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Les cartes, planches, tableaux, etc., peuvent §tre filmds d des taux de reduction diff^rents. Lorsque le document est trop grand pour §tre reproduit en un seul clich6, il est film6 d partir de I'angle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'image j ndcessaire. Les diagrammes suivants illustrent la m^thode. -1, 1 2 3 32 X 1 2 3 4 5 6 f I / AUTHORIZED BY THK COUNCIL OF PUBMC INSTRFrTFO? ►N. CANADIAN COPVRIOHT RDITION. E t' C L I D S Elements of (reoirietry: BOOK r., VmiGmu FOR THE l,8K« oF ..UNIOR CLA8SK8 .N PrBLIC AM. PRIVATE 8t!HOOLS. BT KOBKKT POTTS, M. A. VWK HUNDREnTH THOl'SANl). TORONTO : ADAM MILLER &. ri>. 1876. entered according to Act of ParlUment of Canada, in the year 1876, by ADAM MILLER A CO., in ttie Office of the Miniiter of Agriculture. A FOl A lint The e A 8trn A supi Thees A plan straight li A plan plane, whi A plan( oue anoth( EUCLID'S El-EMENTS OF GEOMETRY. kr 1876, by BOOK I. DEFINITIONS. I. A FOINT i» that which has no parts, or which has no magnitude. II. A line is length without breadth. III. The extremities of a line are points. IV. A Btaraight line is that which lies evenly between its extreme points. V. A superficies is that which has oi^y length and breadth. VI. The extremities of a superficies are lines. VII. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. vin. A plane angle is the inclination of two lines to each other in a plane, which meet together, but are not in the same direction. IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. E'. « ^"^'^'in's KI.KMBNTS. a letter pla^cVd'fA^ ""^^ «"e angle Rt ^ • '^^' M caUeJ the 4 ai -^•*'--'';'°— .«,,. '■"•-"«■-'-.„.«,,, th ^ tenn or ooonaar, ,« ^iae «^*rem,ty ot any tl^ft,^. Afl^isthat.Hch.enoIofe7w one or more bound- arfes, •"ay be expressed bv ,7^^„^eupononeof "^' ^nus theanele « "amed the anf f ^«J«medthea„"f;: '' ''■e. i« called the \"S^l^ IS called a *« other 18 called angle. leSi DEFINn.0N8, ^ XV. A circle » a pmiie UgUfe contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. XVI. And this point is called the center of the circle. XVII. A diameter of a circle is a straight line drawn through the center and terminated both ways by the circumference. • ' XVIII. A semicircle is the figure contained by a diameter and the part of the cucumference cut off by the diameter. f""- "* O, XIX. The center of a semicircle is the same with that of the circle. XX. Rectilineal figures are those which are contained by straight lines. XXI. Trilateral figures, or triangles, by three straight lines. xxn. Quadrilateral, by four straight lines. XXIII. Multilateral figures, or polygons, by more than four straight linee, EUCLID'S BLRMENTS. xxrv. chreTe^Xlde:? ^"'' *" ^^""'^^'^ ^^^^^^^ " ^' -^^ ^ XXV. An isoaoeles triangle is that which haa two sides equaL XXVI. A scalene triangle is that which has three unequal sidca. aid XXVII. A right-angled triangle is that which has a right angle. XXVIII. An obtuse-angled triangle is that which has an obtuse angle XXIX. An aoute-angled triangle is that which has three acute angles. XXX. ^nf^T.t'sriTfrS'"' ^""^- • ''"•" "" "" «•««- 1-' I angles right angles beir are |oini L point Tl a strs from lat which haa !& uigle. DRFTNTTTONg. XXXI. Il\idlrf„:ir* "'"^^ ^ '^^ »^ -8»- "ght a„KK but hiu ail its sides equal. not XXXII. A rhombus has all its sides eq ual, hut i ts angles are not right angles n XXXIII. A rhomboid has its opposite sides equal to each rth^r h»t .n «des are not equal, nor its angles right angles ' its a XXXIV. AU other four-sided figures besides these, are called Trapeziums. XXXV. \.Zl^t}^\ '^"'1'^''' ^'"^!i ^'^ '"'^'^ *« ^'•^ >" the same plane, and which bemg produced ever so far both ways, do not meet. A. areVr?lS'.3";v!' V''"""'^'^''^ ^?"''^' "^^^ich the opposite sides Kto ofTts^^'p^st^^^^^^^^^ ^^ '^'^^^-^^ '« ^^^ '-^^^ '-« mgles. sides equal POSTULATES. poin\Tot';oXfp'oint^* ' ''^^'^' ^^""^ "^^^ ^« '^-'" ^-^ -^ »- n. • eSht w '"''''^ '*'"^^^* ""' "'^y ^'^ P'^^"^'* t° «"y ^«°gth in III. «h,m thattent?^' ""^^ ^' *^'''"^'^^ *'"" "^^ '''''''' ^^ any distance mJOLID'B FT.KMRWTg. AXIOMS. I. Tamos which are equal to the same thing are equal to n. If equals be added to equals, the wholes are equal „ m. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes aie unequal V. If equals be taken from unequals, the remainders one another- are unequal VI. Things which are double of the same, are equal to rne another. VII. Thmgs which are halves of the same, are equal to one another. VIII. same space, are equal to one another. _, IX. The whole is greater than its part exactly X. Two straight lines cannot enclose a space. XI. All right angles are equal to one another. xn. right anglef ; these straiX Unl K • ^'^^ together less than two length ,neet upon thauSe on wh.VK '"^ continually produced, shall at two right ang/es ^'"^ "^ ^« ^"^'1«* *l^cb are less than ROOK I. PROP. I, II. to one another.. qunl. lal. I unequal. le another. > another. »hich exactly ake the two 88 than two ced, shall at ^e less than PROPOSITION I. FFIOUI.KM. 7b deirrih* an a^iilateral triangU upon a given Jitiito itraight Una, Let A li he the given '(traight line. It it required to doHcribe an eciuilateral triangle upon AB. c Prom the center A, at the diHtunco AB, describe the circle BCD ■ (post. ;j.) ' fr< tn the center //, at the disfance BA, describe the circle ^Tff; ami from (J, one of the points in which the circles cut one another draw the .straight linos CA, CB to the i)oiiits A, B. (po8t. I.) ' Then ABCuhnW ha an ef|uilatiTnl triangle. Because the point A is the center of the circle BCD, therefore ylC'is equal to AB; (def. 15.) and because the point // is the center of the circle ACB, therefore ffC.'is eijual to AB-, but it has been proved that AC is equal to AB; therefore A C, BC are each of them etjual to A B ; but things which are equal to the same thing are-ecjual to one another; therefore AC in equal to BC; (ax. 1.) wherefore AB, liC, CA are ecpuil to one another: aiid the triangle ABC is therefore equilalcrul, and it is described upon the given straight line AB. Which was required to be done. PROPOSITION II. PROBLEM. From a given point, to draw a straight line equal to a given straight lint Let A be the given point, and PC the given straight line. It is required to draw from the point A, a straight line equal to SC. From the point A to B draw the straight line AB; (post. 1.) upon AB describe the equilateral triangle ABT), (l. 1.) and produce the straight lines DA, I)B to JS and F; (post. 2.) from the center B. at the distance BC, describe the circle COH (post. 3.) cutting DF in the point O: and from the center D, at the distance DO. describe the circle fiKL cutting A£in the point L. 8 ROCLID's ELKMKNT8. 4 Then the straight line AL shall be equal to BC Because the point ^ is the center of the ch-cie r^« therefore ^Cis equal to BG; (def. 15 ) and because D is the center of the circle GKL .„^ 71 i^/'i »°''^ ^^ ^^ «q»al to DO, .1. n ,*"" '^^' ^''-^ parts of them are eoual. (i l \ therefore the .,en,.i„der It i. equal SZe rSn] ;y„, ,„, , , PROPOSITION III. PROBLEM. From the greater of tu>o given etraigkt lines to cut off a part cgual to the less ,reater.^^ ""^ ^^' *^« ^^'^ ^^'^ straight lines, of which AB is the It is requiredto cut off from AB the greater, a part equal to C, the lens EB Then ^^ shall be equal to C. Uecmse A is the center of the circle I)EF therefore AH is equal to AD: (def 15 T but the straight line C is equal to AD (coLr \ whence ^^and fare each of them equalto.^Ji. .a« bean cut off equal to C, te'lT "'^^VhK^JbVl^L^^ PROPOSITION IV. THEOREM. ^ach other; they snail likZisl haveSr '"'''"'''''^^y ^hose sides equal to the two triangles simll beeZal and !t!il .?'* '"" ^''"'^ ''^^« '9^< «»'i -o each, .... L. ,. ..tat;:1 .tL^Xoff.' """ *^ •'^"''^' ^'^^ ^ Jequdl^'lh^^'o sWef^iTl'?; f t"^ \^^^''^^ *"« «'d- ^^ ^^ to DF, and tL included an^lf'lJ^ '^^^'l ^.^ }^ ^^ ^^^ ! It F. "iciuaea angle BA C equal to the included an£0; one anothf^r IX. 1.) 18 been drawn be done. ual to the less ihABin the to C, the Jeas. BOOK 1. PROP. TV, V. 9 0; (1.2.) circle DEP ) 1.) a part AE be done. / the othfir, des equal tc equal, and equal, each sides AB 3 DE. and •ded anQ-le Then shall the base BC be equa! to the base EF; and tlie triansle ABCio the manple DEF; and the other angles to which the equal SHle« are oppo^ie shall be equal, each to each. viz. the ai-le ABC to the angle i>7i/, and the angle ACB to the angie UF±i E P For, if the tranjjle ABC he applied to the triangle DEF, so that the point A may he on D, and the straijjht lin^ J li on DE- then the point B shall coincide with the point Xi, because AB is equal to Z) '^j and AB coinciding with LE, the straight line A C shall fall on ">P, because the angle BA C is equal to the le EDF- therefore also the point C shall coincide with'the point' F, because ^ C is equal to BF; but the point B was shewn to coincide with the point E: wherefore the base i^C shall coincide with the base EF- .f *v 1- •'^'=flo''t?;r™Z5 ■ C«. ,a, 3, each to each , ■"•*• ^^"^ "i'^ «o the two CO OB. ."d their re^Sra^'S iSt eS'rwf' are opposite , ^ * ' '"' *" "'''> »» "hich the equal side. therefore the angle FSC i. equal to the an,,I« «/>» A..a ™. .fc^ -"^-- . theteforetWmainingangle jlcLeauSt^L '°°'?'' 'I""'' which are thelngfes wThe baw of ?hh ?'"''',"'"«''"«'«-^«'B- «nd it ha, also Seen proved """'S'' -*-"gle CG£, \ CGB; equal side^ CF, lual; ngle.4CB. C} e. lar. also which another. ileACB For, if ABh^' not equal to A C, one of them is greater than the other. , J r ^ . possible, let AB be greater than ytr. and from BA cut off 52) equal to &TlS (i 3 )' and join DC K r.« . "^^"' '" the triangles Z)i?C ^«r ^ '" "^- therefore the base Z)Cil equal to the basfi// /i /v and the triangle 2)5(7 is equal to the tranglfj^C^ There ore S^n^'" '^' ^r^^'' "^^^^^^ is absurd! (1^^) inerelore ^5 is not unequal to A C, that is, ^5 is equal ti ^C. tOB. Hence an equiangular triangle is also equilateral. U...., PROPOSITION VII. THEOREM. triangles Zt'tZ tS Zt Zl'haZr ""^ f. ''' '^''^ ""' ^ '««> other extremity. ' '^'"'"^ '^''" "'^'^^ «»•« <«-»»ma5, that are terminated in B. c D A B oth« San^™ *' "'^^ "' •''* "' «■« ««"ff'«' >. w„ho„. the ♦»,»-„r .L , " ""l""' '•'' -*^ in the trianele ACT) therefore the angle AbCis equal to the angle S CD h li \ but the angle ACD is rrrp«to.fk„„ *u' __ p'^flXT '. (^- ?•) but the .„,,eTcJ,17eSLTl„7e VSS^ii'^ ^ in, because the s d« 7?^,-« «o..„iL d tT-- ^^^^^F ^".^"A"^ ^^•&t-£l^^!^^=^^,,, the angle BBCm both equal to, and greater than thi angle^Ci). .^»k ^'' '^^ -- ^ ■" *^ia^^.71^2'M. within th. 12 EUCLID'S ELEMENTS. therefore tlie JgU^,%in/Tr^ *" ^? '^ ^'^'^ triai.gle^Vz) are equjl to o,fo L^li'^f'lT'' ^^^ "^her sicle"of the base CD, ■».icl. more tl,e„ i, tlla/^fc^^g' ;f." «'»" tl.e mgll BOD; Agam, because BO is ?q J to In" ","""'• "'" ""S^ BCD. ^fzC i„ ^* h "is,:."'""" «™'- ''- ti., " '"'"'"'"'' ^ »'"'-- atde of i,^ ^,., Xi n/« f«''-'«'^V1w^lVr„? '^ '" *» ^i^, AC B c , ^ Then the angle ^^ C sho n k \ ^ For, if the trkn Jp 1 p^l'J"^^ *" the an^Je A*/) i? «o that the point B be on ^^In^l^ ^ ^^''^ ^o J^/^^' then because »?■; ^ ''K'''^^^ ^in« BC on FW. therefore the pSrc^.^ S i£ f ^ [^^ " for, If the base ^Ccoincide vv r!I!"u^ "^"^ ^A i>i^. W c.n.,e with th;St^:^i;;^^t .e^^^ then, upon the s ), aiHerent situatioE - °" "-"""'ngte.i..™ two .ides, fe «..„ 72). f the base (72), ; (ax. 9.) ?Ie BCD; ugle DQD. BCD, X (I. 5.) BCD, ater than the ?Ie is upon a of the other, which is con- sontained by '^sAB,AO, to Da, and BOOK I. PROP. IX, X. PROPOSITION IX. PROBLEM. 13 To bitect a given rectilineal angle, that i», to divide it into two equal tmgles. Let £A (7 be the given rectilineal angle. It is required to bisect it. In AS take any point Z); from AC cut o«AE equal to AD, (i. 3.) and join D^; on the side oi DE remote from A, describe the equilateral trianf>;le DEF(i. 1.), and join AF. Then the straight line ^J^ shall bisect the ans^le BAC. Because AD is equal to AE, (constr.) and ^2^ is common to the two trianj^^les DAF, EAF; the two sides DA, AF, are equal to the two sides EA, AF, each to each} and the base 252^' is equal to the base EF: (constr.) therefore the angle DAF is equal to the angle EAF. (I. 8.) Wherefore the angle BA C is bisected by the straight line AF. q.e.p. OF. iF, SA,AC, '■ situation there can 'd in one ose sides ble.(i.7.) DF; DF, and PROPOSITION X. PROBLEM. To biteet a given finite straight line, that is, to divide it into tuo aaucl parts. Let AB be the given straight line. It is required to divide Al into two equal parts. Upon AS describe the equilateral triangle ABC; (I. 1.) and bisect the angle A CS by the straight line CD meeting AS in the point D. (I. 9.) • Then AB shall be cut into two equal parts in the point i>. Because ^ C is equal to CB, (constr.) and CD is common to the two triangles A CD, BCD', the two sides AC, CD are equal to the two BC, CD, each to each; and the angle A CD is equal to BCD-, (constr.) therefore the base AD is equal to the base JiD. 'I. 4.) Wherofore the straight line AB is divided into two equal parts in the point D. Q.E.F. 14 EUCLID'8 K1,EMENT8. PROPOSITION XI. PKOliLi;:.!. 70 rfrow a Btratpht line at riaht annli, tr. gtoen point in the same. ^ ^^" *° " ^»'"^' *"-'%A/ line, from a lel^5 be the given straitrht line ftn#1 /^„ ■ It IS re.iuiied to draw a sWailhf Hnf f ^''u'" P"'"^ '" ^t. angles to Ji/ ^^^'*>^* ^°® fro«» the point C at right BCF, ECF-, ' "^ ^^'^ *'°'"'"°» ^o tiie two triangles the two sides DC, CF&rp Pm.oi tr. *v^ 4. -j «~ and the'bLe o/Sual to th''«°>>'«^'^^^^^^ ^^, each to each , therefore the anjrle Z>r^l i ? Jf' ^ "^^^ (constr.) Bu. .hen Z'F^r^^^'''°^^^''^i^^' "■ '•' WhtfotlZtJilV"t^,^e^^- f f^i' • Hgh. .„g,a. liaif ^^cHS'i'' "* '^ ''8°""' -^^ '"' '"-"'O-. «. U.. two straight I) ftom the point B, draw BE at right angle, to ^S . f, „ ■, «.«ef„re U,e fgrfriH^^ittd^ie^^ic Li > the less equ"K° If freatS?. 1° "'t?1°'' -^^'^ <"• '■)• " "»o suaight lines cannot have a common segment r. * PROPOSITION XII. PROBLEll. BOOK I. PROP. XIT, XIII. 16 f-aiffht line, fntn a point in it point C at right CD', (r. 3.) 3,) and join CR angles to AB. tlie two triangles ^■^, each to each j Jonstr.) CF: (I. 8.) s. igh^ line makes saoh of them ia ht angle, ajyht line AB, fated that two ■he two straight ; (1. 11.) . (def. 10.) ^BD; Q ^ (ax. l.> ssible. segment *■ it ittltj n-f aimu. I 1 ^t*u^-? ^^ '^"^ "'7f" Straight line, which may be produced any length both ways, and let Che a point without it. It is required to draw a straight line perpendicular to AB from tiic point C. V TI B p^ D B . ,. ^ ^PO" the other side oi AB take any point D. and from the center C, at the distance CD, describe the circle EGF meeting AB, produced if necessary, in i^'and <9: (post. 3 ) Tu .V. ^'T\ ^^ '" ^(^- 10.), and join C^. Then the straight line C7/ drawn from the gi^en point C, shall be perpendicular to the given straight line AB. Join FC, and CG. Because FH is eqxml to HG, (constr.) and JJC IS common to the triangles FIIC GHG- the two Bides /Yi. if (7 are equal to the two GIL, HC, each to each, and the base CF is equal to the base CG ; (def. 16 ) therelore the angle FHCh equal to the angle GHC; (I. 8.) and these are adjacent angles. Jiut. when a straiglit line standing on another straight line makes the adjacent anges equal to one another, each of themis aStTnT 5?ct,^;^t? i?\V'riof ' ^""'^ "^"" '^^ -^- ^« caiK' jeTp^;;: dra;!r[:t ^t4i?\iK:^^i? ^P^pendicular C^ has been PROPOSITION XIII. THEOREM. The angles tohich one straight line makes with another upon one side ^i it, are either two right angles, or are together equal to two rtghTa^es an^^s' CBAZtn'!''' ^^ "^'^ "^^^ ^^' "P- -« -^^ of it, the Then these shell be either two right aneles. or, shall be together, equal to two right angles. D B For if the angle CBA be equal to the angle ABD. each of them is a right angle, (def. 10.) fr^„"L * «. a.n?,le C5^ be not equal to the ande ABD. from th:i point B draw BB at right angles to CD. (i 1 1) Ihen the angles CBF, FBI) aie^wo right angles, (def lOJ .rf 16 KTTrnn's RI,FMKNT8. And because the anele Cnvi. therefore the angles cS^ Fn^^'' '^'^ of these equals ^• .^ . ABE, EBD. (fx^af^"'" ^^""^^ ^° tJ^e tWe 'angles f^. Again, because theanile />j?^ ;„ , ^' add to each nffh. '^"'',^ ^° ^^^ t^o angles 7>7/7? pr, . •ierefore the angles USAZlBn ^ "" '"■'' ngl" ""Kle- '"^■^^°> <"• '-C , '^™"'«^'''"'-"°'»»'4h..ngle.. Wtoef.„,„he„.e.,aigh.,i„e,&, ,.,.„ (ax.^l.) -^re equal to the angles CBA, ABD ' therefore Ve TeZ^Tg trSi?^ '''^T^" «"^^« ^^^. ^,^^5 (ax. 3.) ^ '"^^^ ^^^" equal to the^emaining angle the less angle eoual tn fh ^ Wherelbre.ifatapoint.&c o..B.n es CBA, ABE, luals ; ;h^ee angles VliA, IBC; "^^'*' "•ee angles DBE, 'qual to the same ilto one another! - — <^.ic aiiucneri ^^\JJBA,ABCi ngies ; two right angles. E.S J^pon the opposite fht angles; then ■ line. straight lines djacent angles BC, ith it. right angles ; les; (hyp.) BA^ABTt: 'BA, ainmg angle ible: BC. It no other irefore is in BOOK 1. PROP. XV, XVI. ' j7 PROPOSITION XV. THEOKEM .^Jf*;?^;ir*'*' '"" "*' ^^ ^^^'^^ '^» ^'^-a o, opposite ««.&. anyle C'^'i^ to the angle i^jj. ''^"'' *° '^« *°S^« ^^^ ^^^ the c acijfcr:S,t cS^'yi1>f ^ ""^^^ ^^'^ ^^ »* ^'^^ point ^. the these angles ar. together equal to two right angles ii V<\ iKain, because thfistr<.;.,>,+ i;t.„ r» it _., '8".'' ""g'es. (I. \6.) f at the point J?, A„n;„ u " I "'s^^'ci equal lo iwo rieht an? ea Af,ain, because the straight linei)i4'makes S in the adjacent an^Wes AEI),DEB\ ^ ^^ ' !.ut the ^n^t^EAAFuT 'H *°u'^° "S^* «"&•««! angles ; ' ^^^ ^'''' ^''^^ ^^^^^^ ^ be eqSal to two right «^herelore the angles CEA, ^^D are equal to the angles^^D 7.A-« and the ren.5 rgZl^il^l;^^ ^f'"^ -^^3^^ ^'^^•• (ax. 3.) ^ ^^' ^-^^ '' ^q"^^ to the remaining angle DEB. is eq" aUo rangHl^"^^ '^ demonstrated, that the angle CEB Therefore, if two straight lines cut one another, &c q e n othS the'aSTeT Ith ILTSI' T;^"^' ^traigtlinroS each together equa' to foSj right ^'s P°'"' ^^"'^ '^^^ ^"*' '^^ ber oTiines meetkg7n're^Lt* '" ?' «".?le^made by any num- angles. ^ ° °°® P"'''^' ^^'^ together equal to four right PROPOSITION XVI. THEOREM ^it>^'r°:;:;^iferV:z^^^^ ''* '^'"'''^ ^^^^^^^ ^-- '*«« TCwfe^xlnS^^^^^^^^^^ ^^^e produced to D. interior opposite angles d^io^^j^a ^'''''' *^^° '^'^'' °*'^« A P Bisect y4C'" ^ '1 l(i \ o->^ : • ni* P..*.™ 2,^ .„ ^. „7ki„g ^>:,i^i'.rilV30 'and jota ^a 18 Euclid's elemknts. the tria/is,'les A HE, CFB; ' ^^' ^^'-^^ ^^ «ach. ia and the angle AEB h equal to the anL^le rPf the other, ea^hToHrch^'to wh 0""!; ^JU'r- T'"'^'"^"^ ^"^'^ °^ wherefore the an-le 7/>J T ;? ^"'^f'l"^ ^^es are oj)posite: but the angle ECB or ACnii "''"f ' 'V^" ''"^''^' ^'^'^'' therefore the an.r£^V>n tt ^''T^*"" ^^'^" ^h'^ ''"k'*^ ^^CF^ ^ In the san.?"„?inn1r^5 hf "1"^'.' "be'h" "'^'^. ^^/ "'• ^^^ ^• ducedto G; itmaybedemo/Aratedth«tn ^''''T'^V^"'^ -< ^ he pro- ihereiore. if one side of a triangle/^^c q,b.d. i PROPOSITION XVII. THEOREM » A B Tfc^- V . Produce any side 5C to 7) - '''luai to ^Z/, (I. 3.) and join S/j^*'' thlefbre tfr" ^^J^/,^""' ^" ^^' '" the^tr anflt^^Z) therefore thfeVterLr fnl^V/i"!'^'"' ^^^^7' P'"?'"^^'^ ^^ ^ onposite angle "Tr/?^ (I 1 6 f '' ^'''^'^'''^ '*^''" ^^^ i"^«"«r «'«^' .herel.re.„uch.^^;^^^^i:S?-^^^^^ I liereiore the greater side, &c. Q. a.v. ^ PROPOSITION XIX. THEOREM anglf 2/1>^^ ^' ^ ^^^"^^« °^ ^h^^h the angle ^5C is greater than the Then the side ^C shall be greater than the side A£. B For, if A Che not greater than AB, ACmust either be equal to, or le.s. than^iJJj +>,„« ♦k , >, -r^'I ^ ^' ^^^'^ equal to A B, then the angle ABC would be equal to the 'angle ACB- (i 5^ but It is not equal ; (hvp.) ^ ^^^.(1.5.) therefore the side ^ C is not equal to ^^ .•L , -^"^in, if ^(7were less th'in >* » then the angle ^i^Cwould be les:ihaVthe\1fl'e^e5. (i 18) ^^ ^ but It is not less, (hyp.) ^ ^^^. l^- i».; nnH ^'A r ?' '^^f ^^'' "°t less than ^5 ; and ^C has been shewn to be not equal to AB'. therefore A C is greater than AB. ' Wherefore the greater angle, &c. q. e. d. PROPOSITION XX. THEOREM Any im tides of a triangle are together greater than 'the third side. Let A TiCViQ a triansl- Then any two sides of it together shall he '^^^.y ^hnn the th.V.l '. VIZ. the sides BA, A C greater than the side ic, ''^'' y\) ft'iW.rn'* #fKvi/NT8. •nd JJC, VA Kreater thun AB, " therefore the ar.deTrv/is L''*''i V^' ^^""^^^•) U:v/elore also the anJle'^ir/i •!^"" l^' ">'''' f^^'^^' ('i^- »• and that .;;r^:!::.r^^KSY;^'r^^^^^ the..ret.£^--f;:;,;'>-^^^^^^^ therefore t h's' e 7y ^'I'n J '^ ;?'' ""^ ^ ^' In .he .sun e mamH-r ?t .. ."'■'r''''"'^''" *^'"' ^'' ''•' are "■(k'i)i^,,,Hi.,, oitheseunequBls, Ta towar rnak from (post fro from . F. G- Th straigi BOOK 1. PROP. XXT XXTl. il I II. 1.6.) ..9.) IDC. If D, oin nC, constr.) h'ADCi (1.6.) AC'J); (ax. the anifle Ai iC 'leBDC, ronter sirbj (I. 19.; I' side liC; V, T than nc. stratfd, 111 CA; AB. .1). ;m. tii'iten tteo straight ' than the other two C, the ends of the awn to a point I) CoU\iii triangle CDE h gra.ter than the interior and opposite angle CED- Kra^tr for the same re i.-,,: the exterior an^de 'cED of the triangle ABU « greater than the u.lerior and opposite angle BA C; and Jt has h-r.j dv'.iiionstrated, n ^^^., that the a igle BDCiH greater than the angle CEB; much more therelme IS the an^Ie //Z>C' greater thaS the angle BAC I herelore, if from the ends of the side, &c. q. e. d? PROPOSITION XXII. PROliLEM. T^o make a triangle of which the aides shall he equal to three aiven ,ira.,ht hues, but any two whatever of these must be great r than the'aZ? f u- u^**^ ^' ^' ^^® ^^^ ^^^^^ P*^^" straight lines, 01 which any two whatever are greater than the third, (l 20 ) namely, A and B greater than C; ^ ' '' A and C greater than B; and B and r' greater tlian A. to jXtziizi: * "■'""''"" "' '"'"^" *' "■"" "»" "• «i»^ A- towljdr^.'*'^'^^* ^'"' ^^ terminated at the point D, but, unlimited make />/• equal to A, FG equal to B, and r?// equal to C- li 3 ^ (poTta!"*""'" ^' '^ *^* ^'^^^"^« ^^^' describe^Jhe drci; VkI, h-nlT * K** '"^u"" ^' f * *^^ ^''^^^^^ ^^' describe the circle HLKx JpmX where the circles cut each other.draw KF, KG to the poinds ^^ilv^ltTc^^"" '''" ^''' ^"^ "^^^^ ^'^"^^ *° ^^'^ '^^« Because the point J^-is the center of the circle 1)KK therefore FD is equal to FK\ (def 15.) but FB is equal to the straight line A x therefore iPA' is equal to ^. Again, because G is the center of the circle HKLy thenfore ^//^s eq-ial to GK, (def. 16.) but GH is equal to C\ therefore also GK is equal to C; («. 1.) and ^6" is equal to JB J W'"^, tft ETICT.TD'S ELRTUf^NTS. thwwfoie ihe three strai-ht lines A'J? p^ ^-r, ^quai to the three, a:B C ' ^^' ^^' "« respectively PROPOSITION XXIII. PROBLEM Let ^2. be ^ f L^^^hf^l^^^^^^^^^^ f ^^n point in it. It 18 required, at the given pSntiinh. '"^'^''• -^e an angle that shalf be eff^^^ ^^S^ ^SSSliJ^^i^^ to the three straight 1 nefc^S'/^V^^'' ""^^^''^ ^^^" b« «q«aT therefore?h1a^^S,^J^/-it^^^^^^^^ Wherefore, at th^gH:^;^ „^ointTfn S '^ • '"^^^ ^^^- (^- «•) angle ^^^ is .ade 4uall^X^1verrtSSfS^,^^i^^i:^i^^^^ PROPOSITION XXIV. THEOREM Wan Me angle contained bTthetwol^f, " V"^' '^^ ^"^ "/ ''4«"' i^'-ea^er of the other, ^'^^"'^ «"^'«» ^^all be greater than the base AcXtuoieil'nS '^f^^"^}^^^ f-ve the two sides AB Thenthe ba^e i^^lX frl^ef tCn^tC A BOOK I. PROP. XXIV, XXV. 23 '^ are refipectivelv des KF, FO, 0£, > t/. Q.E.F. EM. > a rectilineal angle ven point in it. le. ^raight line AB, to lineal angle DCE. nDF; :h shall be equa? AF he equal to ?le DCF. to each, F' Ck (I. 8.) fhtline AJi, the leDCF. Q.E.F. des of the other, of them greater f the other; the 'r than the base two sides Ab, , Ah' equal tc leaiiule JE'/jP ie FF. Of the two sides i>^, DF, let D^ be not greater than J)F, at the point A m the line i)^, and on the same side of it as i)W make the angle FBO equal to the angle BA C: (l. 23 ) make DG equal to ^ J' or A C, (i. 3.) and join Fd, Gp. Ihen, because DF is equal to ^J9, and Z)6? to ^ C the two sides DF, DG are equSl to the two ABAcMh\o eacl , and the angle ^Z)G is equal to the angle BA C; therefore the base FG is equal to the base^BC. fi 4 N And because 2)» is equal to DJ^in the triangle DFG, therefore the angle DFG is equal to the angle DGF; (i 5 ) but the angle DGFh greater than the angle FGF; ax 9 therefore the angle DFG is also greater than the angle FGF- much more therefore is the angle FFG greater than th!^J\.EGF the angleT^??/'' ''''"^^' ^^^' "^^ ""^^^ ^^^ ^« grfatef than and that the g;eater angle is subtended by the greater side j (i. 19.) therefore the side FG is greater than the side EfI but FG was proved equal to BC\ therefore 5C is greater than FF. Wherefore, if two triangles, &c. q.e.D. PROPOSITION XXV. THEOREM. «a^A be produced a"H m»Pt ■■*'nr— JM- - • j in the point ^, " " ' " P^^^^^^'-"' "'^aras i/ uuti iy, fben C^-fc'uf' is a triangle, u C6 BtJCUD*S KLKMRNTS. opposite angle £FG. (j le ) ^ *^*" *^^ '"^'^"O'" ^nd there Je"k?Ztilli; Xair tn^" ^^^. ^^^,' ^^^^P-) ^P' when produc^rtowardri' C. ^^'"^"^^^^t^d, that they do nottee, though Xfed'f v'eVri{^?;r'",1 ^'^"^' "^'^^'^ --* '^-ther way e i' uuoeu ever so lar, are parallel t,o one another- Cdef •i'i\ therefore ^5 fs parallel to CD ' ^ ^'^ Wherefore, if a straight line, &c. q.b.d. PROPOSITION XXVIII. THEOREM. make the interior angle, upon XLT-^^'''''^^ '^ ''^ ^^'' ^*'»*' "^ angles ; tJ. tu,o straijt lin:fZZ 'pZllSt oZZhT' '° '''' "^^ -^^'%';SftL^tS:^Lt Ifer^, - «-i.ht lines opposite angle OHD nnnn fK J " • , 9"^^ *° ^^^ interior and t^e two inferior afgles'^l^"^:^ii'^L^;i^^ ^-« ^f ' or ™ake equal to two right angles. ' ""^ ^^^ ^^'"^ "^^^ together T»>en ^5 shall be parallel to CD. Because the angle EQB is equal to the anele Onn rv^ > and the angle EGB is equal to the angSrf/^' W therefore the angle A GH is^qual to the an -le G^t ?.7 , , and they are alternate angles, " tr///>. (ax. 1.) tlierefore ^2? is parallel to CZ>. n 27 ^ n.b1r«iet(Cf' "*'" ''''''■ ""'' "^ •°««*- '^'"l " two "UM a^gUsf ?,' i!,^' ^''^ "' "•» '°8»"- e,ual .o two ♦>,»«<•„ ^^u ^^^^ ^''"'" t^^^se equals, the common ancle 7ir u therefore the remaining anele AGH is Pmml fn fif ^ • • ' <^^i)j (ax. 3.) ^ ^^"^^ ^° *h« remaming angle and they are alternate angles i VT,«. ♦• *.5l''''^^ore .-45 is parallel to CD. (i. 27 ^ ^Tierefore. If a straight line; &c. o.b.d . ^ luced to A, le interior and (hyp.) to, the angle ards B, D. ' do not meet !t neither way ; (def. 36.) ie the exteriot f the line ; or I to two right itraight lines interior and 'F; or make ide together (ax. 1.) ual to two ial to two les BGH. ing angle BOOK I. PROP. XXIX. PROPOSITION XXIX. THEOREM. 27 Tfa straight line /all upon two parallel straight lines, it makes the alter' nate angles equal to one another ; and the exterior angle equal to the interim and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two right angles. Let the straight line ^Ffall upon the parallel straight lines ^5, CD. Then the alternate angles A Gil, GHD shall be equal to one another ; the exterior angle BOB shall be equal to the interior and opposite angle OH I) upon the same side of the line EF; and the two interior angles BGH, GHD upon the same side oi EF shall be together equal to two right angles. B V •D First. For, if the angle AGH be not equal to the alternate angle GHD, one of them must be greater than the other; if possible, let A GH be greater than GHD, then because the angle A GH is greater than the angle OHD, add to each of these unequals the angle BGH-, therefore the angles AGH, BGH are greater than the angles BGH GHD; (ax. 4.) but the angles AGH, BGH ate equal to two right angles ; (l. 13.) theiefore the angles BGH, GHD are less than two right angles ; but those straight lines, which with another straight line falling ujion them, make the two interior angles on the same side less than two right angles, will meet together if continually produced; (ax, 12.) therefore the straight lines AB, CD, if produced far enough, will meet towards B, D ; but they never meet, since they are parallel by the hypothesis ; therefore the angle A GH is not unequal to the angle GHD, that is, the angle A GH is equal to the alternate angle GHD. Secondly. Because the angle AGH is equal to the angle EGB, (l. 15.) and the angle A GH is equal to the angle GHD, therefore the exterior angle EGB is equal to the interior and opposite, angle GHD, on the same side of the line. Thirdly. Because the angle EGB is equal to the angle GHD, ■add to each of them the angle BGH; therefore the angles EGB, BGH are equal to the angles BGH, OKD: (ax. 2.) but EGB, BOH are equal to two right angles; (i. 13.) therefore also the two interior angles BGH, GHD on the samy side of the line are equal to two right angles, (ax. 1.) Wherefore, if a straight line, &o. Q.E.D. 88 : buclid's klements. „. . ., .. """OfOSmON XXX. THEOkEM -^SbeaJsoparallej'to Ci).' „ Then ben«,^pS''ty.'The'';^'^,r; ■'^- ^^^^ ^^- „ „, /iA cut, the parallel ,traisht line, .ij), Ef, j, WOPOSmoN XXXI. PROBLEM ^'■- " "•"*' "-'*-«* «^'»- ^».'" ^~«.«..,™„ ..„^„ . ("' re':i*d\fSrroS,^'^*?«'™» "'"■>''' line- to the .traight line Da' '''' '^^ I"""' "<■ " "'"isM line paraltel 3it« »idetf f^f ■'""" •» ">' »"«'» ^^C, (,. 2i, ,„ ,h. „„„„ '"lC''i?l5',f['"8l" line MA to J. an. i^f parallel ,„ 4e gi;en'at;.igh, line Sc""";^" " ""S"' BOOK I. PROP. xxxn. '29 are paraliel tt iel to EF. PROPOSTTTON XXXTT. TTTFO-RP'M. If a side of any triamjle be produced, the exterior nnqleia equal to the two interior and opposite angles ; and tlte three interior angles of every tnanyle are together equal to two right angles. Let ABO be a triaiij^We, and let one of its sidea BChe. produced to D. llicn the exterior angle A(JD sliall be equal to the two interior and opposite anjries (JA/i, ABU: and tiie three interior angles A pr, BOA, OA li siiail be eciuai to two right angles. ^ IS, EF, in iP> (I 29.) "^F, CD. in ?le II h ij J lb GIIFs . si. ■n Hraight ' parallel \f op];i> ^F. lit; ther. iraigji! BCD Throu,2;h the pohit draw t'E parallel to the side BA. (i. 31.") Then hccnuse (JE is parallel to BA, and AC meets tliem, th( refore the angle ACE is equal to the alternate angle BA C. (i. 29.) Again, because CE is parallel to AB, and ISO I'ulis upon them, therefore the exterior angle ECU is equcd to the interior and op- posite angle y]/;6'; (i. 29.) but the angle ACE was shewn to be equal to the angle BAC; therefore the whole exterior angle ACB is equal to the two inf-Tior and opposite angles CAB, ABC. (ax. 2.) Again, because the angle ACB is equal to the two angles ABO, BAG, to each of these equals add the angle ACB, therefore the angles ACD and ACD are equtU to the three angles ABC, BAC, and JOB. (ax. 2.) but the angles ACD, ACB are equal to two right angles, (l. 13.) therefore also the angles ABC, BAC, ACB are equal to two right angles, (ax. 1.) Wherefore, if a side of any triangle be produced, &c. q.e.ti. Cor. 1. All the interior angles of any rectilineal figure ti ., ether with four right angles, are equal "to twice as many right angles as th© ligure has sides. For any rectilineal figure ABODE can be divided into as many triangrcs as the figure has sides, by drawing straight lines from a point F within the figure to each of its migles. Then, because the three interior angles of a triangle are equal to two right angles, and there are as many triangles as the figure has sides, therefore all the angles of these triiuigles are equal to twice as many right angles ns the figure lias sides -, but the same angles of these triangles are equal to the interior angles of the figure together with the angles at the point F-. 30 Euclid's elements. equal to lom- rjgiit angles. ^ '' ''*'"^ dnvclmi, are togctlicr «"Sles. (ai 3.) '^'™' "S'"^ ol the figure are eqS to four right ■u.d joined to™;* rtel;e'°r??rn''°\'''»«''' '»'». TJioTi I, Join j?^/ tWo^SSV^^iUZ','t'thf2/"^^''»«*'ke„, ,„™d because is is e„ual to ™^,S n^ ""''' ""S'" -BO^i fr 29 ) fceforf E:=b^f,Jtr;:,''trt ,%Te";° the^kte^^ «aa the inaugle AJj3 to the tria,,li: Ig^;'- *■> BOOK 1. PROP. XXXIV. 81 )n verfex of all Cor. 2.) il to the angles es are equal to r right aiijrles, « l)as sides, ij'ure, made by are togctlier cterior anfjle erior angles, ies; .all the iu- ice as mauj irior angles. ) four right id parallel 'dparallel. 1, (I. 29.) triangles ''i', each and the other angles to the other angles, each to each, to whieh the equal sides are oj)posite ; therefore the angle A CB is equal to the angle CBD. And because the straight line PC meets the two straight lines AC, BJJt and makes the alternate angles A CB, CBD equal to one another; therefore ^ C is parallel to BD ; (l. 27.) and A C was shewn to be equal to B D. Therefore, straight lines which, &c. u,K.D PROPOSITION XXXrV. THEOREM. The opposite sides and atigles of a parallelogram are equal to one anotkett rnd the diameter bisects it, that is, divides it into two equal parts. Let A CDS be a parallelogram, of which BC is a diameter. Then the opposite sides and angles of the figure shaM be equal to one another J and the diameter .BC shall bisect iu Because AB is parallel to CD, a. d PC meets them, therefore the angle ABC is equal to the alternate angle BCD. (I. 29.) And because ^Cis parallel to BD, and JBC meets them, therefore the angle ACB is equal to the alternate angle CBD. (l. 20.) Hence in the two triangles ABC, CBD, because the two angles ABC, BCA in the one, are equal to the two angles BCD, CBD in the other, each to each ; and one side BC, which is adjacent to their equal angles, common to the two triangles ; therefore their other sides are equal, each to each, and the third angle of the one to the thira angle of the other, (l. 26.) namely, the side AB to the side CD, and ^ C to SD, and the anele BA C to the angle BDC. And because the an^le ABC is equal to the awgle BCD, and the angle CBD to the angle A CB, therefore the whole angle ABD is equal to the whole angle A CD : (ax. 2.) and the angle B^Chas been shewn to be equal to SDC; therefore the oj)posite sides and angles of a parallelogram are equal to one another. Also the diameter PCbisects it. For since AB is equal to CD, and BC common, the two sides AB, BC, are equal to the two DC, CB, each to each, and the angle ABC h&s been proved to be equal to the angle BCD; therefore the^triangle ABC is equal to the triangle BCD ; (l. 4.) and the diameter PC divides the paralielograia A CDB into two equal part^ t. s» 111 BUCLId's EI.EMEVT8, WON XXXV. THE ' same bate, and between th» PROPOSITION XXXV. THEOREM. Let the \>Mfi\\e\of^r&m» ABCn Kurwy.^ and between the same f.arallels^4F /f^ "P""^*"* same base ^0. ^;^^^n the parallelo,ra.^.yC'i> «hail be e.uai to .he paraile,o,ra. Hut if the sideq >4 /) j?jp 'ated in the same pobt , ' '^P^'^'" '" *^« '^^^^ ^^' he not termi- '^VheWforhV-^^^ i^ « ParalJeIo,.ram. inereloie ^Z> is equal to BC; fi. a4 ^ and ^^ ,8 equal to DC: h 34 1 hence in the triaflgles ^^j5,>7j>; and the ^^J:rTilfFllhT:'Zfi^ '^' ''^'^ '^^' EAB , (I. 29 r ^ ° '^' '"''^"'^^ ^"d opposite angl. an':iTo;n'r^^^^"'"^^^"^^^^^^^^^^ and from the same traj.ezium take the triangle ^^» therefore the para"lt4m7i?S- "^ ^?"^V^- ^ Therei^paralTeYo^alf^i^^^^^^^^^ ParaZZ; ^^^^^^^^lON XXXVI. THEOREM. ^«a. to^^StZZ: "''"" ^^'^' ^'^^^^ '^'"^ *^'-^- '^« -m« ^.«,fe&. ^, and^Wei'^fh^f^^^^^^ "Pon equal bases ^C, FG. ^^™ the parallelogram ^^C2) shall be eaual to th« ., .f BOOK I. PUor. XXXVI, xxxvti. A D ■ B 3a me paralleU, art same base J?C. } parallel ograa D 9 \l 9CF, opposite jf the triangle aarallelogram be not termi- ni le whole, or posite anglf .4.) IB. C, IB, dmEBCF. yalkh, art BC, FG, ii„(_ C F O Join BE, CII. Ihen because BC is eqiml to FG, (hyp.) nnd FG to EU, (i. 34 ) therefore yyC is equal to /;//; (ax. 1.) tnd these mes are parallels, and joined towards the same parts by the straight lines BE, CJI; ^ but straight lines Avhich join the extremities of equal and imrallel 9J-aight hnes towards tlie same parts, ar? themselves equal and parallel • therefore BE, CII are both equal and parallel ; wherefore EBCH is a parallelogram. UM. A.) And because the parallelograms ABCI), EBCII, are upon the same base BC, and between the same parallels BC,AJ£- JPiJc^/T/sS f"^"''^''^''"^™ ^^^^ ^« ^q"'^! to the parallelogram For the same reason, the parallelogram EFGH is equal to the p5,rallelogram EBCJl; ^ therefore the parallelogram ABCD is equal to the parallelogram JbJfGH. (ax. 1.) '^ Therefore, parallelograms upon equnl, &c. q.e.d. PROPOSITION XXXVII. THEOREM. Triangles upon the same base and between the tame parallels, are eaual to one another. ^ Let the triangles ABC, DBChe upon the same base BC, and between the same parallels AD, BC. Ihen the triangle ^ii'Cshall be equal to the triangle DBC. E A D p Produce AD both ways to the points E, F; through B draw BE parallel to CA, (i. 31.) and through Cdraw CJP parallel to BD. ^ ^«^*^^.°^^^'® ^»"^^' ^^<^-^' DBCFh a parallelogram; imd EBCA IS equal to DBCF{i. 35.) because they are upon the same base BC, and between the same parallels BC, EF And because the diameter AB bisects the parallelogram EBCA therefore the triangle ^£C is halfof the parallelogram J&i?C^- fi 34) also because the diameter DC bisects the parallelogram DBCF therefore the triangle DBCh half of the parallelogram DBCF, but the halves of equal things are equal ; (ax. 7.) therefore the triangle ABC is equal to the ti-iangle DBC. Wherefore, trianirlep, &c. q.e.d. c5 64 K'TMn's ELEMENT8. PIlOroSlTION XXXVIII. •TIIEOIiPM , Tnaugln upon cgual buses and betu^eeu A. to one another. oettcten the same parallds, an tquzl Let the trinncles ATlV 7)rK'v.« . b"weea the »„„,^, .^IZu fiyj^ ' " ">""' ■"■»' ^^ ^^. ^"i Ihen .he trianfile ^i^. .„„,•, ,, ^^^^^ ,^ ,^^ ^.^ ^^ ^^^, o A D n Produce ^7) both ways to the no'-its r? 77 tluou.h 7; chaw /yr? {mrallel to^ci ri 3l\ and thoy "a.-e ecu^^afVonf a' ^^^'r! ?,''«- '^^««-. I becanse fhey are upon equal Imses >][• /'> And becaur h :nrtet!.Tr '""""^■'•^ ^^^ ^^^ th:JSi,^s:^i::;;;:;S^.f^?« the i'-"";^^^- nE,,, but the halves of emnl,l,?nlf^^^ Parallelogram DJ^FJI, therefore the trian Je AJiC 1 1" ?? T^^ ' ('^^^ ^O ^Vherero., t.-S.l:ie1fp"oi\5^r^^^^^ , PROPOSITION XXXIX. THEOREM ^etS::l^S:;zit ''^'"^ ''''' «"" ^'^^ ^^^ -- -v. ./ ,, «.. Let the equal triann-le nr^n u and upon the .same sid" of it ^ "P°" ^^^ ^"'"e base J?C Ihen the triangles .i^'c; i>-2,Cshall be between the same parallels. A D '°'" Fofir^i n r^^ '^"" ^^ P'^rallel to i?C. ;p«« •!., . -^"^"^-^ be not parallel tn 7?^ •» tharf„reS'^ilf,?'^/„'';;i '°.'h? 'Angle Mc,- (hyp.) ° .a equal lu the Iriangie ii'i/Ci I. alk!s, art tfuzt ^C, EF, and e DEJ-: ) m GTiCA, am OIiCA\ 1 DEFH, im DFFJIi DBF. .D. '« 0/ t<, arc i base ^C parallels. C, (I. 310 t. 37.) lyp.) iO, BOOK I. rnop. XL, xli. 35 the greater trlnriKlo r<,ual to the less, which is impossible 5 tlicniorc AF is not parallel to liC. In the Mime manner it can he .lemonstratod. tfcat no other h„e -irawn (Vom A hut A I) is parallel to BCi AD IS theivtoio jiarallel to JtC. Wherelore, equal triangles upon, i;c. q.e.D. PliOroSITION XL. TKKOUEM. r.quaUrmnffleivpone>pinl.ba.s,»in (he same straight line, and totcar be not parallel to BF, if possible, throuf^h A draw A(i parallel to i?JS', /j 31 x meetm- j;/;, or ED produced in 6-', and join (iF Ihen tlie trum-ie yi7,'C'is equal to the trian-Ie C/y;/'', (i. 38.) because they are upon equid '>ases BC, EF • and het\ve(>ii the same jjarali^ls BF, AG-' but the t,nan-r!e AliCU equal to the triande DEE; fhvn ^ there ore the tr;angle DEFU equal to the triangle GEI>, fix U the greater tnan-le equal to the less, which is imuossihle :* therefore A G is nt parallel to BF. And in the same manner .c can be demonstrated, t.uit there xs no other line drawn from A parallel to it but AD- AUk therefore parallel to BF. * W herefore, equal ti'iangles upon, &c. q. e. d. PROPOSITION XLI. THEOREM. If a paraUvhgram and c triangle be vpon the same ba.^e, and between Ue same parallds ; the parallelogram sh An,] h ^^'M'^ CEFG is a parallelogram, rfjf 4 ^ And because the triancr «, Aliv a vn """" l''''i- -"^•) EC, and between theTa",^e ;,alSl?C^^^ ''" "^"^^ '^^^^ ^^• they are therefore equal to one another- (i 38 ) because they are upon the same base Vr nn K f I ^^- ^^'^ parallels EC, AG ■ ' "^ between the same "'''and',-; U^''''"'^^''''" ^^"^ ^« «q^-I to the triangle ABCU^&\ ana It has one of its angles TT^/f cn„oi t^ *\ "' ."n"^^^^^. (ax.b.j Wherefore, a paralleloSa,rjfevv> ? i ^^'^ ^''■.?' ^"^''<^ -^^ PROPOSITION XLIII. THEOREM. also BK KJ^ fh« ^fiT ^""",\^ ^' ^'^"f' ". -) . and because a the noinf /a t?"'^ '° '"^ ^-'^'I't an^^les ; straight lines A'Jf, JUlVTZ '" ^¥ ^^^'^'ht line r///, the tw, jacent angles A7/6^ r/// ^"eoualto'/r''- 'f '^^ "^ "' "^^^^^ the Zl therefore 7iA' is in t hf e! ^ ?''^ '"'^'^t angles, , And because the uLJlG TfV"'' ^''^ ^^^^- ^^. H.) therefore the .„gle iV^J/^ ^^f,taU^tfi^! ^^'^^^^'^ ^'^^^- ^'^. add to eipf. nf ,1 ' , ^ alternate anj;le HGF- (i 20 ^ therefore the anglesl;7/r HcP:""^' ''^1 ^"^''« i^Z; ' ^ ' ^^"^ but the angles ^MHG, nGlt^ZlTt^u '^' T^^'' ^'^^^' ^^^-^1 therefore also the angles HGF IWr.V "" 'l^^'' '^"-'*^^' ^'- 2^-) and therefore i^G' i"; in the samf « r-; 4'",^"'''^ '" *"^ ^i*^''^* angles. same ouaiijii!, line wiUi QL, (i. i|.j BOOK I. PItOr. XLV, SLTI, IK', it-BiJ';(i.43.) (constr.) iBM, (I. 15.) (ax. 1.) ?ram LB has angle AUM f /^«rc, and given recti- Pqual to the S9 42>J?, and ual to the B angle H. iired. >'igle E, I/; les ; the tw,-> > the ad- V(i. 29.) ''', ITGL', il. 2!>.) anjf^es, And because JTJ'is parallel to IIG, and //G' to ML, therefore KFh parallel to ML ; (i. 30.) and FIj has been j)n)V(>d j)arallel to KM, whereibre the figure FKML is a parallelogram ; and since the i)arailelogram IIF is equal to the triangle ABB, and the inirallelogram GM\.o the triangle BDC\ therefore tiie whole parallelogram KFLM is equal to the whole rectilineal tigure AliCD. Therelbre the parallelogram KFLM has been described equal to the given rectilineal figure ABCD, having the angle FKM equal to the given angle /i'. q.e.f. Cor. From this it is manifest how, to a given sti- ' (lit line, to apply a parallelogram, which shall have an angle equal to u given rectilineal angle, and shall be equal to a given rectilineal figure ; viz. by applying to the given straight line a parallelogram equal to the first triangle ABD, (i. 44.) and having an angle equal to the given angle. PROPOSITION XLVI. PROBLEM. To describe a square upon a given straight line. Let AB be the given straight line. It is required to describe a square upon AS. From the point A draw -4 C at right angles to AB ; (I. 11.) make ^D equal to ^^; (l. 3.) through the point D draw BE parallel to AB; (T. 31.) and through B, draw BF parallel to AB, meeting BF in E\ therefore ABEB is a parallelogram ; whence AB is equal to BE, and AB to BE; (I. 34.) but ABh equal to AB, therefore the four lines AB, BE, EB, BA are equal to one another, and the parallelogram ABEB is equilateral. It has likewise all its angles right angles; since AB meets the parallels AB, BE, therefore the angles BAB, ABE are equal to two right angles ; (1.29.) but ^JZ) is a right angle; (constr.) therefore also ABE is a right angle. But the opposite angles of parallelograms are equal ; (l. 34.) therefore each of the opposite angles ABE, BEB is a right angle? wherefore the figure ABEB is rectangular, and it has been proved to be equilateral; therefore the figure ABEB is a square, (def, 30.) ftnd it is described upon the given straight line AB. Q.E.F. 40 Euclid's elements. Cor. Hence, every parallelogram that has one of it£ angles a rirfit angle, has all its angles right angles. PROPOSITION XLVII. THEOREM. In any right-angled triangle, the square tchirhis described upon the side tubtmdmg the right angle, is emial to the squares described upon the sides which contain the right angle. ' ^ ** let ABC he a right-angled triangle, having the right angle BAC. Ihen the square described upon the side £C, shall be equal to the squares described upon ^^,^iC. ^ G On ^C describe the square? JiBEC, (\ 46 \ with it at the point ilhl' frV ^°" ^''f ''PP"-''^^ '"^^^ o^ ^S, make ^ And because the ;fo,Mffj'in°^'°*«»''t»"flerac.(ax.2.) No, the p.?alie*;™ ",?i'i^Rr ^^''.'""S''' ^^«^ l'^ BOOK I. PKOP. XLVIII. 41 glesariflit )on the $ide m the aides [ual to the Therefore the whole square BDEC is equal to the two squares QB, nC; (ax. 2.) ^ H , and the square BDECh described upon the strniirht line BC, and the squares GZ>', IIC, -'pon AB, AC: therefore the square upon the side BC, is equal to the squares upon the sides AB, AC. Therefore, in any right-angled triangle, &c. Q.E.D. PROPOSITION XLVIII. THEOREM. If the sqnare described npon one of the sides of a triangle, he equal to the squares described tipon the other two sides of it; the angle contained by lilt se two sides is a right angle. Let the square deseribed upon BC, one of the sides of the triangle ABC, be equal to the squares upon the other two sides, AB, AC. Then the angle Ji AC shall be a right angle. ) B, make t angles; 14.) It Ime. a(ax.2.) ides FB, included r. 41.) te same From the point A draw AD at right angles to AC, (I. 11.) make AD equal to AB, and join DC. Then, because AD is equal to AB, the square on AD is equal to the square on AB; to each of these equals add the square on AC; therefore the squares on AD, A Care equal to the squares on AB, A C: but the squares on AD, AC are equal to the square on DC, (I. 47.) because the angle DA C is a right angle ; and the square on B C, by hypothesis, is equal to the squares on BA,AC\ therefore the square on Z>C is equal to the square on BC\ and therefore the side DC is equal to the side BC. And because the side AD is equal to the side AB, and ^ C is common to the two triangles DA C, BA C; the two sides DA, A C, are equal to the two BA, A C, each to each ; and the base DC has been proved to be equal to the base BC; therefore the angle DAC'i% equal to the angle BAC; (I. 8.) but DA C is a right angle ; therefore also BA C is a right angle. Therefore, if the square described upon, &c. Q.e.d. ;en lie 3. NOTES TO BOOK I, ON THE DEFINiTICNS. Of primary importance to draw°a dtuncLn V?''"'' ^l"^ '^ '' ^ ^"^J'^ct thmgs and the things themselves! These hvn tf'''''^ '^" concq.tion of property contrary to the real nature of he Sh^n^^r ^° "°^ ^"^'«^^'« ^"v be regarded as arbitrary, but in certain r!!^^f' """^ consequently cannot tions which the things themselves Tuo3''' T'' ^''^ theconcep- medium of the senses." The esSuial E'r ° ^\^ J^'"'^'^ through the be.ng inductions from obseryaSrand t not ''' ""^ Geometry therefore evidence of the senses. '""^^""''^ ^"'^ experience, rest ultimately on the vidull\orCiTi;;:^:irude'^"b"ut 17T'''' rl^ *^^ --^-ee of indi- posed uniform and consistent ""''''' "Uellect being sup- Euclid's Elements marbrdSenS'° 'i"^^"'^- '^'^^^ definitions in explain the meaning Vthrtems emnlr.r/''Tv"^°^° ^^^" "^"-^S explaining the meaning oulXZs^SZtT^ ^^"'"' ^^'^''^^ ^^^des described in the definitions. ' '"^'i'°'° ^^'^ existence of ti , things natul?S^;;^rEIrKJf?™JS^/:^?^« ^ - *« -P'a- the marks Avhereby the thin" de ned m.w v •* '' ^.""'"ent that they give of the same kind. It wi la oncTL i^^'^'^'Suished from ever-/otl er Geometry, one of the pure 8ctn"cs bein^Th^f' '?'' 'H definitions o like the definitions in any one of tl.i^ni?" ."^f ^^P^ions of space, are not of any new physical facts^i y rend r7e S'lrl !"''"''f " '^'^'' ^^''^^^^y fication in the definitions of the 1 itter ''^'^ ^'""'^ alteration or modi- definS^.; is. !:s 'i:i^:rx^:^ ^t'^'n °^^ i^-*- ^-hd-s IS no part," or which cannot brpaAed'r ti^T} '' '^''''.' "^'^•^'^*' ^here Proclus The Greek term -,^'L^ ^ er^S^tol ;^« ' "'-'V,' ^^V^-'^inod by on a surface, in other words, « i^S? IL / n ^ ^^^'^^ ^*'>« or nun A means the sharp end of anj^ tldS or /To;i, ^^^^ ^"g^'si^ term ;,.»,/, P'nnt comes from the Tnfin ^ ?' ^ "^^'"'^ '"^^e by t. The word Neither of theseTeim .if "CSZe""^' '''' ^^"^'-'^ woriMS notion of M-hat is to be undo xrnn 1 > ' ^^'?"'^'' ^^ S*^''' » very exact definition of a point mereirex nt'c s a^u" Ev'!' "' ^"""^^'>'- ^^"'i'l'« the proper and literal meanin; of S- GrS to , r^''''-'*r^''i''" ^■^^'^^des physical point, or a mark .vhich I s bS to th ' ^' '''^'^''^ '° '^^""'^'^^ « Pythagoras defined « T^ninf. u ^^^'"^° /o the senses. Sciences, and 1 observation. »w consistent t is a subject ;once])tion of involve any ently canno't the concep- tlirough the ry therefore ately on the nee of indi- abstraction, -he general conception conceptions •nd appear- beins sup- nple ideas, equivalent prms, does initions in ich merely ;h, besides tl things '.•plain the they give 'erv other litions of e, are not discovery or modi- I^uclid's ich there aiued by or nw) k •in ]H>int, he word rd point, rv exact Kuclid's xcludes Icnote a havinr' egativc ry may 1?0TES TO BOOK 1. 43 whiT^f '^ P''^'^' '^'''f intelligible. A point is defined to be that which has no magnitude, but position only. no,^hi;VA ^i?""""^ ^'^"^v^^ ^'?^ ^'''' ^''^^ '""3^^ ^"'^ breadth, and it is im- possible to draw any line whatever which shall have no breadth. Tlie S\ ''Tl""""' the conception of the length only of tlie line to be considered, abstracted from, and independently of. all idea of its breadth! of tip H^HnU- "^ achnition renders more intelligible the exact meaning ot the defanition of a point: and we may add, iliat, in the El«nents! l.utlid supposes that the intersection of two lines is a point, and that two lines can intersect each other in one point only. ^ l)ef. IV. Ihe straight line or riglit line is'a term so clear and intel- hgille as to be mcapahle of becoming more so by formal detinid.m tuchd sdefamtion IS E.;tl,7a rp,,,.^,; ,V?<., .'iru i^ Lo -roi. ^ ' "«"rl , hZy^. '<^oj') between its extremities, and which I'roclus explains as being stretched between its extremities, v iir' «\p«,. -rtra^Uvn It the line be conceived to be drawn on a plane surface, the words deviates either from one side or the other of the direction which Is Hxed c^v d w'wllnr/''^' ^^V ""'^ f^""' '' ""^y ^' distinguished fhmi 2 must L ,fl '^^^.^ conceived to be drawn in space, the words .'g l.oo, between hsSemtties.'^''^ '' ^^"^ '"""^°" '"^ '^'^^ ^'''^ «^" ^^^ ^"- Every straight line situated in a plane, is considered to have two sides • and when the direction of a line is known, the line is said to be^^iJen n K^i^'mtniJud^ ^'^ ^^"°"' '' '"'^"'^ "^ ^^ ^^^-"^' ^* - -'^ - ^: straidiUine in'!!n'>°'' of a straight line, it follows, that two points fix a straight line in position, which is the foundation of the first and second postulates. Hence straight lines which are proved tocoincide nnvoor .n e pouu. are called, "one and the same straight line." Prop. 14. Book? or, which IS the same thing, that "two btruiirlit lines canno ave a common segment." as Simson shews in his Corcdlarv to Prop 11 , Sook i! "Strif /^' °^^^"g dehnition of straight lines has also been proposed Straight lines are those which, if they coincide in any two pointTcoin Une?and Tnl"^ ^^-^P^-^^^e^d." Eut this is rather a c^Jiterion of ^^^gU S;« «i r?""' *° *^'° eleventh axiom, which states that. " all ri<^ht angles are equal to one another." and suggests that all strai^'ht lines mav be made to coincide wholly, if the lines be equal ; or partially if the S desc LroS tKv • f^-'f^'^ should^roperly'be res£icted to he nSn t". ^ ^^ "" '^''^""^' •''' '^ ^''''^'' independently of any com- parison of us Foperties or of tacitly assuming the existence of axioms JJcf. vii. Luclid s definition of a plane surface is 'E^l-rr^So, inrubd- tw v'v!'!" *' 'T """'" "''' ""'^''' *'''^"'«'» '^"-«'. " A plane surface is of whLh's- '' ^T^y^' '^^^"y ''''^ the straight lines ?n uVmus ead bv Hero thP F?? ^^ f T '^' ^^^^"^t^°" ^^^^^^ ^^^« originally proposed i/anv nn^f; -A ,^ Pl^"« superficies may be supposed to be situated D Jf ^vf , ^ i-^"'^ ^"^ ^' continued in every direction to any extent, angles formed blT^f ""^^r ^^^^^ "-^'^ clefinition seems to include the « f hat f J^ 1 7 ^^""^'^^ ^'"^'' °^ ^ ^^ve ind a straight, line, as weU as that formed by two straight lines. ^ =, a» weu (Veomlt^y/'''^' ^' '"""^^ ^' °"-^5"' ^« "^-^^'i "^^ Elementary I 44 Euclid's elements. P; fn'r'"",°^u''^^ term an,ul>!^ su^'' meaning in the coTist?uc^t?on of Pron'^ froni Proclus, as it seems to have a 33. Book in. and Prop. 3. 'Ik Jf * Th^l'^ V- *^^«?^ ^^'^ of Prop. circle is not once alluded Co in Book'i anl , •«'!""'" "^ '^''^ ^'S'^^''*' "fa cussion of the proj.erties of ^e cird ' in 7 T' ''l^r*^ ^^^°^« the dis- this definition ! .'Hence you may cSec^ W t^ "'' ^ 'f '"^ '^"^'-^'^s on for It is either .vithin the li^r? L in tf.^^'''^ \'"' '^^"^^'' ^^« three places :• m the semicircle ; or without the yiea.^nt \ '" ^" ''■' P^^^^eter, as Def. xxrv-xxix. Trian-lesare fivL^nf- ? ''f '^'" '^''"^'^ ""es." to the relations of their side's and into tl "'*° ''j'"'^" ^^'^■^^^«' ^^^ '^f^rence to their angles. A further iSSiofra.hf '^T'^^ ^^ ''^^'^'^"'^^ both the relation of the sides and amfl, L ^l^^- """"^^ ^^ considering In Simson's definition ofthe isoSh , t^^ ^^ *"^"»'"- omitted, as in the Cor. Prop 5 Poo^ *' "? ' *^« ^^^^^ equilateral, and an equilateral trian'^lp,-/' ^". ^''^''^celes triangle may be Book IV. Objection has reenSto^^ '''''''^'' '"^ P^"P ^5. triangle. It is said that it cannot be adn? i "'?".. "^'^'^ acute-angled three angles of a triangle are acute wh h ?/' ^ definition, that alfthe mav be replied, that tlfe definitions onhe^hrL\"PPf'"^ ^'^ P^f- 29- It and seem to supply a foundat on for « 1; three kinds of angles point out Def. xxx-xxiiv. £ SnUion/r.f n f -f'"'"'?^'"" of triangles, objection. All of them, excent he 1^"'- "^"^f ? ^S^^^'^'^ ^^« "able to idea of a parallelogram but^c^ EnSffl"'',^'" ""'^^'^ ^^^ general after he ha'd defined fSoi-sTded figures no o^?"^ P^'^^^^l ^t^aiglft lines adopted than the one he has fo lowed'' Ld f?/ ""^"S^'^cnt could be him, without doubt, some nrohabU. I'oc ^'^'^e^H^J' ^'^^^^ appeared to Seventh Lecture, remarks 5n soSe of S .^'.^ ^enry Savile. in his dissimulandum aUquot harum iS^^nih?/ "^^-^'""^""^ "^ Euclid, "Nee metria." a few verbal eSatio^^rcvK^""^"""^ ^'«« ^^^^^ i" Geo- A square is a four-siS ^.tir « ^^""^.^"^^P- made in some of them, one angle a right anSe • wi > -^"'^ ^L^^'"^ ^^^ i^« sides equal, and angles. ^ * "^t^t angle, aU its angles are right NOTES TO BOOK I. 45 ar aoncf'ptiop . The literal ception of an f two straighf aitude of the which it ie they meet, ia i out in thf leet is called Bconfoundec' xed in mag' h which ali ■T, or which ane another, angle which Jclid always the latter is ' anffle, opQi) IS to have a ise of Prop, "gment of a ire the dis- remarks on ree places :• ;rimeter, as les." y reference i reference onsidering hf must be e may be t Prop. 15, Jte-angled lat all the ;f. 29. It point out ingles. 3 liable to e general :ght lines could be peared to ie, in his d, "Nee I in Geo- if them, [ual, and that if a jre right Ail oblong, in the same manner, may be defined as a plane figure of four sides, having only ito opposite sidea equal, and one of its angles a right angle. A rhomboid is a four-sided plane figure having only its opposite sides equal to one another and its angles not right angles. Sometimes an irregular four-sided figure which has two sides parallel, is called a trapezoid. Def. XXXV. It is possible for two right lines never to meet when pro- duced, and not be parallel. Def. A. The term parallelogram literally implies a figure formed by parallel straight lines, and may consist of four, six, eight, or any even number of sides, where every two of the opposite sides are parallel to one another. In the Elements, however, the term is rnstrictcd to four-sided figures, and includes the four species of figures named in the Delinitions XXX — XXXIII. The synthetic method is followed by Euclid not only in the demon- strations of the propositions, but also in laying down the definitions, lie commences with the simplest abstractions, det.ning a point, a line, an angle, a superficies, and their different varieties. This mode of proceed- ing involves the difficulty, almost insurmountable, of defining satisfac- torily the elementary abstractions of Geometry. It has been observed, that it is necessary to consider a "olid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the defini- tions of a point, a line, and a superficies. A solid or volume considered apart from its physical properties, suggests the idea of the surfaces by which it is bounded : a surface, the idea of the line or lines which form its boundaries : and a finite line, the points which form its extremities. A solid is therefore bounded by surfaces ; a surface is bounded by lines ; and a line is terminated by two points. A poirt marks position only : a line has one dimension, length only, and defines distance : a superlicies has two dimensions, length and breadth, and defines extension : and a solid has three dimensions, length, breadth, and thickness, and defines some portion of space. It may also be remarked that two points are sufllcient to determine the position of a straight I'ue, and three points not in the same straight line, are necessary to fix the position of a plane. ON THE POSTULATES. Thb definitions assume the possible existence of straight lines and circles, and the postulates predicate the possibility of drawing and of producing straight lines, and of describing circles. The postulates form the principles of construction assumed in the Elements ; and are, in fact, problems, the possibility of which is admitted to be self-evident, and to require no proof. It must, however, be carefully remarked, that the third postulate only admits that when any line is given in position and magnitude, a circle may be described from either extremity of the line as a center, and with a radius equal lo the length of the line, as in Euc. i, 1. It does not admit the description of a circle with any other point as a center than one of the extremities of the given line. Euc. i. 2, shews how, from any given point, to draw a straight line equal to another straight line which is given in magnitude and position. 46 euclil's elements. ON THE AXIOMS. contion nf tl,„ t,,./! f u • "i P'-'^'^ to rest on tlie same basis. Thf con- • as ni'ltrS^^d'SfJ'a:!;:,^' ^•'^"--'-^-»; -« ^-h theorems monstratcd An ,v - i / v "" ''^''^^ '■'' capable of being de- demonstration. '*-"^" °^^'''-'"' ''^"^^^^ require a formal majid^ut mrs'a'emir^^^^ '™"^- ^'^^^"-^ -• that every sively till the same Ipaee ^ ' "'"' '''""'■ •"'^o"'^"'I«« "^^y succes- mo^'simpIe^rebSa'e t^iurs^f ""'"f"''' \'- ^"'"'^^-^ ''^^'^tive, and the parison of ma^tSes T,,me «r/.''^'''^'^' V"^ "'}^V'^^^y' I" tlie com- unknown are compted 7i I the kn"'" "^ ^"«^-' "^"^^ ^he cally deduced witl resDPPrfntiV """""• ""' conclusuns are syntheti- which is thus formaliv sUte n^h e 'hth ■ x[. 'm °"^'/'"' • °^•^^"f .'y- comcide with one another tha i, «.i.^v, 'ix'™:" Magnitudes which equal to one another "' ' "''' "''^''^y ^^^ ^he same space, are to be Placed on LT.h^r °»« '""menoal iiiasr.ituje may be conceived this proposition is not uniL^f.'^lf ^^ ^^"^^rked, that the converse of s^ :i;KHSS'lr "°s* -— ^^ 37 • but th 'ir Kof ^^ • ° parallelograms or two triangles, Euc. i 35 NOTES TO BOOK 1. 47 D B C Ihis axiom is the criterion of (Jeometrical eqwulity, ari.l is cssentiallv diffMre.it troin the criterion of Arithmetioal equality, "."wo geometrical magnitudes arc equal, wheu they coincide or may he mad.; to coimndo • two ahstract numbers are equal, u-heu they coiUuiu the «Hme a-.^r.-afe ot units; and two concrete nunil)cr.s are e(iual, wlien tliev cuiuaiiAhe ■arm- number of units of the same kind of niai,'nitu(le. It is at once oh- vioua that Anth.netical representations of Geometrical nia-nitudes are not admissible in Euclid's criterion of Geometrical Eijuality, as he has not axed the unit of magnitude of cither the strai-ht line, tho an^le, or the yuporhcies. Perhaps Euclid intended that the hrst seven axioms should be apjihcaole to numbers as well as to Geometrical ma-nitudcs, and this IS in accordance with the words of i'roelus, who culls the axioms, common notions, not peculiar to tho subject of Geometry. Several of the axioms miiy be generally exemplified thus • Axiom I. If the straight line ylZJ be equal A u ' to the straight line CD ; and if the strai^'ht line EF be also equal to the slrai;,'ht line CD ; E then the straight line Ali is equal to the straight line EF. Axiomii. Iftheline/lZ?bccqualtotheline A. CD\ and if the line EF be ilso equal to the line Gil: then the sum of the lines AU and EF is equal to the sum of the lines CD and UH. Axiom III. If tho line ABha equal to the A_ line CD ; and if the line EFhc also etjual to the line GU; then the difference oi AU and EF, is equal to the difference of CD and GU. Axiom IV. admits of being exemplified under tTi(. I. If the line ^ Zi bo equal to the line CD ; ^ and if the line EF be greater tlum the line GH\ " then the sum of the lines ABtiwl EFls greattr than the sum of the lines CD and GU, 2. If the line AB be equal to the line CD ; a and if the line EF be less than the line GU\ — then the sum of the lines AD and EF is less ] than the sum of the lines CD and Gil. A.iiom V. also admits of two forms of exemplification. 1. If the line AB be equal to the line CD; a B and if the line EF be greater than the line Gil; * then the difference of the lines AB and EF is E 1 greater than the difference of CZ) and G//. 2. If the line ABhii equal to tlie line CD ; A n and if the line EF be less than the line Gil; then the difference of the lines AB and EF is ^ 1 less than the difference of the lines CD and GH, The axiom, ''Ifunequals be taken from equals, the remaindorG are unequal, nuiy ne exemplified in the same manner Axiom VI. If the line .'I Zy be double of the A B line CD ; and if the line EF be also double of ' C o the line CD', 2 j ■ then the line ABi% equal to the line EF. Axiom vii. If the line AB be the half of A B the line CD : and if the line EF be alro the C B hs-If of the liiie C7J ; ' "" ' By then the line .,45 is eijual to the line EF. n two following forir.8 : B C B D H F c D H c D 11 c D G H 48 buclid's elbmbnts. It may be observed thnt when enual mnRnituueg are taken from un- equal magnitudeg, the greater remainder exceuda the leas remainder by as much as the greater of tlie unequal mugnitudes exceeds the less. It unequals be taken from unequals, the remainders are not always unenual ; they may be equal : also if unequals be added to unequals the wholes are not always unequal, they may also be equal. Axiom IX. The whole is greater tlian its part, and conversely, the part 13 less than the whole. This axiom appears to assert the contrary of the eighth axiom, namely, that two magnitudes, of which one is greater than the other, cannot be made to coincide with one another. Axiom X. The property of straight lines expressed by the tenth axiom, namely, " that two straight lines cannot enclose a apace," is ob- viously implied in the definition of straight lines ; for if they enclosed a sp-vce, they could not coincide between their extreme points, when the two lines are equal. Axiom XI. This axiom has been asserted to be a demonstrable theo- rem. As an angle is a species of magnitude, this axiom is only a parti- cular application of the eighth axiom to right angles. Axiom XII. See the notes on Prop. xxix. Book i. ON THE PROPOSITIONS. Whenevbr a judgment is formaUy expressed, there must be some- thing respecting which the judgment is expressed, and something else wh'ch.constitutcs the judgment. The former is called the subject of the proposition, and the latter, the predicate, which may be anyUiine which can be affirmed or denied respecting the subject. The propositions in EucUd's Elements of Geometry may be divided into two classes, problems and theorems. A proposition, as the term imports, IS something proposed ; it is a problem, when some Geometrical comtructwn is required to be effected: and it is a theorem when some Geo- motrmxl property is to be demonstrated. Every proposition is natu- rally divided into two parts; a problem consists of the data, or thinas 9^ven: and the qucBsita ov things required: a th. orem, consists of the inibject or hypothesis, and the mnclusion, or predicate. Hence the di.stinctitm between a problem and a thtwem is this, that a problem consists of the data and the quiesita, and requires solution: and a theorem c( usists of the h}'pothesis and the predicate, and requires demonstration. All propositions fae affirmative or mujative; that is, they ei ler asser Bome property, as Euc. i. 4, or deny the existence of some property as Jiuc. I. 7 ; and every proposition which is affirmatively stated has a con- tradictory corresponding proposition. If the affirmative be proved to ba true, the contradictory is false. All propositions may be viewed as (1) uriiversally affirmative, or uni. versally negative ; (2) as particularly affirmative, or particularly negative Ihe connected course of reasoning by which any Geometrical truth is established is called a demonstration It is called a direct demonstration when the predicate of the proposition is inferred directly from the pre- misses, as the conclusion of a series of successive diuluctions. The de- monstration is called indirect, when the conclusion shows that tlje intro- duction of any other supposition contrary to the hypothesis stated in the proposiUon, necessarily leads to an absurdity. It has been rmarked by Pascal, that " Geometry is abnost the only subject as to which we find truths wherein all men agree . and one cause 01 ijus u», uiat Ueoiueicrg aiuuc itsgard the true laws of demonatratioa." NOTKg TO BOOK 1. n from un- tnainder by i less. not always icquolg the ■PTsely, the 16 contrary ich one is lother. the tenth 5e," is ob- enclosed a , when the able theo- ly a parti- be some- thing else tject of the ng which e divided the term :ometrical ome Geo- is natii- or thincf's Its of the istinction 3t8 of the pend. Otliers have maintained that a definition explains only the meaning of a term, and dues not embrace the nature and properties of the thing defined. If the propositions usually called postulates and axioms are either tacitly assumed or expressly stated in the definitions ; in this view, de- monstrations may be said to be legitimately fou.ul- d on definitions. If. on the other hand, a definition is simpl v xphuuition of the meaning of a term, ^yhether abstract or concret.-, .uch murks as may prevent a misconception of the thing defined ; u will be at once obvious that some constructive and theoretic prindplcs must be assumed, besides the deiini- tions to form the ground of legiti.nue demonstration. These principles we conceive to be the postulates and axioms. The postulates describe constructions which may be adnatted as possible by direct appeal to our experience; and the axioms assert general theoretic truths so simple and self-evident as to require no proof, but to bo admitted as the assumed Krst principles of demonstration. Under this view all Geometrical reasonings proceed upon the admission of the hvpotheses assumed in me dehnitions, and the unquestioned possibility of the postulates, and the truth of the axioms. ' Deductive reasoning is generally delivered in the form of an enthymeme or an argument wherein one enunciation is not expressed, but is readilv supplied by the reader : and it may be observed, that although this is the m-dinarymode of speaking and writing, it is not in the strictly sylI„.a.stio torm ; as either the major or the minor premiss only is formally stated before the conclusion : Thus in Euc. i. 1. Because the point A is the center of the circle BCD ; therefore the straight line Ali is equal to the straight line AC. ifie premiss here omitted, is : all straight lines drawn from the center 01 a circle to the circumference are equal. In a similar way may be su])plied the reserved premi« in every enthv- nSs Jfl%S'!^'.n;^i^^^^ *-m the major and min'or •o , j I r X'" ~.'."°S'='^"' wk; so on, ana thus any iiroeess ot"reasoiun-r IS reduced to the strictly syUogistic form. And in this way it is shown -^)0 Euclid's elements. tlutt the ftoiioral theorems of Geometry are demonstrated by means of syllogisms founded on the axioms and definitions . ^>,P^'^J•'^'""^■'T ?«"\'sts«f three propositions, of which, two are called thre? tTrmf r. ?' f"'*^; '^' f.""^i"«i°"- '^'^ese propositions contain .r^lT^' «^^'^Ject and predicate of the conclusion and the middle X.Tnf'tT 'connects the predicate and the conclusion together. T e subject ot tlie conclusion is called thefninor, and the predicate of the con! elusion 18 called the major term, of the sylIog-»n. The major term appears m one premiss and the minor term in the other, with the middle term term and the major term, is called the major premiss ; and that w lich stratLn of C I'^l.^^'f '"C'*'' ^'^n^^Jf /ake the syllogism in the demon- stration of I rop. 1, Book 1, wherein it will be seen that the middle term is thesubject of the major premiss and the predicate of the minor Major premiss : because the straight line A B is eoual to the straight line .IC- ItZuT''' '' ' ' '^" ''™^^'' ^^' '''' '' ^^^"^ ^° ^^^ ^ "^"'i^* ^°iw '«rM«'tf ^°'' ?■" '*'''f * ^^^ ^^ '' '^"""^ t° t^^ straight line ^C. Here, lit is the subject, and AC the predicate of the conclusion. ^Cis the subject, and Ali the predicate of the minor premiss AU. frP- V ^?^i(^<^U and AC the predicate of the major premiss. " Also, A C IS the major term, JJC the minor term, and Ali the middle term 01 the syllogism. In this syllogism it may be remarked that the definition of a straight Ime IS assumed, and the definition of the Geometrical equality of two Btraight lines ; also that a general theoretic truth, or axiom, forms the ground of the conclusion. And further, though it be impossible to make any pmnt, mark or s,gu (^,,^,7.. which has not both length and breadth and any line which has not both length and breadth ; thc^lemonstraUon in Geometry do no on th.s account beoon.e invalid. For they are pursued on the hypothesis that the point has no parts, but position only : and the line has length only but no breadth or thickness / also that the surface has length and breadth only, but no thickness : and all the conclus one at w-hich we arrive are independent of every other consideration. ihe truth of the conclusion in the syllo^iism depends upon the truth of the premisses. If the premisses, or only one of them be^lot true the conclusion is false. The conclusion is said to follow from the premLes • whereas, in truth it is contained in the premisses. The expression musi be understood of the mind apprehending in succession,\heTut7o tlie preimsses, and subsequent to that, the truth of the conclusion: so thft the conclusion >Wo«,5 /,w» the p.-emisses in order of time tJZZ '■'""'' "" '"''■'^' ^"^ '^'' "^'"'^'^ apprehension of the whole argument. Every proposition, when complete, may be divided into six parts, as i'roclus has pointed out in his commentary. thP^^nnr^-tZT^ffv!""' "V? '"'"^"'L"""'"''""' ^l^^ch States m general terms tne conditions of the problem or theorem. f,- ^^« exposition, or partimdar eimnciation, which exhibits the sulgect IgJL'drcribS."^"''""'^'^ '""^ "^^ ^ '^'^'' ^''^ -^-^ '' »° -^« . 3. The determination contalr i\iG predicate in particular terms as it V^TiT- ^?' f ^"^'""' ^''^ ^''''' attention to the dmoSaSi by prououacmg the tliuig sought. «"o««ti«a, 7 means of are called ons contain the mid die ther. The of the con- rni appi are ddle term, he middle hat which nor premiss he demon- die term is r. It line ,}C; le straight ision. premiss. _ premiss. iddle term a straight ty of two orms the e to make i breadth, nstrationa e pursued : and the e surface n elusions I. the truth true, the remisses ; iion must truth of iclusion ; of time le whole parts, as ral terms le subject to some ns, as it Bt]:atio!^ NOTES TO BOOK 1. «1 " the'dem?r:sSor"'' '^''"'' '^' ^"''"''''^ '° ^''^"' '^' '^'^^'^^ '»' f.„?v ^'f *, ''^r'"'f''^J^"»- is the coTmexion of syllogisms, which prove the oro iem t""^f ""^ '\ '^'°''^'"' ^''^ P"^«ibilityL impossibVliVy of he problem m that particular case exhibited in the dia^rram b. ihe conchision is merely thfe repetition of the "cnerai enunciation wherem the predicate is asserted as a demonstrated Truth '""""*''""* irop. I. In the first two iJooks, the circle is employed ay a mp. ehan.cal instrument, in the same manner as the straij^^ht^inl! and the^s^ made of it rests entirely on the third postulate. No prop^rt^s ot "he postul.ite. W hen two circles are described, one of which has its center in the circumference of the other, the two cirdes boin. each of hemln V each otl" iHw?: T''T' '""^ "^'^-•/heir circumfJI-ences must Silt' c^ ll^^h ., • ^ ".^^ ""^ '^ '" ^b^i^"" *'r«'" the two circles cutting whirh ^J ,r'"- -^"^' ]• -llie Riven line has two extremities, to each of which a line may be drawn from the given point. 2. The enul ateral t angle may be desmhed on either side of this line. .3. AmrtS lilt vh.':i'^>''""^ triangle ^yy;; may be produced either way "^' H„. ^ , i*" ^' '■'^" 1"""^ '•'^ ^''f*"''" '" the line or in the line nro duced the distinction wlu.h arises from j<,ining the two ends ofThe hne .n..,?'' ^•'^"struction of this problem assumes a neater form by first de .r.'izjin ii.';;'r,'.".*5 ««;'-;,'"•"*"""="■'"■ '-<"^»«- t.r„„ are not ri.»i„,.teJ to ..„e paitioular position ofltiZmmihl Ihe consirucaon follows the same order tnl^i,,,, /,, ^.^ '^q'"*^» "es. ^■^■-ait^:"£r',„3r^£Ef- A corollary is a theorem wliirh ro^ui.a fy^,^ .u,. j.^. ... . a proposition." "^ ^^^ Qt.aionsiranon of Prop. yi. ia the conyerse of one part of Prop. y. One proposition d3 6$ Euclid's elkments. IB defined to be the cotiverse of another when the hvpothesifl of ♦>,« forrner becomes the predicate of the latter; and vice vS ^' 1 here 18 besides this, another kind of conversion, when a tlieorem has several hypotheses and one predicate ; by assumi,^ tL preS Tav hf'^'f"""'? ^h^"r« »f t^e hypotheses, some one of"theh/po these' may be inferred as the predicate of the converse. In this inanner that converse theorems are not universally true : as for instance the followmg direct proposit on is universally true; ' in the line ED I^^'t'"^''^''''''^.^'-^''''''' "f ^^^ '''^ ^O' ^^^''5 otherwise three dife.ent cases would arise, as may be seen by forming the d fferen? figures Ihe point G might fall below or upon the base EF prmluced fhe"^/' fr^ '^^''P- ^'''^- a'^d Prop. XXV. bear to eadi other the »ame relation as Prop. iv. and Prop, viiif Prop. XXVI. This forms the third case of the equality of two tri- al;? th'ree^'r^ ''/'-"^^t ^'' '^'''' "'^^« ^"d three Hi I. and when any three of one triangle are given equal to any three of another the tmnglcs may be proved to be equal to one Another, whenever the Pron Z^Tf'- ^T f '^' 5^yP0the.is are independenl of one another sides' and thTlZ',"^^'"' case when the hypothesis consists of two sides and the included angle of each trian-le. Prop. viii. contains the second M'hen the hypothesis consists of the three sides of ea, h triangle. Prop. xxvi. contains the tliird, Avhen the hypothesis cons?st« of two angles, and one side. either adjacent to the^S an'les or opposite to one of the equal angles in each triangle. tS fs another Ina'Z' n'"^ ^y ?"'='-^'^' r^^^" '^' hypothesis consi ts of wo s 5e and one angle in each triangle, but these not tiie angles inclu(?ed bv the two given sides in each triangle. This case however is onlv tnJ under a certain restriction, thus : "u»ever is only true If tico triangles have two sides of one of them equal to ttoo sides of th. fher.e h ^^ ^^^^ ^^^^ ^^^^ y opposite to one oftheaJJt to ^r^dt^^^^rs^aSTt^th^ t^£^^^^^ the angle LCA to the angle EFD, Z the'^n Te'^^^1 1 t'he'^ar'i.W-S" nitiottia:::^:' '''> ^'^ ^^^^^^ ^« ^he equal sii':?^; .i^&, Di^Ff Euc. I, 4. AG is equal to DF, Then in the triangles * NOTES TO BOOK I, 55 and the 8n£.« ',l7 "'''"''' ^' ^^''^ ^' bisect thediagonalsareeqSal iTthp'n./.! 1 P^'"''*"'''^"'^"! be right angled, the diagonals bLecTea^h othef a? 1 S'^ .^ " T"'' '' " ''^"""''- ^'^ryiZm^hiy.^ltr^^^^^^ '« not expressed to considerlwo trapeSs ^the s!^p J demonstration, seems in fact one of them, to tL7^tfiaXABE'''?Jf '"-f itude, and from angle DCF; and then the rpm! Li ' ^""^ ^"'" ^^^ other, the tri- that is, the panUleWram jflr« ^« ^' "^^^ ^^ ^^« third axiom: Otherwise, tfe triangle, whose base is'ifi /fi A' Parallelogram A'/^Cf . the trapezium, which wouM «r>rloi ♦ T ^•^"* ^'^ '« taken twice from Which i„« ippS'ihrrdSs. £ .s\r.si't « .*" «- ^ NOTES TO BOOK I. sv It may be observed, that the two parallelograms exhibited in fig. 2 partially lie on one another, and tliat the triangle whose base is BC is a common part of them, but that the triangle whose base is DE is entirely without both the parallelograms. After having proved the triangle ABE equal to the triangle DCF, if wo take from these equals (fig. 2.) the triangle whose base is DE, and to each of the remainders add the triangle whose base is BC, then the parallelogram A BCD is equal to the parallelogram EBCF, In fig. 3, the equality of the parallelograms A BCD, EBCF, is shewn by adding the figure EBCD to each of the triangles ABE, DCF. In this proposition, the word eqital assumes a new meaning, and is no longer restricted to mean coincidence in all the parts of two figures. Prop, xxxviii. In this propopition, it is to be understood that the bases of the two triangles are in the same straight line. If in the diagram the point E coincide vdth. C, and D with A, then the angle of one triangle is supplemental to the other. Hence the following property : — If two trinngles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles ai'e equal. A distLiction ought to be made between egurJ triangles and equivalent triangles, the former including those whose sidei: and angles mutually coincide, the latter those whose areas only are equivalent. Prop. XXXIX. If the vertices of all the equal triangles which can be described upon the same base, or upon the equal bases as in Prop. 40, be joined, the line thus formed will he a straight line, and is called the locus of the vertices of equal triangles uj^on the same base, or upon equal bases. A locus in plane Geometiy is a straight line or a plane curve, every point of which and none else satisfies a certain condition. With the exception of the straight line and the circle, the two most simple loci ; all other loci, perhaps including also the Conic Sections, may be more readily and effectually investigated algebraically by means of their rectangular or polar equations. Prop. XLt. The converse of this proposition is not proved by Euclid : viz. If a parallelogram is double of a triangle, and they have the same base, or equal bases upon the same straight Une, and towards the same parts, they shall be between the same parallels. Also, it may easily be shewn that if two equal triangles are between the same parallels ; tliey are either upon the same base, or upon equal bases. Prop. XLiv. A parallelogram described on a straight line is said to be applied to that line. Prop. XLv, The problem is solved only for a rectilineal figure of four sides. If the given rectilineal figure have more than four sides, it may be divided into triangles by drawing straight lines from anv angle of the figure to the opposite angles, and then a parallelogram equal to the third triangle can be applied to LAf, and having an angle equal to E : and so on for all the triangles of which the rectilineal figure is composed. Prop. XLVi. The square being considered as an equilateral rectangle, its area or surface may be expressed numerically if the number of lineal units in a side of the square be given, as is shewn in the note on Prop. 1.5 Book II. The student will not fail to remark the analogy which exists between the area of a square and the pi'oduct of two equal numbers ; and between thesid9 of a sq^^ars and th0 sjitars root of a nv,mhsv. There Ls, however* d5 58 Euclid's elements. this distinction to be observed; it is always possible to find the produci thi sli^f « '"'^' ^ ''1'^'? °" ^ «^^^" ^^"'^ ' but conversely, though the side of a given square is known from the fi-ure itself, the exaet f v!^^' "/.^ ''"'-%where me given nuuiher ^s a square number. For root oi 9 or 3, uulicates the number of lineal units in the side of that . square. Again, it the area of a square contain 12 square units, the sS of the square IS greater than 3, but less than 4 lineal units, and thor' no number which will exactly express the side of that square: an apprnx.. mation to the true length, however, may be obtained to any aliign' d degree of accuracy. ' "'"'o"-^ nn„^™^" ^\r\' ,J" a "g^t-angled trinngle, the side opposite to the ri-'ht nng le is called the hypotenuse, and the other two sides, the base and perpendicular, according to the ' r position. f>in t1•^^^'^"'f ^f.? ^ nl*^^^^ ''l''''''^' ^""^ described on the outer sides of tne triangle ABC. Ihe Proposition may also be demonstrated ( I ) when the three squares are described upon the mjier sides of the triangle : (2) rTn tLT ''^'''^r '' fT''^F^ °," *^'^ ""^'^'^ ^i'ie ^""^ the other two squares ?nnei«irh nnH^i ' ° .?' ^"''"^^' '■ (^^ '^'^^"^^ °"« «1"^^« i« described on the inner 6ide and the other two s.iuares on the outer sides of the triangle. nuse be described on the inner side of BU and the squares BG. HC on the outer sides of ^7?, AC; the point D falls on the side FG (Euclid's fag.), ot the square BG, and KII produced meets CE in E. Let LA meet ii6 in M Jam DA; then the square GB and the oblong LB are each double of the triangle DAB, (Euc. i. 41.); and similarly bv ioining EA the square HC and oblong LC are each double of the triangle £^c' W hence it follows that tlie squares on the sides AB, AC are together equal to the square on the hypotenuse BC. By this proposition may be found a square equal to the sum of any given squares, or equal to any multiple of a given square: or equal to the difference of two oiven squares. The truth of t.bis proposition may be exhibited to the eve in some particular instances. As in the case of that right-angled triangle whose three sides are 3, 4, and 5 units respectively. If through the points of division ot two contiguous sides of each of the squares upon the sides, hues be drawn parallel to the sides (see the notes on Book ii.), it will be ob- vious, that the squares will be divided into 9, 16 and 26 small s,,uares, each ot the same magnitude; and that the number of the small squares into which the squares on the perpendicular and base are divided is equal to the number into which the square on the hypotenuse is divided 1 rop. XLViiiis the converse of Prop, xlvii. In this Prop, is assume(5 tlie Corollary that •' the squares described upon two equal lines are equal, and the converse, which properly ought to have been appended to Piop. XLVI, '■'■ _ The First Book of Euclid's Elements, it has been seen, is conversant with the cons luction «nd properties of rectilineal figures. . It first lays down the detiuiti»ns which limit the subjects of discussion in the First Book, next the tiiree postulates, which restrict the instruments by which the constructions in Plane Geometry are effected ; and thirdly, the twelve axioms, which express the principles by which a compai-ison is made between the ideas of the tilings defined. QUESTIONS ON BOOK I. 50 This Book may be divided into throe parts. The first part treats of the ori«in and pronertit-s of triangles, both with rcopecL to their sides and andcs; and the comparison of these mutually, both with regard to equality and inequality. The second part treats of the properties ot parallel lines and of parallelograms. The third part exhibits tlie connection of the properties of triangles and parallelograms, and the equality of the squ nes on the base and perpendicular of a rijjht-ungled Uiungle to the aquore on the hypotenuse. _. ^ , , . i -.v »v^ "When the propositions of the First Book have been :ead with the notes, the student is rccoinnu'nded to use different letttrs in the diagrams, and where it is possible, diagrams of a form somewhat different irom those Pxhibited in the text, for the purpose of testing the accuracy of his kiiow- iVdge of the demonstrations. And further, when he has become Buthci- ently familiar with the method of geometrical reasoning, he may jmtude, ami specify the difTcrcnt kinds of ma-nitude considered in Geometry. What dimensions of space belong to figures treated of in the first six Books of huclid? 3 Give Euclid's deRnition of a "straight line. What does he really use as his test of rectilincarity, and where does he tirst employ it? What obicctions have been made to it, and what substitute has been proposed as an available detiuition? How many points are necessary to fix the position of a straight line in a plane? W uen is one straight line said to att, and when to tiwet another ? . ^ , x- *v 4 What positive property has a Geometrical point? From the defiiiitionof a Btraiyht line, shew that the intersection ot two hues is a ■ ^"'?" Give Euclid's definition of a plane rectilineal angle. What are m limits of the angles considered in Geometry i Does l;,uclid consider angles gr.'atcr than two right angles ? ^ u. 6. When is a straight line said to be drawn at right angles, and when perwJif/'VwAir, to a given straight line? a ,„ „,„ „^ 7 Define a triamjle ; shew how many kinds of triangles there are ac cordin- to the variation both of tho angles, and ot' the sales 8. What is Euclhl's. definition of a circle ? Point out the assumption involved in vour definition. Is any axiom implied m it ? hhew that in this as in all other defiiiitions, some geoiuetncai fact is assumed liS somehow previously known. _ j . t- r i 9. DcHne the quadrilateral figures mentioned by tuclul. 10. Describe briefiv the use and fouiulati(m of dehuitions, axioms-j and postulates : give illustrations by an instance of each. 1 1 WhHi obrectinn may be ma.le to the method >iiid ordor m which Euclid has laid down the elementary abstractions •)! the fecieuce of Oco- meuj' r Whut other method htvs been ang-estcd \ 60 Euclid's elements. Sci L^^^^^^^^^^^^^ aeEnmon, in the 15' nT^hl?^" ?^ P'-in'^'plos of construction assumed by Euclid, ^nfni !? Z'^^ instruments may the use be considered to iLet iiDm'oxi ? X cSl?'^'i^^'''"r^*"^^*'^«^ ^^^j^ only «;;;oiLs ;r' line as ra^iS^'' Unl^^ ''''';,!^''^ *'T ""^ ^''"*^'-' ^''^^ «"v stra^.),, ithlh n? Ws n;.wii • '^°'' *^\'. P?«tulate diliU from Euclid's, and wjucnot nis problems 18 assumed in it? in GeomTtry?'^°'P^'' "' '^' ^''^'^^^ ^^^"^"^^^ correspond to axioms have';pfcTr2oncrtl! n' *^'1'' "'^l""''' ""^ P°"'^- "^* ^^"^^ ^'^^'^h admit SfbelgsoTxp,2se^l"'''^' ^'''''' '''' ''''''''' °^ ^^"«« ^•'^'^»' asaiLS*n/r *'''•' °^ equality are assumed by Euclid? Is the SSSl re onFnTr^^^^^ <-• « )• — tial to al? '''^^H~^«^^^ -n appeal. -iKf ^^^^^^^^^ ^and Theorem. 24* W f n ■ ' " P'°''^'"' '''^^^ *° ^"^ indeterminate Give an example ano'tt^'^Giv^ernST'lTec""'"'^ ^'" ^""^■'^^«'' "^ reci 3 of It-not.under^ir.rcl-EL'^iLanieVaTeTe^SS^^ necessarv to demonstrate converse propos^tbus ' ow a^e they pLved ^o^if::::^:^^ — ^"^« 'iepend a propoSiin ? ^ "^''^ P'"'"' "" '^' '""'^^ ^''^'''^ of establishing ??■ wfi"^'"''' ^^^'''f "^ 'l'^""''' ^"'^ '■"'^''•^^< demonstration. .../J;«) Wh^^hTtf ^^ *^/ '''J^ *^"'''^''*' ^"^ ^^^^^^ ^'v he term neceLJv^tmS ''"'' '^ ^* ^'"^ *^^* *^« conclusions of Geometry a,e p.|ottS.So^s Sr?a^^^^^^- -^"^ - -<^ ^ the of «. g^jn Sf^h;/«-^ss^:^^SS,;; ? ^^^ "^^- ^^^« ^.. .ad;a8 ^u m two i^omts G and // ; shew that either of th^e dis" QUESTIONS ON BOOK 1. 61 u * 1, « .. f>i« radiuB of the iecond circle ; and cftnces DO, DIf may be taken aa the ramus oi «» give the proof in each case. ^ o, 3. are rendered necessary * 3€. Exphunhowthepropo9t,onsEuc.i.i»d^^^^ necessary for by the restriction imposed ^V J^e t^h rd^ostu at . 1^^^ ^^^ ^^^^^^^j, Lit tii;!rs;aSnn£^ei: dSe a^ isoscolca tr.angle on a «^lf S^ie how Euc. r 2. -y be extc^ded^ theJoUo^^^ •• From a wivcn point to draw a Btrai-ht uue may how much upon Axiom ? . j, ^ ^nd state why it 40. Draw the figure for the third case 01 ii-uc. needs no demomtration. , o is It indifferent in all cases on ,1. In the construction ^'icM i. 9. 9^^^^^^ .^ described? which side of the ouung Une the cquilattrai ti „ ^^^^ ^ ^ 42. Shew how a given Btra,,tlinc^^^^^^^ ^^^^ ^^.^^ ^„^^ , plait\rtS"a5sTbirctt:e;^;;^mt%han one of the corresponding ^^"ff1hewthat(fig.Eue.r.n)^yr;;intw^^^ cular drawn from the middle P^ f "^ ^^.^^^"1';^ e! distances from the extremities i^^ that ^^^^ ^ ^^ ^^^ ^ 47. From what proposition may it ou iui«; the shortest distance between two points^ j ^^^^^ ,, 16. 48. Enunciate the P^oPf^^^ ftl^o^Euc i 2 . that the two straight 49 Is it essential to the truth of t-uc. i.Ji, lines be drawn IVom the extremities of tneba^e ^^^^ ^^ ^^^^ 60. In the diagram, Euc. i. ^i, oy ^^^w m B/)C exceed the less UACt gtrai'-ht lines, any two of them 51. To f«'^°^.?:*^X?,hiTd is a sinUar limitation necessary with must be greater than the thira . is a buima respect to the three angles ? , ... ,^^66 lines whose lengths art 62. Is it possible to iorm atr angle ^'"1 tnree iine ^2, Vs ? ,. 24. viz. " tl»t «B is that •«!« '^f 1' -^ "t &™°lity of two »e.* ^t. mS.;^." oVa tnangle must be given ta o,a« U>.t Ih. tria^eU may be ueaciibed ? ^ Euclid's elements. whJfnSon'VSTn"? Provtr""' oa«e of Euc. r. 26? Under -59. .Show that the an-lp "«" "< '"» '"■"^""''"" «" '''"i^'^d ? to two given str!!?, ' S^^h?oh na ^'^7'" l""' P^n>en.licular« dran^ c<...tainod by he lines th"dvc; ' ''' °'*^"' " ''^"'^ ^^ ">« angle each to each ? ^ ^'" *° * ""<^ ^ud two angles of the other, prook. '"rh^^'proSit 'LIL^?.-;:!,^^^^ ^^T" ''"'^'^•'•'•"'' «-^ «3'nthe.KaI 62. Cun it be proper vnmlirrtp'f ''<-'"'""«"«'"l "nalytiVally t npvermeet if indd S^proJuS ' Lhn.^"^ '''° '"■"'«''* ''"'^^ ^^at they K-d«e of .ome other property ofmuhnn^T^'' TT'^"\'^y *" ""' ^^"^^ far«tpr. Heated of thc'm a S-s^Lv' o^ Snt'-'itT'^^'^ ''''' P'^P-^^ 6.5. If lines b;;;n?Sd'"?e""lo'iL"d ^"'"P'^-tary .o theSe otherwise than paralfel'? If-^X whiTc eu';n' ta^nS'*? "" ^"^"^ ^« '-S^^l^f f^"^-----^-^^^^^^ in the an.irtn;;:;;iSt"b:p.^s:Tto;i^r'^r'f s''"^^ ^-^^^^^ -* -« twdlth axiom as a coSv .J'Euc x '^r '''^'''' ^"''' '^^'^^^^^^^M'u straight'?!;;™ is eon 't-ant? ^''^ '''"' ''^'^ '^^'^tance between two parallel filing I Sr^s^i?j?^:s r irr2:;!:ir^2;i;f t^:" ^^^^'"'^^ ^-- /-. T;ikin- as the doHnirinn r^V Vi i ^^ "^ the same an.'le. equally inclined to ,e san e "i.l t^l"^^^^^ ''r^^'' ^'»^'« ^''^t thc^ ^e that ■' bein, produeed evS so ft^t th wa^^^^^ !''' ''""^ ^''''' ' F°ve also Euclid's axiom 12 bv mo.,,,! ^^i ^'^^^ they do not meet?" Prove 73. M'hatis men Mw^ ? "^/ '^ """® definition. ^® angles with,,ut producinrrsl'oi^;S^,!dr.°''^ ^^"'^^ *° ^-° "^'^^* -n;d ^wl^ra^X^llJSL^^^^-f^S^l^i- be cnaji^le are equal to two right angles ^ ^^'"'^^ ^'"S^^^ o^ « cor:ii;ries ^api^nded t S?' x^'s^'^a:!""?!"" '^'^T' ^"""^^^^ *^^ two ^n^les of a tri^n^le, as we^^as tole uV '.* l^' ^"^^'"«^ ^"^ '"^erior -gle8ofaparallelogram.:^nt:ira\%\ttn"le'"'" ^'^ ^"° ^'^terior on/Liht^arrhldltlt^r^ -e equal to Of th. proportion, Aiao .^Sv'S ^v^ SLS^ ^^ire;t a^JSS QUESTIONS ON BOOK I, 6d Undrr Iclogram, when its dingonals bisect each other: and when ita diagonals divide it into four triaiiifles, which are equal, two and two, vi«. those which have the Bamo vertical nriKlos. 79. If two straight lines join the extregaitios of two parallel straight lines, but not towards the same parts, wlicu are the joining liuus equal, and when are they unequal ? 80. If either diameter of a four-sided flfjure divide it into two equal triangles, is tiie figure necessarily a parallelogram ? Prove your answer. 81. Shew how to divide one ot the paiallelo<;rams in Euc. i, M, by straight lines so that the parts when properly arranged fhall malce up the other parallelogram. 82. Distinguish between equal triangles and eouivalent triangles, and give examples from the First l!ook of Euclid. 83. What is meant by the locus of a pi :nt? A.A'uce instances of loci from the first IJook of Euclid. 84. How is it shewn that equal triang ea tpon tl >• same ba«^ or equal bases have equal altitudes, whether tlit/ aio sltuned on the same or opposite sides of tlie same straight line ? 85. In Euc. I. 37, 38, if the triangles are noi . iwards the same parts, shew that the straight line joiiiing the vertices of the trianf^'V'g is bisected by the line containing the bases. 86. If the complements (fig. Euc. I. 43) be squares, determine their rotation to the whole parallelogram. 87. What is meant by a parallelogram being applied to a straight line i 88. Is the proof of Euc. i. 45, perfectly general ? 89. Deline a sqxiare without including superlluous conditions, and explain the mode of constructing a square upon a given straight line in conformity with such a definition. 90. The sum of the angles of a square is equal to four right angles. Is the converse true i If not, why ? 91. Conceiving a square to be a figure bounded by four equal straight lines not necessarily in the same plane, what condition respectuig the angles is necessary to complete the definition ? 92. In Euclid i. 47, why is it necessary to prove that one side of each sqtiare described upon each of the sides containing the right angle, should be in the same straii;ht line with the other side of the triancle ? 93. On what assumj)tion is an analogy shewn to exist betwein the product of two equal numbers and the surface of a square ? 94. Is the triangle whose sides are 3, 4, 6 right-angled, or not? 9.5. Can the side and diagonal of a square be represented simul- taneously by any finite numbers ? 96. By means of Euc. i. 47, the square roots of the natural numbers, 1, 2, 3, 4, &c. may be represented by straight lines. 97. If the square on the hypotenii e in the fig. Euc. i. 47, be described on the other sine of it : shew from the diagram how the squares on the two sides of the triangle may be made to cover exactly the square on the hypotenuse. 98. If Euclid If, 2, be assumed, enunciate the form in which Euc. i. 47 may be expressed. 99. Classify all the properties of triangles and parallelograms, proved in the First Book of Euclid. 100. Mention any propositions in Book i. which are included in mo;o s'STieril ones which iqIIqw*