.Art. il. .Dp fltq UnciS reUx -qd Circulareni; 5? de nicvfiira �;nbsp;nbsp;nbsp;nbsp;' angulorwn, quorum vertex non ejl in circuCi cen

vnpafatione Triangalorum. Art. li. Di cxteris Polygon is.

1. De Polygorus in Geneve.

:�� 2. De Polygon is Symmctricis.

INDEX.

Caput I. De Genesi Solidorum; nbsp;nbsp;nbsp;Angulis solidis,nbsp;nbsp;nbsp;nbsp;at-

INDEX GEOMETRIE PRACTICA

EORUMQUE USU.

II. nbsp;nbsp;nbsp;DE LONGJMETRIA.

III. nbsp;nbsp;nbsp;DE PLANIMETRIA.

De Agrorum Geod^efia.

De divijione amp; redud�one figurarum.

De divifione 5? redu�lione Triangulorum.

n� nbsp;nbsp;nbsp;---- .....- nbsp;nbsp;nbsp;____/-i.. .

De divifione atque reductione QuaBrilaterdm (5fc. 209

IV. SOLIDOMETRIA SINE STEREOME-TRIA. nbsp;nbsp;nbsp;214

Caput I. De Solidorum Constructione atque Dimen-

SlONE.

De Solidorum ConfiruSione.

De Solidorum Geodtejia.

De STEltEOMETRIA DOUORUM.

ETM^

^ ^ tenfionis fpeciem, longitudinem fcilicet , Tl ^ r? latitiidinem atque profunditatem, conft-derat.

2. PiinSiim a geometris confideratur veluti quanti� ^as, cujus dimcnfiones funt infinit� parvsc, amp; eui nulla extenfio finita tribui poteft, definiturque a Wolfio;nbsp;^uod quaquavcrfum ft ipfum terminat, feu quod no�nbsp;i^bet terminos alios a fe diftindbos.

CAPUT P R I M U M

3. Um piinB:iim a termino ad tcrminum moved concipitur, Z/mcfl defcribitur; quai Recta eft, fi.nbsp;in nullam partem dcilectat pundum in decurfu ; Cur-va, 11 continua fit deviatio, amp; Mixta ^ ii partim lineanbsp;delcripta redid atque curva conftec.

4. Quamlibet igitur lineam concipere licet ut fe-' riem continuam infinitorum pundorum; amp; quoniam duo qutccumque punda libi contigua lineam redamnbsp;conftituant oportet; quadibet linea , five reda, fivenbsp;curva, fpedari poteft compolita redis infinit� parvis.

5'. Tinea nullam habet latitudinem , neque profundi-tatem ; ejufque extrema, cum lint punda, extenfa di' ci nequeunt.

Ex line� red� genefi ( 3 ), axiomatis inftar, aflumera licet propofitLones lequentes :

9- Rcfla AB CT_4B.I.j?^. i.) brevior eft re�lis AC BC; Item brevior eft curva aKB.

10. nbsp;nbsp;nbsp;II. Lineas re�la� unica eft fpecies, curvarum ve*nbsp;ro infinita; polTunc dari.

11. nbsp;nbsp;nbsp;III. A pun�lo dato ad pun�lum datum unica recta duci poteft.

12. nbsp;nbsp;nbsp;IV. Bina pun�la fuiEciunt, ut linete re�tai pofitionbsp;determinetur; at pilunbus, quam duobus, opus eft, utnbsp;curvai fitus innotelcat.

�g. Demonstilatur : Si enim duo unius lineac puncta torent alteri lineac communia; quoniam hsccnbsp;pun�la akerius linesc fitum determinarent (12), debe-ret altera in priorem incidere; atqui hoe eft contranbsp;fuppofitum; ergo unicum tantum poffunt habere punctum commune. Q. e. d.

THEOREMA II.

tequaliter a piin�is A amp; B; 6� infuper aliud quod-dam rectix Gi punctum v. g. I, c�am cequalitcr dijiet ab A amp; B, quodlibet punctum luicce.

�4. Demonstr. Ex ftipp)ofito pun�la G amp; I non defle�lunt ab A plus quam a B, ncque a B plus qua�t

ab A ; atqui ut a G ad I reda ducatur , fic moren condpiendum eft pundum G, ut, ubicumque exiftat,nbsp;etiam non defledat ab A nee B (3)? ergo fingulumnbsp;re�tic GI pundum sequaliter ab A amp; B diftat. e. d.

15. nbsp;nbsp;nbsp;Duda AB, quoniam O eft pundum re�te GInbsp;^13) eft AO = OB.

16. nbsp;nbsp;nbsp;Si producatur reda GI Cfig-3')i bscc tranfibitnbsp;per omnia punda sequaliter ab A amp; B diffita; ft enimnbsp;foret aliquod pundum v. g. X extra linealn GI, datamnbsp;aut produdam, quod ab A amp; B acqualiter remotumnbsp;foret, atque adeo effet AX=XB; quoniam eft AL=LBnbsp;C 14), eflet AX=XL AL; quod fieri nequit,feu quodnbsp;implicantiam involvit (9).

17. nbsp;nbsp;nbsp;RedaBXC^^. 4) , una fui extremitate fixi innbsp;X, altera B circumducatur, donee rurfus ad B redeat.

18. nbsp;nbsp;nbsp;Totum fpatium, a linea BX percurfum, Circiilusnbsp;dicitur i amp; pundum X, ejufdeni Centrum audit.

19. nbsp;nbsp;nbsp;Curva BCLFB, ab extremo pundo B defcripta,nbsp;Circumferentia item Peripheria eirculi appellatur.

20. nbsp;nbsp;nbsp;Quoniam pundum B, omni inftanti, d quoli-bet pundo pcripheriae, ad vicinum progrediens, cogatur

^efleclere, tria peripherise pun�ta in eadeni linea re�la reperiri nequeunt; atque adeo ab eodem pundlo ad ean-dem re�lam non poffunt duci tres lineoc recttc acquales.

lt;its vocatur, amp; recta BK, terminata utrimque in peri-phenz, Chorda v�iSubtenJa audit; hecque fubtendit, amp;n fujientat , arcum BZK, item arcum BCAGK.

22. nbsp;nbsp;nbsp;Pars circuli, V. g. KZBK,coinprehenfa inter chor-damBK amp; arcum ^nhten^um, Segmentum circuli dicitur.

23. nbsp;nbsp;nbsp;Chorda per centrum tranfiens, fpeciali nominenbsp;Diameter appellatur : talis eft chorda AB, item CG, finbsp;X fit circuli centrum.

24. nbsp;nbsp;nbsp;Kefte omnes, a centro ad aliquod peripheri�nbsp;pun�tum, duCtx., Radii vocantur. Tales lunt BX, XA,nbsp;XG amp; XC, pofito qu�d in X fit centrum.

25. nbsp;nbsp;nbsp;Omnes radii ejufdem circuli, vel equalium cir-culorum, funt equales : omnia enim peripherie ejufdem circuli, aut equalium circulorum punfta, a centro diftant radio iftius circuli.

27. - SeSor circuli eft pars circuli , duobus radiiSnbsp;AX amp; GX (^fig. g), atque arcu GS, comprehenfa.

fi8. ScHOLiON. In charta circulus defcribitur circino; majoribus autem circulis defcribendis, filum , funiculus auc pertica, ad-hibetur ; una enim eorum extremitate, ftylo fixa, altera cir-cumducitur ( filum atque f�niculum apt� tendendo}, amp; cufpide,nbsp;aeca amp;c. peripheria delineacur.

29. nbsp;nbsp;nbsp;Si chordsc AB amp; GC Cfig-S^ dianietri, punctum X eft centrum.

Demonst. Ut chorda AB, item GC fit diameter, quaclibet per centrum tranleat oportct C 23 ); igiturnbsp;pun�tum illud, quod centrum dicitur, quodque unicum eft, reperitur in finguia cx didlis chordis; atquinbsp;duac rcdae, fiefe interfecantes, unicum poffunt haberenbsp;pundum commune ( 13 }; ergo pun�ham X eft centrum. Q^. e. d.

Diameter peripheriam dividit in diias partes crguales, Et � converfb; fi qua chorda peripheriam dividatnbsp;in duas partes cequaks, illa diameter efl.

30. nbsp;nbsp;nbsp;Dico 1� fi chorda AB Qfig. 6) diameter fit,nbsp;effe arcum AKB = arcui BCA.

Demonst. Fmge peripheriac partem BCA plicari ver-fus alteram BKA; quoniam finguia puncla are�s BCA �cqualiter ab X, centro circuli, diftant, ac quailibetnbsp;are�s BKA pun�ta, centrumque immotum confiftat;nbsp;nullum prioris are�s pundum dari poterit, quin coin-cidat alicui pundo arcus BKA; c�m igitur didorumnbsp;arcuum extrema '^xa maneant in A amp; B � arcus illosnbsp;coincidere , ergo ' amp; jcquales effe, oportet. Quod eftnbsp;primum.

Dico S'*�, amp; � converfo : fi. arcus BKA fit = arcui BCa, chordam AB effe diametrum.

Demonst. Si chorda AB (fig- 6) non foret diameter, centrum extra eam dari, neceffe eflet; pone igitur eentrum alibi,v.g. in O; duda BZ erit diameter C23);nbsp;ergo arcus BCZ eflet = arcui BKAZ C30), quod eftnbsp;eontra luppofitum. Ergo AB diameter eft; quod eftnbsp;ifterum.

31. nbsp;nbsp;nbsp;Dum dus diametri AB amp; CG (fig- 5) fiefe in-terl�cant, eft arcus BC = GA, amp; arcus AC = arcuinbsp;GKZB : nam eft arcus GA4-AC=x\C-l-BC (30), ergo arcus GA = BC. Similiter eft arcus AG AC =3nbsp;AG 4- GZB; ergo arcus AC = GZB.

32. nbsp;nbsp;nbsp;ScHOi.iON. Placuit geomettis peripheriam cujuflibet circuitnbsp;in 360 parces �qnales ( qu� gradus dicuntur ) partiri ; quianbsp;numerus 36a, aecurac� per plures aliss, �c per 2, 3, 4. 5,nbsp;6, 8, 9, 10, 12, ]8 amp;c. divifibilis eft. Quivis gradus in 60nbsp;minores parces sequales, quae minuta prima vocancur; amp; quodlibet minutum primum in 60 minuta J�cunda amp;c. fubdividitur.nbsp;Gradus dcbgnantur per (�), minuta prima per (')? minutanbsp;fecunda per(quot;) amp;c, E. G. 5�, 28', 19quot;, ifquot;', dcnotac 5 gradus, 28 minuta prima, 19 l�cunda , amp; 17 tertia.

33. Si chorda AB Cfig- 7) fit diameter, dico eam tfajorem eCTe chorda Ch',aut quiicumque alia non dia-oietro.

Demonst. Ad X centrum, radios CX amp; FX duc-tos imaginare ; eft CF minor CX FX ( 9 ) ; atqui CX-(-FX=AB ( 26); ergo diameter eft omnium chor-lt;^�itum. maxima. e. d.

Demonst. Si non foret diameter, centrum extra earn elTet C^3)jnbsp;nbsp;nbsp;nbsp;v. g. effe in X; due ra

dios CX, item FX; fequeretur CF effe = CX FX C 26); atqui hoc implicantiam involvit ( 9 ); ergo recta CF per centrum tranfit, feu eft diameter. Q^. e. d.

. Jo : dum duo arcus Jiint cequales, chorda qu�que, a quibus Jufientantur ^ cequales funt.

� 35. Dico 1�, fi chorda AB (^fig. 8) fit scqualis chordae FC, arcum AOB effe = arcui FXC.

Demonst. Concipe chordam FC cum arcu FXC tranf-ferri, atque fic difponi,ut pun�tum F in A,amp; C in B collocetur; nullum puneftum arcfis FXC excedere po-teft arcum AOB (illud ipfum enim non diflaret aequa-liter a centro, ac fingulum pundum are�s AOB); ergo arcus FXC coincidit cum arcu AOB, amp; proinde einbsp;eft acqualis; quod eft primum.

Demonst. � pundo H, medio arcus BC, diametrum HI dudam imaginare; deinde arcum HCXFI plicari;nbsp;coincidet hie cum arcu HBOAICso)^ amp; quoniam arcus HC = HB, pundum C cadet in B; ergo cumnbsp;eXF = BOA, hi duo arcus qu�que coincident; ergonbsp;pundum F in A cadit, unde amp; chordae AB amp; FCnbsp;funt = (ii); quod eft alterum.

36. nbsp;nbsp;nbsp;Chorda; majores fubtendunt arcus majores; amp;�nbsp;� converlo.

37. nbsp;nbsp;nbsp;ScHOLroN. Ne tarnen exiftimes, e^ein ratioije chord.as atquenbsp;�fcus fubtenfos crefccre.

I)um diKZ chorda aquaks JcJe interj�cant, aliqui duo arcus oppofiti funt cequak-s; amp; � converjb.

38. Dico 1�, li chorda AB (fig. 9) fit = CF, eftnbsp;arcus AC = FB, vel areas AF = BC.

Demonst. Nam C35) arcus AFB =;= arcui FAC, BCA = CBF; ergo ar^us AC = FB. Vel arcusnbsp;ACB == FAC, aut AFB = FBC; ergo arcus AF=BC;nbsp;quod eft primum.

Demonst. Erit arcus CA AF='BF FA; vel arcus AF -b FB � CB BF ; ergo femper chordae AB amp; CF fuhenrant arcus tcquales; ergo a;quales funt (35^Jnbsp;quod eft fccundum.

^ puncto dato X (fig. 10), ad reSam AB, duas rc�as ducerc^ quarum fingula re3a GHnbsp;aqualis jit.

B. Tum, un� pede circini fixo in X, (leu ex X tam-quam cencro) defcribe arcum OZ, qui redam AJS

fecet in pundis O amp; Z C fupponitur enim GH lat longa); erunt redsc XO amp; XZ scquales (25), amp;,nbsp;ex conftrudione, fingula acquatur recital GH.

40. Resolutio. Ab extremis pundis A amp; B, ut een� tris, eadem circini apertura, (pro libitu, l�d majo-ri quam eft medietas rec�lcc AB3 ducantur arcus, fe-fe interfecantes in G amp; I; ducSa GI, re�tam ABnbsp;bifariam fecat.

Demonst. Ducendai AG, EG, AI amp; BI funt radii acqualium dreulorum; ergo amp; iliac re�tac xquales funcnbsp;(^25); unde pun�ta G amp; I acqualiter ab A amp; B remora func C 8) ; ergo du'da reda GI, eft AO = OBnbsp;Ci.5gt; 0,-d-

41. ScHOUON. Eyem methodo reftam duxeris, cuju� fingula punfta aequalicer ab A amp; B diftant.

42. nbsp;nbsp;nbsp;Resolutio. Dudam chordam AB Bifariam l�cc.snbsp;per re�lam GI C40), hacc quoque arcum datum innbsp;duas partes aiquales 'partitur.

Pemonst. Quoniam pundum L acqualiter ab A at-que B diftat (41), diordrc AL amp; LB ducendac ccqua� les funt; ergo arcus AL = arcui BL (35); ergo arcusnbsp;ALB divifus eft bifariam. Q. c. d.

�f- Chordam AB bifariam feces C40) per reflam GI, quam, ad alterum ufque peripherioc pun�tum H,nbsp;protrahas.

Ki Chordam LH dein bifariam feces ; eritque punctum ejus medium, v. g. X, circuli centrum.

Demokst. Quoniam ex conftru�fione pun�la G amp; I ^qualiter a puncfis A amp; B diftant, etiam pun�tum Lnbsp;�iitat a;qualiter ab A amp; B, item pun�tum H ajqualiternbsp;ab A amp; B ('14); eft ergo chorda AL=LB, amp; chorda AH = HB; ergo arcus AL=LB, amp; arcus AH=HBnbsp;C35); itaque arcus HA AL = arcui HB4-BL; ergonbsp;chorda LH dividit peripheriam in duas partes acquales,nbsp;amp; confequenter eft diameter (3o);unde pun�tum ejusnbsp;tiiedium eft circuii centrum. t. d.

�Per tria piinSa ABC (fig. 13), (non in eddem reSi conflituta (20) circuli peripheriam diicere.

44- Resol. I. Re�tas AB amp; BC du�las bifariam feces ( 40), per re�tas GI amp; FL.

11. Ex pun�lo X, in quo fefe re�tx GI amp;e FL fecant, ad intcrvallum XA, ducito circulum; dico quodnbsp;ejufdem peripheria tranfeat quoque per pun�ta BC

Demonst. Ex conftru�tione pun�tum X diftat requa-liter ab A ac a B, amp; a B rcqualiter acaCfid); t�rgo diftat acqualiter ab omnibus tribus; atqui finguianbsp;pun�ta peripherise circuli defcripti ex X ut centro, adnbsp;intcrvallum XA, diftant ab X acqualiter, amp; quidemnbsp;pro re�ia XA; ergo peripheria illa per pun�ta BC tran-e. d.

45- ScHOLioK. Pari procedendi metbodo circulus perficitur, cu-jus folum arcus delineatus exiftit : aflumancur enim in eo tria pundta, pro libitu; aci^ue per htec circumferenda thicatur.

46. Ex diSis evidens eft, quod, dum tres recftenbsp;AO, BO amp; CO (fig. 14) a penpheria ad idem eom-mune pundium 0, intra circulum, concurrentes, acqua-les funt,'illud pundlum fit centrum : cliordas enim ABnbsp;amp; BC, bitariam divifas concipe per redtas EG amp; FH;nbsp;tranfeunt hsc per pundtum 0 (iiT); acqui pundtumnbsp;interfedlionis earum eft centrum (44);; ergo amp;c.

47. nbsp;nbsp;nbsp;Sint du3C redsc (fig. 15) AB major, GI minor, tales, ut extrema cujufque ajqualiter ab extremisnbsp;alterius diftent (atque adeo (1,5) fefe mutuo bifariam fe-cent); fic GX, item GZ = AO; eft XO = OZ (15)nbsp;lt;ergo AX=:ZB) eftque XG GZ=:AB. Finge XGZnbsp;ftlum tenuiffimum, fixilque extremis pundlis in X amp;nbsp;Z, ftylum L Cquo conftanter filum XGZ tenditur)nbsp;eircumduci per pundla AIB, donee ad G redux fueric,

48. nbsp;nbsp;nbsp;Spatium , lineis XGZ percurfum, EUipfis di-citur; curva a ftylo L percurfa, feu delineata, Peri-pheria ElUpfios, amp; fimpliciter, ElUpfis audit.

51. Linea, qusc eft tertia proportionalis ad majo-rein amp; minorem axin. Parameter majofis axis dicituf;nbsp;^ tertia proportionalis ad minorem amp; majorem axin �nbsp;Parameter minoris axis.

52. Csetersc recla; omnes.; per centrum ellipfcos duc-^nbsp;amp; utrimque in peripheria ellipfis terminatre. Dia-netri ellipfis vocaotur.

53. Evidens eft fmgula peripheriae ellipfeos puncla,nbsp;g. K amp; L amp;c. a focis X amp; Z, limul fumptis, di-ftare rota longitudine majoris axis AB : ubivis cfiimnbsp;ftylus fuerit, KX KZ = AB.nbsp;nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;,

.S4. Suppone circulum ACFG ():g. 16) in dirc�iio-nem harum litterarum, in reda AB , rotando, movo-fi ufquc in B, dum integram circumvolurionem ab-folverit, ita ut pun�ium A, atung�ns re�tam AB in A, rurfus rectam AB attingat in B.

55. Pundum A curvam defcripferit ALB, qutc Cy-clo�s dicitur j eritque AB aiqualis peripheriai circuii ACFG.

56. Curvam Cyclo�dalem , vel limpliciter Cycloiiem, �efcribunt lingula pun�a peripheriac rots curr�s, innbsp;plano protradi.

III.

57. Finge redam AB (^fig- 17) ? una fui extremi-tate fixam in A, altera B circumvolvi, donee ad B

redeat; atqu� interim pun�tum aliquod , ab A ver-fus B (re�tam AB femper concomitans, atque eidem inflftens) scquabili motu, ica ferri verfus redt� ABnbsp;extremicatem, ut hanc attingat, dum extremitas B pe-riphcriam BGEFB abfolverit.

58. Curva A369B, ab ifto pundto, mobili in linea AB, defcripta, prima Spiralis, feu Hdice audit.

59. Seciinda ^iralis exurgeret fccunda rcvolutione,fi re�la AZ foret duplo longior AB, atque punctum mobile, scqualitcr pcrgeret a centro rcccdcre; defcribcretnbsp;enim fecundam fpiralem BRSV�^Z.

60. ScHOLiON I. Ut fpiralem delinecs, femi-perepheriam BCE divide ( pro libitu ) v. g. in 6 partes squales ; ab hil�ce punc-tis, per centrum A, totidem age diametros, peripheriamquenbsp;diviferis in la partes acquales (31) : divide quoque reftam ABnbsp;in Ia panes sequales; ab A , in linea AG, ponc unicam divi-fionis AB partem �, ab A , in linea AI, pone duas tales, amp;nbsp;ica confequencer. Ab A incipiendo , punfta notata in radiisnbsp;AG, AI amp;c., convenicntc curva connefte , amp; primam fpira-lem delincaveris. Si rebtam AB produxeris, ita ut BZ=AB,nbsp;atque , circumvolvendo AZ , defcripferis peripheriam , eujusnbsp;AZ fit radius, producito diametros ul�que ad peripheriam cirtulinbsp;mmoris concencrici; pone unicam partem re�la; AB in lineanbsp;GL, V. g. a G in O; duas ab I in K amp;c. ,cum a B curvam du-,nbsp;cico per illa pun�ta, donee ad Z pextiseric; cricuue BRSVZnbsp;lecunda fpiralis.nbsp;nbsp;nbsp;nbsp;e b �nbsp;nbsp;nbsp;nbsp;1

6r. ScHOLiON II. Secunda , atque fequentes fpirales, delincarf poffunt abfque radiorum prolongatione ; Icilicec apertus circi-J1US pro AB, uno pedc pofito in A amp; altero in B, fic movea-tut, ut pes untis conllanter primae fpirali infiftat , alter vetonbsp;femper dire�l� centrum refpiciac : etenim dum unus pes primamnbsp;fpiralem percurrerit, aker fecundam delineaverit; amp; profequen-lt;30, dum prior pes fecundam percurrerit, altg: tertiam nota-verit.

62. nbsp;nbsp;nbsp;Cum fims duarum reclarum inter ie fpe�latur,nbsp;turn neceffiirio vei versus fe mutuo inclinant; vel nulla inclinatio feu tendentia unius ad alteram habetur�nbsp;omniaque punfla unius acquali ubique intervallo ab altera diltanc,atque quantutnvi,? produ�tcc nunquam con-currunt, hocque cafu Parallels, vocantur.

63. nbsp;nbsp;nbsp;Dum reda� fe mutuo versus tcndunt, feu incli-nantur, mutua earum inclinatio Aiiguliis dicitur, ejulrnbsp;que Fquot;mex ell: in pun�to concurs�s rc�larunj.

64. nbsp;nbsp;nbsp;Si refta� ad aliam redtam inclinatto (produdlamnbsp;fi opus) utrimque lit scqualis, ea Perpendicular is adnbsp;aliam dicitur : ut AB (Tab. II. fig. i ) ad CF perpen-dicularis eft, fi AB non magis inclinet verfus lineamnbsp;BP, quam verfus lineam BC.

65. nbsp;nbsp;nbsp;Si vero major fit inclinatio in unam quamnbsp;aliam partem.redlas,versus quam tendit,ut AB Cfig.a)

66. ScHOLiON. Contendunt nonnulli, turn fol�m de angulis trac� tanduro, ubi de lineis eft aftum, hoe dufti modvo, angulos,nbsp;lineis effe magls compofuos. Admictimus quidem unicam lineam,nbsp;foUtari� fpeftacam, angulo magls ciTe fimplicem ( hic enimnbsp;ncceffari� ex inclinadone duarum reftarum enafcitur) ; at plu-rium reftarum combinado, aut pofido unius ad alteram colla-ta, profeft� nihil minus habet compofid, quam ipliufmet an-guU confiderado; im�, quoniam primus linearura inclinatarumnbsp;efFeftus clt angulus, ifque primum mend illas confidcrand f�-fe offert, ideo ab angulis poti�s ordiendum duximus.

De Angulis.

67. nbsp;nbsp;nbsp;Lineac, quarum inclinatione, aut concurfu angulus formatur, Latera rel Crura anguli dicuntur.

68. nbsp;nbsp;nbsp;Omnis reda v. g. AB {fig. 2) altferi, non innbsp;extremo pun�to occurrens, cum ea elEcit duos angu^nbsp;los X amp; O, qui Vidni appellantur.

69. nbsp;nbsp;nbsp;Duin duac ret�la; AB amp; FC Qfig. 3 ) fefe ititer-fecant, quatuor efSciuntur anguli, quorum quilibecnbsp;duo oppoliti, V. g. O amp; X, item G amp; Z, ad Verticem.nbsp;oppojiti audiunt.

70- Scholion. Angulus, vel unici nota deflgnatur, vel tribus, quarum media anguli verticem denotat; prima cum fecunda,nbsp;unum latus, amp; fecunda cum terda alterum latus; v. g. angulusnbsp;formatus per latera AB amp; CB (Jig. a), poteft vocari angulusnbsp;X, angulus ABC, item angulus CBA.

Hypothesis I.

71. In re�ta AG 4) immota, aliam BX fuper-po�tam, uno fui extremo in X fixo, circumvolvi ima-ginare, donee pun�lo B ad G pertingence, rurfus ean-dem redlam conftituaiit : qua proportione re�ta BX ab AX removetur, efl proportione crefcic angulus ad X.

C0R.0L-

72. nbsp;nbsp;nbsp;Quoniam tot pun�la pcrcurrerit re�ta, ad CXnbsp;pertingcns, quot funt pun'fla periphcriac in area BC ,nbsp;ideo magnitudo anguli BXC proportionatur numert�nbsp;graduum are�s BC, feu menfuratur area BC.

73. nbsp;nbsp;nbsp;Cum ex vertice cajullibet anguli, �t centro, cir-culus, aut arcus defcribi pollic, qui anguli crura feunbsp;latera, fprodudta, ft opus_) interfecec; quilibet angu-lus tot graduum eft, quot graduum eft arcus circuii,nbsp;defcripti ex ejus vertice,inter crura anguli comprehenfus.

74. nbsp;nbsp;nbsp;ScHOLioN. Anguli magnitudo defumi nequic a laterum lon-gitudinc , fed ab eorumdem divergencia : lie idem manec anguli GXL valor, five latera XG amp; XL longiiis producanturjnbsp;eritque angulo GXF minor, lic�c latus XF latere XL minusnbsp;fueric.

75. nbsp;nbsp;nbsp;Duo igitur anguli aiquales funt, c�m corumnbsp;refpe�div� latera sequaliter a fe mutuo divergunt, feu.nbsp;incUnant.

76. nbsp;nbsp;nbsp;C�tn ad punctum E linea mobilis pertingit,nbsp;ubi non inagis in B quant in G inclinat, linea EXnbsp;Perpendicularis eft ad lineam AG. Angulus EXA eftnbsp;jcqualis angulo EXG fuo vicino, amp; quifque ReBus-audit; quoniam vero prior angulus menfuratur arcunbsp;ACE, amp; alter arcu GEE 072.), duo illi arcus ftmtnbsp;a^quales; cum vero arcus AEG �c 180� (30^� ar-c�s ACE, item GFE eil 90�.

_ 78. Ad idem re�tae pundtum , unica perpendiculars dari poteft ; nam omnis altera redta, non incidcus

29- Anguli omnes CXA, IXA, func recio minores, amp; Acuti audiunt; angaii FXA, LXA funt redo ma-jorcs, amp; Obtufi compellantur.

Demonst. Ex o, tit centro, due mediam periphe-riam CGF; O menfuratur arcu CG, amp; Z arcu GF C72); acqi� duo illi arcus = 180� (30); ergo anguli � amp; Z � 180�. e. d.

81. nbsp;nbsp;nbsp;Quiiibet igitur angulus minor eft 180� : quif-que enim angulus vicinum habet, vel haberet, pro-dudo uno ejus 'latere ; atqui cum illo vicino folumnbsp;= 180�; ergo amp;c.

82. nbsp;nbsp;nbsp;Dum Cfig. 3) AO ad CF perpendicularis eft,nbsp;etiam CO, item F� ad AB perpendicularis eft ; eftnbsp;enim O redus (76), feu po� (77); ergo etiam Z,nbsp;X, item G eft 90� ( 80).

quotquot luot, � 180�; fc Qjig. 6) anguli A, B, O amp; G =nbsp;nbsp;nbsp;nbsp;: etenim fi, ex eorum vertice

communi X, media penpheria CKF dcfcribatur, quil-que angulus menfuratur arcu intercepto inter fua latera (73); ergo omnes fimul menfurantur media peri-�pheria; arqui^hatc = 180� (3o)i ergo amp; illi anguli 'fimul =' 180�.

84. nbsp;nbsp;nbsp;Omnes anguli ad idem pun�lum, tot quot con-ftitui poffiint (fig. 7 amp; 8), �mul faciunt 360� : duc-ta enim, ex eorum vertice communi, �t centro, ia-tegra pcripherid , omnes illi anguli fnnui fumpti men-fu'rantur tota circuli peripheria (72); alcj^ui illa eftnbsp;360� C32); ergo amp;c.

Demonst. 0 G = i8o*; amp; X4-G=:j8o� (80); ergo eft 0=X. Pari ter X4G=i8o'�; amp; X Z=i8o�;nbsp;ergo eft Z = G. Igitur anguli ad verticera oppofitinbsp;funt =. Q. e. d.

86. Resol. I. Ex A, ut centro, intervallo arbitrario,nbsp;circino defcribc arcum ZG;

II. nbsp;nbsp;nbsp;Ex B, �t centro, eadem manente circini apertura,nbsp;defcribc arcum indefinitum RE ;

III. nbsp;nbsp;nbsp;Circino fume intervallum GZ, atque illud transfernbsp;ab R in arcum RL, v, g. ab R ufque in O;

Demonst. AG fic imponatur in CF, ut A in B, ^ G in R coincidant; arcus GZ incidet in arcum

C S

3^.0; Sc qnoniam, ex conftrudione, funt scqualcs, Z incidet in O; ergo AZ coincidet cum BO; ergo angu-Jus A coincidet cum angulo OBF, amp; confequenter einbsp;fft acqualis.' Q^. e. d.

PROBLEMA

JI. Eum divide bifariam per rc'f�am LC C 42); dico C efle divifum in duas panes acqualcs.

Demonst. Eft arcus AG = GB, ex conftrudlione C42); atqui angulus ACG menfuratur arcu AG, amp;nbsp;angulus BCG menfuratur arcu BG (72); ergo ill i duonbsp;anguli funt acquales : eft ergo C divifus in duas partes Bcquales. lt;2,- lt;-�� d-

5. II.

reSce AB difant aqiialiter d quibitjvis diiobus punSis recriE FC, v. g. Z amp; H, hinc indenbsp;aquditer d B difltis.

8�. Demonst. Ex B, tamquam centro, ad interval-lum BZ, mediam peripberiam ZXH dcicribe; quoniam ex fuppofito, eft BZ = BH (25), peripheria per punctum H tranfeat oportet; erit arcus ZX item XH yo�nbsp;(76); chorda ducenda ZX = chord� ducend� XHnbsp;Co.4) 9 proinde recfta AB eft talis,. ut duo eins punc-ta X amp; B diftent �qualiter a Z amp; H ; ergo fingulanbsp;pun�la rea� AB, diftant tcqualiter a 'z amp; H (14).nbsp;Quaecumque nunc pundla, hinc inde in re�la CF,

ttqualiter a B remota, affumere libet , codcra inodo dcmonftrabitur, lingula reftac AB pun�ta ab iis squa-liter effe diffita; ergo quaclibct punda re�lac AB dif-tant scqualiter a quibulvis punctis re�t� FC, hmc inde tcquaiiter a B diliids. Q^. e. d.

Si AB (fig. II) fit od CF perpendiciilarisy amp; X difitet tequaliter d Z amp; H, quodlibet punctum rtetez AB �nbsp;difitat ctqudittr d Z 6� H, tfitque rtaanbsp;HZ dlvifia bifiariam.

== BO; quoniatn CB perpend']cuiarilt;; ad AO^ iteralB ad AO C 82gt;, ell ZX = Z�, amp; XH = OH (88) ; ergo quodlibet punclum rebla; AO dillat aiquaiiter a Znbsp;amp; H C14); ergo ZB = BH. Q^. e. d.nbsp;nbsp;nbsp;nbsp;�-

Si rcSla AB ( fig. ii') Jit talis ^ lit duo ejiis piinUxt tequalitcr dijlcnt d dnobus piinclis recite CF,v.g~

90. nbsp;nbsp;nbsp;Demonst. Singulum pun�lum re�la: AB diftatnbsp;ajqualiter a Z amp; H ( 14); ergo eft ZB = BH; amp; dc-feripta media peripheric ZXH ex B, ut centro, chorda ZX = chorda; XH (14); ergo arcus ZX = arcuinbsp;XH (35); porro hi arcus = 180� C30); ergo erit arcus ZX 90�; jam verb angulus XBZ menfuratur arcunbsp;ZX(7a) ; ergo iilc angulus eft 90�, amp; arqualis fuo vi-cino; ergo AB eft perpendicularis ad CF (76). e. d.

_ 91- Ergo methodo tradita C40) recla A3 perpeq-^iculariter amp; bifariam fecatur.

Demonst. Si enim duci poffet alia , v. g. AX ; fit ZB=BH, erit AZ=AH (88); prscterea'fit ZX=XO,nbsp;deberet quoquc AZ effe = AO (88); atqui hoe im-plicat (20); ergo fola AB poteft efle perpendicularisnbsp;ad re�lam CF, per punctum A dudta, amp; cast�ra; om-nes obliquae funt. �. d.

perpendicularis eji brevijjima, quee � punBo dato ad liiicam. datam. duci potejt.,

93. nbsp;nbsp;nbsp;Si.AB.(j^g. 13) fit perpendicularis ad CF, dito earn eflb breviorem quacumque obliqua, vi g. AO,nbsp;ab A ad CF du�ta.

Demonst. Produc AB ufque in aliquod pundum O, ita ut GB = BA; quoniam OB ad AG perpendicularis, eft GO = AO ( 88 ) ; fi ergo AB maior eflet,nbsp;vel Ecqualis AO; amp; BG major, vel jequalis GO; to-ta AG effec major, vel = AO OG , quod impli-cat(9); ergo perpendicularis AB eft brevil��ma, qus�nbsp;ab A ad rectam FC duci poteft. Q^. e. d.

94' Ergo per lineam perpendicularcm diftantia puno ti dati a linea re�ta data inveftiganda eft. �

55. Si qua re�la ab A ad CF duda, v. g. AB, fi? ^rev�Ema, eft ipfa perpeudicularis ad CF.

96. nbsp;nbsp;nbsp;Inter obliquas, ab eodem pundlo .ad eandem reonbsp;tam du�las, ese longiores funt, magifque ad re�lam,nbsp;lt;iatam inclinata:, qua: a perpendiculari magis recedunt;nbsp;fic AH (�5--12) eft AO major, magifque, quam hxc,nbsp;inclinata ad CF J undc angulus H minor eft angulonbsp;AOBC74).

97. nbsp;nbsp;nbsp;Dum h pun�to A, reda: AZ amp; AH, dudac,nbsp;�quales funt; perpendicularis ab A ducenda ad CF ne-celfario cadit inter Z amp; H.

PROBLEMA I.

98. Resol. I. Hinc inde in redta CF fumantur duo�nbsp;pun�ta Z amp; H, sequaliter.a B diffita Cproducendonbsp;redam CF ft opus; quod necelfari� fieri debet, linbsp;pundum B fuerit in, vel circa redac CF extremum;);

I^emonst. Ducenda Zr= ducend� Hl (a5);amp;,ex conftrudione, eft ZB = HB; ergo IB eft ad CF per�,nbsp;pcndicularis Cpo}. Q^. c. d.

^9. Eadem itaque praxi ad pun6lum lincsc dat� angulum re�tuiU, feu 90�, formabis; quem bifar^mnbsp;dividcndo (87}, angulum 45� habebis.

loa Resol. I. Ex A, �t centro, due arcum ZH, qui , re�tam CF, (produilam fi opus) fecec in duobusnbsp;pun�is Z amp; H.

II. Ex Z amp; H, cadem circitii aperturd, ducito arcus fefc interfecantes iri K. Dico rc�lam AK cfle per-pendicularem ad CF.

! Demonst. ZA eft = ah, amp; KZ = KH (25); ergo AK cil perpcndicularis ad CJF (90). e. d.

105. nbsp;nbsp;nbsp;V. Et Angulos E amp; P� L amp; S uddjacentesnbsp;ternos appellamus.

106. nbsp;nbsp;nbsp;Dico 1�, fi AF item BG 17) fit paral-Jela ad Cl, efle AF ad BG qu�que parallelam.

DemonST. Si AF amp; BG non forent inter fe paral-Jelse, aiterutram partem versus produ�lac, in aliquo pun�to communi, concurrerent; atqui hoe fieri ne-�quit ; nam illud punctum concurs�s diftaret a tinea Cl sequaliter, ac fingula puncta lincaj AF, itemnbsp;' diftaret a tinea Cl xqualiter, ac fingula pun�ta tineasnbsp;BG; c�m AF item BG fit parallela ad Cl (62) ; ergo fingula pun�ta tinea; AF 'diftarent a;quatiter hnbsp;linea Cl, ficut fingula pun�ta tinea; BG; deberent ergo line� af amp; BG coincidere; quod non fuppo'niturjnbsp;ergo AF amp; BG funt inter fe parallel�. Quod eft pri-mum.

Dico 2�, fi AF item Cl fit parallela ad BG, efl@ AF amp; Cl inter fe etiam parailelas.

Demonst. Neque AF neque IC, quantumvis pro-du�t�, concurrent cum BG quantumvis produ�ta (62)� ergo BG, quantumvis producta, femper feparabit AF amp;nbsp;1C, quantumvis produ�tas; igitur AF amp; IC, quantumvis produ�t�, concurrere non poterunt i ergo parallel�nbsp;lunt. Quod eft fecundum.

107. nbsp;nbsp;nbsp;Dum re�la alterutri ex duabus parallelis eftnbsp;pa�-allela, eft etiam ad alteram parallela ; erunt eaftffinbsp;�lt;tu� parallel� eidem terti� (lo�).

�o8. Dico, fi AB amp; CF (fig. i6) fint parallela:, angulos correfpondentes, v. g. E amp; Z efle aiquaies.

Demonst. Finge CF, fibi conlianter parallelam, ita transferri versus G, ut pundtum Z delcribat rectamnbsp;�CE; dum ad E pervenerit, erit adhuc redta CF pa-rallela ad redtam. ABnbsp;nbsp;nbsp;nbsp;arque adeo nulium fit

n'iam igitur unum ejus punctum commune habet ia E, cum redla AB, incidet CF in rectam AB; ergonbsp;re�la GH candcm inclinationem' habet ad CF, exifinbsp;tentem in Z, quam haberet ad eandem, conftitucamnbsp;jn E, feu quam habet ad redtam AB; ergo eit E =nbsp;2 ClS^quot;) I = S, quia prioribus ad verticem refpe�tiv�nbsp;opponuntur. Et quoniam E K = Z P (8^)�

K = P, ergo L = R, quia praacedencium funt rcfpec-tiv� ad verticem oppofiti ; ergo anguii correfponden~ tes sequales funt. e. d.

CoR-OI-^arium III.

III. Si OX fit perpendicularis ad AB, eft ipftnbsp;etiam perpendicularis ad CF : nam 0 = X (lopj);nbsp;ergo, cum, ex fuppofito , fit O redtus ( 7� ), eftnbsp;etiam X re�tus; ergo Stc.

II4. ScHOUON I. OX brevior eft omni alil, a punfto O ad reiftam CF, aut a punfto X ad reiftam AB, ducibili (93 ); amp;nbsp;proinde perpendiculati paiallelarum diftancia eft inquirenda.

itq. ScHOLioN 11. Quoniam qualibet parallelarum punfta scqua-iitM ab alia diftant f 6a), onnnes pcrpendiculares inter eafdera parallelas ftint inter Ie sequales.

Si anguli conefpondcntcs y v. g. E amp; Z (fig. 16), aqiiales Jint, reSce AB amp; CF funt internbsp;Je parallels.

TI4. Demonst. Imaginare rnrs�s recftam CF conftan-ter fibi parallelam moveri, donee pun�lum Z attin-gat pun�lum E; quoniam E = Z, redla ZF incidet in redam KB (75); ergo CF in AB cadet; c�m ergo CF fibi rnanl�rit parallela,eft AB parailela CF, exiF-tenti immotas in Z; ergo li. anguli correlpondentesnbsp;scquales fuerint, redtse AB amp; CF lunt inter l� paral-Jelaa. Q^, e. d.

Dim reamp;a GH (fig- 17)? tre^ (aut phires') parallelas AF, BG amp; Cl, cequaliter d fe miituo dljjitas,nbsp;interfecat, ejl LK = KO.

116. Demonst. Ab O item K dimitte perpendicu-�ares OZ amp; KS; erit OX etiam perpendicularis ad BG,

P a

Elementa Geometric. cum X = Z C ); attenta parallelarum diftanttSnbsp;scquali, eft KS = XO; pr?etcrea, cum S = Z (luntnbsp;enim ambo re�ti), linea SK eft parallela Z[ (114};nbsp;eft ergo anguius LKS = I ( 108); fuppone nunc li-neas SK, LK amp; LS (luu quern ad fe mutuo habenc)nbsp;transicrri ita, ut S in X, amp; K in 0 coincidat; quo-'nbsp;niam anguius LK5 = I, amp; S=:X, incidet re6ta,KLnbsp;in OK, amp; SL in re�lam XB; ergo redte KL amp; SLnbsp;fefe quoque interfecabunt, feu concurrent in pundto K;nbsp;coincidic ergo praccis� re�'ta KL cum recta OK; proin-de Equales font. Hand difticulcer nunc idem fierinbsp;perfpicies, tot quot lubet datis parallelis ccqualitcr afenbsp;�iiuiuo diffitis.

5/ TcUc2 AF (^fig. 17) BG amp; Cl [mt inter fe parallela^ amp; fit KL �OK, ilia parallel^' aqualiternbsp;d fe mutuo difiant.

�I7. Demonst. Du�lis perpendicularibus OZ amp; KS, eft anguius LKS = I, amp;'anguius OKG= angulo KLInbsp;(108); imaginare igitur redtas KL, LS amp;.SK Cpi�outnbsp;font inter fe difpofitsc ) ita collocari, ut pundtum Lnbsp;in K, amp; pundturA K in I conliftat; �(atque adeonbsp;coincidat redia KL cum redta KO); anguius LKS innbsp;angulum I, amp; anguius KLS in angulum OKX, coin-cidet; (cum refpedliv� aequales fint, attento quod OXnbsp;dit parallela KS; amp; XK parallela SL); cadet ergo LSnbsp;in KX, amp; KS in IX; ergo S in X; eft ergo KS=OX;nbsp;atqui heC linesc diftanciam parallelarum metiunturnbsp;(112); ergo parallelae AF, BG amp; Cl diftant aequaliternbsp;B fe mutuo. c- fi-

lig. Resolutio I. Dac redam BL indefinit�, qu� cum reda AB efficiat angulum quemcumque ABL;

II. nbsp;nbsp;nbsp;In linea BL, incipiendo a B, circino fume, arbi-trario intervailo, partes acquales BZ, ZH amp; HS.

III. nbsp;nbsp;nbsp;Due redam AS, amp; ad hanc paral]elas HX amp; ZO;nbsp;dico redam AB efle divifam trifariam.

Demonst. Per pundum B due BK parallelam AS Cii8); eric hscc qu�que parallela cuique ex aliisnbsp;parallelis (lo?); quoniam BZ , ZH amp; HS, ex con-ftrudione, funt tequales, omnes ill� parallelac a fe mu-tu� acqualiter diitant Cii?)? ergo etiam partes line�nbsp;AB , intercept� inter iiias parallelas, aquales funcnbsp;(Il6gt;nbsp;nbsp;nbsp;nbsp;Q^-e.d.

I)e Jim linecz re3.cz ad Circularem; amp; de menfiira angularum, quorum vertex non efi innbsp;circuU Centro.

T2o. Tangens circuli eft, v. g. reda, AB (^Jig. 3 ) , qu� ita peripheriam attingit, ut eandem non interie-t

121. nbsp;nbsp;nbsp;Secans dicitur redla CGF, qua; in pundlo G,nbsp;Ubi peripherias occurrit (produdta fi opus)j eandcin pe-xipheriara fecat, amp; aream circuli ingrcditur. CG dicii*nbsp;tur Secans Exterior; FG Secans Biterior.

122. nbsp;nbsp;nbsp;Si X Cfig- 4 ) centrum , amp; KL tangens in K,nbsp;dico XK effe perpendicularem ad KL.

Demonst. KL fic atdngit peripheriam, ut earn nul-libi fecet ( 120); ergo XK minima eft, qua; ab X' ad KL dud poteft : (omnes enim alia;, v. g. XZ, pe-ripberia egrediuntur; ergo funt radio XK majores)nbsp;ergo XK eft: perpendicularis ad KL ( 95}. Q^. e. d.

123. nbsp;nbsp;nbsp;Ad idem peripheria; pundum , unica tantumnbsp;tangens duci poteft : nam du�to radio ad illud �punctum, illc eft ad tangentem perpendicularis;;^^ 122 );nbsp;atqui ad idem recta; punctum unica poteft�bluci pcr-pendicuiaris (92); ergo amp;c.

124. nbsp;nbsp;nbsp;Demonst. XK minor eft omni ali^ ab X ad KLnbsp;ducibili C93); atqui XK eft radius; ergo omnes alia?,nbsp;sb X ad lOL ducendas, circulo egrediuntur, c�m ftnt

J�^ajores XK; ergo nullum pun�tum re�l� KL circuli aream ingreditur : illud enim minus diftaret d centro,nbsp;quam pun�lum K; ergo redta KL, peripheriac occur-rens in K, quantumvis produ�ta, aream circuli nonnbsp;ingreditur; ergo eft Tangens (120). Q^. e. d.

125. Demonst. Si KH non tranfiret per centrum, radius du�tus a centro ad pundum K, eflet perpendicularis ad CL ( 122); atqui ad idem rec-lai pundum unica poteft duci perpendicularis ( 78 ) ;nbsp;^rgo KH coincidit in radium dudum, amp; confequenternbsp;etiam per centrum tranfit; live eft diameter (23 )

Si Diameter fecet Chordam perpendiculariter ; vel Ckor-dam non Diametrum bifariam; vel iiniim Chordce arcum bifariam; pruediSa omnia fequuntur. Et Ji qua Chorda fecet alteram Chordam perpendiculariter amp; bifariam;nbsp;vel perpendiculariter ^ item uniim ejus arcum bifariam;nbsp;vel Chordam item unum ejus arcum bifariam; Chordanbsp;fecans efl Diameter, atque reliqua omnia fiunt.

12�. Dico 1�, li KH CfiS- s') diameter fit, amp; O redus; eft AO = OB; arcus AR = KB, amp; confequenter arcus AH = HB.

Demonst. Du�li radii AX amp; XB funt �= Ct/^S ) � �ucenda AK = KB; ducenda BH = HA; AO = OBnbsp;C86); arcus AK = BK, amp; arcus AH = BH ( 35).nbsp;Quod eft primurn.

Dico 2�, ft KH Diameter fecet chordam AB non Diametrum bifariam; efle O Redum, amp; arcum AKnbsp;5= KB amp;C.

Demonst. Ex Centro X due radios AX amp; XB, hi funt scquales (zg); ergo duo punaa reaae KH, fei-licet O amp; X, diftant aiqualiter ab A amp; B; ergo XOnbsp;perpendicularis eft ad AB (po); chorda AK=:KB, amp;nbsp;chorda AH = HB C88); ergo arcus AK = KB, arcus AH = HB (35). Quod eft fecundum.

Cum igitur Diameter Chordam quampiam fecat bi-fariam; vel hacc Chorda eft quoque Diameter, vel pcrpendiculariter fecatur amp;c.

Dico 3�, ft KH fit diameter, amp; arcus AK = KB; eft - O reaus, amp; AO = OB amp;cc.

AX = BX (25) : ergo fingula pundta lineac KH difi tant xqualiter abAamp;B(i4);amp; XO pcrpendicula-ris ad ABnbsp;nbsp;nbsp;nbsp;= OB (89) amp;c. Quod eft

Demonst. Ducenda chorda AK = KB C35); ergo punda O amp; K diftanc xqualiter ab A amp; B j ergo Onbsp;reaus C90) amp;c. Quod eft ultimura.

THEOREMA

THEOREMA V.

Demonst. Per Centrum O due ad AB pcrpendicula-rem XI; erit hocc diameter (23), amp; etiam perpendi- � cularis ad CF ( 111 ) , eftque arcus AX = XB; amp;nbsp;arcus Cl = FI (126); jam vero arcus XACI = XBFInbsp;(30); ergo arcus AC = BF. Quod eft primum.

Demonst. Ab O Centro, ad X puneftum tangenti�, due radium OX, erit hic perpcndicularis ad XL (12a);nbsp;ergo etiam eft perpcndicularis ad AB (111); ergo arcus AX = XB (126); quod eft fecunduna.

Dico 3�, fi KL amp;; GH, tangentes in X amp; I, fint parallel�, eft arcus XACI = arcui XBFI.

Demonst. Ab X, per Centrum O, due re(ftam;erit hacc perpcndicularis ad KL (122), amp; produ�la uC-que ad redlam GH, erit quoque ad earn perpendicu-laris Cm); non poteft re�la XO produdta cadcre innbsp;aliud puneftum quam in I : nam fi caderet v. g in Z;nbsp;�u�tus radius 01 effet etiam perpcndicularis ad GH;nbsp;quod fieri nequic C92,); ergo arcus XACI item XBFInbsp;eft 180�, amp; proinde �quales funt. Quod eft ufti-�ium.

128. Duse ergo tangentes parallel� tangunt in punc-tis diametrali�er oppofuis.

Si dim chordaf una chorda amp; altera tangens, aut duet tangentes, intercipiant arcus aquales, illanbsp;funt inter Je parallels.

129. Dico 1�, fi arcus AC Cfig. 6) fit =BF;chor-d� AB amp; CF funt parallel�.

Demonst. ^ pun�lo X , medio are�s AXB, due diametrum XI; efl: S redus (126) ; amp; quoniam eftnbsp;arcus CAX = FBX, eft etiam E redus (126); ergonbsp;E = S; ergo redt� AB amp; CF iunt inter fe parallel�nbsp;(114). Quod eft primum.

Dico 2�, fi arcus AX fit ;= XB; chorda AB, amp; tangens XL funt inter fe paraUel�.

Demonst. Duc diametrum XI; eft angulus EXS rectus (122), item S redus eft (126); ergo line� AB amp; XL funt parallel� (115).' Quod eft fecun-dum.

CttjuJIibet chordce. diftantia d.centra eji cequalls mcdietati chordcz fujlentantis arcum , qui ciim arcu dnbsp;priori chorda Jiibtenjo � 180�.

130. Demonst. Sit X 7) centrum; I reiftus; re�ta XI diilantiam chordae AB a centro metitur (94nbsp;per centrum X due FG diametrum paralielam ad AB;nbsp;prajterea due chordam BL paralielam ad IX,; eritnbsp;OB = OL, amp; arcus BG = GL ( 126 ) ; arcus AFnbsp;= BG ( 127}; ergo arcus AF = GL; ergo arcusnbsp;AB BGL = 180�. Prseterea eft IX = BO (113);nbsp;ergo.Xlf~BL. Q, e. d.

II. In pundo F erige perpendicularem FG C98 ) : dice reclam FG efle tangentem quaifitam (124}.

132. nbsp;nbsp;nbsp;Angulus yld Centrum eft ille, cujus vertex efi;nbsp;in Centro Circuli.

134. Angulus Ad peripheriam eft ille, cujus vertel �ft in peripheria. Si formetur per duas chordas,

E %

angulus A in figuris 9, lo amp; 11; appellatur Infcrip-tus : li vero per fecantem exteriorem amp; chordam, uc angulus ^ fig 12: vel per chordam amp; tangentem, u,cnbsp;angulus A fig. 13, Angulus Segmenti audit.

135. nbsp;nbsp;nbsp;Angulus Excentricus ille eft, cujus vertex eftnbsp;intra circuii peripheriam, non tamen in centro.

136. nbsp;nbsp;nbsp;Angulus Extra peripheriam eft ille, cujus vertex extra peripheriam exiftit; ut angulus A in figuris^nbsp;19, 20 amp; 21. Tunc, ft formetur per duas tangentesnbsp;ixt fig. 19, Circumfcriptus tindit; Semi - circumferiptui^nbsp;vero, ut20, ft per tangentem AB, amp; fecantem AEnbsp;tranleuntem per Centrum, elhciatur.

137. nbsp;nbsp;nbsp;In fig. 19 arcus BGC; in fig. 20 arcus BGE;nbsp;amp; in fig- 21 arcus FGE Concavus anguli A vocatur;nbsp;amp; in qualibcc ex hifee tribus figuris, arcus BC Con-vexus anguli A compeilatur.

/Lngulus infcriptits menfliratur msdictate arci'is compre-henfi inter chordas, qiiibus formatiir.

Demonst. i�. Vel una chordarum, puta AB Cj%. 9) per centrum 0 tranfit; per O due diametrum FL pa-rallelam ad AC; erit arcus AL = arcui FC (127),nbsp;item eft = arcui FB ( 31 ) ? 0 = A ( io8) ; atquinbsp;O menfuratur toto arcu FB C72), ftu | are�s CFB;nbsp;ergo A menluratur | arcus CFB. Quod eft primum.

AB confiftit; due diametrum AF; angulus FAC men-furatur | are�s CF, amp; angulus FAB menfuratur | are�s FB; igitur A, qui duobus illis angulis compo-liitur , menfuratur { are�s CFB. Quod eit feeundum.

3�. Vel centrum O li) extra utramque chor-dam exiftit ; ducito AE- diametrum ; totus angulus CAE menfuratur | are�s CBE; atqui una ejufdemnbsp;pars, fcilicet angulus BAE menfuratur | are�s BE;nbsp;ergo altera , feu .angulus CAB , menfuratur are�snbsp;Bi^C; 'quod eft ultimum.

139. nbsp;nbsp;nbsp;Ergo angulus A fegmenti (fig. 12), per chor-dam amp; fccantem exteriorem formatus, menfuratur 4nbsp;arcuum reilduorum , qui non intercipiuntur a chordis,nbsp;quibus vicinus ejus, feu angulus I, efficitur ; namnbsp;A I=: i8o� C80); ergo fimul furapti menfuranturnbsp;� totius periphcriae;fcd I menfuratur | are�s GC (138);nbsp;ergo A menfuratur 5 are�s CA 5 are�s AG.

O tangentem formatus , menfuratur { are�s AOC, fuflentati ab illa chorda^ tangentem versus.

140. nbsp;nbsp;nbsp;Demonst. 1�. Vel arcus AOC Cfig-if) ==nbsp;AHC ( feu AC diameter ) , ell A proinde redtus ,

2�. Vel arcus AOC Cfig. 14) minor eft arcu AHC; ^ puncto C ducatur CG parallela ad AB; eft arcusnbsp;AHG = AOC f127); A = C flop); atqui C menfu-^^^ur i are�s AHG (138), feu |are�s AOC; ergo Anbsp;��ienfuratur ~ are�s A�C; quod eft feeundum. �

3�. Vel arcus AOC Cfig- 15) major cft arcu AHC; dudaCG parallelfi ad Ali,eft arcus AHC =5arcui AOGnbsp;C 127 angulus LCA = A (109 ); jam ver� angu-lus LCA menfuratur | arc�s ACG (139), feu ^ ��

THEOREMA III.

Jiiratur ^ arc�s AF, ciii injijlit, amp; infuptr 5 arc�s BC, cui injifiit angulus X dnbsp;ad vcrticcm oppofitus.

141. Demonst. i�. Vel arcus AC Cj%-16)eft=:arcui AF; ad pun�lum A duda tangens AL (131) erit parallela ad CF (129); igitur A = O (109);nbsp;A ver� menfuratur \ arc�s ACB (140), feu | AF -f |nbsp;CB; ergo amp; O. Quod efi: primum.

2�. Vel arcus AHC (^fig. 17) eft major arcu AF; �uc AH parallelam ad CF ; arcus HC elt = arcui AFnbsp;( 127) , A = O ( 109 ) ; A au tem menfuratur f arc�s PICB ( 138), feu ^ arc�s AF 5BC; ergo amp; O.nbsp;Quod eft i�cundum.

3�. Vel arcus ASF Cfig- major eft arcu AC; ^ucito AS parallelam ad CF; eft arcus AC = arcuinbsp;rS (127); O = angulp fegmen� LAB (109); porronbsp;hic angulus menfuratur ~ arc�s AS i arc�s ACB (139);nbsp;ergo, c�m arcus AC lit = arcui SF; A, feu O menfuratur i arcus AS i SF-p | CB; quod ell tertium.

142. Quoniam arcus oppofiti inter diametros funt �quales (31), atque adeo angulus in centro etiamnbsp;menfuratur ^ arc�s, cui infiftit, amp; infuper | arc�s,cuinbsp;infiftit angulus ei ad verticem oppofitus; generatim o1gt;

tinet, angitlum intra periphcriam mtnfurari \ arc�s , cui injiflit3 amp; injuper ~ areas, cui infifiit angulus ei ad ver-tkem oppojitus.

THEOREMA IV.

jingulus A extra peripheriam (fig. 19, ao, ai) menjiiratur | excefs�s are�s concavinbsp;fupra convexum.

143. Demonst. Vel A eft circumferiptus Cfig.xg'): du�la CG parallela ad AB, arcus BC eft = BG C127 ) ;nbsp;proiade arcus BGC, qui eft concavus anguli A, fu-perat arcum BC, qui eft eonvexus anguli A, ad ar-'nbsp;cum GC; ergo, c�m A fit = I (loS)� I ver� men-furetur f are�s GC (140), A etiam menfuratur |nbsp;are�s GC, qui eft exceflus concavi fupra convexum.nbsp;Quod eft primum.

Vel A per tangentem atque fecantem conftitui-tur ao); due CG parallelam ad AB; eft arcus BC = BG (127 ) ; ergo arcus BGE, qui eft concavus A, fuperac arcum BC, convexum anguli A, adnbsp;arcum GE; quoniam igitur A = I (108}, amp; I menfuratur I are�s GE (140); A etiam menfuratur fnbsp;are�s GE, feu f excefs�s concavi fupra convexum�nbsp;Quod eft lecundum.

3�. Vel A formatur bina fccante Qfig. 21); ducito CG parallelam ad AF; eft arcus BC = FG 0^7);nbsp;A = I (108); atqui I menfuratur | are�s.GE Cho);nbsp;ergo A quoque menfuratur | are�s GE, qui eft ex-ceifus concavi FGE fupra convexum BC. Quod eftnbsp;ultimum.

145. nbsp;nbsp;nbsp;Et cum toto fuo convexo facit } aggregad exnbsp;concavo amp; convexo limuL

147. nbsp;nbsp;nbsp;Semicircumfcriptus facit cum fuo convexo 90�;nbsp;amp; � converfo; li guis angulus extra peripheriam, pernbsp;tangentem atque Iccantem formatus, faciat cum convexo fuo 90�, ille eft femicircumfcriptus, atque adeonbsp;ejus fecans per centrum tranfit (136).

Demonst. Du�la XC, angulus XCA menfuratur f arcfis AGX (138); quoniam AX diameter eft illiusnbsp;circuli (23)� arcus AGX eft 180� (30), erit angulus XCA 90�; eft ergo XC perpendicuiaris ad CA;nbsp;ergo CA eft tangens in pun�lo C (124). e. d.

CAPUT

CAPUT III.

149. nbsp;nbsp;nbsp;Figura KcBilinca eft fpatium lineis redlis, feunbsp;Jateribiis, (juac in extremis pumitis concurruntj termi-'nbsp;natum.

C 0 R 0 L L A R I U M.

150. nbsp;nbsp;nbsp;Quoniam quaclibet duo vicina figurac lateranbsp;angulum conftituant oportet; tot funt anguli, quotnbsp;latera figurae : nequeunt igicur pauciores anguli effenbsp;figur� redtiline�, quam tres; quia tribus ad minusnbsp;lateribus opus eft, ut fpatium condudatur.

152. nbsp;nbsp;nbsp;Si tribus lateribus redis figura terminetur,nbsp;Triangulum audit; ft quatuor, Quadrilatcrum, five Tt�nbsp;tragonum; ft quinque, Penmgonum; fi 6, 7,8, 9, ic?nbsp;amp;c. lateribus conftet, Hexagonum^ Heptagonum, 03.0�nbsp;gonum 3 Nonagonum, Decagonim amp;c., compellatur.

153. nbsp;nbsp;nbsp;Circulus dicitur Circumfcriptiis Figurce, aut Ft*nbsp;gura Circulo Infcripta, dum peripheria per verticeqjnbsp;cujuflibet .ejus anguli tranftt.

154. nbsp;nbsp;nbsp;Figura rtBilinta eft Circulo Circumjcripta, ye?nbsp;C^irculiis Figurcz rcSilinccc Injcriptuf 3 dum �ngulumnbsp;%ur� latUvS eft drculi tangens^

15^. jingulus Externus figura re�lilineai eft ille, qui vicinus eft alicujus anguli Interni, forinati pernbsp;duo vicina figurae latera. Ac proinue ^ngulus Ex-gt;nbsp;terniis formatur per unutn figurae latus, amp; vicinum la-tus produtfium.

156. ScHOLioN. Hac nota A trmigulum : AA triangula : ? quadrUaterum : ?? quadrUaura denocantur.

De Triangulis.

De variis Triangulorum Jpeckbus, atque proprictatibus.

Ratiore laterum, triangulum eft Mquilaterum, fi habeat tria latera tequalia; ft duo fuerint acqualia,nbsp;Ifofccks; amp; ft nulla acqualia fint, Scakmm audit.

158. In A iibfceli angulus, qui per aiqualia late-. ra formatur, Angulus ad Vtrticem, amp; latus ei oppo-fttum, Bafis vocatur; duo anguli, ad hoe latus con-� ftituti, Anguli ad Bafiii appellantur.

i�9. Ratione angulorum, triangulum dicitur dEqul-anguluni, ft habeat tres angulos acquales; Reclangulum^ vel Orthogomm, fi unus trium angulorum re�lus fit;nbsp;ObtiifanguluTn, ft alterutrum angulum 'h�hezt-Okiifum;nbsp;Acutangnlum, ft omnes tres acuti exiftant.

ifo. In A retftangulo, latus oppofitum angulo ree-' to dicitur Hypotmuja; reliqua duo latera Catheti au-diunt,^

7 nbsp;nbsp;nbsp;G	F
p nbsp;nbsp;nbsp;, /	1 1
/ nbsp;nbsp;nbsp;\inbsp;nbsp;nbsp;nbsp;:\nbsp;nbsp;nbsp;nbsp;/	t
......-L��g nbsp;nbsp;nbsp;I X ; nbsp;nbsp;nbsp;l * / nbsp;nbsp;nbsp;\	i 1 X

^J2 nbsp;nbsp;nbsp;^ ASc---^ nbsp;nbsp;nbsp;-	A

i6i. Altitudo trianguli efi: perpendicularis , dufla ' ab uno ejus angtilo ad lacus oppoficum ( quod etiamnbsp;( Bafis vocari poteft); five iiitra bafin cadac, five ianbsp;eandem produftam.

163. nbsp;nbsp;nbsp;Demonstr. Trium angulorum, cuju�ibet trianguli vertices, tria funt puncfia, qusc in eadem re�tanbsp;non conftituuntur; atqui eis circumlcribi poteft circulus ( 44 ); ergo cuilibet A poteft circumfcribi circulus.nbsp;^e.d.

164. nbsp;nbsp;nbsp;Dico angulos A, B, C (Tab. IV. �g. fimulnbsp;facere 180�

Demonst. Concipe triangulo ABC circumfcriptum circulum; A menfuratur | arc�s BC; B ' arc�s AC;

C i arc�s AB ( 138); ergo anguli A B C men-lurantur -1 totius periphericc; atqui medietas illa= 180�; ergo amp; anguli A B C fimul faciunt 180�. Q^. e.d,

165. nbsp;nbsp;nbsp;Tres igitur anguli fimul furopti unius trian- \nbsp;Enli, tequivalent tribus angulis fimul fumptis eujuf-C�mque trianguli.

166. nbsp;nbsp;nbsp;In quolibec A unicus dari poteft angulus reonbsp;tus, aut obtufus.

167. nbsp;nbsp;nbsp;Angulus extcmus A, v. g. F a) acquiva-let duobus angulis A C internis, hbi oppolitis :nbsp;nam F B= 180� (80); amp; B A C=i8o� 064)�nbsp;ergo F=A-fG

'68. Duo externijV. g. F K, fuperant A, internum utrique oppofitum, ad : nam F = A C; amp; K=A B (167); ergo F K fuperat A ad A Bnbsp; C, qui = 180�; ergo Ka

1�9. Tres extemi, v. g. F O K, = 360� : ete-nini B F; A 0; C K faciunt fimul ter 180� ; atqui A B C= 180� ; ergo rclidui , feu tres ex-terni = bis 180�, feu 360�.

In quovis triangulo lams majus opponimr angulo majori : (y � converjo; angulus majornbsp;opponimr latcri majori.

Demonst. Sit A ABC inferiptuin circulo; c�m A menfuretur f arc�sBC , amp; B f^arcfis ACCi38);ericnbsp;i are�s BC major, quam - are�s AC; ergo arcus BCnbsp;eft major arcu AC; conlequenter chorda BC majornbsp;eft chordd AC (3�); quod erat primum.

Elementa Geometeije. nbsp;nbsp;nbsp;4.^

Demonst. Eft arcus BC major arcu AC (36) ;� ergo | arc�s BC major eft quam | arc�s AC; porronbsp;A menfuratur ^ arc�s BC ; amp; B f arc�s AC; ergonbsp;A eft major B. Quod erat fccundum.

In. IfofceUj anguli ad bajin fiint ccqualcs; 6* � con~ vcrfo : Ji in J\ duo anguli tequales fiat,

171. Dico 1�, ft A ABC (fig. 1) ftt ifofcele, fic Vit A fit ad verticem, elTe B = C.

Demonst. Eft latus AB = AC (158); pone itaque A ABC infcriptum circulo ; erit arcus AB = arcuinbsp;AC ( 35) ; porro B menfuratur f arc�s AC; amp; Gnbsp;menfuratur | arc�s AB (138)1 ergo B = C; quodnbsp;erat primum.

Dico 2�, ft B = C, efle A ABC ifofcele, fic ut A fit ad verticem, amp; latus AB = AC.

Demonst. Sit rurfus A ABC infcriptum circulo ; erit arcus AC = arcui AB, quia B menfuratur -f arc�s AC, amp; C i arc�s AB (138); ergo chorda AC =nbsp;chordaj AB (35); igitur A ABC eft ifofcele ( 157)�nbsp;^uod erat alterum.

Corollahium I.

17 a. A scquilaterum eft scquiangulum ; amp; � con-verfo : eft enim �ndcquaque ifolcele, dum habet quac-iibet duo latera aequalia; proinde duo quilibet anguli Poflunt fpedari ad bafin ; igitur quivis duo angulinbsp;�quaics funt ( 171); ergo quilibet eft 60� ( 164).nbsp;Et fi quilibet duo anguli acquales (feu 60�) fuerint,nbsp;etiam duo quslibct latera xqualia fint oportct (171)*

THEOREMA VI.

Si B, item C C^g- 3) acutus , perpendicularis ^ tz A ad EC duSa j cadit inter BC; ji autem unus ,nbsp;y. g. B (fig. 4f')ifuirit obtujus^ cadet extra Aj in latus CB produSum ultra B y V. g. in X.

174. Demonst. I� pars. Quoniam B iteraC minor cft B cum ccrta parte A = 90�; amp; C cum altera parte A etiam = 90� ; pone itaque in duas tales partes A dividi per redlam AZ, ut angulus CAZ faciacnbsp;cum angulo C 90�; quoniam Z = C anguio CAZnbsp;Cl67), erit Z redtusj ergo AZ ell perpendicularis adnbsp;BC (76); atqui ab A ad BC unica duci potell perpendicularis (9a); ergo ilia cadit intra bafin BC.nbsp;Quod erat primum.

Demonst. c*** pars. Non poterit perpendicularis versus C cadere in lineam BC : nam vicinus 0 rectus foret; quoniam ergo B obtufuseft, ex fuppofito, B Scnbsp;angulus AOB facerent plus quam 180�; quod implicat,nbsp;cum in A ABO tres anguli folum = 180� (164).nbsp;Neque cadere poteft in B : B enim re�lus foret; caditnbsp;ergo in CB produ�lam, v. g. in X. Quod erat alterum.

In A JJo/celi ABC (fig. 5 ), A Jit ad vertkem ; Ji habeatur iinum ex fequtntibus ; 1� vel 0 reSus;

175. Demonst. primum. 0 = C-{-X; amp;Z = B4-I C 1�7 ) ; porro 0 = Z, cum fmt ambo re�li; erg()

C X=B I; jam ver�, ex luppofito, B=C(i7t); ergo Iz=X; quoniam vero AB = AC (158}, amp; Onbsp;rectus, cft B� = OC (89). Quod erat primum.

iuppofito I = X, eft B I = C-hX; porro Z=:B4-I, amp; 0=:C Xnbsp;nbsp;nbsp;nbsp;ergo 0=:Z; atqui � Z=i8o�

(80}; ergo O item Z eft 90� (cu re�tus; igkur rurfus BO = OC. Quod erat �itimum.

1� O reaus : 2� i = X : g� BO = OC; A ABC eji IJoJcek ^ Jic ut A Jit ad verticem; amp;jnbsp;reliqua duo Jequuntur.

Demonst. Z = 0; fedZ = B I, amp;0 �C X C 1�7 ); ergo B 1 = C X; c�m igitur I = X, eftnbsp;B=C; ergo A ABC eft ilbftele (157), A ad verticemnbsp;igitur BO = OC (89); quod eft primum amp;nbsp;fecundum.

Demonst. Eft AB = AC (88); ergo A ABC ifog. cele (157); igitur B=C (171); quoniam ergo 0=:Z,nbsp;�ft I = X. Quod eft fecundum amp; tertium.

Demonst. A ABC fit circulo inferiptum; produca-�iitur AO ufque ad H peripheric pun�tum; I menfa-'

ratur f arc�s BH, amp; X f arc�s CH (138); ergo, cum. 1 = X, eft arcus BH = HC ; igitur chorda HAnbsp;dmdt chordam SC, item arcum BHC bifariam; proin-de O eft re�tus (126). Quod eft fecundum amp; ter-tium.

In A fi linea, du3a ex vertice alicujiis angiili, fit aqudis ciuiibct parti lateris oppofiti; angutus ilk reSus ejt.nbsp;Et fii in A reSangulo 3 ex angulo reamp;o linea ducaturnbsp;ad hypotenufam, ita ut vel ifia linea fit cequalisnbsp;alterutri parti hypotenufie-, vel partes hypotenufix fintnbsp;cequales inter fe; linea ifia efi cequalis fingulce portinbsp;hypotenufix.

Demonst. Quoniam BX = CX, C = angulo XBC; amp; quia AX = XB, A = angulo XBA ( 17O; ergonbsp;B = A C; porro B A C = 180� ( 164); ergo B = 90�, feu redlus eft (76). Quod erat primum.

Dico 2�, fi B redlus fit, amp; vel BX � AX, auf = CX; elTe BX = AX, item == CX.

Demonst. Pone BX �AX, erit A AXB ifofcele Cl57), A = angulo XBA CijO� angulus XBCnbsp;= C Cc�m B = A C}; igitur A CXB ilbfcele, amp;nbsp;CX = XB (171) ; quod erat fecundum.

Demonst. A ABC fit circulo infcriptum Qfig. 7 ); quoniam B, ex fuppofito rectus, menluratur | arc�snbsp;�ALC (138), erit arcus ALC 180�; ergo AC erit diameter (30}; igitur X, punctum ejus medium, eftnbsp;circuli centrum; proinde BX, CX amp; AX fuut scqualcsnbsp;inter fe f 24). Quod erat ultimura.

' nbsp;nbsp;nbsp;THEO-

THEOREMA X.

proinde alter 30�; latus oppofititm angulo reSo, ejZ I 4- latere oppojito angulo 30�.

178. nbsp;nbsp;nbsp;Dico, fi B Cfig- 6) fit re�lus, C 60�, amp; Anbsp;30�, efle AC f BC.

Demonst. Ex B ducito BX ad X, pun�him mediuni lateris AC; eric BX == CX, item == XA C ^77) j igi�nbsp;tur A CXB eft ifolcele ( 157); ergo C = angulo CBXnbsp;(171); quoniam ergo C eft 60�, erit quoque angu-lus CXB �o�, five Ecqualis angulo CBX; ergo A CXBnbsp;eft scquiangulum, adeoque amp; aiquilateramp;m ( 17a); eftnbsp;ergo BC = CX; igitur eft AC f BC. Q^. e. d.

In h f fi fit umim latus � altero, atque inj�iper habeatur iinum ex Jequentibus :

179. nbsp;nbsp;nbsp;Dico fi ifig. 6) fit AC f BC, amp; Bnbsp;�e�tus, effe C 60�, amp; A 30�.

DemonST. AX fit = XC; dudta BX eft =; AX, item == XC (177); ergo A CXB eft gcquilaterum, ac proin-^e ffiquiangulum'C172); ergo C eft lt;5o�; igitur A. 30gt;nbsp;�Quod eft primum.

Demonst. Sit AX = XC; erit CX = CB; ergo A CXB ifofcele ( 157 ) ; angulus GSi == angulo CBXnbsp;(171); quoniam veto C = 60�, duo illi anguli fi-mui faciunt 120�; qaifque igitur eorum = 60�-; igi-tur A CXB eft aequiangulum, ac proinde scquilaterumnbsp;( 172); ergo BX = CX, amp; proinde edam = AX;nbsp;ergo B eft re�tus (177) ; igitur A 30�. Quod eftnbsp;jfecuudum.

Demonst. Imagineris a pundo C ad redara AB 'duci perpendicularem; hxc fit oportet f � redd ACnbsp;(178); igitur elTet = CB; ergo CB eft perpendicularisnbsp;�ad AB C 95 )� proinde B redus; C 30�. Quod eftnbsp;ultimum.

i8o. Resolutio I. Cirdno fume intervallum red� AB , atque eo mediante, ex A amp; B, fiant later-fediones in C :

rfr. ScHOU�N. Quoniam A item B eil 60� (17a), per prtece-� dens problema modum habes conftruendi angulum 60� ; �juem li bifariam feces (87), angulum 30� obtines-, acque hunc deinde hl duas parces squales dividendo, angulum 15� habebis amp;c.

182. Resolutio I. Aflumarur una, v. g. AB, pro ba-fi, aut ipfi d-ucatur tequalis ;

W. Ex B, intervallo GH, ducatur alter arcus, qui priorem interfecct, v. g. in E :

183. ScHOLioN. Si quK�bet diiae ex reftis datis, terdi non fu@--rint majores, problema nequ�c refolvi (i6a).

II. Ab X, pundo concurs�s redarum CX amp; BX, ad lams BC due perpendicularem XO :

Demonst. Ab O ad CX demitte perpendicularem CZ, haneque produc ufquc in G ; quoniam angulusnbsp;Oez = GCZ, amp; Z red�s, A CGO eft ifofcele, amp;nbsp;OZ = ZG ( 176); pari ter due Ol perpendicularem adnbsp;BX, eamque produc ufque in H; quoniam angulusnbsp;OBI = HBI, amp; I redus, A OBH eft ifofcele, amp; 01nbsp;=== IH (176); itaque redm CX amp; BX interfecant per-pendiculariter amp; bifariam chordas OG amp; OH circuli�nbsp;cui punda H, O amp; G forent inferipta; quaclibet ergonbsp;per centrum iftius circuli tranfeat oportet (la�));nbsp;Igitur pun�lum X interfedionis eft centrum iftius circuli Cap); ergo punda H, O amp; G diftanc scqualiternbsp;sb X; funt igitur XG, XO, amp; XH inter fe ajquales;nbsp;Proinde peripheria ex X, intervallo XO, defcripta, tran-Ilt per punda H, O, G; pra^terea A GXO eft ifof-cele C 157), ergo angulus XGO = XOG ( 171 );,:fednbsp;ftuoniam A COG ifofcele eft, angulus CGO = C�G,

(J7t); igitur angulus CGX = XOC; jam ver� an-^ guius XOC, ex eonftrudione, redtus eft; ergo etiamnbsp;CGX eli: redus; proinde CG eft tangens (124). Sj^nbsp;mili modo demonftrabitur angulum XHB = XOB, feunbsp;redum, amp; confequenter BH elfe etiam tangentem. Igitur triangulo ABC circulus GHQ eft infcriptus (154)*

�, 11.

185. nbsp;nbsp;nbsp;Congruere dicuntur figurac, dum, und alterinbsp;fuperpofita, latera iateribus, amp; anguli angulis coinci-dunti five dum fefe mutuo perfed� tegun^.

186. nbsp;nbsp;nbsp;Triangula Mquilatera funt, quorum tria latera unius funt refpediv� acquaiia tribus iateribus ai-terius.

187. nbsp;nbsp;nbsp;Triangula Mquiangula funt , quorum tr^s an�nbsp;guli unius refpediv� funt acquales tribus angulis al-terius.

188. nbsp;nbsp;nbsp;Latera Homolloga in AA scquiangulis ea di-puntur, qujc relpe�tiv� angulis scqualibus opponuntur,

theorema J.

189. nbsp;nbsp;nbsp;Si (Jig. 11 ) AC == ac; AB = ab; BC = be,nbsp;dico A abc elTe congruum A abc^ atque A = a,B=^,

Demonst. Superponatur latus ab in AB, ita ut a in A, adeoque (attenta duorum illorum lat�erum xqua-Utate) � in B coincidat; tum ex A, intervallo AG,nbsp;ieu cc, delcribe arcum LCO; amp; ex B, intervallo BC,nbsp;feu bc, arcum HCS, fel� proindc interfecantes in C;nbsp;Suoniam ac � AC, terminabitur illa in arcu LCO;nbsp;pariter, quoniam � BC, illa terminabitur in ali-quo pun�to arc�s HCS; porro re�te ab amp; bc termi-nantur in extremitate fua c, ubi concurrunt; ergonbsp;pundum illud arc�s LCO, ubi^ terminatur re�la ab,nbsp;idem fit oportet cum pun�to arC�s HCS, ubi terminatur reda bc', five debet effe pundum C, in quodnbsp;proinde cadit pundum c; igitur latera unius A inci-dunt refpediv� in latera alterius, amp; confequenter etiamnbsp;refpedivi illorum AA anguli cojncidunt ergo AAnbsp;congruunt (185). Q. e. d,

H�o AA aquiangula, O habentia uiium. latus homollo� gum cequale , fiint congrua.

190. Dico, fi (fig. II) A =3 fl; B = ^gt;; Sc proinde C = c; atque prxterea fit v. g. AB = ab; duo ilia AA efle congrua; amp; proinde AC = ac, BC = bc.

Demonst. Pone ab in AB, ita ut a in A, amp; 6 in B cadat; quoniam 0 = A, ac cadet in AC; amp; c�mnbsp;^ = B, bc non poteft non incidere in BC (75); ergo ac amp; bc concurrent in idem pundum , in quodnbsp;AC amp; BC fibi occurrunt; feu c cadet in C; ergo Anbsp;ABC congruit A abc ( 185^. Q^. e. d.

THEOREMA HL

Si in duobus AA duo latera unius fint rejpeamp;ivf cequalia duobus lateribus alterius; amp; anguli ^nbsp;kis lateribus comprehenji, isquales Jint^

Demonst. Pone AC in cc, ut C in c, amp; A in ff conliftat; quoniam C=:c, cadet latus CB in cb,amp;t c�mnbsp;iint=, coincident ; ergo B cadet in b, amp;c conrequen�nbsp;ter latus AB coincidet cum latere ab; igitur A in a,nbsp;amp; B in A coincident; igitur AA congrua lunt. Q^-c.d..

THEOREMA IV.

T)um duo AA habent duo latera reJpeSiv� aqualia^ amp; utrimque unum angulum, lateribus cequalibusnbsp;oppofitum., esqualem; ifia AA congruunt jnbsp;vel .anguli oppqfiti reflantibus lateribusnbsp;cequalibus , faciunt fiimul 18o�.

ip2. Dico , fi (^fig. II ) AC = ac; BC = bc, k, A = a, AA congruere; vel B ^ = i8o�.

Demonst. Si AB = ab, AA congruunt (.189); fi infcqualia fint dicta latera ; pone AC in ac, fic utnbsp;coincidant; quoniam A = a, cadet AB in ab; tune,nbsp;fi fit AB majus quam ab (�t in fig- 12), aut ABnbsp;minus, quam ab (�t in fig. 13), confianter A iCBnbsp;erit ifofcele , c�m bc fit = BC; ergo in fig. 12, eritnbsp;B =� vicino A; amp; in fig'- 13, erit A = vicino B ; igitur B i = 180�. �,� c. d.

lia, illud quod habet majorem angulum, per illa la' ter a conjlitutum, ctiam majorem habet bajin :nbsp;illud, quod habet majorem bajin, habetnbsp;qu�que majorem angulum, bajinbsp;ijli oppojitum.

ipg. Dico 1% fiCfig- 14) fit AB~kE; AC=KF; ,amp; .pra:terea fit A major quam K, lams ]3C effe maiusnbsp;latere EF.

Demonst. Fiat angulus EKG aiqualis A; fit AC = KG; ducatur EG; AA ABC amp; KEG congruentnbsp;C 191); ergo BC = EG; KG == KF, ex hypothefi;nbsp;Igitur A KGF ifofcele ( 157 ); ergo G � angulonbsp;KFGnbsp;nbsp;nbsp;nbsp;porro angulus EGF eft minor G; amp; 'an

gulus EJ'G eft major angulo KFG; igitur in A EFG Jingulus EFG major eft angulo EGF; ergo lams EGnbsp;eft majus EF (^70) j fed er at BC = EG; ergo lamsnbsp;BC eft qu�que majus latere EF. Quod erat primum.

Dico 2�, fi AB = KE; AC = KF, amp; preterea fit BC majus EF, efle A majorem K.

Demonst. Si A non foret major quain K; tune vel effet = K, atque A ABC congrueret cum A KEFnbsp;(191); igitur effet BC = EF; quod eft contra hy-fothefin : vel A foret minor K; itaque, juxta priroanxnbsp;partem hujus theorematis, EF deberet effe majusnbsp;Sluam BC; quod rurfus eft contra hypothefin; igiturnbsp;A eft major quam K; Quod erat alterum.

�94. Polygona dicuntur Symmetrica ^ dum terminart-tur quibufvis lateribus oppolitis xqualibus atque pa-rallelis.

196. nbsp;nbsp;nbsp;Polygona Regularia funt, dum amp; orti�iia latera func Ecqualia, 'K omnes anguli xquales. Cacteranbsp;polygona Irregtilarla compellantur.

197. nbsp;nbsp;nbsp;Pun�lum jcqualiter ab omnibus angulis polygon! regularis diftans, Centrum polygon! audit. Recta quaclibet ab alterutro angulo ad centrum dud�anbsp;.Radius Obliqmis; amp; a cencro ad unum latus perpea-dicularis, Radius ReSus vocacur.

s�, vel duo latera tantum parallela funt, amp; Trap-pesyiis vocatur l amp; utroque hoe cafu eft polygonum irregulare.

3�, vel fingula oppofita latera funt parallela, atque ParaUelogrammum audit. Hujus quatuor dantur fpe-�ies : vel enim fingula latera funt xqualia, amp; finguli

anguU

�ing�li scquales ( adeoque rc�li), amp;c Quadratum appel-iatur; Hocque cafa elt polygonum regulare ( 19�): '''�cl omnia latera quidem acqualia funt, fed foil angulinbsp;'^ppoliti tcquales, amp; Rhombus audit. Vel fola oppp-lita latera funt xqualia; ii tune quatuor anguli fintnbsp;recti, Reamp;angulum; It foli anguli oppofiti a;quales fue-rint, Rhombwdes vocatur; atque tribus ultimis cafibus,nbsp;Symmztricum elt.

199. Recta in polygono quocumque, ab uno an-gulo ad alium du�ta, Diagonalis vocatur.

aoo. ScHOLiON. Unguium Procafrtnum vocanc, cujus crura er* trorium concurrunc, atque duobus redis minor elt; uc angulinbsp;A, B, F amp; C ifig. 15). ytngmuni Regramp;dknUm appellant,nbsp;qui gibbofus eft, amp; per latera introrfum concurrentia formatur;nbsp;atque duobus redis major ell; talis ell angulus E, facitque cuninbsp;angulo C 360�. Sed de figuris, quibus anguli regrediences in-terveniunt, 1'pecialiter non agimus; c�m ex priucip�s dads,nbsp;dum occurrunt, facili negotio eruancur.

5- I.

�mnes anguli interni Polygoni [mul faciunt toties 18o\ quot funt Polygoni latera , duobus exc^tis.

201. Demonst. Ad pun�tum aliquod intra polygonum , ad arbicrium, furaendum, � quolibet angulo rectas ducito, atque in tot AA figuram refolveris,nbsp;fluot funt polygoni latera; omnefque anguli, omnium,nbsp;illorum AA, faciunt fimul toties 180�, quot funt fi-guraa latera; jam ver� anguli illi omnes, qui confti�nbsp;ruuntur ad idem illud pundtum intra polygonum, fa-�^'Unt fimul bis 180� (84); reliqui ergo omnes, quinbsp;prscis� comprehenduntur per angulos polygoni, faciunt

202. nbsp;nbsp;nbsp;Igitur in Qnadrilatero anguli omnes intcrni fa-ciunt limul bis 180�; in Pentagono ter 180�; in Hexa-gono quater i8o� amp;c.

203. nbsp;nbsp;nbsp;Omnes anguli extern! cujuflibet Polygon! fa-ciunt limul 360� : quilibet enim externus cum vicinonbsp;fuo, qui intemus ell, = 180� (80); tot autem vi-cinorum funt paria, quot figura? latera; ergo omnesnbsp;extern! cum internis faciunt toties 180�, quot funtnbsp;figurse latera; atqui interni faciunt limul bis 180� minus (201 }; ergo externi faciunt bis 180�.

204. nbsp;nbsp;nbsp;Si quadrilaterum ABCF (^fig. 16), amp; hexago-num ABMCFG (^fig- 17) fymmetrica fuerint C^94)�nbsp;dico effe A = C; B = F amp;c.

Demonst. 1� prs. Du�iis diagonalibus AC amp; BF Qfig. 16); quoniam AB parallela ad FC, eft L = E,nbsp;FI = K; amp; quoniam AF parallela ad BC, eft I = R,nbsp;S = Z Qiogy, igitur L I, feu A, eft =E-|-R, feunbsp;C; amp;H-1-Z, feu B,eft = K S,feu F. Quod cratnbsp;primum..

Demonst. 2^^ pars. Du�lis diagonalibus AC amp; BF CfiS- 17)5 ctim AB fit parallela ad FC, eft L = E,

H = K; amp; c�m AG �t parallela ad CM ^ efl; I = R. (109); igitur L 1, feu A, = E R, feu C; quo-^iam ver� BM parallela ad FG , eft Z = S ( 109);

H Z, feu B, = K S, feu F; ducendo diago-nalem MG, fimili ratiocinio confequitur M �G. Quod erat alcerum.

dirc�c oppojitiim angulum diiScz, JeJe miitu� in eodem piinko bifariam Jccant.

205. nbsp;nbsp;nbsp;Si polygona Qfig. 16 amp; 17 ) fymmetrica fuerint,nbsp;dico diagonales AC, BF, MG fefe mutuo bifariamnbsp;fecare in O.

Demonst. Eft L = E; H = K C 109 ) j AB = CF C194); igitur AA AB� amp; CFO congrua funt ( 190);nbsp;igitur AO = OC ; BO = OF ( 185) ; prsctereanbsp;17) Z = S, X = P (109); quoniam igitur eftnbsp;MB z= GF, AA MBO amp; FGO Hint qu�que congruanbsp;(190); ergo BF dividitur bifariam; itaquc MG tran-fit per O pun�tum medium re�tai BF; amp; ita porronbsp;de diagonalibus in Oclogono, Decagono amp;c. Q^. e. d.

206. nbsp;nbsp;nbsp;In polygono fymmetrico, diagonalis qua^libet,nbsp;put�' AC, dividit polygonum in duas partcs tcqualcs :nbsp;dividit enim illud in' aaqualem numerum AA, quorumnbsp;fingula oppofita congrua funt , proinde amp; squalia.

n a

Demonst. Du�ta diagonali AC, eft L = E, R=rl (109); ergo AA ABC amp; AFC congruunt (190);nbsp;igitur AF = BC, AB = FC. Itaque ? ABCF conftatnbsp;lateribus oppofitis aequalibus atque parallelis; ergo fym-inecricum eft (194). Q^- e. d.

los aaqualcs; vel Jingula refpeSiv� oppojita latera cequalia, efl Jymmetriciim.

Demonst. A B = C F; ergo c�m quatuor di�li anguli fimul = 360� (202); A B = i8o�; itaquenbsp;af amp; BC l�nt parallelai (ii5);paricer A F = B4-C;nbsp;igitur A F= 180�; itaque AB amp; FC funt etiainnbsp;parallelac inter fe (iiSquot;); igitur ? ABCF eft paralle-logrammutn (198); adeoque amp; fymmetricum (207).nbsp;Quod erat primum.

Demonst. Du�la diagonali AC , A ABC congruit A AFC, B = F, L = E, I = R C 189 ) ; ergo ABnbsp;amp; FC; AF amp; BC funt parallels (i 15); ergo ABCFnbsp;eft parallelogrammum C198), adeoque amp; lymmetri-cum (207}. Quod erat alterum.

Quadrilatcrum habens blna oppoj�ta latera cequalia amp; paralkla , fymmetricum efi.

209. Dico, ft AB amp; FC (fg. 16) fint scquaiia atque parallcla, ? ABCF effe fymmetricum.

Demonst. Ductis diagonalibus AC amp; BF, eft L�E C109); ergo AA BAC amp; ACF congruunt (191); igi-tur BC = AF; B = F; R = I; jam ver�, quianbsp;^ = I, BC eft parallela AF (115); ergo ? ABCFnbsp;paral]elogrammum ( ip8 ) , adeoque amp; fymmctri-cum ( 207 }. (2,- c-

Qjtadrilatcrum hahenS duo latera cppofita ctgiialia, amp; duo reflantia latera parallela, efl Jymmctricum,nbsp;vel anguli oppojiti Jimal faciunt 180�.

aio. Dico, fi AB = FC (^fig. 16); amp; AF paral-Jela BC, efle ABCF fvmmetricmn, vel angulos A C, item B F = i8o�.'

Demonst. Quoniam AF parallela BC , eft S = Z (109); ergo A FAB congruit A FBC, vel A C.nbsp;= 180� (192); ft AA congruarst, eftA = C,H=K;nbsp;^fgo AB amp; FC parallelle ( 115); igitur O ABCF eftnbsp;lymmetricum (194); ft A C = 180�, etiam B Fnbsp;= 180� : nam A C B F= 360� (202 ); igt-tur ABCF eft fymmetricum, vel anguli oppoftti = i8o�.nbsp;d.

Ilexcgonum 3 OSogonum amp;c., quibiifvis angiiUs amp; lateribus direS� oppofitis aquaUbus coii-Jians, fyiumctricum eJL

^ 211. Si Cfig- 18) A=C, B=F, H=G; AB=CF, BH = FG, amp;HC=GA, dico iftud hexagonum elTenbsp;fyniinetricum.

Demonst. Duc BG amp; HF; A ABG congruit A HCF C 191); igitur BG = HF; proinde BGFH eft fvmrae- .nbsp;^ncum C208); ergo GF amp; BH fuut parallck OP4)i

fimiliter, due AF amp; BC; AA AGF amp; CHB congruent (191); ergo AF=BC; igitur AFCB eft fymmetricumnbsp;(208) ; igitur AB amp; FC lunt parallelac ( 194); dudisnbsp;AH amp; GC fimili ratiocinio reperies AG parallelamnbsp;HC; itaque hexagonum datum terminatur lateribusnbsp;diredt� oppolitis aiqualibus atque parallclis; igitur lym-metricum dl (194). Q^. c. d.

5. III.

Omnt Polygonum Regitlare, pari laterum numero ter-minatum, ejt quoque fymmetricum.

212. Propofitionis veritas perfpicua fit per numeros 196, amp; 211.

PROBLEMA I.

213. Resolutio I. Divide A item F bifariam C87) per redas, qusc concurrent v. g. in X.

Demonst. Redas FX, EX amp;cc.fig. 19- ducito, fimi-lefque duclas concipe fig. 0.0; quoniam A = F (196), amp; ex conftru�tione quilque bifariam divifus eft , eftnbsp;I=:K, ergo A AXF eft ifofcele (171); igitur XA =nbsp;XF;pra:terea I = H; quoniam igitur AF = FE (196),nbsp;A A XAF amp; XFE congruunt (191); ergo R = K;nbsp;XE = XF j quoniam igitur E = F, eft G = HI ita-

A CXE congruum A XFE (ipi), confequenter CX = XE; amp; fic de cacteris. Omnia ergo pundta A ,nbsp;F, E, C amp;c. diftant acqualiter nb X; elt igitur X polygon! centrum (197); atque adeo circulus ex X, in-tervallo XA, delcriptus, per vertices omnium angulo-rum poiygoni tranfit; igitur huic circumfcriptus cil il-le circulus (153). Qa

2,14. Binos polygon! regularis angulos bifariam fe-tando per reitas, pun�lum concurs�s earumdem dat poiygoni centrum.

Demonst. Polygonum datum circulo inlcriptum Goncipe; quodlibet latus C c�m omnia fint a^qualia)nbsp;scqualem peripberiac arcum fubtendit C3.5)j Wti-dem ergo arcus acquales erit divifa peripheria, quocnbsp;funt polygon! latera ; divifo itaque 360� per nunie-rum laterum, quotiens dat valorem cujufque arc�s ,nbsp;I'eu i duorum arcuum, qui fuftentantur a lateribus,nbsp;qutc iftum polygon! angulum conftituunt; jam veronbsp;quifque angulus polygon! regularis menfuratur l omnium arcuum, exceptis duobus; ergo quotiens ille fa-,nbsp;cum quolibet angulo 180�; quotiente ergo fubduc-ex 180, reliduum dat valorem anguli polygon! re-�blarjs. e. d.

a 16. Igitur quotiens ille eft jcqualis angulo formato in centre polygon! per binas re�tas, � duobus vicinisnbsp;angulis ductas.

Circulo dato Polygonum regulate inferibere.

217. Resolutio I. Subftrahe angulum polygon! ex 180�, reiiduum dac angulum Ibrmandum In centronbsp;(216) ;

II. Ejufdem angul! crura interclplent in peripheria ar-cum, cujus chorda, quoties licet, peripheriac appli-cetur; atque conftrudum cr!t polygonum dcfidera-turn.

Sit E. G. conftruendum Hexagonum regulate; quo-niam qulllbet cjuldem angulus eft 120� (215), hoc fubdudo ex 180�, reflduum �o� dat angulum !n cen-tro X (^fig- 20) formaiidum; eric itaque arcus ABnbsp;60�; chorda AB fexies peripheriac appiicetur, atquenbsp;hexagonum regulate deferiptum erit.

ai8. ScHOLioN I. Chorda fuftentans arcum 60� (feu latus hexa-goni regularis infcripci circulo ) eft ccqualis radio circuli : amp; � converfo, fi chorda fic aequalis radio, fufteiitac ilia arcum 6o� :nbsp;primo enim cafu, cum AX = XB, A XAB eft ifofcele ( 157);nbsp;ergo angulus XAB =: angulo XBA (i7i)v jam ver6, cum Xnbsp;lit 60�, duo illi anguli faciunt 120�; igitur quilibet eorumnbsp;eft etiam 60�; eft ergo A XAB iequiangulum , ergo amp; sequi-laterum (172); igitur AB=: AX, feu radio circuH. Secundonbsp;cafu, dudis radiis AX amp; XB , erit A XAB �quilatcrum, acnbsp;proinde aequiangulum ( 172 )�, quilibet ergo ejufdem angulus eftnbsp;6o�-, fed X menluratur arcu AB (72)-, ergo arcus AB eft �o�.

metric� dividi itr 4, 8, 16 amp;c. partes tequales, atque adeo polygonum regulare 4 , 8, ,i� amp;c. latcrum , per Geometriam ele-tnentarem, circulo iiifcribi,

a�. Angulus 6o�, 30�, 15� amp;c.-, his mediancibus eiEcies polygo-lUim 6, ta, 04, amp;c. laccrura.

3� Augulus 72'^ ( de quo poftea) 36�, 18� amp;c. , quibus con-Irrucre liccc polygonum 5, 10, ao amp;c. laterum. Pro casteris autcm , praxis communis atq'.ic mechanica adhibctur : fcilicctnbsp;leiiLando percircinum , aut I'ranjportorio (de quo poftea), pe-tipheria in partes dcfidcratas dividitur.

St 20. Resolutio I. In A atque B fac angulos, quorum quifque faciat \ anguli polygoni :

III. nbsp;nbsp;nbsp;Eidem applices, quoties fieri poteft, latus AB; atque polygonum quatfitum habebis.

am. ScHOLioN. Super data refta AB(/g. a) facile eft Quadra- � turn delineate : erigendo enitn in A'amp; B perpendicularcs AGnbsp;amp; BF, fic uc qu-jeJibet lit a:qualis AB , atque ducendo GF,nbsp;Quadratum ABFG exuigit.

222. Resolutio I. E centre polygoni (214), ad unuffl, liivLs, iiwz radium rectum :

II- Ex eodem centro, intervallo di�li radii reSi, circulum ducito, eritque ille polygono regular! inlcriptus.

Demonst. Quoniam omnia polygon! regularis latera ^q^ualia funt; dum polygonum eit inferiptum circulo,nbsp;Zinnia latera fant totidem chordae acquales, quae proin-�ngulae aequaliier a centro diftanc (13�) 5 ighur per-pendiculares quaeiibet, � centro polygoni ad quodlibetnbsp;iatus dudtse (ibu omnes radii recti') aquales funt .(54);

ergo peripheria, � centro polygon! ad intervallum nnius ex iitis perpendicularibus, feu radiis re�tis, de-fcripta, tranfibit per extremum cujuflibet; atque adeonbsp;in fingulis illis pun�tis tanget quodlibet polygon! la-tus (124); igitur circulus ille erit polygono infcrip-tus(i54). Q^c.d.

223. C�m polygonum regulare circulo circumfcrip-tum eft, quifque ejus anguius, iftius circuli eft Cir-cumfcriptus; tot ergo erunt arcus convex! , quot an-guli circumfcripti; amp; qaoniam omnes hl circumfctipti a^quales funt, erunt quoque convex! linguli tequales.nbsp;Itaque

Resolutio I. Divide peripheriam in tot arcus sequa-Ics quot funt polygon! futura latera, v. g. quinque pro pentagono :

II. Per pundta divifionis a, b, c, e, f (fig- 3) ducito tangentes , eafque produc, ut fingulse dutc vicina:nbsp;angulos conftituant A, B, C, E, F; eritque illudnbsp;polygonum regulare, amp; circulo circumferiptum.

Demonst. In primis quilibet duo anguli funt sequa-les, c�m quilibet duo convex! tcqualcs fint ex con-ftruclione (146); practerea, quselibet duo latera xqua-iia habet : nam AA aAb, bBc, cCe amp;c. funt ifofcelia amp; congrua; quodlibet eft ifofcele : nam anguius Aaltnbsp;eft = angulo Aba : quifque enim menfuratur are�snbsp;�b (140); ergo A aAb eft ifofceles ( 171) : fimiliternbsp;anguius Bbc eft = angulo BcZgt; : quifque enim menfu-ratur } are�s bc (140)1nbsp;nbsp;nbsp;nbsp;eft quoque Ifof

celes , amp; ita porr�; ergo latus Aa = A/gt; j B� = Bc; Cc = Ce amp;c.; prsterea bafes omnium illorum AAnbsp;, erunt inter fc a:qualgs (35)? fi-cut amp; anguli ad bafes

lilas confhimti (im� amp; AA illa funt omnia inter fc s^quiangula; congruunt igitur AA qusclibet (190);nbsp;oirmia igicur latera aA, Kb, iB, Bc, cC, Ce amp;c. funtnbsp;inter fe scqualia; quoniam ergo bina talia latera com-ponunt fingulum polygon! latus, fing-ula polygon! deli-neati latera tcqualia' fint oportet. fgitur polygonumnbsp;delineatum regulate eft, amp; circulo circumfcriptum.nbsp;e.

224- ScHOLiON I. Latera ejurdem Circutnfcripd, aut asqualium Circumrctiptorum , a vercice ufquc ad pimfta tangentite, sequa-lia funt.

�225. ScHOLioN II. Circulus fpeftari potcft ut polygonum regularc, cujus latera infinit� parva funt ; etenim pcripheria componitucnbsp;reftis infinit� parvis (4), atque proinde inter fe �qualibus;nbsp;ha; qu�que refta; continuo , amp; aqualiter centrum versus incli-nant i ergo anguli omnes, quos' intercipere concipiuntur adnbsp;peripheriam, funt qu�que acquales; ergo �t polygonum regula-re circulus haberi poteft ( 196).

ARTICUL�S II1.

226. nbsp;nbsp;nbsp;Lineac AF, BG, Cl amp; EK (fig. 4) propor�o-naics funt, dum prima eft ad fecundam., ficut! tertianbsp;eft ad quartam; five dum eft AF:BG = C1;EK.

227. nbsp;nbsp;nbsp;7'res linccz, v. g. AB, CF, OG (^fig. 5)nbsp;cuntur proponionaks, dum prima eft ad fccuudam,nbsp;ficut! fecunda eft ad tertiam; five ft fuerit AB:Cb =nbsp;CF:OG;' hocque calu eft CF Media propordonalis inter AB amp; OG.

228. nbsp;nbsp;nbsp;Re�la AC Qfig. 6) divifa dicitur media amp; extrema ratione f dum tota AC eft ad partem majorcm

229. nbsp;nbsp;nbsp;Figures, (jujccumque Similes funt, 11 conftentnbsp;numero lacerum ajquali, latera lingula linguiis homo-logis fint proportionalia, amp; anguli, lateribus homolo-gis coinprchenli 5 aiquales.

230. nbsp;nbsp;nbsp;Demonst. Concipe lineam XK, item KI divi-fas in inlinit� parvas partes, inter fe proinde requa-Ics; amp; a fingulis pun�lis duftas parallelas rcfpe�lu KLnbsp;amp; IC : in quot partes atquales divifa fueric XK, innbsp;totidem partes, inter fe a^quales, OL divifa qu�quenbsp;crit (116); limiliter, in quot partes, inter fe sequa-Ics, divifa fuerit KI, in totidem partes, inter fe a;qua-les, divi�i erit LC (116); igitur eft XKiOL = KI:LC.

232. nbsp;nbsp;nbsp;In A ABC (^fig. 8 ) 11 fit OZ parallela ad BC,nbsp;eft AZ:CZ = A0;0B; patec ducendo per A paralle-lam ad BC.

THEOREMA II.

In A ABC Cfig- 8)yi AZ:ZC = AO:OB ji OZ dlvidat latera AC amp; AB propordonali-ter^, ejt OZ parallda BC.

234. Demonst. Condpe AZ, amp; CZ divifas in partes aliquocas, a;quales inter Ie : paritcr imaginare AB divifara in tot partes aliquotas, in quot AC eft divifa:nbsp;tune a Z imaginare duci parallel am r�fpeau CB, v.g.nbsp;recStam ZL; amp; crit ACrAZ=-AB;AL C233); quo-niam igitur, ex hypothefi. componendo, eft AC:AZ =nbsp;AB : AO; confequitur cfle AL = AO; itaque parallelanbsp;illa cadit in O : igitur ft ZO dividat latera AC, amp; ABnbsp;proportionaliter, ea eft parallela ad bafm BC. Q^-e.d.

amp;c.

236. nbsp;nbsp;nbsp;Demonst. a pun�lo Z due ZX parallelam AB;nbsp;eft OZ = BX (198 amp; 207}; prseterca eft AC:AZ =nbsp;BC.BX (233); ergo eft AC:AZ = BC;OZ; Q-e.d.

tricum, quo cafu B = 0, OZ parallela BC; vel B L = 180� (210); hoe igitur cafu, quoniam L = 0nbsp;lt;^iop), B 0= i3o�. Q. e. d.

238. nbsp;nbsp;nbsp;Si Cj%- 10) A = a; B = b, amp; proinde C=c;nbsp;cft AB:aA = BC;�c = AC:ac.

Demonst. Triangulo ABC imponatur A abc, ita ut a cadat in A, amp; linea ac in lineam AC; ac proinde , c�mnbsp;nbsp;nbsp;nbsp;A, cadet ab in AB; c�m ergo 6 = B,

cft BC parallela ad bc C11.5); habetur igitur AB: ab = BC:ic = AC:ac (233 amp; 236). Q^-e. d.

239. nbsp;nbsp;nbsp;Dum duac tranfverfaj, v. g. IK k. EL (fig. ii)nbsp;l�l� inter parallelas AB amp; CF lecant, fegmenta funtnbsp;proportionalia ; five eft 10 ; OK = EO : OL : etenimnbsp;I = K; L = E C109); O = X ( 85 ) ; adcoque AAnbsp;lOE amp; KXL funt acquiangaila; ergo kc.

Dum tr'ia latera uniiis A funt proportionalia tribus la-tcribus alterius A;duo illa AA funt fimilia.

240. nbsp;nbsp;nbsp;Dico, fi Cfi^� 12) fit AB:ci = BC:ic = AC:flc;nbsp;efle duo iUa AA fimilia, atque adeo A = d; B = b;nbsp;C = c (229).

) Demonst. Sit AL = ab; due LK parallelam BC; erit AB:AL = BC:LK = AC:AK (238); quoniamnbsp;igitur, ex hypothefi, eft AB:�i = BC:k = At-.ac; k

ex conftrudlione ai/ = AL; erit �c=:LK, ac = AK; igitur A aic eft asquiiaterum A ALK; proinde con-gruunt ( 189) ; ergo A � a; L = i; K = c; fed eftnbsp;L = B, K = CC 108)5 ergo B = �y C = c; ergonbsp;AA ABC amp; aic fimilia funt (22^). e. d.

Si in diiobus AA uniis angulus primi Jit aqualis unl aiigulo Jecundi, atque anguli illi aquales formenturnbsp;per latera proportionalia; duo illa AAnbsp;Jimilia fint.

12,41. Dico CfiS� �o)j A = c; amp; fit AB:fl5=; AC:ac; effe B = b, C~c amp;c.

Demonst. Ponatur a in A, ita ut ac cadat in AC; quoniam angulus a � A; cadet ab in AB (75); igitur eft AB : Ab = AC: Ac ; eft ergo bc parallela BCnbsp;C ^35)5 ne proinde B = b, C = c ( 108); igitur AAnbsp;ABC amp; abc fimilia funt (238^. e. d.

Ji duo AA habtant duo latera proportionalia, amp; utrim-qiie unum angulum , lateribus proportionalibus op~ pojitum, cequalem; AA ijla jimilia fint, velnbsp;anguli refiantibus lateribus proportionalibus oppofiti, fimnl f�clutit i8o�.

Demonst. Sit AK = ac , AL � ab; A AKL conduit cum A abc (191); igitur UC � bc; quoniam igitur, ex hypotheft, eft AC-.ac = BC:bc, erit AC: AKnbsp;= BC: LK; proinde vel LK eft parallela BC, velnbsp;JI-i-L=:i8o� (237); primo cafu erit B=L, amp; C=:K

(108); quoniam igicur L = 6', amp; K = c, erit B = ft, C = c; adeoque A ABC erit acquiangulum , ergo fi-mile A abc. Secundo ver� cafu, cdin L = A, B 3nbsp;facient 180� e. d.

In AA Jimilibus, altitudines QreJpeSiv� ad bajes homo^ logas) fuut lattribiis proportionaks.

C = c, dico pcrpendicularem AE, du�tam ex A ad BC (produ�lam fi opus), fe habere ad perpend:cula-rem cl, du�am ex a ad Ac ( produ�tam � opus ), fi-cut AC:ac amp;c.

Demonst. Sint B item C acuti in fig. la; B obtu-fus in 13; amp; C obtufus in 14. Perpendicularcs ex A item. a dudsc cadent ut in figuris (174); quoniamnbsp;C = c;E = I (funt cnim ambo re�li); ent angulusnbsp;CAE=angulo cal ( 165); ergo A ACE eft scquian-gulum, ergo fimilc A cal; eft itaque AC:ac = AE:aInbsp;(238); fed c�m A ABC fimile fupponatur A aAc, eftnbsp;ACtac = AB-.aA = BC:Ac (238), erit igitur etiamnbsp;AB:aA = AE:flI; item BC:Ac=?AE:aL ^ c. d.

244 ScHOLiON. Si V. g. fol�m habeamr C=:c, femper habebicur AC:�c = AE:aI; praterea erit BC:iic fieuti perpendicularisnbsp;ex B ad AC du�;a,ad perpcndicularera cx b ad ac duttam ; atnbsp;altitudines, feu perpendicularcs �Ik , nou funt proportionaksnbsp;leliquis lateiibus AA ABC amp;; abCj quia hsec fimilia non fup-ponuntur.

Dum in A Unea duSa ez vertice alicujiis anguU, hunc bifariam. fecat; ca dividit bajin eadem ratione, �tnbsp;funt inter fe latera., direB� oppofta baf�osnbsp;partibuSf amp; pcr quoe angulus Uknbsp;formatiir.

Demonst. Producatur CA ufque in K, ita ut AK ^ AB; dudla FK, A KAF eli ifofcele (157); F = Knbsp;C171 3 ; led angalus BAC, feu I L, eft = K Fnbsp;C ) ; quoniam, igitur, ex hypochefi, L = I, eftnbsp;F = L; proinde KB eft parallela AO (115); atquenbsp;adeo AC: AK feu AB = CO; OB ( 230). e. d.

anguli dividit bajin in eadem ratione, ac funt latera, qiiibus angulus ille formatur, amp; qute di-reS� oppojita jiint bajios partibus; angu-ius Uk bifariam ep divijus.

Demonst. Producatur rurs�s CA ufque in K, fit-gue AK = AB ; dudta KF; actento l�ppofito, eric CA;AK = OC;OB; ergo AO parallela KB (234) ;nbsp;igitur K = I C 108 ) ; led K = F ( 171 ), adeoquenbsp;F = I; fed quoniam KB parallela AO, eft F=L (109)5nbsp;ergo I = L. 0^. e. d.

247. Demonst. iquot;quot;quot;. B C=90�; ergoXrsB C^ item O = B C; jam vero X = C K ^ 0 = Bq-I,nbsp;Cl67); ergo I = B; C=L; O-X; igitur AA ACX,nbsp;AXB amp; ACB funt acquiangula, ac confequenter fimilia (238). Quod erat primum.

Demonst. 2quot;�- Quoniatn A ACX fimile AXB, eft BX: AX = AX: CX. Quod erat l�cundum.

Demonst. 4quot;�. C�m A AXB fit fimile ACB, eft BC;BA = BA:XB. Quod erat ultimum.

COROLLARI�M.

248. nbsp;nbsp;nbsp;Si Cfig. 17) BC fit diameter, amp; O re61us;nbsp;quoniam etiam eft A reftus (mcnfuratur enim | arcus BC), tres medias proportionales affignarc eft :nbsp;nempe AO inter BO amp; OC; AC inter BC amp; OC; amp;nbsp;AB inter BC amp; OB.

THEOREMA XIII.

249. nbsp;nbsp;nbsp;Dico, ft A Qfig. i8_) re�lus fit, efle BC'=nbsp;AB^-hAC�.

Demonst. Ab A ad BC demitte perpendicularem AX; cadet hscc intra bafin BC (174); igitur AB� = BCxBX;nbsp;amp; AC^=BCxCX (247); porr� BCxBX BCxCXnbsp;= BC=; ergo BC� = AB� -f- AC�. Q,. e. d.

� converjb : fi in A quadratum unius lateris Jit cequalc quadratis rejlantium laterum, angulus qppojitusnbsp;priori latcri reSlus efi.

Demonst. Quoniam BC, ex hypothefi, eft laws Maximum, A eft major quam B, item major quainnbsp;C C170); igitar B, item C eft acutus; itaque perpen-lt;igt;calaris, duda ab A ad BC, mtra bafin BC cadit C i74)-Quoniam X , item 0 redtus eft, AB� = AX� XB�;nbsp;amp; AC-' = AX� CX� C249O; ergo BC� = 2A'X� XB�nbsp; CX�; jam verb BC�= BX� CX� 2CX x BX; ergonbsp;2CXxBX=2AX�; igicur CXxBX = AX�; adeoquenbsp;eft CX;AX = AX:XB; fed eftO = X; ergo AA COAnbsp;amp; BAO fimilia font C 241 ) j igitur B = I; C = L ;nbsp;proinde C B = I L; ergo I L, feu A = po�.nbsp;e- d.

Tgt;um bina chorda fife in Circulo ficant^ earumdem fig�-menta fiint reciproc� proportionalia.

Demonst. Dudis chordis BC Sc AF, eft C = A, B = F, 0 = X; igitur AA BCO amp; AFO funt a:quian-gula; ergo fimilia C 228 ) ; adeoque eft OC: OA �nbsp;OB ; OF = BC : AF. Q^. e. d.

052. ScHOLiON I. Si (fig. 19 ) habeatur unum ex fequentibus: 1� OC = OA; 2� BO = OF; 3� chorda BC = chord� AF,nbsp;vel, quod idem eft, arcus BC=:arcui AF; omnia reliqua fe-quuntur.

053. ScHOLiON II. Si chorda AB fit = chord� CF. quoniam tunc vel arcus AC = BFvel arcus BC = AF (i

ti, V. g. OA, eft O Centrum : nam eft OF � OL ( 25a ); igitur CF = AB ; ergo CF etiam eft diarneter (34); atquinbsp;punftum interfeitionis duarum diametrorum elt Centrum (29);nbsp;ergo amp;c.

K a

theorema XVI.

Tangens AB (fig, 20) eft media proportiona�is inter totam jecantem BC, amp; fecanttmnbsp;exteriorem BO.

255. Demonst. Du�lis chordis AO amp; BC, C men-furatur \ arc�s AO (138); item angulus BaO men-furatur f arctjs AO (140); func ergo duo ilii anguii inter Ie a;quales. Angulus BAC meniuratur | arc�snbsp;AOC (140), item angulus BOA meniuratur ^ arc�snbsp;AOC C ^39)� �^rgo duo anguii l�nt quoque internbsp;fe ajquales; B praeterea eft communis; eft ergo A BaCnbsp;scquiangulum A ABO; igitur amp; ei �mile eft (238);nbsp;ergo eft BC: AB = AI3: BO. e. d.

356. Si itaque forct AB = OC, quoniam haberetur BC: OC = OC: BO, re�ta BC divifa effet medid amp;nbsp;extremd ratione (228).

057. ScHOLiON. Si (y?g. 20) fic AB tangens, amp; arcus AKC 180� i tres mediae ptoportionales habentur-, in primis talis eftnbsp;AB inter BC amp;; BO (255)-, amp; quia, du�is AO amp; AC, angulus BAC, item O reftus eft, eft AO media proportiona�isnbsp;inter BO amp; OC, amp; AC talis eft inter BC amp; OC (247).

THEOREMA XVI L

Jn A ABC (fig. 21 ) fit B reSus : erit reSangulurrt CJeu produSum^ fuper CA-l-AB, amp; fiuper diffk^nbsp;rentia hypotenufie CA amp; catheti BA,nbsp;aqiiale BC�.

258. Demonst. Ex A, ut centro, intervallo AB , due cireulum : produc CA ufque in L; erit CL =nbsp;CA -p AB; amp; CO erit differentia CA ad AB; igiturnbsp;eft CL, feu CA-f AB:BC = BC:CO (255) : ergonbsp;CA-{-ABxCO==BC\ 2st.d.

Demonst. Du�lis CL amp; BK,AA ALC amp; AKB funt sequiangula : nam C = B : quifque enim men-luratur \ arc�s KL ( 138 ); angulus ALC = angulonbsp;AKB; quia quilibec menfuratur | arc�s BLKC C^S?)�nbsp;quociam itaque A communis eft, duo ilia AA I�untnbsp;aquiangula, ergo amp; fimilia (238); eft igitur AC:ALnbsp;= AB:AK. il-cd.

In Ellipji, quadrate Ordonnatarum Qfic audiunt perpendicular es^ a peripheria ad axin quemlibet du3ce, in Omni figure ciirvilinea ) ad majorem axin, pro-portionalia Junt reSangulis Jiiper partibusnbsp;correjpondentibus majoris axis.

260. Tab. VT. fig. i. Sit ACBDA Ellipfis : AB axis major : CD axis minor; adeoque 0 in Centro redus.nbsp;G Item L fint foci : RP item El ordonnatac : dico efquot;nbsp;fe RP*;APxPB = EI*:AIxIB.

Demonst. Due RL : a G per R redam ita produc, ut RZ = RL. Ex R, intervallo RL, mediarp peri-pheriam ZLHQ deferibe; fit GM = MZ; erit MR = �nbsp;differentiai inter GR amp; RL; GQ eft tota differentia �nbsp;five eft f MR. Quoniara GO=OL (47 ), amp; HP==

C126); eft OP I � GH; eft autem GZ: GL = GH: 99 c 259); amp; capiendo terminorum medietates, eritnbsp;GM: OL = OP: MR; cumque fit GZ = AB, affu-mendo medietatem majoris axis, loco GM; babebisnbsp;OB; OL = OP : MR; invertendo OB: OP = OL: MR;nbsp;componendo OB: OB OP (feu AP) = OE :OL MR;nbsp;sut invertendo OB : OL = AP: OL MR; ulterius com-ponendo OB:OB OL (feu ALj) = AP;AP-fOL MR;

led AP = GM OP; itaque eft OB : AL = AP: GM OP OL MR ; quoniatn ver� GM MR = GR;nbsp;amp; OL OP = GP; tandem habetur OB:AL = AP:nbsp;GR GP; atque in hac proportio'ne pottrerous terminus cll fumma hypotenufsc GR amp; catheti GP A reonbsp;tanguli GRP.

Refumendo fuperioTem proportionem OB:OL = OP: MRi habetur dividendo OB:OB�OL = OP:OP � MR;nbsp;feu invertendo OB: OP = OB � OL : OP � MR ; amp;nbsp;ulteri�s dividendo OB;OB �OP (feu PB) =OB - OLnbsp;(feu LB):OB �OL �OP MR; fed OB = GM;nbsp;amp; �OL �OP = �PG; igitur OB:PB = LB:GMnbsp;-f MR � GP; atque ultimus hic terminus eft differentia hypotenufsc GR, amp; catheti GP A rectangulinbsp;GPR ; haccee differentia vocetur x, amp; habebitur OB:nbsp;PB = LB : X.

Refumatur OB: AL = AP: GR GP , atque ulti-mus hic terminus vocetur r, quia ipfe eft fumma hy-potenufe GR amp; Catheti GP; eritque OB;AP= AL:j; amp; multipUcando terminos hujus proportionis, per ter-minos hujus OB:PB = LB:x; habebitur OB�;APxnbsp;PB = AL X LB : r X x; atqui fumma hypotenufa; amp;nbsp;catheti GP multiplicata per earumdem differentiamnbsp;eft sequalis RP� ( 258 ); ergo OB�: AP x PB = AL xnbsp;LB: RP�; igitur eft� RP�: AP x PB = AL x LB: OB�.

Si a G per E redtam agas, donee ipfa tequalis fit ducendaa EL, atque ex E, intervallo EL, circulumnbsp;ducas , fimili ratiocinio reperies effe EI�: AI x IB =nbsp;AL X LB : OB�. Ergo eft RP�; APxPB = EI�: AIx IB.nbsp;Et fic de cacteris. Q^. e. d.

261. Quoniam igitur OC Ordonnata eft, atque adeo habcatur OC�: AO x OB j feu OB� = AL x LB; OB�;nbsp;eft OC� = ALxLB.

a�a. Dudo ex O femicirculo AKB, amp; produfta PR, erit PS ordonnata circuli, cujus AB, axis major,nbsp;diameter ell; eritque PS� = AP x PB (248); eratnbsp;etiam 00quot;� = ALxLB (261); igimr, c�m fupra ha-beretur. RP�: AP x PB = AL x LB: OBquot; ; habebicurnbsp;RP� : PS� = 00=�: OB*; adeoque RP : PS = OC : OB ;nbsp;Piniliter produdta IE, crit \f ordonnata majoris circuli,nbsp;amp; lp = AI XIB ; fupra ver� erat EP : AI x IB =nbsp;AL X LB : 0B=�; itaque eft .EP : !� � = OC*: OB*; acnbsp;proinde EI: !�= OC : OB; amp; ita de cscteris. Itaque qusc-libet ordonnata ellipfeos ad majorem axin, eft ad cor-refpondentem ordonnatam circuli fuper majori axi, ft-cut I minoris axis ad ~ majoris; feu ficut minor axisnbsp;ad majorem.

In. Ellipji, quadrata Ordonnatarum ad ninorem axin, Jiint proportionalia reSangiilis fuper partibusnbsp;correjpondentibits minoris axis.

263. Si VN amp; FX fint Ordonnat� Ellipfeos ad mi-norem axin CD, dico effe VN*: CN x ND = FX*: CX xXD.

Demonst. Duda VT ordonnata ad majorem axin AB, eft AOxOB feu AO*:CO*= ATxTB:VT*;nbsp;eft autem AT x TB = AO� � TO* : (nam, quoniamnbsp;AB dividitur scqualiter in O, amp; insqualitcr in T, eftnbsp;Carith.) AT xTB = AO* � TO*); igitur AO*:CO* =nbsp;AO* � TO*; VT ; amp; permutando, AO*: AO* � TO*nbsp;= CO*;VT=; amp; dividendo AO*: AO* � AO* TO*nbsp;== CO*; CO* � VT* (datur autem fignum quadra-to TO , quia refecando AO*, mmis fuiflet fublatum,nbsp;cum fol�m refecandum fit AO* � TO*; ideoque re-

ftituitur TO�, quod nimis fuiflec fublatum); ver�m jyO� � AO� reducimrad;^ero;itaque habetur AO�:TO*nbsp;= CO�:CO� � VT; at TO� = VN� (eft cnim Onbsp;VNOT reiftangulare, adeoque fymmetricum); amp; CO�:nbsp;CO� � VT� = CO� � NO�; pra;terea C arith.) CO*nbsp;� NO� = CN X ND (quia CD dividitur bifariam innbsp;O, amp; inaiqualiter in N ); igitur AO�: VN� = CO�:nbsp;CN X ND; feu VN�: CN x ND = AO�: CO�.

Simili modo demonftrabitur FX�: CX x XD = AO*: CO�. Igitur eft VN�; CN x ND = FX�; CX x XP.

c. d.

264. Dudla ex O media peripherid CYD; re�te 5N amp; cX, perpendiculares ad minorem axin CD, ordon-nam funt Circuli, cujus diameter eft minor axis; erit-que Zgt;N� = CNXND; item aX�= CXxXD (248);nbsp;at erat fupra VN�; CN x ND = AO�: CO�; ergo eftnbsp;VN�: AN� = AO�: CO�; igitur VN: AN = AO : CO.nbsp;Pariter erat FX�: CX x XD = AO�; CO�; ergo eftnbsp;FX�: aX� = AO� : CO� ; itaque eft FX: aX = AO :nbsp;CO. Igitur Ordonnatcz quaclibet Ellipfeos ad minoremnbsp;axin, ie habent ad correfpondentcs Ordonnatas circulinbsp;fuper minori axi, ficut | majoris axis ad | minorisnbsp;axis, five ficut major axis ad minorem.

PROBLEMA I.

3�5. Resolutio I. Fiat, pro libitu, angulus X per reaas XH amp; XL indefinitas :

in. Ducatur OL; deinde � HK parallela ad OL ; eric LK tertia proportiomalis petita : nam eric XO: 0E[nbsp;= XL feu OH: LK ( 232). Ergo AB; CE = CE:liL

III. Due re�lam BF; tum k pun�lis G amp; K re�his O� amp; XK parallelas ad BF; eritque AB divifa in ea-dem proportione, qak AF (230), feu qu^ CE.

III, nbsp;nbsp;nbsp;In C erige perpendicularcm, quam ufque in punctum O peripheri'32 protrahe; amp; erit CO n^edia pro-portionalis defrderata.

Demonst. Duc chordas EO amp; AO, atque concipe integrann peripUeriam delcriptaro elle; augulus E�A ,nbsp;qui menfuratur f penpheriai (138), elt 90� : ergo,nbsp;cum anguii ad. punclum C fint qu�quc recti, eft COnbsp;media proportion alls inter AB amp; EC (^47)- Qa

tur BF : AB = AB: BG ( 255 ); igitur BF � BA; EA � BA � BG : BG; eft autem BF � BA = BG = EK;nbsp;amp; BA � BK=AK : igitur his fubftitutis, erit BK:nbsp;Ba = AK: BK, feu alternando BA: BK = BK . AK ;nbsp;ergo AB divifa eft nicdia amp; extrema ratione (2.28).

270. Resolutio I. Lineam AB divide media amp; extrema rauone ( 269) in K; litque AK pars minor

II. nbsp;nbsp;nbsp;Ex A item K, intcrvallo aequali BK, fiant incer-fectiones in Z :

Demonst. Ducantur ZK item ZB : crit KZ = AZ C25), item KZ = BK, ex conftrudione; ergo BK,nbsp;KZ amp; AZ'lunt inter Ie aiquales. Quoniam igitur, exnbsp;cnnitructione , eft AB � BK = BK: AK, erit etiamnbsp;AB : AZ = AZ: AK; cum itaque angulus A per lateranbsp;ilia proportionaiia confiituatur, A BAZ amp; .ZAK fimilianbsp;funt (241); ergo K = Z; B = I; led c�m BK = KZ,nbsp;eft B=L (171); ergo 13=1; itaque B | � angulo Z;nbsp;igitur B eft etiam | � A (nam A=K=Z); fed an-guii B-j-Z-f-A= 180� (164); ergo A eft 72�. Q^-e.d.

271. ScHOLiOK. Si angiiilutn 72� bifariam Teces; angulum 36� ob-tinebis; atq^ue hunc bifatiam dividens, habcbis angulum 18'� amp;c.

Super data OX (fig. 8 ) confinicrc A fimile A ABC dato, fiimpto OX pro latere homo logo AB.

272. Resolutio. Fiat 0 = A, X=B, amp; nbsp;nbsp;nbsp;11=0

( 105); ergo A XHO eft tcquianguium, adeoque �-aile A ABC (238;.

�. 11.

De ccetcns Figuris [milibus.

Omnia Polygona regularia ejufdem fptciei Qadcoqiie O omnes circuli')^ Jiint Flgurce jimiles.

273. nbsp;nbsp;nbsp;Patet evidenter ex definitione polygoni regu-laris (196), amp; figurarum limilium io.2g'j.

In Polygonis regularibiis ejufdem fpeciei, latera fint radiis obliquis atque reSis proportionalia.

274. nbsp;nbsp;nbsp;Demonst. Sint v. g. duo Hexagona regularianbsp;A amp; B (fig. 9). E centro cujullibet dudis radiis obliquis, A refolvitur in fex AA congrua inter fe; fimi-liter B in totidem AA inter fe congrua dividitur ;nbsp;critque fingulum A in hexagono A aequiangulum,nbsp;ergo amp; fimile C 238 ) , cuilibet A in hexagono B; igi-tur eft AC: BK = CO; XK. OH item XL fit radiusnbsp;tedlus, erit OH altitudo communis omnium AA polygoni regularis A; amp; XL erit altitudo communis omnium AA polygoni regularis B ; radios autem redtosnbsp;ejufdem polygoni sequales effe inter fe demonftratumnbsp;eft C2.21) atqui ill� altitudines, feu radii red:i,pro-portionales funt lateribus polygoni (243); ergo etiamnbsp;eft AC: BK = OH; XL. Q. e. d.

275. nbsp;nbsp;nbsp;Si A amp; B fuerint polygona regularia ejufdemnbsp;fp�ciei, perimeter polygoni A ad perimetrum polygo-ni B, �t radius obliquus vel redus A, ad radiui�nbsp;obliquum, vel rectum B. Et � converfo.

276. nbsp;nbsp;nbsp;Certa fumma laterum polygoni A ad eandemnbsp;fummam laterum polygoni B, �t radius obliquus, velnbsp;redus A, ad radium obliquum, vel, re�lum B. Et �nbsp;converfo.

277. nbsp;nbsp;nbsp;Quoniam Circuli funt polygona regularia ejulP-dem l'pedei ( 225), peripheria circuli A Qfig. 10) adnbsp;peripheriam circuli B, �t radius A ad radium B; amp;nbsp;� converfo.

278. nbsp;nbsp;nbsp;Si arcus AC tot graduum fit, quot graduumnbsp;eft arcus BL, arcus illi proportionales funt circulorumnbsp;radiis. Et � converfo.

279. nbsp;nbsp;nbsp;Chordae AC amp; BL, quac arcus illos fimilesnbsp;fubtendunt, qu�que proportionales funt radiis illorumnbsp;circulorum : amp; � converlb.

280. nbsp;nbsp;nbsp;Se�lor AXCHA etiam fimilis eft fedori BOLFB:nbsp;tot enim funt latera in arcu AHC, quot funt lateranbsp;in arcu BFL; concipiendo itaque, ex X amp; O, ad illanbsp;latera, dudos radios obliques, quilibet fedor in aequ�nbsp;multa AA fimilia refolvetur.

281. nbsp;nbsp;nbsp;Et quoniam A AXC fimile eft A BOL, feg-wenium ACHA �m�e eft fegmento BLFB.

theorema III.

Polygona guceciimque Jlmilia^ Ji rtfolvantur in pef diagonaks ad angulos homologos duaas 3 eruntnbsp;AA homologa fmilia.

2.82. Si duo Pentagona Cj%- n) fimilia fuerint , ita ut A = a, B = Zi j C = c, E = e, F = �,' amp;nbsp;practerea lit ASf.ab � BC: bc = CE; ce = EF : ef =nbsp;%'A:fa; duCtis diagonalibus AC, ac j AE, ae; �iconbsp;A ABC liiuile A abc; A ACE limile A ace; amp; Anbsp;AEb' limile A aef.

Demonst. Quoniam B = i, hique anguli forman-tur per latera proportiunaiia, AA ABC amp; abc lunt fimilia (^40; pariter , cum anguli F amp; �, inter fenbsp;�cquaies ex hypotheli,'conltituantur per latera propor-tionalia, A AFE limile eft A afe (241); ell; igiturnbsp;angulus ACB = angulo acb; h angulus AEb = an-gulo aef; quoniam igitur, ex hypotiioll, eft C = cnbsp;amp; E = e, ent angulus ACE = angulo ace, amp;c angulus AEC = angulo aec; adeoque angulus EAC = angulo eac (165) ; igitur A ACE eft tcquianguium,nbsp;adeoque (238) limile A ace. Q^. e. d.

THEOREMA IV.

converfo : Ji duo Polygona quaciimque in cequ� multa AA Jimilia rcjolvi qucant, Foly~nbsp;gona fimiiia fine.

' 283. Demonst. Propter angulos aiquales AA limi-lium , ctiam anguli polygonorum aiquaies erunt ; Sc quoniam latera iftorum polygonorum funt qu�que latera iftofum AA finrilium, etiam polygonorum latera homologa proportionalia fint oportet; igitur amp; jnfanbsp;polygona fimiiia funt (nbsp;nbsp;nbsp;nbsp;gt; Qi c. d.

Super-data reSa HG (fig. 12) confiruere figuram fimi-km. ABCF, fumpto HG pro latere homo-logo lateris AB.

H. Fiat angulus HGI = BAC; item angulu^ IGK CAF, ducencio GI amp; GK hue ulque indefinitas; aucnbsp;pottus lineai litai oceuk�, v. g. plumbagine, vel cuB'nbsp;pide, leviter exarenmr.

III. Fiat H = B; atque in I, ubi recla Hl oceurrit diagonali GI, fiat angulus GIK = angulo ACF ;nbsp;amp; crit ? GHtK nhiile ? ABCF : conftat enim duo-bus AA tequiang�lis, feu limilibus, duobus AAnbsp;quadrilateri ABCF.

PROBLEMA II.

Super reSa ce ( fig. ii ), i/r latere komologo ai CE, Fentagoniim conpruere pmilenbsp;ABCEF dato.

385. Resolutio. Du^'is diagonalibus AC amp; AE, fac A cae fimile A CAE; deinde A eba fimile A CBA;nbsp;amp; tandem A efa fimile A EFA j entque abcef pen-^nbsp;tagonum quaifitum

SECTIO SECUNDA

SPe�lavimus h�c ufque Polygona �t figuras, agendo de lineis atque angulis, quibus terminantur : nuncnbsp;de iis agendum, i� quamp; Magnitudinibus, 2,� q�d Pla-nis.

CAPUT PRIMUM

s86. Superficies eft extenfio, in qua bina; Dimenfio-nes, Longitude fcilicet atque Latitude confiderantur; generaturque, dum Linea a termino ad terminum mo-vetur.

287. Nullam proinde Profunditatem habet; nequ* cjus extrema extenfa dici peflunt in latum.

288. u4irea feu Superficies Figura eft quantitas, qu� magnitudinem Spatii, lateribus figursc comprehenfi�nbsp;exprimit.

289. Altitudo Parallelogrammi aut Trappezo�dis, eft Perpendicular is, ab uno latere parallelo ad aliudnbsp;demiffa.

290. Figurae dicuntur habere tandem Akitiidinem� ��m inter eafdem parallelas conftituta funt, aut con-flitui peflunt.nbsp;nbsp;nbsp;nbsp;Corol-

api. Omnia igitur AA vertice comraunicantia, amp; �quorum bates eidem re�te infiftunt (�t AA AOC Scnbsp;AOB fig. i6 Tab. V.) eandem alticudinem habent :nbsp;d��ta enim per verticem A parallela ad BC, in eer eai^.nbsp;dem parallelas AA iila confticuca foren:.

292. nbsp;nbsp;nbsp;Si AB Cfig. 13) ad BC perpendicularis, mo-tu libi parallelo ita moveri concipiatur, ut pun�lumnbsp;fi re�lam BC percurrat; in defcripta Saper�cie ABCFnbsp;rectangulari, toties eft invenire re�lam AB, quo: fuuCnbsp;pun�ta in re�la BC.

293. nbsp;nbsp;nbsp;Igitur Area ? Re�langularis habetur, unumnbsp;iatus in aliud coutiguum ducendo.

294. nbsp;nbsp;nbsp;Quoniam vero in Quadrato latera qusKumqucnbsp;�cqualia funt, ducendo unura Iatus in fe ipium, Areanbsp;Quadrati obtinetur.

295. nbsp;nbsp;nbsp;Schol. Quemadraod�ra rnenfurae Lfinearuin funt minores lineas

nots, V.g. VtrgCL (qua: in Brabanda = ao pedibus); Hexape~ da (quae in Gallia =3 6 pedibus )�,nbsp;nbsp;nbsp;nbsp;adiequans 10 pe

des amp;c.; quas menlurae inl'uper lubdividumur in pedes, pes ia poUicesamp;c.; ita menfura: fupcrficierum funt minores Superficiesnbsp;quadraca� cognic�. V. g. fl 4-BCF (fe. 14)nbsp;nbsp;nbsp;nbsp;fit,

atque AB duos pedes xquet, erit Superficies , leu Area ABCF quatuot pedum Quadratorum : eceniin duclis EG atque KB ,nbsp;qua: latera quadraci bifariam fecent, exurgent quatuor miuoranbsp;Quadraia, laceraque finguli pedem acquabunt : amp; quoniam qui-libet pes hie Ibleat dividi in 10 partes Kquales, qu^ Fcdlises,

audiunt, divifo quolibet latere Quadrat! KBEO in lo polliccs, duftifque tineis, ut vides , patet quemlibei. pedem quadratumnbsp;100 continere pollices quadrates. Quemlibec pollieem iusdivi'nbsp;dimus in to lineas, atque adco quiiibet pnilcx quadratus lOOnbsp;lineas quadratas continet: igitur Pcs quadiatus looo lineas qua-dratas compJedtitur amp;c.

jirea RhomU, ant Rhombo�dls ABEH C ^g- i 5 2 amp; 3 Tab.nbsp;nbsp;nbsp;nbsp;efi it quale ? reSangulari ABCG for-

(utpg. a); vei ultra F (ut pg. 3 ); ? ABCG , quo-niam anguli quiiibet oppofici xquales lunt (funt enim redti omnes^/eii: rymmctricum (^o8); igitur BC=AGnbsp;(1943 : ergo A BCF congruit cum A AGH (ipo):nbsp;etenim anguius BCF eft aequalis angulo AGH (funtnbsp;enim ambo re�ti J; anguius Cl B = H, cum AH fitnbsp;parallela BF (108); itaque anguius CBF = angulonbsp;GAH ( 1659; funt igitur duo ilia A A qu�que sequa-lia; addendo igitur iingulo, in pg. i eandem partemnbsp;communem AbB; amp; in pg. 2 partem AGFB, erit (j rec-tangulare ABCG = parailelogrammo ABF1� In figu-ra vero 3, AA BCF, amp; AGH commune habent Anbsp;GXF; ergo ? BCGX = AXFH; addendo ergo utri-que A AXB, ? redlanguiare ABCG eft = paralleio-grammo ABb'H. Q. c. d.

297. nbsp;nbsp;nbsp;Igitur ducendo bafm AB in ahitudinem AG,nbsp;prodit area ABbH ( 293).

THEOREMA II.

298. nbsp;nbsp;nbsp;Dico aream A ABC Cpg- 4) effe | � produdlonbsp;�bafts BC per pcrpcndicuiarem AO.

Demonst. Duc af parallelam CB, amp; BF parallelam ad AC, erit ACBF parallelogrammum ( 198) f Anbsp;ABC (206); atqui area ACBF efc = fado baleos ABnbsp;in perpendicularem v. g. A� (_ 297 ); ergo area Anbsp;ABC = I BC X AO. e.d.-

299. nbsp;nbsp;nbsp;Si fuerit ABCG (fig. 5) trappezo'is, fic utnbsp;AG amp; EC lint parallelai, litque GO v. g. cjufdcm al-ticudo : area iilius eric = | BC 1 AG X GO.

Demonst. Dudla diagonal! BG, area A BCG = � BC X GO; amp; area A BAG = ^ AG x GO (298);nbsp;ergo area utriufque A limul, feu ABCG = 4 BC nbsp;i AG X GO. e.d.

300. nbsp;nbsp;nbsp;Demonst. � Centro Polygoni concipe ad an-gulum quemlibet Polygoni duclos radios obliquos; innbsp;totidem AA fueric relblutum polygonum, quoc funcnbsp;ejuldera latera; ericque area cajufque A ^ fadti bafeosnbsp;in altitudinem ; jam vero alcitudo omnium AA eftnbsp;= radio recto (patet ex demonftratione ad numcrumnbsp;2.22�); igitur omnium AA, feu polygon! area, eftnbsp;= I product! omnium laterum per radium redtum.

301. nbsp;nbsp;nbsp;Quoniam igitur Circulus polygonum eft regulate C 225}, atque ejufdem radius rpftus ab obliquo

p2 nbsp;nbsp;nbsp;Elementa Geometric:.

305. �ScHOLioN I. Circuli omnes inter fe fimiles Tunt (a73')-, func ergo petipherite omnes radiis feu diametris fuis proporcionales;nbsp;hoc eft, quse eft ratio unius circuli ad radium aiit diametrumnbsp;fuam, eadem fit oportet circuli cujul'cumque ad radium aut diametrum fuam. Quaa vero ea fit, exaft� hue ufque non inno-tuit ; props verum ramen, rationem peripheriae ad diametrumnbsp;ftatuic ^rchbmdas lt;22:7. exaftius alii 314: 100; aut in majo-Tibus circulis, ubi de minimis curandum eft, 3141� 10000. denbsp;qUibus poftea.

303. ScHOLiON II. Dum dicitur Circuli Ouetdraturam adinvtn-tam kite ufque non ejJe;)\oc unic� vult, determinatam non cf-fe in numeris proportionem inter peripheriam amp; diametrum ; neque lineam reftam affignari poffe , quas lit peripherite Circulinbsp;dati requalis : fi enim ea probari pol�it, quasrendo mediananbsp;proportionalem inter peripheriam amp; ^ radii, amp; fuper ea for-mando Quadratum, erit hoc Circulo dato arquale.

jirta Seaeris ejl = | producti arcus fcQoris per I radii.

304. Demonst. Concipe Secloris arcum divifum in infinitas partes tcquales, amp; � circuli Centro ad hasnbsp;partes radios dudos ; in infinit� parva AA fedoremnbsp;diviferis ; omnium illorum AA balls erit arcus Icdo-ris, communis vero altitudo erit circuli radius; atquinbsp;cujufque A area eft = produdi bafeos per altitu-dinem; ergo omnium area, quae amp; fedoris area eft,nbsp;= I produdi are�s fedoris per radium. Q^. c. d.

Circuli Segmentum ABKA Cfg- �) inquirere.

305. Resolutio I. Inquire aream fedoris AXBKA ; II. Dein aream A AXB, haneque fubducito ex area

ARTIC�LUS II.

THEOREMA I.

306. nbsp;nbsp;nbsp;V. G. triangulum A eft ad triangulum B, fi-cuti f produ�li bafeos A per akitudinem A, eft adnbsp;I produ�i bafeos B per akitudinem B; amp; confequen-ter ficuti fadum bafeos A in akitudinem A, eft adnbsp;fadum bafeos B in akitudinem B.

CoROLLAItlUM I.

C0B.0LLARIUM II.

308. nbsp;nbsp;nbsp;AA habentia eandem bafin, vel asquales bafes , funt �t akit�dines.

C0R.0LLAR.IUM III.

309. nbsp;nbsp;nbsp;Parallelogramma scqualis altitudinis amp; bafis,nbsp;scqualia funt.

C0E.0LLAE.IUM IV.

310. nbsp;nbsp;nbsp;Parallelogramma scqualis altitudinis funt ut bafes; amp; fi fuerint acqualis bafeos, funt �t akit�dines.

COR.OLLAR.IUM V.

311. nbsp;nbsp;nbsp;Si binsc dimenfiones, � quarum fado figurscnbsp;area prodit, fint reciproc� propprtionales, aresc jcqualesnbsp;funt. V. G. fi in duobus AA, aut duobus parallelo-grammis, bafis primi fit ad .bafin fecundi, �t akitudo

lecundi ad altitudinem primi; quoniam quatuor illi termini funt proportionales; eft factum primi in ulti-mum = fa�to fecundi in tertium; jam ver� pnmumnbsp;fa�tum dat duplum areac primi trianguli, auc femelnbsp;aream primi paraiielogrammi; amp; fecundum iaclum datnbsp;duplum areac fecundi trianguli, aut lemel aream fecundi paraiielogrammi; ergo amp;.c.

AA habmtia unum angiilum cequaltm, funt inter fh in ratione compojita laterum angulos iflosnbsp;aquales conjlituentium.

Demonst. Sint v. g. AI amp; OK altitudines AA; erin AC: OX = AI: OK ( 244); igitur CB x AC: XZ x OXnbsp;= CB X AI: XZ X OK; jam vero CB x AI = bis areasnbsp;ABC; amp; XZ X OK = bis area; OZX ; ergo bis Anbsp;abc : bis A OZX = CBx ACtXZxOX; igitur Anbsp;abc ; A OZX == CB x AC: XZ x OX. Q. e. d.

313. nbsp;nbsp;nbsp;Dum igitur duo Parallelogramma habent unumnbsp;angulum scqualem , quoniam ea funt inter fe, ficutnbsp;AA in qua; dividcrcntur du�tis' diagonalibus ex re-ftantibus angulis, Parallelogramma illa erunt inter fenbsp;�t re�tangula, feu produ�ta, laterum angulos iftosnbsp;acquales refpc�tiv� coniticucntium.

beant unum angulum acqualem, formatum per latera reciproca, ca = funt. V. G. � fitnbsp;nbsp;nbsp;nbsp;7) X = C;

AA Jimilla funt inter je in ratione duplicata Qfeu �t quadrata ) laterum homologorum.

315. Si ifig- 8 ) fit A = a, B = C = c; diconbsp;A ABC; abc = AB* : ab^ = BCquot;: bcquot;- = AC= ; cc*.

Demonst. Quoniam A = a, A ABC : A abc = AB X AC : aZgt; X cc ( 312 ); fed quoniam AB : AC =nbsp;ab: ac ( 238 ) , eft AB x AC : �3 X ac = AB x AB ;nbsp;ab X ab; igitur A ABC ; A abc = AB*: ab^; fed propter AA fimiiia eft AB* : ab^ � BC* : ^�c*.= AC*: cc*.

316. nbsp;nbsp;nbsp;Quaecumquc figuraj fimiles in scqualem nume-rum AA fimilium rcfolvi poffnnt (282), lateraquenbsp;fingula perimetri homologa Cfitisc ornnia inter lb Hintnbsp;in eadera ratione) erunt bales atque latera homologa illorum AA; igitur quaccumquc figura; fimiles funtnbsp;inter fe in ratione duplicata laterum homologorum.

317. nbsp;nbsp;nbsp;Et quoniam Circuli omnes fimiles funt, atquenbsp;in iis dimenllones homologac funt Peripheria, Diameternbsp;item Radius; Circuli omnes funt inter fe ut quadratanbsp;peripheriarum, diaractrorum, aut radiorum; im� etiamnbsp;ht quadrata arcuum fimilium, aut chordarum eoliiemnbsp;fubtendentium ; illi enim funt proportionales periphe-tiis , ha: vero circulorum radiis.

318. nbsp;nbsp;nbsp;Lineas omnes, ut in eodem Plano conftitutas,nbsp;hue ufque fuppolitum fuit, uti prxmonuimus (6);nbsp;nunc agendum de ficu quocumque Linex aut Plani,nbsp;ad fuperficiem planam.

319. nbsp;nbsp;nbsp;Concipe redtam BC (/ig. 9) liber� in a�re pen-dulam, ad quam XA perpendicularis fit, circa fe ip-fam, iit axi, ita moveri, ut pun�ta B amp; C fitum nul-latenus mutent : re�la AX deferibet fuperficiem planamnbsp;AGKLA, ericque re�ta BC ad illud Planum Perjpendi-cularis.

320. nbsp;nbsp;nbsp;Dum igicur reda, v. g. BX, perpendicularisnbsp;eft ad planum quoddam datum, ea quoque perpen-dicularis eft ad redam quamlibet in piano dudam pernbsp;X. Et � converfo : ft qua reda, v. g. BX fit perpendicularis ad redas omnes, in piano per X dudas, ericnbsp;reda BX ad planum perpendicularis.

221. In eodem plani pundo, unica poteft erigi; amp; k pundo extra planum dato, ad planum unica demitd

322. Perpendicularis breviffima eft , quae a pundo dato ad planum datum demitti poteft; atque adeonbsp;pundi a piano diftantia , perpendiculari inveftigandanbsp;cii

323. nbsp;nbsp;nbsp;Omnes perpendiculares ad idem planum funtnbsp;inter Te parallelac : concipe enim' per eoruradem extrema in plano redlam trajici; erunt anguli correfpon-�ientes c�m redli fint ) scquales.

C0E.�LLAK.1UM V.

324. nbsp;nbsp;nbsp;Re�laj cujufcumque in plano ducfta�, nequitnbsp;pars una elTe in plano, pars altera infra aut fupra :nbsp;li -duo igitur ejufdcm, redtai pun�ta fuerint in plano,nbsp;t�ta re�la in plano ent; quoniam duo illa punda U-gt;nbsp;lius redai fitum determinant (12).

THEOREMA I.

Tria piincla-, v. g. A, H, C ffig- io)gt; noTl conjijiunt in .eadcm reBa, Plani fitumnbsp;determinant.

325. nbsp;nbsp;nbsp;Demonst. Pun�la illa redis AB, BC amp; CAnbsp;connedantur; tum finge redara AB per fingula punp-ta redarum BC amp; AC moveri ; c�m ad C pervenc-rit, Aream feu Planum trianguli ABC luftraverit;nbsp;eruntque reda; iliac omnes, adcoque amp; tota fuperfidesnbsp;A ABC, eidem plano, quo continentur punda A,nbsp;B amp; C, in liften tes; immo in quemcumque fenfum rec-tas illas protraxeris, eidem plano conftantcr inflftantnbsp;oportet (324); igitur tria punda non in diredum po-fita, plani' fitum determinant. e. d.

326. nbsp;nbsp;nbsp;Si ver� in eadem reda confifterent, plani fitusnbsp;non innotefceret ; tune enim per tria i�a punda pof.nbsp;fet reda trajid, atque fecund�m hanc fupponi aliudnbsp;planum attingere; jam ver� pollet alteri plapo fic eflenbsp;contiguum fub inclinatione quacumque; ergo ejufdenanbsp;ficus minim� determinaretur-

327. nbsp;nbsp;nbsp;Tria puncfta, qua; non confiftunt in eadem li-nea re�ta, nequeunt efle diverfis planis communia ; finbsp;enim torent diverlis plains communia; quoniam eujuf-que plani litum determinarent (325)1 aeberet cujuf-que plani pofitio eadem efle, atque inddere cum po-Ikione alterius; quod eft contra hypothefm.

328. nbsp;nbsp;nbsp;Ergo interfe�tio duorum planorum eft lineanbsp;re�ta : interfe�lio enim pun�la erunt diverfis planisnbsp;communia; atqui pun�ta illa non poflunt non elle innbsp;eadem re�ta C327); ergo amp;c.

Demonst. Per pun�ta A, B amp; F determinatur fitus plani, per tria illa pun�ta du�ti (325); quoniam ve-r� pun�ta F amp; B fut�ra funt in ifto plano, tota recta FB m eo qu�que erit (324); ergo amp; pun�tum Onbsp;erit m eodem plano; igitur AC amp; BF funt in eodemnbsp;plano (325). Q^.e.d.

330. nbsp;nbsp;nbsp;Sit Planum ABRL la), atque huic aliudnbsp;FCflG ita luperimpofitum, ut cum ABCF, in initio,nbsp;coincidat (imm�, quoniam profunditate plana defti-tuantur (287 ), unum amp; idem conllituant planumnbsp;ABCF \ Pun�tis C amp; F manentibus immotis, feu fu-per re�ta CF ut axi, planum fuperius volvi finge, donee ad oppofitam partem CRLF pertingat, atque cumnbsp;eo coincidat

331. nbsp;nbsp;nbsp;Vicini anguli, per planum mobile amp; immobilenbsp;formaci, faciunt iiinul 180�.

332. nbsp;nbsp;nbsp;Angulum re�lum, feu 90�, plana conftituent,nbsp;live erit unura ad aliud perpendiculare, dum planumnbsp;mobile ita dilpolicum fuerit, ut non magis in AB,nbsp;quaVn in RL propendeat; l�u dum pundum H mediamnbsp;penpheriara dcl'cnpferit.

333. nbsp;nbsp;nbsp;Dum bina plana fefe intcrfecant, anguli adnbsp;verucem oppofiti aquales funt.

334. nbsp;nbsp;nbsp;Omnes anguli ad eandem re�lam ejufdem plant , tot quot conftitui poflunt per diverfa plana, faciunt �mul 360�.

unum, efi qu�que talis. ad aliud; amp; Ji qua recta ad duo Plana fit Perpendicularis, Plana ilianbsp;inter Je parallela funt.

335. nbsp;nbsp;nbsp;Demonst. 1* pars. Ut duo plana fint inter fenbsp;parallela, ca uhique aiqualiter a fefe mutuo diftent opor-tet, amp;; quantumvis produda concurrere nequeunt; rec-

igitur omnes in �no plano duda�, nunquam pote� runt concurrere cum ulla re�ta, in altero plano duda:nbsp;pone igitur duci re�tas, in eandem diredionem , pernbsp;perpendicularis extrema, erunt ill� inter fe parallelaj;nbsp;atque adeo anguli adiacentes interni facient fimul 180�nbsp;quoniam itaque reda xlla angulum 90� coq-N 2

ftituat cum. quibufvis re�Hs in uno plano per ejus ex-? tremum du�lis (32.0), conftituet qu�que angulum po�,nbsp;feu reiftum, cum quibtifvis re�lis, in altero plano duonbsp;tis, per aliud iftius re�tae extremum, amp; parallelis re-fpe�tu redarum prioris plani ; igitur perpendicularisnbsp;illa erit qu�que talis ad aliud planum ( 320(^uodnbsp;eft primum.

Demonst. 2^* pars.: Re�la illa erit perpendicularis ad rectas quailibet, in quo vis plano per ejus extrcinanbsp;traje�as C320); condpe igitur, hinc inde, duci pernbsp;extrema illa tot lineas, quot opus eft, ut utrumquenbsp;planum tegatur; oppofitsc qua;libet duac, in eandem di-lebtionem du�tae, erunt inter fe parallelae (115); ergo amp; illa plana, quantumvis producla, nunquam concurrent i igitur parallela funt. Quod eft fecundum.'

336. nbsp;nbsp;nbsp;Perpendiciilares qualibct, inter duo Plana pa-ralkla duSce , aquaks funt.

337. nbsp;nbsp;nbsp;Tria vel pliira Plana parallela fecant eandemnbsp;re�am in partes proportionaks difiantice fuce a Jenbsp;mutuo.

338. nbsp;nbsp;nbsp;Dum igitur diverfas re�las interfecant, partesnbsp;. interceptx inter plana parallela proportionales funt inter fc.

339. nbsp;nbsp;nbsp;Si duo Plana parallela Jicent idem tertium, an-guli correjpondentes aquaks funt. ^djacentes = 180�nbsp;amp;c. cceteraqite fiunt , �t in lineis parallelis., quee con-cipiuntur duci, hike inde , per extrema reSa, ab unonbsp;ad nliiid Planum du�te.

Elem�NTA GtOMETRIjr. nbsp;nbsp;nbsp;lOl

SECTIO TER TIA

D E S o L I D I S. Definitio L

^OrpuSj five Solidum, exteufum e�:, tribus^ di-

341. \Angiilus folidiis is eft , qui comp��iruf pluri-bus quam �uobus angulis planis^ in'eodein plaiio n�h confiltentibus, ad. idem tarnen pun�lum ,-fetu/r^�rin:eOTnbsp;confiitutis.

34a. ScHOUON. Duo a'ngiili plani Mucroriem, fcii ^pkem, five Unguium j'olidum. ccu�liiuere iiequeunt; at necelTari� fpatiumnbsp;aliquod ciica vertkem relinquent vacuum quod tprtio planonbsp;occludendum eft; tribus adeoque ad minus angulis planis opusnbsp;eft, ad tonftituendum angulum folidum.

344. nbsp;nbsp;nbsp;Atque adeo angulis planis., amp;� multitudine, amp;nbsp;rnagnitudin^ acqualibus, ac eodem ordine dilpofitisnbsp;contineri debenc.

345. nbsp;nbsp;nbsp;ScHOLiON. De angulis folidis, qui ex planorum inclinado-ne oriuntur, eodem modo eft ratiocinandurn , ac de angulis planis , qui ex iinearum concurfu, feu intiinatione ortum dutuiic,

346. Solida fuperficiebus planis terminata Poly '�dra eompellantur : atque, pro �t vel 4, 5, 6 Scc. planis

347. nbsp;nbsp;nbsp;Polytdra Regularia funt folida, quorum omnesnbsp;anguli funt jcquales, atque polygonis regularibus amp;nbsp;congruis terminantur.

348. nbsp;nbsp;nbsp;Bajis folidi eft fuperficies, cui folidum infifte-re concipitur.

349. nbsp;nbsp;nbsp;Altitudo Girporis eft Perpendicularis � verticenbsp;Corporis ad baiin (produ�lam, fi opus) demiffa.

S50. Duplici modo gencrari Solida concipimus : i* per inotum re�tilineum plani fibi femper paralleli : 2�nbsp;per motum rotationis figurai fuper refta quadam utnbsp;Axi.

351. Si figura quxcumque plana AB (jfg. 13, 14, J5gt;nbsp;nbsp;nbsp;nbsp;17, x8 amp;c.), conftanter fibi manens parallela�

juxta duduni re�tac BC Cquse DireSrix a�dit) movea-tur; Corpus, feu Solidum ,ex�Tgit, quod Prijma voca-lur, ii Bafis AB redi�nea fuem (ut in fig. 13, 14, 15 amp; 16); Cylindrus ver�, fi Bafis AB Circulus ficnbsp;(ut in fig. 17 amp; 18).

35a. ScH�LiON Superficies plana KL, qua; ad alteram AB pa-railela elt amp; congrua, etiam JSafis Corporis compellatur.

353� fol�m oppofitac Bafes in Prilmate atque Cyiindro,. modo progcnitis, congruac funt; fed amp; l�c-tiones quaccumque paralleiai ad balin, congruae bafi-bus fint oportet.

354. nbsp;nbsp;nbsp;Prifma itaque c� Corpus, feu Solidum, bafibusnbsp;re�tilineis, congruis amp; parallelis, amp; lateraliter Paral-lelogrammis tcrminatum.

culari paraiiela amp; congruS , amp; lateraliter convexitate circulari, cujus diametri, ball paraiiela:, ubique dia-inetro bafis acquales funt; fic ut linea, per omniumnbsp;diametrorum Centra tranfiens, quae jdxis vocatur, recta fuerit.nbsp;nbsp;nbsp;nbsp;quot;

356. nbsp;nbsp;nbsp;ScHOLiON. Cylindrum fpeftare licet inftar PrifmatU infini-tanguli : etenim , c�m circulus haberi poffit tamquam polygonum regulate infinitis lateribus re�is conftans; lali modo peri-pheriam cujurque bafis concipiendo divifam, atque a fingulisnbsp;puniflis peripheriae unius ad firgula correfpondentia punfta al-terius, reftas mente ducendo, Corpus cxurget bafibus reftilineisnbsp;congruis amp; parallelis, amp; lateraliter parallclogrammis terrainatum;nbsp;licet ergo illud appellarc prifma infinicangulum ( 354)�

357. Prifma dicitur ReSum, item Cylindrus ReSiis audit, dum linea tliredlrix ad bafin eft Ptrpcndicularisi

Obliqiia ver� vocantur illa Corpora, fi fuerit linea dt�� rectrix ad bafin obliqua.

358. nbsp;nbsp;nbsp;Prifma, a Bafi, fua, fpecialia fortitur nomina :nbsp;dicitur enim Prifma Triangiilare, fi bafis fuerit A;nbsp;'Quadrangulare fi fuerit ?; Pentagonalt fi fuerit Pen-tagoh�m amp;c.

359. nbsp;nbsp;nbsp;Vocatur Parallelepipedum, fi bafin habuerit Pa-rallelogrammuffl; amp; tune fi fuerit bafis re�tangularis,nbsp;atque prifma KeSum fit, Paralklepiptdum Re�langu-liim audit.

360. nbsp;nbsp;nbsp;Si fuerit bafis Quadra turn, amp; Prifma redlum;nbsp;�amp; prseterea linea ^/reSr/x (qua; hic aqualis efi: prififia-tis Altitudini'), fuerit sequalis lateri Balbos; nominenbsp;fpeciali Cubus, vel Hexacdrum. indigitatur.

361. nbsp;nbsp;nbsp;Si figura quxcumque plana AC, feu Re�lilinea,nbsp;feu Circularis (_^g-. 19, 20, 21 amp; au), fibi feraper pa-rallela, ita moveatur juxta du�tum reda; BX (ad punc-

� turn X, medium figurae, perpendiculariter aut obliqu� demiffai), ut pun�tum illud figura;, conftanter rebbenbsp;BX infiftat; amp; praeterea fingulo progreffuum inftantinbsp;1-atera figura; decrefcant tali progreffione arithmetica ,nbsp;ut, cum figura ad B pervenerit, eadem latera pundinbsp;inftar evanuerint; Solidum figura redilinea progeni-tum Pyramis i amp; ball circular! prodiens, Conus com-pellatur.

362. nbsp;nbsp;nbsp;Ubivis ergo Pyramidem, vel Conum, fedioncnbsp;parallela ad bafin, fecari fmgas; erunt fediones figure bafi fimiles.

563. Pyramis ade� corpus eft in cufpidem definensgt; cujus bafis eft Figura Recftilinea, Plana yero lateralianbsp;to�dem AA quot lunt bafeos latera Cfig' 19 amp; 2,0).

364. nbsp;nbsp;nbsp;Conus eft folidum in cufpidem qu�que defi-nens, pro baft Circulum habens; lateraliter vero con-Vexitatis circularis Diametris, bafi parall�lis, continuonbsp;decrefcentibus proportione arithmetic^; ut amp; practereanbsp;linea , per diametrorum Centra tranfiens, re�la fuericnbsp;Cfig- 21 amp; 2,2 ).

365. nbsp;nbsp;nbsp;Scholton. Conum pro Pyramide, cujus bafis figufa eft re�-tilinea infinic� parvis latetibus conftans, Ipcftari pnteft : eteninvnbsp;a vercice ad finguld peripheriae bafeos punfta redas concipieU'nbsp;io dudtas, Pyramidem infinhangtilam habebis.

366. nbsp;nbsp;nbsp;Re�ta BX {fig. 2,1 amp; 22), a Vertice ad Co.�nbsp;ni centrum du�la,.Coni Axis audit; atque, pro�t adnbsp;bafin fuerit perpendicularis; vel obliqua, Conus Rcc-�nbsp;tus, aut Obliquus , dicitur.

367. nbsp;nbsp;nbsp;Pyramis, a bafi, qu�c A, ?, vel Pentagcnnbsp;Hum amp;c. fuerit, Triangularis ^ Qiiadr�ngularis^ PentO'nbsp;gonalis amp;c. denominatur.

368. nbsp;nbsp;nbsp;Si in Pyramidis, aut Coni, genefi, Plani mo�nbsp;tus fiftatur, priufquam ad verticem pertingat, feu an*�nbsp;tcquam bafeos latera evanefcant, Pyramis triincatanbsp;yocatur Corpus progenitum, fi fueric bafis figura ree-tilinea, vit ACHG Cfg- i Tab. YlU}} Conus ver� trim-catiis ^ fi bafis circularis exiftat; �t ACHG (^fig. n) :nbsp;qudd autem Solidum formari concipitur preeter die�

tum Corpus truncatum, motu ad verticcm ufque pro-trado, dicitur Pyramis abfcijja, �t BHG (fig. i), vel Conus abfcijfus, �C BHG (fig. 2

I I.

369. nbsp;nbsp;nbsp;Quadrilaterum re�langulare ABXO (^fig. 3),nbsp;.fuper XO, �t axi, rotecur : redse XB, OAj amp; quai-.cumque intermedia:, prioribus parallelaj, erunt radiinbsp;Circulorum jcqualium amp; paral]elorutn , formabiturquenbsp;Cylindrus redus, cujus ^xis eft XO CsSSj St 3573-

370. nbsp;nbsp;nbsp;Si A ABC (fig. 4 ) , in quo A redus eft ,nbsp;fuper uno Cathetorum v. g. AC, �t axi, gyretur, exur-get Conus reSus ( 364 amp; 366 ). Hypothenufa BC ,nbsp;(quac eft a:qualis cuilibet re�ia:, a vcrtice Coni redinbsp;ad aliquod pcripheria: bafis pundum dudx_), Lotusnbsp;Coni recli vocatur.

371. nbsp;nbsp;nbsp;Si Trappezo�s redangularis ABKX (^fig. 5),nbsp;in qua A item X redlus eft, fuper AX circumvolva-tur, Conus truncatus enafcitur. Et produdis AX amp;nbsp;BK ufque in L, ubi concurrunt (fuppofito qu�d la-tus XK fit latere AB minus) ; circumvolutione Anbsp;redanguli LXK, formatum concipe Conum abfcijjumnbsp;( 368 ). KB Coni truncati Lotus compeilatur.

372. nbsp;nbsp;nbsp;Medium Circulum ACB Cfig- 6) fuper Diametro l�a AB, �t axi, circumvolvi finge, gignitur Cor-pUvS, quod Globus vel Sphcera audit.

373. nbsp;nbsp;nbsp;Sphcera itaque Corpus cft, cujus lingula fuper-ficiei pun�ta, acqualicer diftant a pundto quodam medio , V. g. X j quod Centrum Sphacrac dicitur.

374. nbsp;nbsp;nbsp;Re�la AB, fuper qua circumvolutio fadia po-nitur, Axis fphacrai audit.

375. nbsp;nbsp;nbsp;Redac omnes, utrimque in fphscras fupcrficienbsp;terminatac, amp;'per fphserac Centrum trajedsc, Diametrinbsp;fphairai compellantur.

376. nbsp;nbsp;nbsp;Igitur omnes ejufdem fphsirai, aut xqualiumnbsp;Iphxrarum Diametri, inter fc aaquales fint oportet.

377. nbsp;nbsp;nbsp;Quaclibet Diameter fphn:rac haberi poteft pronbsp;Axi ejufdem ; fuper ea enim fa�lA revolutione mediinbsp;circuli, in cujus plano fuerit illa diameter, eadem fcm-per jphacra prodit.

THEOREMA I.

Maximus angulorum planortim, qiiibus componitur an-gulits Jblidus, minor ejl Jummd rcUquorum.

378. nbsp;nbsp;nbsp;Dico angulum CAB Qfig- 7) c�b minorcm an-gulo CAG angulo BAG.

O a

Demonst. Concipe planum CAG, item BAG plicari ita, ut cadant in planum CAB; fi fbret angulus CABnbsp;�cqualis duobus aliis fimul fumptis , deberent plananbsp;CAG amp; BAG ita tegere planum CAB, ut per reftamnbsp;AG fefe contingcrcnt ; fi vcro foret angulus CABnbsp;major reliquis duobus, plana CAG amp; BAG ita incide-rent in planum CAB, ut tertius angulus planus me-diaret; utrumque autem implicat : etenim h�e pofi-to, quoniam eicvatione planorum GAG amp; BAG i�pernbsp;AC amp; AB manentibus immotis, ea a fefe fepararinbsp;necelle eft, non poffent per AG ftfe mutu� continge-rc. Ergo angulus CAB minor eft reliqujs, quibufeumnbsp;angulum A Iblidum conftituit. Q^. e. d.

Summa omnium angiilorum planorum, quibus idem angulus Jolid�s fofmatur, minor eji 360�.

379. Dico angulos planos CAB, CAG amp; BAG (.fis- 7) facere fimui minus quam 360�.

Demonst. Planum CAG fuper AC, amp; planum BAG fuper AB, �t axibus, ita rotetur, ut in eodem planonbsp;expandantur (�t fig. 8); tali caiu feparatione planorum novum angulum GAG fuboriri necefle eft; atquinbsp;ille fol�m laait 360� cum tribus angulis planis, quibus folidus componebatur (84); ergo amp;c. Tot quotnbsp;lubet fuppofttis angulis planis, quibus componitur angulus folidus; duobus a fc mutuo, feparatis, reliquignbsp;omnibus adhuc ftbi contiguis, omnia corumdem plananbsp;expanfa finge in plano communi : acced�t femper no-vus quidam angulus , qui cum prsefatis tantum faciatnbsp;360� C84); igitur anguli plani conftituentes angulumnbsp;folidum faciunt min�s quam 360�. e. d.

THEOREMA I.

380. nbsp;nbsp;nbsp;Demonst. Tribus, ad inin�s, planis opus efl:.nbsp;Ut unicus angulus folidus eftbrmetur C3423; atqui trianbsp;illa plana, cum a fe mutu� divcrganc, .fpauum ca-vum anguli nequeunt obtegere, amp; ita hue ufque nonnbsp;habetur completum folidum, led fuperficies tantum ;nbsp;igicur unum prjcterea fuperaddendum eit planum, utnbsp;fpatium undique claudatur , atque ita Poly�dro pro-funditas accedat, quo Corpus compleatur. Q^. e. d.

381. nbsp;nbsp;nbsp;Igitur Poly�drum pauciores nequit habere an-gulos, quam quatuor : tria enini plana unicum angu-lum folidum formantia , fpatium triangulate vacuumnbsp;relinquent, quod ubi plano triangular! clauditur, tre�nbsp;practerea enalcuntur anguli folidi.

Quinque folum darl pojjitnt Poly'�dra Regulariq; tria, quorum plana funt AA ctquilatera; unum, cujusnbsp;plana jiint quadrata; amp; unum Penta-gonis mminatum.

382. nbsp;nbsp;nbsp;Demonst. Anguli plani, quibus angulus Po-ly�dri componitur, ornnes inter fe a:quales funt (347),nbsp;faciuntque minus quam 360� (379)� ^ qwniam tres,nbsp;ht minimum, requiruneur ; perlpicuura fit, quinquenbsp;tantum modis, angulum fohdum efformari polTe Po-�ygonis regularibus ejufdem fpeciei :

1� tribus AA jcquilateris; atque tunc angulus foli-dus erit i8o� (cum quifque angulus planus, quo for-matur fit �o�); addendo itaque quartum A tcquilate-rum, prioribus congruum, quo fpatium, � tribus aliis relidum, concludatur, habebimr Tetraedrum.

2� Quatuor anguli triangulorum acquiangulorum,Ver-tice jundi, efficient angulum folidum 240� : atque odto talia AA conjun�ta, efficient folidum o�to piano-rum feu O�da�drum.

3� Quinque AA sequilatera, apice jun6la,efficiunt angulum folidum 300�; amp; fuperaddendo quindecim alia, prioribus congrua, ut fpatium reliquum occludacur, Po-ly�drum enafcitur ao conftans AA aequilateris, quodnbsp;Jcofa'cdrum audit. Non poteft autem fieri angulus fo-lidus ex pluribus quam ex quinque AA acquiangulis :nbsp;fex enim adaequant 360�, amp; plures quam fex nume-rum iftum excedunt; quod fieri nequit (379).

4� Angulus Quadrat! eft 90�; tres tales jundi angulum folidum componunt 270� : atque fuperaddendo tria prioribus congrua fit Hcxacdrum. Quatuor autemnbsp;tales anguli angulum folidum componere nequcunt :nbsp;fumma enim eft 360�; quod implicat C379).

5�. Angulus Pentagon! regularis eft 108�; tres junc-ti efficiunt angulum folidum 324�; amp; tunc � duodedm pentagonis regularibus amp; congruis fieri poteft Dodecac-drum. Quoniam quatuor Pentagon! regularis angulinbsp;= 432� : ex iis angulus folidus fieri nequit (379);nbsp;ergo nec corpus aliquod. Cactera polygona regularianbsp;nequeunt adhiberi : tres enim eorumdem anguli, velnbsp;faciunt fimul 3�o�, �t in Hexagono regulari; Vel fa-ciunt plus quam 3 60�, ut in fequentibus omnibus.nbsp;Jgitur quinque foltim dari queunt Poly�dra regularia,nbsp;fcilicet Cubits.^ Tetratdrumj OUa'cdrum^ Dodeca�drum amp;�nbsp;Jcofa'cdrum. c. d.

383. nbsp;nbsp;nbsp;Solida Jimllia funt, qusc figuris fimilibus jcquinbsp;multis, codemque modo diipofitis,.terminantur.

384. nbsp;nbsp;nbsp;Igitur qui�bet anguli homologi, in Corporibusnbsp;fimilibus, xquales funt.

385. nbsp;nbsp;nbsp;Corpora quxcumque Regularia ejufdem Ipecieinbsp;fimilia funt.

386. nbsp;nbsp;nbsp;Demonst. Omnis femidrculus eft alteri fimilis;nbsp;fed quxlibet fphaira prodit femidrculo fuper diametro,nbsp;ut axi, gyrato, (372); terminatur itaque quxlibccnbsp;iphacra fuperfide fim.ili amp; eodem modo difpofita; talinbsp;nempc, ut fingula pun�la fuperflciei cujufque a Centro fuo diftent radio fphacridtatis fphacrx. Ergo -fph�-rsc omnes inter fe fimiles funt C383).

ks, altitudines pint proportionales lateribus homo-~ logis bafeos. � converfo : p Pripnata reSanbsp;pmilia pierint; altitudines pint proportichnbsp;nales lateribus komologis bapum.

387- Demonst. i� pars. Attends bafibus fimilibus, amp; c�m Prifmata re�la ponantur; plana lateralia cujuf-

que prifmatis erunc quadrilatera reftangularia; amp; alti-tudines funt aquales redangulorum iftorum lateribus; itaque quadrilatera illa re�langularia homologa eruntnbsp;fimilia C 2.29 ) ; igitur prifmata fiinilia funt ( 283 ).nbsp;Quod erat primum.

Demonst, 9? pars. Quoniam parallelogramma late-ralia unius debe^nt effe fimilia pardlelogrammis latera-libus homologis akerius (383); latera unius debent effe ad latera akerius, ficut latera homologa bafeosnbsp;unius ad homologa latera balbos akerius (229); jamnbsp;Ver� in Prifmate redo, latus fingulum plani lateralisnbsp;eft aequale akitudini; ergo akit�dines funt �t latet�nbsp;balls homologa. Quod erat fecundum.

288. Quoniam in Cylindris quibufeumque redtis, bales conftanter llmiles � exiftunt ^ atque illos fpedarcnbsp;licet, �t prifmata reda ; Cylindri quilibet redi filni-les erunt, dum Peripherisc, vel Diametri bafium, funtnbsp;�t akit�dines, feu axes Cylindrorum. Et � converfo :nbsp;in Cylindris re�tis fimilibus, axes funt ut Peripherie�nbsp;aut piametri bafium.

Pfiphata obliqua fimilia funt, aim ad bajes fimilei aqualiter inclinantur, amp; pratered latera Parallel�-grammorum lateralium^ yel altitudines ^ funt propor-tionalia lateribus hornologis bafiium. Et � converfo :nbsp;Prifmata fimilia ad bafes fimiles aqualiter inclinantur; latera Parallelogrammorum lateralium, item al-titudines , funt proportionalia lateribus homologis ba-fium.

389. Dico 1� li in Prifmatibus (ifig. 9), acqualiter inclinatis, fint bafes fimiles, amp; prseterea fit AB : ab :nbsp;AH == ah; vel AB:a� = aEit. Hl: altit. hi; Prifmata fimilia effe.nbsp;nbsp;nbsp;nbsp;Demonst.

Demonst. i^'pars. Quoniam � plana lateralia in fin-� gulo prifmate funt parallclogramina (354) acqualiternbsp;indinata (atque ade� anguTus BAH == angulo baknbsp;amp;LC. ), paralleiogramma honaologa erunc aequiangula ;nbsp;quoniam itaque, ex hypothefi, conftant .lateribus pro-portionalibus, erunt amp; limilia (2,2,9); igitur prifmatanbsp;illa fimilia qu�que erunc (3^3} : quod eft primuianbsp;primac partis.

Si ver� AB : ab = altit. Hl : akit. hi; dudis AI amp; ai, eric angulus I, item i re�tus (320) ; angulusnbsp;lAH = iah, propter xqualem prifmatum indinationem;nbsp;igitur A IAH ccquiangulum amp; fimile A iah; ergo eftnbsp;iH : ih � AH : ah ; led, ex hypotheft, eft AB : abnbsp;= IH : ih; igitur eft AB : ab =i/[H :JJt- Quod eft al-terum primx partis.

Dico 2�, ft bina iife Priftnata fimilia fuqrint; aqualiter ea indinari ad bafes fimiles; latera, itemnbsp;altitudincs effe proportionalia iateribus bafimn homo-iogis.

Demonst. Non folum bafes fimiles, fed amp; paralleiogramma lateralia fimilia fint oportet (3 S3 ); eft adeo AB : ab = AH ; ah; anguli quoque folidi A amp; a xqua-ies funt (384); xqualiter itaque prilmata illa ad bafes indinantur ; dudis igitur perpepdicularibus Hl amp;nbsp;hi ad f�bjectas bafes, amp; deinde r.ectas AI amp; ai, ericnbsp;angulus IAH = iah fed eft I = / , utpore redinbsp;(320); igitur AA IAH amp; .iah fimilia funt; ergo eftnbsp;AH: ah � IH: ih; fed erat AB : ab ^ AH : ah; itaquenbsp;�eft eciam AB : ab = lB.:ih. Proinde prifmata fimilianbsp;non folum ad bafes fimiles xqualiter indinantur fednbsp;amp; latera parallelogrammorum lateralium, item altitu-dines prifmatum, funt proportionalia Iateribus homolonbsp;gis bafium. Quod erat l�cundum.

390. Bafes in Cylindris inclinatis funt qti�que Cir-culij ac proinde inter i� fimiics (273) , amp; quoniam Cylindri ut Prifmata cenferi poffunt (3 56); Cylindrinbsp;quicumque obliqui fimiles funt, dum eorum axesnbsp;ab bafes scqualicer indinati , funt Diamctris bafiumnbsp;proportionalcs. Et � converib : in Cylindris obliquisnbsp;fimilibus axes, acqualiter-indinati ad bales, funt ba-lium Diametris proportion ales.

' altitudines fint propartionales Latcribus homologls ba~ fiiutn; atquc infuper cadant in piinSa bafium homo-loga. � converfo : in Pyramidibus fimilibus, alti-tudines fiiint proponionaks latcribus homologis bafiium,nbsp;caduntquc in puncia fimilia earumdem.

391. Dico 1�. Si Cfig. 10 amp;c ii) ACK fit fimilis ack; nltitudo BI ; alt. bi � AC ; ac; atque altitudincs iilaenbsp;ccciderint in bafium punda fimilia (hoe eft, ab an-gulis homologis proportionaliter ad latera homologajnbsp;diftantia ); Pyramides illas efl� fimiles inter fe.

Demonst. dudis AI amp; ai; anguli I amp; i funt redi (320); quoniam pun�a I amp;; i funt fimilia bafium;nbsp;ell AC: fl�C = AI: ai; cfim itaque , ex hypothefi , fitnbsp;AC : flc = BI : bi; erit AI : ai ~ BI : bi; jam verbnbsp;I = � : ergo AA AlB amp; �/i funt fimilia; igitur eftnbsp;BI : bi = BA : ba; ergo eft AC : cc = BA ; ba. Duc-tis Cl, ci; KI amp; ki, patebit AA CIB amp; cib; AA KIBnbsp;amp; kib, effe inter fc fimilia ; adeoque fingula lateranbsp;AA lateralium Pyramidis ACKB proportionalia funtnbsp;homologis latcribus AA lateralium Pyramidis ackb;nbsp;funt ergo Pyramides illx inter fe fimiles (383). Quodnbsp;erat primum.

Dico 2�. Si Pyramides praidi�t� fimilcs fuerint; eft alt. BI : alt. bi � AC : ac Stc.; atque I amp; i faiic punc-ta bauum fin�lia.

Demonst. Propter Pyramidtim flmilitudinem, eft an-:gulus folidus A =: �; atque adeo AB amp; ab aiqualitcr inclinantur ad luas rcfpcdliv� bafcs ; ergo angulusnbsp;lAB =: iab; led 1 = / (utpotc rccli) ; ergo AA iABnbsp;amp; iab lunt fnnilia; igitar eft IB :/� = AB ; ab; fednbsp;etiam eft AC : ac = AB : ab ; ieaque eft AC: ac = BI:nbsp;bi = AI: a/. Duclis Cl, ci; KI, ki; inteilectu eft facile AA CIB amp; cib; AA KIB amp; kib, inter Ie effe fimi-lia; atque adeo pun�ta I amp; /, proportionaliter ad ho-iTiologa' bafmm latera, diftare ab angulis bafium C, c;nbsp;K, k ; igitur altitudines in Pyramidibus fimilibus fantnbsp;proportionalcs lateribus homologis balium, caduntquenbsp;in bafium pun�ta limilia. Quod erat alterum'.

39a. Coni fpeftantur �t Pyramided; k. quoniam ba-fes eorum, utpote Circuli, femper ftmilcs funt; quan-do Coni limiles funt, altitudines (qui amp; axes funt in Conis re�tis), item re�tae qutecumque, ad bomologanbsp;puncta pcripberiac bafium duftte, proportionalcs fvintnbsp;bafium Diametris.

393. In Conis obliquis fimilibus Qfig- ts), ailes BX amp; bx funt qu�que .proportionalcs Diametris bafium : nam, attenta Conorum aiquali inclinatione; eftnbsp;angulus BXC = bxc; angulus BCX == bcx; ergo AAnbsp;BCX amp; bcx fimilia funt; igitur eft BC: ic = BX:/)ar;nbsp;verum , ex bypothefi , eft BC . bc =� AC : cc ; ergonbsp;eft BX : bx = AC '.ac.nbsp;nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;�

P *

CpK-OLLARIUM. II L

394. nbsp;nbsp;nbsp;In Conis, fi altitudincs, bafium Diametris pr�-portionalcs, cadant in puncta bafium fimilia (produc-ta fi opus); aut axes sequaliter ad bales inclinati, fintnbsp;Diametris bafium proportionales, Coni fimiies fintnbsp;oportet.

theorema V.

lil Pyramids truncata, ahfcljja ejl fimilh Pyramiamp; ex abfdjja amp; truncata compojitce.

395. nbsp;nbsp;nbsp;Demonst. Sit ACK Cfig- 13) bafis major,nbsp;F-FI 'bafis minor Pyramidis truncattc : produ�tis late-ribus, �t vides, prodic Pyramis integra ACKB, com-pofita ex truncata amp; abfciffa EFIB; bafis abfcilTsc fimi-lis eft majori ball truncatai C 353 ); amp; quoniam late^nbsp;ra balls minoris, feu FE, IE amp; IF funt parallela la-teribus hofflologis AC, AK amp; KC; AA lateralia ab-fciflic, memp� BFE, BIE amp; BIF, fimilia funt AAnbsp;BCA, BKA amp; BKC Pyramidis integrtc. Igit�r ab-fdlfa fimilis eft Pyramidi conftanti abfcifsa amp; truncate tol (383). 0^. e. d.

396. nbsp;nbsp;nbsp;Conus abfciffus eft fimilis Cono ex abfciffo S?nbsp;trunca�o fimul.

397. nbsp;nbsp;nbsp;Pyramidis abfciflk�, aut Coni abfcil��, altitudo,nbsp;eft ad altitudincm abfciffi amp; truncati fimul, ficuti di-menliones homologtc bafium amp;c.

398. Resolutio. Unica ejus fuperficies inquiratur, at-que hacc codes fumatur, quot funt Poly�dri facies;nbsp;amp; furama dabic qu^plitum.

399. Ergo in Cubo capiatur fexics quadfatum uniusnbsp;lateris. Inquirendo arcam unici A lateralis in Tetra�'�nbsp;dro, eamque quadruplicando, habetui* fuperficies iftiusnbsp;Poly�dri regularis amp;c.

II. Perimeter bafeos ducatur in latus alicujqs Par-alle-logrammi lateralis, non fpe�lans ad balin, feu ia altitudinem; amp; prodit fuperficies omnium Parallelo-grammorum lateralium ; huic igitur fummsc adden-do duplum unius balls , fumma finalis dabit tota-]em Prifmatis re�li fuperficiem. Etenim ex geneltnbsp;Priftnatis ( 351) manifeftum evadk , toties balls

Perimetrum reperiri in Parailelograinmis kteralibus, qnot funt punc^ta in Linea dircBricc ; hic autemnbsp;acqualis eft fingulo lateri Parailelograminoram rec-tangulorum latcralium , Prilma�s aititudinemnbsp;cool�tuenti.

401. Resolutio. Peripheria bails ducatur in axin amp;nbsp;radium bails; fadum dabic quaciitum.

Demonst. Ex Cylindri genefi C S.'^O P^cet, in fu-perficie convexa totics reperiri Peripherram balcos , quot funt pun�ta in Linea direSrice, quai bic aequa-tur axi Cylindri; itaque ducendo Penpheriam ballsnbsp;in axin , prodit fuperficies convexa; Periplicriara veronbsp;ducendo in Radium, prodit area utriufque bails; ergonbsp;ducendo Peripheriam bails in axin amp; radium, fadumnbsp;sequatur fuperficiei totali Cylindri. Q^. e. d.

402. Erit igitur convexa ad utramque planam fimulnbsp;fumptam, ilcut altitudo, feu axis Cylindri, ad radiumnbsp;bails.

403. Resolutio. Inquirantur ilgillatiin area:, bails amp;nbsp;AA latcralium; addantur in unam fummam ; at-que hxc cric int�gra Pyramidis fuperficies.

4Q4. Resolutio. Peripheria bafeos ducatur in latus amp; radium bails Coni; dimidium fedi integram Coninbsp;redi dabit fuperficiem.

Demonst. Sit Conus redlus ABC 14); mente ducantur iniimtoc re�lai a vertice ad lingula periphe-riac bafis puncla; atque convexam fuperficiem divil�risnbsp;in innnita AA, quorum omnium bales ajquantur peri-pherisc bafis; k. altitudo = lateri BA Coni; atquenbsp;adeo, duceiido pcripheriara in lams BA Coni, dimi-dium fadti scquatur fuperficici omnium AA, feu fu-perficiei Coni convexa;; atqui peripheriam ducendo ianbsp;radium bafis , dimidium facii eft acquale aresc bafisnbsp;(301); ergo peripheriam ducendo in lams, amp; radiumnbsp;balls Coni re�ti, dimidium facti incegram dat ejufdemnbsp;fuperficiem. Q^. e. d.

� 405. Igitur in Cono recto, convexa fufgt;erficies eft ad planam; ut latus Coni ad radium balls.

406. nbsp;nbsp;nbsp;Resolutio I. �triufque bafis (qusc fimiles funtnbsp;C39.5)lt;, fefeque habent ut quadrata laterum homo-logorum) qua�re fuperficies ;

II. Dein cujufque fuperficici lateralis Cfiua; Trappezo�-des funt) aream determines; omnes in unam col-ligantur fummam; eritque hxc totalis fuperficies quxfita.

407. nbsp;nbsp;nbsp;Resolutio. Adde peripheriam minoris atquenbsp;majoris bafis in unam fummam; hanc ducito in latus Coni. Peripheriam bafis minoris multiplicesnbsp;per radium minoris bafis j amp; peripheriam majori�

bafis per radium majoris bafis; tria illa fadta iii unam fummam colligantur, amp; ejus dimidium da-bic quod quairitur.

Demonst. A fingulo pun�lo peripheriac minoris bails GH (j(%. 2) ad correlpondens pundum majoris AC , redas duci concipe; atque convexam Coni trun-cati fuperficiem diviferis in infinitas parvas Trappezo�-des, inter eafdem parallelas (quarum nempc altitud�nbsp;eft = AG), eonfiftentes; quoniam ergo omnium illa-rum Trappezo�dum latera parallela, peripherias minorinbsp;amp; majori aiquentur; fuperficies Coni truncati convexanbsp;acquatur dimidio fa�li utriulque peripheriaj in latusnbsp;AG Coni truncati. Bafis vcro minoris area aequaturnbsp;dimidio fadi peripheriac fuac in radium fuum; amp; majoris bafeos area = dimidio fadi peripheriac majorisnbsp;in radium fuum; ergo i trium fadorura dabic fuperficiem integram Coni truncati. Q. e. d.

Si Conus obliqiius Q fig. 15) fecetur Plano ptrpendicu-iari ad Planum trianguU ABC, cujiis latera funti, AB minimum, BC maximum Coni latus, 6* Diameter AC bafeos; atque ab eodem triangulo auferaturnbsp;cliud A BFE fmile priori, fed ftu cantrario Cfitinbsp;fe�tione fubcontraria), ita ut angulus Bf E = an-gjilo BAC; angulus BEF = BCA; erit fiSio EK.FLEnbsp;Circulus.

408. Demonst. Per pundum G, arbitrari� defump-tura in communi fe�tione EF plani fecantis EKF amp; trianguli ABC , ducatur ad diametrum AC parallelanbsp;Hl, atque per hanc finge duci fedionem balt paral-lelam, erit h8cc,v. g. FIKI, Circulus (362); communisnbsp;fedio planorum EKF amp; HKI eft recta KG (328 ),nbsp;qu� amp; perpendicularis eft ,ad A ABC, c�m communis fit duobus planis ad ABC perpendicularibus, ex

hypothefi; eft ergo KG perpendicularis ad HI item EF ( 320); amp; quoniam quodlibet ex AA BEF amp; BHI,nbsp;fimile eft A ABC; erunt ea inter fe ftmilia; angTilusnbsp;BEF = BIH amp;c.; igitur A A EHG amp; IFGfunt arquian-gula : eft ergo GH : GE = GF : GI; igitur GE x GF =;?nbsp;GH X GI, feu =: GK^. Ergo fedio EKF talibus conftatnbsp;ofdonnatis, quae fingulac funt mediae proportionales internbsp;partes diametri; jam verb hacc eft Circuli proprietasnbsp;(248); ergo EKFLE Circulus fit oportet. Q^.

Si Conus qinfcumque (fig. 16)Jtcctur piano perpmdica* lari ad planum trianguli ABC, formati per minimumnbsp;latus AB amp; maximum BC in Cono obliquOynbsp;6* per axin tranfeuntis; atque a triaagulonbsp;ABC refecetur aliud ei difjmik BEF;

409. Demonst. Conum ABCD fecari fingas piano ^ quod , tranftens inter extrema E, F redse EF ( qu�nbsp;Diameter SeSionis audit) parallelum fit ad bafin ADC,nbsp;ut ita habeatur fb�tio eircularis GLHM, cujus dia-meter GH, quae communis eft fedao plani refeeantisnbsp;amp; A ABC , parallela erit ad bafin ADC. Praitereknbsp;Conum ABCD alio piano perpendiculari ad A ABC,nbsp;cujus diameter fit IK, fecari concipe. Prior fedio rec-tam eliicit LM, amp; altera NO, atque fingula perpen-diculariter amp; bifariara dividitur ab EF ; funt igiturnbsp;EM amp; NO ordonnatce ad EIlt;� axin eurvac ENFME.nbsp;His pramfilfis : in AA GPE amp; IRE funilibus, ericnbsp;GP : IR r= EP : ER; amp; in A A funilibus HPF amp; KRE,nbsp;habebitur HP : KR = FP : FR; igitur termmos_ prio-ris analogiac multiplicando per terminus pofterioris ho-mologos;erit GPxHPdRxKR = EP x FP:ERxFR;nbsp;in 'qua, in locum duorum primorum GP x HP amp;nbsp;IR X KR, fuibfiuuendo PE� amp;-RN% qusc iis jequaa*

Si Cylindrus re3us fccetiir plano ad bajin incUnato; erit J�3io Ellipjis, cujus minor axis Diametro ba-Jis Cylindri aquetur.

410. Si Cylindrus re�lus ABCD (Tab.p i ) (cujus balls eft circulus AEBL, amp;; axis recta TN, ad candein balla perpcndicularis) , fccetur plano GICK ,nbsp;inclinaco ad bafin; uico l��lionem GICK elie Ellipliu ,nbsp;eujus minor axis ccqualis ik Diametro AB baleos.

Demonst. Imaginare plano reftangulari ABCD, tran-feunte per C pun�tum lupremum, amp; G in�mum lec-tionis, atque per axin TN, Cylindruin fecari; ejufdem plani inierfcdio, communis cum fedione GICK, cftnbsp;reda GC, quai curvsc GICK iongicudinem reprxfen-,tat; atque produda, cum diametro BA produda con-currit in S, qui erit angulus inclinationis plani fecan-tis GICK ad balln Cylindri. Ducantur, ad arbitrium,nbsp;in plano GICK, reda; IK, HQ perpendiculares ad CG;nbsp;amp; in fuperficie convexa Cylindri, redac HE amp; IF adnbsp;�bafin AEBL perpendiculares demittantur; denique innbsp;plano Circuli AEBL, pep punda E amp; F, agantur rec-t� ER amp; FL perpendiculares ad diametrum AB, quatnbsp;eas bifariam fecabit in pundis P amp; N. His prscmiliis :nbsp;red� IK amp; FL , quoniam perpendiculares funt adnbsp;idem planum SBC, parallelai funt inter fe (323);nbsp;funt quoque inter fe atquales ; quia IKLF eft Paral-lelogrammum. Paritcr aiquales amp; parallel� funt HQnbsp;amp; ER, cum HQRE fit ctiam Parallelogrammum :nbsp;nam terminatur lateribus oppofitis parallelis. Red�nbsp;IM amp; FN inter fc funt aequales , utpote pcrpendicu-lares inter parallelas T'N amp; IF; igitur KM amp; LN funtnbsp;. quoque ccquales; k. quia FN = NL ^ funt enim ra-

4�� bafeos), eft IM = KM. Similiter oftenditur, ei-ic inter fe tcqualcs EP, PR, HQ, OQ; itaque re�ia CG eft axis refpeftu Ordonnataram IK amp; HQ, internbsp;fe parallelarum : erit itaque CG axis major, amp; IKnbsp;axis minor , fuppofito pun�to M medio majoris axis,nbsp;pro�t in figura eft ; igitur minor Axis aequatur Diametro bafis. Quoniam bina plana HQR.E amp; IKLFnbsp;inter fe parallela funt, item communes interfe�lionesnbsp;OP amp; MN ; re�tae AB amp; CG dividuntur propor-tionaliter ab iftis planis ; crit ergo GO x OC : GM xnbsp;MC = AP X PB; AN x NB : fed ( AP x PB =nbsp;P�^ = OH=; amp; AN xNB = NF* = MP; igkm fub-j�ituendo eft GO x OC : GM X MC = OH� : MP; amp;nbsp;ftc de ca:teris Ordonnatis qu?e duccrentur in i��lionenbsp;GICK ; ergo l�cftio illa Ellipfis eft (260). Q^. e. d.

Ellipfis eji ceqitalis Qrculo , cujus Diamctn efl media, proport�onalis inter axes Ellipjis.

411. Demonst. Concipe � Centfo Ellipfeos duos Cir-culos defcribi concentricos; unum, cujus idem fit radius cum radio majoris axis, amp; alterum, cujus idem fit radius cum minoris axis radio : demonftratumnbsp;fuit C262) omnes Ordonnatas majoris Circuli eflc adnbsp;correfpondentes Ordonnatas Ellipfeos , ficuti majornbsp;axis ad minorem; igitur Circulus fuper majori axinbsp;fe habet ad Ellipfin, ficut major axis ad minorem.nbsp;Similiter demonftratum tuit (264) omnes Ordonnatasnbsp;Ellipfeos fe habere ad correfpondentes Ordonnatas Cir-?nbsp;culi fup(?r minori axi, etiam ficut Uiajor axis ad minorem ; itaque Ellipfis fe habet ad Circulura fupernbsp;minori axi, etiam ficut major axis acf minorem; amp;nbsp;confequenter Ellipfis eft media proportionalis inter duosnbsp;illos Circulos; atque adeo eft sequalis Circulo, cujusnbsp;Diameter eft media proportionalis inter niajorem amp;nbsp;minorem axin. O. e.d.

Q a

4ia. Ut igitur Ellipfeos area determinetur; quacratvir �efta, media proportionalis inter axes Ellipfeos; Cir-culi, cujus illa diameter foret, inveftigetur area; atquenbsp;htcc dabit quaefitum.

413. nbsp;nbsp;nbsp;�llipfiS fe habet ad Circulum quemcumque ,nbsp;ficuti redangulum fub axibus fuis ad quadratum Dia-metri Circuit dati.

414. nbsp;nbsp;nbsp;Ellipfes fefe habcnt inter fe, �t redtangula fubnbsp;fuis rcfpe�tiv� axibus.

Metiri Jiiperfcum Cylindri reSl AGCL (fig. 2)' tnmcatl per planum ad bafiii incUnatiim.

415. Rf.soi.utiq I. Bafis AL Ellipfis (410}, eft adnbsp;planam GC, �t axis AL ad GC diametrum; qua:-rantur eorum arose , atque addantur in primam,nbsp;fumrnam.

II. Peripheriam bafeos GC due in AG LC; amp; | fadi dabit convexam; qu� primae fummai additanbsp;prodibit totalis Cylindri fupcrficics.

Etenim fit LO parallela GC ; peripheria bafeo.? in �G 4- LC dudd , habetur bis fuperficies convexanbsp;Cvlindri GOLC; amp; ducendo eandem peripheriam innbsp;A*0, habetur bis fuperficies convexa partis AOL; ergo amp;C.

THEOREMA V.

Spkara fuptrficies (sqiiatur reSangulo fiiper Diametm amp; Peripherid Circidi Jplicera maximi.

416. Demonst. Sint F, B Cjfe- 3) Circnmfcripti Jcquales : LT, TF amp; BE fint acqualcs inter fe ; critnbsp;Te parallela FB f 129}. Sit LC paralleia FB feu TE;nbsp;erit BE = EC C230); igitur, c�m FT = BE (224),nbsp;re�tae LT, TF, CE amp; EB acquales funt inter fe; amp;nbsp;quoniam angulus L = angulo fTE, amp; C = angul�nbsp;BET (108), atque angulus FTE = BET, eft L = C:nbsp;igitur FBCL eft Trappezo�s ifofcelis. Sit X Centrum;nbsp;dud�i AX, erunt I amp; O re�ti. Si media Trappezo�snbsp;ABCO circumvolvatur fuper AO, �t axi (feu fi Trappezo�s LFBC fuper axi AO mcdiam revolutionem ab-folvat) , fuperficies convexa Coni truncati, qui generator , aequatur re�langulo fuper AO amp; peripheria,nbsp;ATRSEA : nam eft AI = 10 ( 117 ); du�ta FH per-pendiculari ad LO, erit FH = AO; du�la ES, A LHFnbsp;�imile eft A TES ; eft enim angulus HEF = fTEnbsp;C 108), qui eft = S (quifque enim menfuratur | ar-,nbsp;c�s EAT i); H = angulo TES ( funt enim ambo recti } j igitur LF FH feu AO == TS : TE : fed Periplie-ria� funt inter fe, �t Diaraetri; ergo LF: AO � Cir-culus fuper TS : Circulum fuper TE; adeoquc fee-tangulum fuper LF amp; Peripheria Circuli fuper TE =nbsp;redlangulo fuper AO amp; Peripherie^ Circuli fuper I�S,nbsp;feu Peripherid ATRSEA : atqui primum re�5languiumnbsp;aequatur fuperficici convex� Coni truncati, progenitinbsp;revolutione ABCO : nam fupcrflcies illa = i'cdlangulonbsp;fuper AO amp; f aggregati ex Peripher�s Circuloruta ,nbsp;quorum FB amp; LC funt refpedtiv� Diametri (407 ) ;nbsp;jam autem, c�m TE fit media arithmetic� proportio-nalis inter FB amp; LC (nam in quantum TE fuperatnbsp;�3gt; in tantum LC fuperat TE}; Peripheria Circu��

luper TE eft squalis f Peripheriarum Circulorum , quorum Diametri func FB amp; LC. Ergo fa�lum AOnbsp;in Peripheriam ATRSEA jcquatur fuperticiei convexcOnbsp;Coni truncati, prrcfata revoiucione prodcuntis.

Condpe (^fig. 4) Trappczo�des continuari , ita ut LP, PG, CM, MH, tangentes in P amp; M, acqualesnbsp;fint omnes inter fe : dudtis PS diametro, PM amp; MS,nbsp;item LK perpcndiculari ad GH amp;c. Simili modo, quonbsp;ante , demonftrabis , per revolutionem Trappezo�disnbsp;OCHV, formari Conum truncatum, cujus convcxa lu-perficies, cum tcquetur re�langulo fupcr GL amp; Peri-pheria Circuli fuper PM), eadcm quoque scquabiturnbsp;re�iangulo fuper perpcndiculari OV, amp;; Peripherienbsp;APRSMA Circuli maximi ejul'dcm. Quoniam ergo innbsp;media Sphtcrtc luperficie infinitas tales licet concipcrenbsp;Trappczo�des, quarum omnium fuperficies convexte fi-mul conftituunt Sphtcra; mediam fuperficiem ; atquenbsp;earumdem fuperficies ccquatur fa�to Peripherise maximalnbsp;per perpendiculares (feu Sagittas') interceptas, live pernbsp;Radium Sphaerae : media Sphaerx fupcrficics liabcturnbsp;ducendo Radium in Peripheriam Circuli maximi; ergonbsp;tota Sphairai fuperficies prodit ex fa�io Diametri iqnbsp;Peripheriam Circuli maximi. Q^. e. d.

417. Ergo Sphacrae fuperficies eft ^ ared Circuli fuper Diametro , quae aiqualis foret Diametro Sphsc^^nbsp;rac (301 gt;

theorema VI.

418. Dico, ft Sphxra Qfig- 5) fecetur Plano per AFBI tranfmiflb, fedtionem AFBI effe Circulum,

Demonst. Si Planum fccans^per Centrum Sphacr� tranfeac, manifeftum eft fedionem fore Circulum ;nbsp;omnes enim recta; a Centro Spha;ra; ad Pcripheriamnbsp;fedionis duda;, erunt radii Spha;ra;, exiltentes in eo-dem plano, amp; inter fc a;qualcs; ergo Peripheria eritnbsp;curva in omni fui pundo sequaliter diitans a pundonbsp;quodam medio ; igitur Circulus erit fedio. Si nonnbsp;tranfeat per Centrum fedio, pro�c in figura : ad planum fecans , � Centro, ducatur perpendicularis XO;nbsp;amp; ab � ad fuigula Peripheria; 'fedion is punda quot-cumque redic v. g. FO, 01, OA amp;:c., quibus occur-rant radii XF, XI, XA amp;c.; anguli XOF, XOI, XOAnbsp;amp;c. omnes redi funt (320) ; ergo FX� = XO'�-i-OF�,nbsp;XP = XO* 10*. AX* =: XO* OA* amp;c. : fednbsp;XF* = XP = AX* amp;c. Ergo XO* OF* = XO*nbsp;-flO* = XO* -t- OA* amp;c.; igitur OF* = 10* = OA*nbsp;amp;c. Ergo OF = 10 = O A ; amp; ita de ca;teris redis ,nbsp;dudis a Peripheria J3FAI ad pundum O : ergo Peri-pberia illa lalis eft, ut omnia ejus punda xqualiternbsp;diftent k pundto quodam medio O; igitur Circulusnbsp;�ft fedio. Q^. e. d.

419. Si fit CO Qfig. 6 ) perpendicularis ad AB Dia-metrum, fada revolution e AOC fuper AO, exurgic Sphaira; Segmentum; eritque AO ejufdem altitudo, feanbsp;flt;igkta; amp; LC Diameter bafeos, qua; Circulus eft.

�420. Resolutio I. Multiplicetur fagitta AO (^flg, 6) , per Pcripheriam Circuli maximi, Sc fadum ericnbsp;fuperficies convexa (�t evidenter patet ex primanbsp;liarte demonftrationis ad nuraerum 416^.

11. Deinde quaere aream Circuli, cujus LC chorda feg-menti Diameter ell ; hac lummd priori adje�^ft, ha'-bebitur totalis fegmenti luperficies.

421. Si fint CO item FX perpendiculares ad AB Diametrum, fa�iri circumvolutione COXFC,pars Sphas-rai progenita Zona audit.

422. Resolutio I. Convexam dabit fadtum jagittet OX in Peripheriam Circuli maximi (�t manifeituninbsp;fit ex fecunda parte demonftrationis ad numerumnbsp;41�).

II. Inquire pneterea areas Circulorum, quorum OC amp; XF radii exifrunt; priori adjedtac fimul dabunt to-taiem Zonac fuperficiem.

423. nbsp;nbsp;nbsp;Itaque tam fuperficies convexac Segmentorum,nbsp;quam fuperficies convexx 2k)narum, fe habent ad inte-gram Sphacra; fuperficiem, �t fagitta ad Diametrum.

424. nbsp;nbsp;nbsp;ScHOuoN. Nonnunquam Segmencum Splia:rae Zanam com-pellanc : apud Gcographos enim globi cerraquei Tuperficies, Cii-culi* polaribus comprehenfa, Z^na frigida. audit.

Qitorumcumq�C Corporum Jimilium fuptrficies , Jive intt-grtt , Jive parus homologee, fitnt inter J�, �t quadrata dimtnfwnum homologarum.

425. Demonst. Integra: enim fuperficies, five par-tiales homologsc, funt aggregaium es xqu� multis, fi-

guris fimilibus, quas fingulac funt �t quadrata late-rum homologorum; fed in Solidis fimilibus (�t ex demonftratis conltat ) tres folidi dimcnfiones propor-tionales fant; ergo aggregata ex omnibus , vel ex par-tibus homoiogis, c�m aequ� multis figuris fimilibusnbsp;conibent, funt qu�que �t quadrata dimenfionum ho-rnologarum, ncmp� longitudinum, altitadinum, aut la-titudinumv Q^. e. d.

426. nbsp;nbsp;nbsp;D�is�oVstr.' Prifmatis atque Cylindri genefinbsp;(351) manifel�um . eft, toties balm fibi fuperpofitamnbsp;reperiri in fpatio percurfo a bafi, quot funt puncflanbsp;in aititudine; ergo di�lorum Corporum prodic folidi-tas ducendo bafin in altitudinem. Q^. e. d.

CoROLLARIUM I.

428. nbsp;nbsp;nbsp;Prifmata vel Cylindri aequalis bafis amp; altitudi-Tiis, scquantur inter fe.

Cylindri reSi AGCL ( fig. 2), truncati per planum ad bafin inclinatum, invefiigare Jbliditatem.

42'9- Resolutio. Bafin GC due in maximam AG amp; n�ninura LC Cylindri altitudinem, amp; medietas iac-�l dat Cylindri f�liditatein-

Demonst. Du�S OL parallela GC, amp; concipiendo Cylindrum AOUB re�ium abfolvi : bafi GC du�ta innbsp;OG 4- LC, fa�lum acquatur bis foliditati Cylindrinbsp;GOLC; amp; bafin GC ducendo in AO, fa�tum dat Cylindrum AOLB, feu bis medium Cylindrum AOL;nbsp;ergo bafi. GC du�a in AG LC, facilum dat bis fo-liditatem Cylindri truncati per pianum ad bafin in-clinatum. Q^. e. d.

Cylindri reSi (fig. 7), truncati-per planum FEKL, axi OX, parallelum, inquirers foliditatem.

431. Igitur ambo Cylindri truncati, in quos dividi-tur Cylindrus AGCB fe�lione praefata, funt inter fe �t bafium fegmenta KLC amp; KLG.

432,. Demonst. C�m fint scqu� altae, poflunt divi-di in ajqu� multa plana bafi parallela; amp; c�m bafes-a;quales lint, Piana bafi parallela, fibi correfpondentia ^ in utraque Pyramide , eadem proportione decrefcunt;nbsp;adeoque quaccumque Plana in utraque, fibi correlpon-dentia, erunt aqualia; ergo fumma omnium Planorum,nbsp;primam Pyramidem conftituentium, acqualis erit fum-mae omnium Planorum, quibus fecunda Pyramis com-ponitur ; ergo Pyramides illae aequales funt. e. d.

433- Quoniam ergo Coni quicumque �t Pyramides infinitangula: fpe�lanturnbsp;nbsp;nbsp;nbsp;funt ii quoque aequa-

theorema iil

434. Detur triangulare Prifma Qfig. 8) : fecetur plano ab AB du�to per- diagonales AF amp;BF, eritque, ABCF Pyramis prima. Reliqua deinde pars dividaturnbsp;plano ab A per diagonales AF amp; AE trajedlo; atquenbsp;enafcuntur bina: aliaj Pyramides KFEA amp; ABEIF. Di-co tres illas Pyramides effe a:quales.

Demonst. In primis priores duas, fcilicet ABCF amp; KFEA, inter fe sequantur : bails enim primas ABCnbsp;eft acqualis ball fecundrc KFE; amp; altitude primae CFnbsp;acqualis eft akitudini fecundae AK : Pyramides ergonbsp;iliac acquales funt C432). Secunda Pyramis KFEAnbsp;acqualis eft terciac ABEF ; fit enim bafis fecundac AKE,nbsp;amp; bafis tertiac ABE; erunt bafes iliac acquales C206);nbsp;utriufque Pyramidis vertex erit in F, atque communis earum altitude erit perpendicularis ab F ad bafinnbsp;AKEB demiffa; hahebunt adco Pyramides praedidaenbsp;bafes acquales, amp; eandem altitudinem; funt ergo iilacnbsp;dure etiam inter fe tcquales (432)^ adeoque omnesnbsp;tres acquales funt. Qj e. d.

435'. Pyramis triangularis eft tertia pars Prifinatis, fuper eadem bafi amp; aititudine.

R a

436. nbsp;nbsp;nbsp;Et quoniam Prifma quodvis in triangularia re-folvi potefi;; quaelibet Pyramis eft pars tertia Prifma-tis, ejufdem bafis atque altitudinis.

437. nbsp;nbsp;nbsp;ConiTS pro Pyramide infinitangula, amp; Cylin-drus pro Prifmate iniinitangulo habentur; eft ade�nbsp;Conus paps tertia Cylindri, xqualis fecum bafis amp; aEnbsp;titudinis.

P R O B L E M A III

438. Resolutio. Determinata ratione altitudinis Cor^nbsp;poris abfci�i ad truncati C 397 ); qutcre primo foli-ditatem Pyramidis, aut Coni totalis , conftantis ab-fciflb amp; truncato fimul : deinde Ibliditatcra foliusnbsp;abfcifli : hanc lubtrahendo a priori; quod fupereft,nbsp;dabit foliditatenl truncati.

THEOREMA IV.

439. Demonst. C.oncipiatur ^phterte fuperficies refo-luta in infinite parvas fupcrficies planas, v. g. triangulares, quadrangularcs, aut mixtim ; amp; � Ceutro, ad latera cundla planulorum, duci redlas: evidens eft Sphte-ram conftare cx innumcris Pyramidibus , in CentrO;nbsp;coeuntibus, quaruin altitudo eft Radius Sphgcra;; atque''adeo omnes iilsc Pyramides eandem altitudinemnbsp;habentcujufquc autein Pyramidis foliditas xquatur

Elementa Geometei;�:. nbsp;nbsp;nbsp;135

fadlo bafeos io tertiam altitudinis, feu Radii, partenx (436); omnium ergo, five ipfiufinet Sphairac, folidi-scquacur fado omnium bafium, feu fuperficiei

440. nbsp;nbsp;nbsp;Quoniam igitur Sphaerrc fuperficics eft * arcinbsp;Circuli maximi; prodit etiam Sphscrs: foliditas duccn-do areapn Circuli maximi in | Diametri.

441. nbsp;nbsp;nbsp;Dum Sedor Circuli AXB ([fig. 9), fuper unonbsp;Radiorum, v. g. AX, �t axi, gyratur, Spha3ra3 Se�lornbsp;prodit; fic ut ejus bafis fit SphEcrie fuperficics convexanbsp;fegmenti, cujus AO (d�da BO perpendiculari ad AX)nbsp;Sagitta exiftit.

Resolutio. Multiplicata AO per Circulum Sphscras maximum, producfum cequatur fuperficiei .convex�nbsp;iftius fegmenti (420) ; hanc dein ducito in ter-�nbsp;tiam Radii Sph�rac partem, amp; prodibit foliditas. Exnbsp;di�lis (439) demonltratio pater.

442. nbsp;nbsp;nbsp;Eft ergo Sphscr� .Sedor ad Sphacram integram,nbsp;fit Sagitta Segmenti bafeos Sedoris, ad Sphscr� Diame-trum.

44^. Resol�tio I. Qu�re foliditatem Scdor;s, cujus fuperficics convexa fegmenti forec bafis : ponamuf-que BL (^fig. 9) Diametrum fore ejufdem fegmenti;nbsp;X Centrum fit amp; O redus.

11. Inquire deinde foliditatem Coni redi, cujus BL effet Diameter bafis, amp; OX akitudo : hancque fub-trahendo ex foliditate Se�loris, quod reiiquum eit,nbsp;fegfnenti dat foliditatem.

THEOREMA V.

compofta fiiperjicid amp; altitudinis, quorum fado pro~ deunt.nbsp;nbsp;nbsp;nbsp;gt;

445. nbsp;nbsp;nbsp;Quoniam itaque in Corporibus quibulcumquenbsp;fimilibus, dimenfiones fingulcc homologie func pro-portionales, atque ade� fuperlicies illae iunt ut qua-drata, v. g. altitudinum ; ducendo fuperficics illas innbsp;akitudines refpe�tivas , aut earumdem panes ; fadanbsp;Cquae aequantur didorum Corporam foliduatibus) eruntnbsp;inter fe, �t quadrata altitudinum, in akitudines duda;nbsp;hoe eft, in rations triplicata, feu �t Cubi altitudinum ,nbsp;aut laterum homologorum.

SECTIO QUARTA

446. nbsp;nbsp;nbsp;Trigonometria Plana fut a Sphasrica, qua� denbsp;AA, per trium Circulorum arcus formatis, diiferit,nbsp;diftinguatur) eft fcientia ex tribus trianguli redilineinbsp;partibus inveniendi reliquas.

447. nbsp;nbsp;nbsp;Sinus reclus arc�s, vel anguli, qui eo arcunbsp;pienl�ratur, eft reda, duda ab extremitate arc�s, amp;

perpendicularis ad Diametrum, feu radium. Vocatur etiam fimplicicer Sinus. E. G. Cfig- i Tab. X.) lint B,nbsp;O, F amp; X re�ti (ade�que I in Centro) : OG eft Sinus arc�s BG, item anguli I; GX eft Sinus arc�s GF,nbsp;item anguli L.

448. nbsp;nbsp;nbsp;Eft ade� Sinus redus arc�s vel anguli, dimi-dia pars chordsc, fuftentantis arcum duplum arc�s, vd.nbsp;anguli, cujus Sinus redus dicitur.

449. nbsp;nbsp;nbsp;Idem eft Sinus arc�s, ejufque complement! adnbsp;�8o�. Item idem eft Sinus anguli, ejufque Vicini ; amp;nbsp;ita, poflto Centro in K; perpendicularis GO eft aequ�nbsp;ben� Sinus arc�s GF�, quam arcus BG; item scqu�nbsp;eft Sinus anguli GKE, quam anguli I : nam GO eftnbsp;reda perpendicularis ad Diametrum, duda �cqu� ben�nbsp;ab extremitate arc�s EFG, quo menfuratur Vicinusnbsp;anguli I, quam ab extremitate arc�s BG, quo men-furaiur 1.

450. nbsp;nbsp;nbsp;Reda BO, quac pars eft radii, intercepti internbsp;Sinum redum amp; Peripheriam, dicitur Sinus Verfusnbsp;ejufdem arc�s BG, item anguli I.

451. nbsp;nbsp;nbsp;Cojiniis arc�s BG, vel anguli I, eft reda GX,nbsp;quac eft Sinus arc�s GF, vel anguli L, qui prioris arc�s , vel anguli, Complementum eft ad 90�. Eftquenbsp;Coj�nus ille acqualis redae 01, feu parti radii, intercep-te inter OG Sinum redum, amp; Centrum. Vocaturnbsp;ctiam Sinus Complemcnti.

4.5^- Arc�s vel anguli daci Sinus Vzrfus eft l�mpcr �xceflus radii fupra Cofinum. Et � converfo ; Cofinusnbsp;fempe� eft exceffus radii fupra Sinum Ferfum.

453. nbsp;nbsp;nbsp;Sinus totus efi: Radius BI, feu Sinus quadran-tis BGE, ac proinde anguli re�fi, feu p��.

454. nbsp;nbsp;nbsp;Tangens are�s BG, vel anguli I, efi: redta, oc-^nbsp;currens perpendiculariter radio IB, terminata ab unanbsp;parte in B, amp; ab altera parte in punclo, ubi occurritnbsp;redtac, � Centro per aliud are�s extremum du�laj; adeo-que in figura eft AB. AL dicitur Secans are�s BG,nbsp;vel anguli 1. CF Cotangens are�s BG, item angulinbsp;I; CL vero eorumdem Cojecans audit.

455. nbsp;nbsp;nbsp;Arcus, aut angulus 90� nullam habet Tangen-tem, neque Secantem : pofito enim arcu FELE, aut an-gulo K 90� : nulla redla, ad radium KE in pun�lonbsp;E perpendicularis, concurrere poterit cum KF aut KR,nbsp;in quameumque partem producantur.

456. nbsp;nbsp;nbsp;Pro eseteris autem arcubus, aut angulis, eadeninbsp;eft Tangens, eademque Secans are�s, ejufque comple-menti ad 180�; aut anguli, ejufque Vieini : nam recta, v. g. AB , occurrens perpendiculariter radio BI, amp;nbsp;concurrens cum redia, � Centro per G aut S du(fta,nbsp;�cqu� ben� ducitur d B, extremitate are�s SRB, quamnbsp;a B, extremitate are�s GB. Deinde Tangens per Bnbsp;traje�la, cum diametro SG altetutrara partem versusnbsp;produefta , tantum concurrere poteft versus unicamnbsp;determkiationem; igitur AB eft Tangens are�s BG itemnbsp;are�s BRS, ejus Complcmenti ad .180�; item eft Tangens anguli I, ejufque Vieini, feu anguli BIS. Idemnbsp;eft de Secante AI.

Demonst. Chorda� arcuum fimilium ad Radios candem rationem habent (279); fed Sinus funt chor-darum dimidia (448); ergo Sinus illi ad Radios ean-dem rationem habent. e. d.

458. ScHOLioN. In Tabulis Sinuum amp; Tangtritium ordinariis, Radius, feu Sinus totus, concipitur in 10,000,000 parces sequa-les divifus ; amp; ultra has fra�tiones, in dcccrminando Sinuum Scnbsp;Tangentium quantitate, non defcenditur. Secantibus m n uti-mur, cum omnia Trigonometrie Probleinata abfque, illaruinnbsp;ope refolvi poffinc.

459. Chorda fuftentans arcum 60� eft =; Radio C2iS),feu Sinui toti: Sinus ergo anguli 30�=5,000,000.

460. Resol�tio I. Ex quadrato GL Radii, feu Sin�s totius, fubtrahatur quadratum Sinus GO :

Etenim, cum ? GOLX fit redfangulare , eft GX = OL; fed GL* = GO'* OL� feu GX^; ergo amp;cnbsp;E. G. lit GI = 10,000,000; GO = 5,000,000; in-venietur GX = 8660254, Sinus 60�.

461. nbsp;nbsp;nbsp;Resolutio I. Quasratur per praccedens problemsnbsp;Sinus verfus BO :

II. Ex GOquot; OB�, = chordae GB�, radix quadrata tra-hatur : erit hacc a�qualis ducendac BG; adeoque ejus dimidium dabit quaefitum.

II. Deinde ad BX, GX amp; BK quaere quartum pro-portionaleni (arith.); atque bic inventus dabit KE.

Nam AA BGX amp; BKE funt acquiangula, ergo fimi-lia; eft ade� EX : GX = BK : KE; ergo amp;c.

Datis Sinibus FG Cfipi- 3) ^ DE, arcintm FA amp; DA, quorum differentia DF 45' mawr non ejt; invenirenbsp;Sinum quemcumqiie intermedium IL.

463. nbsp;nbsp;nbsp;Resotutio I. Ad DF, IF amp; -differentiam internbsp;^p^DE amp; IL, quxratur quartus numerus proportionalis

Demonst. Cum arcus DF amp; FI paucorum fol�m fint minutorum, pro rcdtis, citra errorem fenfibilem.nbsp;haberi pocerunt. Por^ro FG, IL amp; DE parailelaj lunt;nbsp;li igitur ex F ad DE demitcacur perpendicularis F'H;nbsp;erit HE = PG = KL : aucoque DH cft differentianbsp;datorum Sinuum PG amp; DE ; ob parallelas autem, eftnbsp;DF : IF = DH : IK ; jam vcro IK FG = iL;'ergonbsp;quartum proportionalcm addendo minori Sinui dato,nbsp;prodit Sinus IL defiderams. Q^. e. d.

Datis Sinibus BD amp; FE ( fg. 4) duorum arciiiim qito-rumcumque AB amp; AP j invenire Sinum areas femidijferentiai corumdem.

464. Resol�tio I. Sinus minor BD fubtrahatur ex majore FE, relinquetur differentia = P'K :

II. nbsp;nbsp;nbsp;Ex datis Sinibus BD amp; FE, inveniantur Cofinusnbsp;BI amp; FH C460) = DC amp; EC :

III. nbsp;nbsp;nbsp;Cofinus minor FH fubtrahatur � majore BI; ericnbsp;BK differentia ;

IV. nbsp;nbsp;nbsp;Ex BK� 4- FK� extrafta radix dat chordam BF,nbsp;eujus dimidium dat Sinum qutclitum.

4*^5. Resolutio. Ex dimidio quadrati Sin�s totius ex-tradla radix dat petitum.

Sit enim angulus L f/Fg. O 45� j ctiam angulus IGX = 45^ Ergo GX = ^XL; fed GL� = GX� LX�,

S %

P R O B L E M A VIL

466. Resolutio. Sinum totum divide media amp; extrema ratione, amp; dimidium numeri majoris dabit quae-litum.

Demonst. Sic I Cfig- .5) in Centro g6�; fiat BO = chordae BA : quoniam 1 = angulo �BI, eft 01 =nbsp;OB = AB. AA AB� amp; AIB funt acquiangula, ergonbsp;fimilia; erit ade� AI : AB = AB : AO ; igitur cumnbsp;AB == OI; eft Al : 01 = 01 ; AO; itaque AI divifanbsp;cft medid amp; extrema ratione ( 2.28), amp; 01 = ABnbsp;eft ejufdem pars major ( narh in A A�B, cum angu-lus AOB fit major angulo ABO, eft AB (170) major quam AO); ergo chorda AB fuftentans arcumnbsp;36�, cujus dimidium = finui 18�, eft sequalis ma-jori parti Sin�s totalis, divili media amp; extrema ratio-ne. e. d.

467. nbsp;nbsp;nbsp;Invento Sinu 18�, reperitur Sinus 36� per pro-blema tertium C462).

468. nbsp;nbsp;nbsp;ScHOLiON. Ut datum uumerum dividas media amp; extremanbsp;ratione : lumito quinies quadratum ejus dimidii : ex iilo qua-drau) extrahe radicem quadratam : ex hac radiee inventa fubtra-lie numerum datum; refiduum amp; radix illa erunt partes quaslitte.nbsp;Manifefta fit methodus per Problema V. (269) ; etenim cumnbsp;ibidem AE - � BA; BE* = 5AE*; igitur radix BEquot; dabitnbsp;BE; AE fubtrafta ex BE, rciiquura BG = BK dividet ABnbsp;media amp; extrema ratione..

PROBLEMA VIII.

Dato Sinu FG ( fig. 3) unius minuti, fm 60quot;; invenirc Sinum unius, vel aliquot Jecundoriim MN.

469. Resolutio. Ad AF, FG amp; AM quaere quartum proporcionalem, ille dabit quod qua�ritur.

Demonst. Arcus AM amp; AF perexigui funt, atque ade� AMF pro redta haberi poteft citra errorem innbsp;fra�tionibus radii decimalibus, quibus Sinus exprimi-mus, affignabilcm; quare c�m fit MN ipfi FG paral-lela ; erit AF ; FG = AM: MN. e. d.

470. ScHOLiON. Eadem latione, fi fueiic opus, inveniri poteft Sinus aliquot fcrupuloium terciorum.

PROBLEMA IX.

Ddtfs Sinibus 30� (459); 15� C4�1); 4.5� C4�.5)� �* 36� (467); Cetnonem omnium Sinuum conflruere,nbsp;nonniji iinico miniito ^ aiit denis feciindis,nbsp;imm� unico fccundo inter Je differentibus.

471- Resolutjo I._ Ex Sinu 36�, amp; 18�, inveniantur Sinus 9�; 4� 30'; a� 15' (461). Sinus 54�; 73�;nbsp;81�; 85� 30' ; 87� 45' (460). �Sinus 27�; 13�nbsp;30'; 6� 45'; 40� 30'; 20� 15'; 42� 45' (^461);nbsp;inde Sinus 63�; 76� 30'; 83� 15'; 49quot;� 30'; 69�nbsp;45'; 47� 15' (460) : ulterius Sinus 31� 30'; i.5�nbsp;45'; 38� 15'; 24� 45� (4^0 : hinc Sinus 58� 30';nbsp;74� 15'; 51� 45'; 05� t4' (460) : denique Sinusnbsp;29� 15' (461), amp; ejus Cofinus 60� 4.5' (4*^0).

II. Ex Sinu 45� inveniantur Sinus 22� 30'; n� 15' C461); 67� 30'; 78� 45' (4609; 3 3� 45' (461);nbsp;56� 15' ( 460).

�II. ExSinu 30�, amp; Sinu 54�,mvcniamr Sinus I2�C4^4gt;

IV. nbsp;nbsp;nbsp;Ex Sinu 12�, inveniantur Sinus 6�; 3�;nbsp;nbsp;nbsp;nbsp;30';

45' C46O; Sinus 78�; 84�; 87�; 88� 30'; 89� 14' ('460) : porro Sinus 30�; 19� 30'; 9� 44';nbsp;4^�; 21�; 10� 30'; 5� 1.4'; 43� 30; 21� 4.lt;;nbsp;44� 15' (4�1) : ulterius Sinus 51�; 70� 30'; 80�nbsp;i.lt;; 48�; 69�; 79-� 30'; 84� 45'; 46� 30'; 68�

45� 45' C460) : inde Sinus 25� 30'; 12� 4.4'; 35� 1.5'; 24�; 34� 30'; 17� 1.5'; ?9� 4.5'; 23� 'is'nbsp;C461); binc Sinus 64� 30'; 77� 15'; .44� 45'; 66�;nbsp;55� 30'; 72� 45'; .50� 15'; 66� 45' C460) : hincnbsp;porro Sinus 32� 1^'; 33�; i� 30'; 8� 1.4; 27� 44'nbsp;(461) : inde ulterius Sinus 57� 45' ; 57�; 73� 30';nbsp;81� 45'; 62� 15' ( 460) : porro Sinus 38� 30';nbsp;14� 15'; 36� 45' (461); amp; horum Coilnus 61� 30';nbsp;74� 44'; 43� 44' (460) : dengue Sinus 30� 45'nbsp;(461), amp; ejus Cofinus 59� 14' (4609.

V. nbsp;nbsp;nbsp;Ex Sinu 14� inveniantur 7� 30; amp; 3'' 45' (461) :nbsp;hinc Sinus 75�^ 82� 30'; 86*quot; 15' (4609 : inde 37�nbsp;30 j 18� 45'; 41� 15' (461), amp; horum Cofinusnbsp;42� 30'; 71� 14'; 48� 44' C 4609 : denique Sinusnbsp;26� 14' (4619, amp; ejus Cofinus 63� 44' (4609.

VI. nbsp;nbsp;nbsp;Quod fi Sinus hac ratione inventi in ordinem re-digantur , numero 120; amp; difi�rentiam inter duosnbsp;immediat� flbi mutu� fuccedentes 44' deprehendes:nbsp;inveniantur ergo Sinus intermedii per Problema IV.

VD- Tandem Sinus fcrupulorum fecundorum, ab i uf-que ad 60, inveniantur per Problema prtcccdcns (4699: ita Canon Sinuum erit conftru�tus.

Dato Sinu GO (fig- i) arc�s BG; invenire Tangentem AB, amp; Sccantem AI ejufdem arc�s.

472. Resolutio. Quoniam eft AB parallela ad GO, dicatur : �t Cofinus GX = 01 ad Sinum GO; ita

BI, feu Sinus totus, ad Tangentem AB. Item ut Cofinus GX = 01 ad BI, feu ad Sinum totum; ianbsp;GL, l�u Sinus totus, ad Secantem AI.

473. nbsp;nbsp;nbsp;ScHOLiON I. Conftrufto igitur Canone Sinuum, baud difR-cilis eft conftruftio Canonis Tangentium, atque Secaniium.

474. nbsp;nbsp;nbsp;ScHOLiON II- In ufu ordinario Canonis Sinuum atque Tan-gennium, l�lenc otnitti dua; dexdms Cyphrae, quae nota ab aliisnbsp;feparanturi iil�dera camen ucendum elt, cura calculus exaftiornbsp;defideratur.

475. nbsp;nbsp;nbsp;ScHOLiON III. Quoniam Sinus lt;amp; Tangentes funt numerinbsp;prolixi, qui multiplicaiionem , amp; divifioncm in Trigonomettianbsp;permoleftas redduni �, ideo Geomecrae certos numeros excogica-runc, qui loco vulgarium , non fine infigni calculi compendio,nbsp;poffunc adhi'jcri; quia mukiplicadonem in additioncm ; amp; di-vifionem in fubcractionera convertunc. Dicuncur Logarithmicnbsp;amp; non fol�m pro omsibus Sinibus amp; Tangentibus', Verdm etiamnbsp;pro numeris naturalibus ab i ufque ad looo, nonnunquam, ultra,nbsp;in Tabulis Sinuum amp; Tangencium vulgaribus extant.

47*5. Resolutio. Ut Logarithm! eo accuratiores inve-niantur,aflumendi funt Sinus ad radium 10000000000 conftru�ti. Mul�lantur neinpe Sinus in Canone Pi-Tisci majore 4 ultimis notis. Cum adeo Sinus fintnbsp;numeri 10 �t plurim�m notis conftantes, in Ca-none autem Logarithmorum, qui proftat maxirno,nbsp;numeri naturales ultra 5 notas non afcendunt; Logarithm! eorum inveniuntur per Problema Arithme-tices. Utendum vero, eft Canone Logarithmorumnbsp;niajore.

Ex. Gr. fit inveniendus Logarithmus Sin�s 23�, qui apud Pitiscum eft 39073112.84. Refe�tis, versusnbsp;finiftram, quinque notis 39073, iphs refpondens Logarithmus eft 4.5918768; confequenter Logarithmusnbsp;numeri 3907300000 eft 9.5918768. Differentia tabu-laris eft ui. Quare infertur : ut 100000 ad lU,

ita notac reiidusc Sin�s dati 11284, numeram quaf* tum proporcionaiem 12 : qui fi addatur Logarithmonbsp;P.59187�8, prodit Logarithmus quxiitus 9.5918780 ;nbsp;q�alis in Canone triangulorum artificial! reperitur.

II. A fumma fubtrahatur Logarithmus Cofin�s; amp; re-fiduum eft Logarithmus Tangentis.

�In A reBanguIo unus Cathetorum eji ad alium, ficut Sinus totus ad Tangentem anguli, huic alterinbsp;Cathetorum oppojiti.

478. Si fit A recfius C fig- 8) eft AC ; AB = Sin. tot. : tang. C. Item eft AB ; AC = Sin. tot. : tang. B-

Demonst. Ex C, intervallo CB, due Circulum vel arcum ; produc CA ulque m L; ad pun�tura L fiat

tangens LK; producamr CB, donee concurrat cum tangente LK; c�m KL fit parallela AB ^ erit AC; ABnbsp;= LC ; LK; jam vero LC cft radius Circuli, l�u films totus; amp; LK eft Tangens : ergo AC : AB == fin.nbsp;tot. ; tang. C. Si ex B, intervallo BC, arcum ducas;nbsp;produdia BA, donee concurrat cum tangente ad punctum C ereifia; evidens fiet effe AB t AC = BC l�unbsp;fin. tot : tang. B. Q^. e. d.

479. nbsp;nbsp;nbsp;Demonst. C�m enim omne triangulum Cir-culo infcripcibile fit; erunt latera AC, CB amp; ABnbsp;chordae arcuum cognominum; conl�quenter latera di-midia finus arcuum dimidiorum 044�) � l�d arcus di-midii funt menfurac angulorum oppofitorum B, A amp;nbsp;C (138); ergo �t lams AC ad finum anguli fibi op-pofiti B; ita latus BC ad finum anguli fibi oppofitinbsp;A; ita etiam AB ad finum anguli fibi oppofiti C.

Jn A ( fig. 10) Ji A formetur lateribiis AC amp; AB intequalibus; atque perpendicularis ab A ad BC duc-ta, V. g. AO, intrd bajin BC cadat : efi BC adnbsp;AC -i- AB, [icut differentia inter AC 6* AB ad dif-ferentiam inter fegmenta CO 6* OB.

480. nbsp;nbsp;nbsp;Demonst. Ponamus AC majorem quam AB;nbsp;crit etiam CO major quam OB : nam AC^ = AO*nbsp; CO*; amp; AB* = AO* OB*; quoniam itaque AC*nbsp;fft majus AB*; erit AO* CO* majus AO* OB* ;nbsp;igitur CO* eft majus OB*; ergo CO eft major quamnbsp;OB. His praeniillis : ex A, �t Centroj intervallo late-

ris AB minoris, defcribe Circulum; amp; produc FA uF que in peripheriai pun�tum E; eft CE = CA AB;nbsp;B� = OL C 126); ergo CL elt ditferentia CO ad OB;nbsp;amp; Cb clt differentia CA ad AB : jam vero eft BC: CE,nbsp;feu AC AB = FC : CL (259J. e. d.

Jn A C 11 ) fi A formetiir lateribiis AC amp; AB inaqualibus; atque perpendicularis ex A ad BC demif'nbsp;fa , intrd bafin BC cadat : ej} AC AB ad FC difnbsp;ferentiam , feut 1'angcns medietatis angulonim B Cnbsp;Qproidiais lateribiis oppofitorumf, ejt �(f Tangenten!nbsp;medietatis differentice angulorum B amp; C.

481. Demonst. Ponamus latus AB brevius effe AC. Ex pun�to A, �t Centro, intervallo AB Circulum defcribe : produc FA ufque in E : ducatur EB; amp; FKnbsp;paralleia ad EB : infuper due FB : tune ex F, xxtnbsp;Centro, intervallo FB, due areum BG; amp; ex B, �tnbsp;Centro, etiam ad intervallum BF, ducito areum FI.nbsp;CE eft = AC -I- AB. FC eft differentia inter AC Scnbsp;AB. Chorda BE eft Tangens medietatis angulorumnbsp;B C : nam, c�m F fit Centrum are�s BG, amp; an-gulus FBE redlus (menfuratur enim | are�s FRE);nbsp;erit EB Tangens Circuli, eujus BF eft radius (124);nbsp;item eft 7Fngens anguli EFB; jam vero hie angulusnbsp;Ecquatur dimidio angulorum B -i- C : etenim, quianbsp;AF = AB, eft angulus AFB = angulo ABF; itemnbsp;angulus AFB eft x'qualis angulis C, amp; FBC fimulnbsp;fumptis C 167 )� FK eft Tangens medietatis differentia: inter B amp; C : nam FK eft Tangens anguli FBK;nbsp;B enim eft Centrum are�s F'I; amp; quondam, ex con-ftruCtione FK paralleia ad EB, atque ade� angulusnbsp;KFB = angulo EBF ( T09 )^, qui redus eft; eritnbsp;etiam KFB reiftus; ergo KF eft Tangens are�s Flnbsp;(124), item anguli FBC; jam vero B l�perat C ad

bis angulum FBC : nam angulus ABF = angulo AFB = angulis C amp; FBC. Igitur c�ra FK parallela adnbsp;EB , cit CE feu AC nbsp;nbsp;nbsp;nbsp;: FC = EB : FK. Q. e. d.

In A reSangiilo, ambobus Cathetis AB amp; AC (fig. 12) nous '; invenire Jrjyputhenujam, S� angulosnbsp;acutos B amp; C.

dudo fecundo, feu AC = 54; in tertium = 100000; fadum 5400000 dividatur per primum, feu per 79;nbsp;amp; quotus 68354 dat tangcntem anguli B; cui in Ca-none tangentium refponder quam proxim� 34� 21' :nbsp;Eoc fubtrado ex 90�, refiduum 55� 39'dat proxim�nbsp;angulum C.

Per Logarithmos vero finuum amp; tangentium faci-lius operaberis modo fequenti. Logarithmum fecundi, Ecu numcri 54, amp; Logarithm, finds totalis adde innbsp;nnam fummam; ex qua fuberahe Logarith. primi 79 gt;

Log. tang. B - - 9.8347667, cui in Canonc reli^oGdent etiam quam proxim� 34� 21'.

483. ScHOLiON. Dicitur gudm proxim� : quia numeri 68354 Canoiie cangendura', amp; 9.8347667 in Canone Logarichmorumnbsp;tangencium non exafl� repenunttir; atquc ade� ultra 34� m'nbsp;miniua aliquoc infcriora, feu fcrupula lecunda amp;c. aderunt. Sinbsp;itaque prt�ter fcrupula prima, ukeri�s fecunda delideres; fequen-li invcnientur methodo.

484. nbsp;nbsp;nbsp;1� Operanda per Canonem finuum amp; tangen-tiuin : a tangente v. g. 68354 fubtrahe tabulae proxim� minorem 68343, amp; notetur differentia ii. Similiter proxim� minorem 68343, fubtrahe ex proxim�nbsp;majore 68386, amp; notetur differentia 43; atque dici-to : 43 dant 60quot;, quot dabunt 11? amp; mvenies 15quot;;nbsp;crit ergo B 34� 21' 15quot;; amp; quoniam adhuc datur,nbsp;operatione finita, refiduum; adeffe fcrupula tertia collines at negligi poffunt in communi praxi; � tarnennbsp;ea exoptes ; iacile ex diclis reperies tertia, �t inveniftinbsp;fecunda amp;c.

485. nbsp;nbsp;nbsp;2� Si ref�lutionem trianguli inftituas per Lo-garith finuum aut tangcntium; eodem modo, �t mox,nbsp;proc�d�s; v. g. in cafu dato; a Logarithmo tangentisnbsp;B 9.8347667 , fubtrahe tabulrc proxim� minoremnbsp;9.8346961, amp; notetur differentia 706. Similiter Lo-oarithmum proxim� minorem 9.8346961 fubtrahe exnbsp;proxim� majore 9.8349673 ,amp; notetur differentia 271a;nbsp;�tque dicito : 2712 daut �oquot;, quot dabunt 706? 8cnbsp;etiam invenies 15''amp;c.

PROBLEMA II.

quot;Datis duobus anguUs A amp; C (fig. 13), und cum /lt;z~ ftTc AB; invenire latus BC.

Ex. gr. fit C = 48quot; 35'; A = 57� 29'; AB = 74 pedibus. Per Logarithmos ita opcramur :

487. ScHOLiON. Quod fi 83 pedibus non contentus, etinm digitos, feu pollices, defidcrcsquot;, evolve eundem Logarithmum BC , fub Characleriftica a poft 830 : amp; Logarit. 83a quam proxim�nbsp;ad eum accedere deprehendes ; adeoque, pr;ecer 83 pedes, ad-liuc a digitos effe. Si porr� Lineas defideres; quEere eundemnbsp;Logarit. denu� i'ub Charafteriftica 3 poll 8320; amp; ipli quamnbsp;proxim� Logarit. 8321 refpondentem reperiesproinde fore latus BC 8�, 3', 2'7i Hoe pa�lo femper ratio inftituendanbsp;eft, quando Logarithmus, fub fua Charafterillica, non accu-ratus repcritur.

489. nbsp;nbsp;nbsp;ScHOLiON I.' Nota tamen , per piKcedens Problema dcteimi-nari fol�m A, auc ejiis Vicinum : ponamus cnim in Anbsp;determinates efle angulos, atque latera; li forec C = f; �C = 6c-,nbsp;amp; AB == a6; tantum inferri poteft A = a, vel A = Vici-iio a (_ 19U ).

490. nbsp;nbsp;nbsp;ScHOLioN II. Nonnunquam ver� A ex adjunftis determina-ri : puta fi fueric C reftus, auc obeufus (i�6); vei li fucric Cnbsp;acutus, amp; latus AB majus latere BC ; nam A non pocerit cf-le rectus, auc obtulus, ctim C deberet efle iplo major (170).

491. nbsp;nbsp;nbsp;ScHOLiON III. Ut invenias B : fubtra�tis A C ex 180�,nbsp;reliduura dabic quslitum; �c patec ex adjuncto l'chematej in quonbsp;ponitur A 55� 40' 39quot; j C 64� 33'.

Datis duokis Trianguli lateribus inaqualibus AC o� BC (fig. 13)5 angulo intercepto C;^nbsp;invenire angulos reliquos.

Ponuntar AC amp; BC inaequalia : alias, c�m foret A = B, faciilim� innotefcerent abfque Trigonometna.

ad diirerentiam eorumdem ; ita T(ingcns femiluinm:c angi:dorum A amp; B,nbsp;ad 'I'angentem femiditFcreiJdsc eorumdem.

II. Addatur femidiffcrcntia ad femifummam ; aggrega~ turn erit angulus B, datorum laterum majori ACnbsp;oppofitus. Eadem a femifumma fubtrahatur, rema-nebit angulus A.

P R O B L E M A V.

493. Resolutio I. Ponamus BC CfiS- ^4) tion efle minimum tnanguli iatus; adeoque A non erit minor quam B, neque minor quara C (170); itaquenbsp;B item C erit acutus; amp; pcrpendicularis AO, exnbsp;A ad BC demiffa, cadet intra baiin BC (i743-

Elementa Geometetje. fuerit AC = AB; crit CO = OB; B = C amp;c. Re*nbsp;Iblvatur igitur A AOB per Problema ill. (488 ).

n. Si AC amp; AB inscqualia fine, fiaeritque AC majus quam AB; crit CO majus quam OB. Sit CF dit-ferentia inter AC amp; AB : amp; CL fit differentia inter CO amp; OB.

AC = 45' nbsp;nbsp;nbsp;AB = 3�'

AB AC=8i' nbsp;nbsp;nbsp;FC='lt;?'

- 1.2�0��75, cui in Tabulis proxim� refpondent 18'. Quod fi ulterids qusefiverisnbsp;(487), invenies tandem LC = 1822quot;' :

Tabulis quam proxim� accedic Logarith. 17� 36'; adeo-q�e angulus B = 72� 24'.

Log. Sin. CAO - - 9.8108297, cui in Tabulis quam proxim� refpondec Logarith. 40� 19'; adeoquenbsp;angulus C = 49� 41'.

CS^EOMETm�^

P R A C T 1C JE

SECTIO PRIMA

DE INSTRUMENTIS GEOMETRICIS, EORUMQUE USU.

rp ca exarari folct Calamo, Plumbagine, ^ aut Stylo fcrreo vei xneo, juxta Re-gulam, ad pun�la data, applicatam.

Recta delineacur etiam fine Regula; fi filum, cretJl vel cerufsfl libutum, pun�tis datis appriraatur, amp;nbsp;mediis digitis prehenfum furfum traliatur, moxquenbsp;iterum dcntittatur.

Pracft� fint baculi plurcs, quatuor aut quinque pe-dibus ulti; quorum fummitati muccinium , aut folium chartac albac fit alligatum, ut � longinquo vi-

deri queant. In quolibct ducendic rc�lai cxtremo perpen^icularirer baculus unus defigatur; tum, ocu'nbsp;lo applicato ad eorum altcruirum, vifus in akerumnbsp;dirigatur; dum interim minilter alios, pront necelTenbsp;fuerit, intennedios, m diredtionem lineai vifualis,nbsp;iu figat Ceidem manu lignum dando, num a finif-tra, aat dextra divergant) ut oculo in unum di-reko, eseteri non appareant.

495^ ScHOUON I. Regui� ex Orichalco, aut argento parata:, facile chartam nigranc ; i'is ergo praefcrancur , qua: ex ligno du-liori, puca Ebenino, elaboraiae 1'unc.

ScHOLioN II. Num Regula exafta Cc examinatur; C juxea earn Refta ducatur, amp; deinde invertaiur Regula, duft�que li-ne;e rarlus applicctur, atque umc altera linca datta priori accurate coincidat.

497. ScHOLioN III. Ducendis in charta Lineis, pennse optimte iunt, qua: ex corvorum alis crelluatur. Atramentuin Sinicumnbsp;comrauni eft qu�que prtefcrendu-m.

transferantur partes 10 scquales Bi, 12, 23, 34, amp;c.: intervallum vero 10 partiuni = AB ex B in E,nbsp;CS E in F amp;c. quoties libuerit.

2. nbsp;nbsp;nbsp;In A excitetur pcrpendicularis AC, arbitrarisc lon-gitudinis, in partes 10 aiquales divilii.

3. nbsp;nbsp;nbsp;Per punda divikonum, 1, 2, 3, 4, 5 agati-tur parallelae ad Af�.

4. nbsp;nbsp;nbsp;In ultimam CD transferantur partes 10, partibus,nbsp;ipfius AB acquales.

5- Tandem puncta loamp;pIP ^ 8; Samp;j amp;c., li-'HCis traniVerlls Cp, 98 amp;.c. conneclantur.

V a

Dico, fi AB fuerit decempeda, fore Bi , I2, 23, 34 amp;c. pedes; 99 digituin unum; 88 digitos duos;nbsp;77 tres; 66 quatuor amp;c. Et ft fuerit AB virga;nbsp;331 , 12 amp;c. fore fcrtipula prima; 99; Sic. fcrupulanbsp;fectinda Sic.

Demonstr. Bi = 12 = 23 amp;c. = AB, per conftru�iionem ; fed pes eft decempedsc pars decima;nbsp;citni ergo AB fit decempecia, per hypothefin; eruntnbsp;Bi, 12, 23 amp;c. pedes. Quod erat unum.

Porro quia 99 eft parallela ipfi Ap, per conftruc-tionem ; C9 : CA = 99 ; Ap; fed C9 = ^ CA, per conltrudionem ; ergo 99 =nbsp;nbsp;nbsp;nbsp;-A-P-nbsp;nbsp;nbsp;nbsp;Quare,nbsp;nbsp;nbsp;nbsp;cum Ag

li.t pes; erit 99 digitus. Eodeni modo oftenditur clTe 88 duos, 77 tres amp;c. digitos. Quod erat fecundum.

Ex ditftis perfpeclu facile eft; pofito AB quacumque quantitate integrd, v. g. virga amp;c,; effe Bi, 12 amp;c.nbsp;icrupula prima; 99; 88 amp;c. fcrupula fccuuda. Quodnbsp;erat tertium.

499. Si ergo Circini crus unum collocatur in I, amp; alterum in K; erit intervailum IK = 1� 4' 5quot;; amp; itanbsp;porto.

3. nbsp;nbsp;nbsp;Dein a puneftis, 9, 8, 7 amp;c. ducico 9X, cacteraftnbsp;qtie, fic ut fingula fit parallela �ad AF :

4. nbsp;nbsp;nbsp;Producatur pro libitu AL, in qua, tot quot opusnbsp;erit, notentur partes LC, CB amp;c. aiquales AL.nbsp;Erunt ade� BC, CL, LA partes integrsc; Li, 12,nbsp;23 amp;c. fcrupula prima; pX, amp; catterai parallelsenbsp;erunt fcrupula fecunda.

501. Si igitur oporteat habere 1� 6' 9quot;; Circini crus Unum ponacur in B amp;: alcerntn in 6 ; hocque inter-vallum transleratur in re�tam, in charta du�iam; cuinbsp;deinde adjungas Circini intervallum a 9 in X.

502. Resol�tio I. Quoniam in Brabantia folet adhi-beri menfura 20 pedura^ qua; Virga audit; ideo 20 fiia terrea connedtantur annulis (�t videre eft innbsp;figura 3), fic ut a medio cujufvis annuli ad Centrum vicini pes unus detur. Extremitatibqs Cate-nx, utrimque, manubrium AB aptatur, atque abnbsp;hoe , ad centrum ufque vicini annuli pes quoquenbsp;habetur. Ut autem dccimalium praxi inferviat : annuli C, F, G amp;c. lerrei lint, amp; minores intermediisnbsp;annulis cupreis 1,2, 3 amp;c.; atque ita Bi, 12, 23nbsp;amp;c., live intervallum duorum quorumlibet annulo-rum cupreorum viclnorum, ~~ virgse, live Icrupu-lum primum dabit. Intervallum duorum pedumnbsp;extremorum dividatur in 10 partes aequales ; adnbsp;Icrupula Bccunda erit quoque parata Catena.

503. nbsp;nbsp;nbsp;ScHOUON I. Manubrium _AB, qualc edara habecur ad alte-rum catenae extremum, iemi - cirtulo , caius Diameter sequa�snbsp;fit Diametro Baculorum pedaliuni (de quibus problemate fe-quenci) eft inficxum in fut medio ; notabilis enim'furrepcrccnbsp;error, fi baculorum craffitiei negligeretur ratio.

504. nbsp;nbsp;nbsp;ScHOUON II. Cypbrar, a dexdmisnbsp;nbsp;nbsp;nbsp;fegregat* , deno-

tanc Virgas, fi iis menfura inftiuiatur : dextimae vent, poft riotam pofica?, fcruptila fuo ordine defignant. Ex. gr. 8.564 de-fignantur 8 virga:, 5 fcrupula prima (feu polliccs 10), 6 ftru-pula fecunda (live lineas -12), amp; 4 tertia amp;c.

505. nbsp;nbsp;nbsp;ScHOLioxr III. Menfurs; longitudo, ejufdemque partitio,nbsp;fi�a eadem eft ubivis gentium : varias menfurarum (pecks ic-

prjefentat Tabella fcqucns in particulis iftiulinodi, qaalium pes Rhenanus, amp; Romanus andquus, cft looo.

1000

1050

1036

9�8

960

934

507. nbsp;nbsp;nbsp;Paciunt ergo 878 pedes Lovanienfes , 909nbsp;Bruxellenfes; feu a 8 pedes Lovanienles quam proxi-m� faciunc 29 Bruxcllcnfes amp;e.

508. nbsp;nbsp;nbsp;ScHOUON. Leuca Brabancica compleftitur 2cx)Oo pedes Bra-baniicos , five looo Virgas. Milliare Belgicum, feu l.euca ici-ncris , communicer 1500 pedum Rhcnolandicorum arftimatur.nbsp;Paffus Geometricus fade 5 pedes; Fades verp Cominunis 2 '�nbsp;pedes Kqunt.

509. Resolutio I. in charta : ope Circini, cujus crura ad lineai datac intervallumnbsp;aperiantur , amp;: dein fcalce geometrica: applicentur.

Ad manum fint deni aliquot bacuH pcdales (uti fi-gura 4 exhibet) , ferrea eufpide muni�, quos nii-nilter deierat onmes ; prarccuat hic , una manu Catendm trahens, infequente, cuni altero Catenaenbsp;extremo , ipfo Geometra , qui fcrupulosc caveat nenbsp;aut mtorqueantur fila aut annuii catena;, vel anbsp;recto traraite declinetur. Applieato uno Catena; ex--tremo ad initium linesc menfurandae; intra femigt;nbsp;Circulum manubrii oppofiti figat miniiter unum c

baculis pedaiibus ; quem , ultra progredicns , ibidem relinquat; huic appiicet Geometra l�mi Circu-lum manubrii, dum mierira lecundum b-iculum minifter terra; infigit. Eaculi hi omncs a Geome-tra recoiligcndi, eorumque numero patebit virgarumnbsp;numerus; quem, cum Icrupulis, 11 qua; fuennt, fi-rata operauonc, in charta nocabit.

Awot objeBum qmdpiam, E. G baculus, murus amp;c. ^ perpendiculariter ereSum fit, explorare.

511. Resolutio I. Detur Gubus ligneus A (fig. 5), cujus iatus iingulum, pollicem unum acquat ;

II. nbsp;nbsp;nbsp;Funiculo, mobili per foramen congruum, in medio cubi factum, pendcat Cylindrus ameus B, a

III. nbsp;nbsp;nbsp;Cubi Iatus applicetur ad baculum , vel parietem ,nbsp;Ibramine ejufdem deorfum Ipectante : quoniam per-pcndiculum iftud conftantcr ad horizontem perpcn-dicuiariter delcendit, ejufdem funiculus debebit ubi-que ad objcdum explorandum efie parallelus; atquenbsp;adc� fic uilponi Cylindrus, ut ncquc ab co divcr-gat, neque mnitatur eidem.

51 a. Resolutio I. Si fucrit mediqcris extenllonis. Ex. Gr. tabula;, muri, norologia Iblaria amp;;c.; lolentnbsp;uti inftruraentis cxhibitis fig- � amp; 7, in quibusnbsp;pendiculum (feu globulus, five Cylindrus, plcjffnbsp;heus aut aineus, � tenui filo pendet), juxia lineam

perpendicularem ad BC. bafin, liber� pendet. Ba-fis BC in varies fenlus plano eipiorando applicetur : non enim crit hoe horizonti parallelum, nili con-ftanter perpendiculi lilum exaraca: linese, ad bafinnbsp;BC pcrpendiciilari, refpondeat.

513. SciiOLioN. Perudle quoque eft parallclepipeJum ligneum Cfig: �) pede longum; cui Cylindms Vitreus, fpiricu vini, ex-cepta majori a�ris bulla, rcplcms j fupern� amp; paiallel� affigicur :nbsp;bulla enim mediurn lubi non occupabit, nifi dum amp; tubus amp;nbsp;parallelepipedum in plano requieverint horizonti parallelo.

514. Resol�tio IL Si vero Campi plaga fit libcllanda : utuntur Geometra� inftrumento, exhibito fig. 9. Lagena: Cyiindnaca: AF amp;. BC, diainctri faltem uniusnbsp;po�icis, � vitro limpidiflimo conilata:, per fiphonemnbsp;AB ex Cupro, aut feiTi lamiijis ilannatis, compofi-tum , 7 aut 8 pedibus longum, atque tripedali lufi-tcntaculo innixum, communicant. Infunditur aquanbsp;limpida, donee ea ad mediam lagenularum altitudi-nem affurgac. Quoniam in brachiis fiphonis liquornbsp;fefe ad libellam conftanter componit; perfpicuura fit,nbsp;radium vifualem OX horizonti fore parallelum : atque adeo facile innotefcit dilFerentia, fi qua detur,nbsp;inter plagam datam, amp; horiiontaie planum.

515. nbsp;nbsp;nbsp;ScHOLroN I. Si fuerit montis ptoclivitas, aut longior quae-dam horizoncis plaga , examinanda : diverfis vicibus operationbsp;peragicur : ad boe praefl� fine pettiese nonnullae, ro aut 20nbsp;pedibus alcas. Collocaco inftrumento iibellacorio in montis ca-cumine A (^fig. lo), pertiea BC jubeacur ita defigi perpendi-culariter in montis declivi , ut radius vifualis, per lagenas tra-

. jeftus , incurrat in perticiB extrcmilm ( aut notetur punctum ejufdem in quod dirigitur). Dein crar.sferatur inftnimencum adnbsp;fecundam -lladoriem, ad eum feilicet locum , ubi prima petdeanbsp;BC fuerat erefta; atque collimecur-in pertieam GFj amp; ita por-ro. Prob� tarnen feiendum, aldtudinis inftrumenti rationcinnbsp;negligendam non effe.

516. nbsp;nbsp;nbsp;ScHoi.iON U. Si opetadonem fimilem ad oppofitam montisnbsp;dcclivitatem inftir.uas , innotelect qux dift�renua lit inter pla-

PROBLEMA

�njlrumtntum Goniometncum, feu Graphome-trum conjhuere.

giy. ResolutiO I. Ex orichalco, magnitudinis arbitra-risc.parari folenc ; quac tarnen majoris diam�tri funt, prscfttnt. Exhibetur fig. i Tab. XII.

il. X Centrum eft, per quod trajiciatur refla FH J peripheria lemi-Circuli disdditur exa�lil�mc in 180�.

III. Regula KL, in Centro X mobilis eft; in ejus medio re�ta KL delineatur, quac per Centrum X tran,'^ feat.

tree, feu Pinnules , in quibus rima , ad inftrumen-tum qu�que perpendicularis, exadl� refpondeat dia-metris AB amp; KL.

V. nbsp;nbsp;nbsp;Intra inftrumenti aream collocari poteft Pizis nau~nbsp;tka , Iic UE ejufdem linea mcridiana coincidat cum.nbsp;radio defignante 90�. In Rofa notanda eft lineanbsp;declinationis; hxc autein hodieduin in Brabantianbsp;eft circiter 18�, occidentem versus.

VI. nbsp;nbsp;nbsp;Figitur Graphometrura cneo giobulo intra exca-vata fegmenta PG, ut cochleii Q, pro libitu, plusnbsp;minufve prefsa, Graphometro fitus quifcumque de-tur, atque firmetur : utque tant� confiftat firmius,nbsp;fuftentaculo tripedali totum innititur.

VII. nbsp;nbsp;nbsp;Ut graduum quoque minuta, cx. gr. a 5' ad 5',nbsp;dignofcerc valeas : fic precedes.

tranfit OX radius, qui amp; trajidiur per rcguloc pinnii^ las), fit arcus OE item OG 13�; queinlibec aucem ar-eum divide in 12'^ tanium , ut vides; ut fic quilbuenbsp;gradus regula; faciac 1� 5'. Pone igitur lincam OX,nbsp;cui refpondcc linea villonis, per dioptras traje�'ta, nonnbsp;coincidere cum aliquo gradu periphcriic Graphoroctri,nbsp;�ed ultra eum, ex. gr. 20� vagari. Attent� inquire,nbsp;bui gradui penpheriae refpondeat gradus quidam Secto-ris OE; ponamufque reperiri fecioris gradum,nbsp;j2mo peripheriaj coincidcntem; erit ade� angulus BXOnbsp;�12� periphericC, amp; prxterea 8 graduum fe�toris : quo-niam ergo gradus quilibet ledtoris faciat unum gradum, amp; quinquc minuca peripheriaj Graphometri; oc-to illi fedloris gradus facient 8� 40' peripheriaj Graphometri ; erit adeo angulus BXO ao� 40'. Itaque dignofcesnbsp;graduum quanticatem, numerando in peripheria Graphometri, quot gradus integri comprehendantur, atquenbsp;in figura ao�; turn numeres in fe�lorc, quotus ejusnbsp;gradus refpondeat alicui peripheriaj gradui, amp; totiesnbsp;adde 5'.

518. SciiOLioN 1. Si libueric graduum minuta fingula prima de-terminare majoris diametri fic oportec Graphometrum : fu-manturque tune in feftore Regul� mobilis 61 gradus periphe-tia;, arcumque illura Regulae mobilis divide in 60�.

.519. ScHOLioN 11. Haud difScile eft , ex diftis, intelligere, quo pafto pars qu�libec divifionis eujufeumque (ex. gr. Icalx , Ba-rometri.s, aut Thermometris affixa�). pro arbitrio dividi qucatnbsp;in minores particulas ; etenim juxta gradus inftrumenti fixos,nbsp;adjungatur Regula mobilis, in quam v. g. ti gradus cransferas;nbsp;eorumque intcrvallum in 10 parces aequales pattiaris; liocquenbsp;pafto cujufque gradus dignofccre poteris amp;c. Regula hscnbsp;mobilis, inftrumenco cuicumque affixa, Nonius compellatur.

^20. Resolutio I. Detur lamina ex orichalco AXBCA � Cfg- 3) pro Centro fit Cufpis X. Diameter AB :

circiter 3 pollicum fu. Periphcria accurat� in 180'� gradus dividatur. Aren inftrumenti excavanda eft,nbsp;ut confpici queac debica linearum, amp; verticis angulinbsp;ad Centrum, applicatio.

^11. ScHOLiON. Conficitur quoque CK cornu lamina pellucida ; amp; quoniam per illud facile cranl'parenc exaracse lineae , non eftnbsp;opus excavauone : immo amp; hoe Tranfporcoiiura orichalceo ianbsp;eo prieftac, quod chartam minim� maculec.

Traniportorii Centrum, l�u eufpis X, ad anguli vertieem applicetur; radiufquc inftrumenti coincidatnbsp;altcrutri cruri anguli dati; tum videatur, per queninbsp;peripheriai gradum crus aliud Cpirodudum ft opus)nbsp;tranfeat, v. g. in figura LX; atque perifgt;hcricc arcus,nbsp;inter B, amp; redam LX comprehenfus, delignabit valorem anguli dati.

Sit arcus AL (fig. 4) pars fexta peripheriae Circuli, atque adc� graduum 60. In partes fox primo dividatur; fexta autem pars prima AB, in 10 gradusnbsp;fubdividatur; notenturque partes; ut exhibetur innbsp;figura.

hunc arcum : Circino capiatur are�s intcrvallum AL (eft hoe R'qualc radio Circuli, cujus arcus AL parsnbsp;cxiftit (218); unoquc. pede-in X fixo, altero defcri-hatur arcus indennitus (produclis antea cruribusnbsp;XK amp;'XC, ft opus) : fumatur Circino intervalluninbsp;ZO, atque illud transferatur ad arcum AL, qui valorem dabit are�s OZ, adeoque amp; anguli centralis X.

525. nbsp;nbsp;nbsp;SciioLiON. Si facrit arens ZO major 60 gradibus; prim�nbsp;applicetur arcui ZR, are�s AL totum intcrvallum ^ ex. gr. o

X a

Z in L', amp; deinde inquiratur, quot gradus compleftatiir pars iclidua LO ; hac addita cum �o� in unam fummara, quanticasnbsp;anguli X obdnecur.

Graphometrum figatur, feu difppnatur ita, ut ejus Centrum anguli menfurandi Vertici refpondeat. Signa ponantur in extremis rc�arum, angulum datumnbsp;conllicuentium. Difponatur lie Graphometrum, utnbsp;per dioptras Diametri HF i) linea vifualis oc-currat fignorum uni : dein Regulam KO mobilemnbsp;(toto caeteroquin inftrumento lixo) lie dirige, utnbsp;per dioptras KO radium vifualem trajiciens, aliudnbsp;lignum pofitum detegas ; in peripheria, anguli datinbsp;valorem repenes.

527. nbsp;nbsp;nbsp;ScHOLiON I. Si in Campo angulum datum metiri non H-cea: ; Vcrdcalis , auc ejufdem 'Vicinus inveitigetur.

528. nbsp;nbsp;nbsp;Schol, II. Certus evades te omnes angulos exaft� dimen-fum efle, li, finita oper.acione , TiciiKiS quoque medaris �, hif-que prioribus additis, fumma quorumlibet Vicinojum 180 ad�-quet.

529. nbsp;nbsp;nbsp;Schol. III. Inquirendis angulis regtedientibus , aut procur-rentibus mororura amp;c. infervie inftrumentum exhibitum fig. d:nbsp;confiac reguhs AB amp; BC circa axin, feu centrum B, pro libi-tu aperibiiibus.

530. nbsp;nbsp;nbsp;Schol. ly. Anguli A ( fig. 7) valor quoque inquiritur openbsp;Trigonometrie ; fciiicec ab A in X fume panes squales, pronbsp;libitu , v.'g. 100 ; ab A in Z v. g. 80 talcs; deinde inquirenbsp;quot tales inveniantur a Z in X; acque refolve A per Prbble-ma V. (493).

Data quantitate anguli; ipfum defcribere.

Ducatm- re�la XE s)- In X (punfto ubi for-mari petitur angulus) ponatur Centrum Tranfporto-rii, ita ut radius ejus cum redta XE coincidat : pumcrentur gradus a B versits C; amp; ad gradum

wkimum ex. gr. 45�, notetur cufpide pundum in charta. Re�la, ab X per illud pundum du�ta, v. g.nbsp;XL, angulum defidcratum formabit.

Ducatur redta XZ (fis;. 5); deinde ex X, �t cen-tro, intervallo arc�s AL (fig. 4 ), fiat arcug inde-finitus ZR; turn in aren AL totidem fumantur gradus, quot compledetur formandus angulus; at-que illi ab Z v. g. in O, transferantur; ab X pernbsp;� re�la ducatur, eiiicietque quod quacritur.

Collocetur Graphometrum ita, ut radius ejus, feu Diameter, latcri dato formandi anguli refpondeat :nbsp;Regula raobilis ad gradum datum promoveatur : bacillus ita erigi jubeatur, ut per dioptras coliineantinbsp;occurrat; hajcquc defidcratum efficict angulum.

534. ScHOLlON I. Angulum re�tum in charta , aut quovis ali* rainori plano dato, ducimus NorinA : componitur hxc biiiisnbsp;Regulis ex ligno, orichalco amp;c. ad angulum redlum 8)nbsp;juuftis.

535- Schol. II. Num Norma accurata fit probatur , fi eo mc-diante formetur angulus A (j�g. 7 ) : dein fumantur in latere AH partes tequales 4, ab A in X; amp; ab A in Z tales tres;nbsp;nifi a Z in X, 5 talium partium diflantia inveniatur, reftusnbsp;hand erit angulus A, adeoque nee Nornnz crura ad angulumnbsp;ledtum junfta fuerint.

536. Facile igitur, five in charta, five in Campo, angulum rectum conftruxeris; fi ex tribus re�lis, qua-ruin una 3, altera 4, amp; tertia 5 partes sequales facit,nbsp;triangulum conftruxeris. Circino illud eflicies in char

PROBLEMA XI.

jd punBo Z (fig. I Tab. XIII.), extra reel am AB dato^ ad illam dacerc rectam , qua cum ca angulum �nbsp;determinati valoris effieiat.

537. Resol�tio I. Due redam pro libim ZL; atque anguli L quantitas inveiligetur :

n. Anguluin L com formando , adde in on am fiim-' mam : h:cc fubtrahatur ex 180; reiiduum dat valorem anguli in Z furmandi, Eric ergo angulus Onbsp;defidcratas,

538. ScnoLiON. Ut in Cati^po GcomctrEc a nunfto dato C (fig. 1') ad reftam AB perpendicularcm ducanc ; Regulam Gr.iphonietrinbsp;mobilem diTponunt ad angulum reftum ; atque protedemes curanbsp;inftrumento in linca AB , fie ut Diameter ejus linea; AB coin-cidat (qnod exp�riuntur fepius per dioptras collineando), tamnbsp;diu progrediuneur, tinm per dioptras Regtilae mobilis, vifus oc-currac figno in C coiiftituco.

Per puiiMum C (lig. 3 ), ad datam rcciam AB, paralklam ditcere.

�I. Fiat angulus OCZ = O : amp; rc�la CZ, prout ne-celTe iuerit, utrimque protrahatur : critque ZL pa-rallela ad AB ( 115).

�40. ScHOLioN. In charta nonnunquam utuntnr ParalleUfmo (^fig. 4) ; efi: autem inflrumcntum ex duabus regulis IL amp; CDnbsp;paratum ; recinaculis EF amp; GH tequalibiis, fic ut EG fit �qua-lis FII, in punflis E, F, H, G mobilibus ,coniunguntur. Quo-niarri in ? GEFH latera qutelibec oppolita conftantcr Kquab�anbsp;funt , quocumque inodo regulte ad fefc mutu� admoventur ,nbsp;? GEFH fempet parallelogrammum erit (aoS)-, atque adeanbsp;GE amp; HF conftantcr paralielce fint opoitet (igS ); ergo �; iq'*

- l-ulanira IL amp; CD latera inter fe qu�que parallela erunt Cto6). Ut igittir hujus Paralklifmi ope, per pundittn G datum, patal-lelam agas ad datam AB ; Regul� latus unum applices ad rec-tam AB, alterius ver� Regultc latus ad pundtum C adducas;nbsp;amp; juxta hujus dudtum, per C redtam agas ^ eritque hsc priorinbsp;AB parallela.

�41. Resolutio I. Bins Rcgulac 5 amp; 6) ex ori-chaico, AF Sc VD, medio circitcr pollice lat�, pc-de vero medio long�, in Centro X ita conjungun-tur, ut inftar Circini (non tarnen nimid facilitate aut moleftia) moveri Q[ueant : imm� amp; in eadem re(tanbsp;dilpofit�, R^ul� pedalis vices habeant. Fig. 5 ex-hibet fuperficiem Circini proporiionalis ab una parte,nbsp;fig. 6. ver� oppofitam.

II. X Centrum mot�s quam poteft accuratiffim� dc-terminetur ; nill enim ad hoe redt� omnes (duabus ultimis exceptis) tendanc, totus inftrumenti ufusnbsp;claudicat, k. hallucinatur.

54a. III. Ducantur primo rc�l� du� LX; atque fmgu-la in 200 V. g. partes �quales dividatur : inlcrvienC autem redlarum divifioni , aut earumdem determFnbsp;nand� rationi; vocantur Linete partium aqualium.

543. IV. Secund�, redlas ducito PX, qu� propordo-nibus quadratis re�larum determinandis propri� fint; has autem modo partieris fequenti. Sit XI f � PX:nbsp;erit adeo PX� ~ XP : fiat EK Cfig- 1nbsp;EH f � EK, fit perpendicularis ad �K; fit EI ==:nbsp;EH; erit adeo HF = 2EF ; fiat E2 = Hl; H2 fitnbsp;= E3 ; H3 = E4 amp;c. ka fiet, ut re�la EK divi-datur taliter, ut EP fit |�Ea�; f � Es'; | � E4�nbsp;amp;c. ^ � EK�. Transterendo autem hafce partesnbsp;ad redas EPnbsp;nbsp;nbsp;nbsp;, Planorum Lima erit con'

544. ScHOLiON. Mechanici divilion�m hanc perficere folent ope tabella: fcquentis, pofico PX aequare partes 1000. Porr� inve-niri hse poflunt Analogia, cujus primus terminus fit maximumnbsp;latus , ex. gr. 64; lecundus ver� hom�logum illud latus, quod

terminus fit quadratum numeri partium lateris maximi j quod hic =s 1000000; Radix quadrata quarti termini 78155, quamnbsp;proxim� dabit 179 partes pro latere homologo plani quintupli.

drate ad Dodecagonum inc�ufiv� , conftruendis adin-venta. Affumatur RX pro latere Quadrati inferipti Circulo : atque in 1000 partes divifum ponatur ;nbsp;Tabella adje�ld innotefcunt partes aflumendaj X5 pronbsp;Pentagono ; X6 pro Hexagono amp;c.; eas autem fa-Q�i negotie eruere liceat � Sinuum Tabula.

546. nbsp;nbsp;nbsp;VI. Ex adverfo, ad oppofitam. Circini oram, innbsp;fuperficie delineantur Chordarum, So�dorum, atquenbsp;Mttallorum. Linesc. Primo pro Chordis : diacanturnbsp;re�te CX : deinde dacatur in charta, vel poti�s innbsp;tabula aenea, recta illi scqualis, quEC aOumatur pronbsp;Diametro femi-Circuli, cujus Peripheria dividaturnbsp;exa�t� in 180�. Circino capiatur intervallum abnbsp;extremitate Diametri, ad fingulum ex gradibus, adnbsp;ufque 180; quodlibet autem in Circino proportio-rali transferatur ablt;X, in redtam XC; notenturquenbsp;partes per quinos, aut denos fingulos gradus : ha-bebunmr ade� rr�tata� i Sg� Chordai, quarum primanbsp;fubtcndit unum gradum, ultima vcr� 180� Circuli ,nbsp;cuius Diameter asq^uatur re�5:ai'^datac, qute ailumptanbsp;fu�rit pro Diametro lemi-Girculi diyifi in 180�.

547. nbsp;nbsp;nbsp;VII. Chordarum Liners, proxima efle folet Solido-rum Linea SX; talis autem hxc eft, ut SXquot;* lifenbsp;^ Xi^. Porro ut facili methodo medias divi-Itonum particulas determines, Tabella fequens detur.nbsp;Ponitur XS partium 1000; erit adeo Xi = 250.

X5, pro lacere homologo Solidi ex. gr. quiniuplo ir-ajoris quam eft minimum Xi : hanc inftiiues Ang-logiam, in qua primus terminus fit maximum 1'oli-dum C atqne adeo hic 64); fecundus fit Iblidum fi-milc, cujus qua;ritur latus homologum 5; tertiusnbsp;ver� fit Cubus numeri partium lateris maximi, leunbsp;bic 1000000000; quartus autem invenietur 78125000;nbsp;cujus � 427, quam proxim� dabit latus liomo-logum Solidi quintupli. Et ita de cacteris.

548. Sequitur Lirica Metallorum MX. Sex Metalla numerantur, chymicifque chara�feribus indigitantur :nbsp;nempe Aurum O : Plumbum b : Argentum d : Cuprumnbsp;S : Ferrum $ : Stannum %.

Quoniam duorum Solidorum pondera funt in rations compofita gravitatis fpecificai amp; Voluminis : co�-

fequens eft primo; fi tuerint ejufdem Voluminis, pon' dera effe in eadem ratione, qua funt gravitates fpeci-ficx : fecundo, li fuerint ejufdem gravitatis fpecificac,nbsp;fore pondera �t Volumina : tertio , fi fint ponderanbsp;aiqualia, fore gravitates fpecificas in ratione inverfanbsp;Voluminum. Quod fi ergo cognita fit ratio inter gravitates fpecificas amp; pondera, amp; prxterea datum lit Volumen unius; facil� innotclcit alterius Volumen; cumnbsp;proportionis quartus fit terminus. Ex. gr. quseritur Volumen folidi ex Ferro, ejufdem pondcris cum Solidonbsp;ex Stanno, cujus notum efl� ponitur Volumen, velnbsp;Diameter, quae fit 1000 partium : demus fpecificasnbsp;eorum gravitates effe ut 558 librae ad 516 libras cumnbsp;2 uneiis. Fiat hacc analogia ; ut 558 libr. ad 516nbsp;libr. 4- 2 uncias, ita cubus 1000, ad cubum Diametrinbsp;Solidi ferrei; quac, radice cubica extracta , invenieturnbsp;974 quam proxim�.

549. Tabclla lequens finiftima indicat gravitates Ipe-cificas; immo quot libras amp; uncias pes cubicus cujuf-que Metalli adaequet in menfura amp; ponderc gallica-nis : dextima verb indicat rationem latcrum homolo-gorum, feu Diametrorum in Sphxris; pofita Diametro Metalli leviffimi, feu Stanni, 1000 partium; dum So-lida, ex iis compofita, xqualis funt ponderis.

�50. IX. Ad Oram Regular extimam GD amp; GA rec-� ta ducitur , in qua partes notantur, quibus inno-tefcunt pondera Globorum 1'ormentorum beilicorum: quoniam enim experientia habet gl�bum ferreumnbsp;Diametri 3 pollicum Parifinorum ponderare librasnbsp;� 4 Galiicas; intervallum 3 talium pollicum ponaturnbsp;a 4 in 4 in Linea Solidorum XS; atque manentibusnbsp;ita apertis Circini cruribus, reliqua interval�a ab inbsp;in i; a a in 2; a 3 in 3 amp;c. a 64 in 64, transfe-rantnr a G in re�tam GD-^ Intervalla autem, qusenbsp;re�ta GD majora fuerint, notentur omnia in re�tanbsp;quadam, du�ia in charta; turn apertis omnino Circini cruribus, ita ut regulac ambai unicam redamnbsp;GDAG conftituant, perge a G eadem notare. Sinbsp;dein Circini crura ita coardes , ut intervallum glo-bi unius librai a 2(i in 2A conftituatur in Linea Solidorum ; intervallum ab i in i, dat glob�m ^; a 2nbsp;in 1 globum famp; a 3 in 3 globum libr.; intervalla bacc qu�que notentur in linea GD : amp; Lineanbsp;Globorum conftruda erit.

_g5i. X. .Ad oppofitam Regul� Oram TADT rcda ducitur, in quam transferuntur Tormentorum beilicorum Diametri cavitatis. Reda: hujus partitionbsp;facil� abfolvitur ope Linece Glahorum : ut enim Globus in cavitate Tormenti Cylindrica movendi facili-tatem habeat (fic tarnen ut plus aequ� baud vacil?nbsp;let); folent Diamctrum cavitatis Cydindricic ita af-fumere, ut ea Globorum Diametrum excedat, unanbsp;fcilicet linea, pro Globo 6 libr.; duas lineas pronbsp;12 libr.; tres aut quatuor pro 24 libr. amp;c.: transfe-rantur igitur in lineam TADT ( eodem modo , quonbsp;in prsccedenti) partes Linece Globorum, fic ut quac-libet pars, proportione jam dida, major elSciatur.

55a. Resolutio. Circino, cujus crura acuminara funt, accipiatur reCts AB intervallum; turn Circini pro-portionalis cruribus apertis, prioris Circini crura ap-plicentur, hinc inde, in Linta partium aqiialhim XL,nbsp;in tali numero (potius majori quani minori), quinbsp;dividi poffit exa�t� per nunierura partium , in quasnbsp;re�ta AB eft dividcnda. Ex. gr. fi dividi petatur innbsp;7 partes tcquales ? Ponatur intervallum AB a 70 innbsp;70 : tuiii mancntibus ita fixis Circini pr�portionalisnbsp;cruribus, Circino acuminato fumatur intervallum anbsp;10 in lo; atque hoe fepties continebitur in rectanbsp;AB.

553. ScnoLiON. Qu�d fi tefta AB data longior fuerit, ut vel earn non caperet Circini intercarpedo, vel lakem angulus cru-rum Circini nimium obtufus efficeretur ; operaberis per i, -jnbsp;auc i rectae datae, quam partem proportionate divides ad partes,^nbsp;quas deberet continere; hafque dein transferes ad integtam rec-tam AB.

�54. Resolutio. Primo cafu applicetur intervallum re�tac AB a 140 in 140; amp; Circino acuminato in-^nbsp;venies quaifitnm intervallum a 160 in 160. Secun-do cafu ver� intervallum AB ponatur a lao in 120,nbsp;amp; a loo in 100 reperies. quod quxritur.

^55. Resolutio. AB item CD utrimque applicentur ab X, in Linea XL parthim aqualium (quo inno-tefcec earumdem ratio) : fitque XK item XOi=AB;nbsp;XG item XH = CD : turn ita aperiantur Circininbsp;proportionalis crura , ut intervallum KO fit fcqualenbsp;EF; eritque GH quxfita. Si foret prima re�farumnbsp;major fccundA, v. g. fi darentur ab, cd, ef: fiatnbsp;XK item XO = cd : XG item Xtl = ab : fed tuncnbsp;fieri debet GH = ef: dabitque quarcam KO.

PROBLEMA XVII.

^^6. Resolutio. Adjiciatur tertia linea EF, qux fit jcqualis CD : atque ad re�ias AB , CD, EF, quaerenbsp;quartani proportionaiem; amp; hxc quxfitam dabit.

Ex. gr. quxrendum cft latus homologura, cujus qua-dratum fit ad quadratuin lateris AB ut ti ad 13, vel ut 19 ad 15 amp;c.

�57. Resolutio. Primo cafu applicetur intervallum AB in Linea Planorum XP a 13 in 13 : Circinonbsp;acuminato capiatur diitantia ab 11 in 11, atque haecnbsp;dabit primum quxfitum. Secundo cafu ponatur AB

558. nbsp;nbsp;nbsp;Resolutio. Circino acuminato fumatur interval-lum 6c, hocque applicetur Linea Planorum XP innbsp;partibus aiqualibus, Ex. gr. a 12, ad 12; tenta deinnbsp;quibus partibus, scqualiter ab X diffitis, queat ap-plicari intervallum BC; ponamufque reperiri ab 18nbsp;ad 18 ; erit adeo A ABC ad A c6c �t 18 ad 12 :nbsp;etenim BX^: bX^ � BC*: 6c*; fed BX*: 6X* = 18:13nbsp;Cattenta Awece Planorum conftrudlione); ergo BC*;nbsp;6c* = 18 : 12 amp;c.

Circini proportionalis crura ita aperte, ut Lineac Planorum efficiant angulum reSum in X.

559. nbsp;nbsp;nbsp;R�solutio. Circino acuminato fume in Lineanbsp;Planorum Qfig. 5), ab X, partes aliquot pro libitu,nbsp;ex. gr. = 40 ; dein lie aperiantur ejufdem crura, ucnbsp;intervallum illud = 40, poPfit poni a ao in 20, v. g.nbsp;a K in L; five fit KX item XL = 20, KL veronbsp;40 ex illis partibus : quoniam enim KL* = KX� nbsp;LX�, erit angulus KXL rectus.

Duobus Planis Jimdikts datis, tertium confiriiere prio-^ ribus fimilc, 6* iifdtm Jimul fumptis ceqiiale.

g6o. Resolutio. Circini proportionalis iJnecz Plano-rum ad angulum re�lum difponantur. Ponatur AB ab X in K; amp; CF ab X in L; ent intervallum KLnbsp;scquale lateri, fuper quo facSum planum (v. g.nbsp;triangulum) fimile uni ex duobus datis, acquale hoenbsp;erit iifdem fimul fumptis : nam KL�= KX� XLquot;*,nbsp;feu = AB� CF�; quoniam igitur plana �milianbsp;funt �t quadrata laterum homologorum, planumnbsp;fimile fuper latere homologo KL aequabitur duobusnbsp;planis fimilibus fimul fumptis fuper KX amp; XL.

�01. Resolutio. Circuli dati Radius applicetur Lima Polygonum XR a 6 in 6 : atque tali manente . Circini proportionalis apertura : capiatur intervallumnbsp;a 5 in 5 pro Pentagono; a 7 in 7 pro Heptagono ;nbsp;ab 8 in 8 pro O�lpgono amp;c. Illudque, quoties po-terit, peripheriac applicetur; totidemque chordae duc-tae efficient quod quairitur.

Ex. gr. ponamus Circulo dato Cfig-7^ inlcriben-dum elTe Nonagonum. Lima Polygononim XR , radius OL circuli dati applicetur a 6 in 6; dico intervallum d 9 in 9 fore lams Nonagoni forniandi.

Demonst. Eft X6: X9 = 66:99; quoniam ergo X9 efi: aequale lateri Nonagoni rcgularis, cujus radius ob-liquus foret aequalis X6 (ex conttruclione Linea Poly gonorum); confequens ell re�lam a 9 in 9 fore etiamnbsp;a:qualem lateri Nonagoni regularis inferipti circulo ,nbsp;cujus radius acquatur re�ae a 6 in 6. Q^. e. d.

�62. Resolutio. Intervallum AB Polygononim Linds applicetur, ad numcrum utrimque a;qualem nu-

Geometria Practica. mero laterum, quibus Polygonum conftruendum con-ftave oportet. Capiatur Circino acuminato interval^nbsp;lum a 6 in �, acque co, ex A amp; B, fiant inter-fectiones in O; eritque O centrum Circuli delcri-bendi, cujus peripheriaD, quoties fieri poteft, appli-cetuf chorda AB.

Ex. gr. defcribendum efi: Nonagonum, In prscce-denti figura fit 99 = AB ; dico intervallum a 6 .in 6 fore Circuli delcribendi Radium.

Demonst. Erit X9: X6 = 99:66; quoniam igitur X9 eft xquale lateri Nonagoni regularis infcripti Circulo,nbsp;cujus radius eft aequalis X� (ex conftru�fione Linccenbsp;Polygonorum); erit re�ta a 9 in 9 rcqualis lateri Nonagoni regularis, infcripti Circulo, cujus radius a;qua-tur re�tac a 6 in 6. e. d

�63. Resolijtio. of Circuli radius transferatur ad Lineam Chordarum a 60 in 60 v. g. KR Turnnbsp;Circino acuminato capiatur intervallum a 50 in 50nbsp;V. g. LE, qux, applicata periphcrias, arcum�50� fub-tendet. Nam eft XL : XK = LE : KR ; amp; quoniamnbsp;XL = chorda: arcus 50� in Circulo, cujus radiusnbsp;= XK ; erit LE = chordae arc�s 50� in Circulo-^nbsp;cujus radius = KR.

.5�4. Facil� hinc methodus habetur anguli dati valorem inquirendi; aut valoris dati angulum formandi. Ex- gr. inquirendus eft valor anguli B Qfig. 10') ;

ex B delcribatur arcus KL. BK ponatur in Lines Chordarum a 6o in �o; quaere dein, cui numero, hincnbsp;ind� acquali, congruac intervalium KL; acque iiie an-guli B quaniitatcm delignabit. Si ver� ad pundium Anbsp;L fiS' ^ ^ ) conftruendua fit angulus v. g. 70� : cx Anbsp;fiat arcus indefinitus OZ- AZ ponatur a 60 in 60 :nbsp;Circino acuminato capiatur intervalium a 70 in 70;nbsp;ponatur hoe a Z in arcum ZO, v. g. a Z in F; amp; recta AF efficiet angulum 70�.

Solido dato, ex. gr. Pyramuk, cunts alt kudo aut homo-' logum aliquod latus fit = recicc AB; invenire al-titudinem, aut latus homologum, Pyramidis fi-milis, quee ad datam Pyramidem fit in ra-tione data. Ex. gr. ~ vel | minor.

565. nbsp;nbsp;nbsp;Resolutio. Primo eafu intervalium AB ponaturnbsp;in Linea Solidorum XS, v. g. a co in 20 : amp; inter-vallum a 60 in 60 dabit altitudinem, l�u latus homologum quaefitum. Secundo ver� cafii intervaliumnbsp;AB ponatur v. g. a 50 in 50 Linea Solidorum; amp;

Ex. gr. AB amp; CD (fig. 12) fupponantur latera ho-mologa datorum Solidorum fimilium.

566. nbsp;nbsp;nbsp;Resolutio. Intervalium minoris AB ponatur innbsp;Linea Solidorum, hinc ind� in acquaii diftantia anbsp;Centro X; atque Circini proportionalis crura fic fixanbsp;retineantur ; turn Circino acuminato capiatur CD,nbsp;atque exploretur, quibus partibus Linea Solidorum,nbsp;hinc ind� a Centro X aiqualiter diffius, intercipiatur;nbsp;prior numerus amp; pofterior rationem indicabunt.

Ex. gr. fl fuerit AB appiicata a 26 in 26; CD ver� repenatur a 38 in 38; eric priraum Solidum adnbsp;fecundum ut 26:38.

Invenire Diamttrum Sphcera, ex. gr. ex .xiuro, cequalis ponderis turn. Sphara v. g. ex Argeato, cujusnbsp;Diameter data fu.

^6'j. Resolutio. Sumatur intervallum Diainetri dat�; acque ponatur in Linea Metallorum ab Cl in d : di-ftantia ab O in Q erit Diameter quxfita Sphxr� au-re�.

Demonst. Ex conftruftione Linece Metallorum^ di-ftanti� ab X Centro ad ufque Chara�teres chymicos cujufque Metalli, �quantur Diametris Solidorum limi-lium pondere acqualium, compofitorum ex illis Me-tallis; amp; quoniam intervalla, qu� Metallis illis refpoii-dent, funt in eadem ratione ac Diaractri prsfat� :nbsp;confequens omnino eft, eadem intervalla effe qu�quenbsp;�qualia Diametris Solidorum fimilium, pondere �qua-lium, � Metallis illis compofitorum. Eft ixaque in-tervallum ab O in Q Diameter quxfita in cafu pofito.

568. Resolutio. In Linea Metallorum capiatur Diameter Argenti ab X ulque d ; atque h�c transfera-tur ad Lineam Solidorum , v. g. a 50 in 50 : dein invariata Circini proportion alls aperturd, Circino acuminate capta Diameter Auri in Linea Metallorumnbsp;ab X in G, applicetur hinc ind� sequali numero par-

tium Linece Solidorum : quod in cafu fiet fere a 27 ad 27; igitur gravitas fpecifica Argenti eft ad gravin-tatem fpecificam Auri, fere �t 27 ad 50.

Demonst. Per conftrudtioneni Linece Metalloriim perfpicuum eft, fpecificas Metallorum gravitates elfenbsp;in ratione rcdproca Cuborum Diametrorum fuarum,nbsp;qure in Linca Metallorum notatx fuerinc : amp; quoniaffl'nbsp;Diametri iftae in cafu applicatae funt Linece Solidorumnbsp;apcrtura�; ncmpe Argenti a 50 ad 50, feu ab A in Bnbsp;13), atque hinc reperiretur Auri Diameter a 27nbsp;in 27 quam proxim�, feu ab F in L; cvidens omninanbsp;eft, ob AA funilia ABX amp; LFX, elfe AB^: LF^ =nbsp;BX*: FX^; porro cubi mtervallorum AB amp; LF, quainbsp;Diametri funt Argenti atque Auri, in ratione inverfanbsp;indicant proportionem gravitatum Ipecificarum : igiturnbsp;amp; 27quot;�� folidum denotant quoque, in rationenbsp;inverfa, gravitates eorumdem fpecificas; amp; confequen-tcr Argenti ad Auri = 27 : 50.

369. ScHOLiON. Quoniam Metallorum Diametri, in Linea Metal-loruin notatae, ita longte funt, ut mult�m Circini crura aperiri opnrteat, ad earumdem intervalla ponenda in Lineam Solido-rum : opcrari licoat per earumdem mediam, aut tertiam partcra.nbsp;Imm� amp; facilius , fi pro libitu, data Circini cruribus apertura,nbsp;capia'.ur, ex. gr. in cafu dato, intervallum ab (J in (J i hocquenbsp;in charta notetur : dein intervallum ab � in � transferatur innbsp;Lineam Solidorum a 27 in 27 ( etenim qute ratio datur Dia-metrorum iftorum Solidorum in Linea Metallorum, talis fitnbsp;oportet inter intervalla ([ in (J) amp; O in �) : turn, invariatanbsp;Circini aperture , inquiratur , cui numero refpondeat interval-lum ab (J in ; atque reperies proxim� a 50 in 50 amp;c.

Solidi, ex. gr. ex Stanno, pondere dato = 36 libr.; invenirc pondus Solidi ex Mrgento, ejufdemnbsp;cum priori Voluminis.

570. Resolutio. Circini proportionalis cruribus, pro libitu, apertis, capiatur primo intervallum a 2A in

if, hocque nocecuf. Dein incervallum ab d in d ponatur in Linea Solidorum � 36 in 3� ; atquenbsp;ita fixis Circini cruribus, videatur, cui numero Li~nbsp;necE Solidorum congruat intervallum ab d in d, priusnbsp;noratum; inv'enies autein a 50 in 50.. Igitur fi litnbsp;Stannum 3� libr. ejufdem Voluminis cum Argcn-to, ponderabit hoe 50 libr.

PROBLEMA XXX.

X^otis Diametris , feu laterihus homologis Solidorum fi-milium, cx. gr. Globorum Stanni amp; jirgenti; iii-venire ra�onem eorumdem ponderum.

571. nbsp;nbsp;nbsp;Resolutio. Diameter Globi ftannei applicetur iinbsp;2A in 2^; moxque capiatur intervallum., ab d in .d;nbsp;quod fi hoe luerit majus vel minus Diametro Globi , feu Spha;rac ex Argento; erit qu�quc Globusnbsp;argenteus raajoris, vel minoris ponderis quam Globus ftanneus ; etenim ex conftrudione Linea Me-tallontm, Globi iili pondere aiquales forent, fi argen-tei Diameter eongrueret intervailo ab d in d; ft igitur inquiratur, quibus partibus refpondeat intervallum iltud ab d in d in Linea Solidorum; rationesnbsp;ponderum innotelcunt.

PROBLEMA XXXI.

Sphara, ex. gr. Cuprea, Diametro atque Pondere 10 libr. datis; invenire Diametrum Spharanbsp;Aiirea 15 libr.

572. nbsp;nbsp;nbsp;Resolutio. Quecratur Diameter Sphaira� aureac,nbsp;aequalis cuprex, per Problcma XXVll. (567); Diameter ifthacc applicetur a 10 in 10 in Linea Solidorum : intercarpedine a 15 in-15, qusefita prodi-bir Diameter.

373. ScHOLiON. Porro ut Diametri cavkatum quarumcumque, aut convexicatum live foliditatum j menluientur ; fiat Circinus {fig.nbsp;14 ) ; cujus X fu Centrum motus ; bina crura AX amp; BX cur-va, oppofua fint amp; in aaquali a centro diftancia, cruribus CPnbsp;amp; FX; ica ut in primis AXB amp; BXC re�lam conHituant;nbsp;prxtcrea fint AX, BX, CX amp; FX inter fe- aquales : quo-niam enim AA AXB amp; CXF congrua funt; fi cruribus curvisnbsp;capiatur Diameter convexitatis = AB; crit FC eidem aequalis;'nbsp;vel fi brachiis reftis capiatur Diameter caViJatis = FC; intct-vallum AB eidem �quatur.

574. Rksolutio I. Fiat Tabula redangularis ABCD. (fig. I Tab. XV.) 16 pollices longa, la'ver� lata.nbsp;Partis media; redlangularis EF�GH luperficies, medionbsp;pollice circiter, circumjacente altior eit, ut ei .apta-ri queac re�tangulum, leu Qnadrum amovibile, con-tentum lateribus AB, EF; BC, FG; CD4.GH; DAnbsp;amp; HE; paraturque hoe � ligno buccino, atque icanbsp;applicatur, ut idem efficiat planum cum fuperfidcnbsp;media. Quadrum porro ittud ideo eit '^amovibile,nbsp;ut, d�m. charta tegitur re�langulum EFGH , in-flexis ejufdem limbis infra quadrum, ca e.xpanfa re-tineatur.

II. nbsp;nbsp;nbsp;Latus AD item BC dividitur in 200 v. g. partesnbsp;.acquales ; AB item CD in 150, prioribus acqua-les. Divifio hscc Scalam qu�que fubminiitrabit.

III. nbsp;nbsp;nbsp;Ex punt3:o X, medio FG, ducatur media Circulinbsp;Peripheria FLG; quam cxa�b� in 180� partiaris :nbsp;holque gradus transler in latera FE, EH, amp; HG;nbsp;fcilicct Rcgula, conftantcr appiieata ad X centrum,nbsp;per quemlibet gradum peripheria; ducatur, h juxtanbsp;cara lineola; feu gradus, eorumquc numen notenturnbsp;in limbo FEHG. Operatione finita delcantur cha-

radteres du�i intra arcara reftanguli EFGH. Divi-fio limbi FEHG qu�que facillim� abfolvitur, fic difpofito Tranfportorio , ut ejulilem Centrum re-fpondeat pun�lo X; Diameter ver� incidat in rec-tam FG per X du�lam. Regula dein, ab X pernbsp;fingulos gradus trajecta, indicabit in limbo FEHGnbsp;transferendos gradus.

perpendiculum affixsc funt Pinnulae viforiac KR, qua-rum Rima exafl� refpondeat re�tac OZ, du�tac jux-ta latus unum Regulas. In hac Regula, Scala, atque menluras varia? exarari folent.

V. nbsp;nbsp;nbsp;Foramen acu fiat in O, quod pun�tum, reda?, pernbsp;rimas pinnularum traje�lac, exa�t� refpondeat ; ut ,nbsp;pofui in B aut X Acu, qua? per O trajiciatur,nbsp;circa eam moveatur Dioptra KR.

VI. nbsp;nbsp;nbsp;Menfula tripedali fuflentaculo, �t Graphometrumnbsp;fg. 1 Tab. XII., innititur, ficui cuicumque propria.

575. ScHOLioN. Manifeftum in primis eft Mcnjulam ptaefatatn Graphometri vices habere , fi Regula, feu Dioptra KR, peinbsp;foramen O tranfmittendo acum , figatur in X. Mukiplicernnbsp;feu univerJ�aUm ejufdem ufum ex fequendbus colligere liceat.

SECTIO SECUNDA

Mctiri dijlantiam duoriim objeSorum BC, ex A qiifdem jed non � f�j� mutuo^ accejjbrum.

576. Resolutio I. Commodam elige fiationem A: a B item C per A re�tas age, ita ut fit BA = AL

^78. Illi Vel fk�lo A ABC ; due OX parallelam ad BC : erit AO : AC = OX : BC ; quoniam ergo me-tiri licet AO, AC amp; OX, facil� innotefcit BC.

579. nbsp;nbsp;nbsp;IV. Vel redtae AC amp; AB du�laj, item angulusnbsp;A, menfurentur : ope Trigonometriae primo inveniesnbsp;angulos ACB amp; ABC, amp; deinde re�tam BC.

580. nbsp;nbsp;nbsp;V. Du�tis AB amp; AC, iifque'menfuratis, inqui-ratur valor anguli A : Tranfportorio fiat in chartanbsp;angulus a = A ; fit cA in partibus Icalae = AB :nbsp;ac in talibus partibus � AC : Circino captum in-tervallum �c, atque fcalae applicatum , dabit name-rum menfurse = BC. Etenim AA ABC amp; abc fi-milia funt.

�581: VI. MenfulS Geometrica (fig. 4) commod� diF pofita, ducantur juxta Dioptram, collineando in Bnbsp;amp; C, re�tsc AF amp; AG indefinitae : AC item ABnbsp;menfurentur : fiat Ab, in partibus Scala:, a:qualisnbsp;AB ; amp; Ac = AC : erit bc du�la, in partibus Sca-lae, tcqualis BC.

Invenire dijlantiam ditorum objeSontm AC C fig- 5)gt; quorum itnim tantum cjl accejjibik.

582. Resolutio I. Statione in B eleclS ; inveftigetur quantitas anguli B, mediante Graphometro; deinnbsp;inquiratur qu�que valor anguli BAC; quoniam laws AB notum eft, determinabis AC per Trigono-jnetria: Problema II. C486}.

583. nbsp;nbsp;nbsp;II. Vel in charta dacatur ab = AB in partibusnbsp;icala;; fiat a = A; b = B : erit ac = AC in partibus fcalac.

584. nbsp;nbsp;nbsp;III. Vel collocetur primo Menfula Geometrica innbsp;O i^fig. 6), eamque fic difpone, ut reda, fecundumnbsp;unum cjus latus partium scqualium duda, in A tendat : menfuretur AO, fitque aO ei aequalis in fcal�nbsp;partibus; per Dioptrara collineans ab O in C, ducitonbsp;indefinitam OZ; tune tranflata Menfula fic confti-tuatur in A, ut idem latus Ao incidat in AO,;nbsp;turn ab A Dioptra diredl in C, ducatur Ac: erit-que hxc, in partibus Scalas, aiqualis AC ; efl: enimnbsp;A ACO fimile A Ac o.

585. ScHOLiON. Si opoTteat fluminis (^fig. 7) latitudinem deter-minare; ducatur primo in ripa BC, parallela ad iittus; ad quarrjnbsp;fiat perpendicularis OK (538) : tune inveftigetur anguli B valor; amp; relblve A OBK vel per Trigonometriam, vel in charta formando A obK fimile A OBK, fic ut ob in partibus icala*nbsp;sequetur OB : vel fiat Z = B, producaturque KO, donee con-currat cum ZL : nam hoe cafu AA KOB amp; LOZ funt coa-grua; atque adeo efl; KO = OL.

586. Resolutio I. Ducatur CF Bafis notabilis, re-fpe�liv� ad objedorum diftantiam tum a fefe mu-tuo , tum ab obfervatore ; ponamufque FC = 150nbsp;virg. Inveftigcntur primo in C, mediante Grapho-metro quantitates angulorum FCB amp; FCA, hiqu�nbsp;notentur : in F inquirantur anguli AFC amp; CFB.nbsp;In A FCB per Trigon. invenies BC; amp; in A CFAnbsp;determinabis AC : nofces ade� in A ACB lateranbsp;AC amp; BC cum angulo intercepto ACB : innotcf*nbsp;eet ergo AB per Problem a IV. C 49^ ),

587. nbsp;nbsp;nbsp;II. Dudt�i FC �t in prsccedenti, iifdemque men-i�uraiis angulis : in charta ducatur cf = 150 par-tibus fcalou ; fiant c = C; � = F; angulus /ca =nbsp;FCA ; angulus cfb � CFB : A afc erit limiie Anbsp;AFC ; A bef iimiie A BCF; eit aae� VC:fc =nbsp;AC ; ac; item elt FC : fc = BC : bc : ade�que ellnbsp;AC ; ac = BC : bc; verum angulus BCa = bca :nbsp;ergo AA ABC amp; abc ftmilia llint ; itaque ab innbsp;partibus Icalte elt = AB.

588. nbsp;nbsp;nbsp;III. Duda FC (fg. 2) 150 V. g. virgarum; Men-l�la Geometrica primo collocetur in F, ita ut unumnbsp;ejus latas coincidat re�lx FC : deinde collineandonbsp;a'b F in A amp; B, ducantur FZ amp; FX. Translera-tur inde Menfula in C, atque ita difponatur, utnbsp;lams C/'incidat in re�lam FC, pun�lumque C im-mineat punfto c limbi, diftanti ab � 150 partibusnbsp;fcalaa; five fit fc � 150 partibus fcala;. Tum a Cnbsp;collineando in A amp; B ducantur re�te, quac occur-rant reftis fZ amp;/X in a eritque A FAC limiie A fac : A FBC fimile A fbc : adeoque du�tanbsp;cb; A ABC eft fimile A abc, ut in prtecedentinbsp;demonftratum fuit : igitur eft ab, in partibus Scala: , scqualis AB.

589. Quoniom igitur eft angulus CAB = cab, eftnbsp;ab duda parailela ad AB; atque adeo ft ad ab ap-plicetur Dioptra, atque in ejus dire�lionem reda pro-trahatur, erit hate parailela ad AB. Igitur hic babesnbsp;modum ad AB inacceflam ducendi parallelam. Si ve-ro ea per pundtum datum, v. g. C, ducenda fit : innbsp;Menfula due primo per c parallelam ad ab; dein ap-plicetur Dioptra ad iftam parallelam, redtamque pro-trahes in ejus diredlionem.

590. nbsp;nbsp;nbsp;Resolutio I. Difpofito fpeculo plano horizon-taliter in i (aut ibidem folFala unius akeriulVe pedis quadrad lactd, quac amp; aqu?i repleator), ira re-irocecle, aum turris extremum per radium reflexumnbsp;IE incurrac in oculum : quoniam eft anguius AlLnbsp;= ElF ( opdc.3; L rero = F; AA AU amp; EFInbsp;luniiia 1�unt; ell adeo FI; EF = IL: x\L.

591. nbsp;nbsp;nbsp;II Baculus DZ perpendiculariter fit ereclus; hu-mi dccumbentis fic moveatur oculus, doncc per Znbsp;vifu trajedto extremitas A profpiciacur : erit enimnbsp;DH:DZ = LH: LA.

592. nbsp;nbsp;nbsp;III. Lucente fole erigatur ad perp^endiculum bacu-ius RS; SG fit umbra baculi, LM ver� umbranbsp;turris : erit GS : SR = ML: LA.

594. nbsp;nbsp;nbsp;V. Graphomctro (fig. 4 ) in C ita difpofito, utnbsp;diameter ejus horizond paralleJa fit; planum veronbsp;ad eundem perpendiculare. Per Dioptram collinee-tur in A; notetur angulus AXK, menfurcturquenbsp;LC = KX. Quoniam K re�tus; per Trigon. inve-nics AK, cui adde XC = KL.

595. nbsp;nbsp;nbsp;VI. Vel inveftigato angulo KXA, menfiirataquenbsp;bafi KX = LC : ducito in charta kx, in partibusnbsp;fcalaj, ajqualem LC; fiat a: = angulo AXK; k rectus fit; erit ak , in partibus fcala:, aiqualis AK;nbsp;huic adde XC amp;c.

A a 2

593. nbsp;nbsp;nbsp;IV. Menfala Gcometrica fic difponatur, ut unumnbsp;ejus latus BC ad horizon tem fit perpendiculare ;nbsp;eledhi commoda ftadone, per Dioptram, in B acunbsp;fixam , collineecur in A , notetur CO ; erit OC; BCnbsp;= �K: KA; huic addatur OP � KL.

5p6. Resol�tio L Eligantur ftariones eommod� C amp; H, tanto a fefe incervailo diffitE, ut, debit� dif-pofito Graphometro, anguli AXK amp; ABK notabi-liter difterant. Primo reiblve A BXA (486), utnbsp;ita habeas AX ; dein A re�kngulum AKX (482),nbsp;quo prodit AK : huic addatur XC = KL.

597. 11. Ut in prsccedenti, ex C amp; PI inquiratur valor angulorum AXK amp; ABK, menfureturque CH. Innbsp;charta ducatur bx, in partibus fcalac, = BX = CH.nbsp;Fiant b = angulo ABX; amp; angulus bza = angulonbsp;BXA producatur bx, amp; ab a ad earn demittaturnbsp;perpendiculans aR : erit hsec, in fcalac particulis,nbsp;ajqualis AK amp;c.

^98. III. Menfula Geometric^ primo debit� collocatd in F, ducatur FE; dein tranflatfl in T, fumaturnbsp;pun�lum G, fic ut fG in fcalac partibus fit = FG:nbsp;a G collineando in A, ducatur GO; ab O demittaturnbsp;01 perpendicularis ad fG, feu ad latus inferius men-fulac; eritque 01, in partibus fcalsc, sequalis AK amp;c.

599. ScHOLiON I. Si oportuerit monds inquirere akitudinem; quoniam ad perpendicularem, � vernce ad bafin monds duftam,nbsp;accedi nequit; � duplici ftadone operatio peragacur.

�OO ScHOUON 11. Si altitudo quKpiam fupra alteram conftituta fit ini^uirenda; primo quEeratut altitudo utriufque fimul �, deinnbsp;folius mferioris; fubtrafta hac ex utiaque, lelinquetur qusfita.

�o�. Resol�tio. Perpendiculura XK demiffum dabit al-titudinem primam : Graphometro, cujus diameter

deorfum perpendiculariter fpeclec, inquiratur quan-litas anguli KXL, item anguli LXA. Primo refol-vatur A redangulum XLK (482); quo notum erit latus LX; in A ALX erunt qu�que noti tresnbsp;anguli (etenim eft angulus ALK re�lus; fubtra�tonbsp;igitur angulo XLK ex 90�, refiduum erit angulusnbsp;ALX); refolvatur adeo A ALX (486).

601. Resolutio I. Quoniam nubes notabiliter a ter^ ra diftant promifcu�, iitumque continuo mutant :nbsp;bina Graphometra in F atque E, ftationibus debit�nbsp;dillitis, lint ita difpoEta, ut eorum diametri hori-zonti parallel�, atque inltrumenti planum deorfnmnbsp;perpendiculariter convertantur ; prsft� fint obferva~nbsp;tores duo, in F unus, atque alter in E : dato fibinbsp;figno, eodem inftanti quifque colluieet in mediumnbsp;nubis pundum ; atque exad� notet angulum ; primo refolvatur A ABL (486); ut nota habeas latera AL amp;: BL : dein perpendicularem LX inveniesnbsp;per Problema V. ( 493) : huic addatur XZ=AE.

603. IL Vel affumpta ball FE, omnibufque �t ante pcradis , ducatur in charta ab, in partibus fcal� ,nbsp;= AB : fiat a = angulo LAB ; k b = angulonbsp;LBA; du�ia perpendicularis lx, in partibus fcalaj,nbsp;cft Kqualis perpendicular! LX amp;c.

SECTIO T E R T1A

lt;504. Bonnariiim asquatur quadrato, cujus lacus fm-gulum 20 eft Virgariim, atque adco 400 virgas qua-dratas coniple�ticur. Jiigcntm eft quana pars Bonna-Tii, ac proinde = 100 virgas quadratas.

60fy. Modura dimctiendi agros triangulares atque quadrangulares, ex di�lis (Articulo 1, qui incipit adnbsp;Isj'rum ^ haud difficuitcr coliiges.

606. Resolutio I. Agri menfuraudi, ante omnia , obambules perimetrum, rudique calamo ejul'dem innbsp;charta figuram delinees.

II. nbsp;nbsp;nbsp;Confiderentur praecipu� anguli A amp; E iraxiin�nbsp;difliti, ad quos recta AE ducenda per medium agrinbsp;trajiciatur; re�ta hacc Fundamentalis audit.

III. nbsp;nbsp;nbsp;Turn Graphometro procedens in fundamentali li-nea, ab hac ad quemlibet � reftantibus angnlisnbsp;perpendicularcs ducito; quas redlas oranes in figu-ra , rudi calamo delineata, annotes , fimul cumnbsp;mcafura; quantitate, ut in figura videre eft : erit-que agcr divilus in quatuor AA re�iangula, amp; tra-pezo�dem re�langularera ; itaque cujufquc feorfumnbsp;inquiratur area; addanturque omnes in unam fum-

Geometb-ia Practica. mam : eric iiaic agri valor : �c ex adjedonbsp;Icheraate.

A AOB == 6.4 A AXF = 25.08nbsp;A FXE = ^55nbsp;A EiC = 3.8nbsp;trap. B�lC = 25.74

607. Resocutio. Dacatur ae, in fcalsc partibus, acqua-lis fundamentali lineai AE; five hxc faciat � Icala 11.8; ex quibus ao = 3.2; ox = 5.6; x/ = 1 ;nbsp;ie � 2. erigantur perpendiculares, ob ~nbsp;nbsp;nbsp;nbsp;xf =.

5.7; ic = 3.8 : demum connedantur redis extrema perpendicularium ; eritque abcefa menfurati agri fi-gura fimilis. Solct autem juxia earn delineari fca-la, qua: dcfcriptioni inferviit; ut ejufdem benefi.-cio, amp; latcrum valor, atque totius figurce menfuranbsp;queat dignolci.

6o3. SenouoN'. In medio linese fundamentalis Pyxide nantica difpofica , inqni�i .acque nocari folet,quam plagam latus fingU'nbsp;lum refpiciat inquiiac quoque Icrupulos� Geometra, idque au-.nbsp;R�cet, qui fint ad menfuratum agrum domini vidni.

609. Resolutto. Per curva latera , ab A in B col-lineahdo, nonnulli redam trajiciunt , quac ad cen-furarn Geometrac, ab agro demat moraliter in una, quod eidem ab altera parte addit. Per O ducatur

LR parallela ad AB. Juxca curva latera AL amp; BR ducantur IX amp; BZ perpendiculares ad AB ;nbsp;ad redlas IX, LR amp; BZ (ut videre eft in figura )nbsp;tot ab agri Perimetro ducantur perpendiculares, utnbsp;ejufdem partes interceptae a redis lineis par�m dif-ferant; refoivcrifque figuram in redangulnni BIXZ,nbsp;plurimas trappezo�des redangulares, nonnullaquenbsp;AA redangula; quorum omnium area, in unamnbsp;fummam colle�ta, agri valorem dabit.

610. nbsp;nbsp;nbsp;ScHOLioN L Cavebk fibi Georaetra,ne nimiiimconEdat een-furs, atque ica facil� a vero curpiter aberree; qua propter tu-ti�s operabitur ducendo juxta lacus irregulare reftam v. g. RL,nbsp;ad quam perpendiculares plurimas demittat; quam fi ducat rec-tam AB per irregularia latera; ea enim vix unquam, nequi-dem moralicer, aquam efficiet divifionem.

611. nbsp;nbsp;nbsp;ScHOLlON II. Si lylva , lacus vel ager inundatus live ar-boribus confitus , menfurandus fit : conftitues circa agrum rec-tangulum , au: trapezo�dem reftaugularem; atque ex ejus lateri-bus, ad omnes amp; fingulos agri menfurandi angulos, perpendiculares egrediantur, qu� exceflum , quo circumlcriptum qua-drangulum majus eft quam campi area, in triangula amp; trapezo'tde*nbsp;redtangulares fecabunt: quorum fingulorum are�, fi in unam fummam redigantur , amp; fumma collefta ab eodem re�tangulo autnbsp;trapezo�de reftangulari auferatur; refiduum sequatur ares campinbsp;menfurandi. Ex. gr. detur ftagnum ABCEFG (fig. 5); inclu-datur hoe intra redlangulum HIKL; hujufque aream determines ; ducito dein a quolibet ftagni angulo perpendiculares adnbsp;latus reftanguli adjacens, �t vides; cujuflibet crapezo�dis rec-tangularis inquire aream; in unam fummam redigantur; fubtra-hatur haec ex area redtanguli HIKL : reliquum erit area ftagni.

612. ScHOLiON III. Ut ftagni figura fimilis in chartam projiciatur:nbsp;fiat figura fimilis redlangulo HIKL; debitifque in pundlis eri-gantur perpendiculares, fimiles iis qus in ftagni menfura fuerintnbsp;duftse; extrema perpendicularium reiftis conneilantur; eritqu#nbsp;ftagni figura fimilis delineaca.

613. ScHOLiON IV. Si agri aut campi iindique inacceffi, cujusnbsp;anguli tarnen finguli eminus conipicui funt, fuerit inquirendanbsp;area, aut delineauda figura ; operaberis modo tradito 587 amp;nbsp;588 : quemadmodum enim ibidem primo ex F,amp; dein ex � innbsp;A atqu� B coilineatum fuerit, fic in hoe calu collineetur primonbsp;ex F amp; dein ex C, bafeos affumpta; extremis, in quemlibetnbsp;agri aut campi angulum : defcriptaque figura in partibus fcala*nbsp;inquiratur, exurget agri aut campi quaefui area.

�14. Resolutio I. Per B agatur re�ta BL parallda ad AC : ab aliquo pundo, v. g. A, redai AC ,nbsp;ducatur AZ perpendicularis ad AC; erit hacc� scqua-lis BO ; igitur ducatur AC in AZ , medietas fadinbsp;dabit aream A ABC.

615. nbsp;nbsp;nbsp;II. Si ver� nullus detur extra agrum egreffus :nbsp;in pundo quodam, v. g. A, redae aC, erigaturnbsp;ad hanc perpendicularis AX : quoniam hacc cft pa~nbsp;railela ad B� , erit XC : BC = AX : BO amp;c.

Ex data Circuli Diametro, Peripheriam prope veram invenire : vel ex data Peripheria, Diametrum;nbsp;atquc inde ejufdem Circuli areamnbsp;conciudere.

616. nbsp;nbsp;nbsp;Resoeutio I. Utamnr proportionc tertio loconbsp;conltgnata (302); atquc pro prima parte dicatur :nbsp;ut 10000 ad 31416; ik Diameter data ad quartumnbsp;numerum; qd erit Peripheria qutcfita.

Pro fecunda vcro parte dicendum : �t 31416 ad loooo; ik Peripheria data ad quartum numerum;nbsp;eritquc hk inventus Diameter pctita.

. Diametro atque Peripheria cognitis , una in quar-tam partem alterius ducatur; fa�tum aiquatur Cir-cuii arccC quseiitx.

617. nbsp;nbsp;nbsp;Resolutio. Eft area Circuli ad quadratum Dia-mecri ia proportione Archimedis = ii : 14. Pofitanbsp;latione Diametri ad Peripheriam = 100:314 ; eritnbsp;area Circuli ad quadratum Diametri = 785: 1000,nbsp;In majori ver� proportione allegata (302); ericnbsp;area Circuli ad quadratum Diametri = 7854:10000;nbsp;igitur affumendo ex praedidtis alterutram v. g. fe-cundam , dicito : uti 785 ad 1000; Iic area datanbsp;ad quartum ; qui erit quadratum Diametri quscfua:.nbsp;Inventa Diametro, Peripheriam determinabis per prac-cedens problema.

618. nbsp;nbsp;nbsp;Resolutio. Radium Circuli due in femiffemnbsp;are�s Sedloris; fa�tum dat quaditum C304).

Tabulam conjlruere 500 Segmentorum Semi-Circitli ^ rejpondentium totidem Sagittis ab i adnbsp;500 accrej'centibus.

6ig. Resolutio I. X (fe. 7) in Centro re�lus, fit acqualis O : quoniam KX (ex hypotheit ) = 500,nbsp;atque adeo KF diameter = looo ; erit area lemi-circuli LKZ = 392700. Primo invenietur AO :nbsp;nempe ex AX' = 250000 fubtrabendo OXquot;; extrac-ta ^ ex refiduo dat AO = | x\B. Porro angu-los A amp; OXA inrenies per Trigon. C482}.

II. nbsp;nbsp;nbsp;Nunc notus erit arcus AK Se�loris KXA; atquenbsp;adeo ejufdeiii ratio ad circuli quadrantetn KXZAK,nbsp;qui ia cafu == 196350, innotelcit.

III. nbsp;nbsp;nbsp;AB bafis , amp; perpendicularis OX, qu�que nota�nbsp;funt ; innotcfcit ade� area A AXB; fubtrahaturnbsp;ha�c ex area Se�loris AKBX; remanebit area Seg-mcnti, cujus KO Sagitta datur.

Ex. gr. fit KO = I; erit OX = 499 : hujus qua-dratum = 249001 : fubtrahatur hoe cx AX^ =z 250000; ex refiduo = 999 ^ proxim� dabit 3.16 pro AO :nbsp;quoniam igitur OX =: 499, erit area A ABX =nbsp;15768. Angulus AXK, in Canone majori, invenie-tur quam proxim� facere 3�? 37''� 25quot; : five 13045quot;;nbsp;arcus porro quadrantis KAZ = 324000quot; : itaque dica-tur : ut 324000quot; ad 13045quot;; fic 19�350 ad quartum,nbsp;feu Se�torem KXA; qui reperietur = 7904.5 erit igitur totus Sedlor AKBXA = 15809 : ex hoe fubtrac-ta area A ABX = 15768, reiiduum =: 41 dabit Seg-m.cntuni AKB. Et ita de caeteris.

3Sb 2

ip�

P R O B L E M A IX.

Ope pracedentis Tabula aliam conjlruere^ pojlto Integra Circitlo = looo ; eoque divifo in Seginenta loo,nbsp;totidemque Sagittas^ qua. crejcant �t numeri

621. Resolutio I. Quoniam iftius Circuli, cujus area ponitur facere 1000, radius efit divii�s in 50 partes ajqiiales; evidens eft, Sagittas i, 2, 3 amp;c.�refpon-dcre Sagittis 10, 20, 30 amp;c. Tabulae praecedeiitis.

II. nbsp;nbsp;nbsp;Area Circuli praecedentis Tabulae Integra poniturnbsp;effe 785400; dices itaque : �c 785400 ad 1000 (five �t 7854 ad 10); ita Segmenta praecedentis Tabulae, correfpondentia nunieris 10, 20, 30 amp;c. fefenbsp;habent ad Segmcnca Tabulae conferuendae, i, 2,gnbsp;amp;c. Inftituto calculo, exurgit pro Segment� SagitUc i,nbsp;proxim� 1.�9. Solent aucem fcrupula fecunda ne-giigi in praxi ordinaria; nifi 5 fecunda exccdanc,nbsp;�ut in cafu, atque tune augcri folet unitate cyphrinbsp;proxim� finiftima, quse loco 6 erit 7. Scribendumnbsp;itaque pro primo Segmento 1.7. Simili modo in'nbsp;verlies, Segmencum 3748, Sagittse 20 preecedentis Tabula; , dare pro Sagitta 2 fequentis Tabulse, 4.77 ;nbsp;pro quo itaque fcribes 4.8 : amp; ita de carteris.

III. nbsp;nbsp;nbsp;Inventis 50 prioribus Segnrentis, ceetera fubtraclio-ne innotcfcent : ex. gr. l�btrailo Segmento Sagittscnbsp;49, quod facit 487.2, ex area totius Circuli, feu exnbsp;1000, reiiquura 512.8 dat Segmentum Sagittte 51.nbsp;Subtradlo 474.5'Segmento Sagittae 48, ex 1000 ;nbsp;relrquum 525.5 dat Segmentum Sagittae 52. amp;;c.

PROBLEMA X.

�I. Dicatur : Diameter = 34.6 dat Sagittam 23.6 ; quid dabit Diameter Tabulae = 100 pro Sagitta fi-mili? Proxim� reperietur 68.2; cui proxima eft Sagitta 68; eique refpondet Segmentum 724.1.

III. Dein dicatur : Circulus Tabulx = 1000 pro Seg-mento dat 724.1 ; quid dabit Circulus continens Virg. 940 pro Segmento fimili ? invenietur pro qux-fito Segmento 680.654.

624. ScHOLiON. Quod fl proximior ad verum acceffus defidere-tur , ideo quod Sagitta data, = 68.2, fuperet Sagittam Tabula ,= 68 , ad 2 fcrupula prima, five ad yU ; fubtrahatur Segmentum = 7241, � proxim� fequenti =; 736.0 ; refidui = 11.9, capiantur j\- , quas proxim� = 2.4 : hocque addatur priorinbsp;Segmento = 724.1; eritque fumma Segmenti prop� verum 726.5.nbsp;Turn denu� dicatur : Circulus Tabulae = 1000, pro Segmentonbsp;dat 726.5 : quid dabit Circulus continens Virg. 940, pro Seg-mento fimili ? Exurget 682.91 pro Segmento qutefito. Atquenbsp;ita differentia ad inventum,per prohlema,Segmentum,eft 2.256,nbsp;quam in praxi non folenc magni facere.

I I.

Dividere A ABC (fig. 8) in partes qiiajcumque, ex. gr, in duas, qiiarum una = | alteriiis.

62^. Resolutio. BC bafis dividatur in 5 partes acqua-ies; � quibus OB = 3; atque adeo OC = 2 ; �uc-td AO, crit A AOC = | A AOB (307).

agro ABC (fig. 9) refccare qiiantitatem Virganm datam : cx. gr. = 45, per reBam ab A ductam;nbsp;etiamji totius agri valor hand notiis jit.

626. Resolutio. Ab A ad BC agatur perpendicula-ris AO : menfuretur fiaec; ponamufquc cam in Vir-gis = 3.0. Duplum quantitatis reiecanda; = 90, dividatur per AO = 3.6 : quotus = Virg. 25, datnbsp;bafin BZ fumendam : dudta adco AZ, habetur Anbsp;AZB = Virg. 45.

$27. Rf.solutio T. Si force CF = AF, evidens eft, dudlam BF quxfitam dare partitionem.

II. Si ver� fuerit CF, v. g. minor quara AF : fuma-tur AI = Cl : ducatur FB : dein ab I ducatur IG parallela ad FB; ducla FG divider A ABC bifariam.

Demonst. AA BGF amp; BFI eandem habcnt bafin BF, eandemque akitudinem (c�m conliftant internbsp;eafdem parallelas) : funt ergo ea inter fe sequalia ; at-qui A CBF A BFI erat x'quale ^ A ABC : �rgo etiamnbsp;A CBF A BFG , l�u ? CBGF facit ^ A ABC,nbsp;five jcquatur A AFG. Q^. e. d-

628. nbsp;nbsp;nbsp;Resolutio I. Ab O ad AB ducito perpendicu-larem, atque per earn divide quantitatem Virg. rc~nbsp;fecandam : quotas dabit | bafeos fumendac ab A innbsp;re�la AB.

II. Quod ft fuerit re�ia AB minor quam duplum quo-ti : du�ta OB, inquire aream A A�B : hanc fubtra-he ex quantitate refecanda : refiduum divide per perpendicularem ab O ad BC demilfam : quotus dabit f bafeos a B in latere BC fumendcc; ad quodnbsp;punctum recta ab O duCta efficiet de�deratam agrinbsp;diviftonem.

Dipidere A ABC (fig. 12) in tres v. g- partes aqua-les, per reSas, bajl BC parallelas.

629. nbsp;nbsp;nbsp;Resolutio I. Inter AC amp; | ejufdem quscraturnbsp;media proportionalis = AF : ducatur FG parallelanbsp;ad CB : quoniam eft AC� = 3 AF�; A ABC = 3nbsp;A AIlt;quot;G.

�I. Dein inter AC amp; | ejufdem quacratur media pro^ portionalis = AH : ducatur HL parallela ad CB :nbsp;quoniam eft ACquot; z= | AH'; erit A ABC = | Anbsp;AHL : igitur A AFG = ? GFHL = ? LHCB.

Dividere A ABC ( fig. i Tab. XVni.) in tres partes (equaks,per reSas ab 0 du3as.

630. Resolutio. Trifariam dividatur BC in pundis Z amp; X : ducatur AO, amp; a Z item X agantur XL amp;nbsp;ZK parallelsc ad AO : dudac OK Sc OL delideratamnbsp;efficient divifionem.

Demonst. Duda AZ, A AZB, five A BZK -f- A AKZ = I A ABC : porro A AKZ = A ZKO ( ha-bent enim eandem bafin KZ eandemquc altitudinem);nbsp;ergo A BZK A OZK, feu A BKO, = | A ABC.nbsp;Similiter, dudS AX, demonftratur, A CLO facere | Anbsp;ABC. Igitur ? AKOL facit quoque | A ABC ; at-que adeo tres illa� partes inter fe tcquantur. Q^. e. d.

631. ScHOLiON. Si fueric in agro operandum : poteris primo to-tum nbsp;nbsp;nbsp;menfurare ; acque prolequeris uc diftum ell proble-

jii piincto F (fig. 2 ) intrd arcam A ABC dato , illud dividere in tres v. g. partes cequales.

632. Resolutio I. Faciat CO bis BO : ducatur FO, fitque AZ parallela ad FO ; du��s AF K �Z eritnbsp;? AFZB = I A ABC.

Demonst. Si duceretur AO, A AOB erit = | A ABC : porro propter parallelas FO amp; AZ, A AFZnbsp;efi: = A AOZ : igitur ? AFZB etiam eft = i Anbsp;ABC. Q,e.d.

II. Dein dividatur AZ bifariam in X : ducatur CF : amp; ab X agatur XL parallela ad FC : du�ta LF,nbsp;erit quoque A LFA = y A ABC.

Demonst. Du�l� CX, A CXA = A CXZ ; five A LXA A CLX = A CXA = | A ACZ : ver�m,nbsp;attentis paral. CF amp; LX, AA CLX amp; LFX aiqualia funt;nbsp;ergo ? ALFX = | A ACZ : at, du�i� FX, A FXAnbsp;= I A AFZ : ergo A AFL=: i ? ACZF ; atque adconbsp;A AFL = ? CLFZ. Qua.'libet igitur pars aequaturnbsp;qu�que tertiae parti totius Trianguli. e. d.

Dividere A ABC (fig. 3 ) in. tres partes aqiiaks per reSas � quolibet angiilo diiSas, amp; intrd ar earn,

Resolutio. Sit CZ = ^ BZ : fiat ZL parallela ad AC ; fit X pundtum medium re�fec ZL ; dicanbsp;re�las AX, BX amp; CX efficere divifionem petitam.

Demonst. A ACZ = i A ABC : fed, attentis pa-rallelis AX amp; ZL, A ACX eft = A ACZ; ergo A ACX aequatur parti tertiae A ABC. Qiioniam ZX =nbsp;XL, re�tacque AC amp; ZL parallelaj funt; A CXZ = Anbsp;AXL ; amp; A BZX = A BXL : igitur A CXZ Anbsp;BZX, feu A CXB, eft = A ALX A BXL, five Anbsp;A.XB : ergo redlsc AX , BX atque CX dividunt Anbsp;ABC in tres partes aequales. Q^. c. d.

Pojito, B item C aai tos ejje ; dividere A ABC (fig. bifariam, per reamp;am, qud perpendictilaris fitnbsp;ad bafin BC.

634. Resolutio. Ducatur AL perpendicularis ad BC ; turn inter | BC amp; partem majorem linetc BC, v. g.

BL, quscratar media proportionalis = BO ; in punc-to O erigatur perpendicularis OF; haccque divider A ABC bifariam.

Demonst. AA BLA amp; BOF fimilia funt : igitur eft BL : LA = BO: OF: fi itaque duobus primis terminis detur communis akitudo BG; amp; duobus pofte-rioribus detur communis akitudo BO; habebitur BCnbsp;X BL: BC X AL = BO� : BO x OF ; in qua analogianbsp;patet, qu�d, c�m BC x BL = -zBӒ; debeat qu�quenbsp;BC X AL = 2BO x OF : igitur i BC x AL, = A ABC,nbsp;aiquatur bis medietati BO x OF = A BOF. e. d.

635. Resolutio. Capiantur l�rianguli dati bafis amp; akitudo : inter unam amp; medietatem alterius quicra-tur media proportionalis : Quadratum fuper hac for-matum erit scquale ifti A dato.'

636. Resolutio. Per A due indefinitam, quae lit pa-rallela ad BC ; in B fiat angulus CBG = angulo dato , adeoque in cafu t= 60� ; a pun�to O, m.edio bafis BC, ducatur OF parallela ad BG : eritquenbsp;BOFG parallelogrammum fub angulo 60�; amp; aiqua-le A ABC. Nam A ABC amp;; Parall. BOFG ha-bent eandem altitudinem ; porro area A ABC eftnbsp;scqualis fa�fo akicudinis in | BC = OB; amp; areanbsp;BOFG eft xqualis fado akitudinis in BO : ergo amp;c.

s. il

�. II.

637. Resolutio. Bina latera parallela , cx. gr. BC Sc AF, dividantur in tres partes tequales in punftisnbsp;O, Z, X, G : ducra; OX amp; ZG divident Parallelo-granimum datum in tria Paralielogramma minora amp;nbsp;A�Qualia.

Dividere Parallelogrammum (fig. 7 ) ia tres partes cequales, per reaas, quarttm. unica ab

Dividere Parallelogrammum ( fig. 8 ) trijariam , per rec~ tas, quarutn i/na d punSo G proficijcltur.

639. Rssolutio I. Fiat BO item AX = BC : O.X duCta efficiet Paraliclogrammum XOCF | ABCF.

II. Sit FL = OG : GL du�la dividet parallelogram-mum XOCF bii'ariam; adeoque in tres partes tcqua-Ics divifum eric Parallelogrammum x4BCF.

Dividere Parallelogrammum (fig. 9) trifariam, per rec-tas ab lino angulo, ex. gr. F, ducias.

�40. Resolutio. Sit CO = 2BO; AT = 2BI : dudlse FO amp; FT parcitionem dcfideratam eincienc.

Demonst. Du�ia BF, A ABF = A BCF, feu quodlibet eorum =: f ParaJleiogranxmi ABCF : porr� A AIF = 2 A BIF : ergo A A!F = | Parallelogramiriinbsp;ABCF. Similiter A COF � 2 A FOB; . ergo A COFnbsp;quoque = Parallclogrammi ABCF ; igitur recta FOnbsp;amp; F1 figuram datum trifariam partiuntur. e. d.

641. nbsp;nbsp;nbsp;Resolutio I. Capiatur BO = | BC ; fiat OXnbsp;parallcla AB : fit LX = OG : ducla GL habebiturnbsp;trapezo'is GBAL = f Parallelogrammi dati ; namnbsp;AA G�I amp; LXI congrua lunt, atque adeo lequalianbsp;inter fe ; at OBAX dat ~ Parallelogrammi ABCF ;nbsp;ergo amp; GBAL = f ABCIC

642. nbsp;nbsp;nbsp;Resolutio. Latera parallcla, AF amp; BC triiariamnbsp;divide ; redtte ad oppofita divilionum pundta du�tasnbsp;eflicient quod qua:ritur.

Quadrilaterum quodc'imque ABCF (fig. 12) reductre ad Triangulum, retentis lateribus AP� atque CP�.

64:^. Resolutio. Ducatur AC : per B ad AC duca-tur parallela donee concurrat cum FC produdta, in G : du�ta AG, erit A AGF = Quadrilatero dato.

DemonST. A ACB amp; A ACG inter eafdem parallc-3as eonnftunt, habentque eandem bafin AC; funt' ea ergo ajqualia : porro ? ABCF conltac A AFC Anbsp;ABC : lea A ABC = A AGC; ergo A AFC A AGC,nbsp;fea A AGF cil = ? ABCF. (^. e. d.

645 ScnoLioN. Qaod fi punftum 0, medium reft� GF, extra latus CF repertum fucric , uc in fig. 13-, dneatur AO : ad tee-ta^ Al, lO amp; Cl quite quartam proportionalcm =iL; duCtanbsp;AL divirmuem efficiec propofitam ; nam A AOF = vO ABCF:nbsp;at A OCI = A AIL (314) : ergo ? AFCL � A AFO = Inbsp;,? ABCF.

P Pv O B L E M A I X.

Dividere Trapezim (fig. 14) bifariam, per reHam ab 0, piinS.0 medio reBes. Cb', datlam.

lt;546. Resolutio. Ab A dneatur AL parallda ad FC (quod fi. higt;c area figurcC egrederecur, ea dTct a Bnbsp;ducenda) : fit AX = LX: ducatur OB, ad quaninbsp;ab X ducatur paraliela XK : duda OR figuram fc-cabit bifariam.

Demonst. Dudis OX atquc BX : trarezois LXOC = trapezoidi XAbO : A BLX = A LXA : ergo ?nbsp;LXOC A BLX, leu A BCO A BXO, eft = {- ?nbsp;ABCF : fed, attentis paralleiis KX amp; BO, eadcmqua

ba� BO, A BXO = A BKO : ergo A BCO A BKO, feu ? BKOC = I ? ABCF ; igicur K� du�la trape-zini datam bifariam par�tur. Q^. c. d.

647. Resolutio. Refolvatur Trapezis data in A AGF; hujus capiatur dinvidium per A AOF : ducatur XA :nbsp;dein OL parallela ad XA : ultimo XL ; hsequenbsp;dividet ? ABCF in duas partes a;quales.

Demonst. A AOF, feu A FAX A_OXA, eft = i ? ABCF : attentis vevo �L atque AX paraliclis,nbsp;cademque ba� AX, A OXA =; A LAX ; igitur A FAXnbsp;4- A LAX, jeu ? LAFX, = ' ? ABCF; five eft =nbsp;p LBCX. 0^. c.d.

�48. Resoltjtio. Allijmatur figur� data: latus quod-piam pro homolcgo : tunc primo cafu, inter latus illud amp; quinquies idem latus, queratur media pro-portionalis : fuper hac, fit homologo latere, fiat fi-gura f milis; eritquc htec quintupla figura; datse.nbsp;Secundo cafu vero, inter latus affumtum amp; ejuf-dein , quscratur media proportionalis; fuper �a for-

�49. Resolutio I. In primis totius agri mcnfura per-ficiatur, per reclam fundamentaleiTi HZ, ad quam � quolibet angulo, �t in figura videre eft, pevpen-dicuiares demitrantur; ponamuique .totaiem agri valorem = Virg. 5248; atque adco Bonnaria 13, amp;nbsp;40 Virgas quadratas.

�l. Circa agri medietatem, re�lam conftitue CG, qu?i propoiitum agrum fccundum latitudinem ejus, innbsp;duas dividit partes quaicfcumque A amp; B, ad aefti-mationem invicem sequales; quaram fmgularam, exnbsp;pra?.fcripta meniurandi mechodo, area inquiratur :nbsp;pars A contincat Virgas quadratas 2592: pars veronbsp;B habeac Virgas quadratas 2656, quarum, differentia = 64, iccundum quam pars ll excedit partem

A, nbsp;nbsp;nbsp;cujus dimidiura = 32, addendum erit ipli A,nbsp;amp; idem a parte B auferendum, amp; fic acquaeles crunt;nbsp;�quod id ipfum in fequentem modum pcrficictur :

ni. Longitudinem lineac CG, agri nimirum latitudinem complc�tentem, menfura, qua;, cx. gr. longa litnbsp;Virg. 46.8; coRftat itaque, fi a CG linca conftitiia-tur parallela versus B, diftans und Virga , fiec Pii-rallclograrmrmra Virg. 46.8 (pro tantilla enim a!ti-tudine, feu diftantia parallclarum, obliquitas FG adnbsp;CO contemni poteft in praxi) ; at deiideratur Pa-rallelograinmum = 32 Virg. : porro in hunc modum,nbsp;per regulam proportionis, invenics dicendo ; Vbrg.nbsp;4�.8 acquiruntur , quando ad diftandam i Virg�nbsp;lincam removeo : quot Virgis auc pedibus amp;c. eademnbsp;erit removenda, ut acquiram Virgas 32? Producun-tur cx rcgula proportionis 0.68. Quare fi linca OFnbsp;eonftituatur parallela ad CG, diftans ab hac, versus

SECTIO QUARTA

S- I-

650. Modus rcprxfcntandi, feu delineandi, in plano Corpora Geoineirica, cx iis quaa in Ferfpccliva tra-duntur, facile cruitur : verum non foler Ine in ngo-�c attendi ad PitnBum principalt feu visus : qua denbsp;caufa in Cubi vel Farallelcpipcdi delineatione 'ijip,. 1nbsp;Tab. XIX.); fiat prini� ABCF quadratum vel reCIrm-gulum ; dein GHiK Quadratum vel rectangulum priorinbsp;cengruum ; anguli , ut vides , redtis connedtantur,nbsp;eritque Cubi vel Parallelepipcdi dciincatio peradlanbsp;amp;c. '

6^1. Resolutio. In rectam AB ifig- 2,) latus Cubi quacer transferatur; atque lu}'cr ca quatuor quadra-la clliciantur ; unum hinc inde quadratum eucquenbsp;fiat fuper IL A MK : turn debit� plicencur quadratanbsp;contigua qtiadrato IKML, tegaturque pars luperiornbsp;quadrato �DBN ; fuperficies fingula; glutine ncc-tantur; eritque Cubus conftru'flus.

652. nbsp;nbsp;nbsp;Resolutio. Fiat HEFi rciSanguJura congruumnbsp;bafi Parallclepipedi conftruendi : lint EA,

1�, IK, FG, FM, EL aiqnales alt-itudini Paraiiele-pipcdi; dti�lifque AB, NO, KG amp;; LM complean-tur redlangula : lint GO, amp; KD aiquales HI : quo fict re�kngulum CGKD congruuni ball HEFI; dc-nique plicentur redangula contigua. ballparfquenbsp;fuperior occludatur reCiangulo CGKD; glutine fir-mcntur, eritque conftructum Parallelcpipedum.

653. nbsp;nbsp;nbsp;Resolutio. Conftruatur bafis Prifinatis, ex. gr.,nbsp;pro triangular! A KBD (p^'- 4) : continuetur BDnbsp;in A amp; E donee Fat AB = BK, amp;: DE = DK :nbsp;Super AB, BD amp; DE conftruantur paralldogramiTianbsp;AG, BH, DF, quorum akitudo AG akitudini priLnbsp;niacis tequetur : tandem fiat I'upcr GLi A GIH ipftnbsp;BKD congruum : hilce partibus debic� jundis, glU'nbsp;tineque fixis, conftrudtum erit Prilma triangulare.nbsp;Ncc abfimili modo mulcanguiare quodcumque coii'nbsp;itructur.

654. nbsp;nbsp;nbsp;Resolutio. Eadein diametro dclcribantnr circulinbsp;AB amp; CD Cpg- 5). BG fit = akitudini Cylindri;nbsp;fuper qua confiruatur rcdangulum BCFE,quot; ita utnbsp;CD fit peripherix Circuli CD, xqualis amp;c.

655. nbsp;nbsp;nbsp;Resolutio. Radio AB defcribatur arcus BE, amp;nbsp;ei applicentur tres chorda; BC; ED = DF;CB = CF;nbsp;ducanturque re�te AB, AC, AD amp; AE : tum pli-catis, dcbit�que junais hilce triangulis, conftruc-ta eric Pyramis.

656. nbsp;nbsp;nbsp;Resolutio. Pro bafi, ex Cenrro X fifig. 7) defcribatur Circul�S, amp; diameter producatur in C, doneenbsp;AC latcri Coni squalis fiat. Quxratur ad AC amp;nbsp;AX,. in numeris, atque 360�, numerus quartus pro-porcionalis. Radio CA, ex centro C, delcribatur arcus DE, amp; ope Tranfporcorii fiat angulus DCE, con-fcquenter arcus DE, numero graduum invento sequa-lis. Eric Seifior CDE, cum circulo AB, rete pro Cono re�to.

Demonst. Eric AC ad AX, ficut 360 ad quar-tum immerum = angulo C; feu ficut peripheria Cir-culi, cujus AC eft radius, ad arcum DAE : fed elb AC ad AX , ficut peripheria Circuli, cujus AC efl:nbsp;radius, ad peripheriam Circuli, cujus AX eit radius;nbsp;feu ad peripheriam bafeos AB. Ergo arcus DAE =nbsp;peripheria; bai�os AB. Igitur fic plicando DCE, utnbsp;DAE coincidat in peripheriam bafis AB, amp; Etus CDnbsp;laceri CE conciguum fit ; glutineque nexis iifdem,nbsp;conftrudlus eric Conus redtus. O. e. d.

657. Quod fi ex A in F transferatur latus Coni trun-cati, amp; radio CF arcus GH defcribatur, tandemque ad 360�, numerura graduum arc�s GH, atque FG,nbsp;numerus quarcus proportionalis quaeratur, amp; inde Diameter Circuli IF decerminetur; habebitur rete pro Co-*nbsp;no truncato. Fit enim CDBAF rete pro Cono inte-gro; CGFIH pro Cono ablcilfo : ergo DBEHIG pr�nbsp;truncato.

^.58. Resolutio. Conftruatur A acqmlaterum DEF ifi�- O � fuper fingulo cjus latere conitruanturnbsp;adiiuc alia itidem cquilatera DAE, EBF ik F'CD.nbsp;Ex hoe reti Tetra�drum conftrui poteft.

659. Quod fi BC Cfig. 9) condnuetur in H , donee fiat CH = F'C, amp; fit in refolutione Problematis, conftruantur AA sequilatera CHi, CGH, HLI, DCI :nbsp;ex reti OCta�drum conftrui poterit.

660. Resolutio. Conftruatur A squilaterum ABC Qfiff. 10 ) : in baft AB continuata fiat AB = BEnbsp;== p'Q =:GH = HD. Per C agatur ipfi AB paral-lela CE, amp; fiat AB = Cl = IK = KL = LM = ME.nbsp;Ducantur re�tae CS per C amp; B, NT per I amp; F,OVnbsp;per K amp; G amp;c. Similiter ducantur aliae redsc TOnbsp;per B ik I, SP per F�� amp; K, TQ per G amp; L amp;c.nbsp;Ex hoe reti conftrui poceft Icofa�drum.

661. Resolutio. Defcribatur Pentagonum regulars ABODE C fg. I Tab. XX.) ; iuper quovis ejus latere fiant qu�que Peiuagona congrua ipfi ABODE.nbsp;Ad K aliud fiat S etiam ca:tens congruum; aiquenbsp;huic fimilicer congruum jungatur R : amp; fuper quovis hujus refiduo latere delineentur fimiliter priori-bus congrua; Ut videre eft in figura.

66a. ScHoooN. Delineentur retia in charta ex pluribus folii* eompabta. Delineate exfcini^an�ur, refefta charta luperhua jux-ta eorura perimetros. Exfciffa agglutir.encur chartac co'orata: ;nbsp;hujus fuperfluum ica refecetur, ut partibus perimetri altcrnisnbsp;margines quidam rclinquainur, queraadmodum videre cft in fi-eura 8. Singula retium intra perimetruin lineamenta , ex ar.nbsp;EF, FD amp; DE (fig. ,8) in rete Tetracdri, fcalpello piofun-di�s imprimantur, ut commode complicari queant latera perimetri foiidi. Denique retia complicata marginum ope congluti-nentur.

663. Resolutio. Sumendo Cubum unius lateris pro-dit Cubi Ibliditas (427 ) : Tetra�drum Pyramis eft: �cia�drum Pyramis genfinata ; Icofa�drum vero csnbsp;20 Pyramidibus triangularihus ; Dodeca�drum exnbsp;12 quinquangularibus conftat, quarum bales in fu-perficie Icofa�dri amp; Dodeca�dri lunt, vertices innbsp;Centro co�unt : horum ergo foliditas habetur ducen-do bafes in tertiam aldtudinis partem.

P R O B L E M A II.

66s,. Resoi.utio. Immittatur corpus Parallelepipedo ca-vo, eique aqua, aut arena (ne madefiat} fuper-fundacur; amp; altitudo aqua:, feu arena: notetur. Cor-porc cxtraclo obiervetur denu� aquae, aut arena: complanatac altitudo; inquiratur foliditas Parallele-pipedi, primo ufquc ad primam nocam : deinde partis refiduse, feu Corpore extracto : hac fubtra�a exnbsp;priori; refiduum sequatur foliditati Corporis immerfi.

66^. SenoLTON. Quod (i Corpus in Parallelepipedo iftiurmodi coRimod� deponi nequcat, ex. gr., fi ftatiiam certo loco affixainnbsp;diineciri jubeamur : Prifma quadtangularc, aut Parallelepipedum,nbsp;circa ipfum conftrui debet ex afieribus. Rcliqua peragenda luncnbsp;ut ante.

666. Resolutio. Si Corpus cavum in numero gco-metricorum non contineatur, refolutio eadem eft qusc Problematis pra:cedentis. ,Si Corpus cavum fue-rit Parallelepipedum, Pril'ma, Cylindrus, Sphaera,nbsp;Pj^^ramis, vel Conus; foliditas prim�m totius Corporis, cavitate inclusa ; deindo cavitatis, quae ean-dem cura Corpore figuram habere fupponitur, pernbsp;methodos fupra traditas, inveniatur : hac enim exnbsp;ifta fubtrafii, rclinquitur foliditas Corporis cavi.

Invcnire Cubiim dato Corpori, cujiis foliditas inveniri po-tcfl, aqualcm i vel qiii fit ad hoe 'in data qvacumque ratione, ex. gr. iit ^ ad i, vel �t 1 ad 4.

66'j. Resolutio. inveftigetur foliditas Corporis per Problemaca tradica. Ex ea vel cjus mukiplo, aut

fubmultiplo deiiderato, ex. gr., triplo, aut fubquaf-druplo, extrahatur radix cubica, qua; erit latus Cu-bi defideraci.

CAPUT II.

668. In Stereometria Doliorum famofa eft menlura apud Belgas nota vulgari idiomate em aeme, ime aime,nbsp;continctque pocula 96; hujus pars quarta, vulgo eennbsp;vircndeel, line quartelctte, 24 pocula scquat. Punctumnbsp;vulg� een Jchreve, un point, quatuor comprehcnuicnbsp;pocula. Ut ver� inveniatur, quot pollices cubicos pocu-lum datum capiat : detur Epiftomium in fuperiori parte aheni; impleatur hoe aqua, donee cBluere incipiatnbsp;per Epiftomium. Parallelepipedum ligneum, longuranbsp;3, latura 2 pollicibus, altum ver� pro arbitrio , amp;nbsp;ttt neceffe fuerit , colore oleofo aut vernicc pidturanbsp;(ut aquae in poros irapediatur ingreffus), lento motu,nbsp;pcrpcndiculariter in aquam demittatur, donee effluens,nbsp;per Epiftomium, aqua, poculum exa�lum, quo recipi-tur, adimpleat. Altitudo, ad quara demerlum fueritnbsp;Parallelepipedum, ducatur in baftn Parailelepipedi, quaenbsp;in cafu = 6 poll. quadr.; fa�lumquc dabit pollices cubicos , quos tenet Poculum. Ouoniam ver� Lovaniinbsp;ad to poll. deprehenditur mergi, Lovanienfe Poculumnbsp;conftat �o pollicibus ctibicis.

PROBLEM A I.

669. Resolutio I. Sit vas cylindricura ABDPCfig- 2), cui infundamr menfura una,qua ad fluida menfuran-da udmur ; fitque EG liquoris libella, OX altitudonbsp;= GE. Ducatur HL = ED diametro Vafis ; H7nbsp;indefinita, fit perpendicularis ad HL : fumatur Hi

Geometk�a Practica.

*= HL : Ha fiat = Li : La meter vafis, duas menfuras capientis, fed ejufdeni aLnbsp;titudinis cum vafe, quo4 nonnifi unam capit. H3?nbsp;diameter Vafis, capientis tres menfuras amp;c. In unum

antur divifiones invence in alterum ver� latus KRnbsp;xqualis.

Hi,Ha,H3, . contiguum , altitudo uni menfaracnbsp;KI = OX, quoties fieri poteft.

670. II. Eafdem racnfuras. expedite per. Tabcllam ra-dicum quadratarum inferibes, vit fcquitur : fit Diameter menfurae AB = Pi = loco : erit eius quadra turn joocooo. Ex hujus duplo extra�ta radix qua-drata erit Pa. Si ex triplo , quadruplo, quintuple . amp;c. radix quairata extrahatur, prodibuntnbsp;amp;c. quem in ufum confirudta eft Tabula fcquens.

Demonst. Cylindri eandcm akitudincm habeiitcs funt inter fc ut quadrata Diametrorum C 444 3 : ergonbsp;quadratum Diametri vafis duas, tres, quacuor amp;c.nbsp;inenfuras capieniis , efii duplum , triplum , quadruplumnbsp;amp;c. quadrati diametri vaiis, menluram nonniii unamnbsp;capientis. Quare, 11 inde radices extrahantur, habebun-tur in refolutione fecunda Diametri iplsc. Quuniamnbsp;ver� in prima refolutionis parte AB Diameter vafis ai-fumitur atqualis HL = Hi; erit ipfius Li quadratumnbsp;duplum, quadratum ipfius L2 triplum. quadratum ipfius Lq quadruplum amp;c. quadrati ipfius Hi. Undenbsp;patet eife rectas Ha, H3, H4 amp;c. diaractros vaforumnbsp;quaefitas. Quod 11 itaque has divifiones ad diametrumnbsp;vafis Cylindrici applices; iliieo conftabit, quot rr.cnfu-las capiat vas Cylindricum, eandem cura ifto bafin,nbsp;fed altitudinem illius habens, quod unam menluramnbsp;capit. Quare ft porro, opc altcrius divilionis in Vir-gula fadx, inveltigcs quoties aititudo unius menfurxnbsp;in altitudinc vafis dati contincatur, amp; per hunc nu-merum diametrum modo inventam multipliccs'; prodi-bit numerus menfurarum cavitatem vafis dati adimnlen-tium. lt;2,. c. d.

672. ScHOLioN. Ut autem partes decimales menfiira; integrae de-terminentur; matiente cadem vads akitudine ; posacut Diameter unius menlura� = i , leu tooo partiiim decimalium �, erit ejusnbsp;quadratum. 1000000; CU jus pars dccima =: looooo : ind� exuac-la radix quadrata = 316 continet panes decimales diametrinbsp;unius menfura;, qure conveniunt diametro Cylindri deciir.amnbsp;menfiirte partem cominentis, ejufdem tarnen cum Cylindro, in-tegram menfumm capientc, alticndinis. Si ex duplo hujus deci-mte , nempe aocooo, radix extrahatur , prodic diameter bafis,nbsp;duas dccimas unius menfura: capientis 447; amp; ita porro. Quodnbsp;li quadrato diametri unius menfura: ibooooo, adjicias partemnbsp;decimam lococo, amp; cx fnmma extrahas radiceni quadratamnbsp;1.049; erit ca diameter vafis, qute capit i-t- menfurie.

Ratio perfpicua fit per demonllrationcm Problcmatis prxfcntis. Atque fic patet, quomodo Virgula pithomctrica accurati�s con-llrui poffit, ut iiucrvalla inter menfuras integ�as lubdividanturnbsp;in partes decimales.

TABULA

Diainetrorum pro menj�tris integris amp; carum panibus dedmalibus.

674. ScHOLiON. Affamere folenc Virgam 4 pedibus longam; unum-que ejus lacus in partes 4,, feu pedes, dividicur : pes quilibet in 10 pollices ; amp; quifque pollex in JO lineas fubdividicut ; ac*nbsp;que ita djvil'um crit hocce latus in 400 pattes iequalcs.

Invtnirt capadtatcm Dolii cylindrici (fig. 3).

�75. Resolutio I. Inprimis explorare eam licet pe-dali me�)lura : ninijmm menfuretnr diairister inre' ri�r Cylindri, atque ind� concludatur area bafeos ;nbsp;hfcc inventa ducatur in Cylindri longitudinera : factum dividatur per 60; quocus dabit q�antitatcm po-culorum.

II. Latere PZ (fg. 2) Virgx Pithometricx cylindricx, capiatur diameter interior 01 Dolii : dein latere KRnbsp;Virgx, inquiratur Dolii longitudo interior : una innbsp;alteram ducia, fa�ium dabit numerum menfura-rum, quas capit Dolium.

lt;76. SCHOLION I. Tabulse, ex quibus Doiia conftnii folenc, ultra fundum prominent : menftirctur utrimque hiec prominentia, atque notetur creta in'ipfa fuperficic Dolii in A amp; B Cfig-i): diltan-tia base AB prasterea minucnda eft ad cralTitieni bafium circu-lariurn ; porro crafficies illa haheri folet aqualis tabularam craf-litiei, prout ea ad orifidum O capi poteft.

677. ScHOLioN II. Quod fi bafiS alterutra non fuerit cxaft� cir-cularis, capiatur major atque minor ejufdem diameter : medietas utriufque fimul collectie pro vera alfumatur.

Inquirert foliditatem Dolii (fig. 4).

678. Resolutio. Latere PZ Virgx cylindricx capian-tur primo Diametri interiores bafium EC amp; AF : medietas utriufque habeatur pro vera diametro ba-

fcos

feos circularis lateralis. Dein eadem Virga capiatur major diameter interior OP, per orificium ventris :nbsp;addatur hacc in unam funamam cum diametro mi-nori vera : medietas dabic cequatam Diamitrttm FKnbsp;= OX, Cylindri KLCF; cui aliimilari poicft Do-lium;lic�t hoe in rigore non ita fit. Dein latere IvR�nbsp;partiuin xqualium Virgte, inquire Dolii longitu-dinem internam, ntodo quo fupra di�lum eft (676):nbsp;longitude hscc ducatur in Diametrum sequatam,nbsp;fa�lumque dabit numerum menfurarum , quas con-tinet Dolium.

lt;gt;79. ScHOLioN. Vulgaris ilia vaforum vinariorum ilereomerria, inquin Alecias , qu� per hanc Cylindraceam perticam, ex Colanbsp;mukiplicatione aquatae Diametri in aldcudinem perficitur ,nbsp;fraudulenta eft , amp; in defcftu femper peccat, prsfertim quan-do iiimis ventricofum exiftit vas, amp; Diametri valde funt ina;-quales. Quare, qui eujufvis generis Dolia, quancascunque Dia-mecrorum ina;qualitacis, per perticam, feu Virgam Cylindra-ceam, ver� amp; accurac� metiri velic, correftionem neceffari�nbsp;adhibere debet lit fequicur. Proponatur Dolium menfuranduiT),nbsp;cujus bafeos Diameter decul�atim accepta jucidat in partes inte-quales, ad pundtum f {fig. 5). Altera bafeos diameter cadat in K:nbsp;unde punctum mediam L erit a;quaca extremarum bafium Diameter, quod incidat in 8.a. Vafis Diameter, ad orificium amp;nbsp;medietaiem accepta, incidat in M , partem = 13.3. Inter no-tul.is L amp; M compreheuduntur partes latitudiiium inaequalesnbsp;= 5.1 �, quorum dimidium = n.55, fi addatur miuori diametronbsp;= 8,a', fiec ffiquatio tiumerorura = 10.75-, vel utramque diametrum fimul adde, fiunt ai.5 , quarum dimidium dabit ean-dem numerorum squaciotiem = 10.75. Porro pet cireinum^ ae-cipe inter L amp; M ijotulas, pundhim medium O, delignans innbsp;percica diametrum vafis tequatam = 10 6 , qu� femper minornbsp;erit cequatiqne numerorum , amp;. deficit in hoe �xempio o.ts; cujus pars tcTtia = 0.05 adjiciatur -^qua^a; diametro'� fiet vera amp;nbsp;conic� corre�a vafis diameter = 10.65.nbsp;nbsp;nbsp;nbsp;tandem in lon-

gitudiiiem vafis, per partes iongitudinis percicaj a;quaies accep-tam , mulciplicato, qua: quidem longitudo (dempta lignearum b.aflum amp; marginum abundancia, �t /upra docuimus) fit 15.4;nbsp;fict Dolii capacitas qu^fita == 164.,?-.

Quod fi abfque fcrtipulofa hac correbtione , ex vafis longitudi-ne = 15.4, amp; xquata va-fis diametro = 10.6, fteteometriam abfolvas ; invenies per eorundem numerorum mukiplicationeinnbsp;pro vafis ejufdem capacitate 163.24, qua: a vera deficit unanbsp;l� menfura, l�per qua fuetic Virga conftrubta.

F f

�680. Resoluti� I. Cum vafa, pro quibus Virga hxc paratur , debeanc, moralucr laitcm , effe iiraiiia :nbsp;qaod obcinet, dum inter Diametrum acquatam at-que lon'gitudinem eadem moraiiter rcperitar ratio.nbsp;In Doliis auftriacis lervatur haco maxime commen-aanda : tertia nimirum parte longitudinis tabularum

II. Quoniam, experientia tefte, fabricata hacce Virgil fuper Cyiindro, cuius longitudo ad aeqnacam dia-metrum fefe hal5et uc 6 ad 5 : ea adhibetur, abfquenbsp;crrore notabiii in quibufcumque Doliis, nili in iisnbsp;longitudo non faciat niajoris scquatac Diametri;nbsp;vel fi longitudo � niajoris sequatae Diametri com-ple�latur; aut ii Diameter ventris | faciat Diame-tri bafium. Quod fi reperiatur hoe ita elie : Doliinbsp;capadtas iaveiiigetur Virga pithometrica quadrat�.

lil. Sit Cylindrus olcx (fig- 6_)taiis, ut 0/ altitudo fit ad cl diametrum balbos, �t 3 ad 5 : contincat-que vas iliud Cylindricum menluram unam , ex,nbsp;gr., anum Punctum. Virga Gc tranfverfim in Cylindronbsp;�ilponatur, notecurque in punCto O numerus i. Quoniam Cylindri fimiles funt qu�quc �t cubi diagona-lium : ut invenias notam a pro diagonali ca, de-lerminaiite piincla 2 : pone oc = 1000 : fume ejusnbsp;cubuin ; ex eodem duplicato excradfa dabit i!259inbsp;pro C'2. Ut habeatur Diagonalis Cylindri 3 punSo-rum : tnplicetur cubus 1000; extrada p' dabitnbsp;pro cg; amp;c.

P R. o B L E x\I x\ V.

b .;. P^ESOLUTio I. Dolinm, Jibellic beneficio, it.2 con-i�it�atur, ut axis ejus fit hori'/.ond iiarallelus; ne Icilicct fluidum in una Dolii pane altius lit quamnbsp;in altera. Ejufque capacitas inquiratur.

II. Turn Virga pedali inquiratur Dolii asquara Dianac-meter. Noteturque altitudo liquoris relidui fub Diametro ventris ; cx hac altitudine llibtrahatur i dit-ierenti� majorem inter amp; minorem Doiii Diamc-

F f 2

irum : reliquuin clabic veram Sagittam Segment! Do�i, Cylindric� confuierati. Inquiracur area iltiusnbsp;Scgraenti; hscc inventa ducacuf in Dolii lorgitudi-nem : fa�tum dabk quan�tatcm lluidi reiidui.

Ex. gr. Dolii capacicas fit 190 poculorurn : Diameter ventris = 330 ; aititudo liquoris reiidui, feu pars virgaj madida, = 117 : cujufque bafis Diameternbsp;== 310 : ita operate :

.�quata Diameter Dolii = 320 : vera Sagitta ma-dida =: 112 : primo ieaque dicito : 320 dat 112; quid dabit Diameter loq tabulae C622)? invenicturnbsp;tabulae Sag. 35 cujus Segm. 312.00.

Segmentum 312.00; ^uid dabunt 190 pocula ? Rep�-rientur nbsp;nbsp;nbsp;pro reiiduo.

683. ScHOLiON. Porro ut fiidiiori atqiie prompdori raetbodo (licet non scqii� tuta , nili fuerinc Delia fimilia ei, quod con-Urutlioni VirgK iiilcrviic) quandtas liquoris in Dolio non ple-110 inveftigetur , duo lequenda ex W'olfio , pro Coronide, fint Problemata. �nbsp;nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;�

Virgam pithometricam conjlriiere ad dtterminandam qma�tateni fluidi in Dolio non plena.

684. Resolutio I. Affamatur Dolium aqua plenum, cujus capacitas jam cognita efi : numerus menfura-rum cx. gr. per 20, aut numerum alium minoremnbsp;vel majorem dividatur, prout Dolii capacitatem innbsp;j-�artes majores vel minores dividi comniodum vifumnbsp;luerit.

II, Dolio oollocato, �t in prtecedenti ; Virga per ori-� ficium ventris intrudatur, donee fundum Dolii attin-gat.

UI. Ei quantitate fluidi ex Dolio emifsi, quai numero mcnfurarum rer divifionem, paulo ante N'� i in-vcnto, refpondet; in Virgula notetur decrementum altitudinis in fluido, quod exprimic totios capacica-tis parcein vigefimam.

IV. nbsp;nbsp;nbsp;Eodem modo notabis decrementum altitudinis,nbsp;reliquis particulis vigeiimis quanticatis fluidi, in Dolio content!, refponden�.

V. nbsp;nbsp;nbsp;Horum dccrementorum intervallis in una Virgulacnbsp;facie notatis; altera dividitur in panes quotcunquenbsp;minutas inter fe aequaics, ultra vigefimarum inter-valla inacqualia continuandas; ex. gr. in aoo aucnbsp;plures.

n. Eo, �t ante, debit� difpofito, Virga, per Problema praecedens parata, per oriflcium Dolii O (/g'. 4 ) in-trudacur, donee fundum in P attingat.

III. nbsp;nbsp;nbsp;Ea rurfus extra�ta , notetur quot partes in facienbsp;partium scqualium vino madidae fint.

IV. nbsp;nbsp;nbsp;Hinc inferatur : �t numerus partium ajqualium innbsp;altera Virgula: facie,' profunditati totius Dolii OPnbsp;refpondentium , ad numerum fimilium partium ,nbsp;altitudini fluidi PG convenientium ; ita numerusnbsp;earundem partium, qua: intervallo fcrupulorum vi-gefimorura congruunt, ad numerum quartum pro-portionalem inveniendum.

V. Capiatur circino intervallum tot partium cqualium in Virga, quot numerus inventus exprimit, amp; trans-feratur in Scalam fcrupulorum vigeiimorum, note-turquc eorum numerus, quae ipli congruunt. Pernbsp;hunc dividatur numerus menlurarum, quas Doliumnbsp;integrum capit : quotus erit numerus menfurarum,nbsp;quas fiuidum in Dolio contcntum replere poteft.

Ex. gr. fit OP = i6o; PG = 58, numerus partium tequalium, quse intcgio fcrupulorum vigefimo-rum intervallo congruunt, = i2,o;capacitas deniqueDo-lii 128 menfurarum. Circino captum interval!um tot partium icqualium in Virga, quot numerus inventusnbsp;exprimit, tfanflatus in Scalam fcrupulorum vigeumo-rum, indicet feu j. Quod li itaque. 128 per 5nbsp;dividas, quotus == 25I indicabic numerum menfurarum fluidi in Dolio rciidui.

Sag.	Segm.	Sag.	Segm.	Sag.	Segm.
19�	108624	231	137412	2 66	1�75�1
197	109436	232	138138	267	168458
198	110209	233	138977	268	169300
199	�11056	234	139867	269	170203
200	111835	235	140708	270	171107
201	I12�J�-	23�	141.553	271	17 2007
202	1134'O	237	142397	272	172855
203	�4^.55	238	143234	273	- 173771
204	^15035	239	144092	274	174670
205	1158 '1)	240	144944	275	175.523
206	11��50	241	145795	27�	176433
207	117492	242	14��50	277	177335
208	118269	243	147.505	27 8	178202
209	� I loiog	244	148365	279	17911�
210	119895	245	149224	280	180033
211	120�9�	. 246	150086	281	180895
-1 2	121539	247	150948	282	18181�
213	122341	248	151813	283	182738
214	I2qig8	249	152681	284	183�O2
21.5	123991	250	153548	285	184527
216	124796	251	1.54419	286	185453
217	125764	252	155290	287	18�321
218	126460	253	156166	288	1872�0
219	127270	2.54	157039	289	188123
220	12813-3	255	1.57915	290	189055
221	128948	256	158738	291	190087
222	129764	257	1^:^9622	292	190863
223	130584	258	160508	293	191798
224	131453	2.59	161391	294	.192676
224	13227�	260	162226	295	193615
220	133100	261	1�3102	296	194497
22y	133928	262	163992	297	19.5438
228	14480�	263	1�4871	298	196337
229	135�58	264	1�5774	299	19726�
230	13�469	2�5	166�67	300	198162

s 0 �	Diam.	2 0	Diam.	t? 0 D	Diani. j j
I	I.OOO	17 i	4.123	33	.5-744 ;
	I.4I4	18	4.242	34	.5-831 i
3	1.73a	19	4-3.59	35	.5-916;
4	a.000	20	4.472	36	6.000
S	a.a^�	ai	4.582	37	6.082.
6	2.449	22	4.690	38	6.1641
7	2.645		4-796	39	6.244',
8	a.8a8	24	4.898	40	6.324
9	3.000	25	5.000	41	6.403
10	3.16a	26	5.099	42	6.480
11	3-316	27	5-196	43	6-.5.57
la	3-464	28	5291	44	6-633
1.3	3-605	29	.5-385	45	6.708
14	3-741	30	5-477	46	0.782
15'	3-^73	31	S-5^7	47	6-855
16	4.C00	3^	.5-657	48	6.928

		3-0	1.732	6.0	, 2.449	9.0	2.000
I	31�	I	1.761quot;	1	' 2.469	I	'3.016
o	44.7	2	1.78S	2	,2.489	2	3-033
3	54^^	3	1.816	3	' 2.509	3	3-049
4	�32	4	1.844	4	2.529	4	3.0��
5	707	5	1.871	5	2-549	5	3.082
6	775	6	1.897	6	2.599	6	3-098
7	^37	7	1.923	7	2.5S8	7	3-I�4
8�	894	8	1.949	8	2.�07	8	3-130
9	949	9	1-975	9	2.�2�	9	3.146
o	I.ooo	4-0	2.000	7.0	2.645	lO.O	3.162
r	1.049	I	2.025	I	2.�64	I	3.178
o	1.095	0	2.049	0	2.083	2	3-194
3	I.I40	'3	2.073	3	2.702	3	3.210
4	1.183	4	2.097	4	2.720	4	3.226
5	1.225	5	2.121	5	2.738	5	3.241
6	1.2651	�	� 2.145	6	2.756	6	3-25^
7-	1-3041	7	2.1�8	7	2.774	7	3.271
8	1-34-	8	2.191	8	2.792	8	3.286
9	i-,378:	9	2.214	9	2.810	9	.8-.8or
o	1.414	5.0	2.23�	8.0	2.828	II.O	3-31�
I	1.449	I	2.258	I	2.84�	I	3-331
2	1-4^3!	0	2.280	2	2.864	2	3-34�
3	1-517	3	2.302	3	2.881	3	3-3�I
4	I-.549I	4	2.324	4	2.898:	4	3-37�
S	I-.58I	5	2-345	5	2.9151	5	3-391
6	I.5l2j	6	2.3661	6	2.932	6	3.406
7	I-�43	n i	2387.	7	2.949	7	3.421
8	1.673	8	2.408;	8	2.96�	8	3-43�
9	1.703'	9	2 429	9	2.983	9	3-451

r- I	125	17	515	�	33	718	49	875!
2	177	18	530		34	729	50	884'
3	21�	19	.545		35	739	51	892I
4	250	20	5.59			7.50	52	901'
S	279	21	.573		37	760	53	9101
6	306	22	586		3^^	770	54	9i8j
7	330	23	.599		39	780	55	927I
8	3.53	24	612		40	790	5�	935:
s	375	25	625		41	800	57	944 i
10	395	2�	637		42	810	58	952:
11	414	27	650		43	819	59	9lt;5o,
12	433	28	661		44	829	60	968;
13	4.50	29	673		45	^39	61	9761
H	467	30	684		46	848	62	984!
1.5	484	31	696		47	8.57	63	9921
16	500	32	707		48	866	64	lOOol

I	250	17	643	33	802	49	914
2	3E5	18	655	34	810	50	921
3	360	19	667	35	818	51	927
1 4	397	20	678		825	52	933
1 .5	427	21	689	37	833	53	9391
! 6	4.54	22	700	38	840	54	945
1 7	478	23	711	39	848	55	951
! ^	500	24	721	40	8.55	56	95�
9	520	25	731	41	862	57	962
10	538	26	740	42	869	58	967
II	550	27	7.50	43	876	59	973;
� 12	.572	28	7.59	44	882	60	9781
13	588	29	768	45	8^	61	984,1
iM	602	30	777	46	896	62	989!
\|i.5	616	31	785	47	902	63	9951
1 u----	�30	32	794	48	908	64	lOOOj

	libr.	unc.
Aurum	13 26.	4	Aurum	730
Plumbum	802.	2	Plumbum	863
Argentum	720.	12	Argentum	89.0
Cuprum	627.	12	Cuprum	937
Ferrum	5.58-	0	Ferrum	974
Stannum	51�.	2	Stannum	1000
Sit itaque linca MX =		1000.	lt;?X=974. ?X =	= 937'
CX = 895.	bX = 863.	gX	= 730 : erit rit�	con-

I N D E X

ELEMENTOR.UM GEOMETRIjE.

INDEX.

INDEX GEOMETRIE PRACTICA

�. 2.

ETM^

CAPUT P R I M U M

THEOREMA II.

I)um diKZ chorda aquaks JcJe interj�cant, aliqui duo arcus oppofiti funt cequak-s; amp; � converjb.

III.

De Angulis.

67. nbsp;nbsp;nbsp;Lineac, quarum inclinatione, aut concurfu angulus formatur, Latera rel Crura anguli dicuntur.

Hypothesis I.

C0R.0L-

C S

PROBLEMA

5. II.

PROBLEMA I.

CoR-OI-^arium III.

P a

THEOREMA

THEOREMA V.

CttjuJIibet chordce. diftantia d.centra eji cequalls mcdietati chordcz fujlentantis arcum , qui ciim arcu dnbsp;priori chorda Jiibtenjo � 180�.

E %

THEOREMA III.

THEOREMA IV.

CAPUT

CAPUT III.

C 0 R 0 L L A R I U M.

De Triangulis.

De variis Triangulorum Jpeckbus, atque proprictatibus.

GeOM-.' to/�: 3

Elementa Geometeije. nbsp;nbsp;nbsp;4.^

Corollahium I.

THEOREMA VI.

175. Demonst. primum. 0 = C-{-X; amp;Z = B4-I C 1�7 ) ; porro 0 = Z, cum fmt ambo re�li; erg()

' nbsp;nbsp;nbsp;THEO-

THEOREMA X.

Ga

�, 11.

theorema J.

THEOREMA HL

THEOREMA IV.

anguU

5- I.

n a

5. III.

212. Propofitionis veritas perfpicua fit per numeros 196, amp; 211.

PROBLEMA I.

Circulo dato Polygonum regulate inferibere.

ARTICUL�S II1.

THEOREMA II.

amp;c.

COROLLARI�M.

THEOREMA XIII.

K a

theorema XVI.

THEOREMA XVI L

c. d.

PROBLEMA I.

272. Resolutio. Fiat 0 = A, X=B, amp; nbsp;nbsp;nbsp;11=0

( 105); ergo A XHO eft tcquianguium, adeoque �-aile A ABC (238;.

�. 11.

De ccetcns Figuris [milibus.

theorema III.

THEOREMA IV.

PROBLEMA II.

SECTIO SECUNDA

CAPUT PRIMUM

THEOREMA II.

p2 nbsp;nbsp;nbsp;Elementa Geometric:.

jirta Seaeris ejl = | producti arcus fcQoris per I radii.

Circuli Segmentum ABKA Cfg- �) inquirere.

ARTIC�LUS II.

THEOREMA I.

CoROLLAItlUM I.

C0B.0LLARIUM II.

C0R.0LLAR.IUM III.

C0E.0LLAE.IUM IV.

COR.OLLAR.IUM V.