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(3) ELEMENTS O F. GEOMETRY; WITH THEIR 1 .. «. Application to the Menfuration of Superficies and Solids,. TO Determination of the. THE Maxima and Minima. of Geometrical Quantities,. AND TO THE Geome-. Conftrudtion of a great Variety of. trical Problems. *. By. •. .. THOMAS SIMPSON,. And Member. of the Royal. Academy of. F. R. S. Sciences at. Stockholm. The. FOURTH EDITION, Carefully Revifed,. L Printed for J.. 0. Nourse,. N D. 0. Nf. Bookfeller to His. MDCCLXXX.. Majesty..

(4) ./. .. Digitized by the Internet Archive in. 2017 with funding from. Wellcome. Library. https://archive.org/details/b28782033.

(5) TO THE HONOURABLE '. ’. i. \i. .. Charles Frederick Efq; ,. Majesty’s &c> &C*. Surveyor-General of his. Ordnance,. Honourable. T. Sir,. HE fubjedt of the. much. ftieets. which. beg leave to lay before you, confequence to mankind,. is. I. here. of fo. as juftly to. claim the regard and fandtion of the Great.. Geometry. not only a mod: accurate, but a. is,. very extenfive fcience, whofe application and great utility, as well in the arts of peace as of. war, are well. known. to. You.. But though this work, if the manner in which it is executed be correfpondent to the importance fufficient. of the fubjedt,. merit to render. it. may. not want. worthy of the. approbation of a Gentleman, who, amidft a multiplicity. of public. employments,. pre-. ferves an undiminilh’d ardor for the fciences,. A. 2. and.

(6) DEDICATION.. iv. and a knowledge of the works of nature. yet I have. Sir,. ;. ftill. art. and. farther motives. Your great influence and zeal to promote the good of an inftitution under which I am placed ; and the favours that I have received at your hands, make me addrefs. for this. earned: to. :. embrace. this opportunity. of. teftify-. ing publickly, that I am.. Honourable With. Sir,. great refpect,. Your much. obliged,. and mod: obedient. humble. fervant.. Royal Academy,. March. 3,. 1. 7 60,. Thomas. Simplon..

(7) :. R E F A C. P. M. r. E.. upon the fubjeCt of Geo metry, was to open an eafy way for young beginners to arrive at a proficiency in that * ufeful fcience ; without either being obliged to go thro a number of unnecejfary propofiticns , or having re defign in writing. ungeometrical methods of demonfirathat abound in mojl modern compositions of this. war/tf iion,. nature. /i>i?. .. I was not unapoccurred that were not eafy. ‘The difficulty of the undertaking ,. prifed of ; and objections to be removed : Neverthelefs , reception. I have grounds. to. hope ,. my firfl attempt has met with ,. endeavours have not been entirely unfuccefsful. No pains havejndeedfieen fpared to render the work ufeful And l flatter myfelf, that the fpirit and rigour of demonfiration , fo effiential to the fubjeCt , are alfo tolera bly well preferved j though I have not been fo intent to guard againjl the attack of Criticks, as to lofe fight. of my main defign of furnifioing a plain , eafy infiitution for learners : Tet 1 have ftrong hopes , that there will not be found in thefe Jheets , any inaccuracies ,. or overfightSy that are abfolutely unpardonable.. To eupeCl a faultlefs know that the moft. piece. is. impoffiible. elaborate. and. ;. And I. well. beft- approved fyf-. tems of Geometry extant , are not without many imperfections. Bitty were the fmallefi imperfection to be a real faulty. my ambition would rather. bey to. Jhew fome. de-. gree of judgmenty by avoiding a multitude of fucb faultSy than by expofing and magnifying the flaws of other writers.. avoid one. :. It. And. to dijlinguifh. is. thofe. more eafy. to fee. a faulty than to. men who are the moft fanguine. themfelves at the expence of. A. 3. otherSy. are. generally.

(8) --. PREFACE.. VI. generally obferved to ftand in need of greater indulgences^ than even the perfons whom they unmercifully. But 1 fhall put an end. attack.. pointing out one objeftion , that this. work. •,. which. is ,. to this digreffion. may. by. be brought againjl. that in demonftrations admitting. of feveral cafes , the moft eafy ones are fometimes omitted j and that the converfe of fome propofitions is But this , I conceive , will not at all demonfirated. be found a real advantage to the learner ; without. which ,. it. would have been. the Elements in the compafs they fides,. the great eft part. being fuch as. may. have comprifed. impoffible to. now. Be. take up.. of the demonftrations omitted. be inferred from thofe given, by means. of Axioms only they may, therefore, be eafily /applied by any reader, Jhould they happen to become tie cejfary, which 1 have fcarce ever found to be the cafe. But , even allowing this to be a defect, it is abundantly compenfated by the extenfive application given in the three laft feblions ; which is infinitely more ufeful, in it[elf, and more necejfary to the forming an able Geometrician, than any thing of the kind we have been /peaking of. *,. In. fecond, edition. this ,. new work ) many have been made. pofitions. is. (. which. is,. in a manner,. and. conftderable alterations. The order of fome of. changed :. And fome difficult. a. additions. the firft pro-. propofitions in. more plain. In the fourth book feveral new Theorems on proportions are The folid Geometry is now connected with added. the plane, and is demonfirated with the fame accuracy . the fecond book are rendered. The menfuration of explicitly. handled. ;. and. move and the demonftr ation of the feSuperficies. Solids is alfo. veral rides is here eftablifhed on a better foundation, than even in authors who have wrote profeffedly on the fubjebt. The Maxima and Minima , and the con. ftrubhon of Geometrical Problems, are likewife confi derably extended and improved. And, at the end, ,. Notes.

(9) •. .. PREFACE. Notes geometrical and critical, very ufeful to improve the judgment of young Jiudents , are now added.. But , whilft 1 am. improvements and upon to anfwer to a charge , which , (hould it appear to deferve credit , would indeed leave me hut little room to pafs myfelf upon the world for a judge in thefe matters. As the gentleman by whom I Jland accufed is known to the matters of criticifm ,. talking of. I am. called. ,. world by. his holding one of the moft confiderable. ma-. kingdom 1 Jhall, in order to thematical pofls do all due honour to the manner and importance of his writing , give you his own words in the. •,. 44. 44 44. “ “ 41 4. ‘. 44 44. 46. “ 44 44. There has lately been publijhed a book under the title of Elements of Plane Geometry , defigned for the ufe of fchools , which is an incorred copy of the firjl eight fed! ions of this work , lent the pretended author on a particular occafion , and printed in a fpurious manner , without my knowledge or confent ; an adion too fcandalous for any man of honour to The Editor imagined, 1 fuppofe , that be guilty of. mangling the the changing fome proportions , demonjlrations of others, was a fuffdent difguife *7 pafs for his own performance ; far this will juftify fuch a piece of piracy, mujl be left to the judgment of the publick. Were I. attempt to defcribe the ideas excited in my mind by the lingular modefty of this important and Jolemn appeal to the publick , 1 Jhould be at a lofs for fit words to exprefs them, without tranfgrefiing the bounds of decency. But I hope that I have not de to. ferved fo ill of the publick , to be thought capable of a ding fo very humble a part, as that of copying from this author and of mangling his demonjlrations, in ,. order to make them pafs for feript of his. ( containing. my own. .. —That. a. manu. between 20 and 30 of the principal.

(10) PREFACE.. vii’i. principal Theorems in Geometry , extremely. my bands ,. ill. digejled). it. was. came. into. lent. me, but forced upon me , by himfelf ( the. is. indeed true. ;. but. not 'very. firjt night after my removal to Woolwich) in virtue of an article in the original rules and inftruftions for whereby it is ordered, that the fecond the Academy mafler (hall teach Geometry under the direction of the But this well intended article, which firft mafler. has been made fubfervient to the purpofes of ignorant tyranny , and daring calumny , has fince , in confequence of a publick examination , been annulled by an exprefs order of the Mafler -General of the Ordnance I could mention fame particulars , [upported by good authority , •,. .. —. that occurred in the courfe of that examination , which would but ill agree with the importance he affumes in his confident accufation. while. ;. but 1 do not think. it. worth. This Gentleman has , himfelf, by his different publications , fo well convinced the world of his abilities^ as to render any farther comment on that head :. intirely unneceffary. and. ineffectual.. ADVE i. R-.

(11) ADVERTI. S. M. E. E NT.. every work of this nature, defigned to contain whatever may be moft requifite to the forming of a regular and complete lyftem of Geometry, a number of propofitions muft neceffarily have a place, whofe chief ufe and application lie in the higher branches of the Mathematics;. S. in. £\. and there being many perfons, particularly young gentlemen in publick fchools, who want to learn fo much Geometry only, as is neceflary to give them a proper introduction into the practical and moft common applications thereof ; fuch as Menfuration. Trigonometry, Navigation, Fortification, PerfpeCtive, &V, For thefe reafons, I thought that it might be of fervice, to point out to fuch Readers, what propofitions in thefe elements maybe omitted, as leaft ufeful to them ; without either hurting the connection, or taking away from. The. the evidence of the other demonftrations.. numbers, of thefe propofitions,. in. the. feveral. books, are as follow. In 29th.. Book. In. Book. 2d Corol. In. I.. the 6,. 17,. II. the 4, 5, 10, to the 9th.. Book. 19,. 1. 1,. 21, 22,. 1. 23,. 2, 13th,. and. and the. the 4, 5, 6, 7, 8, 9, 15, 18, 19, 20, 25, 26, 27, and 28th. III.. Book IV.. the 4, 5, 6 , 9, 1 r, 13, 16, 17, 20, 21, 22, 23* 25, 26, 27, 28, and 29th.. In. In.

(12) (. In. Book V.. 10. ). the i, 2, 16, 17,. 18 , 19, 20, 25,. 26, 28, and 31ft. In. Book VI.. the two or three. firft. propofitions. except by thofe who are only, need be read concerned in furveying and dividing of lands j to whom the whole Book will be highly ufeful. •,. Alfo, with regard to the feventh book, if PerfpeEtive be the only application in view (which I have known frequently to be the cafe) the ift, 2d, 4th, and 1 2th propofitions may fuffice. But if a more general idea of the properties of interfering planes fhould be required, fuch as is neceflary in the doctrine of folids and fpheric geometry ; then all the propofitions, to the 12th, ought to be taken.. The. 22d, and 23d propofitions of this feventh Book ftiould alfo be read by thofe who would be able to find the content and proportion of folid bodies ; as (hould, likewife, the whole eighth book* except, perhaps, the firft and ninth propofitions, together with the three firft lemmas which may be thought too plain, by thofe who are not very folicitous about geometrical rigour, to need a demonftration. 17th, 19th, 20th, 2. •,. 1 ft,.

(13) 11 1. 1. 5. 1. XI. (. An. INDEX. TABLE. or. 4 4 5. I. 8. ). referring. the places in. to. thefe. Elements , where all the moft material propofttions in the firjlfix , and in the eleventh and twelfth books of Euclid, are demonftrated.. Theie Euclid.. Book. El.. 1.. .Thefe Euclid. j. P.B.. B.. II.. Prop. 5. 12.. 1. P.u. 6 8. 18.. 1. 12. 14.. 1. 13. 9. 5. 1. 3. 12. 4-5. }. B. III.. 7. s}. >81. 3. 1 I. 9i. 1. 32. 21. •5. 2. 1. I. 8.. 1. 7-. 1. 6.. 25. 3-4. Pr.. 33. 22. 5. 10.. 1. 34. 2. 26.. 1. 35. 2l * 3. 24.. 1. 36. P. B. :. 22.3. 17 18. P.B.. |(. H-7 6.7 13*7 16.7 21.7. 19. 24 25 28 291 30I 1. 32 33. 34 35. 10.. 4 B. XII. 20. 26,. 28.. 4. 24.. 5. 23-5. 22 23 25. >3. 27. 2. 3 °*. 5. 5. 29. 28 J 29. 5. 1. 3°. P. B.. *. 7-. -. 4. 6.. 12. 5. 7-2. 4-4 B. XI. 6.4 Pr. 3. 15. 1.4. 4. 8.8 7-8 9.8. o CJ. 18. 5. S CO. 19.5 29. 4. |. to. 25.4. 28.. i. 3-8 Cor. 4. 24,. 26.. 5. P. B.. I. 4. 5. H 15 16 18. 4. 3. b. v.. 7 8.7 9-7 3-7 7-7 10.7. 12. 25.. J. 19. c. 4- 7 5-. 16. 25.5. 11. 1. 13-5 14 Cor. to. *4 5. 2.7. 12.. 2. 13. 1. Cor. to. 9 10. 3. 15-5. I!. 5-5. 4. -. 9 10. 1. w] !. 18. 4. 19 4 11.5. 1. 8. 2. 3. 2.. 8. 3. 9.. 1. I. 5. I. 10. 5. 7-4. i. 2. !'•. 31 32. 6.6 7.6. P. B.. 16.. 9*5. 3- 2. 4. 7. 18.3 16.3 14*3. 6.6. 5-4. 4. 5 >. 8. 6. 25. 37 ]. }. 21. 5 10. 3 17. 3. 2. 2 Pr. 2. i.. 4 Pr 5 6 3-4. 14.4 17.4 15-4. 22. |. Pr.. B. XI.. 22 24. |. B. IV. Cor. to. 11.. P.B. 2.. 5*3 B. VI.. 2. 20. 1. 2. 2. B.. 2-3. 8*3 7-3 3-3 4-3. 15 17. 8.5 7-5. 1. P. B.. 14. 23.. 28 j. 19. 1. *3-. 24. 9. 6. pTT. 1.. 22 23. 17 18. '. 3-. 21. 10. 2. J. 3. 29. V.. B.. 19.5 P.16. 15. 26 27. Euclid.. El.. P.B.. 14. '. 6.. >. Theie. .. 10. i. Euclid.. El.. P.B. !*7 2.7. cn. o. O •or. to 11..

(14) The. BOOKS. following. are. all. Mr. Thomas Simpson, F. R. for J. Nourse.. S.. by. and printed. S5AYS. ON SEVERAL CURIOUS AND USEFUL Subjects, in speculative and mixed. I. XT'. X-rf. Mathematicks blems of the are explained. firft j. ;. in. which the. raoft. difficult. and fecond Books of Newton. s. Pro-. Principia. in 4to.. Mathematical Dissertations. II.. written. on a variety. Subjects, in 4to. of Phyfical and Analytical III.. Miscellaneous Tracts on fome. curious and. Mechanics, Phyfical Aftrovery interefting Subjects in. nomy, and. jy. #. Speculative Mathematics, in 4to.. t he Doctrine. of Annuities and Rever-. deduced from general and evident Principles ; with ufeful Tables, (hewing tile Values of Angle and joint Lives, &c. in 8vo. 2d. Edition. sions,. y A. Treatise of Algebra. ;. wherein the fun-. damental Principles are fully and clearly demonftrated, and applied to the Solution of a great Variety of Problems, and to a Number of other ufeful 4th Edition, in 8vo.. VI. ions a. ;. The Doctrine and Application containing (befides what. Number. of. Solution of a. blems. is. of Flux-. common on the Subject). new Improvements in Variety of new and. in different. Enquiries.. the. Theory, and the. very interefting Pro-. Branches of the Mathematics, 2 Vols.. Svo. 2d Edition.. VII.. Trigonometry, Plain. and. Spherical,. with the Conftruction and Application of Logarithms, in Svo. 3d Edition..

(15) elements. GEOMETRY. BOOK. li. DEFINITIONS. EOMETRY. x.. S. V.. J. Extenfion. is. that fcience, by. we compare fucn. which. quantities together as. have extenfion 6 is. diftinguifhed into lengthy breadth ,. and. thicknefs .. 2.. A. Lihe. is. that,. which has length without. breadth.. The terms , bounds. or. extremes. of a. Line are. points.. A. Surface is that, which has length and breadth, only, as C. 3.. The bounds of a Surface are. B. lines .. 4.. A.

(16) Elements. 2. A. 4.. Solid. is. Geometry.. of. that,. which has length,. / I). D.. breadth, and thicknefs, as. / The hounds of a 5.. A. R ight. 7 /. Solid are furfaces.. (or ftrait) line. that which lies even-. is. between its extremes, or which every- where tends A B the lame way, as AB.. ly. A. Plane furface is that, which is every-where perfe&ly flat and even, or which touches, in every part, any right line extended between points any-. 6. .. where taken. in that furface.. An Angle. 7.. is. the inclination,. or opening of two right-dines meeting in a point, as D.. When. 8.. one right-line. DC, (landing upon another AB, makes the angles on both. fldes equal, thofe an-. gles. are. called. right an-. and that line CD is laid to be perpendicular to the other AB on which it gles. G. *. ;. _. D. B. infifts.. 9. is lefs. 10.. which. An. Acute-angle. is. that,. 1. than a right angle, as E.. An. Obtufe- angle. greater angle, as F. is. than. is. a. that,. right-. 11.. The.

(17) ^ Book the Firft. diftance of. The. diftance of a point. two points, is the Rightone to the other. from the reaching. 11. line. The. 3. 12.. from a. line,. is. a. Right-line drawn from that point, perpendicular to, and terminating in, the line given.. D. C. 13. Parallel (or equidiftant) right-lines AB,. CD. are fuch, which be* ing in the fame plane-. furface, if infinitely pro-. ^. duced, would never meet.. A. 14.. Figure. is. bounded. a. fpace,. and. is. either. a furface, or a folid.. A. 15. in. right-lined plane Figure. is. that,. formed. plane iurface, whofe terms, or bounds, arc. a. right-lines.. 16. All plane Figures bounded lines, are called Triangles. T 1. 17. is. that,. all. An. r-. r*. equilateral. by three right-. Triangle. whofe bounds or. fides are. equal, as A.. '. '. 18.. two. An. ifofceles. fides are equal,. Triangle. is,. when. as B.. B. 2. 39.. A.

(18) Elements 19.. A. Geometry .. of. fcalene Triangle. is,. when. the three fides are unequal, as. all. C.. A. right-angled Triangle is that, which has one right-angle, as ACB; whereof the fide 20.. £. AB. oppofite to the right angle,. is. cal-. led the Hypothenufe.. 21.. An. A. obtufe- angled Triangle. C. is. that,. which. is. that,. which. has one obtufe angle. 22.. has. An. acute-angled Triangle angles acute.. all its. 23. Every plane Figure bounded by four rightlines, is called a Quadrangle, or Quadrilateral. 24.. Any Quadrangle, whofe. oppofite. fides. are parallel,. called a Parallelogram, as. A Parallelogram,. 25. gles are. all. Re&angle, 26.. whofe. A. right-ones, as. Square. is. Rhombus. gram whofe its. whofe ancalled a. is. a parallelogram. fides are all equal,. A. D.. E.. gles all right-ones,. 27.. is. and. its. an-. as F.. is. a parallelo-. fides are all equal,. but. angles not right, as G.. 28 . All.

(19) Book the Firjl .. 5. befides thefe,. 28. All other four- Tided figures,. are called trapeziums. \. 29. A. right line joining any. of a four-fided figure, 30.. That. fide. is. two oppofite angles. called a Diagonal.. AB upon. which any parallelogram. ACEB,. ACB. or triangle. Hand, is called the bafe ; and the. is. fuppoled to. CD. falling thereon from the oppofite perpendicular angle C, is called the altitude of the parallelogram, or triangle.. 31. All plane figures contained under more than four Tides, are called polygons ; whereof thofe hav-. ing. five Tides,. fix Tides,. 32.. A. are called Pentagons. Hexagons. 33.. A. Circle. all. is. bounded. curye-line. APCD,. thofe having. To on.. Regular Polygon. well as Tides, are. figure,. and. j. ;. is. one whofe angles, as. equal.. a plane. by. one. called. circumference, everywhere equally ditfant from a point E within the circle, its. D. called the center thereof.. 34-* T he Radius of a circle, is the diftance of the center from the circumference, or a right-line. EA. dravvn from the center to the circumference.. b a. AX. I-. 1.

(20) — Elements. AXIOMS,. or Self-evident Truths.. Things, equal to one .and the fame thing, are. 1.. alfo. Geometry •. of. equal to each other.. 2.. Every whole. 3.. Every whole. is. is. greater than. equal to. its. all its. part.. parts taken. together.. equal things, equal things be added, will be equal. wholes the 4. If to. 5. If. from equal things, equal things be taken. away, the remainders will be equal. If to, or from unequal things, equal things. 6.. be added, or taken away, the l'ums, or remainders, will have the fame difference, as the unequal things firlt. propofed.. 7.. All right-angles are equal to one another.. More. than one right-line cannot be drawn from one given point to ang other given point B. 8.. A. 9.. If. two points. D, F, in a rightline MN,are pofited at unequal diffances. ^. *>. T—p. 17. DC, F£, from. an-. other right- line. AB in the fame plane-furface; thofe. two. lines,. being infinitely produced, on the fide of EF, will meet each other.. the leaft diftance. 10. If.

(21) Book the io.. C. two. If. Firji.. 7 1?. right-linesCA, CB, making an angle C, be refpcitively equal. two. to. other. FD, FE, making an angle F, and the angles which they make C, and F, be likewile equal the right-lines AB, DE joining theii extremes will be equal, and right- lines. ;. the. two. triangles. DFE. ACB,. equal in. all. re-. fpefts.. If this fliould not appear fufficiendy evident for. DFE. to be removconceive the triangle ed, and fo applied to the triangle ABC, that the point F may coincide with C, and the fide fall upon the fide then, becaule FD is fuppofed ; equal to CA, the point D will alfo fall upon A. And, the angle F being equal to the angle C, the fide FE will fall upon CB ; and confequently the point E upon the point B, becaule FE is fuppofed equal to CB. Therefore, feeing all the bounds of the two triangles coincide, it is manifell, that not only the bafes AB, DE, but the angles oppofite. an axiom. ;. FD. CA. to the equal (ides, are alfo equal.. When. all. CA, CB, FD, FE. the four lines. are. equal; the triangle DFE, being contrariwife applied to fo that FE may coincide with CA, will,. ACD. ACB. agree with the triangle (as is manifelt from the reafoning above) and fo, the angle E (as did before) now coinciding with the angle A, the two angles E and mult necelfarily be equal to each other, in this cafe, where the triangle. alfo ,. :. D. D. DFE. is. an. ifolceles. one.. B 4. POSTU..

(22) Elements of. 8. POSTULATES,. Geometry or. .. PETITIONS.. That, from any given point, to any other given point, a right line may be drawn. 1.. That,. 2.. a right line. may be produced,. or con-. tinued out, at pleafure.. That, from any ppint as a renter, with a radius equal to any right- line aiTigned, a circle may be defcribed. 3.. That,. 4.. a right-line. may be drawn. cular to another, at any point afligned it. alio pofiible for to. is. make. perpendi;. and. that;. a right-line, or a. right-lined angle, equal to any right- line, or rightlined angle afiigned, or to the half thereof.. This fourth Populate is added, more for the fake of making the proper references , than through abjolute ble. what. here barely affumed as poffieffected, and adlually demonfirated , in the begin-. neceffity is. :. fince ,. is. ning of the Fifth Book, intirely independent of every thing but Axioms and the other Populates, above laid down,.. Jt. though he eajy. may. thefe. alfo. be proper. to. note here , that ,. Populates are not always quoted , it will where , and in what fenfe , they are. to perceive. to be underfiood.. Notes and Observations,. with the Jigni-. fications of Signs ujcd in this Trabf.. A. Proposition. is,. when fomething. is,. either,. propofed to be done, or to be demonitrated, and is either a Problem, or a Theorem. A Problem is, when fomething is propofed to be done. Thf.q-. A.

(23) Book the Firjl .. A. Theorem. is,. when fomething. is. 9 propofed to. be demonftrated.. A Lemma. is,. ed, in order to. when fome premifeis demonftratrender the thing in hand the more. eafy.. A. Corollary. from fome. a. confequent truth, gained. preceding truth, or demonftration.. A Scholium are. is,. is,. when remarks and obfervations. made upon fomething going. before.. >. A -. The. The which. ltands,. The fign. Signs.. denotes that the quantities betwixt. fign it. fignifcation of. \. are equal.. denotes that the quantity preceding it, is greater than that which comes after it. denotes that the quantity precede The fign ing it, is lefs than that which comes after it. The fign +» denotes that the quantity which it precedes, is to be added. The fign denotes that the quantity which it , precedes, is to be taken away or fubtradled. figure, or number, prefixed to any quantity, (hews how often that quantity is to be taken, or repeated as 5 A (hews, that the quantity reprefented tr~,. —. A. *,. by A,. is. When. to be taken 5 times. feveral. angles. are. C. formed about the fame point each particular angle defcribed by three letters, whereof the middle one fhews the angular point, and the other two, the lines that form the angle thus CBD or DBC fignifies the angle formed by the lines CB (as at B), is. :. and DB.. When,.

(24) B. io. Elements When,. Geometry .. of. any demonftration, you meet with fe~ one to the. other continually by the mark of equality (“),the conclufion drawn from thence, is always gathered from the firfb and laft of them which are equal to each other, by Thus if virtue of the firft axiom. B— C — D, (D) be then will the bill (A) and the laft equal to each other. in. veral quantities joined the. ;. A~. you meet with two numbers, the firft fiiews the propofition, and the fecond the book. Moreover, Ax. denotes axiom ; Poll, poftulatum Def. definition Hyp. hypothesis. Note alfo, that, when- ever the word Line ocAlfo,. when. in the quotations. •,. ;. curs, without the addition of either right or curved, ,. and that, when always underftood a line is faid to be drawn to, or from an angle, the angular point is meant. a right-line. is. :. THEOREM A line A (. with. it. two. ). I.. Jlanding upon another line. angles. (ABC),. (. ABD). (CD) makes. which , taken. to-. gether , are equal to two right angles.. 1. Def. 8. equal,. make two unequal,. Poll. 4.. dicular. right-angles let. to. BE. CD. the greater of into the parts. c. Ax. d. Ax. 4.. 3.. plain. is. it. ABD. AEC,. If the angles are. b. they a ;. if. be perpendividing ,. them (ABC) C. EBA. EBC,. ;. then the former part EBC being- a right- ancde a , and the remaining part EBA together with the whole lefier angle ABD, equal to another rightan gl e EBD c ; the whole, of both the propofed angles, taken together, muft necefiarily be equal to. two right-angles. d .. C O R O L-.

(25) .. Book the Firft. 2 I. COROLLARY. Hence fame. all. the angles at the fame point (B) on the. of a right line (CD) are equal to two. fide. right-angles. c. c .. THEOREM (ABC, ABD) which. them. theft lines. ;. BD) with. together are equal to. (BC, BD). 3.. II.. If one line (AB) meeting two others (BC, in the fame point (B), makes two angles. right angles. Ar.. two. will form one con-. tinued right-line.. For, if pojfihle, let BH, and not BD, be the continuation of the right-line CB then the angles ABC being two and :. —. ABH. right angles. ABD mon. f. ;. to. ABH ~. 6. ABC. =:. and. C. from thefe equal quantities, both, be taken away, there ABD * j which is impoffible h if. I). ABC, com-. Ifhe oppofite angles. (DC, BA). :. 1.. r. 1.. Hyp.. remain. will. 2. .. THEOREM lines. T. ’. h. Ax. Ax.. 5.. 2.. III.. (DEB, AEC), made. two. by. interfering each other., are equal. .. DEB + DEA ~ two right-angles = AEC + DEA-, For. 1. whence, by taking away common, there remains. = AEC. 1. DEA,. 1.. \. DEB. k k. .. *. jm. t. t. T. I-I. E O-. Ax.. t..

(26) 2. Elements. 1. Geometry. of. THEOREM 'Two right-lines. (. AB, CD). you fay, they are not when produced out, meet. EA,. iPoft.. fary). let. there. EH ~. be taken EG , and 1. .. m Poft.. i-. n. Def. 8.. *. Ax.. p. Hyp.. 10.. i 2. i. 1. duced. necel-. Ax. 8.. in. them,. then. let. fome point,. as. ;. F. C. pro-. (if. parallel to each other.. parallel. If. In. IV.. perpendicular to one and. (EF), aye. the fame right line. .. G.. d. ~pr. let. FH. be drawn m . The triangles EFIF and EGF, having EHzzEG, the angle HEFzz GEF % and FF common, are therefore equal in all relpedts ° : and fo, the angle being zz p (TFD)zza right-angle , (as well as HEG) muft be one continued right-line q : which is im-. the right-line. EFH. EFG. HFDG. pojfible. r. Therefore. .. AB. CD. and. are parallels,. SCHOLIUM. In this theorem, the poffibility of parallel lines. which being. produced, in the fame plane, can never meet) is demonftrated for EF may be drawn perpendicular to ABk and CFD, which laft, it is deagain, perpendicular to EF monftrated, will be parallel to AB. (or fuch,. infinitely. :. 1. *,. THEOREM Perpendiculars. (EF,. (AB,. CDJ. rallel lines. equal to each other. •,. of the two parallels. For, 3. GH. AB. and. GH) to. V.. one. (AB). of two pa-. terminated by thofe lines, are and alfo perpendicular to the other. (CD).. CD. being parallel to each other,. EF. 5 Ax. 9. can neither be greater, nor lefs than ; and Def. ancj therefore muft .be equal to EF. If you fay,. I3 ' tPoft.. that 4... EF. is. not perpendicular to. CD. be perpendicular to EF‘, meeting. ;. then. let. FM. GH produced (if.

(27) Book the Firfi,. *3. M. : fo (hall neceffaryj in u ; be parallel to. AB. FM. GM —. and confequently EF w GH; which is imx Therefore EF is pojfible. .. perpendicular to CD. And bythe fame argument, is perpendicular to CD.. GH. COROLLARY.. V.. Hence, through the fame point F, more than one parallel cannot be drawn to the fame line given. AB.. SCHOLIUM. the preceding, proportion, the confidence of the twenty-fifth definition, or the poffibility, that all the properties afcribed to a. From. redangle, can fubfid together in the fapie figure, will appear, together with the method of conftrudiom For at any two points C, D iri aTlght line RS, ^ ^ ^ two perpendiculars CG, may be ereded y and a perpendicular to one of y Pod* 4 thefe, at any point E, meeting the other in F, thus condruded may be drawn. The figure * 4* will be a redangle for CE and are parallel Z therefore the angle F (as as are alfo and EF well as C, D, and E) is a right-angle \ If CE be 1 s- 1. made — CD, then will the redangle have all b its Tides equal Which anfwers to the definition b 5.1. and Ax> u of a fquare.. __. .. DH. *. •,. CEFD DF. :. CD. :. CEDF. .. THEOREM Right-lines line. (CD). (AB, EFj. VI.. parallel to the fame right-. are parallel to each other .. For.

(28) — Elements. 14 For. the. let. Geometry. of. line. HIG. .. Cr. AL. .. /. * 5; 1.. be perpendicular to CD: then, that line being alio 1? Xi perpendicular to both AB r* and EF c , thefe lafl: are parallel to each other. T? Ji. I Jb'. JJt. Tr. rt. d. 4. 1.. .. THEOREM A QP). line. (AB). interfering. VII. parallel fines. (SR,. SDC, PCDJ. equal. two. makes the alternate angles. to each ether.. S-. *«. CDE, h. Ax.. Ax.. DE. and. DE. 1.. g. CF. be pere pendicular to QP, and SR ; S. then thefe lines FC and f are likewife parallels ; and To the triangles CFD and Q. Let. having the. fide. CF. —. A/. 7.. — DE % FD — CE % and they will alfo the angle F — E. ,10.. FDCczECD. 5. ,. -. 1.. .. equal to each other. k 7. 1.. 1. 1.. 1. » Ax.. 4.. *. have the angle. Hence, a line interfe&ing two makes the angles (BDR, BCP) on 3. ^. h. COROLLARY 1. y^. :. for. I.. parallel lines, 'the. fame. fide,. BDR ( “CDS j) —BCP. COROLLARY. k .. II.. Hence, alfo, a line falling upon two parallel lines, makes the fum of the two internal angles (SDC-F QCD) on the fame fide of it, equal to two right-* angles: for the angle SDC being PCD, and v QCD two right-angles 1 thence is PC D — alfo to two right-angles m . QCD SDC. —. —. + +. 1. THEOREM. VIII.. If a line (AB) interfering two other lines (PQ, RS), makes the alternate angles (DCP, CDS) equal to each other. j. then are thoje. two. lines parallel.. For.

(29) Book the -. •. >. 15. Birjl. i-. •. .. '. i.. For, if-poflible, let fome ocher line DT, afnd not DS, be parallel to n then muft ;. CDT. PQ. - DCP. which. is. = CDS. 0. mpcjfible. p. n. :. Sch. to 4. 1.. 0. q .. p. CO RO L. s. 7 - «•. Hyp. Ax. 2.. on two others, makes the angles (BDR, BCP) above them, on the fame fide, equal to each other j then thofe two lines. Hence,. a line falling. if. are parallels. becaufe. :. SDC — BDk. THEOREM. r .. IX. i ,/i. •. If one fide ( AB) of a triangle ( ABC) he produced, the external angle (CBD) will.be equal to both, the internal oppoCite angles (A,. For,. let. AC. rallel to. will. CBE. BE. l. ,. together.. be pathen. 5 •,. the angle. C) taken. C. ~. and the angle. A = DBE. “. :. there-. fore C-f A ” CBE + DBE = CBD 1. y. .. COROLLARY.. '. .. Hence. tht external angle of a triangle is greater than either of the internal, oppolice angles.. .. ,. THEOREM The. X.. three angles of any plane triangle. 1. (ABC). 5. ti;jrr '. taken. two right-angles. C/\ For, if A B be produced to. together , are equal to. D, then C. + A — CBD. Z ,. /. to. which equal quantities let the angle CBA be added, then. C + A + CBA = CBD 4-CBA — tworight-angles b. will. \ \. / /. £. *9.1.. \.. 1. a. a. .. CO R 6 L. b i.. 4 1. ..

(30) 1. Elements. 6. Geometry.. of. COROLLARIES. If. 1.. two angles. in. one. triangle, be equal to. two. angles in another triangle, the remaining angles c. Ax.. 5.. will alfo be equal. If. 2.. c .. one angle. one triangle, be equal one. in. angle in another, the fums of the remaining an* gles will be equal. c .. one angle of a triangle be right, the other tWo taken together, will be equal to a right-angle. 3.. If. 4.. The two. acute.. lead angles, of .every triangle, are c. THEOREM. XI.. angles of a quadrangle (ABCDj are equal to four right-angles .. The four inward taken together ,. Let the diagonal AC be drawn then the three angles j. of the triangle u 10. j.. —. ABC. two right-angles. being d. and. ,. ACD. thofe of the triangle equal alfo to two right an1. follows that the. gles'. ;. of. the anglesof both trian-. all. it. furri. which make the four angles of the quadrangle, mult be equal to four right-angles c .. gles, c Aar. 4.. COROLLARY. I.. Hence, if three of the angles be right ones, the fourth will alfo be a right-angle.. COROLLARY. U.. Moreover, if two of the four angles, be equal to two right- angles, the remaining two together will likewife be equal to two right-angles,. SC HO-.

(31) Book the Firft.. 17. SCHOLIUM. from any point P, within a polygon A8CDE, lines be dVawn to all the angles, fo as to divide the whole int<? as many triangles If. APB, BPC, CPD, DPE, EPA, the polygon has Tides ; the fum of all the angles of thefe triangles, (which together make up, or compofe the angles of the. as. ^. polygon, over and above thofe about the poin#P) wjll be equal to twice as many right angles as the polygon has Tides by 10. 1.) Therefore, Teeing all the angles about the point P, whereby the angles of all the triangles exceed thole of the polygop, are equ^l to four right angles, it is manifeft, that all the angles of the polygon, taken together, will be equal to twice as many rightangles, wanting four, as the polygon has Tides.. THEOREM The. angles. (ABC). gle. XII.. (A, B,) at the bate of an. if0 Cedes Irian*. are efsial to each other.. CD. bifedb, For, let the line or divide the angle ACB intwo two equal parts ACD,. BCD, BCD,. CD gle. Aa. and meet the. then. AB. in. having. common, and R 0. :. f. ,. the. ACD — BCDs,. —. D. ACD, AC zr. BC. triangles. f .. -. alfo. have the angles Hvd 0 h. •. COROLLARY. I.. Hence, the. line which bifefls the vertical angle of an iiofceles triangle, bife&s the bafe, and is alfo perpendicular to it h .. C. Def. 18.. anwill. h. A. CORO L-. Ax.. 1 o,,.

(32) 8. Elements. 1. Geometry •. of. COROLLARY Hence angle. is. II.. appears alfo, that every equilateral likewife equiangular. it. THEOREM In any triangle. (ABC). tri-. XIII.. the greatejl fide fub tends. the greatejl angle.. Let A B be greater than AC ; which let there be taken AD — AC ; drawing CD.. c. in. triangle. /. >. y\. A. ADC. *. k. ADC. \. being A ii'ofceles, the angles „ACD and D B 12 *• are therefore equal j whence ACB, which exceeds the former of them, mult alio exceed the k Ax. 2. latter , and confequently, much more exceed Cor. to B, which is lefs than. The. 1. s'. 1. ADC. ADC. 1. COROLLARY. Hence,. any triangle, the fide that fubtends the greateft angle, is thegreateft; becaufe ACB cannot be greater than B, unlefs AB is greater than ffl. AC m. *3* *•. in. .. THEOREM. XIV.. (AB, AC, CB) of one triangle, three fides (DE, DF, FE) of another. If the three fides be equal to the. triangle , each to each refpedlively. \. then the angles op -. pofed to the equalfides will alfo be equal.. Let. BAG =. gle. AG = let. U. Hyp.. •. fhall. the. gles. ABG. being. GC. and. drawn. DEF. D,. DF, and. GB. be. m Ax. io. an-. the. ;. fo. trian-. and. be equal. in. all. = DF = AC%. refpefts. and. n!. BG. :. therefore,. = EF ~. AG BC n ,.

(33) ;. Book the Firji .. ACG. the angle. — BGC. DFE. 0. — AGC. alfo. is. and confequently. ;. BCG. and. ACB — AGB. ABC, DEF are. therefore the triangles. :. °,. 19 p. 0. 12. 1.. — 'Ax.. 4. or 5*. equal. refpe&s m .. in all. SCHOLIUM. The. demonftration of the laft theorem, in obtufe-angled triangles, may admit of another cafe ; which, however, is not neceffary ; becaufe, if the triangle (equal to DEF) be conceived to be formed on the longeft fide of ABC ; then, all the angles CAB, CB A, GAB, being acute q the Cor 4 0 .** line CG will, always, fall within the figure ACBG r , r ^x. 2. AGB. GBA. 15. *. -. ,. as in the prefent cafe. %. THEOREM. XV.. If two triangles { ABC, DEF) mutually equiangular^ have two correfpondingjides ( AB, DE) equal to each other , the other correfpondmg fuxes will alfo be equal.. If you fay. BC. is. (g. great-. EF BC let. er than. from a. BG. parr. be taken. EF. 5. —. AG. and let be drawn.. AB. rr. ,. DE, BG. will alfo. fore. The. have. BAG —. =. ABG, DEF having B zr E (by hyvothfis ),. triangles. EF, and. but D — BAC. BAGzrD BAC w which 4. •,. is impoffible.. j. u j. there-. H. ,. t. u. Hyp.°'. w. r. and. ' 1. 2.. COROLLARY. having any two equal, are equal to each other \. Hence, equiangular correfponding. Tides. triangles,. i. Ci. THEO-. aX* I O*. 2 o.

(34) .. 20. Elements of. Geometry. THEOREM. XVI.. Tf two right-angled triangles (ABC, DEF) having equal hypothenufes (AC, DF), have two other. (BC, EF). fides. (AB, DE). the remaining fides. likewife equal-,. will he equal\. and. the. two. triangles equal. in all refy efts.. In. take. AB produced, BG — GC let. and drawn. then,. :. BCG. triangles 7 z a. b. c. DEF,. = — AC. Cor.. r. to io. a. having. Hyp. and the angle E ED, BC Ax. 7 a D, and Ax. io. will alfo have the angle . y whence, the triangle being ilofceles, b 12. I. the angle G, or D, will be ; and confequentJy. J. F 1. 5* *•. ‘. alfo. =. CBG — CG ~ DF. G~. ACG. :. =A. — ACB. c *,. therefore the triangles. ABC and. DEF,. being mutually equiangular, and having d z= DF, they are equal in all refpedts .. THEOREM. AC. XVII.. If two triangles (ABC, DEF) having two fides (AC, BC) of the one equal to two fides (DF, EF) of the other refpeftively , have alfo the angles (A, D). fub tended by two of the equal fides (BC, EF) equal to and if the angles...(B, £) fub tended by the each other be either , both acute or both obtufe then will the two triangles be equal in all refpefts.. other equal fides. Let. e. '\x.. 7«. DE. :. CG. ,. and. then,. FH. the. be perpendicular to. angle. AGC. being zz. AB. ;. and. DHF % A=.

(35) Book the Firji ,. i*. A =: D, and the fide AC=:DF CG will alfo be Hyp. — FH S whence, CB being — FE the angles ‘SGBC and HEF are likewife equal and fo, the 1116 f. f. ,. 2. f. ;. 1 *. ,. h. 1 *. *. ,. triangles. ABC. and DEF, being mutually equiand having the fides AC and DF equal,. 1. Cor. i. angular ', to IO. are equal in all refpecds s. The demonftration is the fame, when both the angles are obtufe, as in the triangles AbC, DeF for, if Cb (— CB FE) rr Fe t the angles G^C and HeF being equal (as before ), the angles AbC k i. i. and and D<?F will likewife be equal k . Ax. 5. 1. :. —. THEOREM. XVIII.. If two angles (A, B) of a triangle (ABC) be (BC, AC ) fubtending them will like-. equal, the fides. wife be equal.. CD. Let. ACB, then. bifeft the. AB. and meet the. angle in. D. :. ACD,. triangles. BCD. being equiangular k , and having CD common to. both,. they. AC - BC. will. alfo. have. *.. 15. 1.. THEOREM Any two. (AC, BC). of a triangle are greater than the thirdfide. fides. taken together ,. XIX. (ABC) (AB.). In BC produced, there be taken. let ^ ~ CD CA,. and. let. angles. equal. AD be drawn. The D and DAC are. ni ;. therefore. m. BAD,. which exceeds the latter", mult alfo exceed the former D ; and confequently BD (or BC AC) mult exceed. +. c. 3. AB. THE. 12. 1.. n. Ax.. 0. Cor. to. 2.. «•. O-.

(36) Elements. 22. of. Geometry. THEOREM Of. all the right lines. .. XX.. (PA, PB, VC). falling. from. that a given point (P) upon an infinite right line ( RS), (PA) is the leaf which is perpendicular to it\ and , is the near eft the perthe reft, that (PB) which. of pendicular. than any other (PC) at a greater. is lefs. difiance.. For p. Hyp. Cor. tO. I. 4.. 0» I«. angle cute q ,. “. Cor.. to. 13. I.. Cor. to 9.. ,. BP. '. jj. r. BAP being a rightABP will be aand therefore AP r .. Again, when PB and PC are both on the fame fide of the perpend cular. PA. ;. then. is. CBP cr right PC tr“ PB. angle. 5. cr. and. coniequently fide of the perpendicuIt PB be on the eonrrary AB then from AC, let AB be taken lar to PC thereand Ax. 10 the two lines PB, PB will be alfo equal 3 preceding fore PC, which exceeds the one (by the I.. =. •,. s. BCP %. cafe) will alfo. exceed the other.. theorem Of two (AB, BC). •,. xxr.. (ABC, DEF) having two fides one equal to two fides (DE, EF). triangles. , of the refpeblively , the bafe of that each to each other the , of (ABC) will be the greateft , which is Jubtended under. the greatefi angle. Let the angle. ABG. cc:. E,. BG ~ EF — BC) alfo (. XL. let. AG. and. CG. be drawn, upon the. laft. of which,. produced..

(37) Book the produced, Since it is. BG. let fall. “. BC. w. the perpendiculars and, confequently, ,. evident, that. GI (whether. 23. Firji.. BH and. AI. GH =HC. the point. I. u.. u. ". x. 4. *•. *. ,. be confi.. G. G. and K, or between dered as falling between z and H) will be lefs than Cl ; and therefore AG, b a . alfo lefs than or its equal ,. AC. DF. THEOREM. z “. Ax.. Ax. •. -. 2.. 10 *. XXII.. (ABC, DEF), having one angle one angle (EDF) in the (BAC) other and the fides (BC, EF) oppofed to them alfo. Of two. triangles. the one equal to. in. ,. equal, that. {. ABC}. will have the great eft hafe , where-. of the oppofite angle. (ACB). the leaf from. differs. a. right- angle.. EH. be perpendicular to AC and rz HE, GI zz DF, in which produced, take be parallel to GB, and EFI ; alfo let GA, meeting AB, produced if neceffary, in j and let Cl and KF be drawn.. Let BG, and. HK MN. BM —. N. ~. being BCG d , and the latter of c d thefe greater than thence is c Hyp. (or ), e lCBcrKFE; and confequently BIcrEK ; whence Ax * I0 6 2I u d alfoBG(lBl) c-EH(lEK) or its equal ; and therefore being BN, becaufe and parallels, both the points and will fall on the fame fide of AG. But BN (as the triangles NBM, f are equiangular, and have is ) s : therefore BA is alfo greater than DE. % 15. 1.. The. angle. ICG. KFH. EFH. 11. BAr. BM. M. AG N. MN. BM — EH. DEM — DE. C 4. T H E O-. *. *.

(38) : I. Elements. 24. Geometry. of. THEOREM. .. XXIII.. ABD). flanding upon If\ of two triangles (ABC, the fame hafe (AB), the one be wholly included within. two fides (AD, BD"* of the included one taken together , will be lefs , and the angle (D) contained by them greater, rejpeliively , than the two fides (AC, BC), and the contained angle (C) of the other. the other , the. Case be. one fide. in. AD. Then, 19. 1.. CD. Ax.. k. Ax.. 1. 6.. be. j. lels. of. the. of the contained triangle. other. AC +. than whence, by adding. h. than. or than. is lefs. AC + BD. common, *. If the vertex. I.. Ac. will alfo. + BD. -|-CD. AC + BC k ADB C ACB. equal 3. Cor.9.1 But the angle its. Case. BD. .. is. If the vertex be within the other triangle be produced to in E : then [by the. II.. Let AD meet BC. «. fum of AD than the fum of. preceding cafe) the. and. bD is lefs. AE and BE-,. which. laft. fum,. and confequently the former, is, again, lefs than the fum of AC and BC. Moreover, the angle ADB ur BhD C.. c. THEOREM 7 he. oppofte fides. £ £XIV.. (AB, DC). of any parallelogram fABCD) are equal, as are alfo the oppofite angles (b, D) j and the diagonal (AC) divides the paralle-. logram into two equal parts.. For,.

(39) :. 25. Bool the Firft • For, AB, DC, and AD, BC d. BAC. —. n. parallels. being. is. DC A. zr. DAC°,. in all. and. °,. BCA U. therefore the equi-. angular triangles. having. angle. the. ,. ADC. ABC,. A. p. AC common, are equal. reipe&s. q .. COROLL AR1 Hence,. one angle (B) of. if. right-angle, for. all. D, being. zr. B,. is. angle. a right. 7. Every quadrilateral is. and. BCD. is. byTheor. V.. THEOREM are equal,. be a. the other three will be right ones. DAB — D,. zz B, and. a parallelogram. XXV.. A BCD). a parallelogram.. whofe oppofite fides (See the preceding. fcheme).. AC be drawn ; then the trianbeing mutually equilateral , they r s will alfo be mutually equiangular ; confequently t AB will be parallel to DC, and to BC Let the diagonal. gles. ABC,. ADC. r. fjyp.. *. AD. THEOREM The. lines. and d. DC. are parallel. therefore,. DC*. and. BD ,. g.. (AB, DC). be drawn. the angle. ABD. Becaufe is. AB”W Hyp.. — CDB w. 7". j. BA beingO zz BD Common,. the remaining Tides and angles will l.kewife be relpeclively equal. confequently. AD. 7 ;. and. parallel to. f .. the correfponding ex-. tremes of two equal, and parallel lines are themfelves equal and parallel.. u. ^.. XXVI.. (AD, BC) joining. Let the diagor&l. 1. BC. THE. O-.

(40) 2. Elements. ). of Geometry. THEOREM in one fide. If,. (. intercepted by them ,. NGM. AB,. 1GN, b. 1. *. 7- i-. c. 24-. A. RyP-. *•. • *5- 1-. of the other fide{h.Q) will alfo he equal to each other .. be parallel. HI. interfering. and DE in Then, the ‘3-. H. (EG, GI). hafe, the parts. to. of a triangle (ABC,), from at equal diflances (DF, FH), ) HI) he drawn parallel to the. (DEM, FG,. Let. MGE,. XXVII.. AB). (D, F,. three points lines. .. N. and. M.. triangles. having the. EGM % and GN FD. IGN zz ING rz Mb,. angle. (-. GM Gl. FH %. c. d. z=. will. = GE. .). alfo. have. B. e .. COROLLARY Hence. one. I.. of a triangle be divided into any number of equal parts, and from the points of divifion lines be drawn parallel it. appears, that,. if. fide. to the bale, cutting the other fide, they will alfo. divide. it. into the. fame number of equal. COROLLARY. parts.. II.. FG, HI, cutting the fidesof a triangle, be parallel to each other, and another line be fo drawn as to cut off FD Hence,. alfo,. if. two. lines. ~. DE. FH and GE — GI,. this line. DE. will. be parallel to. the two former.. THE. O-.

(41) Book the Firjl. THEOREM in the fides. If. XXVIII.. of a fquare ( ABCD), equally dijlant angular points , there be taken four. four points (E, F, G, H,) the figure other the. from. 27. .. (. -. EYGH) form-. ed by joining thofe points , fhall alfo be a fquare.. AD,. For the wholes. DC, CB, f. qual. and. ,. BA. being e-. alio. the parts. AE, DF, CG, BHs, the remaining parts. ED,. HA muff:. con-. H. h be equal iequently whence, all the angles being equal ; D, C, B,. B. FC, GB,. •,. A. the Tides. 1. GH,. EF, FG,. HE will be equal likewife and the angle DEF — — A qTherefore, becaufe DEH AHE from thefe, the equal angles DEF, AHE A.HE be taken away, there will remain HEF ~ k. ,. k. is. .. J. ,. A — h. if. 1. 9.1.. f. By the fame argument (or by Theor. 25th, and the Corok to the 24th) the a right-angle. .. other three angles wiil be right-angles,. THEOREM all the fides. If. bifebled,. XXIX.. of any quadrilateral. the figure. (EFGH). ('ABCD). formed by. joining. be. the. points of bifefiion , will be a parallelogram.. Draw and BD.. HG. are. the. diagonals. becaufe parallel. AC. EF and to AC n. ,. they are alfo parallel to each other 0 . After the fame. manner is FG F.H therefore ;. parallelogram. to. Darallel. EFGH. is. a. L. A. p .. The End of. H<. the. First Book.. £. i’Def. 24..

(42) ELEMENTS O. F. GEOMETRY. BOOK. II.. DEFINITIONS. t. T N a parallelogram ABCD,. i.. 1 EF, HI,. gram. the. two. right-lines. interfering G, be drawn,. parallel to the fides,. the diagonal. dividing. if. in. the fame point. parallelo-. C. F. B. four other parallelograms thofe two into. •,. GD, GB the. through which r diagonal does not. pafs, are called. ments two,. ;. and. Comple-. the. other. jE. A.. HE,. FI, parallelograms about the diagonal.. 2.. be. Every rectangle is faid to contained under the two. right lines. AB, BC. that are the. bale and altitude thereof.. A. £ The. \.

(43) ... 29. Book the Second.. AB. two right- lines ’The reft angle contained under end BC is often, for brevity fake, denoted by the figure is a fquare , it is ufually repre-. ABxBC.. But when. the letter , or letters fented by placing the number 2 over 1 denotes the fquare expreffing the fide thereof: thus. AB. made upon the. AB.. line. THEOREM The. (BD,. rectangles. FH). I.. contained under. equal. are equal.. lines ,. O. D. For, let the diagonals AC,. EG. be drawn. :. —. AB. becaule. /. then,. X. EF, BC = FG, and B r 1 a the A. I. triangles. F Hyp-. F. B. ,. a. ABC,. EFG a e equal And, in the very fame manner will ADC and EFIG appear to be equal. Therefore the whole redtangle ABCD is alfo equal to the b. b. Ax.. c. Ax. 4. i.. io.. .. whole rectangle. EFGH. c .. THEOREM Parallelograms. (ABCD, BCFE) ftanding. (BC) and between. fame. bafe. AF). are equal.. For, fince (in Fig,. —A. and CDF equiangular'. BE f. :. figure. II.. d ,. i.). the. fame. the triangles. parallel. F FDC,. the angle f. upon the. (BC,. — BEA. d ,. EAB are becaufe CF =. they are alfo equal , therefore, if each be taken from the whole. ABCF,. *,. there will remain. ABCDzzEBCF. d. COROL-. t .. to 7. e. Cor.. 1 • 1. to 10. f. h .. Cor.. 15. 1.. s 24. h. Ax.. I.. 5.. i.

(44) 3°. Elements. of Geometry .. COROLLARY. I.. Hence, triangles BAC, BFC (Fig. 2,) landing upon the fame bale, and between the fame parallels, are alfo equal, being the halves of their refpettive J. 24.. parallelograms. 1.. *.. COROLLARY. II.. Hence. all parallelograms, or triangles, whatwhofe bales and altitudes are equal, are equal ever,. among. themlelves becaufe all fuch parallelograms are equal to redlangles Handing on the fame bafes, and between the fame parallels and thefe laft are equal, by the preceding propofition. *,. *,. THEOREM EA). The complements (EC,. (AC). to the. DAB. e. '. I. k. D. whole triangle and the parts ,. DIE, EFB refpedtiveTy equal to the parts. DHE, EGB 3. of any parallelogram. are equal.. For, the whole triangle DCB being equal k 34. t*. III.. k ,. Ax. s^-maining parts. the re-. EC,. EA. mult likewife be equal. THEOREM A. trapezium. *.. IV.. (ABCD). whereof two fides (AD, BC) are parallele is equal to half a par allelo%ram y whofe hafe is the fum of thofe two fides, and its altitude , the perpendicular dijiance between them.. /. '.

(45) 1. .. Book the Second,. 3. AD. produced, For, in and let take DF zz BC and FE be all CG, parallel to AB, meeting AF and BC produced, in and E. ^ Then G, is a parallelogram of the lame altitude with ABCD, and n Confir. having its bale AF equal to the fum of 0 Cor 2 °, BC n but this parallelogram, becaule BG zz 2* t° z p and zz , is equally divided by the line p ,\ 1 •,. DH. H. AE. AD HF. :. CHD. CGD. CD. 9 j. *. -. and. fo. ABCD. is. the half thereof.. q. Ax. 4.. THEOREM The fum of all the rectangles contained under a given and all the parts’ AH, HG, GB) of line (AD another (AB), any how divided is equal to the redangle contained under the two whole lines.. \. ,. Let. ABCD. be the reftangle contained under the. two. whole. and. lines,. HF, GE. be parallel to. meeting. DC. Then. will. let. AD,. in. F and E.. AF,. HE, GC. be redtangles altitude with. r. of the fame. AC. AF zz AD x AH, HE zz AD x HG, and GC zz AD x BG and confequently AD x AB( AC zz AF + HE + GC) s. ;. f. •,. zz. Cor.. therefore. AD X AH + AD x HG + AD x BG. THEOREM. u .. to. 24. s. 24.. 1. 1.. Ax. 1.. & 1.. 2.. Ax.. 5. .&. 4. i.. VI.. If a right dine (AB) be, any -wife, divided into two parts (AC, BC), the fquare of the whole line will be equal to the fquares of both the parts, together with two rectangles under the fame parts. Let. &.

(46) —. — Elements. 32. ABGI. Let. of. Geometry.. be the fquare. ABGI. of the fquare. in. M. I. of AB, and CBEF that of BC, and let EF and CF be produced to meet the fides. G. F. N. E. M. and N.. From the w and De-CM, EN. equal quantities take the equal quantities CF and EF, and there remains fin. 26. x Ax. x 5.1. therefore, all the angles of the figure being ; y Cor. to z right ones y , is a fquare upon FN ( AC) z Def 26 anc^ AF, FG are equal to two redangles under a * BC and but : rr BF 2. FI AF FG,. A. FN. Ax.. 3. 1.. B. FM ~. NM. *,. AG + + AB = BC + AC + 2 AC X BC AC. *. 6. c. or. i. 1. 1. COROLLARY. +. b .. I.. Flence, the fquare of any line is equal to four times the fquare of half that line.. COROLLARY Hence,. alfo, if. mult be equal. ;. II.. two fquares be equal,. becaufe unequal lines. their fides. BA, BC have. not equal fquares.. THEOREM. VII.. The difference of the fquares (ABEH, AC IK) of any two unequal lines ( AB, ACj, is equal to a rectangle under the fum and difference of the fame lines .. In. AC. i. EH,. — BF'. EB, produced, take let F'G be drawn parallel and. let. both ways, to. Cl be produced meet and FG. EH. D and G. It is DF is a redangle. in c. Cor. 24.. evident that. c. ‘’AX.24.I.QP. eg. of the given. to. d). =. lines. whofe altitude. t. ,. whofe bale. he. AB,. FE. difference. AC. •,. (becaufe. and. BE. 6'L,.

(47) Book the Second.. BA. iz. lines. e. and. ,. but. :. BFn AC. (becaufe the fquare AI.. DK. iszzthe. ). this rectangle. DB+DK. —. f. h. DF zz. 33. fum of. ~ DB. is. THEOREM. Def. 24 .. GB g — H >T* f. = the. GB). the fame. *. fquare. AE. x. S h. f. 3* 2. VIII.. (AC) fub tending the right-angle of a plane triangle (ABC), is equal to both the fquares (BE, BG) made upon the fides (AB, BC) The fquare made upon. the fide. containing that angle.. Let. the. BE,. fquares. duced ther. to in. AB). BG. meet L and. which take each. Tides. equal. KL to. E. of the be proeach o-. D. \. A. D. in. and IG. AE. (or. and let Cl, IK, and KA be drawn. Since and FBC (which are continued right-lines l ) are equal to each other*, EL, DG, ED, and LG will be all equal among themfelves ; and fo the angles E, D, G 2 and L being all right ones' , EDGL will be a ^ Hyp. & 5- *• iquare, and confequently ACIK a iquare likewife n " 2 Now, if from the fquare DL, the four equal 0 tri- 0 J*'** IO angles ADC, CGI, ILK, and KEA be taken away, there will remain the iquare Al and, if from the p fame DL, the two equal parallelograms DB, BL p i (which are equal to the laid four triangles, becaufe two of them *) be taken away then there D3 will remain the two fquares BE and BG. Confequently the fquare AI is the two fquares BE s Ax. 5. q and BG *. ABH. .. 1. 1. 11. .. l ’. “. :. —. ;. .. The fame demonf rated. ctherwife.. Let AD be the fquare on the hypothenufe AC, and BG, BI the two fquares on the fides AB and. D. BC. :.

(48) 34. Elements. .. BC. MBLI. let. :. (produced). meet r. Ax.. if. 7.. GH. of. Geometry . AE, meeting GF. be parallel to H; and let. EA. in. N. from the equal. be produced to. in. GAB, CAN, common to. r. .r.H. angles. the angle. NAB,. be taken away, there will remain 3. Ax.. 5.. — BAC G. gle *. u. Def. 26. fide. is-. NAG. *. ;. to 2. 2. as the an-. ABC. r. ,. and the. AG — AB the Tides AN AC zr AE) are likewife l. ,. (. u ;. and therefore the pa-. rallelogram V. whence,. alfo rr. is. and equal. T -. both,. AM —. AH. logram. w ;. the paralle-. which. laft,. confequently the former,. * 2. 2.. is. and equal to the fquare. BG *. Banding on the fame bale AB, and between the fame parallels. By the lame argument, the paralis— the fquare B'. and, confequenclelogram. CM. the fquare. ly, y. Ax.. 4.. :. fquares. BG. AD. and BI. — AM + CM) —. (. both the. y .. COROLLARY. Hence, the fquare upon either of the. z. Ax.. a. -. 2.. 5.. Tides. in-. cluding the right angle, is equal to the difference of the fquares of the hypothenufe and the other z fide \ or, equal to a rettangle contained under the fum and difference of the hypothenufe and the other fide. a .. The. THEOREM. IX.. ( AC, equal to the difference. difference of the fquares of the. two /ides. of any triangle (ABC) is of the fquares of the two lines , or diflances (AD, BD) included between the extremes of the bafe (AB) and the. BC). perpendicnlar. For, fince 4-. BD. 1. (CD). AC*. of the triangle. zz. DC*. 1. AD% and. -f (by the precedent), it. is. BC*zzDC*. evident that the difference.

(49) ). Book the Seco 7id. difference of. 3. AO and BO will be equal to the difDC + AD and DC+ BD or 1. ference between. b. 1. b. 1. ,. between. AD. 1. common, from. and. c. BD%. by taking away. DC%. c. Ax.. 5.. Ax.. 6.. both.. COROLLARY. I. 1. .. ,. \. Since the redangle under the fum and difference of any two unequal lines, is equal to the difference of their fquares d , it follows, that the difference of j the fquares (or the redangle under the fum and difference) of the two fides of any triangle, is equal to the redangle under the fum and difference of the diftances included between the perpendicular and. 7. 2 , '. the two extremes of the bafe. •. COROLLARY It follows ,. II.. moreover , that the difference of the. fquares (or the reft angle under the. two. /. fum and difference. equal to twice a a reflangle under the whole bafe , and the diftance of the. of. th'e. fides of. triangle , is. perpendicular from the middle of the bafe.. For,. E. be the middle of the bafe, and let AF being BD % the ex- e cefs of above BD (or AF) will {in Fig. 1.) be DF zDE*, therefore the redangle under the * fum and difference of and BD ( s let. EF be. made —ED*, then. ~. AD —. BO). is. —. —. — AO. AD. AB,x 2DE, Again h. being. D. 2. rz. AD-{2ED, and. in. this cafe. 5.. .. 9. 2. .. (in Fig. 2.). — AD + AF — FD — BD — AB, we have, alfo* AC" — BO = ABX2DE.. BD AD. —. ^x. ,. THEO-. h. Ax. 4.. 1.

(50) — Geometry. Elements of. 36. ). THEOREM. .. X.. of one fide (AC} of a triangle (ABC} is greater, or lefs than the fum of the [qua res of the bafe ( AB) and of the other fide [ BC ), by a double reft-. Fhe fquare. angle under the whole bafe. (. A B and the ). d[fiance (. 3. D. the of the perpendicular from the angle (B) the mentioned greater when that is perpenfide firft ; ,. op polite to. ,. dicular falls beyond the faid angle ( as in. when. lefs ,. and. ’. 1.. EG,. Tn Fig. m. ». 2. ‘. El. 9. a*. Ax. 3*. , .. (. in. 1.. X BD). 2. 2.. 1. AC. 1. — BC. to the. ;. 2 BI. nl. — twice the - AH (AB. 1. therefore,. ;. and. if. from the 1. ). +. and. — AB*. laft. 1. -f-. B[. 2. and laft be taken away, then firft. n. .. 1. 3.. AB. rectangle. 1. — 2 AB X BD BC* — AC — 2EI — 2BI. x BD. firft. we have here. E. I.. AC AB. 4... in. of thele equal quantities, AB 1 and AB 1 lefs both BC. 2BG — 2AB .. AB. FH. CD. m. Ax. Fig 2.. and let the perbe continued out to meet. bifedling. = 2EH +. 1. In Fig.. ». but. •,. EH. 1. pendicular (produced). -. in.. }. Let the fquare ABHF, on the bafe AB, be di« vided into two equal redtangles EF and by. 2.. Cor. falls on the contrary fide (as. it. i. 3.). the line. '. Fig. 1. —. and-fo, by adding of thefe equal quantities,. BC". ;. AC — 2AB x BD 1. T. FI. E O-.

(51) Book the Second.. THEOREM. 37. XL. The double of the fquare of a line (CE) drawn from the vertex to the middle of the bafe of any tri-. (ABC),. angle. together with double of the fquare of. (AE),. the femi-bafe the fides ( AC,. BC). equal to the fquares of both. is. taken together.. CD. be perpendiAB: then, becaule 1 {by the precedent) exceeds the fum of the two 2 fquares and CE 1 (or For,. ler. C. cular to. AC. AE. BE. and CE 1 ) by the double re&angle* 2AE x ED (or 2 BE X ED) and becaufe 2 BC is lei's than the fame fum by the fame double redangle; it is manifeft that both AC 2 and BC 1 together muft be equal to that fum twice taken ; the excels on the one part making up the defed on the other. 2. ;. THEOREM The two diagonals (A EC,. (ABCD; fquares. each. bifedl. other. XII.. BED) ;. of a parallelogram. and the fum of. their. equal to the. fum of the fquares of all the four fides of the parallelogram. is. Eor, the triangles. DEC. AEB, being equiangular p , and having AB z= DC % have A E CE, and BE DE r Moreover, bewill. alio. —. (. —. .. caufe. 2AE. 1. + 2ED = 1. the double of thefe,. (DBO. - AD + u. 1. -AD. 1. + CD. BC. -h. D. 3. by taking. ,. 4AE (MC CD + AB 2. we have 1. 15. 2. 2. 2. ). + 4ED. 2. 11 '. Cor. 2. -. J. to 6. 2.. 2. .. THEO. 1.. .. 1. Ax. 4. and 24. 1..

(52) Elements. 3». of. Geometry. THEOREM. .. XIII.. If from any point (F), to the four angles of a rectangle (A BCD) jour lines be drawn \ the fums of the fquares of thofe drawn to the oppofite angles will be 1 Z FB Z equal (I fay, that FA- -+ FC ).. —. For, let the diagonals AC and BD be drawn, bifedting each other in E*, and let then the E, F be joined triangles ABC, BAD being equal in all refpe&s w , thence. + FD. D. •,. w 24 and Ax. Will ,0. *. y ii. 2.. But. AE(4AC)=zDE(4-DB).. FA + FC =. (iDE. Z. z. Z. + 2 EF = l. ). aAE* FB +. y y. 2. FD\. End. of the. Second Bock..

(53) ELEMENTS O F. GEOMETRY. BOOK. III.. DEFINITIONS. A. i.. NY light-line. L\. FD,. T.. C. palling. through. E. the. of a circle, and terminating in the circumference at both ends, is called a Diameter. center. 2.. An. E. arch of a circle,. any portion of the periphery, or circumference, as. is. Gl. ACB.. The. chord, or fubtenfe of an arch ACB, is a right- line AB joining the two extremes of that arch. 3.. A femi-circle. contained under any diameter and either part of the circumference cut off by that diameter. 4.. is. a figure. D. 4. 5-. A.

(54) Elements. 40 5.. A. 6.. A. Geometry. of. .. fegment of a circle is a figure contained under an arch ACB and its chord AB. Sedtor of a circle. under two right-lines EF,. is. a figure contained. EG, drawn from. the. center to the circumference, and the arch EG included betwixt them. When the two lines EE, EG,. Hand perpendicular is. to each other, then the Sedtor. called a quadrant.. 7.. An. angle. ABC. is. fegment ABC, when,. faid to be in a. of a circle being in the periphery. thereof the right-lines. BA, BC by which. it. formed, pafs through the extremes of the chord AC bounding that fegment. is. 8.. An. angle. hended by two. ABC. the periphery, compre-. in. right-lines. arch of the circle,. ADC,. is. BA, BC,. including an. faid to ftand. upon. that. arch.. 9. faid to. A. right- line. AB. A. B. c.. is. touch a circle, when, patting through a point (C) in the circumferencethereof, it. cutteth off no part of the. circle.. xo.. Two.

(55) Book the Third. 4*. .. io.. Two. (TCQ, RCS). circles. are laid to touch. each other, when the circumferences .of both pafs. through one point (C) and yet do not cut each other.. 11.. Two. fame plane,. in the. circles,. are faid to. cut one another, when they fall partly within, and partly without each other ; or, when their circumferences cut each other. 12.. A right-line. be applied, or infcribwhen both its extremes are in the is. faid to. ed in a circle, periphery thereof. 13.. A right-lined. a circle,. when. all its. figure. is. faid to. be infcribed. in. angles are in the circumference. of the circle. 14.. A. circle. is. right-lined figure,. faid. when. to be defcribed about a the periphery of the circle. pafles through aid the angles of that figure. 15.. A. about a. right-lined figure circle,. when. all. is. be defcribed thereof touch. faid to. the fides. the circle. 16.. A.

(56) Elements. 42 1. A. 6.. circle. lined figure,. is. when. of. Geometry .. be infcribed in a righttouched by all the Tides of. laid to it is. the right-lined figure.. A. right-lined figure is faid to be infcribed 17. in a right-lined figure, when all the angles of the former are fituate in the fides of the latter.. THEOREM If the fides (AB, BC,. CD, &c.). I.. of a polygon in-. fcribed in a circle , be equal, the angles (AOB, BOC, &c.) at the center of the circle , fub tended by. COD, them ,. "sill likewife be. equal.. For AO, BO, CO &c. being equal a. to .each. o-. % as well as AB, BC, CD &c.. Def. 33. thcr. of. 1.. the trianglesAOB,. BOC, COD,. are. mutually equilateral ; and therefore have all the. AOB, BOC. angles. &c. equal * 14. 1.. other. to. each. b .. SCHOLIUM. On. depends the divifion of mathematical inftruments for taking and meafuringof angles. For, if, by repeated trials, or any other means, the circumference of a circle defcribed about a center O, be divided into any number of parts AB, EC, CD &c. fo that the chords be equal then it is evident, from hence, that all the angles AOB, BOC, COD &c which make up this propofition. ;. the four right angles at the center,. AOD, DOG, GOK,. KOA. will alfo be equal to each other, let. the.

(57) — Book the Third.. 43. OA. of the instrument be what it will. the radius In the divifion of the circle for practical ufes, the number of parts into which the circumference is. number of equal. thus divided, or the center,. grees. is. 360. ;. fo that a. \. angles at the. which equal angles are called deright angle, confiding of 90 of. be an angle of 90 degrees every angle being denominated, from the degrees and parts of a degree, contained therein; each degree being conceived to be fubdivided into 60 equal parts, called minutes ; each minute again into 60 equal parts, called feconds ; and fo on to thefe equal angles,. is. faid to. •,. thirds, fourths,. fifths,. &c.. at pleafure.. THEOREM. II.. Any chord (ABJ of a circle, falls wholly within the fame: and a perpendicular (CD) let fall thereon from the center of the circle will divide it into two ,. ,. equal parts.. Let C, A, and C, B be joined and thro’ any point E in the ;. chord. AB,. let. the. right-line. CEF. be drawn, meeting the circumference in F. It is evident, becaule CB c , that thefe equal lines are on different fidesof the perpendicular CD d ; and fo, CE being. CA r. Def. 33. 1.. “ CA. or. E. CF. d ,. (take it where you will in the line AB) and confequently the line itfelf, will fall within c the circle . Moreover, becaule the triangles ACD,. the point. 20.. 1.. Ax*. 2*. AB. BCD will. CA. have zr be alfo zz. AD. CB BD. and. CD. common, thence. f .. 16.. COROLLARY. Hence. a line bifefting. any chord. at. rght-angles,. paffes thro’ the center of the circle.. T. FI. E O-. 1..

(58) Elements of Geometry. 44. THEOREM. .. III.. Any two chords (AB, DE) equally dijlant from center (OJ of a circle , are equal to each other.. Let ® f. *. ContVr. t. 16. i.. OF,. B. OC be drawn, and let O, D and O, A be joined, Becaufe. Hyp. Def. 33. the perpendiculars. -. OF = OC. OD. %. OA. iz. f ,. an ^ ^ ant^ ^ are ^Oth r ^ghtangles s, therefore is DF. —. ! 2 3. k Ax. 4.1.. AC. -. h. DE. and confequently. ,. 2D p. i. — 2 AC. k. AB h. zz. THEOREM In a. circle. diameter. \. fAEFB). and. ,. cumjerence, that is. 1.. 19. 1.. of. ail. (CD ). Draw it. will. OC. and. A. B. IV,. the great eft line. (AB). is. the. others terminating in the cir-. which. greater than any other. then. m. the. is. neared the center (O),. (EF) further from. OD. it.. ;. appearthat AB. OC + OD) c- CD m 2. Let OP be the diftance of CD from the cen(or. .. and OQ^that of EF, both taken in the fame radius OR 3 Draw OE and OF. Becaufe the triangles DOC, OFE, have two Tides equal each to each n , and have the contained angle DOC c~ the contained angle FOE 0 ; therefore, alfo, will the bafe DC be ter,. 31. Def. 33. i-. °. Ax.. 2.. Pai.. 1.. <3. 3. 3.. greater than the bafe FE P ; and, confequently, greater than any other chord at the fame dillancc,. with. EF. g .. COROLLARY.. Hence, a right-line greater than the diameter, drawn from any point within a circle, will cut the circumference.. THE. O-.

(59) •. c. Book the. Thir'd.. THEOREM. 45. V.. AFFB \ from. Jf to the circumference of a circle ( any point (D) which is not the center , right. lines. DA,. the greatejl of all (DA) fhali be that which paffes through the center (C) ; and, of the reft , that (DF) whofe other extreme (F) is placed. DF, DE). be. drawn. ,. near eft, in the circumferente , great eft, will exceed any other is. (. DE). extreme. (A ). of the. whofe extreme (E). at a greater diftance ,. From 1.. 2. zr. to the. .. the center C, let. AD. (. Since. DCE. c. ,. CE. and. CF. DC + CF zr DF DC common CF — CE, therefore DF zr DE zr. be drawn.. s. r. r. .. ). is. and. DCF. *. u. is. 3. .. COROLLARY. I.. Becaufe no two lines, DE, DF, drawn from D, on the lame fide of the diameter AB, can be equal to each other w , three equal right-lines cannot pof-w fibly be drawn from the periphery to any point, befides the center of the circle : and, therefore, if from a point in any circle, three equal right-lines can be diawn to the periphery, that point is the center of the circle.. COROLLARY Hence. II.. alfo follows, that no circle can be defcribed to cut another in more points than it. FBG. two: for, if it were pofiible to cut it in three points G, E, F, then right-lines drawn from the center. Q,. Ax. l 9. 4.. *•. 2*.

(60) \. of Geometry *. Elements. Q. Def. 33.. ?. would be. to thofe points,. all. equal. *,. which. is. ^ ewn to ^e impoflible unlefs when the center Q coincides with C and then the circles themfelves y. y. Cor. ,. 1. to 4. 3. .. ;. will neither cut,. come one. circle. nor touch, but coincide, and bex .. THEOREM A. right-line. VI.. (FD) drawn through any. point. (A). in the circumference of a circle, at right- angles to the radius (EA) terminating in that point, will touch the circle. .. From. any point. FD,. in. to the center E, let. b. 20. 1.. B. D. the. BE. be drawn j which being greater than % the point B mud, neceffarily, fall out of the right-line. a. A. F. AE. b : Def. 3 3. circle and Ax. t h e fame 2 ‘ of 1. and therefore, as argument holds. 'good with regard to every. FD. (except A), it is manifeft other point in the line that this line cuts off no part of the circle, but. touches. it,. in. one point only.. THEO REM. .. VII.. If the dijlance (AB) of the centers of two circles, le equal to the fum of the two femi- diameters (AM, BN), the circles will touch each other , outwardly ;. and. the right-line. (AB). joining their centers , will. pafs through the point of contain.. I11.

(61) s. Booh the Third. AB,. In. and. lee. 47. AC =. AM, take be drawn per-. DCE. AB BN. pendicular to being alfo rz. then,. :. c ,. BC. c. and. cumferences of both circles will pafs through the point. C. J :. Conftr.. the cir-. d. DE. but the right-line. Ax. E. -. S'. Def. 33. 1.. b y the precedent) falls wholly above the one, and wholly (. below the other ; therefore the wholly. circles themfelves fall. and without each ocher, touch in one point C only.. COROLLARY. Hence,. two. be placed at a diftance, from one another, lefs than the fum of the two femi-diameters, a part, at leaft, of the one will be contained within the other but, if the diftance be greater than that fum, the two circles will then neither touch, nor cut each other. the centers of. if. circles. :. THEOREM If the diftance. fCAF, DAE). (CD}. of the centers of two circles. he equal to the difference of the. two. (CA, DE),. femi- diameter. and that. then will thofe circles touch radius (CA) of the greater,. inwardly. ;. which. drawn through. is. VIII.. the center. (. D. ). of the. lefj'er. s. will meet the two peripheries in the point of contain.. From any. point. E. in. the. circumference of the. lefier, to. the two centers, let. EC. ED. A. and. CA DE DC — DE 4 DC CA. be drawn. Becaufe exceeds by the line or becaufe. c. £. — DA + DC therefore DA — DE and fo the ciris. g,. h. ;. cumference of the. circle. e. Hyp. f. Ax.. 4.. s. Ax.. 3.. h. Ax.. 5.. ,. D likewife.

(62) :. Elements. 48. likewife paffes through 1. CE. Geometry. of. A. :. but. CA. is. .. greater than. 1. therefore every point in the periphery of the (except A only) falls within the circle C : which was to be demonflrated.. 5* 3*. :. circle. D. COROLLARY Hence,. if. the centers of. two. T.. circles. be placed at. a diftance from each other, greater than the difference of the two femi- diameters, a part, at leaft,. of the one. will fall. diftance be. lefs. without the other-, but,. if. the. than that difference, the leffer circle wholly in the greater, but without touching it. will then be contained. COROLLARY. II.. Hence, and from the 'precedent, it likewife appears, that if two circles touch, either inwardly or outwardly, a right-line, drawn through their two centers, will alfo pafs through the point of contadt. k. becaule they can only touch, when the diftance of their centers is equal to the fum, or to the differCor. of ence of their femi-diameters k . 7. and Cor. ,.. i. THEOREM. IX.. If the diftance of the centers (F, G) of two circles (DL, be lefs than the fum, and greater than. MH). the difference of the two femi diameters thofe circles will cut each other.. (FL,. GM),. For, fince the diftance of the. two. centers. is. fuppofed than the fum of the femi-diamelefs. D. ters,apart of the i. m. Cor. to or *. Def. circle. MH,falls within the. othe,r. DL. 1 •,. and. fince. that diftance. 7- 3-. 0. t. ^ *. one. u. of 3 .. is greater than the difference of thofe femi-diameters, a part of the fame circle alfo n falls withoucthe circle which was to be proved .. MH. DL. THE O-.

(63) ;. Book the 'Third.. THEOREM angle. fj'he. to the angle. (BDC) (BAC). X.. at the center of a circle , at the circumference,. angles Jland upon the fame arch. In. 49. is. double. when both. (BC).. Let the diameter. ADE. the firfi cafe (where. AB. be drawn.. through the. pafies. + C° — 2A In the fecond cafe BDE r: 2BAE, (by cafe 1.) to which adding CDE — 2CAE, we have BDC = 2BAC In the third cafe CDE n 2CAE {by cafe 1.) from whence fubtra&ing BDE = 2BAE, there remains BDC — 2BAC center). BDC rzA. 0. P. 9. r.. .. * 12. t,. ,. q.. s. Ax.. r. Ax.. 4.'. ,. r. .. THEOREM All angles. ('EABF) Case er than. center. I.. (EAF, EBF). in. 5.. XL the. fame fegment. of a circle , are equal to each other. If the fegment be great-. a Jemi- circle. C draw CE. ;. and. 4. __. -g. from the. CF. then / and EBF being each of / f them to half EC'F % they muft neceflarily be equal to each other, gfp-. EAF. —. E. \\. /. \ —. j. *. yE. 1. Case. 10. 3..

(64) Elements. 5°. Case. II.. Geometry.. of. If the fegment he. H. than a femi- circle ; let be the interie&ion of EB and then the triangles AEEI and BFH, having the angle lejs. AF. AHfLzzBHF — BFH (by cafe. 1. *3.1. *. Cor>. **. to 10. I.. E. :. alfo. have. AEH. and. ,. they. 1.). EAH = FBH. will u .. THEOREM Angles. (D,. G). in. the. upon equal fubtenfes (AB,. XII.. circumferences , /landing. EF). of circles. having equal. And. the fubtenfes. diameters, are equal to each other.. of equal angles, in the circumference of. circles. having. equal diameters , are aljo equal.. From. QF w Hyp. x £>ef.33.. ot. .. f. Hyp.. EQ =1 FQ. *. Q,. PA, PB, QE„. let. “. Since AB EF w , and w therefore is P ;. AP = BP x —. =Q. r ,. and confe-. = Q) z=G. 2. Hyp. Becaufe D 1.G, therefore P — Q whence, PA being — QE, and PB n QF w AB V will alfo be = EF. quently. ' •. z /£’. 1.. the centers P, and. be drawn.. D (=. 4.. P. 2. 4-. z. ;. ,. ‘.. 1. COROLLARY. Hence angles in the circumference, {landing upon equal chords of the fame circle, are equal.. THEO*.

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