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Lorsque le document est trop grand pour Stre reproduit en un seul cliche il est film^ d partir de Tangle sup^rieur gauche, de gauche A droits, et de haut en bas, en prenant le nombre d'images nicessaire. Les diagrammes suivants illustrent la mAthode. 13 22X 1 2 3 1 2 3 4 5 6 ♦ ^Qi; *W^^ ^1i!' asssssa 4«S BSJKs: =!=^ '9*' >3^ :.^ E tJ C L I ,*<■ # -J- ^X.*V^' '^- wip^XdiirAiTOt jja^ #;; ^, FRANCF-S Y©UNG ssas ■«■ t 5F" ^^ . , vr-*- TORONTO: JAMES CAMPBELL & "/. •. j'A ■Iff'riip >,iiii;ti 'SSiiiii I ^:^f^1^^K'^' a4!|^ i^' ]^:^>' ! '. • ^.^ of] otb \ , EUCLID'S ELEMENTS OE GEOMETRY. BOOK I., BASED ON SIMSON'S TEXT| WITH EXPLANATORY REMARKS, Etc. ST FRANCIS YOUNG. The itepi vn trnlded by do lamp more clearly, through the dark nuMi ot Kature ; by no thrend more surely, through the mOnite turnings of the hibTrintk qf Philosophy ; nor, lastly, is the bottom of Truth sounded more happUj \i$ tof other line.— Sakbow (on tht Study 9f Hathematia), TORONTO: jiiitered according to Art of the Parlimnent of Canada, in the year one thousand cif/ht hundred and seventy-one, by James Campbell & Son, Toronto, in the office of the Ministei- cj Agriculture, at Ottawa. 1. Ge( tigate th they be ] 2. Th( pronoun ( Gconietr means, " 3. Geo tised by that wer dation of 4. Froi count be Expevim( definite r any prop( reasoniii{! compasse Geometrj and Sui-v intimateh 5. The" establishe flourished Plato, Eu 6. It Wi connected been taug 7. Som( birthplace It is certa: matics at B.C., in th( his death i 8. Hisv his works The fourti been addei monk of I iated the ' Ilc'Tiry Bil riWhJcred t mKODUCTORY REMARKS. 1. Geometry is tho science which enables us to inves- tigate the relations existiry? between parts of space, whether they be lines, surfaces (superficies), or solids. 2. The term Geometry is derived from two Greek words pronounced Ge (g hard), the earth, and Mktrine, to measure) (jcometry, tlierelore, m the simplest acceptation of tho word means, " meamrement of the earth." * 3. Geometry in this form is said to have been first prac- tised by the Egyptians, in order to restore the landmarks that were swept away and destroyed by the yearly inun- dation of the river Nile. / J uu 4. From this germ of practical measurement (if the ac- count be true) Geometry grew into a theoretical science. Ji-xperimental processes gradually indicating and forming dehnitc rules, by which we are enabled to test the truth of any proposition in Theoretical Geometry by mathematical reasoning, and construct or build up, by 'the use of rule and compasses, various fonns and figures in practical or applied Geometry, to which Architecture, Engineering, MaVpinff and Surveying, and other kindred arts and sciences, are so intimately allied. \7:^^ ,^*'^^ schools of Geometry are said to have been established by Thalcs, 600 n.c., and Pythagoras, who flourished sixty years later: tho science was advanced by I lato, Eudoxus, and others. 6. It was Icit lor Euclid to bring into a well-ordced and connected chain the first principles of GeometJ , Uat had been taught by these early geometers. 7. Some historians assign Alexandria, in Egypt, as the binhplace of Euclid ; others assert that he was born at Tvre It is certain, however, thdt he founded a school of mathe- matics at Alexandria, and flourished there circa 323—084 B.C. m the reign of Ptolemy, the son of Lagus : the time of nis death is not known. 8. His writings were numerous ; the most renowned of all his works 18 his " Elements of Geometry," in fifteen books, ibe fourteenth and fifteenth books, are supposed to have been added by Hypsicles of Alexandria, about 170 a.d. A monk of Bath, named Adelard. is said to hnvo first trans- lated the "Elements " into Latin in the reign of Henry L- II c'liry Bilmgsley, afterwards lord-mayor of London, first ' rt:iulcied them into English a,d. 1670. ■ r.\pfiti;t.im,;: • »«f» •'«•««>-'•.■•»•**,_. iv INTRODUCTORY REMARKS. 9. The tmnslntion from tho Greek text, used in the S resent dnv, was made by Dr. Robert Simson, Professor of latbematlcs in the University of (llasgow : the first edition of which was published about 17r)8-9. This is, however, superseded bv the vahiablo annotated edition of the " Ele- ments," by Uv. Potts of Trinity College, Cambridge— a standard work that is indispensable to the requirements of the advanced student. 10. All boys siiould learn and lay to heart Kuclid's reply to Ptolemv, when he asked if there was any easier method of actiujring the science of Geometry than by the "Elements?" " TiiKUK IS NO Royal road to Geometuy," was tho philo- fiopher's answer— and there is no short cut to a knowledge of any branch of learning: wo must follow the track that has been patiently and laboriously trodden out for us by those who have gone before us, remembering that diligence, with thoughtful attention to the first steps, can alone make us proficients in any subject of study, always under the bless- ing of Alnughty God. 11. In studying Euclid, first be sure that you thoroughly un- derstand his meaning; do not attempt to pass on to the second definition or proposition until you have mastered the first. 12. Learn the definitions, postulates, and axioms by rote, and associate them in your mind with the numbers affixed to them as they stand in order, that you may be able to repeat any one without turning back, when reference is made ,o it in any proposition to substantiate the reasoning employed. 13 In going through the Propositions, do not attempt to learri them by rote, and never try to repeat them without following every link of reasoning on your diagram. 14. When you think you are master of a Proposition, lay aside vour book and endeavour to write it from memory, constructing your diagram, as you proceed, with different letters and in a different form from that given with the text, as pointed out in Propositions 1, 2, and 3, where extra diagrams are given which correspond, one equally well with another, withVhe requirements of the text : this will firmly fix in your mind the method of proof or demonstration em- 15. Lastly, remember that your faculties of reasoning and arcumentative powers will be sustained and matured by a ^ .-. .1 *:^-l -«--i-lir • it Tvin "«^ol>lp vriii to '"^ictll^- ' course or mathciiuuiCiil siaay . it ^\i.l •........*•• ;--u .ij ...„l... iruish that whirh is solid and useful from that which is spe- cious and flimsy, gold from tinsel, truth from falsehood. ed in the rofessor of rst edition however, the " Ele- ibridjje — a rcments of :lid'8 reply method of lements?" tho philo- owledge of k that has s by those fence, with le make ns : the bless- •oughlyun- the second . the first, ms by rote, •s affixed to e to repeat made »,o it employed, attempt to ;m without m. osition, lay m memory, th differeht th the text, rhere extra ly well with I will firmly itration em- isoning and atured by a V.1 to. difitin- 'hich is spe- ALSUUOOD. EUCLID'S ELRMENTS OF GEOMETRY. BOOK I. DEFINITIONS. The word DKriNiTiox is derived from tho Latin verb Definiuk, to mark out a limit or boundury ; we may then at first consider a Definition, for the most i)art, as a short des- cription of the properties luelon;,'!!!};; to certain geometrical forms and fipnrcs, giving us marks whereby we are enabled to conceive an idea of tiicm in our minds, and to trace their shapes on any flat surface. TI»e Delininons of Hook I. of Euclid's Elements may be divided into four Sections as follows, the third admitting of further subdivision :— fiection 1. Point, Line, and Surface Dcf. I VII. „ 2. Angles Uef. VIII.— XII. „ 3. Figures Def. XIII,— XXXIV. A. 'J he Circle and its porta. ..Def. XV.— XIX. B. Ilettilineul Figures Def. XX XXXIV. n. TrianpU's Def. XXIV.— XXIX. b Quadrilateral Figures.Def. XXX.— XXXIV n 4. Supplementary Def. XXXV Etc. Section I. Point, Line, and Surface. Def. I— VIL L A roiNT is that which has no parts, or that which has no magnitude. Euclid's point is therefore Imaginary, shewing position only : we cannot make a point, however small, without size or magnitude ; the emallest dot we can make with a pen or pencil must have length and breadth to be visible. II. A LINE is length without breadth. Here a line is merely an imaginary track from one point to an- other, whether straight or curved :— as in the case of the point, aline drawn on paper must have length and breadth to be visible. Euclid's definitions of a point and line apply only to ideal pointa and lines, which can exist only in imagination. ill. xiie EXTEEMiTiEs OF A LINK arc points. The points denote the position of either end of tlie line. aafeiif 6 EUCLID. IV. A STRAiaiiT LINE is that which lies evenly between its extreme points. A straight line, therefore, is the shortest possible distance between any two points or positions. The difleronce between a line and a straight line is this : let us take any two points on tlie surface of a table which is perfectly level, a line may be represented by a piece of wire passing from one point to the other, above, below, or through the table, bending to the right hand or to the left ; but a straight line between the same points is one that may ba traced with tlie aid of a ruler on tliur Hat surface of the table, in a direct course, with- out the slightest turning to one side or the other. SUPERFICIES (or surface') has only length and V. A breadth. Like Euclid's point and line, his superficies can exist in imagina- tion only ; there is nothing in nature that has length and breadth without tliickness ; the superficies of any thing is merely the surface or outside: Sl'peiificiks, the Latin term for surface, is derived from the Latin preposition Super, above, and the noun Facies, a face. "VI. The EXTREMITIES OF A SUPERFICIES are lines. Lines mark or determine the extent of any surface, and clearly define its limits of boundaries. VII. A PLANE SDPERFiciES is that in which any two (or more) points being taken, the strai^rht line {or lines) be- tween them lies wholly in that superlicies. The surface of a level table or iloor is the best example that we can have of a plane superficies : the word plane means even or level ; hence the reason why the instrument with which tlie carpen- ter renders the surface of a rough plank even and level is called a Plane. Bear in mind the difiference between a superficies and a plane superficies ; the former may bo applied to the surface or outward face of any thing in nature, however uneven it may be ; but the latter can only be used wlien we arc speaking of the surface of any thing that is perfectly flat and even. Section II. Angles. Def. VIII— XII. VIII. 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. ' A pbino u p.ri evj'T? ?v.rfRce; ws mny drnw tivo lines of any des= cription meeting each other on the even purface of a table, slate, or black bo> rd ; tlie corner enclosed by the lines bending towards each other at the point of meeting, is called a phtne angle. '^'MfimmMm' •"■X *A„. EUCLID. y between nee between , line and a Hurfcce of a i by a piece 1, or tbrough t a straight ed with the course, with* ength and in Imngina- und breadth y the surface , is derived iVCiES, a /ace. ines. . and clearly iny two (or ' lines) be- iple that we ans even or I tlie carpen- 3I ia called a ind a plane or outward be ; but the rface of any [. t wo lines to are not in of any dea- ible, slate, or owarda eaub IX. A PLANE EECTiLiNKAL ANGLE IS the inclination of two straight lines to cne another, which meet together but are not in the same straight line. The term bectii,ineal means formed by straigM lines, derived from the Latin ac^ective Sectus, straight, and the noun Linea, a line. To form a correct idea of what an angle is, suppose AB and CD to be two very narrow strips of paper, fastened together by a pin thrust through the point E, where they cross each other; the corner BED, formed by the opening of the lines from the point E, is called an angle ; the corner or angle will be smaller or "greater in size as we move the end D of the strip of paper CD nearer to or farther from the end B of tlie strip of paper AB ; thus by moving the strip CD into the position QF, we make an angle BE F, greMer in size than the angle BED, formed by the previous position of the lin^s. Remember that it is the extent of opening between the lhie» that is called the angle contained by the lines; the length of the lines themselves Lat« nothing whatever to do with the size of the angle. N.B. — When several anj^lcs are at one point B, either of them is expressed by three letters, of which the letter that is at the vertex of a the angle, that is, • at the point in wliich the straight lines that contain the oi angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of these straight lines, and the other upon the other line. Thus the angle which is contained by the straight lines AB, CB, is termed the angle ABC or CBA ; that which is contained by the angle AB, DB, is named the angle ABD or DBA ; and that which is contained by DB, CB, is called the angle DBC or CBD. But if there be only one angle at a point, it may be expressed by the letter at that point as the angle at E. X. When a straight line standing on, another straight line makes the adjacent angles equal to each other, each of 8 EUCLID. B the angles is called a nioiix angle ; and tho straight Hue which stands on the other is called c the rERPENDICULAR to it. Adjacent, lying nexUo or n<>ig7ihour- ing, from the Latin preposition Ad, to or near to, and Jacere, to lie, a verb. Perpendicular, from the Latin noun rEUPENDICLLUM, ffl pluinb line. In the figure, CD is perpendicular to •- AB, and the angles ADC, BOC, are right angles adjacent or lying next to each other, formed by the perpendicular line CD standing on the straight line AB. XI. An OBTUSE ANGLE is that which is greater than a right angle. Obtuse, from Obtusus, blunt- ed, participle of the Latin verb OBTUNDERE, tO blunt. In the figure, the angle ABO is an obtuse angle ; the opening formed by the inclination of the straight lines AB CB to each other, is greatei than the inclina- tion of the straight lines AB DB, forming the right angle ABD. XII. An ACDTE ANGLE is that which is less than a right angle. Acute, from Acijtus, a Latin adjec- tive, meaning sharp or pointed. In the figure the angle ABC is an acute angle ; the opening formed by the inclination of the straight lines A B CB to each otlier, is less than the inclination of the straight lines AB DB, forming the rigiit an^le ABO. Section III. Figures. Def. XIII— -XXXIV. XIII A TERM or BOUNDARY is the extrewity of any thing. Term, frum Terma, a Greek noun, so pronounced, meaning I'mU or extent. XIV. A Figure is that which is enclosed by one or more boundaries. Figure, from Fioura, a Latin noun, meaning shape or form. If the liguie is encloned by one or two lines, they must of necessity be curved ; but if by more than two boundaries, they can then be (traight linc«. nm^'<%^,^& raight line med by the e ABD. « than a ::^ IV. my thing. iJiing limit e or more •form. If icessity be 1 theu be EUCLID. 9 A. The Circle and its Parts. Def. XV—XIX. XX. A CIRCLE is a plane (flat or even) figure, contained by one line, which is called the ciRCUMiERENCB, and is such that all straight lines drawn ' - from a certain point within the figure to the circumference are equal to one another. Circle, from the Latin noun CiHcn- Lus, a round figure, ring, or hoop. CiR- cuMFBRENCE, from the Latin preposi- tion CiRCCM, around, and Ferens, bearing or carrying, participle of the Latin verb Ferre, to bear or carry. X,VI. And this point is called the centre of the circle. Centre, from a Greek noun, pronounced Kentron, meaning a point. XVII. A DIAMETER of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. ■ Diameter, from a Greek verb, pronounced Dlvmetrise, mean- ing to measure across. XVIII. A SEMICIRCLE is the figure contained by a diameter, and the part of the circumference which is cut off by it. Semicircle means half a circle, Semi, a Latin particle, meaning hay. In the above figure the line ABODE is the circumference of the circle, F is the centre ; DA and BE, straight lines passing through the centre F. are diameters of the circle, i lie figure ABO, contained by Iho diameter AO, and hall' the circumference ABCD, . is a SF.Micmcr.E. The figures AED, EDB- and BAE, are also eemicircles. All the straight lines FA, FB, FC> FD, FE, drawn from the centre F to the ciraimference, are equal to one another ; they are c.illeil kadii of the circle. A wheel is the best practical illustration that we can have of a ekcltt uad iu various iiarts. XIX. A SEGMENT of a circle is the figure contained by a ■ **... 10 EUCLID. ^ In the circle ACEOBF the figure, by^traighSr^^ ^^^^^''^^^tJ^ose which are contained etraighi line"!''''''''*^'' ^'**^"» «' tbianglbs, by threi^ TiS:Sr^^^^JiJ^;,^- «^e ^?«° «-eral a,«ectlve having tt/xe angle" from t^-"^'**"'^- Thianole, a Sure «OUn .^XG,,„g, /J^^«°» TKES. THU. THREE, aud th^ lIuh QcAT-„oK./o«r.?n^li;";jf^f™'° ^-«° °u«^eral a^'ective 'W-'*^^^^^^^^^^ ^^«^«-« or PO.TOOKS, by.o.« from a Greek adjective Dronn,m«nH V •'^^'■^ '^'^"^ ""'"y a"f//e* «« aye. a Greck'nor;oZnouncei '^* '""'"'■ «"^^^~^ ^.Triavgha. Def. XXIV-XXIX an;4:" '""' «•*» (»"" '^'^ Laths, a *?& ^'**'°*' «*» <»• eauoA and '-nvi [hut not passing b are contained meral adjectire lANGLE, a figure »ud the Latin raight lines, leral ad[|ective ONS, by more Btive MuLTus, 'ff rmny angles, I, and Go-Mu, . • BUOLID. XXy. An ISOSCELES (pronounce C as K\ TBIANGLE ,8 that which h^ two sidL eqZ. ^ Isosceles, having egml sides or legs, from a GrePk ha^.Tj „, tsr '"'"^^^ " *^^* -^^^^ The word Scalene is derivpri fm™ - /^ ._ Miectiv., pronouaced S A-SS Vcrl-S o^^t 11 ''««. by .^Sree ^^^^ A biohx-anolbd ibiaxglb is that which has ^^f^7I^use%?(l^'^-'''"'''^'' '^^^'^^^^^ » that which (/»Vaa,te a„°fe (^.'"'"■^ ^'"•'"■E " that which ha. b *arfnWamb.r«. £.y. XXX-XXXIV. -; "• '--n OuLu;ifS IS tiiat which ho «»^e,, but hasno^ all its sides eTual (2) The oblong has its oj^posUe sides equal to one another. Ig EUCLID. The word Rhombus ia derived ?»««»* ^'"'-'''^ """"• pronounced KHOMBos ; a term applied to a paral- lelogram with equal sidea. act having Its angles right angles. The Khombns may be formed by placing two equilateral triangles of the same size together. 605* to 'wc. nor us nriRles right anffles. " fori,"«r'"V T''"' ^^<^'"n the shape or caSwtLf "'^"''""*''^^*^^-- besides these are TBxrK.cM. from the Greek «oun T„.-p..„.o.. „ ^,,, ,^,,, SECxr0X4. SUPP^KMENXAKV. ^./ XXXV-ctc. the same plin'e^ltnd^whlTh''"" "^? ^'^ «»^h as are in ways, do not meet '^' ^'^"^ P^'^^'^^^d «ver so far both The term parallel ' ' 18 derived from two Greek . "ime .pace Wween them Th. m^l^? ""'""'ng ""netly the Psper, and Uie primed line; „f ti?,, k J ''"" "" » "M^' »f mualo but the lr„„ rail; of ?,««».? heS°°™' «?'^~iW*»>« /»»" .erpeuimeeo«„e,are,»„rj'£°'',""J^„''';,V° ' """^ " From this we infer that a '" f «o»^oi». of Which ^^^^oppos^e^'^r!tmmT^'' i\jfonr.Hded/igr,re, of the figure are also eg,ial Jo ofe ifther ''^''"'' '"^^ *°^ «'^'<» (*^Vm^ft:S:du:rM^ir *'^«^^°''^« -g,esof » paraUel. l^e term r.a.LLKLocKAMis taken from a G,.c* uoua. pro squal, but ita L equal to each 3 these aro ^all table. V— etc. h as are in so far both y the side of nee without exactly the et of music 'aight lines; » curved or ice between PARALLEL mon to the ndedfigtire. and angles a parallel' Doun, pro EUCMD. 12 nonnced PARnALEFLo-cnAMMA. which moans « mrallel drn^ mre. See the derivation of the word pauallkl aCe. Postulates. The term Postulate is derived from the Latin verb PosTiiLARR, to ask Or demand. We are asked to allow that certamassert.ons are true, :vithout requiring any roof of the truth of the statements made in them. ■ On insSon we sec at once that what is demanded is possible, and that ^n^irht'iJatr'^'""^^^'-"^^"'"'' ^« "---" Look at definition IV. ^' A straight line is that xvhich lies evenlv betu^emt, extreme points," it matters not where the extreme poi" 8 of the straight hue may be in position; therefore, wherove" vo C determine the position of any two points, it is possible for ,^n draw a straight line between them. *° n. That a TERMINATED STUMGUT LiXE mav be produced to any lenf/th m a straight line (m //^e same stnucj/a course). We wish to extend the length of a certain straiHa line that w« have drawn. We do thi?. in practice, by ph.cin- the clgo of cm? flat ruler against the line already drawn, and .racing a continuation of the same by passing our pen or pencil along the edge of the ruler to the distance desired, whatever length that di^tanoe may be. III. That a circle may be described from any centre, at any dutance trom that centre. We can place one leg of our compasses on any point we choose and trace arge or small circles with the other leg, accordingly as we bring the legs of the compasses farther apart or closer together. Axioms. The word Axiom is dciived from a Greek noun, pro- nonnced Ax-i-o-ma, which means a statement which claims belief by reason of its sehf evident truth. An Axiom is an assertion worthy of credit, a simple truth which is self-evident, admittinix of no arf,'unient with respect to its correctness, and re(iuiring no proo? The first se%'en are alike applicable to numbers, superficial extent and solids. ' L Things which are equal to the same thine arc canal to one another. ^ In numbers 3 + 2 = 5,44-1 = 6; thorefi.re ,3 4- "> - Since both comiwund quantities are coual to the tanie quantity. 4 + 1, filMplo ■**!*«!««* .34 EUOLIIX '■"';„"• ."^nn """"■'' '"''''l«'lSmr ""'""""'"''■■"I QNaaUUcTa™ po ind ,,u.„iuiM arc ,^u.l to tli™, "", ""■ """> >>«"' com. on='a„«|;ir "■""'' """'"'■'".of tho ™«, „,.ee,ua,,o ■ . ^^■ J''° "'""■■' i' «»•«<"'«■ tlmn its PART than t»o riRlit „„„!,.,, tl,?se J',',, ° ??'"« »>« notions"^ th. require solution orScmolTtmLui ' ""? *'"'' ^^Ht.mc, t wiU founded on the truth? hnn fe?or^.n''''''"'r'^«' '^^^on'^^^^ o. It the Pronositin,, ^ • 4 ^ «««^«M When it puts foAvard so/no/h''^'''"'''.^""' ^^^^> *• ", on the contrary thA p\r ^P'nethinff to be done njent or assertion rSim? ?5^P°«"'o" conveys a stat;. TuEORKM, from a gS ioll nr"''"''''^''' ^'^ ^ called a weaning « M,«y tobetokX P'^""°»"«ed The-o-hee-ma tf^at u^hatyou have Xn^XU'/'' *^'"^'^'«^' ««^ ^«/ro«, */^5« M«. the,, r'esultlZel^ue "^^^ ^'^^'^^ ^^^ you haTtl ^7Proclus.t4.a;K»^^^^^^ f inundation. ■ Tl.y^f ...I.J- . . laoposmoN., 'Problem 'TWOREH I ^'"^ *'^^"'^^' is ffiven. f Proof. V Chnclusian, / Enunciation. I Ifypothesis. ... < ^^'cnce. \ i^tusiruction, I Iknwnatraiion^ V Conclusion, 16 EUCLID. 8. In the l*ro1)lom, the Enitnciatiow, a term derived from the Latin verb Enunciare, to declare, is printed at the head of the Proposition in italic3, and declares Avhat con- ditions are granted or given, and what you are required to effect on these conditions. 9. That which is givek, asserts the conditions that aro prantcd in particular terms, indicating the things given by letters. 10. That wiiioii is sought, shows what is required to bo done on tlic conditions given. 11. The CoNSTKucTioN, Sl term derived from the Latin verb CoNSTRiJEUE, to pile together, to build, is the course adopted to build up, step by step, by the Postulates, a figure or diagram, which will satisfy in every particular that which is required to bo done, or which will shew that the thing which is sought is ed'ected. 12. I'lje Proof, by reasoning founded on the Definitions and Axioms (and by reference to the truths proved iii preceding Propositions as we advance), shews that the con elusion to which the steps of the construction have brought us, is correct. 13. In the Theorem, the Enunciation states what con- ditions we are allowed to assume, and the particular conse- quences that must follow from the truth of these assumptions 14. The Hypothesis, from a Greek word pronounced nuPOTHESis, meaning supposition, asserts the conditions as- sumed or supposed, indicating them particularly bv letters. 15. The Sequence, a term derived from'the Latin verb Sequi, to follow, points out what must follow, the con ditions supposed in the hypothesis being considered correct. 16. The CoxsTROcTioN, in the Theorem, consists of a slight addition to the figure indicated bv t!ie Hvpothesis, to aid us in the demonstration of the truth of the Sequence. 17. The Demonstration, from the Latin verb Demon- strare, to shew or point out, is the chain of mathematical reasoning by which we shew the assertion conveyed in the Sequence to bo true. 18. The Conclusion, in the Problem; states that what was re(iun-0(l to be done has been efl'ected on the given con- ditions : in the Theorem, it is the enunciation repeated as a statement, of wliich the truth has been fully shewn. ^ j9. In some Theorems, in order to shew that the Sequence interred from the Hypothesis is true, we have to assume a i-ALSE IlrpoTiiEsis, aud then, by arriving at an absurd con- Cons: AB, des( 2. Fn circle A( 3. Frc draw th {Postula The tr KUCUOii Tm derived •inted at the what con- required to ms that aro ;s given by quired to bo I the Latin the course ostulatcs, a r particular II shew that Definitions proved in lat the con ive brought what con- ular conse- 3sumptions pronounced iditions as- by letters. Latin verb ', the con red correct, nsifits of a pothesis, to equence. h Demon- ithematical eyed in the that what given con- f)eated as a vn. e Sequence ) assume a tbsurd con- Vf demonstration is^rLdLDiBECT pf''**- -^^^ ^'"^ ^^ Is the first example we mS wUh ^*'''^°«'^«« 6, Book I^ 20. In the course of Book T ♦!,« r n • wil be found, which may need soi?« /v".""^"? expressions ration, &c. :-. ^ ^ ''°™® explanation as to deri- JaifJ^. 'Th^senseVffwor'd "T'^fr^'" '<> *« stood from the meaning of thrLaH^^ ^ T".^^ ^°^«^- rAD.PLiCARE),fromwhichit sHoLni ° ^^'■^ Applicarb laying or pladng on^tlZi clZfiT ^' conveying the idea of another, as when we foK pS ^JI'T^'^^ ^^ '''« ^'''^ «/ the edges may coincide or toich in L'°'^ '"'^^^''^ «° that to each other. ^^ ''^ ^^^^^^ Part, fitting exactly ■on ;vith those about thoSSr'ST''"''"' '^'"'J°""=- « denved from .he Latin vercS'L *^ «%™7"°'"" 1— PEOBLEM. GiVEN.-Let AbI^HJ e-"^- ^'•'"■'^"''' *"''"^'" ^^ PROP SonGHT.-.It is XQoWKAa • J^'^h* '"^«- upon it. requnWTo describe an equilateral triangle AB describe the circle BCD. VoSl"^^ .^* the distance .2. JTrom the centre B «7 f i, i- "°'^ ^0 circle ACE. iPoSthf '^' '^^^^^"^^ ^A, describe the d^aw^r^^^iKlin'e^^ - one another, ^Postulate 1/i '^'^, v^tJ, to the points A an'' B a perftaps be a better word than *^crafi. 1$ aUCLlD. > •^^?!!:"r'* ^^^'^"sc ihc point A is the centre of tlie circle WGV AC ifl equal to AB. (/>^'it.»^'on 15.) a. Bi^mse the point B is the centre of the circle ACE. BC 18 eijual to BA. {Definition 15.) 3. Therefore AC and BC are each of them equal to AS. 4. But things which are equal to the same thinff are equal to one another. Therefore AC is equal to BC. {Axiom l.) 6. Wherefore AB, BC, and CA, are equal to one another. t.ONCLD8ioN.— Therefore the trianple ABC is an equila- teral tnang e, and it is described on the given itraicht line AD, Which wot to be done. PROPOSITION 2— PROBLEM.. From a given point to draw a straight line equal to a yivm straighi line. Given. — Let A be the given point, and QC the given straight line. SocoiiT.— It is required to draw from the point A a straight line equal to BC. CoNflTRucTioN-l , From the point A to B, draw the straight line AB. {Postulate 1.) • ^ '\ Upon AB describe the equilateral triangle DAB. (Prop. 1, jUook I.) -^ V / 3. Produce the straight lines DA, DB, to E and F. (Po>- • *i ■^i^^.*^®/^"*''® ^' '^^ ^^^ distance BC, describe *'w circle CGH. {Postulate's.) 5. From the centre D, at the distance DG, describe the circle GKL. {Postukite 3.) AL shall be equal to BC. ^/?i''^^>r!' I^«cause the point B is the centre of the circle CGH ; O IS equal to BG. {JJpfinition 15.) -.,"'. *^' " ' ^ point D is- the centre of the circle GKLi -OLiseru. oJG. (,Z»^t»uw« 15.) ro of tlie circlo e circle ACE, equal to AB. Iiing are equal /. (Axiom 1.) ) one another, is an equila- i straight line tal to a given 3C the given te point A a fcncLiD. 19 w the straight DAB. (^Prop. ,nd F. (Pr?>- describe +/-.i' describe the of the circle ;iicie GKLj BO; (Slf" '--"O- AL, i. equal ,» ,he remainder line BG. Which wa, to 6e S.' ^ *° ^^® «'^'^'^ "'"^'ght apex qf the trilateral tHangie.Z Z'.^ y^"J^,'^''^'d tfuit the Im^ and the base qfthe ewilJJ^alMnn 1 f*"^ '^^^^^^ »<''av« ^ «/.^ dr.... .m aLys Zl^J^^I^^^^ Point <,^ Prom /A ^^OPOSITION 8.-PROBLE3L Prom the. greater of t^o yiven straight lines to cut ofT Given- ^'^^ to the leis. ''^ ^"^ "^^ <'ff a part Of wS'AB^fs^the ^"eltw^' *^' '^'^ «^^«" "'"'^ight lines, . paTe^JaT?^ '47ff ISl '° ^'^^ °^ ^-- AB, the greater, — B circie^^^ ^^« <^-nl AD, describe the PKOo'/-!fV/'^^' 'Y^ be equal to C. ™oF.--i. Because the point A is the centre of fi,« • , (Dp.fir,ifi.^ "e centre of the arcle DEF AF i71'« "«^ause tne point A is the ( 5»f « u^itvfi 1, !ua! A r% ^ v^ •-: {^JJy UoTf ' fe^rt!.?.i4r,r';.rif4LTc!^*° xww 1.) so EUCLID. ^fr^?l?r^'''^'~''^^^liT^ f""' ^^' *^e greater of two straight Imes, a part AE has been cut off, e^ual to C, the less. Which was to be done. PROPOSITION 4.--THE0REM. If two triangles have two sides of the one equal to two sidei oj the other, each to each, and have likewise the angles contained by those sides e(jual to one another, they shall likewise have iheirbase^ or third-sides, equal; and the two triangles shall be equal, and their other angles shall be equal, each to each, viz., those to which the equal side^ are opposite. ha^e-"^™^*"'"^^* ^^^' °^^' ^® ^'''^ triangles which 1. The two sides AB, AC, equal to the two sides DE, DF. each to each ; ' ' vi«., AB equal to DE, and AC equal to DF. 2. And the angle BAG equal to the angle EDF :— then— Sequence.- 1. The base BC shall be equal to the base EF ^"V"u*^"^^® ^^^' ^'^^^^ '^® *^^"*^ *« t'le triangle DEf" 3. And the other angles to which the equal sides are opposite, shall be equal, each to each. Ar>oX":'J^^ angle ABC to the angle DEF, and the'angle ACB to the angle DFE. o , ^ « Demonstration.—!. For if the triangle ABC be applied to {or placed upon) the triangle DEF. 2. So that the point A may be on D, and the stiaightline AB on DE. " 3. The point B shall coincide with the point E, because AB IS equal to DE. {Hgpothesis 1.) 4. And AB coinciding with DE, AC shall coincide with DF, because the angle BAG is equal to the ancle EDF. {Hypothesis 2.) * 5. Wherefore also thpi nninf C\ eliaii nn:reen demonstrated that the whole an- (z?JroS;;;-:ritf ^'^ ^"^'^^^ ^^^^ ^^"^^ ^^« «>«° «q-^- 14. Therefore the remaining angle ABC. is eaual to tha totLtlGcy^T/r^'^^*^'''^^"^^^^ i« «q««i •mu- [^ *Jt-B. (iJemonstration 12.) v.ON€MisioN. Therefore, the ang es at the base &p (See Lnnnaation.) Which was to be sfmvn. ' ^' CnW^^r''^"''' '''"■^ eqrnlatcral triavgh is also eguu ««::«»|««feB««K*w *1^« -ngle ACB (^'Il'S L^ '"« ^G " equal to the base AB. (^op'^tZk'ir'' ^^°" «^-r to the triangle ABO e'^Therefore" aI"?-''"' 7^''^ '' ^^'^'^• equal to AC. ^^ " "°' ""^^u^l to ^Cj that is, AB fe feiiilKK 24 EUCLID. ! Conclusion -Wherefore, if two angles, &c. (See Entm^ tuiMn.) Which was to be shewn. ^ "^ fo/mr''''^''''""^'"'' '"'"^^ "i'''-^''9ula^ triangle i, also ^i. This Proposition and Corollary are the conversA t^f th^,^ .»• r PKOPOSrnON 7.-TnE0KEM. t>on Me 5fl?«e base, and on the same side of it ther^ mn^., fhlT r r^ ^'^ ^''^. ^"*"^' ^'^""' <« ««« another, and likewise those which are tcnnmated in the other extremitl Hypothesis-I. Let the triansled ACB ADB unnn fh. the^itr^quano'^o^^e^Hlttr:""^^' " '""^ ^''^ ^ ^^ 3. And their sides CB. DB, terminated in the extremitv B of the base, hkewise equal to one another. ^^^"^einity B ^^L t7 'T"f ""^ P^^>^^l'ifif!f of the equalit,, of the sides m order to demonstrate the iunossibilitu of s'Lh nLJfi' arnvmj at an absurd conch ion, wZl^wm hU^^n^ ^ reasoning on a false supposition. -^ "''"' ^'■^"» CoNSTuucTioN. — ,Toin CD. Cask L First let tiie vertex of each trianjjle bo without the other triangle. (See rtgurel.) 3)kmonstra.tion— i.BeociusG AC is equal to AD. (//voo- ttiesis 2.) ^^ , 2. The triauirle ADC is an *" '^'''V'mmfensmiffr^^jti^i, &c. (SetEnun- 'ianghisahovqui' ■se of Proposition 5 •osltion 5 becoming In the former the and the resulting ii" required to ba angles at the base consequent reality I termed negative are led by reation- he first byputhesis. it, there cannot ire terminated in 'ler, and likewise nity. ADB, upon the ■ve, if possible, extremity A of he extremity B '% of the sides, such a case, by 'II follow from EtTCLlD. 23 Fig 1. efore equal to An (/.^»»"9.f"°«'° *°° ^ Sroat^r than .Uo ogle BOO. i Much''°Z,Jj°r*''!*°°- ""'«'> R'"""!!.." BCD. o. iviucn more, therefore, is the amrio Ron / / • i • greater ihan rte angU ADC, ix,'™' 9)°X,er S,°„ itn " 8. But the anfflo BDC has been shown 7^ k ^ ''^ than the angle BCD. (De Jn'.S^s ) " *' ^° ^'^'^^^^ 9. Ihereforo the angle BDC is both pm,..,! f^ i I J) The figure is constructed withiJie vertex D of the triangle ABB, within the other triangle Construction.— Produce AC AD, to E and F. pKMONSTRATION— 1. BccaUSO AC^s equal to AD. {Hypothe. ^_ an'J ECdTdc Z"; L'o^^^ -^ the eaual to one ^^:rit^X^{:^ ^^^ '^^ ^D, are (Z'i'.f''"^"^^ ^^^ " greater thai the angle BCD, 4. Wherefore the ancle FDO io i;i.„ • J. The triangle BDC i» ,n . i (/%'""'«" 3.) .nsle BDC i, e?„all°5iie"'.„tie'BCD'"(S«^„™',*V i if" .4 J I If >€« w*"-1V^ 26 E170LIO. PBOPOSITION^S.-TIIEOREM. J-Jtwo triangles have two sides nffh. the other, each to each, and htliilT' '^"f ."> '^o 'n'des of f^ mgU which is contained hrthe^ZV^'K ?«^« ^^"4 h^e^jualto the angle contamed L,i ''^^' ''f *^^ '>^^ shall them, of the other. '^«'«*««^ h the two side^, equal to ilVPOTHESls.— Let ABO ncc- , '• Ihe two sidp.rot ♦-;„_,. r^-.. . _ """■"i-^nwewww^s^si**^ EUCLID. SO each. fie BCD. (Con- the base DB ■ AB is divided was to be done. en straiijht line and C a given line from the \. Ij¥E, (i^o- Phoof.-,. Bec.n8TDcl ejn* to CE rr"'« " "'!° '*^- and FC c„m„,„„ ,„ .^ ,^„ trirgl s°Dcf/E^CF'™""'° ">' CF;S.*::.ch" °°' ""'• ^ '"-"^ <» 'i» '"« »iao. EC. J.^A„d the base DF i, e,„al to the ba.e EF. (,C«mruc. And u;er.5'/:ij:^e';.t?S'^ ■' »'-' '» *« "8.0 ECF. each of them is called a rwht^^^l ^9^*L*?.one another, 6. Therefore each of the JngfefD^F fcf'^r ^V , Conclusion.— Wherefore from fj^f' • ' '^ ^ "='" ''"^le. gven straight line AsTa s raS? li^e^C^'hl'rr "" ? ^^^ at nght angles to AB.' Which wJ7o be done ''° ^^'^ strSXr-^^ ^^^P ^^^'^^ Probfe;f;rma,be demon- ABCy;^B5ria;rSe ,eS^ '^^ *^« *-^ ^^'•-•ght lines CoNSTRucxiON.-FroTthe noint T°h" *° ^^1? ^^ *''^'« angles to AB. (Prop.U,BookT) ^' ^'^^ ^^ ** "-'SJ^' Demonstration. — i Be- cause ABC is a straight' line. * the angle CBE is equal to the angle EBA. (Dejinition 10.) 2. But because ABD is a straight line, the angle DBE IS equal to the angle EBA. {,JJeJimtion 10.) - „ ^ (^tfol^fr'"''' "" '-S'^ "SE - equal to the angle CBE. segment. "raigh t lines cannot have a common PEOPOSITION l2._PK0m,EM. Given.— Let AB be thp .rivon c*.„,-.,u. ,, t, ■ produced to any length both ^rlnd ,"* ?l*^'^ ™^y.^« Without it. ''^' *"^ ^6* C be a point Mi4tSKi«»' 30 sacLio. .ido'^fAr"""-'- '^''"'« -^V"-' O »m tt. o«.e. 2. From the centre C, at the distance CD, describe the cir- cle EG F. (Postulate 3.) 3. Let the circle E G F meet the straiprht line AB in the points F and G. 7 4. Bisect FG in H. {-Prop. 10, Uook I.) pofnt'c '^l,?in'~^^° '^'''}."^^, "°^ ^"' ^'^^^ from the civen ^T J^oin CF, cg"'"'"'^""''' '' *'" «^^*^° ^^^'^^g^^ "«« aS" GHC"^ HC common to the two triangles FHC HC; e?ch trealf ''"' "^' "'^ ^^"^^ *° *^« *- "^^ GH, ^.^3. And the base OF is equal to the base CG. (Dgini. fptJg'S/^' ^7^? ^^^ " «q"^^ *° the angle CHG. (^-P- »' ^ook I.) and they are adjacent ancles * ^""» line, makel the adf" '^'1" ^*"t' '^'^"'^'"^ ^'^ ^««^her straight bKQDENCK.— These «nrrlo= .hi!l 5m - . . or shall together bo <^^iu^^-^r^;;t^r:^''^'''''^' JKUCLID. 31 ihe point C, a pon the other rom the given aight line AB to HG, (Con- angles FHC wo aide? GH, angle CHQ, ther straight nother, each rhich. stands definition 10.) int C, a per- ;ht line Aa ^er upon on6 'her equal to e with CD, ight angles, Qition 2, Book L) *» ^^ ^ ^^- (.^91^ B ^ Th''' ■""■• "f "'° ""8l« EBD to each otthcc cgimN .0 on * *X;""' °™ '«""' •» '^' --^ """«. «« e,„al .olw-„T!S,Ttj,*= -f-^l^ ABC,ar=,„,e.her e,„a. iiBri»''°?T-7"'''?''°''"'? 'V; "?'»« '■l'i"'> o-e straight PKOPosrrioN 14— theorem. i/ala fomt m a straight line, two other ,trai„ht lines «»™ ,k. oppome sAsof it ,„„ke the adjacent angle! ogeTc Zmlt irx^^:'"-^ "'■° """'"•' ''"''•"■" "''" - s 'ie° Hii-oiKEsis.--At the point B in the straight line AB let the two straight hnos BO, BP, upon opp„,ig, 'X o?'a^ SCrCLID. two 8S roake the adjacent right anples. ^|E<,aKNCB.-BD shall be in the same .traight line with (Falsr HrpoTHnsis.)— For if BD be not in the same straight line with BC, let BE be in the same straicht line with it. Demonstration.—.!. Now, because the straight line AB makes, with the straight line CBE upon one side of it, the angles ABC, ABE, these angles are, to- gether, equal to two right angles. {Prop. 13, Book I > 3. Iherefore the antrles ABC ARF «-« - i ,. i angles ABC, ABD. (Z^mV.) ' ' ® ^^^^'^^ *** ^^^ 4. Take away the common kngle ABC, anjie ABD^MSa"?!!.:?^^' ^' '^^^^ *« the remaining wStsimV^.;^^^^^^^^^ ""«*' '^'^''^ to the greater! 6. Therefore BE is not in the same straight line with BC oth;rtai«h"t ht'Ja'nr'•^r'^^ ""' demoUrST'that^'; otner straight line can be m the same straight line ^ith BC, Tr, , . /?OPOSITION 15.-THE0REM. If Uvo straight hnes cut one another, the vertical, or opposite „ angles shall be equal. ^^ Sequence, — The c angle AEG shall be equal to its opposite angle DEB, and the angle CEB to its op- posite angle AED. luoLn). ', eqaal to two ight liae with sa 3ook I.) her equal to qual to the e remaining tlie greater, newith BC. ted, that no ne with BC, ith BC. raight line, or opposite ^0, cut one ■■1. Wh,.,Tlb,„ ,1,6 n„Kl", CEA AED """'; '-^ oi,t-l..» AED, DEB. (,Uii', , P '^^°' "■■« 'I""! «" the 4. Tuke awin- the common nncle AED ceWaei:;;™^;";;!;"""™" '» "■«■- «!.« .„e.„g,e. Conclusion. — Therefnrf if f,.,^ o» • l .• to four right a^kJ! "'''''"^^^^ Potnt, ore, together, eguai PROPOSITION 16.-TJIK0REJI. // one side of a triamjk be. produced, the exterior nvnia /. greater thar. either of tl>^ interio; o;po:^:;ir^ «* be^rXX- D."^". ^^"^ '^ " ^-"^'«' -d let its side BO Seqcjjnce.— The exterior ancle Ann c),„ii i. Bisect AC in E. (Prov. 10, i^r;o/(,- I.) ^ 2, Join BE and pro- duce it to F, ami maice EFe(|naIto BE. (Pro- position 3. Book I.) 3. Join FC. Bemonstration. — 1. Because AE is equal to EC, and BE equal to EF. ( (Jonstru<-t inn 1, 2.) 2. AE, EB, are equal to CE, EF, each to each. 84 EUCLID. (/>™;fv&t)"*''*^^ " "-' •<> "« Wangle CEF, 6. And the remaining angles of thp nno ♦« ^u a^"et;s ^^'^^' -^' — hft?wS r :,rsf (/vTSt) ^"^^^ '^^ '^ «^-^*o the angle ECF. it can be shewn in the samP tn^ .? ^^ produced to G. that is, the angle ACD (W 2ZII '^^' -'^^ ^"^'« ^^G is greater than the angle ABC ^ ''^^''''' «er^tm/ a«^/e.j Conclusion. — Thereforp if ««« -j (See Enunciation.) wS^iTfc itV * *^'"«'^ *"• J.„ , , ''''°''0«"''0N ".-THEOREM. M ,u,o angles of a ,ri„n,h are, u^Mer, hu Aon ^ rigM angles. ^ HTPOTHEsis.-Let ABC be any triande i^trZX\^t:iT ""'''' ^*« *^?^'^- «^all be lesa Construction.— Produce A a dO to D. /I » Demonstration. — i Be cause ACD is the exterior angle of the triangle ABC it Js greater than the interior and opposite angle ABC {Prop. U, Book l^ .,2- To each of these add the angle AC B. 4ie™AEg;°Ac'a 7i:» AOB..regre,.er.h.„.he 4. a„X 7pZ%°:i2^\ r- '°^»«'-' ^'"■"' 'o 'wo ^^^ le CEF, because 5, Book I.) > the base CF. i triangle CEF, the remaining the equal. sides he angle EOF. he angle ECF. the angle BAE. »roduced to G, 10 angle BCG, vertical angles) a triangle, &c than two right shall be less EtrcLux 85 i ' M ^ter than the qual to two er, less thao also CAB^'lRr^""'''! ^* 'T ^^ ^'^^^ «'«* BAG, ACB. and ajso UMd, ABC, are less than two right anelps CoNCLnsiON.-Therefore any two amS A« /o Enunciation:) Whi,:h was to belhewn. ^ ' ^"^ ^^' _, . PKOPOSITION 18.— THFORFM AC i, greater "h„/'.he''sMe Ab' "'°"«''' "' '"'''^'' ""= '^e «>|e'ecr-'''''° "«" '*^'= *"" "= H'^'e' than .he Construction. — 1. Because AC is greater than AB, make AD equal to AB.(Pro/j.3.iSoo/fcI.) 2. Join BD. Demonstration. 1. Because ADB is *- for the side AbIs eoull t^?i •^^a'^'^" ^'««««'«« triangle, 3. The angle AdH In w "^f ^°' if^^^^truction.) ' i5oo>t I.) ^^ ^'^^ '" ^KQt'KNCK.— Any two sides ofit toffether fibnll ha /«.«^* BC, greater than AC; and BC, CA, greater than AB. Construction. 1. Prodiioo BA to the point D. and make AD equal to AC. (Prop.'H, Book I.) B^ 2. Join DC. ■ — ^** {Alrnml^ ^ ' ^^° " «'«'*'«' ^^"^ the angle ACD. gri^t^^'tLmr^ngi: BS'i ^tilt^^L^rf "^?' j^ subtended by the gfeator side »!'« greater angle is (i^;/l9"Jr.i'; ^^'^ ^' '^ ^^-^^ *^- *^« -^e BC. 7. The straight Une BD is eaual to BA and AO. ABC would be he angle BCA. then the anglo p. 18, Boo^:l.) le angle BCA. [ual to AB. ■ . ;le, &c. (See tttr than 0\e all be greater greater tlum 'maouia. to AC, the 5, nook I.) angle ACD. in the angle sle DCB, is Iter angle is he side BO. n 1.) 3. 87 tN'tllnS" *'^ "^'^^ ^^ -'^ AC are greater than BC, .Vr«v/r<'r ««,//«. •' "^ "tanyle, hut shall contain a Hyi'otiik.sih Let ABC hi\ points B uu,i C, the ends of the shlc Br '^ !' ..""^ ^''•'^"' "'''e lines BD, CD, bo drawn to the "« n. n ' •' ''" *^« '*f''"K''t AC of t,lu> triunKle BAG * '''' ^""'^ *'"^» ^he sidc8 BA 2. But BD DC shall contain an anirl« Run the iinj^^ie BAC. ™ *°*^^®' ^°0, greater than CONSTIUICTION. — Produce BD to E. A 1)kM<)N8TK\TION. (!•) 1. Uecaiise two ,-Mesof atri- angle are greater than the third side U'rofKm,Bookl.\ the two sides BA ———_>___ AE of the triaiiirie BAF ,.«. \ ' ~~ ^<* g>.Add,toe,t.hoAl:^l^^g«^-thanBE. ' 8 Therefore the sides BA Ar iAxtoM 4.) "-^ '*'^' AC, are greater than BE, EC 5. Add to cnol, of these DB ^' *"' ■®°°* !■) CD.- D£r"[Sj;4"- ^^ ^"' '^ ««»'- ""» .be .,-d» B| fa ■(;;::i,t7) *«' ^^ '*"- - «-.er .,„„ o. Inerctoro !» IHiMONSTHATioN.—ni ^ 1 A ' • ' f^eatcr than BD, DC 38 EUCLID. f|i 11 '!l ""u,%IS°' "'" °'"°"" °°«'° '"'"■° '"""Kl" CDE I, create. PROPOSITION 22.-rROBLEM. 2b make a triainjU, of winch the sides shall be emml a, //,.... to A, B, and C, ea.^h to each. <-i>NSTKUOTION.— !• Take a straijjht Ji..e, DE, tcrmi- Dated at the point D, but unlinuted towards E. 2. A fake DF equal to A, FG eijual to ^ _ — B, an.l GH equal to C. (^Prop. 3, Ihuh I.) cirHe'HL;i;'?/?::X?3r '" ''"'^"^^ GH, describe the ft; Join KF KG. (or /.MZG^.) i».>^fc««;,..i. sCDEi'sprcutei *,) the exterior \C (or /J A A'), e ai!;;Io BDC is router than tho !_ equal to three f these, mtuit be 'aiiTlir lines, of tliini, viz. :— B, und B and of which the A ' B describe the describe the nil have its of the circle struction 2,) {Axiom U) G given EUCLID. og HLK^rw"' ^'''''"r° *!lo P"i"t G ''■" the centre of the circle HLK GH IS c(|nul to QK. {bvtivhum 15 ) tioiAT " "'"''^ '° '•'" "'■'^'^ht lino C. (CWfruc- ^^.^ Therefore GK is equal to the straigi.t line C. {Axi. 7. And FG 18 cqunl to tlicstrai^'ht line B. (Comtrnrtion2\ 8. Ihercforetho three stnii^r|,t |i„e.s KF Vfi r2 "^ straight n,JsXli,C: ' ' ' ''^""' '" ''"* "''''9ivi PJlOroSITION 23._PUOI5LEM. At a given point in a aiven s/might line, to wake a rectilineal angle equal to a given rectilineal angle. *"""^ GivKN— I Let AB be the given straight line and A tha given point in it. ' "" '^ ^°® 2. And let DCE be the given rectilineal ande ^ou(uiT.-~It IS required to make an nru-l^ pV ♦>,„ • point A, in the straight line AB, thu'hu t tllZZ given rectilineal angle, DCE. * "* ^"® ik?"",'::"^''-'- '" °°' °^ ^"^ -"^ «■*« -"<■» ttlineal anqle I^OE), take any points, D, 2. Join DE. 3. On AB, construct a tri- angle AFG, the sides of which ffi^xiiV"' '"''' "''"''' "°" ''''' °^' ^^• to GAr'^'' ""^""^ *° ^'^' ""^^ ^^"'^J *o DE, and EC equai pLlil"*", ^^ «"an be equai to the angle DCE. I'Hoor.-l. Because DC, CE are equal to FA, AG. each to *1i.'W^*i«':j'.;i 40 EUCLID. iw';*'r'; ""'^''' °°^ '^ "I""' "> ft" «"«'<■• FAQ. (P„;,. ,, B.VCU rcctiUueal un«l„',;l^£^,r*^,;:;;;;"^;2;r '" "° riMirOHiriON 2t._TItKOKEHf. V (WW trtt' tl,^ „..., , ; . ^_^Uy,.„r„.s„.-l,„, ABC, DEF, „. .w„ .,,„„/„, J^^; ^. I wt the an,. BAG grc-atcr t.l.,.n the ,uHe EDF base EF. ° ^^ ''*'*^' •^^ greater than the CuNSTKUCTION. — I. Let the side DF ofthctriaiijrle DEF, be Kieater than its side DE. -• 'I'lieii at the point D, in the straight line ED, muko the angh> EDG e(iual to tlie angle BAG. (Prop. 23, Book I.) ^ t ^t^^^ ^° ^^ or AC. (/Vop. 3, Book X.) BA, AC, 1 ",,^1 to the?;o^ED Dg''"" f'^ '''' l^° «^^^ 3. rheictoro the hAOA Rr* ;« i (i^r,v..4, Z^ooX:I.r ^^ *' ^^"*^^ ^° t^^^ base EQ. ^. And because OG is equal to DF, iConstruction 8,) ^in^-^ FQ. (ConstruO' FAQ. (Pro;,. 8, pniiit A, in the adi! o(]iiiii to tho > be done. \L fual to two sides fiiincil h;i thv two fotititiiinl 1)1/ the xp '.;/■ that which (iitxc of the olhiT. riiuiglcs, which f. DE, DF. -ach ^'io EDF. iater than the EUOUD. 41 , Book I.) to DE, (lly. the two sides ) each. EDG. (Om- lie base EQ. mtniction 8,) ^^5.^B. .he an,.eDC?F is g.^ati/ ll^ \£''l^ eQF. ^^«.Jhercf«re the an.le DFG is greater than the ungle 8. A„.i because t'^'^anSrEFG 1^^^^^ . greater than its un^Ie EGF and th t t h '""*'''*' ^^^^ '^ Buhtcnded by the pn-ater side *" *'"""""'' ""«'« i» (i?;v'"irJ^;::(.io "''" '^^ '^ «^"^^- ^"- ^^^^ ^ae ef ll'" r\uJ? ^vas proved to be ecpml to BC 11.1 herefore BC is creator than EF IJ two tnmi(/ks have, two .viV/^s- of tL> n i the other, each to e.a.h, bJl^S^y;,y'^ '' '"'•" ^^^'^ of base, of the other, the annh roZ^ , """, "'"}''''' ^'""' '^« ^hichhasthe.,reuerl^T.h^^^^ T^ 1"' *'^^-^ "/that ^^Hv.oxaKs.s.^Lct ABC, DEF,- be two t'rian.les, which anple EDF. ^^"^ ^^^ '*'"^" t>e greater than the pKMoNSTUATioN.— 1. For if it ho r,^* either equal to it or less than L ' *^'^'''^''' '* '°"«t be 2. But the angle BAG is not equal to the angle EDF becau_se then the S would be 43 BUOLIDi, bwaiise tlion the Lo BC 1,,, i f ", "'"" *» nii«l„ EDF, (/V„/,„,„v,™ 24, /wi ) " ''" ''="' "'"" ""= I""" Ef! ,»1"'".)''° '"'" °° '' »»• '<=» «'»■> *o baso.EF. (Zr^ ^ J. Therefore ,he an^le BAC U e„. le« ,h.„ .ho angle -tt' tor -fcX^r ."r"' «*<= " - «"-' « ^ie EDF. '™ "" '"K'" BAC must b. greater than the PKOrosiTION 26.^TnEOREM. Me c«« to the t/!o^'::^^Zt^f "^"^ "'' ''''-' ''"^^' «/ ^^lWuEsis.-Let ABC, DEF, be two triangles, which i-<^^-^to\T':ii^^^^^^^^ are ad- viz., BC equal to EF ^"''^ '° *^« '^« triangles; -/.And the third angle BAG, shall be equal to the third angle EDF. Hypothesis.— (ir.) For ifAB be not equal to Dt, one of them must be greater than the other- let AB be the greater of the two. CONSTBUOTION. — 1 " b Make BG equal to DF. (Froposntu>n ^ r^vt. theanjfloEDF. tho aiijjjlo EDF, lu the base EF. ttSO.EF. {Uy. han tho angle is not equal to atcr than the c. (JSeeEniOh' to two anfflea one side,- viz., 0), or the then shall the third angle of igles, which DEF, EFD, FD. ich are ad> triangles; lal each to BUOUD. 43 2. Join GO. inin^'llif^ ""«'" '^°<= " «l'.al .0 .he .„g,„ DEF. v^i2utl:'°k l':? '"'" ""^ " "I""' «> '"0 b«e DF. (^Pn. (ivi'ST/;,!;,!. t";«'° "'^^ " -I'ml .0 .he .„g,e DFE. poiii".;''" ""«'" °" '» «'I'"" to «ho angle BCA. (i/y possil)lc. '"" '" ""= greater, which is im- .0 DE ';';;',;;f •^'b'^? i-;« ;7,i,«»' 'ode , ,„. i. ab is »,„., ^ 10. 'I'lKreloie the „o AB Br W'*'""'"'' 2-) EF, eucl, ,„ eaeh, "™' '*^> BC, are equal to th (4/';./tl ;!'; ""^'^ '^^^ « «1"»' '» .he angle DEF (P!i:;i>ri:7w''l)*'"° '^^ « =<»■"' '° «he ba,e DF. S»^^i'i'i!'(i/!t'r;;"S.neS t r •;«'» i°i- . Skquknce —1. Liiic- wiso HI this case the other sides shall be equal; viz., -AC to DF, and BC to 2, And also the third anple BAC to the third angle EDF. Hypothesis TH.) „ ForifBCbenoteq^lto tr. one nf t)i«»r> rn---- i-- greater tiwaUis^tier. 1« BCbettegrea«,rfu«,^ equal to the two, DE, 44 BUCLID. •«• •iom AH. J-)KMf)NSXRATTr»W 1 T> AB BH, are equal to the wo sidel^Tf p"''^' l^^ ^^^'^ «idea 2. And the antrle ABU ,-I , ^ ^^» ^ach to each potheds.) "*''° ^^"^ »s equal to the angle DEF (//„. 3; TliLTcfore the base AH {» ^ , pomion 4, Book I.) " " ^'i"^^ to the base DF. (P^o- F^«///W. 4, /.W;T)"^'^ ^^" *° *^® triangle DEF. (Pro- ,»,4r,'.r "«'» ^■'D « e,„al ,„ ,he ang,, ecA ( ff/ impossible rp,. •:• " opposite ano-le RPA , u- u V' equal ,„EP a„dAB'i"e°q„"arrDE'%?' ;?". '». BC is 11. therefore the two AB Rr {hypothesis.) EF, each to each, ' ' ^^' ^^^ equal to the t^vo DE iBypoi^l {'; ^"«^« '^SC - equal to the angle DEp' 13. Wherefore the basft An • ' iProposUlon 4, ^oo;?: I.) '^^ " ^qual to the base DF 14. And the third angle BAG ,•« EDF gle BAG ,s equal to the third angle Conclusion. -Therefore if ^ ^""""«^^«''-) Which JZl bes^Zn. *"'"^'^^' *«• (^ ^--^a// be parallel. ^ '^"^ ""^"^^«'-' ^'^e^e two straight lines the tC^sS!frff^ thestraijyht line EF, which f«ii« AEF cpn ^^'" /'"es AB. CD. maH .u! ^?^'^^ ^^"s upon ' fcFD, equal to one another aiteraate angle* EP. iProporitio,, ^lal ioEr,(Con. w/. the two sidea each to each gleDEF. (//^. Jase DF. (Pro- le DEF. (Pro- Sj each to each, 2;he angle EFD. leBCA. {tly. il to the ar^gle triangle AHC, BCA, which is that is, BC is thesis.') > the ttvo, DE, angle DEF. 'he base DF. le third angle 8. &c. (^See tnes make the straight lines h falls upon mate angles EtJCLlD. 40 8EQtTEWE.-AB Shall be parallel to CD CD being prodii^d, "^ ' '^ '^^^ ^« »«« Parallel, AB and w»l meet either towards B, D, or towards A, C; let them be produced, and meet towards B and D in the point G. Demonstration. which is impossible. ' ^^''^ '^"^^ *° ^^^ (hypothesis), towards b'd" ^^ ^°' ^° ^^^ P-duced, do not meet towar'dVi^ c"'""" '' "^^ ^^ ^h^-'^ that they do not meet 7 Therefore AB is pTraffit'cD ^°^ ^^^^-^ (^«/- 35.) Conclusion.— WherefnrP u' • ,. Tfn . -, ,.^«^POSITION 28.-THEOREM he parallel to one anotC, ^ ' ' '"'^ '^'"'3^^ ^'««-' ^haU the'^tSrhtli^S;];^^^^ EF, which falls upo..- angie?BG"^l5rGtl'D^/teXV:"Sirto^^^^^^^^^^ ^''^^ ^^^ SBQCENCE.-AB Shall be pSlei to cS''^''' '"^^''• Demonstration —ri ^ i R»; to uu. the anrrl^ o.u^'Th^:^ h Because the anffle EGB i. ^n„„] ^^:-i«wi,aiKAnON. to the ano-lA nun 15, itilo" ■"*" ^°^ " '='"" •» *» ""g'- AQH, ^Pro,. ^'^'i'fte* 46 EUCUD, V' 1 3. Therefore the anple AGH is cqiiftl to the iiri- ple GHD (Axiom Ij. and these angles are alternate angles. 4. Therefore AB is pa- „ rallel to CD. (J'ro,,. 27, Bonk I.) ^ ' ' ♦nVJ? f-/p"'"- ^^''•^'/•"e the angles BOH, GHD, are equal to two right angles, Uiipnthn^h ") ' 4. lake away the common aiiL'le, BGH. 5. Therefore the remaining angle, AGH. is eonal to thp 6. Therefore AB is parallel to CD. [Pum. 27, Book I ) Conclusion.— Wherefore, if a strai. 'lit line &c r'4* Enuncmtian.-) Which wo. tohe .hervl ' ^ PROPOSITION 2!).-TnE0REM. ^^^fhTnT^'^ !''' '^f ''^""/^"> P^'-^f'"^ xfraif,ht fines, it makes the alternate amjle, e.p.al to one another, and the exterior an- ate equal to the interior and ommsite upon the same sid,: and likewise the two interior angles upon thf same side, tuneth^ equal to two right angles. •' «,Sn!?T''-'!-T^'^ "^'i'- ^^'"^'^"^^ "°e ^^ fall upoTr the parallel straight lines AB, CD, Sequence.— 1. The alternate angles, AGH. GHD, ^ lall be equal to one another. 2. The exterior angle, EGB, shall be equal to GHD, the interior and opposite angle upon the same side. 3. And the two inte- rior angles, BGH, GHD, upon the same side, shall be together equal to two right angles. Htpothe8is.-(II.) For if AG H be not equal to G HD, one Of tbemmustbegreaterthantheother; let AGH be the greater EnCLID. 47 ), are efjual qual to two ual to the Uml to the t;Ies are al- Bonk I.) &c. {See ?.«, it makes xtfirior an- ? side, and, if, together upon the ), sliall be DKMON8TRATTON.— 1. Now, bccanse the ancle AGH is grmcr than the angle QHD, add to each of these the angS ri^lu InKhf '^^ ''"^''^ ^°"' ^"°' ""'^ ^''' '^^^ t^° 5 Hut those straight lines which, with another straiHil 1 ne fallm.|r upon them, make the interior angles on the sa7ne fhlv ;'^' T ''"> !"'" ''"" "f^''^ ""P'""' ^i'l °>eet together"? they he j.rodiiced tar enough. {Axiom 1" ) ^ ^' " du'c^d'lren^'gh" ^'"'^'^ '""^ '^^' ^°' ^"^ -««* 'f ?">- iino;/'r/v:Ms:)* "''''' '""""^^ *^^^ ^^•"- p'^^^"^^' ^t^-^gi^t rwn'lT't-"'''.,^^'' an«le AGH is not nncmal to the angle GHD; tha IS. tiie angle AGH is equal to the angle GHD lb ' Book I ^ ^°" '^ ^^"^^ *" ^^''^ ^^^^'^ ^^^' (^'*'^- Axiom\T^'^''^ *^^ *"^^^ ^^^ '^ '^'^"^^ **^ *^® ^°^'^ ^"°* 11. Add to each of these the angle BGH 12. Therefore, the angles EGB, BGH eanal to fJi« angles BGH, GHD. {Anom'2.) eqnai to the two'n JhtrgiS. "''^^ ^'^ ^"^^" '°^' ^"'^' -« ^'i"^ *o CoNcr.usioN.~Wherefore, if a straight line, &c (Sa Enunciation.) Which was to he shewn. *5"* "°^ *<^ ^^ »HD,one 3 greater. PKOPOSITION SO—THEOREM. Straight lines, which are parallel to the same straight line, are parallel to each other. HTP0TnKsi8.-Let AB, CD, be each of them parallel to EF Sequence.— AB shall be parallel to CD J^.-^riSK''''''--}^lV^^ ^^'•aight Une GHK be drawn. Demonstration.— 1. Becanse GHK cuts the parallel graight Imes AB EF the angle AGH is equal to the I^ GHF. {Proposition 29, Book t) ^*^ 46 EUCLID. 2. Again, because the straight line GK cuts the parallel straight lines EF, * U- S CO, the angle GHF is equal to the angle GKD. (Prop. 29, Booh I.) 3. And it was shown that the an- gle AG K (or 4 6^/7) is equal to the an- gle GHF. {Demonstration 1,) 4. Therefore the angle AGK is equal to the angle GKD {Axiom 1), and they are alti-rnate angles. 5. Therefore the straight line AB is parallel to the straight line CD. (Prop. 27, Book I.) Conclusion. — Wherefore, straight lines, &c. (See Enuu' station.^ Which was to be shewn. PROPOSITION 31.— problem:. To draw a straight line through a given point, parallel to a given straight line. Given. — Let A be the given point, and BC the given straight line. SooGHT. — It is required to draw a straiglit line through the point A, parallel to the straight line, BC. F -1. In BC take any point D. g ^ _ _ CONSTRTTCTION. 2. Join AD. 3. At the point A, in the straighi line AD. make the angle DAE equal to the angle AD'O. (Projt. 23. JJuok I.) 4. Produce the straight line EA to F, EF shall be parallel to BC. Proof. — 1. Because the straight line AD meets the two straight lines BC, EF, the alternate anglos EAD, ADC, are equal to one anotlier. 2. Therefore EF is parallel to 80. (^Prop. 27, Book I.) (■«SIS?%-« be parallel ^ BUCLID. 49 ingle GKD ;he straight (5ee Enun- araUel to a the given ne through e the angle be parallel ets the two I. ADC. are Book I.) C0KOLU8ION —Therefore, the straight line EAF inrtr-™ through the given noinr A T.ovaii^i * 11 .^' "* "™^^ Une, ic. #>44 X ,?6ti *' *^' ^^^"^ «'^"«^ PEOPOSmON 32.-THEOREM. .id!,"BTb:'^^;i^»:aTD'» » >-8le, and let one of i» Sequence.— I. The exterior an^Ie ACD .h«n K« i ABC BCA PAR T,l""'°.' «"gles of the triangle viz to AB. (ft»73 .ifcSfel I '"'"" *=' '^'"' <== J^^'l {Prop. 29, i?oo^- 1 ) ^ ^^' '^^^» are equal opposite angle, ^1^!^}^°°^^ Zl\) '"^ '"'"" -<■ posite „„^-le,, CAB, ABo! ^"-J ° *" "™ '"'"''"■ "d oP" CBA, BAC, AOB. Mx,t° I) ° '«""' •" "■" """^ «"g'« 7. But the Hnrrloc Ar>n Ar^^. angles. (Propri^^Bookl.) *"'^°' ''''^ ^*^"^ *° *^« "g^« 8. Therclbre, also, the angles CBA BAP aod to two right angles. {Axiom 1.) ' ' ^^^' "® ^"i^ 60 SUCLID. C0NCI.CS10N. — Wherefore if a side of any triangle, 4c. (See Enunciation.) Which was to be shewn. Corollary. — I. iAll the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides. 1. For we can divide any rectilineal figure, ABCDE, into as many triangles as the figure has sides, by drawing straight lines from a point F, within the figure, to each of its angular points. 2. Now, by the preceding proposition {which shews us that the three interior angles of a triancjle are ei/unl to two right angles), we see that all the angles of these triangles must be egual to twice as many righf angles as there are triangles. 3. Or, in other terms, that all the angles of these triangles must be equal to twice as many right angles as the^ figure has sides. 4. But all the angles of these triangles are equal to the angles of the figure, together with tiie angles at F, the common vertex of the triangles. 5. And the 2nil Corollary of Prop. 15, shews us that the angles made by any number of lines meeting together at one ppint, ai'e equal to four right angles. 6. Therefore the angles made by the meeting of the lines AF, BF, CF, DF, and EF, in the point F, are equal to four right angles. 7. Therefore all the angles of the triangles are equal to the angles of the figure, tbgether with four right angles. 8. And, consequently, all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides. Corollary. — II. All the exterior angles of any rectilineal figure, are together equal to four right angles, 1. The interior angle ABC, with its adjacent exterior angle ABD, are equal to two right angles. {Prop. 13, Book!.) 2. Therefore all the interior, together with all the ex- terior angles of the figure, ai'o e^uul (o twice as many right angles hs the figure has sides. iangle, &c. rectilineai to twice as CDE, into le triangles the^ figure 3[ual to the at F, the us that the iogether at jf the lines ual to four re equal to angles, e, together many right y rectilineal at exterior 13, BooA; I.) all the ex- many righc WtTCLID. 61 8. Therefore, by the foregoing corollary, all the interior- with all the exterior angles of the figure, are equal to all the interior angles of the figure, together with four right angles. 4. Take away the in- terior angles of the figure which are corn- mon to both, and w« find that the exterioi angles of the figure: remaining on one side are equal to the four right angles remaining on the other. 5. Therefore all the exterior angles of any rectiUneal fisvae are equal to lour right angles. PROrOSITIOK -THEOREM. The straight lines which joir -tr^mities of tioo equal and par alkl straight lines towu,.^ uie same parts, are also them- selves equal and parallel. Hypothksis.— Let AB and CD be equal and parallel H r^- AC Bd' ''""^^^ towards the same parts by the straight Sequence.— AC and BD shall also be equal and parallel. Construction.— Join BC. Demonstkation,- 1, Because AB is parallel to CD, and bC meets them, the alternate angles ABC, BCD. are equal. (Proposition 29, Book I.) 2. Because AB is equal to CD, and BC common to the tAvo tri- angles, ABC, DCB, the two sides AB, BC, are equal to the two sides BC, CD, each to each. 3. And the angle ABC is equal to the aneie BCD {JJemonstration 1.) ^ 52 BCCLID. 4. Therefore the base AC, is equal to the base BD. {Prop. 4, Book I.) 5. And the trian l-j ABC, is equal to the triangle BCD. {Prop. 4, Book I.) ^ • 6. And tho other angles are equal to the other angles, each to eaph, to Avhich the equal sides are opposite. 7. Therefore the angle ACB is equal to the angle CBD. 8. And because the straight line BC meets the two straight lines AC, BD, and makes the alternate angles ACB, CBD, equal to one another. 9. Therefore the straight line AC is parallel to BD. {Prop. 27, Book I.) A' d AC has been shewn to be equal to BD. {Demonstration 4.) Conclusion.— Therefore, straight lines, &c. {See Enun- ciation.) Which was to be done. PROPOSITION 34.— THEOREM. The opposite 'sides and angles of parallelograms, are equal to one another, and the diameter bisects them, that is, divides them into two equal parts. Hypothesis.— Let ABCD be a parallelogram, of which BC IS a diameter. Seqdence.— 1. The opposite sides and angles of the figure shall be equal to one another. 2. And the diameter BC shall bisect it, DEMONsinATioN.— I. Because AB is parallel to CD, and BC meets them, the alternate an- a gles, ABC,BCD, * ~" ~~ are equal to one another. {Prop. 29, Book I.) 2. Because AC is parallel to BD, and BC meets them, the alternate an^rles ACB, CBD, are equal to one another. [Prop. 29, Book I.) 3. Wherefore the two triangles ABC, CBD, have two angles, ABC, BCA, in the one equal to two angle;?, BCD, CBD. in the other each to each 4. And thay have one side, BC, common to both triangles, adjacent to the equal mgUn in each. J J ■mm^'^^^ STTOLID. ca base BD. ngle BCD. ler angles, e. ;le CBD. J the two ite angles il to BD. ) be equal See Enun- 'e equal to is, divides of which is of tha lave two e;?, BCD, !;riangle8, 6. Therefore their other sides are eqnal, each to each- VIZ., the side AB to the side CD, and the side AC to the side BD. (Prop. 26, Book I.) 6 And the third angle of the one is equal to the third B"cfi! 1?lV.t6:ffilt '"''' ''''^' ^^"^^ *° ^ ^"^'^ Rcn^I'^ because the angle ABC, is equal to the angle bCD, and the angle CBD to the angle ACB «,,^i7i'nn^'"/ f *? '^^?\^ *°S^® ^^°' " e^l'^a^ to the whole angle ACD. (Axiom 2.) ^•, ^"r?^^® fll^^® ^'^^ *»*" ^®®" shewn to be equal to the angle BDC. (Demonstration 6.) h^iuioihq Therefore the opposite sides and angles of parallelograms are equal to one another. fe *"" ♦« rn^^'S o^**" dJameter bisects them ; for AB being equal two^ Bct^CD'^eaTtreach'^ '"°' '^"^ "^' ^^ ^^^'^^ ^he angf; B^'D.^Vi^^'Sn'l^) '"" ^"^^' ^^"^ *° ^^'^ BCD. "'t %loflT' ""^"^ " ''"'' *' *'' *"''^^^' AB''Dc'L^^t^^^^^^^^ PROPOSITION 35.-THEOREM. Parallelograms upon the same base, and between the same parallels, are equal to one another and between the same parallels, AF, BC. Sequence. — The parallelogram A BCD, shall be equal to the parallelogram EBCF. Case I.— If the sides AD, DF, of the paral- lelograms A BCD, DBCF, opposite to the ^se^C, be terminated in the same poiivt D, it.is plaip *-':''KHf 54 EUCLID. 1. Each of the parallelograma ii doable of the triangle, BDC. {Prop. 34, Book I.) 2. And that they are therefore equal to one another. (^Axiom 6.) Case [I. — But if the sides AD, EF, opposite to the bass BC, of the parallelograms ABCD, EBCF, be not terminated in the same point ; tiien — 6 B FIQ. 1 FIC..A Dp^mon^tkation.— \ Because ABCD is a parallelogram, AD is eciual to BC. ^Prop. 34, Book I.) 2. For the same reason, EF is equal to BC. 3. Wherefore AD is equal to EF {Axiom I), and DE is common. 4. Therefore the whole {Jig. /.) or remainder {Jig. II.), AE is equal to the whole (Jig. ./.), or remainder (Jig. II.), OF, (Axiom 2), (Jig. I.), (Axiom 3), (.,%. //.) 5. And AB is also equal to DC. (Prop. 34, Book I.) 6. Therefore the two, EA, AB, are equal to the two, FD, DC, each to each. 7. And the exterior angle FDC, is equal to the interior, EAB. (Prop. 29, Book I.) 8. Therefore the base EB, is equal to the base FC. (Frop. 4, Book I.) 9. And the triangle EAB equal to the triangle FDC. 10. Take the triangle FDC, from the trapezium ABCF, and from the same (or from a similar) trapezium ABCF, take the triangle EAB. 11. The remainders are equal. (Axiom 3.) That is to say, the parallelogram ABCD, is equal to the parailelogram EBCF, Conclusion. — Therefore, parallelograms upon the same base, &c. (See Enunciation.) Which was to be shewn. The latter pnrt of this domonstratlon wmilrt be rendered more intclH- Rible to tlif learner's mind, it.tlie o|ic>ration of taking away the triangles from the trnpeziums were actually perfdrmed on two similar trapeziums, cutout iu paper or card-board. This method is also useful where super-poaltioa is required In the demoustrution, as iu Proouituw 4 , J I l«'4;iei%'^ the triangle, 3ne another. J to the base it terminated irallelogram, ), and DE is 3r {fig. IL), er (fig. II.), Book I.) the two, FD, the interior, le base FC. i FDC. zium ABCF, 'Aura ABCF, That is to arallelogram on the same shewn. (I more intelH- B trianglfg from eziuins, cutout uper-positioa is J EUCLID. 55 PROPOSITION 36— THEOREM. ParciOehgrams upon equal bases, and between the same parallels, are equal to one another. Htpothesis.— LetABCD, EFGH,be parallelograms upon equal bases, BC FG, and between the same parallels, AH, BG. Sequence. — ^The parallelogram ABCD shall be equal to the parallelogram EFGH. CoNSTUTTCTioN. — Join BE, CH. Demonstration. — 1. Because BC is equal to FG (Hypothesis), and FG to EH. (Proposition 34. Book 1.) 2. BC is there- fore equal to EH, (Axiom 1,) and they are parallels, (Hypothesis,^ and joined towards the same parts by the straight lines BE, CH. 3. But straight lines which join the extremities of equal and parallel straight line?, are themselves also equal and parallel. (Prop. 33, Book I.) 4. Therefore the straight lines, BE, CH, are both equal and parallel. 5. And EBCH is a parallelogram. (Definition 35, Note.) 6. And it is equal to the parallelogram ABCD, bscause they are on the same base BC, and between the same parallels, BC, AH. (Prop. 35, Book I.) 7. For the like reason, the parallelogram EFGH is equal to the same EBCH, (being on the same base EH, and betvreen the same parallels, EH^ BG.) 8. Therefore the parallelogram ABCD is equal to the parallelogram EFGH. (Axiom 1.) Concision. — Wherefore, parallelograms, &c. (See. Enun- ciation.) Which was to be done. PROPOSITION 37 THEOREM. Tvic^nles wnon the same base^ and between the same parnJlfils are equal to one another Htpothesis. — Let the triangles ABC, DBC, be upon the 6ame base, BC, and between the same parallels, AD, BC. .^S^H?" 06 EUCLID. E,f"^SST)^- ''"'"^^ ^° '"^' ^^>'^' *° *^« p-"^ 2 Through B draw BE, parallel to CA, and through O draw CF parallel to BD. (^l^rop. Ql, Bookl.y ^ 3. rhorefbre each of the figures EBCA, DBCF, is a paralle- logram. (JDef. 35, Note.) 4. And EBCA 'is eqnal to DBCF, be- cause they are upon the same' base BC, and between the same parallels BC, EF. (Prop 35 Bool- T\ 5^ And the triangle ABC is the h\alf o AheTaSldogra^^ EBCA because the di.nneter AB bisects it. (Prop. 34, Bookie 6 And ^,e triangle DBC i.s^the half of ihe parallelo^S D6CF, because the diameter DC bisects it equaL"(iw to" '' '^"^' *''"^^ are' themselves also ^ S^Therefore the triangle ABC is equal to the triangle ii^f''wn7-~V'T^?'^' triangles, &c. (See Enuncia. twn.) Which was to he shewn. ,. PROPOSITION SB—THEOREM. Triatiffko upon eqnal hases, and hetu-een the same parallek an equal to one another. ' Htpothests.— Let the triangles ABC, DEF he uTinn equal bases BC, EF, and between the snmeparalle'ls BF AD angro^EF *"'""''" ^^^ •'''^"" b J equal to the tri-' <^?rTpo7tT^^ ^"'"^^ ^^ ^^^'^ -•- *« *he points 2. Through B draw BG parallel to CA. and throurrh P draw FH parallel to ED. (Prop. 31. Book I.) ^^ ^ 3. Then each of the figures GBCA, DEFH is a nRr-ill^ logram. (Uefrmtion S5, Note.) ^'^'■", is a paralle- ■ ^uaU>"i-*^'?r '-'^"^^''", ''"'^ °*^''^' ^^'^"^'^ t^ev are on EUCLID. lual to the 3 the points through O 67 35. Book I.) rallelograra 34, Book I.) rallelogram selves also be triangle e JSnuncia- rraJlek, art , be upon Is BF, AD. to the tri- the points through F a paralle- ley are on aliels, BF, diameter DF bisects c « it. (Prop. 34, Book ^ " »^— » I.) 7. But the halves of equal things are eiiual; therefore the triangle ABC is equal to the tri- angle DEF. {Axiom. 7.) CoNCLusiON.-Wherefore triangles upon equal bases &c (See Enunciation.) Which was to be shewn. ' PROrOSITION 39.- THEOREM. Equal triangles vponthe same base, and vpon the same side ofiL are between the same parallels. Hypothesis.— Let the equal triandes ABC DRr k« upon the same base BC and upon the^^same side'of i^ ' . hEQUENCE.-They shall be between tl>e same parallels • nr tn other words-Join AD, then AD shall be p3e to BC CoNSTHucTioN.-l. For if AD is not pamllel to BC 2. Join EC. Dkmonstratiov.— 1. The triangle ABC is Pnnoi ♦« ♦!,« triangle EBC, because they ^ ®^"^^ *° ^^ are upon the same base BC, ^ - , andbetM-een the same paral- "^ *— ** lels, BC, AE. (Prop. 37, Book I.) 2. But the triangle ABC is equal to the triangle DBC. (IJi/pofhesis.) 3. Tlierefore the triangle" DBC is equal to the triangle EBC {Axiom 1), the greater equal to the less, which is impossible. 4. Therefore AE is not parallel to BC. 5. In the same manner it may be demonstrated that no other line but AD is parallel to BC 58 EUCLID. 6. AD is therefore parallel to BC. Conclusion. — Wherefore, equal Enunciation.^ Which was to he done. triangles, &c. (5ce PROPOSITION 40 THEOREM. Equal trinriffffi^ upon equal bases in the same straight line, and towards the same parts, are between the same parallels. Htpothesis. — Let the equal triangles ABC, DEF, be upon eqnal bases BC, EF, in the same straight line BF, and towards the same parts. Seqi'enck. — The triangles, ABC, DEF, shall be between the same parallels ; or, in other vwrds — Join AD, AD shall be parallel to BF. CoNSTKOCTioN. — 1. For if AD is not parallel to BF, through A draw AG parallel to BF. (Prop. 31, Book I.) 2. Join GF. Demonstra- tion. — 1. The triangle ABC is equal to the tri- angle GEF, be- cause they are upon equal bases, BC, EF, and between the came parallels BF, AG. (Prop. 38, Book I.) 2. lint the triangle ABC is equal to the triangle DEF, (Hijpothei'is.') 3. Therefore also the triangle DEF is .qual to the tri- angle GEF (Axiom 1), the greater equal to the less, which is impossible. 4. Therefore AG is not parallel to BF. 5. And in like manner it can be demonstrated t'lat there is no other parallel to it but AD. 6. AD is therefore parallel to BF. Conclusion. — Wherefore equal triangles, &c. (See Enunciation.) Which was to be shewn. PROPOSITION 41.— THEOPi^-M. If a parallelogram and a triangle he upon the same base, and betiVren the smm parallels, the paranehgram shall be dou^- of the triangle. Hypothesis. — liQt the parallelogram ABCD, and tb^ UUCLID. 69 &c, (See iyht line, and KUCLID. 1. For they nre upon eqnal bases BE, EC, and between the same parallels BC, AG. 3. Therefore the triaii;.'lc ABC is donhlc ofthc triangle AEC 4. J3utthepuralK.|..frrain FPCG is likewise double of the tnatif.'! AEC. {Prop. 41, fhuh [.) 5. i'or th. y are upon the same base EC, and between the same parallels EC, AG. 6. Tl ercfore the paral- lelop:ram FECG is eqnal to the triangle ABC. {Axium 6.) 7. And it has one of its angles CEF, equal to the given anple D, (Construc- tion 3.) CoNCLirsiON. — Wherefore a parallelogram FECG has been described eqnal to the ^ given trianfxle ABC, having b one of its angles, CEF, equal to the given rectilineal angle D. Which was to be do7t«. PROPOSITION 43.— THEOREM. The complements of the paralhhprams which are about the diameter of any parnJklncjram, are equal to one another. Hypothksis.— Let ABCD be a parallelogram, of which AC IS the diameter (1), and EH, GF parallelograms about AC, that IS, through which AC passes (2), and BK, KD the Ao^''rN P'^'"^'^'^'^^'*^'"^' ^''^^^^ ^^^^ "P ^^'e whole figure ABCD. which are therefore called the complements (3^ hEQUKNCE.— The com- plement BK shall be equal to the complement KD. Demonstration. — 1. Because ABCD is a paral- lelogram, and AC its dia- meter, the triangle ABC is equal to the triangle ADC. {Prop. 34. Bonk I.) 2. Again, because. EKHA is a parallelogram, thediamefer of which is AK, the trianrfe AEK IS equal to the triangle AH K. (Prop. 34, Book I.) tria ■^j^ypi^'^™® reason the triangle KGC is equal to the i'^"^ KITCUD. and between rianple AEO. loubl(! of the md between 01 IS to be dom. 8 ahovt the another. I, of which iims about IK, KD the lole figure Its (3). e triangle ik I.) aal to tlie 5. The triiiTisrlcs AEK KGC am n«.,oi .,. ^i AHK, KFC. (Acio>n2) '^^'^' *'" '''1"^' «o the tnanglc. 7. Iherelorotho n'riiairiiii;„^. sl.^^'i^ «» '"K'*" 62 EUCLID. 2. Wherefore the angles BHF, HFE, are togetter lees than two right angles. But straight lines, which with another straight line make the interior angles upon the same side less than two right angles, do meet if produced far enough. {Axiom 12.) 3. Therefore HB, FE, shall meet if produced. CoNSTKucTioN,— (II.) 1. Pvoduce HB, FE, towards BE, and let them meet in K. 2. Througli K draw KL parallel to EA or FH. 3. Produce HA, GB, to the points L, M. 4. HLKF is a parallelogram, of which the diameter is HK, and AG ME are parallelograms about HK, and LB BF are the complements; LB shall be the parallelogram required. Proof. — (II.) \. Because LB BF are the complements of the whole figure, HLKF, LB is equal to BF. {Prop. 43, Book I.) 2. But BF'is equal to the triangle C. {Construction 1.) 3. Therefore LB is also equal to the triangle C. {Axiom 1.) 4. And the angle QBE is equaf to the angle ABM. {Prop. 1.5, Boohl.^ 5. But the angle QBE is equal to the angle D. {CoU' struction 1.) 6. Therefore the angle ABM is also equal to the angle D. {Axiom 1.) Conclusion. — Therefore the parallelogram LB is applied to the straight line AB, equal to the triangle C, and having the angle AE?M, equal to the angle D. W7uch was to be dom. PROrOSITION 46.— PROBLEM. i'o describe a parallelogram equal to a given rectilineal , figure, and having an angle equal to a given rectilineal angle. Given-. — Let ABCD be the given rectilineal figure, and E the given rectilineal angle. Sought. — It is required to describe a parallelogram equal to ABCD, having an angle equal to ". CONSTUUCTION. — 1. Joiu DB, {dividing the rectilineal figure ABCD into two triangles, ADD, DBC.) 2. Describe the parallelogram FKHG, equal to the triangle ADB, and hiving the angle FKH equal to the amrle E. (Prop. 42, Jionk L) 3. To the straight line as to be done. Corollary. — From thi'«K'"ieA/^.xij 2. On BA, AC, describe tbe squares ABFG, ACKH, (Prop. 46, Book 1.) ,^ ,„ 3. Through A draw AL parallel to BD or CE. (^Prop. 31, Book I.) 4. Join AD, FC. , ,it Proof 1 . Because the angle BAG is a right angle {ni/po- thesis), and that the angle BAG is also a right angle. {DeJ 30.) 2 The two straight lines AC AG, upon opposite sides of AB, make with i , at the point A the adjacent angles equal to two right angles. . u a^ 3. Therefore CA is in the same straight line with AU. (Prop. 14, Book I.) , , . , . . 4. For the same reason AB and AH are in vho same Btraight line, , ^ ^ , ^ tt • ^i (ie< the pupil fully sliew why AB and AH are in the same straight line.) . , ■, \ 5. And because the angle DBG is equal to the angle FBA (Axiom 11), each of them beuig a right angle {JJeJint- tion 30), add to each the angle ABC. , , ■, 6. Therefore the whole angle DBA is equal to the whole FBC. (Axiom 2.) ^ i . ^-u 7. And because the two sides AB, BD, are equal to the two FB, BG, each to each, and the angle C A equal to the angle FBC. , , , _^ , 8 Therefore the base AD is equal to the base Fo, and the triangle ABD to the triangle FBC. (Prop. 4, Book I.) 9 Now the parallelogram BL is double of the triangle ABD, because they are on the same base BD, and between th same parallels BD, AL. (Prop. 41, Book I.) 10 And the square GB is double of the triangle FBC, because they are on the same base FB, and between the name parallels FB, GC. Prop. 41, Book 1.) 11 But the doubles of equals are equal to one another, therefore the parallelogram BL is equal to the square GB 12. In tlie same manner, by joining At, BK, it can bo shewn that the parallelogram Cl is equal to the square HC. (Let the pupil prove that the parallelogram LC is equal to the square HC.) _^ . , . ., _ 13. Therefore the whole square BDhU is equal to inc two squares GB,HC. (Axiom 2.) 14. And the square BDEC is described on the straight line BC, and the squares GB, HG, upon BA, AG. Stm. KDCLI0. lare BDEC, ■G, ACKH, (^Prop. 31, ngle (JTi/po- e. iDef.'-iO.) site sides of ingles equal le with AG. 1 vho same e tn the same o the angle agle {Defini- to the whole equal to the equal to the »ase FC, and 4, Book I.) the triangle and between •i angle FBC, between the one another, square GB. (K, it can bo e square KG. LG is equal to equal to tho L the straight C. 67 15. Therefore the square described upon tlie side BC is equal to the squares described upon the sides BA, AG OowcLusTON.— Therefore in any riglit angled triknele, &c. (See Enunciation.) Which was to be shewn. ^^ PROPOSITION 48.~THEORE7/r. If the square described upon one of the sides of a triangle he equal to the squares described upon the other two sides of it the angle contained by these two sides is a right angle. ' Hypothesis — Let the square described upon BC, one of the sides of the triangle ABC, be equal to the squares described upon the other sides, BA, AC. Sequence.— The angle BAG shaU be a right anrie CoNSTHucTioN.— 1. From the point A draw AD at right angles to AC. (Prop. \l, Book I.) 2. Make AD equal to BA. (Prop. 3, Book I.) 3, Join DC. DEMONSTRATION. — 1. cause DA is equal to AB, square of DA is equal to square of A B. 2. To each of these equals add the square of AG. 3. Therefore the squaies of DA AC, are equal to tho squares of BA, AG. (Axiom 2.) ■ 4. But the square of DC is equal to the square of DA AC (^Prop 47, Bookl.\ because the angle DAG is a right aAgle. (Construction 1.) & e »* 5. And the square of BC "is equal to the squares of BA. AC. (Hypothesis.) ^ 6. Therefore the square of DC is equal to the square of t$C (Axiom 1.) 7. And therefore the side DC is equal to the side BC .•®'oA J'a*^^"^*'' ^^^ ^^'^^ ^'^ ^^ ^qual to ^^ (Construc- tion 2), and AG common to thn two tr^nntr^oa nar* oa^ the two sides DA, AG, are equal to the two BA, AC, each to each. Bc' "^^ *^^ ^^^^ ^^ ^^ ^^^^ proved equal to the baa« 68 EUCLID. 10. Tucrcforc the angle Q^AC is equal to the angle BAG. {Prop. 8, Book I.) 11. But DAC is a right angle. (^Construction 1.) 12. Therefore, also, BAG is a right angle. (Axvm 1.) ^ Conclusion.— Therefore if the square , Ac. iSca /..^iWicior Hon.) Which was to be dont. END OF BOOK I i ic angle BAC. n 1.) (Axi-irn 1.) •i-i'^' '\m -m CaiTabian ^xm, oi Mool -^^m^. mH(?RI2£D IT THE COO|«CIL W PUBUC F TOR OMTARli). A|S2f'I>'S FIRST L iTIN BOOK ARNOLD'S SECON LATIN BO^If ' ^fil^OLD'S LATIN PKOSE COMPOSirrov CANADIAN TmRD R,?r>^ nP i,S?^^'^^ LESSOxNS. CANAblAN f OURTH R?u .1? n^ oi*^^^^ LESSONS. CANAD AN Fi™ Soo^L^L^.l^S'^'^ ^ "'^^^^^NS. CANADIAN SPEUIN? Pr>n^ BEADING LES.,0N3. HORACE BVr COHJiBiiM ' POTT'S ELEMENTS DP Ge6meTB SMITH'S SMAL1.SQ LATm DK-rif)VABV l^» 3?BENCii AUD / 5' ■ JAMES CAMPBELL ^ SON, PUELISHEM. TOROJfiO.