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Ko. 2. 
 
 CENTS 
 
 ' ' '^ I l|i 'i 'iii' I I I I I I PRICe. 20 r«r» 
 
 (^ o (^ (T ; .' 
 
 «nt,«d «oordlng to Art 01 th. P„„.„.„, „, 
 
 '-.. .» .. r.r ,„. .„„„.„„ .,,^ .„;,::;::;:;;;;:;:,^,^ ;:;:; ;;; - 
 
 ""• "" "•""■'•"■• '•»• 0' '»'«««o. .t ih. Department ol 
 
 Agriculture. 
 
INIRODUCTORY REMARKS. 
 
 All work in Practical Ociinu^try roquiros to bo rlnne nu'chanicalU', 
 exc('|it in llie case of ciirvts wliicli c.innot Ixi drawn liy niuans of enni)iasse8. 
 
 The 
 
 followiiif,'- are the nccesHiiry instnunentH ; — 
 
 PENCILS -cithiT H or HFr, shariK-m-.l to a wcclg(>-slia|ic<l jxiiiit, tho 
 Hat si.lc (if which »!iciiil<l rest aK.iiiist tlio riili . -i ilrawlii},' lim-s. A iiicoo 
 of tine sami paper is ali.,ut the lie,t tiling' for Iceepinj; tlie point of tlie pencil 
 sliarp, and saves the lilade of tlie jiocket knife. 
 
 RULER-niade of hard wood, at least six iiu'Iii-s l.>nj,', witli a straigln 
 I'dtfe, an.l divided into indies, and halve ., (piarters, eightlis, and sixteenths 
 tit an inch, 
 
 COMPASSES with steel, iH'ncil, and pen points whicli fit into a 
 socket in one of tlie legs. The stati..naiy Ir;,' sle.idd liave a mi-dle point if 
 IM.^silile, BO that its length may lie allei-.d to eoirespond to whichever one 
 of the moveable point.s is in use. The Ktatioii.iry leg should lie a tritle 
 longer than the other leg when the ]HMieil or pen jsiint is in use, and I'xactly 
 the same length when the steel point is in use. The pencil used ill tliH [len- 
 cil point should be a little .sofliT than that used with the ruler, i^d F or H, 
 and should be sharpened in the same way. In drawing circles it- -I • 
 sih.iild always be perpendienlar to the radm.-^ Properly constriuli d com- 
 pa.sses liave a hinge joint in each leg, so that when the jiencil or jien point in 
 in n.se, it can be kept peruendicnlar t.i the siirfai-e of the paper. If tins is 
 not attended to in the cai- ■ of the pen point, the pen will not work pro|K'rly. 
 The joint of the compasses can be tightened or loosened by means of a little 
 ne tal key which accompanies them. The joint shoidd not be so loose tli.at 
 the' legs will change their relative jiosition when the compasses are being 
 used, nor should it be so tight as to reipiire any exertion to separate the 
 leg«. Practice will teach just how tight it should be. Tlie compassea 
 shoidd b« held loosely by the joint only, lietween the thumb and first finger, 
 with the steel orneeille ]Biint restinir on the paper, without any pressure, 
 and thi) otlier leg maile to revolve around it. The stuih-nt should practise 
 until lie can draw several concentric circles without punctining the pa|ier 
 with the steel point. It is absolutely necessary that the steel point should 
 be a.s sharp as it is possible to make it. India ink only should be sed ii. 
 the iH'iis, as other inks corrode and spoil the |ioiiits. The two steel isiints 
 are iiserl together when it is necessary to measure or to set olf distances very 
 accurately. 
 
 A DRAWING PEN for "inking in ' straight lines. Its i«iints 
 should be ex^ully the same length and ground to a sharp roiindeil edge. I- 
 use it should be held nearly veitical, with the handle slightly ineliued in the 
 direction of the ed:,'e of the ruler, and drawn along the paper at a uniform 
 rate of sjieed witlioiit any stopiiages. It should be wiped out with a rag or 
 piece of chamois skin every time it is tilled, and before being put away. 
 
 PROTRACTOR, made of litlier metal, horn, ivory or wood ; used 
 for mea.-uring angles. It is not absolutely necessary as the student can use 
 iirubleiu xiv. for tliia purpose, but most boxes of mathematical instru- 
 
 ments contain a protractor. Its .' irin and instructions for constructing 
 one are give,, ,„ an exeicise on iirolilem xiii. In using it the centre of the 
 semi-crcle ,s placeil over the point whore the angle is to be constructed with 
 the diameter comeidiug with one line of the angle, and a pencil mark made 
 atthecirciiinfeieiieeopiH,«ite the proper miiuber. A line is then drawn 
 through tins point from the centre. In the form of the protractor shown in 
 fig. lit, mside the semicircle, the point corresijonding to the centre of the 
 semicircle is in the middle of the lower edge. 
 
 A SET SQUARE, being a triangle of thin wo,h1, will W found use- 
 ful, th .ugh not necessary, for drawing lurallel lines ali.l ereetilig perpen- 
 diculars. The ruler is held in position and the set sipiare sli.l along, 
 with one edge Hrmly pressed against it. A s piaie about five inches high, 
 having angles of 30°, 00° and 'JO will be most convenient. 
 
 Whin working the exerci.ses it would be well to work them first on separ- 
 ate sheets of p..,|,er until the best form, and t'le amount of space re.piired 
 for each IS a.scertaiiied ; then to work tlieni in pencil in the pi-oper |,osition, 
 in the book, and afterwards " ink in " all the lines. Each step should be 
 w..rkedoutbynieansof the methods given in the problems involved, but 
 witlh.ut Inferring to them unless nec.-.ssary. When every probl,.m is 
 thoroughly uielerstood mechanical methods may be adopted, that is. per- 
 pendiculars, parallel lines and angles may be drawn by me.aiis of the ruler 
 and .set s,piare, and the protractor. One of the objects in view sli..iild be I,, 
 commit the different methods to memory, and this becomes ...isy if the iva- 
 sons why each p,articular constniction is employed, are understood. The 
 proofs of problems given will bi. helpful in this diir-ction, besides satisfying 
 the eiirioiis of the truth of the results. 
 
 Diawiiigsonagiveiimi/f arc sometimes .a,sked for. Wlien an object is 
 represented as being A its natural size, it is evident that (i inches in the 
 drawing will represent 1 foot in the object. This scale would be indicated 
 by the words : (1 indies to the foot, or by a fraction, J. la - same waN- a 
 scale of IIU Would be one of 1 inch to thi^ foot, or a "scale .' ..W oni! of ( 
 inch to the f.M t, or a scale of IKi (3-48) one of :j inch to the foot. The 
 fraction indicates the pro|H,rtioii which every measurenieiit i,i tin- drawing 
 bears to the corresponding measurement in the object drawn and that for 
 every foot in the object, 1-2, 1-12, l-Ki or 1-48 of a foot must be taken in the 
 drawing. 
 
 When measurements are ivsked for in the exercises, the distances between 
 the proper point.s, or the length of the proper line should be taken by the 
 coni|,assi s, the compasses appliinl to the pro|i,r s,-ale, and the measurement 
 carefully ascertainerl and written under the .solution of the problem. 
 
 Till' sign ' attached to a figure signifies /.»)< or./Vrf, and the sign ", inch or 
 i>. ■Ii. K : thus, ;' d" reads 1 foot (i inches, and i" J" le.ads, 2 feet 1 inch, and 
 *Vn/f 4' III I" reads, Scale 4 feet to the inch. 
 
 In the exercises, bisecting lines are siijiiKised to Im> tenninated by the 
 points of intersection of the arcs employed. 
 
 Students are supisised to be familiar with Kuclid's Elements. 
 
 Points ore supposed to be joineti by straight lines, and when the word 
 line is used, a straight line is understood. 
 
 I 
 
 are 
 < > 
 
 and 
 
HIGH SCHOOL DRAWING COURSE. 
 
 PRACTICAL GEOMETRY. 
 
 PROBLEM L 
 
 To Bisect a Given Line. A B.-r/v,. i. 
 
 I 
 
 " itii A (IS a centre, 
 "siiig as a radius any 
 ciistaiico greater tliaii 
 lialf tlie length of A Ji, 
 as A X, dravv an arc 
 C X D. Witli n ns a 
 centre, nsin,- the sanio 
 •« radius, draw anotlier 
 arc cuttiMR the first 
 one in C and D. Join 
 CD. This line will 
 bisect A Jt in E. 
 
 Proof.— Join. -1^. AD 
 11^ ;™V''''- Then ni 
 Hio A8^Ci>and7.'t'y; 
 AC, AD, lie rM bJJ 
 
 wo all equal, because thev »,.„ , i- f ^^' A ^' ^' '^ a''^' -'>'i> 
 <^CZ>Lt/,,^'^''y«^t'>«yao fu c^ e<,„a circles, and the 
 
 < ^ Oi'.' ,. the ilt^AK ': .7™:"'""- ?"1 *"? < .-«. C ^ = the 
 
 fi,„ 1 ' T ;; "-"iiiinun, and t lo < A O f, 
 the ba«e A E .. the base £ A (eZcM i 4 ) 
 
 EXERCISES. 
 
 • J)raw a horizontal lino U inches long and bisect it. 
 
PROBLEM 11. 
 
 To Bisect a Given Arc, A'B.—(Fig. s.) 
 
 Fio. 2. 
 
 With A and B ns 
 centres, using any 
 radius greater tlian 
 Imlf tho lengtli of 
 A Ji, draw arcs to in- 
 tersect in C and 3. 
 Join C £>. This lino 
 will bisect the arc in 
 £. 
 
 Pkoop.— If ^ and S 
 he joined this problem 
 will become similar to 
 the preceding one. (Eu- 
 clid iii. 30.) 
 
 PROBLEM IIL 
 
 To Erect a Perpendicular to a Given Line, A B, at 
 one of its extremities.— r/ty. 3.j 
 
 Select any point, as C. Using this 
 as a centre and C A, tho distance 
 from it to the nearer extremity of 
 tho given lino, as a r.adius, draw 
 an arc cutting the given line or 
 tho given line produced in D. From 
 B draw a line through C, cutting 
 the arc in £. Join A E. This is 
 the required perpendicular. 
 
 Proof.— Tho arc E A D, containing 
 the < E A D, ia a aemicircle, ,". this 
 
 Fio. 3. 
 
 <isaL. {Euclia iii. 31.) 
 4 
 
 \ 
 
 EXERCISES. 
 
 . (a) Draw an arc of IJ inches radius and bisect it. 
 
 {/)) I5isect tho bisecting line. 
 . (a) On a horizontal line IJ inches long draw a semicircle 
 of f inch radius and bisect it. 
 
 (b) Bisect each half of it. 
 
 . (a) Draw a horizontal lino 1| inches long and at its left 
 hand extremity erect a perpendicular 1^ inches long 
 (by method shown in fig. 3). 
 
 (c) Bisect the perpcndienlnr 
 
 
irole 
 
 left 
 oiig 
 
 Fid. 4. 
 
 PROBLEM IV^ 
 
 To Erect a Perpendic^^to a Given Line A B 
 from a point, 0, within it. -r/Vy. ^.J' ' 
 
 With C ag a centre, 
 usiii- any radius, draw an 
 'i'o D E. With I) as a 
 ci.'ntif, and D C as ladiu.s, 
 draw an arc to cut the 
 first arc in /'. With /' as 
 a centre, and F D as a 
 ■f radius, draw an arc, If, 
 to cut the iirst arc in // 
 With // as a centre and 
 
 ///'Msrndius, drawanarc 
 to cut tlio iast one in K 
 Join K C. Tliis is tlio re 
 
 This problen. „.ay be u.ed for eret"' '""'P''"'''^'"'--"-- 
 the extremity of a line. '"'« " I^'^^P^^Ji'^ular at 
 
 Pboof.— The AoCFD, OF 11^,,,] S- p n- 
 <B D Cr, CFJl and KFU are <s c.f S^'^'T "'("""to^.-'I- .-. the 
 IS isosceles, having a vertical < <,f ]9o' • ir ^f V"- "'" A CFK 
 {Bx^M i. 32.) iJut />0i/' is an > A', ': DCKi^l '^ "^ ^'''• 
 
 PBODLEM IV.— ANOTUEB MBTHOD.— ('/'i^ ..;j 
 
 With C as a centr, 
 and with any radius, 
 draw a semicircle, cut- 
 ting AS in J) and £. 
 With £> and ^ as cen- 
 tres, and with D H as 
 radius, draw arcs to 
 intersect in/'. Join C/'. 
 Tills is the required per- 
 Jiendicular. 
 
 'PKOOF.-Join DF^EF 
 
 SFC, DC ^ nc FP i. . J hen in the /^^ i) },^ c and 
 
 base FF, ^'^reZ^'Z'^ht'J^riiJ':^^^/^ j!/. = "'o 
 
 these are adjacent <8 < i C f (iwlid i. 8) and 
 
 6 
 
 EXERCISES. 
 
 7. (a) Drawa vrtical JinrUl, "'"■'' ^"^''■ 
 
 i"it .1 iu^htintsfi";:;:? '•''^'"'''' ^""" -M-int 
 w withthep,,i,.,.n;:^i;^rs ,,s::u:;^''«-,:'>-, 
 
 nsa ocnhe, ,|t;nv an .,,•-. .,f i • V Pt'i'^'^ndioulars 
 
 and bisect tl,isa,v "" '' ■■"'""" '° ""-"' f'^n., 
 
I'liUlJLEAI V. 
 
 To Erect a Perpendicular to a Given Line, A B, 
 from a Point, C, lying away from it-rriff. 6.) ' 
 
 With C as a cpiitre 
 
 draw an arc to cut A B 
 
 in D and E. ^Vith D 
 
 a and A' as centres, and 
 
 witli 
 
 imy radius, diaw 
 
 arcs to intcrsoot in /'. 
 Join C /■; This is tlic 
 required perpendicuhir. 
 
 < il 7/ f (/!«.•/„/ ,.4); and tlif«ea.o adjacent <a. 
 
 PKOULEM V. — A.VOTIIKl! METIKIU. 
 
 //■vy. -.; 
 
 in A li fake any 
 points /> and C/. AN'ith 
 />as a c<>ntro and D C 
 as radius, draw an arc 
 C K F. AVith r,' as a 
 eentri! and 6' C as ra- 
 dius, draw an arc tu 
 cut tho first arc in C 
 and /A Join C If. This 
 is the required j'crpen- 
 dicular. 
 
 Proof.— .Toin^/r*, OIT, 
 IX'tmaJtll. In tho As 
 
 - j3r J. ''i^" ''"'' '^^ '■'' common. .-. tho < f,'(; 1) L tlm 
 ^ ir'r, ^y^-"^'- ^'>- -'^«'^'" '" t''" As COK and i/f/A', CV; 
 
 the < G A C = the < GKa{Eudid i. 4), and these are adjacent 
 
 EXERCISES. 
 
 9. (rt) l>iaw an oblique line 1} inches long and from a point in 
 It, [j meh from eitlier extronuty, erect a perpendicular 
 H inelies loni; (liy method sliown in f]fr. r>.) 
 
 {h) .Toin the mori' distant extremities of these perpendiculars 
 and dix iiUi this lino into four ecjual parts. 
 10. iJrawa verti.al lino 1 J J inches lon^ and select a point j 
 inch away from it and about opposite to its centre. 
 From this point erect a perpendicular to tiie lino. 
 
 "■ ':','} V'"''' "" '''^'''* '"'"'" ^"'""''' ''y ''"™ '* ""'"'3 long. 
 (A) \\ ith their point of contact as a centre" and a radius 
 
 of LI inches draw an arc to join their extremities 
 and bisect it. 
 (c) I'Voin the centre of this are draw a lino pertx'ndicular 
 to one of the original lines. 
 
 ar 
 to 
 
 Si 
 
It in 
 ular 
 
 lars 
 
 •It I 
 tre. 
 
 )nn;. 
 liii.i 
 ti('s 
 
 liar 
 
 PROBLEM VI 
 
 To Draw a Line Parallel to a Given Line AB at 
 ■' ^^^^'^ °i«tance, O, from it.-r^Z'.^ 
 
 JnA n t;,k„,iny 
 t«(j]„)iiit<. Z;,.,,,]^' 
 I'^om th.-so points 
 erect pcrpeiulieu- 
 Inrg DF andi'(7 
 
 (ri'oi)i,.iiiiv.). -Witii 
 
 ^^nn.l/.'M.s.'ontn.g, 
 »'"1 with a r,„li„s 
 
 -- to cut tl,„ ,K.,.pcn,lic„l,.,, ;,. ,. „„, 'T''W C'. /'-- 
 
 t'Tough these points wi!n,ep,„,„;/toTi ' '''""" 
 
 ^^ ^' (A«<..^»^ i. 5 ^ '^ "™ -iual and ,.,.all„I, ,. i^ a i« p,,a.lol 
 
 toiV, 
 
 KXERCISES. 
 
 12. (n) T),wv a vortical j;,,,, ] i ;,„,. ,^ , 
 
 t'xtromiti,.s,.,..,,t a .,,-,,"" "'"' '^f'""' <'f i(s 
 
 . . „i"-'i-n,ii,.„i,. ., ,„„i hi!::;" t^ "'" ^^"--t-^ of -^ho 
 r;:ii.uiarto,hov;'K;:;:ni:;:'':^;^^ 
 
 '. >"<iMg inotlio,! shown 
 "';;';;- 1;.MK ami at a, iistanco 
 
 J" tig. 7. 
 13. (a) I)mwanol,li,|u,. lin,. ■' 
 
 of i inch from it ,lra«- a linn u"^ ","" ? " "'"t'lnco 
 to it. ali.u, 1^ inches long p,ir„],^,i 
 
 It. Draw two ol>li,,uo lines U ;,„K , , . 
 
 raw a linn j.araljol 
 
 "":!;S;ni;'';,^li;;'^''-'r.-''''tapoint3inch 
 
 PROBLEM VIL 
 
 To Draw a Line Parallel to a Given Lin« A p ■ 
 pass ^through a Given Point o^/^S .;' '" 
 
 ^\ ith Casaccnti-o 
 
 <l'aw an arc to cut 
 
 ^JiinB. With O 
 
 as a centre, and 2) C 
 
 Pm. 9. o -' ,is ,.,^j|iyg^ j_,,^^^ ^^^ 
 
 With 7) as a centre and F f as r„ r ^ "'''' *" ''"'^ ^ '^ '"" ^'• 
 
 -from A in ^. Aliu'dr^r "^ -^"" "" '""■ '" "'* -'- 
 to .1 Ji. ""•'"" *'"■""«'' C-uiul ^ will be parallel 
 
 Proof. — ,7,,;,, nn m^ , , „ 
 (£W?" «/ .-vre eMUal, .-. tho < A}' '' .'l'"' '''» >"'»''« 
 
 
]'li()j;!j;.M VI 11. 
 
 To Divide a Given Line. A B, into any number ot 
 
 equal parts. -r^'Vy. lo.) 
 
 Dniw CD immllcl 
 to A II lit iiiiy (listuiiro 
 fiiJiii it (indlilciii vi,). 
 Ori C n N,t otr any 
 (lisfiimc, ns C /, n'n 
 niiiiiy tiiiicH ax tliii 
 imniliiT of cli\iNioiiM 
 
 ' '■'•i|'iii''''i".( /-'. 'I'liiH 
 
 n..iy 1,0 con., l,y nwuMS ..f tl„. .livi.l.Ts, or, l,y „„.,.„. .,f „ 
 .".■asm,., ior lauiM.h ,n,:y l.u «..t olF ,!,„ „.,,ui,v,l „n,Ml,fr 
 of tn,„.s. ll,ro„gl, C ,in,l .1 ,1,,.^, „ ]!„„ of in,l,.rmito I,.,,,,,), 
 and tluough 2> and /,' draw a li„„ to .,„vt it in E. Fro„i ] 
 ^, .■?, 4aml S draw Ii„..s to i; to out yl /y i„ „, J, ^, ,/,„„, ^ 
 rJiesn points will divide A li into ciual parts. 
 
 a,.,,li.d to any of' thu dh-isi'l: " '' ' • '^ ^ • -^ ^- Tlu« proof „,uj. |,„ 
 
 A.VOTHER MKTHOn (F'oj. 11.) 
 
 
 
 At ono nxtreniifv 
 of A Ji, as A, draw n 
 lino A C, forming an 
 angle will, J 7j',andoii 
 it M't ofT any convrn. 
 iiMildistanic, as iiianv 
 liiiii'S as tl;o niiiiilicr 
 of (livi>i()ii.s riM|iiir(.|| 
 in A Ji. IVoni tlio 
 
 EXERCISES. 
 
 Fio. 11. 
 
 la.st of tl.pso points, as /7, dr„w a linn to Ji. Pr,-,,,, t|,„ 
 other points, .;, ,J, Q ,,„d J, draw li„,..s parall-l to /; JI (proll,,,, 
 vu ). Thoso linos will div ido A Ji o.p.dly in tho points u, k a 
 and d. (Euclid vi. 2.) i . . 
 
 10. Tiraw a liorizonlal lino 2 incla's long and divi.lo jt into 
 Bi'Viii ocjiial parts hy mctliod kIiowh Ui lig. 10. 
 
 17. ]>rawav..rti,alIinolJincli..a long and divide it into ton 
 
 I'lpial par ts liy inctliod shown in lig. 1 1. 
 
 18. A person is sitting out cabl.ago plants^at r<iual distances 
 
 in rows -M JVet Jong. Ho putg 1 7 plants in oaoh row 
 How far aro tlioy apart ? Sculo J imh tc 1 foot 
 
 eq 
 
iito 
 tun 
 coa 
 
 JW. 
 
 I'KOBLEM IX. 
 
 To Divide a Straight Zi^^^A B. into parts whi-l. 
 will have a given ratio. - (n, i^) 
 
 Lit, it lio rtijiiiri!i\ 
 to ilivido A Jt into 
 t«'0 p,irt.s, wliicl, 
 will l)..,ir to ona 
 "iiiUlicr tlio lutio 
 "f I to 0. Draw 
 <'' T> ,,,„„!1M to 
 ■'' A lit niiy dii). 
 Ki.i. ]o ' ' t""™ fiom it (pro'l,. 
 
 "^^ °«: -'^ -- i-tf ^listanc, f,„n. tinn-IZm J;':' r ""^ 
 
 miio tiiiii'M fi„ni /•'(„ /; Ti.„ 1 . '"""'"Ill to /,nii(l 
 
 ^^.n^>^-z>i.to\^tc'^,,r:fi:^'^^"-fr''-v''r"«'. 
 
 p.j^r-'^'"' '-""<• ""- '■• -w. n .,., ,. .„„;,„, ,^„^„.,^ ^^^^ 
 
 PROBLEAI " 
 
 Angle, D E.-r/^ /.?.; ^° 
 
 "^ "li ^ ns a contiu 
 nii'I with any mdius, 
 diiuv an (1,0 to cut D C 
 in // and D E in F. 
 Usinif tlio siiino riidiiin 
 and 7/ as a ccntic, draw 
 an arc (o cut A B in 
 A'. With A' as a contra 
 and i--// as radius, dniw 
 
 A 
 
 an arc to cut the arc at K in T , ll f '"'''""' '''''* 
 from Ji TI,„ ? .^ r ' "'"' *'"'°"-''' ^' draw a lino 
 
 from £. Tho angle ABLh equal to the angle £DC. 
 
 (Hwlui i, 8.) ' • • "" < ■tt-L-.B' = tho < LJiK. 
 
 KXERCI8KS, 
 
 '''"'';;v::;:;;'i:;;:;;S^'r"«' •.--■ 
 
 20. D.vrid,, ,1 vcrti.al iiro 3 iM,'i„.M i,„ • . 
 
 „, ,,, '7<— t,„.ra:;;^:;':::^r''"""^^- ''^^'" 
 
 w-'.todnid,,itHothattl S,'S,V''''%^ 
 I'lii'S a.s nuHh as the (jr.t ■„ l '"' "'"'"•■ three 
 
 'lalf as n„n I, as the in V'' ""■" ^"^^ «'"«" '""'o 
 
 Biiow I.W the ciivij^J';tii^;;;;;;: ::;:.:;: ''^'"^'-" >- 
 
VnoiiLEM XL 
 
 To BiHoot a Given Angle, A B O.-r/'iy. ^.j 
 
 .A 
 
 A\"itli 11 IIS a cpii- 
 tio, (Iiaw ;ni .-uo to 
 I'lit A Jl iu J) iuiil 
 
 cii iu ]■:. Aviti, /) 
 
 .111(1 /,'ns cciidvs iiiiil 
 with any radius, draw 
 aivsto intersect in /'. 
 J'liu J', F. Tliis lino 
 will bisect tliu iiiiglo 
 A 11 C. 
 
 BdTf ^'lt"n ''nr ^;: '!'"As«/>Fa,„l lUCF, thomaos 
 
 KXEKCISES. 
 
 22. At tlie oxtreiiiitios „f a linmontal line ] J i,H-I„.s 1„„„ .„„! 
 
 '"lopi'osite sides of it ennstniet e,|ual au.dc.s wi'iid, 
 ^vlil be tegetlier le.ss tliau a right anVde. 
 
 23. Const met a right angle and biseet it. 
 
 24. (a) AVilh lines one iiieh lung construct an ngle c,,ual lo 
 
 Halt a I'igiit angle. 
 (/-) Ou the opposite side of one of then, .•onstr-iet an aiede 
 ocjual to one-,piarter of a right an-h'. 
 25. („) Draw two lines 1 ineh long forming a„ ,.„,,, , ,,,i t„ 
 two-thirds of a right angh'. "ji'.u to 
 
 (/') At the other extremity ot^one of them on the opposite 
 side of It construct an angle e,i«al to one-thin of 1 
 nglit angle. 
 
 PKOF'.LEM XIT. 
 
 iiO'(Emli,l i, 
 
 of 30'. 
 10 
 
 To 'JVi.M.Hit; a Right Angle—fF!^. j.;.j 
 
 Construct the right angle 
 A Jl C (iHo\)U;in iii,). Withi)';is 
 acentrodraw an arc to culT? J 
 in n and IlC in 7;'. With D and 
 ^ as cen- , ^s and .vith J) Ji as 
 radius, draw arcs to cut the 
 are /; /; iu /' and G. Join 
 Jl I , I! <;. I'heso linos trisect 
 the light angle ^ /y (7. 
 
 PifioF.-^^ ,Joiiii>^7, jcji'. Tlicu 
 J< I'' Jl is nn eciuilateiiil A 
 
 n "I'd tlxi < h' Jl /•• iu T,. - t 
 
 '■' '■«' . "ImI '. IIJ, an < of 30°, /. FB(! is also au < 
 
 mt 
 the 
 a;i 
 fori 
 lias 
 
 a sc 
 
 I 
 
I'l^OBLEM Xlir. 
 
 1 
 
 ! 
 
 '^^ -—(I'l'j. 16.) 
 
 ConstnictanViitanglo 
 nnd ti-isoet it (prohlem 
 -'^"•)- J^y foiitinuiiig 
 tli<Micsji;issiiigthrou,nli 
 ^^ iuul A' till tlioy meet 
 "1 /', and joijiiiig yy /,'^ 
 t'l" Hsht angl(^ Hill be 
 I'isiocted. In this way 
 angles of 30°, 45" and 
 «0' ^^•ill bo obtained. 
 Bisecting the arcs A D 
 and EL (problem ii ) 
 
 and 75'. By means of the dividers divif" '";■" '"'"'^' '^^ ^^° 
 eqiml parts. Tho.se divisiorof v t ""■'' "'''' "* ' ''' ''"t° ^ 
 
 "'-;"■" without meCni:;; a^^ ?, J" '~II enough to be 
 quadrant or arc of -JO" will l,e di i , ' ^'' '' '""' *''"-^ tf>« 
 "f which will ..ep,.ese„t 1 A li i ""^'° P'"''^' --'' -" 
 i{ will n,,..ko (7/y A' an angi o ,' '"? '■'"" '^ "^ "'--•- 
 will n>ake Ci?//an angle of 19°. ' '^ " '""^ *" ^'^''^i"'^ ^^ 
 
 Note.— It 
 will be a use- 
 ful exercise 
 to draw on a 
 piece of thin 
 but. still" card- 
 board a .senii- 
 <;'rele, and di- 
 ' vide the cir- 
 into 180 parts, as .shown in //„ /^ cnmfercnce 
 
 l.a.s^already been explainerl ' "'^' ^'°°'' O"- ''orn, Its use 
 
 EXEKCISES. 
 
 26. («) AtonocvtremityofTH^n ;, i , 
 
 ^^^ ■i"f;leof,5i>°. ^ "^'-'"^'"'^ long construct an 
 («') At the other extremity nn ti 
 
 -teen o^:l^:'^,'\tZT:iT'' '"^'^ - ''-' 
 «..xteen feet fron, the ^ven"; ntl t"""' l^f"' "'"' 
 d'-'iwn,g on a scale of J ,ol t ' f I' .,^^"'"' '^^ " 
 
 cSm«r''^^'-^-"'''-thi^::ii'St 
 
 ''■ '" '-i^:;^;;::; --S^t ^ -»^^'-'ar protector 
 
 
 ■ 
 
PROBLEM XIV. 
 
 To Construct a Scale of Chords. 
 
 (J'ig. 18.) 
 
 Oonstruct a 
 right angle 
 CA li (problem 
 iii.). "With ^ as 
 a centre draw 
 "11 arc to cut 
 A B in B and 
 ^CiuC. Tri- 
 sect tlie right 
 ■•iiigle, and fur- 
 ther sub<livide 
 it as (sxplaincd 
 in jirobleni xiii. 
 •'oiii/iC. Witli 
 B as a centre, 
 radii i; 5, B 10, 
 
 Fio. 18. 
 
 Bt5. Bsx,,^., „.p»,h,., ,,„, ""i " 5. mo. 
 
 .mmbred 10, iO, M, w L eU, , ' ° ™" " '» "'« P»nt. 
 .1..UM .» „i„«y ivi;.' ,„ '.h*- J° e Xw "•" """ 
 
 .l.o.n i„ U,. 1„„, p„, „, jtjV"' '"'«"'■ ""I ""B"! «. 
 
 EXERCISES. 
 
 29. Construct a scale of chords with the chord of 60' 1 1 in. 
 
 30 At tl >^.T'' T '' ■'^ ^""^'^""^^'^S the fdloiingU"; r 
 
 30. At the left hand extremity of a horizontal !,„„ i • 7, 
 
 construct an angle of 04° '""^ ^ '""'' ^°"fe'' 
 
 31. At the right hand extremity of a horizontal lin,. lU u 
 
 long, construct an angle of 75° ^^ '""''^" 
 
 a 
 /I 
 L 
 
 13 
 
 ar 
 
mciies 
 ingles, 
 h long, 
 
 inches 
 
 PROBLEM XV. 
 
 
 With ^ and/; as centres ana 
 ^ ^ as radius, draw arcs to in- 
 tersect in a Join AC, £0. 
 The triangle Cl if is equilat- 
 eral. (Sudid i.l.) 
 
 Fw. 19. 
 
 PEOBLEM XVI. 
 
 To °°°"™ot - E,„.a^,, Triangle „, a Given 
 * o 
 
 Draw a straight 
 line, 7/ C. Prom any 
 po-nt, D, in it, erect 
 I perpendicular D E 
 equal to A (problem 
 IV.) Througli^draw 
 "• 'ine, /' r/, parallel 
 to ^C (problem vii.). 
 With 7) as a centre 
 drawa semicircle cut- 
 ting .fii)6' in J5 and 
 C-. With li and C 
 
 as radius, draw arcs to cut the semicircle i:;CT; Th "^ f 
 A and 7/ draw lines from D to out /■ P. r i 7 J,, r''rf' 
 Dj-Gis the triangle required. ^'"'" 
 
 probt'.ir)i".^ |^,f "^^^^^^^^ of 00- (.e proof of 
 
 A DFO „ equilateral, and ita iititude^i'/i"/ "' «°^ ""J the 
 
 EXERCISES. 
 
 ''• ^"^ '':^:SttJr^ ^'^- '-« --truct an 
 
 ^*^ ^Zo^:'^'' '"''' "'"^^ '^ "- P«^P-"->- to the 
 
 ''• ^^^ "^JilaSSaiS: ^^ -^^- ^-^ --truct an 
 
 (b) Bisect two of its angles 
 34. (a) C-;-t an^^equnateral t^^^^^^^ 
 
 (b) Divide one side .so that its two carts, will K *i. 
 
 ratio as th« «de bears to tIeEuJe ""' *'' ^""' 
 
 
 i< II 
 
PEOBLEM XVII. 
 
 To Cfonstruct a Triangle, its three sides A, B 
 and C being given.— ('/Vy. qi.) 
 
 Wake D E oqual to 
 A. "W'jtli E as a cfintro 
 and a radius equal to 
 H draw an arc. WitJi 
 D as a c(^ntre and a 
 radius e(]ual to C draw 
 anotlier arc to intersect 
 ^ tlie first one in /'. 
 Join DF, FE. Then 
 FDE is t!ie triangle 
 required. (£'Mc«(i'i. 22.) 
 
 Fiu. 21. 
 
 PROBLEM XVIII. 
 
 On a Given Line. A B, to Construct a Triangle 
 Equiangular to a Given Triangle, CDE.-r/Vy. J; 
 
 At A make the angle 
 FAB equal to the 
 angle J'7)C, and at i5 
 make the angle F BA 
 equal to the angle 
 ■^.fi'C (problem x.). Pro- 
 duce the lines forming 
 
 •, . n . ■ . ■ , ""■"•' "' F- Then t,."o 
 
 AB\% equiangular to the triangle E C D. 
 
 triangiv 
 
 EXERCISES. 
 
 35. Construct a triangle the sides of which will be U 13 . 
 2 inches rcs|irctively 'i' '*' "i'"' 
 
 " <^-«atriangleth;sidesofwhichwillbeas5,7 
 ^ '" -I'i"-? ar':::,'S'^,;/f f-'" the end of 
 
 wall l.y a 1 e . d H I '"?'"" '>' " '^""'* '^•"' "- 
 
 .n...eltai elu 5 tilr, ''"rr"'"' '""i" ^'•°"> *'"' 
 to 10 feet. '"• ^'"' "• ^^-^l" of 1 inch 
 
 36 
 
 37 
 
 14 
 
WiOnLEM XIX. 
 On a Given Line A vt 4-^ r, 
 
 r/'Vy*. i'3 and Slj 
 
 /''im.~u^t C l,o tl,., 
 K^l^en vortical angle 0^^. 
 
 Prn,l„ce /fyj to J). 
 ^t ^ make tlio nnglo 
 l> A E equal to C 
 
 (l>''"I.Ion, X.). Bi.s,,<,.ttl,o 
 
 "iiikn Uio mif^lo A li !■• 
 '•<|iml to th,. angle yyy</'. 
 'a ■nicn(li,!t,i,ingl<. /'^/y 
 <■« tlif. isoscplos triangle 
 I'lvjiiircd. 
 
 l''i(i. 23. 
 
 p ii-ijiiircd. 
 Sl!|'<'^+ <')'^.l &;"f./"^^' ^^-'. The < 
 
 '■"' ""«'" I'., an angl,, „f 
 '' "iil<o tlio nnglo C/1^ 
 
 ''"'■'' '"< •" 'i'st ta«„. 
 
 EXERCISES. 
 
 '"^ ?.^*,S/;;:!^;;is^^;^^_2.j feet f.. ti. foot of 
 
 v^-i«t,.eLeig,;n^1,!-^:--b^touc,^^ 
 
 .39. („, 
 
 i moll long. ' ^'* ^ ""•'' Jong and base 
 
 
 u 
 
PEOBLEM XX. 
 
 To Construct a Square on a Given Line, A B- 
 
 (l'i<J. 25.) 
 
 At ono extremity, A, of 
 the given line erect a perpen- 
 dii'ukr, A C (problem iii.), 
 "lid make it equal to yl ^ by 
 means of an arc drawn with 
 ^ as a centre and ^ ^ as a 
 radium. With the same radius, 
 and B and C as centres, draw 
 arcs to intersect in D. Join 
 liD,DC. Then A CD 2! is 
 the square required. 
 
 PROBLEM XXI. 
 
 Fui. 2J. 
 
 To Construct a Square on a Given Diagonal - 
 
 Bisect A It in (7 by the line 
 -0 -£" perpendicular to it (prob- 
 lem i.). With C as a centre 
 and C^ as radius draw a 
 circle to cnt D C E in D and 
 >,g E. Join AD, DB, BE, EA. 
 Then ADBE is the square 
 required. 
 
 Proof.— The linos CD, CH 
 (A and CA being ucjual and the 
 
 Km. 26. -■' '\ " '/' ^ ^ "'"i -t" ^i are ul»„ 
 
 equal. (Enclid i. 4.) The <a 
 
 -n «on,,circ.l.s are Ls (E,u^UU iii. 31), .•' ^' nlli'i:f;^^^_ '"''" 
 
 EXERCISES. 
 
 41. On an oblique line IJ i,;;^o„g ,„„3,ruct a square. 
 
 (6) On efich side construct a square 
 43. Construct a square with a diagonal 2 inches long. 
 '' ;nrc^S:5i:;t:«^^--''-^« its diameter 
 
 (6) On each side construct an equilateral triangh. 
 
 are 
 ch( 
 
inches 
 
 imeter 
 
 PROBLEM XXII. 
 
 Make CD equal to 
 A. At C erect a per- 
 pondicular, CE Corob- 
 lem iii.), and from it 
 cut off C Z' equal to ]i. 
 
 ■VVith/Jasacenti^and 
 a radius equal to B, 
 draw an arc and cut it 
 by another arc drawn 
 T • r,^, "'"^ -f as a centre and 
 
 Jom/T/.G-z,. ThenC/'(7Z,i,e,;e 
 
 Fkj. 27 
 
 a radius equal to A. 
 oblong required. 
 
 anir; L^^EuZ I 'S'S^r ""' ^^^ '» » L. .-. all it. 
 
 PROBLEM XXIIL 
 
 are given. -(/'ij,. ^.y.) 
 
 Make C i) equal to A 
 Bisect CZ) in ^(problem 
 !•)• "VVitli ^ as a centre 
 and^C-or^iJasnulius 
 draw a circle. From C and 
 ^ as centres, with a radius 
 equal to Ji, draw arcs to 
 cut the circumference of 
 the circle in /-and //. Join 
 (^ ^. -F" D, J) If atulJI C 
 Then CZ-i)//;, tteob^ 
 long required. 
 
 J'lo. 28. 
 
 chord, of these equal arc, aVo 'airefual ° n^t.^ " ''' »"d the 
 J^^ or are Ls J?uc-/W iii. 31), • c'V/ W • "^ f''« <» '' ^^ J^ and 
 17 h..tfJ>B,B an oblong (iW W i. 34) 
 
 EXERCISES. 
 
 45. Construct an oblong wh^j,, ^j„ , , . , 
 
 inche.s. "'^'^ ^'" !»" 1 inch and IJ 
 
 40. Make a line 2 incheq ln„„ 
 
 :ii 
 
PKOBLEM XXIV. 
 
 To Find the Centre of a Given Oircle.-(/V 5.'>.) 
 
 Draw any cliord A li. Bi- 
 sect it (prol)l(Mu i.) and pro- 
 duce the biscctinjj lino to cut 
 the circumference of tlie circle 
 in two points Fund G. Bi- 
 sect F e m E (problen. i.;. 
 Then Eis the centre of the 
 circle. {Euclid iii. 1.^ 
 
 ANOTIIKU MKTIIOl). — ('/•Vy. SO.) 
 
 Draw any two chords as 
 A II and C D. Bisect thum 
 (prol)leni i.) and j)ro(luco 
 tlie bisecting lines to meet 
 in E. Then E is the centre 
 of the circle. 
 
 PitooF.— The lines biseet- 
 "ig tlio cliords A n ami vu 
 both contain tlio centre (A'k- 
 <■';■'' "i.- 1). .". the point Ji', 
 ivhioh is conniion to l>oth of 
 these lines, is the centre. 
 
 'IS method can be used for finding the centre 
 
 EXERCISES. 
 
 49. Dn-vw a circle 3 inches in diameter, and find its centre 
 
 JO. Diy a penny on the paper and trace a line around its cir- 
 cumference. iMnd the centre of this circle by n.eans 
 
 r. ,,.. , ' """""' '*''°'"" '" ^'i- -'■'• ««-'" ""ti^^ <m page 3.'). 
 
 Jl. With a radius of 1 inch, draw an arc less tha.t a sen.i- 
 crcle and ascertain the position of its centre by 
 means ot the method shown in lig. 30. 
 
 52. {a) By means of some circular object about 2 incites in 
 diaineter, trace a circle and divide it into four equal 
 
 (6) J5isect one of the quadrants. 
 
 (f) Trisect another of tiio quadrants. 
 
 18 
 
 I 
 
! cir- 
 eaiis 
 3 3r). 
 
 emi- 
 I by 
 
 s in 
 jual 
 
 PROBLEM XXV. 
 
 To Find the Centre ofl^ven Arc. A B.-r/V,. .,/.; 
 
 l>rnw any chord C D. Bi- 
 Bcot it by the line /'^(pro- 
 blc.ni.) cutting tho arc in 6'. 
 Join C 6'. At C, make the 
 ai'glo G C II equal to the 
 '"'fe'loCr/i? (problem X.) and 
 \s produce the line (77/ to cut 
 G E in JI. Then 7/ is the 
 centre of the arc. 
 
 1 he lino i li bisocts tho arc VGD 
 in 6- and tho lino C'i)in.Vaml 
 In theA8C^Xand7)GT. Or/ /f/fr^'^V^^ tl'e chord G ZJ. 
 base O A' ^. tiio boHo DX ' thoV>7 ' v u " «','""»"" and tho 
 '■ 8.) Again, in th. A^clP f ^,A n'v T, ')< <-? ^' \\i- <^'"'"^ 
 common and tho <cD// -^tL^Dr'ir'-^nii ^>?-<'}l i» 
 cliil 1. 4.) But CII ~ /' If \ </■"'-'/, . . C II = D II. (Ku. 
 
 (JiUc7/a 111 n \ » 
 
 i«thoce„t.eo£tl.a;."t^:Sall:r 
 
 , JT 
 
 PllOBLEM XXVI. 
 
 Points. B and 0.-(Fig. 33.) 
 
 With 7/ .and C as centres and a 
 radms equal to A, draw arcs to in- 
 tersect in A With/) as a centre 
 and the same radius as before, draw' 
 a circle. 
 
 Vm. 32. 
 
 is not 
 
 iiXEKOlSES, 
 
 •>'•''■ With a radius of 2" draw n„ „ 1 
 
 .'*.%~ of .oine circular ol^ect draw an arc and ,ii,di,s 
 
 '''''^:;,:;r?;;;rJ'''---*-'otopassthro^ 
 
 13 
 
PKOBLEM XXVII. 
 
 To Draw a Circle who"irc[rcumference wiU Pass 
 
 through three Points, A, B and C. not in 
 
 the same straight line.~( Fi;,. as.) 
 
 bisect tlicso liiuis I)y tlio 
 Wnea F O MuX n E {wo. 
 blemi.). Tlicn the point 
 of intersection of these 
 bisecting lines is tlie 
 eontro of tlie circle re 
 quii-i'd. AVith G as a 
 centre and G B as 
 •iidius draw a circle. 
 
 Note.— This problem 
 is derived from cor. to 
 Fio. 3.S. Euclid iW. 1). 
 
 TROBLEM XXVIII. 
 
 To Draw a Tangent to a Circle from a point, B, 
 in the Circumference.— f /'('-/. 34.) 
 
 Draw a radius from 
 B. At B erect a 
 perpendicular, B C 
 (T>roblem iii.). Then 
 B C is tlie tangent 
 ■ required. (Eueliiiiii. 
 17-18.) 
 
 ANOTHER METHOD. 
 (^iSI- S4.) 
 
 Braw any radius 
 A D, luid produce it, 
 making D E equal to 
 D A. Bisect A E 
 (pr.-.)>lvm i.). Then 
 /' //■ is the tangent 
 recjuired. {Eurlid'm. 
 
 10 
 
 EXERCISES. 
 
 ru. Draw a Irinngh! whose sides will bo 1 J', IJ' and 1 J' long 
 respectively, and draw the smallest circle 'hat will 
 contain it, 
 
 5S. Draw a tangent to a circle 1 f in dinmeter. 
 
 51). Throe upright posts are placed so that the distance from 
 the fir.st to the third is twice the distance of the 
 second from the third, and the distance from the 
 first to the second is two-thirds of the distance be- 
 tween the first and third. Show the position of a 
 fourth post which will be oquitlistant from the other 
 three. 
 
 CO. A circle L'" in diameter is touched by a lino forming an 
 angh> of 24^ with a horizontal line. Show the point 
 of contact. 
 
 th 
 rai 
 
 the 
 
lit will 
 
 from 
 of thn 
 
 -tin tlm 
 ice be- 
 n of n 
 
 1 other 
 
 inf( (in 
 ) point 
 
 I 
 
 PBOlil.KM XXIX. 
 
 To "-w^ T.„,ent y: „„„ o,„„ ^„^ ^ 
 
 point, A, without it. -//Vy. ./,7.; 
 
 Join tlio given point, J, 
 with tho ,-,.ntr,., yy, of the 
 cirulo. J!i,,,ct ^ J, t„ C 
 (problem i.). AVith C as 
 a centre and CA as nuliua 
 draw un arc to cut tho cir- 
 fumfercnco of the circle in 
 O and A'. Join a D and 
 ^ ■£■. Then either A I) „r 
 ^ -8? will bo the tangent 
 required. 
 
 «*dm» of tho circle, .-. i> A L r'tiii''*'''' '"• «!•) Butii />";': 
 
 Flu. 38. 
 
 ANOTHER MKTIlOD.-Y/'iy. .J^.^ 
 
 r>'-Hw a line from the 
 «''■''" r'm.t, ^, to the 
 ^ w'tre, y;, of the circle, cut- 
 ting the circumference in 
 ^- AVith yj as a centre 
 ">'cl Ji A as .adius draw 
 an arc, ^C Prom/) erect 
 a perpendicular to cut this 
 arc in ^(problem iv.). Join 
 ■£"-» cutting the circumfor- 
 !■"><'. 36. ei'co of the circle in /'. 
 
 the t,in,7«, I ■ I •^"''^ ■^■^- Then FA is 
 
 t,ingbu(, rc(iuircd. (jS-„c«(/ jji, 17^ ' -^ ^ is 
 
 31 
 
 KXMRCrSES. 
 
 its axis. .Show 1 1, ..ni. f . "■ "■"* r"''-'^""' t". 
 
 ^^ *-'"-'-."-wc::;ii:irsrr^^^^ 
 '"xrs;t"Shtrr^^'°^'' '■"'-- 
 
 cumleronce. '" ''" tangential to iu cir- 
 
 t'!9l 
 
PROBLEM XXX. 
 
 On a Given line, A B, to Oon8truc<. a Roffular 
 
 Pentagon— .Sywii'.i/ .\f,i/„„/,*-( /.•;,, ,;;) 
 
 AVith A itnd /( nn rtm- 
 tri'H and A tt im mdiiiH 
 ilraws arcH iiilm'HiM'liiijj 
 in // 1111(1 A', With A' 
 lis a (Miiitio mid A' A hh 
 iiiditm (Iniw III! iini iMit- 
 tiiif,' tlio (itlinr (w(i iki'cN 
 ill D ttiid /■■. .1 lit, // A', 
 ciittiiij;tlio liiHtuniiu /,. 
 From /) mid /', lliniii),'ti 
 /., dniw liiii'H (o cut (lid 
 fii-xt two (U'CK ill C mid 
 -A". Join A C and // Ji, 
 With C imd A' iih ctMitroH 
 mid C A iiH niiliiiH draw 
 arcs to iiiti'iNcit in M. 
 Join r J/ and AM/, 
 
 H'lii. 37. 
 Tliis Mii'thoil cannot lie jirovcti 
 
 ANOTJIKII MKTllOI)— r.VHBn//. (fiij, ,V,S'J 
 
 With A iiM u colli i'i> 
 and A It an nuliuH dniw 
 asriiii-circh«,cii(linK/l/i 
 ^ l)i-oduccd. in C mid di 
 vide tho I'irciiinfci-oiico 
 into llv<i (•(|iii;l jiarlH. 
 I''roiii A thioujjh I ho 
 sufdiid diviHioii draw a 
 lineal;:'. Drawacirchi 
 topassthroiii,'h //./Imid 
 ~^ (lirol)lcin xwii,), anil 
 on its ciri'iiiiifcrcin'o not 
 oir tho, distaiico A U iiH 
 nirtny times as is possible. Join tho p.in's tiius ohtainod. 
 
 eciual to SOO" divul. ,1 by thu number ..f hc ( ,V, ■■(;,/ i !•' ..J\ 
 
 Tho < CA 2 =2(180° -^ 5) or 72', whio; i. ] , ■(., ' '^•' 
 
 • See note on Polygcm on third page of cover. 
 22 
 
 Kiii. ;«. 
 
 EXERCISES. 
 
 f)l. 
 
 65 
 66. 
 
 On a vortical lino J long construct a rogular jxintagon by 
 II s|iccial niotliod. 
 
 Construct a pentagon whoso sides will bo 1| " long. 
 
 Construct an isoseolos trian^jlo whoso baso will Im J " and 
 \frlical nnglo IT. This trianglo may bo used as ono 
 of tho triangles around tlio ciMitro of iv jicntagon. 
 Tho baso will bo tho length of ono sido of tho pon- 
 tiigon, and tho vertex will bo tho centre of tho 
 circumscribing circlo drawn with a radius eiiual to 
 ono of tho equal sides of tho t>ianglo. Tho length 
 of ono bido of tho pentagon n' jst be measured oiF 
 aiouiul tho circumference of tho circle. 
 
I'UOliLK.M \x\r. 
 
 On a Given Line, A B. to Construct a Regular 
 Hexagon -.s>c.m;j/.M.,/_r/Vi/. so). ^^'^ 
 
 i''iu. ;io. 
 
 'VVith A /luH radiuHdmw 
 ■i .iivlo wlioso circuiiuor- 
 '■"•" will pa.s8 tlirougii A 
 '"'J yi UmMvux xxv.'.j. 
 Around tl,o ci.vumfcrc.ico 
 I'liifo clior.ls A D, Dh', K r 
 /•V,' ,111(1 (,'// oquill i^AJt 
 
 Tli-n ADEFGIi^xW be 
 
 tlieiwjuiml hexagon. (JF„. 
 cHd'w. 15.) 
 
 WiOBLEM xxxir. 
 
 0^2^:::^ ^^"^^V "^ '' l^o required to 
 
 inscnhoiuthegivrnoirdo 
 a reguliu- lioj)tag„„. Draw 
 adia»i..tL.r^yya„jdi,,ij^ 
 
 It lafo as many o.jual parts, 
 J", 'istlioroarasi.ieaintho 
 required polygon (prol,le,„ 
 ^'"•)- With^ a„d7?„, 
 
 centres and .J/yas radius 
 draw arcs to intersect \xx 
 C- ProM. C through the 
 f"-'"'"^ division, '2, draw a 
 line to cut the circunifer- 
 ence of the cimle in J) 
 Join A D. Around the 
 circumference j.lace chords 
 "qual to A D. Tlien A D 
 
 I'lo. 40. 
 
 iiii» method cannot be proved. 
 
 KXEEC'JoKS. 
 
 Cr. On a linn I' long cont 
 '''*■ •"""■''■"•'. r..gconKt 
 
 t a regi.lar hexagon 
 
 method iXniui. ';;;''«"'"•'"'-«""•'>•>".-..» of 
 
 69. Construct the greatest „„ 
 .,, „, *'y " circle I" in di,u, 
 '"• -"rtco seven points so th 
 fruui tlino others. 
 
 'H w 
 
 "^ will be contained 
 
 "0 will Lo;j'diHt„^t 
 
 33 
 
PROBLEM XXXIIT. 
 
 Octagc 
 
 On a Glvon Line, A B, to Construct a Regular 
 
 '^■■'--'iOn—Sj.ennl Jr,-/hod.—{Fuj. 4I.J 
 
 On A Ji construct an 
 isosceles tn'anglo A C B Iiav- 
 iiig a vertical an;,'le of 45' 
 (prol)Iem xix.). AVitli C aH 
 a centre and C A as radius 
 draw a circle and produce 
 A C and Jl C to meet its 
 ciicuiiiference in D and E. 
 Through C draw diameters 
 FG and //A', perpendicu- 
 lar toAE and B D respec- 
 tively, and join A 11, H F, 
 Vm. 41, FD, DE, EK, KG and GB. 
 
 Vumn. Tin, <B ,( (,'« and DOE are <a of 45° (Euclid i. 15), 
 amlhe , « /> ( v ,uh1 (I <J A nro<8ot 90^ .-. the <8 E C K Jd 
 (, (.Wftni. H„f.li> ftncIf(,rasiiniI.irre,i8.mtho<8 about Care all 
 i-MUftl anil iiro Hul.lciKhMl by equal chorda (Endid iii. 26 and 2!l) • 
 thiiocuwm loimii-ucled on J Ji is eipiilateral. Ayain • ' CK, CG, 
 O/., ( , ,.(,■,, nn- nil e.|Mal and the <8 contained by them are 
 oqua ; tho. «rA,V, ((/ K, 0(!Js, C BO, CB A, C A B, etc., ^nc 
 OMual (r.i„'l,d I, . ) ;, (1,0 ^8 K(fB, G B A, etc., are also emial. 
 lhooct.;aoii on .1 B m Ihoroforo regular, being both e.iuilateral and 
 
 I'I,M)BLEM XXXIV. 
 
 Within a Given Circle, H G, to Inscribe a Regu- 
 lar Octagon— Special jfei/iud.-fFi;/. J,;:;. J 
 
 Draw any diameter, 
 A B. Bisect it by the 
 diameter C D perpendicu- 
 lar to it (problem i.>. Bi- 
 sect the right angles ai;ou t 
 the centre of the circle, 
 by the lines E G and U F 
 (problem xi.). Join A E, 
 EC, C F, etc. Then 
 AECF n a D II will be 
 the octagon required. 
 
 S4 
 
 Ki.j, 42. 
 
 Note. — This problem 
 requires no proof. 
 
 EXERCISES. 
 
 71. Construct a regular octagon whose sides will bo |" long. 
 
 72. Inscribe a regular octagon in a circle of 1" radius. 
 
 73. Inscribe a regular octagon in a circle 3" in diameter by 
 
 means of method shown in fig. 40. 
 
PROBLEM XXXV. 
 
 To ^-J - EI w^ the Transverse Axis 
 
 AB, and Conjugate Axis. CD, are 
 
 Given.-r/% 44-) 
 
 Draw A B and C D, 
 
 ■;i-t right angles, bisect- 
 ing one another in E. 
 ■With C and D as cen- 
 V *''<"* and ^ ^ as radius 
 draw arcs to cut A li 
 i'l/V and /';.'. These 
 points will be the /oct 
 (plural of ybcu«) of the 
 
 F,„. 43 ''"'P'"- Between one 
 
 markotfonvl5anvnun,hn.„f • °^*''«^°".'»s^^and ^ 
 
 care to ha^•o the r d^s n^ ^r'?"' "' '' ~' '' -^' ^- ^> ^'-^king 
 PI- With /' n 1" :7"' '"■""•'■^''' ''^ *'-^ "PP--h 
 
 centres an.I 1 Jl „, radius Wi^i a' f™"'" ^'''' *''« «••'"«' 
 
 ^' j. and . /, .., Jl\ J^'^J^^ ';^^^ as centres and 
 
 radii draw ares to interse;* ,„ 7^ ^ ^' .l"'/!^ "^•'"^'' '' 
 
 d;s^;.!;-;rL-s^:- 
 
 Place a pi„ i, .^h of t o fo 1 "P°" "'" P^"'^"''-''*^' = 
 -tre„.ties of the oo,:; / ;'"^-°"- - one of tL 
 Bt--ing tightly around the three 1« ^" " P'^'*^" "^ ^^'-^'ad or 
 -dof the conjugate axis.td^St^rjV''^""^ ^^ '"^ 
 If the pencil be moved aroun.I tl,« "f ^™"' '''" ^™:>'°"- 
 
 EXERCISES. 
 
 '' '^ '~ ISt^^ll^r -/>"P--- trans, 
 r.^. By the practical metlJd rdX '^^'^ I-- ^°"^'- 
 
 '- will be 2' and s'tngrtptS; ^""^^^ ^''-" 
 
 ' i 
 
 SB 
 
AN'OTlIKIl MKTri(ll). — (FiiJ. ^4.) 
 
 Draw A li aiirl C D aX 
 riglit an<,'lps bisecting one 
 another in ^(problem i.), 
 With K as a centre and 
 radii E A and EC draw 
 circles. Divide these 
 circles into any number of 
 equal jiarts, say 16, by the 
 lines«A, <!(/, e/, etc. From 
 the points a, c, e, n, I, ii, 
 etc., in the outer circle 
 draw lines parallel to CD 
 and cut them by lines parallel to A B drawn through the 
 points a, c', e', n, /', A', etc., in the inner circle. A lino drawn 
 through the points of intersection of the perpendiculars will be 
 the ellipse required. 
 
 See note on tlie ellipse on third page of cover. 
 
 PIJORLEM XXXVI. 
 
 Fio. 44. 
 
 To Draw an Elliptical Curve when only tho Trans- 
 vers Axis, A B, is Gv7en.~( J'iy. V^) 
 
 Divide A B into 4 equal 
 parts (problem viii.) in the 
 points C, D and E. With 
 C and E as centres and 
 C A as radius draw circles. 
 With C and E as centres 
 and C E as radius draw arcs 
 to intersect in F and 6". 
 From F and G draw lines 
 tlirough C and E to cut the 
 circumferences of tho circles 
 in //, A', L and M. With 
 
 Fir.. 4,'). 
 
 .^and C as centres and F L as radius dniw the arcs, H K a.na. 
 Z J/ which will complete the curve. 
 
 2fi 
 
 EXERCISES. 
 
 76. By means of intersecting perpendiculars draw an ellipse 
 
 whoso axes will bo as 2 to 3, tiie transverse axis 
 being 3| " long. 
 
 77. Draw an elliptical curve whose createst diameter will be 
 
 3" long. 
 
n ellipse 
 !rse axis 
 
 r will be 
 
 
 PROBLE.\r xxxvri 
 
 , l^'^iw ■■my cl,o,.(l, ^ 7i. 
 Select a„_v j,„i„t, C, ill 
 'lie curve of tl.o ellipse 
 "■id tl,rou-l, C draw a 
 l'"(>, i)amll,.l to A n 
 , (l"(,l.l,.m vii.), cutting 
 tliucurveiuA Jiis^ut 
 '}'• ''"''l CD (pioMeiu 
 i),nii<Itl,n>ugli^',i,icl/' 
 tli'.iw ,1 line to cut the 
 lunoiu r,'nii,I //. Bj. 
 "•ct a H (i>,„ble,n i.). 
 . AVitli K as a cniitro 
 
 mf,,ui.p..„t,/,.,;,^ ,,, 
 
 I'lu. 40. 
 
 rinuv a ciivlo to cut tlio oliipso 
 
 ■•"'tl .lo„i tlioso points. Bis vt /" i/ '■'l/"v""r /,'' ■"', "' '■""' ^'. 
 ln-ou,;li t!K.i,c,;ntres T ?' t , 1 7^ s WI,. r' / '■""' ^' ^^ -^"'J 
 
 ami >'^ the conjugate axi.s. "'" ti-ansvcrso a^is, 
 
 PROBLEM XXXVIII. 
 
 Let A' and }' 1„, 
 two given ix)ints. 
 l'"iiul till? axes and 
 tliofiH'ioftlioellip.so 
 (l'iol)lonis.\.\.vv.ancl 
 xxxvii.). Draw 
 lilies from A" and }' 
 to cacli of the foei. 
 Pi'odueo one of tlie 
 lilies, as /'.< ): to (1 
 and bisect tlieamde 
 /'Vrr,'. Tlie bisect- 
 ing line will bo tlie 
 tiiugent re(|uired. 
 Jiisect the angle y.'/ 
 
 i"ii line will be the perpendicular re.iuirH^'''' '•^'"-' ^'^'•''■^ 
 27 
 
 EXERCISES. 
 
 '9- («) By method shown i„ fig 45 draw an elii,,,! I 
 
 w ,,se .ransv<.,.soaMs will be 3/ Ion-' """' "'"''^ 
 i)]'ind us conjugate axis, and its fok " 
 
 ^^^ tt:;n::^*"^,irs,^,:;'-r-r-»^-..onedraw 
 
 alineperpe,Hl!cS:;:'t:'::'t«''''^"^''°--'''''''-'' 
 
PEOiiLPLM XXXIX. 
 
 To Draw an Oval of a Given Width, A 'B.—(Fhj. 4s.) 
 
 r 
 
 Draw ,-1 cii-clo witli a 
 radius ('(lual to oik^ lialf of 
 A li. Draw the ciiaiiictor 
 CD porpondiculai- to A IS 
 (prolilom iv.). From A and 
 Ji draw linos through D. 
 With A as a coiitro and 
 -I II as ladiiis draw an arc 
 from /,', cutting,' A D jiro- 
 diufd, ill //. AVitli n as a 
 cciitio and li A as radius 
 draw an arc from A, cut- 
 ting BD inoduccd, in F. 
 ■\Vith D as a centre and 
 D /'as radius draw the arc 
 Fi,;. 4,S. /'//. 
 
 peoble:\i XL. 
 
 To Draw the Involute of a Given Circle, A B — 
 (Fig. 49.) 
 
 Divide the tirclo into any 
 iiumlicr of equal parts, say iL', 
 ill the ])oiiits J, J, ,/, 4, etc., and 
 frciM these points dniw taii- 
 K<'nts to the circle. On the 
 tangent from 1 measure the? 
 len.ijth of one of tlie divisions 
 of the eireuinferenee nf the 
 circle, as J, J'. On tlio taiij;ent 
 from ,.' measure tlie leii.:;th of 
 two of th(^ divisions, as X.', !.''. 
 On each of the tanfjent;; mea- 
 sure the length of the iiumlier 
 of divisions indicated by the 
 iiuinljcrat the point from which 
 the tangent is drawn, and 
 through these jioint.s, F, S', S', 
 Fia. 4'J. 4'i t-'tc,, draw the curve, 
 
 28 
 
 EXERCISES, 
 
 so. The transverse axis of an ellipso is -j" long. ]ls foci are 
 .} ■ from eacli end. Find tlu. length of the ctnju-atf 
 axis. '' 
 
 til. Draw the involute of a circle J" in diameter. 
 
 11 "^^ 
 
 '< 
 
 To 
 
 HCl) 
 
 Not: 
 than or 
 line am 
 
 2n 
 
Is foci am 
 tTnju,'j;,-ito 
 
 PROBLEM XU. 
 
 I'l^iccfc tlio given 
 
 If )• In ono of tlio 
 'i'i"H no ov DC hikn 
 ['"y V'int, J'\ nml trom 
 it erect a pfrpcndicu- 
 J'lr, /•//, (problem iv.) 
 c^liml in ]ongth to A. 
 J'V/,iM // draw a lino 
 piirallcl (o D C (prob- 
 '"'1 vii.) to out C K 
 '" 1^- With K as a 
 '■'•iitronnd A or ///'as 
 'ifiMliuHdniwacircIc. 
 
 ^'Ki. 50 
 
 PROIiLKM A'J.li, 
 
 . ^^"'^ '"'y '•'"lii-M of tl,o given 
 ""•^l", .iK r/ A, and p,.odu..o it 
 <<ml<ii,g // // ,,,,,„.| t^ ^_ ^^,.j,^ 
 
 ^'a«H,c,(,r„a„dr///a8 radius 
 
 •'•"«"m arc ///.'. In ^/"select 
 
 '"'y I"'int., A and fro,,, it draw a 
 
 I'll", LM |"Tpe,„lic,ilar to EF 
 
 !""'''"•» !*•■) "Md ,.,,„al to A. 
 
 ',""" •'/ '''■"«' 'I 'i»'3 parallel to 
 
 ,,;/'"■"'''"'" Vii.) to c„t thn arc 
 ///. .u.V. Wi.l, ^„.,. .,,,,, t,„ 
 
 '""I ^(, or/, ,)/„« radi„sdrawa 
 
 ^j^;'^^;.>^t.ost,.aigbtii::;;';^;^;;;'r'' '""' "■•" ^'^"'" 
 
 thanoiie-halfof'tbo'leastdisfl.','!''''',''''. "''''" '"""•*■ ""* 1"' ''■»» 
 line and tho given circle ""'■'■"" *'"' «'^"" "f^ig'-t 
 
 2!) 
 
 Fin. ni. 
 
 EXEECISES. 
 
 ^_^. I|r„«. an oval, ,l,e ,,„„,,,t ,,idtl. <,f .-bich vill bo "^ 
 -'.iW_aen.loofr,.adiustot.uebbotbJinesofa;;„g,e 
 
 Hfunv tbeir relative iorio,:!"'' " ''""^'^"*''' -*-■ 
 
PROBLEM XLIII. 
 
 Within a Given Circle, A B C, to Inscribe any Num- 
 ber, six, of Equal Circles, each one to Touch 
 two Other Equal Circles and the 
 Given Circle.— (Fi^. r,:^.) 
 
 Draw any dianictor AD 
 and at A diviw a tangent 
 /lF(pi-oblonixxviii.). Di- 
 vide the circlo into twico 
 as many equal jiarts as 
 tlio nuiidii'i- of ciivlcs 
 n-ijuirt'd, in this casij 
 twelve (in'oblcni xii.). 
 Produco one of the dia- 
 niotors closest to A D, as 
 G II, to nicet the tangent 
 in S. Bisect the angle 
 AS (problem xi.) and 
 produce the bisecting lino 
 
 EXERCISES. 
 
 Fio. 52. 
 
 to meet A D in T. On the alternate radii N, B, d"o C, 
 and K measure from the distance T. AViUi the points 
 a, b, c, rf,/and T aa centres and TA as radius draw circles 
 which will fuKil tlio conditions required. 
 
 I'ROBLEj\r XLIV. 
 
 To draw a Circle Touching the Three Sides of a 
 Given Triangle, A B G.-~(Fi(j. r,s.) 
 
 Bisect any two of the 
 angles, as C i? ^ and 
 II A C, and jiroduce the 
 bisecting lines to meet 
 in D. With B as a 
 centre .and its least dis- 
 tance, as D E, from any 
 one of the sides, as a 
 
 Fia. 63. 
 
 SO 
 
 radius, draw a circle. 
 {Euclid iv. 4.) 
 
 86. A tin.sniith has a circular sheet of tin 24" in diameter and 
 
 wislies lo cut out of it eight ecjual circles. Sliow by 
 a drawing on a .scale of \ how lie would proceed to 
 cut the large circlo. 
 
 87. What will be tlio diameter of the largest circlo which can 
 
 be drawn in a right angled triangle tht; sides contain- 
 ing the riglit angle being 1 J" and iV lung respectively 1 
 
 eq 
 as 
 an 
 eqi 
 
 In 
 
PROIiLEiM XLV. 
 
 About a Given Circle Ann* « 
 
 _ Klltl tho CPutl-O, X, of the 
 Circle (i.i'oblPin xxiv.). Sflecfc 
 I'lypoiMt, ^.intliooircunifer- 
 enco a.ul witl, A as a centre 
 ■wl A X as r.ulius draw an 
 arocnttiM-the circumfcrenoo 
 1" /) and ^^ J„i|, 2) j; ■yyjj.j^ 
 
 -O and B as centres and D E 
 as radius draw ares to inter- 
 sect in F and join D F and 
 
 equilateral triangle inscribed in l^^Jt^^^^^^ 
 as centres and /> A' as rn,!,-,,. i *> "tn -y, A and /' 
 
 and K. Join ^ ^ ^.'^Vr ^l^/^^^^^ ^ ^/^ 
 equilateral triangle rcjuired. ^^ "'" '^'^ *''" 
 
 PROBLEM XL VI. 
 
 r^''V. nr,.) 
 
 Draw any radius 7/A- and at 
 A draw a tangent, Zjy (problem 
 ™.). At A', on Z J/ make 
 the angle Z A' .V equal to the 
 ""glo /' G D, and n,ake the 
 
 angle JZA'Pequal to the angle 
 ''-OG (problem X.). Join JT/' 
 
 TiieuA-iVP will be the triangle 
 required. {Euclid iv. 2 ) 
 
 Fio. 56, 
 
 EXEKCISES. 
 
 Sa W,at will ,,e the length of tte sides of tho smallest e • 
 ;;;-;;ljnang,e wbieh will contain'T^S^^l^^;; 
 
 q;i Tr „ ■ . ,,, : °-^'"'-^"'''n'«wui-ement? Scale 4 
 
 IS 
 
 i 
 
 I 
 I 
 
 .J 
 
 81 
 
TKOBLE.M XLVII. 
 
 About a given Circle, A B 0, to Construct a Tri- 
 angle Equiangular to a Given Triangle, 
 DGF.-(jy.6tJ.j 
 
 Produce ono side 
 i) /' of tliu given 
 triangle, indftlMitcly 
 to // and A'. Draw 
 any diameter, /. M, 
 of the given circle. 
 At 2f on L Jf niako 
 the angle Jf N P 
 ,, equal to the angle 
 A' /' G, and make 
 the angle M N It 
 equal to the angle H DO (problem x.). At M, ]' and Ji draw 
 tangents to the circle (problem xxviii.) to intersect in S, Taud 
 V. Then 5 2" K will be the triangle required. {Euclid iv. 3.) 
 
 PROBLEM XLVIII. 
 
 Flu. 56. 
 
 Within a Given Square, A B C D, to Construct a 
 Regular Octagon.— ("/Vy. .77.) 
 
 *fv 71 R 71 B 
 
 Draw the diagonals, A C and 
 B D, of the square, intcr.secting 
 in E. With ^(,y;, C'and /) as 
 centres and A E as radius draw 
 arcs to cut the sido of tlie 
 square in F, O, //, K, L, J/" and 
 N, P. Join F^r, lf1\ TAUviA 
 N K. Then N K FMN PLC 
 will be the octagon required. 
 
 Fig. 57. 
 
 32 
 
 EXERCISES. 
 
 91. About a circle 1" in diameter construct a triangle equi- 
 
 angular to tiie ono mentioned in exercise '.)0. 
 
 92. A carpenter lias apiece ..J timber 12" square and wishes 
 
 to make it octagonal in form. How would the 
 change be efl'eoted t Scale J. 
 
 93. What will be length of the sides of an octagon whose 
 
 diameter is 2^" 1 
 
i'liOBLE.AE XLIX. 
 
 To Find a Third PropoJ^^al to Two Given Lines. 
 
 First.— Lot it bo w- 
 'I'lii-i'fl ('. find a third 
 pi'oiioi'fioiml wliicli will 
 ho groiitcr tliaii A or Ji. 
 -{I-'i'J. o!/.) 
 
 Draw any li„n C F 
 inakiiifr C D eiiual to A 
 and D F equal to B. 
 From C draw a line, C6' 
 
 lien 7/A wdl bo tl,e proportional re 
 • (^ ^^ ■: ClI : II K. {Euclid vi.n.) 
 a 
 
 SEcojfD.-Letitborocjuir- 
 ed to find a tliird propor- 
 tional wliicli will bo less tiun 
 A or n.—(Fi,j. GO.) 
 
 Draw any lino, C^, mak- 
 ing 6'2> e(]ual to A and D F 
 equal to i.'. From C draw a 
 line, CO, of indefinite length, 
 
 FiQ. 58. 
 
 quired, that is CD 
 
 Flo. 59. 
 
 forming any angle with CF and "'":;^'"°*""'''fi"ite length, 
 
 B. Join 113 Fro n^^if' ^,. " '''""'"''' ^ ^^ "'l"-! to 
 
 .. , ■" .1^. X loni ji draw a nie parallel to // /) /^ 1 1 
 
 vii.) to meet CG in K. Then // K J !? ^^ "'''" 
 
 required, that is, CD : C }■ CU • // /'° '^ '^'"''P-"''--^' 
 
 Ml 
 
 EXKRCISKS. 
 
 04. D.-aw two lines V and J" Ion. respo,.,ively an,l fin.l n 
 th.rd proportional wl.ieh will ,„! ,,,.„„.,;!' , ^ '^ 
 
 J6. iinda thinl proportional (-reateri t,. tw . l' i • 
 
 tiM! ratio of a and \ ^''"^'"•^'^ *" *"" '"»'•'< bearing 
 
 37- Find a third Jn'oportionnl (less) to two Jines 1 1 ' a. -1 V 
 long respectively. ' " Jints i^ a. .1 I 
 
PR0I5LKM I.. 
 
 To Fiiul a Mean Proportional Between two Given 
 Lines, A and B.— (/Vy. .;>'.) 
 
 Draw any line, f ' /■', iii;il<- 
 iiij,' CD oi|iial to A and 
 D F ei|ual to Jl. liiscct 
 C/'in^'(|inibli'mi.). With 
 6' ns ,a orntro and C as 
 radius draw a somicinlo. 
 Friini J) eii'ot a jipi'pen- 
 dicular (prolilcni iv.) to cut 
 the circuMifercni'o of tlio 
 i-cniicirilo in //. Tlion D 11 
 
 Fic. 60. 
 
 will !)(> tli(( moan innportional required, tliat \y; C D \ I) 11 \' 
 DU-.DF. (Euclid \i. \Z.) 
 
 Note. — Thi.s problem is used when it is rponircd to con- 
 struct a siiuaro ciiual in area to a given oblonj;. For instance : 
 Let it bo rciiuircd to construct a scpiarc wliidi will be p(iual 
 in area to an oblong whoso sides are eipial to A and li respec- 
 tively. (Fi<j. til.) 
 
 Draw any line C /'eciual 
 in length to the sum of A 
 and U. Bisect this lino in 
 C. On it draw a semicircle, 
 and from Z* erect aperpen- 
 flicular to cut its circuin 
 feivnce in If. On D If con- 
 struct a S(|uaro 1) L t\ If 
 which will b(! the scjuarc 
 required. 
 
 Fl.:. 01. 
 
 84 
 
 EXEltClSES. 
 
 !VS. Find the nie.in proporlion;.! between two lines 1" ;ind 1!" 
 
 long, respei^tively. 
 no. AVhatwilllm the length of the sides of .a .squarn ecpialin 
 
 area to an oblojig ;,i " x 1 " '( 
 100. A farmer has a lield RiO' .v .-iOO' a,, 1 exchanges it for a 
 
 .square one of thi^ s.-iuie area. What will be the 
 
 length of its sides? Scale L'UO' ^ V 
 
ghni.:ral notks. 
 
 INKINOIN.-Thn toachnr »I,„„M 
 
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Tim in(;ii SCHOOL 
 
 DRAWING COURSE. 
 
 1. 
 2.- 
 
 TIIK FOLLOW |N(i AKK TKK y.ooKS IX THIS CdCUSK: 
 
 -Freehand. :>,. — Linear Perspective. 
 
 -Practical Geometry. 4.— Object Drawing, 
 a— liidusti'ial Design. 
 
 These Hooks are fully illustrated, and printed on heavy drawing paper. They are sold at 20 cents 
 each, at all bookstores. Iva.li Hook is a Te.xt-Book and Drawing- Hook combined. 
 
 The Grip Printing and Publishing Company, Publishers, 
 
 26 & 2,S rkON'I' STREET WEST, TORONTO.