.
374.i;:'.6
■%
3:5.421
132.
.57t)..07
■%
376 992
133
377.777
■y9
37.-^.563
134.
379.348
•H
380.134
135.
380.919
■Yz
381.704
136.
3,^2.49
■H
3rt3.-<;75
137.
384.061
M
384.846
138
385.631
■y^
Circum.
3(56.417
387. -.^02
387.9r'»
39i'.773
389.558
3110.344
391.129
391.915
39-J.7
394.271
395.842
397.412
398.983
400 ..54
402.125
403.696
405.-i(i6
406.. -'37
4(if<.40.'S
409.979
411.55
413.12
414 691
416.262
417.833
419401
4-J0.974
42J.545
424.116
425 687
427.2.58
428.H->8
4:!0.399
431.'.»7
433.541
435.112
Diiim.
■'A
139.
14o!
1
I4l/
1
143!"
.1
144.'
145'.
146.
147!
118!
149.
15o!'
•>2
M
Vr,
Circum.
%
43ti.6p2
4 38. -.'53
439.824
441.395
4 12 966
444.536
446 107
447.67rt
449.249
4.'.0 r^'l
452 39
453.961
4.')5.532
457 103
4.5H.(i74
460.244
4i;l.-15
463.3nt)
464 9.57
4H6.5-ja
46.r U9J-'
4ti9.669
471 24
472.811
To Compute ihe rirc nm of r Diam: tpr gpfatcp th;in any in the prpp, il n? Table.
RuLK — Divide the dimention by two, three, four, etc., if practicable to do ao. until it is reduced
to a diiimeter to be found in the t ible
Take the tabular circumference for this dmontion. mult ply by 2, 3, 4, 5, etc, according a» it
W:is divided, and tlie product will give the c rcurnference rei| ired
Example — What is he > ircumfereiice fir a diameter ot 1050 ?
1050-^7=150; tab. circum. 150 =47!, 239, wh.ch X 7=3299.073, circum. required.
Toi'ompiite the lippumfer ncf lop an Inli'Sf ami Fra lion not give '■ in Ih** Table.
RuLB. — Double, treple, or qiiadr pie ih" dim ntion given, until the fraction is increase I to a'
whole nurcber or to one f those in the able, as i, {, etc , provi led it is nractical to do so. '
Take the circiirafer -nces for i is di;im ter ; ami if it i< double of th u for which the circiimference
is requir d. tak one hi If of it ; if t cble. ta'e one third fit ; and if quadruple, o .e fourth of 11
ExA.MPLK. — llequired the circ mfcr tice of i il875 inches
2.21875 X 2 =4.4375 =4 ' , which < 2 =8. | ; tab c rcum =27 8817, which-4-4=6 9704 ins.
To('ompiile Ihe Circnm of a Ilium Irr in Pec an<l Inches tie. by ih pp tedinf Tablr,
RuL''; — Reduce the d m ntion to inches or eigliths, as the case may be, and take the circumfe-
rence in that te m from t e tahle fo. that number.
Divide this number by 8 if it is in eighths, and by 12 if in inches, and the quot ent will give the
area! in feet.
ExAMPLs. — Required tl e c'rcumforence of a circle of I foot 6J inches.
Hoot 6S ins =185 ma. =147 eighths. Circum. of 147=461.815, which-j-8=57. 727 tncAe* • and
by 12=4.8l/«rt.
TADLM in.
AREAS AND CIBCUMPERENCES OP CIRCLES, PROM ,'o TO 100,
[Adviincing by Itnlhs.]
Diam
Area.
Ciicum.
Di m
Aija.
Circnm
Diini.
Area
Cirtum
5.
19.635
15.708
'.0.
7. ^..-,4
31.416
i
.1
.oo:8:.4
.31416
.1
20 4282
l(i,0j2]
.1
^0.1 |,-^(i
31.7:!ol '
.2
.0:U4I6
.6->832
.2
21-J372
I6.:;3.i3
•J
81.713
:5j.oii;;
M
.UTdi).-^*)
.94248
.3
2-J.Oiil-
lii.i.504
Si
^:\.:>-S.i
3:.:!.58
.4
A'2Mti
l.-J56t)
.4
22.9022
1() '.'616
. i
h4.:m^8
:!2.(i726
.5
.VM:\'i
1 ..■)70rt
.5
•i3.7:8:»
17.-J78.i
..5
f^<) 5903
:5-.'.;».-()-
.6
.28-'74
1.8.^5
.6
•-'4 (i3iil
l7..".9-.",»
.6
HH..'t75
:5:5 :!oo',i
.7
.384^5
2.1991
.7
25.5 1:6
i7.'.»o;i
. #
89.9 J04
:5:5.615l
.8
.r)0-.'()i>
v'..".l.;:!
.8
•J(i42(H
18.-'21-i
8
9l.(;o9
33.;»-j<»-i 1,
.9
.6;!«i:
2.8274
.9
•-'7 :i:;'.i7
ls..,:{.,i
.9
93 :!l;!3
:54 24:54 ;!
1.
.7.<.-.4
3.1416
6.
•-'8.2: 14
IS.S496
11.
95.0334
31.5576;,
.1
.9.".0:!
3.45.)7
.1
•J9.2247
19.U»:;7
.1
96 7r.91
:i4..-7I7
.2
1,1309
3 7(i.)9
2
3(t. 1907
19.4779
2
98..5-JII5
35.18.59
.3
i.:;-.'73
4.0-4
.3
:;i i;25
19.792
.3
100. .,-77
:55.501
.4
l.r):!93
4,3982
.4
3'.'.169;»
■.•0.J06J
■ j
1 02.07 05
:i5.8!42
.5
i.7i;7i
4.7124
.5
3:.!s:a
V0.1-J04
..5
lo:?..-(;91
:56.1284
.6
2.01 06
5.(»265
.6
31.21-.'
•J0.7::45
.6
1 05.68:: 1
:t6.44-j5
.7
2 2t;i)8
5.3407
.7
35.".'5i;6
21.0487
. /
I07.5i:u
:i6.7567
.6
2.5446
5.6518
,H
36.3168
•ji.3t;.8
.8
!0'.>.:r>9
37.(»708
,9
2.rt:{.')J
5.9<)9
.9
37.3'.fJ8
•.'I 677
.9
1 1 1 .2204
:57.:{84
2.
3.1416
6.V-32
7.
3« 4846
•Jl.9;il2
12.
li:!.(»i»7ti
37.6992
.1
3.4<i3r.
6.5973
.1
39.5itv!
22 ::053
.1
114 9904
:58.0l:53
.2
O.801M
6 9l!5
.2
40 7151
22.6195
•}
116.^989
:58.:55>75
.3
4.1.-)47
7.2256
.3
41.8539
22.9336
.*>
11H.823I
.38.61:6
.4
4.r>-i:{;)
7..-.::.98
.4
43.0OS5
23.2478
.4
I24.76::l
38.955-
.f>
4.9087
7.854
.5
44.1787
23.. 562
.5
122 7187
:59.27
.(i
r>.30i»3
8.1681
.6
45.3647
23 c-<76l
.6
124.6901
:5: 1.5841
.7
5.72.');')
8.4823
.7
46..'>()«:5
24.1903
.7
126.6771
39.8983
.8
6. 1 r>75
8.7964
.8
47.7837
24.5044
.8
12"'6799
40.2124
.9
6.60.')2
9.1105
.9
49.0168
24.8186
.9
1:10.6984
40.526')
3.
7.0686
9.4248
8.
50.J6.-)6
25.132-i
13.
13-.'.7326
40.W40:<
.1
7.r)476
9.7389
.1
51.53
25.4469
.1
1:14.7S.'4
41.1.549 1
.2
8.0424
10.0531
.2
b'i.'*W2
25.7611
.2
1:56.84-
41.4691 ;
.3
8.r>r)3
10.3672
.3
54 1662
26.07.52
.3
1.38.9294
41.7H3i !
.4
9.0792
10.6814
.4
55.4178
2- -..3891
.4
111.0264
41.0974 ;
.5
9.6211
10.9956
.5
56.7451
26.7036
.5
14:5.1:591
42.4116
.6
10.1787
1 1 3097
.6
58.0881
27 0177
.6
145.26)75
42.7257
.7
lOwiVil.
1 1 6-.'39
59.4469
27 3319
.7
147.4117
43.03,t9
.8
11.3411
11.93S
.8
60.8J13
27.616
.8
149..57J5
43:554
.9
11.94..9
12.2.->22
.9
62.2115
27.9602
.9
151.7171
4:5.6'.8.'
4.
12.r,664
12.5»i64
9.
63.6174
28.2714
14
1.5:!.9:5<4
43.9824
.1
13.20»r,
12. '•805
•I \
65.0::89
28.5H,S5
.1
l,')ii.l 1.53
44.2965 ;
.2
13.8r>44
13.1917
.2
66.4762
28.90-27
158.:!fi8
44.6107
.3
14.522
1 3 5088
.3
67.9.^92
29 2168
.3
160 6061
44 9248
.4
15.2().->3
I3.s-i3
.4
69.:{;»79
29.5:51
4
162.. -'605
45.2:19
.5
ir).9043
14.1372
.5
70.88:^3
29 >^4.".2
.5
165 1303
45.5.5:12
.6
16.619
11.45; 3
.6
72.3824
30.1.593
.6
167.11.58
45.867:5
^
.#
17.3494
i4.765.T
.7
73 39"'2
:J0.4735
.7
169.717
46.1815
.8
18.01(56
'.5.0796
.8
75.4298
30.7876
.8
172.034
46 4956
.\f
18.8574
15.3938
.9
76.977
31 10!8
.9
174.3()66
46.8098
20
AREAS AND CJIRCUMFERENCES OF CIRCLES.
TAB LE.— ( Continued. )
Diam.
Area.
Ciicum.
Diiim.
Area.
Ciicum.
Diam
Aifti.
(ileum.
1.-).
I7t>.71.')
47.124
.6
3:!: •.:9J3
64.7161
^•>
53'.t. 12,19
82.3(199
.1
i:y U71>
47.43-1
330.536
()5.0311
3
.54.!.-.5:;3
82.(i24 *■
.2
181.1i>rt
47.7523
.H
33;t.7954
65.34.52
.4
547.3.I-J3
82. :{8J
.3
lH3.K-,42
4rt.Oh64
.9
343.070.'.
65 6:.94
5
541.5471
KV2.,24
.4
IrH. •.'()..■!
4i-.3Hl6
21.
:;46.3t;i4
65.;t7:;6
.6
555.7176
83. 5. i( 1.5
.5
l-'^.();cj3
IH (iiMn
.1
;ii;i.6ii7i>
()6.287 7
.7
5..9.'.'o;i.''
(-3.f-07
.«
r.tl.lol'.t
49.00.-59
2
352.1';«01
66.1 '1019
8
5()I.10. 6
-4.1948
»
./
I'Jii.iV.KW
49 3231
.3
35(i.:;-.'H|
66.916
.9
56-. 3232
^4...09
.H
iyt).(i()7j
4y.tp372
.4
359 ().~ 1 7
67.2;;02
27.
.572 5. .66
84 ;<-.;32
.9
l9H.;-,.-)r>y
4:».1'.'I4
.5
3(.3.(I5I1
67..-.444
.1
57().-ii5C.
H5.1373
IG.
201 (l(i-Jl
.50 26.. 6
.6
3(i6. 4362
()7 rt5-5
.2
5-1.0703
85.4515
.1
•^03..-)H;!ri
50.5711?
.7
361t K!7
(;f*.1727
3
585.3. 03
85. 7 ().'.*>
.2
2()ti.l2(i:5
.•.o.rticiy
.«
373.2.5:i4
6-.48.;8
.4
5 :'.6)69
86 0798
.3
2(irt (;:-,".>
51.20(<
.9
376.6-56
(.8.^01
.5
593.95."<7
h;.394
.4
211. --Mil
51 5224
22.
3-o.i3::«.
(i9.1152
.6
.59f*.28C.3
*6.7lt8l
.5
213sv;5l
5I.KM4
"a
38;!..5'.t72
69.4293
.7
602.6j'.'5
87.0223
.6
21ti.4->lrt
52.1505
.2
3-7 {»765
(i'J.7 435
.8
6ii6.98.-<5
87.3361
.7
•-•1U.(M()2
.")2.4647
.3
39(1.5751
7 0.0;.7ii
.9
6I1.;;()32
87 .6506
.«
2-il (1712
52.7788
.4
394.0823
70.3718
•JH.
6 15. 7. .36
.87.9618
.9
2-.M 3H
53.(t9.{
.5
3l)7.6(W7
70. (H()
.1
620. 1596
88.2789
17.
2v<i ;iH(i(i
53.4072
.6
401. 1.509
71.(i00l
.2
624.5814
88.5931
.1
2-".' (i.")H."^
53 7213
.7
404.70."!7
71.;i43
.3
628 019
fc8.it072
.2
232.3.-.27
54.03:,5
.8
40ri.2ci23
71.6284
.4
633.47J2
89.2214
.3
23r).(M)-J3
54.34'.K>
.9
411.-716
71.9426
.5
637.9411
89.5:!56
.4
2:!7.7.-'77
54.603ci
23.
415 47(56
7J.25()8
.6
642.42.'.:
89.8497
.5
2J<t 5-JH7
54.978
.1
4lrt0972
72.5709
.7
646:926 1
90.Hi39
.«
243.•J^:)^.
55.2921
.2
422.73:;6
72.f-f-51
.8
651.4421
90.478 1
.7
246 0.-.7i)
55.61163
'.\i
42(i.:;85H
73. 1992
.9
»)55.8731)
90.7922
.8
248 ^^61
55.9204
A
430.0.536
73.5134
29.
660.:.214
91.1061
.9
251.6r>
56.2346
.5
431!. 7371
73.8276
.1
9«>'..0845
91.42(»5
18.
•jr)4.4():M>
56.54.-^8
.6
437.4363
74.1417
.2
669.6(.34
91.7347
.1
257:5018
56.8629
.7
441.1511
74. 4. -.59
.3
«)74 •j.".8
92.04-8
M
2t;o. ir.ort
57.1,71
.8
444.rtdl9
74.768
.4
678 8t;K{
92.:i()3
.3
263.0226
57 4912
.9
448.62."^3
75.0d82
.5
&-3.4943
92 3772
.4
26.').y(»5
57.H)54
24.
4:.2.3904
75.3<JH4
.6
688.136
92.9913
.5
2«W.W031
5«.1196
.1
456. 16.-il
75.7125
.7
692 79:!4
93.30.55
.6
271.7169
5-^.4337
.2
4.".;t.99l6
76.(»267
.8
697.4(i6>i
93.(iii»6
274.()46r)
58.7479
.3
463.7708
76.3408
.9
702 1.554
93.93:i8
.8
277.r>917
59.062
.4
467.5957
76.6523
30.
706. '-6
94.248
.9
280.r):)27
5i).37t52
.5
4;1.43()3
76.96t>2
.1
711.5.-02
94. .562 1
19.
283.5294
59.6904
.6
475.2926
77.2e33
.2
716.3162
94.8763
.1
286..V2I7
60.0(i45
.7
479. 1646
77.5i>75
.3
721.0678
95.1904
.2
2'^9.529-i
60 3187
.8
4^3.0524
77.9116
.4
725.8352
95.5046
.3
2-.12.5.536
60.6:'.28
.9
4rf6.".'558
78.2258
.5
730.6183
95.8 ia8
.4
295.5931
60.947
25.
490.875
78.54
.6
735.4171
96.1329
.5
21W.64.S{
61.2612
.1
494.rt098
78.8541
.7
740.2316
96.1471
.6
301.7192
61. .5753
.2
49-S.7604
78.1(i93
.8
715.0(il8
96.7612
.7
301 '-06
61.8895
.3
502.7266
79.4H24
.9
749.9077
97.0754
.8
3l)7.<)0-(2
•i2.2()36
.4
50t).7O86
79.796<)
31.
754.7694
97.3896
.9
311 0252
62.517H
.5
510.7063
^0.1108
.1
759.6467
97.7U37
20.
3N.)6
62.832
.6
514.7196
8*i.424S
•>
• •
764.5397
98.0179
.1
317.3094
•63.1461
• <
518.74H,8
80.T:{;U
.3
769.4485
98.332
.2
320.4746
63.41)03
.«
522.7936
8l.0;'.32
.4
774.;:729
9-'.6452
.3
323 1)554
637744
.9
526.854 1
Kl.3()74
.5
779.3131
98.9(i04
.4
326.H52
64.0H86
26.
530.9304
f 1.6^ 16
.6
781.2689
99.2745
.5
330.0643
t)4.4028
.1
:)55.022:'.
81.9976
.7
789.2406
99.5887
AREAS AND ClRCTTM?RRE5Cra OP ClftdtlSS,
TABLV:.— (Continued.)
21
Diam.
Are; I.
Circum.
Diam.
Area.
Circum.
Diam.
Area.
Circum.
.8
79A.'2'Z7rt
99.9028
.4
1098..5H0-,'
117 4958
43.
14.5-2.-2016
l:55 0-^MH
.9
7y'.».-j:<0rt
100.217
.5
i 104.40-7
117.nl
.1
I4.'.8.;t608
1:55.40-29
32.
804.2-4'.>tJ
100.5312
.6
1110.:J07l
1 18.) -241
.2
1405.7148
1:55.7171
.1
80.».2rt4
1oo.h4.'>:j
• •
nio.-j.^ii
118 4:583
.3
1472.-5:585
1:50.0:5:52
.2
Hl4.:W4l
10 1.1.595
.8
1122.2109
1 18.7.-)-24
,4
1479.:548
i:50.:5».54
.3
81'.'.39y9
101.4730
.9
ll2rt. 1.504
119.0000
.5
14-0.1731
1:5(5.0.596
.4
«-J4.481.T
101.7478
:{8.
11:54 1170
119. tHO^
.6
14.»3.0l:59
1:50.9737
.5
H2y..")7'^7
102.102
.1
1140 > (40
1 19.0949
,7
I4.n'.."'705
l:!7.-2879
.6
834.H9I7
102.4101
2
1140.0"<7
l-20.Mii;ii
,8
!.50t5.742:
1:57.602
.7
S3y.'i-i03
102.7:U)3
.3
1 1.52.0954
1-20 :5-2:52
.9
151:5.0-2.-7
13;.91(J2
.«
84 -4.9(147
J 03.0444
.4
115H.li;»4
l-20.(;:574
44
1.5-20.53 n
l3-'.*2:!04
.9
850. 124^
lo:{.3.')eO
..5
1104.1. ".9 1
1-20.9510
.1
I5J7.4537
l:»-'.5415
33.
855.3UU«)
103.0728
.6
117 0.2 145
121.-20.57
.2
1.5;;-! 3.8M<
l:5H.-.5H7
.1
8G0.492
103.9869
.7
1170.2-.57
|-.'l..57;'9
3
1.5:!1.:5:5<»0
1:59.17-28
.2
rtti5.69y2
104.30! 1
.a
llrt2.:5725
121..S94
.4
l54f<.;!(K)I
1:59. l-'7
.3
870.9-J22
104.0151
.9
11;^H4051
l-.'2.-J0-'2
.5
i.'..)5.-2''8:5
l;;9.-oi-2
.4
87b.h)UH
104.9294
3l\
1194.54:54
1-22.. 5-224
.0
1.502.28(52
140 11.5:!
.5
881.4151
105.2430
.1
12(K».72T3
l-22.-i;;o5
,7
I..09.2998
1 40. 1-295
.6
8>-^.6H5l
105.5577
.2
1200.877
1-2:!. 1.507
.8
1570 :5292
140 7i:;o
.7
891.9709
105.rt7l9
.3
121:5.0421
1-234. .4-1
.9
1.5.-';i.:!742
141. ('578
.8
897.-.>7->3
100. 1-0
.4
1219.224:!
12:5.779
4.5.
l.V.K).i:!5
1 1 ! :572
.9
9U-i.5'^95
100.5002
.5
1225.4211:5
l-24.09:!2
.1
1.)97..".I14
1 1 1 t)-01
34.
907 ifi->4
I00.dl44
.0
1231.0:528
124 40:3
•)
lOOl.OiKiO
14 2. 00(13
.1
9i:).-J709
107. 1285
.7
I237.>i)l
1-24.7215
Si
101 1.7! N
14 2.3144
.2
9l8.ti:{.Vi
107.4-J72
.8
1244.121
125.0:5.50
.4
101:5.8:55
14.'.(i280
.3
9J4.0il5
107.7506
.9
125(1. :5u40
l-2.).:549"^
.5
1025.9713
142.94-28 ]
.4
929.4109
lO.i.07 1
40.
1250.64
1-25.004
.0
l(.:53. 1-293
1 4:!.-2.5(>9
.5
9a4.rt2v;3
h>fi.\ir*:>2
.1
I2r.2 931
1-2...97-1
.7
IO40.:!02
li:5..5711
^'{
940.-J494
10.-S.0993
•_>
1209.2:5.<.-S
l-2(). -292:5
.8
1047.4.-40
143.8-52
.7
945 (iiCi-i
109.0352
.3
127.'..5t)0J
l-20.OO()4
.9
l0.54.J)8-5
144.1994
.8
951.1508
109.3070
.4
12«1.^9H4
l-2t'). 9-2(16
4().
1001 '.fOfU
144.51:50
.9
956 &>b
109.0418
.5
12-iH.2523
1-27.-23 18
,1
l(i09. 1:599
144.8-277
35.
902.115
lUt>.C5.)0
.6
i -294.02 1;»
lJ7.r)48;>
/2
1670.:5-i»l
14.5.141.)
.1
907.0200
110.2701
.7
i:5o:.oii7l
127.80:51
!3
li 183 0511
145.1.50
.2
973,142
110..-.-43
.8
I:5o7.40n2
1-28.1:72
.4
1090.9:! 17
145.7702
.3
978.079
110.8984
.9
l313..-^-J49
12-.4914
.5
1098.-2:!! 1
1 ir..O-44
.4
984.2318
111.2120
41.
i:!->o.-i571
1-2.-!. 80.50
.6
170.'.. -4:52
llO.:59-5 i
.5
9'"i9.800.5
HI. >20S
.1
i:;j().7().55
129.1197
.7
1712.87!
14ti.71-27
.6
995.:iS45
111.8109
.2
i:{:5:!.ltu»:5
I29.4:;2:5
.8
17-20.2141
l47.0-2u8
.7
l000.9«43
112 1551
.3
1:::51>.01H'.)
1-29.74'*
,9
17-27 57:10
147.311
.8
looo.o
112.4092
.4
1310.14U
l:5(».(Mi-22
47.
17:54.91-^0
147.(15.52
.9
1012.2313
112.7fS34
.5
1:5.52.0.551
l:5(i.:5704
.1
1712.:5:!92
147.9093
30.
10l7.87.-^4
113.0970
.6
i;j.59.1-18
l:50.0',t05
/>
1749 7455
11 -'.-28:55
.1
10v!3 54n
113 4117
l:!«)5.7242
13l.(t047
!3
I757.1(i75
i48..5i»76
.2
10-'9.2195
113.7-259
.8
1:572.28-22
1:51.:5I88
.4
1704.0045
!4M.i)ll8
.3
1 034.9 i:U
114.04
.9
l;57f'..-<.50
131 032
..5
1772.()5H7
1I9.-J26
.4
1040.02:!5
1!4.:{542
42.
r.5H5.1450
l:;l 9472
.0
1779.5-279
14.»..5:5,!l
.5
1040.:U91
114. 00-^4
.1
i:592.05(W
132.21113
.7
17-'7.0l-27
149.8.543
.6
1052.0904
114 9825
.2
l:59-'.07!7
1:52.. -.7.55
.8
1794.51:53
1.50.10.84
.7
1057.8474
1 15.2907
!3
ll0.5,:{('-'3
l:52.88,tO
9
l-'02.0-29()
150.4^-2.5
.8
1003.02
115.0UH
.4
1411.9007
1:53. -20:59
4-^.
l-iO'.»..-.r,16
150.7968
.9
1009.4081
11.5.;«25
.5
1418.()-i.-^7
l:53..M8
1
1817.1092
I51.U09
37.
1075. -J 126
110.2392
.0
14-25.:;r2.-.
1:53. -;52l
•1
18-21.07-20
151.4-251
.1
1 OS 1.0324
1 10.55:'.:!
• /
14:5-J.(U19
1:54.1103
.3
18:52.-2518
151.7:592
2
10-0.8079
110.^)75
.S
14:58,7271
l:{4.4004
.4
1 -i:59.8400
152.05:54
.3
1092.7191
117.1816
.9
U 45. 4.58
134.7746
.5
1847.4.570
152.3676
22
ABEAS AND CIRCUMFERENCES OF CIRCLES.
TABLE.— (Continued.)
Diiitii.
A re; I.
Circum.
Diam.
Area,
Circum.
Diam.
Area.
Circum.
.»'.
K..-..o-:w
hVJd^l?
.2
•.':{(l7.-2'224
170 '2747
.8
2f^0''.6218
187.H576
.7
1 >').'. 7-.'..;;
i.V.'.. '.tiV.t
.:{
.:li...7t4
170.588:5
.9
-Jr'l-.023
1-.-!. 1^*18
.«
H7(t :i-j;'
l.v.{.:{i
.4
•.•;{j».-2-i:j
lio.'.io:5
(10.
•2^27 44
H-.496
.!»
l>;rt.u;)ti;!
!.'):< »ijj-.'
..5
•2:!:V2 >:u:<
171.2172
.1
•2'-:56 h;-2()
l:rr.HlOl
•I'J
1 'r^;> 7 1.") I
l..:{.'.':{-»
.()
2:ui.4o:{
171.5:54:5
.2
•2-4(5.321
I89.r24:i
.1
l>ii;;.-iani
I5J.-2.VJ.".
• t
•.':n9.;(rt74
17l.84."i5
.3
28.-|5.7K)
189.4:584
.'2
ii»t»i 1:01;
I.'i4.."irt'>7
.r(
2:r>8.5rt76
172.1596
.4
:28<55.-26l8
lK.t.75^.'6
:.\
l'.)(H.'.l(»(i-^
l.".1.8>t)8
.9
•2:{(i7.-2o;!4
17-2.47:5^
5
•2ri74.7«5(»:5
190.0(56^
A
i.ti().ti").-i;
15... hi.-)
55.
•2:J75.HX.
17-2. 7 8ri
.6
•2''8 4.-J(51.-
l;i0.:5809
.;">
I'.iji <'ji;:<
15.'... 092
.1
•j:in4.4-^2^2
17;« 1021
.7
289:5.7984
190.69.M
.<i
i'j;!-.'.-jO'.tt)
l."....rS .'•.{:{
'2
•2:i',t:{. 14.52
17:5 4hi:5
,8
•29o:;.:54i
191.0092
.7
r.M<).(lU-6
I5(i. 1:575
.:i
24ol ..-illH
I7:5.::!(i4
.9
•2-.M2."993
191 :52:54
.H
I1M7.>-.':U
l..«.4..1f.
.4
J 110 51H2
I7i ol I'i
(11
29^2-2.4 731
191 6:576
•U
1 ;».').').()").!«
|. lb. 7 5.58
.5
2I19.-J^2-':'.
i:4.:!"'-^8
.1
293-.'.( 16:51
191.9517
r)0.
I'M ;;{..'.
157.08
.6
•2427. ..5 11
1 7 4. 1 17^29
•2941.(5(5-5
192. -.659
.1
l;»7l.:{riM
l5:.:",tMl
. 1
•.'i:{''.(59.5"'
1 7 1 977 1
!:1
.'95l.-2-'97
192.58
o
VJlUS.i '4
l.'.7.7ii8;!
.r
•2445.-i:.2H
175.:!092
.4
•2; '(50 9-J65
i93.''942
!:1
19-^7. l:!-Jti
L.>.0J-24
.9
•245 J. •2-2.57
ir.-,.6l.54
.5
•29a)..".791
19:!. ■.'0-4
.«
l'.t9:) ojio
l.')H.:};U)()
5(5.
•24(.;:.oi4i
175.9-29.5
.6
•2:'f'0.2474
l9:i.5-2--'5
.;")
•.'90J. '.•'><);;
i:.8.6.">()-
.1
•247l.^l«7
17(5. •24:57
.7
•29ff9.93l4
I9:5.K5t57
.tt
•JO 11 (.90(17
15- ui;i<)
•2
24-(I.(.:J-7
176..5.,79
.8
•2999.(5:5
1.14.1508
.7
•J(tli-I.^li-J-'
1.. 9. -2 791
!:5
•24-9 47 15
176.-72
.9
:5(t(t9.:5l64
194.465
,h
•j()-i;.-:ui;
15.).5.':?2
.4
•24.ir<.:i2.5'.l
177.1^62
62.
3019.077(5
194 ; 792
.!»
•2o:{ 1.^77
1..9.9074
.5
-2.-.()7.1'.f:{l
177.5001
.1
:5()-28.82l4
195.09:53
f)!.
•J04-J.."j-.':>4
l.i02J6
.(5
-251-,).(I7()
177..- 145
.2
:50:5n.5869
1;»5.4()75
1
•i(l^^().^4 4:!
UiO ..:5.)7
,7
-2.52 1.'. '7. ;6
178.1-2-7
..{
:504-.:i65i
195.7216
.'.i
•^0.').S..-^>4
1(50.8499
.fi
"25 ;:!.>H.>^,-i
17r'.4 428
.4
:}(i.--'.l.v9i
196.0:5.58
a
•.>o(ii.'.»'j:>;{
IGl ICl
.!»
•25r2.8l>8
178. 7. .7
.5
:5(K57.9»i.-7
196.:<5
.1
•j()7i.;>9."):'.
}(;i.47rt-2
57.
•2.'.5 1.7(5 1(5
17;).o;i2
.6
:5077 7.M1
19(5 61.41
.5
•j-)h;;.();71
l(i 1,7924
.1
•2.-|60. 7-2*5
179.:5S5:5
.7
:50-'7 6311
li>6.97.-<3
.<i
•il)91.i7l()
i()-2 I()ii5
•2(5(5i).:0:U
17'.'.6;t95
.8
:!(Kt7 4919
1 97.^2924
.7
•.'U99.-J."^7ri
IGJ.4207
•>
...
•2.-)7"< (>9.V.>
18ir.or,;()
.9
31o7.:5641
197.60(5(5
.rt
•ii(i7.4it;ti
1()2.7;U8
.4
•2587.7045
l8o.:5-2;8
(5:5.
31I7.^2.V26
19J. 9-208
.9
•jii.'> T)*)!.:
1():5 049
.5
•2.5'.»6.7^287
l.ri'.642
.1
31-27. '..■.64
19-s. -2:549
r>2.
•>i-.'.{.:-jiii
l(i:;.:5():{->
.6
2605. 7 6S7
180.. '.■.61
.2
3i:i7.07.58
198..5491
.1
•.'|:!1 >97i">
)('•:{. ():7:{
. 1
•2614.824:5
IH|.'280:5
.3
3147.0114
198.^(5:52
.>
•214(1.089:?
1():V99:?.'>
.8
•2<i-2:{. 89.57
181.5-44
.4
31.56.9664
199.1774
Ji
•2 1 4. -i. -2:67
lt;4 :i05t)
.9
•.6:!2.9H2H
181.8986
.5
31(56.9-291
199.49 L6
.4
•J !.■)()..') 199
ItU ()198
58.
•2(54^2.(»8.56
182.-21-28
.6
3176.9115
199.80.57
.r»
•il()4.7.,)S7
l»i4.9.!4
.1
2()5l.-204(!
18-2.5-269
.7
318(5.9097
•200.1199
6
•Ji7:50i:5:i
I(;5.-2I81
'}
•2(.(54.:{:!82
182.8411
.8
3196.9-2:55
•200 434
.7
•2I8I.j8;5.".
i(>5 5(;2:{
J',
26()9.4.-l^<2
18:5.1:..52
.9
:5^20(5.9.->31
•200.7482
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•il!<9..'.ti9r)
l(!5.,'»7til
.4
2('i78.6.5:?f
18:5.4(^94
(54.
32lt599-i4
•201.0(5-24
.9
•>l:i7..-71J
lt;6.iy(«)
.5
•2«5K:.8.;5i
18:5.78:16
.1
3227.059:5
•201.:i765
5S.
•-i-.'(K).l-r:()
165.. 5(148
.6
•2697.o:i2l
184.0977
.•2
:5-23:.i:!(5
•201.6907
.1
•2-2l4..'>-Jl(i
I6ti8189
.7
•27ti6.2449
= 84.4119
.3
3247.2284
•202 ( (048
.2
•^•i-.'-'.-7U4
lH7.i:{..l
.8
•27l5 47:;:i
I.H4 7'2()
.4
3257.3:5(55
202.319
,:{
•2^:51. •.>:?.')
lt;7.447-2
.9
•27 ^24. 7175
18...0402
..5
:5'267.460:5
202.(.:i:52
.4
•2-j.:9.<)ir.-i
1()7.7614
59.
27:i:{.9774
185.:5514
.6
:5277.51)98
•202.947:5
.r>
•^•J4H.(tlll
l(i8.075(>
1
274:5."25^29
1,-5.6(585
.7
3287.755
203.^2615
.6
•i-i.".t). i'2-J7
?G-.;i897
/2
27..-2.5142
I85.98-J7
.8
:5-297.92i5
203.57.56
.7
•.'•J()4.8.01
1 6-5.7049
.:i
•2761.8512
186.-2t59(5
.9
:i: 108. 11-26
203.r'.-^98
.8
•i-i7:i.-,>9:{l
!n'.'.018
4
•2771. 17:59
l-(5.iill
65.
:i;il>'.3i5
•204.204
.9
:iJKI.7r)19
I69.:'.:i^22
.5
27^0.51 •2:5
lr6. 9-25-2
I
:5:52-.5:54
•204.5! 81
54.
2-i.'0.'iStU
169.(5464
.6
2789.8(564
187.-2:{9;J
•>
3:i:59.7668
•204.8:523
.1
2-298.7165
169.9605
.7
2799.2^62
187.5535
.3
3349.0162
205.1464
AREAS AND CIRCUMFERENCES OF CIRCLES.
Z5
TAB\E.~ (Continued.)
Diam.
.4
Area.
Ciicum.
Diiim.
Area.
Ciicum.
Dium.
i
Area.
Circum.
3:{r>;>.2.«'l4
m').46or.
71.
39.59.2014
2-i3.053(i
.6
4C.<)8.:5816
240.64(ih
.5
3:Mi'.».:)»)2:;
2(».').774H
.1
3;»7ti.;i619
22:! :{(;77
.7
4620.4218
240.9607
.6
:{:{7i».-.>y
20<;.tt~'^9
.2
39rti.,-.:{rtl
22:{.6n:9
.8
46. .2.4776
24l.-.'74H
.7
:j;W(».17I2
2(16.403 i
.3
3'.t'.)2.73()l
2-j:{.i»'.'t)
9
4644 .5492
241.6987
.8
:{4()(i 4:n>2
206.7172
.4
4tM)39373
2J4.310-J
77.
46.56.6: :66
241.90:52
.9
:{41(>.f*42U
•J07.03;4
.5
4015. hill
2-M.<.244
.1
4(;ii8.7:59)i
242.2173
m.
:M2120J4
2(ir.:u.')6
.6
40'.'6.4002
224 93-^5
.2
46.-0. rt5-^3
242.5:515
.1
:{j:5i.r)77r.
2{)7.6:.97
.7
4037. ♦i.55
225.2527
.3
4692.9927
242.-4.56
.2
:{-i4i.w.{;»
207.1>::il>
.8
4018.9-'.54
2-.'r..:.«>68
.4
47(15.1429
24:5.1.598
.3
:i4.")2.:;749
208.288
.9
41160.21 16
2-.'5.o-l
.5
4717.;5087
243 474
.4
:}4t)-i.7y7l
20.-*. 6022
72.
4(171.5136
220.1 952
.6
4729.4903
243.75V-1
.;')
:547:5.j:{5l
2U8.ltJti4
.1
408.'.8332
2-.'6..".093
. 4
4741. (1875
244.1023
.6
;)4.->3.»W'!8
201) r>;{0.'.
.2
1094.1645
226.82:J5
.8
475:5.9605
244.4 i64
.7
:i41»4.l64
20y.544«i
.3
4!(»5.5)-^
2J7 13:6
.9
476»i.l292
244.7:506
.8
:«t)4 .6-1:52
209.-5-8
.4
4116.0793
227.4518
78.
477rt.:i7:i6
245.0 148
.9
:ir>ir>.i4:{
210.173
.5
4lJ8.2i587
2-J7.7tifi
.1
4790.6:5:56
245.:5589
67.
3.')2.'>.6«i06
•J 10.4^7 2
.6
4l31).(i5J4
'Zr>!i 0801
.2
4p02.9()1)4
245.67:51
.1
:{r):«).l9-J(?
210.rt(»13
.7
4l5l.(J6ti7
22^ :{94:{
.3
4815.201
245.9872
.2
:{o4ti.7404
•.ill. NT).-)
.8
4lti2 4943
2-Jrt 7081
.4
48J7.5082
246.:5014
.3
■.\:>:^7 :m:i
211.4-29(;
.9
4l7:<.i);!76
2-il). 022(5
.5
48:59.8311
246.615H
.4
Iir)H7.MS.>7
211.7438
73.
4lrt5.39b6
2J9.:5:i68
.6
485:. 1697
246.9297
.5
3r)78.4787
212.0.58
.1
4196.8712
229.«)5(»9
.7
4864.-524 1
247.2439
.6
3r,8y.o.ii)r>
212 3721
.2
4208.::614
2-.'9.965l
.8
4876.8973
247.548
.7
:{^.yi^.7l;V.>
212.tW63
.3
4219.0678
2:W2792
.9
48h;».2799
247.8722
.8
:it)io.3."..-Ji
2 13. 00' 4
.4
4231 3896
2:{0..59:{4
79.
4901 6814
248.1864
.9
:w-ji.ui6
2l3.:;i4ti
.5
4-42.9271
2:!0.9()76
.1
4914.01)85
248.5005
68.
:{():{ i.e-'ittt
2!:{.6-J8.-i
.6
42.. 4. 4803
2:51 2217
.2
4926.5.514
248.8147
.i
:5642.:{788
2t3.;)429
.7
4.:(i6.t)49:>
23l.5:!.5'.i
.3
49:589-^2
249.1288
.2
3H.):{.(to38
214.2.571
.8
42;7.6;!39
2:!l.-5
.4
4951.4443
249.443
.3
3tk»:{.'S((4
214 ..712
.9
428, •.234.5
2:!2 1642
.5
496:5.9243
249.7.572
.4
3H74.r)4l
211.8454
74.
4300.8.>()4
2:52.4784
.6
4976.484
2.50.0713
.5
3Hs5.-il»;U
215 1996
.1
4;il-.'.-|.-'-,'l
2.;2.7925
.7
4988.9314
250.3855
.6
3691 i. 01)6
215.5137
.2
43-24. 1 •j;)6
•J3:!.l((67
.8
5001.4.586
250.6996
.7
370ii.844.')
2:5.8si79
.3
4335.792-'
2:5.;.42()- 1
.9
5014.0014
251.01:18
.0
3717.6437
216.142
.4
4347.4717
2:54.7:55
60.
5026.56
251.:{280
.9
37J8.4r)87
216.4562
.5
43.59.16ti3
.1:54.0492
.1
50:i9. 1:542
251.6421
69.
373y.-^894
216.7704
.6
4370.8766
2:54.:5633
.2
5051.7242
251.9.563
.1
37r)0. 1357
217.0^45
.7
43rt2.602b
2:{4.6775
.3
5064.3258
252.2704
o
3760.9;t78
217.3987
.8
4394.3448
234 9916
.4
5076.9552
252.5846
is
;]77 1.6756
217.7128
.9
4406.1018
2;{5.3058
.5
5089.5883
2.52.8988
.4
.3782.7H91
218.027
75.
4417.875
235.62
.6
5102.2411 1 53.2129
.5
3793.6783
218 3412
.1
4429. 66:«
2:J5.9341
.7
5114.909^ ' 253.5271
.6
3804.6032
218.6.553
.2
4441.4684
2:56.2483
.8
5127.59:{»
253.8412
•^
.«
38)5.54;{&
218.9695
.3
4453.2886
23ti.5624
.9
51^0.2937
254 1554
.8
:iH26r)002
219.2836
.4
4465.1246
236.8766
81.
5153.0094
254.4696
.if
384:.4722
219.5978
.5
4476.9763
237.190O
.1
5165.7407
254.7837
70.
3848.46
219.912
.6
4488.8437
237..5049
.2
5178.4877
2;.5.0979
.1
:«.-»y.4s>.-,2
220.2261
.7
4500.7268
2:57.8191
.3
5191.2505
255.412
.2
:W70.4d26
220.5403
.8
4512.6256
2:58.i:5:J2
.4
5204.0285
255.7262
.3
3881.5174
220.8544
.9
4524.5401
2:58.4474
.5
5216.8231
256.0404
.4
3892.56S
221.1686
76
4536 4704
238.7616
.6
5229. 6:«
256.:i.545
.5
39' (3 6343
2214828
.1
4.548.4163
2.39.0757
.7
5242.4586
256.6687
.6
3!)14.7I63
221.7969
.2
4560.37-7
2:J9.38i»9
.8
52.55.2998
256.9828
.7
:i9-.i5.Hl4
222.1111
.3
4572.3553
2:59.704 1
.9
5268.1568
257.297
.8
3936.9274
222.42.52
.4
4584.:i583
240.0182 1
82.
5281.0286
257.6112
.9
3948.0565
222.7394
.5
4596.3571
240.3324
.1
5293.918
257.9253
24
ABEAS AND CIRCUMFEREIfCES OF CIRCLES,
TABLE— (Continued.)
Diam.
.2
Area.
Circum.
Diam.
Area.
Circum.
Diam.
Area.
Circum.
r):{0«>-^2l
2r>8.2395
.8
(5054.5149
275.8324
.4
6851.4840
29:5.42.54
.3
r):U9.74:{U
25H..V)3H
.9
6068.3224
275 14(56
.5
(W06.1631
29:5.7:596
.4
r>:!:w.()7:.'.
2:.8.-i;40
88.
6o82.l:«70
27(..4(508
.6
68.S0.rt579
•294.<»5:n'
.5
r);M:..«-,'rt7
25;». 182
.1
601»5.y(584
270.77 19
.7
6''95.5(585
'294:5679
.6
r>:{:.8.r)iir)7
•2.')9.J90l
.2
0109.^15
2r7.0«9l
.8
(5908.-2947
•21M.(5H2
.7
r):}7i.r)98:{
•J.59.810:',
.3
61-23.6774
277.40:i2
.9
69-25. 0;i(>7
294.9962
.8
5:w4.r)7«y
200.1-244
.4
6137.5554
277.7174
94.
(59:59.7944
•295.3104
.9
r):5y7..')9(»M
2()(t.4380
.5
6151.4491
278.0:516
.1
6954.r.077
295.6245
83.
5410.6'.'()(;
20»».7.V.'H
.6
01(55.3585
278.:{457
.2
69(59. :55<5'-'
2t(5.93-(7
.1
54-.':{.tiG»)
201.0009
.7
bl79.2e37
278.(5.-,99
.3
6it84. 1(514
•2'.»6.24:{0
.2
i')4:{»).7-.'7"-i
•itii :mi
.8
0193.2245
-27r<.975
.4
099S.9821
•2".»6 507
.3
5441>.MH-.'
201.0'.».2
.9
0207.1811
279.2H82
.5
701:5.8183
•29«i.8Hl2
.4
(.4l)2.8'«W
-2(;2.00'.»1
89.
6-221.1534
279 0024
.6
70-28.6702
297.1953
.r>
5470.00.-) I
20-2.3-230
.1
(5-235.1413
279.9105
7013.50-i5
-297.5095
.6
54-^9. 1-4M
2(i2.<):{70
2
6-249. 145
280.2:507
.8
7058.418
297.^2:56
.7
5r>o2 'i«j8;»
-202.95 lit
M
(5203.1(544
2-'0.5448
.9
7073.:J-202
'298.1:578
.8
5:)l5.4-.>4;5
2t;3.-20»
.4
(5-277. 199.-,
280.8.')9
95.
7088.'2:55
'298.452
.9
55'^8.5958
203.5802
.5
(5-2i' 1.203.-)
2Hi.;7:v2
.1
710;J.1(554
'29H.7661
84.
5.')4i.7rt-^'4
203.8i)44
.0
O30.-).3l0ci
2^1.4H73
.2
7118.1110
299.0723
.1
5r.54.9847
204. 20<)
.7
6319.399
28l.882.->
.3
71:53.07:54
•299.:5944
.y
i>bt}6.W.\2
2(»4.5-2J7
.8
(53:53.497
2H2.1ir)6
.4
7148.051
•299.7080
.3
55«1.4:57-J
204 83on
.9
():U7.6813
2rt2.4-298
.5
7163 0443
:500. 0-228
.4
5594.(i8til>
205 151
90.
6:5(51.74
282.744
.6
7178.05:53
:5()o,:5:569
.5
r>tio7.9:.2:}
2()5.4052
.1
6375.885
•28:5.0581
.7
7193.078
:50O65ll
.6
;)0-jJ.'2:{:{4
265.779:!
.2
0:!90 04.")8
2rt3.3723
.8
7'208.1184
:500.9()52
.7
5G:{4.56-'ii
2(i6.0lt35
M
(5404.-2-2-22
2«3.68(54
.9
7-22:5. 1745
:50l.-2794
.8
5(547.84-^8
200.407ti
.4
(5418.4144
2M ()0t;(5
96.
72:58.2464
:50i.r)9:50
.9
5»il) 1.171
200 7218
.5
64:{2.6-22;{
284.3148
.1
7-25:5.3:5:59
:50l.9O77
85.
5«)74.5I5
207.03(5
.6
6440. H474
•2H4. (5-289
.2
7-2(58.4371
:!02-22l9
.1
56-*7.Hr4G
2t 57. 3501
.7
(54(51. 08.-)2
284.9431
.3
7-2S3.5561
:502.5:5(5
.2
5701.25
2(57 .(504 ;5
.8
6475.3402
2rt.-).-2572
.4
7298.6907
30'2.8.")02
.3
5714.t)41
2(57.9784
.9
64rt9.01(J9
2-5.5714
.5
73 13.8411
-.503.1044
.4
57r.'8.0478
208.29-20
91.
6.-)(»:{.«^(574
285.rtH.-)6
.6
7:529.0072
:50:5.4785
.5
5741.4703
2()8.(5(»0.x
.1
(5518.1995
280.1997
.7
7:544.189
:503. 79-27
.6
5754.9085
268.i>-2(i9
')
05:V2.5173
286 5139
.8
73.->9.:5804
:504. 10(58
.7
57(58. 36-.'4
■269.-2351
.:{
(554(5.^^9' »9
2-6.829
.9
7:574.5996
:504.4-.'!
.8
578l.8:!2
•269..')492
.4
«V>01.-2081
287.14-22
97.
7:5<9. 8-286
:50» ::i.52
.9
5795.:517:{
2(59.8034
.5
6575.5(5.-) 1
2-7.4.-.64
.1
7405.07:52
:505.0493
86.
5f 08.8184
270.1770
.6
65h<).9458
•2f?7.7705
.2
74-20. :5:5:55
:i05.:5(5:]5
.1
58-2-i.3:551
270.4917
.7
66(»4.:52-22
288.0847
.3
74:55 6095
:5O5.(5770
^2
5835.8G75
270.8059
.8
6(5 IH 7542
•288.:59r>8
.4
7450.9013
:505.9918
!3
5&49.4157
271.12
.9
6(5:5:5.182
2-8.713
.5
74(56.2087
:50i;.:50(5
.4
586-i.97i»5
271.4342
' 92.
(5647.0:550
289.0272
.6
7481.5319
:5O6 6-i0l
.5
5876.5591
271.74M4
.1
(56(52.0848
2H9.3H3
.7
7496.8707
:}06.93(53
.6
5890.1541
272.0()»i5
.2
6(570.5597
•289.(5:)55
.8
7512.2-253
:507.2484
.4
5903.7(i54
272.37(57
.3
0(591.0101
289.9096
.9
75-27.. 5950
:507. 5(5-26
.8
5917.392
27-2.6908
.4
6705.5507
290.2838
98.
7.542.9816
:507.^708
.9
5931.0344
273.005
.5
67-20.0787
290.598
.1
7558.:58:;2
:508.1',t()9
87.
5944.69:i(i
273.3192
.6
67:!4.6l(55
-290.<J121
.2
7573. fO'O
:508.5051
.1
59r>8.3(J44
273.(5333
.7
6749.1699
291.-2-2(53
.3
7589.--'3:58
:;0S.8l<»2 ,
.2
597-2.0559
273.987.-,
.8
0763.7:591
•291.5404
.4
7(504.682(5
30i).r.534 1
.3
5985.7091
274. -261 (5
.9
6778.:524
291.8546
.5
7(5-20.1471
:509 4470 ;
.4
5999.4821
274..5758
93.
6792.9-246
•292.1(588
.6
76:55.627:5
:509.7017 !
.5
6013.2187
274.89
.1
6807.5408
•2i>2. 48-29
.7
765l.l9t53
310.0709
.6
6020.9711
275.-2041
.2
68-22.173
•292.7971
.8
7(566.9349
310.:5H5
.7
6040.7391
275.5183
.3
6836.8296
293.1112
.9
1
7682.1623
310.7072
AREAS AND CIRCUMFERENCES OF CIRCLES.
S5
TXBl.E.— (Continued.)
Diatn.
Area.
Circum.
1
Diara.
Area.
Circum.
Dium.
Area.
Circum.
W.
761)7. :o:)4
311 0184
.4
7760.0347
312.275
.8
7H22.61.-4
3i:5.51l6
.1
77l:!.-i»i4l
311.:{:!25
.5
7T7.').6.'>«i:'.
3IJ..>'92
.9
7H:{r!.299H
313.H458
.2
7728.r:5:5t)
311.6467
.6
779l.2i>:j.;
3I2.90:13
100.
7.-54.
314.16
.3
7744.4^S8
31l.%0i
7
T«(»6.91<iti
.;i;;.-ji:.-.
To Cooipu
te the Arei or rirniniferrnrc
of a Diometer greater than any
in the preceding Table.
1
8e(
i Rulen, pag
sn 176 and 181.
,
1
Or
J[/ the DtameUr exceeds 100 and is leiis than 1001.
R
Re
move the de
cimal 1 oint, and tiike > ut the art*a orcircumfierence as for a Whole Number
by removing the decimal point, it' for the area, iwo
plaws to the right ;
and if for the circum- {|
ferenc<
;, one place
III
U8TRATI0N.-
-The area of 96.7 is 7344.18!) ; hence lor 9t>7 it is 734418.9 ; and the circum-II
feireaw
iof96.7ia3(
)3.7927, and for 967 it 18 3037.9^7
TJ^BI^V.
IIIX.
ABEAS AMD CIECUMPERENCBS OP CIBCLES
PROM 1 TO 50 FEET.
(Advancing by an Inch.)
-
Diam.
Area.
Circum.
Diiim.
Area.
Circum.
Dium.
Area
Circum.
Feet.
Feet. Ins.
Feet.
Feet. Ins.
Feet.
Feet. Ins.
1ft.
.7854
3 m
'Aft.
7.0686
. y =''
1 oA
19.635
15 8}i
I
.9217
3 4%
1
7.4666
9 81^
1
20.2947
15 11^
2
1.069
3 8
7.8;5:
9 11>^
2
20.96:>6
16 2g
3
1.2271
3 11
3
'<,295r
10 Jla
:<
21.6475
16 5^
4
1.39fi2
4 -^H
4
rt.72t)5
10 5%
4
22.34
16 9
5
1.5761
4 5%
5
9.16rt3
10 S^
5
23.0437
17 H
6
1.7671
4 6^
6
9.6211
10 11%
6
23.7583
17 32
17 6^
7
1.9689
4 11^
7
10.0846
11 3
7
24. 48:^
8
2.1816
5 2%
8
10..v>;»l
11 6)^
8
25.2199
17 9^
9
2.4052
5 5%
9
11.04 46
U 9%
9
25.9672
18 S
10
2.6:598
5 9
10
11.5-IO;t
12 }4 i
10
26.7251
18 :{Jg
11
2.8852
6 2>i
11
12.0481
12 :{%!
11
27.4943
18 7>|
2/'.
3.1416
6 3%
4A
12.5664
12 63^4
6ft
28.2744
18 10>|
1
3.4087
6 6)4
1
13. 09.. 2
1-^ 9J8:
1
29.0649
19 \H
2
3.6869
6 9%
2
13.6:5:.3
13 1
2
29.ri668
19 4%
3
3.976
7 H
3
14.1^62
13 i}i
;5
;50.6:96
19 7i|
4
4.276
7 3J^
4
14.7479
13 7ii\
4
31.5029
19 10^
5
4.5869
7 7
5
15.3-^W)
1:5 lOig I
5
32.:«76
20 ig
20 4%
20 8)1
20 llH
21 2%
21 8^
6
4.9087
7 lOH
6
15.'.)()43
14 l^
6
33.1831
7
5.2413
8 1%
7
16.4986
14 45^
7
34.0:591
8
5.585
8 4^
8
17.1041
l-l '^^
8
34.9065
9
5.9:595
8 7%
9
17.720..
14 11
9
:i5.7H47
10
6.:!049
8 10^
10
18.:i476
15 2^
10
:56.6T35
11
6.6813
y IVb
11
i.'.a.-58
15 51^
11
37.5736
se
AREAS AND dKCUMFERENCES OF CIRCLES.
TABLE.— (Conftnoerf. )
Diam.
7/t.
1
3
4
5
6
7
8
9
10
11
8/<.
1
y
3
4
5
6
7
8
9
10
11
9/<.
1
2
3
4
5
6
<
8
9
10
11
10/t.
I
2
3
4
5
6
7
9
10
11
ll/K.
1
2
3
4
5
6
Areiu
Feet.
3."j.4846
:<i».40<>
40.3388
41.2rt25
42.2367
43.2J2J
44.1787
4o.l6r)6
46.1H38
47.173
48.19-J6
49.2236
r.1,3178
f.2.3816
5:5.4r)62
54.r)412
r)5.^377
56.74r)I
57.^628
58.992
60.1321
61.2826
62.4445
63ol74
64.W006
65.99r)l
67.2007
68.4 lt)6
69.644
70.8823
72.1309
73.391
74.662
75.9433
77.2362
78.i>4
79.854
81.1795
82.516
83.8627
85.2211
86.5903
e7.9«>97
89.360.i
90.7«;27
92,1749
93.5986
95.0334
96.4783
97.9347
99.4021
100.8797
102.3689
103.8691
Circum.
Feet. Ins.,
H
11^
21
22
22
22
23
23
23
23
24
24
24
24
25
25
25
25
26
26
26
26
27
27
27
28
28
28
26
29
29
29
29
30
30
30
30
3».
31
31
31
32
:52
32
:?2
33
33
33
34
34
34
34
35
35
35
35
36
ll^«
i
23^"
9^
H
«•%
y>2
'A
3^
5
2%
5>o
•-^
11%
H
^%
%
'%
lit'
Di'im.
7
d
9
10
n
12A
1
'I
3
4
5
6
7
8
9
10
II
13/i!.
1
2
3
4
5
6
7
8
9
in
11
Wft.
1
2
3
4
5
6
7
8
9
10
11
l.-./i.
1
2
3
4
5
6
7
8
;>
10
II
16/)!.
1
Aieii.
Feet.
I05.::794
106.;'013
io.-'.4;;42
lUi».9772
111.5319
113.0976
114.6732
IJ6.2»)07
117.rt.59
119.4674
121.0876
122.71^7
I24.359rt
126.0127
127.6765
129.3.")(»4
131.036
132.732«)
134.4391
136.1574
137.8S67
139.626
141.3771
143.1391
144.9211
146.6949
148.4rt96
150.2943
152.1109
153.93-4
155,7758
157.625
159.4852
161.35.-.3
163,2373
165.1*13
167.0331
168.9479
170.8735
172.8091
174.7565
176.715
17-^.6-32
l-0.6()3l
182.6..45
lfr4. 65.5.5
lJ^ti.6684
lS?<.6'.t23
4,0.72()
192.77 1()
194. -282
i9(i,-'.»4t;
19?^. 97 3
201.(1624
203.1615
Circum.
Feet.
Ins.
:«)
^%
36
'%
36
lO'^B
37
'^/^
37
.^K
37
^%
37
n\^
:{8
'^%
38
'-%
\\6
^%
39
:i9
■^H
39
«%
39
91^,
40
'A
40
:i%
40
^%
40
10
41
1^
41
4%
41
:i-o
41
i^»%
42
1^8
42
•»%
42
8
42
llfft'
43
^M
43
-h
43
«^
43
11%
44
->8
44
6
44
y^
44
H
45
SH
45
45
4t>
46
46
46
47
47
47
47
48
4^
48
48
49
19
49
50
51 1
50
y%
4
IlK
IK
*\
'1%
2%
5^-
^^
Di.'im.
2
3
4
5
6
7
8
9
10
11
17/i;
1
*.»
.4
3
4
5
6
7
8
9
10
11
18/it.
1
2
3
4
5
6
7
8
9
10
11
v^ft.
1
2
3
4
5
6
7
8
9
10
11
20/K.
1
2
3
4
5
6
7
8
Area.
Circum.
Feet.
205.2726
2(»7.;{'.>46
209.5264
211.6703
21:>..-!251
215.9896
21-!. 1662
220.3537
222.551
224.7603
226.9«06
2-i9.2l05
231.4525
2;};;.7055
2:5.5.96-2
23y.243
240.5287
242.8241
24.5. l:U6
247.45
249.7781
252.11-4
254.461)6
256.rt303
259.20:i3
261.. 5872
263.9807
2t)<).3S64
26-!.b031
271.2293
273.6678
276.1171
278.5761
281.0472
2S3 5294
286.021
288.5249
291.0397
293.5641
296 1107
298.6483
301.20.54
303.7747
.306.355
308.9448
311.5469
314.16
316.7824
319.4173
322.063
324.7182
327.3&58
330,0643
332.7522
335.4.525
Feet. Ins.
50 9>^
51 \^
51 'i%
51 6>^
51 10
52 1^
52 41^
52 7%
52 I 01^
5 4%
53 8
.53 11^
04 'Z%
o\ 5%
r>4 8K
54 11^
05 2%
;)5 6
55 9>^
56 y^
56 :\%
nb 6>^
56 9%
57 4
57 7^
n7 log
58 1%
08 4^-^
58 7%
58 10%
58 2
59 5>^
59 Hl^
5y iiK
6it ^ly
60 5%
60 8%
60 U%
60 3)^
61 6W
«>1 >2^
61 3%
62 6%
62 9%
62 1>^
63 4^
63 73^
63 11>^
«3 \%
64 4%
64 7%
64 \\%
AREAS AND cmCTTMFERENCES OF CIRCLIS.
2T
TABLE.— {Continued.)
Diam
1)
10
H
-n/t.
1
•J
3
4
5
6
7
8
9
10
II
2-2/t.
1
2
3
4
5
6
4
8
9
10
11
23/lt.
1
2
3
4
5
6
7
8
9
10
11
24/!!.
1
2
3
4
5
6
7
8
9
10
11
2r,yi.
1
•>
3
Area
Feet.
:MO.f'rt44
3J3.t.i;4
:i4ti.3(il4
:i4'j.iiJ7
:{.') 1.^-04
:ir)7.44:w
;?t30.-J417
:5(W.05ii
3H8.T011
:571.r>4:}'2
374.;iy47
3T7.-.'587
3.-0.i:53t)
:ia3.oir7
385.9144
:i88.ri-2-^
3^1.7389
394.t)(J83
397.6087
400.5583
403.W04
406.4935
409.4759
412.4707
415.4766
4184915
421.5192
424.5577
427.6055
430.H658
433.7371
436.8175
439.9106
443.0146
446.1->78
449.2536
452.3904
455.5362
458.6948
461.8642
465.0428
468.2341
471.4363
474.6476
477.8716
481.1065
484.3.")06
487.6073
490.875
494. 1516
497.4411
500.7415
Ci.cuin
Feet. Ins.
t-5 53 R
6;, .->4
65 115
m 57,„
66 9
66 i,i
67 3%
67 ti>^
67 9^^
68 3/
68 m
6s 7
68 H)i^
m 1%
69 4}4
69 7%
69 10%
70 1%
70 5
70 8>^
70 11)^
71 2^
71 5%
71 8%
71 11^^
72 3^"
7'i 6U
72 90
73 y,
73 3%
73 6g
73 9>^
74 1
74 4^
•74 7>^
74 10^
7o \%
75 4%
7;> il
76 51^
76 8>^
76 11.^
77 2^
77 9 '
7H K
78 31^
78 6K
78 9^,'
79 %
79 3J^
Di itii.
;>
t>
7
'.)
10
h
26 y<.
1
2
3
4
5
6
7
8
9
10
II
27/i!.
1
2
3
4
5
6
7
8
9
10
11
28/c.
1
2
3
4
5
6
7
8
5»
10
11
29/(!.
1
2
3
4
5
6
7
d
9
10
A.ea.
Feet.
5(»l.0..1
.M', :.\',:\2
.,!'). ;<)():!
:)14.(l4.-^4
517.4034
.V.'0.769->
5-.'4.l441
5v:.53H
53(1.9304
534.3:579
53/ .7583
541.l-i)()
54 4.1 29.t
548. ('i'3
551.5471
555.0-JOl
5.")8.5)»59
56J.0027
56.").5084
569.027
572.5.)66
576.0949
579.6463
5H3.'J085
586.779ti
590.3()37
593.95-57
597.5625
601.1793
604. R07
60'i.4436
612.931
615.7536
6J9.42-J8
623.10;.
6-^6.7982
H30.r.002
634.v;l5'2
637.9411
641.6758
645.4-J35
649. 1-^21
652.9495
6.56.73
660.5214
664.3J14
668. 1346
r>7l.95^7
675.7915
679.6375
6-3.4943
687.:;598
691 --'385
<i9.5.I2rt ■
6'.n' I»2ti3
''ileum
Feet Ins.
79
1 4
f^O
80
81
Hi
81
81
82
82
82
82
83
83
83
84
84
84
84
85
a5
85
85
8t)
86
86
86
7
87
87
87
88
%^
88
89
)?9
89
89
90
9«i
90
90
91
91
91
91
92
92
92
9-J
93
93
9.3
'38
or,
7%
1034
^)i
\\%
•>3/
%
11.^/8
3
^%
%
3>^
9%
4M
^h
11%
\%
*/^
11
•-^
"^
«%
11>2
9
¥^
3W
«%
9>^
1^
4%
10%
Di iin.
Area
II
'Mft.
1
2
3
4
5
6
7
8
9
10
11
31/i!
1
2
3
4
5
6
7
8
9
10
n
32/«.
1
2
3
4
5
6
7
8
9
10
11
33//.
1
2
3
4
5
6
7
8
9
10
11
Uft.
1
2
3
4
Feet.
70j.!t:i7:
71 ().-(;
7ii).:;:i9
714.7:>5
7 1 -.69
7-j2 t)537
7j6.r,:;it5
730.6183
734.6147
73"'.t;242
74-J.6I47
74H 67:'.i-i
750.7 161
754 7(>9l
7.",f<.831 1
762.906-.'
7(it).992l
771.08()6
775.1914
779.3131
783.4403
787..580.-i
791.732*i
795.8922
800.0()54
804.2496
808.4422
812.6481
816.865
821.0904
825.32t)l
829.5787
833.8:}68
838.1082
842.3095
846.6813
850.9855
855.3006
^^59.624
863.9608
868 3087
872.6649
877.0346
881.4151
885.804
890.2061
894.6196
899 0113
903.4763
907.9224
912.3767
916.(^415
921.3232
Jt2.5.-103
930.:51(t8
Oin uni
Feet Ins.
!'3 117^
94
'8
94 6
91 «tj^ :
95 %
95 3U
95 6^1
95
9^
96
■'A
96
4
9)
714
96
1034
97
IJ-o
97
4%
97
"!%
97
K'%
98
2
98
0^
98
^%
98 111^
99
2%
99
5%
99
«%
loo
100
^%
loo
<i%
loo
9>^
101
%
101
m
101
6%
101
10
102
^H
102
4%
102
^%
102
10%
103
i^
103
4%
103
8
103
11^
104
2J4
104
fi'4
104
H%
104
11%
105
2%
105
6
105
9J^
106
.^
106
106
6%
KKi
9%
107
0%
107
4
107
71^
107
\(\%
108
1%
t8
AKEAS AND CIRCUMFERENCES OP CIRCLES.
TABLE.— (Continued.)
Diam.
6
7
8
9
10
11
35/i!.
1
y
3
4
5
6
7
S
9
10
11
36/it.
1
2
3
4
5
8
9
10
11
37/it.
.t.
2
3
4
5
6
7
H
9
10
II
1
2
3
4
5
6
7
«
9
10
n
Area.
Feet.
934.8223
93y.34v!l
943. (-753
9484195
9r.2.972
957.538
9t)2.115
9()«.770l
971.2989
975.9085
980.5264
985.1579
989.8U03
994.4509
999.1151
1003.7902
1008 473(5
1013.1705
1017.B784
1022.5944
1027.324
1032.0646
1030.8134
1041.5758
1046.3491
l051.i:{U6
10.'....9J57
1060.7317
101)5.5459
1070.37:{8
1075.2126
lU~0.0594
lU84.9vi01
10,-97915
1094 6711
l()99.5ii44
1104.4687
1109.3'- [
1114.. 071
1119.241
1121.1H91
Il2;i.l478
il::4.ll76
i!39.(i95;{
ll44.ii>tW
Il49.(t892
11. "4.09.17
ll.".9.lj:i9
1164 1591
Ilti9.-J02:!
1174.2592
1179 3271
1181.103
1189 4927
il94.;)9.{4
Circum.
Feet. Ins.
108
108
108
109
109
109
109
110
110
110
HI
HI
ill
111
112
112
112
112
113
113
113
113
114
114
114
J 14
115
115
115
115
116
116
116
117
117
117
117
IH
118
118
118
119
119
119
119
1-^0
120
120
120
121
121
121
in
122
122
4^
2
^%
11%
-'%
y%
4
iS^
'\
11>8
•;%
il>8
'^%
h
'^%
4^
l^il
%
-%
11%
61^
Diam.
1
2
3
4
5
e
7
8
9
10
11
4oy<.
1
3
4
' 5
6
7
8
9
10
II
41/i!.
1
3
4
5
6
7
8
9
l"t
11
42/;;.
1
2
3
4
5
6
7
8
9
10
11
43/if.
1
2
3
4
5
6
7
Area.
Feet.
1199.7195
1204.8244
1209.9577
1215.099
1220.2..42
1225.4203
1 -.'30. 5943
223.").7^^22
1240.981
1246.1878
1251.4084
1256.()4
l261.^7:t4
1267.1327
1272.397
1277.6092
12^2.9553
12'^8.«2;V2:i
1293.5;.72
I298.87ti
1304.2057
1305.5433
1314.8949
13v:0.v;574
I325.(i-J76
1331.0119
1336.4071
1341. .-101
1:547.2271
l:{.")2.6551
1:558.0908
1:563.5406
1:569.0012
1:574.4697
l:579.95Jl
1:585.4456
1:590.2467
1:596.4619
1401.9.-8
1107.5219
141:5.6098
14)8.6287
1424.1952
1429.7759
14:55.:5675
1440.9668
1446.5802
1452.2046
1457.8:565
146:5.4827
1469.1397
1474.8044
14^0.48:53
14-6.1731
1491.8705
Circum.
Feet. Ins.
122
12:5
123
123
123
124
124
124
124
125
125
125
125
126
126
126
126
127
127
127
128
128
128
128
129
129
129
129
130
i:50
1:50
l;50
131
131
131
131
132
i:52
132
132
1:53
133
i:j:5
1:54
1:54
134
134
135
135
1:J5
1.35
1:56
1.36
136
136
9>^
•IK
'%
10>.4
f
•/8
11
2W
-%
8>2
11^8
i*>8
9
!>>8
\^
'■%
I'Vs
1%
%
1>8
'^^
%\
^K
^%\
1.1%
fi%:
%|
:^%!
9%
1
10%
1%
4^
Diam.
8
9
10
11
1
2
3
4
5
6
8
9
10
11
45/if.
1
2
:5
4
5
6
7
8
9
10
11
46A
1
2
3
4
5
6
7.
8
9
10
11
47/;.
1
2
3
4
5
6
7
8
9
10
11
48/t.
1
2
Area.
Circum.
Ffiet.
1497.5821
1503.:5046
1509.0:548
1514.7791
1520..-):544
1526.2971
15:52.0742
15:57.8622
154;5.6.")78
1549.4776
15.55.-.!883
1561.1165
I5t;6.9591
1572.8125
1578.6735
1584.5488
1.590.4:^5
1596.:52.-^6
1602.2:566
1608.1555
1614.0819
1620.0226
1625.9743
16:)1.9:!:54
16:57.9068
1643..8912
1649.. -831
1655.8.<92
16.11.9064
1667.9:508
1673.9698
16800196
1686.0769
1692.14-5
1698.2311
1704.:521
1710.4254
1716..->407
1722.66:54
1728.9005 147
Feet. Ins.
137
137
137
137
138
138
138
139
1:59
139
1:59
140
140
140
141
141
141
141
141
142
142
142
142
143
143
14:5
143
144
144
144
145
145
145
145
146
146
146
146
147
2%
8%
11%
■-^^
'»%
9
%
'%
fi%
9%
3%
7%
10%
4%
^^
1%
5
«%
IIM
•^%
h%
^%
11%
3
6%
y%
17:54.9486
1741.10:59
1747.2738
1753.4545
1759.6426
1765.84.52
1772.0587
1778.2795
1784.5148
1790.761
1797.0145
1803.2826
1809.5616
1815.8477
1822.1485
147
147
148
148
148
148
149
149
149
150
150
150
150
151
151
3%
6%
9%
1%
4%
'0%
IJ-a
4%
li^
2%
H%
UK
2%
5%
8%
3W
6%
3%
SIDES OF EQUAL SQUARES.
TABLE.— (ConHnued.)
29
Diam.
Area.
Circum.
Diam.
Area.
Circum.
Diam.
Area.
Circum.
Feet.
Feet. Ins.
Feet.
Feet Ins.
Feet.
Feet Ins.
3
18-iH.4«0-2
151 6%
11
1X79.3355
153 81^
7
1930.9188
155 9W
156 ^
4
!«:{4.77»1
151 1U>^
49/^.
1885.74.54
153 1V4
8
1937.3159
5
!841.17-J7
152 ii^
1
1892.1724
154 2%
9
1943.914
156 3>^
6
1847.4571
I'o-i 4%
'2
1898.5041
154 5>^
10
1950.4:»2
156 65-^
7
18:.:{.80rt7
152 7^4
3
1905.0367
154 8^
11
1956.9691
156 9%
8
I. ->(>(). 175
15'i 105^
4
I911.4lt(i5
154 11%
5oy?.
1963.5
157 %
9
iHtlti.fwVJl
15:; 1^
5
J 9 17.9609
155 2%
10
l»7-i.y3t)5
153 3>g
6
1924.4263
155 6
TABLE OP THE SIDES OP SQUARES-EQUAL IM ABEATO
A CIRCLE OP ANY DIAICETEB.
FROM 1 TO 100.
Diam Side of Sq
1.
■'A
5:^
6.
.88()2
1.1078
l.:;293
1.55U9
1.7724
1 9m
2.2ir,6
2.43ri
2.65h:
2.f*f<02
3.1018
3.:'.233
3..0449
3 7665
3.988
4.2096
4.43il
4.6527
4.8742
5.0958
5.3174
5.5389
5.7605
5.9-^2
6.2036
6.4251
6.6467
6.8683
Diam. Side of Sq. Diam.
8.
9.
■H
10.
11.
•y^
13.
14.
•Va
'Va
7.0898
7.3114
7.5329
7.7545
7.976
8.1976
8.4192
8.6407
8.8623
9.08:{8
9.3054
9.5269
9.7485
9.97
10.1916
10.4132
i 0.6347
10.>'563
11.0778
11.2994
I ! .5209
11.7425
11.9641
l2.18.-)6
12.4072
12.6287
12.8503
13.0718
15.
16.
18.
■Va
■%
■Va
■Va
■Va
19.
20.
■Va
I
Side of Sq. Diam. Side of Sq
21.
•Va
13.2934
13.515
13.7365
13.9581
14.1795
14.4012
14.6227
14.8443
15.06.59
15.2874
15 509
15.7;:05
15.9521
16.1736
16.3952
16.6168
16.8383
17.0.-)99
17.2814
17.503
17.7245
17.9461
18.1677
18.3892
18.6i09
18.8323
in o:.39
19.2754
22.
■Va
23.
24.
'•Va
25.
■Va
Ya
26.
•Va
27.
■Va
28.
19.497
19.7185
19.9401
20.1617
2U.:W32
20.6048
•J0.8263
21.0479
21.2694
21.491
21.7126
21.9341
.'2.1557
22.3772
22.5988
22.8203
23.04 19
23.2634
23.485
23.7066
23 9281
24.1497
24 3712
24.5928
24.8144
25.03.-)9
25.2575
25.459
Diam.
29.
"6
■Va
30.
31.
32.
■Va
■Va
■Va
%
I
Va
Side of Sq.
33.
:J4.
■Va
■Va
35.
•Va
2 .7006
25.9221
26.1437
26.3653
26.5868
26.8084
27.0299
27.2515-
27.473
27.6947
27.9161
28.1377
28.3593
28.5808
28.8024
29.0239
29.24.55
29.467
29.()886
29 9102
30.1317
30.:J533
30.5748
30.7964
31.0179
31.2395
31.4611
31.6826
30
LENGTHS OF CIRCULAR ARCS.
TAB LE.~(Conttnued. )
Diam.
Side of Sq.
Diam.
Side of Sq.
Diam.
Side of Sq J
Diam.
Side of Sq.
Diam.
Side of Sq.
36.
31.9042
49.
43.4251
62.
54.9461
75.
66.467
88.
77.9)58
H
32. l-^.")?
•Va
43.6467
-Va
55,1676
■h
66.6886
y
78.2095
A
32.3473
'%
43.8682
■h
55.3892
y
t)6.9 1 04
y
78.4316
%
32.r)t)ti8
rA
44.0898
■H
55.6107
■U
67.(317
y
78.6526
37.
3VI.7904
50.
44.3113
63.
55.8323
76.
6:.:5532
89.
78.8742
M
33.0112
■Va
44.5329
■H
56.0538
■H
67.. 57 48
y
79.0; '57
%
33.23:J5
■%
44. 7.545
4
56.2754
y
67.7964
y
79.3173
Va,
33.45.-)l
■%
44.976
■%
56.497
_u
6''.0179
y
79.5389
■38
33.<>7H6
51.
45.1976
64.
56.7185
7<.
68.2395
90
79.7604
%
;;3.dy82
•Va
45. ' 9!
'Va
56.9401
y
68.461
y
79 982
%
34.1197
-'A
45.6407
•y^
57.1616
y
68.6820
y
80.2035
\
34.3413
■Va
45.r622
-Va
57.3832
■h
68.904 1
y
80.4251
39
34.r)628
52.
46.0838
65.
57.6047
78
69.1257
91.
80.6467
Va
3 i. 7884
■Va
46.3054
•^
57.8263
y
69.3473
y
•'0.8682
'A
35.006
■%
46.5J69
■A
58 0179
K
69..5t) -8
y
81.0898
%
35 2275
•Va
46.7485
'Va
58.2691
■^l
69.7904
y
-1.3113
40
35.4491
53.
46.97
m.
58.491
7 '.
70.(»iI9
92.
81.5329
H
35.6706
■^A
47.1916
yA
:)8.7125
■'4
70 2335
y
81.7544
A
25.892i
■%
47 4131
■H
58.9341
.1.,
70.455
■A
-1.976
%
36.1137
'%
47.6347
■Va
59.15.56
?4
70.6766
y
82.1975
41
3«).3:i53
54.
47.K562
67.
59.3772
r-u.
7o.8;i81
93.
8i.4191
Va.
36.5569
■Va
48. ,-778
■%
59.. 5988
■H
71.1197
y
82.6407 ■
A
36.7784
■%
48.2994
•A
59 8203
1/
71.3413
■A
82 8622
%
37.
■Va
48.5209
■Va
60.0419
•?4
7 1.. -628
y
83.0-38
42
37.2215
55.
48.7425
68.
60.2()34
1
7!.7i<44
94.
83 3053
Va
37.4431
•Va
48.964
•K
60.485
.I4 72.0059
y
rt3.5269
%
37.6649
•H
49 1856
•K
6i'.70ii.")
>^ ; ; 2.227.-.
y
n.1.7484
A
37.8862
•%
49.4071
•Va
6(/.928l
■H
72 4191
■A
83.970
43
38.1078
56.
49.6287
69.
61.1497
&i.
72.671/6
95.
e4.l9i6
Va
38.3293
■ Va
49.- 503
Va
61.3712
y
72.8.'21
y
84.4131
%
3rt.5;.09
A
5(1.0718
•y^
61...92-<
y
7.;. 1137
y
84.6317
^
38.7724
J/a
5(».2934
•Va
til. ."^143
y
73.:{3.>3
■A
-4.8.5«!2
44
38.994
ot .
50. ."1149
70.
62.03.59
8.!.
73.55(>8
m.
85.0778
Va
39 2155
M
50.7365
•K
62.2574
y
73 7784
y
85.2993
%
39.4371
■¥
5U.958
•>2
62.179
y
73...9it9
y
8.>.5209
Va
3J.65-7
■Va
51.179t)
■%
t)2.700i»
y
71.221..
y
85.7425
45
3i».8802
58.
51 4(»12
71.
62.9221
rt4.
74.4431
97.
85 9616
:»
40.1018
■Va
5L.6227
■yA
(i3. 1 ):;7
y
74.6647
y
86.185
40.3233
■H
51.8443
■%
63.36.52
y
74.'^8.)2
y
86.4071
%
40 5449
•Va
52.06.58
■Va
!>;. 5-6^
%
75. 1077
y
86.6289
46
40.7664
59.
52.2874
72.
..3..-'o.-!;;
85.
75 3293
98.
86.8502
•M
40.9.-^8
■Va
52.50-9
•M
64 029.*
.14
75.5.508
y
87.0718
y^
41.2096
■'4
52.7305
•>2
64.25 1
y
75.7724
y
87.2933
■Va
41.4311
n^
52.9521
Va
64.47.;0
.'4
75.9934
y
87.5449
47
41.9527
60.
53.1736
73.
64. 91
76.21.55
99.
87.7364
■Va
41.^742
■Va
53.3952
•>4
64.9 itii
H
76.4371
y
87.958
•>2
42.0958
%
53.6167
■%
()5.1377
A
76.65-6
y
88.1796
.%
42.3173
.•%
53.8383
■Va
65.3.592
y
76.8rt02
y
68.4011
48
42..->839
61.
54.0598
74.
65..580H
t
77.1017
100.
88.6227
'Va
42 7604
1
54.2814
yA
65.802:1
■H
77.3233
y
88.8442
14
42.982
54.503
•K
66 0239
■A
77.5449
y
89.0658
'Va
43.2036
■yA
54.7245
•Va
66.2455
i y
77.7664
y
89.28?4
TABLK VI.
TABLE OP THE LENGTHS OP CIRCLAB ARCS.
The Diameter of a Circle assumed to be Unity, and divided into 1000 equal Parts.
Hght.
Length.
: r-1
H'ght.
Length.
H'ght.
Length.
H'ght.
Length.
H'ght.
Length.
.1
1.02645
.148
1.0^743
.196
1.09949
.244
1.15186
.292
1.21381
.101
1. 02098
.149
1.0.5819
.197
1.10U48
.245
1.15:508
.293
1.2152
.10-2
l.<>27.")2
.15
1.05896
.198
L. 10147
.246
1.15429
.294
1.216,58
.103
1.0'>806
.J5l
1.0.5973
.199
1.10247
.247
1.1.-.549
.295
1.21794
.104
1.0286
.152
1.06051
.2
1.10348
.248
1.1,567
.296
1.21926
.105
1.0-2914
.153
1.0613
.201
1.10447
.249
1.1.5791
.297
1.22061
.lOt)
1.0297
.154
1.06209
.202
1.10548
.25
1.15912
.298
1.22203
.107
1.03026
.l.n5
1.06288
.203
1.10<i5
.251
1.160:53
.299
1.22347
.108
1.030H2
.1.56
1.06368
.204
1.10752
.252
1.161.57
.3
1.22495
.109
1.03139
.157
1.06449
.205
1.108.55
.253
1. 16279
.301
1.22635
.11
1.03196
.1.58
1.06.5:;
.206
1.109.58
.2:4
1.16402
.:502
1.22776
.111
1.03254
.159
1.06611
.207
1.11062
.2.-.5
1 . 1(k'>26
.303
1.22918
.112
1.03312
.16
1.0.i693
.208
1.11165
.•-'56
1.16649
.:504
1.23061
.113
1.03371
.161
1.06775
.209
1.1126S'
.257
1.16774
.305
1.23205
.114
1.0:i43
.162
1.06'-.58
.21
1.11:574
.2.")ft
l.)()8'.t9
.306
1.23349
.115
1.0349
.163
1.06941
.211
1.114/9
.2.59
1.17024
.307
1.2:5494
.116
1.03551
.164
1.07025
.212
1.115H4
.•J6
1.1715
.308
1.23636
.117
1.03611
.165
1.0710'.»
.213
1.11692
.261
1.17275
.309
1.2:578
.118
l.03()72
.166
1.07194
.214
1.11796
.■1&2
1.17401
.31
1.2:5921
.J 19
1.03734
.167
1.07279
.215
1.1J9(I4
.2t):5
1.17,527
.311
1.2407
.12
1.03797
.168
1.07365
.216
1.12011
.264
1.17655
.312
1.24216
.121
l.0:i86
.169
1.07451
.217
1.12)18
.265
1.17784
.313
1.2436
.122
1.0:5923
.17
1.075:^7
.218
1.12225
.266
1.17912
.314
1.24506
.123
1.03987
.17]
1.07624
.219
1.12334 ,
.267
l.ln04
.315
1.246,54
.124
1.04051
.172
1.07711
.22
1.12445
.268
1.18162
.316
1.24801
.125
1.04116
.173
1.07799
.221
1.125.56
.269
1.18294
.317
1.24946
.126
1.04181
.174
1.07888
.222
1.12663
.27
1.18428
.318
1.25095
.127
1.04247
.175
1.07977
.223
1.12774
.271
1.18,557
.319
1.25243
.128
1.01313
.176
1.08066
.224
1.12«85
.272
1.18688
.:52
1.25391
.129
1.0438
.177
1.08156
.225
1.12997
.273
1.18819
.:521
1.25539
.13
1.04447
.178
1.08246
.226
1.13108
.274
1.18969
.322
1.25686
.131
1.04515
.179
1.08337
.227
1.13219
.275
1. 19082
.323
1.25836
.132
1.04584
.18
1.08428
.228
l.i:«3i
.276
1.19214
.324
1.25987
.133
1.04652
.181
1.08519
.229
1.13444
.277
1.19345
.325
1.26137
.134
1.04722
.182
1.08611
.23
1.13.557
.278
l.t9477
.326
1.26286
.135
1.04792
.183
1.08704
.231
1.1367]
.279
1.1961
.327
1.26437
.136
1.04862
.184
1.08797
.2:^2
1.13786
.28
1.19743
.328
1.26588
.137
1.04932
.IfeS
1.0889
.2:53
1.1:590:5
.2ol
1.19887
..329
1.2674
.138
1.05003
.186
1.08984
.234
1.1402
.282
1.20011
.33
1.26892
.139
1.05075
.187
1.09079
.235
1.141:56
.283
1.20146
.331
1.27044
.14
1.05147
.188
1.09174
,2;]6
1.14247
.284
1.20282
.332
1.27196
.141
1.0.522
.189
1.09269
.237
1.14:563
.255
1.20419
.333
1.27349
.142
1.05293
,19
1.09:i65
.238
1.1448
.286
l.20.->.58
.334
1.27502
.143
1.05367
.191
1.09461
.239
1.1459T
.287
1.20696
.3:55
1.27656
.144
1.05441
.192
1.09557
.24
1.14714
.28^
l,2(i8-i.><
.336
1.2781
.145
1.05516
.193
1.09654
.241
1.14H3i
.2-9
■..2091 '.7
.337
1.27964
.146
1.05591
.194
1.097,52
.242
1.14949
.29
1.21202
.:538
1.28118
.147
1.05667
.195
1.0985
.243
1.15067
.21)1
1.212:39
.339
1.28273
32
LENGTHS OF CIRCULAR ARCS.
T ABLE— ( Continued. )
H'ght.
Length.
H'ght.
Length.
H'ght.
Length.
H'ght.
Length.
H'ght.
Length.
.34
1.284-28
.373
1.3373
.406
1.39372
.439
. 1.5327
.472
1.51571
.341
1.-28583
.374
1.33896
.407
1.39548
.44
1.4.5512
.473
1.51764
.342
1. '28739
.375
1.34063
.408
1.39724
.441
1.4.5697
.474
1.51958
.343
1.'28895
.376
1.342-i9
.409
1.399
.442
1.45883
.475
1.52152
.344
1.29052
.377
1.34396
.41
1.40077
.443
1.46069
.476
1.52346
.345
1.29209
.378
1.34.5b3
.411
1.40254
.414
1.46255
.477
1.. 52541
.346
1.29366
.379
1.34731
.412
1.40432
.445
1.46441
.478
1.52736
.347
1.29.523
.38
1.34899
.413
1.406
.446
1.46628
.479
1.52931
.348
1.29681
.381
1.35068
.414
1.40788
.447
1.46H15
.48
1.53126
.349
1.29839
.382
1.35237
.415
i.4ii966
.448
1.47002
.481
1.53322
,35
1.29997
.383
1.3.5406
.416
1.41145
.449
1.47189
.482
1.. 53518
.351
l.oOJ.')6
.384
1.35575
.417
1.41324
.45
1.47377
.483
1.53714
.35-2
1.30315
.385
1.35714
.418
1.41503
.451
1.47.565
.484
1.5391
.353
1.30474
.386
l.;i5914
.419
1.41682
.452
1.47753
.485
1..54106
.354
1.30()34
.387
1.36084
.42
1.41861
.453
1.47942
.486
1.54302
.355
1.3(»794
.388
1.36254
.421
1.42041
.454
1.48131
.487
1.54499
.3.56
1.30954
.389
1.3642.->
.422
1 .42222
.455
1.4832
.488
1.54696
.357
1.31115
.39
1 .36596
.423
1.42402
.456
1.48509
.489
1.54893
..358
1.31276
.391
1.36767
.424
1.42583
.4.57
1.48699
.49
1.5509
.359
1.31347
.392
1.36939
.425
1.42764
.458
1.488.'.9
.491
1.55288
.36
1.31599
.393
1.37111
.426
1.42942
.459
1.49079
.492
1.55486
.361
1.31761
.394
1.37283
.427
1.43127
.46
1.49268
.493
1.55685
Mm
1.31923
.395
1.374r)5
.428
1.43309
.461
1.4946
.494
1.55854
.363
1.32086
.396
1.37628
.429
1.43491
.462
1 .49651
.495
1.56083
.364
1.32249
.397
1.37801
.43
1.43673
.463
1.49842
.496
1.56282
.365
1.32413
.398
1.37974
.431
1.43856
.464
1.50033
.497
1.56481
.366
1.32577
.399
1.38148
.432
1.44039
.465
1.50224
.498
1.5668
.367
1.32741
.4
1.38322
.433
1.44222
.466
1.. 504 16
.499
1.56879
.368
1.32905
.401
1.38496
.434
1.44405
.467
1.50608
.5
1.57079
.369
1.33069
.402
1.38671
.435
1.44589
.468
1.508
.37
1.33234
.403
1.38846
.436
1.44773
.469
1.50992
.371
1.33399
.404
1.39021
.437
1.44957
.47
1.51185
■-"
.
.37-2
1.33564
.405
1.39196
.438
1.45142
.471
1.51378
To Ascertain the Length of an Arc of a Circle by the preceding Table.
Rule. — Divide the height by the base, find the quotient in the column of heights, and take the
length of that height from the next righthand column Multijly the length thus obtained by the
base of the arc, and the product will give the lenth of the arc.
Example. — What is the length of an arc of a circle, the base or span of it being 100 feet, and
the height 25 feet ?
25_t_i00=.25; and .25 per table, =1.1 5912, <Ac length of the base, which, being multiply by 100=
115.912/ee<.
Note. — ^When, in the division of a height by the base, the quotient has a remainder after the
third place of decimals, and great accuracy is required
Take the length for the first three figures, subtract it from the next following length ; multiply
the remainder by the said fraction al remainder, add the jiroduct to the first length, and the sum will
be the length for the whole quotient
Example. — What is the length of an arc of a circle, the base of which in 35 feet, and the height
or versed sine 8 feet ?
8h-35=. 2285714 ; the tabular length for .228=1.13331, and for .229=1.13444, the difference
between which Is .00113. Then .5714X. 00113= .00064568^
Hence .228=1.13331.
and .^--i^rrrj— i — ;i;'. c .0005714= .000645682
1.133955682, the sum by which the base of
the arc is to be multiplied ; and 1. 133955682 X :i5=39.68845/£e;.
TABLE VTI. 'TT
TABLE OF THE LENGTHS OF SEMI- ELLIPTIC ABCS.
The TroYisverse Diameter of an Ellipse assumed to be Unity, and divided into 1000
equal Parts.
H'ght.
Length.
H'ght.
Length.
H'ght.
Length.
jH'ght.
Lergth.
H'ght.
Length.
.1
1.04102
.1-18
1.09119
.196
1.14531
.244
1.2038
.292
1.26601
.101
1.04202
.149
1.0922f<
.197
1.14646
.245
1.20506
.293
1.267:M
.'xO->
1.04362
.1.^.
1.0933
.198
1 14762
.246
1.20632
.294
1.26^67
.103
1.04462
.151
1.09148
.199
1.14888
.247
1.20758
.295
1.27
.104
1.04r.62
.152
I 09558
2
1.15014
.248
1.20884
.296
1.27133
.105
1.04662
.153
1.09669
.201
1.15131
.249
1.2101
.297
1.27267
.10(>
1.04762
154
1.0978
.202
1.1.5248
.25
1.21136
.298
1.27401
.107
1.04862
155
1.09891
.203
1.15366
.251
1.21263
.299
1.27535
.108
1.04962
.156
l.lOOO-i
.204
l.l54c«4
.252
1.21.39
.3
1.27669
.10 J
1.05063
.157
l.lOliS
.205
1.15602
.253
1.21517
.301
1.27803
.11
1.05164
.158
1.10224
.206
1.1572
.254
1.21644
.302
1.27937
.111
1.05265
.159
1.10335
.207
1.15838
.2r,5
1.21772
.303
1.28071
.llii
1.05366
.16
1.10^47
.208
1.15957
.256
1.219
.304
1.28205
.113
1.05467
.161
1.1056
.209
1.16076
.2.57
1.22028
.305
1.28339
.114
1.05568
.162
1.10672
.21
1.16196
.258
1.221.56
.306
1.28474
.115
1.0.5669
.163
1.10784
.211
1.16316
.259
1.22284
.307
1.28609
.116
1.0577
.164
1.10896
.212
1.164:]6
.26
1.22412
.308
1.28744
.117
1.05872
.165
1.11008
.213
1.165.57
.261
1.22541
.309
1.28879
.118
1.05974
.166
1.111:<;
.214
1.16678
.262
1.2267
.31
1.29014
.119
1.06076
.167
M1232
.215
1.16799
.263
1.22799
..311
1.29149
.12
1.06178
.168
1.11344
.216
1.1692
.264
1.22928
.312
1.29285
.121
1.0628
.169
1.11456
.217
1 17041
.265
1.230,57
.313
1.29421
.122
1.06382
.17
1.11569
.218
1.17163
.266
1.23186
.314
1 .29557
.123
1.06484
.171
1.11682
.219
1.17285
.267
1.23315
.315
1.29603
.124
1.065ri6
.172
1.11795
22
1.17407
.268
1.23445
.316
1.29829
.125
1.06689
.173
1.11908
.221
1.17529
.269
1.23575
.317
1 .29965
.126
1.06792
.174
1.12021
.222
1.17651
.27
1.23705
.318
1.30102
.127
1.06895
.175
1.12134
.223
1.17774
.271
1.23835
.319
1.30239
.128
1.06998
.176
1.12247
.224
1.17897
.272
1.23966
.32
1.30376
.129
1.07001
.177
1.1236
.225
1.1802
.273
1 .24097
.321
1.30513
.13
1.07204
.178
1.12473
.226 .
1.18143
.274
124228
.322
1.3065
.131
1.07308
.179
1.12.586
.227
1.18266
.275
1.24359
.323
1.30787
.132
1.07412
.18
1. 12699
.228
1.1839
.276
1.2448
.324
1.30924
.133
1.07516
.181
1. 12813
.229
1.18514
.277
1.24612
.325
1.31061
.134
1.07221
.182
1.12927
.23
1.18638
.278
1.24744
.326
1.31198
.135
1.07726
.183
1.13041
.231
1.18762
.279
1.24876
.327
1.31335
.136
1.07831
.184
1.13155
.232
1.18886
.28
1.2501
.328
1.31472
.137
1.07937
.185
1.13269
.233
1.1901
.281
1.25142
329
1.3161
.138
1.08043
.186
M3383
.234
1.19134
.282
1.25274
.33
1.31748
.139
1.08149
.187
1.13497
.2:J5
1.19258
.283
1.25406
.331
1.31886
.14
1.08255
.188
1.13611
•236
1.19382
.284
1.255;i8
.332
1.32024
.141
1.08362
.189
1.13726
.237
1.19506
.285
1.2567
.333
1.32162
.142
1.08469
.19
1.13841
.238
1.1963
.286
1 .25803
.334^
1.323
.143
1.08576
.191
1.139.56
.239
1.19755
.287
1.25936
.335
1.32438
.144
1.08684
.192
1.14071
.24
1.1988
.288
1.26069
.336
1.32576
.145
1.08792
.193
1. 14186
.241
1.20005
.289
1.26202
.337
1.32715
.146
1.08901
.194
1.14301
.242
1.2013
.29
1.26335
.338
1.32854
.147
1.0901
.195
1.14416
.243
1.20255
.291
1.26468
.339
1.32993
34
LENGTHS OF SEMI-ELLIPTIC ARCS.
^ TABLE.— (Con/inued.)
ffght.
Length. '
H'pht.
Length, 'h
!
ght.
Length.
H'ght.
Length.
H'ght.
Length.
:,u
1.33132
.39t;
1.4l2il
452
1.496i'i
.508
1.5KU9
.,5154
1.(570157
.341
1. 3:5272
.397
l.4l:(57
453
1.49771
.509
158474
.565
1.67245
.\U'2
1.33412
.398
1.41504
4.54
1.4" •92 4
.51
1 5>-6-*9
.566
1.67403
.343
1.3;'.552
.399
1. 41(151
4.-.5
1.50077
.51 1
l.,58784
.567
1.67.561
.344
1.33692
.4
1.41T98
4.56
I 5023
512
1.5894
.568
167719
.34.5
1.33833
401
1.41114.'.
457
1.. 50383
.513
1.59096
.5(59
1.(57877
.346
1.33974
.402
1.4209J
458
1.. 50536
.51 1
1 .592rvj
.,57
1.6803(5
.347
1.34115
.403
1.42239
459
1.50689
.515
l.,59408
.571
1.68195
.348
1.34256
.104
1.42:M)
46
1.50842
516
1.. 59564
.572
1.68:^54
.349
1.34397
.405
1.42. .33
461
1.50'.'96
.5 1 7
1.597-J
.573
1.68513
.3
1.34539
,40(>
1. 42681
Wl
1.5115
.518
1.. 59876
..574
1.68672
.351
1.34681
.407
1.4v!"<29
4<)3
1.51304
.519
1.60032
..575
1.(58831
.352
1.34823
.408
1.4-".tT7
161
|.5i45.-s
5-.'
1 (■.0188
..576
1.6899
.353
1.34965
.40.'
1.43I-.';.
465
1. 5161 -J
.521
1.60344
.577
1.69149
.354
1.35108
.41
I.13J7:!
166
l..'.176ii
522
1.605
.,578
1.(59308
.355
1.35251
.4!l
1.424-.']
467
1.5192
..523
1.60(ii-6
..579
1.69467
.356
1.35394
.412
1.425C>9
468
1. 5207 J
..524
i 60812
.,58
1.69626
.357
1.35.-i37
.413
1.43718
169
1.5-J-J29
.5-.'5
1.60968
.,581
1.69785
.358
1.3568
.414
1.43SC.7
47
1..5-.>:',->4
-..•J6
It; 1124
.582
1 :59945
.359
1.35823
.415
1.4401ti
471
1.5 .'5; .9
-5-J7
1 6128
.583
1.70105
.36
1.35967
.416
1.44 Mm
472
1.526;a
5-i8
:.6U:;6
.584
1.702(54
.361
1.36111
.417
1.44314
473
] .52.-'49
.5-.>9
I 61592
.585
1.70424
.362
1.362,55
.418
1.44163
474
I 53004
-53
1.61748
..586
1.70.584
.363
1.3«)399
.419
1.4-1013
475
1.53 159
.531
1 61904
.587
1-70745
.364
1.36543
.42
1.44763
476
1. 533 14
5 12
1.6206
..588
1.70905
.365
1.36688
.421
1.41913
4T7
1..534t)9
-533
1.622)6
.589
1.71065
.366
1.36833
.422
1.4.->064
47>
1.53625
.5:!4
1.62372
.59
1.71225
.367
1.36978
.423
1.4.-)214
479
1 53781
.535
1.62.'.2H
.591
1.71286
.368
1.37123
.424
1 45364
48
1.53.137
-536
1 .6-J6-4
.592
1-71.546
.369
1.37268
.425
1.45515
4'^1
1 54093
-537
1.6284
.593
1-71707
.37
1.37414
426
1.45665
4^2
1 5424.t
-538
1.6-996
..594
1 71868
.371
1.37662
.427
1.45815
4^3
1 54405
.539
1.631.52
.,595
1-72029
.372
1.3770f<
.42^
I.4.V.t66
484
I 54561
-54
1,63309
.,596
1.7219
.373
1.37854
.429
1.46167
4«5
1 54718
541
1 .6:;-l65
.597
1-7235
.374
1.38
.43
1.4626H
486
1.54875
.542
1 636J3
.598
1-72511
.375
1.38146
.431
1. 461 19
48T
1.55032
-543
1.6378
.599
1-72672
.376
1.38292
.432
1.4657
488
1.55189
544
l.63i)37
.6
1-72833
.377
1.38439
.433
146721
489
1. 5.5346
-.545
1.64094
.601
1.72994
.378
1.38585
.434
1.46ri72
4.»
1 .55503
-546
1.(54251
.602
1-73155
.379
1.38732
.435
1.4/-023
491
1-5566
.547
1 .(54408
.603
1-73316
.38
1.38879
.436
1.47174
492
1.55817
■548
1.64.565
604
1.73477
.381
1.39024
437
1.47326
493
1.55974
-549
1.(54722
.605
1-73638
.382
1.39169
.438
1.47478
494
1..56131
•5
1 61879
.606
1-7:i799
.383
1.39314
.439
1.4763
495
1 56289
-551
1 .6.503(5
.607
1-7396
.384
1.39459
44
1.47782
496
1..5t)447
-552
1.65193
.608
1-74121
.385
1.39605
.441
1.479:!4
497
1 5660ri
-553
1.C.535
.609
1-74283
.386
1.29751
.442
1.48086
498
1. 56763
-554
l.6.";.507
6
1.74444
.387
1.39897
443
1.48238
499
1-56921
-.55,5
1. 6.5665
.611
174(505
.388
1.40043
.444
1.48391
5
1-57089
-556
1 .658-.i3
.612
174767
.389
1.40189
.445
1.48.544
501
1-57234
,5.57
1.6,5981
.613
1.74929
.39
1.40335
.446
1. 48697
502
1-57389
.,558
1 66139
.614
1-75091
.391
1.40481
.447
1.4885
503
1-57544
-559
1.66297
.615
1-7.52,52
.392
1.40627
.448
1.49003
504
1-57699
-56
1.66455
.616.
1-75414
.393
1.40773
.449
1.49154
505
1578,54
-561
1.66613
.617
1-75,576
.394
1.40919
.45
1.49311
506
1-58009
•562
1.66771
.618
1.7,5738
.395
1.41065
.451
~ — —
1.49465
507
1-58164
.563
1.66929
.619
1-759
LENGTHS OF SEMI-ELLHTIC ARCS.
TABLE.— (Continued.)
35
H'Kht.
Length.
H'ght.
Length.
H'ght.
. ength.
H'ght
Length.
H'ght
Length.
.6-2
1.76062
.676
1.H5215
.732
1.94.5.52
.788
2.04117
.844
2.1:59,-6
.6-21
1.76224
.677
1.8.5379
.733
1.94T21
.789
2.04-29
.845
2.14155
.t^':-2
1.76386
.678
1.85544
.734
1.9489
.79
2.04162
.f46
2.14:534
.6-2:^
1.76.548
.679
1.8.5709
.735
1.950.59
.791
2.046:15
.847
2.14513
.H24
1.7671
.68
1.8.5e74
.736
1 9.5228
792
2.04809
.848
2.14692
.6«»r>
1.76H72
.681
1.8(i(l31? ^
.737
1.9.5397
.793
2.049f^3
.849
2.14871
.6->6
1.77031
.682
1.86205
.738
1.9.5.5«i6
.794
2.051.57
.85
2.ir>05
.6-27
1.77197
.6r*3
1.8637
.739
1.95735
.795
2.0.5:{3l
.851
2.15229
.6t»8
1.773.59
.684
1.8t».535
.74
1.9.5994
.796
2.0. -.505
.8:)2
2.1.5409
.6:29
1.77521
.685
1.867
.741
1.96<J74
.797
2.0.5679
.8.5:{
2.1.5589
.6:^
1.77684
.686
1.86866
.742
1.96244
.79r<
2.05853
.8:>4
2.1577
.6:u
1.77847
.687
1.87031
.743
1.96414
.799
2.06(1-^7
.8.55
2.1595
.6:w
1.7H009
.»)88
1.87196
.744
1.9t)5K3
.8
2.06202
.85<)
2.1613
.>i:i:i
1.78172
.689
1.87362
.745
1.96753
.801
2.0():i77
.857
2.16:509
.634
1.78335
.6.>
1.87527
.746
l.t»6923
.802
2.06.552
.8.58
2. 16489
.63.5
1.78498
.691
1.87693
.747
1.970:<3
.803
2.06727
.859
2.16668
.636
1.7866
.692
1.87859
.748
1.97J62
.81(4
2.0ti90l
.86
2. h)«48
.637
1.78823
.693
1.88024
.749
1.97432
.S05
2.07076
.H61
2.17028
.638
1.78986
.694
1.8819
.75
1.97602
.806
2.07251
.862
2.17209
.63S»
1.79149
.695
i.88:?;.6
.751
1.97772
.807
2.07427
.863
2.17389
.64
1.79312
.696
1.88522
.752
1.97943
.808
2.07tiU2
."'(M
2.17.57
.641
1.79475
.697
1.88688
.753
1.98113
.809
2.07777
.WW)
2.17751
.642
1.79638
.698
1.88854
.754
1.98283
.81
2.079.53
.866
2.179:52
.643
1.79801
.699
1.8902
.755
1. 9*^453
.811
2.08128
.867
2.18113
.644
i.79i)64
.7
1.89186
.756
1.98623
.M2
2.0s:W4
.868
2.18294
.64.5
1.80127
.701
1.893.52
.757
1.98794
.813
2.0848
.869
2.18475
.646
1 8029
.702
1.89519
.758
1.98964
.811
2.086.56
.87
2.18656
.647
1.804.54
.703
1.89685
.759
1 99134
.K15
2.088:52
.871
2.188:57
.648
1.8061T
.704
1.89^51
.76
1.99305
.816
2 09(108
.872
2.19018
.649
1.8078
.705
1.90017
.761
1.99476
.817
2.09198
.873
2.192
.65
1.80943
.706
1.90184
.762
1.99647
.818
2.09:36
.874
2 19;<82
.(•)5l
1.81107
.707
1.9035
.763
1.998i8
.819
2 095:56
.875
2.19564
.652
1.81271
.708
1 .90517
.764
199989
.82
2.09712
.876
2.19716
.6.")3
1.81435
.70it
1.90684
.765
2 0016
.821
2.09888
.877
2.19928
.654
1.81599
.71
1.90852
.766
2.00331
.822
2.10065
.878
2.20U
.655
1.81763
.711
1.91019
.767
2.00502
.823
2 10242
.879
2.20292
.656
1.81928
.712
1.91189
.768
2.00673
.824
2 10419
.88
2.20474
.6.57
1.82091
.713
1.91355
.769
2.00844
.825
2. 10.596
.881
2.20656
.658
1.82255
.714
1.91523
.77
2.01016
.826
2.10773
.882
2.208:59
.6,59
1.82419
.715
1.91691
.771
2.01187
.827
2.1095
.883
2.21022
.66
1.82583
.716
1.91859
.772
2-01359
.828
2.11127
.884
2.21205
.661
1.82747
717
1.92027
.773
2.01531
.829
2.11304
.885
2.21388
.662
1.82911
.718
1.92195
.774
2.01702
.83
2.11481
.886
2.21571
.663
1.83075
.719
1.92363
.775
2 01874
.831
2.11659
.887
2.21754
.664
1.8324
.72
1.92531
.776
2.02045
.832
2.11H37
.888
2.2*937
.665
1.8:^404
.721
1.927
.777
202217
.8:»
2.12015
.889
•2.2212
.6t56
l.8;",£G8
.722
1.92868
.778
2.02389
.834
2.12193
.89
2.22303
.667
1.83733
.723
1.93036
.779
2.02561
.8:35
2.12371
.891
2.22486
.668
l.K^97
.724
1.93204
.78
2.02733
.836
2.12549
.892
2.2267
.6^9
1.84061
.725
1.93373
.781
2.02907
.837
2.12727
.893
2.22854
.6/
1.84226
.726
1.93.541
.782
2.0308
.838
2.12905
.894
2.23038
.671
1.84391
.727
1.9371
.783
2.03252
.839
2.13083
.895
2.23222
.672
1.84556
.728
1.93678
.784
2.03125
.84
2.1:5261
.896
2.23406
.673
1.8472
.729
1.94046
.785
2.03598
.841
2.134:}9
.897
2.2359
.674
1.84885
.73
1.94215
.786
2.03771
.842
2.13618
.898
2.23774
.675
1.8505
.731
1.94383
.787
2.03944
.843
2.13797
.899
2.23958
LEN0HT8 OF SRMI-ELLIPTIO ARCS.
TABLE.— (Conrmued.)
H'ght
Length.
H'ght
Length.
H'gh;.
Length.
H'ght.
Length.
H'ght
■
Length.
.9
2.24142
.921
2.27987
.942
2.31fc.)2
.963
2.3.581
.984
2.39823
.901
2.2433r>
.922
2.2817
.y43
2.32038
.964
2.36
.985
2.40016
.902
2.24508
.923
2.28354
.!)44
2.32224
.96^5
2.36191
.9,-6
2.40-.'08
.903
2.24H91
.924
2.28537
.945
2.32411
.966
2.363HI
.987
2.404
.904
2.24874
.925
2.2872
.946
2.32.'>y8
.967
2.36.571
.9H8
2.40592
.905
2.25057
.926
2.2."'903
.947
2.327W5
.9»i8
2.36762
.989
2.40784
.906
2.2524
.927
2.290«6
.948
2.32972
.969
2.36952
.119
2.401)76
.907
2.25423
.928
2.2927
.941)
2..3316
.97
2.37143
.991
2.41169
.968
2.25606
.929
2.294.53
.95
2.33348
.971
2.37334
.992
2.41362
.909
2.25789
.93
2.29636
.951
2.33537
.972
2.37525
-.993
2.41556
.91
2.25972
.931
2.29H2
.9.52
2.33726
.973
2.37716
.994
2.41749
.911
2.26l.-)5
.932
2.30004
.953
2 33915
.974
2.371)08
.995
2.4 11(43
.912
2.26338
.933
2.30188
.1)54
2.34104
.1)75
2 3-1
.996
2.4Ji:i6
.913
2.26521
.934
2.30373
.955
2.34293
.9Tt)
2.38291
.997
2.42329
.914
2.26704
.935
2.30557
.956
2.344^3
.9t7
2.3M482
.998
2.42522
.915
2.26rt88
.936
2.30741
.957
2.34673
.l»7fS
2.3H673
.999
2.42715
.916
2.27071
.937
2.30926
.958
2.34862
.979
2.38864
1.
2.42908
.917
2.27254
.938
2.31111
.9.')9
2.35051
.98
2.31(055
.918
2.27437
.939
2.3121)5
.96
2.3.5241
.;t8l
2.31»-.'47
.919
2.2762
.94
2.31471)
.961
2.3.5431
.982
2.39439
.92
2.27803
.941
2.31666
.96-J
2.35621
.983
2.31)631
To AseertaiD the Length of n Semi Kiliptic Are (right Semi-Eilipse)
by the preceding Table.
Rule. — Divide the height by the base, find the quotient in tlie column of heights, and take the
length of that height from the next iighthand column. Multiply the length thus obtained by the
base of the arc, and the product will be the length of the arc.
E.XAMPLK. — What is the leagth of the arc of a semi-ellipse, the base being 70 feet, and the
height 30.10 feet
30.10-^70=.43 ; and .43 per table, =1.46268.
Then 1.46268 X 70=102.3876 feet.
When the Curve is not that of a Right Semi-Ellipse, the Height being half
of the Tranverse Diameter. ' -
RuiiB. — Divide half the base by .twice the height, then proceed as in the preceding example ;
multiply the tabular length by twice the height, and the product will be the length required
ExAMPLB. — What IS the length of the arc of a semi-ellipse, the height being 35 feet, and the
base 60 feet ?
60-4-2=30, and_30-H35X2=. 428 the tabular length of which IS 1.45966.
Then 1.45966X35 X 2=1 02. 1762/ee^
Note. — If in the division of a height by the base there is a remainder, proceed in the manner
given for the Lengths of Circular Arcs, page 32.
■f^^^y-
TABL13 Vin.
TABLE OF THE AREAS OF THE SEGMENTS OF A CIRCLE.
The Diameter of a Circle aaaumed to be Unity, and divided into 1000 equal Part$.
Vereed
Bine.
Seg. Area.
VerHcd
Siue.
1
Seg, Area '
Versedl
Sine.
Seg. Area.
Versed
Sine.
Seg. Area,
Versed
Sine.
Seg. Area.
.001
.00(104
.048
.01.3-2
.095
.0379
.142
.0()M22
.189
.10312
.(tO-J
.00012
.049
.01425
.096
.03849
.143
.06892
.!.»
.1039
.003
.00022
.05
.0146^
.097
.03908
.144
.069()2
.191
.10468
.004
.000:14
.051
.01512
.098
.03968
.145
.07033
. 192
.10.547
.005
.00047
.052
.0i:.56
.099
.04027
.146
.07103
.193
.10626
.006
.00062
.053
.01601
.1
.04087
.147
.07174
.194
.10705
.007
.00078
.054
.01616
.101
.04148
.148
.07245
.195
.10784
Mti
.00095
.055
.01691
.102
.04208
.149
.07316
.196
.10864
.009
.00113
.056
.0173;
.103
.04269
.15
.073^7
.197
.10943
.01
.00133
.057
.01783
.104
.0431
.151
.07459
.198
.11023
.011
.001.-)3
.058
.0183
.105
.04::91
.152
.07531
.199
.11102
.012
.00175
.0.59
.01W77
.106
.044:.2
.153
.07603
2
.11182
.013
.00197
.06
.0l;»24
.107
.04514
154
.07675
.201
.11262
.014
.0022
.061
.01972
.108
.04575
155
.07747
.202
.11343
.015
.00214
.062
.0202
.10;!
.04638
.156
.0782
.203
.11423
.016
.00268
.063
.02068
.11
.047
.157
.07892
.204
.11503
.017
.00294
.064
.02117
.111
.04763
.1.58
.07965
.205
.11584
.018
.0032
.065
.021»S
.112
.04826
.159
.08038
.2'J6
.11665
.019
.00347
.066
.02215
.113
.04889
.16
.08111
.207
.11746
1 .02
.00375
.067
.02265
.114
.04953
.161
.08185
.208
.11827
.021
.00403
.068
.02315
.115
.05016
.162
.08258
.209
.11908
.022
.00432
.069
.02336
.116
.0508
.163
.08332
.21
.1199
.023
.00462
.07
.02417
.117
.05145
.164
.08406
.211
.12071
.024
.00492
.071
.024(58
.118
.05209
.165
.0848
.212
.12153
.025
.00523
.072
.02519
.119
.05274
.166
.08.554
.213
.12235
.026
.O0."k)5
.073
.02571
.12
.05338
.167
.08629
.214
.12317
.027
.00.587
.074
.02624
.121
.05404
.168
.08704
.215
.12399
.028
.00619
.075
.02676
.122
.05469
.169
.03779
.216
.12481
.029
.001)53
.076
.02729
.123
.05534
.17
.08853
.217
.12563
.03
.00686
.077
.02782
.124
.056
.171
.08929
.218
. 12646
.031
.00721
.078
.02S35
.125
.05666
.172
.09004
.219
.12728
.032
.00756
.079
.028(?9
.126
.05733
.173
.0908
.22
.12811
.033
.00791
.08
.02943
.127
.05799
.174
.09155
.221
.12894
.034
.00827
.081
.02997
.128
.05866
.175
.09231
.222
.12977
.035
.00864
.082
.03052
.129
.05933
.176
.09307
.223
.1306
.036
.00901
.083
.03107
.13
.06
.177
.09384
.224
.13144
.037
.00938
.084
.03162
.131
.06067
.178
.0946
.225
.13227
.038
.00976
.085
.03218
.132
.06l:i5
.179
.09537
.226
.13311
.039
.01015
.086
.03274
.133
.06203
.18
.09613
.227
.13394
.04
.01054
.087
.0333
.134
.06271
.181
.0969
.228
.13478
.041
.01093
.088
.03:^87
.135
.06339
.182
.09767
.229
.13.562
.042
.01133
.089
.03444
.136
.06407
.183
.09845
.23
.13646
.043
.01173
.09
.03501
.137
.06476
.184
.09922
.231
.13731
.044
.01214
.091
.03558
.138
.06545
.185
.1
.232
.13815
.045
.01255
.092
03616
.139
.06614
.186
.10077
.233
.139
.046
.01297
.093
.03674
.14
.06()83
.187
.10155
.234
.13984
.047
.01.339
.094
.03732
.141
.06753
.188
.10233
.235
.14069
AREAS OF TIIE SEGMENTS OF A CIRCLE.
T A BLE.— (Continued. )
1 Vewd
hiuu.
1'
Hog Arun.
.tj.lt)
.I41.')4
.'2:n
.142.J9
.UM
.14:«4
.•j:w
.I44(il»
.'i4
.14494
.•24 1
.1458
.21'2
.!ii ;().'.
.'U:i
.14751
.'M
.UK{7
.24.')
. 1 4923
.246
.15(109
.247
A 'Mb
.24H
.15182
.249
.l.'>2()8
.2.)
. 15U55
.251
.l.'>44i
.2r>2
.I.-.528
.2.'>:i
.1.-.615
.2.S4
.157(12
.2;')..
. 15789
.2r)«
.ir>e7(i
.257
.l.")96J
.258
.l()05l
.259
.161:59
.26
.161*26
.261
.KKJU
.■:6>
.16402
.2(i:J
. 1649
,264
.ltw7-i
.265
.1(J666
.266
. 1675.)
.267
.16'-44
.268
.16iKU
.269
.1702
.27
.17109
.271
.17197
.272
.17287
.27:<
,17376
.274
.17465
.275
.175.54
.276
. ! 7643
.277
.17733
.278
. 17822
.279
.17912
.28
.18002
.281
. 18092
.282
.18182
.2o3
.18272
.284
.18361
.285
.1^4.52
.286
. 18542
.287
.18633
.288
.18723
'5
^*^'seg Ami.' ^'/^ Hcg. Area.
iinc.
2H9
.1^814
2<>
.l-'.Mi..
2lt 1
.18i>95
2'.>*.>
.l'.HJ86
2i>;{
.19177
291
.19268
295
. 193(;
21M;
.I9J51
297
.19.-.42
298
.ll'63»
299
.19725
3
.19817
301
.19908
302
.2
303
.200;»2
304
.20184
:{U5
.20276
306
.2o;'.68
307
.2016
30-(
,20.")53
309
.20645
31
.20738
3tl
.208 5
312
.20923
313
.21015
3i4
.21108
315
.21201
316
.21294
317
.21387
318
.21.i8
319
.21573
32
.21667
321
.2176
3->2
.21853
323
.21947
324
.2204
325
.22134
32(;
22228
327
.22:521
32"-
.22415
329
.22.'.09
3;;
.22603
331
.22697
332
.22791
333
.22c86
3;i4
.2298
335
.23074
336
.231(i9
337
.23263
338
.23359
339
.23453
34
.23.547
341
.23642
.342
,343
.344
.315
.346
.347
.348
.319
,:5.-)
,351
.3.52
.:i5;i
.3,->4
.3.55
.356
.3.57
.X>H
.3.59
.m
.361
.362
.363
.364
.365
.366
.367
.3(58
.369
.37
.371
,372
,373
.374
.375
.37(5
,377
.378
.379
.38
.381
.382
.38;{
,384
.:i85
.38(5
.387
.388
.3''9
.39
.3i»l
.392
.393
,394
.23737
.23h:'.2
.23927
.24022
.24117
.24-.' 12
.24307
.21103
.•:441W
.21..93
.246-9
.24784
.248rt
.24;(7()
.2,5071
,25167
.25263
.25:;59
.2545.)
.25551
.2.5647
.2.5743
.2.">839
.25936
.2(^032
.2C.128
.2622r.
.2(;;52l
.2(5-1 18
.26514
.26(511
.2670>
.2f5^04
.2(5:'01
.26998
.27095
.27192
.2T2rt9
.27386
,27483
.27580
.27677
.27775
.27872
.27969
.28067
.28164
.28262
.283.59
.284.57
.2f^.554
.2-552
.2878
8iDc. ^- ^"^
.395
.:v.m
.397
.39"^
.399
.4
.401
4(r2
.103
.404
,4 05
.406
.407
.4yf5
.409
.41
.411
.412
.413
.411
.415
.416
.417
.418
.419
.42
.421
.422
.423
.424
.425
.42(5
.427
.428
.42;>
.43
.131
.432
.133
.434
.435
.4:56
.437
.439
.44
.441
.442
.443
.444
.445
446
.447
.2rt« 18
.2^915
.29043
.29141
.292::9
.29337
.2: '4 35
.29533
.2:M531
.29729
.29-27
.21K>25
,30024
.30122
.3022
.30319
.30417
,30515
.3(»il4
.30:12
.30811
.309119
.31008
.31107
.31205
.313iM
.31403
.31.502
.31(5
.31(599
.31798
.31897
.319'.«i
.32095
.32194
.322; (3
.32391
.3249
.3259
.3268i(
.32788
.32887
.329H7
.33086
.331S5
.33284
.33384
.33483
.33582
.336-2
.33781
.33'-8
.3398
Sine. ^ ^"^
.448
.449
.45
.451
.4.52
.4.53
.4.54
.45.5
.4.56
.4.57
.4.58
.4.'>9
.46
.461
.462
.4(53
.464
.4(55
.466
.467
.4(58
.469
.47
.471
.472
.473
.474
.475
.476
.477
.478
.479
.48
.481
.482
.483
.484
.485
.486
.487
,488
,489
.49
.491
.492
.493
.494
.495
.496
.497
.498
.499
.5
.34079
.34179
.:»4278
.34378
.:M47r
.:u.>57
.34676
.34776
.34H75
.3497JS
.3,"i07i.
.;t5!?4
.:{.5274
.3.5374
.3,-)474
.3.'k573
.3,^.«573
.35773
.35872
.35972
.3(5072
.3<)172
.3»)2:2
.36371
.36471
.3(5.571
.3(5671
.36771
.3(5871
.36971
,37071
.3717
,3727
,3737
,3747
,3757
.3767
.3777
•3787
.3797
.3807
.3817
.3827
.3837
.3847
.:i857
.3867
.3877
.3887
.3897
.3907
.3917
.3927
AREAS OF THi: ZONES OF A CIRCLES.
39
To Aserrl'iin thn Area uf a Segment of a Circle by tlie preeediug Table.
Rui I — DiTifle the hei^rht or vpned gino by the diamoter of the circl«' ; find the quotient in the
coluiiiii ol vfised «ine>). Take the area noted lu the next column, niulii|ily it hy the ^•l|uare of the
dianieier, and it will kivc \\u- area
bx AMFLK. — Kequiitd ihe urea ot a dcgoieut, iu height being 10, and the diumeUsr of the circle
fill feel.
I(i_}..V)=-.2, and .2. [ler table,=-.lll82; then .ni»lX^0^='2l9.:>r,Jeel.
N'lTic. — If in the division of a height i)y tiie base, the quotient has remainder after the third
liict- ot de< jiiaiii, and great iicciiracy id required.
Tnkf the ai«a f >r the fir.-t tliree fi.'un's. subtraet it from the next ftdlowing ar a. multiply the
letiiiiiiidcr by the said fiaciioii. and add the product to the firat area ; tlie itum will be the area for
Uii- who e (|iiotient.
'/ W hat li the area of a sigmcnt of a circle, the diameter of which is 10 feet, and the height of
it 1.675 fe.t
1.67:)-*-l<» .1575; the tabular area for .157=. 07892, and for .168=07906, the difference between
which la 0()07.S.
Then .5 X. 00073=0003(35. ^
Hence " .157 =.07892
.0006=. 000365
.079275, the sum by which the square of the dia-
meter of the circle is to be multiplied ; and .079285 X 10 -=7.928G/«'e<.
r»-r^-rvj- w-vj -i-r^-0-rn-i-L ro-0.r i^^ ■ ■■■•1'
TABLE IX. i. ^
TABLE OP THE AREAS OF THE ZONES OP A CIRCLE.
The Diameter of a Circle assumed to be Unity, and divided into 1000 equal Parts.
H'ght.
Area.
H'ght.
Ares.
H'ght.
Area.
H'ght.
Area.
H'ght
Area.
.00 1
.001
.029
.02898
.057
.05688
.085
.084.-)9
.113
.11203
.00-2
.002
.03
.02998
.058
.05787
.086
.08557
.114
.113
.00:{
.003
.031
.03093
.059
.0.'>886
.087
.0H656
.115
.11398
.001
.004
.03i
.03198
.06
.0.'>986
.08H
.08754
.116
.11495
.oor>
.00.')
.033
.03298
.061
.0t=085
.0^9
.08853
.117
.11592
.OOrt
.006
.034
.03397
.0<i2
.06184
.09
.08951
.118
.1169
.007
.007
.035
.03497
.063
.062^^3
.091
.Oi'05
.119
.11787
.008
.008
.036
.03.-)97
.064
.(it)3-^-^
.o;»2
.09148
.12
.11884
.009
.009
.037
.03697
.065
.06482
.093
.01>-J46
.121
.11981
.01
.01
.038
.03796
.066
.0658
.094
,09344
.122
.12078
.Oil
.011
.039
.03896
.067
.0668
.095
.09443
.123
.12175
.012
.012
.04
.03996 '
.06.8
.0678
.096
.0954
.124
.12272
.013
.01:5
.041
.04095
.069
.0<)878
.097
.09639
.125
.12:^69
.914
.014
.042
.04195
.07
.06977
.098
.09737
.126
.12469
.oir,
.015
.043
.042.t5
.071
.07076
.099
.09835
.127
.125<)2
.016
.016
.044
.64394
.072
.07175
.1
.09933
.128
.126.^)9
.017
.017
.045
.04494
.073
.07274
.101
.10031
.1-J9
.127.55
.OIH
.Old
.046
.04593
.074
.07373
.102
.10129
.13
.128.52
.019
.019
.047
.04693
.075
.07472
.103
. 1 0227
.131
.12949
.0-J
.02
.048
.04793
.076
.07.').')
.104
.1032.')
.132
A.mh
.0-21
.021
.049
.04892
.077
.07()69
.l.t->
.104'>2
.IXi
.13141
.022
.022
.05
.04992
.078
.07768
.106
.1052
.134
.132:W
.023
.023
.051
.0:.091
.079
.07867
.107
.10618
.1:^5
.133:i4
.024
.024
.0.V2
.0519
.08
.0796*)
.108
.10715
.136
.1343
.025
.025
.053
m->d
.081
.08064
.109
.10813
.137
.i:i^.27
.026
.02599
.054
.05:J89
.082
.08163
.11
.10911
.138
.1362:^
.027
.C2o9b
.055
.0.^489
.083
.08262
.111
.11008
.139
.13719
.02«
.02799
.056
.05588
.084
.0836
.112
.11106
.14
.13815
40
ABEA.S OF THE ZONES OF A CIRCLES.
TABLE.— (Continued.)
-«v.
ffght.
Area.
H'ght.
Area.
H'ght.
Area.
H'ght.
Area.
H'ght
Area.
.141
.i.-jgu
.W7
.19178
.2.53
.24175
.309
.28801
.;565
.32931
.142
.14007
.198
.1927
.254
.24261
.31
.2«88
.366
.32999
.143
.14103
.199
.19:!61
.255
.24:547
.311
.28958
.;<67
.3:3067
.144
.14198
.2
.194.53
.2.56
.24433
.312
.29036
.:}68
.331:35
.145
.14294
.201
.l'.^545
.257
.24519
.313
.29115
.369
.3:5-203
.146
.1439
.202
.196:56
.258
.24604
.314
.29192
.37
.3:527
.147
.14485
.203
.19728
.2.59
.2469
.315
.2927
.:571
.33337
.148
.14581
,204
.19819
.26
.24775
.316
.28:348
.•572
.:33404
.149
.14677
.205
.IWl
.261
.24861
.317
.29425
.:373
.:3:3471
.15
.14772
.206
.20001
.262
.24946
.3 la
.29.502
.374
.3:3537
.151
.14867
.20?
.20092
.263
.25021
.319
.295H
.375
.3:3604
.1.52
.14962
.208
.201r'3
.^64
.265
.25116
.32
.29656
.:376
.3367
.1.53
.150.58
.209
.20274
.21201
.:321
.29733
.:377
.3:3735
.154
.151.53
.21
.20365
,266
.'252.85
.322
.2981
.378
.:3:3801
.1.55
.15248
.211
.20156
.267
.2537
.323
.29886
.379
.3:3866
.156
.15343
.212
.20546
.268
.25455
.:{24
.29962
.38
.33931
.157
.15438
.213
.206:!7
.269
.2.5.5:^9
.:525
..50039
.:38l
.33996
.1.58
.155;?3
.214
.20727
.27
.25623
.:}26
.:50114
.3»2
.34061
.1.59
.15628
.215
.20818
.271
.25707
.:i27
.3019
.383
.341-25
.16
.15723
.216
.2(>908
.272
.2.5791
.:i28
.30266
.:384
.3419
.161
.15H17
.217
.20998
.273
.25875
.329
.30:341
.:385
.34253
.162
.15912
.218
.21088
.274
.259.59
.33
.:30416
.386
.34317
.163
.16006
.219
.21178
.275
.2<.U43
.:53l
.:50491
.387
.3438
.164
.16101
.22
.21268
.276
.26126
.3:52
.30.566
.388
.34444
.165
. 16195
.221
.2i;j5»
■ .277
.26209
.:i33
.30«)41
.:389
.34507
.166
.1629
.222
.21447
.278
.2629:5
.:«4
.30715
.39
.34569
.167
.16384
.223
.21537
.279
.26376
.3.55
.3079
.;391
.34632
.168
.16478
.224
.21626
.28
.264.59
.336
.30864
.392
.34694
.169
.16.572
.225
.21716
.281
.26541
.337
.30938
.393
.34756
.17
.16667
.226
.21805
.282
.26624
.338
.31012
.394
.34818
.171
.16761
.227
.21894
.283
.26706
.:i39
.31085
.395
.34879
.172
.168,-.5
'.228
.21983
.284
.26789
.:54
.31159
.:396
.3494
.173
.16948
.229
.22072
.285
.26871
.341
.31232
.397
.3,5001
.174
.17042
.23
.22161
.286
.269.53
.:542
.31305
.398
.35062
.175
.17136
.231
.2225
.287
.270a5
.343
.31:378
.399
.35122
.176
.1723
.232
.22335
.'2f^6
.27117
.344
.:3145
.4
.:55182
.1/7
.17323
.233
.22427
.289
.27199
.345
31,523
.40 J
,3,5242
.178
.17417
.234
.22515
.29
.272'^
.346
.31595
.402
.35302
.179
.1751
.•^35
.22604
.291
.27362
.:547
.31667
.403
.3,5361
.18
.17603
.236
.22692
.292
.27443
.348
.3U)39
.404
.3542
.181
.17697
.237
.227S
.293
.27524
.349
.31811
.405
.3.5479
.182
.1779
.23rt
.228«)8
.294
.27605
.35
.31882
.406
.:3.5538
.183
.17883
.239
.22956
.295
.27686
.351
.319,54
.407
.35596
.184
.17976
.24
,23044
.296
.27766
.3,52
.32025
.408
.3.5854
.185
.180<)9
.241
.2:a3l
.297
.27847
.:55:5
.32096
.409
.35711
.186
.18162
.242
.23219
.298
.27927
.:i54
.:32167
.41
.3,5769
.187
.1b254
.243
.23306
.299
.28007
.:555
.32-237
.411
.358-26
.188
.18:U7
.244
.23394
.3
.2rt088
.356
.3-2:507
.412
.3,5H83
.Irt)
.1844
.245
.234^1
.301
.28167
.3,57
.3-^:577
.413
.3.59:39
.19
.18532
.246
.2:5568
.302
.28247
.:i58
.:52147
.414
.35995
.191
.18625
.247
.23655
.303
.25:327
.3r)9
.:3-2517
.415
.:36051
.192
.lf^717
.248
.23742
.304
.28406
.36
.:{-25«7
.416'
.36107
.193
.18809
.249
.23829
.305
.28486
.361
.3-2656
.417
.36162
.194
.18902
.25
.23915
.306
.2856.5
.:562
.327-25
.418
.:56217
.195
.18994
.251
.24002
.307
.2^-644
.3()3
.32794
.419
.36-272
.196
.19086
.252
.24089
.308
.2^723
.364
32862
.42
.36326
ABEAS OF THE ZONES OF A CIRCLE.
TABLE.— (Continued.)
41
ffght.
Area.
H'ght
Area.
H'ght
Area.
H'ght.
Area.
H'ght.
Area.
.421
.3638
.4:w
.37202
.453
.37931
.469
.38.'>49
.485
.39026
.422
.36434
.438
.3725
.454
.37973
.47
.3H5-'3
.486
.3905
.423
.:i6488
.439
.37293
.455
.38014
.471
.38617
.487
.39073
.424
.36541
.44
.37346
.456
.38056
.472
.3865
.488
.39095
.425
.36594
.441
.37:J93
.457
.38096
.473
.38683
.489
.39117
.426
.36646
.442
.W44
.458
.38137
.474
.38715
.49
.39137
.427
.36698
.443
.37487
.459
.38177
.475
.38747
.491
.391.56
.428
.3675
.444
.37533
.46
.38216
.476
.38778
.492
.39175
.429
.36802
.445
.37579
.461
.38255
.477
.38808
.493
.:^9192
.43
.36853
.446
.37624
.462
.38294
.478
.3n838
.494
.39208
.431
.36904
.447
.37669
.463
.383}2
.479
.38667
.495
.39223
.432
.36954
.448
.37714
.464
.38369
.48
.3889.%
.496
.39236
.433
.37005
.449
.37758
.465
.38406
.481
.38923
.497
.39248
.434
.370.^4
.45
.37802
.466
.38443
.482
.3895
.498
.39258
.435
.37104
.451
.37845
.467
.38479
.483
.38976
.499
.39266
.436
.37153
.4.o2
.37888
.468
.38514
.484
.39001
.5
.3927
J%w Table is computed ordy^or Zones, the longest chord ^ which is diameter^ .
To Aseertain the Area of a Zone by the preeeding Table.
Rule 1. — When the Zone is Lexs than a Semicircle, Divide the height by the diameter, and find
the quotient in the column of height. Takeout thearea opposite to it in tlie next column on the right
hand and multiply it by the square of the longest chord ; the product will be the area of the zone^
ExAKPLB. — Required the area of a zone the diameter of which is 50, and its height 15.
15-^^0=. 3 ; and .3; as per table,==. 28088.
Hence .28088X502=702.2 area.
RuLB 2. — When the Zone is Greater than a Semicircle : Take the height on each s'de of the dia-
meter of the circle, and ascertain, by Rule I, their respective areas ; add the areas of these two
portions together, and the sum will be the area of the zone.
Example. — Required the area of a zone, the diameter of the circle being 50, and the heights of
the zone on each side of the diameter of the circle 20 and 15 respectively.
20-1-50=. 4; .4, as pertable,=. 35182 ; and .35182Xo0-'=879.55.
15-s-50=.3; .3, as per table, =.28088; and .28088X502=702.2. '^'
Hence 879.654-702.2=1581.75 area.
Rule 3. — When the longest chord of the zone is les^ thin diameter, Take the height or distance
from the diam. to each of the chords respectively ; find the area corresponding to each height and
deduct the lesser from the greater area ; the result will be the area required.
Note. — When, in the division of a height by the chord, the quotient has a remainder after the
third place of decimals, and great accuracy is required.
Take the area for the first three figures, subtract it from the next following area, multiply the
remainder by the said fraction, and add the product to the first area ; the sum will be the area for
the whole quotient.
Example. — What is the areacrf'a zone of a circle, the greater chord being 100 feet, and the
breadth of it 14 feet 3 inches ?
14feet 3incl:es=14.25 and 14.25-t-100=.1425; the tabular area for . 142= 14007, and for 14S=a
.14103, the difference between which is .00096. r
Then .5X.00096=i.00048. , rr;^ y ItT:??^-
Hence .142 =.1400Y
.0005 =.00048
.14055, the Bom bj which the square of the greater chord.L3 to be multiplied ;
.14055 X lOOa =1405.5/e«<.
and
VA.Bt^E
SPECIFIC GRAVITIES. ""
The Specific Gravity of a body is the proportion it bears to the weight
of another body of known density.
If a body float on a fluid, the part immersed is to the whole body as
the specific gravity of the body is to the Hpecific gravity of the fluid. ' '^
When a body is immersed in a fluid, it loses such a portion of its own
weight as is equal to th U of the flui<l it displaces.
An immersed body, ascending or desc nding in a fluid, has a force
equal to the diflerence between its own weight and the weight of its bulk of
the fluid, less the resistance of the fluid to its passage. - *:'.''' '•■ ' '•' 'i: ;'
Water is well adapted for the standard of gravity ; and as a cubic foot of
it weights 1000 ounces avoirdupois, its weight is taken as the unit, viz : lOOU.
To Ascertain the Specific Gravity of a Body heavier *
than Water.
RuLE. — Weight it both in and oat of water, and note the difference ;
then, as the weight lost in water is to the whole weight, so is 1000 to the
W X 1 OOP
specific gravity of the body. Or, ^ _„, = G, Vf representing the
weight in water, and G the specific gravity.
EzAHPLR. — ^What is the speoific gravity of a stone which weighs in air 15 lbs., in
water 10 lbs. ?
,, ^ 16—10=5 ; then 5 : 15 : : 1000 : : 3000 tpec. grav.
To Ascertain the Specific Gravity of a Body lighter !!5
.^ ..,. ^ . than Water.
RtTLE. — Annex to the lighter body another that is heavier than water,
or the fluid used; weigh the piece added and the compound mass sepa-
rately, both in and out of the water, or the fluid ; ascertain how uuich each
loses in water, or the fluid, by subtracting its weight in water, or the fluid,
from its weight in air, and subtract the less of these differences from the
greater; then,
As the last remainder is to the weight of the light body in air, so is
1000 to the specific gravity of the body.
ExAHPLE. — ^What is the specific gravity of a piece of wood that weighs 20 lbs.
in air ; annexed tx) it is a piece of metal that weighs 24 lbs. in air and 24 lbs. m
water, and the two pieces in water weigh 8 lbs. ?
20-+-24 — 8=44 — 8=36=ioM of compound nuus in water ;
24—21 = 3=dos8 of heavy body in water.
33 : 20 : : 1000 : 606=24 tpee. grav.
• To Ascertain the Speoific Gravity of Fluid.
RuLK. — Take a body of known specific gravity, weigh it in and out or
the fluid ; then, as the weight of the body is to the loss of weight, so is the
Sfieciflc gravity of the body to that of the fluid.
SPECIFIC GBAVITIES 43
Example. — What is the specific gravity of a fluid in whi'dl a piece of ooppar
(spec. grav. =9000) weighs 70 lbs. in, and 80 lbs. out of it ?
-,,- 80 : 80— 70=10 :: 9000 1125 4pcc. yra».
To Compute the Proportions of t"WO Ingredients in a
Compound, or to discover Adulteration in Metals.
iivi.K. — Take the diflerences of «'ach specific gravity of the ingredients
and the specific gravity of ti»e compound, then tnultiply the gravity of the
one by the dirterence of the other ; and, as the sum of the products is to the
resptctive products, so Im the specific gravity of the body to the proportions
of the ingredients. ... if '•> ' .
Example. — A compound of gold (spec. ffrav.=lB.888) and silver (spec, grao.^^
10.535) has a specific gravity of 14 ; what is the proportion of each metaL ''
18.888—14=4.888X10.535=51.495 . .„ . ' '~ .^f
:^^'' C"^^^, '♦•""" 14—10.535=3.465X18.888=65.447 '*^'''0*« ^-a v'^^'^^.' ;
-H"'" !:^ '^ • 65.447+51.495 : 65.447 :: 14 : 7.835 ^ro^c^. "''^^^^/
vCii^v, •*--» ... 65.447+51.495 : 51.495 :: 14: 6.165 ai'iuer. """'*. ,, v
To compute the Weights of the Ingredients, that of the
M«' . is^fi*: •;..,„. compound being given. - ■••* t.^i-=.y
"^ " RuT-K.— As the specific gravity of the compound is to the weight of the
compunnd, so are eacli of the proportions to the weiglit of its material. > "^ ;.
f ,^ Example. — The weight, as above, being 28 lbs., wbatare the weights of the
'ingredients? _^
■ --14^28 •• I 7-835: 15.67 ^0^ ""*
...J'^j^i^
Proof of Spirituous Iiiquors. '^^'^
"^ A cubic inch of proof spirits weighs 2.S4 grains; than, u an immersed
cubic inch of any lieavy body weighs 234 grains less in spirits than air, it
shows that the spirit in which it was weighed is proof.
If it lose less of its weight, the spirit is above proof; and if it lose
more, it is below proof.
Illustration. — A cubic inch of glass weighing 700 grains weighs 500 grains
when weighed m a certain spirit ; what is the proof of it ?
700 — 500^200=grains=weiffht lost in the spirit.
Then 200 : 234 : : 1. : 1.17= ratio of proof qf spirits compared to proof tphittf
or 1.=».17 cAove proof.
Solids.
Rule. — Divide the specific gravity of the snbstance by 16, and tbe
quotient will give the weight of a cubic foot of it in pounds.
44 SPECmO GRAVITIES.
i^^' OP DIFFERENT BODIES AND SUBSTANCES.
METALS.
Alluminu..!
Antimony
Arsenic
Barium
BiRmuth
Brass, copper 84
" tin 16
" copper 67 \
zinc 33 J
plate
wire
Bronze, gun metal
Boron ■
Bromine <
Cadmium...
Calcium
Chromium
Cinnabar
Cobalt ~
Columbium
Gold, pure, cast
" hammered
- ? " 22 carats fine
« 20 carats fine
Copper, cast
" plates ..•«•
" wire
Iridium
" hammered
Iron, cast
" gun metal...
hot blast
cold "
wrought bars....
" wire..,.
rolled plate
Lead) cast >
« rolled
Liithium. ..<)...... ......a
Manganese >..
l£agDe8ium.....ww..«.*><
Mercury— 40°
«< + 32«
« 60°
«« 2I2<»
Molybdenum..
Nickel.
*' cast
Osmium
4(
U
U
U
t€
Speci-
fic fp'a-
vity.
» •*.. « <..
2560
6712
6763
470
9823
8832
7820
8380
8214
8700
2000
3000
8660
1680
5900
8098
8600
6000
19258
19361
17486
16709
8788
8698
8880
18680
2.3000
7207
7308
7065
7218
7788
7774
7704
11362
11388
690
8000
1750
16632
13698
13580
1.3370
8600
8800
8279
10000
Weight
of a cu-
bic inch
.0926
.2428
. 20H4
.017
.355o
.3194
2828
.3031
.2972
.3147
.0723
METALS.
Palladium
Platinum, hammered.
*• native
*> rolled
Potassium, 59<^
Red-lead
Rhodium
Ruthenium
elenium
Silicium
Silver, pure, cast
•• '• hammered.
Sodium
1085 Steel, plates.
.3129
.067
.2134
.2929
.3111
.217
.6965
.7003
.63
.5682
.3179
soil
" tempered
hardened.
and
IT ire.... .... .........
Strontium ..*.....<
Tin, Cornish, ham nierd
" pure
Tellurium
Thalium
Titanium.
Speci
flc gra-
vity
Wolfram ....
Zinc, cast ..
rolled.
((
WOODS (DRY.)
Alder.,
Apple
Ash...
Tungsten.
3146 Uranium
.3212
.6756
.8319
.2607
,264
,2655
,2611
.2817
.2811
.2787
.4106
.4119
0213
.2894
.0633
.5661
.4918
.4912
.4836
.311
.3183
.2994
.3613
Bamboo
Bay
Beech...........
••«••••••••
11360
20337
16000
22069
865
8940
10650
8600
4600
10474
10511
970
7806
7833
7818
7847
2540
7390
7291
6110
11850
6300
17000
10150
7119
6861
7191
Birch
Box, Brazilian
" Dutch M..**
" French
Bullet-wood
Butternut »
Campeachy
Cedar
** Indian
800
793
845
600
400
822
852
690
567
1031
912
1328
928
376
913
661
1316
Weight
of a cu-
bic inch.
.4106
.7.356
.6787
.7982
.0313
.3241
.3862
.3111
.1627
..3788
..3902
.0.361
.2823
.2833
.2828
.2838
.0918
.2673
.26.37
.221
.4286
.1917
.6149
..3671
.2575
.2482
.26
Cubic
foot.
50
49.562
62.812
43.125
26.
51.375
63.25
43.125
36.437
64.437
57.
83.
58.
23.6
57.062
35.062
82.157
SPECIFIC GRAVmBS.
45
WOODS, (Dry.) t^
{Continued.) vity.
((
(<
Charcoal, pine.... ,
" freHh burnen
van ••••••••••!
soft wood ...
triturated ...
Cherry ,,
Chesnut, sweet
Citron
Cocoa
Cork
Cypreas, Spanish
Dog-wood
Ebony, American-
" Indian
Elder
Weight I WOODS, (Dry.)
Filbert
Fir (Norway Space)....
Gum, blue
" water.
Hackmatack
Hazel
Hawthorn u,....
Hemlock
Hickory, pig-nut ,
" shell-bark..
Holly...
Jasmine ,
Juniper
Lance-wood »
Larch }
AjcmoD ...•.•••••••. ........
Lignum-ritas
Lime
Linden
Locust
Logwood
Mahogany... .M. ...... <
** Honduras...
** Spanish
Maple
*' bird's eye .........
Mastic
Mulberry \
Oak, African ...
" Canadian.
441
380
1573
2Hn
1380
715
610
726
1040
240
644
756
1331
1 209
6;»5
570
671
600
512
843
lOOO
592
860
910
368
792
690
760
770
566
720
544
560
703
13.33
804
604
728
913
720
1063
560
852
750
676
849
561
897
823
872
ot a cu-
bic I'ool.
27 5621
23.75
98 312
5
26
687
125
375
17
86
44
38
45
H5
15
40
47
83
75
43
35
41
37.
32
52
62
37.
•^3.75
56.875
23.
49.5
43 125
47 5
48 125
35 375
45.
34.
31.
43
83
50
{Continued )
Speci
tlognk-
vity
••••»••«•«•■
. 25
26
187
562
437
625
937
5
687
5
Oak, Dantzic
' En<rlifih
' green
' heart, 60 yearn....
' live, green
' " i^easoned......
white
Orange
Pear
Persimmon
Plum
Pine, pitch
" red
" white.
" yellow
Pomegranate
Poon
Poplar
" white
Quince m...
Ro.se-wood....
Sa-ssafras
Satin-wood
Spruce
Sycamore
Tamarack
Weight
uf a oa-
bio foot.
Teak (African oak).
Walnut
" black
Willow
Yew, Dutch. . .
" Spanish.
( Well Seasoned.*)
937
312
25
37.75
45.5
57.062
45.
66.437
35.
53.26
46.876
36.
53.062
35.062 White Oak, upland.
56.062
61.437
64.6
Ash
Cherry ........„.....,<... .
Cypress «....«
Hickory, red...
Mahogany, St. Donig.
Pine, white..*,
** yellow...
Poplar.
•••«««.•.•«.«
«
James iUver.
759
932
1446
1170
1260
1068
860
705
661
710
785
660
590
554
461
1354
580
383
529
705
728
482
885
500
5-23
383
657
745
671
500
486
585
788
807
47.437
68 25
71.625
73.125
78.75
66.75
53.75
44.062
42 312
44.375
49.062
41.25
36 875
34.625
28.812
84 626
36.25
23 937
33.062
44 062
45.5
.30.125
55 312
31.25
3^.9.37
23.937
41.062
46.562
41.937
31.25
30.375
36 562
49.25
50.437
722
624
606
441
838
720
473
541
687
687
759
45.126
39.
37.876
27.562
52.376
46.
29.662
33.812
36.687
42.937
42.437
*Ordiuuioe nuuuud 1841.
46
SPECIFIC GRAVITIES.
Stones, Earths, &o
Agate
Alabaster, white.........
" yellow
Alum
Amber
Ambergris
Asbestos, starry...,
Asphaltum
Barytes, sulphate.... \
Basalts.
Borax ..
Brick...
it
fire
work in cement.
" " mortar <
Carbon
Cement, Portland
Roman
«
Chalk.
Chrysolite
Clay...
" with gravel
Coal, Anthracite.,
* »' Borneo
" Cannel.
((
<<
((
a
it
t(
ii
n
Caking
Cherry
Chili
Derbyshire. . .
Lancaster....
Maryland
Newcastle....
Rive de Gier.
" Scotch-
Splint
Wales, mean.
<(
u
Coke
" Nat'l, Va.
Concrete, mean
Copal.
Coral, red
Speci
flc t;ra
v<ty
2590
27.S()
2ti99
1714
1078
866
3073
905
1650
4000
4865
2740
2864
1714
1900
1.367
2201
1800
1600
2000
3500
1300
1560
1520
2784
2782
1930
2480
1436
1640
1290
12.38
1318
1277
1276
1290
1292
1273
1355
1270
1300
1259
1300
1302
'315
1000
746
2000
1045
2700
Weight
of a cu-
bic foot.
Stones,Earths,&c
170.625
168.687
107.125
67.375
.062
562
.125
192
56
103
250.
304.062
171.25
179
125
75
437
562
107
118
85
137
112.50
100.
125.
218.75
81.25
97.25
95.
174.
120.625
155.
89 76
102.6
80.626
77.375
82.. 375
79.812
79.75
80.625
80.75
79.562
84.687
79.375
81.25
78 687
81.25
81.375
82.187
62.5
46 64
125.
65 312
" white
Hornelian
Diamond, Oriental
" Brazilian....
Earth, * common soil.
** loose ,
moist sand...
mould, fresh..
rammed ,
rough sand....
with gravel...,
it
i(
a
i(
Emery,
Flint, black...,
'* white...,
Fluorine
Glass, bottle .
Crown.
(i
a
flint,
a
(<
n
n
green
optical
white
window
Garnet
•♦ black
Granite, Egyptian red.
Patapsco
Quincy
Scotch
" Susquehanna
Gravel, common
Grindstone
Gypsum, opaque
Hone, white, razor
Hornblende
loiiine
Jet.
Lime, hydraulic
" quick
Limestone, green
" white,
Magnesia, carbonate...
Marble, Adelaide.
Africain.
Biscayan, black.
Carara
common ..........
Egyptian
French
Italian, white....
hpeci-
ficiH'a
vity.
2550
2613
3521
3444
2194
1500
2050
2050
1600
1920
2020
4000
2582
2594
1320
2732
2487
2933
3200
2642
3450
2892
2642
4189
3750
2654
2640
2652
2625
2704
1749
2143
2168
2876
354t)
4940
1.300
2745
804
3180
3156
2400
2715
2708
2695
2716
2686
2668
2649
2708
Weight
of a cu-
bic foot.
137.
93.
128.
Vi8
100.
120.
126.
250.
161.
162.
82.
125
75
125
125
25
375
125
5
170.75
155.437
183 312
196.
165.125
215.625
180.75
165.125
165.875
165.
165. '^5
164.062
169.
109.312
133.937
135.5
179.75
221.25
171.562
50.25
198.75
197.25
150
169.687
169 25
168.437
169 75
167.875
166.75
165.562
169.26
Spec srav. of the earth in vaxioosly eatimated at from, 5, 400 to 5,600.
SPECIFIC GRAVITIES.
47
Stones,Earths,&o
•••••••••
Marble Parian.
" Vermont, wliite
Marl, mean
Mica
Mortar
Millstone.. . ....,
Mmi ,
Nitre
Op.l ,
Oyster t^hell. . .,
Paving-stone. ..
Peal, Oriental.
Peat
Phosphorus ,
Piaster i)t'Pari>!. ,.
Plunjbago ,
Porphyry, red. ...
Porcelain, China
Pumice stone
Quartz
Rotten-stone
Red lead
liesin ,
Speci
vity
Rock, crystal
Ruby
Salt, commun
Saltpetre
Sand, coarse •
common
damp and loose..
dried and loose.
dry
mortar, Ft. Rich.
" Brooklyn
sillicious ,..
Sapphire
Shale
«
it
II
II
it
Slate
Slate, purple
Smalt
Stone, Bath Engl..
«« Blue Bill
" Bluestone (basalt)
«« Breakneck.. N.Y.
" Bristol Engl.
" Caen, Normandy
Common
tt
177.
165.
lOJ.
175
«6.
109.
155.
IG^O'lOl.
2838
•J 050
1 7.j(i
2800
1750
2184
Weight
oi a cu-
bic lUUt.
li)00
2114
2092
2410
2650
600
1H29
177(1
1176
2l'i0
2765
2;'.ou
915
2660
1981
8940
1089
27H5
4283
2130
2090
1800
1670
1392
1560
1420
1659
1716
1701
3994
2600
2900
2672
2784
2440
1961
2640
2625
2704
2510
2076
2520
118.
130.
151.
37
83.
110
73
131.
172
143
57.
166,
123.
558.
68.
170.
133.
130.
112
IU4.
87.
97.
88.
103.
107.
IU6.
162.
I8i
167.
174.
152.
122.
165.
164.
169.
156
129.
157.
375
57
37o
5
37o
25
875
75
75
5
062
62
5
25
812
75
187
25
812
75
062
93
125
625
5
375
5
75
66
25
33
5
25
662
062
875
75
5
Stones,Earths,&c'Sa
viTy.
((
«
((
<(
Stone, Crai;;leth..Engl.
Kentish rag ''
Kip'HBay...N Y.
Norfolk (Parlia-
ment House) .
Portland. ..Etigl
Sandstone, mean
" Sydney
Staten Wd. N.Y
Sullivan Co. "
Schorl
Si>ar, calcareous
'" Feld, blue
<« >< green.,...,
" " Fluor
Stalactite
Sulphur, native
Talc, mean
Ta'c, black. ...
Tile
Topaz, Oriental
Trap ,
Turquoise ,
•••••••
Miscellaneous.
A.sphaltum
Atmospheric Air.
Beeswax
Butter
Camphor
Caoutchouc
s
Egg
Fat of Beef.
" Hogs
" Mutton
Gamboge
(jum Arabic
Gunpowder, loose
'* shaken.
ti
Gutta-percha.
Horn.
Ice, at 32" ...
Indigo
Isingiads
[vory
Lard <
solid.. ,
2316
2651
2759
2304
23(18
2200
223
2976
268
3170
2735
2693
2704
340(>
2415
2033
250(1
2900
1815
4U11
2720
2750
905
16.30
«
965
942
988
903
1090
923
936
923
1222
1452
90;i
1000
1550
1800
980
1689
920
1009
1111
1825
947
■Weight
uf a cu-
bic foot.
144.75
165.687
172.
744,
148,
137,
139
186,
168,
198
170,
168,
169
215,
150
127,
156
181,
113,
o
812
125
937
312
5
937
062
25
25
437
170.
56.562
103.125
.07525
60.312
58.875
61.75
56.437
57.688
58.5
57.687
90 . 75
56.25
62 5
96.875
112.5
61.25
105.662
57.5
63.062
69.437
114.062
59.187
(•) .001905.
4S
SPECIFIO GRAVITIES.
Miscellaneous.
Mastic
M^'rrh
Opium
Soap, Castile.
Spermaceti....
Starch
Sugar
.66
t(
Tallow.
Wax....
Liquids.
Acid, Acetic
^' Benzoic
" Citric
" Concentrated
" Fluoric
" Muriatic
" Nitric.
** Phosphoric
" " eol'd..
" Sulphuric
Alcohol, pure, 60°
95 per cent
80
It
(<
<(
((
((
50
40
25
10
5
" proof spirit, '50
per cent 60''.
" proof spirit. 50 .
percent 80". J
Ammonia, 27.9 per ct.
Spec!
flc ItTA
vity
1074
1.360
1.336
1071
943
950
1606
1326
972
941
964
970
1062
667
10.34
1521
1500
1200
1217
1558
2800
1849
794
816
863
934
951
970
986
992
934
875
Welfsht
of a cu-
bic loot.
67,
85.
83,
56.
58,
59.
100.
82.
60.
58.
60.
60.
125
5
937
937
375
375
875
25
812
25
625
Liquids.
Aquafortis, double,
" single..
Beer
66
41
64
95
93
75
76
97
175
115
49
51
53
58
59
60
61
62
.375
687
.625
062
,75
062
375
562
622
937
37.0
437
625
625
.58.375
54
8911 55
.687
687
Bitumen, liquid
Blood (human)
Brandy, f or 5 of spirit
Cider
Ether, acetic
" muriatic
** sulphuric
Honey
• ••••a ...
Milk
Oil, Anise-seed.
" Codfish
" Cotton-seed.
" Lipseed
" Naphta
" Olive
" Palm
" Petroleum .
" Rape
" Sunflower...
" Turpentine.
«« Whale
Spirit, rectified
Tar
Vinegir
Water, Dead Sea
. ,- 60"...
2120...
distilled, 39ot ...
Mediterranean ...
rain
sea
Wine, Burgundy
" Champagne
" Madeira
" Port
u
It
it
<(
((
((
Sppci-
fic gra
vity.
1300
1200
10.} i
848
1054
924
1018
866
84
715
1450
1032
986
923
940
848
915
969
878
914
926
870
923
824
1015
1080
1240
999
957
998
1029
1009
1026
992
997
10.38
997
Weight
of a ca-
ble foot.
81.25
75.
64.625
53.
65 875
57.75
63.625
.54.125
52.812
44 687
90 625
64.5
61.625
57 681
58 75 •
53.
57 187
60.562
54.875
57.125
57.875
54.375
57.687
51 5
63.437
67.5
77.5
62.449
59.812
62 379
64.312
62.6
64.125
62.
64.. 375
62.312
62.312
Compression of the following fluids under a pressure of 16 lbs. per
square inch : .
Alcohol 0000216
Ether 0000158
Mercury „ 00000265
Water 00004663
* Specific gravity of proof spirit according to Ure'a Table for Sykes'a Hydrometer, 920.
tl cubic inch = .252.69 Troy grains.
WEIGHTS AND VOLUMES OF VAKIOUS SUBSTANCES.
49
Elastic Fluids.
It Cubic Foot of Atirtosfjheric
Its asBumed Uravity of 1 t«
Atmospheric air, 34°.
Ammonia.. •
Azote ,
Carbonic acid
•* oxyd
Carbureted hydrogen.
Chlorine
Chloro-curbunic
Cyanogen
Gas, coaI....M
Hydrogen ,
Hydrochloric acid ,
Hydrocyanic " ■
Muriatic acid
Nitro^^en
Nitric oxyd
Nitrous acid ,
Nitrous oxyd.
Oxygen
I.
.589
.976
1.52
.972
.559
2.47
3.389
1.815
.4
.752
.07
1.278
.942
Air weighs 527.04 Troy Grain*.
tht Unit for Elastic Fluids.
Phosphuretcd hydrogen 1
Sulphureted " 1
Sulphurous acid 2.
Steam, * 212"...
247
972
094
638
527
102
Smoke, of bituminous coal.
coke.
wood
Vapor of alcohol
bisulphuret of carbon
Vapor of bromine
chloric ether
ether
hydrochloric ether....
iodine
nitric acid
spirits of turpentine..
sulphuric acid
" ether
sulphur.
water
u
((
((
t(
t(
l(
((
((
<(
77
17
21
48S3
102
105
09
613
64
1
44
586
255
675
75
763
7
586
214
623
"M^eiu^lits ancl'Voliiincts ofvarious Substanoes
in Ordinary Use. <- ...'v
Substances.
^.« .....
Metals.
Brass^^PP""!
( zinc 33 )
gun metal...
i-heetfi
wire....
Copper, cast
'• plate
Iron, cast
•* gun metal
** heavy forging
'* plates
*' wrougiitbar.«.
Lead, cast
" rolled,
Mercury, fiO",
Steel, plates
*' soft
Cubic
Cubic
Foot.
lucli.
Lbs
Lbs.
488.75
2829
543.75
.3147
513.6
.297
524.16
. 3033
547.25
.3179
543 625
3167
450.437
.2607
466.5
.27
479.5
.2775
i8l.5
.2787
486.75
2816
709.5
.4106
711.75
.4119
S48.7487
.491174
487,75
. 2823
489 . 562
. 2833
Substances.
Metals.
Tin
Zinc, cast....
" rolled.
Woods.
•••••••••
Ash
Bay
Cork
Cedar
Chestnut..
Hickory, pig nut.
" shell-bark.
Lignum-vitse
Logwood
Mahogany, bon-
duraa
Cubic
Foot.
Lbs.
455.687
428.812
449 437
52.812
51.375
15.
35 062
38.125
49.5
43.125
83.312
57.062
35.
66.437
Cabio
luch.
Lbs.
.2637
.2482
.2601
Cub.Fpet
in a Ton.
42.414
43.601
149.333
63.886
58 754
45.252
51.942
66.886
39.255
64.
33.714
t Equal to .0752914.3 lbs avoiidupois. * Weight of a cubic foot, 257,333 Troy grains.
50
WEIGHTS AND VOLUMES OF VARIOUS SUBSTANCES.
Sul>r,Lances.
Oak, ranailinn....
'* Kii;;li-li
♦* live.seuxoiied
" white, (iry...
'• '• upluiicl
Pine, pitch
r"ii» • •••••••••
" white
" well Keas-oiieil
" yeilow
Spruce...
Walnut, black, drv
Willow .'.
*' dry..
Miscellaneous
Air
Basalt, lueun
Brick, tire
" mean
Coal, anthriicitc I
" bituiiiin.,iiieari
" Canriel.
" Cuinberlaud...
Cubic
I'OOt.
Cub Fpet
iu a Tuu
54.5
58.25
()G 75
5S.75
42.9:57
41.25
:{d.875
;U.<J25
21). 562
H:{ 812
31.25
31.25
Hi).6«2
30.375
07:)291
175.
137.562
102.
Hi) 75
102.5
80.
94.875
84 687
41
38
33
41
52
54
CO
64
75
66
71
71
61
73
12
16
21
24
21
28
23
26
Substances.
Cable
Fuut.
Cnb.Fee*
ill a I'oa.
.10!
.45;'.
.55h
.674
.16;)
.30;
.745
693
.773
248
.68
.68
265
.74»
.8
.284
.961
.958
.854
.609
.45
Coal, Wel.«h,!iican
'oke
Cotton, bale, mean
** " pre.ssed <
b^arth, clay
((
((
it
common hoi
grave
dry, eand....
loose
moist, sand.
mold
mild
with gravel .
Granite, Quincy...
Su.-queirna
H,iy, bale
' pre.ssed
India rubber
*' vulcanized .
liiinestone
Marble, mean
Murtar, dry, mean
Water, fresh
" salt
^Steam
81.25
63.5
14.5
20.
25.
120.
137.
109.
120.
93
128.125
128.125
101 875
126.25
163.75
169.
9.52o
25
56.437
625
125
312
75
.25
875
98
27 56
35 . 84
154.48
114.
89.6
18.569
16.335
20.49
18.667
23
17
17
21
17
13
89 S
482
.482
987
.742
514
197
167
97
62
64 . 1 25
.036747
I3.254
2:15. 17
89.6
39.69
11,355
13 343
22 . 862
35.84
34.931
Application of* the Talkie—
When the Weight of a Siil)stance is required. Hi'i-k. — Ascertain the
volume of the ^uiist;itice in cubic feet ; multiply it by the unit in the second
column of tallies, and divide the product by 16 ; the quotient will give the
weiirht in pou'iiifi.
When til e Volume is given or ascertained in Inches. Rule. — Multiply
it by the unit in the thud column of the tables, and the product will be the
weight in pouMd.«.
Example.— What is the weight of a cube ofltalian marble, the sides bieng 3 feet ?
3^! X 2708 = 73116 oz., which H- 16 = 45 ,9.75 lbs.
Or of a sphere i f cast iron 2 inches in diameter ? f-.*,-
2- X -5236 X .26 weight of a cubic inch = 1.089 lbs.
Comparative Weig-lit of" Timber in a Oreen
au<l Seasoned State.
Weiglitof a Cub. Ft.
Timber.
Weight of
a Cub. Kt.
Timber.
Green.
Seasoned.
Green.
Seasoned.
American Pine
Lbs. Ox.
58.3
60.
Lbs. Oz.
;io u
50.
5:1.6
Cedar
Lbs. Oa.
71.10
48.12
Lbs. Oa.
28-4
Atb
English Oak
43.8
B«ech
UigaFir
35.8
BALLOON. — WEIGHTH OF PATTERNS. 61
To Compute the Capacity oFa Oalloon.
Rrtic.-From ppeciflc gravity of the air in grains par cubic foot «mr
ftract that oftlie gaz with wliich it it inflate I ; muitipiy the reniaimler by
the vo ume of the balloon in cubic feet ; divide the product by 7000, an(^
from the quDtient nub-tract lite weight of tlje balloon and its attachtnenla.
Example —The diameter of a balloon is 26 6 feet, its weight in 100 Ibg and the
spec fie giaviiy of the gaz with wh.ch ills inflated is .06 (air bemflr assumed at n •
what ii its capacity ? B » cu.n;,
^ 527.04—31.62X26 6^ X.523 6 495.42X 9354.726 ' ' " ' ' ''
7000 —100=- 7000 —100 =597. 461 /i«.
To Comjint^ tlie Dlametei* of a Balloon, tlie
Weifflit to l>e raliseU beiuip fg-lveu.
By inversion of the preceding rule, • ^
g/W X 700^»-«'
\ — i52HtJ ^d, 8 and «' feprenntittg fAe weight nfnir and gag in
grains per cubic fool, and d the diameter of the balloon in feet.
Example. — Given the elements in the preceeding case
{/597 46-f 100X7000-5-527.04— 31.62
Then :523"6 =v/l8821.09=26.6/«e<. ' ' *•"
To Connputo tlie TTeitrht of Cast 3£otaI l>v
tlie WtiiKht ol tUe JPatteria.
^ ' "^ • ' When the Pattern ia of \fhite Pine.
Rule.— Multiply the weight of the pattern in pounds by the following
multiplier, and the product will gve the weight of the ca-»titig:
Iron, 14; Brass, 15; Lead, 22; Tin, 14; Zinc, 13.5.
When there are Circular Cores or Prints.— Multiply the square of the
diameter ol the core or print t.y its length in inches, the product by ,0175,
and the result is tlie weiglit of the pattern of the core or print to be deducted
from the weight of the pattern.
It IK customary, in ihe making of patterns for castings, to allow for
shrnkage per lineal foot of uHilern.
Iron and Lead ^th of an inch, Brass and Zinc -^jth?, and Tin ^jth.
52 KEY TO THE TABLEAU
_- n ;:• y '.-.fit '•■.■..-..• ir, oVf ')■ i
PROBLEM.
.■'■■• "■ .
To determine the accurate solidity of any irregular bady of
small dimensions or of a body composed of several
elementary parts -with different dimensions ,, ,•
■'-"■ and forms. *- ,'
(1) RVUE. If it is the capacity of any vase or vessel
which we want to measure, the idea generally suggest itself of an-ivin-f
at the result by detenniniiifj the number of times which such a vessel ca i f/ive
place to or contain the contenis of any other vessel of an eleinentiiry form of
which toe know the capacity.
(«) But if it is the solidity of the substance itself of ths
vessel, &c., which we desire to measure, the manaer of operating
does not ituaiediutaly preseut itself to the miud of any one wishing to obtain
tlie result.
(3) RULE. If the solidity to be measured is that of a non
absorbent substance, ^i^^ immerse it in a vessel full of water or any other
liquid of which we will, measure the displacement by meanf of another vend of
known capacity ; or if the first vessel is large enough and it form rectangular or
cylindrical and of easy gauging, we will first put in it enough liquid to cover the
object to be measured ; having afterwards observed the height of the level of the
water in the vessel, we will immerse in it the object in question and observe again
tJie level of the liquid ; if now we suppose that each fraction of a metre, inch
line or any other unit of the height of the containing vessel corresponds to a cubic
metre, foot, inch, or line, &c., we will have but to count the number of such units
in the height of the displaced level of the water to obtain immediately the solidity
of the proposed object.
(41) If the body is absorbent, we may for instance use sand or any
other fluid substance, of the kind, that toe can level the surface of by means of a
rod with a rectilineal edge.
In this manner we woald arrive at the solidity of the most deversified
bodies of the animal, vegetable or mineral kingdom and of the thousand aud
cue raw or mauafactared objects wMch we have constantly under o
MENSURATION OF SOLIDS 08
and of which it wonUI often bo im possible to luenBiire the ■olidities by the
ordinary rules of geometry.
It is well to remind also that we may arrive by a simple proportion til
the solidity of a body by coinpiiring its weight with that ot another body of
the same substance and of determined solidity, that is by th^ system of upe-
cifie gravities which nhowsiit the sun o time how to obtain the solidity of a
body from its weight : which will form the subjects of the next (iroldem.
Ex. I. The weight of an irregnliir block of stone i^ 13 pounds? ounces :
required to determine with the help of the given piece the weight nearly of u
cubic foot of such stone.
" Anf. First cube the block of stone; to that eflfi-ct get a rectangular
vessel, say 10 inches square or I0«) inches in hiuizontal area, and the height
of which is divided into inches and hundreths of an inch ; having poun^l into
the vessel water enough to cover the stone to be cubed, I note the height of
, the water which I find 8. 53 inches, I then immerse the stone in the vessel and
* I note a;;ain the height of the water which is now 9. Hi) inches ; ihe diff<;rence
of these heights is I.36inche8. Since the vessel is 10 x 10 inches, it is pl.iia
that every inch of its height corresponds to 100 cubic inches and conse-
quently, each hundredth of an inch of such a height to one cubic inch ; there-
fore the observed height 1 .3G, of the displaced level of the water correspouda
' to I3«) cubic inches ; therefore the solidity of tiie stone is 13), and we will
now obtain the weight of the cubic foot by making I;}6:2I5 ounces (weight
of the stone) : : 1728 cubic inches (that is a cubic foot) : 2732 ouuces, or, di-
'' Tiding by 16,17()i pounds, the required weight. < '
2> In a cylendrical vessel such that each inch of its height corresponds
- to 1 cubic inch of space or solidity, we have immersed a piece of silver which
' Las displaced by 73 hundreths of an inch the level of the liquid in the vase •
required the solidity of the iugot of silver ?
Ans. 73 of a cubic inch.
3. Having filled with water any vessel, we have immersed in it au
object the solidity of which we want to know ; we have gathered in another
vessel, the water overflown, the quantity of which is 3 gal. 2 quaita and i
pint ; what is the solidity of the proposed object, the galloa made use of
being 231 cubic inches ?
Ans. I gallon + 2 quarts + | pint =231 + 115i + 14/y = |f cubic
inches.
4. Required the solidity of an absorbent subs.tance placed in a vessel one
foot square filled with sand; after having removed the object to be measured,
we find that the uniform height of the sand in the vessel, first levelled to
that effect, is .3 of a foot, the height of the vessel being 1.5 feet ?
Ans. 1 .5<--3=1.2 fe6t= height of the displaced level of the sand, and as
64 KEY TO THE TABLEA.XJ
the vessel is I sqnare foot in horizontal section, it follows that the solidity of
the object is 1.2 cubic feet.
ft. In a vessel having the form of Ihe frnstuin of a cone is a quantity of
liquid of which the diameter at the surface i.s 10 iurhes : we immerse ia itaa
object which increases by 9 inches tlie heiglit or depth of the liquid in the
ve.seel and which gives to its displaced surface a diameter of 14 inches ; re-
quired the solidity of the propo.sed bo;ly ?
Aia§. The volume of water displaced which is at the same time that of
the object, is that of the frustum of a cone of which the parallel bases mea-
sure respectively 10 and 14 inches and of which tlio height is 9 inches 5 this
22 2
8ol.= (iri. T.) (10 +14-1- 4 times 12) x 7354x94-6 =872 x .7854 x 1.5
684.8688 X ] .5= 1027.3032 cubic inches. ., ,
THEOREM. ''
"*o-
To determine the solicity or "weight of any body or substance,
. by comparing the volume or •wei^jht of such body "with
that of a body or substance of the same nature of which
. "we know beforehand the weight and volume.
(•>) REm. The weight of a ciibie foot of water at the temperature of
40° Faliieiiluir (at wliich water nearly reaches its greatest density) is 1000
ounces avoir du ^oids nearly, or 62i pounds (engli>h w i^ht) and we denomi-
nate weight or specitie gravity of any body or substance, the weight of a
volume otsuch body or substance equal to iliat of the water tiken f.>r com-
parison ; whence it results that if in advance we know the weight of a cubic
foot, for inst mce, <»f each of the ditferent substances that we may bo called
on to measure or value, us stated in table X, we wi I at once determine by a
simple proportion the volume of any other weight or quantity of the same
substance or the weight of any other vo.ume of such substance, by the fol-
lowing rules. i -,:■' •'.
(6) RULE. To determine the solidity of a body from its
weight • wnfee the proportion : the specific weight of the proposed body is to
( : ) its weiyht in ounces or pounds, c&c , as ( : : ) 1 cubic foot or 17i8 cubic inches,
is to (:) the solidify cfthe body infect or inches, as the case may be.
Ex. 1. The weight of a shell or cast iron ball or of any fragment of such
a solid is 4.^ pounds : required the solidity of the proposed body?
Ads. It is seen by table X of specific gravities that the weight of cast
iron is 450 pounds nearly, per cubic foot j we will then obtain the required
solidity by making 450 pounds : 1728 cubic inches : : 45 pounds : 172.8 cubic
inches.
MENSURATION OF SOLIDS 66
2. Required the volume of a rrnrble statue the weight of which is 1000
pounds, thn spccidc gravity of ih(i mill ble from wliich the statue is drawn
being 170 pounds nearly to the cubic foot ?
Ans. 170 pounds : 1 cubic foot : : 1000 pounds : 5 9 cubic feet nearly.
3. A quantity of sand weighs 13 pounds : wh;it is its solidity ?
AnN. Fiom table X, the specific gravity of sand is 1.520, that is, 1.520
times the weight of an equal volume of water or lo.'O ounces to the cubic foot
(since the weight of a cubic foot of w.iter is 100.) ounceK) ; we will therefore
make 1520 ounces : 1728 cubic inches :: (13 x 16 = ) 20S ounces : x :=
1 728 X 208 =23Gi cubic inches.
r526
4. The weight of a tu.^k or tooth of an elephant is 23 pounds ; what is
its solidity?
Ans. Ivory is 1825 ounces to (he cubic foot; we will th('ref)re obtain
the solidity of the tooth by making 1825 : 1 : : (25 pounds or) 400 ounces : .22
nearly of a cubic foot, or 1825 ounces : 1728 cubic inches :: 400 ounces :
378.74 cubic inches. », .-. ,:., ^^^w*. . ? ,, . -f ' ^irt-'i^i-^^t
5. It is required to determine in advance the probable weight of a cast
iron grating which must be cast according to a carved model of piue wood
the weight of which is 7 pounds I
Ans. We will first obtain the solidity of the pine model by making, as
per rule (the |»ine being considered in this case as of 25 pounds to the cubic
foot) 25 ponnds : I cubic foot : : 7 pounds; .28 of a cubic foot. Now, as the
solidity of the cast iron is 450 pounds per cubic foot, wo will obtain the
weight of the proposed gratiiig = 450 x .28 = 126 pounds.
(7) RULE. To determine the "weight of a bot^y from its
volume; make the proportion : as one cubic foot is to ( : ) the volume of
the proposed body, so is ( : : ) its specific giavity to ( : ) its weight.
Ex. 1. The volume of a heap of sn<tw on the loof of a building is 7000
cubic feet, the weight of a cubic foot of this snow, made heavy by rain, &c.
is 30 pounds required the total weight which bears on the roof ?
Ans. 7000=210,000 pounds.
2. What is the weiglit of a piece of pure cast gold the dimensions of
which are 3 inches by J x i inches ?
Ans. The Polidity=3xt X ^=2| cubic inches; the ppecific gravity of
pure gold is 19.258; the rule gives : I cubic foot or 1728 cubic inches : 2^
cubic iuches : : 19.258 : x - 1 9.258 x 2 25 =2.5.07552 ounces
1728
3. One desires to know the weight of a firkin of butter the volume Of
which obtained from the rule to article (112), is 1830 cubic iochea f
56 KEY TO THE TABLEAU
Ans. The specific weight of the butter is .940 of thnt of water, that is, of
940 omicea to the cubic foot ; we will therefore obtain the reijuired weight
= 18;iO X 940=995i ouiicet»,-j-id=62 poundn Siouuces.
4. What is the weight nearly of a stick of english onk half-dry, the
volume of which is J50 cubic feet ?
Ana. The half-dry «)ak, from the table, is 66 pounds nearly per cubic
foot, whence the required Wfight, is 150x66=9900 pounds. . ^.
- .5. What is the weiglit nearly of a box (if bound books the volume of
which is 15 cubic feet?
Ans. 15 cubic feet x 43 pounds nearly =645 pounds.
' PROBLEM.
,.. ...^ ^
To determine the specific gravity of any body or substance.
''**^ (8) RUIjIJ. I. Oube and weight the proposed body, and afterwards
malce this proportion j as the solidity of the body is to { : ) its weight in ounces,
so is {::) a cubic foot ofsvch body to ( : ) the icdght ofonefcoi of it in ounces ;
that is, by cutting off three figures for deci-its specific gravity. ,^. ., ,,.,.
Ex. 1. What is the specifio weight of seasoned black walnut, if a simple
of this wood the dimensions of which are ll/<7x9 inches, weighs 24
oances T
Ans. 11 X 7 X 9=69.3 cubic inches=:8ol. of the proposed body; now, from
the rule 69,3 inches : 24 ounces : : 1728 inches : 598 ounces or 37.4 pounds j
the required specific gravity is therefore .598 of that of water the weight of
which is J 000 ounces to the cubic foot. , . .. ., ,,, ^
3. An irregular piece ef chalk of which the solidity has been obtained,
=432 cubic inches, by the method of exeuiple 4 of the lust but one problem,
weighs 43i pounds : required the specific gravity of that substance.
Ana. 432 inches : 1728 inches : : 43i pounds : 174 pounds : wheiice, the
required specific gravity is 174 x 16=2.784 times the weight of an equal
volume of water. ,' '"
3? A bateau or pontoon of 100 feet by 20 x 10 feet and the total volume
of which is consequently 20,000 cubic feet , required in its construction 5000
feet of white pine half-seasoned, the weight of which is estimated at 40
pounds for the cubic foot, 500 cubic feet of elm computed at 50 pounds to the
cubic foot, and 5000 pounds weight of iron spikes : required the draught of
water of the proposed body?
Ans The weight of the pine=5000 x 40 - 200,000 pounds, the weight of
the elm = 500 X 50 - 25000, the iron 5000 pounds; the total weight of the
bateau is consequently 230,000 lbs; the average weight or the specific grav-
MENSURATION OF SOLIDS 57
ity of the pontoon is 2"l!), 000 pounds -f- 20,003 cttbic ffcet= 11.5 pounds te the
cnl»ic toot, that is 1 1.5 x IG - 184 ounces per cabic foot, say .184 of the wt'iglifc
of an equal volume of water. The weight of the pontoon is 10 feet, there-
fore thi* drauglit will be .184 of the height of the pontoon or 1.84 feet, that is-
I foot 10 inches and .96 of an inch=l foot 11 iucL«s nearly.
. 4. By what quantitj' can the bateau or pontoon of the last example be
loaded without cau>iug it to founder or sink beyond its deck or superior
surface f
AnSi. Since water weighs 62.5 pounds to tbe cubic foot and the total
volume of the pontoon is 20,000 cubic feet, the total weight o'' the water which
the pontoon must displace before sinking to the lever of the water is 20,000
X 62.5 = 1,250,000 pounds; now the weight of the boat is but 230,000
pounds ; whence it follows that we might still without causing the bateau
to founder load it with a weight equal or nearly equal to the difference
between 1250,000 pounds and 230,000 that is 1020,000 pounds.
(9) RULE II- If the body to be computed is heavier
than "Water • fi>'>^t weigh the body in air, then in water, by means of a
hydraulic balance ; the difference between the results will be the weight lost in
water, or the weight of a quantity of water equal in volume to that of the body.
Make now the proportion : as the weight lost in water {: ) is to the weight of the
body in air (::) sois the specific gravity of water (:) to the specific gravity of the
body. ]':"" \ .. ■• . .; vj; "^;^»
Ex 1. A piece of tin weighs 183 pounds, its weight in water is but 158
pounds : what is the specific gravity of tin ?
Alls. 183—158=25 : 183:: 1000 : 7320=required specific gravity.
2. A block of granite weighs 21 ounces iu air and only 13 ounces in
water : what is the specific gravity of the granite ?
I "' " " Ans. 2625
(lO) RULE HI. If the body to be computed is lighter
than water ; tie to the proposed body by a thread the weight of which is
relatively null, another body heavier than water, so that both of them taken
together may penetrate or sink in the water ; having first weighed each body in
air, and the heavier in water, weigh then in water the compound body, and from
fhe weight lost by the compound body, substract the weight lost by the heavier
body as weighed alone ; the remainder is the weight lost by the light body. Then :
as the weight lost by the light body in water. {:) is to the weight of that body in
air, {::) sois the specific gravity of water ( : ) to the specific gravity of the body.
Ex. 1. To a piece of elm which in air weighs 15 grains, we have tied a
piece of copper the weight of which is 18 grains in air and 16 grains in water
and the compound in water w<;ighs but 6 grains : what is the specific gravity
of the elm ?
58 KEY TO THE TABLEAU
Ann. 18 — 16 = 2=tlie number of grains lost by the copper t»j (he
water.
18+ 15 — 6=27=the number of grains lost by the compound trt
the water.
27—2 =25=tlie number of grains lost by the elm in the water.
^5: 15 :: 1000 : 600 = tlie specific gravity of the elm.
3. A piece of copper, weijijliing in air 27 ounces ami in water 24 ounces,
is tied to a piece of coik weighing in air (J ounces, and the coiupound weighs
in water but 5 ounces : what is the specific gravity of coik? . . .^^ .•v
Ans. 0.240. •.
\ u-^<\X- I ■. ^ . PROBLEM. '■ -.- i-v> ^ ' ■ -V. ,i-:-^n,;?v';.-..
To determine the quantity of each ingredient or element in a
compound of two substances or elements. ^
(11) RULE. Find fir>t the specific weif/ht of the compound, mixture or
alloy, and of each of the component elements and multiply the difference of every
two of these three specific weights by the third. Make then : the (jreatest product,
(: ) is to each of the other product, (;:)«» the weight of the atloy, ( : ) is to the
weight of each ingredient. -.^ , ^.-^ ,- •,., .,. ; .^^ > . _ .. ^ .- ■... . ■■ ,_^^ ,
Ex. 1. A mass of gold and silver weighs G'2 ounces, and its specific
gravity is 16J26 j what is the quantity ()f each ingredient, the specilic gravity
of gold being- 1LI640, and that of silver 11 091 ?
Alls. (19640— 11091) X 1612fi=137,86l, 174. Alloy. t,,^? -
(I9()40 — 16126} X 11091 = 38,973,774. Silver, i .| ,
(16126— 11091) X I964() = 9d,88/,4U:). Gold. ,; ...
137,861,174 : 98,838,400 :: 63 : 45 ounces, 3 penny weight,s, 19 grains of gold.
137,861,174 : 38,973,774 .. i3 : IT ounces, 16 penny wi-iglits, 5 grains of silver.
ft. A mass of copper and gold weiyhs 48 ou-:ces, and its specific gravity
is 17150, the specific gravity of gold is I9u40 and that of copper 9J00 : what
is the quantity of each element of the mixtiiie ?
An§- Gol(l=42 ounces 2 pennyweights 2 1°5^§ grains, copper =5 ounces,
17 penny weights 21 f^eia g'^'i'is.
3. .^n alloy of silver and copper weighs 60 ounces, its specific gravity
being I0.)35 : required the weight of each ingredient, their respective specific
gravities being 11091 and 9000 ?
Ans. 46 ounces 7 penny- weights 9 1 Jftsy^ grains silver, 13 ounces 12
penny-weights 14 rWFsVff "^ copper.
4. An alloy of copper and tin weighs 1 12 poands and its specific gravity
is 8784, what is the quantity of each of the ingredients of the mixture, their
respective specific gravities being 9000 and 7320 ?
An*. 100 pounds copper, 12 pounds tin.
MENSURATION OF SOLIDS 59
5. Hf'qninMl thp •weifjlit. of poM, in n compound of quartz ami golil the
specific gravit}' of which is 3300, that uf gold being 1964IJ and that of quarts
3'JOO !
Ans. 19640- 3000= 16640 x35;i0 -58,240,000=
Factor for the coiiiponnd body.
V'- • 19640--3500 = 16I40, 10140 x3f)00 = 48,42n,000 =
r ; : Factor for the qnartz.
3500 -3003 -- 500, 500 x 19340 = 9,823,030 =
Factor for the gold.
58240000 : 9820000 :: 100 : 10.86.8612638 -ounces of gold ; if this result .be
correct, the weight of the qnartz must be equal to the diflfereuce between the
weight of the gold and till t of the alloy, and in fact 58210000: 48420000 ::
100 : 83.13i73li2 ) ounces of quaitz; the sum of these numbirs= 100 j there
fore, &.C.
-■-»■ --w;J:--
ki:^j ■:..t-<.-^r !.;■,'/- -^t'^-,-'' PROBLEM. ' ^-iU-A'rui
To determine the solidity of the largest piece of squared
timber that may be got out of a round log", or
,y, . out of felled or standing tree.
(1*2) RUEiE. MnltipJij the diameier of the tree or log ty the half- diameter ,
and this proiiict by the Un'jh : the result wdl he the required solidity.
In fact, it is plain that the diara. AB multip'ied by the
lialf-diameter OC (oi- ^ AB) sjiven for piodnct the. area of the
inscribed sipiare ABCi), tint is, the area of a section, of the
timber to be computed, by a plane perpirmlicnl ir to it-* kU
lenirth, and thit area multiplied by the length of thsj log
gives (78 T.) th*. reipiired solidity.
RI^M. This rule supposes tint the diara. of thu tree is
every win r.; the same or that we make use of a mean diameter, as taken at
middle of the length, aiid this gt-neially done when there is not too much
diflFcreiice lulween the diameters of the oppo>^ite ends; but to be precise
(148, T.) Wf must as already stated (91, T.) add to the sura of the areas
of the ends of the log or tree to be measured f mk- times the area of a section
taken at the centre ami multiply the whole by ths sixth ^»art of the length,
or which. is the same thing, multiply the sum of the areas by the whole
length and take the sixili part of the result.
1.x. I. The circumference of a log, the length of which is 12 feet, is
6.23 feet, dedui-ti(»n being ma le of the bark if nece-*sary : h iw many cubic
feet of wood will there be in the stiik of siiuared timber to be got out of the
log ?
Ana. The circ. 6.23 correspouds to a diam 2, the section of the timber
^ KEY TO THE TABLEAU
Tvill th^reforp be 2 X 1=2 square feet in area, anil as the length is 12, the
solidity will be 21 cubic feet. jrar.' ,^h.^. ^<i ;;
2. A tree th ^ hei<,'ht of which is 50 f 'et, has for its sup. diam. 30 inches,
nnd for its inf. diam. 3i inches, for its interni. diam. 83 iiicliea ; wliat is the
Bulidity of the piece of sijiiare tiniWer that luiy be got out of it.
Ans. Ana small end = 2i < \\ feet =-- 3.1 -'b sup. feet, area large enil=3x
1^=4,5 sup. feet, intermediat area=2.75 < 1.375= i7dl25, 4 interuiediaie
flrea=15.l25, the sura of the area8=22.75 and that sum x 50 -r 6= 189.6 cubic
eet. - •'•
' 3. We have measured at 5 places nearly equidistant by means of a fhick-
ress compass, the «liain. of an irregular tre^ just fell, d ; these diaujeters are
respectively 3D, 30.i, 38, 37i and 3 5 inches, and ihe length of the tree 40 feet;
what will its solidity be after it has biien sipiared.
An§. The sum of the diameters 190 iiu'he8-T-5=38 inches=mean diam.
= 3^ feet, 3.163 x 1.583 - 5.012 nearly = area of the section ; multiplying this
latter by the length 4'J, we get200i cubic feet.
PROBLEM.
To cube a stick of timber AB -which is but partly squared, or
of which ths edges or angles are 'wanliing,
,4; called '* w^aney timber." « v- n^
a;**
■ ■^*<*^'v 'y-ht.^p_.
(13) RUILtE. Square the diam. AB of the timber, and from such square
subtract that of the diam. ab of the sapwood, the difference of these squares multiplied
by the length of the timber, will be the required solidity. >•■
I« fact, it is plain tliat the surface wanting at
each of the four angles, corners of edges of the tim-
ber, to complete tlie square A B, is the triangle abo,
or a triangle equal to abo, when as it is supposed, ef
z=gh=kl=ab ; now the square on a6 is woith 4 abo ;
therefore, &o.
REM. 1- If the sides a&, e/, &c. are not equal
to each o.her, we may take one fourth of the sum of
these four sides for a mean diameter a&, or for greater
accuracy, we will make separately the squares of ab, ef, Sec, and the fourth
«f the sum of those squares will be, or the sum of the fourths of those squares
will be the quantity, nearly, to be subtracted from the square AB to obtaiu
the net area of the section of the timber.
KIS9I. II. Let us observe as in the last problem that if the timber is uot
throughout its entire length of equal size, its ."iection must be taken at about
the middle of its length, and this is generally what is done (148 T.) or, we
will determine several sections of the timber and then take their mean, or
MENSUKATION OF SOLIDS 61
finally we will mnke the sain of the areas of the oppositfl ends plus four times
that of the inteniu'diatc^ section and afterwards multiply the v?hole by the
length nud take the sixth part of the result.
RI291. III. We must also observe that we may arrive at the area of any
regular or symmetrical octagon or of the kind liere illu.strated by subtracting
from the square of the perpendicular distance AB which separates any two of
its pai allel sides, the square of one ab of the sides aiijucent to the first.
Ex. 1. An eight sides piLir is .'ifeet wide or thick AB, the side ab <>t the
chamfer aob is (i inches : what is the solidity of the pillar, its length or height
being 10 feet ?
Ans. (3 + 3— (.5 X .5) = 8.75 superficial feet, and 8.7 i x 10—87.5 cubic
feet=required solidity.
a. A log of timber the edges of which are waney, measures 30 inches
square and .'JO feet long, the average «»f the sides ab, ef, &c., of the wane is 9
inches ; what is the solitiity of the timber /
Ans. (30 X 30) minus (9 X 9) =919 square inches -area of the section of
the timber = 6.382 feet very nearly, and 6.;«2x30= I7j.4f) cubic feet.
3. We h:ive reduced to 30 inches square at the large end a tree the
diam, of which w;is at that point 36 inches ; at the small end the diain. 30
inches has been reduced to 25 inches ; the w.ine, sapwood or defect from a
true 8(|uarea6is from 7 to 6 inches respectively at tlie two ends, such as
obtained by a direct measurement of the piece of wood to be cubs'd, or by
means of a sketch made from a scale of equal parts : what, is the solidity of
the timber, its length being 60 feet ?
Alls. Area at the large end =(30 x 30) - (7 x 7)=851 square inches,
area at small end = (25x25) — (6x6) = 589 sq. f., the intermediate area
/30 - 25 30 ' 25\ /7 + 6 7 + 6\ 22
(^— g- X— 2-J~(~2~^~irj =(27ix27i)-(6Jx6i) =27.5 -6.5 =
756.25-42.25=714 ; 851 +859 • 4 times 714=4296 square inches, dividing by
144 we obtain 29.83^^ square feet, multiplying by ^ of the length or by 10 we
obtain 29S.33 cubic feet.
Am8. Area section at the centre = 714 square iitches, 714-4-144=4933
square feet, 4. 9583 x ()0=2y7.498 ciildc feel, th;it is, equal to the ai^curate soli-
dity by Jess than one foot neatly, or by ie.s.s than one 30i)th neaily, or by
less than one thiid nearly of 1 per cent, suflaeieut accuracy (148. T.) iu
practice.
REM- IV. A conipaiison of the two answers of the last problem indi-
cates suflBciently thit the ordinary practice of cullers, who take the dimen-
sions of a log at the middle of its length, and afterwaids multiply the area of
th.! section at th t place by the length «»f the timier, to obtain thus its soli-
dity, is, cuuaideriug all things, (148 T-) suucliuued by oiiuuiuatuuces.
.■ i »
; ^ INDEX
■:-,. , ..,--..•■,'■.-:/ '-.T -*r ■: 7:^'-!.-: ::( iu >..,','-':^ :;.'.v
The Stcrpotncti icon : ?iompnclntnro mid jt,.iipim1 fcahire of ^adi of thf^
200 Kolidg on the board ; see the diiigr.ira at the beginning of lliis
pamphlet 5
The Areas of Spherical Triangles & Polygons to any radius or dia-
meter : a paper read before the Iloyal Society of Canada iii 18i3. 55
On the general application of tlie ptisin »i.lal formula : a paper read be-
fore the Koyal Society of Canada in I8:i2.... 61
TABLES
I. Squares atid Square Roots of numbers from 1 t<> IGOO .■... 4
II. Circumferences and areas of circles of diameter ^'j to 150, advaii-
tiiig by ^ 11
III. Circumferences and areas of circles of dimneter ^'^ to 100, advan-
«""ghy I'a 19
IV. Ciicumferences and areas of circles of diameter I to 50 feet, ad-
vancing by 1 inch or xV 25
V. Sides of Squares equal in area to a circle of diameter 1 to 100 ad-
vaucing by \ 29
VI. Lengths of circular arcs to diameter 1 divided into 1000 equal
parts - 31
VII Lengths of serai-elliptic arcs to transverse diameter 1 divided
into lUOO eipial parts -. 33
VIII. Areas of the segments of a circle to diameter I divided iuto
lOOOequal parts 37
IX. Areas of the zones of a circle to diameter 1 divided iuto 1000
equal parts 33
X. Si>ecific gravities or weights of bodies of all kiuds : solid, fluid,
liquid and gazeous >.. 22