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CIHM/ICMH 
 
 Microfiche 
 
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 CIHiVI/ICMH 
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 Canadian Institute for Historical Microreproductions institut Canadian de microreproductions historiques 
 
 1980 
 
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 1 
 
 2 
 
 3 
 
New system of de 
 
 Mr. Baillar;:6's lecture on WcJnrsday ovi'iiii 
 bctbrt' the Litei'iirv iiiid lli>tiji'it'iil Sucicty ot QiU'lu'c, 
 once mure liu'v vci'y intcrc.-tinir, even in a p ipiila 
 an otlierwise dry and ali-trife i^nliject, may liecuiiu 
 ably iiaiidled. 
 
 'J'lie lectnrer sliowed tlio relaliDnsliip (if L'oonieii 
 the indnslrie-i nf hie. He traced its (iri,i.'in Irnni 
 aiiti(|tiity, its jiraihial ilevelopeni'iil up tii the pR'-e' 
 He shdued h<<\\ it is tlie ha>is iil'all unr pnlihc wnri 
 how wc are indeliteil to it lor all the cijnstrnctive a 
 relaliun diip to ineciianics, liydranlic>, dptie-, and 
 physical ''icnces. The t'airer portion of nianl<in 
 Mr. I!., have the keenest, most appruiiative p'roc] 
 its advanlaixes and hea.ities, as evidenced in llu 
 varymi: coinliinatioMs so c(iMniii;:ly lievised in their 
 for needle tracery, lace^^ and Ctnlirii;clery. He sho 
 relationship to cheii\istiy in crystalhzaiiun and pdari: 
 to botany and zouloj^y in the laws of murpholi 
 tlieoloj^y, and ^'o on. In treating nf the circle an^ 
 c<jiiic sections, lie drew ipiite a jjoetical c (niparisi 
 ween the engineer who traci's ont his cnrves aim 
 woods aid waters nf the earth, and the asirinioiiK 
 sweeps lilt his miirlity ciicuits amidst the starry t'.ii 
 the heiivens. The parabola was Inlly illnstruteil 
 application to the throwing ol' projoc'ides oi' war, 
 eviijenceil in jets of water, the spcikinir trmiiii 
 mirror and the rellector, which, in iight-hoiises, 
 the rays of light, as ii were, into a Imndle, and send 
 forth together on their errand of humanity. In trt 
 the ellipse, this almost magic curve which is' trace 
 the heavens by every planet tluit revolves aliout 
 by every satellite about its primary, lie alhidtd 
 most beautiful of all ovals — the face of lovely worn; 
 showed how the re-appearance of a comet ma; 
 predicted even to the very day it heaves in sigi 
 though it has lieen Jiliseiit tor a eciitiirv, and how in 
 ages, when these phenomena were unpredicted, tli( 
 upon the world in une.xpectel moments, carryin; 
 everywhere and giving rise to the iitmo.-t an.xii 
 consternation, as if the enil i>t'all things were at li 
 a word, Mr. Baillarge wentoverthe whole tlidd ol'g 
 and mensuration, both plane and spherical : a diltii 
 uitliiii the limits of asiiis-'le lecture; and kept the ai 
 So to say, entrance.! with interest l()r two whole 
 wiiioh the president. Dr. AiMJersoii, remarked : weri 
 as but one ; and no doubt it must have been so t' 
 since Mr. Wilkie, in seconding the voti- of thank-- | 
 by Capt. .\she, alluded to the pieasiiro with whieh 
 listctied to the lecitnre as if, he said, it were liki 
 him, instead of the unpromi-iiig matter torcsiia I. 
 the title. Mr. Haillargc next explaineil in u( 
 Ptereometrical tableau, wiiicli we hope to see soon ini 
 into all the scliools of this I)omiiiion. He show 
 Conducive it i, ill be in slK.'tening the. time lu'relol'n'c 
 to llie .study of solids and 'Veii to liiat ot plane an. 
 
Te'w system of determining the solid contents of a body of any shape, by one and the same rule. 
 
 {lixtracl from (he " Oin-bcc/hviv Mciriiiy" o/'.b/h Miur/i, '"^Z--) 
 
 
 on Wc'.liiesilay fvciiiiig lii>t, 
 ji'ifiil Society of <.iiU'l>t'i;. pr.ivcil 
 "tiiiL', even in a p ipuliir scnso, 
 i-^L- siilijfcl, iniiy lit'Oonu', wiien 
 
 I'oliilioii.-'hiii lit' i^conicli'v to iiil 
 
 tr:ie("l its urijiin iVoiii I'tiiiote 
 
 [ji'mMit ii|) to tlic prc-eiil time. 
 
 IS iii'iill our pnlilic wiii'k^!, ainl 
 
 ull the con^trnctivo arts ; its 
 
 liyili'aiilios, optic-, anil all the 
 
 fi'f portion oi' nianl^inil. suiil 
 
 i.i>t ai>pri ciative p'ir.cption ot' 
 
 es. as evidence.! in I lie dvpr- 
 
 nniniily devised in tlieir desiu^ns 
 
 III ciiilii'i>;dery. He slidwed its 
 
 crystallizaiiiin and pnlari/.atiun ; 
 
 I the laws ot' incirpholojry ; to 
 
 :reatin.i^ of the circle aiiil other 
 
 lite il poetical c miparison liet- 
 
 ne.i's out his curves anion;: the 
 
 arth, and the a^ironniner who 
 
 iits amidst the starry I'lrests of 
 
 la was tnlly illustrated in its 
 
 ig of projiKUdes (if war, al~o as 
 
 '!•, the spcakin'i trmiii)t, the 
 
 which, ill hj;lit-hiMises, ji;aihers 
 
 , into a bnndle. and sends them 
 
 md of hnmaiiity. I ii treatinj; of 
 
 ijic curve which is traced out in 
 
 et that revolves almnt the sun, 
 
 s ])rimary. he alindid to that 
 
 -the face of lovely woman. He 
 
 ranee of a comet may now be 
 
 y day it heaves in si;.'lit, and 
 
 or a century, and Imw in former 
 
 la were nnpredicted, they Imrst 
 
 'ctel moments, carryiii^r terror 
 
 <e to the utmost an.xirty and 
 
 1 of all thing's were at hand ; in 
 
 overthe whole tudd ofne mietry 
 
 e and spherical ; a dillicnlt (eat 
 
 lecture ; and kept the andieiice, 
 
 interest lor two wlnde h.iiii's, 
 
 idersoM, remarked : were tu him 
 
 it must have lieen So tu others, 
 
 liiii: the Vdte of thaiik-i prop ised 
 
 he pleasure with which lie ha I 
 
 lie said, it were like poetry to 
 
 iii-inj; matter foresiiadowed in 
 
 ne.vt explained in detail liis 
 
 h we liiipe to see soon iiitroduccil 
 
 is l)(iniinioii. He showed hinv 
 
 ■nini; the time heri'luf.ire devuted 
 
 I'eii til thai I f plane and cmivex 
 
 snperlicies, spherical tri,u:onoiii(-try, ireoiiictrica! |)rojeotio:i 
 pcrsprcilve dravMii.'. the develnpment of siirt'ices, shade-* 
 and shadows, and the like. Mr. Wiikie, so t^ir as upp iri- 
 unity ha I lieeii all'ii-iled liim nl' pruviii;; the e.i.enlaliniis, 
 corrnhiiraled .Mr. H.'s statement in re'ati'in tu the iinmeiisi' 
 savin:.' in time, where many ali~truse prolilcms which 
 ;,'eiicral ri'ipiired hours nr days to solve, cm iiow(it'ihi; 
 ruleh', as .\Ir. liaiilarvc asserts, su i.:eneiMlly applic.-iMe, 
 and, as ha-; IiCl'Ii (iertitied by so many per-ons in (estimunials 
 over their own si'_'natures. ) with the heipnfihe new formula 
 aiiil tableau, be perfurmed in as miny miiiutes; to -ay 
 iiothiiiLr <-'f the use the mmlels are in imparling at a glance 
 a Knowledu'e of llieir nomenclature or tianies, and an ac- 
 (]uaintancediip witii their varied shape-* and li.'nr-s. \l ■ 
 shiiwed hiiw. ti the architect an 1 en-.'ineer, the builder and 
 mechanic, the inodijs are sn-.'^estive of the funns and 
 I'clative pr.pp irtiiiiis of liiiildiii.s, mt's, domes, piers and 
 ipiays, (-isierns and reservoirs, cauldrmis, vats, casks, tubs 
 and other vessels o|' oa|).u;ity, eatlnvurks tA all kinds, 
 compri-iii;.' r.iiiroad and uther ciUiiiL's an I embankments, 
 the shafii.i' the (Jreek an 1 Itoitim cihimn, spiire an i 
 w.uii'V limber, saw-Kii'-^, the cinipiii2 'ent, the -ipiare m- 
 sphiyed iipen;n_' of a door or window, nica or loophole in a 
 ivall, the v:oi'.t or arched ceihnj; of a church or hill, the 
 billiai-d or the eannon iiall. or. on a laru"r scale, the moon, 
 earth, sun and p!ani-ts. .Mr. I)aiilari.'i!, we may .-lild, has 
 receive I an orler for a tableau t'lMin the .Minister of 
 lOlucaiion ot' .Vrw Iirunswick, with the vieu- of imro iucing I 
 it into all the s^li mis of that Province: an 1 .NIr. V'annier. 
 in writin;; to .Mr. jiaillairiio, from l-'ranee. on lie- lOihof I 
 ilannary l.wi, to aivisc iiim ol the .irrantm .' ot' hi-; letters- 
 patent t'lr that Country, says ijiat .Me-~rs. Hiiinberl k N'oe, 
 the l=re-i lent and secretary of the society lor the gem-rahza- 
 tion of education in j-'rance. have intiimitel their intention, 
 at their iie.M general meetiiiiL'. of hiving some mark of 
 distinciic.in coiifi-rred on liim '.ii' the beiielit which his imen- 
 tioi: and ili-co\i-i'y are liki-ly to eonl.-r on ednr ilioii. Mr. 
 (iiard. in writing lo Mr. lViiihiiri;c. mi tiie pirt of tlie Hon. 
 .Mr. ("hanveaii, Minis'i r of Puhiic Insiriiciion, says ; •• Ii 
 ■• -e t'la nil dr\oii- d'en rec unman ler ['adoption d,ui> 
 •■ tonic.: le-i mai-ijiis d'<idiicatii.n et dans toiites les eooles,'' 
 From the Si'iniiiary aiin Laval University, Mr, .Ma.iiL'ni 
 wriiesi : '• I'liis mi etudie, p'ns on appr- I'm 1:1 c tie I'onnuli' 
 '' dii cubage lies corps, pins on est eiichante (the more 
 " one marvels) de sa siinpheile, do sa elarle et siirtout de 
 " sa iir.in le gcnerilile." Kcv. .Mr. .Ml'Qu irrie-, Ii. A. 
 "shall III- deli-hied to -ee Iheuldaiid lelious procsses 
 " supersede I bv a finiiula so simple an I so e.-vact, 
 Xewioii, of Yah- Collegr, (nitel Suites: •• cmsidrrs the 
 " tableau a miisi u-eful arr.ui'.'ement for showing the variety 
 "and e.xtoiil of the applications of the loriiiuhi.'' \\i^' 
 College r.Vssoinpiion •• will adopt .Mr. Mailiairges system 
 " as part of their course of iiis(i-iiction." .Mr. Wilkie li is 
 written to the aiiihor that •• the rule is precise and simple, 
 '• and will -r.-.tilv sh irie i lie' prooesse.s of c.Ukiulation. 
 
 "The tableau," says thii< competent judiie, "comprising 
 '• a- It doesagreat variety of elementary inodi-ls, will serve 
 '■ adniir.ibly tii educate the eye, and must grc.itly f.icilitate 
 '•the study of Solid meusuration."' •' .\gaii)," .-ay-s Mr. 
 Wilkie. •• the (loveriinient would conl'er a boon on scliools 
 •• of the mid lie and higher class by aUbrding access to so 
 " su'.':.'estive a c illcciion,'' 'I'liere are others wlio, irres- 
 jii'ctive life, iii-idi-ratimis as to the comparative accuracy of 
 the firniiila, or of its advantages, as applied to mere 
 ineii-iiraiioii, ai-i' awake of the fact that the models are so 
 much more siiL'-'eslive to the pupil and the tc.iclier than 
 their mere representation on a lilackboard or on paper, and 
 who, in their written opinions, liave alluded especially to^ 
 tiiis leatiire ofthe prop )sed .system. M. July I'resident of 
 the Quebec IJraiich ot tiie Montreal School ot^ Arts and 
 DeM.'ii, in a letter on the subject to Mr. Weaver, the 
 I're-i lent ofthe Jioar 1, and after hiving himself witnessed 
 its aivantaires on moic than one occasion, says, in his 
 expressive stvle, •■ the diU'eivnce i.s enormous." Professor 
 Toii-aini, of the N'ormal .School, nufresne. of the Mont- 
 in i.oiy Acaleiiiy, Boivin. of St. Hyaciiithe, and many 
 olliers, are ul the same opinion ; among them M.M. K. .S. 
 .\I. Hoiichette, (,)'l''arivll. Fletcher, St. Aiibin, .Meckel, 
 .iiinean, Veiiiier, (i.ilhe.'her. l.afrance, an . the late 15rotlier 
 .V.iih mv, &e., &c. Neither will it be t'orgoiten that the 
 professors of the Laval University, after reading the 
 enunciation of .Mr. H.'s firmul ., as given in iiis tieatise of 
 bSiJti, e.xpressel themselves thus : " I'll ilmtte involontairu 
 •' s'emp.ire d'almrd de I'esprit, lor- pi'on hi le No. l-")'-'; 
 •' iiciis nil e.xanien atteiitif des paragniplies suivaiits, dissipe 
 '• hieiitoi ce d«i,;te et I'oii re-te itoiine ii la vue i'liiie 
 " formule, si claiie, si aisee i^ .eieiiir et doiit rapplicatioii 
 " est si genende."' Mr. fleiclui-, of the Crown Lands 
 |)( nartineiit, says: " I have compared, in the case of 
 •• s veral solids," the results obtained by your mode of 
 •' I iipntation witli those resulting from the ordinary and 
 " more lengthy processes, and congratulate you .sincerely 
 " on your enunciiUioii of a formula so brief and simple iii 
 •• its 'chai-iicter, an I so precise and satisfactory in it.s 
 '■ results," .Mr. B.iillairge also tool: occasion dnrhig hi.s 
 lecture 10 allude, in other relations, to his tieatise on 
 geometry and meiisur.ition, in which he showed he !ia.^ 
 introduced many important modiliciitioiis iii the usual 
 mode of treating the subject of plane and spherical geometry 
 and iriionomeiry. In conclusion, we must add that the 
 Coiitici! of Public instruction, at its last meeting, appointed 
 a Coiiimittee, composed ofthe Lord Mishop of Qneliec, and 
 of Bishops Langeviii and Larocipie, to reiiort to theCouncil 
 at its nf.\t general meeting in June, and who, il may be 
 taken tor granted, after 'he many tl.itttring testimonials iu 
 r>lat;oii to the utility and many advantages ol the stereo- 
 metrical talileau for purposes of e incatioii, cannot but 
 
 it,; neiid and direct ild adoption in all the MchouLs of 
 
 the Doiiiiiiion. 
 
MoN'CKAl^V Ml'MliKR OF 'lid-: SiUll TV FoR rili: (il.XFKAI 1/ Al |n\ "V 1". 1 H i \ IK i\ i\ 1''ran(i:, 
 
 New system of measuring all bodies, segments, frustums and ungiilas of those bodies, by one and 
 
 ( Patculcd ill Canada, in titc L 'nilcd States ol America , and in /in rope. > 
 
 This is a Case 5 ti'cl lonj^", 3 iV'ct wide and 5 inches dec]), with a hinged ("dass C()\er. under Lock and Ke\-, 
 exhibiting- and alTordini;- free access to sonu! 200 well-hnislied Ilardwocul .\h)dids oi'ever) conceixahle i'denientl 
 form, each of which l)eing merel)- attached to the board, 1)\- means of a wire-peo; or nail, can l)e reiiio\ed and r(j 
 Student or Professor. 
 
 Tlifi nso of tlic Tiihl.'au 
 
 To fliiii tha "O.lfS ooq| 
 
 
 ,i,c study „r „ ,c„. ,„ ,!„„ «.;.-.,£-;; t-J::. '■""" "' ■. ■ ■■^"- ,, — „ „„, 
 
 of ii .];,v or two, iind so sun- ,,,rM;iilcs, njouLT .t'l^'i'-" -^^ ^ ,., - ^,„,, ,a lluno-. v ^>« "*'"'' .,,^,,^,0.. 
 Ot .Solid (lOOllli'tiy, tllC JNo- .'.„„. „iin\e lie I'l 11:111- en Iviiiice. || 
 
 iiiviMilul on. I Jiri'ii 
 fnir tune- tliu iiiid'-ll 
 iiii.l iimlii, ly t ic 
 \,v tin' sixtii imit 
 h^i^lit or lo.igtu 
 bo ly. 
 
 iiioiicliiturc of 'iiMiiiotric'd 
 find otluT forms, tlie devclop- 
 iiK'nt of siirficc's, tjooinctri- 
 c;il I'rojcctioii iind perspect- 
 ive, pliine jiiid ourvod nreas 
 and Splierical Geoiiicfrv. 
 and Triixononictry. and men 
 juration of siirfices and 
 HiliiJs, tliat the several 
 brandies liercinbt'fore nu'ii- 
 tioned may now be taiii;lil 
 even in the most elementary 
 schools, ami in coiiv<'nts, 
 where such study could not 
 even liave been dreamed of 
 lierefofoi'c. 
 
 Kacli Tableau is accom- 
 panied by ii Treatise explan- 
 atoiy of the mode of nuvis- 
 nrenicnt by the " I'risiiioida 
 Formula, " ami an explana- 
 tion of the solid, its natur.\ 
 t^h.ipe, opposite bases, and 
 middle section. 
 
 Afrcnta Willi ..' for the snli 
 i<f the. 7'(ihl((ni in Cdiuida, 
 i/ie Uniltd ,Sliite.t, <ifc- 
 
 ^ X"^ \:w ^y \ H ^^ 
 
 ^^^# (!^(^ (I % 
 
 
 
 i 
 
 ^ 
 
 C 
 
 For the use of Atxhitects, Hnoineers. .Surveyors, .Students and .Xpi.reniices. Customs and Excise Officers IVo 
 \latliematics,_ L mversities, Colleoes. Semmaries, Convents and othc-r luhiraiional Kstal)nshments, Schools of Art a 
 jNleasurer.s, ( ,auo(,-rs, Shii)d>uilders, Contractors ,\rti/;ms and others in Canada, and e]s(n\her(> 
 
<fr^;z^ - 
 
 Ml TMMEMM 
 
 p\ I\ I-'r\X('F., F.!f., ETC. 
 
 les, by one and the same rule. 
 
 Jin rope. ) 
 
 Lock and J\c\-, so as lo cxcliuU' dusl while 
 t'i\al)lc i'll(.'inciUar\ . (ic^oiiKlrical or other 
 :,' rcnuncd and replaced at pleasure, by the 
 
 \K\ 
 
 I'.S 
 
 To null tha BO Id oontoni ol 
 Buy body. 
 
 llUlif'^ : 'I'.i \'m'. ^lltn iil'tlu' 
 
 |,|VlMil<'l «!"' "''"■il". il'''l 
 
 f,,\r tune- thu mitKllo iiiea, 
 mill miilii, ly I lu wli'.lu 
 l,y (in • i'ix'ii I'l"'- "' ''"• 
 hii.^lit or lo.igtii of the 
 lioiy. 
 
 Aiijiriivcil !)}■ tin' Couiieil of Pub- 
 lic lii-truoliipii III' tlio I'roviricu of 
 tjii liH', nini iilrciily ii'lo|it«il and 
 iTiloiol liy iiiiiiiy Iviui'uliiiriii! iiml 
 (itlior I'.'stiibli-liLMiioiils in C.innlii 
 iiml cl-uwlii'i-i!. l-'o.- infi<riiiutiou 
 uiid testiiuoiiiali! a|i|ily Id. 
 
 0. B.MLL-.RQE, 
 
 QCKBEC, 
 
 Ihtniiriii-it Meitihii- nf tlie Svnetil 
 fill- till (inti-ni'iuitidu >;/' h'dii 
 ( iHiiii III t'lUiiiii', etc., etc. 
 
 ^© 
 
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 Tlie .Arclib .»!iii;i ol' C^iit'lic, the 
 lii.-llnp I 1' l(i.iioii-l<i, tlio iJijihoh of 
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 (Millic, tlie iHim iiion lio.inl of 
 Woi-kf. the .^■olioolr ol .vrt iinj Uj- 
 .Mj;ii. the Laval l.niv rsity, Ihu 
 J^u.nin.iry, l^., the Collui^o at Olti- 
 w I, Xii'ilet, liimuiiski, Moiitin i- 
 gii\, S(. Mii'l'al, 0., I'Keule Nnr- 
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 .M.iicotio, l.uililers, the Cioineil of 
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 n iicnt Hoar Is of Works, i)„ ilie 
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 the dooil Shepherd, (ircy .Nuns, 
 Sib ir- do la Co igrd^aCn n, .^oeuia 
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 Etc. Mc. Etc. 
 
 lise Officers, Pro fe.ssor.s of Geometry and 
 l-Schools of Art and Desijrn, Mechanics, 
 
 KEY 
 
 / 
 
 TO 
 
 BAILLAIRGE'S 
 
 STEREO^IETIUCAL TABLEAU, 
 
 NEW SYSTEM OF MEASURING 
 
 ALL 
 
 PA' ONE AND THE SAME RULE 
 
 FOR THE USE OF 
 
 ..o.„..™,.-i-"--rro'.r,!rrs":^^^^^^^^^^^^^ 
 
 I5v CHS. BAILLAIRGE, 
 ,„cn.T«T. ,,»<■..—. s..v.vo«, 
 
 HONORARY MEMBER 
 
 ..r/ATlOS OK KI.VCATION IN FUANCE 
 
 e ,„UV.At,. K^'i'^OuiWuBt.SHBU, 
 
 187G. 
 
THE TABLEAU, 
 
 Registered, COD formally to the act of Pailiainont of Canada, by C. 
 P. P. Baillaiugk, the 2.'3i'd Febniary, 1871, iu the Office of the 
 Minister of Agiicultuie; at Ottawa. 
 
 Patented in Canada, tlic Uaiited States and Europe. 
 
 Registered according to the act of PiulianuMit of Canada, in tlic year 
 one thousand eight liundred and seventy four, (1874) by the 
 author C. P. P. Baillairge, Esq., in tlie Bureau of the Minis- 
 ter of Agriculture, at Ottawa. 
 
 
RKAD 
 
 THE PRP^FACE. 
 
 Tlio '[iicstidii iiiiiy 1)0 jislu'd '' If the system be so simple, why 
 so vol'imiiioiis 11 "Kiev" ? Now, il will be immediiitely seen tlini, tlio 
 preseiit- worl: is ill iciiliiy, for the iiiosf ]>iirt, u iiieic, " Meiisiirutiou 
 of Aro.'is " wiiieli minlit [)erliM[)s have been omitted, since thei'c are 
 already many works wliich treat, on that subject, and that the mode 
 «)f measuring;- I Ik; surface of area (»f any solid is,;ii]>|)()sed to 1)(> known 
 before its cubical contents can be arrived at. It is however mor<'satis- 
 factory for 'I'cachers in gcnenil, Professors and Stiidoiits to find thus 
 brou<;ht together in a sin;j;le volume, all that they require, than to 
 have- to seek it elsewher". Tlie mensuration of areas is not at all 
 suiterlluous, even in llie " K<'y " ; since, in p)iul, of fact the whole 
 ditticulty and labor of computing; the solid contents of any body, con- 
 sists in determining- the areas (»f crtain of its couiponcuit faces ami 
 sections. 
 
 That which also contributes lar,;;-ely tc swell tiie <limensions of 
 the " Key ", is the o-reat number of examples, f lly worked oiii, of 
 the author's system as applied to the com[)utation of the most intri- 
 cate solids, aad the nunuMoiis tables of which theyroat utility will 
 become apparent, when, having to comi)ute the capacity of any boiler 
 tub, vat or cask— the volume of a cylimler, sphere, spheroid, conoid 
 or of any segnu>iit, frustum or uiigula of such bodies, the calculation 
 ■will be found, so to say, fully worked out, since it will sulUce to take 
 out the re(piisite areas, add them ami multi[)ly tlieir sum bv the 
 sixth part of the length or altitmhi of tlie body ; after which ii simple 
 multiplication or division (as the case maybe) ofth(^ units so ob- 
 ^lined, will leduce them to inches, feet, metres, gallons, litres, &c. 
 or to any otiier units greater or less than the lirst. 
 
 At page XXIX, however j that is, after tlie testimonials will bo 
 fouud an 
 
 ABRIDGED OR SYNOPTICAL KEY TO THE TABLEAU. 
 
 and, to any one -, ho understands tlie nomenclature of solid forms 
 and the mensuration of areas, this Abridged Key contains all that is 
 essential to tlio full and entire intelligeuce of the author's system. 
 
SDBSCaiBERS TO THE STEREOMETSICAL TABLEAU. 
 
 I'lic Ari!il(isli(i]irit' of l^)iiflicc. 
 Tlic I'lMiliiics ('^»1IV(•^^. 
 Tlic liisliojiiic of Kiiiioiislvi. 
 TIic Iii>.lio|)ri(' <il' Kiimstoii. 
 The r.islioinic (if St. ll.viicinllK^ 
 Tlic Liiv.'il I'lii Vfisity, ii>iicl»cc. 
 'I'ln' l<iiv;il X'HiiimI .'<c1i(»i)1, <,!iU'l»c'C. 
 'I'iic ^' niihiny, (^tiiclK'c, 
 'J'lif Society for tlie viil,;;iirisiiti(>ii 
 
 ol'cdncjltioil, l'"!.'!!!!"*'. 
 
 Tlic Collc^'O, ()tl;i\v;i. 
 
 Tlif College, K'iiiioiiski. 
 
 TIic Collcj^r, Sto. Amu; Luixxm- 
 
 tiric. 
 Tile scliool of Arts & Design, 
 
 Qneltec. 
 Tlio J?(t;ii(l of Land Siuveyora, 
 
 QiK'liec. 
 Tlie C'oll(\^(% rAssoinpfioii. 
 Till! Depi'. orruhlic Works, Ot- 
 
 tawii. 
 Tlie folleo-e, Nirolet. 
 Tlie Coll'ge, St. Hyaeiiitlio. 
 DiUVesiie's Aciulemy, Moiitiiiagiiy. 
 Tln! Academy, St. Micliel. 
 J. F. IVaeliy. Aiclit., (Quebec. 
 J. Lepage, Ai'clit.. (iiiebee. 
 Tlie Education Ot1ico,Ne\v Biuus- 
 
 Aviclc. 
 
 E. Tlaniilton, Esq. Quebec, for a 
 
 school. 
 The City Hall, Quebec, lor tlic 
 
 Dept. of Works. 
 Godin & Devarennes, builders. 
 N. Pitoii, builder. 
 
 F. N. Martin, Surveyor, liiniouski^ 
 The Government, I'rovinceof Que- 
 bec, for Model Schools & Aca- 
 demies. 
 
 C. Roy", Surveyor & Engineer. 
 J. Maguire Plunibor, «fcc., Quebec. 
 J. Marcotte, iron founder, Qtiebec. 
 The Jaccpies - Cartiev Normal 
 School, Montreal. 
 
 .M. I'itoii, Manitoba. 
 
 Th(^ Ministry of i'ublic Lis|nicli<ni, 
 
 IJelgium. 
 
 The Convent of the (!ood Shep- 
 herd, (^tuelx'c. 
 
 Th(i Con\('nl of the Sisters of Cha- 
 rity. 
 
 Tlie Convent of Jesus-Marie, Cap 
 Iioiige. 
 
 The J).pt. of Public Works, 
 
 (Quebec. 
 
 The lit).!) (1 of Arts & Trades, .Mon- 
 treal. 
 
 .). II. Clint, Lumber Mercht. Que- 
 bec. 
 
 P. Cote, builder, Quebec. 
 
 The College, Aylmei', Ottawa. 
 
 S. W. Towiisend, Ilaniilton, C. W. 
 
 The Freres Schools, Canada. 
 
 The Freres Scliools, France. 
 
 G. Bisset, iron founder, machinist? 
 Quebec. 
 
 T. Archer, Lumber Merchant, 
 Quebec. 
 
 R. Steckel, Civil KngineiM-, Ottawa. 
 
 \'. \'annier, I'aris & Caiuida. 
 
 The C»)nujiercial Academy, Que- 
 bec. 
 
 G. 11. P.aldwin, Civil Engr. Boston. 
 
 Tlie Litteraiy & Hist. S. Quebec. 
 
 The Worci'ster Free Inst, Alass. 
 
 P. V. T)u Tiemblay, Surveyor & 
 Engineer, Bai<- St. Paul. 
 
 C. .Jobin, ship builder, Quebec, 
 
 J. Bacine, iron smith, Quebec. 
 
 A. Beanme, Lumber Merc. Quebec. 
 
 Le College Melbruu, Haute Py- 
 renees, France. 
 
 A. Humbert, artist, Parin, Franco. 
 
 The Academy of Science, Pari.s, Fr. 
 
 The Conservatory of Arts and 
 Trades, 
 
 &c., &c.. &c. 
 
THE 
 
 STEREOMETRIGAL TABLEAU. 
 
 TESTIMONIALS. 
 
 No. '1112-11.— Ministry of Public Instruction, 
 
 Quebec, this 18tli September 1871. 
 C. Baiu,airq£, Esq., Quebec. 
 
 SiR,-Iani instructe.l by the Honorable, the Minister of Public Instruc- 
 tion to acknowledge receipt of your letter of the 8th instant, transmitting copy 
 of the prospectus of your Stereometrical Tableau. 
 
 He will make it a duty to recommend its adoption in all educational 
 establishments and in all schools, persuaded that he is of its practical utility. 
 The tableau and accompanying formula reduce to an operation of the most 
 simple kind the measurement of every description of solid, which required 
 according to the old method a calculation long and often very difficult for 
 persons especially who were not in the daily practice thereof. 
 
 I lave the honor to be, Sir, your obedient servant, 
 
 Louis Giard, Secretary. 
 
 Paris, 8 Juillet 1873. 
 
 Monsieur,—" II est bien decide que votre tableau aura le premier prix 
 delaSociete libre d'iustruction et d' education populaire; vousenserez 
 averti ofKcieliement a la rentree des vacances, fin Octobre, et vous serez en 
 meme temps convoque pour la distribution solennelle des recompenses qui 
 aura lieu dans le couraut du uiois de Mars prochaiu. 
 
 V. Vannier. 
 
 C. Eaillairge, Ecr 
 
VI THE STERKOMKTRICAIi TAHLEAtT. 
 
 Quebec lindqet 12 Aprd 1873. 
 
 TOK UUSl'I.INK CONVENT. 
 
 Wc learn willi piitislaclion ami Iciiitimati' priilothal thin ailniirahlo insti- 
 tiition lian onk'iTil one ul'Mr. I!iiillaii'j;6'.s Sterconietrical Talilfaiis, ami tha' 
 this gentleman, during (i oingle sitting ol' a few lioiirri devotion, generouwly 
 granted him by the Nunn, managed to render them thorongiily ci)nver.«ant 
 with his system of nomenclature and mensuration. It seems almost incre- 
 dible, and yet, we arc ensured that tiie Reverend Superior (the talentea 
 Miss Cimon, of St. Paul's Bay) accompanied by Sister St. Croix and Sister 
 St. Raphael, at once mastered and perl'ectly understood Mr. Raillairge's 
 fcystem in all its <letails. Paying comparatively little attention to the more 
 ordinary forms of wiiicli the mode of measurement was ti; tliem apparent 
 at a glance, they selected for their questions the more couipiex forms, sucii 
 as the sections of the sphere and sjiheroidH and the numerous and varied 
 prismoids, to be found ainimg the 200 models of tiie tableau. The edu- 
 cation given by the Religious Ladies to their pupils comprises the geometry 
 of lines and surfaces, and from tiiis to the mensuration of solids by Mr. 
 Baillairge's system, there is but one step, a simple addition of certain sur- 
 faces and Ihe multiplication of their sum by one sixth part of the heii 
 or length of the body under consideration. The Nuns intend to C(jmmence 
 immediately the teaching of this branch to their pupils, who will be exam 
 ined on the tableau at the next examination which takes place in June. 
 The noble example thus given by the Ursnlines has been rapidly followed 
 by another important educational estabiisiiment, the Convent of the Relig- 
 ious Ladies of Jesus Marie, on the Cap Rouge road, and we are, moreover, 
 informed that the Soeurs de la Congregation do St. Roch, togetiier with the 
 Nuns of tl\e Good She) herd and the Sisters of Charity intend forthwith to 
 add to the already var.cd programme of their tuition, this study of Stereo- 
 tomy, which, up to the present time, could not be even dreamed of, but 
 which Mr. Bailiairge's system now renders possible by reducing, as it does, 
 the study of a year to that of a day or two, so to say. 
 
 " Mr. Baillairge next explained in detail his stereometrical tableau, 
 which we hope to see soon introduced into all the schools of this Dominion. 
 He showed how conducive il will be in shortening the time heretofore de- 
 voted to the sti'.dy of solids and even lo that of plane and convex superficies, 
 spherical trigonometry, geometrical projection, perspective draving, the 
 development of surfaces, shades and shadows, and the like. Mr. Wilkie, 
 K -) far as opportunity had been afi'orded him of proving the calculations, 
 pofrobor^ted Mr, B.'s statement in relation to the immense saving in time, 
 
TESTIMONIALS. VII 
 
 wIhto many nl)struf-p problems which gcncrp'Iy rcrjtiir.'d hours or days to 
 solve, can now (if the riili' be, aH Mr. Haillar^e asserts^, ho (generally appli- 
 cabli', anil, a.s been ccrtilied by so many jiersonH in testimonials over tbeir 
 own Hij^natiiroH,) with the help ot'tlie new ibrmula and tableau, be pertbrm" 
 (mI in as many minnt'.'s ; to say iK^tliing of the use the models are in im- 
 j)artinji; at a glance a kn<jwledj:e of their numencrlature or names, and an 
 ac(iiiaintanceMhip with their varied shapes and tigures. lie sliowed how, to 
 the architect and engineer, the builder and nieciianic, the models are sug- 
 gestive of the forms and relative i)roportions of buildings, roofs, domes, 
 piers and quays, cister';'^ and reservoirs, cauldron^*, vats, casks, tubs and 
 other vessels of caj-acity, earthworks of all kinds, comprising railroad and 
 other cuttings and embankments, the shatts of the Greek and Roman 
 column, square and waney timber, saw-logs, the camping tent, the square 
 or sjilayed opening of a door or window, iiich or looi>hole in a wall, the 
 vault or arclied ouiling of a churo' or iiall, the billiaril or the cannon-ball, 
 or, on a larger scale, the niO' ...irtli, sun and planets. Mr. Baillairge, 
 we may add, has recrived an oaler for a tableau from the Minister of Edu- 
 cation of New-J5ru.iv,>!ck, with the view of introducing it into all the schools 
 of that Province ; and Mr. Vannier, in writing to Mr. Baillairge, from 
 France, on the ICth of January last, to advise liim of the granting of his 
 letters-patent for that country, says that Messrs. Humbert & Noe, the Pre- 
 sident and secretary of the society tor the generalization of education in 
 France, have intimated their intention, at their next general meeting, of 
 Jiaving some mark of distinction conferred on him for the benefit which his 
 invention and discovery are likely to confer on education. Mr. Giard, in 
 writing to Mr, Ikiillairge, on the part of the Hon. Mr. Cliauveau, Minister of 
 Public Instruction, says : " 11 se feri un devoir d'en recomniander I'adop- 
 " tiou dans toutes ies maisons d' education etdans tcutes les ecoles." From 
 the Seminary and Laval University, Mr. Mainguy writes : " Plus on 6tudie, 
 " plus on approfoiulit cctte formule du cubage des corps, plus on est en. 
 " chante (the more one marvels at) de sasimplicite, de saclarte et surtout de 
 " sa grande generalite. " Rev, Mr. McQuarries, B. A., " shall be delighted 
 " to see the old and tedious processes superseded by a formula so simple 
 and so exact. " Newton, of Yale College, United States : " considers the 
 " tableau a niost useful arrangement for showing the variety and extent of 
 (' the applications of the tbrmula." The College I'Assomption "will adopt 
 " Mr. Baillairge's systen; ns part of their course of instruction. " Mr. Wil. 
 kie has written to the author ihat " the rule is precise and simple, and will 
 " greatly shorten the procs^es of calculation. The tableau, " says this 
 competent judge, " comprising as it does a great variety of elementary 
 •' models, will serve admirably to educate the eye, and must greatly facili- 
 " tate the study of solid mensuration." *' Again," says Mr. Wilkie, the 
 *' Government would confer a boou on schools of the middle and higher 
 
i: 
 
 VIII THE STEREOMETRIOAL TABLEAU. 
 
 " class by afibr<lin<^ acces-s to bo susxgcstivo a collection." Tiiorc arc 
 othora who, irrespective ofconsideratioiia as to the comparative accuracy of 
 the forniula, or of its advantaf^es, as applied to mere uicnsunitioii, are 
 awake to the fact that the models are so much more siig,i;estive to tiie pupil 
 and the teachei than their mere representation on a blackboard or (;ii paper 
 and who, in their written opinions, have alluded especirdiy to tiiis feature 
 of the proposed system. Mr. Joly, President cf the Quebec J5ranch of the 
 Montreal School of Art and after having himself witnessed its advantages 
 on more than one occasion, says, in his expressive style, " the dilference is 
 enormous. " Professeur Toussaint, of tiie Normal School, Dufri'sne, of the 
 Montmagny Academy, lloivin, of St. Hyacinthc, and many others, are of 
 the same opinion : among them MM. II. S. M. Uouchette, O'Farrell, Flet- 
 cher, St. Aubin, Steckel, Juneau, Verner, Gallagher, Lafrance, and the late 
 brother AniJiony, &c., &c. Neither will it be forgotten that the professors 
 of the Laval University, after reading the enunciation of Mr. Ij.'s formuhi, 
 as given in his treatise of ISGG, expressed themselves thus : " Un doute in. 
 " volontaire s'empare d'abord de I'esprit, lorsqu'on lit le No. 1521 ; mais un 
 " eyamen attentif des paragraphes suivants, dissipe bientot ce doutc et Ton 
 " reste etonne a la vue d'une forniule, si claire, si aisee a retenir et dont I'lip- 
 " plication est si generale." Mr. Fletcher, of the Crown Lands Department, 
 says : "I have conipared, in the case of several solids, the results obtained 
 " by your mode of computation with those resulting from the ordinary and 
 *' more lengthy processes, and congratidateyou sincerely on your enunciation 
 " of a formula so brief and simple in its character and so precise anil sati jjuc- 
 " tory in its results." Mr. Baillairge also took occasion during liis lecture to 
 allude, in other relations, to his treatise on geometry and mensuration, in which 
 he showed he has introduced many important moditications in the usual mode 
 of treating the subject of plane and spherical geometry and trigonometry. 
 In conclusion, we must adil that the 'Council of Public Instruction, at its 
 last meeting, appointed a Committee, composed of the Lord J'ishop of 
 Quebec, and of Bishops Langevin and LaroC(iue, to report to the Council at 
 its next general meeting in June, and who, it may be taken for granted, 
 after the many flattering testimonials in r':iation to the utility and many 
 advantages of the stereoinetrical tableau for p.irposes of education, cannot 
 but recommend and direct its ailuption in all the schools of the Dominion. 
 — From the Quebec " Daily Mercury " march 20 1872. 
 
 f Our progress and success are of course attributable, before all, to your 
 
 • perfect mastery of the subjects of which you treat, and to the clear and 
 
 j concise manner in which you enunciateii and demonstrated the several pro. 
 
 I positions introduced to our notice ; but neither must we omit to siiy tha 
 
 our intelligence of the problems and demonstrations has been greatly fajili. 
 
 tated by our access and rele.reuce to the models of your " I'ci.'deau. " 
 
TESTIMONIALP. IX 
 
 Wo (Iccin il not ont of place to remark tliat in onr opinion the word 
 " SlereoDietricul " wiiicli you liave prcfixeil, as qualitative of the uses tliat 
 your " Tahlciiu " (;;in ho applied to, is not Hu;.';^'estive ennujjh of tlio many 
 ailvantages wiiicli such a varied colieclion of models presents; for, not 
 only is it of paramount importance and utility, as illustrative of your system 
 of mensuration, by one and the same invariable formula, iniplioci in the 
 title at the head of the hoard : hut, we hesitate not to say that to the use of 
 the " TahledU " we are indoli> ' in an eminent deiiroe for the singularly 
 rapid progress wo have been enabled to make since the Itn of January last 
 (only 30 k'ssons) not only in Geonietjy proper and in the Mensuration of 
 surfaces and solids, botli plane and spherical ; but also in the study of 
 geometrical I'rdjeclion and jjcrspective, shades and shadows, the develop- 
 ment of sui'iaces and the lines of jicnetralion of divers solids, i'c, &c. — 
 From an address to Chs. IJaillairge, professor of the school of Arts. 
 
 Quebec, G Scptemlire 1S72. 
 
 MoNsiicii!, — J'ai le plaisir lie vous annoncer <pie le Conscil de I'lns- 
 truction Pnbliipie vient d'approuver voire Tableau et suis heureux do vous 
 en teliciter. 
 
 Bien sinceremont, Votre tout devoue, 
 
 C. r>Aii,i.Aii:(;K, Ecu. P. J. O. Ciiaivk.vu. 
 
 Quebec, 0th January 1S72. 
 G. W. Wkavkk, Esq. 
 
 Presidtnt nf Board of Arts dj- Manufdcturcs. 
 
 ]\[y Dkah Sill, — Our evening class began kv-t week. I am happy to say 
 every thing looks promising and we have been fortunate enough, to secure 
 again Mr. C. IJaillairge's invaluable services, as teacher, for this winter. 
 
 We luive procured Irom him, for our school, the '' Steremnelrical 
 Tableau " which is his invention, and J am so delighted with it, that I send 
 you a photographic representation of it, and a number of letters and other 
 documents printed which will serve to explain the tableau, and show at the 
 same lime huw useful it ia considered by the most eminent aulhorilies in 
 this country. 
 
 You ought to get the tableau for your schools at Montrcah You sliowed 
 nie last year when i visited your schools, several wooden models of 
 geometrieal ligures ; I was struck with their usefulness at the lime, and 
 thought of procuring some for our schools, but there are only a few of them 
 and their p'ric^.' is very higli. Mr. P>aillairgc's Tableau costs only tifty 
 dollars, and it contains ttvo hundred geometrical figures. I lancy the col- 
 lection embraces every variety ol figure that can ever be reipiired lor practi. 
 cal use. 
 
, 
 
 X THE STEREOMETRICAL TABLEAU. 
 
 They are solid figures made of wood, eacli fixed on a nail, ?o that they 
 can be taken oil iiy tiio loaclicr for deniuiistration and handed to the pupils; 
 who are enabled to understand and master their divers hluipes and hjrnid 
 with much greater ease, lluui if they saw ihcm drawn un a black board, or 
 in a book ; the ditl'crence is enormus. 
 
 In addition to the great help they afford, for the study of geometry, 
 these figures are very useful as models for earthworks, piers, reservoirs, * 
 castings, roofs, domes, columns, cauldrones, &c., &c., &c. 
 
 The tableau is most useful too for the working out of that wonderfully 
 simple rule, which has been applied by Mr. Baillairge, for the first time, 
 to the measurement of the solid contents of all botlics. It was known pre- 
 vious to his discovery to apply to a certain number of boihes, but he has 
 found out that it apjjlied to all ivithout exception. You will find that rule 
 in the jiajjers I send you, and in his treatise on geometry. I will soon let 
 you know, what i^rogress the school is making and remain, my dear Sir. 
 
 Yours truly, 
 (Signed,) II. G. Joi-y. 
 
 President Qufbec School of Art. 
 
 Cefte formuk est veritablement curieuse par sa ge- 
 
 neralite et merite d'etre recomnuuulee. 
 
 Enlin j'ai accompli cette f tiche, et voici le resultai : 
 
 — L'anteur demontre sa formule comme rigoureusement exacte pour lui 
 grand nombredes corps enoncea, et comme aussi approximative qu'nn vou- 
 dra pour eeux auxqucls elle ne a'applicuie pas d'une maniere absolument 
 rigoureuse. J'ai verifie soigneusement la demonstration de la formule pour 
 les lers corps, ce\ix auxqnels elle s'applique rigoureusement. La proposi- 
 tion et la demonstration sont exades et vraies dans tons ces cas. 
 
 Quand aiix autres corps, il est vrai que plus on multipliera les sections, 
 suivant le besoin, plus I'approximation sera proche de la verite. 
 
 Je finis en disant que j'approuve et recommande la 
 
 methode de Monsieur IJaillairge telle que jn-ojjosee par lui. — Lettredu Rev. 
 L. Billion professeur de Math. Seminaire de St. Sulpice, Montreal, d Mgr. 
 Larocque Evcque de St. Hyacinthe. 
 
 "Montreal Gazette " 13 November I8Y3, 
 Eni'CATiONAT.. — Last evening, Mr. C. l^aillairge, C. E., and Mr. J. 
 Carrel, both identified with tlie Quebec School of Art, visited the rooms of 
 our An School iu St. James street. Mr. Baillairge there exhibited his 
 
TESTIMONIALS. XI 
 
 stereomctrical tableau, ami explained his system of teaching solid geometry 
 by a iiietliod so simple as to bring tliis hitherto difficult subject within the 
 <'rajsp of ordinary students 'I'lie tableau cannot fail to make easy and popu- 
 lar tlie study of solid forms and the mode of measuring their surfaces and 
 their solidities and volumes. Tiiis tableau or board, which is made up of 
 some two hundred models, comprises almost all the elementary forms which 
 it is possible to conce'ive. To comjiuto the volumes or cubical contents of 
 the solids, Mr. liaiiluirge has found that it is necessary only to apply the 
 following rule: To tlie sum of the parallel end areas add four times the midille 
 area, and multiply the whole by one sixth part of the height ov length of the 
 solid. The whole difficulty is, therefore, reduced to mea:?uring the areas of 
 the opposite bases and middle section, the remainder of the work being a 
 mere multiplication. This system, it appears, has attracted considerable 
 attention among mathematicians and educational authorities both in thi.s 
 country and in Europe, and its introduction into several of the colleges and 
 schools ill Quebec has led its author to seek its adoption in similar institu- 
 tions in Montreal. The stereomctrical tableau is well worthy of the attention 
 of those engaged in educational work. 
 
 The extraordinary progress made by the pupils, in the 
 
 short space of three montiis, in stereometry or the mensuration of solids, is 
 attributable to the grand and important discovery by their professor, Mr. 
 Baillairge, of a rule, one and the same, applicable to every known torm, 
 from a pyramid to a sphere, from a stick of timber to a vessel or other body 
 of any shape or dimensions. 
 
 We have seen them in a very few minutes by the help 
 
 of Mr. Baillairge's new an(' beautifully simple and accurate rule arrive at 
 the number of gallons in a cask of any size or shape. We have seen them 
 determine by the same rule the exact weight of a shell, the true contents 
 and weight of a hollow cast-iron colum, the size and weight of a pontoon 
 and its draught of water. — From the Saturday Budget of May G, 1871. 
 
 In the instances which I have subjected to critical 
 
 analysis, I have found the rule to work most admirably — combining com- 
 prehensiveness, utility with simplicity and great exactness. It will render 
 a study heretofore charged with difficulty and abstruseness at once easy and 
 acceptable — modernizing that which was ancient and which from its multi- 
 tudinous formulaj had becume an isolated branch of Jlathematies. Believ- 
 ing it to be of universal use, I shall heartily lend myself to the iniroducliou 
 of your system. 
 
 IIOIUTIO R. N. BiGELOW, 
 
 Quebec deer. 20, 1871. M. j\. 
 
XII 
 
 THE STEREOMETRIOAL TABLEATT. 
 
 ;i 
 
 I have coniparcil, in tlic case of several (soli-ls, the rcsnlt'i ohtaiiicd by 
 your mode ol'cttiuiJiUutiini with lliosp resulting; iVoiii t!ic ordinary and mure 
 lentliy procossey, and cungralulatc you sincorly on your enunciation of a 
 formula so brief and sjinple iu its cliaracter and so precise and satitil'actory 
 in its results. 
 
 E. T. Fletcher, 
 
 Inspector of Surveys Dopt. of Crown Lands. 
 Quebec 27 deer. 1871. 
 
 J'ai !u nttcntivement les appreciations quo nombre d'liommes compe- 
 tonts out faites de votro donnee reellcmcnt mcrvcillou.-<o, tians les solutions 
 Stcrcometriipies quelconqucr', el j"y donnc mon plcin a'^sciitinRiit. 
 
 J'ai ea occasion il'en rcmanpu'r la jnstcs.«e, il y a un an, en pveparant 
 quclquert elevcs pour Tetude de rarpontaL'o, et je puis dire que loutes les ap- 
 plication que j'on ai faites ont ete des plus salisfai^antes. 
 
 Veuillez bien me considerer comnie votre souscripteur. 
 
 CdE. Dl'KRKSXB. 
 
 Professeur de Mathematiques, etc. 
 College de M'Mitinagny 4 Janvier 1872. 
 
 T sball be dcliibted to see the old tedious processes 
 
 superseded by a formula so simple and so exact. 
 
 A. N. McQl'aruie, 13. a. 
 Professor of Mathematics, etc., at the florin College. 
 Quebec .31st Jany. 1872. 
 
 What pleases me most is your Slereometrical Ta, 
 
 6/eaw as, under the direction of the Revd. Superior, I am now en;^aged in 
 certain theoretical studies relating to this formula which your are about to 
 render famous. 
 
 The more one examines, the more deeply one looks into this formula of 
 the " cubing of bodies, " the more one nuirvels (plus on est enchante) at 
 its simplicity, its clearness and espi'ciully its great generality. 
 
 Nor oiiii I ilo otherwise than hope it may, as well in thmry as in 
 practice, assume the place which it is entitled to and that thus your etlbrts 
 may be fully crowned with success. 
 
 L. F. N. MAixiar, Ptiie. 
 
 Professor of Mathematics, 
 
 Seminary of Quebec 25th November 1871. 
 
TESTIMONIALS. XIII 
 
 Je V0U8 declare done, monsieur, que je recommande 
 
 vivement I'adoption de votrc Tableau Stereometrique, et voue prie de m'en 
 
 faire connaitre le coflt probable. 
 
 T. BoiviN, Ptre. 
 
 College de St. Hyacinthe 8 Janvier 1872. 
 
 Please to accept my sincere thanks for the cony of your excellent 
 Treatise on Geometry, Trigonovietry, Ac, which yon kindly sent me some 
 time a!^o. I have perused its pages with deep interest. In my humble opi- 
 nion it is a complete success ; whether we consider its admirable arrange: 
 ineiit, the clearness and conciseness of its demonstrations, or the amount of 
 valuable knowledge it contains. Your Book on Mensuration is a boon to 
 every practical matiieiiiatician, and your remarkable Theorem for finding 
 the contents of any solid, is sufficient, of itself, to immortalize its originator. 
 In a word, your work merits a distinguished rank among the ablest produc- 
 tions on the exact sciences. The flattering and well-merited praise lavished 
 on it by the press, both English and French, is sufficient guarantee that it 
 meets the approbation of the public. 
 
 Your Treatise is us^ed as a book of reference by the pupils of our 
 Academy. 
 
 Bro. Anthony. 
 Commercial Academy, 25 Feb. 1868. 
 
 The January Session of the Board of Examiners Jar Land Surveyort 
 Jor the Province of (Quebec. 1^12.— Extract from the minutes. 
 
 Moved by the P.'esident, Adoiphe Larue, Esq., and seconded by E. T. 
 Fletcher, Esq., and resolved : 
 
 " That the Board of Examiners for Land Surveyors, having taken into 
 consideration the Stereometrical Tableau of Charles Baillairge, Esq., Civil 
 Engineer, and the very ii- at d-ml precise formula conn' cted therewith, desire 
 to record th^ir opinion of the utility and importance of this formula, and 
 coincide wholly with the opinions expressed by those to whom it has been 
 already eubmi'ted, and further they would recommend that the Board be 
 provided with one of these Tableaux. 
 
 Algzani/ER Sewell, 
 
 Quebec, January 2nd, 1872. Secretary of the Board of Surveyors. 
 
 Paris, 10 Janvier 1872. 
 Monsieur Charles Baillairge, a Qu6bec, 
 
 Cher Monsieur, — /vujourd'hui seulement, j'ai pu d6poser rotre de- 
 mande de brevet et je nx'empresse de vous en envoyer le bulletin de depot 
 qui vous donne le droit d; brevet a partir de ce jour, 1 Janvier. 
 
XIV THE STKREOMETRICAL TABLEAU. 
 
 Envoycz-moi niaintenant votre lableau-modele le plustot possible, j'en 
 ai parle d plusieu'a inembres de racpJeniie, entre autrend Littie qui desire 
 beaucoup le voir. 
 
 Messieurs Humbert et Noe, president et secretaire de la societe pour la 
 generalisation de I'instruction en France sont *"es-sinipi>thiques d votre 
 oeuvre et eeproposent de vous faire recompenser d leurprocl'aine assemblee 
 generale. 
 
 V. Vannteb. 
 
 No. 1)4. rue de Lev s, Pauis. 
 
 Ser. 2, No. 255. Education Office, Province of New Brunswick, 
 
 FiiEUEiiicTON, January 25tii 1872. 
 C. BAiLi-AiRGfe', Esq., Quebec. 
 
 Dear Sir, — I am instructed by the Board of Education for this Pro- 
 vince to apply to you for a set of your Stereometric Tableau and your 
 text-book on Practical Mathematics. The Board desire these articles for 
 inspection, with a viewof prescribing them for general use in all the Schools 
 of this Province, should they be deemed suitable for the purpose. Should 
 there be any charge for these articles, the same will be met by this Depart- 
 ment. 
 
 Your Obdt. Servt., 
 
 Theodore H. Raud. 
 
 Mr. Baillairge's Stereometrical Tableau seen^" tome to be a very useful 
 arrangement for showing the variety and extent of the applications of the 
 Prismoidal Formula. Where demonstrations are given in the study of 
 Mensuration of Solids, it will aid a teacher in illustrating the rules, but it. 
 would probably be much more valuable to those who try to teach that study 
 without introducing demonstrations of the rules. 
 
 H. A. Newton, 
 Yale College, Feb. 6th 1872. Prof, of Math, in Y. College. 
 
 No. 13567. Subj. 995. Ref. 20814. Department of Public Works, 
 
 Ottawa, Feby. 7th 1872. 
 
 Sir, — In reply to your letter of the 26 ulto., I am directed by the Minis- 
 ter to request you to furnish the Department with one of your "Tableau 
 Stereometrique " at the price of Fifty dollars, together with your account 
 for the same. 
 
 I have the honor to be, Sir your obt. servant, 
 
 Che. Baillairge, Esq. F. Braun, 
 
 Architect, etc., Quebec. Secretary. 
 
TESTIMONIALS. XV 
 
 New Haven, Feb, 7th 1872. 
 Cus. Baillaiiig'^;, Esq., 
 
 Bear Sir, — I have been much interested in looking over tho papers 
 descriptive of your usefiil, vfvluable and (as it very plainly appears) univer- 
 sal ai)plication ol' a rule for tlie mensuration of polids. I sincerely congra- 
 i. -'nte you on the success which your discovery has met with in all quarters 
 in whicii IL hits at present been introduced. It mupt have been a great 
 labour to work it out to its present state of perfection and you have the 
 satif^faction of knowing that you cktq a benefactor and staunch pilot in that 
 sea of difficulty : Geometry. 
 
 Yours very sincerely, 
 
 E. B. Barber. 
 
 High School, Quebec, 7th Feb. 1782. 
 Cns. Batllairg^, Esq., 
 
 My Dear Sir, — I beg to acknowledge with many thanks the receipt of a 
 
 number of papers exjtlanatory of your new formula for finding the contents 
 
 of solid bodies. 
 
 The rule is precise and simple, and being applicable to almost any 
 variety of solid, will greatly shoiten the processes of calculation. I have 
 proved its accuracy as applied to several bodies. 
 
 Tlie Tableau compi sing a great variety oi elementary models will 
 serve admirably to educate the eye and must greatly facilitate the study of 
 solid mensuration. 
 
 The Government would confer a boon on schools of the middle and 
 higher class by affording access to so suggestive a collection. 
 
 I have the honor to be, my dear sir, your obedient servant, 
 
 D. WlLKIE, 
 
 Rector. 
 
 Votre travail est d'une utilite superieure. Votre 
 
 formule est destinee, ce me semble, a siniplifier dc beaucoup les operations 
 dans le toise des corps, et a rendre, par li, des services signales'a I'ensei- 
 gnement comme i I'application de cette partie importante des math6- 
 matiques. 
 
 Aussi mofl plus grand desir est-il de voir adopter votre formule et votre 
 tableau par nos Maisons d'education. 
 
 En finissant, j'ai I'honneur de vous informer que nous adopteroua votre 
 syeterae, comme partie de notre enseigncment. 
 
 J. C. Caisse, Ptre., 
 Prefet des Etudes au College de L'Assoruptioa. 
 College T-'Assomption, 27 Janvier 1872. 
 
XVI 
 
 THB HTEREOMETRICAL TABLEAP. 
 
 1 'i 
 'I 
 
 K! 
 
 From Formula} evolved by the Calculus, I find tha 
 
 you'»' Theorem holds ^ and " voi. all solws produced by the REvoi.uTioif 
 "o/'a«y Si RAiGHT LINED FiQUKE, or oi any curved lined figure of the 
 ** second degree, around ant axis within or without tlie figure in either case." 
 Ab to its applicntinn to aV. regular Polyhedrons, you give in your Treatit-e 
 a clear Demonstration of ihe faoir I bc»'.e reason, moreovt;r, to suspect that 
 your Theoreir holds good, with matheniaticai exa'jUiCPs, in more cases than 
 you give it credit for. I hope soon to be able to present you with an analytical' 
 Demonstration of your theorem as applied to solids of Revolution. 
 
 Needless to say that your Stereometrical Tableau should be in use in 
 every school where Mensuration is taught within the Domini jn. 
 
 In conclusion, let me congratulate you on the highly remarkable, and 
 deeply useful discovery you have made. 
 
 Quebec, 26 deer. 1871. J. O'Farrell. 
 
 II n'y a pas besoin d'une longue dissertation pour faire voir quelle im- 
 mense ntilite ofFre un pareil tableau. Tons les professeurs qui ont enseigne 
 la Giometrie dans I'espace et la Giometrie descriptive savent que, dans 
 les classes lea mieux composeea, il y a toujours un certain nombre d'eleves, 
 tres intelligents d'ailleurs, qui eprouvent dea difficultes oouvent insunnon- 
 tables, a s'imaginer exactement, d'apres des lignes tracees sur un tableau 
 noir ou flur le papier, la furine exacte d'un solide. Le tableau de M. B. 
 eupprime totalement ces diificultes. Quand on voit il faut bien croire, et 
 dans toutes lea classes ou I'on emploiera ce tableau, tons les eleves seront A 
 m£me de suivre aisement et de coniprendre les explications du professeur. . 
 
 II serait trop long de detailler ici les avantages qu'oflTre le meme tableau 
 pour calculer le volume d'un corps quelconque. Ce corps fut-il de la forme 
 la plus bizarre, on trouvera, dans le tableau de M. B., une ou plusieura 
 figures qui representent approximativement ce corps ou les parties dans les- 
 quelles on peut la decomposer, et il deviendra des lors facile de calculer, du 
 moins avec une erreur tres faible et inappreciable dans la pratique, le vo- 
 lume d'un corps de forme quelconque. 
 
 Ottavra, le 27 x bre 1871 E. B. de St. Aubin, 
 
 (Extrait du Courrier du Canada, ler Octobre 1873.) 
 Nous apprenons avec plaisir qu'une nouvelle medaille d'honneur vient 
 d'ltre decernee ^ un canadien par une societe fran9aise. M. Chs. Baillairge 
 de cette > ille, a re^u cet honneur de la " Societe pour la generalisation de 
 1 'instruction en France, " et il est en meme temps nomine membre hono- 
 raire de cette societe. Nos felicitations d M. Baillairge qui par sou travail 
 coatribue 4 taire connaitre son pays d I'etrauger, 
 
TK8TIMONIAL8. XVII 
 
 "^ )reometrical Tableau.'^ — In a large number of schools io Germany 
 the puj are taught, from the tiin'^ they commence the alphabet^ to judge 
 of color . nd geometrical form by the rej/n'ai use and comijariHon of colored 
 slips and small blocks of wood representiiij.' ihe flriiuntary p'iiieiploH which 
 will, in after years, be called into stufiy. The advantages of litis early 
 training are very manifest, as all are aware of the superiority of this nationa- 
 lity in that branch of science. An exceedingly ingenius device i'm the study 
 of forms has been invei;::ed by Mr. C. 13ailiair';e, a native of Quebec, and 
 patented in the Ijnite'i States, Canada, and Europe. It consists jf a board 
 about five feet long, and three feet wide, on which are placed some two 
 hundred models, comprising, so to say, all the elementary forms, their 
 segments and sections and numerous other solids, both simple and compound. 
 The instruction conveyed by this tableau, appealing as it does to the une- 
 ducated eye and mind, is, the inventor thinks, destined to be of great use la 
 developing the intelligence of the beginner and the untaught masses of 
 mankind, M. Baillairge is in possession of a mass of printed testimonials 
 from high officials and otlier distinguished men in Canada and Europer 
 together with reports from educational institutions, all highly complimentary 
 to him and the invention. A specimen of this stereometrieal tableau may 
 be seen at No. 7 Park street, of this city, in the possession of Mr. S. W. 
 Townsend I and we would recommend our teachers to call and inspect it, 
 believing as we do that this method of training should be considered as & 
 subject of great importance. — Hamilton Daily Spectator 19 Sept. 1873. 
 
 baillairoe's stereometrical tableal. 
 Our engraving is a perspective view of the above named educational 
 device, which has been patented for its inventor, Mr. C. Baillairge, of 
 Quebec, in the United States, Canada and Europe. It consists of a board, 
 about live feet long and three feet wide, with some two hundred wooden mo- 
 dels, comprising, so to say, all the elementary forms, their segments, and 
 sections, and numerous other solids, simple and compound. 
 
 The tableau is set in an appropriate frame, with glass covering, so a8 
 to exhibit the models while excluding the dust. The front can be opened at 
 pleasure so as to afford access to the models, each of which is merely sup- 
 ported on the board by a round nail or wire, which admits of its easy removal 
 and replacement by teacher or pupil. The instruction conveyed by this 
 tableau, appealling, as it does, to the uneducated eye and mind, is, the in- 
 ventor thinks, destined to be of great use in developing 'he intelligence of 
 the untaught masses of mankind. He expects to introduce it into all the 
 educational institutions of the United States and elsewhere, as it is now 
 being disseminated in Caaada ; and he has no doubt that the tableau will 
 
'I '' ' 
 I 
 
 t , 
 
 d 
 
 XVm THE STEREOMETRICAL TABLEAU. 
 
 also find its i)lace in the stiulio of the engineer and architect, to whom the 
 i iiiodels will be 8ug;:estive of various forms and relative proportions which 
 
 cannot fail to aid them in their piirmiits. The rapid success attained by a 
 f, pchool in Quebec, in mensuration of all kinds of surfaces and yet higher 
 
 mathematics, including conic sections, was attributed to the use of this 
 tableau. Every tableau is inscribed with a rule for finding the solid con- 
 tents of any body, called " the prismoidal formula." This formula has 
 been shown, by Mr. IJaillairge in his treatise on geometry and mensuration 
 p\iblishe 1 in 18G(j, to be Uas restrictive than supposed, and he has added to 
 the known solids, measurable thereby, a long list of ^Jiers discovered by 
 him, the whole of which are given in the tableau. Each tableau is also 
 accompanied by a printed treatise, explanatory of every use to which the 
 models can be put. Mr. Baillairge is in possession of a mass of testimonials, 
 from high officials and other distinguished men, both in Canada and Europe 
 together with reports of various educational and other institutions, all liighly 
 complimentary to him and his invention. 
 
 Dr. Wilkie, of Quebec, thinks " the government would confer a boon 
 on schools of the middle and higher classes by affording access to so sugges- 
 tive a collection ; " and Professor N«iwion, of Yale College, considers the 
 tableau "of great use for showing the variety and extent of applications of 
 the prismoidal formula. " — Scientific American June 1st 1872. 
 
 1^; 
 
 Worcester Free Institute. 
 
 IVorcester, Mass., 24 July, 1873- 
 This is to certify that 1 have carefullv examined Baillairge's models as 
 applied by him to the teaching of mensuration by the prismoidal formula 
 and I consider them eminently useful in all schools where mensuration is 
 
 taught. 
 
 (Signed,) C. 0. TnoMPSON, 
 
 Principal Wr. Free Inst. 
 
 — Geometry, Mensuration and St er comet rical Tableau, by Coaules 
 Baillaiuge, civil engineer, &c. ; Middkton et Dawson, edileur, Quebec, 
 L872. 
 
 M. E. Blain de St. Aubin, doune I'appreciation suivante du travail de 
 M. Baillairge: 
 
 — On salt quelle sfirie interminable de regies ou formules, dont plu- 
 eieurs trcs-compliquees, les anciens traites de geometric donnent pour le 
 mesurage des solides. M. B. n'en aqu'une qu'il enonce comme suit et 
 denioutre clairement etre applicable H toute espece de solides, si bizarres 
 
TESTIMONIALS. XIX 
 
 quo puisspnt etre leurs fornica, — " A hi somineileH stirfacc^ doR hafcn pftral- 
 leli'9 ilii suliiie uevaluer, ujuiiter 4 (WiH hi siirl'iice au ci'iitrc vl imiltiiiliiT le 
 tout pur la sixieme pariie de la liaiilcur on longueur du solide." 
 
 C'est dans le but do populariner ruHugo de cotte re;;le que M. R. a ou 
 rccourrt i son Tableau Stcrcu^tctrhjuc. ^' Ct.' tahlr.an, dit M. Ihiillairi^e, 
 est un cadre oil .-out pliicen environ 200 nioilcioH diHercnts do Holidusi 
 chaque niodcle pent etre deplace li volonte, en .sorte qu'on pout le nicttre 
 entre lea mains de I'eleve pour (ju'il I'cxaniino. Le t.ibleau coinprcnd 
 ioutcs lew f'ornics cleincntaires inuigihibles de solides, depui.s le prisnie or- 
 dinaire jiisqii'au cone concave, cic , etc " Sur 
 
 chacpie inodele, — dit jiius loin M. H. — est tracee une lii.^ne qui indii|ue la 
 nature et les diniensiona de la section du milieu " 
 
 On con9oit ais6nient les avantagea que presente I'emploi du ce Tableau. 
 L'eleve doit apprendre en fort pen de tenqys hi maidere d'applicpuT hire- 
 ment I'unique Ibrniule, eiionceu tout a i'hcnrc, au calcul du volume de 
 cliacun des 200 rtoliiie.s contenuH dans le Tableau; ct. phu turd, dans la 
 pratique, il s'habituera vite a decomposer un solide quelconqu" en parlies 
 se rapprochant, par la forme, des modeles qu'il a ainsi etudiea. 
 
 Quant aux solides de formes comparativement regulieres, tels que 
 pieces debois, blocs de marbre ou de pierre, reservoirs et cliaudieres dans 
 les usines A vapeur, les distilleries, etc., I'appiication de la foiinule de M. 
 B. offre des facilites et des avantages qui deflent tout concurrence, et nul 
 doute qu'elle se repandra universellement au grand avantage de tousles 
 praticiens. Telle est, du reste, la prediction que n'ont point liesite a fuire 
 plusieurs savants etrangcrs qui ont eu connaissance de la decouverte de M. 
 Baiilai»'ge; et nos meilleurs professeurs canadiens sont tous du meme 
 
 avis 
 
 — Journal de I' Instruction Puplique, Nov. 1S73. 
 
 Je me reserve de revenir bientot eur le tableau stereometrique de M, 
 Bailie xge, ou nouveau systeme de toiser tous les corps, segments, troncs 
 et onglets de ces corps par une eeule et meme regie. Ce travail, d'une im- 
 portance majeure, comme tout ce qu'a produit M. Baillairge, est marque 
 au cachet de I'utilito pratique. C'est un esprit progressiste, un jugcment 
 rare et une forte conception, qui out preside a son execution. En le cou- 
 ronnant d'un premier prix et d'un diplome, le jure a su rendre ten)oignage 
 au merite, et nul iloute que nos colleges et autres in: titutions de premiere 
 classe ratifieront cette appreciation en I'introduisantdans leurs classes pour 
 aiuer I'enseignenieat arda des matheiaatiques. — ^Opinion Publique. 
 
 Sept. A. N. MONTPKTIT. 
 
XX THE STEREOMETitlOAL TABLRATI. 
 
 No. 2272-71. MiniaUude V Inatvuctinn Publique. 
 
 Qi'KBKc, cc S('i)t. 1872. 
 C. lUiM.AiiuJK, Ecuycr, Quebec. 
 
 Moiif^ifur. — .I'ui I'lioiiiicur ile vous transmettrc, Kur rnutrc fouillct, 
 
 copie lie la resolution u<lopteo p.-ir le CoiiHoil de 1' Instruction Pul»li(iue, up- 
 
 prouvant voire " Tableau etoreonietrique pour toiser tou8 les corps, seg- 
 
 I inentf, tronc.i et oiiglets ile ct's corps," ainsi jue votre " Nouveau traito de 
 
 geometric ct ile tri!,'onon)6trie rectiiignr et (^plierique, suivi dii " Toi.-6 deg 
 
 surfaces et des volumes." 
 
 Loi'i8 GiAKD. Secretaire-Arcliiviste. 
 
 Journal de l' Instruction Publiqne, Nuv. 1872. 
 
 TABLEAU BT£uKOMi:TIlIQl'E VE M. BAIM.AIUGE. 
 
 Nous avons deji eu occasion de purler dc ce tableau, etde I'impulsion 
 extrpordinaire qu'il doit donner I'etude du toise. L'auteur a, depuia ob- 
 tenu les certilicats les plus flutfeurs de tons les hommes competents sur 
 c«'*'e matiere. Le tableau, usee la lurniule (pii i'accompagne, est appele, 
 au dire de tons, ^ taire une veritable revolution dans les methodes de me- 
 Burage pour les solidej. Le conseil do rinstruction publique, 4 sa derniere 
 eeunce I'a approuve, avec le " Traite de geometric" du nienie auteur. Ce 
 tableau, de cinq pieds par trois contient deux cents modeies en bois, com- 
 prenant toutos lea varietia de formes, depuia lea corps lea plna simples 
 jusqu'aux corps les plus bizarres et les plus ditficiles i toiser. Ces modeles 
 eont mobiles et ne sont fixes au tableau que par une petite tige en f'er, de 
 sorte (pie les elevea peuvent les examiner et les etudier de main en main. 
 L'auteur espere que son oeuvre, tout en simpliflani et en facilitant les 
 calcula du savant, aura surtout pour resultat de meltre A la portee do tous, 
 une science demeuree jusqu'ici, par ses dilHcultes presque insurmontables^ 
 en dehors des atteintes du plus grand nombre. Tousles colleges et les 
 ^colea trouveront dans le "Tableau etereometrique " un puissant moyea 
 de progres surs et rapides. 
 
 Nous en publions ci-dessous une gravure, et nous renvoyons le lecteur, 
 pour de plus amplea details, a notre bulletin bibliographique, ou uft 
 homnie expert en cette matiere, M. Blain de St. Aubin, en fait une excel- 
 lente appreciation, dans le conipte-rendu qu'il donne d'une conference lue 
 par M. Baillairge, devant la Societe historique de Quebec. 
 
 Extrait du Courrier du Canada du 20 Aout 1873. 
 
 TABLEAU STEREOMETBIQUK UE CHS. BAILLAIRGE, ECUYER. 
 
 Monsieur le Redacteur, 
 
 M'accorderiez vous dans lea colonnes de votre bienveillant journal), 
 quelques lignes touclianl un object bien paisible, malgre que nous eoyiona 
 dans un temps de grande agitation ? Je desire appeler I'attention du 
 
 
TESTIMONtAf-S. XXI 
 
 •jouvcnicmont loc.il sur iiii siijct qui conccrno l'i'iHeij,'notncnt piililic : jo 
 viMix |);irli'i' ilii tahlcim st<'!rc.iin6trii|ne ile M. Clis. l)uillaii''_'6 it cxiiriiiicr 
 ijKiii limiililc (i|)iiniiii sur Ic.i lo-iillutH iiraiiiiuec (jue jh'iU iivuir cello eluJo 
 jiiiiir los jiri)|'e-sciir.-< il K's cloven. 
 
 Si I'liii I'll ju'jc |i:ir ic-i (eiii(ji;;nnj?p« partuiit DJitciiUM, ce tableau est 
 • ie.^tiiie a uiu' ^rriiiitie reiioiiiinec. D'aillciirs ci'tte reimmirieu N'cten I au- 
 jijiirii'liiii iiDii-Henlciiu'iit dans ics iliilerciite.s Provinces Je iu l'ui.'<Hanco, 
 mairt mcMie mix Ktuts-L'nis, en 1mum|h', en France Hurtuut. t'oii cunt 
 vonus ii M. Iiaillairj,'e les eloge.s et les recuminaiiJation;^ des plus iiautes 
 uiuurite.s CDinpcleiites. 
 
 riii^ipur.s il( niis niaisonf (1*6 liication I'ont mis en nratique, et en ont 
 'iitcnti (les re<inltat:i ((iii depussiMit tuiUes previsions. Les datncs Ursulines 
 (ie (jneliec, les pn-mieres, ont mis le taiilcau et la f'rmnio de M. Baillairge 
 eii usage, awe taut de bonlieiir, ([iie lunles les aiitris ^.'nniniunantes reli- 
 gieii.-es (lilt decide de suivre cet exeiiiplc. coiivainciU's (pi'elles sunt de 
 rmilile et dos grands avantages (piViU're ce precieiix travail de notre dis- 
 tingue iiiatlieiiiaticieii. Outre les Reveivndes dames Ursulines, les lleve- 
 retides sduirs de la Congregation de Si. Hucli, les s(Biir8 de la Charite, cellea 
 dn lion Pasteur, et le > dames du couvent de Jesus-Marie sur le cliemin du 
 ('a[) Rouge, ont toutes (lc>;ide que leurs eleves seraient iiiierrogees sur ce 
 lal.leau anx prochaitis cxamcns. 
 
 Un de mcs amis, in-titutcur diplome d'Ecole academiquc, nc craint 
 pas d'allinner (pi'ii I'aidc de la t'oriiiule et du tableau stereometri(iue, il 
 f'erait coinpreii liv en qui'l((ues le(jons seulcinent, a un eleve, ayaiit 
 lies aptitudes (jnlinaires, les toise de toutes espeees de aolides avec plus de 
 succes qu'il ne I'a jamais pu obtenir, en un an, et meme en dixdiuit mois, 
 avec les lurmules t^uivies jusqu'a present. " C'est si simple, si clair, dit-il, 
 que 9a saute aux yeux meme des enfants." 
 
 Quant ^ moi, M. le Redactcur, lie d'amitie avec plusieurs instituteurs 
 possedant leurs diplonies d'Et:ole Modele, je connais leur opinion et sais 
 leur desir d'enseigner, p<vr le inoyen i.le cette nomenclature. Tous s'aocor- 
 dent a dire que Touvrage de M. J^aillairge repandra necessairement logout 
 des matliematiques dans cette province. Ce sorait done un bien grand 
 service a reiidre au pays, a la jeuuesse canadienne, que de distribiier cet 
 ouvrage reconnu d'une sigrande importance. 
 
 Comine la plupart des instituteurs sont peu renumeres et quo leurti 
 ressources pecuniaires eont tres limitees, nous ne pouvons meme pas desi- 
 rer que ces homines de sacrilice aclietent, de leurs propres deniers, ua 
 tableau qui, par les etudes e*. le temps qu'il a coutes, est audessusde leurs 
 moyens. 
 
 Deja le gouvernement du Nouveau-Brunswick a repandu ce tableau 
 dans toutes les ecoles de cette province. L'Honorable M. Cliauveau, mi- 
 
f 
 
 I'l 
 
 hi 
 
 
 XXII THE 8TERE0METR1CAL TABLEAU. 
 
 nistre de I'lnstriJCtion Pnbliqne, en en reconnaissant toiite la valour, s'etait 
 engage envera I'autciir a le f'aire repandro dans nos eooles ot no-i nmisons 
 d'education, et le bureau dea Travaux Pulilic'< en a distriime un grand 
 nombre d'exeinplairea, pour guider sea employes dans tons lea ineau- 
 rages. 
 
 II eat done A eaperer que le gouvernenicnt local prondra en conside- 
 ration le desir unaniine dea honimes qui s'occupent de I'instruction pubil- 
 que, et pourvoiera liberalement les instituteurs d'ecolea acadeniiquea et 
 d'ecoles tnodeles de ce tableau ai precioux et destine a toule une heureuse 
 revolution dans renseignenient dea sciences matheinatiquea. Ce sera en 
 nienie tenipa un bienfail pour la jeuneaae studieuse et un honneur pour 
 notre gouvernement, qui le premier aura encourage ce mode d'enseigncment 
 simple, pratique, et fructueux. 
 
 Un ami de 1,'educatiox et du proores. 
 
 {Extrait d'une letire de Aug. Humbert d V. Vannier.) 
 
 Paris, le 4 Aout 1872. 
 
 J'ai faitrecevoir M. Baillairgfe Membre Tilulaire de la Societe de Vul- 
 garisation pour I'enaeignement du Peuple. 
 
 A defaut de son tableau que nous n'avonapas, j'en ai expose la photo- 
 graphie a Paris et a Lyon. 
 
 Son tableau appelle tous les regards d I'exposition, on m'interroge, on 
 me demande des explications que j'ai donneea de mon mieux, et cela in- 
 teresse beaucoup. J'ai done tout lieu de compter aur un aucces. 
 
 Extrait d'un Journal de Paris du 16 Aout 1873. 
 
 TABLEAU STEREOMETRIQUE 
 
 Nouveau systeme de toiser tons les corps, queltes que snient leurs Jnrmes 
 par une seule et meme regie : par C. BA/LLAlIiGE, Archiiecte, etc., 
 
 a Quebec {Canada.) 
 
 M. C. Baillairge, architecte du gouvernement a Quebec (Canada), 
 nous a adresse un tableau qu'il appelle stereomctrique. 
 
 Ce tableau, espece d'armoire, contientdeux cents petits aolidea en bois 
 affectant deux cents formes ditterentes, spheres, demi-spherea, segments 
 c6nes, tronca de cdnes, pyramidea, tronea de pyramides, polyeirea les 
 plus varies, onglets, etc. ;— enfin figurant toutes lea formes elementaires de 
 Bolides quon peut rencontrer dans les arts, la construction et la nature. 
 
TKRTIMONIALS. XXIII 
 
 « 
 
 M. C. oaillair;;^ dutiiu', inscrite on tete do ce tahloau, ])onr trouver 
 
 le volume il'iin de ces deux cents corp.'*, une fornmle qui poiirra s'appli- 
 quer A mi ct)ri)s qnelconque de la nature, cur son V()lmne ponrra toiiJDurs 
 se decomposer en un de ceux conipria dans le tableau ou s'on rapprocher 
 iidiniment. 
 
 Void cette regie : 
 
 A la somnie desf .surfaces des exlremitcs paralleles ajouler quatre 
 fois la surfdcc an centre, et multiplier le tout par la sixieme parlie. de 
 la hauteur ou longueur du soiide. 
 
 Cette formule est evidemment d'une simplicite extreme et abregera 
 considerablement les calculi jjour les liomnies qui s'occipent npecialeinent 
 de niesurage et de inetrag", — tels que les toiseurs-verificateurs, les mesu- 
 rcurH, jangeurs dus contributions imiirecten et des douanes, les arpenteurs, 
 geometrcs, coMfstrncleurs, arcliitectes et ingenieurs. 
 
 Quant aux eleves des inaisons d'education, des ecoles d'arts-et-metiers, 
 il est bon de ler.r I'aire cuntiai.re une niethotle abregee et se rapprochant 
 infininient de I'exactiuide. ilais nous pensons qu'il convient do leur en- 
 seigner comment on arrive par les donnees de la science a formuler cette 
 regie universelle. 
 
 Si Ton doit eviter de donner trop de temps aux etudes theoriques, il 
 faut bien prendre garde a I'exces contrairo. Le jeune hoinme dont toute 
 la Gcience consiste en une menioire bien farcie de formales dont il ne sail 
 pas retrouver les elemetits' est toujours un praticien bic embarraese, lora- 
 que les conditions du travail qu'il a a remplir sortent des donnees or- 
 dinaires. 
 
 Ce sera un bon caboteur, ei vous le vo..lez, qui manoeuvre bien oon 
 navire en vue des cotes ; mais, une fois en pleine mer, il sera desoriente, 
 ne sachant ni faire le point, ni determiner ga position par le calcul. 
 
 y. Baillairge, du resie, parait etre completement de notre avis, car, 
 anteridureiient an tableau dont nous parlons, il a public un excellent 
 ouvrage sur la geometrie la trigonometrie, dans lequel il a developpe 
 une foule de questions pratiques, de theoremea etde fornmles remarquables 
 par leur nouveaute, parnii lesquelles se trouve celle que nous venons d'ex- 
 poser concernant la mesure des solides de forme quelconque. 
 
 Cet ouvrage a attire a M. Baillairge les eloges mentis des hommeB 
 competents du Canada. 
 
 Nous regrettons vivement que I'ouvrage n'ait pas accompagne le ta- 
 bleau qu'on nous a adrcsse ; le compte-rendu que nous avons entrepris y 
 aurait iTai'ne en interet ct en clarto. Reduits aux elements ordinaires de 
 la science, nous soiiimes entraines d des calculs trop longs, et probable- 
 inent par des seutiers mujns directs que ceux iudiques par I'auteur, pour 
 
XXIV THE STEREOMKTRICAL TAIU-EATT. 
 
 ■i 
 
 'i 
 
 que iiou- vous invitioriH A verifier avt'c nuns I'exactitiuie de la regit' <l(jnnee 
 piir M. I]ii',liair^'e. A conx a qui la trijrniKiinetrif cf la genn.etrif df.-crip 
 ' livi' .-out fainiliert'.", iioiis (.liroii.-i : ]• aitt-s cuinim' noii.s, chireht z et vtnis 
 
 tri.iuvcvi'z. — A cjiix qui travaiilfiii iiraiitiiuincnt, uous coiit'eilleruiis tl'ap- 
 
 pliqiier la regie que nuus indi(iuoiis a un suliile (luelcoiique, duiil ils f-aveiit 
 
 ' ; liiesurcr le vclumc [)ar une autre inetliode, et ils .<e r( ndnml cuinpte de 
 
 I I't'Xactitudk' lie la liiriuule j)ar la siiiulitude dcs deux resultiits; il'uii lis 
 
 l • tonclurtint que ce qui est vrai pour uti cas le sera i)our I'autre. 
 
 I Oil recounaitia ainsitjue la loriuuie est iiiatlieiiiati(iui'nieiit cxaetc, pmir 
 
 i les prismes el prisinoides, cyliiuireset cvliiulruides, droits ou ineiiiies, pyra- 
 
 ' nudes regulieres, et irrcgulieres, et les troncs de ces solides eiitre baserf 
 
 j paralleles, pour le cone droit ou oblitpie et sou tronc ciitre bases paralleles, 
 
 pour la sphere ou le spheroide et tout segment ou iroiic de ce corps t-epare 
 *■, ' du solide eiitierparun plan incline d'unc inaniere queloouque aux axes 
 
 ou diametres, ou coinpris entre deux i)lans parallele.'^ (pielcoiniues, pmir le 
 cou< ides paraboliques et liyperboiiqiies droits ou iucliiics et les tri ncs t 
 ces solides coinpris enlre plans paralleles; et ces divers solides ccin;'liiuent 
 dana leur ensemble la presque totalite des solides elemeMaires que Tun 
 puis?e etre appele a evalner. 
 
 II n"y a que pour les ftiseaux seiils et les on;r!ets (pie la fornmle n'est 
 paw matliematiquemeut exactc, mais encore se rapproche t-on sensiblement 
 de la veritc. 
 
 Du reste, il est toujours facile, quand un corps affecte une forino par 
 trop fantai-iftte, de le decumposer en jilusieurs ,-olides se ra{)|)nicliant des 
 formes plus geoiiietriques cl auxquels on pent appliqiier la Ibrmule en 
 toute securite. Le tableau construit par M. ]5aillairue doiine i)recisemeiit 
 les formes lea plus diverses ipii ijermettent, en rapprocliant les dilferents 
 Kolides, il'en construire de nouveaux de furines composees. On fiiit ainsi 
 I'analyse et la synthese de la lyetilode, il'iine part, en appliquant la me- 
 tliode au volume total compose des dillerents solides; de Tautre, en calcu- 
 lant le volume de cliaque morceau separe et en additionnant les resultats; 
 I'addition de cette scconde maniere d'ojierer doit donner le meme chiflVe 
 que I'aiJplicalion de la regie au volume total. 
 
 Est-il necessaire d"indi(iuer les applications qu'on pent faire de cette 
 meiliode tlans la pratique? Cela nous parait inutile, nous remplirions 
 toutes les colonnes de la Revue ; les liommes pratiques sauront bieii le mo- 
 ment de e'en servir. 
 
 Quant aux eleves, je laisse aux profesyeurs le soin de Kur faire tou- 
 cher dw doigt ces applications, multiples et si journalieres. 
 
 Pour me resumer, je dirai que la methode de M. C. liaillairge est 
 d'une eimplicit6 et d'une exactitude remarquablcs, qu'elle est appelee d 
 rendre de trus-gruiids services aux praticieus, ea leur evilant des pertea 
 
 ■ 
 
TESTIMONIALS. 
 
 XXV 
 
 -uti-iiienil.l.- .U' t.-miH d en l.ur ,H_.nn.'itant a'nl.tiMiir K-s result it. clurcho-' 
 
 aiicifiiiu's t'lriiiulcs iiKiiiiem ihiiu"- : <; c-i 
 
 III' III I Mil |l. 
 ,]. Ml) lAM). 
 
 aii--i 'XI' I' iiiriii qMe I'ur 
 
 ,K,MCuna.v,.,i- i..i.rt.u- .10 i.ruiMg.r Cf sy.ieine: K |. .ur n. . mi, |..ai.-ir 
 
 (i'tii feliciter I'tuiuiir. 
 
 
 Pakis, K' Icr A. ut ls72. 
 
 4^>^^ 
 
 ' % 
 
 
 -1 
 
 / 
 
 
 SECRKTAKTAT 1':NE1UL 
 
 iu:k TKrKK.UT, It-, 
 
 AFFUANCIIIR 
 
 A 
 
 i'I{i:sii)i:ni.s ii()N<ii:aii!i;.s. 
 Al>.. ,, . 
 
 Ildiioro Ariioul. dfifi'i- iV Ariidrmw 
 J'ri'xiiind (If III Swiiic lUnr. il tn- 
 slrili-finil rl il'i'ihirilliiin imliilhl irif. 
 I.Duis VxiiMvyMiil'in- ih I'aiiufi' scim- 
 liii<lii,-. hi krre armil If ili'liiije. ''<'•. 
 <•/'■. , , 
 
 (.'iiiiiili' Fliinim.inim. ""'""■ ''''"■ 
 j}liiridili' lies moiiilc-1 Imhilen, rff.< Mcr- 
 V'illfs 11 li'itf!!. '''■•• '■'''• 
 J,fi(iii 1,1 bun, Clii Tilii nn-i-iri' (Irliii- 
 structioujiriiiMiif en Urliji(i>ic. 
 
 Va'IM'KKSlDKNT.S HON'- 
 MM. , ., ,-^ 
 
 A do liiinpi'^. I'lidnir rn <lrf»t. ill- 
 ficitr il'iii 'I'mle. ineiiihri' /)/v;/i:ssi.h/' 
 'dc /'((.■.•.-■'<■ .1"" )>liilchcliiii'liii'- 
 11. CIuim';, uii'mhir liliilaiiv (''- hi 
 Si«-iil( irmili-cii'ihKiii' ih- J'lin'.i. Hi- 
 <l(it!''>ir-/<iii<l:ilmr iln la Jiti'uc liii- 
 
 iniUliiiiii ■ 
 
 VUKSIDKNT. 
 
 M. I'ranz-MiLi'stli'. Anlu'li-clf, Vefi- 
 liailmr I'l hi /.t'l/ioii irhiniiifiii: nirni- 
 'lire <l II Ciiiixt il .iiijiiri' iir ih hi Sm-iiU: 
 li\ii-e iVinftnicliiiii ft iVi-'hii-iiliiiii po- 
 puhiiri'i. 
 
 sr.cuKrAiiii; cknkk.vt.. 
 
 JI. AuKUsti. lIiiMilx'i''. tifvntuirr HP,- 
 nerol </'■ hi .vii-ii'li' Hhrr iriiistnicliun 
 et irnhiiuiliiiii iii'imhiifs. 
 
 i i;ks(ikii;i{ 
 
 JI. J. I'diitaiui', iiiihiir ih- hi comp- 
 tabiliti uptnise cit '2 lieiirig. 
 
 Meiubres tituhiiri'S. 
 
 F. 
 
 Kroits il' •iitrfc 1 
 
 Cutisuti'ju uiuiUi-'Uu ^ 
 
 Muii.-ifiir l^ail!:iirL:e, 
 
 Architocte. otc. 
 A Quebec, i,^ aiuuta;. 
 
 Muii?icur. 
 
 J'ai riiomiour lie vouj< don er avi.s (jiie 
 Ic Council Snpericur \ lent do \ , us admettre 
 ii liiirc partie do la Souiete do Vulgaiisatioii 
 pour rKu.«ei;.:;ueiucMit du pcuple ii litre de 
 Meiubro Titulaire. 
 
 Nous somuiesliourcux il'une deci.sion qui 
 assure a noire CEuvre voire precieux con- 
 coui'.s et nous p^perons les nioilleurs etlcts 
 de voire propugande active en laveur de 
 rinstruction et de 1' Education populaires. 
 Veuillez agreer, 
 Monsieur et tres honore Collegue, 
 Tassurance de nics sentiments de haute con- 
 sideration. 
 
 Le Secretaire General londateur, 
 
 (Signe,) Alg. liuiiBtuT. 
 
!||if 
 
 XXVI THE STEREOMETRICAL TABLEAU. 
 
 Eutract from a Idler of Urnlhrr litn^ilian of St. Thomas, to liroi/wr 
 Paidiiin ofl/te Lasullc hi.<litute, Sew York. 
 
 Mr. l!iiill;\irj;c's sysK'in liiis alM) rt'Cfiveil 
 
 tlio gciKTul approliatuiii of tin- principiil lluiiscs of EJucutioii iii^tlio Duiiii- 
 nioii of Caiiaila, und iti foreign liiiul. It surpasses, by its facility and its 
 eimplicily, all that ever appeared of the hko on the continent. 
 
 1 am very anxious to have you examine Mr. IJ's sy-^tini, as no one is 
 liotter alile to jiid,L'e of its merits than yourself, heiag, as every one knows, 
 so distingnisiied a matliematician. 
 
 E.iirail d'ltne lellre du 7'ey. Frcn; fiasilinn, cid( rant de Montrail, an 
 
 Frere Lii^otiri, Vi.^iti'.ur dc-i Frcres din Eadca Chrcticnues, a 
 
 Londres (Angleterre) et uu lieu. Frcre Patrick, a Paris. 
 
 , Mr. liaillairge est re9n menihrc 
 
 lionoraire de la Societe d' Education de France. Honneur que lui a inerite 
 son nonveau .systeme de stereometrie el pmir U'qucl il est mande a Paris, 
 par le I'resident de la Societe, pour y recevoir le premier prix de concours. 
 
 Ce systeme a aussi re^u rapprobaiion des pnncipaux etablissementa 
 d'ediication de la Puissance du Canada et il I'etranger ; il surpasse par .sou 
 extreme simplicite tout ce qui a jamais paru en ce genre, coinme vous 
 pourrez vous en convaincre par ce que v.>us en dira ce monsieur. 
 
 Je Huis tres aisc. Ires cher Frere visiteur, que vous preniez connais- 
 pance de ce systeme, par ce que je ne coniuiis personne qui puisse niieux 
 que vous apprecier le merite de cette nouvelle decouverte. 
 
 Ottawa, 17 Decembre 1873. 
 
 J'ai cnseigne, uuii-meme, depuis de 
 
 longues annees, cette f-pecialite d'evahiuiion des surfaces ct solides, et, vu 
 rexlreme complication que comporle ceite etuiie, j'applainlis vraiment au 
 mode somviaire que vous venez do imu reveler, lecjuel parait evideniment 
 destine a suiiplanter le systeme suivi jusqu'ici. 
 
 Sans elre revolutioiuiaire, j'aime ce-i pelits bouleverseinents tendant a 
 reformer certains regimes parfoi- trop conservateurs, lorsqne surtout cea 
 benins lutadysmes, n'ont d'aiure etfet (juc de vuigariser uue branche d'en- 
 seignement qui devrait etre accessible ii tous. 
 
 Voire devoue, 
 R6vd. 1'f ere des Ecoles Chretiennes. F. Andre. 
 
 IP 
 
TEST1.AI0NIAL8. XX VI 
 
 ( Tfiiimlnlinii). 
 
 St. Thomas, Di'ceinlicr, 1.j 1"^T;5. 
 Siu, 
 
 I ln'fT to acknowledfrc vtccipt ol 3onr Utter < ''tlip 4tli instant int;irniin<r 
 me tli;ittlie first prize liail been awanieii yuii, in Fiaiioe, tnr Vuur Stereu- 
 iiietncai Tableau. Tiiis T cnnsicU'r a hujipy presaire o( its I'utnre siicceH.s. 
 
 It pleasfs nie as ii must ail l^vt r- uf s ience, ili. t, notwitlisian.lin'j; the 
 preji'.ilices of tlie aj^e, you have lintiwn llo^v to trium|iii over tlie (iitiirulties 
 v.'liicii naturally presented tiiemselvi's in tlie reseaicii ol'lhe I'unilamental 
 principles of sucii a systeu). Iniiecd. witli the jirace ol" (JimI, vnur elliirls 
 will Sunn lie ernwue 1 with C'lmplete ^uccess ; and I hfjie vuu will be rendered 
 the gbiriuus testinmny of beinj^ a Hfiieliictur of tin Iiuman race, liv the 
 liisciivery of a new system whi<;h will jiive a I'rt.-h impetus to popular 
 education, and be the means ol' giving to the less eihu-ated an oppurtnniiy of 
 understanding its commercial value. It i.» to be hopc(l that the prumnter.4 
 ol science will recognise the immense benefit you are cont'erring on Society 
 at large and will render yi u full justice. 
 
 Had I remained in M ntreal, I should have labored to introduce 
 your system in our classes there, lint be assured, Sir, that very soon, 
 I hope, you will iuive the satistliction of seeing your new system inlr. duced 
 to tlie greater numlter of our institutions in Canada. It mu<t be ackinv- 
 ledged that until recently it was not much known, but owing to ytjur kindness 
 its reception in our institutions will soon make it so. 
 
 It camiot be otherwise than that, after .<o many testimonials have been 
 awarded yon from the principal educational establishements of this I'ldvince, 
 approving of the superiority uf y .. -y-tem — not only for its facililv but f>r 
 its simplicity — you will soon, I say. have the sati>factioii (jI seciii" it in n-e 
 in all educational institutions, even the uio.st elementary, tiironi'hout the 
 Province. 
 
 I herewith send yon a letter of introduction to our venerable Ihother 
 Patrick, in Paris, lately a visitor to the idiri-tian .-clmols in the I'nitcd 
 States, a nuin of great merit and a very great lover of Science. The sfcon.l 
 letter is for the Rev. Brother Li.'onti, vi'siior to Kmrland and resiiim ' in 
 London (.'rem h) and who recently vi.-it'd (.'aiiada. When jiere he saw voiir 
 work on (reomelry, ami pas.-ed high encoinnims mi it, [ have no doubt but 
 that he will most willingly adopt yonr sy-item. The third Ktter is fir the 
 Rev. Ikuther Paulian, visitor to Xew-York, a matheinatician olthe first (mier. 
 
 I stopped at Quebec with the intention of seeing you, but wa-i infirnnd 
 th:it yon were ont of Tnwn. i think that the llev. Uruthcr .\i)liraates 
 Diivclor of Quebec, who is so well known in France, will do himself tJu. 
 pleasure of giving you several letters of intr iduction. 
 
 I have the honor to le, Sir, Ytiur very humble Servant 
 
 (Signed) FiiEiiE Basiliav. 
 
? 
 
 
 XXVIII 
 
 THE STEREOMETKrCAL TAHLEAU. 
 
 (TniiLiliiliiiii). 
 
 Ktk. Axxk ]n: \.\ I'kuadk, I^.'mI Driciiil.cr l-T.'!. 
 
 Sir,— I liiivi' llio lioiior to ;ickii(i\vK'l.;i' ihi' receipt ol' yoiir iiit' lestiiii; 
 letter (pftlu- 22.1 iiistHiit. 
 
 Willi Vuii, I slioiilil very miicli like ti.i we ymir MereniiKtiie'iil 'I'ti'-leau 
 fisriiriiiir in iny Selinol, nii'l if tin- (rovermiicnt will Imt li.i'l a livl|)iii(j 
 liaiid tiicrc if* imt little liniiiit it will lie al"pi"il \>y all lur Acitli'iiies 
 aiui Moiji'l S.OiiK)!.-'. IJclieve me, Sir, m the I'licii 1 nf vuiiiii, tn whose 
 iiiscniction i have devoted myself for 28 years past, lui cxt.rtion will he 
 waiiliii'j; on iiy part, \<j secure for them the advaiitai^es to In- de-ired irum 
 A our iiiipon I'll iliscovcry. 
 
 I i.itlK •ize you to add my name to tliose petitioiiin;: the ITonnniiile The 
 MiMi>ter of J'ulijic Instruction to suhscrihe for your TaMiai; and intmliice 
 it it) our Sc.i nils, Acaih^mie^! and M"di'i ^rl,.' 1-. 1 lou nf i!;c opinion that 
 the yoiuli of tiic Country will ilci i vt' iircat au vanta;_'t' truiu it and that it will 
 contrilmle n t slijililly to di-eminatc amoiiL' tin' pei p!c a ta-l'> t'^r tiie 
 iiuehanical .'I'ls, a taste whieii, so to say, only l>.'.in-- t" lia\c txi.-!ence 
 umoiiii us. 
 
 I liave no <iciulil -ut lliat tic 2o\-. rnnicnt u '1 L'i\ i- ;i "U -uidi I'nciuiaiic- 
 tnint as will, at least in part, ind"niiiity Vnn i'r y.'iir '.'irai lu'-.r and ilu- 
 t'Xi)cnse> vnu have lieen at in th' advancenicnt cf n-e|nl seieiii- tlii'iiULiliiiUt 
 the Country. Tiie Ihitterinjr t( tiinnnial.- eonfei led np"n y^u hy !•' ranee ami 
 the Uniie 1 States ^In^iild iniipre>s up.>n i1i..m.' at ili" liead dlUnr .^I.or.- the 
 ju-^tie.e ot'ciiininj- III yi.iir aid ly a-kin:ol' ilie Lclt s aiiire i,i v. le ;i >iiin 
 wdi;oh w.iuld renninerale you f^ry'iir iiiv:i!uali!e lal'nr and reiinliiii-c yon 
 in tiie amount vou have expended. 
 
 I am ihei'eliire prt pare I t<i si^n any petiti..n i hai wiil lia\r V>f it- ulijecl 
 tlu' introduclicui ot'your excellent Tahleau in all our Sell' ml-, or of ih man- 
 dim' tif the l,e::i^lature to indi'iunify you tor yuur expendiinre and lali.ir. 
 
 T will \Nrite to Mr. lietiairneau jiir a eopvut'ihe pftitioii I ein.: si.Mied 
 and think 1 can assure you of ohtainiiiu' tlie siLMiatures df a j;""d m;iny 
 persons ot inthience in this locality. 
 
 With consl(U'ralion, \ remain, Sir 
 Clis. Baillairj^e Esq. Your devoted Servant 
 
 Architect &c. Quebec. (Si<.Mied) 1). N. St. t.'vii. 
 
 QfEBKC, Sept. 2Ttii is";!. 
 The un lersiirned who have witnessed the many advantages of the 
 Stereoincirical Tahleau as applied in the teaching' of (li'Mmetry and .Men- 
 suration, eel., to the pujiils ot' the (^nidiec Schoid of Arts and .M.innlaeiures. 
 Would reijomniend thai the .Montreal Sclmid he also provided wiili one 
 or more of these useful adjuncts. 
 
 (Signed,) Hon. I'L Ciiixic, 
 
 J. WooDI.KV, 
 
 Jj. J. JJoivix, 
 Jievd. U. At liET. 
 
 Member of the Board of Arts & Manntactures for the Province of Quebec. 
 
SYNOPTICAL OK ABRIDGED 
 
 KEY 
 
 TO TIIK AUTIIOU'S NKW SYSTEM 
 
 OF MEASURING ANY SOLID, 
 
 SEGMENT, FRUSTUM OR UNGULA OF SUCH SOLID. 
 ^ BY ONE AND THE SAME KULE. 
 
 (1.) To the fium of the areas of the opposite avd parallel ends or 
 Ixtses of the hoOij to he measured, add four times the area of a 
 section thereof parallel to these hases and eijitidistaiit from eaeh 
 of tlicm. and mnltiphj the whole hij the sixth part of the heiijht or length 
 of the solid. 
 
 (2) To be brief, wo will call ''intermediate or half-icay aection^^ 
 the section in qnostion in (he formula; or .ti^ain, and at will; 
 " centre section'''' "middle section,'''' and we shall alwiiys dcsif^nato 
 this section by the letter M, initial letter of the word middle as wo 
 desi<j;nate by Vt and 15' the opposite bases or ends of the solid, and 
 by L or 11 its len/ijth or height. 
 
 (S) T]w lenffth or heii/ht of t]\p> solid nnder consideration, shall 
 always be the distance between its jjarallel bases or ends, that is 
 the perpendicular drawn from one of these bases to the other or to 
 the plane of this base, produced if necessary. 
 
 Then the formula will write : 
 Vol unie = (area IJ + 4 .irea M + area B') x ^ L or H. 
 
 or : 
 V.= (B + 4M + B')^LorH. 
 or, 
 ^ to dispose the areas so as to facilitate their addition : 
 
 C + area TO ^ + ^ 
 
 V= < + 4 area M V x J L or H, or I + 4.M 
 ( + area B ) 
 
 (+ J5'> 
 
 Sum of the areas 
 
 X i L or H. 
 
XXX SYNOPTICAL OR ABHIDOKD KKY 
 
 Nature and value of the bases B, B'. 
 
 (1) Soiiutinu'8 OIK! of tlie (MhIs or bases of the solitl, ;is with the 
 
 ItjrjMiiid, cone, eoiM»i(l, ne^^nieiit or iiiiyulii of a Hi<iiere. splieroid, or 
 
 8i)in(lle, &e., will be but u i>(>int and its area, eonseciuently, null 
 
 or equal to zero ( 0), Sonietinies, eaeb of tii<! bases will be null as 
 
 to area or =0, as in tlie case of the sphere and sjtlieroid; n(tw, one 
 
 of the bases will bo a 8ini])le line, as for the wed^e and eertain pris- 
 
 inoids and un^uhT, and its surfaee a<,fain null ; at otlier times, eaeh 
 
 of the bases, as with eertain ]»risnioids, will be but a line and the 
 
 L surfaces null, as before ; but in all cases, the author advises the 
 
 i, pu]>il to maintain entire the formula and to write, as the case 
 
 I ■ may be : 
 
 areas areas 
 
 (+ H) ^+ '»^ 
 
 V.=: , + 4 M V or, V.=- < + 4 M ^ 
 
 Sum X J L or H. Sum x ^ L or II. 
 
 areas 
 
 ( + 
 or,V.= 
 
 Sum X J L or H. Snm x ^ L .r II. 
 
 (5) REiV. It is clear from what ])reced( s that the respective 
 
 surfaces in question are all i)hine surfaces, or must be considered as 
 
 such, and that, with the author's system, every surface is null, to 
 
 which a plane surface or a plane can touch but in one point, as in the 
 
 sphere, splu'roid and conoid ; which does not prevent one from 
 
 measuring in the same manner by the formula, and with the same 
 
 j accuracy, a spherical cone or pyramid, or any frustum of such a body 
 
 I' c(»mpris('d between paralled or c(uicei trie bases, one of which is 
 
 I consequently concave and the other convex. 
 
 ■i 
 
 (C) These enunciations would l»e quite suflicient to give a 
 
 perfect understanding of the autlun's sj-stem, but some observations 
 
 concerning nnne particularly, if not each of the solids of the tableau, 
 
 ( at least every category or class thereof will perhaps not bo useless. 
 
 (T) We say "class" or •'category" and in fact it is pi()i)er to 
 
 observe that the solids are disi)osed, on the tableau, by grou[»8 or 
 
 families, each in one or several vertical rows. I'hese rows are 20 
 
 1 iu number and the horizontal rows 10 in nundiei-, forming 2(K) jiieces. 
 
TO TUB 8TEKF.0.MRTRICAL TABLEAU. XXXI 
 
 Tlu^ first row to tin* rij^lit (it woulii l>c iinlifliTf-nt to rovovso tJio 
 onlor (iiul lic.Lciii at the left) compiisos the priHiu under suiue of its 
 varied forms. 
 
 (H) The four foHowiiifj raiif^es oft'er the ])riainoi(l, under several 
 diversified aspects (see inhoductioii, i)af;e (!) ineliidiii^ tlie re/jfuhir 
 or i»latonio. solids, (dodecahedron, iconahodron, &c.,) ivnd certain 
 unguhe of prisms. 
 
 {\}) The sixth row, still ;^oing towards the left, is the pyramid 
 aiul tlie frustum «>f tliat solid. 
 
 (10) Hows 7 and H sliow the ri<jht, inclined, truncated cylinder, 
 and tlie numerous unguhe, ami frusta of ungului, of thiu solid, with 
 also some cylindroids. 
 
 (11) !> and 10 are the right and inclined cones, their frusta and 
 unguiie. 
 
 (12) II is the concave cone with its varieties and sections. 12 
 and I'-i are tlu'i right and ini lined parabolic and hyperbolic conoids, 
 with their frusta, ungula! and trunciited unguhe. 
 
 (i;i) 14, l."} and 1(>, the flattened and elongated spindles with 
 their decomposed parts and varieties. 
 
 (14) 17 and 18 are the sphere and its segments, frusta, ungula), 
 &c., spherical cone and i)yramid and frusta of these bodies betwc ■; 
 parallel bases. These solids ofl'er also to the appreciation the 
 spherical, tri-rectangular, tri-acutangular, tri-obtusangular, &c., 
 triangle, and facilitate to the pupil, tlie understanding of spherical 
 geometry and trigonometry, and to the professor, the teaching of 
 these sciences. 
 
 (15) 1!) and 20, finally, are the flattened and elongated 8i)heroiJ 
 with the decomposed parts of these bodies. 
 
 See again on this subject ** The Introduction " page 7. 
 
 Let us first consider the 
 
 PRISM OR CYLINDER, 
 
 Right, Inclined, Tvristed. ^ 
 
 (JO) The prism is a body whose breadth or size is every where 
 equal or uniform ; it is, in other terms, a solid which throughout its 
 whole length or height is of invariable diameter or thickness, and 
 the opposite and parallel bases or ends of which, as well as each 
 
 I. See tlie liilroduction, page 11, laat, |iiuagnipli, letter of the Uevd. M. Uilliou 
 tuuthtiiuutiuiHU ut the St. Sulpice tjeuiiuuiy, Muutreul. 
 
' ■ !* 
 
 XXXII 
 
 RYNOPTIOAri Oil AiminOEI) KKY 
 
 :( 
 
 .t ' 
 
 1 -'■> 
 
 section parallel to tli'iso bancH, are coiiHJ'ciiiciitly.Kiinilar and <'f(iial 
 plane fi^iin'H; llicso U/^iirt;s may Indid'tTciitly bo nTtilincar, rinvili- 
 near or nuxtilincar. 
 
 We will tkeu (d)tain the solidity or volume by making 
 
 C + area W ) 
 
 V.= ^ + 4 area M \ 
 
 ( f area \ V ) 
 
 Sum of areas x ,\ L or II. 
 
 and, Hn]>posiii<r 
 the base 
 = i, 
 
 Sum of areas x J L. or H. 
 
 f (Ml. 15 
 
 = < or (i a. M 
 or 6 a. IV 
 
 I L. or II = 
 
 S 
 
 a. n 
 or a. M 
 or a. H' 
 
 X L. or II 
 
 (17) That irt : in the case of tlio prism, the ;4eneral tbrmula is reduc- 
 ed to the simplilied ("xprcssion : B or IV or M x L ; but we advise the 
 pupil not to endeavour to remember this fiu'inula, simplilied thou;;h 
 it be, sino(( he Avill always (see the introduction, ]y,iiH' !>) return to 
 it of himself; for one soon sees that it is the siune thing to multiply 
 any luunber by another number, or to multiply <j times the lirst by 
 the sixth part of the second. 
 
 PRISMOID 
 
 Hight, Inclined, Tvristed. 
 
 (18) The prismoid of which we treat at length, from page 
 161 to 1(57 of tliis work, has for its opposite and paralled bases or 
 ends, any plane figures, equal or une(|ual, similar or dissimilar, 
 rectilineal, curvilineal, (U- niix'tilineal, and one of which, as in the 
 case of the pyramid or the wedge, may be a simple point or a line, 
 or each of the bases a p-ere line as already stated (4). 
 
 We nuiL- icn write, according to the case : 
 
 V= 
 
 C + a. B ) 
 } + 4 a. M } 
 ( + a. IV ) 
 
 ( + 0) 
 
 or < + 4 M V 
 ( + \V) 
 
 C + B) 
 
 or ^ + 4 M V 
 
 ( [ 0) 
 
 ( + 0) 
 or ^ + 4M V 
 
 Sum of the a. 
 X J L or H. 
 
 Sum of the a. 
 X J L or 11. 
 
 Sum of the a. 
 X J L or II. 
 
 Sum <»f thea. 
 X J L or II. 
 
 PYRAMID, CONE 
 
 Regular, Irregular. Right, Inclined. 
 
 (19) In the pyramid, the base, or one of the ends is any plane 
 figure and the intermediate section a figure similar to the base and 
 equal in area to the fourth part of the base (95, T.). 
 
 OM 
 
TO TIIK STKRKOMK.TRir.VI, TAni,KAtr, XXXIII 
 
 Tlic Hcclioii (if the loiK', as (if llic |iyiiiiiii(l. li.v a plane 
 
 passiii;;- (liroiiuli ils axis ami apex, is a liiaii;;lc. and tiic liicadtli of 
 
 tliis ttiaii^'lc, fak(>ii at tli(^ half of its altiliidc is (|ia,i;c H.'i, icni.) 
 
 lialf tliat of llic liasc. Now, this same half-way incadlh of tiiC! 
 
 tiiunKh; fiirnisii«'s till' concsiKtndiii^ diainclcr of the pyramid oi of 
 
 thu cone ; tliat is. llic diamclcr of the iialf way scclioii of ihc 
 
 solid by ii plane parallel to the plane of its Imse. The cone, if 
 
 i'i;>ht, has for liasc u circle ; if inclined, an ellipse, and for its middle 
 
 Ht'ction parallel to the base, a circle or ellipse siinilai- to this hase and 
 
 eiiinil in snrface to the fourth part of it ; tiie other hase or end, of llie 
 
 (>()ii(^ or pyramid, is a nieic point, and its area in coiisnineiicc is null 
 
 or = 0. 
 
 Which ^ives us: 
 
 M. 15,) i <>) 
 
 a. M ^ = { AM [ 
 
 Sinn of the areas x^Loill. Sum of ihe areas ■ \ J,()iH. 
 
 Ami supi)os. t t ) 
 
 the l»as(? = S + 'I ^ \ > 
 
 =\, (+ jS 
 
 S. of the a. X J L or II. S. of the a. x 
 
 iLor H. 
 
 That is : for the pyramid, the cone, the formula icdnces to mul- 
 tiplying thi^ surface or aica of the liase liy (lie i. nf \\\c hciolit. 
 PARABOLIC, HYPERBOLIC CONOID 
 Right, Inclined. 
 (30) Hero the base is a circle or an ellipse, accordiii;^' ns tin* 
 solid is rifrht or inclined, and the half-way section biitweeu tlie base 
 ami the ap(>x or the opjiosite ends, is, as any othei' section 
 parallel to the base, a li,i;ure similar to such base and in tin parabo- 
 loid, equal { 7 ) in area to the half of it; or, which is tlic same 
 thing, the diameter of this section is equal to tlie square root (see 
 the tables) of half the s(|uare of the corresponding diameter of 
 the base. The other base or end of tlu! solid is but a point, since 
 we liave agreed to consideiMis such every curved surface wliich a 
 plane surface or a phuu* can touch, at if time, but on an inlinitely 
 small extent ; that is, a i)oiut. 
 
 Whence : 
 ( + area B' ) 
 y.= ] + 4 area M [ 
 r + area 15 N 
 
 Sum of the areas Sum of the areas 
 
 X > L or 11. X ^ L or II. 
 
11' ' 
 
 XXXIV 
 
 8YN0PTICAL OR AnRinnRD KK.Y 
 
 ';[ 
 
 ( r 
 
 11(1 SlippON. 
 
 S ' 'V 
 
 (+0) 
 
 
 (;$<,', H or L 
 
 lIlC 1»IIS»! = 
 
 < ^ 4 X i C 
 
 = +-f 
 
 ^Z. 
 
 < til' 
 
 = 1, 
 
 + 1 
 
 + 1^ 
 
 
 (l:<i 11 <.!• L 
 
 X }f L or H X i I-" o^" I^ 
 
 (21 1 So Mint for tlic^ paruboloid, tho general formula is reduced 
 to tliiit of iiiiiltipl\ iiig the ana of tlu* base Ity half the Iiei^ht ; 
 l)iil as this expi('ssi(»n, siinplitied tlioiijfli it lie, ditlers tVoiii the ^'eii- 
 eral forimiia ami may confuse th(^ memory, (Intiodnction payo !)) 
 the pupil will do well n<»t to endeavour to retain it ; but instead of 
 (liat, and to remove all doubt coneerinng the simplilied formula, to 
 resoit immediately although, it is true, with a few additional fi- 
 gure's — to the sole and universal formula <»f the autiiorj for, it 
 cannot lie denied that a longer process under a less tension of the 
 iniiul, is loss toilsome, and causcM less anxiety as to the accuracy of 
 the result, than a shorter but nmrc arduous operation. 
 
 SPHERE, SPHEROID 
 
 Flattened, Elongated. 
 
 (183) In the sphere and spheroid, the only area to be computed 
 
 is that of the e( ntral or Imlf-way section, each of tlie two otlier areas 
 
 being, as tliat for the top of the conoid, null or = 0. The central 
 
 section is either a circle or an ellipsis. 
 
 Thence : 
 
 C + area W ) 
 
 V= ^ +4 areaM [ 
 
 i + area 1> ) 
 
 four great circles ^ 
 
 or (ellipses as the ? x ^ R. 
 
 case nuiy be. ) 
 
 Sum of the areas x^ 
 diam. or II. or L. 
 
 Sum of the areas. 
 X J^ D or 3 K. 
 
 Ci'.l) REITI. As for the spheroid or ellipsoid, it is indifferent 
 under which aspect it be considered, respecting its half-way section 
 and its heiglit, lengtli or diameter : but as it is more simple, to find, 
 either by calculation or from the tables at the end of this treatise, the 
 area of a circht than that of an ellipse, nnitters can be managed 
 so that its central section be a circle, which will be done by per- 
 forming the imaginary section of the solid by a plane perpendicular 
 to the fixed axis. The solid would eqiudly be measured in an inclined 
 position (171, R.) being attentive however, as has been said (it) 
 to tak(! for tlie height or length a perpendicular to the plane of sect- 
 ion and terminated on both sides by planes parallel to such a sect- 
 ion and both of them on opposite sides tangential to the solid under 
 consideration. 
 
 ! ,-f 
 
 Uil 
 
TO THE STKnEnMF.TRICAL TABLEAtT. XXXV 
 
 SEGMENT 
 of Sphere, Spheroid. ' 
 
 (•21) 'I'lic scjjiiiciit Iiiiviii;,Htiit one <'(Hii|)iitil»li' liiisc, (lie loriiiiilii 
 to nu'iiHiiic it (Iocs not tlill't'r in any way tVoiii tiiat of tlit> cone ■ 
 cuiioid, pxcupt lio\vcv(!r that th« relation between the ai«a of i(,s 
 Imse and that of its intennediate Keetion vmieH witli ilic liei;;lit 
 of the wegnient. 'J'he radiiiH of thi,s Heetion in tlie Kcynient of ii 
 Kplieie "small eircle of the spiiere " \h tMpnil (JI71, <■.) to the 
 H'lnaie root (sec tlie tahle) of the jiroduet <»f the haif-vers«'d sine 
 ^neight) of tlie se;^nieut, by the remainder of llie dianicler of liic 
 sphere of wliicJi lliese/inient is a pari, and wlien necessar\ iliis diame- 
 ter is obtained i>y dividin;;' the .^cpiaie (see the tables) of tiie radins of 
 the base of the segment, by its lu'lyht, to get the reiiiiiiiidcr of the 
 diameter. 
 
 ( + area !>' ) ( + " ) 
 
 V= ] -t 4an"a M[ = ] I 4MC 
 
 f + area US i i l^S 
 
 Sum of tlie arms x I II. 
 
 Sum of the areas x }. II. 
 
 FRUSTUM 
 of Pyramid, Cone, Conoid, Sphere, Spheroid. 
 
 (25) 111 all these solids with two jtarallel bases, the bases 
 and half-way section are similar liynres : circles, if (he fnis- 
 tnm be that of a right cone or conoid, sphere or spheroid cut by 
 planes peri)eudicular to its fixed axis ; similar regular polygons, if 
 the frustum is a part of a regular i)yramid of th(^ same name ; and, 
 similar rectilineal, mixtilineal or curvilineal figures, if the pyramid 
 is irregular. 
 
 (20) In each of these cases, the vertical secti(ui of tlie solid 
 by a plane parallel to its axis, presents a trai»ezium. Now, the 
 mean b eadth of the trapezium is obtained by taking the hail'-suni of 
 its i)aralled sides, that is, their arithmetical mean ; and this mean is 
 precisely the diameter of the frustum at half-height between its two 
 bases ; whence it is easy to arrive at the fivctors of the lialf-way 
 section of the solid, and consequently at the area of such a section 
 (see the tables.) 
 
 1. We do not add: " fegment of pyramid, cone and conoid" simjily becaiiso all 
 fucli8egnicn(F, tha* is, all sucii ])arts cut off fr^m tlio aj.ioes of tlicse Fulids by a ])liine pa- 
 rallel or not to the base, is ftill a pyramid, a cono, a conoid and us voliiuie 6iibj.?t to 
 the formula already girun. 
 
I"1 
 
 XXXVI 
 
 6YN0PTICAL OR ABRIDQEI) KEY 
 
 V = 
 
 + aroa B' 
 4- 4 aiTii M 
 + avcn 1? 
 
 Sum oftlic areas +,\ h. 
 
 UNGULA 
 
 of Sphere, Spheroid, comprised between planes of sect- 
 ion passing in any direction through the 
 centre of the solid. 
 
 (S^^ In ciii'li (>r tlicKo .solids, flic ojniositi.' Itasos or cuu.: are null as 
 to area or==(i ; tlu' ot'utral section aloiu^ lias any value and this Kectiou, 
 ill llie spliei'e. is a sector of a circle (a jtart of a circle comprised 
 between au arc and two ladii) Avliiist in the sjilieroid, the same sect- 
 ion is circular, if the planes of section have their common intersection 
 in (lie lixed axis of the solid, in the other case it is elliptical. 
 
 Whence, (he ciihic content is : 
 
 ( -' area 15' ) 
 Vz^ \ ^ 4 area M \ 
 
 ( r^ an'., r. ) 
 
 Sinn of areas x ^ 11. or L. 
 
 (2S) K • y^. in practice, the leiii;' 
 
 = 5+4 area M [ 
 .'^iiin of areas x -J^H. or L 
 
 of the arc of the sector 
 may lie (Urectly measured, by means of a metallic libboii or the like, 
 or of a tliin rod that can be titled to tlie curve of the solid, to deter- 
 mine its circular or elliptical circumference, or any part of such 
 circiimfefence. 
 
 UNGTJLA 
 
 of any Prism, Prismoid, Cylinder, Cylindroid, Pyramid, 
 Cone, Conoid, comprised between planes of 
 section having their common inter- 
 section in the axis of the solid. 
 
 (3ft) It is clear that tlie ungula of a prism or prismoid, cylinder or 
 eylindroid'is nothin.u,' else itself but a solid of the same name ; that the 
 unnula of a ])yraniid or of a cone is simply a pyramid having for base, 
 in the case of the pyramid, any plane tlgiire, and in the case of the 
 cone, a circular oi elliptical sector, according as the cone of which the 
 unnula forms part, is right or oblicpie. As for the unguhi of a conoid, it 
 will be considered, with respect to its measurement, as the segment 
 
TO I'HE STEREOMETRICAL TABLEAU. 
 
 xxxvii 
 
 of i\n uiifTula of a spliere or spheroid (see the following paragraph). 
 It is ili'iir that the apex or one of tlie bases of the uugula is but a 
 line or point, as the case may be, and tliat 
 
 lu all such cases the formula is : 
 
 V.= 
 
 ( area IV 
 J + 4 ar«'a M 
 ( + area B 
 
 Sum of areas 
 X J H. or L. 
 
 that is, as 
 tiie ease V = 
 may be. 
 
 + 4 M>orV.= 5 +4M > 
 + B ) (+ oS 
 
 Sum of areas 
 
 Sum of areas 
 X J L. or H. 
 
 SEGMENT, FRUSTUM OF AN UNQUIA 
 
 in the conditions of the enunciation, par. 127, of the 
 
 treatise ; that is, in the conditions enumerated 
 
 in the two last paragraphs (27 and 29). 
 
 (30) It is plain that if the segment in question be that of an 
 ungula of a sphere or spheroid, this segment will have but one 
 base of an}- value, the other base being a mere point. The base \nll 
 be a circular or elliptical sector and the section at half-height 
 and parallel to the base, will be a sector similar to the base. We 
 will then have for the expression of the volume of the propc^ed seg- 
 ment : 
 
 C + area IV ) 
 
 V.= < + 4 area M v 
 
 ( + area B ) 
 
 Sum of the areas 
 X ^ II. or L. 
 
 (31) If it be a frustum of an ungul' > sphere or spher- 
 oid between parallel bases, the expression will be : 
 
 C + 
 
 Sum of the areas 
 X ^ H. or L. 
 
 V= 
 
 + 4 
 
 + 
 
 area B' ) 
 area M > 
 area B ) 
 
 Sum of the areas + J H. or L. 
 
 (.12) Finally if it be a frustum of an ungula of a prism 
 or prismoid, pyramid, cone or conoid (for the segment of an 
 ungula of a pyramid, coue or conoid, is evideutly a solid of the same 
 
. i 
 
 ' ! : V: 
 
 i ■ 
 { ' 
 [ I 
 
 r III 
 
 I 1 
 
 I "1' 
 
 i ' 
 
 i J: 
 
 XXXVIII 
 
 SYNOPTICAL OR ABRIDQED KEY 
 
 niiiuc as that of whicli the ungiila loinm part) the loiiinila will be, 
 U8 always: 
 
 ( 4- area IV ) 
 V= J -f 4 area M [ 
 
 ( -^ area 15) 
 
 Sum of areas x ^ II. or L. 
 
 SPHERICAL CONE OR SECTOR, SPHERICAL PYRAMID. 
 
 (JilJ) To an-ivo at the voliniu' of these bodies, wc must do pre- 
 cisely as for the ordinary cone and pyramid, .sive that the base and 
 middle section will be convex or concave surfaces which will be 
 measured according to the rules fomul (105, 107), the volume being 
 
 always ; 
 
 ( area IV ) 
 
 V.= ] 4 area M [ 
 
 i area 1} S 
 
 Sum of the areas x J 11. 
 
 Sum of the areas x J II. 
 
 FRUSTUM 
 of a spherical cone or pyramid between parallel bases. 
 
 (Jll) Will 1)0 expressed as the frustum of the ordinary cono 
 and pyramid by : 
 
 area IV ) 
 
 V.= < +4 areaM [ 
 
 + area B ) 
 
 Sum of the areas x J il. 
 
 ■) 
 
 FRUSTUM OF A TRIANGULAR PRISM 
 
 that is, having its opposite bases or ends not parallel 
 
 to one another. 
 
 (35) The frustum of a triangular prism, considering any of its 
 lateral faces as one of its bases, and the edge or opposed side as tho 
 other base, is nothing el.,e but a ])rismoi(l ; such is the wedge when 
 tho edge of tluit solid is of uue(iual breadth with the heatl. Under 
 this view, the edge or side in (piestiou being but a mere Hue and 
 consecpiently null as to area, we will have as an expression of tho 
 volume : 
 
 C area IV ) ( ^ ) 
 
 V= < +4 areaM } that is V.= < 4- 4M } 
 
 ( + area B > ( + \\ < 
 
 Sum of thu areas x ^ H. 
 
 Sum of tho areas x ^ II. 
 
TO THE STEREOMETRICAL TABLEAU. XXXIX 
 
 Tf ilio frustum of the triangular prism (of tlio last para- 
 ,a;rai>li) is itsclC truncated by a jtlaiic i>arallcl to one of its lateral 
 faces, we will still have a piismoid whose volume will be: 
 
 ( area H ) 
 
 V.= ) + 4 area M \ 
 
 ( + area ]} ) 
 
 Sum of areas x ^ 11. 
 
 SPHEROID WITH THREE AXES. 
 
 (JIO) This solid, as also any scfjnient, frustum, oruiiffula thereof, 
 seijnient or frustum of such uu;;ula, is exactly measured by the for- 
 mula, whatever tlu^ direeticm of the ])laues of sectiou may be. There- 
 fore, as tli(^ case may be : 
 
 or < + 4 31 > or 
 
 8inn of tlic a. Sumofthea. Sumofthea. 
 
 X ^ 11 or L. X A L" *>!■ il- X ^ L or H. 
 
 COMPOUND BODIES. 
 
 (117) The tableau jtresents a certain number of these bodies; for 
 instance a cylinder terminated at one end by a segment of a sphere or 
 R])heroid (such would be a mortar) ; a frustum of a cone ending in 
 the sauu' way (a ;;uu for instance) ; a cylinder or frustum of a cone 
 crowiKMl willi a cone (a hay-stack or ciicular tower with a conical 
 roof) ; a cone ending at its base by a segment of a sjjhere or spheroid, 
 like certain kinds of buoys. It is jdain that to measure tliese com- 
 pound bodies or ;uiy other forms that can be decom])osed into ele- 
 ments of the kind niveady treated on, the composing parts thereof 
 must be separately coinputed, in order to nmke nj) afterwards the 
 sum of such i)arts, according to the rules which luive just been given 
 
 APPROXIMATELY. 
 
 (See the genei'al expression, par. 127). 
 
 (ft8) And very nearly, say generally at .00.5 or at about (i) one 
 half 1 'r cent, more or less, often (see the detailed problems of the 
 treatise) witii perfect accuracy or very near an exact result j is 
 the volume obtained of 
 
Wl 
 
 ' lr 
 
 
 
 XL 
 
 SYNOPTICAL OR ABRIPQED KEY 
 
 ANY FRUSTUM 
 
 of a Prism or Prismoid, Cylinder or Cylindroid, Pyramid^ 
 
 Cone or Conoid, Sphere or Spheroid, comprised 
 
 between non parallel bases. 
 
 (39) By dcconipoKiiin; it, by an iina.!;'' iry section ])iniill('l to 
 one of its bases and i)as8ing throu^li tlie nearest point of the otlior 
 base into a frustum with parallel bases (tlie exact volume of which is 
 obtained by the rules already given) and an ungula. 
 
 ANY UNGULA 
 
 of a Prism or Prismoid, Cylinder or Cylindroid, Pyramid, 
 
 Cone or "yonoid, Sphere or Spheroid. 
 
 (40) In this solid, as in the regular ungula of paragrai>h8 
 (27 and 29) the apex or one of the bases or ends, is but a mere line 
 or point, and its volume is very nearly. 
 
 (See the detailed ungulic of the treatise). 
 
 ( area B' ) ^ i 
 
 V.= •{ +4 area M \ That is V.= < 4 M V 
 
 ( + area B ) ( + \\) 
 
 Sum of the areas >< ^ H. Sum of the areas x } H. 
 
 REM. As will be seen (H40) if the base of the cylindrical ungula 
 be not truncated, that is, if this base is a circle or an ellipsis, 
 the formula gives the exact volume of tlie solid, and in the sanie 
 manner under the same conditions, the exact volume of an ungula of 
 a prism will be arrived at. 
 
 FRUSTUM OF AN UNGULA. 
 
 ?a B' ) 
 ill M } 
 iii B S 
 
 If the ungula of the last paragraph is cut off by a plane parallel 
 to its base, of which the tableau offers examples, the volume will 
 not the less be, as usual : 
 
 ( area B' 
 V.= < + 4 are; 
 ( + area 
 
 Sum of the areas x J H. 
 
 ELONGATED SPINDLE, FLATTENED SPINDLE. 
 
 (41) The spindle considered, as a whole, is not a usual solid ; 
 it has little importance, and to be convinced that it cannnot be mea- 
 sured at once, as with the elongated or tiattened spheroid, it is 
 sufficient to compare it in one's mind to an exact spheroid having 
 
 • M 
 
TO THE STF.nF.OMF.TRICAI, TAHT-KAH. Xhl 
 
 /lio '^aiiu'' iixos or (liiinictcis. It is tluMi seen liow iiiiicli ils voIimik^ 
 is less (liiiii tliiit of the corrcsiioiMliiiii' splicroid \\lii<'!i is inoic swollen 
 townnls (lie fiids of its jixis in the clonuiitcd sidicjoid, iuid in the 
 opposite dii'cctioti, if it 1»«' a tiattciicd splicroid. 
 
 ('12) Unt if it 1)(" inii)ossihlt' to iirriv<' at oiwc at the volnnic of 
 the s'lindle, one succeeds almost innne<liMtcly, l»y measuring- the 
 half of tliis solid, and ai'eiwards doiiblinn' the result, since 
 then, by takini;- its secticni liall"-\vay between the centre of tln^ 
 spindle and its a])e\ or end, the veiy element mImCIi contributes 
 especially to m:ike its v<diime vary, is considered, and this process 
 ai»plied to the li '<teiied siiiiidle, will nive the exact volume if the 
 peritneter of a cm lioii of the liall'-spiiidle by a i>Iaiie ]>assinj;' tiirou^h 
 jts fixed axis, is an arc of a conic srction, as will generally be 
 the case, the llattciied spindle beinn tlnii eoiisidered as two e(|iial 
 se<;'mcnts of a sjiIk re or s)»lierni(|. niiiird Ity their bases or planes per- 
 pendicular to the lixed axis <»f the solid, of which the composing- 
 segments of the spindle form part. 
 
 It will be seen (pnd). LI) that it is sutlicient to divide tin' 
 spindle into two parts which will be measured separately an.i the 
 sum of which wili be afterwards taken, to arrive at a result which 
 shall dilter from the truth but by the Hth part to the quarter of one 
 per cent. 
 
 CENTRAL FRUSTUM OF ELONGATED SPINDLE. 
 
 (Cask). 
 
 (43) This solid which gives its form to the thousand and one 
 varieties and dimensions of casks, throughout the whole world, is, res- 
 ])ecting the measurement of its capacity or voliune. of great impor- 
 tance, on account of the generally high jtriceof the contents. Well! 
 as will be seen (prob. LII), it is sufhcieut to measure at once the 
 half-cask to arrive at the exact volinne, within the (juarter to tlie 
 fortieth part of oiw jier cent; niaxintum error of one (piart on 
 a hundred gaUoiis or of one litre on -100 lities ami which does 
 not exceed generally l\^ to ^\jof a gallon or litre for every J 0(1 gallons 
 or 100 litres, and can, besides, by that itself, be rectitied, that the 
 errcn- is known to be always in excess and that conse(piently the result 
 may be; diminished by so much, if required. 
 
 ( area 15 ) 
 
 V.= < + 4 area M J 
 
 ( + area M' ) 
 
 Sum of the areas x J L or II. 
 
" f! 
 
 
 
 
 ,! 
 il 
 i 
 
 
 i 
 
 ' li' 
 
 ■ i 
 
 1 :■' 
 
 t 11 1' 
 
 1 ■ 
 
 XLTI 
 
 STNOrTICAL OR AnRinOED KF.T 
 
 CONCAVE CONE. 
 
 (11) Tlic coiiciivo cone is ;iii;il(>^()u>*, ns (o its voliimo, to the 
 f'l<>ii,<,'at(Ml lialt'-spiiidlc, wliicli iiiiiv also he ciilk'd a convex cone ; and 
 in liie same way as we vciy nearly arrive at tlic volume of tlie lialf- 
 s])ind1e, by nicasnrinjf it in two slices; so, if the liollow (»i convex 
 cone is deconi]tosed into two parts, by a jdane i)arallel to that of its 
 base, to nieasnre separately each of these ])arts and add them to/i,<'ther 
 alterwards, the volume will be obtained at less than one halfi)er 
 ccut loss. ^ 
 
 FRUSTUM OF CONCAVE CONE 
 
 "between parallel bases. 
 
 (■15) A ureat rnany vessels of capacity assume the form of this 
 solid and as the iioUow or concave cone is analogous to the half-spin- 
 dle or c<mvexcone, so the frnstiim of the concave cone may be con- 
 siderc<l as analogous to the central half-frustum of the elongated 
 sjtindle or half-cask. Then by nu-asuring it at once, provided its 
 curve be uniform Lhroughout its height and especially if this curve is 
 iiot considerable, the volume or desired content will be very nearly 
 arrived at, and if this curvatnie of the lateral face of the solid or 
 vessel t)f cajtacity in (lucstion, is considcr.'iide or of une(|ual radii in 
 A'arious ))aits of the height of the frustum, a nearly perfect accuracy- 
 can be secured, by decom]tosing it mentally with a view to its mea- 
 surement, into two or at most into three parts or slices by planes of 
 section i)arallel to the bases. 
 
 The volume of each of the com])onent slices will be: 
 
 V.= 
 
 
 are 
 
 a 
 
 15 
 
 •t- 
 
 4 are 
 
 a 
 
 M 
 
 + 
 
 are 
 
 a 
 
 15' 
 
 Sum of the areas x I II or L. 
 
 COMPOUND BODIES. 
 
 (46) These bodies nmy assume many varied forms. The tableau 
 presents some of them: foi' instance, a central oi' eccentric frustum 
 of a sphere or spheroid surmounted by a concave cone (kind of 
 
 1 For forms with coticave siJoB tlio volu ne is leu*; ni for cotiTcx bodies the 
 Tolume il mora. 
 
TOTIIE STEREO.METllIOAI. TABLKAtT. 
 
 XLIII 
 
 (loiiUM)r luiiiiirc't) ; scuiiiciit of ;i sphcvr or s])Ii<'roi(l siiiiiioiiiittMl liy 
 a st'^iiu'iit of an elongated spiiidlc! or convex vttiu', or of a hollow 
 or concave cone ; two frusta of ri^'ht cones united liy their broader 
 bases; two others, by their smaller bases j two frusta of coiu'ave 
 coiu'S and two others in the same conditions. And it is clear that 
 other forms may be (conceived, in almost iidinite variety, but of 
 which tlu^ rules already ,niven are sutticient to deternnne the res- 
 pective and conii)osing volumes. 
 
 SUNDRY. 
 
 (-l?) Resides the solids which have; Just been enumerated, it is 
 proper to say a word of certain fdinis which the tableau preseuts 
 and of which the origin or the whole solid of which the body under 
 consideration forms a jtait. mi^lit not at once su;j,<!;est themselves to 
 the mind. Thus, the ecceiilric solid riiij; may be, considered 
 as the central frustum of a veiy elon<;ated spiutlle bent on itself, 
 Then will it l)e measured by addin,i;' to the sum of the areas of both 
 its less and greater secticms, 4 times the area of the half-way .section 
 between tlie.so first, to multiply t!ie whole by the length of the 
 half-circumference used as the imaginary axis oftluMiug. 
 
 (48) Th<; bent roue or half-spindle in fonu of the hoin 
 of an ox is measured like the inclined com-, considering as its height, 
 the perpendicular drawn from i'^s apex to the plane of its base. 
 
 (49) There is the eccentric fi'iistiini of an elongated 
 
 spindle, which nuiy represent tlu( shaft of the roman column, swol- 
 len as it is, about the third part of its height, ami the volunu; of 
 which may be had by taking separately that of each of its com^tosing 
 half- frusta. 
 
 (50) The regular polyhedrons, as it is seen, may bo de- 
 composed into as many regular and e([ual pyramids as the soli 1 lias 
 faces and be easily measured in this manner, each pyramid having 
 for base one of the faces of the polyhedron and for height the half- 
 height of the polyhedron, that is the half-diameter or ladius of the 
 inscribed sphere. 
 
 (51) The decomposed parts of the flatfenedand elon- 
 gated spindles and of certain other solids furnish the idea 
 and in consequence the manner of measuring, or gauging any 
 sailing vessel, steamer or other, by decomposing it, if necessary, into 
 elements of the kind already treated of. 
 
i 
 
 ii ■• 
 
 XLIV 
 
 SYNOPTICAL OR ABRKGKD KtT TO IIIE TAIILEAD. 
 
 i 
 
 i 
 
 m 
 
 1 5(1 
 
 il- 
 
 (H'2) ICI'i^I. Tho regular jtulyhodrons coulj equally bo at onco men«i!red, by 
 tnking tlio trouble of liiiiiinj; ihu aroii u( tUo cuiitral section of each of (hum. All these 
 soliiis hiivo two i>arallol bast'.-*, oi-y of tliu biit^os boin;;, fur the lotrahodoii, a |)oiut — 
 for tilt! totialiiUron is iiolhiUj; hut ii |)yramiil. — The octabudrou may b,- coiisidfrod as a 
 double jiyraiiiid or a coiii|i(iuiid ol two i>yraniid.", bn.-o to liase, ind bo uioasuroii in this 
 manner. As to iho dodecahedron, it will be seen that while each of its parallel bases is 
 a re"ular i"nia>;oii, it." half-way section between these based, is a regular doca^'on or 
 a ten ^^ided ngular jiolvgon, and each side (T which is c(|ual in length to thii half-Jia- 
 jjonal of the ounta^ion. As to the niiddlo foction of the exahedion, if it betaken 
 parallcd to two opposite and paiallcd sides of the solid, it will bo a twelve sided roi^ular 
 iiolvKoii.tho pcriinct T of which it would lio too long to deterniint^ hero. If on the contrary 
 the half-way section is supposed p irallcl to or tMiuidistaiit Irom two o|)iiosite vortices of 
 the solid, that is, jierpondicular to tho axis or diameter uniting two opposite points or ex- 
 tremities of tho Miliil, this -ectioii will bo a regular doi'agon each hide of which will be 
 euual lotholialf-.'ido of the iriangle forming tho face of tho polyhedron. Finally, if any two 
 Opposite sides or eilges be taken for the parallel bases of tho icosahedron, the half- 
 way section parallel to th(8c ed-es and perpendicular to tho piano which unites them 
 will bo a six sided synnnelrical pil.vg' n, two opposite and pariilled sides of which, each 
 euual to the side of tho triangle forming the face of tho polyhe<lron and being one of 
 thfso fides, and tho other sides of the exagi'n, parallel, two and two, and rotpoetively 
 equal to tho height or right radius of the said component face. 
 
 REDUCED TABLEAU, 
 
 (7S;j) Klv>l. It is hardly necessary to say that, in this treatise and in tho 
 abiid"ed key, every thing which rohiles to tho so called tableau of 2110 models, is 
 equalled applioalde to the reduced tableau of 105 models which tho author is preparing 
 for elementary schools and with a view to reduce tho price of it in order to place it 
 within the means of persons less capable of ordering it. 
 
 In tho reduced tableno the moilols will be 195 in number, disposed in 15 ver- 
 tical rows and ou 7 horizontal rows (15 x 7 = 105;, Then beginning for instance by 
 tho left, tho 
 
 1st vertical row would be the prism, its frustums and ungulas. 
 
 2nd, 3rd, 4lh row, the pri-moid ai;d vaiious frusta and uugulo). 
 
 6th row, the pyramid, its frusta and ungulas. 
 
 (Uh row, the cylinder, its frusta and ungulte. 
 
 7th row, tho cone, its frusta and ungula.>. 
 
 8th row, the conoid, its frusta and ungula. 
 
 Stlh nw, the flattened spindle, its segments, frusta and ungulre. 
 10th row, lite ol'mgated spindle, its segments, fr,.sta and unguhe. 
 11th, 12th and 13th row, the sphere, its segments, frusta and ungulro. 
 14th row, the flattened spheroid, its segments, frusta and ungulse. 
 15th row, tho elongated spheroid, segments, frusta and ungulaD. 
 
 And if in the tableau any segment, frustum or ungula is wanting to complete 
 tho number of th<'se included in tho nomenclature of tho solids to which 'ho formula 
 relates, it ca» easily bo mentally supplied in the same way, as, if required, any com- 
 pound solid may equally be decomposed by imaginary jdanes of section, into elemen- 
 tary forms, to submit its volume to tho required computation. 
 
KEY TO THE 
 
 BAILLAIRGE 
 
 STEREOMETRICAL TABLEAU. 
 
 INTRODUCTION 
 
 TO TUB 
 
 NEW METHOD OF MEASURING 
 
 ALL 
 
 B iDIES-SEGMEXTS, FRUSTUMS AND UNGULAS OF THESE BODIES, 
 BY OXK AND THE SAME RULE. 
 
 Tlio iiiillior of till' new in"!lio(l lias llic honor to iuibrni Lis ool- 
 ]('a;;ii( — Aicliitcots, Eii<;iiu'ors, Surveyors, Professors of Geometry 
 anil Matlicnialics, the Direetoi's of Uuiversities, CoUeges, Semiiuiries, 
 Convents aiul other Educational Establisliineuts, the I'rofessoi's and 
 rMl)ils of Schools of Art and desii^ii, M(>chanics, Measurers, (juagers, 
 Custom House and Excise Olhcers, Ship-Builders, Contractors, Ar- 
 tisans and others of Canada and Elsewhere. 
 
 That he has perfected his *' Steroometrical Tahlean " in order to 
 reduce to jnactice his universal fonnnhi for finding the solid content 
 *if a hodv of anv form or dimensions. 
 
 Advertiskjient. — With tho view of protecting his discovery and invention and of 
 guarding against any iBl'i-ingoDicn; of liis right?, either here or elsewhere, the author has 
 taliun out Letters Patent of his tableau, as well in Canada as in the United States and in 
 Europe, 
 
2 
 
 THE STEREOMETRICAT, TAIU.KAIT. 
 
 
 ^\^ 
 
 If 
 
 k 
 
 ! 
 
 ii 
 
 On the subject of tliis foi'mulii the (^iiclicc Sciiiiiiiuy iil'tcr Im\ iiiii' 
 eiibiniltcd it to compelciit jwrsoiis, v\]uvsava itself tliiis : "Of tin- 
 tlK'oicins iiiid foiiiiiiliit' r(iiiiiik;ililc for their novelty, tlie most strik- 
 ing' is the <;eneriil rule for the content of iiny known solid. An in- 
 volnntiiry doubt at liisf tiikes |iosses>iun of the nd.id ; luit a caiil'iil 
 cxatniiiiition soon dispels tliis lioul»t and one rcDiiiiiifi itslmntihil til 
 siijht ofa formula su cleiu; tio cdxif to retain, (ciil <>/ irhich the (ijipHat- 
 iiou is so (iciufdl. " 
 
 An article in the '', Journal d'l-ducatiiMi '* s]ieaks of the .si((/</c;( 
 inqmlsc which this discovery imparts to science. 
 
 His nietliod of explanation, says another journal, is \ 'w, and S(, 
 simple as to be within tlie ;;iiisp of every mind. 
 
 The tableau, the dinunsions of wliicli are about '{ feet in 
 hcifjjht by 't feet in leni;lh, is of hard wood, dyed or painte<l of snilalde 
 colour. Fixed to tlie board ar<' sonu^ xJOO models of hard wood, 
 polished, oiled or varnishe.l asre(|uir< d. Ka<h little model is adjusted 
 to the tableau by means ofa pej;' or pin ef wire attached to llie lio 1 
 eo that the model may be easily removed and replaced at plea>ur( . 
 
 T/fC »;r)Jc/,s com]uise almost all the elementaiy forms that it is 
 possible to conct'ive. or to wiiich any compound Itody, l>y division or 
 de<'om])osition, could lie reduced. Amcui:; them will be Ibund ail the 
 prisHLs and prisnioids, <'vlinders and cylindroids ri,i;hr and oblicpie, 
 and the frustums and uu;;ulas (»f tlicse bodies; pyramids, cones and 
 conoids, ri^dit and oltlicjue, witli their fiustums and uu.i;ulas; the 
 sphere with its sul»divisions into iiemispliere, (piarter siihere, half- 
 quarter or tri-rectangular spherical pyramid, seniuenls, /.ones, jiyra- 
 inids, frustums and unj^ulas; the prolate and oblaie spheroid and 
 ellipsoid, Avith their se^i'ments, half, (|uarter. fnistums. etc.; in line, 
 the s])indles and their frustums, etc., includiu,;;' casks of all sorts ; 
 the regular polyhednms, concentric and eccentric riitiis. and a num- 
 ber of other varied jmietical forms too many to enumerate here. 
 
 The new rule or formula dispenses with all consi- 
 deration, all preliminary calculation as to the nature, 
 form or dimensions of the entire solid of w^hich the body- 
 to he estimated forms a portion. I'iius, when it is required 
 to lind by ordinary rules, tlie tMibical contents of a segment, fruy- 
 tum or zone ofa spheroid, for instance, it istirst necessary to tind the 
 axes of the solid so as to take them into account; luit l»y the new 
 system :ve proceed immediately to estiumte the solidity reipiired, by 
 applyiuij directly to it the formula (IV ) U i- -1 M) }. II. In like man- 
 ner, if one supposes he has to deal with the frustum of a pyramid 
 
 '.;iJ! 
 
INTllODnOTION. 3 
 
 fur ox.itnpl*', llic fisl tliiiin- to asccrlain is tliiit i) is one, in order to 
 apply to it till' nnliiiar.v lulcs of iiMiisMralion ; for iliis itiirp(»s(', it is 
 iniiiiri'ii lo iiirasiirc llic rcsin'ctivc lfii;ft!is of tlic cdiccs or sides of 
 till' iijiprr and lower liases, ^.o as to estaldisli tlie pioportioii between 
 them, ami lie a^'SIl^ell lliereliy tliat tliey are, or are not pi'opoiti<»nal, 
 ■..itiiiiMt wliieli the solid to Ite estimated is not the frustum of a pyni- 
 iiiid : whilst, on the contrary, l>y the new fonnula, it in siitlieient to 
 lie ecitain of Ihe jiiiiallelism itf the sides or ed^es (wldeh is seen at ii 
 j;Iaiiee and without any measurement) to ])roeeed direetly to tlio 
 aiiplicalion of the rule ; for, from the authors jioiut of view, the frus- 
 tum of a pyrandd, reput<'d sucli, is re;j;ard; d as a prisnioid, and sub- 
 ject, on that account, lo the general fonnula, in tlic same way as tlio 
 whole pyramid which is also a prismoid. 
 
 But, setting aside all preliminary consideration as 
 to the nature of the solid to be measured, and of tlie Ion;; 
 and alistruse calcilatioiis which must he made for that puri>ose when 
 the ordinary rules are followed -it is to be reuuirked that the calcu- 
 lations wiiich remain to be made to llnd tl)o content of tlie solid or 
 liijnid in ([uestion are laborious, dillicidt, lon^, and sonietinios never- 
 ('udinn', as for instance. Avheu they relate to tlu' j^najjii-.ti; of a cask, or 
 the measurement of the, frustum of a spindle. There are very 
 tVe(piently aljicbrac formulae, differential antl integral calculus, and 
 a simple nds-print in the formula of which the mind can not pene- 
 trate ov follow the mysterious intricacies, — an error which may not 
 be detected and woubl cousetpuMitly r«'main uncorrected — a simple 
 error of this sort is enou,nh to render the whole calculation nselesa> 
 and necessitate^ itsbcuu!; <jono over aj;ain ; instead of which, by the 
 system now jtroposed, as there is but one rule, one sinipb> formula to 
 learn and retain, and that of the simplest nature, the mind can 
 follow it step by step, and tin; eye itself perceive immediately it 
 there be error. 
 
 At present, there are as many difterent rules as 
 there are solids : one for the i»rism or cylinder; one for the 
 l)yramiil or cone : another for the frustum of the cone or p.yramid ; 
 a third for thi> si»here ; then three others for the segment, zone and 
 nnnula of this body; another for the spheroid, with additional 
 formulae and in eipial number for the segment, frustum and ungula, 
 according as the intersecting plane is parallel or inclined to the small 
 or great axis or to any diameter whatever of the solid. How many 
 otiiei' formulae dilfcring always from one another, and each of them 
 from all tiiose already enumerated, when it in necessary to jinive jit 
 
• 9 
 
 
 J: 
 
 I 
 
 .i 
 I 
 
 . I 
 
 ill 
 
 ■ 
 
 i 
 
 i 
 
 r\ 
 
 '^ii 
 
 ^Ml^ 
 
 4 TUB STEREOJIKTIUOAT, TAHr,P..\(r. 
 
 (lie cniitciil of a i>aralM»lic or Iivpciltnlic foiioid. ri,:;lit or ol>Ii(|iif, <.!" 
 a sc;;iiicnt (*f a I'iu'ulai', cllipliial or oilier spiiidlc, ol" an imniila of a 
 (•.vliiitlcr, cone, conoid or spiiKlli". Weill we can now l;i.v aside all 
 these rales, all these varions forniiilae wliicli it is iin|M>ssil>le for the 
 nieiii(>r,v to reliiin, antl foi' wiiicii we alwa,vs need a liooiv ai onr 
 dis|)osal J we «an set aside all tliese ninltifarioas fornnilae with llie 
 books c(»ntaiiiiii;;' them, and armed with tiie new system, take hohl 
 of any solid whatever to lind its .•onliiit l»y llie aid of a quite simple 
 rale that may by renieudiered like the Lord's prayer, namely : "to 
 thesnmofthe parallel end areas, add four limes the middle area, 
 and multiply the whole by one sixth part of the hei,';lit or length of 
 the solid.*' 
 
 The calculation is thus reduced in every case, by 
 the authors system, to that of the areas of the opposite 
 bases and of the middle or interznediate section, *<» attain 
 Avhicli end is j»recisely the object of tiiC taldeau in ((ueslioii, w i>.eie. 
 (Mie may imnusdiately see the foini of tin' solid, the nature of the 
 surfaces which foini it •; bases, and, by means of a stroke or line, the 
 nature and dimensions of the section, surface or ba>^e half-way 
 between the bases or o|)poslte extremities of tlu^ solid to be estimati'd. 
 Ill this manner it will be. seen thai in the [tyramid, the cone, the 
 conoid, tlu^ segment of a sphere or spheroid, the upper suri'ace is 
 jeduced to zero or is nail, whilst in thi> s])here, the spheroid, and 
 certain prismoids, ('ach of the opposite surfaces or areas l>ecomes 
 notluiijj;, v.iiich reduces tile calcnlaiion lo multiplyinn' 4 times the 
 middle section by the si\tli part of tlu; height of the solid. 
 
 The formula is mathematically exact for pi i^ 
 
 ms 
 
 ami 
 
 prismoids, cylinders and cyliiidroids ri^ht or olili<pn\ rei^ular or 
 lrre;;ular pyramids ami the frustums of thes(^ solids betwei-n parallel 
 bases ; for the ri<;iit (U' ol)liquo cime and its frustum between parallel 
 bases, the H])here, the spherical se^nient and /one, the spheroid and 
 every sej;'ment, zime or frustum of thi-' body cut from the whole solid 
 by a plane inclined in any manner whatever to the axes or diamclers 
 or comprised between any two parallel i»lanes ; for ri;;lit or obliipie 
 parabolic or hyperbolic conoids and the frustums of tla se bodies 
 comprised between parallel planes ; all of which vai'ioirs solids cons- 
 titute nearly the total number of the elementary forms thats one can 
 be called on to estimate. 
 
 It is for spindles only, and certain ungulas of these 
 bodies, and of other solids, that the formula is not ma- 
 thematically correct, nuJ yet even in these latter cases can we 
 
rNTiioiti:r'T!f'\. r> 
 
 <il)l;iiii vcsiilt^ more ('(irn'ct , iiii>r'> s:\lis('iU'lory Ity iirMiH of tlio ',r,Mif'- 
 I ill r< nil I II la 1 1 1:111 (Mil he i|ii:m- liy miy of 1 lie oi licr iiiiMii.-i In w Iiii-ii \' i- 
 roiiiiiiinily have t'ccoinsc in praclic*'. It saDit'cs t'lir iliis (o divitl* ilic 
 halt' spiiKlIc or fnisiiMii (if (Iii.H soliil info two, or at most into ilircc 
 |»ai't;-<, liy sections or |i'aii!'-< paralli'l to llic Iti-;"-*, aJid iipoly tlii' for- 
 mula separately to eieli of these pai'Is. to arrive at a result ajiproi- 
 chlii'^ almost to peil'eei aei'iiraey, and so ol' tli • unnula^ of jnisnis am! 
 prismoids, ol" pyraiiiiils, cones, conoid-, ami spindles, of sp'ii'res iind 
 spheroids ;inl the IViisI inn < of Ihes" inr^Mlas helwren pai',iil<l Uiscj, 
 to divide th" solid into '^, .'5. oi' .i p iris at ino-;|, loap]ilyne\t ihe 
 forni'.ila lo < adi oi't he I'nmpmient se^ineiils sejiurately. as il is liesi^les 
 evidently neeess;i,y to do so fiU' the euhie.il contents of eoaiponnd 
 solids such as ships, si'Iiomu'i's, yacht. ;, lioats, .S,-c., anil other simil ir 
 foliiis, composed as they are in every case of some of t he'elemeiitaiy 
 foiius w liicli will hr fn; I lid aiiioai;' tlie nixl'Is of tiie lablrau. 
 
 And ia tho majority of case 1, even thii subdivision 
 of the solid into 2, 3, component se[,nient^; willbe useless 
 since, for tin' uie4,ii]as of prisms, cylimlers, pyramid-*, ami cones, the 
 inaxiiiinin error l)et\veeii the iriU' content and that ohlaiiied liy the 
 propsised formula does not exceed .lll)"> or the half of one jicr cent. 
 
 Let UB considsr al30 th3 iintncn-J3 advantage of 
 such a tableau for tha mere nomenclature of bodies 
 By it, the least advanced pupil'3 in CdUo-jcs, Convents, 
 and other schools, are enabled tc prosecviie a study from 
 which they have hitherto been debai-red. In i'i'''i. for I'lo 
 mind io seize and iindi'rstaud, on pip.'r, thv. t^rapliir repres -at itioii 
 of a hitdy, ir nee led a i)!'evious iciio\vledu-<' of drawing' and ]tersp;n'rivs' 
 which is only acquired ;^('nerally l»y advanci'd pupils ami in the last 
 ;, ears ofcolle!;-e ; wliilst to-day ilie yoiin;;'est, tiu' least advanced may 
 detacli the mitde! tVoiii the talilean, take it, held il, and liamlle it ; 
 the masier or i>rolessoi', tlie nun or mistress will tell him or her tin? 
 name of ir and poinr out an "xamide in the thousand and one objects 
 met witli in our every day pursuits and re(piirenieiils. 
 
 What the several models of the tableau may 
 
 represent. 
 
 The segment of an erect or reversed cone will, 
 according to circumstances, represent ;i l"\\<'i> liuhthouse ,1 
 tirmbler, saltinu," tni), Imtter lirkin, Uucket, an ordinary tuh oi' vat or 
 of 8iieh a.s are hvhu in breweries luid elsewhere ; tlie flat or dei>redsed 
 
V' 1 
 ■I [ ■ m 
 
 ,■'1 
 
 .1' ■% 
 
 m 
 
 S 
 f 
 
 I 
 
 6 TTTE STKnrnMr.TRTOAL TAr.LF.Ar. 
 
 coil!' will Fn'^f'^^i'st tlif' i'lci! of tli;^ (ovcv or hottoni ot"n raiiMmii. of a 
 I'oii", ct;'., til!" pyiiiiiiil will hi' t'l ' iin;m\' of'a iiiiiiiirlc nr of tlic s]>ire 
 of a i^.'c 'p! ' (■!:;•. ; tli ' ri'^'!u cicii: 1, t!i!' .si'^'m-iit ol' a .sphere or splio- 
 roiil will h;' a (l:)ai > m )r>' o)' l;'s-i ('Icvateil or <li'pv,>s;so(l, wlii'st the 
 sud:' s)li I, ]>■/ I'.'V ■rsiii.i; if, will pr(v-;;'tii ,,> the iiiii'^'ination a basin, 
 a r; scrvoir, a boiler savh as is I'ouiid la the sii,i^':ir luU, the (li.stilkny 
 or oilier inainirachny. 
 
 The iiioliue:! and reverssd conoid, ths sajment of a 
 sphoroid equally inclined and rcveraed, will r'^ present 
 the space occ'Upi(>!l by aay liquor, liquid, or tluid whatevor at llio 
 bottoai of a vessel whieh iVoin o'.'.i cause or a: other has become 
 inclined ti> the ]iori/:)n. 'i'he piisni, th<' prisniuid will liic-'wlsc bo 
 the laoihl of those ihoasaail :iad one ]ilai!i oi' claitoratc roofs which 
 crown onr douicstic habiiaUoiis, our public buil(liu;;s, oui' palaces. 
 Tiie roof of tlif comniou doi'ni >i-v. iadow, if it lie sloped, will b(> a 
 tri insular piisin obIi(jue o: inclined : il'it be not s >, it will be the 
 frastuni of a i)risn), and th" body of the d.ivaier will be. imliilerently, 
 according' to the aspect 1 ador wliieh ir is viewed, a ri'^hl triau,uular 
 pri-ni, or the nn^i'ula of a quadran.iiular prism. Amouu'st the pi'is- 
 mnids will also l)o found the stitdc of tinibei' o.' saw lou', the trunk of 
 a tree, the I'aiiroad culiiun' and embankn!C^^, the leservoir. ([uay, 
 ])iliar. caniiiin,!^' tent, the spl;iy(Ml or scpiai'e o]»cnin;;' for a window 
 doc). uich Ol' loop-hol • in a wall. The(iuarter of a sphere or si»he- 
 roid, the half seu'm-'iit will be tlie vault of the apsis of a clinch or of 
 a hall tei'ininateil in the same manuer. Tiie whole s])here. the sphe- 
 roid will lu' the billiard ball, the ball of a ^tee]de. the earlli wo 
 inhabit, the luoon, the sun, the jilaiuts. In a word the niodcd of 
 e:ieh of the elementary forms which it is possible to conceive will be 
 foauil on th^ t.ibLviu, and can !»;■ iiaudled at pleasure so as to exa- 
 mine it under eveiy possible asjject. 
 
 The base!?, lateral faces, central or intermediate 
 sections of the inodcls of the tableau, also offer to the 
 pupil, in the previous study of the inensuration of sur- 
 faces, the repT-esentation of each of the plane figures, 
 and of e)ver,v sort of convex or concave surface of either 
 single or double curvature, 
 
 '['he sipiare, rectauule, lozenp,e, ijarallelorrrani, trapezium — 
 the ciinilateral. isostades, scalene, riiiht-aiiiiied. a(aife-an,uled. obtu- 
 se-augled triangle; — the regular, irregular polygon — the circle, 
 
 ' 'l 111 
 
 
INTRODITTION'. 7 
 
 scHii-circlc, <|n;i(liiiiit . sector, sc^iiicnt. Zdiif, l'iiii\ coLccntiic iiml 
 cNcciitiir liiii; — the ciliiisc. m mi ellipse, senmeiit of elliii.se less or 
 gre:iler lliiiii ilie half' - lli<' other eoiiic .".ectioiis. ]>;iliiliolii, iiviieilioia 
 — tlie ('(luiljiltial, (! i-reehiii,'4i lar, I -olitiise-aii.uulai- spln-rieal Iri- 
 an, i;le. - l!ie sjilieriral se^iiieiil, xoiie. .ne.— .se,i;'iiieiirs ami zones, &c., 
 ol'lirolatc ami oiilati' s;ili"roiils or elli|is(ii(ls. \f. 
 
 The tableau will Lave the cif: A of interesting the 
 pupil and of makings attractive a study, heretofore dry 
 and almost impossible. I-y ni;ai)s of llie IrJdean ami llie 
 iU'eiieral I'oriiiiila, sleicoldiiiy ami st; reiiimtry. iliai i.^, i!ie iiomeu- 
 clatiire. i)i()[»ci'iies and measiireiiienl ol' hodics can lie tau,^lil in e\eii 
 e]( iiieiilai'V s.'iionls ^Vlle|■e siillieieiil ,n'e!ini<'Iiy lias iii>! !u en taui;lit 
 to enable iIh' pnitils to deteiniine llie area of any plane limire wlial- 
 ever : since, in icality, llie proposed system rednces the ni;'i!sural ioii 
 of bodies or \oliinies to this — tiie leniaimlrr ol'tlie work Ixiu'^ merely 
 an additi"ii of the areas thus fonnd j'.nd the nndli[ilicaiiou of ihiir 
 snni by tlie sixth i)art ol the lui,i;lit (.f ijie solid. 
 
 THE FOKMULA. 
 
 Til!' iiri>m()id il formnla proper, is, of eonrse not m-w, -iuce it 
 has been sonu'iimes used in ilie eoniputation of :;r!ii\vori;s on rail- 
 ways and otherwise ; but it {\:)cs not appear to liave sn^^'^ested itself 
 in the ease of the frustum of a eoue ov of a i)yramid. an imdined co- 
 noid or segment of a spheroid, tlie IVus'.um of an elli[is()id (Muitained 
 Itelween parallel i)lanes inclined in any way to tlie axesol'tlie solid 
 and in .general to most of tin' S!)lids of my Tableau, and <vrn, liad 
 the idea su,!;;',u:ested itseii'so to do. siill would tiie formula nexci' have 
 become of ^■euelal apidiraliiui either in piactice oi' iii th<' teaehiu:;' ef 
 Stereometiy witiioui the lielj) of the Tableau, any nn)re than steam 
 without the steam (Ui^ine, or electricity witlioul the tcl('^i;rapli. 
 
 Tiie author must share with othe;s Ihe niLuit of the discovery as 
 applied to certain solids to whicii its ap[)licnbility has been shewn, 
 thouyii in a varied and more tu' less indirect way: but lureiii lie is 
 not singular, and peiha.ps it is as well th.il it siioulil be so, as ii iiut 
 adduces additional prof in sup[»ort of hiss 'heme ; norshouM he take 
 it to heart any more than Leverrier aiul Adanrs in relation to Xep- 
 tuuo, Newton or 1/ ibnitz com/ernin,:;' the discovery of ///M('y;/*'. 
 
 " It is pi, liu, say> Ih:' ilvd. Mr, liillion, in Ins letter to IJi.shop 
 *' Lai'oc((ue, that the author docs not uquiie tiiut oue should maku 
 
T 
 
 w 
 
 a 
 
 TMX ' T'/VKOM: TK'CAT, TAIJI-KAI'. 
 
 I 
 
 I 
 
 111 
 
 '■ us (• (iriiis rnniiiili in ininiv cases Aviicrc a movi- sini])l(' cxiucs- 
 '• sidii iii:iy ic|tlaic il. '' 
 
 'i'nic. IIh re aii'. iinl '• ma iiy ca.- ;'s. " lail some lliicc or roiir cases, 
 wlicic till' ■.•,(ii( ral lormiiTa may lie i(])!ac(«l liy one o!' a mcic simple, 
 iialuii': !)iil (ill' simnlilieil e\]ires.-i<iiis llnw diriclly and witlioiit 
 etVnit iVoni tlic p nera! forniula. Few peisons ; ;is yel, liave in lliis 
 ves|)e!-t nndristdnil the aiii iini s (ilij;;! in regard lo tlie ne(■es^ity, the 
 adva.nta.ue oldiie and only one and llir s;;m. lotninla. I'of all |;ossil>lc 
 solid-.. A iiern-;il ol'llie loiio'.vin.u' lelt •!• iVoni jnol'rs.M))' LalVaiice will 
 liowever siiow the wi.-ilom ol'liis views. 
 
 OiuIht II (/(■(•;■. l,~;i. 
 
 To M. ('. r> \ii.i.\[:;i.i:. 
 
 " >ii, - 
 
 '■ As to the coiicctiH'ss ol' the I'oi inula, none can doulil it, since 
 w lien nromul.uat in;; it. as ycm !iav(> don<' in your treatise iMii!, you 
 liavc youisrll' ;;iven all the necessary prool's to l>ear it out. 
 
 " With res]»ect to tjie prism or cylinder, to lieuin ^^ itli. (Article 
 Ivi.ll oi'yi'iir lreati->) il" o'ljecred to, tliat for ihi'se solids, at least, 
 your rormula far tVom oileiiuu; any ailvaiita.m'. Imt complicates the 
 i-ali'iihiiioii : 1 >houlil reply thai it is not so, since in this case tin- 
 loiiuiila is inini'. dialcly leducrd to the ordinary tide. Thnstiie pupil 
 will) !i;'-. already I;'ai ihd that every seetiou oi'a pi ism or of a cylin<UT 
 liy a plaiii' jiarallel to its liase is a li^iire similar ami e(|Ual to the 
 base, will say to himself at once " i'.ut the sum of (he bases, plus 1 
 times tile iiileiinediate section, is e(]iial to d times the base, amid 
 time the lia-e muitiidilieil iiy the>i\iiiol' liic height is simply tlui 
 same as muUijilyin^' once base by the whole li< ijilit : " and in 1'act; 
 this is the case, but by appl\ ii!,u' the Ibrmnia as well to the prism 
 iind cylinder, its use becmnes p'lieral f(M- all solids and oliviates the, 
 necessity of learning' or lememberiiiL;' one single additional rub'. Ibr 
 in line therein is the immense advaiita.^c <d' tlu' system \ on pioposo 
 " one ai.d the same formula " Ibr all possiI)!e solids, and the moment, 
 you iiitrodi a sicond. aye e\eii a third Ibr the jiyramid and cone, 
 iilbuith for t.ie pa.raboloid. to ciiiie which by the ordinary rules 
 would appeal more simi)le than by yours, you then iiitiodiice conl'ii- 
 sioii into the mind of the ]ui])il, id' the measurer ; from that moment 
 theri' is dan.ncr of conlbiiiidin.t;' these rtiles, of mistaUin^ii' tlie one for 
 the othe, «d' for^fltinu all of them or not lemeMibcriiin' them and <d" 
 the iieci ssily, for thai veiy veaxm, <d' a book of v<'rerciK'e. a theing 
 wUiih your t>ystem cuUiely dispenses w itli. 
 
INTRODUCTION. 9 
 
 Ajjain, for tlio pyramid or flic oono, as you give \t (1;')2.'> of your 
 r.oouH'try) the surl'acc of thcintoriiiediato section is a quarter of that 
 of tlic bas«> ; now t ^ 4 = 1, and I + 1 ~ '2, and it is dircetly seen 
 that tlic litli i»ait of 2 is tlic sanu^ tliin^f as tli(^ tiiirdpart of I ; wlienoo 
 it follows tliat Mic ;:>('iu'ral forniu' i ')rin;^s oiio back iinincMliatcly to 
 the ordinary rule, and that, without tlio necessity of knowing this 
 rule beforehand. 
 
 " Again, let the ])risni, th<' cylinder, tin; pyiauiid, the cone, as 
 lia])pens in practice, be ever so litth' convex or concave ; where should 
 we be with the ordinary rules, whilst on the contrary in this case 
 your fcunnila is the only one wliich can correctly cube the Ik y in 
 question ; and for the spindle (l.v'il) the jiaraboloid (l.'SfU) be its 
 lateral surface the least in the world too niiich or too little convex or 
 bulged, the ordinary rule which consists in innltipiying its base by- 
 its height and taking half the product, will give eitlier too great or 
 too little a vohnne. How them in this case arrive at the truth, 
 if not by y(»ur formula which, as in the case of the cask, introduces 
 into the calculation the versed sine (or very nearly) of the greater or 
 lesser convexity of the body to be cubed, that is to say the very ele- 
 ment which tends to vary its cubical contents. 
 
 " I have just pointed out the articles or paragraphs of your trea- 
 tise containing the ])roofs of the correctness of the formula for the 
 above mentioned Ixtdies ; paragraph l"))!! also proves it in the case of 
 the predate or oblate spheri>id or ellipsoid ; — l.'ifiS, for the segment of 
 this solid,— 15* ;t), for the hyperboloid, and 1581 to l.')U2 for the pris- 
 moid in general. 
 
 " The fact is that too few persons yet have taken the trouble to 
 examine your l>ooks, so great is the t«'ndency with us to remain al- 
 ways in the rut of the old routine. In fact, it has taken us 20 years 
 to substitute for pounds, shillings and ])ence the infinitely more 
 simi)le and exjieditious decinnil calculation of dollars ami cents, 
 where the mere shitting of a point works wonders : it will take 10 more 
 yet to appreciate your work, to introduce your formnla, your tableau 
 indispensible ixa it is, into the general educaticm of this country. 
 
 " 1 firmly believe, however (and it is too often the case, so little 
 is a juoidH't heeded in his own country) that you will i>e immediately 
 appreciated abroad, and I do not (h)ubt but that your tableau, once 
 known in the United States and iMirope where, I am told, you have 
 had your invention i)jitented : I do not doubt, I say, but that as soon 
 as you have publishe*! your prospectus in a foreign country, so soon 
 will ycm have orders and in great numbers for the introduction of 
 the tableau in all the Universities, Schools and other institutions 
 
10 
 
 THE STERKOMF-TRICAIi TAHLEAIT. 
 
 I 
 
 I 
 iti!' 
 
 !!f'' 
 
 f 
 
 I 
 
 
 (It'voii'd to tlio tpiicliiiij;', iint only of \ ontli, but al-o of niaturor joarK 
 Miiioii'j; all iialioiis. 
 
 " ('. J. L.-L>! i.-VNTK. 
 
 1 'ri>/cnn(>r. 
 
 'i'lic Inllowiiit;- letter Ti'diii a distinuuisliei! iiiallieiii iiii iaii of Al- 
 sace, I'laiice, set'J fortli visy coiicisely and j;eiieraIi/(> the nonien- 
 clafure of solids t(» wliicli tjie Stereonietricnl fonmila lipplies. 
 
 " \Vi;sTM(.i:i:i.ANi) Point, \. T... Itli X<>v. 1871. 
 
 " ^^y Dear Sir. — f see tliat yon are hent on (loin;.;' all you can to 
 render your rule foi' (indinii- the solidity of Jiny body \vha(«'V('r really 
 useful and praetieal. I <-(tnsid;'r that a Stereonu'trieal i ablean .inch 
 as lliat of which yon -^end nie the prnspeclus is (lie inuisiciisihlt* 
 coniplentent of tli(> ride for the majority (d' persons who may rc(|niro 
 to use it, and in coJlercM, etc., this will also be a j;reat liel]i to tlio 
 jinpils who are .-^tndyiui;' analytical geometry of three dimeiisicm.s and 
 sphericid triijononiel ry. 
 
 " Ilavinji had <»ccasion to verify tlio exactness of your formula 
 relatively to diltei-eiit kinds of bodies, I liave found in fact, as you 
 say in other terms, that it is strictly ai)plicable to all polyhedrons 
 without exception, the sanwofall solids <:feiierated l)y the revoln*^ioii 
 of curves of the second order around one or other of their ])rincipal 
 axes as w(dl as to sej;inents of these solids, Avliatever be th(> direction 
 of tlu' plane of section. In this respect, it seems to nn- you mi.niit 
 add to y()ur prospectus, that in general the rule is ai)plicable in a 
 manner rigorously exact : 
 
 " 1° To all solids generatetl by tlie revolution of a straight lino 
 around two plaiu's parallel to one another limited in extent by con- 
 tinous outlines of any form whatever and irrespective of the varia- 
 tions of the relative vtdocities of the two extrennties of ihe moving 
 line. 
 
 " 2° To every s(did generated by a i)lane of dclinite (tutlino 
 moving witii uniform velocity in a straight lin<' frttni one point to 
 auotliei', whilst its surface varies in a corre.sponding maiiner as tlio 
 square of the generating cord of a zone of any conic s» iiioii. 
 
 " As to .stdids generated by regular curves of an i.rder higher 
 than the conic si'ct ions, it is generally easy enough to sulrdivide th<'m, 
 with an exactness luoic than sufticieut for all practical purposes, — so 
 that tlie resultant partial solid may bc! classed in one or other of 
 the two categories of solids which I have just dc-cribed. 
 
 " Moreover, aiiart from certain kinds of ungulas of eleineiitary 
 s(dids, I see few bodies of regular forms which may b«' nu^t with in 
 l)ractical nieasurenu'ut of guaging or be in u.se in tiie arts and trades, 
 for which a subdivisi(ui repeated beyond two oi- three times is neces- 
 sary ; and as for irregularly shaped ungulas it w ill be very easy for 
 you to refer to them iu your Key to the tableau. 
 
IKTRODUrTIOV. 11 
 
 " I foiii:'! il very ('(tiivciiinit, lately to .■>j)])ly lln' rule (j^'occcd- 
 
 inu; by siil>(li\ isioii) ti> a tiiiiik of a liy pi'ilxilic ronoid yciu'ratcil I»y 
 
 tlic revolution of a iiyix-rboiii of I lie "jtii order aromitl one of its 
 
 asymptotes, witli (lie view of veryfyiiiLj liie dej^ree of itieeisioii 
 
 broii.iflit to Ix'ur on the coiislrnetion of a eonver^iii^f eojiperlnlie 
 
 havinji; that ]»arli( nlar sliajx^ wliieli I liave employed in a hydraulic! 
 
 e.\l)eriment. 
 
 "R. Stfckki.." 
 
 Mr. Steekel, .■'''((>)• havln.n' jtroved the nia11ieniati<'al eoneetiiess 
 of the foiiiiiila for all bodies iiienlioned by the author in his ]»ros- 
 jtectds. has m:ide ioni;' and laborious analyses respeetin.if the •ipi)lie- 
 alion ofllic formula to the un^culas of the cylinder, cone, eonoids, 
 and sphei'oids, &e., &(•., r.nd lias demonstrated that in faci, as <lm 
 author says, the maximum erior for these soli<ls which are moreover 
 rarely met with in ]>ractiee, does not ,i,'enerally exceed .Oil.') oi- the 
 half of one p'M' cent, taking;' the whole body at one measurement, and 
 (hat thiserror. lit ■ Ic as it, is, is easily eliminated and is reduced to 
 nothin,!;'. so to say, by subdividin;.'; the uii^ula. the half frustum of 
 si)indle, i\.c.. into (wo or at most into three parts, and a}ii)lyin;i' tint 
 formula to lach of (hem and afterwards takinjj; the kuui of the com- 
 jioneut i>ar(s. 
 
 'rhelir\(!. Ml'. Billion (Madiematician of (he seminary of St, 
 Se.lpice, .Mniu real.) says that "the formula is ajtplicable to a whole 
 series of other bodies \\hich the audior has n(>t sjioken of" audi here 
 mean boilies to which it apjdies exactly. In fact, let us sujtpose any 
 body wliatcvr mentioned by (he author, for example the frustum of 
 a pyramid. This boily may be considered as formed by (he juxtapo- 
 sition of an iiiliidty of jihines ])ai'allel to two bases. Now, let us sup- 
 ])ose all tlir ,ilau<'sstrun,i; from one base lo the other by any strai.^lit 
 or curved ,■ .. \\U;\[ ver, an^' susceptible of l»«'in!4' curved or bent in 
 any manner whatt^ver. liy incliuinu'. bendin;^, cnrvin;j; (M twistin;i; 
 this kind of <lir<>xtrix, every section of (he said body is similarly af- 
 fec(ed and a new solid results, of the sanu- bases and same heiiiht as 
 the llrst, perlVfily e(|uiva1<'nt in volume, but curved to the arc of a 
 circle, parab(>!:! or other curve under any law whatever, or bent of 
 any an;;le ^v!latev^r, the sections r«'niainin.i,' always in the same 
 plane. If the direction is changed into spiral and tliat the base is a 
 circle, we have a a\ readied pillar, &:c., &c. There then are a multi- 
 tude of bodies to which the same formida apjdies, for the reason that 
 tliey are eipiivalent to the first ones. 
 
 "The author demonstrates his forniuhi to be strictly correct for 
 a great number (if bodi«>s enumerated an<l as approximate as couhl 
 be desired ior tliose to which it does not apply in a measure absolu 
 
12 
 
 THE ST KREO.M ETHICAL TABLEAIT. 
 
 
 ^11 
 
 ■ 
 
 tely exact. The proposition iiiul the deiuonstration, in every case, 
 
 are exact nud true. 
 
 *' As for other bodies, it is true tliat tlie more niiineroiis the sub- 
 divisions, if need be, tlie closer the approximation to the true ans- 
 wer. " 
 
 " Tlie works published \\\\ to the present time, says Mr. Bhiin, 
 c<mtain ijuite a numlier of ditlcreut rules lor liudiuif tiie contents of 
 solids formed by tlu^ revolution of a curve of the second order around 
 its axis. These rules which f;ive rise to as many formulas must bo 
 remembered ; that is, Ave must overload the memory, witiiout beiii<r 
 certain of its readiness and faithfulness at tlie moment we need any 
 particular formula. Mr. IJaillairge's formula is ijtncml and dis- 
 penses with this somewhat painful effoit. 
 
 " Moreover, this formula is apjdicable when a solid is j;enerated 
 by a curve revolvin;.!; around an axis not its own, and that is a case 
 whidi lias rarely, if ever been foreseen by tlie authors who liave 
 written on the subject and whose habitual fault, witiu)ut inteiuling 
 to lessen their i)r()fonnd knowledge, has been to have always kept 
 theory too much in view to tlu' great detriment of ])ractice. 
 
 " I have established with Mr. Ste<kel, that in the greater num- 
 ber of works on mensuration no nuntion is made of a s])heroidcut by 
 a jdane in any direction obli(pie (in«lined) to its axes, a case provided 
 for by Mr. Baillaiige, NO. lotitlof his treatise. No more is anything 
 said of a paraboloid in tlie same conditions, (1,5G4) and still less of a 
 liyjarboloid, (1,5()G). 
 
 "Guagers, partieularly, may derive immense assistance from 
 Mr. 15ail!airge"s formula, since the great majority, it might almost be 
 said, the wliole of the tuns, puncheon.s, barrels, boilers, kettles, res- 
 ervoirs, &c., and all ves.s<'ls generally u.sed for holding li<[uids, are 
 notliing but segments or frusta of circular, parabolic, hyperbolic, 
 or elliptical spindles, si)heroids or frusta thereof, spherical hyperbo- 
 lic (calottes) (h)nu's, frusta of cones or of conoids with concave or 
 convex surfaces, &c. 
 
 " Mr. Baillairge's formula also dispenses with the difficulty of 
 classifying the solid to be measured, an operation subject to nniny 
 cirors in practice. 
 
 "In tine, the same formida supplies the nutans of finding the 
 qinmtity of liquid in any vessel only partly tilled, in any positi(Ui the 
 vessel may be and without being obliged to change this itosition, 
 (1,577, 1,.')78, &:c.). 
 
 «' I should far exceed the limits of a newspaper article, if I wish- 
 ed to enumerate all the advantages of the discoirry nnule by Mr. 
 Baillairge, for it is really a discovery of great importance. 
 
 I say frankly, I at tirst doubted the exactness of Mr. Baillairge's 
 
 I ,K 
 
iNTiinnnrTiov. 13 
 
 falniliitinii.'*, and boforc cxprcssiiij^ an opinion. I ininlc tlif ciilcnln- 
 tions niVM'lf, tlicn coiisiiltcd clt-vri' men and well vt'iscd in praiticc. 
 I Imt rccoi'd licro tiicir opinion '.vliicli will l»c contirincd later hy all 
 those who shall <>Mip'o,v the ni'W system of ineasnienient. 
 
 Now as to the correefncss of tlie results ^fiven by the formula in 
 tin- ease of frusta of spindles, for instanee, and of vessels of eapaeity 
 of this kind coniitared with those l)y tiie ordinary rules, the author 
 could not do Ixtter than inihlish at length the pertinent remarks »>f 
 professor Gallafi;her on this sultjeet. 
 
 '' That this formula is matliematieally correct, as applied to al 
 the solids enumeiated by you in y(»ur prospectus, there is <d" coursel 
 uo doiiltt. You iiave fully demonstrated this in your valiialtle work 
 on Geometry and Mensuration published in lH(!t!. Mr. Steckel has 
 uot been slow in slutwing this in a nn)st comis' mannei- in his letter 
 to you on the suliject, to say nothinix of the letters of the !>!{. MM. 
 Metliot and .Main^ui on part of the inofessors of mathematics <»f the 
 Quebec Semimiry and Laval I 'ni versify, where tlu' expressions 
 " etonne*' and "enchaute " sutlicieiitly show the hi;;h estimation in 
 Avhicli your discovei'y is held by these comix'tent Jadiifs ; but, in my 
 opinion, you do not sutUcii'ntiy insist on the ^leat value, the many 
 aiul manifest advantages of your ride as applied to spimlles, the 
 middle frusta of which are met with every day and in every i)art of 
 the civilized world under the thousand and one forms of casks of 
 every conceivalde size and variety, and the necessity of nu-asuring 
 which with promptness, on account of their number, and with ac- 
 curacy, on account of the generally valuable nature of their contents, 
 renders some simple, easy and commodious rule, like tiie one now 
 proposed by you, of the lirst importance to all maidvind. 
 
 Now, Sir, that your rule embraces these valuable re(piisites, let 
 me compare it, in its working and in its results, with the rules laid 
 down l)y some of our best nuithematiciaus and authors such as Houny- 
 castle for iustauce, see Kev. E. C. Tyson's edition of his mensuration, 
 page 147. 
 
 Problem XX\TI (for example). 
 
 " To liud the solidity of the middle frustum of an elliptic spindle ; 
 " its length, its diameters at the middle and end being given ; also 
 *' the diameter which is half way betweeu the middle and eud dia- 
 *' meters being known." 
 
 llule, " 1° From tlie sum of three times tlu^ Sijuare of the middle 
 *' diameter, and the s(iuare of the end dianu'ter, take four times the 
 " square of the diameter betw^eeu the ndddle ami eml, and from four 
 *' times the last diameter take the sum of the least diameter and 
 ■' three times that of the middle, and 4 of the quoticut arising from 
 
14 
 
 THE STKREOMCTniCAL TABLKAU. 
 
 }]■■ 
 
 
 § 
 
 '' (lividinj;- (lie i'oiiiicr »lill( rtiicc l.;\- tlic latter will ;j,ivc the cciilral 
 '* ilifltiiKr, 
 
 *' i" Kind tliciixcs nf ilic .ilipso liy I'loblcni IF. niul llic area 
 '* of llie elli|iliciil se;;nieiU, wlidse eord is the leii^^tli of tlie iVilstmii, 
 " l).v I'u.ldein V. 
 
 " .'{'^ Divide tlir(c t'. - tlie area tliiis I'Diiiid 1'v tlie leii;;tli of 
 '' tlie fnistiiiii, and from t lie "laotient suliliai't the di,.k i. nee lietweeii 
 " tlie middle diameter and that of the »'iid, and niullii)ly the ro 
 *' inainder lt,v ei<;!it times the eentral distance. 
 
 '• P Then iVoni the sum of the s(|Mare of the hast diameter, 
 " and twice the sipiare of that in the niiddh'. take ilie inodiict last 
 " found, and this ditl'erence miiltiidie<l liy ihe lenulh and the pro- 
 " dnct a.nain hy .i2(il7!i!t. «SkC.. will ,i;ive the solidity i((|nired.'' 
 
 llei-e, tlie mind is absolutely Itewildcrcd at the nieic recital of 
 the mnltifarioiis operations to be i)erformed (not less than 27 in 
 iinmber) and the meie res-iti-; of (>ach of these operations, irresi»ec- 
 live of the details of the ui iltiplications, divisions and other com- 
 ]»nta(ions necessary to ari'iv'' at them, take up twi» whole payi-s of 
 the book, 
 
 Ajtiilied, say to a cask oT'^S inches in len.^th. biinu diameter 121 
 inches, hea<l diameter 'Jl.(! inches, and dianu'ter half way between 
 In'ad and ban.!;- •-.'•'{.lOfMdt inches, the result, as t'ully wniked out at 
 pa^e l-IH, 14i>of said book, ;^ives 11,S,")-1^ cubic inches, very nearly, 
 or 51 f;allons and .1 half-pints. 
 
 Now, the same example, Sir, by your fornmla, brings out 11,- 
 d'h).2 cubic in., which dilfers from the last result by only .<»()()( lOl.i 
 or less than half an inch on nearly I'iOOO inches, or the :24()th part 
 of one jter cent in excess, tiie 11th ]iait of a nill. 
 
 Not only then, is your foiniula in this case to b' considered in 
 every respect as accuratis as that of IJonnycasth'. b:;t it is really 
 ornios ill practice; for, even if the error in excess attained tlie- 
 nnixiniuin of .01).") or i of one per <<mi!^, where is the practical 
 mea.siirer or <;auger who, for the .sake of a quart on a .")(• jrallon keg 
 or half a gallon on a hoghshead, would, could (Uivote hours of hi.s 
 time to calculate l»y tlu; old method what can be done with greater 
 accuracy and in less than 2 minutes by the new : for. very merchant 
 will tell you that in i)ractical cask gauging tlier. s ..••■nerally au 
 error in excess or in defect of from one to two gallons on .i hogshead. 
 
 And oven this comi)aiative accuracy oflheoM rule, can only 
 l)e arrived at by taking in all the decimals, which no one would be 
 likely to do, on account ol' the immense labour of the comimtations ; 
 whereas, by the new formula, by reason of its great simjdicity and 
 c(Miciseness, all the decimals may easily be taken in and no harm 
 can result at some of the last decimals being neglected, since tb« 
 
iNTunnircTio.v. 15 
 
 rcsiilf !is s]m\vii above is, iiiid. for cdiivcx loi'iiis, iilwiiys is, (liou^'Ii 
 tvcr s(i .slii;;;.!, . in cxi'i ss of I lie triu- conffiit. 
 
 I iiMi wicii;' liowcvcr ill iissiiiiiiii; t!i!it the ni;i\iniiiin error in 
 disk ^Mii^iii^' li; your rule is .(M),") or tlie iialf of one pel' cent, neither 
 do you K!iy so in your iMospeetiis. iiiiil on l!ie eonliiiry you sliow 
 iiio>t satisliielorily at pa.n'e 7(H, 7()!» of .\ our said treatise, in tli(< 
 iiiinier(»use\ain|i]es y;iven hy yoii and I'lilly worked out and coin pa red 
 ill eaeii ease wiili the results .id veil liy i'oiiiiyeastle's rules, tiiut the 
 iiiaxiiiiiiiii enoi' in exeess <loes, in your lirst and •?ii(l (^xaiiipW s, not 
 exceed \ of one per cent or one (piarl on a Iio.l^s! ead : in ex. S it is 
 4, of I per eent ; I'.x. 10 ,!;ives f lie iiiaxiinani error as ,', of I per «'eiit ; 
 ') ;;ives 1 of 1 per eeiil ; Kx. 1 and l!J ('■! . ,',, of I per «'e.it ; Kx. !), i',y 
 of 1 per eent ; I'.x. vJ and IvJ ( I and ."'). ' <<\' I per e<'iit ; and ex. 7, ^\ 
 (if 1 per eent. and these exaiiijiies eo\cr all varielii's and sizes of 
 cireiilar, (dliptie. and jiaraholie casks. I'lat is of tlie three varieties 
 ^jeiierally met with in practice. 
 
 I)iil ill dwelliii;;- oil the foriniil:!. I !; i 1 liaveas yet said notliinj; 
 of the all im|ioi taut '• Stereoineti ical i'aldeau "' without which, a.s 
 ,vou i>ertiiieiitly icniaik, the rule would lie almost as useless in 
 leaeliin;n- iiien>iiiation in schools, if imt in tlie i»iactice (d' it, as 
 steam without the steam eii.uine or « leelricity without the tcde^'iapli. 
 
 There are many other advantages, ap.iit from the mere meii- 
 nuratioii of bodies, wliicli your tableau po>sesses, as enumerated by 
 you in ycnir iHosjKM'tus and which it is useless for me to dwell 
 upon, a.s I lully eoiieur in all that .\<iu el. dm for it; th(Uij;'h J think 
 you iiiiglit liav. t'lirther insisted on the ' .aiitaj;*' of siu'li a tableau 
 ill tlio studio of the apiuciitice, ; : veil of tlie professicmal 
 
 architect, who will, aiiiont;' the niod'ls. tiiid that of almost every 
 conceivable shape er proportion of roof, dome, &e., Avhicli lie may 
 b« called up(!ii to desiyn; the Civil Kn.niiieer, every description of 
 prisinoi'd to be met with in the cuttings or embankinenls for rail- 
 roiuls, canals, docks, \c., or in the piers or abiitnieiits of brid,i;cs or 
 other structures ; the ineclianical enj;iiieer, every variety of boiler, 
 copper or olln ' ve.ssel iiud the eomponent parts of ail sorts of 
 machinery. 
 
 Quebec, 9lli December 1871. 
 
 Extract frnui .• D'n^ciission of liuillniriir's Sirndnictricdl /<>r»iiil(t In/ 
 
 the Rei\ X. Muhiijiii, pvofexHor of MailtfmatkH at tin; Laval i'lii- 
 
 rersHi/. 
 
 It is easy to conclude that, in practice, unless it be known before- 
 hand that it is reqnind to cube a sjihere, an (dli}»soid or se^nieuts 
 of those bodiv's. it would be extremely loiij;- and dilbciilr to establish 
 
 1° the kind of ciirve to w Iiich the 2 directrices belong and 
 
 2° the posiliou of their axes. 
 
• "'"T 
 
 IG TIIK STERKOMETRrCAL TABLKATT. 
 
 Tlicvcrorc it is nnicli simpler to siipiMtsc tlie body to lie ciilicd 
 iiividcti into !i Cert ill II niiinlifr of seel ions .so tlml the ciirvrfl side iiiiiy 
 lie snisihly a sirninht line. Tlicsc scclioiis, like triiiicjitcd coni'H, aro 
 ciibrd very readily l>y the stereoiiielricai Ibrniula. 
 
 'I'liis is moreover, the (tidy resort for all solids that the sten>o- 
 metrical formula tan not ciilie at once. The same lemark apjilies << 
 fofHiiri to solids terminated laterally, partly l»y planes and partly liy 
 a eiirved surfa«'e. 
 
 Hecause, in praetiee, tln^ stereometriea! foiinnla eannot ffive, at 
 one attempt, the exact contents of certain Ixxlies, we nnisf not heiico 
 dednce an arunmeiit ajiainsi this formnla ; and for the very siniph* 
 I'eason that, in these cases, tiie measurement of the IkkIi/ as n irtittle 
 is imi»ossil)le. And if in some exeeedin;;ly rare «'ases, certain very 
 <!omplicated formulas exist, pvaviirnUii they will ;;ive a resnlt less 
 exact thiin the stcreometrical foinuila. 
 
 Up to the present time, a eeitain niimltei' of bodies were worked 
 ;, by easy i-nles ; otheis by very complicated ones; others a,u:ain had to 
 
 i' ijj, bo <livii\ed mentally Into different parts or wvw icdiiced by approxi- 
 
 mation ; now the stcreometrical formwhi ajiplies with advanta^^^e to 
 all these cases 
 j*;, 1° It is as easy to apply as any one whatever of the old rules. 
 
 ?' 2° In application it is miicii simpler than a numlter of others. 
 
 ;j^ It can compare very advantajieously with all tlie others on 
 
 account of its nre.it exactness, accordinj; to the case in hand : the 
 
 precediiiji' demonstration and discussion lieiu^- intended to show tho 
 
 ,j(i conditions in wliicii the result is iinoiirously correct in ordei' the bet- 
 
 •I,' ter to (loint out what course to follow for resolving certain problems 
 
 in a satisfactory manner. 
 
 'Die object of the accompanyiiitr formula is and tlie result of 
 its employment will be to itopiilari/.e and ;;enerali/,e the study of 
 solid ucometry. The author is free to jidmit that in I'liiversities 
 and other establishments of lii.iiher education, the advanta^^e of tliis 
 one and only one iindeviatin;;' formula for the solid contents of all 
 bodies miiy not at lirst si,i;ht ajtpear to offer the same advantages as 
 in elementary and other institutions of the kind; for in the tirst, 
 other riib's are taiii^ht and the study of al.u'cbra and the hi,i,'lier cal- 
 culus art'oid facilitii's not jios.sessed by the second, of arrivini;' at sa- 
 tisfactory c(Uiclusioiis; but, onco out of collef>e, all this al.ycbra and 
 calculus, all these ever varyin.H' formulas are soon, very soon forj;-ot- 
 Yf- ten, and the piofessional, the iii;;lily educated man will lose; all trace 
 
 of his mensuration of solid forms ; while the mere artizan, the me- 
 chanic, the eufjineer ami architect, the measurer, gna;ncr and juacti- 
 cal man of every jurade, who has never learnt aujj;ht but to computo 
 areas of all kinds by simple rules not easily forgotten, need only- 
 have recourse to sim|)le addititni, multiplication and division to work 
 out the contents of the most complicated solid, or of a ves.sel of ca- 
 pacity of any kind whatever. His less fortunate^ employer, the 
 learned professional, will ajtpear to a (li.sadvantago in the eyes of 
 the world at large from his incaitacity to do the same, unless he alsa 
 shall learn how to use the formula in every case where college and 
 univer.sity rules and formulas have been long ago forgotten. There 
 are now a days too many other sciences to learn and college life i» 
 5 too short to allow of devoting yeais or even months to a study which 
 
 can be mastered in a day, if old fogyism and conservatism will but 
 I Htand aside. 
 
 r' 
 
 i 
 
KEY 
 
 TO 
 
 BAILLAIRGE'S 
 
 STEREOMETRICAL TABLEAU. 
 
 NEW SYSTEM OF MEASURING 
 
 BODIES-SEGMENTS, FKUSTA AND UNGUL.E OP THESE BODIES 
 BY ONE AND THE SAME KULE, 
 
 (1) It is tiscful to <,Mtln'v inul jtrcscMit iiiidcr a conciso form tlie 
 various rorinulii' or rulcvs Avliicli relate to the caleiilation of tli(! super- 
 lieiesauil vohiinosoftlie various Ixxlios and figurcB previously Hi)okeu 
 of.i A synopsis of tliis kind will allow one to refer more easily 
 to tliose rules, in order to find at a glance tlie one required, con- 
 cerning llie problem to be sol /ed ; and some practical examples of 
 the vaiious cas(!s, will better teach the pupil the course to be followed 
 to arrive at the result required. 
 
 ("i) To determine an areji or a volume is, as Inis been seen, 
 (itlltl iind Jl<MI Cr. ^) to find the number of times that this area 
 «)r volume contains another area or volume, which is taken as the 
 
 1 (New Troatisc on rectilineal and spherical geometry and trigonometry, Ac., by 
 the same author). 
 
 2 liKMARK— The iiuirbers in black print and in paronthoees, as (333 and 
 1014 Q.), (^4 0.), (1018 G.), &c., refer to the "new treatife on rectilineal and gpheri- 
 oal geometry and trigonometry, ifec," by the same author, and tlie numbers also in 
 black |iriiit nn.l followed by a T., rofur to the present treatise in which the correspond- 
 ing numbers of the paragraphs or articles relate to the defiiiition, demonstration orsolu- 
 tiuo, as the case may be, of the thing enunciated in the treatise in question. 
 
'~r 
 
 ;l:!f„ 
 
 18 
 
 KEY TO THE TABLFAtT. 
 
 :.s: 
 
 il 
 
 I'll': 
 
 jiKMNiiiiii;^ unit (81 C!.)- 'I'Iium, avIm ii it Ik snid tlint ii sfinnif fof^o 
 foiitiiiiis :jil sqiiiirr t'cct, it iiiiist Im> iinticisltiotl tliat tlic iiicnsiir' 
 iii^ unit iH tilt' squiii'i- toot an<i tliiit \\[\^ unit is contiiintd 'M* 
 tinii's in llic scimirc toisc, tltc liiitMl toisr bi'ifijj (! t'tct. iind (> < (> 
 = ^. Likrwisc, if tlic cultif toise, ('(uilMiuN '^l(! cultic i'ctt, it l» 
 that tlu! <iil)i(- loot i.s in tlnit case tlir unit ol' niciiHiirc, :inii tluit \h\>^ 
 niiit i.H rontiiincd 2U> tinit's in llic toicc, \vlii<'li Itcinji- (> lineal tVct;, 
 its yoiuiii*' is (lOlN («.) «i (i x(i — UKI ; and if llit- niMc im tio 
 coHtainrt 1(H)(( culiic (U'ci-irictrt'w, it is tliaL tlic nlt'a^ml•illy unit is tlie 
 ded-UK'lro and that JOx It) < lO^KMtO. 
 
 (Jl) 'I'lu' nicasiirinu unit whicli it is proper to einplov is nsmilly 
 tlio K(|iiaro or tliticHhe (as tlw case may he) whose side is (:i;i:i and 
 loll G«) the lineal unit which st'ived to estahlish the lineal dinn-n- 
 sious of the li<,n»ie to be measured j but it is clear that notliinj;' jue- 
 vents one from couiputiu;; in scpiiiic metres or yards the snifaee of 
 H lij;uic v'hoHe dimensions are expiessed in feet or inches, &(:. j 
 and in tliu 9i»m« way it will be indillereul to express in cubic feet, 
 in nu^tres oi toises, &('., the content of a body or solid, whose lineal 
 dimensions niij^ht !)'• <;iven in yards, feet, or inches, &,e. ; paying 
 attention only to the reductions necessary to ciian^^c the.ijiveii ele- 
 luents into elements of another ivame, that is, of a dillcrent valne. 
 
 (4) The fornuila of the author, to find at once, or by decomposi- 
 tion, the volume of any body, is as follows : • 
 
 ** To the sum of the bases or opposite and pamllel iii<l cucr/N <>/ 
 " the 1)0(111 to he meo^ufed, or of aini one of Us eotrpotuntt slices, add 
 ^^ J'oar-ti. (('« the urat of a seetion juirollel to these Ixtses and situated 
 ** half-wan t>etn'een them, and mnttipUj the sum of these areas Ini the sixth 
 " 2mrt of the height or length of the solid."' 
 
 (5) The new system rlien reipiires but the simph* nieasMrruunit 
 of certain surfaces and sections of the body under consideration, 
 Bince what rcnniineto be done to arrivcat theproposed vuhinie is but 
 a simple addition of these areas and the multiplication of their sum 
 by the height or length of the solid, to take uftevwiirds the sixtik 
 part of the result. 
 
 («) But there is often a superficies or area to be measured iude- 
 pendant of every consideration relating to the volunu' of the body of 
 wliich such area forms the total or partial surface. 
 
 For these various reasons^ il is then proper to treat (irst of the 
 
 • mil 
 
MENStlRATION OP SURFACES. 
 
 19 
 
 MENSURATION OI^' SURFACrlS. 
 
 PROBLEM I. 
 
 To determine the area of any square, rectangle, 
 lozenge, rhomb or parallelogram. (') 
 
 E 
 
 II 
 
 1'' : 
 
 (7) IHJI.K. I Mi(Uii>hi the hrv^ (182 G.) hi/ the heiijht (ISO «.) 
 and the product will l)c the required area (33,1 and 341 €».)• 
 
 Kx. 1. What is tlio iircii of a sciiiaro wlioso Hide iiioasures 204.3 
 feet? Ann. 417."38.4i) s<iuaie foot. 
 
 2. Wliat is tlio iiiuulicr of squares (rlie square is 10x10 = 100 
 «(inart' feci) in ai'e('taiif,'iilar iloor, ceiling, partition, wainscot^voof, 
 &,o., wliosc k'nglli=:GO foot and bieadtli 35 feet 1 Ans- 21. 
 
 3. Wliat is the area of a parallelogram whose base is 12.2.') 
 and liei-;lit S.r> ? Ans. 104.125. 
 
 4., How many s({uare yards of painting, in a rectangle whose 
 base is (HiM fert and hcigjjt 33.3 feet t Ans. 245.31. 
 
 5. To deternnne the area of a rectangular Iloor whose length is 
 12i feet, and breadth !> inches ? Ang. 91 sq. inches. 
 
 (1) Sec tho component fftcog and pootions of the prisms and other mrdels of the 
 Tablraij. Tlioso. figuros are met with every whore in tho praotio<< o/ thu measurer, 
 geometer, surveyor, .fen. ; thus, tho floor or coiling, or one of the walls of a room or apart 
 loent will be generally a square or a rectangle, Tho ?ame for a door orwInJ.ow, part of 
 which at least will be rectangular, and this figure will be met again in the developed 
 surface of a door, or any other opening which would be arched without being splayed, 
 as well as in tlio development of tho circumference of any apartment the plan of which 
 were a circle or any other curvilineal figure and of which it will olways be ea«y to 
 obt.iin with snfRcient accuracy tho curvilineal dini^.,jions with the (lid of a ribbon, if tho 
 surface to be inoa-^ureil bj convex, or by means of a rod, thin enough to be fitted to 
 tho concave surface to 1)0 measured. As for tho oblique-angled parallelogram, such 
 surfaces will often be met with wlicre two superposed coursos of stairs of tho same in- 
 clination occur. The subdivisions of territories into cantons, Iota and parts generally 
 affect for tho most part figures of this kind. 
 
20 
 
 KEY TO TUB TABLEAU. 
 
 '1 ■!, 
 
 ■I 
 
 I 
 
 I 'I 
 
 li 
 
 6. Required tlic nnmbor of square yanls of tapestry necessary 
 to cover a puiuUelnyrani, wlioso base is S7 feet, and lieiglit 5 feet 3 
 inches ? An«. 21 j\. 
 
 ?. How njiuiy square feet of glaziiii^ in a rectaiij;uliir window 
 being 75 inches liigli by 37i inches broad '! Aiin. 75 < J37i-f- 
 
 144 = 19 square feet 7Ui square inche8=19[24« = 19.53125 feet, or G'. 
 3" X 3'.U" = J9.G5 = 19'i-J^-'i= 11)^^ = 19.53 or about 19i sq. feet. 
 
 8. How many square inche^i of fjildin<>- are reijuired to cover a 
 6urlac(! wliose h^Dgth is 3 feet 3 inches and deveh)ped breadth pr 
 perimeter 13 inclies ? An§. 507. 
 
 9. Wlnit is the number of superficial feet in all the mouldings 
 of a stone, wooden or plaster cornice, &c., whose; length is GO leet 7 
 inches high and developed breadth or contour 3 feet 3i inches ? 
 
 A US. 199^^j (very nearly) sq. feet. 
 
 REITIARK. Tiiese developed breadtiis, contours or perimeters 
 are obtained by means of a thread or /.lybon whicli is b(Mit round tlw 
 various mouldings, in a direction i)erpendicuhir (S>00,998^G.) to 
 their length. 
 
 10. Required the number of square yards of varnish on a door 
 wliose lieight is 7^ feet and developed breadtli (measured around all 
 tlie mouldings, «fec.) 3 feet 11 inches ? Alls. 3sq, y. 2}.p s. f. = 3sq. 
 y. 2.375 sq. 1=3^^1-^ sq. y.=3.2(j39 sq. y. say nearly 3^ sq. y. 
 
 11. How many square metres in a piece of ground being ll3.7o 
 metres long by 10.5 metres broad ? Aiis. 1194.375. 
 
 33. Determine in scpiare arpents and perches, the area of a 
 farm 40 arpents 5 perches deep or long, by 3 arpents 7^ perches 
 front or broad (10 lineal perches forming a lineal arpent and conse- 
 quently 10 X 10 or 100 square perches, a square arpent.). 
 
 Ans. 151 arpents 87^ perches. 
 (8) Rl^LiI] II. Find the product of the two adjacent sides of the 
 pnrallelotjram, and midtiply this product by the natural sine of the 
 included angle. 
 
 It has been seen (1231,1'^ «.) that 
 •when R=l the perpendicular D E of 
 the right angled triangle AED is equal to 
 tlioi)roduct of the hypolhenuse AD l)y the 
 sine of the angle A ; bat DE is the height 
 of the parallelogram AC, and since area 
 AC = AH X DE and that DE=AD x siu. A, it is clear then that area 
 A(; = ABx ADxsin. A. 
 
 G D 
 
 C 
 
 V 
 
 
 1/ 
 
 A J 
 
 K 
 
 a 
 
 ! ; 
 
MENSURATION OF SURFACES. 21 
 
 Ex. 1. What is the area of arliomb or lozenge whose aide is 25 
 cliains and iiicluded angle 57° 3;}'. Ann. 25 x 25=G25, and 
 
 025 X .84380 (nat. sin. of 57°33) = 527.41 25 sq. ch. 
 
 (9) To solve this same problem by logarithms ^ 
 Avhere R-- 10, we have (1221) 1° G ) li. : .sin. A :: AUDE ; whence, 
 DE -ADxsin.A . ^^^^^ areaAG = ABxDE and by substituting to 
 
 DE, its value ^ ^<i'i. A^ ^^ obtain for area AG, tlie expession 
 
 AB X AD X sin. A ^^. ^vhich is the same thing, area AG = 
 
 A T> w AT") X '^i M A 
 
 _ '■ — ^ ; that is, we must add together the logarithms of the two 
 
 adjacent sides and the logarithmic sine of the inclndcd angle ; this sum, 
 
 diminished hy the logarithm of the radius, will be the logarithm of the 
 
 rc<i Hired area. 
 
 / + log. AB 25 1.397940 
 
 + log. AD 25 1.397!)40 
 
 Log. area AG^-j + log. sin. A 57°33' 9.926270 
 
 -log. K 10. 
 
 Log.area.AG= 2.722150 
 
 -"ei 
 
 The next less log. 2.722140:= 527.41 clis. j the ditt'erence between 
 this log. and the log. found is 10, to which adding (I28ft O.) two 
 cipliers and dividing by 82, we have ( , cry nearly) 22 which is added 
 to the right of the ligures 527.41 already found, making as before, 
 527.4122, 
 
 Ex. S. lletjuired the area of a farm the sides of which are res- 
 pectively 40J^ arp. and 3 arp. 7i per, and the enclosed angle 57°33'. 
 
 Ans / + log. 40^ arp. or 405 jier 2.()07455 
 
 + log. 3 arp. 7i^ per. or 37.5 per 1.574031 
 
 + log. sin. enclosed angle 57" 33'. .. 9.926270 
 ^-log. K .... 10. 
 
 Log. required area. = • 
 
 Log. required 8urfacc= 4.107756 
 
 The next less log. .107549 corresponds to the number 1281 ; the 
 difference between this log. and the log. found is 207 ; adding tlie 
 and dividing by the dif. (D) 338, wo obtain 612426 which are written 
 (1286 CJ.) to the rigiit of the number already found 1281 to get 
 128161212() ; but the characteristic of the found log. is 4, which cor- 
 responds (1273 *! ) to 5 integer figures ; then the n-quired area 
 is 12816.12426 peiche^j . ■ 128 arp. 16.124 (or 16^) perclic , nearly. 
 
 (1) For the tables of logarithms, boo the " New treatise of Geometry and Trigo- 
 nometry, &a," by the eame author. 
 
22 
 
 KkY TO iIIE TABLEAD, 
 
 PROBLEM II. 
 
 To find the area of a triangle. ^ 
 
 fh 
 
 E H 
 
 When the base and altitude are given. 
 
 I 
 
 
 I'll 
 
 !!?, 'I, 
 
 
 (10) RUL,r;. jVitUiplji the hasr hj/ the (tUUiidc <iu(l take half the 
 product. Or, mitltiply o)tc oj'lliese (rntwusions by half the other, (184 1 or 
 :M8 Ci.) 
 
 Ex 1. Wliat is tho area of a triaugle whose base is 025 and al- 
 titude 260 ? Ans. ](;t2.-,()0. 
 
 2. How many square yards of plastering ar(i there in a trian- 
 gular surface whose base is 40 feet and altitude 30 feet ? Aiis. ()<;|, 
 
 3. How many square metres in a trianguhir |)ieoo of yrouiul 
 whose base measures Hi) metres 7 dcei-mctics, and altitude 17 metres 
 39 centi-metres ? Ans. The required surfaee=:30.7 metres 
 X 17.31) metres=2(J6.03G5 sq. m. 
 
 4. How many squares of clap-boarding are required to cover a 
 gable whose base is 39 feet 9 iuelies and lieiglit 23 feet 4 inclies '/ 
 
 Ans. 4033^ sq. ft. =4 scpiares (i3J sq, ft. 
 
 5. To determine tlie number of squares of roofing witli tliateli, 
 tile, slate, sliinglcs, ziiie, lead, copper or otlier metal, &.c., in a hip- 
 roof whose base is (55.4 feet and lieight 37.3 feet ? 
 
 Ans. 12 squares 19.71 sq. ft. 
 
 (1) Soe tho eomj)onont or limiting faces of the pyramids and other models of the 
 tahltau, nnd ttio sections or paniHol plnuos iis indioatod by tho lino half-way between 
 tho parallel bases or f loea. Tho triangle, like tho parallelngratn, is otten mot with in 
 the practice of the monsurer, ito. Tho gables of a building, the hips of roofs, tho sides of a 
 garret-window, &o., afsunie that kind of figure ; .and it is nut uncommon cither to have 
 to dotormino tho area uf a triaugular piece of ground. 
 
 ! *i 
 
MENSURATION OF SURFACES. 
 
 23 
 
 2ni) cask. 
 
 When there are two sides given and the included angle, 
 
 (11) IIV^LI^. Find the coiiliinicd prodiiil (-11 (i) of the firo (liirit 
 sides and of the nat.^ sine of the included anyle ; the half of this j>io- 
 diict will lie the required area. 
 
 AVi! iiiivc (ri:ji r «.) us in 
 
 tlic case (H T ) of tlio panillolo- 
 
 i^nim, CD=AC x sin. A or 15C x 
 
 • It Af.n AM < ("I) 
 Kin. J 5 ; or urea ACIJ= 
 
 and since CD=AC x sin. A or 
 lUJ y sill. 1), \v(> obtain lor tlic 
 area of liic triaii,ij,le the cxprtss- 
 hion i (AU X AC x sin. A, or i (AH x 1?C x sin. ]>). 
 
 \ 
 
 V 1) 
 
 i; 
 
 Kx. !• What is tlM> arcaoC a trian;j,lo two sides of wliicli are 
 30 and 40 and the oncloscd angle liO^ ? Ans. -JOO s(i. in. 
 
 a. Dt'tonniiK^ tlio area of a trianjijle ono side of wliicli is 
 45 yards, anotlier side '.i7 yards and (lie enclosed angle <J0° ? 
 
 Alls. 7:20.! lUGl. 
 
 J8. Tlie other data remaining the same, to determine the area 
 for an onclo.sed angle = 45° ? Ans- ,588.(i(Jt)4. 
 
 (13) By Lo^uriniiilS. Add together the logarithms of the 
 two sides and the lo(/arith)nie sine of their enclosed anijle ; from this 
 sum tai.e JO. /(/»/. of the radius^ and the remainder H'ill he the lo;f. of 
 doable the area of the triangle. 
 
 For, («. 1229 1") R : sin. A : : AC : AD or K : sin 15 : : ]?C : 
 
 /<i-w I .,,,_ACxsin. A BC x sin. H, , . ,,r, 
 
 CI); whence, CJ>=— — — =. and as area AI5C 
 
 = 15A X CD, we have area AnC = ^" ^ ^^ ^ ^"'- ^=^l^J^i''-l- 
 
 li It 
 
 Ex. 1. Kequired tlie area of a triangle who.se sides are AB= 
 
 J 25.81, AC = 57 .05, and the enclo.scd angle A=57°25' if 
 
 (1) For tho tables of nntural 8ine.«, io., soo the " Now treatise of rectilineal 
 and s|)horical gcoinetiy and trisjuiionietry, Jcc. " by tho same author. 
 
24 
 
 KtT TO THE TABLEAU. 
 
 I 
 
 i| 
 
 ' 
 
 ■ f 
 
 lit 
 I? 
 
 An§. 
 
 Log.2Al]C = 
 
 /^log. AR 125.81 2.000715 
 
 + !()«. AC 57.(i5 ].7ti079!) 
 
 4 1().<X. sin. A 57°25' 0.02502G 
 
 l + loj;. li , 10. 
 
 Log. 2 ABC = y.7d() 140 
 
 And 2 Al}C=()lli.-l, or AnC = 3055.7=aica required. 
 
 S. How many square yards in a tiiangle whoso sides are 25 feet 
 
 and 21.25 feet and the enclosed angle 45° ? 
 
 yiii) CASH. 
 
 Ans. 20.8()04. 
 
 When the three sides are kno'wn. 
 
 (i;i) RULE I. Add the three sidcit together and tale half their 
 sum. From the hal/snm take the three sides scvcralbj. Find the 
 fontinnc.d product of the holf-siini and the three remainders, lids pro- 
 duct will l)e the sqnare of the area of the triantjle, and the square 
 root ^ of this product Up area required. 
 
 Let ACB be the triangle. Take CD 
 equal to the; Hide CIJ and draw 1)IJ j draw 
 AE parallel to 1)15, to meet at E the pro- 
 longed side C15 : CEAvill tlien be equal to 
 to VA. Talv-e CEU perpendicular to 1)15 
 and oouseipiently also to AE wliicli is par- 
 allel to DR ; CFG will bisect DE, AE at 
 Faiid G. Draw Fl parall.'l to AH, Fill 
 
 wliicli will meet C'Aat H andEAjtrolong- 1*-:"'>k '.'"ti" E 
 
 ed at L Finally, t'roni tlie ce.itre II, with 
 radius FII, deijeribe the circuuiferenco '^f a circle ; this circum- 
 ference will meet at K the prolongation of CA, pass through the 
 point I, on account of AI=:FH=DF (wlienre, IIl^IIF), and pasa 
 also (-144 CS) througli thii point G, becauj^e FGl is a right angle. 
 
 Now, since IIA=IID=i AD and CD=CB=i CD + i CB, it is 
 clear that CII is equal to the half sum of the sides AC, BC of the 
 triangle ; that is CIi=i CA + i CB ; and since HK=i lF=iAB, it 
 follows that Clv = iAC + iBC-i-iAB — iS, if the sum of the sides is 
 represented by S. 
 
 Moreover, IIK=HI=:iIF=iAB, or KL = AB; whence, CL=CK 
 — KL=iS— AB, AK=CK— AC=iS— AC, and AL=DK=CK— CD 
 
 (1) See Ihc tables at tho end of this treatise. 
 
MENSURATION OP SURFACES. 25 
 
 =iS— BC. lint, ArixOG = ar('n Af'R, find AGxFa«aroft ARE, 
 wlionco AGx('F=tiii'ea ACI5; jiiul hy similar liiaiif^lcs, AG:CG:: 
 DF : CF, or as AI : CF ; tlicnifon^ AG x CF (area of ACiJ) = CG x 1)F = 
 
 CG X AI ; tlioii AG x Cb^ x CG x AI or, wliicli is tlio same thing, AG x 
 (JF X CG X AI is equal Ut tlio sqiiaro of the area ACI5. 
 
 But CG X CF = (57« «.) (iv x CL= JS x (JS— AR), 
 and AG x AI - (573 « ) AK x AI. = (i.S— AC) x (iS— BC) ; 
 
 whciico AG X CF x CG x Al = iS x (JS-AB) x (iS -AC) x (JS— 
 BC) = area ACB x area ACB = (arca ABC)-. 
 
 Ex. I. Say to liud tlio art-a of a triangle whoso sides are 20, 
 .■^O an<l 40. 
 
 20 4.'j 4.'j 4^ 
 
 ;«> 20 :H) 40 
 
 40 — — — 
 
 — 2.'>=lst remainder. ir)=2ud rem. 5 = 3rd rem. 
 
 2)J»0 
 
 45 = half-Hum. 
 
 Now 45 X 2r) X 15 X .5 =- 84.^75. 
 
 The square root of this product is 200.47117, <1ie required area. 
 
 2. The tlirec sides of a triangle being 24, IW) and 48 ; what is its 
 area ? Ans. 418.282. 
 
 3. Required the area of an equilateral triangle Avhose side 
 is 25 ? Alls. 270.r)82. 
 
 (14) By l<ogai*iniiii!ii. After having dctcrmitml the three re- 
 mainders, make the additiou of the htijarUhmsof the half-sum and three 
 remainders ; the half-sum of the four hujarilhms will anstccr to the 
 required area. 
 
 Ex. 1. How many square yards of plastering in a triangular sur- 
 face whoso sides are 30, 40 and 50 feet ? Anis- (>()§. 
 
 3. The three sides of a piece of laud measure '15.3, 330.7, and 
 
 402.5 metres. What is its area ? 
 
 .505.3 (31!».2.^>— .505.3 = 1 l.^r>5=1at remainder. 
 
 330.7 (il!>.2.5--3;{0.7=2S8.55=2nd remainder. 
 
 402.5 (J15>.25— 402.5=21 0.75=3rd remaiuder. 
 
 1238.5 
 019.25 = half-sum. 
 
 + h)g. half-sum 019.25 2.7018000 
 
 + log. l.st. remainder 1I3.!)5 2.0.)(!7I43 
 
 -)- log. 2Dd remainder 288.55 2.4(10221 1 
 
 + log. 3rd remaiuder 210.75 2.4350.-)! U 
 
 2)0.0447005 
 4.82238025 
 
 This log. corrosponda to CG432.447 which ia tho required area. 
 
 2 
 
sissssssasm 
 
 20 
 
 KRY TO TTTF, TAni.FATT. 
 
 P '^: 
 
 '1. ,: 
 
 (15~^ The same example by natural numbers ^vill 
 filiow tlie iV(lvaiifii';(^ rfsulliii;;. ia the iircsciil cmsc, iVoiii rlic use of 
 l(),i:,i lit Inns to (liiniiiifli llic wmk ; but, on their sidi-. n;iliiriil 
 iiumbtM's liiivf this ;iilviiMt:i,u;i' ov»t h\ij;,irithiiis, th;it hy tiikinif 
 in all tlio di'cini.ils, with even the ailiiition (»!' ciphers lo (•(nitiuiii' it' 
 xhmmI he (ho division nr the «'xtra<iion of tin- t'mrt ioiiMJ part of the 
 refiuirfd root, |>n'cisi<»n may be canicd to any (ieniee of ai>i)ro\- 
 iination recjiiired. whilt^t wet-annot with exactness give to theanswer 
 obtaiiH'd by lonarirhins, a .irr<'ater nnml»er of liniires than contained 
 in the frtictionai part of the loi^. iisi'lf, as shown by the in(<xiictue88 
 of the last fignre (7) of tiie answei' tlnis obtained. 
 
 61D.25 
 11:5. 95 
 
 ntiHH 25 
 65732 5 
 185775 
 61925 
 61925 
 
 7056.15:^ 75 
 
 2 H^.5') 
 
 352817 OS 75 
 S5'JS]7G «7 5 
 5f,450f^30 00 
 504508:500 
 1411270750 
 
 203G1108.7456 25 
 21(; 75 
 
 101805543 7281 25 
 1425277012 1937 5 
 1221000524 7 3750 
 20301108745 025 
 407222174912 50 
 
 70503.53 75 20301108.74 50 25 
 
 C)J 113270320.01 4218 75 
 30 
 
 m 
 
 Proof. 
 00432.4493 + 
 00 132.4493 -) 
 
 199297 3479 
 5978920 4 37 
 20572979 72 
 205729797 2 
 1H2804898G 
 18929734 79 
 2657297972 
 3985940958 
 3985940958 
 
 4413270319.9970 7010 + 
 
 12,0) SI 3 
 750 
 
 132,4) 5727 
 5290 
 
 1328,3 43103 
 398 19 
 
 13280,2 325420 
 205724 
 
 1.32801,4 5909001 
 5314570 
 
 V = 60432.449304 + 
 
 1328648,4) 65508542 
 53145930 
 
 13286-188,0)1230200018 
 H957S4001 
 
 1.12861898,3) 4047001775 
 3985940949 
 
 1328G']898C,0,4)G17M826O000 
 
^MENSURATION OP SORPACES. 
 
 27 
 
 (1«) KUijtiit. 'I'hIm- foi- Ixiftr i)/ itnij ijirrn Iriftnulf ADV., iln 
 
 ilimkr .s,Wr liE ; Jii'<( (.»7j* €i.) Di: : AD I AH :: AD— AK : CD, dif- 
 
 f'rrnicc of tlif sci/iuvnifi Mlt, WE oj' tlic Ixit^c I'lj llic i)i iinndicitldr AH ; 
 
 "///(■»(:i«"? C«.) •>!{-- i 1)1::+ i DC or I'.K - ilM-: -iDC ; now f/oii will 
 
 h((i'iilWH iti.) the iH! ixiidiriihir or iiltifinlr AI5 of Ihf triaiKjIc — 
 
 V A D '' — l^^or, «( f / /, c ( I a-l S> « . I i 1 1 1 . « /■ « iJiS •> « . ) A D : sin . J} ( = li ) 
 
 ;: MD : sin. HAD, lo ol>l,iin (iy!:jll ii., 2') AM ■= AD x c().s 
 
 i;AD, trill II K- I, //('// is, '■/■ ijDH aorl,' hi/ iKiliinii numbers or, Al} = 
 
 AD X t'os. 15AD ./• Ill SI I I It 111 7.'- ;; 
 If i/i)u irurk hi) hx/drtUni')-', where lo(j. Iv = J(). binaluj 
 
 ijou will obtain area ADt: -^ J(DE x AH). 
 
 A 
 
 D C 15 K 
 
 Ex. 'JIm' (l;itu liein^' sdlltlio same as in the last example j wo 
 sliall Iiave accoiiling to llit' lulo 
 
 AD = 4()-:2.r) AD = 402.5 DE=r)()r).;{ = biiso 
 
 fAE = ;W(l.7 — AE=;{;V).7 ~2=^252.()5 = half-baso 
 
 = sum 7:3:5.2 =(lir. 7l.rt 
 
 DC -^ 104. |.<-U78--(i:r. of tlic scg. 
 
 •4- s= 52.oi)ir)rii)=^iiaif-air. 
 
 iDE-=2r)2.(r) 
 + ^DC-= r)2.0!>ir)8!) 
 
 - scs,'. 15D- ;}04,71I,^)S!» 
 Nat. sill. I...,(i(l=-.7r>7l220 coircspomls to 40^ 12' 10^)737 "--BAD. 
 DE : AD-f- AE :: AD-AE : HI) -»E (oi- DC) 
 50r).;{ : 7:{:{.2::71.H : 104.l8:M7d + --DC 
 
 71.8 
 
 7:{:« 
 51324 
 
 r^{)r)M)'y2tm.7(\ ( lO l. I83I78 1 I 
 505;j 
 
 1 It is bociiiiso thid qiioliDiit niint oiiloi iiili> III • i- ilenl iiiiiii (o Ix) iiia le to (ind the 
 6iue of tho aiiglo HAD tliivt it ia iipon-siiiry U> r;irry tlio duoiiirilB lar eiiuugli to .-oyuro 
 sufficient oxaotnoss in tho last tiguros of tliiH sine, 
 
28 
 
 KEY TO THE TABLEAU. 
 
 2.137 AD :R:: 
 
 20212 402.5:1 
 
 28175 
 
 BD :ein.BAD 
 304.741589: .7571220- 
 
 i)2;")G 
 5053 
 
 42030 
 40424 
 
 IGOOO 
 151;VJ 
 
 yoio 
 
 5053 
 
 39570 
 35371 
 
 22'.)!tl 
 20125 
 
 2S(;i]5 
 2S175 
 
 41)08 
 4025 
 
 8839 
 8050 
 
 7890 
 
 , 1. i. 
 
 1 i; :(j,i, 
 
 41990 
 Nat. sine fouiul-=.75712i() 
 
 Next loss siuo.-=.75(il)l).51=49° 13' 
 
 Dif. of COS. for 60"=2202 
 
 GU" : 2202 : 4(M»737" : 14707 
 2202 
 
 Diffeicnw==- I2(i!) 
 Dif. for ()()"= IJKH) 
 
 rJUU: GO" ::12G1): 40.0737" 
 G 
 
 HOI 474 
 801474 
 801474 
 
 l<)0)7(il4 
 7G0 
 
 -r G)882422 
 -^147070 
 
 1400 
 AB— AD X iiat, cos. BAD 
 
 BAD-=49'' 12' 40.0737" 
 
 Nat. cos. 40" 12' == 
 
 Dif. for 40.07"^= — 
 
 Nat. COS. of 49° 12' 40.0737" 
 X AD 
 
 .G53420fi0 
 14707 
 
 .(15327353 
 402.5 
 
 .'J2<)G3G7G5 
 1.30G54706 
 2(il30!)412 
 
 AD X uiit. COS. BAD-=AB^=2()2.042.5!)5825 
 X DE .505.3 
 
 788827787475 
 131471 i::}7Ui2G 
 13147J2!l7:il25 
 
 AB X DE--2 area ADE-- 
 i AB.DE== 
 
 1328G48!)3()703 
 GG432.44G83 = area ADE. 
 
MENSnRATION OF SURFACES. 29 
 
 (11) Tlio aroa foiuul ai-cordinjc; to tliis nilo in (i(i l.'K.l 1(18 square 
 iiietrcH. Tlio cxiU'tiicss (il'iliis rosiiU in yet, as it is hl'CM, t'anicd bat 
 to tlio 7tli (ii^iiic, and i( camiot hv otlicrwisp sIikm* (he iiataial .siucs 
 ascil aad wliiili cuter, as cicnicnts, into tlic solution of tlio i>rol)li'Mi, 
 oxtond but to 7 li^iux's, tlic last of wliicli ovou is always too great or 
 too small accordiufj; as It lias been, or not, increased by unity wLeu 
 the foUowinj;' ligiire is greater or less than 5. 
 
 (IS) Ijet usreniaiklicre that this example, the calculation of which 
 wehavejust made, in tliieedilfereiit iiianiiers, allows one to com[)are 
 the amouut of work required I>y .-ach mode of solution, and enables 
 one to choose when required, whether the most expeditious means 
 (the first) or the one admit liii!,' of the greatest precision (the second) 
 or the one requiring no extraction (.,(' a root (the tiiird). 
 
 (19) It is hardlj' nec<>ssary to renmrkthat this problem, like tho 
 preceding one, and the one following, may also be solved by means 
 of a graphic construction allowing one to establisli with the help of a 
 sufficiently subdivided scale, the lengtli or value of the perpendi- 
 cular Al? in terms of the base or sides ; and that is often enough 
 the shortest, though uot tho moat precise mode of arriving at the 
 required result, 
 
 PROBLEM III. 
 To find the area of a trapezium. ' 
 
 A/ lU 
 
 S 
 
 (30) UI;l.E Find (;J1« G.) the sum of llic (wo ^HtraUel sides ; 
 muUipli/ this sum hi/ the Iniijlitoy hreadih of the trajjczium, and half the 
 product will he the required area. 
 
 'i 
 
 I Soo tho ooraponoiit f.tcos (bases and latoral faces) ami tho sections and paral- 
 lel planus of tho prismoids and other juodels of tiio Tibleau. The trapezium (173 Cii.) n 
 presents itself often enough, in practice, to tho calculation of tho inea>urer. ThuB, 
 the interior tablo of a window whoso sidoa aro generally splayed, presents tho form of 
 a trapezium ; so for tho coiling of a window door or othur splayed opening ; and it is 
 plain also that the developed surface AliCD, (part of a concentric ring, see tho parallel 
 bases of the hollowed cylinder of liio Iablkau and tho latoral faces of tho Bectiona of 
 
 the hollowed sphero),ol'thoj;uiib of a curvod as well as splayed J^ - — —--11 
 
 opening may also be considered as a kind of trapezium with \ / 
 
 parallel curvilineal baseg,but whose area is erjually dotormin- V J 
 
 edby tho rule horo given, since that figure is nothing olso but a 1* v 
 
30 
 
 KKV TO TflE TABLF.AO. 
 
 Ex. 1. In a tniiK'ziiiiii, IIh) pavnllol sides am lOi iiiid 121 feet, 
 
 anil flu- nt r])('n<ru'iiliir (lislaiice iu'twecii these Hides ;{ IVcI "^ inclies. 
 
 Wiiai i.s til.' aiva / Ans. i (.l(»i I i:t) x -Vi- i(l<>."» I \-i:2:>) ■< -\MM 
 ~i.lM7ri X .'M(iti = ;{t!.<IIMiJd aq. teet. 
 
 J2. Keqiiii'od tlie urea of a jiiece of oidiiikI wlioso i»aiallel sides 
 measntt) i-eHpoctively 7") and l:iii liulvs, and tlie perprndieidar l.')4 
 liulis 1 Ails. I.")i(i!i nq. linlca. 
 
 :t. flow niaii.v square feet aren in a Ixtard wliose Ien,i;tli is l"Ji 
 feet, brealli al one end !."» inflit.'s and at llie ollnr end II inilns? 
 
 Alls. i:}=^;i;j.r)ii(i(;(if 
 
 4. ifow nianv s(|nare yards in a Iraptv.iuni wliuse parallel sides 
 are 210 and ;i2(> feet, and lieiylit (i(! feet ? AiiN. ^D.'i.'J/j. 
 
 a, TIm" parallel side.s of a farm are 12. T)! and S.22 eliains, and the 
 perpondicnlar 5.1.") fhaius j what is the area in siiuare chains ? 
 
 Alls. r);}..-jri)75. 
 PKOBLEM IV. 
 
 To find the area of a quadrilateral.* 
 
 <,:U) R»-'t.E. MnUiphj (:151 H.) ntlicr of the diuf/oiials (IT.t CJ.) 
 i)/ ll'.r ,jH'uliiln(cml, liji tlie lidl/siim of the pojtcndiculars dratcn/i-ovi 
 the opposite (inijles to the eummon base. 
 
 il\. 1. What is tlic area of a 
 quadrilateral BD whoso diagonal 
 AC is 42 feet, and peri)endieular8 
 BF=18 aud DF= Ui feet ? 
 
 Ans. 7J4 sq. f(. 
 
 3. ITow many scpiari^ toises of 
 
 paving ar(> there in a (quadrilateral 
 
 wluKSo diagonal is (i.j feet and the 
 
 twoiwrpendiculava 2rf and .'{;}i feet ? 
 
 Ans. r)r).52()ri:>. 
 
 frustum or part of a oiroular ring, and that the modo (1145 d.) of arriving at the 
 area of that figure is like the one that shows how to dotormino the ■.ti;ii of the trapezium 
 go called. Tho trapezium is ;ilso ofton mot with in tlio tionr or ooiling of a room, two 
 gidoa of which only uro parallel, ia tlio roof of a dormer-window, lliglit of stairs, roof or 
 ceiling of a garret, and tho sides or jambs of a rectangular window a!<sumo also thia 
 form when tho ceiling or table is inoliued or splayed. Finally, wo aro very often called 
 on to detonuine the area of a lot of ground having the form of a trapo/iiura, 
 
 1 iSeo tbo bases, laleial faoes, seotions or parallel pianos of certain models of 
 the TAVLEiiu. 
 
MF.NPURATIOS OP SUnFACES. 
 
 31 
 
 Jl. Il<»\v niiiny s(|iiiii<' niclrcH iircii in t\ (|niHlruiij;uljir piece of 
 p[VOUiMl one til' wliHsc I )iii<{i)iials is *i I iintrcH iiiid the pel piiHiieiilnr 
 (listiiiices fioiii tiiis i>i;iMni);il to III), two <»pp(»sile unfiles, 'JH iinii A'i 
 iiietieM ? Alls. l!''-if) Nq. in. 
 
 ■I. Delerniine (lie nnnilter of H(initre» of (l(>orin;j; t<» cover a 
 (|Uiitlriliileijil s|)ii(e, wlio.M' (iiii^iiniil is lOH feet (I inches, iunl the i)C'i'- 
 pendicnliuK ')(! feet M inches, iiiul (JO feet !• imiies .' 
 
 Alls- <•'■{ xi(u<ivcH, 47|'Sj Kt|. l't„ 
 5. T?e(piire(l the nninlier oriirponls in n jiiece ol' land one ol'wli.iso 
 »li!i;,n)na]s ineasnres/d.,' ix-iclies, :in<l (lie pet pendicniars '^{'\.'t iind MO.'J 
 peirlies ? 4IIN. 1!» arp. !>8.(i73 i»er. 
 
 PKOBLEM V. 
 
 To find the area of an irregular polygon. ' 
 
 (22) lWt4V,.— Mr(t>iuyv (he diittiounls trhirli irill ilirldc Iht' (jirfi} 
 pohji/oii iiilo (jKdtlrihilcrnIx mid tria»(ih'n. Determine sejxtratehi the, 
 (irvttH of tlu'KC eoniponenl JiifiiirH ; their sum iriH t>e Itie iireii miiiired, 
 
 Kx. 1. neterniine (lie area of 
 tlie polygon ]{K, in wliicii HD — 
 m, CK=l2l, AD = ^7^, m.='j.-,, 
 EH=il4, AEr::4(), and l'\i = S. 
 
 Alls. 3 (151) X {'K)~l (IH.nx 
 12.8)=llS.|()::=area r.Cl). .l(HL l 
 KIl) = i (V.).r, \ 14) = 11.7.-. and 
 quadrilateral area AliDK = AD x i 
 (BL + EII)==^7..^ X \\.7r,=:K'.i.\\ir,, area AEF - AE x iF^J =40 x 4 = 
 HiO. Area AB(IL)EF=:J 18.40 + ;{i2;J.iar>+ l(J(»=:GOi..'-)±-). 
 
 2. Kequired the luiniber of acres (ilic acio is 1(10,000 square liidvs) 
 ill a }»oly;u;onal piece of nround WE whose diiigoiials HD, AD aiul AIO 
 meiisnre, respectively 1. 'J (diains ((lie lineal chain is J 00 links)— .'W links, 
 i;} chains !>!• links, and II chains |:{ links, and whose ])erpeiidiculars 
 CK=:I7;{ links, JJL^ti chains, E1I = ^! chains and V(i !{} chains. 
 AiiN. I{Dx(:K = I:«;{x 17;{-'J;{0(;0!> : 2=II.ia04i = aroa BCD. 
 AD X BL=:1.3'J!) X i]00=27080O^2=:l.'}(>200 =area ABD. 
 AD X Ell — l'.iW X 220= 1307780 -> 2= l.WiiOO =area ADE. 
 AE X VG = I4i;{ X 37r,=r)2!»875-:-2=2()4!»:J7^ = arca AEP. 
 
 2)ia.480(54 fi.740;i2 =areaABCDEF. 
 G.74032 that is G acres and 740:^2 so. 1. 
 
 I. 
 
 1 Sfio (he bsBeB, lateviil faces, eeclions or parallel pluuea of certaiu models of the 
 Tableau. 
 
32 
 
 KEY TO TUG TAlir RAH. 
 
 or (5 iiori'H 2 roods jiml ;Jl(i.'!--J links (tlm road hciiio I lie loiirth part of 
 
 the acre, tliiil, U KMMMtO-r -|~!;'r)(l()(» links) 
 or(» ncroH, 2 hmmIw, ;{H immtIics, and 'JHvI links (the lim-al |>ti(li bcinj^ 
 tli(^ four'.li pai t of a cliiiin, llial is 'J.'t links, autl I he si|m;ii<! perch con 
 H<'(pu'n|lv — '^."i X 2't~i')2't si|. links.)' 
 
 PROBLEM VI. 
 
 To determine the area of a long and irregular figure 
 bounded on one side by a straight line. ' 
 
 (2JI) IH^IiE. I Miiisiirv, <il iiivh cml of lliv utraifilit liiir, the 
 pcrpcinlinilitr hrctnllh of llic fhiiire ; hicasiirc also litis hraidlh at scleral 
 intrniivtiiiile pldvis njiiiilislmil Jfniii <inh ullicr. 
 
 2*^ To the luil/'Smii of the crtrnnr htrndlhs oilil the sum of the in- 
 tcrme<li(ite lireadths ; miiltiitli/ then the sum Ihiis olttained tnj one of the 
 equal partH of the line of /ucc ; the jinxho I ivill lie veeij uearUj the 
 area reijnired. 
 
 1 Oiintci'M I'hiiiii is lili ciiulisli feci, (liviilt'd into IdO liiil<i", pmcIi of which Ir con 
 BequtMilly =ii'') -f- 100^^ ".!•■,' fiii,'liHli iiiclii'ti. Tlit» iicti! \a t'((iiiv!ileiit to I <[i;iiii X '" 
 flmiiin= I(t s!iiiiii(( <liiiiii«= I ppiclicM X "^ (if'rclica = IliO H(|iiiir(' |i(Mi'li(m= 100 liiikR 
 X 1000 liiiI{H= 100000 nqiiiu'c liiil<f<. 'I'Iih :iilvuiitii^'B nl llii.-! tiiviHioii ol' (IciiitHi's cliiiiii 
 iiitd 100 ]):uiH coiiRiHlK in iIhh, tlmt, nil llie dimnnsions wiiicli i(. liclpH to <;Htiil>li.sli, mo 
 iiuiii(!(li!it(dy aiitiliciilili' iinil without iciiiiclion to lii'ciiniil ciilcnhition. 'I'lm o|ierii. 
 tio'i hfiii^ (ioiKi, Ci ili'cinuilt' uni oMt, oti'. Ilio iciii!iiiiin.n lit,'iiioH lo Iho Ivft bt>iii>{ ucreH 
 Hinre thorp are 1001)00 links in the ucro nn<) that tocnt ollT) li<,'iires is eijnivnlHnt to di.' 
 vitiini; hy 100000. It in ['lain .ilso ttiMt I'lir llin roodn wo liavo l)Mt lo niiilliply lirat 
 the rtmiaindcr hy I and aL;aii\ nit nIlT) fii^nrcM, wliicli is crpiivMliMit to dividing' at onco 
 l)y yr>000 (nnniher of links iuarodd) and i^ liy far niorocxpt'iiiiioiiH. Kor perclies, the 
 aecond remainder iM tlii'ii ntnilijilii'd liy 10, cnlliriL;oir 5 liLTinx's an hcforc, ai'iCH th» 
 peri'h is tho 10th pail ol' tin' rood : or if dcsirahlc to nciilcct tlin loods, iho first romain- 
 iler may ho at once iniiltiplii'd hy KiO (I X '") =1"'* !^> liu'iiros oijnally cut 
 off. Tho last romainder .ITiI'JO is (!vidi'ntly a fraction of a piMvh, that is, ^'''''f J\ of ii 
 perch ; and tlie p(]nare perrh hcini,' CrJj links, .j.^_i^^^^ .>f(i'.'.') = .00(>-!r), t'lis nnmber 
 multiplied hy the numerator .irjIL'O j^ivcs the tiS'i links of the answer; that, is: 
 for the links the last remainder is simply multiplied hy (i'.'S and 5 decimals fut oil' 
 na before. 
 
 2 The tracts of land which are near, and are hound- 
 ed on one sidehy the wiiulinus of a road or river, &c., 
 often present to calcnhition li^inros of this kind; or, after 
 haviiiK determined hy the method of the last problem the 
 areii of the rectilineal polyj^on AIUJDK whidi forms part of g/ 
 the irregular polygon AtUUDEcdA, the ni(!thod of tho pre- 
 sent problem will be made use of to obtain the seoondiiry 
 aud iiregulur parts AahcB, Ad«E. 
 
mkn.si;rath»n oi' si/ukacks. :{3 
 
 Ia'\ AV.ra an iirc^iilnr <i<;iir(' Iiiiviii;; loi' £ 
 
 ifrt l»jiK(^ (lie Hliiiij>lit liiii! AK. .\t llif pdiiils f^l ^^t^^^ 
 
 \, \\, (', I) iMxi K, <'iiiii<list:iiil t'loiii (;a<'li i : I i : 
 
 other, (hinv tlio pcrjiciKiir 'iiiis A», IW>, (Ic, \ H T' — 11 — il 
 I)(/, V.c and d('si;,nnlc tlicsc |mi |)cii(li(iil)iis 
 by tli«! letttilrt (I, h, r, tl, v, 
 
 1'licn Ct'^.l «) llir nicji nf iIh- tiapoziinii AII/hj =".*-'' x AB, 
 
 tlir;iica i>f (iic Irapcziiiiii \\(',h^~~ x HC, 
 
 «* 
 
 the arc'ioCllicfriipczimii ('I)f/r=^- x CD, 
 \\\\{\ I lie aicii of I lie tiiipcx.iiiiii \)V.rO~ — ~ x DE ; 
 
 tJien, tlM'ir s.ini, or I lie iirca of llw wliolc limine is fMinai to 
 
 Ciiiico Ali, 1>C, Xc. arc ripial lo cmIi ollici. I»uf, this siun 
 
 , . ( " I /' I (• ■■■ (I >■■ '- I < A I?. 
 
 13 equal l<» V>2 'jy 
 
 oxpiTsmion \vhi<'li aj-rccs wilh ilii; cmiiiciatioii of the iiilc. 
 
 (44) ffA^f IxTomcs vorv sniiiU. we will nunc I C r 
 
 'oh X ^""^^L 
 
 the le8« havoarea. AIVk>— ^^ x A15 ai.d il <( ^^ 
 
 and A auM'oiit'oiHMlcd in one <.V tiic same point. A il C D X 
 or that tlu> trapczinin Wihn iMcontcs ( lie !ri.in,t;le .\])h, we will have 
 
 ILJL '~1; in lliiscnsc it isplaintlial I lie i-\picssion tor the area of the 
 
 //) /> ; <• e ! (/ (/ '■ t'\ . ,, , . 1 . 
 
 the same thinf^. (I> <■!(/■ .\ r) AP.. Ami if Rrheconies also=o, the 
 ftxpi-esaion for the area A1'.(7/'A will lake tlie lonn (/) I r \-f1) x AB. 
 
 Ex. !• Tlie breadths of an irre<<ulai li^are at 5 equidistant points, 
 being H. '2, 7. 4, !♦• -. '"• -• i""^ ^'^ ''""^ <^''^ length of the basc=40 ; 
 wiuit is its area ? 
 
BSM 
 
 34 
 
 KEY TO THE TABLEAU. 
 
 m> 
 
 I' 
 
 ii>« 
 
 » .: 
 
 f 
 
 Ono of tiic extreme l)vea(Uli8= 8.2 
 The other extrei\io l)reiiilth= 8.(> 
 
 Siunot'tlieextrcine breadths— 1(5.8 
 
 Half sum = 8.4 
 
 Ist in termedmte breadth =7.4 
 
 2d intermediate l>readth = !).2 
 
 IW intermediate l>rcadtli =IU.2 
 
 Sum of the breadths 
 
 =:J5.2 
 
 the uhole ba.s(i 
 One ()fthee<iniil ]>:nK 
 Sum of the b 
 Multiplied by 
 
 =:aveii, wanted 
 
 = 40 
 :40-:-4=H> 
 
 =;ir).2 
 
 10 
 
 = :irr* 
 
 S. The length of an irregular figure being 84 metres and the 
 breadths, at six equidistant points 17.4, 20.0', 14.2, U'uii, 20.1, and 21. 
 4 metres ; required the area. Ans. ir).50.(]4 sg. m. 
 
 3. Tlie length of a strip ofland is 125 perches and its breadth in 
 15 different and equidisliint points, is 5.2, 4.(;, 7.2, 8..'?, !).4, 8. 1, 7.;j, 
 7.9, 0.0, 7.2, 7..% 8.4, 7.4, 0.5, and 5.8 perches. What is its content '/ 
 
 Ans. Tlie sum of the extreme halt-breadths and of the interme- 
 diate breadth8= 101.7, tlie length 125-^- I4=:8.f>2857 and 8.!»2857 x 
 101.7=008.0:^)0 square perches. (^) 
 
 (25) REHI. Some ai'thors teach how to determine! (he area 
 of the (igure of thw pioblcni by tinding the product of tlie wliole 
 base AE by the mean of t'.ie breadths wliicii is obtained by juldiug 
 together all these bioadtlis and then dividing tlieir.sum by their num- 
 ber. This rule is erroneous, and the more so the less the number of 
 breadths or divisions in the figure to be conijjuted. Tlie error of 
 this method, in case there were but tliree«.'oinponent parts and con- 
 
 1. If the lineal [)erck ill qiieslioti here is \S /reach fce(, tliiiliti, the. tenth ofaii ai'peiif. 
 the iiieii just, foimd will Ixi e([iiiviileiit. to !) scjiiiU'e iiipButs, SOiiSlJ fc()iiiiifi pHirhetf, 
 for, H8 h ht«.8 ulreiKiy been remarked, the wqciure uipeiit ia 10 X I') j)er(ilie8=l00s([iiaro 
 p' of'.Gfl, Hill an the a(|iiai'B perch Is 18X'8=H-I square feet (or the square arpeiit 
 w-iX '00==;J'^100 aqiiar:, feet) th .i.^cliual .OOod of a Hcjiiare peich may he rednceii if 
 need he info hqiiarelef' ' y iiiiiitiplyiiig hy Itt), which ^ives in thia exa\iiple tl.,');{ 
 square (eet. //", on the contrary the lineal perch irere lii.J enijlish feet which In ihat of 
 I liutf'.i chain, we wonld have after d'vidlnj; l)y lti0,5 acres, 108. 0:);")!) percliH, ami if 
 we desir.^d afte- wardrt tore.iacc, into S({nare feet, the decimal of a perch, it Is plain that 
 .lie minare perch heing Ki.i X l''i=27~'.~.'> sqnaro feet (or the acrH='J7~'.'J5 X "'0 or 
 fit) X *>')«= 1!*''>'>0 aq.iare leet) it would ho siiHicieul to lunliip., ,0',id(i by •J7ii.','5 to Inivo 
 9.ti!) eiifjlion t<qi'-ire feet. 
 
MENSntlATION OF SUUKArJES. 
 
 35 
 
 s<M|iicii(ly foir iii'iglits or bicadtlis, mi<i,lil icacli U} vi.") ptT e'(!iil sliovt 
 ol'llio exaol juca. It gives for tlm luciiii brcmltii. in tliis «>xaiuj)le, 
 ]{)7.2~]rj=7.]4m ct 7.14G<I x 125-893.825 stpiare i,ercli<;s instead of 
 (KIH.O.T); or an iiror of nearly 15]'(r(]i(,s of ground. 
 
 PROBLEM VII, 
 To find the area of a regular polygon. ' 
 
 (Sfi). RUTE 1. Midliplii (Mili «) Uw. pcnmclev <>/ the pohif/on 
 hji its- f'Kjlit htiJj'-rddius, (tiid the [irdthiit iriU he the rctjiiircJ <ire<i. 
 
 KEIW. It' (ill- polygon be known Ini. by i(H side, determine tirst, 
 its riglit ladins in tlie following manner : Divide '.M)^ by the iiuni- 
 liei- of sides of I lie |)roi»ost'd ]iolyg<ni, and the (|ii()tient, will be (O'-IOG.) 
 tlie angle at the centre ; that is, Mie angle subtended by one of tlie 
 equal sides. Now the right and oblique radii of th; polygon ft vm 
 with the half-side a reetangiilar triangle in which the Itase is known, 
 that is the half-side, and tlie oi)posite acute angle, that is, tiie l.'alf- 
 anglo at the centre, to find the perpendicular or light radius of the 
 polygon. 
 
 Ex. 1. Say to find llieareaot a ii'gulai 
 hexagon whole side is 20 feet f 
 
 Ans, ;i«)0'-f-(;=()U and (iO-; 2 = :iO jingir 
 AOa, lialf ofAOli, We have also OAG^ 
 90°— AOG=60<^andA(4=:l();theu(132i>G.) 
 sine AOG : AG.: sine OAG : OC : whence, 
 
 Sin, AOG 30° a), conip, log, O.mium 
 
 is to sii), OAGtiO" 0.937.')3I 
 
 as AG r l.OOOOOO 
 
 18 to 
 
 OG 17.32(».-)2 1 .2335G1 
 
 Now as there are (I sides, each eqiiiil to 20, we will have the 
 pm-inieter 20x(i=120and the area=12() ^ A (I7..320J2) or which is 
 the 8anie=]. •120.52 x A(I20)=:: 17.32052 x dd^ II),3!>.23I20 s. f. 
 
 Ex. JJ. What is the siipcrlici.il coiite'it of'.iii octogon whose side 
 i8 20 ? Ans. I!)3l.3()r). 
 
 I. Seo IIk" Ji'li' ■>'. Lj-JC's (if llic ii"lil |irisiii^ nwii iiii-'iiliiiiis ol' llic Tiitili'iin ;Hi>t 
 tlinir [miiillt I .seciiioiin, <ii' |i|jiiii's. 
 
36 KEY Td Till-, TABLEAU. 
 
 For the aiiuli- ;ii tlic (•ciitio^MGU-' :-i:<— 15 the Imll' of wliivli 
 
 22° 30' in tlio iuinlc AOC. jHl.jiici'iil to tlif lij^ht nuliiisi, and itsi'oiiii»l»'- 
 
 mentOAGcoiisotiiiniH.v ^Il0''-;22S-Ur=:(i7'^:j0: Imt we huvc (1'l.tl, 
 
 3^ G) ()i;=:.\(; X iiat. tiiii.u. UAfl ::|(l •J.4llt>l -rrJ-l.-Jil^il suulivicu 
 
 ^, =24.U1'J1 X H(i (lusir-piT.)- Ih:ji.;joS. 
 
 I, ;{. lit'(iuir('.(l llic Mien of :i U(>iia.i;()ii uliiisi' sidi' iiiiMfSUit'S t< feet 
 
 I luul the iK'ipeiuliciilar <lia\vn fiom tlie leiilic— IU.9!> leet. '! 
 
 ■^ ' • Aiisi. *I5.(»4 s- f. 
 
 j ij . -1. Find tln^ area nf a it'unlai Ih'iiImlioii \\ hose sid»' - IK.IjHsnul the 
 
 ' viyht nuliiis=='J8 1 AnN. l8S){>.;i4. 
 
 5, The 8i(h-' of a pen I a^" a "■-*■"> nniif> ami tlie distance IVum the 
 
 side to the cciitie-- 17 'J nifties ; whni is iK coulciil / 
 
 k - Ams. 1075 s(|. ni. 
 
 (•27) With tJu- ln-l|» of this I iile. it is casN lo ohtain the area of 
 
 any polyj^ou ' that is of ;i polygon of any iiiunlier of sides. Having 
 
 t'iilcuhited and disposed nncU'r liie foi ni of the followinu; table, tho 
 
 vehitive areas of the various ]iolv,i;ons haviai; foi sicU- unity or I ; 
 
 namely : 
 
 A. L'dtliiis o/' ilic ... I Uitiliua tit' IIk' 
 
 jSaiiie. . •: , Shirs. . 
 
 vn'vuiu.virvit'. in.scr. ciitir 
 
 Triangle U,577M5(i:{ ,. M (l.'.'HS(;75l 
 
 Hquiire l».7()7l(Kis 1 (».:,( i( H km it i . 
 
 Pentagon ll.85(H;5Un 5 O.dHSlUlO 
 
 Hexagon l.tlOUOdild .. (; (».8titi(»'J54 
 
 Heptagon J.15;2;{H',M .. 7 I.((.is-Jti()7 
 
 Octagon 2 ...i.M0{;5t;-.»8 .. ^ l.'i()7l(l'ii- 
 
 Enneagon....l.4t;UH);.':2 .. !> \ .•.\r.i7'Ar'7 . 
 
 Decagon l.OIHO.'JK) .. hi l.5;{c«H14S . 
 
 Undecagon...l.7747:]-M . 11 l.7()--i.-<|:{(; . 
 
 Dodecagon... I. !»MI 85 1 7 .. I'J I.H(i(i()-J,-)| 
 
 And hecai'se (5<3'» <*•■* <Iie a eas of similar rolygons ai'e to each 
 i^ "; ' other as the ..lUares of tlieir hoiMoh»g(.iis .sid.is, the area of any 
 
 ? given polygon will have to tiie square oi' its , ide. tlie same relation 
 
 i '' that the area of tlie polygon of the same ranie and w liose side is J, 
 
 ; has to the s(iiiare of unity ; wheiu-e, we have : 
 
 i I ' (2*') ^HJl'''* "■ Stjitdic Ilic side of tlir </irtii j>(tli/(ji>ii ; iniiltiplif 
 
 : ' then this square b,< (he nfva of tlic poli/fiiiii of the soiiic ikiiih' irhosr side 
 
 t i': /.v 1 : the product irill be the i'ciiiiii'cd tiieo. 
 
 A rtii. 
 
 '/'//<' lllUlIf 
 
 UAB'. 
 
 (1. i:(:{(»l-jr 
 
 
 :«)^- 
 
 l.(KH)(IOi)t> 
 
 
 45 
 
 l.7-J()4774 
 
 
 54 
 
 •J..5!)8<l7(i4 
 
 
 (III 
 
 :{.t);{:t}»it:4 
 
 
 .54,^ 
 
 lf<'2H\'27\ 
 
 
 iuh 
 
 ti.lHI8'^|-i 
 
 
 70 
 
 7.(;!»4:i088 
 
 
 72 
 
 !».;u;5ti:i!IJf 
 
 
 'Vr 
 
 l|.!!t(i|52l 
 
 
 75 
 
 I Seellie U.isck aiui (laialli'l .scclidiiM of llii- [iiisins ami (n i.--iii(iiiiH, Ate., of tlie 
 Tableun. 
 
 'J. Ill lli« I'atf'f i)f llii'. it'i;iiliu- III- »-vt!i sviiiimhIi iial (iiilondii lln; aioa in iiiiiiiBdiulieiy 
 olitiiiiieil l>y liikiii.H Iiihii tlio .si|uaie of tlio iIkiiIiIh rij"!!!, rii(iiiii< iii- m|iii|Ii(>iii on oiifl of 
 .sidcH, lliti MCpiaic! olllm oIIkm' .liiii?, as will ln' si.-cii in i.lie iiiHiisiualioii of Holiils*. 
 
MENSUn.ATtON OF f«tJBF.\rES. 37 
 
 Ex. I. Wliiil is tiu' ;ir«'ii of n vi'gulnv hexa^'oii, wlios*' »i(l<' \u 
 1M» ? 
 
 Ant. 20-— lOK, the mwi of the liexiijjion of the table =ii.r>9H07t)2, 
 
 iiiul 2.59e07ti;i -^ im^WM.^immi), m before. 
 
 2. Determine the siipiiTuial eoiiteiit of a peiiiagou whose side 
 is 25 yards ? Aiis. l075.!-n>8;J75 sq. y. 
 
 Jl. Tiie .side of a deeaiioii measure!* "20 metres : what is its urea ? 
 
 Ann. ;^077.(i8352 sq. m. 
 
 4. Fiud the aiea of a duo tb'canoii wlio.se side is G ? 
 
 Ans. 40a(K)148()4. 
 
 5. Tiu sid(! of a piece of i; round liavin.ij; tiie form of au equilate- 
 ral triangle measures li arpen ts 7per(;hes and (! feet, wlia»' is its content ? 
 
 Antj. y7,'. per. x:J7.\ per= l,'J!»;r, or l:{l»a77777, x {).433()I'27=G0;J. 
 r)l;i847»7 o'- 'i square arj^ents, '.i!, sipiare ])erelies nearly. 
 
 PROBLEME VIII. 
 
 To find the circumference of a circle ' -whose diameter 
 
 is known, or the diameter of a circle of which 
 
 the circumference is know^n. 
 
 (a9) UlJf.E. Miiltiiilii («.S5 O.) (he diameter !>;/ :{.i4l(), and the 
 product will he the eitxuin/'eience ; or dicide {0S7 W.) the circHi.i/ercnce 
 by 3.1416, and the (inotieiit will he the diaviet r. 
 
 Ex. 1. What is the tireumferenee of aclrele wliose diameter is 
 25 / A US. 78.54. 
 
 2. If the diainclcr of the earili be 7!tI2 miles, what is its circum- 
 ference If Alls. 24884.6136. 
 
 3. Determin<' the dianiiU'r, to oircumfcreiiee il«)52.K)44 ¥ 
 
 Alls. 3709. 
 
 4. Required tiie circuniftrenee, when tiie diameter is 17 metres 1 
 
 Ans. 53.4072. 
 
 5. The eireumfe/ nee of a riicle is given -354 feet, determine 
 its diameter ? Aiis. 112.681. 
 
 I. See the l)!i8en lunl mntioiis or |iiiiiilltil ciifliii^' plmies of the cylit)ilei'H, coiien 
 mill tViisfti of liirhr (■(iiiHs, t'liiBlii iim' st'ijiiniiits of sphoioH, cVc, iiuionn thu luoileU of 
 Ihe Tah/eaii. 
 
38 
 
 KKY T(» TriK J'Alll.DAr. 
 
 UI'ilTI. Tlic iflntidii 7:','-2 wttiild ji-ivf I'oi' iliis (liiinittci I J'2.(;:{.(; 
 This lust result, is too siiinll b.v 1;*,;^, (»tii iiiiil or j,,f,|-j,, of tlit- whole, 
 and enables one to.jii(lj;e of Ihc; leliilive cxiiclness ot'tlie two ratios. 
 
 PROBLEM IX. 
 
 To find the area of a circle. 
 
 ■* 
 
 i 
 
 (JIO) RULE 1. Milltij)lii (..|:6I €i.) ///'■ <-irciniiffirii(r hi/ lidl/'tlir 
 ItVt.K II. MuUipJji (lO'il «.) ///r .«.•,/»„,•(■ (-/■ ilu' nidiiis hii 
 
 ai4J(). 
 
 KULit: III. Miiltijili/ (deiii. olONI t;). ///r s.jiiiin: ti/tlic diumv- 
 dr hji .TS.l-l, 
 
 Ex. 1- Wha( is the aiea of a circle ot which the (liaiiuttei' is 10 ? 
 
 All**. 7H.5-1. 
 
 If the (lianietei' were 100, the, area would he 7844 
 
 If tlie diameter wove 1000, (he area would he 785400 
 
 2. The diameter is 7, and the eircuiiifer<'nc(!!21.!»!ll2, what is the 
 area of the eireh' f Ans. .■W.484(). 
 
 ti. llow many square yards are there in a circ]<' w iiose diameter 
 is ai feet ? Alls. l.()(i!)01fi. 
 
 4. The diametei' beiii;.; 7, wliat is the area of tlu' circle 1 
 
 Alls. ;j8.484(i. 
 
 5. Find the area of a eireh; whose radius is ',\{)] perchs ? 
 
 An«. i2!i'2'^.47;J4 s(iuai'e i>erelie.«(. 
 
 (Rl'I..E iV.) Multijih/ the s<iiiiire of the rircioiifo-citcchy .{)7Dri8 : 
 tlic product irill be tlic area 0/' the eirele. For, let e the jL^iven circum- 
 ference, d the diameter and 7: = 'lUir^9 ; then (086 «) c=7:d, and 
 
 C687 G.) (7:=^; theiice the area of the circle=— .'^inee (1034 G.) 
 Tt 4 
 
 the area of a circle whose radius is r—Tir and tliat r7 = 4r ; but since d 
 
 4 4 
 
 = .-, we have d = ( _ ) =— ; and as—- = ! ~d . we have tt _ =:j;r 
 
 2 2 
 
 ■■c X 
 
 I 
 
 - = c X 07!.>58. 
 
 7:2 47: 4x3.1415!> ]'2.r)(J(;;i(i ia.5(U)3(; 
 
 Ex I. Find the area of a circle whose circumference is 10.75. 
 
 Ans, {».l[)G46375fl. 
 
MENSUIIATION OF SURFACES. 
 
 39 
 
 »1. Dct<'riiiiii(', ill jirrcs, Hit- iin'ii of a pioce of j^roiuul whoso 
 t'irciinift'i'CiKMi m«;usuio.s one iiiik' (s;iy "^O cljiiiiisof (>untcr=OG x 80 = 
 r)2ri() on <,'lisli fcc-t) ? Ann. 50.r)3l'2. 
 
 PROBLEME X. 
 
 To find the area of a circular ring or the space com- 
 prised between two concentric circles. ^ 
 
 CW) Kl'LB:: I. Fiiiil (1111 «.) hij the lust proldem the areuH 
 of the two cirelos : their di(fervnce will he the area of the riiiij. 
 
 HULl-: II. .ynliipiji C'i'7liM.) the sum (f the diameters hi/ their 
 dijj'ereuee : this produef iimliiplied hij ./d.lJ will he the reiinired area. 
 
 KUff^E 131. .)fHti:pli/ the half-sinii of the rireiiinferences of the 
 two eireh's hij the half-differeiiee of their diameters, that is hi/ the hreadth 
 (f the riiKj, and the proditel leill Iw. the rei/nired area. 
 
 For oiic'li unit of (.lie «li;im(^t(.'i' coiiesponds 
 to3.i4l(i units of the circunifoiciicc, ; tlicn if(« 
 C = a A — a unit ov any part of tlie dianietcr 
 A 15 or C I>, tli(! (!xccss of tlic ciicuniferonce a !> 
 over tiie file. (,' D will l)<'(M|ual to tlu' excess o!" 
 A 15 on (I h ; wluMice <i h is an aiilliuielic mean 
 (ia<»5 <Ix.) '»t't\v('en eirc. A and eiir. ('. Now, 
 (12N in.) A !•: : a e : (J F ;: ein,-. A. cire. :« h : cire. (J D ; thoroforo 
 a (is an arithmelie mean l)elw<'en AK, (J F ; and sine(i tho arc A E, in- 
 (U'lijiitely .small, may \w cnnsidered (ISO (i.) as l)eing sensibly a 
 straight line, the part .V K I' C of th(^ circular I'ing may be considered 
 as a trapezium ; but, area trapezium A E F C = (341 €«.) ac ■<. 
 A C ; then also, area ring A CJ — cire. a, l> /, A C 
 
 Kx. 1. How many s(juare inclu's in the area of a circular ring 
 
 whose exterior diamefci' is :{() iiiclies and the breadth 'ik inches ? 
 
 A UN. ^I5.udr>. 
 
 2. The diameters of two concentric circles are l.")and 10 : what is 
 the ar(>a of the ring formed Ity these circles ? Alls. 08.17."). 
 
 I. SiH'li wciiiM \>i: -.w iillcy iirniiii.l .i ciicMitii' i,'iii'iii'ii, ilit) liorl/.ontiil section 
 <ir 11 liolloweii cdliimii, l.lii) giomiii pliiii ot I In; WhII oI a lowni', ii Herlioii piM-peii- 
 (liciiliir to lliH iixiaof 11 pliiu nc tulio, iVc,, iV.*.-. Set; llu^paniik' biwes «t Ilie liollow 
 cyliiutoi- of till! Tuh/ian. 
 
" I 
 
 40 
 
 KEY TO THE TAIJLEAU. 
 
 Ans. 24.3474. 
 
 3, Required the area of tlic liiijn wliosr coiilaiiiiiis (vircles 
 liavo for dianH^tcrs !) and ."i ? Ana. 'Ui.W'il. 
 
 4. The two diaiiu'lcrs nl' a ciiciilai riii^ ;iie 21.2.5, and !),7.) j 
 what is its superficial <M)nt(^n) ? Aiin. 27i).!)!)o]. 
 
 9. Deterniiiio the aica of the space conijjrised hetwee.i two 
 concentric circles whose diameters are l.'j and K! ? 
 
 (3:i) If tlio circles An, a h, had not tlie saiiK^ 
 centre, as is the case for an eccentric wheel, it 
 is clear that tlie area of tlie annular space compri- 
 sed between thecircles wonhl in the snme wiiylx' ob- 
 tained by (iniiin;.;- (llnh' 1) Ww ditVerence of area of 
 each of tliem. ' 
 
 li -'/I. 
 
 h 'v 
 
 PROBLEME XI. 
 
 To find the length of an arc of a circle, " 
 
 (214.) RUIjE 1. MiiHipli' tlir uiimbvr of (h'ljm's in llir jjioposcd 
 arc In/ .0087266 and ffiis prodiicf lii/ the didinclrr of llic <irclr. 
 
 REM. I. Since the eircumft'icDce is ."Mil (i wlicn Mm^ diameter 
 Is 1, it follows that :j.l4Jn-f-;}(>() =().(l()872(i<; = h'n^Mi •' of the arc of one 
 degree, to a diameter (Mpial to nnity. 'I'liis (jnotient multiplied 
 by tlie number of decrees in an arc, will he the length of this arc in 
 the circle whose dianieter= I ; and this product multiplied by any 
 diameter will give the length of the arc in n circle of (hat diameter. 
 
 
 tti ,' 
 
 1. Hiist ■ c,(!i.lral ^cclion of the ecct>iitiif liiii: <<\' tlip Tahli'an ; projectidii 
 oil H plati" ot tlie opposite liiiseR ot a riii.-tiiin (il'iiii ciMi-]iii> ('(inc. 
 
 y. Sop Hiiioiii; tlip models (if lliH Tahhov. llie limiting,' mics nf Mie spj^tnptita iiik) 
 R.-'flton of iicirclp, Itjisps of llic iiii^'iilii of ii cyliintpi-. coups Hiiii fnistaof riglitcoiiea, la 
 leral H'uie8» of sptierica! pyimniiis mik) of seclioiis of hollowed eplici'ea, &c. 
 
 ■1. ttliJiH ulreiiciy lieeii.oliserved iiiid iM'»i<leR it i- clpiir tliMl tlip exiictiiess of a fe- 
 «nlt ia limited liy tliut of the pliuiients coiueiiied ; it is llieii h;iidiy nefieHSiiry to 
 reinarlt tliat i\i'Poidiii>,' lo tlip dp).'ipp of piecisioii wmited, if, miiy hocouie necessary to 
 »i«B a liirjj[er or sinnllm- niimliri- of the ilecimal.s of tiie unit of siicli element ; thus it i« 
 deiifthiit tite solution ofiiie prol)lcuii in question here niiiy leqiiire to feplHoo the 
 i-ntio 7r»= 3.1416 genoiiilly used, by tlie inorw exact ratio 77 = ;i,l ll.')!» or by the mtio 
 etill move iippmximative n = It. M l.WJ, r=3-HI.> DOti, r: = :i. 1 4 1 59^)5, &c., with an 
 •tddivional defitiml of llip tpiin or factor ;r foi' (lacli additioiiai deciiiiHl of the unit of 
 (ht result. 
 
 f!!u 
 
JIKNSIIRATION OK SIJRKACKS. 
 
 41 
 
 RtM. SS. Siin'i« !< iiiiutit** is rli(^»iO(li part oC a dHgrue, aad » 
 socoud tlio 60tli of II minute oi f,lu' ((lOxUO) ."kJOOtli of a degree; if 
 the arc propust-d coutiiiiis ii)iiiutes, they will be reduced, by dividing 
 tliom bv ()(), to (lii> (l(>cini;il of ;\ (li'gr<ie, Jind if tin- arc also contain se* 
 rouds, tlu^ niiuntcrt will thsi he rcdutCMl lo seconds and tlio whole 
 afterwards divided b.v -M^W whicli will chaufto as before into decimals 
 of a decree the fractioual part of tjie arc. 
 
 Ex. i. The diameter bein;.; \f* feet, what is the length of the 
 fUTof;«) • Ans. 4712364. 
 
 8. Find the leu,:;tli oi'an arc of J-<iMO or I'-i^'"*, with a diametev 
 .tf20? Ans. 2.12.3472. 
 
 3. fii a circle wlioso dinnieter is (i8, what is the length of the are 
 of 10°. 15 or 10.2.>^ 1 Ann. t^.0823f)6. 
 
 A. Required tli(> length of an arc ot'.>7'l7 14' ; the radius of the 
 circle being 25 feet ? Ans. 25 feet. 
 
 For .57^ 17' 44" is the ;5.l4l.>!)2()ih ]^.u■{ of lf^n\tli!it is the length 
 of the radius in terms of the circunifer«Mice. 
 
 a. Determine in a circle whose radius is 2t), the length of an 
 arc of ,50^ 30' 3 ' ? Ans. 15.885. 
 
 REM. a. If the number of degrees in the required arc were not 
 known, it would be easily found by the nic^thod of par. (785 G.) 
 where the chord and height of the arc are given to tind the remain- 
 der. 
 
 (35) RULE U. Defenninc (1S5 «.) the length of the whole 
 tircumference. of which ihc tjiren arc formR apart and establish then the 
 foUowiitg proportion, ri: : 3(i0^ ; ihc length of the fdrcumference :: the 
 number of dcgrcc)^ in the ore : the length of the are. 
 
 Ex. I. Under a radius 14, what is the length of the arc ofGO° f 
 
 Ans. 146607720 
 3. The cluird AH of an arc ACB is 30 
 feet, and the heiglit or versed-sine EC is S 
 feet ; find the length of the arc ? 
 
 An»i. 35J feet, nearly. 
 
 3. What is the length of the arc whose 
 chord is 48i and height IS\ ? 
 
 Ans. t;4.767 nearly. 
 
 4. If the chord of an arc measures 
 
 20.386 perches, and its versed-sine 4 percli-^s ; what is the length of 
 the arc t Ans. 'i:?.402 perches nearly. 
 
 5. Required the length of an arc of circle whose chord is 40 anA 
 tli« height 15 ? An*. .W.JW nearly. 
 
42 KKT TO THK TABLEAU. 
 
 (SH) RVIjE III. ft h also nhowii fhat : Ike leiii/th of mt arc i*i 
 i>enj nearly obtained, hy aubtracUngfrom eight tinwn the chord of half the 
 mc^ the chord of tfie whole arc, and thcu taking one third of the dif- 
 
 Ex. 1. Tho chord of an arc is .'}6,7.'> and Mm chord of half th« 
 arc 2t3.2 ; what is fho length of tho arc ? Ana. lO.HKi nearly, 
 
 Ex. 3. What ifi the length of »n arc whoso chord is .■)0.8 and 
 the chord of half the arc .'W.fi ? An». (i4.(iti neiirly, 
 
 REM. When tho chord and the height only of the whole arc are 
 known, the chord of half tho arc, if need he, is obtained equal 
 (30ft Cr.) t^ the square root ofthesnm of the squares of the vor- 
 iied-frine and hclf-chord. 
 
 PROBLEME XII. 
 
 To find the arc of a sector of a circle. ' 
 
 (St). Rr£.E I. .¥«///;% (4»0 2^ G.) the mr of the i^eefor {[\n\X 
 i« the length of the arc) bg half the radius. 
 
 RlTIiE II. Find the area of the whole circle, and then muke 
 //«« proportion : .'WO degreett : degrees in the are of the sector :: the 
 area of the ivhole circle : the area of the sector. 
 
 Ex. 1. Required the area of a sector, wl.osp arc is 18 degrees 
 and diameter of the circle .3 feet ? Ans. 0.35843. 
 
 S. What is the area of a sector of which the arc is 20 and the ra- 
 •iins 10 ? Ans. 100. 
 
 S. The arc of a sector is 147'^29' and its radius 25, what is the 
 •nperticial content ? Ans. 804.Jin86. 
 
 i. Determine the area of a sector, when the chord of the arc 
 =28 and the chord of half the arc = 16 ! Ans. 275.39. 
 
 5. The radius of the circle being 10, what is the area of the 
 •eet/or of which the cord of the arc is 20 ? Ans. 157.08. 
 
 «. The chord of the arc is 16 and its height (5 : what is the area 
 of the sector ? Ans. 88.873. 
 
 y. To find the content of a aector of which the height of the are 
 =4 and the radiu e=8 ? Ans. 66.858 nearly. 
 
 1. 8«« KMOHg the modsla ot thu Tabltau^ ih« lateral fttoet o^' th« tri Rcntangntiir, 
 kri-ffetftnr«l«r andl tii-«btaMngaUu', ipharJcNJ pyittinidt 
 
iiKNHnmATioN or bvrvaoer. 
 
 43 
 
 PROBLEM Xin. 
 
 To And the area of a sector of a circular ring or tlie spaoe 
 comprised betvreen tTvo arcs of concentric circles. ' 
 
 (38). RULE 1. MulUylij (clem, of 8il, H. 111. T.) Th» UaiJ- 
 unm of the interior and exterior ureti of the sector by its breadth j tluit 
 ia by the breadth of the ring ofwhivh the sector forms apart, or, which 
 »N the same thing, by th^ diff'inTnce of the radii of the convvntrie arct 
 ifhich contain it. 
 
 RUJLE II. Find by the last problem tlie area^ 
 of the two concentric sectors : their difference will 
 he the retjaircd area. 
 
 Mix. 1. The arc AEB or CFD of a sector AB of a uiicului' riug i« 
 30" the breadth A C of the riug 2i ami the radius AC) of the exterior 
 arc 15 inches If Ans- The vrea= 17.99875, say 18 sq. iii. 
 
 Sj. The two radii >.>£ a sector of a circular riujj; are 10.625 aud 
 4.875 aud the angle at the centre (J or A B that is tJie am A E B Lb 
 270" ; required the area of the sector 1 Ams. 209.99(i, say 2U). 
 
 3. Tlie arcs which comprise a section of a circular riug are II 
 feet 9 inches and lU feet 3 inches, and the breadth of the riug Ki 
 inches ; what, is its area t Auii. llfV sq. feet. 
 
 4. Determiue the area of the space comprised betweeu two 
 half <;ircle8 having a common centre, and whose diameters measure 
 20 aud 30 ? Ann. :i9.270 x ."i =. 19ti.;i'5. 
 
 (39.) REin. If the componeuL sectors 
 ABO, CD^ had not tiie same ceutre ; the 
 area of the space CFDO wt)uld first be 
 found by adding to the sector C'FDo, or 
 taking from it, as the case may be, the 
 sum of tli( triangles C'Oy, DOt/, and then 
 taking the difference befweei; .\KB<J 
 and CFDO ; which is plaiu 
 
 I. See oil the 'I'ubleaii llie coiicmiliic riiiK, bii«e nl' ilut hijlliiwmi ryliuii(>r. 
 ;ilao t.U« liil«nil taces ol the »ecliuiiJ< (if ill*! liolluwKii ijilinif 
 
 Mat 
 
44 
 
 
 
 KKY TO TUt TATlTi«AIT. 
 
 PROBLEM XIV, 
 
 To find the area of a segment of circle. ' 
 
 (40) RULE 1. VFind (4:W G.) 1>II inohlfin \ll the ami 
 of the Hector of the name are. ii-" Find af'tenr(trils the area of tlte 
 triangle formed bi/ the chord of the Kenineitl and the radii of 
 the sector. 3" The mtiu of these areas irill be (-1.14 <«. ) tfiat 
 of the segment, if the segment be greater ihau a semi-circle, and if the 
 segment ie less than half a circle, its area irlll l>e eijnal to the difference 
 of these areas. 
 
 Ex. 1. Find the iii»'u ol tin- se^incnt AKH 
 whoHe clioid A B is 12 and tlif nulius AC --10. 
 AD 
 
 : AD=iAB 
 :; Sin. D 
 
 10 
 
 (i 
 90' 
 
 ;ir. comp. lo.ir. !t.O(HMKKt 
 
 7rHir.i 
 
 lO.OlHHIOO 
 
 Sin. ACD m° o2'=;W5.87' 
 
 i>.77Hir)| 
 
 = 73.74'^' -1 lie dtiiU't'K in tlir aic AEI?. 
 
 Then 79.74 < (34 REM. I T.) (».()()H7'2(it! x 20 = 12.87 --= k-ngMi 
 (nearly) of the arc AEB and AKB x ,iAC= 12.87 x 5=--G4..'{5=ar(a of 
 the sector AEBC 
 
 Now C I) = VA( ' " — aT> , — VTOO".'}* ; = V «4 ~ f< ft (i X 8=48 area 
 of the triang!'^ ACB. Thence, nect. AKBO- . v BC-t)4.35-48= 1().:]5 
 rrseg. AEB. 
 
 2. Required the iirt'ii of (iic 8('<;iufnt wliowe lieiglit is 18 an<i 
 diameter of the cirelt! 50 f An*. (>M6. 3138. 
 
 3. The chord of a st','j,nient^ Ki, llie diaineter=20 ; what is its 
 surface f Ann. 44.7(»4. 
 
 4. Tlie arc of a N('<>inent contain.s !'0~ witli a radius=f> ; wliat is 
 its area? Ans. 23.1174. 
 
 5. Determine llie area of a Hi-gnienl of wliich the cliord of the arc 
 18 24 and the chord ol" luilf the arc =^ 13 ? See (530 or r»3« G.) 
 
 Ans. 82.533,'ftJ. 
 (41) RULE •»• V Diridc the height or vcr.'^ed-sine by the 
 diameter and find the tfuolient in the table of rersed-sineii at 
 tfie end of this culiiwr. 2-' Miiltijilg then the iinml>er at the right of 
 versed-sine hi/ the .si/iiarc of the diameter, and the result irill be the re- 
 quired area. 
 
 Hce muoiii; IIip uiodclM tif iIh' 'I'ahlfini llie !<:i*i'S iiiul |>iU';tlUI »et;tiuii:< ol «»r 
 •liiwa imj^'*'* ot'cTlhiiitiiis, i-niies. sjiinillps, &t'. 
 
 *iiAl 
 
MKNSITRATrON or BltRrArHfi. 4fi 
 
 (42.) The talil<- ill qiioisfioii t'oiitiiiiiM Uk> aifuiH of tli(> noguicnlx 
 ol'a cii'i'lc wliosf (liii!iKt»r is I and which is snppoM'il to be Uivideil 
 into KKKI t'liiial parts. 'I'ht'ir will tic IouimI the ari-a of a st'KUieut 
 whose liei;;lil is ihf oiii- ihnusaiitltli of the iliameter. that of a 
 m'giueiit whose hfiglil is 2 thousamlths of the (Uameter, that of 
 a segment who:^t' hfi<j;ht or versed-siiie is jn'oo of the diameter and 
 so on up to the segiiiciit whose heiglit is {'^'^''^ of the diameter, thftt i» 
 up to thetintirt' half-ciirie. 
 
 (43.) It is plain that tiiis rule is similar to rule II of problem 
 VII and that it (Idcs not re(iiiire a speeial demonstration, for it 
 is suHloieiit to ituiciuliei . to show its exactness, that in two differ- 
 ent eireles simihu He^iments are (till CJ.) those that eorrespond to 
 equal anjiles at the eentre and whose ehoids (double sines (1210 O.) 
 of the halves of lliose angles) and the vc is(id-sine,s have consequently 
 to eucli otiiei' the ratio of the diaiui'tirs of those circles and that 
 (557 GO such siinilai- tigurcs are to each other as the squares of 
 these diameters. 
 
 (44.) It is hardly ncccs.sai.v to add tliat if we had to do with a 
 segment greater than a semi-circle it woukl sutlice to operate on 
 the other segmenl, and then subtract it from the wliole circle, and 
 if tlio quotient of tlie given versed-sine by the diameter is not found 
 in the table, it will be easy to determine by a aim[»le proportion th« 
 difterence of area corresponding to the fractional part of such sine. 
 
 Ex. 1. The versed-sine of a segment of a circle being 10 and tlie 
 diameter .'j(» : find the area of the .segment ? 
 
 Ans. ]0-=-.")0=i;;=l — .J2 = versed-sine of the table: tlie area 
 which corresponds to tliis vorscd-siue is ,11182;) wliich mnltiplied by 
 S.'jOO the square of tlie diameter gives for tins area of the proposed 
 ficgment 27!'..jr)7.'j. 
 
 2. Required the area of Die segment whose height is ti and dia- 
 meter of the circle 21 ? 
 
 Ans. (i-;-21 - ,!\ = .28'y^ = versed-sine of the table to 
 
 which oonesponds area 184521 
 
 The area which coire.<pon(ls (o the next greater versed-sine is .185425 
 
 '!< 
 
 The difference bet ween these areas is , 000904 
 
 This difference :; i thai is ■: 5 et-r7 give.s foi' area coi-. to ^ .00064(> 
 To which ladd the area wlii.li ((.r. to 285 184521 
 
 To obtain the whole area of Llie segment 28.J7' of tlie table 185167 
 
 Now, multiplying by liie square of ihediam. 21 x21 = 441 
 
 We obtain for ari'a of the proiioscd segment .,..., 81.(i38()47 
 
IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 V 
 
 / 
 
 O 
 
 (./ 
 
 A 
 
 
 
 v.. 
 
 
 ^ 
 %' 
 
 «^ 
 
 1.0 
 
 I.I 
 
 1.25 
 
 1^ 
 
 12 8 
 
 III 2.5 
 
 
 liae 
 
 
 1^ 
 
 
 
 
 
 1^ 
 
 IM 
 
 2.2 
 
 
 1^ 
 
 = 
 
 us 
 
 us 
 
 1^ 
 
 2.0 
 
 u 
 
 
 IHUI- 
 
 
 i^ 
 
 
 
 1-4 IIIIII.6 
 
 ^^ 
 
 m 
 
 ^ ♦•^s 
 
 ^Q> 
 
 V 
 
 
 ■%^ 
 
 V 
 
^ 
 
1 
 
 46 
 
 MT TO THE TABLEAtT. 
 
 3. Fitnl tlie area of a Hegiuenl, wIiohp height is 2 and diama- 
 Ui 52 ? ^i>»- 26.88. 
 
 4. The versed-sine is 5 nud the diameter 25 : what is the area 
 of the aeginent ? Ans. 69.889375. 
 
 a. The height of a segineut ia 9 iuchea aud the diameter 3J feet : 
 find the area ? Aii«. 205.4118 square inches. 
 
 PROBLEM XV. 
 
 To find the area of a zone of a circle, or the space c n- 
 
 prised hetw^een any tAVO parallel chords and 
 
 their intercepted arcs. ^ 
 
 (45). RULE I. Flint find bij Hit mrtltod of par. (574 «•) 
 <tc., the diameter or radius of the circle auJ the other elements of the 
 cahulatiun to be made. Determine then (135 G.) separately by the 
 problems already given the areas of the component sectors and triangles, 
 und tahs their sum, if the zone be ceniral ; or if the zone be either central 
 or lateral, determine by the lust problem the areas of the two segments 
 having for chords the chords of the zone ; the difference between these 
 segments, or between the whole circle and the sum of these segments, will 
 be the required area. ♦ 
 
 Ex. 1. The twopiinilleljchordtiof a 
 /one are 12 and 2U and their perpendi- 
 cular di.stauce is 13 ; what isj^the area ? 
 
 AUH. 
 
 .87859. 
 
 2, Find the aiea of a zone of a 
 eircle whose jtarallel chords measure 12 
 and Hi and tin distance between them 2 f 
 
 An§. 28.379. 
 
 3. iJetermiut tlie siiperdcial con- 
 tent of a zone whose sides are 96 aud 60 
 and tht! breadth 26 '! An§. 2136.82. 
 
 I. See iduuiig ...e modelu ut' llie TabUan llii.' Inixeii iiiu) pmiilliii tittt^ljoim ui'uertiiiii 
 UiigulO) oj' a ('yUii(l«>i', toliu, uplieie. Scv. 
 
MRNSURATrON OF 8URPA0KH. 
 
 47 
 
 4. If two pariillel chords of a circular zone are 20 and 15 an<l 
 their porjiendiciilar distance 17.5 : What is tlie area ? 
 
 An». .•W5.4.'}6J). 
 
 5. Required tlie ar(*a of a zo.ie each of whose parallel chords 
 Ik 40 and the breadtli 36 ? 
 
 W. One of the parallel chords of a zone of a circle is 30 and 
 pasKCis throuj;Ii the centre of tlie circle, the other is 16 ; required the 
 area ? 
 
 RElfl. I'he given segnicni nuvy also be considered as com- 
 posed of the trapezium AHFD and of the two eqiuvl segments AD, 
 UF and its area detr "ned in this manner. 
 
 PROBLEM XVI. 
 
 To find the area of a lune, or the space comprised 
 
 between the arcs of two eccentric circles 
 
 ■which intersect each other. ' 
 
 (16) RUIxE. Find (4»« G.) hi; fhc prnhltm before the last, the 
 area/< of the tiro .ictjwi)il-^ iriiicli form the tiinr : ifteir elijferenee will be 
 the required area. 
 
 L L L 
 
 V "■"•i.^,..--" 
 
 Ex. 1. The chord AB of a lune AEBLA is 20 and heights of the 
 composing segments AEH, ALH, are 5 and 8 ; what is the area of the 
 '"HP ? Ans. 49.892704. 
 
 •1. The chord = 20 and tlie heights of the segments 10 and 2; 
 what is the area of tlie hine ? Ans. 130.204. 
 
 3. Determine the area of a lune whose length of the chord is 48, 
 and the heights of the segments 18 and 7 ? AnS- 408.608, 
 
 4. The base AB of a lune is 10 and the radii AC, AD of the two 
 containing arcs AEB, ALB, are 7 and 6 ; And the area. 
 
 5. The chord of a lune being 10 and the heights of the segment* 
 IS and 13 ; what is the area ? 
 
 I. S«e Rnirtiig the ihoHbIb of th« rai/««»nlie oppo»it« I>«8m iin«i Ui* t<iir«tl«t ■•«- 
 \i** •! tk* nngnla of th* h«ll«w«<l «Ttiail*i . 
 
48 KKY TO THE TAULKAir. 
 
 PROBLEM XVII. 
 To find ' the circumference of an ellipsis. 
 
 (47) This flgiire shown by ;itiy soction FI,AD (9»7 G.) or FE, 
 RN (1099 G.) ofii cylindor, or br. (I0r»r» « ), ac (1050 G.) ofii cone 
 by a plauo which being incUntjd (o the axis of ihom) Kolida meets 
 their two sides, is often mot with by tho nu'asiirci- - It is found in 
 the circus, amphitheiitre, garden plot, &•(•., and on a smaller scale in 
 the oval of a window &c., bat it is especially tlio semi-ellipsis whioli 
 is met with, in the sections of vaults of all kinds, in the arched head of 
 a door or window, or of an aiclicd opening betwtMMi two apart- 
 ments, &c. 
 
 (48) One might perliaps think nt fiis( that the circumference of 
 the ellipsis should bo an arithmetical moan bel wcenthe circumfereucea 
 of two circles having for their respective diameters the greater 
 aod less diameters of the ellipsis, or which is the same thing that this 
 circumference siiould be eiinal to tlnit of a circle, whose radius were 
 equal to the half-sum of the great and small radii of the ellipsis, 
 that is whose radius would be of an arithmetical mean between the 
 half-diameters of the ellipsis; and it is very nearly so for ellipses 
 whose diameters diti'er from each otlnM- but from 25 to 20 per cent, 
 but to be convinced that it is not always so, it is sutlicient to resort 
 to an extreme case, (as we have already done at par. (Oa§ O). For 
 instance, let us suppose that whilst the small a\'is of the ellipsis is 1, 
 the great axis be 1,000,000 ; it is evident that the circumference of 
 such an ellipsis will be sensibly equal to the double of its groat 
 diameter, that is 2,000,000 while the half-sum ')00000 + .5 or 500000 
 (for the 1.5 may be neglected) of the axes x 3. 141(5=1,570809 ; and if 
 thesmall diameter were intinitcly small with regard to the greater axis 
 supposed to be equal to 2, the exact circumferenc(! would be 4 (double 
 the great axis) while the arithmetical mean circumference would be 
 
 I. Although it is not, possil)li! witli the nriiicipje.s hithorio meiitioiied 'o given 
 aeniouRtrHtioii of tliis rule imd llwj four tbll()wiii.4, we liiive howevei- Mioin,'ht proper 
 to insert them lioro in order to ooitiplHte ilie niles iiei^Kni^iiry to the lueurtiu'eineiit of 
 plane surfnces, or of itiosH (1140 G.) wliicli ImvIiij; simple curvnture imiy be 
 dflveloped into plane anrfaces. 
 
 'J. Sea tiniong the models of the 7\ib/Mii, the I)iir>es and sections of the ohiiqtitt 
 ejlinders, cones Hiid conoids, (Stc, and li nstn ot .inch bodies. Tliese oilipsus are of 
 VArioni dograaiof «ec'«ntri(:iiy or hav« tliair dinmutiir.. in VMried rMtio«. 
 
MENSURATION OP SURFACES. 40 
 
 3.14159, &c., tlie orror being iii tliat case 4— 3.1410=8584 or nearly one 
 fourtli. 15ut if tlic circiiiiifeienco of an ellipsis canuot be correctly 
 obtaiiKul in this niivmu'r, it is demoustrable that it can be arrived at 
 by tiie following' nictliou : 
 
 (49) RrJLE 1, Multiplij the square root of the hnlf-stim of tho 
 sguares of the two diameters of the cllij)sis by 3.141G, and the product 
 will be the re(^uircd circumference. 
 
 Ex. 1. The greater diameter AB of an 
 ellipsis is 15 and the smaller diameter 13 ; 
 what is the circumference : 
 
 AlVS. '"'' 
 
 583 et 3.416 x 13.583= 42.()723528. 
 
 ( ^ ) i- (a9 G.) =V184,5=13. 
 
 3. The greater and snialler axes being respectively 25 and 20 j 
 determine tho periphery of the ellipsis ? Ans. G9.397!? 
 
 3. The semi-diameters of an ellipsis are 121 Jind 7\ ; what is the 
 perimeter ? Ans. 04.7607. 
 
 50. It is plain that the semi-ellipsis CBD is ecjual in perimeter 
 and area to the 8emi-ellii>sis ACB, and that each of them has for mea 
 sure the semi-circumference and the half-area of tlie whole ellipsis. 
 This rule and the following which show how to find the circumfer- 
 ence and area of the whole ellipsis give then also tlie means ofar- 
 rivinff at the iKremeter ACB or CBD or area of the scmi-cllipsis of the 
 same name. 
 
 It is moreover evident that any other diameter E 11 divides the 
 ellipsis into two parts of the same area and jierimeter. 
 
 (51). There is an important propertij of the ellipsis which permits 
 one to trace it withfaciliti/ or to discover ivhclhc a citrvilincal figure re- 
 sembliwj an ellipsis is reallji one or not ; it is that the sum FC f F'C, 
 FG + F'G, of the radii drawn IVom two points, F'F' on the greater dia- 
 meter and which are called foci or centres of the ellii)sis, to any 
 third point C or G, &:c., on its circunUVncnce, is constant and equal 
 to the greater diameter AB ; then it is dear that this very property 
 permits ns to establish tlie foci. Indeed, tiic two diameters of any 
 ellipsis being given, from the point C oiD end of tho smalhiraxis, aa 
 centre and with radius Cl'^-('F' = OA or UU--iAlJ, AB will bo 
 intersected in the re(inired i'oci 1"'' iiiid F', fr<»in tlio points F and 
 F' as centres, with radii FG, F'(ji of wliu li liic sum = AB, that is, witU 
 any radius FG less than FU and another radius F'G equal to t^e 
 
mm 
 
 60 KEY TO THB TABLBAD. 
 
 difference between the first radius FG and the diameter AB, arcs may 
 bo traced tlio intersection of which in Gr will ^ve a point, and by 
 repeating tlic oi)eration a series of points through which may bo 
 drawn a curve which will be the required ellipse. 
 
 (5'J) Oi-, there may he fixed at F and F' needles to which will be 
 fastened the ends of a thread of a litngth such as to give FC + F'C or 
 FG + F'G = AB ; it will tlien be suificient to hold the thread tight by 
 means of a pencil or point which can be moved round tlie two foci to 
 complete the outline of the ellipsis. 
 
 (53) To perform the same operation on a large scale ; after having 
 taken FG or F'G at pleasure, less than AF'or HF' but greater than AP 
 or BF', knowing Llie otiier radiu8 = AB — FG or AB— F'G, as the case 
 
 may be, and FF' being also known = 30F=2VGF^^=^0^= 2 
 V0A2_0C'^ one will have but to compute eitlier FF'G or F'FG of the 
 two angles at the base of the triangle GFF' and draw either of tiie two 
 radii of tlie ropiired length and with the required angle to give a 
 point G of the pn>i)osed circiinircreiice ; this operation repeated will 
 giveaseriesof points through which may be traced aline which will be 
 the recpiired circumference. Let us also observe that the measuring 
 of the radius G V or (>F' maybe avoided, by computing each of the 
 angles at F and F" and afterwards making an intersection G of the 
 directrices FG, F'G. 
 
 (54) Let us add that a geoinctrioal or graphic construction on a 
 small scale would have the advantage of giving in a more expedi- 
 tious manner and often accurate enough all the angles GFF', GF'P, 
 &c. necessary to determine tiie intersections or points G of tlie requir- 
 ed perimeter. 
 
 (55) The ellipsis is also I meed as/olUrwi : Let ac— AO or BO the 
 semi-greater a.vis,«f<= CO or DO half-smaller axis. In moving the right 
 line ac so iia to keep tlie point e on tlie dlamet(!r DC and the point b on 
 the diameter AB, the i>()iut c will describe the required ellipsis. In 
 practice tlie right line ac is any rod with projecting points at a,b and 
 c, and along thci diameters A 1?, C D are disposed rods, grooves 
 or elides to guide the points /) and c. 
 
 (50) UIJI^i: II. When the diameters are not very unequal, the 
 circumference of the ellipsis is pretfhi correcthj obtained by multiplying 
 the hat/-sum of these diamaters by Ji.j lJt>. 
 
 Thus the three last examples computed in this manner will rea- 
 pectively give for answers 42.41 instead of 42.67, 69.11 instead of 
 69.40. and 62.83 instead of 64.76 ; so that when the difference be- 
 
MENStJRATION OF SURPAOES. 51 
 
 tween the diameters docs not exceed } or } or when the ratios between 
 the diameters are 5 : 6 or 4 : 5, the error in the result does not ex- 
 ceed Y5TJ o"" jhsi ^"'^ when thedifforenco between the diameters is | or 
 when these diameters are to eacli othcn- as 15 : 25 the error be- 
 comes nearly ^^ of the whole result. When tlio diameters are to 
 each other as 1 : 2, the circumferences obtained by the two 
 rules are to each other as 47.12 : 49.GG, the error being in that case 
 ^ nearly. The diameters being as I : 3 the circumferenots are 
 nearly :: 6.3 : 70, the error being in that case r^d nearly. Wlien the 
 diameters are :: 1 : 5, the circumferences are :: i'l : ll.'3 and the error 
 ^ nearly. Finally if the diameters to each otiier :: 1 : 10 the pe- 
 rimeters would be :: 173 : 223, and the error .{'., or \ nearly. Which 
 will enable one to choose either of the rules according to the degree 
 of accuracy required in the result. 
 
 RE9f. Besides it is plain that we might also, after having found 
 the circumference, according to this second rule, correct it bj' the ad- 
 dition of the error or d(^ti('ienc3' jtropoitioned to tlie ratio Ijetween 
 the diameters, and as established above. 
 
 PROBLEM XVIII. 
 
 To determine the area of an ellipsis. ' 
 
 (51) RULE. MuWphi the product o/tlictwo diumcters hi/ .7854 ; 
 the result will he the re<piin'(l area. 
 
 Ex. 1. What is the area of an <;lli]).sis win -ii' dinmettn-s are 24 
 and 28 ? Ans. 24 x 28-432=AB < CD, and 4.M2 x .7854= 
 
 33!>.2!»28-area ABCD. 
 
 2. If the axes of an ellipsis are 35 and 25, what is its area ? 
 
 Ans. 687.225. 
 
 3. Required the area of sm oval whose length is 70 and breadth 
 50? Ans. 2748.0. 
 
 4. The greater axis of an ellipHis measures 840 links, the siuiiller 
 axis 612 links ; required the number of acres within this enclosure T 
 
 A us. 1 acres 6 perches. 
 (58) REin. Since the rule gives for area of the ellipsis the ex- 
 pression AB.CD X .7854 or wliich is (§7r«)ihe8ame thing[ AB.CdV 
 
 I. The component, f.icos of aevfli.il of (lie rood' t« of tho Tahle.au present ellipaes 
 of various degieea of ecoentricity, or whoHe diameters to eiicli other in viirioua 
 ratios. 
 
w^ 
 
 62 KEY TO THE TABLEAU. 
 
 X .7854, it evidently follows that the ellipsis is equal in area toaeirdc 
 mhose diameter would he a mean proportional between the two diame- 
 ters of the cllijfsis. Let d this moan diamctev. we have AH : d:: d : 
 
 3 ■ 2 2 2 
 
 CD and since (104 G) AT? : d:d : C J) it is plain also that the area 
 of the elU2)sis is a mean proportional between *hosc of the inscribed and 
 drcumseribed circles, that is, between those of two circles having for res- 
 pective diameters the two diameters of the clli2)sis. 
 
 (50) REin. The two rn'os which show how to determine the cir- 
 cumference and area ofau elUi),sisniay l»e witli advantage substituted 
 to the less precise and longer mtithod of par. (13'? G.) in the com- 
 putation of the perimeters and areas of the curvilineal, tliat is 
 (47 T.) elliptical bases of the oblique cylinder and of the frustum 
 of a cylinder (997 and 1099 G.) 'if' ^^ell those of tlie oblique cone 
 and frustum of a cone (1055, 1065, 1067, 1140, &c.G.) 
 
 PROBLEM XIX. 
 
 To find the area of an elliptic ring. 
 
 (60) RUIiE I. Determine separntch/ the areas of the two concen- 
 tric ellipses, and taJce their difference which will be the required area. 
 
 RUliE 11. Mnltiphj the half -sum of the parallel circumferences of 
 the two limiting ellijiscs bj the breadth of the ring, 
 
 Ex. 1. What is the area of an elliptic 
 ring whose interior diameters are 10 and 20 
 and the exterior diameters 12 and 22 ? 
 
 Ans. 10 X 20 X .7854 = 157.08,12 x 22 x 
 .78.54=207.34.50; the difference .50.2()5(} of these 
 two results is the required area of the ring. 
 
 2. The exterior circumference of an ellipsis is 100, the interior 
 circumference 90, the breadth of the intermediate space being 3.5 ; 
 reqni. ^'1 the area of the ring ? Ans. 332.5. 
 
 3. Determine the area of an elliptic half-ri;ig, whose parallel 
 perimeters measure 93 and 77 inches and breadth 10 inches ? 
 
 Ans. 8.50 square inches or 5.9028 sq. feet. 
 
 4: Compute the area of .any part Art cC of an elliptic ring, whose 
 exterior arc AC is 15, parallel Jirc a c 12, and breadth 3? Ans. 40.5. 
 
MENSURATION OP SURFACES. fiS 
 
 REW. It is lianlly necessary to retnark tliatiftlio brondth of 
 tlio aniiulav spaco were not everywlicre ociual, or even if the interior 
 ellipsis had any other position relatively to its exterior env<'lo])e, or 
 any ratio whatever between its diameters, the required area would 
 none the less be obtained by the first of the two rules of this problem. 
 
 PROBLEME XX. 
 
 To find the area of a segment of an ellipsis ■whose base 
 is parallel to either of the axes of the ellipsis. ^ 
 
 (61) Dmde the hriijht of the serjmcnt hj that of the two diarnvtcrit 
 of which this hoicfht forms apart, and Jitid in the laldr annexed to this 
 treatise the segment of a eircle u-hose versed-sine is e(iiial to the qnolieni. 
 Next find the eontinned prodnet of the se(jment thus fonnd and of the two 
 axes of the ellipsis ; this prodnet will be the required area. 
 
 Ex. 1. Compute the area of the el- A 
 
 liptic segment AGH whose heij^lit AK= ,.— ■^— -..,,^ 
 
 10, and the two axes AB, CD, lU and '2r. ? B/^^ — F-— ^T" 
 
 A.1S. 102.02. t B C~0— jk 
 
 2. What is the area of the segment of an ellipsis, whose base GH 
 is at 36 from the centre 0, the axes being 120 and 40 ? Aiis. .'5.30.7.5. 
 
 3. Determine the area of an elliptic segment whose heiglit CL 
 is 8 inches ; the two axes being 4 and .3 feet ? Ans. 
 
 (03) REM. If the segments of ellipses ACD, acd, aee, of the 
 6gure of par. (1140 *«.) answer to tiie definition of the enun- 
 ciation of this prob. we may if need be apply the rale here givcm 
 to express their areas. Tlie area of the elliptic scgnuMit whieli 
 forms the upper surface of the uugula fig. 2 of par. (1143 O.) 
 could be computed in the same way if required. And if the segment 
 to be computed were the zone or part AEFB, C(tPlD, tlie required 
 area would be equal to the ditference between the semi-ellipses ACB, 
 CAD and their respective segments EOF, GAH. 
 
 1. Several of the nngnlas of u cyliiuier, roue iind spheroid of the Tab/mu present in 
 their sections segmentH of ellipses, some greiiter, others sinuller thiui the semi-ellip- 
 818, others, semi-ellepses, luid others zones of ellipses. 
 
64 KEY TO THB TABLtAU. 
 
 PROBLEM XXI. 
 To find the area of a parabola. ^ 
 
 (63). Tliia fif,'ure is that of tlie rfection of a cone by a 
 piano parallel to its inclined side. (ADC, fig. of par (1140 G.) 
 gives an ideaof it). It has this peciliarity that any point E, H, &.C., of 
 tho curve is eciuidistant from a I'oiiitF called the focus and from a 
 straight lino MN pcrix^ndicixlar to tho axis CU) called the directrix 
 and whoso distance SC from tho apex C of the parabola is equal to 
 the distance FC from the focns to the apex ; so that one has always 
 EF=EM, HF = 1IX, &e. Now it is proved that the position F of the 
 focus is found by bisecting B D in T, joining CT and drawing TR 
 perpendicular to CT to obtain DR=CF=CS. The focus F found and 
 the position of the directrix MN determined, the curve is traced 
 by drawing a series of indefinite s'raiglit lines GH (called ordiuates) 
 parallel to A B or perpendicular to the axis CD ; tUen, from the focus 
 F as a centre and with a radius US equal to the distance between 
 the pariillclfi Gil, MN intersect GH in G and H, which determines 
 two points in the perimeter of the parabola. This operation sufficient- 
 ly repeated will give a series of i)oints, through which may be traced 
 a curve which will be tho figure required. 
 
 (64) The parabola is also traced with a square abc, whose branch 
 be is equal to the distance between the directrix MN & the base KL of 
 the proposed parabola. At the extremity c of the square and to the focus 
 F, tied a thread cGF equal in length to ch. The branch ab of the square 
 is then made to slide along the directrix MN holding at the same time 
 the thread tight along tlie brancli be, by means of a point or pencil 
 whose motion describes the required parabola. 
 
 (65). RUIjG. AfuUiply the base by the heightand take tico thirds 
 of the product for the required area. 
 
 Ex. 1. Find the area of the parabola 
 ACB whose base AB is 20 and height 
 CD 18 ? Am. 240. 
 
 2. The biise of a parabola is 1.'J.5, and 
 the height J 1.25 ; what is the area ? 
 
 Ans. 101.25. 
 
 3. CD = 10,AD.-8 ; what is its area 1 
 
 Ans. 10G|. 
 
 ThiB figure, like all the other tigiires, treated of iu the " mensuration of areas," is 
 to bo foiiiid amongst the coiupouHiit faceflor KideHof the models of the Tableau, See the 
 couicHi and oonoidai nngiilaa of the 'Tableau. 
 
MKN80RATI0N OF 8URFA0ES. 
 
 66 
 
 (66). BEm. It follows from tlio drfinition of the jmrabola that 
 any part GCH, ECV of the parabola ACB toniiiiiatod hy a base GH, 
 EV, parallel to A B, is still a parabola, and not a more K<'<;inci)t, 
 aa in tlie case of the ellipsis ; for the cone may be c'on«iih're«l cut 
 by a plane parallel to its base and this as well on the one as on the 
 other Bide of that base at KL without ceasing to be a cone and conse- 
 quently, without the definition of the section KCL or ECV, &.c, 
 being in any way thereby altered. 
 
 Whence it results that to arrive at the area of a zone, or segment 
 A£VB of a parabola by any line EV parallel to its base, one will 
 have but to tiike the diflfereuco between the entire and i)artial para 
 bolas ACB, ECV. 
 
 (67). There is still the hyperbola > (seel ion of a cone by a 
 
 1. The hyperbola ACH in hIiowii liy HeclioiiB ofceilaiii inigiilati Oi <:i)ncR iiiiiJ coiioidH 
 which will be foiinil .inioiig the niodelB of the Tableau. 
 
 This ciii've ift, but. in iicout.mry hpiiko, luia- 
 logoiiB to the ellipHlH. 'I'liiia, whilRt. in ihn 
 ellipBiH(51 X.)i 't' >BtliR Biiinof the raiiii iliawii 
 from the two fuvi wliich ia roiiHtaiit or iiiva 
 riable : in the hyperbola, on t,ti« coiitrai}', it, in 
 the (tiffereuce of tliene Hauia imiii whicli reuininH 
 constant ; which cmiseH tliH two halves, paita 
 or brHnclins of the curve (conjiijjaled liypeibo 
 Ihb afl they are called) to prentMit to one aiioilier, 
 not their concave HideB an in the uliifiHiH, but 
 their convex end«, apices or sides. To trace llii.s ciiive without any con.lition of 
 dimeusions, that is, without any condition as to the diun-nf'ionri of the cone or the 
 position of the plane of bection : having taken at will any two points, K, K, at 
 any distance from each other, from one of tliese points or foci willi any radian, 
 described au arc on each side of the axis (that in of the litiH wliich connects 
 the foci), and describe, from the other focus as a centre and wiili a radius 
 greater than the first by a given ditterence, two other area whicli at the point of 
 their iutersection with the two first arcs will deterniine 'J points of the required 
 curve. This operation repeated with two new radii, lalting care however that 
 the second radius be always greater than the first by the given difference (which 
 as has been seen, must remain constant) will give two other polniH in !ii« 
 curve to be described ; and other points in the curve may also he determined 
 till their sequel and direction render plain the course of the hyperbola. If 
 now, the radii he transposed, it is plain that we will have a new series of points. 
 that of the conjugated hyperbola. 'J'de point O which is half-way between ll.e two 
 foci IB called the centre of the hyperbola as of the ellip«i8. 
 
 The hyperbola may also as the parabola and ellipsis, be tra.-ed by a mechanical 
 operation. Taking a ruler fasteueil at one end to one of the foci of the curve to be des- 
 cribed, and 80 that the ruler may be movable round the said focus, the other 
 end ot the ruler will be fastened to the second focus by a string or line whicli 
 must be shorter thau '.h« ruler, by the required difference between tha radii ; then a 
 
56 
 
 KEY TO THE TABLHAO. 
 
 piano which moots its base at an angle greater tlian tliat made by 
 the side of the cone with tliat l>ase) the cjclold (wliich is dos- 
 cril)ed by a point place<l <fn llie circiiiiiforence of a circle maintained 
 in (he same [tlinie, <liirinj^'iiii culiie rcvoliilion ol" the said circle along a 
 strai,i,'Iit liiK' called (li(M)a.s(' of tlic^ curve, which is very much like 
 a semi-elliitsis, lig. of i)aragra|th (<*H) au<l several other cur- 
 vilincal li>?iireN, whoso areas and perinuiters ono may have to 
 i'onipule, and for which thei-eare special rules which allow of estab- 
 lishing with all the required precision their relative or absolute areab 
 and <'ircuniferences;but it hi to be remarked here as has already been 
 done (113 I €1.) that generally it will bo necessary to en(|uire lirst as 
 to the natuie of the proposed figure ; and the luere labour entailed 
 by this preliminary operation will often be sutlicienl to cause one to 
 deeide on resorting immeiliatcly to the method of the following pro- 
 blem. 
 
 (ON). A practized eye will often find it ditficult to umlerHtand 
 tbe nature of the figure to be computed, and nniy sometimes 
 make ])retty grave mistakes thereby. 
 There is for instance the curve AKCFH, 
 called" flat arch " {a iise-<lc-]mnier) and 
 otheis of the sann; kind often met 
 within the arched heads qf o[>eiiings, 
 and which one may be sometimes dis[»os- 
 
 ed to consider ;is an ellipsis, so as to compute its superficial content 
 by iho. rule ai>plical>le to that figure; now, it is seen that in this 
 case tlie dilference Al'.Cc t- r.F('/(or::,' AECc) between the two figures, 
 may be too grijal to allow it to be neglected. 
 
 PROBLEM XXII. 
 
 To determine the area of any curvilineal figure. 
 
 (<$9). KIII^I'I nividc the whole JUjiin\ if it he irri'fjular, (that is, 
 if the corrcspo)tdin<f iHirtu are not si/mmetrieal) the half or fourth part, 
 if rcxjular, into tr((pc::iiims of the same l>r€adtk or hci(jht, and proceed 
 then in the ntanncr of prolilem VI., douhlimj or quadrupUhj if need 
 be the area thus found to obtain the whole area of the fujure. 
 
 pencil or iioiiit wliicli will hold tlift Mtiiiig liff' iuul at tlmsiune tioie in coutHct. with 
 the niler, will describe, the ruler jjoing round the tirwt focus, the required hyperbola 
 iiud the trauijpoHitiou ul'tlie rulur uud utriug will allow of dencribiug at will the otiier 
 bnvucli of the curve, 
 
MENSURATION OF SURPACKS. 
 
 67 
 
 (*) 
 (TO) The iTKithoil of compntation by trapezium*, will be the movi 
 accurate as there will be in the figure to be computed conca^ ' .^ 
 and convexities adb, hcc, compensatory of each otlitr such ah are 
 
 *. Ainon^ t.liHHn fiKiirea, a ig ilie ovum or ovnl ; (hiicIi is t,l/e vmticHl nectioii 
 of thfl e^;g, fc<',.,) 6 iiftliH clIipHis, (sudi ia tln^ .•^tii'tiou of tli»* riii'Imi, &,c.,) or iuiy 
 of.liei' (iliiiloj{oiiH (ij{iiiH, till! hiillH evu wimlow. IIih HHCtioii of tlio h|)1ihi()1), t.lio mii- 
 phit.lieiiter, &c. ; c is the 'f-HllipsiM, oi- flit-uicli, cycloiii, llio tiiit siiclied Ijeiid of uii 
 opening, llie s^clioii of u vdiilt., &,c, ; d, hiiv iirHK'i'''-'' *'"iviliiit!iil ligiiio ; e ii piiiultoiii 
 oriiiiy oilier Hiiiiiliir lij^iire, liypi-iliolii, tlm iiiiscii iirc'icd IumiI of iiii opm iiij; ; lliiiHec- 
 tion of II vault, tlie verlicul nee? ion of u coimid, iIoiuh, &(!. ; / is the niking iiroli or 
 the section of iin inclined viuilt. g is llio develoiied concfiive or convex snifnco of an 
 iingiilH of )i riglii, cyliniinr; /t, the developud l.itHiiil HiirliU't) of no nn^'iiln of 
 a rij(lit. cone ; vi flio dHvulopnient. of tlie Hnrface of tlie iin^Mlii of lui oliiiqne 
 cylinder or cone. Tlie luneitKH or int"rrtei'iioii.s of vaults, already iiieiitioniMJ at aiticle 
 (ll'lS C) pi'eseiU alrtO surfaces the devi'lopinHiit of which (.ffer:' to the (•on«ldera- 
 tioii ofthe iiieasiirer the l.hioe last, lii,nires which have jiMt, heen deiined. The develnp- 
 ed lateral snrt'ace of i he Irnniiiiii of a riyht cylinder pri\si'iii« i In; loi m /-, and it. followa 
 frotn par. (997 G.) and froai the dnin. of pie. (I0J>9 C) 'hat it, Hiifiices 
 to multiply the half-siiin of its less and ^'ivaler hei^hi tv, is, hy the lei)).Mli oj) 
 perpendicniar toj'Jin- i)<. this breadth heiiifj evidently eipial to the dtivelopHd cir- 
 ciiinference of the section of a cyllinler by a )>laiie perpiMidiciilar to its axis or side. 
 The development, of the lateral surface of an ol)li(£ii(! cylinder (f)97j prt-si ills the 
 figure 71. the heij^lil, of which r.» which is that of i Iih inclineil side ot llie cylinder, is 
 everywhere iinifonii, the area of the envelope being consequently equal to the (irodiict 
 ofi'jhy the breadth oj), the perimeter of a section per|ieiiiiicnlar to the axis or 
 Bide of the solid. 
 
 It is useful to state also that if the iinirnlaof the rigu cyliinler of which the liiinre^ is 
 the envelope, instead of heini; [laiiial asKIiX IC orKLlil'' | iit;e 401). (J is entiitMU- com- 
 plete »8 ADrf, page 388 (i, iIim area of fj will he ohiaiiied hy lindini; f.lie product of op 
 by the lialf of r.», for in ihiscaae (/ will he bur- the envelop.- /■; of the friistum of a cylinder 
 whose less height w< would he e(jual t.nzeiii. Hy iiiijiistiiig to tlie lateral face of an 
 llllgiilaor frnstum of a cylinder or cone, etc. of the Tableau, a slieetof paper, to trace or 
 out it out afterwards at will, the pupil will get an excellent idea of the njitnre of the 
 developed Biirfaces her* mentioned. 
 
 .'4 
 
 1 '■■■ 
 
 
68 
 
 KEY TO THE TABLEAU. 
 
 seen in tho figiiros g, h, m, k, sinco the BOgmcnt bee which is 
 neglected hy cousldering as a h-apezium tho 
 part BCccb of the area to bo computed, will 
 be compensated by the segment adb Avhich 
 is in exce88_,iu the trapezium Aliba. 
 
 \B 
 
 (71) But when the figure is e.-tirely convex 
 one will add to the prfision by taking in the 
 sum of the segments ahd bee, &ic., determin- 
 ing by mere inspection or otiierwiso the mean 
 breadth which will bo multiplied l)y tlie cor- 
 responding perimeter adbec to obtiiu its area. 
 
 (Tli) Let us observe also that instead of considering as null the 
 initial height of the figure, at the pointAoftlie springing of the 
 curve, which would give for the area of the part AMhdaA of the figure 
 the triangle AB6, one will obtain more accuracy by considering as a 
 strfjght line the almost vertical part A« of the curve, which will 
 then give for a more approximate area of this component part of 
 the figure, the trapezium Art^B instead of tlie triangle A6B. 
 
 It is also plain that a continued subdivision Y>d, Ee, suificient to 
 allow of considering the parts ml, bd, be, &c., of the convex or con- 
 cave circumference of the fig. as being sensibly straight lines will 
 also have for result to add considerably to the accuracy of tlie opera- 
 tion. 
 
 (T3) There is also a pretty correct and 
 expeditious mode of arriving at the area of 
 an irregular figure ABCD : that of reducing 
 it to finy equivalent rectilineal or regular 
 figure by compensatory lines ah, be, that is, 
 such that the sum of the parts cut off by 
 these lines be equal in area to the sum of 
 
 the parts comprised in their inclosure, a graphic or mechanical ope- 
 ration for the accuracy of which one must often trust to an ocular 
 estimation. 
 
 (1'4) Finally, as to the evaluation of the developed lengths of 
 the perimeters of the figures in question here, let us remark again as 
 was done at page 596 Gr, that often the most expeditious manner 
 and not the least exact, of arriving at them, will consist in the use of a 
 thread or ribbon, or wooden or metallic rods thin enough to allow of 
 a^usting them to the perimeter to be computed, in order to immtj- 
 diately deduce from them the required dimensions. 
 
MENSURATION 
 
 OP 
 
 BODIES OR SOLIDS. 
 
 (See the luotlels of the iStemumetrical Tableau) 
 
 (T3) The meusiinitiou of solids, ooiii|nis('S that, of tlieir surfaces * 
 as well as that of their volumes or solidities. 
 
 It has already been seen (5, T.) that tlie unit of measure for 
 plane surfaces is a square the side of whicli is the unit of lens^th. 
 
 Any curve line is also referred to a unit of length, and its nu- 
 merical value is tlie number of times the liim contains tliis unit. It 
 is also to be observed here tliat the rule already given (page 177, 
 2° CJ.) to find the numerical relation between two straight lines or 
 to determine tlieir common measure or tlie greatest common divi- 
 sor, applies equally to any two curved linesof the same radius, since 
 that equality of curvature will allow of the supcrpositio?i and entire 
 and perfect coincidence of these lines in th(^ same manner as if tliey 
 were straiglit. Now if it is supposed that the lineal unit he reduced 
 to a straight line and that a S([uare be constructed on this line, this 
 square will also be tlm unit of niejisure for curved surfac(!S. 
 
 C^O) Thounit of volume is (1011 G.) a cube tlie component face 
 of wliich is equal to the superficial unit which serves to compute the 
 area of the solid, and the side equal to the lineal unit used to express 
 its lineal dimensions. 
 
 1. The opposite bases, the liiteiul fuce.-i inid llio t:e(^ti<iiis of iIih viirioiis niO(?el8 of 
 the stercometr'cal 'ahleau proHeiit, iiinniii; oilieirt, nil tlio pliiiit! lif^nires iilremly U'eateii 
 of iu the " Mensunitiou of niirt'iice.-i," (•(mi|)jirtirig \\w siiaate,reclani)le, paralkUoijrani, 
 triuiigle, ;^olijjoii, circle, x'-cior, set/meiU, zone, lane, elli^isis, parabola, lojinrbola, dsQ., Jiq- 
 
60 
 
 KRY TO THE TABLEAU. 
 
 PROBLEM I. 
 
 To find the area of a right prism ^ (946 G.) 
 
 (See the strreomctrical tableau.) 
 
 C7V KIJI^E. Mulfiplii (ft»3 G.) the perimeter » ABFCA or 
 ABCBEA (fig. of the follomiuj page) of the base b\j the height AE or 
 AF as the case ma;/ be, n)nl the product will be the lateral area. To 
 this area, add those of the two bases when the whole area is required. 
 
 Ex. 1. Wliat is the area of a cube whose side is 20 ? 
 
 Ans. '-^400. 
 
 2. Determine tlie whole area, of a triangu- 
 lar ])risni, the base of wliich is an eciuilateral 
 triangle liaving 18 indies side and the height 
 20 feet ? Ann. i)I.!)4!> sq. feet. 
 
 a. Required (iu^ weight of copper ne- 
 cessary to cover tiie inside of acistern the length 
 ofAvhich measures 10 feet, breadth 5 feet and 
 heiglit or dei)th 4 feet, the (•()i>per to be used being 5 pounds to the 
 square foot ? An§. 850 pounds. 
 
 I. 'I'lic pri^<Mi, wliicli comprisPM iil^o (liR piil)p hik) pMnillplopinftdoii, prettentfl itHetf 
 evurv ilay tollie tiieiisnKM's (•i)ii,<i.|(Miiti()ii. Il is see-n in the boiiy iitiij \viii><'i of Imild. 
 ii({8 of nil kiiiiiuMS wt'lliisiii tlie tiuiiic of the variniirt ni),ii-tn;ent,Hf(>nniiiff piirt of l.hein- 
 It in fomul iiifiiiii in tliH wiillrt. piiliiiH imd piei-s of hM kinds of l)Mililin^rt kikI oa 
 a Hiiuillei- nciili' in I'acli of tlw coiiiiHiMinf.; sIoiihs or bricks of tliertf> bodiMR. Giibled 
 roofs most ofien pieweiil tlie li'/nre cif liu; ri^lit triangliii- prism imd tlie giililew of the 
 Willis forniiii>; tlii'ir pill :illel liases are also prisms of tlie same name. 'I'lie body oi' 
 B<iimreof 11 (.'H'l'^'t-window is geiieriilly iiolliiiiit Iml ii tiiaii>,'iilar prism or right, half- 
 piirallelopipedon and the roof of a garret wimiow. if i: ho hip'd, is an ohliijiiH tritiugil- 
 liir prism, provided the iii(diiiiiiioii of the hip he ei^iiiil to that of the roof, and if 
 the plane of the hip lie not pirallel to that of the roof, it is then h frnstiira of H 
 prism the solid and snperlicial coiilent of \vhi<'h is to be valued. 'I'here are also in the 
 iirtaaiid triidesathoiHiiiid undone olijects atf-oting the form of a cube, right, obiiqiieor 
 truncated parallellopipedon, right, oblique ortrnnoated polygonal prism or which mny 
 be deeonipoaed into solids of this kind. 'I'lie onttingH and eitibankmentH of iHilroada 
 Hiid other roads, ofl en enough present to thecoiisiiieratioii of the meaaurerqiiHdi'uiigiilar 
 prima having for parallel bases triipeziumo. 
 
 '2. Each of the edges or Hide.s (,\H,(;K, 1st lig. or AF, IJG, Cil, &c., Und fig.) of 
 the prism being of the same length, it is evidently the Rame thing to multiply succes- 
 sively each aide or edge (base of the parallelogram which goea to make up the lateral 
 sinface of the prism) by the length of the corresponding parallelogram or to add 
 according to the rule all these breadths ao aa to multiply them at ouce by the length 
 of the aide of the prism. 
 
MENSURATION OF SURFAOES. 
 
 61 
 
 4> How many square motres ar« thoro in 
 the lateral area of a building tlio length of 
 which J8 100 metres, breadth 2J1.3 metres and 
 haight 17 metres ? Ans. 4192.;i 
 
 5. A room mea8Xire8 40 feet by 2'^, and its 
 height is 15 feet; how many square yards of 
 plaBtering will be required to cover tlu; tour 
 walls and ceiling 1 Ans. 82795. 
 
 6. What would be the cost to line witli 
 
 leadof Tpounds to the foot and at 4 pence a poiiiul, tlie inside of a rec- 
 tangular vessel the length of wliich is '3 feet 2 indies, breadth 2 feet 8 
 
 inches, and height 2i feet. 
 
 7 .'133 X .5 
 
 Ans. Area to be covered— 37. ' ^ souure feet, =263-— pounds 
 
 12 * 'Id' 
 
 = £4.7.9J=$17.55.185, 
 
 T. What is the lateral area of a deal of 10 feet, by 12 inches, and 
 3 inches ? Ans. 2.5 sq. feet. 
 
 8. How many supei-ficial fee cat stone in tlie lateral area of 
 an octiigonal pillar whose side is i.» inches and lieiM;|it 10 feet ? 
 
 Ans. 100. 
 
 9. Ho\V many squares of wainscot to cover the lateral area of a 
 hexagonal building whose oblique radius is 20 feet and height C3 feet? 
 
 Ans. 39.G0 
 
 10. What is tlie lateral area of a polygonal post 3 f<'ef [K'liineter 
 and 10 feet high ? Ans. .30 s<piar(^ feet. 
 
 11. The perimeter of an iron bar is 3^ inches, its length 7 feet ; 
 
 what is its lateral area ? 
 
 Ans. 3.7.5x74=315 square inches. 
 
 PROBLEM II. 
 
 Find the volume of a right prism. 
 
 (See the tabled h.) 
 
 GENERAL FOREIULA. 
 
 T8. To the sum of the areas of the paniilel ouls or base/t (A F, El) 
 or AD, FT) of the solid (see the fuj arcs of the last problem and the fol- 
 lowing) add four times the area of the section halfway between them; mul. 
 tiply tU whole by the sixth i^art of the hciaht {AF or AE) or length of the 
 
 'J ; 
 
62 KEY TO THE TABLEAU. 
 
 boihi (or, li'hh'li is the smne, mulfiph/ this sum hij the whole height and 
 then take tlie sixth inirt of the prvduct) the result will be the required 
 volume. 
 
 (7».) REITI. Tlie i)iisiii (9-IO U.) bciiij,' a hotly whoso size 
 (hrciulth, thicknc.is or <liiim(^l(ir) is every where the same, it is plain 
 that any sci tioii <tt' this solid by a plane 
 ahcd p.ualhl to tlie hase lU) or FIl, is 
 or ual to the base, which tlusretbre in the 
 case ot'tlic i>risiii rcchiccs tlie rule or ijc-ntMal 
 J'orimila at the top of the t,tcreumcirical 
 tableau to the more simple following ex- 
 pressictn : (see : Iiitroiliietioii, page 8, last ^| 
 j)aragraph.) 
 
 RUfiE. Ihti'i luiuv first the area of the 
 base ; mulliplti this area bij the hvitjht, the 
 j)roduct will be (1020) the rolume of the 
 prism. 
 
 Ex. 1. What is the solid content of ii. cube whose side is 24 
 inches ? An§. 13.824. 
 
 tS. How many cubic feet in a block of iiu'irble whose length is 3 
 feet 'I inches, breadth 2 feet eight inclies aii'I lieight or thichness 2^ 
 feet? Ans. 21 J. 
 
 :i. IIow many gallon.s of water contained in a cistern having the 
 dimensions of the preceding example, the gallon being 282 cubic 
 inches ? ^ Ans. 129||. 
 
 4. What is the volume of a triangular prism who.se height is 
 10 feet, and tlireo sides of its triangular basis 3, 4 and 5 feet ? 
 
 Aus. 60. 
 
 5. Required tiie number of cubic feet of stone in a pillar 15 feot 
 high and whose base is a regular liexagon the side of which is 1 foot 
 4 include ? Ans. 69.282. 
 
 O. Determine the numl)c-r of toi.ses of masonry (the toise being 
 6 X ♦) X 2=72 french cubic feet) in an octagomil prism of J 2 feet high 
 and 3 feet side ? Ans. 7 toises 17.47 cubic feet. 
 
 7. The pier separating two splayed windows, and whose base is 
 consequently a trapezium, measures 13 feet high, 2 feet thick, 9 feet 
 broad outside and 7 broad inside ; required the number of bricks that 
 have been required to build it, at 20 bricks to each cubic foot f 
 
 Ans. 4.160. 
 
 I. N.H. T)ie eii^MiHh iiuperiiil ^ullnii is 'J77.'J74 en^lisli cubic iticlieR, the old beer 
 giilloii=°J8!:i cubic inches and the wiue i;h1Iou Hctuully used in CauudH is 1231 english 
 cubic iuchea. 
 
MEN8ITRATI0V OP SURFACES. 
 
 63 
 
 §. A stone gablo 3 foot tliick, measures 40 feet at its base and 
 20 f(M't liigli 5 how many cubic yards of masonry does it contain f 
 
 Ans. 44if 
 
 9. The facade of a building is TW metres, its Iwiglit 17 inetros 
 and tlie tliickness of tlie wall /'•'} ceutiniotres ; wliat is the vujunie in 
 cubic metres ? Ans. 3M ^ |7 x .73=40M.r)3. 
 
 10. Required the number of cubic metres in an <'nibanknient 
 whose length is 100 metres and eacli of tlie jianillel planes consti- 
 tuting its ejids is a tiapezium having for ])arallel bases ."{ metres and 
 13 metres for its height 3.3 metres. Ans. riCIO. 
 
 11. A well must be 27 feet deep, and iti-', jdane must be a regular 
 hexagon whose radius ofthecircuni.scribed circle be.") feet; how many 
 cubic yards of rock are to be mined to give it its required dimen- 
 sions ? Anil. The side of the hexagon is (Hitt G.) ."i feet ; ") — .") x ,■) = 
 25, and 25 X 2..'>9d07U2 (area (27 T.), of tlie hexagon whose .side 
 is J) =04,9510 square feet = area of the given 
 64.0510 X 27 
 
 lexagoii, and, 
 
 27 
 
 :G4.051!) cubic yiirds. 
 
 Ex. 12. What is the solidity of a rectangular iron bar 4^x1 
 inches and 14 feet h)ng ? Ans. 7'}G cubic inches. 
 
 13. Required the volume of an eight sided post wjiosc height is 
 10 feet and breadth of each side 7 inches 1 
 
 Ans. 7 x7x 4.828427 1 := (3T T.) area of the base = 23fi..-,02f>279, 
 and X 120=28391.15 cubic inches, and-M728(or 12 x 12 x 12)= 10.43 
 cubic feet. 
 
 PROBLEM III. 
 
 To find the area of an oblique prism. 
 (See the tableau) 
 
 (80) RULE. MuUq)lii (996 G.) Ilie 
 
 length AF, BO, CII &c., of the side by the pe- 
 rimeter of a section L^INliSL perpendicular 
 to the side. 
 
 Ex. I. What is the area of the face and 
 two sides of an inclined beam or rafter with 
 parallel bases the length of which is 12 
 feet, the breadth of the face 9 inches, and 
 that of the sides 13i inches ? 
 
 Ans. 36 square feet. 
 
 % The length of a cor ice under a 
 
 (i 
 
 '! ; 
 
 ]•• 
 
64 
 
 KEY TO THE TABLEAU. 
 
 flight of stairs hotwoen parallol walls is 20 feet and the circumfei-ence 
 or periiiH'ter of a section of tlio cornice porpentlisular to iti direc- 
 tiou is 27 inches ; what is its doveloped area I 
 
 Ans. 45 square iset. 
 
 PROBLEM IV. 
 
 To find the volume of an oblique prism. 
 
 See the tableau. 
 
 REiTI. Tlie oblique i)risni, being, as the right prism, of inva- 
 riable diameter thiongliout its whole length, every section of this 
 solid by a. plane i)arallel to the base would form a surface equal to 
 that of tlie base ; whence it is plain tliat for the oblique prisna, as for 
 the right prism, the general formula is reduced to the following sim- 
 plified exjjression. 
 
 (SI) KUI.E. .¥»//(>?// (I«20 G) the area of the base ABODE 
 or FGHIK hu the height IPperpendicular to this base ; the product 
 wilt be the required volume. Or, which is the same thing. 
 
 MuUiphj (lO'l.-i G ) the side A F, BQ, 
 CM, dV., oj'llic solid hji the area of a section 
 abcdea perpendicular to this side. 
 
 Ex. 1. How many cubic feet of oak 
 will be required for the carriage of a tlight of 
 stairs 17 feet long and 15 x 4 indies square. 
 
 Alls. Gi 
 
 JJ. Tlio horizontal base of the breast 
 of a chimney built obliquely, measures 
 7 feet by 18 inches, tlie perpendicular 
 height being 7 feet 3 inches ; how many 
 bricks does the parallelopipedou contain at 
 18 bricks to the cubic foot ? 
 
 Ans. 76\ cubic feet x 18=1370i bricks. 
 
 3. Tlie triangular sidi^ of a garret or dormer window has for its 
 horizontal length 7 feet, for vertical height 5 feet, the breadth of th6 
 dormer being 4 feet; the roof of the dormer window is hip'd pa- 
 rallel to the roof (/f the building ; the heiglit of the triangle which 
 constitutes its vert a' section is 2 feet ; what is the total volume t 
 
 Aiif- The body or square of the dormer (right triangular 
 
MBNBtiaATION O* SUttFACFS. 
 
 06 
 
 prism) 0)=i (7 X 5 X 4)=70 cubic f«et, the roof (olilique triuDKular 
 j)ri8m) = i (7 X 4 ■< 2) = 23 cubic feet; the required vol unie is couse- 
 quent'ij [)8 eubi'; feet. 
 
 PROBLEM V. 
 
 To determine the area of the frustum of a prism. 
 
 (Seethe tnhlean.^ 
 
 (S2) Riri.K. Find sepamtehj (]0.»« €J.) the area of eaeh of Hh 
 companent faces ; their sum will he the area reijuired. 
 
 lix. What is the number of siiperliciiil IVet of cut stone in the 
 circumference oftlu^ toj) of u chimney situated oliliqui l.v on an inclin- 
 eu roof, that is tiie component facesof wliich are not i>arailel to those 
 of the building ; the \Anuv of the chinim'y bciiifj a rectangle.'? feet 
 by 4 feet and the respective Iieights of its four sides or edges '/ , 8, 9i 
 and Si feet ? 
 
 Ans. i (7 t- 8) X 3=22 + KS f 9i) x 4 = 35 + -KIH + Si) x 3=27 + i 
 (8i + 7)x4=3l = lloi. 
 
 PROBLEM VI. 
 
 To fihd the solidity of the frustum of a triangular prism. 
 
 (See the tableau.) 
 
 GENERAL FORMULA. 
 
 (83) To the sum of the areas of either (f its three pairs of parallel 
 bases or faces ACFD- BE, ABED-CF, BVFE~AD) add four 
 times the area of a parallel section or plane h((lf-wai/ between them. 
 
 I. Ileie the (irimii in (jm^hlioii ii()("< iiol rust ciii diu- nt it.s piiiMllel bil^^es ; liiil, 
 tliis circiiuiHlaiicH cuiirim pieveni <)ii« fVom Hmiiiiiis; iinmediately on llie imtiirH of tho 
 nolid to be lueHsured ; foi', it ip pviilnntly inditfeicMl, rt»s|iH(!tiiig the volume re()iiire(l, 
 whether the poBition of the polyhedron be VBrtioal, horizontal or iiiclliied. 
 
 1; 
 
 i! 
 
cc 
 
 KEY TO THE TABLFAIT. 
 
 This sum inul(ij)Ued hifthc sixlli iHirtof Ihc (or.c<poHdtnii heiijht of the 
 solid f null be the required roliinii'.. 
 
 (80; III 'Hct, tlioii^fli tli<' IViistiim ol'i'. tiiaii^iilai' ptisii, wlitMi con- 
 sidcri'd wilh ifsiicct lo its iioii paialU'l Imses or eiulH ABC, GHK, or 
 ABC, 1>EF, cannot 1m» nicasuicd at once ]>y tlic ji't-noral formula and 
 must br (k'coniii()s<'d liy a jtlant' of section iiaralici to one oftlioso 
 bases and passing' tlirou^li tbe nearest point of t'lc otber base, tliat 
 is, tlirouyli tlie end of its edge or (»f its siiortest side, into a prisui 
 and a jn raniid, wliicli may be measured sejtaratitly by tlie general 
 formula, to take ai'terwaids tlie sum of the results ; iiowever, if 
 att(nition be paid to the nature of the solid, to wit, that the faces 
 and edges or opposite sides are parallel, an*' that tlie siib's or edges 
 being lout simplo lines, the area of each «)t' tliem is I'lual to zero (0) 
 it will immediately be si^en how to aiiply the rule t(» measure at 
 onco the proposed prism. 
 
 (§5) Example. Let ABC-DEF (tig. of the following page) a 
 frustum of a prism where AD=:8, ¥C = 7, IJE = iJ, CK=4, ]ieiglit,=5, 
 (the base GIIK being considered perpendicular to the sides or edges 
 AD, FC, BE). We will have by the formula : volume = area AGFD 
 + 4 times thc^ area (irj'd i tlus area BE, which it null, the wUulo mul- 
 tiplied by I of the height. 
 
 The upper base BE is but a line and is equal to 
 
 Tbe lower base=-^i^ x r,K = -~ x 4=7* x 4= 30 
 
 The section acj'd (Imagine such a section ((c/d parallel to the base 
 ACFD and half-way between this base and the apex BE) 
 
 , ADfBE 8 + ()q. . 
 gives ad— ^^ =_.^_-8i, c/ 
 
 FC + BE 7 ^ 9 
 
 =8, ami 
 
 the breadth of the trapezium acfd isfy/i = KG-f-2=2, whence 
 area «/hi=8J I 8, that is l(ii-.--2 or 8.ir x t2= IfiJ and four 
 times thisarea=l(ji x4= 66 
 
 The sum of the areas is 96 
 
 which multiplied by the 6th part of the height or by JJ, gives 80 
 units for the volume of the proposed frustum, w'lich agrees with the 
 result given hero below of tiie calculation of the same frustum by 
 another method and proves evidently the accuracy of the formula. 
 
 Here again, as for the prism, the general formula is reduced to 
 the following more simple expressicm : 
 
 (86) RU1.E 1. j¥m/</2% (1093 G.) ihc base of the frustum by 
 one third of the sum of the heights of its three paraUel sides or edges. 
 
MENSURATION OP RURFACEP. 
 
 67 
 
 RULE II. MitUipIji the third of ihe 8iim of itn three parallel 
 sides bif the area of a section perpeiidicii >• to these sides. 
 
 KEi^. Tliiw Hccoiid iiilc, it, has ' leii said (lOIHI G.) 'lorivps 
 
 evidpiiM.v IVoiii Ihiit of paraniMpli (I035 «.) but slioiild tiiiM oon- 
 
 clnsiiui, pciiiaps too iiiiiiM'diatc ('t»r tlii' j»iii»il to pciroive its 
 
 truth, not be Coiind ri^'oroiis and satisfactory t'uuugh, it is liowevor 
 
 easy to show its oxactiioss, in did'cn'ut 
 
 ways, the followinp; of wliich l!ioii.i;h llu; 
 
 mo8t-«.\l»(Mlitioiisis not the least foiicliisiv*!. 
 
 Let then AHC— DKF tho fnisfum of an 
 
 obli(iu(> tiian!j,idai' i»risni, dividtul into two 
 
 frusta of ri.i-lit prisms aiIiv-AlU\ (UIK - 
 
 DEF by a plains (iHK pcipiMidiciilar to the 
 
 parallel sides ;* ' ', HK, CF of tiie solid. The 
 
 volume of eaclieoinposinif frustum is ecpial Cl01>fl<}.) to the jirodnct 
 
 of the eommon base(}lIK by one (hiid of the sum offhe peipendiculara 
 
 GD, JIK, KF GA, Jlii, KV : bui (illK x -i((iD i HE |-KF) + 
 
 GHK X J(GA ) II1? + K(')=GI[K x i(GD ! GA t HE -I- IliJ i KETKC) 
 = GIIKxKAD + BE i CF) : tlierefore. &c. 
 
 Ex. I. The bas(! of tlie iViistiini of a ii,i;ht liiaiigular i)rism is 
 10 square feet, the sides are 7, H ami !> feet ; what is its solidity ? 
 
 Aim. 80 cubic feet. 
 
 'i. The tliree sides of the frustum of an obli(iMe i)rism are 7js, 8| 
 and Defect; tlie base and heii>lit of a section ])eiit('Mdieular to the 
 side are respectively' .') iind •'} i\-i'i ; wliat is the solidity of the solid ? 
 
 A«s. 8 J < 7}> = ii:i'l cubic feet. 
 
 3. The three sides of tin; base of an inclined prism measure res- 
 pectively li, 4 and .') metres and tlie heii;hts of its three ai)ices are G, 
 7 and 8 metres j what is its solid content l Ann. 42 cubic metres. 
 
 PROBLEM VII. 
 
 To find the solidity of tho frustum of a prism whose 
 base AD or section perpendicular to the side 
 is a regular polygon or having symmetri- 
 cal halves (1097 G ) 
 
 (87) BiK^J. Here a.'j; liii llie n^neral lormulu is re<lin'ed to and 
 maybe replaced by eilhir of ilie Coll'iwiML; simplitied expressions, 
 or one n)ay at will compute sepir.ilely tlie volume of each of tli«i 
 
68 
 
 KRT TO THR TAnT,FAU. 
 
 c(»mi»oii»>iit fiuata of triiui<4iiliir prisms, hh in tlie following prohleni, 
 and then t:iU(> tln-ii' snni. 
 
 (MS^ MirtAi I. M ititi phi (tonil «.) the bns'^ bif half the sum of 
 the he'ujhh of tiro oppositv hhIcs ; the product will be the volume re- 
 fjuired. 
 
 KI'l^l-] II. MitUipIji (hr half sum of two of the opposite fiides or 
 cfh/cs Iff Ihr fnixlui'i bij (he una of a avetion perpendicular to thote 
 parallel »/(/(•«. 
 
 Itl'^ITI. 'riii>' HcciMid nilfdcrivcKUf^Min from 
 par. :^0*.KI <«.) ninco one may Huppose the fniH- 
 tnm of the polvnoiial ini.sin diviilcd into fiiista (»f 
 tvian;nular prisms, and for each of tlicsc compo- 
 nent ftnstamakc tlicsainc proof uh for tlic frus- 
 tum of tiic triaii^Mdar prism of I lie last jjroMcm. 
 
 Kx. I. IFow many ciihic foci of stone in I lie 
 top of a chimney havinjf for horizontal section a regidav hexagon 
 whose 8id«^ is 2 feet, the hei<;hts or lenytha of tvro opposito edges of 
 the frustum heinj; IM and 17 feot f 
 
 Alls. 2. -)!i8(»7f;2 X 2 X /'--t ''^"j = 15:1.884572. 
 
 S. Find the luimher ol'cubic inches ofbircii in a staircase baliiRter 
 having for liorizontal section a regular octagon 13 inches diameter and 
 whose least and greatest length or height measure re.spcctively 37 
 and 2i' inches. 
 
 Aii*<. 'I'lu^ rccpiired side of the (tctagon is pretty accurately ob- 
 tained in this case liy describing a circle 3 inches diameter thence 
 to find the choid of one eighth of its circumteretjce. This opera- 
 lion givis for the bieadlh of one (>!" tli<' sides of the baluster l/^j inches 
 
 nearly, say 11.5 ; or (l.l.j) = 1.322.). and I. ;1225 + (tJH T.) 4.8284.271, or 
 to abridge 4.H.'5 l..'<2 -(I.M75 s(|uare inclies=:;iiea of the section of 
 the baluster; linally, ti.-'i75 x ii27 I 29) - (i. .'575 x 28= 178^ cubic inches. 
 
 PROBLEM VIII. 
 To determine the solidity of any frustum of a prism. 
 
 (See the tableau.) 
 
 (89) RULE. B'rst find separately (10«8 €i.) by the preceding 
 rules the volume of each of the frusta of the component triangular 
 prisms, and then take their sum. 
 
MRNStlRATIOff OF StJRrAOr,*. 
 
 69 
 
 Ex. An fmbaiiktiHMit ol' enrtli pn-snits tlio form of the fiUH- 
 tuin of a liglit priHiii liavhig for itslmsc tlit» |n»l_vj^oii AIU'DI'.A ; tlu* 
 area of the comi»oiipnt liartc AIIC is SI) s(|iiiii'f yards, Miat of tln> Itase 
 ADC = 73 yards and tliat of llic Itasi- ADK — im yards ; tlitt liciKlifrt 
 or longtliH of tlic paralli'I sides A, K, (', iVc, aic vsitcctivcly 7, H, <», 
 18 and II ft't't ; wliat is IIki mmilM'r of niliic yards in tli*- itropost'd 
 BoHd ? AON. (1 lOS.'-JO" O.) loO X i (Y T H ' G) I i}57Ti~(7T9Tl[3 
 + 585 X 1(7 + 13 I rr=3l)0() Mi.T) 1 i (i(>r)= llJ.itlKi cubic feet; divid- 
 ing by 27 \v(« liave r)!»2i^ cubic yards. 
 
 REHI. lUm'i w«i have reduced into square fe(*t, Hi<^ areas of tlie 
 basos given in S(|nare yards, and divided i>y 27, bnt it is |»Iain lliaf, 
 since 3 times !>--27, it woidd be ilie s;iine thing to multiply imme- 
 diately by the yards an<l then divide them by 3. 
 
 PROBLEM IX. 
 
 To find the solidity of a w^edgo. 
 
 (See tlie (tthlran.) 
 
 (90) RUliK. To the 8<tmo/ the ((iras o/rillirr o/' its fhrrr pairr. of 
 parallel bases, add 4 times the cenlnil or iKil/'-inn/ xrvfinii iiiiil lunltiphj 
 the 6um by the 6th part of the height eoirrsjKniiliiiii to .viirh Imsrs. Thv 
 reatUt ivill be the required roliime. 
 
 e o f 
 REHI. Tile wedge, 
 as alreaily stated 
 (1000 G.) is but a 
 triangular prism or 
 the frustum of a prism, * 
 as the edge f/ is eqiuii 
 or unequal to tlie two /) m. e 
 other sides; til us the general formula will be replaced as the case may 
 be, by the fiimplitied exj)ression which is derived fiom it in the cas** 
 ofthepriam, as of tliefrustum ofa iism. Yetiii the ciiseofthc wedge 
 of which the base is generally a rectangle and consequently very 
 simple to be measured, the student will perlmps lind ii more advan- 
 tugeotis to keep to the method of the gen«'ral foniiida. 
 
 Ex. The rectangular base of a wedge is 20 < 40 {\'v[, the edge 
 '95 feet and the height JU feet ; what is its solidity ? 
 
 (40 f 40 ' 3.^)) y 20 x lOj or (10J>4 U. It EM.) ^ 
 
 Ann. 
 
 6 
 
 40 + 40 + 35) X i20) X 10) = 3833.33. 
 
70 KEY TO THK TABfiEAtJ. 
 
 3. Wliat is tlio Hulid coiitciit ofji wt-dxt' t\ni IdiHt* of wliifli mt'a- 
 Hur«'H "t Iccl I iiiclits liy I' iiitlics, (In- U'liglli oftluf ('(lg«';H ftct iiiid the 
 |MTlK'ii(liciilar lici^flit 24 tVct ? Ann. 4.i;JI!i cubic feet. 
 
 ii. All incliiM (t pliiii*' iiK'ots a liorixoiitiil plane aiulloriiis witli 
 tliis Inst a wcdnc the rdj^c of wliicli measures 1(11) feet; tli«» ifctan- 
 jiuliir liasc HO feet liy~(( fct-t and tlic pi'ipciidiciilar tlistaiK'c Itctweoii 
 tliL' edge and tlu' hum' '.M) f«'et ; what i.s tlu) voliinu' of tiie solid f 
 
 AUN. SCO.OOO cubic feet. 
 
 PROBLEM X. 
 
 To find the solidity of a prismoid. (') 
 
 REiH. A siinidc exantiiiatiou of the nindols of (he tahlcau nhowB 
 at once tlic iiatnic of tlie intcrnicdiatc .section or phme parallel to the 
 ba.-K s and )ialf-\vay Itctween tliciii. This (lucstion lio\'«'vor is treated 
 of indeliiii iiiidfr tin.- licadinj; of proldeiii LIX to which one may refer 
 for any thiiij; concerning the jiriHinoid. 
 
 (!M) IIUI^E. To tluiiioii ofthi'oi-eaHoJ'thcUvoponiJlc}ha8e8,AV 
 nv, (uhl fitur fiiiics llic (lira of a intntlld section EG equidistant from 
 
 I. 'I'liis Hciliil. lilfn lliH piiHTii, is vmy oI'iimi iiihI, with liy ;lie niHiiHiirer. liectHii- 
 >.rtiliir viilK willi iiirliiiHcl xiiicH in'K of iliis lorm ; h tint liipi'our proHHiilH I.Iih mime 
 tljiiiit) ; liii>,'fi leriHivoiis iiin iu)t,liiii>{ Iml, iev(M'HP(l priaiuoiilB; if in found ii>,'aiii, in the 
 ImRiiiH, wlmi'ves. |iilliin», iilinlin*-ntH and consirnctirMm of riicIi kimi ; excMvationa iind 
 em 111 will ]».•<, riioMii'ix 1111(1 fMiliiiiiUinenlH, iVc, K'''"''i'l'y iixHiiinn liiin form ; (he conti- 
 nuHil (Miilmiiluiiciit of 11 liiilwiiy is siiltdiviiiiMi liy viM-iii^d Hi-cMions into prinrnoiilH each 
 reHliiuK <in one of iln lateral facen iind wlioHe paiiillel liaRHs lire conNeqnenliy per- 
 peniiiciilai- ro ilin lioi i/.oii ; liif prismoid ii» iiinl airain in every piece of squared timber 
 the eiuih of wliirli ari! iin('<|iiai ifctaiiKleH, ii is ai(ain Heen in llie piiiiigH of balls, and 
 hIicIIs, and ii is alsooliiMi itqieareci on variouR ticalen in llie arts and trudea, &c. It hu8 
 bciMi reinarld'd (note paire Ij-j) llial one iiiiisf take care not to (tonfoiind the prismoid 
 \Vilh tlic fniHiiiMi ol a pyi.imid, or ratiit>r, slioiiid it liave been said, tlie fniRMiin of a pv- 
 lamid wiili llif prismoid, for it evideii' y follows from the delinitioti of the prigmoid 
 that any frnstnai of a pyramid with parallel liases is at the same time a prismoid am! 
 can 111' nn'asnrcd by till' rule applicable to Ibis latter; but the so called prismoid in 
 in)t the friistiiin of a pyramid, and (Mie cannot conRer(nenlly determine its solidity 
 by the nile applicabl" to the frnsinm of a pyramid, fboiijrh however in certain cases 
 this last rule nuiv jjive an approximation very near the truth. I^et ns add also that 
 since wheo it is reiiuired liist to dHlerinine the iialnre of the solid to iiemeasiired, one 
 must ill the case of the friistniu of a pyramid ascrertain the [iroporlionateness 
 of the sides as well as their parallelism, and that their parallelism alone is siiffl- 
 cienl to consliiiite the prismoid; one will also often save a useless work by consi- 
 deiiii); as a prismoid any solid the lateral faces of which were inclined to one another 
 Hiid the sides of the opposite bases parallel to each other. 
 
MENSUHATION OP SDRKAOES. 
 
 71 
 
 theiie banes : thru mitlliph/ thr sum thus ohttiinrd hi/ our nixth of the 
 heif/iit or iH'riundicKhtr iliHldncc l>ftuceii the paralUl yWujics (IIOI <«.) 
 and the renxlt irill Iw the rciiiiircd roliime. 
 
 Ex. One of tlici bastes of u icftiiii;;iilar 
 primiiuid in !;20 x ti') t'tet, tli«'» *l»ev if* 10 x 15 
 the lieiglit is 12 fee t ; wliaL i's solidity if 
 
 Alls. C-iS X 20) + (l."» - 10) I k'M X 13) X 
 
 1^ = 1850x2= .1700. 
 (i 
 
 2. A wiiatfor pillar <ias tor its {tarallci 
 bas«'s, n;ctaiigl«is uu'aHiuiiig icspci'tivcly 
 100 X 50 feet and 80 x 40 feet, the lieiglit is 
 30 feet ; wh.it is its content in iMil)ic yards ? 
 
 AllM. 4518,;^. 
 
 3. A i)ilc of broken stone measures 100 x 20 feet at the bottom, 
 !»7x 10 feet on the top, and .'3 feet high or tliick, wliat is its content 
 in cubic toises f 
 
 Ans. (100 + 20) 4 {!«; X l(i) ■ 4 X i)d X 18) x g(or by h) -f- 210 = 
 24J|o cubic toises. 
 
 4. A cutting or excavation presenis the form of a reversed i)ris- 
 moid ; the inferior area >♦' the excavation is 10, ()()(» metres, the up- 
 per area 14,400 metres, the half-way area between the parallel bases 
 is 12,100 metres and th(< heigjit or de[)t]i of tlie excavation is !> metres ; 
 how many cubic metres liave been dug out i Ann. 109200. 
 
 5. How numy cubic feet of water can Ix; contained in a reser- 
 voir the base of whieli is a rectaii .le 1(10 x '>0 feet, its upper 
 base a rectangle 180 x 13a feet and depth 20 feet 1? Ans. 202,G00|. 
 
 6. What is the cubic pace contained withiu a roof the base of 
 which is a rectaugle 40 x GO feet, the upper part a rectangular flat 
 20 X 40 and height 12 feet ? Ans. 18,400 cubic feet. 
 
 7. What is the solidity of a piece of square timber who.se length 
 is 24 feet and the ends parallel and rectangular [)laue8 30 x 27 
 inches and 24 x 18 inches ? 
 
 Ans. 102 cubic inches, 
 
 8. A trough the depth of which is 20 inches, has for its parallel 
 bases rectangles of 30 x 30 inches and 30 x 24 inches 'I 
 
 O. Aus. 10.34/5 cubic feet. 
 
 
 t^ 
 
. I 
 
 ii 
 
 KEY TO THE TABLEAU. 
 
 O. An pnilmiikinoiit for 
 arailwny iiuiismcs 300 \ jirds 
 in Iciifith, its ends arc liiipc- 
 ziiiiiis, tlu; pariillcl sidfsof 
 one are 4 and 34 yiuds juid 
 tlio lu'iglit JO yards, llio sides 
 of the otlu'r iirc 4 iiiid !!• 
 yards aud the lieiglit 5 yards ; 
 liow many cuhic yards does 
 it contain '! 
 
 Ans. Area of one end = ^ (4 \ .'34) x 10 = 190 yards, area of the 
 other end=i(4 i 19) x 5 - 57i yards, intermediate area 
 -i{4 t 4) ' i (84 i 19) X i(lO ' 5) = 15.25 x 7.5 = 114..J75 square yards. 
 
 114.37.') ^4 = 4r)7..')00, 190 r .57.5 f 457.5=705, aud 705 x i(300)=705 
 X 50 = 35,XJ.")0 culiio yards. 
 
 10. A causeway on a sloping or iuclined ground measures 100, 
 metres in lenj^tli ; the areas of tlie quadrihiterals with parallel sides 
 forming the vertical ends or bases of the prismoid perpendicular to 
 its lengtli, are I 'JO and 80 s(]uare metres, aud the area of a half- 
 way section between these latter is 100 metres; how many cubip 
 metres have been requiicd to form it ? Alls. 10,000. 
 
 11. What is tiie cubic space occupied by u pile of cauuon balls 
 whose recta r)gular base is .30 feet by 10, the upper piano 25 feet 
 by 5 and the hcnght 4 fet^t '•' AuK. 833| nearly. 
 
 12. The pedestal of an eiiuestrian statue the height of which is 10 
 feet, has for its parallel bases rectangles of 15 x 7 feet and 12=4 
 feet ; what is the solidity of the ma.ss of stone which it is formed of t 
 
 Ans. 750 cu. feet. 
 
 PROBLEM XI. 
 To find the area of a regular pyramid. 
 
 (See the pyramids of the stereometrical tableau.) 
 
 (ft2) RULE. Miilliplij {I039 CJ ) the perimeter {ABODE A) 
 of the base lii/ the inclined half-height (SF) ; the product will be the la- 
 teral or convex area. To the lateral area add that of the base, when 
 the whole area is required. 
 
MENSURATION OP SURFACES. 
 
 73 
 
 Ex. 1 . Wliat is tliP lateral area of a regu- 
 lar triaiigiiliir pvnimid, the iucliiied hoiglit of 
 wliicli is^Oaiidciicli side oFtlio basoU. Alls. 90. 
 
 '2. IJ('i|uir('d tlH'wiiolcaii'aofa regular py- 
 ramid liic iiicliiu'd lieiglit ot'wliicdi is 15 iiietres, 
 and t he base a pciitiigou the side of which is 25 
 
 metres ? 
 
 Ans. 2012.778 square nu'tres. 
 
 Sj^IIow many squares of siiingle, zinc or 
 otiier metal, &t'. would bi' required to cover a 
 roof liaving the form of a regular i)yramid 
 the base of wliieh is 200 feet perimeter, and 
 
 the iuclined height '33 feet i? 
 
 Ans. 33. 
 
 PROBLEM XII. 
 
 To find tho lateral area of ths frustum of a re^lar py- 
 ramid with parallel bases. (Fig. to Prob. XI.) 
 
 (Sec ou the tableau the models of this solid.) 
 
 (ft.'J) Stl'LK. Find (lOlO G.) the product of the half sum of the 
 2)cri)in't('rs (A BCDEA, abcden) of the two bases by the inclined height 
 (//<') of the fr 11^1 inn ; ijou will have the required area. 
 
 Ex. 1. What is the lateral area of the frustum of a heptagonal 
 pyramid, tlie inclined height of which is 55, each side of the inferior 
 base 8, and cacli side of the superio, baso 4 ? Ans. 2.310. 
 
 2. An eight sided roof, terminated by a platform has for mea- 
 sure of its iiK'liiu'd height 17 feet ; the length of the side of the regular 
 octagon constituling its base is 20 feet, aiul the side of tho superior 
 polygon is 10 feet ; required the weight of the lead that covers it, 
 lead being pounds per square foot ? Ams. 12240 pounds. 
 
 3. How many superficial feet of cut stone are there in the late- 
 ral area of of a polygomil tower the inferior and superior perime- 
 ters of which measure respectively 100 feet and 80 feet and the 
 
 inclined height of which ia 40 feet ? 
 
 8 
 
 Ann. 3600. 
 
V* 
 
 KEY TO THE TABLEAU. 
 
 PROBLEM XIII 
 
 To determine the area of a pyramid, or of any frustum 
 of an oblique or irregular pyramid. 
 
 (See tlio inodols of tlio tableau.) 
 
 RfJIjE. Get (1059 CJ.) Hrpnratetij the area of each of the connw- 
 neni faces atidlakc their sum fur the n(piired area. 
 
 PROBLEM XIV. 
 
 To find the solidity of any pyramid. 
 
 (See tlie models of tlio ial>leau) 
 
 GENERAL FORMULA. 
 
 To the sum of the areas of the tiro 1)ascsor ends of the pyramid add 
 four times the area of a sect i or, holf-iraij l>etwecn them ; the sixth jtart 
 of the product of this sum by the heiijht of the solid, will be the required 
 volume. 
 
 95. REM. Here,tlieanpenorbascS, 
 (apes of the pyruniid) in but a point iiiid 
 its area is therefore mill, or=0, Tlui 
 areaoftlichalf-Wiiy section Iiiis exactly 
 and in all cases, the value of tlie fourth 
 of that of tlio base, since its linear di- 
 mensions arc * the Inilves of those of 
 the base, and the figures AV, «<■,— X V/, 
 xijg are (1033 ©.) similar polygons 
 of which the areas are as the scpiares 
 cf the homologous sides ; which gives, 
 as already stated, ixi=}. Now the 
 
 1. Ill tlift Bimiliir triiiii),'l('s AlUS, ab S, tlio lioiiioloironH sidna uip (,"5»20 Cw. ) pro 
 portionul. Therefore, -ince ah irtlmlf-wny belweeii Aniinil S, wo Imvo A <x=^u S=J 
 AS iiiul couBefiueiilly AU ; AS ■.: a b : aH, Mint it\ a b = AB -j- 'i. 
 
MENSURATION OP SURFACES. W 
 
 snpci'ior aroa Ixnii/r, as just seen, ociual to 0, tlio formula is, 
 for tlio pyramid : the area of the ham plus 4 times the area of the 
 haff-iriiji i^ccHoii iiiiilliplicd hij the aixlh part of the hcitjhf. But thcluilf 
 lu'ii;lit area hciiii^' Mm fourth j>art of tluit of (lie. base, and as 4 times 
 i arc 1, tlic formula is furtliersiniplilK'dandlicc'omes : twice the area 
 of the Itane hij the iUh part if tlie hei'jhf, expression Avliich reduces 
 at last to tlie following'. {Iiitroductiou pa<,'o 9.) 
 
 (06) Bri^K. ,¥«//;;)^/ (lOIrt «.) the area of the hase hy one 
 third of the hci(jhl (©57 <*.) and the prod net will be the required so- 
 lid it ij. 
 
 E.v. 1. What is the solidity of a pyramid, of which tlie base is a 
 square :}() feet each side, aiul the lufight 2a feet ? Aitg. 7500. 
 
 3. TIk! siil(!i of the e(iuilateral triangle which forms the base 
 of a pyramid is ,'} feet, its height is 30 feet ; what is its solidity ? 
 
 An§. 38.9711. 
 
 ;i. Wl!." is V olid content of a hexagonal pyramid of which 
 the lieigl't is G.4 feet and etach side of its base G inches ? 
 
 Ana. 1. "38564. 
 
 4. The lieight of a pyramid is 12, and each side of its pentagonal 
 base is 2 ; its cubic content is reciuirod ? Alls. 27.527(). 
 
 5. AVhat is tlie V(dume of tlie space occupied by the roof of an 
 octagonal tower of which the side is 5 metres, the height of the roof 
 being 10 metres ? 
 
 Ann. 02x4.8284271 (2ST.) = 120.71 0(]77.') metres is the area of 
 the octagoual 1 isc of the roof and 120.71 x 10 -^ 3 = 402.3G6 cubic 
 metres. 
 
 PROBLEM XV. 
 
 To find the solidity of the frus .am of a pyramid "with 
 
 parallel hases. 
 
 (See the frusta of a pyi'amid of the tableau) 
 
 («r) KITPi 1. To the sum of the areas of the two bases add 
 (llO;2 a.) four times the area of a seetion half-way between them, 
 tha: in, of a seetion of whieh the lineal factors are arithmetical means 
 (l*i«5 fi.) I)etween those of the two bases ; multiply then the sum thus 
 obtained hy one sixth of the heiyht of the frustum j the product roill be 
 the ^•eipdred solidity. 
 
 (i}S) Kllf-.li: II. Find [%(*VA) firrt a mean proportional be- 
 tween the two bases ; then add toyclhcr this incanxn'oporiiotuUandiiM 
 
78 KEY TO THE TAHrRAU. 
 
 twohases of the frustum ; and multiplti Ihis sinn In/ one third a/ the 
 height of the frustum ; the 2>>'<'ducl irill l>c the nijiiirrd sulidihi, 
 
 REM. In the case oftlic tVnsruiii ola pyiiimitl tlic sltrcdiiutri- 
 cal fonnulii canuot be 8inii»lii'uMl ; it ciiiiiKtl lu' rtdiiccd iis in ilic 
 case of tlie prism or of the wliolc i>.\rami<l, to a iimrc siinplt i\|>r»s- 
 Bion. Ou tlie coiitrtny, even other nih- to olitiiiii the voliiiiic of 
 the frustum of a pyramid is inoic lompliciilcd aiui rcijiiiics iiiorc^ 
 work tliaii the formuhi of the tatdvau, whether we wmilil ciiIm' 
 the frustum by taking the (lifVereiiee of the whoh' ami the jiaitial 
 pyramids (AS, as, i\g. to \y,\<n' /.'}, 74) or arrisc at tin- result 
 by Legendre's formula which will he IouikI here below (Kl'lilu II). 
 In fact, we must in the first case conipule by rules of |>roiiorlioii. tlio 
 lineal dimensioue of the i)artial i»yrami(l wliic li is winiliii.i; in the 
 frustum under consideration to form the whole ]tyraniid of which 
 the frustum is a part, afterwards calculate each of tiiese pyramids ; 
 and in the second case there is to Ix^ found a mean iiroiiortional be- 
 tween the two bases, that is, between the aicas of tliese bases, ope- 
 ration \tliich requires the extraction of the s(|Mare root of t lie pro- 
 duct of these two areas the one ]>v the otlx r ; whilst by tlie rominla 
 of the author, we arriTe immediately and without any dil1i<iilty at 
 the area of the intermediate secticui the factors of which are each an 
 arithmetical mean between those of the opposite l)ases of the solid. 
 
 Ex. I. What is the number of cubic feet in a i)iece of timber 
 the length of wliich is 24 feet and the ends are squares of !."> and t! 
 inches the side ? 
 
 Ans. ^13 + 6 =90.22.'S + 3Gf 00=n.->l=(4-144) 2 feet .')1 square 
 inches, which multiplied by one third of 24 ;;ives l!t.o (tabic feet. 
 
 2. Required the volume of a pentagonal ])edestal the height of 
 which is 5 feet, each sidi' of the inferior base Irf inches and each side 
 of the superior base 6 inches ? Aiin. lt.;{l!>2."). 
 
 3. A fort, the height of which is 15 metres, has fin- base a \v<^\\- 
 lar octagon the side of whicli is 10 metres, the side of the supeiior 
 polygon is 9 metres ; what is the solidity of the tower ? 
 
 Ans. Area inf. oct.=(28 T.) 4.8281271 x 10=482.84271 square 
 inetres, area sup. oct. = 4.8281271. x 9 = ;191.102.")!>.51, proport. mean 
 area=V 482.84 x 391. 10=434..%, ihv: sum of the .'3 areas = 482.84 + y91. 
 1+434.56=1.308.50, and 1308.5 x \ (1.5)=(J542.5 cubic nu'tres. 
 
 Ans. By the rule (1101 C) of the prismoid, we have for area 
 half-way of the parallel bases (10 + 9)^-2=9.5, and (9.5) x 4.8284271 = 
 
MFNSUIIATION OF SURFACES. 
 
 < < 
 
 1 i:r).7(!, X 4:=7i:5.(l|. I7I:U»U-- IHQ.SI + .'JlM.I-'^nin.lH.iiiuier.Kl.HS X }f 
 (I. ■)) = ().") 1*^.1." iis Iifrurc. I'dillic (iilV('r«'iict> .(•."> Ix'twci'ii the two ic- 
 isiilts {•oiiics only rmiii oiii' not having' t.iUcn inio tlic Iwo calciilaiinn.s 
 ji .ifrcaiir nimilirr ol' dcfiiiials. 
 
 1ll:I.'>l. Ill this last rx:;iii|t!f, tlic area of (lie smaller Itasf — 
 4.f<iSl'-271 < !»'aii.l that of tluMiiralcr - 4 Hv;,-i4x»71 x J(l ; the inodiuM of 
 
 2 'J 
 
 tliosi'twoaicas. thi'oiH'hy Ihi- ol!MTisl.8'2."<|-J7l < !> x 4.Hv!r'|-,'71 ■; 10 = 
 4.H-^"<4t27i.'x !) •; Kfilir si|iiaiv root (.rwliiih is l.S-J- |-J7I : !» l()-tli(i 
 ])i'o|Mii'tioiial iiicaii area rciiiiiicil. it is tiicii plain thai in llic calcit- 
 laMoii of the soli(!ii\ of the (■nisiiini of a pyiaiiiid ityllic fii>l of llio 
 two vulfs lu'ic .uivcn. one will save a Ion.i;aii(l usclfss worlc by usiiijj; 
 the lUftliodjiist indicatctl to (Ictcriiiinc llic piopoitioiial iiifaii area ic- 
 (|iiii('il, instead of in ;ill i ply in;;' (he one l»y the other t lira I ras iS:i.H |'^71, 
 31 • 1 . 1 ( ••J.")! T) I . afterwards to extract its sipiaic root. This reinaiU aji- 
 plii's also to the fiustitin of a cone, jirol). XWTII. 
 
 PROELEM XVI. 
 
 To find the solidity of the frustum of any pyr miid, that 
 is with non parallel bases. ' 
 
 (See iiiiioiig tlie luoiit Is of ihe /r(/)/(7n(. tlie fnisdiin of a tiiaii^i;iilai' 
 lt\ianiid with non paralhd liases.) 
 
 (Of>) IISILI-^ I. IHriilr the f nisi urn to hr nilwd hi/ a jilaiir of 
 section c F '^ h ixinillil to <»ir af tlir Imscs ;i 15 e d. iiiiil pitssiii'j thromih 
 the iicdrcst point f o/' tin' other thine, into tlie f'rnxtnni oj'ii jHiedniiil irilli 
 inirnUi't tmxen <in<l a sutiit iel:ieli trill tie deeompnaeit into ii.' iikuiji pi/ni- 
 mids on the fiiren f'nistnin hits .v/i/cs lexs tiro. Fiml nepunilelii ,. xo- 
 liilitij o/'ettih of thenc elcnu'iits lii/ the preecdini/ riilcft : their .siun irill lie 
 the reipiircil soliiliti/. 
 
 (I0») ItllI.E II. Determine iVUi'7 ii.) 
 scpdriilelij the renjtevtire soliilitiesoj' the irhot^ 
 and partiiil punnnids ; the di(j'erenee of these 
 volumes irill he tlie required soUditij, 
 
 Kx. The inferior and superior or oppo- 
 site areas of the friistuiii of a pyramid wi'li 
 uon parallel liases, are •"ill and'^O metres, the 
 resjiective liei<;hts of the wlioli' and partial 
 pyramids are '^'•^ and 17 metres j what is 
 the solidity of the frustum ? 
 
 Ans. ;{()TT (•■«)— ''^'^^TTr7) = .'ivI() cubic 
 metrcd — 11;$^ cubic uietres — ^iUi^ cubic 
 metres. 
 
 /■-I 
 
 I. Tliti tigiiru in iioL liiut uf tiie IVusluiu uf a |iynimiil, but iuiiigiiiti it to hf one. 
 
78 KEY TO THE TABLEAU. 
 
 PROBLEM XVII. 
 
 To find the area of a right cylinder. 
 
 (.Sco tlio <\vliiuU'iH of tlic tableau.) 
 
 (MM). MiilUpJii (Dft.t ii.) ihc clrnim/cn'ncc of the has<' hij the 
 hvi'ilil to obtain the lafcntl ((rvit. To tliis area add lliasc of the two tmscs 
 ij' the lehole area is injitired. 
 
 Ex. I. W'lint" is tli(' lateral area of a fvliiulfi' of wliirli (lie dia- 
 meter of tli(> base is vIO, and liei,Ltlit .")() ? Alls, ."i I 11.(1. 
 
 2. Wliat is the iiiuiilier of sajx-riirial feet of eul stoiu! in the 
 convex siiiface of a circular pillar the, height of whicii is 7 feet and 
 circnnif( renee 8 fcot 4 inches ? Aiis. .")8J. 
 
 :t. How many yard.s of coatinji: aro thero in th" eircnnif»'renco 
 and ceilin^'of a circular room which is 20 feet in diameter and 10 feet 
 high ? 
 
 Alls. Cir. = ;{.l 11(5x20 -02.8:32, convex area=()2.H:l2 x ]0 = (j28. 
 :{2 area of the ceiling ="2(Tx' 20 x .78(J4 = 314.1(j, rectuired area = 
 G2S..'{2 • Ml 1.10 -104.72 square yards. 
 
 4. Wiiat is the coat of polishing the convex area of a inarhlo 
 column ofwliich tlie diameter is 12 inches and length 10 feet, at 
 one dollar a snperlicial foot ? Alls. >':J1.42. 
 
 3. A cylindrical tower the height of which is 10 metres and dia- 
 meter also 10 metres, has for lateral ar(>a ? 
 
 Ans. 314.10 square metres. 
 
 6. IJequircd liow many feet in area there are in a running foot 
 of the interior surface of a cylindrical drain the diameter of which 
 is 3 feet ? Ans. 3. 14 [:>'.) x 3 - 0.42 177. 
 
 7. A cut stone vaidt is semi-cylindrical, its diameter is 10 feet 
 and length .")0 feet ; what is its concave area ? 
 
 Ans. 78.").4 square feet. 
 
 8. What is the umnber of square Indies of gilding in the surface 
 ofan iron bar the length of whicli is 14 feet and the diameter It 
 inches ? Ans. circ. .3.027 x 1(k'^-(m:).37. 
 
 J>. ilow" many superficial inches of silvering would be required 
 to cover the interior, that is the concave surface and tin* bottom of 
 a cylindrical vessel 7 inches in diameter and inches high ? 
 
MENSURATION OP 8URFACKS. 
 
 79 
 
 Aii">». 'I'lio Itottoin =■ 7 X 7 x.7'^.">4 -38.481(J s((U;»r<' inches, tlio 
 
 coiiciivc iuca --."J.! IK) x 7 x l) = l!»7.UvJ0"i 8<iiiaro imiics, in nil r.»:j(i.4 
 sqiiaie inches? 
 
 PROBLEM XVIII. 
 To find the solidity of a right cylinder. 
 
 ^ (See the tahlcdu.) 
 
 RE.^I. The ^jcneral fornuiln is reduced here, as in the case of 
 llio vi;;lit inisni, since the cylinder is nofhinfj else liut a prism having 
 an inlinili- ninnhcr of sides, to tlie lollowin.!;' expression. 
 
 (10-2) RdLI::. M III tipl II (lO'i-.i U.)ilt<' area »/ the banc h;/ the 
 heiijlit ; tlic i>ri>ili(rt trill he the snlidili/. 
 
 Ex. I. Heqnired the solidity of a cylinder 
 of which Ihe height is 20 and cireuniferenco 
 of the base 5 J? 
 
 Ann. (.").')) X (31, T.) .070.18 = 2.407.'} 
 •—area of the base, and 2.4073 x 20 = 48.14(i. 
 
 2. A bucket or other cylindrical ves- 
 sel is 1,") inches diameter and 12 iiu'lies high ; 
 liow many gallons of wine will it contain, the 
 gallon being 2.'J1 cubic inches ? 
 
 Alls. I.") X I.-) X .78ri4 X 12 = 2120.58 cubic 
 inches, -^ 2."U = !».18 gallons or gallons, ] 
 pint and half a pint, nearly. 
 
 3. A bar of wrought iron is 14 feet buig and 11- inches diameter, 
 what is its solidity in cubic inches ? 
 
 Ans. 1.2.'> X 1.2.') X .7854 x 1(58 (or 14 x I2)=20(;.ir;rr>. 
 
 4. A .stone column is one foot dianioter and 10 feet high ; what 
 is its solidity ? Ans. 7.8.")4 cubic feet. 
 
 5. What is, per lineal foot, the capacity ofapip(M)r drain of 3 
 feet diameter ? Ans. 7.(Mi8tJ cubic feet. 
 
 U. The foundation of a chimney is a cylindrical ma.ss of which 
 the dianuter is 10 teet and height 10 feet j how many cubic yards 
 of masonry do«'s it contain ? 
 
 Ans. 78."). 4 cubic feet-f-27— 20 cubic yards, 2 cubic feet. 
 
 7. 'I'he a>;le-tree or inMi spindle of a inill-wluM'l is 10 foot long 
 and !• inches diameter ; what is its .solidity in cubic feet ? 
 
 AUs. i) ;< Ox .78o4-T-r/28=4.4iti cubic feet. 
 
80 
 
 KKY TO TIIK TABLEAU. 
 
 PROBLEPfT XIX. 
 
 To fiad the area of an oblique cylinder. 
 
 (Sec (lie tal)lrau.) 
 
 (Uy.l) Itrri:. Mnllli^ln {UnjU.) the IvihjIIi AV or Wo/llic 
 side h'l llir circiiiiifcrcnrv i>/' <( scrlion LU i)trj)riHl!fiil(ir lo llic nidc or 
 axis I'O o/'tliv iijlhitlcr ; llir proihict will he Ihc htlcrtil (irta, 
 
 l!\. f. 'I'lic s<iiii-cvliii(lriiMl viuilt, 
 of IIm' o]t(iiinu' III' ai ell ol" a hi iduc cros- 
 siiiji' a rivci' ol)li(|Uc!_v, is MO iVcl diaiiH'- 
 tcr ami "•JO loct Idiijjf ; wliat is ils con- 
 cave area ' 
 
 Alt'*. !M'2i s(]iiar(' ('('(•(, nravly. 
 
 '2. A staii' laiiiiii;- trniiinatcd at 
 eacli end l»y the verti<Ml fai-es of tin; 
 newels, ineasiires l(L^ inclies ciieuiiire- 
 venee iiiul I.") I'eet loii^' ; wlnit in IIh- 
 imniher of siiperlieial yards of vaiuisli 
 laid over V 
 
 Au«. I"l; iiielies=:.Hr.'5 I'eet, and l.lx .r;7'.'>= I'l.l"^.") sciuaro fe(;t= 
 1 yanl 1,1, feet. 
 
 a. ^\ hat is ilie area of the ziiip in a jtijie (lie diameter of wliicli is 
 !) ini'lies, and of wliirli tlie ieiij;lii, ."> feet, is lerniinalvd by tin' jiaialh'l 
 plain's of two alleiiiate elbows (l.>Jl €r.; or faciiiy in opposite direc- 
 tions. 
 
 Asts. eir.: 
 feet nearlv. 
 
 ;J.1410 X •J = :28.-J7M, oir. x CD and-:-l4 1 = 11^^ s'lii'H'e 
 
 PROBLEM XX. 
 
 To find the solidity of an oblique cylinder. 
 
 {S(>e the Itthlcdii.) 
 
 IttlTI. 'I'he general forninla is reduced here, as for the oblitpio 
 prism to liie following siniplllied expression. 
 
 RULE I. :\lt>ltq)lii ihc (trcd of tliv base hi/ the hvi<jht of the solid ; 
 
 the proiliicl irill be the rvipiirvd solidili/. 
 
 (101) KUCI:: II. Miillipl!/ (lOiG) ihc lonjih of ihc side hit the 
 area of a section pcrpendicidar to the side or axis ; the product will be 
 the required solidity. 
 
MEVSURVXrON OP 8DRPA0EP. Bl 
 
 Vx. What is tlio solid content of tlie stair railing of the Instpro- 
 hUxv. ? 
 
 Aim. Area soct. ]»(Mp. = (31, T.) 10.5 x 10.5 x .07058=8.7737 
 sqiiaie iiulics, hikI H.77;{7 x 180 (I lie lengtli in iuclie8)= 1579,20 cubic 
 iuclu's, or .014 cubic foot, or y"^ nearly. 
 
 Ex. Tlie area of tbe base of a cylinder is 3.33 square metres and 
 the pcriK^ndiciilar distmice which separates its two bases, is 10 
 iiietres ; what is its solidity ? Aiis. 33.3 cubic metres- 
 
 PROBLEM XXI. 
 
 To find the area of the fVustum of a right cylinder or of 
 
 the frustum of an oblique cylinder -whose large 
 
 and small axes CD, FE or GH, LK of the 
 
 opposite bases, are (1099 G.) in the 
 
 same plane CDEF or GHKL. 
 
 (105) IIUL.E 1. If the frustum he right, muUiphj (dem. o/1099 
 and 1097 ^0 the hal/'-Kum of the ijnuitest and least heights UP, 
 Fl* of the frustum by the in'i'tmctcr of the base of which PP is the axis ; 
 the 2>rodu(t a-ill be the lateral area. 
 
 RULiE II. Jf the frustum is oblique, inultiphj the half-sum of 
 the lengths of the least and greatest sides DE, (JF of the frustum 
 [ the perimeter of a section AB j)erj)endicular to the axis of the cylin,' 
 dvr. 
 
 Ex. 1. The diameter of a cylinder is _ 
 
 10, its least height is 0.4 and its greatest p 'y^7\~~~^-^^^ 
 
 height 10.(i ; what is its convex area ? /^^rv^^T^"; 
 
 2- A half-elbow of a stovo pipe -P/iClSll^H^/ : 
 
 EP, FP or of any conduit (the rectilineal /VtOx / •' 
 
 olbow is nothing but a double frustum of / Ia^cTTSvV^ : 
 
 a right cylinder, that is two frusta of right /^J'\-~t-^^^ : 
 
 cylinders meeting at any angle what- ^(r—^^-t-^- • 
 
 ever) of which the diameter is 7 inches, ^-4>_jZ.-yD p 
 lias for its least and greatest lengths, 4 H. 
 
 and 11 inches respectively ; what is its lateral area ? 
 
 Ans. 3.M16 x 7 x i (4 + 11)=1G4.9 or say 1G5 square inches, or 
 (-:-144) = l square foot and 21 square inches, or \\ square feet nearly. 
 
 3. Between the two component frust'\ of the rectilineal elbow of 
 a cylindrical han,d railing, is a third frustum of which the greatest 
 length is 3 inches and least length null. What is its area, the dia- 
 meter of the rail being 9 inches 1 . . ... 
 
 9 
 
 r: 
 
82 j:ey to the tableau. 
 
 Ann. 11" is i)liiiii that the proposi «1 frustum i>i notliitip 1>ut n 
 douMt' iiii^ulii ol'ii ri^lit (ylindcr, that is two inii^nla tiiii(«-*l li,v th«-ii' 
 peipomlicular haws ; the nniuired uiou of which = 3.1110 < 9 a i(;{)= 
 28.2744 X J. 5= 42.4 Hquaie iiulM-rt. 
 
 4. In .1 cyliiuln(ial vessel iiicliucd t;i> the horizon in n liquor of 
 wlileh the least distance IVoin tin- surtaee to the hottoni is <>7 deei- 
 mftres and tlie greatest is ].',V.i uu-tres, th«' diameter of tiio vessel 
 being 1 metre ; required the urea of the surface exposed to tlic action 
 of the liquor f 
 
 AiiN. lx3.14H) X i(Ct7 > l.a3)— 3.141H square metres,=latcral 
 area, the Itottom z= I x .7854 = .78.>4 square metres, tlie whole 
 area=;}.141G + .7854 = M.t)27n s.j. m. 
 
 5. A Hemi-cvlindrical vault is terminated hy two Avails, un- 
 equally oblicjue to the axis or direction of the vault ; tlu^ diann-ter is 
 20 feet and the least and ;j;reatest lengths '>M\ and \V) teet, what is the 
 area ? A lis. HJ:J(J.'/.J s(juure feet. 
 
 6. The drum of a circular stair of whidi tin- dianu'teris 10 feet, Is 
 tevminjited by the inclined roof of tlie building ; its least height from 
 the level ofthetloor of the last story is 7 fet^t, its greatest height 
 13 feet ; what is its lateral area in square yards ? 
 
 AmN. :U4.iG-5-9=-.a4§ nearly. 
 
 PROBLEM XXII. 
 
 To find the solidity of a frustum of a right cylinder, or 
 
 of a frustum of an oblique cylinder of which the 
 
 great or small axes CD, FE or GH, LK of 
 
 the opposite hases, are (1099 G ) in the 
 
 same plan CDEF or GHKL. 
 
 REm. To apply hero the general formula, it wou»d be neccs^ 
 saiy to decompose the solid by a plane pandlel to one of its bases, 
 and passing through the point F (* ) the nearest of tiie other base, into a 
 right or oblique cylinder as the o'vse may be and the ungula of a cy- 
 linder with a circular or elliptical base and of which the general rule 
 gives the exact volume. 
 
 (106) nVL,K I. Multiply (1099 G.) the base CHDG, that is 
 the area of the base, hij the half-sum of the least and greatest heights 
 EF, FP of the frustum ; the product will be the required solidity. 
 
 (1) See ihi- lif,'iii-e of IIib IhhI proltloni tiiui iiiiii>;ii)(! ii pliiiiH of section piimll "I to 
 Cl> or UN mid iniKKiiif; lliioiigli iIih jioiiil, K ; then llie Boclioii KNh of lli« iiiigiila 
 )<)tnill<'l to tliH pliiiiH |iiiKHiiig ihiongh K will be llie uectiouof the luigulii lialf-wuv be- 
 tweuu ita base uud ila upux U. 
 
MENSCRATfoN OF SCRFAORS. 8i 
 
 ItllLE II. MullipJn (1090 0.) the area of a section AB per- 
 l)*u(licuUu' to the axiitW of the qitimlvr, hij the half-8um of the lentjths 
 of the leant and tjtratest nidett DE CF of thefrmtnni. 
 
 T.\. I. Ill II cyliiKlricjil vj'h.s(i1 tlic bottom of wliicli sunk niul dls- 
 ttii'btMl itrt vcrlical ji'isitioii, tlie Iciist im-liiicd lui^^litot'tlu! liquid cou- 
 tuiiu'diH l;Me(>t atid tlicfjieatt'st liciglit 15 fti't, tlic diiiincterofthe vat 
 \mu\i 120 loet ; rrciuircd tin* iiimibcr ot'gulloiis ol'liiiuur («ay 7i gal- 
 lons to the cubic toot) in tl»') vat f 
 
 Alts. 4.i!»8.24 cubic fwt=. 32} W5.80 gallons. 
 
 *i. Tlic sriiii-cyliiidriciil coping ofa wall niectiiig two otiicrs at 
 tincqiiiil obli(|ii<; angles, incaHiircH^rcet diainctcr and its lucuu length 
 is 1()0 feet ; what is its solidity I 
 
 Ans. area perp. sec. = 3 x 3 x .7854 x 100=353.43 cubic feet. 
 
 2 
 
 PROBLEM XXIII. 
 
 To find the area and solidity of any frustum of a cylinder. 
 
 (See the tableau.) 
 
 (107) RUI^E I. Tmaifme the frustum cut (at AB,fi'g. to pro- 
 blem XXI) bif a plane perpendicular to the ajcis of the cylinder, liefer 
 to that common base, the two component frusta ; yet by the two last pro- 
 blems the area or solidity of each of them, and take their sum; or, which is 
 the same thiny, multiply the common base or the circumference, as the 
 case nuty be, by half the sum of the two yreatest and two least sides AC, 
 JiK-AF BD of the two frusta : the result will be the solidity or arca^ 
 as the case may be, of the proposed frustum. 
 
 RUIiE II. The area of the ba6e CD multiplied by the half-sum 
 of the least and yreatejt heights FB, EB of the frustum, will give its 
 solidity. 
 
 PROBLEM X;XIV. 
 
 To find the area of a right or regular cone. 
 
 (See the tableau.) 
 
 (108) RULE. Multijdy (ion O.) the circumference of the base 
 BE by half the side, or inclined height AS, or BS, or dc. ; the 
 
 product will be the convex area ; to t^w area add that of the baa&y if 
 the vohole area ia required. 
 
84 
 
 KEY TO THE TABf,KAU. 
 
 El. 1. Wliiif, in tlio latcml aroii of ii 
 c'onctlio Hidrofwliich isSO iiiid «liiiiiutcr of 
 tliu biiHo Bi 7 A UN. (;()7.ry.). 
 
 5. Whnt in iho oniivcx urea of a coiio 
 tlie si(l« of which is 'M and (liaincU'r of the 
 baaolSf An«. 1272.348. 
 
 8. The bottom of a boih'v is a rcvcr- 
 RP(l cone tlio (lianicter ofwhicli is 10 feet 
 ard side (5 foet j what is its hit«'ral area ? 
 AiiH. D4.248 Hqunro feet. 
 
 4. A veHsel the diameter of wliioli is 
 10 inches has a conical lid the bide of wiiich is 5J inches, what is the 
 area of the latter I Ann. 10 x .3.1416 x 2.875=(l().;}-il sctnare Indies. 
 
 ft. A reservoir the plan of which is circular ind tiie. vertical sec- 
 tion of which drawn thrv)Ughthe centn! is an i^lO■^celes triangle, is (M) 
 metres wide and the length <»f its inclined hmU; is ;3.'5 metres; iiow 
 many bricks would bo required to line its area, at 75 bricks per square 
 
 metre 1 
 
 AUfi. Diam. 00 y 3. 14 It] x Kii x 75=2.'«,2(;4. 
 
 6. A tower is 150 feet circumference and the inclined side of its 
 conical roof measures .'10 feet ; how many squares of shingle roofing 
 would be re(iuired to cover it ? Aiim. 22i^. 
 
 1. Wlnit will be tht weight of the conical cover of a gazoineter 
 the circumference of which is 180 feet and the inclined side HO feet, 
 the iron being 5 pounds to the square foot ? Aun. 13,500 pounds. 
 
 PROBLEM XXV. 
 
 To find the area of the frustum of a right or regular cone 
 with parallel bases.— (^t'c the tableau.) 
 
 (109) Multiply (lOl* G.) thclialf-mm of the circumferences of 
 
 the two bases by the inclined height of the frnntum ; you will have the 
 
 convex area ; to which add the areas of the two bases, to obtain the whole 
 
 area. 
 
 Ex. I. The side of a frustum of a cone is 12^, and the circum- 
 
 erences of its bases 8.4 and 6 ; what is its lateral area 1 Ans. i)0. 
 
 "i. What is the whole area of the frustum of a cone the side of 
 which is 16 feet and radii of the bases 3 and 2 feet ? 
 
 Ans. lat. area=251.328, area inf. ba8e=28.2744, area sup. base 
 = 12.5664, whole area=292.1688. 
 
 3. The conical part of a funnel has for greater diameter 10 inches, 
 the smaller diameter 1 inch, and the inclined side 15 inches j what is 
 its hiteral area t 
 
 Ans. 259. square inche8=rl.8 square feet. 
 
MENSIJUATION OF SFRFACFB. 
 
 85 
 
 4- TlKMncliiipd roof of a oirciiliir towtM' tlic (liiinictcr of wliicli is 
 30 ft'ct iiiid tli<i «i(l<! SiO f»Mit is t*>riiiiimtc(l lit tho top by iv platform 
 tlut circiiiiifiM'eiircof whirli is '.].\ i\>A\t ; rriiuiicil llio iiiiiiilxtrol' nqnarcit 
 of zinc to cover it, incliKliii;; tlic platfunii. 
 
 Ann. lilt. .•\r<"ii-li27'^,lf', arc.'i siij). l.iisr^fMM)' x .n7!i5R = 8r..nG, 
 required afeii^: IMr»i».l4 s(iiiare feet= l^i siiiKtim !l sipiare feet. 
 
 5. How many sipiare inches of j^ildin;; will Ite refpiired to cover 
 tlie inteiior of a j,'ol)let the inferior rirciinilerence of which is C 
 inches, the sup. circ. 7 inches and the side MJ inchoH f 
 
 Antt, TIm^ lateral surface 'JH i (ti f 7> = 23.7.') sciuare inches, 
 thebottoui = l> X tj X .{)7*X)ti = 2.tiG^>, the whole =2.^.015 square inches. 
 
 PROBLEM XXVI. 
 
 To determine the area of a cone or of any frustum of an 
 oblique or irregular cone. — (.See the tableau.) 
 
 (IIO) KULi:. Dirulethclaterclan'aof 
 the cone, bi/ liitea drawn from the iijiex to the 
 base, into trianylcs or sectors, and the lateral 
 area of the frustum of the cone bij lines drawn 
 between the two bases, into trapeziumt^, dc. ; 
 calculate separately the area of each of the 
 component parts and take their sum for the 
 required area. 
 
 PROBLEM XXVII. 
 
 To determine the soliciity of a right or oblique cone. ^ 
 
 (See the models.) 
 
 RCm. The general formula is reduced fortlio 
 right cone, as witli the regular pyramid, to the fol- 
 lowing simplified expression, for oo = iAOand con- 
 sequently half-way area o = i area and as 4o= 
 0, it follows that 4o~rO x ^S0=0 x JSO. 
 
 (Ill) RULK. Multiply {MHO G.) the area of 
 the base by one third of the height, and the product 
 will be the required solidity. 
 
 1. Ueiict the nPiiei'Hl foiiimlii mid the lenmrk reUitiug to the BolUlity of the 
 pyramid, problem XIV. 
 
8f 
 
 KEY TO THE TABLEAU. 
 
 Ex. T, What is Mio solidity of a cone 
 tlie Iioiylit of whic.li is 27 foel; iiiitl huHci n ciiclo 
 10 tWt (liuiiicter ? Aim. /Oii.f^H. 
 
 58. Tlici circmiiftMt'nce of tlie baso of ii ri;;lit 
 cone i.s 9 feot, its height bciiig lOJ feet, what is 
 its solidity ? An*. i22..")0. 
 
 3. Tiio .'Ilea of tlu^ base of an obIi([iie 
 cone is ]()()() nietre.s and its lieiglit 30 metres ; 
 ■what is its solid cont(;nt ? 
 
 Ans. 10,000 cubic metres. 
 
 4. A roclv or hillock haviii,^;' the form of an irrej^nlar cone lia.s for 
 its base a figure tlie area of which is il.'WO scjinue yards, the JKright of 
 tile body being 105 yards ; how many cubic yards of stiitt' would it be 
 necessary to take away to remove it ? Aiis. 185, .lOO. 
 
 5. What is the volume of the space included uiuhu' a conical 
 roof the height of which is 30 feet and diameter 30 feet ? 
 
 Alls. 70()8.G cubic feet. 
 
 6. How many cubic inclies of sugar i)lum8 can a cornet 3 inches 
 diameter ami inches long contain ? Ans. 21t. 
 
 T. The circumference of the conical bottom of a boiler is 10 feet 
 and the height of the cone 1 foot ; Jiow many gallous will that part 
 of the vessel contain ? 
 
 Ans. 10 X 10 X .07958 x J=:2.G52GGf) cubic feet, x 1728 an(1^231 = 
 19.843 gallous. 
 
 PROBLEM XXVIII. 
 
 To determine the volume of a frustum of a right or 
 
 oblique cone, that is, of a frustum of any cone, 
 
 with parallel bases. — (See the models.) i 
 
 (113) RUI.E I. To the sum of the 
 
 areas of the two bases, add four times the 
 area of a section half-icay between them, that 
 is of a section whose lineal factors arearith-^ 
 metical means (1365 €».) between those of 
 the two bases ; then multiphj the sum thus 
 obtained by one sixth of the hcifjht of the 
 frustum; the product icill be the required 
 so^.Ulity. 
 
 BiJLE II. Find (1063 «.) first 
 a proportional mean between the two bases ; 
 get afterwards the continued sum of the 
 
 1, 8«3e the lemtirk wiiicli roltitea to ufruatiiin ot'a p^ruoiid, probltiui XV. 
 
MENSURATION OF 8URPACKS. 
 
 87 
 
 wean and the two luiftos of the fvuHtum ; thrn multiph/ Ihis eum hy 
 one third of the heiijht of tlie fniHtnin and the product will be the re- 
 quired solidHij. 
 
 Hx. I. liciiuiretl the solidity of a frustuin of a ri;;lit cone, the 
 height of wliich is 18, the diaiiKitiir of tiie inf. base d, and that of 
 the sup. Ifiiso 4 7 
 
 An!4. Inf. haso — sTHx .7851 = 50.2050, sup. haso. = 4x4 < .7854 
 12..5G()4, the aritli. iiu-.an factor between 8 and 4:^i (8 i- 4)=:0, < x 
 .7754 X 4=1 i;3.()l)7(), tlie sum of tliese areas= I75.!>2!'(), umltiplying 
 by y (the sixth partof tlie hcij^ht 18) we obtain 527 7888. 
 
 "i. How ni.iny cubic feet of water may be ootiiuin<;d in a reser- 
 voir luivin;; th(! form of (he iVustuin of an inverted tone tlie greater 
 diameter of which is 200 feet, .smaller 100 feet, and dci)th 25 feet ? 
 
 Alls. 458.15;J cubic feet. 
 
 3. A conical ])ipe unites two drfiins of 10 and 20 inches circum- 
 ference, its length or tlie ])er«>endiciilar distance between its two 
 bases is 25 inches ; what is the capacity of that part of the drain ? 
 
 Ans. Area small eud = (3l T.) (10> x .07!i58 = 7.058, urea large 
 end=(2(r) x .07058 — 31. 8;J2, the arithmetical mean circ. = K10 + 20) 
 = 15, (15/x.07058x4=:7i.G22, the 8um = ll]. 412, this sum x J(25) = 
 464.510G(j cubic inches. 
 
 4. What is the capacity of a firkin the height of which is 20 
 inches, the inf. diam. 10 inches, and the suj). diam. 10 inches ? 
 
 Ansr. 2701. 87G cubic inchest- 1728 = 1.55 cubic feet. 
 6. A vessel presenting the form of two frusta of cones, united by 
 their greatest bases, measures 40 inches long, 28 inches at the bun" 
 or ceutre and 20 inches at the head or ends ; how many gallous will it 
 coutain ? 
 
 An»». 20 x 20 X .7854 = 314.10, 28 v. 28 x .7854=0. i.7530, 24 x 24 
 X -7854 < 4=1800.5010, the sum of the areas = 273!».4752 ; x J (20) 
 =9131.584 cubic inches = the content of t»n»M)f the component frusta, 
 X 2=18203.1080 cubic inches, -^ 231 = 70.00133 gallous. 
 
 Ans. By the 2nd rule wo obtain : 
 
 a 
 area lesser base = .7854x20 =314.10 
 
 2 
 
 area greater base = .7854x28 =010.7536 
 
 area mean i>roport. = (98 REi^. T.) .7854x20 x 28 = 4.30.824 
 
 multiplying by cue third of the height of the frustum 
 
 1300.7370 
 
 
 we obtain for sol. of the frustum 
 doubliug, wo have for total sol. as before 
 
 9131 ..5840 
 S 
 
 18263.1680 
 
 ' -r^ 
 
88 
 
 KEY TO THE TABLEAU. 
 
 RfJ.'Hf. It is liiwdly nocossiiiy to say t1i<at instciid of multiplying 
 Rcparntely l)y .7Hr)4 or by .((7II58, iis tlie caso may Itc, tlio squares of 
 the (liam. or circ. of tlic oi)|)osite bases, and 4 times the 87H(7rc oftlio 
 diiun. or ciic. of the interinediiite base then to tai<e llieir sum ; 
 the sum oftiiese scjuares may be muUiitlied at ouce by the factors 
 .7854 or .07'J5d. 
 
 PROBLEM XXIX. 
 
 To find the volume of any frustum of a cone Avith non 
 parallel "bases-— ("See the tableau.) 
 
 (113) RUL.r] I. Decompose Ihe i/iven 
 frustum into a frustum of a voue with paral- 
 lel bases anil an atKjulx of a cone, b)/ a plane of 
 seetion parallel to one of the bases anil passing 
 through the nearest point of the other base. Now 
 get the volume of the frustum of the eone with 
 ■jyarallel bases bg the rule of problem A' AT///, 
 th"n that of the uugula of the eone bg the fol- 
 lowing ploblem ; the sum of these volumes will 
 he that of the proposed frustum. 
 
 (114) KULjI:] II. J>eterminc separatclg 
 
 (10117 <*.) the respective volumes of the entire 
 and partial cones ; the difference of these vo- 
 lumes will be the required soliditg, '^' 
 
 Ex. The inf. and su]). areas of the frus- 
 tum of a cone with imn parallel bases are 30 
 and 20 metres, the respective heij^hts of the 
 eniiio and partial cones are ;3.'} and 17 metres j 
 what is the volume of the f ustum f 
 
 .(30 X ^ 33)— (20x ^ 17) = 330-113J = 2IGf cubic metres. 
 
 PROBLEM XXX. 
 
 To find the volume of the ungula of a cone. 
 
 (See the tableau.) 
 
 (115) REM. Say the ungiila DAd with entire bases ABDEA 
 or AbdeA, that is, each of the bases of which is an ellipsis, or the one a 
 circle and the otln'r an ellipsis, or the ungula AliC-D with i)artial 
 bases, that is, each base of which is the segment of an ellipsis or the 
 one the segment of a ciirle and the other the s<'gmentof an ellipsis, a 
 parabola or a hyperbola, according as the uugula iu each of the two 
 
MENsrnATtoN o: suhfaces. 
 
 89 
 
 \^^=r-^ 
 
 c;is«'S ln' iliiit ofii vi,nlil Of (if an ol)li<|nc vnur. Ima.uiiH' a plane of scc- 
 (i(»ii paiallrl lo cidii'i of ihf I wo liases anil riiawn lialT-uay l)el\vee)i 
 siirli lta>e and the a|H\ 1> <n' '/. II oi I), as 1 he rase niav he. of I lie nn- 
 ^ula, that is. passiiiii; tliroiiuh llie niidtlle oC Dd oi' 1)15 ; ilw seclion 
 will lie in 111*' lirst case a seuii-ciicle ii' llie l»ase to wliicli it is pa- 
 ra 1 lei is a ciiele, a senii-ellipsis if the hase 
 to w hieh il is )>aiallei is an elli|p-is. and sneh 
 s<'nii-ellijisis will alsd lie similar to th<' 4in<' lo 
 wliii'li it is parallel : in the second case the 
 section oi'the iin.uula A!»t" —1) will lie the sei;- 
 inenl ofaciirle. ifihe liase jiarallel to il is 
 a circle, or the se^^inen' of an ellipsis if the 
 plane of seclion i- parallel to an elliji- 
 tical lia-e. OI linall.\ a [lara liola oi a hypei- 
 li(da. 'rio' a|iex ol iIh' n i.unia is evidi'htly 
 liiit a point d or 1>. Doi \> and its area is 
 ronse(|aenll_\ nnll or 0. Therefor*' the ;;('- 
 lieral forninhi " sum of the areas of l he t\\<i 
 
 «>|»|Mi>iie hases or ends, pins | times the a lea <d" the half-way section, 
 iiiiilti|ilie(l l»y ihesivlh pari of the lieinht of the iiiijiiila '' is itMliiced 
 as for the cone or pyramid to the following expression : 
 
 (ll<>i 'MLS: I. 7'" llii' (irr<i of flii' Ixixc (nld Jniir tiiiirfi thai of 
 tlic liiili-iroji xirliiiii jHinilh'l lo .sur/y ln(sr aiiil iiuiiiijtiii till' sum of 
 ilivse (irros hii tin- ni.iili /tarl of Ilic liiiijlil of llic soUil • liif ri'.siilf icill 
 Itc llir soUiViIji ii(ur!ji of till' jn'ojuixril iiiKiiihi. (ScciIh' pro'olems re- 
 la tin i; to liie areas of the sei;Mieii|s (if circles and elliiises, to that of 
 the paralxila, \c.. [tidlihins W and \.\l, ^ic. 
 
 WL\. I. Ciiiie the finslnm of a ri;;lit. 
 cone the heiuhtof which 'JO. llie inferior 
 <lianietei = lo. snperioi diameter == !>.(!, 
 inlerinediale diam. (lo ; !).()) -. 'Z ^=. 
 \'Z.'-). Imagine a plane of section passing; 
 lielWeeii lllc op|.osile ends of the inferior 
 and snpeiior diaineiers ; this section will 
 <livide the ,t;iven frnstiini into Iwonn- 
 jiiila'. the one lia\ inu' for its hase the infe- 
 rior lia^e of tin- frasi inn < if the cone, t ho 
 otln-r, the superior hase of the friistiim. 
 'I'lie intei nndiate section of the imnnla, 
 tlu'(lianieter<il whose liase is I."). will hi^ 
 th<'seii-mi'nl ofa circle whose \ crsed-si'it! — 
 Jo : rj — /..land the inlermediate s( tioii 
 of the oilier iri,u,nla will also he the se;;- 
 mcut of il circle haviiiy for its ver6cd-siiic 12.3—7.5 = 4.8. The area of 
 
 10 
 
90 
 
 KEY TO THK TABLKAn. 
 
 tills l;itff'r s(';fm(>iif — ^11, T.> 12.JM iiiid tliiit of tlic nllici scirinoiit is 
 eciuiil to lliiit (>ril(c fiiclt I he »1 in III. orwliii'li is |-J.:{ iiiiims lt2.!M - I \t*.t*'Z 
 — 42.y4=7r>.ri8. Now. voliiiiii' iiil'ci ini- miuiiIji = 7.").HS X 4 t 17(i71 Oircu 
 of tlitM-ir<-lc I lie (liimi. ol'wlii.li is I.")) "JO =-()=r h;()(>.7."). Ilic \vn\ solidi- 
 t.v = l.'>!M;.!iS, tlic (litliriiiLf — 3.77 — less t'liiii the lV>iirtli part of one 
 per cent in excess. 
 
 (117; 'i'lie otliiT Mii-iilii liiis tor its solidity I'^MM 4 : 7'2.36 
 (nrea oftlie ciiele whose diiiiiieter is!i.(ii~ I7I.7()(! 7'J.MH=r->44.1 4ri({ x 
 20-f () r HI.'}.H-J!I. Tills voliniie added to l( KK 1.7") o Ives lor tlie sum of 
 the solidity of tlie two iini;iil;e "J II 4..")7!'. 'llie solidity of (lie fnistiini 
 = jirea circle, diani. !•.•• i- nreji. circle diani. iri-r4 area circle diam. 
 I'Z.'S. the whole niiilliiilied liy 20 : (I 'J 1 1 l.ti;{;{7(i which dilfeis fioni 
 the last result only l>e<'aiise we ilid not take in ail i lie decimals. 
 
 Ex. 'l- With tlie same finsinm of a cone as in ilie l,i>r «\aiiiiile, 
 suppose the pliiie of section pas-^es tiirouiih the end of liic supe- 
 rior base and > ills the inferior Itase at ."> from the o|ipo>iie end ot that 
 base ; then (41, T.) for i lie area of tliesenineiil of I lie i>ase tlie versed - 
 sineof which is ."> we will have r)i.,l(;:i7,aiid as ilie an-a of ihe circle to 
 diam. 15 isl.") x ]."> ■. .7S,")4 = 17<l.7ir). we will oliiain, d<(liiclin^' frcnn it 
 51.r)(IM7, the area of the orher sci^iiifiit oI'iIh' l>ase the versed-sine of 
 Avhichis Kt^lli.'). I.")|."{. Now fur the resjicclive inteiniediale h.ises of the 
 two un^ilhe. we oltlain for tlie one w hose versed-sine is ."> (half of 10) 
 45.8 1!*'.2 and for the oilier semneiit whose versed-sine is l'J.:{— 5=7..'i 
 weolttain area circle i.. diameter I'J.M -4.").:M!iVj= I IS.r^*':?'^— 45.:il!l"2 = 
 73.474. We coiise(iuenlly obtain f(M' the solidity of one oftlie two 
 unjj;ula' (the one haviiij; for its Itase the int'. hase of the frnstnni) lx!5. 
 J513 I 4 times 45.31!»"^ -3(l(i.548 and 3()t;.5IS --jo^fir- 10-^1. ?<•,>( i? ; the 
 real solidity= 1015.701. the dilference is (! » , say the (! tenths nearly 
 of one per cent in exctss. 
 
 Ex. Jl. Another frustum ofwiiich the heij;ht~40 and of which the 
 inf. and siijt. diani. are (>(> and 3>-<.4 and the intermediate diam. conse- 
 quently = 4!).'J = tlO ^ 38.4 -r-2, the friisi n in divided into t woiiii;;nhe with 
 bases n(»t truncated as in the example n. 1, j;ives for resjx'ctive areas 
 of the segments conslitntiii,;; the iiitei mediate bases of the twoun- 
 guhe whose versed-sines aie 1!».'^ ami 3(1, (;87. 05(1^51 and I'Jl l.J:J()105 
 whose 8uni = l!H>1.17(Kj5() (aica oftlie circle forming; the intermediate 
 section of the IVnstum) ; area sup. ba.se.— 1158.1 1!M^J4, area inferior 
 base = 28*27.44, solidity inf. nii;4iila = 5!.22li. th«^ solidity oftlie siijie- 
 rior ungiila = 2(>.()l"i, the sum of the two -77. 2(i8- solidity of the 
 frustum. TIm! lirst scdidity 51.22(1 is .123 more than the real solidity 
 which is 51,1U3, the error beiiiy 0.24 or i per cent nearly, the second, 
 
MENSUaATION OF !»UR7A0KS. 
 
 91 
 
 volnini- iil.Ol"^ is ]cs^ ilnni tlic ri'iil solidity wiiidi wliicli is 'ZGACht, hy 
 A'S-i oi iifiiil.v (lie iiiilf uf (Hi<> per ci'iit in dctt'ct. 
 
 'I lie suliilily of liic ((IIht iiii;L!:iiliV = .")i.r)<iM7 • 1 timfs7."M74 • 72. 
 :JH-^").||i. wi^olc - -JO (;-|;}!lv».H(»7()(;. wliirli added to |(iei.H'Jti7 volume 
 of llifotlifi iiii,!;iilii !;ivcs toi' the sMlidit,\ of ilic tViisi mil ^J4I l.(iM.'J7(5« 
 lliit rlir iVii>iiuii (•aii'iiliilt'<l scpaiiKi'l.v — as in tlic las'. fxaiiii»le 
 
 ^lij.ti.i;{7(;. 
 
 (IIS) KOI. since llic fonnnla a|»|ilic<l to titc tVnstiim of the 
 I'onc ,uivcs an accurals icsiiit . imd t in' solidity of i lie iinunla iiaviiij? for 
 inferior base tlie liase of I lie fi usiiiin. is iiicater than the real solidity, 
 as itiias just lieeii sn'.w, by alunit A per cent (more or less) it fid- 
 lows that the solidity oftlie oilier iiiii^iila is less than the real solidity 
 piccisely by t he ()iiaiil ily which is in excess in tin; first iiiiyulii, which 
 is evident. 
 
 What is certain, is that in practice one would prefi'V disrc- 
 jiaidini;- as the case may be. the half percent of error in (|iieslion 
 to taking the trouble, of arrivinuat a more accurate result, through 
 the loiiL- and ditliciilt cab iilaiion re((iiired by the accnratt' formula iuul 
 lose a time which nfiieially would he more valna'hle than the .stake 
 ol'half a unit in a hundred, or of one iiiiir in *JIIO. 
 
 Ex. 1. 'I'lie segments of ellipses beiiii;- the bas( .s of the iinguhi of 
 n cone, are (Ul T.) respectively of "Jll and 1.1 feet area, and 
 the lu'i,<;hts of the whole and partial cones perpendicular to those 
 bases are .lOtH G.j :{•> and -i'-i feet ; what is tiie volume of the 
 «ii<;nla .' 
 
 Alls. (^Ox yjll)— (i.-)x .'j-j;iy = ;iOU— ll.j - It^."} cubic feet. 
 
 PROBLEM XXXI. 
 
 To determine the solidity of the ungula of a cylinder. 
 
 (See the several tiiiyiihe of a cylinder of the SU'iHoincf rival Tableau,) 
 
 (HVtt) XOTK. 'riuMUignla of a cyliiHler, like the uiiguhi of the 
 foiH' is .scmeiiiiies met with in the p'aclice of the measurer at the 
 intersections of vaults. &;c. ; and in ilie jiract ice of the gaiiyer who 
 lias sometinie-s to calculate the (juanlity of li(|Uoi in a cylindiioal or 
 conical vessel inclined to the horizon. '1 h • compmient jiaits of cer- 
 tain bodies ma;, also oll'er to calculation solids of tlii> kind, as when 
 we dec<Hin>ose with respect to its nieasur<nieut, the frustum ol' a 
 cylimler or a cone with ihhi parallel bases into a finstnm with [yarallel 
 bases anil .III nii;;iil,i, to deiii mine separately tlirii solidity and get 
 at'tciwiiido the um of lhc6c suliclitics. 
 
92 
 
 KKY TO Tni; TAnr.FATT. 
 
 (1'20) RTTI. ir ilic li.isc (ifllic I'liynli isintiic or iini liiiircntctT, 
 tliJit is, iC this l);is(' is, ciilii 1 ji ciiclc (»r mi clliiisi-;. jiccokIjiim ns lli»» 
 nii^'iilii or I iiccviiiidfr ol'w liicli I Ik- iiM,i;'iil;i Iniriis ii |t;iM is rinlit <n' in- 
 clined, tlicii I lie ini';iii,i iiiny Ix- considcird ns iln li ii>Miin ol'.-i cylindci- 
 will) n(in]):iiii!l(l li;isrs ;ind whusc Iim>I li(ii:iil t(. :iihI whose so- 
 lidity, ;is it Iims l»('en se<-ll ( I07, T.) is e(|ii;il lo 1 lie |H<>dllct ol'lh(5 
 iireii ol' its h;isc 1)\ h;ili liii' height ol'liiesulid: I'or I he iii c.i <it' llic 
 hillt'-Wiiv seelioi) |i;inillel to llie Itii-e is in tii;it eiise eijii il to li.ilt' the 
 base. i)eiiiL; evideiii ly li.iH' a circle or half iiii fl)i|i>!s ;,h ilie case may 
 he; now llinii's the hall'-ciicle or iiall'-ellipsis in ipiestion — t uice 
 the itasc, anil I w ice the liase plus the hasc. , t!ial isihree limes the l)as(') 
 iinilli|)lie<l l>y the (ilh part ol' ilx' li< iL;iil ol I lie >oii<i is ((pial lo miil- 
 tiplyilli;' the l)ase liy hail' the height, 'i'lic iicneral e\pie-sion is 
 tlieieloif siniplilied in ihe pii'scnt case and .Ljives I'or 
 
 IfiUlvli. Mi(lli/,li/ llif hdsr III' Ihr ilii'Jiilii. hjl linlf il.< ln'nilit. Ilil'ino. 
 duct icill lir the I'ljinriil sdliililji. 
 
 (V-il) ICi:>l. II. ll'lhe nni;n]a is truncated l>\ a plane parallel 
 to its liase we iiiiist^ resort to the ^ciieial rominla. 
 
 I£L'Ij1I. To IhcKiiiu n/tlic iirnix nf llir iqijiDsile liiisr.s iif llir I'ninliiin 
 of the KiMfitlK (tilil \ lii)i(!i till' tn III fij' n ^itIIhh linli'-iiiii/ lirlirmi Ilir tiro 
 Olds iif the siitid mill Ihr iinnhul uj thin siiiii liy the iillt intit uf the heiijht 
 of the ^(ilid irill lie Ihe reijiiind xuHililii. 
 
 ilfOTi;. '!'!ie liases will lie accordini; to da(a, circh" and tlio 
 Hejiiiieiit of a ciicle, or an < llipsis and ihe se;iiiieiil olan ellip-is ami 
 the half-way section will he either the se:;nient of a ciicle or llie 
 segment of an ellipsis. 
 
 (I'4'i) ICI..1I. lll.\\'liin the niiLi'iiIa td' a cylinder to he coiiiim- 
 ted is an nnunla jMopnly so called, lli.'il is, with partial hasc.ii i.s 
 liieiisnred exactly like the iiii,niihi of the cone (jiroh. XXX; that is, hy 
 addinfi; to the area ol'ils liasc I limes ihe area o\' {\\v half-way section 
 to multiply afierw arils the whoie Ity I he (ii h p n i of the heii;lil of I ho 
 solid. If the 11 nun la is that of a i i^hl cone, the iiase is the seL^meiit of ;i 
 circle and the parallel section also a semiiein of a circle. The 
 same if it i' lie nnunla of an ohlitpie cylinder, tlie half-way sec- 
 tion parallel to the hase will he, lihe ihe liase. the segment of 
 an elliitsis. In each of the two cases, the lieiMlil or versed -sine of tli(5 
 sejiineiif the area of w liicli is to he valued is half that of the hasi'. As 
 to the chord of the sejiinent if this seunieiit is of a circle we will liavo 
 («S«m, <".) the half-chord = the sipiare ro>>l of the jtrodint of tlu^ 
 vt'ised-siiie hy the rest of the ilianieter and if the seiiimnl is of an 
 ellipsis, we ^'i':! ohtaiii the ha!! chord hy niakiiii; : ' the uiealer axis oi" 
 
 I. See, in iiilMlioii lu llii.-<, lliu ailicli-s on lliu t^Iiifduiii ;iii(i cf ijil IimI ."jiiinHe. 
 
ME.VPURATION OP SURFACES. 93 
 
 (li,nnr(cr of ilic dliiisis is to ( : ) its minor .-ixis oi (li:iMH'l< r ;is (::) tlio 
 ,s(|iiiirc root ol'tlH' vtisid-siiir oi" the scuiiiciit iiiiiil ipliril i»y liicrcst 
 of till- nMJitc^ ii\i> is to ; ilic liair-rlionl. licsiilis Wf iiiii\ iilso in prnc- 
 licc olitain iliicctly lin' cliord ot'tlir sc^nirnt liv liMcing and iiica- 
 
 snriii^ liial <Im>i(| on (he solid itself. 
 
 (ia;J). Itl'..^l. IV. If llic nn-ula of Ihr I.ist |>,iiin,^iai>li is tinnc- 
 .'d<'d l)\ a jilanr |iarallrl to its liiisc, foiiuini^' tlins wliai may iir 
 called llie l'in.>tnm of an nn^iila <d' a cylindi-i' witli p.'iiallei liases, \v«^ 
 may still easily obtain either l>y direct measaiemi'nt of tiie model or 
 solid to lie cnlied, or liy calciilal ion. llie chord and \cisi'd->ine of 
 the intei mediate section, llie \ (■ised->ine IieiiiL; t he iiall"-siiiii of tliost* 
 of the ciicalar or elliptic segments of the opposite ha^es. 
 
 (I'll) lli:'^l V. If the nnunia is ceiuial. its ajiev, diameter 
 of the cylinder, will lie a nicic line ;ind I he area -II. In that case, t ho 
 central oihalf-w .ly sec lion will lie the cent lal /.niie of a ciicle or cf an 
 ellipsis, according as t he iiii'^iila lie riuhl or inclined. It' the nn.niila is 
 eccentric, its ape\. cliord of a seciion. will still Ue a liix' .ind its 
 area=l(. In sncli ca-e its half-way section will he the eccenli ic /.duo 
 of a circle or an ellipsis as the case ma> l>e. 
 
 (ia.">) lll-'^l. V!. If the nn?;iila. central or eccentric, is'lriinc- 
 ated Iiy a plane parallel to the base, it will then l>e the frustum 
 of a central or eccentiic iinuula of a ri^ht oi' olili(|iie cylinder one ot" 
 whose liases, if entile, will lie a <'ircle or ;in ellipsis, the other liase 
 a central or excentric zone of a circle or ellipsis and thi' hall-way 
 section also the /.one of a circle or ellipsis either cent la I or ecceiit lic as 
 the case m.av he. If the liase of theiin^nla is not eiitii'e. ir will Ik^ 
 cither a central or eccentric zone of a circle or ellipsis, this se!;nieiif 
 beinn" as I he case may he, more or Ii-ss than half the whole liase of 
 the cylinder of which tin- miL^nla I'oinis a jiarl. 
 
 I',x. I. I'Ct it '«' rerpiired to cnhe a lateral iiii:.;iiia of a ri;;hf 
 cylinder, its lieiiilit = "^l. the versed-sine of the sinment of a cil•clt^ 
 coiislil iitini; its liase = ^; and ihe diameter oi the cylinder of which 
 the nnniila foi ms a ]iart = 10. 
 
 The area of tin- liase = ( H, T.) V.'^ 11)=:. -^ - versed-sine of tht^ 
 table at the end of this volnmc, and tin- tabular area correspondin;;- 
 to this sine = .1 1 IHvi.'{ w liicli beiiin multiplied by ID or by Kit) irives 
 I'or re(piii'ed area ll.l.~",';!. Thus ;ilso we obtain tlu' areji iif the seg- 
 ment at half-heiuht <»f the unj;iila= I : 10=^.1 — .(Ml H/.l — tabular iircii 
 which 1(10— l.tlri?.") .,,„| this x 4=:l(;.;r), .iddiin- II.JS'j;}. we have 
 27..');{"JM = suni of the areas of the base ami I timi's the half-way sec- 
 tion, this sum X 4( = xJl ; (!) j;i\es for the volume of the proposed 
 uiiyuhi llU.l:JL>:i. The icul bulidity is lUL/. 11531, iho diil'crciico .(iUo6= 
 
94 
 
 KEY TO TIIR TAnr.EAU. 
 
 .(5;{(», tliiit is : tlic liitc of cnoi' is h«ss tliiiii tlic jj (»f 1 per rent in excess. 
 
 K\. 'I. Tile iiei^lil ol'tlie l.ilei'iil ilii;L;'illa of ii eviiiiilei' is 'J i iiiid llie 
 (liiiiii. of llie e\ iimler HI as ii« tiie jjisl ( \;Mii|tli'. I'lie liase of I lie aii- 
 ;;iila is a sciiii-ciicle. Kcijniied llie xiiidil v. Area Itase 10 10 x 
 7t^'>A : Vj — . ■{!•.. ")7. tJM' versed-sine (if tli«' half-way sennient =.'i : vi-'^..!, 
 !i..') I |((--.'J,-) <oi respondiii^ in tlie laities to .l.').T)l(i wliicli < 100 - !,')_ 
 .Tihl. Now i:).:{:)|(i l <;|.IIH|, adding ;{;t.-J7 we olilaiii lOO.tiHril 
 niullilil\ ill.!; 1>\ l,oiiesi\lii ofllie liei;;lil, welia\c j'ortlie plisiimi- 
 <lai volume lOvi.7.'»M() ; the aiciirale solidily is |00. ijic dirfciciiee is 
 Xl./.')."!!) etinal to a lale ofeiioi' of .(I-^-' oi a little inoic tlnin the two- 
 tliirds of one per emt in e\ee,«.s. 
 
 Vj\. tl. NVilli tlie Mime ijiam. (10) of a evliiidci' and llie saiint 
 liei;;lit of an insula CJ I) as in the two last exam|tles, llie l» ise of the 
 Ulifiiiia wliieliiii I he 1st example is less than a semi-cinle, in tlii-:;in(l 
 <'xam|>le e(|nal to a scmi-iindc, is in this ."{id example ^lealei' than 
 a semi-eiicle, its veised-siiie hein;;' - rt ~ JO— Vj; we lii\e seen 
 tliiit tin' area of the se^iini'iit having; xi for its verscd-siiu' o; liei;;lit = 
 1 {.{.'^•iM and as the whole base o!' the e\ liiidei— ■ 10 < .7p'.")4 = 7H,.') I we. 
 will olitain the aica of tins leiinired sej;inent — 7;^..")4 — 1 l.lr!"^;} — 07 
 ;{r)77. ''I'lie half-way section at lialf-hei^iht has f<M' lieiyht, or veised- 
 hine t^-^2 = -\, 4 : |0=.4 — tali. v. >. .•i!(:i;{(i!»-tal.. area ^ iOO-!;*!). 
 ii'M> andx 1 = II7.;M7(i,ll7.;M7(; 07.;{:)77 = lsi.7o:>:< = sum of tlici 
 areas, wliieh < 4, (itii pari of the lieinlit ; we oht aiii 7.'}-<.H'i|-j for tlio 
 volnnie l>y the iicneral fonniila, the K-al voliinn' =7.'n.'~ilH, ihedill'cr- 
 ence is I.OOM-J eipial to a rale of error of .(j;i7 per cent or less than 
 the two thirds of o'.e per cent. 
 
 lix, I. A lateral mi;;iila <if a cylinder whose diameter ~2."», lian 
 for its Ininlit (iO and for base a .segment of a circle whose versed- 
 Hiiie = ."). \\ hat is the volnme of tin? unf;iiia V 
 
 Since .'):Vj.") and 2i : 2'> in this example as 2 : 10 and as 1 : JO in 
 example N' I, w«' have for tab. area as in the first »!xam[ile 
 .J llHVI.'i and .0I0H7.") which multiplied each by 'i."). or by ()2'> ;;iv<; for 
 area of the base tilt. HI* ami for .irea of the hal'f-way section •2r)..")4(iH7."). 
 This last V 4 times-10;J.|S7.'», addin-;' (l!>.!^!l w<> have I7-^Ml77 whicii 
 X JO (thehei-;ht . (I) = 17:20.77. The real soliility = ]70I>.liO, the dif- 
 ference= I O.H7^. <»:{.") = less than two thirds of I per cent. 
 
 E\. J>. r^et lis compute now the complemeiital ntii^iila to that of 
 example 4, that, is, ihe nn^-iila which, with that of the last ex- 
 ample, make lip the cylinder of which each of them forms a part. 
 
 The, area of the base of the cylinder = VJ."> x xJ.l • .78.") t=l!)0.H7.') 
 anil this base ,<■ the height (iO — :JL>4.W.5= .solidity- of the cylimhT. 
 
MENSURATION OF SURFACES. 95 
 
 Tilt' iiiiunlii t;i 1k' coinpiiifd Ims lor ils iiilriior biisf tin- Iiiisn of (ho 
 cyliinln loss tin' li.isc of llir iiiijinlii iilrcjHly foiiipiilcd, lli.il is, IMO, 
 H?.') -Illicit — 4"J(l.!lrt.'», llu' supciioi' liasc Ix-iii;;- »'?iliu' - IIM07"» ami llio 
 liiiif-Wiiy «»'i'lioii=: l!l(I.H7.'» -•jr).r»|(iS7."i (aic;! oC tlic h<'<;iiiciiI wIiosd 
 V. s.='^..'))-l(;r).;{:2r<lv!:). I'liis last area laUiii -I tiims I Htil. Ml ::»."), 
 afliliii;;' tilt' an-iis of llii' opposite i)as«'s, we obtain V,'77:M7^."» tor tlio 
 siiMi ol'ilicaifas, niiiiliplyin,!;' l>y ll>(<)tli part ot'tlir liri;;lii) we oidaiii 
 for the vol. ot' I lie 11 MX II la •J77;tl.7'i."), if to this vol. we a<l<i I7'jfl.77 vol. of 
 llie Isl iiii^iila we li.ive "jl*. I.VJ. l!»."> - volii me of I lie e_\ liiider as Iiereiii 
 aluive (letei mined, llie dilt'ti (lice liet ween .."> and I.!'.") Itein^ caused 
 by our takin.n' (!1».^1) lor (!!t.."^HlK{7.') for the area of the inferior segment 
 of the 1st iiiiniila. 
 
 If I7(l!>.!» is ihc real vid. of llie Isl iin.nnla ; then, as 2!»l.V^>.."i istho 
 real \ ol. of (he ryliiiiier, it follows thai I lie aeeiirale solidity <d' tho 
 I'ompleiiienliil aiiiiula jiisi eompnted in thi.s exuinple to *^77.'{|.7'^.'» is 
 XJIII.VJ..") I7l»:>.!» -:J77l'^.<i. riieref'ore the vol. of the eompleiiK'ntal 
 iiM^iila is ill this exam[»le too small by II nearly — .1)1 = j,',,, per cent 
 or J, <•!' I per cent. 
 
 THEOREM. 
 
 (126) General expression for the lateral area (convex or 
 concave I of any solid of revolution, or of a segment 
 or frustum of such a solid Avith a single base or 
 two parallel bases, and whose plane of sec- 
 tion is perpendicular to the axis of the 
 generating curve. 
 
 Divide llie ueiier.iting curve into e(pial parts so small that each of 
 them be sensibly a li.^lil line : draw ihroiinh eacli point of division a 
 i'ircnmfeiciice parallel to the base or perpendicular to the axis of 
 the sidid. These parallel circiimfeii-nces will divide the area to be 
 compiiled into zones of ('(jiial breadth ; each of these zones will be a 
 continued trapezium the area of which will be <»blaiiied by mnlti- 
 piyiiif^ the half sum of its parallel bases or circumferences by tho 
 heii;ht^ oi' bi<'adth of the zone, and tin- whole area of tin' ])ioi)osed 
 solid will bc^ equal to the sum of the areas <t\' its component zones. 
 
 The rcifinred area irill the re /'on' he tthtaiiieil hj/ (uhliiif/ fo the half- 
 sum of the cireitDi/erences of the oppitnife Ixtsrs or ends of the mlid, the 
 Sinn of the iiiterDiettiiite eireitm/ereiices of all the rompoiieiit zones, ami 
 aJ'lcncanU muUqili/imj the whole by the breadth of one of these zones : ex- 
 
^(\ 
 
 KK.Y TO THK TVUl.KAt!. 
 
 ,n..ssi.M. ;.ni.I..o..nis lo lliil of pn-. < till uuil «» T. ) lor ill.' an-u 
 «»r iiMV pliiiii' limine. 
 
 Ol >-.£ 
 
 rA A /^. 
 
 r/ 
 
 ^i) 
 
 6 n 
 
 7/ 
 
 'lliiis A l>- (' licin^ iiiiv ('(Hidid, m liiilt'-spiinllc. ;i liciiii^iiilicn', a 
 liiilt'-s|ilici<ii(l Ol iiiiv s('<;iii<'iil oi' ii s|i)ii'ic, splicioiil or spiiiillc, wilii 
 a siiijilc lijiAc AH, we will Imvc llic lateral area -(A ciic. .\\\ r ciic. 
 ab I ciic. (■(/) A Au =^/r -<■('. 
 
 If tlic ;^iv«'ii sc.i;iiu'iil or rrasliiiii AT. dc lias i wo ha^fs Al>. ril, 
 tlic aica will he- (A ciic. AH ciic. <il> ciic cic. . .|i ciic. r(/) x A 
 a ov (If. Ifllic opposiif Imlvcs of the ,<oli(l arc s\ iniiit'lricai as in 
 tile cask AH or any oilier cfiidal fi iisl iiiii o.' semiieiii ofa s]»iii(llt' «tr 
 speioid. il is I'lnlly iieetssary lo oliseive lliat it will siitlice lo ope- 
 ra I e on one of llie syiiiiiielrical halves ami aflei wards doiil)le I lie re- 
 Bult. 
 
 If llie solid A!>-(" in (piestion is with a concave surface, that is, 
 fi'dieialed li\ tlie levoliilion of a curve AC or A// w liicli presents its 
 convexity lo tiie axis CD of the « 
 
 s()li(l, it is |dain lliii we will / ^ 
 
 in tlie siiine in.iiiner ol)taiii tlie /V^TlV — t'^'fC A ^^/h<f 
 
 aiva-..--(iciic.AH rcirc.<</; . ciic. ^' ItM 1/1^ /^ 
 
 tlie«'ase of tliecoiiuid <m' seg- 
 ment with asinj;U! Iia.se, or— i/i 
 ciic. AH i ciic. <ih i cii«-. \c., i i ciic. ////.) A <i or (fc, ».^»'.. in the 
 cuse of tin.' Irustiim or scynienl with two parallel bases Aii, <jli. 
 
MENSnUATrON op POMDS. 
 
 07 
 
 It rolli>\VH«'vi(l('iii!y from wliiil prrccdcH lliiit if tln' (jcinMiitiiiK lino 
 1)1 (lir sill t'.'U'f Id Itr ('iiiii|iiil<'il is iiiixttl, lliiil i>, piiiilv roiiVt'X mid 
 |)iU'il,v coiK-iivc, or il'iliis lin*' is |i;iitlv iii;lil iiiiil pitrlly ciiiMd. llli^ 
 Htiiiif pKx-css will <|iiitc lis simply liiitl tt» l)u- ilctii luiiiiiiioii ol' ItH 
 iii.'ii or siipciticii's. 
 
 ItlMI. Ii is to lie rcniiiikt'd llial llic ;;<')U'iii1 loi iniil.i Just vntn- 
 lili -licii will yciit r,ill.\ {ii\ c. lor iiiiy cnnx «'X sin riicr. a rrsiill w iiicli will 
 1i( ill (It I'll I ot'iiic i('ipiii('<l iiit'ii of iIm- solid, iiiitl in tlif siiiiMMViiy, 
 the ifsiili oiil;iiii<(l in liu- ciisc ul'ii coiiciivc sin fiicc will lie in excess 
 ♦il'llir real ana of die proposed iiody. 
 
 Ill f. (I, ill piaclice, tlic Ineadlli AaC. of one ol the eoniiH»nent 
 zones of I lie snifaee lo he nn'asnied, 
 will «liller nnnt' or less fi oin the. 
 hliaii;Iil line Aet', aecoidin^ as AC 
 is a jiieatertM' smaller ptnlion ofihe 
 geiieratin^ciii ve. I'lierelore instead 
 
 ofconsiderinj'AC -a sliaijj;lit line with a length — AeC we will add to 
 the aeeinaey of tin- result liy laUiii^Cor thu breadth of the zone the 
 developed liveadlh A«(' of that zone, which will he ohtaiiied pretty 
 accurately with the help of a scale of equal parts snllicieiitly siiluli- 
 vided and thin enoiij;li to be adjiisied to the convex or concave di- 
 icction oltlie length of the arc to be computed. 
 
 However, though wc shall have added to the precision of the ope- 
 ration by siibstitnting tor Ihereclilineal breaiitliAcC of the zone, its icul 
 breadlli A(jt" ; w.' will still be in defect or in excess (d"the re»iiiired 
 area, although by a very small quantity, with regard to the whole 
 area. This <iiiantity will be, very neii'ly, equal to (rfo I 'bd) (or2rtc) 
 >■ ."J.l IU> X ^ A«l' or to 'fi:, times tin; doubh? of the area of the space Ac 
 C(»A, orto]"^^ times the area of Ihe triangular space having 'ic for its 
 base and for its height the developed Ungth of lln^ arc (iV ;>r 2<ic k 
 y. 1 ll(» iseviilentlythe dilfereiice between the circumference oft and the 
 mean cd, of tht^ circumferences AH, ("D, and it is jnecisely- by 
 the product of that difference by Ihe length of the arc aC or (tA, or 
 wiiich istlui same, tiling, by the product of tlii^ half ditTerencc tiC by 
 ihe whole arc AaC, that thti retinired convex an-a is slight or want- 
 ting, or that the concave area to be deteriniiied is great or in excess j 
 but on account ofAl' being very small, tin; dilfereiice, either in excess 
 or in defect, between the accurate area and thean^a obtained by the 
 formula, will always be, jis we have just said, a relatively snnill and 
 insigniticant quantity, which will soon be seen with the problems 
 and solutions shortly to be submitted in order to couipure their 
 
 11 
 
98 KKY TO THK TAIU.BAn. 
 
 nccmncy willi tluil of the results wliicli the ordiiiiiry iiilcs fmiiisli, 
 mill to jiidgo at till) Hiiiiu; liiiu; of lUu aiiiDiuit ot' lalioiU' iuciil'Mit 
 thereto. 
 
 THEOREM. 
 
 General expression for the volume of any solid. 
 
 ( St)c tlio luoili'ls* of tho stvrcuiHctrical tableau.) 
 
 (lai"). 0/cirt'n r'ujhl or ohl'mnv prhm or vulinder—o/cverif rrffu- 
 lur or irri'iiiil((r j>iiritiii:i! or of crcni ri'ilit or ohliijiic cottc —of vvrrtj 
 fnistuiii of It pi/ruinid ur cone voinprisitl hcticeeii pardlUI Ixttics—tf (lie 
 sphere —ofccerjf sjilwriviil sector or ptjramhl—of ercrij Hpkeroid —ofeveri/ 
 sctjuwnt of (I Hjflwrc or spheroid trilh (i siiiijlc Ixtsc or of crcriffrnsiion 
 of these hodics irith liro ptinillcl Ixtscs inclined in an if inn/ to the axes 
 of the soliil — if creri) rii/ht or inclincil pandndoid or parabolic conidil — 
 vfereri/ riijht or inclined hi/jwrhidoid or hi/perlndic conoid —oj every 
 aetjment of a paralndoid or hi/perholoiil irilh (( sinijlc hnse or of ercri/ 
 frnstnm of Ihcsc Ixniics with tiro parallel bases inclined in anii wai/ to 
 the <«.<v« of the solid —o/' ercrif lerdije or other Jrnstuin of a triani/nlar 
 prism —of ever ij part ofsnch wedije or ofsnch a Irnncated prism separa- 
 ted from the whole solid hi/ a plane parallel to either of its lateral faces — 
 of everif other pristnoid or cjiliiidroid—of cverij Hmjula of a sphere or 
 spheroid comprised between planes of section passin;/ in ani/ direction 
 thronijh Iht iCnire of the soliil -of cvcrj/ nnijnUi if a prism or prismoid, 
 cylinder or cijiindroid, / 'mid, cone or conoid comprised between planes 
 of section having thei' n intersection in the axis of thcsolid and of 
 
 everij scfiment of .da with a sini/lc base, or of aniifriislKin of 
 
 such nnijida bei attel biises(anfi upproxliuiitlvclj',) o/ //te 
 
 semi-spindle o, .-central frnstnm of a spindle (cask) comprised be- 
 
 tween parallel planes perpendicnlar to the fixed axis ofthesolid, — ofa:y 
 vnijula irhatcverofaprism or prismoid, cijlindcr or ciilindroid, pijramidf 
 cone or conoid, sphere, spheroid or sjfuidle, and of ercrii frnstnm of an, 
 unyula comprised between parallel bases) : the volnmeis eqnal to thesnm 
 of the area of its base, if there is but one, or of its parallel bases, if there 
 are two, and of four times the area ofasection half-way between the bases, 
 between the base and apex, or between the opposite apices, as the case 
 may be, multiplied by one sixth of the heiylit of the solid. 
 
 Let A and B tliu cpposito buses, base aud apex, or opposite 
 
MENSURATION OP SOLtlW. 
 
 99 
 
 a|)ict's of (my one (»f tlic bodies jiHi cminn'ratiMl. let S a ]mra]1(>l hcc- 
 tloii liiilt'-Wiiy III! wt't'ii A aixl It, {ind il l!i*- lii'i!4lil of llif soliil ; we 
 will oid.iiii, us iIhm'ii.s«' iii;iy \ni, voliiiiic = (ai«'a A f ana IJ t-4 area S) 
 X ,', H, or (Mii'ii A I 4 area S) x ,', If, or (4 area S) < J II, acx-oriliii^ as 
 aivuaitcx II -0 or art'.i apcfX A i aica apex M — 0. 
 
 (I!4H^ Now, oI' tlic live i('>,niliir iiolylu'droiis, tlie tctralicdi'on i.s a 
 pyraiind, tilt' hcxliacdroii is a <-ul)(« that Ih a prism, and cachortliu 
 tlircc oiIm-i's is a coiiipoaiid of pyramids t'(|iial to «-!icli otiit-i' ; every 
 iViistiini of a piily;;oiial prism is a t-ompoiiiid of I'liista of tiiaii;;iilai' 
 jirisms cacii liaviiii;' foi' its liiise one of the liiteral faces of the j;iveii 
 fnistiimaiid the ed^'es or anises of which nil unite and hecomeeoii- 
 fonnded in one oftho parallel edf;es ot'tliesoiidtHin any iij;lit linn ]>a- 
 ralhd tothesidesof ihefrnstiim, situated in its inteiioraiul which may 
 lie ciHisidered as an avis of the prism of which the frustum forms u 
 pait ; «'veiy frustum of a cylinder may also l»e considered as a com- 
 pound of frusta of triaiigu. v prisms, .sincethccyliiideriUelfislmt an 
 inlinitary prism ' ; every ci ular, elliptical, paralxdic, &.c., spindle, 
 t'lon^'jited or ll.ittened, as the case may he, will l»e, decomposed, a.s 
 luis already been shown, by sections perpendicular to the lixed axis 
 of tlie solid (Hits G.) into cones and frusta of eones, or, if possible, 
 into frusta of a sphere or spheroid, or of a parabolic or hyperbolic 
 conoid, subdivisions to which may be added if required, the cylin- 
 der and spherical segment. 'I'he conoid orspheroid whose f,'enerating 
 curve* is not that of a section of a c(»ne, can be decomi»osed 
 (ll!ll> <j«.)like the spindle, into I'rusta of coiies or conoids, segment'- 
 of aspluire, segments of spheroids, of paralioloids or of hyperboloid^, 
 &-C. ; the ungnlaofa cylinder, cone or coiuiid, vVc. will be consider- 
 ed asacompoundof rectilineal or spherical pyramids, tfcc, and every 
 otiier body will bo subdivided, as the case may be, into elemeuta 
 (1113 ii.) of the kind, just enumerated. 
 
 The expression is therefore general, as has been said at the 
 liead of tills article, and will serve at will to dttermiue the solidity 
 of any body. 
 
 (129) Used till now (1103 €>.) to the consideration of so varied 
 a number of expressions lor the volume of the several solids iu 
 
 1. We hnve seen elsewhoro (S3 *o S8 TO that as regards the frustum 
 ofarogiiliir prism, that is, the bases of which are regular or symmelriiHl polygons, 
 and (lO't *o lO? T-) US to the frustum of a cylinder, this ^i b livigion or iinagt. 
 nnry dueoin|iositinti by p'lno.'; of section is not at all Uduessarj, giuoe such bodtet 
 wre oiiiy iuuusuroii wiliUoul lUaU 
 
100 
 
 KEY TO TUB TAHI.nAU. 
 
 qiiostioTi, iind that, •widioiit cxim iiu'lii<liii<j tlic sjiIk roitl, jiaiiiixtloid 
 and tlie segnuMits of those bodies, wliitli still ■•ivc iis(! to ndditioMa! 
 formulae; tht^ pupil will at tirst jtcihaps be astonisiicd and will cvi ii 
 doubt the existence of a tbiuiula wliicli may be applied at onee, to 
 bodies as dissimilar to eaeh other as are the i>rism or eylindei', the 
 spliere, tlie sej^'ment of a, sphere, the pvraniid oi' cone and tin- \ve(l<;e, 
 &c., and ivhose limit iuif surfaces are iiidiil'erently plane oi curved ;>r 
 both ; but the followiu;;' retlections will ite siitUcieni to piove the 
 accuracy oftlu! enunciation of the proposition. 
 
 (130) In I lie first place, the/>r/.s';Hor cv//)((7r/- has foi' its soli- 
 dity (1103 1° and (('» «.) th(i area of its base midliplied by its 
 heiglit ; but the opposite bases of ;i prism oi' eylindei' are e(piiil and 
 every section of such solids paiallid to the base (!)i;i fi.) is equal lo 
 the base ; the sum of the 2 bases plus 4 times the half-way section 
 between * is then espial to six times the base, and it is the same 
 
 thin<^- '.ultiply G times the base by one sixtli ofthehei;;ht or to 
 simply multiply tlie base; by the w hole height. 
 
 {131) In the sec'oiMl place, the solidity of the ptfrauiid or 
 cone, (inlinitary jiyramid) is 1 1103 ;i^ and 7"^ C) tuu' third of the 
 product of its base by its heiiihl : but the i)arallel section half-way 
 between the base and theajx-x iscipial lo ihelbiiith |iai t of tie' ba.-e, 
 since tlie sides or other honiologoiis lines of that secti:)ii are halves of 
 those of thebaseand the areas ;ire as the stpiares of (he li'iniologoas 
 Bides, that is, I : 1 when the sides are 1 : :.'. Therefore in thiscMse 
 the base i)lus 1 times the section between the b;:se and the apr\ is 
 equal to twic*; the base, and it is the same thing to miiltii>ly twice 
 the base by oric sixlli of the height or to simplify the formula by 
 uiultiplyiug the base by one third of the height. 
 
 (133) firesides, as it is shown CIIOaG.) by the def. of tlie 
 prismoid, //(C /'/•(/«/»») of n piirnmUl is at the same time a prismoid 
 and {\w frustHm of a cone (frustum of an iiilinilary i»yramid) is still 
 a prismoid, and tliese frusta, sui>i»osing that their lieight be imh li- 
 nitely increased, will at last become the very solids of which they 
 at first formed but a i)ar( ;and the formula (area A I area B i 4 area S) 
 will always hold good, whatever may be thearea of tlu^ apex or of I lie 
 superior base li, and when B is but a point and eonseipiently its area 
 become ecpial t' 0, the foiiiiiila will become : (area A ; 4 limes area 
 S) X ^ of the height. 
 
 (133) III Ihe third place, tbe solidity of the sphere is 
 (I07'> <".) ecpidl to its area m.iltijtlied by (uie third of ifs radius ; 
 now tills .I'.'ea is precisely equal to four of its great circles, that is 
 to lour times the area of a section of llie sphere eqiiidistiiut from any 
 
MENSURATION OP SOMnS. 
 
 101 
 
 two o])|»t)sito iijiiccs or iJdiius of its coi vex area ; tlioiico thercforo 
 flic acciinicy ofllw I'onnnla, since, llic sixlli part of tiie liciyht of the 
 splicK', tlial is of ilsdiaiiictcr, isoiio tliiid of the nidius or scmi-dia- 
 iiictcr. 
 
 (151-1) Witli rcsjx'ct to I he Jicniisiilu're, its so- 
 lidity is cc] mil (1077 (i.) tot lie <'oiivcx a lea by 
 one third of the radins ; hnt its convex aij'u 
 is ecinal to two ;;reat circles, sini'c the area 
 of the entire sphere is ecpial to 4 yreat circles, 
 and we have (1 f,neat circles x ,^ E F; = (*J 
 
 {^rea"^ circles -f^Y KK) ; bat area section (i\)h (or ED=FD)=: J are.i 
 hiisc A15, since D//- = /d''-—I)F^- KB- ~(iFli/-^l—] =.f and as 
 four times 'i'^'^ weobtain 4 area <tb i area AH - 4 area AH ; tlierefore 
 4 area AH v ,[ EF or 2 urea AH x ^ KF -- (area AH (4 area <th) < I 
 EF ; therefore, tScc. 
 
 (i:t5) And in i^c>ncitil, had \\v to do 
 
 Avith any (iin/ Kcijment V.\) of the sjilicre, its solidity 
 is equal (lOSS H.) to the sum of the solidities of 
 the truncated cone El) and seyineiit HD ; but 
 the solidity of HI), that is tsf tiie solid generated 
 by tlie revolution of tiie senmeul HI) is (IOSJ> O.) 
 the dilference between the splierical se<'t()r gene- 
 rated by the revolution of the sectcu' lU'I) and the solid jreneratetl 
 by the revolution of the isosceles triangle JJC'I) ; this dilfer- 
 ence is e(puil to(lONJ> ii.)'i~(Vli''—V<l)''EF-l7:(Cc'^-V(hKV ; but 
 Cc — Vd ~ Vb—Vd —<th —ad because u[' <t(' being common to the rec- 
 tangular triangles ab(.\ udV ; therefore! the solidity of iliesoliil gene- 
 rated by HI) (and which with the trun<'ated cone generated iiy the 
 revolution of the tiapezium EHDF forms the spherical segment we 
 liave to do witli)=:'^~(,«^ — ad ) EF. Now, ~«/> =r:(i0'21 €ii.;area 
 
 'i , J 2 
 
 circle ah^ ~ ad —area circle ad and conse(iueiily " (ah — ad ) = are!i 
 of the circular ring <//>. It is plain also that we nniy write ~f«//— 
 ad ) § EF or 4 ~ (ah' -ad' ),l EF, sinc< l-^-i^l ; therefore the solidity 
 of HI) -(4 area db)x}. EF or 4 times the area of the; ring generated 
 by the revolution oi' hd, muliii»iied by one sixth of the height EF of 
 the segment. JJiit the solidity of the comi)onent truncated coiie = 
 (lis T.) (area biise FD 4 area, base EH i 4 times aicji parallel sec- 
 tion or/^ X ^ EF ; therefore the <'ntire solidity of the segment of a 
 sphere = (aiea base FI) -t area b;,se EH + 4 area section, «fc equidis- 
 tant from l^B and FD) x J EF ; therefore, Stc. 
 
 
102 
 
 KF.T TO TITR TABLKATT. 
 
 (130) In till" foiirlll place. Al'icr li.ivini; sliowii llicaccii- 
 riK'y of (he " i/eiicriil i:rjiri'ssi(iii " in tlic o.isi' of the splicif niiil cdiio, 
 solids uciHT.itt'tl l)y tlii' re vol ii I ion of i lie ciiclc ii.id 1 1 iiiii,!;lc, the two 
 «'Xtr»'iiu' seel ions of lilt,' colic (iiiitl tiif most (lissiiiiiiiii j tlu; one by ii 
 pliiiu'piiiiillcl !o ilsliiisc, tlic oilier l»y a jiiiinc |)ci |iciMliculiir toits linso 
 jiimI i»assiii<i,' tliroii;;ii tlic iii>c.\ of llic cone, uc an' imliiccd to JMlicvc 
 that it will he I lie same liy anaio^^y, with I lie liodics ;icncralc(l l»y the 
 revolution of the lliicc otiier conic scciions |Hd|)cily so cailctl. that 
 is: llicclli|isis (jjeiU'iatiiiH' the cili|isoi(i tn s|>iiei(titi), the jiarabolii 
 (jiciii'iatinj; tlie |iaral)oloi(i> and liic hypciliola, (iiciiciatinn' tiic liy- 
 Iterlioloid;, and tins on account of the inteiiiicdiate position wliich 
 these tiiree sections occupy hetween the two otlicis, eacli of tliese 
 latter havin:; to pass successively to the state of hypeiliola, Jia- 
 lalioia and ellipsis, or vice-veisa, in order to liecoine from a tri- 
 aii,i;l<', ii circle or from a circle a tiian;;le ; or which is the .-same 
 thin;;, the i-oiie liaviii^- to jiass successively to the state of liy|tei- 
 Ixdoid, par 'udoid and ellijisoid, to liecoine a sphere, or the sphere 
 l»y the reverse jirocess to become a cone. 
 
 And in fact, the expressions furnished by the '' dirt'ereiitial and 
 integral I'alculns" for the respective soliilities of the spheroid, jiiul 
 parabolic and hyperixdic conoids, or of the se,:;Mients of these bodies, 
 are easily reduced to and translated into those contained in tlie enun- 
 ciation of this articli' and from wiiich they (litter but by the form. 
 
 {1117) Filially. Ii remains lo demon- 
 strate tiiat when \\n- sctiiiiciit AC of a spindle, 
 f'.u instance, orofdiii/ other soliil of rcriilt((i(ni, 
 &,c., is not that of a s[>here, spheroid, ie<;ular 
 conoid or c(»ne, we none the less (d)lain its 
 s(didity, at least very nearly, by the formula 
 (E I F i 4(ib) X I V.F. In fact we always have sol. truncated cone 
 AC = (HreJi K * area F t 4 area ed) < ^ KF, which generally otters a 
 very near approximation to the required solidity. 
 
 We have a<;;ain (by the formula) for the volume of the solid gene- 
 rated by the revolution of the segment HbV. about the axis EF, 4 times 
 the area <d"tlie ring, the breadth of which is (Hi, multiplied by one 
 sixth of the height KF; and, drawing th "ght lines bC, bli, the solids 
 generated by the revolution of the triangles bdH, bdV, considering 
 them lis continued trianguhir prisms, will have for solidity the area 
 of the ring bil, their common base, by half the height lOF, or which 
 is the same thing, three times the area of the annular base db —ac by 
 one sixth of the height KF, or H(d. \\b{.] = '^ area Mx^ EF, which 
 uddcd tu that of thu cuiiipuuuut truucutud cuuu AC ul' thu solid to bo 
 
MF.NSTTRATION OF SOLID!'. 
 
 103 
 
 roiiiput<'tl, t'liiiiislics !i new iippioxiiiiMti m still nearer tli.iii the tiisj, 
 to tlie ic(iiiir<(l solidity, 'riicrc still niiiiiiiis to coniplfie tlir solidity 
 givMi liy the roiiiiiilM (K !- F \ 4 ab) k }. KF, the imidiict of ,1 KF liy 
 oii('(! tlie aicii of tlif liii^' dcsciilKMl by hil, and to ri»v» r or nirct this 
 lust product \vc have tlic solids ^icnciatcd l>y tlic Kiohition of the 
 st';j;ui('iits hcV, hf/W. Now, il is plain that tin- sum of tl . .>•■ latter is 
 to the solid generated l»y the .se;;nient IU>V, in tlu^ ratio, neiiil_\ of 
 the respeetive ai«'as of the sum of the se;;n)ents /)|{, />('to the sej;- 
 iiient lU"; but these areas are to each other, very nearly, as 1 is to 4 ; 
 whence il follows that the remainder (area hit ■- ,1 FF; just spoken of 
 •will sensibly correspond to tin; solidity of the sum of the solids MS, 
 bC whicli goto imike np the <riven segment AIJCD ; therefore, &:v. 
 
 (ViH) Let us remark that ihe dilVerence between the accu- 
 rate solidity of the proposed sennient and its approximate solidity 
 by the foinnda (F F 4 (th) ,'. FF, is alwa,\s in excess, whiili is 
 ])artly owing to the tact that (•on>ideriMg the solid gener iled by die 
 revolution ofthe segment li/»(' about the axis Ml' as a continuous 
 ])rism. (or as a solid ling having for sectimi the segment i)/»l';wiih 
 a mean length < <pial to l he half->iim of the circumfeiences nh. cil. we 
 tak(! this length a litllr too great, since the continued prism in ques- 
 tion loses nu)r<' of its length in C than it gains in l> ; which in- 
 (liwes lis to observe also that since the solid ring ;;eneiated by the 
 revolution of the segment IW'C is rather the coMtiuued t'l irsmni of a 
 prism or a sei ies of frusta of prisms, we miiihl obtain its >olid it y with 
 sullicient acciiiacy l>y making (IO!>»> ti.) tin- pioduct ofthe gcner.it- 
 ing- aica UhV (secti<m ofthe piism by a plane )>ei pendiciilar to its >ides 
 or edges) by one tliird of the sum ofthe ciiciimferences at H, b and (' 
 (respectivi' lengths ofthe edges ofthe ring or frustiinri and we might 
 still !m1(1 to the accuracy of the solidity to be obtained by multiply- 
 ing the generating area H/iC ofthe ring by onetiflh ofthe sum (»f 
 the live circiimfei'ences at \\, (j, b, c and C or by the sum of ai.y num- 
 ber of ciiciiinfereiices (taken at e(iiiidistaiit points frt>m each other) 
 divided by the number of tliese circumfeiences. 
 
 (i:i9) Tlie rule which has. just been given 
 to obtain tlie solidity of the segment of a solid 
 with a convex Huiface, is also applicable to the 
 srfjment of a noUd irith a cotircr siirfUcc, the same 
 demonstration lieing adniissable in both cases, 
 as indicated by the letters in the figure ; 
 with tins exception only that tlie ditVerence be- 
 tweeu the accurate and the appioximatc solidities will evidently bo 
 
104 
 
 JfET TO TTIR TABLKAr. 
 
 in ildVct iiisti'iid of 'iciiii; in ('>:coss. for in tliiscusc iIk' nuMU lciii;tli of 
 tlic coiitiiiiird friistimi of a !»risni or of tlic solid rin;,' .u;iMifratcil l>y 
 tlio ri'voliilioii of till- scuiiii'iil iW* (" is Ic^s ili in liic iiumii to In- oh- 
 tiiiiicd l)y lakin'4 into considniition ilic ciiciinifi'ifnccs at li and 
 iit C. We will thci.'foM' oltinin V(i\ niMiiy I he solidity of the 
 .st';^incnt AC. »'i|niil to the (lilfncncc of llic solidiiics of llic frustum 
 of a cone AC and of I hi' solid rinu prodnrcd l>y tlic rc\(dntion ot' the 
 8c,i,nnt'iit \U)C. til It is hy iiiiikin.:;- tin- prudiicl of tin' >ixtli part of the 
 lu'iuht KF hy tin- snniofllic airas of the l>;iscs Ai5, CD and four 
 times thu si'clioii a h half-way helweeii those base.s. 
 
 (1 10) The saini^ rule will also i^ive with siifH- 
 fieiit aci'iiiiicy in practiee, the solidity of the cointiil 
 AVA\ irilli (I ciiitr.'trt' snr/'tirc, ami often we will indefi- 
 llltely adil to the aci'iiiaev ol'tlie volume of the solid 
 to he ohiained liy a eoiitiniied snltdivi>i.»ii of tint l»ody 
 to he eominited, into parallel segments, sm iller and 
 smaller aiul of eiiual hei,:;'hl. or thiekness. However 
 ill most eases, it will not be necessary to eaiiy the nimiher of tho 
 subdivions lieyond 3 or o to obtain a sullieient preeisicm in the 
 l-e.sult. 
 
 T 
 
 L^r'" 
 
 i'lT 
 
 Vl\ for the solidity of the n«!Xt slice IJC we 
 
 will obtain (AH > CI) 4- Acd) ,| I'Q, and so on ; «/!. JpL .nj J 
 
 whence it isjdaiu that the entire volume of the ' ' ™ -i^ ' 
 
 g„lid = () t 4 a^ ( 2 AB f- A cd + 2 CD + 4 
 
 cf, 2 EV • &C. + MN) - ,1 01', &e., that is : 
 
 to the sum of the areas of the ends (), T, of tiie given solid, or of the 
 
 <»xterior bases of the first and the last slices, we will add twict^ the 
 
MENSORATION OF SOLIDS. 
 
 105 
 
 (1-12) It is plain nlso that to arrive at the 
 polidity of aujf frustum or sc^jfiiiciit A B ah, of a 
 sphere, sjiheroid or coiudd with iioii ])aranel 
 bases .\.15,n/), it will siini('et()eoiiii)iite separately 
 the voliiiue of tin- whole solid AH— E and (liat 
 of the. partial solid ab—enud then take tiie dif 
 fereiiee of those solidities. We will thus ohtaiu 
 vol. AH /)a = (areii AH t 4 area iiiterinediate 
 
 fie«'tioii between AH and K x ,\ EF) less (area ab + 4 area iuterine- 
 diale section between ab and c x ^ ef.) 
 
 (1 13) Let us now apply this general formula to the solution of 
 tlie several problems relatinj^ to it, (excepting h<)wever the prism 
 or cylinder, tlui pyramid or cone, the frustum of a pyramid or cone, 
 and tile prisinoid, which have ;dready been treated of), and let us also 
 take tlie opportunity of comparing, in ccitain cases, the results thus 
 obtained and those furnished by the ordinary rules, so as to judge of 
 their comparative accuracy ami of the amount of labour necessary 
 to work out tlie result. 
 
 PROBLEM XXXII. 
 
 To find the area of a sphere. 
 
 (144) RIJL.I: 1. Mult iphi (1071 G.) the circumference of ont 
 of its (jratt circles bij its diameter AF. 
 
 RSJLE II. Mnltiiyhj (107S *••) the square of its diameter or 
 four times the square of its railius by .7854 and by 4, or at once 
 by 3.14 IG. 
 
 Ex. 1. What is tlie area of a 
 sphere the diameter of which is 7 ? 
 Ans. I.'.y.!iyd4. 
 
 2. The diani. of a sphere is 24 in- 
 ches ; what is its area f 
 
 Alls. 1809.5616. 
 How many scjuare inches of gild- 
 ing would be re(iuired to cover a 
 spherical ball the circumference of 
 which is 78.54 inclies? 
 
 Ans. 78.54-5-y. 1416=25= diam. 
 and 78.54 x 25=11)63.4 sq. inch. 
 
 4. What is the area of the earth if tlie diani. is 7912 miles T 
 
 Ang. 196,663,355.7504. 
 
 12 
 
106 KEY TO THE TABLEAH. 
 
 5. How many Hiiportlfinl ft-ct of lead or other nictiil would he 
 requircHl to cover a licmisiilu'iical douiu the iliaiiieter of wliieh is :]:i 
 feet 4 iiichois ? 
 
 Ans. H;}j X ;};i\ \ 787}^ -< 2 — 17'y't s(|iiai(> fci't ; f'oi', it" the area of 
 the whole sphcic is (■(|Mal to I ji'icat ciich's, it is plain tiial that of 
 the heinisplii'ie is equal to 2 nieaf circles. 
 
 O. The vault of th(> apsis of a cliui<'h is in the Conn of a 
 quarter of a sphere the radius of which is ir» feel ; required the iiiini- 
 ber of yards of phistc^riu^ iiecessaiy to cover its siirfaee ? 
 
 Alifi. ;M) X 30 X .78r)4-7-l) = 78.r)4 or 78^ yards ; for, since the eii- 
 tii'o sphere is equal to 4 great circles, the qiuirter of a sjihere is equal 
 to one. 
 
 7. What will be. at T) pounds p(>r sfjuare foot, the weiijht of a 
 hemispherical copper boiler the circuiuference of which is lrf8i 
 inches I 
 
 Ans. 188.5^3.l416=:diani. =00 and 188.5x00^11310 scpniro 
 inches of which the half oO.").")-^ 144 -3I).^i7 square feet, this area mul- 
 tiplied by 5 (the weight perscj. foot) ;;ives lLtO.3.") [)ounds. 
 
 (I'lfi) RL'IjK III. (\»isi(lcr the siir/dcc of llie t^'pliere as <i com- 
 pound of couiiinivd {r(ipv:ii()tis or zones of lujiKtl fireadtli AH— HC=: 
 (JD=DE—Eb' and pnteeed in the manner of paraijraph. (I;i0 T.) 
 
 Ex. 1. What is the area of a hemisphere the diauieter of which 
 is 263 i 
 
 Ans. The circuniference=203x 3.1410=:8-20,2408, the fouith part 
 of the circumference 20(J.r)0t 12 divided into 5 etpial i»aits, gives for llio 
 developed width of one oftlu' component zoni's 11.31204. The inter- 
 mediate diameters of tlies(! zones obtained by mean.s of a scale of 40 
 units to the inch., measure resjjectively, conipiiting from thi; base to 
 the apex, 200, 213, 145, and 82 : th»; sum of tliese intermediate dia- 
 meters i)lus the half (131.5) of the diameter 2(i3 at tlu; base, is 830.5 ; 
 this sum X 3.141G gives the sum 2ii01>.01»88 of the circumferences to en- 
 ter into the computation ; this latter x 41.31204, width of one of the 
 zones, finally gives for answer 107,787 uuits of urea. 
 
 REIII. The two first rules give each of them for area of tlic pro- 
 posed hemisphere 108,050.0(J units. Tlie dilt'ereuco between these 
 results is 803.5, 803.5-:-108.050=()08 nearly, whereby the rate of 
 error is ^ of 1 per cent nearly. Wt! theretbre concliule that iu every 
 analogous ciiae, it will sutlice to increase by .008 or. 01, nearly the 
 result obtained by this rule, to come very near the required area. 
 
MRNSnnATION OF SOLIDS. 107 
 
 V.\. 1. Let il lie r(Miiiir((l now to opcriito willi 10 sections or 
 y.Diifs instciid of ."), tlit' (liiiiiictcr of tlio lieiui.si)licro iTinainiiig the 
 saiiic ? 
 
 AiiN. 'I'lic !» iiiici inidiiitc (liainclfis bciii^ as follows: 200,2.^0, 
 2:lt. 2i:{, It^d, l.')). Ill), H',';iii(l 12 : their smii t IMJ) (iialf tlic diaiueter 
 2(i;} at tlu' l»asf) is 1(17 1.."), tills sum s ;{. Hid = r)2r)l.l844, Kum 
 of tlic <'ircuiiift'r<'iic<'s to lie used as an clmirnt of tlic projiost'd 
 <M)iiiputati(in : tlic hnatltli of one of tlit> coniponiiit zones will in this 
 ras(> lie i',, of the ipiartcr of tlic ciiennifcrcncc, that is 8()22t()8-f- 
 4 4 10 or at once by 10 = 2(l.(ir)(i(l2 ; hut, r)2.")l .1H44 x 20.(jr)(J02 = 
 108, lOS.,")?. It has already heen seen that the accurate resnlt is 
 I08.().")0.(i(i ; the ditl'erence of tiicsc results is now hut 182, equal to 
 .0017, that is : the delicieiicy is no more than ,1 of I per cent. 
 
 This rate of error added to the resnil of any other aiialo<jfous 
 operation would therefore give a pretty near approximation to the 
 trnth. 
 
 K\. 3. Let ns see now in how far the precision of tlu^ resnlt will 
 he added to, hy workinir ont the solution of the same prohlem, hy 
 means of an additional numher of subdivisions, say 20 for instance. 
 
 Alls. The iniernii'diat.' diameters are 2(52, 2G0, 27^6, 250, 243, 
 2'.U, 224, 2i:J. 200, 18(5, 171, 154, i;J8, ]1'», 101, 82, 62, 42, 21 ; the 
 
 sum of the inteiniediate diameters i the half-diameter at the l)aae = 
 yiUO.a ; multiplyinj; by ;iHl(i and by 10.:j,»80l (breadth of one of the 
 sections) we will obtain I08.ti79.5 against 108,050.60 tluj true area. The 
 difl'erence is in this case in excess instead of less than the required 
 area, as it shcnid be (ISO) and as it would Ix^ in fact if we had com- 
 puteil the intermediate diamcteis of the componeiit Ziuies instead of 
 obtaining them graphically or mechanically, as has been done 
 with the aid of a small diagram on paper and a scale of equal parts. 
 This diiVcreiicc! or excess is but of 2!) units in 108, 0.50, say .0027 or 
 less than ,\, of oni' per cent ; it is due to our neglecting, in measuring 
 the iiiteriiiediatedianiet(ns.ihe fractions ofunits which ifm;eded could 
 he taken into consideration ; but it may he a<linitted that in practice 
 a resnlt which, like this, deviated from the truth hut hy ,jj'(jg in 
 excess or delicieiicy, would be e([uival(>nt to perfect accuracy. 
 
 (116) KL:JI. Ifwcput this third rule among those that may 
 be used to determine the area of a si)her(> or p.irt of a sphere ; it is not 
 that we should think proper to apply it to arrive at the area of a 
 sphere proi»erly so called or at the solution of any anologoiis 
 problem that can be solved hy more simple and direct rules ; hut it 
 is because in practice, it is rare uuuugh that wo huvo to duid with a 
 
108 KEY TO THE TABLEAU. 
 
 perfect Bpliero, part of a perfect sphere, a s])li(>roi(l or part of si 
 spheroid properly so calkMl, a true paraboloid or l»y|M«rl)oloid, 
 or ill general with a solid of revolution, the j;('iieratinif eurve of \vhi(;li 
 18 an exact section of acone, hiicIi as the cireh'. tlieeilipsis, the i)ara- 
 bola and hyi>erl)ola. It is therefore evident that in all cases in which 
 we niiglit not have to operate on a perfect sph(>roid or conoid, or tho 
 kind of which conld only bo established bnt by innch prelimi- 
 nary labonr, it wonld b«ibetterto pnceed ininiediately by Huh! Ht 
 than to have reconrso to another rulo wliicii did not accurately 
 apply to tho pioposed problem. 
 
 (14'7) Let ns add that if the surface to he mcamtred, instead of 
 being every where of eqmd curvature, as that of the aphcre, were, as that 
 of the paraboloid, cCc, of unequal currafurc, we mi«;ht, liefore proceed- 
 ing to the 8\ibdivision into zones of <;(iMal bieadtli, lirst divide tho 
 area to l>e compnted in two or several paits which would afterwards 
 be subdivided into a less or greater number of zones accoiding to tho 
 lesser or greater curvature in the corresponding part of the geiierati tin- 
 arc. One might then calculate separately the parts of unequal cur- 
 vature and afterwards take tho sum of those parts. 
 
 (148) Usualh/ also, the n^easiirer or (jeometriiian, in considerintj 
 the degree ofxyrccision to he brought to bear in the practice of the details 
 of his art, will not lose sight of the importance of not devoting to the 
 solution of a problem, a labour and time which would notbojusli/icd bij 
 circumstances. It would for instance bo idle, we may even say- 
 unjust, that to establish to within amillLontli, thousandth, hundredth 
 or any other unit near of tho accurate result, a i)roi)()sed area or vo- 
 lume, one should devote to it a time which wonld cost those inter- 
 ested more than a fra(;tion of the value of such unit. We say *' usually ;'^ 
 for it is plain that there may be circumstances, either in a question 
 or cause in litigation where the cost of doing justice to the parties 
 may be more and in fact is often more, in an unlimited proportion, 
 than the value at stake. 
 
 PROBLEM XXXIII. 
 
 To find the solidity of a sphere. (See the tableau.) 
 
 REOT. A piano or piano surface RS touches the sphere but in 
 one single point D ; therefore tho areas of tho opposite and 
 parallel ends or bases D, M are each of them null or=(), which re- 
 duces the formula iu the case of the sphere to multiplying 4 times 
 
MEN8URATI0N OF KOLIDS. 
 
 109 
 
 B 
 
 D 
 
 
 
 ■^ x^^-^ 
 
 h? 
 
 tlio aroii of a fjiofit^ circle, lliiit is. of a section passiiij,' lliiouf;li the 
 cciitn^ C, by the sixth paitof the height DM peiijeiuiiciihir to that 
 seel ion. 
 
 (IIO) RULE I. Miill!i>lii 51 
 
 (1075 Cir.) //((' (trcd by vue third of / 
 
 the rdiliits. 
 
 RL'LE II. ('i(he (1103, 10°) 
 the (liduictcr and multlphj the num'ocr 
 Ihiis found III/ I ^ ; that is, bij U-.V^^W 
 or the soliililii o/d s])hcre thediitmeter 
 of which is I ; for (10«4 CJ.) the 
 s.)Ii(litie8 ov volinn(!.s of any two 
 spheres are aa the cubes of their diani. 
 
 RULE III. Multiply 4 times 
 the arcd o/'itsectiou of tlic sphere eipii- M 
 
 distdut from its opjiosile ends or a^iiees hi/ the sixth of the heiifht perpen- 
 dicular to that scetion. Tliis rub', in tlie case of the sphere, is evi- 
 dently analoi,M)iis to the first, for tlie area of the spliere is equal to 4 
 great circles, tli(^ jfreat circles is the section of the spheie by a plane 
 passing througii the centre C, that is, equidistant from two opposite 
 points D, M, of i(s surface, and the fith of the lieiglit D.M is but the 
 sixth of tlie diameter or the third of the radius. 
 
 Ex. 1. Wliat is the solidity of a spliere the diameter of which is 
 J2 I Alls. Jlil^'ia xl2 X .r)-j;5(i = It()4.7e08. 
 
 3. Tf the mean diameter of the earth is 7018.7 miles, what 
 is its solidity in cubic miles ? 
 
 Ans. (7913.7)' x.523G = 2.-)9,092,792,fl82.G374nn8cub. m. 
 
 3. The top of a steejdc is terminated by a spherical ball the 
 diameter of which is 2j feet, ; what is its solidity ? 
 
 Ans. 2|x2§=7^-7.11inil, 7 x 2|=:rl8.()nn(3(i()(i, 2§ x J or 
 2.GGfiGG(;G -^ 9=2:>G29r)2,J8.()G()G()t!(] x .29G29G2=18.9(529G29 = (2§)^, 
 and 18.9G29()29 x .523G=: 9, 9290074 cubic feet. 
 
 4. What is the solid content of a cannon ball having a diameter 
 of 10 inches ? 
 
 Ans. lo'^lOOO, ai'.d 1000 x .523G = 523.G cubic inclies. 
 
 5. How many cul)ic inches of gunpowder to till a shell tlic in- 
 terior diameter of which is 12 inches f 
 
 Ans. 12 X 12 X .7854 x 4 or 12 x 3.141G = 4r)2.3904 = area of the 
 sphere and that area x^ radius or J diam., that is, by2 = 904.G808 
 cubic inches. 
 
110 KY TO TIIK T.\Hr,KA\T. 
 
 O. How iii;in\ cubic feet of nil' iiiJiv he (•(Hitainod in n buoy of a 
 
 Bphcriciil torni willi an int. diainctcr of H) feel ? 
 
 Ah*. .")'^.'{.() ciiliic t'cof. 
 
 7. .V sloiu' hall is ;n'('cl (lianirtcr ; wlial is ils \\«-i,i;lil at I.'jO 
 
 ])OIIIm1s pel' ciibie fool .' 
 
 Aii«, .'{ X ;{ X :{ -. ..Vj;{(l > l.')(» - XJI-iO..').-' poniids. 
 
 ><. How manv ;;alloiis(ir li(|iior (2:i\ cubic iiiclics pcr;,'alloii) iiiay 
 bc contaiiicil in a licniisplici i<-al boiit r 10 I'cct (liaiiictcr f 
 
 AiiN. 'I'lic content of tlic vcssr'liii cubic feet - 10 x ..VJ.'Ui -: 2 = 
 201.8, (lie number of ^^allons per cubic foot" 17"JH enliic inclies-f-231 
 = 7. ISO.')l!i.'), sa,\ 71. and "^'(il.H x 7i = I'Xt'Mj {Gallons, or iikuc correctly 
 2()l.p*x7.J8==]!>.")H.',»ti oallons. 
 
 !>. A lu'iiiisplicrical vault ol'tlie imiforni lliickuess ofone foot, 
 ineasuics 10 feet inteiior diameter : liow manv bricks have been re- 
 quired to built it. al 20 bricks per cubic foot ? 
 
 An*. Ft is plain that the recpiiied solidity is equal to tlicdifter- 
 eucc <»f the solidities of the exterior and interior hemispheres ; but, the 
 ('Xteri«)r heuiisphere= 12 x .,")2."}0-=-2 ^ 4r)2.;i!» cubic feet, the interior 
 liemisphere -- 10 x ..")2;{(i4^2 - 2(>I.H cubic feet, (he din'ereiice of these 
 solidities is litO.")!) cubic feet and l!IO.(»x 20=:)812 bricks. 
 
 lO. 'I'he thickness of a bomb is ,"> inches aud its exterior circiun- 
 fcrence (!2.8;J inches; whal is itsweiuht, at 480 pounds per cubic foot ? 
 
 Aiis- Wc have for the cxteiior diameter of (he bond) (i2.8;}-;-3. 
 141<j-20 inches; tln'relore the diain. ot" the hollowed i)art is 10 
 inches ; tu)w the solidity of the bomb Is the ditVerence of the solidi- 
 ties of the exterior and interior spheres. The sol. of the exterior 
 sphere, = 2(r X .,-}2.W = 4188.8, the int. sol. = lo' x ,.5230 =.")2;}.li, the 
 ditVerence of these solidities is ;{00.">.2 cubic iinhes ; now, I cubic 
 foot or 1728 cultic inches : 4l^0 pounds weiylit :: ;jtjd.j.2 cubic inches : 
 1018 pounds weight. 
 
 PROBLEM XXXIV. 
 
 To determine the convex area of a spherical segment 
 or of any spherical zone ^ 
 
 (See the tableau.) 
 (150) RULE I. MiiUiplif (10T3 €J.) the hcujht oC, 00 of the 
 
 1. The gplhn-ionl segment is any pnrt aeft", of the sphere, cut off from the whole 
 ■pbere by a]ilano ofsc(?tion aeh, u sinnll circle of tho sphere. 
 
 The spherical zone is any part (((7;-AEB of the fphero comprifed between two pa- 
 rallel planes ab~-Ali, It is, a^ tho case may be, lateral, ce;itral, exoeutrio- 
 
MENSl'llVTION OV R'lMDS. 
 
 Ill 
 
 8Pfimeut, or the hriiihf Od of the zone hi/ the vh'cumferencr of a ijrcdt 
 lirch'of the uplicrc : llif jtrinliicf will he the rei/iih-ed (ire<i. 
 
 III'^M. Il'tlic (liaiiM'tcr of tlic s|)lM-ru 
 irt not jiivcii it iiiii.v cnsilv lie rmiiiil l>v 
 tli<- iih-iIiimI of par. (5IO <«.) \t\ *li- 
 vidiii^r till' si|iiiiic of i;iiliiiH of till* 
 Itasi^ of lli«' si'niiHMil Ity tlic liri;;lit, to 
 obtain tli(^ rcniaiiiilrr of tlir ilianirtrl' ; 
 till' rcniainilcr thus fniinil ' the ;i;iv('n 
 hcinlil. will lie liif iiiiiiiitd (lianiett-r of 
 the splicic. 
 
 Ex. TlitMliainctt'i' of a s|ili('r(' luini^ 
 42 (h'linu'trcs, wliat is llic toiivcx aica 
 of a .sn^intMit the Ik i^lit of wliicli is !• drcinH'tics ? 
 
 Alisi- 4-Jx;M11(J = (ire. i:U.!»47'i which x U= 1187.5248 R<iuaio 
 dcc'inu'trt's. 
 
 "i. TIic lailiiis of the I>asi> of llic roof of ;i lantcni in th« 
 foiinofii s|»li( rical si'^nicnt, is iOfi'ct, the lifi^lit of tlii^ roof is 4 ffct. 
 How many supeilicial feet of loud or other metal would bo required 
 to cover it if 
 
 Aus. 'w ~ 4 = 2.'), 2.') I 4 - diam. of the sphore=2n, 21) x 3.1416= 
 circ. !.tl.I(»<i4, now !i|.l()(;4 = 4;ki4.42r)(i square feet. 
 
 3. IJequircd the area of the lid of a boiler in llic form of a sphe- 
 rical .seginent the circiinifereiu'e of which is iil.l inches and hei^i'ht 10 
 inches ? 
 
 Alls. J)l.lH-3.l41() = 2!)=diani. (»f the lid the radius of which is 
 consequently 14.5 inches ; to obtain the diam. of the sphere of which 
 
 the segment fornisa part, wo have (14.5)-;- 10=2l.025=the rejiiainder 
 of the diam. of which the height of the lid forms a jtart ; therefore the 
 required diam. = 21.025 + JO .= ;3I.(»2."), this diam. x 3.I41(U !»7.4(idl4 
 =:circ. ofa great circle, this latter x 10 gives 074.0814 for the re- 
 quired convex area in squaro inches. 
 
 4. A hemispherical dome, a segment of which has been taken offto 
 lay the base of the lantern which crowns it, presents con.so- 
 quently the form ofa spherical zone or of a s[dierical segment with 
 two bases ; ic is required to determine its convex area, its height 
 being metres and diameter of the sphere of which it forms a i)art 
 20 metres ? 
 
 Au§. 20 X 3.141(5=circ. 02.832 and 02.832x9 = 605.488 square 
 metres. 
 
112 KEY TO THE TAnr.EAU. 
 
 (I.TI) Kini. Tl'tlio liidiiiH or «liaiii. of tlio f»ii1ioro ofwliiclia 
 zoiK^ lornis II |),'irt is not kmnvii and tliiit tlit' only dnhi he tlic radii 
 or diamctt'is of tiu^ inf. and hii|). Itascs oI'IIm^ sc;r|||,.||| jhhI tin; 
 ]ti'i<.<,'litor ])cr|M'iidi(-ular distaiict' wliicli HcitaratcM I hem, iiara;;;raii]i 
 (571 ii.) may fiirnisli tlic method of arrivinj; at llu^ rccinirt'd ra- 
 dius ; but one may attain it <|uit4' an well and in a m(»n> cxpcditiouH 
 manner and a<'<iirately c iion;;!! in practice l>y a simple ;;rapliic process 
 ■\viiieli \voiiId allow of to detenninin;n at once; the required radiuH 
 or diam. of llie spliere, with tlic »amo scalo luadtMiHo of to llx on 
 the paper thc^ relative proportions and positions of the data, tho 
 centre of tlu^ circli' heinjj then easily ('oun<I l>y repeated trials on tho 
 ])eritendiciilar (prolonged if necessary) which unites tho centres of 
 tho two <(iven chords. 
 
 S. The diainoters of tho inf. and sup. hasos of a roof in the form 
 oftlio sejjjnient of a sphere measure respeotiv(!ly JO and 12 nietros 
 and tht^ heij;ht 2 metres ; what is tho area of the zone forming tho 
 lateral or convex surface of the .'oof 1 
 
 Alls. We obtain (574 O. ) either by calculation or by 
 construction tbo <lianu'ter 20 of tho spliere of which the segment 
 forms i)art. This diam. {^ives for circumference (J2.8H2, this circ. x 
 2, height of the roof, gives for its convex area 1 .'}r).(j()4 s(iuare nu'tres. 
 
 O. What is the convex area ofa8egmont21| inches high taken oflf 
 from a sphere G feet diam 1f 
 
 Aus. 18 10.577 square inches. 
 
 T. If tbo diameter of the earth considered as a perfect sphere la 
 7970 miles, the height of the 1 igid zone must be 252.3C12d3 miles j 
 ■what is its area If 
 
 Anti. 7970 x3.MlGx2.J2.3G128,7Gl = 0,318,701 .square miles. 
 
 8. What is tho area of one of thd 10 component sections or comr 
 partmonts of a vault or dome in the form of a spherical segment, the 
 interior diameter of the segment or of its base being 40 feet and its 
 height 10 feet f 
 
 Ans. The remainder of tho diameter of the sphere of vi-hich tho 
 
 height 10 of the segment forms a part is (5390.) (i40) -^10 = 40 and tho 
 entire diameter consequently =40 + 10 « ."SO, the circumference = 50 x 
 8.1410 = 157.08 and the entire area of tho 8egment = 157.08 x the 
 height 10 = 1570.8 ; therefore tho area of tho proi)osed section is 
 1570.8^ 10= l.')7.08 square feet. 
 
 (153) RULE II. Divide the surface to he computed into zones 
 of equal breadth, and proceed afterwards in the manner of par. 
 (126 T.). 
 
MENSDBATION OF SOMDS. 113 
 
 K\. 1. 'I'lH'intVr. (■ir(MiinlV'roj)r<'orus|»|iciical/niH',()i' wliicIi!ip|)ciirH 
 to lie (die, luciisuK's •,'()() led, its Hiip. circmiiri rciicc ','(.'{ I't'cf, and two 
 iiilci iiu'iliiitc <-ir<Miiiir('i('iiccH (MiiiidiKliiiif !».'>() and '-i'H Irct, tlic length 
 of (lie ;;«Mi('ratiiij; arc is 1.") I'c<'f, and tin- d(vtIo|t('d liicadtli of one of 
 the llncc coniiMtncnt /ones is fonHcijiicnf l.\ '> fct't, ; Avhafc is the urea 
 of till' cntiio zone I 
 
 % 
 
 Aw*, i'-im i^.-)(» I 5>;M I i'*i:]-7'2().r), (ids sum < 5 rr :j(i()o.r, sq„ft,.e 
 inolit's nearly. 
 
 a. 'I'lie vault, or arclied eeilin;;' of a eircidar room in tlio foim 
 of a spherical H('<{ment has for its inf. diaii. Irtd decimetres, and for 
 the inlermediate diani. of live component; zones 154,11!), H;J and 42 
 decimetres, th(?len,i;lli of IIk^ ■^eneralini,' curve, that, is, the eiirvili- 
 neal distance from tlMwenlrci ofllie vault to its s[)riug is KW deci- 
 metres i!H ndllinu'tres ; what is its concave area ? 
 
 An«i. I(i;{.t],-' decimetres-; rj = 2().G:)li = breadth (sf one of tlic com- 
 ponent zones, i inf. diam.nri80-f-2=y3, !):J t- 1.14 f llf) f 83 f- 4*1 = 41M), 
 i!»() ■; :}. 1 'i(» ~ l");J!i,;{f44 sum of tlie circumferences to enter into the 
 comi>ntation, ii«»w, 1.").'{!».HS 1 < 'JO.dKJ = .•{|,7!>7..'") atpiare decimeties 
 or ."(17 s(|iiare metres !»7,l s(|uare decemetres, since the s(iuare metre 
 is 10 < 1(1^ 100 s(piar»' decimetres and that by putting the decimal 
 point 2 places back wc divide by 100. 
 
 PROBLEM XXXV. 
 
 To determine the solidity of a spherical segment 
 or of any spherical zone. — (i^ei^ the models of tlie tableau.) 
 
 (15:1) RI'J.TI. 1. The spheiical segment aehC or a e b D 
 (See the li.i;ureof the paragraph (ItiO) may be smaller or greater thau 
 a 111 unsi)here or equal to a hemisphere if the idane of section passes 
 through the centre of the sphere. In every case the general formula 
 gives its accurate solidity. In the same way, the splierical zone may 
 be lateral, central or excentric. We will call it lateral when it is the 
 zone of a liends[»here like the one which in the figure is comprised 
 l)etween the jdanes of section, parallel circles AEli, a e b. It may 
 be central if its phmes of section, opposite or limiting bases, are equi- 
 distant from the centre O of tlu; sphere, and exceutric, if these bases 
 are unequally distant from the centre. 
 
 13 
 
114 
 
 KIT TO Tfl?, TABI.KATT. 
 
 (15'i) REin. If. Toobtiiiii in tlic spliciicnl sconiciit, tlic diu- 
 metcr of tlie central or liiill'-way section, it sntliccs to icnicnihcr 
 (530 G.) that liic Iialt'-cliord a o (sec the lij^iin! of parajfrapli (1*50) 
 is a mean jjiopoilictnal between the versed-sine o (! hei^lit of the 
 se^iucnt and the remainder o I) of tli( diameter. L(^t tlieiefore 
 AEH-(' any s(>gment of a spliere, and <i c h its iialf-way section, tliat 
 is sucli that we may obtain ()f> = f>0=iO(," : llien as (>(', lieii^ht of 
 the solid, is known, we will obtain <>V=^OC and we will find oa or 
 oh = i ab = \i'o C X o I). Letai>ain (i e h 1) tiie segment to be measured, 
 and AEH its half-way suction passing; tinon.t^li a point lialf- 
 ■way between o and 1). Knowing o 1) and cousiMpiently ()!):= V <> I)^ 
 we will (djtain OIJ or OA - i.VB - VOl) ^ OC!, or by measuring directly 
 the required diameter of the body to bo eoinpnted. 
 
 (153) RLi^I. III. To obtain in 
 tlie spherical zone AI51)f', for instance, 
 the diam. of its intermediate section ; 
 we will at lirst find, if it isnot ah'eaily 
 known, the radius OH or OF of the 
 sphere of which the zone to be com- 
 puted forms a ])art. To tiiis etl'ect 
 (574 «.) the plane figure ABCD being 
 the vertical section of the splitMical 
 segment in question, we will obtain 
 
 EF^CE X ED-^AE, then AF - AE or dU i EF and diam. 1?F- 
 VAJi-^ I AF'-'. The radius OH bei ng now lvnown = .li?F, we will ob- 
 tain Oa = VO"H^— HO'-^or Oil — Vt)(.J-L'lT-, or after liaving found OG 
 orOHweAviU obtain OH = GlI-Ori or OG^ Gil— OH ; nc.w, if we 
 suppose the line GH prolonged on both sides to tlie circumference at 
 XY, we will (d)tain G.\, = OH-OG and IIV=OH— Oil and theiu'e 
 we will easily obtain, as in HEW. II, the intermediate diam. half- 
 Avay between AB and (T). 
 
 (156) KIJL.i': 1. MiiltiphiilOHHU.') 
 the half-siim of the areas of the 2)araUe^ 
 bascfi hij the hei(/ht of the se;imeiit ; add 
 to this jjfoditet the soliditj/ of a .sphere the 
 diauieter of ichieh is equal to the heifihl 
 of the seipnent; tlic sum of these two so/i-Af 
 dities icilll be the reqitireil volume. 
 
 REi^I. Wlicn the segment has but 
 one single base, the other is considered 
 = 0. 
 
 RrLE *t:\To the sum of the areas 
 of the hi f. and sup. bases of the seymeut, add i times the area of a 
 
BIENSURATION OF SOLIDS. 115 
 
 section equidistant from those Ixh^cfi, and mnUipUj tlie whole hij the 
 sixth imrt of the hvitjht ; the result will he the required soliditi/ (135 T.) 
 ICx. i. Wliiit is tlic solidity of ii scf^iiiciit loriniiin ]»iiit of ji 
 Rplii'i-c tlic (liiiiiH'tcr (»f wliicli is 10, tin' ;<'sp('ctivf (lisiaiices from the 
 cuiitrc to ciicli of tin- pliiiics of section being 1(5 iind 10 ? 
 
 Ans. \\'(' must tiist (ioterniiiie tlic iiroas of tlie i»arjil lei bases 
 of ilic, given segment ; but, tli(i diameters of these bases are parallel 
 <'lioids of a great ciiele of tlie spliere, distant fiom the. centre of tlie 
 eirele, tlie one by 1(1 and the other by 10 units of ii 'asnre, tlie seg- 
 ments of the diameter of the great circle peri»endicular to tliese 
 choids are respectively, ol'one of tliiMu, lti4-20 = 8(i and 40 — ;j()=4, 
 of the other, 1() + 20=:.'J0 and 2(»-10 = 10 ; now we havt> (.ItOG.) 
 ;{(; X 4-114 ^the s(iuare of one of the iialf-chords and ;]() < 10 = ;^()() = 
 the s(juare of the other lialf-chord ; these s(|iiares multiplied each 
 by .78:)4 and by 4 or at once by a.l41(], give 452.8004 and 942.48 
 for the re(|uiied areas of the parallel bases. The sum of these areas 
 = i;j!U.8704, this sum x'.i, the half-height (KJ — 10) of the segment, or 
 the half-.-^um of these areas x (3^4184. t)112 = part of the required so- 
 lidily ; the remainder of the re(iuired soli<lity - (J x ..')2.'3r) — 113.0976 
 — sol. of a sphere the height of which is (i. 'J'hese two .solidities 
 united give 42!)7. 7088 for the solidity of the i»ropo.sed segment. 
 
 2. The same example by Rule It gives for area half-way 
 between tlie parallel bases -i-] < 7 = 2-"lI — the s(piare of the radius of 
 the base or intermediate section, this square x 4 gives the .sq..are of 
 the diam. of snchltaseor section, and this last s(piare x .78.)4 gives 
 its area = 72.).70U(J, 4 times this area = 2902.8.']S4 to which adding the 
 .sum of the areas of th(i bases we obtain 4297.7088 for the required 
 solidity, for J height = l aiul iuultii)lying b}- 1 does not alter the 
 value of tlie multiplicand. 
 
 3. How many cubic feet of liquor may a hemispherical boiler 
 10fe(;t diameter contain ? 
 
 Alls. We have seen (134. T.) that in the hemisphere th j area 
 of the intermediate section equidistant from the base and apex 
 ofth'3 solid is equal to the ^ of the area of the base or of a great 
 circle of tiie sphere; now, we have for area of the sup. baoC of the 
 boiler 10 n 10 x .7854=78.. 14 scpiare feet ; but 4 times f =3 and three 
 times 78.54 (-78.54 = 4 times 78.54=314.10 and 314.16 x J height= 
 314.10 X 5^n = 2(il.8 cubic feet. 
 
 (I5T) RE.^I. In the case of the hemispiiere, as in the entire 
 sphere, I\ule II does not offer any advant,- and on the con- 
 trary, it gives more work, since it is more isiaiplc to avtivo at the 
 
116 KKY TO Tlli: TABLKAU. 
 
 requii'tMl result, to ciibo at oik-c tli ^ (linnictcv, inultiply tliis <'ii1k' l>y 
 .52;{6 sind take liall' the proiliicl foi' IIk^ solidity of iIk; liciiiis- 
 l)htir(!. 
 
 4. How many ^tjalloiis ofwatci- may (iiid looiii in a reservoir in 
 tli(? form of a splierical seij-meiir wiliia JOI) I'eet diameter, ami '20 iVet 
 deep, iit 7i galhdis per euliie foot V 
 
 AiiM. By the first rule, we ohtiiiii tlie re(|iiired vol.^area of tlie 
 base of tiie segment (tiiat is, tlio sup. area of the reservoir)- tlu^ 
 lieight (vertical dei)th of the reservoir) r !2, pins the vol. of a sphere 
 liavin^ for its diameter sneli height ; that is, the re(pnre(l vol. - 
 ( 101) . 100 X .78.54 X 2i) ^'2 = 7p<.-)4(») f (20 < 20 ^ 20 x ..•)2y(; — 4 1 88.8) — 
 82,728.8 cubic feet a 7.r)=(;20,4()(! /.allons. 
 
 Alls. l>y the .second rule, we hav<' lirst (5 !<► H.) for the le- 
 niainder of the diam. ofthe si)here or of the ;;reat circle of which 
 tlui heii;ht of tiu^ reservoir foiins a part (.} 100) ^ 20= 12.'). 12.') : 10 
 (lialf-distuuce from tht? surface! to the, bottom) = I.T), IIJ.! v 10 = I-'J.'jO 
 = rectaiigh? of tlie .segments of tlu! diam. = s<puire of th(^ half- 
 diam. of the intermediate section, this s(piare ^ ;{.141() -4241.10 = 
 area iuterin. section, 4 times this aica !- the area of the l)ase ofthe 
 Begment — 24.818.04, this .sum .<. 20 .-0 = 82,728.8 cubic feet, as before. 
 
 (IS*^) SSK.TI. Tiie clioice to be mad(! b(;tween th> two rules for 
 the solution of this pioblcm will sometimes depend on the nature 
 of tlie data, but especially on the doiiltt that might exist as to the 
 particular kind (»l the (igiire to l)e computed, and tlii^ use of this 
 formuhnvill dispense with the necessity of eiiqiiiring first of all as to 
 the exact nature of the proposed solid. Tims, if the reservoir to 
 be measured were the segment of a spheroid, hyjierboloid, or any 
 other figure resembling nearly thos(! just enumerated, rule H 
 would in any cast; give its accurate solidity (I'2T T.), or very nearly, 
 ■while if we treated as part of a sphere proper a figure which 
 Averc! not such and calculated it by the rule ai»plical)le to the si>heie, 
 we might be deeply mistaken in the result. 
 
 5. A basin the form of which .seems to be that of spherical seg- 
 ment, has for its sup. diam .15 inches, for half- way diam., 12 inches, 
 and for depth or luuglit 7 inches ; what is its capacity in gallons of 
 231 cubic inches? 
 
 Asis. Sup. area= 15 •< 15 X .7854 = 170.715 s(piare inclu>s, inter- 
 mediate arca=:12 < 12 x .7854= ll.'M)l)7(), area lia.se (- 4 intermediate 
 area = 020 1054, this sum 7 H- = 7114 cubic inches nearly; divid- 
 ing by 2;ll we obtain 3.18 or 31 gallons nearly for the capacity of the 
 proposed vessel. 
 
MENSURATION OP SOLIDS. 
 
 117 
 
 0. rii<- viicnity or spncc midcr !i (loiiic or iirclu'd cciliiii" of a cii- 
 cnliir room, ]»r('.scMls ilic aspect ol" llic sc^^iiK'Ht ol'a splicrc willi paral- 
 lel Imscs the (liaiiM'tcis ol' wliicli iiicasiir<' H'SpcciiN rly I!*. 1* metres 
 aiui H.7IH iiH'Ires ; the diameter of Ilit! dome ('(iiiidistaiit I'rom its 
 l>ases is I7.;{"J mctics ; reipiired the iiiimher of nihie metres of air 
 to he iiealeil. the iiei,i;hl l)eiiix f^ metren ! 
 
 Alls. (I!t.!>/ < .7."<.14=::;«i(jx .7Hr)4 = yil.(l-J, (8.718) =7(1 and 7(> 
 X .78.")-! = .')'!( ii», {\7:.ti) =;}()(( and :«)(» x .7854 x 4=:i)4^.48, the sum 
 KM.'}. 1!» of those areas ■. 8 ;-()=- 17.")(i.!»'J <iil»ie melics, orAvliicli is (he 
 same thiiij; and moic simi)le (!!».!•)- i (8.718/- +4 (17.."5'.J)'- -< .78.")4 x 8 
 -T-tir-.sol. 
 
 7. A vessel !iavin,n' (lie form of the frMstum of ji cone is terminat- 
 ed by altottdUi liaviii^i'' the apjiearance of a si»herieal segment. The 
 inferior diameter of (he vessel is IvJ feet, the in termediate diameter 
 <jf the segment is 8. 7".2 feet, and its heij^lit 2 feet; how much must 
 be added to th(^ content of the body of the ves.sel to obtain its whole 
 capacity ? 
 
 All*. (12) l-4(8.7-i) X .78.")4 xj> .•-(1 = 117.. '5 cubic feet, (when) 
 
 we liave taken (S.72) = \^; for, <»r (i 12)- -f 2= 18, l8 + 2 = diain. of 
 (Jie spher«' of which (he scju'ment forms a part. Now the interme- 
 diate half-diameter=v'l!' ■■- 1 <»' 'li<' ••'mn . -\A times li>^v'7^> = S.72 
 nearly) and 1! 7.3 x7i = 88!) <^alloiis nearly. 
 
 PROBLEM XXXVI. 
 
 To determine the solidity of a spherical ungula, and the 
 area of the lune -which forms its base. 
 
 (See the tdhlain.) 
 
 HK^i. The spherical uiigida is a 
 part of the solid sphere eomi)ri.sed be- 
 tween two half fireat circles WC, EDC 
 meeting at any acute, rij;lit or obtuse 
 iiujrle AOE and its solidity is evidently j^\ 
 to (hat of \]w entire si)Iiere as its aii.i;le 
 AOE is to ;H\{)^ or as its arc AE to the 
 entire circ. whence it evidently follows 
 that its solidity will immediately be 
 obtained by the following : 
 
 Rl^rK I. MuHiphiMhnrn the area 
 of the mctor of a tinlc AOE bij I of (he diamckr Vl>, hiKjtk or Iniijht 
 
lis 
 
 KKY TO TIIK TAIM.EAU. 
 
 n/lhc splii'i'lral si'ipin'iil : I'm- llic iirciis of tlic uppositt' ba.s(!.s oi' ciitl.s 
 (' iiiul I ), iu<' liolli < (|ii;il to /(TO. 
 
 ttl'M'. 11. (,'rt JirsI (1075 H.) tlir <imi. Ihni llir sol!,]!!,/ of llio 
 entire sjtiirrc <■/ irliirli the iiininhi /'(iniis a jxni. IHriilv llicii ihnt 
 urcit (iiiil siiilililji hji thr rulii) hrhrccn thv (Uii/lc of tlic nntjida ami 
 yOl)^ ; the rcKiilf irill he Ilic mini red <ircit niul xnliiVilji. 
 
 Ex. 1. IJriiuircd llic iirci and solidilv ol' tlir iin^nilii ol'ii sj)liero 
 tlu' iin,i;lc of w Incli is til) and t lie diaaiflcr 10 .' 
 
 .41BS. '1'Ih' aira ofllic cnliir s].li("rc = (l I I «.) H» x 3.1 IK) x 10— 
 MM.K! units, tin- ratio of (id' to :5(;i» — ,\,, tlicrcloic .Ml l.l(;-:-l() = 
 1- (iiiiiiMl ar»'a = .'?l. IK). 
 
 'I'lic solidity ot tli<" entire sii1i(Te = ( I !!»<».) l(i' .r)-j:}(; = :)2:}.(;, 
 tliis solidity divided liy tlie ratio iV,!""*' <'slal»iislied, j;ives W.-'itJ 
 I'or tlie solidity of the jtroposed uni;ula. 
 
 "2. One of the com part men ts of the interior vanlt or of the exte- 
 rior roof of ii dome, presents tlie ti^iire of a spherical semi-hiiie, tlio 
 diameter of ilie dome is IIHIfeet, and its circninfereiice is divided 
 into 1(! paits or se«'tions by riiis dr.iwii from the apex to the 
 sprin^in.i;' ; re(|nired tin' area of one of the component semi-limes ? 
 
 An*. The entile area of llie s]iliere of wliicli the dome forms a 
 part= I()()x;5.1 !1() x l()(» = l(lirx .'M-IKI = lOOOK x ;},l4I(i — .'JLaiO 
 (sipiare feet, this area divided by .'i'^, since there are \\l semi-limes in 
 tlio entire area, ^ives for the i'e(piir<'(l area \)i\{ stpiare feet. 
 
 (a<»0) KUI^i^ BBI. Miiltiph/ lite leiitifh of tlie urcirltirli measures 
 thehreailth of the lime Inj the diameter af the sphere of irliieli it forms a 
 pitrt ; the prodiief irill lie the reipiired area. The area thus obtained 
 (or as established hji the lirsl rule) midtipJied bij one third the radius 
 vill ijire the reijiiired solidilj/. 
 
 Kx. 1. How many sipiare metres of silk arc^ there in one of tli(^ 
 component sections of a spherical balloon of which the diameter is 
 10 metres, and the iinmber of the component breadths 3l) ? 
 
 .ins. The \vh<)b> circumference of (he balloon beinf<- 10 x 3.1416 
 = 31.41() mi'tres and the number of comjiartments 3(), it folbtws that 
 trte bi-eadth of the gore will be 31.4 KJ-i- 3(i = .87:2:|inetres, llien, .d72| 
 X diam. 10 = 8. 7^1 sipiare metres = re(piired area. 
 
 3. There is to be replaced one of the 10 component ungnlae of ji 
 wooden ball 30 inches diameter, required the soiidity and convex 
 area of the ungula. 
 
 Ans. The circ. of the ball -30 x 3.14IG=:!I4.218 wlieiice it follows 
 that the breadth of the ungula=U4.348-;-10:=9.421d, this breadth x 
 
MKNSURATION OF S(M,!r)>. Ill) 
 
 <liiiiii. .'{Olives fill' llic iiicii of (lii' iin.niil;i 'iSl2; s(|iiiir«' iiiclics. 'rii(> 
 s()li(lity = tli(' juv;! v one tliird llic iii(lins = v>'-J,rM - 1.') :- .•{-•'JH-J./l I 
 x:}(»^(; = '^-<-J.744 < 10^ 'J = !2H'27.-41 : •J = J li:{;riil.ic iii.lhs or 14i;{.7-i 
 -^ 17vJf^ (iinmhcr of chIhc iiiclics iu ;i ciiltic loot ) = >"2 iiciirly otiu-iiltic 
 foot, let tlif lour lit'dis of II culiic loot. 
 
 It. IJ('(iiiiic(l llic iiiiiiihiT ot'toiscs (M7 ciihic c ii.nl isli feet per toisc) 
 ol" iiiiisom y in out' of lIic S coiiipaitiiiciits of ii li('iiii-si)li('rif;il Viiiilt; 
 ofwliicli llic int. (lianii'lrr is MlllVct imd tin- tliickiicss of tlir vjiiilt 
 'S feet .' 
 
 ylil*. Il is pliiiii (I0S;5) Miat we will oht.iin (lif r«'(iMin'(l soli- 
 dity liv tiiUiiii; the dilltTcMic of llic coinponcnt scini-iiiiniiliU' of tlio 
 iiiti'iior .iiid exterior licMMsplicrcs of tin' proposed \iiult. Now, t ho 
 int. diaineler heini; ."{O, tile solidity <>{' the spinre = ;il) x .,">^';{(|^ 
 I li:{7, llie sol. of IJH' ext. spliere=;;}(i x ..VJMt)=:-J 1 1*2!». the difl'eicncci 
 (!iM':2!>— Ui;{7— l()-i!lri) of these solidities divided l)y the iiiinil»er 
 (l(i) of the (-(Mil pollen I senii-iiii,i;iilae of the entire sphere, nives for llm 
 solidity of the compart nient dlM', cnhic feet, dividing tiiis latter 
 nnmlier liy ."^7 \vc get 7 toiscs :U\ cnldc t'vvi. 
 
 Or, aj)iiroxinialively, l>y innitiplyiiiL; the liaU'-snni of tlie ext. 
 and int. areas ot' liic eoii>.i)ai tnieiil by the thickness of liie vault f 
 
 \vc have area of the int. s[>licre .'51) s .'{() x .7f*r)4 ■, 4 or ."JD x;il4l(> 
 = ^■^■-27.44 of which the half lH;5.7'-i is the interior area ol' the entirts 
 vault, tlie area of the ext. sphcre=::ji! x -'M 4U! = 4()71..")I."{(! ofwliicli 
 the half ^J()M').7.')(i8 is ihcextciior area ol' tin; entire vanll, t he sum 
 y44!).47().S of these areas :-8 is the sum of the ext. and int. areas of 
 tile section of the vault to he. measured, and this lat ter sum 4-'?l .l^lt! x 
 li (half-thickness of the vault) or half that sum ninitiplied l»y tlio 
 tMitire thickness of the vault, gives Jor the cnliical content of tlio 
 compart nieiit lilti,' culiic feet, (tr 7 toises .'57 j cubic feet. 
 
 (lUl) atHni. We say " appro.ximati'ly,"" and in fact, the solid 
 to l»e measured is notiiing hut the frustum of a spherical pyramid, 
 comprised between parallel bases. The spiierical |»yraniid. like the 
 ordinary pyramid, lias for its solidity (lOf^'i <«.) the third of the 
 product of its base by its height ; but, were it true thai one could 
 jurivc at the solidity of a frustum of a pyramid l»y multiplying llio 
 half-sum of its [»arallel bases by the height of the frustum, it would 
 liajtpcii also that one would correctly obtain tins s(tlidity of the entire 
 pyramitl e(iual to the, lialf product vA'ita i)ase by its height ; for if it 
 be supposed that tlie height of the frustum increases iiidetinitely, this 
 height must ut last become e<pial to that of tlie (Miliri^ pyramid, and 
 its superior base will by the fact cease to exist or become cipiu!. to j 
 
120 KK.Y TO THE TABLRAtJ. 
 
 in tliiit Oiisc tlic li;ilf-siilii of tlic oiijtositc bases will be, tlic liall'-baso 
 of llic pyiaiMiil, and llic riilt; WDiild tlicii j^ivc Ibr tlic soliiiiiy ot'llio 
 ]iyariiiiil, the liair-|in>ilii( I (if its base by ils li('i;;lit ; l)iil tliK solidity 
 of lii(^ pyramid is on tlif contrary tiit- liiird oI'iIr' product of Itsbiiso 
 by its lici^Iii ; and tin- difffii'ncc Ik'I ween ,V and \ is ,V ; liicrcfore 
 tiic t'lior of tlic apinoxiniativc nictlmd nii^^iit in an cxti«'nic caso 
 iTMicli hllj per cent. In the ubovt; cxanii»lc the error in excess is bnt .'H 
 feet for (i l."5 feet or i per centi nearly, and would be still less if the 
 the di'nicicr of the vault were ;nreatcr I'clativcly to its tlii(d\n(!ss, or 
 whic ■; the same thin^-, if the hciiilit or tliickness of tlie frnstnmto 
 be mcasnrcd formed a smaller i>arl of thi; entile lieight (»f the pyra- 
 mid of which the frnstnni forms a part. 
 
 PROBLEM XXXVII. 
 To find the solidity of a spherical sector. 
 
 (Sec tlie models t)f the tdhleaK.) 
 
 (1<»2). SIB<1I. I. The scctoi' or spherical cone is, as indicated 
 by ils name, any part of the solid si>hcre comprised between and 
 liaving for its base a splicrical sc;j;nie,nt and foi' its lateral wall the 
 surface generated by the revolution of the radius of the sphert^ about 
 the circumference of the suialhu' ciicle of tin; spiiere serving as a 
 base to the segment. Wt> may consider th<^ spherical cone under 
 two aspects and measure it in consecpicncc : 1^ as a spciical sector or 
 as an intinitaiy sjihcrical jiyrandd to obtain its solidity by adding to 
 tlie urea of its spherical base 4 times the area of the inuiginary sphe- 
 rical Itase parallel to the lirst and situated half-waw between the 
 exteri(U' surface ami the centre or otherwise; between tlui base and 
 apex, asi for an ordinary pyramM or cone. Now it is (-vident that as 
 in the ca.so of the pyramid and cone properly so called or with plane 
 bases, the area of tiie half-way section is equal to the fourth part of 
 that of the base, which reduces the method of cubing the cone or the 
 spherical sector to that enunciated in tiie above given rule. 2^ as 
 ji comiiouud of a sjiherical segment and a right coue winch may be 
 measured separately i)y the rules already given to take afterwards 
 the sum of the compmient solidities. This nmthod evidently dis- 
 penses with the necessary knowledge to arrive at the conve.c area of 
 the l)asc of the sector to be measured and reduces tiie whole work 
 to that of obtaining tlie resi>ective areas of the circle answering as a 
 common base to the component cone and segment and of the circle 
 parallel to this lattei and situated half-way between it and the apex 
 pf the segment. 
 
SIKNSURATION OF SOMUH. 121 
 
 flG.D K?vtF II. r<i't lis siiy iilso. rcspcclinir t]i(> spliciiciil (*<uie 
 tliiit ir il wcir r((|iiiri'(l to ciiln' iiii v t'l iisi iiin of a spin rii-al coik! com- 
 I)risc(l liclwcrii iiiiiallcl liases, such as llic section ol' a slicll for 
 iiislaiici" or (lie vault, of a eiiciilar loom of imifoiiii deiitli, w« 
 Avould aiii\e al ii, as in tlie case of ilie fiusinin of an oidinary cone 
 Ity adilini;' to t!ie sum of I lie comn e\ and ion<'av(' a teas of its iiaral- 
 lel liaxs I times the area of a section paiailel to tlie Iuihcs imdlialf- 
 way between them, ami alli iwirds mullijiiyin.'^' tlie wholt^ l»y tlie 
 six til pai fof iIh- height olthe frustum ; (tr Ity comimtiiii;- tlie respective 
 eoli<lities of 1 lie component spheiical coiu's, llieii to take their dif- 
 f eve nee. 
 
 (I«l) tli't/i:. After hiirnuj c^fiihlisJinl In/ the method of pro- 
 hlon .'M tlic <irc(( ol' tlir Ixisr oj' liir ticclnr , irr iinisl )uiillij>li/ (KUT tJ.) 
 that <irc(t I'D one third oJ' the ritdiii>i to ijrt the reijitired solittitj/, 
 
 K\. 1. 'i'lie lu'inlit of tlie senMi'Mit, foi'iniii.if (OT^i H.) tlie base 
 of a spliiiical .sector, is I.V metres, and the radius of tlie sphcfe of 
 which the sector forms a part is 5 metres ; wiial is the solitlity of 
 tlie sector ? 
 
 Alls. The area of the base — ciic. of a yi'eat circle >■ tlie height 
 of the seu'imuit, tlie circ-diaui. !(» :Mli<; ol.llt) < l..>=17,lii4 
 8iiaar<' nieiies. this area I radius or by .").•-.'{ — 7?^.,') I cubic metres. 
 
 "i. \\ hat i- the solidity of a buoy having- the form of a spherical 
 secioi. the length of the side being lU feet and the diameter of the 
 biis(! o feet ? 
 
 Ans. With those diita we obtain liist the height of thoaeginent= 
 
 10- MO --J..") -10- !».<)S-^') — .air,') of <it, the ciic. — ditnn. 20 x 
 y.i.41(;=:():2.!?:}^ which < .:{I7.')- i:t.!M!>l(i >.iuaie feet — area of the con- 
 vex base, this latter ■■ lO-r--'}— tid. ii)7 cubic f(!et. 
 
 ii. A circular tower of whii-h the int. diam. is HO feet, hnn for its 
 out stone vault the frnstnm of a sector with parallel ba.ses, (lie tlnck- 
 ness of whicli i> .'> feet, (he lieight of the cap of the vault is 10 feet ; 
 what is the c.iucave area and the solid conreiit of tin; vault ? 
 
 Alls, 'riiti solidity of the frustum is (IfJSS G.) equal to the 
 dilTerence of the eonipoiieiit entire and partial sectors=e.\t. area or 
 of the exlradosx .\ I!, less the int. area or of the iiitradosx^ >• where 
 II. and r are the respective radii of the ext. and int. s[)h(ires of which 
 the sectors of the same name torni ii jiart ; now, we first obtain 
 (.140 4ji.) ftir tlio remainder of the diameter of the great circle of 
 
 14 
 
122 KKY TO THE TABI-EAH. 
 
 which tlio lifi^'lit of tlio v.nilt r'ums a pint aii<l of wliich thn 
 diaiiH'tcr oftlic vault is a clioni l."> -^10 (Ihc si|iiai'c (iltlic semi- 
 chord -:- till' vcisimI sine, thai is, thr tliaiii. of Ihc vault : its li('iu,ht ) 
 = 2:2r)7-l()^'JC.r» ; then we olttaiii the diaiiirh'i^'J'J..') | |(l t;!'J.."> and 
 th(* ia(liiis= Iti.'jri, and tin- (h-ptli of I he vaiill hcin;; r> feet, we ohtaiii 
 for Ihd radins of llic «'\lia(htH |(i.r»t*-f- r) = i!l.vJ."» ; now, \vc will ohiaiii 
 the interior art-a, oftlic vanll l)y niakin.t^ Ihc ciiciinifi rcnrc |(I^!.|(I2 
 (=:r3.|.||() <;{>,>..")) and liv iiiiilliplyinj;- if, h.v the lici^lif !(», wliich will 
 give H)2l sqiiari! fccf tor the rc<|iiircd area. 
 
 "\Vo will ohfaiii (107 1. 1^ G.) tho area of tlie exfrados hy mak- 
 ing /• : i{ :: inf. ar< a : ext. area, or lil.'^.') : iilv!.". :: !()i»l : j; let 'ZiVl : 
 452:: 1021 : .' = 171^ ; finally the required solidily = cxt. area x ?, 1{ - 
 int. areax J /--(iriH x 21.2.")-^ ,*}) — (1021 x l(i.25 -; 3) = 123d2— nS-'W 
 = 6852 onhic feet of cut stone. 
 
 KU.H. I, The a]»pr()xiinrtto riih' in (|iicstion in llie rem. to 
 last ])rohlein, would yive in the actual case i d/I.S r 1021) i 5 = n!>22 
 tliat is an excess of 70 cuhic feet, the error heinir consc(]iiciitly K',, per 
 cent. 
 
 4. A reservoir, the lateral Avail of which is the zone <tf a sphere 
 and flic bottom a ])lane surface, is lined in all its concave suil'aeo 
 ■with a tliickness of einht inches hrick niasomy radiatiii,:;' to llio 
 centre of the sphere of which the reservoir is a sei^ineiit. 'I'lie. supe- 
 rior diamt^ter of the reservoir, which is at the same tim*; that of tho 
 sphere is 100 feet an«l the de])tli of the reservoir or height oftlic zone 
 is 20 feet. Required the number of Inicks in the IVusf iim of u .s]'hc- 
 rical sector formed by the lateral liiiin,^' of the basin ? 
 
 Aim. The. circ. of the int. siihere or of a great circle is 100 < 
 3.141() = 814.1(), this circ. x the height 20 of the interior zone, gives 
 for the area of that zone (j2H;{.2 sfpiare feet, and the solid sector of 
 which that zone is the base or convex ansa is (i28;j.2 x 1^ r =(528;}.2 x 
 50-=- 3=104,720 cubic feet, the i.n-a of the ext. zone of the brick lin- 
 ing is obrained 1074. 2" O.) by making 100 "': 101.\*:: 0283 2 : G451. 
 8687, this latter x \ R or by ,', (101^ - 1()8,!»64.8:)4 cubic feet-- V(d. oi: 
 the ext. solid sector, the ditt'eience 4244.804 of the int. and ext. sec- 
 tors is the solidity of the lining in cubic feet, niiiltii)lying hy 20 we 
 obtain 84,898 for the niunber of bricks emidoyed in the work. 
 
 RE.^. II. In this latter example, the .'Jiini of the ext. and int. 
 parallel areas of the lining is 1273.1.0687, this sum x fhc; half-fhick- 
 ne.ss, 4 inches, or by ;^ of a foot, giv(,'s 4245.022L> cubic feet, x 20 ^ 
 841)00^ bricks, or a dillerenco de 2^ bricks only in the resuU j proving 
 
MEN.SUUATlON OK TFIK SOMDS. 123 
 
 <lins, iiH iiln'iidv stilled, tliiit willi ii viT.v siniill tliirlciicsx rclnfivcly ?;; 
 
 t<» llic lii'liiis, \vr (ililiiiii vciv iiciiily llic sididilv of tlic tViisliiiii oCu 
 
 s|>lirriiiil sictoi-, liv miillipl,\ in;;- its lii'ii^lit l>y Ilic !iiill'-sum of its pii- 
 
 riilltl liiiscs. Iliiwcvcr. with ic'^fiird to iIhi iiinoiml of work (MitailtMl . 
 
 l>y tlic two iiiodrs of ('III CM lilt inn, iIm- so-oikI iiicllind olVcrs no iidvan- ; .' 
 
 tiij^t' over the lii.>l wliicli it, is tlii'iftort' licllcr to employ in all cases. 
 
 l-lvti III. We may also in piaetiec (and this is wliat is soiiui- 
 tim«'s (lone) when llie thickness oi a vault is iiiiirorm and its ladiiis 
 orcmvatiire relati\cly <;reat. simidily the operation and arrive at 
 a siillicieiitly approximate result hy mull iplyintf at onco tho * 
 
 inl. or e.\t. area of the vault Ity its tliiekness. In the last, <'xam]t]« 
 this maniii r d' proceeding- irives, hy usin<; the area of the intnulos of 
 the brick lining-, <>vJS.'{,:j ■• H inches (»r by the '.J of a foot =41HH.H culiic 
 feel < '^D^^J^M/Ttl, this result is delicient by ll*J;i bricks or It per 
 cent, if on I lie conliary wf take the exi. area (i ir)2 x § we have 4301 
 iMiliic feet, or i^fl.OviO bricks, result which is in excess of tho truth by 
 1 l:;2::i bricks or 1 1- iier cent as before. 
 
 PROBLEM XXXVIII. 
 
 To liiid the area of a spherical triangle ^ 
 
 (See the tableau.) 
 
 (105) lfilir..i'l 1. Comjitilc Jirst llie aim of the splicrenf which the 
 iriaixjlc fi>n)is a pari, and diridr the urea hy 6 to obtain (I19!S €}■) 
 that of the Iri-rectanyitlar trianylc. 
 
 Cinnpiitra/'lerirards (ISOO ii.) the Kiim of the three anf/lcfi,from it 
 siihtraet ld()^f/»r/ diriile the remainder hifUO^ ; multipli/ then the quo- 
 tient bij the Iri-reetantjalar triamjlc and the result will be the required 
 area. 
 
 RIJT..E II. 'MnUiphj, as for the triangle with a plane surface, the 
 derelojicd Iciifith of the base hi/ the developed heiijlit perjtendieular to 
 that Ixt.'ie ; the result will bo the area nearly of the 2>>'02}osed trianyle. 
 
 1. We will find among tho models of the tableau, pyramids and frusta of spheri- 
 cal iiyriiinidti tho bnse.s of which preiont tho aoutc-aiigled, right-angled and obtuso- 
 an^lod spherical triangle, inoludin<i tho tri-rootangular spherical triangle in queat^o^, 
 
124 
 
 KIT TO THE TAnrrATT. 
 
 ill, I 
 
 » 
 
 Q 
 
 
 
 El 1. l\i'(|iiii('<l tlic ;ii»'ii (»r;i tiiiiii.i;lc dcsciilM'd on ;i siilicic the 
 dinm(;t«'r of wliidi is ;j() Trfl, tin; nn^jilcs Itciii;;' I K) , i^ ainl iH' ? 
 
 Alls. Tlio iiK'ji of llic entire si)lioro -diiuii. MO x 30 x .7851 x 1 = 
 
 30 X M.inC^aH^r.'Jl sqiian- feet of wliii'li J - H.");}. K} -arcii of llic tri- 
 rectuii/^iilar (i iaiii;lt' wliicli iimsl en (eras an clrnK-nl into 1 !i«' ralcnla- 
 tioii to l)c made. 'I'lic sum oi'iIh^ lliice au.ulfs is .'JOd'', ."iiKI ' — IdO^- 
 120°, ]20°-M)0='-=i;, an<i I', limes tiicarea y.Kl.jaortlic tii-ieetan-ii- 
 lar tiiangle i^ive.s -171.21 tlie k <iuii(d area. 
 
 2. 'I'lie ani^les of .ui e(|uilal(ral spliciical ti ian;;Ie are eacli of 
 120°, and (lie diani. of tln^ sjdu're of wliicli (lie triarinie form.-i a part 
 is 20 metie.s ; what is tlie area of the triangle ! 
 
 Alls. 20 X 3.Ml(i-r 8-^l.")7.()8 - tri-rec't. triangle area, iIkssiimi of 
 the juigles=3(i0'', 8(JU^ — J8()° = J80, 180^ -^-im"=:r2and 157.08 a 2-=:' 
 314.10 square metrcH. 
 
 3. Otieof theScoinpartnientsof the siirfaeeof a <lonie or vault in 
 the form of a hemisphere is an isosceles splu^ical Irianyks of whieh 
 each of the angles at I lit; base is aright angle, and the angli- of which 
 at tlio apex - ;JtiO''-:-8- 45^, the length of the arc; measuring the 
 breadth of tlie eompartinent at liie springing' of the dome is ;{:t.27an(i 
 the w hole circnmference is c<»nse(piently ==y'J.27 x 8--;}l I.IU, wheiiuo 
 tlie diani. is 100; what is the area of the compartment ? 
 
 AiiN. The uhole area of ihe sphere of which the seini-liine to bo 
 
 computed forms a part =100 x 3,] ll()--.'}NI(! square units, tlie tri- 
 rect. triangle -31 ll(!-:-8=^.'3n27, the sum of Ihe angles exceeds by 45° 
 two liglit angles, 45"-f-!X)=-.i, tlierefore the areii or Bolidity=-3U27 
 -i-2=-lUtJ3^ — lecpiiied area. 
 
 Bcsl(l4^!4, in this exain))]o Avhere the triangle to be computed 
 forms a known aliquot ])artof the entire sphere, the calculation be- 
 comes si mplitied amhcdiu'edto that of obtain ing the area of t. he splieie, 
 afterwards to take its Itith part. The examphi has iu'verthel(^.ss the 
 advautage of showing the accuracy of the rul« (the area of the entire 
 
MKNSDRATlOiN OF POLIDS. 
 
 «|)li('n'.'{1 III) (liviilctl l»y III ;;iviiii(;is Ix'Tmc l!t(!.'l', far the ('(tiiv-cx iiic;v 
 of I lie |>r<»|Mtsc(l iiii^iilii) iiMil iiidiciilivs I lie iiiiiiiiitc nf proceed in;;' in ;iiiy 
 otiicr iiiMtlonoiis cusi', 
 
 4. The simi ot'tlic llircc iuinlc^ ot'ii (iiiiii;;I(' Irjit'i'd on tlu' siii'- 
 fiirc of tlic tcri'fsliiil splific, cxccid-i (J BUt <".)'•> oin- second (I") 
 IriO^, wliiit is ils iire;!, siipposiiiL; tlie carlh to lie ii perfecl s|!ii( re willi 
 a (liniiuU r of 71)1^ cii^lisli miles ! 
 
 Alls. TIk^ iuvii of tli(! oiiith^ (7I>1'^) x.'M IKr l!i<;.f;i;:l.:i,M.7.'., 
 dividing' li.v i'*. wf <d»tjiiii foi' I lie iiica of llie (ri-rec(. tiiaiijilc :J l..")."*"^, 
 l»~l>. 17 s(iiiiire miles : now I"-;-!lO' -1" : ;{;! l.dOK \iiiiiid)er of secoiids 
 ill !IO°:^.-{K)=' X CO' .; CO") -r: ,j._, ;,,,,,, ^.()(»()(H):M,-^(;1v> iie.ulv inid (lie iirca 
 
 of llic tii-nrtiiii-iiliir triiiii-le '-il.r.r-^i, !U!».17x .0()nO(;;i((SC|-jo,-, whieji 
 is till! same tldii.n, divide<l liy il;. coiivci'se .'{-JlOOIt -ivcs 7.").,'^7."!;ll for 
 the area ofllie piopo-^id I riaii,!:;le in sqiiaic miles. 
 
 {!!<»«) IRir.TI. g. Il is plain from the iiile thai the area ofcveiy 
 spherical tiiaiinlc oltlii- same ratlins, that is, ofeveiA 1 1 iani^le tiaeed 
 on Ihe same sphere lias a direct lelaiioii to the excess of the sum of its 
 tliree ani^h's on IHO . l"or instance, if the spherical excess wero 
 10 seconds instead of (me. ihi' area of I Ik- triani^le wonld lie 7. Ir. 7. "'.•.' I 
 scpiare miles instead of 7.").i"^7;!"JI : in t!ie same wav if the exi'css of 
 the lIiro(} an;^Ies on IHO ' w a i hut oI'oim' tenth of a second, the, .'ii'Ciiot' 
 the trian,i;'le to 1»e compnted wonld he Iml om- tenth of wlial il is 
 foronc second, \ i/ : 7.r>.-^7;i"JI. An «'Xcessofom^ ndnnle v.ouhl ;;iv<} 
 for the area of I he 1 riari'.le to he c(nn|mt,ed a, nnniliei' of miles (!0 I inns 
 gr(!ater tlnm that uiven h,v ii .sec<md, that is, the .34001 h part oflho 
 tri-veef. Irian-le, since ;i-J 1000 : (iO-.l-lOO or that OO'^ CO -.^lOO ; so 
 1° would .'^ive the fMltii i«art of the, tri-iec(. triaiii^h; and so on ; 
 Avhence it evidently loUows that in every ,i;('odesical survey of 
 part of the terrestrial sphere, il will snltice, alter haviii,<;' estahlished 
 the area cori'esp(m(ling for insiance to a second, oi' to on(; lOth, 
 lOOtii JOOOth, cVc, of a seconil, to ninltiply this area l»y the nninher 
 of tifconds or tenlhs of a see<md, \sc., in t he exce.s.s of t he sum of tin* 
 tlu'oe angles of any trian,i;le on IHO^, to obtain at onc(! the aica of 
 snch tiiangle, and we liave seen (1 J l.l U" i.i.) the manner of estab- 
 lishing if required such spherical cxiuvss. 
 
 KF]rfl. 11. It is plain that if tlie splnuical tiiangle AC'l', for iu 
 stance, is tlio tri-rectangiilar triangle orihe Hih jiart of tin; sphere and 
 it 1)0 divided into any numbeiof I'lpial parts, that is, of isosceles 
 sidierleal triangles (Mfiial to each other, A(JE, Et'E, &c., all the 
 angles at C will be equal to each other and the ares AK, lOE also 
 equal to each other ; the areas of all these triangles will eonsu- 
 queutly bo equal. Let then the angle AUE at the ape:; or 
 
IL'G 
 
 KKV TO TUK TAnLKAU. 
 
 jMili' ('= 1" and Itccausf tlir spiicc nil jintmxl (' i.s dividr.l into 300", 
 (lir nrcii <»t' till- l)i-r('(Mini;liir splicriral 
 isosccli's tiiiiM^lc A('K, oi' dl' w liiili tilt! 
 sides roiiiiiiiNini; llir iiii.iilf of I ' ill llio 
 iipcx or pole (' ;in' i<|iiiil tt» onrli olliii 
 nnil (>:icii ot'tlnni to one (piiirur of a 
 cirt'iimtrrciii'i', iliis iiicji, ^.•l^ we, must A' 
 t'Viilcilll.V I'f llii' "^i'll!! li.lll of 111, it of tlic 
 hcniisplicre AlIi)A -(' or wliidi i.s tlic 
 same thin;;' tlic i'l'tii pari of that of tln< 
 (piai l( r ol'a licniisplifrf oi' tii-icclanuidar 
 trian;;lc. if llic aii,;;l«' at (' is luil 1 or the 
 (JIM !i parr of I -\ tli.' arcaol' the tiian-!<^ ACKwill he hut the (WP < (10)' 
 54()(ltli partot'fliat of the scmi-qiiartcr of the splKii'c. If tlie, an;;Io 
 at I' is hui a second, the same iU'ea will he hut: {\h\ (!HP x (i!)' s liO" 
 ;}v!l.()()()lli |)a.t of thai of ihe liair-(|Uarler of ihe ,s[iiiere, (tr as shcnvu 
 in the ali!i\e example, of 7r).S7.'};JI sipiare ennTish miles ; whence it; 
 is plain, that an aii;;le (' or Al'K of .1" woidil jfive 7J>r^7i'il 
 K(pi;ire miles ; an annle A('K of .til" would <;ive .T.Vv.'i'il of a siiuaro 
 mile ; an aii,i;le AC'li ot .001" an area of .07.>H7;J'.21 of Ji s(pnii»i mih* 
 and so on, or, a-, just said. .O/l-^/iJiJI for eaeli thoiisandtii ofa second. 
 Now I Miiiai-e niiU'---.")'JS(l ■; .VJ>0 eimli>h feet --tJ/.S/,-*,-!!)!) sipiaie feet; 
 and mnllipl\ iuij hy .07.")S7:{'JI we ohtain t2. 1 1 ."),'J'i.'{. 4 I'eet which then 
 coiTcsponds also ti> the area of a spherical Iriannle in which tlie ex- 
 cess of the sum of its an;;!i's on ISO^ is of .001" or of the thousandtli 
 part of a second; whence, if tlie excess it hut of 0001" the corros- 
 pondin,!; area of ihe trian^li! will he 2il.."i>"-i.:U ; if tlie spherical ex- 
 cess is of .00001" or of one hundnul thousandth part ofa second, tlio 
 spherical area will he Itut of '^l,l5"i.xj:{l and linally if the sphe- 
 rical excess is of.OOIIOOl" or of the niilliontli part of a second, 
 the ar<'a of the spherical triauLfle corresi)on(lin'>- to such an excess 
 Avill lie of 21 l.").'i"2:! 1 square feet or of an extent of ground not ex- 
 ceeding a s(iiniie of lii I'eet in tlie side. ^V'hence : 
 
 RL'Ll] I. To (hivn)iiiic the spherical area of antj trianf/Ie <les- 
 cribal on ihe surfarc of lite terrestrial spheroid ' {sphere Jlatlencil at the 
 poies hji iieiirh/ one three hiindreiUh (;y,\,i) of its iliam.) we have but to 
 muUipli/ each millionth ofa second of the excess of tlie sum of its three 
 
 I. As tho are;>s of simil.ir flu:ures ato to ench other .as tho squares of their homo- 
 logous sides, we will arrive nt tho !iroa of a sphorioal triangle desoriboJ on a sphere of 
 any nidiua by making the ro'j^uired iirojiortion. 
 
MRN-TTRATION 01' SOLIDS. 
 
 127 
 
 a»f}}efi o» \fi)'^ prr '2\\'> xijit'irr Jilt, or ritch tliDiisKHiUh of n sermnl l>;/ 
 i?l ir»,'^'J.'{ «7»(//'«' /'cr/ (>*■ /'»/ .(I7.'>."'7;{*21 siiHiirc luilcn, or aocli ,01" dniit- 
 (Irrdllt o/a siruiiil) hi/ ,7')^7'-V2\ xt/iittrr milts, or ctirli A" (tntlli ol' (i 
 Ki'coii(l) l»y 7,'>'-\7'i'i\ siiiiiirc niilrs of (•■•icli 1.' (hi-coikI) /*// /.'..f^?;}'^! 
 nijiiorv ittiirs', rtihiiiiiii lo Untt rffnt llir ijnircin ( \ (iihI iiiiinitrs (') in 
 the i/ircii I'.rccsK nf the mini iij' Ihr time iiinilix of aiicli .i/ilicricitl trhnujle 
 orer vJ rii/iil (iiniltx, into sciaiiils, iimt tiinllijilii njlfrintriln tlii:ir scvomla 
 hij 77>.^7'A'1\ mill llir fhiclloitu ofsct'oiulH iis linx Jiisl l)ccii saiil. 
 
 *iiil C'oa* OTJiC'ly, to ilrtrniiiiir till' siilirrii-id cxcrss of llir sum 
 of llir llirrr mi'ilcs nf oiii/ sjtlirririil IriitiKjIr on 'Z rlijlil (UhjIis, irc miiif 
 dii'i'lc tlir iirrii jirrrioiisli/ ohUiinnl in an opjiro.i'iinalc inunner (I'l/ 
 coiisiihrin;/ [ttiS, II. 2.) ///(' Irnijllis ofllwarrs conslitnlinii its siilca as 
 those of I ir i*iilrs itf <i rrvliliiiral trimuilr) hii 'l'l\7) sijiinrr frvt to ol>- 
 tuin Ihr iiuinlirr oj' inillionlh^ of' ii smnnl iKDII.OlH ") conliiiiinl in sinh 
 ejrcess, or hij '2Al'),'^'2'-i siiiHircfcrf i>r .(t7."»Hr.'{,J| sijimrc niilrs to ob- 
 tain Ihc niinihrr of tlionronillhs of (I srronil (.(HM ') vonlninvd in sold 
 excess, hij .7.")H/;{'.'I siiinirr inlirs fir the IninilreiKhs of a sreonil dH"), 
 liij 7.')T<7''VZ\ siinure miles fir /Ac tciillis i)f' a sccoiid (.1"), liiially by 
 7.'i.^»^7•'{■2l square miles I'lir tlie seeomls {\" > miil Hie sn-tiiuls if reiinirnl 
 reiliieed inio ininiiles hi/ iliriilliii/ hi/ (ill, uml the niiiniles into dri/rees hij 
 (llridln'i hii i'l'), irill still i/ire the sjilirrleid <:rress rniiilrrd, 
 
 IWaM. hi. TIk' spliericul triaii;;le ACIC on wliicli we liavo ar- 
 gued is, as sta.ed lii-redaii.u'alar at A ami K. lliat is, tlu' aii.!;les at A 
 uml Kiiie, aiitlevitleiilly arc, vi;;lit ; wlieiiee it foMows fliat Iho aii.i;l() 
 at C at tlic apex oi at tliepole is the splierieal excess or the (|iiaiitity i)y 
 Avhidi ihc .'5 aii;;lcs exceed 2 ri.iflit aii,nles and in the saau^ way as this 
 spherical excess furnishes the area in liiecase ol'the l)i-recliin<^iihir 
 isosceles triangle, (l*400 G.)so does Unit exi-css allord ihc means of 
 arriving; at tlie reipiired area, or tiic area at tiic reci'.ilred excess in 
 any other spiierical triangle. 
 
 PROBLEM XXXIX 
 To determine the area of a spherical polygon. 
 
 (See the tahlean.) 
 
 {Hi'7) RlfliE. Find as in the last })rolih'm 
 tlicarcaofthctri-rcetamji'lar triiin;/le (1301 CjJ.)- 
 From the sum of all the amjles of the poh/jon 
 subtract asmany times il ri/jht amjles us there arc 
 sides less two. Diride the remainder hij W^ and -nl 
 mulliiihj the tri-reet. triaiujie hi/ the quotient thus 
 obtained : the jjroduct will be ihc required area. 
 
128 
 
 KET TO THE TATILKAU. 
 
 I'Jv. I. AVlinli is Uio iiH'a ofiv nvuiiliii' i)oly,i,Mn o^c•i,^'llt .sides dos- 
 crilted on llic siiifiicc of :i splicrc of wliit'h tin,' iliaiiieter is oO, each 
 ai)gl<'. of tln^ pol.vi^-oii Itciiig \ M° f 
 
 Ajis. ! 40° X 8— I :'10'"^ = Hmii o:' (ln^ !in.:,'l(>s ofllio uoly-^oii, 180° 
 X (',= |0S()^ — as iiiiiny tiii'ics 2 ri,u,iif aii'^lrs as sidivs less two, 11:20— 
 l()S()=4(),40-r-!tO =,; ; the, area of the. iJioposcd poly.^oii will tlioii be 
 the i of Miiit of th • tri-roct. lnaii;j;l(', the ai'i-a of the sphen; - ;M) x 
 30 V ai4l() = 8.141(; X !M)0rr28'i7.1-l wliich -r-8 = 3r»3.-IM area of the tri- 
 reet. tiiaiigle, this latter x 4 -h9 ^ JtJ/.Od the lequiied area, of the 
 polygon. 
 
 2. 1\e()iiired tlu* av(>a of an irregular 
 polygon of 7 sides descrihed on a splicro ^« 
 
 of 8^ metres radius, th<' sum of the angles 
 being IU80-' I 
 
 2 
 
 A US. Area of the sphere^ 17 x .'3.1410 
 
 — ;)07.!f2--*4 of which tlie eiglilh i»a,rtll.:{. 
 4!>0;i is the area of the tri-rei't. triangle, 
 J()80°— .-) times 180"=I80'\ ]80^-f;M)=r2 
 and 1 i:l !;M).'J X 2 -- 2--20.!»8U0 area of tlu pro- 
 posed polygon. 
 
 It. The sum of the Mangles of a polygon of a geodesieal trian- 
 giilatiou is'J;5IO^ I'.jO", what is the. area o(' tiie jjolygon in s(iaaro 
 miles, supposing the diameter of (he eailliai tlie, survey to be 7tMiI 
 - nglish miles, that is supposing- tiie. trigouomet^rical opcu'ation to 
 have taken plaee. on Ji spiiere of that, diaiu(^ter. 
 
 Aais. We obtain us in the la^t probhuu, for tlie area ('ori'espond- 
 diug to au excess of 1", 75.87.'J'-Jl sipiaii; ndles.anl wo luvve seen that 
 the area to be measured is iu direct relation wllli th(> number of 
 units in llu^ given excess ; now ti:;- siiui of the allgh^s is in this (^x- 
 an>ph^ U:3I0- 1' .W wliich diminished by i;l times 180^ or by 2;M0= 
 leaves for the excess l' '>{)" or 110" ; tlie recjuired area will tiicu be 
 110 times 75.874.'5'2l that is 8340. 17.1.'} 1 square nnles. 
 
 (KJH) lil'^M. The supposition just made seems to indicate tliat 
 the eartn is not in all ii.s "xtent of the same curvature, that is of tlio 
 Binnr radius or diameter, or that it is not a perfect sphere, and in 
 fact the terrestrial globe is a s[)heroid of which the tluttening towivrds 
 tlio poles is nearly Tf,\5f .• the diiim. at the e(iuiit< r or about vJ() nules i 
 now th(^ areius of two opheres of ditt'ere'-t radii or of any two homo- 
 logous parts of the sphere.-, are to eacli otlu r (iOTl G.) as the 
 squares of the radii of ;! ose spli.'res. 
 
 Let it bo reqnir 4 tlien t»» »inu the relation of tlie areas of two si- 
 
MENSURATION OP SOLIDS. 
 
 129 
 
 miliir fi'j,nrcs I ivirod on tlio forvestrial s])lieie, one at n point, wliero tlie 
 osculatnry dianiclt'r is 7lM;i, the otiicr in alatitiidi; wlieic lliis diame- 
 ter is 7^:50 miles, avm will make 7m2 : ,',m: I : l.n()4.>r)52; multiplying 
 by tliis last niinihii' (lie H^lO.O.l.'U si|uare miles of the last example, 
 Ave obtain S:{~ 1.071 s<iuare, miles for the U"ea of the same polygon 
 at a iK)int wliere the diameter of the earth ■would Ix^ 7n.']() in.^tead of 
 7!)I"2, riiat is a diiTcrenee of .'38 S(}uare miles, ((uantity whieh though 
 relatively small, eonsiderinp,- tlll^ total area of the extent of territory 
 coi prised in the survey, is none the less very large iu itself, equiva- 
 lent as it is (o Ihat of a eity or canton of (5 miles diameter; which 
 {shows the im[)ortanee of eonsidcring the relative dimensions of each 
 part of llii' terrestrial si)here iu theoperatious to be performed to de- 
 termine its area 
 
 PROBLEM XL. 
 
 To determine the solidity of any spherical pyramid. 
 
 (See the fahJeau.) 
 
 *0 
 
 k.f.-' 
 
 (196) SllFLE. F'nul first hi/ the preceding) rnhnthc area of the 
 hasp of llic ijircii jii/rdmidj multijih/ then {HiS'^ti.) this area hi/ one 
 third the h''i(iht of the pyramid ; that is In/ one third the radius of the 
 sphere of irhich the pi/raniid forms ipari and the result loill be the re- 
 quired S /lid ill/. 
 
 Kx. I. AVIiat is the solidity of ,'-, spherical pyramid the base of 
 ■which is 10 square metres aud the height 30 metres ? 
 
 Alls. 100 cubic metres. 
 2. Among the component parts of a polyhedron to be cubed, is 
 .1 sidierical ])yr;!mid or a jiart e'" a sphere bounded by planes meeting 
 ill the centre oi the sphere of which the py. imid forms a ^'art j what 
 
 5 
 
130 SET TO Till'! TABI.KATT. 
 
 is its solidity, tlio rjidins Ikmuu; !■') iiiclics mid flic nrcii ol' llic (ri;iiiylo 
 or polygon const ihitiiii;' its biisc lUO inelics ' 
 
 -tjis. ;■)()() ciilii^' ini'lics. 
 
 Jl. Tlicro is to bo Diii.dc a v;iiilt or ^vivt of ;i viiult of wliicli tlio 
 radiiiH of tin iiifriidos is ;i() fcot, the dcplli of tlic v.'iiilf.M frcl iiiid its 
 foiiii lliat of an irr(',i;iiiar jKily^on of wliicli the int. area or snpcrli- 
 cics is 100 sqnaro feet ; what is its solidity ? 
 
 Alls. The solid to be conipiitcd is tho finslnniof a si)!icric;i! 
 pyramid witli jfa^'nllc'. bases ; this solidity is eiinal dOS-S ii-) lotiie 
 diiiference of the .solidilii'S of tlie coniiioiient entire aiul |t;irliiil or 
 ext. and int. jtyianiids. We will then obtain for the riMjuired solilidy, 
 the expression u'xt. area x I U) — (int. area :< .', r) ; we have Iherelbre 
 to iind the ext. area whieh must enter int(»flie ealeiiiatioit to bi' mtnle ; 
 to tlsat elfeet we obtain (1071, 2'' «.) .-{l)": '.]:)':: 100 : l'JI=r:;,rea of 
 the extra.los ; now, (1:21 < ll) — (lOOx 10) ■- i:J3l — 1000 -:;:WI cubic 
 feet of nuisonry. 
 
 4. What is the weight of the fi'a,i;inent of a shell or bomb of 
 ■\vhich the int. diam is 10 inches, tiie thickness ."> iin-hes, and ili(>, 
 int. and ext. or concave^ ami convex areas (iOamI i^ 10s(|uare inclies, (he 
 planes of section of the fra.unu'nt bein.i; (liii'cted towards the, ciMitrc 
 of the sphere, of whicii the solid to be nu'i'sared forms a part, and 
 the Aveiyht of the cast-iron beiM,<;' -ISO pounds to the < iibic loot .'' 
 
 Ans. 02JO X IO^;j)- (00 : .") :-3) = 800- I00=::700 cubic indies, 
 the cubic foot— 12 ^ IXJxlO — 172^ cubic inchi's. whence we obtaii* 
 the reipiired weight by malviii;;' 1728 : 4>":^0 :: 700 : U)-li pounds. 
 
 PROBLEM XLI. 
 
 To find the area or solidity of any regular polyhedron. 
 
 (See the ."> regular polyhedrons of th(^ tahlcdH.) 
 
 (lOft) 161'I^B-i 1. E'or Hio ai-c.^ : mlciilaic the urcn oforcoj 
 its component /(KCK, ninl niitlliiili/ (lllH fi.) tlud ana l^tj lite nnmlier 
 o/fiices i)i the proj>ose(l jxilj/lietlnni. 
 
 For the Holiday : Mullljilj/ {IVilli.) the area of the poli/- 
 Jiedroii till the third of the ratJinfi (»P of the iiiserituil sphere, that is In/ 
 the third of the perpendieular tet fall fr:<m Iheeentre to one of the/ares 
 of the solid, the product nill lie llie reijiiiretl siilidili/. 
 
 IlKin. Wehaveseen (;ll3a ;iml 1 BCM €J.) that to determine 
 iij the cas(M)f the Dodecahedron and Icosahedion, the ladius of the 
 inscribed sphere, we mu.st lirst Iind flu; angle formed by two of tUi* 
 
MEVSDrtATrON OP SOI,TD^^. 
 
 131 
 
 ailj;ic(>nfc faces of tlicso Holid.s, uiid wo. liavo indiciilod tlio iiiaimor of 
 establisliiiig tliis aji^lc. Wo may also by meaii.s of tlio same angle, 
 
 oalciilalc (lie pci'ixMKliciilar in eacli of Hit' three otlior polylie- 
 «lr()ii:< (of uliieli l!ial of tlu' liexahi'dion is e([ual to tlio half-.sido 
 oftliatbodyj or obtain tiiat ixipendic-ular by tlie iiietliod of par. 
 (lias «,) or (I i:J1 G.) a.s the case may l)e. 
 
 X ^ 
 
 (I'yO) It is well to calcid;it(^ and dispose umhu- the form of a 
 table, aiJ at (2Y T.) for the iCLjnlar poly.u'ons, the areas and soli- 
 dities of the live polyliedions havini;- for their sides the unit, in order 
 afterwards to mak(! us(^ when reinured oft hose areas and solidities, to 
 determiut! tlie area or the soU<lity of any other regidar polyhedron of 
 the same name. 
 
 Table of the regular polyhedrons of which the side is 1. 
 
 NA;\tK. 
 
 X^ OF FACES. 
 
 ANOI,!'; OF TllK 
 FACES. 
 
 70^ 31' 42" 
 
 !,'0° 
 101)'^ 28' 18" 
 
 110^ :v.]' .-)4" 
 |;J8^ 11' 23" 
 
 AIMCA. 
 
 SOLIDITV. 
 
 Tetrahedron 
 .Hexahedron 
 Octaiiedron 
 
 i)ii(l('ciilic(li(in 
 Icosahedr(tn 
 
 \ 
 
 
 
 s 
 
 12 
 
 20 
 
 1 .7;>2or)08 
 
 (>. 110000(10 
 
 .'i.llillOlO 
 
 20.(i4.')7288 
 
 ^.0002.j40 
 
 O.I 178.-) 13 
 1 .0000000 
 0.4714045 
 7.0:i3ll8!l 
 2.1810050 
 
 (111) RULE II. 1° r<U' Uie Jii-ca : Siimtrc the side of the 
 given poli/hcdyoii and mn1tii)li/ thin siinarv be the area of the pohj- 
 hcdrou of the name name of which the side is 1. 
 
 MM 
 
132 
 
 KKY TO THE TABLEAU. 
 
 For tlie smfiirrs of similar i)()l,vlic(lr()iH nrc ruiii|)osc(l of a like 
 number of similar poly. nous, aii'i tlic arras of liicsc polyidos or llicir 
 smns arc to each other (550, CJ.) as tlu; squares of thtir lioniolo^ous 
 sides. 
 
 oo Foi- j|j(^ solidity : eitlic the side i>/' tJic (jlvcn poli/hcilrou and 
 mnHij/h/ lliat vnhc bij llic nulidili/ of lite ih>lijhedivn of the siDiic xanie 
 of wliicJi I lie side is 1. 
 
 For, simihir polyhedrons arc composed of a lik<' mimher of siiiii- 
 larpyramids and the solidities of tliese pyramids or their sums are to 
 each other (1070 CJ.) iis the cubes of their liomologous sides. 
 
 S^ (U 
 
 Ex. 1. Wbat is the area of a tetrahedron the side of whieh is 12 ? 
 
 Ans. VI X 12 X 1.7;i;2()iJl)8=24L».4ir);3J52. 
 
 ft. The area of a Lexalicdrou or cube the side of which is .30 ? 
 
 Amjj. .1400. 
 
 3. llequired the area of au octahedron the eido of whieh is 10 ? 
 
 Ans. 10 X !0 X 3.4G41010-y4(J.4l010. 
 
 4. Determine the area of a dodecahedron tlio sideof which '\sSl 
 
 Ans. 33 X 20.(i4.-)728d:= 1 8.'>.8 1 1 .")92. 
 
 5. "What is the area of an ico>ahedrori the side -^vhich is 20 ? 
 
 Ans. 8.G(;02.54x20^ = 34G4 1016. 
 
MENSITIIATION OF SOLIDS. 
 
 133 
 
 6. What is tlie solidity ofa U'lriilu-dioii llic si<l(' of \vlii<'li is !.">? 
 
 .iHs. !.">' ; 0.1 I7,s-)|:}=:;}:)7.7J8. 
 
 7. Tin; solidity of a cube llii- side of wliicii is I'-J ? 
 
 S. IFtlic sido of ill! octalicdioii is 10, wliat is i(s solidity ? 
 
 A US. 171.-1015. 
 
 1>. Till! side ol'ii d()d<H'aln'dr<)ii is 2 ; wlial- is its solidity ? 
 
 Aits. Iil.:i0l!i.'>l2. 
 lO. AYliatis lliesuliiiily ot'aii icosaliedioii tiu! .^idc ofwliicli is '20 ? 
 
 4j «. J7-ij;j.j(). 
 
 11. A iiioiiiiiiuMit lias lu'oii tcrniiii.'itt'd by a ball or a ofowiiiiitr 
 cut stoiK! having" the roiiu of a dodccalicdioii of wlucli tin; ('d,!4(M)r 
 side nieasmes I'il inches : ro(|iiircd llic solidity ortlic block ol'stoiie 
 iu cubic fool, and its area in s<iiiaie feet '? 
 
 Ans. 'I'lic area = l;J..>x i;j..")x20.(J4.")7i28S = :{7<i-2.(i"'l07:;>i siiuaro 
 iuclu's. Wc woul(UM[Ually obtain this aica without usin;j,' that of the ta- 
 bic by c()nii)iitini^' scpaiatcly liy the iiielhod of par. (28, T.) tho 
 jii'ea ofoiie of thecoinponeiit i>oly!.i;oiis and niulliplyiiii;' afterwards by 
 I'-l the element thus obtained. Thus the iirea of a pentagon (he side of 
 wliicliis 1— 1.7ii04771, multiplying by 1 8iJ.2.) (square of the, niveii side) 
 we obtain for theaiea of one of the faces of the pro[)osrd polylx-droii 
 3i;J.r).')7(MH)l.") siptaie inches ; then, muitiplyinn' by \2 (number of 
 faces of the dodecahedron) wo obtain as before 37(i'J.(idl07.'!d .■<(|uaro 
 incites, w ich proves also tlie accuracy of tlio tabular niulliplier. 
 Now we 1... %e but to divide the number of inches just found by 14-4 
 (the square inches in a square foot) to obtained square feet lrf.tJ84 
 square inches, the re(iaired area. 
 
 Alls. The solidity = 13.r) X 13..> x i:j..-) or (la..!) or x!(il0.rj75 x 
 7.(i:3;311,>0:=:187eo..'3:nt) citbic inches, dividing by 1728 (jinmber of 
 cubic inches iu a cubic foot) we obtain 10.87 cubic inches nearly. 
 
i:^4 
 
 KEY TO TIIK T.\Br,KAU. 
 
 PROBLEM XLII. 
 
 Being giv^en the diameter of a sphere, to find the side of 
 any of the rcgulal- polyhedrons, which may be in- 
 scribed in the sphere, circumseribod about the 
 spliere, or v/hich is equal to the sphere. 
 
 (Sec tlic .'» rc.n'iil.ir polvlicdioiis ofilic luhlcdn.) 
 
 (172) HULU- Jfidlipl'i llic ijircii iliiiDicIrr !>!/ Ilic nintihrr irlilrji^ 
 i)i lln' Ihlloiriii!/ tiililr, <iii'<irrrn h> llic ijur^llint. <niil llic pnxliicl irill be 
 i/iv ulilv of I lie fniiiireil jxihjJiednui. 
 
 !l siiniics IVoiK wiiiit IiJis jilrculy ix'cii s.iid conccriiinn- flic regu- 
 lar polylu'droiis (jiancs \'l'A to A'l7) lo show iiuiiiciliatrly liow tiiis 
 table lias hccii calciilalcd. 
 
 77;.' illiniidi'r I Ciijuihle of 
 of tift[ihrre.lii:in(j \\hcii)(i ■iiiserihed in 
 the. -side of (I j (he xplicie, «s 
 
 Ciijiiitile i\l' liiitKj' hjiiiiiil ill Kuli- 
 cireiniiserihril \dilij lit that of the 
 
 'J'('tralirdn>ii....i (I.HKillHH; 
 
 iii-xaiirdioii ! (i.:.;;;r)u:{ 
 
 nctaliiMl.oii I (».7o7l()(iS 
 
 Dodf'calKdroii. . (l.ood.-^-i'JI 
 
 Icosalicdroii ()..VJ.")7;)li:> 
 
 (i!)iint Ihesjiltere, in 
 
 ^;.-ll!t|S!t7 
 l.lllJ(IIMilii) 
 1.'J'JI7II7 
 ().llil(l:i7!» 
 (•.(Jiilodl.") 
 
 sjilicre, is 
 
 l.*i|:i;M-'() 
 
 i.o:r)(;;j()i) 
 
 ()..|()SS|!I() 
 
 S;\. Ii is r('(|uii"d to locast iiillir form of a ])i'rf('ct ciibc of 
 <Miiial snlidily. a caniioii hall ofwliicli ilic diaiiK'Icr is 10 inches; 
 what will lie (he length of ihc si(h of the i('i[iiiicd hcNahcdroii I 
 
 Asis. O.HO.VJlt.KS X 1()-H.().')i'l»:.8 iudica. 
 
 '2. Uy liow iiiiich will the W( ii;ht of a. sloiie. sjuhcre o fi'ct dianie- 
 tcr lie diiiiiiii>!icd, \t\ r'cdiiciii;;' it the ^icaJest re.!;ulav ])olyhcdroii 
 of ^t) sides tiiat may be ,L;ot out of it. the wci;;ht of liie wtoiio bt.-iiig 
 siipitosed c([iial to 1.1(1 iioiiiids per cubic foot ' 
 
 Ans. The solidity of the Ji;iveii splieve^o x ,5"2."5n=:()5.4.5 cul)ic 
 feet, or (!.)..")."» n I.")0=:!)S j7i pounds in weiglit. The side of the ic- 
 (piircd icosaliedron will be, from tin? vnle, ()..")'2.')7;5()r) x i')=2.t!28(!.')-15 ; 
 eubini;- this laKer ir.imber, Ave obiaiii Id.lb.'} and miiliijilyiiii;' tins 
 cube by (lie solidity '■J.l^Ui!!.") of the polylunlion of the same iiaiue the 
 8id(M)f which is I, we obtain for t!ie solidity of the sphere reduced 
 to !in icosaliedron .'}!». (!•,'() cubic feet or iJl'.ti-tJ < l."i() = .")ltl.M.!» pounds 
 iu weight J tlie ditteiuuce 3d73.G poundfi is the requiiod weight. 
 
MENSURATION 01' SOLIDS. 135 
 
 PROBLEM XLIII. 
 
 Being given the side of one of the five regular polyhe- 
 drons, to find the diameter of a sphere that may- 
 be inscribed in the polyhedron, circum- 
 acribed about the polyhedron or 
 equal to it in solidity. 
 (l^co tlic .") ic^'iiliii' polyhedrons ol' llio li(hlcati.) 
 (IIU) -I'S'S^S^ M((l:r the Jhlloii'luii /irojxtiiioii : As the irspiciire 
 number (if tlir aharc lalilr, innlrr llic lilli: " iiisciihed," " circniiiscrihcd,'^ 
 " o/iKil." is h) I. so in the si'lr of llir ijircn jioli/ln Jroii lo iJic (litniictcr of 
 ilic iiiscrihrjJ, circuiitserihrtl orequdl sphere, as lh<: case, nuiij be. 
 
 In ollirr words : iis ilw side of tlic iiiscrihcd, (■irciiniscriltcd or ('(innl 
 polylicdroii Cjis \\w cmsc may be) "'"tlic lahlc, is to tlic diiinutcr 1 of 
 its iiiscrilx'd, circiniiscrihcd or <'(|UiiI siilicrc, so is \\\v side oCtlir .i;iv('ii 
 ])olylicdroii to liu' diiiiiu'tcr v)!' its iiisi'rilit'd, fircumscrilicd or <'(iual 
 sphere. 
 
 Silx. J. 'i'lie side of an ieosaliedron is 0.ti-^Hiir>, it is re(jnired to 
 reduce it to a sjthei'c of the i;'reatest possii>lc diaiiietci', wliat will 
 its diaiiieler lie ? 
 
 Ans. .ii(;i,')^ IT) : I :; 2.(iQ.'^rir» : ;5 07M, nearly the i^ccpiircd s(didi(y. 
 The area of the i;iven icosahedron is ('iSi T. ) 2. <)">'(!.") - •JiJ'.'.'^d.') x 
 .l:i;!(ll:i7x20-r)f).S!!'>j,-,r). Ods area x:!.!>7:i-:-(i Oliat is hy (he sixth 
 part of the diiniieler or \\\v tiiird ol't he i ad ins oi' the inserilied sphere) 
 iiives for the solidity of tlic icosahedron :!!>.( i"2.')l» cnltic feet or '.V,)Xrl{\^ 
 as in example '2 of \\\v pn cediiijj,' problem, each of the two resiiUs 
 heinn' in this way a veriiicafiim of Ihe accuracy of the other and at 
 8aine time a ])roof of the ace n racy of tie' fictors of t lie lal)le. 
 
 JJ. I!ei|nired the diameter of a cannon hall lliat can lie obtained 
 l>y recaslinin' a mass of iron under tiie f-irin of an ociahedron I','. 
 indies side, V 
 
 4ns. I. {»:],")( !:5 : I :: VI: ILar-Tir*, that is, (he diam. of the bail 
 A\ ill be 1 I.C inches nearly. 
 
 PROBLEM XLIV. 
 
 To find the solidity of any spheroid. 
 
 (See on the lableau.ilu' that lened.ehm^aled and .'iaxed spher'oids.) 
 
 (17-1). BIl'ff.K S. Miilliiilii ihe Ji.red (i.ris hji ihc si/iutrc of the rc- 
 rohiiKj ((.lis' [or hij Ihc rcet(())(jle or jiroduct of the tiro ((.res of rerohdion, os 
 the case (mty he) (in' tite iroduct Inj ..""J;}!; ; llie result will b(j the rc(piirc(l 
 solidify. 
 
 
130 
 
 KIY TO TIIK TAIU.KAU. 
 
 ie!:i^I. It is plain lliiil tliis iiilo 
 is in tvi'iy Wiiy jin;il(),!;(>iis to llic oiui 
 j;iv»'ii (BOSW <«,) I'or csiahJisliin;.;' Ilic, 
 Holidily ol' ;i spill IT ; ;ni(I in lad, llio 
 Bjilicniid. liKc ili(> snlicic, is ciiinil t<» 
 llic .; ii;' its circniiiscrilu'd c.vlindc)' : lor 
 it is (I('ni(>n-;tr.il(il in '• cnnii's'' IJial if 
 \\v liavi' \o : r/( » in llu' ellipsis :: A(.' : oi) 
 in tlicciiclc ; we \\ ill also olilain ociiK' 
 in tliccHip'-i ■ :: c : <)(' in ihc cirtlc ; tlicncc, siiu'c v:v may (lOOll O.) 
 con side r llu pi' . c and Ilic splici'did as carli (•(uu|K>scd ol' an inlinily 
 of thin sli<'<'s or .^n per posed smlaees <j,('nciaUd liy i lie re vol n I ion oft ho 
 Hann^ iniinher of onlinales oc ))erpendii'nlar to ihe lived axis Ali of 
 the two solids, an<l that tliese eoniponenl. sniiaees are (o each otiior 
 as the sipiares of Ihe _;■(' n e ra i i n ;j,- ladii, i; is e\ident IhaL the. two 
 solids will also he to each other as tin,' sipn;res (JOl^ii.) of ijutso 
 orc'inales, or which is the sanu' thin,!;, as ihe anas of Ihe liases or 
 coiresitoiidiiiH- se<Mions of the cylinders of eqinil height AO cir- 
 (.'uniscrihcd alionl those solids. 
 
 What W(^ have Just said of ihe elon,!;-afed (U- i)i()late si)heroid AIJ 
 i'.nd of its cireani.-ciihed sphere, is (Mpially undeislood of ihe llat- 
 tened or ohlate spheroid (!1) and its inscrihed sj>here, I'or, whatever 
 tlie relation ol"())i; to niC, may bii in each of these, two last solids, we 
 Aviil have between <(jji and AO of the oniMiie same relation as be- 
 tween itin a-id A() of liie other. 
 
 (STrj) KSJJf^K IJ. ]\[iillii,l!i (127 T.) I ///»« Ihratra of a,n/ sec- 
 tion (A!), CD, (HI, «Jcc.) i)((sxin(/ lhr(»i(jli tlicci.-ntre (O) <if Ihc KjJin-oidj 
 hy } (if lite pcrpotdiciihir hc'ujht (CD, Al! «/■ KT, v\ c.) of lite .solid cor- 
 resjiDitdiiu/ to siicli feetinii. 
 
 For, ill lli<? Jirs* plsice, in 
 the c.'ise of the s]dieroid j^cnt!- 
 liited by the vevolnlion of Ihe 
 semi-elliitsis ACJJ abont its 
 axis AI5, the factors in Ihe two 
 rules are rednced to the same. 
 In fact the tirst rnhi gives for 
 the solidity Al. x CDx CD k 
 .r)'^.'](J and the second rule gives 
 CDx CD x.7SrA x4 K } AB ; 
 iftiu'se expressions are e([ual or equivalent, we must get (neglect- 
 ing the factnis .\!',, CD, common to Ihe two foiinulas) .78.")4 x 4x J :r; 
 .0-2M ; or .7oJ4x4=a.l41ti-H(j=,52.{i; ; therefore &c. 
 
MKNSURATION OP SOLIDS. 137 
 
 8»» IhcNorond plaro, I'Ih' section Ali (il'llic saincsplioroid ia 
 an «'lli|)>ii.s ('(|iiiil ill cvcrvtliin^- l(t the cilipsiH ACitI) and its area is 
 ('»•(( T.)=:A15 X CD X .7ri.'j4 ; if tlio second rule Ik^ e(»iri>i'.t, wc will then 
 nl.liiin AU - (!I)x .7H.-)1 xj X ,\ (;i)::r:A!$ vCl) >: CI) x .re'Jd ; and in 
 i'iU't by eliniiniilin^ Hie Ciietois AH, CD and CD coniiiioii to tiio two 
 e.\i)re>*sions, vliere still leiiiiiins ./H.")! x 4 x J = .,W.*J() ; tlit'r<'f«)ie, &,c. 
 
 Ill lli4! third plHco, it is to be denioiistrat(Ml that 4 area section 
 OH X 1 Kl'isstill «"(iiial to CD x AI$ x ..W;^); now, the " conies "show 
 that w iiiitever Miii.v Im- tlu^ axes oiconjiij-iite ' diameters Gil, KFinado 
 use of, the |)aralleIo,L;r;inis cireiiiiiscrilx^d to the ellipsis aiidof which 
 the sides are jiarallel to these conjugate axes, an^ all eciiial in area to 
 the rectaii;;le AlJxCl) ; hilt (S, T,) the area of tlie paiall(^llo<;iam 
 li.ivini; for its sides VAl, KF is (ill x KF x iiat. sin, an^de EOC or EFP 
 = (illl X EF. 'J'lie area of theseetion (JJl = (for any s<'ction of a sphe- 
 roid is J n ellii)sis) CIl x CD x .78.14 and we have Just seen that (ill 
 X EF = A 11 X CD ; tlienslore Gil x CD x .78r)4 x 4 x J EF = AH x CD x 
 CD X .52:{(!, CD heiiiy; conmion to both formulas, Ali x CD = Gil x 
 EFaud .78.") I x 4 x ,'; — .rr-i.'JiJ ; therefore, &,c. 
 
 KKiTI. Tn the case of the flattened spheroid j^enerated bj' tlio 
 vevolnlion of the senii-ellii»sis DAC round the axis CD, the proof is 
 niialo^';oiis to tli.; one* just ^dven. 
 
 *" Ex. 1. ^^'hat is tlio solidity of an ellipsoid of which thcaxis of re- 
 volution is (!(), and tiie lixed axis 80 ? 
 
 Ans. uO x (!0= 3(JU0,:j(J(J() x 80 =- 288000,288000 x .5230 =150700.8 
 units of solidity. 
 
 3. With the same data, what will bo the solidity of the flattened 
 spheroid! ,4un. 80x80 = 0100,0100x00=384000,384000 x .5230 = 
 20J 002.4 units of solidity. 
 
 3. A prolate spheroid has for its axis 100 and 200 j what is its so- 
 lidity ? 
 
 Ans. 1 ntfx 200 X ..'3230= 1 ,047,200 = the required solidity. Now 
 let EF in this exanijile any dianieter= 100, we, will obtain its conju- 
 gate Gil = V AJ^M^C D '^ — EF~( for it is deni'.nstrated in " conies " 
 that the sum of the squares of any pair of eonjii<fate diameters is 
 e(iual to the sum of the sipiares of the niajoi ,ind minor axes) = 
 14i).8i;j22, 4areaGiI = Gllx CD x .78.")4 x 4 = 47005.3212. Since All. 
 CD = EF.G11 X nat. sin. EOG, wo obtain nat. sin. EOG=AB.CD-^EF. 
 Gll=20000-r 24800=.8042141 = 53° 32',aud. 8042141 x 100=EP= 133. 
 
 I. The diam. Gil, conjugate of KF, is tho one parallel to the tangent PF to the 
 ellipsis at tho point F, where tho diametor EF meets the curve. 
 
 16 
 
138 
 
 KKY TO THE TABLKATI. 
 
 49n,'54, and l70(M.;Wli,' x I.-IU.-IDO.-)!^ 0=1,0 l7,inn.R, tlio (liircrcncp 
 .2 between tlie two results beii)/^ due to tlio dei'luials neglected in 
 the cidciilatioii. 
 
 4. Ittlu' two iixcH oftlic cnrtli are to eacli other iiR Mill and MO'J, 
 what will Ite the solidity ol" ilu' siilicroid (it is llatteiied, tlie polar 
 diain. IxMng less than tlie eiiiiatorial diani.) and by how much will 
 tliis solidity diller Croni that of a sphere on the great axis. 
 
 Amn. 'I'he solidity ofthe spheroid = MOI x ;J()4 x .'JtCJ x Syim = 
 imMf<72.li:V2!^ \h(' solidity ofa sphere on tlie greater fixis=^ 1171()^J(]1 
 .3504 aud the dill'erenee of these solidities is 4r<;ic!J.017(J. 
 
 PROBLEM XLV. 
 
 To determine the solidity of any segment of a sphere 
 
 or spheroid Avith a single base or of any frustum 
 
 with tAvo parallel bases, perpendicular 
 
 or not to the axes of the solid. 
 
 (See segments and solid zones of tnhlcan.) 
 
 (170). RIIIjE. To the arcn of the base of the scfimenf or to the 
 sum (if the (treas of the l)((ses of the fruntiim, add 4 tlmei^ the area ofa 
 section half-way between the Ixtue and a}}ex or l)etwcen the parallellxiseit, 
 as the case maijbe, and miilliiil// thewholeby J of the hehjht ofthesolid : 
 the produet will be tlie required noliditij. ^ 
 
 In the first place, as to what con- 
 cerns the semi-spheroid (of w hioh we may 
 besides obtain the solidity by eom]»iiting 
 that of the entire splieioid and taking the 
 lialf thereof) we Inive seen ( 174 T. ) 
 that area section cd : aiea section CD in 
 the spheroid :: area section cd : area section 
 CD in the sphere ; now it has been demons- 
 trated (131, T.) that in the spheie, area 
 
 cd half-way between A and = J area CI3 ; therefore also in the sphe- 
 roid, area c(i!=:| area CD ; therefore area CD + 4 area c(7=4 area 
 CD, and by the last problem, solidity ACD = 4 area CDxJ AO ; 
 therefore solidity ACI) = (area CD ■)- 4 area t'(?) x I AO. 
 
 Wow, for the semi-8i)heroid of which the base AH is an. ellipsis 
 = ACBD and section ab also an ellipsis sinnlar to the base 
 (for all parallel sections whatever of the spheroid are similar 
 ellipses) we again obtain area ab ; area AB :: area ab : area A15 in the 
 the sphere ; for as ah : AB :: ab : AB in both solids and (lOl. G.) 
 
 1. The figure of this paragrajih represents any segment ACB or nC/> or cCiZofa 
 sphere or spheroid, or any frustum AB — ah or ah—cd, or AB — cd of a sphore or sphe- 
 roid ; but the demonstration talies [laco by the iiguio at tho top of page 136. 
 
MENSURATrON OF SOLinS. 
 
 139 
 
 ah : AIJ :: (th : Al! in luitli Kolids, and the nrciiH of siiiiil;ir oIlipHos 
 as of nil oilier .similiir li,;,Mii('M, an- to cacli oilier as the Hiiuarcs of Hieir 
 diaiiictcrs or oIIm r lioiiKthi^oii.-, lines ; tlniret'oro area ellipsis afr= J 
 area ellipsis AM ; now, solidily Iiali'-splKiroid ACI5==liy tlio last pvo- 
 bleiu I area Al> x ,'f CO ; therefore also I liesanie soli(lity = (area AB -(- 
 4 area ah) :■: ^ CO. 
 
 llEiYI. I. It is also a property of th« ellipsis that every diam. 
 EF of that ii.i;iire l»is(!ct8 every chord or doiilde-ordiiiate fjh paralhd to 
 the coiijiififalc <liaineler Cill, which ;j;ive8 nh=ii>j and wo diuiions- 
 trat(^ that in the same way that we olttain ((•(•iiics) AV> : CD :: ^JAo.oH : 
 or, and CD : Al> :: VC'»t. m D : ml>, so wv also obtain EF : Gil :: 
 VEh.. hF : iih and consetiiiently that uh : Olf :: mh : OIJ :: oo 
 : OC when On, Oni and Oo hav(! to OF, OC and OA or to iiF, 
 mC Sc, (>\ the same relation. We then obtain area section ;ili = i area 
 section Oil and as it is already denionstratiul that soli<lity seiui- 
 si»lieroid OFIl = i (4 area C4i x ,\ El*) we will also obtaiu solidity 
 GFII=(area Gil )- 4 area yh) x I Op or by J, EP~ 2. 
 
 (ITT) III the second plaee, as to any segment of a spheroid 
 otluM' than the semi-spheroid, it will siillici^ after what we have just 
 .said and the demonstiation given at par. (lJt5 T.) of the accuracy of 
 tlic rule in (he case of any segment of a sphere, to show its e(iual 
 accuracy in tin; actual case ; which will also dispense from adding 
 unnecessarily to the alniady voluminous dimensions of this treatise. 
 
 E\. 1. What is the solidity of a seg- 
 ment M\A of a spheroid with one base MX 
 perpendicular to the lixed axis AB, the 
 height Ao of the segment being 10 units 
 and the lengths of the axes AB = 100, CD 
 = ()0 'I 
 
 Ans. ATI : CD :: VAo. oB : oM, whence 
 o^I-lS and iMN-=3(i ; rm=Ar.rh x CD -f- AB = 13.()/<)()98r) and mn = 
 2G.l-md7 ; area MX + 4 area jh» = (MN -f- 4 mn ) x .7854, Mx' = 129G, 
 ?>()('= 684 very nearly, (129(; + 4 tim(;sG34) x .78r)4 = 31()().7328, multi- 
 plying by ^ Ao, or by J 10, we obtain 5277.888 units of solidity iu 
 the proposed segment. 
 
 2. Required the solidity of a segment MNB of a spheroid by a 
 plane MX perpendicular to the fixed axis A15, oB being=t)0 and AB, 
 CD 1(10 and (!0 respectively ? 
 
 Ans. If ))('*•' is iu)t given we find it=Ar'.r'B x CD-^AB (since 
 AB : CD :: VAr'./'B : rm' or 100 : (jO :: V55 x 45 : r'»t') =29.8490208 or 
 
J^.'.W 
 
 IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 
 
 1.0 
 
 I.I 
 
 11.25 
 
 1^12^ 125 
 U£ I2£ 112.2 
 
 2.0 
 
 ;!f m 
 
 U 11.6 
 
 ^ 
 
 ^%. 
 
 /: 
 
 
 
 
 /A 
 
 


 140 
 
 KEY TO THE TABLEAU. 
 
 r 
 
 
 "liil 
 
 m'H'= 59.6992410, ^FN t- 4m'ii' x .7854 x ' «>B=soli<lUy ^iX15 --■ 18;J2I8. 
 112; tlie sum 188,49(> of tlieso solidities in the solidify of the entire 
 splieroid ACHD, for (171 T.) GO x (50 t- 3(MJ0,:}(i()0 x 100=;5(iO,()()0 -.unl 
 3t)0,(K)0 X .5230=188,49()^ which also proves the iicciiiacy of the rulo 
 of this problem, 
 
 REiH. II. In the two htst examples wo have suppos<Hl the 
 axes AB and CD to be kiiowu : but that kiiowled;^-e is in no way es- 
 sential, since the intermediate diameters hj», m'li' are considered 
 known or besides since they may be directly obtiiined by niea- 
 Buring, in practice, those diameters ; and it is one of tiie advantiii^cs 
 of the rnle of tliis problem, tliat it d«)es not riMpiire our knowing 
 what spheroid the segment to be measured belongs to. 
 
 3. A segment MXC of a spheroid by a 
 plane MX perpendicular to the axis of re- 
 volution CD, and of which the bas^e is con- 
 sequently an ellii)sis, has for its height oC 
 12 units, the axes AB, CD being respective- 
 ly 100 and GO : what is tlie solid content of 
 the segment ? 
 
 Ans. CD : AB :: -JCa'aJ) : oM or GO : 100 :: V48 x J:2~: 40, and be- 
 cnnse the parallel sections MX^, AB are similar, we have AB : CD :: 
 MX :p(7 conjugate diameter of the elliptical base MX of the given 
 segment : tliereforejj^ = <*" ^ 80^100=48 and area Vp X'r/ = 80 x 48 x 
 .7854; since t>C= 12 we obtain r C=Gaiid /-D — .'54,GO : 100 :: V oTTlT; 
 30= nn, the conjugate diam. of nn=18 (for 100 : (50 :: '.i\) : 18) and 
 the area of the secticm mn=60 x 3(i < .7854 ; that admitted, we obtain 
 
 2 'Z 
 
 Bclidity lVIXC = (area MX + 4 area mn) x '^ oC = MX + 4 mn x .7854 
 X 2=19(503.584 units of solidity. 
 
 4. What is the solidity of the other segment of the same si)he- 
 roid? 
 
 Ans. Wehaver'D=oD— ioC=24, and >''C = 3G, whence we ob- 
 tain as before m' w' =-.97 .9796 ; the other diameter or axis of the ellip- 
 sis ?»iM=58.78776 ; whence area «i'ii'=4.");7;}.904 and area MX i 4 area 
 m'n'~2111 1.5.52, this simi x ^ 48 or by 8= IG88!»2.41G the required 
 solidity. 
 
 The two segments united give 188,496 which is in fact the soli- 
 dity of the entire spheroid as it has been seen at the 2d example. 
 
 5. What is the so^:,iity of a central frnstnm 
 AD of a spheroid the i)avallel buses of which 
 are equal circles of 40 inches diameter, the 
 greatest diameter of the frustum =50 inches 
 and the height or distance between the parallel 
 bases 18 inches ? 
 
MENSURATION OP SOLIDS. 
 
 141 
 
 All*, (area AlUaroa CD + 4 area EF) x J = OP=(40^-l 4(Uor 
 twice 10 ) + 4 times 50' ) x .7854 x 3 = 31101.84 (Miljic iiiclies or 18 cu- 
 bic fcit iicnrly. 
 
 «. The rospoctivo diamotcrs of tlio laiallcl bases of tlio fnistum of 
 a spheroid are 10 and 20, tlie diameter of a section equidistant to tiiose 
 bases is 30 and tlie lieiglit of tiie frustum is 40 : Avikat is its solidity ? 
 
 Ann. (10" ; 2o' i 4 limes 3o' ) x .7854 x 40 -f- = 3220.14 < 404-0 
 =64402.8 cubic inches. 
 
 7. One of the component parts of a cnl-de-lamp bearinj^ 
 against a w;dl, ])rese«ts the form of the sei)ii-sr'4;ment or frnstum of a 
 sjdieroid witli e!iii)tical i '1(4 bjises. The dinmcn r.", of tlie ellipses 
 or rather of llie inf. and t .^ semi-ellips(>s measure respectively 30 
 and 31) inches, the intermediate diameter is 30 and tlie tliree conju- 
 gate seini-diameters or projections of the ciil-de-lamp measnre 10, 
 13 .Mild 12 inches, tlie heiybt of the frustum is IS inclu's ; uliat is its 
 solidity ? 
 
 Alls. (30 >. iO + 30 X 13 f 4 times 30 x 12) x .7854 x 3 = 5972!).07 
 cubic inclies or 3.4 cubic feet lu-arly. 
 
 8. It is desired to kncMV iiow many <>-allons there •!r»'>(231 cubic 
 inches to a .^;illon) in a cask of wine tlu^ len,i;th of which is 40 inches 
 and the dianu-ters at the centie and at each end 32 and 24 inches ? 
 
 Ans. (twice 24" -I 4 times 32" x .78.54 x ■!() : = 27478.5 cubic 
 inclies, dividing;' by 231 we obtain 119 f;all()ns minus a lialf pint nearly. 
 
 O. In an inclined vessel, the form of whicli seems to be that ofa 
 semi-spheroid, is found a (pianlity of li(iiu)r, tlu^ >i,ieatest depih of 
 the li([uor is 15 inches, the respective diameters of its ellipti»'al area 
 are 48 and .30 inclii>s and the corresi>i)n(lin,n- diameters of the inter- 
 mediate paralh4 ellipsis between the surface and the bottom are 30 
 and 221 inches ; what is the quantity of litjuor in the vessel ? 
 
 Aus. (48x304-4 times 30 x 22.5) x .78.")4x 2.5 = 8t);)4.378 cubic 
 inches, say 37^ gallons nearly. 
 
 PROBLEM XL VI. 
 
 To determine the solidity of the frustum of a spheroid 
 •with non parallel bases. '— (See I'rnsta of tableau.) 
 
 1. Wo will obtain with accuracy, as in tho case of tho 
 Bphere, tho solidity of any unj;ula ADli.VI A <>( a flattonod or 
 elongated sjilioroid or of a S|iher(iid with 3 axes, provided 
 the common intersection D.M of the contdiniiig or liiuitini» 
 planus DA.\5, DB.M passes in any direction througH tho centre 
 of tho folid of which tho unguis fi^rms put; and if tho 
 edge D.M of tlie iiiii;ula does nut pa*s lhrou;;h tho centru (t of 
 the ^[lheloid, we will uone th'j iers obtain very nearly the soli- 
 
142 
 
 KEY TO THE TABLEAU. 
 
 (1'9'8) RIFIil-2. Con\mdc the soUditij of the segment of a spheroid 
 with a sintfle base of which the (jiveu ffuslum forms part, cahulate also 
 the soliditjf of the ticgment whieh is wanting to given frustum to 
 comph'tc the sctpiicnt ; the difference of these solidities will be that of 
 the proposed friislii m. 
 
 Ex.. 1. Lot it bo requirfdlofiiuT tlie solidity 
 of tlie part CDrtc ofa spIiiToid coin prised between 
 a plane CI) passing tliroiigh the centre perpen- 
 dicularly to All and any other plant; ea not pa- 
 rallel to the first. 
 
 Ans. We must for this purpose determine 
 the unknown axis ABol'tliesplieioid ofwiiich the 
 hi'ight AO of the segni(>nt C"I)A foiins part. Hav- 
 ing measured any ordinate ab and th(i abstiissae 
 Cb, bD or rather dD=ab, ad = b\) and a> = CD—bT>, Tve will make 
 (conies) \/Cb.b D : ab :: CI) : Ali and Ave will obtain the solidity of 
 C1)A=4 area CD x i. AG. We will then measure ac, Oil parallel to 
 ae, oO drawn from the centre to the middle ])oint o of ac (oO form- 
 ing part of the diameter EF the conjugate of Gil) and the abscissa 
 j> II of the ordinate ap parallel and equal tooO ; with those data, "WO 
 ■will make V^i'-i'II : <'P " GH : 1"^F ; avo Avill then obtain oF and con- 
 sequently the ix'rpendicular Fr, as at i)ar. (175, T. Ex. 3). It 
 remains to establish the diam. mn of an intermediate section between 
 ae and the apex F of the segmentrtc F; now we will obtain mqov Hf/half 
 (17«, T., UE3I. 1.) of »HH by making EF : GII :: ^/E<j.tjFjjmq. Fi- 
 nally we will obt ain t he HMjuiri'd solidity CDoc=(4 area CD x J Ao) 
 minus (area ac h 4 anvi uin y. I Fr). 
 
 2. If the solid to be measured was the frustum KLac, Ave Avould 
 operate for the segment KL15 as it has been done for rtcF and the 
 sum of the solidities oF these segments deducted from that of tho 
 entire spheroid AC HD, Avould Icavo the solidity of the proposed 
 frustum. 
 
 PROBLEM XL VII. 
 
 To find the solidity of a right or oblique paraboloid or 
 of any frustum or segment of a paraboloid com- 
 prised between parallel bases, perpendicu- 
 lar, or not, to the ajcis of the solid. 
 
 ( Se(! models of the stereometrical tableau. ) 
 (179) If ULE. To the sum of the areas of the opposite bases, add 
 
 dity of tho propoaod ungula, by muUipIyin(» tho proJuot of 4 times tho area of its half 
 • ay section AOBA (a kind of sector otaoirclo or ellipsis) by tho sixth part of the 
 length DM of the ungula. 
 
MENSURATION OF SOLIDS. 143 
 
 4 times the area of an iud meiliatewctinn lialf-wdij hctwecn Ihcni ; mill- 
 tiiivj the. whvlo hi/ ,1 of the heif/hf of the body to be measured and Ihcjti'o- 
 duet will he the required solidilij. 
 
 In fjict, the^eiKuatiiif^ I);U'iil)olii A15C 
 is a. cmv(( such that the jibscissao aro \no- 
 portioiial to tin; .s((Uar('.s of the oidiiiates 
 that is : wo always liavcs ed : CD :: 
 
 2 3 
 
 db : DH , and it is tlic same for every 
 other [tair or system of axes or ('oiijup;ate 
 tliainetera KF, Gil whieh still give G/j: 
 
 Gil :://t^ FII^ then if C(Z=-i CD, db will l)o=U)B antlin the same 
 
 2 '2 
 
 way if G/( = i Gil, we will have eh = lEH ; now it is (lemoiistrat'Ml that 
 the solidity of the paraboloid is i'(jual to half its circnmsrri!>ed cy- 
 linder, that is, this solidity = area base AIJ x J CD, or solidity 
 
 2 2 
 
 FEG— area base EF x -J Gp ; but \\ bd =i lU) we have area ab= 
 \ area AiJ (since the parallel sections ah, AH aie circles and the si- 
 milar lignres are to each other as the stpiares of their homcdogons lines) 
 and area AB 1-4 area o/^ = 3 area AH ; therefoic area AH x JCD=3 
 area AH x I CD = (area AH + area ab) x ^ CD. Similarly area ellip 
 tical base EF=2 area similar elliptical base c/'aud areaEFx^ Gj)= 
 (area EF f- 4 area ef) x JCJjj. 
 
 Ill the Nccoiid place, Let AH ba any segment of a paraboloid 
 with ]»arallel bases, we demonstrate that the solidity is obtained by- 
 multiplying by the height <ZD of the frustum, the half-sum of the 
 areas of its parallel bases ; now on account of C(? : CVy : CD :: db : qn : 
 
 2 
 
 DB , it is plain that the intermediate area mn is an aritmetic mean 
 betwee?» area AH and area ab ; wiience it follows that area AH + 
 area ai f- 4 area m}t=0 area ?HH ; therefore the solidity ofAH6«= 
 (area AH t- area ab -l 4 area mn) x ^ dD. 
 
 (180) KEIU. In the case of the paraboloid c frustum ofa;/a- 
 raboloid properly so called, it is plain that this rule otters no advan- 
 tage and on the contrary it is more simple to arrive immediately at 
 the desired solidity by computing the i»roduct of id) by area AB, 
 or of i GfZ by area EE, or of \ rZD by the sum of the areas of AH and 
 ab, as the case may be ; but in practice it is rare that the solids to 
 be measured are perfectly geometrical, and were they, we Avould 
 not know it without much preliminary labour which Ave might as well 
 devote at once to thecalcuhition of the required solidity accord- 
 ing to the rule here given ; whilst if (13T, 14.0, T.) we took for a 
 par aboloid, a solid which on the contrary were a segment or 
 the frustum of a spheroid or of a hyporboloid, or which only 
 
f 
 
 144 
 
 KEY TO THE TABr<EAn. 
 
 (..::! 
 
 rosciiililtMl Hiosc solids \vit]l()ll^ boiii.iif idciilifuiltlo will) any oftlicrn, 
 tlio viil(> of this i)r«)l>lfii! is tliiit ()ii(> wliicli \v(»ul(I oll'or (lie giianiiiteu 
 of ill! aiKMiiiU'.v very near tlio truth. 
 
 Kx. 1. What is the solidity ofari,i;lit paviiltoloid the lioiyht of 
 wliicli i.s 84, nnd tlio radius oftlus base; 2il 
 
 Ans. diaiii. -18 x 48 x .7SrA x ^84 - 7(;()01.r)873 tlio voquiv. solidity. 
 
 2. Wliat is, in ij^alloiis ()f'^:j| cubic inches, the capacity of a pa- 
 rabolic boiler the (h'pth of which is liG metres and the diameter (JO 
 inches ? 
 
 .4 IIS. r.o" X .7854 X 18-f-231 = ."50,893.!)2 cubic inches -f- 231 = 220.32 
 or 22().\ yallon.s nearly. 
 
 3- A vault whicji a]»pears to be parabolic, is GO inches lii,y:li, the 
 diameter of its base is 40 metresand its intermediate diajiu'ter is 28 
 metres 2'^.') millimetres ; uhat is the sididity of tiie included space ? 
 
 Ans. (40" +4 times 28.285"') x .78.54 x 00 -•;- = 37/)L>y.2 cubic 
 metres. 
 
 ■i. In an inclined vessel whicli may be a paraboloid or the seg- 
 ment of a spheroid, is a (]uantity of iifpior tli<! suri.ict! of wiiich iscon- 
 scfjuently an ellipsis having for iis diameters .50 and 30 inches, the 
 gn-atest depth of liie licpior is 18 inches, and oiu'. ofthc! diameters 
 (the least) of the e]li|»tical section taken at the middle of that 
 depth is 22.5 inches : wiiat is tlie content ? 
 
 Ans. The iutei'mediate section being similar to th(^ l)ase or surface 
 ■we will (d)tain its greater diamet<'r by making 30 : 50 :: 22.5 : .37.5 j the 
 solidity = (50 x 30 h 4 times 22.5 x 37.5) x .7854 x 3 = 11-J8G cubic 
 indies. 
 
 5. One of the componrnt parts of a solid to lie measured, seems 
 to be the frustum of a parabolic conoid with parallel bases, the res- 
 pective circumferences of its two circular bases and of a section half- 
 Avay between them, are 182.2, 041,145.15 inches and the lieight is 
 48 inches ; what is its solidity in cubic feet ? 
 
 Aus. Dividing each of the circuniferenc(!S by .3.1416 we obtain 
 for the diameters of the respective sections 58, 30 and 4G.2 inclies, 
 
 2 2 2 
 
 ■which gives for the solidity (58 +30 -f- 4 times 4G.2 ) .78.54 = 10054.5, 
 and 100.54.5x48-^-0 that is by 8, = 80,430 cubic inches, -f- 1728 = 4G..55 
 cubic feet. 
 
JIENSURATION OP SOLIDS. 
 
 PROBLEM XLVIII. 
 
 145 
 
 To determine the solidity of any frustum ABEF ^ of a 
 right paraboloid ABC, -with non parallel bases. 
 
 (Sec the tableau.) 
 
 (ISl) flTJff^i;. Oct, htf the last problem, the respective solidities 
 of the entire paraboJoid AIJC of which the frustum fornts part, and of 
 the i>((ylial iiarabohid or srijinent EFfr wanting/ in the t/iren frustum to 
 complete iho entire paraboloid : the dijf'erenee of these solidities tcill be 
 the required solidih/. 
 
 Let ABEF (figuro oftlio last pro 1>1 (Mil) a section of the given 
 finstuni l>v a ])laiH' innpciidiciilar to tlie centre D of its base ; take on 
 tlie axir* ])d of 1 lie section any Icn^ftii Dd, measure Dli, (16 and since 
 
 (lt«),T.) \veIiaveCD:C(Z::DIi"(Z/>'niaki'(l)«,<liv. G.)CD— cd : ed :: 
 DI?" ~db : db , or Avliich is the same thing DB —db : (76 :: Dd : dC, 
 ^vhicIl Avill give for the liciglit of the generating parabohi JD + tZC — 
 DC. Now, through the middle point II of EF, draw HG parallel to 
 DC (for in tlie parabola the centre is infinitely distant aim any diame- 
 ter Gil, that is any bisectrix Gil of the clumls or parallel double 
 ordinates EF, ef, re, is conse(inently parallel to tin* axis CD), mea- 
 
 'J 2 2 
 
 sure any lie, HE and re and make, as before, re — HE : HE :: Hr : HG ; 
 AvithHGand the angle GH p or G HE, we easily find (175 T.) the 
 perpendicular lieight Gy> oftlui segment FGE, to compute afterwards 
 th(! respective solidities of the entire ami partial conoids and their 
 dillereuce, which will resolve the problem. 
 
 (IS'l) REin. If the frn<«tum to bo measured ABEF (See 
 figure to problem L and suppose it to be the frustum of a parabo- 
 loid) is tliat of iin oblique paraboloid ; draw from A to F .any 
 stiiiiglit lino V)b parallel to FE, bisect in /(', li' these double ordi- 
 nates and draw G/i' H Avhicli will pass through the apex G' of the 
 segment FEG' ; draw afterwards Ee iiarallel to AB, bisect these pa- 
 
 By^^B 
 
 We inny n!so obtdin ns for tho ungula ot the prism, 
 pyrnmid, i^ono or cylinder, tlio solidity very nonrly of 
 tho ungula ABC-D, AB(J-D', AOU'-D of a right or in. 
 clined parabiilic-couoid, and th.it, absolutely in tho 
 eaino manner a9 for those varioas Bolidd. (See pro- 
 blem XXXI, &0.) 
 
 77 
 
146 
 
 KEY TO THE TABLEAU. 
 
 ralleU in 11', h ajid diinv tlio dimnotor G/( IT wliich will iiicct tin- 
 apex G of the obliciiic pjuaboloid AlWl ; wc will ciiliiiliilc. iis Itdor*', 
 the lioightH GP, (Vp ot'tlio ontiic aiul i»;iitiiil conoids, witli the iiid of 
 the t'.iigles ir'li'E, (MI15 iiiid of tlu* strai'^Iit lines (■// iili' of wliicli wo 
 "will establish tlu^ leiif^tlis as already staled, and we will obtain tlio 
 solidity of the fiustiiin==soIidity AUG— solidity FKG' ^ ar«'a Alix i 
 GP — area EFx^G'j). To obtain if reiiuired, CI), we will draw from 
 any point between A and F a stiai.niit line m o « (suppose a stiaij;lit 
 lino m o n perpeudicnlar to the ])araliels G'li, (ill and of wliicli <> in 
 the middle point) perpendicular to Gil or to G'H', llie perpendicular 
 CD, where m = ii Avill bo the reciuired axis. 
 
 PROBLEM XLIX. 
 
 To find the solidity of a right or oblique hyperholoid, 
 
 or of any frustum of a hyperboloid, comprised 
 
 between par'fllel bases, perpendicular or 
 
 not, to the axis of revolution. 
 
 (See the tableau.) 
 
 (183) RIJIjR. To the sum of the areas of the opposite Imses of the 
 tolid, add 4 times the area of a .icclion half irai/ between them, multlpJji 
 ike whole by \ of the height and the product will be the required sulidity. 
 
 In the case of the right hyperbo- 
 loid ABC or of tlie frustum A]\ba of a 
 right hyi)erboloid with parallel bases, 
 this rule is, in other terms, the very 
 cue given by the ** ditl'erential and in- 
 tegral calculus " aud since the inter- 
 mediate diain. is here essential to the 
 calculation to be nuvde, it is to be de- 
 monstrated how it can bo obtained 
 
 when it is not among the necessary data. The hyperbola is such that 
 its centre is outside the circumference of the c.n-ve, and as in the 
 circle, the ellipsis and parabola, so in the hyperl)ola any diameter 
 produced OC, OG bisects the chord or double oidinate AB, <«/>— EF, ef 
 parallel to the tangent tg, t'(f drawn through the i)oint C or G where 
 such diamt'or meets the curve. It follows then that to deter- 
 mine the centre of the hyperbola, it sutfices to draw and bisect lu 
 D, d, — fii, /t, any two pairs of parallels AB, ab, — EF, ef, and to pro- 
 duce outside the figure the straight lines D(?, lI/( uniting the points 
 of section, to their nu^eting at O which Avill be the retpiired centre j 
 pr, if the direction OD of the axis is known, the iuteraectiou of that 
 
MEN«nRATION OP soLins. 147 
 
 nxisbytlio 8friii/j;-1i( lino H/f ])io(lnci'(l will lU'torniino the required 
 <'(Mitr('. Now, l)y tlio nature of tlie liyiMnbolii, we demonstnito in 
 
 "conies" tliMt 2()C.CD\-V.D': 2 OC.VdbQd:: !)»': r76 , or that 
 iiOr,.f!II I (;il":i2(KT.(i/( I (ill':: H^:^ la'; this is tlien the manner to 
 <»l)t!iin the intt'iiiicdiiitc (liiiiu. «6 or </l)y taking C(?=(?l) or Gh=h 
 IV as the case may be. 
 
 Kx. I. TIk^ hoif-lit CD of a ri^ht hyperholoid ABC is 10 inches, 
 and AD Iho radius of its base is I'-J inches, the intermediate diame- 
 ter (lb is ir).874r) inches ; wliat is its solidity ? 
 
 Alls. (2t''+4fois 1.5.874.'/) X I0^(i-{]|0?3.4.')4(>!)1 cubic inches. 
 
 2. A vessel which seems to be a right hyperbolic conoid, has 
 for its heii,dit or depth .50 inches, for su|>. diam. 104 inches and for 
 intermediate diam. (!8 inclu's : what is its capacity in wine gallons. 
 
 Alls. 104" X .78.54-- 8404.88(5 l=:area of the superior base, 4 times 
 
 08% .78.5 ! = 08' X .•]. 14 11)^14.5^0.7584, the sum of these areas is 23021. 
 0448, this sum x \ oO or, whicii is the same thing .< 50 and the pro- 
 duct^-l]= 101847.04 cubic inches, -r 231 = 830i gallons. 
 
 3. Ilow many cubic metres of space are there under a vault 
 
 whicli ajtpc^ars to l)e hyperbolic and the height of whicli is 1;5 metres, 
 
 the diameter of (he base .■'.2 metres and tlu) intermediate diam. 20 
 
 metres 1 
 
 Ans. 51.52.224 cubic metres. 
 
 4. A boiler in the form of a hyperboloid, contains a quan- 
 tity of liciuor ; it is askcil how many more gallons would be re- 
 (jiiiicd to till it, the part of tlie vessel to be tilled having conse- 
 quently the foini of the frustum of a hyperboloid Avith jiarallel 
 bases ; the dianicteis of th(;se bases are 24 and 32 inches, the inter, 
 diam. 28.1708 and (he height of the frustum 20 inches ? 
 
 All*. (21 i 32' f 4 tinuis 28.1708') x .78,54 x 20-=-0=12499i cubic 
 inches or 54.108 gallons, or 7.2334 cubic I'eet. 
 
 ft. One of the component parts of a cul-de-hunpe or other object 
 
 to be njeasured, presents the appearance of the frustum of a hyper- 
 
 'lM)loid of whicli the height is 12 inches, the smaller diam. 6 inches 
 
 the grcatei' diaju. 10 inches and the intevm. diam. 8i inches : what is 
 
 its solidity ? 
 
 A IIS. GG7.59 cubic inches. 
 
 (1§'1) UE!tI. For the oblique hyperboloid or the segment of a 
 right hyperboloid by a [»lane not perpendicular to the axis, the *' cal- 
 culus" shows li(»w to obtain the solidity by making the following 
 proportion : Gil r 2(jrO :: § Gil i 2G0 :: i cylindroid of tlie same 
 base and lieight : required solidity. 
 
148 
 
 KEY TO THK TABLEAU. 
 
 6. Let iMi«^ v<H|uir('il fo cuhc tlio liypciboloid EF(! of wliicli tlip 
 givator axis EFof I lie •■lliptical I) isc mciisurcs 7d units iiutl its roiijn- 
 giite diameter m.2, let ()a:=4I7, Gll-I!).8, (Jp^l.l.H; w,. Inivo 
 Been that to find </tliepe is to l>c iniulc 2(iO.GU + Gil': 2 GO.G/( I- 
 GA':: he'': /«/, or (wliieli (73, Ax. G.) is t'le .siiiik- tiling) ::EF't/ , 
 that is, in fignies, 2()4:].M() : lej.C? :: (108 1 : '^7'i().l8=<f, Avheiice ef= 
 52.44317 ; now the parallel section r/l)('in<r a similai' ellipsis to EF, 
 wo will obtain area <;/l»,v niakinj;- EF : «;/' ::area. EF : area r/'or()Odl : 
 2750.18 :: 42;J!».275 : l!»l().:}=aiea cf ; linally (area EF f 4area cf) "< J 
 Gj)=solidity EFU, or (42;J!».275 ! 4 times VJUV.i) x 15.8 and-^CJ = 31, 
 348.4508 units of volume in the solid to he measured. For ])roof, the 
 rule given in tluj remark i)rccediiig tin's (>xan)plc, ,!;ivi's J{K5.2 : 'J(\,G 
 :: 33490.2723: 31,348.4525, the ditlerence sj^i'Jir.zij <'>i .0(HH)(I(H)5 being 
 due to tlie neglected decimals. 
 
 
 PROBLEM L. 
 
 To determine the solidity of any frustum ABEF of a 
 hyperboloid -with non parallel bases AB, EF. ^ 
 
 (See the Uihleaii.) 
 
 (185) RTLE. Get nei'anilcl'/ fhe irspcctire solUlilics of the entire 
 hyperboloid AUG and of the partitd hijperboloid EFG, mid take the dif- 
 ference of these solidities vhieh u'ill he the required Holidity. 
 
 Draw B6 parallel to EF, Ec ivirallel to 
 AB, bisect these two pairs of parallels and 
 through the points of bisection draw the 
 straigljt lines HO, H'O of Avliich the intersec- 
 tion at willl)e the centre of the 'c<'iierat- 
 ingcurve. Througli the points of intersec- 
 tion G, G' draw the ]>erpendicuiars GP, 
 Qi'p to the bases AB, EF and the re(iuired 
 solidity will be (area AB i- 4 area internte- 
 diato section between AB and G> -< JGI*, 
 minus (area EF I- 4 area inter, sect, be- 
 tween EF and G') x \ G'j^>. 
 
 Item I. To fix the direction of the axis CDof rerolutiou : from 
 the centre O with any radius, intersect the oi)posite sides of the 
 
 1. We may niso obtain as for the ungnln of a prism, pyramid, oono or cylinder^ 
 the solidity nearly of the ungula of a right orinolrned hyperbolic conoid, and that, ab- 
 Bolately iu the same manner as fur those various solids. (Sue the problem XXXI, &e 
 
MENSUHATION OF SOLIDS. 
 
 1 19 
 
 curve, join lli('s(> iiilcrsrctioiis 1»y :i slriii,i;lit liin'. iiiid 01) dnnvii pcr- 
 pciuliiMilar IVom tl;«' ccntii' O to tliis lath r will lie tlir r('<niiiitl <Ii- 
 rcctioii. 
 
 llKi>l. II, To tiiid tlic points (i jind ('.', Unit is, the liirtois (il>, 
 Pi';* niul llic other clcniciits nccessiiiy to (lie ealeiilation of the areii.s 
 of tilt! iiilerniediale Kcclions ami oftlie solidities of the enliic aiul 
 
 jmilial solids, we have seen (iS:j T.) that:2 OG.iill I- (ill : XJOU.l.J/t + 
 G/j':: Aw'v H"', whieli pves (JMi <liv fi.) (2()(!.(1H f (;H')-(2 0G. 
 G/i + 0/j ) : 2 CHJ.GA : dlt':: AT.' ~cK' : cK. In liiis pK.noilioi), wo 
 know (2 OG.GII I Gil")— (2 OCrAil, ■\-{}l,')-2 Wi. Ao.II// (as a 
 
 •J ■) 
 
 simple dia.niani of 2()G.G/( l (4/* superposed lo 2oG.(iII t-CilTsliows 
 at once) ; we also kiu)\v AU — cK and <'E j (hat is, '3 terms to 
 
 liud thojourth 20(i.GA i- dh" ; now (:85!> €1.) //()— (20G.G/t i G/iO = 
 GO'VGO^^GO, IIU— GU = GI1 and with Gil and tliu ungk GHIJ wo 
 determine GT, \:c., ''cc. 
 
 PROBIEM LI. 
 
 To determine the solidity very nearly of any spindle, 
 either circular, elliptic, parabolic or hyperbolic. 
 
 (1S5) Um.E. 7)ir!(lc the svini-npintVe (A("l) or r,("l)) into iwo 
 parallel seclioiis or slircs- (AKF, E('i)F) oftldchiiess or liei'jht (AL, LK) 
 equal or nearlij eijiial, bti iilaiiex j)erj>citilifiilar to the axis of revolution 
 (AH) of the (leiierafiitif eiirre {\V,\> <»/• AI)1>;) <jvt sej)aralel;i the soliditif 
 of eaeli of these slices, In/ addiiif/ to the sum of tlie areas of its 
 [HirtdUi or oj)i)osite h((ses, 4 times Ihe area of a seelioii (ef, eii) eijiddis- 
 iaiit from those liases, and multipli/ the whole hi/ \ of the heii/ht of the 
 sliee ; male afterieards the sum of the solidilies of the two compouent 
 sliees uud double the result for the soliditij nearly of the in-oposed sinndlc. 
 
 (I§7) Kxl. Il<><iiiifod tliofiolidlly 
 n<^arly ni a <!ir<uilui' spindle (that 
 
 is generat<'d by tlu^ revolution of tin; ai'o 
 of a circle) of which (he length AB is IS, 
 and the. diameter CD .'JO ? 
 
 Ann. If the inteiinediate diamoters 
 EF, ef, ed are not j>'iven or may he direct- 
 ly obtained by Ww uieasurement of tlie 
 solid to be computed, it will be easy to determine (Jieni by cal- 
 
150 
 
 KKT TO TnE TATII.RATJ. 
 
 t''i 
 
 dilation ; fliiis we will obtain iniincdintcl.v llio liMliiis oC ofllie nrc 
 Af'll li.v (lie iiutlmd of ] tar. (.110 ii.) : 2J V 18:=;ii2 - tlio ir- 
 maiiitlci' of llic di;iiii, of wliicli ("Iv forms part", flic diani. — l]^ i 18 = 
 50 and the ladiiis constMiiicnlly^iJ."). Now we will ohiain op, oij and 
 «)/• ri'Sjicciivrlv ( (|iiiil to llic s(|iiai'c roots (if llic dilVcrcnccs lictwocn 
 tlic s(iuan' of the radius and lln) s(|iiarcs of ip Iv/ ""d <•*•, wliicli is 
 ('\-idi'iit ; now, if we sii[iii(is(' AL -■ Kii \\v, will iiuvr A/( = Hli -- Lm = 
 «hK, or ry - t), Iv/=1^, ryHH, r*y> -ov — vp - (1-2.")— .'}'2I - .'{()l oFwIiich 
 tluu' it 17.yH'.'i.'>',J from wldfli sulitrartiM-in/. — 7 = 11.") — Mtlicic remains 
 Kji or /'« = J()..'}l!>;r"2 and oonscciiicnlly <;/' or 2 r»— ijl).(ll)87(ll or say 
 StJ.dlir*?, for, !is tll«^ ditVcriMKu^ of Volume aceordini,'- to this rnlo is 
 always in I'xcess w(^ may ne,!;leel at. least the Ia>.l deeimals ; in tho 
 sanii' manm^r we liiid diaiii. KF = iJ!t.f«()3 and vil -''-\\J)'M{\, 
 
 The solidity of EC - (I)C^ 4 4 cd^ v EF ') x .7d:)4 x ,\ KL or by 2 
 .^lOlt.'H.e'-i, the solidity of f/A=(EF" -I 4 r/) x .78.11 x ;J = 4()!»2.72, 
 lliese solidities added to,i;'ether and the Tvliolo x !2, ^ives for the soli- 
 dity of the entire spindle MOO ID cubie niiits. 
 
 ill-]i>l. The accurate solidity of the spindle ol thoiast examido is 
 i2;)!)l(J.()7J4, Ihatis tlu^ ai»proximat«i solidity exceeds by .ii,;';'j:T or .01)44 
 (less than the half Imndicdth; the real solidity, which in practice is 
 <j;eneriillyeiinivalent to perfect or .it least siiilicient acciiiacy.coiisider- 
 iii,if theadditioiiallabonr which must he devoted to the calculation by 
 theordinaiy rules ; and besides as already slated (81;$? 4«t, 1 Jltt T.) 
 ■we may willi the rule yiven here carry the precision to any decree re- 
 quired by a subdivisoii of the semi-spiiidh' into nioi<' immerousslices 
 and of which the sides come nearer the .straight Yww. 
 
 
 {\SS) Ex 2. To find flic 
 Molidity iiearly of anvlilp- 
 ti<'iil Ni>iii<llc (that is iH'cne- 
 
 rated by tins revolution of the .' [ i ""jgr-iSLjD.:!^ 
 arc of an ellipsis round a chord M "JP "JT'^^'if "o ""~' 
 perpendicdhir to one of its axes) ■*■' 
 
 of which tlie lenj^tli AB is 80 decimetres, tho greater diameter CD 24 
 decimetres and a diam. EF eiiuidistantiroiii A and CD 18. 9!J0!J4 deci- 
 cimetres ? 
 
 Ans. Let AECGrB tlio geiieratincf cnrve ; to And its centre, 
 draw (170 T, K. I.) any two iiarallel chords \\V, ah and thron.uh tlio 
 middle jioiiits of these chords draw a stiiiii;ht line (iU which Avill 
 intersect CD, produced if necessary, in > centre of tlie ellipsis. Let 
 now CO = 30, VkO, have a diameter of the i'llipsis = 2C0, an ordinate 
 AK or KD=iAL>=40, au abscissa CK or segment of the diam. = J2 
 
MENSUn \T10N OF SOLIDS. 151 
 
 iiii<lfoiis('(]iH'iilIy (lie (>tlicr,-.«'L;iiUMit='i C'( )— Olv^fiO — 12 -4.'^, fo (iud 
 (l^ii i\ le. I.) iIk' oiliiT (liaiiiclcr MX of (Imi cllijisis i>y iiiiiUiiij^ 
 VCiv A C-i ^JO-CK; : IvB :: :» CO: MX m /l^ x 4ri : l(» :: tlO : |(»l» -Mx" 
 MX iM'iii.i,' tlu; siiiiillcr ov tlic •,'i'c;iicr<Iiiim oflliccllii)^;^. iiccoiilinj;' us 
 tlic rccluii^Ic of the sfj-iin'iils is ;;i'c:ilcr or rtiuiillcr tliiui lliu siiiiiifi) 
 oF tlio ortliii;it(^ or [xrpiMHlic'iiliir IvlJ. 
 
 Ti* olttjiiii <■/', \\i' will lirsl nuikf II10 i>ro|»()rlioii ]\[N':iJ('() or 
 (wliicli JKtlicaimu'tliiii.i;) MO : Co :: v'Mv-'/X : ./('or.")!) : :{() :: ^xJOxHO: 
 P7 = i24 iiiid iis »7~K<)-C()-CK — .•«! -1-2 -1"^, wv, will olihiin nr~ 
 21— IH-ri; iiiiddiiiin vJ'='*(n-\2 ; wo will (••piiilly liiol (w-v.':'-'!:) Ili, 
 cs'— ))(,v r^l |:}!)4l2--c;« and C c»i~-=^(li!iiii. (•(Z=22.7dd;J4. It'KF wi-rt; not 
 givou it could ht.' equally ilctci-Miiiicd. 
 
 Diuiu. EF18.!)!)0IM = .'{(lO.lM.l-i Diaiu. KF IS.DIHMn'' = .'KiO.(!5.)ri 
 4 Diaiii. n? 227Hd:i4" =;»()7'7.'il.")." 4 Diaiii. if. l±mwu = o/li.OOOO 
 Diain. CD 24.0()(»(l()" = .Wi.OOOl) 
 
 — — Sn!i>=!);{(;,(irM. 
 ^ .;><r>4 
 
 Prod net =7y.j.(J41M')t; 
 
 rr()diict=2:3U7.0!i:!') x 40 
 X 40 
 
 -^r»)ji!th';..!»7rt; 
 
 -4-0) 01(i-i:!.7l Qiioticiic= l:M)|.:{-2It7(; 
 
 Quotioiit -^ V>7r{}Al-2 —'2 vA. EFA 
 
 -2 vol. ECDF 
 
 sol. 2 FC =1.-7H0,(;2 
 Bol. 2 EFA= 4'I01.;W 
 
 sol. AI5 = 20Ud4.0r> 
 
 The sum 20,084.85 of tlicso solidities is that of tho i>i'oposed 
 spindle and dilfcrs but hy 57 units in excess, or the, fourth i);ut of I per 
 cent, from the accurate sol. 20iJ2S.;Jl ofwliich the calculation hy oidi- 
 uary rules requires as much nioie labour, and otlers in cons(!- 
 quenco of the diversity of the operations to be performed (as de- 
 tailed in an enunciation of 15 lines of text) many more chances of 
 error, as of cours(! it is ahvays the case, more or less, when tho 
 process to be followed is not so simple and dir((ct that we may 
 easily account, by following the details of tho calculation, for each 
 of them. This is the enuutiation in question. 
 
 I'' From three iiincs ilic s(piarc of the diameter at the centre (CD) 
 subtract four times the square uf the diameter (V.IP) hetvcca the middle 
 and the cud ; a/so, from, four ti men this last diameter^ subtract three 
 
152 
 
 KEY TO THE TABLEAU. 
 
 -'■i 
 
 •X, 
 
 ■ >•' 
 
 tiiiirfi till' ilidmrtcr (if the mil re ; <iiiil otic four Hi of Ihc ijiioliciif pro- 
 crciliiifi from Uw iliristiiin of llir Jirat ililfcrciicc hji tlic last, will ijirc ihc 
 cciilnil (liKlaiicr {( )\\.) 
 
 a' Fiii'l I'll llw mclhod of pur. (I7ii, T. Bt. I.) flic a.rin of the 
 cUlpn'iti mul hfi the method of i)(ir. (<;i T.) the a red of the «/eiier<ilinif 
 se(/itieiit (WW.) 
 
 3' Piriile three tinief) the itrea thits/oiiiKl hij the leiiijth (AI5) ()/" 
 the sjiiiidle, (iiul snititrael J'roin the ijiiotieiit the didiiieter <it the eeiitre ; 
 ))iiiltipli,' then the remainder hij four I'mex the eeiilnil disliii:ee, and tiihe 
 this proditet Jroiii the sijiiiire of the ditimeter <il the eeiilre ; this latter 
 reiiuiiiider mnUiplieil bij one third the leii'ith of the sjdiidle, mid thepro- 
 diiet iti/din bij 1..")7(I71*, leiil (jire the solidilij of the sjtindle. 
 
 Tlic ciilciilatioii to 1)0 uiiulc IVoiu tliLs t'liiiuciiitioii is at Iciist, of 
 two or llircc |):il;i's. and that Avitlioiit oven vouipn^liciKlini;' llu' de- 
 tails of the niiiliiiilicalioiis. divisions iVc. ; \\iiil.>t all t!ial is cs.M'ntial 
 to tlic ciuincialioii of Ihc iiilti ,i;ivt'ii Ikto is ifsnnu'd in (iu'sc words : 
 Miillijili/ the sixth jKirt o/ Hie heii/ht ofeaeh of the eomponeiit sliees bij 
 the sum of the areas of its Ixises plus four times the area <f a seetioii. 
 hdlf-icdii hetireeii them ; the sum of the soliililies llius found will he the 
 solidiiii if Hie spindle renj nedrlij ; and as, in prartico, tlu! ncccssiuy 
 diamcti'rs ('/', EF, <■(/, CD, arc ,i;('nc'rall.v olttaint'd by a direct nica- 
 snrcuicnt oftliosc diani.'tors or of their rcspceiive <'ireiinit'erenees, all 
 tho calculation to lie iiiadt' is rednced, fxccptiny tiio mullipUcatious, 
 to till one.jn^-t indicated at tlio bottom of last page. 
 
 (is!» a:x. ',1. To fiBKi t.ii« 
 sol. of a put ii!>oiic .spindle, 
 
 (that is geiieratetl by th«^ revolu- 
 tion of a i)aral)olii ACH or AD 15 
 about a <loiible ordinate AK1> per- 
 pendicular to tiie axis (."K.) of which 
 tlic lcnj;th Al> is (JO and the great- 
 est diam. CD 34 ? 
 
 Ans. Ill lilt' t'ii'^i^ <^f <1it- I'iirabola and oqu il distances A?/, 
 nL, Li», )nM, the iuteriuediate diameters (;/', EK, ed, if not given, are 
 very easy to be deternuiu'd, since as seen (B7J>, T.) the abscissae or 
 segments Cj), C//, Cr, of the axis are sa the squares of the corres- 
 ponding ordinates ep, El], Cr and when these ordinatesare equal mul- 
 tiples ov siib-iniilliples of one another, tlu* segments or abscissae ai'c 
 ab'.o simple multiples or siib-nndliples of the entire iixis Civ ; now 
 
 2 2 
 
 (215 W.) on account of Er/ — i AK Ave will obtain E7 — JAK and 
 consetiuently C(/ = i CK, we will equally obtain C<'= IC7 or j\, CK 
 since cp : AK :: 3 : 4 and that 3:4 :: !) : l(J ; we will 
 
 (> 
 
 1 J 
 
MKNSITRATION OF SnLins. 
 
 153 
 
 ilicii liml in ihi^ iii.iiu'.cr f'/ \7 f -I 4. ',!."», L'y 4.2r)4-4 or l7-:-l(i-= 
 J.dC:^*'). I'/' ," 1'" }f,\i ■■^.•'> • 1 ()(ivJr)-!t.:;(i-.>r., fmiii wliicli wcoMaiii 
 ^liniii. (;/'- Vi;/K— 1 l.r?."), Kl' — •Jl\'/ - '-J.").."), cJ "J l\/-~;ll.r^7r) ; now sol. 
 
 A1;F=::(;uvi i:i't4 iiivii (;/')x,', AL=ii;K ■ Irf) ^.7.-'r.4x,\ AL — 
 ('J.').."> ! I limes I I.S7.') )■•: .7Sr»l vjl oi' ill onct' liy ."> (since tlicic 
 iirc I w (I ('(MKiids oittpnil scunnnls in llic s|iin(ii(' to lie cnlicd) 
 ~<)(>.'{;{.l cnliic nniis ; ilic soliditv (.I'tlie tViisliini FC — (.'M I 4 times 
 .•{1.^7.")" i 'j:..:,') X 7^.')I-^A or l.v ."i to oUlain •^KC:=i2.'{(».VJ.7 n-^'--, 
 nnils : the sum •,'l'iH.").S of ilicsv solidities is llie solidil v of the jut. 
 ])os<'d spindle : it dil'leis iVom tlie iic( urate solidity ■■Jli((r);{.4 lint \>y 
 rV2 units, tlnit i-<. .;,,',;; <»i' ."iHl, .-av ,', ol 1 per rtii! in excess. 
 
 KEiYI. However coinitlicali'd ordinary rnles lor the solidify 
 of tlu' circular and elliplic spiud'es may he, the rnle lor the para- 
 holic spindle is on t he contiary wvy simple; it consists only in ninl- 
 liplyini;- the siinare ol' the central <liameler l>y the len,:;lli (»f the. 
 s[)indle and tlie prodiui a^aiii hy .!|SS7!I( ;M4l.">!»-f-7i) ; hut, there 
 is always this to he considere(J that if the spindle were not properly 
 paiaholic, this last rnle nii,i;hl he jtretly far from giving an accmaK^ 
 resnlt, wliilst wilii the i;('iit'i<il ••i'*-' iomid hero for all the elemen- 
 tary solids, Ave havt' not lirst to consider tho natnrc? of tiie solid 
 lit he measured. ( Nce|»l h(»u<.ver w hell we have lo iluleriuiiie by cal- 
 cdlation the r((|uiied inlerniediate dianii'ters. 
 
 (!i)0) r.\. t. A^2Miun<' AlU'Dhavin'illie 
 apper.reiice of hcin;;- ll.V|M>ri>(»li<' (that 
 is, L^ciuMated hy the revcdntion of a iiyper- 
 bo!a ACK or ADI!. about a eord or donhlo 
 oidiiiate AKll i>er|>endicuhir to its axis ("K 
 or K 1 )) and of w iiich iheureater diamelei t'l) 
 = 71 inches, measures JIMJ inches in leni;(li 
 AI), and its intiiinediile diameters taken at 
 3 places III, li, n, eipiidistant iVoni ejich other 
 and each di.-taiice «'ciual to the fourth part 
 of the half len-lh AK of the si)indie, are resfpectively </^2G.8, EF = 
 4!>, <■(/ = (;."). 4 : what is its solidity ? 
 
 Aii»i. (CI )" - 4 c<f + i:v' ) X .7854 x ^ LK and (FF^f- 4 </) x .78.j4 
 X .\L, or w hieli is the same tliiiiL;-, since AL LK, sol. ^ (Cl>" i 4 
 (■(/' • t2i;F' , 4 (;/•") • .7,'<.")4 - }, LK or AL, or by J, LK or AL to obtain 
 at once llie solidity of the entire s[)indle = (7l + 4 tinu s (m.4 ' x 
 twice 4I> ■ 4 limes '2(i.S ) x .7d.">4 -- 53.-0 = :i0tl,!i 14 cubic inches or 
 IID.74;^ cubic i'cul. 
 
 18 
 
154 
 
 kj;y to tfib tableau. 
 
 To find Op or Vp and consoqncntly plv ~ Civ — Q) - cii=^=i inter. 
 diam. ef, avo first obtain AK : cp 1: 'JOC'.Clv + VK ': 20C. Cp • f>", llicn, 
 as we said (Rii:]VI. II.) (!>()C.CIv + Cic')— (20('.(> -\- Vp')=-'-2Kp.p (- 
 Kj)'j now it is jdain (»»»,«.) that 2Kp.2)() i Kp^ + jiO ==K0^ ; 
 
 wlience, pO ==K0^—{2Kp.j}0 + Kp ) or j)0"==KO— (20C.CKT"CK 
 
 2 2 
 
 — 20C.C;) + Cp ) and Op==^{)p . Wo will tiicn equally obtain //o by 
 
 finding fust SOCXVjf + C// ==:( 20C.('K + C k') x Iv/ and by extracting 
 
 AK2 
 
 afterwards the square root of the ditfcrence or remainder KO - (200. 
 CK + 6'K'— 20 C.(Vjr ! C//'-), H'*'ii tl»'i« will eonie ()/--=v'K(J---(20CJ.' 
 (Ck f- CK'^ — 20iJ.Cr I €/•■•'), and eonse(|iientIy the other necesssury 
 diameters EF, cd. AVe have already shown that to find the centre 
 O, and consequently OC or OK it sulHces to draw and bisect any two 
 parallel chords cli, Ct of the generating curve and then unite 
 the points of bisection by a straight line the prolongation of which 
 will intersect the axis of the cuive (produced if necessary) in a point 
 which will be the required centre. 
 
 (191) REin. If we have devoted to the study of tlie spindlo 
 considerable space, it is not because this solid propeily so called 
 offers itself very often to the valuation of the measurer ; but it is with 
 the viewof arriving at the consideration of the fiustum of a spindle 
 which forms the subject of the following problem and which presents 
 itself every day under the thousand and one forms of casks, bar- 
 rels, tuns, puncheons, (luaiters, &c., such as iu-e used to contain and 
 transp.H't tobacco, sugar, tiour, pork, oils, molasses, beer, brandy, 
 liquors in general and a thousand «)tlier substances capable of adapt- 
 ing themselves to the form of such vessels. 
 
 PROBLEM LII. 
 
 To determine the solidity very nearly of the cen- 
 tral frustum of any spindle, that is of the frustum of a 
 spindle of which the opposite and parallel bases EF, 
 GH are equidistant from a plane CD parallel to the 
 bases and passing perpendicularly through the centre 
 O of the axis of the spindle of -which the frustum forms 
 part. 
 
 (193). Kf'i^E To the area of one I'lP of the equal hoses, add 
 that of a parallel section CI) taken at the centre of the 1 rnsi nm and\ timen 
 the area of an intermediaie parallel section cd e<iHidistant frotn, the 
 
MENSURATION OP SOLIDS. 
 
 155 
 
 eenfre o and hnsc A, and ninUiplif tlie whole hij I of the heh/Jit, length 
 or dcplli of tlir/ninliDn ; (he rci^iilt will be the fciiuired soliditi/, 
 
 KK.TB. 'f is liiinlly iiccossiiiy to say 
 tliiit to obtiiiii l»y ciilciiliition iIm^ iiit(!riiio- 
 (liiite diiiinctcr cd, wo liavo ('/> t'<]iial to 
 tlic liiiir-diUViciu'C! Ix'twt'Cii tlio diameters 
 CI), EF and tliat wcarterwardrt liiid C</ and 
 <'onse(nien(ly ed~~CD — 2(Jq, iti tlu) same 
 manner as in tli(i various cases of tlie last 
 j)fol>lein. If we. have to ineasme tlie eapa- 
 city ol'ii oasic we will obtain its interior diameter CD by introducing 
 tlironj^li the buny liole a sealo ofinciies or otlier c(iiial parts. We will 
 obtain tlie intermediate diani. trfby measurinf^ the distance rtc between 
 the cask and a rectilineal rod <v/ tan^(Mit at (', and afterwards make cfi 
 ==Cl)-~2(tc. From the whole length measiirtid outside, the sum of 
 the thicknesses of the two ends must then be subtracted, for 
 the interior leii,<;tli or height to bc! tak(^n into the calculation. To (d)tain 
 €(/ tangent at C, it is plain that it will sulHce to see that eE — yGr 
 or A6'=I5;/ ; iinally cij exterior length ()f the cask is the distance in- 
 tercepted on the straight liiu' <■// ^-" two other straight lines 
 Ihj, Ve resting on the j)aiallel bottoi o as to meet c(j. We could 
 also arrive (^1, G.) at the required areas of tlu^ respective sections 
 CD, ed, EF by measuring on the outsid<! of the cask the circumfe- 
 rences of thos(^ sections from which should be deducted the double 
 thickness of the staves multii»lied by 3.14IG or by 3}. 
 
 Kx. I. What is the solidity of the central frustum of a circidar 
 spindUi the length of which is 40 inches, the greater diameter 3(5, the 
 smaller Ki, an<l the intermediate diam. 31.8i2G inches 1 
 
 2 2 
 
 Ans. (areaCD + 4 area c(? + areaEF) x J 4()=(3G F 4tiines 31.82G 4- 
 Id) X .7854 X 40 T- ((-=29,340 cubic inches, exceeding but by .002 8or a 
 little more than one fourth of 1 per cent the accurate solidity 29,257.3 
 cubic inches-=l2(!j gallons nearly. 
 
 JJ. The length of tlie frustum of a circular spindle is 3 feet 4 
 inches, the diameter at the centre 2 feet 8 inches, the extreme 
 dianu'ter 2 feet and tlie intermediate diam. 30.0.'388 ; wllat is its 
 solidity ? 
 
 An«. 27,.30l cubic inches, to 27 ,287i cubic inches the accurate 
 solidity, or an excess of .0005 or 2V ^^ ^ P^i" cent. 
 
 :{. Ke»iuired the capacity of a vessel having the form of the 
 frustum of a circular spindle, the length 50 inches, the smaller and 
 greater diameters 25 and 35 indies and the interm. diam. 32..574 ? 
 
 Alls. 3;),887 cubic inches-^ 1728=-=23.083 cubic feet, to 39,782 
 
156 
 
 KtY TO TUB TABLEAU. 
 
 11' 
 
 m. 
 
 
 cnliic iiidics or "3 022 ('ul)i<.' feet, say an excess of .()il2i! or iicaily oii<; 
 loin til of I per cent. 
 
 -I. Tlie central zone: of a ciii'ular spimllc nn asiiies :{ feet in 
 ]e?i,!;tli, tlie extre. lie diameters are 2 feet, aii<l Hi im lies ami t lie cal- 
 culated interni. diaiii. is '2'2A)T'22 : what is iis >()li(lity '! 
 
 Alls. 1.'{,I()4 cnliic inclies or 7.rii^'V17 cnliic feet, (lie acenrate soli- 
 dity accoidi:>i;' to tlie <;(neial rules lieiiii; l.'MlliO. I cnliic inclies or 
 7, .">7.")-lG cubic feet, say an error in excess of .tltllOo or j'„ of I pi'r 
 cent. 
 
 5. ^Vllat is tlie caiMcity of a lio^sliead the len^lli of wliicli is ."> 
 feet, the cxtreiiu; diii.iieters .")!) and JiU inches and the interiu. diain. 
 45.3D4 ? 
 
 Alls. f>l,-13!».89 cnhic inclies, to HI, .'302.7.') the accniate solidity, 
 tlui ditt'erenc(^ in excess Iteiiiu,' .OOi.l or i of 1 per cenl. 
 
 O. A cask appeal i II, n' to form jiart of an elliptical spindle is 2S 
 inclies long, its greatest diani. is 2 I inches, the diam at t he head 21 .(> 
 and the interin. diaiii. 2.'5.40itOll inches : what is its capaciiy in wiiu; 
 gallons of 2.'JI cubic inches to the gallon '] 
 
 Alls. (24* i 2i.()*-! 4 times 2a.lO!)(l!)% x .7!-r,l 28-M>---] JjS.'j.'j.a 
 cubic inches, to J J,c^.">1.7,'j the acciirale s(d. tlie e\ce> 'icin;;' in this 
 case but of .OOOdO,") which shows that the pio]>osed cask i \ cry nearly 
 tlu' frnstniii of u sjiheroid, I lie rule in such case giving as seen ^S7<>, 
 U.) the accniate solidity. The reipiired capatily in gallons is .")!.. "UO. 
 
 7. How many gallons may be contained in a tmi of elliptic cur- 
 vature the greater diaiiu ter of which is ."52 inches, the smaller diame- 
 ter 24 inches, the diam. at 10 inches from llie lu:ad o0.1(>7.><j inches 
 and the length 40 inches ? 
 
 Alls. 27,42.'5.7 cubic inches or(-^2;5l) J 16.72(), .say lb-',' gallons 
 nearly ; the accurate capacity is 27,41!t.(i cubic inches, liie diileiencc 
 iu exccs.s being but (! cubic inches or one 40ili of a gallon. 
 
 8. The central zone of a parabolic spindle is JJti inches long, its 
 diameterat tli(> centre is also ;j() inches, that of the apex 20 iiiclie.'* 
 and the interm. diam. ;J2 inches; what is itssolid conleiil in cubic feet i 
 
 Ans. 27,204 cubic inches, to 27.2.'}3.!) aceui'iit<i sol. or an excess 
 of .0022 ; say an eriiu' in excess of I of 1 jier cent. In ciiliic feet the 
 solidity is l.").70.> to 1.j.7(J. 
 
 9. Determine the <apaeity of a tun the length ofwhicli is 40 
 inches, the great and small <lianieteis ;i2 and 21 inches and tln' in- 
 termediate diameter 30 inches if 
 
MENSURATION OP SOLinP. 
 
 157 
 
 it 2 
 
 Asii. 0^2 I 1.M > -I tiiiu's .-JO )x.7r^.'>l • -ll> ~ (I •;27,2t.'7.Q ciihic 
 iiU'li(!S or I I7.S7 v:;iil()iis iicirly ; the acriiralc soiiiliiy is 27,2\U.fi 
 cubic inclics, siiy an citki' nt' .OOOil-J or ,', of I per cent, ('(iiiiN-.-ilciit to 
 15 ot'ii .gallon or n lit lie more llian liall' a pint. 
 
 1<>. How niany<'ul)ir red arc tln'ic in a hogshead tin' (li.inictcr 
 of wliicli at tlic ccniic is ;") t'cet, at tlic head 3 feet, its intermediate 
 tliamcter 4.5 I'ccl and len,i;tli 7 feet .' 
 
 Ann. I(»."),:}71."), to 10,1. l!)l:3l the aceiiiatc solidity, or an c.\ccs.s 
 of J of 1 i)cr cent. 
 
 11. IIoM- many i;allonsof salt can be juit into an em[)ty llonrbar- 
 rel the hci,i;lit of wliieli is 2'i inches, the inf. or sup. diani. 17 
 iniln's, the .greater diam. Xll) inches and the inteiniediatc diam. be- 
 tween the bottom and the centre WK'-i inches 1 
 
 Aii«. (17" I L'O' . f -! times l!».;f ) or •.217;» ■< .78.'»1 x ^>r)-^(i---7!;}r» 
 cnbic inch's, divi<lin,i;- by ^.'.'11 we obtain ;{l» ,!4allons 2 (piart.s and ;5 
 pints nearly or (-:-::2:W[); li J„ baslicls nearly. 
 
 J'i. 'J'licic aic three varietu's of casks in whicli llu'cxtreme dia- 
 meters are 21 and ■i2 inches, in one tiie interm. diam. is ;i(l.'-2 inches, 
 in another this diam. nie.isnres ;5() inches and in tiurd "Jii.-J incln's, 
 tln' Icn^ytli is ■|!2 iiu'lies, what is ijie content of each cask in imiierial 
 gallons of 277.'J7.1 cnbic inclies ])er .gallon ? 
 
 Alls. (24' + y2' + 4 times-mii"^) X .7d."j4 x 42 -f 6 ^277.274== 1 04.0(3, 
 to 104, ditr.=-.,',T<)ri.. oallon. 
 
 (24% 32' r4 times:}0') < .784.") v 42-^n~277.274==103.!0(), 
 to 103, difr.=:,v, ofa .gallon. 
 
 (24^ + 32* X 4 times 2!>.2*) x .7854 < 42 ^(J :-277.274 = !iy.3.j 
 to 99.3, aui: = 2'^ oi n sidUni. 
 
 PROBLEM LIII. 
 
 To find the solidity nearly, of any frustum of a spindle 
 EFHG- or ed GH, with parallel bases perpendi- 
 cular to the axis of the spindle. 
 
 (See the tahlvdn.) 
 
 flO.I) Rriii:. Compute acpdnitrli/ the nolidih/ of cdcli of the 
 slices KFIX,', (illDC sititated <tt opposite sides 0/ the centre on ijeeiiter 
 diam. Vi> of the <jiveu finslnm, lnj addiin/ to the suin 0/ the Ixises VI), 
 EF, CD, Gil of each of them, foHV times the (tre/( of an intcrmediote 
 section cf, cd, and vinllipli/ these sums In/ one si.rt!' of the heii/ht of the 
 respective slices ; the sum of those solidities will be the required soiidifi/. 
 

 158 
 
 KEY TO THE TABLKAU. 
 
 n 
 
 Ml 
 Hit 
 
 REM. T( is ])laiii tliiil ifllK* iViisfnm is liito- 
 I'jil as cdim Of (Iocs not cxlciid Itcyoiid the rciitio 
 CI), \v«'. will liiivf liiit out' opcijitioii to perform to 
 (Ictcuiinnc its solidity = (aroii cil + arcii ( < 1 1 + I iiii'a 
 yli) X v oB. 
 
 Kx 1. One of the coinixnieiif , nts of a eiil- 
 de-lam]) presents the form of the hiteral fnistiini 
 of a spindle. Its lliiee diameters are Si4,:]() and 
 S2 inches and its hei,<;ht 21 indies : wiiat is its solidity f 
 
 Jins. (2l''+S-y + A times 30 ) k .7854 x 21 —(J =1-1.21)4 cubic 
 inches or 8.272 cubic feet. 
 
 2. A tun phu'ed on end and the lieijilit of wliieh is 42 inclica aud 
 sup. diani. 24 inehes, contains wine to tiie ihree fourths of its height ; 
 tlie entire c.ipacity of t!ie tun is 104 imperial .gallons (277.274 cubic 
 inches per gallon) how many {"allons arc there renuuuiug in tlu; tun ? 
 
 AiiM. Here, since tlie en tin' solidity of the frustum of tlie spindle 
 te be uieasured is known, then, instead of computing' separately 
 the solidities of the two component slices of the frustum to <;(!t their 
 sum, it will sulliee to cube the emitty part of tiie tun andai'terwards 
 subtract its solidity from that(»rthe entire tun. The diam. of the 
 tun at the lieii;ht to which the wine reaches is;{0.2 inches ami tht^ in- 
 termediate diam. Ix tweeu this latter and the top is 27. (J. Then the 
 s<d. of the frustum to be deducted is (24 +4 times 27.(5 -f-40") x 
 .78.14 < i-42-r-(I^fi2''{.'lcubic inches : 277. 274=22^ ^;alIons nearly, there 
 remains then in the tun 104— 22^=81^ gallons. 
 
 PROBLEM LIV. 
 
 To determine the approximate solidity of any seg- 
 ment of a spindle (Adc, aCb Acb) -with a single base paral- 
 lel or not at the axis { AB) of the spindle or to its diameter 
 (CD ; or the solidity of a frustum (ABba, dcbe, AbBf ) 
 ■with parallel bases inclined or not to the axes of the 
 solid ; or the solidity of any ungula ABC— D, ABC — 
 D', AEC— D of a spindle. 
 
 (104). RUI.1E. To the sum of the arean of the parallel or oppo- 
 site bascH (if there is hut one hase,we eonnider the ofher'—i)) of the frus- 
 tum, add A times the area '\f'-i, st'ciio/i einiidisiantfroiii those bases and 
 
MENSURATION OP POLIDS. 
 
 159 
 
 .^ 
 
 muliiphj ilic vlmle hi/ one sirlh of fhc hchjltt of the se(/)iienl, friis- 
 liiut III' ini<iiil(i iix tlie ease maij he. 
 
 KL,rtJ. H'tlic ;j,iv('ii i'nistmn contiiiiis tlio 
 
 cciilic ot'tlic si»in(lU' of wliicli it loriiis ])ai(, 
 
 diiiw tlir()ii.!;li till' ocutro a soclioii ijOh parallel 
 
 t<» tilt" Iri.s.'s and calc'iilato sciiaiatcly cadi of 
 
 tlio <•> iiiixiiM'iit i)aits of the fnis'uiii and add 
 
 tlicii tonctlicr ; hut it" tlic i)ii;:illt 1 liases Ah, 
 
 /15 are e(|iii('...;i lilt (Voiu the eeiitro 0, lii( ii it is jdain lliat we will 
 
 liiive but one ojiemtioii to peifonn ard afrerwuids ilouhle the result. 
 
 PROBLEM LV. 
 
 To determine the solidity nearly of any frustum of an 
 ungula ■with non parallel bases. 
 
 (See the tahlcau.) 
 
 (103) RDI^K I. J^ecainposc the fniMnni KFlKi hi/ mi iniiu/iiiiirij 
 f.laiic imraUel U> cither Gil, J'^F (if its Imses and /inssiiii/ tlinmyh the pmiit 
 1' or 11 {(IS the ciise iiiiii/ he) the nearest to its other hiise, into the friisliim 
 of a spinilh' irith puriiUel liases anil un unijnla ; then eoinputc tiepuraiehj 
 their solidities and add them together. 
 
 (IIMJ) Kl'I^U II. Ciinijiiile hij the hist prohlem the resjjccti re solidities 
 of the tiro SCI/ rents of a sjiindle leith a sini/le hase 
 GliU, V'VM of irhich the f/iirn frii^tinn forms part ; 
 the difference of these solidities will be the required 
 solidity. 
 
 Rl^]n. An inclined ton or cask containing 
 liquor and whi<'h one would not displace to la- 
 cilit.ite its .nauuint;- will soinetinu's jiresent to 
 the valuation of the measurer a solid of this 
 kind. 
 
 PROBLEM LVI. 
 
 To value the content, nearly, of a ton or cask laying 
 on its side and being but partly full. (See the tableau.) 
 
 (107) RUEiE I. Afterharinf/ obtained the chords and versed-sines 
 of the seijiiicnts of a circle forming the opposite bases /Ff hllh of the frus- 
 tum FII to be valued, muUiphj the sum of those bases jyliis 4 times the area 
 of a central section puralUi to those liases, lii/ ]. hi'i;iht oi- length Al? of the 
 caslc, or to lie still nearer the accurate content, operate onlij on the half of the 
 fruitum and aficnvards double the result. 
 
1(50 
 
 KEY TO THR TABLEAIT. 
 
 ^=m::v 
 
 {iVfH) HI ma: II. Ifllirlv,unr 
 ill Uic Ion iliii:-! jiiit ri Hill till' licdiJs o)' 
 finis V.V, (III (if til'- rr'ificJ, as at cij <ir 
 ril, ire iHIl mine its cmifciif hi/ the 
 iiirlliii(l(iftJie hist liiitoiil' prolilcii', and 
 the same if the llijiinr <iiilreii<lies lli(i<e 
 eiiils, as afFAi oral), irc will eiileidate 
 ill the same wainicr the sef/ineiit KCfl 
 oraVh, tailiiiiiiiisli hi/asiiiucli tliccoii- 
 
 triil III' llie entire tun. If t lie area i if the liivr>r is at AR, axis of tht ton it 
 is plain that ire irill obtain the content \V\)\l\\ — V\W\)—\VV',. If on 
 the eoiit'-arijthesnrfiiee is at cdor fh, ire will first ailenlntebij theiinilniil of 
 the last tint one prolikiii, the fnistnin of n spindle rsi/Dpr or vinlJ (as the 
 case mill/ he) of iiiiicli. the eorrrsjiiiniliiir/ segment of ;lie ton forms jinrt, to 
 subtract afterwards the niajnlae, or ki)ids of pi/ramids hnh\l, fmfi'' or the 
 Kidids crpF, dsijM composed, of the semi-conoids Wp, 1511'/ and of the 
 frusta re A}), sdW') of whirh ire irill prelti/ correclli/ obtain the solidities hij 
 the fieneral rule nf the smn of the parallel bases jt\, re r 4 times the inter- 
 mcdiate section, iiudliplicd bij one sixth of the heiijht. 
 
 PROBLEM LVII, 
 
 Determine the solidity nearly of a convex or concave co- 
 noid Aa Cb B, AalbB or of the segment of a spindle 
 terminated by a convex or spherical base ADB. 
 
 (Sci! (he tidjleait.) 
 
 (991) ICl'f^E 8. Decompose the solid hij a plane pnf;sinri throiiifh 
 AB into the sriinnuit of a spindle or conoid ABC and the seijmcnt of a 
 sjiliere A 1)11 irhich i/oii irill ralneseparrteli/ tnj the rules alreudij ijieen and 
 ufterirards tii];e the sum of the component solidities. 
 
 (200) KCJI.I'} II. To the area of the conrex base (ADB) add 4 
 times that of a /inraliii conree section a'db' equidistant from the base and 
 apex and mnlUpl;/ the snin liy one sixth of the helfjht CD of the solid. 
 
 AVhiit hiis hc'cii alrciuly stiit(Ml (1977 «, 
 Jii7, '*'.) ('(mccriiiiii; llu! si»li('ric;il sector ami 
 ordiiiiuv coiHiiil. siitliccs to show liiiMicdiatcly 
 that we must h;, llial lulc anivc at a pretty 
 eoirecl dctei uiinat ion of ihe solidity of the pro- 
 posed solid. 
 
 ItlMl. If the heipht, CI) were unequal to 
 AC or 15C, that i.s, yicater oilii8.s tliiiii AC or bC wo would ovidoutly 
 
MENKUR/VTION OF SOMDS. 161 
 
 havotoincivnaeitlK' sol.oftlu^ proposiMl conoid, or to iliininisli itbjthe 
 uliCl'ci'Mii'c of tlid solidities of llic icspcclivc K«'<j;iiM'iitH ADI5 liiiving 
 for iiidii CD — AC «»r lU' and CI) < or > A or IK.', iis tho case 
 may Iw. 
 
 The same if t\w hasp of llic ronoid were coiicavo, wo would 
 coinputo (lie sol. of tli(i conrspoiidin^ conoid with jdaue base, aud, 
 aftcrwaids (hsdiict from it llic- sol. of (ho hollow sc^Mncut. 
 
 (201) BSUIjI'] 15. Vompnti; the solhUt;! of the component spheri- 
 cal sector AHH-C, then the solidilii of the coiitiitucd frustum of apriani 
 of which llie ijenerfttiu;! set/men t .\oCa'A »c l»f*(7/'I} is the section ; the > 
 sum of these i.>licUties uill he the required solidity. 
 
 PROBLEM LVIII. 
 
 To determine the solidity of any vault of -which the 
 thickness is not uniform. 
 
 (202) RlJLr. Caleuhtte separatehf (pa(ic 431 G.) by the pre- 
 
 ccdiiuf prolileins, the volumes of the comp:>ucut exterior and interior 
 solids {that is of the prisms or eiflinders, hemispheres, hemi-spheroids or 
 conoids, or of the seijments (f those solids) and afterwards take their 
 dijj'erenee which will be the solid content of the proposed vault. 
 
 PROBLEM LIX. 
 
 To determine tho solidity of any prismoid or 
 
 cylindroid. 
 
 (Sco tho numerous and varied prismoids and cyliudroids of the 
 tableau.) 
 
 (203) RULE. To the sum of the areas of the two parallel bases, 
 add four times the area of a section equidistant from those bases, and 
 multiply the whold by one sixth of the heiyht of the body; the result 
 will be the required solidity. 
 
 UKM 1. It is said (1102 G.) that "tho prismoid is a solid 
 having for its parallel bases, any phme ligurcs with parallel sides." 
 This definition does not exclude the equality of the parallel sides ; 
 therefore, any prism or cylinder {infinitary prism) is at the same time 
 aprismoid. 
 
 Keither does the definition exclude the proportionality of tho 
 parallel sides ; therefore, any frustum ofapyramidor cone (frustum of 
 an infinitary pyramid) with parallel bases, is a prismoid. 
 
 19 
 
162 
 
 KBY TO THE TABLEAU. 
 
 Nor does tlio definition nasijnrn limits to tlic inequality of tlio 
 pamlk'l siiUjs ; tlicierorc, ciicli of (lu^ sides of ()!ie of tlie iiiUiiliel 
 bases inny dinniiiNli indefinitely, even so iis to become <lniilly = f(; that 
 baso will tluMefoie. also be rediieed to zeio or to asiii<;!(! point, as in 
 tlie case of tli(^ pyramid ; tlierefoi(^, the putuimid or cone (injhtitarif 
 pyramid) is aluo a j't'ixmoid. 
 
 If the sum of all tin; sides, but one. of on<i of the bases, bec<mies 
 equal to tlie side thus excepted, that bas(! Hill be but a line or ed;;i! 
 parallel to the idane of the oilier liase, as in the ease of the \ved;;(' ; 
 therefore, any wedije or other solid h((rinii/or one of its Ixtses any plane, 
 figure and f'^r the other base a line j^niralUi to tlie first, is still a 2)ris- 
 nioid. 
 
 (2U4) It uould ai)p(!ar that in tliis inanner of reducinjj; to asini|) lo 
 line any plane ii<iure, we have ne^^lected the necessary paialllelism 
 of the opposite sides ; i»ut it is not so, for if the base to l»e reduced 
 is a rcctan<?le for instance, the two sides pcnpcMidicular to the ex- 
 cepted side 1)econie each of them =0 ; the sum of the sides less one, is 
 the side of the rectangle parallel to the excepted side, and Aviiich, when 
 the perpendicular sides become null, ends by ajtproachiiijj the ex- 
 cepted side 80 as to form with this latter but one and the same 
 line or edge. If the base to bo ro<luced is any polygon, there 
 will be, or not, in the perimeter of that base a side parallel to the 
 excepted side ; if there is a side parallel to it, this side may dimi- 
 nish or increase so as to become of eiiiial length to that; of the 
 excepted side, and all the other sides beconiiiig each - 0, the 
 two parallel sides will unite to form but one ; if there is not in 
 the base on which we operate a sid<^ parallel to the excepted side, 
 ■we will interpose between two of the sides of that base, a side which 
 shall be the reqnin-d parallel ; for in the same way as one of the 
 sides of the inismoid nniy, without affecting the deliiiitioii, become 
 equal to 0, so may a side at hrst ecpial to zero acquiie any develop- 
 ment, and that, iu any i)roportion as in any direction. 
 
 (805) It is plain that if one of the two bases of any figure, 
 may become a line, it is the same of the other base which may also, 
 from any figure, becoraealine. If the two lines which now form the op- 
 posite bases are i)aiallel to each othei', it is plain that the solid will 
 have ceased to exist or shall hav«) become equal to zero or to a mere 
 surface ; but if the lines or edges answering as opposite bases to the 
 body in question are not parallel to each other, though in i>hines 
 parallel to each othei', the solid will not have ceas<'d to exist ; there- 
 fore a jnismoid may he stick as that its opposite bases he both of them 
 pvnpU lines or edges. 
 
MENSURATION OF SOLID!*. 163 
 
 (JJOtt) Let IIS siiy, to sum ni), tliat : npmmoid uinij have for itspm allel 
 Hiiises : any tico jUinrcs eqiuil or simiJur^ aun tiro Jitfiires unequal or not 
 mniiJitr ; ainj Ji'/are and a line purtdlvl to the plane of that fif/ure, any 
 Jlffxrc and a /uiint, any tiro linen not parallel, hat sitaalcd in phmes paral- 
 lel to each otiicr ; siiy, Cor iiisliiiu-n : two oiiiinl oi' ini('(|Uiil HqiiaroH ; u 
 siiiinrii and any recta iijfl(! ; any two r«H;taii;;U's or iiaralli'logninirt ; 
 nny two ('([iial oi' siuiilur, unequal or dis.siiuilar polyiifona, of which 
 tli(! sich'sol'tlie one eonespdiul either to |)aiallel Hides or to nn<ruhir 
 points of tin- otiK'r ; a s(|uare, reetanj^le orotht^r polygon and a circle 
 or ellipsis (inlinilary polygon); a circle and any ellipsis or any two 
 illij/ses (this latter prisnioid with eiirvilineal parallel bases is some- 
 times known uiid(>r the name oi' cylindroid) a sqiuire, rectangle, pa- 
 rallellogiam, polygon, circle, ellipsis and a line ; a Hipnire, rectangle, 
 paralh^logram, jtolygon, circle, ellipsis and a point ; two lineaof any 
 lengths not parallel but situated in planes parallel to each other. * 
 
 (30?) ISI^rfl II. There is now to bo considered the kind or 
 nature of the figure answering as an intermediate section between the 
 o])poKite bases of the priamoid to be measured. Thus, it is plain 
 that if the opimsite bases are rectangles with parallel sides, the in- 
 termediate ]>arallel section will also be a njctangle or a square ; if 
 tlie two bases are ])arallellograins with parallel sides, tlie section will 
 also be a parallellogram ; if the bases area square, rtsctangle, paral- 
 lelogram and line jtarallel to one of the sides of such rectangle, 
 &c., the section v.ill still be, in the first case a rectangle, in the second 
 case a rectangle or a square ; in the third case a i)arallelogram ; if the 
 bases are any figure and a point, the section will be a ligure similar to 
 the base and equal (131, T.) in area to the fourth of the base ; if the 
 
 '1. All thenefortns arc raetwith in practice, and more especially amnng the diversified 
 roofs of buildings of all kind?. A tower or square turret for instance, will bo often ter- 
 mioatod by a roof crowned by a cireular or octagon platform, or the tower 
 will have for ground plan a circle, and for platform to roof i square or other polygon, 
 or nguin there may be two squares of which the sides of the one are parallel to the 
 dingoniiKs of tho oiher, that Is for prisinoids of which the parallel bases are any 
 figures. If a building of which the horizontal srction is a square, rectangle, or poly- 
 gon, is covered wUh a roof tonninated by a ridge long or short, we will obtain 
 the pi'ismoid of which one of tho barges is any figure and the other base a line. The 
 wedge is also a solid of the kind. Neither is it rare to find among the component 
 parts of a roof or other object to be measured, prismoids of the kind of that of par 
 (20S, T.) that is, of which tho bases AB, OD are both simple lines, without tha 
 area of tho intorraediato section abed having any tho less for that a real and 
 easily determined value. In this latter case the factor " 4 area alcd " =BAXCD 
 (SOS, T.) whence, the sol. = AB X CD X height -i- 6. 
 
164 
 
 JCET TO THK TAHT-KAH. 
 
 .£'■ 
 
 two hiiftos nrc liiicn (J>07, €4.) tlic! oiui [xTpcinliciiliir !<» tix" (lircctiou 
 of tim other, tin* Hcctioii will br ii hihuiic or a vcctiuij^Ic ; if tlu^ buHcs 
 aro liiK'8 not prrpciKru'iilar tlic one to tlic diicctioii of llic otliiT, tho 
 section will bo ji lozt'iigo or pinaUclognun. 
 
 (JIOS) Notliin;!? is I'asicr in all tlu-Mc ciiscH 
 tlian to (IctrrniiiK! iIm^ area of I lie intt'inicdiato 
 section of which the niiiliiplicis oi' fiu-tors ar<» 
 ench an arillimotical mean between tlie parallel 
 sides of the opposite bases or lielwetMi tlie sides 
 or edges and op|)osite points orapieesas the ease 
 maybe. For inslanee, in the i)ri'^ni(>i(l A15-('I) 
 where eaeh of the bas(>s is a sinipic; line or «'(li;(! 
 AH, CI) and of wlueli the area is eonse<iuently 
 null, "WO obtain for intermediate section the s(]ii;H'e, rectangle or 
 parallelogram ahcd in wliicli ob ~i \]\ i I) or=i AH, since I):=:0, the 
 same, dc=i A\i = ah, and <«l — i CD — he ; whence area section abed 
 = ab X. ad or i A15 x ^ CI) if it is a rectangle, or (8, 'T.) = a'j x ad x 
 nat. siu. angle bad if it is a parallelogram. 
 
 (200) If one of the bases is 
 any polygon and the otiier base 
 also any polygon, and if all the 
 lateral faces of the prisinoid are 
 triangles, that is if each of the 
 sides in one of the bases corresponds to a point in tho other base, 
 the number of sides in the intermediate section will bo equal to tho 
 sum of the numbers of sides in tin; two bases. 
 
 (210) Ifone of the bases is any 
 rectilineal fignr<^, and tin* otlua- 
 base a line not parallel to the sides 
 thereof the number of sides in 
 the int. section Avill be equal to the 
 number of sides of the bnse i»lus 2 j and if the line or one of tho 
 sides, or more than one, of tlie llgure forming one of the bases, is 
 parallel to one or to more than one of the sides of the other base, the 
 number of sides of the interm. section may vary indeflnitelj', as the 
 case may be, but will nevertheless be always easy to determine with 
 the help of a simple sketch of the figure. 
 
 The summary just made may still be simplified, abridged or 
 translated as follows, to wit : The imsmoUl or eylindroid (infimtary j>ris- 
 moidj is a solid with parallel bases ofwhieh the planes (lateral faces) pas- 
 
MKNsrRATION OP SOLIDS, 
 
 165 
 
 siiifi flinuifili the s'/V/rv or rthirs of nvr nf tfir hnsc^, rirr trrmiii<>l"(( hj junnts 
 or bij iiiinillcl sii'rs or v(iijf:< in Ihv other Ixtsr. 
 
 Ill oilier tn iii.-t : Tfir jirl'^inolil or ci/h'ndrniil is such that all its hife- 
 riilfiKcs (irr, or its lafrnil snrfavr innii tic tlcconiposal into triinii/lcs or rtvti- 
 liueol lr/ipc:iiiiiis, Ihift /<, iritlt plune siir/ncvs, or 2Mrts cttpal/le uj'ltdinj de- 
 i'elo2)c(li)ttoj)lfnicmir/<iven. 
 
 Lotus iiil<l lli;it, every i;risiii ',.1 iiijiy lu! decomposed, ifoiioof 
 it8 l)ii.se.s is (I!;;', to \y.\y. {'2VJi'V.) ii ly li-iire niid tli<' other hnse n 
 line, iiilo (wo pyriiniids jiiid iiii I'leiiieiitiiry prismoid liiiviii;:>' lor it.H 
 buses l!:ies, (ll-'. hi par. !iO,S, T.) ; if its two biises nre any li^'iires, 
 into Four pyraiiiid.s liaviii.i'' tiieir buses t,wo iiiid t wo in (lie opposlto 
 buses of tlie solid, and a prisnioid liavinj;- lines lor its bases ; or, nt 
 l»leasnre,into pyramids, wedi-es, &:e.,and prisinoids willi lineal liases, 
 iiccordiiii;- to (lie manner of operaliiiii' (lie division of (lie solid by 
 planes of which the iiiimlier iiiid position may be viiricd. 
 
 (211) Il'oiie <if (he bases is for iiis(anco a 
 circb.' or elliiisls and the other base an elliiisis, 
 th(» iiiteini. base will also bo an ellipsis of wliicii 
 wo will obtain tiie dinm. <;/— i (,AV> \ <tl>) and 
 the dial. . (jh = i {CD + id). 
 
 (3BSS) If one of the, baa(!s is a circb^ or an 
 ellipsis AIJ and the other base a line KF, the, 
 iiiterm. base nbj'vdc will be ■' inixtilineul li^^iire 
 of which the parts (d> and vd will be strai,nlit 
 lines pai'allel to KF, and the parts oed, hj'c simi- 
 lar fi<-ures (103:1, ii.) to CDA, CHD. To calcu- 
 late the area of the interni. section, we h.ive (lOIiS, 520 G.)ab and 
 rfceach = i FF, «(? and ^c each = ^ CI), and as the component i)arf8 
 ACD-E, 15CD-F of the prismoid aie evidently pyramids with mixti- 
 lineal bas(>s, we obtain (IJtl T, ) the area a('d = 
 i area ADC, area h/e, =i area IJCI) ; that is we 
 will obtain area section ef=ab or i EF x ad or 
 be or i CD + i area AH, and if EF is not jiavallel 
 to AB or perpendicular to the direction of DC, 
 we will moreover multiply (S T.) the product 
 ab, be by the nat. sin. of the angle bad or substi- 
 tute for the factor ad or be the perpen<licular 
 breadth of the parallelogram ab cd. 
 
IGG 
 
 KEY TO THE TABLEAU. 
 
 CHJl) II'oiH^ of tlic liiiscs is :i circle, or an 
 <'lli|»si.s All iiiul tii(^ oilier ii s(|iiar(! or iec(iiii,nl(} 
 E(!, t lie )-;i veil piisiiioiil will lie dticomposed iiil;) : 
 1°, ii |i!isiiioi(l KFtJlI -CI) (liiiviii/;' for ils bases 
 ('<J07 'I'.) a .s((iiai(5 or rcctaiinhs and a line, and 
 lor intdrni. section a reel allele, t'fiih where, v/=<ih 
 = -i KF or(;il, uinl c;i=/l,^l VAl i- } ('])) ; 2^ 
 two prisnioid.s AO-KII and 1>() — V(\ (liavini'Ciacii 
 lor ils Itases lines A(), KII and MO, V(i and 
 ^tiiiTl T.) for iiitcriii. l»ase iirectaiinle ((/*//i' wliei-e 
 «(/> — /./mi i;ii iind (tic or/(/-~i Ao, and iiral 
 vliere linn vil =i F(i and lut or jt— A <)1>) ; .'5' I'onr pyramids AOC 
 -II, A()l)-E, I'.OC-(; and 1!()I)-F (having' each I'orils inb-rni. section 
 a lii^nre *)///, ru'/.', ifh and (((//'respectively equal in area to one Ibiirlli 
 of the correspondiiii;' base A()(', AOI), 1>()C and liol), or their sum 
 e(iUid in area to one fourth of the base Al>. 
 
 (iJl I) It is jilain from the ]»rismoids or cylindroids just men- 
 tioned that those bodies may indeiinitely vaiy tlieii' forms, but the 
 in'ccediii;;- coiisidi-rntioiis will siillice to indicale, the manner of pro- 
 ceedin;;' in each case to the, determination of the iiilermediatu area 
 to enter as an element into tin; culcnlalion of the solidity to bo es- 
 tablished ; or if necessary, to determine, immediately whether tlio 
 jn'oi)osed solid is, or not, sncli a prisinoid or cylindroid that its soli- 
 dity may b(( measured by the i^cneral lule here ^i;iven. 
 
 Fx 1. A tent or tester (.f wlii<'li the sup. base is a circle orellip- 
 sisoi«ine metre in area, and its inf. base a reclaii/^le of;5 rietrcrs area, 
 has for its intermediate section ii niixtiiineal linnro of which the area 
 is two metres, height or ptsrpendicular distance between the parallel 
 hases 2i metres ; required the solidity^ of the sjiace or air com- 
 prised between the cm tains i 
 
 Alls. (1 I 3 I 4!ines2)x2.r)-|-0=:5 cubic mefcroa. 
 
 2. A cain]»in;:;' tent of whidi the top or sup. ba.sc is a ridge, 
 that is a simple line or edge 2 yaids long, and of wiiich the 
 inf. bas(^ is composed of a rectangle of 2 xJ^yaids and two semi- 
 circles of y yards diani. is 2 yards in height ; what is its solidity ? 
 
 AiiM. In tins example it is plain that the prismoid to be cubed 
 is composed of a Avedge with cipial edges (that is (mOO, C}.)ofa 
 triangular prism) and two semi-cones, 'i'he area ol" I he, base; of the 
 tent is composed of that of the rectangle— 2 x ;ir:r(; s(]nare yards, and 
 of tlittt of two Bomi-circlcs, that is of u circle \i yards dium. =3 x 3 x 
 
^rF.NSURATION OF SOLFItS. 167 
 
 .785-1= 7.0(180 s(|iiiii-(' ,var<l.<. in all laOCiHO sriiiiirc yiinls ; tlio lu'ca of 
 tlir iiitciiiM'dialc scclion is ((iiial l<» half the icctaiinlr at llic l>as(* 
 ]»Iii.s one foiirlli CJiia, T.) of tlu; two scini-ciirlcs, and lias coiisc- 
 • liiciitly llic value of ."{ I I.7ii7]r)—A.7i'>7irt, 'I'lic aica, of the sup. 
 base l)ciii,y,' null in ilic actual case, Mic sol. — (area Itast^ i I intcrm. 
 area) xhfiuht :(;,-( i:5.()(;ri(; i- 4 limes J. 707 1. i) x ,1 lifi^lit=;W,i:{7xi x 
 2:C>---W.7l2ii'nWH' yards. 
 
 I15MI. irtlic ('\f. sill lace orilic fcst-cr or of (lie caiiipiii;;- tciil of 
 llic two last fxaiiiples, instead of Iicintj taut, that is plane or rajiaolo 
 of l)eiii.i;' dt'veloped (|li'14M».) into a plane siiit'ai'e, were concave or 
 not, laid, we would eipially obtain the rerpiired solidity, at Iciist 
 veiy nearly (ltV.i, l-IO, T.) by (lie same vnle ('HKt, T.) 
 
 Kx J$. An ohservatory of wliicli tlui j;roiind plan is an octo/^ou 
 of !((() metres area, is crowned witli a roof terminated by a circu- 
 lar |)latform of wliirli tlie area is ^J.l nietr<'s, the area (d'tlie interme- 
 dial(^ seclion is .")'! metres ; wlnit is the solidity of the spaci' occupied 
 by tli(! roof the height of which is (I metres '/ 
 
 Aiis. lot) ■ 12.")+ I limes .''.|}:=3 1!) cubic metres. 
 
 PROBLEM LX. 
 
 To determine the accurate solidity of any irregular 
 
 body of small dimensions or of a body composed 
 
 of several elementary parts with different 
 
 dimensions and forms. 
 
 (215) UL'I^B]. I( it iH tlic rapacity off any vase oi* vcm- 
 sel wiiJtBi AV<; livaiit (o measure, the idea (/nicrnJl;/ sii(/fjcf>ts 
 itself of anieiiifj at llie result hi/ ilelenniniiiri the niimher af times irhieh 
 such (I resscl ani ijire jilicce to or contain the contents of nmj other vessel of 
 an eleinentori/forni of nleieh we know the cajxieiti/. 
 
 (Sl<») K tit it' itisllie !«oIl<lity ct^'tfli'O KiiibHlancc ifNcIf 
 of the vf NMcl, &.<'., wiiicli we duMirB to iiioa.Mire, I ' inaii- 
 nor ofoperatin.n' does not immediately present it.self to the luind of 
 any one wisliinj;' to ;.blaiii liie result. 
 
 (217) 8ICUI'<I^ If tlic solidity to bo iiica.siircd i«( tiiat of 
 a noil :il»!<(ori»riitsnl»!ijtan€e, zee unmersc -it in. a vessel full of wnler 
 or anil other liiiuid of u-hieli, we will mcnsurc the dispbicmcwcnt hij means of 
 another vessel of knoirn capaciti/ ; or if the Jirst vessel is large cnouijh 
 and its form rectaniiular or ci/iindrieal and of easi/ ijaiK/inr/, ire will first 
 ])ut in it enoiif/h lii/nid to cover the object to he measured ; havitaj after- 
 wards observed tlie height oftlie level of the water in the vessclj we luill im~ 
 
us 
 
 KKY TO Tin; TAULIvAU. 
 
 li 
 
 I- 
 
 virrse in it tlir ohjucl: in qncsHon ami ohxerrc dfiain tlir Irrd of fhr liijiiid ; 
 ■if noir ire siijijiD-ic. Hull etch ff<i<:lii)H of a iiiclrc, uicli, line or diii/ ullic)' 
 unit of Ike h>:i(jld of the. contdiiiinfi rcfsel corrcxpondx lo a cidnc metre, foot, 
 inch, or line,, iC'c, ire will hare hut lo count llm iinnihrr of siirji miils in 
 theliei'jht of tlie lii^iihiceil Icrcl of the water lo ohtaiii iinincili'iteli/ the soli- 
 dity of the proposal ohject. 
 
 ('11^) 11° the Riixly is al>'<»i'!»f;iii, ire niai/ forinstiuice use sand 
 or anij other Jlniil siihstiDtce, of the hiiul, that ire can heel the snrjacc 
 of III/ means of a rou tcilh a, rectilineal cdije. 
 
 Ill tliis iiiiiimcr \V(>, would arrives al, \hv. solidify ofllic most; dc- 
 voisilicd bodies ol'tlKi iiiiiiiial, vcj^ctablc or iiiiiicral kiii.ii(loiii and of 
 the tlioiisaitd and oii(( raw or nianiirat'turcd ohjccls wliicli we have 
 coiislaiilly under our cy^ and of wliicli it would often Ik; inipo.ssUde to 
 iiieasuie tlic soljditie.s l»y the ordinary rules of geometry. 
 
 It is well (o I'eniind also tliat we may airive l>y a simple pro- 
 portion at the solidity of a body by coiniiariui;- its wei^lit with that 
 of anotliia- body of the same substance and of det«Minined solidity, 
 that is l»y the system of s|)e('i(ie gravities whieli shows at the samci 
 time how to obtain the solidity of a body from its weight : which will 
 form the, subject of the next i>rolilem, 
 
 Ex. 1. TluMveigiit of an irregular block of stone is 13 pounds 
 7 ounces : lequired to det»'rmin(! witii the help of the given piece tho 
 weight nearly of a cubic foot of such stone f 
 
 Alls. First cube the block of stone ; to that effect get a rec- 
 tangular vessel, say 10 inches square or 100 inches in horizontal area, 
 and the height of which is divided into inches and hHndr(!t]is 
 of an inch ; having jtourcrd into the vessel water enough to c(»vei' the 
 stone to be cubed, I note the height of the water which I find 8.53 
 i»;ches, I then immerse the stone in the vessel and I nctte again tho 
 height of the water whichis now 9.8i5 inches ; the dillerencc^ of these 
 heights is 1.31! inches. Since the vessel is 10 x 10 inches, it is plain 
 that every iiuih of its height corres|>onds to 100 (Mibic inches and conse- 
 quently, each hundredth of an inch of such a height to one cubic inch ; 
 therefore the observed height 1.3(), of thedisidaced level of tho water 
 Corr(>si)onds to 130 cubic inches ; therefore the solidity of the stono 
 is 13(i, and we will now obtain the weight of the cubic foot by nmk- 
 ing 13G:21.'i ounces (weight of the stone) :: 17:i»8 cubic inches (that 
 is a cubic foot) : 273:i ounces, or, dividing by 10,170;^ pounds, the 
 required weight. 
 
 a. In a cyliudriciil vessel such that each inch of its height cor- 
 
MENSURATION Ol'' SOMDS. 169 
 
 V('s|)(»ii(l< lo I ciiljic incli nf s|);i('(' (ir -iili'litv , we liavc iiiiiii('vs(>(l a 
 piece ofsilvci' w liii'ii lias (lisjil^cd li\ 7;{ liimdrcllis of an iiicli iho 
 level ufliif liqiiitl ill llic vase ; ii'(iiiire'l llic solidily of (lie ingot of 
 .silver / 
 
 Ann. .7'-) of a eiihic iiicIi. 
 i$. Ila'.'iii.n' I'illed widi waler any vessel, we lia\'e iinniersed in it 
 an ol'jecl I lie suliiliiy of wliifli we want to know ; we have, j^alliered 
 in a no! her vessel, llu^ water overllow n, the (|iiantily of wliicli is^J <^al. 
 Xi (piarls and A pint ; what, is the solidity of the propoHed object, tlio 
 f;'a]ion made use of iieiiii; -•{! ciihic iiiclM-s ! 
 
 AtiS. I Million I -i <iiiarts . i pint-'JMl i IMi (- \.i^,;=zm)[} 
 
 cilhic iiii-hes. 
 
 4. Keipiired the solidity of an alisor'oent siihstance i»la(.'ed iti a 
 vessel one loot siiiiarc tilled with sand ; after having- removed the 
 ohjecl to he measured, we find that the iinifoi'm height of tli<' s.iiid 
 in the vessel, tiisi levelled ♦■>tliat etfcct, is .M of a foot, 'he height (»f 
 tile ve>sel lieing !.."» I'eel 
 
 Alls. !..") — .;j:- \ .'2 lee I —height of (he <1 is placed level of (he Hand, 
 and as the v-ssel is I s(inai'e foot in horizontal seeiion, it follows 
 that the solidity of the olijecl is l.'i cnl)i(,' feet. 
 
 5. In a \( ssel ha\ing tin' form of l];e friistnm of a cone is a 
 (piantity of li(iuid of which the di.a meter at the surface is 10 inch(^s ; wo 
 immerse in i( an oliject which increases hy !' inches (he height or depth 
 of the li(pii<l in (he vessel an<I which gives to its disjilaced surface a 
 diameter of II inches ; ■.(Mpiind (he solidily of the |)roposed body ? 
 
 Alls. TIh! volnine of waler displaced which is at the same tiiiio 
 ('ia( of (he object, is ('at of t he fiiis( iim of a cone of which the pa- 
 rallel ba.ses measure r.six'clively ID aad I 1 inches and of wiiich (luj 
 height is!) inches ; this sol. — (Il'i, T.) (ID '(- I I ''-h 4 times 12 )X 
 .7Sryl X !) -;- :^ f:72 x .7^.14 x IS) = (i,-^4.8t)S8 .< 1.5 = V<)27.'M:i2 cubic 
 inches. 
 
 THEOREM LXI. 
 
 To determine the solidity or -weight of any body or. 
 stxbstance, by comparing the volume or weight of 
 such body -with that of a body or r^ubstance of the 
 same nature of which we know^ beforehand the 
 ■weight and volume. 
 
 (Slll>) UJ-.TI. The weight of a- enbic foot of water attlieteni- 
 y)eratiins oi' 40'^ Fahnniieit (at which water nearly reaches it.s 
 greatest density) is IDOD ounces arvir <hi 2)oi(ls nearly, or O^i pounds 
 
 20 
 

 r)!i 
 
 170 
 
 KKY TO THE TABLEAU. 
 
 (cnglisli weiglit) and avc (IciioniiiiiiU' wcimlit or spt'cilic niavit.v of 
 any body or substiuicc, the. wciglit of ii Yoliinic of siuli body or 
 substance t'quiil to tliiit of tb<; water taken for oinpaiison ; wlicnco 
 it results tliat if in advance wo know tie wei.i;ht of a eiibie loot, 
 for instance, of eacb of tbe different substanees thai we may be 
 called on to measure or value, as stated in table X, we will at once »le- 
 teriuine by a sinijile proportion the volume of any oilier weight or 
 quantity of the same .substance or the weight of any oilier volume 
 of such substance, by the following lules. 
 
 (320) ilUtjK. To determine flae Nolulity of ii body 
 from its wei^lil ; )it(thc tlw iiniixn-lidii : (lie .'<iHriJic n(i(jlit of 
 the proposed hoilji is /o (:) its irii<jlit in <iini<rs or jioiiiids, dr. as (::) 
 1 cubic foot or 17'2S citlnv inches, is /(>(:) the soliditif of the bodij in 
 feet or inches, as the case muij l>c. 
 
 Ex. 1. The Aveigiit of a shell or cast iron bailor of any fiag- 
 nieiit of such a solid is 40 i»oiiiids : re(iiiired the solidity of the 
 proposed body ? 
 
 Alls. It is seen by table X of sjk cilic gravities that the weiglit 
 of cast iron is 450 pounds nearly, ])er cubic foot ; we will then obtain 
 the re(iuired solidity by making 4r)l) pounds : \7'lr^ cubic inches :: 4.) 
 pouuds : il'Z.ri cubic inches. 
 
 3. Required the volume of a marble statue tiie weight of which 
 is 1000 pounds, the six'cilic gnivily of tlit; niaiMe fidiii which the 
 statue is drawn being 170 pounds nearly to thv ciil».'c loot '. 
 
 Alls. 170 pounds : 1 cubic foot : : lOOO pounds : 7).\) cubic feet 
 nearly. 
 
 3' A quantity of sand weiglis l.'J i>ouii(Is : what is its solidity ? 
 
 Ans. From table X, the specilic gravity of sand is J..')20, that 
 is, 1.5!20 tinu's the weight of an e(pial volume of water or loiJO 
 ounces to the cubic foot (since tlm weight of a cubic foot of water is 
 lOOO(Minces) ; we will therefore make I.V^I) ounces : 17'^^ cubic inches :: 
 (iyx|G=) 208 ounces : X = 17'JH x -JOr* =s",*;5(jA cubic inches. 
 
 i.v:io 
 
 4. 'Yhv weight of a tusk or looth of an elephant is 'ZTt pounds ; 
 what is its .solidity ? 
 
 Alls. Ivory is 1825 ounces to (he cubic foot ; we will therefore 
 obtain (he scdidity of the tooth by nndiing 182.'> : I :: (25 imunds or) 
 400 ounces : .22 nearly of a cubic foot, or 1825 ounces : 1728 cubic 
 inches :: 400 ounces : ;}78.74 cubic inclx's. 
 
 .1. It is required to determine in advance the juobable weight 
 of a cast iron grating wliicii must be cast ai-cording to a carved mo- 
 del of pine wood the weight of wJiich ia 7 pounds 7 
 
MKNSURATrON OP SOLIDS. 171 
 
 Ans. We will fiist obbiiii tlio solidity of tho pine model by 
 lunlviiisj, iis i)('r iiih^ (Mio ])irio beiiiif considered in tliis case as of 25 
 pounds fo ( lie eubic foot) 2.") ponnds : I cubic fi)()t :: 7 pounds : .28 of a 
 cubic foor. Xow, as tlie solidity of the cast iion will also bo = .28 of a 
 cubic foot and the wei.n'ht of cast iron is 450 pounds jter cubic 
 foot, we will obtain the weight of the proposed grating=:4oO x .28= 
 126 pounds. 
 
 (S'11) KITLR. To determine the weigrlit ot a body 
 from its volume: make the proi)orti(Mi : as one cubicfoot is to( : ) 
 the volume of thi" in()[>oscd body, so is ( :: ) its specific gravity to ( : ) 
 its weight. 
 
 K\. 1. Tiic vohinie ot a heap of snow on the roof of a building 
 
 is 7000 cubic I'eet, the weigiit of a cubic foot of this snow, made 
 h(%'<vy l)y rain. &c. is .'JO pouiuls : required the total weight which 
 bears on the roof l 
 
 An§. 7000x30=210,000 pounds. 
 3. What is the weiglit of a piece of |)ure cast gold the dimen- 
 sions of which are 3 indies by J x i inches ? 
 
 Ans. Th<' solidity=3 < 1 x -J^-2f cubic inches j the speciflcgravity 
 of pure gold is l!).2.')8 ; tiu> rule gives : 1 cubic foot or 1728 cubic 
 inches : ■^r cubic iuciies :: 1!>.258 : x = 11).2.")H x 2.25=25.07552 ounces 
 
 ~ 1728 
 
 3. One desires to know the weight of a lirkin of butter the vo- 
 
 Innie of which oittained from the rule to article (112), is 1830 
 cubic inches ? 
 
 An?*. The specific weight of the butter is .940 of that of wat i-, 
 that is, ofiMO ounces to the cubic foot ; we will therefore obtain the 
 required weight=JI830j<!)4()=y!)5i ounces, -f-l()=(32pounds3i ounces. 
 ~r728 
 
 4. What is the wc'ight nearly of a stick ofenglish oak half-dry, 
 the volume of which is 150 cubic feet ? 
 
 Ans. The half-dry oak, from the table, is 60 pounds nearly per 
 cubic foot, whence the recpiired weight, is 150 x 66=9900 pounds. 
 
 5. What is the weight nearly of a box of bound books the vo- 
 lume of wiiich is 15 cubic feet ? 
 
 Ans. 15 cubic feet x 43 pounds nearly =645 pounds. 
 
172 
 
 KRT TO THE TABLEAU. 
 
 PROBLEM LXII. 
 
 I jl 
 
 To determine the specific gravity of any body or 
 
 substance. 
 
 (222) Rt'fjK. I. Cnhc an\l trcifih the pyopoxnl IkhIi/, (UkI a/'trr- 
 wards niiilce fliis propoiiiDii ; as tin' Noliilili/ of lint haih/ /.v l'> {:) //s- 
 irrifilit ill oiniccs, so is (::)(( vnhic ftxit of such hixhj /'»(;) the iniijlit 
 of one foot of it in ouncvs ; that in, Inj cuttinfj off three Jitjiircs for deei- 
 iis sjfveife (irariti/. 
 
 Ex. 1. Wliiit is (lie specific \vri!;li(- of scMsoiicd l)];ick \v;il- 
 iint, ifii sniiipl(M)l' tills wood tlic (liiiiciisioiis oi' wliicli lUf 'I ■; 7 x .!> 
 iuclios, Avoiylis 2i oiiiucs t 
 
 A1I8. II X 7 '< .*.)-(')'.). ^^ cnh'ic inclics - sol. ol'tlio proposed body ; 
 HOW, from tlie rule (!!>.•'{ iiiclics : 21 ounces :: 17^JS iiiclies : .")!)H ounces 
 or;J7.4 jiounds ; tlie reipiircd spe( itlc j;ra .-ity is lliei< Core .')\>6 (d'tliiit 
 of Wiitev tlie weight ofwliicli is KMKI oi;«c<'s (o IJie cubic loot. 
 
 2. An ii'n\<;iiliir [Mice oi' chalk of wliicli tlie solidily ]i;is 
 l)Con obtained. = ]MvJ «'uhic inches, liy the method of example 1 of 
 the lust but one jnohleui, weijihs -1;{A pounds : ic(piiied (he s[K'ciUc 
 gravity of tiiat substance. 
 
 AnS' -J-'J^ inches : 17;JH inches :: 4.'}i ])ounds : 171 pounds : 
 ■whence, tlio re(pured s|iecilic .iniaviry is 174 :< l(Jr:r*2.7c; J tjuu^s tlie 
 weighlof an equal vohnne (»f water. 
 
 3. A bateau oi- pontoon of 100 feet by 20 x 10 feet and tlie total 
 Yolume of wliicli is consecpUMitly 20,000 cubic feet, i-e((uired in its 
 coustnu'tion .")000 feet ol'white pine half-seasoiii'd, the wei,i;ht of which 
 is estimated at 40 [K)unds to the cul)ic foot, MOO cubic feet of elm com- 
 puted at oO pounds to the cubic loot, and .")000 pounds wei.^iit ofirou 
 spikes : required the draui^hft of water of the proposed body ? 
 
 Ans. The weight of the pine-.')000 < 40 = 200,000 pounds, the 
 weight of the elm = ."»00 x .")()=; 2.j000, (lie iron ilOOO pounds ; tiie total 
 weight of the bateau is conse(iuently 2;{0,000 lbs ; (he avenige weight 
 or the apecilie gravity of tlie pontoon is 280,000 pouiids-^20,000 cubic 
 feet = 11.5 poH'i 'Is ^^^ tl'<' cubic foot, that is ll.."> ; 1<)= 1H4 ounces ])er 
 cubic foot, say .184 of (he weiglit of an ''(jual voliimt! of watei'. 'I'lie 
 height of the iioontoon is 10 i'eet, therefore! the draught will be .184 of 
 the height of the pontoou or 1.84 feet, that is 1 loot 10 inches and .96 
 of an inch =^1 foot II inches nearly. 
 
 4. liy wha,t <iuantity can the bateau or poontoon of the last 
 example be loaded without causing it to founder or sink beyond its?. 
 deck or superior surface ? 
 
JIKNSUlJA'l'IdN OF SOI.IIiK. 173 
 
 Ann. Since wahT wcinliH t'rl.r, iioiiiids fo (lie ciiliic fool .-iikI \]w 
 total volume lit" (lie ]i()iit(>oii is •JO.OOO cuhic i',.,'r. the lolal wii^ht of 
 tlie wuhT vliicli the ]»()iit.Miii iiinsl .lis|ilace hefoie .siiikiii-- to llic 
 level of the wahT is ^id.dllO x Cj.r.iir |.^r,(),()()0 pomnis ; ii(i\v the 
 weij;iit (.f the iM.at is IxH :i:j(l,()()0 puiiiids ; wiieiiee it follows lint w »>, 
 iiii.nlit slii! wiliioiii caiisiii,!;- ihel.ateaii to l'oiiii(lei' load ii wiih a, 
 wei.uhl e(]i!al or luaily eqiiai to the (iilfereiiei- hetwceii i-J,";i),ni((» 
 pounds and ^.';i(».()l(lt. that is l():jO,()(K( poinnis. 
 
 (22:S> Ct3 L5] II. i!nh«>i»o<l) to Ik> coiispii!l«>«l in hea- 
 vier utiaii w;;S<"B' : _//;w/ ((•(■■■;/// Ihc hu,Iii ill air, thru 1,1 ir,,i,r, 
 hji wcdiix <>f(i li:/(lnnilii' hahnirr ; llic (lij/rrnirc httirrrn l/ic nsiilln 
 will lie ihc iriiijJil lost in iriilrr. or lliv irritjlit ,if<i ijiniiilil;/ n/ iralrr cninl. 
 ill rijliniir Id IIkiI of ll'r lioil:/. Mole iioir !lir jiroihii'fioii : ii>i Ihr nrii/lif: 
 losi ill ir(ilrr{:) is lo llic irci<ilit of llic hadji in air ( :: ) xa is llic 
 specif c iirnrilij of iniirr { :) to llic s-prcifc r/rai-iti/ if Ihc lu-ilc. 
 
 Biix. 1. A piece of till wei^ilis \t^.\ pounds, its \vei,i;ii(: in \\al(i' 
 is but loS pounds : what is tlu' speeiiie .uravit.v of tin .' 
 
 Ans. IS:{ -1.18 = 2.1 : IHM :: l(t(t(» : 7:5-J()=re(piired speeiiie _i.i';ivit v. 
 
 a. A liloek of oranite vx'i'^hs -Zl (niiiees in air and only l;j 
 ounces in water : Vvliat is llie speeiiie giavity of t lie ^lani.'e .' 
 
 All?.-. i^-M-.M. 
 
 (221) KII.8: 3 23. 14 SS»o J>o«1y lo be : Oisipiiitd ix 
 lig'liicf Uiasi AVJiJrr : lie lo Ihc /irojioscd hodjihij a llinml Ihc rniqhl 
 of which is irl<ilircli/ null, another boili/ header Ih'in irali r. so thai 
 both of Ihein luken loijelher iikiij penelrnle or sink in the iratrr ; harin,/ 
 frsi ireiijhed each hoih/ in air, and the hciieier in iniler, ini(/h then in. 
 Wilier Ihc compound hodij, and from the wciahl lout hij Ihc comjioniid 
 J>od!/,siilislr(icl the wciiihl lost hi/ Ihc hairier iiodi/ as wci<jhed alone ; 
 the remainder is the wrii/hl losi hi/ the lii/hi hodij. Then : us ,'hc irrii/ht 
 lost hi/ Ihc iii/hl hod i; in irater, (:) is lo the wrii/ht of' that hod,/ i„ 
 air, ( :: ) so is Ihc specif c ijraril;/ of water {:) lo the specif c i/rarifi/ of 
 the hodij. 
 
 liX. I. To a piece of elm which in air Avei^i^lis I.") i>iaiiis. we. 
 liavo tied a l)iec(^ ofcoppei' liie weight of which is 18 grains /// o/>- 
 uud IG grains in water, and the conii)oun<l in water weighs hut (5 
 grains : what is tiie specilic gravity of tiie elm ? 
 
 Alls. 18— If) —'i =tlie number ((fgraias lost by I Ik; co]ii)er 
 
 ill Ihc water. 
 18 I- 1.")— r):=r27 = t!ie number of "grains lost by tlie coin- 
 
 ])oiind in Ihc water. 
 27—2 =:2.} = tlie number of grains lost by the ehu 
 
 in the water. 
 2". : 15 :: lOOO : li()0.:::tlie speciOo gravity of llie elm. 
 
till 
 
 V'ii 
 
 Y Ij.!. ' 
 
 174 
 
 ICi Y TO TIIK TAnI>F,A'^ 
 
 U. A pi'-cc orcoii])*']'. wci^Iiiiii; in iiir 27 oiinoos ntid in wnh-r^l 
 oniKM's. is tied Id ;i picci- of cnrk \vrii;liiii'j,- i'l ;iir <> (hiiicc.-j, iiinl llid 
 coiiipoiind \V('ii;!is in wMtcr hut.") (Ciiicc; : wliiit is (lie spec! lii; ;^Tii- 
 vit\- (if cork ? 
 
 Aj»n. ().t>lO. 
 
 PROBLEM LXIII. 
 
 To d teriTiiae the quatitiby of each injrvoclient or clement 
 in a co.npound of two sub^tanoes or elemDnts. 
 
 (ti'i'rl) \ii,V ijVj. P^iii'l JJ::-:! Ihe '<iifi-ill(: irri')lit (ifthr nuiiiiinniil, mi.vtnre 
 oi'(iIli)f/, oihI (tf rai'li itfthc c'tm}>i)iu'ti^ clemi^nt^mcl iiitillipli/ thciliji'cir.ncr, of 
 ci^nri/ tiro of tlK"<e thira s/trcllic irrl'/ht-i hji Hirlliiftl. Mahw fhni : tlir i/rrtd- 
 efit proilK'f, ■ : ) ■/.•;■ to mch of the ollwr j/roiliicts, ( :: ) as the wciijht of the 
 (illoi/. ( : ) v's' /" III'' ii'i'ii/lil of raii'i im/rcilifiif. 
 
 D'x. 1 A m.Ks of i^old iiinl silver \V!'i'j,iis (i*.* ounces. ;ui(l its s]»fi- 
 cific .yr;i\ iiv :s |(!|'^:> ; what is llie (|iiiuitily of t'iic'i iiiureiliciit, tiio 
 sju'ciHc ,i;iavity of uold l)eiii,<x liXJlO. jiikI liiat of silver JlOiU i 
 Aas. (i:>:;i(l-ll<»:M) -. liil'^i)- i:}r,S(;i.174. Alku-. 
 
 (i:>(;!()-i(;i^(;) X IH):>I— ;h.i»7;{.77I. silver. 
 
 (Kll-^O ^ll<»!»|) - li»(ii()^!>S,887,4()0. (iold. 
 i;37,8(il,17-l : ;»8.88S,4lM) :: (i;{ : !.'> ounces, -l penny wei.niiis, ID ^^rains 
 
 of.'Aold. 
 |;]7.8()i,17i : :j-^,i»7;],774 :: (JM : 17 ounces, l(J penny weiglits, "> gniins 
 
 of silver. 
 
 S. A mass of copper and <;()1.1 wei!j;]is 18 ounces, and its specific 
 gravity is I7l.'j0, (lie siiecilic .-iravily of j^old is I!)(M() and that of cop- 
 jter !>()()() : what is tlie ([Uantity of each element of the mixture f 
 
 A31!«i. (i<>l<l ^-42 ounces 2 p»'nny wei,i;hts 2 ^',',',11; grains, copper 
 = .> ounces. 17 penny weights^! 'i;^,',!" g''"''"*- 
 
 ;5. An alloy of silver and copper weighs (!l) ounces, its specific 
 gravity l)eing lO.VT) : vequived the weiglit of each ingredient, tlieir 
 respective six'cidc gravities being ll(l!M andDU'X) f 
 
 Alls. M ounces 7 penny-weights !» jilalirTa g''Sii"^ silver, 13 
 ounces V2 penny-woights 14 ^^,,^."0 of copi)er. 
 
 l. An alloy ofcojiper and tin weighs llxJ i)ouik1s and its specific 
 gravity is 8784, what is the (piantitj- of eacii of the ingredi(Hits of tlio 
 mixture, tlieir respective specilic gravities being !)()()() and 7'320 ? 
 
 Alls. 100 pounds copper, 12 pounds tin, 
 
 5. Kc(piired the weight of gold, in a compouml of quartz and 
 gold the specific gravity of which is 3500, that of gold being 19(540 
 and that (.f quartz 3000 ? 
 
AlKN-UllATKiN OK St'Mlis. 175 
 
 Ans. incio-Mooo— uidio, locio < ;r»()()— nn.Q'o.oodrr 
 
 Fiicltii' lor I he cdiiijoiiiiiI liody. 
 
 l!itil()--;rj()i) it;i !u, ini lo v ;!()()(>=:. I8,4vJ(»,(H)()= 
 
 Fiictor for tlic (|ii,irlz. 
 
 F'U'tor tor tlic u<i!(l. 
 r)S'^l()Ui)0 : !',-i(M)(in :: 100 : lO.Hll.Siirjii:}-— oiiiiccs of .^old ; if tliis rc- 
 silU Ik- (•on»'ct. (lie \Vfi;;lir of tlic <|ii;ii{z iiiilsl lie cqiiill to llir dilVcl'- 
 ciicc hct wcMi llii' wri^lil ol' tli('U,'ol(l iiiid llj:if (»!' tliciillov, ami in 
 fiict .>:J 10(10(1 : -l.-^i-JOOOO :: 100 : S:i.|:i.-r;!i;vl T oiiiicis of (iii;iit/. ; llio 
 KMiii of tlicse iiiinilifis:^ 100 ; lliciL'forc, iVc. 
 
 PROBLEM LXIV. 
 
 To determine the solidity of the largest piece of squared 
 
 timber that may be got out of a round log, or 
 
 out of felled or standing tree. 
 
 (22«) IHjL.K. Mnltiidii the duniirta- <>f l!,<: tree nr loj h;/ Ihr lialf- 
 dutmrtcr, and tlil'^ prudnd bij the Icnijtk : the ranlt wiU be the rc'iiiircd 
 soUditi/. 
 
 Ill fact, it is plain lliat the diaiii. AI5 iiiiiUi|ilit'(l 
 l>y tlif liair-<liaiii('t«'r (>(" (or A Al'o </wrn for 
 l)i'odiict tli(^ aica of tlio iiisciihcd s(i!iai(' AlK'D, 
 tiiiit is, tli(! an'iiofa section, of tho linilicr to In; 
 couip.-^cd, l),v ;i i)liuu' ii('i|n'ii(iicular to its liii^ili, 
 and that urea inii!tiiili((l by the Scii'^lh of tlic lo,n' 
 gives {"is, T.) (lie re(|iiiied solidily. 
 
 iCI'Ml. Tiiis rule supposes that the diaiii. of llietice is evciy 
 ■\vliere. the same or (hat we make use of a uieaii (!iaiiie!( r, as taken 
 jit the. middle of thi^ Ieiii;lh, and this is .^-enerally done when iIk re is 
 not too iiuicli dili'ereiice hetweeii tlu^ diainete'.s of the opposiie ends; 
 but io be precise^ (i I>^, 'i'.,) we must as already stated (JJi, i'.) add 
 to (he sum of the areas of (lie ends of iiie ioi; or l\n- lo he 
 measured four times the area of a section fiiken at llieceiitie and 
 jiiiiUi[)ly (lie whole by llie sixth jiarr of tht^ length, or which is (lie 
 same tiiiiij;', multi[)ly the sum of (he areas by tlie whole length and 
 tako the sixth part of the result. 
 
 Ex. 1. The circumference of a ht.n', tlie K'H^tii of whicli is 12 
 f«'et, is (!."2t^ feet, deduction beiiif^' made of the bai k if necessary : 
 how many cubic feet of woo;! will there be in ihe slick of sijiiared 
 tiiubur to bu yot out uf the log I 
 

 170 
 
 KI.Y TO Tin; TAlN.IAn. 
 
 •' ,t 
 
 >%IIS. 'I'lic circ. •!,■-'-' ('((ncspoiKl-* (o ;i diiiiii. 'i, t lie seel ion of tlio 
 t iiiiltcr will llicrdurt' !ir '.' . I -"j sqiiaii' I'fi'l in area, and as I iif l('iij;tli 
 is l-J, tlic s((li(lily will lie :2\ cnbic I'tfl. 
 
 !5. A tn'c tlic li('i.'-'.!it kT wiiicli is '0 iVi'l, lias fi>r its snj*. diain. 
 Mf) inclii's, and foi' its inl". diaiii. -'{il imiics. lor its intctni. diani. :{:{ 
 iiiclics : wlial is the solidil.v ol'tlic jtiici' (if .•^inarr tiniWiT tliut may 
 1m' ;;()I oul (if it. 
 
 Ans. Area small ('ml=2ix \\ fVct •-= 3.12.') sup. feet, invix liiif,'o 
 i'lid- :{ l.v— l,.j s(ii>. feet, iMiciincdiatc arca^JJ.'.') < l.:!7.1 -."{.THl'-J.'), 
 ■1 inli'inii'diatc ai« a - I.'). l'J."i, tht- sum of li»c UU'U.S=;^~.7.> and lliut 
 sum X .')(• -(l-~- ISi'.i) ciiliic fc'l. 
 
 ;5. W'c have iiicasuicd at. "> places nearly eijiudislant liy nieanH 
 of a tiui'Iviiess eomiiass, the diam. of an irrenidar ticejnst felled ; 
 these (li.-.meleis art- respectively ;5!», .Mi'V. 1)6, ;j7i and -M inches, and 
 tie' 1( ii-lh of ihe tree 40 feetj what will its .solidity l»o after it has 
 been si|iiai('d. 
 
 Aois. The Sinn of the diametx-rs lf)0 inches -f-.'jzr.lS inclieH= 
 mean diain.^JJ,'^ feet, ."{.I (it! x i.."ir':J=^."».(ll'~I nearly = ai'ea of the .section ; 
 inuliiplyiny this latter liy theleiiyth 10, we, ;;-et '-JOOi cubic feet. 
 
 PROBLEM LXV. 
 
 To cube a stick of timber AB -which is but partly 
 
 squared, or of which the edges or anglss are 
 
 ■wanting, called " waney timber." 
 
 (•i'^iJ) DIUI^E, HijHiu'c till' (limn. Al> oj' titc (iiiibcr, (iiid from 
 such sijicirc siilitntct thdt nf (Iw (Hkiii. ah of the ndpiroiul, the difference 
 of flie'<e s'jiiiircs iiiitllijjlied by the lenijth of the tiiidwr, trill lie tl'". re'itiired 
 ,^uiidit;/. 
 
 in fact, it is jilaiu tliat tlu! sni{ac(! 
 wanting- at each of the four an,i>I(!s, corners 
 of edu'es of l!ie timber, to coniph't(^ tlie 
 s(piare A 15, is tiie triani^h! oho, or ii tri- 
 an.ule e(jnal to (tbo, when as it is snpi»osed, 
 ('J'—ilh = l,!--<(b;\\t>\\ the sijiiare on ai is 
 worth i (tbo ; tlierel'oie, &c. 
 
 iiL.'ti. B. If the sides ((/;,(/, i^'Lc. are 
 not e{[iial to caili other, we nniy tak(.- one 
 foartii of til". Slim of tliesi four sides for u mean diameter (ib, or for 
 <;-rcater acuMiracy, W(i \iill ma.'vc separately t!ie sipiares ol' ab, ef, &e., 
 and llie ioui ill of the, sum of llioise .S(pia,rcs will be, or tlie, sum of tho 
 
 
MENSURATION OP SOLIDS. 177 
 
 fourtlia of tliortdfltiiiiucs will 1»' tho qiiuiitil^y, noarl.v, Id lie MiJjiraci d 
 frum Mic Hqiiiuo AB to obtain tli(^ net area of tlio h( ctioii uf tiio 
 tiiiibor. 
 
 niKin. II. Lot U3ol),s(Mvraain tli«' laHtproblfiii liiat ifdic tiinbor 
 is not tliroii;j;lioiiL its ciitiitt length of rtiiial hv/m, its Hcotioii must bo 
 tiikeii at about tho middle of its Iciij^tli, and tiiis is ^jfc no rally what is 
 <lon('(l IS T-)or, \V(* will (Ictcruiini! stivcral sections of ibi- tiinbi rand 
 tlieu take their nieiin, or linali.v we will make tlio sum of the areas 
 of tlie oi>[>osi(e ends pins four times that of the intermediat*- seetion 
 and afterwards multiply tUo \vliolo by the leoj^tii and take the sixth 
 part of the lesult. 
 
 IlEIVI. III. We must also observe that we may arrive at the 
 area of any re,t,nilar or symnuitrical oela^on or of llie kind here illiis- 
 tratcd by snlttraeliii'jf fi(»m tli<^ .sqmini of the [xsriiendieular distance 
 AH which separates any two of its i)arallel sides, the scpnuo of 
 one ah of the sides adjacent to the first. 
 
 Ex. I. An ei;,flit sided i)illar is .'{ f«'et wide or thick Al?, the side 
 ah of tiie chamfer aoh is (J inches : what is the solidity of the pillar* 
 its leugth or height being 10 feet f 
 
 An.s. ('i + 3— (.5 X ..'))=8.75 .superlkial feet, and 8.75 < 10=87.5 
 cubic feet=re(iuired solidity. 
 
 3. A log of timber the edges of which are waney, rn(%'isnre.s .'30 
 inches square and JiO feet long, the average of the sides ah, vf, &c. 
 of the wa!ie is !> inclies : what is the .solidity of th(! timl>er ? 
 
 Ans. (30x;{0) minus (!)x!))=OI9 square inches=area of the 
 section of the timber=e.y82 feet very nearly, and G.'362 x 30=191.46 
 cubic feet. 
 
 3. We have reduced to 30 inches square at the large end a tree 
 tlio diam. of which Avas at that point 30 inches ; at the small end the 
 diam. 30 inches has been reduced to S-l inches j tlie wane, sapwood or 
 defect from a true s(iuare ah is from 7 to (> inches resjiectively 
 at the tw() ends, sucih as obtained by a direct measurement of the piece 
 of wood to be cubed, oi' by means of a sketcli made from a scale of 
 equal parts : what is the solidity of the timber, its length being 60 
 feet ? 
 
 Ans. Area at tlie largo end = (30 x 30) — (7 x.7) = S'A square 
 inches, area at small end = (25 x i>.j) — (6x())i=581) sq. f., the inter- 
 mediate area (^' x ^£±^)_(^?>6^7^N 3.(27ix27i)_ 
 
 (6i X Gi)=27.5'~6..5^ =750.25 — 43.25 = 714 j 851 + 859 + 4 times 714 
 = 4296 square inches, dividing by 144 we obtain 29.83J3 square feet, 
 multiplying by ^ of .the length or by 10 we obtain 298.33 cubic feet 
 
I'! i 
 
 17H 
 
 K1,Y T<» TIIK TAnr.KAU. 
 
 III 1 1 
 
 
 ivi 
 
 AiiH. Aiiii Mciioii at llic roll* (^^ri-l H(iiiar«* iiioliis, 711 : 11-1 = 
 .lltKl s«iiian« fiof, l.!».")>S;{K(i(» -2i»7.l!W ciihic (Vet, llial is, ^'^u^.^\ to 
 tlu' accurate solidity Itylcss than one loot nearly, or by Icsm than niid 
 .■{(lOili nearly, or hy Ichh tlian t)no lliinl nearly of 1 pur cent, Hiilli- 
 cient iiccuracy (11^ T.) in pnictict). 
 
 ItI'MI. IV. A coini»aris»»n (tf the two (iiiMwtUM of tliu hiHt pro- 
 hh'ia indicate.-! siilih'ieiilly that the ordinary pratMice of cullers, 
 who taki> the dinn-n.sion.s of u lo^' at. tli(> uiiddl(>. of il.s leni^th, and 
 afterwards niulliply the area, of the Miction at that place l»y the 
 leii!?tli of the timber, to (diiaintlniH it s soiidit \-. is, «'oii.sid(>rlng all 
 thiug.-*, QiA*, 'II'.; —inciiiMKd '»> cm iiiu..i.iiue,> 
 
 m 
 
 'js;' 
 
 ^r^<S*^T^i 
 
I« 
 

 TABLES 
 
 OF 
 
 1. Squares and Square Roots of numbers from 1 to 1600 
 
 II. Circumferences and areas of circles of diameter v\ to 
 150, advancing by J. 
 
 III. Circumferences and areas of circles of diameter tV to 
 
 100, advancing by ^. 
 
 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 diameter 
 1 to 100 advancing by |. 
 
 VI. Lengths of circular arcs, to diameter 1 divided into 
 1000 equal parts. 
 
 VII. Lengths of semi-elliptic arcs to transverse diameter 
 1 divided into 1000 equal parts. 
 
 Vlll. Areas of the segments of a circle to diameter 1 di- 
 vided into 1000 equal parts. 
 
 IX. Areas of the zones of a circle to diameter 1 divided 
 into 1000 equal parts. 
 
 X. Specific gravities or weights of bodies of all kinds 
 solid, fluid, liquid and gazeous. 
 
 !'i 
 
2 
 
 TABLE OF SQUARES, SQUARK ROOTS 
 
 No. 
 
 Square. 
 
 S(ire. root. . 
 
 No. 
 
 Square. 
 
 Sqrc. rof>t. j 
 7.8102497 ! 
 
 No. 
 121 
 
 Square. 
 14641 
 
 Sqre. root. 
 11.0000000 
 
 1 
 
 1 
 
 1.0000000 
 
 61 
 
 3721 
 
 2 
 
 4 
 
 1.4142131; 
 
 62 
 
 3844 
 
 7.8740079 
 
 122 
 
 rH31 
 
 11.0153610 
 
 3 
 
 9 
 
 1 .7320.J0S 
 
 63 
 
 3969 
 
 7.9372539 ' 
 
 12.! 
 
 15129 
 
 1 1 .0;i053(;5 
 
 4 
 
 16 
 
 2.0000000 
 
 61 
 
 4096 
 
 8.0000001) ! 
 
 124 
 
 15376 I 
 
 ll.l;i552S7 
 
 5 
 
 25 
 
 2.2360(180 
 
 65 
 
 4225 
 
 8.0622577 i 
 
 125 
 
 15625 \ 
 
 ll.lso3:;i)!) 
 
 6 
 
 36 
 
 2.4494897 
 
 66 
 
 4356 
 
 8.1240:{.S4 
 
 126 
 
 I.5S76 
 
 11.2249722 
 
 7 
 
 49 
 
 2.6457513 . 
 
 67 
 
 4489 
 
 8.1853528 1 
 
 127 
 
 16129 
 
 11.2694277 
 
 8 
 
 64 
 
 2.8284271 j 
 
 68 
 
 462 1 
 
 8.2462113 i 
 
 128 
 
 16381 ! 
 
 1I.31370S5 
 
 9 
 
 81 
 
 3.0000000 
 
 69 
 
 4761 
 
 8.306(i239 
 
 1 29 
 
 h;641 
 
 11. .•{578 167 
 
 10 
 
 100 
 
 3.1622777 
 
 70 
 
 4900 
 
 8.3(i(i(;oo:! ; 
 
 130 
 
 16900 
 
 11.4017513 
 
 11 
 
 121 
 
 3.3166248 
 
 71 
 
 5011 
 
 8.42(;i4!)S 
 
 131 
 
 17161 ! 
 
 11.4455231 
 
 12 
 
 144 
 
 3.4641016 
 
 72 
 
 5184 
 
 8.4852S14 
 
 132 
 
 17424 1 
 
 11.4-<912.53 
 
 13 
 
 169 
 
 3.6055513 
 
 73 
 
 5329 
 
 8.5440037 | 
 
 133 
 
 \1C>^'.) 1 
 
 1 1 .5325626 
 
 14 
 
 196 
 
 3.7416574 
 
 74 
 
 5476 
 
 8.602325:! ! 
 
 134 
 
 17956 
 
 11.57.5s;569 
 
 15 
 
 225 
 
 3.8220833 
 
 75 
 
 5625 
 
 8.0(102540 
 
 i:!5 
 
 1 S225 
 
 11.6189,)00 
 
 IG 
 
 256 
 
 4.0000000 
 
 76 
 
 5776 
 
 8.7177970 
 
 136 
 
 1S4I)(; 
 
 11.6619038 
 
 17 
 
 289 
 
 4.1231056 ■ 
 
 77 
 
 5929 
 
 8.7749644 
 
 137 
 
 18769 
 
 11.7016999 
 
 18 
 
 324 
 
 4.2426107 
 
 78 
 
 6084 
 
 8.S3l7(i09 
 
 i;:8 
 
 moil 
 
 11.7173401 
 
 19 
 
 361 
 
 4.3585989 
 
 79 
 
 6241 
 
 8.8881944 i 
 
 139 
 
 19.!21 
 
 11.7.S08261 
 
 20 
 
 400 
 
 4.4721360 
 
 80 
 
 6400 
 
 8.9412719 ■ 
 
 140 
 
 19i;00 
 
 11.8321596 
 
 21 
 
 441 
 
 4.5825757 
 
 81 
 
 6561 
 
 9.0000000 
 
 141 
 
 lossi 
 
 11.8743421 
 
 22 
 
 484 
 
 4.6904158 
 
 82 
 
 6724 
 
 9.0.j53,s51 i 
 
 142 
 
 20164 
 
 11.9163753 
 
 23 
 
 529 
 
 4.7958315 ' 
 
 83 
 
 68S9 
 
 9.1 1 01. 336 
 
 143 
 
 20449 
 
 1 1 .95^2607 
 
 24 
 
 576 
 
 4.8989795 
 
 84 
 
 7056 
 
 9.1(551514 
 
 144 
 
 207.((; 
 
 12.0000000 ! 
 
 25 
 
 625 
 
 5.0000000 
 
 85 
 
 7225 
 
 9.219:)4I5 
 
 145 
 
 21025 
 
 I2.i:i41594() 
 
 26 
 
 676 
 
 5.0990195 
 
 86 
 
 7396 
 
 9.273,; 185 
 
 146 
 
 21316 
 
 12.0830460 
 
 27 
 
 729 
 
 5.1961524 
 
 87 
 
 7569 
 
 9.3273791 
 
 147 
 
 21(109 
 
 12.12 i:i.^)57 
 
 28 
 
 784 
 
 5.2915026 
 
 88 
 
 7744 
 
 9.3H08315 I 
 
 148 
 
 21904 
 
 12.16.-)525l 
 
 29 
 
 8-U 
 
 5.3851648 
 
 89 
 
 7921 
 
 9.4339811 
 
 149 
 
 22201 
 
 12.2065556 
 
 30 
 
 900 
 
 5.4772256 
 
 90 
 
 8100 
 
 9.)^i(i8330 
 
 150 
 
 22500 
 
 r2.24744S7 
 
 31 
 
 9(51 
 
 5.5677644 
 
 91 
 
 8281 
 
 9.5393920 | 
 
 151 
 
 22801 
 
 12.2SS2057 
 
 32 
 
 1024 
 
 5.6568542 ; 
 
 92 
 
 8464 
 
 9.5916630 1 
 
 152 
 
 23104 
 
 12.32^8280 
 
 33 
 
 1089 
 
 5.7445626 
 
 93 
 
 8649 
 
 9.613650-: 1 
 
 153 
 
 23109 
 
 12.36:13169 
 
 34 
 
 1156 
 
 5.8309519 1 
 
 94 
 
 8836 
 
 9.6953597 
 
 154 
 
 23716 
 
 12.4096736 
 
 35 
 
 1225 
 
 5.9160798 ■ 
 
 95 
 
 9025 
 
 9.7467943 
 
 155 
 
 24025 
 
 12.449.Si)96 
 
 36 
 
 129G 
 
 6.0000000 
 
 96 
 
 9216 
 
 9.7979590 ' 
 
 156 
 
 24336 
 
 12.4S999(;0 
 
 37 
 
 1369 
 
 6.0827625 
 
 97 
 
 9409 
 
 9..848S57S 
 
 157 
 
 24619 
 
 12.5299641 
 
 38 
 
 1444 
 
 ' M14U40 i 
 
 98 
 
 9604 
 
 9.>i9949l9 
 
 15-^ 
 
 24961 
 
 12.5698051 
 
 39 
 
 1521 
 
 6.2449980 
 
 99 
 
 9801 
 
 9.949S744 , 
 
 159 
 
 2528 1 
 
 12.6095202 
 
 40 
 
 1600 
 
 6.3245553 
 
 100 
 
 10000 
 
 'O.OOOUOOO 
 
 160 
 
 25600 
 
 12.6491106 
 
 41 
 
 1681 
 
 6.40312-12 
 
 101 
 
 10201 
 
 10.049875t; 
 
 161 
 
 25921 
 
 12,6885775 
 
 42 
 
 1764 
 
 6.4807407 : 
 
 102 
 
 10404 
 
 10.0995049 ; 
 
 162 
 
 26214 
 
 12.7279221 
 
 43 
 
 1849 
 
 6.5574385 
 
 103 
 
 10609 
 
 10.1488916 ' 
 
 163 
 
 265(19 
 
 12.7671453 
 
 44 
 
 1936 
 
 6.6332496 
 
 104 
 
 10816 
 
 10.1980390 
 
 164 
 
 2r.8'J6 
 
 r2.b;Mi2485 
 
 45 
 
 2025 
 
 6.7082039 
 
 105 
 
 11025 
 
 10.2469508 
 
 165 
 
 27225 
 
 12.8452326 
 
 46 
 
 2116 
 
 6.7823300 
 
 106 
 
 11236 
 
 10.2956301 
 
 166 
 
 27556 
 
 12.8840987 
 
 47 
 
 2209 
 
 6.8556546 
 
 107 
 
 11419 
 
 10.3440804 
 
 16? 
 
 278.S9 
 
 12.9228480 
 
 48 
 
 2304 
 
 6.9282032 
 
 108 
 
 llGGt 
 
 10.3923018 
 
 168 
 
 28224 
 
 12.9614814 
 
 49 
 
 2401 
 
 7.0000000 
 
 109 
 
 11881 
 
 10.4403065 
 
 169 
 
 28561 
 
 13.00000''0 
 
 50 
 
 2500 
 
 7.0710678 
 
 110 
 
 12100 
 
 10.4880885 
 
 170 
 
 28900 
 
 13.0.3SJ048 
 
 61 
 
 2601 
 
 7.1414284 
 
 111 
 
 12321 
 
 10.5356538 
 
 171 
 
 29241 
 
 13.0766968 
 
 52 
 
 2704 
 
 7.2111026 
 
 112 
 
 12514 
 
 10.58.50052 
 
 172 
 
 295S4 
 
 13.114.^770 
 
 53 
 
 2809 
 
 7.2801099 
 
 113 
 
 12769 
 
 10.6301458 
 
 173 
 
 1 29929 
 
 13.1.-)29464 
 
 64 
 
 2916 
 
 7.3484692 
 
 114 
 
 12996 
 
 10.6770783 
 
 174 
 
 i :i027() 
 
 13.1909060 
 
 65 
 
 3025 
 
 7.4161985 
 
 115 
 
 13225 
 
 10.723S053 
 
 175 
 
 30625 
 
 13.22,S7566 
 
 66 
 
 3136 
 
 7.4833148 
 
 116 
 
 13456 
 
 10.7703296 
 
 176 
 
 30976 
 
 13.2664992 
 
 67 
 
 3249 
 
 7.5498341 
 
 117 
 
 13689 
 
 10.816()5;i8 
 
 177 
 
 31329 
 
 13.3041347 
 
 68 
 
 3364 
 
 7.6157731 
 
 118 
 
 13924 
 
 10.8627805 
 
 178 
 
 31 ('.8 4 
 
 13.3416641 
 
 59 
 
 3481 
 
 7.6811457 
 
 119 
 
 14161 
 
 10.9087121 
 
 179 
 
 32041 
 
 13.3790882 
 
 60 
 
 3600 
 
 7.7459667 
 
 120 
 
 14400 
 
 10.i»544512 
 
 1 ISO 
 
 1 32400 
 
 13.4164079 
 
OF NUMBERS FROM 1 TO IGOO. 
 
 3 
 
 No. I Sqnnrc. 
 
 181 
 182 
 I8;{ 
 181 
 18-) 
 18f; 
 
 187 i 
 
 188 I 
 18'J 
 190 
 15)1 I 
 l'.)2! 
 
 lii;?! 
 
 11)4 i 
 
 111.-) i 
 i;)(; j 
 
 197 I 
 
 1 98 i 
 
 1 99 j 
 2110 ! 
 
 201 I 
 
 202 I 
 
 2();{ 
 
 204 
 20") 
 
 2oi; 
 
 207 
 208 
 209 
 210 
 211 
 212 
 2l;! 
 214 
 21;-) 
 21() 
 217 
 218 
 219 
 220 
 221 
 222 
 22;) 
 224 
 225 
 220 
 227 
 228 
 229 
 2;iO 
 2:il 
 2:{2 
 
 2:i;i 
 2:i4 
 2:i5 
 2;5(; 
 2;{7 
 2;!,s 
 
 2:!9 
 240 
 
 :r27(;i 
 :iri2i 
 s;{».^9 
 :{:i8.-)(; 
 ;it22r) 
 
 .'M.V.li! 
 
 ■ii9i;9 
 
 3-)721 
 ;!('.! 00 
 
 ;i(i48: 
 :i(;8i;4 
 
 37219 
 
 ;i7(;:if> 
 •is(i2:) 
 ;!8ii(; 
 
 38809 
 :'.9204 
 391101 
 40000 
 40101 
 40^01 
 41209 
 4U;!(i 
 42025 
 
 424;;(; 
 
 42819 
 
 4. rid I 
 4:ii;si 
 
 44100 
 4 1.VJI 
 
 4 1914 
 
 45::(i9 
 
 4J79r, 
 4(122:) 
 
 4(;(ir)(i 
 
 470S9 
 47524 
 479(11 
 4.S400 
 488U 
 492S4 
 49729 
 r)OI7(i 
 50(!25 
 51(i7ti 
 51529 
 5P>4 
 5244 1 
 52900 
 5:i:{(il 
 5.;>24 
 5I2S9 
 
 5 IT 5(1 
 5.'):;25 
 
 5:(;9(; 
 5(ii(;9 
 5t;(;i4 
 
 57121 
 57(i00 
 
 Sqrp. root. 
 
 i;!.45:;(;2io 
 
 1;{.490737(1 ' 
 13.527749:i ' 
 
 i3.5(i4i;(;oo '■ 
 i;{.(;oi4705 ! 
 1 ;).(;;;-< 1.^17 . 
 13. Cm 1791:; 
 
 13.7113092 : 
 13.7477271 
 
 i:!.7si()i^>! 
 
 13.^202750 
 13.85(1 10i;5 
 13.S924400 
 13.92S3SS3 i 
 13. '.Ill 12100 
 1 J. 0000000 ; 
 
 I4.035i;(;ss 
 
 14.0712173 
 
 1 l.lOCM.'idO 
 
 14.1 121 3,-)(; I 
 
 11.177 1 109 
 
 ll.212(i70i 
 
 14.2l7.'^('(i-< 
 
 14.2S285(19 
 
 11.31 7S2 11 
 
 I4.;i52700l 
 
 U.3s7l!»I.i 
 
 14.4222051 
 
 I4.45(;s,{2:! 
 
 14.491 37ti7 
 
 ll.525-!;!90 
 
 M.5.;0219-' 
 
 14.5915195 
 
 l4.(;287:!^-i 
 
 i4.(;(;287s3 
 I4.(i9(;9;;85 
 
 ll.7.!09199 
 14.7ii4.^231 
 
 1 i.79st;is(; 
 M.s;;2;!97u | 
 I4.8(;(i0(;s7 1' 
 
 14.H9Si(it;44 II 
 14.9331845 li 
 14.9(;:'(;295 I! 
 15.0ii00000 1! 
 15.03329i;i |. 
 
 I5.()(;(;5i92 !i 
 
 15.099t;i;89 |i 
 15.l:t274(iO I 
 15.1057509 ! 
 
 15.23154(12 ti 
 1 5. 21143375 I ; 
 15 29i05^5 ;. 
 15.329709; , 
 15.31122915 i 
 15.3948043 
 15.11:72480 
 15.459(1248 
 15.4919334 
 
 Xo. 
 
 211 
 242 
 243 
 214 
 245 
 
 2u; 
 
 217 
 
 248 
 
 219 
 
 250 
 
 251 
 
 252 
 
 253 
 
 254 
 
 255 
 
 250 
 
 257 
 
 258 
 
 259 
 
 2(10 
 
 2(11 
 
 2(12 
 
 2(13 
 
 2(14 
 
 2(15 
 
 20(1 
 
 2(17 
 
 208 
 
 209 
 
 270 
 
 271 
 
 272 I 
 
 273 
 
 274 
 
 275 
 
 270 
 
 277 
 
 278 
 
 279 
 
 2^0 
 
 281 
 
 2H2 
 
 2S3 
 
 284 
 
 385 
 
 2S0 
 
 287 
 
 288 
 
 289 
 
 290 
 
 291 
 
 292 
 
 293 
 
 294 
 
 295 
 
 29il 
 
 297 
 
 298 
 
 299 
 
 300 
 
 Square. I Sqre. root. 
 
 580-11 
 
 585114 
 
 59049 
 
 595.1(1 
 
 (10025 
 
 (1051(1 
 
 01009 
 
 01504 
 
 02001 
 
 02500 
 
 03001 
 
 03504 
 
 0I0O9 
 
 0451(1 
 
 05025 
 
 0553(1 
 
 (l(!0l9 
 
 00504 
 
 070s I 
 
 07000 
 
 0S121 
 
 (ISO 1 4 
 
 091(19 
 
 09(19(1 
 
 70225 
 
 70750 
 
 712S9 
 
 71824 
 
 72301 
 
 72900 
 
 73 1 1 1 
 
 739St 
 
 74529 
 
 75070 
 
 75025 
 
 70170 
 
 70729 
 
 7 7 2^4 
 
 77811 
 
 78100 
 
 7H901 
 
 79524 
 
 800.^9 
 
 80050 
 
 81225 
 
 81790 
 
 82:!(19 
 
 829 1 1 
 
 83521 
 
 84100 
 
 84081 
 
 8520 I 
 
 85S19 
 
 801.10 
 
 8(025 
 
 87010 
 
 88209 
 
 8880 1 
 
 89401 
 
 90000 
 
 1; 
 
 :I174( 
 
 15.550.3492 
 15.58S4573 
 15.0204994 ; 
 15.0524758 ! 
 15.0S43871 ! 
 15.7102330 I 
 15.74S0157 I 
 15.7797338 
 15.811.3883 
 15.S429795 
 15.J<745079 
 15.9059737 
 15.9373775 
 15.9(1^7194 
 10.0000000 
 10.0312195 
 10.0023781 
 10.0934709 
 10.1215155 
 10.1551944 
 10.1^<(11141 
 10.2172747 
 10.248(4708 
 10.27SS2OO 
 10.30950(14 
 10.;!IOi;!40 
 10.3707055 
 10.4012195 
 10.4310707 
 10.4020770 
 10.4924225 
 10.5227110 
 10.5529454 
 10.5831240 
 10.0132477 
 10.0433170 
 10.078.3320 
 10.7032931 
 10.7332005 
 10.7030540 
 10.7928550 
 10.82200.38 
 10.8522995 
 10.^,s 194.30 
 10.91 15:!45 
 10.9110743 
 10.9705027 
 17.0000000 
 17.029.3804 
 17.0587221 
 17.0SS0O75 
 17.1172428 
 17.1404282 
 17.1755040 
 7.2010505 
 17.23.30879 
 17.2020705 
 17.2910105 
 17.32050H1 
 
 No. 
 
 301 
 
 302 
 
 30.3 
 
 301 
 
 305 
 
 300 
 
 307 
 
 30.S 
 
 309 
 
 310 
 
 811 
 
 312 
 
 313 
 
 314 
 
 315 
 
 310 
 
 317 
 
 318 
 
 319 
 
 320 
 
 321 
 
 322 
 
 323 
 
 324 
 
 325 
 
 32(1 
 
 327 
 
 328 
 
 329 
 
 330 
 
 331 
 
 332 
 
 333 
 
 334 
 
 335 
 
 330 I 
 
 3.37 , 
 
 338 
 
 3;!9 . 
 
 340 
 
 311 I 
 
 342 ' 
 
 343 i 
 344 
 345 i 
 340 
 
 347 ! 
 
 348 1 
 
 349 i 
 
 350 I 
 
 351 I 
 
 352 i 
 
 353 , 
 
 354 I 
 
 355 I 
 350 
 
 357 1 
 
 358 ! 
 
 359 I 
 300 
 
 Sqnaro. 
 
 90001 
 
 91204 
 
 91809 
 
 92410 
 
 93025 
 
 9303(5 
 
 94219 
 
 94804 
 
 954^1 
 
 90100 
 
 90721 
 
 97341 
 
 97909 
 
 98590 
 
 99225 
 
 99850 
 
 100489 
 
 101121 
 
 101701 
 
 102400 
 
 103941 
 
 1030S4 
 
 104329 
 
 104970 
 
 105025 
 
 100270 
 
 100929 
 
 107584 
 
 10M24I 
 
 1 08900 
 
 109501 
 
 110224 
 
 110S,S9 
 
 111550 
 
 1 1 2225 
 
 112890 
 
 113509 
 
 114214 
 
 114921 
 
 115000 
 
 110281 
 
 11G904 
 
 117019 
 
 118330 
 
 119025 
 
 119710 
 
 120409 
 
 121104 
 
 121801 
 
 122500 
 
 123201 
 
 123904 
 
 1 24009 
 
 125310 
 
 120025 
 
 120730 
 
 127449 
 
 128164 
 
 128881 
 
 129000 
 
 Sqre. root. 
 
 17.3193516 
 
 17.3781472 
 
 17.4068952 
 
 17.4355958 
 
 17.4042492 
 
 17.4928557 
 
 17.5214155 
 
 17.5499288 
 
 17.5783958 
 
 17.0008109 
 
 17.0351921 
 
 17.6035217 
 
 17.6918060 
 
 17.7200451 
 
 17.7482393 
 
 17.7703888 
 
 17.8044938 
 
 17.8325515 
 
 17.8605711 
 
 17,88854.38 
 
 17.9104729 
 
 17.9443581 
 
 17.9722008 
 
 18.0000000 
 
 18.0277504 
 
 18.0554701 
 
 18.0831413 
 
 18.1107703 
 
 18.138.3571 
 
 18.1059021 
 
 18.1934054 
 
 18.2208672 
 
 18.2482876 
 
 18.27560)69 
 
 18.3030052 
 
 18.3303028 
 
 18. ,3575598 
 
 18.3847763 
 
 18.4119526 
 
 18.4390889 
 
 18.4(161853 
 
 18.40.32420 
 
 18,5202592 
 
 18.5472370 
 
 18.5741756 
 
 18.6010752 
 
 18.6279360 
 
 18.6547581 
 
 18,6815417 
 
 18.7082869 
 
 18.7349940 
 
 1«. 76 16630 
 
 18,7882942 
 
 18.8148877 
 
 18.8414437 
 
 18.8679623 
 
 18.8944436 
 
 18,9208879 
 
 18,9472963 
 
 18,9736660 
 
4 
 
 TABLE OF SQUARES, SQUARF. ROOTS 
 
 No. 
 
 Square. 
 1 30:12 ! 
 
 Sqre. root. 
 
 No. 
 
 Squiirc. 
 177241 
 
 Sqre. root. 
 
 No. 
 481 
 
 SquiM'o. 
 231.301 
 
 Sfjrc. root. 
 
 19,0000000 
 
 421 
 
 20.5182815 
 
 21.9317122 
 
 ;{(;2 
 
 131044 
 
 19.0202970 
 
 422 
 
 1 78084 
 
 20.54 2(i38(; 
 
 482 
 
 232324 
 
 21.9544984 
 
 :^(;8 
 
 1317t;9 
 
 19.0525589 
 
 423 
 
 178929 
 
 20.5009038 
 
 483 
 
 2332S9 
 
 21.9772010 
 
 3G4 
 
 1 32-19(1 
 
 19.0787840 
 
 424 
 
 179776 
 
 20.5912003 ! 
 
 484 
 
 2:54256 
 
 22.0000000 
 
 3(i5 
 
 133225 
 
 19.1049732 
 
 425 
 
 180625 
 
 20.0155281 
 
 4S5 
 
 2:55225 
 
 22.0227155 
 
 ;W() 
 
 13395(i 
 
 19.1311205 
 
 426 
 
 181476 
 
 20.0:!97074 
 
 480 
 
 2:56196 
 
 22.0151077 
 
 :^(;7 
 
 131(189 
 
 19.1572441 
 
 427 
 
 182329 
 
 20.(iO:!97S3 1 
 
 4S7 
 
 237109 
 
 22.0i;s07t;5 
 
 H()8 
 
 135t24 
 
 19.18;!3201 
 
 428 
 
 183184 
 
 2().0>!81i;09 ' 
 
 4S8 
 
 2:5-!l4l 
 
 22.0907220 
 
 ;{ti'j 
 
 i:>(;i(;i 
 
 19.2093727 
 
 429 
 
 181011 
 
 20.7123152 : 
 
 4.S9 
 
 2:19121 
 
 22.li:;:5U4 
 
 ;J70 
 
 13t;90(l 
 
 19.235:;8I1 
 
 430 
 
 1MJ900 
 
 20.7:i01414 
 
 490 
 
 240100 
 
 22.1:559130 
 
 371 
 
 137041 
 
 19.2013003 
 
 431 
 
 185701 
 
 20.7005395 
 
 491 
 
 241081 
 
 22.1585198 
 
 H72 
 
 138384 
 
 19.2873015 i 
 
 432 
 
 180024 
 
 20.7840097 
 
 492 
 
 212004 
 
 22.1810730 
 
 373 
 
 139129 
 
 19.:U3207'> 
 
 433 
 
 187489 
 
 20.8080520 j 
 
 49:5 
 
 24.3019 
 
 22.20:50033 
 
 374 
 
 13987() 
 
 19.3390790 
 
 434 
 
 188350 
 
 20.8320607 ; 
 
 494 
 
 244036 
 
 22.2201108 
 
 375 
 
 140(;25 
 
 19.3049107 : 
 
 435 
 
 1 89225 
 
 20.85005:)0 1 
 
 495 
 
 245025 
 
 22.2485955 
 
 37G 
 
 14137(i 
 
 19.3907194 
 
 430 
 
 1 90096 
 
 20.8,<^00130 i 
 
 490 
 
 2I()016 
 
 22.2710575 
 
 377 
 
 142129 
 
 19.4104878 
 
 437 
 
 190909 
 
 2(1.9015150 ! 
 
 497 
 
 217009 
 
 22. 29:; 4 908 
 
 378 
 
 142,S84 
 
 19.4422221 
 
 438 
 
 191814 
 
 20.!t2>il495 
 
 498 
 
 218001 
 
 22.3159i:;0 
 
 37 i) 
 
 143041 
 
 19.4079223 , 
 
 439 
 
 192721 
 
 20.952;{2(;8 
 
 499 
 
 21901)1 
 
 22.:5:i,- 5079 
 
 3.S0 
 
 144400 
 
 19.4935887 i 
 
 440 
 
 1 93600 
 
 20.9701770 
 
 500 
 
 250000 
 
 22.;5(iO(.798 
 
 381 
 
 145101 
 
 19.5192213 ! 
 
 441 
 
 194481 
 
 21.0000000 
 
 501 
 
 2.-, 1001 
 
 22.38:50293 
 
 382 
 
 145924 
 
 19.5448203 
 
 442 
 
 195:564 
 
 21.02:!7900 
 
 502 
 
 252001 
 
 22.40,V5505 
 
 383 
 
 14G089 
 
 :9.570:!S58 
 
 443 
 
 190249 
 
 21.0475052 
 
 50:5 
 
 25:;i)i)',) 
 
 22.4270015 
 
 384 
 
 147450 
 
 19.5959179 i 
 
 444 
 
 197i;« 
 
 21.07i:i075 
 
 501 
 
 25|i)l,i 
 
 22.449941:5 
 
 385 
 
 148225 
 
 19.62141(;9 ; 
 
 445 
 
 198025 
 
 21.0950231 
 
 505 
 
 255025 
 
 22.4722051 
 
 38(1 
 
 148990 
 
 19.(;4(;s827 
 
 440 
 
 198916 
 
 21.11.S7121 
 
 500 
 
 2500:50 
 
 22.4944438 
 
 387 
 
 149709 
 
 19.0723150 
 
 447 
 
 1 99809 
 
 21.1123745 
 
 507 
 
 25 < 049 
 
 22.5100005 
 
 388 
 
 1 50544 
 
 19.0977150 ' 
 
 448 
 
 200704 
 
 21.1000105 
 
 50S 
 
 2.')8004 
 
 22.53sso;j:5 
 
 38'J 
 
 151321 
 
 19.7230,s29 
 
 449 
 
 201001 
 
 21.1S90201 
 
 509 
 
 2590-<l 
 
 22.50 102S3 
 
 390 
 
 152100 
 
 19.7484177 i 
 
 450 
 
 202500 
 
 2 1.21 320:! I 
 
 510 
 
 200100 
 
 22.5S31790 
 
 391 
 
 152881 
 
 19.7737199 1 
 
 45. 
 
 203401 
 
 21.2367000 
 
 511 
 
 201121 
 
 22.(ilt53091 
 
 392 
 
 153004 
 
 19.7989899 
 
 452 
 
 204:{0t 
 
 21.2002910 
 
 512 
 
 2.12141 
 
 22.6274170 
 
 393 
 
 154449 
 
 19.8242270 
 
 453 
 
 205209 
 
 21.28379(;7 
 
 513 
 
 203109 
 
 22.61950.53 
 
 394 
 
 155230 
 
 19.8494.332 
 
 454 
 
 200110 
 
 2I.:5072758 
 
 514 
 
 2(54190 
 
 22.0715081 
 
 395 
 
 150025 
 
 19.8740009 
 
 455 
 
 207025 
 
 21.3,307290 
 
 515 
 
 265225 
 
 22.(59:50114 
 
 39t; 
 
 150810 
 
 19.8997487 
 
 456 
 
 2079,36 
 
 2l.:i54!505 
 
 51(i 
 
 2()625(; 
 
 22.7150.331 
 
 397 
 
 157009 
 
 19.9248588 
 
 457 
 
 208819 
 
 21. 377558:5 
 
 517 
 
 207289 
 
 22.7370340 
 
 398 
 
 158404 
 
 19.9499373 
 
 458 
 
 209704 
 
 2l.4009;i4(; 
 
 518 
 
 2(58324 
 
 22.759(5134 
 
 399 
 
 159201 
 
 19.9749844 
 
 459 
 
 210081 
 
 21.42 I2S53 
 
 519 
 
 2093(51 
 
 22.7S15715 
 
 400 
 
 100000 
 
 20.0000000 ! 
 
 400 
 
 211600 
 
 21.4470100 
 
 520 
 
 270100 
 
 22.8035085 
 
 401 
 
 100801 
 
 20.0249844 ; 
 
 401 
 
 212521 
 
 21.4709100 ' 
 
 521 
 
 271 Ul 
 
 22.8251244 
 
 402 
 
 101004 
 
 20.0499377 j 
 
 462 
 
 21;M44 
 
 21.4941853 j 
 
 522 
 
 272484 
 
 22.8473193 
 
 403 
 
 102409 
 
 20.0748599 ! 
 
 463 
 
 214,309 
 
 21.5174348 
 
 523 
 
 27:5529 
 
 22..S0919:« 
 
 404 
 
 1(!3210 
 
 20.0997512 1 
 
 464 
 
 215290 
 
 21.5400592 
 
 521 
 
 274570 
 
 22.89104(53 
 
 405 
 
 104025 
 
 20.1240118 ' 
 
 405 
 
 216225 
 
 21.5038587 ; 
 
 525 
 
 275025 
 
 22.9128775 
 
 40i) 
 
 1G4830 
 
 20.1494417 i 
 
 460 
 
 217156 
 
 21. 58703:; 1 i 
 
 526 
 
 270(570 
 
 22.9:540,><99 
 
 407 
 
 1(;6049 
 
 20.1742410 
 
 407 
 
 218089 
 
 21.0101828 
 
 527 
 
 277729 
 
 22.9504800 ' 
 
 408 
 
 166404 
 
 20.1990099 i 
 
 408 
 
 219024 
 
 2l.(;:5:5:5077i 
 
 528 
 
 278784 
 
 22.9782500 
 
 409 
 
 107281 
 
 20.2237484 ' 
 
 409 
 
 219901 
 
 21.6504078 i 
 
 529 
 
 279841 
 
 23.0000000 
 
 410 
 
 108100 
 
 20.2484507 I 
 
 470 
 
 220900 
 
 21.0794834 i 
 
 5:10 
 
 2S0900 
 
 2:5.0217289 
 
 411 
 
 108921 
 
 20.2731349 
 
 471 
 
 221841 
 
 21.7025:541] 
 
 531 
 
 281901 
 
 23.01:54372 
 
 412 
 
 109744 
 
 20.2977831 
 
 472 
 
 222784 
 
 21.7255610 i 
 
 5:52 
 
 28:5021 
 
 23.0051252 
 
 413 
 
 170509 
 
 20.322J014 
 
 473 
 
 223729 
 
 21.7485632 
 
 6:53 
 
 28408!, 
 
 23.0867928 
 
 414 
 
 171390 
 
 20.3409899 
 
 474 
 
 224676 
 
 21.7715411 
 
 5:54 
 
 285150 
 
 23.1084400 
 
 415 
 
 172225 
 
 20.3715488 
 
 475 
 
 225625 
 
 21.7944917 
 
 5:S5 
 
 286225 
 
 23.1.300670 
 
 416 
 
 173050 
 
 20.3960781 
 
 476 
 
 220570 
 
 21.8174242 
 
 530 
 
 287290 
 
 23.15107:58 
 
 417 
 
 173889 
 
 20.4205779 
 
 477 
 
 227529 
 
 21.8403297 
 
 6:i7 
 
 2><8309 
 
 23.1732005 
 
 418 
 
 174724 
 
 20.4450183 
 
 478 
 
 228484 
 
 21.80:52111 
 
 538 
 
 2S9444 
 
 23.1948270 
 
 419 
 
 176561 
 
 20,4694895 
 
 479 
 
 229441 
 
 21.8860080 
 
 539 
 
 290521 
 
 23.21037;« 
 
 420 
 
 176400 
 
 20.4939015 
 
 480 
 
 230400 
 
 21.9089023 
 
 510 
 
 291000 
 
 23.2379001 
 
OF NUMBERS FROM \ TO IGOO. 
 
 No. 
 
 Sijuarf- 
 
 Sqre. root. 
 
 No. 
 
 ! fioi 
 
 S (iinrc. 
 
 Sqre. root. 1 
 24.515:501:4 
 
 No. 
 
 Square. 
 
 S \\-('. rtif t. 1 
 1 
 
 '25.7099'2j'i^ 
 -25.729:1.1^7 
 
 541 
 
 292tiSl 
 
 2:!. 259 1067 
 
 ;4(H201 
 
 661 
 
 4:56921 
 
 512 
 
 29:i7(i4 i 2:-!.2.'<(l>-9:{5 
 
 (i02 
 
 :;(;2404 
 
 24.5:55(188:4 ' 
 
 662 
 
 4:5.s244 
 
 5<:{ 
 
 294^49 ■ 2;l;io2.S(;o4 1 
 
 i (;o:{ 
 
 :;(;:56(i9 
 
 24.55(;058:5 
 
 66:4 
 
 4:59569 
 
 25.74,s78!,>i 1 
 
 , 514 
 
 2959;!(; ' 2;{.:!'J:^(i7(; i 
 
 ' (;o4 
 
 :5(;isi(; 
 
 24 5761115 
 
 664 
 
 440896 
 
 25.7(lsl97a 
 
 545 1 
 
 297(125 
 
 2:!.:!i52:!5i i 
 
 005 
 
 ;5(;6025 
 
 24. 59(17 I7S 
 
 665 
 
 4422-25 
 
 25.78759'ji^ 
 
 54 1; 1 
 
 2'.)Slli; 
 
 2;{ ::i;i;(ii29 ! 
 
 (lot; 
 
 ;i(;72:5t; 1 
 
 2l.6170i!7:5 
 
 666 
 
 44:5556 
 
 25.>^0(197^'^ 
 
 647 1 
 
 2992(19 
 
 2:;. :!->():! 11 
 
 007 
 
 :5(iS4 19 
 
 24 (;:57:!700 
 
 667 
 
 444S89 
 
 25.826:5 l-^l ' 
 
 548 i 
 
 .•iud.'ioi 2;..i(i'.i;;99s : 
 
 (108 
 
 ;S(19i:64 1 
 
 24 .657 (.560 
 
 668 
 
 446224 
 
 25.84569*^0 , 
 
 54 'J 
 
 :iOiioi 2:{.4:iii7i90 
 
 (509 
 
 :570.s8l ; 2I.677!)'254 
 
 669 
 
 447561 
 
 25.8650:5-i;^ 
 
 550 i 
 
 .•{02500 2;i.l5207S8 
 
 (ilO 
 
 ;572100 121.69.-1781 
 
 670 
 
 44S900 
 
 25.8H4:55'^2 ; 
 
 551 
 
 :!o.'!i;oi 
 
 2:5.4 7:f:!s92 
 
 (111 
 
 ;57;4:52i ^ 24.71^1142 
 
 671 
 
 450-24; 
 
 25.90:16677 ! 
 
 552 
 
 ;iOi7(ii 
 
 2:!.49l(;s02 
 
 612 
 
 ;5745i4 ; 2i.7:5si;:5:i8 
 
 672 
 
 4515st 
 
 25.92296-S 
 
 55H : 
 
 ;{(i.")S09 
 
 2:!.51.")!i,')20 i 
 
 6i;5 
 
 :5757(;9 1 2l.75ss:-l6S ; 
 
 67:5 
 
 452929 
 
 25.91224-^5 ,| 
 
 554 
 
 :{(iii9i(; 2;!.5;;72(i4(; I 
 
 614 
 
 ;!76996 
 
 21.77!i(i2:;4 
 
 674 
 
 454276 
 
 •25.9615100 
 
 555 
 
 ;;()>'()25 2;!.5.')Si;{so | 
 
 615 
 
 ;57s225 
 
 21.79919:55 
 
 675 
 
 455(125 
 
 25.9S076-1 
 
 55(i ' 
 
 .•{(lOll'.C. 2:5.579(;522 
 
 616 1 
 
 1579456 
 
 21. Si 9:54 7:5 
 
 676 
 
 456976 
 
 26.0(10(1000 ' 
 
 557 1 
 
 .•',10249 2;{.t;00S474 
 
 617 
 
 :580(;^9 
 
 24.s:59i,^ir 
 
 677 
 
 45.s:i29 
 
 -26.01 9'2^2-:|7j 
 
 558 i 
 
 ;;ii:!i;4 2;!.ti2J02:ii; 
 
 618 
 
 ;581921 j 
 
 24.S596058 
 
 678 
 
 459(lS4 
 
 26.0:!sj:5-^l ,1 
 
 55i» 
 
 .•U24S1 2;{ (;i.;i80s 
 
 619 
 
 ;5h:-5161 ' 
 
 2 I. .^797 106 
 
 679 
 
 461041 ' 
 
 26. (157(12^'! !i 
 
 5(i0 ' 
 
 :ii;;(;oo 2;!.i;i;i:u9i < 
 
 620 
 
 :SH4400 21.S997992 
 
 680 
 
 462400 
 
 2(1.076-'0yt> ! 
 
 5til 
 
 :il472l ; 2:!.(i>^5i:;su i 
 
 021 
 
 :!s56 11 ' 24.919-^716 
 
 6,Sl 
 
 46:1761 ' 
 
 26.09597*>7 1 
 
 5(12 
 
 ;!15,s|l 2:!.7iMi.-,:;92 
 
 (122 
 
 :iS(i>>si ■ 2l.!i:i9927S 
 
 682 
 
 4651-24 
 
 26.1151207 , 
 
 , '•■* 
 
 :!i(i9i;9 2:5.727(;2io i 
 
 1 62;^ 
 
 ;5SS129 • 24.9599679 
 
 68:5 
 
 4664^<9 
 
 26.1 154 26f'7 , 
 
 5ii4 
 
 •11«09(; 2.i.7lNlS42 i 
 
 1 624 
 
 ;5S9:576 i 24.9799920 
 
 684 
 
 467856 
 
 26.15:1:19:57 ! 
 
 505 
 
 :{19225 . 2:i.7(i9728(; : 
 
 : 625 
 
 :5!)0625 ; 25.0O0()(l0O 
 
 6«5 
 
 4692-25 
 
 '26.17250-47 i 
 
 5li(; 
 
 :!20:!.".i; ' 2:!. 79117545 1 
 
 {i-H\ 
 
 :59l876 1 25.0l',i!l920 1 
 
 686 
 
 470596 
 
 26.1916017 j 
 
 5ti7 
 
 ;'.214^9 1 2;!.Sll7(il8 | 
 
 1 627 
 
 ;h9:5129 ; 25.o:;99(;8i 
 
 687 
 
 471969 
 
 26.21 068»f^ , 
 
 5(i.S 
 
 :i22(;2i ' 2;!.8.".27.")0i; ' 
 
 t 628 
 
 :59i:584 
 
 25.05992S2 1 
 
 688 
 
 47:5:514 
 
 26.22975-41 ,1 
 
 5(1 ',» 
 
 ;;2;!7i;i 2:i.>-.-,:iT-jo9 ! 
 
 i 629 
 
 ;595611 
 
 25,079S724 
 
 689 
 
 474721 
 
 26.24^8005 
 
 570 
 
 ;!2 19(10 ' 2:{.s74i;72S , 
 
 I 6:50 
 
 :596!»00 
 
 25.(i99S(lOS 
 
 690 
 
 476100 
 
 26.2678511 J 
 
 571 
 
 ;;2(;oti ' 2:{.8m:muiip:! : 
 
 ; 6:; I 
 
 ;;98i6i 
 
 25.11971:54 
 
 691 
 
 477481 
 
 26.2S68780 i 
 
 572 
 
 :!271S4 ; 2:i.91(;5215 i 
 
 ! 6:52 
 
 :^9942 1 
 
 25.1:19(1102 ■ 
 
 692 
 
 47X864 
 
 26.:505'^929 
 
 57;i 
 
 H'28;i29 ! 2;5.9;!7ii84 ; 
 
 1 6:5:4 
 
 4006S9 : 25.159191:5 
 
 69.4 
 
 4S0249 
 
 26.:524S9:52 
 
 574 
 
 ;i2947(; 1 2:i.9,-.s2971 
 
 j 6:54 
 
 401956 1 25.179:1566 
 
 694 
 
 4H 161-56 
 
 26.:5J:i'^7i)7 
 
 575 
 
 ;i:;o(;25 '■ 2:i.979i;-)7(i 
 
 ! 6:!5 
 
 40:5225 ; 25.199206:5 
 
 695 
 
 4815025 
 
 26.:562S527 
 
 57il 
 
 ;>;;i77i; • 2i.()(i(i()(ioo 
 
 i 6:56 
 
 404496 
 
 •25.2190104 
 
 696 
 
 481416 
 
 26.:5818110 
 
 ! 577 
 
 ;);;2929 ■ 21. 020^21:; 
 
 6:!7 
 
 4(15769 
 
 25.2:5SS5S9 
 
 697 
 
 485S09 
 
 26.4007576 i 
 
 1 678 
 
 •!:!40S4 1 2I.OHli;!(Hi 
 
 6:is 
 
 407011 
 
 25.'25S(1619 
 
 698 
 
 487201 
 
 26.4196896 j 
 
 ! 579 
 
 ;?;'.524l 1 2t.0(i2(lH8 
 
 ' 6:59 
 
 40S:52l 
 
 •25.27S4I9:5 
 
 699 
 
 48S601 
 
 26.4:586081 1 
 
 ; 580 
 
 ;'.;!(!400 i 24.0s:ils91 
 
 640 
 
 409600 
 
 25.29S221;5 
 
 700 
 
 490000 
 
 26.4575i;-5l 
 
 1 581 
 
 :i;575(;i i 24. 10:19 110 
 
 641 
 
 410.-'Sl 
 
 25. ;5 179778 
 
 • 701 
 
 49I40I 
 
 '26.4761046 
 
 : 582 
 
 8:58724 1 •24.124i;7ti2 
 
 642 
 
 412164 
 
 25.. •5:57 7 189 
 
 1 702 
 
 492801 
 
 -26. 495-2826 
 
 : 683 
 
 :4;;9.^S9 ; 24.145:!929 
 
 ' 64:4 
 
 4i:54l9 
 
 ■25.:557I447 
 
 7o;5 
 
 494209 
 
 -26.5141472 
 
 i 581 
 
 :141()5(; 24.1C>ti0919 
 
 i 644 
 
 4147:16 
 
 25.:5771551 
 
 ; 704 
 
 495616 
 
 26.5:52998;} 
 
 585 
 
 :M2225 24.1^il77:i2 
 
 645 
 
 416025 
 
 25.:596-^502 
 
 ! 705 
 
 4970-25 
 
 26.551«:}61 ' 
 
 58t; 
 
 ;44:i:i9i; 24.2(i74:i(;9 
 
 64ti 
 
 417:516 
 
 25.41(15:501 
 
 706 
 
 4984:16 
 
 26.5706605 
 
 1 587 
 
 :il45(i9 1 24.22><0S'J9 
 
 647 
 
 418609 
 
 25.4:5111947 
 
 707 
 
 499849 
 
 26.5S94716 
 
 588 
 
 ;i45744 
 
 24.2if^7ii:; 
 
 (548 
 
 419904 
 
 25.455SJ41 
 
 708 
 
 501-264 
 
 26.6082694 
 
 689 
 
 :ii()92i 
 
 24.2(;9:i222 
 
 (149 
 
 421201 
 
 25.4751784 
 
 709 
 
 50-2681 
 
 26.62705149 
 
 590 
 
 :ii8ioo 
 
 24.-2.'^9915i; 
 
 1 650 
 
 422500 
 
 25.4950976 i 
 
 710 
 
 504100 
 
 '26.6458-252 
 
 591 
 
 ;ii928i 
 
 2t.:ilOI91(i 
 
 651 
 
 42:«01 
 
 25.5147016 i 
 
 711 
 
 505521 
 
 26.66458:5:4 ; 
 
 592 
 
 :;50i(ii 
 
 24.:!:iio,')Ol 
 
 i 652 
 
 4-25101 
 
 25..-):542907 , 
 
 712 
 
 506944 
 
 26.68:5:5281 
 
 59;> 
 
 :!51i;49 
 
 24.:!5i:)9i;! 
 
 : 65:5 
 
 426409 
 
 25.55:18647 
 
 713 
 
 508;S69 
 
 26.7020598 ' 
 
 594 
 
 ;!,v2h:!(; 
 
 24.:5721152 
 
 654 
 
 427716 
 
 '25.57:1 r2:57 
 
 714 
 
 509796 
 
 26.7207784 : 
 
 595 
 
 :{54025 
 
 2t.:{92t;2iH 
 
 I 655 
 
 429025 
 
 25.5929678 
 
 715 
 
 511225 
 
 26.7:5948:49 
 
 59G 
 
 ;:«52i(; 
 
 24.4i:iiH2 
 
 1 656 
 
 4:^0:4:56 
 
 25.6124969 
 
 716 
 
 512656 
 
 26.758176:4 
 
 597 
 
 ; ;i5(;409 
 
 24.4;i:55s:{4 
 
 ! 657 
 
 4:51649 
 
 • 25.6:520112 
 
 717 
 
 514089 
 
 26.7768557 
 
 598 
 
 :-i57i;o4 
 
 "24.4540:585 
 
 j {}M 
 
 4:42964 
 
 25.6515107 
 
 718 
 
 515524 
 
 '26.7955-2'20 
 
 599 
 
 :{,'i8soi 
 
 24.474 47(J5 
 
 i 659 
 
 4:54281 
 
 i 25.670995:4 
 
 719 
 
 516961 
 
 •26.8141754 
 
 (iOO 
 
 i i^C.OOOO 
 
 •24.49»S974 
 
 ! 660 
 
 4:55600 
 
 ' 25.6904652 
 
 720 
 
 1 518400 
 
 26.8:5-28 157 
 
TABLE OF iJQtTARES, SaiTARR ROOTS 
 
 No. 
 
 Square. 
 
 519841 
 
 1 
 Sqre. root. ; 
 
 26.8514132 
 
 No. 
 
 781 
 
 Sqimrc. 
 609961 
 
 Sqrp, i't»ot. 
 
 No. 
 
 Square. 
 
 Sqre. root. , 
 
 7:^1 
 
 27.9163772 
 
 811 
 
 7072S1 
 
 29 0000000 
 
 722 
 
 521284 
 
 26.8700577 
 
 782 
 
 611524 
 
 27.!t612il29 
 
 812 
 
 70-<96l 
 
 2!». 0172:563 
 
 72:5 
 
 522729 
 
 26..S^'S(;593 
 
 783 
 
 6130,x9 
 
 27.9-<21372 
 
 813 
 
 710619 
 
 29.03111123 
 
 724 
 
 521176 
 
 26.9072181 
 
 78 1 
 
 614656 
 
 2-<.00()(i0ll0 
 
 841 
 
 7123:16 
 
 29,051(17Sl 
 
 72iJ 
 
 525625 
 
 2(;.925S210 ' 
 
 785 
 
 616225 
 
 2S. 01 7-^515 
 
 845 
 
 7 11 025 
 
 29.0i;,s.s-i:57 i 
 
 72G 
 
 527076 
 
 26.9M3872 
 
 786 
 
 617796 
 
 2.-<,0:55tl915 
 
 816 
 
 715716 
 
 29,0-<i!0791 1 
 
 727 
 
 52S529 
 
 26.9629375 
 
 787 
 
 619,169 
 
 2^,05:15203 
 
 817 
 
 717409 
 
 29,10:52641 
 
 728 
 
 5-/9'JS4 
 
 26. 9S 11751 
 
 788 
 
 620944 
 
 2S. 07 1:5377 ! 
 
 818 
 
 719101 
 
 29.12(11:196 
 
 729 
 
 531441 
 
 27.0000000 
 
 789 
 
 622521 
 
 2-<.0S914:i8 1 
 
 849 
 
 720801 
 
 29.1376016 
 
 730 
 
 532900 
 
 27.0185122 
 
 790 
 
 624 1 00 
 
 2S. 1069:586 
 
 850 
 
 722500 
 
 29,1517595 
 
 7;il 
 
 534361 
 
 27.0370117 
 
 791 
 
 625681 
 
 28.1217222 
 
 851 
 
 721201 
 
 29.1719(113 
 
 7a2 
 
 535824 
 
 27.05519M5 
 
 792 
 
 627624 
 
 2S. 1424946 
 
 852 
 
 72590 1 
 
 29, 1.^90:190 
 
 73;{ 
 
 537289 
 
 27.0739727 
 
 793 
 
 62.S819 
 
 28.M02557 
 
 853 
 
 727609 
 
 29.206 l(i37 
 
 734 
 
 538756 
 
 27.0921.114 
 
 794 
 
 6:101.36 
 
 2S.I7s()056 
 
 854 
 
 729:516 
 
 29.22327VS4 
 
 735 
 
 540225 
 
 27.1HWs;{i 
 
 795 
 
 6:12025 
 
 2S. 1957444 \ 
 
 855 
 
 7:11025 
 
 29.210:5,s:50 
 
 736 
 
 541696 
 
 27.1293199 
 
 796 
 
 633616 
 
 28.2131720 1 
 
 856 
 
 7:127:5.; 
 
 29.2574777 
 
 737 
 
 543169 
 
 27.14774;!9 
 
 797 
 
 6:55209 
 
 28,2:11 I S,S4 ! 
 
 857 
 
 7:11419 
 
 29.2715623 
 
 738 
 
 544614 
 
 27.1661554 
 
 798 
 
 6:16801 
 
 28.2I8S!):58 1 
 
 858 
 
 7:56164 
 
 29.2916370 
 
 739 
 
 516121 
 
 27.1815514 
 
 799 
 
 6:58101 
 
 28.21165881 , 
 
 859 
 
 737.S.-S1 
 
 29.:10.^7018 
 
 740 
 
 547600 
 
 27.2029110 
 
 800 
 
 640000 
 
 2S.2812712 
 
 860 
 
 7:59600 
 
 29.3257566 
 
 741 
 
 549081 
 
 27.2213152 
 
 801 
 
 641601 
 
 28.3019134 ■ 
 
 8i;i 
 
 741321 
 
 29.:il2>^0lo : 
 
 742 
 
 550564 
 
 27.239iI7ti9 
 
 802 
 
 6i;i201 
 
 2S.:51!'6015 
 
 862 
 
 7 1:50 14 
 
 29.:559.>^365 : 
 
 743 
 
 552019 
 
 27.2580263 
 
 803 
 
 644S09 
 
 2>^,33725I6 ] 
 
 86:5 
 
 741769 
 
 29.:576S,;i(5 , 
 
 744 
 
 55353(1 
 
 27.2763634 ; 
 
 804 
 
 646416 
 
 2S.351S0:5S 
 
 864 
 
 71619,1 
 
 29.:i93-^769 
 
 745' 
 
 555025 
 
 27.2!M68,S1 
 
 805 
 
 64M025 
 
 2-^.:!7252l9: 
 
 865 
 
 74S225 
 
 29,410'^-<23 
 
 746 
 
 556516 
 
 27.3130006 : 
 
 806 
 
 6496:5i; 
 
 2-<, 390 1391 
 
 866 
 
 74995(5 
 
 29,427.-<779 
 
 747 
 
 558009 
 
 27.331.1007 : 
 
 807 
 
 651249 
 
 2H.. 1077454 i 
 
 867 
 
 75l6-!9 
 
 29,414.^637 
 
 748 
 
 559501 
 
 27.34958^7 : 
 
 808 
 
 65286 I 
 
 2^,425:5108 
 
 868 
 
 75342 1 
 
 29,461.s:i97 
 
 749 
 
 561001 
 
 27.3ti7S644 
 
 809 
 
 651IS1 
 
 2s. 44 2925:1 
 
 869 
 
 755161 
 
 29.17S.^059 
 
 750 
 
 562500 
 
 27.3,siH279 
 
 810 
 
 656100 
 
 2x. 160 10^9 
 
 870 
 
 756900 
 
 29,4957621 
 
 751 
 
 564001 
 
 27,4013792 
 
 811 
 
 657721 
 
 2^.47s0ill7 
 
 871 
 
 758011 
 
 29,5127091 ' 
 
 762 
 
 565504 
 
 27.4226184 
 
 812 
 
 659344 2S. 1956 137 
 
 S72 
 
 76o:is 1 
 
 29,529iI16l ' 
 
 753 
 
 567009 
 
 27.4408155 
 
 813 
 
 66091.9 2M.5I31519 
 
 873 
 
 762129 
 
 29.51657.34 
 
 754 
 
 568516 
 
 27.4590601 
 
 814 
 
 662596 1 2S.5:!06H52 
 
 874 
 
 76:1876 
 
 29,56:14910 
 
 755 
 
 570025 
 
 27.4772633 
 
 815 
 
 661225 2S,54S201S 
 
 875 
 
 765625 
 
 29.5,^039.>^9 1 
 
 756 
 
 571536 
 
 27.4954512 
 
 816 
 
 W>:'>Hb\> 
 
 2S.5657137 
 
 876 
 
 767:576 
 
 29,5972972 ] 
 
 757 
 
 573049 
 
 27.51363:10 
 
 817 
 
 6(J74Si» 
 
 2S, 58:12 11 9 
 
 S77 
 
 76:tl29 
 
 26,i!14!^.-)8 
 
 758 
 
 574564 
 
 27.531 799.>< 
 
 81 S 
 
 669121 
 
 2s.f;ooi;99:i 
 
 878 
 
 770.S81 
 
 29.631 0i;i8 ' 
 
 759 
 
 576081 
 
 27.5499516 
 
 819 
 
 670761 2M.(;i,^ 1760 
 
 879 
 
 772611 
 
 29,6179:112 
 
 760 
 
 577600 
 
 27.5(;,S0975 
 
 820 
 
 672100 2S.6:{56421 
 
 880 
 
 774100 
 
 29,66179:19 
 
 761 
 
 579121 
 
 27.5862284 ' 
 
 821 
 
 674041 2^,65:!i)!)76 
 
 881 
 
 776161 
 
 29,6816142 
 
 762 
 
 580644 
 
 27.6013475 ; 
 
 822 
 
 6756S1 2^,0705421 , 
 
 8,h2 
 
 777921 
 
 29,69,^ ts!4 8 , 
 
 763 
 
 582169 
 
 27.6221546 ' 
 
 823 
 
 677329 28.fiS79766 
 
 883 
 
 7796>S9 
 
 29,715:5159 ' 
 
 764 
 
 583696 
 
 27.6405499 
 
 -i2t 
 
 67M976 . 28.7051002 
 
 8,><1 
 
 781 156 
 
 29.7:121375 
 
 765 
 
 585225 
 
 27.6586331 
 
 825 
 
 680625 2S. 72281:52 
 
 885 
 
 788225 
 
 29. 71 s9 196 '■ 
 
 760 
 
 586756 
 
 27.6767050 
 
 826 
 
 682276 2s'. 7102157 
 
 8m6 
 
 784996 
 
 29,7657521 ■ 
 
 767 
 
 588289 
 
 37.691764.^ 
 
 827 
 
 6S3929 
 
 2^.7576077 
 
 887 
 
 786769 
 
 29,7825152 
 
 768 
 
 589824 
 
 27.7128129 
 
 828 
 
 6.S55S4 
 
 28,7749S91 
 
 888 
 
 78S514 
 
 29.79932-^9 , 
 
 769 
 
 591.361 
 
 27.7308492 
 
 829 
 
 6S7241 
 
 28.792:5601 | 
 
 889 
 
 790:521 
 
 29.8161030: 
 
 770 
 
 592900 
 
 27.7488739 : 
 
 830 
 
 688900 
 
 28.S097206 ! 
 
 890 
 
 792100 
 
 29,S32.^678 ■ 
 
 771 
 
 594441 
 
 27.7668868 ] 
 
 831 
 
 690561 
 
 28.8270706 
 
 891 
 
 793SS1 
 
 29.8li)(;23I I 
 
 772 
 
 595984 
 
 27.7848880 i 
 
 832 
 
 6!)2221 
 
 28.«4I1102 
 
 892 
 
 795664 
 
 29,8(;r)3690 ! 
 
 773 
 
 597529 
 
 27,8028775 i 
 
 833 
 
 69:5'<89 
 
 28.8617394 
 
 893 
 
 797419 
 
 29,8831056 
 
 774 
 
 599076 
 
 27.8208555 
 
 834 
 
 695556 
 
 28.8790582 
 
 894 
 
 7992:56 
 
 29,899S328 
 
 775 
 
 600625 
 
 27.8.388218 
 
 835 
 
 697225 
 
 28.8963666 ' 
 
 895 
 
 801025 
 
 29,9165506 , 
 
 776 
 
 602176 
 
 27.8567766 
 
 836 
 
 698896 
 
 28,91.36646 1 
 
 896 
 
 802816 
 
 29,9:532591 : 
 
 777 
 
 603729 
 
 27.8747197 
 
 837 
 
 700569 
 
 28,9309523 ; 
 
 897 
 
 801609 
 
 29,9499583 i 
 
 778 
 
 605284 
 
 27.8926514 
 
 838 
 
 7022 14 
 
 28,9482297 \ 
 
 898 
 
 806101 
 
 29,9666481 i 
 
 779 
 
 606811 
 
 27.9105715 
 
 839 
 
 703921 
 
 28,9651967 1 
 
 899 
 
 808201 
 
 29.98:53287 , 
 
 780 
 
 608400 
 
 27.9284801 
 
 840 
 
 705600 
 
 28.98275351 
 
 900 
 
 810000 
 
 30.0000000 ; 
 
or NTTMBERS PROM 1 TO 1600. 
 
 Xo. 
 
 901 
 
 Sqimre, 
 
 Sqre. root. 
 :^0. 01 00021 
 
 No. 
 
 Sii'iiij'o. 
 
 Sqrt'. root. 
 ;5 1.0000000 
 
 No. 
 1021 
 
 Siiimre. 
 
 Sqre roof. 
 ! 31.95:50900 
 
 
 811S01 
 
 901 
 
 92:i521 
 
 : 1042441 
 
 
 902 
 
 8i:;(iO( 
 
 ;-:o.o:!;!:ii4s 
 
 902 
 
 9254 14 
 
 :51.Oi0124S 
 
 1022 10 44184; 141. 9087:547 
 
 
 9(t;{ 
 
 si 5 101) 
 
 ;jo.oii»:i5wi 
 
 90.1 
 
 927.10!) 
 
 ::!l.o:!22ii:i 
 
 ' 102:! 
 
 1040529; :51.9S4:5712 
 
 
 9(1 i 
 
 8i72ii; 
 
 ;!(!. 11005928 
 
 904 
 
 929290 
 
 :5!.ois:549i 
 
 1021 
 
 10H570 
 
 , ;!2. 0000000 
 
 
 90,'i 
 
 SI 902.') 
 
 .■U).os:;'jl7!) 
 
 905 
 
 9.11225 
 
 ;5I.O0 4M91 
 
 10'J5 
 
 1050025 
 
 :52. 0150212 
 
 1 
 
 90(1 
 
 S20.s.!ii 
 
 ;{0.0!»9s:i;i9 : 
 
 900 
 
 9:i;5i5t; 
 
 :51.0SO54()5 
 
 1021. 
 
 1052070 
 
 ;52. 0:512:548 
 
 1 
 
 907 
 
 822(;i9 
 
 ;i0.1104l07 I 
 
 907 
 
 9;S50s9 
 
 :5!. 09002:50 
 
 1027 
 
 ' 1051729 
 
 :f2. 0408407 
 
 
 908 
 
 H244(il 
 
 :{0.i;i:!(t.iH:{ 
 
 908 
 
 9:i702 1 
 
 :S1. 11 20984 
 
 : 102; 
 
 ; 1'|-)07S4 
 
 :52. 002 4:591 
 
 
 909 
 
 S2(;2S1 
 
 MO. M 90209 i 
 
 909 
 
 9:i'<901 
 
 :{1.12S7048 
 
 1029 
 
 ! I0,')88 4l 
 
 .'52.0780298 
 
 
 910 
 
 828100 
 
 :io.l0O2O0:{ i 
 
 970 
 
 910900 
 
 ;5i.i4is2:!0 
 
 lo:50 
 
 1000900 
 
 ;{2. 09:50 i:ii 
 
 
 911 
 
 829921 
 
 :U).l.s'27705 i 
 
 971 
 
 942S41 
 
 :-!l.I00s729 
 
 lo:ii 
 
 1 00290 1 
 
 :^2. 109 1877 
 
 
 912 
 
 8:!1741 
 
 M0.19:):;:!77 
 
 972 
 
 914 7SI 
 
 .■51.1709145 
 
 lo:52 
 
 1005021 
 
 ;52. 1247508 
 
 
 9l:{ 
 
 8;;;{5i;9 
 
 :!0.215s-<91l 
 
 97:! 
 
 910729 
 
 :il. 1929479 
 
 10:5:1 
 
 10070S9 
 
 :!2.i4o;5i7;^ 
 
 
 914 
 
 8;i5.t9(; 
 
 .■{0.2:{2i:i29 '. 
 
 971 
 
 91.S070 
 
 :il.20^!»7:5i 
 
 10:54 
 
 10091.50 H2.15.')8704 
 
 
 915 
 
 «;!7225 
 
 :{0.24M»i;(;9 
 
 975 
 
 9.>0025 
 
 .'51.2219900 
 
 1 0:55 
 
 107r2'25 
 
 :52. 1714159 
 
 
 91(1 
 
 M;i905(i 
 
 ;!0.205l9i9 
 
 970 
 
 952570 :il.21099s7 
 
 ' 1050 
 
 107:5290 
 
 ;52.1S095.'49 
 
 
 917 
 
 8108-9 
 
 :!0.2s'.'()o79 ; 
 
 977 
 
 951529 :!1.250:i992 
 
 10:57 
 
 1075:509 
 
 :52. 202 1844 
 
 
 918 
 
 842721 
 
 :i0.29<5!4S ' 
 
 978 
 
 9504S4 :il .2729915 
 
 10:58 
 
 1077414 
 
 :52.21S0074 
 
 
 919 
 
 8(4501 
 
 ;{0.;!!50!28 
 
 979 
 
 95SI4I :!1 ,2-!-<;i757 
 
 1 0:19 
 
 1079521 
 
 :12. '2:5:55229 
 
 
 920 
 
 8lt)40l) 
 
 :{0.;{;;i50i8 ' 
 
 9 SO 
 
 90010(1 ;il.;{0l9517 
 
 1040 
 
 I OS 1000 
 
 :i2. 2 190:^10 ' 
 
 
 921 
 
 848241 
 
 :^(».;!i79si8 
 
 9-'l 
 
 902:101 :ii.;!2o:n95 
 
 1011 
 
 10s:50Sl 
 
 ;52. •204,5:5 1(5 
 
 
 922 
 
 85U081 
 
 ;io.:;o 11529 1 
 
 9S2 
 
 904;!24i ;^1.;!:!0S792 
 
 1012 
 
 10S570 4 
 
 :52.2S00248 
 
 
 92;{ 
 
 85l!)29 
 
 :^0.;i8o9i5i 1 
 
 98;{ 
 
 900289i :{i.;!52.s:ios 
 
 104:5 
 
 10S7S19 
 
 :42. 2955105 
 
 
 924 
 
 S5.)77() 
 
 ;io.:i97:;o.s:! ' 
 
 9^1 
 
 90^250' :!1.;!0S77I." 
 
 1041 
 
 10'^99:50 
 
 ;^2.:5I098,S8 
 
 
 92;-) 
 
 S55li25 
 
 :{o.4i:?.si27 1 
 
 985 
 
 970225 :!1.:!SJ7097 
 
 1 045 
 
 1092025 
 
 :^2.:4-20 4598 
 
 
 92ti 
 
 S5747i; 
 
 ;!0.4;i02181 1 
 
 980 
 
 9721 '.to: ;!I.40O0:iO9 
 
 1010 
 
 109111(5 
 
 ;52,:54l92:w 
 
 
 927 
 
 859.129 
 
 ;iO, 4 100747 
 
 9^7 
 
 971109 :i 1.4 10.5501 
 
 1047 
 
 1090209 
 
 :^2.:557:5794 
 
 
 92S 
 
 801 ls4 
 
 ;!o.4(;:ioit24 
 
 988 
 
 970114 :!1.4:!2 107:5 
 
 lots 
 
 109S:504 
 
 :52.:i7'2S281 
 
 
 929 
 
 SOiiOll 
 
 ;!0. 47950 l:i 
 
 989 
 
 97.S121 :il.4is:!704 
 
 1049 
 
 1100101 
 
 :{2.:58S2095 1 
 
 
 9;)0 
 
 8(11900 
 
 ;i0.4!i590l4 
 
 990 
 
 9.S01()0 :il. 40 12054 
 
 1 050 
 
 1 1 02500 
 
 :52. 40:570:55 
 
 
 9;;i 
 
 8(ii)7(il 
 
 :)0. 5 122920 ' 
 
 991 
 
 9820811 ;!1. 4801525 
 
 1051 
 
 1104001 
 
 :r2.4i9i;«i 
 
 
 9.S2 
 
 8(;8.;2i 
 
 :^0.52.>^0750 
 
 992 
 
 9840(;4' :!1. 1900:115 
 
 1 052 
 
 1100704 
 
 :^2. 4.545495 
 
 
 9:!:{ 
 
 870 1S9 
 
 :!0.5I5()187 
 
 99;^ 
 
 9S00I9 
 
 :51.511!)025 
 
 105.! 
 
 110SS99 
 
 H2. 44990 1 a . 
 
 
 9;i4 
 
 872;{5i; 
 
 ;!0.5i;i4i;!o 
 
 991 
 
 9sso:io 
 
 .'51.5277055 
 
 1051 
 
 1110910 
 
 ;{2. 405:5002 i 
 
 
 9;!,-) 
 
 871225 
 
 :iO. 5777097 : 
 
 9!l5 
 
 990025 
 
 :51.5i:!0200 
 
 1055 
 
 111:50-25 
 
 .■52.1S070:^5 
 
 
 9:5G 
 
 87i;o9(; 
 
 ;i0.59lll71 
 
 990 
 
 992010 
 
 .'51. .1594077 
 
 1050 
 
 11151:50 
 
 ;^2. 49015:^6 
 
 
 9;i7 
 
 8779(19 
 
 :!0. 01 04557 i 
 
 997 
 
 991009 
 
 :il.575:5O0S 
 
 1057 
 
 111 7^249 
 
 :^2.511.5:{64 
 
 
 9:!8 
 
 879<44 
 
 .10.0207.^57 , 
 
 998 
 
 99t;004 
 
 :^l.59li:5so i 
 
 1 0.)8 
 
 1 1 1 9:50 1 
 
 .'52.5209119 
 
 
 9:59 
 
 881721 
 
 ;{o.(i4;;ioo9 1 
 
 999 
 
 19»J,S001 
 
 :-5i.ooo9i;i:5 
 
 1059 
 
 1121481 
 
 ;52.542'2802 
 
 
 940 
 
 88;!(i00 
 
 :U).05lt|!94 
 
 looo 
 
 1000000 
 
 :51. 0227700 
 
 1000 
 
 ir2:!,;oo 
 
 ;!2. 55704 12 
 
 
 941 
 
 885481 
 
 ,•50.07572:!;! 
 
 1001 
 
 1000201 
 
 :!l.o:5.s.-)840 
 
 1001 
 
 1125721 
 
 ;^2.57299 49 
 
 
 942 
 
 8.s7:i(;i 
 
 ;!0.0920!,s5 ' 
 
 1 002 
 
 1004004 
 
 ;5!.054:5S:!0 
 
 1 002 
 
 1127844 
 
 :i2.58S;5415 
 
 
 i 94 ;i 
 
 889219 
 
 ;!((. 708:1051 
 
 lOlC! 
 
 1000009 
 
 :S 1.070 1752 
 
 loo:^ 
 
 1129909 
 
 ;12. 00:50807 
 
 
 ! 944 
 
 89ii;{ii 
 
 ;^0.72i5>^:!(i : 
 
 1001 
 
 loosoio 
 
 ;si.o-'5:i590 
 
 l(i04 
 
 11:42090 
 
 :52. 01 90129 
 
 
 1 94.-> 
 
 89.S025 
 
 ;io. 740.^52:1 ; 
 
 1 005 
 
 1010025 
 
 ;51.7017:519 
 
 1005 
 
 ! 1:54225 
 
 .'52.0.'U;{:577 
 
 
 9ir, 
 
 89491.; 
 
 .-10.7571 1:!0 
 
 looo 
 
 loiO(i;;o 
 
 :5 1.7 1750:50 
 
 1000 
 
 11:50:550 
 
 :12. 0490554 
 
 
 94, 
 
 890808 
 
 ;!u.77:i:!05i : 
 
 1007 
 
 1014049 
 
 :!l.7.i:52i;:5:! 
 
 li'07 
 
 li:58l,i9 
 
 :12. 00 19659 
 
 
 948 
 
 89.S701 
 
 :!0. 7890080 
 
 10(H 
 
 101(5004 
 
 •51.7190157 
 
 1 008 
 
 1140024 
 
 :12.080209;{ 
 
 
 919 
 
 900001 
 
 ;io.8058i;{0 
 
 1009 
 
 101 SOS 1 
 
 ;51.7O170O:5 
 
 1009 
 
 1112701 
 
 :52.0955054 
 
 
 950 
 
 902500 
 
 ;10.S220700 
 
 1010 
 
 1020100 
 
 .'51.7801972 
 
 1070 
 
 1144900 
 
 :«.7 108544 
 
 
 9.t1 
 
 90I401 
 
 :i0.s:ix'.'s7!t 
 
 1011 
 
 1020121 
 
 :!1. 7902202 
 
 1071 
 
 1147041 
 
 ;«72(>i:^6:i 
 
 
 952 
 
 90(;;!()1 
 
 .so. 8.544972 : 
 
 1012 
 
 10241441 
 
 :^1. 81 19174 
 
 1072 
 
 1149184 
 
 ;«. 74141 11 
 
 
 95;i 
 
 90S209 
 
 ;10. 87009^1 i 
 
 101 ;5 
 
 1020109 
 
 ;5 1.8270009 
 
 107:5 
 
 1151:529 
 
 :^2. 7506787 
 
 
 954 
 
 910110 
 
 ;50. 8808904 i 
 
 1014 
 
 102S190 
 
 :^i.84:i'!()00 
 
 1074 
 
 115.5470 
 
 32.7719:192 
 
 
 955 
 
 912025 
 
 ;5o.9o;!074;^ i 
 
 1015 
 
 I0:i0225 
 
 ;^ 1.8590040 
 
 1075 
 
 11556^25 
 
 ;r2. 787 1926 
 
 
 950 
 
 9!a9H0 
 
 :^0. 9 192477 
 
 1010 
 
 10.T2250 
 
 HI. 8747549 
 
 1070 
 
 1157770 
 
 32.8024398 
 
 
 957 
 
 915849 
 
 ;^o.9:>5tioo 
 
 1017 
 
 10:54289 
 
 :^l. 8904.174 
 
 1077 
 
 1159929 
 
 ;r2. 81 76782 
 
 
 958 
 
 9177(i4 
 
 :10.95 15751 
 
 1018 
 
 1 o:5o:r2 1 
 
 ;^i.9oon2:5 
 
 1078 
 
 1102084 
 
 32.8329103 
 
 
 959 
 
 919081 
 
 :!0,9077251 
 
 1019 
 
 io:k^oi 
 
 :S1. 921 7794 
 
 1079 
 
 1104241 
 
 .32.848i:«4 
 
 
 ! 9i;o 
 
 921000 
 
 ;40.98:-i8008 
 
 1020 
 
 1040400 
 
 :^1.9:^74H88 ; 
 
 1080 
 
 1100400 
 
 32.8633535 
 
 
8 
 
 TACr-E OF t^QUAUliS, SQUARE ROOTS 
 
 No. 
 
 lOSl 
 1082 
 
 ioh;{ 
 
 1084 
 1085 
 108G 
 1087 
 1088 
 1089 
 1 0'JO 
 1091 
 1092 
 109;t 
 109-J 
 lOUo 
 109li 
 1097 
 1098 
 1099 
 1100 
 1101 
 1102 
 llOii 
 1104 
 1105 
 HOC 
 1107 
 1108 
 1109 
 1110 
 1111 
 1112 
 1118 
 
 nil 
 
 1115 
 lllli 
 1117 
 1118 
 1119 
 U20 
 1121 
 1122 
 U2H 
 1124 
 1125 
 11 2G 
 1127 
 1128 
 1129 
 11 HO 
 
 ii;u 
 
 1132 
 11H3 
 1134 
 1135 
 1136 
 1137 
 1138 
 1139 
 1140 
 
 ?(ini.ire, Sqrc. root. No. j Sqniirn. I ?(jrc. root. || No. j .S(|Uiirc'. i?qrp. ror/t 
 
 11085(11' 
 
 1170721 
 
 1172S.S9 
 
 1 1 750.')(; 
 
 1177225 
 
 11793:i(i 
 
 1I815G9 
 
 118;!7I-I 
 
 1185921 
 
 1188iU0 
 
 119l)2sl 
 
 11 92 KM 
 
 1191(119 
 
 119fis;;G 
 
 1199025 
 
 120121(i 
 
 i2o;;iu9 
 
 12(»5(;04 
 
 1207.^01 
 
 TilOdOO' 
 
 1212201 
 
 1214101 
 
 121Gti09 
 
 121>-^8l(i 
 
 1221025 
 
 12232;;(i 
 
 1225-1-19 
 
 1227(1(;4 
 
 1229Sf<l 
 
 1232100 
 
 123-ri21 
 
 12;-\,54! 
 
 12.!87ti9 
 
 121099G 
 
 12-13225 
 
 12-1515(i 
 
 1247089; 
 
 1249924! 
 
 1252101 
 
 1254J00 
 
 125tili41 
 
 1258881 
 
 1201)29 
 
 1203370 
 
 1205025 
 
 1207870 
 
 12701:-: 
 
 1272384 
 
 1274041 
 
 1270900 
 
 1279101 
 
 1281-124 
 
 128.3089 
 
 1285950 
 
 1288225 
 
 1290490 
 
 1292709 
 
 1295044 
 
 1297321 
 
 1299000 
 
 32.8785044 I 
 32.89370S4 • 
 32.9(H!»t;5:! , 
 32.921155:; 
 32.93:»;i:!82 
 ;!2.95I5!41 ! 
 :!2.9i;9ii830 
 :i2. ((848150 
 :!3. 00(10000 
 33.015! ISO 
 33.()3()28iM 
 33.04512.33 
 3:{. 0005505 
 33.0750708 
 :)3. 0907842 
 :!3. 105^907 
 
 :;3.i2o:'903 
 ;;3.i;!);(i830 ' 
 
 .33.15:1089 
 33.1002479 
 33.1813200 
 .33,190: 1853 
 .33.21144;!8 
 :!3. 2200955 
 ;^;i.24 15403 
 3:!. 250578:1 '•: 
 33.2710095 i 
 
 :{3.28(;(;3:;9 ! 
 ;« .30 105 10 : 
 
 33.31(,'0(i25 
 33.3310000 
 :',3. 3400040 
 :i3.3(; 10540 
 33.37('i0385 
 33.3910157 
 :>3.4005f^02 
 :^3.4215499 ' 
 ;«.4:i05070 
 33.4514573 
 33.4004011 
 :i3.4813:Wl r 
 :13.4902084 I 
 33.5111921 : 
 :«.5201092 ' 
 .33.5410190 
 33.55592;i4 
 •i3.570s20(; 
 :«.5857H2 
 33.0005952 
 ;{3.6 154726 I 
 33.6303434 [ 
 :13. 6452077 ! 
 33.6000053 ! 
 ;!3.0749105 : 
 ;43. 68976 10 i 
 ;53. 7015991 ! 
 :-13.7 194:506 i 
 33.7340556 | 
 :{3. 7490741 ' 
 33.7638860 I 
 
 1141 
 II 12 
 
 111;; 
 
 1144 
 1145 
 1146 
 1117 
 11 18 
 1149 
 1150 
 1151 
 1152 
 1153 
 1154 
 1155 
 11, -.0 
 1157 
 1158 
 1 1 59 
 1100 
 1101 
 1102 
 1103 
 1104 
 1105 
 1100 
 11(;7 
 1108 
 1109 
 1170 
 1171 
 1172 
 
 117;; 
 
 1174 
 1175 
 117(; 
 1177 
 1178 
 1179 
 11 so 
 1181 
 1182 
 1183 
 1184 
 1185 
 1180 
 1187 
 1188 
 1189 
 1190 
 1191 
 1192 
 1 1 93 
 1194 
 1195 
 1190 
 1197 
 1198 
 1199 
 1200 
 
 130I,s81 
 
 i:;oiioi 
 i:;(h;ii9 
 i-3087:!t; 
 
 1311025 
 
 I3l3:;i0 
 
 131 50 09 
 
 i:; 17904 
 
 1320201 
 1322500 
 1321. SOI 
 
 ;;3.7."80915 ' 
 
 :;:;.79:;i9()5 
 
 3:;-80s2s:;o 
 
 :i;;.82;;o09i 
 
 :;:;.837s4st; 
 
 :;:;. 85202 18 
 
 :;:;.807:;s84 
 
 :;:;.ss':i487 
 
 3:;.s'.<o'.ii)25 
 
 ;;:;.9iioi:i9 
 
 3:;.92(i:;909 
 
 1327101 ;;;;.9iii 
 
 •'-,•, 
 
 1329 1 0!> 
 
 i:i;;i7io 
 
 13:14025 
 
 I33o:!;;o 
 
 1338049 
 1.34 09041 
 
 i;;4:;28i! 
 
 13I560OI 
 1347921: 
 
 i;;5(i2i4! 
 
 i:;525(;9^ 
 
 I:i54s9ii 
 l:i57225; 
 1.359550: 
 1301889; 
 
 i:;0422i! 
 i:;oo5(;r 
 i:;os9o(): 
 
 13712411 
 
 i:;t:;5s4: 
 
 1375929 
 
 i;;7827o| 
 
 i:i80025i 
 1382970 
 i;;s5329 
 
 i:-187084! 
 
 i;;9ooiii 
 i;;92ioo: 
 
 1394701; 
 
 13971241 
 
 13994S9| 
 
 14018501 
 
 14042251 
 
 1400590 
 
 1408909; 
 
 I41l:444' 
 
 14137211 
 
 MlOlOOi 
 
 14184811 
 
 1420.S04! 
 
 1423249| 
 
 i425o:;oj 
 
 1428025! 
 
 1430416: 
 
 1432809 
 
 l4:-;5204 
 
 1437001 
 
 1440000 
 
 33.955S5:)7 
 
 :;:;. 9705755 
 :;:; 9-52910 
 :;4.(i(H)(i(Miii 
 
 34.01 17027 
 
 :;4.029:;9:io 
 
 34.0I40S90 
 34.0587727 
 
 ;;4,o7;;45oi 
 
 ;il.0ssl211 
 
 31.i027s5s 
 
 :;i.ii74i42 
 34.i32o:;(;:; 
 
 31.1407122 
 •;4.1013sl7 
 34.1700150 
 
 :;4.ii>oo42o 
 
 ,34. 2052027 
 34.219S773 
 34. 2;; I 1-55 
 ;!4. 2 190,-75 
 
 34.2o:;o8;;4 
 
 34.27s27:;0 
 :)4.292-,>04 
 
 34.';o7i;;;;6 
 
 :;4.3220040 
 
 :;4.:;3i;5(;94 
 
 34.35112S1 
 
 :;4.:;o5(iso5 
 34.:;s022(i8 
 :;4 .39 17070 
 :;4.409:;oii 
 
 34.42:;s289 
 
 :;i.43s;;507 
 
 ;14 .4528003 
 
 34.407:1759 
 
 34.481 s793 
 
 :;4.49o;;700 
 
 ,34.5108078 
 34.525:1530 
 34.539S321 
 34.554:;051 
 34.5087720 
 31.5832329 
 34.5970S79 
 :{4.012i;U16 
 34.0205794 
 ! 34.0410102 
 
 1201 
 
 1202 
 1211:1 
 120 I 
 1 205 
 1200 
 1207 
 120s 
 1209 
 1210 
 1211 
 1212 
 1213 
 1214 
 
 1218 
 1219 
 1220 
 ]■'•>] 
 
 122 1 
 i 225 
 1220 
 1227 
 r22s 
 1229 
 
 i2;;o 
 I2:;i 
 1 2:12 
 
 1233 
 
 1 2:14 
 12:15 
 
 12:10 
 
 1237 
 
 1 2:18 
 12:19 
 
 1240 
 1241 
 124 2 
 1243 
 1244 
 1245 
 1240 
 124 7 
 1248 
 1249 
 1 250 
 1251 
 1 252 
 1253 
 1 254 
 1 255 
 1 250 
 1 257 
 1258 
 1259 
 1200 
 
 1412401 
 
 1 1 1 ISO I 
 
 1 14720'.) 
 
 14191110 
 
 1 152025 
 
 1454 130 
 
 I45(;si9 
 
 1459201 
 
 1 lOlOSl 
 
 14 01100 
 
 1400521 
 
 140S944 
 
 147131)9 
 
 1473790 
 
 14711225 
 
 14 7S(150 
 
 I4S10S9 
 
 148:1524 
 
 1 185901 
 
 1488100 
 
 1490S11 
 
 149:i2S4 
 
 1 195729 
 
 1 198170 
 
 1500025 
 
 15(1307(1 
 
 15(l552!t 
 
 1507984 
 
 1510411 
 
 1512900 I 
 
 1515:101 ; 
 
 1517824 ' 
 
 15202s 9 
 
 1522756 
 
 1525225 
 
 15270!K; 
 
 1530109 
 
 15;;2044 i 
 
 15:15121 ' 
 
 I5;i70oo i 
 
 15I00S1 ' 
 1542504 I 
 151504 9 
 1547530 ; 
 1550025 
 1552510 
 1 555009 
 1557509 
 1500001 
 1502500 
 1505001 
 1507504 
 1570009 
 1572510 j 
 l57.-i025 ' 
 15775:10 
 15S004 9 ' 
 15S2504 
 15S5081 i 
 1587600 ' 
 
 34.6551160 
 3I.(109S716 
 31. OS 12904 
 
 :;i.O!»s7o:hi 
 
 ;;4.713i099 
 34.7275107 
 34 .74 1 9055 
 ;;4. 75029 14 
 34.770-1773 
 ■54.7850543 
 31.7991253 
 :i4.si:i79()4 
 34.8281 195 
 34 8425028 
 ;;4.850s50l 
 34.8711915 
 34. s- 55271 
 ;;4 .8998507 
 34.91 4 ls05 
 34.9284984 
 :;4.9I2S984 
 3 1. 94 2S 104 
 :i 1. 957 1166 
 31.97! 1109 ; 
 ;; 1. 9,-57 114 
 
 :V). 000(1000 
 
 35.0142828 j 
 :;5.02s5598 | 
 :;5.0i2s:;()9 
 
 :;5.057O903 
 35.07 13558 
 :;5.0s50096 I 
 :i5.099-575 
 ;55. 1 140997 
 
 :;5.!28:i:ioi 
 
 :15.I4 25508 
 35.15il7!fl7 
 :i5.17!0108 
 ;15.!S52242 
 35.1994318 
 35.21:10:137 
 ;!5. 2278299 
 35.2420204 
 35.2502051 
 35.270:1842 
 35.2845575 
 145.2987252 
 35.31 2SS72 
 35.3270435 
 35.;illl941 
 35.355:i:i91 
 ;55.:i094784 
 
 ;55.;;8:ioi20 
 
 :;5.3!»77400 
 35.4118024 
 :55. 4259792 
 35.4400903 
 35.4541958 
 35.4082957 
 35.4823900 
 
OF Ni;.MBKU« FIIOM 1 To 1(500. 
 
 No. I Squiire. SiH'm. root 
 
 2ill 
 •Ju2 
 2ii:i 
 •-'d I 
 •2(iJ 
 2(1 1 i 
 
 2ti: 
 2i;s 
 
 270 
 271 
 272 
 27:5 
 271 
 27J 
 271) 
 277 
 27S 
 27'J 
 2S0 
 2Sl 
 2.S2 
 2S8 
 •1^1 
 2-j 
 
 2st; 
 
 2S7 
 2.-!.S 
 2.S'.) 
 2'JO 
 2'Jl 
 2112 
 2'J.! 
 2'.); 
 2;»J 
 2'Ju 
 2!) 7 
 21)8 
 2'.)S) 
 ;{0J 
 
 ;ioi 
 
 .102 
 ■Mi 
 
 :!i) I 
 :{0.) 
 :ioiJ 
 :)07 
 ;{()s 
 
 ;^io 
 .'{11 
 ;{I2 
 ;^i;{ 
 
 Hi 4 
 
 ;ii5 
 ;!i7 
 
 .'tis 
 ;{ii» 
 ;{2o 
 
 I.V)i)l21 
 l.V.)2ull 
 
 i.v.)7(i:ii» 
 
 l'il)(l22.j 
 ltil)27,')ti 
 i(il),">2-!'J 
 1(!(>7.S21 
 IiilO.liil 
 li;i2i)UO 
 
 iii;:)iu 
 it;i7:Hi 
 
 li;20.')2!) 
 
 i(i2.;()7ti 
 
 li)2Jii2J 
 lilJslTd \ 
 
 lii:;ii72:> '■ 
 
 l(i:i.;2si 
 
 li;:;.)Sii 
 
 Jd.iSIt);) 
 
 lGi():)iU 1 
 i(;(;!:)2i ' 
 
 llililO.S'J I 
 llj ISil.jii 
 
 lt;:>i2:i.'i 
 
 lii.'):;7:M 
 
 l(;."it>.M;) 
 l(iai."y'.i ' 
 
 l(i(JllUO , 
 
 li!iilili"<l 
 
 iijii'.)2i;t : 
 hi7i.srj 
 
 lii7H:{G 
 l(i77t)JJ i 
 
 llJ7;ililii 
 lox2J()'J [ 
 lilHIrtOl ' 
 
 i(;s7h)i 
 iii'.)ouao : 
 iii'.»ji;oi ' 
 ii;;).j2o-i 
 
 ll)'.)7.S0;) 
 17UU11G 
 
 17i):!()2.j 
 
 1 (U.Ml.iii 
 170s2l'J 
 
 l71U«(il i 
 
 I7i;ii.si : 
 
 1710100 I 
 
 1718721 
 
 1721;i41 
 
 l72;{iMi'j 
 
 172t;i5'Jtl 
 1721)225 
 l7;il8otJ 
 17.!M8!) 
 i7:!7l21 
 
 i7;;o7iii 
 
 1742i00 
 
 No. Sfiiiiu'o. Sqre. root 
 
 I 
 I 
 
 ;;.■)..•) lojii is 
 .i.K.vjii;. )'.).! 
 .i.K.Vis;! I;; 
 :!.■). .V')27777 
 
 il.).;)iiiiS^IS,"» 
 
 ;;.-..:.si),s:).!7 ,1 
 
 .■I.KolMKi;!! I 
 
 .•;.■>. (;(»s:)s7(; ! 
 
 ;i.j.(i2.i()2i;2 
 
 :;.).';:i:o.v.t.! 
 
 ;i.').(;.')io-^ii!) 
 
 :!.>.(;ii.iioin) 
 
 ;!,'.. (;7iM2.V') 
 
 :'..j. 7071121 
 
 ;:.). 72 11122 
 
 .'!.). 7^>.)I .!i!7 
 
 ;i:..7i.ii2:.s 
 .;.').7i; ;io;).) 
 
 ;>.'i.7770S7iI 
 
 ;{.•). 70 loio.i I 
 
 ;!.'). 80J027(! , 
 
 ;;:•.>; is;)s:u 
 
 ,!,■). ^.120 t.j 7 , 
 o.ks |i;s;IiIiJ 
 
 ;!.j.si;i)-(i2i ' 
 
 :m.s7 17822 I 
 .i.j.^ssTlO',) ! 
 
 ;i,j.'.n)2i;ii;i 
 
 .{.j.'.IJu.'iilOO 
 .!.■). ',).;0 1881 
 
 :!!.oinoij 
 .■;j.'j:)>;!002 
 
 .i.> 0722 11. J 
 
 :;:).o-;(;i08t 
 ;i.;. 00000(10 
 :U\ ■i;;8»i;2 
 ;!il. )277(J7l 
 
 .•>i).oii(;i2(; 
 
 :ii;.0.V)Jl28 
 ;)().()ii'.).!77i! 
 ;iii.0i.!2.{71 
 .It) 007001.! 
 
 ;il;. 1 100102 
 :u;.i2i78;{7 
 ;!i;.i.!8(;220 
 
 .Hi. 
 
 IJ.'ili 
 
 .■!(;.ii;o2S2i; 
 ;{i;.is()io.-)0 
 
 ;!G.10:i0221 
 ;M.2077.M0 
 Ht;.221oi0(i 
 
 :5G.2;{j;!4io 
 
 H(J.2 101370 
 3t).2()2t;287 
 ."iO. 27071 i;{ 
 
 .•{(;. 200 104; 
 
 .Sli.:iO 12007 
 
 ;m..)18o:;oi) 
 
 3t;.3.U8042 
 
 i;:2i 
 
 l.!J2 
 i32.i 
 I.! 2 J 
 l:!2.V 
 I32il, 
 I. 327 
 1328, 
 1320! 
 l.VM) 
 
 1331; 
 
 1332: 
 1333 
 I334i 
 133.J, 
 133.;, 
 1.337 
 133S 
 1330 
 I310i 
 13U| 
 13121 
 1313; 
 
 13I.J 
 
 I3ia 
 i;!i7i 
 
 13 IS 
 1.3(0 
 13.-)(»' 
 
 1351; 
 
 1352: 
 
 1353: 
 
 1.35(1 
 13)5^ 
 1350: 
 135 7 1 
 135S1 
 1350, 
 13301 
 
 i:v;: 
 
 13021 
 13i;3| 
 
 I3i;(i 
 
 13051 
 
 i30i;i 
 
 1307; 
 
 1.3(i8! 
 
 13001 
 
 13701 
 
 1371 
 
 1372i 
 
 1373] 
 
 1374 
 
 1375 
 
 1370: 
 
 1377 
 
 I37S: 
 
 1370J 
 
 13801 
 
 7(5011 1 
 
 7i:os( j 
 
 750320 
 752!t7t; I 
 755025 
 75S270 I 
 700020 
 7i;3584 ! 
 7i;02ll I 
 70s;)i)0 
 771531 ■. 
 774221 i 
 770880 
 770550 i 
 7S2225 I 
 
 78(8:); I 
 
 7S7500 j 
 7002(1 I 
 702021 I 
 705000 I 
 70S281 I 
 80000 I 
 s0:!i;(0 I 
 8i)i;330 j 
 800025 I 
 811710 
 81 (100 
 817101 
 810S01 
 822500 
 825201 
 8270:K 
 s:io.iOO 
 !^:i]Mi] \ 
 830025 I 
 8 18730 I 
 811(10 
 
 8(4101 ; 
 810SS1 j 
 8 40000 
 852321 
 8550 4 4 
 857700 
 800 lOi; 
 803225 
 80505t; 
 808i;h0 
 871424 
 874101 
 
 87i;ooo 
 
 870041 
 882,38 [ 
 885120 1 
 S37870 i 
 800025 
 80337i; I 
 800120 
 S0SS8 4 ' 
 
 0010 11 ; 
 
 004400 j 
 
 30.3(55(;37 
 30.3503170 
 3i;. 3730370 
 30.3SOS10S 
 3i;. 1005 (01 
 .3ii.ll 12820 
 30.(JSiH12 
 30.4(17313 
 30.4,>5(523 
 30.4001050 
 30.182S727 
 30.40 ;5752 
 .30.5102725 
 ■.2300(7 
 ).{70518 
 ■.5133SS 
 )05,)103 
 
 No. S.iiiare. 
 
 r80S2:i 
 
 ;{0.5023(si» 
 
 3i.(;o;oio( 
 30.oiooi;i;8 
 
 ;{0.(;333H1 
 
 .io.t;(>iooi ( 
 30.i;i;o,;o5i; 
 
 30.0 7 12 (It; 
 30.087872(; 
 
 30.701 (osi; 
 
 30.7151105 
 30.72S7353 
 3i;.7(231i;i 
 :{i;.75505l!) 
 30.7005520 
 30.7.131483 
 30.7007.(00 
 30,81 0.32 (i; 
 30.8230053 
 30.837(800 
 30.8510515 
 30.80 k; 172 
 .•{0.8781778 
 30.8017335 
 30.00528(2 
 .30.018S200 
 30.0323700 
 30.04500,; 4 
 .3i;. 050 (372 
 30.0720031 
 30.0801840 
 37.0000000 
 37.01.35110 
 37.0270172 
 37.0(05184 
 37.0540140 
 37.0675000 
 37.0800924 
 37.00(4740 
 37.1070500 
 37.1214221 
 37.13 1«803 
 37.1483512 
 
 .381 ! 
 3s2i 
 3S3! 
 381, 
 3 ■'5 
 3S0, 
 387 i 
 ,388 ' 
 
 3 so: 
 
 300 
 
 .301 : 
 
 302 i 
 303 
 30 (j 
 
 ,i;)5; 
 
 30(i I 
 
 3o;: 
 
 .308 
 
 300 : 
 
 (00, 
 401 i 
 402: 
 40.3 i 
 401, 
 (05 
 
 40 ; I 
 
 107, 
 (08 
 400 
 (lOj 
 
 (11 
 
 412 
 41.! 
 414 
 415 
 410 
 417 
 118 
 (10 
 420 
 421 
 422 
 42.! 
 424 
 425 
 120 
 427 
 428 
 420 
 430 
 431 
 432 
 433 
 434 
 435 
 430 
 437 
 438 
 CiO 
 440 
 
 1007101 
 100002 4 
 1012080 
 1015(.)0 
 1018225 
 IO2OOO1; 
 1 0237i;0 
 10205K 
 1020321 
 1032100 
 1034 18! 
 I0370il( 
 10(01(0 
 10 (.3230 
 10(0025 
 10(8H10 
 1051000 
 1 05 1 (0 1 
 1057201 
 1000000 
 1002801 
 1 00.500 ( 
 100S(00 
 1071210 
 107 025 
 107i;S!0 
 
 io7o;(o 
 
 10S2104 
 I0<,2S| 
 1 0S8 1 00 
 1000021 
 10037(4 
 1000500 
 1000300 
 2002225 
 2005050 
 2007880 
 2010721 
 2013501 
 2010100 
 2010211 
 2022081 
 2024020 
 2027770 
 2030025 
 2033 470 
 20,30320 
 2030181 
 20420(1 
 2014000 
 20477GI 
 2050G24 
 2053489 
 2050350 
 2059225 
 200200G 
 2O040G0 
 20(i78i( 
 2070721 
 2073000 
 
 Sqre. root. 
 
 ;!7.1(;i808» 
 37.1752000 
 37.1887079 
 37.2021505 
 37. 21. ".,5881 
 37.22.)0200 
 37.2224480 
 37.2.>58720 
 37.2002003 
 37.2827037 
 37. 20,; 1 124 
 37.3005102 
 37.32201.)2 
 37.330.309 1 
 37.3100088 
 37.3030831 
 37.37i;4032 
 37.3S08382 
 37.40,32084 
 47.41057.38 
 37.4200345 
 ,37.4 432004 
 37.4500410 
 37.40008S0 
 .{7 .4833200 
 37. 4000005 
 37.5000087 
 37.5233201 
 37.5,300187 
 37.5(00007 
 37.5032799 
 37.570.5885 
 37.5808022 
 37.0031013 
 37.0104857 
 37.G207754 
 37.0430001 
 37.0503407 
 37.G0001GI 
 37.G828874 
 37.0001530 
 37.7004153 
 |37.722G722 
 ! 37.7,350245 
 137.7401722 
 137.7024152 
 37.7750535 
 37.7888873 
 37.8021163 
 37.8153408 
 37.8235606 
 37.8417759 
 37.8549864 
 37.8681924 
 37.8813938 
 37.8945906 
 37.9077828 
 37.0200704 
 37.0.3(1535 
 37.'.)4;33I0 
 
io 
 
 TABLK OF SQUARES, SQUARE ROOTS 
 
 No. 
 
 nil 
 
 1112 
 I4i:t 
 1441 
 1445 
 144t; 
 1447 
 1441 
 1449 
 1450 
 1451 
 1452 
 1453 
 1454 
 1455 
 145G 
 1457 
 145S 
 1459 
 1460 
 1461 
 1462 
 1463 
 1464 
 1465 
 1466 
 1467 
 1463 
 1469 
 1470 
 1471 
 1472 
 1473 
 1474 
 1475 
 1476 
 1477 
 1478 
 1479 
 1480 
 1431 
 1482 
 1483 
 1484 
 1485 
 1486 
 1487 
 1488 
 1489 
 1490 
 1491 
 1492 
 1493 
 1494 
 
 8<innr(-. 
 
 2076 18 1 
 
 2(>7'.>:!ti I 
 
 2IIH22I9 
 
 20H5i:U) 
 
 2088025 
 
 2090!II6 
 
 20t»3H()l> 
 
 201)6701 
 
 2001)601 
 
 2102500 
 
 2105401 
 
 2108301 
 
 2111209 
 
 2114116 
 
 2117025 
 
 21190:!(; 
 
 2I22S49 
 
 2125761 
 
 21 2868 I 
 
 2131600 
 
 2131521 
 
 2137444 
 
 2140361) 
 
 2143296 
 
 3146225 
 
 2149156 
 
 2152089 
 
 2155021 
 
 2157961 
 
 2160900 
 
 2163841 
 
 2166781 
 
 2169729 
 
 2172676 
 
 2175625 
 
 2178576 
 
 2181529 
 
 2184484 
 
 2187441 
 
 2190400 
 
 2193361 
 
 2196324 
 
 2199289 
 
 2202256 
 
 2205225 
 
 2208196 
 
 2211169 
 
 2214144 
 
 2217121 
 
 2220100 
 
 2223081 
 
 2226001 
 
 2229019 
 
 2232036 
 
 Sqrt'. nxit. 
 
 3:.9iio:.i).')S 
 
 37.97:ti;7.'il 
 
 37.9S6><;!1)S 
 
 3H.()I)II(I0I)() 
 
 3X.0131556 
 
 3M 026:iO(i7 
 
 3-<.():{9l5;!2 
 
 38.0525:)r)2 
 
 3-^. 0(1573 J6 
 
 3S, 0788655 
 
 38.01)19939 
 
 .38.1051178 
 
 3S. 1182371 
 
 38.1313519 
 
 3-<.l4ll622 
 
 3.S.15756S1 
 
 38.170661);! 
 
 38.1837(;62 
 
 3S.196S,)H5 
 
 38.209916;; 
 
 :W. 2230297 
 
 38.2361085 
 
 38.2491 S29 
 
 ;-{8.2ii2252i; 
 
 3S. 2753 184 
 
 38.2883791 
 
 ,38.30113(10 
 
 :iS. 314 1^-81 
 
 :^8.;!275;{5s 
 
 38.3405790 
 3S.:{536178 
 38.3666522 
 :58. 3796821 
 ;H8. 3927076 
 :«.4057287 
 38.4187454 
 38.4317577 
 38.4447656 
 :^8. 457 7691 
 38.4707681 
 38.4837627 
 :^8,49675:J0 
 38.50973110 
 ;^8. 5227206 
 38.5356977 
 :^8.5 186705 
 :iS.56l6;<89 
 38.5746030 
 38.5875627 
 :^8.6005181 
 38.6134691 
 33.6264158 
 33.6393582 
 38.6522962 
 
 No. 
 1 ID.-) 
 
 149:i 
 
 I 197 
 
 1498 
 
 1499 
 
 1500 
 
 1501 I 
 
 1502' 
 
 1503 
 
 1504 
 
 1 505 
 
 1506 
 
 1507 
 
 1 508 
 
 1509 
 
 1510 
 
 1511 
 
 1512 
 
 1513 
 
 1511 
 
 1515 
 
 1516 
 
 1517 
 
 1518 
 
 1519 
 
 1 520 
 
 1521 
 
 1522 
 
 1523 
 
 1524 
 
 1 525 
 
 1526 
 
 1527 
 
 1528 
 
 1529 
 
 1530 
 
 1531 
 
 1532 
 
 1533 
 
 15,4 
 
 15;{5 
 
 1536 
 
 15:57 
 
 I5;i8 
 
 15.59 
 
 1540 
 
 1511 
 
 1542 
 
 1543 
 
 154 4 
 
 1545 
 
 1546 
 
 1547 
 
 Si|iiiiro. S]rc'. root. 
 
 2-.!;5')025 : 
 
 22;!Mi)i(; ' 
 
 2211009, 
 
 2241001 
 
 22170(11 ! 
 
 2250000 j 
 
 225.3001 
 
 22511001 ' 
 
 2259009 i 
 
 22620 1 lij 
 
 2265025 ; 
 
 22<i8031 
 
 2271019 
 
 2274 06 J 
 
 2277081 
 
 22-^0100 
 
 2283121 
 
 22^61 14 
 
 2289169 
 
 2292196 
 
 2295225 
 
 229.s2.">(i 
 
 2:101289 
 
 2;;oi:w4 
 
 2307:561 
 
 2310100 
 
 2313111 
 
 2:516181 
 
 2:! 19529 
 
 2322576 
 
 2:525(;25 
 
 2328676 
 
 2331729 
 
 2:534781 
 
 2:537811 
 
 2340900 
 
 2313961 
 
 2317024 
 
 2:550089 
 
 2:553156 
 
 2,556225 
 
 2359296 
 
 2362369 
 
 2:^65444 
 
 2:^68521 
 
 2:571600 
 
 2374681 
 
 2377761 
 
 2:580849 
 
 238:59:56 
 
 2387025 
 
 2:W0116 
 
 2393209 
 
 No. 
 
 3-<. 6652 299 
 
 :!-<.(;7><l59:5 
 
 38.6'.)10><4:5 
 
 ;5y. 70 10050 
 
 .38.7 1 6921 4 
 :5S.729>^:5:!5 
 .38.7427412 
 :i8.755(i|l7 
 :18. 7685 139 , 
 :S8.7.sll3'<9 
 3,>^.71)i:,29l ' 
 3^^.^0:52158 ; 
 
 ;5s.820():i78 
 
 :!8.s;!'J:)7.-)7 ! 
 3H.?<15s49l I 
 ;!S.M5,>^7IS1 I 
 ;58. 87 15831 | 
 .5S. 8811 142 
 3-i.8;)7:!ii06 
 :58 9101529 
 3S. 9230009 '■ 
 
 :i'^.9:;5>!i47 1 
 
 ;^8.9i,S68ll I 
 3S, 96151 91 ! 
 3x. 974,3505 1 
 .38.9871774 ' 
 39.0000000 1 
 39.0128181 ' 
 39. 025(5326 
 :i9.0:H8JI2(M 
 39.0512483 I 
 39.0640199 
 39.076817:5 
 :!9. 08964 06 
 :!9. 1024296 
 :59. 1152114 ; 
 ;r.). 1279951 i 
 39.1407716: 
 
 ;59. 15:551:^9 I 
 
 :{9. 1663 120 ' 
 
 39.1790760 
 
 39.1918:159 : 
 
 39.2045915 
 
 39.2173131 
 
 39.2300905 
 
 39.2128337 
 
 39.2555728 
 
 39.2683078 
 
 39.2810:587 
 
 39.2937651 
 
 39.3064880 
 
 :59.3 192065 
 
 39.3319208 
 
 1518 
 1519 
 1550 
 1551 
 1552 
 1553 
 1551 
 1555 
 
 1556 I 
 
 1557 ! 
 1558 
 1559 ! 
 
 1560 
 1561 
 15(52 
 
 156:; 
 
 1561 
 
 1565 
 
 1566 
 
 1567 
 
 1568 
 
 1569 
 
 1570 
 
 1571 
 
 1572 
 
 1573 
 
 1571 
 
 1575 
 
 1576 
 
 1577 
 
 1578 
 
 1679 
 
 1180 
 
 1581 
 
 15-12 
 
 1583, 
 
 15S4 I 
 
 1585 
 
 15,86 
 
 1587 
 
 1588 
 
 1589 
 
 1590 
 
 1591 
 
 1592 
 
 1593 
 
 1594 
 
 1595 
 
 1596 
 
 1597 
 
 1598 
 
 1599 
 
 16 yO 
 
 8i|iiurr. 
 
 2:i96;50 1 
 2:599101 
 2102500 
 2I0,'m;0I 
 210^70 4 
 2lll-<09 
 211 1916 
 2I1S()25 
 2l2Ii:5(i 
 2 12 12 19 
 2127:564 
 24:50 1>1 
 2i:5:;600 
 
 2436721 
 2i:!l)>^l4 
 2112969 
 21160% 
 
 2119 
 
 ::!5(i ! 
 2|,V,|S9i 
 245S621 ! 
 2.|.il76I I 
 2tr,l900 
 2l'i80ll I 
 2171184 i 
 2174:519; 
 2177175 
 2|S0(;25 
 2|8:;776 
 2l8(i929 
 2I900-S4 
 2493211 
 2196400 
 2199561 
 2.->02724 
 2505S89 
 2509056 
 2512225 
 2515:596 
 251S569 
 2521714 
 2521921 
 2528100 
 2531281 
 25 5446 1 
 2,537619 
 25108156 
 2511025 
 2517216 
 2550109 
 25,53604 
 2556S01 
 2560000 
 
 Siit'i". mot. 
 31).:51I631I 
 
 :!9.:;573:!73 
 
 39.;57()0:591 
 39. .3^27373 } 
 :!9.395I3I2 I 
 39.1081210 
 3'.i. 420^067 I 
 39.433 MH3 I 
 
 :;9.4ii;ii;58 I 
 
 39.458s:59:5 | 
 :!9. 4715087 
 ;59.4SII7I0! 
 39.1968:55:5 ! 
 :!9. 509 1925 ; 
 :59.522M57 
 39.5 517918 ! 
 :!9.5l7i:!99! 
 :59.5i;OOS09 ; 
 39.5727179 
 
 :;9.5,s,-,:5508 | 
 :;9. 5979797 i 
 
 ;59.(;;o604i) { 
 
 39.62:52255 | 
 39. 6:!.)8 124 
 ;59.6I8|552 
 39.(5610640 
 :!9. 157:56688 
 :59. 6^62696 
 
 :!9.(598si;i;5 
 
 :59.7 114593 
 
 ,39.72 10 181 
 
 39.73(56:529 
 
 :59. 7492138 
 
 :59.76 17907 
 
 39.774:5636 
 
 39.7S69325 
 
 39.7991976 
 
 :S9.8l205-<5 
 
 39.8241155 
 
 159.8376646 
 
 39.8497177 
 
 ;59. 81522628 
 
 39.8748010 
 
 39.8873413 
 
 39.8998717 
 
 39.9121041 
 
 :i9. 92 19295 
 
 .39.9374511 
 
 :59. 9 199687 
 
 ;59. 9624824 
 
 39.9749922 
 
 39.9874980 
 
 40.0000000 
 
TABLE II. a. 
 
 AREAS OF CIRCLES, FROM c', TO 150. 
 
 [Adiandni; by an Eighth.] 
 
 Diam. 
 
 Area. 
 
 .00019 
 
 Uiam. 
 
 Aren. 
 
 Dinm. 
 
 Area. 
 
 Dlam. 
 
 Area. 
 
 Dinm. 
 
 Area. 
 
 Vl 
 
 4. 
 
 12. .500 1 
 
 10. 
 
 78.54 
 
 16. 
 
 201 .002 
 
 22. 
 
 380.134 
 
 1 
 
 An*!*" •? 
 
 ■H 
 
 13.304 
 
 ■H 
 
 80.5157 
 
 ■'6 
 
 204.210 
 
 H 
 
 384.465 
 
 ■S3 
 
 .OOOi 7 
 
 ■'^ , 
 
 14.1802 
 
 ■U 
 
 82.5101 
 
 M 
 
 207. .391 
 
 .34 
 
 3H8.822 
 
 iV 
 
 .00:!07 
 
 ■H 
 
 15.0331 
 
 H 
 
 84.5409 
 
 .% 
 
 210.597 
 
 .?«' 
 
 393.203 
 
 1 
 1 
 
 .0'227 
 
 
 15.9013 
 
 10.8001 
 
 % 
 
 80.59 
 
 88.0043 
 
 
 213.825' 
 217.073 
 
 ■'I 
 
 397.008 
 402.038 
 
 I'a 
 
 .02701 
 
 ■H 
 
 17.7205 
 
 •H 
 
 90.7028 
 
 '% 
 
 220.353 
 
 ■% 
 
 400.493 
 
 J 
 4 
 
 .01909 
 
 ■i 
 
 18.0055 
 
 7 
 •^8 
 
 92.8858 
 
 ■jj 
 
 223.054 
 
 7 ■ 
 
 440.972 
 
 s 
 
 
 5. 
 
 19.0.55 
 
 11. 
 
 95.0334 
 
 17. 
 
 220.981 
 
 23. 
 
 410.477 
 
 A 
 
 .07(17 
 
 ■H 
 
 20.029 
 
 ■H 
 
 97.20.55 
 
 ■)i 
 
 230.33 
 
 ■H 
 
 420.004 
 
 t 
 
 .11015 
 
 
 21.0175 
 
 • 7tt 
 
 99.4022 
 
 A 
 
 233.705 
 
 A 
 
 424. .557 
 
 ■,'', 
 
 
 22.0907 
 
 101.0234 
 
 .% 
 
 237.104 
 
 .% 
 
 429.1.35 
 
 Iff 
 
 .15038 
 
 A:. 
 
 23.7583 
 
 .hi 
 
 103.8091 
 
 .).. 
 
 240.628 
 
 .».. 
 
 433.731 
 
 i 
 
 .1!)G;;5 
 
 ■■^8 
 
 24.8505 j 
 
 .^l 
 
 100.1394 
 
 ^% 
 
 243.977 
 
 .,^& 
 
 438.303 
 
 i°j 
 
 .2H5 
 
 4 
 
 25.9072 1 
 
 •'■{ 
 
 108.4343 
 
 ■H 
 
 247.45 
 
 ■H 
 
 443.014 
 
 « 
 
 ..•50079 
 
 4 
 
 27.1085 
 
 ■jI 
 
 110.7536 
 
 ■Jb 
 
 250.947 
 
 •% 
 
 447.099 
 
 8 
 
 c. 
 
 28.2744 
 
 12. 
 
 113.098 
 
 18. 
 
 254.467 
 
 24. 
 
 452.39 
 
 u 
 
 .;ni22 
 
 •H 
 
 29.4047 
 
 ■H 
 
 115.400 
 
 ■H 
 
 258.016 
 
 ■)i 
 
 457.115 
 
 1 •> 
 
 .41178 
 
 •i'li 
 
 30.0790 
 
 ■li 
 
 117.859 
 
 •U 
 
 201. .587 1 
 
 •\\ 
 
 401.804 
 
 
 J« 
 
 31.9192 
 
 .% 
 
 120.270 
 
 .?8 
 
 265.182 ! 
 
 .% 
 
 400.038 
 
 ii 
 
 .51848 
 
 • '.» 
 
 33.1831 1 
 
 ■H 
 
 122.718 
 
 •••J 
 
 268.803 1 
 
 .>., 
 
 471.430 
 
 7 
 8 
 
 15 
 
 .001.12 
 
 ■'4 
 
 34.4717 
 
 ■H 
 
 125.184 
 
 .^8 
 
 272.447 1 
 
 .>8 
 
 476.259 
 
 .09029 
 
 •/8 
 
 35.7847 
 37.1224 
 
 •'b 
 
 127.076 
 130.192 
 
 A 
 
 v'8 
 
 270.117 
 279.811 I 
 
 V8 
 
 481.106 
 
 485.978 
 
 1. 
 
 .7851 
 
 7. 
 
 38.4840 
 
 13. 
 
 132.733 
 
 19. 
 
 283.529 
 
 25. 
 
 490.875 
 
 •M 
 
 .99402 
 
 •3s 
 
 39.8713 
 
 ^^ 
 
 i:!5.297 
 
 .3^ 
 
 287.272 
 
 .38 
 
 495.796 
 
 1.2271 
 
 •M 
 
 41.2825 
 
 •H 
 
 137.886 
 
 .Yx 
 
 291.039 
 
 
 500.711 
 
 •?'t 
 
 1 .4848 
 
 .% 
 
 42.7184 
 
 .% 
 
 140.5 
 
 .% 
 
 294.831 
 
 505.711 
 
 • 1.^ 
 
 1.7071 
 
 .>., 
 
 44.1787 
 
 .1..' 
 
 143.139 
 
 .v.. 
 
 298.618 i 
 
 .^.> 
 
 510.700 
 
 •^/8 
 
 2.07;'.9 
 
 ■•)a 
 
 45.0030 
 
 ..^ 
 
 145.802 
 
 .fi 
 
 302.489 1 
 
 .,^8 
 
 515.725 
 
 ••^i 
 
 2.4052 
 
 •% 
 
 47.173 
 
 ■% 
 
 148.489 
 
 •% 
 
 300.355 ! 
 
 .?4 
 
 520.709 
 
 
 2.7011 
 
 •J/s 
 
 48.707 
 
 .% 
 
 151.201 
 
 A 
 
 310.215 
 
 •Ja 
 
 525.837 
 
 2. " 
 
 3.1410 
 
 8. 
 
 50.2050 
 
 14. 
 
 153.938 
 
 20. 
 
 314.' 1 
 
 26. 
 
 530.93 
 
 •H' 
 
 3.5 Ilia 
 
 •;& 
 
 51.8480 
 
 .% 
 
 150.699 
 
 .38 
 
 318 >l 
 
 ■^8 
 
 530.047 
 
 •>4 
 
 3.970 
 
 •h 
 
 63.4502 
 
 ■\i 
 
 159.485 
 
 M 
 
 322.063 
 
 •34 
 
 541.189 
 
 •>t 
 
 4.4302 
 
 3a 
 
 55.0885 
 
 .% 
 
 102.295 
 
 3^ 
 ./8 
 
 320.051 
 
 ■A 
 
 546.350 
 
 
 4.90S7 
 
 . '.. 
 
 50.7451 
 
 .1., 
 
 105.13 
 
 .U 
 
 330.064 
 
 .>.; 
 
 551.547 
 
 •H 
 
 5.4119 
 
 ■''^ 
 
 58.4204 
 
 .'^ 
 
 107.989 
 
 .}l 
 
 .334.101 
 
 '■% 
 
 556.762 
 
 '% 
 
 5.9395 
 
 ■% 
 
 00.1321 
 
 \ .% 
 
 170.873 
 
 .% 
 
 338.163 
 
 t 
 
 562.002 
 
 ■l\ 
 
 0.4918 
 
 •A 
 
 61.8025 
 
 1 J-8 
 
 173.782 
 
 7/ 
 V 8 
 
 342.25 
 
 56'. .267 
 
 3. 
 
 7.0080 
 
 9. " 
 
 03.01 7J 
 
 15. 
 
 176.715 
 
 21. 
 
 346.301 
 
 27. ' 
 
 572.557 
 
 •'8 
 
 7.0099 
 
 •'-8 
 
 05.3908 
 
 •^8 
 
 179.672 
 
 .Vb 
 
 350.497 
 
 ■}i 
 
 577.87 
 
 
 8.2957 
 
 
 07.2007 
 
 .'4 
 
 182.651 
 
 ■3t 
 
 351.657 
 
 A 
 
 683.208 
 
 8.9402 
 
 09.029.3 
 
 .■'» 
 
 185,601 
 
 358.841 
 
 .% 
 
 588.571 
 
 ■)i 
 
 9.0211 
 
 M 
 
 70.8S'23 
 
 .1., 
 
 188.692 
 
 • »' 
 
 363.051 
 
 •3'o 
 
 593.958 
 
 .% 
 
 10.3206 
 
 ■^ 
 
 72.7599 
 
 ■% 
 
 191.748 
 
 ■H 
 
 367.284 
 
 •;^ 
 
 599.376 
 
 ■% 
 
 11.0446 
 
 •% 
 
 74.662 
 
 •% 
 
 194.828 
 
 ■\ 
 
 .37l.54.'i 
 
 3-' 
 
 604.807 
 
 ■H 
 
 11.7932 
 
 •Ji 
 
 76.5887 
 
 ■A 
 
 197.933 
 
 'A 
 
 375.826 
 
 610.268 
 

 12 
 
 
 
 AREAS OF riRPLKM. 
 
 
 
 • 
 
 
 
 
 
 TABLE.— {Continued.) 
 
 
 
 
 I 
 
 Ditiin. 
 28. 
 
 Aivii. 
 6l5.7f)t 
 
 Diiiin. 
 35. 
 
 Aroii. 
 9G?.115 
 
 ; Diam. 
 
 Aien. 
 1.385.44 
 
 Diiiin. 
 
 Aron. 
 
 Diain. 
 56. 
 
 Area- 
 2163.01 
 
 
 42. 
 
 49. 
 
 18-5.74 
 
 i , 
 
 •}a 
 
 ■■J I -JtiH 
 
 •'h 
 
 9(i8.i>l)l) 
 
 : •'..' 
 
 i;;'.i;i,7 
 
 'H 
 
 18! 6.17 
 
 •'a 
 
 2171.02 
 
 
 ,? 
 
 A 
 
 ('.2'; Tiif< 
 
 .'.t 
 
 '.)7."i.90S 
 
 .', 
 
 1 101.98 
 
 •\\ 
 
 19(5.01 
 
 .'.I 
 
 2185.05 
 
 
 
 :f 
 
 (i:<'2.:r>7 
 
 .^. 
 
 981'. 8 12 
 
 .••'a 
 
 1410.29 
 
 .^l 
 
 1914.7 
 
 •'" 
 
 2196.11 
 
 
 
 (iHT.'JIl 
 
 •'i 
 
 989. S 
 
 ; •'=-• 
 
 MI8.G3 
 
 .'., 
 
 1921.42 
 
 • •« 
 
 2507.19 
 
 
 ;| 
 
 ■fi 
 
 G4;t.5rj 
 
 .'I 
 
 9y(i.7fl3 
 
 .'i 
 
 1 12G.9S 
 
 •'^l 
 
 1931.15 
 
 • •'b 
 
 2118.3 
 
 
 
 ■'^4. 
 
 (i4'J.lH2 
 
 ■li 
 
 I003.7y 
 
 ■% 
 
 1 1.35. 3G 
 
 ' -H 
 
 1913.91 
 
 ■% 
 
 2529.43 
 
 
 !' 
 
 ■Ji 
 
 G5i.h;{<j 
 
 ■ji 
 
 1010.822 
 
 \ ■% 
 
 1143.77 
 
 •% 
 
 J 195;!.G9 
 
 A 
 
 2510.51 
 
 
 ) 
 
 29. 
 
 Gti0.o2I 
 
 3G. 
 
 1017.878 
 
 43. 
 
 1152.21 
 
 50 
 
 1 19113.5 
 
 57. 
 
 2551.76 
 
 
 
 ■K 
 
 Cl>().227 
 
 •}a 
 
 1024.959 
 
 : .'a 
 
 I JG0.G5 
 
 •H 
 
 ' 1973. .33 
 
 •)8 
 
 251)2.97 
 
 
 
 A 
 
 t)7l.!>58 
 
 
 1032.0G5 
 
 •k^ 
 
 14G9.I3 
 
 vt 
 
 1 198.3.18 
 
 ..'4 
 
 2574.2 
 
 
 
 .H 
 
 G77.7H 
 
 1039.195 
 
 .h 
 
 1477.G3 
 
 ..^-, 
 
 1993.05 
 
 .H 
 
 2585.45 
 
 
 
 .».< 
 
 GH;h4<Jt 
 
 .•.. 
 
 I04G.3)9 
 
 .'.. 
 
 148(1. 17 
 
 .'.. 
 
 20(12.97 
 
 .'.. 
 
 2596.73 
 
 
 i 
 
 ■fa 
 
 G8i).2'J8 
 
 .'^ 
 
 1053.528 
 
 .:^o 
 
 1191.72 
 
 ,?8 
 
 2012.89 
 
 .'0 
 
 2i;08.03 
 
 
 > 
 
 
 G!)j.l28 
 
 ■•ll 
 
 IOGO.7.12 
 
 ■'ii 
 
 1503.;t 
 
 ■'^■i 
 
 2022.85 
 
 •■■'.1 
 
 2619.36 
 
 
 ^ 
 
 7O0.l)Sl 
 
 .^• 
 
 10G7.9G 
 
 44. 
 
 1511.9 
 
 \ -'l 
 
 20,32.82 
 
 ■>8 
 
 26.10.71 
 
 
 ; 
 
 30* 
 
 70G.8G 
 
 37. 
 
 1075.213 
 
 1.J20.53 
 
 51. 
 
 2012.82 
 
 .•)8. 
 
 2642.09 
 
 
 
 •% 
 
 712.7G2 
 
 .'8 
 
 1082.49 1 
 
 ■/B 
 
 1529.18 
 
 .'a 
 
 2052.85 
 
 ■H 
 
 2653.19 
 
 
 
 -H 
 
 718. Gi) 
 
 •'■i 
 
 1089.792 
 
 ■.k 
 
 1537.8(; 
 
 ■k^ 
 
 2062.9 ' 
 
 .'4 
 
 2GGI.91 
 
 
 i 
 
 .?» 
 
 724.G41 
 
 .■••« 
 
 1097.118! 
 
 15 IG.55 
 
 .'I 
 
 2072.98 
 
 .h 
 
 2676. .36 
 
 
 • 
 
 .!.< 
 
 ".•{0.G18 
 
 •M 
 
 11U4.4G9 
 
 ■}.i 
 
 I5.)5.2s' 
 
 ' .1.. 
 
 20s;{.08 
 
 .'-2 
 
 2687.8 4 
 
 
 
 .'52 
 
 7:iG.Giy 
 
 .5o 
 
 1111.844! 
 
 .h 
 
 15GI.0.{ 
 
 ! ..^8 
 
 2093.2 
 
 .''I 
 
 2699.33 
 
 
 
 :li 
 
 742.GI4 
 
 •••4 
 
 U19.2U; 
 
 ■:U 
 
 1. -.72.81! 
 
 •'■* 
 
 2103.:!5: 
 
 ■U 
 
 2710.86 
 
 
 
 748. Gi) I 
 
 ■K 
 
 112G.GG8 , 
 
 •7-8 
 
 158l.Gi 1 
 
 ■h 
 
 2113.52 
 
 •h 
 
 2722.4 i 
 
 
 
 31. 
 
 7iJ1.7Gi» 
 
 38. 
 
 11.54.118 1 
 
 45. 
 
 1590.431 
 
 52. 
 
 212.1.72 
 
 59. 
 
 2733.98 
 
 
 if 
 
 .;J^ 
 
 7G0.8G8 
 
 .^B 
 
 1141.591 i 
 
 ■}b 
 
 1599.28, 
 
 • « 
 
 21.33.91 
 
 ..'b 
 
 2745.57 
 
 
 <* 
 
 Ai 
 
 7GG,l)i)2 
 
 •H 
 
 1149.089' 
 
 •^^ 
 
 IGO8.I5: 
 
 ••^ 
 
 2141.19' 
 
 •M 
 
 2757.2 
 
 
 r 
 
 77:!. 14 
 
 M 
 
 115G.GI2 ' 
 
 .?8 
 
 1GI7.01 
 
 .•*8 
 
 2I5I.4G: 
 
 .;^B 
 
 2768.81 ! 
 
 
 
 . l:i 
 
 77i).Hi;j 
 
 M 
 
 |iG4.159l 
 
 .'.. 
 
 1G25.97 
 
 .'.. 
 
 21GI.76 
 
 .1.. 
 
 27X0.51 i 
 
 
 
 '.fa 
 
 78.J.51 
 
 .5t 
 
 1171.731 1 
 
 .fu 
 
 lG.il.92' 
 
 .?ii 
 
 2175. OS 
 
 .>6 
 
 2792.21 i 
 
 
 
 • '^1 
 
 7yi.7;v2 
 
 •'i 
 
 1179.327 1 
 
 ■h 
 
 1G43.S9 
 
 ••'.I 
 
 2185.42: 
 
 •■At 
 
 2S03.93 
 
 
 ' 
 
 
 
 71)7.1)78 
 
 •^a 
 
 118G.91S 1 
 
 •Ja 
 
 1G52.88 
 
 •ll 
 
 2195.79 
 
 ■jI 
 
 2815.(:7 1 
 
 
 
 32. 
 
 804.2.-) 
 
 39. 
 
 1194.593 ; 
 
 4G. 
 
 1GG!.91 i 
 
 .53. 
 
 22()(;.19 
 
 60. 
 
 2827.44 ! 
 
 
 
 -.^a 
 
 810.515 
 
 •'^ 
 
 1202.2G3I 
 
 ■H 
 
 1G70.95 
 
 •H 
 
 2216.61 
 
 ■H 
 
 2,8;!9.2;! i 
 
 
 
 ■ 
 
 .;'a 
 
 81G.8G5 
 
 .'5 
 
 12(19.9.18 i 
 
 ■k 
 
 1G80.01 ; 
 
 •'^ 
 
 2227.05 
 
 ■:4 
 
 2851.05 
 
 
 
 82H.20i) 
 
 • ••'n 
 
 1217.G77 
 
 1G89.1 ! 
 
 .?a 
 
 22,37.52 ; 
 
 2S62.S9 1 
 
 
 
 .1.. 
 
 82D.578 
 
 .'., 
 
 1225.42 
 
 .1.; 
 
 ■"S.23 
 
 .'.'. 
 
 2218.0' 1 
 
 M 
 
 2871.76 
 
 
 
 •■'^ 
 
 8;i5.<J72 
 
 .'I 
 
 1233.188 
 
 .^l 7 
 
 •% 
 
 22.58.5;; 
 
 •/a 
 
 2886 65 
 
 
 1 ^ 
 
 842.;{S)0 
 
 •■'4 
 
 1240.981 
 
 .4 
 
 '% 
 
 2269.07 
 
 •'yi 
 
 2898.57 I 
 
 
 
 848.8;i;{ 
 
 •Js^ 
 
 1248.798 ' 
 
 0.V3 
 
 7^ 
 
 2279.64 
 
 ■A 
 
 2910.51 
 
 
 
 • o 
 
 33. 
 
 855.:',01 
 
 40. 
 
 125G.G4 1734.95' 
 
 54. 
 
 2290.23 
 
 61. 
 
 2922.47 
 
 
 
 •Jb 
 
 8G1.792 
 
 •'^ 
 
 12G4.5 
 
 1744.18, 
 
 ■H 
 
 2300.84 
 
 .-•a 
 
 2931.46 
 
 
 
 / o 
 
 8G8.309 
 
 ■ H 
 
 1272.39 
 
 ..H 
 
 1753.-15 
 
 
 2311.48 
 
 .^4 
 
 2946.48 ! 
 
 
 .4 
 
 874.85 
 
 .% 
 
 1280.31 
 
 .% 
 
 17G2.73 
 
 2322.14 
 
 .% 
 
 2958.52 ; 
 
 
 
 . • o' 
 
 881.415 
 
 .1.; 
 
 1288.25 
 
 .1., 
 
 1772.05 
 
 1 / 
 • . 
 
 2332. ><3 1 
 
 .'.^ 
 
 2970.58 i 
 
 
 ■ 
 
 >« 
 
 888.005 
 
 •;''8 
 
 129G.21 
 
 .^l 
 
 1781.39 
 
 •?-^ 
 
 2313.55' 
 
 :% 
 
 2982.67 1 
 
 
 
 •'*i 
 
 894. G2 
 
 •'■1 
 
 1304.2 
 
 •% 
 
 1790.7G 
 
 1 
 
 2.351.28 
 
 •■'.i 
 
 2991,78 j 
 
 
 
 34. 
 
 901.259 
 
 ■i 
 
 1312.21 
 
 ■A 
 
 1800.14 
 
 23(;5.05 
 
 ■% 
 
 .3006.92 1 
 
 
 ! 
 
 907.922 
 
 41. ' 
 
 1320. 2G 
 
 18. 
 
 1809.5G| 
 
 -^ 
 
 55. 
 
 2375.83 ' 
 
 62. 
 
 3019.08; 
 
 
 ' 
 
 'A 
 
 914.GI 
 
 •M 
 
 1.328.32 
 
 H 
 
 1818.99 
 
 ■H 
 
 23SG.65 i 
 
 'H 
 
 30.31.26 
 
 
 i; 
 
 921.823 
 
 •H 
 
 l3;iG.4 
 
 ■li 
 
 1828.46 
 
 ■i 
 
 2397.48 ! 
 
 A 
 
 3043.47 
 
 
 928.0G 
 
 .^ 
 
 1344.51 
 
 .h 
 
 1837.93 
 
 .% 
 
 2408.34 1 
 
 ■H 
 
 3055.71 
 
 
 
 \/ 
 
 934.822 
 
 '}'! 
 
 1352. Go 
 
 .1.; 
 
 1847.45 
 
 M 
 
 2419.22 
 
 
 3067.97 
 
 
 '.H 
 
 941.609 
 
 ■h 
 
 13G0.81 
 
 ■■% 
 
 1856.99 
 
 '^ 
 
 24.30.18 
 
 .'1 
 
 3080.25 
 
 
 
 948.419 
 
 '% 
 
 13G9. 
 
 ■% 
 
 1866.55 
 
 ■% 
 
 2441.07' 
 
 .3092.56 
 
 
 955.255 
 
 •>8 
 
 1377.21 
 
 ■% 
 
 1876.13 
 
 ■'X 
 
 2452.03 ; 
 
 3104.89 
 
 
 i 
 
 ■ i 
 
 i- 
 
 .» 
 
 
table: II. b. 
 
 CIRCUMFERENCES OP CIRCLES, PROM l^ TO 150. 
 
 [Adcaucimj by an Ei(jlith.\ 
 
 
 Dium. 
 
 Cii'cum. 
 .04909 
 
 Diiuii. 1 
 4. 
 
 Circiim. i 
 
 1 
 
 12.5ii(;4 
 
 Dium. 
 
 10, ' 
 
 Cii'cum. 
 31.416 
 
 Dium. ! 
 10. 
 
 Circum. 
 
 Dium. 
 22. 
 
 Circum. 
 69.1152 
 
 50.2656 
 
 
 1 
 
 .09-S17 
 
 ■ii 
 
 12.9591 1 
 
 •'a 
 
 31.8087 
 
 •^8 
 
 50.658;^ ■ 
 
 •38 
 
 69.5079 
 
 
 M2 
 
 ■!^ 
 
 13.3518 1! 
 
 ■:i 
 
 32.2014 
 
 .'i 
 
 51.051 ] 
 
 .H 
 
 t;9.9066 
 
 
 iV 
 
 ,19G;!j 
 
 13.7U5 
 
 32.5911 
 
 ..•'a ; 
 
 51.4437 
 
 :h 
 
 70.2933 
 
 
 \ 
 
 .3927 
 
 ■:4 
 
 14.1372 
 14.5299 
 
 
 ;i2.9S(;8 
 33.3795 
 
 •fa 
 
 51.8:i61 i 
 52.2291 i 
 
 ■k 
 
 70.686 
 71.0787 
 
 
 A 
 
 .589 
 
 •;''4 
 
 14,9226 
 
 •^4 
 
 .33.7722 
 
 •^4 
 
 52.6218 
 
 'H 
 
 71.4714 
 
 
 \ 
 
 .7851 
 
 •J8 
 
 15.3153 
 
 •Ji- 
 
 3 1. 1(119 
 
 ■fi 
 
 53.0145 
 
 'fi 
 
 71.86J1 
 
 
 *■ 
 
 
 5. 
 
 15.708 
 
 ll. 
 
 31.5576 
 
 17. 
 
 53.4072 
 
 23. 
 
 72.2568 
 
 
 1% 
 
 .98175 
 
 •}a 
 
 16.1007 1 
 
 'H 
 
 34.9503 
 
 ■ 'h 
 
 53.7999 • 
 
 ..s; 
 
 72.6195 
 
 
 
 1.1781 
 
 il 
 
 16.1934 , 
 
 yl 
 
 35.313 
 
 .Ki 
 
 54.1926 ; 
 
 is 
 
 73.0122 
 
 
 X 
 
 l.;;7415 
 
 16.8861 ■ 
 
 35.7353 
 
 .?., 
 
 51.58,53 
 
 73.4349 
 
 
 Iff 
 
 .1.,' 
 
 17.27.S8l 
 
 .'., 
 
 36,1284 
 
 1 - 
 
 51.978 
 
 .'.; 
 
 73.82^6 
 
 
 i 
 
 1.5708 
 
 ■'C 
 
 17. (17 15 
 
 ■'^ 
 
 36.5211 
 
 '.9^ 
 
 55.3707 
 
 .>a 
 
 74.2203 
 
 
 A 
 
 1.7G715 
 
 '% 
 
 18.0612 
 
 '% 
 
 36.9138 
 
 ■li 
 
 55.7631 
 
 •% 
 
 74.613 
 
 
 
 ■}i 
 
 18.4569 
 
 ■jI 
 
 37.3065 
 
 ■jI 
 
 56.1561 
 
 .% 
 
 75,0057 
 
 
 9 
 
 1 .9o35 
 
 6. 
 
 18.8496 
 
 12. 
 
 37.6992 
 
 18. 
 
 56.5188 
 
 24. 
 
 75.3984 
 
 
 U 
 
 2.159S5 
 
 -'^ 
 
 19.2423 
 
 ..'a 
 
 38.0919 
 
 .,^8 
 
 56.9415 
 
 .H 
 
 75.7911 
 
 
 a 
 
 2.3502 
 
 it 
 
 19.635 1 
 
 ■l-i 
 
 38.4846 
 
 >l 
 
 57.3312 i 
 
 ■H 
 
 76.1838 1 
 
 
 4 
 
 
 20.0277 
 
 .■V 
 
 38.8773 
 
 57.7269 i 
 
 76.5765 
 
 
 15 
 
 2.55255 
 
 .1., 
 
 20.4201 
 
 .',. 
 
 39.27 
 
 .'..' 
 
 58.1196 1 
 
 .!.< 
 
 76.9692 
 
 
 7 
 
 2.7489 
 
 ••'0 
 
 20.8131 
 
 .91 
 
 39.6627 
 
 .9'a 
 
 58.5123 
 
 ■^' 
 
 7 7. .36 19 
 
 
 B 
 1 ■•, 
 
 2.91525 
 
 M 
 
 21.2058 
 21.5985 
 
 
 40.0554 
 40.4481 
 
 •78 
 
 58.905 
 59.2977 
 
 •78 
 
 77. 546 
 7i' 1473 
 
 
 1. 
 
 3.1416 
 
 T 
 
 21.9912 
 
 13. 
 
 40.8408 
 
 19. 
 
 59.6904 
 
 25. 
 
 78. ul 
 
 
 1/ 
 
 ;{.5343 
 
 •H 
 
 22.3839 
 
 •M 
 
 41.2335 
 
 .H 
 
 60.0831 
 
 .H 
 
 78.9327 
 
 
 3.927 
 
 % 
 
 22.7766 
 
 ^ 
 
 41.6262 
 
 a 
 
 60.4758 
 
 it 
 
 79.3254 
 
 
 4.3197 
 
 23.1693 
 
 42.0189 
 
 60.8685 
 
 79.7181 
 
 
 .1.^ 
 
 4.7121 
 
 .1.; 
 
 23.562 
 
 .In 
 
 42.4116 
 
 .KV 
 
 61.2612 
 
 M 
 
 80.1102 
 
 
 
 5.1051 
 
 ■'I 
 
 23.9547 
 
 .91 
 
 42.8043 
 
 .9^ 
 
 61.6539 
 
 :9i 
 
 80.5035 
 
 
 5.4978 
 
 •'/a 
 
 24.3474 
 
 ■K 
 
 43.197 
 
 •H 
 
 62.0166 
 
 .H 
 
 80.8962 
 
 
 ■ 'a 
 
 5.8905 
 
 24.7401 
 
 .% 
 
 43.5897 
 
 •Ja 
 
 62.4393 
 
 .% 
 
 81.2889 
 
 
 2." 
 
 6.2832 
 
 8.-^" 
 
 25.1328 
 
 14. 
 
 43.9824 
 
 !20. 
 
 G2.832 
 
 26. 
 
 81.6816 
 
 
 •^H 
 
 G.6759 
 
 H 
 
 25.5255 
 
 •!« 
 
 44.3751 
 
 .,^8 
 
 63.2247 
 
 - •¥ 
 
 82.0743 
 
 
 
 7.0G86 
 
 >-A 
 
 25.9182 
 
 
 44.7678 
 
 :i| 
 
 63.6174 
 
 •K 
 
 82.467 
 
 
 7. 46 13 
 
 ■H 
 
 26.3109 
 
 45.1605 
 
 91.0101 
 
 .% 
 
 82.8597 
 
 
 ■', 
 
 7.854 
 
 .1.^ 
 
 26.7036 
 
 M 
 
 45.5532 
 
 .i.V 
 
 61.4028 
 
 .1.,' 
 
 8;}.2524 
 
 
 yl 
 
 8.2167 
 
 -ya 
 
 27.0963 
 
 .91 
 
 45.9459 
 
 .91 
 
 64.7955 
 
 •% 
 
 8;i.645l 
 
 
 ■'^'A 
 
 8.6391 
 
 
 27.489 
 
 .% 
 
 40.3386 
 
 •■*4 
 
 65.1882 
 
 ■% 
 
 84.0378 
 
 
 3. 
 
 9.0.321 
 
 27.8817 
 
 .% 
 
 46.7313 
 
 ■Ja 
 
 65.5809 
 
 84.4305 
 
 
 9.4248 
 
 9."^' 
 
 28.2744 
 
 15. 
 
 47.124 
 
 21. 
 
 65.9736 
 
 27. 
 
 84.8232 
 
 
 •H 
 
 9.8175 
 
 •■^8 
 
 28.6671 
 
 .'a 
 
 47.5167 
 
 ■Vs 
 
 66.36G3 
 
 •M 
 
 85.2159 
 
 
 .'4 
 
 10.2102 
 
 A 
 
 29.0598 
 
 >l 
 
 47.9094 
 
 ■\^ 
 
 66.759 
 
 .Vs 
 
 85.6086 
 
 
 A 
 
 10.6029 
 
 A 
 
 29.4525 
 
 48.3021 
 
 .% 
 
 67 1617 
 
 86.0013 
 
 
 .'., 
 
 10.9956 
 
 .1..' 
 
 29.8452 
 
 .Vo' 
 
 48.0948 
 
 .'., 
 
 67.5444 
 
 •>.< 
 
 86.394 
 
 
 ■■'l 
 
 1 1 .3883 
 
 •>8 
 
 30.2379 
 
 •€ 
 
 49.0875 
 
 • •'a 
 
 67.9371 
 
 ■^ 
 
 86.7867 
 
 
 •?4 
 
 11.781 
 
 •^4 
 
 30.6306 
 
 •'4- 
 
 49.4802 
 
 .^i 
 
 68.3298 
 
 •?4 
 
 87.1794 
 
 
 .■4 
 
 '■ 
 
 12.1737 
 
 ■i 
 
 31.6233 
 
 .^ 
 
 49.8729 
 
 •Ja 
 
 68.7225 
 
 ■Jt 
 
 87.5721 
 
12 
 
 (;ii!0(impe'u;ncks of circles. 
 TAWhE.— (Continued.) 
 
 Di.uu. 
 
 28. 
 
 29. 
 
 ■J 8 
 
 
 30. 
 
 
 •/o 
 
 31. 
 
 ■n 
 
 ./8 
 
 32. 
 
 ■k 
 
 M 
 
 •/u 
 
 .>8 
 
 1,-; 
 
 •/8 
 
 33. 
 
 34. 
 
 
 .^8 
 
 .38 
 
 Ciiciim. 
 
 87.9filH 
 S8.S570 
 88.7502 
 8!).142'J 
 «!).">;!")(; 
 80.1)283 
 00.32 1 
 'JO. 7 137 
 Dl.lOC-i 
 yi.'I'J'Jl 
 U 1.89 18 
 92.28-10 
 92.G772 
 93.0(;99 
 93.4026 
 93.8553 
 94.248 
 94.0107 
 95.0334 
 95.4201 
 95.8188 
 90.2115 
 90.0042 
 90.9909 
 97.3H!)0 
 97.7H23 
 98.175 
 98.5077 
 98.9004 
 99.3531 
 99.7458 
 100.1385 
 100.5312 
 100.9239 
 101.3100 
 101.7093 
 102.102 
 102.4947 
 102.8874 
 103.2801 
 103.073 
 104.000 
 104.458 
 101. SSL 
 105.244 
 105.030 
 100.029 
 100.422 
 100.814 
 107.207 
 107.0 
 107.993 
 10S.;{,S6 
 108.778 
 109.171 
 109.503 
 
 Diiim 
 
 35. 
 
 Circuni. i' Diani. 
 
 .J 8 
 
 ■k 
 
 .'A 
 
 30. 
 
 ^7. 
 
 
 
 18. 
 
 39. 
 
 .78 
 
 
 40. 
 
 41. 
 
 .'A 
 
 
 .h 
 
 109.950 
 
 no.:; 19 
 
 no 711 
 
 111.134 
 
 1 1 1 .527 
 
 111.919 
 
 112.312 
 
 112.705 
 
 113.098 
 
 113.19 • 
 
 113.883 
 
 114.276 
 
 114.008 
 
 115.001 
 
 11^ ■ \ 
 
 ': 0.239 
 
 110.032 
 
 117.025 
 
 117.417 
 
 117.81 
 
 118.203 
 
 118.595 
 
 118.988 
 
 119.381 
 
 119.774 
 
 120.100 
 
 120.559 
 
 120.952 
 
 121.344 
 
 121.737 
 
 122.13 
 
 122.522 
 
 122.915 
 
 123.308 
 
 123. 01 
 
 124.093 
 
 124.480 
 
 124.879 
 
 125.271 
 
 125.004 
 
 126.057 
 
 120.449 
 
 120.842 
 
 127.235 
 
 127.027 
 
 128.02 
 
 128. 'x 1 3 
 
 128.806 
 
 129.198 
 
 129.591 
 
 '29.984 
 
 130.376 
 
 130.70'> 
 
 131.102 
 
 131.554 
 
 13. 
 
 08 
 
 ■k 
 
 /8 
 
 .'a 
 1 : 
 
 Circuni. 
 
 44. 
 
 ., 8 
 .'.1 
 
 15. 
 
 ■hi 
 
 ■k 
 
 1 
 
 46. 
 
 ■A 
 
 ■'a 
 
 
 47. 
 
 !48. 
 
 k 
 
 .1., 
 
 'U 
 -A 
 
 I • 
 
 « >i 
 
 % 
 
 131.917 
 
 132.31 
 
 132.733 
 
 133.125 
 
 133.518 
 
 133.911 1 
 
 134.303 
 
 1;M.090 
 
 135.0S'J 
 
 135.481 
 
 135.871 
 
 130.207 
 
 l.".(i.00 
 
 137.052 
 
 137.415 i 
 
 137.838 
 
 138.23 '51 
 
 138.023 |: 
 
 139.010: 
 
 139.408' 
 
 139.801 
 
 140.191, 
 
 140.587, 
 
 140.979,: 
 
 141.372 5:: 
 
 111.705 
 
 142.157 
 
 142.55 
 
 142.913 
 
 143.330 
 
 143.728 
 
 144.121 
 
 144.514 
 
 144.900 
 
 115.299 
 
 145.092 
 
 110.081 
 
 140.477 
 
 140.87 ! 
 
 147.203 i 
 
 I47.055 
 
 ' Diiiiii. 
 
 '19 
 
 ! 
 
 
 1 •' ' 
 
 
 St 
 
 , 
 
 I ' 1 
 
 
 . 4 ' 
 
 
 3-' . 
 
 
 .78 1 
 1 
 
 1 
 1 , 
 
 fi~ ! 
 
 
 ..8 
 
 1 
 
 '3.-' 
 
 
 •A 
 
 
 ■'« 
 
 j 50 
 
 
 
 
 1 
 
 v8 
 
 1 
 
 v^ 
 
 ii 
 
 1" 1 
 
 1! 
 
 •"!' i 
 
 1 : 
 
 •, 8 1 
 
 C'irciiin. 
 
 148.018 
 148.441 : 
 148.833' 
 U9.226j 
 149.019 
 150.0111 
 150.404 1 
 150.797 1 
 151.19 
 151.582; 
 151.975 I 
 1.52.3681 
 152.76 I 
 153.153 I 
 153.5 to I 
 
 54. 
 
 :?: 
 
 ■A \ 
 
 
 • 8 ; 
 
 
 k 
 
 , 
 
 
 
 
 ■ 5;^ 
 
 
 1 
 
 00. 
 
 1. 
 
 % 
 
 •A 
 
 k 
 
 •A 
 
 I5;i 
 
 151 
 
 .938 
 .331 
 
 Diani. 
 
 ;>(). 
 
 154.724 
 
 155.117 
 
 155.509 ; 
 
 155.902 I 
 
 150.295 I 
 
 150.687 ■ 
 
 157.08 
 
 157.473 
 
 157.805 
 
 158.258 
 
 15-:<.(i5l 
 
 159.044 
 
 159.436 ' 
 
 150.823 
 
 160.222 
 
 100.014 
 
 101.007 
 
 101.4 
 
 101.792 
 
 102.185 
 
 102.578 
 
 102.971 
 
 103.30;{ 
 
 10:!. 750 
 
 101.119. 
 
 104.541 
 
 104.9;i4 
 
 105.327 
 
 105.719 
 
 100.112 
 
 100.505 
 
 100.898 
 
 167.29 
 
 107.083 
 
 108.070 
 
 108.108 
 
 108.801 
 
 169.254 
 
 109.046 
 
 170.030 
 
 170.132 
 
 170.82rj 
 
 171.217 
 
 171.61 
 
 172.003 
 
 172.39,; 
 
 172 Tbj 
 
 173.181 
 
 173.57H 
 
 173.906 
 
 174.,359 
 
 174.752 
 
 175.144 
 
 175.537 
 
 •x8 
 •^1 
 
 57. 
 
 I 
 
 I 
 
 ■58. 
 
 .-1 
 
 •/8 
 
 •'8 I 
 
 ■■A 
 
 ■J'J. 
 
 60. 
 
 ■.'8 
 
 .^8 
 
 I 
 
 61 
 
 •.'8 
 
 k 
 
 . 1 . 
 
 V8 
 . •') ' 
 
 •A 
 
 •l8 
 •'4 
 
 .1.; 
 
 '% 
 ,3, 
 
 62. 
 
 ■A 
 
 k 
 
 i 
 
 Circum. 
 
 175.93 
 
 170.322 
 
 170.715 
 
 177.108 
 
 17.7.5 
 
 177.893 
 
 178.280 
 
 178.679 
 
 179.071 
 
 179.404 
 
 179.857 
 
 180.249 
 
 180.642 
 
 181.035 
 
 181.427 
 
 181.82 
 
 182.213 
 
 182.000 
 
 182.998 
 
 183.391 
 
 183.784 
 
 184.176 
 
 184.569 
 
 184.962 
 
 185.354 
 
 185.747 
 
 18(M4 
 
 180.533 
 
 180.925 
 
 187.318 
 
 187.711 
 
 188.103 
 
 188.196 
 
 188.889 
 
 189.281 
 
 189.674 
 
 190.007 
 
 190.10 
 
 190.852 
 
 191.215 
 
 191. ''38 
 
 192.03 
 
 192.123 
 
 192.810 
 
 193.208 
 
 19.'!. 601 
 
 193.991 
 
 191 ..387 
 
 194.779 
 
 195.172 
 
 195,505 
 
 195.957 
 
 190.35 
 
 190.743 
 
 197.135 
 
 197.528 
 
AREAS OF riitCLES. 
 
 TABLE— (Coiitmted). 
 
 18 
 
 Diiim. 
 
 G3. 
 
 Diam. 
 
 64. 
 
 
 65. 
 
 1 
 
 % 
 
 I 
 
 5" 
 
 66. 
 
 67. 
 
 ■'I 
 
 68. 
 
 .1.. 
 
 •fa 
 
 69. 
 
 I 
 
 .1., 
 
 70. 
 
 :7i. 
 
 ;72. 
 
 7.3. 
 
 Aiva. 
 
 :51 17.25 
 ;{12!).0;{ 
 
 .•ii i2.o; 
 
 ;5l.'yl.l7 
 
 :!l66.'.):f 
 :U7;).ti 
 
 :U91.91 
 
 ;r204.M 
 :i2i7. 
 
 .'{229. oS 
 ;^212.18 
 .,2.5 1, HI 
 :i267.1(; 
 
 •{2K0.LS 
 
 :r292.8t 
 
 :{;!05.;J6 
 
 ;«;n.09 
 ;^:m;{.89 
 :?:i5(;.7i 
 
 :!;!(;!)..')(; 
 
 ;{:;9o.;{;{ 
 
 H408.26 
 :U21.2 
 
 Hi;u.i7 
 
 H4-I7.17 
 ;{4iiO.!9 I 
 :{47;!.24 i . 
 
 H4h(;.;{ i! . 
 
 :U99.4 '] . 
 
 ■}^\^-'^\\ • 
 
 ;{ij2u.66 i;74. 
 
 ;{5H8.8;! 
 
 ;{552.02 
 8565.24 
 H578.48 
 8591.74 
 Hu05.03 
 3618.;i5 
 36:51.69 
 3645,05 
 3ii58.44 
 3671.85 
 3685.29 
 3698.76 
 3712.24 
 3625.75 
 3739.29 
 .3752.85 
 3766.43 
 3780.04 
 3793,68 
 3807.34 
 3821.02 
 3834.73 
 
 ■a 
 
 
 •/8 
 
 I 
 
 ■ 'h 
 
 1, lO. 
 
 I' • 
 
 i 
 
 i 
 
 Area. ! Diiirn. 
 
 •}8 
 
 ! 
 
 ■ .1 
 
 76. 
 
 
 % 
 
 r. "\ 
 
 3848.46 
 
 3862.23 
 
 3876. 
 
 3889,8 
 
 3903.63 
 
 3917.49 
 
 3931.37 
 
 3945.27 
 
 3959.2 
 
 8973.15 
 
 8987.13 
 
 4001.13 
 
 4015.16 
 
 4029,21 
 
 4043.29 
 
 4057.39 
 
 4071.51 
 
 )0S5. 66 
 
 4099.83 
 
 4114.01 
 
 4128.26 
 
 4142.51 
 
 4156.78 
 
 4171.08 
 
 41M5.4 
 
 4199.74 
 
 4214.11 
 
 4228.51 
 
 4212.93 
 
 4257.37 
 
 4271.84 
 
 428(i.33 
 
 4:100.85 
 
 4315.39 
 
 4329.96 
 
 4311.55 
 
 4;!59.17 
 
 4373.81 
 
 4,388.47 
 
 4403.16 
 
 4417.87 
 
 4432, K) 
 
 4447.37 
 
 4462.16 
 
 4476.98 
 
 4491.81 
 
 4506.67 
 
 4521.56 
 
 4536.47 
 
 4551.4 
 
 45(16.36 
 
 45Sl,35 
 
 4596.36 
 
 4611.39 
 
 4626.45 
 
 4641.53 
 
 77. 
 
 78. 
 
 •'a 
 
 79. 
 
 
 80. 
 
 SI. 
 
 83. 
 
 Area. 
 
 
 4656.64 
 
 4671.77 
 
 4686.92 
 
 4702.1 
 
 4717.31 
 
 4732.54 
 
 4747.79 i 
 
 4763.07 I 
 4778.37 I 
 479;!.7 I 
 4809.05 I 
 4821.43 
 4839.83 
 4855.26 
 4870.71 
 4886.18 
 
 4901.08 I 
 4917.21 I 
 4932.75 
 4918,33 
 4963.92 , 
 4979,55 
 4995.19 
 5010.87 
 5026.56 
 5012.28 
 5058.02 
 507:{.79 
 5089.59 
 5105.41 
 5121.25 
 5137.12 
 51u3.01 
 5168.93 
 5184.87 
 5200.83 
 5216.82 
 5232.84 
 5248.88 
 5264.94 
 5281.03 
 5297.14 
 5313.28 
 :.„29.44 
 5345.63 
 
 5;i(;i.84 
 
 5378.08 
 0394.34 
 5410.62 
 5426.93 
 5443.26 
 5459.62 
 5476.01 
 5492.41 
 5508.84 
 5525.3 
 
 Diani. 
 
 84. 
 
 Area. 
 
 Diam. 
 
 85. 
 
 
 
 36. 
 
 .'■4 
 J- 
 
 '8 
 
 .'4 
 1 - 
 
 87. 
 
 88. 
 
 ..4 
 •, 8 
 
 89. 
 
 % 
 
 ■H 
 ■^ 
 
 % 
 
 90. 
 
 t 
 
 .h 
 
 I. 
 
 5541.78 
 
 5558.29 
 
 5574.82 
 
 6591,37 
 
 5607,95 
 
 5';24.56 
 
 5641.18 
 
 5657.84 
 
 5674.51 
 
 5691.22 
 
 5707.94 
 
 5724,69 
 
 5741,47 
 
 5758,27 
 
 5775.1 
 
 5791.94 
 
 5808.82 
 
 5825.72 
 
 5842.64 
 
 5859,59 
 
 5876.56 
 
 5893,55 
 
 5910.58 
 
 5927.62 
 
 5944,69 
 
 5961,79 
 
 5978,9 
 
 5996.05 
 
 6013.22 
 
 60.30.41 
 
 6047.63 
 
 6064,87 
 
 6082,14 
 
 6099.43 
 
 6116.74 
 
 6134.08 
 
 6151,45 
 
 6168.84 
 
 6186.25 
 
 6203.69 
 
 6221.15 
 
 6238.64 
 
 6256.15 
 
 6273.69 
 
 6291,25 
 
 6308,84 
 
 6326.44 
 
 6344.08 
 
 6361.74 
 
 6379.42 
 
 6397.18 
 
 6414.86 
 
 6432.62 
 
 6450.4 
 
 6468.21 
 
 6486.04 
 
 91. 
 
 92. 
 
 '8^ 
 
 
 .1.; 
 
 93. 
 
 I 
 
 1 ' 
 
 194. 
 
 % 
 
 95. 
 
 
 •fa 
 
 i 
 
 96. 
 
 97. 
 
 
 \\ 
 
 .1.; 
 
 Area. 
 
 6508,9 
 
 6521.78 
 
 6589.68 
 
 6557.61 
 
 6575.56 
 
 6593,54 
 
 66' 1,55 
 
 6629.57 
 
 6647.63 
 
 6665.7 
 
 6683,8 
 
 6701.93 
 
 6720.08 
 
 6738.25 
 
 6756.45 
 
 6774.68 
 
 6792.92 
 
 6811.2 
 
 6829.49 
 
 6847.82 
 
 6866.16 
 
 6884.53 
 
 6902,93 
 
 6921,35 
 
 6939.79 
 
 6958.26 
 
 6976.76 
 
 6995.28 
 
 7013.82 
 
 7032.39 
 
 7050.98 
 
 7069.59 
 
 7088.24 
 
 7106,9 
 
 7125,59 
 
 7144.31 
 
 7163.04 
 
 7181.81 
 
 7200.6 
 
 7219,41 
 
 72.38,25 
 
 7257,11 
 
 7275,99 
 
 7294.91 
 
 7813,84 
 
 7332,8 
 
 7351.79 
 
 7370.79 
 
 7389.83 
 
 7408,89 
 
 7427.97 
 
 7447.08 
 
 7466.21 
 
 7485.36 
 
 7504,55 
 
 7523.75 
 
14 AREAS OF CIRCLES. 
 
 TABLE — (ContimmJ). — [Admncing hy a Quarter and a half.] 
 
 T>- un. 
 
 98. 
 
 99. 
 
 
 
 Vb 
 
 100. 
 
 101. 
 
 
 1,, 
 
 102. 
 
 103 
 
 •2^ 
 
 'h 
 
 104. 
 
 'h 
 
 
 Area. 
 
 7.542.98 
 
 7502.24 
 
 7581.51 
 
 7000.82 
 
 7620.15 
 
 7039.5 
 
 7658.88 
 
 7678.28 
 
 7697.71 
 
 7717.10 
 
 77HG.0a 
 
 7750. IH 
 
 7775.00 
 
 7795.2 
 
 7814.78 
 
 7834.38 
 
 7854. 
 
 7993.32 
 
 7932.74 
 
 7972.21 
 
 8011.87 
 
 8051.58 
 
 8091.39 
 
 8131.3 
 
 8171.3 
 
 8211.41 
 
 8251.61 
 
 8291.91 
 
 8332.31 
 
 8372.81 
 
 8413.4 
 
 8454.09 
 
 8494.89 
 
 8535.78 
 
 8576.77 
 
 8617.85 
 
 Dium. 
 
 il05. 
 
 100. 
 
 .h 
 
 107. 
 
 •H 
 
 108. 
 
 109. 
 
 .h 
 
 110. 
 
 1 
 
 111. 
 
 112. 
 
 .h 
 
 :^4 
 
 113. 
 
 Areii. 
 
 8G59 
 8700 
 8741 
 8783 
 8S24 
 8860 
 8908 
 8950 
 8992 
 9034 
 9070 
 9118 
 9160 
 9203 
 9245 
 9288 
 
 9;i3i 
 
 9374 
 
 941" 
 
 9luo 
 
 9503 
 
 9546 
 
 9589 
 
 9033 
 
 9070 
 
 9720 
 
 9704 
 
 9808 
 
 9852 
 
 9896 
 
 ;)9-l0 
 
 9984 
 
 10028 
 
 10073 
 
 10117 
 
 10162 
 
 Dium. 
 
 03 
 32 
 
 7 
 
 ,18 
 75 
 43 
 2 
 
 ,07 
 04 
 ,11 
 ,28 
 ,53 
 .91 
 .37 
 ,92 
 ,58 
 ,34 
 .19 
 14 
 ..9 
 .34 
 ,69 
 .93 
 .37 
 .91 
 .73 
 .29 
 .12 
 .00 
 .09 
 .22 
 .45 
 .77 
 .2 i 
 
 114. 
 
 115. 
 
 116. 
 
 3 
 
 ..-4 1 
 
 •A^ 
 
 117. 
 
 ■¥ 
 
 118. 
 
 119. 
 
 120. 
 
 74 
 
 121. 
 
 122. 
 
 Areu. 
 
 10207.00 
 
 10251..S8 
 
 10296.79 
 
 10311.8 
 
 l()3S(;.9l 
 
 101,32.12 
 
 10177.43 
 
 10522,84 
 
 10568,34 
 
 10613,94 
 
 10i;5!»,64 
 
 10705.44 
 
 1075 1.3-1 
 
 10797.31 
 
 10843.43 
 
 10889,62 
 
 I0;I35.9 ■ 
 
 10',>82,3 
 
 1102S,7Hl 
 
 11075,37 
 
 11122,06 
 
 11108.83 
 
 11215.71 
 
 11262.69 
 
 ll;:09.70 
 
 11. 356. 93 
 
 11404.2 
 
 11451.57 
 
 11499.04 
 
 11540.61 
 
 11594.27 
 
 11642,03, 
 
 11(189,89 
 
 11737,85 
 
 11785.91 
 
 11834.06: 
 
 Diain, 
 
 123. 
 
 Area. 
 
 124 
 
 
 1 
 
 k 
 
 1 
 
 ''i~: 
 
 125. 
 126! 
 127." 
 128! 
 129! 
 13o! 
 131 ! 
 132! 
 133! 
 134! 
 135! 
 1.36! 
 137! 
 138! 
 
 i„:i 
 
 11882.32 
 11930.67 
 11979.2 
 12027.66 
 12076.31 
 12125.05 
 12173,9 
 12222.84 
 12271.87 
 12370.25 
 12169,01 
 12568,17 
 12667.72 
 12707,61; 
 12867,99 
 12968.71 
 13069.81 
 13171 •'■ 
 13273 
 13.')75 
 13178 
 13581 
 13(i81,81 
 13788,67 
 13:192.94 
 3997.54 
 114102,64 
 i 14208,07 
 11313,91 
 14420,14 
 14526.76 
 14633.76 
 14741.17 
 11848,96 
 14957.16 
 15065.73 
 
 Diain. 
 
 !l39. 
 
 ,140." 
 
 ' .^. 
 
 141. ' 
 
 
 Area. 
 
 142. 
 143! 
 111! 
 145! 
 146! 
 147. 
 1 148! 
 149! 
 150! 
 
 15174.71 
 
 15284,08 
 
 15393.84 
 
 15503.98 
 
 15614,53 
 
 15725,47 
 
 I 15836,8 
 
 ! 15948,52 
 
 !l60r,0,64 
 
 16173,15 
 
 : 16286,05 
 
 ; 16399.34 
 
 U 65 13. 03 
 
 (16627.11 
 
 116741,59 
 
 i 16856.44 
 
 I 16971.71 
 
 17087,36 
 
 17203,4 
 
 17319.83 
 
 1V4;!6,67 
 
 17553.89 
 
 17671,5 
 
 17789.51 
 
 To Compute the Area of a Diameter greater than any in the preceding Table. 
 
 RiTLE.— Divide the dimension by two, three, four, etc., if practicable to do so, until it is reduced 
 to a diameter to be found in the table. 
 
 Take tiie tabular area for the diameter, multiply it by the square of the divisor, and the product 
 will give the area required. 
 
 ExRUPLR. — What is the area for a diameter of 1050 ? 
 
 1050-{.7 = 160; tab. area, 150 = 17671.6, VFhich x 7 =865903.."^ -rea requited. 
 
 To Compute the Area of an Integer and a Fraction not given in the Table. 
 
 RoLK. Double, treble, or quadruple the dimension gi^'cn, until the fraction is increased to a 
 
 whole number, or to one of those in the table, as J^, \, etc., provided it is practioiibio to do i-o. 
 
 Take the area for this diameter; and if it is double of ihat for wich the area is required, take one 
 fourth of it ; if treble, tiiko one 9 th. of it and if quadruple, take one sixieouth ot it, etc., eto. 
 
 ExAMPLK.— Required the area for a circle of 2. ^^ inches. 
 
 2. ^^ X 2=4|, area for which =15.0331, which -^ 4=3,768 im. 
 
f:rr.rvT.MF.:v:!N('; s ov circlks. 
 TAWLK. -(CoDfiininl.) 
 
 17 
 
 IXmv.. 
 
 Ciivuiii. 
 
 Uiiiin 1 CiiTiun. ' 
 
 Diaui. 
 
 ('irc'iini. Dunn. 
 
 Circ'iini. 
 
 1 Diaui. 
 
 Circnni. 
 
 (,:',. 
 
 I!l7.i)21 ' 
 
 10. 
 
 219.012 
 
 77. 
 
 2 1 1 .'M)-\ ! 8-t. ; 
 
 2(11!. 894 
 
 ;9l. 
 
 285.880 
 
 .'J 
 
 l'JS.,",l 1 
 
 ]„' 
 
 22().:U)5 
 
 .'a 
 
 212.29(1 1 .'ij' ; 
 
 2(1I.2S7!^ .1,; 
 
 280.278 
 
 .^.. 1 
 
 lilS.TOli 
 
 •!,' 
 
 22:!.(I',I7 
 
 ..', 1 
 
 2 12. (ISO :i .'i ' 
 
 2;ii.(iH H .'!., 
 
 280.071 
 
 '•''.. i 
 
 l:i;).(ii)'.' 
 
 • ";'. 
 
 221.0;) '; 
 
 .;':; 
 
 2i:!.0Sl ' .-;■ 
 
 2(15.07: .?,•! 
 
 287.004 
 
 y.l \ 1 ;)'.). r.)2|'i 
 
 .1.. 221. -is:! : 
 
 .'., 
 
 24:1.474 .1.. 
 
 2(15.4(15 > .1., 
 
 287.450 I 
 
 .•f,i ' 1 '•»'.). >-^l il 
 
 /•:. 221.87(1 
 
 "1 , , 
 
 2i:{.,Sl7 .■';■ 
 
 2(15. <5S 1 .fa 
 
 287.849 
 
 • '*. 1 
 
 2(10. 'JTTi! 
 
 :■■■, 222.2>H 
 
 •■■''.' 
 
 214.259 ."^Y 
 
 2(1(1.251 j .:ij 
 
 288.242 
 
 •Ja 
 
 2ii().(;7 i 
 
 .;(, -222.(;(il 
 
 •J a 
 
 2M.(152 1 .'.. 
 
 2(1(1.(148 .7|j 
 
 288.0:44 
 
 G4. 
 
 2oi.,i(;2 ; 
 
 71. ' 22:1.0 Jt ' 
 
 78. 
 
 245.015 85. 
 
 2(17.o:!tl 92.' 
 
 289.027 
 
 ■hi 
 
 2()l..|.;.j 1 
 
 .!„' i 22:i.-j-it; . 
 
 .'a 
 
 2I5.4;!8 : .',: 
 
 2(17.129 .!„' i 
 
 2s;».42 
 
 .Kv "-ioi.si.^ 
 
 .'n i 22:!.8:i!> ; 
 
 • ,',. 
 
 245.S:! ' .1.1 
 
 2(17. S21 : .'., : 
 
 2S9.813 
 
 .•',•1 ; 2(12. -J-! I 
 
 .-J! 22i.2:;2 
 
 
 24(1.22:; .'v, 
 
 2I1S.214 :•';.. ' 
 
 290.205 
 
 .1.. 2i)2. (;;;:; 1 
 
 1 , . 
 
 22 1.(12 t 
 
 1 I 
 
 21(1.(11(1 .'.. 
 
 2r,s.(i(i7 .'., 
 
 290.598 
 
 ..^j i 2();{.()2(; 
 
 '.^ i 225.017 ; 
 
 >a 
 
 217.(10>^', /\; 
 
 2(ls.999 |i .fa 
 
 290.991 
 
 .:5, : 2();!.41i) 
 
 .3,- [22-,. 4 1 
 
 .'ti 
 
 217.401 .■■!,• 
 
 2(19.1192 1 .3,'- 
 
 291.:48:{ 
 
 .)., 2.)::.8ili 
 
 .7,, , 225.S0^ 
 
 •-'a 
 
 217.794 .;„ : 
 
 2(19.7S5:| .7 
 
 291.770 
 
 G5, 204.201 
 
 72. ! 22(1. lit., 
 
 79. 
 
 2IS.lS(l 80. 
 
 270. 17S 9:?. 
 
 292.109 
 
 .}.a 20l.r,!)T 
 
 .3,,' ;22G.5SS 
 
 I ^ 1 
 
 2 IS. 579.' .1., 
 
 270.57 .1„- ; 
 
 292.502 
 
 .h 20J.«)8I! 
 
 .'.I 22(1. 'J.sl ! 
 
 •',' 
 
 2 IS. 972 .', 
 
 270.9(1:1 .1, 
 
 292.954 
 
 Ja 
 
 20,1. •is2i 
 
 .h 227 .:i7:! i 
 
 .'■'a 
 
 24;>.:!(15 .■•;• 
 
 27i.:;5i; ' .p., 29:i.:^47 
 
 .}... 2(lo.7TA| 
 
 .1., 227.7(1(1 
 
 1 , 
 
 21:'. 757 .1.. 
 
 27 1.7 IS .1.. 29:1.74 
 
 .;'u 2(h;.i(;s 
 
 .•"V i 228.15'J 
 
 ".■i.' 
 
 250.15 .•>[. 
 
 272.1 11 .•'>:. 294.1:42 i 
 
 1 .-''i : 2(l(;.;-,li 
 
 .'■■■i 22s. 5.M 
 
 •'*i 
 
 250.5 i:i .a, 
 
 272.5:14 .:). 294.525 j 
 
 Ja , 2()il.'j.j;!l 
 
 .'-,. 22s. '.ti4 i 
 
 7 
 v a 
 
 25o.;i;;5 .'',, 
 
 272 !I2(1 .7,, : 2iM.!)18 
 
 GO. 
 
 2tl7.:MG 
 
 7;i. 
 
 22i).:i:i7 i 
 
 80. 
 
 251.;!2S\S7. 
 
 27:i.:;i9 94.' ' 295.:ii 
 
 .'. 
 
 207.7;!S 
 
 .,'a 22!).7:! | 
 
 ■^l 
 
 251.721 .!(, 
 
 27:1.712 .(y i 295.70:5 \ 
 
 .'.I 
 
 2(i,s.i;ii 
 
 .'-t 
 
 2:;o.l22 
 
 •.'.' 
 
 252.1 i:i .'i , 
 
 274.105 .'i_, 1 29(1. 09(1 
 
 .>'» 
 
 20S.r)2.4 
 
 .•'a 
 
 2:10.515 
 
 1 ".» 
 
 252.50(1, .■';., 
 
 27l.l!l7 .'v, 2!>(1.489 
 
 1 , 
 
 20.s.iii(; 
 
 .'•> 
 
 2:10.908 
 
 . '.► 
 
 252. S99 1 .1., 
 
 27l.s;) .1., 
 
 2!t0.8Sl ! 
 
 '.% 
 
 20'.).:io;) 
 
 •?" 
 
 2:iL.:{ 
 
 ."j^^ 
 
 2511. 2;i2 .•'>,"; , 
 
 275.2S:5 .f 
 
 297.274! 
 
 ■?i 
 
 201). 702 
 
 ■ "4 
 
 2:il.G9:! 
 
 i ••■'1 
 
 25:;.(;s4 .:'( , 
 
 275.(175 .;»! 
 
 297.007 ( 
 
 Ju 
 
 21(l.0'.l.J 
 
 •Ja 
 
 2:12. osd 
 
 7 
 V 6 
 
 251.077 .J„ ' 
 
 27(1. 0(1H ' .7^ 
 
 298.059 ■ 
 
 G7. 
 
 210.-l.s7 
 
 74. 
 
 2112.478 
 
 81. 
 
 254.47 88. i 
 
 270.4(11 '95.' 
 
 298.452 i 
 
 i,; 
 
 210.SS 
 
 .'a 
 
 2:12.87 1 
 
 •'a 
 
 25l,s(I2 .1,,' 
 
 270. S5:! .1,: 
 
 298.845 1 
 
 ! ".'I 
 
 2ii.27;i 
 
 .'.I 1 2;!;i.2(i4 
 
 ' •./^ 
 
 255.255 .1.1 
 
 277.21(1 .', 
 
 299.2:17 
 
 .3?a 
 
 2ii.(;t;r) 
 
 i .■■;: ! 2:>:!.(i')7 
 
 ' .'-'a 
 
 255.(1 IS .■•"■;i 
 
 277.il29 .^S, 
 
 299.0:5 1 
 
 .'.. 
 
 212,0.j,s' 
 
 .1.. :2:i4.0ii) 
 
 .'.. 
 
 25(1.04 .'., 
 
 278.0:12:; .1., 
 
 300.02:? ' 
 
 
 21 2, .I;")! 
 
 :': ;2;;4.4-i2 
 
 1 ^ 
 •- a 
 
 25(1.4:''{ .5- 
 
 278.424 ;| .5J, 
 
 :ioo.4io ; 
 
 .:'., 
 
 212.H-i:{ 
 
 .:i :2:M.^';^5 
 
 1 
 
 25(1 820 . .-!,' 
 
 27S.817 i .3. 
 
 1400.808 
 
 • ' 
 
 2i:!.2:;(; 
 
 .;., ; 2:15.227 
 
 ; ' 
 
 257.219 .-■■ 
 
 279.21 .7- 
 
 1 :?oi.2oi 
 
 08. 
 
 2i;;.()2i) 
 
 75. i 2:15.02 
 
 S'> 
 
 257.(111 89.' 
 
 2T!l.(102 90.' 
 
 ' ;i01.594 
 
 1 .?»' 
 
 211.022 
 
 1 .1,;' 1 2;i(i.oi:5 
 
 '".'a 
 
 258.001 .1;) 
 
 27;).9;t5 .(,; 
 
 1 ;?0I.98G 
 
 : .'.1 1 2N..ii.i 
 
 , .''1' 1 2:1(1.405 
 
 . -.'.i. 
 
 25S.;!97,, .ii 
 
 2so.:is8 ' .4', 
 
 ;io2.:?79 
 
 ; .?a 1 211.MJT 
 
 :'..• ;2:i(1.7;)8 
 
 .:'a 
 
 25S.7s9^ .■:,, 
 
 2S0.781 ' .3,, 
 
 :i02.772 
 
 .', 
 
 215.2 
 
 .'., 12:17.1111 
 
 . ' .. 
 
 259.182: .1. 
 
 2sl.l7:i .'.. 
 
 :4(i:!.l04 ; 
 
 • ''l! 
 
 21;J.iV.)2 
 
 .■"a 
 
 2:^.5^•i 
 
 1! •i'a 
 
 259.575 i .-V 
 
 2s 1.500 , .•■;,; 
 
 ;!o:?.557 
 
 ■3' I 
 
 21ij.!)^;> 
 
 ••'.I 
 
 2:17. 97(j 
 
 \\ .;;., 
 
 259.9(17;, .3 
 
 2s 1.959 ..'! - 
 
 :io:5.95 1 
 
 ■J a 
 
 21G.:!78 
 
 •Ja 
 
 2;is.:i(i9 
 
 1 » . > 
 
 ' -.'a 
 
 2(10. :i(l i .: 
 
 1 •-^■-2.:i5i , .7 
 
 :4oi.;?4;? ! 
 
 GO. 
 
 21(;.77 
 
 7G. 
 
 1 2:18.7(12 
 
 '^:;. 
 
 2tl(i.75:! '.90. ' 
 
 2S2.744 97. 
 
 1404. 7:?5! 
 
 .'i( 
 
 217.1(1:) 
 
 •.'a 
 
 1 2.19.154 
 
 1,, 
 
 2(11.11(1 ; .',, 
 
 2s:).i;:7 .1,; 
 1 2.^:1.529 ; .4" 
 
 ' 2s:!.922 j .3,-; 
 
 1405. 128 j 
 
 J 4 
 
 2i7.r)r)G 
 
 .,',, 
 
 ! 2:19.547 
 
 1 y.i 
 
 2(il.5:;s 1 .(', 
 
 :405.521 1 
 
 .H 
 
 217.9IH 
 
 
 i 2:19.94 
 
 1 ■',, 
 
 2(11.9:11 .:'.. 
 
 :)05.9i:?! 
 
 1 , 
 
 2lf^.;Ml 
 
 ".1,! 2-io.:i:!2 
 
 . '.. 
 
 2(12.1124 , .1 , 
 
 : 2si.;!ir, 
 
 J 
 
 :!0(i.:ioo ; 
 
 ".•^'a 
 
 2! 8. Till 
 
 .•57, 240.725 
 
 .") ^ ^ 
 
 2(12.71(1 />:: 
 
 ; 2SI.708 
 
 ••'^'a 
 
 .•iOO.099 1 
 
 :?: 
 
 211».127 
 
 y. 241.118 
 
 i •■'1 
 
 2(11!. 109 .a" 
 
 ' 2s.-,.l 
 
 •■' 
 
 1407.091 1 
 
 2PJ.U1'.) 
 
 .lil 241.511 
 
 i -Ja 
 
 2G;J.502,i .7. 
 
 j 285.49:? , 
 
 •Ja 
 
 ;?07.4«4 
 
18 
 
 TXmA-: (CoiitiiinalJ. 
 
 Diaiu. 
 
 CiiTiim. , Diiiiii. 
 
 CiiTiim. ; Diiiin 
 
 Circiun. ' 
 
 DiMiii. 
 
 Circnin. llliiin. ' 
 
 1 
 
 Ciiiimi. 
 
 98. 
 
 307.877 jlOj. 
 
 32!).S(;,s' 111. 
 
 :{,-.s.l 12 
 
 12.3. 
 
 380.117 13:i. ' 
 
 i:;.;.os2 
 
 ..^i' 
 
 :{!I8.'27 1 .' 
 
 ;!:)0. (;,").■! ' .1 
 
 ,' ."..'jS.il-jS 
 
 •'.;! 
 
 ;;s7.2(i2 .1., I 
 
 i:;-.2r)3 
 
 •'•'! 
 
 :MH.(;t;2 ! .'.i 
 
 .?;;i..i:;;): .' 
 
 .' ;!.v.».7i;; 
 
 . '■''■ 
 
 3s7.!iss IJO. ' 
 
 •130.821 i 
 
 •'••'i 
 
 80:!. o:..") .■', 
 
 :!:{2.22i ! .■' 
 
 1 i'riO.I'.i;) 
 
 • "1 
 
 ;)ss.77;! .'.. ' 
 
 111. 30.-) ! 
 
 .I.J 
 
 :!Oi).iis 100. 
 
 :;.■;:;. 01 ! 11.). 
 
 i ;;.;i.2si 
 
 121. : 
 
 3^'.). ,"..>! III. i 
 
 Il2.!lti0 
 
 ''l\ 
 
 .".():>. SI .1 
 
 .■'):'.;!. 7:»:)/ .' 
 
 .: ;;i;2.oi;',) 
 
 '.,: 
 
 :!:)0.3ii .'■2' 
 
 111.')!'". 
 
 
 :!io.2:;:! .'. 
 
 ;;:m.,-)S '! j 
 
 ! :\\i.<:>:, 
 
 .'.. 
 
 ;!'.ii.i2'.) 112. 
 
 1 Hi. 107 
 
 7 : 
 
 :;io.<i2i; .''., 
 
 ;!;;.■). lioc, ,■' 
 
 1 .'iiWi.tii 
 
 • i 
 
 ::\)\AKo .'-' 
 
 ■tl7.07S 
 
 99. 
 
 .•{ll.Ol.s 107. 
 
 ii.so.ir.i no. 
 
 :•,; 1.12 1 
 
 12.-). : 
 
 31)2.7 ; 1-13. i 
 
 110.210 , 
 
 •'., 
 
 :!ii.4ii .', 
 
 :;.'io.!):;7 .' 
 
 ; ;!0.').211 
 
 _ 1 
 
 31)1.271 .'.. ! 
 
 ■IT)!). 82 i 
 
 .'.> 
 
 Hii„-»i 
 
 :i;!7.722 .', 
 
 V .".ii.'.. !»;)>; 
 
 120! ' 
 
 .3;)-).s|-.> Ml. 1 
 
 -i.-2.:io ■ 
 
 _:!,, 
 
 .■{12.11'ti .''i 
 
 :!:!s.i-,()7 
 
 1 ;!00.7S2 
 
 1 - 
 
 ;;:)7.H2 .'.,. 
 
 i,'):;.iioi 
 
 .'.. 
 
 ;!12..")S;) 108. 
 
 :;:;'. 2;i:; 117. 
 
 1 !)ii7.."i()r 
 
 127! ' 
 
 :\[)<.\)<\ 115. ' 
 
 i.-,,-. r.32 
 
 _,57 
 
 :U2.!)S2 .' 
 
 ;i 10.07-^ .' 
 
 .".('.s.;!';; 
 
 1 
 
 {:)().-,:, \ .'-: 
 
 ■1,'. 7. 10.3 
 
 '-■U 
 
 :ii:{..H7.-i .'. 
 
 ;uo.so.i 1 .' 
 
 ■J :;,;;). i;!s 
 
 !2s'. ' 
 
 102.12.') 110. 
 
 -ir)-!.(i74 
 
 ^7_ 
 
 :il:!.7(M .'i 
 
 iMI.Ol'.t ' 
 
 1 :!r.'.).:i2.; 
 
 1 . 
 
 -i():!.0',ii; .'.. 
 
 .ioi».2n 
 
 100/'' 
 
 :u-i.i«i lOD. 
 
 :ii2.i:ii n^. 
 
 , :i7().7(i'.t 
 
 120". ': 
 
 ■10-.. 20,; 117. 
 
 -i.-.i.sir) 
 
 .1 . 
 
 :{14.^M;-) .'i 
 
 :ii;i.22 1 .' 
 
 ' :i7i.r.ti 
 
 .'.. 
 
 100.^37 .'j 
 
 403.:'.s0 , 
 
 '. ' .' 
 
 ;!i;j.7:u ' .'.' 
 
 :i4i.oi),v .' 
 
 ■., ;!72.2.s 
 
 130. 
 
 ■108.|()-< 1-18. ! 
 
 404.0:)7 
 
 •;'4 
 
 31(;.51(i }: .'.J 
 
 ;wi.7i)i .■ 
 
 1 ,'57i>.(IO."> 
 
 1 
 
 ;0!I.'.)7'J .'j| 
 
 4i;i;.r.28 
 
 101. 
 
 :!17.:{02 ! no. 
 
 .•u:).57o no. 
 
 : :;7;!.s.-. 
 
 131'. '' 
 
 111..').") 110. 
 
 4i;s.(i'.)s 
 
 
 ills. 087 '.' .h 
 
 :mi;.:;.;i ; .' 
 
 ij :;7!.o:;i'. 
 
 1 
 
 -Ii3.12 .'- 
 
 400. Olio , 
 
 .'.> 
 
 :!IS.872 ' .'j 
 
 :!17.117 
 
 _ 
 
 ■:'■ -MfyAll 
 
 \-vi. ' 
 
 iii.cDi i:;o. 
 
 471.21 '■ 
 
 ' ,1 
 
 :!l!).(i.")S .'■'■i, 
 
 ;ii7.!):;2 
 
 , ' 
 
 'li 371"). 207 
 
 '.. 
 
 -110.202 .'.. 
 
 472.811 
 
 102. 
 
 ;i2i).-ii:i 111. 
 
 :u-!.7is :i20. 
 
 i 37t;.i);)2 
 
 !:;:;! ' 
 
 ■117.S33, 
 
 
 .'.1 
 
 :!2i.22;t .'4 
 
 Mio.fjo:; 
 
 .> 1 1 . 1 ( ( 
 
 1 
 
 -11;). 10 I ! 
 
 
 .'.. 
 
 322. OM .!-j 
 
 ; 800. 2SS 
 
 •-•; 37s. ,ji;:; 
 
 13l'. ' 
 
 -I2ii.;»7i 
 
 
 •■'.'i 
 
 ;r22.7'.)'j '' .-1 
 
 • ;i:)0.07i 
 
 •I' 37l>.:il.S 
 
 .'.. 
 
 -122..". 1.") 
 
 
 io;5. 
 
 :!2;!.o85 112. 
 
 ; :!.-)!. S;V.) 121. 
 
 ' 3<i).i:;i 
 
 13:»". ■' 
 
 121.110 
 
 
 ■U 
 
 ;!2 1.157 l .'., 
 
 :i.">2,0(.j 
 
 ; 3-'l).'.)l'J 
 
 1 
 
 ■12.').0S7 ' 
 
 
 
 1 , 
 
 82,-).ir)(;' .■'.! 
 
 :!;^;!.i:! 
 
 ■-• 3S 1.701 
 
 i3i;'. " 
 
 ; I27.2.JS 
 
 
 
 ■f^ 
 
 ir^.K'Jtl ; J.i 
 
 , :!5I.21') 
 
 1, :!s2.i',) 
 
 . '., 
 
 -i2-^.s28 
 
 
 
 lOi. 
 
 82(1. 720 "113. 
 
 1 :{;■).->. 001 122. 
 
 ! :!s;'..27.-) 
 
 137". ' 
 
 .i;;o.3;)'j 
 
 
 
 ■^:. 
 
 :!27.r)i2|: .>, 
 
 :'')r).').7.so , 
 
 'r :{S1.0il 
 
 .'.. 
 
 -i:;i.07 
 
 
 
 1 
 
 :{2s.2'.»7 '! .'. 
 
 :i,")i).rj72 
 
 '■J :')Si.sio 
 
 13s! " 
 
 ■I3:!..-)I1 
 
 
 ':-^i 
 
 :{2i).ns:! 'i .,:*.^ 
 
 :!o7..'!.J7 i 
 
 'ij 38.-). 03 1 
 
 .'.> 
 
 -t3,) 112 1 
 
 ■_.. 
 
 To Compute lliR I'ircuin. of ii DiiiiiU'tci' ^itiUit than any in the precciliiii Tiibli;. 
 
 K I.R. — DiviJi! the iliiiicntion hy two, three, lour, etc., it' pruu'.ioaljlo t:) do so, until it i.i reduced 
 to a (iiMiiiotor to !'0 found in (ho tiible. 
 
 Take Ihu tiibuhir oircutntenMhi; tor this dimuntion, multiply it by 2, 3, 4, b, ete , according ;is it 
 was divided, and the jjtnduct will ffive the circunilV'ri'iicc re jiiire 1. 
 
 Ex.\Ml'l,K. — What is the circiiiiii'ereiice tor ii di;inieter id' Kl.'jir.' 
 1050-7-7 = 150; tab. cireuui., 150^^471, '-'.'JU, which x 'irraiau.UT.'', ci'/i";);. nt/nlrcd. 
 
 To (lumpiite the (MrcumfiTcncc for an Integer and Friietion not given in the Tabic. 
 
 liui.K. — Double, treble, or q'liidruplo the dimeiition given, until Iho frnctto'i i-; inreised to a 
 whole number or to o'lo of those in the t ibK", it.< i. }, etc., provided 't is priflicil to do .-o. 
 
 Tnko the circumfi'rcnce for thi.s iliiimetcr ; and it' il \f iluub'c of thai foi- which the ir'umforenco 
 is required, t.iko one half of it ; if treble, t:ike o:ic t'lird of it ; anJ if q ladropU', o lo lourth of it. 
 
 KxASIPliK. — Required the circunifeience of 2.21875 iiichoii. 
 2.21876 X 2 r= 4.4376= •Ij-jy, which x ''^ = ^■1 ; tab. oircum. = 27.S5l7, wh;ch-^4=- f,.0704 ,•„^^ 
 
 To Compute the Circum. of a Diameter in Feet and Inehes etc., by the preceding Tabic. 
 
 Kur.E. — Reduce the diinention to inches or eighths, as the casa may be, and takj the circumfo- 
 rence in that terra fiom the table for thiit number. 
 
 Divide this number by 8 if it U in eight'.is, and by 12 if in inoho?, and tho quotient will givj tho 
 area in iwK 
 
 ExAMPLK. — Requited the oircumforenoo of a circle of 1 foot (ii{ inehe.s. 
 lfoot6;j ins == is;; ins. = 141 eighths. Circum. of 147 = 461.815, which -^-^=5 57.727 Inches; and 
 
 by 12 = 4.81 feet. 
 
TA.15L1^ III. 
 
 AREAS AND CIRCUMFERENCES OF CIRCLES, FROM i\ TO 100. 
 
 [A(lr(tiirii)rf h;/ 'J'cnt/is.] 
 
 
 Diam.l 
 
 A ca. 
 
 Circuui. 1 
 
 Diam. 
 5. 
 
 Ar.'ii. 
 
 Ciiriini. 
 
 Diam. 
 
 Area. 
 
 C:ireiim. ' 
 
 
 
 
 ]9.<i.S5 ! 
 
 15.708 
 
 10. 
 
 78.54 
 
 H1.41fj 
 
 
 .1 
 
 .nnTs.-.i' 
 
 .:n4]c! 
 
 .1 
 
 20.I2S2 i 
 
 111. 0221 
 
 .1 
 
 80.1 18d 
 
 :^i.7:!0l 
 
 
 .2 
 
 .(ru no; 
 
 .(■.2>;;2 
 
 •> 1 
 
 21.2:!72 \ 
 
 IG.Ii.ld.'J 
 
 .2 
 
 8l.7i:{ 
 
 :r2.oi4;i 
 
 
 .:i 
 
 ,(l7(hi-<tij 
 
 .012181 
 
 .^ 
 
 22.0i;i,S ^ 
 
 Id. (1504 
 
 .8 
 
 88. 112:! 
 
 :i2.;!.')8 
 
 
 A 
 
 .I2'>'iti 1 
 
 1.25 k; i 
 
 A ! 
 
 22.0(122 i 
 
 Id.OdK) 
 
 .4 
 
 84.9488 
 
 :52.d72d 
 
 
 .0 
 
 .l'.)i;:;,5 i 
 
 1.5708 :', 
 
 .5 ' 
 
 2.S.75-;^ '•■ 17.2788 
 
 ..5 
 
 8d. 590:1 
 
 :i2.o.8d8 
 
 
 .(i 
 
 .2.^274 1 
 
 l.^iS5 ;1 
 
 .(5 
 
 24. •;;!() I 1 17.5020 
 
 .(; 
 
 88.2475 
 
 8:!.:i009 
 
 
 .7 
 
 .:i-i|^,") ! 
 
 2.1 001 '! 
 
 ■7 ! 
 
 25.517(1 
 
 17.0071 i! .7 
 
 89.9204 1 
 
 :i:5.di5i 
 
 
 .8 
 
 .502il(i 
 
 2.5i:;:; ': 
 
 .8 ' 
 
 2t;.l20S 
 
 lH.2212 1 .8 
 
 Ol.doo i 
 
 :i:!.9292 
 
 
 .'J 
 
 .(;:!(; 17 i 
 
 2.S274 
 
 .0 
 
 27.:!:;o7 
 
 1.^.5:154 ' .9 
 
 o:!.:il:!:! 
 
 :! 1.2 1:54 
 
 
 .1 
 
 .7S,-, [ 
 
 :!.iiir. 
 
 c. 
 
 2!-;. 2744 
 
 l8.S40d 11. 
 
 95.():i:!4 
 
 :!l.,557d 
 
 
 .1 
 
 .'Joo:; 
 
 ;!.I557 ' 
 
 •1 
 
 20.2217 
 
 lo.id:i7 1 .1 
 
 9d.7d01 
 
 ;!4.8717 
 
 
 2 
 
 1.1 :;()'.) 
 
 :!.7(;oo ' 
 
 .2 1 
 
 :;o.ioo7 1 
 
 10.1779 i .2 
 
 98.5205 
 
 85.1859 
 
 
 •1 
 
 1 .:!27:5 
 
 AA)M 
 
 .:! 1 
 
 :!1.1725 1 
 
 10.702 .:! 
 
 100.2^77 
 
 85.501 
 
 
 .4 
 
 1 .a:;ii:! 
 
 4.:!082 
 
 .4 j 
 
 :!2.i(;o9 ' 
 
 20 10,12 .4 
 
 102.0705 1 
 
 85.8142 
 
 
 .5 
 
 1.7.;7l ' 
 
 4.7124 ! 
 
 .5 1 
 
 :]:',.\<M 1 
 
 20.1204 \' .5 
 
 10:?. 8(10 1 1 
 
 8d.l2S4 
 
 
 .('. 
 
 2.010,; 
 
 5.02ii5 
 
 .(■> 1 
 
 :!I.212 1 
 
 20.7:115 1; .d 
 
 105.(l.s:i4 
 
 8d.4425 
 
 
 .7 
 
 2 2t;i)S 
 
 5.:il07 
 
 .7 j 
 
 :!5.25i;(; \ 
 
 21.0187 1! .7 
 
 107.51:!4 
 
 8d.75d7 
 
 
 .H 
 
 2.r.Mii 
 
 5.(15 i 8 
 
 •^ 
 
 Hii.:!Uis 1 
 
 21.:i(;28 ;i .8 
 
 109.:!59 i 
 
 87.0708 
 
 
 .<J 
 
 2.S;i;.2 
 
 5.'.)i;o : 
 
 .0 
 
 ;;7.;!928 | 21.(177 s .9 
 
 111.2204: ' 
 
 87.884 
 
 
 ') 
 
 ;i.iii(; 
 
 (;.2-:i2 
 
 ^ 1 
 
 ^ * i 
 
 ;;^.4.si(j 
 
 21.0012 i 12. 
 
 li:!.007d 
 
 87.d992 
 
 
 "!l 
 
 :!.4i;:ui 
 
 (;.5'.)7:) 
 
 •1 
 
 :!0.502 
 
 22.:!()5:{ ! .1 
 
 114.9904 
 
 88.0188 
 
 
 .2 
 
 3.siii;5 1 
 
 0.0 11 5 
 
 .2 i 
 
 40.7151 
 
 22.(1105 
 
 1 -'^ 
 
 lld,80S9 
 
 88.:i275 
 
 
 ■1 
 ••* 
 
 4.1.V17 ' 
 
 7.225ii 
 
 .:! I 
 
 41 85:19 
 
 22.o:!:id 
 
 \ .:-! 
 
 I1S,H2:!1 
 
 88.(141 G 
 
 
 .4 
 
 4.:)2:;'j ■ 
 
 7.5:508 
 
 A 
 
 4:1.008') 
 
 2:1.2478 
 
 .4 
 
 120.7(1:11 
 
 88.9.558 
 
 
 .;") 
 
 4.;)087 
 
 7.S54 
 
 .5 
 
 4 1.1787 
 
 2:{.5d2 
 
 ■-> 
 
 122.7187 
 
 80.27 
 
 
 .() 
 
 5.:-i()i):i 
 
 8.11181 
 
 .« 
 
 45.:Uil7 
 
 2:i.H7dl 
 
 A\ 
 
 121.(1001 
 
 89.5841 
 
 
 .7 
 
 0.72').') 
 
 8.182:'. 
 
 .7 
 
 4(1.5(i(;:! 
 
 21.100:! 
 
 .7 
 
 12(1.(1771 
 
 89.8988 
 
 
 .8 
 
 (i.I.')7u 
 
 8,7 Oil! ; 
 
 .8 
 
 47.7.s:i7 
 
 24.5014 
 
 i .8 
 
 128.(1709 
 
 40.2124 
 
 
 .'J 
 
 ('..dO'.-! 
 
 0.1 lOi! 
 
 .0 
 
 40.0i(i>! 
 
 21,S.l^'d ' .9 
 
 i:io,dO,s4 
 
 40.52dd 
 
 
 •1 
 *•> 
 
 7.0i;s(; 
 
 ;m2is : 
 
 8. 
 
 50.2il5o 
 
 25.1:! 28 i i;i. 
 
 l:!2.7.S2d 
 
 40. S 108 
 
 
 .1 
 
 7.r)47(; 
 
 o.7:;.^o 
 
 : .1 
 
 51.51! 
 
 25.41(19 !' .1 
 
 1:14.7824 
 
 i 41.1519 
 
 
 .2 
 
 8.0124 
 
 l().ii5:!i 
 
 .2 
 
 52.S102 
 
 25.7dll 
 
 i •) 
 
 1 'Id .848 
 
 ! 41.4(191 
 
 
 .;{ 
 
 8. ,■);■>;■! 
 
 10.;!(iT2 
 
 
 51.10(12 
 
 2d. 0752 
 
 1 "."5 
 
 i:;s.0204 
 
 1 41.7882 
 
 
 .4 
 
 0.071)2 
 
 lo.d^M 
 
 .4. 
 
 55.1178 
 
 2d.:iH04 
 
 ; A 
 
 141.02(14 
 
 i 42.0074 
 
 
 .0 
 
 <).(121l 
 
 U). 005(1 
 
 .5 
 
 .■i(;.7451 
 
 2il,70:!d 
 
 .5 
 
 l4:i.i:!oi 
 
 i 42.11 Id 
 
 
 .() 
 
 10.1787 
 
 1 li.:;oo7 
 
 .() 
 
 ! 5S.08S1 
 
 27.0177 
 
 .d 
 
 I45.2d75 
 
 ! 42.7257 
 
 
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AREAS AN1» CIKCl MI'Klll'.NCES OF OIHCLE? 
 
 21 
 
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 1.3(55.7242 
 
 131.0047 
 
 .3 
 
 1757.1075 
 
 148.5976 
 
 ! .2 
 
 1020.2105 
 
 113.7250 
 
 .8 
 
 1372.2822 
 
 i:;i.3i88 
 
 .4 
 
 17(51,0045 
 
 148.9118 
 
 !;i 
 
 lOlii.OlDl 
 
 114.04 
 
 .0 
 
 1378.85(5 
 
 131.032 ■ 
 
 .5 
 
 1772,0587 
 
 149.226 
 
 .1 
 
 10U).(;2.".") 
 
 114. .35 12 
 
 42. 
 
 1:585.4450 
 
 131.9472 
 
 .15 
 
 1779,5279 
 
 149. 5:10 1 
 
 .") 
 
 10l(i.34f^l 
 
 114.(i081 
 
 .1 
 
 1302.0508 
 
 l:;2.20I3 
 
 .7 
 
 1787,0127 
 
 149.3543 
 
 .1) 
 
 10,V2.000I 
 
 11 I.0S25 
 
 .2 
 
 1308.(5717 
 
 132.5755 
 
 .8 
 
 1794.5133 
 
 150.1084 
 
 .7 
 
 10.')7.8474 
 
 115.21)(;7 
 
 .3 
 
 i405.:i083 
 
 1:52,8890 
 
 .9 
 
 1802.0290 
 
 150.4826 
 
 .S 
 
 1 Oil:; .02 
 
 11 5. 01 OS 
 
 .4 
 
 1411.9007 
 
 133,2038 
 
 48. 
 
 1809.5016 
 
 150.79(58 
 
 1 .;) 
 
 10i;0.4081 
 
 115.02;") 
 
 .5 
 
 1418.0287 
 
 1:53.518 ■ 
 
 ,1 
 
 1817.1092 
 
 151.1109 
 
 ! ;'.7 . 
 
 107-). 2 1 20 
 
 1 11(5.2302 
 
 .(5 
 
 1425.3125 
 
 1:53,8:521 
 
 .2 
 
 1824,0726 
 
 151.4251 
 
 ; .1 
 
 10sl.0:!24 
 
 1 11(5.55:13 ! 
 
 .7 
 
 1432.0119 
 
 134,1403 
 
 .3 
 
 1832,2518 
 
 151.7392 
 
 .2 
 
 I(l8(i.8ti70 
 
 i 110.8075 
 
 .8 
 
 1438.7271 
 
 1:11,4004 
 
 .4 
 
 1839,8406 
 
 152.0534 
 
 .3 
 
 1002.7101 
 
 1117.1810 1 
 
 .9 
 
 1445.458 
 
 134,7740 
 
 .5 
 
 1847.4576 
 
 152.3676 
 
w 
 
 AREAS AND riRCtrMPKHKNCKS OK » Ill.'LF.? 
 
 'TABLE— (CoHtimicd). 
 
 Di; 
 
 iarn.i Aivu. 
 
 .G 
 .7 
 .8 
 .9 
 
 49. 
 
 50. 
 
 .1 
 .2 
 .3 
 .4 
 .5 
 .6 
 .7 
 .8 
 .9 
 ). 
 .1 
 .2 
 .3 
 .4 
 .6 
 .6 
 .7 
 .8 
 .9 
 
 51. 
 .1 
 .2 
 .3 
 .4 
 .5 
 .6 
 .7 
 .8 
 .9 
 
 52. 
 .1 
 .2 
 .3 
 .4 
 .5 
 .6 
 .7 
 .8 
 .9 
 
 53. 
 .1 
 .2 
 .3 
 .4 
 .5 
 .6 
 .7 
 .8 
 .9 
 
 54. 
 .1 
 
 iHa.-j.os:?;] 
 ls(;-j.72"):{ i 
 
 lH70.;iS'J',) 
 
 l878.o.)(;:! : 
 
 lSS.j.74.')l ' 
 
 1H1);{.I5()1 ' 
 
 liU)1.17()ii 
 
 ii)i(i.t;r)S7 1 
 ii)2i.42t;:i 
 ii);;.'.2()'.M 
 r.iH).o(»si; 
 
 l!t»7.82;il 
 
 iu.")r).(M:is 
 
 ii)7i.;!(;is 
 i;)7'.).2;!i)i ; 
 
 l!)S7.l;i2(i : 
 l!)i)5.04li; j 
 
 2i)02.%(;:{ I 
 
 2010.iiO(i7 • 
 
 20 18. 8028 
 
 202(1. 8:5 u; 
 
 20:54.877 
 
 2042.82,>t 
 
 2050.844:! 
 
 20.j8.8784 
 
 2()(i(;.i)2i):! 
 
 207l.i)ii.):i 
 20S:{.0771 
 
 2o;)i.i7it; 
 
 20i)9.2878 
 2107.41i;G 
 
 2ii">.r)i;i2 
 
 212:!.721i; 
 
 2i;n.8;)7(; 
 
 2N0.0s'j:i 
 2148.2i)(;7 
 
 2i5(;.r)i:)S) 
 
 21 (14. 7.587 
 
 2i7;i.oi;{:{ 
 
 2I81.28:{5 
 
 2l89.oi;;)5 
 
 2!'J7.8712 
 
 2200.1 880 
 
 2214.02 10 
 
 2222.8704 
 
 22:51.2:55 
 
 22:59.0152 
 
 2248.0111 
 
 2250.4227 
 
 2204.8701 
 
 227:^.29:51 
 
 2281.7519 
 
 2290.2201 
 
 2298.71 05 
 
 Clrciim. 
 
 i 
 
 1.52.0^17^ 
 1.') 2. 9 95 9 
 15:5.:51 
 
 i.-):5.(;2i2 
 1.5:5.9:581 
 
 154.2.V25 
 
 151.5007 
 
 15I.SS08 
 
 155.195 
 
 155.5092 
 
 1,55. 82:!:! 
 
 i.-,(;.i:!75 
 
 150.4510 
 
 15ii.755-< 
 
 157.08 
 
 157.:!!) II 
 
 157.70s:! 
 
 15S.0221 
 
 I5s.:-):!i)() 
 
 158.050>^ 
 
 15S.90I9 
 
 15!t.27'.)i 
 
 i59.59:!2 
 
 159.9071 
 
 100.2210 
 
 100.5:157 
 
 100.8199 
 
 1(;1.104 ' 
 
 101.1782 : 
 
 101.7924 
 
 I02.10«i5 
 
 102.4207 
 
 102.7:518 
 
 10:!. 019 
 
 io:!.;!ii:i2 
 I0:!.ii77:^ 
 lo:i.9!):55 
 ioi.;!05(; 
 
 10Li;i98 
 
 ■ l(i4.9:!4 
 
 105.2481 
 
 105.502:5 
 
 j l(i5.8704 
 
 i 100.1900 
 
 I(;0.5048 
 
 li;ii.8189 
 
 1 107.1:5:51 
 
 i 107.4472 
 I07.7(il4 
 108.0750 
 108.:^897 
 108.7019 
 109.018 
 109.:5:522 
 109.GI04 
 109.9005 
 
 Diain. 
 .2 
 
 .;i 
 .4 
 .5 
 .0 
 
 .7 
 
 .8 
 .9 
 55. 
 
 1 .1 
 
 Aivii. 
 
 .:! 
 
 .4 
 .,5 
 
 .0 
 
 .7 
 .8 
 .9 
 
 50. 
 
 .4 
 .5 
 .() 
 .7 
 .8 
 .9 
 57. 
 .1 
 .2 
 
 :^ 
 .4 
 .5 
 
 58. 
 
 59. 
 
 .0 
 .7 
 .H 
 .9 
 
 .1 
 .2 
 
 .;{ 
 
 .4 
 .5 
 
 .0 
 .7 
 .8 
 .9 
 ). 
 .1 
 _2 
 
 .3 
 .4 
 .5 
 .0 
 
 .7 
 
 2:507.222 H 
 2:!15.71i : 
 2:521.281:5 
 2:!:!2.s:5i;5: 
 2:! 1 1.10:! 1 
 
 2:! 19.9^7 I 
 2:!5<.5>70 I 
 2:!07.2ii:;i 
 2:575.8:55 
 2:!^ 1. 1822 
 2:59:5.1452 
 2IOl.X2:!8 
 2U0.51S2 
 2119.22-^.! 
 2127.9511 
 2i:!0.(i95i> 
 2115.152-^ 
 2151.2257 
 210:5.0141 
 2 171. 81 ■<7 
 2IS0.0:!S7 
 2H9.47I5 
 2l9-!.:!259 
 2507.19:11 
 2510.070 
 2521.97:50 
 25:5:5. 8SS8 
 25 12.81 ^S 
 2551.7(510 
 2500.720 
 2509.70:51 
 2578.0'.i5'J 
 25,s7.7015 
 251)t;. 72.^7 
 2(iO5.7087 
 20 1 4. .^2 4:5 
 202:5.«'J57 
 20:52. 0^28 
 2012.0-^50 
 I 2051.2011; 
 
 2O0O.:5:i.s2 
 
 12009.4882 
 2078.05:18 
 2ii87..s:!5l 
 2097.0:5.^! 
 2700.2 t49 
 2715.47:1:5 
 2724.7175 
 27:5:5.9774 
 274:5.2529 
 2752.5442 
 2701.8512 
 2771.17:59 
 2780.512:5 
 2789.8001 
 2799.2302 
 
 Circutn. 
 170.2717 
 
 UO-.iS''':! 
 170.90:5 i 
 
 171.2172 
 
 17 1.5:11:! 
 
 17I.8l.-)5 I 
 
 172.151),; 
 
 172.47::s 
 
 172.7SS 
 
 17:1.1021 
 
 17:1.110:1 
 
 17:1.7:101 
 
 171.0110 
 
 171.:!:.-^ 
 
 171.07211 
 
 171.9771 
 
 175.:!092 
 
 175.0151 
 
 175.929 1 
 
 17i;.2l:i7 
 
 17i;.5579 
 
 17(;.872 
 
 177. H02 
 
 177.5001 
 
 177.8115 
 
 178.12S7 
 
 178.4128 
 
 178.757 
 
 179.0712 
 
 l79.:i-5:! 
 
 179.0095 
 ISD.OIIIO 
 180.:127S 
 180.012 
 180. 9511 
 
 181.2m):i 
 181. 5 -in 
 I Si. 89 -ill 
 1S2.212S 
 1S2.5209 
 l.->2.ht-ll , 
 18:5.1552. 
 18:5.11191 
 
 ls:!.7sio 
 
 Mi. 0977 
 1^1.4119 
 Is 1.720 
 1S5.0M)2 
 I85.:!54l 
 
 I85.(;i;s5 
 
 1S5.9827 
 
 1 81;. 2090 
 
 180.011 
 
 180.9252 
 
 187.2:19:5 
 
 187.5535 
 
 Diuin. 
 
 .9 
 
 00. 
 
 01 
 
 iiZ. 
 
 0:5. 
 
 01. 
 
 G5. 
 
 Area. 
 
 2808.';2IS 
 28 IS. 02:1 
 2S27 II 
 28:10.8720 
 
 
 2-<l";.:i21 
 
 • > 
 
 2S55.7S5 
 
 1 
 
 2-!05.2i!|s 
 
 .5 
 
 2S7 l.7(;o.l 
 
 
 
 2S8 1.20J5 
 
 i 
 
 2S'.) 1.79-54 
 
 .8 
 
 290! Ill 
 
 9 
 
 21)! 2.899 1 
 
 .1 
 
 29:12. o;:!! 
 
 •) 
 
 21)ll.00'^5 
 
 .:! 
 
 21)51.2897 
 
 A 
 
 21).;().9205 
 
 .,) 
 
 •Jl»7(). 571)1 
 
 .0 
 
 29S().2l71 
 
 -r 
 . 1 
 
 29-<9.9:il 1 
 
 .s 
 
 •ji)!)i).i):5 
 
 .9 
 
 :!Oi)i).:!iOi 
 
 
 :!()! 9.0770 
 
 .1 
 
 :102-<.S2II 
 
 •J 
 
 :!0 !8.5si;:> 
 
 .:5 
 
 :50i8.:i05i 
 
 .4 
 
 :505 s. 1591 
 
 .5 
 
 :io.;7.9.;s; 
 
 .0 
 
 :5077.791l 
 
 . 1 
 
 ' :;i)>7. 0:511 
 
 ,-•! 
 
 ! :iOi)7.i9r.) 
 
 .0 
 .7 
 
 .8 
 
 .1 
 .2 
 
 • > 
 
 .4 
 .5 
 .G 
 
 .7 
 .8 
 .9 
 
 .1 
 .2 
 .3 
 
 .Hi! 7.2520 
 :; 127.1501 
 :il:57.075s 
 :il 17.0111 
 :il 50.900 I 
 
 :^l0ti.92;)i 
 
 :51 70.911 5 
 :il -'0.9097 
 :5 19.1.92:55 
 :52()0. 95:51 
 :i210.99-^l 
 :5227. 059.1 
 
 :52:57.i:io 
 
 :!217.22St 
 
 ;5257.:5:j05 
 
 32i!7.40'):! 
 
 :5277.591)S 
 
 :52S7.755 
 
 3297. 92*; 
 
 3:108.1120 
 
 15:51 8.. -5 15 
 
 :5:!28.5:54 
 
 :53,1s. 70t;8 
 
 3:149.0102 
 
 C'inn 11. 
 
 1S7.8070 
 H^.ISM 
 
 ;8s.490 
 
 ISS.SIOI I 
 
 is;t.rji;{ 
 isD.CNti 
 I -1). 75211 1 
 
 ll';I.O0.1S , 
 
 I90.:is09 j] 
 
 1-90.01)51 
 
 1111.0092 I 
 
 I91.:!2.5l 
 
 I9l.ti:i70 
 
 191.9517 
 
 11)2.2';59 
 
 192.58 
 
 19:i.S912 
 
 19,1.2081. 
 
 19.1,5225 
 
 I9!.s:!07 
 
 191.150-! 
 
 191.105 
 
 191.7792 
 
 11)5.09,1:! 
 
 195.1075 
 
 195.7210 
 
 19; .0:158 
 
 190.115 
 
 19.1.0011 
 
 ll).1.97.s.i 
 
 197.2921 
 
 i 197.11000 
 197.9208 
 
 ' 19s. 2:1 19 
 19S.5I91 
 
 i 19S.8.i:52 
 
 ' 191).: 771 
 
 \ 101). 191.; 
 
 199.si)57 
 200.1199 
 I 200. 4:5 4 
 . 200.7482 
 201.0024 
 2i)l.:5705 
 201.0907 
 202.0018 
 202.319 
 
 202.i;:5:;2 
 
 1 202.947:1 
 |20:!.2tll5 
 ' 20:5.5750 
 20:5. 8S9S 
 20 1.20 1 
 201.5181 
 20 4.8323 
 205.1401 
 
Aiu'.As AND (MKrtrMi.'i<;aKNrKs mV circles. 
 TABLE.— (Continued.) 
 
 23 
 
 Dinm. 
 
 Aron. 
 
 :<;i:)9.28i4 
 
 Cii'cniii. 1 
 
 2n:).4G0G t 
 
 'Diani. 
 71. 
 
 Avon. Circmn. 
 
 Diam. 
 
 Area. 
 
 CiiTuni, 
 240.G4G5 
 
 .4 
 
 ;59.-.9.201 1 22:5.0.j;5G 
 
 
 
 .G 
 
 4'108..1-^1G ; 
 
 .5 
 
 .i:ir.ii.r)(i-j:! ' 
 
 2().".771S 
 
 .1 
 
 ;!;)7o.:!t;i9 22:!.:i(i77 ] 
 
 .7 , 
 
 4G20.I2IS 
 
 24(1.9007 
 
 .t; 
 
 ;i:;T;».><.".'^'.> '■ 
 
 •joii.dss:) 
 
 .2 
 
 ;;;)■< i.".:;-^i 22;!.Gsi9 
 
 .8 
 
 4G:52. 177(1 '' 
 
 211.2748 
 
 .7 
 
 ;i;iii(».i7i2 
 
 2(ni.io:u 
 
 .:5 
 
 ;!992.7:;oi 22:!. 99.; i 
 
 .9 ' 
 
 4(1 1 1.5492 
 
 2H.5!t87 
 
 .H : 
 
 ;iuio.r.);)2 
 
 20i;.7172 
 
 .4 
 
 .|(io:;.9:i7:! 22i.:!l02 
 
 77. 
 
 405(1. G.'ltlO 
 
 211.90:52 
 
 .1) 
 
 ;i410.>!|29 
 
 2(»7.();!1 1 
 
 .5 
 
 101.'). I Gil 224.G24r 
 
 •I i 
 
 .1008.7:590 
 
 212.217:5 
 
 GG. 
 
 ;;»i!i.2(fji 2i>7.;!i:.G I 
 
 .G 
 
 i(i2i;.ioo2 22i.9;w:) 
 
 .2 
 
 4(l<0.85s:i ! 
 
 2 52,5:515 
 
 .1 
 
 ;ii;il..'>77.'i •,!li7.i'.J97 ' 
 
 .7 
 
 |0:;7.ti.').'. 22.'>.2.'.27 , 
 
 ■^ : 
 
 4092.9927 ' 
 
 212,84,50 
 
 .'2 
 
 ;ii4i.'.)i;:i:; 2(17.97:59 
 
 .8 
 
 •KlH.'.l^.".! 22.').J(;i;8 
 
 .4 ! 
 
 4705,1429 
 
 24:5,1598 
 
 '1 
 
 ;il.V2.:!7l'.» 2!l-^.28-; 
 
 .9 
 
 ■|(li;o.21iii 22.'i.'^.>^l : 
 
 .6 j 
 
 4717.:i0S7 
 
 21:5.174 
 
 •-t i 
 
 ;{m;2.71)71 •j()h.(;o22 ^ 
 
 72. 
 
 |()7I..')l:!(; 22i;.19,")2 ■ 
 
 .G I 
 
 4729.190:i 
 
 24:5.7881 
 
 .5 ! 
 
 ;ii7;;.2.;:)i ' 20-^.'.M(;i i 
 
 .1 
 
 1(H-J..><;!;!2 22(;.r)09:5 <\ 
 
 .7 i 
 
 471I.0S75 
 
 241,102:5 
 
 •'• 
 
 oiS.'i.Gs^S 'iOO/i;!!),) 
 
 2 
 
 10:) 1.1(1 IT) 220.82:55 1, 
 
 .8 ; 
 
 475:1.9005 
 
 2I4.41G4 
 
 •7 
 
 ;ii9i.iiw i 2();»..JiiG i 
 
 "..H 
 
 1105.5125 227.l:;7G! 
 
 .9 ; 
 
 4700.1292 
 
 214.7:100 
 
 .8 ! ;!.')() t.tii;!2 
 
 2(t'.t.s,')'^8 ; 
 
 .4 
 
 ■lll('..S79:i 227.151s: 
 
 78. 
 
 .177S.:57:5G 
 
 215.0148 
 
 .9 1 
 
 i).')!.-).! i:; 
 
 2llt.l7:! i 
 
 .0 
 
 H2-<.25^7 227. 7Gt; 
 
 .1 
 
 4790.(1:5:50 
 
 245,:5,VS9 
 
 G7. 1 
 
 ;i.vj.').i;i;or. ■ 2i<».is72 1 
 
 .G 
 
 .l,:i9.t;521 22-<.OS01 ' 
 
 .2 
 
 4S02,9()9l 
 
 245,07:51 
 
 ■1 ! 
 
 ;i.'>.lb.l'.)2s 'jKLsoi;! 
 
 .7 
 
 4i5i.(ii;(;7 22s.:59»:5| 
 
 .3 
 
 4S 15,201 
 
 245,9872 
 
 .2 
 
 ;).J4(;.7iui 
 
 •iii.li.'i:) 
 
 .8 
 
 I1(;2.4;m:I 22S,70-^4 
 
 .4 
 
 4827.50.-:2 
 
 2I0.:5014 
 
 .'^ 
 
 ;ij.')7.:!()i;! > 
 
 211.4291; 
 
 .9 
 
 Il7:!.9;!7(i 229.022G: 
 
 .5 
 
 4>^:i9.8:!ll 
 
 240,0156 
 
 .4 
 
 ;),)(>7.8s;>7 ! 2ll.74.i8 
 
 7:5. 
 
 (is5.:;9,;(; 229.:{:;(;8 
 
 .« 
 
 4852.1097 
 
 240,9297 
 
 .5 
 
 O.J7S.47.S7 i 212.0.')8 
 
 .1 
 
 4r.)t;.>i712 229.G509I 
 
 .7 
 
 4804.5211 
 
 217.24:59 
 
 .G 
 
 ;).ys9.o,sii.-. ; 2i2.;i72i 
 
 ■) 
 
 -I20-'.:1GN 229 9i;5i 
 
 .8 
 
 4S70.S97:5 
 
 217.548 
 
 .7 
 
 ;;,vj9.vi59 2i2.G^(i:{ 
 
 .:5 
 
 .i2l9.S(;78 2:50,2792 
 
 .9 
 
 48SSK2799 
 
 247.8722 
 
 .s 
 
 ;m;io.;!.")s| 
 
 2i:i.oo(u 
 
 .4 
 
 ij:il.:!>^'.iG 2:50.59:11 
 
 79. 
 
 4901. 0Sl4 
 
 248,lS0(: 
 
 .9 
 
 ;ii;2i.oiG 
 
 •ji:i.:;i Ki 
 
 .5 
 
 1212.9271 2.!(l. 907(1 
 
 .1 
 
 4914.09S5 
 
 24s,a()05 
 
 G8. 
 
 MG;il.GS9G 
 
 2l.'!.G2-<8 1 
 
 .G 
 
 .|25l.-ls(i:! 2:51.2217 
 
 .2 
 
 4920.5:514 
 
 248,8147 
 
 .1 
 
 ;}GI2.H788 
 
 2i:i.9J29 
 
 .7 
 
 -i2(Ki.oi9:) 2:51.5:559, 
 
 .3 
 
 49:58. 9S2 
 
 249.1288 
 
 .2 
 
 ;iG:):!.Os:;s 
 
 214.2.-)7l 
 
 .8 
 
 4277.G.!:{:» 2:51.80 
 
 .4 
 
 4951.414:5 
 
 2 59,443 
 
 .;? 
 
 oi)t);{..'so4 
 
 21I..J712 
 
 .9 
 
 42-9.2:', 1:! 2:52.1042' 
 
 .6 
 
 490:5.921:1 
 
 219.7572 
 
 .4 
 
 ;<i)74.ull 
 
 2l4.8ir)l 
 
 74, 
 
 .i:!ll0.>501 2:)2.17Sl 
 
 .G 
 
 4970.484 
 
 250,071:5 
 
 .5 
 
 :;.;>;>. 29;ii 
 
 2l.'>.l'.»9ii \ 
 
 .1 
 
 I:il2.1s2l 2:52.7925 
 
 .7 
 
 4988.9:514 
 
 250,:5855 
 
 .G 
 
 :i(;9ti.oui; 
 
 2ir).r>i:i7 
 
 . .2 
 
 i:!2l.l2:)t; 2:1.;. 10(17 
 
 .8 
 
 500I.458G 
 
 250.6996 
 
 .7 
 
 ;{7u<;>ii') 2i:).8279 
 
 .•^ 
 
 i;;:;5.792S 2:5:5.4208, 
 
 .9 
 
 5014.0014 
 
 251.01:58 
 
 .8 
 
 .■;717.(;i;!7 21ii.l42 
 
 .4 
 
 i:; 17.471 7 2:51.7:55 
 
 80. 
 
 5020.50 
 
 25l.:52.^0 
 
 .9 
 
 ;i72s.-loS7 ; 2l(i.4,M;2 i 
 
 .5 
 
 -1:159. IG(">:5 2:11.0 592 
 
 .1 
 
 50:59.1:542 
 
 251.6421 1 
 
 G9. 
 
 ;;7.{9.2sl(4 2IG.77Ut 
 
 .G 
 
 -i:i7o.s7(;(; 2:it.:^(;:;:5 
 
 i -'^ 
 
 .5051.7242 
 
 251.9503 : 
 
 .1 
 
 ;i7.')().i:i')7 
 
 21 7. us 1.) 
 
 1 *- 
 
 i:5-<2.G02(! 2:11.(1775 
 
 .A 
 
 5004,:525S 
 
 252,2704 
 
 ') 
 
 ;!7G0.9978 
 
 217.:i9.S7 
 
 1 .8 
 
 4:19 1.:! 11-^ 2:11.991(1 
 
 .4 
 
 5070,9552 
 
 252.5846 
 
 ".:? 
 
 ;!77i.87r)ii 
 
 217.7128 
 
 ! .9 
 
 4 1 Oil. 1(1 18 2:15.. '5058 
 
 .5 
 
 o0<9.588;l 
 
 252,8988 
 
 .4 
 
 ;;782. 7(191 
 
 21^.027 
 
 7a. 
 
 4I17.S75 2:55. G2 
 
 .G 
 
 5102.2411 
 
 25:5,2129 
 
 .5 
 
 ;i79;].G78:i 
 
 218.:'. 112 
 
 .1 
 
 4129.(1(1:18 2:15.9:541 
 
 .7 
 
 5114.909G 
 
 253.5271 
 
 .G 
 
 ;;hu j .go:!2 
 
 •iis.t;.-.;-);! 
 
 .2 
 
 .H41.4(H4 2:iil. 248:5 
 
 .8 
 
 5127.59:58 
 
 253.8412 
 
 .7 
 
 ;{8i.-).r)i;{8 
 
 21 S. 909:) 
 
 • > 
 
 1 l5:5.2ssG 2:1(1.5024 
 
 .9 
 
 5140.29:^7 
 
 2.54.1554! 
 
 .8 
 
 ;{S2G.ol)(i2 
 
 ' 219.28:!(; 
 
 .1 
 
 ■1405. 124(5 2:5(1.87(1(5 
 
 81. 
 
 515:5.0094 
 
 254.4696 
 
 .9 
 
 ;i847.4722 
 
 1 219.r)978 
 
 .5 
 
 447G.97(i:5: 2:57.1 90S 
 
 .1 
 
 5165.7407 
 
 254.7837 
 
 70 
 
 :isis.4G 
 
 j 219.912 
 
 .G 
 
 4H8.84:^7 2:17.5049 
 
 .2 
 
 5178.4877 
 
 255.0979 
 
 .1 
 
 :i859.l9')2 
 
 1 22().22G1 
 
 .7 
 
 4500.72(18 2:17.8191 
 
 .3 
 
 5191,2505 
 
 255.412 
 
 .2 
 
 ;{S70.4S2G 
 
 220.510:5 
 
 .8 
 
 4512.0250 i 2:58.1:5:52 
 
 .4 
 
 5204.0285 
 
 255.7262 
 
 .:^ 
 
 ;{ssi.r)i74 
 
 ■ 220.8.114 
 
 .9 
 
 4524.5101 ! 2:58.4474 
 
 .6 
 
 52 IG, 82:51 
 
 250.0404 
 
 .4 
 
 ;),-<92.r)(i8 
 
 i 221.1(18(1 
 
 7G. 
 
 45:5G.4704' 2:58. 7G1G 
 
 .6 
 
 5229.G:« 
 
 256.3545 
 
 .5 
 
 ;!90:?.g;uh 
 
 •221.4S28 
 
 .1 
 
 4548.4 1G:5 2:5i».0757 
 
 .7 
 
 5242. 458G 
 
 256.6687 
 
 .G 
 
 :59l4.7lG:i 
 
 221.79(19 
 
 .2 
 
 45(10.:5787 1 2:5y.;4S99 
 
 .8 
 
 5255.2998 
 
 256.9828 
 
 .7 
 
 ;;!)2.-).8u 
 
 222.1111 
 
 .3 
 
 4572. :555:? ' 2:h;).704 
 
 1 .8 
 
 5208.1,568 
 
 257.297 
 
 .8 
 
 ;!9:;g.927-J 
 
 222.42r)2 
 
 .4 
 
 4584.:558;5' 240.0182 
 
 82. 
 
 5281. 029G 
 
 257.6112 
 
 .9 
 
 ;^948.05u5 
 
 222.7:594 
 
 .5 
 
 459G.:i571 i240.:5:524 
 
 i •! 
 
 529:5.918 
 
 257.9253 
 
24 
 
 AREAS AND C'lUCIIMl-'KUENCKS OF CIRCLES, 
 
 1 Diain 
 .2 
 
 1 Arm. 
 
 Ciriiiiii. 
 2.")8.2395 
 
 1 Diaiii. 
 
 j 5;!nG.822l 
 
 .;i 
 
 ij;ii!).7i;ii» 
 
 25s. 55; 111 
 
 ' .9 
 
 A 
 
 r>;{;{2. (;":.> 
 
 25S.sOI(; 
 
 S8. 
 
 .5 
 
 o;Mr).G2s7 
 
 I259.1S2 
 
 .1 
 
 .G 
 
 o;{rj8.r)!)-)7 
 
 i 259. 4901 
 
 •> 
 
 .7 
 
 5;{ 7 1.5983 
 
 ' 259.,^ 103 
 
 .•> 
 
 .s 
 
 o;{H4..'J7(;2 
 
 1200.1214 
 
 .4 
 
 .!) 
 
 .'j;{'J7.5908 
 
 200.43c^0 
 
 .5 
 
 8a. 
 
 5.JI0.G20G 
 
 200.7528 
 
 .0 
 
 .1 
 
 5423. G(1G 
 
 201.000!» 
 
 i .7 
 
 .2 
 
 543G.7272 
 
 201.3S1I 
 
 .8 
 
 .3 
 
 5419.8012 
 
 201.0952 
 
 .9 
 
 .4 
 
 51(i2.89G8 
 
 202.0094 
 
 89. 
 
 .5 
 
 517);.0(ir)l 
 
 2t;2.32:{(; 
 
 .1 
 
 .G 
 
 5489.1291 
 
 202.{;.;7o 
 
 ,2 
 
 .7 
 
 551)2. 2(;89 
 
 202.9519 
 
 I ..'! 
 
 .8 
 
 5515.4213 
 
 203.201 
 
 1 -t 
 
 .9 
 
 5528.59.')8 
 
 203.5802 
 
 ' .5 
 
 84. 
 
 5541.7824 
 
 203. s 1)4 4 
 
 i .0 
 
 .1 
 
 5554.9847 
 
 204.20S5 
 
 ; .7 
 
 .2 
 
 55(;8.2032 
 
 204.5227 
 
 .S 
 
 .3 
 
 5581.4372 
 
 '204.s;)08 
 
 .9 
 
 .4 
 
 5594.0809 
 
 205.151 
 
 :9o. 
 
 .5 
 
 5007.9523 
 
 205.4052 
 
 ! .1 
 
 .6 
 
 5021. 2;m 
 
 205.7793 
 
 i -2 
 
 .7 
 
 5034.5082 
 
 200.09;{5 
 
 .3 
 
 .8 
 
 5047.8428 
 
 200.4070 
 
 .4 
 
 .y 
 
 5001.171 
 
 200.7218 
 
 .5 
 
 85. 
 
 5074.515 
 
 207.030 
 
 .G 
 
 .1 
 
 5087. 8740 
 
 207..3501 
 
 .7 
 
 .2 
 
 5701.25 
 
 207.0043 
 
 .8 
 
 .3 
 
 5714.041 
 
 207.9784 
 
 .9 
 
 .4 
 
 5728.0478 
 
 208.2920 
 
 91. 
 
 .5 
 
 5741.4703 
 
 208.0008 
 
 .1 
 
 .G 
 
 5751.9085 
 
 208.9209: 
 
 .2 
 
 .7 
 
 5708.3021 
 
 209,2351 : 
 
 .3 
 
 .8 
 
 5781.832 
 
 209.5492 ■■■ 
 
 .4 
 
 .9 
 
 5795.3173 
 
 209.,S034 
 
 .5 
 
 86. 
 
 5808.8^84 
 
 270.1770 1 
 
 .G 
 
 .1 
 
 5822.3351 
 
 270.4917 ■ 
 
 .7 
 
 .2 
 
 5835.8075 
 
 270.8059 1 
 
 .8 
 
 .3 
 
 5849.4157 
 
 271.12 ' 
 
 .9 
 
 .4 
 
 5802.9795 
 
 271.4342 
 
 92. 
 
 .5 
 
 5870.5591 
 
 271.7484' 
 
 .1 
 
 .G 
 
 6890.1541 
 
 272.0005 i 
 
 .2 
 
 .7 
 
 5903.7054 
 
 272.3707 ! 
 
 .3 
 
 .8 
 
 5917.392 
 
 272.0908 
 
 .4 
 
 .9 
 
 5931.0344 
 
 273.005 
 
 .5 
 
 87. 
 
 5944.0920 
 
 273.3192 
 
 .G 
 
 .1 
 
 5958. 3G44 
 
 273.0333 
 
 .7 
 
 .2 
 
 5972.0559 
 
 273.9875 
 
 .8 
 
 .3 
 
 5985.7G91 
 
 274.2G1GI 
 
 .9 
 
 .4 
 
 5999.4821 
 
 274.5758 
 
 93. 
 
 .6 
 
 0013.2187 
 
 274.89 
 
 .1 
 
 .6 
 
 0020.9711 
 
 275.2041 
 
 .2 
 
 .7 
 
 G040.7391 
 
 275.5183 1 
 
 .3 
 
 Area. 
 
 0051.51 49 
 
 0()0s.;;224 
 
 Ol>2.l:i70 
 
 0(l'.»5.9(;.S| 
 
 01 09. si 5 
 
 0123.0771 
 
 0i;{7.555l 
 
 0151.1191 
 
 010.").;!,")SJ 
 
 0179.2S37 
 
 0193.2215 
 
 0'J07.!S11 
 
 0-J21.15;;i 
 
 02:;5.i ii;i 
 
 (12 19.145 
 
 O^O.'l.iOl I 
 
 0277.1995 
 
 0291.i;o;!5 
 
 (i;i()5.3i0.s 
 
 0.11 9.399 
 
 0:i;i.!.497 
 
 o;;i7.(;si3 
 
 03111.74 
 
 0.'!75.SS5 
 
 G3:)U.0158 
 
 0404 ,2222 
 
 0418.4144 
 
 0132.0223 
 
 0410.8474 
 
 Cirt iiiu. Diaiiv 
 
 iK 
 
 0401.0^ 
 
 0475.3402 
 
 04Slt.0109 
 
 050;{,S074 
 
 0518.1 095 
 
 0532.5173 
 
 0540.8909 
 
 0501.21)81 
 
 ti575.5051 
 
 05>!).'J158 
 
 0004.3222 
 
 0018.7542 
 
 00;{3.1S2 
 
 0047.0350 
 
 0002.0848 
 
 0070.5597 
 
 0091. 0101 
 
 0705.5507 
 
 0720.0787 
 
 07.'! 1.0105 
 
 0749.1099 
 
 070.".. 7391 
 
 0778.324 
 
 0792.9240 
 
 0807,5408 
 
 0822.173 
 
 0830.8200 
 
 275.s;!21 
 275.1 100 
 270.1008 1 
 270.7749 
 277.0801 
 277.iO:i2 
 277.7174 ' 
 27s.():;io 1 
 27s.;;i57 
 278.(i5i)"J I 
 27.>.|,975 I 
 27!t.2SS2 I 
 279.';024 I 
 279.!) 1 05 I 
 2.-0.2:;07 
 2-0.5 148 I 
 2^0. .-59 
 2S1.!7;!2 
 2S1.4.S73 
 I 2S1.S,S25 
 I 2S2.115(; 
 , 2S2.I29S 
 j 282.744 
 i 2-3.0581 
 2s;!.;i723 
 I 2s;;.o,-0 I 
 i 284.0000 
 284.;il48 
 
 .■;s4.(;289 
 
 281,9131 
 2s;i,2572 
 2,-5,5714 
 
 i 2S5.s»5(; 
 
 ■ 2s0.;'197 
 2,-0.5139 
 
 i 2S0.829 
 
 i 287.1422 
 
 j 287.1504 
 2S7.TT05 
 
 I 288.0847 
 288.398si 
 
 1 288.713 
 2,-9.02:2 
 289.;i4]3 
 2,S9.0555 
 2,S!),9090 
 2'J0,283,S 
 200,598 
 2:)().9121 
 291.2203 
 291,5104 
 29 1. 8540 
 292.1088 
 292.4829 
 292.7971 
 293.1112 
 
 A 
 .f> 
 .G 
 
 .7 
 .8 
 .9 
 
 94. 
 
 .2 
 .3 
 .4 
 .5 
 .G 
 .7 
 .8 
 .9 
 95. 
 .1 
 2 
 
 !.3 
 .4 
 .5 
 .0 
 .7 
 .8 
 .9 
 90. 
 .1 
 2 
 
 .3 
 .4 
 .5 
 .0 
 .7 
 .8 
 .9 
 
 .1 
 .2 
 .3 
 .4 
 .5 
 .0 
 
 .T 
 . I 
 
 .8 
 .9 
 98. 
 .1 
 .2 
 .3 
 .4 
 .5 
 .0 
 .7 
 .8 
 .9 
 
 97. 
 
 Area. 
 
 0851.4.^10 
 
 O.-'OO.IO;!! 
 i (i,-.s(i.s579 
 ; O.S1I5.50.S:) 
 (;9I0.2947 
 (;925.0,!07 
 0l);i9,79ll 
 0951,5077 
 
 (;'.)(;9,.'!50s 
 
 09SI.1014 
 099M.!l,-<21 
 
 7oi;;,8is;{ 
 
 702s, 0702 
 
 7(1 1. 'i. 5025 
 
 70.-..S.I1.S 
 
 707;!. 3202 
 
 70S.-.2.'15 
 
 7103.1054 
 
 7118.1110 
 
 7133.0734 
 
 7 MS. ();-,! 
 
 7103.014:; 
 
 7 17s, 053;! 
 
 7 1 9.!, 078 
 
 720S.11SI 
 
 7223. '745 
 
 72.;s.24(;i 
 
 7253.;i3:i9 
 
 7208,4371 
 
 ;2s;i,55(;i 
 72!!-, 01)07 
 
 73i;;,siii 
 
 732:1.0072 
 7:!t4.1S9 
 7;!59.:iS04 
 7371.5990 
 
 7;;.-9,s2so 
 
 7I05,07.'!2 
 
 7i2(i.3:!;;5 
 
 7i;!5.00'.)5 
 
 7150.9013 
 
 7400.2(187 
 
 7IS1.5;)I9 
 
 7490.S7O7 
 
 7512.225,3 
 
 7527. 5iJ5)! 
 
 7512.9810 
 
 755S.;is32 
 
 7573, sOOO 
 
 75si),2338 
 
 7004 .0S20 
 
 7020.1471 
 
 7035,027.3 
 
 7051,193,{ 
 
 7000, 9:;4'.) 
 
 7082.1023 
 
 CirciiMi, 
 
 293.4251 
 
 293. 7300 
 
 291,05,37 
 
 294, .1079 
 
 29I,(;S2 
 
 291.9902 
 
 295,;; 104 
 
 295,<;215 
 
 295,9;;S7 
 
 290.2i:i0 
 
 290.507 
 
 20(;.8812 
 
 297.195;; 
 
 297.5095 
 
 2I)7,S2;;0 
 
 29s.i:!78 
 
 2l)S.I52 
 
 29S.7001 
 
 29!l.072;! 
 
 29;i.:;944 
 2'.):>.70so 
 
 I 300.0228 
 
 i :ioo.;!;;(;i) j 
 
 ;ioo.(;5ii I 
 
 i :!oo.9(;52 I 
 
 30|.27:'4 j 
 ;101.5!I30 
 ;!01.!»077 
 i;i02.2219 
 
 ;!02.5;;o 
 ; ;io2.s;j()2 
 
 303.1044 
 303.47.S5 
 I 303.7927 
 304.1008 
 I 304 .421 
 i301,7;!.52 
 i 305,0493 
 
 \ ;;o5.;{035 
 
 i 3i)r),O770 
 i 305.9918 
 
 , ;!oo.;;o(; 
 
 ! 300,0201 
 
 3oo,9;!o;{ 
 
 1307,24-4 
 
 j :;o7,5(;2o 1 
 
 ! ;>07.8708 ! 
 
 1308.1909 : 
 308. .505 1 
 ;i0s.«192 
 309.1334 
 309.4470 
 ;{09.7017 
 310.0759 
 310,390 
 310.7042 
 
AREA8 AND CIRnUMFKIlKNOES nv riRrLFi». 
 
 TAWLE.—fCuHtmued.J 
 
 2ft 
 
 Dinni, 
 
 U'J. 
 .1 
 
 .2 
 
 Aroa. 
 
 77i;!.2(;U 
 
 772s.8;{;!(; 
 
 7744.42HKi 
 
 Circiini. 
 
 Diam- 
 
 :!1 1.0184 1 
 
 ;{ii.:!;i25: 
 .•ui.c.ic.tI 
 :ni.960Hi 
 
 1 .4 
 
 .:> 
 
 .0 
 .7 1 
 
 Arcn. 
 
 77(10. o:] 17 
 777i"). ().">().'{ 
 
 Clrrum, 
 
 :!1 2.275 
 ;!12.:>h!t2 
 
 Diani, 
 
 77!)i.'j;»;!(;i;!i2.!to:t;i 
 
 7H0(;.'J4tJ(Ji3i:{.2176 
 
 100. 
 
 .3 
 .9 
 
 Aroa. 
 
 7822.01 ")4 
 7H;{S.2'J'J8 
 
 7KiJ4. 
 
 Cirfura. 
 
 ;ti:{.5116 
 :UH.8458 
 .'{14. 16 
 
 To Compute the Area or f irniiiifiTcnr!^ of n Dinmpfcr greater than any 
 
 in tiic proffdiiig Tnlile. 
 
 Poo Ruloc, pftgos 176 nnd 1^1 . 
 
 Or, /I'l/" />i,iiiii/, ,■ f.'-rnil.t lOi) iiinl i.t /i:i.* l/idli lOUI. 
 
 Uunidvo 111!) (]<'ciiiial jx int, and tnko mit tho aroii or ^irniimforonoe as for n Wholo Number 
 by removing tlio decimal point, it lor the uru.i, two (ilucoa to tiio right; iind if for tho oiroum- 
 fcrcnpo, mio jihu'o. 
 
 iLLDSTHATKiN.— Tho area of 1)6.7 is 7.';44.I«S) ; honco for y67 it ia 734418.9: and the oironm- 
 fcrenco of i)U.7 is ^03.7927, ond for 967 it is :ii):i7.927. 
 
 TABLIO HIT. 
 
 AEEAS AND CIKGUMFERENCES OF CIRCLES. 
 
 rniiM 1 TO 50 FEET. 
 [Adrnncinfj by an Inch.] 
 
 Diam. 
 
 Area, 
 Feet. 
 
 Circtun. 
 
 Diam.' 
 
 i 
 
 Art ,1. 
 Foot. 
 
 C'irciiui. ! 
 
 Diam. 
 
 Area. 
 Feet. 
 
 Cin um 
 
 
 Feet. Ins. 
 
 luet. Ins. 
 
 Feet. luH. 
 
 I ft. 
 
 .7f^o4 
 
 :^ i-^.. 
 
 ■oft. 
 
 7.0686 
 
 9 5 
 
 5 ft. 
 
 19.6.35 
 
 15 8H 
 
 I 
 
 .9217 
 
 '^ 4% 
 
 1 
 
 l.UXC. 
 
 9 Sk 
 
 1 
 
 20.2947 
 
 15 U-H 
 
 2 
 
 1.069 
 
 3 8 
 
 2 
 
 7.8757 
 
 9 IP,^ 
 
 2 
 
 20.9656 
 
 16 2H 
 
 .3 
 
 1.2271 
 
 3 11 
 
 '1 
 • > 
 
 8.2ii.>7 
 
 10 2'i 
 
 3 
 
 21.6475 
 
 16 5^^ 
 
 4 
 
 1 .:{it(52 
 
 4 2 In 
 
 4 
 
 8.7265 
 
 10 5-'';i 
 
 4 
 
 22.34 
 
 16 9 
 
 .5 
 
 l.r)7til 
 
 4 bhl 
 
 6 
 
 9.1683 
 
 10 i<l^ 
 
 5 
 
 23.0437 
 
 17 )^ 
 
 fi 
 
 1.7671 
 
 4 8 '.J 
 
 6 
 
 9.6211 
 
 10 llJa 
 
 6 
 
 23.7583 
 
 17 m 
 17 eH 
 
 7 
 
 l.i)r.89 
 
 4 ll'V 
 
 7 
 
 10.0846 
 
 11 3 
 
 7 
 
 24.4835 
 
 8 
 
 2.18K) 
 
 5 2^.. 
 
 8 
 
 10. .5591 
 
 11 6U 
 
 8 
 
 25.2199 
 
 17 9,^8 
 
 9 
 
 2.40.52 
 
 5 5.^H 
 
 9 
 
 11.0446 
 
 11 9)„^ 
 
 9 
 
 25.9672 
 
 18 H 
 
 10 
 
 2.G.S98 
 
 5 9 
 
 10 
 
 11.6409 
 
 12 }" 
 
 10 
 
 26.7251 
 
 18 .3% 
 
 11 
 
 2.8802 
 
 6 21. 
 
 11 
 
 12.0481 
 
 12 .3% 
 
 11 
 
 27.4943 
 
 18 7.& 
 
 2 ft. 
 
 a.i4iG 
 
 6 3:', 
 
 ^ft. 
 
 12.5664 
 
 12 6V 
 
 Gft. 
 
 28.2744 
 
 18 10)^ 
 
 1 
 
 .^.'1087 
 
 6 6i.> 
 
 1 
 
 13.0952 
 
 12 9% 
 
 1 
 
 29.0649 
 
 19 iK 
 
 2 
 
 ,s.(;8(;9 
 
 6 9-'i) 
 
 2 
 
 13.6M.53 
 
 13 1 
 
 2 
 
 29.8668 
 
 19 4^ 
 
 .S 
 
 .3.976 
 
 7 i'( 
 
 3 
 
 14.1862 
 
 13 41^ 
 
 3 
 
 30.6796 
 
 19 7M 
 
 4 
 
 4.276 
 
 ? P' 
 
 4 
 
 14.7479 
 
 13 7I4 
 
 4 
 
 31 ..5029 
 
 19 loM 
 
 5 
 
 4. .5869 
 
 5 
 
 15.3206 
 
 13 lO'-i 
 
 5 
 
 .32. .3376 
 
 20 ig 
 20 4K 
 20 sH. 
 
 6 
 
 4.9087 
 
 7 101^ 
 
 8 1?| 
 8 4'.< 
 
 6 
 
 15.9043 
 
 14 I'^H 
 
 6 
 
 .33.1831 
 
 7 
 
 5.2413 
 
 7 
 
 16.4986 
 
 14 4% 
 
 7 
 
 .34.0391 
 
 8 
 
 5.585 
 
 8 
 
 17.1041 
 
 14 7?8 
 
 8 
 
 34.9065 
 
 20 11,5 
 
 9 
 
 5.9395 
 
 8 7% 
 
 9 
 
 17.7205 
 
 14 11 
 
 9 
 
 35.7847 
 
 21 2% 
 
 10 
 
 6.3049 
 
 8 io;^.i 
 
 10 
 
 18.3476 
 
 15 2)6' 
 
 10 
 
 .36.6735 
 
 21 5|^ 
 
 11 
 
 6.6813 
 
 9 1% 
 
 11 
 
 18.9858 
 
 15 5H' 
 
 11 
 
 .37.67.36 
 
 21 8^4 
 
2G 
 
 ARKAH AND t'MU'UMKI.IlKNt.'KS i)V C'lROLE^. 
 
 T\\nA'].~(('initinitcit.) 
 
 Diaiii. 
 
 Avon. 
 
 TJt 
 1 
 2 
 3 
 4 
 5 
 C 
 7 
 8 
 
 y 
 
 10 
 
 11 
 
 2 
 3 
 
 4 
 6 
 G 
 7 
 8 
 9 
 
 10 
 
 11 
 
 9//. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 lOJt. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 i ^ 
 10 
 11 
 
 1 
 
 2 
 3 
 4 
 6 
 
 6 
 
 F.H-I. 
 
 38.I81G 
 39..I0G 
 40.3;Wrt 
 4I.'2.S'<J5 
 
 .I2.i!:;ti7 
 
 43.2022 
 
 44.1TS7 
 
 45.IGr)G 
 
 4G.1G:{8 
 
 47,173 
 
 4S,1!)2G 
 
 49.22;{fi 
 
 50.2iirjG 
 
 f)i.;;i7H 
 
 52.381(1 
 fi3.45G2 
 54.5412 
 55.(1377 
 5G.7151 
 57.8(;28 
 58.9!)2 
 G0.i:!2I 
 G1.2S2G 
 02.4445 
 G3.(iI74 
 G4.800G 
 G5.!)1)51 
 07.2007 
 08.4100 
 09.044 
 70.8823 
 72.1309 
 73.391 
 74.002 
 75.9433 
 77.2302 
 78.54 
 79.854 
 81.1795 
 82.510 
 83.8027 
 85.2211 
 80.590:! 
 87.9097 
 89.3008 
 90.7027 
 92.1749 
 93.5980 
 9.':i.0334 
 96.4783 
 97.9347 
 99.4021 
 100.8797 
 102.3089 
 103.8691 
 
 (?irrntn. 
 
 I'm. 
 
 im. 
 
 Area. 
 
 Fret. IllH. 
 
 21 
 
 22 
 
 22 
 
 22 
 
 23 
 
 23 
 
 23 
 
 23 
 
 24 
 
 24 
 
 2 ' 
 
 24 
 
 25 
 
 25 
 
 25 
 
 25 
 
 20 
 
 20 
 
 20 
 
 20 
 
 27 
 
 27 
 
 27 
 
 28 
 
 28 
 
 28 
 
 28 
 
 2'.> 
 
 29 
 
 29 
 
 29 
 
 30 
 
 30 
 
 30 
 
 30 
 
 31 
 
 31 
 
 31 
 
 31 
 
 32 
 
 32 
 
 32 
 
 32 
 
 33 
 
 33 
 
 33 
 
 34 
 
 34 
 
 34 
 
 34 
 
 35 
 
 35 
 
 35 
 
 35 
 
 36 
 
 11 ^, 
 3 
 
 2'.. 
 Oj. 
 9 .t 
 I'.i' 
 4'.'.' 
 7'. 
 lO-'i, 
 
 is 
 
 11 
 
 2'h 
 
 : 7 
 8 
 9 
 
 : 10 
 ii 
 
 I2jt. 
 
 I 
 
 2 
 ' 3 
 
 4 
 
 5 I 
 
 i 
 
 ! 8 
 9 
 
 10 
 I 11 
 
 I3yz 
 I 1 
 
 ir-. 
 
 5'. 
 9 
 
 9'.. 
 
 . 11 
 
 :?;'{ 
 
 7 
 
 10' H 
 
 4?« 
 
 7'. 
 
 ■IS 
 
 5 
 
 nil 
 
 5'.. 
 
 8^a 
 
 11^ 
 
 2H 
 
 9>4 
 
 H 
 
 i 
 
 4i8' 
 
 7.^4 
 lOflJ 
 
 ?. 
 
 1; 3 
 4 
 5 
 
 6 
 
 7 
 
 !i 8 
 
 9 
 
 ' 10 
 
 '! 11 
 
 l\Jl. 
 
 1 
 
 I 2 
 
 3 
 
 , 4 
 
 5 
 
 'I ' 
 
 ' 8 
 
 ' 10 
 ,' 11 
 1,15 ;7. 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 11 
 16 /It 
 1 
 
 Kcct, 
 
 10,V3791 
 100.',H)13 
 108.i;U2 
 109.9772 
 111.:.:! 19 
 llil.OUTO 
 
 ii4.(iT;;2 
 
 110.2007 
 
 117,S59 
 
 11 9.4074 
 
 12),(IS7(; 
 
 122.7187 
 
 12 1. .15:18 
 
 121'). 01 27 
 
 127.0705 
 
 129.11501: 
 
 I3l.0;!t; 
 
 132.7;;20 
 
 131.4:191 
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 141.3771 
 
 143.1391 
 
 114.91 11 
 
 140.0949 
 
 148.1890 
 
 159.2913 
 
 152.1109 
 
 153.9:184 
 
 155.7758 
 
 157.025 
 
 159.4852 
 
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 103.2:173 
 
 105.1:103 
 
 107.0:131 
 
 108.9479 
 
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 174.7505 
 
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 178.08:12 
 
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 182.0545 
 
 184.0555 
 
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 188.0923 
 
 190.720 
 
 192.7710 
 
 194.8282 
 
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 198.973 
 
 201.0624 
 
 203.1615 
 
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 209.5201 
 
 211,0703 
 
 213,8251 
 
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 218.1002 
 
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 222.551 
 
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 220.9800 
 
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 203.9807 
 
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 271.2293 
 
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 270.1171 
 
 278.5701 
 
 281.0472 
 
 283.5291 
 
 280.021 
 
 288.5249 
 
 291.0:197 
 
 293.5041 
 
 290.1107 
 
 298.048:1 
 
 301.2051 
 
 303.7747 
 
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 308.9448 
 
 311.5409 
 
 314.10 
 
 316.7824 
 
 319.4173 
 
 322.003 
 
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 327.3858 
 
 330.0643 
 
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 6 
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 07 
 
 07 
 
 07 
 
 08 
 
 08 
 
 08 
 
 08 
 
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 70 
 
 70 
 
 70 
 
 70 
 
 71 
 
 71 
 
 71 
 
 71 
 
 72 
 
 72 
 
 72 
 
 7;! 
 
 73 
 
 73 
 
 7.'! 
 
 71 
 
 74 
 
 74 
 
 74 
 
 75 
 
 75 
 
 75 
 
 75 
 
 70 
 
 70 
 
 70 
 
 70 
 
 77 
 
 77 
 
 77 
 
 78 
 
 78 
 
 78 
 
 78 
 
 79 
 
 79 
 
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 551.5171 
 
 555.0201 
 
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 572.5500 
 
 570.0019 
 
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 052.0195 
 
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 075.7915 
 
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 79 
 
 70 
 
 80 
 
 80 
 
 80 
 
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 81 
 
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 81 
 
 Si 
 
 82 
 
 82 
 
 82 
 
 82 
 
 83 
 
 83 
 
 83 
 
 84 
 
 81 
 
 84 
 
 84 
 
 85 
 
 85 
 
 85 
 
 85 
 
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 80 
 
 8(i 
 
 80 
 
 87 
 
 87 
 
 87 
 
 87 
 
 88 
 
 88 
 
 88 
 
 89 
 
 89 
 
 89 
 
 89 
 
 90 
 
 90 
 
 00 
 
 00 
 
 01 
 
 91 
 
 91 
 
 91 
 
 92 
 
 02 
 
 92 
 
 92 
 
 93 
 
 93 
 
 93 
 
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 8 
 9 
 
 10 
 
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 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
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 1 
 2 
 3 
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 Area. 
 
 Foot. 
 
 702.9377 
 
 700.80 
 
 710.7009 
 
 714.735 
 
 718.09 
 
 722.0537 
 
 720.0305 
 
 7.30,01 S3 
 
 731.0147 
 
 73S.02I2 
 
 7l2.(ill7 
 
 74(i.(i7;i8 
 
 750.7101 
 
 754,701»4 
 
 758.8311 
 
 7(i2.00(i2 
 
 7(i(!.0021 
 
 771.0SO(i 
 
 775.1014 
 
 770..;!31 
 
 783.1403 
 
 7S7.5S08 
 
 701.7322 
 
 7 05. s 92 2 
 
 800.0(i51 
 
 S0J,2l0ii 
 
 808.4422 
 
 812.0181 
 
 8io.S(;5 
 
 821.0001 
 
 825.3201 
 
 829,57S7 
 
 8;;3.8308 
 
 838.1082 
 
 812.3095 
 
 8IO.tis!3 
 
 850.9S55 
 
 855.3000 
 
 859.02 1 
 
 803.9009 
 
 8li8.3087 
 
 872.0(il9 
 
 877.0340 
 
 881.4151 
 
 885.801 
 
 890.2004 
 
 891.0190 
 
 890.0413 
 
 903.4703 
 
 907.9224 
 
 912.3707 
 
 910.8445 
 
 921.3232 
 
 925.8103 
 
 930.3108 
 
 (.'iiTum. 
 
 Feet. Ills. 
 
 93 
 
 91 
 
 91 
 
 91 
 
 95 
 
 95 
 
 95 
 
 95 
 
 90 
 
 90 
 
 90 
 
 90 
 
 .97 
 
 97 
 
 97 
 
 97 
 
 98 
 
 98 
 
 98 
 
 98 
 
 99 
 
 90 
 
 99 
 
 100 
 
 lOo 
 
 100 
 
 100 
 
 101 
 
 101 
 
 101 
 
 101 
 
 102 
 
 102 
 
 102 
 
 102 
 
 103 
 
 103 
 
 103 
 
 103 
 
 104 
 
 104 
 
 104 
 
 104 
 
 105 
 
 105 
 
 105 
 
 106 
 
 106 
 
 106 
 
 106 
 
 107 
 
 107 
 
 107 
 
 107 
 
 108 
 
 11^8 
 
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 94 
 
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 4 
 
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 6 
 
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 4 
 
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28 
 
 AREAS AND CIRCITM PEBENCES OF CIRCLES. 
 T ABLY,- (Continued). 
 
 Diiim. 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 35// 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 G 
 
 9 
 
 10 
 
 11 
 
 36// 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 37//, 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 38// 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 39//. 
 
 Area. 
 
 l<\fft. 
 
 Cirfum. 
 
 Feet. Ins. 
 
 9;U.8223 
 
 939.3121 
 
 943.^753 
 
 948.4195 
 
 952 972 
 
 957.538 
 
 9G2.115 
 
 9Gti.7T01 
 
 971.2989 
 
 975.9085 
 
 980.52j;4 
 
 985.1579 
 
 989.8003 
 
 994.4509 
 
 999.1151 
 1003.7902 i 112 
 1008.473Gi 112 
 1013.1705 
 1017.8784 
 1022.5944 
 1027.324 
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 103G.8134 
 1041.5758 
 104G.3491 
 1051.1306 
 1055.9257 
 10G0.7317 
 1065.5459 
 1070.3738 
 1075.2126 
 1080.0594 
 1084.9201 
 1089.79)5 
 1094.6711 
 1099.5G44 
 1104.4687 
 1109.381 
 1114.3071 
 1119.244 
 1124.1891 
 1129.1478 
 1134.1176 
 1139.0953 
 1144.0868 
 II49.0892 
 1154.0997 
 1 169. 1239 
 1164.1591 
 1169.2023 
 1174.2592 
 1179.3271 
 1184.403 
 1189.4927 
 
 108 
 !08 
 108 
 109 
 109 
 109 
 109 
 110 
 110 
 110 
 111 
 111 
 111 
 111 
 112 
 
 ,112 
 
 !U3 
 
 I 113 
 
 ; 113 
 
 1113 
 
 ill4 
 
 114 
 
 114 
 
 114 
 
 115 
 
 115 
 
 115 
 
 115 
 
 116 
 
 116 
 
 116 
 
 117 
 
 117 
 
 117 
 
 117 
 
 118 
 
 118 
 
 US 
 
 118 
 
 119 
 
 119 
 
 119 
 
 119 
 
 120 
 
 120 
 
 120 
 
 120 
 
 121 
 
 121 
 
 12! 
 
 121 
 
 122 
 
 4H 
 IOJb 
 
 5Ja' 
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 54 
 878 
 
 33,; 
 
 10 
 
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 1194.6934 1122 
 
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 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 
 I 9 
 I 10 
 ^ 11 
 40//. 
 
 : 1 
 2 
 3 
 4 
 5 
 
 6 
 7 
 8 
 9 
 
 10 
 
 11 
 41//, 
 1 
 2 
 3 
 4 
 6 
 6 
 7 
 8 
 9 
 
 10 
 
 11 
 42//. 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 II 
 43 //. 
 l 
 
 2 1 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 Area. 
 
 Feet. 
 
 1199.7195 
 
 1204.8244 
 
 1209.9577 
 
 1215.099 
 
 1220.2542 
 
 1225.4203 
 
 1230.5943 
 
 1235. 7S22 
 
 1240.981 
 
 1246.1878 
 
 125I.40S4 
 
 1256.64 
 
 1261.8794 
 
 1267.1327 
 
 1272.397 
 
 1277.6692 
 
 1282.9553 
 
 1 288.2523 
 
 1293.5572 
 
 I29h.876 
 
 1304.2057 
 
 1305.5433 
 
 1314.8949 
 
 1320.2574 
 
 1325.6276 
 
 1331.0119 
 
 1336.4071 
 
 1311.8101 
 
 1347.2271 
 
 1352.6551 
 
 1358. 0;08 
 
 1363.5106 
 
 1369.0012 
 
 1374.4697 
 
 1379.9521 
 
 1385.4456 
 
 1390.2467 
 
 1396.1619 
 
 1401-.98S 
 
 1407.5219 
 
 1413.0698 
 
 1418.6287 
 
 1424.1952 
 
 1429.7759 
 
 14;!5.3675 
 
 1110.9668 
 
 1446.5802 
 
 1452.20 Hi 
 
 1457. 83C5 
 
 1463.4827 
 
 1469.i:!97 
 
 1474.8044 
 
 1480.4833 
 
 1486. 1731 
 
 1491.8706 
 
 Circuni. 
 
 Feet. lu.s. 
 
 Diam, 
 
 122 
 
 123 
 
 123 
 
 123 
 
 123 
 
 124 
 
 124 
 
 124 
 
 124 
 
 125 
 
 ! 25 
 
 125 
 
 125 
 
 !26 
 
 126 
 
 126 
 
 1 26 
 
 127 
 
 127 
 
 127 
 
 123 
 
 ri8 
 
 128 
 
 128 
 
 129 
 
 129 
 
 129 
 
 129 
 
 130 
 
 130 
 
 130 
 
 130 
 
 131 
 
 131 
 
 131 
 
 131 
 
 132 
 
 132 
 
 132 
 
 132 
 
 133 
 
 133 
 
 133 
 
 134 
 
 KM 
 
 131 
 
 134 
 
 135 
 
 1.35 
 
 135 
 
 1,35 
 
 136 
 
 136 
 
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 136 
 
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 9 
 10 
 11 
 
 44// 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 
 45/--. 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 
 11 
 
 46/. 
 1 
 
 2 
 3 
 4 
 
 5 
 
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 7 
 
 8 
 
 9 
 10 
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 1503.3046 
 
 1509.0348 
 
 1514.7791 
 
 1520.5344 
 
 : 1526.2971 
 
 i 1532.0712} 
 
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 i 1590.435 
 
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 I 1602.23b6 
 
 I 1608.1555 
 
 ; 16 II. 08! 9 
 
 I 1620.0226 
 
 i 1625.9743 
 
 1631.9331 
 
 1637.9068 
 
 1643.8912 
 
 1649. ^',^'31 
 
 ir)55.8S92 
 
 1661.9004 
 
 1067.930^ 
 
 1673.9698 
 
 1.680.0196 
 
 1686.0769 
 
 1692.1485 
 
 1698.2311 
 
 1704.321 
 
 17 10". 4 251 
 
 1716.5407 
 
 1722.6634 
 
 1728.8005 
 
 ; 7.54.9 (86 
 
 1741.1039 
 
 1747.2738 
 
 175:). 4545 
 
 1759.6126 
 
 1765.8452 
 
 1772.0587 
 
 15 78.-795 
 
 1784.5148 
 
 1790.761 
 
 1797.0145 
 
 1803.2826 
 
 1809.5616 
 
 1815.8477 
 
 1822.1485 
 
 Circum. 
 
 Feet. Ins. 
 
 2K 
 
 137 
 137 
 137 
 137 
 138 
 
 138 
 
 138 
 
 139 
 
 139 
 
 139 
 
 139 
 
 140 
 
 140 
 
 140 
 
 141 
 
 141 
 
 141 
 
 141 
 
 141 
 
 142 
 
 l-'2 
 
 142 
 
 142 
 
 143 
 
 143 
 
 143 
 
 143 
 
 144 
 
 144 
 
 144 
 
 145 
 
 145 
 
 145 
 
 145 
 
 146 
 
 146 
 
 146 
 
 146 
 
 147 
 
 147 
 
 147 
 
 147 
 
 148 
 
 148 
 
 148 
 
 148 
 
 149 
 
 149 
 
 149 
 
 150 
 
 150 
 
 150 
 
 150 
 
 151 
 
 151 
 
 11,?^- 
 
 5J- 
 
 7'..' 
 
 10J8 
 
 IM 
 
 7'o 
 10^ 
 
 IJ" 
 
 1 
 5 
 
 iiH' 
 2% 
 51.; 
 
 «P 
 iiJ^ 
 3 
 ^\ 
 
 9>4 
 
 &% 
 
 9% 
 lf« 
 AH 
 Vi 
 WH 
 
 I'e 
 
 4% 
 7/4 
 11 
 
 5'.i 
 
 sH 
 
 11\' 
 
 8 
 
 75 
 
 8 
 
 3% 
 6% 
 
 3H 
 
SIDES OP EQTTAL SOXTABES. 
 
 29 
 
 TABLE.— (Continued.) 
 
 Uiiiui 
 
 4 
 
 5 
 ('• 
 7 
 8 
 9 
 10 
 
 Aren. 
 
 Foot. 
 IS28jr,02 
 
 Cirs 
 
 inn. 
 
 Foot. Ins, 
 151 
 
 1 .^;! 1 
 
 7t;)i 
 
 isii 
 
 1727 
 
 1>'I7 
 
 Jo7i 
 
 1 S.).'! 
 
 S('S7 
 
 Is'CO 
 
 ]7r, 
 
 isin; 
 
 :V)21 
 
 IS72 
 
 i).li;;> 
 
 152 
 
 l.-)2 
 1.32 
 15.! 
 
 I5:i 
 
 7'., 
 1()^,{ 
 
 iniiun 
 
 11 
 11)// 
 1 
 2 
 
 4 
 5 
 
 () 
 
 Area. 
 
 Cii 
 
 Tiiin. 
 
 Foot. 
 
 i87!).;{.15.-) 
 1>*S5.7I5I 
 IX!)2 1721 
 IS!)-!. 5011 
 l!K)5.0.1(i7 
 ll»ll.l!)i;5 
 11)17. 0'iOI) 
 
 i;)2i.i2>;.i 
 
 I Foot. Iii.i 
 
 I !5:{ 
 
 151 
 151 
 151 
 5 4 
 1 55 
 i 55 
 
 1 1 H 
 '1% 
 
 I! 
 
 'n 
 
 Diiiin 
 
 2^-^ 
 
 7 
 8 
 !) 
 
 10 
 
 11 
 \QfC. 
 
 Areii. 
 
 Foot. 
 
 1[):?0.01,SS 
 
 1!».!7.:!15!) 
 
 IIICMIN 
 
 1!)5().4;{!)2 
 
 l!.>5ii.;>iJD] 
 
 IIIG3.5 
 
 Circum. 
 
 Foot. Ins. 
 
 155 
 
 1 5i; 
 
 1 50 
 
 1 5(; 
 !5r, 
 
 157 
 
 % 
 
 TABLE OF THE SIDES OF SQUARES -EaUAL IN AREA TO A 
 CIRCLE OF ANY DIAMETER. 
 
 FROM 1 TO 100. 
 
 Dinm.|Si(l6 of Sq. ^ Diiini 
 
 if 
 
 g 
 '% 
 
 .S8tl2 
 
 l.ior.s 
 i.;!2'.){ 
 
 1 .550;) 
 1.772t 
 
 1.11!) ^ 
 
 2.2 1 5i) 
 2.4:{71 
 2.(;5H7 
 
 2.8-<l)2 
 
 ;i.ioi8 
 
 ;i.;{2:!:{ 
 
 .1.5111) 
 
 .'i,7ili)5 
 
 ;i.l)88 
 
 4.201)(J 
 
 4.4:ill 
 
 4.0527 
 
 4.8742 
 
 5. 01)58 
 
 5.:il7t 
 
 5.5380 
 
 5.7005 
 
 5.982 
 
 0.20:!G 
 
 0.4251 
 
 0.0407 
 
 (i.80S;{ 
 
 m. 
 
 ■\i 
 
 X 
 
 ■H 
 
 11. 
 
 
 12. 
 
 
 H 
 
 •^4- 
 .'.. 
 
 M 
 
 13. 
 
 U. 
 
 X 
 
 .Side of .Sq 
 
 7-081)8 
 
 7.;!114 
 
 7.5:{29 
 
 7-75 15 
 
 7.1)70 
 
 8.1970 
 
 8.4192 
 
 8.0407 
 
 8.S(;2:^ 
 
 9.08:!8 
 
 9.:ii)5t 
 
 9.52';9 
 9.7485 
 9.97 
 10.1910 
 I0.1l:i2 
 10.0:!17 
 10.850:{ 
 11.0778 
 .'99 1 
 
 5209 
 7425 
 9041 
 12.1856 
 12.4072 
 12.0287 
 12.S50:i 
 l;i.07l8 
 
 Diam. 
 
 .1.. 
 
 16. 
 
 17. 
 
 1 - 
 
 18. 
 
 ./4 
 
 I - 
 
 19. 
 
 20. 
 
 .% 
 
 .x4 
 
 21. 
 
 
 .Side of Sq. iDiam.;Sicle of Sq. 
 
 l:!.29.!4 
 l:i.515 
 1 ii.7,>05 
 
 i;i.95si 
 
 14.1790 
 
 14.4012 
 
 14.0227 
 
 14.814:! 
 
 15,0.;59 
 
 15.2M71 
 
 15.5'I9 
 
 ]5.7;!05 
 
 15.9521 
 
 10.17:!0 
 
 10.;!952 
 
 10.0108 
 
 lO.H.lS.i 
 
 17.0599 
 
 17.2814 
 
 17.503 
 
 17.7245 
 
 17.9401 
 
 18.1077 
 
 18.8892 
 
 18.0103 
 
 18.8:i2:i 
 
 19.05;i9 
 
 19.2754 
 
 22. 
 
 23. 
 
 •/4 
 
 24. 
 
 >4 
 
 ■H 
 
 25. 
 
 20. 
 
 a? 
 
 •/4 
 
 
 27 
 
 28. 
 
 •x4 
 
 •5f 
 
 ./-^4 
 
 19.497 
 
 19.7185 
 
 19.9»0l 
 
 20.1017 
 
 20.3832 
 
 20.0018 
 
 20.8203 
 
 21.0479 
 
 21.2091 
 
 21 
 
 21 
 
 21, 
 
 22. 
 
 22 
 
 .491 
 7120 
 ,9311 
 1557 
 .3772 
 22.59.S8 
 22.8203 
 23.0119 
 2.3.2034 
 23.485 
 23.700(5 
 23.9281 
 24.1497 
 24.3712 
 24.5928 
 8144 
 1^5.0359 
 25.2575 
 25.459 
 
 Dliiiri.iSide of Sq. 
 
 29. 
 
 30. 
 
 3i 
 
 31. 
 
 3> 
 ./4 
 
 32. 
 
 
 .33. 
 
 .1.. 
 
 .x4 
 
 34. 
 
 M 
 
 ./'4 
 
 35. 
 
 ■11 
 
 :lf 
 
 25 7006 
 
 25.9221 
 
 20.1437 
 
 20.3653 
 
 26.5808 
 
 20.8084 
 
 27.0299 
 
 27.2515 
 
 27.473 
 
 27.0910 
 
 27.9 Kit 
 
 .28.1377 
 
 28.3593 
 
 28.5808 
 
 2.-S.8021 
 
 29.0239 
 
 29.2155 
 
 29.407 
 
 29.0886 
 
 29.9102 
 
 30.1317 
 
 30.3533 
 
 30.5748 
 
 30.7904 
 
 31.0179 
 
 31.2395 
 
 31.4011 
 
 31.6826 
 
30 
 
 SIDES OF EQIIAI, .SQUARE.- 
 
 TA\\LE~( Co ntimied). 
 
 Ditiin. 
 
 Sido of Si|. 
 
 Diiini. 
 49. 
 
 Side of S(i.; 
 
 Diniii. 
 
 .Side of Sq.'JDium. 
 
 1 
 
 Side Of Sq. 
 
 Diuni. 
 
 : 88. 
 
 Side of. Sq. 
 
 77.088 
 
 36. 
 
 81.1)012 
 
 43.1251 
 
 02. 
 
 54.940! 75. 
 
 06.407 
 
 •H 
 
 wi.vi-a 
 
 'H 
 
 43.0107 
 
 .^i 
 
 55.1070 
 
 .'., 
 
 00. (;8S6 
 
 1 -v 
 
 78.2095 
 
 M 
 
 H2.;M78 
 
 1 - 
 
 •i:).80s2 
 
 * 
 
 55.3892 
 
 1 
 
 00.9104 
 
 1 .'.. 
 
 78.i:{10 
 
 Xi 
 
 H2.ri(;,*-is 
 
 '■^ 
 
 44.0898 
 
 .^4 
 
 .55.1; 107 
 
 • 1 
 
 07.1317 
 
 A, 
 
 78.0520 
 
 37. 
 
 ;;2.Ti)(i[ 
 
 50. 
 
 -11.3113 
 
 03. 
 
 55.8:;23 
 
 j70. 
 
 07. 35:12 
 
 89. 
 
 78.ST42 
 
 •M 
 
 3.;. 01 12 
 
 •H 
 
 41.532;) ; 
 
 ..u. 
 
 5ii.05;;8 
 
 .'., 
 
 07.5748 : 
 
 , .', 
 
 79.0957 
 
 .H 
 
 33.2;;:'.;) 
 
 .'., 
 
 11.7515 
 
 . .> 
 
 50.2751 
 
 1 
 
 67.7904 
 
 i .'.. 
 
 79.;; 173 
 
 S 
 
 33. 1;').-)! 
 
 •:*.i 
 
 -il.97ii 
 
 .^4 
 
 50.497 
 
 1 'A^ 
 
 08.0179 
 
 i A^ 
 
 79.5:!s9 
 
 38. 
 
 3.'i.(i7(it; 
 
 51. 
 
 ■J5.I970 
 
 64. 
 
 50.7185 ;77. 
 
 08.2395 
 
 i 90. 
 
 79.7004 
 
 -)i 
 
 33.Si).S2 
 
 ■H 
 
 45.4191 
 
 •K 
 
 50.9401 j .1, 
 
 08.401 
 
 ■\{ 
 
 79.982 
 
 .'.. 
 
 34.111)7 
 
 , 1 , 
 
 45.0407 
 
 .'., 
 
 57.1010 
 
 . .'., 
 
 08.0826 
 
 .1 ; 
 
 80.20:15 
 
 :fi 
 
 31.3113 
 
 ••*4 
 
 45.8022 
 
 •i?4 
 
 5T.;!^:;2 
 
 ■ • ■ '. 1 
 
 08.9011 
 
 'A, 
 
 80.4251 
 
 39. 
 
 3l.rjti2,S 
 
 52. 
 
 40.08:i8 
 
 65. 
 
 57.0017 78. 
 
 09.1257 
 
 ; 91. 
 
 S0.0I07 
 
 •M 
 
 34.7884 
 
 ■'i 
 
 40. 3051 j 
 
 ■a 
 
 57.82i;3 
 
 ! -'l 
 
 09. .34 73 
 
 :!! 
 
 8O.,«0S2 
 
 l; 
 
 3'j.00(; 
 
 J 
 
 40.52:i9 1 
 
 5S.0479 
 
 •t 
 
 09.5088 
 
 81.0898 
 
 ■>4 
 
 35.2275 
 
 •■'^ 
 
 40.7485 1 
 
 :-f. 
 
 58.2091 
 
 A 
 
 09.7904 
 
 A 
 
 81.;!113 
 
 40. 
 
 35,li:)l 
 
 53. 
 
 40,97 i'G6. ' 
 
 58.491 !;79. " 
 
 70.0119 
 
 1 92. 
 
 81.5:^29 
 
 .'.. 
 
 35.()70() 
 
 ■H 
 
 47.1910 
 
 ■H 
 
 58.7125 ; .1, 
 
 70.2.3.35 
 
 .1:,' 
 
 81.7514 
 
 1 , 
 
 35.81)22 
 
 .'. 
 
 47.4131 
 
 .1.. 
 
 58.9.341 ! 
 
 i 1.. 
 
 70.455 
 
 .'.; 
 
 81.970 
 
 :;ii 
 
 30.1137 
 
 .^'4 
 
 47.(i3l7 
 
 .;'4 
 
 59.155t; 
 
 ! :r 
 
 ■ 1 
 
 70.0706 
 
 1 ■■■'1 
 
 82.1975 
 
 41. 
 
 3(1.3.353 
 
 54. 
 
 47.8502 ' 
 
 07. 
 
 59.3772 
 
 80. 
 
 70..^9>1 
 
 1 93. 
 
 82,4191 
 
 .,^-1 
 
 3(j.55t)l) 
 
 'H 
 
 48.0V 78 ' 
 
 ■H 
 
 59.5988 
 
 i -'l 
 
 71.1197 
 
 ■u 
 
 82.0107 
 
 1 
 
 3G.7784 
 
 A 
 
 48.2994 ' 
 
 .C 
 
 59.8203 
 
 A 
 
 71.3413 
 
 A 
 
 82..^022 
 
 :?.! 
 
 37. 
 
 ■^ 
 
 48.5209 
 
 ■^ 
 
 00.0419 
 
 1 .".1 
 
 71.5628 
 
 A 
 
 83.08;;,s 
 
 42. 
 
 37.2215 
 
 55. 
 
 48.7425 
 
 08. 
 
 00.20.34 '81. " 
 
 71.7844 
 
 94. 
 
 83. .•■1053 
 
 •M 
 
 37.14;il 
 
 •H 
 
 48.904 1 
 
 ■ H 
 
 00.485 
 
 .'., 
 
 72.0059 
 
 • "^ 1 
 
 83.5209 
 
 • - •> 
 
 37.tl()4() 
 
 .1., 
 
 49.l8,-,0 1 
 
 .'0 
 
 00.7005 
 
 .1.. 
 
 72.2275 
 
 ij 
 
 8;!. 7 184 
 
 .i?4 
 
 37.8.s(;2 
 
 •'4 
 
 49.4071 1 
 
 Sa 
 
 60.9281 
 
 i •'4 
 
 72.4491 
 
 A^ 
 
 8:(.970 
 
 43. 
 
 38.1078 
 
 50. 
 
 49.0287 ■ 
 
 69. 
 
 01.1497 
 
 ■82. 
 
 72.0706 
 
 95. 
 
 81.1916 
 
 •K 
 
 38.321)3 
 
 •^-1 
 
 49.8503 . 
 
 •H 
 
 01.3712 
 
 .1-4 
 
 72.8921 
 
 •H 
 
 84.4131 
 
 Jc 
 
 .38.5501) 
 
 .'..1 50.07:8 i 
 
 .': 
 
 01.5928 
 
 .'., 
 
 73.1137 
 
 
 84.0:U7 
 
 ..^4 
 
 38.7724 
 
 :fi\ .00.2934 
 
 •ill 
 
 01.8M.5 
 
 •>'4 
 
 7:i.3353 ! 
 
 A, 
 
 84 .8502 
 
 44. 
 
 38.1)1)4 
 
 57. 1 50.51 19 
 
 70. 
 
 02.0359 ■■ 
 
 83. 
 
 73.5508 
 
 96. 
 
 85.0778 
 
 •M 
 
 39.2155 
 
 .^i \ .'■>0.T3(;5 ; 
 
 .>a 
 
 02.2574 ! 
 
 1 - 
 
 73.7784 
 
 .^1 
 
 85.299:) 
 
 .'* 
 
 31).437I 
 
 .'.. 
 
 50.958 ! 
 
 .'.. 
 
 02.479 
 
 1 
 
 73.9999 ' 
 
 I 
 
 .'^5.5209 
 
 .).i 
 
 39. 058 7 
 
 .h 
 
 51.1790 ! 
 
 .3*4 
 
 02.7000 
 
 .^'4 
 
 7-1.2215 . 
 
 ' A, 
 
 85.7425 
 
 45. 
 
 39.8802 
 
 58. 
 
 51.1012 i 
 
 71. 
 
 02.92:: 1 
 
 84. 
 
 7-1.1431 ' 
 
 97. 
 
 85.9046 
 
 :« 
 
 40.1018 
 
 ■H 
 
 51.(i227 
 
 ■H 
 
 03.1437 
 
 ..'4 
 
 71.0047 ' 
 
 •^:| 
 
 8li.l85 
 
 40.3233 
 
 1 
 
 51.8113 1 
 
 A 
 
 03.3052 
 
 1 , 
 
 7.1.8802 
 
 1 
 
 80.1071 
 
 «;'^ 
 
 40.5419 
 
 31 
 
 52.0058 1 
 
 M 
 
 03.5808 
 
 ! 'A 
 
 75.1077 ; 
 
 A, 
 
 80.628!) 
 
 40.7004 
 
 59. 
 
 52.2-74 : 
 
 72. 
 
 03.8083 
 
 85. 
 
 75.:r293 : 
 
 98. 
 
 80.8502 
 
 :« 
 
 40.988 
 
 .'•4 
 
 52.50SH 1 
 
 'H 
 
 (i4.0299 
 
 ■H 
 
 75.5508 ' 
 
 ■y^ 
 
 87.0718 
 
 41.2090 
 
 1 
 
 52.7:!05 
 
 t .> 
 
 04.2514 
 
 \ 
 t •> 
 
 75.7724 i 
 
 .1.: 
 
 87.29;:3 
 
 4,:^^ 
 
 41.4311 
 
 '.h 
 
 52.9521 
 
 .^4 
 
 04.4730 
 
 .'.'l 
 
 75.99:{4 ! 
 
 Ax 
 
 87. .54 19 
 
 41.0527 
 
 GO. 
 
 6:;.i7;iO ■ 
 
 73. 
 
 O4.(i940 
 
 80. 
 
 76.2155 i 
 
 ' 99. 
 
 87.730.1 
 
 •!<< 
 
 41.8742 
 
 -'4. 
 
 53.3952 
 
 .\{ 
 
 04.9101 
 
 .':, 
 
 70.4371 
 
 .).! 
 
 87.958 
 
 ■'1 
 
 42.0958 
 
 .,'..< 
 
 511.0107 1 
 
 A 
 
 i'i5.1377 
 
 1 
 
 70.0586 ' 
 
 •'■" 
 
 88.1790 
 
 48:* 
 
 42.3173 
 
 X 
 
 53.8:js;i 
 
 ■fi 
 
 05. .3592 
 
 1 :-'4 
 
 76.8802 ■ 
 
 \ 
 
 88.4011 
 
 42.5839 
 
 61., 
 
 64.0598 
 
 74. 
 
 05 580S 
 
 87. 
 
 77.1017 '■ 
 
 ■my 
 
 88,0227 
 
 .'^ 
 
 42.7004 
 
 M 
 
 51.2814 
 
 .H 
 
 05.802;; 
 
 , .'.. 
 
 77.:!2.33 
 
 .'a 
 
 88.K.M;i 
 
 .''. 
 
 42.982 
 
 4> 
 
 51.50.-; 
 
 1 
 
 O0.02;'9 
 
 '., 
 
 77.5419 
 
 .'.. 
 
 89.0058 
 
 ■^.1 
 
 43.2036 
 
 .^4 
 
 54.7245 
 
 :;'4 
 
 06.2455 
 
 Ai 
 
 77.7604 
 
 A 
 
 89.2874 
 
TABLE VI. 
 
 TABLE OF THE LENGTHS OF CIRCULAR ARCS. 
 
 The Diiiiiictcr of a Circle assumed to he Umlij, and divided into KKK) eqmd Parts. 
 
 nviit. 
 
 I.02(;(.'3 
 
 Il'Kl't- 
 
 Length. 
 
 H'ght. 
 
 Lungth. 
 1.09949 
 
 IlVlit. 
 
 Lnifrtii. 
 
 HV'ht, 
 .292 
 
 Lfuglh. 
 
 .1 
 
 .us 
 
 1 .05743 
 
 .196 
 
 .244 
 
 1.15180 
 
 1.21.381 
 
 .101 
 
 l.()2i;!)8 
 
 .149 
 
 1.05.S19 
 
 .197 
 
 1.10048 
 
 .245 
 
 1.15;!0S 
 
 .29;j 
 
 1.2152 
 
 .102 
 
 1.02T.V2 
 
 .15 
 
 1 .05S9() 
 
 .198 
 
 1,10147 
 
 i .240 
 
 1.15429 
 
 .294 
 
 1.21058 
 
 .io:{ 
 
 i.02n0(; 
 
 .151 
 
 1.0597;{ . 
 
 .199 
 
 1,10247 
 
 .247 
 
 1.15549 
 
 .295 
 
 1.21794 
 
 .101 
 
 1. 02^11 
 
 .152 
 
 1 .OilOul ; 
 
 .2 
 
 1,10:!48 
 
 j .248 
 
 1.1507 
 
 .290 
 
 1.21920 
 
 .lor. 
 
 1.02914 
 
 , .15:5 
 
 l.OOl.'i 
 
 ! .201 
 
 1.10447 
 
 i .249 
 
 1,15791 
 
 .297 
 
 ; 1.22001 
 
 ,lOtJ 
 
 1.01197 
 
 ■ .154 
 
 1.00209 - 
 
 i .202 
 
 1. 1 0548 , 
 
 1 .25 
 
 1.15912 
 
 .298 
 
 1,22203 
 
 .107 
 
 i.o;;o2(; 
 
 .155 
 
 1.0028S ; 
 
 .20:] 
 
 1.1005 ; 
 
 ! .251 
 
 Lioo;;;! 
 
 .299 
 
 1 .22;i47 
 
 .los: 
 
 1 .o;;o,-^2 
 
 .15(i 
 
 l.Oii.'lOS . 
 
 i .204 
 
 1.10752 
 
 .252 
 
 1.10157 
 
 .3 
 
 1 .22495 
 
 .109 
 
 i.o;;i;!9 
 
 .157 
 
 1.00149 
 
 .205 
 
 1 , 1 0855 
 
 .25;! 
 
 1.10279 
 
 .;!0i 
 
 l,220;i5 
 
 .110 
 
 l.o;;i9(i 
 
 .15S 
 
 1 .005;! 
 
 .200 
 
 1,1095S 
 
 .254 
 
 1.10402 
 
 1 .302 
 
 1 .22770 
 
 .111 
 
 l.o:{2.-.4 
 
 .159 
 
 1. 000 11 
 
 .207 
 
 1,11002 
 
 .255 
 
 1.10520 
 
 .3(t;! 
 
 1.22918 ! 
 
 .112 
 
 l.o:{;!i2 
 
 .10 
 
 1 .ooo'.i:! 
 
 .208 
 
 1,11105 
 
 .250 
 
 1.10049 ' 
 
 \ ,304 
 
 1.230(il 
 
 .IIH 
 
 i.o;i;{7i 
 
 ; .Kll 
 
 1.00775 
 
 .209 
 
 1.11209 
 
 .257 
 
 1.10774 i 
 
 ,3ii5 
 
 1.2:1205 : 
 
 .in 
 
 1 .o;i4;{ 
 
 : .I(i2 
 
 i.(i(;s58 
 
 .21 
 
 I.Ii:!74 
 
 , .258 
 
 1 , 1 0899 ; 
 
 1 ,300 
 
 1.2:!349 ; 
 
 .Uo 
 
 1.0:!49 
 
 .iti:{ 
 
 1.00911 
 
 .211 
 
 1.11479 
 
 ' .259 
 
 1.17024 
 
 i .307 
 
 1.23194 ! 
 
 .lie 
 
 l.O.-i;).-)! 
 
 ; .104 
 
 1.07025 
 
 .212 
 
 1,11584 
 
 .20 
 
 1.1715 
 
 1 .308 
 
 1 .23030 
 
 .117 
 
 i.o;{(iii 
 
 i .105 
 
 1.07109 
 
 .2i:{ 
 
 1,11092 , 
 
 .201 
 
 1.17275 
 
 .309 
 
 1.2:i7S 
 
 .lis 
 
 i.o;!(;72 ' 
 
 : .kk; 
 
 1.07191 
 
 .214 
 
 1,11790 
 
 .202 
 
 1.17401 
 1.17527 1 
 
 .31 
 
 1.23921 
 
 .119 
 
 i.o:!7;!4 1 
 
 i .107 
 
 1.07279 
 
 .215 
 
 1.11904 
 
 .2(;;! 
 
 .311 
 
 1.2407 
 
 .12 
 
 I.O:!797 
 
 : .itis 
 
 I .o7;;o5 
 
 ,210 
 
 1.12011 
 
 .204 
 
 1.17055 
 
 .312 
 
 1.24210 
 
 .121 
 
 i.o:;s(; ' 
 
 ■ .109 
 
 1,07451 
 
 .217 
 
 1,1211s ! 
 
 .205 
 
 1.177S4 : 
 
 .313 
 
 1 ,2430 
 
 .122 
 
 l.o;i92:! 
 
 .17 
 
 1 . 075:57 : 
 
 .218 
 
 1,12225 
 
 .200 
 
 1.17912 
 
 .314 
 
 1.24506 
 
 .12:5 
 
 l.o:;9s7 
 
 .171 
 
 1.07024 
 
 .219 
 
 \.\ZVA\ ' 
 
 .207 
 
 1.1804 
 
 .315 
 
 1.24054 
 
 .124 
 
 l.()40.')l 
 
 i .172 
 
 1.07711 
 
 .22 
 
 1,12445 ; 
 
 .208 
 
 1.18102 
 
 ,310 
 
 1.24801 
 
 .12.^ 
 
 1.0411(J 
 
 .17H 
 
 1.07799 , 
 
 !22l 
 
 1,12550 ! 
 
 .209 
 
 1.18294 
 
 , .317 
 
 1.24940 
 
 .12(1 
 
 I.OIISI 
 
 .174 
 
 1.07SS8 i 
 
 ,222 
 
 1,1 200;^ 
 
 .27 
 
 1.18428 i 
 
 ; .318 
 
 1.25095 1 
 
 .127 
 
 1.01247 
 
 ! .175 
 
 1.07977 ' 
 
 .22;! 
 
 1,12774 
 
 ,271 
 
 1.18557 
 
 .319 
 
 1 .25243 
 
 .12S 
 
 i.oi:!i;^ 
 
 1 .170 
 
 1.08000 
 
 .224 
 
 1.12885 
 
 .272 
 
 1.180^8 i 
 
 ! .32 
 
 1,25391 
 
 .129 
 
 1 .Ol.-iS 
 
 .177 
 
 1 A)^ 1 50 
 
 .225 
 
 1.12997 
 
 ,27.'! 
 
 1.18819 ! 
 
 i .321 
 
 1.25539 
 
 .i;i 
 
 1.04447 1 
 
 .17S 
 
 1.08240 
 
 .220 
 
 1. Kilos 
 
 .274 
 
 1.1S90-J 
 
 , .322 
 
 1,250.S(; 
 
 .i;ii 
 
 1.04."ii.') ■• 
 
 .179 
 
 i.os,s;<7 ' 
 
 .227 
 
 l.i;!219 
 
 .275 
 
 1.19082 
 
 .323 
 
 1.25830 
 
 .1:^2 
 
 l.()l.')S4 
 
 .18 
 
 1.0V42S 
 
 .228 
 
 i.i;!;i;Hi 
 
 .27(1 
 
 1.19214 
 
 ' .324 
 
 1 .25987 
 
 .i;i;^ 
 
 1 .04(;r)2 
 
 .181 
 
 1.0S519 
 
 .229 
 
 l,i;il44 ; 
 
 .277 
 
 1.19;!45 
 
 ' .325 
 
 1.2(;i37 
 
 .i;!4 
 
 1.04722 
 
 .182 
 
 1.08011 
 
 ,2;! 
 
 l.i;)557 
 
 .278 
 
 1,19477 
 
 .320 
 
 1 .20286 
 
 .i:!;-) 
 
 1.04792 
 
 .18:5 
 
 1.08704 
 
 .2;!1 
 
 i.i;;o7i 
 
 .279 
 
 1.1901 
 
 .327 
 
 1 .20437 
 
 .i;!(i 
 
 1.0!S(;2 
 
 .184 
 
 1 .0S797 
 
 .2;!2 
 
 i.i;!7so 
 
 .28 
 
 1,1974;! 
 
 .328 
 
 1,20588 
 
 .i;!7 
 
 1.0l9.i2 
 
 .185 
 
 1.0889 ■ 
 
 ,2;!;! 
 
 \.\:m)\\ . 
 
 .281 
 
 1.19887 
 
 .329 
 
 1,2074 
 
 .i;{.s 
 
 1 .o.")Oo:i 
 
 .18(1 
 
 1.08981 
 
 .2;!4 
 
 1.1402 
 
 .282 
 
 1 .2001 1 
 
 .33 
 
 1,20892 
 
 .i;i9 
 
 1 .orj07r) 
 
 .187 
 
 1.0!KI79 : 
 
 .2,!5 
 
 1.14i;5G 
 
 .2s;! 
 
 1,20140 
 
 .331 
 
 1,27044 
 
 .14 
 
 1.0.-) 147 
 
 .188 
 
 1.09174 
 
 .2;!0 
 
 1.14247 1 
 
 .284 
 
 1 .20282 
 
 .332 
 
 1.27190 
 
 .141 
 
 1 .0.^)22 
 
 .189 
 
 1 .09209 ' 
 
 .2;!7 
 
 i.i4;!o;! I 
 
 .285 
 
 1.20419 
 
 .3:^3 
 
 1 .27;i49 
 
 .142 
 
 1.0,-)29;^ 
 
 .19 
 
 1.09:;05 
 
 .2;!8 
 
 1.1448 
 
 .280 
 
 1 .20558 
 
 .334 
 
 1,27502 
 
 .M;i 
 
 1 .o.-);it;7 
 
 .191 
 
 1.09101 
 
 .2;!9 
 
 1.14597 I 
 
 .287 
 
 1 .20(;90 
 
 .3;« 
 
 1 •270.50 
 
 .144 
 
 1.0.-) Ill 1 
 
 .r,)2 
 
 1.0!)557 
 
 .24 
 
 1.14714 1 
 
 ,288 
 
 1 .20828 
 
 .330 
 
 1,2781 
 
 .M.-) 
 
 i.o,j.M(; 1 
 
 .19;{ 
 
 1.090.y4 
 
 .241 
 
 1.14S;!1 
 
 ,H«9 
 
 1,20907 
 
 XM 
 
 1 ,27904 
 
 .14G 
 
 l.orwoi 
 
 .194 
 
 1.09752 
 
 .242 
 
 1.14949 
 
 .29 
 
 1.21202 
 
 .3:48 
 
 1.2S11.S 
 
 .147 
 
 1.05GG7 
 
 —— — f 
 
 .195 
 
 1 .0985 
 
 .243 
 
 1.15007 
 
 .291 
 
 1.212;i9 
 
 .339 
 
 1 .28273 
 
32 
 
 LENGTHS OP CIRCULAR ARCS. 
 
 TABU:.—(Coi)tinucd.) 
 
 :m 
 .;m2 
 
 .315 
 
 .317 
 
 .3lcS 
 
 .311) 
 
 .35 
 
 .351 
 
 .352 
 
 .:i53 
 
 .354 
 
 .355 
 
 .35(i 
 
 .357 
 
 .35S 
 
 .359 
 
 .3t) 
 
 .ai;i 
 
 .3()2 
 .303 
 
 .3(;t 
 
 .305 
 
 .3(iii 
 
 .3(17 
 
 .308 
 
 .30'J 
 
 .37 
 
 .371 
 
 .372 
 
 Length. 
 
 1.28-128 
 1 .28583 
 1 .2^!73;) 
 1 .288<I5 
 1 .29052 
 1 .29209 
 1 .29300 
 1 .29o. i 
 1 .29081 
 1 .29839 
 1 .29997 
 1.30150 
 !.;{(»315 
 1.30-174 
 1 .30(i34 
 1 .30794 
 1 .30954 
 1.31115 
 1.31270 
 1.31437 
 1.31599 
 1.31701 
 1.31923 
 1 .32080 
 1.32249 
 1 
 
 32413 
 
 32577 
 
 32741 
 
 32905 
 
 330(p|) 
 
 1 .33234 
 
 1 .33399 
 
 1.33561 
 
 Hgl.t. 
 
 .373 
 .374 
 .375 
 ..i',0 
 77 
 ■■<78 
 .-,79 
 .38 
 .381 
 .382 
 .3s3 
 .38^ 
 .385 
 .380 
 .387 
 .388 
 .389 
 .39 
 .391 
 .392 
 .393 
 .394 
 .:'9o 
 .390 
 .397 
 .398 
 .399 
 A 
 
 .401 
 .402 
 .403 
 .404 
 .405 
 
 Length. Il'/nht. Length. 
 
 1 
 I.3|; 
 
 1 .3373 
 
 1 .:<:\>[h\ 
 1.3j()i;;! 
 
 !|229 
 
 :9(; 
 1 .;i-i5i;.'! 
 
 1.31731 
 1.31^99 
 1 .3500S 
 1 .352;;7 
 1. .35 1 00 
 1.35575 
 1.35744 
 1.359M 
 1 .30084 
 1 .30254 
 1.. 30. 1 25 
 1 .:!0590 
 1.3i;7(i7 
 1 .30'.I39 
 1.37111 
 1.37283 
 1 
 
 !7455 
 1 .37(;28 
 1 .37801 
 1 .37974 
 1.3S1 18 
 1 .38.322 
 1.3S490 
 1.38071 
 1 ..38840 
 1.39021 
 1.39190 
 
 .100 
 
 .107 
 
 .408 
 
 .409 
 
 .41 
 
 .411 
 
 .412 
 
 .413 
 
 .414 
 
 .415 
 
 .410 
 
 .417 
 
 .418 
 
 .4 1 9 
 
 .42 
 
 .421 
 
 .422 
 
 .423 
 
 .424 
 
 .425 
 
 .420 
 
 .427 
 
 .428 
 
 .429 
 
 .43 
 
 .131 
 
 .432 
 
 .433 
 
 .434 
 
 .4:15 
 
 .430 
 
 .437 
 
 .438 
 
 1 .39372 
 
 l..;i51S 
 
 1.3!)T24 
 
 l.;!9!t 
 
 1.40077 
 
 1.40251 
 
 1.40432 
 
 1 .400 
 
 1.4078,S 
 
 1.40!>0i; 
 
 1.411 15 
 
 1.41321 
 
 ■ .4 1 503 
 
 1.41082 
 
 1.41801 
 
 1.42011 
 
 1.42222 
 
 1.424(12 
 
 1.425s:; 
 
 1.12701 
 1.42!I42 
 1.43127 
 1.43309 
 1.43491 
 1 .43073 
 
 l.i3S5(; 
 
 1.4 4039 
 1.4 1222 
 1.44-115 
 1 .44589 
 1.4477.3 
 1.44!i57 
 1.45142 
 
 light. 
 
 Lc'n,y:th. 
 1 .15:127 
 
 H'ght. 
 .472 
 
 1 Length. 
 < 1.51571 
 
 .4.39 
 
 .41 
 
 1.4.1512 
 
 .473 
 
 1 .51 7(; 1 
 
 .III 
 
 1 .45,;97 : 
 
 .4 71 
 
 1.5! '.158 
 
 .112 
 
 1.45s,s;!, 
 
 .475 
 
 1.52152 
 
 .113 
 
 1.4(i()ii9 
 
 .470 
 
 1.52::40 
 
 .444 
 
 1.40255 
 
 .477 
 
 1.52511 
 
 .445 
 
 1.40141 
 
 .478 
 
 1 .527:!0 
 
 .440 
 
 1 .40028 1 
 
 .479 
 
 1.52931 
 
 .447 
 
 1.4(;.-^I5 
 
 .48 
 
 1.5:! 120 i 
 
 .4 18 
 
 1.470(121 
 
 .481 
 
 1 .5;!322 
 
 .4 19 
 
 l.47!:-9: 
 
 .482 
 
 1.5:1518 ' 
 
 .45 
 
 I.l7:i77' 
 
 .483 
 
 1.. 537 1 4 
 
 .451 
 
 1 475i;5 
 
 .484 
 
 1..5391 
 
 .452 
 
 1.47753 
 
 .485 
 
 1.54100 i 
 
 .453 
 
 1.47912 
 
 .480 
 
 1.54302 
 
 .454 
 
 1.4Sl:il 
 
 .487 
 
 1.51499 
 
 .455 
 
 1. 4.^^32 
 
 .488 
 
 1.54090 i 
 
 .150 
 
 1.4S50',l 
 
 .4.'-'9 
 
 l.5l>93 ' 
 
 .457 
 
 1 .48011!) 
 
 .49 
 
 1 .5509 
 
 .I5,S 
 
 1.4s,v89 
 
 .491 
 
 1.55288 
 
 .459 
 
 1.49(179 
 
 .492 
 
 1.55480 j 
 
 .40 
 
 1.49209 
 
 .493 
 
 1 .55085 
 
 .401 
 
 1.4910 
 
 .491 
 
 1 .55854 
 
 .402 
 
 1.4!)h51 
 
 .495 
 
 1 .50083 
 
 .403 
 
 1.I!I8|2| 
 
 .49i; 
 
 1 .50282 
 
 .404 
 
 1 .50():i:; 
 
 .497 
 
 1.50181 
 
 .405 
 
 1 5022 1 
 
 .498 
 
 1 .5008 
 
 .400 
 
 1.5(1110 
 
 .499 
 
 1 .50879 
 
 .-107 
 
 1 .50008 
 
 .5 
 
 1.57079 
 
 .408 
 
 1 .508 
 
 
 
 .409 
 
 1 .5(l!)!)2 
 
 
 
 .47 
 
 1.51185^ 
 
 
 
 .471 
 
 1.51378] 
 
 
 
 To Ascertain the Length of an Arc of a Circle by the preceding Tabic. 
 
 Rm," — Divide tha height, by Iho Ivt=c, ft h1 thf> quotient in th- folnmn oF hoi:ht.«. ,nnfl tnki the 
 length (if thit height from thn nu't riiihtiiaml ciihiinn. ,\liiiti|,ly the length thus ibtninoj hy the 
 base I f the nrc, nml the proiJiict will '^iv,> iho I 'ng;ii nf the inc. 
 
 ExAMPi.R — Whatis the length of an arc of a circle, the base or Fjian of it being 100 feet, and tho 
 height •J.'i feet? 
 25-=-](i0 T-- .25; and .25 per table, = M5912, t.'.f l.-m/th vt' the heme, vhich, hdix/ mnltqilial hi/ 100 = 
 
 NoTK.—Whon, in tho division of a hoig'if bv the b^so, tho quotient has a rcm.iinJor after the 
 third ilai'e if dooin !■<, an I frrent acciiritcy i.< refniToil. 
 
 Take the lengt'-. tir tlio rirt three t'liinre-, giii.tri'-t it, finin tho next filh.w'n',; length: mnltiply 
 the remainder bv the ,.nid fra^'UDna' reni;iinjcr, add tlie produ t to ll:e lir.-t length, and the s-uin will 
 bo the Inngh fi.r the wholu qnoient. 
 
 Exnu'f.if.— Wiiat is tho length of an arc of n cir. lo, tlic base of whicii in 35 feet, and tho heii'ht 
 or vorseJ Mto 8 fcot ? 
 
 8-f- 35 = .228)714; the taV^nlar I?ngth for .228 - I.13.;;n. nn 1 for .229 = 1.13144, the difTerenco 
 letwecn which 13.00113. Then .6714 x -00113 = .00(045032. 
 
 Henoa ' .2l!8 =: l.i:r;3.. 
 
 and .01.05714= .0()nri450,S2 
 
 }.V.V.n>r.Q52, tho sum by which tho base of 
 th« arv ii to ba muiUpUed; aad 1.13395i6S3 x 35 a 29.Q&Sib/ut. 
 
XAIJLI-: VII. 
 
 TABLE OF THE LENGTHS OF SEMI-ELLIPTIC ARCS. 
 
 The Transverse Diameter of an Ellipse assumed to be Unity, and divided into 1000 
 
 equal Parts. 
 
 nVht. 
 .1 
 
 Len!,'tli. ! 
 
 Il'glit. 
 .148 
 
 1.09119 
 
 Iiy.t. 
 
 LiMiKth. 
 
 IlVlit. 
 .244 
 
 Ll.Ml,l,'t]l. 
 
 H'Kht. 
 
 Lcnifth. 
 
 i. 01102 1 
 
 .190 
 
 1.14531 
 
 1 .2o:;8 
 
 .292 
 
 1.20601 
 
 .101 
 
 1.01202 i 
 
 .119 
 
 1 .09228 
 
 .197 
 
 1.14010 
 
 .215 
 
 1.20500 ! 
 
 .293 
 
 1 .26734 
 
 .102 
 
 i.oi3i;2 ! 
 
 .15 
 
 1.09.;3 
 
 .198 
 
 1.11702 
 
 .240 
 
 1.20632 
 
 .294 
 
 1 .26867 
 
 .io;{ 
 
 1.0 1102 ! 
 
 .151 
 
 1.0914- 
 
 .199 
 
 1.N8SM 
 
 .217 
 
 1 .20758 
 
 .295 
 
 1.27 
 
 .101 
 
 1.0!.5i;2 
 
 .152 
 
 1.09558 j 
 
 .2 
 
 1.15014 
 
 .218 
 
 1 20881 
 
 .296 
 
 1.27133 
 
 .105 
 
 1.01002 ' 
 
 .153 
 
 I.0!)009 i 
 
 .201 
 
 1.15131 
 
 .219 
 
 1.2101 
 
 .297 
 
 1 .27267 
 
 .IOC) 
 
 1.01702 : 
 
 i.l54 
 
 1.0978 
 
 .202 
 
 1.15218 
 
 .25 
 
 1.211.36 
 
 .298 
 
 1.27401 
 
 .107 
 
 1,()I.S02 i 
 
 . A 00 
 
 1.09891 
 
 .203 
 
 1.15300 
 
 .251 
 
 1.21203 1 
 
 .299 
 
 1 .27535 
 
 .10,S 
 
 1.011102 
 
 .15i; 
 
 1.10002 \ 
 
 .201 
 
 1.151-Sl 
 
 .252 
 
 1.2139 
 
 .3 
 
 1 .27009 
 
 .lOi) 
 
 1 .0.')003 
 
 , .157 
 
 1.10113 
 
 .205 
 
 1.15t;02 
 
 .253 
 
 1.21517 
 
 .301 
 
 1.27803 
 
 .11 
 
 1.05104 
 
 .158 
 
 1.10221 1 
 
 .206 
 
 1.1572 
 
 .254 
 
 1.21011 : 
 
 .302 
 
 1 ,27937 
 
 .III 
 
 1 .052(i5 
 
 ' .159 
 
 1.10335 ■ 
 
 .207 
 
 1.15-<38 
 
 .2,-)5 
 
 1.21772 , 
 
 .303 
 
 1.28071 
 
 .112 
 
 1 .05:i(;(; 
 
 .10 
 
 1.I0H7 
 
 .20^ 
 
 1.1 5:157 
 
 250 
 
 1.219 ! 
 
 .304 
 
 1 .28205 
 
 .113 
 
 1.05107 
 
 , .161 
 
 1 . 1 050 
 
 .209 
 
 1.10070 
 
 .257 
 
 1.22028 ; 
 
 .305 
 
 1.2S339 
 
 .114 
 
 l.()550S 1 
 
 ; .102 
 
 1.10072 1 
 
 .21 
 
 1.10190 
 
 .258 
 
 1.22150 1 
 
 .306 
 
 1.28174 
 
 .115 
 
 1. 051109 
 
 : .103 
 
 1.10781 ' 
 
 .211 
 
 1.10310 
 
 .259 
 
 1.222f^4 ; 
 
 .307 
 
 1 .28609 
 
 .IIG 
 
 1.0577 
 
 ! .164 
 
 I.IOSOO 1 
 
 .212 
 
 1.10130 
 
 .20 
 
 1.22 112 ^ 
 
 .308 
 
 1 .28744 
 
 .117 
 
 1.05872 
 
 ; .105 
 
 I.IIOOS 
 
 .213 
 
 1.10,557 
 
 .201 
 
 1.22511 
 
 .309 
 
 1.28879 
 
 .118 
 
 1.05974 
 
 .100 
 
 1.1112 ! 
 
 .214 
 
 1.1(;078 
 
 .202 
 
 1 .2207 
 
 .31 
 
 1.29014 
 
 .119 
 
 1 .0 1070 
 
 : .107 
 
 1.11232 1 
 
 .215 
 
 1.10799 
 
 .203 
 
 1.22799 : 
 
 .311 
 
 1.29149 
 
 .12 
 
 1.011178 
 
 .168 
 
 1.11314 i 
 
 .210 
 
 1.1092 
 
 .204 
 
 1.22928 
 
 .312 
 
 1 .29285 
 
 .121 
 
 1 .Oi;28 
 
 ' .169 
 
 1.11450 ' 
 
 .217 
 
 1.I70U 
 
 .265 
 
 1.23057 
 
 .313 
 
 1.29421 
 
 .122 
 
 1 .00382 
 
 ' .17 
 
 1.11509 1 
 
 .218 
 
 1.17103 
 
 .266 
 
 1.23186 ; 
 
 .314 
 
 1 ,29557 
 
 .12:5 
 
 1.00181 
 
 .171 
 
 1.110S2 
 
 .219 
 
 1, 172^^5 
 
 .207 
 
 1.23315 
 
 .315 
 
 1.29603 
 
 .124 
 
 1 .005SO 
 
 i -1^- 
 
 1.11795 '. 
 
 .22 
 
 1.17407 
 
 .208 
 
 1.23145 i 
 
 .316 
 
 1 ,29829 
 
 .125 
 
 l.()!;089 
 
 I .173 
 
 1.11908 ' 
 
 !221 
 
 1.17529 
 
 .209 
 
 1.23575 ' 
 
 .317 
 
 1,29965 
 
 .120 
 
 1.0i;792 
 
 I .174 
 
 1.12021 I 
 
 .222 
 
 1.17051 
 
 .27 
 
 1.23705 j 
 
 1 .318 
 
 1,30102 
 
 .127 
 
 1 .0(;8;)5 
 
 ; .1.5 
 
 1.12131 
 
 .223 
 
 1.17771 
 
 .271 
 
 1.238,35 
 
 .319 
 
 1 ,30239 
 
 .128 
 
 1 .01)998 
 
 .176 
 
 1.12247 ! 
 
 .221 
 
 1.17897 
 
 .272 
 
 1.23900 
 
 .32 
 
 1 .30376 
 
 .129 
 
 1.07001 
 
 i.l77 
 
 1.12.36 
 
 .225 
 
 1.1802 
 
 .273 
 
 1.21097 
 
 .321 
 
 1.. 30513 
 
 .13 
 
 1.07204 
 
 .178 
 
 1.12173 I 
 
 1 .226 
 
 1.18113 
 
 ,274 
 
 1.24228 
 
 .322 
 
 1 .3065 
 
 .131 
 
 1 .07308 
 
 .179 
 
 1.12580 i 
 
 1 .227 
 
 1.18200 
 
 .275 
 
 1 .24359 
 
 .323 
 
 1.30787 
 
 .1:52 
 
 1.07412 
 
 |.18 
 
 1.12099 ' 
 
 1 .228 
 
 1.1839 
 
 .270 
 
 1 .2448 
 
 .324 
 
 1.30924 
 
 .133 
 
 1.07510 
 
 .181 
 
 1.12S13 
 
 .229 
 
 1.18514 
 
 .277 
 
 1.21612 
 
 .325 
 
 1.31061 
 
 .134 
 
 1.07021 
 
 i .182 
 
 1.12927 
 
 .23 
 
 1.180.38 
 
 .278 
 
 1.21744 
 
 .326 
 
 1.31198 
 
 .135 
 
 1.07720 
 
 ! .183 
 
 1.130H 
 
 .231 
 
 1.18702 
 
 .279 
 
 1.24876 
 
 .327 
 
 1.31335 
 
 .13(5 
 
 1.07S31 
 
 .184 
 
 1.13155 
 
 i .232 
 
 1.18,S86 
 
 .28 
 
 1.2.J0i 
 
 .328 
 
 1.31472 
 
 .137 
 
 1.07937 
 
 i .185 
 
 1.13209 
 
 1 .233 
 
 1.1901 
 
 .281 
 
 1.25142 
 
 .329 
 
 1.3161 
 
 .138 
 
 1.0S0t3 
 
 .186 
 
 1 . 1 3:H3 
 
 i .231 
 
 1.191.34 
 
 .282 
 
 1 ,25274 
 
 .33 
 
 1. 3 1 748 
 
 .139 
 
 1 .08149 
 
 .187 
 
 1.13197 
 
 .235 
 
 1.192.58 
 
 .283 
 
 1,25106 
 
 .331 
 
 1.31886 
 
 .14 
 
 1 .08255 
 
 .188 
 
 1.13011 
 
 .236 
 
 1.19,382 
 
 .284 
 
 1 .25538 
 
 .332 
 
 1., 32024 
 
 .141 
 
 1 .08362 
 
 ' .189 
 
 1.13726 
 
 .237 
 
 1.19506 
 
 .285 
 
 1.2567 
 
 .333 
 
 1.32162 
 
 .142 
 
 1.08169 
 
 .19 
 
 1.I38U 
 
 .238 
 
 1.1963 
 
 .286 
 
 1 .25803 
 
 .334 
 
 1 .323 
 
 .143 
 
 1 .03576 
 
 .191 
 
 1.13956 
 
 .239 
 
 1.19755 
 
 .287 
 
 1 .25936 
 
 .335 
 
 1 .324.38 
 
 ,144 
 
 1.08684 
 
 .192 
 
 1.14071 
 
 .24 
 
 1.1988 
 
 .288 
 
 1 .26069 
 
 .330 
 
 1 .32576 
 
 .145 
 
 1.08792 
 
 .193 
 
 1.14186 
 
 .241 
 
 1.20005 
 
 .289 
 
 1 .20202 
 
 .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. 
 TXliLE.— (Continued.) 
 
 .84 
 
 .841 
 
 .34'2 
 
 .848 
 
 .844 
 
 .845 
 
 .34t> 
 
 .347 
 
 .348 
 
 .849 
 
 .3J 
 
 .851 
 
 .852 
 
 ,358 
 
 .854 
 
 .855 
 
 .35t) 
 
 .857 
 
 .858 
 
 .859 
 
 M 
 
 .861 
 
 .362 
 
 .868 
 
 .364 
 
 .365 
 
 .866 
 
 .367 
 
 .368 
 
 .369 
 
 .37 
 
 .371 
 
 .372 
 
 .373 
 
 .374 
 
 .375 
 
 .376 
 
 .377 
 
 .378 
 
 .379 
 
 .38 
 
 .881 
 
 .382 
 
 .388 
 
 .384 
 
 .385 
 
 .386 
 
 .387 
 
 .388 
 
 .389 
 
 .39 
 
 .391 
 
 .392 
 
 .393 
 
 .394 
 
 .395 
 
 L'Uirtl). 
 
 1.. 88 182 
 1.88272 
 1.88412 
 1 .88552 
 1 .88692 
 1 
 
 8.8888 
 88974 
 84115 
 84256 
 84897 
 1.84589 
 1.846H1 
 1.84828 
 1.. 84 965 
 1.85108 
 1.85251 
 1 .85.S94 
 1.85587 
 1.8568 
 1 .8.J828 
 1.85967 
 1.. 861 II 
 1.862.')5 
 1 .86899 
 1.8(;.)48 
 1 .86688 
 
 ,86888 
 86978 
 ,87128 
 ,87268 
 87414 
 1.87662 
 1.87708 
 1,87854 
 1.88 
 1.88146 
 1.88292 
 1 ,88489 
 1.88585 
 1 .88782 
 1.88879 
 1.89024 
 1.89169 
 1.89814 
 1 .89459 
 1.89605 
 1.89751 
 1 .89897 
 1 .40048 
 1.40189 
 1.40385 
 1.40481 
 1.40627 
 1.40778 
 1.40919 
 1.41065 
 
 IlKlit. 
 
 .896 
 
 .8!»7 
 
 .898 
 
 .899 
 
 .4 
 
 .401 
 
 .402 
 
 .408 
 
 .404 
 
 .405 
 
 .406 
 
 .407 
 
 .408 
 
 .409 
 
 .41 
 
 .411 
 
 .412 
 
 .418 
 
 .414 
 
 .415 
 
 .416 
 
 .417 
 
 .418 
 
 .419 
 
 .42 
 
 ,421 
 
 .422 
 
 .428 
 
 .424 
 
 ,425 
 
 .426 
 
 .427 
 
 .428 
 
 .429 
 
 .48 
 
 .481 
 
 .432 
 
 .488 
 
 .484 
 
 .485 
 
 .486 
 
 .487 
 
 .488 
 
 .439 
 
 .44 
 
 .441 
 
 .442 
 
 .448 
 
 ,444 
 
 .446 
 
 .446 
 
 .447 
 
 ,448 
 
 .449 
 
 .45 
 
 .451 
 
 LuiiKlh. 
 
 1.41211 
 1.4i;i57 
 1.41504 
 1.41651 
 1.41798 
 1.41915 
 1.42092 
 ' 42289 
 12886 
 .42588 
 1.426SI 
 1.42829 
 1.42977 
 1.48125 
 1.48278 
 1.48121 
 1 .48569 
 i. 487 18 
 1.48867 
 1. 14016 
 1.441(;5 
 1.44814 
 1.44 1 (;8 
 1,41618 
 1.44768 
 1.44918 
 1,45064 
 1,45214 
 1,45864 
 1.45515 
 1,45665 
 1.45815 
 1 .459(i6 
 1.46167 
 1.46268 
 1.46419 
 1.4657 
 1,46721 
 1 ,46872 
 1,47028 
 1.47174 
 1 .47826 
 1 .47478 
 1.4768 
 1.47782 
 1,47984 
 1 ,48086 
 1.48288 
 1.48891 
 1,48544 
 1.48697 
 1 .4885 
 1 .49003 
 1,49157 
 1.49811 
 1,49465 
 
 Hgbt. 
 
 ,452 
 
 .4,: 8 
 
 .151 
 
 .455 
 
 .456 
 
 .457 
 
 .458 
 
 .459 
 
 .46 
 
 .461 
 
 .462 
 
 .468 
 
 .464 
 
 .465 
 
 .46i; 
 
 .467 
 
 .41)8 
 
 .469 
 
 .47 
 
 .471 
 
 .4 
 
 .478 
 
 .474 
 
 .475 
 
 .476 
 
 ,477 
 
 .478 
 
 .479 
 
 ,48 
 
 ,481 
 
 ,482 
 
 .488 
 
 ,484 
 
 .485 
 
 .486 
 
 .487 
 
 .488 
 
 .489 
 
 .49 
 
 .491 
 
 .492 
 
 .498 
 
 .494 
 
 .495 
 
 .496 
 
 .497 
 
 ,498 
 
 ,499 
 
 .5 
 
 .501 
 
 .502 
 
 ,508 
 
 ,504 
 
 ,505 
 
 ,506 
 
 ,507 
 
 LPiiKlh. ;!irj.ht. Len>{t!i. | H'glit 
 
 111 
 
 1,19618 
 1,49771 
 1.49924 
 1.50077 
 
 5028 
 
 508,S8 
 
 50586 
 
 50689 
 
 50842 
 
 50996 
 
 1.5115 
 
 51. -101 
 
 51458 
 
 51612 
 
 51766 
 
 5192 
 
 52074 
 
 52229 
 
 1 .52884 
 
 1 .52589 
 
 > i 1.52691 
 
 1.52819 
 
 1 .5; 
 
 1 
 1 
 
 5;;ooi 
 
 .58159 
 )8814 
 )8l69 
 )8625 
 1.58781 
 1.58987 
 1..M098 
 1.54249 
 1.54405 
 1,5(561 
 1,54718 
 1 .54875 
 1 .55082 
 1.55189 
 1,55846 
 1,55508 
 1,5566 
 1,55817 
 1.55974 
 1.56181 
 1. 511289 
 1.56447 
 1.56605 
 1.56768 
 1,56921 
 1 .57089 
 1.57284 
 1.57889 
 1.57544 
 1 .57699 
 1.57854 
 1.58009 
 1.58164 
 
 .50'^ ; 
 
 .509 I 
 
 .51 I 
 
 .511 I 
 
 .512 I 
 
 .518 I 
 
 .514 I 
 
 .515 i 
 
 .516 
 
 .517 I 
 
 .518 , 
 
 .519 
 
 .52 
 
 .521 
 
 .522 
 
 .528 
 
 .524 
 
 .525 
 
 ,526 
 
 .527 
 
 ,528 
 
 .529 
 
 .5,8 
 
 ,5.81 
 
 .5.!2 
 
 .588 
 
 ,5,8 4 
 
 .585 
 
 .586 
 
 .587 
 
 .588 
 
 .589 
 
 .54 
 
 .541 
 
 .542 
 
 .518 
 
 .514 
 
 .545 
 
 .546 
 
 .547 
 
 .548 
 
 .549 
 
 .55 
 
 .551 
 
 ,552 
 
 .558 
 
 .554 
 
 .555 
 
 .556 
 
 .,557 
 
 .558 
 
 ,559 
 
 .56 
 
 .561 
 
 .562 
 
 .568 
 
 1.5'<819 
 
 1.5S471 
 
 1.58629 
 
 1 .58784 
 
 l.5s!).l 
 
 1 .59096 
 
 1.59252 
 
 I.5ii-I08 
 
 1.59564 
 
 1 .5:i72 
 
 1 .5;i876 
 
 1.6(1082 
 
 1.60188 
 
 1 .60844 
 
 1 .605 
 
 1 .6(1656 
 
 1.60812 
 
 1 .60968 
 
 1.61124 
 
 1.6128 
 
 1.61486 
 
 1.61592 
 
 1.61748 
 
 1.61904 
 
 1 .6206 
 
 1.62216 
 
 1.62872 
 
 1 .62528 
 
 1.62684 
 
 1,6284 
 
 1.62996 
 
 1.68! 52 
 
 1.68.i09 
 
 1.68465 
 
 l.68ii28 
 
 l.(i878 
 
 1 .68987 
 
 1.61094 
 
 1.64251 
 
 1.64408 
 
 1.64565 
 
 1.61722 
 
 1.64879 
 
 1.65086 
 
 1.65198 
 
 1.6585 
 
 1.65507 
 
 1.65665 
 
 1 .65828 
 
 1.65981 
 
 1.66189 
 
 1.66297 
 
 1.66455 
 
 1.66618 
 
 1.66771 
 
 1.66929 
 
 I . 
 
 .564 
 .565 
 .566 
 .567 
 .568 
 .569 
 .57 
 .571 
 .572 
 .578 
 .574 
 .575 
 .57() 
 .577 
 .578 
 579 
 58 
 .581 
 ,582 
 
 .584 
 .585 
 
 .587 
 
 .588 
 
 .5^9 
 
 .59 
 
 .591 
 
 .592 
 
 .59.8 
 
 .594 
 
 .595 
 
 .596 
 
 .597 
 
 .598 
 
 .599 
 
 .6 
 
 .601 
 
 .602 
 
 .608 
 
 .604 
 
 .605 
 
 .606 
 
 ,607 
 
 .608 
 
 .609 
 
 .61 
 
 ,611 
 
 .612 
 
 ,618 
 
 .614 
 
 .615 
 
 .616 
 
 .617 
 
 .618 
 
 .619 
 
 I -. 
 
 Length, 
 
 1 .67087 
 1.67245 
 1.67408 
 1.67561 
 1.67719 
 1.67877 
 1 .680.86 
 1.68195 
 1.68854 
 1.68518 
 1 68672 
 l.(;888l 
 1 .6899 
 1.69149 
 1 .69808 
 1 .69467 
 1 .69626 
 1.69785 
 1.69915 
 70105 
 70264 
 70424 
 70584 
 1.70745 
 1.70905 
 1.71065 
 1.71225 
 1.71286 
 1.71546 
 1.71707 
 1,71868 
 1.72029 
 1.7219 
 1.7285 
 1.72511 
 1.72672 
 1.72888 
 1.72994 
 1.78155 
 1.78816 
 1.78477 
 1.7;;688 
 1.78799 
 
 1.74121 
 1.74283 
 1.74444 
 1.74605 
 1.74767 
 1 .74929 
 1.75091 
 1.75252 
 1.75414 
 1,7.5576 
 1.75788 
 1.759 
 
LEXG1H8 OF SEMt-KLMPTIC ARCB. 
 TABI '\.— (Continued.) 
 
 85 
 
 HV'lit. 
 
 Lcn^'th. 
 1 .76062 
 
 H'Kht. 
 
 Li-iiKtli. 
 1.85215 
 
 ; ii'-iit 
 
 ii 
 .7;{2 
 
 I.,piijk'th. 
 
 HV'ht 
 
 LeiiKtli. 
 
 IllVht 
 .844 
 
 Length. 
 
 
 .62 
 
 t .676 
 
 1.94.552 
 
 .788 
 
 2.04117 
 
 2.13976 
 
 
 .G21 
 
 1 .7ii224 
 
 1 .677 
 
 1 .S5;i79 
 
 .7.!;! 
 
 1.94721 
 
 .789 
 
 2.0429 
 
 .845 
 
 2.141.55 
 
 
 .022 
 
 1.76;)86 
 
 ' .678 
 
 1.85544 
 
 .7;!4 
 
 1 1.9I>9 
 
 .79 
 
 2,04462 
 
 .84 (i 
 
 1 2.14.8:84 
 
 
 .62;! 
 
 1 .76548 
 
 ! .679 
 
 1. 857 09 
 
 .7;!5 
 
 i 1.95059 
 
 1 .791 
 
 2.046.85 
 
 .847 
 
 1 2.14513 
 
 
 .624 
 
 1.7671 
 
 i.68 
 
 1 .85874 
 
 .7;!6 
 
 ! 1.95228 
 
 .792 
 
 2.04809 
 
 .848 
 
 2.14692 
 
 
 .625 
 
 1 .76872 
 
 i .681 
 
 1 .8(;0.'!9 
 
 .7;i7 
 
 ! 1.95;!97 
 
 .798 
 
 2.01983 
 
 .849 
 
 2.14871 
 
 
 .62(1 
 
 1.77084 
 
 .6-12 
 
 1 .86205 
 
 .7:{8 
 
 j 1 .95566 
 
 .794 
 
 2.05157 
 
 .85 
 
 2.1505 
 
 
 .627 
 
 1.771i»7 
 
 i .6s;: 
 
 1 .86;i7 
 
 .7;i9 
 
 1 1.957.85 
 
 : .795 
 
 2.05881 
 
 .851 
 
 2 15229 
 
 
 .628 
 
 1 .T7;!;V.) 
 
 ' .684 
 
 1 .8(;5;!5 
 
 .74 
 
 1 1.95994 
 
 .796 
 
 2.05505 
 
 .852 
 
 2.15409 
 
 
 .629 
 
 1.77521 
 
 .685 
 
 1 .867 
 
 .741 
 
 1.96074 
 
 1 .797 
 
 2.05679 
 
 .853 
 
 2.15,589 
 
 
 .6H 
 
 1 .77684 
 
 , .686 
 
 1 .86866 
 
 .742 
 
 1.96244 
 
 : .798 
 
 2.058,53 
 
 .854 
 
 2.1577 
 
 
 .6^1 
 
 1.77S47 
 
 I .687 
 
 1.870;!1 
 
 .74;! 
 
 1.96414 
 
 .799 
 
 2.06027 
 
 ' .855 
 
 2.1595 
 
 
 A-^2 
 
 1.7!>00!) 
 
 ' .688 
 
 1.87196 
 
 .744 
 
 1 .9658:8 
 
 :.8 
 
 2.06202 
 
 ; .856 
 
 2.1618 
 
 
 .6h;^ 
 
 1.78172 
 
 ' .689 
 
 1 .-^7;i62 
 
 .745 
 
 1.9675:8 
 
 ' .801 
 
 2.06377 
 
 .857 
 
 2.16309 
 
 
 .6;u 
 
 1 .7.><;w5 
 
 .69 
 
 1 .87527 
 
 .746 
 
 1 .9692:! 
 
 ' .802 
 
 2. 06 ,"1 5 2 
 
 , .858 
 
 2.16489 
 
 
 .635 
 
 1.784!)8 
 
 .691 
 
 1.87i;!t;! 
 
 .747 
 
 1.9709;^ 
 
 : .808 
 
 2.06727 
 
 ' .859 
 
 2.16668 
 
 
 .6:!6 
 
 ] .7866 
 
 .692 
 
 1 .S7859 
 
 .748 
 
 1.97262 
 
 i .804 
 
 2,(16901 
 
 ; .86 
 
 2.16848 
 
 
 .6.S7 
 
 1 .7882;! 
 
 .69:! 
 
 1 .88024 
 
 .749 
 
 1.974:12 
 
 \ .805 
 
 2.07076 
 
 .861 
 
 2.17028 
 
 
 .6158 
 
 1.78<)86 
 
 .694 
 
 1.8819 
 
 .75 
 
 1 .97602 
 
 ' .806 
 
 2.07251 
 
 .S62 
 
 2.17209 
 
 
 .6;{'J 
 
 1.7'J14!» 
 
 .695 
 
 1 .88;!5i; 
 
 .751 
 
 1.97772 
 
 ! .807 
 
 2.07427 
 
 I .868 
 
 2.17:^89 
 
 
 .6-1 
 
 1.7'J;!I2 
 
 .(;96 
 
 1 .88522 
 
 ' .752 
 
 1.97948 
 
 i .808 
 
 2.07602 
 
 .864 
 
 2.1757 
 
 
 .611 
 
 1 .7'J475 
 
 .697 
 
 1 .886,^^8 
 
 .75;! 
 
 1.981 1;8 
 
 1 .809 
 
 2.07777 
 
 .865 
 
 2.17751 
 
 
 .642 
 
 1.796;i8 
 
 : .698 
 
 1.88S54 
 
 .754 
 
 1 .98288 
 
 .81 
 
 2.07958 
 
 .866 
 
 2.179:12 
 
 
 .64:i 
 
 1.79801 
 
 .699 
 
 1 .8902 
 
 .755 
 
 1.98458 
 
 1 .811 
 
 2.081 -JS 1 
 
 ; .867 
 
 2.18113 
 
 
 .644 
 
 1.79;)()4 
 
 .7 
 
 1.S9186 
 
 : .756 
 
 1 .98628 
 
 ' .812 
 
 2.08804 , 
 
 ' .868 
 
 2.18294 
 
 
 .645 
 
 1.80127 
 
 1 .701 
 
 1 .^9;!52 
 
 ! .757 
 
 1 .98794 
 
 .818 
 
 2.0H48 i 
 
 .869 
 
 2,18475 
 
 
 .646 
 
 1 .8029 
 
 1 .702 
 
 1.89519 
 
 .758 
 
 1 .98964 
 
 .814 
 
 2.08656 ! 
 
 .87 
 
 2,18656 
 
 
 .647 
 
 1 .80454 
 
 . .To;! 
 
 1 .89685 
 
 .759 
 
 1.99i;!4 
 
 .815 
 
 2.0H882 
 
 .871 
 
 2.18837 
 
 
 .648 
 
 1.80617 
 
 i .704 
 
 1.89,<51 
 
 '.76 
 
 1 .99805 
 
 .816 
 
 2.09008 
 
 .872 
 
 2.19018 
 
 
 .641) 
 
 1 .8078 
 
 .705 
 
 1.90017 . 
 
 .761 
 
 1.99476 
 
 .817 
 
 2.09198 
 
 .873 
 
 2.192 
 
 
 .65 
 
 1.8094;! 
 
 .706 
 
 1.90184 
 
 .762 
 
 1.99647 
 
 .818 
 
 2.0986 
 
 .874 
 
 2.19:^82 
 
 
 .651 
 
 1.81107 
 
 ; .707 
 
 1 .90;!5 1 
 
 1 .76;! 
 
 1.99S18 
 
 .8!9 
 
 2.095:W 
 
 .875 
 
 2.19,564 
 
 
 .652 
 
 1.81271 
 
 1 .708 
 
 1,90517 i 
 
 ; .764 
 
 1.99989 
 
 .82 
 
 2.09712 
 
 .876 
 
 2.19746 
 
 
 .65;{ 
 
 1.81 4 ;!5 
 
 .709 
 
 1.90684 i 
 
 ■ .765 
 
 2.0016 
 
 .821 
 
 2.09888 
 
 .877 
 
 2.19928 
 
 
 .654 
 
 1.81599 
 
 .71 
 
 1 9^)852 1 
 
 I .766 
 
 2.00881 
 
 .822 
 
 2.10065 ; 
 
 .878 
 
 2.2011 
 
 
 .655 
 
 1.8176:! 
 
 .711 
 
 1.91019 ' 
 
 ' .767 
 
 2.00V • 
 
 .823 
 
 2.10242 ■ 
 
 .879 
 
 2.20292 
 
 
 .656 
 
 1.81928 
 
 .712 
 
 l.;ai87 ! 
 
 ' 768 
 
 2.00 
 
 .824 
 
 2.10419 : 
 
 .88 
 
 2.20474 
 
 
 .657 
 
 1.82091 
 
 .71;^ 
 
 1.9l;!55 
 
 ■ .769 
 
 2.00.S44 
 
 .825 
 
 2.10596 , 
 
 .881 
 
 2.20656 
 
 
 .658 
 
 1 .82255 
 
 .714 
 
 1.9152;! 
 
 .77 
 
 2.01016 
 
 .826 
 
 2.10773 ; 
 
 .882 
 
 2.20889 
 
 
 .651) 
 
 1.82419 
 
 .715 
 
 1.91691 
 
 .771 
 
 2.01187 
 
 .827 
 
 2.1095 : 
 
 .883 
 
 2.21022 
 
 
 .66 
 
 1 .8258;^ 
 
 .716 
 
 1.91859 
 
 .772 
 
 2.01:859 
 
 .828 
 
 2.11127 
 
 .884 
 
 2.21205 
 
 
 .661 
 
 1.82747 
 
 .717 
 
 1.92027 ! 
 
 .77.8 
 
 2.01581 
 
 .829 
 
 2.11304 '• 
 
 .885 
 
 2.2i;888 
 
 
 .662 
 
 1.829U 
 
 .718 
 
 1.92195 j 
 
 .774 
 
 2.01702 
 
 .88 
 
 2.11481 i 
 
 .886 
 
 2.21571 
 
 
 .66;{ 
 
 1 .8:1075 
 
 .719 
 
 1.92;i6;! 
 
 .775 
 
 2.01874 
 
 .881 
 
 2.116.59 
 
 .887 
 
 2.21754 
 
 
 .664 
 
 1 .8:!24 
 
 .72 
 
 i.925;!i : 
 
 .776 
 
 2.02045 
 
 .882 
 
 2.11887 ; 
 
 .888 
 
 2.21937 
 
 
 .665 
 
 1.8:}404 
 
 .721 
 
 1.927 : 
 
 .777 
 
 2.02217 
 
 .838 
 
 2.12015 ; 
 
 .889 
 
 2.2212 
 
 
 .666 
 
 1 .8:>568 i 
 
 .722 
 
 1.92868 : 
 
 .778 
 
 2.02.889 
 
 .884 
 
 2.12198 1 
 
 .89 
 
 2.22:!03 
 
 
 .667 
 
 i.8:!7;i;{ i 
 
 .72;! 
 
 1.9;!0;i6 
 
 .779 
 
 2.02561 i 
 
 .885 
 
 2.12871 i 
 
 .891 
 
 2.22486 
 
 
 .668 
 
 1.8:!SU7 ] 
 
 .724 
 
 1.9;i204 ' 
 
 .78 
 
 2.027:!8 ; 
 
 .886 
 
 2.12549 \ 
 
 .892 
 
 2.2267 
 
 
 .669 
 
 1.84061 1 
 
 .725 
 
 i.9;i;!7;! ' 
 
 .781 
 
 2.02907 1 
 
 .837 
 
 2.12121 1 
 
 .893 
 
 2.22854 
 
 
 .67 
 
 1.84226 
 
 .726 
 
 1.9;^541 
 
 .782 
 
 2.0:i08 1 
 
 .888 
 
 2.12905 ; 
 
 .894 
 
 2.280,88 
 
 
 .671 
 
 1.84;rJl 
 
 .727 
 
 1.9;!71 
 
 .78:8 
 
 2.08252 1 
 
 .839 
 
 2.i:'r/83 i 
 
 .395 
 
 2.2:8222 
 
 
 .672 
 
 1.84556 
 
 .728 
 
 1 .9;!878 
 
 .784 
 
 2.08425 
 
 .84 
 
 2.18261 
 
 .896 
 
 2.28406 
 
 
 .67H 
 
 1.8472 
 
 .7'<;9 
 
 1.94046 
 
 .785 
 
 2.0:8598 
 
 .841 
 
 2.13439 
 
 .897 
 
 2.2:859 
 
 
 .674 
 
 1 .84885 
 
 ,7;i 
 
 1.94215 1 
 
 .786 
 
 2.08771 
 
 .842 
 
 2.13618 
 
 .898 
 
 2.23774 
 
 
 .675 
 
 1 .8505 
 
 .7.^1 
 
 1.94;i83 1 
 
 .787 
 
 2.08944 
 
 .843 
 
 2.13797 
 
 .899 
 
 2.23958 
 
 
36 
 
 LENUl'HS OV SKMl-F,L!,lPriC ARCS. 
 
 TABLE.— (Cuntimied.) 
 
 
 Len)j;th. 
 
 .021 
 
 Iii'n>,'th. 
 
 HVht. 
 .912 
 
 LciiKtli. 
 2.31852 ■ 
 
 U-Kl.t. 
 
 Loiijith. 
 2.3581 
 
 iH'gl.t. 
 i .981 
 
 Lenjfth. 1 
 2.39H'?:5 1 
 
 2.2(112 
 
 2.27987 
 
 .9(1:5 
 
 .!)01 
 
 2.2i;!25 
 
 .922 
 
 2.2817 
 
 .913 
 
 2.:i20:5S 1 
 
 .9(54 
 
 2.311 
 
 1 .985 
 
 2.4001(5 i 
 
 .<J02 
 
 2.24508 
 
 ,92:5 
 
 2.2s;{5| 
 
 .941 
 
 2.32224 
 
 1 .9(55 
 
 2.:5t;i91 
 
 ■ .98(1 
 
 2.40203 
 
 .903 
 
 2. 2-1 () 9 1 
 
 .924 
 
 2.28,-)37 
 
 .915 
 
 2.32111 
 
 .9(5(5 
 
 2. .3(5.381 
 
 ■ .987 
 
 2.401 
 
 .904 
 
 2.2-1874 
 
 .925 
 
 2.2S72 
 
 1 .94(1 
 
 2.32598 
 
 .9(57 
 
 2..3(;571 
 
 ; .9^8 
 
 2.40592 
 
 .905 
 
 2.25057 
 
 .92(5 
 
 2.2.^903 
 
 I .917 
 
 2.327.^^5 
 
 ' .9(58 
 
 2.:!(i7(12 
 
 .9S9 
 
 2.40781 
 
 .90(5 
 
 2.2524 
 
 .927 
 
 2.290.SG 
 
 .9-18 
 
 2.;i2972 
 
 ; .9ii9 
 
 2.:5(;952 
 
 .99 
 
 2.IO'.l7i5 
 
 .907 
 
 2.2542.S 
 
 .928 
 
 2.2927 
 
 I 949 
 
 2.331(5 
 
 i .97 
 
 2.:57143 
 
 .991 
 
 2.411(59 
 
 .908 
 
 2.25r,0(; 1 
 
 .929 
 
 2.29153 
 
 1 .95 
 
 2.:i3;i-18 
 
 ! .971 
 
 2.37.331 
 
 .992 
 
 2.41.3(12 
 
 .909 
 
 2.25789 ' 
 
 .9:5 
 
 2.29();;G 
 
 .951 
 
 2.33537 
 
 .972 
 
 2.37525 
 
 .9!»3 
 
 2.115.5(5 
 
 .91 
 
 2.25972 . 
 
 .9:51 
 
 2.29S2 
 
 .952 
 
 2.:!372(5 
 
 .973 
 
 2.3771(5 
 
 .994 
 
 2.11749 
 
 .911 
 
 2.2(il55 
 
 .9:52 
 
 2.:;oooi 
 
 .953 
 
 2.:5;!;)15 
 
 1 .974 
 
 2.37908 
 
 .9!I5 
 
 2.41913 
 
 .912 
 
 2.2(i;;:i8 ■ 
 
 .9.5:^ 
 
 2.:;oi>8 
 
 ; .954 
 
 2.3)104 
 
 .975 
 
 2. .381 
 
 .99(1 
 
 2.421.3r) 
 
 .91 ;{ 
 
 2.2(;521 
 
 .'XM 
 
 2.:!03:3 
 
 j .955 
 
 2.31293 
 
 .97(5 
 
 2. .38291 
 
 .997 
 
 2.42329 
 
 .914 
 
 2.2tl704 
 
 .935 
 
 2.30557 
 
 ■ .95(1 
 
 2..344S3 
 
 ; .977 
 
 2.:i8l82 
 
 .9!t8 
 
 2.42522 
 
 .915 
 
 2.2(1888 
 
 .9.S() 
 
 2.30711 
 
 1 .957 
 
 2. .34073 
 
 , .978 
 
 2..3.si;73 
 
 .999 
 
 2.42715 
 
 .91(5 
 
 2.27071 
 
 .937 
 
 2.3092(5 
 
 i .958 
 
 2.:{48(;2 
 
 ' .979 
 
 2.:is8(;4 
 
 1. 
 
 2.42908 
 
 .917 
 
 2.27254 
 
 .938 
 
 2.31111 
 
 .959 
 
 2. .3505 1 
 
 j.98 
 
 2.39055 
 
 
 
 .918 
 
 2.274:57 
 
 .939 
 
 2.31295 
 
 .9(5 
 
 2. .352 II 
 
 .981 
 
 2.39217 
 
 i 
 
 
 .919 
 
 2.27(52 
 
 .94 
 
 2.31479 
 
 .9(11 
 
 2.:i5431 
 
 1 .9H2 
 
 2.39139 
 
 ; 
 
 
 .92 
 
 2.2780;-5 i 
 
 .941 
 
 2.31 (5G() 
 
 .9(52 
 
 2.:^5(;21 
 
 i .983 
 
 2.39(531 
 
 1 
 
 
 To Ascertain the Length of a Semi-Ellipllc Arc (right Scrai-L llipse) 
 by the preceding Tabic. 
 
 RuLF. — Divide the height by tho hn?e, find the quotient in the cnlunin of hpi;'hts, nnl tnke the 
 length of thnt heij;ht from the ne.xt ri!;htli!ind colniiin. iMiilii|iiy the length thus obtuinod by the 
 bate of iho ar^-, and tho product will be tlie lunj^th of the nrc. 
 
 Lxamplf;.— What is the length tf tho arc if a aeini-ellipse, the ba.-:o being 70 feet, and tho 
 height 30.10 feet. 
 
 30.10-^-70 = .43; and .43 per tablo, = 1.462fi8. 
 Then 1.46268 x 70=::1U2.3376 feel. 
 
 Wlien tlie Curve is not that of a Eight Semi-Ellipse, the Height being half 
 of the Transverse Diameter. 
 
 Buf.E. — rSvido half the bnso by twice tho beii^ht, then proceed ns in the preceding esample ; 
 multiply tho tabular 1 njrth by twii-o tho height, .-iinl thu jiro mot vlll I e the loiisth lequirel 
 
 hUMi'i.K. — What is tho length of thu uro of a souii-ollipse, the height being 3j fuut, and tho 
 base 60 fuct? 
 
 eO-i-Szz.^O, and .30- r-:i5 x ^ = ,428, tho tabular length of which is 1.45966. 
 
 Then 1.45966 x 35 x :i= 102.1 7fi■2/«c^ 
 NoTK. — 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. 
 
TuVliT^TZ VIIT. 
 
 TABLE OF THE AREAS OF THE SEGMENTS OF A CIRCLE. 
 
 I'he Diameter of a Circle assumed to he Unity, and divided into lOOO equal I'arts. 
 
 VtsoJ 
 
 
 Vor.-cd 
 
 
 jWrpeil 
 
 1 
 
 i Versed 
 
 '. . 
 
 Vor?cd 
 
 
 Sine, 
 
 ,Se;^ Aroa. 
 
 SUie. 
 .018 
 
 Scg. Aron. 
 
 ■ 095 
 
 Srg. Area. 
 
 1 Sine. 
 
 .Sog Area. 
 
 C-ine. 
 
 Scg Area. 
 
 .001 
 
 .00004 
 
 .01882 
 
 ■ 0879 
 
 .142 
 
 .06822 
 
 .189 
 
 •10312 
 
 .002 
 
 .0(1012 
 
 .049 
 
 .01425 
 
 ■090 
 
 .0:!849 
 
 .1.18 
 
 .00.S92 
 
 ! .19 
 
 • 1089 
 
 .00;! 
 
 .00022 
 
 .05 
 
 .OlIOM 
 
 ■ 097 
 
 .08908 
 
 .144 
 
 .009ti2 
 
 i .191 
 
 ■ 10468 
 
 .00 1 
 
 .00081 
 
 .051 
 
 .((151 2 
 
 ■ 098 
 
 .0;i908 
 
 .145 
 
 .07088 
 
 .192 
 
 • 10547 
 
 .005 
 
 .00047 
 
 .052 
 
 .015.50 
 
 ■ 099 
 
 .04027 
 
 .146 
 
 .07108 
 
 ,198 
 
 • 10626 
 
 .OOC) 
 
 .0001)2 
 
 1 .058 
 
 .01001 
 
 • 1 
 
 .04087 
 
 .M7 
 
 .07174 
 
 .194 
 
 .10705 
 
 .007 
 
 .OOOTS 
 
 .05 1 
 
 .01040 
 
 ■ 101 
 
 .04148 
 
 .1(8 
 
 .07245 
 
 .195 
 
 .10784 
 
 .008 
 
 .0001)5 
 
 .0."i5 
 
 .01091 
 
 i .102 
 
 .O420H 
 
 .149 
 
 .07816 
 
 .190 
 
 .10864 
 
 .009 
 
 .00118 
 
 .0."0 
 
 .01787 
 
 ■ 108 
 
 .01209 
 
 1 .15 
 
 .07887 
 
 .197 
 
 .10943 
 
 .01 
 
 .001.88 
 
 .057 
 
 .01788 
 
 ■ 104 
 
 ■0481 
 
 .151 
 
 .07459 
 
 i .198 
 
 .11023 
 
 .011 
 
 .00158 
 
 .058 
 
 .01 S8 
 
 ■ 105 
 
 ■ 04891 
 
 .152 
 
 .07581 
 
 1 .199 
 
 .11102 
 
 .012 
 
 .00175 
 
 .059 
 
 .01877 
 
 ■ 100 
 
 ■04452 
 
 .1.58 
 
 .07008 
 
 .2 
 
 .11182 
 
 .01;? 
 
 .00197 
 
 .00 
 
 .01924 
 
 ■ 107 
 
 ■ 04514 
 
 .154 
 
 .07075 
 
 .201 
 
 .11262 
 
 .014 
 
 .0022 
 
 .001 
 
 .01972 
 
 ■ 108 
 
 .04575 
 
 .155 
 
 .07747 
 
 .202 
 
 .11343 
 
 .OliJ 
 
 .00244 
 
 .002 
 
 .0202 
 
 .109 
 
 .040.88 
 
 .156 
 
 .0782 
 
 .208 
 
 .11423 
 
 .OIG 
 
 .002(;8 
 
 .008 
 
 .02008 
 
 ■ 11 
 
 .047 
 
 .157 
 
 .07892 
 
 .201 
 
 .11503 
 
 .017 
 
 .002!,'4 
 
 .004 
 
 .021)7 
 
 ■ 111 
 
 •04708 
 
 .158 
 
 .07965 
 
 .205 
 
 .11584 
 
 .OlS 
 
 .0082 
 
 .0(;5 
 
 .02105 
 
 ■ 112 
 
 .04S26 
 
 .159 
 
 .08088 
 
 .206 
 
 .11665 
 
 .019 
 
 .00817 
 
 .OtiO 
 
 .02215 
 
 ■ 118 
 
 .04889 
 
 .16 
 
 .08111 
 
 .207 
 
 .11746 
 
 .02 
 
 .00875 
 
 .007 
 
 .02205 
 
 ■ 114 
 
 .04958 
 
 .161 
 
 .08185 
 
 .208 
 
 .11827 
 
 .021 
 
 .00408 
 
 ! .008 
 
 .02815 
 
 .115 
 
 .05016 
 
 .162 
 
 .08258 
 
 .209 
 
 .11908 
 
 .022 
 
 .00482 
 
 .009 
 
 .02880 
 
 ■ 110 
 
 .0508 
 
 .108 
 
 .08882 
 
 .21 
 
 .1199 
 
 .02:5 
 
 .001 02 
 
 .07 
 
 .02417 
 
 ■ 117 
 
 .05145 
 
 .104 
 
 .08406 
 
 .211 
 
 .12071 
 
 .024 
 
 .00492 
 
 .071 
 
 .0240S 
 
 .118 
 
 .05209 
 
 .105 
 
 .0848 
 
 .212 
 
 .12153 
 
 .025 
 
 .00528 
 
 .072 
 
 .02519 
 
 ■ 119 
 
 .05274 
 
 .100 
 
 .08554 
 
 .218 
 
 .12285 
 
 .02(i 
 
 .00555 
 
 .078 
 
 .i)l'571 
 
 ■ 12 
 
 .058;;8 
 
 .107 
 
 .08029 
 
 .214 
 
 .12817 
 
 .027 
 
 .005.S7 
 
 .074 
 
 .02024 
 
 ■ 121 
 
 .05404 
 
 .108 
 
 .08704 
 
 .215 
 
 .12899 
 
 .028 
 
 .00019 
 
 .075 
 
 .02(i70 
 
 ■ 122 
 
 .05409 
 
 .169 
 
 .08779 
 
 .216 
 
 .12481 
 
 .029 
 
 .00058 
 
 .070 
 
 .02729 
 
 ■ 128 
 
 .05584 
 
 .17 
 
 ,08858 
 
 .217 
 
 .12563 
 
 .08 
 
 .OOO.sO 
 
 .077 
 
 .02782 
 
 ■ 124 
 
 .050 
 
 .171 
 
 .08929 
 
 .218 
 
 .12646 
 
 .o;5i 
 
 .00721 
 
 .078 
 
 .02885 
 
 ■ 125 
 
 .05666 
 
 .172 
 
 .09004 
 
 .219 
 
 .12728 
 
 .0:52 
 
 .00750 
 
 .079 
 
 .02889 
 
 ■ 120 
 
 .057.88 
 
 .178 
 
 .0908 
 
 .22 
 
 .12811 
 
 .08;} 
 
 .00791 
 
 .08 
 
 .02948 
 
 ■ 127 
 
 .05799 
 
 .174 
 
 .09155 
 
 .221 
 
 .12894 
 
 .o;]4 
 
 .00S27 
 
 .0,^1 
 
 .02997 
 
 ■ 128 
 
 .05806 
 
 .175 
 
 .09281 
 
 .222 
 
 .12977 
 
 .o;}.T 
 
 .00X04 
 
 .082 
 
 .08052 
 
 ■ 129 
 
 .05983 
 
 .176 
 
 .09807 
 
 .228 
 
 .1306 
 
 .080 
 
 .00901 
 
 .OcS8 
 
 .08107 
 
 .18 
 
 .00 
 
 .177 
 
 .09884 
 
 .224 
 
 .13144 
 
 .087 
 
 .009.88 
 
 .0H4 
 
 .08102 
 
 .131 
 
 .06007 
 
 .178 
 
 .0946 
 
 .225 
 
 .13227 
 
 .088 
 
 .00970 
 
 .085 
 
 .08218 
 
 .182 
 
 .00185 
 
 .179 
 
 .09587 
 
 .228 
 
 .13311 
 
 .089 
 
 .01015 
 
 .080 
 
 .08274 
 
 .188 
 
 .00208 
 
 .18 
 
 .09018 
 
 .227 
 
 .13394 
 
 .04 
 
 .01054 
 
 .087 
 
 .0888 
 
 .1.84 
 
 .00271 
 
 .181 
 
 .0909 
 
 .228 
 
 .13478 
 
 .041 
 
 .01098 
 
 .088 
 
 .088H7 
 
 .185 
 
 .00889 
 
 .182 
 
 .09707 
 
 .229 
 
 .1.3662 
 
 .042 
 
 .01188 
 
 .089 
 
 .08444 
 
 .180 
 
 .00407 
 
 .188 
 
 .09845 
 
 .23 
 
 .1.3646 
 
 .048 
 
 .01178 
 
 .09 
 
 .08501 
 
 .187 
 
 .06476 
 
 .184 
 
 .09922 
 
 .231 
 
 .13731 
 
 .044 
 
 .01214 
 
 .091 
 
 .08558 
 
 .1,88 
 
 .00545 
 
 .185 
 
 .1 
 
 .232 
 
 .13815 
 
 .045 
 
 .01255 
 
 .092 
 
 .08010 
 
 .189 
 
 .00614 
 
 .186 
 
 .10077 
 
 .233 
 
 .139 
 
 .046 
 
 .012;>7 
 
 .098 
 
 .08074 
 
 .14 
 
 .00083 
 
 .187 
 
 .10155 
 
 .2.34 
 
 .13984 
 
 .047 
 
 .01889 
 
 .094 
 
 .08782 
 
 .141 
 
 00768 
 
 .188 
 
 .10283 
 
 .235 
 
 .14069 
 
88 
 
 AREAS OF THE 8F.GMENT8 OP A CIRCL. 
 
 TAhLE.— (Continued.) 
 
 - 
 
 Versed 
 
 ~ 
 
 Versed 
 
 . 
 
 Versed 
 
 
 Vcrtted 
 
 ■ ■ ■ 
 
 Verred 
 
 
 
 Siiio, 
 
 Sag. A It'll. 
 
 Sine. 
 
 Sen. Aroa_ 
 
 :.SiDe. 
 .342 
 
 8eg. Area. 
 
 Sine. 
 .395 
 
 Seg. Aroa. 
 
 Sine. 
 
 Seg. Aroii. 
 
 
 .230 
 
 .14154 
 
 .289 
 
 .18814 
 
 .2;!737 
 
 .2SSI8 
 
 , .4 18 
 
 .34079 
 
 
 .237 
 
 .14239 1 
 
 .29 
 
 .18905 
 
 .343 
 
 .2.".832 
 
 .390 
 
 .28915 
 
 ,449 
 
 ..34179 
 
 
 .238 
 
 .14324 1 
 
 .291 
 
 .18995 
 
 .34 1 
 
 ,2;;927 
 
 .397 
 
 .29043 
 
 1 .45 
 
 ..34278 
 
 
 .239 
 
 .14409 
 
 .292 
 
 .19086 
 
 .345 
 
 .21022 
 
 .;J98 
 
 .29141 
 
 .451 
 
 .34378 
 
 
 .24 
 
 .14494 
 
 .293 
 
 .19177 
 
 ..".1 6 
 
 .24117 
 
 ,399 
 
 .292;:9 
 
 .4.52 
 
 .34477 
 
 
 .241 
 
 .1458 
 
 .294 
 
 .19208 
 
 .3)7 
 
 .21212 
 
 -4 
 
 .29;;37 
 
 .453 
 
 .34557 
 
 
 .242 
 
 .14665 
 
 .295 
 
 .19.30 
 
 .348 
 
 .243(17 
 
 .401 
 
 .29435 
 
 .454 
 
 .34676 
 
 
 .243 
 
 .14751 
 
 .296 
 
 .19151 
 
 ,349 
 
 .24103 
 
 ,io-j 
 
 .29.".33 
 
 .455 
 
 .34776 
 
 
 .244 
 
 .14837 
 
 .297 
 
 .19542 
 
 35 
 
 ,24498 
 
 .403 
 
 .290:;i 
 
 I .450 
 
 .34875 
 
 
 .245 
 
 .14923 
 
 .298 
 
 .19034 
 
 .351 
 
 .21593 
 
 .404 
 
 .29729 
 
 .457 
 
 .34975 
 
 
 .24(1 
 
 .150O'» 
 
 .299 
 
 .19725 
 
 .352 
 
 .2-10.S9 
 
 .405 
 
 .29827 
 
 .458 
 
 .35075 
 
 
 .247 
 
 .15090 
 
 .3 
 
 .19817 
 
 ,.353 
 
 .24784 
 
 .40(i 
 
 .29925 
 
 ,459 
 
 .35174 
 
 
 .248 
 
 .15182 
 
 .301 
 
 .19908 
 
 .354 
 
 .2488 
 
 .407 
 
 .30024 
 
 ,40 
 
 .35274 
 
 
 .249 
 
 .15268 
 
 .302 
 
 .2 
 
 ..355 
 
 .21970 
 
 .408 
 
 .30122 
 
 .401 
 
 .35374 
 
 
 .25 
 
 .15355 1 
 
 .303 
 
 .20092 
 
 .350 
 
 .25071 
 
 .409 
 
 .3022 1 
 
 .402 
 
 .35474 
 
 
 .251 
 
 .15441 1 
 
 .304 
 
 .201S4 
 
 ,357 
 
 .25107 
 
 .41 
 
 .30319 
 
 .403 
 
 .35573 
 
 
 .252 
 
 .15528 1 
 
 .305 
 
 .20276 
 
 .358 
 
 .25203 
 
 .411 
 
 .30417 
 
 .404 
 
 .35673 
 
 
 .253 
 
 .15615 j 
 
 .30(5 
 
 .20368 
 
 .359 
 
 .25359 
 
 .412 
 
 .30515 
 
 .405 
 
 .35773 
 
 
 .254 
 
 .15702 
 
 .307 
 
 .2046 
 
 .36 
 
 .25455 
 
 .413 
 
 .30014 
 
 .400 
 
 .35872 
 
 
 .255 
 
 .15789 
 
 .308 
 
 .20553 
 
 .361 
 
 .25551 
 
 .414 
 
 .30712 
 
 .407 
 
 .35972 
 
 
 .256 
 
 .15876 
 
 .309 
 
 .20645 
 
 .3(;2 
 
 .25047 
 
 .415 
 
 ..".O-ll 
 
 .408 
 
 .36072 
 
 
 .257 
 
 .15964 
 
 .31 
 
 ,20738 
 
 ,303 
 
 ,25743 
 
 .410 
 
 .30909 
 
 .409 
 
 .30172 
 
 
 .258 
 
 .16051 
 
 .311 
 
 .2083 
 
 .364 
 
 .25839 
 
 .417 
 
 .31008 
 
 .47 
 
 .30272 
 
 
 .259 
 
 .16139 
 
 .312 
 
 .20923 
 
 .305 
 
 .259:^0 
 
 .418 
 
 .31107 
 
 ,471 
 
 .30371 
 
 
 .26 
 
 .16226 
 
 .313 
 
 .21015 
 
 .300 
 
 .26032 
 
 .419 
 
 .31205 
 
 1 .472 
 
 .30471 
 
 
 .261 
 
 .16314 
 
 .314 
 
 .21108 
 
 .307 
 
 .26128 
 
 .42 
 
 .31301 
 
 .473 
 
 ,.30571 
 
 
 .262 
 
 .16402 
 
 .315 
 
 .21201 
 
 .308 
 
 .26225 
 
 .421 
 
 .31403 
 
 .474 
 
 .30071 
 
 
 .263 
 
 .1649 
 
 .316 
 
 .21294 
 
 .309 
 
 .20321 
 
 .422 
 
 .31502 
 
 .475 
 
 .30771 
 
 
 .264 
 
 .10578 
 
 .317 
 
 .21387 
 
 ,37 
 
 .20418 
 
 .123 
 
 .310 
 
 ,470 
 
 .30871 
 
 
 .265 
 
 .16666 
 
 .318 
 
 .2148 
 
 .371 
 
 .26514 
 
 .424 
 
 .31099 
 
 .477 
 
 .30971 
 
 
 .266 
 
 .16755 
 
 .319 
 
 .21573 
 
 .372 
 
 .20011 
 
 .425 
 
 .31798 ' 
 
 ,478 
 
 .37071 
 
 
 .267 
 
 .16844 
 
 .32 
 
 .21667 
 
 .373 
 
 .20708 
 
 .420 
 
 .31rt97 
 
 ,479 
 
 .3717 
 
 
 .268 
 
 .16931 
 
 .321 
 
 .2176 
 
 .374 
 
 .26804 
 
 .427 
 
 .31990 
 
 .48 
 
 .3727 
 
 
 .269 
 
 .1702 
 
 .322 
 
 .21853 
 
 .375 
 
 .26901 
 
 .428 
 
 .32095 
 
 .481 
 
 .3737 
 
 
 .27 
 
 .17109 
 
 .323 
 
 .21947 
 
 .370 
 
 .20998 
 
 ,429 
 
 .32194 
 
 .482 
 
 .3747 
 
 
 .271 
 
 .17197 
 
 .324 
 
 .2204 
 
 .377 
 
 .27095 
 
 .43 
 
 .32293 
 
 .483 
 
 .3757 
 
 
 .272 
 
 .17287 
 
 .325 
 
 .22134 
 
 .378 
 
 .27192 
 
 .431 
 
 .32391 
 
 .48+ 
 
 .3767 
 
 
 .273 
 
 .17376 
 
 .326 
 
 .22228 
 
 .379 
 
 .27289 
 
 .432 
 
 .3249 
 
 .485 
 
 .3777 
 
 
 .274 
 
 .17465 
 
 .327 
 
 .22321 
 
 .38 
 
 .27386 
 
 .433 
 
 .3259 
 
 .486 
 
 .3787 
 
 
 .275 
 
 .17554 
 
 .328 
 
 .22415 
 
 .381 
 
 .27483 
 
 .434 
 
 .32089 
 
 .487 
 
 .3797 
 
 
 .276 
 
 .17643 
 
 .329 
 
 .22509 
 
 .382 
 
 .27580 
 
 .435 
 
 ,32788 
 
 .488 
 
 .3807 
 
 
 .277 
 
 .17733 
 
 .33 
 
 .22603 
 
 .383 
 
 .27677 
 
 .430 
 
 .32887 
 
 .489 
 
 .3817 
 
 
 .278 
 
 .17822 
 
 .331 
 
 .22697 
 
 .384 
 
 .27775 
 
 .437 
 
 .32987 
 
 .49 
 
 .3827 
 
 
 .279 
 
 .17912 
 
 .332 
 
 .22791 
 
 .385 
 
 .27872 
 
 .438 
 
 .33086 
 
 .491 
 
 .3837 
 
 
 .28 
 
 .18002 
 
 .3,33 
 
 .22886 
 
 .386 
 
 .27969 
 
 .439 
 
 .33185 
 
 .492 
 
 .3847 
 
 
 .281 
 
 .18092 
 
 .334 
 
 .2298 
 
 .387 
 
 .28067 
 
 .44 
 
 .33284 
 
 .493 
 
 .3857 
 
 
 .282 
 
 .18182 
 
 .336 
 
 .23074 
 
 .388 
 
 .28164 
 
 .441 
 
 .33384 
 
 .494 
 
 .3867 
 
 
 .283 
 
 .18272 
 
 .336 
 
 .23109 
 
 .389 
 
 .28262 
 
 .442 
 
 .33483 
 
 .495 
 
 .3877 
 
 
 .284 
 
 .18361 
 
 .337 
 
 .23263 
 
 .39 
 
 .26.159 
 
 .443 
 
 .33582 
 
 .496 
 
 .3887 
 
 
 .285 
 
 .18452 
 
 .338 
 
 .23358 
 
 .391 
 
 .28457 
 
 .444 
 
 .33082 
 
 .497 
 
 .3897 
 
 
 .286 
 
 .18542 
 
 .339 
 
 .23453 
 
 .392 
 
 .28554 
 
 .445 
 
 .33781 
 
 .498 
 
 .3907 
 
 
 .287 
 
 .18633 
 
 .34 
 
 .23547 
 
 .393 
 
 28652 
 
 .446 
 
 .3388 
 
 .499 
 
 .3917 
 
 
 .288 
 
 .18723 
 
 .341 
 
 .23642 
 
 .394 
 
 ,2875 
 
 .447 
 
 .3398 
 
 .5 
 
 .3927 
 
ARRAfi OP TMR ZONKS OP A CmrLR. 39 
 
 To AsctTlftiii llie Area of a Segment of a Circle; by llic prtci'diiig Tabic. 
 
 RuLK.— Diviili t'lo hoiijht or vor.'Oil siiio )>y Iho dianuilor of tho oir>!lo ; find tlm <jiiotiont in the 
 floliinin of v(n'so(l i-ii u-". 'I'nk i tho iiroa liotoil in tlio next column, iniilti|ily it by tliu fujuaro of tlio 
 (liiiiiintor, nn I it will ;^ivc' tho nri'a. 
 
 Kx vmi'm;.— llcHiuiidd tlio aroa of a sogniont, it^ hoijjht beini» 10, and tho diainotcr of tho cirolo 
 50 foot. 
 
 l()_L..'in-=.2, iind .'2. |inrtiblo,--=.niS2; thon .111S2 X :>02^'i7'J.bb/,;/. 
 
 Niu'i.; — If, in tho divisj.in of a lioi'^lit by tho lia-o, tlio ijuotiDnt has roiuaindor after tho third 
 place of do'iinals and i;-uat ii'i.iiriiiy is rofiuirod, 
 
 Taku tho area for 111) lir.'t throe ligiii'iM, t'.ibtrnot it fruin Iho next following nrcn, multiply tho 
 romaindoi by th'> fnid fraction, and add the product to tho llrst area ; tho sum will bo tho area for 
 tho wholo (jiioticnt. 
 
 2. U'h.it is tho aroa of a 8"gtnoiit of aoirolo, tho diamotor of which is 10 f^ot, and tho height of 
 it 1.575 f.'ot ? 
 
 1.5"5H-ltl— •'•''7'> ; tho tabubir area f-r .157=. 07Slt2, and for .loS=.U70fld, tho diiforenco between 
 which is .')lli)7:!. 
 
 Thon .5 X .00i)7a=.00(t3()5. 
 
 Ilonoo .)57=.07H!)2 
 
 .U0U5=.0UII3f;.'> 
 
 .079275, the Bum by which tho square of tho dia- 
 motor of the circlo is to bo niultipliod ; and .079285 X 10^=7.1)28t)/a7. 
 
 TA-ULli: IX. 
 
 TABLE OF THE AREAs OF THE ZONES OF A CIRCLE. 
 
 The Diameter nfa Circle assumed to he Unity, and divided into 1000 equal Parts, 
 
 ! Il'ght. 
 
 Arei. 
 
 H'ght. 
 .0:19 
 
 Area. 
 
 ir-ht. 
 
 .057 
 
 Area 
 
 .05088 
 
 H'ght. 
 
 Area. 
 
 li'ght. 
 
 Area. 
 .11203 
 
 .001 
 
 .001 
 
 .02898 
 
 .085 
 
 .0S459 
 
 .ii;i 
 
 .002 
 
 .002 
 
 .o:i 
 
 .02998 
 
 .058 
 
 .057,S7 
 
 .0,S0 
 
 .0f<557 
 
 .114 
 
 .113 
 
 .oo;{ 
 
 .ou;j 
 
 .o:;i 
 
 .o:io9;{ 
 
 .059 
 
 .05S,S0 
 
 .OS 7 
 
 .0S050 
 
 .115 
 
 .Il;i98 
 
 .001 
 
 .004 
 
 .o:!2 
 
 .0;il98 
 
 .00 
 
 .05980 
 
 .088 
 
 .08754 
 
 .110 
 
 .11495 
 
 .00.5 
 
 .005 
 
 .Oil,! 
 
 .0:1298 
 
 .001 
 
 .00085 
 
 .0'^9 
 
 .0885:1 
 
 .117 
 
 .11592 
 
 .OOG 
 
 .000 
 
 .o;{4 
 
 .o;i;i97 
 
 .002 
 
 .00184 
 
 .09 
 
 .08951 
 
 .118 
 
 .1109 
 
 .007 
 
 .007 
 
 .o;!5 
 
 .0.5497 
 
 .oo:{ 
 
 .0028:1 
 
 .091 
 
 .0905 
 
 .119 
 
 .11787 
 
 .oo.s 
 
 .008 
 
 .o;{0 
 
 .0;i597 
 
 .004 
 
 .00:i82 
 
 .092 
 
 .09148 
 
 .12 
 
 .11884 
 
 M'J 
 
 .OOi) 
 
 .o:!7 
 
 .0;)ti97 
 
 .005 
 
 .00182 
 
 .09:1 
 
 .09210 
 
 .121 
 
 .11981 
 
 .01 
 
 .01 
 
 .o;{8 
 
 .0;i790 
 
 .000 
 
 .0058 
 
 .094 
 
 .09:141 
 
 .122 
 
 .12078 
 
 .Oil 
 
 .011 
 
 i .o.so 
 
 .o;i890 
 
 .007 
 
 .0008 
 
 .095 
 
 .0944:1 
 
 .12:1 
 
 .12175 
 
 .012 
 
 .012 
 
 .04 
 
 .0:5990 
 
 .008 
 
 .0078 
 
 .090 
 
 .0954 
 
 .124 
 
 .12272 
 
 .01.5 
 
 .OKI 
 
 .041 
 
 .04095 
 
 .009 
 
 .00878 
 
 .097 
 
 .09(;:59 
 
 .125 
 
 .12309 
 
 .014 
 
 .014 
 
 .042 
 
 .01195 
 
 .07 
 
 .00977 
 
 .098 
 
 .097:17 
 
 .120 
 
 .12469 
 
 .Oh". 
 
 .015 
 
 .04;; 
 
 .04295 
 
 .071 
 
 .07070 
 
 .099 
 
 .0!>8:i5 
 
 .127 
 
 .12502 
 
 .OIG 
 
 .010 
 
 .014 
 
 .04;i94 
 
 .072 
 
 .07175 
 
 .1 
 
 .099;i3 
 
 .128 
 
 .12659 
 
 .017 
 
 .017 
 
 .015 
 
 .04494 
 
 .07;i 
 
 .07274 
 
 .101 
 
 .100:11 
 
 .129 
 
 .12755 
 
 .018 
 
 .018 
 
 .040 
 
 .01o9;i 
 
 .074 
 
 .07:5. ;i 
 
 .102 
 
 .10129 
 
 .13 
 
 .12852 
 
 .010 
 
 .019 
 
 .047 
 
 .0109:1 
 
 .075 
 
 .07472 
 
 .10:1 
 
 .10227 
 
 .131 
 
 .12949 
 
 .02 
 
 .02 
 
 .048 
 
 .0479:1 
 
 .070 
 
 .0755 
 
 .104 
 
 .10:525 
 
 .132 
 
 .1.3045 
 
 .021 
 
 .021 
 
 .049 
 
 .01892 
 
 .077 
 
 .07009 
 
 .105 
 
 .10422 
 
 .1:13 
 
 .13141 
 
 .022 
 
 .022 
 
 .05 
 
 .04992 
 
 .078 
 
 .07708 
 
 .106 
 
 .1052 
 
 .134 
 
 .132:18 
 
 .02:5 
 
 .02H 
 
 .051 
 
 .05091 
 
 .079 
 
 .07807 
 
 .107 
 
 .10018 
 
 .1:15 
 
 .i:i;i;i4 
 
 .024 
 
 .024 
 
 .052 
 
 .0519 
 
 .08 
 
 .07900 
 
 .108 
 
 .10715 
 
 .136 
 
 .1343 
 
 .025 
 
 .025 
 
 .05:1 
 
 .0529 
 
 .081 
 
 .08004 
 
 .109 
 
 .10818 
 
 .137 
 
 .l:i527 
 
 .020 
 
 .0251)!) 
 
 .054 
 
 .05:189 
 
 .082 
 
 .08103 
 
 .11 
 
 .10911 
 
 .i;i8 
 
 .13023 
 
 .027 
 
 .0209!) 
 
 .055 
 
 .05489 
 
 .08:1 
 
 .08202 
 
 .111 
 
 .11008 
 
 .139 
 
 .13719 
 
 .028 
 
 .02799 
 
 .050 
 
 .05588 
 
 .084 
 
 .08:10 
 
 .112 
 
 .11106 
 
 .14 
 
 .13815 
 
40 
 
 ARFAH OP TTTF, ZONKS OP A CIRCLP. 
 
 TABLE.— {Continued) 
 
 nVht. 
 
 Area. 
 
 ll'ght. 
 
 Arua. 
 
 1 li'fe'l'f. 
 
 1 .._ __ 
 
 ' .25;! 
 
 Arou. 
 .21175 
 
 li'ght. 
 
 .;!09 
 
 Area. 
 
 n'ght. 
 
 Area, 
 
 ' 
 
 .Ill 
 
 .i;i9ii 
 
 .197 
 
 .19178 
 
 .2-v^'Ol 
 
 .:i65 
 
 •;!29;!1 
 
 
 .112 
 
 .14u;>7 
 
 . 1 98 
 
 .I!t27 
 
 ' .251 
 
 .21261 
 
 .;!i 
 
 .2^8,S 
 
 1 .'Mt\ 
 
 .;!2999 
 
 
 .I4:t 
 
 .1 IK),". 
 
 .199 
 
 .I9;i61 
 
 1 .2.55 
 
 .21:! 17 
 
 .;!ii 
 
 .2''958 
 
 ' .;!67 
 
 .;!:>067 
 
 
 .141 
 
 ,1419S 
 
 .2 
 
 .194.W 
 
 .256 
 
 .244;!;! 
 
 .:!12 
 
 .2!lO;i6 
 
 .;!68 
 
 .:^;;i;!5 
 
 
 .1-1.') 
 
 .14294 
 
 .201 
 
 .19545 
 
 .257 
 
 .21519 
 
 .;ii;i 
 
 .29115 
 
 .;i69 
 
 .;!:i2o;! 
 
 
 .14(; 
 
 .1 i;!'.) 
 
 .202 
 
 .196;!6 
 
 .258 
 
 .24604 
 
 .;!14 
 
 .29192 
 
 .;!7 
 
 .:^;;27 
 
 
 .147 
 
 .1448.'. 
 
 .20;{ 
 
 .19728 
 
 .259 
 
 .2169 
 
 .;m5 
 
 .2927 
 
 .;!7i 
 
 .;!;!;!;!7 
 
 
 .14H 
 
 .14.5^1 
 
 .204 
 
 .19819 
 
 .26 
 
 .21775 
 
 1 .:ii6 
 
 .29.'! 18 
 
 .;!72 
 
 .;!;!404 
 
 
 .lli> 
 
 .14677 
 
 .205 
 
 .1991 
 
 .261 
 
 .21.^61 
 
 .H\7 
 
 .29125 
 
 : .;!7;! 
 
 .:!;!47i 
 
 
 .1.') 
 
 .14772 
 
 .206 
 
 .20001 
 
 .262 
 
 .2I94(; 
 
 .;!ls 
 
 .29502 
 
 1 .;!74 
 
 .;!;^5:i7 
 
 
 .1-)1 
 
 .14St,7 
 
 .207 
 
 .2011112 
 
 .26:! 
 
 .25021 
 
 .;!19 
 
 .295^ 
 
 ' .;!75 
 
 .;!;)6()4 
 
 
 .ir>2 
 
 .1 l!i(;2 
 
 .20*^ 
 
 .201.^:5 
 
 .264 
 
 .25116 
 
 ..•!2 
 
 .296,>6 
 
 .;!76 
 
 .;!;!67 
 
 
 . 1 ,•):; 
 
 .l.''.0."ih 
 
 .209 
 
 .20271 
 
 .265 
 
 .25201 
 
 .;!2l 
 
 .297;!;! 
 
 .;;77 
 
 .;!;^7;!5 
 
 
 .i.'ii 
 
 .1515;'. 
 
 .21 
 
 .2o;!(;5 
 
 .266 
 
 .25285 
 
 .;!22 
 
 .29^1 
 
 .;!7s 
 
 ,;i;!soi 
 
 
 .1.').') 
 
 .1524S 
 
 .211 
 
 .20156 
 
 .267 
 
 .25;!7 
 
 .;!2;^ 
 
 .29.>^86 
 
 1 .:^79 
 
 .;i'!8(;6 
 
 
 .if)*; 
 
 .I5;ii:5 
 
 .212 
 
 .2(1546 
 
 .2(;h 
 
 .2.5455 
 
 .;i24 
 
 .29962 
 
 i .;i8 
 
 .;!;!9;!i 
 
 
 .!r)7 
 
 .I5i:w 
 
 .21;! 
 
 .206;!7 
 
 .269 
 
 .255;!9 
 
 .;;25 
 
 .;;oo;;9 
 
 ! .;;si 
 
 .;!;!996 
 
 
 . 1 "iS 
 
 .i.'-.o;!:; 
 
 .214 
 
 .20727 
 
 .27 
 
 .25(i2;! 
 
 .;!26 
 
 .;!0114 
 
 1 .;i82 
 
 .;;i06i 
 
 
 .1;VJ 
 
 ArnviH 
 
 .215 
 
 .2(I,S|K 
 
 .271 
 
 .25707 
 
 .;!27 
 
 .;i0i9 
 
 • 'm>^ 
 
 .;!4125 
 
 
 .it; 
 
 .1572.-; 
 
 .216 
 
 .20908 
 
 .272 
 
 .25791 
 
 .;!28 
 
 .;;o266 
 
 i .;i84 
 
 .;!419 
 
 
 .1(11 
 
 .15817 
 
 • 217 
 
 .2091H 
 
 .27;! 
 
 .25.S75 
 
 .;!29 
 
 .;!0;i4l 
 
 .;!85 
 
 ..^425:! 
 
 
 .11.2 
 
 .15PI2 
 
 .218 
 
 .21 OSS 
 
 .274 
 
 .25959 
 
 .:^;! 
 
 .;!0U6 
 
 1 .'AW 
 
 .;^4;-ii7 
 
 
 .Kl.S 
 
 .16006 
 
 .219 
 
 .21178 
 
 .275 
 
 .2604;! 
 
 .;!;!i 
 
 .;i0491 
 
 1 .;i87 
 
 .;!4:^8 
 
 
 .164 
 
 .16101 
 
 .22 
 
 .21268 
 
 .276 
 
 .26126 
 
 .;!;!2 
 
 .:!05(i6 
 
 .;!88 
 
 .;!4444 
 
 
 .165 
 
 .161115 
 
 .221 
 
 .2i;!5S 
 
 .277 
 
 .26209 
 
 .;!;!;! 
 
 .;!0641 
 
 .:!89 
 
 .;!4507 
 
 
 .166 
 
 .1629 
 
 .222 
 
 .21447 
 
 .278 
 
 .2621»;! 
 
 .:^:i4 
 
 .;!0715 
 
 .;!9 
 
 .;il569 
 
 
 .167 
 
 .16H><4 
 
 .22:! 
 
 .215;i7 
 
 .279 
 
 .2ii,>16 
 
 .;!;i5 
 
 .;i079 
 
 ^ .;!9i 
 
 .;;46;!2 
 
 
 .168 
 
 .16478 
 
 .224 
 
 .21626 
 
 .28 
 
 .2()459 
 
 .;};!6 
 
 .;^0864 
 
 ; .:^92 
 
 .;i4694 
 
 
 .169 
 
 .16572 
 
 .225 
 
 .21716 
 
 .281 
 
 .26541 
 
 .;^;!7 
 
 .;!09.S8 
 
 ..•lit;! 
 
 .;!4756 
 
 
 .17 
 
 .16667 
 
 .226 
 
 .21805 
 
 .282 
 
 .26624 
 
 .:!;!8 
 
 .;!1012 
 
 • .:m 
 
 .;S4818 
 
 
 .171 
 
 .16761 
 
 .227 
 
 .2 181' 4 
 
 .28;! 
 
 .26706 
 
 .;!;i9 
 
 .;iios5 
 
 .;!95 
 
 .;!4879 
 
 
 .172 
 
 .16S55 
 
 .22.S 
 
 .219,s;i 
 
 .284 
 
 .26789 
 
 .;!4 
 
 .;!1159 
 
 .;-!96 
 
 .;i494 
 
 
 .17.'{ 
 
 .16948 
 
 .229 
 
 .22072 
 
 .285 
 
 .26871 
 
 .;!4i 
 
 .;!12;^2 
 
 ..'i97 
 
 .;i50oi 
 
 
 .174 
 
 .17012 
 
 .2.'! 
 
 .22161 
 
 .286 
 
 .2695;! 
 
 .;!4 2 
 
 .:!i;>(i5 
 
 ,;^9,s 
 
 .;i5062 
 
 
 .175 
 
 .171:^6 
 
 .2;^i 
 
 .2225 
 
 .287 
 
 .270;{5 
 
 .;!4;! 
 
 .;!i;!78 
 
 ! .:!99 
 
 .;!5122 
 
 
 .176 
 
 .172;^ 
 
 .2;)2 
 
 .22;«5 
 
 .288 
 
 .27117 
 
 .;!14 
 
 .;ii45 
 
 1 .4 
 
 .;^5l82 
 
 
 .177 
 
 .I7;i2;{ 
 
 .2:t;! 
 
 .22427 
 
 .289 
 
 .27199 
 
 .;ii5 
 
 .;!152;! 
 
 1 .401 
 
 .;!5242 
 
 
 .178 
 
 .17417 
 
 .2;!4 
 
 .22515 
 
 .29 
 
 .2728 
 
 .;^46 
 
 .;!1595 
 
 ' .402 
 
 .;i5;^02 
 
 
 .179 
 
 .1761 
 
 .2:^5 
 
 .22604 
 
 .291 
 
 ,27;-i62 
 
 .;!47 
 
 .;S1667 
 
 .40;! 
 
 .;!5;^6i 
 
 
 .18 
 
 .1760;! 
 
 .2;;6 
 
 .22692 
 
 .92 
 
 .2744;! 
 
 .;!48 
 
 ..■!i7;r.) 
 
 ^ .404 
 
 .;!542 
 
 
 .181 
 
 .17697 
 
 .2;!7 
 
 .2278 
 
 .29;-! 
 
 .27524 
 
 .;!49 
 
 .;!i8ii 
 
 1 .405 
 
 .;o5479 
 
 
 .182 
 
 .1779 
 
 .2:i.s 
 
 .22.^68 
 
 .294 
 
 .27605 
 
 .;^5 
 
 .;!1882 
 
 ' .406 
 
 .:!55;!8 
 
 
 .18H 
 
 .I78.s;i 
 
 .2;'.9 
 
 .2295(; 
 
 .295 
 
 .276S6 
 
 .;!5i 
 
 .;!11154 
 
 ' .407 
 
 .;i5596 
 
 
 .184 
 
 .17976 
 
 .2.1 
 
 .■:;!04.|: 
 
 .296 
 
 .27766 
 
 ,;«2 
 
 .;!2025 
 
 : .4(),s 
 
 .;;5654 
 
 
 .185 
 
 .18069 
 
 .241 
 
 .2:^1 -ii 
 
 .297 
 
 .27847 
 
 .;^5;^ 
 
 .;!2096 
 
 .409 
 
 ..•!5711 
 
 
 .186 
 
 .18162 
 
 .242 
 
 .2:1219 
 
 .298 
 
 .27927 
 
 .;!54 
 
 .:i2167 
 
 1 .41 
 
 .;^5769 
 
 
 .187 
 
 .18254 
 
 .24;^ 
 
 .2;!;;o6 
 
 .299 
 
 .28007 
 
 ..•!55 
 
 .;r22:!7 
 
 ' .411 
 
 ..>5826 
 
 
 .188 
 
 .18;M7 
 
 .244 
 
 .2;^;^94 
 
 "1 
 
 .28088 
 
 .;-!56 
 
 .:!2;^07 
 
 i .412 
 
 .;^588;^ 
 
 
 .189 
 
 .1844 
 
 .245 
 
 .2:^481 
 
 .;^oi 
 
 .28167 
 
 .;^57 
 
 ..•!2;i77 
 
 .41.-! 
 
 .;!59;^9 
 
 
 .19 
 
 .185;{2 
 
 .246 
 
 .2;;568 
 
 .:^02 
 
 .28247 
 
 .:!58 
 
 .;!2447 
 
 ! .414 
 
 .J!5995 
 
 
 .191 
 
 .18625 
 
 .217 
 
 .2;-!(;55 
 
 .:^o:! 
 
 .2.^;-!27 
 
 .:i59 
 
 .;!2517 
 
 i .415 
 
 .;^605I 
 
 
 .192 
 
 .18717 
 
 .248 
 
 .2;;742 
 
 .M04 
 
 .28406 
 
 .;i6 
 
 .;!2."i87 
 
 .416 
 
 .;^6107 
 
 
 .19;{ 
 
 .18809 
 
 .249 
 
 .2;!829 
 
 .;^05 
 
 .28486 
 
 .;^6i 
 
 .:!2656 
 
 .417 
 
 .;^6162 
 
 
 .194 
 
 .18902 
 
 .25 
 
 .2;!915 
 
 .;^06 
 
 .28565 
 
 ..362 
 
 .;^2725 
 
 .418 
 
 ,:^6217 
 
 
 .195 
 
 .18994 
 
 .251 
 
 .24002 
 
 .:!07 
 
 .28644 
 
 .;w.'? 
 
 .:i2794 
 
 .419 
 
 .:i6272 
 
 
 .196 
 
 .19086 
 
 .252 
 
 .24089 
 
 .H08 
 
 2872;^ 
 
 .;^64 
 
 .:!2862 
 
 .42 
 
 .;^6;!26 
 
 
ARFAS OF TMK ZONES OK A f!IKCr,F. 
 'VAVAA'l.— (Continued.) 
 
 41 
 
 |lH'ght. 
 .421 
 
 Area. 
 .3038 
 
 1 
 
 .437 
 
 AlCM. 
 
 .372'i: 
 
 nviit- 
 
 - . 
 
 .453 
 
 Area. 1 
 
 IlVht. 
 
 .371)31 
 
 .4 (It) 
 
 .422 
 
 .30434 
 
 .438 
 
 .3725 
 
 .451 
 
 .371)73 
 
 .47 
 
 .423 
 
 .3(;-iys 
 
 .43i) 
 
 .372118 
 
 .455 
 
 .3,^01 1 
 
 .471 
 
 .424 
 
 .3(;.vii 
 
 .14 
 
 .37310 
 
 .450 
 
 .3.^()5(; 
 
 .472 
 
 .425 
 
 .30o!)4 
 
 • 111 
 
 .3731)3 
 
 .457 
 
 ..'isoik; ' 
 
 .473 
 
 .426 
 
 .300(0 
 
 .4-12 
 
 .3714 
 
 .45S 
 
 ..•'.8137 i 
 
 .474 
 
 .427 
 
 .3i;0l)H 
 
 .443 
 
 .37187 
 
 .45!) 
 
 ..38177 
 
 .475 
 
 .428 
 
 .3(;7r) 
 
 .141 
 
 .;;t53;( 
 
 .40 
 
 .38210 
 
 .470 
 
 .429 
 
 .30H02 
 
 .4 15 
 
 ..")757li 
 
 .401 
 
 .3^<J55 1 
 
 .477 
 
 .43 
 
 .30s:>3 
 
 .440 
 
 .37021 1 
 
 .402 
 
 .:;.^2i)4 1 
 
 .478 
 
 .431 
 
 .30;)04 
 
 ,447 
 
 .3700!.) i 
 
 .403 
 
 .3s;;32 1 
 
 .471) 
 
 .432 
 
 .30'J")4 
 
 .4 IS 
 
 .37711 
 
 .401 
 
 .3s;;oi) 
 
 .18 
 
 .433 
 
 .37005 
 
 .4 1'^ 
 
 .37758 
 
 .405 
 
 .3SI00 ! 
 
 .4Wl 
 
 .434 
 
 .37054 
 
 .45 
 
 ,37M)2 
 
 .40t; 
 
 .3,S.|-I3 
 
 .4^2 
 
 .435 
 
 .37101 
 
 .151 
 
 .378-15 
 
 .407 
 
 .3S.17;) 
 
 .4.^3 
 
 .430 
 
 .37163 
 
 .452 
 
 .378i5« 
 
 .408 
 
 .38514 1 
 
 .484 
 
 Arcn. M Il'ght. I Area. 
 
 I .3,«51!» 
 I .3H5.M;i 
 ; .3h017 
 
 .3m;5 
 .3sos;{ 
 .3h715 
 .3ts747 
 
 .3,S778 
 .3.^H08 
 
 ..■>^'S07 
 .3;^M)5 
 
 .3.s;»:i:{ 
 
 .3M)5 
 
 .381)70 
 
 .31)001 
 
 .485 
 
 .41^0 
 
 .4>s7 
 
 .4SS 
 
 .4 Si) 
 
 .4i) 
 
 .4'.) I 
 
 .4112 
 
 ,493 
 
 .4'J4 
 
 .41)5 
 
 .41)0 
 
 .41)7 
 
 .41)8 
 
 .41)1) 
 
 .6 
 
 .31)026 
 
 ..".1)05 
 
 .3',)i)73 
 
 .31)01)5 
 
 .31)117 
 
 .31)137 
 
 .31)1.50 
 
 .;;!)175 
 
 .3511)2 
 
 .31)208 
 
 .31)223 
 
 .31)230 
 
 .31)218 
 
 .31)258 
 
 .31)200 
 
 .3927 
 
 This Table ;> comi>uted only for Zones, the lonjcst chord nf which Is diameter. 
 
 To Ascertain the Area of a Zone by the preceding Table. 
 
 Ri'LR I. — Whrii Ihr Xone is Jj,'ss than a Si-micirc/c, DiViJo thu lloi^•'lt by tlio diameter, nn I find 
 the quotient in ilio roliiiiiti ol' liint;ht. Tiiko mit tl\u .aioa n|i| isito to it in tlio next I'oUium on thu riijlit 
 hand, and multipl)' it by tho quiire of the lon;^0!tt chord ; tbu piodiict will bo tho artM of tho zone. 
 
 ExiUPLG. — Koquired tho nre-' 'if a zone tho di;iinetor of whioij is 50, and its height 15. 
 
 15t-o0=.3; and ..'), us pur table, = .:iSOSS. 
 Hence .28086 X 50=^ = 702.2 arm. 
 
 Rule 2. — Whi'ii the Zuie is (j, ■eater than n ffemirirr/e : Take tho hoi;;hton each sidp of tho dia- 
 meter of tho cirein, and ascertain, t^.y llulo 1, their ro.ipoctivo area* ; aid tho ai'oaa of those two por- 
 tions together, and the sum will bo tho area of the zone. 
 
 ExAMPLK. — Requirol tho area of a zono, tli ' diameter of tho oirclo hoin:j 50, rad tho hoights of 
 the lone on each siue of tho diameter of tho circle 20 and 15 re-poi'tivly. 
 
 20-^i0=:.4 ; .4, as per tal.lei = . 05182 ; and .35182 ■; 50-' — 87'.).55. 
 15-f-50 = .;i; .3, as pur table =-28088; and .28088 x 50- = 702.2. 
 Hence 879-55 + 702.2 --= 1581-75 area. 
 
 RPLR 3. — ]Vheii the toiiifrsl chord of the zmu' is /e.<s 'lia,i dimtietfr. Take thu hoijjtitor distance 
 from tho diam. to each of the chords respeoiivoly ; lin I tho are i oorresprnding to Otioh height and 
 deduct the leaser from the greater area; thu result will be tho are i reijuired. 
 
 Nora. — When, in the division of a height by the chord, tho quotient has a remainder after tho 
 bird place of decimal?, and great accuracy is required, 
 
 Take tho area lor the first three figuroj, subtract it from tho ne.\t fullorin;; area, multiply tho 
 remainder by the said traction, and add the product to tho Grst area ; tho sum will be tho area fur 
 the whole quotient. 
 
 ExAMi'i.t:. — What is tho area of a zone of a circle, tho greater chord bring 100 feet, and tho 
 breadth of it 14 feot 3 inches ? 
 
 14 feet 3 inches = 14.25 and 14.25-4-in0 = -1425 ; tho tabular area for .142 = .11007, and for .143 
 = .14103, the diflference between which is .000'J6, 
 Then. 5 X.0009fi = -00048. 
 Henco.142 =-14007 
 .0005 = . 00048 
 
 .14055, the sum by which the square of tho greater chord is to be multiplied; and 
 .14056 X 1002=l405.5/ee^ 
 
TABLE 
 
 SPECIFIC GRAVITIES. 
 
 Tlie Sj)ccil]C Gravity of a I ody is tlio proportion it liearH to t!ie weight 
 of anotlior hoily of known density. 
 
 If a boily <loat on a flnid, tiie part ininicrsod is to the wiiole body aa 
 the ppecific g-.-avity oFtlie h-xiy is to the specific gravity of tlie (hiid. 
 
 When a lirxly i« ininierscd in a thiid, it Iohcs Hnch a portion of its own 
 weiglit as is eqnal to that of tiie ihiid it displaces. 
 
 An iInnler^^ed body, ascending or descending in a (laid, lias a force 
 equal to the difference lietween its own weight and the weight of its bulk of 
 the fluid, less the resistance of the fluid to its passage. 
 
 Water is well adapted forthe standard of gravity ; and as a cu')ic foot of 
 it weights 1000 ounce's avoirdupois, its weight is taken as the unit, viz : 1000. 
 
 To Ascertain the Specific Gravity of a Body heavier 
 
 than Water. 
 
 Rri,i!. — Weigh it both in and out of water, and note the dtfl'erence ; 
 
 then, as the weight lost in water is to tlie whole weigiit, so is 1000 to the 
 
 W X lOliO 
 specific gravity of the body. Or, ~iy~jz = Q, w representing the 
 
 weight in water and G the specijic gruriiij. 
 
 I"". X a:\iim.i;. — Whiit, i.s iIk; siM'cific, ginvity ofii Htmie w'licli weijxhrt in air lij His., in 
 wiiter 10 lliH. I 
 
 15—10= 5 ; tlien 5 : 15 : ; 1000 : : 3000 apec. (jrav. 
 
 To Ascertain the Specific Gravity of a Body lighter 
 
 than Water. 
 
 Kfi.i;. — A'liiex to the lighter body another that is heavier than water, 
 or the fluid used ; wcigli the piece added and the compound mass separa- 
 rately, botii in and out of tiie water, or the fluid ; ascprfain how much each 
 loses in water, or the fluid, by subtracting its weight in wat^r, or the fluid? 
 from its weight in air, and subtract the less of these diflerences from the 
 greater ; then, 
 
 As the last remainder 's to the weight of the light body in air, so is LOCO 
 
 to the specific gravity of the body. 
 
 ICxA.MiM.i;. — Wluit. Is tlie apci'il!^' i;i-iivity of ii piftCfi of wood tluit wfii^lia 'JO Hi8. 
 in ail ; iiiniexeii to ii is ii piece ol mutiil rlmt. wei^liH xJl ilia, in nil' iiii<i "-il H)h. in water, 
 mid t.lie two piecHH in waiiM- wi'iyli H lli-i. ? 
 
 'JO -|-- 1 —8= 1 1 — 3~;fo=r/os.5 ofrompound mass in loater j 
 
 yi — "Ji = '\--lii,i!> (if Ik in/ hmh/ ill ii-atei: 
 
 :?:; . jo : : lOOO ': (;0l)='J4 spec, gm- 
 
 To Ascertain the Specific Gra.ity of Fluid. 
 
 Rn.K.— Take a body "'"i. town s}ie,'.ilic gravity, weigh it in and out of 
 the fluid ; then, as the wei"ht of the liedy is to the Lea of weight, so ia the 
 ^pecifio graritjr of the body to liiut of the fluid. 
 
RPEOIFrO aRAVlTlES. 4t 
 
 lCx\Mri<K.— Whiit is ttm Rpccilic uriivity of ii Hii'ni in which a pince of copper 
 {spec. (/i(ii'.=!tOOO) W(!ixli« *<) ll>n- ill. imd SO llis. out. ol it ? 
 
 80 :80— 70=10 : : !)l '■> •. I I'-T) .'jyw. ^rj-aw. 
 
 To Compute the Proportions of two Ingrsdionts in a Cora- 
 pound, or to discover Adulteration in Metals. 
 
 lliLK. — Take the difUTences of eacli specific gravit}' of the ingredients 
 and the specific gravity of the compound, then nmltiply tiie gravity of the 
 one \)\ tiie ditFcrence of the other ; and, as the hiuii of the products is to the 
 respective products, so is tlie specific gravity of tlie body to the proportions 
 of tlie ingredients. 
 
 1<'XAMIT.K.— A (•()m|>oiiii(l of^'iild {xpi'r. -? vr?' =I8.HS8) iiii(i Bilver (spue. grav.= 
 I0.53.">) liiiH 11 sp(!(i:i(' ;j;iiivity of I J ; wluir, is tlm piopoilion of hucIi metiil ? 
 I8.8SS- I 1=I.888X I0..0;i5=r)l. I!);") 
 
 i4._io.r);3'j=:i. i(;.jx i8.888.=()r). 1 17 
 
 65.417+51. 1!).') : (;5.'M7 : : I J : 7.8:)5^oW, 
 (15. 147+51. ■1!)5 : 51. 195 : : 14 : t;.l(;5 silver. 
 
 Tocompute the Weights of the Ingredients, that of the 
 compound being given. 
 
 Rn.E.— Art the specific gravity of the compound is to the weight of the 
 compound, so are eacli of tlie proportions to the weigiitof its material. 
 
 KjcAMi'Li?.— The weiglit, iw iibove, being 'JS lbs., wlmt, ure the weights of theiii- 
 giediieiits ? 
 
 14. ~a. •^c.itiy. \-2,xi silver. 
 
 Proof of Spirituous Liquors. 
 
 A cubic inch ot proof spirits weighs 234 grains ; then, if an immersed 
 cubic inch of any heavy body weighs 2:U 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. 
 
 Ii,i,i;sTPAiiON.— A cubic inch of glitss -.veiglMiiK 700 gniiiia weighs 500 graiitH 
 V7heii weighod in u cert.nin spirit, ; wliiit is Mie p. ■ if of it ? 
 
 7QQ—rm=:^'MQ=gyaui.s=weiylt.t lost i,i ike spirit. 
 
 Then UOO : lU : : 1. : i.l7==ralL0 of proof of spiriU compared lo proof spiriU, or 1. 
 ==. 1 7 above proof. 
 
 Solidt 
 
 Rut, K.— Divide the specific gravity of the subBtance by 16, and the 
 quotient will give the weight of a cuMc foot of it in pounds. 
 
44 SPECIFIC ORAVITrES. 
 
 OF DIFFERENT BODIKS AND SUBSTANCES. 
 
 METALS 
 
 Aliiititiniini 
 Antimony 
 Arsenic. . 
 Bariinn. 
 Bisniull) 
 Brass, ooppe 
 " tin 
 
 METALS. 
 
 
 copper ( 
 zinc ',i 
 pi lite, 
 wire. 
 
 T 84 ) 
 
 Bronze, gun inetul . . . 
 
 Boron 
 
 Bromine 
 
 Cadmium 
 
 CalcMim . . . . 
 
 Clirominm 
 
 Ciniuihar. . . 
 
 Coliait 
 
 Columl)iuui 
 
 Gold, pure, cast 
 
 " liaminered. . . .. 
 " 22 carats fine. . 
 " 20 carats tine.. . 
 
 Copper, cast 
 
 plates 
 
 wire 
 
 Iridium.. . 
 
 " hammered ... 
 
 Iron, cast. 
 
 " gun metal. . 
 hot blast-. .... 
 
 cold " 
 
 wrought bars. . . 
 
 " wire 
 
 rolled plates.. . . 
 
 cast . 
 
 " rolled 
 
 Lithium 
 
 Manganese 
 
 Magnesium 
 
 Mercury— 40" 
 
 " +;52" 
 
 " CO**, , 
 
 ♦• 212° 
 
 Molybdenum... 
 
 Nickei. - 
 
 " cast 
 
 Psmium 
 
 
 (( 
 
 n 
 
 <( 
 
 <( 
 
 Ijeac 
 
 8s;!2 
 
 7821) 
 
 8:iso 
 
 8214 
 8700 
 2000 
 
 ;{iioo 
 8(;:.o 
 i;)8(i 
 
 oi)00 
 8(i'J,-< 
 8000 
 0000 
 
 10:501 
 
 174S(; 
 1,)700 
 
 8788 
 
 800- 
 
 8S8(i 
 
 1 8(;8() 
 
 2;i(U)0 
 
 72t)7 
 7;i08 
 70()j 
 72 is 
 7V88 
 7774 
 7701 
 I :;>.-. 2 
 
 .Spcci- ! Wi'iKlit 
 lii' nr;i- of ii cii. 
 1 viiv. ibieiiKjli. 
 
 ll:!88 
 
 500 
 
 8000 
 
 I7r)0 
 irio:{2 
 
 i:{;V.)8 
 
 i;{.-)80 
 i:5;!70 
 
 8000 
 
 8800 
 
 8270 
 
 10000 
 
 .002(1 I'alliidiiim 
 
 L'hitinum, hammered.. 
 
 •' native .... 
 
 ■♦ rullcd 
 
 Potassium, f)!)* 
 
 .Hli»4 
 
 282; 
 . -MVA 
 
 ;U47 
 
 .072;; 
 
 .los:, 
 
 ;{120 
 
 0.'>7 
 
 •J i H J 
 
 .2020 
 
 .3111 
 
 .217 
 
 . 000.- 
 
 .70 '.'I 
 
 .6:525 
 
 ..-)C82 
 
 .;jl70 
 
 i{i'({li';id 
 
 khuiliiitn 
 
 Iiutii('i:inm i 
 
 ... I 
 
 .^'.'U'liiiim ' 
 
 Silic.'ium , ' 
 
 Silver, pure, cast . . . . ! 
 
 " hammered. I 
 
 Siiiiium I 
 
 Steel, plates. | 
 
 ' soft . . 
 
 .8:510 
 
 .2007 
 
 . 204 
 
 .2r)5.-) 
 
 .201 
 
 .2'^1 
 
 .281 
 
 .2787 
 
 .410( 
 
 .4110 
 
 .021:5 
 
 .280 I 
 
 .0(;:^:-5 
 
 .;")00 
 
 .4918 
 
 .4012 
 
 .48:^0 
 
 ..•5111 
 
 .:518:! 
 
 .2004 
 
 I i:5:.o, 
 2(i:5:57i 
 
 lODOllj 
 
 220 Oil 
 
 80;')j 
 
 8040| 
 lOOodj 
 
 SO no, 
 
 4 .".00: 
 
 temp 'reii 
 liar.lei 
 
 and! 
 
 ...1 
 
 wire. 
 
 Strontium 
 
 Tin, Cornish, hammerd 
 " " pure 
 
 Telluiium 
 
 Thalium 
 
 Titanium 
 
 I'uiigsten . 
 ;^14i) l^raiiiun 
 :5212 
 G7r)0 
 
 Wolfram.. . . 
 ;ast . . 
 rolle,!. 
 
 Ziiio, cast 
 
 WOODS (Dry). 
 
 10474 
 I Of) 11 
 
 070 
 78(10 
 7.s:5:; 
 
 7818 
 7817 
 2.V1I) 
 7:500 
 7201 
 OIK) 
 1 1 850 
 
 r):]oo 
 
 17l)U0 
 
 lOl.-iO 
 
 7110 
 
 0801 
 
 7101 
 
 Alder.. 
 Apple. . . 
 
 A.sh .... 
 
 Bamboo 
 av .... 
 
 Beech 
 
 iJirch . . . 
 
 Bu.\, Brazilian. 
 
 " Dutch... 
 
 " French.. . 
 Bullet wood.. . 
 
 Butternut 
 
 Campeachy . . . 
 Cedar . 
 
 " Indian. . . 
 
 ,410.) 
 
 .7:5:)() 
 
 .".787 
 .7082 
 .o:5i:{ 
 .:;24l 
 .:5-<:)2 
 .:;iil 
 
 .1027 
 
 .:5788 
 .:5002 
 
 .o:^.H 
 
 .282:5 
 
 28;5:i 
 
 2828 
 .28:^8 
 .0018 
 .207:{ 
 .20:!7 
 .221 
 .42<(\ 
 .1017 
 .0119 
 .:5071 
 .2o7.) 
 .2482 
 .20 
 
 (Millie 
 t'llot. 
 
 800 fiO 
 
 T03;io 
 
 84;)1.V2 
 00014:5 
 40( ):;;") 
 822Jrjl 
 
 8.-)2!r):5 
 000 '4:^ 
 
 507 
 
 lo:5i 
 
 012 
 
 i:52s 
 
 928 
 ;570 
 
 9i;^ 
 
 501 
 KSlf) 
 
 ;io. 
 
 04. 
 ;J7. 
 
 8:5. 
 
 58. 
 
 2:5. 
 
 57. 
 :55. 
 82. 
 
 562 
 812 
 
 i2.:« 
 :^75 
 
 25 
 1 25 
 4:^7 
 
 4:^7 
 
 002 
 002 
 157 
 
si'EciFrc auAvri'iES. 
 
 45 
 
 WOODS, (Dry. 
 
 ' Conllniicd.) 
 
 Sppc.i- 1 ^\^■i!.'■llt 
 
 ifu; ifi'ii- 111' It I'll 
 viiy. I liif fiHit. 
 
 I i 
 
 WOODS, (Dry), 
 
 {Cunlimied.) 
 
 Oluircoiil, i)iiH' 
 
 '• tV(',-h iiiii'iu'il 
 
 '• (liik '.! 
 
 " Sufi W.IUii. . . 
 
 " trimiateil . . . 
 
 4Hi-J7. 
 ir)7:!!'.)s 
 
 Clion-y 
 
 Clie~imt. sweet... . 
 
 CiifoK 
 
 Cociia 
 
 Cork 
 
 Cypress, SpMiii-h . 
 Dug-\vuo:l. ...... 
 
 Ebony, Atiu'ricaii. 
 " Iiiiliaii . .. . 
 
 Elder 
 
 E!in 
 
 Filbert 
 
 Fir (Norway Space). 
 Gmii, blue 
 
 " water 
 
 Hiickiiiataoic 
 
 Hazel . 
 
 Hawthorn 
 
 Hemlock 
 
 Hickory, pig-niit 
 
 " she 11- bark.. 
 
 Holly 
 
 Jasn)ine 
 
 Juniper 
 
 Lance-wood 
 
 Lari'li < 
 
 Lemon 
 
 Lignuni-vitiL' , 
 
 Lime 
 
 Liiuien 
 
 Locu.st 
 
 Lojiwood . . . 
 
 Mahogany 
 
 *' Honduras.. 
 
 '' Spa.ii.sh — 
 
 Maple 
 
 " bird's eye.. 
 
 MaHlic 
 
 Mulberry ..... 
 
 Oak, African , . 
 " Canadian. 
 
 i:-i.s(i 
 
 715 
 
 iilll 
 
 72(1 
 
 17. 
 
 ;ii;! 
 f) 
 
 :!s 1 
 i:).;57." 
 
 iot(i!(;.j. 
 
 2(0:15. 
 
 (;m|io. 
 
 i:>ii,n. 
 
 i;}3i|H3. 
 
 r2()!)j75. 
 G'.!') I. '5 
 570|;i5 
 
 G7i|n. 
 
 600 ;57. 
 
 512;J2. 
 
 84;i.)2. 
 
 loot! (!2. 
 
 51)2 ;n. 
 
 HGU'5:5. 
 !)I0'5G. 
 86^^i2:5. 
 71)2 11). 
 C HO 4:5. 
 7r>();47. 
 770:48 
 
 r)6() :i5 
 
 720i45. 
 544 ;u. 
 5!;o ;55 
 
 703,4:5 
 
 1:5:5:58:5 
 
 804i50 
 (5041:57 
 728U5 
 9i;5;57 
 720!45 
 lOG:i,G6 
 5G0i;« 
 
 25 
 
 25 
 
 187 
 
 5G2 
 
 4:57 
 
 G25 
 
 i):57 
 
 5 
 
 GS7 
 5 
 
 75 
 
 875 
 
 5 
 
 125 
 5 
 
 125 
 
 ;^75 
 
 Oak, Dantriic 
 
 " ■ Engli.-h 
 
 " green. , 
 
 heart, GO year.s. . , 
 live, green. .... 
 " .seasoned..., 
 white 
 
 Orange 
 
 >, 
 
 
 Pear 
 
 Persimmon . . . . , 
 
 PInm 
 
 Pine, pitch 
 
 " red 
 
 " white 
 
 " yellow. 
 
 'omegranale. . . . 
 
 >.. .„ 
 
 iplar 
 
 " white... 
 
 Quince 
 
 Hose-wood . . , 
 .SassiitVas ... 
 
 Satin-wood. 
 Spruce. . . . 
 Sycamore 
 Tamarack. 
 
 Teak (African oak).. < 
 
 Walnut 
 " black. 
 
 852 
 750 
 57G 
 84i) 
 5G1 
 897 
 82:i 
 872 
 
 5;^ 
 
 4G 
 
 ;^G 
 
 53 
 
 :i5 
 
 56 
 51 
 54 
 
 9;i: 
 
 812 
 
 25 
 .75 
 .5 
 ,062 
 
 !4:^7 
 
 .25 
 
 .875 
 
 062 
 ,062 
 .062 
 
 .4:i7 
 
 Willow 
 
 Yew, Dutch . 
 " Spanitili. 
 
 (Well Seasoned.*) 
 
 A,sh 
 
 Beech 
 
 Cherry 
 
 Cypress 
 
 Hickory, red 
 
 Mahogany, St. Domg. 
 
 Pme, white. 
 
 yellow 
 
 Poplar 
 
 White Oak, upl. '. . . 
 " Jamea iiiver, 
 
 Sp>'('i- 
 
 fie Hill 
 
 viiy. 
 
 759 
 
 9:52 
 
 1446 
 
 1170 
 
 1260 
 
 1068 
 
 86(! 
 
 705 
 
 661 
 
 710 
 
 7S5 
 
 66 U 
 
 590 
 
 554 
 
 461 
 
 1:551 
 
 580 
 
 ;58:5 
 
 529 
 705 
 
 728 
 482 
 885 
 500 
 
 62:^ 
 
 ;583 
 657 
 745 
 671 
 500 
 486 
 585 
 788 
 807 
 
 722 
 624 
 606 
 441 
 838 
 720 
 473 
 541 
 587 
 687 
 759 
 
 Waiijlit 
 
 (if II <M1- 
 
 lili; I'uot, 
 
 17.437 
 
 58.25 
 
 71.625 
 
 73.125 
 
 78.75 
 
 66.75 
 
 53.75 
 
 44.062 
 
 41 312 
 
 375 
 062 
 25 
 875 
 625 
 812 
 625 
 25 
 937 
 062 
 062 
 5 
 
 125 
 55.312 
 31 25 
 :58.9;^7 
 23.937 
 41.062 
 46.562 
 41.937 
 31.25 
 30.375 
 36.562 
 49 . 25 
 50.437 
 
 45.125 
 
 39. 
 
 37.875 
 
 27.562 
 
 52.375 
 
 45. 
 
 29.562 
 
 ;«.812 
 
 36.687 
 
 42.937 
 
 42 437 
 
 " Orduiiuue mauual 1841. 
 
46 
 
 SPECIFIC GRAVITfES. 
 
 Stones, Earths, &c.'fi.;ifrii'.,i.i cnbi.' 
 
 vity. ] lout. 
 
 2r)!t-0 — 
 27;-!() 170 625 
 2G'J'.»il(i8.tiS7 
 17111107. 120 
 
 Agate 
 
 Alabaster, white... 
 " yellow... 
 
 Alum 
 
 Aiiiher il07.s! (i7.;57r) 
 
 Ambergris |8<>(i | — 
 
 Asbestos, starry |;!07o 1 92 062 
 
 . , ,, ( 00.') i>ti.i)t\2 
 
 -^^Pl^^'t"'" j 1 16:,0 103.125 
 
 Barytes, sulpl'ate... J | ^.^,;5|.^„4 o,;2 
 R u ( l2740;17l.25 
 
 Borax |171 4;i07. 125 
 
 Brick (II1K)0|118 75 
 
 «< fire !220ljl37.562 
 
 " work ill cement.. ;i800 112 50 
 ., ,, ,, ( 160(1 100. 
 
 " ""'«"»>'il2000il25. 
 
 Carbon .S5(i0 
 
 Cement, Portland. 
 " Roman... 
 
 Stones, Earths, &c. ii<'!rniuiM cuiiio 
 
 I viiy . Font. 
 
 " while... 
 
 ('ornclian 
 
 Duimund, Orientiil.. . , 
 '■ iJruzilian. . 
 Kartli, t coniirion soil 
 
 " loose , 
 
 ini'ist .«au(l . . 
 moiilii, t're.'^h. 
 nuMMied . . . , 
 njiigh .sand. . 
 witli oravel . . . 
 
 2550 
 26i:{ 
 .1521 
 
 Chalk 
 
 Chrysolite 
 
 Clay 
 
 " with gravel . . . 
 
 Coal, Antracite 
 
 " Borneo 
 
 " Cannel 
 
 Caking 
 
 Clierry . . . . . 
 
 Chili 
 
 Derbyshire... 
 Lanca.ster 
 Maryland. . . . 
 Newcastle . . 
 Rive de Gier. 
 
 Scotch 
 
 i;]U0i 
 
 ft 
 II 
 it 
 (( 
 i( 
 tt 
 (< 
 
 xt 
 
 (I 
 (( 
 
 Coke 
 
 Splint , 
 
 Wales, mean.. . , 
 
 " Nat'I, Va. . . 
 Concrete, mean. 
 
 Copal 
 
 Coral, red. 
 
 156i/| 
 1520J 
 2781 
 2782 
 
 i!);U) 
 
 2480 
 
 i43(; 
 
 1640 
 1290 
 12;5S 
 
 i;5is 
 
 1277 
 1276 
 1290 
 1292 
 1273 
 1355 
 1270 
 1300 
 1259 
 1300 
 1302 
 1315 
 1000 
 746 
 2000 
 1045 
 2700 
 
 25 
 
 218 
 
 81 
 
 97. 
 
 95. 
 174. 
 
 120 625 
 
 155. 
 «9.75 
 
 102.5 
 80.625 
 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 
 
 :;44 1 — 
 
 u 
 u 
 a 
 ii 
 
 Emery 
 
 Flint, black. . 
 
 " white . , 
 Fhionne. ... 
 Glass, but lie , 
 
 " Crown 
 
 " Hint.., 
 
 gref n. . , 
 o[)lioul. 
 white . . , 
 window. 
 
 u 
 
 ii 
 ,1 
 11 
 (( 
 
 Garnet 
 
 " black 
 
 Granite, I'^gyptian red.. 
 Patapsco. . . . 
 
 Quincy 
 
 Scotch 
 
 " Sn.-(iuehanna. 
 
 Gravel, common 
 
 Grindstone 
 
 Gypsum, opaque 
 
 Hone, white, razor. . . 
 
 Hornl)lende 
 
 iodine 
 
 Jet 
 
 Lime, hydraulic 
 
 " (piick 
 
 Limestone, green 
 
 " whUe 
 
 Magnesia, carbonate. . . 
 
 Marble, Adelaide 
 
 African 
 
 Biscayan, black, 
 
 Carara 
 
 common.- 
 
 Egyptian 
 
 French 
 
 Italian, white.. 
 
 12194 
 1500 
 2050 
 205(1 
 16(10 
 1920 
 202(1 
 4000 
 2582 
 2591 
 1320 
 27;!2 
 24si7 
 12933 
 j;l200 
 
 3450 
 2892 
 2642 
 41H9 
 3750 
 2654 
 2(i40 
 2t^■)2 
 
 (1 
 (I 
 
 <• 
 
 2625 
 2701 
 1740 
 2143 
 216S 
 2876 
 1354(1 
 4940 
 1300 
 2745 
 804 
 3180 
 3156 
 2400 
 2715 
 2708 
 2695 
 271i; 
 2686 
 26 6 S 
 2649 
 2708 
 
 137. 
 
 93 . 
 128 
 128 
 100, 
 120. 
 126. 
 250 
 161 
 162 
 
 82, 
 170, 
 155, 
 183 
 196 
 165 
 215, 
 180, 
 165 
 
 165 
 
 1 65 . 
 
 165 
 
 164. 
 
 169 
 
 109. 
 
 133 
 
 135, 
 
 179. 
 
 221 
 
 171, 
 50 
 198, 
 197 
 150, 
 169, 
 169, 
 168, 
 169. 
 167. 
 166 
 165, 
 169. 
 
 125 
 75 
 125 
 125 
 
 375 
 
 1 25 
 
 5 
 
 75 
 
 437 
 
 312 
 
 1 25 
 (125 
 75 
 125 
 
 87.'^. 
 
 75 
 062 
 
 312 
 
 937 
 
 5 
 
 75 
 
 25 
 
 562 
 25 
 75 
 25 
 
 687 
 25 
 437 
 75 
 
 875 
 75 
 562 
 25 
 
 
 t Spec. grav. of tlie earth U variously estimated at from 5.450 to 5.600. 
 
8PE01PI0 GRAVITIES. 
 
 47 
 
 Spcci I Wi'iulit 
 
 Stones, Earths, &c.,ii.>i!ii. otatniii. 
 
 viiy. I Tdoi. 
 
 
 ^larble Pariiiii 
 
 " X't'i'iiioiit, white 
 
 Marl, iiic'iiii 
 
 Mica 
 
 Miirtar. ... 
 
 Millstone. 
 
 Miiil . . . 
 
 Mure 
 
 Opal 
 
 Oyst(M-->liell . . . 
 Pavinji-stoiie. . . 
 Peal, Oriental.. 
 
 Peat 
 
 •1 
 
 Phosplionis 
 
 Plaster (if Paris . 
 
 IMiiinlia;;!.) 
 
 Porplivry, red.. . 
 J-'urcelam, China. 
 Pumice-stone. . . 
 
 Quartz , 
 
 llutteii-stotie 
 
 Red- lead 
 
 Resin 
 
 Rock, crystal. . . 
 
 Ruby 
 
 Salt, ooinmoii . . 
 Saltpetre 
 
 Sand, coarse ... 
 " conunon . 
 
 danip and I io.se 
 dried and loose. . 
 
 dry 
 
 mortar, Ft. Rich. 
 " Jh'ooklyn 
 
 " silliciou.s 
 
 Sapphire 
 
 Sliale 
 
 n 
 
 u 
 u 
 
 Slate 
 
 Slate, purple 
 
 Smalt 
 
 Stone, ]}ath Engl. 
 
 " Blue Hill .... 
 
 " Biiu'stone (basalt) 
 
 " l]reakneck..N.Y. 
 
 " l!ri>tol ...Engl. 
 
 " (^aeti, Normandy 
 Ci'imnon ...... 
 
 2S:{S 177 
 
 2t;r)0 lo.) 
 
 1750:1119 
 2S()0|17.> 
 li^SJi Hi) 
 1750 Id!) 
 2 IS r 155 
 
 ItlSlMlUI 
 
 liXlO 118 
 2.14| - 
 2(!)2!lH0 
 2I1(;:151 
 2«;50i - 
 
 (;()() I :{7 
 i;^2i»! k;5 
 
 I77»'|ll0 
 
 ii7(;! 7H 
 
 2100 
 
 27G5 
 
 2800 
 
 '.) 1 5 
 
 2(i(;o 
 
 i:!l 
 
 172 
 
 57 
 lOG 
 
 ly.si 128 
 
 s;»4o 
 
 lo;<9 
 2785 
 42.S8 
 2180 
 20!)0 
 ISOO 
 1070 
 18 'J 2 
 1560 
 II 1:0 
 1(159 
 1710 
 1701 
 8994 
 2000:102 
 
 2900I18! 
 
 2072|107 
 2784:174 
 
 558 
 08. 
 170. 
 
 18(1. 
 
 112 
 
 104 
 
 87, 
 
 97 
 
 88 
 
 108 
 
 107, 
 
 100, 
 
 .875 
 
 57 
 
 .875 
 
 .5 
 875 
 25 
 
 <75 
 .75 
 
 .75 
 
 
 
 002 
 025 
 5 
 25 
 812 
 75 
 1-7 
 25 
 ,812 
 75 
 ,002 
 987 
 
 .125 
 ,025 
 
 5 
 
 875 
 
 5 
 
 75 
 
 Oi; 
 
 25 
 .88 
 
 n 
 
 2441) 
 1901 
 2010 
 2025 
 2704 
 2510 
 2070 
 252" 
 
 152 
 122 
 105. 
 104 
 
 109. 
 
 15(;, 
 
 129 
 
 157. 
 
 .25 
 
 5 
 
 502 
 
 002 
 
 .^75 
 
 75 
 
 5 
 
 Stones, } arths, &c. 
 
 .Stone. rraigletli..Enj;l. 
 " Kentish rag " 
 •' Kip's i?ay..N.Y. 
 " Norlodc '(Parlia- 
 ment House;. 
 Portland . .Engl. 
 Sandstone, mean 
 " Sydney 
 " Staten Isl'd. N Y. 
 " Sullivan Co. " 
 
 Schorl 
 
 Sjiar, caieareou.s . 
 
 " l''eld, blue 
 
 " " green.. . . 
 Fluor 
 
 .S|iftM , Wfijriit 
 
 licKra lit u Culiio 
 vilv. I l'"(iot. 
 
 (I 
 
 <t 11 
 
 Stalactite 
 
 Snl])hur, native.. . 
 
 Talc, mean 
 
 Tale, black. . . , . 
 
 Tile.'.... 
 
 Topaz, Oriental. . 
 
 'J'rap 
 
 Turquoise 
 
 2810 
 2051 
 2759 
 
 2804 
 280cS 
 2200 
 2287 
 2970 
 2088 
 70 
 2785 
 2098 
 2704 
 H400 
 2415 
 2088 
 2500 
 2900 
 1815 
 4011 
 2720 
 2750 
 
 MISCELLANEOUS. 
 
 Asphaltum 
 
 Atmospheric Air.. . 
 
 Beeswax 
 
 iJutter. 
 
 Camphor, 
 
 Caoutcliouc. ...... 
 
 Kgg 
 
 Fat of Beet' 
 
 " Hogs , 
 
 " Mutton ..... 
 
 ftamboge 
 
 (jum Arabic 
 
 Gunpowder, loose . 
 " fliaken 
 
 Gutta-percha. 
 
 Horn. 
 
 Ice, at 82°... 
 
 Indigo 
 
 Isinglass. . . 
 
 1 vory 
 
 Ijani 
 
 lid., j 
 
 905 
 
 1050 
 
 « 
 
 905 
 
 942 
 
 988 
 
 908 
 
 1090 
 
 928 
 
 980 
 
 928 
 
 1222 
 
 1452 
 
 900 
 
 1000 
 
 1550 
 
 1800 
 
 980 
 
 1089 
 
 920 
 
 1009 
 
 nil 
 
 1825 
 947 
 
 75 
 
 087 
 
 5 
 
 812 
 
 125 
 987 
 812 
 
 5 
 
 987 
 002 
 25 
 
 2.T 
 
 487 
 
 144. 
 105. 
 172. 
 
 144. 
 
 148. 
 
 187. 
 
 189. 
 
 180. 
 
 108. 
 
 1 98 . 
 
 170 
 
 108. 
 
 109. 
 
 215. 
 
 150 
 
 127. 
 
 150 
 
 181. 
 
 118. 
 
 170. 
 
 56.562 
 
 108.125 
 
 .07529 
 
 60.312 
 
 58.876 
 
 ei-'i^ 
 
 56.437 
 
 57.687 
 
 58.5 
 
 57.687 
 
 90.75 
 
 56.25 
 
 62.5 
 
 90.875 
 112.6 
 
 01.25 
 105.562 
 
 57.5 
 
 08.062 
 
 69.487 
 114 062 
 
 59.187 
 
 (") .OOiaoii. 
 
48 
 
 SPKCIPIC GRAVITIES. 
 
 MISCELLANEOUS. n'^™'„f>.c..in. 
 
 il\. I 1mm, 1. 
 
 Mastic 
 
 Myrrh 
 
 Opium . . , . . 
 Sua]). Ciiftile. 
 Rpcrinaci'ti . 
 Starch . . . . . 
 Suijar 
 
 .60. 
 
 Tallow 
 Wax .. 
 
 LiaUIDS. 
 
 Aciii, Act'tic 
 
 " lU'iizoic 
 
 " Citr,c.. . . 
 
 " CoiiccMitrati'il 
 
 " Fhioric .... 
 
 " Muriatic. . . 
 
 " Nitric . 
 
 " Pl)usphoric. 
 
 " " solid 
 
 " Sulpiiuric .... 
 Alcohol, j)iirt'. C){)°. . 
 \>') per cent. 
 
 
 i),J 
 
 
 ,vn 
 
 
 !)0 
 
 
 40 
 
 
 25 
 
 
 10 
 
 
 5 
 
 1071 
 
 );'.(; 
 
 I :i:;i; 
 
 1071 
 
 :»i;) 
 
 it.'iO 
 
 |i(;o(i 
 !;!•.!(; 
 
 j ".'7 -J 
 
 I !t-n 
 I ;t(ii 
 
 'J70 
 
 ('.'(i7 
 
 t;7.i2:) 
 
 s '> . 
 h;{ -) 
 r.C). ;i:i7 
 i,H '.i:;: 
 .v.» ;;7.". 
 1oo.:j7.' 
 
 s2 s7.-. 
 00 IT) 
 M 812 
 00.2i"i 
 00. 02.'' 
 
 Oil. .'}7"i 
 -II OH 
 
 !0;il 01 (i2."r 
 ''"•" '.l,"),(.02 
 
 |.,V'0 
 ,1200 
 
 liriaH 
 
 |2h()I' 
 
 I.SII' 
 
 O.-J 7,-) 
 7.') . 
 
 70.002 
 
 !)7 :!7:. 
 
 1 7 ") 
 
 1 1.'. ,"j(;2 
 
 LIQUIDS. 
 
 Sp.Ti j \Vri(,'l,r 
 flr^ni'dl'li Cllliio 
 viiv. i Foot. 
 
 .\quali>rii~. (Imililc . . . 
 
 " siiii;!e 
 
 Pxcr. .. ..". 
 
 liiliiiD^'M, li(]u:(l 
 
 r>iciii(| (iiumaii) . 
 
 I 
 
 Itraiiily. ;": or .) ol'f--piriti 1)2 1 
 
 ''iiicr 
 
 iMher, ,n;ciic . . . 
 
 ijiiiriaiic. . 
 
 r-r.ipliunc. 
 
 HI. 2.5 
 
 7o. 
 
 01.025 
 
 05.875 
 57. 75 
 0;i 025 
 
 Midi .vi- . iii5 
 
 l.'iOO 
 1200 
 1 0.". I 
 
 s I .^; 
 
 1 05 1 
 
 (1!^ 
 
 S 1 5 
 715 
 1 150 
 
 i(i:r2 
 
 ilHO 
 
 '.rj:! 
 
 '•I 
 
 701 lit 02L 
 
 " proof f^pirit, '50 > 
 per cent OO'^ | 
 
 " pruu!' .spirit. 50 ) 
 
 per cent.HO"* \ 
 
 Anmionia, 27. D per ct. 
 
 CoMipre.a.sion of the 
 square inch : 
 
 AlCdliol 
 
 Ether 
 
 HIG 
 
 ho;; 
 '.>:>, I 
 
 051 
 1/70 
 
 51 
 
 5H i);'.7 
 
 5 H . 1 1 7 ; I 
 5!) A'M 
 (III (i2."i 
 1)^0 (il.0 2.'. 
 02. 
 
 DO 2 
 
 i 
 
 j 875 
 
 I HIU 
 
 58.:-575 
 
 54 OH 7 
 55.6H7 
 
 1) 111 
 Hl,-< 
 II I 5 
 001) 
 h7> 
 I) 1 4 
 
 :i20 
 
 IldMeV 
 
 .Milk." 
 
 (.»;l. .\iii.M...-(.c,i. 
 •' CoJIi-li .. . 
 •• ('uiliin-M'Cii 
 " l.in.-^' cil ... 
 
 '• Xapiilu 
 
 •■ <) ive 
 
 " I'alin 
 
 •• I'rlrolt llhi 
 
 " l!:ipe 
 
 •' HniillMwrr 
 
 ■' i Miiiiiiliiie 
 
 •' \VI,,-,|r 
 
 >(i!ni. i( e.iitk'd. 
 
 Iiir 
 
 Viiu'u'ur 
 
 Water, Dia.l Sea . . 
 00". . . 
 " 2)2" .. 057 
 
 '• (li^lll!(Ml. ;il)"t .' 01' 
 '' Mcuitirranean . 1020 
 " ram. . . . . . li'OO 
 
 " sea . . 
 
 Wine. nur,uiiM(ly. . 
 " (,''anijia;:ne, 
 " ^ .ira ... 
 " i'ort 
 
 52 HI 2 
 
 4 1.087 
 00 025 
 
 1 5 
 
 01 .(i25 
 57.0,s7 
 
 5H.75 
 
 5;! 
 
 57,1H7 
 00.502 
 54.H75 
 57 125 
 
 ."m >'75 
 
 5 1 ;i75 
 
 1)2:. I .'7 0H7 
 H2li 5i 5 
 111 I.. I t;:! i;;7 
 liiHOl (iT .5 
 
 1240 
 1100 
 
 following iliiiii.s umlir a prc^-ure 
 
 n . .) 
 02. 14!) 
 
 50.H12 
 
 02 ;>7i) 
 
 04 ■.',\2 
 
 02 5 
 
 04 125 
 
 1)112 02. 
 
 11071 01 ;i75 
 
 11 ;iHi 62.;) 12 
 
 I 0071 02.;512 
 
 lOliO 
 
 of 15 11).-. per 
 
 .0000210 
 . 000015H 
 
 Mercury 
 Water. . 
 
 .00000205 
 (ioo(!40ii;i 
 
 * K|ifcili<; gravity ut I'-mit'.Hpiiit iicconiiii^f lo l.^rc'H 'l'iil)K' Cui- ;Sylit-h'« llyiivcuiu'lfi, 'Ml 
 t 1 cubic iiicli =.i.'ul3.tiil Tioy gniiiis. 
 
"WEIGHTS AND VOU'MES OF VAIUOTJS SUBSTANCES. 
 
 49 
 
 101a wti<? FluiclN. 
 
 It Ciihic Font of Atiiiospliciic Air weufJts oii?.!)! Troi/ Grains. 
 Its asKumcd Grarity of I in the Unit for Elastic Fluids. 
 
 AtniiKphcric uiivU" . , 
 
 Aimiionia 
 
 A ante ■ 
 
 Cdi'lmiiio nciil 
 
 " (i.N.V 1 ...... 
 
 Ciirlmretcd livilnigcri . 
 Chl.iriiit'. .... 
 
 (."liliii'n-cailiiiiiic 
 
 Cvimii^ieii. 
 
 Ga.i, c .al 
 
 Hviiruixi'ii ... - . 
 llyli''cliliiric aciil 
 Hy>lr.ic.yaiiic " 
 Muiiaiic aciil . . 
 
 Niiro;j:i'n 
 
 Ni;i'.i; I'xvii .... 
 Nilroii.-. aciil . . . 
 Nin-.iii^ li.xvd. 
 Oxygen..: 
 
 1. 
 
 , r)S',) 
 
 '.•TO 
 .'.172 
 
 ri.'i'.* 
 
 2 Al 
 
 3 :!s:i 
 l.sl.-, 
 
 . ( .1- 
 
 07 
 
 I.27S 
 
 .1)12 
 
 I -'IT 
 
 1JT2 
 .0'.)l 
 2 . (l.'.s 
 1 ,r,2T 
 1.102 
 
 Pliiisi)liiirc'tc':i liydni^Tcii 
 
 1 
 
 77 
 
 •>ii!))lmrcl 
 
 .■.1 
 
 1 
 
 17 
 
 Sulpmii'ij 
 Sleaiii,* 
 
 IS acitl 
 
 
 21 
 
 212" 
 
 ■1H8:5 
 
 Sinokf, o 
 
 Iiitiiiniiiijii.* coal. . 
 
 
 102 
 
 ii 
 
 Coki' 
 
 
 1 o.-, 
 
 (( 
 
 \V( X 11 1 
 
 
 0!) 
 
 Vajmrof. 
 
 llciilldl ... ... 
 
 I 
 
 (ilH 
 
 .4 
 
 isiiiplmreiot'carlioii 
 
 2 
 
 04 
 
 Vaifir (if 
 
 liiMimiio ..... 
 
 5 
 
 1 
 
 ti 
 
 cliliific ether . . , 
 
 A 
 
 4t 
 
 u 
 
 CI her. . 
 
 2 
 
 ijSG 
 
 n 
 
 hy.lrucliluric etlior 
 
 ■) 
 
 2.)5 
 
 u 
 
 iii.iini' 
 
 K 
 
 07;j 
 
 II 
 
 nitric acid 
 
 ■■> 
 
 7;-) 
 
 << 
 
 spirits (.flurpentine 
 
 t 
 
 .7o;{ 
 
 n 
 
 Milpliuric ucid 
 
 2 
 
 .7 
 
 It 
 
 " cilier . . 
 
 •) 
 
 .aSG 
 
 11 
 
 pulphur , 
 
 •) 
 
 .211 
 
 a 
 
 water 
 
 
 .02;! 
 
 ill Oviliiisii'v XJwo. 
 
 SUBSTANCES. 
 
 METALS. 
 
 ('iil)i(' Fiiiil '(^iit). Iiirh 
 
 Lbs. 
 
 '^--1: 
 
 me .->■> y 
 i.nin metal .Vi;? 7') 
 sheets. ... |51.'> 
 wire . .ja24 10 
 
 Co})per, ca^f, . . . 
 
 " i)late.s. . 
 
 Iron, ca^^t 
 
 " gun metal. . . 
 
 '* heavy t'orgini, 
 
 " plates 
 
 1/^47.20 
 :.)4:{ 02") 
 i4.j().4:!7 
 |4()0 5 
 1 179 .') 
 ISl.a 
 
 " wrunirht burH.i4S0.75 
 
 Lead, ca-t . . 
 
 '■ rolled. 
 Mercury, 00" 
 Steel, plates. . 
 
 " ,-oft,.., 
 
 709 
 711 
 
 S4S, 
 
 187 
 
 4S9. 
 
 7487 
 
 75 
 
 502 
 
 Lbs. 
 
 2829 
 
 .SI 47 
 
 297 
 
 ,3 179 
 
 :;io7 
 
 ,2t;07 
 .27 
 
 2775 
 .2787 
 
 2s 10 
 
 41(10 
 . 4 1 1 9 
 
 491 17. 
 
 2828 
 .2s;i:! 
 
 SUBSTANCES. :c;,bicF(.ut. cuii.incii. 
 
 METALS. 
 
 Till.. 
 
 ZiiiC, 
 
 cast ... 
 rolled. . 
 
 WOODS. 
 
 As], 
 
 I!av .. 
 
 C^i-k 
 
 Cedar 
 
 (.;he.-tnut.. . .. 
 
 iiicktirv, i)i^ mit . 
 
 '• ' i-helM)ark. 
 LiLMiuiu-Vita' 
 
 i'l.nuWCHid 
 Malicijiany, Ho 
 tluras . 
 
 '■'-\ 
 
 Lbs. 
 
 455.087 
 428 812 
 t49.4:i7 
 
 52.812 
 
 51..S70 
 
 15. 
 
 35.002 
 
 38 125 
 
 49 5 
 
 43.125 
 
 SI! 312 
 
 57.002 
 
 35 . 
 
 00.437 
 
 Lbs. 
 
 . 2037 
 .2 Is2 
 .2001 
 
 Cull. I'fet 
 it . Tun. 
 
 12.414 
 43.001 
 1 19.333 
 03.880 
 5.S.754 
 45.252 
 51.9J2 
 20 ^^(y 
 ;i9.255 
 1. 
 33 714 
 
 ^ Kqmil tu .U7u!.lUJ43 lbs. avtiiidiijiiiiii. 
 
 * Wfiglit lit' B culiio font, y07.353 Tro^' giiiiiiu. 
 
50 
 
 WRIGHTS AND VOLUMKS OP VAIUOUS SUH^TVNCKS. 
 
 SUBSTANCES. 
 
 Oak, (\iii!iili;ui . . . 
 " Eii^rlisli. . . . . 
 " livcseasotied 
 *' white, ilry.. 
 " " nplatiil 
 
 Pine, |iitcli . ... 
 
 " re.l 
 
 " wiiite 
 
 •' well sea.^onecl 
 " yellow .... 
 
 Spruce 
 
 Walnut, lilack, dry 
 
 Willow 
 
 " dry 
 
 Miscellaneous. 
 
 Air 
 
 Jja.-ialt, mean. . . 
 
 JJrick, fire 
 
 " mean . . , 
 
 I ,, _ . Cnl>. I'Vft 
 (Juiiic Foot. |„ „.,.„„ 
 
 G(j 
 
 o;{ 
 
 42 
 •11 
 .'iG 
 
 :u 
 HI, 
 
 .'IG, 
 •M). 
 
 li 
 
 25 
 
 75 
 
 7> 
 
 WM 
 
 2.-) 
 
 ^75 
 
 G25 
 
 .,<;:' 
 
 812 
 
 25 
 
 25 
 
 5G2 
 
 375 
 
 Coal, anthracite 
 
 " hi turn ill., mean 
 
 " Cannel 
 
 " Cumbcrlaiul . . 
 
 .(175291 
 175. 
 l;i7.5G2 
 102. 
 
 81). 75 
 102.5 
 
 80. 
 9 4.875 
 
 84.G87 
 
 41 
 
 :v.] 
 
 41 
 52 
 5V 
 (it) 
 Gl 
 75 
 
 Gi; 
 
 71 
 71 
 
 i;i 
 
 7.{ 
 
 .101 
 455 
 
 .55s 
 G7J 
 l(i9 
 
 .74; 
 .G'.t;', 
 
 .77S 
 
 .24 
 
 .GS 
 
 .GS 
 
 .2G;^ 
 
 .744 
 
 12 
 
 IG 
 
 21 
 
 21. 
 
 21. 
 
 28, 
 
 2:?. 
 
 2G 
 
 8 
 
 2s4 
 
 9i;i 
 
 9.", 8 
 
 STJBSTANCBS. 
 
 Coal, Wel.-il), mean 
 
 Coke 
 
 Cotton, hale, mean 
 
 " " pre.'Sfied | 
 
 Hartli, clay 
 
 " Common .«i>il 
 U'ravcl 
 
 u 
 
 dry. sand. 
 loii(;e. 
 
 
 molil. 
 
 '■ mud... 
 
 " with gravel 
 (Jranite. Quincy . . 
 Susqueh'nii 
 
 Hay, hale 
 
 " pres-^^ed 
 
 India ruhlier 
 
 viUcanizeil 
 
 Linie.<tone , 
 
 .Marhle, mean . . . . 
 
 854JMiirtar, dry, mean. 
 jWater, fresh 
 
 G09| " salt 
 
 45ll.Steani 
 
 C'liliic 
 
 Foot. 
 
 81 
 
 25 
 
 G2 
 
 5 
 
 14 
 
 5 
 
 20 
 
 
 25 
 
 
 120 
 
 625 
 
 i;)7 
 
 125 
 
 109 
 
 312 
 
 120 
 
 
 0:5 
 
 75 
 
 128 
 
 125 
 
 128 
 
 125 
 
 lol 
 
 875 
 
 TiG 
 
 25 
 
 1G5 
 
 75 
 
 1G9 
 
 
 9 
 
 525 
 
 25 
 
 
 5G 
 
 437 
 
 197 
 
 25 
 
 1G7 
 
 875 
 
 97 
 
 98 
 
 G2 
 
 ;> 
 
 G4 
 
 125 
 
 .0:)G747i 
 
 Oiil). KmhI 
 III II 'I'on. 
 
 27 5G9 
 35.84 
 
 154.48 
 
 114. 
 89.6 
 18 569 
 1 6 . 335 
 
 20 49 
 18.GG7 
 23.89.^ 
 17 482 
 17.482 
 
 21 987 
 17.742 
 13.514 
 1 3 . 254 
 
 235.17 
 89.6 
 39.69 
 
 1 1 . 355 
 13.343 
 
 22.862 
 
 35.84 
 
 34.931 
 
 A-pplicjition ol" tlio TiiLll>loH. 
 
 When the \\'ei(fit of a Sidi^ttdnce is rcqniicd \iv\.v.. — Ascertani tlio 
 volume) of the .suh.stauce in cuhic feet; mulii|ily it hy the unit in tlie second 
 column of tables, and divide the jiroducjt hy Hi; the (piotieiit will giv(i tlio 
 weight in pmnids. 
 
 \V licit tlie Volume is r/iren or (ificerfained i)i Inches. liin,i':.— Multiply 
 it hy the unit in the thiid eolunin of the tahles, and the product will bo tho 
 weight in pounds. 
 E.KA.Mi'i.K.— Wliiit is the wci^lii olii ciilio of Itiiliiiii inarhln, the sides being 3 feetl 
 
 :i;) X '.iros^nii m „:.. wiiicii -h iii= i5ii'.i.75 ibs. 
 
 Or of :i HplniVH of (Mst, iron ",' iiiclii'.i in (Ibinii'ier / 
 
 ■j3 X •5'-!:U) X •■-!'' n-fijlit ii/a cubic t/ir/t== l.08!t lbs. 
 
 Coitipsirtitivo "Weijylit of* Tiiiilx?!* in n, Green 
 aiKl S«'^s*.!«*oiietl tStiite. 
 
 
 TiinbtT. 
 
 VVvijflit of 
 
 ii{;iib. Ft. 
 
 'I'iiiiber. 
 
 N\ Hifflit ot iiCul fH. 
 
 
 Green. 
 
 >S«u8oneil 
 
 G leeii. 
 
 Seasoned 
 
 
 
 I.bs, Oz. 
 
 44. la 
 
 Llm. Oz. 
 30.11 
 
 :,Ma i 
 
 Cmlar 
 
 l^ii^lis 
 Kiga !• 
 
 
 Lbs. Oz. 
 
 'l\. 10 
 
 4H.I',' 
 
 Lbs. Oz. 
 :.'t* 4 
 
 Ash . 
 
 
 1 Ouk 
 
 43 8 
 
 ».-, oil 
 
 i r . . , 
 
 T) 8 
 
 
 
BALLOON. — 'WRianT OP PATTRRN'S. 61 
 
 To Oomimtti tlio Csii>n(Mty oTn Unllooii. 
 
 Rdl.n. — Fniiii siiccilic i,fr,ivity uC the riir in M.-;iins )i,ir culiir luot siili- 
 strjict tliat <•(' till' iriiz witli wlii'.-li it is iiilljitcd ; iiiiilti|i!y tlic rciri,iiiiil(,'r hy 
 the Vdlilliic (i('tl)(( biilldiill ill ciiliic feet ; (liviilf the lUiiduct liy 70()((, ;inil 
 from tlio (|iioti('iit, siilistnict tlio wcigiit of tlic Imilnoii niui its iilt.iclmiciits, 
 
 Kx.VMI'I.i: — 'I'lii^ (liiiliii'lHi- <if ii lialliKiM i.-i •.'•;.(! fi-i-i, jiM W(!ii,'li( is 100 Ioh., iiihI iIib 
 S|)f{'i^'i' uiiivilv (it ihe f^MZ with wliicli it is ililliilcii \n .0(1 (air buic^' iirtHiiiiirii ;it, I) ; 
 wliHl. iH iu capiici'y I 
 
 5'K^ ~ ''' ''' '""^ ''' '■''i"''^-*'^"'»"fl' ^^'^ ireirjht of air and gas in 
 
 -Wo ' "" =— 7000 '*'*'=•'■" •"' ' '^•'■ 
 
 To Coiiii>ii<o tlio OianK'tei' oFa. liiilloou. tlio 
 Weijyflit to I>o i-jiissetl l>eiiij»" jfivou. 
 
 By inversion of tlio procoding rnlo, 
 g/\y X 700 -^7^s' 
 
 grins per cubic foot, and d I he diameter of the balloon infect. 
 
 EXAJU'I.K. — (liven llu' clcitKMit.s ill tliH jiiHCHiiiiig ciisH. 
 
 ^'5!t7.'l(i+ino X 70nO-^.rJ7.0l-'!uT-.' 
 Uien 1 , , _ ^^'ISS^JI.O'J^M.fi feet. 
 
 To Ooiii]>iito tlio ^V<'i;»-lit or Caw t ]\Ietal by 
 tlio Woij»lit ot tlio Piittei'ii. 
 
 When the Fattern is of White Pine. 
 
 Rui,r.. — ^rnltipiy tlic wciiilit of tlio luittcrii in jionmls iiy tlie following 
 inultiplior, iiiid tlic iinxlnct will yivo tlic wcinlit of the casting : 
 Iron, 14 ; Urass, I.) ; i.oad, '^2 ; 'I'm, II ; Zinc, !:{.5. 
 
 When there are Circular Cores or Prints. — Mnliijdy tlic S(|naro of tlio 
 ilifiiiictcr of tlio core or print by its iongtli in inclios, the product hy .0175, 
 and tlio result is tlio weight of tlu^ liattoi.i of the core or print to he deduct- 
 ed from the weight of the pattern. 
 
 Jt is cnstomaiy, in the making of patterns for castings, to allow for 
 shrinkage per lineal foot of pattern. 
 
 Iron and Lead -Jtli of an inch, JJrass and Zinc I'^ths, and Tin j^2tli-