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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ r - il 3S i f I (^.ZUftfJ^m^J^l^u^^ n.au/raaie rum y/vWif Bene ^erenutl, Hominnza enim Teftig>ia -video. FIRST VOLUME OF THt INSTRUCTIONS ». GIVEN IN THE DRAWING SCHOOL sstablishe:d by the DUBLIN-S.OCIETr, Puritiant to their Resolution of the Fourth of February, 1768; To enable Youth to become Proficients in the different Branches of that Art, and to purfue with Succefs, geogra- phical, NAUTICAL, mechanical, COMMERCIAL, and MILITARY Studies. Under the Direfition of JOSEPH PENN, heretofore Profeflbr of Philosophy in the Unrverfity of Nauts. ^id munut Reipuhlie0 majut aut meliut afftrre poffumut, juamft Jw ventutm b*ne BrUJUamut f Cicero. DUBLIN: Printed by Alex. M'Culloh, in Henry-ftreet, M,DCC>LXIX. "1 1 w i A \ \ V ir^j%M^4» AUSPICIIS FREDER/CI HARVETf Episcopi Derrensis Supreme Curiae, &c. Promovemte Societate Dublikensk FAVEN TIBUS JO SEP HO HENRr, ROGER PALMER et GULIELMO DEANE, ArMIG^RIS* OMNIGENiE ErUDI'TIONIS MJECENATIBUS. Jojipbus Fenn olim in Academia Nanatenfi Philofophiae Profeflfory purs et mixts Matnefeos Elementa digeiEt et publicavit^ in ufum Scholae ad propagandas Ar- tes in Hibernia fundatc. Anno Chrifti M,DCC,LXVIII, die iv Menfis Februarii. A. Rt. Hon. Earl of Antrim Rt. Hon. Lord Annaiy Rt. Hon. Earl of Ancram Hon. Francis Annifley Clement Archer, M D. Merryn Arcbdall, Efq ; Benedid Arthnre, Efq; Mr. John Atkinion MTilliamAnfHcU.Efq; Mr. Hillary Andoe Mr. }ohn Auftin Mr. Thomas Atidm. Rt. Hon. Earlof Bcdive Rt. Hon. Earl of BeUamont Rt. Hon. William Brownlow Sir Lucius O'Brien, Bart. Sir Charles Bingham^ Bart. Rev. Dr. Benfon Rev. Dean Bourke Conftantine Barbor, M. D. David M'Bride, M. D. John Bourke, Efq; Bcllingham Boyle, Efq ; Walter Butler, Efq; Dominick Bourke, Efq ; John Blenherhaflet, Efq; Thomas Burroughs, Efq ; David Burleigh^ Efq ; John Bonham, Efq; Francis Booker, Efq ; Robert Birch, Efq : Matthew Bailie, Efq ; John Blackwood. Efq ; Rev. Mr. John Ball Mr. Richard Bartlet Mr. John Boulder Rev. John Bowdcn^ D. D. I' r f>i^ hi SUBSCRIBERS Wiliiam Bury, Efq^ Mr. Patrick Cullen Rupert Baibor, h!4 ; Mr.Maurice Coliiii Hon. Georgt Barncwall, Efq; Mr. Samuel Collins Mn Thomas Broughali Mr, John Gafper battier Mr. JoJm Bloomfield Md Edward Beaty Mii William Becby M4 Henry Blenerhaflet Mil H.Bradley Mr^ Thomas Brow n Mr. George Begg Mr« Jofeph Barecroft Mr; Richard Bolton Mr. Richard Blood Mr, Lawrence Brync Mr. Chriftopher Brigg?. Rt. H. Lord Vif. Clanwilliam Mr. Richard Cranfield Mr. Richard Cowan Mr. George Carncrofi Mr. }ohn CarroO Daniel Cooke, M. D. Mr. liaac Ctmon Mr. WiUiam Cox Mr. Richard Connel Mr. Hugh Chambers. D. Rt. Rev. Lord fip. of Down Rev. Dr. Darby Nehemiah Donnellan^ Efq ; William Dunn^ Efq; Arthur Dawfon, Efq : Henry Dillon^ Efq; Rt Rev. Lord Bp. of Clonfert Edward Denny, Efq ; Sir James CaldweU, Bart. William Deveniih, Efq; #Sir Paul Crofbic William Doyle, Efq; fion. Francis Caulfeild Edward Donovan, Efq; John Cro(bie Dennis Daly, Efq ; Lomas Quffc, Efq; Henry Doyle, Efq; Rev. Maurice Crolbie, D. D. Henry Dunkin, Efq ; Rev. Henry Candler, L.L. D. George Da wfon, Efq; Rev. Dean Coote Matthew Carter, M. D. George Cleghorn, M. D* Jojbn Curry, M. D.. Rev John Conner, P. T. C. D. Rev Angnftus Calvert A. M. Arthur Craven, Efq; * Capt. St. Claire John Cook, Efq; John Carden, Efq ; Andrew Caldwell, Efq-; Robeg Clements, Efq ; Stratford Canning, Efq ; Andrew Crawford, Efq James piulfeild, Efq ; Lawrence Crofbie, Efq ; Hugh Carmichael, Efq ; John Coningham, Efq ; John Conway Colthurft, Efq ; Henry Cope, Efq; Edmond Coftclo, Efq ; Thomas Caulfeild, Efq; Mi. Edward Cnllcn, T.C. D. Robert Day, A. M. Dennis Doran; Efq> James Duncan, £fq ; Mr. Purdon Drew Alea^der M 'DonncI, Efq ; Mr. Jofeph Dioderici Mr. Henry Darjey Mr. Robert Deey Mr. SiiTon Darling Mr. John Dawfbn Mr. Hugh r&niel Mr. George Darley. £. John JEnfor, Efq ; JohnEnery, Efq; John Edwards, T. C. D. John Evans, Efq; J. Echlin, Efq ; F. NAMES. Thomas Fitzgrrald, E(q$ Robert Fit^j^cfa*d, Efq; Thomas Fic/.gibbon, Efq; Thomas Fofter, Efq j Thomas Franks, El'q ; Auguftine Fitzgjiald, E(q; . George Faulkner, Eiq; , Capuin Feild John Ferral^ M. D. Rev. Mr. Fetberfton Rev. Dr. Thomas Fofior Mr. William Feild Richard French, Efq; Thomas Forfeith, Efq. G. Sir Duke Gif&rd, Bart. Luke Gardiner, Efq ; Sackviile Gardiner, £(q; Benjamin Geal Efq ; ' Thomas ^Soodlet, Efq; WUUamGun, £{q; Thomas St. Gwige, Efq ; William Grogan, Efq; Henry Gore, Efq ; Thomas Gledftane, EXq; Rev. Mr. Gtattan Rev. Mr. John Graves, A. B. Mr.Ponf.Gouldft>ury, T.CD. Mr. Luke George, A. B. Mr. Anthony Grayfon Mr. Charles Giilefpie Mr. Thomas M'Guite Mr. John Grant Mr. Daniel M'Gufty. H. Rt. H. John Hely Hutchinfon Hon. Mr. Juftice Henn Rev. Dean Harman Claude Hamilton, Efq; Peter Hohnes, Efq; KaneO'Hara, Efq; Edward Herbert Efq ; WiUiam Hamilton, Efq; Charles O'Hara, Efq; John Hobfbn, Efq ; John Hatch, Efq ; "1 Rt. H.Sir WiUiam Fownes, Bt. Mr. Guftavus HamUtoo Henry Flood, Efq;. Henry Hamilton, E^; John Fitzgibbon, Efq ; Thomas Hartley, Efq ; John Fofier, Efq; Francis HamUton, Efq; F" ■"" SUBS Sackville HamilCoii, Efq; Gorges Edmund Howftfd, E<q ; Rev. Mr. Richard Hopkins Mr. Willtani Holt, A. M. Mr. TKonas Hoieit T. C. D. James Edward Hamilton, Efq ; Mafter Chaifet Hamilton Mr. William Hickey Sanael Hayes, Efq ; Mr. Robert Hunter Mr. William Huthchinfon Mr. F. Heoey Mr. Thomas Harding MeC }of. and Ben. Houghton Mr. John Hardy Mr. James Homidge Mr. David Hay David Hartley, Efq; Mr. Kobert Hunter. I. 9t« John Jeflfi!ryes. Biq ; Rev. Mr. Daniel Jkduou ficnjamin Johnibn, Bfq ; Charles Inncsj £i^ ; Bdr* RobcSft Jaffiay K. Rt. Hon. Earl of KingAon Maurice Keating, Efq; Redmond Kane, E(q : Thomas Kelly, Efq; I>ennis Krlly, Efq ; Anthpny King Efq ; Jofeph Keen, Efq; Rev. Mr. Andrew King Rev. Mr. Kerr Mr. Gilbert Ki)bee L. Rt. Hon. Lord Lifford, Lord High Chancellor Rt. Hon. Earl of Lanefborough Rt. Rev. Lord Bp. of Limerick Rev. Dean Letablere Edward Lucas, Efq ; Walter Laurence, Efq ; Richard Levinge Efq ; IXivid Latouche, Efq ; John Latouche, Eiq; Guftavus Lambart, Efq ; Robert LongficU, Efq, CRIBERST NA William Ludlow, Efq; Thomas Lee, Efq; John Lee, £iq ; Charles Levinge, Efq; Charles Powei Leflie, Ffq ; Henry L' Eftrange, Efq ; Hugh Lyons, Elq ; Thomas Litton, Efq ; William Lane, Efq ; David Diggi Latouche« A. M. John Lamy, A. B. Mr. Charles Lbam, M. Right Hon. Earl of Miltown Rt. H. Lord Vif. Mountgarrtt Rt. H. Lord Vif. Mount-Caihel Rt. Rev. Lord Bp. of Mcath Rt. H. Sir Thomas Maud, Bt. Sir Capel MoUyaeaux, Bart. Hon. Barry Maxwell Colonal Maiba Dr. George Maconchy Paul Meredith, Efq : JuftinMacCarthy, Efq; Thomas Maunfell, Efq ; Ftancis Matthew, Efq; Alexander Montgomery, fifq; Arthur Maguire, Efq; Charles Mofs, Efq; John Monk Mafon, Efq; . Arthur Mahon, Efq ; George Monro, A. B. Rev. Edward Moore Rev. R. Murray, S. F. T. C. D Mr. Thomas Moife William Mofle, T. C. D. Mr. Chriftopher Moyers Mr. Thomas Morris Mt, Hugh Murphy Mr. Robert Moifct, A. B. Mr. Thomas Mulock A. M. Mr. Dominick Mahon Mr. John Moran Mr. John Maddock Mr. George Maguire . Mr. Richard Mellin Mr. George Maquay. N. Sir Edward Newenham MES. Edward Noy, Efq ; Braughill Ncwburgh, Efq ; Mr. Walter Nngent. O. George Ogle,Efq; Cook Otway, Efq ; AbleOnge, Efq; P. Rt Hon. Lord Vif. Powcrfcourt Sir William Parfons, Bart. Rev. Dr. Kene Pciceval Roger Palmer, Efq ; Chriflopher Palkce, Efq f John Prefloo. Efq ; Park Pepper Efq ; Robert Phibbs Efq; William Pleafants, A. B. Mr. William Penrofe Mr. James Paynofe John Prendergaft, £(q ; Edward Pfgot Efq; Mr. Jofcph Parker Mr. Richard Pike. a Henry Quin, M. D. R. Hon. Mr. Juftice R<A)infon Colonel Rofs Rev. Dean Ryder John Rochfort, Efq ; George Rochfort, Efq ; Andrew Ram« Efq ; Richard Reddy, Efq ; Thomas Rynd, Efq : Mr. James Rynd, T. C. D. Mr. William Rynd, T. C. D. Mr. James Reed, T. C. D. James Rainsford, Efq; Richard Robbins, Efq ; Mr. Chriftophcr Rielly Mf . Thomas Robinfon, T.C.D. Mr. John Read Mr. Henry Roche Mr. John Reilly Mr. William Reilly Mr. Jofeph Rooke. S. Rt. Hon. LK>rd Southwell Rt. Hon. Lord Stopford SUB Hon. Mr. Jufticc Smith ^iT George Savillc, Bart. Sir Annclly Stewart, Bart. Hon. Rob. Hen. Southwell Hon. Hugh Skeflington Bowen Southwell, Efq ; John Smyth, Efq; Chirle3 Smyth, Elq ; Ralph Smyth, Efq; William Smyth, Efq ; William Smyth, Efq ; Thomas Smyth, Efq; Jofeph Story, Efq ; . William Swift, Efq ; John Stewart, Efq; Henry Stewart, Efq ; Charles Stewart, Efq ; Mark Sinnet, Efq ; Ge. Lewis Shewbridgc^ A. Mr. Edward Strcttell Mr. John Sheppey Mr. Patrick Sherry Mr. Thomas Sherwood Mr. Frederick Stock Mr. William Sweetman Mr. Thomi^ i^mifke Mf. John Sewitd SCRIBERS NA Mr. William Sliannon Mr. Samuel Simpfon Mr. Edward Scriven Mr. William Sweetman Mr. John Seat on. T. R. Hon. Philip Tifdal William Tighe, Efq ; Richard To wnihend, E(q; William Talbot, Efq ; John Tunnadine, Efq ; Wentworth Thewlefs, Efq; Charles Tottenham, Efq; Robert Thorp, Efq ; Riley Towers, Efq ; Ed. Badham Thorhhill, Efq ; Eyre Trench, Efq; . Richard Talbot, Efq ; ^' Charles Tarrant, Efq ; Mr. Theophilus Thomfon Mr. Arthur Thomas. Agmondifham VcCcy, Efq ; Rev. Dr. Vance John ViLars^ M. D^ M E S.. John Ufher. E(q; Mr. Henry Upton. W. Rt. Hon. Earl of Weftmeath Rt. Hon. Earl of Wandesfbrd Sir Richard Wolfely, Bart. Rev. Tho. Wilfun, S.F. T.C.D. Bernard Ward, Efq; Charles WiUiam WaU, Efq; Edivard Wilmot. Efq; Hans Wood, Efq ; Ralph Ward, Efq ; Robert Waller, Efq; Mark Whyte, Efq ; John Wetherall, Efq; Meredith Workman, Efq; John Whitingham, Efq ; Stephen Wybrants, Efq ; Rev. Mr. John Wync Rev. John Waller, F. T. C. D. Jofeph Walker, Efq ; Mr. John Wilfon Mr. Samuel Whytc Charles WaU Efq ; Mr. William Williamibn. Z. Mr. Mark Zouch; PLAN of the Instructions given in the Drawing-School eflablijbed by the DUBLIN SOCIETT^ to enable Youth to become Proficients in the different Branches of that Art, and to purfue nuith Succefs geographical, nautical, mechanical, commercial or military In^ qutries^ VOiftvete y /' Ignorance font les deux Sources empoijonnees ditourier Dei- fordres, y les plus grands Fleaux de la Socieie. THE EdacatioiT of Youth is conddered in all Countries as the Ob- je3 which intereds mod imn^ediately the Happinefs of Families, as well as that of the State. To t^his ]End, the, ablei^ Hanc)^ 4re errlptb)!- ed in forming Plans, of Inftrui&ion,^ > ^the belt calc;u)ated. ^r the .various Pirofeflions of Life> atid Societies are formed> coo[ipoied of Men'diftinr giiUhed, as well by their Birth and Rank, as by their Experience and Knowledge, under whofe InfpeAion> and by whofe Care they are carried into Execution, by Perfons of acknowledged Abilities iji their different Departments: And thus the Education of Youth is conduced, from their earKeft Years, in a Manner the beft fuited to engage their Minds in the Love of ufeftil Knowledge, t<s^ improvo their Underftandings, to form their Tafte and ripen, their Judgments, to fix in them an Habit of Thinking with Steadinefs and Attention^ to promote their Addrefs and Penetration, and to raife their Ambition to excel in their refpedive Provinces. However neceffary fuch Regulations may appear to every reafpnable Perfon, however wifhed for by every Parent who feels the Lofs of a pro* per Education in his own Pra3ice ; never thelefs they -had not been even thought of ID this Country^ where that Extent of Knowledge^ requifite Wife Regu* laciont rtlar tive to the Education of Youth, in England, Scotland, and other Parts of Eu^ rope. Fatal Conic auencci re- niltina from the Ncgle€l ofthisOhica IV COURSEOF to prepare Youth to appear with Dignity in the virioas Enoployments of Life» or to enable them to bring to Perfe^on the different Arfsfbr whtefi they are defigned^ being not attended to ; Education was regarded as a puerile Objed, and of Courfe abandoned to illiterate Perfon^y who from their illiberal and mechanic Methods of teaching gave Youth little or no Information. To remove fo general and well grounded a Complaint, it wa&propofed that the Youth of this Kingdom fliouki receive in the Dcawiiig^Scbool eftabliihed by the DuBLiN-SociETTy the Inftrudions neceOkry to ena- ble them to become Proficients in the different Branches of that Art, and ^^[^"^ to purfue with Succefs, geographical, nautical, mechanical, commercial School eftal or military Enquiries : in this View, an Abftrad of the following Plans a^^in" were delivered to their Secretaries and Treafurer in the Month of Ofto- IjpHon of ber, 1 764, to be laid before the Society ; and to prevent an Undertaking of Xe Dublin- National Utility^ to be defeated through the Suggeftions of .Defign or Ig- oTifmmr "^^rance, the Plans were printed ; which being received by the Public FoMmcjuM with general Approbation, the DuBLTN-Soci£TY,purfuant to the Report TupplMdthii of their Committee appointed to examine into the Merit of the Plans, ^^•^' and the Charader of the Propofer, refolved, the 4th of February, 1 768, that they fliould be carried into Execution by the Author, under their immediate Infpedion. Tbe PLANS an as follow. I. PLAN of a Courfe of pure Mathematicks, abfolutely neq^flary for the right underftanding any Branches of pradical Mathematicks in their Application to geographical, nautical^ mechanical, commerciaU and military Enquiries. !!• PL AN of the phyfical and moral Syftem of the Worlds including the Inftrudions relative to young Noblemen aud Gentlemen of For- \ tune. ni. PLAN of the military Art, including the Jnftrudions relative to Engineers, Gentlemen of the Artillery, and, in general^ to all Land<- Officers. IV. P l^ A N of the merchantile Arts, or the Inftru Aions relative to thofe who are intended for Trade. PLAN MATBEMATCKS. V V. PLAN of the naval Art| including the InftruAions relative to Ship-Suiders, Sea-Oflicers, and to all rhore concerned in the Bufinefa of the Sea. VI. P L A N of a School of Mechanic Arts, where all Artifts» fuch ai Architedst Painters* Sculptors, Engravers, Clock-maken, bfc. receive ThcYomk the Inftmdions in Geometry, PcrfpeSive, Staticks, Dynamicks, Phy- of this King ficks, f^c. which fuit their reipefiive Profeflions, and may contribute to ^TtT ofThe improve their Tafte and their Talents. moftimpor- Thofc Flaws have convinced the Noblemen and Gentlemen of For- **J?}'Ji"?i tune of this Kingdom, that their Children, and in general, the Youth l^^} \ '' of this OmntTy, were deftitute of the moft important Means of In* ftrudton, and would ever be deftitute of them, until they had refolved that Men of Grenios and Education fliould be encouraged to appear as Teachers. PLAN »fa'C^ur/e of pupe MafbematickSf abfoliitely neceffary for the right undirftunding any Bp^fucbes of pruHhal Matbematicks in tbeir Ap- plication to gt^grapbicaly nautnal, mecbanical, commercial, and military Inquiries. FiX futcfuamin univerfa Matbefi itaiificile aUt arduum occurrere poffe, quo non in^enjo fete per banc Methodum penetrare liceat. 1. PURE Mathematicks comprehend Arithmetick, and Geometry. Pradical Mathematicks, their Application to particular Objtds, as the Laws of Equilibrium, and Motion of folid and fluid Bodies, the Motion of the heaveqly Bodies, (jfc. they extend to all Branches of MaiS^- ktuiMm Knowledge, and ftren$;thenihg our inteltedual Powers, by form- ticks tng in the Mind an Habit of Thinking clofely, and Reafoning accurate- ly, ferve to bring to Perfeaioti, with an entire Certitude, all Arts which Man can acquire by his Reafon alone. It is therefore of the higheft. Importance, that the Youth * of this Country (hould be me- thodically hrou^t acquainted with a Courfe of pure Mathematicks, to -^ ferve as an Introdudion to fuch Branches of Knowledge as are requifite to qualify them for their future Stations in Life. The Noblemen and Gentlemen of Fortune, therefore, have iinanimoufly refolved, that fuch a Courfe (hould be given on the rlioft approved Plan, in the Drawiko School eftablifhed under their Infpedion, by a Perfon, who, on ac- count of the Readinefs and Knowledge he has acquired in thefe Matters, during the many Years that he has made them his principal Occupation^ is qualified for making the Entry to thofe abftrufe Sciences, acceflabte to the meaneft Capacity. * The proper Ace to conmcace thii Coivfe is 14* -^ yi . COURSEOP II. Method of As to ihe Method of teaching Mathcmaticks, the fynthctic Method thcmitiM^** being neceflary to dircover the principal Properties of geometrical Figures, which cannot be rightly deduced but from their Formation, and iuiting Beginners, who, little accuflomed to what demands a ferious Attention, fland in Need of having their Imagination helped by fenfible Objeds, fuch as Figures, and by a certain Detail in the Demonflralions, is fol* lowed in the Elements (a). But as this Method, when applied to any other Refearch, attains its Point, but after many Windings and per- plexing Circuits, viz, by multiplying Figures, by defcribing a vaft many Lines and Arches, whofe Pofition and Angles are carefully to be ob- ' TW <5 h ^^^^^^* ^"^ ^y drawing from thefe Operations a great Number of in* 't^k Method cidental Propofitions which are fo many Acceffaries to the Subjed ; and fliouU not very few having Courage enough, or even are capable of fo earned an Se^^hi^'he Application as is neceflary to follow the Thread of fuch complicated fimplc EJe Demondrations : afterwards a Method more eafy and le(s fatiguing to raenci. the Attention is purfued. This Method is the analitic Art, the inge- nious Artifice of reducing Problems to the moft fimpie and eafieft Calculations that the QueAion propofed can admit of; it is the uni- verfal Key of Mathematicks, and has opened the Door to a great .Num- ber of Perfons, to whom it would be ever fliut, without its Help ; by its Means, Art fupplies <jreniu8, and Genius, aided by Art fo ufefuU has had Succefles that it would never have obtained by its own Force alone ; it is by it that the Theory of curve Lines have been unfold- ed, and have been diftributed in .different Orders, Clafles, Genders, and Species, which as in an Arfenal, where Arms are properly arrang- ed, puts us in a State of chufmg readily thofe which ferve in the Re- The Anali- ^o^u^^^'n ^^ ^ Problem propofed, either in Mathematicks, Afironomy* tick Method Opticks, ijc. It j» it which has conduced the great Sir I/aac Newtam ••^'^of to the ^wonderful Difcoveries he has made, and enabled the Men of ticarpifcwe Genius, who have come after him, to improve them. The Method of ries. Fluxions, both dirtGt and inverfe, is only an Extention of it, the firfi be* (t) It is for ihe(e Reafont that in all the puKGc matJiematical Schools eftabliihed ia Ei^gf^ad, Scodaod, &c. the Mafters commeocc their Courfes by the ElemenU of Geometry ; we /hall only inftancc that of Edinbnrgh, ^here a bandred young Gentlemen attend from the 6rft of Kovember to the firft of AugnA? u>«l are divided into 6ve dalTes, in each of which the Mafter employs a full Hour every Day. ^n ^^ ^^^ o^ lowed Claft, he teaches the firft fix Books of Euclid's Elements, plain Trigonometry, pra^cal Geometry, the ElemenU of Fortification, and an IntroduAion to Algebn. The fecond Cbfi ftudies Algebra, the i itfa and itth Books of Euclid, fpherical Trigonometry, conic SeAions, and the general Principles of Aftronomy. The third Cials goes on m Aftronomy and PerfpeAive, read a Pirt of Sir Ifaac Newron's Principia, and have a Courfe of Kxperimcnts for illuftrating them, performed and explained to theiA : the Mafter afterwards reads and demonftrates the £leme«:s of Fluxions. Tho.e in the fburth Clah read a Syftem of Fluxions, the DoArme of Chances, and the reft of Newton's Principia, .with the Improvements they hate received fnm the united Efibrti of the ftrft MaUiematicians of Europe. MATHEMATICKS. VJI ing ihe Aft ef finding Magnitudes infinitely fmall, which are the Elc-* ments of finite Magnitudes;^ the feccnd the Art of finding again, bv the Means of Magnitudes infinitely fmally the finite Quantities to whicfi they belong ; the firll as it were refolves a Q^antityi the lad reilores it to its firft State ; but what one refolves, the other does not always reindatey and it is only by anaiitic Artifices that it has been brought to any Degree of Pcrfedion, and perhaps, in Time, will be rendertd univerfal, and at the fame Time more umple. What cannot we ex- ped, in this RefpeS, from the united and conftant Application of the firft Mathematicians in Europe^ who, not content to make ufe of this fublime Art, in ail their Difcoveries, have perfeded the Art itftlf, and continue r9 to do. This Method has alfo the Advantage of Clearnefs and Evidence, and HuthcAd- the Brevity that accompanies it every where does not require too ftrong vanugeor' an Attention. A few Years moderate Studv fttflices to raifc a Perfon, IJ^Jnw' of Ibme Talents, above thefe Geniufes who were the Admiration of aoi J^IX^. Antiquity ; and we have feen a young Man of Sixteen, publifh a Work, CTrait^ des Courbes d double Courbure par Clairaut) that Arabimedet would have wiihed to have compofed at the End of his Days. The Teacher of Math«maticks, 'therefore, fhould be acquainted with the difFerent Pieces upon the anaiitic Art, difperfed in the Works of the moft eminent Mathematicians, make a judicious Choice of the mofl ge- neral and eiTentlal Methods, and lead his Pupils, as it were, by the Hand, in the intricate Roads of the Labyrinth of Calculation ; that by this Means Beginners, exempted from that clofe Attention of Mind, which would give them a Diftafte for a Science they are defirous to at- tain, and methodically brought acquainted with all its preliminary Prin- ciples, might be enabled in a fhoit Time, not only to underfland the Writings of. the mofl eminent Mathematician?, but, rei!eding on their Method of Proceeding, to make Difcoveries honourable to themfelves and ufefu! to the Public. III. Arithmetick comprehends the Art of Numbering and Algebra, confe- ^^^ ^^. . quently is diftinguiihed into particular and univerfal Arithmetick, becaufe meckk du- the Demonflrattons which are made by Algebra are general, and nothing ^^^ ^^^ can be proved by Numbers but by Induaion. The Nature and Forma- Veiled?" tion of Numbers are clearly dated, from whence the Manner of ptr- forming the principal Operations, as Addition, Subtrafiion, Multipli- cation and Divifion are deduced. The Explication of the Signs and Symbols ufed in Algebra follow, and the Method of reducing, add- ing* fubtrading,' multiplying, dividing, algebraic Quantities fimple atS compound. This prepares the Way for the Theory of vulgar, algebraical and decimal Fradions, where the Nature^ Value^ K&n^ VIII CaURSEOP Manner of campnriitg ihtm, tnd their OperatiofiiSy ire carefidtj nil- folded. The Compofition and Rerolution of Qyancitics conies after, including the Method of raifing Quantities to any Power, extracting of Roots, the Manner of performing upon the Roott of imperfed Powers, radical or incommenfurable Quantities, the various Operations of which they are fufceptible. The Compofition and Refolution of Quantities being finilhed, the Dodrine of Equations prefents itfelf next, where foWn«Eqt- ^^^^^ Genefis, the Nature and Number of their Roots, the difierent tiom. Redu£bions and Transformations that are in Ufe, the Manner of foiving them, and the Rules imagined for this Purpofe, fuch as Tranfpofition, Multiplication, Divifion, Subftitmion, and the Exlradion of their Roots, are accurately treated. After having confidered Qyantiticfs in themfelves, it remains to examine their Relations ; this Do3rine comprehends arith- metical and geomerrical Ratios, Proportions add Progreflions: The Theory of Series follow, where their Pormationi Methods for difcorer- ing their Convergency, or Divergency, the Operations of which they are fufceptible, their Reveriion, Summation, their tJfe in the Irtveffi- The Natiiff S*^^^" ^* *^^ Roots of Equations, Conftnidion of Logarithms, Wr. are ana^Lawiof taught. In fine, the Art of G>mbinations, and its Application for de- chtnce. termining the Degrees of Probability in civil, moral and political Enqpit- rics are difclofed. Ars cujus Ufus et NeceKus ita unhtrfale ejty utjine ilUf nee Sapientia Pbllofopbiy nee HiRwici Exa^itudo, nee Medici Dex^ teritas^ aut Politici Prudentiaf conjifiere queut. Omnis enim borvm Lahr in conjedando, et omnis Conje^ra in Trutinandis Caufarum CompUxiam^ bus aut CoMbinationibus verjatur. tv. Divifionof GEOMETRY is divided into Elementary, TRAKseiKPE>7TAt> Geometiy and SuBLIME. u^, Tiin- '^^ ^P^" '^ Youth an accurate and eafy Method for acquiring a rcendentcl Knowledge of the Elements of Geometry, all the I^ropofitions in Euclid and Su- (a) in the Order they are found in the beft Editions, are retained with blimc, (0 '* PMTfffCUityititht'MMhodaikrFortti of Iteafonifig, it the ptfeiilitf Okar^ftcnftlc €f •* Eddies Etfeaiciic<« TttX, is iiitdrpolac«d by CXbpfttiUi and CUtiilt, ttmMkVM by Herigoile mA << Barrow, or dmraved by T-a(f<{uet and Derdialei. but of the OH|iiMl^ kaaded down t<^ ut by ** Antiquity. His Cemonftrations being conducted with the molt exprefs Defign of reduciow *• xht Principles ailbinedto the fewf^fk KniMber, atid moft e?ident thirt ndght be, and in 1 1i^ ** tbod the moft nanirali as it U the trtc^ condocive towirdta inft mid cdrnpieteCdtaMVlaiioiL " of the Sabfed, by beginnina with fuch Particulars as are -moft eafily coooehred, and flow aoft ** reiulily from theTrincipIes laid down ; thence by gradually proceeding to (iich as are more ob- ^ (hire, arid reqdh^ a longftt dhaiii of AtgvfMent, tt^ have Ai^t^M^beeii regafM hi all A^dbL *' as tfato moft ^xMk in their Kind." Such-i* the Jadgmem Hit the &0YAJU^80Cl£TY, Ao have expreif*d at the lame Tine thar Ditfrke to the new modelled £lemenci that aCpreleat every wh«re abbund ; and it the illiberal and ffttfdunic Mc!thods of teadfiilg thoft! moft fet^ Aftf- whicKistabeho)>ed, wfltiH^er be coUktciMhcad ia dit niUHcMoblt i&£DcU^:ad€M- land, Aec. MATHEMATICKS. IX atl poffible Attention} as alfo the Forniy Connedion and Accuracy of his Demonftrations. The eflential Parts of his Propofitions being fet Methodical forth with all the Cicarnefs imaginable, the Senfe of his Reafoning arc ^J^*^ J|J^ explained and placed in fo advantageous a Light* that the Eye the lead Eiements^f attentive may perceive them. To render thefe Elements ftill more eafy, Eufl'<l •« the different Operations and Arguments eflential to a good Demonftra- *^«*^***- tion» are diftinguiflied in feveral feparate Articles ; and as Beginners, in . order to make a Progrefs in the Study of Mathematicks, fhould apply themfelves chiefly to difcover the Connexion and Relation of the differ- ent Proportions, to form a juft Idea of the Number and Qualities of the Arguments, which ferve to eftablifh a new Truth ; in fine, to dif- cover all the intrinficalPartsofa Demonftration, which it being impofftble for them to do without knowing what enters into the CompoHtion of a Theorem and Problem, Firft, The Preparation and Demonftration are diftinguiihed from each other. Secondly, The Propofition being fet down, what is fuppofed in this Propofition is made known under the Title of Hypothecs, and what is aifirnaed, under that of Thefis. Third- ly, All the Operations neceffary to make known Truths, ferve as a Proof to an unknown one, are ranged in feparate Articles. Fourthly, The Foundation of each Propofition relative to the Figure, which forms the Minor of the Argument, are made known by Citations, and a marginal Citation recalls the Truths already demonftrated, which is the Major : In one Word, nothing is omitted which may fix the Attention of Be- ginners, make them perceive the Chain, and teach them to follow the Thread of geometrical Reafoning. V. Tranfcendental Gcometrv prefuppofes the algebraic Calulation; it com- Tranfcen- mences by the Solution of the Problems of the fecond Degree by Means of ^"^y.^^*^ the Right-line and Circle : This Theory produces important and curious Remarks upon the pofitive and negative Roots, upon the Pofition of the Lines which exprefs them, upon the different Solutions that a Pro- blem is fufceptible of; from thence they pafs to the general Principles in what it of the Application of Algebra to curve Lines, which confift, Firft, ^^^^ ^ In explaining how the Relation between the Ordinates and Abciffes of a Curve is reprefented by an Equation. Secondly, How by folving this Equation we difcover the Courfe of the Curve, its different Branches, and its Afymptots, Thirdly, The Manner of finding by the direft Me- thod of Fluxions, the Tangents, the Points of Maxima, and Minima. Fourthly, How the Areas of Curves are found by the inverfc Method of Fluxions. The Conic SeSions follow; the bcft Method of treating them is to Beft Method confider them as Lines of the fecond Order, to divide them into ^^^^f^ their Species. When the moft fimple Equations of the Parabola, tions. COURSE OF The differ- rnc Ord.n of Curves. Sublime O^ometry. Its DivifioD. / What the firft Part compre- hends. Ellipfei and Hyperbola are found, then it is eafily ihewn that thele Curves are generated in the Cone. The Conic Sedions are terminated by the Sohition of the Problems of the third and fourth Degree^ by the Means of thefe Curves. The Conic Sedions being finiihed> they pafs to CurretW a fuperior Order* beginning by the Theory of multiple Points> of Points of Inflec^ tiont Points of contrary Infledion* of Serpentment, (Jc. Thefe Theo* ries are founded partly upon the fimple algebraic Calculation^ and partly on the dired Method of Fluxions. Then they are brought acquainted livith the Theory of the Evolute and Cauftiques by Refledion and Re« fra£^ion. They afterwards enter into a Detail of the Curves of diflFerent Orders, ailigning their Clafles, Species, and principal Properties^ treat- ing more amply of the bed known, as the Folium> the Conchoid, the Ciflbid, e^r. The mechanic Curves follow the geometrical ones, beginning by the exponential Curves, which are a mean Species between the geometrical Curves and the mechanical ones ; afterwards having laid down the ge- neral Principles of the Conftnidion of mechanic Curves, by the Memos of their fluxional Equations, and the Quadrature of Curves, they enter into the Detail of the beft known, as the Spiral, the Qyadratrice, the Cycloid, the Trochoid, i^c. VI. Sublime Geometry comprehends the inverie Method of Fluxions, and its Application to the Quadrature, and Redification of Curves, the cubing of Solids, (Jc. Fluxional Quantities, involve one or more variable Quantities ; the natural Divifion therefore of the inverfe Method of Fluxions is into the Method of finding the Fluents of fluxionary Qgaatities, containing one variable Quantity, or involving two or more variable Qjiantities ; the Rule for finding the Fluents of fluxional Quantities of the moft fimple Form, is laid down, then applied to diflFerent Cafes, which are nxxe compofed, and the DifSculties which fome Times occur, and which em* barrafs Beginners, are Iblved. Thefe Refearches prepare the Way for finding the Fluents of fluxional Binomials, and Trinomials, rational Fra£iions, and fuch fluxional Qinuw tities as can be reduced to the Form of rational Fradions $ from thoice they pafs to the Method of finding the Fluents of fuch fluxional Quan- tities which fuppofe the Re&ification of the Ellipfe and Hyperbola, as well as the fluxional Quantities, whofe Fluents depend on the Quadra- ture of the Curves of the third Order ; in fine, the Refearches which Mr. Newton has given in his Quadrature of Curves, relative to the Qjja- drature of Curves whofe Equations arc compofed ef three or four Terms ; MATHEMATICKS. XI And this firft Part is terminated by the Methods of finding the Fluents of fluzionaU logarithnoieticaU and exponential QyantitieSf and thofe which are affed^ with many Signs of Integration^ and the various Me- thods of Approximation^ for the Solution of ProblemSf which can be reduced to the Qjiadrature of Curves* The fecond Part of the inverfe Method of Fluxions* which treats of fluxional Qjiantitiesy including two or more variable Qyantities* com- mences by ihewing how to find the Fluents of fuch fluxional Quantities as require no previous Preparation; the Methods for knowing and ^ fiiftinguiihing thefe Quantities or Equations ; afterwards they pafs to the Methods of finding the Fluents of fluxional Quantities, which have f^oSd ftrt need of being prepared by fome particular Operation, and as this Oper- comDrc* ation confifts moft commonly in Separating the indeterminate Q^ntities, ^^ after being taught how to conftruft diflFerential Equations, in which the indeterminate Qgantities are feparated, they enter into the Detail of the different Methods for feparating die variable Quantities in a propofed Equation, either %j Multiplication, Divifion, or Transformations, be- ing Ihewed their Application, firft to homogeneous Equations, and after being taught how to conftnid thefe Equations in all Cafes, the Manner of r^ucing Equations to their Form is then explained. How the Me- thod of indeterminate Co-eflicients can be employed for finding the Fluents of fluxional Equations, including a certain Number of variable ^^antities, and how by this Method, the Fluent can be determined by certain Conditions given of a fluxional Equation. Fluxional Qjiantities ^f diflFerent Orders follow ; it is fhewn, firft, that fluxional Equations of the third Order, have three Fluents of the fecond Order, but the laft Fluent of a fluxionary Equation of any Order is fimple ; then the vari- ous Methods magined by the raoft eminent Mathematieians for finding thefe Fluetits, fuppofing the Fbxion of any one variable Quantity con- fttaitf are explained, and the Whole, in fine, terminated by the Applica- tion of this Do£b'ine to the Qsiadrature and Rectification of Curves, CttlMng of Solids, &!r. vii. Such is the Plan of a Courfe of pure Mathematicks traced by New^ Condufion. t^Hj improved by CoteSfBernffulfyf Euler, QairauU D^Jiemiett, M*Laurin, Stff^fon, Fontain, * &c. which ferves as a Bafis to the Inftrudions re- qoifite to qualify Youth to appear with Dignity in the different Employ- ments of Life, or to enable them in Time, To bring to Perfedion the various Arts for which they are intended. * Qtisdratiira currarmn, kamoiiia mcDilvanmii ftc XII SYSTEM OF THE PLAN of the Syflem of the Pbyftcal and Moral Worlds including the Infirufiions relative to young Noblemen and Gentlemen of Fortune. P L AN of the Syflem of tbe Pbyftcal World. Nubem pellente matbefty UtiJity of the Study of the Sy- ftem of the World. Is t Prc- iervative againft the PaifioDi. Leads to Virtue. Clauflra patent calif rerumque immobilis ordo: fam fuperum penetrare domoi, atque ardua call Scandercf fublimis genii concefjit acumen. I. STUDY in general is necefiary to Mankind, and eflentially contri« butcs to the Happincfs of thofc who have experienced that adive Curiofity which induceth them to penetrate the Wonders of Nature. It is, bcfides, a Prefervative agatnft the Diforders of thePaf&ons; a kind of Study therefore which elevates the Mind, which applies it ciofely, confcquently, which furniflies the moft affured, arms againft the Dangers we fpeak of, merits particular DiftinSion. " It is not " fufficient, fays Seneca^ to know what wt owe to our Country, to our " Family, to our Friends, and to ourfelves, if we have not Strength of " Mind to perform thofe Duties, it is not fufficient to eftablifli Precepts^ ** we muft remove Impediments, ut ad prcecepta qua damus pofpt animus " ire, folvendus eft. (Epift. 95.) Nothing anfwers better this Purpofe than the Application to the Study of the Syftem of the World ; the Wonders which are difcovered captivate the Mind, and occupy it in a noble Manner; they elevate the Imagination, improve. the Underftand- ing, and fatiate the Heart : The greateft Philofophers of Antiquity have been of this Opinion. Pytbagoras was accuftomed to fay, that Men ftiould have but two Studies, that of Natutc, to enlighten their Undcrftandings, and of Virtue to regulate their Hearts ; in eflFca to be- come virtuous, not through Wcaknefs but by Principle, wc muft be able to reflea and think ciofely ; we muft by Dint of Study be delivered from Prejudices which makes us err in our Judgments, and which are fo many Impediments to the Progrefs of our Reafon, and the Improve- ment of our Mind. Plato held the Study of Nature in the higheft Efteem ; he even goes fo far as to fay, that Eyes were given to Man to contemplate the Heavens : To which alludes the following Paflage of Ovid. Finxit in effigiem moderantum cunffa deorum, Pronaque cum fpeUant animalia cetera terram, Os bomini fublime dedity ccelumque tueri Jufptf et ereflos ad fidera toller e vultus. r PHYSICAL WORLD. XIII II. The Poets who have illuftratcd Greece and ItaJy^ and whore Works J* cdcbm- arc now fare of Immortality, were pcrfcSly acquainted with the Hca- pj^,^* vens, and this Knowledge has been the Source of many Beauties in their Works : Homer, Hejiad, Aratusy among the Greeks : Horace^ Virgilf Ovidf Lucretius, Manilius, Lucan, Claudian, among the Latins ; make life of it in feveral Places, and have expreflfed a fingular Admiration for this Science. Ovid after having anounced in his Fafti, that he propofes celebrating the Principles on which the Divifion of the Roman Year is founded, enters on his Subjed by the following pompous Elogium of the firft Difcoverers of the Syftem of the World. Felices MnimoSf quibus bac cognofcere primis^ Ihque domos Juperas fcandere curafuit, Credibile eft illos pariter vitiijque locifque, Altius bumanis exeruiffe caput. Ncn venus out vinum fuhlimia pe^ora f regit ^ Officiumvefori fnilitiaque labor. Nee levis ambit io perfufaque gloria fuco, JUagnarumve fames Jollicitavit opum, , Mmvoere oculis diftantia Jydera noftris^ • JEtberaque ingenio fuppo/uere Juo. • Sic petitur cctlum, Claudian in the following Verfcs, celebrates Arcbimedes on his Inven- tion of a Sphere admirably contrived to reprefent the celeftial Motions. Jupiter in parvo cum cerneret atbera vitro, Rjfitf et adjuperos talia di^a dedit : Huccine mortalis progrejfa potentia cura I Jam meus ihfragili luditur orbe labor. Jura poli, rerumque fidem legejque deorum Ecce Syracujius tranjiulit Arte fenex ; Inclufus Variis famulatur fpiritus ^ftris, Et vivum certis motibus urget opus ; Percurrit proprium mentitus JignUer, annum, Etjknulata novo Cyntbia menfe redit : Jamque Juum volvens audax induftria mundum Gaudet, ct bumanajidera mentc regit. XIV SYSTEMOFTHE yirgil feems defirous of renouncing all other Study^ to contemplate the Wonders of Nature. Me vero primum dukes ante omnia mufttf ^arum facra fero ingenti percujfus amore, Accipiant, calijue vias et Jydera monjirent DefeBus foils varies, lunaque labor es, Unde tremor terrisf qua vi maria alta tumefcani Ohjicibus ruptis, rurfufque injeipfa refidant^ ^id tantum oceano properent ft tingerefoles Hybernif vel quee tardis mora noBibus obftet Felix qui potuit rerum congnojcere eaufas. Ceor. n. 475. La Fontaine imiutes the Regrets of Virgil in a niafterly Manner^ where he fays^ ^andponrront les neuf/ieurs loin des cours et dee villes, i^occuper tout entier^ €t m' ^prendre des deux Les divers mouvements inconnus d nosyeux, Les noms et les vertues de ces clartes errantes. Songe dun habitant da MogoL Foltaire, the firft Poet of our Age^ has teftified in many Parts of hji Worksi his Tafte for Aftronomy» and his Efteem for Aflronomersy whom he has celebrated in the fineft Poetry. What he fays of Newton b worthy of Auention. Confidens du Tres Haul, Subjlances eternellesp ^i parez de vos feux, qui couvrez des vos aUes, Le trone ou votre maitre eft ajfis parmis vous : Paries I du grand Newton n^etiez vous point jakax. To which we can only oppofe what Pope has faid on the fame Sub- jeft: Nature and Nature's Laws lay hid in Night ; God faid» Let Newton be^ and all was Light The great Geniufes of every Species have been furprized at the In- difference which Men fliew for the Spedacle of Nature* Tajo puts Refledions in the Mouth of Rsnaldo, which merit to be recitedfor the Inftrudion of thofe to whom the fame Reproach may be applied ; it is at the Time when marching before Day towaixls Mount Olivet^ he con- templates the Beauty of the Firmament. PHYSICALWORLD. XV Cm gU occbs alzati contempUndo intorno, $uinci notturne r quindi matutim Bellezze^ incorruptibili e divine \ Fraftftejfo penfavOf o quanta belle Lucif il tempio celefie in Je ragunat Ha iljuo gran carro il de^ Pauratajtelle Spiega la notte^ e Pargentata Luna \ Ma non i cbi vagbeggi o quejlaf o quelle ; E miriam not torbida luce e bruna, Cb*un girar ffoccbi, un balenar di rifo Scopre in breve confin difragil vifo. Jerus. Cant, xviii. Si. I2, 13. HI. The Knowledge of the Syflem of the World has delivered us from E^reOi the Apprehenfions which Ignorance occafions ; can we rccal without 7***^nit!ce Compaifion> the Stupidity of thofe People, who believed that by making ofthTsyf. a great Noife when the Moon waa eclipfed> this Godders received R^'ic^'*^^^ from her Sufferances^ or that Eclipfcs were produced by Inchantments (a) ? JJ^^j^ Cvmfrufira refonant JEra auxiltaria Luna. Met. iv. 333. Canfuj et e Curru Lunam deducere tentant, Etfaceretft non JEra repulfa fonent. Tib. El 8. The Knowledge of the Syftem of the World has diflipated the Errors of The Koow- Aftrology, by whofe fooliih Prediaioos Mankind had been fo long abufed. s^l^^^^ The Mbmture of 1 1869 fliould have covered with Shame the Aftrologcrs the World erf Europe i they were all, Chriftians, Jews and Arabians, united to **f*^*%*' anounce, fevcn Yean before, by Letters pobliihed throughout Europe^ ^n In a Conjundion of all the Planets, which would be attended with fuch Aftrok>gy. terrible Ravages^ that a general Diflblution of Nature was much to be drettded^ fo that nothing Kefs than the End of the World was expeded : this Year nttcwithftanding pafled as others. But a hundred Lies, each M well atteftedy would not be fufficient to wain ignorant and credulous Men frem the Prejudices of their Infancy. It was neccflary that a Spi- rit of Philofophy, and Refearch, ihould fpread itfelf among Mankind, ■open their Underftandings, unveil the Limits of Nature, and accuftom them not to be terrified without Examination, and without Proof. IV. The CometS) as it is well known, were one of the great ObjeSs of Terror which die Knowledge of the Syftem of the World has, in fine, (a) ScBCCSy Upolit, 7S7. Tacit Ann, Fhitarch In Fcridc, et de dcfcQa Oracalorum. XVI The Know ledffcofthe Svftem of the World ufeful in Geography and Naviff a tioD, ana confequent lyofthe greateftim portance to there King domi. SYSTEM OF THE removed. It is not without Concern wc find fuch (Irangc Prejudices in the fineft Poem of the laft Age, whereby they are iranfmitted to the iateft Pofterity. ^al colle cbiome fanguinoje borende% Splender comfta fuol per Varia adufiay Che i regni muta^ ei fieri morti adduce^ Ai purperei tiranni infaufia luce. Jerus. Lib. 7. St. 52. The Charms of Poetry are aduaily employed in a Manner more phi- lofophical and ufefuU witnefs the following fine Pafiage. Cometet que Pon craint a legal du tonnerref Cejfez d*epouvanter les peuples de la terre ; Dans une Ellipfe immenje acbevez voire courst Retn9nteZ9 de/cendez pres de V afire des jours ; Lancez vos feux, volez, et revenant fans cejfe% Des monies epuifez ranimez la vielleffe. Thus the profound Study of the Syftem of the World has dlflipated abfurd Prejudices, and re-eftabli(bed human Reafon in its inalienable Rights, V. To the Knowledge of the Syftem of the World, are owing the Im- provements in Cofmography, Geography, and Navigation ; the Obfer- vation of the Height of the Pole, taught Men that the Earth was rounds the Eclipfes of the Moon taught how to determine the Longitudes of the different Countries of the World, or their mutual Diftancet from Eaft to Weft. The Difcovery of the Satellites of JupiteTf has contri- buted more effe&ually to improve geographical or marine Charts, than ten thoufand Years Navigation ; and when their Theory will be better known, the Method of Longitudes will be ftill more exafi and niore eafy. The Extent of the Mediterranean was almoft unknown in 160O9 and To-Day, is as exadly determined as that of England or Ireland. By it the new World was difcovered. Cbrlfiopber Columbus had a more intimate Knowledge of the Sphere, than any Man of his Time, Hnce it gave him that Certainty, and infpired him with that Confidence with which he dire&ed his Courfe towards 4he Weft, certain to rejoin by the Eaft the Continent of Afia^ or to find a new one. And nothing feems to be wiftied for, to render Navigation more perfe£l and fecure, but a Method for finding with Eafe, the Longitude at Sea, which is now ob- tained by the Means of the Moon : And if the Navigators of thi$ Kingdom were initiated in Aftronomy, by able Teachers^ as is praftifed PHYSICAL WORLD. XVfl in other Parts of Europe, their Eftimation would approach within twenty Miles of the 'JVuth, whilft in ordinary Voyages, the Uncertainty amounts to more than three hundred Leagues, by which the Lives and Fortunes of Tboufands are endangered. The Utility therefore of the Marine to thofe Kingdoms, where Empire, Power, Commerce, even Peace and War, are decided at Sea, proves that of the Knowledge of the Syftem of the World. VI. The adual State of the Laws, and of the ecdefiaftical Adminiftra- The Refor tion, is cffentially conneded with the Syftera of the World ; St. A- "^'^*'^ gujline recommended the Study of it particularly for this Reafon; St. dJaepend* Hypp^Ute applied himfelf to it, as alfo many Fathers of the Church, cd on it. notwithftanding our Kalendar was in fuch a State of Imperfedion, that the Jews and Turks were afloniihed at our Ignorance. Nicholas V, Lton X, Cffc. had formed a Defign of re-eftabli(hing Order in the Ka- lendar, but there were at that Time no Philofophers, whofe Reputa- tion merited fufficient Confidence. Gregory the Xlllth, governed at a Time when the Sciences began to be cultivated, and he alone had the Honour of this Reformation. VII. Agriculture borrowed formerly from the Motions of the celeftial iinfeflilia Bodies, its Rules and its Indications ; Job, Heftod, Varro, Eudoxus, Agrkukuir* Aratujf Ovid, jMmy» Columella, Manilius, iurnidi a thoufand Proofs of it. The Pleyades, Arflurus, Orion, Syrius, gave to Greece and Egypt the Signal of the different Works ; the rifing of Syrius anounced to the Greeks the Harveft ; to the Egyptians the overflowing of the Nile. The Kalendar anfwers this Purpofe adualy. VIII. Ancient Chronology deduces from the Knowledge and Calculation of is the Fomi Eciipfes, the moft fixed Points which can be found, and in remote Times ^^^^ ^^ we find but Obfcurity. The Cbinefe Chronology is entirely founded up- ^^ ^ on Eciipfes, and we would have no Uncertainty in the ancient HiAory of Nations as to the Dates, if there were always Philofophers. (See the Art of verifying Dates.) It is from the Syftem of the World we borrow the Divifion of Time, Fnrnlftef and the Art of regulating Clocks and Watches ; and it may be faid, the Meuis that the Order and Multitude of our Affairs, our Duties, our Amufe- ^J^,^jf inents, our Tafte, for Eicadtiefs and'Precifion, our Habitudes have ren- dered this Meafiire of "Time almoft indifpenfabie, and has placed it in the Number of the Keceflaries of Life; if inftead of Clocks and Watches, Meridians and folar t)ial$ are traced, it is an Advantage that the Knowledge of the Syftem of the World has procured us. Dial- XVIII SYSTEM OF THE Is i:^erui in ling being the Application of fpherical Trigonometry ; a ProjeAion of the Sphere upon a Plane, or a Se^ion of a Cone» according to ^he Forms given to a Dial. X. The Knowledge of the Changes of the Air, Winds, Rain, dry Wea- ther, Motions of the Thermometer, Barometer, have certainly an eflfen- tial and immediate Relation with the Health of the human Body ; the Knowledge of the Syftem of the World will be of fenfible Utility, when, by repeated Obfervations, the phyfical Influences of the Sun and Moon upon the Atmofphere, and the Revolutions which refult will be diP> covered. Galen advifes the Sick not to call to their Afliftance Phyfici- ans, who are not acquainted with the Motions of the celcftial Bodies, becaufe Remedies given at unfeafonable Times are ufelefs or hurtful, and the ableft Phyficians of our Days are convinced, that the Attradions which elevate the Waters of the Ocean twice a Day, influence the State of the Atmofphere, and that the Crifis and Paroxifms of Diforders cor^^ refpond with the Situation of the Moon in refped of the Equator, Sy- figies, aijd Apfides. See Mead, Ho/man, &c. XI. Thofe Advantages which refult from the Knowledge of the Syftem of the World, has caufed it to be cultivated and held in fmgular Efleem by all the civilized People of the Earth. The ancient Kings of Perfia, tiou of the and the Priefts of Egypt f were always chofen amongft the mofi expert ^^^*^ in this Science. The Kings of Lacedemon had always Philofophers in their Council. Alexander was always accompanied by them in hist mtli^ tary Expeditions, aiyl Jrijlotle gave him &r\€t Charge to do nothing without their Advice. It is well known how much Ptolemeus the fecond King of Egypt^ encouraged this Science ; in his Time flouriftied Hypar* cbuSf Cdlimacbut, ApolloniuSf Aratus^ Biorif TbeocriteSf Conon, yulius Cafar was very curious in making Experiments and Obfervations, as ii appears by the Difcourfe which Lucan makes him hold with Acbore^ Prieft of Egypt, at the Feaft of Cleopatrs. Caltivated in all Aeei by aU the cmlized Na ' Media inter preliafemper Stellarum ccelique plagis fuperifque vacavi. Nee meus Eudoxi vincetur faflibus annus. Phar. Has beenF the favorite- Study of €rcat Pri rnacet. The Emperor Tiberius applied himfelf to the Study of the Syftem of the World, as Suetonius relates ; the Emperor Claudius forefaw there would be an Eclipfe the Day of his Anniverfary, and fearing it might occaiion Commotions at Rome, he ordered an Advertifement to be pub- liftied, in which he explains the Circumftances, and the Caufes of this Phenomenon. It was cultivated particularly by the Emperors AdriQ^ PHYSICALWORLD. XIX msd S^erus^ by Cbarlemagnet by Leon V, Emperor of ConJLantmopIe^ by Alpbonfo X, King of CafliU^ by Frederick II, Emperor of the • Wejty .by CVi//y> AlmamoTiy the Prince Uluheigbi and many other Monarchs of Among the Heroes who alfo cultivated it, are reckoned Mahomet II, Conqueror of the Greek Empire; the Emperor Charles V, and Lewis XIV. In fine, the Eftabiifliments of diflFerent Philofophical Societies in Eng^ Jandf Scotland, France, Italy, Germany, Poland, Sweden, Rujpa, &c. have ^iven the Monarchs, Nobility, and Gentrv of thofe Countries, a Tafle for the more refined Pleafures attending the Study of the Sciences, and particularly of the Syftem of the World, an Example worthy to be imi- tated by thofe of this Kingdom. XII. PabUck Befides thofe renowned Societies which have all contributed to the f^JJ^.^ Progrefs of every Branch of human Knowledge, and particularly of the b the dif- Syftem of the World, there has been eftabliftied in the different Parts of ferentPaftt Europe public Schools, conduced by Men of fuperior Talents and Abi- for^SftSft lities, who make it their Bufmefs to guide and in(lru3 the young No- log young bility and Gentry in this noble Science, and furnifh thofe who difcovcr Noblemen fingular Difpofitions with every Means of Improvement. men^fFo- An illuftrious Englijbman, Henry Saville, founded in the Univerfity of tune in what Oxford two Schools, which have been oi vaft Utility to England \ the J^?'^*^ Mailers have been Men all eminent in this Science, John Bainbridge in ilLworid. 1619, John Greaves in 1643, Seth Ward, Chrijlopher Wren, Edward Foundation Bernard \Xk 1673, David Gregory in 1691, Brigg^ Wallis, and 7. Caf- of Henry well in 1 708, Keill in 1 7 1 2, Hornjby, &c. ^ Savilfc. The Schools eftablilhcd at Cambridge, among whofe Mafters were Founda- Barrow, Newton, Cotes, Wijion, Smyth, and Long, all celebrated Aftro- ^^^ ^^ tlOmtTS. and^ncaj. The School of Grejham at BiJhops^Gate in London, which has cflen- college of ttally contributed to the Progrefs of Aftronomy ; among the Mafters of Grefliam. this School were Doftor Hook, and other eminent Men. The Royal mathematical School at Cbrift's-Hofpital, where Hodgfon, *^'^["*1 Rohertfon, &c. have bred up a great Number of expert Navigators and ©f chrift^a Aftronomers. Hofpitai. The Schools of Edinburgh, Glafgow, and Aberdeen, are known all Mathcmati ever Europe \ the Nobility, and Gentlemen of Fortune of Scotland, fu- cal Schools perintending them, and taking every Method of encouraging both Maf- ^ Scotland. tcrs and Students to Affiduity and Attention, to go through their refpec- tive Talks with Alacrity and Spirit; the Names of Gregory, M^Laurin, Stuart, Simp/on, &c. the famous Mafters, will never be forgotten. * He ordered the Works of Ptolemey to be tranflated into Latin, and publickfy to be taught alKapkt. XX The Royal College. Obrerrato ries and SchooUof Experimen tal Philofo phy. Of CafTel. Of Urani bourg. OfDantzick Of Copcn lugen. Of Pekio. SYSTEM OF THE The Royal School of France^ founded by Francis I, has cfTentially con- tributed to the Progrefs of the Knowledge of the Syftcm of the World. Orancet Fine\ Stadiusj Morin, Gajfendu de la Hire, de Lifle^ who were fucceflively Mafters of it, have been celebrated Aftrononiers, lie. XIII. Experiments and Obfervations are the Foundation of all real Knonr- ledge, thofe which ferve as a Bafis to the Difcoveries relative to the Syftem of the World, are made and learned in Experimental Schools and Obfcrvatories : The firft Obfervatory of any^^lebrity, was built by William V, Landgrave of Heffe, where he colled^d all the Inftm- ments. Machines, Models, l£c, which were known in his Time, and put it under the Diredion of Rotbman and Byrgius, the firft an Aftroso- mcr, the fecond an expert Inftrument-Maker : The Duke of Bro^lio, General of the French Army, having rendered himfelf Mafter of Cafftl in 1 760, took a Copy of the Obfervations and Experiments made ia this Obfervatory, and depofited it in the Library of the Academy. Frederick L King of Denmark^ being informed of the fingular Merit qf Ticbo Brake, granted him the Ifland of Venufia, oppofite Copenhagen^ and built for him the Caftle of Uranibourgb, fumiihed it with the larg- eft, and the moft perfeS Inftruments, and gave Peniions to a Number of Obfervers, Calculators, and Experiment- Makers, to aflift him, which enabled him in the Space of 16 Years, to lay the Foundation of the Sys- tem of the World, in a Manner more (table, than was ever before ef- feSed. The moft eminent Men took Pleafure in vifiting this incom- parable Philofopher : • The King of Scotland going to efpoufe the Prin-. cefs Jnne, Sifter of the King of Denmark, pafled into the Ifland of VertufU with all his Court, and was fo charmed at the Operations and Succets of T!^ycho, that he compofed his Elogium in Latin Poetry : So much > Merit raifed him Enemies, and the Death of King Frederick II, (iimilh- ed them the Means of fucceeding in their Machinations. A Minifter called Walchendorp, (whofe Name ftiould be devoted to the Execration of the Learned of all Ages) deprived him of his Ifland of Venujia, and forbad him to continue at Copenhagen his Experiments and Obfervations. XIV. The firft Obfervatory of the laft Age, was that of Hevelius, eftab- liflied at Dantzick ; it is defcribed in his great Work, intitled, Macbina CeU/tis. The Aftronomical Tower of Copenhagen was finifhed in 1656, built by Chrijlian IV, at the Solicitation of Longomontanui. There has been an Experimental School and Obfervatory at Peiin thefe 400 Years, built on the Walls of the City : Father Verbiejl be- ing made Prefident of the Tribunal of Mathematicks in 1669, obtained of the Emperor Cam-hj, that all the European Inftruments, Machines, PHYSICALWORLD. XXI Modelsj Efff. Ihould be added to thofc wiih which it was already furnlfli- cd. (See the Defcription of China by DubalJ.J There has been made there a vaft Colledion of ufeful Experiments and Obfervations^ a Copy of which is depofitcd in the French Academy. XV. The Royal Obfervatory of England was built by CbarlesIL under the The Royal Dtr^ion of Sir J, Moore, four Miles from London, to the Eaftward Obferruoiy upon a high Hill : It will be for ever famous by the immortal Labours JJlitS*^" of Flamjlead, Halley, and Bradley ; Flamftead was put in Poffeffion of this School at Obfervatory in 1676, where, during the Space of 33 Years, he made ^'?°*j'* a prodigious Number of Obfervations contained in his Hiftory of the ftmou. by Heavens: HaUej fuccetded him, and was, without Doubt, the greateft (he Labours Aftronomer England produced ; at the Age of Twenty he went to the ifJliy'^d'* Ifland of St. Helen, to form a Catalogue of the Southern Stars, which Btt^t^ he published in 1679 ; then he went to Dantzick to confer with Hevelius, he travelled aHb through Italy and France for his Improvement; in 1683 he publiflied his Theory of the Variation of the Magnetic Needle ; in 1686 he fuperintended the Impreffiori of the Prineipia Matbematica Phi- lofofia Naturalis, which its immortal Author could not refolve with him- felf to publifli. The fame Year he publilhed his Hiftory of the Trade Winds; in 1698 he received the Command of a Vcffel to traverfe the Atalantic Ocean, and vifit the Englijb Settlements, in order to difcover vrhether the Variation of the Magnetic Needle, found by Experiment, agreed with his Theory, and to attempt new Difcoveries ; he advanced as ht as 52 Degrees South Latitude, where the Ice impeded his further Progrefs ; he vifited the Coaft of Brajil, the Canaries, the Iflands of Cdpe Verde, Barbadoes, fire, and found every where the Variation of the Compafs comformabte to his Theory; in 1701 he was commiflioned to traverfe the Englijb Channel, to obferve the Tides, and to take a Survey of the Coafts ; in 1 708 he vifited the Ports of Triejie and Boccari in the Gulph of Venice, and repaired the firft, accompanied by the chief In- gincer of the Emperor; he publiflied in 1705 the Return of the Comets of which he was the firft Difcovcrer; and we have feen in 1759 the Accompliftiment of his Prediftion ; in 171 3 he was made Secretary of the Royal Society ; he examined the difFerenL^ethods for finding the Longitude at Sea, and proved that thofe whiJi depend on the Obferva- tions of the Moon were the only pradicable ones, and as 'thofe Me- thods required accurate Tables of this Planet, which did not differ from Obfervation more than two Minutes, he fet about reSifying them, hav- ing difcovered that to obtain this Point it was fufficient to determine, every Day during 18 Years, the Place of the Moon by Obfervation, and to know how much the Tables differed from it, the Errors every Period afterwards being the fame, and returning in the fame Order : It was XXII Other Obfer vatories and Experimen tal Schools in £DgUa<l. Thofc of .Edinburgh, The Royal Oblervatory of Paris. Other Ob (crvatories and Expcri mental Schools in France. Of Nurem berg in 1678. Of Leiden in 1690. STSTEM OF THE in 1 722 that this courageous Aftronomcr, in the 65th Year of his Age, undertook this immcnfe Work, and after having completed it, and pub- liflied the Succefs ot his Lab( urs for foretelling accurately the Moon's Place, and deducing the Longitude at Sea; we loft this great Man the 25th of January 1742. Bradley fuccccded him, who inriched Allrorcmy with his Difcoveries and accurate Obfervations. He dc,parre. hi^ Life ihe I3ih of July 1762, in the 70th Year of his Age. M. Majkelne, his Succeflbr, continues his Obfervations witli the moft adive 2^eal and happy Difpofitions. The Royal Obfervatory not being fufficient for all thofc who parfuc the Study of natural Phiiofophy, there has been formed feveral Obfcrva- tories in London and the different Parts of England^ for Fxample, the Obfervatory of Sberburn near Oxford, where the Lord MacUsfieldy late Prefident of the Royal Society, M.HornJby^ &c. have made Experiments and Obfervations for many Years. The Experimental School and Obfervatory of Edinburgh, built by the Subfcription of the Nobility and Gentry of that Kingdom, has been rendered famous by Af * Laurin, The Royal Academy of Sciences de- puted in T747 the King's Aftronomer* LeMonier, to obferve there an «nnulary Eclipfe of the Sun. XVI. The Royal Obfervatory of Parts y the moft fumptuous Monument that ever was confecrated to Aftronomy, was built under the Diredion of the great Colbert, immortal ProteSor of the Arts and Sciences. It is near 200 Feet in Front, 140 from North to South, and 100 in Heighti the Vaults are near eighty Feet deep ; there are alfo feveral others in Paris, and in other Parts of Frjtnce, as that of M. Lemonier at the C^- pucbines of St, Honore, that of Af. Delijle at the Hotel de Cluny, that of M. La Caille at the College of Majarin, that of the Palace of Luxem^ hurgb, that of M de Pouchy in Rue des Pojles, and that of M. Pingre it St, Genevieve j the Obfervatory of Marfeilles which F, Pezenas has ren- dered famous, that of Lyons where F. Beraud made Experiments and Obfervations for a long Time, that of Rowen and Touloufe from which M. Bowin and M. Dulange, M. d^ Auguier fend annually to the Academy a great Nimiber of ufef^and curious Experiments and Obfervations; that of Strajburgb wher Af. Brakenafer has made fome. XVII. The Senate of the Republic of Nuremberg, ereded an Obfervatory in 1678, and put it under the Diredion of Geo, Cbrifiopber Eimmart, Phil, IVurzelban built another in 1692, defcribed in his Book Uraniet Ncrica Bafts, The Adminiftrators of the Unive^fity of Leyden, eftab- liflied in 16909 an Experimental School sind Obfervatory. Frederick I, King of PruJJia, having founded in 1 700, an Academy of Sciences at *, .PHYSICAL WORLD. XXHI Birlin, built an Experimental School, with an Obfcrvatory. The pre- 9^ ^^^ fent King of Pruffia^ added a fupcrb Edifice, where the Academy aftu- "* '^*^' ally holds its Aflfemblies. The Inftitution of Bologna a famous Academy, P^l««ly eftabliihed in 1 709, by the Count of MarftgU^ with the Permiflion of J^j' JlJ'j. Clement XI. has a fine Experimental School and Obfervatory, which Manfredi and Zanotti have rendered famous. There are four Experi- mental Schools, with Obfervatories, at Rome ^ that of BUncbini, that of the Convent oi Ara Cali, that of the Convent of Minerva, and that of Trinite du Mont. There is alfo one at Genoa, founded by the Mar- quis of Salvagi; one at Florence, which Ximenet has rendered famous ; cist zt Milan, ereded in the College of Brera, in 1713. The Supe- riors of the Univerfity of Altorf, in the Territory of Nuremberg, ered- of Akort ed an Experimental School, arid an Obfervatorv, and furnifiied it with >" 1714* all the neceffary Implements. In 1 714, the Landgrave of Hejfe, Cbarlet I. Heir of the States and Talents of the celebrated Landgrave we have al- ready fpoke of, built a new Experimental School and Obfervatory, and put It under the Diredion of Zumback: In 1722, the King of Portugal^ OfUQion^ JobnV, ereded an Experimental School and Obfervatory, in his Palace *° ■7«». at Lijhon ; there is alfo one in the College of St. Antony. The Expe- rimental School and Obfervatory at Peterjbourg, is one of the moft mag- f^^*^?^ nificent in Europe, it is fituated in the Middle of the fuperb Edifice of ,y^."* the Imperial Academy of Peterjbourg, it Is compofed of three Flights of of Utrecht Halls, adapted, for making Experiments and Obfervations, and is 150 "*■'*** Feet high. In 1726, the Magiftrates of the Republic of Utrecbt, built an Experimental School, and an Obfervatory, in which the famous Mujcbembroek made his Experiments and Obfervations. In 1739, the King: of Sweden trt&jtAont at Upfal, and put it under the DireSion ofUpfiJ of Wargentin. In 1740, the Prince of HeJfe Darmfiad, ereSed ano- "* *739- ther at Giejfen, near Marborougb. There are two Experimental Schools and Obfervatorics, at Vienna, where P. Hell^ and F, Liganig, diftinguifli of Vienna; themfelves aftually. There is one at Tyrnaw in Hungary ; one in A- knd, at JVilna, &c. &c. Of Wibui. Such are the renowned Eftablifhments to which we are indebted for our Knowledge of the Syftem of the World, and 'the Improvements it receives every Day ; but there are a great many Branches, which require fuch long Operations, and fo great a Space of Time, that Pofterity will always have new Obfervations and Difcoveries to make. Multum igerunt qui ante nos fuerunt,fed non peregerunt, multum adbuc re flat Ope- ris multumque reftabit ; nee ulli nato pofl mille Sacula pracludetur Occafio aliquid adbuc adjiciendi. (Sxnec. Epif. 64.) XVIII. Thofe great Exahiples of all the civilized Nations of the World,. have at length brought the Noblemen and. Gentlemen of this Country,,, XXIV SYSTEM OF THE to a true Senfe of the Importance of procuring to their Childrtflf thofc Means of Inftrudiont which may prevent their regretting in a more advanced Age, the mif-fpent Time of their Youth ; which is the only Period of Life in which they can apply themfelves with Succefs^ to the Study of Nature : In this happy Age, when the Mind begins to think, ' and the Heart has no Paffions voilent enough to trouble it. Shortly} 4he Faflions and Pleafures of their Age will engrofs their Time» and when the Fire of Youth is abated, and they have paid to the Tunralt of the World the Tribute of their Age and Rank, Ambition will gain the Afcendant. ' And though in a more advanced Age, which will not however be more ripe* they ihould apply themfelves to the Study of the Sciences, their Minds having loft that Flexibility which they had in their youthful Days, it is only by the Dint of Study, they can attain what they might acquire before with the greateft Eafe. Publick To improve therefore the Dawn of their Reafon, to fecure them from €ftibUfli*ain Ignorance, fo common among People of Condition, which expofes them die City of daily to be fcandaloufly impofed upon, to accuftom them early to the DobliD for Habit of thinking and afiing on rational Principles, a School ha« been Yourb^ln^ eftablifhed on the moft approved Plan, where, after having fpent fome evry Bnnch Time in learning Elementary Mathem aticks, they are initiated ^^tMadw *" ^^^ Miftcries of Sublime Geometry, and of the Infnitesimal raaticks pw CALCULATION; from thofe abftraft Truths, they are led to the Dif- fuant to the covery of the Phenomena of Nature, they are taught how to difoeni of ^^^N^ their Caufes, and meafure their Effeds ; from thence they are con* bicmen and duded as far as the Heavens, thofe inunenfe Globes which roll over our Gentlemen Heads with fo much Majefty, Variety and Harmony, letting themfelves of tbefing- ^^ approached ; they are taught how to obferve their Motions, and io- domofire- veftigate the Laws according to which this material World, and all o?Feb^i* Things in it, are fo wifely framed, maintained and preferved. zytfs! "^'^ To relax their Minds after thofe Speculations, they are brought bade to Earth, where, free from all Spirit of Syftem and Refearch of Caufes, they are taught how to contemplate the Wonders of Nature in detai]. But as it prefents an immenfe Field, whoTe whole Extent the greateft Genius cannot compafs, and the Inquiries the moft valuable, and the only worthy of a true Citizen are thofe by which the Good of Society is promoted, they are confined particularly to the Study of what may contribute to the Perfeftion of ufeful Arts, fuch as Agriculture and Commerce, that thus initiated in the true Principles of the dif- ferent Branches of Knowledge fui table to their Rank, having completed their Studies in this School, far from being obliged to forget what they have learned, as hitherto has been the Cafe, thev may be enabled to purfue with Succefs, fuch Inquiries as are beft adapted to their Genius. r r ^ " ' ^ — "^fir PHYSICAL WORLD. XXY PregTifs of tb€ DifcvoerUt relative to the Syflem of tbe VTortd. L X H E firft Views which Philofophcr* had of the Syftcm of the World, rf ' PhTbS^ were no better than thofe of the v ulgar, being the immediate Sugeeftions ^*g a •^ of Senfe; but they correded them; thus the firft Syfttm fuppofed theofWworl4 Earth to be an extended Plane^ and the Center round which the Heaven- ly Bodies revohred. The BaAjhnians from examining the Appearances of Sence were the of the B^bf- firft who difcovered the Earth to be round, and the Sun to be the Cen- loaiut, and ter of the UniYerfe (m) in thefe Points they were followed by Pvtbagoras and «'/y'*«8** his School TIL The true Syflem of the World being difcoirered, it may appear fur- Sizing that the Notion of the Earth's ^ing the Center of the Celeilial otions (hottld generally prevail: for.tho' on « fuperficial Survey it feems to be recommended by its Simplicity, and to fquare exaCtly with the Ap- Ethtu ihte pearances of S^ence, yet on Examination it is found entirely infufEcient to btve bcca. explain the Phenomena, and to account for the Heavenly Motions : This "^^ conftrained Ptolemy and his followers to incumber and embarrafs the Hea-|^e£artli Tens with a Number of Circles and Epicycles equally arduous to be con- to be tt reft 4 ceived and employed, for nothing fo difficult as to fubftitute Error in thel^^"^"^ - room of Truth* i-toiomy. Probably the Influence of Arifiotle^ Authority^ whofe Writings in Ptoh^ inf% Time were held in the higheft Efteem, and confidered as the Standard of Troth, lead this Philofopher into Error : But why did not Ariftotle de- clare in favour of the true Syftem, which he knew, fmce he en- deavoured to overthrow it: this Reflexion is fufficiently mortifying to the Pride of tbe Human Underftanding, whatever was the Caufe, thus much is certainy that the Ptdomaic Syftem generally prevailed to the Time of O- pomieusk IV. This great Man revived the ancient Syftem of the Saiyhnlanx, and of Co^eraicdi] Pffb^orof which he confirmed by fo many Arguments aiod Difcoveries revive« che that Error could no longer maintain its Ground againft the Fvidence of J^'^^ jf^;| Demonftration ; thus the Sun was reinftated hvCoperhicus in the Center of thnoru. * the World, or to fpeak more exa£tly, in the Center of our Planetary Sj&eau (n) NawTov b his Botfk ps Systkmatk Mvitdi fttiribvtei ihli Opiaida to Noma Poapilius, tad iayi, (Pa^ 1 .) it wm to reprereot the Sua io the Ceater of tbe Celeftiri Oihict that Noma caufod t rouid Temple to be built ia hQoOw of Tefti, the Goddefc of Fire « the Middle ^ vhkh s petpetml Fire wst preferred. 1 XXVI SYSTEM OF THE V. Syftem of ^^^ Copcmican Syftem eafily accounts for all the Celcftial Phenomena^ TithoBraheand tho' Obfervation and Argument are equally favourable to it, yet 7/Vitf- Brabe an eminent Philofopher of that Age refufed his aflent to th^ Evi- dence of thefe Difcoveries, whether deluded by an ill- formed Experiment* (b) or carried away by the Vanity of making a new Syftem, he compofcd one which fteers a middle Courfe between ihofc of Phlomy and Copernicus \ he fuppofed the Earth to be at reft and the other Planets which move round the Sun, to revolve with him round the Earth, in the Space of 24 Hours ; thus retaining the moft exceptionable Part of Ptolomy's Syf- tem, viz. the inconceivable Rapidity with which tht primum Mobile is fuppofed to revolve, from whence we may learn into what dangerous Errors the mif- application of Genius may lead us. . . The Difco- Tho^ Tycbo erred in the Manner he made the Celeftial Bodies move, \«"«» ^^^ yet he contributed very much to the Progrefs of the Difcoveries relative to Syft«n of* *^^ Syftem of the World, by the Accuracy and long Series of his Obfcrva- the World, tions. He determined the Pofition of a vaft Number of Stars to a Degree of hI!^T*^df exaSnefs unknown before ; he difcovered the Refraction of the Atmofphere, ^ ^ °' by which the Celeftial Phenomena are fo much influenced ; he was the fird who proved from the Parallax of the Comets, that they afcend above tho Moon ; he was the firft who obferved what is called the MoorCs variation \ and in fine, it is from his Obfervations on the Motions of the Planets, that Kepler who refided with him, near Prague, during the laft Years of his Life, deduced his admirable Theory of the Motions of the Heavenly Bo- dies. yi. How mvch Copernicus undoubtedly rendered important Servicts to Human Reafon remaiDcd to by rc-eftablifliing the true Syftem of the World : It was already a great ]rd i^«r^o^ P^***^ gained that Human Vanity condefcended ta place the Eirth in the Num- peroicBt. her of the fimpte Planets; but much ftill remained to be difcovered : neither the Forms of the Planetary OrbitSj^ nor the Laws by which their Motions are regulated, were known \ for thefe important Difcoveries we are in- debted to Kepler. (b) It WIS objedled to Coperaictts, that the Motion of the Earth would produce EftA* which did not take Place ; that, for Example, if the EartK moved, a Stone dropped from die Top of a Tower, ought not to &11 at the Foot of it, becauCe the Earth moved during the Tidi •f the Stone*s defcent^ that notwithftandiog it falls at the Foot of the Tower. Coriivicoi replied, that the Situation of the Earth with rel)>e£l to Bodies that fait on its Surface wai tbe fame as that of a Ship in Motion, with refpefk to Bodies that are made to fall in \i\^ afTerted, that. a Stone let hW from the Top of the Mad of a Veflel in Motion, woold fiUl i* the Foot of it. This Experhnent which is now inconteflible was then ill-made, and wm CheGn^ «r the Pretext which made Trcho reluf^ his ai&nt to (he PifcQveries of CopenucQi. r PHYSICAL WORLD. XXVII This eminent Philofophcr found out, that the Notion which generally pre- ^ bailed before his time, that the Planets revolved in circular Orbits, was cf-ofKe^eV Toncous; and he difcovered, by the means of Ticho's Obfervations, that tbc el pticiif the Planets move in ElHpfes, the Sun refidtng in one of the Foci : and ihat ^^^ oibiti. ihty move over the different Parts of their Orbit, with different Velocities, fo ^][i^'**Jf that the Area defcribed by a Planet, that is, the Space included between theiheftrcmitad firaight lines drawn from the Sun to any two Places of the Planet, is always ^« '«»«•• proportional to the time which the Planet employs to pafs from one to the other. ' Some years afterwards, comparing the Times of the Revolutions of the j^^j^^^ different Planets about the Sun, with their different Diftances from him, he which Tub- found that the Planets which are placed the far theft froni the Sun to move fift» between iloweft, and examining whether this Proportion was that of their Difttnces,j|^"^*^ he difcovered after many Trials, in the Year i6i 8, that the Times ofthe diftta* their Revolutions were as the Square Roots of the Cubes of their mean ^^ Diftances from the Sun. vn. Kepler not only difcovered thefe two Laws, which retain his Name, and which regulate the Motions of all the Planets, and the Curve they defcribe^ but had alfo fome Notion of the Force which makes them defcrtbe thia Curve; in the Preface to his Commentaries on the Planet Mars, we difcover the firft Hints of the attradive Power ; he even goes fo fiir as to fay, that the Flux and Reflux of the Sea, arifes from the gravitv of the Waters towards the Moon: but he did not d^luce from thb Principle what might be expeded from his Genius and indefatigable Induffay. For in his Epitome of Aftrono* niy(c) he propofes a phyfical Account of the planetary Motions from quite different Principles; and m this fame Book of the Planet Mars, he fuppofes in tiie Planets a iriendly and a hoftile Hemifphere, that the Sun attrafis the one jindrepek the other, the friendly Hemifphere being turned to the Sun in the nianets defcent to its Perhihelium, and the Hoftile in its Recefs. VIIL. The Attradion of the Celeftial Bodies was fugeefted much more clearly ly M* Hook, in his Treatife on the Motion of the Earth, printed m the Year 1674, twdve Years before the Principia appeared. Tbe/e are bis Wordsf Page 27» ^^ I fliall explain hereafter a Syftem of the World, diffierent in ma* M ny Particulars from any yet known, anfwering in all Things to the com* «< roon Rules (^Mechanical Motions. This depends on the diree following tf Sufpofitions, ^ (c) Sse Gxtfoiy, Bosk i^ h|« #>; XXVIII SYSTEM OF THE ''*S!!r ^*^ ^^ ^^^^ ^^^ cddlial Bodies, whatever, have an Attradton, orgravitatti^ ^rnuH*!^ " Power towards their own Centers, whereby they attrad, not oiJy their irsai^A* ** o^n Parts and keep them from flying from them, as we may obferve th« '* Earth to do, but that they do alfo attrad all the other celeftial Bodies that ** are within the Sphere ot their ASivity ; and conrequently not only the '' Sun and the Moon have an Influence upon the Body and Motion of the- ** Earth, and the Earth on the Sun and Moon, but alfo* that Mercury, Ve« ** nus, Mars, Jupiter aud Saturn, by their attradive Powers, have a confi* ^* derable Influence upon the Motion of the Earth, as in the fame Manner '' the correfponding attradive Power of the Earth hath a coniiderable inflo-^ ** ence upon the Motion of the Planets/' '' ad That all Bodies whatever that are put into adired and (imple Motion,^ ** will fo continue to move forward in a ftreight Line, till they are by fome ^* other effedual Power defleded and turned into a Motion, defcribing a Cir«^ ** cle, an Ellipfe, or (bme other more compounded Curve Line."^ ** ^d That thefe attra&ive Powers are fo much the more powerful in ope^ << rating, by how much the nearer the Body wrought upon is to their own *« Center." ** Thefe feveral Degrees I have not yet experimentally verified, but it is. ** a Notion which if fully profecuted as it ought to be, will mightily affift the ** Allronomer to reduce all the celeftial Motions to a certain Rule, which i «< doubt will never be done true without it. He th^tt underftands the Na-- " tureof the circular Pendulum and circular Motion, will eafily underftand << the whole Ground of this Principle, and know where to find Diredions << in Nature for the true ftating thereof. This I only hint at prefent to fuck <* as have a Capacity and Opportunity of profecutiog this Enquiry, &c/* IX. We are not to ima^ne, that this Hint thrown out cafually by Hpoi, de» trads from the Glory of Niwton, who even took Care to make Mention oC it in his Book tU S^tmati mundi (d)» the Example of H«oi and KepUr makea us perceive the wide Difference between having a Notion of the Truth, aixf being able to eftablifli it by irrefragable Demodlration; it alfo fliews us how little the greateft Sagacity can penetrate into the Laws and G^nftitution o£ Nature^ without the Aid and Diredion of Geome^y. X. Scrtate ao- Kephff w^o made (iich important Difcoveries» whilft he fbltow^ thu qn« tkttsoflUperring Guide, affords us a convincing Proof of the Errors into which the ler. brighteft Genius mav be feduced, by indulging the pleafing Vanity of in* venting Syftems j who could believe, for Inftancci.uuit fuch aMu couldt MflA4itioa tf 1)31. ' P H YSICAL WORLD. XXtt idopt the wild Fancies and whimfical Reveries of the Pythagoreans, eon- €erning Numbers: yet he thought that the Number and intenral of the pri* mary rlanets bore fome Relation to the five regular Solids of Elementary Ge- ometry (e), imagining that a Cube infcribed in the Sphere of Saturn would touch the Orb of Jupiter with its fiJt Planes, and that the other four Kgular Solids, in like Manner, fitted the Intervals that are betwixt the Spheres of the other Planets: afterwards on difcovering that this Hypothefis did not iquare with- the Diflances of the Planets, he fancied that the celeftial Moti- M38 are performed in Proportions correfponding with thofe^ according to which a Cord is divided in order to produce the Tones which compofe the Odave in Mufic (0 ; Kepter having fcnt to Ticho a Copy of the Work, in which he attempted' to eftablifh thofe Revcries.Ticho recommended to him, in his An- Wife cmw- fwer(g), to relinquifli all Speculations deduced from firft Principles, all ^^'%^^J^ ibning a Priori, and rather fiudy to efbbHfh hisRefearches on the fure and *^ firfld Ground of Obfervation. The great Hugbens himfelf (h) beHeved that the fturth Satellite of Saturn, ^'»««fica which retains his Name, making up with our Moon and the f6ur Satellites of HMhce^ Jupiter fix fecundary Planets, the Numbenof the Planets was complete, and It was labour loft to attempt to difcover any more,, becaufc the principle Planets are alfo fix in Number, and the Number Six is a perfeS rlumber^ as being equal to the Sum of its aliquot Parts, i> % and 3. XL It was by never deviating from the moft profound Geometry, that NeW'^ Un dffcovered the Proportion in which Gravity ads, and that in his Hands ihe Principle of which Kepler and Hook had only fome faint Notion, became the Source of the moft admirable and unhoped for Difcoveries. . AdYanisgcc One of theCaofet which prevented Kepler from applying the Principle ^f^lIpUr •F Attradion to explain the Phoenomcna of Nature wirh Succefs, was hisin hit time* fenorance of the true Laws of Motion. Newton had tho Advantage over**»«»*»«®nf®f Kepler of profiting of the Laws ofMotion, eftabliftied by Hughens, which £221^ n1" he has carried to fo great a Height in his Mathematical Principles of Natu-derftood. stdPhilofopby. xn, . The Maihemalfcal Principle of Natural- Phifefophy confift of three ^^^gjjj^ Books, befides the Definitions, the Laws of Motion and their Corollaries ; tlie fi^ Book-is compoted of fourteen Sedions^ the fecond contuns nine, (^) M]rfteriiiin Cofinogi^iciBB* • (f). MyOvivn CoTmogniphiaini^ (g) Uti fnfpeofit fpeGaUuiooilms ^ priori ^eTcei^MtiNi tmiirem pettw «d «lbf«rvitiOfk€t ^^M finml oUcrcbtt confidenndM idjiccrcai «(it U Kepler who fpeaks) oou» in fcauidtm< «iitMQcm os](fteru cefmognphsd XXX SYSTEM OV THE and the third, the Application of the two firft to the Exjrfication of the Phoenomena of theSyftemof the World. XIII The Princtp'm commence with eight Definitions 9 Newton (hews in the ^fiuiiom. t^Q fir (I ho^ the^iwtn///; of Matter and tbe ^antity of Motion fliould be meafurcd ; he defines in the third, the Fis intertt^e, or refitting Force,whKh all Matter is endued with ; he explains in the fourth what is to be undcrftood by a^ive Force ; he defines in the fifth 4be centripetal Force, and lays down •in the fixth, fevenih and eighth the Manner of meafuring its abfolute ^anhtj/f its motrix ^antity, ^nd itr accelarative ^antity ; atlerwardsheclUblilhcs the three following Laws of Motion. XIV. rtwiofmoift. That a Body always per reveres of iifelf, in its State of Reft, or ot •sioji. uniform Motion in a ftraipht Line. ad. That the change of Morion, is proportional to the Force imprcff"* and is produced in the ftraight Line in which that Force a&s. 3d. That Adion and Readion are always equal with oppofite Di« regions. XV. t^e'^afcai! Netvton having explained ihofe Laws, and deduced from them fcvenl oa coottin* Corollaries, commences his firft Book with eleven Lemmas, which com- the princi- pofe the firft Seftion, he unfolds in thofe eleven Lemma; his Method of ^MRml^^' Prime and ultimate Ratio f ; this Method is the Foundation of infinilcfljnttl geometry Geometry, and by its Affiftance, this Geometry is rendered as certain 11 that of the Ancients, the other 13 The thirteen other Sedions of the firft Book of the Princlpia, are employ lofiriont'^on^*' in demonftrating general Propofitions on the Motion of bodies, Abftrac- £e im!tion ^tng from the Species of thefe Bodies and of the Medium in which tbef ^fbodiet. move. It is In this firft Book that Newton unfolds all his Theorv of the graviu* tion of the celeftial Bodies, but does not confine himfelf to examine tbe Queftions relative to it j he has rendered his Solutions general^ and has givett a great Number of Applications of thole Solutions. XVI. h^tftf In the fecond Book, N^ton treaU of the Motion of Bodies in refilbS cfae motioiior Mediums. bodiei io re- fhi, fecond Book which contains a very profound Theory of Fluids, mm diral "** ^^ ^^^ Motion of Bodies which are immeiied in them^ feenois to have beefl to deftined to over throw the S vfliem of Vortices,though it is only m the Scholi- ^J^***^^;^ urn of the laft PropoCtion^that Newton openly attacks Dejcgrtitp and ffVV^ ^Di^m^^ ^ celeftial Motions are sot produced by Vorticei. r PHYSICAL WORLD. XXXI XVIL ^ In fine, the third Book, of the Principia treats of the Sy aem of the World ; ™j2,^Jf In this Book, Newton applies the Propcfltions of the X^o firft : inihcr^ftem .this Application we (hall endeavour to follow ^(rw/on, and point out theofthtwwUL. Connexion of his Principles, and (hew how naturally they unravel the Me« chanifm of the Univerfe. xvin. The Term, Attraftion, I" employ in the Senfe inVhich Newton has defined Whtt iV it, underftanding by it nothing more than that Force, by which Bodies tend^^^*>y^^* towards a Center, without pretending to aflTign the Caufe of this Tendency. ^ *"^^' Principal Phenomena of the SyJIem of tbe World. X HE Knowledge of the Difpofition and Motions of the Celeftlal Bo- dies muft precede a juft Enquiry into their Caufes. It will not therefore appear unneceflary to prepare our Readers by a fuccinddefcription of our planetary Syflem for our Account of the manner ^^ti;/0/i demonfirates thepowers which govern the Celeflial Motions and produce their mutual Inffuences. This De- fcriptionmuft neceffarily comprise fome Truths, difcovercd bythatilluflrious Philofopber^ the Manner he attained them with be defcribed in the Sequel. The celeftlal: Bodies that compofe our plknetary Syftcm, are divided into of "the celt ' Primary Planets^ that is, thofe which revolve round the Sun, as their Gf»/^r '*** *>«*»«• and Secondary Planet i^ othcrwife, called Satellitety which revolve round their 2^y['^ rcfpeaive Primaries as Centers: There are fix Primary Planets whofc into priod- Names and Charadera are as follows,^ i«i •^ <ccoa i# JL£ ^rypltnet*. 9 Mercury^ <^, ' ^ Fenuff Kamei tod ^ Tbe Earthy, •?!j:'^"' o iWtfr/, fiJptJpMaet*. 3. Jupiter^, 1) Saturn*. In eniimcratihg tBe Primary Planets, wc follow the Order of their Dif-^^^J[7 tances from the Sun, commencing with thofe which are neareft te him. thtt iitve The Earth, Jupiter, and Saturn, are the only Planets which have been^«**»^«^. difcovered to be attended by Secondaries : TheEafthhas only one Satellite, ^^J^^^° namely, the Moon ; Jupiter, has four, and Saturn five, exclufive of his Ring, ftiai bodies of fo that our Planetary Syftem is compofed of eighteen celeftial Bodies, in»<»or pUneu. eluding the Sun and the Ring of Saturn. * , . ' SecooTdi- ^''' vifioa of the The Primary Planets are divided into fuperior and inferior Planets, the pUaeu into iflferior Phntti are thofe which are nearer the Sun than the Earth is j thcfe^^^VJ*^ '■*• XXXn S V S T E M OF THE ^ which tre are Merciiry and Venus ; the Orbit (a) of Venus includes that of Mefcnfy ^^''^'iLj' *"^ •^'^ ^^* ^""» *"^ ^^^ Orbit of the Earth is exterior to thofc of Mcrcuqr what 't^T ^nd of Ve nus^ and inclofesthem and the Sunalfo. tmiofe- This Order is difcovered, by Venus aiid Mercury fometimes appearing to "•"'• be interpofed between the Sun and us, which could never happen unlcfs how chi« or-^hefe Planets revolved nearer the Sun than the Earth, and it is very percciv* ier hat beett able that Venus recedes farther from the Sun than Mercury ^oes, andam- difcovcrcd. fgquently its Orbit includes that of Mercury. which trt *^^^ fupcrior Planets are thofe which are inoVe diftaift from the Sun than theVuperior ^^^ Earth 18, thefe are three in Number, Mars^ Jupiter %xA Saturn \ we pitiMu and know that the Or'bitsof thefe Planets indole the Orbit of the Earth, be* Trrli *****' caufe the Earth is fometimes interpofed between thciti and the Sun. ^et".**" The Orbit of Mars incWes that of the fearth, the Orbit cS Jupiter thst of Mars, and the Orbit <rf Saturn that of Jupiter f^ fo that of the three fu- perior Planets Saturn is the remoteft from the Earth, and Mars is the neareft. hdw ft hat This Arntingement b difcovered by thofe Planets which are nearer the beeo difco- Earth (b) fometimes coming between the Eye and the Remoter, and intcrr vcred, ccpting ihem from our View. IV. AU tfaePlanets are opaque Bodies ; thb apoears of Venus and Mercury, Th« plaoett becaufe when they pafs between us and the oun, they refemble black Spots -flre opi^ae traverfing his Body, aud aflTiime all thofe various Appearances which are *^^ called Phafcs) that is, the Quantity of their Illumination depends on their Pofition in refped to the Sun and us. For the fame Reafon, fmce Mars has Pbafes we Infer his Opapity, and the fame Conduiion is extended to Jupiter and Saturn, becaufe their Sate- lites do not appear illuminated while their Primaries are between them* and the Sun which proves that that Hemifphere of thofe Planets whidi is tunn The plaaeti ^ ^^om the Sun is opaque : Laftly, we know that the Planets are fphcri' w«rphcrical cal Bodies, becaufe, whatever be their Pofition, in refped of us, their Sur« face always appears to be terminated b]^ a Curve. We conclude that the Earth is fpherical, becaufe in Eclipfes her Siaddw, always appears to be bounded by a Curve, and when a Ship faib out of fight, it gradually difappears, firft the Hulk, next the Sails, and laftly the Maft, finking to the Eye and vaniihing, and moreover, \\ the Earth was an extend* cd Plane, Navigation would luive difcover«I its Limits and Bbandaries the contrary of which is proved by manv Voyagers, fuch as Drake, ForbiA# and Lord Anfon, who have failed rouno the World. (a) Ori>it b tlieCwfewUdia^aactdtlMiw itimltiaf riwd (he Bodf which Aim It ai a Center. (b) Wolf *« ElcmMs «f AOrgmar* PHYSICAL WO R LT). XXXIII V. All that vre know therefore concerning the primary Planets, proves that J***'^,'*^^ they are opaque, folid and fpherical Bodies. tUof'thc The Sun appears to be a Body of a Nature entirely different from the Pla- famcMtorc, nets ; we know not whether the Parts of which it is compofed be folid or ^ .fluid ; all that we Can dtfcover is, that thofe Parts emit light & heat, and burn ue that the when condenfed and aflembied in fufiicient Quantity ; hence we may probabl; tli« 6«o iira •conclude, that the Sun is a Globe of Firerefembling teire(lrialFire,*fince thc«^*^*^^** Effeds produced by this<and the folar Rays, are exskQly the fame. VI. All the celedial Bodies compleat their Revolutions round the Sun in Ellip- in ^htt ^ fes (c)y more or lefs excentic, the Sun refiding in the common Focus of all curve thece their Orbits j hence the Planets in their Revolutions fometimes approach J*^"|J^^^t' nearer, and fometimes recede farther from the Sun ; a right Line pafling bout the fun. through the Sun and terminating in the two Points of the Orbit of a Planet, ^^^ {, ^^ . which are neareft and remoteft from the Sun, is called the Line of the Apfides^ line of the the Point of the Orbit which is neftreft the .Sun is called the PeriUlium \ 'pfi^J" ^^ itnd the Point of the Orbit which is remoteft from the Sun is called the |[^ pj^eli Apbeliunt. vin. The primary Planets in their Revolutions round theSon, carry alfo their i^ ^htc di- Satellites, which at the fame Time revolve round them as their Centers. itAionthe All fhcfe Revolutions are performed in a direaion from Weft to Eaft (d),Pj,"^' ''" There appear from Time to Time Stars that move in all DircQions, and r ' -with aftonifliing Rapidity, when ibey are (ufficiently near to be vifible, ihtkj^ ^®* are called Comets. Wehave notyetcolle8ed Obfervations fufficienttadetermjne their Num- ber, all that we know concerning them, and *tis but lately that the Dif- Icovery has been made; is that they are Planets revolving round the Sun like '^* f **"**' the other Bodies of our Sy ftem, and that they dcfcribe Ellipfes fo very cxcen- "* ^ "*^* trie as to be vifible only while they are roovieg -over a very fnuU Part ot * their Orbit. vn. All the Plaiiets'in their Revobtionsrouvd the 'Sun, dbCsrve the <wo Laws Theplmets M>f K^er. •o'^ ^^^* ObferTfttions evince, that the Comets obferve the firft bf thfefeLaws,^^^[''J[l5j^p *jian*e!yi that whi*h makes the celeilral Bodies --(e) defcfibe eqcral Areas in e-kr (0 A Species bf Curve, which is (he fame "with what Is commonly called an Oval, the foci are ' the points in which Gardeners ia, their pegs in order to trace this curve of which they make t ' frequent nfe. (d) The SpeAator is fuppofed to be placed on the Earth. (e) By (he Word Area, in general is underftood ^ Stkrface, here it (ignifies the Space sfi- ciadcd between tv/b Uaf » dr'awo irom tb« Ceoier co iwo foists wtier e the Pi«D9t i« fbiuzdi XXXWr SYSTEM OF THE qual Times ; and in the fcqoel it wifl be flmwir, diat all the Obfemtions that have hitherto been niadc, concerning their Motions, render it highly proba- ble that they arc regulated by the fecond Law, that is, that ihar pcnodic (i^ Times are in the fefquiplicatc ratio of thdr mean Diftances. VIII. . f^fpMiht Admitting thefc two Laws of Krplrr, confirmed by all aflronoroical Ob- T*^* /^ fcrvations, from them we may derive fcveral convincing Proofs of the Mo- • •^ tion of the Earth, a Pobl which had been fo long conteftcd ; for fuppofing the Earth to be the Center of the Ccleftial Motions, thefc two Laws afc ' not obferved ; the Planets do not defcribc Areas proportional to the 1 im« around the Earth, and the periodic Times of the Sun and «]f Moon, tor inftance, round this Planet, arc not as the Square Roots of the Cubes Gttnw mean Diftances from the Earth ; for the periodic Time of the Sun aroundiDc Earth, being nearly thirteen Times greater than that of the Moon, il$ i^"- tance from the Earth would be, according to Kepler's Rule, bctwcenfavc and fix Times greater than that of the Moon, but ObfenrationsdemonlWK, that this Diftance is about four- hundred Times greater, therefore, admitn^ the Laws of Kepler, the Earth is not the Center of the celcftwl i^*- volutions. ^ , , p.,,fg The centripetal Force(g) which Newton has demonftrated.to.be the UJ^ of the Revolutions of the Planets renders the Carve they defcribe »«^"° r" Center concave (h) towards it, fince this Force is exerted in drawing tne off from the tangent (i) ; now the Orbits of Mercury and Venu^ m opj Parts, arc convex to the Earth ; of confequence, the inferior Planets not revolve round the Earth. . ^ The fame may eafily be proved of the fuperior Planets j forthetc ♦hofc Areas trc proport'tonal to the Times, that is, they art greater Or lefs, « ^^ ^*"** which they are defcribed are longer or Ihorter. , , ^^^ (f ) Periodical Time, is the Time that a Planet employs in corapleating ita Revolution m iW ' An Example, of Sefqaiplicate Rauo wilf render it more intelligible than a Defini"<»» ^^ then the mean Diftance of Mercury from the Son, to be 4, that of Venus 9, ^ P* ^^ Time of Mercvy 40 Days, and let the periodical Time of Venns be required, ^^^'^^^^^^^ firft Numberi 4 and p, there wUl refult 64 And 7 2p ; afterwards extraaing the ^'^f^Tt^ tbefe two Numbers, there will be found 8 for that of the firft, and ay for that of the ^^^^^ | by th« Rule of three you will hive 8 : 2 7 : : 40 : 1 3 $> That is the Square-Root of the ^^^y mean Diftance of Mercury from the Sun, is .to the Square Root of the Cub^ •f the mean p» ' Venns from the Sun, as the periodic Time of Mercury round the Sun is to the periodic ** . ^,t of Venus roun<|. the Siip,which is found to be 135,. accordipg to the Suppofitions w<iic been made, and this is what is called Sefquiplicate Ratio. . ^ , ^^ (g) The Word Cemthfetal Force carries its Definition along with it* /®^ *' ^ ■o more than that Force which makes a Body tend to a Center. ^^. (h) The two Sides of the Cryftal of a Watch may ferve to expUia ihofe ^. j^. CAVE and Convex^ the Side exterior to the Watch is convex, and that wbicb i' Side of the Dial-plate is concave. (i) A Tangent is t right LiM which touches % Corvti without cutting it. PHYSICAL WORLD, XXXV Ametimei observed to heJire^(k), {omttimti JlMtionarf, and afterwards retrograde i all thofe Irregularities are only apparent and would vanilh if the ^rth was the Center around which the heavenly Bodies revolved^ for none of thefe Appearances would be obferved by a Spedator placed in the Sun, fince they refult only from the Motion of the )£arth in its Orbit combined with the Motion of thofe planets in their refpeflive Orbits ; from hence wc may fee the Reafon why the Sun and the Moon are the only heavenly Bodies that appear always dired; tor as the Sun defcribes noOrbrt^ its Motion can* not be combined with that of the Earth, and as the Earth is the Center of the Moon's Motion, tons (he fliould always appear direft; as would all the Planets to a Spedator placed in the Sun. When Copernicus firft propofed his Syftem, an Objc6Hon was raifed againft it, taken from the Planet Venus by fome who alledged, that if that Objeftioft Planet revolved round the Sun ihe (houid appear to have Phafcs as the Moon, "*«*f «® c» to which Copernicus anfwered, if your Eyes were fufficiently acute youJ^^J.^^^^ would adually obferve fuch Phafes, and that perhaps in Time fome Art maypiaaetveaae be difcovered fo to improve and enlarge the vifual Powers, as to render thofe Phafes perceivable: This Prcdiaion of Copernicus was firft verified byjlj^i^"^^^ Galileo, and every Dtfcovery that has been made fince on the Motion ohioa the heavenly Bodies has confirmed it. IX- The Planes (I) of the Orbits of all the Planets interfeA in right Lines pafling through the center of the Sun, fo that a Spedator placed in the Center of the^"^' f ^ Sun would be in the Planes of aU thofe Orbits. Ort>iu imer The Right Line, which is the common Sedion of the Plane of each Or- feet bit, with the Plane of the Elcliptic, that is, the Plane in which the Earth v^hat h nm moves, is called the Une of the nodes of that Orbit, and the extreme Points^crftood by of this Seaion, are called the Nodes of that Orbit. Jj^ j|^^«^«e The Quantities of the Inclination of the Planes of tlve different Orb!ts,the m^ with the Plane o{ the Ecliptic, are as follows, the Plane of the Orbit of^' ^ orbit Saturn is inclined to the Plane of the Ecliptic in an Angle of 2<i f, that of. ^ . Jupiter Id t« that ^f Mars in an angle fome what lefs than id, that of Venus o/tbc'oT fome what more than ^^ \, and that of Mercury about ^d. bits to the ^ Ecliptic The Orbits of the primary Planets being Ellipfea, having the Sun in one of their Foci, all thefe Orbits are confequently ezcentric, and are more or lefsfo,according to theDiftance between their Centers and the Point where the Sun is placed. (k) A Plinet it fatd to be DimtCT when it appears to move tccordiag to the Order of the Signs, -that is, froai Aries to Tanros, from Tavrvs to Gemini, &c. which is alfo faid to move ja conTeqnentia, it is ftatiooary when it appears to correfpond for fome Time to the fame Poiou of the Heavens, and in £ne it is RiTJkooaADa when it appears to move contrary to the Order of the Signs, which is alfo faid to move in Anteccdentia, that is, from Gemini to Taums,^ from Tanms to Aries, &c. (I) Tkf phuie «f the Orbit of a Fiaiist if the fwfiice oa which it isfiippoicd fm990. XXXVl yrSTEM OF THE' •xccotricitf Theeicentrictty of aU tboTe Orbits have been oKtiiired, indhDve ban ILr^tt l«mi^^"'^ ^' foHows, in dccimtl Parts of the fcmidiimrter of the Earth's orbit, ditmcicn fuppofed to be (kTidei) iiilo 100,0^^ Parts, •fiht ttrth That of Sanim^ $4207 Patts* That of Jupiter, 25058 Thtt of Mars, 141 15 That of the Harib, 469a Th*t of Vtnos, 500 And in fine, that of Mercury, 8149 Psrt$. The excentricity oi the Planets mcafured in decimal Parts of the femidi* o*MhV'X' ameter ot their Orbits, fuppofed to bt divided into 100,000 Parts, sie nc(» io femias fotloWS, .'itrntifrtoi rhut of Saturn, 5683 Psiti. urn^iftit »j-,^^^^^f Jupiter, 4822 That of Mars, 9263 That of the Earth, 5700 That of Venus, 694 Thst of Mercury, 21000 Fwls Whence it appears that the Excenlrictty of Mercnry is aknoft infenfiUc. XL Proportion The Planets are of different Maenitodes; of the Earth alone we know the »r ihf (liftahfolute Diameter, becaufe this Planet is the only one whofaCircumfierencc j^**^*^^^,^ admits of adual Menfuration, but the relative Magnitudes of the Diame- *ters of the other Planets have becndifcoyered, and the Diameter of the Sua being taken for a common Mcafure, and fuppofed to be dividcdinto 1000 Parts: That of Saturn is 137 That of Topiter 181 That of Mars 6 That of the Earth 7 That of Venus 12 That of Mercury 4 Hence we fee that Mercury is the leaftof all the PlaoeCs» for Spheres are as the Cubes of their Diameters. xn. ^ The Pknets are phced at different Diftancaa from the Siob taking the ^ ^il^, Diftance of the Earth from the Sun for a common Meafuro, and fuppofinff firfDthcfaott divided inta ioo/KX> Parts^ the mean Diftnacea of the Pfanela are as follows. That of Mercury is 38710 That of Venus 7333 That of the Earth loooo That of Mars 1 5 2369 That of Jupiter jaoi loi In fine^ that of Salona. 9$38po PHYSICAL WORLD. XXXVII The mean Diftancesof tjic Pun and the Planets from the Earth, have al- Dj^^^y^,^ fo been computed in Semidlametcrs of the Earth; the mean Diftances of thcti^ pUoct^ Son, Mercury and Venus from the Earth are nearly equal> and amount to^rom tbt aaooo Semidiametcrs of the Earth, thai of Mars 1533500, that of Jupiter*'"** 1 15000, and that of Saturn 21 0000. XIII. The Times of the Revolutions of the Planets round the Sun, are lefs in Periodic Proportion of their Proximity, thus Mercury the neareft revolves in 87 Days,**"** ^^ ^^ Venus next in Order revolves in 224, the Earth in 565, Mars in 686, Jupi-'tJ^^****^ terio 4332, and Saiurn the remotcil from the Sun in 10759, the whole in round Nuspbers. XIV. The Plancis, befides their Motion of Tranflation round the Sun, have a-Rotitlop of Bother Motion Qf Rotation round their Axis, called their Dt urnal RevolutionJht phatu We only know, the diurnal Revolution of the Sun and of four Planets, Mctot em namely of the Earth, Mars, Jupiter and Venus ; this Revolution has beenP??^*^ ^? difcovered by Means of the Spots obferved on their Difcs, (m) and which *'*'^**'^*' " facceiEvely appear and vanifli 5 Mars, Jupiter and Venus having Spots on Inwhtcplt their Surface, by the riegular Return and fucccffive Difappearance of the fame "".• '*»'»• ^ Spots it has been foi^nd, that thefe Planets turn round their Axes,and in what^*^^" Time they qompteat their Rotation; thus it has been obferved, that Marsceived makes his Rotation in 23!^. aook and Jupiter in 9!^. ^Gm, Aftronomers are not agreed about the Time m which Venus revolves loceititnde round its Axis ; moft fuppofe the Time of rotation to be abput 23 h. But J^'jljj"*?*'*' Sign. Btanchini who obferved the Motions of this Planet with particular iahe'iSLr Attention, thinks fl)e employs 24 Days in turning round; but a? he was ti<» of ve compelled to remove his Inftniments durine the Time he was obferving,**** ao Houfe having intercepted Venus from his View ; and as he loft an Hour m this Operation, 'tis probable that th^ Spot he w:as oblerving during this Interval changed its Appearance; however this be bis authority in Aftrono- intcal Matters deferves we Ihould fufpend our Judgment till more accurate Obfervations have diicided the Point. M. de la Hire obferved with a Telefcope 16 Feet long. Mountains in Venus higher than thpfe of the Moon. The extraordinary brightnefs of Mercury arifing from his proximity toTheroteti6n . the Sun, prevcnta our difcovcring by Pbfervation i\s Rotatiqp; ^nd Satqfn^^^'^'y is too remote to have his Spots obferved. urn**cwinot In the Year 17 15 Cqffini obferved with a Telefcope 118 Feet long ;bcdifcovcr three Belts in Saturn refembling thofe obferved in Jupiter, but probably*^ !*y^ t}ioie Obiervatioas could not be purfued with accuracy fuiScient to con-^b/^'^ cUide the Rsotation of Saturn abput its JUis» («) By the Difk of •Pkoet U ttiderftood Uut Part of its forface whidi ii vifible (0 w, . XXXVIIl SYSTEM OF TftE bat tailogy As McrcarjT and Saturn are fubjed to the fame Laws that dired the tToScfSte Courfes of the other Planets, and aj far as has been difcovered appear tfMt cbofe to be Bodies of the fame Nature, Analogy authorizes us to conclude plcjMtf re- that they alio revolve, round their Axes ; and perhaps future Aftronomers AflrAjti^ may be able to obferve this Motion, and to determine its Period, XV. There appear from Time to Time Spdts upon the Sun, which have ferved to difcover that it has a rotatory Motion about its Axis. How the ro It was long after the Difcovery of thofe Spots^ before Aftronoiliers couM ^*«<«^^ obferve any, fufficiently durable and permanent, to'enable them to determine iu^iithu the Time of his Revolution. Keill in the 5th Lefture of his AArondmy» been difco relateSrthat feme Spots have been obferved to pafs from the Wefiern Limb ^^^ of the Sun to the Eaftern Margent in 13 Days and half, and after 1 3 Daft and half to re-appear in the Weftem Verge of his Diflt, from whence he in- fers that the Sun revolves 'round its Axis in the Space of about 27 Days from Weft to Eaft, that is in the fame Diredion of the Planers ; by means of . thofe Spots it has been difcovered,^ that the Axis round which the Sun it- vo Ives, is tnc'ined to the plane of the Ecliptic in an Angle of yd. Jaquier^ in his G)mmintary on 'Newton^ has made feme Refledions on thefe Spots that deferve to be remarked; as no Obfervations prove the ' Times of their Occultation to be equal, but on the contraty, all the Ob- servations he eould colled, provethem to be unequal ; and, that the Time . during which they are concealed, has been alwavs longer than that, during which they have been vifible, from hencd he concluded (as atfo tP*f Art. 41 1, of his Aftronomy) that thofe Spots are not inherent to the Sua, but removed from his Surface to fome diftance. The Solar Spots were firft difcovered in Germany ^ in the Year 161 1, bf ^Jobn FabriciUfy (n) who from thence concluded, the diurnal Revolution of the Sun. They were afterwards obferved by Scbeiner^ (o) who publifted the Refult 6f his Obfervations. The fame Difcovery was made by Galih in Italy. JScbeifur obferved mdre than fifty Spots on the Surface of the Sun; tbij may ferve to account for a Phenomenon^ related by many Hiftorians, tbit the Sun, fometimes for the Space of a whole Year, has appeared very Pale, as this Effed would natutally follow from a Number of Spots fufE- *ciently large and permanent, to obfcure a confiderable Portion of hii Surface. (n) Wolf. Elemeau Aftroiioimat Cip. 1. (o) Scheiner htving infenncd hit Sapcrior thtt he bid difcovered Spett in the Sun, he gft««lf replied, <« thtt it impoffible, 1 hav« retd Arittotk two or thi«e tiioei «ve£« aad htvc f^i ^ V At leaft ncaiioB ©f it. »» P H Y S I C A L W O R L D. XXX» It U no longer doubted that the Earth turns round her Axis in 23h 56m which compore our agronomical Day; Aom this Rotation arife the changes of Day and Nighty which all the Climates of the Earth enjoy. XVL thedM* This Motion of the Celeftial Bodies about their Centers alters their Fi-o^^«f<>- guresy for it is known that Bodies revolving in Circles, acquire a ForceJJ^J^ ^. which is fo much the greater, the Time of their Revolution being thetb«plaiictai. fame as the Circle which they defcribe is greater. This Force is called ^^fi'^^" . Centrifugal Force \ that is, the Force which repels tbem from the Center ;'J^!^^J^^" wherefore, from their diurnal Rotation, the Parts of the Planets acquire a u,, j j^ Centrifugal Force, fo much greater as they are nearer the Equators of thefeccntriptuL Planets: ((ince the Equator is the greateft Circle of the Sphere,) and fo force. much lefs as they are nearer the Poles (p) ; fuppofing therefore the Heaven- ly Bodies in their State of Reft, to have been perfect Spheres, their Rota- tion about their Axes muft have elevated their equatorial and depreflfed their polar Regions, and of Confequence changed their fpherical Figures into that of Oblate Spheroids, flat towards the Poles. The Theory thus leads us to conclude, that all the Planets, in Confe- J^^**^^^*^^ quence of their Rotation, Ihould be flat towards the Poles, but this is only in which the fenfible in Jupiter and the Earth. In the Sequel it will appear, that theel«vttion of Proportion of the Axes (q), in the Sun, is aflignable from Theory, but is|jLre5!|*Ji'. too incondderable to be obferved. The Meafures of Degrees of the Meridian, taken at the Polar Circle in France, and at the Equator, fix the Proportion of the Axes of the Earth to he as 173 to 174. By the Help of Telefcopcs the oblate Figure of Ju- piter has been perceived And the Difproportion of his Diameters. is much greater than that of the Earth, becaufe this Planet is a great deal bigger, and revolves with greater Rapidity about its Axis than.the. Earth ; the Propor- tion of the Axes of Jupiter is efteemed to be as 13 to 14. obferTttloa XVII. pr»v«th*t As the Spots of Venus, Mars and Jupiter are variable, and frequently ^^^^^^^ change their Appearance, it is probable that thefe Planets, like our Earth, ler, Venus* are furrounded by denfe Atmofpheres, the Alterations in which, produce thefe •nd the Sua -. Phenomena in refpeft of the Sun, as his Spots are not inherent on his Difk, J^Jj^"^-. and as they frequently appear and difappear, it is manifeft that he is furround-i^^Qiofpbercs s cd by a grofs Atmofphere, contiguous to.his Body, in which thefe Spots arc fucceffively generated and didplved. (p) The Poles tre the Points tboat which the Body revolves, and the Equator, th« Circle eqoi diftant from thofe Pointo dividing the Sphere into iwt> equal Parts. (q) Axis or Diameter, in general, ia a Line which paiTes through the Center, and is tenni- Bated at the Ciicumterence, In the prcfent Cafe, the Axes are two Lines which pafs through tb» QcaUtf 0D»«f wbtch is terowMtcd at the Polf s, iod the other at the £qaaior». XL SYSTEMOFTHE xvni. What has hitherto been fet forth was known before the Time of Newfon^ bot no one thought before htm, that it was poffible to difcover the Quan- titles of Matter in the Planets, their Denfittes, and the different Weighu of one and the fame Body futceflively transterred to the Surtaces of the dif- tmiCm^ik^ftrtni Planets. How Newton attained to thofe aftonifhing Difcoveries will §m^ !•»«' be explained in the Sequel ; at prefent it fuffices to fay, that he found out ^^^'•'*Mhat the Maffes of the Sun, Jupiter, Saturn, and the Earth, that is the ft«#fiJ* C^iiantities of Matter thofe Bodies contain, are to one another, as i t^f ifi^^ lyfTii. foppofing (r) the Parallax of the Sun to be lo' 3' ; that their >cnrities are as too, 94, 67, and 400; 6c that the \^ eights of the Tame Body, ^w^tirU* p''C*^d fuccefllively on the Surfaces of the Sun Jupiter, Saturn, and the Earth, ^ tht *«w?woiild be as 10000,943, 529, and 435 ; in determining thofe Proportions, ^^••^•^*' AVw/tf/i has foppolcd the Semidiameter&of the Sun, Jupiter, Saturn, and the wCff»^« Rarth,tobeas 10000,997, 791, and 109.it will be (hewn hereafter whynci. ffs^pMU'^ ther the Denfity, nor the C^antity of Matter of Mercur), Venus, sod Mt€ '^t«»»j**Mars, or the Weights of Bodies at their refpcSivc Surfaces, are known. ^f$f$4 tft ibtf VI V It follows from all thofe Proportions that Saturn is fiearly 500 Times lefs pft>f0ft,iiM than the Sun, and contains 3000 Times lefs Matter, that Jupiter is 1000 omU > uik« Times lefs than the Sun, and contains 1033 Times lefs Matter. Com- •mirfit(Tc«arp3^^j with thc Sun the Earth is only as a Point, being 100,0000 Times lefs; IndoAbc* and In fine, that the Sun is ii6Times greater, than all the Planets togeher. , $M« XX. Corfiparing the Planets with ohe another^ we 'find that Mercury sod Mars are the only Planets lefs than the Earth ; that Jupiter is not only the biggeft of all the Planets, but is bigger than alt the Planets together, sod that this Planet is two thoufand Times bigger than the Earth. XXI The Earth befides her annual and diurnal Motion, has alfo a third Mo- '^*/^^tion, by which her Axis recedes frohi its Rarallelifm, (fj& alter a certain Time iqiihioxei. is direded to different Points of the Heavens, from this Motion arifes whtt . is called tbe Preceffion of the Eqnint>xes that is, the R^greflion ot the equi- ' ItftToo*it u "<^^al Points, or thofe Points in which the lereftrial Equator cuts thc ' perfbinied Fcliptic. The eqiiinodial Points mbve contrary to tbe Order of the Signs, tnd iowhfttand their'Motion is fo very flow, that they do not compiear a Revolution J^JJ^Vilj^S^in lefs than 25920 Years, they recede a Degree in 73 Yeais, and thetn- iti annual ' nual Quantity is about, 50''. qtuotity (r) The parallax of the 6un, is the Aftgle, 'andcr which the SemidkRifeter of the Eattb itftea fHmithe fiun, ind in general £hd pataltaz of cny celeftial Body, with refpeft to the £iitb, i9 the Angle vhder which the i'efniditimetcr of the Earth wonld be fecn from that Body. (0 A tine is faid to be psYattel wten it alwayi prefemttlw ftme pofitkir with itfttA^t^ . P«k( fvppofcd fixed* PHYSICAL WORLD. XLI. Ahif/Mi firand, is will ftpfear in the Sequel, the Ciufe of this Motion in Ike Attrtdion of the Sun anrf Moon oti the Elevation of the eqoetorial Partb of the Esrth. The Preceffion of the Eqainoxes hai caufed a Diftiodion of the Yetr Tropicd into the tropieai and fydinaL The rropical Year is the Interval of Time y*y- elapM hetween two fucceffive vernal Or autummri Equinoxes, in two annual ,^'* Revolutions of the Earth. This Year is fomewhat (horter than the fydereal Year, or the Time intervening the Earth's Departure from any Pofot of her Orbit, and her Return to the fame* XXIN It remains to defcribe the iecondary Planets, which exclufive of the Ring ThefeeoMk of Saturn, are lo in Number ; namely, the 5 SateHires of Saturn, the 4 '^ P>*««**- of Jupiter, and the Moon, the only Satellite attending the Earth. Obfervation proves that ihefe Satellites in revohring round their Primaries. J^ <*• oMerve the Uws of Kepler. ^'J^ ^ The Satellites of Jupiter have been but lately difcovered : The Difcovery Kepler, bafore the Invention of Telefcopes was impoffible. Galliko difcovered the l^'^coverjof four Satellites of Jupiter, which in Honour of his Patron, he termed the ^\^^^^ Medicean Stars. Thefe are of the greateft Utility in Geography and Aftro- * nomy. Hugben/'w^s the firft who difcovered one of Saturn's Satellites ; it ftill re- AadoMoft tains his Name, and is the fourth* Afterwalds Cajlm difcovered the four ^^ ^*^* others^ XZIII. Takine the Semidiameter of Jupiter as a common Meafure, his 4 Satd- Diftta^of lites revoltc at the foHowirtg Diftancet ; the firft at the Diftance of 5 Semi- ^*|"'?*~ diameters, the fecond of 9, the third of 14, and the fourth of 25, ne^d- ^^ '^^u ing Fradions. Thefe Determinations have been deduced by Cgjtni from his pbact. OUervalions of their Eclipfes. Their periodic Times round Jupiter are fo much the longer as they are TkaiirfcrloA remoter from this Planet. The firft revolves in 42 Hours, the fecond in bmjmitt^ 85, the third in 171, and the fourth in 400, negleding the Minutes, The diurnal Rotations, Diameters, Bulks, Malfes, Denfities, and attrafitve Forces of thefe Satellites, have not as yet been difcovered ; and the beftTele^ Icopes reprefent them fo vaftly fmall, that there is no Hopes of ever attaiii- tng Certainty m thefe points ; the fame is the Cafe with regard to the SateN lites of Saturn : Thefe are placed fliU* further beyond the reach of our Rdearches. XXIV. Taking the Diameter of Saturn's Ring^ for a common Meafure, the {JJf*^^ Dift'ances of the Satellites of Saturn commencing v?ith the innermoft, ^re of* uvan in the following Proportions. from fetora. XLII. SYSTEM OF THE Ic their p€ri The firft 18 cxpreflcd by i , the fecood by 2, the tbiid by $, the fearth by ^uJlSl 8, aod the fifth by 24, negleding Fraaioos ; and their periodic Tifliea, la* plaiitc cording to Cajini, are 45^ 65^ 109^, 382^ and 1903* refpefibtdy. The Mooos of Saturn, all revolve b the Plane of the Equa tor qt thit Planet, except the fifth, which recedes from it about 15 or 16 Degrees. Several Philofophers, and among them Hugbens^ have fufpcded^ tbain •f flwh^ Tekfcopes were once brought to perfedion, a fizth Satellite of Situra te| MDccrniAga tween the fourth and fifth would be difcovered, the Diftance between tbofe filth fuel- two Satellites being two great in Proportion to that which feparates the Urn*'' ^'others ; but there would then occur, thia other Difficult v, that this SateUitf, , ' which would be the fifth, notwithftanding muft be lels than any of i^ four interior Mooos, fince with our mofl perfed 1 elefcopcs it cannot be perceived. The Orbits of the Satellites of Jupiter, and of Saturn, are awrly con- centric to thofe Planets. Ohfervfttioo Maraldi has obferved Spots on the Moons of Jupiter, but no Coniequm- ^f MaraMi ^cs could as yet be derived from this Obfervation, which if properly poniiw SefueTlUci ^ accurately repeated, might condud us to the Knowledge of feversl> •r Jupiter, terefting Particulars refpeding the Motions of the Satellites. XXV. Of the Hog Saturn, exdufive of his five Moons, is alio furrounded by a Riflgi ^ oCSttafn. where adhering to his Body; for through the Interval which fepaniesw idhw t^ Body from the Ring, we can view the fixed Stars : The EHameter of thi the hody of Ring is to the Diameter of Saturn as 9 to 4, according to Hughent^ tbit» n'lifttwie ^^^^^ ^*^*" ^^^ Double of the Diameter of Saturn ; the Diftance of the B»7 ^omthcb«! of Saturn from his Ring, is nearly equal to his Semidiameter ; lb that W 4y of Uic Breadth of the Ring is nearly equal to the Diftance lietweeo its ioteiwr 1 1 dr*' tter ^'"^^ ^"^ ^^^ Globe of Saturn. Its Thicknefs is very inconfiderable, wf ltibf«adtL' ^^^^ it ttiins its Edge to the Eye, it is no longer yifible, but only appesr»» lt» thick- a black Line extended acrofs the Globe of Saturn. Thus this Ring under- J*^/ goes Phafcs according to the Pofiiion of Saturn in his Orbit, which prows ptqoeVidJi it to be an opaque Body ; and which like the other Bodies that compo'e^'^ fabjca to planeury Syftem, ftiines only by refleding the Light it receives from tht Fh**«. Sun. . We cannot difcover whether the Ring of Saturn has any Motion of R^JJ* tion^ as no Changes in its Afptd are obferved to authorife us to coocluoe this Rotation. * r * The Plane of this Ring always forms with the Plane of the Eclip^.'^ an Angle of 23^ i, hence its Axis remains always parallel to itfelf ^ » its Revolution round the Sun. . Of the dif- The Difcovery of the Ring of Saturn, the only Phenomenon of the K«» cofer|r of obferved in the Heavens is due to Hugbens. Before his Time, AftronointfJ o"io'iMcoo obferved Phafea in Saturn, for they confounded Saturn with his Ring; but tbofc ^''^Vitbe Phafcs were fo different from thofe of the other Planets as to be utterly '^^* PHYSICAL WORLD. XLIU pKctUe* In Hevilius mty be Teen the Nimet he gives to ihofe AppeArances ^ He- of Seturn, and how far (t) he wu frorti afligning the true Caure. '^'^ Hngbftu compariM the different Appearances of Saturn^ found they were produced bf a Ring furrounding his Body ; and this Soppoilition is fo confor- — bic with all Teldcopic DifcoTerics, as to be now generally received* Gffimry defcribing the Notion of HalUj^ that the terreftrial Globe is g^^** ^ only an AJTemblage of Shells concentric to an internal Nucleus, propofes a cer^agSlt Conje£bjre concerning this Riog» that it is formed of feverid concentric rii«. Shells detached from the Body of that Planet, whofe Diameter was former* ly equal to the Sum of its adual Diameter, and the Breadth of the Ring. Another Cbojefiure has alfo been propoGed, that the Ring of Saturn is on- i^^ ^^^\, ly an Aflemblage of Moons, which from the immcnfe Diflance appear toiiteiorjnpi be contiguous ; but thofe Conje£hires are not grounded en any Obfervation. <*' ^^ ^*' By the Shadows of the Satellites of Jupiter and Saturn projeded on Sfi^^rieVTbo their Prinuries, it has been difcovered, that they are fpberical Bodies, dlf t. XXVI. The Earth has only one Satellite, namely the Moon ; but her Proximity of themota has enabled us to puflb our Enquiries coocerniq; this Satellite much further than about the others. The Moon performs its Revolution round the Earth in an Ellipfe, the Whtt carve Earth being placed in one of the Foci i The Form and Pofition of this El- ^^^^^ lipTe 'li continually changing ; thefe Variations are caufied by the Adion of the ^^'tl^ Sun, as will appear in the Sequel. The Moon in her Revolution round the Earth obferves the firft of the two Laws of Kepler^ and recedes from it onl v by the Adion of the Sun upon her $ ihe oompleats her Revolution round the Earth from Weft to Eaft in 27 d. J^JJJ"^'* 7 h. 43 m. which is* called its perMical Montb. The Difc of the Moon is (ometimes totally, and at other times partially^ illuminated by the Sun. The illuminated Part is greater, or lefs, according to its Pofition with rerped to the Sun and the Earth ; thefe are called her Her phtfet. Pbajei. She aflumes ail thofe various Phafes during the Time of her /jnodic ^J^]^"^'^ Revolution,' or the Interval between two fucceffive Conjun^om with the Son. This fynodic Month of- the Moon confifis of 29 Days i nearly. The Phafes of the Moon prove that ihe is an opaque Body, Ihining only Tbt 1 by reBeaing the Ught of the Sun. , !iS"ffl! We know that the Moon is a fpberical Body, becaufe (he always ap^ cti bod/T'* pears to be bounded by a Curve. The Earth enlightens the Moon during her Nights, as the Moon does the The etrth Earth during ours ; and it is by the refleded Light of the Earth that we fee ^^^^^ the Mppn, when ihe is not illuftrated by the Sun. dsrinf het* nights. (I) HarelMt !a ofrarcnlo de Sttvroi Nttirt fiicie diftiogniflief the different Afpe€fct of Satorn hj tbe NeoBet of MoMrphericiini, TrirphericniDf Spherico-tsfatniDi eUipti-CQt&fttvtBy fphert* ^ecn^idstviii, sad febdiTidet them sgua into other Pbsfct. XUV. SYSTEM OF THE bfT^IS ^ '** SiirfiKe of the Earth it about 14 tiittcs gr«Mer thrn flat of the '* '^^ Moon, the Eirth faen from the Moon would appear 14 times Wighier, tud rcfloft 14 timet more rays to the Moon, than the Mom doet to ut, fap- pofiflg both equally capable of refle£bng Light. lacGMtiom The Plane of the lunar Orbit forms with the Plane of the Ediptii^ la •fthcoMt Angk of about 5<^* •fdicvoo. .j^ ^^ ^^^ ^ ^1^ Q,.^ ^^^j^ ^^ ^^^^ defcribet ronad (he Earth, it called ibe Line 0/ the Apfides (0) •/ the Mqqh. The Moon accompanies the Earth in her annual Rcfolutioo renad du Sun. If the Orbit of the Moon had no other Motion but that by which it ii carried round the Sun along with the Earth, the Axis of this Orbit wooU always renflain parallel to itfelf ; and Moon being in her Ap9gee^ aad ia her Perigee^ would he always at the fame Diftances from the Ewrth, and wcoU always corrcfpond to the fame Points of the Heavem ; but the Line of the Timeof the Apiides of the Moon rcYolves wiih an angular Motion round the Earth, sc- 7fXl£ ^rding to the Order of the Sigm ; and the Apogee and Perigee of the Mtx» of Uic tp do not return to the fame Points ia lefs than 9 Years, which is the Tiiaesf £dct. the Revolution of the Line of the Apfidcs of the Moon. Xcv«lation The Orbit of the Moon inierfe£b the Orbit of the Earth in two Pointty ^f^"^^ which are called her Nodes ; tbefe Points are not always the fame, but cbsnp " perpetually by a retrogreffivf Motion that is contrary to the Order of the Timeof itt Signs,, and this Motion is fuch, that in the fpace of 19 Years the Nodei rcTolatiMi. perform a whole Revolution, after which they return to the fame Poiots sr the Orbit of the Earth, or of the Ecliptic. tl'^Ua y^^ Excentricity of the Orbit of the Moon changes alfo continaslln \n^^ this Excentricity fometimes increafes, fometimes diminiihes, fo that theDif* ference of the greateft and lead Excentricity exceeds half the leaft. It will be explained in the Sequel how Newten difcovered the Caufe of lu tbofe Inequalities of the Moon. roBiITrtr ^^ ^y uniform Motion that the Moon haa, is its Motion of Rrtstioi uis. abput her Axis ; this Motion is performed exadly in the fame Time as itt Revolution about the Earth, hence its Days coofift of 27 of our Days, 7^ In wlist 43*' time it it This equality of the lunar Day and the periodic Month makes the Mooi pcrfcnncd. ajways prefent to us nearly the fame Difc. The uniform Motion of the Moon about its Axis> combined with the la- equality of lis Motion round the Earth, produces the apparent Ofcillttioo LibrttioQ ^f of the Moon abckut her Axis, foroeticnes Eaftward, and at other tiaMi Weir ^e. mo^ii. ward, and this is what is called ker Liiraiien ; by thb Motion ib^ pftfeD" {u\ Tht luKt of the Apfidet of the Moob is the Use whk& ptflet Uiroagh the Afot^^ Perigee *, apogee it ^he Point of the Orbit the Remote^ from the Berth, tad the Perigee h the Peint of the Orbit the neareft to the Earth \ and io geifefal^ the Ap6de« of aoj Orbil vt^ f oiotf the ReoMteft froxDt aad ae«reft to> the ceatral Potjit. PHYSICAL WORLD. XLV to M fomeiiiMt PWi which w«rt coooeftM, nni conoeah others thit wtre ▼ifible. Thit Librtiion of the Moon srifes from her Motion in tn Elliptic Orbir^ Iti c««l«. lor if ihe revolved in t cirovlar Orbit, hftving the Eftrtb for its Center, Mid turned about her Axis in the Time of her periodic Motion round the Earth, ifae would in all Pofitiom turn the fame D.fc ezafily towards |he Eartb. We are ignorant of the Form of the Surfiice of the Moon, which is on lh» other Side of her Difc with Refped to us. Some PhiloTophers have eves attempted to explain its Libration, by affigning a conical Figure to that Part of its Surface, which is concealed from us, ai^ who- deny W Rotation round her Axis. The SurfsRce of the Moon is <ull of Eminences and Cavitiea, for which wt$km ihe fttk&s on every Side the Light of the Sun, for if her Surface w«s even and poliflied like a Mirror^ flic would only refled to us the (mage of the SuQ. ^. The mean Diftance of the Moon from the Earth is nearly 60 i Semi- thll™^ diaroeten of the Earth. from tl# The Diameter of the Moon is to the Diameter of the Earth, as 100 to !?^^* 365^ its Ma& is to the Mafs of the Earth, as i 10 39, 788 and its Denfity itl mtSr*' is to the DenHcy of the Earth, as 1 1 to 9. Its den6tr. And laftly, a' Body which woqld weigh 3 Pounds at the Surface of the Whatbodiet Earth, transferred to the Surface of the Moon would weigh one Pound. y'^t!^ on All thefe Proportions are known in the Moon and not in the other Satel- '^' ^'^*^* litcs, becaofe this Planet fupplies a peculiar Element, namely her Adion on the Sea, which Newtm knew how tomeafure aixl to employ for determining bcr Mafs, the Method he purfued in this Enquiry will be unfolded in the Se- quel Theory ef the Primary Ptantts. I. In accounting for the celeilial Motions, the firft Phenomenon that occurs to be explained is the perpetual Circulation of the Planeu round the Center of their Revoltuions. By the firft Law of Nature every Body in Motion perfeveres in that rec- ticlioear Courfe in which it commenced, therefore that a Planet may be defleSed from the ftraight Line it tends to defcribe inceflantly, it is Neceflsry that a Force diflPerent from that which makes it tend to defcribe this Araight Line flioold inceflantly A3 on it in order to bend its Courfe mto a Curve, in the fame Manner as when a Stone is whirled round in a Sting. The Sling inceflantly reftrMns the Stone from flying off* in the Diredion of the Tangent to the Circle it defcribes. Hew tke To explain this Phenomenon, the Ancients invented their folid Orbs f^f '«?* p^* end DifiarUi Voriiees, but butb one and the other of tboifc Expircati^ns Ind o^fcV XLVL SYSTEM OF THE tet expUiD were mere Hypothefes devoid of Proof, and though DefcarUs tkpUmtioii tion^o"the ^^ ^^^^ Philofophical, it was no lefs FiSiitious and Imagiaary. pUoeti in II. their orbrtt. NewtoH begins with prorag in the firft Proitofition ^a), that the Areas defcribed by a Body revQlting round an imtrtoveaWc Center to which it it tripui^***" continuaHy urged/ are pnoportional to tlie Times, and Teciprocilly iti the fore- xvhich Second, that if a Body revolving round a Center dcfcribcs about it Areas liirjtrj tnc proportional to the Times, that Body is aSuatcd by a Force diredcd frot^'flying *®. ^^^^ Center. Since therefore according to Kepkr^s Difcoverics, the Pla- cff bythe ncts defcribe round the Sun Areas proportional to the Times, they are ac* tangent. tuated by a centri()etal Force, urging them towards the Sun, and retaining them in their Orbjts. Newton has alfo (hewn (Cor. i . Prop. 2.) that if the Force aOing on a Body, urges it to difierent Points, it would accelerate or retard the Dc(crit>rion df the Areas, which would confequently be no longer proportional to the Times : Therefore if the Areas be proportional to the Hmes, the revolving Body is not only aduated by a centripetal Force, direfied to the central ^ody, but this r orce makes it tend to one and the fame Point. HI. As the Revolutions of the Planets in their Orbits prove the Exiftance of ^ centripetal Force drawing them froni the Tangent, fo by their not dcfccnding in a ftraight Line towards the Center o\ their Revolution, we may conclude that they are ad;ed upon by another Force diiFerent from the Aodthepro Centripetal. Newton has examined (b) in what Time eKh Planet would jeaile force Jefcend from its prcfcnt Diftance to the Sun if they were aSuated by no them "rem ^^^'^^ Porce but the Sun's Aftion, & he has found (P.36) that the different Pla- ftUing to nets would employ in their Defcfenr, the Half of the periodic Time of the the center Revolution round the Sun of a Body placed at Half their prefent DiftanceS, and confequently thefe Times wbuld be to their periodic Times, as i to 4\/2. Thus, Venus for Example would take about 40 Days to defcend to the Sun, for 40 : 224 : : i : ^^/^ nearly; Jupiter would employ two Years and a Month in his Defcent, and the Earth and the Moon fixty-iix Days and nine- teen Hours, &rc. fince then the Planets do not defcend to the Sun, (bme Force mufl: neceflarily counterad the Porce which make them tend to the Sun, and this Force is called the Proje^ik P$rce, IV. Of the cen- The EflFort exerted by the Planets iri Confequence of this Force to re- force^oftbe ^^^ ^^^^ the Center of their Motion, is what is called their Centrifugal pltaett. Fffrce^ hence in the Planets, the centrifugal Force is that Part of the projec- tile Force, which removes then! dirediy from the Center of their Revolu- Mon. (a) When the Propoiiciont are qaoted without qvotiog the Bdok, they are the Propbfitiont of the firft Book. (b) Dt fyftcmate mondi, ptgc 3 1 . cditioa 1 ;^3 1 . PHYSICAL WOULD. XLVIL .'V. The projeQite Force has the Tame DrreAion in all the Planets, for they all revolve round the Sun from Weft to Eaft. Snppofing the Medium in which the Planets move to be void of all Re* Mance, the Confervation of the projeAile Motion in the Planets, is %c^ « counted for from the Inerria of Matter, and the firft Law of Motion, but its Phyfical Caufe, and the Reaion of its Diredion are as jet unknown* VI. Kewtoadlf ' After having proved that the Planets are retained in their Orbits by a covert the Force direded lo the Sun, Nfivhn demonftrates (Prop. 4,) that the centri- Jj^.'^pJi^JI* petal Forces of Bodies revolving in Circles are to one another as the Squares to the Saa of the Arcs of thofe Circles defcribed in equal Times, divided by their 50 1>« >a tha Rays, from whence be deduces (cor. 6) that if the periodic Times of Bo- JJ^ J^^5 "J^JJ dies revolv'uig in Circles be in the fefqutplicate Ratio of their Rays, the cen- of their dif tripetal Force which urges them to the Center of thofe Circles, is in the tancei from inverfe Raib of the Squares of thofe fame Rays, that is of the Diftance of Jjj/'^^ ihofe Bodies from the Center : But by the fecond Law of Kepler, which all die timet tad the P|anets obferve, their periodic Times are in the felquiplicate Ratio of diftmoet. their Diftaaces from their Center ; confequently, the Force which urges fei^fiJi^J^ the Planets towards the Sun. decreafes as the Square of their Diftance of thair or- from the Sun increafes, fuppofing them to revolve in Circles concentric to bjtt Mag the San. "'^»^- yriu * The firft and moft natural Notion that we form concerning the Orbits of the Planets, is that they perform their Revolutions in concentric Circles ; B«fcraiU|i- but the Difference in their apparent Diameters, and more accuracy in the i^tJjJJ!^^ Obfervations^ have long (ince made known that their Orbits cannot be con- that the pi«* centric to the Sun ; their Courfes therefore, before Kepferh Time, were ex- ««• revolt plained by czcentric Circles, which anfwered pretty well to the Obfervations g^J^^cw OP the Motions of the Suit and the Planets, except Mercury and Mars. trie cirdci. From confidering the Courfe of this hft Planet, Kepler fufpeaed that the But Kepler Orbits of the Planets might poffiUy beEllipfes, having the Sun placed in one JU'tJl^ of the Foci, and this Curve agrees fo exadly with all the Phenomena, that ^oWt in ci it is now univerfally acknowledged by Aftrdnomers, that the Planets revolve Hp^* round the Sun in elliptic Orbits, having the Sun in one of the Foci, VIII. Affumtng this Difcovery, Newton examines what is the Law of centripe- tal Force, required fo make the Planets defcribe an Ellipfe, and he found {P»op. ti.) that this Force muft follow the inverfe Ratio of the Pianet'a Diftance from the Focus of this Ellipfe. But having found before (cor. 6. Prop. 4.) that if the periodic Times of Bodies revolving in Circles be in the fefquiplicate Ratio of their Rays, the centripetal Forces would be in the in^ verfe Ratio of thofe fame Diftances ) he bad no more to do to invincibly XLVIIL SYSTEM OFTHE prove that the centripetal Force which dire£b the celeftial Bodiea in their Coorfet, follows the inverfe Ratio of the Square of the Diflaooet| but to examine if the periodic Tiaies follow the fame ProportioD in EUipfet m m tb*t tfi eliip But Ntwhn demonftrates (Prop* 15.) that the periodic Times in EUtpAv iMChcpcrio gre in the fefquiplicate Ratio of their great Axes \ that is, that tfaofe TtMoo^ «r!o'thc* ^^^ in. the fame Proportion in Ellipfes^ and Grcles whole DiaoKters are equal IkiMpropor to the great Axes of thofe Eltipfes. tjpa M ia This Curve which the Planets defcribe in their Revolution is endued with Coai^'iieot ^^^* Property, that if fmall Arcs defcribed in equal Times be taken, the lytbtceotrt Space bounded by the Line drawn from one of the Extremities of this Arc, prtti force aiK] by the Taqgcnt drawn from the other Extremity increafes in the fame uioi'^the ^^^^ ^* ^^^ Square of the Diftance from the Focus decreaies; from piwtf in whence it follows, that the attradive Power which is proportional to this their orbitt. Spscc^ follows alfo thb fame Proportion* tbt fijosreL . **• of the diT Newtmt^ not content with examining the Law that makes the Plaoela de» taoce. (cribe Elites; he eliciuired further weather in confequence of this Law: Tkm tmiri ^1^ mif^t not defcfibe other Qirves, and he found (Cor. i* Prop, ty) dial ptui Sm^ this Law would only make them delcribe a conic Se£bon, the Center of tfae b«^ ID this Forpe being placed in the Focus, let the projedile Force be whal it wouU. the^tMU Other Laws, by which Bodies tnigbt defcribe conic Sediims, would maka can only de them defcribe them about Points difrartent from the Focus. Newt§m (bund, ^be conic fc^ example, (Prop. 10.) that if the Force be as the Diflance from the Center^ H^M^ it will make the Body defcribe a conic Sedion, whofe Center wouU be the ^acad in Center of Forces, tlms Newton has difcovered not only the Law which the oMof tiM centripetal Force obferves in our planetary Syftem, bnt he hu alio Iliewn ^^' thai no other Law could fubfill in our WorU in iu prefent Sute. Maana^ of Newtm alterwafda examines (Prop. 17.) the Curve e Body wouUdeicribe doecmAoiof with a centripetal Force decreaftng in the inverfe Ratio of the Square of the iTplmtfuo Di&BBCt, fuppefing the Body let go from a gi^ Poi^r, with « Difec< p^Qf the . tion and Vieloqity i^nned at Pleafure. Uw of cen, 7*0 foWe th|s Problem^ he feA out with the Remark he had made^ (Pr^p^ for^^tobe i^*) that the.Velfxjiiesof Bodies defcribing conic SetQiona» are in eaoh P^ giveo. of thofe Curves, as the Square-Roois ,of the principal Parameten, divided by the Perpeadiculars, let fall from the Foci 00 the Taii§eots to tkofe Points. This Propofition is not only very interefting, coniidered merely as a geo^ metrical Probleoi, but alfo of great ufe in Aftronomy; for finding by Obfervation the Velocity and DireSioft of a Planet in any Part of its Orbit, by th<: Afllilance of this Propofition, the Remainder of its Orbit is found out, and the Petermination of the Orbits of Comets, naay in a great Meafiira i^ be deduced from this Propofition. PHYSICAL WORLD. XLIX XI. It is eafy to conceive that in coniequence of other Laws of centripetal l^^^ Force different from that of the Square of the Diftances Bodies would ^^'^^^^ defcfibe other Curves, that there are ibme Laws by which notwithftan- of other ding the pn^eiEHle Force, they would defccnd to the Sun, and others by Uwtofcen which notwithftanding the centripetal Force, they would recede in infini-J3dbL*d«^ turn in theHeavexily Spaces; others would make them defcribe Spirals, ^Cf^nbed. and Nnuten in die 4ad Proportion, mveftigates what are the Curves de-: fcribed in all Sorts of Hypotliefis of centripetal Forces. XII. It evidently appears from all that has been faid tliat the perpetual Circula- f^e perpe- tion of the Planets in their Orbits depends on the Proportion between thetwi clrcuk- centripetal and the projedtile Force, and thofe who afk why the Phnet$|;^° ^^ }^^ arriving at their Perihelia, reafcend to their Apbelia, are igncNrant of this^^j.^or^t9 Proportion ; for in the higher Apfis the centripetal Force exceeds the Cen-refnits from trifiigal Force, fince in defccndihg the Body approaches the Centre, and int**«Pf^ft»- thc lower Apfis on the Contrary, the centriAigal Force fiirpailes in itsJJJ^^^"^^ turn the centripetal Force, fince in reafcending the Body recedes from thet«i and pro. Centre : A certain Combination between the centripetal Force and the cen- j«^i'« ^'ce- trifugal Force was therefore requifit, that they might alternately prevail and caufe the Body to defcend to the lower, and reafcend to the higher Apfis per- petually. Another Objedion was alledged with regard to the Continuation of die Heavenly Motions, derived from the Refiftance they (hould undergo in the Medium in which they move. This Obje^on Newton has amwered in ^^^ ^^^^ (Prop. 16. B. 7.) where he (hews that the Refiftance of Mediums dimmifhaminwhich in the Ratio of thcif Weight and their Dcnfity ; but he proved in the Scho- «»»« hwvrn- Ihnn of (Pftyofiiion 22. B. 2.) that at the Height of two hundred Miles a-Jji^^^'^^^^ bove the Surface of the Earth, the Air .is more rarified than at the SurfiM», of Ju reiiii, in the Ratio of 30 to 0,0000000000003998 or nearly as 7 5000000000000 ancc. to I, ftiom whence he concludes (Prop. 10. B. 3.) iupponng the Refiftance of the Medium in which Jupiter moves to be of this Denfity, this Planet defcribing five of its Semidiameters in 30 days, would from the Refiftance of this Medium, in loooooo years fcarcely k>fe looooooth Part of its Mo- tion ; from hence we fee that the Medium in which the-Planets move may be fo rare and fubtilc, that its Refiftance may be regarded as Void ; and the Proportibnality conftantly obfervcd, between the Areas and the Times, is a conrincmg Proof that tfris Rdiftance is aihially infenfible. XIII. As we have ftiewn that the Proportionality of the Times and of the A- reas which the Planets deicribe around the Sun, proves that tliey tend to the Sun as to their Centre, and that the Ratio fubfifting betwQpn their periodic Times and their Diftances, fliews that this Force decreafes in the invenc L SYSTEMOFTHE Ratio of the Square of the Diftances. If the Planets which peribrm their Revolutions round the Sun be furrounded by others which revolve round them, and obferving the fame Proportions in their Revolutions, we may con« elude that thefe Satdlites are ureed by a centripetal Force direded to their Primaries, and that this Force decreaies as that of the Sun in the duplicate Ratio of the Diftance. We can difcover only three Planets attended with Satellites, Jupiter, the Earth, and Saturn; we know that the Satellites of thofe three Planets de- fcribe around them Areas proportional to the Tunes, and omfequently aie urged by a Force tending to thofe Planets. Thecompa- XIV. rifoii of the Jupiter and Saturn having each feveral Satellites whofe periodic Times Snwf and ^^' Diftances are known, it is eafy to difcover whether the Times of their dUUocM of Jlcvolution about their Planet, are to their Diftance in the Proportion difco- thefsteHiteaveredby KifUr; and Obrervations evince that the Satellites of Jupiter and of Stturn Saturn obfcrve alfo this fecond Law of Kfpltr in revolving round tiicirPri- pVovM tiwt "paries, and of confequence the centripetal Force of Jupiter and of Saturn the centri-decreafe in the Ratio of the Square of the Diftances of Bodies from the petal force Centre of thofe Planets. of thofe pit- yy^ uTWi'in*^'^ As the Earth is attended only by one Satellite, namely the Moon, itap- vcrfcratioof pears at firft View difficult to determine the Proportion in which the Force the Tquate a<£ls tliat maRes tlie Moon revolve in her Orbit round the Earth, asindiis tlJt^ **'^' ^^^'^ ^^ ^^^ ^° '^^"^ ^f Comparifon. How^New. Nswhn has found the Means of fupplying this Defedl; his Method is as ton difcove- foUows : All Bodics which fidl on the Sur&ce of the Earth, defcribe accord- *<^^ ^V ^***ing to the Progreffion difcovered by Galliko^ Spaces which are as the Squares foJce of "the o^*e Times of their Defcent. We know the mean Diftance of the Moon Earth hi- from the Earth which in round Numbers is about 60 Semidiameters of the low. the Earth j and all Bodies near the Surface of the Earth are confidered as cqui- fame piopor- j-^^^^ ^^^ ^^ Centre ; therefore if the fame Force produces the Defcent of heavy Bodies, and tlie Revolution of the Moon in her Orbit ; and if this Force decreafes in the Ratio of the Square of the Diftance, its Action on Bodies near the Surface of the Earth ftiould be 3600 Times greater than what it exerts on the Moon, fince the Moon is 60 Times remoter from the Centre of the Earth ; we know the Moon's Orbit, becaufe we know it prefent the Meafure of the Earth, we know that the Moon defcribes this Orbit in 27 Days, 7 Hours, 43 Minutes, hence we know the Arc (he de- fcribes in one Minute ; nowby(Cor. gProp. 4.) the Arc defcribcd in a gi- ven Time by a Body revolving unifbnnly in a Circle with a given centripe- tal Force, is a mean Proportional between the Diameter of this Circle anJ x\it right Linedefcribed in the Body's defcent during that Time. j PMYSICaLWORLD. LI It is true that the Moon does not revolve round the Earth in an cxaft Circle, but we may fuppofe it fuch in the prefent Cafe without any fcnfiblc Error, and in this Hypothefis, the Line exprefling the Quantity of the Moon's defcent in one Minute, produced by the centripetal Force, is found tobeneariy ic Feet. But the Moon according to the Progrefiion difcovered by Gallileoy at her pnefent Diftance would defcribe a Space 3600 Times lefs in a Second than in a Minute, and Bodies near the Surface of the Earth defcribe, according to the Experiments of Pendulums, for which we are indebted to Hu^hens^ about 15 feet in a Second, that is, 3600 Times more Space than the Moon defcribes in the fame Time ; therefore the Force caufing their Defcent ads . 3600 Times more powerfully on them than it does on the Moon ; but tliis is cxa^y the inverfe Proportion of die Squares of their Diftances* By this Example we fee the Advantage of knowing the Meafure of the Earth ; for in order to compare the Verlc Sine which exprefles the Quantity of the Moon's defcent towards the Earth, with the cotemporary Space dc-(\,re of"^ fcribed by Bodies falling by the Force of Gravity near the Earth, we muft Birth* wm know the abfolute Diftance of the Moon from the Earth, reduced into Feet, ncccfrtry for as alfo the Length of the Pendulum vibrating Seconds j for in this Cafe it is?!***°s *!• not fuiEcient to know the Ratio of Quantities, but their abfolute Magni- *^^*'3r* tudes» Xvf» Jupiter, Saturn, and our l£arth therefore attra& Bodies, in the famegQthorifetut Proportion that the Sun attrads thofe Planets, and Indudtion authorifes us to conclude, to conclude that Gravity follows the fame Proportion in Mars, Venus, and^!>«^ nkUJi Mercury ; for by all that we can difcover of thefe three Planets, they appear J*^",mepro! to be Bodies of tlie fame Nature with the Earth, Jupiter, and Saturn ; from portion in ipvhence we may conclude, with the higheft Probability, that they are cn^ «be pi*"<^« dued with the attra<ftive Force, and that this Force decreafes as the Square ^'^J.^j^yj*^^^ of theDiilances. xvli. It being proved bv Obfervation and Induftion that all the Planets arc en- From dued with the aittra«ive Power decrealing as tlie Square of the Diftances j J**^^'^* ^.^ and by the fecond Law of Motion, A^on is always equal to Re-adkion, ciude/Sie** wc (hould conclude witli Newton^ (Prop. 5. B. 3.) that all the Planets gra-mututi as- Titate to one another, and that as the Sun attradls the Planets, he is rcci- trtaion of procally attnufted by them ; for as the Earth, Jupiter, and Saturn ad onj"^^^**^" their Satellites in the inverfe Ratio of the Square of the Diftances, there is '^ no Reafbn why this Adlion is not exerted at all Diftances in the fame Pro* portion ; thus the Planets ftiould attract each other mutually, and the £f- fe&s ot this mutual Attn^Aion are fenfibly perceived in the Conjundion of Jupiter and Saturn, LII SYSTEMOFTHfi XVIII. As Analogy enduces us to believe that the fecondary Planets ai« in all Refpeds Bodies of the fame Nature with the primary Hanets, it is highly prob^le that they aie alfo endued with tlie attnuStive row^, and confe- quently attraft their Primaries in the fame Manner they are attnKSled by them, and that they mutually atttad each other. This is further coniirmed l^ the Attra<aion of the Moon exerted on the Earth, the EffeSls of which are vi- fible in the Tides and the Preceffion of the Equinoxes, as will appear in the Sequel : We may therefore conclude that the attra<ftive Power belongs to all the Heavenly Bodies, and that it aAs in all our planetary £yftem in the ihverfe Ratio of the Square of the Diftahces. XIX. But what is the Caufe which makes One Body revolve round another ? for Wtiat caufejj^ftance, -tlie Earth and the Moon attrading each other with Forces decrea- Wiy?evoUcf*^g^^ the duplicate Ratio of their Diftances, why Ihould not the Earth round ano-rcvolve round the Moon, infteadof caufing tlie Moon to revolve round the thcr. Earth ; the Law which regulates Attraftion does not therefore depend on the Diftance alone, it muft depeod alfo on fome other Element, in order to account for this Determination, for the Diftance alone is infui&cient, iince it is the fame for one «nd the other Globe. This canftf Erom examining the Bodies that compofe our planetary Syftem, it is natuml appears tube to conclude that this Law is that of their Mafles ; the Sun, round whom all the mafa of^^ Heavenly Bodies turn, appears much bigger than any of them 5 Sa- ^e^ centra ^^^^^ ^^^ Jupiter are much bigger thah their Satellites, and our Earth is much bigger than the Moon whidi revolves round it. But as the Bulk and Mafs are two different things, to be certain that the led*"^ ^"^bJ Gravity of the Celeftial Bodies follows the Law of their Maffes, it is necef- roaflfes*l>fthef^ to determine thofe Mafles. planets ne- But how Can the Mafles of the different Planets be determined ? this ceflary to jVh^^^;^ haS ftlCWn. aetermme _.__ point. rp^ ^^^ ^^ p^^^ J ^j^^^ condu&d him to this Difcovery. Since the Attradrion of all the Celeftial Bodies on the Bodies which fur- vioad that round them follows the inverfe Ratio of the Square of the Diftances it is N^i'Jf^'^«^^y probable that the Parts of which they are compof«d attrad'each . this dVcovc-^*^^ i" *^ ^^^^^ Proportion. . ry. The total attiaaive Force of a Planet is compofed of the attradive Forces of its Parts ; for fuppofmg feveral fmall Planets to unite and compofe a big • one, the Force of this big Planet will be compofed of die Sum of the For- ces of all thofe fmall planets ; and Netvtort has proved in (Prop 74 75 and 76,) that if the Parts of which a Sphere is compofed, attrad each other mutually in tlie inverfe Ratio of the Square of the Diftances, thefc t>HYSICALWOftLD. LUI entire Spheres will attract Bodies which are exterior to theiH, at whatever Diftance they are placed in this fame inverfe Ratio of the Square of Diflan- oes I and of all the Laws of Attradtion examined by NewUn^ he has found only two, namely, tliat in the inverfe Ratio of the Square of the DUlan- ces, and that in the Ratio of th« Ample Diftances, according to wliicli Spheres attra& external Bodies in the fame Ratio in which their Parts mu* tually attract each other j from whence we fee the force of the Reafoning "which made NiWton conclude that fince it is proved on one Hand from Theory, (Cor. 3. Prop. 74.) that when the Parts of a Sphere attradl each o- ther with Forces decreafmg in the duplicate Ratio of the Diftances, the en- tire Sphere attrafts external Bodies in the fame Ratio, and on the other, Obfervations evince that the Celeftial Bodies attraft external Bodies in this Ratio, it is obvious that the Parts of which tlie Heavenly Bodies are com- pofed, attradl each other in this fame Ratio. Newtcn examines (in Prop. 8. B. 3.) what the lame Body would weigh at the Surfaces of the different Planets, and he found by means of (Cor. „ ^ . c 2. Prop. 4.) in which he had demonftrated, that the Weights of equal Bo- "eiKht of dies revolving in Circles, are as the Diameters of tliofe Circles, divided the ftmebo- by the Squares of their periodic Times, therefore the periodic Times of ^>' "P^" ^^^ Venus round the Sun, ot the Satellites of Jupiter round this Planet, of thei^''^^."^ Satellities of Saturn round Saturn, and of the Moon round the Earth, andthcftmedif- the Diftances of thofe Bodies from the Centres about which they revolve ^"!^c from being known, fuppofing alfo that they defcribe Circles, which may be fup-^^**'^**^** pofed in the prefent Cafe, he difcovers how much the fame Body would weigh transferred fuccelfively on the Surfaces of Jupiter, Saturn and of the Earth. Having thus found the Weights of the fame Body on the Surface of the different Planets at the fame Diftance from their Centres, NewUn dedu- An<* prove* CCS the (^lantities of Matter they contain, for Attradtion depending on the q^j^atittw of Mafs and the Diftance, at equal Diftances the attractive Forces are as thcmittcr are Quantities of Matter in the attracting Bodies; therefore the Mafles of theP'^P<'^^'«»*l different Planets are as tlie Weights of the fame Body at equal Diftances ^^Jg^[' from their Centres. XXI. We may difcover after the fame Manner the Denfity of the Sun and of thof^ Planets which have Satellites, that is, the Proportion of tlicir Bulks F««n and M^^, for Newtoriy (Prop. 72.) has proved, that the Weights of e- J^«°^«^« qual Bodies, at the Surfaces of unequal homogeneous Sphere ;, arc as their^denfi- the Diameters of thofe Spheres; therefore if thoie Spheres were heteroge-ti«t. neous and equal, tlie Weights of Bodies at their Surfaces would be as their .Denfity, fuppofing the Law of Attraftion to depend only of the Diftance, LIV SYSTEMOFTHE and the Mais of die attrading Body; tfaerdbre the Weights of Bodies the Surfaces <^ unequal and heterogeneous Spheres, are in the compound Ratio of their Denfities and Diameters ; coi^uently the Dcnfities are as the Weights of the Bodies divided bv their Diameters^ XXIf.* Tkcrimikft From hence we find, that the fmaller Planets are denfer and placed near- and deoTcftcr the Sun, for where all the Proportions of our Syftem were laid down, Mu^ JJ^we faw that the Eartli, which is lets and nearer the Sun than Jupiter and fta. Saturn, is more denfe than thofe Planets. ZXIII. Newton deduces from thence, the Reaibn of the Arrangement of tlic Cdeftial Bodies of our planetary Syftem, which is ad^^ted to the Denfitj of their Matter, in order that each might receive a Degree of Heat more or tefs according to its Denfity and Diftsmce ; for £]q>erience fhews us that The rf«ronthe denfcT any Body is, the more difficultly does it receive Heat ; fixim aflifaed bjy^hcnce Niwton concludcs that the Matter of which Mercury is compofed NewtoB. Qxoxild be feven Times denfer than the Earth, in order that Vegetation might take place ; for Illumination, to which, ceteris paribus. Heat is proportional, is inveifely as the Square of the Diftance^ but we know the Proportion of the Diftances of the Earth and Mercury from the Sun, and from this Pro- portion we difcover that Mercury is feven Times more illuminated, and confequently feven Times m(H% heated than the Earth ; and Newton dif- coverod, fiom his Experiments on the Thermometer, that the Heat of our Summer Sun^^ feven Times augmented, would make Water boil ; ttiere- fore if the Earth was placed at the Diftance of Mercury fix>m the Sun, our Ocean would be diiTipated into Vapour; removed to the Diftance of Sa* turn finom the Sun, the Ocean would be perpetually frozen, and in both Cafes all V^etation would ceafe, and Plants and Animals would perifti. XXIV. Tke dtnfi- It cafily appears, that the Mafles and Denfities of fuch Planets only as hUnrtl **2"^ attended by Satellites can be difcovered, fince to arrive at this Difcove- ^hich htTery we muft compare tlie periodic Times of the Bodies revolving round thofe fatciiitea on. Planets, the Moon alone is to be excepted, of wliich mention will be made moon cz~ a a v . ccpttd. Having determined the MaflTes of the Planets, we find that thofe Bodies Why tbertm which have lefs Mafs, revolve round thofe which have a greater, and the "^^jjj^^lj.' greater Mafs a Body has the greater is, ceteris paribus, its attradivc Force; till rtrolalthus all the Planets revolve round the Sun, becaulc the Sun has a much greater Mafs than any of the Planets, for the MafTes of the Sun, Jupiter, and Saturn are refpedtively as i, i loo and .3000 ; fmce therefore the Mai- (t% of thefe Planets exceed thofe of any other in our Syftem, it follows tint the Sun (hould be the Centte ot the Motions of our planetary Syftem. 1 rr PHYSICALWORI^D. LV xrvi. If AttraAion be plt>pc»tionaI to the Mafles, the Alteration caufe4 by the T^ •!<««• Adkmof Jupiter on the Orbit of Saturn in their Coniundticn, ^"^^J^^JJ^J*^ much to exceed that produce4 in the Orbit of Jupiter bv the Adion of &i. mutvdjj * turn, fince the Mafe of Jupiter is much greater than tnat of Saturn, an^F^^'^ >a this Obfavation evinces ; the Alteration in the Orbit of Jupiter in its Con-^^J^*^ junction with Saturn, though fenfible i$ Qoi^derabljr leis than wha^t is ob- ntk»Tf thti? ferved in tbe Orbit of Saturn. malTn, xxyii. But if the Effe<a rf Attraftion, or the Space defcribcd by the attraded Body, depends on the Mafs of the attracting Body, why (hould it not alfo depend on the Mafs of the attraft^d Pody ? This Point furcly deferves to be examined. Experiment proves that all Bodies near the Surface of the Earth, when the Rcfiftancc of the Air is removed, defcend with equal Velocities ; for in the Air-pump, after exhaufting the Air, Gold and Feathers fall to the Bot- tom in the fame Time. Newton has confirmed this Experiment by another, in which the fmallefl Di£Ference becomes obvious to our Senfes. He relates (Prop. 14. B. s. and Prop. 6. B. 3.) that he compofed feveral Pendulums of Materials en- tii^y dmcrent 5 for inlfance of Water, Wood, Gold, Glafs, &c. and ha^ ving fufpended them by Threads of equal Length, for a ^nfiderable Time their Ofcilhtions were Synchronal. XXVIII. It admits therefore of no Doubt, that the attraftive Force of our Earth ^t^rtAioa is proportioned to the Mafles of the Bodies it attracts, and at equal Difbn- •„ proportki* ces it depends folely on their Mafles, that is on their Quantities of Matter ; nil to the hence if the tcrrcftrial Bodies were transferred to the Orbit of the Moon, ™*^** '*'***" it havu^ been proved ;lready that the fame Force z&s on the Moon andf^'£^*°beii^ on thofe Bodies, and that it decreafes as the Square of the Diftances. The had to the Diftances being fuppofed equal, it follows, that fuppofing the Moon de-*?™* <>J n>«- prived of her projedile Force, tliofe Bodies and the Moon would fall in^j'^^jnj * the fame Time to the Surface of the Earth, and would defcribe equal Spa- bodies. ces'in equal Times, the Refifbmce of the Air being taken away. The feme Thing is proved of j 11 the Planets having Satellites, for in- (lance, of Jupiter and Saturn 5 if the Satellites of Jupiter, for example, were all placed at the fame Difbnce from the Centre of this Planet, and deprived of their projcftile Force, they would defcend towards it and reach its Surface in the fame Time; this follows from the Proportion be- tween the Diflances of the Satellites and their periodic Times. LVI SYSTEMOFTHE XXX. From the Proportion between the periodic Times and Diftancei of the primanr Planets from the Sun, h may be proved in like Mvuier, that the Sun a^s on each of them proportionally to its Mafs, for at equal Diftanccs their periodic Times would be equal, in which Cafe, fuppofmg their pro- jefiilc Force deftroyed, they would all reach the Sun at Ae iamc Time-, therefore the Sun attrads each Planet in the dired Ratio of its Mafs. XXXI. This Truth is further confirmed by the Regularity of the Orbits whidj the Satellites of Jupiter defcribe round this Planet, for NexuUn has proved i[ Cor.3. Prop. 65. ) that when a Syftcm of Bodies move in Circles or regu- lar Elfipfcs, thcfe Bodies cannot be a£ted upon by any fenfible Farce but the attrad^ive Force which makes them defcribe thofe Curves ; now the S>i tellites of Jupiter defcribe round that Planet circular Orbits, fcnfiUyr^- lar and concentric to Jupiter, the Diftances of thefe Moons and of Jupi- ter from the Sun fhould be confidcred as equal, the Difference of tlidr Diftanccs bearing no Proportion to the entire Diftancc ; therefore if any 0* the Satellites of Jupiter, or Jupiter himfelf, were more attrafied by the Sun in Proportion to its Mafs than any other Satellite, then this ftnwg- cr Attradlion of the Sun would dillurb the Orbit pf this Satellite j aw Ncivtcn fays, (Prop. 6. B. 3.) that if this Aftion of the Sun on oncot the. Satellites of Jupiter was greater or lefs in Proportion to its Mais than that which it exerts on Jupiter in Proportion to his, only one thoufandth part of its total Gravity, the Diftance of the Centre of the Orbit of this Sa- tellite from the Sun would be greater or lefs than the Diftance of the Cen- tre of Jupiter from the Sun, by the two thoufandth part of its whdc Dif- tance, that is by a fifth Part of the Diftance of the outermoft Satellite of Jupiter from Jupiter, which would render its Orbit fenfibly excentric; fmce then thofe Orbits arc fenfibly concentric to Jupiter, the acceleratitig Gra- vities of the Sun on Jupiter and on its Satellites, are proportional to their Quantities of Matter. The fame Reafoning may be applied to Saturn and its Satellites, wbofe Orbits arc fenfibly concentric to Saturn. Experience and Obfervation therefore leads us to conclude, that the At- traftion of the Celeftial Bodies is proportional to the Maflos, as well in the Aitrtaionattraaing Body, as in the Body attracted ; tliat it is the Mafs which dcter- fiilwiyi re- mines a Body to revolve round another, that every Body may be confidtf- ciproc . ^ indifferently, either as attrafting or attr^ded ; in fine, that Attraffion is always mutual and reciprocal between two Bodies, and that it is the Pro- portion between their Mafles which decides when this double AttniSBon mall or ftiall not be fenfible. PHYSICAL WORLtf. tVlt llivre is afiother Property of Attradiofi^ by which it aQs ^udtly 6n Attnfttai Bodieswhethtr atReftorinMotion, oiid produces e<}ual Accelerations in ^y*^*°^* equai Ttmes, from whence it fellows that m AQion ts continued afid unt- mtiDudix form. Wbidi fufficiently appears f#on(i the Manner gravity accelerares whether the falting Bodies, ahd from the Motion of tht Planets, which as we have ^^'** ^5 «< fliewn before^ Are only greater ProjedHes regulated t^ the fame Laws. mociM."^ ixxin. Since the Proportion fubfiflii^ between the Ma^ of Bodtte which at- >MI« of tra£k each other determines how much one apprxMches towards the other, *^A^^'"-^' it is evident that the S6n having a much greater Mafs than the Planets, [hcpiunt their Aflioo on him ihonld be infcnrible. Hovfrever the Adi6ri of the oa Uu faa Planets upoll the Sun, tho^ too tnconfiderable to be fenfible, p^ciduces its Ef- fe& ; and on Examination we find that the center round which each Planet revotves is not the center of tM Sun, but the Point which is the common center of Oravity of the Siin and Planet whofe revolution is conddered. Thus the Mafs of theSm being (o that of Jupiter sis x to ^^^ and the diftance of Jupiter firom the Sun being to the Sun's femi diameter in a Ratio fomewhat greater^ it follows tb«t the common Center of Oravity of Jupiter and the Sao ia not far diftant from the Surface of the Sun. By fhe fame way of reafoning we find that the common Center of Gra- vity of Saturd and thaSoif h\lt within the SorfiMre of the Sun, and making the fame Calculation for all the Planets, Nittiton fays (Prop. 1 2!, B, 3.) that if the Eartfi and all the Planets were pfaced on the fame Side of the Sun, the common Centes of Gravity of the Sun and all the Planets would fcarce be one of his Diameters diftant from his Center. For thor* we cannot deter- mine the MaBes of Mercury, Vends and Mars, yet as thefePlariets are ftill lefs than Saturn and Jupiter^ which have infinitly lefs Mafs than the Su0^ we may conclude that their MaOesdo not alter this Proportion. XXXlV. It is about this common Center of Gravity that the Planets revolve^ and ^^'^ «^l^ the Sm hinafelf ofeillates round this Center of Gravity in Proportion to the ^^^' ^^ AAsonsof diePlaneu exerted on him. ^Vtien therefore we connderthe ^n /fcii.* Motion of two Bodies whereof one revolves round the other, rigott)ufly !»(« ro«o^ ^peakiag we ihoukt not regard the central Body as fited. The two Bodies, *^* ^^"^7 '^noB, the central Body and that which revolves rouml it, both revolve round griTi^ of their oonMnon center of Gravity,but the fpaces they deicribe round this cork our piaoctt- mon Center being in the inverfe ratio of their Mafles, the Curve defcribed ^ ^J'^*^ by the Body which has the lead Mafs is almoft infeniible: Vot this Reafon the Curve deferibed by the Body wlK>fe revolution is fenfible is only con- fUered, and the fonall Mitiofi of the central Body, which is regarded as fized^ sB^aegleaed. LVni. SYSTEM OF TH^ xxxy. The Earth and the Mton therefore revolve round dietr oomiAoo Coit^ df Gravitypand this Center f evolvet round the Center of Gravihrof the EartB and the Sun* The Care is the fame with Jnpiter and hb Moons^ Saturn and hit Satellites, and with the Sun and ait the Planets. Hence the Sod according to thedimrent PoTitions of the PUnets fhould move fucceffively on every Side around die common Center of Gravity of our planetary Syftem.' TMi csm- xnvk •fTrsWcr^U '^^ common Center of Gravity is at reft, ht the different Pi^rts of this m ftfT^ Syflem conftantly correTpDndrto the Ikme filed Stars y now, if this Center was not at reft hot nn^ves imifennly in a ftrsight Line, during fe many fhoufan^ Years that the Heavens haive beerobftrved, there muft have bc^ remarkiid feme Alteration in the Relation that the difierent Parts of our planetarjr Syftem bear to tiie fixed Stars ; biit as no Alteration has been oh* lervedf it is natural to conclude that the common center of Gravity of our Syflem is at rcflL This Center is the Point where all the Bodies of our pta« *Mc« this netary Syftem would naect if their proje£ble Forces were dcftroy*d. ^^ *S« ^ ^^^ Center of Gravity of our planetary Syftem is at rc^ the Center of flcntir«rc)M the Sun cannot be this Center of Gravity fince it moves according to the Aioi whitli different PofitionscMT the Planets, though on Account of thefmall Diftnnce ^i\ P*** between the Center of the Sun and the conunon Center of gravity of 0u' ^ ^' planetary World it never fenfibly recedes firom its Place. XKZVIK Since Attradion is proportional to the Ma&of the attra£Ung Body, and that of the Body attraded,we ftiould conclude that it belongs to every Par* tide of Matter, and that all the Particles of which a Bmiy is compofed attrafi each other ; for if Attraction was not inherent in every Particle of Matter it would not be proportional to the Mafs; XXX VI II. Asfirtr fo Thu Property of Attraction, of being proportional to the Mafles^fupplys ^foundH "' ^^^^ ^" Anfwer to an Objection which has been alledged againft the Mthtittrte- mutusl Attraction of Bodies. If all Bodies k is faid are endued with tfab ffoDof lercf- Property of mutually attracting each other, why is not the Attraction which trial borfiti fereftrial Bodies exert on each other ftnfible ? but it is eafty perceived that ftoaut!'*^ Attraction being proportional to the Maflei of the Attracting Bodies,the At- traction exerted by the Earth on ttreftrial Bodies is far more intend than what they exert on each other, and of Confequence thefe partial Atrat^ tioos are ftbrorbed and rendered infenfible by that of the Earth.. XXXIZ. ic ii fcnfi- the Academicians who mcafured a Degree of the Meridian in Peru, ioK bu ia fomc agined they perceived a fenfible Deviation in the plumb Line occafioned br thl^ir^ia^ flie Attraction of the Mountain Chimboraco the higheftof the Cordiliers ilia aim ^'tiif certain from Theory that the Attraction of this Mountaiif ftiouldafiicct the fHYfllCAL WORL9. MX. numb Line and til Bodies in its N«ghberhood^: l^t it remains tobiowplMil^ Has whether iheAuanrity of the oMerved Ekvktion corresponds with that which "* *yJ^ flioald refolt mm the Vulk of the Mountain for befides that thefe Obferyati- .ko. ^OQs do notdeaeraiine theprecife Qoantity of the Devttatiofi^oh account of the ' ^ eiTors infeperable horn practice, Theory does not fumifh ahy Method of ef- jdmatifig exactly the quantity of this Devitation^as the entire Magnitude. Penfity Joc. of tl» Moostavi are unknown. XL* The famereafoa that hinders us froni perceiving the mutual Attraction of Bodieson the furhce of the Earth, renders alfo the mutual Attraction of the ^leavenly Bodies very feidom fenfible. For the more powerhil Action that the Sun exerts on them, prevents this mutual Atovctipn from appearing^ However inCooiecsfes it is perceivable, ferinftancein the conjuncdon of Saturn and Jupiter their Orbits are feniibly difturbed, the Attraction of thofe two Planets being too ftrong to be abforbed by that of the Sun. As to the fennUe Attractions of certain tereftrial Bodies, fuch as Magne- Magnctilitt tifm and Electricity, they follow other Lawa and probably arife fronp Caufes ^^ *'^' different from the univerfal Attraction of Matter. aiArciu Niwtw demooftrates (Prop. 66.) that the mutual Attraedons of two cMfM ftm^ Bodies revolvii^ round a Third, dtfturb lefs th^ Regularity of their motions t^ «Biv«r yrhen the Body round which they revolve is a^tated bjr their Attractions^ m ollbo^ riumtf it was at reft; hence tbeinconfideraUe Irregularities obferved in the planetary Motions, is a further Proof of jthe mutual attraction of the celefti* jalBodiea. Thelrregolarilies in the Motion of any Planet arifing firpm the Aftiom nanmr oT #f the reft, are more or left eonfideraUe, in Proportion as the Sum of the dcccmiiiiiaf Tradioflscpmpofed each of the Nfafs and Square of the Qiflance of each of ^ '''.^* the other Planets, is more or lefs confiderable with reTped to the Mafs of ^q^Tot * the Sun divided by the Square of its diftance from the Planet, but as the the piwett Planes in wUch the Pbnets defcribe tKeir Orbs are differently fituated with '''f "< ^'^ neaped to each other, the Directions of the Central Forces of which the ^^"^j Planets are the Origin, are each in different Planes, and they cannot be all reduced to fewer than Three, by the Rules of the Compofition of Forces ; feach Planet therefore fliouid be confidered as actuated every inftant by three Forceaal the fame Time, the firft is a tangential Forpe, or a Force fueling in dieDirection of the Tangent of the planets Orb, which is the Re- pj\i of the Compofition of all the Motions which the Planet was affected irith the precedent Inftant The fecond is an accelerating Force, com- pounded of all the central Forces of the Planets, reduced to one in a right line in a Plane whofe Pofition is determined by the Center of the Sun, and )}y jtbjc Direction^ of the tangential Force} the Difference between thi^ 1 IX SYST.F^M or. THE compdUiKlcd' Force and tbe fimple ceotMl Fbrce which h« no oAer Semrce but the Sun, is called tbe peituprbadng Force. The third Foice is the de- tur bating Force, compounded of all the fame central Forces of the Ptaoets reduced to one in a Direakm perpendicular to the Planes of their Orbits ; this Force is very (mall in comparifon of the two othen, un acoouot ti the fmall Inclination of thofe Planes to one anotbjcr, and becaufe the Son Ab(tniAing pl^^^d in the Interfection of all thofe Planes does no vay contribute to the from tiic Production of this deturbating Force. If the Planets were only actuated by pmitaai ac- th^ t^o firft Forces their Combination would fenre to determine their th^hmett Trajectories which would be each in a conftant Plane^ and iftfiepeftor- fheir aphelia bating Force vaQiflied then they would be regular EllipfeSy and confequeat* arc at left. ly the Aphelia and Nodes of the Planets would be fixed (^rop. 14. E 3. 4r Prop. I . & 1 1 . E I .) if not ; the! e Trajectories might be confidered as mo- veable ElUpfes on account pf the prodigious excefs of tbe central Force o( the Sun over the perturbating Force, it is thos Newtw invefttgated the ?uantity and direction of the Motion of the Line of the Apfidks of the lanets occaGoned by tbe Action of Jupiter and Saturn, which according tQ his Determination follows the Seiquiplicate Proportion of the dtjilances of the Planets from the Sun, from whence he concludes (Prop. 14. B. 3.) that fuppofmg the Motion of the line of the Apiides ci Mars in which this Nb* tion i? the mod fenfible to advance in a 100 Years ^y^ 20^ in oonfequcatiit The flow the Aphelia of the Earth, Venus and Mercury would advance 17* 4^ motion of iq^ jjf & 4™ 1 6* refpectively inthefitmellme* of*the^ to- ^^^ ^^^ Motion of the Aphelia confinns the Law of univedal Qt^ XuUtk^ vitation, for Newton has demonftrat^ (Cor. i. Prop. 45.) that if the proof that Proportion of the centripetal Force would recede from the Duplicate to ap- attra^oii proach to the Triplicate only the 60th Part, the ApGdes would advance 1 iavVrft ra*io Dcgrccs in a Revolution, therefore fince the Motion of the Apfides isal? #fthef<]uare moft infenfibU, Gravity fpUowsthe inverfe duplicate Proportion of tbe of the Jif- diftancc?. ^^^ But the deturbating Force which afis at ihefame Time canfes the Planes of thofe moveable Ellipfesto Change contintxally their Fofition; let there be fuppofed in the Heavens an immoveable Plaoe, in a mean Pofitioi! between all thofe the Trajectory of the Earth would take in coniecpjeiiceof the deturbating Force, which may be cgUed the true Plane of the Ecliptie^ it is mantf^ft that this Plane being very little encltned to the Plane of the Orbit of Each Planet, it is almoft parallel to it, and confequenily the Dired* ion of the deturbating Force is always fenfibly perpendicular to the tnie ^lane of the Ecliptic, and it is nfy to conceive that the efied of this Force produced in the Direction in which it acts, is either to remove the Pbnet ^rom or to make it approach the true Plane of the FxKptick, confequenth to c^ufe a Variation in the Inclination of thefmall Arc which the Planet det i«|-W- PHYSICAL WORLD. LXL ciibct that tnflttit with the true Plane of the Ecliptick^ the Pofition of the Planes of the TrajeAories of the Planets varies therefore in Proportion of the Ifltenfity of the deturbating Force, and in the Diredion in which this Force a£b ; if for Example the Force tends to make the Planet approach the true Plane of the Ecliptic the Node advances towards the Planet with a Velocity, which tho' iinail increafes dtminiihcs or vaniflies according as the intenfity inf the deturbating Porcie increafes dtmintfhes or vanifties, but in this Caie ^ Node cannot a,dvance or go meet the Planet without moving in an op- posite Dtredion to that of the Planet, if therefore the heliocentric Motion is retrograde as in a great Number of Comets, that of the Nodes will be di- refty the contrary would arrive if the deturbating Force tended to remove the Planet from the true Plane of the Ecliptic. N^wfon fays that fuppofing the Wane of the EcKpttc to be fixed the Regreffion of the Nodes is to the fo^Sf thV Motion of the Ap helium in any Orbit of a P)anet as i o to ai nearly ( c ). node* of th^ It is therefore only by this Compofition of Forces that all the Ir*^ pitnettac- jegttlarities of the celeftial Motions can be inveftigated, it is by difcern- Ncwtoa!'* ing thepartkidar FJfedsofeaeh of thofe oompounfled Forces, and after- wards uniting them, that not only thofe Inegularities that have been (cMerved can be determined, but thofe which inrill be remarked here- after will be foretold. But it is eafy to perceive how much fagacity and addrels to handle the (iiblimeft Anatyfia thefe Refchearches require, and as it it ahnoft impoffible to combine at once the central Forces of.more than three Bodies placed in different Planes, in order to difcover the in^larities of the Motions of a Planet or Comet it is neceflary to calcuUte fuccefively the Variatiomthat each Planet taken feperately can caufe in the central Force 6f which the Sun is the Focus. The Suceefs that has attended the united Efibrts of the firft Mathematicians in Europe (hall be expUined hereafter. ' Tiiory c/ the Figure 9f tbe Planet f. I. The Planets have another Motion viz. their Rotation round their Axes, we havefeen already,that this Motion of Rotation has only been difcovered orT£frec!!nI in the Sun, the Earth, Mars, Jupiter and Venus, and that' Aftronomers do motion of not agree about the Time in which Venus turns round tho' they are onani- ^^ pi«B«tt mous with refped to its Rotation. But tho' it has not been difcovered from j^ bMn*dir Pilfer vation that Mercury, Saturn and the SateUities of Jupiter and Saturn cover«4. * torn round their Axes, from the uniformity that Nature Obferves in her Operations, it is highly probable that thofe Planets revolve round their Axes, and that all tbe cdeftial Bodies partake of this Motion. (i) Dt SyaeoBSlt n«adi Pi|e z€ £4itioo, 1731. •fJ^P SYSTEM OPTHt This Rotatioo of the Pltnetp foond tinif Axes is the odIt cdeftiaf Motion which is anifern^: this Motion does npt appear to trife mm Gra- vity, and its Caufe has not as yet been diffovered^ The mjutual Attra£Uon of the Parts of which the Plaoeu aur compoled JJ'J^*' binds them together, and prevents their being difperfed by this RotatioiL of the parts Fof it is weil ^Bown that aU Bodi^ nwrnng round acquire a centrifsgal Poroe whick C019 by whiph they endeavour to recedie from the Center of their Rci^uttoiis 4 Sm«tt pr*. hcpcc, were not the Pari* of the Planets held together by their mutual An Veau them tradions, they would be difperfed apd fcattered by their Roution. For from htiDg .fuppofiog the Gravity of luiy one Part of the furface ot the revolving Bodj tbe roo^M <J«ft">vcd, this Part uiftead of revolving wijth the Body would fly off in the ^ 'diredion of the tangent; therefore if Gravity did not cpunteraft the Eflbrta of the centriftigal Force which the Parts of the cdeiKal Bodies acquire 19 revolving round their Ax.es, this force vrould diiperfe their Parts. Ill* Thp' this mutifal Attrai^on of the Parts of a PUnet, counterafis the The nxpts /centrifugal Force, yet it does not deftroy it, this Force ftill productng itf ry motion, ppe^, in rendering the diameten of the revolving Body unequal, fupposr !^'«tori of m it to ^ 'Ividi for the SUnets being compofed of Matter iirhofe Parttder fi e pUseti. at equal Diftaoces are equally urged to the Center, they wpuid be cxaS Spheres if they were at reft- But in ooofequeoce of the Motion of Rota- tion the Parts acquiring $ centrifugal Force endeavpur to recede from their Centers with Forces which increase as they are placed nearer the Equator of the reyolving Body, fioce the centrifugal Forces of Bodies revolving m .Circles,, are as their Rays (upppfiog the Time of Rev)t>lution to be equal: therefore f^ppofinp the Planets to be fpherical and compofed of fluid Mat- ter, before they acquired a Motion of Roution, that the Equibliriuai of th4r Parts may be preferved during thisRotation, and that they nuy afiiime | permanent form it was neceflary that theColumn whofe weight vraa dinu% lihed by the centrihigal Force flioold be longer than the ColuaHvshofe W^eight is not altered by the centrifugal Force, and therefore the Bquatniirial ptameter mud exceed the Diameter paffing thro* the Pblcf. IV. Ne^iw in (Prop. 19. B 3) determines the exceiii «f the equatorial Metltpd |jx)ve the polar Colump of the Earth, fuppofing as he <ses all thro' the Prin* ^e^fer dS^ cipia that the Gravity of Bodies near the fiirfac^iDf the Earth U the refuhof f «- miniog the Attra&ion, of ail the Particles of whkfa the E!arth coofidered as Homo* ^ ^^"^5 ^^ geneoys is compofed: he employs for Data in the Solution of this Problcink T"" ^'^P 1 ft the Semidiameter of the Earth confidered as a Sphere and determji^ by Picard to be 1961580a Feet Q,\ the Length of the Pendulum ^ibrating feconds ifi th^ Latitude of Paris which i^ 3 Fcpt 8f Liiiea. \ P*[tSltAL WORLl9. iXIft; ^rom the llieory of OTcillations and this Meafure oFa Pendulum vibrating fecondsy he proves that a Body in the Latitude of Paris making the neceflary Corredion for the refiftan^ of the Ai^, defcribes in a fecond 2174 Lines. A Body revolviAe \h a Qrde at ihc Diftattee of iffSi <6oo Feet from tlhe Center; which is the Semidiameter of the Eartb,^ in iy 56* 4' which is fh^ exadTime of thediufnal Revolution, f6pp6fingits lOfotioft uniform, delcrbes in a fecond; an Arc of .1433, 46 Feet; of whidi the verfe, Sine is^; 6,05236^6 Feet, o^ 7. J4064 Lines ; therefore ih the Latitude .of Paris the torte of Gravity is to the cenb-ifugjaf Force, which Bodies at the Equator derive ffom Aie murnal Rotation, as 2174 to 7, 54064. Adding therefore to the Force of GraVitv. in the Latitude of Paris, the Force detraded there- from by the centrifugal For^e in (hat Lfititude, in order to obtain the total Force 6f G^vrfy \ii the LMfude of Paris, Ife^oA finds that this total Force is to tie ceh^trifiigat Force under the Equator as 2^9 to i fo that unde^ die E^iiat6r the centrinigal Force ^imiaiflies the centrifugal F^cc by ,|» N^ton determines (Cor* 2. Prop. 91.) the Proportion of the At- tra^^on of a Spheroid upon a Corpufcule phced in its Kkt produced, to that of a Sphere, on the fame' Corpuicule, whofe Diameter is equal to the iefler Axe of the Sphercnd; emptying therefore this Proportion ind fuppof- rng the- Earth hombgeneod^ and at rdl^ he finds (Prop. 19. B. 3.) that if its Form be that of a -Spheroid whofe Iefler Axe is to the greater as lod fo loi, the Graving (g) at the Pole of this Spheroid^ will be to theGravit^r fX) at the Pole 0/ a Sphere, whofe Diameter is th^ \tSkx Aie of the ^herbidas 126 to I2^V In' the (kme Manner foppb^ng if Sf^emd tyhofi^ equatorial Dtameter is the Axe 6^ Revolution, the Gravity (V) at th^ Equatbr which h the Pole of this new Spheroid, will be to the Gra^ty (T) of a Sphere at the fame Place having the fame Axe of fl!eVolmion; as i 25 to i 25. Nenutm ihews afterwards that a mean pro^rtio^al (d) between thefe two Gravities (V, t) exprefle^ the Gravity at theEqttator of theEarthr ^oniequentiy theGravitv (G) at the Equator of the Earthy is t6 the Gravity (f) of a Sphere at the iame Place, having the fafme A^e of Revolution, as 1254, to 12& and having demonftrated (Prop. 72) that the Attradion of homogeneous Spheres at their Surfaces is proportional to their Rays, it follows that the Gravity (» at the Surface of the Sphere whofe Diame- fer is the le(ter Axe cf the Spheriod, is to the Gravity (T) at the Surface of the Sphere whofe Diameter is the |^reat Axe of the Spheroid, as 100 to 101 wherefore by the Compofition of Ratios g X 7 X r it to y X G X t or the Gravity (g) of the Earth, at the Pole, S3 to the Gravity (G) at fhe Equator as 126 X 126 X 100 to r25 X 125I X xoi that is as 501 to 500. Buft he had demonftrated, (Cor. Prop. 91.) that If the Corpufciile is ptacfd wkhiii the Spheroid, it would be attra^M in the Ratio of its diflaaoe LXIV. SYSTEM OF TI^E from the Croter; thererore the Gravities in each of the Canals ooirrefponii- ing to the Equator and to the Pole will be a* the DilUnccs firom the Cen« ^ ter of the Bodies, which are placed in thofe Canals; therefore iuppofing theTe Casals to be divided into Parts* proportional to the Wholes^ owfc- quentely at Dtftances from the Center proportional to each other, by Tranfverfc Planes, which pafs at Difiances proportional to thofe Canals. { The Weights of each Part in one of thofe Canals, will be to the Wo« ghts of each correfpondent Part b the other Canal, in a oonftant Raiio^ coniequeotly thefe Weights wilt be to each other in a conftint Rttio of each Part, and their accelerative Gravities Conjointly, that is as lot to 100, and 500 to 501, that is, as 505 to $ot i therefore if the ceo- trifugai Force of any Part of the Equatorial Canal be to the abiblot« Weight of the (anie l^art as 4 to 505, that is, if the centrifiig^ Feroc detra£ls from the Weight of any Part of the Equatorial Onal ^ Parts, the Weights of the Correfpondent Parts of each Canal will be- come equal, and the Fluid will be in Equilibrio. But we have fees that the Centrifugal Force of any l^art under the Equator, is to its Wei- f;ht as I to 289, and not as 4 to 505; the Proportion of the Aaxs therefore muft be different from that of 100 to loi, and fuch a Pro- portion muft be found as will g^ve the Centrifugal Force under the Equator^ only the iSgih Part of Gravity. J^J^hM But this is eafly found by the Rule of Three ; for if the Proponioo ooDcMe* of 100 to 1 01 in the Axes has given that of 4 to 505 for the Pwf^ the rfttio of portion of the Centrifugal Force to Gravity, it is manift^ that the Pnn tbe cardi^to P*^*"^*^" ^^ ^^9 ^ ^3^ ^ rcquifite to give the Proportion i to 389 of iw't htt of (he Centrifugal Force to Gravity. aa9 to 130, ♦• Tbe flat. '^^^ Conclufion of Newton, that is, the Quantify of the Deprei&oa of aeftorthe* the Farth towards the Poles, which he has determtnM is grounded on cftrth to- i^is Principle of the mutual Attra&ions of the Paris of Matter. Boi poVel'ionld ^^^5 Depreffion towards the Poles would olfo rcfult from the Thcor]^ alwiyi re. of FluIds, and that of Centrifugal Forces, tho' NewiofC^ Difcoverica t«it from the concerning Gravity werd rejc&edyiinlefsvery improbable Hypotbefes con* te^rTfiTgii ^«^rning the Nature of primitive Gravity were adopted. forces and V* thatoffluida Noiwithflanding the Authority ot Nauton^ and although Hugbens ia uitfiiofli'^m alTuminga different Hypoihefis of Gravity arrived, at the fame Condufioa ▼icy it af. ot the Dcpreffionof the Earth towards the Poles ; and tho* all tbe £a* rucued. periments made on Pcndulun^ in tbe different Re^ons of the Eartl^ The met- confirmed ihe decreafe of Gravitv towarda the Equator, and confe* lure of the qutntly favoured the opinion of the Flatnefs ot the Earth towarda tb^ degree, of p^|^g^ ^^^ ^j^^ Mtafures of Degrees in France, which feemed to de» tke meridi as ifi TraBc, crcaft as the Laiiiude increated ftiH rendered the Figure of the Earth. t^HYSlCAL WORLD. LXV. tittcertaiiu Hypothcres were formed on the Nature of primitive Gra- •ceifion««i vitjr, which gave to the Earth, fuppofcd at reft, a Figure whofe Alter- ''••^* ^«* atkm agreed with the Theory of centrifugal Forces, and with the ob-ZSITJJti! long Figure towards the Poles refulting from the adual Meafures. £nH. For the Queftion of the Figure of the Earth depends on the Law ac- cording to which primitive Gravity afts, and it is certain, fcr Example that it this Force depended on a Oiufe which wouM make it draw fometimes to one Side 4iiid at other Times to another, and which increafed or dimintfli- cd without any conftaot Law, neither Theory nor Obfcrvation ever could determine this Figure. VII. To decide this Queftion finally it was Meceffary to Meiforfc a Degree un- The m^ ' der the Equator^and another wi*«?*h« polar Circle; if the French Af- faren ofSl litn iatth€ rcirck _ ^ „.^ ^..„ sttht Tbwry of Newton^ with Refped to the Figure of the Earth, whofe De- «l»»»r preffion towards the Poles k now uni verfally allowed. S^^* Vni. ^( Newti In determining the Ratio of the Axes of the Earth, Newton befides the mutual AttraAion of the Parts of Matter fuppofes the Earth to be an Elliptic Spheroid, and that ks Matter is Homogeneous ; Maclaunn in his '^^ ^"PP* cxccHem pjice on the Tides which carried the Prife of the royal Aca-?*^'"*** demy of fiances in 1740, was the firll who dcmonftratcd that the Earth fup- in delT^ii pofed FWd Aod Homogeneous, whofe Parts attrad each other mutually and iaiche a. are befid«i Attra£Ud by the Sun and Moon, revolving about its Axis, would ^"'J^^^* necfflkfily aflume the Form of an EHiptic Spheroid, and demonftrated fur- Mid«ri« Hner^ liuit In this Spheroid not only the Diredion of Gravity was pcrpendi- vcriiicd tho oidlM' 10 the Surface, and the Central Columns in Equilibrto, but that any ^^ Foiit^ wliatfeever within the Spheroid was equally preflTed on every Side; vriiicli laft Point was no lefs Neceflfary to be proved than the two firft, in Order to be afliired that the Fluid was in Equilibrio, yet had been negleCted ^by all tbofe who before treated of the Figure of the Earth. The Cafe is not the fame with regard to the fecond Suppofition viz* it i« probi fhe Homogeneity of the Matter of the Earth, for it is very poflibk Ue that Um (andAfirw/^nhinifelf was of Opinion Prop, ao B, 3) that the Denfity of^UJ?"**** fhe Earth increafes in approaching the Center, now, the different Den* ^' ItHiee of the Strata of Matter compoiing the Earth fhould change the L»a«^ according to which the Bodies of which it is compofed Gravitate^ and oi Confequence ibouUl alter the Proportton ot iu Axes. '^ LXVL SYSTEM OF THE IX. Ctairaut improving on the Rerearches of Macliurin has fhewn that a* ^'j^ '•**• mong all the moft probable Hypothefes that can be framed coocemingthc •f ch« «!!tb Dennty of the interior Parts of the Earth cofidered as an Elliptic Spheroid, dccrctr«t io that adopting Attradion, there always fubfifts fuch a Connexion between fropoitioii |])^ Fra^on expreiEng the Difference of the Axes^ and that which ex« lUtu^t^Mi'VtS^ the Decreafe of Gravity from the Pole to the Equator, that if one chff poles, of thofe two Fra&tons exceeds ^fv by any Quantity, the other will be ex- actly fo muchlefs; Io that fuppofing, for Inftance, that theexcefs of the equatorial Diameter above the Axe is ^fi-, a Supposition conformable with the adual Meafuf es, we (hall have ^ -^ «fe or ^jt for the Quantity to be fubtraded from rfv in Order to obtain the total Abreviation of the Pen- dulum in advancing from the Pole to the Equator, that is to fay, that this Abreviation or what comes to the fame the toul Diminution of Gravity, will be TJ^ — rir?; or j\t nearly. Now, as all the Experiments on Pendulums fliew that the Dimination of Gravity from the Pole to the Fjquator, far from being lefs than x\^ as this Theory requires, is much greater, it follows, that the adoal Mea> fures in this Point are inconfiftant with the Theory of the Earth confix dered as an Elliptic Spheroid. It follows from the Theory of Clairaut, that admitting, the Soppofi- tions the moft natural we can conceive or imagine with regard to the internal Strudure of the Earth confidered as an oblate Elliptic Spheroid, that the Ratio of the Axes cannot exceed that of 229 to 230 ftnce thk Ratio is what arifes from the Suppofition of the Honoogeneity of the Eaj-th, aad that it refults from this Theory, that in every other Cafe Gravity in- creafing, the DepreiTion towards the Poles is lels. Tho' the Earth fuppofed Fluid and Heterogeneous whofe Parts at- tract each other mutually, alTumes an Elliptic Form conliftent with the Laws of Hydroflaticks, yet it might equally aflume an infinite 'Number of other Forms conliftent with the fame Laws, as Dalambert has demonftrated^ and as a Variation in the Form would neceflarily produce one in the Decreafe of Gravity from the Pole to the Equator, and confequentty in the Ratio of the Axes, it is highly probable that a Figure will be found that will condudl toa Refult fuch as will reconcile Theory with Obfervatka The Recherches of this eminent Mathematician fliall be explained hereafter. Newton having computed the Ratio of the Axes of the Earth, detcr« mjnes the Excefs of its Height, at the Equator above its Height at the rules, in the following Manner. The Semidiameter (b + c) at the Equa- tor being to the Semidiameter (b) at the Poles, as 230 to 229, c ^ -^ and 2b =458 c. and the Mean Semidiameter according to Picart's incnluration^ bein£ 19615800 Paris Fcet^ or 3^a3| (6 Milei^ PHYSICALWORLD. LXVII. (rec1u>ning 5000 Feet for a Mile,) 2 X 1961 5800 = ab + c. confeqtientljr 459. c. = 2 X 1 961 5800 and the Excefs (c) of the Height of the Earth at the Equator, above its Height at the Polet, is 85472 Feet or 17 Miles itt and Subftituting in the ^uation 2 X 19615800= 2b + c. for c its Value, there will refult 459b =r 2 X 1961 5800 X 229, wherefore the Height (b) at the Poles will be 19573064 and the Height (b+c) at the Equator 1 9658536 Feet. z. After determining the Relation of the Axes of the Earth fuppofed Ho- yj^^t ^rt tuogeneous, Newton inveftigates after the following Manner (Prop. 20 B. 3) the frcigbti what Bodies weigh in the different Regions of the Earth. Since he had jJ^^JJJJ't proved that the Polar and Equatorial Grfumns, were in Equilibrio when their fcgioot of Lengths were to each other as 229 to 230 it follows that if a Body (R) be the earths to another (b) as 229 to 230, and the one (B) be placed at the Pole^ and th^ other (b) at the Equator, the Weight (W) of the Body (B) will be equal to the Weight (w) of the Body (b). but if thofe two Bodies be placed at the Equator the Weight {fF) of the Body (B) will be to the Weight (w) of the Body (b) as 229 to 230^ wherefore the Weight [W] of the Body [B] at the Pole will be to the Weight [ff^ of the fame or of an equal Body at the Equator^ as 230 to 229, that is reciprocally as thofe Columns, we fee by the fame realbning, that on all the 0>lumns of Matter compofing th^ Spheroid, the Weights of Bodies flioutd be inverfely as theie Columns, that is ais their Diftances mm. the Center : therefore fuppofing the Diflance, of any Place on the Surface of the Earth, from the Center to be known, the Wdght of a Body in this Place will be known, and confequently the Quan- tity of tbe Increafe or Decreeife of Gravity, in advancing towards the Poles or the Equator: but as the Diftance of any Place frmn the Center decreafes nearly as the Square of the Sine of the Latitude, or as the Verfe Sine of double the Latitude as may eafly be proved by Calculation, we fee how Nffvton formed the Table given (Prop. 20 B. 3) where he lays down the Decreafe of Gravity in advancing from the Pole to the Equator. Example. The Latitude of Paris being 48' 50* that of Places undef the Equator 00* 00" and that of Places under the Poles 90^1, the verfe Sines of double thofe Latitudes are 1 1 34, ooooo,and 20O0o,and the Force of Gravity (g)at the Poles being to the Force of Gravity (G) at the Equatoi* aa 230 to 229, the Excefs (g — G or E) of the Force of Gravity at the Pole, is to the Force of Gravity (G) at the Equator as 230 ^ 229 to 229, or as I to 229 but the Excels (e) of the Force of Gravity in the La- titude of Paris is to the Excefs (E) of the Force of Gravity at the Poles as t r 334 to 20000,wherefore by the Compofition of Ratios, e x E is to ExG, or the Excefs [e] of the Force of Gravity in the Latitude of Paris is to the Force of Gravity [G] at the Equator as 1x11334 to 229X20000, LXVIIL SYSTEM OF THE that IS, as 5667 to 2190000, and the Force of Oravity [e-f G] ib the LstK> tude of Paris ib 10 the Force of Gravity [G] at the Equator as 5667+22900^ o» that is, as 2295667 to 22900a By a Uke Calculus the Force of Gravity IQ any other Latitude is determined. The? MT9 ^ Gravity is the fole Caufe of the Ofcillatioiis of Pendehifm, the proportioQai (lackning of thefe Oicillations proves the Piaiinution of Gravity, aad *^ f? their Acceleration proves that Gravity wEti more powerfully ; but it is de^ fhronti pl^ n>onftj'*^cd that the Celerity of the Ofcillations of Pendulums is inverfely 4iaaQ»» as the Length of the Thread to which they are (ufpeiided, therefore when in Order to render the Vibrations of a Pendulum in a certain Latitude fynchro* nal with its Vibrations in another Latitude, it muft be fliorteoed or lesgthiw ed, we ibould conclude that Gravitjr is \eb or p-eater in thisRe^oa than in the other ; Hugbens has determined the ReUti<m which fuUifts be« tween the Quantity a Pendulum is lengthned or (horten^d and the Di^ sninutipn or Augmentation of Gravity ; fo that this Quantity being pco» portional to the Augtpentation or Diminution of the Weight, itfew^ $em has given in his Table the Length of Peoduluma taAoKl of die Weights. Example. The Length of the Pendulunp in the Latitude of Paris being }£ Z\ 561/ the Gravity in the Latitude of Paris [2295(667] is to the Gravity aft tba Equator [2290000] 29 the Length of the PeiuJukim in the Latitude of Paria [3^* 8*» S6i J to the Length of the Pendulum at the Equator \iK 7 ^684} By a wt Calcuhtf the tength of the Pendulum in aoy other Latitude ia 4e« fenpip^ Th^ Degrees of Latitude decreafingin the Spheroid of JS&wIm ia tiio S^utltSr '^"^ Pr<)^rtion as the Weights, the fame Table gives the Qspaiity ef tre in che the Degrees in Latitude coQinnencing from the Equator where the Lamude H^e pro* i« o^ to the Pole wherf it is 90'. fortioiL Example. Th^ Length of a Degree [d] at the Poles^ beiag to the Leogtb of a Degref [D] at the Equator, as the Ray of the Circle which has the fame Curviture aa the Arc of the Meridian at the Pole, is to the Ray of the Grcle which has the fame Curviture as the Arc pf the Meridian at the Equator of the Earthy that iSy by the Property of the EUipTis^ as the Cube of 230 to liic Cube of 229^ that is, as 12167000 to 12008989, the Excefs [d^D or E] of ihe Degree at the Pole is to the Degree [D] at the Equator, as 1 5801 1 to a 2008989 ; but the Excefs [e] of a Degree in the Latitude of Paris, is to the JLxcth [E] of the Degree at the Pole, as 1 1 334 to 20000 verfe Sines of Dott« Ibleof thofb Latitudes. Wherefore by the CompoTiLion of Ratios eXE is 10 ExD^or the Excefs [e] of a Degree in the Latitude of Paris is to the Leagtb of the Degree [D] at the Equator, as 89544.8337 n to 12008989000 ; £id the Length [H-DJ of a Degree in the Latttftidf of Pari( ia to tlie Length of % PHYSICAL WORLD. Diogrce [D] ai tlieEqnator, at 120985338337 to 120089890000; but tht Leni;th ol a Degree in the Latitude of Farit, according to PjV«rrf*i, Menfura- tion is 57061 Toires^ wheretore the Length of a Degree at the Equator h 56637. By a like Calculus the Length of a I>egree in any other Latitude is Dect^rmiaad. XII. LXIX LstihuUrf PlmdJum. iittfimrf umDtg OMPltce. in tbe Mtriditn. Deg. Feet Lines. Toifej. 3 • 7,468 5«637 S 3 • 7,48a 56642 10 3 ♦ 7,526 56659 56687 »5 3 • 7,59« 20 3 • 7,69a 56724 as 3 • 7,8it 5^769 30 3 • 7.948 56823 35 3 • 8.099 5688a 40 3 . 8,a6» 56958 I 3* 8,394 a 3 • 8.327 56971 3 . 3- 8,361 S6984 4 3 • 8,394 3 • 8,448 56997 4S 57010 6 3 • 8yt6i 57024 I 3 . 8,494 3. 8,528 5703$ 57041 9 3 • 8,561 57o6r 50 3 • 8,594 57074 S5 3 . 8,756 57>37 «o 3 • 8,987 57195 «S 3 • 9,044 57250 70 3 • 9,i6a 5729s 80 3 • 9.329 573«<> «5 3 • .9,37a 3 • 9,387 57377 90 5738a JLtlU Ntwt9rf%TM^ givesthedecrea&of Grsvkyivoin tbePoIetotheEquator fame what left than what refiibs from aAual MeafinrcSy but this Table is only a4cQb|todforiiipCaieof Homoipiieity^ ami be informs oa ariJi^ ^d of LXX. SYSTEM OF THE tbe ProBofifcion where he gives this Ttble^ that fitppofingtheDchfity of the Parts ot the Earth to increafe from the Circumference to the Center, the Dimiaution of Gravity from the Pole to the Eqaator would alio increafe. XIV. , Ahho Newton Teems inclined to believe^ from the Obrenratioiis he rektei btttea thU* ii^ Vxo^, 20 on the lengthning of the Pendulum occafioned bj the Heat in diflb^Mtco'the Regions of theEquator, that thefe Differences arrife from the different chthcttat Temparature of the Places in which the Obrervations have been made, the which^l«« S^^( C'**^ ^"^ Attention employed in prerervtng the fame Degree of Hnt thfiit tht by means of the Thermometer in the experiments made fince Newtm^t P«*^^"B Time on theLengthof PenJulums in the different regions of the Earth provci itioM bar* ^^^^ ^^^^^ Differences do not arife from this Caufe^ and that the Dc- iMMr ex- creafe of Gravity from the Pole to the Equator exceeds the ProportioD af- Dtrimcnu fign'd by Newton in his Table. thrt ttoS" ''" ^^^ *^ Lengths of the Pendulum Correded by the Barometer and 4iSEbrcncct reduced to that of a Pendulum ofcUlating in a Medium urithout Rcliftance ciMoc uik are under the Equator^ 439, ai Line/p wbiM ^* Portobdlo Latitude, 9 Dq;rees» 439, 30 o, 09 Difirmea. of the pea- At title Goave Latitude, 18 Degrees, 439, 47 o, a6 Mamrio At Paris Latitude, 48'' 50* A¥>» ^1 l> 4^ ^H^h Vi. At Pello Utitude, 66* 48n> 441, ay a, 06 tb^tT<^i- Nowthe differences proportional to the Squaresof th« Sues of the Latitude^ 0Bi. are 7, 14, 138, ao5^ which are lefs than what refu^ts from Experimeat XV. Method At the End of Prop. 19. B. 3. Ntwion (hews how to find the Proportioa (ivea bf of ihe Axes of a Planet whofe Denfity and diurnal Rotation are knownk, em- for1^4iiit ploying for Term of Comparifon the Ratio difcovered between the Axes of the ratio of the Earth ; for Whether the Bulk or Ray (r) of a Planet be greater or left the txet of than the Bulk or Ray (R) of the Earth, if its Denfity (d) be equal to the Den- my piMct. ^1^ p^ ^f ^1^^ £^^{^^ ^^j ji^^ j^i^^ ^^y ^ j^, diumal Roution be equal to the Time (T) of the diumal Rotation of the Earth, the fame Proportion will fubfift between the centrifugal Force and Gravity, and confequently between its Diameters as was found between the Axes of the Earth : But if its <&- urnal Rotation is more or lefs rapid than that of the Earth, the centrifugal Force of the Planet will be greater or le& than the centrifugal Force of the Earth and confequently the Di£Ference of the Axes of the Planet will be great- r R. er or lefs than the difference of the Axes of the Earth in the Ratio of --to rp r^ (Cor. a. Prop. 4.) and if the Denfity of the Planet be greater or lefs thsm the Denfity of the Earth, the Gravity on this Planet will be greater or lefs than the Gravity on the Earth, in the Ratio of d r to DR, and the DiflRn^* ence of the Axes of the Planet will be greater or left than the DiffistCKe of PHYSICAL WORLD. LXXI- ttitAxesof the Earthy in Proportion as the Gravity on the Planet is lefs or greater than the Gravity on the Earth confequently in the Ratio -^ ^^^tf wherefore if the Hmt of Rotation and Denfity of a Planet be different frora that of the Earth, the Diflerence of the Axes of this Planet compared with its lefler Axis, it to ^ the difibrence of the Axis of the Earth compared r R Di^TT with its leffer Axis, «»7;^^ '^ t TxD k "^^^^^ ^""^ ^^ ^ "dxTT '^^ the exprei&on of the Difierence of the Axes of the Planet. XVI* 6Kt#rfiiiiM Hence the Difierence of the diameters of Jupiter, for inftance whofe di- tionefthe* nmal Revolution and Denfity are known will be to its leflcr Axis in the com- ntio of tl^ Eund Ratio of the Sauares of the Times of the diurnal Revolution of the ^^fj^^ irtb and Jupiter of the Denfities of the Earth and Jupiter, and the Difference iag to thii OMthod* of the Axes of the Earth compared with its lefler Axis, that is, as ^^ X 122 x-^ to I. that is, as I. to 9 I needy : Therefore the Diameter of 49* »«9 Jupiter from Eaft to Weft is to 'its Diameter paflittg thro' the Poles as lO f fo 9 f neerly. Niwton adds that in this Determination he has fuppofed that the Matter of Jupiter was Homogeneous, but as it is probable on account of the Heat of the Sun that Jupiter may be denfer towards the Regions of the Equator than towards the Poles, thefe Diameters may be to each other as 13 to II, 13 to la, or even as 14 to 13, and that thus Theory agrees with Obfervatimi^ fince Obfervation evinces that Jupiter is depreflcd towards the Poles, and that the Ratio of his Axes it lefs than that of xoj> to 9^ and is confined between the ratios of i x to 1 2 and 13 to 14. XVIt Awrrim- This Method that Newton takes to explain a Depreflion towards the Poles Sf^ig^S of Jupiter lefs than that which refults in the Cafe of Homogenity feems hy Mtwwn very improbable,it is furprifmg that in Order to explain the flatnefs of the Fi- "^l^^ sure of Jupiter, he has had recourfe to a Caufe whofe Effeft would be much ^^ ^ j, more fcnfibly perceived on the Earth than in Jupiter, fince the Earth is much pita i. kft nearer the Sun than Jupiter. \. .. . .^ .,. t% r^- ^oiufSm The Propofition of aairaui that the Flatnefs dimmiflies as the Denfity m- j,^^^ ^ creafes towards the Center, furnHhes a natural Explication of this Phenome- nofi infuppofing Jupiter denfer^towards the Center than at the Surface^ an yrhf the Hypotfaefis entirely confiftent with the Laws of Mechanicks. t^o of a^ XV II I* TaDitcr the As the two Principles ineceflkrj fiwr determimng the Axe» namdy thei,rtfc^ Aonul Rewlutioii and the Denfity, ere known only in Jupiter, the Etrth, A.fi» «« and the Son, thefe are the only celeffitlBodiet the Proportion of whofe Ax- fc««»»0. ct can be ^covered. How this Proportioa hm been dUcoTcred u the Earth LXXir. SYSTEM OP THE and Jtipiter has been already ihe wn ; the Difler^nce of the Axes of the Sfhi tiSJ'ofThr ** '° '*' ^^^'^^ ^*'* *" ^^^ compounded Ratio of the Square of i to 27^ diur- txen of the nil Revobition of the Earth to that of the Sun, of 4xx> to 100 Denfity of fan if too the Earth to that of the Sun, and ai^ Difference ot the Diameters ot the iTc^^to^V ^^'■^'^ co.Tipared to its leffcr Axe, to i, that is, as ^^V? ^® I, a Differ- obrerTc<l. ^"^^ ^oo inconftderable to be obferved* Theory of tbe Fncijjion of tht Efuimoxes. I. It vas For many Ages it had been thought that the Axis of the Farth dur- aTon*\*i^e ing Its annual Revolution prefcrved the fame Pofition, and this Suppofiti- th^Ti'hJ"* on was very natural. For Theory (hews that this Parallelirm (hould refuk tsit of the from the two known Motions of the Earth, the annual and diurnal Motion ; Teft'i'dYt'i *"**'" Faa for a Number of Years this Parallelifim isfenfibly prcfenrd. pti«Ut^(tBi* ^^^ fi^om the Continuance, and accuracy ot Aftronomical Obferrations it has been difcovered that the Poles of the Earth are not always direded to the fame fixed Stars, and of Confequence that the Axis otthe Earth does not always remain parallel to itfelf. 11. "«Vhc" . This Motion of the Axis of the Earth wasfirft perceived by il^r/^- fin'twbo otufi and afterwards eftablifiied by Ptolomey who fixed this Motion to a perceived Degree in a hundred Years, fo that the entire Revolution of the Sphere of on^onhe*** ^^^ ^^^ ^^^^ ^^^"^ whettcc Ptolomey derived this appearance, was com* pTles of the pleated in 36000 Years; and it was generally believed in his Time that etrth. at the Expiration of this Revolution called the gnat rear, die ccleffiat fi«d°the*'^ Bodies would return to their primitive Pofition. d^stioa%f '^c Arabs dlfcoTered that Ptolomey had made this Motion too flow, UU thit r^wal^ higbieig fixed it to a Degree in 72 Years, and Modern Aftronoroersby w^sttltJ *5^'"gi'*^ 51" annually have confirmed the Difcovery of Uilugbieig% fc che'g^eac <^^^ ^^^ Revolution of the Poles of the Earth is compleated in 25920 yetr. Years. Wfai&ht>cis „Y. t*ha7ui!e "^^^ equinoctial Points chance their Places in the fimc Time and by the •ffignodbf. fame Quantttv as the Poles of the World, and it is this Motion of tbe rt*iomty Equinoctial Points which is called the Preceffion of the Equinoxes. ^Jt!oI»!*^ Tho' the fixed Stars are immovable, at Icaft in refped of us, yet ts the coo* Thi»fetPt^ nion interfcftion of the Equator and Ecliptic Recedes, it is neceflary that fion C8u(e» the Stars which correfpond to thofe Points (hbold coAtimially appear to •* W^l^"' change their Places , and that they ihould fcem to advance edwartl, from the fi^ei whtnce it arrives, that their Longitudea^ which ts reckoned on the Eclspcic PHYSICAL WORLD- LXXIII. from the Beginning of ifrw, or the vernal Interfcdtion of the Equator nrd Ecliptic, continually increafes, and the fixed Stars appear to move in Confc^ ^tis tht funtia \ but this Motion is only apparent and arifcs trgm the Regrcflion of caafc wbf the Kquinoftial Points in a contrary DireSioiu tUm* of thT ^^ ^ equator and In Confequence of this RegreiTion, all the Confiellations of the Zodiac tbe cci:>fic have changed their Places fince the Obfervations of the firft Aftronomcrs;^**"^**®'" For the Conftellation ^riV/, for Example, which in the Time of Hipparchus the^fame* coirefponded to the vernal InterfeSion of the Equator and Elliptic, is now ftar^itdtd advanced into the Sign Taurus^ and Taurus has paffed into Gemini^ &c. and ^'"*jjj^** thus they have taken the Place of each other, but the twelve Portions of the confteUati- Ecliptic where thefe Conftellations were formerly placed, dill retain the ons of the fame Names they had m the Time of Hipparcbm. SlUi'td*** Vlf chcit p^acfgj Btfort Newton the phyfical Caufeof the Preceffion of the Equinoxes was utterly unkown, and we ihall now proceed to (hew how be deduced this Mck tion from his Principle of univerfal Gravitation. Wehavefeen that the Figure of the Earth is that of an oblate Spher<»dj| Flat towards the Poles and elevated towards the Equator. In Order to ex- plain the Freceflionof the Equinoxes, Newton premifes3 Lemmas, from tsmmm whence he deduces (Prop. 39. B. 3.) that this Revolution of the equinoctial ^^ ^^^ Points is produced by the combined ASions of the $un.and Moon on the pro- ^Modc luberaat Matter about the Earth's Equator, d«ct thia Vn. Motion froai In thjB firft Lemma he fuppofes all the Matter by which the Earth con- ^ of ^uS- fidered as a Spheroid would exceed an infcribed Sphere, to be reduced to a Ycrfal srafi' Ring inve(ting the Equator, and coUe6ls the Sum of all the Efibrts of the uUm< Sun^ on this King, to make it Revolve round its Axis which is the conunon Section of tbe Plane of the Ecliptic with the Plane paffing thro* the Center of the Earth, and Perpendicular to the ftraight Line conne^ing th6 Centers of the Earth and the Sun. In thefecond Lemma he inveftigates the Ratio between the Sum of all thofe Forces, and the Sum of the Forces exerted by the Sun on all the protuberant Parts of the Earth, exterior to the infcribed Sphere. In the third Lemma he compares the Quantity of the Motion of this Ring, placed at the Ec^uator^ with that of all the Parts of the Carth takea as a Sphere. vni. To determine the Force of the Sun upon this Protuberant Matter about the Equator of the Earth, Newton aflumes for Hypotbe/ts, that if the Earth was anihilated, and that onl^ this Ring remained, defcribing round the Sun the annual Orb, and revolvmg at the fame Time by its diurnal Motion rwnd its A^^ inclined to the ^liptic in an Angle of 23^ 30", th« Motion LXXIV. SYSTEM OF TtlE of the Equinoctial Points would be the fame, whether the Ring was imd ot vompofed of folid Matter. Newton after having tnveftlgated the Ratio of the Matter of this fuppo&d Ring, that is, of the Protuberant Matter about the Equator, to the Mattel of the Earth taken as a Sphere, and having found itxaflfuming the Ratio of the i Axes of the Earth] to be as 459 to §2441 , he proves that if the Earth and thb Ring revolved together about the Diameter of thi. Ring, the Motion (R) dl the Ring would be to the Motion (T), of the interior Globe, or to the Morion of the Earth round its Axis, in a Proportion compounded of the Proportion 459 to 52441 of the Matter in the Ring to the Matter in the Earth, and of the Number looooootothe Number 800000, or as 4590 to 419528, (a) and confequently that the Motion (R) of the Ring would be to the Motion (R+T) of the Ring and the Giobe^ in the Ratio of 4590 to 4241 18. He found (Prop. 32. B. 3) that the mean Motion of the Nodes of the Moon in a Circular Orbit, is 20', 11*, 46*, in Antecedentia,\n aSydereal Year ; and he proved (Cor. 16 Prop. &S) that if feveral Moons re- volved round the Earth, the Motion of the Nodes of each of thofe Moons would be as their periodic Times, from whence he concludes that the Mo- Kcwtoa tton fn) of the Nodes of a Moon revolving near the Surface of the Earth •Mfidcrtthe in 23^ 56™. would be to 2G* 1 1" 46% Motion (N) of the Nodes of our m^^^t ^^" '" * Y^^T^ as 23^ 56", the Time of the Earth's diurnal Rotation, thteqmior to 27' 7^ 43% the periodic Time of the Moon, that is, as 1436 to 39343; of the ta cb and by the Cor. of Prop. 66 the fame Proportions hold for the Motion of ** * ^'^^^ the Nodes of an Aflfemblage of Moons furrounding the Earth, whether thefe h!»iii^Mth« Moons were feparate,and detached from each other, or if they coalefired tiebeof the fuppofmg them liquified and forming afluid&ing^ or that the Rin^ be- •■tdi. came hard and inflexible. He dedac ct Therefore, the protuberant Matter about the Equator of the Earth bei^; fromdiis confidered as a Ring of Moons adhering to the .Earth, and revolving along i^^ withit, fincc the Revolution (n)of the Nodes of fuch a Ring, is to the that the Revolution (N) of the Nodes of the Moon, as 1436 to 39343, (according lo attraaion Cor. 1 6. Prop. 66) and that the Motion (R) of the Ring is to the Sum of •olhe^eie- the Motions (T+R) of the Ring and the Globe to which it adherca, at vatiQoatthe4S90 to 424118; nxRiBtoNx T + R, as I43<SX4590 to 39343 c2«f«'the X 4241 1 8, or ^r^ is to N, as 1 43^X4S9<> to 39343 X 4141 1 8 5 bat preceaioa of T' '^ the equi* it is demondrated that the Sum of the Motions T+R of the Ring and the uoxes. Globe to which it adheres is to the Motion (K) of the Ring as the RevolutioQ (n) of the Nodes of this Ring to half the annual Motion [|P.] of the Equi- noctial Points of the Body compolcd of the Ring aud Globe to which it ad- (a) The ratio, of the anot Ion of the riiif to the motton of the interior |lgfec afiiacd bj KcwM^^ it 4550 to 48$«i3. which if eyrpneyoi as ilisU be Aewq bcrcaftcr. J PHYSICAL WORLD. LXXV. Iiercs, (b) wherefore the annual Motion (P.) of the equinodial Points of the Body compofed of the Ring and Globe to v/hichjt adheres, will be to the an« aual Motion of the Nodes (N) of the Moon, in the compounded Ratio of 1436 X 4590 X a to 39343 X 4241 1 8. But Newton found (Lem. 2. R 3.), that if the Matter of the fuppofeil Ring was fpread all over the Surface of the Sphere fo as to produce towards the Fxjuatorjthe fame Elevation as that at the Fquator of the Earth, the Force of the Matter thus fpread to move the Earth, would be lefs than the Force of theequatoral Ring in the Ratio of 2 to 5; therefore the annual Regrefs of the equinoctial Points is to the annual Regrefs of the Lunar Nodes, as 1436 X4S9oX2X2to 39343x424118X5, and confcq le.itly in a Sydereal Year it will be 22* , 58^ , 33' without anv Regard being had to the Inclination of the Axis of the Ring, which Confiaeration caufes dill a Diminution in this Motion in the Ratio of the CoAne [91706] of this Inclination (which is 23 i) to the Radius (i 00000.) The mean annual Prepeflion of the Equinoxes produced by the Ac tion of the Sun will be therefore 21*6' nearly, fuppoflng the Earth Homoge* neous and the Depreifion towards the Poles tIt* Simplon found from his Theory 21* 6* (Mlfcellanequs Tra^s) D'Alambert fl3« nearly (Recbercbet Sur la Preceffion des Equinoxes) Euler 22* (Mem. dg Berlin Tom, 5. 1749^. And if this Quantity is greater by a third than what Obfervation indicates, it probably ariies from the Earth's not being Homo* ^neous, BS was fuppofed, the Refearches of Simpfon^ Euler^ and U^Alaii)*' \mt relative to this Obje^ fhall be explained hereafter, IZ. brcfttlaritiet In this Manner ATwu/ow determined the mean Quantity of the Motion JJ^^^^**; of the equinodial Points. But not without examining the different Varie- ^Inoftiu*' ties of the Adion of the Sun on the protuberant Matter about the Equator points pro- foppofed to be reduced to a Ring. **Jl"''**Vk* He (hews in Cor. 18, 19 and 20 of Prop. 66 that by the Aftionof the f^',"*®"^* Sun the Nodes of a Ring, fuppofed to encompas a Globe as the Earth, would reft in the SyfigieSj in every other Place they would move in if«- tecedentia^ they would move fwifteft in the Quadratures, that the Inch- fiation of this Ring, would vary, that during each annual Revolution of the £arth^ its Axe would Ofcillate, *and at the end of each Revolution would jeturn to its former Pofition, but that the Nodes would not return to their former Places, but would ftill continue to move in Antecedentia. (b) K^wtoo (vppofea tbtt the Sam of the Motiont of the Riosind the Globe to which it adherct is to Che Motion oftbe Ring, as the Revolution of the Nodes of this Ring it to the tnnatl Mo« tioa of the Eqvinoaial Points of the Body compofed of the Ring and Globe to which it tdherei, |a wUcb h^ \% mifttkca as (htli be Ihewo hereafter. LXXVI. SYSTEM OF THE X. The grcateft Inclination of the Ring ihould happen when its Nodes are e7iht ^n'^ ^" *h^ Svfigies, afterwards In the Paffagc of the Nodes to the Quadratures, on the pro- thfs Inclination ftiould diminifti, and the Ring by its Effort to chamgc tu tubcrant Inclination, imprefTes a Motion on the Globe, and the Globe retains this Mo* l^VequiTo"/ tion, till the Ring, or the protuberant Matter about the Equator, (for it is caufe^ an thc fame Thing according to Nekton) by a contrary Eflfnrt deflroys tlus aoauai nut»- Motion, and impreffes a new Motion in a contrary Diredion. "Tsof the Hence we fee that thc Axis of the Earth (hould change its IncHnatioii Mrth, with Refpeft to the Ecliptic, twice in its annual Courfe and return twice If the earth (q its former Pofition. JTwardTthe* Newton has (hewn in Cor. ai of Prop. 66 that the protuberant Mat- pdkV and ter about the Equator making the Nodes retrograde, the Quantity of this deprcffcd to Matter increafmg, this RegreiTton^ would increaie^ and would diminifli when ^'uato/thc this Matter diminiftied ; hence if there was no Elevation towards the Eqoa. ^ulnoAiai^ tor, there would be no Regreflionof the Nodes, and the Nodes of a Globe, points which inflead of been Elevated towards the Equator was deprefied, and would td- confequently would have its protuberant Matter about its Poles, would fteaJ of re- move in Lonfequentta. tfog adinj. And he adds, (Cor. 22 of Prop. 66) that as thc Form of thc Globe r^^'the ^"^'^Ics US to judgc of the Motion of the Nodes, (o from thc Motion of deplcffion^of the Nodes we may infer the Form of the Globe ; and confequently if the Che earth Nodes move in Jntecedentia, the Globe will be elevated towards thc Eqoa* Hwardf the j^^.^ ^^^^ ^^ ^j^^ Contrary deprefTed, if the Nodes move in C^nfequentU^ ^Vbe mooa which is a further Proof of the Flatnefs of the Earth towards the roles. contri buret Xt* ?uOiw'''oV ^^^* ^^^^ hitherto confidered only thc Aaion of thc Sun in cxphunmg the fnotjon ^^^ Preceffion of the Equinoxes, and we have feen that in Confequcnce «" of (heenuiQ' this A6iion the equinoctial Points would receede annually 21* 6^ B^ ^Thlrthc' ^^^ ^^^ by her Attraction Aftson the Earth and influence very fenftfaiy aftion^f the ^^'^ Phenomenon, its Aftion being to that of the Sun as 2t to i (c) if the* moon on the Inclination of its Orbit to the Equator was always the fame as that el protuberant (he Ecliptic'to the Equator, the Regreffion thencc rcfulting would bc lo tbU Ihe"qu'ator' arifincj trom the Sun's Action as 2 i^ to i. But becaufe its Nodes Ihift con* iifDorepow- tinually their PIaces,ithappcns that the Inclination of its Orbit to thc Equator, crfui than on which depends its EfFeift varies continually, fothat when the afccnding fan!^ *^* Node is in Aries, the Inclination of the Moon*s Orbit to the Equator a- (c) The Proportion of the Foree of the Son to that of thc Moon, affigoed by Newtm t to Ay 481 S- which be alfo afiigns for tho Proportion of thc Proceffion of the Equinoxes n • daced by the San to that produced by the Moon but this Proportion doea not agree with t : Theories which depend on the Determination of the Mtfs or the Mooo, and it appccrs fn i Computation aa (hall be (hewn hereafter, that the Preceffion of the fiqninoxea prodoccd by I \ Sun and that prodiiced by thc Moon «re not in the ftmc Proportion M the Forces nf thofe L« • misarics. 1 PHYSICAL WORLD. LXXVIl ttmnts to a8<* {, but when the afcending Node nine Years after, is in Libra it fcarce amounts to i8. ir in each Revolution, which renders the PreceiTton ari(ing from the Adion of the Moon very une<}ual during the Space oi 1 8 Years, and Caufes a Nutation in the Axis o^ the Earth, whereby its Inchnation to the Ecliptic varies during the Revolution ot »hc |£"J||iu^*^ Nodes of the Moon; after which it returns to its former Pofition. This the ewth Nutation from Theory, amounts to 191, agreable to Obfervation, the P'odpced by mean Preceffion arifing from the Adion of the Moon, to 35*, 5, conle-^ '^'''"•' quently the Preceffion arifing from the Afiionof the Sun to 14s 5, and the* * "*^^" greateft Difference between the true Preceffion arifing from the AAion of the Moon, and the mean Preceffion amounts to 17s 8, Theory of tbt Ehhing and Flovoing of tbe Sia. '• . , The ^pD It IS very eafy to perceive the C)nnedion between the Ebbing and Plow«> cation of thb ing of the Sea and the Preceffion of the Equinoxes. Newton deduces his Ex- J^j^ *^f plication of the Ebbing and Flowing of the Sea, from the fame Corollaries of tbTfei, \% Prop. 66, from whence we have feen he drew his Explication of the Precef* deduced lion of the Equinoxes ; thofe two Phenomena are both one and the other a^*?^ iJJ^V' jieceflary Confequence of the AttraAions of tbe Sun and Moon on the Parts ,. it^hiit* which compofe the Earth. of fhe pre If^ ceilioaoftli* Galileo imagiiied that the Phenomena of the Tides might be accounted '"^B^^of for, from the Motion of Rotation of the Earth, and its Motion of traniUtionGeiiieo cos round the Sun. But if this great Man had more attentively examined the ^^^^ Circumftances attending the Ebbing and Flowing of the Sea, he would have flowing •£ perceived that in Confequence of the diurnal Motion of the Earthy the Sea the fci. indeed would rife towards the Equator, and that (he Earthwould aflfume the Form of a Spheroid depreflfed towards the Poles, but this Motion of Rotation would never produce in the Waters of the Sea a Motion of Flux and Reflux, as Newton has demonftrated Cor. 19. Prop. 66. Newton Proves in this fame Corollary^ applying what he had demonftrated in Cor. 5 and 6 of the Laws of Motion, that the Tranflation of the Earth round the Sun has no Effisd on the Motion of Bodies at its Surface, and confequently the Motion of Tranflation of the Earth round the Sun, cannot Produce the Motion of Flux and Reflux of the Sea. '^^» . Tlieebbiof On examining the Circumftances which attend the Ebbing and Flowing of tad flovw^ the Sea, it was eafy to perceive that thofe Phenomena depended on the Po-^.**» J^ fition of the Earth with Refped to the Sun and Moon; but it was not fo, to {{[« ^{o^ dlfcover the Manner thofe two Luminaries Produce diofe Phenomena anduietsBaoA LXXVIIi; SYSTEM OF THE moon 00 the the Quantity that each contributes to their ProduAion: we fee but theE&ds wittfi. in which t[jc A£tioii& ot ihofe two Luminaries are fo confounded, that it is only by the AffiAs^nct of Newton* s Principles we are enabled todi(bnguiihone from the other, and affign their Qua itiiy. It was refervcd for thii great Man^ to diTcover the true Caufe of the Ebbing and Flowing of the Sea, and to reduce thofe Caufes to Computation ; we ihall now trace the Road which conducted him to thofe Difcoveries. IV. K«ad which ^^ begins by examining in Prop. 66. the Principle Phenomena whidi cooductcd fliould Refult from the Motion of three Bodies which attrad each other NcwtoQto mutually in the inverre Ratio of the Squares of the Diftances^ thefmall «»ttty* ^"^* Revolving round the greater. tiuc each of After having (hewn in the firft 17. Corollaries oF this Prop, the Irregulari- Uwfc lami ties which the greater Body would Caufe in the Motion of the lefler, v^rich ?tttrto^' itfelf revolves round the third, and by this Means having laid the Fouwiiti. dace thofa on of the Theory of the Moon, he conflders in Cor. 18 feveral fluid Mp phcoomeui* which revolve round a third, he afterwards fuppofea that thofe fluid N'cs all become contiguous fo as to form a Ring revolving round the central Bo- dy, and proves that the Adion of the greateft Body would produce in the Mo- tions of this Ring the fame Irregularities as in thofe of the folitary Body io whofe Place the Ring was fubftituted ; infine Cor. 1 9. he fuppofes the Body found which this Ring Revolves to be extended on every Side as farasthn Ring, that this Body which is folid contains the Water of this Ringioi Channel cut all round its Circumference, and that it revolves unifonnly round its Axis, he then proves that ttie Motion of the W^ter in this Chanod will be accelerated and retarded alternately bv the Action of the greater Body and that this Motion will befwifterin the Syiigiesof this Water, and flower in its Qjiadratures, and finally that this Water will Ebb and Flow aft^r the Manner of the Sea, Newton applies this Prop, 66 and its Cor. to the Phenomena of the Set (Prop. 24. B. 3.) and proves that they are a neceflary Confequence of the com- bined Adions of the Sun and Moon on the Parts which compofe the Eardi. V. He afterwards inveftigates the Quantity, each of thofe Luminaries contri* bute, to the ProduQion of thofe Phenomena. As this Quantity depends od their Diftances from the Earth, the nearer they are to the Earth, the greater the Tides (hould be, Cxteris Paribus, when their Adions, confpire together: and according to Cor 14. Prop. 66, thofe EflFeds are in the Inverfc Ratio of the Cubes of their Diftances from the Earth and the fimple Ratio of their ^'a(^es. Neuton examines firft the Aftion of the Sun on the Waters of the Sea, becaulc lis Qjjantit) of Matter with Refped to that of the Earth is known. Ke obferves that th^ Attradion of the Sun on the Earth is counterbalanced PHYSICAL WORLD. LXXIX as to the Totality by the centrifugal Force arifing from the annual Motion of the Earthy which he confiders as uniform and circular: But what is true at to the Totality is not To as to each particle of the Earth^that is/hat the centri<A fugal Force of tzth of ihofe Particles cannot be fuppofed equal to the Force with which the fame Particle is Attra£led by the Sun, finceeach Particle has the fame centrifugal Force, and the Particles of the Earth which are nearer the Sun are more attra£ted than thofe which are remoter. Thus the Dif. tance df the Earth from the Sun, being aaooo Semidiametefs of the Earthy and the Law of Attradion, the inverfe Ratio of the Squares of the Diflances, the AttraSive Force correfponding to the Point of the Earth nearefi the Sun, to the Center of the Earth, and to the Point of the Earth remoteft from the Sun, will be nearly as I looi, i looo and 10999, and as the Sun's AttraAion balances the centrifugal Force of each Particle of the Earth, this Force will be Pro]k)rtional to 1 1000; if from the attraftive Force of the Sun on each of thofe three Points, the centrifugal Force be Subduded, there will remain J, o,— i; which proves that the Center of the Earth is at ktA with Refpe^ to the Motions of the Waters of the Sea, and that the two Extremities of the Diamieter of the Earth dtreded towards the Sun, are aduated by equal Forces with oppofite Diredions^ whereby the Parts tend to recede from the Center of the Earth. If ^1 the fame Diameter (here be taken two Points equally diftant from the ^^^^^ ^^^^ Center, tho/e two Points will be like wife aAuated by equal Forces with op- of die cb^ pofite Diredions^ whereby they tend to recede from the Center; but this ^. And Farce will decreafc as the Diftance from the Center of the Earth. thisDi-^iJ^'*^ Ameter of the Earth dtre£ied to the Center of the Sua may be called the Solar Axis of the£arth,tf we now conilder the EclaatOr cofrrefponding to this Axe,it is evident that each Point taken in the Plane of this Equator may be fuppofed eqtially diilant from tt^ Center of the Sun, and cbnfequently that none of th^ Points of this Plane are affe^d by the Inequality between the eentrifugal Force and attra&ive Force,and confequently their Gravity towards the Center of the Earth will Hot be diminiihed, therefore if we conceive two Canals full of Water the one pailing thro' the demi folar Ate^ and the other thro* a Ray at its Equator, which communicate at the Center of the Earth, the V/ater will afcend in the firft and defcend in the other, this will happen both in the one and the other demi folar Aze^ and is the firfl Source of the £bbing and Flowing of the Sea. Each Particle of Water in the Canal of the demi folar Axe b attraded Secoa<i towards the Sun in the Dire£lion of the Canal, but this Force ads on the Source <>f Particles of Water in the other Canal, obliquelv, it therefore ihould be re- ^J^^^iSf iol ved into two, one perpendicular to the Canal, and the other parallel to it. «f thtftak nrhe firft may be conhdered as perfefiUy deftroied by the centrifugal Force ; Vot the other Force adds to the Gravity of each Ptrticle ia tlua Caoal, this LXXX. SYSTEM OF THE Imall Force docs not exift in the Canal of the demi Iblar Axe, and for thii Reafon the Water will defcend in the Canal ol the folar Equator, and will fuftain that of the Tolar Axis to a greater Height. This b the fecond Souice of the Ebbing and Flowing of the Sea. From whence it appears that the Afcent of the Waters of the Sea doo not arife from the total Adion of the Sun, but from the Inequalities in that Adion on the Parts of the Earth. Newhn obferves that in Conleqnenceof this Adion the Figure of the Earth (abftracting from its diurnal Motion) ought to be an elliptic Spheroid having for greater and leOer Axes the fobr Axe and the Diameter of its Equator, and determines in the following Man- ner the Force of the Sun which produces the difference of thofe Axes. Dcttrmiaa He confiders the Figure of the Earth (abftracting from its diuinal Motion) tionof the rendered Elliptic by the Action of the Sun, as a ftmilar Effect to the Figure SoVprodoc ^^^^^ Or'>i^ o{xht Moon, (abftracting from ita cxcentricity) which he hid iBK tbccicva ihewn (Prop. 66. Cor. 5) to be rendered Elliptic and'tohave its Center in the tion or dc Center of the Earth, by the fame Action. He demonftrated (Prop. 15. £f^u^ ^- 3) ^^^^ *^ f'^f" (^) "^^^^ ^^*w« ^^^ M<»n towards the Sun, i$ » •fchtrca fai the centripetal Force (g) which draws the Moon towards the Earth, asthe two poinrt Squar^ of the periodic Time (tt) of the Moon round the Earth, to the iwiti!*"^ Squarcof the periodic Time (TT) of the Earth round the Sun, accordi»g» Cor. 17 of Prop. 66 ; but the Inequality (V) in the Adion of the Sun on the Parts of the Earth being to ita Action (G), as the Ray (r) of theEtfth, to the Ray (R) of ita Orbit, and the Force (G) of the Sun which rettunsihc Earth in its Orbit^ being to the Force (g) which retains the Moon in itiOr* bit, as 'f'f Ray of the Earth's Orbit divided by the Square of its P' iodic Time, to ~ Ray of the Moon's Orbit divided by the Sqtttreof its periodic Time (Cor. 2 Prop. 4), V X G is to G X g, or the Ine- quality (V) in the Action of the Sun on the Parts of the Earth, it tothe centripetal Force (g) of the Moon towards the Earth as^Y^^'^' that is, as the Ray of the Earth divided by the Square of its perio- dic time round the Sun {f^) to the Ray of the Moon's Orbit,dividcd bythc Square of its periodic Time round the Earth (-— ) Wherefore by the Compofition of Ratios, g X V is to F X g, or the Force (y) of the Sun diflurbing the Motion of Bodies on the Surface of the Earth, is to its Force (F) with which it difturbs the Motion of the Moos, w -"TPf^ to HA± or as the Ray (r) of the Earth, to the Ray (b)of the Moon's Orbit, that is, as i to 60 J. J^riYSICAL WORLD, tXXXl; To ccMTiparcnow thofc two Forces with the Force of Gravity at the Surface of the Earth. Since the Force (F) which draws ihe Moon towards the Sun, is to the centripetal Force (g), which would retain the Moon tn an Orbit, defcribed about the Earth quieleent at its prefcnt Diftancc (60 { Se- Jnidiamcters of the Earth) as the Square ofay**, jh. 43". to ^6^^. 6^. g^. or as 1000 to 178725, or as I to 178 ^J ; and that the Force which retains the Moon in its Orbit, is equal to the Force (y) which would retain it in an Orbit defcribed about the Earth quiefcent in the fame periodic Time, at the Diftance of 60 Semidiameters, according to Prop. 60, in which it has been demonftrated that the adual Diftance (60 i Semidiameters) of ih,c Centres of the Moon and Earth, both revolving about the Sun, and at the fame Time about their common Centre of Gravity, is to the Diftance (60 Semidiameters) of their Centres, if the Moon revolved a- bout the Earth quiefcent in the fame periodic Time, as the Sum (1+42) of the Maffcs of the Moon and Earth, to the firft of two mean Propor- tionals (42 i) between that Sum and the Mafs of the Earth. Confe* qoently that the Force {*y) which retains the Moon in its Orbit is lefs than die Force (g) which would retain it in an Orbit defccribed in the fame periodif Time, about the Earth quiefcent at the Diftance 60 i Semi- diameters, in the Ratio of 60 to 60 |^, (Cor. 2, P. 4); by the Compofi- tion of Ratios FXg is to gXV or the Force (F) which draws the Moon towards the Sun, is to the centripetal Force (7) which retains the Moon in its Orbit, as 1X60 i to 178^1^x60. but this Force (y) which retains the Moon in its Orbit, (in approaching the Earth) increafing in the in- verfe Ratio of the Square of the Diftance, is to the Force (G) of Gra- vity as I 1060X60, wherefore VxF istoyxG^ or the Force (F) which draws the Moon towords the Sun, is to the Force (G) of Gravity as iX^-r to 60X60x60X178 i% or as i 10638092,6. From whence Newton concludes [Prop. 36. B. 3.] that fincc the Afcent of the Waters of the Sea, and the Elliptic Figure of the Lunar Orbit [ab- ??P**a-^* flraiSing from itsExcentricity] are fimilar Phenomena arifingfrora the Solar of the fan Force, and that in defcending towards the Surface of the Earth this Force on the wa- decreafes in the Ratio of 60 ^ to i. the Force of the Sun which deprefles J^'^^^ ^*»« . the Waters of the Sea in the Quadratures, or at the Solar Equator, is to the force**ofKa- Forccof Gravity as i to 638092,6x^0 ior as i 1038604600. But thisYity. Force is double in the Syfiges, or in the DireQion of the Solar Axis of what it is in the Quadratures, and afts in a contrary Direftion [Cor. 6. Prop. CC"]^ -^irhcreforc the Sum of the two Forces of the Sun on therWaters of the Sea, in the Quadratures and Syfigies, will be to the Force of Gravity as 3 to -38604600 or as I to 12868200. thofe two Forces united O^mpofe the total yorcc which raif<?s the Watcw of the 5w in the Solar Canal, their EffetS r txXX41- SYSTEM OF THE beiog tiie Tame as if they were wboly employM in raifiog the Waters lA thi Syfigies^aod had no EflFcd in the Quadratures. VI. . Newton after having invcftigated the Force of the Sun which produces coocluaM ^^^ Elevation of the Waters in the Solar Canal^ deiei mines in the following from hit Manner the Quantity of this Elevation. He confiders the Elevation of the theory that Waters of the Sea arifing from the Adion of the Sun, as an Effed fimilar t^thrJatli ^o t*^« Elevation of the Equatorial Parts above the Polar Parts of the Earth, of the Tea arifing frono the centrifugal Force at the Equator. Now the centrifugal Force la a Mck (p) ^^ ^^^ Equator being to the Force of Gravity (G) at the Surface of the Earth as i to 289^ and the Force of the Sun (F) exeitedon the Waters of the Sea being to the Force of Gravity (G), as i to 12868200, by the Com- pofition of Ratios, FxG is to CxG, or the Force (F) of the Sun exerted on the Waters of the Sea, is to the centrifugal Force (C) at the Equator, as 1X289 to 1X1286S200 or as i to 44527 ; confequentlv theElevatioo (85472 Feet) at the Equator produced by the centrifugal Force, is to the Elevation of the Waters in the Solar Canal produced by the Adion of Sun, as I 1044527. which (hews that the Elevation of the Waters in the Places direSly under the Sun and in thofe which are dire^ oppofite to them is 1 Foot, 11,4^ Inches. andfl^win/ The fluid Earth would prefarye a Spheroidal form its longeft Diameier •fihe fca pointing to the Sun without any Ebbing or Flowing of its Waters, if it bad Sl^matioT "^ Motion of Rotation. It is therefore the Rotation of the Earth round its of^otatioo Axis joined to its oblong Figure which caufes altematly a Depreflion and of the e«rth Elevation of the Waters of the Sea. If the Axis of Rotation and the Solar "iMaftiOTt ^^'^ ^^^^ ^^^ fame, the Waters of the Sea would have no Motka •f the ftw of reciprocation, becaufe each Point during the Rotation of the Earth aoaoooo. would be conftantly at the fame Diftance from the Solar Poles. But as thofe two Axes form an Angle, it is eafy to perceive that each Point of the Surface of the E!arih approaches and recedes ahemally from the Solar Pdes and that twice in a Revolution, and the Waters will continually rife in this Point during its Approach to, and will fall continually during its Recefs froffi ..<. thofe Poles. Newton invefiigated the Relation whith fubfifts between the cftimating Elevaiion of the Waters in any Place above that at the Solar Equator and thcaAioasf their Elevation in the Solar Canal; and found that the Square of theRacfius Ihtwrte" £5^ '* '^ '^^ Square of the Sine [ss] of the Altitude of the Sun in any •f\b7rcain rkcc, a& the Elevation [S] of the Waters in the Solar Canal to their Ele- wmj i^iiice. vation [bsS] in that Place. viir. It is Manifeft that what has been faid with Refped to the Sun Ihould be applied' without Reftri^ion to the Moon and all the Phenomena of the Tkfer fHYSICAL WORLB. UXXIlf jptOTt evidently that the A&lon of this Luminary on the Watcn is confiJeri- «•* " ^ tij greater than that of the Sun, which at firft View ftiould feem the more ^^^^ /urprifing, as the At^radive Force of the Sun arifing from its immenfe Bulk um of cht is fo powerful as to Force the Earth to Revolve round it, whilft the Irregu- «n««i «» parities produced in its Orbit by the Aaion of the Mpon are fcarce fenfible, fXJlJSr^ butif we confi^er that the Motion of the Sea proceedes from its Parts be- ^ thtm- ing differently attraded fjroni thofeof the reft of the Earth, bepayfe their tmof tk« Fluidity makes them recdve moreeafily the Impreflionsof the Forces which f"}*ie*ru Aden them, it will appear, that the Adion of the Sun which is very pow- tentiont iii ^ul on the whole Earth attraj£b ^11 its Parts almoft equally on Account of its the motion great Diftance; but the Moon being much peafer the Earth Afts more une- •ft*>«*«^*' qually on the different Parts of our Globe, and that this Inequality Ihould be much more fenfil^le fhan that of the Sun ; thefe inequalities being in the Inverfe Ratio of the Cufaies of thje Di(tancef of the Lunitna|rie; from the Earth, and in the Ample Ratio of their Quantities of Matter. The Elevation of the Waters of the Sea arifing from the A£Uon of the Moon, in the Dtre^on of the lunar A:|i$, abpve their Height at the lunar Equator, being once determined, the Elevation o\ the Waters of the Sea in any Place above their^ Height at the lunar Equator, will be found, for in this Cafe, as in that of the Sun, the ^uare of the Radiij^s (i) is to the Square of the Sine [tt] of the Altitude of the Moon in any Place, as the IClevarion [Lj of the Waters in the Dire^ion of the lunar Axis, above their Height at the lunar Equator,to their Elevation [tt L] above the fame Height^ in that Placi^ From <he Combination of the Adions of the Sun ^nd Moon o^ th^ Wafers Tfce nrU- of the Sea there refult two Tides, viz. the folar Tides and lunar Tides ^ luk, 'rift vHiichare produced independently of eacho^her. Thofe two' Tides by be- fhxDthe ing confounded with each other appear to Form but onie, bqt fubje^ to great coojoinc tc- Variations, for in the Syfigies t}ie Waters are elevated and dcpreffcd at the f^^JJ',^''^* fame Time by both one and the other Luminary, and in the C^ad^atures the moos. Sunraifesthe Waters where the Moon deprefle^ thpm, and feciprocalty the * Sun deprefles the Waters where the Moon raifes them, [one being in the Horifon when the o^her is at the Meridian] fp that from the Anions of thofe Luminaries fometimes confpiring and at other Times oppofed, there refuft very fenfible Variations both with refped to the Height of thf Tides and their Time. X. It is demonftrated that the Elevation of the Waters, produced by the ^njoint Adions of the Sun and Moon, is fenfibly equal to the Sum of the Elevatipns produced by the Actions of each feperately, wherefore the fvhole Elevation produced by the united Anions of the two Luminaries will^ LXXXIVj SYSTEM OF THE be Expreffed bv ssS+ttL; which (hews that the Elevation of the Wy ters in any Place will continuallv increafe until they attain their grcateft Height, and then it is high Water, after which it will continually dccreafc during fix Hours, and then it will be low Water ; the Difier- cnce between thofe two Heights is called the Height of the Tide ; from whence it appears that the Height of the Tides depends upon a great Number of Circumftances, viz. the Declination of each Luminary, the Age of the Moon, the Latitudes of Places and the Didance of the two Luminaries from the Centre of the Earth. xr. To examine the Variations in the Height of the Tides according to all thofe Circumftances, let us firft fuppofe the Orbit of the Moon and that of the Sun in tlie Plane of the Equator, and let us further fuppofe them ^^ ^,^^ pcrfcftly Circular, and let a Place be chofen at the Equator^ in which ton fame Cafe we may fuppofe s=i and t=:i, which will happen at the appulfe to eilimate of the Luminaries to the Meridian in the Syfiges, and the whole Elcvatioo theiftion ^jii i^g cxprefled by S+L; about fix Hours after s=o and t=o nearly on the wi. and the Waters will have no Elevation confequently the Height of the ttrtofth* Tides in the Syfigies will be expreffed by S+L; but in the Quadratures ^"* at the appulfe of the Moon to the Meridian t=i and s=o, and the E- Icvation ot the Waters will be exprcfled by L, about fix Hoars after 6=1 and t=o nearly, and the Elcvntion of the Waters will be evprefled by S and the Height of the Tide will be expreflcd by L— S, confequently the Hei?,ht of the Tides in the vSyfigies and Quadratures will be as S-J-L to L— S. if therefore the Height of the Tides in the Syfigies and Quad- ratures at the Time of the Equinoxes was determined from Obfervatkm, on the Coaft of an Ifland fituatcd near the Equator, in a deep Sea, and open on every Side to a great extent, the Ratio of L toS, the Effeds of the Forco of tJie Sun and Moon, or the Ratio of thofe Forces which are proportion- al to thofe Efre£ts, would be found. As no fuch Obfervations have been made, Newton employs for de^ termitiing the Ratio of thofe Forces the Obfervations made by Sturmy three Miles below Briilol. this Author relates that the Height of the Al- fcent of the Waters in the vernal and autumnal Conjunftion and Oppofiti* on of tlic Sun and Moon, amounts to about 45 Feet, but in the Quadra* tnres to 25 only, wherefore L+S is toL — S as 45 to 25 or as 9 to 5, con- fequently 5L+5r=9li— 9S, or i4S=4L and S is to L as 2 to 7. To reduce iliis Determination to the mean State of the variable Circum- ftances ; it is to he obfervcd I* that in the Syfipjicr the conjoint Forces of the Sun and Moon being the greatcfl:,it has I>ccn fuppofed that tlie correfponding Tidcisalfo the p;re.iicft, but the Force impreded at that Time on the Sea being incrcafcd by a new Though a |pfs Force fiill afting on it until it be- comes too weak to raife it any more, the 7 ules do not rife to their greateft Height but fomc Time after the Moon has paflcd the Sjfigics, Nezvtom PHYSICAL WORLD. LXXXV. trom the Obfervations of Sturmy concludes that the greateft Tide followg next after the Appulfe of the Moon to the Meridian when the Moon is di(- tant from the Sun about 1 8^ {. the Sun's Force in this Diftance of the Moon from Syfigies being to the Force [S] in the Syfigics, as the Coflne [7986355] of double this Diftance, or of an Angle of 37 Degrees, to the Radius [ 1 0000000] in the Place of L-f-S in the preceding Analogy L+Of 79^^355 ^ is to be Subftituted. In the Quadratures the conjoint Forces of the Sun and Moon being leaft, it was alfo fuppofed that the kaft Tide happens at that Time, but the Sea loofes its Motion by the ReduAioa fame Degrees that it acquired it, fo that the Tides are not at their leafl ^f '**».» «^- Height until ftmc Time after the Moon has pafledthe Quadratures, and 1^'^^n** fffwton from the fame Obfervations of Sturmy concluded that the leaft ftate of the Tide follows .next after the Appulfe of the Moon to the Meridian when ''"'•We cir the Moon is diftant from the Quadratures i8<» i. Now the Sun's Force "'P^^^f"' in this Diftance of the Moon from the Quadratures being to the Force £S] in the Quadratures, as the Cofine (7986355) of double this Diftance or of an Angle of 37 Degrees, to Radius (i 0000000) in the Place of L— S in the preceding Analogy, L— o, 7986355S is to be Subftituted, It is to be obfcrved 2^ that the Orbit of the Moon was fuppofed to Co* infide with the Plane of the Equator, but the Moon in the Quadratures, or rather 1 8^ i paft the Quadratures, declines from the Equator by a« bout 2Z^ 131% now the Force of the Moon in this diftance from the Equator being to its Force (L) in the Equator, as the Square of the Cofine (8570327 ) of its Declination 22<! 13m, to Radius (i 0000000) in the Place of Lr-o, 7986355S in the preceding Analogy 0,8570327!^ 0^7 9863$$$ is to be Subftituted. It is to be obferved 30 that the Orbits of the Sun and Moon were fuppofed to be perfectly Circular, and confcquently thofe Luminacies to be In tfacur mean Diftanccs from the Earth. But Newton demonftrated that the lunar Orbit (abftrafting from its Excentricity) ought to be an Ellip- tic Figure, havingUs Centre in the Centre of tlie Earth and the ftiorteft Di* ametef direded to the Sun; and determined (Prop. 28. B.3.) ihe^Ratioof this {boTtcd Diameter to the longeft or the Diftance of the Moon from the Earth in the Syfigies and Quadratures to be as 69 to 70. To find its Diftance when 18 -I Degrees advanced beyond the Syfigies, and when 18 i Degrees pafs- cd by the Quadratures, it is to be obferved that in an EUipfis if the longeft Scmidiameter be exprefled by (a) its fliorteft by [b] and the DiflFercnce of the Squares of the longeft and fliorteft Scmidiameters by [cc] and the Sine of the Angle which any Diameter [y] makes with the longeft ScmidiT funeter by [s] yy = — ^ — : — wherefore fubftituting fuccefllvely in tlus Expreffion 69 for [a] 70 for [b] for [s] 3173047 and 9483236 the Sines of 18 |- Degrees and 71 i Degrees : thofe Diftanccs will be 69,098747 and 69,897345 and the mean Diftance will be 69 i as equal to halt the Sum LXXXyi ; SYSTEM OF THE pf the the longeft and fliorteft fcmidiametcrs. But the Force of thc'Moon tf move the Sea is in the reciprocal triplicate Proportion ot its Diftance,and therefore its Forces in the greateft and lead of thofe Diftances are to iu Force in its mean Diftance, as 0,9830427 and 1,017522 to i. confeqoeotly c?theSlILV" ^^^ preceding Analogy, in the PJapeof L+o, 7986355S,wcmuft put 1itto*httofi,oi7522L + o,7p8|5355 S, and in the Place of 0,85703271- thcfaoM 0,7986355 $i we mud put 0,9830427X0,85703271-— p,7J9863Ss8; from ^» 5 to I. whence we have 1,0175221^+0,79863558, to 0,9830427X0,8570327 L Th« force —0,79863558 as 9 to 5, confequcntly 1,017522 LX5+ 0,79863558x5 Bf thcrna =0,9830427X9X0,857032 1^^-0,79863558X9, and by tranfpofuion, S is rf!t3°Sif«i fo U 050,9830427X0,8570327X9 r- Of«75aaX5 to 0,7986355X5 + th« wateri 0^798,6355X9, that is, $ is to L as I to 494815 nearly. erthefeato Zfl of*io*fe« Nfwton having thus fleterminied ihp Forc« pf the Moon to raife the mn6 erm to Waters of the Sea, afligns the Quantity of this Ellevatipn. The Force (0 of ' Lf*^S i^^ ^"" '^*"8 to ***« Forc« (4,48 1 5 ) of the Moon, as the Elevation (i Foot moM Ufc- n»% Inches) arifingfrom the Adion of the Sun, to the Elevation (8 Feet f igc«« ' 7^1 inches) arifing from the Adion of the Moon. So that the Sun ami Mooo together may produce an Elevation of about i of Feet in their mean Dit tances'from the Earth, and an Elevation of about la Feet when the Mooo is neareft the Earth. HowWcw. '^^^ Influence of t(ie Moon 00 the Tides has enabled Niwhn to EftiiM* ton iBTcfti- her Denfity, her Qyantity of Matter, and what Bodies weigh on her Sur- tatcd Che face. Compared with the Denfity and Quantity of Matter of the Earth, ^ ^^Jf^y^/f theWeightsofl^^ For fince the Force (v) oftbcMooa 2r matter of to move the Sea is to the like Force (V) of the Sun as 4^ 48 1 5 to i,aod v is to whtTbodiet ^ as ^abfolute Force of the Moon divided by the Cube of its Difiance ftom hi^'furfice the Earth to g^-abfolgte Force of the Sun divided by the Cube pf itsW- t^txi^L tance from the Earth (Cor. 14 Prop. 66); 4, 48 15 Is to i as ^ to -^^ tat roiTttt^of the abfolute Force (g) of the Moon is to the abfolute Force (G) of the San, the etith, as the Denfity of the Moon and Cube of its Diameter conjointly (dXq*) » we1th*of the Denfity of the Sun and Cube of its Piameter conjointly (DXp*)i *"* )>odiet on the apparent Diameter(3i ». 1 6t)of the Moon being to the apparent Vm»^ 1^' '^^'' ter (32« i2«) of the Sun as^to^,^ is to ^, as^HLlSi to ^'^^ wherefore by the Compofition of Ratios S ig to ^ts dX, 141 5^3 ^ » X ,1 545o8f confcquently 4, 481 5 is to i as d x ,»4tS83 to D X ,»S4S^ that 18, as theDcnfitiesof the Moon and Sun and the Cubes of their s|v p^arent Diameters cpnjunaiy, from whence it foUoiys that the Denfity (<•) « i> H Y S I C A L W O R L ft tXXivri ihcMoonis to the Dcnfity (DJ of the Sun, as4tif 'i i^ — I — ^^ ^ ,14*583 *» 54508 4891 to 1000, but the Denfity (D) of the Sun is to the Denfity (c) of the Earth, as 1000104000, confequently D><d is to DXc, or the Denfity (d) of the Moon is to the Denfity (c) of the Earth as 4891 Xiooo to 4000x1000 iJu - ^ or as 1 1 to 9, therefore the Body of the Mo6n is more Denfe and more the mooo. Earthly than the Earth its felf. Andfmce the trud Diameter of thel^n [fronithc OWervations of the Q„„ti| ^f AftronomersJ is to the true Diameter of the Earth as 100 to 365, the Quan- mitter ia tity of Matter in the Ekrth, is to the Qyantity of Matter in the Moon as ^« °><x"* i^oCfoooX 11 to 48627125 X9, that is, as 1 1039,788; And finc^ the* accelerative Gravity on the Surface of the Kfoon is to the accelerative Gravity 6h the Surface of the Earth ast&e Quantity of •^eithtdl' i^atter in the Moon, to the Quantity of Matter in the Earth, diredly, and bodieionict ^ the Square of the Diftances from the Center inverfely, they will be fMif««-' to each other as I X 13324 1039,788X1000 that is as I to ^nearly: con- fequently the accelerative Gravity on the Surface of the Moon virill be about three Times lefs than the accelerative Gi'avity on the Sui'face of the Earth. Ganicl Bernoully,!n his Piece on the Tides which carried the Prize - y 6f the Academy of Sciences in the Year 1 738, obferves that the Method jfj?^^ of eflimating the Proportion of the Force of the Sun to that ot the Moon by fcrcatppiali the greateft and lead Heights of the Tides as employed by Newton is very ®" * "^^T* uncertain \ bccaufe in the Ports of Englanid and France the Tidies Art. not immediately produced by the Adionsof the two Lurninaries,put are rattier a Confequence of the great Tides of the Ocean,as the Tides of the Adriatic Sea' areaConfeqqenceof the Small Tides of the IVfediterranean, and that the primitive Tides may cfiffer very ftnfibly ift'every RefpeSfrom the fecondary Tides which is confirmed by Obfervation; the Proportion of the Spring and Neap Tides being found to be very different in the different Ports, At Sr.' Malo's, for Example, the greateft artd leaft Height of the Waters are to one another as 10 to 3, and below Briftol according to Sturmy they are to each' other as 9 to 5. He obferves further that the Motion of Rotation of the Earth being very rapid vnth Refpe^ to the Motion of the Sun and Moon ; The Sea cannot every Inftant aflume its Figure of Equilibrium without any fenfible Motion^ hence the Waters which were raifed by the combined Anions of the Lumi- ^ oaries tending on one Hand to conferve as much as poflible by their Force f>{ inertia the Elevation they had acquired, and on the other tending as they recede from the Moon to loofe a Part of their Elevation, they will be lejs tkvated than they would be if the Earth was at Reft, and confequently the Mfeap Tides are greater suid the Spring Tides lefs than what refulu from n LXXXYIIL SYSTEM Of THE Computiti#n rounded on the Laws of Bquilibrium, wherefore the gfdt Spring Tides and Neap Tides are in a greater Ratio according to the Lawi ot Equthbrium than that of 9 to 5. Bernoully fuppofes them to be to each other as 7 to 3, confequcntly thit thI*^'S^*^ the Force (L) of the Moon is to the Force (S) of the Sun as 5102. Apro- according to portion which anfwers better to the Oblerved Variations in the durati- Straooliy. on and interval of the Tides (Variations which receive no AkeratioD from the above mentioned fecondary Caufes) and to the other Theories which depend on a Determination of the Force of the Moon. Hence the Denfitj of the Moon is to the Denfity of the Earth as 7 to 10, the Quantity of Mai- tcr in the Moon is to the Quantity of Matter in the Earth as 1 to 70,ani finally the accelerative Gravity at the Surface of the Moon is to ihcaccclcr- ative Gravity on the Surface ot the Earth as 1 to 5. XIV. g. j^ If the Moon's Body were Fluid like our Sea it would be dcvatcd by the fifweofThc Adion of the Earth in the Parts which arc neared to it and in the Partsop- moon. poiite to thefe, and Newton enquires into the Quantity of this ElevaW He obferves that the Elevation (8 \ ) of the Earth produced by the Aft* of the Moon would be to the Elevation (E) of the Moon (if it bad tb^ fame Diameter as the Earth) produced by the Adion of the Eirth as the Quantity of Matter in the Moon to the Quantity of Matter ^ the Earth, or as i to 39,788. and the Elevation (E) produced by the Adion of the Earth in the Moon if it had the fame Diameter as the Birth, is to the real Elevation (x) produced in the Moon by the Aflion of the Earth, as the Diameter of the Earth to the Diameter of the Moon oris 365 to too. wherefore by the Compofition of Ratios 8 f X E is to EX« or the Elevation of the Earth (8 \ ) produced by the A&ion of the Mo* is to the real Elevation of the Moon produced by the A&ion of the Earth* I X 365 to 39,788 X 100 or as loSi to 100 or x =93 Feet, conleqoen^ the Diameter of the Moon that paflcs through the Centre of the Earth, niup exceed the Diameter which is perpendicular to it by 186 Feet Hence ^t is, that the Moon always turns the fame Side towards the Earth. , In ElFea La Grange in his Piece which carried the Prize of the rojff* Effeftof Academy of Sciences in the Year 1764, iuppofing with Newton ^^ fpheroid"? M^^" ^5 a Spheroid having its longed Diameter dircQed tovicards the Earth, fifu e of ihc has found that this Planet fhould have a libratory or ofcillatory Motion ab«» j moon. its Axis, whereby its Velocity of Rotation is fomeiimes accelerated andoOif Times retarded, and that then the Moon fliould always turn the hiOt ** nearly towards tlie Earth, though it did not receive in the Beginning* y tion of Rotation whofe Duration was equal to that of its Revolution. J^ Grange has demonftrated alfo that the Figure of the Moon might be'"^ that the Prcceffion of its cq^ulnoCtiul Points or the Retrogradatioii ?f ^ PHYSICAL WORLD-# LXXXIX Nodes of the lunar Equator^ would be equal to the retrograde Motion of the Nodes of the lunar Orbit; and in this Cafe he found that the lunar Axis would have no fcnfible Nutation. The Afiion of the Sun in all thofe Inquiries, is almoft infenfible with refpefi to that of the Earth ; it is the Earth which produces the Motion of the Nodes of the lunar Equator* by ading more or lefs obliquely on the lunar Spheroid ; hence the Preccrfion of the lunar Equator* and the Law of the Motion pro- duced in the lunar Spheroid* diflPer very much from that which is obferve d in the Equator of the Earth. The Kefearches of this eminent Mathe- matician of Turin, ihall be explained hereafter. XV. Newton having fliewn that the Tides proceed from (he combined Ani- ons of the Sun and Moon* and determined the Quantity that each of thofe Luminaries contribute to their Produdion* enters into an Explanation of the Circumftances which attend the Phenomena of the Tides. There has been obferved in all Times, three Kinds of Motions in the Thrfckindi Sea, its diurnal Motion* whereby it ebbs and flows twice a Day* the °^ variati- regular Alterations which this Motion receives every Month, and which ^^^ ob- follow the Pofition of the Moon with refpeS lo the Sun, and thofe fcrvcd in which arrive every Year and which depend on the Pofition of the Earth jjj-*jjj°fc'r." with refped to the Sun. To deduce thofe Motions from their Caufe, we are to obferve that Diumal the Sea yielcfing to the Force of the Sun and Moon impreffed on it in variatioai. Proportion to their Quantity, acquires its grealeft Height by a Force compounded of thofe two Forces ; hence this greateft Height (even ab- ftrafting from the Force of Inertia of the Waters) Ihould not be im- mediately under the iVfoon, ilbr immediately under the Sun,, but in an intermediate Point, which correfponds more exaSly to the Motion of the Moon than to that of the Sun, becaufe the Force of the Moon on the Sea is greater than that of the Sun. To determine the Pofition of this Point, it is manifeft that at High-Water in any Place, ssS-^-ttL is a Maximum^ and at Low* Water a Minimum or Ssds-\-Ltdtz=o. But the inftantaneous Increment (ds) of the Sine of the Altitude of the Sun, i$ to the correfponding Increment (dz) of the Sun's diurnal Arc, as the Cofine (Vi — ss) of the Altitude of the Sun to Radius (i), or <//= •^i— //X^« ^n^ 'he correfponding Decrement (— ^'^ of the Sine of the Moon's Altitude, is to the corre^onding Increment (dx) of the Moon's diurnal Arc, as the 0>fine (Vi — tt) of its Altitude to Radius (i), or — £//=:dxX|/i — tt=.^dzXv^i — //, dx being to dz as 29 to 30, on account of the Motion of the Moon. Subftituting thofe Values of df and df in the Expreffion Ssds'\'Ltdt=i:09 we willjiave Siv^i — //=^|XL Xti^l — //, or'iliIlii=iii:from whence it appears that at the Time tyi^tt 306' I xc SYSTEM OF THE ■^ of high and low Water the Quantities syi — st and //i — tt arc always in the conftant Ratio of 29 L to 30 5, or of 20 X 5 to 30X2 ; but the Quantity j\^i — // can never exceed ^. confequently tVi — // can or ^V ; and of courfe one of the Fadors / or {^i — n never exceed 3^x 1 29x5 The waters orchcSca ought twice to rife and twice to fall every day. Figh water doe« 00c iro* meJiatelv foMow the Appulfe of the Moon to the Meridian. muft be always very fmalU which proves that the Moon is near the Meridian at High-Water, and near the Horizon at Low- Water. The Waters of the Sea therefore (hould be elevated and deprefled twice in the Space of a lunar Day, that is in the Interval of Tinae elapfed between the Paflage of the Moon at the Meridian of any Place, and its Return to the fanne Meridian ; for the conjoint Force of the Sun and Moon on the Sea, being greateft when the Moon is near the Meridian, it fliould be equal twice in 24 Hours 49 Minutes (a), when the Moon is near the Meridian of the Place above and below the Ho- rizon ; wherefore in each diurnal Revolution of the Moon about the Earth, there (hould be two Tides diftant from each other, by the fanoe Interval that the Moon employs to pafs from the Meridian above the Horizon to that below it, which Interval is about 1 2h- 24'"* hence the Time of High- Water will be later and later every Day. XVI. Since t\^i — // can never exceed ^V> ^^^ confequently the Diftance of the Moon from the Meridian 12 Degrees, the greateft Elevation of the Waters in any Place can never happen later than 48 lunar Minutea, or 50 folar Minutes after the Appulfe of the Moon to the Meridian, if the Waters had no Inertia^ and their Motion were not retarded by their Fridion again3 the Bottom of the Sea. But firpm thofe two Caufes this Elevation ftill happens two Hours add a Halt or three Hours later (a) Whilft the Heavens Cecm to carry the Sun and Moon round from Eaft to Weft rvety Day^ thole Luminaries moTC in a contrary Direction, the Sun 59 m. 8s. ,3 the Moon 13^ ic m. 35 s. in a Day, confequently after (heir CoojunAion the Moon continually recedes lad. 11m. 16s. ,7 from the Sun towards the £aft each Day, until (he is i3oD^rees from the Sun, oc in Oppcfition. after which being to the Weft of the Sun, (he cootinuafiy a^roaches, and K Icnsth overtakes him in 29 Da)S and an Half. From whence it appears that this Planet, the Day o-^ the new Moon, ri(ef, pafTes at the Meridian and (ets about the fame Time as the Sun; the following Days fhe rifes, paiTes at the Meridian, and fets later and later than the Sun, Co that the mean Quantity of the Ketar^^atioo of one rifing compared with the following, of one^AppoUeto the Meridian compared with the following, &e. is aU^ut 40 Minu:es. Seven Dajra and One-cliitd af cr the ConjunOion, the Moon being 90 Degrees to the Eaft of the Sun, cr in itsfirft Quattei; (he riies when the Sun is in the Meridian, palies at the Meridian when the Sun fcta, andlcts at Midnight. The following Days (he comes (boner to the Meridian than the Sun to the oppofice Meridian, but the Difference continually dccreales to the OppofiJon, and then (he riles when the Sun fets, pafTes at the Meridian at Midnight, and fets when the Sun rifes. The follo«4w Days (he comes later and later to the Meridian than the Sun to the oppofite Meridian, the Di^ ference increafmg to the laft Qiiarter when the Moon being 90 Degrees to the Weft of the Sua, rifes at Midnight, pafTes at tly Meridian at Six of the Clock in the Morning and fets at Kooii» The following Days (he rires,^afl'es at the Meridian, and fetf fooncr than the Sun, the LnccndL decreafmg to (he Conjun^on. r PHYSICAL WORLD. XCI in the Ports of the Ocean where the Sea is open ; for the Waters in confequence of their Force of Inertia receiving but by Degrees ihcir Motion* and retaining for feme Time the Motion they have acquired, the Motion of the Sea is perpetually accelerated during the fix Hours which precedes the Appulfe of the Moon to the Meridian, by the com- bined Actions of the Sun and Moon on the Waters, which incrcafes in pro* portion as the Moon rifes above the Horizon, and by the diurnal Motion of the Earth which then confpires with that of the Moon. This Mo- what trc tion impreffed on the Waters retains during fomc Time its Acceleration, theCaufci {o that the Sea rifes higher and higher until the diurnal Motion of the SjJ|!xia«.'^ Earth which becomes contrary after the Appulfe of the Moon to the Meridian, as alfo the combined Adions of the Luminaries which be- comes weaker and weaker, diminifhes gradually the Velocity of the Waters, in confequence of which they fink. It is eafy to perceive that the Prison of the Waters againft the Bottom of the Sea fliould alfo contribute to retard the Tides. In the Regions where the Sea has no Communication with the Ocean, the Tides are much more retarded, in fome Places even 12 Hours, and it is ufual to fay in thofe Places, that the Tides precede the Appulfe of the Moon to the Meridian. In the Port of Havre-dc-gracey for Ex- ample, where the Tide retards 9 Hours, it is imagined that it precedes bj^ 3 Hours the Appulfe of the Moon to the Meridian ; but in Reality, <this Tide is the Effe6t of the precedent Culmination. The Waters falling to the loweft when the Moon is near the Horizon, Low-witcr her Aaion on the Sea being then moft oblique, it is manifeft that Low- **®" ?*** water does not exadly fall between the two High-waters which immedi- f^twtoi the ately fucceed each other, but is lb much nearer to that which follows, as two Eleva- t he Elevation of the Pole in ihe'propofed Place is greater, and the Moon jj^^^JJ^Jjjf has a greater Declination ; that is, in proportion to the Interval between rucceed the ridng and fetting of the Moon and the horary Circle of fix Hours ««ch 6thcr. after her Culmination. *°* "^^^^ XVII. Thcfe are the principal Phenomena which attend the Tides depend- The men- ing on the Pofition of the diflFerent Parts of the Earth in its diurnal Re- ^^„y*" volution with refpeft to the Sun and Moon. We fliall now proceed to explain the Variations in the Tides which happen every Month, and -which depend on the Change of Pofition of the Moon with Refpefl to the Sun and the Earth. XVIIX. In the ConjunSion of the Sun and Moon, thofe Luminaries coming xhc ^rcat- to the Meridian at the fame Time, and in the Oppofition when one eft Tid« oomes to the Meridian the other coming to the oppofite Meridian, they {JI^ ILa cronfpire to raife the Waters of the Sea. In the Quadratures on the fuUMocn. XCiI SYSTEM OF THE Theleaftin contraiy the Waters raifcd by the Sun, arc deprcfled by the Moon, the tweS'"*'^" Waters under the Moon being 90 Degrees from thofc under the Sun ; confequently the greateft Tides happen at full and new Moon, and the lead at firft and laft Quarlcr. XIX. The great- The greateft and Icaft Tides do not happen In the Syfigies and Qua- «^ -«»<*Jcaft dratures, but are the Third or the Fourth in Order after the Syligiel notprecUe- ^^^ Quadratures, becaufe as in other Cafes fo in this, the EfFe^k is not ly hsppea the greateft or the leaft when the immediate Influence of the Caufe is Tim!^*^uid g^'c^^^ft or leaft. If the Sea was pcrfedly at Reft when the Sun and why. Moon z6t on it in the Syfigies, it would not inftantly attain its great-* eft Velocity, nor confequently its greateft Height, but would acquire it by Degrees. Now as the Tides which precede the Syfigies arc not the greateft, they increafe gradually, and the Waters have not acquired their greateft Height until fome Time after the Moon has paflfed the Syfi«> gies, and ftie begins to counteradt the Sun*s Force and deprefs the Waters where the Sun raifes them. Likewife the Tides which precede the Quadratures are not the leaft, they decreafe gradually and do not come to their leaft Height until fome Time after the Moon has pafled the Quadratures. t XX. j The great- The gfcateft Height of the Waters which by the fmgle Force of the on of7he Moon would happen at the Moon's Appulfe to the Meridian, and by f Waters hap- the fingle Force of the Sun at the Sun's Appulfe to the Meridian, ab« af"* ^r"!*^ ftrafting from the external Caufes which retard it ; by the combincil pulftof the Forces of both muft fallout in an intermediate Time, which corref- Mcontoihc ponds more exaSly to the Motion of the Moon than to that of the Sun, wWl'ft*nie w'^crefore when the Moon paflfes fronrl Conjunftion or Oppofttion to pailesfrom Quadrature, this greateft Height anfwers more to the fetting of the theSyfigics Moon. The Sun in the firft Cafe coming fooner to the Meridian than dratur^"*" the Moon, and in the latter the Moon coming later to the Meridiaa and later than the Sun to the oppofite Meridian ; and when the Moon pafles Moon pa^et ^^^"™ Quadrature to Oppofition or Conjunftion, this greateft Elevatico from^tfe anfwers more to the rifing of the Moon. In the firft Cafe, the Moon QaadFatures coming fooner to the Meridian than the Sun to the oppofite Meridiaiiy Syfigiw. *"^ '" ^^^ latter, the Moon coming fooner to the Meridian than the Sun (^). 'Jo calculate thofe Variations in the Time of High-water which arife from the refpedivc Pofitions of the Sun and Moon, let us fuppofe on a certain Day, the Sun and Moon to be in Conjundion at the Ap- pulfe of the Moon to the Meridian of any Place, and confequently that It is High- Water there at that Inftant. The following Day at the (b) See preceding Nott PHYSICAL WORLD. XCIII Time of High- Water in faid Place, the Sum of the Diflanccs (z^-ix'J of the Sun and Moon from the Meridian will be i2<*. 30™. and the In- lenral between the two Tides will be cxpreffed in folar Hours by 35o<*.+Arc z'. Since the Arcs z' and *' are very fmall, they may be fup- pofed without any fenfible Error to coincide with their Sines (v^i — //) (/I—//) and /I— //-f|/i— // maybe fuppofcd equal to Sin. 12^. 30"'. =:o,2i643» and confequently yi — //=o, 21643 — yi — //, we may fup- pofe aUb /=i and /=i : after thofe Subftitutions the Equation illillf' == ^Xtt will be transformed into —ilTif :=z^x — ; and fubfti* 30 S 0,21643-/1^// 30 6* tuting J- for -- we will have t~—: =— which gives for VI— // or for the Sine of the Arc z' required J^X 0,21643=0, 15308 or z'=8^. 48™. or 35f folar Minutes, fo that the whole Interval is 24h. 35"- 1- Let us tiow fuppofc on a certain Day, the Sun and Moon to be in Quadrature at the Apputfe of the Moon to the Meridian at the above memloned Place, and confequently that it is High-Water there at that Inftartt ; the following Day at the Time of High-water the Sum of the Diftances (z'+x'J of the Sun and Moon from the Meridian (if it be the laft Quadrature) will be 77! Degrees, and the Sum of the Diftances (z+z^ of the Sun from the Horizon and Meridian being 90 Degrees, «p— jif'=i2d. 30m, that is, s—y i — tt=io, 21643 and Vi — //=/ — o, 21643. But in this Cafe yi — // mc.y be fuppofed =1 and /=i, wherefore ., . ■■ = =:— *^..^^ r-*-~ — which e;ivcs /=o,'?6o20 anfwer* ing to 2id.*0"^. or to ih .6« ?vT- \ites, fo that the whole Interval (36o«5.+Arcz) is 2< Hou]s> 2f ^ IVT nutc?. From whence it appears tl / Hiel. -Water (hou!d precede the Appulfe of the Moon to the MeriHui vIilH iUe i? puffing from the Syfigies to the Quadratures, and ftiou..; cllow the Ajjulfe of the Moon to the Meridian whilft flie is pal^.i,: from th: Quadratures to the Syfigies; that the greatcft Anticipation • ♦ RetirJ Men fhould be about 50 folar Minutes, and that the Diftarcc cfthe Sun and Moon from eaoh other at the Time of the greateft Aniiripation or Retardation is about 57 Dc-- gees. But from external Caafes Hirh-Water happens in the Ports of the cean three Hours later, cou'equently in »hofe Ports it (hould precede the third lunar Hour, and that I7 the greateft Interval the ninth Tide after the Syfigies, and this greateft Anticipation being repaired in the five fiibfequent Tides, it fliould follow by like Intervals the third lunar Hour, vrbilft the Moon is paffing from the Quadratures to the Syfigies, XCIV SYSTEM OF THE The Tides are greater ceteris pari- bus, when the Moon is in Perigee than when (he is in Apogee. The anuiil Variations, the Tides • are greater in Winter than in Summer. TheTidei depend on the Declina- tion of the Sun and Moon. The Time and Height oftheTides depend up- XXI. Finally, all other Circumftanccs being alike> the Tides are gretteft is the fame Afpeds of the Sun and Moon, when they have the fame De- clination> when the Moon is in Perigee than when ihe is in Apogee. The Force of the Moon on the Waters of the Sea decreaiing in the triplicate Ratio of her Diftance from the Earth. XXII. The annual Variations of the Tides depend on the Diftance of the Earth from the Sun, hence it is that in Winter the Tides are greateff all other Circumftances being alike, in the Syfigies, and lefs in the Qya- dratures than in Summer, the Sun being nearer to the Earth in Winter than in Summer. XXIII. The Effeds of the Sun and Moon upon the Waters of the Sea de- pend upon the Declination of the Luminaries, for if either the San or Moon was in the Pole, any Place of the E^th in defcribing its Parallel to the Equator, would not meet in its Courfe with any Part of the Water more elevated than another, fo that there would be no Tide in any Place ; therefore the Adions of the Sun and Moon on the Waten of the Sea become weaker as they decline from the Equator, 9X^Newt9ti found (Prop. 37. B. 3.) that the Force of each Luminary on the Sea decreafes in the duplicate Ratio of the Cofine of its Declination ; hence it is, that the Tides in the folilicial Syfigies are lefs than in the equi- noSial Syfigies, and are greater in the folfticial Quadratures than in the - equinodial Quadratures, becaufe in the folfticial Quadratures the Moon is in the Equator, and in the other the Moon is in one of the Tropics, and the Tide depends more on the Adion of the Moon than that of the Sun, and is therefore greateft when the Moon's Adion is^pfeatefl. The Spring Tides therefore ought to be the greateft, and the Neap Tides the Icaft at the Equinoxes, and becaufe the Sun is nearer the Earth in Winter than in Summer, the Spring Tides ate greateft and the Neap Tides the leaft in Winter ; hence it is, that the greateft Spring and leaft NeapTides are after the autumnal and before the vernal Equinox. Two great Spring Tides never follow each other in the two neareft Syfigies, becaufe if the Moon in one of the Syfigies be in her Perigee, (he will iri the following Syfigie be in her Apogee. In the firft Cafe her ASion being greateft and confpiring with that of the Sun, the Waters will be railed to their greateft Height ; but in the latter Cafe her Aftion being leaft, the Tide will be lefs. xxiv. The ebbing and flowing of the Sea depends alfo upon the Latitude of the Place ; for the conjoint A&ions of the Sun and Moon changing the Water upon the Earth's Surface into an oblong Spheroid, one of the ^^' ' PHYSICAL WORLD. XCV Vertices of its longer Axis dcfcriblng nearly, the Parallel on the Earth's on the Ltti Surface, which the Moon, becaufe of the diurnal Morion, feems to p^*ei.* (fercribe, and the other a Parallel as far on 4he other Side of the Equator. The whole Sea is divided into two oppofite hemifpheroidal Floods, one on the North Side of the Eqaator, the other on the South Side of it, which come by Turns to the Meridian of each Place after an Interval of 1 2 Hours. Now the Vertex of the hemifpheroidal Flood which moves on the fame Side of the Equator with any Place, will come nearer to it than the Vertex of the oppofite hemifpheroidal Flood which moves in a Parallel on the other Side of the Equator ; and therefore the Ti^es in all Places without the Equator, will be alternately greater and lefs ; the grcateft Tide when the Declination of the Moon is on the fame Side of the Eqaator with the Place, will happen about three Hours after the Appulfe of the Moon to the Meridian above the Horizon, and the lead Tide about three Hours after the Appulfe of the Moon to the Meridian bdow the Horizon, the Height of the Tide in the firft Cafe, being exprefT- ed by a Semidiameter of the elliptic Sedion of the Spheroid nearer the tranfverfe Axe than in the latter Cafe, and confequcntly is greater ; and the Tide, when the Moon changes hei Declination, which was the greateft wiH be changed into the leaft, for then the hemifpheroidal Flood which is oppofite to the Moon, moves on the fame Side of the Equator with the Place, and therefore its Vertex comes nearer to it than the Vertex of the hemifpheroidal Flood under it. And the greatefl Difference of thofe Tides will be in the Solftices, becaufe the Vertices of the two hemif- pheroidal Floods in that Cafe defcribe the oppofite Tropics, which are the fartfaeft from each other of any two paFallel Circles they can de- fcribe. Thus it is found by Obfervation, that the Evening Tides in the Summer exceed the Morning Tides, and the Morning Tides in Winter exceed the Evening Tides ; and we learn (Pro. 24. B.^3.) that at Plymouth f according to the Obfervations of Coleprefs this DiflFerence amounts to one Foot, and at Br/flol, according to thofe of Sturmy to 1 5 Inches. Newton (de Mundi SyJlematCi page 58.) found, that the Height of the Tides de- S^^K^^ creafes in each Place, in the duplicate Ratio of the Cofine of the La- decreafe's in titude of this Place. Now we have feen, that at the Equator, they the dupji- decreafe in the duplicate Ratio of the Cofine of the Declination of ofj^f cofine eaeh Luminary; therefore in all Places without the Equator, "half the of the Sam of the Heights of the Tides Morning and Evening, that is, their Latitude. mean Height decreafes nearly in the fame Ratio. Hence the Diminu- tion of the Tides arifing frorri the Latitude of Places, and the Declina- tion of the Luminaries may be determined* Th€ Height of the Tides depend likewife upon the Extent of the Jflh^Til^ Sea in which they are produced, whether the Seas be entirely fepa- depend on. m XCVI SYSTEM OF THE tke Fxrcnt rated from the Ocean, or communicate with it by a narrow Channel; for vf the Seas, jf jj^^ g^^^ j^^ extended from Eaft to Weft 90 Degrees, the Tides will be the fame as if they came from the Ocean, bccaufe this Extent is fufficicnt that the Sun and Moon may thereby produce on the Waters of the Sea their grealeft and leaft EflFea ; but if thofe Seas be fo narrow, that each of their Parts are raifed and deprelTed with the fame Force, there can be no fenfible EflFea, for the Water cannot rife in any one Place without finking in another ; hence it is, that in the Baltick-Sea, the Black Sea, the Cafpian-Sea, and other Seas or Lakes of Icfe Extent, there is neither Flood nor Ebb. * XXVI. The Tides In the Mediterranean-Sea, which is extendi(i from Eaft to Weft only in the Mc- 6q Degrees, the Flood and Ebb are fcarce fenlible, and Euler has given ^X^r a Method for determining their Qil^ntity. Thofe fmall Tides are ftiU fenfible. rendered lefs by the Winds and Currents which arc very great*in this Sea ; hence it is, that in moft of thofe Ports, there are fcarce any re- gular Tides, except in thofe of the Adriatick Sea, which having a greater Depth, the Elevation of the Waters are rendered more fenfible \ hence it is, that the f^enetians were the firft who made Obfervations on die Tides of the Mediterranean. XXVII. Befides the aflignable Caufes which ferve to account for the Phenomena of the Tides, there are feveral others which produce Irrq^laribes in thofe Motions which cannot be reduced to any Law, becaufe they de- pend on Circumftances which are peculiar to each Place ; fuch are the Shores on which the Waters flow, the Straits, the different I>epths of the Sea, their Extent, the Bays, the Winds, i^c fo many Caitfes which alter the Motion of the Waters, and confequently retard, in- creafe, or dimini(h the Tides, and are not reducible to Calcolatioa. Hence it is, that in fome Places, the Flood falls out the third Ho« after the Culmination of the Moon, and in other Places the lath Hoar; and in general, the greater the Tides are, the later they happen, becaufe the Caufes which retard them ad fo much longer. If the Tides were very fmall, they would immediately follow the Culmination of the Moon, becaufe the Adion of the Obftacles ivhkh retard them would be rendered almoft infenfible ; this is partly the Reafon why the great Tides which happen about the new and full Moon, foUov later the Appulfe of the Moon to the Meridian, than thofe which hap- pen about the Quadratures ; the latter being lefs than the fornner. XXVIII. Euler relates that at St. Malos, at the Time of the Syfigtes, it is High-Water the fixth Hour after the Appulfe of the Moon to the Meridian, and the Retardation increafes more and more luitil at JDtm- Caures which in- fluence the Tide* that arc indeter- minable. Velocity of Ihe Wateri of Che Sea, PHYSICAL WORLD. XCVII M asd Ofitndj it happens at Muloight. From this Retardation the Velocity of the Waters may be determined* and EuUr concludes from tbofey and other Obfenrations, that they move at tlie Rate of eight Miles an Hour ; but it is eafy to perceive, that this Determination can- not be general. XXIX. The Tides are always greater towards the Coails than in the open 'Hie Tides Sea, and that for feveral Reafons ; firft the Waters beat againft the 1*^^*0!^ Shores, and by the Re-adion, are raifed to a greater Height. Secondly, Coafts, nd they coihe with the Velocity they had in the Ocean where their Depth ^'^y* was very confiderable, and they come in great Quantity, confequently meet with great Refiftance whilft they flow on the ^'hores ; from which Circumftance, their Height is ftill encreafed. Finally, when they pafs over Shoals, and run through Straights, their Height is greatly encreaf- ed, becaufe being beat back by the Shores, they return with the Force they had acquired from the Effort they had made to overflow them. Hence it is, that at Briftoly ^the Waters are raifed to fo great a Height ^t the Time of the Syfigies, for the Shores on this Coaft, are full of Windings and Sand-Banks, againft which the Waters beat with great Violence, and are much imp^ed in their Motion. XXX. Thofe Principles ferve to account for the extraordinary great Tides EjcplictUon which are obferved in fome Places, as at Plymouth f Mount St. Michael^ Phcnomoia the Town of Avrancbes in Normandy, df c. where Nrwton fays, the Wa- of the cers rife to 40 or 50. Feet, and fome Times higher. Tii,^ It may happen, that the Tide propagated from the Ocean, arrives at the fame Port by different Ways, and that it paffes quicker through fome of thofe Ways than through the others ; in this Cafe, the Tide will ap- pear to be divided into feveral Tides, fucceeding one another, having very different Motions, and no ways rerembUng the ordinary Tides. Let us fuppofe, for Example, that the Tides propagated from the Ocean, arrive at the fame Port by two different Ways, one of which is a readier and eaiier Paffage, fo that a Tide arrives at this Port through one of thofe Inlets at the third Hour after the Appulfe of the Moon to the Meridian, and another through the other Inlet, fix Hours after, at the 9th Hour of the Moon. When the Moon is in the Equator^ the Morn- ing and Evening Tides in the Ocean being equal, in the Space of 24 {fours* there will arrive four equal Tides to this Port, but one flowing in as the other ebbs out, the Water mufl flagnate. When the Moon it- cVines from the Equator, the Tides in the Ocean are alternately greater and lefs, confequently two greater and two leffer Tides would arrive at this Port by Turns, in the Space of 24 Hours. The two greateft Tides would make the Water acquire its greatcfl Height at a mean Time XCVIII SYSTEM OF THE betwixt them, and the two lefler would make it fall lowed, at a mean Time between ihofe two leaft Tides, and the Water would acquire at a mean Time betwixt its greateft and leaft Height, a mean Hright \ thus in the Space of 24 Hours, the Waters would rife, not twice, u ufual, but once only to their greateft Height, and fall loweft only once. If the Moon declines towards the Pole elevated above the- Horizon^ its greateft Height would happen the third, the fixth, or the 9th Hour after the Appulfe of the Moon to the Meridian ; and if the Moon de- clines towards the oppofite Pole> the Flood would be changed into Ebb. • XXXI. Explication All which happens at Batjbam in the Kingdom of Tonquin, in the cumftknccs l^^titudc of 20^. 50™. North. The Day in which the Moon paffes the attending Equator, the Waters have no Motion of flux and reflux : as the Moon '** B^ih" removes from the Equator, the Waters rife and fall once a Day, and inthe K?ng- come to their greateft Height when the Moon is near the Tropics ; with dom of this Difference, that when the Moon declines towards the North-Pole> Tunquin. ^^it Waters flow in whilft the Moon is above the Horizon, and ebb out whilft ftie is under the Horizon, fo that it is High- Water at the fettiag of the Moon, and Low- Water at her rifing. But when the Moon de- clines towards the South-Pole, it is High-Water at the rifing, and Low-Water at the fetting of the Moon ; the Waters ebbing out dar- ing the whole Time the Moon is above the Horizon. The Tide arrives at this Port by two Inlets, one from the Cbimfe Ocean, by a readier and fhorter Paffage between the Ifland of LeucwU and the Coaft of Canton, and the other from the Indian Ocean, between the Coaft of Cocbin-Cbina and the Ifland of B^me^, by a longer and leb readier PafTage ; but the Waters arrive fooner by the readieft and fhorteft Pafiage ; hence they arrive from the Cbineje Ocean in fix Hours, and from the Indian Ocean in 12 Hours, confequently the Tide arriving the third and ninth Hour after the Appulfe of the Moon to the Meridian^ there refult the above Phenomena. XXXII. At the En- At the Entrance of Rivers, there is a Difference in the Time of the Simt Sbe Tides flowing in and ebbing out, arifing from the Current of the River, £bb lafti which running into the Sea, retards its Motion of flux, and accelerates its !S?Flooir ^*^^^®** ^^ reflux, confequently makes the Ebb laft longer than the Flood, «Bd whyT ^hich is confirmed by Experience ; for Sturmius relates, that above Briftoh at the Entrance of the River Oundal, the Tide is five Hoon flowing in, and feven Hours ebbing out. Hence it is alfo, that all other Circumftances being alike, the greateft Floods arrive later at the Moutbs of Rivers than elfewhere. PHYSICAL WORLD. XCIX xxxni. It has been foundy as has been already mentioned, that the Tides At the Pole* depend on the Declination of the Luminaries, and the Latitude of the au^j °*^ Place ; hence at the Poles there is no diurnal ebbing and flowing of the Tideibut Waters of the Sea ; for the Moon being at the fame Height aU)ve the ^^ Horizon during 24 Hours, cannot raife the Waters ; but in thofe Re- tbe^oL* gions, the Sea has a Motion of flux and reflux depending on the Revo- tion of the lution of the Moon about the Earth every Month ; in confequence of JJ^^"^ which the Waters are at the loweft when the Moon is in the Equa- tor, becaufe (he is then always in the Horizon with refped to the Poles ; and as the Moon declines either towards the North or South Pole, the Sea begins to ebb and flow, and when her Declination is greatefl, the Waters are raifed to their greateft Height at the Pol^ towards which (he declines; and as this Elevation, which does not exceed ten Inches, is produced but by a very flow Motion, the Force of Inertia increafes it very little, confequently is fcarce fenfible. XXXIV. It IS only at the Poles that the Waters have no diurnal Motion ; in ButUU the Frigid-Zone, there is one Tide every Day inftead of two, as in the po^^h,^* Torid-Zone, and in our Temperate-2k>ne6 ; and it is eafy to (hew, that there U no this Paflage of two Tides to one, is not effeSed fuddenly, but like all <*»?[na^ other Effeds of Nature, is produced gradually. For we have feen, that ji^thc FrU- the Morning and Evening Tides in our Temperate-2^nes are unequal, gid-Zone not only as to their Height, but alfo as to the Time of their Duration ; ^'^u***** that the remoter the Place is from the Equator, the greater is this In- therTare equality between the two Tides which immediately fucceed each other, pot two u both as to their Height and the Time of their Duration, for the greateft '^^l^^ Tide (hould laft longer than the leaft ; and notwithftanding which they the £anh. bothceafein 12^24"^. nearly; therefore, in thofe Regions where the Moon after her Appulfe to the Meridian above or below the Horizon, returns to it in this Interval, the leaft Tide will entirely vanifli, and there will remain but the greateft Tide^ which alone will fill up the Interval of XXXV. The Force of the Sun and Moon are fufficient to produce the Tides, why the but are incapable of producing any other fenfible EfFefts here below ; Sni and for the Force (SJ of the Sun in its mean Diftance, being to the Force aw?ng^hc /GJ of Gravity, as i to 12868200, and the Force fS) of the Sim being Tides, pro- to the Force (LJ of the Moon, as i to 4,48 15, by the Compofition of ^^^^'J?^. Ratios LXS is to 5'XG, or the Force (L) of the Moon in her mean jMeEffeOf Diftance, is to the Force (GJ of Gravity, as 4,4815 to 12868200, or hcrebdow.* as I to 2871400. And fince S+L is to L as 5,4815 to 4,4815, S+L . XL is to LXG or the Sum of the Forces (S-\-L) of the Sun and Moon C SYSTEMOFTHE when they confpire together, and in their mean Diftances from the Earth, is to the Force (G) of Gravity as 5,4815X1 to 4,4815X2871400, or as I to 2347565, and the Sum of the greatcft Forces of the Lu- minaries, or at their lead Diftance from the Earthy is to the Force of Gravity, as i to 2032890. From whence it appears^ that thofe Forces united, cannot deflea the Direftion of Gravity, nor confequently the Pendulum, from the true Vertical the loth Part of a Second, nor caufe a Variation in the Length of the Pendulnm beating Seconds, which would exceed the ^os of ^ Line, i^c. THEORY of the Refraction <?/ Light. I. Explication T^ H E EfFeQs )vhich Bodies exert on each other by their AttraSion, of the Re- JL become fenfible only when it is not abforbed by the AttradioQ of L?^t°dcrfv *^^ Earth, and it appears that this mutual A ttrafiion of Bodies becomes ea^rom^the fenfible only when they are almoft contiguous, and that then it Principle of afts in a Ratio greater than the invcrfe Triplicate of the Diftances. Attraftion. j^^^ ^j^^ Atmofphcre, or the Mafs of Air encompafling the Earth, aft- ing on Light in a very fenfible Manner, it is certain, that if Attra&ioi be the Caufe, it fliould follow this Ratio. The Advantage of the Principle of Attradion coniifts in having no Need of any Suppofition but only the Knowledge of the Phenomena, and the more accurate are the Obfervations and Experiments, the eafier it is to apply this Principle to their Explication. 11. It is well known^ that Light ^raverfing Mediums of different Den- fities, changes its Diredion. Snelliusy and after him DefcarUs^ fouiMt from Experiment, that the Sine of Incidence and that of Refn6kion are The Sine of always in a conftant Ratio; ^nA Newton employs the 14th and \A iSa R^fAc. Sedion of the firft Book of the Principia in explaining the Reafon why cion are al-* thofe Sines fliould be in a conflant Ratio, and proving that this Ratio ways in a depends on the Principle of Attradion. It is in this Explication we ^^ fliall follow iV^/<^«. Every Ray of Light which enters obliquely into any Medium, is to be confldered as a Body aded on at the fame Time by two Forces, in order to apply to the Explication of their Effeds the Principles of Mechanicks. Defcartes and Fermat considered Light as a Body of a fenfible Magnitude on which the Mediums ad after the fame Manner as they appear to do on other Bodies: and finding that the Mediums which Light traverfes> pro- duce in them Effeds quite contrary to thofe which fhould refnlt from the Principles of Mechanicks, they invented each an Hjrpothefis in or- der to reconcile, in this Cafe, the Laws of Mechanicks, which are in- conteftable> and the phificial Effeds which are almoft at certain. PHYSICALWORI. D. CI III. It 18 well known, that the dcnfer the Mediums are, the greater Re- fiftance Bodies which penetrate them meet with in feparating their Parts. Now, in this Cafe, the Angle of Refradion is greater than the Angle of Incidence, becaufc the vertical Velocity of the Body being diminiflied by the Refinance of the Mediums, the horizontal Velocity influences '^^ ^*ws more the DircSion of the Diagonal which the Body in obeying the onofBodlL two Forces into which its Motion is refolved, defcribes j hence it is, that ofz fcnfibic when the Refiftance of the Medium is infurmountable, the Body, inftead ^^S^^^^^- of penetrating the Medium, returns back by its Elafticity, and the Pro- portion between this Refiftance and the vertical Velocity of the Body may be fuch, that the Body would lofe all its vertical Velocity, and would Aide on the Surface of the Medium if it had no Elafticity, and if the Surface of the Medium was a perfeSly fmooth Plane. IV. Now quite the contrary happens to the Rays of Light, the dcnfer the The Laws Medium is which they traverfe, the more the Sine of Incidence ex- ^^Vr^^h cccds that of Refradion; therefore the vertical Velocity of the Rays is arffcrent^ ^ increafed in this Cafe, which is quite the Reverfe of what the Laws of ^^^"^ ^^^^^ Mechanicks feem to indicate. of ffennwe Defcartesj in order to reconcile them with Experiment, which he MEgnitudc. could not evade, maintained, that the denfer the Mediums were, the eafier Paflage they opened to Light ; but this Manner of accounting for this Phenomenon was rather rendering it doubtful than explaining it. Fermat, finding the Explication of Defcartes impoffible, thought it more advifable to have Recourfe to Metaphificks, and the final Caufes. He afferted, that fmce Light does not arrive to us by the fliorteft Faf- 5^efcart« fage, which is the firaight Line, it was becoming the Divine Wifdom, and Fermac. it ihould arrive in the fliorteft Time ; this Principle, once allowed, it followed, that the Sines of Incidence and Refradion are to each other as the Facilities of the Medium to be penetrated. v. It is eafy to fee how Attradion folves this Difficulty; for this Prin-- ciple evinces, that the progreflive Motion of Light, not only is not lefs retarded in the more denfe Medium, as Defcartes pretended, but is really accelerated, and that by the Attradion of the more denfe Medium when it penetrates it. It is not only when the Ray has arrived at the refract- ing Medium and at the Point of Incidence that it ads on it ; the Incur- . ▼ation of the Ray commences fome Time before, and it increafes in accounts" proportion as it approaches the refrading Medium, and even within for every this Medium to a certain Depth. nlacTat- Attradion accounts for every Circumftance attending Light in its tenaiog the Pai&ge through one Medium into another ; for the vertical Velocity of ^/^^^ CII ' SYSTEM OFTHE the Ray is Increafed in the more denfe Meditiin» which it tnverfef until it arrives at the Point where the fuperior and inferior Parts of this Medi- um aft With equal Force on it, then it continues to advance with the acquired Velocity until being on the Point of quitting it, the fuperior Parts of this Medium attrad it with a greater Force than the mfcrior Parts. The vertical Velocity of the Ray is diminiflied thereby, and the Curve it defcribes at its Emerfion, is perfedly equal and limiiar to the one it defcribed at its Incidence, (the Surfaces which bound tbctefrta- ing Medium being fuppofed parallel) and the Pofition of this Curve is diredly oppofite to that of the firft. In fine, the Ray paffes through Degrees of Retardation which are in the fame Ratio, and in the fame inverfe Order as the Degrees of Acceleration which it paffed through at its Incidence. vi. Newton^ who was as fuperior in the Art of makingExperiments as j^2tt'J|. in that of employing them, found on examining Che Deviation of the Newton Rays of Light m different Mediums, that Ihe Attradion exerted on JtodiST/ ^^^ Particles of Light follows the Ratio of the Oenfity of thofe Ifc- fnOion of diums, if we except thofe which are gretfy and fulphurous. Since tha Light dc- the different Denfities of ihofe Mediums is the Caufe of the Refinfliofl 5w DtnCty ^^ Light, the more homogeneous Bodies arc, the more tranlparent thej of the will be; and thofe which are moft heterogeneous will be leift ffH th*^*"hLh ^^^ ^^^ Light in traverfmg them, being perpetually rcBe&d '» Jt Jilfc^ different Diredions within thofe Bodies, the Cfyantity ot Light whiA arrives to us is thereby diminifiied ; hence it is, that when the Sky » clear, the Stars are'fo diftinftly perceived, but when clouded, thcwj* are intercepted, and cannot reach (he Earth. vn. Newton alfo found, that every Ray of Light, however fmati, is^ofli- Th« Rayi pofed of feven Rays, which as long as they are united continue white, hot have Sot aU ^^^^^^ ^•^^'r natural Cotour when they are feparated, and that thofe the (ame Rays have not all the fame Degree of Refrangibility, that is, inp^^^ Ref™ ^' through one Medium into another of different Denfity, are infleflw biUt^^*' fome more and others lefs; fo that when they pafs through a Lens thofe Rays do not all meet the Axe at the Tame Diftance, bat fetfic nearer and others farther off, and thus form as many diilind Pidor^ of the Objeft as there are Colours. The Eye only perceives the mw vivid, but as the PiSures are not equal, the greateft form round thofci feveral coloured Circles, Which is called the Crown of Aberration. Tte Aberration is quite diftind from that which arifes from the Defefi ^ JReunion of the Rays caufed by the fpherical Figure of the Lcnfcs- The Aberration of Refrangibility in the Rays of Light is not fcnfibk when their Refradion is inconfiderable ; now the Rays parallel to the I was overcd. P H YSICAL WORLD. CIII optic Axe of a Lens^ and thofe at a fmall Diftance from this Aze^ are very little \ntie6ted, and the Pidure they form may be confidered as fimple^ as not being furrounded by any coloured Circles. Hence it is^ that Artifts arts under the Neceflity of giving to the objedive Glafs an Aperture of a very fmall Number of Degrees of the Sphere of which this Glafs forms a Part, and confequently of increaflng the focal Di- ftance of this Glafs, and the Length of the Telefcope, as often as they change the Proportion of the objeSive and ocular Ulafles, in order to incrcafe its magnifying Power. Thofe Obftacles to the PerfeQion of refraSing Telefcopes ariiing from the Nature of Light, and the Laws of RefraSion, Newton was on the Point of removing ; an Experiment he made opened the Way which leads to this Difcovery, but he did jj^^l not purfue it: the Experiment is as follows : As often as Light, tra- Method for verfing diffgrent Medsumf, i^ Jo corre^ed by contrary Refra^ionsy that it correfting emergetb in Lines parallel to thofe in which it was Incident, continues J^J-o^***'' ever after white. Optics, Firft B. Part II. Exp. 8. arifingfrom EuUr in 1 747, meditating on this SubjcQ, demonftrated, that this Afler- ^*-<*'"*r*P'- tion was falfe, and confequently that the Experiment was ill made. Mr. Do- ]jIJ^^^ lond, an eminent Englifh Optician, well verfed in the Theory and Pra3ice K^i of his Art, repeated this Experiment after the fame Manner that Newton ^*^^ defcribed it ; he conftruSed for this Purpofe, with two Plates of Glafs, a Kind of Port-folio, which being filled with Water, formed a Prifm of Water, that by clofing or opening the Glaffes, was fufceptible of all Kinds of Angles ; he plunged into the Water of this Prifm, whofe Angle was turned downwards,^ another Prifm of Chryftal, whofe Angle was Cumed upwards,, and by moving the Pliates of Glafs, he found that In- clination which was neceffary to make the ObjeSa obferved through the two Prifms of Water and Glafs appear exadly at the fame Height as they did to the naked Eye ; it was then manifeft, that the RefraSion of one Prifm was deftroyed by the Refraftion of the other, yet the ObjeQs were tinged with various Colours, which was quite contrary to Avhat Newton had aflerted. Mr. ZJo/oni/ afterwards tried^ by moving the Plates of his Prifm of Water, whether there was not fome poflible Proportion between the Angles of the two Prifms capable of deflroying the Colours, and found that there was fuch a Proportion^ which widely differed from that which deftroys the abfolute Refradion. The Objeas not coloured viewed through the Prifms thus combined, not appearing at the lame Height as when viewed by the naked Eye. From whence it was eafy to conclude^ that the Aberration of the Rays arifing from their different Degrees of Refrangibility, might be correSed by employing tranfparent Mediimis of different Denfities, and that the Rays would be refraded,. but in a diflPerent Manner from what they would be in paff- iog through one Medium. Mr. Dolond in i759> difcovered a Method CIV SYSTEMOFTHE 1 anfwering this Purpofe, which he has employed with Succefa in the Con- firudion of achromatic Telefcopes, and the mod eminent Mathemati- cians have fince exerted all their Skill in inveftigating the different Onn- binations for the focal Diflances^ and the Quantity of Curviture requi- fite to corred at once> the Aberration ariiing as well from the different Degrees of Refrangibility of the Rays» as from the circular Figure of the Lenfes, THofe Refearches (hall be explained hereafter. VIII. ThePrm- The Principle of AttraSion ferves to explain why the Refradion is Attrad>ion ^h^^ged into Refledion at a certain Obliquity of Incidence^ when the ferves to ex- Rays of Light pafs through a more denfe Medium into a lefs denfe one; plain how for in the Faffage of a Ray through a more denfe Medium into another it cfculngcd ^l^^t is lefs, the Curve it defcribes is inflefted towards the more denfe into Kcilcc- Medium it has paffed through ; now the Proportion between its Obli- ^*^"' quity and the Force which draws it towards this more denfe Medium may be fuch, that its Diredion may become parallel to the Surface of the Medium which it quits, before it has paffed the Limits within which the Attradion of this Medium is confined ; and in this Cafe, it is very eafy to fee, that it (hould return toward the refra&ing Medium it had quit- ted, defcribing a Branch of a Curve equal and iimilar, to the Curve which it defcribed in paffing through this Medium, and reaffiune after having again entered this Medium the fame Inclination it had before it quitted it. The ASion of the Medium which Light traverfes, may give Ae Rays the Obliquity they require in order to be reileded, and as the more the Mediums differ in Denfity the lefs is the Obliquity of Inci- dence requifite that the Rays niay be refleded, the Rays will be refleded at the lead Obliquity of Incidence when the contiguous Space or refhid- ing Medium will be purged of Air, and when the Vacuum will be mod perfeft. And fo it happens in the Air-Pump, in which the more the Va- cuum is increafed, the cjuickcr a Ray is refleSed at the fuperior Surface of a Prifm placed therein. The Refradion is therefore changed into Reflexion at different Incidences, according to theDeniity of the different Mediums, Diamond which is the mod brilliant Body known, operates an entire Refle£kion when the Artgle of Incidence is only 30 Degrees, and it is according to this Angle Jewellers cut their Diamonds, that they may lofc the leaft Quantity of^^the Light they receive. IX. It is eafy to perceive, that when a Ray of Light paffes through a left denfe Medium into a more compa£b one, the Refradion cannot |>e changed into RefleQion let the Obliquity of Incidence be ever fo great* for when the Ray is on the Point of quitting the lefs denfe Meditin/ the other Medium which is contiguous to rt, begins to a6t on it, and PHYSICALWORLD. CV iocreafes continually its vertical Velocity, the Rays of Liglit therefore ia their Paflage through the different Couches of the Atmofphere» whole Denfity continually incieafes in approaching the Earthy are more and more curved; in confequence of which the celeftial Objeds appear more elevated than they really are, and that by how much the more their Rays are curved from their Entrance into the Atmofphere until they arrive to us, the Eye receiving the ImprefTion of Light in the Di- redion which the Rays have when they enter the Eye. This apparent Elevation of the heavenly Bodies above their true Rcfra^rion Height, is called Aftronomical RefraSion, and is greateft near the Ho- i"^n"{|"^"^ rizon, where repeated Obfervations prove, that it amounts to 33'; hence the Day? it is, that in our Climates, the Sun appears to rife 3 Minutes iooner, and fet 3 Minutes later than it really does, whereby the artificial Day is in- , creafed 6 Minutes by the EiFed of Refra£iion. This Effed gradually increafes in advancing towards the Frigid-Zone, and at the Pole, by the RefraAion alone, the Day becomes 36 Hours longer ; hence it is alfo that the Sun and Moon at their rifmg and fetting appear oval, the infe- rior Margin of thofe Luminaries being more refraSed than the fuperior one, or appear higher in Proportion. Newton has fhewn how to determine the Law according to which Rule for Refradion varies from the Zenith to the Horizon ; from his Theory it ^^*"^.'*** refu!ts,\hat the Radius (R) is to the Sine of 8 yd. as the Sine of (z) «1ny d?^ the Diftance from the Zenith, to the Sine of (z — 6r) of this fame Di- unce from fiance diminilhed by fix Times the Refradion at this Diftance, where- ^^^ '^'^^^' fore R — Sine 87 : Sine 87=Sine % — Sine (z — 6r) : Sine (z — 6r)'^ and R — Sine 87 : Sine z — Sine (z — 6r^ = Sine 87 : Sine (z--6r) ; but R — Sine 87 is to Sine «— Sine (z—^r) as 3d.XCof. 88^ to 6rX Cof. (z — ^r). Differences of the Arcs multiplied by the Cofines of the Arcs, which are the arithmetical Means between 90 and 87, and between 25 ajad z — 6r. Therefore the Sine of 88^.^, that is of 90^. dimimfhed by the Triple of the horizontal Refraction, is to the Sine of the Diflance z dimtnifhed by the Triple of the Refraftion at that Diftance, as the horizontal Refradion, is to the Refra£lion at the Diftance z, and as the Coiine of 88d. \ to the Cofine of the Arc z diminiflied by the Triple of the Refradioii ; therefore the Refra£lton at the Diftance z, is equal to the horizontal RcfraSion multiplied by the Tangent of z diminiflied by the Triple of its Refradion, the whole divided by the Tangent of 88<i. 21™. from whence it appears, that the Ref rations eire proportional to tbi Tangmt^ of tbe Diflances from, the Zenith diminifbed by three Times the Refra^ioH. EMomph, Let the Rcfiraaion at the Diftance of 45 Degrees from the Zenith be required, which i^ known to be about i"^. the Tangent of 88^. ai*". is to the Tangent of 44d. 57m. as the horizontal RefraQion 33™. is to 57', the RefraSion ajt 45 Degrees Diftance from the Zenith. By this Rule the following Table was conftruded. CVI SYSTEM OF THE Tabic of Agronomi- cal Ref'rac- tiuD. »_ o 3028.22,3 o 32*28. 4,8 o 4026.59, o 50 I o I ic 1 20 1 1 40 30 ZI 1 50 2 O 2 10 2 20 2 30 2 40 2 50 3 o 3 10 3 ^o| 3 30 3 40 3 50 25.41, fc 24.28,6 23.19,^ 22.IS,2 4,7 20.17,9 19.24,8 18.35,0 17.48,- 17- 4,5 16.23,8 15.45,4 >5. 9 A 14.35 14. 3,9 »3-34 13. 6 12.39,6 12.14,6 'APP- Refrac. Alc D. M. M, S. o 033. 0,04 o O 532.10,44 10 O 1031.22,^4 2C o 1530-35,44 S^ o 2029.49*74 40 D. M.'M. S. 1.51,1 Refrac. 5 o 10 20 30 40 50 o 10 20 46 6 30 6 40 50 7 o 7 10 20 30 7 40 7 50 8 o 8 10 8 20 8 30 D. M.M. S. 8,0 II. 7,9 8.3. (1.28,9 8.40 6». 1,3 8.50 0.4^,0 9. o 10.29,2 9.10 10.11,3 9-54>3 Q.38,2 i.22,8 8.0 5.54,« 5.48>5 5-42,4 P. M. M. S. 15-30 3.23,7 16. o;3.i6,9 16. 30^3.10,5 17. o;3, 4,5 i7.3o'2.58,^ Ai D.M.'M, 3b o 58 o 39 ^ .0 o 9.205.36,5 18. o|2.53,64i 9.305.30,9 18.302.4^,6 9.4c 9.50 »o. o 5. 25, 4! 1 9- 0,2.43,9 5.20,019.302.39,4 5.14,820. 02.35,1 8.54,010.15 8.40,6 10.3c 8.27,810.45 8.14,911. o 8.14,9 8. 2,8 11.15 7-51,1 7.40,3 7.30,2 7.20,5 7. 2,1 6.53>4 6.45,1 6.37,1 6.29,4 6.22,0 6.14,8 6. 8,0 5. 7,320.302.31,0 2.27,2 2.23,6 2.20.3 4.40,323. 02.13,7 5. 0,1 21. 4.53,221.30 46,6 22. o 304. 11. 'I.45 12.00 12.20 12 404. 203 13 13 13.40 14. o 14.20 14.40 15. o 15.30 34,3 4.28,6 4.23,3L 4.16,1 9>4 24. 25- 26. 27. 28. 14. 3,0 56,9 3-51,1 3-45,5 3.40,1 3-34,9 3-»9.9 3-a3>7 34. o 35. o 36. o 7,- 1,6 1,56,2 1.51 1 .46,6 ,4 5 1.42,4156 1.38, 1.34,6 1.31,0 1.27,6 1.24^ 1.21,4 1.18,5 Refrac. t. p. M, 1. 1 8, 5 63 o|o.29^ 1.15,764 00.27,8 D.26,5 1.13,065 o 1.10,466 o 1. 7,9 5,5 1. 3.3 1,1 o.59>' 00.57,0 46 47 48 49 50 45 I 52 53 54 55 7 S8 59 60 461 00 63 o 67 c 68 69 07 0.45,978 o 0.44,2 0.42,6 80 00.41,1 00.38,2 0.36,8 0.35,5 00.34,2 00.33,0 00. Rcfnc. M- S. ^.25,3 0.24,1 O D or .22,9 0.21,7 .20,6 0.19,5 .18,4 o|o.55,o|73 00.17,3 0.16,2 15,1 14,0 13.0 53,1 0.51,2 0.49 ,4]; 6 ojo. 00.47,677 00 74 o 275 00. 79 81 00.39,681 00. 8,0 8800 o 31,7 30,489 0.29, ijgo ojo. 0.12,0 OO.lJyO 0.10,0 o|o. 9,0 \ 0. 0- 00. 00. 00. 1 6,0. 5,0; 4,0 3-0 00. 0.0 THEORY of the Secondary Planets. THE firft Phenomenon which the Secondary Planets offer to natii. ral Philofophcrs, is their Tendency towards their Primaries, in obferving the fame Law as the primary Planets towards the Sun. This Tendency has been fufficiently eftablifhed in treating of the primary pb- netsi abfiradiog at firft^ as was neceflary in order to (implify the QsefticMi, PHYSICAL WORLD. CVll from al) the Irregularities which the PlanetSf by their mutual Attra6lirns produce in each others Motions^ or which ariie from the Adion ot the Sun. Having afterwards examined the Irregularities in the Motions of the primary Planets ; but the Irregularities in the Motions of the fecon- dary Planets deferve particularly to be confidered» in order to Ihew after a more fatisfadory Manner, the Univerfality of the Principle of At- tradion> and the Harmony of the Syftem to which it ferves as a Bafis. The different Kinds of Motions obferved for many Ages in the Moon^ and the Laws of thofe Motions difcovered by eminent Aftronomers, fumiftied Newton the Means of applying his Theory with Succefs to this Planet. This great Man, who had made fo many Difcoveries in the other Parts of the Syftem of the World, was refolved not to leave this Part unexamined ; and though the Method he has purfued on this Occa- fion, is lefs evident, and lefs fatisfadory than the Method he employed in explaining the other Phenomena ; we are however much indebted to him for having made it the Objed of his Inquiry. II. It is^afy to perceive, that if the Diftance of the Sun from the Earth Manner ot" and the Moon, was infinite with the refped to their Diftance from each ^™g '«■ other, the Sun would not difturb the Motion of the Moon about the Earth ; fneaudlty becaufe equal Forces, whofe Diredions are parallel, which a£l on any of uie two Bodies, cannot affeS their relative Motions. But as the Angle jhe'^suiLoE formed by the Lines drawn from the Moon and the Earth to the Sun, the Earth though very fmall, cannot be efteemed as having no Quantity, from this *^^^^ Angle therefore is to be deduced the Inequality of the Adion of the ^^°* Sun on thefe two Bodies. Taking therefore, as Newton has done, (Propof. 66.) in the ftraight The Force Line drawn from the Moon to the Sun, a Line to exprefs the Force ?^ '|^^ ^"'^ with which the Sun attrads it ; let this Line be confidered as the Dia- |iJo others?* gonal of a Parallelogram, one of whofe Sides will be in the ftraight Line drawn from the Mo^n to the Earth, and the other a Line drawn from the Moon parallel to the ftraight Line which joins the Sun and the Earth, One urges it is evident, that thofe two Sides of the fame Parallelogram will ex- ^^* ^<k^c prefs two Forces which might be fubftituted for the Force of the Sun karth. on the Moon; and that the firft of thofe two Forces which urges the Moon towards the Earth, will neither accelerate nor retard the Delcrip- tion of the Areas, nor confequently prevent her from obferving the Law of Kepler, y\z, the Areas proportional to the Times, but will only change g^j^jn^hc the Law of the Force with which . the Moon tends towards the Earth, Direftion of and confequently will alter the Form of her Orbit. As to the fecond j}**^*"; Force, that which afts in a DireSion parallel to the Ray of the Orbit ihc^Earth of the Earth, if it was equal to the Force with which the Sun aQs on to the Sun the Earth, it is eafy to perceive that it would produce no Irregularity in the Motion of the Moon ; but thofe Forces arc only equal in thofe ~^ CVIII SYSTEM OF THE Points of the Moon*s Orbit, where her Diftance from the Sun becomes equal to the Diftance of the Earth from the Sun at the fame Time, which happens in the Qs^adratures ; in every other Point of her Orbit thofc two Quantities being unequal, their Difference expreffes the pertur- bating Force of the Sun on the Moon, not only preventing her from de» fcribing equal Areas in equal Time6, but alio from moving always in the fame Plane. HI. We find in Prop. 66 of the firft Book, only the general Expofitioa of the Manner of eftimating the perturbating Forces of the Sun on the Moon : But in Prop. 25 of the third Book, we find the Calcuktioo Meafurc which determines their Quantity ; we learn that the Part of the F^rce of the of the Sun which urges the Moon towards the Earth, is 'in its mean prtnrbating Quantity, the jrr?^ of the Force with which the Earth a&s on her the Sun. ^^gn ftie is in her mean Diftance. The other Part of the fame Force of the Sun which afts in a DireSion parallel to the Ray of the Orbit of the Earth, is to the firft Part, as the Triple of the Cofine of the Angle formed by the ftraight Lines drawn from the Moon and the Earth to the Sun. TV. Newton employs this Determination of the perturbating Forces (Prop. 26, 27, 28, 29.) for computing the monthly Inequality in the Moon's Motion, called her Variation, whereby flie moves fwifter in the firft and Accelera- third Quarter, and flower in the Second and Fourth, and which becomes tioQ of the moft fenfible in the Oftants or 45 Degrees from the Syfigies. todby^*be" Newton, to determine this Inequality, abftrafts from all the reft ; he Moonoro- further fuppofes the Moon's Orbit to be circular. If the Sun was away, *h"f ^ and he inveftigates the Acceleration in the Area which the Moon de- t 1$ orcc. f^j.j|jg5^ produced by that one of the two perturbating Forces which afis in a DireSlon parallel to the Ray drawn from the Earth to the Sun. He found that the Area defcribed by the Moon in fmall equal Portions of Time, to be nearly as the Sum of the Number 219,46, and the verfed Sine of double of the Moon's Diftance from the neareft Quadra- ture, (the Radius being Unity) ; fo that the greateft Inequality in the Areas defcribed by the Moon, arrives in the Oftants or 45 Degrees from the Syfigies, where this verfed Sine is in its Maximum. V. TheAaion To determine afterwards the Equation or Correftron in the mean remkw Sic Motion of the Moon arifing from this Acceleration of the Area dcrcrib- Crbit of the ed by the Moon, he has Regard to the Change in the Form of the lunar ^^'^raft^d'* Orbit, produced by the perturbating Force. He inveftigates the Quan- bctwcen the ^ity which the perturbating Force would render the Line pafling through the Quadratures longer than that which traverfes the Syfigies. The P H Y S I C A L W O R L D. CIX DftU which hc^ employs in folving this Problem, are the Velocities of Syfigief the Moon io thofe two Points, which be had (hewn how to determine ^ ^Z in the fofegoing Propofitton* and the centripetal Forces corrcfponding (^udrm-^ to the frme Points, which are both one and the other compounded of ^^^^ the Force with which the Moon tends towards the Earth, and the per* Ivrbating Forces of the Sun, which in the Syfigies and Quadratures ad in the Diredion of the Ray of the Orbit of the Moon. Now the Cur- vatures in thofe Points, being in the dired Proportion of the AttraQons, and in the Inverfe of the Squares of the Velocities, by this Means he obtaines the Ratio of the Curvatures, and from thence deduces the Ratio of the Axes of the Orbit, afluming for Hypothefis, that this Curve is an EUtpicy hsving its Centre in the Centre of the Earth, if the Sun be fisppgyfoi to have no appafeot Motion round the Earth ; but when Re« gard is had to this Motion of the Sun, becaufe the lefTer Axe of the Ellipfe 10 alfo earried about the Earth with the fame Motion, as being always direded towards the Sun, that it is a Curve whofe Rays are the fame as thofe of the Bllipfe, but the Angles they form are in- creafed in the Ratio of the periodic Motion of the Moon to its fynodical Motion* The firft of thofe Motions being that in which the Moon is referred to a fixed Point in the Heavens ; the other in which (he is com- pared with the Sun. By the Means of thofe Suppofitions, Newton found that the Axe which pafies through the Quadratures, is greater than that which pafles through the Syfigies by yV VI. He afterwards computes in the fame Hypothefis of the Moon's Or- Compuu- bit being circular, if the Sun was away, by the Principle of the Areas varU^on^ proportional to the Times, the Equation or CorreQion in the mean Mo- of the tion of the Moon refylting not only from the Acceleration found in the ^oon, foregoing Problem, her Orbit being fuppofed circular, but from the new Form of this Orbit. From the Combination of thofe two Caufes, he finds an Equation or Corredion which becomes mod confi- derable in the Odants, and then amounts to 35m. 10' when the Earth is in its mean Diftance ; and in the other Points of the Earth's Orbit, is to 35™. 10', in the inverfe Ratio of the Cube of the Diftance from the Sun, becaufe the Expreilion of the perturbating Force of the Sun, which is the Caufe of all thefe Irregularities of the Moon, is divided by the Cube of the Earth's Diftance from the Sun. This Correflion in the other Points -of the Moon's Orbit, is proportional to the Sine of double Cff the Diftance of the Moon from the neareft Quadrature. VII. Newton pafles from the Examination of the Variation of the Moon, ^^n^^f^hg to that of the Motion of the Nodes, (Prop. 30, 31.) In this Irquiry potion he fuppofes the Moon's Orbit to be circular if the Sun was away, and ^^^ attributes to the Force of this Luminary no other Effect than to change ^'' ex Wluchof the two per- lurbating forces of the Sun he cmployi. SYSTEM OF THE this circular Orbit into an EUipfe, whofc Centre is in the Centre of the Earth, or rather into the Curve whofe Conftrudion we have already given by the Means of an Ellipfc. Of the two perturbatihg Forces ot the Sun, that which urges the Moon towards the Earth, afting m the V\m of the Orbit, cannot produce any Motion in this Plane ; he thcrctore only confiders that Force which ads parallel to the Line drawn from the Earth to the Sun, which he had ihewn to be proportional to the Cofine of the Angle formed by the Lines drawn from the Moon ^ the Earth to the Sun, and we (hall now explain how he employs thii Force. • At the Extremity of the little Arc which the Moon ^J^f*^"*^? ^ ^J^ fmall Portion of Time, he takes one, equal to it, which would beaut which the Moon would defcribe if the pertuf bating Force of the Mooo ceafcd to ad on her ; and at the Extremity of this new Arc, hedw a Line parallel to that which joins the Centre of the Earth m the Sun, and he determine* the Length of this ftraight Line, by tw Meafurc already determined of the Force which aSs in the fwoe Ui- reSion as it ; which being done, the Diagonal of the Paralleloprtnji one of whofe Sides is the little Arc which the Moon would dcfcnbe t the perturbating Force ceafed to aft, and the other, the Arc the Mow would defcribe if this Force aded alone, is the reil Arc the Moon wouW defcribe. There remains therefore no more to be done than to deter- mine, Iww much the Plane which would pafs through this ''"*"^ and the Earth, diflFcrs from the Plane which pafles through the firftSi« and the Earth. The two Sides already mentioned, being produced until they tneet the Plane of the Orbit of the Earth, and having drawn from thrirPoifltt of Concourfe with this Plane, two ftraight Lines to the Centre of the Earth, the Angle which thofe two ftraight Lines form, is the Motions the Node during the fmall Portion of Time which the Moon employs i^ defcribing this fmall Arc, which wc have been confidering. And flf^ ton finds that the Meafurc of this Angle, and confequently the Velocity or the inftantaneous Motion of the Node, is proportional to the I^ AuGt of the Sines of three Angles^ which cxprefs the Diftance of th« Moon from the Quadrature, of the Moon from the Node, and of the Node from the Sun. Ke reffio ^^"' and ProgJef I^ follows from hence, that when one of thofe three Sines becomes fion of the negative, the Motion of the Nodes which before was retrograde, b^ j^ «ch comes direft. Wherefore when the Moon is between the Quadrature iUvoliition, and the neareft Node, the Motion of the Node is dired ; in all otbef . Cafes, its Motion is retrograde, but the retrograde Motion exceeding Law of Che MoCton of the Nodes. PHYSICALWORLD. CXI the i'wtQt Motion, it happens that In each Revolution of the Moon, At the Ena the Nodes arc made to recede. RwoUiuon When the Moon is in the Syfigics, and the Nodes in the Quadratures, the Nodes that is, 90 Degrees from the Sun, their Motion is 33" 10'" 37*^ 12^, wcede. wherefore the horary Motion of the Nodes in every other Situation, is Formula to 33'' 10"' 27iv lav, as the Produd of the three Sines already mention- Jh^hoSlr* ed to the Cube of Radius. . Morion of IX, the Nodes Suppofmg the Sun and the Node to be in the fame Situation with situauoo. refptd to the fixed Stars, whilft the Moon pafles fucceffively through -^ its fcveral Diftances with refpeS to the Sun. Newton inveftigates (Prop. J^n'^JhV J32. B. III.) the horary Motion of the Node, which is a Mean between meanMoti- all the different Motions refulting from the foregoing Formula, and ^"q^^^*** this mean -Motion of the Node is 16' 33'' 16-^' 36^, when the Orbit is fuppofed circular^ and the Nodes are in Quadrature with the Sun ; in every other Situation of the Nodes, this Motion is to 16" 33"' i6»^' 36s as the Square of the Sine of the Diftance of the Sun from the Node, is to the Square of the Radius. If the Orbit of the Moon be fuppoHcd to be an Ellipfe, having its Centre in the Centre of the Earth, the mean Motion of the Nodes in the Quadratures is only 16" i&" ^T"' 42^'> and in any other Situation of the Nodes, it depends likewife on the Square of. the Sine of the DiAance from the Sun. In order to determine for any given Time, the mean Place of the Nodes, Newton takes a Medium between all the mean Motions already mentioned. He employs in this Inquiry, the Quadrature of Curves, and the Method of Series. By this Means he finds that the Motion of the Nodes ill a fydereal Year, fhould be 19° 18' i" 23''', which only differs 3' from that which refults from aftronomical Obfcrvations.. X. The fame Curve the Quadrature of whofe Area determines the mean Dctermina- Velocity of the Nodes, ferves alfo for finding the true Place of the IJ.^cPlac'' Nodes for any given Time, (Prop. 33. B. III.) of the The Refult of his Computation is as follows : Having made an Angle ^®**^? ^'^^ equal to the Double of that which expreffes the Diftance of the Sun Tmic7^" from the mean Place of the Nodes, let the Sides of this Angle be to each other, as the mean annual Motion of the Nodes, which is i9<> 49' 3" 55'"» ^^ ^^ Half of their true mean Motion, when they are in the Qy^c^^^fcs, which is Qo 31' 2" 3'", that is, as 38,3 to i, which being done^ and having completed the Triangle which will be given, fince this Angle and its two Sides are given, the Angle of this Triangle oppoiite to the leafl: of ihofe Sides, will exprefs with fuiEcieat Accuracy, the Equa- tion or Corre£[ion in the mean Motion of the Nodes for determining the true Motion required. cxn SYSTEM OF THE Variation of the In- cliaatlon of the Moon^t Orbit. Honry Va- riatioo oF the loclina- tion. Method for finding the Inclination of the Moon*f Orbit for Determina- tion of the Latitude of the Moon. XI. From the Inveftigation of the Motion of the 'Nodes, Newton paflfes (Prop. 34. B. III.) to the Determination of the Variation in ihe Inch-, nation of the Orbit of the Moon. By employing that one of the two perturbating Forces of the Sun which does not ad in the Plane of the Orbit of the Moon, he obtains the Meafure of the hprary Variation in the Inclination of the Orbit of Jthe Moon ; this Variation* when the Orbit is fcrppofed circular, being to the horary Motion of the Nodes, 33" 10'" 3*»v 12% (the Nodes being in the Quadratures, and the Moon in the Syfigies) diminished in the Ratio of the Sine of the In- clination of the Orbit of the Moon to the Radius : as the Produd of the Sine of the Diftance of the Moon from the neareft Quadrature, the Sine of the Diftance of the Sun from the Nodes, and the Sine of the Di- llance of the Moon fiom the Nodes to the Cube of Radius. And this Quantity dimtnifhed by «>V >s the Variation correfponding to the Orbit rendered elliptic by the perturbating Force of the Sun. XII. The horary Variation of the Inclination of the Orbit of the Moon being thus determined, Newton employing the fame Method, and the fame Suppofitions by which he found the true Place of the Nodes for any given Time, determines (Prop. 35. B. III.) the iBclination of the Orbit for any given InAant of Time ^ the Refult of his Computation b as follows. Let there be taken from the fame Point of a ftraight Line, afium- ed as a Bafe^ three Parts in geometrical Proportion, the firft expreff- ing the leaft Inclination, the third the greateft ; let there be afterwanb drawn through the Extremity of the Second, a Line making with this Bafe an Angle equal to double the Diftance of the Sun from the Node for the propofed Motion let this Line be produced until it meets the Semicircle defcribed on the Difference of the firft and third Lines in geometrical Proportion ; which being done, the Interval comprifed be> tween the firft Extremity of the Bafe, and the Perpendicular let fall from the common Se£kion of the Circle and the Side of the Angle juft men- tioned, will cxprefs the Inclination for the propofed Time. From hence is deduced the Moon's Latitude correfted ; for in t Right-angled fpherical Triangle is given, bcfides the Rrght-angle, tfce Hypothcnufe, viz. the Moon's Diftance from the Node, the Angle at the Node, vi%, the Inclination of the Plane of the Moon's Orbit to the Plane of the Ecliptic, confequently the Side opposite to this Angle, which exprefles the Latitude correQed, will be be alfo given. But there is a more fmiple Method for finding the Latitude of the Moon correded. For the mean Latitude being computed, the Incfifii- tion of the Moon's Orbit to the Ecliptic being fcppofed conftant and equal to 5*^. 9'. 8", the Equation or CorreQion of the Latitude will be PHYSICAL WORLD. CXIII 8' 50'' multiplied by the Sine of double the Difiance of the Moon frcm the Sun lefs the Diflance from the Node. XIII. Newton, after having expofed the Method by which he calculated that ^v: at New- Inequality in the Moon's Motion, called her Variation, and the Method JJ-Jj^*^* j he had followed in determining the Motion of her Nodes, and the Va- to'chc^^xher nation of the Inclination of her Orbit to the Ecliptic, gives an Account ^rres«>l"'i- of what he fays hie deduced from his Theory of Gravitation, with re mooJt*^* fpeft to the Motion of the Apogee, the Variation of the Excentricity, Motion, and all the other Irregularities in the Moon's Motion. It is in the Scho- lium of Prop. 3<. B. m. he delivers thole Theorems, which fervc as a Foundation to tm Conftnidion of the Tables of the Moon's Motion. The Subftance of which is as follows. XIV. The mean Motion of the Moon ihould be correded by an Equation Annua] depending on the Diftance of the Sun from the Earth. This Equation or ^^Jtiow CorreSion, called the annual one, is greateft when the Sun is in his Peri- Son ouZ gee, and leaft when in his Apogee. Its Maximum is 1 1' 5 1", and in the other Moon, of Cafes, it is proportional to the Exjuation of the Centre of the Sun. It is to ^j^f^S? be added to the mean Motion of the Moon in the fix firft Signs, counted Nodci. from the Apogee of the Sun, and to be fubtraded in the (ix other Signs. The mean Places of the Apogee and of the Nodes fliould be alfo each correded by an Equation of the fame Kind, depending on the Diftance of the Sun from the Earth, and proportional to the Equation of the Centre of the Sun. The Equation of the Apogee in its Maximum is 19' 43", and is to be added from the Perihelion to the Aphelion of the Earth ; the Equation for the Node is to be fubtraSed from the Aphe- lion to the Perihelion of the Earth, and in its Maximum amounts to XV. The mean Motion of the Moon requires a fecond Corredion, depend- Firft femiA ing at once on the Diftance of the Sun from the Earth, and on the Situ- ^^ %^- ation of the Apogee of the Moon with refpeft to the Sun ; this Equa- meLTMo* tion^ which is in the dired Ratio of the Sine of double the Angle ex- tion of the prefling the Diftance of the Sun from the Apogee of the Moon, and in **^^"' the inverfe Ratio of the Cube of the Diftance of the Sun from the EUrth, is called the Semeftrial Equation ; it is 3' 45'' when the Apogee of the Moon is in Odants with the Sun, and the Earth is in its mean Diftance. It is to be added, when the Apogee of the Moon advances from its Quadrature with the Sun to its Syfigie : and is to be fubftraded when the Apogee paftes from the Syfigie to the Quadrature. ^■^ CXIV SYSTEM OF THE Second feme ft rial Equation. Determina- tion of the Place of the Apogee, and ot the Ex- ccntricity. Equation of the Centre, or fourth CorreAlon of the Place of the Mood. XVI. The mean Mption of the Moon requires a third Correaion, depend- ing on the Situation of the Sun with refpea to the Nodes, as alfo on the Diftance of the Sun from the Earth ; this Correaion or Equation, which Newton calls the fecond Semeftrial Equation, is in the dired Ra- tio of the Sine of double the Diftance of the Node from the Sun, and in the inverfe Ratio of the Cube of the Diftance of the Earth from ibc Sun : it amounts to 47'' when the Node is in OSant^ith the Sun and the Earth in its mean Diftance ; it is to be added when the Sun recedes in Antecedentia from the neareft Node, and is to be fubtraSed vhen the Sun advances in Confequentia. ^ XVII. After thofe three firft Equations of the Moon's Motion, follows that which is called her Equation of the Centre ; but this Equation cannot be obtained as that of the other Planets, by the Help of one Table, bc- caufc her Excentricity varies every Inftant, and the Motion of her Apo- gee is very irregular. In order therefore to obtain the Equation of the Centre of the Moon, the Excentricity and the true Place of the Apogee of the Moon is firft to be determined, which is effeaed by the Helpot Tables founded on the following Propofition. A ftraiglit Line being taken to expref^ the mean Excentricity of the Orbit of the Moon, which is 5505 Parts of the looooo into which the mean Diftance of the Moon from the Earth is fuppofed to be divided; at the Extremity of this ftraight Line afTumed as a Bafe, an Angle is made equal to double of the annual Argument, or of double the Diftance of the Sun from the mean Place of the Moon once correScd, as has been already direaed. The Length of the Side of this Angle is after- wards determined by making it equal to 1172J, half the Difference b^ tween the Icaft and greateft Excentricity. The Triangle being then com- pleted, the other Angle at the Bafe, cxpreffes the Equation or Cor- reaion to be made to the Place of the Apogee already once corrcfied; and the other Side of the Triangle which is oppofite to the Angle made equal to double of the annual Argument, will exprefs the Exccntriciiy correfponding to the propofed Time. The Equation of the Apogee be- ing added to its Place already correded, if the annual Argument be lew than 90, or between 180 and 270, or being fubtraaed in every other Cafe, the true Place of the Apogee will be obtained, which is to be fubduaed from the Place of the Moon correaed by the three Equations already mentioned, in order to have the mean Anomaly of the Moon. With this Anomaly, and the Excentricity, the Equation of the Centre by the ufual Methods will be obtained, and confequently the Place of the Moon correded for the fourth Time. The Equation of the Centre may be obtained without fuppofing the Excentricity variable^ or a Motion in the Apogee, by applying to doubk PHYSICAL WORLD. CXV of the Angle at the Moon fubtended by the mean Excentricity» or to the mean Equation of the Centre, the Equation 80' Sin (aDif.^© — m. An.(£) exprcfling the Variation produced by the Change of Excentricity, and Li- bration ot the Apogee. XVIII. The Place of the Moon correded for the fifth Time, is obtained by The 6M1 applying to the Place of the Moon corrcQed for the fourth Time, the ^the*** Equation called the Variation which was already found, to be always m Moon*t the direft Ratio of the Sine of double the Angle exprcffing the Diftance ^*?^^ ^ of the Moon from the Sun, and in the inverfe Ratio of the Cube of the riatfon.*' Diftance of the Earth from the Sun ; this Equation, which is to be add- ed in the firft and third Quadrant (in counting from the Sun) and fubtraded in the fecond and fourth is 35' 10" when the Moon is in OSant with the Sun, and the Earth in its mean Diftance. XIX. The fixth Equation of the Motion of the Moon is proportional to S>xih Equa- the Sine of the Angle which is obtained by adding the Diftance of the *^*"*' Moon from the Sun, to the Diftance of the Apogee of the Moon from that of the Sun. Its Maximum is t! 20'', and it is pofitive when this Sum is lefs than 180 Degrees, and negative if this Sum be greater. XX. The fcvcnth and laft Equation, which gives the true Place of the Scwmh Moon in its Orbit, is proportional to the Diftance of the Moon from Equation. the Sun ; it is 2' 20'' in its Maximum. XXI. It IS (icarce poffible to trace the Road which could have conduSed ThcNUthoi Newt9n to all thofe Equations, except fome Corollaries of Prop. 66, ,„J5c°ufc where he Ihews how to eftimate the perturbating Forces of the Sun. It of in invef- is eafy to perceive, that of thofe two Forces, the one which afts in the ^h^^^% ^^c. Dire&ion of the Ray of the Orbit of the Moon, being joined to tl>e corrcftioni Force of the Earth, alters the inverfe Proportion of the Square of the has not at DiAances, and confequently ftiould change not only the Curvature of the JJfcoJcrcd. Orbit, but alfo the Time which the Moon employs in defcribing it :- But how did Newton employ thofe Alterations of the central Force, and ivhat Principles did he make ufe of to avoid or furmount the extreme <Ioniplication and the Difficulties of Computation which occur in this Inquiry is what has not as yet been difcovered, at leaft after a fatis- faSory Manner. We find, it is true, in the firft Book of the Principia, 2l Propofition concerning the Motion of the Apfides in general, by which we learn, that if to a Force which aSs inverfely as the Square of the Diftance, another Force which is inverfely as the Cube of the Diftance be joined, the Body will defcribe an EUipfe whofe Plane revolves about the Centre 1 Cxvi SYSTEM OF THE of the Forces. In the Corollaries of this Propofition, Newton extends his Conclufion to the Cafe in which the Force, added to ihe Force which follows the Law of the Square of the Difta nee, docs not vary in the 'Jrlplicate, but In the Ratio of any Power of the D^ftance If therefore the perturbating Force of the Sun depended on the Di- Aance of the Moon from the Earth alone, by the He p of this Propofi- lion, the Motion of the Apfides of the Moon couW be determined; l)ut as the Diftance of the Moon from the Sun enters into the Exprcflioti of this Force, it is only by new Artifices, and perhaps as difficult lo be found as the Determination of the entire Orbit of the Moon : the Propofition of Newton concerning the Motion of the Apfides in general, can be applied to the Moon. Senfible of which, the firft Mathemt- ticiansof the prefent Age, have abandoned in this, as m every other Point that regards the Theory of the Moon, the Road purlued by ihc Commentators of Nevjton, and have refumed the whole Theory from its very Beginning ; they have inveftigated m a di red Manner, the Paths and Velocities of any three Bodies which attraa each other mutjially. The Succefs which has attended their united Efforts fhall be explained hereafter. XXII. Theory of It is manifeft, that the Satellites of Jupiter, confidered feparatcly, iheSttcUiteiji^^^lj ^^ affeaed by the three Forces which aduate them, in the fame !K&of Manner as the Moon; but their Number introduces a new Source of Saturn. Inequalities, not only each of them is attraded by Jupiter and the Sun, but they attrad each other mutually, and this mutual Atlraaion fljould produce very confiderablc Variations in their Motions; Variations fo much the more difficult to be fubjcacd to exaa Computations, as thev depend on their different Pofitions with refpea to each other, vhich their different Diftances and Velocities continually alter. However, the Laws of their Motions difcovered by Bradley, Wargentin and MarMh have enabled the eminent Mathematicians of this Age, to furmount thofc Difficulties, and to apply the Solution of the Problem of the three Bo- dies to the Inveftigation of the Inequalities of the Motions of thofe Sa- tellites, with almoft the fame Succefs as they had already done to thofc of the Moon. • t . , As to the Satellites of Saturn, Aftronomers have not been able to de- termine the Phenomena of their Motions with any Degree of Accuracy on Account of their great Diftance ; hence the Theory of thofe Planets is reduced to Ihew, that the Forces with which they aa on each other, or that with which the Sun aas on them, and difturbs their Motions, are very inconfiderable when compared with the Force with which they tend towards their principal Planet ; and that this Attradion b ^ inverfely proportional to the Squares of the Diftances. PHYSICAL WORLD. CXVII THEORY ^J tie Comets. I. THOUGH the Comets have in M Ages, drawn the Attention ofThcPer'p** Philolbphersy yet it is only fince the iaft Century and even fincc ^'^^J^JT ffewhn, they can be laid to be known. Senua feenned to have forefeen the ^tmtum Difcoveries which one Day would be made concerning thofe Bodies* but Mccon. the Germ of the trae Principles which he had fown> wtie ftiflcd by the Dodf ine of the Peripateticks, who, tranrmitting from Age to Age, the Errors of their Mafler^ maintained that the Comets were Meteors ct tianiient Fires. II. Several Aftronomersy but particularly Ttcbo, proved this Opinion to Tichopro?. be erroneous, by (hewing by their Obfervations, that thofe Bodies were «<* tl^tthey fituated far above the Moon, they deftroyed at the fame Time, the folid ^/^^^^^ Heavens, invented by the fcholaftic Philofophers, and propofcd Views Mooa» concerning the Syftem of tf.e World, which were much more conforma- ble to Reaibn and Obfervaiion. But their ConjeQures were yet very fiir from that Pointy to which the Geometry of Aewfon alone could attain. III. Defcartetf to whom the Sciences are fo much indebted, did not fucceed Dcibanet better than his Predeceflbrs in his Enquiries concerning the Comets ; 'jTS*^'** . he neither thought of employing the Obfervations v^hich were fo eafy nctl^J^Maer" for him to collet, nor Geometry to which it was fo natural to have Re- ingtVom courfe, and which he had carried to fo great a Point of PerfeQion; he ^^Jl"** confidered them as Planets wandering through the different Vortices, which^ compofed according to him, the Univerfe ; and did not imagine that their Motions were regulated by any Law. IV. Newlortt aided by his Theory of the Planets, and by the Obfer- Newton dlf- Tations which taught him that the Comets defcended into our planetary covered that Syftem, foon perceived that thofe Be dies were of the fame Nature kvowT^^* V7ith the Planets, and fubjed to the fame Laws. about the Every Body placed in our planetary Syftem, fhould, according to the f^iJ'-^^" j*" Theory of Newiortf be attrafled by the Sun, with a Force reciprocally ihcVame froportional to the Squares of the Diftances, which combined with a Laws ai the brce of ProjeSion, would make it defcribe a Conic SeQion about the ' Sun placed in the Focus. According therefore to this Theory, the Co- mets Ihould revolve iii a Conic Sedion about the Sun, and defcribe Areas proportional to the Times. V. Calculation and Obfervation, the faithful Guides of this great Man, enabled him to verify his Conjefture. He folved this fine Aftronomico- geometrical Problem. Three Places of a Comet which is fuppofed to CXVIII SYSTEM OF THE He deter- movc in a parabolic Orbit, defcribing round the Sun Areas proportional oibuV^l to ihe^Timcs, being given, with the Places of the Earth in the Ecliptic Comet from correfponding to thofe Times, to find the Vertex and Parameter of this threobicr- Parabola, its Nodes, the Inclination of its Plane to that of the Ecliptic, vaaons. ^^j ^j^^ Paffage of the Comet at the Perihelion, which are the Ele- ments necefTary for determining the Pofition and Dimeniions of the Parabolar This Problem, already of very great Difficulty in a parabolic Orbit, was fo extremely complicated in the Ellipfe and Hyperbola, that it was ncceffary to reduce it to this Degree of Simplicity. Befides the Hypo- pothefisof a parabolic Orbit, anfweredin Pradice, the fame End as that of the Ellipfe, becaufe the Comets during the Time they are vifible, defcribing but a very fmall Portion of their Orbit, move in very excentric Ellipfes, and it is demonftrated that the Portions of fuch Curves which are near their Foci, may be confidered without any fenfible Error as parabolic Arcs. VI. Hulct for ^^^ Refult of his Solution of this important Problem is as follows. decermiDiDg From the obfervcd Diftances of the Comet from the fixed Stars, whofe thcEkments ^ight Afcenfions and Declinations are known, deduce the right Afcenii- omc . ^^ ^^j Declination, and from thence the Longitude of the Comet re- duced to the Ecliptic, and its Latitude, correfponding to each Obfer- Preliminary vation: Compute the Longitude of the Sun at the Time of each Obfer- CompuuLi- vation, take the Difference (A, A', A") between the Longitude of the ^^ Comet and that of the Sun, correfponding to each Obfervation» which. is the Elongation of the Comet reduced to the Ecliptic. Compote alfo the Diftance (B, B', B") of the Earth from the Sun at the Time of each Obfervation. FxkstHy- Thofe preleminary Calculations being performed, afluming by Con- FOTHfiis. jeQure, the Diftances ^Y and Z^of the Comet from the Sun, reduced to the Ecliptic at the Time of the firft and fecond Obfervation, deter- mine the true Difiances by the Means of the two following Proportions, as the ajfumed Diftance (Y or XJ $f the Comet from the Sun in the fir ji or fecond Ohjeroationy is to the Sine of the objeroed Elongation^ (A or A') fi is the Diftance (B orB') of the Earth from the Sun at the Time of the firft Comctf** <^r fecond Obfervation, to the Sine of the Angle (C or C) contained by the ftraight Lines drawn from the Earth and the Sun to the Comet. Add this Angle (C or CJ to the Elongation (A or A') their Sum will be the Supplement qfthe Angle of Commutation (D or D'). And then fay as the Sine of the Angk X Elongation (A or A') is to the Sine of the Angle of Commutation (D orT>*)% is the Tangent of the obferved geocentric Latitude of the Comet correj- Heliocentric pending to the firft or fecond Obfervation, to the Tangent of the corref Latitude, pj^iing heliocentric Latitude of the Comet (E or E'). PHYSICAL WORLD. CXIX Each of the curt DIftances Y and Z diviftd by the Cofinc of the Vcftor corrcfponding heliocentric Latitude E and E' gives the true Diftance» (V, ^*^*' V) of the Comet from the Sun. Find the Angle contained by thofe Diftanccs thus : Add to (a) or fub- ftraS from the Places of the Earth, the correfponding Angles of Com- mutation (D, D') which will give the two heliocentric Longitudes (L,L') of iheComet, whofe Difference (F) is the heliocentric Motion of the Comet in the Plane of the Ecliptic. Then fay, As Radius^ is to tbeCoJine of the Motion (F) of the Comet in the Ecliptic^ fo is the Cotangent of the great efl of the two heliocentric Latitudes ^ to the Tangent of an Arc X. Subftradt this Arc X from the Complement of the leaft heliocentric Latitude, and [jJ^cTm« call the Remainder X'. Then the Cofine of tbefirji Arc X, will he to the in ju ors^r, Cofine oftbefecond Arc X', as the Sine of the great efl of the two Latitudes , to the Cofine of the Angle contained by the two ve^or Rays of the Comet. Which being done, determine the Place of the Perihelicn by the fol- lowing Rule : fubftraft the Logarithm of the leaft veSor Ray from that of the greateft, take half the Remainder, to whofe Chara6ieriftic, lo be- ing added, it will be the Tangent of an Angle, from which fubduSing 450, the Logarithm of the Tangent of the Remainder, added to the Log. of the Cotangent of i of the Motion of the Comet in its Orbit, will be the Logarithm of the Tangent of an Angle, to which J of the Motion of the Comet in its Orbit being added, the Sum will be the Half of the greateft true Anomaly, and their DiflFerence will be Half the leaft of the two true Tiji« An«- Anomalies. Double thofe Quantities to obtain the two true Anomalies, which will be both on the famfc Side of the Perihelion, when their DifFcTCi\ce is the whole Motion of the Comet, but on different Sides of it, when it is their Sum, which is equal to the whole Motion of the Comet. Find the Perihelion Diftance by adding twice the Logarithm of the Pcrlhtelion Coline of the greateft of the Halfs of the two true Anomalies, to that I^i^^ncc. of the greateft of the two veSor Rays, which will be the Logarithm of the Perihelion Diftance required. ^Determine the Time which the Comet fliould employ in defcribing the Angle contained by the two veSor Rays, by the following Rule ; y<7 the canjlant Logarithm 1,9149328, add the Logarithm of the Tangent I^^^^^cm-^ 4ff half of each true Anomaly. Add the Triple of this fame Logarithm of ployed in the tangent to the conflant Logarithm 1,4378116, the Sum of the two aeicribing ^umbers correfponding to tboje two Sums of Logarithms^ will be the exadi \^^.^^^l Uumber of Days correfponding to each true Anomaly in a Parabola whofe by the two perihelion Diftance is i. Take tht Logarithm of the Difference or Sum vcftorRays. of tbofe two Numbers J according at the two Anomalies are fituated on the fame Side, or on different Sides of the Perihelion. To this Logarithm add the I of tbe^ Log^ of the perihelion Diflance^ the Sum will be Log. of the (a) Aecordoig to the Fofition of the Comet with tefpe^ to the Signs of the Zodiac^ cxx Second Sup* pofitioD or the firft Hypothefif. SiCOKD Ktpothb- • IV Faflage •ethc Perihelion. Place of the SYSTEM OF THE Time tbi Cmet fiould eUtploy to defer ibe the Angle contained If tie tm ve^or Rays, If the Time thus found, does not agree with the obferved Time, another Value is to be aflumcdy for the curt Diftance (ZJ corref- ponding to the fecond Obfervation, retaining the tflfumed Diftance (Y) correfponding to the firfty and the heliocentric Longitude and Latitude of the Comet from thence deduced^ and all the Operations indicated in the foregoing Articles being repeated, another Expreflion will be found for the Interval of Time between the two Obfervations. Which if it approaches nearer the obferved Time, the fecond Value af- fumed for the Diftance (Z) is to be preferred to the firft ; if not» a thicd Value is to be aflumed for this Diftance, and by the Increafe or De- creafe of the Errors, the Value to be aftumed for it, fo that the Inter- val of Time calculated may agree with the obferved one, will eaiily be difcovered, and confequently a Parabola will be found, which anfwen the two firft Obfervations, which may be called ^rft Hypotbefis. This Parabola anfwering the two firft Obfervations would be the Or- bit fought if it anfwered likewife the third Obfervation ; but as this ne- ver happens^ another Parabola is to be found which anfwers the two firft Obfervations, by increafing or diminiihing, at will, the curt Diftance (Y) preferved conftant in the firft Hypothefis, and preferving it ftill conftant, but varying the fecond afliimei Diftance CZ) until this fecond Parabola is obtained. The third Obfervation calculated in thofe two Parabolas^ will (hew which of them approaches neareft the true Orbit fought. To calculate this third Obfervation in each Hypothefis, the Time of the PaflGige of the Comet at the Perihelion, the Inclination to the Ecliptic, and the Place of the Nodes of each Parabola is firft to be determined. To determine the Time of the PafTage of the Comet at the PeriheUoOf find the Number of Days correfponding to one of the two true Anomalies ; for Example, to that which correfponds to the firft Obfervation in the Parabola whofe perihelion Diftance is i, as before direded, the Logarithfli of this Number of Days added to | of the Logarithm of the perihelioa Diftance, will be the Logarithm of the Interval of Time elapfed be- tween the firft Obfervation and the Paftage of the Comet at the Peri- helion, which is to be added to or fubtraSed from the Time of the Ob- fervation, according, as it was made before or after the PafTage of the Comet at the Perihelion. To determine the Place of the Node, fay. As tbe Sine of the fecund Arc X' is to tbe Sine of tbefirfi Arc X, fo is tbe Tangent ^ftbe Mottpm §f tbe Comet in tbe Ecliptic^ to tbe Tangent of an Angle (R). Tben tbe RseU* ius, is to tbe Sine of tbe leafi latitude, as tbe Tangent of tbe Angle R, /p tbe Tangent of tie Diflancefrom tbe Node. By the Means of this Dif- PHYSICAL WORLD. CXXI tance from theNode» and the heliocentric Longitude of theCometi the heHocentric Longitude of the Node is obtajned. With which and the Diftance meafured on the Orbit of the Comet, the Place of the Periheli- inclinaticn. on is Determined. To find this Diftance fay. As the Sine of Angle R, to Rudiusy fo is this Dijiance meajured on ibe Ecliptic^ to the Dijiance required. To determine the Inclination fay. As the Radius is to the Sine of the Angle R, fo is the Cofine of the leafl Latitude, to the Cojine of the Angle of Inclination. The Elements of each Parabola being determined, the Place of the Comet feen from the Earth, anfwering to the third Obfervation, is com- puted in each, by the following Rules. Firft, Take the Logarithm of the Difference between the Time of the third Obfervation, and the Time of the PaflTage of the Comet at the Perihelion ; fubtrad from rt J of the Logarithm of the perihelion Diftance, the Remainder will be the Logarithm of the Difference be- Ruktfor tween the Time of the third Obfervation and the Time of the Paffage finding thr of the Comet at the Perihelion of the Parabola, whofe perihelion Di- Lon^^^^j^ ftance is i. Secondly, Find the true Anomaly correfponding to this and°fllti-* Time, by folving the Equation /H3^= ^ ^^ g (*») in which / cxpreffes ^^^^tl^ the Tangent of half the true Anon^aly, and b the Time employed in defcribing it. Thirdly, When the Motion of the Comet is dired, add this true Anomaly to the Place of the Perihelion, if the third Obferva- tion was made after the Paffage of the Comet at the Perihelion ; But fubtrad it from the Place of the Perihelion if the Obfervation was made before the Paffage at the Perihelion. And when the Motion of the Comet IS retrograde, add the true Anomaly to the Place of the Perihe- lion, if the Obfervation was made before the Paffage at the Perihelion ; but fubtrad it from the Place of the Perihelion, if the Obfervation was made after the Paffage at the Perihelion ; by this Means, the true he- liocentric Longitude of the Comet in its Orbit is obtained. Fourthly, Take the Difference between this Longitude and that of the afcc nding Node, which will be the true Argument of the Latitude of the Comet. Fifthly, fay. As the Radius is to the Cojine of the Inclinationyfo is the Tangent of the Argument of Latitude^ to the Tangent of this Argument meafured on the Ecliptic ; which added to the true Plate of the Node,gives the heliocentric Longitude reduced to the Ecliptic. Sixthly, fay. As the Radius is to the Sine of the Argument of Latitudes fo is the Sine of the Inclination of the Orbit of tbe Comet y to the Sine of its heliocentric Latitude, which, when the Mo- (b) The £«iutioii /^ 4- }/= «-^ may be fbWed chiu : Make. a Kjght-angled Triangle, one of whole Sida U cxpreflcd hy i . and the other by * i calcnhue the Hypocheoeufe (H),fiiiat«onie»PlropordoBBlibttwCfDHH — aft^H— — 4 tt^thtirPifo^ .•.u-v.^1 r ^SAMT 54,«077 CXXII Rule for findiog the cu.tDi- ftaDce. Kulet for finding the eeocentrlc Longicnde and Lati- .cude. SYSTEM OF THE tion of the Comet is direck, is North or South, according as the Argu- ment of Latitude is lels or greater than fix Signs ; and when the Mo- tion ot the Comet is retrograde, it is North or South according as the Argument cf Latitude is greater or lefs than fix Signs. Seventhly, Add the Logarithm of the Ccfme of the heliocentric Latitude to the Leg. of the perihelion Diftance, and fubtraci from this Sum the Lop. of double of the Ccfine of half the true Anomaly, the Remainder will be the Lo- garithm ot the curt Diflancc correfponding to the third Ob "ervat ion. Eighthly, Take tire Difference between the Lv garithm of the curt Di- ftance, and that of the Diftance of the Earth from the Sun, add lo to the Chara£teriftic of this Difference, and it will be the Logarithm of the Tangent of an Angle ; from which fubtraS 45^'^. and to the Logarithm of the Tangent of the Remainder, add the Logarithm of the Tangent of the Complement of half the Angle of Commutation, the Sum will be the Logarithm of the Tangent ot an Arc, which add to this Com- plement, if the curt Diftance of the Comet from the Sun exceeds the Diftance of the Earth from the Sun, but fubtraS from this Comple- ment if the Diftance of the Comet be lefs than that of the Earth ; in or- der to obtain the Angle of Elongation, which added to or fubiraSed from the true Place of the Sun, according as the Comet fecn from the Earth, is to the Eaft or to the Weft of the Sun, will give the geocentric Lon- gitude of the Comet. Ninthly, and laftly fay, As tbe Sine of the Angle 9/ Commutation^ is to tbe Sine of tbe Angle of Elongation^ fo is tbe Tangent of tbe beliocentric Latitude of tbe Comet to tbe Tangent of its geocentric Latitude, The Longitude and Latitude thus found ought to agree widi the obferved ones, if the Parabola obtained was really the Orbit de- fcribed by the Comet. VII, Example, Let it be propofed to find the Elements of the Parabola de- fcribed by the Comet which was obferved in Europe ; the beginning of Marcb 1 742, with a very remarkable Tail,c6ming with extraordinary Ra- pidity from the fouthern Hemifphere, and afterwards advancing towards the North Pole, its heliocentric Motion being retrograde, and its VcJo- •city ancT Splendor decreafing to the 6th of May^ when it difappeared. 174*- Tinw. h. m. i. 4Marciiat itf 9 50 t8 . at 13 39 o t4 April tt 9 39 o Obf. LoBff. of the Comet. • o / // 9 16 o 40 a iS 5* 45 3 ' $ 31 Obferv. Lat. Nofth of the Cemct. ^ I n 34 45 37 «3 S 55 50 3» 50 Long, of the Sim calcnU- ted. » M ay 44 8 It aS 4«7 «• Log. of thelElong. t^ tlie DiCoftheE. Comec from (romiheSuB the Sun. 9.99^910 9.999840 o*oo309a 5« *7 4 W. S« St 17 fi- r PHYSICAL WORLD. CXXIII I Suppofition, ¥=0,879, Z =0,957 of ihe mean Diftance of the First Hy- Earth Irom the Sun =1, then Angle C =105° 42' 48", C=6i° 31' o'', "'thius, C-(-A=i640 9' 52'^ and C-f-A'=ii8« 9' 17", wherefore Angle D= ^«*io«n- 150 50' 8', and Angle 0=61050' 43", coniequcnily the heliocentric IndLonaf/ Latitudes, £=12031' 42'' North and £'=520 3' 38", and the Log. ot ludcofOie the veftor Rays, V=9,954455 ¥'=0,192159. Comet. The Angle of Commutation D=i5«> 50' 8", being added to 5^ 140, 27'44", and Angle D'=6 10 50' 43" fubtraded from 7* 40 27' 16'', the corrcfponding Longitudes of the Earth, gives the heliocentric Longitudes ©f the Comet, L=6» 00 1 7' 52", and L'=5» 20 36' 33" ; tlieir Difierence ^Pg*;j ^«*"- F=27o 41' 19" is the Motion of the Comet in the Ecliptic, the Arc {h"two ve<> X will be found =340 37' u", and Arc X'=42o 51' 7''; confequently tor Rays. the Angte contained by the two veftorRays =45® 22' 8". The Log. of the greateft vedor Ray, 0,192159 lefs the Log. of the leaft, 9»954455==o*237704> and its Half 10,118852, 10 being added to its CharaSeriftic, is the Tangent of 520 44' 38', from which 450 being fubtrafted, and to the Log. of the Tangent of the Remainder 70 44' 38", the Log. of Cotangent of i r© 20' 32'', the i of the Motion (450 22' 8",) of the Comet in its Orbit being added, the Sum will be the Loga- rithm of the Tangent of 340 8^ 5" i, whereby the Halfs of the two true Anomalies are found to be 220 47' 33'' J, and 450 28' 37'' I, TmcAno. confequently the leaft true Anomaly =450 35' 7", and the greatefl =900, roalics. 57' 15"; and their Difierence being equal to the Motion of the Comet in its Orbit, thofe two Anomalies are on the fame Side of the Peri- helion. The Log. of the perihelion Diflancc will be found =9,883835. Perihelion To determine the Time the Comet employed to defcribe the Angle ^'^»«»<^«- contained by the two veftor Rays, to the conftant Log. 1,9149328 adding 0,007233 Log. of the Tangent of 450. 28'. 37"^, and to the con- ftant Log. 1,438112 adding 0,021699 Triple of the Log. of this fame Tangent. I find 83,592 and 28,808 for the Numbers correfponding to 1,922166 and 1,459512 Sums of ihofe Lagarithms, confequently 1 1 2^400 Days is the Time correfponding to the true Anomaly 90°. 57' intCTvalof 15", in a Parabola whofe perihelion Diftance is i. By a like Procefs, I Time be find the Number of Days 36,579 correfponding to the true Anomaly *^^q^{?* 450 35'. 7", in the fame Parabola, I take the Difi^erence 75,821 of iadonscai- thofe Times, becaufe the two Anomalies are fituated on the fame Side cuUtcd. of the Perihelion, whofe Logarithm 1,879780 added to 9,825752 the j of the Log. of the perihelion Diftance, is the Log. 1,705541, to which correfponds 50,762 Days, Time employed by the Comet to defcribe the Angle contained by the two veftor Rays. Comparing this Time with the Interval 50,728 i between the two Obfervations , I find it exceeds it by 0,033 , I theretore make a Varia- tion of 0,001 in the Diftance (Z), in order to difcover which Way, CXXIV SYSTEM OF THE and by how much the Elements of the corrfponding ParaboU wlil be changed. Second Sup- n Suppoiition, Y=: 0^879) Z=: 099569 and repeating the lame fhffiJftHy- Calculations as in the firft Suppofition, I find the hehocentric Latitudes poiheiii. E= 120 31' 42"f E' = 52<» 1' 54" i» the Log. of the veflor Rays, V = 9,954455> V'=z 0,191424, the heliocentric Longitudes, Lnfr a> 1 7' 52", L'=5» 2° 43' 1 1". The Motion of the Comet in the Ecliptic = 270 34' 41"* and the Motion of the Comet in its Orbit =4.$^ 18' 13" the true Anomalies 450 32' 3", and gcP 50' 16'', the corresponding Days 36.529 and 112,056, the Log. of the perihelion Diftance =9*883997 ; finally the reduced Time employed in defcribing the Angle contained by the two veftor Rays 50,594 Days. Frem whence 1 find that by increafingZ by the Qijantity 0,001, I diminilh the Time by 0,168: And I fay, 0,168 : 0,001 :: 0,0334: 0,0002. I diminilh therefore Z by 0,0002 to obtain a Parabola anfwering the Conditions required. III Suppofition, Y=o^79, Z=:o,9568, and I find the heliocentric Latitudes, E=iao 31' 42", E'=52o 3' i&'i, the Log. of the veaor Rays, V=9,954455, and ¥'=0,192009 ; the heliocentric Longitudes, L=6« o«> 17' 52", and L'=58 2© 37' 53" ; the Motion of the 0>met in the Ecliptic, 270 39' 59" j and the Motion in its Orbit 450 21' 22"; the true Anomalies 45® 34' 28", and 90** 55' 50"; the correfpondtng Times 3655684, and 1 1 2,330 Days: The Log. of the perihelion Diftance 9,883870, and the Time reduced employed in defcribing the Angle con* tained by the two veSor Rays, 50,7284 Days, agreeable to Obferration. Having found a Parabola anfwering the two fir(l Obfervations, I ieardi SicondHt for another, anfwering the fame Obfervations,. bv making a Variatioa TOTHnn. jn the Diftance (Y) preferved conflant in the farft Hypothefis. IV &ippofition, Y=o,878, Z=o,957, and I find the heltocentrie rirftSuppo- Latitudes, E=i2« 42' n", E'=:520 3' 38", the Log. of the veaor fitionofthe Rays, V=9,954257, ¥'=0,192159, the heliocentric Longitudes L= ?oiff ^* 6» o' 31' 54^ and L'=5« 2^ 36' 33'' ; the Motion of the Comet in the Ecliptic =270 55' 21''^ the Angle contained by the two veftor Rayi =450 17' 56", the true Anomalies 45'> 44' 56" and 910 2' 52", the cor- refponding Times 36,743 and 112,680, the Log. of the perihelion Dif- tance. 9,883 11 5, the reduced Time employed in defcribing the Angle formed by the two veSor Rays 50,714, which differs by 0,014^1 from the obfcrved Interval, confequenily by diminifliing Y by 0,601, the Time isdiminiftied by 0,048. I fay, 0,048 : 0,001 : : o,oi4|. : 0,0003. V Suppofition, 1=0,8783 Z=o,957, I find the heliocentric Lrati- Second SuD- *"^^^ E=i20 39' 2'' E=520 3' 38" the Log. of the veaor Rays, fofitionof V=9,9543i6 ¥'=0,192159, the heliocentric Longitudes, L=:6* o* thcfecond 27' 40", 'L'=55 2^ 36' 33", the Motion of the Comet in the EclipHc «ypothcfi$. 270 51' 7" the Angle contained by the two vedor Rays 450 19' 20''', the true Anomalies 450 41 '45" and 910 i' 5" the correfponding Times 36,689, P H Y S I C A L W O R L D. CXXV and Ii29590> the Log. of the perihelion Diftance 998833449 and the Time reduced employed in defcribing the Angle contained by the two vedor Rays =50,729 agreeable to Obfervation. Having found two Parabolas anfwering the two firft Obfervations, we are next to examine which approaches neareft the Orbit of the Comet fought, by calculating the third Obfervation in each ; for which Purpofe I calculate the Place of the Perihelion, the Time of the Paflage at the Perihelion, the Inclination to the Ecliptic, and the Place of the Nodes of each Parabola. To determine thofc Elements in the firft Parabola, I find the Angle R=:23o 40' 15'', then the Difiance of the Comet reduced to the Ecliptic Elemenuof at the firft Obfervation from the afcending Node 50 25' 45", which added ^ ^j^"*? to the heliocentric Longitude of the Comet, the 4th of March, which rfi/firft anS is 6 • OP 17' 52", becaule its heliocentric Motion is retrograde, gives the fecondHy- Place of the Node, in 6- 5^ 43' 37''. The Diftancc of the Comet ^'^^• from the Node meafured on its Orbit, which I find to be 130 38 14", fubtraded from the Place of the Node, gives the Place of the Comet in in its Orbit, at the Time of the firft Obfervation: and becaufe it had then 450 34' 28" true Anomaly, I add them to its Place in its Orbit to obtain the Place of the Perihelion in 7'» 70 39' 51". I add ^ of the Log. of the perihelion Diftance to that of 36,568^ Days, Time correfponding to the leaft true Anomaly 450 34' 28", which gives 24,486 Days, for the Interval of Time elapfed between the firft Obfervation, and the In- ftant of the Paflage of the Comet at the Perihelion, which being fub- traSed from the 4th of March at i6h 9' 50", or at 0,673^, the Time of the firft Obfervation, fixes the Inftant of the Paflage at the Perihe- lion to the 8th of February at 0,188. In fine, I find the Angle of In- clination of the Plane of the Ecliptic, and that of the Comet to be 660 56' 14''. The fame Elements in the fecond Parabola are, the afcending Node in 6* 5** 59' 6", the Place of the Perihelion in 7» 7o 53' 42, the Incli- nation, 66^ 47' 14", and the Time of the Paflage at the Perihelion, February the 8 th, 151 J. From thofe Elements I calculate the geocentric Longitude for the '28th of March, at 0,569 of the Day, in each Parabola. The Interval of Time elapfed between the Pafl*age at the Perihelion in the firft Pa- rabola> and the Time of the Obfervation 28th March 0,569 is 48,381 Days. The Log. of the perihelion Diftance, 9,883870, its Triple is, 99651610, its Half, 9,825805, which being fubtrafied from 1,684675, Log, of 48,381 gives 1,858870, Log of 72,255 Days, which corref- ponds to 73** ii' ^", or 2» 13® 11' 7" Anomaly^ which fubtraSed from the Place of the Perihelion 7« 7** 39' 51", becaufe the Comet being re- trograde, the given Inftant follows, that of the Paflage at the Perihe- lion, which gives the true heliocentric Place of the Comet in its Orbit, CXXVI SYSTEM OF THE ed in th^ firft and (econd Hypocbclif. 4S24« 28' 44", from 45 24^* 28^44", fubtraaing 6^5^ 43' 37", the Geocentric Place of the afcending Node, the Argument of Latitude io« iS** 45' 7" oft^cCo- is obtained, which meafured on the Ecliptic is ii» ii** 2^47"; confe- inetcalculat quently the heliocentric Longitude of the Comet is y 16° 46' 24", and the heliocentric Latitude, 37® 20' 41" North becaufe the Argument of Latitude of the Comet, which is retrograde, is greater than fix Signs. The true Place of the Sun the 28 of March, at 13^ 39m is o« 8"" 1 1' 28", and the Log. of its Diftance from the Earth, is 9,999^41 ; therefore the true Place of the Earth feenfrom the Sun, is 6* 8** 11' 28", which ex- ceeds 58 16® 46' 24'' by 21° 25' 4", which is the Angle of Commuta- tion. I find the Log. of the curt Diftance, correfponding to the third Obfervation =9^974915, I fubtraS 9>9749i5 from 9,999841, Log, of the Diftance of the Sun from the Eearth : The Remainder is 0,024926, which by adding 10 to its CharaSeraftic, gives 10,024926, Log. of the Tangent of 460 38' 42"!, from which fubtraSing 45, the Log. of Tan. of Remainder, 1 o 38' 42", added to that of the Tangent of 79^ i f 28', 42' 32", half of the Angle of Commutation ThiidHy' yOTNIIIt. (Complement of 10" ^ 210 25' 4'') the Sum is the Log. of the Tangent of 8** 37' 39'', which fubtrafted from 79^ if 28"; becaufe the Diftance of the Comet from the Sun, is lefs than that of the Earth from the Sun, gives 70'' 39' 49'', or 2» 16** 39'49''» for the Angle of Elongation. By Means of a Fi- gure reprefenting the Ecliptic divided into 12 Signs, in which I place the Sun, the Earth, and the Comet, according to their Longitudes found by the above Calculations, I perceive that the Comet feen from the Earth, is to the Eaft of the Sun. I therefore add the Angle of Elongation to the true Place of the Sun, which gives the true geocentric Longitude of the Comet, in2M8^ 51' 17", which is lefs than the obferved Longitude 2« 18** 52' 45'' by I ' 28"; by a like Procefs I find the geocentric Longitude of the Comet in the fecond Parabola, the 28 of March, in 2» 18® 45' 14", which is lefs than the obferved Longitude, by f 31"; confequenly neither of the two Parabolas, is the Orbit of the Comet. But becaufe the Variations of the Orbits, are fenfibly proportional to thofe made in the curt Diftances, to obtain the two curt Diftances which correfpond to the Orbit fought. I make thofe two Proportions ; (c) As 6' 3" Difference of the two Errors — i ' 28" and —7' 3 1 ", Is to the leaji of the two i' 28" : So is 0,0007 fl«^ 0,0002, Corrections made to the two curt Dijlances Y andZ, to obtain two Parabolas anjwering the twofirft Ob^ fervations, to 0,000235 «nrf 0,000065, Corre^ions to be made to tboje Dif tames Y and Z, to obtain the Orbit required. To apply thofe Corrcdions, I obferve, that fince Y, fuppofed = to o>879, gives an Error of — 1'28", and Y fuppofed = to 0,8783, gives an Error of — 7' 31", by diminifliing Y, the Error is increafed; from whence I conclude, that 0,000235 ^^ ^^ ^^ added to 0,879, to obtain I would haTC faid ai the Sum of the £rron dec. if the one was by cxceli sad the other bj PHYSICAL WORLD. CXXVII the true Value of Y, which confequcntly will be 0>879235 ; in like Manner^ I find that Z ftiould be fuppored —0,956735. VI Siippofition, Y=:Oi879235, and Z-:iO,956735, and I find the heli- ocentric Latitudes, £=^12^29' 17'' j, E'=^i2^ 3' lo'J ; the Log. of ihe veftor Rays, Vi=9,954504, aud ¥'—0,191963; the heliocentric Lon- gitudes, L-^6^ o' 14' 37'', and L'=5* 2^ 38' 19''; the true Anomalies 45' 32^0" and 90® 54' 4''; the cortfponding Times 36,528 and 1 12,243 Days ; the Log. of the perihelion Diftance 9,884049 ; and the Time employed in describing the Angle contained by the two ve6tcr Rnyi-, 50,729; the Place of the Node in 6« 5° 38' 29''; the Place of the Pe- rihelon, 7» 7^35' 13", the Inclination of the Orbit, 66"* 59' 14" ; and the Time of the Paflage at the Pcrhelion the 8th of February, at 4H 48' : In fine, from thofc Elements, I calculate the geocentric Longi- tude and Latitude the 28lh of March, at 13^39', which I find, the one in 2* 18** 53' 18", the other 6^^^ 3' 57^' North, agreeable to Obfcrvaticn. By thefe Rules the following Table was calculated, containing the Ele- ments of all the Comets which have been obferved with dny Degree of Accuracy. Geocentric lor.gi.i.d^ an<lJLav tuJe o the C o- rrct ia!culi.t (d in the rhirJ Hypo- thcfis. I Place of the Years, afccndioe Node. IndinaCioo 837 123 1 1264 1299 1301 ^337 1472 iS3a 1533 1SS6 1577 i$8o 1585 1590 »593 1596 1618 1618 1652 166 1664 i66s 1672 • o / „ 6.26.33.00 0.13.30.00 7.28.45.00120.25.00 o / 1/ 1 2. 00. CO 6. 5.00 3.17. 8.00 0.16.00,00 2. 6.22.00 9,11.46.20 2.20.27.00 4. 7.42.00 5.25.42,00 0.25.52.00 0.18.57.20 I. 7.42.3^ 5-l5,30-4 5. 14.15.00 10.12.12. 3055 12.00 9.23.25.00 2. 1 6. 1. 00 2.21.14.00 7.18, 2.00 68.57.30 70.00.0c 32.11.00 5.20.00 32.36.00 46.30.00 32. 6.30 74.3^-45 64.40.00 6. 4.00 29.40.40 87.58.00 21.28*00 Place of the Perihelion. • o 9.19 / // 3.00 Pcrihe- [Titnc of the Paflage j lion DiPat the Perihelion aij tance. Paris. 0,5 8oo|M arch. 1 1 30 « . »7 o,3i79piiarch. 31 22, 4.14.48.00 0,9478 Jan 9. 5.45.00 0,4108 July o. 3.20.00 9.30.00.00 0,4467 Odk. 0.20.00.00 0,6445 Jane 0,5427 Feb 1.15.33-30 - . 3.21. 7.00 0,5092 Od 5. 6.38.000,1525 May 9. 8.50.000,4639 April. 4. 9.22.000,18350ft. 3.19. 5.50 0,5963Novem.28.i5 o. 8.51.00 1,1094 Od. 7'i9 7. 6.54.30 0,5767 Feb. 8. 3 4.26.19.000,8911 July. 18.13 7* 1 8.1 6*000,51 30 Aug. 10.20. 0.18.20.000,5131 Aug. 17* 3 , 37.34.00100. 2,14.00 0,3798 Novem. 8.12 2.28.10.0079.28.0000.28.18.400,8475 Noveni.1 2.15 2.22.30.3032.35.50 3.25.58.400^^86 Jan. 26.23 21,18.30 4.10.41.2511 ,1026 Decem. 4.12 76. 5.00 2.1 1« 54.3010,1065 April. 24; 5 9.27.30.30 83. w, 10 h 12. 7. 6. 7. o. I. I. 28.22. 19122. 25.10. 21.20. 26.18. CO rctr. 00 dir. dir retr. rctr. 00 rctr. 3* .21 3» 12 54 retr. dir. dir. dir. rctr 9 dir. dir. rctr. dir retr, 1 2 dir. 32 dir. 49 dir. 50 dir. 3 rctr. *.«.. 7.^.3 w|w,.«.w;,«»^.... ..f. 7.24|retr. i.i6.59.30|o^6975lMArch. i. 8.46ldir. Table of the Eleneots of the Comets. CXXVUI SYSTEM OF THE I Place of the Tears, afcendine Node. Inclination 1677 1678 x68o 1683 1684 1686 1689 1698 7.26.4^1079. I'lS S.x 1.40.00 3. 4.20 9. z, 2.0060.56.00 5. 23. 23.0083. IX. 00 8.28.is.oc6$.48.4o 11.20.34.40 xo.23.45.20 8.27.44.15 1699 10.21.45.3569.20.00 1702 1706 1707 1718 6. 9.25. X5 0.13.1 x.4055. 14.10 1.22.46.35 4. 8^1.3.00 1723' 0.X4.16.00 1729 10.10.32.37 ^737] 7.16.22.OV 1739 6.27.25.14 ^742 6. 5.38.2906.59.14 X743 2.18.21. 15 31.21.40 69.17.00 11.46.00 4.30.00 88.36.00 30.20.00 1.9.59.00 76.58. 4 18.20.45 55.42.44 vu^ r.f tu^ P^rihc- jTiroe of the Ptffagil PUceofihe lionDif-atthePerihelioiiaa Perihelion. 4.17.37. 5 10.27.46.00 8.22.39.300,0061 2.25.29.30 7.28.52.000,9601 2.17.00.300, 8.23.44.450,0168 9.00.51.15 0,691 7. 2-31. 6 4.18.41. 3 2.12.29.X0 2.19.54.56 4.01.30.00 1.12.52.2? 10.22.40.00 1,4261 10.25.55*00 tance. 0^2806 May. 1,1238 0,5602 July d 6. Aug. 26. Decern. 18 June. ^. ' 16. Occem. 1. 18. an. 13. ,6459|March.i3. Jan. 30, Decern.! I. an. 14 3250 Sept. 7^40 J' 30a. 0,4258 0,8597 1,10 o,9876jSept Jttnc 0,2229 Jan. 3.x 2.3 8.40 0,6736 June. iFeb. [an. 0,7657 7. 7.35-13 2.19.33, 3. 2.4i.45jO,«35ojJj X743' o. 5.16.2545.48.201 8. 6.33.520,52 -*^ 6.17.10.000,2225 5.18J 6.20' 1744' 115.46.ii 1747! 4.27.18.50. , 1748I 7.22.52.1685.26.57 17481 I- 4-39-43 56•59■ l757' 7. 4- 5-50«a-39' 1758! 7.2. .50. 9168.19.00 1759 4X9-39MJ78.59'** 1759 2.19.50.45 4.51.32 1762 11.19.00.008^.20.00 1763J1 1.26.17.00 72.42.00 1764] 4. o. 7.0052.47.00 I766J i'.' 4.10.5040.5 \2~)J X766' 1.17.22.191 8.18.45) 9. 7. 2.00 7. S 0.50 9. 6. 9.2^0,6553 4. 2.39 00 8.27,37.45 1.23.24.20 4,18.24.35 3.14.00.00 2.24^.3.00 0.15.26.00 4.23.15.25 6.26. 5.13(0.6368 1,2198 3 ' 54 J' ,7985 N( 339> 0,21 o_ 0,9660 Dcc( 1,0090 May 4991 Novcm 5567 Feb. 0,5053 F«b Parit. 5 Sept. 27. 25. 30. 17. 8. 10. 20, March. I. March. 3. ■ 28. June. 18. oa. 21, une. 1 1 . bvem«27. em. 16, X. 12. 17. April. 17. 8407 April h / 0.46 retr. I4.i2dir. 00.1 5 dir. 2.59 rctr. 10.25 dir. 1 4.42 dir. 15. 5 rctr. 17. 6 rctr. 8.32 rctr. 14.22 dir. . 4.52 dir. ,23.39 [iir. .23.48 rctr. .16.20 retr. .11. 6 dir. > 8.3oldir. 10. 9kcU. . 4.48 retr. ,20.35 iir. 2i.2o]rttr. , 8. 1 3 dir. , 7.2itctr. .19.34 rctr. . 1.33 dir. . 9.42 dir. . 3.27 dir. . 2.28 din. ,21.13 ret. .00.00 dir. .18.39 dir. ,13.40 retr. 8.50 rea XO.26 dir. Elemnti 0/ tbi Comet of Hal ley, in its difftrent Revolutions. 1456 »53i 1607 X682 I75i 1.18.30.00 1.19,25.00 1.20.21.00 1.20.48.00 1.23.49.00 17.56.001x0.1.00.0010,5856 17.56.00 10.1.39.0010,5670 17. 2.00 17.42.00 l7»39«oo io.2.i6.oo;o,5868 10.1.36.00,0,5825 10.3.16.0010.583^ June 8. 22. lo.jrecr. Aug. 24. 21.27. fctr. Oa. 26. 3* 59. retr. Sept. 14. 21.31. rctr. March 12. 13*41. rctr. r N PHYSICAL WORLD. CXXIX VIII. Newton having thus folved the above-mentioned Problemy and applied Newton vr- it to all the Comets obferved, deduced from thence a complete Confir- jJ^kJlSoii ination of his Conjedure. For all the Places of the Comets calculated by the ob- in the parabolic Orbits, whofe Elements were delivered in the foregoing relations ©f Table, compared with thofe immediately deduced from Obfcrvation, \^^\ ***" never differed feniibly, which will appear fo much the more fur- Comcu. prifing, when we confider how difficult it is to attain to Precifion in Obfervations of this Nature. IX. As to the Duration of the Periods of the Comets, it cannot be de- The Dun- daced from the fame Calculation, becaufe as we have already hinted, p^,"oJ*^!f their Orbits being fo excentric that they may be taken for Parabolas not be de- without any fenfible Error, very great DiflFerences in their Duration *«€«<* ^ would produce fcarce any Alteration in the Arc of their Orbit, which HuE,ryrf they defcribe during the Time they are vifible. However, it no lefs the Apari- confirms the Theory of Newton^ to have (hewn, that in this Por- comctf ** tion of their Orbit, they obferve the Law of Kepler, that of the Areas tlwfamc being proportional to the Times, and that the Sun attraSs them in the circnmftan- fame Manner as all the other celeftial Bodies, in the inverfe Ratio of eSa "Iter- the Squares of the Diftances. riu. X. Halley^ on examining the famous Comet of 1680, having found that Halleyem- the Obfervations of a Comet recorded in Hiftory, agreed with it in very |'°![*j'^f remarkable Circumflances, and that they had appeared at theDiftance of the Comet ^75 Years from each other, conje3ured, that it might be but one and the of 1680 to ikme Comet, performing its Revolution about the Sun in this Period, he oS>*u! "' tlierefore fuppofed the Parabola to be changed into an EHipfe defcribed by the Comet in 575 Years, and having the fame Focus and Vertex with the Parabola. Calculating afterwards, the Places of the Comet in this elliptic Orbit, he found them to agree p^rfeSly with thofe where the Comet was obferved ; fo that the Variation did not exceed the Dif- ference found between the calculated Places of the Planets, and what arc immediately deduced from Obfervation, though the Motions of the Planets have been the ObjeS of the Inquiries of Philofophers for thou- sands of Years. XI. Befides the Comet of 1680, Halley found three others, >vhich nearly agreed, thofe of 15319-of 1607, and of 1682, the three Parabolas were fitu^ted after the fame Manner, the perihelion Diftances were equal, and the intervals of Time 75 or 76 Years ; he conjeSured that it might be but one and the fame Comet, and that the Differencce in their Inclinations and Periods, might arife from the AttraSions of the fu- / cxxx traction oa the DHfknm Opiniont concerning the TtWs of Cometf. Kewtonifof Opinion chat they are Va poun exhal- ed from the Body of the Comet. Con6miati- on of rhit Opinion. SYSTEM OF THE pcrior Planets ; for he obferved, that the Comet in 1681, paffed very near Jupiter | and it is certain, that the Comets receding farther from the Sun than the Planets, their Velocity and Tendency towards the Sun (hould thereby be confiderably leflened in the fuperior PmrU of theii Orbits, and confequently (hould be more fufceptibfe of the Modifications and loipreffions of the Attradions, which the Planets in their Ap- proach exert on them ; from whence he concluded, that the following Apparition would be retarded, and anounced the Return of this Comet for 1759. But thefe Confiderations were too Yague to be depended upon. To attain to Certainty in this Point, it was neceflary to calcu- late the Situations of the Comet, and the Forces with which Jupiter and Saturn attrad it during feveral Revolutions, and by the Hdp of thofe Forces, exprefled in Numbers, to determine the total EflFed of the Attradions of thofe Planets on the Comet. This Clairaut, and after him the firft Mathcnuticians in Europe have efFeded, and have denxMK ftrated that this Comet obferved in i53i> 1607, and 1682, (houM have the unequal Periods of 913^ and 898 J Months and that the Period after which it would appear again in this Age, would be 919 Montlis, wUch the Event has juftibed. Thefe Refearches ihall be explained hereafter. xii. The Tails of Comets which formerly occafioned the Apfiarition of thofe Bodies to be regarded as portentous Omens, are now ranked in the Number of thofe ordinary Phenomena which raife the Attention^ of Philofophers alone. Some would have it, that the Rays of the Son paf> fing through the Body of the Comet, which they fuppofe to lie tranf- parent, produced the Appearance of their Tails, in the fame Mamer as we perceive the Space travcrfed by the Beams of the Sun paffing through the Hole of a darkened Room : others imagined that the Taib were the Light of the Comet refraded in their Paflage to the Earth, and producing a long Spedrum, as the Sun does by the Refradion of the Prifm. Newton having mentioned thofe two Opinions, and refuted them, expofes a Third which he adopted himfetf : it confifts in regaid* ing the Tail of a Conriet as a Vapour which rifes continoaHy from tke Body of the Comet towards the Parts oppofite to the Sun, for the laane Reafon, that Vapours or Smoke rife in the Atmofj^ere from the Eartlb and even in the Void of the Pneumatic Pump. On Account of the Motion of the Body of the Comet, the Tail is incurved towstrds the Place through which the Comet pafltd, much in the fame Manacr as the Smoke proceeding from a burning Cole put in Motion. What confirms this Opinion is, that the Tails are found greatcft when the Comet has juft pad the Perihelion or lead Diftance from the Sun, where its Heat is greateft, and the Atmofphere of the Sun is mnft denfe. The Head appears after this, obfcured by the thick Vapour thai to [ i PHYSICALWORLD. CXXXI rUet plentifully from it» but about the Centre, a Part more luminous than the reft appears, which U called the Nucleus. A great Part of the Tails of the Comets (hould be dilated and diffufed ufe of the over the Solar Syftem by this Raiefadion : feme of it by its Gravity TaiUofCo- may fall towards the Planets, mix with their Atmofpheres ^nd repair "*J^ '" the Fluids, which are confunied in the Operatjoas of N^ure. vUtln, The Rcfiftancc which the Comets meet with in traverfing the At- mofphere of the Sun when they defcend into the lower Parts of their Orbits, will affcfl them, and their Motion being retarded^ their Gravity comtr.may will bring them nearer the Sun in everv Revoljution, umil at length f*^ into tht they are iwallowed up in this immenfe Globe of Fire. ^"°' The Comet of 1^80, paflfed at a Diilance from the Surface of the Sun which did not exceed the (ixth Part of his Diameter, and it is iiighly prafeable» that it will approach nearer in the next Revolution, nod at length will fall into his body. XIII. Let the Diftance of any one of the primary Planets from the Sun Addition to =1 its periodic Tiroe=^i the Force of the Sun exerted on it=i, the Article %x Diftancc of any Satellite from its Primary =/, and the periodic Time of «>^thcThc- ibe fame Satellite =r ; the Force (F) of the Sun on the Plajiet being to plfn^Vy pli Ac Force (/) rf any Planet on its Sateltate as i to Z. (Cor. 2, Prop. 4,) S^^s^dTn ' fcow New juid the Force (VJ of this Pknet on its Satellite if it was juft as far from t«>n deter. it as the Planet is from the Sun, being to its Force//; exerted on it at i^s K^^rtlona a&jual Diftance fwm it, as r» to i ; by the Compofition of Ratios FX/of ffeMat* i* -to VX/, or the Force (F) of the Sun on the Planet, is to the Force .^/^w*^***. (V) of a Planet on its SajrHite juft as far from it as the PUnet is from ;^^'^^7 theSun,«sita^. Sti^ ' E3e0taplf» The Revolution of Venus nound the Sun (5393*1) bei^gto that 4>f tjbe fourth SalelUAe of Jupiter (400^^1) ^ i to 0,074271 ^^^0^7427 16 itsnA ihe Difiance of VWniu from the Sun 72333 being to the Diftance 4^ Jupiter frtom ti^!S«ii jaoopfi :^s j to 7>19P3 > ^"^ Radius being to «|ie 8ine of 8' ^&' EkwigafciAn of the Saielite»>or its Diftance from Jupiter ^e^i^d from Ae Son, as 7,1903 'to 0)01729^ rs:;o,oi729; wherefore !L-= OjO00937 or --i— , confeqaently the Weight of equal Bodies at equal ■gf 1067 9>iftaiices from the Centre of iche Sun and Jupiler, are to ivie another as f X to — —. 1067 The Revolution of 'Venus round the Sun 5393^ being to that of the Saoxih Satellite of Saturn 362^5 >a$ x t0 4o672475, /^^o672475> «od ^e 1 CXXXII SYSTEM OF THE Diftancc of Venus from the Sun 723339 being to the Dtftance of Sattini from the Sun 954006 as i to 13,18909 and Radius being to the Sine of the Elongation of the Satellite or itsDiftance from Saturn^as I3>i890 r' I to 0,1144, r=o, 11 44, wherefore — =0,000332 or ,conrequem1jr the Weights of equal Bodies at equal Diftances from the Centres of the Sun and Saturn are to one another as i to . 3021 The Revolution of the Earth round the Sun 355d, 256 being to that of the Moon 27d, 3215 as i to 0,748008, and the t>iftance of the Earth from the Sun being to that of the Moon from the Earth, as the Sine of the Parallax of the Moon to the Sine of the Parallax of the San, wherefore — ~ — confequently the Weights of equal Bodies at equal Diftances from the Centres of the Sun and Earth are as i I to — — ^• 169282 Aadition to To determine the Weights of Bodies on the Surfaces of the Sun, onhe The- J^P'^^r, Saturn, and the Earth, or at the Diftance of their Semidiameters ory of the* from their Centres, thofe Semidiameters are to be inveftigated. Firft the priwary PU apparent Diameter of the Sun in its mean Diftance being found to be 22'9f 2"';,^;^ and that of Jupiler 37'' J (as determined from the Pafliije of thofe Satellites howNewton ovcr iis Di(k) and the mean Diftance of the Sun from Jupiter, being to ^lermined jj^^ ^^^j^ Diftance of the Sun from the Earth as 520096 to 1 00000, tioni ofKe and the true Diameters of Spheres, viewed under froall Angles, being in Penfitics of the compound Ratio of thofe Angles, and the Diftances conjointly, the P^^r''sJurm t»'uc Diameter of the Sun will be to the true Diameter of Tupitcr as 192?' •■/the *'* Xiooooo to 37"X52oo96, or as loooo to 997. Secondly, the apparent t^rth. Diameter of Saturn being found to be 16", and the mean Diftance of Saturn from the Sun being to the mean Diftance of the Earth from the Sun as 954006 to 100000, the true Diameter of the Sun will be to tte^ true Diameter of Saturn as i928"Xiooooo to l(5''X9540o6, or m 1 0000 to 791. Thirdly and laftly, the apparent Semidiameter of the Earth being found to be to" 30'" as being equal to the Parallax of Hie Sun^ the true Diameter of the Sun will be to the true Diameter of the Earth as 1928 to 21, or as 10000 to 109 nearly. Now if we fuppofe a Body placed at a Diftance from the Centre of the Sun equal to its Semidiameter, or on its Surface, the Force (i^ of the Sun on this Body being to the Force (V) of Jupiter on an equ^ Body at the fame Diftance from its Centre, as i- to — r- and the Force. (V) of Jupiter on this Body, being to the Force (/), it would exert on it if it was placed on its Surface, inverfely as the Squarea of the PHYSICAL WORLD. CXXXIIl Diftance^y that isi inverfely as the Squares of the true Semidiamcters of I 1 the Sun and Jupiter, or as a to -:=^a- ; by the Compofiiion of loooo 997 Ratios FXV is to VX/ or the Weight (F) of a Body on the Surface of the Sun is to the Weight (f) of an equal Body on the Surface of Jupiter, I I i as 2 to 7rTr"X — - , -or as loooo to 043, and confequently that 10000 ^^^1 997 theDenfity of the Sun is totheDenfityof Jupiter(theDenfities being in the dired Ratio of the Weights and inverfely as the Diameters) as 100 to 944. In the fame Manner it will be found fecondly, that the Weight of a Body onahe Surface of the Sun is to the Weight of an equal l^dy on the Surface of Saturn as , to rrirx % — or as loooo to 529, I 0000 30»* ^^1 confequently that the Denfity of the Sun is to the Denfity of Saturn as 100 to 67. Thirdly and laftly. That the Weight of a Body on the Surface of the Sun, is to the Weight of an equal Body on the Surface I I. 1 . of the Earth as -» to .a^^^q^ 'X a or as 1 0000 to 435, -con- 10000 I09?02 jQj^ ^^-^ fequently that the Denfity of the Sun is to the Denfity of the Earth' as. 10010400. Which Determinatioii on examining the Procefs of' the Computation will appear not to depend on the Parallax of the Sun but. on the Parallax of the Moon^ and is therefore truly defined. XIV. Such is the Flan of the immortal Difcoveries of the moft eminent Cpwctusi- I^bilofophers, and of Sir Ifaac Newton in particular, whofe Efforts and ®** SLXkd Sagacity we cannot fufEciently admire, which fliine through the Whole oif thofe Strokes of Genius, which charafterifc an Inventor, and Recapitii- skr Mind fertile in.B!efources, that no Man poirefTed in fo eminent a De- lationof gree/ Aided by the Succours that the analitic Art fur'niflies in greater iJ^otTS^^* ^bund^nceir it fsmiot furprizing that fome more Steps have been made Prindfia ix% a vaft and diiBfcult Career t^at be has opened, to us, that all the Irre- '^f^J'?'^ ^ularities that liave been perceived in. the Heavens,- have been explained pgy]' and dcmonftrated ; that a great Number of others, which on Account of their SmaJIneis and Complication had efcaped the moft exad Ob- £^r^^^9 h^ve been forefeen and unfolded ; that it has been proved, that ^l^cr Return of the Comet which was obferved in 1531, 1607, and 1682,' ^^j^ht to have had the unequal Periods of 913} and 898^ Months, which ^^^^t^ found to be fo, and that the Period after which it would appear ^^^ixi in this Age, would.be 919 Months ; which the Event has juftified. ^That the Courfe and Laws of the Winds, the ebbing and flowing of ^jy^ Sea, as far. as tbey depend on the attra^ive Action of the Sun and CXXXIV SYSTEM OF THE n New Edition of the Prin- ciptM^ with the Improve meoCs they have receiv- ed to this Diy. Courie of Experi- roencs for illuftratin^ pis. Moon, have been accurately determined. That the Narare and Laws of Magnetifm, the Theory of Light and Laws of Vifion, the Theory of Sound and Laws of Harmony, &c. have been accurately inveftigated. Such is the Plan of the Mathematical Principles of natdkal Phylosophy, which the Mobility and Gentry of the Kingdom of /re/tfiuf purfuant to their Refolution of the 4th of February 1 768, have ordered to be publiHied for the Ufe of the Mathematical School «ftabliihed nnder their immediate Infpedion. Previous to whichfin the Month of November, 1764, a Copy of the Chapter of the Theory of the primary Pianets» as a Specimen of the whole Plao, was delivered to Dr. Hugb Hmnikoth tohcve his Opinion of thefame» which he returned in fix Months after, widi this Anfwer^ That the above Piece was printing hj Sobfcripdon at Ctm- bridge t under the Title of Excerpt a quadam ex Newtorti Principiis, with Refcrrences to the Doftor's Tr^tifc on Conic SefiHotis. PLAN if the Art 9/ making Experiments ^md tJbsi <tf employing them, Experimenta rerum naturalium itafuht exbibenda, ut in bis Mobiles ad^ %Jcentes Jludio fuavijfimo at que utilijfimo bumanse mentis bifioriamy precUrM* que ^rtium inventus quibus natur4m et ^rnare et 4idjuvmr49 ^difare poffiinL rri O illuftrate Sir Ifaac Newton*$ Principiay and thereby to enable X Youth to make a Progrefs in the Knowledge of the Works of Nature, to improve to Advantage Its Powers and Forces, and render them fubfervicnt to th6 Purpofes of Life, they are initiated iti the Art of making Experiments and Obfervations. For thefe Purpofes the Schod is fumiihed with a complete CdlefHon of the beft executed Machines adapted for experimental ttiquiries; they are inftrufited in the Manage^ ment and Ufe of thefe Macliines; they are tanghthow to afcertahi the Dif- ference between the Refult from Theory and from Experiment, and how to employ this DifFerence5( for dcterming the Alterations arilii\g fitmi external Caufes, in order to fliew them how Experiment not only fervesto Confirm Theory, but conduSs to new Truths, to which we can- not attain by Theory alone. As to the Phenomena for tbeDifcoveij of whofe Caufes Theory affords little or tio Afliftance, for Inftance, thoie of Chimiftry, Elieftricity, &c. they are taught how to examine and con- Kider them in different Lights, arrange them in CUiTes, and explain the one by the other as far as the Nature of the SubjeS wifl allow. First Class. Machines for making Experiments on Motion, Gravity, and the Equili- brium of folid Bodies. \ y A Machine for demonftratJng the *tt>eorv; olf Central Forces. ^if Machine is fo contrived ^ t bat By its 2fff!Jtance may hejhtved experi* mentaWy^ tie Problems wbicb appear ibe leajl JujceplthU ofjucb a Solution • tbe Velocities and Majfes may he varied at willj Fri^ion is fo diminijbed PHYSICAL WORLD. CXXXV «r /« (»ule no ftnfibU Errett tb* Tinut art marktd bj Seundj, tad the Expai^ spaces defcribed by an Index. ^^tg '^ A GUfs Globe mounted on an Axis fo that It may be turned round the Tbeoiy wiih any Degree of Velocity, jf ccotrol This Machine Jbews the Effect of central Forces $n Fluids of different * fptcific Gravities^ and an Solids^ which circulate in the fame Medium, A terreftrial Globe which turns on its Axis with any given Velocity. The Surface of this Globe is flexible % its Concavity is filed with a Mat^ ier Jomewbat fluid, and is fo contrivedp that its two Poles are capable of moving towards each other, fo that by turning the Globe, the centrifugal Force raifu the Equator of the Globe, andft>ews the Figure which modern Difcweries attribute to the Earth. A graduated Rule adapted to a Gtafs Tube within which a fmall Cy- linder is put in Motion. Second> A Plane upon which two Bodies de- scribe in the fame Tine unequal Spaces. Third, A Globe of Cork of of three Inches Diameter, with a Ball of Lead of the fanoe Weight. By the Afpftance of the three loft Articles are explained the Properties of Motion, viz. Dire&ion, Velocity, Quantity of Motion, ifc. A fmall Cyftern divided into two equal Parts by a Partition upon which is mounted a double Pendulum, mewing in what Ratio different Mediums exert their Refiftance. A Machine with which is demonftrated the Dodrine of the Collifion Experi- of Bodies. Eing"' The Parts of this htachine are made with the utmoft Care, the Maffes thebo^nne are in given Proportions, and the EffeBs remain vifiUe after the Experiment of the Colli- by the Means of an Index. Sf^;^ ^ A Chronometer orlnftrument for meafuring fmall Intervals oi Time. ^be Pendulum which conjiitutes the principal Part of this Injirument may be lengthened or fljortened according to a Scale accurately divided for tie vibrating Minutes, Seconds^ Thirds, and the different Times of Mufick. A fmall Billiard-Table with its Appendages. The Appendages of this Machine are Hammers Jufpended infuch a Man- ner, tbat the Sgantity of Motion may be regulated by the yelocity, or by tie Mafs, and fo as to exhibit the Motion of a Body impelled by Forces n^ing in different Direffions, and known Proportions. A Machme for (hewing the Motion of a Body which receives at the Experi- fame Time an Impulfe in a perpendicular and horizontil Dire&ion. hiftraiing'' Another Machine for (hewing the Motion prodoced by two Forces the Compofi ading on a Body in Dire£kioiis forming an Angle, but which conftantly Jj^lj^j"^^/ "' remain in the fame Ratia Forvci!* ^ A Machine for (hewing the Acceleration of Bodies which fall freely. Secondly) a Kind of Balance for making the fame Kind of Experiments. Tbefe two lafl Machines not only fluew that the Motion of Bodies is ac» eelerated in their Defcent, but alfo renders fenfible the Law of this Accele- restion* CXXXVI £xpcri- meou for iUuftn^ing tbcDoArioe of die Mod- OB of heavy Bodies. Expcri- rocDts for iUuftratiog the Namre tod Proper- ties of the Center of G rarity Expert- ineDCB for illnftratiog the Theory of fimple Machines, . che inclined Pltne, the Wedge, the Screw, the J-crer. SYSTEM OF THE A Machine for (hewing the Line a Body defcribes when abandoned to its Weight after having received an Impuifion in an horizontal DiredioB. A Machine for fhewing the Motion of a Body abandoned to its Weight after having received an Impuifion upwards^ but oblique to the Horizoii. As the Curve vtbicb refults from this Motion depends on the Obliquity of the Dire^iony tbe Macbine is conftruBed fo tbat tbe Degree of Obliquity may be varied at wilL A Machine which ferves to compare the Velocity of a Body which in its Difcent defcribes a Cyclovd with that of another tending to the fame Point along an indinoi Plane. A Machine for (hewing in what Ratio fevend Forces ad on the fame Body. A Machine for explaining the Laws of Elafticity. Two Cones joined together by their Bafes> and which afcend an inclined Plane. 2d. A Cylindar which afcends all inclined Plane. Tbofe two Macbines ferve for proving experimentally, tbat a Body can* not remain at refl wben its Centre of Gravity is not fupported. TbePUm on wbicb tbe double Cone moves is formed ^of two Rulers inclined to eaei otber and to tbe Horizon, and tbis double Inclination may be varied at fttoi^ fure as tbe Experiment may require, A fmall Carriage with its Appendages. Tbis Model with tbe Parts wbicb accompany it, Jbews the refpeitive Advantages of broad or narrow IVbeels, of large or fmall ones, and vtfbat renders Carriages more or lefs liable to be overturned. A Machine for (hewing the Properties of the inclined Plane. Tbis Macbine is fo conflru^ed tbat tbe Inclination of tbe Plane muty be varied from tbe borizontal Line to tbe vertical, and thai tbe Power ■may aSi in any defired Dire ff ion. A Machine for (hewing the Nature and Properties of the Wedge. Wbat forms tbe Wedge in tbis Macbine are two Planes inclined to eacb otber, tbe Degree of Inclination can be varied at pleafure^ as alio tbe Power, tbe Weigbt and tbe Bafe of tbe Wedge. A Screw which can be taken to Pieces to fhew tbe Principles of its ConftruSion. A Machine for (hewing the Nature ^f the three Species of Levers. A large Beam accurately divided^ mounted on a Foot^ for (hewing the Properties of the Lever. Tbe Power, tbe Weigbt, and tbe Prop or Fulcrum are moveable^ and may be eafily placed Jo as to be to eacb otber in any given Proportions, Two Figures in Eqilibrio on a Pivot, for (hewing the Art of Choid or Wire-dancing. A large Brafs Pully, in which the Circumference and the diametnd 1 PHYSICAL WORLD. CXXXVII Lines bav« oidy been left, in order to (hew that the PuHy may be con- fidered as an AUembUge of Levers of the firft Species. ' At tbi Back of tie Su^orter, there is fixed a Lever of the Jame Specter with tboje which conjiitute the Diametert of the Pulfyt to/erve as a Proof hj the Application of the fame Power and Weight. A Pully wbofe Axis is rapveabte in a perpendicular DireSion, and which fcrves to fliew the ASion of the Power, and of the Weight on this Axis, in diflFerent Cafes. A Block with two Putlies. ad. A Block with four Pullies ; another Block whofe Pullies are fixed on the fame Axis. All thofe combined Pullies are of Metal or Ivory, turned on their Axis with great Prnifion^ snd alt poj[J$ble Can has been taken to diminijb the Fri^ion. An AflfemWage of feveral Toothed Wheels and Pinions, for ihewing Modds for that both the one and the otbf r like the Pullies, may be confidered as j!*]^°ff[^„ Levers. of ^impilT" At the Back of the Supporter, are fixed an Affemblage of Levers which Machines io cvrrefpwd in the fame Manner as the Diameters of the Wheels on the comwmndcd other Side, to ferve as a Proof by the Application of the fame Power and ones. The ffTeigbt. Sn^r^he" A Model of Arcbimedes^s Screwj whofe EfFefls are rendered fenfible by Piic4river* the Motion of feveral fmall Balls of Ivory, which are raiied fucceifively. WiDdmUlf, A Model of an Endlefs Screw, which drives an Axis. 2d. A Model ^^»t"-""»»*' of a Prefs. 3d. A Model of a Capftan . 4th. A Model of a Crane. 5th. A Model of an Engine for driving Piles. A Jack* of a particular Conftru^ion, for raifing great Weights. A common Balance, for ihewing the Defefib to which this Machine is li- abte» and how they may be reipedied. A large Roman Balaace, contrived for naaking the Experinients of San^orius. This Machine is fa conflru^dj that a Perfon may weigh bimfelf with- aut the Affiflance of another. A Model of a Screen for winnowing Corn by the Means of an arti- ficial Wind, sind feveral Screen^ of a particular ConftruSion. A Model of a Saw for culing at the fan:ie Time feveral Flints, Agates, Cornelians, &c. and at one Stroke, to form Tables for SnufF-Boxes, and other Works. An horizontal Turning Leath, adapted for grinding Glafles for Telefcopes, Microfcopes, &c. A Model of a common Wind-Mill. ad. A Model of a Polifli Wind- ^ Mill. 3d. A Model of a Watcr-Mill for extrafting Oil. 4th. A Mo- del of a Water-Mill for winnowing and grinding Corn, drawing up the Sacksi and boulting the Flour. CXXXVIII SYSTEM OF THE As sail thofe Models are intended to Jbew tbe Application ofjsmplt Ma-^ chines in tbe more compounded ones. Care has been taken to leave exp^fed or to cover with Glafs, the Pieces dejlined for Motion^ and the Proportion of tbe Parts have been carefully obferved, ^ A Machine for (hewing the EfFcds of Friction, in Machines more correft, and of a more extenfivc Ule than any hitherto invented. Second Class. • Machines for making Experiments on theMotion^ Gravity and Equi- librium of Fluids. IT. A large Ciftern lined with Lead, with a Cock to it, %hith ferves foe making feveral hydrpftaiical Experiments. Two large cylindrical Glaffes mounted on a common Bafe, betwe^ which is creded a St«m vrhich carries a Beam of a Balance. This Machine is very commodious in feveral Operations which regard tba ^nu for ^^^^^^^ ^^ Equilibrium of Fluids, Scwing*A« ^ ^^^^^ Bottle with a Glafs Stopper, and heavier in this- State than Propcrtiei of a QiI2i"*'ty of Water of the fame Bulk. Irluidi. ^ Glafs Tube, a Part of which rifes perpendicularly, and the other forms feveral Flexions for ihewing the Height ot Fluids in Veflels which have a Communication with each other, A fmall Barrel with a Cock to it, and a bent Tube which ferves for demonftrnting the fame Principle, with fome curious Applications. A Glafs Veffel, partly filled with a coloured Fluid, to which is ad-- jufted a large Glafs Tube, and a fmall fucking Pump, which (erves to fhew that Columns of the fame Fluid are of the fame fpecific Gravitj. A long Tube of Glafs with a Cock at the lower Extremity, and. mounted on a graduated Ruler, to which is adjufted a Pendulum whkh beats Seconds. • This Machine ferves tojhew bow the Parts of a Fluid prefs each other, and in what Ratio the Effluxes thereof are performed* A Bladder filled with a coloured Fluid, to which is fitted a Glafs Tube, which ferves to (hew that Fluids exert their Preflure in all Diredions. A Veffel whofe Bottom burfts by the Prcffure of a fmall Quantity of Kxpcri- a Fluid. mcnts for A large Machine, which ferves to (hew the Pre(rurc of Fluids on the ^I'^^h^of Bottoms and Sides of VelTels which contain tli^m. Fluids upon This Machine conftfls of feveral fine Veffels of Glafs, which are sd^ the the Bot- jujledjucceffively on a common Baje, the Pijion which ferves as a Bottom, Side! of the is Jufficiently moveable as not to caufe any fenftble Error by Fri^ion, tbe VefleU that Columns of the Fluid remain always at the fame Height, and tbe P'cmfer ^ *^/ uniformljf. 1 PHYSICAL WORLD. CXXXIX An Hydrometer with fix fmall cylindrical Vafes^ which are filled with different Fluids. Two finall Cruets, mounted each on a PedeftaU which ferve for die Experiments by which Water is apparently changed into Wine^ and Wine into Water. Two Vafes of different Formsy which rerve to make a heavier Fluid ^uroe the Place of a lighter in the fame VeffeU without mixing. A Veffel perfedly cylindrical of Copper^ with a Solid of the fame Experi- MetaU and of the fame Figure, which fills it exadly, for (hewing how ™«"* ^^ much a Body immcrfed in a Fluid, lofes of its Weight. ihe vutlon A Vafe of Glafs fufpended to the Arm of a Balance, for making Ex- of Fluids periments of the fame Kind. • KcS Two Balls, one of Ivory, and the other of Lead of the fame Weight, thim. prepared to be fufpended to the Arm of the Balance juft mentioned, for fhewtng, that what a Body lofes of its Weight when immerfed in a Fluid, is proportional to its Bulk. A cylindrical Vafe of Glafs filled with Water, with fevcral human Figures of Enamel, of wjhich feme are lighter and the others heavier than a like Portion of the Fluid in which they are immerfed. A -Machine for (hcwing-that the relative Gravity of a Body immerfed jn a Fluid, is changed when the Fluid is condenfed or rarified. This Machine renders palpable By a very quick Operation^ the Effe^s ^vbicb the different Temperatures of the Air produce in tbe different Kinds of Thermometers hitherto indented, A human Figure of Enamel, which is made to move Jn Water by Compreflion. 2d. Two large Tubes of Glafs! mounted in a Frame, in i^hich two Figuves move by a Compreflion which is not perceived by the Spedator. A Model of the Diving BcM, and the Appurtenancesf of a Diver. Ad hyd:roftatic Balahce, wi\h all- its Appendages. A Model of a curious Machine for raifmg up Veflels that are funk, A Water LeveK A fimplt Syphon; ad. A Fountain Syphon moun- ted on a Pedeftal. 3d. A Syphon with its Vafe to be placed in Vacuo, ^^^^-^^ 4th. A double Syphon. '5th. A Syphon^ wbofo Branches are moveable mcnjfor by. the Means of a Joint. . 6ih\ Tantahis^s Cup. . 7th. Aiarge Syphon {{JjJ^'iij.'Jfi T^hofe Branches are moveable, neceflary in Experiments made with the produced by Air-Pump. ihePrcflure, JJI thofe different species of Syphons are of Glafs, that the Motion e/^pf^y/ tbr Ftuids mfi) he more eafify perceived. ... . . > A Model of a Sucking-I^ump. ad. A Model of a Lifting-Pomp. 3d. A Model of k SuAftig ahd Lifting Pump. . 4th. A- Model of the Engine under London-Bridge, ihat^ raifes Water by Fordng-Pumpi, CLX SYSTEM OF THE 5th. A Model of a new Pdmp whore Sucker has no Fri^oni an in termitting Fountain^ Hiero*% Fountain. All tb^ Models if Pumps uni FoiMains are rfGlafsf in ell tbofe Paris in which tie A^ion f^e^t ^^nd the Motion of the f^ahes and Suckers, are eaftly perceived. Expert - Several Cifterns and other Vafes for making Expcrimerits on Ice» ""?^^^5^-^1 and artificial Congelations. 2d. An Aflbrtment of different Salts and "oD^lftb.' Fluids for congeaUng Water with a Vafc, in which without Icc^ a Cold •ni. capable of freezingy may be produced. Third Class. Machines for making Experiments on die Air. III. A double barrelled Air-Pump mounted on a v^y folid Bafe. T:bi Piftons are put in Motion by a Handle. Irdbad of Val^s Stop^Cbeii are made Ufe of which are opened andftuitf and that by the fame Mmtion wbieb raifes and lowers the Pifiems ; iiire is affixed to the Pump a Vfbiriing Machine^ for the Experiments where it m neoeffary. ^ ^^. A fmgle barrelled Air-Piunp» mounted-:oA » folid Bife. ment/for In tie Cotdlru^ion of the wbiHing Maeiine, wUeb firvis at onAfen^ ihewing the dage to this Pump, Care hat been taken, that the Axis of tie great Wheel So^i€$tf ^^ ^^* ^'^S '^^ Frame, in order to firaiUn the Chord, and that the the Air. horizontal Pulley, which receives the whirling Axis, may be raifed or lowered as the Height of the Receiver may require, A large Receiver fitted for making Experiments on Bodies put in Motion in Vacuo, ad. A Receiver of lefs Sixe fitted for the fame Ufes. 3d. A long and narrow Receiver fitted aMb for the fiime Ufes. Tbofe Vafes are fitted jor the above Ufes, by the Means of a Braft Box, filed with a Sort of prepared Leather, through which paf/et a Steel Axle* Tree, which communicates the Motion within the Receiver without letting the Air enter, meSuon ^" Apparatus neccflfary for naaking the Experiincnti on Pii^ in Fire in Vi- Vacuo. Ei«ftr' al ^^ Apparatus for making eledrical Experiments in Vacuo. £xp<ri^ ^ ^^^^ Receiver fitted for operating in Vacuo; a tall narrow Re^ menu ia ceiver fitted for the fame Ufes. Vacuo. y-^^y^ fr^j-^^ are fitted for the above Ufes, by Means of a Braft Box prepared as above, through which paffes a Shaft of Metal, wbafa Bjetre^ mity is fitted for receiving different Sorts of Pincers, and other he/hu^ ments,^ Four Cruets mounted on one comnnn Pedef{a1» and Aifpended f^ as to have their Contents poured out in Vacuo, which ferve for aaixing different Fluids therein. 2d. Two Cruets fufpeoded in the fame Manner. 1 PHYSICAL WORLD. CXLI This Machine is fo c^ntrived^ that the Cruets may be raifed or lowered^ nnd brought nearer to each other f as may be required. An Apparatus for dTaying Inflammations in Vacuo. A Receiver oompofcd of feveral Pieces^ very tallj at the upper End Exp«ri- of whkhy a Machine \& adapted with which may be repeated fix Times, ^Jwll^^th the Experiment of the defcent of Bodies in Vacuo, when the Air is but De^m of once exhaufted. Bodittia A large Vafe of Glafs adjufted to a Receiver, and difpofed for depriv- ^■*"*** ing Fiflies in Water of Air. A large Globe of Glafs, joined to a Receiver by a Neck, to which £xperi- is adapted a Stop^Cock, for making' Expreriments on the Vapours in Jh^in'^hat the Air. ad. Two Vafea of Comparifon having for a common Bafe a the Air itfi!- fmall Receiver, for fin^ilar Ufes. kdwithVa A Receiver^ to whidi are adapted two Barometers, one of Mercury '^"^'' and the other of coloured Water. Two large Receivers with a hoHow Button at Top. ad. Two Re- emtxh of s middle Size. 3d. Four fmall Receivers. 4tlu A Machine very commodioiis for fealiiig up Va(u hermetically, &(. Six fmaH truncated Bnvmeters of different Lengths, mounted each Expcri- on a fmall Bafe, to which a Scale is adapted, ad. Six fmall gage Tubes, "ic"c< ^or forcompreffed and rarified Air. ?hc'Sgree* Theft Gage fnjirwaenis rare More commodious for Ufe than any hitherto of Comprcf- maief and it is tveii known of what Importance it is in making Experimehts^ refhrfVo "f So bf affured^of the Dagree of RatefraBiony or of the Cmdtnfation of the Air." ^ the Air. A Rerelver tot making ExperiraeBts on burnt, or infeCked Air. £xpcri- Two large Copper Hemifpheres, to one of which is adapted a Ring, "**"*» <*" . and to tjte other a Siop^Cock. . f^SdAtl" A Fountain Bottle, and a Vafe to place it in, with feveral fpouting Pipes, which are Jkioceffively adjufted on it. A fmall Receiver for applying the Hand to the Air-Pump. Experl- A Receiver of very tbkk Glafs for burfting a Bladder. ^^ ^^^^ A Supporter, and a fmall Vafe of Glais to pbce Eggs under a Re- Sprin^^of ^ ceiver of the Air-Pump. J^c Air and A fmall Receiver with a (harp edged Brirti, to cut an Apple, or any ^^pphcau- like Body. A large Glafs Tube, at the Top of which is adjufted, a Wooden Vafe for proving the Porofoty of Vegetables. A Tube of Cryftal wbofe Bottom is of Leather, covered with Mer- cury, to (hew that animal Subftancet are porous. A Bladder iufpended in a Reciver. ad. A Bladder in a* cylindrical Vafe of Metal charged with a great Weighs A Machine for compreiTing Air. CXLU Experi- ments for ibewing the Preflbrc of the Air. Expcri- ments for illuftrating the Theory . ef SoudJc. Experi- ' ments for Oiewinv the Operations of Chiraif- tiy. SYSTEM OF THE "1 This Machine is of fufficient Strength to remow all Apprebenfions of Danger 9 and is fufficiently large to place all fucb Bodies with which ExpB- riments are made by the Means of an Air-Pump ; it is conflwu^ed in fucb a Manner y that what paffes within, mi^ eafily he perceived, and the Air is comprejfed with great Eaje by Means of a Lever which puts the Pifion of the Pump in A^ion, A fniall forcing Pump with Valves for compreffing Air in certain Experiments. A Glafs Vafe prepared for compreffing Air on Liquors. A Fountain of Compreflion of Copper. A Tube which contains Water without Air. A Kind of round Bellows, furniihed with a long Tube for ihewing the powerful Efforts of Fluids, Two Hemifphercs of Copper for the Machine t>f Compreffion. An Air-Gun. TA// Air-Gun is furnifbed with a condenfing Syringe in the Butt, and is charged with Balls by a Jieceiver which contains lo. They uiay eafily he taken out without letting the Air efcape. At each Shot only one goes off and one Charge of Air is Jufficient for tbem allf and the loft pierces am Oak Plank half an Inch thick. A Model of a Bellows^ in which the Air is excited hj the circular Motion of feveral Vans. 2d. A Model. of a Bellows whofe Effefik de- pends on a Fall of Water. A Glafs Bell fufpended, with a fmall Hammer put in Motion by a Screw, adapted for Experiments on Sound. A fmall Bell mounted on Qock-Work> with a Tricker« for Experi- ments on Sound in Vacuo. An accouftique Tube of a parabolic Figure. :ad. A Speaking-Trum- pet. A graduated Monochord. 2d. Glafles of feveral Tones. A large Column which imitates the Noife of Rain .and Hail. Glafs Tears> with fome Inftruments neceffary far the Experitoents to which they are applied. Capillary Tubes of different Sizes and Lengths. Fourth Class. Machines for making Experiments on Fire. IV. A Lamp Furnace for (hewing the ordinary Operations of Chimiftrr. IVitb this Machine DifiiUations are performed in Balnea Mars^jt, the Sand Bath, with ■ the Cucurbit and with the Retort. An Affortment of Veffelsof Glafs for the Lamp Furnace. A Table of an Enameller with a Bellows and Lamp;^ Pieces of Ena- mels and Tools, requifite for this Art, in r PHYSICAL WORLD. CXLIll Inclined Planes which turn round by the Adion of two lighted Can- dels. Second) a Lantern which turns round. Several Fluids which ferment with Heat and Ebullition. Second, fe- Cxpcri- vcral Fluids which ferment without Heat. Third, fcveral Fluids which JJJ^^enu- fermenting, burft into Flame, and the Vafes neceflary for thofe Ex- cion. periments. Fulminatory Subftances and Inftruments, neccffary for performing Ex- periments on them. Burning Powders. 2d. Powders for accelerating the Fufion of Mc- E*P«'f" tals. Third, feveral Difolvents of Metals. the DiffLiud The Urinous Phofphorus. ad. Urinous Phofphorus difolved in dit- onohvinal*. ferent Kinds of Oils. 3d. Luminous Calcinations. A Glafs Veffel, by which may be exhibited a Shower of Fire, pro- duced by the Fall of Mercury in Vacua Papin\ Digefter. A large Copper .£oltpile with a long Neck, to which is adapted an accurate Stop-Cock^ which ferves for condenfing Air in Vafes, when there is Reafon to apprehend that the Moifture of other Air introduced Expert- TOkj hurt the Experiment.. 24 A fmaller ^olipile for ordinary Ufes. J^^"^;,,^^'^ 3d. An .£olipile for forming a Fountain of Fire, with the Spirit of fhrKffca^ Wine. 4th. An .£otipile irsoiinted on a Carriage which recoils during of Fire arm« the Experiment. Fire-w<rk., A fmall recoiling Cannon for explaining the Nature of Rockets. Fifth Class. . Machines for making Experiments on Light and Colours. v.. A large Cafe, the Sides of which are of Glafs adapted for the Ex- iperinnents on Refradion. Expcri- In tie two lejfer Sides of this Cafe are adjujiedf concave and convex- Sur- ^il^^^li^^ faces. It can be raifedf lower ed^ or turned round on its PedefiaU and is the Theory furnijbed with a Lamp which in caje of Neceffityy fupplies the Place of the °|,^/?f^7 Rjr^s of the Sun, **" ° '^ * A triangukr Box of Glafs, whofe Sides form with each other different . Angles, mounted on a graduated Circle, with an Index for determining. tlie An^es of Refradion. Two Prifms of foUd Cryftal. ad. A large folid Prifm mounted on a Exofri- Pedefta), fo that it can be raifed, lowered, inclined, and turned round imentsfcr its Axis. 3d. A Prifm fimilar to the former, mounted vertically on a ijJ^^J'',5|."^ pedeftal, fo that it can be raifed, lowered, and turned round its Axis. of^Colouref 4.tfa. A Right-angled triangular Prifm. ^h, A large triangular Pri&n of Rock Cryftal mounted on a graduated Circle, with an Index. A large folding Table with its Appendages^ adapted for making Ex- pjeriments on Light. Six Frames covered with waxed Cloth, for rendering a Room per- CXLIV SYSTEM OP THE n fedly darky with a Tablet and Circles of Metal for opentog Paflages - to the Rays of the Sun» of different Magnitudes and Figures. A plain Mirror of Metal mounted on a Stem which can be lengtbened and (horted) and on which the Mirror can be raifed, lowered^ inclinedf and turned rounds for introducing the Rays of the San into a dark- ened Room. 2d. A Mirror of Glafs mounted as the former* and for the fame Ufes, Four GlafTes of different Colours, mounted in Torteife Shell, ad. Four Mirrors of Glafs mounted in the fame Manner. A large Glafs Lens of fix Feet Focus Length, nKMinted on a Pe- deftal whofe Stem can be lengthed or ihortened. 2d. A Glais Lens of a fhorter Focus mounted, fo that it can be ratfed, lowered or inclinecL Experi* A Frame, in which is adjuftedaGteft Lens between two ▼eitkal S"ingThe Plane** for Shewing that fome Rays of Light unite in a Ihortef Foan diiTercnt than Others. ^c^n^ffl>i* Tbit Machine is fo contrived, thai tbe Experimcni may be nude tefton rVoF ^^y ^^y fiparaiilyf and may be adjufted to the Mot ten of the Sun. Light. A large concave Glafs mounted. 2d. A large multikterat Gl^ mounted. 3d. Two Polyhedrons of very pure Ohb, 4th Two coo* cave Mirrors of Glafs. A very large convex Glais, compoled of two coffved'Glafles mounted on a Pedeftal, for making Experiments on the Refradion of Laght through different Fluids. A large vertical Plane for receiving the Image of the Sun when it has paflfed through the Prifm. 2d. A fmailer Plane, to which is adapted, an excentric Circle for making the Rays of Light of different Coioursy pafs fucceffively. A Cloth fix Feet fquare fpread on a Frame, which can be nufed and lowered for receiving the Images produced by the Magic Lantfaatn, and the Camera Obfcura. £xp<tl- An artificial Eye with Spedacles for different Ages, for fhewing how !J}^^^«j; the Defeds of Sight are remedied by the Help of Gltffes the Lawi of A Cornea of an Infed: adapted to a fmall Microfcope fior ihewiog that Yifion. the Eyes of thofe Animals, for the moft Part, are Mukipiien. An Aflbrtment of Fluids for Experiments on the Colours which re* fult from their Mixture. Invifible Ink, the Writing of which appears and di&ppears feveni Times, when heated at the Fire. 2d. Sympathetic Ink. £xperr A large Mirror of Metal, concave on. one Side, and convex an the muftratin ^^^^^9 mounted on a Pedeftal. Two convex Mirrors of Pafb^board lilver- thcDoftmc cd over, with their Appendages, for ftime catoptrical Expennnents. ofthcKe- A cylindrical Mirror of Metal, with thirty Anamor]^ies. adL A Lidlt!"*^^ conic Mirror of Metal, with fix Anamorphofes. A pyramidal Mirror of Metal, with four Anamorphofes. PHYSICAL WORLD. CXLV To aH tbofi Mirrars is adapted a Machine for reguhUing tie P^ini of Fiew, A Pi&ure, commonly called the mtgicid one, oa account of the Effed of the multilateral Glafs^ for dioptrical Anamorpbofe*. A Magic Lanternt enlightened by the Ray^ of the Sun. 2d. A Experi- Maffic Lantern enlightened by a Laxnp and a conc&Ye Mirror. uiuft^uTna Although this Machine is become very cemmoftf it is tut however defpi^ the 1 beory eahle ; the moft eminent Pbihfopbers of the prejent Jge, have net thought it ^ ^ ^ on- unworthy of a Place among their Machines ^ and have given ample Defer ip-^ opI*icaf?ii^ tions of it. The ahove mentioned one, prefenii a Sight fi much w more ftromeDu, agreeable f as the Obje£ls appear animated^ and are perfeeily well deftgned. Jwitera ^ A Camera Obfcura of a sew Cooflrudton> with a Stool aiKi Table, Camera bb- an4 other* Conveniencies for dedgBiiig. |h]ra,ireAcet A kind of Telefcope for obferving Qbjeaa which prefent th«i»felves ^f^„^'4;^ at Right-angles to the Tube. 2d. A Newtonian Telefcope» with which lefccpeNMi the Objeds are viewed Tideways, or in a Line which foftns aji acute Jof<:op«»» Angle with the incident Rays of thofe Objeds. 3di A catoptrtcal Te- leicope two Feet long» which magnifies the Objeds 300 Timesv 4th. An Achromatic Telefcope 12 Feet long, A portable Micjrol'crope» with the Inftruraents neceflary for obferving. 2d. A larger Microfcope, with a greater Number of Inftruments and Lenfies for increasing or kfiening its magnifying Power. 3d. A Micro- fcopa which has its different Degrees of magmfying Power, with Mir« tors of Refe^on and LeniSes for incveafing the Light ; it is mounted fo that it can be moved in all Diredions. wi& great Eafe, and has a Ma-> chine of a new Cootrivance for fixing it at ils true Pdn*. The Drawer of ita Cheft contaiBs every Thing neceflary for the different Obfervations to which i« may be applied. A double Lens mounted in Tortoife Shell for Obfervations on Ilnffefls, assd other Qperatioos where the Micnoibope is not cominodiofis. An Apparatus for making Experiments oa thoTranfpapency and Opa- city of BkniieSf. confiAing in Squares of poKihed Glafs^ limpid Liquors of different DenTuies, bfc. Sixth CcAda. NfachiiieS' for malung magnetic aad eledrical Experiments. VT. A. foiaJi Tabfeonn Etpot long, and eight Inches broad A Magnet €Ut> b«t not mounted: 2d. A Magnet cut and' fufpended £yp^f2^ io a I^tle Boat of Ebony. 3d. A Mingnet mounted and adjufted to a mentson wliirliing^ Maflhittei 4idi. A» artificial Magnet mounted on a Pedtftal of MignctUm. Ebonr. A 60X filled 'with the Fileings of Iron. ad. A Bafon with little Swms and Frogs of Enamel; 3d; A Box filled with fmall Ends of Iron and Brafs Wire. 4th. A Box filled with fmall Iron Rings. 5 th. A Box containing feveral Iron Balls, and fome Cylanders of the fame Metal. CXLVI SYSTEM OF THE Experi- mcn;s on £learici- Two' large magnetic Needles of poliflied Iron, placed one at the Top of the other, and mounted on a Pedeftal. 2d. A Dipping-Needle rUbunted on a Pedeftal. A fqtiare Rod of pollflied Iron two Feet and a half long. 2d. A round Rod of polifhed Iron two Feet long. ^d. A thin Plate of poUihed Iron eighteen Inches long. 4th. A Stand of vamiflied Wdod: A Brafs Circle garnifhed with Pivots, for placing twelve fmall Stcd Needles. A Glafs Vafe mounted on a Pedeftal for placing a magnetic Needle in Water. A Machine which ferves for trying the Force of a Magnet. A Dial Cqmpafs. 2d. A truncated Compafs for determining the Meridian of a Place, &c. 3d. A Sea Compafs, feveral Steel Needles of different Sizes adapted for magnetic Experiments. A large Tube of Cryftal. 2d. Two fmaller ones and not fo thick. 3d. A large Glafs Tube very thick, two Feet long. 4th. A Glafs Tube three Feet and a half long, with a Stop- Cock, to be applied to the Air- Pump. A thick fquare Rod of polilhed Glafs, about eighteen Inches long. 2d. A round folid Rod of Cryftal. A large Globe of Cryftal adjufted to a whirling Machine. 2d. A Globe of Cryftal, the Inftde of which is laid over with Sealing- Wax, to which is adapted a Stop-Cock to be applied to the Air-Pump, and afterwards to a whirling Machine. A large Stand, whofe Tablet is made of Sealing- Wax. 2d. A Glafs Stand fourteen Inches high. 3d. A Stand of Cryftal of a different Form from the preceding one, for containing Fluids, and Bodies of a round Figure. A Stick of Sealing- Wax one Inch Diameter, and one Foot long. ad. A Tube of Sealing- Wax of the fame Diameter and Length as the Stick. A Stick of Sulphur one Inch Diameter, and eighteen Inches kmg. 2d. A Globe of Sulphur three Inches Diameter. 3d. A Cone of Sulphur covered with a Vafe of Cryftal of the fame Figure. 4th. A Cone of Sealing- Wax covered as the former. 5th. A fmall Globe of Amber and another of Gum. Six fmall Cups of Ivory. 2d. A fmall polilhed Copper Pyramid for making Experiments on the Communication of Eledricity. A Sufpenfory garniflied with Ribbands of different Colours, sd. A Sufpenfory garnilhed with (ilk Twift for communicating Eledricity to living Bodies. 3d. Thread Twift, with a Wooden Ball, for communi- cating Eleftricity a great Way off. A Cake of Rofin and Gum weighing eight Pounds. A Cake of Rofin weighing twelve Pounds. 1 PHYSICAL WORLD. CXLVII A Pallet of Pafte-board covered with Gaufe* and garnilhed with Gold Leaff Balls of Cotton and the Down of Feathers. £j«cri- A Receiver without a Bottom for the Experiments of TranfmiiEon. mmo on A Box containing fix Rackets of Gaufe of different Colours, ad. A tlw Tru^- Box containing Plates of different Metals, Wood, Pafte-board and Glafs. ^^JL^ A Gla(s gamiihed with a Circle of Metal for containing Waten A Bar of Iron one Inch fquare and three Feet long. A fmall Globe of Chriftal mounted fo that it can be rubed in Vacuo, to^hich is adapted a Stop-cock to be applied to the Air-pump. A compleat Aflbrtment of every Thing neceflary for eledrical Experi- ments, either in Air or in Vacuo. Plates of Brafs, Part of which has been beat cold, the other when tempered in Fire. A large Pafte-board covered on one Side with Leaf Gold, and on the other with Leaf-Silver, for (hewing the Dudility of thofe Metals. A Metal compofed of Iron and Antimony, the Filings of which burft into Flame by the Fridion of the File. id. Sounding Lead. 3d. An Amalgama of Tin and Mercury for colouring the fnfide of Glafs- Vcffels. Seventh Class. Machines of Cofmography. VII. A large Planetarium five Feet and a Half Diameter, with all its Ap- £»p«n- pendages for (hewing the different Motions of the Planets, and the jiiuft^tbL Relations of the celeftial Bodies with the Earth. the Theory A Box containing the Pieces ncceffary for explaining what concerns ^^^f^. the Motions and Relations of the Sun, the Earth and the Moon. conTaxyPla- yAiV Box only fuppofes a liable five Feet Diameter, in the Middle of netf. ^tvbicb it is Jaflened. < Two Globes, one celeftial and the other tereftrial, one Foot Diameter, c^nftnifted on the latefi Obfervations, coloured and varniftied, mounted on four pillared Pedeftals, with Meridians and Horizons of a particular Kind of Pafte-board. Two Armillary Spheres, of the fame Diameter as the Globes, the one according to the Ptolemaic, the other according to the Copernican Syftem, coloured and varniftied, mounted on Pedeftals of Ebony. A fmall tereftrial Globe, three Inches and a half Diameter, eoloured and varniftied, with a Meridian and Quadrant of Altitude. Two Globes, one tereftrial and the other celeftial, 18 Inches Diame- ter, coloured and varniftied, mounted on pillared Pedeftals, with Meri- dians, horary Circles, Compaffes of Brafs, engraved and poliftied. The fame Globes varniftied and poliftied, with Meridians, horary Circles, Brafs Compaffes, mounted on a turning Pedeftal of a new Con- ftnidion. CXLVm SYSTEM OF THE Experi- Tbe ceJeJiial Globe is of an azure blue. The Figures of tbi ConfteHa- Sufti^bL ^'^^^ ^^^ perceived as Shades ^ the principal Circles oj the Sphere are mearl- theDoariiK ed in Silver ^ as aljo on the terejirial Globe \ the Stars are raijed in Gold, %^^t ^^^^ '" ^*^"' proper Size, fo that at one View, tie natural Stale of tbe ^ ' Heavens is perceived without Confufton. Two large Planifpheres, mounted on a Frame with Gold Stars, ind garniihed with Meridians and Horizons. A white Globe one Foot Diameter, mounted on a Stand, with fomc Inftruments belongrng to it. A new Dial, which fervcs for tracing the Meridiian of a Place. An aftronomical Quadrant two Feet Radius, ivith two Divifions oF Nonius ; a moveable and immoveable Telefcopc, and an exterior Micro- meter. 2d. An aftronomical mural Quadrant four Feet Radius. A Sextant fonr Feet Radius. 2d. A Sextant one Foot tladius ior Obferfaii- talcing cotrcfponding Altitudes. th^ Uf^if^ ^ Quadrant two Feet and a half Radius, with a Tranfom and dwble ■nrouomical Joint, for meafuring Angles on Latld. inftnimcnu A tneridian Telefcopc or a paffagc Tnftfument, four Feet longi drtntfihc ^^^ »^s Axis two Feet. 2d. A parallatic Telefcopc with its Axis, whidi Scxiam, the ferves for following the Parallel of a Star. 3d. An equatorial TdcfcOpe krc'''*'*^hc "^o^eable by the Means of feveral graduated Circles, with its objefliw Panllitic-t^ Micrometer. 4th. A Telefcopc moveable on an Axis, with an horiMU- tcfcope, die tal and vertical Circle graduated, and an Heliofcope. M^cromtttr, ^ Micrometer, to be applied to a moveable Telefcopc for mctfuring the Diameters, the Differences of the right Afcenfions and Declinations of the celeftial Bodies. 2d. A Micrometer to be appUcd to an aftrooo- mical Quadrant. 3d. An achromatic Micrometer. An Oftant 18 Inches Radius, for obfcrvifig the Altitudes and Diftinco of the Moon from the Stars on Sea. A Clock adapted for aftronomical Obfervations, whofc Penduhffl « fo compofed as to correft the Dilatation to which Metals are liable, ai A Telefcopc condufled by a Clock for defigning the Spots of the MooBi ic- Eighth Class. Machines of Meteorology. Metcorplo- A large Thermometer, conftrufted on the Principles of Reaumur, ji Smm. "^ A Thermometer conftrufted on the fame Principles mounted to accoo- pany a Barometer. 3d. A Thermometer, conftrufted on the famcPriB- ciplcs, to be expofed in open Air. A portable Thermometer one Foot long, conftiuded on the fan* Principles. 2d. A portable Thermometer contrived fo as to be P'^' into Fluids, in order to determine the'r Degree of Heat or Cold. 3» A Thermometer conftruSed with Mercury, for Ezperimeota where th« Heat exceeds that of boiling Water. I PHYSICAL WORLD. CXLIX The Thermometer of Florence, a. A Thermometer of Air with Mer- Obfcmii- cury. 3d. A Thermometer of Air, with coloured Liquor. wKn'thl"* A kmd of Pyramid, garniftied with feveral Thermometers of Water, Denfity of Oil, Spirit of Wine, fall Watcr> Mercury^ for flic wing the Dilatabihty '**5A''H'** of each of thofe Fluids. .r^h%" A large Thermometer filled with coloured Water, for (hewing the Expanfioo Dilatability of Glafs. ?iea"o?b*'^ A double Baronieter4 ad. The Barometer of BtrncuUy, 3d. A Ba- Ctufc? ^ rometer bent in its upper Part. «^hi(^ dlml- Tbofe tbref Machines ferve for /beting the Means ^mpUif^ for render- ^mL ing the Variation in the Wei got or Spring ef the Air mmre Jtnfible, The Barometer (hortened, by the Oppofition of the tWD Column* of Mercury to one Column of Air. ad. The Barometer Jhortenod, by a ^^.^T*''' {lemaiader of Air in the upper Part. 3d. The Barometer of Ambhton. \nl when Tbo/e Machines, ferve for Jbewing the Metbodt employed for rekderinit ^® *>nfi y The fimple and lummous Barometer mounted^ to accompany th^ by the Cad^ Theniiometer> conftruded on the Principles of Reaumut. fe* which di- This Barometer differs from the commtn ones ty the Manner it is filled^ \vVighI!* ty the Form of the Vafe in mhicb it is pltmged^ and th^ Exaffitude of its Effeas. The fame Barometer reodemi portable in a<iy Direfiion, or in &ny kind of Carriage* ad. The fikne Barometer rendered portable in s Walking Cane« ^his Barometer has this Advdnt0ge% tb^t Ae inferior Surface of the Mercury is feen, which is well known 4^ be of UJe, A Dial Hygrometer very fcaiibleL 24. An Hy^^m^ter of another ConftrudioB; A Pfrometer, or Nfachine for meaiuFilig the Adion of Fife ofi. Bo- Experi dies, whofe Dilatation is not immedittely perceived. ihcwh/^the In the ConftruSfion of this MaobtnCi every Imp^fe^ion to which it has Dilatation of. ^m hitherto liable is removed, the Degree of Heat is eaftly regulated, and Meuli. every Precaution neceffary, has been taken t^ hinder the. Duji or the HuMidity to fpoil the Pokjb or the Motion >/ the Pieces. An Anemometer, er Machine for difeoveriftg the Diredion aAd Ve- locity of the Wiadj with the Time during Which it continues. . ' Conclusion; Such 16 the Plan of the Colleaiemof Madhines which the Nobility afld <3rentry of the Kingdom of Ireland bave purchafed, and whofe Conftrudi^ Oil and Application to Experimental Inquiries, they have ordered to be de» JSrribedy and publilhed, for the Ufe of the Mathematical School eftablifh ^d under their immediate Infpe&iODi ptfrfoant to theif Refolutien t>i CL SYSTEMOFTHE 1 PLAN of tbi Syftm of the Moral World. Servare modumf finemque tuerif Naturamque fequif patriaque impendere vitarn, Nonfibifed toti genitumje credere mundo. LUCAN. I. MEN in the State of Nature, being apt to allow no oihcr Rule for determining the Difference which might arife among itetn> but what is common to the brute Creation, namely, fuperior Sircngth. The Eftabli(hment of civil Society fliould be confidered as a Cwn- pad againft Injuftice and Violence, a Compad intended to form a Kind of Balance between the different Parts of Mankind ; but the moral Equilibrium, like the phifical one, is rarely perfed and durable. 9^Rn ?^ Intercft, Neceffity, and Pleafure, brought Men together, but the fame cm ociety. ^^^jy^j induce them continually to ufe their Endeavours to enjoy the Advantages of Society, without bearing the Charges neceflary lo te Support : and in this Senfe, Men, as foon as they enter into Society» may be faid to be in a Sute of War; Laws are the Ties, more or W efficacious, intended to fufpend their Hoflilities, but the proiigiow Extent of the Globe, the Differerence in the Nature of the Rcgi«B of the Earth and its Inhabitants, not allowing Mankind to live m^ one and the fame Government, it was natural that Men (hould divide themfelves into a certain Number of States, diftinguiflied by the dif- ferent Syftems of Laws which they are bound to obey. Had ail Man- kind united under one Government, they would have formed a tangaiu Body, extended without Vigour on the Surface of the Earth. T^k different States are fo many ftrong and adive Bodies, which lending each other mutual Afliftance, form but one, and whofe reciprocal KSoff^ fupports the Life and Motion of the Whole. II. 2it VlJfm' ^'' ^^^ States with which we are acquainted, partake of three Fonw of GoTttla- ^f Government, viz, the Republican, Monarchical, and Defpotic. 1» anemia the fome Places Monarchy inclines to Defpotifm, in others the Monarchical Worid. jg combined with the Republican, &c. Thofe three Species of Govern- ment are fo entirely diftind, that properly fpeaking, they have nothiag in common: We (hould therefore form of thofe three, fo many diftinft Clafles, and endeavour toinveftigate the Laws peculiar to each; it will be eafy afterwards to modify thofe Laws in their Application to a«y Government whatfocver, in proportion as they relate more or le^ to thofe different Forms. In the different States, the Laws (hould be conformable to their Na* ture, that is, to what conftitutes them, and to their Principle, or to ! MORAL WORLD. CM that which fupports and gives ihem Vigour. The Law relative to the Th« Liwa Nature of Democracy is firft explained ; it is (hewn how the People in froJJJ'teNa fome refpeds are Monarchs> and in other Subjeds ; how they eled and ture of De- judge their Magiftrates, and how their Magiftrates decide in certain ™o^"«»«»* Cafe», i^c. Then the Laws relative to the Nature of Monarchies are — ^ - unfolded ; the Degrees of delegated Power and intermediate Ranks derived** that intervene between the Monarch and the Subjed, the Duties of the 'VoRicheNa Body to be appointed, the Guardian of the Laws to mediate between n"7c^***** the Prince and the Subjeft arc properly fettled : In fine, it is proved, **** that the Nature of Defpotifm requires, that the Tyrant (houid exert his The La«i Authority, either in his own Perfon, or by fome other who reprefcnts ^'civcd him ; afterwards the Principles of the three Forms of Governments is JjS^of i2^ pointed out ; it is proved, that the Principle of Democracy is the Love potirm. of Equality, whereby is meant, not an abfolute, rigorous, and confe- quently chimerical Equality, but that happy Equilibrium which renders all its Members equally fubjed to the Laws, and equally interefted in in what their Support : That in Monarchies, where a fingle Perlon is the Dif- confift the pencerof Diftinaions and Rewards, the Principle is Honour, to wit JJl'uirce*^^ Ambition and the Love of Efteem ; and in Defpotifm, Fear. The Formrof more vigiroufly thofe Principles operate, the greater the Stability of the Go^"*- Government ; and the more they are relaxed and corrupted, the more it "**^^' inclines to Deftrudion. The Syftem of Education, fuitable to each Form of Government, follows: It is proved, that they ought to be conformable to the Princi- ple of each Gi)vernment : That in Monarchies, the principal Objeft of Education (hould be the Art of pleafing ; as produSive of Refine- The Laws ment of Tafte ; Urbanity of Manners , an Addrefs that is natural, and ^^ Educate J-et engaging, whereby Civil Commerce is rendered eafy and flowing. toihcPrinci n despotic States, the principal Objeft fliould be to infpire Terror and pic of each implicit Obedience ; in Republics all the Powers of Education are re- rof»no^go^ quired ; every noble Sentiment (hould be carefully inftilled ; Magnani- '""*"*" • ^xiityy Equity, Temperance, Humanity, Fortitude, a noble Difintereft- ^dnefs, from whence arifes the Love of our Country. The Laws relative to the Principle of each Government next TbeLawf rxrcur; it is fhewn, that in Republics, their principal Objeft <i«"vcdfrom fhould be to fupport Equality and Oeconomy; in Monarchies to p|c^"c'*a^h rvmJiintain the Dignity of the Nobility, without opprefling the Pcrople ; Form of Go ss I>efpotic Governments, to keep all Ranks quiet. Then the Dif- vemmcm. ^^ences which the Principles of the three Forms of Government l^^yuld produce in the Number and Obje3 of the Laws, in the Form of fn^^gments and Nature of Punifhments is explained; it is proved, l^tfftt the Conftitution of Monarchies being invariable, in order that g^^icc may be rendered in a Manner more uniform and lefs arbitrary : CLII SYSTEM OF THE More civil Laws and Tribunals ar^ required» which are aceantely it- fcribed ; that in teoiperate Qovernnients» whether Monarchical or Re- publicany criminal Laws cannot be attended with too Biany Formalities; that the Punilhnnents (hould not only be proportioned to the CrimC} bat as moderate as poflible ; that the Idea annexed lo the PiUiithipent, {re< quently will operate njore powerfully than ita Intenfuy ; thM in Repab- licks» the Judgnient (hould be (conformable to the Law, becaufe no Is* dividual has a Right to alter it ; in Monsiscbies, the Clenency of the Sovereign may abate its Rigour ; but the Crimes (hould be always judg- ed by Magiftrates appointed to take Cognizance of thejpa. Id &Ki that it is principally in Democracies, tbit the Laws flaould be fcvere againft Luxury, Diflblutenefs of Manners, and the ScduSion ot iheSex. Advanuges y^^ Advantages ) eculiar to each Governmentt is» in fine, caw*" n^h Form rated ; it is proved, that the Republican is better fuited to finaB Siateii of Govern- the Monarchical to great Empires ; tybat Repubiicks are more fybje^ ^ **'**• Exceffes, Monar<?hies to Abufes ; that in Republicks tfce Laws arc tx^ cuted with more Peliberatioiu in Mona^rchies with more Expe(ii4foa. As to defpotic Governments, to point out the Meatia pec^flary tor it» Support, is in effed to fap its Foundation ; the PerfeQion of thii 0^ vernment is its Ruin ; and the exa£k Syftem of Defpotifm is at oad the fevered Satire, and the mofl formidable Scourge of Tyrj^its. Liberty is The general Law of all Governmoits* at leaft temperate ono, m the Prerogi confequently juft, is political Liberty ; the full Enjoyment of whi** lim^rtS'^ fliould be fecured to each Individual: This Liberty is not the abftai Govern- Licence of doing whatever one pleafes, but the Privilege of daiaj ment. whatever is permitted or authorifed by Law; it may be confidcredo* Is not to be ther as it relates to the Cojiftitution or to the IndividuaL It isihewsy M^^t^ that in the Conflitution. of every Sute, there axe two Pow^ *c JJn<bn€yI Legiflative and Executive, and that this latter has two ObjeSs, tke internal and external Policy; in the legal Diftribulion of thofe diSoici^ ConMetU Sorts of Power, confifts t^e greateft Perfeaion of political Liberty^ trihcCo^ ^»^*^ r^'P^^ ^o *^« Conflitution ; in Proof of which are explaiwi ^ Aitution. Conflitution of the Republic of Romcf and that of Grtat-Britm^ '^ is fhewn, that the Principle of the letter is fpunded on the fundamc&til Esuftsj^ind Law of tl^e ancient Germans ; nai?fielyj that AiFairs of fipall Conieqveic^ 'SoiLmd ^^^^ (Jetermined by the Chiefs, and thofp of Import^nc^ wwp wxff^ *^ ' to the General Aflembly of the whole Naiipn, aftf^r hfmg pf<?^J examined by the Chiefs, Political Liberty qonfid^red, with, rcfp^ ^ Codfidered Individuals, confifls in tbo Security which the Law %flfbrds thiemp "^^^ wJinSiwd?. ^y^^^ Individual is npt in Dread pf afiptber, It is fliewn, thi^tili .als. principally by the N^ati^f* and Proportipn of PuiHfl^mei>ts thit »« Liberty is eAabUfhed. or^ ddlroyed: That Crimea againft Religion MORAL WORLD. CLIII Aooid be f unifted by the Privation of the Advantages which Religion drocurei ( the Crimea againft good MoraU> -by Infanoy ; Crimes againft the public Trao<)uiiity9 by Prtfon or Exile; Crimes againft private Security, by corporal Punifhments : That Writings are lefs criminal than Deeds ; meer Thoughts are not punifliable ; Accufation without a regular Procefs^ Spies^ anonymous Letters; all thofe Engines of Ty- taony* equally infamous with refped to the Inftniments and the Em- l^oyersy fliould be profcribed in every good Government, that no Ac- cufations fiiould be urged but in Face of the Law, which always pu« * jiiftes Guilt or Calumny : In every other Cafe^ the Magiftrate Ihould fay, wr fiould t^folvi frim Sufpicion, the Man wbo wants an jtccufer, witbout wanting an Enemy. That it is an excellent Inftitution to have puUic Officers appointed, who in the Name of the State may pro- fecute Criminals : This will produce all the Advantages of Informers, without their Inconveniencies and Infamy. The Nature and Manner of impofing and collefting Taxes Is after- Libcrtycon- wards explained : It is proved, that they fliould be proportioned to Li* fidcxed with berty ; confequently in Democracies they may be heavier than in other [g^^^'o?* Oovernments, without being burthenfome; becaufe each Individual tYxm and condders them as a Tribute he pays himfelf, and which fecures the *« P«Wic Tranquility and Fortune of each Mem'ber : Befides, in Democracies, *^^"*^' the Mifapptication of the public Revenues is more difficult, becaufe it is more eaiily difcover^d and puniflied; each Individual having a Right to call the Trcafurer to an Account. That in every Form of Government, thofe Taxes that are laid on Merchandizes are leaft bur- thenfome, becaufe the Confumer pays without perceiving it : That the cxceflive Number of Troops in Time of Peace, is only a Pretext to The Aug- overcharge the People with Taxes ; a Means of enervating the State, JfX'Num- and an Inflrument of Servitude. In fine, that the colleding of the b^of Duties and Taxes by Officers appointed for this Purpofe, whereby Troop cncr the whole ProduQ enters the public Treafury, is by far lefs burthenfome suS, ' to the People, and canfeauently more advantageous than the farming out of the fame Duties and Taxes, which always leaves in the Hands of a few private Perfons, a Part of the Revenues of the State, IV. The Circumftances independant of the Nature of the Form of Go- Particular vernmenty which fliould modify the Laws, arife principally from the ^jj^jh^*^ Nature of the different Regions of the Earth, and the different Charac- ftouldmodi* tcrs of the People which inhabit them. Thofe arifing from the Nature |7 the cjif« of the Regions of the Earth, are two-fold ; feme regard the Climate, oToovcn^* others the Soil. No Body doubts but the Climate has an Influence on ment. the habitual Difpofition of Bodies, confequently on the Charaders, the Ltiwa ihould be therefore conformable to the Nature of the Climate in CLIV SYSTEM OF THE 1 re^rs and Palfions of Men. SUvenfitia- coofiltent with the Uw of Ni- tare ind the cWil Law. Countriei TheClimace indifferent Matters^ and on the contrary check its vicious Effeds ; an S?ff re"?^ exa£t Enumeration of which is made, and the Laws for correcting them in the^cha- explained^ it is fhewn^ how in Countiies where the Heat of the Climate inclines the People to Indolence, ihe Laws encourage them to La- bour ; where the Ufe of Spirituous Liquors is prejudicial, they are dis- couraged, (^c. The Ufe of Slaves being authorifed in the het Countries of JJia and Americaf and prohibited in the temperate Climates of Europe, the Law- fulnefs of civil Slavery is next enquired into ; it is proved, that Men hav- ing no more Power over the Liberty than over the Lives of one another^ Slavery m general is inconfiftent with the Law of Nature ; that there has never been perhaps but one juft Law in Favour of Slavery, v/z. the Roman Law, whereby the Debtor was rendered the Slave of the Credi- tor ; the Limitation of this Servitude, both as to the Degree and as to the Time, is pointed out. That Slavery at the utmoft can be tolerated in defpotic States, where free Men, too weak againft the Government,, where it my feek for their own Advantage, to become the Slaves of thofe who ty- be tolerated. ^^^^\2^^ ^yer the State} or elfe in Climates where Heat fo enervates the Body, and weakens the Spirits, that Men cannot be brought to undergo painful Duties only by the Fear of Punilhment. From thence we pafs to the Confideration of the domeftic Ser- Domcft'ic vitude of Women in certain Climates : It is (hewn, that it ihould dendionthe ^^^^ P\^cc in thofe Countries where they arc in a State of cohabiting Climate. with Men before they are able to make Ufe of their Reafon ; mar- riagable by the Laws of the Qimate, Infants by thofe of Nature. That this Subje^ion is (lill more necelTary in thofe CDuntries where Poligamy is eftablilhed, a Cuftom in fome Degree founded on the Nature of the Climate and the Ratio of the Nunber of Women to that of Men ; then the Nature of Repudiation and Divorce is examined, mnd it is proved, that if once allowed, it (hould be allowed in Favour of Women as well as of Men. In fine, political Slavery is treated of ; it is pro^d, that the Climate which has luch In&uence in producing domeftic and'civil Servatttde^ has not lefs in reducing one People under the Obedience of another ; that the Northern People having more Strength and Courage than thofe of Southern Climates, the former are deftined to preferve, the latter to lofe their Libertv ; in Confirmation of which, the various Revolutioiis which Europe^ Alia, Csfr. have undergone, is unfolded ; the Caulcs of the Rife and Fall of Empires is pointed out, particularly thofe of the Roman Empire; it is proved, that its Rife was principally owing to the Love of Liberty, of Induftry, and of Country ; Principles inftiUed into the Minds of the People from their earlieft Infancy ; to thofe in- teftine DifTentions^ which kept all their Powers in A€tion^ and which Political SlaTcry. ItReigaa principally an hot Countriei. MORAL WORLD. CLV ! ccftfcd at the Approach of an Enemy ; to their intrepid Conftancy under Enwneriti- Misfortunesf which made them never difpatr of the Republick ; to that ^afej'of Principle from which they never receded* of never concluding Peace theRifctnd until they were vidorious ; to the Inftitution of Triumphs, which ani- .^"^^g^. mated their Generals with a noble Emulation ; to the ProteSion they ^\^^ "' granted Rebels againft their Sovereigns ; to their wife Policy of leaving to the Vanquiihed their Religion and their Cuftoms ; in fine, to their Maxim of never engaging in War with two powerful Enemies at once, fabmitting to every Infult from one, until they had cruihed the other. That its Fall was occafioned by the too great Extent of the Empire, which changed the popular Tumults into civil Wars ; by their Wars abroad, which forcing the Citizens to too long an Abfence, made them lofe inrenftbly the Republican Spirit; by the Corruption which the Luxury of Jjta introduced ; by the Profcriptions of Sylla, which de- bafed the Spirit of the Nation, and prepared it for Slavery ; by the Ncceility they were in of fubmitting to a Mafter, when their Liberty became burthenfome to them ; by the Neccflity they were in of change ing their Maxims, in changing their Form of Government ; by that Succeflion of Monfters, who reigned almoft without Interruption, from J'iberiut to Nerva^ and from Ctmodus to Conftantine ; in fine, by the Tranflation and Divifion of the Empire, which was deftroyed, firft in the Wejty by the Power of the Barbariant ; and after having languifhed many Ages in the Eajl^ under weak or vicious Emperors, infenfibly ex* pired. The Laws relative to the Nature of the Soil is next explained ; The Infla- ft is (hewn, that Democracies arc better fuited than Monarchies to jf^^jf^J* barren and mountainous Countries, which require all the Induftry theSoUon of their Inhabitants ; that a People who till the Soil, require more ^ !-«*•• L^aws than a Nation of Shepherds, and thofe more than a People who live by Hunting ; thofe who know the Ufe of Coin, than thofe who are ignorant of it. The Laws relative to the Genius of the different People of the Elarth The Laws at length is difclofed, and it is proved, that Vanity which mag- confiicred nifies ObjeSs, is a good Rcfort of Government ; Pride, which deprcffes Siuic*deni- thenriy is a dangerous one; that the Legiflator, in fome meafure, fhould us of the in refpcft Prejudices, Paffions, and Abufes ; as the Laws ihould not be the {jj]'£^'J[^ ^^ befl, confidered in themfelves, but with refpeS the People for which thcyVrc made; for Example, a People of a gay Charadcr require eafy Laws ; thofe of harlh Chara^ers, more fevere ones. The Manners and Cuftoms are not to be changed by Laws, but by Recompcnces and Ex- aznples: In fine, what the different Religions have, conformable or contrary to the Genius and Situation of the People who profefs them, is explained. CLVI The Rftkli- onsofwbkh the daScra^ Foiins of Govern- sneotarefoT* ccpuble. Viituet which Com- merce in- troda«Cf. The Liberty ofTiraae not to be confounded with the Li • berty of the Trader. Should be interdiOed to the Nobi- lity in Mo- ntfchiet. Mairiaigtto be encourage cd. Inecftuous Marrisgcf to be pro. fcribrd. How Popu- lation is pro- moted. SYSTEM OF THE V. The different States confidered with refpca to each other, may yield mutual Aflidance, or caufe mutual Injury. The Afiiftance they afford is principally derived from Conunercc, its Law& are therefore to be un- folded ; it is proved, that though the Spirit of Conrunerce naturally pro- duces a Spirit of Inlereft* oppofed to the Sublimity of moral Virtues, yet it renders a People naturally juft, and banishes Idlenefs and Rapine. That free Nations, who live under moderate GovernmentSf fliould ap- ply themfelves to it more than thofe who are enilaved ; that one Nation fhould not exclude another from its Commerce without important Rca- fons ; that the Liberty however of Commerce does not coxiiift in allow- ing Merchants to a^ as they pleafe ; a Faculty which would be very often prejudicial to them, but in laying them under fuch Reftraints only, as aret neceflary to promote Trade ; that in Monarchies^ the Nobility (hould not purfue it, much lefs the Prince : lo fine, that there are Na- tions to whom Commerce is difadvantageous ; it is not thofe who want for nothing, but thofe who are in want of every thing ; as PoUndy by whofe Commerce the Peafants are deprived of their Subfiftence, to fatisfy the Luxury of their Lords : The Revolutions which Coionieice has undergone, is next difplayed, and the Caufe of the Impaverilhraent of Spain by the Difcovery of Jmirica, pointed out : In fine* Coin be- ing the principal Inftrument of Commerce, the Oferatioas upon it are treated of, fuch as Exchange, Payment of pubUc Debts* &c. whole Laws and Limits are fettled. Population and the Number of Inhabitants being immediately con- neded with Commerce, and Marriages having for their Objed Popu- lation, every Thing relativelhereto is accurately exphuued ; it is flacwa, that public Continence is what promotes Propagation^ that in Marriages, though the Confent of Parents is with Reafon required* yet it fihould be fubjed to Reilridions, as the Law ihould be as favourable as pofllble to Marriage ; that the Marriage of Mothers with their SoBSt on account of the great Difparity of the Ages of tha Contradors* could rarely have Propagation for Objed, and confidered even in this Light* Ihould be prohibited ; that the Marriage of the Father with the Daubster might have Propagation for ObjeS, as the Virtue of engendering caafies a great deal later in Men, and has in confequence been authorifed in fome Countries, as in Tartary ; that as Nature of herCelf inclines to Marriagi^ the Form of Government muft be defeftive, where it ftands ia Necil of. being encouraged ; that Liberty, Security, moderate Taxes* th« Prafimp* tion of Luxury, are the tru^ Principles and Support of Populatiett; that Laws nptwithftanding may be made with Succefs, for encouragii^ Marriages, when, in fpight of Corruption* tha People are attached t» their Country ; what Laws have been made to this Purpofe^ particulaily 1 MORAL WORLD. CLVIl thofeof Juguftui, are unfoMcd; thai the Eftablifliment of Hofpitals f^f*^^^' may either favour or hurt Populalioo* according to the Views in which rfch Sulci. tbey have been planned ^ that there flnNild be Hofpicak in a State where the greateft Paft of the Citizens have no other Refeoice than their In- duftry; but that the Aflkftance which thofe Hofpitals give ihoukl be ^^J^CI temporary ; unhappy the Country where the Mukitnde of Hofpiuls and condnOcd MQna(lerte8> which are only perpetual Hofpitals* feta every Body at their Eafe, except thofe who labour. To prevent the mutnal Injuries which States nay receive from each other* Defence and Attack are rendered neceflary ; it k fliewn, ihat RepiAUcks. by their Nature being but fmall States* cannot defend tkesn- Uves but by Alliances ;, but that it is with Repubtkks they fliouk) be formed. That the defeo&ve Force of Menarchiea confifts principally in having their Frontiers fortified. That States as well as Mea» have a Right to attack each other for their own Prefervation^ firona whence, is derived the Right of Conqyeft> the general Law of which is to ck> as little Kuct to the Vanquiiied a> poffible. That Repnblicks can make kfs confidendble Conquefls than Monarchies; that immenfe CoB<{nefts ^^^o?'^!!^ introduce and eftabliih Defpotifm ; that the great Principle of the Spirit of f, not"^w Conqueft (hould be to render the Condition of the conquered People bet- 7 but coo- ter* whkh is fuelling at mce Che natural Law and the Maxim of State> ^^'^^^^ how far iht^Spamards receded bem this Principle^ in esterminating the Americans, whereby their Conqueft was reduced to a vaft Defert, and they were forced to depopulate their Country, and weaken themfelves for ever> even bjr their Vidory» is explained. That it nny become neceflary to change the Laws of a vanquiflied People, hot never their Means of pre Manners and Cuftoms. That the rooft aflured Means of preferving a c^^feft. Conqueft, i» to pot the Vanquiflied and Vidors on a Level if poffiUe, by granting them the fame Rights asd Privileges ; how the Ronmns coodufted theaafelves in this Refped, is rekbed ;. as aUb how Cejar with Feijpcft to the Gauls. VL After having treated in particular of the different Species of Laws, ^|?« ^«w>^ ab«^ remaisM m» more to be done, bnt to conapare them together,, and /vomthl to- examine them^ with refped to the Objeda on which they are en- Nature, cir- ^Stsd. Men are governed by different Kinds of Laws, by the natural ^^^^^f^^J."" . Lavr common to each Individual ; by the divine Law^ which, is; that of ons, of \he Religbn; by the ecdefiaftical Law, which is that of the Policy different of Rdigion ; by the civil Law, which is that of the Members of ll"^^ the iSune Community ; by the polttical Law, which b that of menr. the Govenmenfi of the Communi^ ; by the Law of Nations, which is tbat of Commnnites confidered with reijpeft to each other ; each of tbefe hnvc thek cUftinft Obje^^ which are not to be confounded, nor CLVIII PLANOFTHE wh«t belongs to one be regulated by the other; it is neceilkry that the Principles which prefcribe the Laws, reign alfo in the Manner of com- pofing them ; the Spirit of ^foderation ihould as nauch as poiSble direft all the Difpofitions : In fine, the Stile of the Laws, fhouM be fimple and grave, it may difpenfe with Motives, becaufe the Motive is fappoied to exift in the Mind of the Legiflator ; but when they are afligned, they fliould be founded on evident Principles. vir Conclafsoii. guch is the Plan of the Syftem of the Moral World, where the In- habitants of this Earth are confidered in their real State, and under all the Relations of which they are fufceptible ; the moral Philofepher without dwelling on mere fpeculative and abftrad Truths, in pointing out the Duties of Man, and the Means of obUging him to discharge them, has lefs in View the metaphifical Perfe£tion of the Laws, thao what human Nature will admit of; the Laws that are exifting, than thofe which ihould be eftabliflied ; and as a Citizen of the World goq- fined to no Nation or Climate ; he makes the Laws of a Mrticidar People lefs the Objed of his Refearch, than thofe of all the People of the Univcrfc. PLAN of the Military Jlrtf including tbe InfiruBims rtUtioe It Engineers, Gentlemen of tbe Artillery, and in general /• alt Umd^ Officers. 1 S' bitenti expeHant Signum, exultantiaque baurit Corda favor pul/atu, Laudusnjuc arre&a Cupids. I. I N C E the Revolution which the Invention of Gunpowder has _ I produced in Europe, but above all, fince Philofophy bom to confole Mankind, and to make them happy, has been forced to lend its Lright to teach Nations how to deftroy one another, the Art of War forms a Science as vaft as it is complicated, compofed of the Aflemblage of a great Number of Sciences united and conneficd together, lending each other mutual Ai&ftance, and which the Youth of this Country who are intended for the the Military State, could never acquire but in a Miiitsuy School, eftabliflied bv public Authority, and conduced by a Man of fu« perior Talents and Abilities. II. There the young Officers are firft brought acquainted with Algebra Matbema* and Geometry, elementary, tranfcendental and fublime, to teach them tick*. xht general Properties of Magnitude and Extention ; how to calculate the Relations cF their different Parts ; how to apply them for determin- ing accei&ble and inacceflible Angles and Diftances^ tracing of Cainps» MILITARY ART. CLIX furveying of Landy drawing of Charts, cubing the Works of Fortlfi* cations, (rTr. and to infufe into them that Spirit of Combination, which is the Foundation of all Arts, where Imagination does not predominate, as neceOary to the Military Gentleman as to the Aflronomer, which has formed Turenne and Coborn, as Archimedes and Newtan. III. Thefe abftrad Notions ferve as an Introdudion for attaining the Art which teacheth the Properties of Motion, to meafure the Times and Mecanicki Spaces, to calculate the Velocities, and to determine the Laws of Gravi- andDyna* ty, to command the Elements by which we fubfift, whofe Forces it n*»ckf. teaches to fubdue, and learns how to employ all that is at our Reach in. Nature, in the moft advantageous Manner, either to aflift us in our En- terprizes, by fupplyi»g our Weaknefs, or to fatisfy our Wants, and pro- cure us all Kind of Conveniencies. They are taught the Application of this admirable Art, more partl- calarly for regulating the Dimenfions which fuit the Linings of the Military Ati^ Works of Fortification, that they may refift the Preffure of the Earth, chitefture.. which they are to fuftain, by determining the Law according to which. this Prefliire aSs. For eftimating the Refiftance that Counterforts are. capable of, accocding to their Length, Thicknefs,. and their Diftancf s £rom one another, for calculating how the Efforts of Vaults ad> in order to deduce general Rules for determining their Thicknefs, according to the Forms that are to-be given them in the different Ufes that are made of them in Fortification, either for Subterraneans, City-Gates, Maga- s&eens of Powder* iic* for afligning the Form of Bridges, relative to the ipreadingof the Arches, determining the Strefsand Strength of Timber,. the Proportions of the Parts of Works,, that- they may have an equal relative Strength with refped to tha Models, according to which they are executed in large Dimenfions. v. Then is unfolded the Theory of the Force and Adion of Gunpowder, aa it ferves to regulate the Proportions of Cannons, Mortars, Guns, £^c. £^111^;^. that of elaftic Fluids, as it teacheth to determine the adual Degree of the Refiftance of the Air to Shells and Bullets, and to affign the real. Trad defcribed by thofe millitary Projedilea. VI. Then the Ufe that can be made of the Dilatation and Condenfation of the Air, as of the Force that it^ Spring acquires by Heat, to move Ma- Punmadcki*. chined, is explained, by (hewing the Effe^ of Pumps, defcribing the Properties of all the Kinds that have hitherto been invented ; pointing aut their DefeSs and Advantages ; to what Degree of Perfedion they «an be brought ^ determining the moft advantageous Proportions and CLX PLAN OF THE Forms of their Parts, and of all the Machines contrived to nake them move, either of thoTe intended for tiie Ufe of private Peribns, {or ex- tingutihing Fires, for fupplying public Fountains, &c. unfoldiiig the Conftrudion of all thofe that have been hitherto executed in the di&reot Parts of Europe, which are put in Motion either by Animals* by the Courfe of Rivers, by the Force of Fire, explaining how this Agent» the moft powerful in Nature, has been managed with the greateft Art ; afterwards is (hewn how to calculate the Force of the Wind, the Advan- tages that can be drawn from it, (or draining an aquatic <Mr noaracageous Land, or to water a dry Ground ; exemplified by what has been praAifed in the different Parts of Europe in this Way. VII. The Art of conducing, raifmg, and managing Water> is next dif- clofed ^ it is (hewn how to raife Water above the Level of its Soupoe by Means of its Gravity, without making Ufe of the Parts which enter Hydfinljckf jpto the ordinary Compofition of Machines j how to difcover by Calculation, if a Water of a given Source, or raifed to a given Height, by any Machine, can attain to a given Place, either by Tren- ches, Aqueduds, or Pipes ; how to conftrud Bafons, Water-HouieSf and Ciilerns to preferve it ; how to diftribute it through the different: Parts of a City, determining die moft advantageous Dimenfions and Difpofitions of the Conduits, and defcribing the moft ufeful and inge« nious hitherto executed. As nothing is more agreeable to the Sight than Water* Works, the Manner of laying them out, and the Conftnidion of the Machines imagined to raife the Water into the Refervoirs, which are the Sool of all thofe Operations, are unfolded, in order that the Engineer may be able to point out to thofe who are willing to embellifli their Gardens what futts them as to the) Expence they are willing to be at, or the Situation of the Place ; and that the Officer may be able to judge of tbe Beauty of Objefts of this Kind. Water, being of all Agents, that from which the greateft Advantage can be drawn for animating Machines, it is (hewn how to apply it to the Wheels of the different Kinds of Mills ; what Velocity they fliraU have relative to the Current which moves them, in order that the Ma* chines may be capable of the greateft Effed ; entering into the Detail of all their different Species ; calculating the Force neceflfary to pttt them in Motion ; the EffeSs they are capable of, by Calculations^ com^ prehending the Fridion of their Parts, and the other Accidents inlepe<-> rable from Praftice ; determining when they aft upon inclined Planes, the Angle they ihould form with the Horizon. In fine, comparing fucla Machines as are contrived for the fame Purpofe, in order to difcover which are to be preferred, according to the local Circumihinces ■r rl Conveniencies for their Execution. 7 MILITARYART- CLXI VIIl. The Art of rendering Works capable of reiifting the violent or flownyanniick Adion of Water, prefents itfelf next ; the various Machines made ufe AfchUec- of in draining, and of linking Piles, is defcribed ; then all that concerns ^"^' the Conftrudion of Sluices, as alfo the Manner of employing them, according to the different Ufes to which they are applied, either in level- ling the Canals of Navigation ; draining of Marflies ; rendering Rivers navigable $ forming artincial Inundations ; making of Harbours, ^c. IX. In order to tender thofe Refearches of real Ufe to the young Officers, Dnnghtin^. they are initiated in the Art of delineating Objeds, as it teacheth how to reprefent all the Parts of Works already conftruded, or that are intended to be conftruded by Plans of them taken parallel to (he Horizon, which Ihcw the Diftribution of all iheir Parts, their Dimcnfions, lie, by Pro- files or Cuts of them taken perpendicular to the Hori2:on, which ftiew the Heights, Situations, lie, of all the Parts, by Plans of Elevation, or Cots of the exterior Parts of the Work ; in fine, by perfpedive Plans or Cuts, which reprefent the Objeft as feen at a certain Dillance, which will enable them to judge of the Effed tha^t all the Parts together pro- duce. X. Thcfe Studies prepare the young Officers for attaining to a Proficl- Attack ana ency in the Art of defending and attacking, which comprehend the Me- I>c^«cc. ihod of fortify'mg regular Poligons, according to the different Syftems, (hewing their Advantages with regard to the local Circumftances, and how far they have been followed with Succefs in the Fortiiications of the mod celebrated Towns in Europe \ the Conftrudion and Difpofition of Batteries, the Management of Artillery, the pointing of Mortars and Cannon, the conduSing of Trenches, the Manner of diftributing the Afferent Stages of Mines, the Form of therr Excavation, the Range- ment of the Chambers, the beft contrived for the huftwinding the Ground and tfcfc Annoyance of the Enemy, the Conftm&ion of Lines :ind the Menfijratfon of therr Parts, the tracing of Camps, entrenched or not efttrenched, in crrcn or uneven Oround, the tracing of the Cstraps of Armies which befiege, included in Lines of Crrc^mvallatton and Contra- vallarion, the Attack of a regular or irregular fortified Place, fitoated in ^rn equal or arn unequal Ground, exemplrfied by the Plans of the moft ^celebrated Sieges, joirrrng Theorr to Praftici?, negleftrtrg not one Derail that may be of Importance. All thefe Operations being made in large qDitneiffions, and "a Front of Fortification being raifed accompanied with pflie other detached Works to be attacked and defended as in a real CLXIl PLAN OF THE XI. Hiftory. Ttcticks* Order ofthe SCttdict. Geography. Geographyy as an Introdu&ion to Hiflory, is ufeful to all Perfons, but the Profeflion for which Youth is intended fliould decide of the Manner more or lefs extenfive, it is to be taught ; the young Officers lliould have an exad Knowledge of the Countries which are commonly the Theatre of War, they are therefore intruded in Topography in the greateft Detail, employing the Method of refering to the diflFerent Places, the Haflages in Hiflory which may render it remarkable, prefer- ing military FaSs to all others ; by this Means their Notions arc ren- dered more fixed, and their Memories though more burthened, will become ftronger. XII. The Life of Man is infufficient to (ludy Hiftory in Detail, the Man- ner of teaching it fhould therefore be adapted to the State of Life for which Youth is intended : Thofe who are deftined for the Law, fhould be taught it, as it ferves to difcover the Spirit and Syftem of the Laws of which they will one Day be the Difpenfers; thofe who are intended for the Church, as it relates to Religion and the ecclefiaftical DifcipUne; the young Officers are taught it, as they may draw Inftru£tion from the military Details, as it furnifhes Examples of Virtue, Courage, Prudence, Greatnefs of Soul, Attachment to their Country and Sovereign ; they are made to remark in Antient Hiftory that admirable DifcipUne, that Subordination which rendered a fmall Number of Men the MaHen of the World ; they are taught how to gather from the Hiftory of their own Country, fo necefTary and fo negleSed, the prefent State of Affairs, the Rights of their King and Country, the Intereft of other Countria and Sovereigns, &c. XIII. The Theory and PraSice of the different Parts of the Military Ser- vice being necefTary to all Officers, they are inftruSed in what reganh '. the Service of Camps, the Service of Towns, Reviews, Armaments, Equipments, &c. As to military Exercifes, and Evolutions, all wt»> are acquainted with the aSual State of military Affairs, know how necefTary it is to have a great Number of Officers fufficiently inftruded in the Art of exercifing Troops ; it is manifcft that a continual Pradice is the fureft Means to attain to a Proficiency in this Art ; the young Of* ficers therefore are taught the Management of Arms, and trained up to the different Evolutions, which one Day they will make others execute xiv. The Order that is followed in the Employ of the. Day is fucb, that the Variety and SuccefTion of ObjcSs may ferve as a Recreation, which is the moft infallible Means to haften Inftru3ion. The Leflbas of Algebra, Geometry, Mechanicks, Hidroftaticks, Hydraulicks, Geo- MILITARY ART. CLXIII graphy^ Hiftory, &c. are firft giveoy and thofe on the various Branches of Drawing fucceed. XV, As Youth is liable to take a Difguft agatnft abftrad Knowledge, when PrsAical its Application is not rendered fenfible, the Teachers of Mathematicks Opcmion«. and Drawing frequently put in Pradice in the Field, the Mathematical, Mechanical, &c. Operations which are fufceptible, and which have l>een already delineated on Paper, Defign at fight. Views, Landfcapes, &c. this Nlethod has the Advantage of procuring the Pupils an Amufe- ment which inftruds them, and rendering palpable the Truths that have been prefented them, it infpires ihem at the fame Time with a Defire of learning new ones, and making them execute after Nature agreeable Operations, it is a fure Means of forming their Tafte. XVI. As the Inequality of Ages and Genius, ^nd even of the good and p^^.^ _, ^^ bad Difpofitions of the Pupils, caufe a great Difference, the State of minactoiu. * the Examination is divided into three Clafles. In the firft are thofe who diftinguiih themfelves the mod by their Application ; in the fecond are comprifed thofe who do their bed ; the third comprehends thofe from whom little is expeded. This State is laid before the Society, in order that it may have an exaS Knowledge of the Progrefs pf each. xyii. Such arc the Means, my brave Countrymen, which the DUBLIN SOCIETY have purfuant to their Refolution of the 4th of February, 1 768, procured you, to enable you to ftudy with Succefs, how to efta- bltfli a Concert and an Harmony of Motion amongft thofe vaft Bodies ililed Armies ^ how to combine all the Springs which ought to concur to« ^ether; how to calculate the Aftivity pf Forces, and the Time of Exe- cution ; bow to take away from Fortune her Affendant, and to enchain her by Prudence ; how to feize on Polls, and to defend them ; how to profit of the Ground, and take away from the Enemy the Advantage of theirs ; not to be dejeded by Dangers, nor elated by Succefs ; how to retire, change the Plan of Operation ; how in the Glance of an Eye to Form the mod deciffive Refolutions j how to feize with Tranquility the rapid Inftants which decide Viftories, draw Advantages from the Faults of the Enemy ; commit none, or what is greater, repair them, in which confifts the Art of War. CondufiOD. 1 CXLIV Dignity of the Trader. W The Difad- ' ▼wU^et in Point of Education thofe of the coramercial ProfcifioR U b«iir under. PLAN OF THE PLAN of tie Mercantile ArtSf including the InftruSions relathe t^ thofe who are intended for Trade. Docuit qua maximui Jtl^t. I. I S E Regulations and well concerted Eiicouragemem$ will con- tribute very little to promote Trade, unlefs they be rendered pria'icable, operative, and ufeful, by the $k\\\ and Addrcfs of the judi- cious and induftrious Trader ; it is he who employs the Poor, rewards tjie iqgei^ious, encourages the Induftrions, interchanges the Produce and Manutaftur^s of ow Country for thofe of another, binds and links to- gether in one Chano of Intereft, the Univerfality of the human Species and thus becomes a Blefling to Mankind, a Credit to his Country, a Source of Affluence to all around him, his Family, and himfclf. The Extent of Knowledge ^nd Abilities notwifchftanding, rec^uifite to fit Youth for fo gre^t and valuable Purpofes, have not been attended to in this Country, and thofe of the commercial Profeffion have laboured under tlic fame Pifadvantages in Point of Education, as the different Cplaffcs of Men we h^vc already fpoke of. II. A Number of Years arc fpent and frequently loft in drudging tbrcu^^ the oommon Forms of a Craipmer School, where Youth arc oblipd to learn what is dark and difficult,, and what muft afterwards coil th^ni much Pains to unlearn, and if long purfued muft in the End retard the quickeft Parts, and go near to cclipfc the brighteft Genius: whilft on the contrary, if the Grammar School Studies were properly direded and carefully purfued, they would learn to pafs a proper Judgment on what they read, with regard to Language, Thoughts, Refledioos,. Principles, and Fads, to admire and imitate the Solid more than the Bright, the True more than the Marvellous, the perfonat Merit and good Senfc more than the external and adventitious j thcirTaftc for Writing and Living might be in fome Meafui* formed, their Judgment refiified, the firft Principles of Honour and Equity inftiljed, the Love of Virtue and Abhorrence of Vice excHed in their Minds: J^re ergo lib^xalibus Studiis Filios erudimus f non fuia Virtutem dare pojfunt, Jed quim Anhmm ad accipiendam Virtutem praparant, quemadmedum prima ilia ut jimtigw vocabant, Literatura, per quam Pveris Elementa traduntur, non decet libe- rales ArteSf fed mox percipiendis Locum parat^ fie liberales Artes n^n per- ducunt Animum ad Firtutem, fed expediunt, III. At a certain Age, not after certain Acquifitions, a Maftcr of Mathc- maticks is looked out for, and in this Cafe great Pretentions, attefted by his own Word, and low Prices, are fufficient Credentials to recommcjid him, although neither the Teacher nor the Student reap much Advan- MERCANTILE ARTS. CLXV ties feom It. When the Round of this Tcachcr^s Form is once finifli- tip the Student is then turned over lo the Compting-Houfe, where he U employed during the Time of his Apprenticeihip^ in copying Letters, going of Meffages, and waiting on the Poft-Office. The Matter, though he hath Talents for communicating, hath not Time for attending to the Inftrudion- of an Apprentice, who, on the other Hand, hath been fo little accuftomed to think, that this Improvement by Self-Ap- plication will be very inconfiderablr, bcfides his Time of Life, and eonftant Habit of Indulgence, render him more fufceptible of plea- fprable Impreffions, than of Improvement in Bufinefs, the more efpc- «ially when he was not previoufly prepared to undcrftand it ; where- fore it is not at all furprifing, if many, who having no Foundation in Knowledge to qualify them for the Compting-Houfe, profit little from the Expence and Time of an Apprcnticeflvip, and from feeing Bufinera aonduSted with all the Skill and Addrefs of the moft accomplished Mer- cl^ant: The Confequence muft no Doubts be fatal to Numbers, and the public Intereft, as well as private, muft fuffer greatly by every In- ftance of this Nature. It is true, that there have been, and flill are, €kmlenien, who, defKtute of all previous mercantile Inftrudion, with- out Money, and without Friends, by the uncommon Strength of natural Abilities^ fupported only by their own indefatigable Induftry and Appli- eation, and perhaps favoured with an extraordinary Series of fortunate Events, have acquired great Eftates ; but fuch loftances are rare, and rather to be admired than imitated ; for we fee nuny fet out with large Capitals, who have (hone in the commercial World while their Capitals ta^dd, as Meteors do in the natural, but like them^ foon deftroyed them- felvest and involved m their Ruin all fuch who were fo unhappy as to be within the Sphere of their Influence. N^vimus Novithtf qui cum Je I^fercatupa vix dederuntf in magnis Metvimoniis fe implicantes^ Rem fuam maU giffiffe ; et frtfOla impiritos Mercatores^ mult is Captionihus fupptjfiUsp , muUisque infidiis ixpafitiys expsrientis videmus. Commerce is not a Game of Chanee, but a Science, in which he who £(!ab]i(h- ii moft (killed bids faireft for Succefs, whereas the Man who flioots at "^"^ °^ .. Random, and leaves the Diredion to Fortune, may go mifcrably wide Scbooh"' of the Mark; of which the People of thisCountry at length made fcnffble, Have come to the Refolution of no longer truftmg the future Profpeds ef their ChiUren in the World to a Foundation fo weak and uncertain : but fetting a proper Value on Education, are determined to be as careful ifi bkaving the Minds of their Children adorned with Virtue and good Senfe, as they are in fetting off what relates to their Bodies. A School is erc&ed tn thi9 Kingdom for training up Youth to Bufinefs, where every Kfefterhas a Salary proportioned to the Difficulty of his Department :: CLXVm PLAN OF THE tf to mode- ^ en, and EfFe&s, how to blend Self-Love with Benevolence^ rate his Paflions, to fubjeS all his A6Hon8 to the Teft of Reafon, and that it is his Duty and Intereft to found all his Dealing on Inregnty and Honour^ as he that accuftoms himfelf to unfair Dealing wilU by De- gree$> be reconcilled to every Species of Fraud> 4ili Ruin and Infamy become the Confequence. The Principle of Law and Crovernment like wife conftitute a Part of the mercantile Plan of Inftru3ion> by which they learn to whom Obe- dience is due, for what it is paid, and in what Degree it may juftly be ' required ; and to give proper Inftrudions to their Reprefentatives in the great Council of the Nation when they are deliberating on any AGt which may be detrimental to the Intereft or the Community with reipefik to Commerce, or any other Privilege whatfocvcr. IX. The Study of Compofltion not only teaches bnt accoftoms the yom^ tio" Merchant to range his Thoughts, Arguments, and Proofs, in a proper Or- der, and to cloath them in that Drefs, which Circumftances render moft natural ; by this Means he is not only enabled to read the Woriu of the beft Authors witn Tafte and Propriety^ to obferve the Elegsincej Juftnefs, Force, and Delicacy of the Turns and Expreffions, and fttH more the Truth and Solidity of the Thoughts ; hereby will the Connec- tion, Difpofition, Force* and Gradation of the different Proo& of a Difcourfe be obvious and familiar to him, while at the fame Time be is led by Degrees to fpeak and write with Freedom and Elegance, whick will infalliably raife the Opinion of the young Merchant in the Eye of his Correfpondents, and of the Public. I. Beok-K«ep- A Merchant ought to Icnow upon allOccaffions what is in kts Power to "^& do without embarrafing himfelf, and have fnch an Idea of bis Dealings, and thofe with whom he deals, that his Speculations may be always with- in his Sphere, to effed which the Method of arranging and adjuftrng Merchants Tranfaftions is, like other Sciences, communicated in a rad« onal and demonftrative Manner, and not mechanicaHy by Rules depend- ing on the Memory alone. TTie Principles upon which the Science is founded is Ijkewife reduced to Praftice by proper Examples in foreign and domeftic Tran^dions, foch as Buying and Sellitrg, Importing* fix- porting, for proper Company, and ijommrflion. Account, Drawing and Remitting too, freighting and hiring Veffels for different Parts of the World, making In furances and Under-writing^ and the various otliet Articles that may be fuppofed to divcrfify the Btifinefs of the pra^cal Compting-Houfe. The Nature of all thofcTratifafiions, and the Man- ner of negociating ihem, are particularly explained as they occur, the Forms of Invoices and Bills of Ssfles, together with the Nature of all MERCANTILE ARTS. CLXIX intermediate Accounts^ which may be made ufe of to anfwer particular Purpofesy are laid open ; and the Form of all fuch Writs as may be fup- pofed to have been conneded with the Tranfadions in the Waftebook» are rendered fo familiar, that the young Merchant may be able to make them out at once without the Afliftance of Copies* XI. In order to accuftom the young Merchants to think, write^ and aS PnAical like Men, before they come upon the real Stage of Adion» dn cpiftolary ^^^^' Correfpondence is eftablifhed among them, in order to accuftom them to digeft well whatever they read, and improve their Stile under the Corredion of an accurate Mafter, to that clear, pointed, and concife Manner of Writing which ought, particularly, to diftinguiih a Merchant. Fi£titious Differences among Merchants are likewife fubmitted to their Judgement, fometimes to two by the Way of Arbitration, and again to a Jury, whilft one aflfumes the Charader of the Plaintiff, and another that of the Defendant, and each gives in fuch Memorials or Reprefenta- tions, according to the Nature of the Fads difcnffed, as he thinks mod proper to fupport the Caufe, the Patronage of which was afligned him. xit. Thus the Education of the young Meh:hant is conduced, that his Conclufion. Knowledge may be fo particular, and his Morals fo fecured, that he may be Proof againfl the Arts of the Deceitful, the Snares of the Difingenu^- ous, and the Temptations of the Wicked ; that he may in a ihort Time be fo expert, in every Part of the Bufinefs of the pradical Compting* Houfe, that when he comes to aft for himfelf, every Advantage in Trade will lie open to him, that his Knowledge, Skill, and Addrefs, may carry him through all Obilacles U> his Advancement, his Talents fupply the Macre of a large Capital, and when the beaten Track of Bufinefs be- <;omes lefs advantageous, by being io too m^ny Hand$> he maydrike out iitiew Paths for himfelf^ (tnd thus bring a Balance of Wealth, «ot only to himfelf, but to the Community with which be is conjie£led, by Branches of Trade unknown before. JPLANoftbe Naval Art, Including the Iriflru^ions relative to Ship* , Builders^ Sea-Officers, and4n general to alt tbo/e wbo are. any ^iy cqh^ cerned in the Bufinefs tf the Sta. t • . ^i duUis. aufut commit t ere fluff thus Alnum^ ^as Natura negat, prcebuit Arte Vias, CtAtJD. I. AS nothing is executed in the Militarv Way> but by the Direaion of Geometry and Mechanicks, no leCs indifpenfibfe is the Ufe of 4h€fe Sciences in Naval Operatiwis^ vii. Ship-building, flowing, work- CLXX PLAN OF THE ing» and conducing Veflels through the Sea. A Ship Is fo complicated a Machine, its various Parts haive fo clofe and fo hidden a Depandance on one another, and the Qualities it ought to be endued with, are fo many in Number, and fo difficuk to be reconciled, the Mechanifin of its Motions depends upon fo many Inftnunehts, which have an eflendal Relation to eacti other, E^r. that if is only by Experience, aided by the fublimeft Geometry, it has been difcovered, that all its Adions are fub- jeded to invariable Laws, and that we can atiain to certain Rules, which coutd enable the Mafter SUp-builders to give their Veflfelt the moft ad- vantageous Forms, relative to the Services for which they are deftined, and inftruS the Navigator how to draw from the Wind the greateft Force, to difpofe of it at Pleafure, and to traverfe the vafMl Seas without* Danger and without Fear. NotwthftanHding which, Mathematicks reduced by the Teachers of them in this Kingdom,, to a few grofs pradicJil Riules, t4etr Application to Sea Affairs, and to all other ufefol Enterprifes, has not is yet been introduced; this Neglea has not only retarded the Progrefs that the Study of the Ma- thematicks otherwife would have made, by hindering it from being knowji that they are the Means the moft proper to fupply the Limitation of our natural Faculties, and that it is from them that all ufeful Arts are to receive their Perfe£^n. But in the prefent Cafe, cannot but be attended with the moft fatal Confequences, and the Difafters that hap- pen but too often at Sea, ate undoubtedly, in a great M<^^re» owing to it. II. The conffruding and repairing of Veflels is entirely abimdoned to Naval ArchI the Direftion of Ship*Carpemers, whofe Knowledge is confined to a ttOoxt. f^^ groft obfcure Rules, which leave the Dtfpofition of almoft all tlie Work to Chance, or to the Cafprice oif Workmen; ihey rely in the moft important Circumftances, on the btindeft PraAtce, on that whidi is the moft liable to Error ; they change th^ upper Part of the Shipi they add a new Deck, or take one away, they alter totally the Form trf her Bottom, i^c. Making all thofe Changes, without knowing what EfftGts will enfiie, even thofe that would manifeft themfelvea in the Harbour, though they could determine them after the moft infiillible and precife Manner, in employing the leaft' Knowledge of Geometry, and the (impleft Operations of Arithmetick. It was therefore neeeflary that a Marine School Ihould be eftablifhed, where the Youth who are intended for the Bufinefs of the Sea, fliould be taught the Nature of Fluids, and the Mecanifm of floating Bodies, how to confider the Ship as a phyfical heterogeneous Body in all its dif- ferent Situations, and relative to its diflferent Ufts ; reprefeming it to themfelves not only when it is Ioaden» and at Anchor, but Alfo when it (ails, when it goes well, doubles a Cape, gets difficultly clear of a Coaft, NAVAL ART. CLXXI ffc. (o that Geometry tnd Mechanicka taking the Place that Chance and blind PraSice had ufurped* Mafter Shipbuilders may exercife their Em- ployments with Diicernment ; fubftituttng luminous and precife Rules in the Place of their imperfed pradical ones; they may be no more ex- pored to the Trouble and Shame of attempting any thing ralhly> but maT be enabled to aflign and forefee the Succefs of their Enterprifesi and producing no Plans but what are fupported by juftifiable Calculationsi in which each Q^lity the Ship ought to have, are difcufled and eftimat- * ed with Exadnefs ; we can feef in verifying their Calculations, what Streft can be laid upon their Promifes ; we may have infallible Means of deciding in Favour of the different Plans propofed for the fame Ship, and the Multitude of their Opinions, hr from bein^ hurtful, may on the contrary be profitable, fince it will often furniih an Occafion of making a better Choice. in. The Ship being built, it is the Bnfinefs of the Navigator to diftribute Mechanical the Loading in fuch a Manner, that flie may fail without Danger, and ^«^S^oa* at the fame Time receive with thegreateft Facility whatever Motions are to be given her, that is, he is to difcover her moft eligible Pofuion in the Water, he is to difpoft her Sails after a Aiitabte Manner to oblige the Veffel to take the Route he intends to follow upon all Occafions, and to make her go well in fpight of the Agitation of the Sea, and the Violence of the Wind, which often oppoles ; for this EffeS, in a Glance of an Eve, he muft be capable of rendering frefent to his Mind all the tnoveabfe Parts of the Ship, which he rtiuft look upon as a Body which he animates as he does his own, and that it is as it were an Extention of it ; feize the adual State of Things in their continual Change, and form the moft decifive Refolutions, which he muft draw from no other Fund but bis own Breaft. This is without doubt, the moft difficult Part of the Napvi^tor's Art^ but at the faitie Tiiiie, the moft important for him to poflefs, as it fumifhes him with die fiirtfft Refources in immer* gent Occafions, arid r^Ad^rs him fuperior in Battle^ It is Airpriiing with what Readiiliifs, the Ship well difpo&dV obe^, as it wcrtf^ the Orders of the ilalful Seaman; btfC on the contrary^ if he does not know all the Nttety of thl^ Part of hii* Art, his Ship, though, excdienty is no more than' a heavy Mafs^ whidh receites all its Motions ft6m the Caprice of "Winds and Weaves, which in ^ight of his Courage and.defperate Ef- forts^ becomes but too furcly a Prey to the Enemy, or ends very foon its Defttey by Shipwreck. Notwithftanding which, no Attempt had been made in this Kingdom to lefifen the Diffiailties 6( attaining to a Proficiency in this Branch of the Na- val Art^ by inftniding Sea-Officers in it after a methodical Manner. Il was entirely ab^doned to blind Pradice, as if it could not be fubjeded to txzGt CLXXII PLAN OF THE Rules in the Employment of the phyfical Means which it makes ufe of td move the Veffel. When a Maneuvre is executed in the Prefcnce of a young Sea-Officer, he docs not know very often for what it is done, or how the Inflruments that are made ufe of ad ; he is furrounded with PcrfoDS too bufy to give him the leaft Eclaircifement ; we may judge from thence how much Time he muft lofe to learn thefe grofs hfotions, which are to ferve him inftead of Theory : The iraperfed Knowledge which the young Sea-Officer will attain to> willbe(totheDifgrace of human Rea- fon,) the Fruit of many Years unwearied Labour ; and neverthelefs, as it will favour of its defedive Origin, it will not give him fufficient In- fight, and W4II leiave bim without exad Rules, whuch.hQ.can abfolutdy rely upon ; he will give, for Example,, a. certain Obliquity tq the Sails; he will receive the Wind with a determined Inctdenc]e> but will he know whether there is nothing to be changed in one^Senfe or the other, in one or the other Difpofition, his only Rule is fervily to copy what he has feen pradifed perhaps erroneoulW by otheis on like Occailons ; it was therefore necefiary that the Youth intended for the Sea, ibould be methodicaljv intruded in the ufi^ Maxims of the Doflrine of the moveable Forces, applied to the Bufincfs of the Sea, fo that rendering them familiar to themfelves in taking Share in all the Maneuvres they will fee executed, in order to apply them mechanically, without the painful Help of Refledion ; they might fee nothing for which they were not prepared beforehand, and of which they could give an Ex- plication to themfelves ^ and as they would not be obliged to execute any Maneuvre blindly-, they might be fenfible of the happy Effeds that a re- flected Exercife can produce, and the Qi^ality of a good PraGtitiooer would be lefs difficult to acquire. IV. The Art of The Navigator not only ought to know how to produce the diflfereot Piloting. Motions of his Ship, but he is to obferve all the Particularities oi its Route, efteem its daily Pofitiony and the Courfe he is to fteer» to arrive at the Harbour where he i» to go: This. is the only Branch of the Naval Art that is taught bv Rufe ; but it is.a general Complaint among Seamen, that very little of what is learned in Schools, is .of real Ufe ; which contributes very much to confirm them in. the dangerous Error, that Theory is of little or no Service ; this proceeds from the Genciality of Teachers having not fufficient Skill to copform their Plans of Teaching to the Exigencies of Seamen, in (hewing them how to modify their Rules of Navigation, according to the different Cafes of Sailing ; how to reduce to the fmalleft Compafs, the Errors to which the Meafures made uie of for determining the Courfe and Diftance, are liable to, and how to nuike proper Allowances for them, which would enable them, as often as the Reckoning would not agree with the Obfervation^ to judge on wkidi 1 NAVAL ART. CLXXIII Side lay ihe-Errory and confequently how to corred them; all which fuppoies in ihe Teacher a profound Knowledge of the Theory of the Art, and a perfed Knowledge of all the Circumftances of the Ship's Motion> in all Caies of Wind and Weather.' Their not being fufEciemly cxcrcifed in Aftronomy, and agronomical Obfervaiicns, make them negleQ inftruQing Sea-Officers how to chufe the mod favourable Circumdancts for obfeiving either by Night or Day, The only Obfervations praSifcd by Sea-Officers, arc the Sun's meridional Height, and its fctting ; they are entirely unacquainted with the Stars, though their Obfervations could be of great Ufe, particularly when the Snft docs not ferve, being obferveable at all Hours of the Night, and the Incertitude tp which the Reckoning is liable demands that the Sea* Officers ihould let no Occafion flip of taking Obfervations every Day ; moreover the moil reafonable Hopes of determining the Longitude at Sea, is founded on the Obfervation of the Diftance of the Moon from a Star, or from the Sun ; this Method gives aflually the Longitude to half a Degree, and has the Advantage of being as eafy put in Praftice as that foT'deCenAining the Latitude. If they had a little Skill in aftronomical Obfervations, they could determine the Pofitions of fo many Places, even of this Kingdom, which are placed in Charts after an uncertain £{limation ; but on. the contrary, they do not know even how to verify the Inftruments that are in ufe at Sea, particularly their Compafles and Quadrants; for want of fuch a Knowledge, they arc obliged tojaike tbem upon the bare Word of the Workman, who is interefted to get them off his Hands at any Rate ; and though they ought to be verified ev/ery .Voyage, on Account of the Accidents that might arife to them,, it is not done. This Particular, however minute, neverthelefs is worthy of Al^tention, 6nce nothing fliould be negleded in the prefent Cafe, fee- ing, in fpight of all the Care that can be taken, the Errors that are committed being but too fenfible, and as great ones may be occafioned <in the Reckoning by the Imperfedion of the Inftruments, as in Deduc- tions deduced from Calculation. We may conclude from thefe Confiderations, that the Ship-builders .and Navigators of this Kingdom were no way apprifed of the important Refources they could draw from-Geometry and Mechanicks, though in fK> Profeffion fo eminent as in theirs, and that they could never be fuffi- Eftablifti-. ciently (killed in their refpeftive Arts, until a Marine School was eftab- 55vinc liihed, conducted by a Perfon exercifed fufficiently in the fublime School.. Mathematicks, as to be able to underfland the different mathematical TraQs that have been publifhed in great Number of late Years, upon the different Branches of the Naval Art, fuch as Ship-building, Stowing, working Veffels at Sea, i^c. by the moft eminent Mathematicians of Eur^9 who (hould make it his Bufinefs to communicate to them after. '1 CLXXIV PLAN OF THE a methodical Manner^ all the Improvements their refpe6dye Art) htye received* and receive daily from Mathematicks. V. Dnnghtins* ^^ ^^ ^'^^^^ *° ^^^^ important Employment bv Drawing-mafiersy it the Ship-builders cannot finiih properly their Plans* without a Tinc- ture of this Art* and fome Proficiency in it* may enable the Navigator to take Views of Lands* draw fuch Coafts* and plan fuch Harbours, as the Ship (hould touch at* which will contribute verv much to render the Geography of our Globe more corred* and leffcn the Dangers of Ni^i- gation ; but what is perhaps of more Confequence* it will make them acquire the Habit of obfervmg Obje6b with Diftinanefs* and reccdteft exaaiy every Part of them, and recall all the Circumftances of di«r Appearances. In one Word, as the Science* which is entirclj occojW in weighing* meafuring and comparing Magnitudes* is neceffiuy iniH Stations and Occurrences of Life* fo the Art which teaches how torcpr^ fent them to the Eye is indifpenCble. AN EXTRACT* /ram tbiPlanpf tte ScBmI of Micbimc Arth wbiTi AnbifiSff PMinttrtf Sculpurs^ mul in general dil Artifis m ManufaBurers reteive the InfiruaUm in Geometry, PerfpeBt^t ^ tiekty Dynamieh, Pbyfickt^ &c. wbiebfuit tbeir reJpeBhe /Vs/(^i and may contribute to improve tbeir Tajte and tbeir Talents. * Rem quam ago, non opinionem fed opus effe^ eatnqu/s nan Seffat ^^^ aut placitif fed utilitaiis effe et amplitudinis immenfa fundaments, BacOH. 1. _ _ odi ^ . chMjic Aru J^ Hand of Man, it Ts capable of producing 1 In the me- TT O W E V E FL figorous, indefatigable* or fupj^ is the mW chttic Aru J^ Hand of Man, it is capable of prwllicing but a fmall Number « KdSSST" Effeds- He can perform great Matters but by the Help of Inftnmientt Theory aud and Rulcs, which are as Mufcles fuperadded to his Arms. The difero* ^^^•^^ Syftems of Inftruments and Rules confpiring to the fhme End, hitherto invented to imprefs certain Forms on the Prodtiftioas of Nature, e^ to fupply our Wants* our Pleafures* our Amufements> our CuHofity> &c. conftitute the mechanic Arts. Every Art has its Theory and Praftice ; its Theory k grounded on Geometry* Perfpcaive* Staticks, Dynamicks* virhofe Precepis cott«fi^ by thofe of Phyficks* as it procures the Knowledge <rf the Materiih their Qyalitics* Elafticity* Inflfejfibillty, Fridloif* the BtfeOi of ^ Air* Water* CoM, Heat* Aridity, arc. produce the Rules and Iflft«- ments of the Art. Praftlc^ is the habitual Ufe of Ihofe Inftramefltt and Rules. * Thit Plan being too exteofife is omitted for the pieicnc MECHANIC ARTS. CLXXV It is fcarce poi&ble to improve the Pradice without Theory, ftnd re* ctprocally to be Matter of the Theory without Pradice, aa there is in every Art a great Number of Circumftances rebtlive to the Materials, _ to the Iffi(lrumenl6» and lo the Operation which can be learned only by ledge of^the Ufe. It is rhe Bufinefs of PraSice to point out the Difficulties, and to Theory tb- fomifli the Phenomena. It is the Bufinefs of the Theory to explain the ^^^/^^ Phenomena, to remove Difficulties and to open the Road to further Im- eroy Anift. provement ; from whence it follows, that only fuch Artifts who have a competent Knowledge of the Theory^ can become eminent in their Pro« feffion. But unfortunately fuch is the Influence of Prejudice in this Country, that Artifts, Mechanicks, ice, are confidered as incapable of acquiring any Knowledge in the Principles of their refpe^ivc ProMTions, and our Youth deftined to receive a liberal Education, are laught to think it beneath (hem to give a confiant Application to Experiments and particu- lar fenfible Objeds, for to pradice or even to fludy the mechanic AriSy IS to ftoop to Things whofe Kefearch is laborious, the Meditation ignoble, ihe EjEfN>(kiop diificylr, the E^ercife diflionourable^ the Number end- left, god th^ Value incQ9&(kr^ble. Prejudice which has debafed an pfefHl #i}d .efHo^bic Clafs of iA^nxp and peopled our Towns with arro< •gftiit:H«ftff^1^ny ^klpf^ ConuwifUtors,. and the Country with idle and JhM^ty t^andlprd^. . Tjbe Judicious, fenfible of the Injufltceand of the fatal Confequences^ alteodiof; ibis Contempt for the mechanic Arts, the Induftry of the People and Eldabliflunent of Manufadures being the moft aiTured Riches of thia Comtry^ have come to the Refolution that the Juftice which is due.to tlMi A^ts end M^nufa&ures, ihall he rendered them ; that the me- cknaick Af n (hgjl be caifed from that State of Meanefs, which Prejudice has hitherto kept them ; that the Protedion of the Noblemen and Gen- tlemen of Fortune (hall fecure the Artifts and Mechanicks from that Indi- gence in which they languifh, who have thought themfelves contemp- tiUk becaufe they have l^en defpifed ; that they (hall be taught to have a better Opinion of themfelvea, as being the only Means of obtaining from them more perfed Produ3ions. A School of mechanic Arts is eftablifhed, where all the Phenomena of The Efta- the Arts are colleflked, to determine the Artifts to ftudy, teach the Men ^*^^Xf^ of Genius to think ufefiilly, and the Opulent to make a proper Ufe of mcchaaic their Authority and their Rewards. There the Artiffs receive the In- Am, ftrudions they ftand in need of, they are delivered from a Number of Prejudices, particularly that from which fcarce any are free, of imagin- . ing that their Art has acquired the laft Degrees of Perfedion ; their narrow Views expofing them often to attribute, to the Nature of Things, Defeds which arifc wholly from themfelves ; DifBcuUies appearing to CLXXVI PLAN OF THE, Stc them unfurmountable, when they are ignorant of the Means of removmg them. They are rendered capable of refleding and combining, and of difcovertng, in fliort, the only Means of excelling ; the Means of faving the Matter, and the Time, of aiding Indufiry, either by a new Machine, or by a more commodious Method of Working. There Experiments are niade, to advance whofe Succefs, every one contributes, the Ingenious dired, the Artift executes, and the Man of Fortune defrays the £x- pence of the Materials, Labour and Time. There Infpedors are appoint- ed who take Care that good Stuff is employed in our Manufadures, and that they are properly fupplied with Hands ; that each Operation em- ploys a diflferent Man, and that each Workman ihall do, during his Life, but one Thing only ; from whence it will refult, that each will be well and expeditiouily executed, and the bed Work will be alfo the cheapeft. Thus, in a (hort Time, our Arts and ManufaSures will be brooght to as great a Degree of Perfedion, as in any other Part of Europe. GENERAL CONCLUSION. Such isvthe Plan of the new Scene of ufefiil and agreeable Knowle^ calculated for all Stations in Life, which the Nobility and Gentrr of the Kingdom of Ireland^ purftiant to their Refolution of the 4th of Febmaij 1768, have opened to Youth, in the Drawing-School eftaUiflied under their immediate Infpedion. Encouraging Men of Geniuran^ Educaiicay from all Parts, to appear as Teachers, inviting the Artifts and Connoificars to devote their Attention to excite the Emulation of the Pupils by adjudg- ing and diftributing the Premiums granted to engage them to advmnoe more and more their Studies to the Point of Perfedion, and taking un- der their Patronage fuch young Citizens favoured by Nature more thas by Fortune, who difcover happy Difpofitions and fuperior Talents for the Service of their Country. Errata. Page LXin Line 15, for the Centrifugal Force diminilhes the Cen* -trifugal Force, read the Centrifugal Force diminilhes the Centripetal Force. ^ Page LXXI Line 14, for 1?2 read 1^ Page LXXXV Line 41, for this Expreflion 6g for (^l), 70 for (T>^, tread, this Expreffion, for (z) 70, for (hj 6q. The ELEMENTS of EUCLID. BOOK I. DEFINITIONS. L A P^ita, is that which has no parts, or which hath do magnitude. Fig. r. J[N this dffinitiwif as well as in tieficondamljl/tbp Evclid Jtmpfy explains tbi manner of conceiving the jirjl objeits of Geometry^ a Point, a Line, and a Super6cies ; be dors not demanftrate that there arefuch objeffs in the clafs of red beings. 7hefe notions, though very ufeful in geometry^ are only abflra^i^ ens which are not to be realifed, bj being reprefented as exifling independent of the mind, where they took their rife. There areno mathematical points in nature^ (at kafl what Euclid y^^i does not prove it) ; but there exift things which have axtenpoOf which may he treated asfimpk marks without magm'tudcfas often as they are confdered not as compofed of parts, but merely as the limits of fomfi other wuLgnitude. Thus, when it is required to meafure the diflance of two flars, tbe Aftronomer proceeds, as if thofe flars were indivifble points : and he is in the rsght\fince he does not propofe to determine their magnitude, but the diflance toatfeparates them, of which they are looked upon as the terms. Tbe fame is to te widerflood with refpe^ of the other notions of this kind. We reprefent under tbe form of a line, or ofaitngth without breadth, every magnitude wbofe length ddane is the obfe/l of our confederation, whatever may be its breadth, its depth, mr its other qualities. The imagination, always difp^fed to transform into re^ aslsties what has none, forms of thofe abflraSiions a clafs of beings which feem. to axifi independent of tbe mind. The Geometer has a right to adopt thofe beings, 4U they mayferve to render his f peculations on magnitude, con/ideredin different points of view, more intelligible \ but it is by no means allowed to him, to form *u/rong notions as to their origin and their real ufe, II. A Line is Length without breadth. Fi^. 2. The ELEMENTS Book I ] DEFINITIONS. m. H E Extremities of a Line, are points (A, Bj). Fig. 3. * IV. hJiraightLitu, Is that which lies evenly between its extreme points (Ai Pi)' f'i' 3- 7*4/1 drfinitiw/i is imferfeBy fince it prefents no ejjential cbarM^ln tf i fiiraight line; for wbicb reafon, Euclid could mate no u/e of it: it is m wfn fuoted in th My of the work. He is obliged to bave recour/e to otber primi" fles (for example 9 to tbe iitb axiom) as often as be bos occajion of emflaj'^ Srutbs, wbicb depend on a perfe^ definition of aflraigbt line, V. A Superficies, is that which hath only length and breadth. Fig. 4. VK The Extremities of a Superficies, are lines (AB, C D, A C, B D,). /?;• 4* vn. A Plare Superfces^ or fimplv a Plrn^, (AD) is that which lies ewci] **• tween its extremities (AB, CD, AC, BD,). Fig. 5. Tbis definition is liable to tbe fame exceptions as tbe four fB. Book I. Of EUCLID. DEFINITIONS. VIIL Jf\ Plane An^le^ is the inclination of two lines (AB, BC,) to one another^ which meet togetherj and v^hich are fituated in the fame plane. Fig. 6. IX. A Plane Re/f i It neal Angle, is the inclination of two ftraight lines to one ano* thcr. Fig. 6. N. B. When feveral angles are at me point B, err^ one of tbem is expr^Jfed by three letters,' of wbicb tbe letter tbat is at tbe vertex of tbe angle, tbat is at iJbe point in wbicb tbe ftraigbt lines tbat contain tbe angle meet one anotber, is put between tbe otber two letters, and one^ of thefe two is fomewbere upon one of tbofe ftraigbt lines, and tbe otber upon tbe otber line. X. When a ftraight line (AB) ftanding on another ftright line (CD) makes the adjacent angles (ABD, ABCj) equal to one another^ each of the angles is called a ri^bt angle ; and the ftraight line (AB) which fUnds on the other (CD) iscalied a/ey^^fitt&Wor. Fig. 7. XI. An Obtufe Angle, (ABC) is that which is greater than a right angle (EBC). Fig. 8. ^ XIL Aniffw/f-<f«ri>,(ABC)isthatwhichislefsthanarightanglc(EBC). Pig. g. XIIL A TVn* or Btmndary, is the extremity of any magnitude. The E L E M E N T S DEFINITIONS. X|V. J\ Figure, is that which is inclofed by one or more boundaries. Fig. lo. XV. A Circle f is a pknc figure contained by one line, which is called the circuM' ferrnrey and is fuch that all ftraight lines (CB, GD,) drawn from a certain point (C) within the figure to the circumference, are equal to one another, Fi^. 1 [. XVI. This point (C) is called the tetder of the circle, and the ftraight lines (CB, CD,) drawn mmi the center to the circumference, are called the/^^ Fig. II. xyn. «, A Diameter of a Circk, is a ftraieht lin^ (DB) drawn thro* the center, fod terminated both ways by the circumference. Fig. 12. XVffl. A Semicircle^ is the plane figure (DEB) contained by a diameter (BD) and the part of the circumference (DEB) cut off by the* diameter (DB). Fig. U* MX. A Segment ofn Circle, is a figure contained by a ftraight line (AF) called i Chords and the part of the circumference it cuts off (AGF, or AEF) called an Ar^, Fig. 12. D EF INITIO M S. XX. JtCEffilinealFigureSf are t^iofe which are contained by ftrgight lines. Pig. ij^ I4j i5> J6, 17. XXI. Trilateral Figures^ or trumgles, are thofe which are contabed hj three firaight lines, ^ig. 13, 1 6, 17. XXII. Quadrilateral Figunip are thoffc whifJi are contained by four ftraight lines. Fig. 14. xxm. Multilateral Figures, orpolygonsi are thole which are contained by nnore (han four ftraight lines. Fig. 15. XIV. As to three Tided figures in particular: ^ AnBquilateral Triangle, is that which has three equal fides. Fig. 16. XXV. An /fi/aks Triangk, is that whidi ha&only two fides equal Fig* 1 7* The ELEMENTS BooVL DEFINITIONS. XXVL j^\^ Scalene Triangle ^ is that which has three unequal fides. Fig* i8- XXVII. Likewife, amorg thofe fame trilateral figures : A Right angled Triangle ^ is that which has a right angle. Fig. 19. XXVIII. An Obtufe angled Triangle^ is that which has an obture angle, (A). Fig. ao. XXIX. An Acute angled Triangle, is that which has three acute angleSj (A, B, Ci)« Fig. 21. XXX. After the fame manner in the fpecies of four fided figures : A Square, is that which has all its fides equals and all its angles right angles. Fig. 22. XXXI. An Oblcffg, Is that which has all its angles right angles^ but has not all its ildes equal. Fig. 23. XXXII. • A Rbomhis, Is that which has all its fides equals but its angles are not fi^^ angles. Fig. 24. r^ Book I. Of EUCLID. Fg.2S Fig^ Fif.27 \ ■;: V, " \ DEFINITIONS. XXXIII. xiL Rhomboid, is that whkh has its appofite fides eqi^al to one ail€fther> bat aD its fides arc not equals nor its angles right angles. Fig. 25. xxxiv. AH other four fided figures befides thefe, are called Tropefiumi. F'g. 26. XXXV. Parallel Jiraigbt Linefy are fuch as are in the faroe plane^ and which being prodoced ever fo£ar both ways, do not meet. Fig, 27. // if for this reafon that every quadrilateral figure wbofe appojite ftdet are faralleU is called a ParallelogranL , Fig. 25. The ELEMENTS BookL 1 POSTULATES, I. Let it be gnmted, thttaftraight Une may be dnwn from any one poirt to any other point. n. That a terminated ftraight line may be proAiced to any lengdi m a ftnig^ line. .„ in. And that a circle may be defcribed from any center, at any diftance from that center. X M S; I O OR, COMMON NOTIONS. I. X W O magnitudes^ which are equal to the fame third, are equal to one another. If the lint A // tqutd to the line B» and tbe line C equal to the fame Um Bf the line kwll he equal to the line Q. Fig. i. n. If to equal magnitudes be added equal magnitudes, the wholes will be equal. If to the line AD be added tbe part DE, and to tbe line BF, which is equai to the line AD, be added the part FG, equal to tbe fart D£» tbe wholes AEp BG, will be equal to one another. Book I. Of EUCLID. Fig.3 Fig.4 F'g.5 A BCA BCA BC ' I ' " - ■' • ' T E F D E F D I AXIOMS. III. F equals be taken from equals, the remainders are equal. If /row tbe whole line AC, be taken the partBC, and from the whole lineDF^efual to AC, he taken fhe part EF, equal to BC', the remainders AB, DE, will he equal. Fig. 3. IV. If equals be added to unequals, the wholes are unequal. If to tbe line AB, be added the part BC, and to the line DE, lefs than AB, he added tbe part EF, equal to tbe part BC j the wholes AC, DF, ^vill be unequal. Fig. 4, V. If equals be taken from unequals, the remainders are unequal. If from tbe line AC, be taken tbe part BC, and from the line DF, Ufs than AC, he taken tbe part EF equal to BC ; the remainders AB, DE, are unequal. Fig. 5. VI. Magnitudes which are double, or equimultiples of the fame magnitude, are equal to one another. VII. Magnitudes which are halves, or equifubmultiples of the fame magnitude, are equal to one another. B jp; tl' p. n lO The ELEMENTS BookL Th AXIOMS. vin. E whole 18 greater than its part. Tbe whole line AC, // greater than its part BC. Fig 6. IX Magnitudes, which coincide with one another, are equal. This axiom it called the principle of congruency j the notion of congrueticjf includes tbe notion of terms, and the notion of the poflibility of their c(Hnci- dence. Two magnitudes coincide^ when their terms perfe^ly agree ; or when they may he contained within the fame bounds, Euclid regards tbe principle •/ congruency as a common notion: be is autbori fed from tbe univerfal praBice of determining be equality of magnitudes^ by applying one to the other^ os in the menfuration of magnitudes by thefooty cubit, pearch, &c. or by inclui-^ ing them within tbe fame bounds, as in tbe meafure of liquids, of grain, ond the like, by pints, gallons, pecks, bujbels. Sec. So that, we judge by the eytf or band, bow one agrees with the other, and accordingly determine their equality. It would be wrong tofuppofe, that fuch a principle could only conduH to a pra^ice purely mechanical, incompatible with geometrical preciffi^n. Euclid has found the means of converting this maxim, into a very JcientiJicJl principle. On congruency he lays down but a few obvious truths, from which be rigouroujly demonjlrates the more complex ones which depend on this prin- ciple. Thofe obvious truths are as follow* BookL Of E U C L I D. Ill AXIOMS. 1. xjLLL points coincide. 2. Straight lioes, which are equal to one another coincide ; and reciprocally, ftraight lines whofe extremities coincide are equal. 3. If in two equal angles (ABC, abc,) the vertexs (B & b) coincide, and one of the fides (BA) with one of the fides [ba] the other fide (BC) will coincide alfo with the other fide [be). Likewife, all angles whofe fides coincide are equal. Fig. 7. Euclid has not feparately enounced , tbofe particular axioms fubordinaie to tbt general one ; be nevertbelefs makes ufe of tbem^ as will eajily appear in am-' ly zing fever al of bis demonfratiens, X. All right angles are equal to one another. XL If a ftraight line (AB) cuts two other iTraight lines (CD, EF,) fituated in the fame plane, fo as to make the two interior angles (DGH, FHG,) on the fanie fide of it, taken together, lefs than two right angles ; thefe two lines (CDy EF,) continually produced, will at length meet upon the fide (K) on which are the angles which are lefs than two right angles. Fig. S. Tbis trutb is not ftmple enougb, to be placed among tbi axioms 'j it is a confe^ guence of tbe XX VII propofition of tbefrfl book j // // only tbcre^ tbat it can be properly ejiablijbed. Two ftraight lines cannot inclofe a fpace. If tbe two flraigbt lines EF and EXf* inelofe a [pace \ tbofe two lines (annot be botb firaigbt lines \ one of tbem at leaft as £XF mufl be a curve line. Fig. 9. B 2, '^ 12 The ELEMENTS Book I 1 EXPLICATION of the SIGNS. Perpendicular. Jg^ L X - - < - - - Greater than > - - - Lefe than + - - - More. - Lefs. V - -• - Angle. I ^TV^S D © O - - Right Angle. - - Triangle. - - Equal. - - Square. - - Circle. - - Circumference. ABREVIATION& Pile. - - - Parallel. Pgr. - ■ • - Parallelogram. Rgle.- - • - Redlangle. r Book! Of EUCLID. '3 jj PROPOSITIDN I. PROBLEM/. \J P O N a given finite ftniight line (AB) ; to conftru^ an equilateral tri- angle (ABC). Given Sought tbejlraigbt line AB. the conftruBien •Jan equiiateral A upQn the finite firaight line AB, Re/olution. 1. From the center A, at the diftance AB> defcribe © BCD. Pof, 3. 2. From the center B, at the diftance BA> defcribe © ACE. P^f, 3. 3. Mark the point of interfedlion C. 4. From the point A to the point C, draw the ftraight h'ne AC. Pof. x, 5. From the point B to the point Qy draw the ftraight line BC. Pof^ i. Demonstration. B E C A U S E the point A is the center of © BCD (Ref, i.), and the lines AB, AC, are drawn from the center A to the O BCD (Ref. 4.}, 1. Thofe two lines AB, AC, are rajs of the fiune ©. 2. Coniequendy, the line AC is =: to the line AB. Likewife, becaufe the point B is the center of ® ACE (Ref. a.), and .the lines BA, BC, are draws from the center B to the O ACE (R^f SX 3. Thofe two lines are rays of the fame circle ACE. 4. Confeqnendy, the line BC is alfo = to the fame line AB. 5. Therefore, AC, BC, are each of them = to AB (Arg, 2. and 4.). But if ttJiiO magnitudes are equal to a fame tbirdy tbej are ejuaf to one another. Ax, 6. The line AC is therefore =: to the line BC. But each of thofe tvro lines =: to one another (Arg. 6,)% is alio = to the line AB (Arg. §.). 7. Wherefore, the three lines AB, BC, AC, which fbnnthe three fides of A ABC, are =: to one another. 8. Confeqnendy, the A ABC conftru<led upon the given fimte ftraight lineABi is an equOateral triangle. Which was required to be dose. A 15, A' I. A 16. A I. />. is.-ff. I. 2).24.Af. 14 The E L E M E N T S 1 BookL i I N M /A ^ . G . ! / ;f PROBLEM II. jp PROPOSITION II. jt; RO M a given point (A), to draw a (Iraight line (AL), equal to a giTen Itiaightlire (BC). Given Sought I. Tk point A. AL = BC. f . The^rafght lint BC. Refolution. 1 . From the point A to the point B, draw the (Vraight h'ne AB. Pof, 2. Upon this ttraight line AB conftrufi the equilateral A ADB. P. i,B.\, 3. Produce indefinitely the fides DA and DB of this A. Ptf. 4. From the center B, at the diftance BC, defcribe © CGM. Pif, 5. And from the center D, at the diftance DOy defcribe © GLN j Pof, which cuts the (Iraight h'ne DA produced, fomewhere in L. B Demonstration. E C A U S E the lines BC and BG, are drawn from the center B to the O LGM (Ref. 4.). 1 . Thofe two h'nes are rays of the fame ® CGM. D, 2. Confequendy, BC = BG. Z>. And becaufe the h'nes DG and DL, are drawn from the center D to the O GLN (Ref. 5.). 3. Thofe lines, are alfo rays of the fame © GLN. D. I. J?. 2. 3. J- 16.5. 15. A being the fides of an equilateral 16. A 4. Confequently, DG = DL. But the lines DA & DB, A ADB (Ref. 2.). S- The line DA, is = to the line DB. 2). 24. B. 1 Cutting off therefore from the equal lines DG, DL, (^fg. 4.) ; their equal parts DB, DA, (y^rg. 5.). 6. The remainder AL is = to the remainder BG. Ax. 3. Since therefore the h'ne AL is = to the line BG (Arg, 6.), and the line BC is alfo = to the fame h'ne BG (Arg. 2.). 7. The line AL is = to the line BC. Ax. i. Put it is manifeft tliat this line AL, is a line drawn from the given point A (Ref 3.). — 8. Wherefore from the given point A, a ftraight h'ne AL, equal to the given ftraight line BC, has been drawn. Which was to be done. Book I. Of EUCLID. 15 A ■v. \ \ C ': * T PROPOSITION III. PROBLEM III. W O unequal ftraight lines (A & CD) being given ; to cut off from the greater (CD) a part (CB) equal to the lefs A. Given Sought ihe line CD > line A. from CD to cut off CB = A. Refolution. 1. From the point C draw the ftraight line CE =: to the given one A. P. 2. B. i. 2. From the center C and at the diftance CE, defcribe © CEB \ Pof. \, which cuts the greater CD in B. Demonstration. T HE ftraight lines CB, CE, being drawn from the center C to dieOBEF (Ref, 2.). I . They are rays of the fame © BEF. Z>. 16. B. i. a. ConfequenUy, CB=CE. D. 15. A i. But the ftraight line A being = to the ftraight line CE (Ref, 1.) • and the ftraight line CB being likevvrife = to CE (Arg, 2.). 3. The ftraight line A is = to the ftraight line CB. Ax. I. And fince CB is a part of CD. 4. From CD the greater of two ftraight lines, a part CB has been cut off = to A the lefs. Which was to be done. n i6 The E L E M E N T S Bookl. I PROPOSITION IV. THEOREM I _ F two triangles (BAC, EDF,), have two fides of the one, equal to tvo fides 6f the other, (i. e. AB = DE, & AC — DP), & have likcwife the angle contained fa) equal to the angle contained (dj : they will alfo have the bafe(BC), equaKto the bafe (EF) ; & the two other angles (i & /J equal to the two other angles ^^ &/^ each to each, viz. thofe to which the equal fides are oppofite ; and the whole triangle (BAC) will be equal to the whole triangle (tDF), Hvpotbeiis. Thefis. /. AB = DE. /. BC = EF. //. AC = DF. //. V^=V^&Vr=:V/ ///. Vtf = VJ. ///. A BAC = AEDF. Preparation. Suppofe the ABAC to be laid upon the A £DF» in fiich a man- ner that I . The point A falls upon the point D. a. And the fide AB falls upon the fide DE. SDemonstratiok. INCE the line AB is =: to the line DE (Hjp. i.), & the point A falls upon the point D (Prep, i.), & the Une AB upon the KncDE (Prep. 2,). I. The point B will fall necefiaril/ upon the point E. Ax, 9. Becaufe the "iaz=. \/d (Hyp. 3.), & the point A falls upon the point D (Prep, i.), & the fide AB upon the fide DE (^Pr^^. a.). a. The fide AC will fall neccflarily upon the fide DF. Ax, 9. Moreover, fince this fide AC ts = to the fide DF. 3. The point C mud fall alfo upon the point F. Ax. 9. .4. Wherefore, the extremities B and C of the bafe BC, coincide with the extremities E and F of the bafe EF. S. And confequendy, the whole bafe BC coincides with the whole bale EF| for if the bafe BC did not coincide with the bafe EF, though the extremities B and C of the bafe BC, coindde with the extremities £ and F of the bafe EF ; two ftraight lines would indoie a (pace (EXF or EYF) j which is impolEble. Ax, la. Since thereforei the bafe BC coincides widi the bafe EF (Arg, 5.). n' fiookl. Of EUCLID. »7 EBS&aSBBBBaiHBPHHIHiMHBBBBaBBBBBBaiV ThisbaieBCwiirbe = totIiebaftBF. i£r. 9. Wfcidivas.tQbf dtoonftrMcd I Amhh. tlk btfe BC coiackUi^with th< bafc^ CF (Ti^^r^. J.), & tbe ova 9tbpr iicks AB* AC» of A oAC cotqckiiDS vitb the the tyrQ odior fides DB, DF, of A BDP f /V<^. :^ A^, 2.1. Thofe two A BAC» EDP, «re iieoeflfLrit7 eqad loeMMetl^. ^;r. 9. WhichwaitobedemonftNrttd Bft In fii«, fincctte W & V' t« ¥^ich the equal ikl^» AC,. DFaw o»" poAte r^. ^); U^ewift, the- Vc & / t» wKjiel^ tlie e^ idM AB, DEy are oppofite ^/(p/. 1.), eoiookli feoth as to their vertices and their fides ^yfr^. 1, z, 5*. 5.X bMow^ that the V^&V^ asalfo the Vc & V/,. IQ^ wUcis tie, equai ft^aaaoppaite^ axe equal t* qm tsotles. 4x 9. WUrkwi^tp be deinonftrated. II. i8 The E L E M E N T S Bookl !IB!9ai I PROPOSITION V. THEO REM IL ^ N every ifofcclcs triangle (BAC) : the angles (a & ij at the bafe (BC) are equal, & if the equal fides (AB, AC,) be produced: the angles fc+eSc d +/J under the bafe (BC) will be alfo equal. Hypothefis. Thefis. /. The A BAC is an i/ofcfles A. /. Vtf & V^ are e^aL U. ABBcAC are produced indefniufy, II. >^c+eSc "id-i^farealja ejUMl Preparation. I. In the fide AB produced take any point D. z. Make AE = AD. P. 3- A »• 3. Through the points B & £, as alio C & D, diav BE, CD. Pof. 1. Demonstration. B. __ » E C A U S E in the A DAC the two fides AD, AC, are equal to the two fides AE, AB of A EAB, each to each (^Pr^/. 2. Hyp. i.) j and the V A contained by thole equa^ fides is common to the two A. 1. The bafe DC is = to the bafe BE; &thc tworcmaining'V«i& ^+^ of A DAC, are equal to the two remainine V« & « -f- ^ *^^ ^ EAB, each to each of thofc to which the equal Udes arc oppofitc. P. 4. A '• And becaufe the whole line AD is =r to the whole line AE (Prep. 2.), and the part AB = <b the part AC (Hyp. i.) $ cutting off &c. 2. The remainder BD will be == to the remainder CE. Ax. 3. Again, fince in the ADBC the fides DB, DC, are equal to the fides CE, EB, of A ECB, each to each (Arg. 2. and i.), & likewife V contained m is equal to V contained n (Arg. i.). 3. The two remaining V of the one, arc ^ to the two remaining V of the other, each to each, ^iz. Vc + < = Wd+f & V^= Vc P. 4. A i. The whole V« + f&^+^being therefore = to one another, as alfo their parts Vf & V^/ ^-^r^. i. &3.); cutting ©^ &c. 4. The remaining Vtf & ^ are likewife =: to one another. Ax. 3. But thofc V are the two V at the bafe BC. 5. Therefore Vtf & V^ at the bafe BC are = to one another. Which was to be demonftrated. I. Moreover, fince Ve + f =r Vd+f(Arfr. 3 . ) are the V under the bafe. 6. It is evident that the V^ + ^ ^ V^+ /uncier t»^e bafe, a^e alfo =r to one another. Which was to be demonftraicd. II. Book I. Of E U C L I D. »9 I PROPOSITION VI. THEOREM III F a triangle (ACB) has two angles fadch+cj equal to one another; the fides which are oppofite to thofc equal angles, will be alfo equal to one another. • , , Hypothefia. Thefis. In th€ A ACB, Va = V* + c. Tbejiik CA c= t9 tbejidf BA, Demonstration. If not, 1. The fides CA, BA, will be neceiTarily unequal, C. iV. 2. Confequently one of them, as BA, will be > the other CA. C. N. Preparation. I. Cut off therefore from the > fide BA, a part := to the < fide CA. P. 3.5. i. I 1 . KMi oir increrore rrom tne ^ nae c/\, a part := to tne <;. noe vJA. r. •^, B 2. Draw from the point C to the point D, the ftraight line CD. Pof. i. _ Nihe A ACB, DBC, the fide BD = to the fide CA (Prep, i.), the fide BC is common to the two A, & V contained a = W con« tained ^ + f (IfyP- ')• I. Confequently, the two A ACB, DBC, have two fides of the one equal to two fides of the other, each to each, & V ontained ii = V con- tained h'\- c. 7.. Wherefore the A ACB is = to A DBC. P. 4. ^. 1. . But the A ACB being the whole, & the A DBC its part. 3. It follows, that the whole would be ^ to its part. 4. Which is impoflible. ^ Ax. 8. Therefore as the fides CA, BA, which are oppofite to the equal \f ah h •\- Cy cannot be unequal. 5. Thofe fides are equal to one another, or CA = BA. C M Which was to be demonftrated. C 2 The ELEMENTS SodkL PROPOSITION Vn. THEOREM If^. ^ R O M ihe eflttremkics (A * B) of s ftimight line (AB), ftwft vkidi hivt been drawn to the fame point (C), two ftn^ght Hues (AC, BC,) : tl«^ cannot be drawn to any other point (D) (ittiated on the fame fide of tfuiHn^ two other Au^ lines (AD, BD,)^ equal to the two firft each to each. Hypothcfe. Thefo. 1 . AC, BC, «(/• AD, BD, t^nftraifbt lines i * iV imp^gAU tbmi AC = Aft 2. Dra^n f nmtbt fame points K&^ I E?BC=;BD. 3. Tfl /w(> dfftrent points D W C, fituattd otf th/a^tJiJe ^ tbi Hnf AB. Demox^stratiok. If not, Tkeie U (Mithe&Lmeiidc ofiSiefine ABapornt D fe fitir- at^, tfcat ACrp AD, %BC ;=:Ea>. Oifeqnendy tto point will be placed, Casb I. EiAcrinAefidcAC, fn'&C. tig, 1. C A s E 3. Or wkhm the A ACB. fig, j. Ca s E 3. Or laftly without the A ACB. Fig. 3. CASE I. Iftkepoiat D ht fappoftd to te in one tf the iitfts, ti in AC. fig. I. Ol^CAUSE thepemxt D is fhjppored tt> be a point in ACififfeieflt from ihe pomt C. 1. The Hnc AD 13 either > or*< the line AC. \ 2, Conftquemly it it impoflible that AD lir AC. 3 Which was t6 be demoaftrnted C A fi E n If the point D be feppofcd to be fituated within the A ACB. Fig, a. CAT. Preparation, I. FramthepointDtothepoint C« draw the ftraight line DC P*/^ r *. Produce at will BD to E & BC to F. Ptf, 2, SoDkl. Of E UC LID. 21 B ECAUSE AC is fiippofed = AD. •1. The A CAD Mrili l>e an ifofedes A. - D,2$.S,t. )si. Coni^nendytfae Vatthel>afe« + ^&^w>llbe^qiMltooii^aAother. P. 5. 1^. 1. And becaufe BC is ffippored =: BD. 3. The A CBD will be Ukewire an ifofeeles A. ' D. 2$. B, t . 4. HencetheYimderdiebafe^9^r-|-f/,wiUbeairoeqiialtoon^anoth|r. P. 5. ^. I. Wherefore, if from Vr + i be taken its pait V^. $. V*Waibe> Vc. C M And 7 to the lame V^ he aTtervmtds added Vtf . 7. ConfequentljT V« + ^ & Vc are not equal. C N. But it hasbeeademonlbaved that m c«niequeiice of the liippoitiioli ^ thisoafe, V«4- ^ & Vf flionld be equal (Ai^g. %X 8. t^rom whence it follows that this iuppoCtioB cannot ikbfift, linlefa thofe angles at the faiti^ time he equal «nd unequal. 5. WWdi fc impoffitte. t, S. 10. Therefore the fuppofiiion which majces AC 1^ Ab tlr BC =f BB, is in itfelf ipipoflbk:. Which was to be demonMated. C AS£ IIL If the point t) be fuppofed to be lyidiopt the A AGS. iF%-. 3, Preparation. B Prom the point D to the point C let tfaerjp he dotwa the ftnuffht EneDC. * ^ Pa/ ECAUSE AC isfuppoMttAD. TheACADwmbemailbicelesA. t>.%%,B.X 2. Confequent||r V^ & ^/-f r«t the iwdfe axe equal t««npei|aiMtec il |. ^, t Again, betaufe 6C is likewife fuMnftd «z fiXl i 3. TheACSDwiUheMiMcefesA. «.t(.B. i 4. Hence Yc * V^ + ^ atthehde wiUbecqaal loooeflftKHhec. P. 5. i?. i " If therefore we take £roia Vi +« its part V«, 5. The Vr will be > the remaining Vi. C. M And if 4» this fiuye y<r hendded V^ 6- Muchmofe then will the whole Vf + ^ be > Vi. C ^. 7. Wherefore V^ + i & V^ «re not equal to one aaaiher. C. AT. But it hasbeenpi^ved ihatin conieq|ttenee af tfae f«ppafiticl||tif this cafe, "id-^ c ft V^ nre «f|md to Dae another. /^w<y^. a.y. . 8. Froai whence it follows that this fuppo(ition<ailoiot fnbfiftiUtiMs tboie angles be at the fame time equal and iinequaL 9. Whichisi^poiSMe. C. AT. 10. Therefore, tiie fwp^iiricii idikhjaakoa AC si AD Ir fiC =:: BD is inipofllUe. Whicli was to he deifiMiftratxrd. 22 The E L E M E N T S J PROPOSITION VIII. THEOREMS. 1 F two triargles (FHG, ACB,), have the three fides (FH, HG, GF,) of the one equal to the three fides (AC, CB, BA,) of the other, each to each, they are equal to one another, & the angles contained by the equal fides are likewife equal, each to each. Hypothefis. Thefis. /. FH=:AC. C^^FirVA. //. HG = CB. A FHG =: A ACB, anJi VG= VB. ///. GF=BA. [VH=VC. Preparation. Let the A FHG he applied to the A ACB, fo that, I. The point F may coincide with the point A. a. And the bafe FG with the bafe AB. Demonstration. B, BECAUSE the point F coincides with the point A (^Prf^. i.),&the lineFG with thcline AB/^/*r^/. 2.), & thofe Unes are tqusA fffjfi.^.). 1 . The point G muft coincide with the point B. >^. 9- The extreme points F & G of the fide FG, coinciding therefore with the extreme points A & B of the fide AB C^rep, i. Arg. i.) . & the ftraight lines FH, GH, being equal to the ftraight line AC, BC, each to each. 2. The ftraight lines FH, GH, will neceifarily coincide with the ftraight Unes AC, BC, each with each. If not ; then from the extremities A & B of a line AB, there may be drawn to two different points C & D, on the fame fide of AB, two ftraight lines AC, BC, equal to two other ftraight lines AD, ED, each to each. Which is impofllble. P. 7- '• 3. Thofe fides therefore coincide. 4. But the bafe FG coinciding with the hafe AB (Prep. 2.), the fide FH with the fide AC, & the Me GH with the fide BC, Cj4rg. 2.). 5. It follows, that the A ACB, FGH, are equal to one another ; as like- wife their V contained by the equal fides, each to each. Ax, 9- Which was to be demonftrated. Book I. Of EUCLID. 23 T. PROPOSITION IX. PROBLEM. IV, O divide a given reQilineal angle (ECF), into two equal angles (ECD, FCD,). Given Sought A rtaiUneal V ECF. V ECD =r V FCD. Refolution, I. Take CA of any lei^th. a.MakeCB=:CA. P. %. B, i. 3. From the point A to the point B, draw the ftraight line AB. Pof. i. 4. Upon the ftraight line AB, conftru6t the equilateral A ADB. P. \. B. i. 5. From the point C to the point D, draw the ftraightline CD. Pof, i. Demonstration. B. • ECAUSE AC = BC (Ref. a.), DA= DB (Ref 4), and the ikie DC common to the two A CAD, CBD. I . Thofe two A have the three fides of the one equal to the three (ides of the other, each to each. 2,, Confequently the V FCD, ECD, contained by the equal ^dt% CA, CDi &CB, CD, are equal to one another. i>, 8. 5. i. Which was to be done. 24 The ELEMENTS Book I. 1 1> A^ A B T PROPOSITION X. PROBLEM f^. O divide a given fioite ftnight line (AB) into twQ •qfoi puf CAC,-Be,). Gtve» Squ^t Afiwttir*i^p /f M AB. AC :s BC RtJbbitioK. I. UpontheftrAightUneABconfbuathee^vilMevsIA AM. /*. i.'-'* a. Divide mto twoeqval parts V ADB by the ftrai^ IkM DC. /. 9^ '• '* B. DU MOJrS T t A T!ON. ___fECAUSE AD=rBD (Ref. i.)> & the fide QC is commoato the two A ADC, BDC, & V contained ADC s V conttined BDC (Ref, 2.). J. Thofe twa^ ADC, SDC, htvotwo fides ii» the opo tqwd » fm% fides in the other, eadi to each, & V contained ADC =s V cmt^ uined BDC (R»f. a.). z. Confequently, the bafe AC = to the bafe BC. 1^ 4, K i. Whid^wastobedovr. BookL Of E U C L I D. ^5 ^ PROPOSITION XI. PROBLEM VI. Jt/ R O M a given point (C), in an indefinite ftraight line (AB)) to raifea perpendicular (CF) to this line. Given Sought 7*^ indefinite ftraight line AB, ^ Vbe ftraight line CF raifei from the point C in this ftraight line. the point C -L upon AJ^. Re/oluiion. T. On both fides of the point C take CD, CE, equal to one ano- ther. P, 3. B. I. 2. Upon the ftraight line DE, conftrua the equilateral A DFE. P. 1. B. 1. 3. From the point P to the point C» draw the (Itaight line FC. Pof. i. Demonstration. JDECAUSE CDis = toCE (Ref. 1.), FD = FE (Ref. a.), & the fide CF is common to the two A DFC, EFC. I . It is evident that thofe two A have the three fides of the one, equal to the three fides of the other, each to each. jSt. Confequently, the adjacent V FCD, FCE, (contained by the equal fides FC, CD, and FC, CE,) are equal to one another. P. 8. B, i. But it is the ftraight line FC, which falling upon AB, forms thofe adjacent V = to one another. 3, Wherefore, the ftraight Hne FC is ± upon AB. D. 10. B. i. Which was to be done. 2.6 The ELEMENTS x)kl I ./ P B F( PR OPOSITION XIL PRO BLEM VII R O M a given point (C), without a given indefinite ftraight Kne (AB) \ to let fall a perpendicular (CF) to this line. Given Sought tbe indtfiniu firMgbt tine AB, U 7bfjiraigbt HneQf^ let fall frm tbt point C tuitbout tbis line, tbe point C X upon AB. RefoJution. I. Take any point G, UfKHi the other fide ofthe ftraight line Afit with reiped to the point C. From the center C» at the dsftance CG, deicribe an arc of DGE cuittng the indefinite lane Ai in two points D & E. H- S- Divide the h'ne D£ into two eqaal parts in the point F. P. lo. B. i' From the point C to the point F» draw the ftraight line CF. PoJ, i. Preparation, From the point C to the points D & E» draw the ftrai^ 2. 3- 4- lines CD & C£. Demonstration. FoJ.i. C to X5 EC A USE the UnesCD, CE, are drawn from tfte center the O DGE (Ref, a. andPrr/.). 1 . Thoie lines are ra7S of tbe fame 0. 2. Confequendy, the line CD is = to the L'oe CE. Since therefore CD is z= to CE (Arg. a.), DF=FE (Ref. 3.). the fide CF is common to the two A DCF, ECF. 3. Thofe two A have the three fides of the one equal to the three fides of the other» each to each. 4. Wherefore the. V CFD, CFE, contained hjr the equat fides FC, FD, and FC, FE, are zzrto one another. But diofe two V CFD» CFE, == to one another (Arg. 4.), are the adjacent angles formed bj the line CF which falls upon the line AB. 5. Therefore, each of thofe two V CFD, CFE, is a L. , and the line CF is ± upon the line AB. D. to. B. 1. Which was to be demonftrated. P.%. B. I. PookT, Of EUCLID. 27 i 1 / /' A « !X -B 6_ T PROPOSITION XIII. THEOREM Fl H E angles which one ftraight line EC makes with another AB upon pre fide of it, arc either twp righf, angles, pr are together equal to twp right angles, Hypothefis. Thcfis. EC is a ftraight line meeting I Either each 0/ V ACE, ECB, is a L. AB in the point C. //. Or their fum is = to two L. SUP. I. If V ACE 18 = tQ V ECB. B Demonstration. E C A US E the adjacent angles ACE, ECB, formed by the ftraight lines CE& AB, are equal to one another (Su^,), 1. It follows, that each of them Is a L». Which was to be demonftrated. SUP. II. If V ACE is not = to V ECB, Preparation. From the point of cpncurie C, raife upon AB |he X CD. Demonstration, X>ECAUSE DCiaXuponAB (Prep,). t Thetwo VDCA&DCBareU. But as V DCB is = to the two V « + « j if the V DCA or V f^s be added to each. 2. The two V DCA + DCB, are =r to the three V « + « +•• ^x. Again, becaufe V ECA is = to the twp V|i» + « | if the V ECB or Vo be added to each. 3 The two V ECA, ECB, are alfo= tp thofc fame three V « + « + p. Ax, 2. 4 Confcquently , the two V ECA & ECB are = to the two V DCA & DCB. Ax, i . ' But the two V DCA & DCB, being two L (Arg, i.). 5. It is evident that the fum of the two V ECA ^ ECB, isalfo=:to ' twoL. Ax. I, Which was to be demonftrated. Da D. ip. -8. I. iP. ii.-ff. I J>. 10. B, I. 2. 28 The ELEMENTS Book I. A E X ■ c _J I PROPOSITION XIV. THEOREM Vn. F two ftraight lines (AC, BC,), meet at the oppofite fides of aftra'# line (EC), in a point C, making with this ftraight line (EC) the fum of the .two adjacent angles (ACE, ECB,) equal to two right angles ; thofc two ftraight lines (AC, BC, will be in one and the fame ftraight line. Hypothefis, Thcfis. I The two lines AC, BC, meet in the point C. The lines AC, BC, are in one V W II. The adjacent V ACE 4- £CB are = t$ Jamejlraigbt line AB. tnjooi^. DfiMONSTRATlON. Irnot, AC may be produced firom C to D, (b that DC & AC fnaj make hut one aixl the fame ftraight line ACD. Preparation, Produce then AC from C to D. Fof.i. B Poll. 4- 5. 6. EC A USE ACD is a ftraight line upon which falls theUneEC. It follows, that thefumof thcadjacent V ACE+ECDis = totwoL. -P. «3-*^' But the V ACE + ECB being ajfo = to two L (Hyp. a). The two V ACE + ECB arc therefore 3i to the two V ACE+ ECD. Ax, 1. Taking away therefore from each the common V ACE. The remaimng V ECB, ECD, will be equal to one another, Ax, 3. But V ECB being the whole & V ECD its part. It follows, that the whole is equal to hs part. Ax. i. Which is impoflible. Ax. 8. Coniequently, the lines AC & BC, are in one & the fame ftraight hfie« Which was to be demonftrated. I PROPOSITION XV. THEOREM Fill F two ftraight lines (AB, DE,) cut one another in (C), the vertical or oppofite angles (ECA, DCB, & ACD, BCE,) are equal. Hxpothefis. Thefis. AB, DE, art ftraight lines nnhicb I. V ECA =r V DCB, cut one another in the f9int C, //. V ACJ? = V BCE,' B. Demonstration. P. 13. A I, P. 13. A I. ^X, I. BECAUSE the ftraight line ACIalls upon the ftraight line DE^/(y/.^. I . The fum of the two adjacent V ECA + ACD is =r to two L.. Again, iince the ftraight line DC Ms upon the ftra^ht ifaie AB (HypJ, z. The fum of the adjacent V ACD -f DCB is alfo = to two L, 3. Confequently, the V ECA + ACD are = to V ACD + DCB. Taking away therefore from thofe equal fums (Arg, 3.) the com- mon V ACD. 4. The remainmg V ECA, DCB, which are vertically oppofite, are equal/ ^jf. 3 Which was to be demonftrated. I, In the iame manner it will be proved : 5. That V ACD is = to V BCE, which is vertically oppofite to it. Which was to be demonftrated. U. C O RO LLA RT 1. J^R O M this it is manifefty that if two Jiraight linescut we another ^ the mgles they make at the point where they cut, are together equal to four rtgbt angles. COROLLART II. j^ ND confequently^ that all the angles made by any number of lines meeting in one point, are together efual to four right angles^ so The E L E M E N T S Bookl 9i ^E C [ N A B ' J PROPOSITION XVI. THEOREM IX, J[f one fide as(AB) of a triangle (ACB) be produced, the exterior angle (CBF) is greater than cither of the interior oppofite angles (ACB, CAB,). Hypotheiis. Thefis. /. ACB is m A. The txterior VCBF > iht inte II CriFw an exterior^ IS formedby tbt rior ^fpofitt V ACB wCAB. fidt AB produced. III. V ACB li CAB are the interior oppofite ones. Preparation. 1. Divide CB into two equal parts at the point D. (Tig, i,) P. lo.A !• 2. From the point A to the point Dy draw the line AD, U pro- duce it indefinitely to £. P«/. i* 3. MakeDE = DA. P.?.«» 4. From the point B to the point E, draw the ftraight line BE. Po/. 1. T Demonstration. HE ftraight lines AE, BC, (Fig. i.) intcrfea each other at the point D. (Prtp.%,). 1 . Confequently, the oppofite vertical V CDA, BDE, are = to one another. P. i S- '• *' Wherefore fince in the A ACD, DEB, the fide CD iszr to the fide DB (Prep. I.), AD = DE (Frep, j.), & V contained CDA is = to V contained BDE (Jlrg. I.). 2. It foUows, that the remaining V of the one are equal to the remaining Vof the other, each to each ofthofe to which the equal fidesare oppofite. P. 4. ^- ^ But the V ACD, DBE, are oppofite to die equal fides AD,DE, (Frep. 3.). 3. Therefore V ACD is = to V DBE. But V CBF being the whole, & V DBE its part. 4. It foUows, that V CBF > V DBE. Ax. 8, 5. Wherefore the exterior V CBF is alfo > the interior V ACB. C ^. In the fame manner, dividing the fide AB into two equal parts in the point D (Fig. a.) it will be proved. 6. That the exterior V AB/ is> the interior V CAB. But this V AB/ is venically oppofite to V CBF. 7. Wherefore V AB/ = V CBF. P. 1 5- * »• 8. Confequently, the exterior V CBF is > the interior V CAB. C V. Which was to be demonftr^ted. BookL ya PROPOSITION XVII. THEOREM X. XTlNY two angles as (ABC, ACE,) of a triangle (BAG), are lefs than two right angles. Hjpothefts. Thefis. ABC is m A. the V ABC + ACB art < /wa L. Preparation, Produce the fide BC (upon which the two V ABC, ACB, are placed) toD. Po/ a. B Demonstration. _ BECAUSE V ACD 18 an exterior V of the ABAC I . It 18 > Its interior oppoitte one ABC. Since therefore V ACD is > V ABC ; if the V ACB be addedtocach. P. i6. 5. f: a. The V ACD + ACB will be > the V ABC + ACB. Ax. 4. But the V ACD + ACB are the adjacent V, formed by the ftraight Kne AC, which fails upon BD (Prep J, 3. Confcquently, thofe V ACD -j- ACB are = to two L. P. But the V ACD + ACB being = to two L (Arg. 3.) & thofe lame V being > the V ABC + ACB (Arg, 2.). 4. It foUowi, that the Y ABC + ACB are < two U. C Which was to be demonftrated. 13. A I. M. The E L E M E N T S I PROPOSITION XVIII. THEOREM. XL N every triangle (ACB); the greater fide isoppofite to the greater angle. Hypothefis. . Thcfis. ACB 1/ tf A, 'wbofeJiJe AB is > AC V ACB, o^^/u to >fidiM, isff^f^ than V ABC oppofiu to tbelejfirjdi AU Preparation, Bccaufc tbe fide AB la > AC (ffypj. 1. Make AD = AC. 2. From the point C to the point D, draw the ftraight line CD. Demonstration. P. 3. A'. Pof. I B D.ii-B.i ECAUSE theiJdeADis=:totliefideACf/VfA I.A 1. The A ACD is an ifofcelcj A. *..-, 2. Confequently, the V iw & « at the bafc CD are = to one another. P. J. ^' *• But V iw being an eirterior V of A DCB. 3. It follows, that it is > the interior oppofite V D3C. P. >6.^. '• But V « is = to V « fylrg. 2.) 4. Therefore V ft is alfo > V DBC. C M Andif toVnbcaddedV/. 5. Much more will V« +/ or V ACB, oppofite to the greater fide AB, be > V DBC, or ABC, oppofite to the lefier fide AC. C iV] Which was to bedeinonftrated. J Of EUCLID. I PROPOSITION XIX. THEOREM XII. N every triangle (BAG), the greater angle, has the greater fide oppofite to it. Hypothefis. Thcfis. In the A BAC, 'iQis> V A. ^e fidt AB f»ppofit€ io VC w > tit fidiO^oppofitt to VA. Demonstration. If not» The fide AB is either equal, or lefs than the fide CB. CASE I. Suppofe AB to be =: to CB. B. BECAUSE the fide AB 18=: to the fide CB (Sup, i.). 1 . The A BAC is an ifbfceles A. 2. Confequendy, the V C & A at the bafe> are = to one another. Buttho(e V C & Aare not:= to one another (Hyp.), ^. Therefore neither are the fides Afi, CB = to one another. C. N. D.2$.B.K P. 5. A I. B. CASE n. Suppofe AB to be < CB. _^ > EC AUSE the fide AB is < the fide CB (Sup. a.). I. It follows, that V C oppofite to the lefifer fide AB, is < V A oppofite to die greater fide CB. P. 1 8. >R r. But V C is not < V A (Hjp,J, a. Confequcntly, the fide AB cannot be < the fide CB, The fide AB being therefore neither = to the fide CB (Ca/e i.) ; nor < the Me CB (Cafe a.f 3. It follows, that this fide AB is > the fide CB. C. N, Which Mras to be demonilrated. 34 The ELEMENTS Book I A PROPOSITION XX. THEOREMXIIL ^ ^N Y two fides (AB, BC,) of a triangle (ABC) are together greater than the third fide (AC). Hypothefis. ABC is a A. Thefie. Jnj t*w9jtdlesy as AB + BG are > ibt thir J AC, Preparation. 1. Produce one of the two fides, as AB, towards D indefinitefx. Prf- ^* 2. Make BD = to BC. -P. V ^- '• 3. From the point C to the point D, draw the ftiaight line CD. Pv- i* Demonstration. I) ECAUSE intheABDCthefideBDis=:tothefideBC (Pre^. a.). 1 . This A is an ifofceles A. D.i^^.B, 1. 2. Confequently, the V at the hafe ii & / are = to one another. P, J. ^- *• But V « -f- » heing the whole, & V « its part. 3. It follows, that V « -f- « is > V ». -Ar. 8. But V « + « being > V « (^rg, 3.), & this V n being 13: to V/» (Arg. 2.). 4. It is evident that V w -f « is > V ^. C. iV. Since therefore in the A ADC, V«w+«is>V/ (Arg. 4.). 5. The fide AD oppoltte to the greater V m + ^ is alfo > the fide AC oppofite to the iefler V/. P. 19. A »• But becaufe the ftraight line BD is = to the ftraight l4neBC(^7V</. 2.), if the fide AB be added to both. 6. It follows, that AB -f- BD or AD is = to the fum of the two fides AB + BC. Jix,t. But AD is > the fide AC (Arg, 5.). 7. Whercforei the fum of the two fides AB + BC is alfo > the third fide AC. C.U. Which was to be dcraonftcated. Book I. Of E U C L I D. 35 J PROPOSITION XXI. THEOREM Xir. j[ F from the ends (A & B) of the fide ( AB) of any triangle (ACB) there be drawn to a point (D) within the triangle, two ftraight lines (DA, DB>) ; thefe ftraight lines will be lefs than the other tiyo fides (CA, CB,) of the triangle ; but will contain a greater angle (ADB). Hypothefis. Thtiis. DA, DB, are fvop ftraight lines dranun I. DA + I?B < CA + CB, from the points A & B /• the point D, //. V ADB > V C. within the A ACB. Preparation, Pnxluce the ftraight Une DA» until it meets the lide CB m E. P^, a. Demonstration. ;^ JOECAUSE the figure ACE is a A (D,z\.B,i,), . 1 . The two fides CA + CE are > the third AE. P, ao. B. u If the line EB he added to each of thefe. z. ThefidesCA+CB(thatisCA4-CE+EB)arc>theline8AE+EB. Ax, 4. Again, the figure DEB being alfo a A ^Z>. a 1 . A i .). 3. The two fides EB + ED are > the third im, P. ao. B. i. If we add to each of thefe the line DA. 4 The lines AE + EB (that is DA + ED + EB ) are > the lines DA + DB. ^^.4. But it has been proved that the fides CA + CB are > the lines AE+EBr^r^. 2,). 5. Much more then will the fides CA + CB be > the lines DA 4-DB. C. N. Which was to be demonftrated. I. x\g AIN, bccaufe V ADB is an e«erior V of A DEB (Prep J, k the V DEB is its interior oppofite one. 1 . It foUows, that V ADB is > V DEB. P. 16. B. 1, 2. For the fame reafon ; V DEB is > V C. But fmcc V ADB > V DEB (Arg. i.), & V DEB > V. C(Arg. a.). 3. It is evident, that V ADB is much > V C. C. K Which was to be denoonftrated. II. Ea ^6 The E L E M E N T S Bookl F ■apavasaBBBBB rj^ PROPOSITION XXIL PROBLEM VUL X O make a triangle (FHE) of which the fides (hall be equal to thitc given ftraight lines (A, B, C,) ; fuppofing jiny two whatever of thefe gi^cn ftraight lines to be greater than th^ third. Given Sought "Thi ftraight lines A,B,C* /u<h thai The cpnftruai^Ho/aATHRficbyttst A-l- B > C, A + C>B, C+B>A, EHa»^*e=:A,FE=B, yFH=C. Re/olution. I. Draw the indcfinhe ftraight line DM, ^•Z '• a. Make ED = to the given A, FE =: to the given B, & FG = to the given C ^. 3-^'' 3. From the center E at the diftancc ED, defcribe the © DR 7 p^r , 4. From the center F at the diftancc FG, defcribe the © GH. ) 5. From the points E & F, to the point of interferon H, draw the ftraight lines EH, FH. N- »• T Demonstration. H £ ftraight lines ED, EH, being drawn from the center E to theODH rRef. tScK.l 1. Thofe two ftraight lines ED, EH, are rays of the fame © DH. D. \b,o. ». a. Confequentiy, the ftraight line ED is = to the ftraight line ER ^. >S- ^' *' Since therefore ED is 2= to EH fjfrg. a.), & & given ftraight line A is alfo = to the fame line ED (Kef. 2,), 3. It follows, that pH is = to the given A. ^' ?• After the fame maniier it will be prov^, that 4. The line FH is = to the given C. But the fide EH being = to the given A (j^rg. 3.), the fide FH = to the given C (j^rg, 4.), & in fine the fide fE = to the given B, (Rel a.). $, It is evident, that the three fides EH, FE, FH, of A FHE, are p: tQ th^ thrpe giyen ftraight lines A, B, C. Which was to be done. rR E M A R K. . HE condition added^ that any two of the given lines Jhould be treater than tvf third, is effentialy in confequence of the XX prop. oftheLBook; voithouttin reftriBion the circles defer ihed from the centers E &f F would not cut o^i attfthtfl defed which would render the conJiruSion imfojfihle^ Book I. Of EUCLID, 37 M PROPOSITION XXm. PROBLEM IX. JtjL T a given point (A) in a given ftraight line (AM) to make a redili-* neal angle (BAC) equal to another given redilineal angle (HDG). Given Sought /. An indefinite flraight line MA, 4n angle BAC made •n AMf //. ne point A in thejfraight line AM, at the point A CS /• V HDG. ///. The redilineal angle HDG. Refolution, 1. In tlie fides DG, DH, of the given V HDG, take any two points E & F. 2. From the point £ to the p(Mnt P, draw the ftraight Kne £F. Pbf. t. 3. Upon the indefinite ftraight line AM & at the point Ay con- jftfu£l a A ABC whoi^ three Ades ibalj be z=: to the three fides of A DFE. /». ^2, B. 1. Demonstration. JjECAUSEthe three fides AB. AC, BC, of A ABC are 2= to Che three fides DF, DE, FE, of A DFE, each to each (Ref. 5.). 1. It follows, that the VBAC& HDG, oppofite to the equal fides BC, FE, arc = to one another, P. 8. t. I, But V BAC being = to the given V HDG ; as alfo made on the the given ftraight line AMf at the given point A fRef, 3.). a. It follows, that at the given point A, in the giyen ftraight h'ne AM, the rectilineal V BAC {s made c= to the given reailineal V HDG. Which was to be done. 38 The ELEMENTS Bookl 1 / A \^ E / • \p Tj^ -■■■ -r^ A^ "" 1^ PROPOSITION XXIV THEOREM. XV. J[ F two trianrics (ABC, DEF,) have two fides (BA, BC,) of the one equal to two fides (ED, EF,) of the other, each to each ; but the angk contained (B) greater than the angle contained (DEF) ; the bafe (AC) op- pofite to the greater angle, will be alfo greater than the bafe (DF) oppofite tp the leflcr angle, Hxpothefii, Thcfis /. BA = ED. i:b€bafe\Qn>tb$hafi\>Y, II BC =: EF. JIIWB>>/DEF. Preparation. I. At the point E, in the line DE, make V DEG = to the given VB. P.23.A1. 2, Make EG = to BC or to EF. P. 3 4. From the points D & F to the point G, draw the ftra^ht f I. KncsJJG, FG, Demonstration. P9f. I. JjECAUSE in the AABC the fides BA, BC, are = to the fides ED, EG, oWi DEG (Hyf. 1, Prep, a.), k V contained B = to V con- tained DEG r^ni:^. 1). I . It follows, that the bafe AC is = to the bafe DG. P. 4- '• '• Again, becaufe EG is = to the fide EF (Pi^ep, a, Hjp, %.). t. The A FEG is an ifofcelcs A. 2>. a$. ^' ^• 3. Confequently, V « = V r + ^. P. 5. A »• Since therefore V « = V r + f (^rg, 3.) | if ftom the Uft be taken Its part f . 4. ThVV«wmbe>Vr. CM And if to V M be added V n. Much more will the whole V « + « ^^ > V r C. !f. 6. Confequcntly, the fide DG oppofite to the greater V ap + *» » > ^ fide DF oppofite to the leifcr V r. P. i> B. »• But the ftraight line DG being > DF fjirg, 6.}, & this fame ftraight line DG being = to the bafe AC fj^rg, ij. 7. It is evident that the bafe AC is alfo > the bafe DF. C. N. Which was to be demonftrated. 5. r BookL Of EUCLID. B9 I PROPOSITION XXV. THEOREM XVI j^ F two triangles (BAC, EDF,) Juvc two fides of the one equal to two fides of the others each to each, but the bafe (BC) of the one greater than the bafe (EF) of the other; the angle (BAC) oppofite to the greater bafe (BC), will bealfo greater than the angle (D) oppofite to the Icfler bafe (EF). Hypothecs. Thefis. /. AB = DE. 7*f angle A •ppafite U the greater tl AC = DP. hafe BC, « > V D o//#//r U the Uff^r //A BC>EF. V-fEF. Demokstration. Ip nott The angle A is either equal or kfs than the togle D. C. N. B. CASE I. Suppofc VAtobe = to VD. BECAUSE V Ais = toVDr5«/. i.),& the fides AB, AC, & D£, DFy which contain thofe V, are equal each to each, (Hyp. i & 2.). I. The bafe BC 13 = to the bafe EF. P. 4. But the bafe BC is not = to the bafe EF (Hyp, 3.). a. Tberefbre V A cannot be = to VD. B, I. 6 CASE II. Suppofe VAtobe<VD. E CAUSE VA is< VD (Bup, 2.), & the fides AB, AC, & DE, DF, which contain thofe V are equal, each to each, (Hyp, i & 2. ). The bafe BC is < the bafe EF. But the bafe BC is not < the bafe EF (Hjp, 3.). Therefore V A 13 not < V D. But It has been fhewn that neither is it equal to it (Cafe, i.). Confequentlj, V A, which is oppofite to the greater bafe BC, is > V Di which is oppofite to the lefler bafe EF. Which was to be dcmonftrated. tj^.B, I, 40 The E L E M E N T S Bookl ft J PROPOSITION XXVI. THEOREM XVIL Y F two triangles ( ACB, DFE,) have tWo angles (A & B) of one, equal to two angles (D & FED) of the other, each to each, & one fide equal to one fide, viz. either the fides, as (AB & DE) adjacent to the equal angles; or the fides, as (AC & DF) oppofite to equal angles in each : then (M the two other fides (AC, BC, or AB, BC,) be equal to the two other ftte (DF, EF, or DE, EF,) each to each, & the third afigle (C) eqwl » the third angle (F). Hypothcfis. CASE I. Thcfis. /. V A = V D. When the equal fides AB, DE, are /. AC = DF. //. V B = V FED. adjacent to the equal angles A&D, //. BC = w- ///. AB =DE. B&FEDr/f^. »&*). //A VC=VF. Demonstration. If not. The iides are unequal, & one, as DF will be > the other AC. Preparation, I . Cut off from the greater fide DF a part DG = to AC. ^- J- ^- *• a. From the point G to the point E, draw the ftraighl line GE. -P*/ »• X5 EC AUSEinthe A ACB,DGE, the fide ACisrrto the fide DG, rPr#/.i.),AB = DEr/i'A3.)»^VAis = toVD. (Hyp. i.). I. The V B & GED oppofite to the equal fides AC & 1>G are equal. P, 4. ^' »• But VB being = to V GED (Arg, i,), & this fiune V B being alfo = toVFEDr^/.a.). %. It follows, that V GED is = to V FED. Ax. i. But V FED being the whole & V GED its part : 3. The whole would be =: to its part. 4. Which is impoilibe. Ax. S. 5. The fides AC, DF, are therefoie not unequal. . 6. Confequently, they arc equal, or AC = DF. C H. Which was to be demonftrated. L Since then in the A ACB, DFE, the fide AC is = to the fide'DF, (Arg, 6.), AB = DE (Hyp. 3.), & V A is = to V D (Hyp. i ). r . The third fide BC is alfo = to the third fide EF, & the V C & F, op- pofite to the equal fides AB, DE, are alfo = to one another. P, 4. B. '> Which was to be demonftrated. II Sr HL Book I. Of E U C L I P Hypothcfis. /. VA= VD. //. VB= VE. ///. AC=:DF. CASE II. -When the equal fides AC, DF, are oppofite to the equal angles B&E. (Fig, 1.&3.) Thefis. /. AB = DE. //. EC = EF. ///. VC=VF. Demonstratiok. If not. The fides AB, DE> are unequal ; and one, as DE, will be > the other AB. Preparation. 1. Cut off from the greater fide DE, a part DG = to AB. P. x, B. i. a. From the point G to the point F, draw the ftraightline GF. PoJ, i. B, BECAUSE then in the A ACB, DFG, the fide AC is = to the fide DF (Ifyp. 3.), AB = DG (Prep, i.), & V A is = to V D, r^/. I.). 1 . The other V B & DGF, to which the equal fides AC, DF, are oppo- fite, are = to one another. P, 4. B. i. The angle B being therefore = V DGF (^rg. i J, & this fame V B being alfo = to V E (Hyp. 2.). a. It follows, that V E is =r to V DGF. Ax, 1. But V DGF is an exterior - V of A GFE, & V E, is its interior oppofite one. 3. Therefore the exterior V will be equal to its interior oppofite one. 4. Which is impoffible. P. 16. B. i. 5. Confequently, the fides AB, DE, are not unequal. 6. They are therefore equal, or AB = DE. C N, . Which was to be demonftrated. I. Since then in the A ACB, DFE, the fide AC is = to the fide DF, CHjp, 3.), AB = DE fjlrg. 6.), & V A is = to V D C^iyp- i). 7. It is evident, that the third fide BC is = to the third fide EF, & the V C & F, to which the equal fides AB, DE, are oppofite, are equal to one another. P. 4. B. u Which was to be demonftrated. II. & III. F 42 The E L E M E N T S Book!, 1 A m. .... N \ B c D • , "\ I PROPOSITION XXVII. THEOREM XVIIL Fa ftraight Iine.{EF), falling upon two other ftraight liftcs (AB, CD,) fituated in the fame plane, makes the alternate angles (mic />, or n ^ 9^) equal to one another : thefe two ftraight lines ( AB> CD,) ihall be parallel. Hypothcfis, Thefis. /. AB, CD, are t*vj9flraigbi lines in the fame plane. 7be lines AB, CD, //. The line EF cuts them Jo that >/ mzr^i p, or "i n=z\/ ^. are pile. Demonstration. If not, The ftraight lines AB, CD, produced will meet either towards BD or towards AC. D. 3$. B,, i. Preparation, Let them be produced & meet towards BD in the point M. Pof. 2. i5eC AUSE the V « is an exteiior angle of A GMH, & V# its interior oppofitc one. , 1. The V«i8> V#. P. 16. A i. But V « is = to Vtf (Hyp. 2.). 2. This \f nln therefore not > V «. ^ ^ C. i^. 3! Confequently, it is impoflible that the ftraight lines AB, CD, ihould meet in a point as M. 4. From whence it follows that they are pile ftraight lines. D. 35. B. 1. Which was to be demonftrated. BookL Of EUCLID. 43 ■Pi^sws 1 A \. .. B '\ \. D c — \ J PROPOSITION XXVIII. THEOREM XIX. X F a ftraight line (EF) falling upon two other ftraight lines (AB, CD,) fituated in the fame plane, makes the exterior .angle (m) equal to the interior & oppofite (n) upon the fame fide, or makes the mterior angles (^ + n') upon the fame fide equal to two right angles ^ thofe two (Iraight lines AB^ CD* ihall jbe parallel to one another, case; I. Hypothcfis. Thefis. "i mz^'i n, AB, CD, are pile lines. Demonstration, JP EC AUS E the V in & / are vertical or oppfite V. I. They are =: to one another. P. 15 The V p being therefore = to V « (^^rg, 1.), & V « being =r to the fameVi-r^M a. It is evident that V / is alfo =: to V «. But the equal \fp&n (Arg. a.), are alfo alternate V 3. Confequentljy the ftraight lines AB, CD» are pile. CASE U. Hypothecs. ne V -f- « are =: /9 2 L». Demonstration. B.\. Ax, I. Thefis. AB, CD, are pile, limes. B. BECAUSE the ftraight line EF fallii^ upon the ftraight line AB, forms with it the adjacent V 9 &/, I. Thofe V p +/ are = to two L. The Vo+/ being therefore =: tqtwoL.. (Arg, i.), &the V^+'t being alfo = to two L. (Hyp,),, a. Itfijlows, thatthe Va+/are = toVfl + if. And if the common angle be taken away firom both fides. 3. The rcmai(iing ^p & n will be equal to one another, Aje, j. But thofe ^oal Vp & n (Arg. 3 J, are at the fiune time s4ter|)ate V, 4. Confcqucntly, th^ ftraight lines AB> CD, are pile. * /* 27. B. i. Which was to he demonftntt^ Fa P.13.B. Ax. I, 44 The E L E M E N T S BookL IL E M MA. F a ftraight line (EF), meeting two ftraight lines (LN, CD,) fituated in the fame plane, irakes the alternate angles fp'\-n ^ o) unequal ; thofe two ftraight lines (LN & CD,) being continually produced, will at length meet in (M), upon that fide on which ie the lefler of the alternate angles (oj. Preparation, For fince V / + « is Aipofed > V «. I. There may be made in the greater V/ + », on the ftraight Ifnc EF, at the point G, an angle n = V <?. P. 23. B. 1. ^. And AG may be produced at will to B. Pof. z. Bi Demonstration. > E C A U S E the two lines AB, CD. arc cut by a third EF, fo thai; the alternate V « & • are = to one another (Prep, i.). I. Thofe two lines AB, CD, are pile. But the line LN cuts one of the two piles, vik. AB in G. a. Therefore, if produced (ufficiently, it will cut alio the other CD ibmer where in M, upon that fide on which is the lefter of the alternate V 9*. Which was to be demonftrated. P. 27. B, I. C. N. w. COROLLARr. HEN V • < V/ -j- », the two interior angles # -f- m are ne- ccflarily < two L 5 fmcc the two anries / + « & « are equal to two L. /^ 13. B. i, Confcquently, when the two interior V, are < two L ; the lines LN, CD, which form ihofe angles with EF, will meet fomewhere on the fide of the line EF, where thofe angles are fituated, provided they are produced fufficiently. • Euclid regards as a felf evident principle tbatf a ft|^ight line (EF), which cuts one of two parallels as (AB) will neceflarily cut the other (CD), pro- vided this cutting line (EF).be fufficiently produced. See the prep. 0/ pr^ pofttions XXX y XXX FJI, and of feveral others. BookL Of EUCLID; 45 J PROPOSITION XXIX. THEOREM XX. J^ F a flraight line (EF), falls upon two parallel ftraight lines (AB, CD), it makes the alternate angles (n & mj equal to one another; ^nd the exterior angle frj equal to the interior & oppofite upon the fame fide (mJ ; and like wife the two interior angles upon the fame fides (^^ -(- »i^ equal to two right angles. Hvpothefis. AB, CD, are tnvo pile /iif«, cut hy the fivne Jiraight lint E)P. Demonstration. /. Thefis. If not. The V « & w are unequal, And one of them as V « will be < the other V «, C. A\ 5. 6. x5eCAUSE the V«is< Vif; if the V/ be added to both. I. The V«+>willbe<thc V«+/. Ax. 4. But fince the V « & V ;^ are adjacenjt V, formed by the ftraight line EF which falls upon AB< Z. Thefe V « + ^ are = to two L. P. 13. S. 1. 3. Confequcndy,the Vw+/(lcf8 than the V«+/>^ are a]fo< two L. C. N. 4. From whence it follows, that the lines AB, CD,' are not pile. Ccr. of Urn, But the ftraight lines AB, CD, are pile. (Hyp.), Confequently, the V « & « are not unequal. P.zy.B.i, They are therefore equal, or V » ^ V «. C. N. Which was to be demonftrated. I. Moreover, V r & V « being vertically oppofite. 7 . Thefe angles are = to one another. P, i^,B.i, But V w bemg=: to "i n (Arg, 6.), & V r being = to the fame V»> (Arg. 7.). • 8. It follows, that V r is = to V «t. Ax, i. Which was to be demonftrated. II. Likcwife, V n being = to V iw (Arg. 6.) ; if V/ be added to both. 9. The V » +/ will be = to V « +p, , Ax, a. But the V w + / *^re == to two L. (Arg. 2.). 10. From whence it follows that the Vm +^ are aifo =: to two L. Ax. i. Which was to be demonftrated. III. The ELEMENTS BookL ■BUS ^^p^ PROPOSITION XXX. THEOREM AU X HE ftraight lines (AB, EF), which arc parallel to t|ic fame ftraight line (CD), are parallel to one another. Hjrpothcfis. Thefis. AB, EF, anftraigbi lines, pile /• CD. ntftraigbi lines AB, EF <r# pile to one another. Preparation. Draw the ftraight line GH, cutting the three lines AB, CD, EF. Demonstration. l!>ECAUSE the ftraight lines AB, CD, are two piles, (ffypj cut by the fame ftraight line GH. (Pref>), 1. The alternate V « & « arc = to one another. P. 29. B. i. Likewife fuice the ftraight lines CD, EF are two piles, (ffyp.) cut bv the fame ftraight line GH. (Prep), 2. The exterior angle » is ;= to its interior oppofite one on the fame fide/. P. 29. B. i. ^ But V « being = to V « {y^rg. i.) & the fame V n being ai(b = to V / (y^r^. 2). 3. The V m &p will be =r to one another. jfx, i. But th^k W m&p (Arg. 2.) are alternate V, formed by the two ftraight lines AB, EF, which arc cut by the ftraight line GH. 4. Confcqucntly, thcfe ftraight lines AB, EF are pile. P. zj. B I Which was to be demonftrated Book I. Of EUCLID. 47 T PROPOSITION XXXI. PkOBLEM X. O draw a ftraight line (AB), thro' a given point (E), parallel to a given ftraight line (CD). Given *rhe ftraight lint CD and the p^int E. Sought ^be ftraight line AB, pile id CD, ^ pajftng thro* the point E. Refolution. B I. In the given ftraight line CD take any p6int F. a. From the point F to the point E, draw the ftraight line FE. Pof, i. 5. At the point E in the ftraight line FE, make V « = to V «. P. 23. A i. 4. And produce the iide EB to A. Pof, a. Demonstration. lECAUSE the alternate V »*&«, formed by the ftraight l?ne EF, which curs the two hnes AB, CD, are = to one another {Ref. 3.). I. The ftraight lines AB, CD, are pile. P, 27. B. i. Which was to be demonftrated. n 48 The E L E M E N T S Bookl. J PROPOSITION XXXII. THEOREM mi J^F a fide as (AC) of any triangle (ABC) be produced, the exterior angle (c-\-p) is equal to the fumof the two interior aitd oppofite angles (« + «); and the three interior angles (n--^ m + rj are equal to two right angles. Hypothefis. Thefis. ABC is a A, one 0/ tvBo/e^Jes I, V r + / w = /a V « + ». AC, is produced indefinite Ij /» D. //./** V « + w + r are =r ta iL- Preparation, Thro^ the point C, draw the ftraightlincCE, pile to thcftraight lineAB. . P.ji.^.^ Demonstration. B, _ BECAUSE the ftraight lines AB, CE, arc two piles (Pr<r/.) cut by the fame ftraight line EC. 1 . The alternate V « & r are = to one another. P, 29. ^- '• Likev/ife becaiife the ftraight line AB, CE, are two plies (Prep,) cut by the fame ftraight h'neAD. 2. T he exterior angle / is =: to its interior oppofite one jb, on the fame fide. P. 29- ^ '* The V c being therefore = to V « (^^g^ i.), & V/= V «» (Arg. 2.). 3. The V <• +/ IS = to the V n S( pi ^afeen together. Ax, 2. Which was to be demonft rated. 1. Since then the V f -f-/ is = b V « + « (Arg, 3^ j if the V r be added to both lides. 4. The Vf + / + iwillbe = iotte;ttrce V«+ « +rofthe A ABC. j^x.Z. But ihefe "i c + p ^ rnr^ the adjacent V, fortned by the line BC, which meets AD at the fam.tf pomt C. 5. Ccnfequenly, the V r + / i f are =: to two L. P. "J. ^- '• Wherefore, the three V w -j- « +'r, which are = to V^+/ + r, C'^rg. 4.) arc alfo = to two L. -'•»• '• Which was to be demonftrat^d. 11. J BookL Of EUCLID. 49 PROPOSITION XXXIII. THEOREM XXIII. Jt H E (Iraight lines (AC, BD,) which join the extremities (A, C, &B, D,) of two equal and parallel ftraight lines, towards the fame parts, are aifo them* fehes equal and parallel. HTpothcfis. Thefis. ACfhD^ are itvojfrasgbtlines^ ^hichjoin I. The flrmight lines AC^BDy are efuni towards the fame partly the extremities II, And thofe ftraight lines AC, BD, 0/ /wo = 6f ^\\t ftraight lines AB^ CD. are pile. Preparation: From the point B to the point C, draw the ftraight line BC. B. Demonstration. > E C A U S E the ftiaight liaes AB, CD, arc two plies (Hyp.) cut hj the fame ftraight line BC (Prep.). I. The alternate V « & « are = to one another. P, 29. S. 1. Since therefore in the two A CAB, BDC, the fide CD is = to the fide AB (tfyp.)y the fide BC is common to the two A, & the V w is = to the V « (-^^g. >.). Z. It follows, tliat the bafe AC is = to the bafe BD. Which was to be demonftrated. I. 1 3. Likewife that the V ACB, DBC, to which the equal fides AB, CD, are oppofite, are alfo =: to one another. But thofe equal V ACB, DBC, (Arg. 3.) arc alternate V formed by the ftraight lines AC, BD, cnt by the ftraight line BC. 4. Confcquendy, the ftraight lines AC, BD, are pile. P. 27. B. Which was to be demonftreted. II. P. 4. -5. f. so The£LEM£Nt^ gbokt ^t I ....-••• B ■iB^aaBBKaHBBaHHiBBBBaBaBaBaaasssaaiMBUai^iBsssBaaa PROPOSITION XXXIV. THEOREM XXIF. T. HE oppofite fides (AC, BD, & CD, AB,) and the oppofite angles (A, D, & m+ r, n +'») of a paraltelognun (AD) are equal to one am>t]ier»& the diagonal (BC) divides it into two equal parts. Hjpothcfis. Thcfis, I. XDisa Pgr. /. Tht Jidet AC, BD, & CD, Aft //. BC is tbt diagonal of this Pgr. are = /• 9nt another^ & V A =: D. //. V«i + r=V» + x. ///. rhi A CAB, BDC, firmed ly the Jiagona/, art =: t§ we aneiier. Demonstration. B, I E C A U S E the firaight lines AB, CD, are two piles CHjf. i .) cot By thefiune ftzaight line CB fHjp. a.). 1. The alternate V « & n are = to one another. P, ag. B. i. Again, becanfe the ftraight lines AC, BD, are two piles (Ify^, i.) cut hy the fame ftraieht line CB fHjp. a.), a. The alternate vr. Sr / are ^ to one another. P, 29. B. i. But the A BDC, CAB, have two Y m & s =totwo\f n& r, (Arg. I & a.), & the fide BC adjacent to thofe equal V is common to the tvro A. 3. Confequcntly, the fides AC & BD, oppofite to the equal \f n &m^ aJfo the Mc5 CD, AB, oppofite to the equal V J & r, are = to one P. 26. B. u another, & the third V A is = to the third V D. Which was to be demonftrated. I. But V m being = to V « (Arg. 1.), & V r = V x (Arg, 2.). 4. The whole 7 « -f" '^ *s = to the whole V « + j. Ax. 2. Which was to be demonftrated. II. In fine, becaufe in the A CAB» BDC, the fide CD is = to the fide AB, C-^^g' 3)' the fide BC is common to the two A, and V m is =: to Y n(Arg. I.). 5;. Thofe two A CAB, BDC, formed by the diagonal BC, are == to one another. P, 4. B. 1. Which was to be demonftrated III. F Book I. Of E U C L I D, PROPOSITION XXXV. THEOREM XXV, Jr AR ALLELOGRAMS (AD,ED,) upon the ftme bafe (BD) & be. tween the fame panrilelt (AF, BD^) } are equal to one another. Hypothefia. Thefis. /. AD&EDiw<«wPgr«. f^ifPn ADw = //»/i$r PgrED. //. And tbofe tvat Pgrs, art uptn thtfam hafi ' BD, li ittvieen tbtfam piles AF, BD. B Demokstratiok. ^34 At. P.34- Ai. I. E C AU S E the figure AD is a Pgr (H^p. i.). 1. The oppofite fides AC, BD, & AB, CD, are = to one another. Likewiie, becaufe the figure ED is a Pgr (Hjp. i.). ^. The oppoiite fides £F, BD, & BE, DF, are ^ to one another. But the fhaight line AC being = to thie ftraight lineBD (Jrg. i.), &^ the ftraight hne £F bein^ alfo = to the fame ftraight line BD (Arg. i,). jl It follows, that the ftraight line AC, is = to the ftraight line EF. Ax. Since therefore AC is = to EF (Arg. 3.) , if CE beadded tp both. 4. The fbaight line AE is necefTarfly = to the ftzaight line CF. Ax. t. Therefore in the A ABE» CDF, the fide AB is s= to the fide CD, (^rg, I.), the $de BE is =: to the fide DF (Arg. 2.), & the bafe AE is = to the bafe CF (Arg, a.). $. Confequentlf, the A ABE is = to the A CDF. ?. 8. B, Taking away therefore from thofe equal A ABE, CDF, (Arg. 5.) their common part CM£. 6. Tht remaining trapeziums ABMC, MDFE, are = to one another. Ax. 3. Addii^ m fine to thofe equal trapeziums ABMC, MDFE» ("^r^. 6.)the common part MBD. 7. The Pgrs AD & ED will be = to one another. Ax. x. Which was to be demonftnted. G% u 5^ The ELEMENTS Bookl. I H PROPOSITION XXXVI, THEOREM XXVL r ARALLELOGRAMS (AC, GE,) upon equal bafcs (EC,DE,)«f between the fame parallels (AH, BE/), are equal to one another. Hypothecs. Thcfis, /. AC, GE, are /«;• Pgrs. fh Pgr AC « = to tbt Pgr GE. //. And tbofe two pgrs are upon e^ual hafes BC, DE, li between the fame piles AH, BE. Preparation. 1. From the point B to the point G, draw the ftraight line BG. ) » a ^ ^. From the point C to the point H, draw the ftraight line CH. J ^' ' B, Demonstration, EC AUSE the figure GE is a Pgr (Hyf. i.). 1 . The oppofite fides DE, GH, are =: to one another. But the ftraight line BC 18= to DE (Hyp, a.), & GH is = to the fame ftraight line DE (Arg. i.). a. Therefore BC is = to GH. But fince BC is = to GH (Arg, a.) ; & they are piles (Hyp a.) who(e extremities are joined hy the ftraight lines GB, HC, (Prep, i & a.). 3. It is evident that thofc ftraight lines GB, HC, arc = & pile. 4. ConfeqUcntly, the figure GC is a Pgr. Moreover, the Pgrs AC, GC, being upon the fame bafe BC, & be- tween the fame plies AH, BE, (Hyp^ a.}. 5. Thofe Pgrs AC, GC, are = to one another. It will be proved ^fter the &me manner. 6. That the Pgr GC is = to the Pgr GE. Since therefore the pgr AC is = to the pgr GC (Arg, 5.), & the Pgr GE is = to theiamePgr GC (Arg. 6 J. 7. It follows, that the Pgr AC is ±= to the Pgr GE. Which was to be demonftrated. P. 34.^.'. Ax, I. P. 33.^. I. Z>.3S.A'. P.35.^'' Ax. I. Of E U C L I D. S3 PROPOSJITION XXXVII. THEOREM XXFIl 1 RI ANGLES (ACB, ADB,) upon the fame bafc (AB) & bctweea iht fame parallels (Afi, CD^) su-e equal to one another. HTpothcfis. Thefis. / ACB, ABD, are t<tvc A. The A ACB is = to the A ADB. //. And tbofe t*wo t^are upon f he fame AB, W iefween the fame piles AB, CD. Preparation. I. Produce the ftrajglit line CD both ways to E& F. Pof, t. Z, Thro' the points A & B, draw the ftrafght lines AF, BE, plJe to the fides BC, AD; which will meet the produced CD P.^i.B.h fomewhere in F & m E. B, Demonstration. ^ECA USE in the figure BF the oppofite fides AB, FC,&AF,BC, are pile (Hyp, 2 & Prep, 2.). «. ThcfigureBFisaPgr. D. 35.-5.I. It will be proved after the feme manner, a. That the figure AE is a Pgr. But the Pgrs BF, AE, are upon the feme bafe AB aad between the lame plies AB, FE, (Hyp. 2 & Prep, 1.). 3. Confequently, the Pgr BF is = to the Pgr AE. P. 35. B. t, But the ftraight lines AC, BD, are the diagonals of the Pgrs BF, AE, ^l (Prep. I & 2.). ^ j^ Wherefore thofe diagonals AC, BD, divide the Pgrs BF, AE, i»to two equal parts. P, 34. B» I, 5, Confequentlj, the A ACB is the half of Ae Pgr BP, & Ac A ADB the half of the pgr AE. Since then the whole Pgrs BF, AE, are equal to oneanother (Arg. 3.), & the A ACB, ADB, arc the halves of thofe Pgrs (Arg, $.), 0. It Iw evideat that the A AQB> ADB, are alfo z=: to one another. Ax. 7. Which was to be demonibated. 54 The E L E M E N T S Bookl PROPOSITION XXXVin. THEQREMXXFiil T. _ RIANGLES (ADB,EGF,) upon equfd bafcs (AB,EF,)&b^ tween the ffinie parallels (AF^ DG>) are equfd to one another. HypoAcfis. Thefii. /. ADB,EGF, -ire/w A. TJt AApB« = ^•''•^2^• //. Andtboft Uvo A are ii/#ic = jtf/r/ AB» EF, & iettveen the /am piles AF, DO. Preparation. I. Produce the ftraight line DC both ways to the points H, C ^ *• a. Thro' the points A&F» draw the ftraight lines AC, FH, pile to the tides BD. EQ ; which will meet the produced line P. 3'' ^- '* pG, (bmewhere in C & in H. B Demonstration. E C A U S £ in the figure BCi the oppofite (ides AB,CD> fr AC»BD, are pile (Hjp. z & Prep, a.), ,. . a , 1. ThcfigureBCisaPgr. D.JS'^'' It nuiy be proved after the iame manner. 1. That the figure EH is a Pgr. But the pgrs BC, EH, (Arg. i Bt a.) are upon =: bafes AB^ EF, fr between the feme piles AF, CH, (ffyP^ *-). - • , 3. Confequently, the Pgr BC, is =: to the Pgr EH. -P. 3^- ^•'' But the ftraight Imes AD, FG, being the diagonals of the Pgrs BC| flH, (Pnf. I fr a.). 4. Thofe ftraight lilies AD, FG. diWde the Pgrs BC, £H» mto two equalparti. F.Jf**- 5. Wherefore, the A ADB, is half of the Pgr BC, & the A EGF is the half of the Pgr EH. Since thenthe whole Pgrs BC, EH, are = to one another (Arg, 3.). and the A ADB, EGF, are the halves of thofe Pgrs (Arg, 5 J. p. It fbllowsi that thofe A ADB, EGF, are alio ^ to one another. A. 7- Which was to be demonftxated. icoki 6# E U C L I D. 55 PROPOSltlON XXXIX. THEOREM XXIX. Jjj QJJ A L triangles (ACB, ADB>) upon the fame btfe- (AB) & up- on the feme fide of it, are between the fame parallels (AB, CD,). iTiefis. Tti A ACB, ADB, are hti^an Hjpotlieiisi I.Tbft^ ACB» ADB, an equal. II. And ibefe A are upen the fame hafe AB« the fame piles AB, CD, If not. Demonstration. B. The ftiaight lines AB, CD, are not pile, & there may be drawn thro' the point C, fome other ftraight line CO, pile to AB« Preparation. I. Draw then thro' the point C» the ftraight line CO pile to AB i P. 31. B, i. which will cut the ftraight Une AD, fomewhere in £. a. From the point B, to the point of interf^ion £, draw the ftra^ht line BE. Fo£, i. , ►ECAUSE the two A ACB, AEB, are upon the fiune bafe AB, {Hyp, 2.), & between the fiune plies AB, CO, (rrep, i.). I. The A ACB is = to the A AEB. P. 37. B. i. But the A ADB being = to the A ACB (Hyp. 1.}, & the A AEB being = to the feme A ACB (Arg, i.). a. The A ADB is = to the A AEB. Ax, i. . But the A ADB being the whole, & the A AEB its part. 3. It follows, that the whole is equal to its part, 4. Which is impoflible. Ax, 8. 5. Coniequently, the ftraight line CO is not pile to AB. It may be proved after the fiune manner, that no other ftraight line but CD, can be pile to AB. 6. Conlequently, me ftraight line CD, drawn thro' the vertices of the A ACB, ADB, is pUe to the bafe AB. Which was to be dempnfirated. 56 The E L E M E N T S Bookt I A V^ B V \ \ / ••••, / / I. C E __P PROPOSITION XL. THEOREM. XXX. JCjQU a L triangles (BAG, EDF,) upon equal bafes (BC, EF,) & up- on the fam6 fide, are between the fame parallels (BE, AD,). ThcCs. TAe A BAG, EDF, an htven the fame piles BF, AD. HypotheGs. /. Tbe A BAC, EDF, are equal, n. And tbafe A are upon = hafes BC, EP. Demokstratiok. If not. The ftraight Unes BF, AD, are not pile, ft there may be drawn thro' the point A (bme other (ha^ht line AO pile to BF. Preparation. I. Draw then thro* the point A the ftniigfat line AO pile to BF, P. 31. ^* '• which will cut the ftimight line £D (bmewhere in G. a. From the point F to the point of interferon G, draw the ftraight line FG. Pof, i. B. > E C A U S E the A BAC» EGP, are npon the equal bafes BC, EF, (^yp> a.), & between the fiuue piles BF, AO, (Prep. i.). 1. The A BAG is = to the A EGF. P, 38. B. r. But the A EDF is = to the A BAC (Hyp. i.), ft die A EGF Is = to the fame A BAC (Arg, i.). 2. Wherefore the A EDF is = to the A EGF. Ax. 1, But the A EDF being the whole & the A EGF its part. 3. It follows, that the whole is = to its part. 4. Which is impoflible. A». 8. 5. Confequently, AO ia not pile to BF. It will be proved after the fame manner that no other ftraight line but AD can be pile to BF. 6. Confcquently, the ftraight line AD, chtiwn thro' the fummets of the A BAC, EDF, is pile to the ftraight line BF. Which was to be demonftrated. ' r Book!. Of EUCLID. 57 I PROPOSITION XLI. THEOREM XXXI. Fa parallelogram (BD) and a triangle (BEC) be upon the fame bare (BC), and between the fame parallels (BC, AE,) * the parallelogram fhall be double of the triangle. Hypotheiis. /. BDw^PgrbfBEC^zi. Jl 7bofe figures are upon the fame hafe BCilf M^veen the fame plieiECt AE. Preparcn'on. From the point A to the point C, draw the ftraight Hne AC. Demonstration. X5 E C A U 8 E the A BAC, BEC, arc upon the (ame bafe BC, & be- tween the (amc piles BC, AE (Hjp z ) I . The A BAC is = to the A BEC. P. 37. B. i. But the ftraight line AC being the diagonal of the Pgr b£) (Prep J. X. This diagonal divides the Pgr into two equal pRrts. P. 34. B, 1. 3. ConfequentJy, the Pgr BD is double of the A MC. But this A BAC being = to the A BEC (Arg, 1.). 4. ThePgrBDisalfodoubleof the ABEC. Ax, u Which was to be demonftrated. Thefts. T:he Pgr BD is double of the A BEC Pof I. H ^ 58 The E L E M E N T S BxkL 1 I. 2. 3- 4- 5- P, 10. B. I. PROPOSITION XLII. PROBLEM XL Jt O defcribe a parallelogram (ED), that fliall be equal to a given triangle (BAD), & have one of its angles (DCE) equal to a given redilineal angle (M). Given Sought /. The A BAD. 7be conflruaion of a Pgr = to the A BAD, //. A reailineal V M. W having an V DCE = to the given V M. Refolution. Divide the bale BD into two equal parts, at the point C. Upon the ftraight line BD at the point C, niakc an V DCE = to the given V M. Thro' the point A, draw the ftraight line AF pile to BD. Produce the fide CE 6f the V DCE, until it meets the ftraight line AF in a point E. Thro' the point D, draw DF pile to CE, & produce it until it meets AF m a point F. Preparation, From the point A to the point C, draw the ftraight line AC. BDemonstratiok. E C A U S E the A BAC, CAD, are upon equal bafes BC, CD, (Ref. I.), & between the fame piles BD, AF, (Ref, 3.). I . The A BAC is = to rhe A CAD. * P. 23 JP.31J Pof.i. Pof. 2. Pof. I. B.I 2. Confcquently, the A BAD is double of the ACAD. But in the figure ED the fides CD, EF, & CE, DF, are pile (Ref. 3 & J .). 3. Confcquently, ED if a Pgr. But this Pgr ED & the ACAD, arc upon the fame bafe CD, & be- tween the fame piles BD, AF, (Ref i. 3. & Prep J, 4. From whence it follows, that the Pgr ED is double of the A CAD. Since then the Pgr ED is double of the A CAD (Arg. 4.), & the A BAD is alfo double of the fame A CAD (Arg, 1.). ' 5. It is evident, that the Ppr ED is = to the A BAD. ^ And as its V DCE is al& =: to the given V M '(Ref 2.). 6. This Pgr ED is =: to the given A BAD, & has an V DCE = to the given V M. \\'hich was to be demonftrated. P. 38. B, r. A 35 B.\. ,B.\ ^x. 6. J Scokl. PROPOSITION XLIII. THEOREM XXXII. Tv H E complements (AF, FD,) of the parallelograms (HG, EI,) about the diagonal (BC) of any parallelogram (AD), are equal to one another. HTpoihefis. /. AD is a Pgr, luhofe diagonal is BC. 77. HG, EI, are the Pgrs aho^t the diagonal. Thefis. We Pgrs AF, FD, which are the complements of the Pgrs HG, EI, fire = to one another. I. 2. 3- 4. 5- 6. Demonstration. X/ECAUSEApisa Pgr, whofc diagonal is BC (Hyp, i .). ♦ This diagonal divides the Pgr into two equal parts.. P, 34. J?, t. Confequenily, the A CAB is = to the A BDC. Ukewifc, El being a Pgr, whoie diagonal is BF (ffyp. 2.), It divides alfo the Pgr into two equal parts. P. 34. B, i. Wherefore the A FEB is := to the A BIF. In fine, HG is a Pgr, whofe diagonal is FC (Hjp, 2.). Which confequentlj divider it into two equal parts. P, 34. B, i . Confcqucntly, the A CHF is = to the A FGC. Since then the A FEB is = to the A BIF (Arg, 4.), & the A CHF =:totheAFGCr^r^.6.). 7. The A FEB, together with the A CHF is = to the A BIF, together with the A FGC. Am, z. But the whole A CAB, BDC, being = to one another (Arg, a.) ; if there be taken away from both, the A FED + CHF, & the A BIF -f- FGC, which are equal (Arg. 7.). The remaining Pgrs AF, FD, which are the complements of the Pgrs HG, EI, will be alfo = to one another. Ax, 3. Which was to be demonftrated. G 2 8. €o The ELEMENTS BookL PROPOSITION XLIV. PROBLEM XII. vJ P O N a given ftraight line (AB), to make a parallelogram (BC) wW fhall be equal to a given triangle (T), and have one of its angles as (bA^j equal to a given redilineal angle (&f ). Given Sought /. rheftrmigbt lim AB. A Pgr made upftn afiraght bni AS JL The AT. =to the A T, bwing one of Us ^ III. The reailineat V M BAC = to tbe given V M. Refohtion. A/ a. B. I . Produce the ftraight line AB indefinitelj. %. Take AL = to one of the fides of the given A T. 3. Make the A AKL = to the given A T. 4. Dcfcribe the Pgr EH = to the A AKL, having an V HAE := to the given VM. P.^i.^-'- 5. Thro' the point B, draw a ftraight hne BF pllp to EA or GH. P. 31. ^- *• 6. Produce GH indefinitely, as alfo GE, until it meets BF in F. N- *• 7. Thro' the points F & A, draw the ftraight line FA, which Pof. u when produced will meet GH produced, (bmewhere in I. 8. Thro* the point I, draw the ftra^ht line ID pile to HB or GF. P. J«. *• *' 9. Produce FB, EA, until they meet ID in the points D& C. iV- ** Demonstration. I E C A U S E In the figure DG the oppofite fides GI, FD, & GF, ID, are pile (Ref, 5. 6. 8. & 9.7. ^ I . Tbj figure DG is a Pgr. D, JS- ^ '■ ^ J Book I. Of E U C L I D. 6i Again, the oppofite fides EA, FB, & EF, AB i alfo HI, AC, & HA, IC, of the figures EB, HC, being pile (Ref. 5. 6. 8. & 9.). a. Thofe figures EB, HC, are Pgrs. • D. 35. B. i. But the ftraight line FI is the diagonal of the Pgr DG (Rtf. 7.), & EB, HC, are Pgrs about this diagonal (Arg, 2, & a^/! 7,). 3. Coniequently, the Pgrs BC, £H, which are the compliments, are := to one another. •''•43* ^»\» But the Pgr EH is = to the A AKL (Rif, 4.), & the given A T is = to the fkmc A AKL (Rtf. 3.). 4. From whence it follows, that the Pgr EH is = to the given A T. Ax. !• The Pgr EH being therefore = to the given A T (Arg, 4.), U this fame Pgr EH being = to the Pgr BC (^-^fy. 3.). 5. The Pgr BC is = to the given AT.- A^.v. Moreover, becaufe the V MAE, BAC, arc verticall/ oppofite. 6. Thofe V are ;= to one another. i^. I J. A !• Wherefore, V HAE being == to the given V M (R^f, 4.). 7. The V BAC is alfo = to this given V M. Ax. i. 8. Therefore, upon the given ftraight line AB, there has been made a Pgr BC = to the given A T (Arg, 5.), & which has an V BAC 3= to tie given VMf^^if^ 7.). Which was to J)C done. 62 The ELEMENTS BookL PROPOSITION XLV. PRO BL EM XIII JL O defcribe a parallelogram (AF), equal to a reSilineal figure (IH)} and having an angle (n) equal to a given redilincal angle (N). Given ' Sought /. A reSilineal fgure IH. The " confiruSion of a Pgr = to the reSilineal II. Areaiiineai^ N. f gu re IH, l^ ia<v i ng a n >/ n:=2 to a givtn'ili,^ Refolution, 1 . Draw the diagonal GK. Pof. i . z. Upon an indefinite ftraight line AP, make the Pgr AE = to the A GHK, having an V « = to the given V N. P. 42. B. 1. 3. Upon the fide BE of the Pgr AE, make the Pgr DF ±= to the A GIK i having an V r = to the given V N. P. 44. ^. ^ Demonstration. B EC AUSE V N is = to each of the V « & r (Ref. 2 & 3.).. 1. The V«isr=to the Vr. ^ Ax,\. If the V iw be added to both. 2. The V « + « will be = to the V r + «. Ax. 2. But becaufe the fides AD, BE, are piles (Re/, 2,) cut by the fame ftraight line AB. 3. The two interior V « + «, are = to two L.. P. 29. ^. ^• 4. Confeqnently, the adjacent V r -}- «, which arc = to them (Arg. 2.), are iaifo = to two L.. Ax. i. The ftraight lines AB, BC, which meet on the oppofite fides of the line BE at the point B, making with this ftraight line BE the fum of the adjacent V r-|- *« = to two L. (Arg. 4.). 5. Thofe ftraight lines AB, BC, "form but one & the fame ftraight Une AC. P. if ^- ^' Moreover, the ftraight lines DE, AC, being two piles (Ref. 2.) cut by the fame ftraight line BE. Book J. Of EUCLID. 6i 6. The alternate V r 8r j, are = to one another. P. 29. B. i . And if the V 1/ be added to both. 7^ The V r 4- I/, will be z= to V j + w. j^x. 2. But becaufe the fides EF, BC, are two piles (Ref.^.) cut hy the fair.e ftraight line BE. 8. The interior V r -f- w, are ^ to two L. P, 29. B. i . 9» From whence it follows, that the adjacent V x -(- ^, which are = to them (^rg. 7.), arealfo= to two L.. Jx. i. The ftraight lines DE, EF, which meet on the oppollte fides of the h'ne BE at the point E, making with this ftraight line BE, the futa oi the adjacent "i s ^ uzzi to two L (^rg. 9 J. 10. Thofe ftraight lines DE, EF, form but one and the fame ftraight lineDR But fince the ftraight lines AD, BE, & BE, CF, are the oppofite fides of the Pgrs AE, BF, (Ref. 2 & 3.). 1 1 . The ftraight line AD is = & pile to BE, & B£ is = & pile to CF. 1 2. Confequently, AD is = & pile to to CF. ------ Moreover, thofe = and pile ftraight lines AD, CF, are joined by - the ftraight lines AC, DF, (j^rg. 5 & 10.). 13. Confequently, the figure AF is a Pgr. ------- And becaufe the Pgr BF is = to the A GIK (Re/. 3.), the Pgr AE is = to the A GHK, & V « = to the given V N (Re/. 2.). 14. The whole Pgr AF is := to the reftilineal figure IH ; & has an V « = to the given V N. Jx. 2. Which was to be demonftrated. P. 14. P. I. P. 34.2?. I. P. 30.^. Ax, I. 1 2?. 35. B. I. B.\, 64 The ELEMENTS Bod^I. PROPOSITION. XLVI. PROBLEM XIV. \J P O N a given ftraight line ( AB) to dcfcribe a fquare (AD). Given Sou^t 7bi ftraight line AB. Afquar* made up^n the ftraight lint fA, Refolution' 1. At the pomt A» ere& upon tin ftraight line AB the perpendi- cular AK. P, II. B.u, 2. From the ftraight h'ne AK cut off a part AC = to AB. P. 3. B. i, 3. Thro' the point C, draw the ftraight line CO pile to AB. 7 » jj , 4. And thro* the point B, draw the ftraight line BD pile to AC, J ^' ^ ' ' ' whidi will cut CO (bmewhere in D. BDeMONSTR A TION. E C A U S E in the figure AD the oppofite fides AB, CD^ & AC, BD, are pile C^^f- 3*4)- 1 . The figure AD is a Pgr. /), 35. B. i. 2. Confequently, the oppofite fides AB, CD, & AC,BD, are = to one another. P. 34.iJ.i. But AC is =: to AB (^/?^/ z.). 5. Confequently, the four fides AB, CD, AC, BD, arc = to one ano- ' ther. Ax, i. Again, becaufe the ftraight lines AB, CD, are pile (,Re/. 3.), 4. The interior oppofite V A & ACD, are = to two L.. P, 29. B. 1. But the V A being a L r^tf, 1 . ). '5. It is evident, that V ACD is alfo a L. C N. Moreover, becaufe AD is a Pgr ^-^^. 1.). 6. The oppofite V are =r to one another. P. 54. B. u 7. Wherefore, the V BDC & B oppofite to the right V A & ACD, are al(b L.. The figure AD being therefore an equilateral Pgr (Arg. 3.), &* rec- tangular ^^r^. 7.). 8. It follows, that this figure AD defcribed upon the ftraight line AB» . 18 a fquare. D, 30. B. 1 . Which was to be done. J Book! Of E U C L I D. ^S COROLLARr I h. VERY paralUlograMy thai has two equal fides AB, AC, including a right angle^ is a f quart ; for dran^ing thro* the points C W B the ftraight lines CD, bD, parallel to the ttvo fides AB, AC, the Jquare AD 'wUlhe dejcrihed (D. 30. B, i.). COROLLARr IL Jh^ VERT parallelogram that has one fight angle , has all its angles right an^ gles. For fince the oppofite angles A W BDC, are equal (?, 34. B. i.), W thi angle A h a right angle^ the angle BDC ijuill he alfo a right angle : moreover ^ the lines AB, CD, W AC, BD, heing parallels ^ the interior angles A W ACD, HJ^e^ •wife A &B, are equal to ttvo right angles (P, 29. ^. 1.) ; tut the angle A heing a right angle^ it is manifeft that the angles ACD & B, are alfo right angles, COROLLARY III. JL H Efquares defcrihedon eoual ftraight lines ^ are equal to one another ^ li r#- eiprocally^ equal fquares are defcriied on equal ftraight lines. J I- 66 The ELEMENTS Book I. PROPOSITION XLVn. THEOREM XXXIII. J[ N any right angled triangle (ABC) ; the fquare which is defcribcd upoB the fide (AC) fubtending the right angle, is equal to the fquares made upon the fides (AB, BC») including the right angle. Hypothefis. Thefi«. ni A ABC is Rglc, or V ABC « a L. rheU^tbeJide AQiszz u tUU^f AB, together 'wUb the D »/BC. Preparation. 1. On the three fidei AC, AB, BC, defcribc (Fig, i.) the D AG, AM, CD. P. 46. A', a. Thro' the point B, draw the ftraight h'lie BH pile to CG. i'. 31. ^- >• 3. From the point B to the point P, draw the ftraight line BF. \ M i^ 4.. From the point C to the point N, draw the ftraight line CN. ) '^' Demonstration. ECAUSE thefigurcAMiaaDr^rf/. i.). The V ABM is a L. , D.lO.B^' But V ABC being alfo a L (Myp), 2. The two adjacent V ABM, AI3C, are zr to two L. ^m, a. The ftraight lines MB, BC, which meet on the oppofite fides of the line AB at the point B, making with this ftraight line AB the fum of the adjacent V ABM, ABC, = to two L. (-^rg. '2.). 3. Thefe ftraight Hnes MB, BC, are in one and the fame ftraight line MC, P. H-^- *• which is pile to NA. P.iZ.^-^- In like manner i( may be dcmonftrated. 4. That AB, BD, are in one & the fame ftraight line AD, which it pile to CE. Moreover, becaufe AG, AM, are D (Prep. 1.), 5. The V FAC, NAB, are = to one another, (being right angles) & the fides AF, AC, & AB, AN, are alfb =1: to one another. • D.%o, BA' Therefore, if to thofe equal V FAC, NAB, V CAB be added. B 1 Book I. Of EUCLID. 67 s. to. N IX D X E ylx, 2. The whole V FAB will be = to the whole V NAC. Since then in th^ A AFB, ACN, the fides AF, AB, & AC, AN, arc = each to each C^rg. $.), & th? V FAB is == to the V NAC, r^rg. 6.). The A AFB will be = to the A ACN. But the A AFB & the Pgr AH, are upon the fame bafe AF & be- tween the lame piles AF, BH, fPre^. 2.). From V. hence it follows, that the Pgr AH is double of the A AFB. P. 41. 5. i. Likevnicp the A ACN & the D AM being upon the fame bale AN, and between the feme plies AN, MC, C-^^g- 3-)« The D AM IS doable of the A ACN. The A AFB, ACN, beij^g therefore = to one another fj^rg. 7.). and the Pgr AH & the D AM their doubles (Arg. 8 * 9.). It follows, that the Pgr AH is = to the D AM. II. 12. '3 P. 4. B. I. P. 41. B.K jfx. 6^ In the fame manner, by drawing fFig. 2.) the lines BG, AE, it is demonftrated, that the Pgr CH is =;= to the D CD. But the Pgr AH, together with the Pgr CH, form the DAG. Wherefore, this D AG is = to the fum of the D AM & CD. But fince the D AG is the D made upon the fide AC, & the D AM and CD the Q upon the (ides which include the L. ABC. The Q made upon the fide AC is =: to the JDmade uponAB&BC taken together. Which ¥ras to be demonftrated. Ax. 68 The ELEMENTS Book I 1 PROPOSITION XLVIII. THEO REM XXXIV. J^F the fquare defcribed upon one of the fides (CA) oJF a triangle (CBA) be equal to the fquares defcribed upon the other two fides of it (AB, BC,)} the angle (ABC) included by thefe two fides (AB, BC,), is aright angle Hypothcfis. Thcfis. The D ofQK is = to ibe D ^/AB, ^he V ABC included hj th together wtb the Q of BC. Jides AB» BC, is L. Preparation, 1. At the point B, in the ilrairht ihw BA| ere6i the perpendi- cular BH. ?. iiJ.!. 2. MakeBH=BC. P.y^.^- 3. From the point H to the point A, draw the ftraight line HA. Pof.i, B Demonstration. ECAUSE Wi\%=iohQ(Prep,%.), ThcDof BHwillbe = tothcaof BC. If the a of AB be added to both. The D of AB & BH, will be = to the D of AB & BC. But the A HBA being Rgle in B (Prep, i J. It follows, that the |J of HA is = to the D of AB & BH. Since then the Qof CA is = to the D of AB &BC (Hyp, i,), the D of HA = to the Dof AB & BH (Arg, 3.), & the D of AB&BH, are = to the D of AB & BC, (Jrg. aj. - The D of CA muft neceflarily be = to the D of HA. ^ pr £ ,. Conft quently, CA is = to HA. --...•..- i 'J^' But in the A CBA, HBA, the fide CA is ;= to the fide HA, ^ ^ ' ^' (Arg. 5.), AB is conunon to the tyro A, & the baft BC is == to the bafe BH (Prep. 2.). Wherefore, the V ABC, ABH, included by the equal fides AB, BC, and AB, BH, are = t<5 one another. ?. 8. B, I But the V ABH is a L (Prep. i.). Confequently, the V ABC will be aifo a L. Which was to be demonftrated. I. P. 46 J. J. Cor. 3. ifo. 2. Book II. Of EUCLID. 69 DEFINITIONS. Jjj V E R Y right angled parallelogram (DF), Is faid to be contained by any two of the ftraigfit lines (AD, DE,) which include one of the right angles (ADE). J. J right angled parallelogram may be thus denoted f hecaufe a right angle li the twojsdes which include it, are what determine this figure. When the length of the Jides AD, DE, including the right angle is fixed^ the mag^ nitude of the re^angle is determined^ its conftru^ion being compleated by drawing thro* the extremities A i^ E of tho/e fides, the lines (AD, DE,) parallel to them, according to D. ^S ^ P* Z^*B. I. 2. A right angled parallelogram DP. is for brevity fate of ten denoted by the three letters about the right angle, in this manner ; the Rgle Pgr ADE, // // alfo reprefentcd thus: The Rglf Pgr AD, DE, that is, the Rgle Pgr refulting from the two fides AD £:? DE, which form a right angle \^ is exprejfed thus: The Rgle Pgr under AD fj DE, or the Rgle Pgr ?/• AD y DE. H 70 The ELEMENTS M III 'M Bookn. DEFINITIONS. 3* Sometimes the parts 9/ a /Iralgbt line/irve to denote a rigU angled faral/elogramy for example (Fig, l ,), tie Jlrp/gbt line AB fc- inc^ divided in C, tbere may be defer i bed (P. 31. B. I.), with thft two lines AC, CB, a rigbt angled parallelogram^ by joining tbem atri^ht anflrs, fcf tbis parallelogram is exprejfed tbus: Tbe RglePgr AC, CB» cr Jimply tbe Rgle Pgr ACB, tbe letter that marks tbe point which u common to tbe two lines, being put between tbe other two letters ; tn like manner J by tbe Rgle Pgr ABC, is to be under flood tbe paralkla- gram defcribed according to tbe fame rules, one of vibofeftdes is AB w the other Y^Q, 4. When tbe lines AD y DB, including tbe rigbt angle, are equal (Fig^ 2.)» tbe parallelogram DC // a fquare (D. 30. B, i.). Js in tbis cafe one of tbe fides DB with tbe right angle, determine the fquare, which moj be defcribed from tbofe data by P. 31. B, I. This fquare may he ^^' preffed tbus : Tbe G of DB, or the D gf AD,. J Book II. Of E U C L I D. 71 DEFINITIONS. n. X H'E figure (ABCGDH) compofed of a parallelogram (DB) about the diagonal (BE), together with the two complements (AD, DC,) is called a Gnomon. The Gnomon Is marked by an arc of a circle (ahc)^ which pajfes thro^ the two complements (AD, DC,) W the Pgr about the diagonal. There may be formed in every parallelogram two different gnomons ; one^ by taking away (Fig. I.) from the whole Pgr, the greater Pgr ED about the diagonal \ the ^ther, by taking away (fig^ 2*) the leffer Pgr ED about the diagonals 72 Tlie ELEMENTS BookiL AXIOMS. L H E whole is equal to all its parts taken together. The whole Pgr PQ^ (Fig. 2.) // equal to all its parts, the Pgrs PR, TS, VQj taken together. 11. J[\, I G H T angled parallelograms contained by equal fides, are equal The Rgle Pgr DF (Fig. i.) // contained by the flraigbt lines AD, DE; confequentlyt if the Jlright line N // eaual to AD, (^ the firaight line M /' equal to DE, the Rgle formed by the firaight lines N y M, wiT/ he Mcejf^ r/ly equal to the Rgle DF. PROPOSITION I THEOREM I. X F there be two ftraight lines (AD & N), one of whicti (AD) is divided into aiiy number of parts (AB, BC, CD,) ; ^hfi redangle cortained b} ihefe ftraight lines (AD & N) is equal to th6 reQangles contained by the undivided line (N), and the fevet^l parts (AB, BC, CD,) of the divided line (AD). Hypothefis. Thelis. A.'Di^'^ are tivo ftraight lines ^ ofieofnvbich 'the Rgle AD . N is 2=: to the Rgles AD is di<uidedinto federal parts AB, BC, CD; AB . N + BC . N + CD . N. Preparation. 1. At'the point A m the ftraiaght line AD, ereft the ± AK. Pi 1 1. 5. r. 2. From AK, cut off a pan EA = N. P, y B. i* 3. Thro' the points D & E, draw the iiraight lines DH, EH, jjlle") toAE,AD. Cpai^i 4. And T hro' the points of diviTion B & C, draw the ftraight lines \ ' ^ * BF, CG, pilctoAEorDR J Demonstration, 1, X H E Rgle AH .18= to the Rgles AF, BG, CH, taken together. Ax, i.B.t. But becaufe the Rgle AH is contained by the ftraight lines EA, AD, (Prep. 3.), & AE is = to N (Prep. 2.). a. This Rgle AH is contained by the ftraight lines AD & N. Jx. 2. B. 2. Likewife, becaufe the Rgle AF is contained bv the ftraight lines EA, AB, (Prep. 4.), & EA is == to N (Prep. 2.). ' - - ■ .^ - *^ j^ ^ jj ^ i by the ftraight lines FB h BC, & that FB = N. P. 34. B. i. And fo of all the others. Confequently, the Rgle contained by the ftraight lines AD & N is = to the Rgles contained by the ftraight lines AB & N, BC & N , CD & M, taken together. That IS the Rgle AD . N is = to the Rgles AB . N + BC . N + CD. N. ^^' ^B.i, Which was to be demonftrated. K *^ 74 The ELEMENTS Bookll. I PROPOSITION II. THEOREM II X F a ftraight line (AC) he divided into any two parts (AB, BC,) ; the reftargle contained by the whol^line (CA), and each of the parts {AB, BC,), are together equal to the fquare of the whole line (AC). Hypothcfis. Thcfis. AC ij a ftraight line divided inf Jbe Rgle CAB + Rgle ACB, two parts AB, BC. ore = to tht D o/AC Preparation, 1 . Upon the ftraight line AC, defcribe the D AF. ' P. 46. B. i. 2. Thro' the point of fedion Bs draw the ftraight line BE pile to AD or CF. ^ P.31J.V T Demonstration. H E whole Rgle AF is = to the Rgles AE, BF, taken together. Ax. i. -B.J- But this Rgle AF is the Q of the line AC (frep. i.). 2. Confeqncnily, the Rgles AE, BF, taken together, are = to the D of the line AC. Ax.\M 3. But the Rgle AE isconra'ned by the ftraight lines CA, AB. becaufe it is contained by the ftraight lines DA, AB, of ".hich DA = CA, (Prep. I.). Ax.i.B/L 4. Likcwifc, BF is aRgle conta'ned hv the ftra'^ht lines AC, C^. l.e- caufe it is contained by the ftralghi lines EB, Bl , of which EB =: AC, (Prep. I &2.). P. 3V^'* 5. Wherefore, the Rj'lc coi:»^nined by the ftraight lines CA, AB, too- ther with the Rvlc coatalp. 1 by the ibaignt lines A\J, Ci;, i-? rr to the D of the ftra-jjht h'ne AC j or the- Rgle CAB -f the R rle ACB, are = to the D or" AC. ' Ax, i.BJ- Wl.ich was to be clcmonft rated. •*•;*-* I!^ Book II. Of E U C L I D. 75 •. PROPOSITION III. THEOREM III. j[ F a ftrai^ht line (AC) be divided into two parts in (B) ; the reSangle con- tained by the whole line (AC) &- of one of the parts (AB), is equal to the reftangle contained by tlic two parts (AB, BC,) together with the fquare of the aforcfaid part (AB). Hypotliefis. Thefis. AC is a firaight line divided 7be Rgle CAB is rr to tie ini0 any I wo parts AB, BC. * Rgle ABC +/^^De/'AB. Preparation. 2, 3- P. 46. B, I. PoJ, 2. Ptif,2. j0c,j.S,2. Upon the ftraight line AB, defcribe the D AE. Produce the line DE indefinitely to F. Thro' the point C, draw the ftraight line CF pile to AD or BE and produce it, until it meets DF in F. ^ ^ Demonstration. I . X H E Rgle AF is s= to the Rgles AE & BF taken together. But the Rgle AF is contained by the ftraight lines CA, AB ; becaufe 1 it is contained by CA & AD, of which AD ±: AB (Prep. 1.). VAr. 2. B. ^. And the Rgle bF is contained by AB, BC ; becaufe it is contained J by EB, BC, of which EB = AB (Prep, l^, Moreover, the Rgle AE being the D of the ftraight line AB, (Prep. I.). The Rgle of CA . AB, is = to the Rgle of AB . BC together with the D of AB i or the Rgle CAB is = to the Rgle ABC + the D of AB. ^^^ Which was to be demonftnited. K 2 2. 3 . I. B. I. M 76 The ELEMENTS Eooktt ., -. PROPOSITION IV. THEOREM IV. J_ F a ftraicjht line (AC) be divided into any two parts ( AB, BC,) ; tht Iqjarc of the whole line (AC) is equal to the fquares of the tn-o parts (AB, BC,) together with twice the reSangle contained by the parts (AB, lC,). Hvpothefts. The (is. AC is a firaight lin§ divided TAfQ 0/ AC is = to tif€ D of AB + into atij fwo parts AR, Bf , tJ^ D of BC + 2 Rglee ABC. Preparation, 1 . Upon AC, d«fcribe the Q AI, P. 46. B. \. a. Thro' the point of divifion B, draw BH pile to CI or AD. P.^i.B.i. 3. Draw the diagonal CD, which v/ill cut BH foinewhcrc in E. Pof. i. 4. Thro' the point E, draw GF pile to the oppofite fides Dior AC. P. 31. -5.1. Demonstration. likcwife AC, GF, DI, arc X5 EC A USE the lines AD, BH, CI piles fPrep. i. a. & 4 ). 1. 7 he four figures AE, EI, DF, GH, are Pgrs. D. 3$. S. 1. And fince each of thgfe figures include one of the right aisles of theD AI. r p ,^ J? I 2. Thofe Pgrs are alfo Rgles. i ^ +^- ^ Moreover, bccaufe the fiijes DA, AC, of the D AI, arc equal, ^ (D. 30. B. I.). 3. The V**is= tothe Vf. And becaufe the ftraight lines AD, BH, are pll^s (Prep, 2.) cut by the ftraight line DC (Prep. 3.). 4. The interior V r is =: to its exterior oppofite V /. 5. Confequently, \/ c =z \/ p, 6. Wherefore, the fide BE is = to the fide BC. 7. And the Rglc BF is a D, ^i«. the D of BC. 8. It may be proved in the fame manner, that the Pgr GH is a D, *viss. the D of AB, becaufe GE = AB. Moreover, BE being = to BC C^rg. 6.). 9. The Rgle AE, or the Rgle of AB. BE, will be = to the Rgle of AB . BC. But the Rglc AE is = to the Rgle EI (P. 43. B, i.) From whence it follows, that die Rgle £1 is alfo == to the Rglc ofAB.BC. * ^x.i.B.Uj Cor. 2. P. $. B. t. P. 29. B. I. jix,i.B.u P, 6, B. I. D,^o.B.u P.34.B.1. Jtx,^. B,2, B(;ok II. Of EUCLID. 77 ssaaiQSB o H I F D \ E • • \ A ] B C 11. ConTcquently, thetwoRglejs AE, £1, taken together, are ;= to twice the Rgle of the parts AB, BC. Since then the two D GH & BF are the fquares of the two parts AB &BC fylrg, 7. & 8.), & the Rgles AE, EI, taken together, are = to twice the Rgle of the parts AB, BC. 12. It follows, that the D of the whole line AC i« ;= to the D of AB-f- the Q of BC + 2 Rgles ABC. Which was to be demoi^ftniteci COROLLART. L yV HEN tnvoftraight lines HB, DF, pile to the fides of a fquare interfed each 0ther in a point E of the diagonal ^ the Rgles BF, DH, formed about the diagonal ^ tirefquares, COROLLARY IL J F the line AC he divided into t*wo equal parts in B, the complements AE, ET, are fquares, fcf thofe complements equal to ^e another, are alfo equal to the Jquares about the diagonal y \^ the the fquare of the njohole line AC is four times the fquare of one of the parts AB or BC. For BF, DH, arefquares (by the precedent Corollary), £sf ari equal to one ano* tber^ hecaufe BC = AB = DE. Moreon^ery AE being = to BF, y EI being z=z to BF (P. 2,^, B. I.), the complements AE, EI, are alfo fquares -, tjf fince they mre equal to one another ^ the D of AC = 4 D 0/ AB =: 4 D ©/ BC. 78 The ELEMENTS • PROPOSITION V. THEOREM V, J[ F aftraightline (AB) bedivided equally in (C) & unequally in (D) ; thercc* tangle contained by the unequal parts (AD, DB,) together with the fquarc of the part (CD), between the points of feaion (C & D), is equal to the fquare of the half (AC or CB) of the whole ftraight line (AB), Hypothefis. Thcfis, •AR is a ftraight line dMded ^ T^< Rgle ADB + /^<? D ff/CD, equally in C, ^ unequally in D. are =r to the D •/ CB, Preparation, 1. Upon the ftraight line CB, defcribe the D CF. 2. Th.o' the point of feftion D, draw DG pile to BF or CH. 3. Diaw the diagonal BH. 4. Thro* the point cf feftion E, draw IL pile to BC or FH, thro' the point A, the ftraight line AK pile to CL. Demonstration. FoJ, 1. Jl5e CAUSE the figure CF is a fquarc (^/^rf/v. 1.). The Rglei LG, DI, ahoui the (jiagonal are Q XCor, 1. Nar .. !v DI iheD '^^'DI3, & LG theDof CD ; becaufe LE = CD, P.l\X^' Moreover, ihe coniplea;ent CE is = to the complement EF. P. 43. -5- *• Let the fquare DI be added to bofh. * The Rgit CI will be = to the Rgle DF. Ax. 2. B. i. But hecaufe AL is = to CB (Hyp,), The Rj^le AL is = to the Rgle CL vfjr.a.BA 6. Confequently, the Rgle AL is = to the Rgle DF. Ax. i.B.i^ Therefore, if the Rgle CE be adued to b«th, 7. TheRgleAEwilIbe = totheRglesPF,CE, f. r to the Gnomon «^r. Ax.i.Bj. 8. But the Rgle AE is contained by AD, DB ; becaufe it is containrd by AD, DE, of which DE = DB C^rg. i .). Ax.i. B.i. 9. Ccnfequently, the Rple of AD. DB, is alfo z= to the Gnomon ahc. Ax. i.B.i' Atiiiifig to both the D LG, which is the D of CD (Arg. 2.). The Rgie AD. DR, together with the D of CD, will be = to the Gnomon aScy together with the Q LG. Ax. 2. B. !• But this Gnomon ahc together with the D LG, is = to the D CF, which h the D of the half CB, of the whole line AB fPrep. i.). 1 1. Wherefore, the R&,ic ADB + the D of CD, are = to the O of CB. Ax. i.B. i. Which was to be demonftratcct 5 la Book II Of E U C L I D. 79 Fa ftraight line (AC) be bifeQed in (B), & produced to any point E; the rcdangle contained hy the whole line thus produced (AE), & the part of it pro- duced (EC), tocrether with the fquare of the half (BC), is equal to the fquare of the ftraight line (BE) niade up of the half (BC) & the part produced (CE). Hypothefjs. Thefis. AC is a ftraight line hifeaed- in B. 7he Rgle AEC + the D of BC/ And ^jobicb is produced to the point E. is = to the □ of BE. Preparation, Upon the <}raight hne BE, defcribe the D BN. P, 46. B. 1. Thro' the point C, draw CL pile to EN or EK. P. 31. 5. 1. Draw the diagcnalEK. Pof, i. Thro' the point G, draw FH pile to EB or NK. ') p ^ I. 11 1. 2.. 3- 4- 5- And thro' the point A, draw the ftraight line AI pile to BK. j ' ^ ' Demonstration. Jl5 E C A U S E the figure BN is a fquare C^rep. 1 .). I. The Rgles CF, HL, about the diagonal are fquares. 34. B. I.). 4. Jor. 2. 4. 6. 8 Vc. ? Z'. 46. i i Cor. 3. ^. 2. B. I. And becaufe HG is = to BC r^. The D HL is z= to the D of BC. Moreover, /^Bbein?=to HQ (Hyp. 1.). The Rgle AH is rz to the Rgle BG. ' yfx.z.B.z. But the Rgle BG is= to the RgJe GN fP. 43. i?. i.). Therefore, the Rgle AH is alfo= to the Rgle GN. Ax.i. B.i. And if the Rgle BF be added to both. The Rgle AF will be = to the Rgles GN,BF, i. e. to the Gnomon nhc. Jx.2. B. i. _ But this Rgle AF is contained bvAE, EC J becanfe EC = EF (y^r^, i.). 7. Confequently, the Rgle AE. EC, is alfo = to the Gnomon ahc. Ax. \.B.\. Therefore, if the D HL, which is the D of BC (Arg. 2.), be added to both. The Rgle AE. EC, together with the D of BC, wlil be = to the Gnomon ahcy together with the D HL. Ax. 2. B.\. But the Gnomon tf^r & the D HL form theDof DE, (Trep. i). Conieqnentlj, the Rgle AEC + the D of BC is = to the D of VAl. Ax, t.B.i. Which was 10 be dcii.onftiaieJ. 80 The E L E M E N T S Book It. PROPOSITION ^HEOREMVIl 1. 2. 4- X F a ftraight line (BE) be divided into any two parts (BC, CE,) ; if« fquares of the whole line (BE) & of one of the parts as (Cfi^ arc equaJ to twice the reftangle contained by the whole (BE) & that part (EC), to- gether with the fquare of the other part (BC). Hypothefis. Theirs. BE is a ftraight line divided 7he Q of RE + the D a/ CE, art ^ . unequally in C, io z Rgles BEG + the D »/ BC. Preparation, Upon BE, dcfcribe the D BN. P. 46. B. i. Thro' the point C, draw the ftraight line CL pile to EN or BK. /*. j i. ^- »• Draw the diagonal EK. Po/ '• Thro' the point G, draw the ftraight line FH pile to EB or NK. P. 3 1 . ^- '• B Demonstration. E C A U S E the figure RN is a fquare (Pnp, 1 .). C ?. 4. i?. i- 1. The Rgles about the diagonal CF, HL, are Q ( Car, i. 2. Namely CF the D of CE, & HL the D of BC ; becaufe HG = BC. P. 34- ^ '• But the Rglc BG being = to the Rgle NG (P, 43. 5 i.) j xi the Q CF be added to boih. 3. The Rgle BF will be z= to the Rgle NC. Ax. 2, B. i. 4. Confequently, twice the Rgle BF is ^ to the Rglea BF & NC. And becaufe the Rglei BF, NC, are = to the Gnomon aic together wiih the D CF. c. This Gnomon ah( together with the D CF, will be alfo double of the RgieBF. Ax,i.B.i. But the Rgle BF is = to the Rgle contained- by BE, EC, becaufe F.F = ECr'^'Vf- '•)• 6. Wherefjre, the Gnomon /i^r together with the D CF i& = to twice the Rgle contained by BE . EC. Jx. l.-B.i. If the DHL which is = to the G of BC fyfrg. 2.) be added to both. 7. The Gnomon ahc + the D CF + the D hlL will be =: to twice the Rgle BE . EC -f- the D of BC. ^ Ax. %. B. i. Since then the Gnomon ahc -j- the D HL are = to the'D of BE, • and the O CF is the D of CE (Arg. 2 ). 5. It IS mamfeft that the G of BE -{- the □ of CE, are = to a Rgles BCC + theGof BC. ^ Ax.uSx Which was to be deraonftrated; Bookll< Of EUCLID. 8r diB O R £ c \ K i^N 1 K "■••••.. '....«••• ^ .N H sstfb PROPOSITION Vin. THEOREM nil X F a ftraight line (AB) be divided into any two parts (AC, CB,) ; four times the reSangle contained by the whole line (AB) i one of the parts (BC), together with the fquare of the other part (AC), is equal to the fquare of the ftraight line (AD), which is nnade up of the whole (AB), & the part produced (BD) equal to the part (EC). Hvpothefis. Thefis. AB is aflraigbt line di<videdin C, fcf Four times the Rgle ABC + the D produced to D,fo that BD = EC, of AC are =z to the Q of AD. Preparation, 1 . Upon AD, defcribe the D AN. P, 46. B. i. a. Thro* the points- B & C, draw BR& CO pile to DN or AP. P. ^i. S. u 3. Draw the diagonal DP. Pof, i. 4. Thro' the points L & K, draw GE & HF pile to DA or NP. P. 31. B. i. _^ Demonstration. OECAUSE the figure AN is a fquare fPrep. i,). 1. The Rgles about the diagonal CH, ER, FO, are fquares. And becaufe in the D OH, the fide CD is bifeded in B (Hyp J. 2. The Rgles BG, CL, LH, IM, are four equal fquares. 5. And the D CH is = to four times the D CL. Moreover, becaufe ER is a fquare (j^rg. 1.). 4. The Rgle EK is = to the Rgle KR. But iince IK = IC (Jrg, 2.), & CO pile to AP (Prep. 2.). 5. The Rgle AI is 1= to the Rgle EK. 6 Confequently, the R^e AI is alio = to the Rgle KR. Ukewiie, becaufe KM=MH(j^rg.z.)y & HF pile to N?CPrep. 4.) 7. The Rgle KR is = to the Rgle MN. 8. Wherefore, the Rglei AI, EK, KR, MN, arc.= to one another. {S cJ: B.2. I. Cor, ; ^.43. B, 2. B. P.36.B.1. Ax, I. B, I. P. 36.-5. I. Ax.i.B. I. 8a The ELEMENTS Book U p O R N * F E \ K i^. H G * l\ il \, ••< \ / I C fl D 9. Confequenily, their fum is =z to iour times ihc Rg j AL If the U CH which is = to four times the D CL C^^g- 3 J be added to both. 10. The Gnomon al^c which refults on one fide, is = to four times the Rgle AI & to four times the Q CI., 1. e, to four times the Rgle AL, the Rgle AI + th; D CL being = to the Rgle AL. Jx. i.B.i. Adding to both the D of AC, which is = to the D FO, becaufe AC = FKr^. 34. B. I.)'. 1 1. Four times the Rgle AL & the D of AC will be = to the D AN. Ax.2. B. i. But the Rgle AL is = to the Rgle contained bv AB, BC, becaufe BC = BL (Arg. 2.), & the D AN is z= to the t) of AD (Prep, i.). 12. Wherefore, four times the Rgle ABC + the D of AC, are = to the D of AD. Ax. i. B. i. B5ok n. or E U C L I D. 83 t / A^'-' F. CD Pi'vOPO:)IT10N IX. THEOREM IX. J[ F a ftra'ght line (AB) be diviikd into two equal parts (AC, CB,), & in- to two une(|UHl parts (AD, DB,) ; the fquares of the two uneqiid parts (AD, DB,) are together double of the the fquare of the jialf (AC) of the whole lire (AB) & of the fquare of the part (CD) between the points of fcQion (C ^' D). Hypothecs. Thefis. AB is aftrai^ht line Ji^sJed The D 0/ AD + the D o/DB, art equally in C ^ unequally in D. Jouhle of the G 0/ AC + the D of CD. Preparation, ^ 1. At the pomt C in the line AB, eredt the ± CE. P, 11. B. i. 2. Make CE = to AC or BC. P. 3. B. i. 3. From the points A & B to the point E, draw AE, BE. Pof i. 4. Thro' the points DiiG^ draw thci ftraight lines 1>G & GF plletoCE&AB, P. 31. A i. B Demonstration. EC A US E CE is = to AC (Prep. a.). »• The V CAE is = to the V iw. ^ P. 5. B. \, But the V ECA isa L f^rep. i.). >** ^' Wherefore, the two other V CAE & m toother, make alfo & IL. P, 32, JJ. i. 3- Coniequently, each of them is half a L. ; becaufe they arie ^ to one anoiher (^rg. 1.). It TttsLy be proved after the (ame manner that : 4. Each of the V CBE & n is half a L. 5. Coa(equcntly, the whole V )w + « is = ^o a L. ^a(, Z,B. i. Again, V n being half a L. C^^'g- 4)» ^ V EFG ^ L j being = to Its interior oppofite one ECB (P. 29. B, i.), which is a L^, rPrep. i). 6. The V EGF is alfo half a L. P, 3a. B. i. 7. Conlequently, EF is= to FG. P. 6. ^ 1. It is proved in the fame manner that : 8. The V BGD is = to half a L, & DG = DB. Since then the Dof AE is = to the D of AC together, with thcQ of CE r^. 47. ^. ».)» & AC = CE (Prep, 2.). 9. The D of AE is double of the D of AC. L2 i 84 The ELEMENTS BookIL 1 1 10. e. of the D of CD, For the (ame reafon : The D of EG is double of the D of FG, 1. becaufe FG = CD. P. 34. B. 1. u. ConfequentJy, the D of AE & the D of EG taken together, arc double of the D of AC & of the D of CD. Jx. 2, B. i. And becaufe the D of AE & the D of EG taken together, are = to the D of AG (P. 47. B. i. & Ar^^ 5.). 12. The D of AG is alfo double of the U of AC & of the D of CD. Ax, i. B. i. But V ECA being = to a L (Pr^p. i.), & V GDC = to V ECA, (P. 29. B. I.). The U of AG is = to the D of AD & to the D of DG. P. 47. B, i. Cr the D of AG is = to the D of AD & to the D of pB taken together, becaufe DB is = to DG (Ara 8.). W herefore, the D of AD & the D of DB taken together, are dou- ble of the D of AC & of the D CD ; or the D of AD + the D pf DB, are double of the Q of AC + the D of CD. Ax.i.B.i, Which w£8 to be dcmonftratc4. »3 IS Bookl!. Of EUCLID. 85 E — G /^. •^ In A r V" ^^ 1 I PROPOSITION X. THEOREM J^. __ F a ftraight line (AB) be bifefted in (C) & produced to any point (D), the Tqiiare of the whole line thus produced (AD) & the fquare of the part of it produced (BD), are together double of the fquare of the half (AC) of the whole line (AB), & of the fquare of the line (CD) made up of thp /ulf (CB) & the part produced (BD), Hypothcfis. Thefia. AB is a flraigbt line hifeded in C 7*^ D «/ AD + the D </ BD, Hre dow and produced tQ the paint D. hie of the D «/ AC + the D of CD. • Preparation. 1 . At the point C in the line AB, cre£t the ± CE. 2. Make CE = AC or BC. 3 . From the points A & B to the point E, draw AE & BE. 4. Thro' the points E & D, draw EG, 1}G, pile to AD & CE, and produce jyG until it meets £B produced, in F. Demonstration. P. \i,B. I. P.%. B, I. Pof, I. P. 31. A I. Pof. 2. XJE CAUSE in the A ACE thp fide AC is =c to CE (Pref, 2.), 1. The V CAE is =to V «. But V ACEisaL^P'-^A. i). 2. Hence each of the V CAE Sc mis half a L. It is proved in the lame manner that : 3. Each of the V / & « is half a L.. 4. ConfequcntJy, V « + « will be = to L.. Moreover, V/ being half a L- f^rg. 3.). 5. The V r will be aifo half a L. But the V BDF being a L ^P. 29. B, i.), becaufe it is the alter- nate of V ECD which is a L (I'rep. i.). The V ^ is alfo half a L. Conieqnendy, the fide BD is = to the fide DF. Likewise, V f being half a L C^rg, 6.), & V G a L., as being di- agonally oppofite to V ECD Y P. 34. B. I.). l^e V « is half a L. Therefore EG = GF. P. 5. B. I. P. 32. B I. 6. 7- 9- Jx,2.B.t. P.i$.B. I. P. 32. P. I. P. 6. -8. I. P,^t.B, I. P. 6. i?. I, I 86 The ELEMENTS Book .1 Alfo AC being = to CE (Prep, 2.). 10. The D of AC is = to the D of CE. 11. Confequentlv, the D of AC & of CE are double of the D of AC And thofe D of AC & CE being = to the D of AE (P. 47. B, i,\ 12. The D of AE wiU be alfo double of the D of AC, It is proved after the fitme manner that : The D of EF is double of the D of EG j i . e. of the D of CP, becaufe EG = CD. Confequentlv, the D of AE together with the D of EF, arc dou- ble of the d of AC & of the D of CD. But the D of AE & the D of EF being = to the D of AF, (P. 47. ^. '•)• The □ of AF is double •f the D of AC & of the D of CD. And this fame D of AF being alfo = to the D of AD & to the Q of DF (P, 47. B, 1.), or of BD, fince DF = BD (Arg. 7.). 16. It follows, that the D of AD + the D of BD, are double of the Dof AC + thcDof CD. Which was to be demonftiated. P. 46. 5. 1. p.46. »3 »S Ax.e.B,!. P^l\^B,x, .^^ ^ -i^ or EUCLID; 87 ^— ^ P R O P O S I T I O N. XI. PROBLEM I. X O divide a given Araight line (AB) into two parts, fo that the rcSan- gle contained by the the whole (BA) & one 6f the parts (AC) ihall be equal to the fqiiare of the other part (CB). Given Sought The Jlraight line AB. ^he point of in t erf e^ ion C, puch that the Rgle BAC Jball be = to the D of CB. Refolution, 1. Upon the faalghf line AB, defcribe the D AE. P. 46. B. i. 2. Bifea the fide BE in D, & draw thro' the point D to the Z'. 10. ^. i. point A the ftraight line DA. Pof i. 3. Upon EB produced, take DH = DA. P. 3. B, i. 4. Upon the ftraight h'ne BH, defcribe the D GH. P. 46, B. 1. 5. And produce the fide KC to F. Pof. 2. BD£MONSTRATIOy. E C A U S E the ftraight h'ne BE is bifedted in D & produced to the point H. p r 1 . The Rgle EH . HB + the O of BD is = to the D of DH. . V \ 2. And thisDof DHis = tothenofDA,becaufeDH = DAr/?c/.3.). \ r^ J. Confequentlj, the Rgle EH . HB + the 'H of BD is = to the Q ^ ^^''- 3- of DA. Ax.\.B.\. But this fame D of DA is = to the D of AB + the D of BD (P. 47. B. i). 4. Wherefore, the Rgle EH . HB + the D of BD is = to the D of AB + the D of BD. Ax. i. J5. i. Therefore if the D of BD be taken away from both fides. c The Rgfe EH . HB will be =z to the D of AB. Ax, ^.B.i. And if from the Rgle EH. HB which is = to t^e RgleFH^/?^/ 4.5.) and from the D of AB which is = to the Q AE (Ref 1.) the Rgle FB be taken away. ^ 6. There virill remain the D CH = to the Rgle GC. Ax/^.B.i, This □ CH being therefore = to the D of BC (Ref 4.), & the Rgle GC = to the Rgle BA . AC ; becaue AG = AB (Ref i.). 7 It follovrSy that the ftraight line AB is divided in C, fo that the Rgle BAC is = to the D of CB. Ax.j.B.t: Which was to be done. B.2. B. I. 88 The ELEMENTS I PROPOSITION XII. THEO REMXL ^ N any obtufc angled triangle (CBA) ; if a perpendicular be drawn from one of the acute angles (B) to the oppofite fide (CA) produced ; the fquarc of the fide (EC) fubtending the obtufe angle (A), is greater than the fquares of the fides (AB, CA,) containing the obtufe angle, by twice the reSangle contained by the fide (CA), upon which when produced the per- pendicular falls, & the ftraight line (AD) intercepted between the perpen- dicular & the obtufe angle (A). TTiefis. The D i/BC w= /# th^ '/A? + /i^D«/AC + aRglejCAP. Hypothefis. /. CBA is an obtuft angled/^, II. ED thf A^ dratvfi from the ^otrttx of the VB /o tb€ oppofite fidi QK produced. Dbmonstratigk. IJECAUSEthe ftraight line CD is divided into two parts CA, AD, (HyP, a.). I . The D of CD is = to double tbe Rgle CA . AD together with the U ofCA&ofAD. P.4. *•*• Therefore if the D of BD be added to both fides, a. The G of CD + the D of BD, will be = to double the Rgle CA. AD +thc Dof CA + the Oof AD ^theDof BD. Xir.2.^.»- But the D of CD together with the Q of BD is = to the D of BC, and the D of AD together with the Q of BD is = to the D of AB, (?, 47. 2?. I.). 3. Confequently, the D of BC is = to double the Rgle CAD + the U of CA + theDof AB. AxAt^' AVliichwas to be deraonftrated. Book II. Of EUCLID. 89 I A^ r ^C D PROPOSITION XIIL THEOREM XII XN every acute angled triangle (CB A); the fquare of the fide (BA) fub- tending one of the acute angles (C), is lefs than the fquares of the fides (CB9 CA)) containing that an^le, by twice the redangle contained by One of thofe fides (AC) & the ftraight line (CD) intercepted between the perpendicular (BD) let fall upon it from the oppofite angle (B), & the acute angle (C). Hypothcfis. Thefis. /. CBA is an acute angUd A. ^the D »/B A + tviice the Rgle ACD //. BD the ± let fall ufion AC is = to the D </ C A + the D of CB. from the oppofite angle B. Demonstration. J3 £ C A U S E the ftraight line CA is divided Into two parts CD, DA, (Hyp, 2.). I, The D of CA together with the D of CD is £= to twice the Rgle AC. CD together with the D of AD. P. 7. B. 2. Therefore if the D of DB be added to both fides: s. The D of CA + the D of CD + the □ of DB will be == to twice the Rgle AC . CD + the D of AD + the D of DB. Jx. z. B. i. But the Dof CD -f the Dof DB is=: to theD of CB, & the Dof AD + the D of DB is = to the D of BA (P, 47. B. i.). 3. Wherefore the D of BA + twice the Rgle ACD is = to the D of CA + theDofCB. . .*p. i.Ai. Which was to be dcmonftratcd. M 50 The ELEMENTS Book II PROPOSITION XIV. PROBLEM IL J[^ O defcribe a fquare that (hall be equal, to a given redilbeal figure (A). Given Sought Tbi ndilintal Jigurt A. W# conJfruSion •f afyuan = /• a gi*u€H rt&ilineml fg^rt A. Refolution. I. Defcribe the Rgle Pgr CE = to the figure A. P. 45, 1. 1. %. Produce the fide BE, & make EF == to ED. P. 3. ^. i. 3. Bife^t die ftraight line BF in H. P. 10. i. i- 4. From the center H at the diftance HB, deferibe the BGF. P«/ 3. 5. Produce the fide DE» until it cuts the Q BGF in G. F9J, i. Preparation. From the point H to the point G, draw the ftraight line HG. F^f. i. B Demonstration. EC AUSEBF isdividedequalljinH&unequally inEf'^f/; ^ &2.). 1. The Rgle BE . EF together with the D of HE is = to the Dof HF. P. J. ^- ^« a. AndiecaufeHF=HG (D, 15. i?. 1.), dieDof HF = the Dof HG, the I^leBE . EF + the Q HE i3j==p the O of HG. ^ ^ \ £^3 ^i. .A I. But theDofHG beii^=: to the D HE + the Dof EG (P.^J.B, 1.). 3. The I^leBE.EF + theDof HE is alfo=to the DofHE + theDofEG. Ax, Therefore, if the D of HE be taken awaj from both fidei : 4. The Rgle BE . EF will be =: to the D of EG. Ax. 3 And tlus Rgle BE EF being moreover =: to the Rgle BE . ED 1 be- caufeEF = ED (Ref. 2.). 5. The Rgle BE . ED will be alfo = to the D of EG. Ax. \.B,h But the Rgle BE . ED is =: to thj given figure A (Ref, i.). 6. Confequently, the D of EG will be alfo = to this given figure A. Ax. i,B.u Which was to be done. J REMARK, X P ihepciiit U falls upon thi point E, thefiraight lines BE, EF, ED, wil ^ each equal to EG, & tbi Rgle Pgr CE iffil/, tMilh the fquun fought (Cor. i. ^ 3. ofF. 46. -B. I.). r Book in. Of EUCLID. 9« DEFINITIONS. J^ Straight line (ADB) is faid to teucb a circU when It ipeet* the circle it beiug produced does not cut it. Fig. i . n. Circhs are faid to toutb one another when their circumferences (ABC^ CEF, pr ABC> GBH) meet but do not cut one another, fig. 2. ni. Two circles touch each other externally, when one (CEF) ftUe without tht other (ABC) : but two circles touch each other mternalljr, ^htn 9iu (GBH) falls within the other (ABC). Fig. 3, Ma 9Z The ELEMENTS Bookm. 1 P E F I N I T I O N S. IV. ^ H E diftancc of a ftraight line (FB) from the center of a circle, istbeper- pendicular (CM) let fall from the center of the circle (C) upon tbisjlraiibilini '(FB); for "wh'ichrtSLhn two Jlraigbtlinei (FB, DE,) are faid to heemtdlj diftant from the center of a circle^ when the perpendiculars (CM, CN,) kt fall upon thofe lines (FB, DE,) from the center (C), are equal. -F// »• hxA a flraigbt line (KQ) is faid to he farther from tie eenter of thecirck than (BF or ED), when the perpendicular (CH) drawn to this line from * ^«nter (C), is greater than (CM or CN). Fig. i. VI. The angle of afegment. Is the angle (CAB or DAB) formed by the 9tA (CA or DA) of the fegment (ACB or ADB) & by its chord (AB). P'ti^^ Book IIL Of EUCLID. 93 DEFINITIONS. ' vn. A N angle in a ftgmenty is the angle (BAC) contained by two ftraight lines (AB, AC,) drawn from any point (A) of the arch of the fegment, to the extremities (B & C) of the chord (BC) which is the bafc of the fegment. Fig, 2. Wbtn the firaight lines (AB, AD,) are drawn from a point (A) in the circumference of the circle^ the angle (BAD) is an angle at the circumference : but when the firaight lines (CB, CD^) art drawn from the center y the angle (BCD) // an angle at the center. Fig^ i. VIII. An angle is faid to infifi or fi and upon the arch of a circle y when the ftraight lines (AB, AD, or CB, CD,) which form this argle (BAD, or BCD,), are drawn ; cither from the fame point (A) in the circumference ; or from its center (C;, to Ac extremities (B& D) of the arch (BED). Fig. i. * IX. Afe^or of a circle^ is the figure contained by two rays (CAf CB,) & the arch (ADB) between thofe two rays. Fig. 3. 1 A ^-\ 9^\ 54 The ELEMENTS Bookm. n W> QU AL circles (ABD, EGH,), are thofc of which the diamcten ', EH,) or the rays (CB, FG,) arc equal. Fig, i. If tbe eircUi be applied to one another, fo that their eenUn eoineide, wk^ their raye are efualf tbe circks muft Ukevjtfe eoincida* II. Similar fegments of circles (ABC, DEF,), are thofe in which the tngkt (ABC, DEF ,), arc equal. Pig. a. Circks arefmiUr figures. If then the two fegments (ABC, DEF,) ^ ^^ ken av^ay hy Juhjlituting the efual angks (ABC, DEF,), tbofe fegments en fimilar. r Book III. Of E U C L I D. P5 r^ soBeaaam L essaBBMaBcssassssssassBBBBssssssasBM^^ PROPOSITION I. PROBEEM I. O find the center (F) of a given circle (ACBE). Given Sought Tbe © ACBE, The anter^ ^f this 0. Refolution. 1. Draw the chord AB. Pof, i, 2. Bifedt it in the ^int D. P, lo. S, i. 3. At the point Din AB» ere^ the X I>£ & produce it to £. P.ii.B.i. 4. BiTcdt CE in F. A 10. M. u Tbe point F will be the center fought of the given ACBE. Demonstration. If not. Some other point at H or G taken in the line> or without the line EC> will be the center fought of the ACBE. Cafe I BSuppofe the center to be in EC at a point H different from P. £ C A U S £ the center of the is in the line EC, at a point H dif- ferent from F J Sup, i J. 1. The rays HE&HC are = to one another. D, i^.B, i. But FE being = to FC (Rff. 4.) & HC < FC (jise. 8. B, 1.). 2. HC will be alfo < FE, & a fortiori < HE. 3. Therefore HE i« not = to HC. 4. Confequently, the point H taken in the line EC different from the point F, cannot be the center of the ACBE. Cafe II Suppoie the center to be without the line EC in the point G. Preparation. BDraw from the center G, the ftraight lines GA, GD, GB. Pof i. E C AUS E in the A AGD, DGB, the fide GA is = to fideGB (Prep, if D. 15. B, i.), the fide GD common to tbe two A, & the bafe AD = to the bafe DB (Ref 2.}. J 96 The ELEMENTS Bookm "^^ 1. The adjacent W M-^t& cto which the equal fides 0A» GB> are oppoiite, are =: to one another. P. 8. B. u 2. Therefore V « + ^ isa L. D. loJ*. But V A being alfo a L (Rff. 3.). 3. It follows, that V «+ A is = to V 41, which is impoflible. ^* 8- ^- '■ 4. Therefore the point G taken without the line EC, cannot be the cen- ter of the © ACBE. Conlequently, fince the center is not in the line EC, at a point H diffe- rent from F/^C«/> I.) nor without the line EC in a point G(^Ctf/>. 11.) 5. The center fought of the © ACBE, will be necelfarily in F. Which was to be done, ' COROLLjfRT. J Fin a circle ACBE, a cborJ EC hifeds anbtber chord AB ai right an f lis ; this thord CE is a diameter^ fef confequtntfy pajfes tbrg* tbe center of the circht (D. 17. B, I.). Book in. Of E U C L I D. 97 -. PROPOSITION II. THEOREM L J[ F any two points (A & B) be taken In the circumference of a circle (AEB) ; the ftraight line (AB) which joins them^ ihall fall within the circle. HTpothciis. Thefis. The tw0 points A &f B art tahn fbeftraiebt line PA falls in ibe O AEB. nnitbin tbt AEB. Preparation^ . 1. Find theccnterCof ©AEB. P. i. B, 3. 2. Draw the ftraight lines CA, CD» CB. iV^ 1. Demonstration. x3 EC AUS E in the A ACB, the fide CA is = to the fide CB, (Prtf. 2. & D. I J. B. 1.). I The V CAD, CBD, are = to one another. P. 5. P. i, " But V CDA being an exterior V of A CDB. %. It is > than its interior CBD. P. 16. P. i. And becaufe the V CBD is = to the V CAD (Arg. i.). 3. This V CDA will be alfo > than V CAD. 4. Confeqnently, the fide CA oppoiite to the greater V CDA, is > the fide CD oppofite to the ieflcr V CAD. P. 19. P. u 5 From whence it follows, that the extremity D of this fide CD falls * within the ©AEB. And as the fame may be demonftrated with reipedt to any other point. in the line AB. .6. It is evident that the whole Jine AB fails within the © AEB. Which was to be demonitrated. N g% The ELEMENTS Book in PROPOSITION IIL THEOREM IL J, F a diameter (CD) bifeds a chord (AB) m (F) ; it Aall cut it at nght angksy & reciprocally if a diameter (CD) cuts a chord (AB) at right anglesj it (hall bifed it. HTpothcCi. TIicCs. CD is a diamier a/ tbt AOBD, Th Marnier CD is 1, «/m %nbicbHfeasARinY. ibe cb^rd AB, Preparation. Draw the raya EA, EB. ?•/. i. Y Demonstration. In the A AEF, BEE, the fide EA is = tt> the fide EB (Prep, fr D. 15. B. I.), the fide EF is common to the two A» & the bale AF i8=tothcbafeBFr^/./ I. Cbnfequently, the adjacent V « fr ir* to which the eqaal fides EA» EB, are oppofite, are = to one another. P, 8. A r. %. Wherefore, the ftraight line CD» which ftaads upon AB oiakiiy the adjacent V ^ & ^ = to one another, is X upon AB. Z>. 10. if. 1. Which wa« to be denpoftrattd. H. Hypothefis. Thefis. CD is a diameter tf tbe ACBD, J- ypon AFm = /«FB. tbe cbord AB ; er vubicb makes \/ m^i^ n. Demonstration. _ H E fides EA, EB, of the A AEB being = to one another (Prep. & D. 15. B. i.). I. The V EAF, EBF, will be alfo r= to one another. P, 5. B. u Since then in the A AEF, BEF, the V EAF, EBF, are=r (jfrg. i.), as alfo the V »i & « (Hyp-)* & the fide EF common to the two A. 0. The bafe AF will be = to the bafe FB. />. a6. J5, i. Which was to be dcmonftrated, Bookm. Of E U C L I D. 99 PROPOSITION IV. THEOREM III jL F in a circle (ADCB) two chords {jkC, DB,) cut one another, they ai« divided into two unequal parts. Hypothefia. Vhe itoo chords AC, DB, of tbe © ADCB £ui om another in the foint £. Demonstratioi^. If not. The chords AC» DB» hiied one another. Thefis. Theft chords an divided rff» $p itvo unequal parts. B. Preparation. From the center F to Ae point E, draw die portion of thedia- meter F£. Prf.t. lECAUSE the diamctcr,or its part FEybife^ls each of the chords AC, DB, of die © ADCB (Sup.). 1. This ftiaight h*ne FE is ± upon each of the chords AC, DB. P. $. B i Z. Confequently, the VFEB, FEA, are = to one another i which f^ir.io.^.i. isimpoifible. ^ X^^x.S.B.u 3. Wherefore, the two chords AC> DB» aie divided into two unequal partt. Whidi.wastob|r4einopftiatai . ^^mmd n too The E L E M E N T S SOB BooUm PROPOSITION V. THEOREM IF. y F two circles (ABE, ADE,) cut one another, they fliall not have the bm center (C). HTpothefis. ABE, ADE, are two © iviicb cut Me another in tbefoints A £* E. Thefis. ^ 7bofe /w« © have diffennt centers. DEMONSTRATION. Ir not, The elides ABE, ADE» have the fiune center C Preparation. I. FrooithepointCtothepointof fedionA, drawtherayCA.)^/- , a! Andfrom the fame point C, draw the ftiaight line CB i which j ^' cuts the two ® inD & B. Because the ftmight lines CA, CD, are drawn from the center C totheOADEri''•^^l.&0. , . Thcfe ftraight lines CA, CD, are = to one another. It improved in the fame manner, that : 1 The Wight lines CA, CB, arc = to one anoAer. 4* Confcqucntly , CB will be = to OD; which is impoffible. 4, TherefoiB, the two circles ABE, ADE, have not the fame center. Which was to be demonftiated. D.iSJ.i. Book in. I PROPOSITION VI. THEOREM V. F two circles (BCA, ECD,) touch one another internally in (C) ; they ftiall not have the fame center (F). Hypothefie. rhe © ECD mcbes the © BCA internally in C. Demonstration. I? not, ' The © BCA, ECP, have the feme center F. Thefia. Vlfff fwQ ® bave different centers. Preparatinn. B Draw the rays FB, FC _ BECAUSE the point F is the center of the © BCA (Sup.). I. The rays FB, FC, are := to one another. * Again, the point F being alfo the center of © ECD (Sup,) Z. The rays FE, FC, are = to one another. 3. Confeqnently, FB = FE (Ax. i, B. \.) -, which is impoffible. 4. WhcrdforC; the -two © BCA, EjCD, have not the feme center. Which was to be demonftrated. D.i^.B.i. Ax.B.B.i. 102 The ELEMENTS Bookm. PROPOSITION VII. THEOREM n I F any point (F) be taken in a circle (AHG) which is not the center (E)i of all the ftraight lines (FA, FB, FC, FH,) which can be drawn from it to the circumference, the gre^teft is (FA) in which the center is, &^ part (FD) of that diameter is the leaft, & of any others^ that (FB or FC) which is nearer to the line (FA) which paffes thro* the center is always grwW than one (FC or.FH) more remote, & from the fame pobt (F) there can be drawn only two ftraight lines (FH, FG,), that arc equal to one another, one upon each fide of the fhorteft line (FD), Hypotheiis. /. The point F taken in tie © AHG /| not the center E. //. The ftraight lineT Af drawn from the point F, paffes tbr^ the center^ tf the © AHG. Thcfis. /. FA is the greatef •/ «« ^ ftraight tines which can he dre^ from the point F t9 f^^ OAHU //. FD w the leaft. .., Ill JnJ of any others FBirFC^M is nearer t$FAis>FCirYn»^' remote, fF. From the point? there can he dre^ only two ^ftraight /»««P^»,^' one upon each fide of the ftftfttfm Jll And the ftraight lines FB, FC»FH, are drawn from the point F to the OAHG, /. Preparation Draw the rays £B, EC, £H, &c. Fig. t. T Demonstration. H E two fides FE + EB of tlie A FEB are > the thiid FB. P.ao. Ai. But £B is = t«» EA (D. ic. B, i.). a. Therefore, FE + EA, or FA is > Fa It is proved in the fiiiiic manner that : 3. The fhaight lint FA, is the greateft of ail the ftraight itnea drawn from the point F to the O AfK}. Which was to be demonftrated I. 4. Again, the two fides FE + FH of tlit A FEH are > the third ER P, to. B. i And ED being =: to EH (D, 1$. B. i.). Bookni. Of E U C L I D. 103 5. The ftraiglit lines PE + FH are llfo > ED. Therefore^ takingaway from both (Ides the part PE : 6. Th* ftraight line FH wiU be > FD i or FD < FH. Ax. 5. B, u It is proved in the (ame manner that : 7. The ftiaight line FD, which is the produced part of FA, is the kaft of all the ftia^ lines drawn from the pomt P to the O AHG. Which was to be demonftrated. II. Moreover, the fide P£ being common to the two A FEB, FEC, the fide EB = the fide EC (T>. 15. B. i.), & the V FEB > VFECr^*. 8. B. I,). «. The bafe FB will be > the bafe PC. P, 24. B. x. For the feme reaibn : 9. The fhaight line PC is > FH. 10. Confeqnently, the fhaight line FB or PC which is nearer the line FA, which pafles thro' the center, is > PC or FH more remote. Whichwastohedeoumftiatcd. III. //. Preparation. Fig. 2. 1. Make V PEG =: to V F^H, & produce EG un^ k meets the O AHG. P. aj. B, t, 2. From the point F to die point G, draw the ftraight line FG. Pof, i. Then, EP being conunon to the two A FEH, PEG, the fide EH = the fide EG r^. 15. B. i.), fc the V FEH= to the V PEG (H. Prtp, I.). 11. The bafe FH will be = to Ae bafe FG. P. 4. B. i. But becaufe any other firaight line, difierent from FG, is either nearer the line PD, or more remote from it, than FG. 1 a. Such a ftraight line will be alfo < or > FG (Arg. 10.). ij. Wherefore, from the (ame point P, there can be drawn only two ftraight lines FH, FG, that are = to one another, one upon each. Qde of the fliorteft line FD. Which was to be demonftrated. I\^. ^ 104 The E L E M E NT S JBooklD. v^m^l'^^ir ^ M \. JL .^••^ ^•^r7~|A ^ i,-^" ~"" 3§/^ G^^s:^ ^%^ PROPOSITION VIII. rUEOREMVIL X F a point (D) be taken wkhont a circle (BGCA), & ftraigfat lmc9 (DA, DE, DF, DC,) be drawn from it to' the circumference, whereof one (DA) pafles thro* the center (M) ; of thofe which faH upon the concave circumference, the greateft is that (DA) which pafles thro* the center ; & of the reft, that (DE or DF) which is nearer to that (DA) thro* the center, is al- ways greater than (DF or DC) the more remote : but of thofe (DH, DK, DL, f)G,) which fall upon the convex circumference, the lead is that (DH) which produced pafles thro' the center : & of the reft, that(DKor DL) which is nearer to the leaft (DH) is always lefs than (DL or DG) the more remote: & only two equal ftraight lines (DK, DB,) can be drawn from the point (D) unto the circumference, one upon each fide of (DH) the leaft. Hypotheiis. /. The point D is taken tvitbout a BOCA Iff tbtfawu plane, IL 7b€ ftraight lines DA, DE, DF, DC, are dranun from this point to the concave part of the © BGCA. ///. jlnd thofe ftraight lines cut the con<vex part in the points H, k, L, G. Thefis. /. DA ^bich pajjes thro' the center M is the greateft of all the ftraight Una DA, DE, DF, DC. //. DE or DF, ^bich is nearer to DA is > DF or DC, the more remote. Ill, DH tvbich ^hen produced pajfes tbr^ center M is the leaft of all the ftraigi^ lines DH. DK. DL, DG. IV. DK or Dl, *which is nearer to the iisu DH, is < DL or DG the mere resmte, V, From the point D only tivo equal ftraigbi lines DK, DB, can he drafuM, 9iu fc^it each fide of DH the leaft. L Preparatioju Draw the rays ME, MF, MG, MK, ML. ;^— «^ Demonstration. I. 1 H E two fides DM+ ME of the A DME are > the third DE» P. lo, B. And becaufe ME = MA (^Z>. 15. B. i.). 1. Book Itt Of E U C L I D. 105 ii. DM + MA or DA will be > D£. It is demonftrated af:er the fame manner that : 3. The ftraight line DA, which pafTes thro' the center M, is > any other ftraight line drawn from the point D to the concave |)art of the BGCA. Which was to be demonflrated I. Moreover, DM being common to the two A DME, DMF, ME =z MF (D. 15. B, I.), & V DME > V DMF (Ax. 8. B, i.). 4. The bafe DE will be alfo > the bafe DF. P, 24. B, i. In li)ce manner it may be ihewn that : 5. The ftraight line DF is > DC, & fo of all the others. 6. Confequently, the ftiaight lines DE or DF, which is nearer the line DA, which pafies thro' the center, is > DF or DC more remote. Which was to be demonftrated. 11. 7. Again,* the fides DK + KM of the A DKM are > the third DM. P. 20. B. 1. If the equal parts MK, MH, (D, 15. ^5. i.) be uken away. 8. The remainder DK will be > DH, or DH < DK. It may be proved in the iame manner, that : 9. The (ha^ht line DH is < DL, & fo of all the othcfrs. 10- Confequently, the ftraight line DH, which produced pailes'thro' the center M, is the leaft of all the ftraight lines drawn from the point D to the convex part of the © BGCA, Which Was to be demonftrated. III. Alio, DK, MK, being drawn from the extremities D & M of the (ide DM of the A DLM to a point K, taken wiihin this A (Hyp, 3.). n . It follows, that DK + MK < DL + ML. P. 21. B, i. And taking away the equal parts MK, ML, (D, 15. B. i.), la. The ftraight line DK will be < DL. In like manner it may be (hewn, that : 13. The ftraight line DL is < DO, & (o of all the others. 44. Confequendy, the ftraight lines DK or DL, which are nearer the line DH, which produced paiTes thro' the center, are < DL or DG the more remote. Which was to be demonftrated. IV. //. Preparation. 1. Make V DMB = VDMK, &produce MB 'till it meets the O. P. 23. B. i. 2. From the point D to the point B, draw the ftraight line DB. Pof. i. Then, the fide DM being common to the two A DKM, DBM, the fide MK=thefideMBrZ>.!S.^.i.).&VDMK=VDMBrn.PrfAi.). 15. The bafe DK will be = to the bafe DB. ' P. 4. B, i. But becaufe any other ftraight line difterent from DB, is either near- er the line DH or more remote from it, than DB. i6. Such a ftraight line will be alfo < or > BD (^rg. 14.). J 7, AVherefoie, from the point D, only two := ftiaight lines DK, DB, ' can be drawn, one upon each fide of DH. Which was to be demonftrated. V. O io6 the ELEMENTS Book III. PROPOSITION IX. THEOREM VIII If a point (D) be taken within a circle (ABC), from which there faB more than two equal ftraight lines (DA, DB, DC,) to the circumference; that point is the center of the circle. Hypothecs. Thefis. . From the point D, taken within a © ABC» Tbe point D is ibtctnttrt} there fall more than tijuo equal Jlraight lines the ABC. DA, DB, DC, to the O ABC. Demonstration. If not, B Some other point will be the center. ' E C A U S E the point D is not the center fSupJ^ & from this point D there fall more than two equal llraight lines DA, DB, DC, to the O ABC fHypJ. 1. It follows, that from a point D, which is not the center, there can be drawn more than two equal ftraight lines ; which is impoflible. P- ?• ^* ^" 2. Confequently, the point D is the center of the © ABC. Which was to be demonftrateiL Book ra. Of E u e L I D. 170 PROPOSITION X. THEOREM IX. i^NE circumference of a circle (ABCEG) cannot cut another (ABFCG) in more than two points (A & B). Hypothciis. Thcfis. ^bt ttvo © ABCEG, ABFCG, cut nej cut one another only in twf one another. points A W B. Demonstration. If not. They cut each other in more than two point?, as A, B, C, &c. Preparation. 1. Find the center D of the © ABCEG. P. i. j?. 3. 2. From the center D to the points of feflion A, B, C, &c. draw the rays DA, DB, DC. Pof. u j3 E C A U S E the point D is taken within the © ABFCG, & that more than two (Iraight lines DA, DB, DC, drawn from this point to the circumference of the © ABFCG, arje equal to one another, (Prel^. I.&D.iS.B,i.), 1. The point D is the center of this ®. P.g.B.^. But this point D being alfo the center of the © ABCEG (Prep. i.). a. It would follow, that two © ABFCG, ABCEG, which cut one ano- ther, have a common center D ; which is impoflible. P, 5. ^. 3. 3. Confequently, two © ABCEG, ABFCG, cannot cut one another in more than two points. Which was to be demonftrated. Da io8 The ELEMENTS BookE PROPOSITION XI. THEOREM X. J_ F two circles touch each other internally in (A) ; the ftraight line which joins their centers being produced, fliall pafs thro* the point of cortad (A). Hypothcfis. Thcfis. Vhe ftraight line CA joins tbe centers of 7bis ftraight hne CA hein^ff the two ® AGE, ABF, which touch ^ ductd, paps thro' tbe point •/ each other internally in A. contad A of thofe two 0. Demonstration. If not, The ftraight line which joins the centers, will fall otherwife» as the ftraight line CGB. Preparation. B From the centers C&D to the point of contact A, draw the lines CA, DA. /"•/ »• E C A U S E in the A CDA, the two fides CD & DA taken toge- ther, are < the third CA (P. 20. -B. i .), & that CA = CB (D. 15. B, 1.). I. The ftraight lines CD + DA will be alfo > CB. Therefore, if the common part CD be taken away from both fides. The ftraik;ht line DA will be > DB. * -^jr. 5. B. i But the ftraight line DA being = to DO (Pref. & Z>. 15. B, i.). DO will be alfo > DB, which is impofliblc. jfx, 8. t, ^ Wherefore, the ftraight line CA, which joins the centers of the © AGE, ABF, which touch each other internally, being produced, will pafs thro* the point of contadt A. Which was to be demonftrated. Book m. Of EUCLID. 109 I PROPOSITION XII. THEOREM XI. __ F two circles (DAM, GAN,) touch each other externally ; the ftraight line (BC), which joins their centers, ihall pafs thro' the point of contad (A). Hypothefis. ne Jlraigbt Ijne ^Q joins the cent en #/ the fwo © DAM, GAN, lobUb $9ucb eacb otber external^ in A. Thefis. nitftraigbt line 'BCpaJfes thro^ tbe point of contad of tbe t'WO ©. Demonstjiation. If not» This ftraight line, which joins the centers, will pafs otherwiie, asBDGC. Preparation. Bi Draw from the centers B & C to the point of contact A, the rays BA, CA. A/ i. BECAUSE BA is = to BD, & CA = to CG (1>. 15. B. 1.). I. The ftraight lines BA + CA are = to the ftraight lines BD -f CG. Ax. 2, B, i. And if the part DC be added to the ftraight lines BD + CG. 3. BD + DG + CG, or the bafe BC of the A BAC is > the two fides BA + CA, which is impofllble. P. ao. B. i. 3. Therefore, the ftraight line BC, which joins the centers, will pafs thro' the point of contadt A. Which was to be demonftrated 1 4 i i no PROPOSITION XIII. THEOREM XII _ W O circles (ABCD, AGDF or ABCD, BECH,) which touch each other ; whether internally ; or externally : cannot touch in more points than one, Hypothefis. Thcfis. I © ABCD touches © AGDF iniernally. The © ABCD, AGDF, or ABCD, //. © ABCD touches © BECH externally. 6ECH, touch only in one point. . If not. Demonstration. 1. Either the © ABCD, AGDF, touch each other internally in more pointy than one, as in A & in D. 2. Ox the © ABCD, BECH, touch each other externally in more points than one, as in B & in C. /. Preparation. I. Find the centers M & N of the © ABCD, AGDF. P, \. B. 3. B2. Thro' the centers, draw the line MN, & produce it to the O. Pof, i.^ z. E C A U S E MN joins the centers M & N of the two © ABCD, AGDF, (Prep, a.) which touch on the infide (Sup. i.). 1. This ftraight line will pafs thro' the points of contaft A & D. P. 11. -ff. 3. But AM is = ioMD(L Frep. 2. & />. 15. B. i.). i. Therefore, the ftraight line AM is > ND, & AN is much > ND. Ax. 8. B. 1. But fince AN is = to ND (I. Prep. 2. & i>. 15. J?, i.). 3. The line AN will be > ND & = to ND ; which is iinpoffible. 4. Confcquently, two © ABCD, AGDF, which touch each other in- ternally, cannot touch each other in more points than one. //. Preparation. Thro* the points of contad B & C of the © ABCD, BECH, Bdraw the ftraight line BC. Pof. i. E C A U S E the line BC joins the two points B & C in the O of the © ABCD, BECH, (II. Prep.). 1. This ftraight line will fall within the two © ABCD, BECH. -P. i. A 3- But the © BECH touching extemaUy the © ABCD (Sup. 2.). 2. BC, drawn in the © BECH, will fall without the © ABCD. D. 3. B. 3, 3. Confcquently, BC will, at the fame tinje, fell within the © ABCD (Arg. 1,), & without the fame © (Arg. 2.) j which is impoilibe. 4. Wherefore, two © ABCD, BCEH, which touch each other exter- ' nally, cannot touch each other in more points than one. Which wa£ to be demonftrattd. r" BooklU. Of E U C L I D. Ill I PROPOSITION XIV. THEO REM XIII. N a circle (ABED) the equal chords (AB, DE,) arc equally diftant from the center (C) ; & the chords (AB, DE,) equally diftant from the center (C), are equal to one another. Hypothefis. CASE I. Thefis. The chords AB, DE, are tquaL 7bey are equally diftant from the center C. Preparation. I. Find the center C of the © ABED. P. i. B. 5. a. Let fall upon the chords AB, DE, the ± CF, CG. P. iz. B. i. 3 . From the center C to the points E & B, draw the rays CE, CB. Pof. i . T Demonstration. H E chords AB, DE, being = to one another C^Jp) & bifefted inF&GCPre^,2.&P.^,B.3.). Jx 1 B t Their halves FB, GE, are aJfo equal. -. p k j> ,' Confequently, the D oj: FB is = to the D of GE. \ rJ \ But becaufeDof CB = Dof CE (Prefi. x.& P. 46. Cor. 3X} p :J'p , It follows, that D of FB + D of FC is = to U of GE + D of CG. \ 1 ^\ ^- '* Therefore jthe equal Dof FB& of GE (^rg, 2.) being taken away. ^ p'^' ^' \' Cor. 3. 4. ^- 3. I. 2. 3- 4. 5. The Dof FC will be = the Dof GCr^;f. 3.B. i.). or FC = GC. ( \ Confequently, the chords AB, DE, are equally diftant from the cen- ^ ter C of the © ABED. Which was to be demonftrated. P. II. Thefis. Tbefe chords are equal. Hypothefis. CASE The chords AB, DE, are equally diftant from the center Q of the © ABED. B Demonstration. ECAUSE FC = GC (Hyp. & D. ^ B. 3.), &CB = CE CPre^- 3. & ^. i5> ^- '.)• y ^-46. S. I. I. The Qof FC = die D of CG, & the Q of CB = the D of CE. ( Cor. 3. :i. Confequently, Q ofPC + D of FB, =0 of CG + D of GE. C P. 47. B. i. Therefore, the equal D of FC& of CG (Jrg. i.) being taken away. \ Ax. i.B.i. 3. The D of FB. will be =;: the D of GE (Ax. 3. B. i.) or FB = GE. C P. 46. B. i. 4. Confequently, FB, GE, being the femichords (Prep. a. P. 3. B. 3.), \ Cor. 3. tbc whQle chords AB, DE, are alfo = to one another. Ax. 6. B.i. Which was to be demonftrated. JI2 The ELEMENTS BookHI. —^ PROPOSITION XV. THEOREM XW. X H E diameter (AB) is the greateft ftraight line in a circle (AIK) ; & of all others that (HI), which is nearer the diameter, is always greater than one (FK) more remote. Hypothelis. Thcfis. /. AB M the diamtter •/ the © AIK. /- The diameter AB « > tad »/ //. The chord HI // nearer tbe diame- the chords HI, FK. ter than tbe chord FK. 77. The chard HI is > the chord FK. Preparation, 1 . From the center C let fall upon HI & FK the X CG, CN. -P. «. B. u 2. From CN, the greateft of thoTe J., take awaj a part CM = to CG. P, 3. B. I. 3. At the point M in CN, creft the J. DM & produce it to E. P. n. B. i. 4. Draw the rays CD, CF, CE, CK. Pof. 1. Demonstration. B E C A U S E the ftraight lines CD, CE, CA, CB, are = to one another ("Pr^/. 4. & D. 15. B, i.). 1 . It foJJows, that CD + CE is =: to CA + CB or AB. Jx. a. B. U But CD + CE is > DE (P, ao. B, i.). 2. Wherefore, AB is alfo > DE or > HI, hecaufe HI = DE C D, 4. B, 3. (Prep. 2.). XP.hB.}^ 3. It may be proved after the fame manner, that AB is alfo > FK. Which was to be demonftrated. I. Moreover, the A CDE, CFK, having two fides CD, CE, = to the two fides CF, CK, each to each (Prep, 4. & D. 15. B, 1.), & the V DCE > V FCK (Ax. 8. B, 1). 4. The bafe DE will be > the bafe FK. 5. And bccaufe HI is = to DE (Prepj,^^\ is alfo > FK. Which'was to be demonftrated. II. P. 24. B. f. -4.^3. 14. B. J. j^.^,., fiookin. Of EUCLID. "3 PROPOSITION XVI. THEO REM XV. X H E ftraight line (AB) perpendicular to the diameter of a circle (AHD) at the extremity of it (A)» falls without the circle ; & no ftraight line can be drawn between this perpendicular (AB) & the circumference from the extremi- ty, fo as not to Cut the circle ; alio the angle (HAD) formed by a part of the circumference (HEA) & the diameter (AD), is greater than any acute reSi- lineal angle ; & the angle (HAB) formed by the perpendicular (AB) ^ the fame part of the circumference (HEA)» is lefs than any acute redllineal angle. Hypothefis. /. AB is dra-wn perpendicular to the extremity A of the diametr. IL And makes tvitb the arch HEA tbe mixtilineal V HAB. ///. The diameter AD makes wtb the fame arch HEA tbe mixtili- neal V HAD. Thefis. /. The ± AB falls ^without the © AHD. //. No firaigbt line can he drauun be-- ttveen the ± AB tf the arch HEA. ///. The mixtilineal V HAD is > any acute redilineal V. IF. Tbe mixtilineal V HAB is < any acute redilineal V. Demonstration. I. If not. The ± AB will fall within the AHD, & will cut it fome- where in £» as A£. Preparation. From the center C to the point of fedion £» draw the ray C£. Pof i. OECAUSE CAis = toCEri>. 15. B. I.). I. The V CAE will be = to the V CEA. P. 5. B. i. a. And becaufc the VCAE is a L (Sup.) ; VCEA is alfo 2lL., Ax.i. B. \. 3. Wherefore, the two V CAE + CEA, of the A AEC will not be -< 2 L. ; which is impoilibie. P.ij.B.i. 4. Therefore, the JL AB falls withoutthe circle. Which was to be dcmonftrated. I. P 114 The^LtMElSrtS Bookm. 6 If noCy Thexe maj be drawn a ftraight line* aa AG> between tbe ± AB & the circumierence of the © AHD. Preparation, From the cenUr C, lat fall upon AG, the ± CG. P.M.B.h AxXB.\. EC A USE VCGAisaU, & V CAG< a L r^x. 8. A i.) as being but a part of the L. CAB (Hjp. i.). 1. It fblknirs* that the fide CA is > the ftde CG. But CA bcji3^= to CE (I>. 15. B. 1). 2. The ftraight line CE will be alio > CG ; which 19 impoffible. > Thecefore» no ftraight line can be drawn between the ± AB & the O of the © AHD. Which T¥as to be demouftFatcd. H. III. & IV. If not, There may he drawA a fttaight line, as AG» which make». with the (Uametes AD & with the J. AB, an acute rediKa^ V GAD > the mixtilineal V HAD» & an acute reaili- neal V GAB < the raixtilincal V EAB. 15 EC A USE then the ftiaiglit line AG, drawn from the extrami* Xy A of the diameter AD, makes with the diameter & with the X AB, an acute re^ilineal V GAD > the mixtilineal V HAD, & a redUineal V GAB < the mixtilineal V EAB (Sup.). I . This (baight line AG will neceflarily fall on- the eztcemitj A of the diameter AD, between the X AB & the circumference of the © AHD ; which is impoffible. //. G?/i. z. Therefore, the mixtilineal V HAD is >, & the mixtiliiMDal V HAB < any acute rectilineal V. Which was tabe dcoBonftrated. III. & IV. COROLLARr. ^ Straight line^ Jranun at right angles to the diameter •/ a circle from the eMttf mty of it, tqucbes the circle •nlj in one point. Booklll.^ Of E U C L ID. 115 G F PROPOSITION XVn. PROBLEM II. ROM a given point (A) withoat a circle (BEF), to draw a tai^gent (AE) to this circle. , Given Soi^ht fjlfi fmnt A without $bt © BEF. TCr tangent AE, drawn from tbt point A to tbi © BEF-. Refolution, 1. Find the center C of the © BEF, & draw CA. P.i. B, 5. 2. From the center C at the diftance CA, defcrbe th^ ® ADG. Pof. 3. 3. At the point B in the line CA, where it cuts tlie O BEF, ercfttheXBD. P,il,B, f. 4. From the center C to the Mint P, where the JL BD cuts the O ADG, draw the ray CD. 5. From the point A to the point E, where CD cuts the OBEF, B. O ADG, draw the ray CD. Pof. 1. From the point A to the point E, where CD cuts the OBEF, draw the ftiaight line AE, which will be the tangent fought Demonstration. BECAUSE in theA CBD, CEA, the fide CB is :== to the iUe CE, the fide CA = to die fide CD (D, 15. B, i.), & the V BCD commoB to the two A. I. The V CM), CEA, oppofite to the equal fides CD, CA, are = to one another. P. 4. ^. I. a. Wherefore, V CBD bcinga L {Re/, 3.), V CEA will he alfo a L. Jfx. i.B,i. 3. Confequently, the firaight line AE, drawn from the given point C P. 16. i?. 1. A, is a tangent of the ® BEF. I ar.i^.i.^.j. P a ii6 The ELEMENTS • Bookni /^ K ( F ^' I 1 , V y^ D ^^=^ B G -E PROPOSITION XVIII. THEOREM XVL J[ F a ftraight line (DE) touches a circle ( AFB) in a point (B) ; the raj (CB), drawn from the center to the point of contaft (B), (hall be perpendi- cular to the tangent (DE). HVpotliciis. Thefis. /. ne ftraight line DE touches the The ray CB isX^ufinm © AFB in the point B. tangent DE. //. And the ray CB pajfes thro' the point of contad B. Demonstration. It not, There may be let fall from the center C, another ftraight line CG -L upon the tangent DE. Preparation. Let fell then from the center C upon the tangent DE, the ± CG. P. U. B, i. B _ lECAUSE theVBGCof the ABCGisaLr^rr/.;. 1. The V CBG will be < a L. 2. Confequently, CB is > CG. And CF being zrCB (D. 15. B. i.). 3. The iUaight line CF is alfo > QG ; which is impoflible. 4. Wherefore, the ray CB is JL upon the tangent DE. Whidi was to be demonftrated. P. 17.B.1. P. 19 A I. Ajc.^.B.u Bookni. Of E U C L I D. 117 A D B E PROPOSITION XIX, THEOREM Xm. J[ F a ftraight line (DE) touches a circle (AGB in B), & from the point of contaS (B) a perpendicular (BA) be drawn to the touching line; the cen- ter (C) of the circle, fliall be in that line. Hypothefis. Thefis. /. 7hejlrasgbt line DE touches the © AGB. 7beftratgbt line BA paffes thr%^ II. And^K is the ± ereded from the point the center C of the © AGB. •/ contaS B in this line. Demonstration. If not. The center will be in a point F without the ftraight line BA. Preparation. Draw then from the point pf contgQ B to the center f , the ftraight line BF. Pof. 1, JJ £ C A U S E the ftraight hne BF is drawn from the point of conuft B to the center F of the © AGB (Prep.), 1. The V FBE is a L. P. 18. B. 3, But V ABE being alfo a L (Hyp. 2.). f jf^ rr. n , z. The V ABE is = to the V FBE ; which is impoffible I ^^- ^^•^- '• 3 . Whercforp, the Renter C will be ncceflariljr in the ftraight Ime BA. ^ ^' o.JS.i, Which was to be demonftratcd. 1 1 i ii8 The ELEMENTS BookUL Tliefis. rbe V BCD tf » tht tt^f tbiO. rj^ PROPOSITION XX. THEOREM XVIIl A PI E argle (BCD) at the center of a circle, is double di the angle (BAD) at the circumfere.icey when both angles ftand upon th^ fame arch (BD). HypoUiefis. /. The V BCD is at the center W V BAD at tbf Q. //. ne fides BC, CD, W BA, AD, of tbofe V, fi and upon tbe Jam arch BD. Demonstration. CASE I. If the center C, is in one of the fides AB 6f the V at BtheO (Fig, 1.). ECAU6E m theACAD the fide CA{s=rto the fide CD (D, 1$. B, I.). tP ^ l\. 1. The V w is = to the V «, & V «» + « is double of W m. < mJ\ d ,• But Vois = to V«» + #tr^.3a.^. I.). lAx,z,D, . 2. Therefore, V • is double of V «, or V BCD is double of V BAD. Jx.6.B.i^ CASE a If the center C falls withia the V at the O (Fig. a.). Preparation. Draw the diameter ACE. Pof. i. XT may be proved as in the firft cafe. 1. That the V • is double of the V «•, & V/ double of the V it. 2, From whence it follows, that V o + / is double of the V « + "» or V BCD is double of the V BAD. Ax. S. B. I CASE ni. If the center C fidls without the V at the O (Fig. 3.). X H E diameter ACE being drawn, it is demonftrated as in the firft cafe, that : 1. The V / 18 double of the V «, & V • +^ is double of the V « + «. Therefore, the V / being taken away from one fide, & the V n from the other. 2. The V « wUl be double of the V m, or V BCDis double of V BAD. Ax. 3. B.\. Which was to be demonftrated. fiookril Of EUCLID. 119 TheTw. \f mis z=z toy n. PROPOSITION XXI THEOREM XIX. HE angles (m & n) In the fame fegmcnt of a circle (BAED), are equal to one another. Hypothefis. The y mti n are in the fame fegment rf tbe © BAED. Demonstration. CASE I. If the fegment BAED is > the femi (Tig. i.). Preparation. 1. Find the center C of the © BAED. P. i. J?. 3. 2. And draw the rays CB, CD. Fef. i. /Because VBCDisdonblcofeachofthe W m&f^(P.zo,B.^), J. It follows, that V « is = to V «. Jx, 7. B. i. CASE n. If the fegment BAED is < the femi ® (Fig. a.). Preparation. Draw the ftrai^t line AE. Pof. I. J[^ H E three V« + « + f of the A BAG, arc = to the three C P, 32. B. 1. V^ + « + ''ofAe^ ^ED. 1 yifc. i.B.i. But V qis = toy r (Cafe iX * Vo= to V/fi*. ij. B. i.). Tlicrefore, the V f + • being taken away from one fide, & their equals y p -{-r from the othen Tiie remaining y m8c n will be == to one another, jfx. j. A i. Which was to be demonflrated. 120 The ELEMENTS EookllL . rp PROPOSITION XXII. THEOREM XX. X H E oppofitc angles (BAD, BCD, or ABC, ADC,) of any quadrila- teral figure (D ABC) infcribed in a circle, arc together, equal to two ri^t angles. HTpothefis. Thcfis. 7be figure DABC is oT quadrilatital The opp^fite V BAD + BCD, or ABC figure infcribed in a®, -f" ADC, are =1/0 a L^ Preparation. Draw tbc diagonals AC, BD. PpJ. I. Demonstration. X>ECAUSE the V« + a are the V at the O, in the fame feg- mentDABC. 1. Thefe V 1/ & « are = to one another. P.zi.E.y It is proved in the fame manner, that : 2. The V / & M are = to one another. 3. Wherefore, the V 11 + / are = to the V « + « or to the V BAD. jfx. a. B, i. Therefore, if the V r -j- ^ or BCD be added to both fides. 4. The V « +/ + r^ + y; are = to the V BAD + BCD. jfx.i.B.i^ But the three V « + / + (^'" + f^' of the A DBC being == to a L. (P. 3*. B. 1.). 5. The two oppofite V BAD + BCDof the quadrilateral figure DABC, are alfo = to 2 L.. ^x, i. B i* It may be demonflrated after the (ame manner, that : 6. The V ABC + ADC arc=: to 2 L. 0, Which was to be demonftrated J fiooklll. Of E iJC Liti. i2i . (/:, L A ' PROPOSITION XXIII. THEOREM XXI. Upon. the there cannot be t (ame firaight line (AB) & ufx)n the fame fide of itf two fimilar fegments of eircles (ADB, ACB,) rot coincide iDg with one another. HTpothefis. Thefis. The fegments ADB, ACB, «/ circles^ are uphn nefe/egmenis art diffimtaf the fame ftraigbt lint H up$n the fame fide of it, Demonstratiok. If noty The fegments ADB, ACB, upon the fkme chord AB, k npoii the £une fide of itf are fimilar. Preparation. 1. Praw any {bajjr]|i Upe AC* which cfita the fegments ADB^ ACB, in the pomts D & C. 1 2. Draw die ftraight Knea fiD, BC. J Jj ECAUSE theV BDA, BCA, are conuined in the fimilar feg? jncnts ADB, ACB, (Hyp. & Prep, i. & ^.). I. Thefc V are t=. to one another. Ak.l. B.t, a. Therefore, the exterior V APB of the A BDC, Tsrill be :i= to its in- terior oppofite one BCD ; which is MnpofTibic. P, i6. B, f, J. Confequenthr, there cannot be \Sfii i)miW fegoientsof ® ADB, ACB, upon the fame fide of the &q[ie f^tiiight line AB, which do not coincide. Which was to be demonftrated. 112 The ELEMENTS BookllL PROPOSITION XXIV. THEOREM XXll OlMILAR fegments of circles (AEB, CFD,) fubtcndcd by equil chords (AB, CD^), are equal to one another. Hypothefis. Thcfis. /. The fegments of © AEB, CFD, The fegments AEB, CFD, are =: t9 arefimilar, one another. IL Tbefe fegments are fuhtendedhy equal chords AB, CD. Demonstration. If not. The fegments AEB, CFD, arc unequal. B E C A U S E the fegment AEB is not == to the fegment CFD (Sup.)^ & the chord AB is = to the chord CD (Hyp, 2.). 1. Upon the (ame (Iraight line AB or its equal CD, there could be two fimilar fegments of ©, AEB, CFD 5 which is impoffible. P. 23. B.J. 2. Therefore, thefe fegments are = to one another. Which was to be demonftrated. J Book III. Of E U C L I D. t2i M PROPOSITION. XXV. PRO B LEM III x\ Segment of a circle (ADB) being given; todefcribe the circle of which it is the fegment. Given Sought 7be fegment of © ADB. W* center C of the ©, of'wbicb ADB is the fegment. Refolution. 1. Divide the chord AB into two equal parts ii^ the point E. z. At the poinnt £ in AB» ereft the JL ED. 3. Draw the ftraight line AD. And V ADE will be >» or <, or = V DAE. CASE I. & II. If V ADE be diher > or < V DAE (Fig. i. & a.). 4. At the point A in DA, make V DAC = to V ADE. 5. Produce DE to C (Fig. i.), & draw BC (Fig. i. & z.). B Demonstration. EC AUSE inthe A ADC the VDAC is = to V ADC (Ref^.): 1 . The fide AC is =; to the fide DC, But in the A AEC» BEC, the fide AE is =: to the fide EB, theiide EC conunon to the two A, & the V AEC = to the V BEC (Ref 2. & Ax. 10. B. I.). z. The bafe AC will be = to the bafe BC. P. 4. B. i 3. Confequently, the three ftraight lines AC, DC, BC, drawn from the point C to the O ADB, are =: to one another. Ax. i. B.i. 4. Wherefore, the point C is the center of the©, of whiqh ADB is the P, 9. B. 3. iegment. . C A S E in. TIf V ADE be = to V DAE (Fig. 3). HEN the fide AE is = to the fide ED. P. 5. B. 1 ^. Confequently, AE be^ngri: EB (Ref. 1.), the three ftraight lines AE, ED, EB, drawn from a point E to the O ADB, are r= to one another. Ax. i. S.i. 3. From whence it follows, that the pomt E is the center of the © of which ADB is the fegment. P. 9. B. 3 Which was to be demonftntted. p. 10. B. I. P. II. B. I. P9f I. P. 23. B. I. Pof.z.Se I. P. s. ^ «• 124 The ELEMENTS Bookm n PROPOSITION XXVI. THEOREM XXIIL IN equal circtes (BADM> EFGN,), equaUnglcs, whether thejr bcrt the centers as (C & H) or at the circumferences as (A& F)} ftanl upoo equal arches (BMD, ENG,). Hypothefis. / TA^VC, H, an ^ at the centers y^ equal II. The V A, F, art V 4f the O, ftf #fwi/. ///. I'befe V are contained in tht tfumi BADM,EPGN. Preparation. Draw the ch6rds BD| EG. Thefis. The arches BMD, ENG, u^M which theft >IJlf**i are tS: ip gnt anther. DEMONStRAtlOK. _ HE two fides CB, CD» «f the A BCD beiiMf :=: to the two fides HE, HG, of the A EHG (Hyp. 3. & Ax. 1. B. 3.), * tht V C=s tothc VH r^/. a.). I. The bafe BD will be 3= to die bift EG. ?. 4. 1 >. And becaufc V A is = to V F (Hyp. 1.). 3. The iegment BAD is funikr to the fegmenc £FG. ' * Ax.%, B.3. 3. Wherefore, the bafe BD bek^ =s to the bafe EG (Arg. i.)» theft Segments will be := to one another. F. 14, ' . }• Therefote, if the e^nal fimients BAD» EFG, (Arg. 3.) be taken away from the equal ©BADM, EFGN, (Hyp. 3.). 4. The remainii^ arches BMD, ENG^ wiU be siib = to one another. Ax.^.i*^ ^ Which was to be denoiftxaied. Book in. Of E U C L I D. 125 PROPOSITION XXVII. THEOREM JCXir. In equal circles (BAG, D£F>) the angles, whether at the centers at (BCG & H) or at the circumferences as (A & £)^ which ftand upon equal arches (BG, DP,) ; are equal to one another, Hypothefis. Thefis. /. The © BAG, DBF, art 2:, as alfi their /. 7*# V BCG tfKattbi anun. arches BG, DF. an =: i9 onf another. 11. The V BCG liViai the centers, as alfi II The\/ A^Eai the O, areai-. theWAi^Eat the O, /an J up9n = fi^^to 9ne anther, jtrches. Demonstration. If not. The V BCG & H at the centers will be unequal, Hr one> af BCG, waibe> the other H. Preparation. At the point C in the line BC, make die Y BCK zs to V H P. aj. J. i. ..T. _ HEREFORE thearchBKisrztothcarchDF. P.a6. -^.J. But the arch DP bcmg =i to the arch BG (Hyp. zj € jt t S t a. The atch BK will be alfo = to the arch BG ; which is impoffible I '^' i' ^ ' 3. Confcquently, the V BCG a? Hat the centers, arc = to one another. ^•^•' ' Which was to be demonftrated. -L - And thefe V being double of the V A & F at the Q ("F. 20. B, 3.). 4. Thefe VA&EatAf Of tifalb=tooiieanothtr. Jx.j. B,i. Which was to be dcno^fttat^d. II. t26 »The ELEMENTS BookUL '1 I PROPOSITION XXVIII. THEOREM XXV. _N equal circles (ABDE, FHMN,) ; the equal chords (AD, FM,) fubtend equal arches (ABD, FHM or AED, FNM,). Hypothefis. /. Tbt © ABDE. FHMN, are equal. 11 7he cbards AD, FM, are equal. Thcfis. 7he chords AD, FM, fuhtend efu$l arches ABD, FHM or AED, FNM. B Preparation, I. Find the centers C & G of the two © ABDE, FHMN. Z. Draw the rays CA, CD, alfo GF, GM. Demonstration. P. x.B.y EC AU S E the © ABDE, FHMN, are equal (Hyp, i.). 1. The fides CA, CD, & GF, GM, of the A ACD, FGM, arc equal, j^x. i. E.y And the chords AD, FM, being equal (Hyp. 2.). 2. The V ACD, FGM, are = to one another. P. 8. B. i. 3. Confcquently, the arches AED, FNM, fubtended by the chords AD, FM, will be alfo = to one another. P. 26. B. 3. 4. And moreover, the whole Q being equal (Hyp. i.), the arches ' ABD, FHM, are alfo equal. Ax.^.B.i. Which was to be dcmonflrated. J Book III. Of fi U C L i D. 127 r - PROPOSITION XXIX. THEOREM XXFl \ N equal circles (BADM, EFHN,) ; equal arches (BMD, ENH,) arc Aibtended by equal diords (Bt), EH,). Hypothefis. Thefis. /. rbe © BADM, EFHN, are equal. 7be chords BD, EH, nvbicb fuh^ II, Tbe arcbes HMD, ENH, are equal. tend tbefe arcbesy are equal. Preparation, 1 . Find the centers C & G of the two © BADM, EFHN. P. t B t 2,. Draw the rays CB, CD, GE, GH. Pof. i. Demonstration. EC A USE the ©BADM, EFHN, are equal (Hyp. i.). 1 . The fides CB, CD, & GE, GH, of the A BCD, EGH, are = to one another. Ax. i. B.^.' But the arches BMD, ENH, being alfo equal (Hjp. a.), a. The V C & G, contained by thofc equal fides, will be = to one another. P. 27. ^.3. 3. Confequently, the chord BD is = to the chord EH. P. 4. ^. i. Which was to be denacnftrated. B ■^'^ 128 The£tfeM£NtS Bookm. (, 7\ i /L" c -^D PROPOSITION XXX. PROBLEM If^. O <fivtde an arch (ABD) into two equal parts (AB, BD,). Given Sought Tti flfcb A 8 D. Tbi JM/9M pf tht mh A© ini$ iw0 equal parN AB, m Re/olution. 1 . From the point A to the point D, draw the chord AD. A/ '• 2. Divide this chord, info two equal pans at the point C. P* ^^- 1- *' 3. At the point C m the ftraight line AD, e^a the X CB, which P- n. ^. >* when produced, will divide the arch ABD into two equal parts at the point B. Preparation. Draw the choids AB, DB. H '• X5 EC A USE the fide AC is r? to the fide CD fX^ a), CBcoflH mon to the two A ABC, DBC, &: ths V ACB = lo cht V DCS (Jx. 10, «. I. kRef. 3.). I. The baft AB is == to the hale DB. P. 4 *•'• 1. Confequendy, the arches Afi k DB, fubtended by the equal choids AB, DB, are = to one onother, and the whole arch ABD, is di- tided into two equal parts in B. P. *8. J.3* Which was to be done» Book Itt Of E U C L I D. 129 """""" /" ^^ i c )• PROPOSITION XXXI. THEOREM XXVIl J[ N a circle, the angle (ADB) in a femicircle (ADEB), is a right angle ; but the angle (DAB) In a fegment (DAB) greater than a femicircle, is lefs than a right angle, Sr the angle (DEB) in a fegment (DEB) lefs than a fe* micircle, is greater than a right angle : alfo the mixtilincal angle (BDA) of the greater fegment, is greater than a right angle, & that (BDE) of the leflcr fegment, is lefs than a right angle. CASE I. Hfpothefis. Thefis. The V ADB is in tbefemi © ADEB. This V ADB is a L. Preparation, 1. Draw the ray CD. P9f. i. 2. And produce AD to N. Fof. 2. Demonstration. B. ► E C AU SE in the A ADC the fide CA is == to the fide CD (D, IS. B. I.). 1. The V CDA is = to the V CAD. Again, in the A CDB ; the fide CD being = to the fide CB. a. The V CDB is = to the V CBD. 3. Confequcntlv, V ADB is =: to V CAD + CBD. But V NDB is alfo =: to V CAD + CBD (P. 32. B, i.). 4. Wherefore, this V NDB is =^ to V ADB. 5. From whence it follows, that V ADB is a L.. C A S E n. HTpothefis. Thefis. ^h€ V DAB is in the fegment DAB > afemi ©. ♦ This V DAB is < 41 L. p.?. B. I. /)..?. B. 1. P.s. B. I. Ax, 2 .A I- Ax,\ .B. 1. D. 10 ,B. I. B Demonstration, E C AU S E In the A ADB, the V ADB i« a L fd'/e I.J. The VDABwiUbe<aL. . P.ty.S.i. R 130 ThcELEMENTS CASE m. Hypothcfis. 7he V DEB is in a figment DEB < afomi ©. Thefis. 7bis V DEB « > i L I. JL HI Demonstration. _ : E the oppofitc V DAB + DEB of the quadrilateral h^xxt ADEB are = to 2 L. P.2aM 2. Wherefore, V DAB being < a L (Cafe 11.), DEB wiU be necefla- lily > a L. CASE IV. Hjpothefis. Thefis. , ni mxtilineal V BDA, BDE, are The V BDA // > tf L, ^ "* formed by the ftraight line BD W V BDE is <« L. ih$ arches DA, DE. Demonstration. i3 E C A U S E the reailineal V ADB, NDB, are U (Cafe I.). I . The niixtih'neal V BDA will be neceflarily > a L.> & ^he inixtilineal VBDE<aL. /x.8J.»- Which was to be demonftrated. Bookin. Of E U C L I D. »3> E ssBsteasa: /^ >^^^ / > i A° ( i /\ \ i / |/ > V \/ J^ * >w : / '"j^ ^v^ 5/-!>^ A" ' ' F ""B PROPOSITION XXXII. THEOREM XXVIIL \y ^ ftraight line (AB) touches a circle (ECF), & from the point of con- tad (F) a chord (FD) be drawn; the angles (DFB, DFA,) made by this chord & the tangent, (hall be equal to the angles (FED, FCD,) wl^ich are in the alternate fcgm'ents (FED, FCD,) of the circle. Hypothefis. Thefis. / BA 15 n tangent 0/ the ® ECF. /. 7*# VFED w cz: /o V DFB. // And FD is a chord of this ® //. Tbf VFCD is = tfi V DFA, drawn from tpe point of conta&. Preparation, I. At the point of contact F in AB, efedl the ± FE. P. 1 1. 5. i. z. Take any point C in the arch DF, & draw ED, DC, CF. Fof, i. B Demonstration. EC A U S E the ftraight line AB touches the ® ECF (Hyp, i.), %vA FEisa-Lerefted at the point of contaft F in the line AB (Prep, i.). 1. The ftraight line FE is a diameter of the ECF. P. 19. J?. 3, ». Confequently, V FDE is a L. P. 51. B, 3. 3, Wherefore, the V DEF + DFE are = to a L. P. 32. B, 1. But V FJ^B or V DFE + V DFB being alfo = to a L (Prep, i.) The V DEF + DFE are<= to the V DFB + DFE 4* S Ax, I. B, I. Wherefore, the V DEF is = to V DFB, or the V in the fegment ^Ax,yB.\. ~ ~" lP.21.A3. DEFis= : to th« V made by the tangent BF & the chord DP. Which was to be demonftrated. I. Ax. I. B I. The V FED + FCD being = to 2 L (P. 22. A 3.), & the adja- cent V DFB + DFA being alfo = to 2 L (P, \y B. i.). 6. The V FED + FCD arc = to the V DFB -f DFA. 7. Wherefore, V FED being = to the V DFB (Arg, j.), the V FCD 18 alfo = to the V DFA ; or the V in the fegment FCD is =: to C Ax.yB. i. the Vcootained by the tangent AF & the chord DF. ( P. 21. B. 3. Which was to be demonftrated. II. R2 132 The ELEMENTS Bookia y^ PROPOSITION XXXIII. PROBLEM F. IJ PON a given ftraight line (AB), to defcribe a fcgment of a drck (ADD) containing an angle equal to a given redilineal angle (N). Given Sought ne/iraight^ineAB togttbtr wiib V N. Vhtftgment ADB defcrihedufa AB, containing an^^ti^H, J C A S E I. If the given V is a L. (Fig. i .). J.T Aifficcs todefcribc upon AB a fenri © ADB. * A/ 3. I. This fcmi © will contain an V = to the given right V N. i^.Ji.^J- C A S £ II. If the given V is acute fFig. 9.) OTchtvL€i{Fig. 5.) Re/olution. I. At the point A in AB, make the VBAErzto the given VN. P. 23. B.i. a. At the point A in AE, credt the JL AG. P.iuB.i. 3. Divide AB into tv, o equal parts in the point F. P. 10. B. 1. 4. AtthepointFinAB,ereathe±FC, whichwillcutAGinC. P. iiJ» 5. From the center C a^ the diftanceCA, defcribc the ® ADG. A/ 3. Preparation. Draw the ftraight hne CB. A/ i. B Demonstration. ECAUSE intheAACF, BCF, the fide AF is = to the fide BF ^i?'/ 3.), FC common to the two A, & the V AFC = the V BFC (jIx. 10. 5. I. &Re/. 4.). I. The bafe CA is = to the bafe CB. P. 4-^- '• a. ConfeqTiently, the © defcribedfrom the center C at the diftance CA, ^ jy ,^ ^. 1. will pafs thro* the point B, & ADB is a fegment deferibed upon AB. { j^' .^n / But AE touching the © ADB in A (Ref, 2. & F. 16. Cor. B. ^Jt ^ ^ ^ ' ' and AB beins; a chord drawn from this point of contad A (^rg. 2.). 3. The V contamctl m the alternate fegment ADB is = the V BAE. P.ja.^. J- 4. Wherefore, V BAE being = to the given V N (Re/, i.), the V con- tained in the fegment ADB deferibed upon AB^ is alfo = to the giyen VN. Which was to be done,' Ax.iJJ' J Bookiir. Of EUCLID. 133 PROPOSITION XXXIV. PROB LEMVI. X O cut off a fegment (BED) from a given circle (BDE), which fliall contain an angle (DEB) equal to a given redilineal angle (N). Given Soudit 7bf (5 BDE, y the reailineai V N. 7be fegment BED cut cj from tbU ©, containing an V DEB = to tbegi<ven V N. Refolution. 1. From any point A to the © BDE, draw the tangent ABC. P. 17, B, 3. 2. At the point of contact B in the line AB, oiak^ the V DBA = to the given V N. P. zj. B. i. Demonstration. 13 ECAUSE the given V N is = to the V DBA {Ref a.), 8c V DEB = to the V DBA (P. 32. B. 3.). I. The V DEB & N are = to one another. Jx. i, A f . Z. Wherefore, the (egment BED is cut off from the BDE, contain- ing an V DEB :;= to the given y N, ?. 21.S.3. Which was to be done. »34 ir TJie ELEMENTS Bookm. PROPOSITION XXXV. T H E R E M XXIX. X F in a circle (DAEB) two chords (AB, DE,) cut one another; the rec- tangle contained by the fegments (AF, FB,) of one of than, is equal toilic rectangle contained by the fegments (DF, FE,) of the other. Hypotliefis. Thefii. /. AB, DE, are tnvo cUrds of the fame © DAEB. ^be Rglc AF . FB « = » //. And tbefe chords cut one another in a point F. ihe Rgle DF . FE. C A S E I. If the two chords pafs thro' the center F of the ®. ^ig- '• T Demonstration. HEN, the ftraight lines AF, FB, DF, FE, are = to one another. 2>.'5f' %. Confequcntly, the Rgle AF . FB is = to the Rgle DF . FE. -^- * ^'^ C A S E 11. If one of the chords AE, partes thro* the center & cuts the other DE which does not pafs thro' the center at L (Fig, 2.). Preparation. Draw the ray CE. N- '• Demonstration. E C A U S E the ftraight h'ne AB !s cut equally in C & unequally B in F. , , I . The Pgle AF . FB + the D of CF is =: to the D of CB, or is = f P. 5 * ^ tothcDof CE. XAxaM But the D of FE + the D of CF is alfo =: to the D of CE (P. 47. B, I.). ». From whence it follows, that the Rglc AF . FB + the D of CF _ IS = to the D of FE + the D of CF. -^. «•* '• 3. Conilquenrlv, the Rglc AF . FB is = to the D of FE. ifx.J.l^-^ And fuice DF is = to FE (P. 3. B. 3), or DF • FE = to the D of FE (Ax, a. B. 2,). 4.TheRglcAF.rBisaIfo=totheRgleDF.FE. Aj'-^' Book III Of E U C L I D. -P. tt.B. r. p. s. B. 2. C A S E IIL If otie of the chords AB, pafTes thro' the cen- ter 8c cttts the ot|ier D£ which does not pafs thro' the center, obliquely fFig, 3.). Preparation. 1. From the center C, let fall upon DE, the X CR 2. And draw the raj CD. B Demonstration. ECAU&E DHisS=toHEr/'''^A i. ^-P._J. B. 3.). 1. The Rgle DF . FE + the D of FH is = to the D of DH. a. Wherefore, the Rgle DF . FE+ D of FH + O of CH is s= to the D of DH + D of CH. Ax.z.B, i. But the D of FH + D of CH is = to the D of CP, & the D of DH + the D of CH is = to the D of CD (P. 47. B, i.). 3. Therefore, the Rgle DF. FE + D of CF.is = to the Q of CD or to the D of CB. ^ Ax.i.B,\. Moreover, the Rgle AF . FB + D of CF being = to the fame D of CB (P. 5. B. 2.). 4. The Rgle DF . FE + D of CF is alfo t= to the Rgle AF . FB + DofCF. Ax.i.B^i. 5. Or taking away the common D of CF, the Rgle DF . FE is =r to the Rgk AF , FB. Ax. 3. B. i. C A S £ IV. If neither of the chords AB, DE, pafTes thro' the center (Fig, 4.). Preparation, Thro' the point P, draw the diameter GH. Pof, 1 . Demonstration. 1^ I B. BECAUSE each of the Rgles AF . FB & DF . FE is = to the Rgle GF.FHrCfl/r ///.;. 1, Theft Rgles AF . FB & DF . FE are alfo ~ to one another. Ax,i.B.\. Which was to be demonftrated. 136 The E L E M E N T S Jlookin. PROPOSITION XXXVI. THEOREMXXX, X F from any point (E) without a circle (ABD) 'two ftraight lines fe drawn, one of which (DE) touches the circle, & the other (EA) cuts it; the redangle contained by the whole fccant (AE), & the part of it (EB) without the circle, (hall be equal to the fquare of the tangent (ED). Hypothciis. Thciis. /. The point E is taken nvittout tbt © ABD. The Rgle AE . EB m = W * //. From this point E, a tangent EDfJafe^ Oof ED. cant EA, have teen dranun. C A S E I. If the fecant AE pafles thro' the center (Fig, i.)« Preparation, From the point of contaft D, Draw the ray CD. /V^ »• Demonstratiok. I. X H E ray CD is then ± to the tangent ED. P. 18. A J- And becaufe the ftraight line AB is bifeSed inC, & produced to the point Ew a. The Rgle AE. EB + the D of CB is = to the D of CE. P. 6. B.%^ Moreover, the D of CE being alfo= to the D of DE+ the D of CD ('P.47.-B.i.),or lotheDof DE+theD of CB (P. 46. Cor^ 3.B. 1.). 5. The Rgle AE . EB + the D of CB is = to the D of DE + theD of CB. The D of CB being taken away from both fides. 4. The Rgle AE . EB will be = to the D of DE. CASE 11. If the fecant AE does not pafs thro' the center. Preparation, 1 . Let fall from the center C upon AE, the JL CP. a. Draw the rays CB, CD, ^ the ftraight line CE. Fig, J. P. 12. .5.1. Pe/;i. J t Book Illt Of E U C L I D. »37 Fig 3 . V^ — -.^D ( JSf^ >E G^^^ ^^ 3. 4. Demonstration. x5eCAUSE the ftrtight Kae AB i« biftaed io P (Prtp. i. & P. 3. ^. iJ and produced to the point E. I. The Rgle AE.EB -f Q of FB ia = to the D of FE. P, 6 B t a. Cbnfequentlv, the Rgle AE. EB + Q of PB + D of FC ij = , ^ " * to the D of FE + C3 of FC, or is = to the D of CE. ( ^•*- *- f '• Bat fincc the D of DE -f Q of CD is = to the D of CE, and * ^'^7- B. i. the D of FB 4- D of FC is = to the D of CB (P. 47. B, ij, or is = to the D of CD (^D 15. & P, 46. Cor, 3. A ij The Rgle AE.EB + D of CD is = to the D of DE + Q of CD. Coflfcqucatly, the Rgle AE.EB is = to the D of DE. ^^p. j. £^ ,; Which was to be demonflrated. COROLLARTL Jl^ (fig- 3 ) f^om any point (E) tnithout a circle (ADBF), /Bere if drawn frvernl firmgbt lints (AE, EG, &c). cutting it in (B & P, &c): the reaangles contained by the whole fecants (AE, GE)r and the tarts of them (EB, EF) ivithout the circle, are equal to one another \ for dra'uoing frow the point (E) the tangent (ED), thefe reSangles ^mll be equal to thefquare of tbeja me tangent (ED). COROLLARr IT. Xf from an^ point (E), hvithout a circle (ADBF), thire Be dranjon to this circle two tangents (HD, EC), they 'will he equal to one a^othe^, iecaufi thefquare of each is eqaaito the fame re&angle (AE.EB). 138 The E L E M E N T S Bookfll 1 1 PROPOSITION XXXVII. THEOREMXXXl F from a point (E), without a circle (ADH), ihcrc be drtwn two (bight Jincs, one ot which (AE) cuts the circle, and the other (ED) meets ii; n theredangle contaioed by the whole fccant (AE) and thcpartofitwithont the circle (EB), be equal to the fquare of the line (ED) which raeetsit: the line which meets (hall touch the circle in D, Hypothcfis. ThcfiJ. /. AE cuts the © AJDH in a * 7he flraigbt line ED mchti tk 11, ED meets the O. © ADH in tbep$int D. ///. Tie RgU AE.EB is = to the D of ED. Preparation. I. From the point E to tBe © ADH draw the tangent EH. P- '7- ^r a. Draw the rays CD, CH and the Hraight line CE. Paf !• Demonstration* ECAUSE theRgleof AE.EB is = to the D ofEDfHy^. 3.)and the Rgle AE.EB is alfe = to the D of EH (Prep. * ^ P- 36. B. 3) ^ p ,(^ n v 1. The n of ED i« = tothc O of EH (Ax. i. B. i.)or ED = EH. \ V^ , ' And moreoyer. fince in the A CDE, CHE, the fide CD is =r to ^ ' ^' the iide CH fD. 1 5. B. 1), and CC is common to the two A. 2. The V CDE is = to the V CHE. P. 8. S. J. 3. Wherefore, V CHE being a L (Prep, i . £«f P. 18. B, 3), V CDE is alfoaL. Axi.Bx 4. And the ftraight line ED touches the © ADH in the point D. { (>. 3. B J Book IV. Of E U C L I D. »39 DEFINITIONS. A Re^ilineal figure (ABCD) b faid to be infcribed in another re^ilineal figure (EFGH), when all the angles (A, B, C, D) of (he infcribed figurci are upon the fides of the figure in which it is infcribed (fig, i). 11. In like manner a re^ilineal figure (EFGH) is faid to be dtfcribed about another reailineal figure (ABCD); when all the fides (EF, FG, GH, HE) of the circumfcribed figure pafs thro' the angular points (A, B, C, D) of the figure about which it isdefcribed^ each thro' each (Fig. i), A reailineal figure (ABC!D) is faid to be infcribed in a circle^ when all the angles {A» B, C, D) of the infcribed figure are upon the circiunference of Ibe criclc (ABCDE) in which it is infcribed (Fig^ 2), IV, A reJiilineal figure (ABCDE) is faid to be defcribed about a circle, when each of the fides AB^i BC> CD^ DE^ EA) touches the circumference of the circle (Fig. ^). I40 The ELEMENTS ?<»kIV. DEFINITIONS. V, A. C''^'^^' (ABCD) iffaiJL to be infcribe4 in a reaiUneaJ Jigurt gFGg "when th? circumferpnce of the circle touches each of the fi^ ^Fj FG» vtt* HE) of the figure in wh^ch it is infcribed (Fig. i). VI, ActrcU (ABCD) is dercrtbed ahout a reaUinealJl^ure (ABOj ^^^^^ circumference of the circle paffes thro* all the angular points (A| Bi ^) tli^ figure about which it is defcribed (Fig. 2). VII. Aflraigbt line (AB) // faid to be f laced in a circle (ADBE), ^5^*^ ^^trcnaitics of it (A & B) ar^ ia the circumference of the circle (fig^ i)* ipoklV. Of E U C L I a HI PROPOSITION L PROBLEM I. I N a given circle (APD)* to place a ftraight line (AB) equal to m given ftraight fine (N), not greater than the diameter of the circle (ABD). Gi?£n. A ABD together wtb tie Braigbt Hne Ny not ^ ^e diameter of SoQghC. ne ftrmigbt line AB pkceJ in the ® ABO &= /a the given ftraighi line N. Refoluthn. Draw Uiie ^Vuncter AD of the ABP. CASE I. IfADiszrtoN. Prf.u _ HERE has been placed in the g^yen Q ABD a ftraight lioe = to the given N. D. 7. B. 4* CASEH. IfAPi»>N. I. Make AE =r to R P. 3. Jf. r* a» From the center A at the diftance AE dercribc the EBF, . and draw AB. Pqf. j. Demokstratiqk. Jd EC AUSE AB is = to AE (D. i $. B. i), and the ftraight line N lz=iohZ(Ref.i.) %. The ftraight line AB, placed in the ABD, will be alfo 2= f^. i.^. f. toN« \D. j.B.u Which was to be done. '"1 i4« The E L E M E N T S BooklVj £ M PROPOSITION II. PROBLEM n. Xn a given circle (ABHC)» to infcribe a triangle (ABC) equiangular to i given triangle (DFE). Gjven. Souglit. A © ABHC together nvitb tie A TA# A ABC infcrihtd in the ® ABHC, DF£. equiangular to tie A DFE. Re/olutUn, p.nM I. From the point M, to the ABHC draw the tangent MN. a. At the point of contad A in the line MN make the V BAM = to the V FED, and the V CAN =: to the V FDE, P. «5-^'*- 3. Drawee. P^.i. Demonstration. JjECAUSE the V BCA is = to the V BAM (^P. jt. J?. 3;, and the V FED is = to the fame V BAM rRef. z) j alfo the V CBA is = to the V CAN (P. 3a. A 3 J and V FDE is = to V CAN (Rif. 2. I. It follows that V BCA is =: to V FED, and V CBA = to V FDE. Ax. I. B. i. H. Conreqoentl)r, the third V BAC, of the A ABC, is alfo to the third V DFE of the A DFE, and this A ABC u infcribed in the C P. 31. B. 1. ©ABHC, ID.S^B.** Which was to be done. j !F Of E U C L I D. ^ PROPOSITION III. PROBLEM TIL x\B0UT a given circle (EFG) todcfcribe a triangjc (ABD), equiangular to a given triangle (HKL). Giyen. Sought neQ EFG» tegetierwitiiie£i The A ABO Jefctihfd ahmH the © HKL. EFG, tfuiangular to the A HKL. Hefolution. I. Produce the fide HL» of the A HKL, both ways. Pof. z. a. Find the center C of the © EFG, and draw the ray CE. F. i. B. 5. 3. At the point C in CE, make the V ECF = to the V-KHM, and V ECG = to V KLN. P. 23. B. i. 4. Upon CE, CF, CG, ereft the ± AD, AB, DB produced. P. 1 1 . ^. 1. Preparation. Draw the ftraight line PE. Prf, i. BDemonstratiok, EC A us E the V CEA. CFA are L (Ref. 4. J ' 1. V FEA -f EFA are < a L. & AD, AB meet fome where in A Ic may be demonftrated after the fame manner, that, a. J\D, DB & AB, DB meet fomewhere in D & B. And (ince AD, AB, DB are -L at the extremities E, F, G of the rajs EF,CF,CG(Ref.4J 3. Thefe ftraight lines touch the © EFG 1 and the A ABD formed r » ,r » by Ihefe ftraight lines is defcribed about the © EFG \ C.' ^7 Moreover, the 4 V CEA + CFA + ECF + FAE of the qua- ^ ^''•^4.^ (Ax.S.B.i. " \Ax.ii.B,i. 4 (trilateral figure AFCE being = to 4 L (P^ 3*. B. ij, and the V CEA +CFA = to a L (Rrf- 4)- 4,. The V ECF -}- FAE are alfo = to a L. ^^^ J. B. r. 5. Or = to y KHM + KHL as being alio = to a L. Kj1x,t.B,i. But V ECF being = to V KHM (Rtf. 3). I /». 1 3. 2?. i . ^. The V FAE is = to V KHL, and V GDE = to V KLH. . Ax. 3. A i. 7. Hence the third V FBG of the A ABD is = to the third V HKL of A HKL. P.ZZ.B.I. S. Therefore the A ABD defcribed about the © EFG is equiangular to the giren A HKL« Which was to bt done. 144 The ELEMENTS fiooklV. PROPOSITION IV. PROBLEM IF. O infcribe a circle (EFO) in a gtven triangle (ABD). Given. The^iABD. Souglit. n# ® EFG iJ^criW i> A AABD. 'tfolutton. t. BifeQ tbe V BAD, BDA b^ cde 4lrAt|lK liiie» AC, DC ^xth duccd ttbfii thc7 meet one'aaofher \fi C. ?• 9- ^' '* a. From the point C let fall upon AD tbe JL CE. P. »i-^- »' 3. And from the ceilter C at the diftance CB, dtfcribe the EFG. P4y Preparatm. Ffom the poiot Clet fall upon AB & M the J, CF^ CG. f. i* • *' Demonstration. IJECAUSE in tbe A AFC. ACE, the V FAC it= to the V CAE (RtJ. O, V CFA = to V CEA (Prep. Re/. % & Ax. 10. A i)» & AC tommon to the two A. I . The ft raight line CF is =: to CE. P. t6. * »• In like manner it mat be demonftrated, that The ftraight Kne CG is li: to CE. Confeqventi/, tbe ftraight lines CF, CE, CG are = to one another ; and the defcribed from the center C at the diftance Cft will ^ alfo pafs thro* the points ^ ^ ^_ _ - I a. And (ince the fides AD, AB, DB are ± at the extremities E, F, ' G, of the taya CE, CF, CG (Ref 2 ijf Pnp.). J 4. Thefe fides ^ill touch the in the points £, F,G. | 5. Therefore the EFG is infcribed in the A ABD. Which wa»tobe done. J Book IV. Of E U C L I D. HS ^.^ PROPOSITION V. PROBLEM T, X O dercribe a circle (ABDH)^ about a given triangle (ABD).* ^■^"'— . Sought. TArAABD. Th © ABDH defcHMaiout /iStAABD. Kqfolution. I. difed tlie fides AB» t>B to the point! £ & F. P. lO. i?. i. a. At the points E & F in AB, DB, tn& the X EC, PC, , prodnced until they meet in C. P.ii.B.i. 3» And whether the point C falls within (ig. tj or without C/i' 5) or in one of the fides ffy. zj. of the A ABD^ troin the center C at the dittance CA defcribe the GABDH. ^ Pofs. Preparaiion. Draw the ftAiight lines CD, CB. P^f, \. BDbmonstratiok. ECAUSE in the A AEC, BEC, the fide AE is = to the fide EB rReJ. i), EC common to the two A, & VAEC cs to VBEC (Ref. a V yfx. lo. A I) 4. The ftraight lineCB is st to CA. P. 4. ^. i. It may be demonftrated after the fame maaner, that a. The ftraight line CB is = to CD. 3. Confequently, the ftraight lines CA, CB, CD are =: to one another 1 and the ABDH defcribed from the center C at the diftanceC^. tiS.t. CA, will pafs alfo thro' the points B & D. ( D. 1 $. 2?. 1. 4. ThereAre this ABDH is defcribed about the A ABD. D. 6. 3. 4. Which was to be done. COROLLART IF the iriangU ABD ie acute atigkd, the p^int C falls luithin it (fiflr« 1)1 hut if ihit$rian0e he ohtufe angM, the point Q falls n»itheut it (fig. 3} j in fine if it he n right angka trianglft the point C is in one of the fides (fig. a). T 1^ H< The ELEMENTS Book IV. PROPOSITION VI. PkOBLEMVL O bfcribe a Squm (ABDE), in a gt?ctt Circk (ABDE). Gives rbi D ABDE iM/cfUidin this ®. Refolutkn^ I . Draw tbe DiamcCers AD, BE, fo aa to oit each other at L- '• * '- '* ^' jt. JoiotlMixExtrcfnitioby theftraiglitLinuAB,6D,D£)£A. H >• Dkmonstratiok. Because in the AABC, DdC tha Me AC it =: to the fide CD {lUf. 1. hD.\S'B\)t BC comoioii to the two /^ & the VBCA= to V BCD (V I. ^ ^*. «o. ^. «)• 1 . The ftiaigbt Une A6 ia = to BD. P. 4- ^ >• It may be demooftrated after tbe faiae manner, that 1. The ftraight line BD is = to DE» DB =:: to £A & EA = to AB. 3. Copfequentl/, the ftratght lines AB, BD, D£, £A ar» = to one another, or the qnadriTaterai fiffure ABDE is eouikiteral. Ax.\*l»^* And foecAufe each of the V ABD^BDE, DEA, EAB it placed in a fenii.0. 4. I'hefe V are L., h the equilateral qadrilateral« figuie ABDE is reaangular. P.Ji.i^J' c. Wharefore this qaadrilateral Gaure iaarquart iofctibed in the(Z).3P*^*^' . ©ABDE. ID. }M Whidi was 10 be done. J Book IV. Of EUCLID. 14/y B E t - c 1 K ^^ D PROPOSITION VH. PROBLEM VIL O defcribe « Square (ABCD) about % giTcn Grcie (HEFI). •Given. Sought Iht © HEFI. 7ht D ABCD JefcrOidahiit the © HEFI. tte/oiutm. u Draw the diameters EI> HP to as to cut each other at L^ Pax. B,ii 2. At the Extremities H» E, F, I of thofe diameters ered the JL AD, AB, BC, CD. P.ii. Ba. Demonstration. 1. 1. H E lines DA, AB, BC, CD, are tangents of the © HEFI. 2. And the ftraight line AD, is Pile, to EI, as alfo the ftraight line BC 5 bccaufe the V HGE + GHA, & V FGE + GFB are = to z\^(Ref,\.li2y P,z6.B.u 3. Confequentlv, AD is Pile, to BC, lilcewife AB, HF, DC ate Piles, i* 30. B,t. 4. Wherefore the quadrilateral figures AI, EC, AF, HC, AC are Pgmes. D.35. B.i» 5. From whence it follows, that the ftraight lines AD, EI, BC, alio AB, HF, DC, are = to one another. . P.34. Ri.' 4 And fince EI is = to HF (D. 1 $. B. i.J, the ftraight lines AD, BC, AB, DC are equal. ifr.l. *.!• But V EID of the Pgme. AI being a L (Ref. 2). J, The V A, which is oiagonallj oppofite to it, is alio a L^ ' ^.34. Ba. It may be proved after the fame manner, that g. The V B, C, D are L. ^ Confequentlv, there has been defcribed about the © HEFI t quadrilateral figure ABCD equilateral (Arg. '6J & reda]q;ttfaur (Jrg. 7. (^ 8) 3 or a fquare. J D. 4. B.u iD.iQt^B.u Which was to bt doae. ^ The StfiMENTS BookXn. aattKateaaaaiBaeBBseasesauaammaae^aBm B r' N I ^ y / L PROPOSITION vni. Problem m O iifenbeiKSfde (ABDE) to i tifea l^iure (FGHf). OffWl. Songnt. ^ t«» D FGHI. Tie « ABQE <«M^ « tit D (FGHI). t. Bififi the lUn FI, FG of the O F<i«I. f >» I^' 2. 31iio' ^esobH of fcakw A & e» drMr AD File, to FG w IH & BE PUe to FI or GH. /* 3'- *' J. From the center C, where AD, BE interieft each other, at the diftance CA deioibe the ^ <ABD£. Ny B. DftMOMfi'mATlOK. f E C A U^£ i^ f^uret F£, BH» FD» AH. FlC, A£» BD, CH are Fktatn. (Ref. i- W D. 35- *• " )• , ., I . The ^ftniigbt liac F A ia xx to BC & FB s lo AC. ^34*^^ Butihe whole lines FI, F6 being equal (D. 30. A i.^ and FA» FB being tbe halves of thofe ftraight lines (Rtf. i). . 9. The ftraight line FA is = to FB. AT-*'* 3. Confeqoentlf, BC4s aMb ;:^ to AC ; aad Ukewife AC is s to €E & ^ ,BC = toCD. i*.»**' 4. JProm whence it follows, tfaatt^e ftraight lines AC BC, CE» CD ares: to one anotli(BE» and the •deicribcd ftotfi the oentec (^^ ,' C at the diftance CA i paftes aUb ^hro' the f oinu fi, D, E. (Di f'-* But the V QAP, EBG, ADH, BEI being L. (P,t^ B. i.) as betng interior oppofiie to ^he L. GFA^ UGB, UiD, flE /D. so, B. i> , . |. The ftraj^t lines FI» TG^ GH, IB are taafenu of the fPi^^i ©ABDEr iCr. . t^. t^lijrtfore this ts inftribed to the fqwe FOHI. i^* f^^ Which wtt to be done. B»ok tf. Of E u c 1. 1 a 149 L saniMKfiBsas«HBaBaHsssssaanHBBaBHBsaBn fROPOSITION IX P&OBLBU JM. O dercribe « circle (ABDE)^ iftpat « ghren i<iitBre {ABDE). Given. 7}f □ ABD£r •Sought. tbt D ABDE. RefoluU^n. %. Draw the dkgoaab AD, BE. /V<' s. From ahe oenier £, where the diagoiuls inteffeft eiic1& other* wl »t the 4iflttMe CA« defcribe the ® ASDE, 1^1- P£MONSTRATIOK. J3^ C AU S E in A ABE, EBD the fide AB is ss to the fide BD AE =:.to ED (D, jo. B. .1.), & BE common to the two A. l. The V ABE ts to Y EBD, fr Ae Whole Y ABD is biicAed by BE. P. 8. £.!• It maj be demonftrmted after the fiime manner, that %. The V BAE, BDE, AED, are bifeaed by AD, BE. But the whole Y ABD, BAE being = to one another (D. 30. ^. i )• 3. Their halves, the Y CBA, CAB will be alfo eqnal. ifx.7. B.i. 4. Cenfequently, QA is ;=: to CB, likewife CA is sz to CE, and CB ss to CD. P. 6. B.I. 5. Hence CA, CB, CE, CD are s to one another, & the defcribed fiom the center C at the difbuice CA, will alfopafi thro* the fAx.i.B.u points B,D,E. lD.i<.B.u 6.. V^herefbre the Q> ABDE is defcribed about the 13 ABDE. D. i. BJ^. Which was to be done. The ELEMENTS T PROPOSITI ON X. problem:!. O dpfcribc an Ilofcclcs triangle (ACB), having each of the tngtealtW hafe (AB), double of the third angle (ACB). Gi?en. Sought. . A line AC taktn at mlL Th Ifi/alis A ACB. *««« * VCAB*rCBA=/*aVA» Rtfoltaion. I. Draw any ftraight line CA. H^' a. Divide this h'ne in the point C, fo that the Rgle. of CA . AD . be = to the D of CD. /" »/• ^ 3. From the center C at the diftance CA defcribc the © ABE. Hi - 4. Place in this © the ftraight line AB =: to CD ft draw CB. P. »*4- Preparation. I. Draw the ftraight line DB. ^»/'' - a. About the A CDB dcfcribe a ©. ^^ S ^ B Demonstration. E C A USE the RgJe. CA . AD it= to the D of CD (Ref. 2.) * the D of AB is = to the D of CD (Ref. 4. V P. 46. Qtr. 3. B.\). t. The Rgle. CA . AD will be aifo = to the D of AB. Ax.h^^- a- Confequentlj, the ftraight line AB is a ungent of the © CDB. P-37* ^^' 3. From whence it follows that V DBA is =: to V BCD. ^5* ^> Therefore adding to both ikies V DBC. , 4. The V ABC will be = to the V BCD+DBC. i**- *«• But V BDA beini alfo = to the V BCD+DBC (P. %%. B^ i. 5. Therefore the V BDA is = to V ABC. A%\ ' »' Lifccwife, fincc CB ia = to CA (Ref, 4. W Z). ic. ^. i). » ^ »r. 6. The V BAC is = to the V ABC. . 'j^X !!». 7. Wherefore, V BDA is = to V BAC, & DB is = to AB- I % I a,. And becaufe CD is dfo = to AB (Ref, 4). r jixx l\ 8. The ftraight line DB will be z=: to CD & V CBD = to V BCD. < p c 1 1. Adding to both fides V DBA or its equal V BCD (Arg. 3/ i r. o. • 9. The V CBD+DB A or V CAB is = to 2 V BCD ; and tlierc haa been defcribed an Ifofceles A CAB having each of the V at the bafe doubVe ix.a. l-^' of the V at the vertex. Which was to be done. Booktv: Of E U C L I D. JSt T PROPOSITION XL PROBLEM XI O infcribe an equilateral & equiangular pentagon (ABCDE) in a given oircle (ACE), Gifcn. Soufbt 7iir ACE. The equilateral ti equiangular pentagem ABCDE, infcribed in the ACE. Refoluiion. I. Defcribe the irofcelcs A f GH ha?ing each of the V at the bi^fe FH double of the V at the vertex G. P. lo. J?. 4. a. Infcribe in the ACE a A ACE equiangular to the AFGH. F. a. B. 4. 3. Bifea the V CAE & CEA at the Bafe, by the ftraight lioes AD,EB. . . P. 9. fi. I. 4. Draw the ftraigfat lioet AB, BC, CD, DE, Pof.i. Demonstration. B. BECAUSE each of the V CAE, CEA is douUe of the V ACE (ReA I. W a.;, & thcfe V arebifcaed (R^, 3.;. I . The five VACE, CAD, DAE, BEA, CfiB arc z^ to one another. Ax.T.B.u ^ From whence it follows that the arches AE, ED, DC, CB, BA ^ p ^ • are = to one another, likewife the chords AE, ED,.DC, CB, BA. \ « Zt »' f ' Bat if to the = Arches AE, CD (Arg. 2 J, be added the arch ABC. ^ '^' *9' -«• 3 • 3. The whole arch EABC is ::;= to the whole arch ABCD, and ^ . « d . ^ ^ CDE is = to the V DEA. J p^'Z' J* It maj be demonfirated after the f«ine manner, that C • 7 • - j^ Each of the V EAB, ABC, BCD is = to the V CDE or DEA. ^. VVhereforeihere has been infcribed in the ACE, an equilatetal CAtg^, zj & e^aianguhr ^Arp /^) peatagone. D. 3. B» 4« Which was to Be done. 15^ The ELEMENTS BodkE PROPOSITION Xn. PROBLEM XU. j[ O dercnbc an equilateral & equiangular pentagone (ADFHK) aM* given circle. Sought 7A# imtihugral petamne ADFHK Given. rAr©LEa Refokthn. P.tiM 1. IntheOLEGyinfcribeanequilateralftequiaapiIatrpemagone 2. To the point B, E, G, I. L. draw the nrf • CB» CE, GO, CI, CL. /y^ t. At the extremttiec of thefe rafl ere€l the JL Modne^d A0, 0F, FH, HK, KA. P"**' Preparation. DtAW the ftraight lines CA» CD> CP, CH» CC iV > Pemoitstratiok. x3e:CAUSE theftnightltnesAD.DFsPH.HK^KAareXatthe extremities of the rays CB, CE, CG, CI, CL. (H^f. %,) u Thofeftraig ^ " " ~ And the V J ABL4.ALB, a. Therefore thefe ftraight lines AD, DP, FH, HK, KA wilt meet in the points D, F, H, K, A. Um i '^ But fincein the A CEP, CGP the fide PE is = to the 6de FG (P. 37. C9r. B, 3. £* Rtf. 3;, CE =; GC. (D. 1 5. B. t.) ft CP com- mon to the two ^. P, i^iy I luc rajTB v-o, v^c, V.VF, ^1, v,!-.. ( t%9f, yj ^p i^^ir. J, light lines will touch the ® in the points B, E, G, I, L { /v ^I)BE4-DeB, FEG+PGE. HGI + HIG,KIL + KLI, ^"^^ _ ^ LB, taken two hy two are < a L». i«f. «•* Book IV. Of E U C L I D. »55 3. The VCPEi« = totlic VCFG* VECF = to VGCF. P.S. B. t. 4. CoofiqpeotI/* V £FO, it doabit of the V CFG» & V CCG double of the V PCG I likewifc V <^HI is double of the V CHO ^ V OCI doubieof VGCH. 5. Moreover, V £CG it := to V GCI» on account of tlie equal trchet EG,Gl(Ref.iJ ^ RzB.B.s. «. Coofequenclr, V FCG u s= to V GCH. jlx.i.B.i. But the V CGF» CGH, of the ACFG» CHG being alfo equal (Rif: 3. (f Ax, 10. /^ I,) &CX} comiBon to the two ^. 7. The ftraight line FG it =: to GH » V CFG is — to V CHG. P. a6. IT. 1 . k Wherefore FH^it double of FG, & likewifc OF is double of £F. Ax. z,B.t, And beaiufe FG is = to EF (P. 37. O. J. 3^. 9. The ftraight line FH it alfo = to DF, (Ax. 6. B. ijy k likewifc the ftraight liaesHK, KA, AD are =:c to FH» or DP. Again V £FG or DFH being double of the V CFG, the V GHI or FHK doable of the V CHG and alfo V CFG = to V CHG; .«? 7/ to. Tiw V OPH. FHK ucsstooM tnoiber, ud likcwUe tbe y HKA. KAD. ADF are = to DFH or FHK. II. Confeqaeativ there has been defcrit^ aboat tbe © LEG a peota* goa ADFFHK (jfrg. i). equilateral (Arg. ^J, aod cqniaogolar (j^. toj, D. Which wa< to b« doae. 4.*. 4. ^^ U The ELEMENTS'^ Book kn. 1 ^.-^ PROPOSITION XIII. PROBLEMJin. ' X O infcribc a circle (GHIKL), in a given equiUtcral wkI cquiiflpltf Pentagon (ABDEE). GiTcn Sought ^ Tbi equilalfralfj ep,iaf^larf€ntag9H «f © GHIKL fii/WW m « ABDLF. femin^H. Refolution. I. Bifea Che two V BAP, AF£ of the peotap>a ABDEF b/ ' the ftraight lines prodtttedCA^CF. *'''•*», *. From the point of concuKe C let fall upon AF the ± CI* P. «*• *' *' 3. From the point C at the diftanoe CL, defaibe the © GHIKL. ^4- 3* Prtparation. I . Draw the ftraight lines C5p. CD, CE. H ^' a. From thr poinc C let fall i^pon AB. BD, DE» EF the ± CG, ^ CH,C1,CK. ^.««''^- Tj Demonstration. XSeC A U S E in the A ACF, ACB the fide AF it = to the fide AB. the fide CA common to the two A & V CAF = to V CAB '(Ref, I li given). ... I. It follows th«t V CFA Is = to V CBA. ^- 4- ^- *' ' But V AFE heing = WVDBA tnd double of V CFAfRef. l). . , , a. Hence, V DBA is alfb double of tSc V CS A, or V CBD = to V CBA Xr. 0.* ' ' It' may be'c'einonftiared after the' fame manner, that 3 The V CDB is 2^ to V CDE h V CED = to V CEF. - Tiiereforc in the A CBG, CfiH, V CBG = to V CBH f^. t). V CGB =: 10 V CHB (Prep, a W Ax, \6.B,\ .), & CB common 10 ^ , the two A. ' ' P,7b.B-^ 4. tonfcquently, CG is = to CH ; likewife CI, CK. CL arc = to CH ^ or to CO. ' '•* 5. Therefore t,be © defer ibed from the center C at the diftance CL will . " ilfo pafs thro' the points G.H, I. K. D.\<i^»'^ Attd'bdCauf^i the ftraight lines- ABrBD, DE, EF, FA areX at the extiemiiici of the lays CG, CH, CI, CK. CL (Prep, a &f Ref, 2). D.iS''^' 6 Jho^e jfralght lines wUl touch the © GHIKL (P- 16. C»r. B. iJ i ^ and iWi is infdibcd in the pentagon ABDEF. 2). $. *♦• ,.. tis^ ;:.:ij .. . :,. WWch was to bc done. BooklV. PROPOSITION XIV. PROBLEM XIF. O defcribe a circle (ADF) ; about a gtTcn equilateral and equiangular pertagon (ABDEF). on -^ e CStco Sottglic TBiipnIaiiralfJifuiangular 7h © ADF» d^criied aimt ibis finUtpn. pentagon. Rifolution, I. Bifeft die V BAF» AFE hj the ftrtight lines CA, CF P. 9. B. 1. produced. S. r rom the point C» where thofe ftraight lines interfeQ each other» at the diftance CA defcribe the ADF. Ptf. J. Preparation. Draw the ftraight lines eB, CD, CE. Ppf. t. Dbmonstratiok. I. jPhE ftTaightlinetC3,CD,CEbifcathe VABD.BDE,DEF.(^^ «3.'-4- 4. Aadbecaufethe VBAF ia=: to the V AFE, the V CAF wiU be ( ^^* . ftlfi) = to the V CPA. Am. 7. P. i. 3. WhereforeCA 18= to CF- P. 6, B. i. It BMj be demonftrated after the fiune manner, that 4. Each of the ftraight linet CB, CD, CE it = to CA or to CF. $• From whence it followa , that the deiciibed from the center C at the diftance CA will paft thro* the points B> D,E, F. D. 1 $. P. i. 6. Confeoueatl/ the © ADF, is delcribed about the given pentagon AHDEF. D. 6* B. 4. Which was to be done. i5<5 The EL E ME NTS Bdcnr. PROPOSITION XV. PROMLSMXr^ X O infcribe an equilateral and equUngular Her^gon (ABDEFG.) h > given Circle (BEG), Giren SoirghC Tbi BEG. Tie iouilaHralli fpiis$u¥larHe»^ ABD£FG» infcrihedin tbt 0BE& Re/olution. • I. Find tbe center Cofthe BEG, and draw anj diameter AE. P. i. ^' 3- a. From tbe center A, at the diftaQce AC defcribe an aich of a0BCG. Prf.y^ 3. Draw tbe rayi CG, CB produced to D & F- P^. i.y** 4. Draw tbe ftraigbt lines AB, BD, DE» BF.>F6,GA. P^ <• B Demons T& A TICK. ECAUSB in the A BCA, tbe fide BC ir=7 to the fide AC, A13 it alfo = to AC (Ref 3. W D. 15. B. \). This A is equilateral & equiangular. .. Wherefore, V BCA is =r to the tSird part of a L* & Hkewire V ACG is alfo =: to the third part of % L* '• 3^'''' Bttl tbe V BCA + ACG + GCF being = to a L- (P^ 13. B. t), |. Tbe V GCF is alfo = to tbe third part of a U ; & the VBCA, ACG , GCF are = to ope another, i jtxJ'B^^* 4. Confiequentljr, the V FtE, ^CD, DOB which are =r to them at being tneir vertical oppofite ones, are alio 2= to one another. P. i$* f' '* 5. Hence» the arches BA, AG, OF, PE, ED, DB aee = to oneanothef, ^ p 16 P t as Iikwife the chords BA, AG, GF, FE, ED, DB. \ p ^ a 1 6. Therefore the Hexagon . AjBDEFG infcribed in the © BEG b^^'^^ equilateral. Moreover the arch BA being = to the arch ED fArg. J^ 1 ff tbe common arch AGPE be added to both. 7. the arch BAGFE will be ^ to the areh AGPBD. Jx. 1. 1. ^ 8. From whence it follows, that V BDB is = to V DBA, aid-Hkewlfe each of the V FEP, GFE, AGF is = to the V EDB, or to the VDBA. Ei7.B,i* 9. Therefore tbe equilateral Hexagon ABDEFG, infcribed in the BEG is alfo equiangular. D. }• '• 4- Which was to be done. Book IV. Of EUCLID. 457 1. B. I. 2. B, 4. 11.^.4. 1. B. 4. PROPOSITION XVr, PROBLEM XFI. X O mfcribean equilateral and equiangular quindecagon (EAFG &c.) in a given circle (EBI), Girett Sought fbi ® E8I The eauilaterarig efuianguiar fuiiJec^gon EAFG f^c, Refolution. t . Defcribe ta equilateral AN. P. z. Ififcribe in the EBI, a A ABD, equiaogukr to the equilat<BraI AN. P. J. And an eqailateral 8c eauiangular pentagon EGBHI. P. 4. Draw the chord EA & place it 1 5 times around in the EBI. P, BDbmonstratiok. E CAUSE the AABD is equiangnlar to the equilateral A N I. This A is alfo equilateral, or AD is =s to A6 = to ED. P. 6. B. t. a. And the arches AD, A6, BD are = i^ ooe another^ or each is the third part of the whole O. P' a8. B. 3. Again, hecaufe the pentagon EGBHT is equilateraU (Rif. l). 3. Bachofthearche«£G,GB.BH,HMCisthehfthpartofthewholeO. P. aS.lT.j. But the arch AB beins the third part (jlrg. 2 ) and the arch EG or GB the Bfth part of the O (Arg 3J. 4. There maj be placed in the arch AB fi?e (ides of the quindecagon, and in each of the arches EG, GB three fides of the quindecagon, or in the arch EGB fix fides of the quindecagon. 5. Confequently one of chefe fides may be placed in the arch AE,and the eauilateral quindecagon EAFG &c. will be infcribed in the EBI. D. 3. B. 4. Moreover, fince each of its V FA^ Is contained in an arch FHE which is = to thirteen parts of the fifteen, ioio which the circum- ference IS divided, 6. Thefe V will be = to one another. P. 27. B. J, y. Therefore there has been infcribed in the EBI, an equilateral & equiangular quindecagon EAFG. Which was to be done. 1 BookV. TlieELEMENTSofEUCLID. 159 B^IM^I M * N - Fig.a N D N 'n' K Q > UK R r ■ N' N' N "N • N' N'N DEFINITION S. x\ Lefs magnitude is faid to be zfari of a greater inagnitudej when the left meafures the greater. {. I. By the expreffioH of meafuring a magnitude Euclid mmu to be contained in it a certain number of times without a remainder, that if a left magnitude N (fig. I.) meafures a greater ^f when the magnitude N h contained in M without a remainder twice, tbrice, four timer, and in general, any number 9f times vjbatfoever, or wbicb cofkes'to^tbefame, wben tbe left magnitude N repeated twice, tbrice four -times,' and in general any number of times pro-^ duces a mbole, equal to tbe greater M. ^ 4. a. Tbofe parts wbicb meafure a wbole without a remainder^ are called aliquot parts, andfucb as are not contained in a wbole exaffly, butaremea^ furedby fome other determined quantity wbicb meafures alfo tbe wbole, are called aliquant parts. Thus tbe numbers ^ 3> 4> 6 arefo many aliquot parts of tbe number la ««- Jideredas a wbole ; as eacb of tbe numbers i, 3, 4> 6 is found repeated in i z a certain number of times witbout a remainder. But tbe numbers^, 7, 9 lie. are aliquant parts of tbe fame wbole ifi ; flx tbey do not meafure it a but with a remainder : alt bougb thenar e all medfured by unity as well as lii wbicb often bappens in ofberhumbirs different from unify, as in tbe number 9 wbicb is com^ menfurable to I a by tbe nnmber 3, as' alfo by unity. - Likewife tbe magnitude N (ffg. a.) /*> an aliquant part of tbe magnitude at r meaft M t€o TheELEMEKTS BookV. N M N N N N N ■ N D E F I N I T I ON S. {. 3»JL ^ general numlert are f aid U he commenfuraHe to each other f»bifi mgf refultfrom unity or one of its aliquot parts repeated a determined numter if times : or what amounts to the fame that which ie meafured by mniiy or me of its aliquot parts. Thus the numbers 6, 9> 1 7, and the fr anions *, | are commenJurabkimmitrs\ betattft thefirfi map be conceived to refultfrom the determined andfuccefive addi* tion of unity ; and the loft from that oj tie jr actions 4 £f 7 aliquot parts of wdty* §, 4* According to this definition^ a commenfurable quantity^ // tbdt which refu Its from the determined repetition of any determined quantity, A quantity is therefore commenfurahUf when it contains one of its parte ^ ess ofMk as a determined number contains unity, f. 5. Commenfurahility is therefore fomething relative. The m^gmtudes M and N are commenfurabief as having a common and determined meafure t which can be taken for unity 9 and meafure them bothexaSlly \ or, as thofet^sso magnitudes may artfe from the determined repetition of the fame quantity R^ be it what it wilL f. 6. It follows from this notion of commenfurable numbers, that they areaO multiples of each other, or aliquot parts, or aliquant parts. For if the quantities M and N> are commenfurable, N meafur^s M> or M meafures N« or fome other determined number r meafures them both. In thefirfi cafe, the number M» // a multiple of N, in thefecond cafe M, is an aliquot part of }i, and in the third, the lejfer of the two is an aliquant part of the Uajl, Thefuness true with refpefl to rational magnitudes in general, f . 7, 716* number which cannot refultfrom a determined repetitibn of" unity or of one of its aliquot parPs is called, irrational or inconvnenfurablej tu/f^ re/pe/i to unity. And in general f magnitudes which cannot refultfrom the determined repel it ion of the fame determined quantity confidered as unify ^ are, incommenfurable to one another, or irrational. BookV. Of E U C L I D. i6t DEFINITIONS. HUS thtfiit (AD or DC) of tbefquare (ABCD) // incommenfurahk t9 its diagonal (AC), ar bow much one contains of the other is inajpgnable (Fig. I ), §, 8. From whence it follows^ that if two nftgnitudes M and N, are incom* menfurable to .each other, M cannot be a naultiple of N ; nor an aHquot part, nor in fine an aliquant part of this fame N, fhr if it was, the magnitudes M and N could he meafured by the fame determined magnitude, which is re^ pugnant to the notion of incommenfurability (Fig* 2 ) A greater magnitude is faid to be a multiple of a lefs, when the greater ii meafured by the lefs. 7buSf the number 12 is faid to he a multipUt of the number 4, hecaufe 4 mea^ fares 1 2 without a remainder, To the. term of multiple corref ponds that of fubmultiple, which ftgnifies, that a Ufs magnitude is an aliquot part oj a greater \ thus 4 is a fubmultiple of 12% as 12 is a multiple of a* Ratio, is a mutual relation of two magnitudes of the fame kind to one another in refpeS of quantity. ^bis definition is imperfeB^ and is commonly believed to pe none of EuclidV» but the addition of fome unjkilful editor ; for though the idea of ratio includes a certain relation of the quantities of two homogeneous magnitudes, yei tbisgjmeral cbaraHer is not fufficlcent ; hecaufe the quantities of two magnitudes arefuf* ceptihle of feveral forts of relations different from that of ratio. Thus, when in a circle the fquare of the perpendicular let fall from the circumference on Sbe diameter, is reprefented as cohfiantly equal to the difference oftbejquares of ibe ray, and of the portion oj the ray intercepted between tbe center and the perpendicular, without doubt, this perpendicular is conftdered as bearing a cer^ tain relation to this portion of the ray, hut it is manijefl that this relation is not a ratio, ftnce tbe quantities are compared only hy the means of tbe ray wbicb is a tbird homogeneous magnitude different from tbe quantities compared. y^ The E I, ^ M S N T S 90Qk ;i D E E I N I T I O N 5- »vr ^''• JVlAGNITUPKSarc liik|t^hftf« t: mlifr to om mOicr ; lafs can be muiliplic4 Tq a^ to €«cqc4 itieothcr* . {, I. Tbi Une$ kijh b4fQe a ratio to om ^mtbir^ if^€Hfi tbi Kmt B^ /«r fxampUs taHfn ttrff timfs 0nd a k^Jf^ h egM^l /• tbgi line A>. muLt^bm par times exceeds it. Tbi RgUi M Cs? N bitpe 0(fa a ratir Ip mi^mtbtrp ttagnfk tbe Rtle N teiken tbree times and a balft is = to Rgle M, and repeated oftner gficeea* it. But tbe line Bf and tbe Rgle M isiffe no ratio to one anotber, becauja ibc Hsu & repeated erorr Ji v/tcn, can nivor produce a mofnitndf ^bUb mtouU. e^ssl or exceed tbe Rgle M. Tberefore^ only magnitudes oj tbo/amfMsfd cstn-hav^ 4 rjitjo^p oneanotboTt nt numbers to number s% lines to lis^s, furfaca^ ioi/urjmvt andfoUds tofolids.. fi 2. In confequence of tbis dejlnition^ a Jlnite magnitude andan infinsiromepiaot no ratio to one anotber, tbougb tbey befupfofed of tbe fame kind. For a n^^ nitude c4nc4ived infinite^ is conceited witbout boundr, confequentlj ajhdte tmtg* nitude repeated ever fo often (provided tbe nutpber of repetions be sUiermim^ c§n,neiter become equal or exceeds an infinite magnitude. |. 3« A ratio // cdnimenrurahle, fuben tbe terms of tbe ratio M Ar N are i mmfurable to eacb otlir, iJ a ratio isfai^ to be sncommenfurablensfbm tbe terms of tbe ratio are incommenfurable. f 4. Tbezntectdttitof tbe ratio of M to Ns is tbe Jhft of tbe tw term wbicb are compared^ and tbe other is called its confequentx The firft of four niagnitixie» is Taid to have the fiime ratio to the fetondf whkk the third has to the fourth^ wheo any equimultiples whttlbever of the firft«B< third being tai^en» and any equimultiples whatfoeverof the fecond aod-fiMinlis frif^r- lodk?. OfEUCLltD. ^63 ^■■■■■■■■■■■■■■■■■■■■■■■■■■■^^ bEPlKlTIOMl If the multiple of the firft, be left thin that of the f^ecdrtd, the fAiiUrpte 6f th* third is alfo lefs than that of the fourth ; or if the muhiple of the firft be equal to that of the fccofld, the rtiuhiple of the third is rffo eq jal to that of the fourth, «r if tke ilraiti|)«B «f tbt ftrtt be gfeafer (hin that of the (tcehd^ tht multipie df tte tUrd 18 Ulib gt^attr than that bf the fourth. §. I. The ratio of tbenumkera to tht humher 69 is the fame as that of tht numlct % to thi nuMbir 24, fot if the two antecedents ±(f6 he fhuttiplie'd . hy the feme rmmher M, afid the fivo cOnJequeHts 6 Csf 24 3> ahothef hUhil^Y N : thi multiple iMof thejlrjl antecedent cAnHbt be±:or>or<^ the inittihU 6 N tfits confeauent^ unlets the multiple of the fecohd antecedent t iK, *# ai th'efdfhk tlm 2S ^^ > ^r < the multiple 24 N of its tonfequent,fof it sJ eifiJefit that aM + aM + »M+iMiial&tt6N^<IN+6N4^ttN>cBifMMtta«Ni Likewife, /f2Mbe>6V, then aM4.iM+iM+aM»airoS6N4.6 hi 4.6N4-6N, that 11,8 M>a4N. Jnd infing^ . 1/2 M if < 6 N, then sM+aM+aM + 2Mi8airo<6N4-6N+6N + 6N,tliatii,8!ii<24N. I. 2. dn the contrary^ the numh&t 4, ^ <^ :f, 8 HfeMt th thifa^i Mh \fii^. If 9ht aMitedenit he multiphid iy 3, attdtbi tdnfeqtsMU hy i, thtrk kvUlr^t/mlt the four multiples 6,6, 21, 16, where the multiple 6 of the IJl fihtetidHii ii J^^dAl id tht muHipii 6 of h/ ttrnfequefit, ^hitjl 41 mkRipU if thi //. dhtMdeni it gteatit than 1 6 muttiple if its ennfefutftt. f» 3. IncommenfuralU ffkigflituJes can ne^r Bms4 itoir ifuimuftiples equals othermifi thiy would- he cOmmenfufMo to ona 4nother§ wherefore in^ commenfurables arefhewn to be proportional only from the joint excefs or de* ftH of their e^imtltipUs ; h)hereas commenfuratle magiiitudis ieing capable of a joint equalitj/, and in^ualfty of their equimuUiples, afe fhewn to be propor-- tionalfrom toe joint equality or excefs of their equimultiples 9 hence it is that ' thefigns in this definitidrtby ^hicb proportionality is dif covered, are applicable /# ai Ikiii of magntttidt ^batfoetret. §« 4. Whai is true with refpeH to the correfpondenei of tKultiphs, Is alfo irae^ with refpea to tiat of fubmielUplee\ But it is pMethle that Eudtd preferred ibt ufe of mul4ipfes to that of fuhmuhiples, becmtfe he eould not prefer iie to^ tmka fmtmi/ltipks without flrfl /hewing, how to divide Magnitude into equal psrtif K»hilfi the formation 6f multiples required no fudh principle. This Geometer had aright to ajfumefor granted, that the double triple, or any multiple of a magnitude could he taken, bnt was under the necejpty of Jbev^ing by the i64 The^ ELEMENTS Book ook"" " Re/olution ofaprohlimy bow to Uke dvHiy -an alffuot part from a given line, and the refolution of this problem fuppo/tng the do^rim of ftmtUtude^ cPuU notbt g iven but in tbe IX. Propcfition ojtbe VL fiwi. VL Magnitudes which have the fame ratio, are called proportioiials. . IVbenfour magnitudes A^B^QD are proportionaU it is ufuattf expreft thus, A : B = C : D and in words, tbefirfl is to tbefecond as tbetbird to tbefourtb. vii. When of the equimultiples of four magnitudes (taken as in the 5 th definition) the multiple of the firfl is greater than that of the fecond^ but the multiple of the third is not greater than the multiple of the fourth ; then the firft is faid to have to the fecond a greater ratio th^n the third magnitude has to the fourth* ' and on the contrary^ the third is faid to have to the fourth a lefs ratio than the firfl has to the fecond. §. f. Sucb are tbe ratios 3 : a (2^ 11 : g for if tbe antecedents be multiplied by ^p md tbe eonfequents by 13, tberewUl refuU2^ : 2$; 99: 117. • . 3:2(11:9 9 ^3 9 '3 a7 :26 ; 99: 117 *" IVberi tbe eorrefpcndenee of tbe multiples does not bold, tbefirfl antecedent 27 being greater tban its conjequent a6 wbilfl tbefecond antecedent 99 is lefs tban its confequent 117. §. a. ^0 dif cover by infpeffiqn tbe inequality of two ratios A : B iir C : D I j tbie cbaraBer of tbe non correfpondence of multiples^ it fuffices to cbufe fer multiples, tbe two terms of one of tbe two ratios, Jor Ex. C : D, and to multiply tbe antecedents A 6c C by tbe confequent D of tbis ratio ; and tbe <tv* eonfequenfs 'Bic'D by tbe antecedent C of tbe fame ratio, in tbis manner. ^ • 5 • E • 5 3:557:9 DC;D:C 9797 AD : BC } CD : D. C a; : 35 ; 63 •, 6^ Wbicb being done, tbe two produffs CD &r D.C will be found equals vubil/l tbe two otbers A,D Sc B.C are unequal, and in particular, if tbe multiple ef one of tbe antecedents be greater tban that of its confeauent, wbilft ibe multiple of tbe otber is equal to its, tben tbe terms of tbe lejfer ratio bavebeen cbofenfor multipliers. On tbe contrary^ if the multiple ej one ef fbe esnfe^ cedents be lefs tban tbat of its tonfequent, whilji tbe multiple of tbe otber it equal to its, tben tbe terms of tbe greater ratio bave been cbofenfor mmltif leers. J Book V. Of E U C L I D. iSg VIII. Analogy or proportion^ is the fimilittide of ratios. Jt ajtgn and cbara^erof proportionals bat been already then (in Dcf. 5,) tbis is afuperfluous definition^ a remark of fome fcboliaft joufled into the text fvbicb interrupts tbe coherence of Euclid'/ genuine definitions. IX. Proportion confifts in three (enns at leaft. S. I. Proportion conftfting in the equality of two ratios^ and each ratio bavinS two terms, in a proportion tbere are four terms f of wbicb tbe firfi and fourth are called the extreames* and thefecond and third the means, thofejour term' are confidered as only three, when the confequent of the firfi ratio at thefam^ time holds the place of tbe antecedent of thefecond ratio : it is for this reafonf that proportions are difiinguifhed into difcrete, and continued. J proportion is difcrete when tbe two means are unequal, and it is called continued when thefe fame terms are equal, thus this proportion Jl : 4 = 5 : 10 // difcrete ^rrtfii/^ the two mean terms 4 Csf 5 are unequal^ on the contrary, the proportion 2 : 4 =r 4 : 8 // a continued proportion on account of the equality of the mean terms 4 £!f 4. ' §. 2. Aferies of magnitudes in continued proportion, forms a geometrical pro» pcilion, fifch are the numbers i, 2, 4» 8^ 16, 32, 6^,ifc. X. When -three magnitudes are proportional the fiffl is faid to have to the third the duplicate ratio of that which it has to the fecohd. XI. When (bur magnitudes are continual prdportionalsf the firft is faid to have to the fonrth the triplicate ratio of that which it has to the fecond, and fo on quadruplicate, &c. increafing the denomination ft ill by unity in any number of proportionals. - XII. In proportionals, the antecedent terms are called Homologous to one another, as alfo the confequents to one another. ' XIIL Proportion is faid to be alternate when the antecedent of the firft ratio is com« pared with the antecedent of thefecond, and the confequent of the firft ratio with the confequent of the fecond. y/*A : B = c : D7 ,, . ., ,. rA:C=B:D 4:5 =16 : 20} '^'^h alternation. ^^ .. ^g ^ 5 . ^o Iflfen the proportion is difpofed after this manner, it is faid to be done by .per- mutation or alternately, permutando or alternando. j t^ Tte ELEM^ENTS loakT. KIY. But when the caufcf ueocs vt cbai^ into AatecedeDU^ and tk antecedeob into confequoatoin the hmtordtr^ it is faid that thecomparifonof thetenv is made by invtrpon pr invirt^nJo. A: B = c J ^lti^rsrar,in^.i^j. fB : A = D I C XV. But the comparifon is made by comp^fithn otc$mpomnJ9^ when the funoftte coofec^uents and antecedents is compared with their refpeStve cdnfcqaoift. A : B = C t Df tbenf^t >A + B;B=:C + D.'0 3:9= 4 : 12X comp^endai 3 + 9 • 9 — 4 "'^ ^***^* XVI. The comparifon is made by dhxifion of ralio» or JividstiJ^ whet At tuA of fchi aatcfifdem above its oonfequaot^ia coa^**^^^ its orafequat xvn. Tho conif ariibnis made by #i# c^wtirJUn of rado, #r €%w9iri^% wta Af mntccedeni it corefared to the excefs of the atttecedtnt above its cenfcquc^* yA S B=c C : Df iheref^^ \k \ A — Bs:C:C-l> 9 : 3 = 12 : 4 1 convertendo.y 9 • 9 •— 3 = Xl ;i»'*4 XVIII. A conclufion is drawn from equality of raiio ojr tx afuo^ whcfn comparingtvo leriesof magnitudes of the faqie number, fuch that the ratios of the firll d| equal to the raiios of the fccond, each to each, (whether the compsfifos w made in the fame order or in an inverted one), it is concluded that thecxtrcaaw of the two feries ace in proportion. Tbefen/eof tbh defiftitkii it AtfoiUwi, if A^B^C,^ h ^ ferief •//•'^ magnitudes^ and a» b> c, d a fir its of four otbir metgnitudetj fueb iht A:B = a;b7 fA:B = c:d B : C = b : c>or inan inverted orderA^ : C = b : c C :D = c : d) IC : D= % :b Bo0kV. Of E U C L I D. $$f In tbi Mi or tbe 9tber cafi it is alhwid to infer ex squo^ wbin ibi ratio of ibkoMiruma kiDof tbe Lfiriee is ejuai to tbe rgiio of tbe extregmes a { d ^Hrllferioe^ifrtiaAtD acer a t d. II. OLf by Cy d io» 2» 3O9 6 XIX. fbi iquMliiy of ratio is called ordinate ratio, when the ratio of tbe firft feries are «I«|lto tb^alioMf lh#4iMMd iivitt each t^ At BMomph i^A A t B ^ • t b B : C= b : c C I J>:sa c i d Ihre tbe ratios are equateacb to eacb in tbe fame direff order, hecaufe tbe frtft^ magnitude is to tbefecondoj tbe fir ft rank, as ibefirft to tbefecondof tbe other ranK ^nd as tbejfcmd is to tbe tbirdof tbefbft rank, fo /# ibefecond to tbe tbird of tbe other, aeid fa orir ia order. If fberefrrrit i^ iirferred that tbe axtriasaes are proportional, -osp tk^ A 1 Yy-sas r 1 d. fbrrnfertnte fsfatd to be made. from disreit fqpality, or ex mqa0 ordinate. XX> CVy^o^coatarj^ aquaU:y of ratia kcoiM imerud^ or ponariafe affahgy, in' the fecond cafe, that is when the ratios of thr iirft l«h«flr ave equa- to-thofe o tbe fJBQODdTenQB vacb^ to eadi^ takng thoft iaft in- an intvrt^d ohler^ {.1. Let tbe two feries of mefgttitudee be. ARCD? fA:B = c:d o K r ^\^bereitisfuppofed\^ J C = b i c ^^^^ ^^ ^S ' '"^^ CC:D = at b fecond is to tbe tbird of tbe firfi^onk,. fo is tbtlt^hut two to tbe laji uuh one of the fecond rank ; and as tbe tbird is totbefourtb of the fir fi rank, fo is tbe tbird from tbe laft to tbe lafi hut two of tbe 0cond rank, and Jo in a crofs order. If therefore it be inferred that A J D = a J d. i'his inference is fM to he made ex sequo perturbate. -1 i6S The ELEMENTS BookV. w §• 2. Beginnertmay eafily dijtinguijb, the cafe ofJirtHeqwiltj fnrnty of periurbate efuality, if tbey remember that when two terms are commn t$ iw§ proportions, and tbat tbey occupy indifferently eitber tbefirjl and thirds if tbe fecond and Jour tb place, tbat it is always tbi cafe of direct efuslitji For Example. A:B = a:b B:A = b:a A:B = a:b B:C = b:c or B:C = b:c or C:B = c:b A:C = a:c A:C = a:c A:B=:»:c Here are always two proportions wbicb bave in common tbe two terms id^ occupying tbefrji and tbird, or tbe fecond and fourth places ; tbetwutkr terms A W C are proportional to tbe two otbers a £^ c taking tbem in tbe Jem order. §. 3. On tbe contrary when tbe 4wo terms wbicb are common to tbe t^Qpriff- iions, are eitber tbe means or tbe extreatnes, it is tbe cafe of pertttrbate ejuJitji for example If A:B = h : c B:A = c:b A:B = b:c B:Cz=:a:b or B:C=a:b or C:B = b:t A : C = a^ c A : C rza : c A : C = t:c Intboje tbree cafes tbe terms B y b wbicb are common to tbe two pr9porti9Ut are eitber tbe extremes or tbe meansr, confeauently tbe otter terms are in /*> portion, fo tbat tbe two terms, wbicb arije from tbe fame proportion A & C 0r a & c remain extreams or means. Tbe/eare tbe denominations given to tbe different ways of concluding Ijatuhlh Euclid now proceeds to demonjtrate tbat tbey are jufl. fiook V. Of B U C L I D. 16^ ■i , ■SSggggBggagBBBP POSTULATEa I. LE T it be granttd, that any magnitude may be doubted, tripled, qua* drupled, or in general, that any multiple of it may be taken. B. That from a greater magnitude, there may be taken one or fenral parts e^ial to a Icis magnitude of the bme kind. ABRBVIATION& Mgn. M^nitade. Mult. Multiple. Equianik. EquimulUpIc •79 The £LEMENTS BeokV, s PROPOSITION I. THEOREM I. \f any number of magnuudes {aM, aH, aO Arc) be - eauimoltipla <i > mtnV(M, N, O Src) each of etch, thcfum (j M+tf N + «0«c)^*7 the fit ft is the fame fnultiale <^ Oic fttflfiiM rf-,N+P 9^^} of all the feco4 is any one of the^(KfiM) is df iliiwt ]fM>: ^ : Hypotheiis. Thefis. , . . iAi-) «rv fMMttI tfM4^N^H(>^^^'>*^'^^S N V equimuliipUs < N of. M+N+O 4*#/ iiM « t/M, »«« 03 0/ |0^^* t/^NW^. . : . Preparation. i Tke mgn. a M being the fiune niiiltiple of M, that « N is of N (tijp')y as many magnitudes A, B, C, &c. as can be uksn out of a M each equal to M, fo many X»Y»Z, ftc. can be taken out of « N, each equal to N. A 7 be each X) e«ch Let then B V equal teM & YVtqual to P^a-l^ cj z3 N B Demonstration. ECAUSE « M is the fame multiple of M, that « N is ofN (lf^)y As many magnitudes X. Y, Z» &c. as are in « N each equal to N» fe many A, B> C» &c. are there in « M each equal to M. But Therefore Likewife It follows that Again, becaule It follows Chat A=M &X=N (Prep.)^ A+X = M-|-N B being = M & Y=N (Preph JS+Y = M+N C = M&Z=:N (Prtph C+Z == M+N Confequently there is in a M as roany'Magnitudes =: M» as there arc in tf M -f fl N = M + N. From whence it follows that <t M + « N is the fame multiple of M+N, that a M is of M, or that i> N U of N, & likewife « M-f«N + « O IS the lame multiple of M + N + O, that a M is of M or uNofN, &c. Which was to be demoaftratdT Bnky, or iccLia »i« I PltOPOSITIOMn TBSOXBU Ji. P tkfirft.niifrindetcM) Ik ife fane ndnple of the ftcnrf (M), dM AedudtehQ 'u0[Aefomrh (N); Ik the fifili fr M) tiie fune mriiipfe «r the fanrf (BD. tli« the fizili (cN) vof dK fawth fN) ; Am ImH the fiift togedier with the fifth (a M-t-rM) he the fane iiMlii|4e of the fccMd QyQ^ l|i«tthctbii4(agedicrwkbtfaefixth(aN-i-'N) it of d* favth (H). IBOB. < ft tnimalt^Kh UN ^ 3 N MMt« Thdk HjpodMlis. aW) Demohstratioit. i3eCAUSE«M k tbeteie wildplt oFM* tlMt^MitoTM 1. There are lu onor mmiinides in «M ^ to M aa there tB( in « N = to N. In like mannsTt becauie ^M is the ftme multiple of Mt thai cN b of N (Ifyph 2. There are as many mafmtiides in ^M =; (o M as there are in c I^ = to N. 3. Coiiieqnemlyyaaiiianjasare in the wholes M + ^Mt<|iitl to M> ib manjr are there in tlie wholenN + ^N ;sto N. 4$* a* #• i 4. Therefore mlA -^cWip the (ame multiple of M that «N + cM is of N. Which was to be demonftrattd. Ifl The ELEMENTS BookV. f«M mN 1 uM «iM uN «iN N. IV PROPOSITION UL THEOREMIli J.F the firft magnitude (« M) be the ikme molriple of tfae fecond M, thit^ 'ibird («N) is of the fourth (r N), and if of the firft (« M) and thini («W theie be t^ceo cquifflultiplea (e«M, «aN)$ thefe (««M, («N) b^lK «quimuhiple«, the ooe of the fecond (M) and the other of^the fourth (N) /. «M Hfpothefis. I Ml are tt» fS ytfuimtUtii .Nj ./ r«M7 f«N3 Thciia. , , Preparation. Divide # « M into its parts i a M, a t Mv &c. etch = « M« , A94 # >i N .into* its parts i a N, « 1 M, &c. each = « N. B DEMONSTRATION. CCA USE #aM IS the fiune multiple of a M, that #iiN isof 1. There are as tosmj magnitudes in^aM = totfMai Ihere are in e « N = tp « N 2, Therefore the miiteber of parts t « M, « i M, &c. in ra M, is = to the number of parts i a N, « i N, &c. in # « N. But becaufe « M Is the fame multiple of M, that « N is of N, and that 1 di M = « M, 1 tf N =: If N. The magnitude 1 diM is the fame multiple of M, that i diN isofN. In like manner dii M is the fame multiple of M, that « 1 N is of N. Since theft I mgn« i di M is the fame multiple of the II mgn. M. that the III mgn. i di N is of the IV nign. N & that the V mgn. di i M is the fame multiple of the II mgn. M that the VI mgo. dr i N is of the IV mgn. N. 5, It follows that the magnitude rdt M, compofed of the I & V mgn. I diM+^ I M,is the fame multiple of the II mgn. M, thatthem^i. r di N, compofed of the III & VI mgn. i a N+di 1 N is of the iV nign. N> Which was to be demonftrated 3' 4- P.t^lr Book V. Of EUCLID. 173 I, PROPOSITION IV. THEOREM IF foor magnittidei rM,N| O, P,) are proportional : then any equimultiple! {a M, a O) of the firft (M) and third (O), Aall have the Tame ratio to any -^quimuUiples (c N^ r P) of the fecond (N) and fourth (P). Hvpothefis. Thefis. /. M:N = 0:P. aM : cN =: flO : cP. f fM cN7 me (N » tfmmUf. <k alfr 9c \ tfuinmbA ^ fflM7 y (o ' cP) '0/ (p- B. Preparation. I. Take of «M &of «0 any eqnimult. R«M, RiiO 7 blt. a« a. Likcwifc of tf N & of f P any equimdt. S c N, S r P ) ^- «• ^ ?- Demonstration* lECAUSE aM 18 tlie &me mult, of M, that « O ia of O (Hyp, a)» & the mgns. R a M, R « O are eqvimuit. of the mrns. oM* «0 ^Pri;^ 1/ 1- The magnitude Ra M is the (aoie multiple of M^ that the magni- tude R a O is of O. • 2- In like manner, the magnitude S r N is the fiune multiple of N thatSrPisofP. And as M : N = O: P (Hyp. 1 J & R« M, R tf O are any equi- multiples of the I term M and of the III O ; -and S r N, S c P any equimultiples of the II trrm N and of the IV P {^rg, t & 2)« 3. If R«M be >,c=or <SfN, RdOwiM be >,=:or < S fP. But the magnitudes RaM&RAOare any equimultiples of the ma^itudes « M & « O9 and the maguitodea S r N, S r P are any equimultiples of the magnitudes c N& c P (Frep. i & zj. 4.. Coafequently, the ratio, of « M to c N is = to the ratio of a O torP; oraM : cN=flO : c? Which was to be demonftrated. D. 5. B. , D. s M. s. 1 COROLLARY. 7 it mamfefi that if S cH bt>, = er < R« M j Itkevoift S c P vmH he >, = «r <R«0 (Arg.^.)i btneteH /<»M=f P.- «0 (D.^.B.^.). Tbenfire, if fmr magtutudtt It frtftrtimtd, thy an tdfi ly iwvtrfim t/r inverttmU. UfA- The BL EM EN T S BookV. «M .N i M V-P N I PROPOSITION V. THEOREM V. J? a magnitude (a M) be the fame muUtpk of tndther (M}» wbtdi t Ml* nitude (j N) taken from the firft, is of a magnitude (N) taken from the otkfi the remainder (a?) (hiJI be the bme multiple of the remaiiider (V), ibat the whole {fi M)« u of the whole (M). Hypothecs. I f TAf mgns. aMfJMari Mi^ nMe^ < Tbf ffigns. tf N & N tinr parfi tddten a^fif ^Andibe mpu a?i/V tbe nmainjtr^ Preparation. Take a manitode P fuch^ that u P may be the fiune aiiiK- . tj^le of P, that 1^ N 18 of N, or «M of M. H^^^ DSMOKSTRATIOK. E C A U S £ ff N It the tame multiple of N, that «T jt of P B , (Pnp) u The _ fum 4 N 4- a P» or a M, of the firft, it the fiune multiple of the fum N + P of the ia(t that .0 N b oi »! ButaM is the fiuue multiple of M» orofN^Vi thatvNitof N (Hyp. zj. Confeqttently» the mgft. 4 M it equimulctple of the mtfna. N + P» &N+V. And of courfe N+ P == N + V. Taking away the common mgn. N. P.i.i^J 4. It follows that ^le m^. fh:ss; to the mm. V. ^. Confequently, a P beine the fame multije of P, that «M it of M (Pnp J, A P is alfo the fame multiple otV, that «r M is of M, Which was to be demooftiatcd. J?ok Yi Of 5 U C L I D. . 175 aM ^ ^ ^ ^ cM ian ^N tN ; • PROPOSITION VI. tSk6REM>t. ^^ Ac two magnitudes (a M, tf N) be equimultiples of two others (M & N) & if ^^n^olt^les (c}A Br r N) 6f tfiefe, be taken from tfie (irA two, the remtin- ders (e M & e N) are either equal to thefe others (M it N), or equimultiples iJF them. Hypothefis. Thefis. CeMf^ef^ the nmmnders ^oM hi gMimuUifU tjf N« CASE L iffM^aiM* * Preparation. Let iN = N. /V:a.JlS- DBMONSritaiTlOK. x3 EC AUftB rM isthe Anettohif^ of M, chfll rM k •f N (Jf^. %.), k that «M == M. (^5ir^ ij, & 1 N = N ^Pr*;^.;, 1. ^he mgn. elk *^«M» ot «M^ wiM be tte fiune multiple of M thatcN+ 1 Nisof N. But a M bemg the fiune multiple of M» that tf N or # N + e N is The two mens, cN+tN&#N + rNare equimukiples of the fiime mgn. N. 3. VThcrcfore the mgn. c JI + i N i= r N + ir N ^*- 6. A i. rTaking away the commion mgn.' tfN» ^ ^ It follows that 1 N is =: rtf -Af. 3. A 1. ^' But I N is = N fPr^:) , If Confequentlvy e N is = N iAr. i. A i. 1; TlieTcfore if eMbc =z M, ^N is'=iN, WUch was to be demonftrated. 1. 176 The ELE M E N;T S BookV. 1 ^M « N t.M c M 1^ N : * <*, B. CASE II. IftMbcmitltipleorM. PreparMM. Take I #N the fiune multiple of N, tliat# M U of M. A/^i'^i DEMONSTItATIOK. ^^ B C A U S E #M is the fiuaie multiple of M, that i # N is of N (Pnp~), k thatcM is the fiune mulupleof M, that cN is of N 1. The magnitude #M4-cMor«M9 wiU be the fiune multiple of M, that I f N + f N IS of N. /^* ' ^ But a M being the fiune multiple of M that «Nor#N+^^>' a. Therefore, the two mgns. i r N -f- rN h # N + c N are equi- . multiples of the fiune mgn.N. A.^*^ 5. Confequently, i#N + fNis=fN+cN Taking away the common mga. # N 4. It follows that the n^n. i r ^f is = # N iir. ]• f*^ But I tf N is the fame multiple of N that # M is of M (fnp.), 5* Therefore* if # M be an equimultiple of M> # N will be an equi* multiple of N Which was to be demottftrfttcd. 1 1 . Book V. Of- EUCLID, »77 • "lA c m «iM M m iM 'v _^ PROPOSITION VIL THEOREM VII. Xl/ QJJ A L magnitudes (M & i M), have the fame ratio to the fame magnitude (m), and the Tame (m) ; has the fame ratio to equal magnitude (M & I M), Hypothefis. M C^ I M ^^ t*w9 uptal mffisy 13 mU a ilnrd. Thcfis. /.M : « =: iM: m //. M : M =: m : iM. Preparation^ I. Take of M & of i M anv equimultiples a M & a x M. ) «. r » - a. And of « any multiple wnatever cm. 3 '^J* i. if* 5 Demonstration* X3 E C A U S £ A M & A I M are equimultiples of M & of 1 M {Prtf. ij> &M = i M r/i'A/ g. Themsn. iiM is=r ii I M. .ifx.6* &!« a. Therefore, if « M be >, =, or < f « ; « i M will likewife be >, nr, or < f «. But M & tf I M are equimultiples of the I term M and of the ' III term iM, as r « and c m are of the II term m and of theJV term «v, ^5. Cwiiequently M : m = 1 M : i». D. 5. A 5* Which was to be demonftrated. i« And becaufc aM:=zaiM (Arg, \) j J. It alio follows that, if r m be >, =, or < a M, likewife c m will be >, =, or < ill M. ^ • ^ ^ . Therefore w : M =j « : i M. P. 5. -^» $' Which was to be demonftxatfd, 11. "TW" 178 The ELEMENTS BookV. «N L*. JL^ 1. 3- 4- O PROPOSITION VIII. THEOREM Vlll F unequal magnitudes (M & N), the greater (M) has a greater ratio to W feme (P),thah the Icfs (N) has; And the fanle rtw^itude (P) has a grcitc^Titt to the lefs (K), than it haj to the greater (M). Hypothefw. Thcfis. /. M > N. /. M: P>N:F //. ?uart^ nuignituJi. //. P : N > P » / Preparation. I. Take from the greater M a part i N == to the lefs N, and the remainder K will he eitner <, or > or infine = N ; Suppofe firft this remainder to he < N. Tale tf R tt iflultirile Of this retafirtder > P 1 take i«N & tfN the fame rttik. of 1 N & N that 4Ris of R. P*/! » * ' Take the oigji. 2 P double of P^ the mgn. 3 P triple of P and fo on until the multiple of P be that which firft becomes greater than a N» arid let 4 P be that multiple. B Demonstration. E C A U S E 4 P 18 the multiple of P which firft becomes > aH I - The next preceding mult. 3 P is not > « N, or tf N is not < 3 P- Moreover aR and i^N being equimultiples^of R &of iN (^r^/.0> a. The mgn. ^R + 1 a N, or a M is the fame multiple of R -{- i N or M, that <i R is of R. Or that a N hoi N (Prep, ^). 3. Therefore a M and a N are equimultiples of M and of N. Moreover, a N and laN being equiniuliiples of the = mgns. N aad 1 N (Pnp. 3 W 1;. 4. The mgn. a N i« = i a N But fl N is not < 3 P (Arg, i), . 5. Confequently, 1 a N is not < 3 P 9ul fl R is > P (Prep. 2). Therefore, by adding, flR4- » aNorflM>4P* Since then « M i» > 4 P, and r?N < 4 P (P^ep. ±Jt and aM, a N ...... .. .^ . (Prep.{h P.ul^' Ax. 6.1^ 6: are equimultiples of the antecedents M and N aiid 4 P, 4 P cqui- iiiultiplca of the confequents R and P {Jrg. 3 y Prep, 4J. It follows that M:P>N;P Which was to be demonftrated. i. Z).7. ^5 Book V. Of EUCLID. 1.79 «vwM M ' R iN n a - BSBacasBii 8. •od a N, A M ^uimHlttf4e« of the oonfeqaeDts N and M» 9. It foUwa that P : N > P : M. u. 7. jr. 5. Which was to be demonftrated. 11. II, Pretaration. If R be fijppofed > i N. or N. 5. Take i u N a multiple of 1 N > P. 6. Take aR & aN the fame multiples of R &of N that i«N isof iN. Pof.i, B. 5. J. Let 4 P be the firft otttltiBle of P > is R ; Qonfocpcatly the. next preceding multiple 3 P willnot be > tfR, or /iR will not be < 3 P. I Demonstration. T may be proved as before (Arg, 1. 2 W 3A that I. The mgns. a M and ^ N are equimultiples of the mgns. M & N. Moreover, aR & aN being equimultiples of R & of N (Prep, 6J, and R being > N (^up.)y 2, It follows that flR is > « N But /tR not being < 3 P (Prep. 7^, Andthemgn. laN being > P (Prep, 5^, 3. Then by adding, tfR-|-i/iN, or<iM> 4?- But iiR being < 4 P (Prep.^), & this fame aR being > «N {Arg.i)^ 4. Much more then « N is.< 4 P. But «M & aN are equimultiples of the antecedents M Sr N {:Arg.i) and 4P, 4 P equimultiples of the confcqucnis P & P, & moreover <flM>4P&aN<4p (Arg. 3 y 4/ 5. Confequently M : P > N : P. £>. 7. B. 5. Which was to be demonfhated. 1. Moreover, without changing thePreparation, it may be demonihated as in the precedent cafe (Arg, 8 EiT q)y that >. The ratio of P : N is > the ratio of P : M. Which was to be demonlhated. 1 1. III. And applying the fame preparation and fiime reading to the laft cafe ivhen R =: i N» \. The demonftratton will be completed as in the two precedent cafes. Which was to be demonitrated. 1 & i u I So The ELEMENTS BookV. ' M N iM PROPOSITION IX. THEOREM IX. ]\4 AGNITUDES (M & i M) which have the fanie rttb to the te magnitude (N) : are equal to one another. And thofe (M & i M) to«ii»i the Tame magnitude (Nj has the fame ratio^ are equal to one another. Hypothefis. M : N = I M : N. Demonstration* I. ^M^ I^ not» thf two mgns. M & i M are unequal I X HEN the two mgna. M & i M have not the fame ratio to the fame mgn. N ' • ?. 8.^-5' But they harp the fame ratio to this fame mgn. N (ffyp.) ; 9. Therefore the nign. M is :^ to the mgn. 1 m. ThcBi rA<«ga.M=t» Hypothefis. N:M = N: iM 111 A Demonstration. n. If not, the two mgns. M & i M are unequd. 1 . A HEN the fiime mgn. N has not the fiime ratio to the two mgns. M&iM. '•*'*^ But it has the lame ratio to thofe two mgns. (Mjp.). a. Therefore the mgn. M is :s to the mgn. 1 M. Which was to be demonftrated. .BookV. Of EUCLID. i8i ^—^^■■■if- M PROP'OSITJON X. THEORElii. J(. _ H A T magnitude (M) which has a greater ratio than anpther (P) hat unto the fame magnitude (N) b the greater of the t wo, and that magpitude (P) to which the fame (N) has a greater ratio than it has unto another qiagnitude (M) is the kfler of the two. Hypothefis. Thefis. M .N« > P : N, ^b€mgn,lAis>f. Demonstration. r If not f M is 3= P, or < P. TC A S E I. If M be :^ P. HEN th(f mgns. M & P have the fame ratio to the &ine nign.N. P. 7. B. $; But they haye not the (ame ratio to the fiune mgn. N (Hj^ /^ a. Therefore the mgn. M is not = to the mgn. P. TC A S E II. If M be < P. HE ratio M : N would be < the ratio P : N (Hyp,) ; ^. 8. A 5, But the ratio M : N i^ not < the ratio P : N (Hyp.) ; j^ Therefore the mgn. M is not < the mgn. P. But neither is the mgn. M ^ P (Arg. a^» ^. It remains then that M be > P. Hypotheiis. N : P > N : M. Dbmonstration. II. If not, P is == or > M. ^Tp\ C A S E I. If P be = M. r. J[. HE ratio N : M would be =: to the ratio of N : P u Which being contrary to the Hypothefis, P cannot be ;= M. r-— , CASE II. If P be > M. « 1 HE ratio N : M would be > the ratio N : P. . Wliich beifl^ alfo contrary to the Hypothefis, P ca^mbt be > M. Bnt neither is P =: M. (Arg» z.) ; . Therefore P is < M. Which was to be demonfirattd. Thefis. 7bf mgn. ?is<M. F. 7. B. P.t.f.i, iSz The ELEMENTS BookV. I ji'mmmmamm C •.A ][ • c « B B tE • CD I [^ B cD « F aai u PROPOSITION XL THEOREM XL Jv ATIOS (A : B & E : F) that arc equal to • fame third ratio (C:01, are equal to one another. Hypothefis. Thefii. CA:B ^ ^ fheratm< & are = f the fame ratio C :D. A:B=:E:r. CE :F Preparafion. I. Take any equimultiples aA% ada^ 6f the thfse ante- cedents A, C, E. a. And any equtipultiples c B» c D> c F of lUe thiee coi4fe- quents ft, D, F. Demonstration. ! hf^i.lS- JjECAUSE A : B = C ; t) (HjfJ, I. If the muhipie a A be >• = or < the niultiDle cB, Ae equimul- tiple a C is likewife >, = or < the equimuftiplc CD D. S-^S In like manner fince C : D = E : F (ffypj t. . If the multiple « C be >, = or < the multiple c J), the equimul- tiple a E wm be likewife >, = or < the equimultiple c F. 3. ConiiBqveDtly if the multiple a A be >, = or <, the multiple < B ; the equimultiple nE is likewiiie >, =or < tke equimultiple c F. 4. Confequently, A : B = E ; F- Z). $• ^ 5- I Which was to be demonftrated. MU. Jook V. Of EUCLID. 183 -' i n^ «D «F PROPOSITION XII. THEOREM XIL [f any mimber of nlagnitudtt^ (A, B» C, D, E, F, &c) be proportionals lie foni of all the antecedents (A + C + E &c) is to the fum of all the onfequents (B + D + F ^c), as one of the antecedents is to its confequent. Hjpothefis. Thefis. be mrm. A, B, C, D, E, F are proportionaU A+C+E : B+D+F=A : B. -A:B=iC:D = E:Feff<:. Preparation, 1. Take of the antecedents A, C, E the equimultiples m A,"] M C9 M E { p J. - • . 2. And of the confequents B, D, F the equimuhiplcs « B, r ^V- *• •^- 5' « D, « P Di:monstrXtion, , j />•/. 5!NC£ thcnAcB = C.:D=E:F (Myp.) i . If « A be>, = or<n B, like wife «C is >, = < nT>i icmt, is >, = orif F D. 5. B. J, Therefore adding on both fides the mgns. >% =, or <. . The m««. iwA+,arC + «fE will be conftsuitly >,=:, or < the mgns. II B + * D 4- « F according as « A is >, =» or < « B. But the mgns. m A -]- mC -^ mE ii m A zre equimultiples of the mgns. A+ C 4- E & A (frep. 1 W P. i. S. 5.; ; alfo the mgns. i»\B -f * D +« F & « B are equimultiples of the mgns. B + D -{- F &B (Prep.ifgP, i.i?. 5Jj Confequcntly A+ C +£ : B + D + F = A : B J^- 5. i?. 5. Which was to be demonftratcd. i84 The ELEMENTS ••C «E ni IV VI «B "D «F I PROPOSITION XIII. THEOREM XIIL F the firft magnitude (A) has to the Tecond (B), the fame ratio, which tk third (C) hai to the fourth (D) ; but the third (C) to the fourth (D) a grater ratio than the fifth (£) to the ftxth (F) : the firft (A) fhail have to thefaoil (B) a greater ratio than the fifth (E) has to the fixth (F). /. A //. C Hypothefis. B = D. F. Thcfis. :B>E F. ]^- iM Preparation. ' I. The ratio of C : D being > the ratio of E : F (tfyf. %) there may be taken of the antecedents C & E, the equinnilt. M C & Hi E ; and likewife of the confequents D & P the equimult. n D & « F, inch, that at C ia > » D, but «t E u ( i^ i ^ $' noi>ifF; lD.7^S a- Take m A the fiiai^ multiple of A that «tC is of C» 3. And ff B the fame multiple of B that » D is of D. Demonstratiok. OINCE then A: B=:C:D (Hyp, 1.;, and that m A, at Care equimultiples of the antecedents, & ir B, n D equimultiples of the confequents (Prep- 2 W 3^, 1. The mgn. m A will be >> := or < » B ; according as ot C is >» =:or<»D. i>$*5 2. Therefore « A is alfo > ji B. But HI E is not > If F (Prep, i), & the mgns. « A ft aiE are equimultiples of the antecedents A&E, &iiB»iiF equimultipiet of the confequents B & F (Prep. 1 li %). 3. Confequcntly the ratio A : B> is > than the ratto£ : F.* D, J- 1- $' Which was to be demooftr&ted. t BookV. Of E U C L I D. >«5 I 1 , A III C B IV P 1 PROPOSITION XIV. THEOREM XIV. J[F ^r namiitudet {A» B, Q D) be proportkmab^ then if the firft (A) be greater, equal, or lefs, than the third (C), the fecond (B) ihall be greater^ equal, or l^fs, than the fourth (D). HyBOtheiis. Theiis. 7. A : B = C : D According as A is >, = or < C* IJ. A « >, = cr< C. ^wllbe >,= 9r<l}. CASE I. If A be > C. 3 DEKfONSTRATIOir. X H E N the ratio of A : B !> > tke mtip C : B. ButA:B = C:D (Hyp, ij. TTicTcforc the ratio qf C : D is > the ratio C : B. From whence it follows^ that D is < B orB > D. It may be denMnilraXed a&er the ianie manner* if A= C* that B wiU he = D ; & if A be < C, that B will be < D. Confequently, according as A is >» :=: or < C> B will be >» = Which was to be demonftrated. P. 8. S, J. P. 13. B. 5. A a i96 the ELEMENTS Bookt * A mm ISBBi tllbPOSiTION XV. tHiolkEMxy^ j^AGNITUDES (AfrB) htve the (aim ratio to cue noditf which their equimuhiples (m A Ar m B) have. Thciis. A : BsiiA:-! HTpothefis. TA^ JV'Kf . M A & » B ar# equimutt. 4ftb€mgns,hfJB, Preparation. I. Diyide m A iato its parts P» Q^ R each :^ A. 1. And m B inro ks parts /» ^» r each :± B. D£M0NST&ATIOK« jfljfli.} J3£C AUSE the mgns. « A, «i B are equimultiples of the mffns. kh%(HypJ. I. The number of parts P, (^R &c. is= to the miinber of parts /, f , r &c. And P being rs Q^±f R (Prtp; i),8cp = q:=ir (Prtj^ %), 2. The men. P:/ = Q^:j» = R:r&c» 3. Wherefore P + Q^+ R, or«A:/4-fvt-roriniB±=P:/. But fmce P =A & ^= B (Prep. \ ii %), 4. The mgn. P : /► = A : Bj '^ '' ' * Bs=iivA:inB. Which was to be demonftnted. $. Confequentlj A BookV. Of EUCLID. 187 • A 9 C n 111 lYwm n D PROPOSITION XVI. THEOREM XVI J. F four «iagnitudies (A, B, C, D) of the Tame kind be proportionak, thej ibail alfo be proportionals when takeq altfrnateiy. . Hjpothefis. Thefis. A : B = C : 9. A : C = B : B. Preparation. I. Take of the terms A & B of the firft ratio, any egnimult.'^ Ml A> CC M O* L P /*• Hi* jfc. Take of the terms C & D of the fecond ratio any equimult. r ™«»- -S- if€> ffD. , Demonstratiok. ' B ECAUSE otA & otB are equimult. of the mm. A & B 1. Then But Therefore Likvwife Confequently A: B=«vA:fiiB. A : B = C : D (Hjp). C: D=i:mA:iiiB. C ! D = « C : « D. iA:«B = «C:«D. P..5.A5. P.ii.B,^. P.11.-8.5. Wherefore, if «i A he >, = or < « C, iwB will he >, = or < iiD. P.14. -8.5. But M A & M B being equimult. of the terms A & B coniidered as antecedents (Prep, ij^ & nCf nD equimult. of the terms C & D coniidered as confequents fPrep. 2J9 ' 6. Confequently A : C = B : D. Z>. 5. J5. 5. yri^ich was tQ be demonftrated. ICOROLLART. T frUtfWs from this propofiiion thai if four mgm, arw proportionals ^ according as the firji is greater y equal or lefs than thejecond^ toe third is like*uiife greater y equals or lejs than the fourth. Forjince A : B = C : D (Hjp.), 1. Then A : C = B : D. ^.16.5.5. a. Therefore, according «* A 1/ >, = «r < B, C mfill he likevfife >, = »r<D. I PM'9% i88 The E L E M E N T S Book V. ' " . _ — -* M ■......"'. j «.c rw] cp • mmmm I PROPOSITION XVII. THEOREM XFIL F two magnitudes tc^ether (A -f- B) have to one of them (B), the famC ratio which two others (C + D) have to otie of thefe (D), the rcMi^Nmng one (A) of the 6rft two (A+ B) (hall have to the other (B), the fame ratio which the remaining one (Q of the lad two (C + D) has to the other of thele (D). Hypothefis. • Tbefis. A + B:B2=:C+D:D A:B=:C:D. PpefaraliM. 1 . Take of the mgns. A,B,C,D any equimult. mA, iviBy iiiC, «^ 2. And of the mgos. B & D any ^quinluk. ii B, » IX Ptfi. B. g. T Demonstration. HEN the whole nxgn. as A -f: «v B will be the fame mult, of the mgn. A + B, that arA is of A, oriw C of C. P. i. B. 5. 2' In like manner» the whole mgn. at C -f" ^ I^ is the fiune mult, of the Bga C + D, that m C is of C. F.t. B. $. 3. Coniequently, ai A -f m B is the Ame mult, of A + B, that ai C + «DiaofC + D. 4. Alfo the mgns. aiB-|-iiB, mD-\-nD are equimult. of the mgns. B&D. ButA+B:B = C+D:D (Hyp.), &iiiA + a.B. aiC + aiD * are equinmlt* of the antecedents A + B & C -f D (^rg. 3^ > alio «f B + ^ B, « p + » D are equimult. of the confequents B & D (Arg. 4/ 5. Confequentlyv ifafA+«Bbc>,z=or<aiB + «B, aiC + af D is alfo >, = or < «iD 4- n D. D, 5. JJ. 5. But ifaiA-|-mBbe>, :=or< mB + ^B; taking away the common part m B. 6. The remainder m A will be >,= or < the remainder n B. In like manner, if «C4-*« O be >, = or < mD + « I^ ; taking away the common part m D. 7. The reaoainder m C will be >, =r or < the remainder iiD. 8. Wherefore, if « A be >, s=, or < « B j aiC will be likewMe >» = or < H D. But ay A & OT C are equimult. of A & of C coniidered as antecedents (Prep, I ) ; Sr ff B, ;i D emumult. of B & D coniideted as confequents (Pnp- 2}. 9. Confequently, A : B ;;= C : D. 1). j. /?. 5. Which was to be demonftrated. Book V. Of EUCLID. it9 d B A B 1 PROPOSITION XVm. THEOREM xyilL \? four magpitiidet (A3tC»D) be proportionab, the firft and fecond together (A+B) (ball be to the fccood (6) m the third and fourth tosetber (C+D) to» the fourth (D). Hypothefis. Thefii. A : B=cC : D. A + B : B»=C + D : D. DbM0N8TRATI0K« If not, A+B : B = C+D : another mgn. M < or > D. CA^E I* LctM<D, orM + R=*D (^F^. i;. Since then A + B : B = C +D : M,or A+B: BssC+M+R :M 1. Dividendo A : B = C + R : M •But A : B = C : D (Hyp.) j 2. Hence, C-fR:M = C : D But C -f R ii > C (Ax. 8. B. i) ; J. Therefore M is > D, & the fuppofition of M < D, is impofible CASE II. LetM>D, orM=:D+R ^Fi^.a/ xJeCAUSE A+ B : B = C + D : M, or A+B : B =:C+D : D+R jL. Dividendo A : B = C — R : D + R Fa-j.B.c. C : D. {Hyp,), P.ii. Aj. P.I I. J. 5. P.i4.2?.5. But A:B = «. Hence, C— R : M= • C : D. But C — R is < C (Ax. 8. B,i)i 6« Therefore M is < D, & the fuppofition of M > D, is impoifibie. Since then M is neither < I> (Ai^. 3^ nor > D (Arp, 7^, It follows that Mis :xD«r A + B : B = C+ D : D. WhFch was to be demonftrated. P.14. B. 5. 190 The ELEMENTS BookV. PROPOSITION XIX. THEOREM XIX. J.F a whole magnitude (A+B) be to f whole (C+D), as a magnitude (A) taken fronti the firft is to a magnitude (C) taken from the other, the ramia- der (B) (hall be to the remainder (D)*, as the whole (A -f B) is to the vboie (C + D). Hrpothefis. Thefis. A + B : C + D=: A : C B:D = A + B:C + B Dkmonstratioh. JjECAUSE A + B I . Therefore Alteroando A + B a. Then Dividendo " B 3. Altern&ndo again B Butiince A + B 4. It follows that B C + D= A A=C+D: A= D: D= A; C + D= A D = A + B: C (Hjp.). C C. c. C. (Hy,.). C + Dt /".le.*? P.17.A5' ?.\xM Which was to be demonftrated. COHOLLART. \F magnitudes taken jtinlly he frtftrti»nals, th(U «> i/ A + B : A =: C + 1^ ' ^ // may fe inferred by ctnverjion that A + B : B ;^ C + D : D (D.xn.B.^- f«r A-{-B:.C4-D=iA:C (Hyp.(JP.i6J. Wherefore A + B : B + D = B : D (P. lo). Cenfefuentfy A + B:B =C + D:D(P. 16). J Book V. Of EUCLID. igt PROPOSITION XX. THEOREM XX. jI P there be three tnftgnitudes (A, B, C) and other three (^ybi c) whicli taken ivro and two in ^ dired order^ have the fame ratio; if the firft (A) te greater than the third (C), the fourth (a) (hall be greater than the fixth (cj and if equal, equal^ anid if lefs, lefsj Hypothcfis. ^ * Thefis. /. A : H = a : 'b * According as Ais >f^=z t '<C,Q. //. B r C xt A : f ais aIfo>fZ=zor <, a Bi Demoi^stratiok. CASE!. Let A be > & _fECAUSE A i8>C. , I. The ratio A : B is > C : B. But A : B =a:h (Hjf. tj. And G:B =3 c : * (Hyt. %li P. ^C^.B. ^. 2' Therefore, the ratio a : ^ ii > r : ^. 3. Coofequently, a is alfo > r. 4. It may be pro? ed after the fame maaneir> tl)|it if A be =: C» a fliall be =S:f, & if A be < C, fl Aall be < c. 5. CpnfequeDtl/v according as A* is >,':^'0r < Cf a will be alfo >» B3 or < r. Which was to be demonfttatcd. P. 8. A S, P.io. Bi J. l^Z The ELEMENTS BookV. PROPOSITION XXI. THEOREM M 1 P there, be three mtgnitudca (A, S^ Q, and other three (tf» (»«J> ^^ iMve the fane ratb taken two and two, but in a crors order ; if ihe firftn^' nitiide (A) be greater than the thiid (C), the fourth (a) ftall be greittrthtt the fixth (cj, and if equal^ equal ; and if k&^ k6w Hf^thefii. Thfc X A : B :?» : « Ace9rdingtu Ais>i^^<y //. B : C ^ a : i «ttair»>i^«^<'* CASE I. UtAbe>C xJecause I. The ratio of A But A &r invertendo C Demovstbatiou. 2. 3- 4- A 18 > C B > C : B B = * : c (Ffy^. ij. Coitfemientlj the ratio ^ : c > ^ : « Therefore c is < « , or « > r It may be denoelbated after the ^unt fliMuner» if A be sx B» alfe tf flail be = n and if A be < C, afludl be < c Confequeatljr, aeooidiag «iAit>>sor<C># ikaU be >«a or < r. WUch was to be deqpkonftrated. t.lM i Book V. Of EUCLID. 193 M m A _■■' ma n B 1 m - rb 1 r C re PROPOSITION XXII. THEOREM XXII. j[p there be any number of magtiitudes (A, B, C, &c.) and as many others ^a, h c, yc J, which taken two and two in order have the fame ratio, the Srft flia!l have to the lad of the firil magnitudes, the fame ratio which the Srft of the others has to the lad, by equality of direA ratio, or ex aquo or" iinate. Hypothciis. Thefis. 7. A : B = fl : ^ Pi i Q ^ a : c. U. B : C = h : c Preparation. 1 . Take of A & a afiy equimult. m \& m d 2. And of B & ^ any equimult. nB 8c n B 3. And of C Sc c any equimult. r C & re. 1 Po/.l.S.i. \i B Demonstration. ECAUSE A : B = ii : * (Hy^- ij. I. It follows that in A : nB znma : nh And becaufe B : C = ^ : r (Hyp. 2). u It follows that «B : rC :=zn6=zrc |. Therefore, m A, n B, r C & av «, nhy re form two feries xif magnitudes which taken two by two in order have the fame ratio. ^. Wherefore, by equality of ratio, according as the firfl: in A of the firft (eries is >, r=: or < the third r C, the firft ma of the other P. 4. 2?. J. feries will be >, = or < the third r c. Confequently, A : C == tf : r. Which was to be demonftrated. P.ao. B. $. D. J. A 5. ■J '? '^ B b 194 The ELEMENTS Book V. - A B n C ■BHi .. t nn rb M n e 1 PROPOSITION XXUI. THEOREM XXIII. Jj^ P ther« be any number of magnitudes (A, B, C, &c.) and as many otfans (a^ bf Ty i^t.) which taken two and two, in a crors order, have the lame ratio; tho firft.ihall have to the lad of the firft magnitudes the fame ratio whkh the firft of the others has lo the laft, by equality of perturbate ratio or or gequ9 perturbatCn Hypothefis. Thefir. /. A : B = ^ : f. A : C :=r 41 ; r. //. B : C = ii : *. Preparation^ I. Take of A, B,tf>an7 equimult. « A,«J3jflltf.- 24 And of Cy b^ Cf any equiomlt. n C» nb^ n c* }P./I.»5. Dbmonstratiok. JDECAUSE MA&MBareequimwit.ofA&B ('Pr^^. 1/ I. It follows that A : B = nvA : Jit B. P.15. JR5. a* And ^ : c = ^r ^ : n c; But A : B = * : f . (Hyp. \). J. Therefore^ » A : m B ^ n^ : n c. P.n. jr.5. And becaufe B : C = ax b, (ffyp. x/ 4. It follows that «iB : n C = ma : nb. P.A.AC. 5. Wherefore, iw A, « B, 11 C, & « a, <w *, « c form two fcifcs of mg^s* which taken two and two in a cro6 ord6t have the (aaK ratio. 6- Confequentlyf by equality of ratio* according as the firft m A of the firft feries is >, := or < the third n C, the firft ma of the other fcrics will be >, = or < the third nc, P. ai. J9,c 7. For which rcafon A : C s: « : c. Z>. j. A 9^ Which was to be demonftrated. J Book V. Of EUCLID. PROPOSITION XXIV. THEOREMXXIF. JL F four magnitiKles (A, B, C, P) be proportionals and that a fifth (£) has to the Cecond (B) the fame ratio which a iixth (F) has to the fourth (D)^ the firft and fifth together (A + E) (hall have to the fecond (6), the fame ratio vrhich the third and fixth together (C+ F) have to the fourth (D). Hypothefis. /. A : B = C : D //. E : B = F : D. A + E Demonstratiok. Thefis. B = C + F;a Because E:b = f : d (Hyp,z). 1 . It follows invertendo B : E = D : F And becaufe A : B =5C : D (Hyf^, i), 2.. Ex aequo ordinate A : E == C : F 3. Componendo A + E:E = C+F:F But fincc E:B=F : D (Hyp. z). P.tS.B.f. It follows, £x «quo ordbate A + ^ ^ B :C+F:D Which was to be demonftrated. P.a^. B, 5. 196 The ELEMENTS Book V. PROPOSITION XXV. THEOREM XW. X P four magnitudes (A, B, C, D) are proportionals, the greateft (A) aad leaft of them (D) together, %Tt greater than the other two (B & C) together. Hypothefis. /. A : B = C : D //. A is the gnateft term^ {? Confe^uently (•) PibfUaft. Preparation. Take I C = C & I D = D. Thefit- + P > B+C B Demonstration. _> EC A USE A:B3=C:Dr/^/.i;&0=iC&D=:iD^/Vf/.> I. It follows that A : B=iC:iQ ^. Wherefore A :B = M: N But the mgn. A being > B (Hyp. 2 J. r P ,#: » ^ 3. The mgn. M is alfo > N \ V:*^ ^^ Moreover, becaufe C =iC &*D =: iD (^rep. \ti 1). ^ "^• 4. It follows that iC-+D=iD+C And fince M is > N (Arg, 3^. 5. It follows that iC+D+M>ip+C+N, that is A+D is > B+C- Ax^Mu Which was to be decnonftrated. (♦) Euclid fuppofes the confequence of this Hypothefis fuficientiy e^fideml frm the foregoing truths ; for fince A : B : C : D (Hyp. ij, & A > C (Jfyp^ a./ B is > D (P. I A. B. 5/ Like^ife A being > B (Hyp. zj Cis>U (F. i& £V- B' 5v?> Confequently D is the leaft of the Iff terms. BookVL Of EUCLID. «97 DEFINITIONS, I t^IMJLJR rtaiHiteal figures (Fig. \.) are thofe (A B C, a be), whicH ha« thdr feveral Angles (A, B, C, and a, b, c) equal, each to each, and th« fides (AB, A C, B C, and a b, a c, be,) about the equal anglet, proportion nals (that is A B :AC?=abia(^aMbAB:BC^ab:bc, andACi BC = ac:bc;. H. •X HE Figures (D A B,d A b) are reciprocal (Fig.%.)t when the antecedent* (A D, A b; and the confequents (A d, A B) of th^ ratio^ are in each of tb^ figures, (that is A D : A d ;= A b : A B. Or the figures (D A B, d A b; are reciprocal j vAen tbt twtfdf* (AD A B tfjirf A d, A b), in tacb of tbofe figures, about tbe fame angle (A), w equal angles, are tbe extreams or means of tbe fame proportion, tbat is, a fid^ (A D) in tbe firft figure is to a fide (A d; of tbe otber^ as tbe remaining fide (A b^ of tbis otber is to tbe remaining fide (A B) of tbe firfi. III. A Straight line (A B) is faid to be cut in mean and extrtam ratio, (Pig. y,) when the whole (A B), is to the greater fegnitent (B C), as the greater foment, is to the lefi (A C). * 198 The ELEMENTS BookVI. DEFINITIONS. IV. J[ Hp atsiiu4e of any figure (A BC) (Pig.iJ^ U Ae perpcndi(»k(BD)fa &I1 frpm ibe vertex (fi) upon the b^fe (A C). IT fQlfoyjf from this De/mfiion, that if two figure^ fiacd vpon tkf^ ffrqigf>t fine, tape the fame Mfude, they 4re k^tuetn the fame p$M\ hecaufe from the nature of parallels the perpendiculars let /i// /r#« •« ^« « ether are always equal. V. A f^^*'^ (AB. BC. CD : DE. EF. FG) is cginpounded of ftveral »/fc^* (AB:PE-f-pC:EF + CD:FG) >vh€n its terms rclult from *« multiplication pf the tcrnw of ijiofc comppupdiog railo«. VL j\ Parallelogram (AB) (Fig. %) is faid tp he qpplifd to (tftraigh\line[0\ when it has for its bafe or fof its fide this propofcd flraight line (CD). A VII Deficient parallelogram ^ F), (Fig,y) is that wbofe bafc (AB) tfl* than the propofed line (C D) to which it is Taid to bf applied. vm. JJuT the deficiency of a deficient parallelogram (AF), (FigA) is • HJ logram (BG) contained by the remainder of the propofed flraight line (CW und the other fide (B F) of the deficient parallelogram. &ok VI. L or EUC LIIX E G i A H • ■-- ' D B 1^ li E F I N IT* I O N S. K. Xj}N' excetdh^ parAllthgnm (A F) ir that, whofe b»fe (A B) is greater than tht pro(>«re4^Imfr' (C D), t6 which it is faid to lie apfXtoA. A> I^D- tbe excejs of an exceeding' parallelogram (AF) is a parttlMogiani (HF) contained by the excefs of the bafe (A B) above the propofed (Iraight line (CD) and the other fide (BF) of the exeeedilig parallelogram. Aod The ELEMENTS Book VI. ■1 ,' 4 F B ] 4^:>\ H G A C D L t 1 PROPOSITION I. THEOREM I. T^RIANGLES (ABC, C B D), and parallelwtms (C F, CE),of the fame altitude, are one to another as thfir bafes (A C^ CD). Hypothefis. Thcfo. 7*e AABCf, Q^-D^tipgrns. /. Ti^e A A B C : ACB D = AC : CD. C F, C E, bave the fame altituih. //. 7be pgm. C F : pgm. C E =: A C : C D- Preparation. I. Produce A D indafinitelv to H & I. 3. Take AG = AC=GH, alfoDL = CD = LL 3. Draw BG,BH,BL,BL P. x,B,t. Pofi,BA. B DEM6NSTRATI0K. E C A U S E the A A B C, G B A, H B G, are upon equal bafes A C> A G, G H, (Prep, z), & between the fame piles. HI, F£, (Hyp. ^ D. 35. B. 1. W Rem. D. 4. B.6.J. 1. Thofe A are = to one another. P.38. ^.i. 2- From whence it follows, that the A H B C, & the bafe H C, are equimult of the A A B C, & of «che bafe A C. It may be demon(bated after the fame manner; that , 3. The A C B I, & the bafe CI, are equimult. of the ACB D, Sc of the bafe C D. 4. Confequently, the mgns. H B C & H C, are equimult. of the mgns. A B C & A C fy/fjr. aJ!, & the mgns. C B I & CI are equimult. of the mgns. C B D & C D, (Jrg. y). But if the A H B C, be >, = or < the A CB I, the bafe H C is alfo >, = or < the bafe C I, (P.x%, B, ij. 5. Confequently, the A A B C : A C B D = A C : C D. D, 5. ^. 5. Which was to be demonftrated. i. But the A A C B, C B D, bemg the halves of the pgms. C F, C E, (P^i.B.i.) r. It follows, that AACB : ACBD = pgm. C F : pgm. C B. Pa^.B, 5. e. Wherefore the pgm. CF : pgm. C E = ACiCD. Pai.B.\. Which was to be demonftrated. 1 1. BookVL Of EUCLID. 201 ^■""^t.'^fi: Pof,i.B,i. p.y.B.^. PROPOSITION II. THEOREM IL X F « ftraight line (D E) be drawn parallel to one of the fides (A C) of a triangle (ABC): it fhalV cut the other fides (A B, B C) proportionally, (that is A D : DB = CE : EB); and if the fides (AB, BC) be cut proportionallj, the ftraight line (D E) which Joins the points of fedion fliill be parallel to the remaining fide (A C) of the triangle. Hypothcfis. Thefis. 7bi /raigbi line DE is pile, to AC. AD : DB =:C E : EB. Preparation. Draw the ftraight lines A E» CD. BI. Demonstration. ECAUSE DE IS pUe.to AC 1. Th« A D A £ is 2. Confequentlj, ADA£:ADBE = AECD:ADBE Bucthe ADA£:ADB£= AD:D6. &the AECD:ADBE= CE : EB {P.i.B,6.) 3. Therefore AD: DB= C E : E B. P.ii.J.j Which was to be demoiiftrated. Hypothefis. Thefis. AD : PB == CE : EB. The Jiraigbt line DE is pUi. ti AC, BII. D£MONSTltATION* ECAUSE the ADA E, DBEa^e between the fiune piles*' as alfo the A £ CD, D B E. 1. It follows that A D A E : A D B E =1 AD &the /iECD:ADBE=± CE But AD: DB = CE a. Therefore the ADAE:ADBE = AECD 3. Wherefore the A D A E is = A E C D. 4. CMiequently, the ftraight line D £ is pile, to A C. Which was to be demonftrsted* C c DB. } E B 1 E E. (H9p.), ADBE. P.i.B.6. P.11.A5. P. 9. B. J. P.39- *'«• I PROPOSITION UI. THEOREM IIL _F the sngle (B) of a irimgle (A B C) be diirided into two eqittl flQgks by a ftraight line (B D) which cuu the btTe in (D), the fcgmenta of the bale (AD» D C) (hall have the (ame ratio which the other fides (A B, B Q of the triangle have to one another ; and if the fegnnents of the bafe (AI>, D Q have the iame ratio which the other fides (A B, B C) of the triangle have la one another, the flraight line (B D) drawn from the vertex (B) to the point of fc£tion (D) divides the vertical angle (ABC) into two equal ang^ Hypothefis. Thefis. 7heftraigbtlin€^'Ddivuhs the ^ ABC AD: DC = AB : B C into Miw equal parUy •r V • ^ V «• Preparation. Thro* the point C draw C E pile, to D B. Prodttre A B until it meets C £ in £. P.ti.Bi. «^ 'I. DEMONSTRATIOlf. 15 £ C A U S E the ftraight lines D B, C £ are pile. (Prgp. i). 1. It follows that A D : DC s= A B : B E. P.%.B^ 2. Andthat V«= Vai, & V«= V/- P.ao. ^.i. But, V • being ^ to V It r^// C A.i. B.t. 3. The V«itftifo=rto V/» &BC=toB£. IP. 6.M.U 4. Wherefore AD:DC = AB:BC. -P.7.& ii. ^.5. Which was to be demonftrated. Hypothefis. The6s. DC = AB : BC. BD*i>iJx V ABC ar Vt= V«- BIL DjSMONST&ATIOK. E C A U S E the ftraight lines D B, C E are pile. fPrep. ij. I. It follows that A D : D C = A B : B E. P^ z. BS. But AD:DC = AB:BC (Hjp,) a. Wherefore AB:BE = AB:BC. P.ii Jc. 3. Confequemly, B E is = B C, & V «» = V /• i^- ^bX* But V«i»alfo = toy.if,«V J»=V* r^.a9Ai/ IpX.Bu 4. Canft^uently,V n « s= to V •, or B D bOeas V AB C Ax.i. B.u AD Book VI. Of EU CLia zQg PRpPDSITION IV. THEOREM ir. _ H E fi<tt (A C, A B & CE, C D, &c) about the equal angles (m & n, &c} of equiangular triangles (ABQ CDE) are proportionals ; and thofe fides (AB, C D, &c) which are oppofite to the equal ^glet fr ic s. Sic) are ho- mologous fides ^ that is, are the antecedents or fonfequents of the ratios. Hypothefit. Thefis. Tie A ABC, CDE are equiangular^ f AB : A C = CD : C E. •rV«=:V«, V^ssVj, /.^AC:BC=;CE:DE. £^V/ = V#. tAB:BC?:;CD:,DE. r AB , C D7 oppofite iQ the II The fides <AQ,C^\ equal >t are b%. (B C , D EJ ml^us. Preparation. 1. Place the A AB C, C D E» fo tbat the bafcs Ap,;C E may be in the fame ftraiffbt line. 2. Produce the fides A B, D E indefinitly to f. Pff, i.B.i. BDEMO^SJRATIOlf. EC AUSE the V « + r of AABC are< 2 L (P.i 7. B.i .) tc^fr ;=;V/. ( Hyp. I 1. The >/ m+s are alfo < 2 L.» & AB> DE meet fomewhere in F. Lem. B.i. But V m bciug =:toV«&Vr==toV/ (ffypj^ 2. The ftraight lines A F, C D alfo B C, F E are pile. P.28. Ba. 3. And the quadrilateral figure C F is a Pgrm. ^-SS- -^i* 4. Confequently, B C, F D ; alfo CD, vF arc ::; to oae another. 'jp.34. B.t. But B C being pile, to the fide F E of the A F A E (Arg, 2/ Therefore A B : B F =r A C : C E. -P. 2.116. Or altemando AB:AC = BF:CE. P.16.B.C. 7. Or AB : AC = CD : C E, C D being = toB F. (jlrg. 4/ F. 7. B.^, Likewife C D being pile, to the fide A F ofthe A F E A. B. It maj be proved in the fame manner, that AC : BC ::;: CE : DE. 9. Confequently, AB:BC = CD:DE jp.22. Jl. Which was to be demonftrated. i. But the fides A B, C D, alfo A C, C E & B C, D E are oppofite to the equal Y r Sc s^'p ho^mh n. jaConfeqUentlv, the fides A B, C D ; AC, C E; B C, D E oppofite TO the equal V are homologous. D. 1 2. B.^^ Which was to be demonftrated. 11. Cbr. Tbereftre equiangular triangles are alfifimilar (D. i. if. 6.^ I; 3«H The E L E M E N T S Bwk W I PROPOSITION V. THEOREM F. F the fides of two triangles (A B C, D E F) be proportionals, tliofe rriangkt Ikall be equiangular, and have their equal angles (A & m, C & n, &c) oppofitc to the homol(^ous fides (B C, E F & A B, D E, &c). Hypotltefi;. Thefis. rbi A A'B C, D E F hmvt their /. 7i^ A AB CD £ F m^eftMttgml^. fid€S'jr»poriiouals% that is, JL The V ippcjtte t9 the h^mdMis fidts fAB : AC = DE :'DF. flre=; .rVAiirV*. VC=Va /.^AB : BC cr DE : EF. ejfVB = VE. (AC : BC = DF : EF. irneJii/esBCrE F, A B, D E, ^ Cy D F. «nr lyfMhgpus, Treparatiofi. I . At D in D F make V ^ = V A & at F, V f = V C. P.23 B. r %. Pcoduce the fides D G, F G until they meet in G. Lem, B. 1 BDbmonstratiok. ECAUSE in the equiangular A A B C, D G F (Prep, i.^P. 3a. B.\), V C=3 V y & V B = V G. 1. AB:AC=:DG:DF, &AB : AC = DE : DF. (Hjp.i). P.^B.6. 2. Therefore, DG : DF z= DE : D F. & DG issiio D E. ^Pii.B,f;. 3. It may be proved after the fame manner, that G F = E F. i 7*. 9. A 5 Since then in the two A DEF, DGF, the fides DE, EF = the fides DG, GF (Ar^. a. If 3^, & the bafe DF is common to the two A. 4. The V » & « arc =i 10 to the V y & / each to each. 1 p o ji 5. And the A D E F, D G F are equiangular. ^ r; ». /f. 1. But the A D G F, is equiangular to the AABC {Prep.i.VP.^2,B.i), 6. From whence it follows that the A A B C, D £ F are equiangular. Jx.i.B, i. Which was to be demonftrated. i. 7. Moreover, the V A, C & B oppofite to the fides B C, A B, A C, be- ingequal each to each, to the V w> 9 & E oppofite to the fidea E F» P £, D F I homologous to the fides B C» A B, A C9 each to each» becaufe the one & the other of thofe V> are equal each to each to the V P* f. G (Pnp. i. P. 3a. B. i, V ^g-^J- 8. It follows, that the V A, m j alfo C, n & B, £ oppofite to the homo- logous fides ace equal. Which was to be demonftrated. ii« Cor. therefore ibefe triangles are alfefimilar. (P. 1. ^1 6) Boojt VI. Of EUCLID. 205 I PROPOSITION VI. THEOREM FL F two triangles (A B C, D E F) have one angle (A) of the one equal to one angle (m) of the other, and the fides (B A, A C, & ED, D F), about the equal angles proportionals, the triangles (hall be equiangular, and (hall have thefc angler ^C & «, alfo B & E) equal «vhich arc oppofit^ to the homolo- gous fides (B A, E D & AC, D F). Hypothefis. • Thefis. /. V A = /(» V «. / "^he A AB C, D EF are equiangular, //. B A : AC = E D : DF. IL 7be "^ Q li n, edfi the \t ^ W 1 are ' III. B A, EDi AC, DF are homologous. z=, to one another. Preparation. I. At the point D in the ftraight line P F make V/ =cto P.aj.Ai. Lem. B.\. V A, or = to^V mkox the point F, V f = to V C. 2. Produce the fides D G, F G until they meet in G. B Demonstration. EC AUSE thcA ABC, D G F are equiangular (Prep.i. & P. %2,B. i)>& particularly VC = Vf&VB=VG. . I. BA : AC = GD : DF \ But B A : AC = ED : DF.fHjf. z). z. Wherefore, GD : DF =. ED : DF. 4. Confequently, G D is =: to E D. Therefore the two A D E F, D G F having the two fides E D, D F = to the two fides GD, D F C'^g-}) & V « = to V / (Prep. \). 4. The V «. f & E, G arerr, & the A DEF, DGF are equiangular. But the AABC, DGF beiagalfo equiangular (Prep.x.h P.^2,.B,iJ, 5. It follows, that the A A B C, DEF are equiangular. Ax.i. B,t. Which was to be demonftrated. i. Moreover, each of the angles C&n being = to V^ C^rep, i . & Arg.j^). 6. The V C is =r to V «. ^n. i. Ba. 7. Confequently, V A beinff = to V « (^fyp-^h V B is alfo = to V E. P. 32. An . And the fides B A, E D & A C, DP oppofite to thofe angles being homologous (Hyp, 3. fcf D. 12. B. t.), S. It follows, that the V C & if, alfo B & E oppofite to thofe homolo* gous fides are ss to one another. Which was to be demonftrated. 1 1 . Cor.- therefore thofe triangles art affijimilar to each other. (P.^ Cor. B.6J- P.ii.B.%. P.9.B.S. P. 4. B.t. 2o6 The ELEMENTS Book VI. 1 PROPOSITION VII. THEOREM l^IL J[f two triangles (A B C, D E F) haye one angle of the one (B), cqail M one angle of the other (E), and the fides (B A, A C & E P, D F) abow t^ other angtes (A & D), proportionals; then if each of the remaininraPglci (C & F) lie either acute, or obtufe, the triangles (hall be cquiaog;aury tai have thofe angles (A & D) equal, about which the fides are proportionak HTpothefis. Thelis. /. >ihisz;zu>f JL Tit A ABC, DEF Mr efwUngAtt 11. B A:AC=::£D:DF &/iS« VBAC&rD«fv = /amMiier. ///. The VC &F «r# bi^b titbtr ^cuttf $r obiuft. Demonstration If not» the VBAC & D are unequal, and mie as BAC ' is > the other D. B Preparation. At the point A in the line A B, make V o = V D* CASE I. If the VC & F are both acute. ECAUSEV»'8 = toVDrPr«^.J,& VB=toVEr/fy/.i;. It follows, that V/iis^ to VF; & the AABG, DEF are equiangular. Confcquentlj, BA;AG = ED:DF. But B A : A C = E D : D F. (Hyp. z). Confcqucntlj, BA:AG = BA:AC. From whence it follows that A Q is =7 to A C. Wherefore, V C is = to V «. And bccaufe to this cafe y C is < L.. ?.23.1J.i. P. 5. Ai^ C. The V «» will be alfo < L 1 & V« which is adjacent to it > L- iP.iv*! But this V n being = to V F (Arg.i), which in this cafe is < L- 7. This fame V n will be alfo < L. ; which is impofiible. g. The V B A C & D are therefore = to one another, & the third V C IS =f to V F, or the A A B C, DEF arc equiangular. Which was to be demonftrated. P.3a.At Book VI. Of EUCLID. ao7 CASE IL If the V C & F are both obtufe. jS Y the fame reafoniiig as in ^he firft Cafe (Jrg.i, to Arg, 5.) it may be proved, that 1. The V C is ±± to V «t. 2. Therefore V «i is alfo > L> <t the V C + at will be > a U> which is impoifible. f.ij.B.i. 3. CoafequeDtJy, the V B A C & D are s: to one another & the third yCisrrtoVF, orthe^ABC, D E F «re equiangular. P.i%.B.il Which was to be demonftrated. REMARK. M^Ftbt^iQlif mn btb rifht mf^ks ibg A A B C Be DEF mt iquUmmt^ lar (Hyf.i.liP.iz.B.2). Con Tbenfyri thfi trumgkt art fiuiUur /• %n9 ankthtr (P. 4. Or. Bk 6)» aoS" The ELEMENTS Book VI / f ^ PROPOSITIOi^l tut ttiEOREM Vai- Ij5r a right Aigled tfiAgfc (A B C), If* pvpendicvlhr (B D) b« iriwiW the right angle (A BQ to the bafe A C, the tt'm^i (A D B, BDQm each fide of it are rimitat to iht v»liole trNM|l« (ABC) M^ M> one vt^' Hypothefis. Thefis. " /. T*# A A B C « rgU. in B. TAr A AD B, BDC ««^ //. BD«±i«/««AC. & V A comftion to the two A- 5 two A A B C, A D B are cquiangolar. /'ji J ' arcalfofimilar. {^-4*^ Uhc fame maamcr, that X^- io one am^tker^ W ^ " 'r /** iart^ibewMe A ABC. B Demonstration* EC A USE in the two rgie. A A D B, A B C, the V « » = to V A B C, (Ax. IO. B. I.;, & V A comftion to the two A- I. The V«« = to VC&thctw< ' -- a. Confcquently, thofc two A arc i It may be demonftrated aftef the fame manner^ 3. The A B D C is fimiiar to the A A B C Likewife in the two rgle. AADB, BDC, y«l BefBgS^io V n (Ax. 10. i?. 1.) & V « = to V C (Arg, 1). _ 4. The V A 18 =: to V /> & the two A A DB, BD Ci art eqluanguUT. Pjii-J 5. From whence it follows that thefe A are fimttao |^-4*^ 6. Confequentlj, the X B D dirides the A A B C into t#o A A D B,' ( C» B D C fimiiar to one another (Arg. <.) & fimiiar Jo the whole A ABCr^/y.a.W3). Which was to be demonftrated. FC O R O L L A R r. ROM this it is manifefi that the ffrpendicuhr B D AntwHfnm tht Vtrtff •f a right angled triangU to the hafif is a mean fropftiend heiwee* «' fegments Pl l> % D C ef the hafe ; for the ^iangles A D B, B D C fcixg <f<^ angular, AD:DB:=DB:D Q(P, ±. "^ tie B. 6.). _. .^.y.«,, , ^ ^ vy .^. triangle A B C » « mean proporti^^. ^ ^J and theftgment A D tr D C adjacent to thai Jidg. for ftna «if*? the triangles A D B» B D C fx equiangular nvith the whole A A B C» A C : A» = A B : A D, & AC : B C = B C : D C ri*. 4. i. 6). Zilfo, each of the fides A BorBCofi t^een the hafe H i k Book VI. Of E U C L I D. 209 A ^. D F. PROPOSITION IX. PROBLEM I. R O M a given ftraight line (A B) to cut off any part required. (For example the third part). Given. Soueht. 7befiraigbt line A B. 7be .abfcinJed fraigbt line A D, which may be tie third part of A B. Refolution. 1. From the point A draw an indlfinite ftraight line A C> mak- ing with A B any V B A C Pe/A.B.i. 3. Take in AC three equal parts A£, EF, FC of any length. P. v B.i. j.JoinCB. Po/i,B,u 4/ And thro' E, draw ED pile, to C B, which wiii cut the P.31.S.1. ftraight line A B fo that A D will be the third part. Demonstration^ 13 £ C A U SE ED is pile, to the fide CB of the A CAB (Fr^ 4}. I. CE:EA = BD:DA. P. z. B.6. But C E : is double of E A (Ref, %) i z. Coniequently, B D is alfo double of D A. • D. 8. ^.5. 3. Wherefore, A B is triple of A D. 4.. And the abfcinded ftraight line A D is the third part of A B. Which was to be demonftrated. Dd ^lo The E L E M E K T S Book VL • .0/ A F/..\n LjU. • • \ ^. ss^ -- >L. MSB ....\Q gg-aig^ PROPOSflTION X. PROBLEM n. _. O divide H gketi itraighc line (A it)* fimihrly to a §lfta ftnrij^t Eoe (AQ divide in the poinu (D^ E &c) Given. Sought. /. Vh firai^t lint A B. To Jk^f^« A BJmitarfy A A C //. fbijiraight tine AC Jvuided in tbe points F C^ G, /• /ioT in tiepointsD, E &c, A F : FGrr A D: D BE^ito FG;GB=DE:EC. RefoJutton. r. Joih tfee gftvii ^alght Knes A B> AC lb sts to coMloh tav ,VBAC, /V:i.^.i. i. BtftW C B^ lb firbin the pdints D Sr E, the fti«iglkt Unet D F, E 6 ^1«. tb C B, idfo D H pile, to A &< P.^u J.i. JljECAU^E Df IspIYe. toth^ikleEOoftlie AAGE/'/?^a« A P.ao. -B.i;, and KE pile, to the fide rtC of the A DHC (Rtf,zJ. I. AF : FG = AD : D fi. And 1)K : KH = DE : EC. P. j. B£. But the figures KF, H G being pgrms. ^Jt^ 2. ?rf />• 35. JJ. 1./ a. It follows, that F G is :=r tb D K ft C D = K H. ^.34. Ai. 3. Therefore, F G : G B a: D E : E C. P.7. » ii. JI5. 4. Confequently, the given ftraight line A B is divided in the points F&G; foihat AF:FQ=AD:DE/kFG:GB=:DE:EC, Which was to. be done J Book VL Of EUCLID. PROPOSITION XL PROBLEM III O frnd a nhird proportionall (C E) to two givea ftraigbt Yiim (AO, hC\. Given. JTA^ tvoQ ftraight lines A B, A C Sought. The ftraigbt line C E, « third fnporti^nal to the ttvo ftrmght lines A B, AC tbaf isfucb that AB: AC ^ AC :CE. Rcfolution. X.B.I. B 1. Tom the two ftraight lines A B, AC b ^s to contnui any V B A C. 2. Produce them, & make B D 5= A C ' P.x.ii.%. 3. JoinBC. Pofi.B.i. 4. And from the extreinity D of the (haight h'oe A draw DEpUe.toBC. ^ frSi^f Pemonstra Tioir. ECAUSE BC18 pile, to D E (ReJ. aJ. AB :BD = At : CE. But a. Confequently, P. 2. B.6, BDi8 = toAC (Rer.z)i A B : A C = A C : C E. i'?- &f 11. B.j* Which was to be doive^. 212 The ELEMENTS BookYl Given. Jbeftraigbt Isms M, N, P. PROPOSITION XU. PROBLEM ir. _ O find a fourth proportioiuil ( C E ) to thfce givea ftraq^bt Son (M, N, P). Sought. The ftraigbt Tine C E* « fitirA pr^fntrtional /• M, N, P ; tb^ is fucby tbat M : N = P:CL Re/oltUipn. l. Drew the two (Ireight lines AD, A £» containing any V D A E. F. X. S.i. a. Make AB = MjBD=:NjAC = P. P^/.r B.h 3. (oinBC. 4. From the extremity D of the ftraight line AD, draw DE, pile to BC. Pji.iti. Demonstration. B lECAUSE BC is pile, to D E (Ref, 4/ I. AB : BD =z AC : CE. P. 2^ BS But A B = M, B D =: N, & A C = P (Ref. z) j %. Confequentlj, M : N == P : C E. P-y.^u.^S- Which was to be done. Book VI. Of EUCLID. ai3 ^^—.,,0 A" g X' PROPOSITION Xm. PROS LEM F. J[ O find a meaa proportional (B D) ; between two giVen ilraight linet (AB, BQ. Given. Soug[htt ^be tnnoftraigbt iines A By EC, The finught line B D, a mean fr^pwr^ tional hifwetn A B & B C, that i$ >cA /*«/ A B : B D » B D : B C. Refolution. 1. Place A B, B C in a ftraight b'ne A C. 2. Defcribe upon A C the femi ©ADC, P^j. B.u 3. At the point B, in A C> eie^ the X B D meetbg the O in D. F.w.Bi, Join AD, &CD. Preparation^ PoJa.Ba. Demonstration. 13e cause the V a D C is in a femi © (Ref, a. W Prtp.). 1. It is a right angle. F.^x.B.%: a. .Wherefore, the A A D C i« right angled in D, & B D is a -L let fall 'from the vertex D of the right angle, on the bafc AC (Ref, 3^. 3. Confequently, AB:BD = BD:BC. Which was to be done. Ji*. 8. ff,6. \Cor. ai4 The E L E M E N T S Bode Vf. E PROPOSITION XIV. 'fHEOREM LX. QU A L pftrallelograms (A B» BC>, which have x>oe angle of tiie «oe (FBD) equal to one angle of Ihc other (G BE), have their Wes (FB, B D & G By B E), about the equal angles reciprocally proportional, (that is, FB:BE=OB:B D). And parallelogranns ihia have one -aqglc of the one (P B D) equal to one aogjfe of the other (G B E) and the fides (F B, BD 0r O fi^ B £)^ about the equal angles reciprocally proportional, are equaU Hypot"hefis. / Thefia. /. Tbeppr. AB is =: to ihe pgr, B (2. FB : BE = GB : BD. //. VFBD« = /o VGBE. iBrefparatifm. I. PJacedieiW9 3>gf8. AB« BC lb as the Adba FS, BE may be in a ftraight line F E. a. Complete the pgr. D E. Pof.z^B.i. BI. DfiMOirSTRATltN. ECAUSE the VFBD» GBE are equal (ifyp.%)i J^FI, B E are in a ftraight line F E (Prep, i). ! • Therefore, G B> B D are in-a ftraight ltne«0 D. P.14. £. i. But the psr. A B being := to the pgr. B C (Hyp, \). 2. The pgr. A B : jgr. D E = pgj. B C : .pgr. DE: - P. 7. Blj. But the pgrs. AB, D£ alfo BcrOE have the fame altttud<: fD^B.6), J. Hence pgr. AB : j>gr. DEisFB : BE ftf^r. BC :fgr.SE=GB : BD. P.i. B.& 4- Confeqiiently, FB:BE=:GB:Bfi (Arg. %), P.iu J.5. Whic|i was to he demonftiated. Hypothefis. The6s. /. FB; BEsGB: BD. '^ie pgr: AB is = f tbepgr.hC II, VFBDw=:/* VQBE. In. Demonstratiok. T may be demonftrated as before, that GB, BD are in the line GD. But the pgrs. AB, DE, & BC, DE, have the fame altitude (D.^B.6). 2. Hence, pgr. AB : pgr.DEsdPB : BE, &pgr. BC : pffr.DE=GB : BD. P. i. JA But F*S : BE ^ G B : BD (HypX 3. Wherefore, the pgr. A B : pgr. D E = pgr. B C : pgr. DE. P.ii. B.j. 4. Confcquently, the pgr. A B » =: to the pgr. B C. P. 9. ^.5. Which was to be demonftrated BookVi Of EUCLID; 215 _ PROPOSITION XV. THEOREM X. JC/jyAL trifo^ks (A C R, E C D) which h«ve one angle of the <nie (m) eqaal 10 one angle of the other (n) : have their fxles ( A Q C B, & E C, C D), about the equal angles, reciprocally proportional ; & the triangles (ACB, ECD) which have one angle in the one (mj equil to one angle in the other (nj, and their fides (AC, CB, & EC, CP), about the equal wigWa reciprocally pni- portional| are equal to one another. C A S E I. Hyoothcfis. Thefis. L The A ACB is = to AECD. ThfiJts AC, C^£* E C, CD» JL yf m is ^m^i^ \^ H. are recitrqcalfy proportional^ tr AC : CD =: EC : CB. DriparaMn. ' I. Place the A A C B, E C D fo flat the fides A C. C D may be in the fame ftraight line A O. 2. Draw tbr ftuight Ibe B D. Po/a.S.i. B Dbmonstration. As. »ECAUSEV«=V« r^^ 2-)y & the ftiaight lines A C, Q D are in the (ame flraight line A D (Prep, 1), I. The Imes E C, C B are alfo in a ftra%Rt Une E B. ^.14. B.i Bat the A A C B being = to the A KClD rHyp. i). ^ The AACB: ACBD = AECD: ACBD. P. Bnt the A A C B, C B D aHb £ C D, CB D have the fame altitude (Prep. z. Arg, \.\i D.a^ Rem, B, 6). 3. Wherefore the A ACB : ACBD = AC : CD. \ j, p^ & the AECD: ACBD == EC : CB. >/'.!.-»& ^ Confequently, AC : CD = E C : CB (Arg,zSiP.ii.B.^). Which was to be demonftrated. a,i6 The ELEMENTS BookVt Hvpothefis. AC : CD = EC //. ?* V < .1 C A S E IL Thefis. CB. 7bt A ACB, is 7=^ $9 iht AECD. = v«. Preparation. f lace the two A A C B» E C D fo that the fides A C, CD, may be in the fame ftraisht line A D. Draw the flraight line B D. Demonstration. .T may be demonftrated, as in the firft Cafe» that E C, CB are in the fame ftraight line E B. And becaufe the A A C B, CBD, alfo the A E C D, C B D have the fame altitude (Prep. 2. Jrv. i.li D.a Rem. B. 6). ». The A A C B : A C B D = A C : C a Likewife AECD:ACBD=EC: CB. But AC: CD =rEC : CB. r^.iA 3. WhereforeA AB C : ACBD = AECD: ACBD^ 4. Conrequently* the A A B C is = to the A E C D. Which was to be demonftrated. [ P. I. BS P.ii.Bz. P. ^ B,s. I ^iooic VI. Of E U C L i D. iif N^ M E, mH B CJ- r PRpPpSITION XVI. THEOREM Xt |P four (Iraight lines (A B, C D, M, N) be proportiomit, ihe reOtngle con^ ained by the extremes (AB. N) is equal to that of ihi means (C D. M). And F the redangle contained by the extreames (A B. N) be e<iiial to the redan- le conutned by the meatis (C D. Ki)s the four ftraight lines (AB, CD, M, N) n pnwortionak Hypothecs. The&. kB:CD=M:N. J^i^. A B. N s: J^^ilr. C D. M. Preparation. I. At the extremities Aft C, of AB,CD,ereQ the XAE,CF. P.ii.Bx. a. MalceAE = N,&CFs±M. P.yB.u 3. Complete the rglcs. E B, F D. f-ix.B.u 5 1. ]DbM0N8TR ATIOH ECAUSE AB : CD=M : N (Hyp.) : & Nk=CF St N==AE (Pnp.%). AB:CD = CP:Ak P.j.liii.B.^. . Therefore the fides of the rgles E B, F D about the equal V A & C, (Prep, i.ffjtx. 10. S. I J are reciprocal D, a. AS. Confequently* the i^le. EB = rgle. FD,or the xgle under AB.AE ftfVl4<f i9.6. = the rgle. under Cf D. C F. I Z). %^.u Confequently, A E beii^ == N & C P = M (Prep, zji * The rgle. under AB. Nis alio =: to the rgle. underCD. M. Ax.%.B,t. Which was to be demonibated. Hvpothefis. Thefis. k #ij/f . AB. N M = w the rgle, CD. M. A B : C D = M : N, > D. Demokstratiok. BECAUSE the rgle. AB.Nis=:to the rrieCD.M (Hyp.)'.U AE^K St C1P = M (Prep. 2j. The rgle. under A B. A E is = to the rgle under C D, C F. Ax.z. B.u But thefe fides being about the equal VEAB, FCD {PrepAMjfx.io.kiJ. AB:CD=rCF:AE. i'.iA. Bj6^ And C F being = M & A E =?^N (Prep. a). A B ; CD =: M : N. P.j.^ii, Aj, Which was to be demonfiraitd. E e i tiS The ELEMEI^TS Book VI 1 PROPOSITION XVII. THEOREM XIL F three ftraight lines (AB» CD, M) be proportionals, the reaaosle (AB.BD coatsified by the extremes is equtl to the fquare of- the men (CD) : And iiF ifce reAsngie cootamed by the extreams (AB.M) be equal to the iquare of the mean (CD), the three ftratght lines (AB, CD, M) are proportionals. HypothefiB. Thefis. AB : CD :&CD : lil ri# r^i^. AB.M if = ts /Ar O •/OX Prtp^raikm. 1. At the exHemitics B ft D of AB, CD ereft the XB£,DP. 2. Make BE = M&DP==DC. 3. Complete the rglei. £ A, P C Bi DSMONSTRATIOK. ECAUSE AB;CDr=M (Hyf), ftCD=:DFftM = B£ I. AB:CD=sDF:BE. -P^tfiilV iVrefere the (ides of the rgles. E At F C aboHt the equal V B ft D (Prtf. I. tS Ax, 10. A \) wrt reciprocal. . D. a. Oniaquently, the rgle. E A is = to the rgle. FC« or the rg)e. under AB. B E = the rgte C D. DF. J F,i^ Bi. y Whtacfbre» B E Scing =MftDF = CD (Prtp. 2}, the igle* ID. 1.14 A B. M is aUbssto the O of CD. Ax,2.B.t. Hypoihefis. Thefis. Tbef^U,At.Mu::^ftbeao/CD. AB:CDsCD:M BIJ, IDlMONSTRATION. E C A U S £ the rgle. uader ABM it 3C to the O of CD^^.^t ft that BEis = M&DF = CD (Pre^. zX ^. The i^lc. under A B. B E 18= to the rgle. under C Di DF. Jx^zJU But thoft fiOi^s aie about the equal V £B A, F DC (Jm. 10.^. 1. ft Fnp. i). a. Therefore, AB:CD=;:DF:BE. P.iAp BS Andface DF = CD&BE:*:M(f?i^.iJ!. 3. AB:qD=^CD:H PlUii.Bi. Whii:h wu to be demonftrated. P.ii.Af. P. j.Ai. P.ji.Jli. Z.B& Book VI. Of E U C L I D. tt9 PROPOSITION XVIII. PROBLEM ri VJPON a given ftnight line (AD) to ddcribe a reaiiinaal figure (M) fimilary and fimilarly fituated to a g^ven reQiKneal figure (N). Given. Sougkt. /. The/rmbt line AD. The r^mtinealfipirtfAfimiim' IL TbirtSdimalfifure^i ioareaitineklJigurt}^(t fimi' larfyfitumttd. Jiefolution. 1. Join HP. Pof.i.Bi. 2. At the points A & D in AD, omke VA = V^ & V** =*^ V rtf wherefore the remaiiung y A B D will be r= to iP-z^, 32. the remaining V E F H. . L frf 3. At the points D & B in D B make V • rs V/ & V 9^ \ V r, confequentiy the remaining V C will be 9 to the Xlem* B.i, remaining V G. J B DEMOVSTRATIOKr D is equiangular to tl A D B C equiangular to the A H P G (Ref- 2 EC A U S £ the A A BD is equiangular to the A E F H» & the ogular to the A H F G (Ref- 2. £* %). BD : FH a BA : FE = AD : EH. & BD : FH«= DC : HG=CB : 6 F. \^ 4- ^-^ z, Confequently, BA:FEc=AD:EHi=DC:HG=s CB : GF. P.\i,B^. But Viwbeing=sY« (Rtf.t), k\f 9:=zS p (R^ i)^ 3. The whole V «• + • " =5? to the whole V « + /• ^^-Z- ^J • 4. LIkewifc VABC=VEFG. Moreover, V A s= V E (Ref. 2^1 & V C = V G (Ref, 3/ 4. Wherefore, the redtilineal figure M is equiangu^ to the redilineal firare N, & thjsir fides about the equal V are proportionals. 6. Inercfere, the reCtilioeal figure M defcribed uDon the given Ime AD is (imilar to the rectilineal figure £Q, & is fimtlarlj fituated. Z>. 1. ^.(S, Which was to he done. The ELEMENTS ^ PROPOSITION XDL THEOREM XHI. OIMILAR triani^ (ABQ DEF) are lo one another in the dupli- cate ratio of their homoiogoin fides (C B, F E or A C, D F. &c). Hypothefif. Tbefis. 7bi irimtigUs ABC* DEF m^iJtmUar. Tit A ABC u i9 the ^ DEFci S^ thai V C == V F, e^ ihi/Jis ihi dm^Ikate ratU 9/0 3 t9 PI AC, DP & CB,F£«rf ^M^/. tbaiisasCE^: Ft** Prtparation. cPn es. T^eC6athirdpcoportioiialtoCB»F£, ftdrawAG. l/v:r.^.j. PE (ffy^tiD.i.B.6J. FE. CG (Prep.). CO. P.ii.B.^* Demonstratiok. Because ac:Cb = df I. Alternaodo A C : D F = C B But C B : F E = P E a. Confcmicntly, Ap:DF=PE 3. Therefore, the fides of the A AGC, DEF about the equal V C & F (ffyp.) are reciprocal (J>. 2. B. 6). 4. Hence the A A GC is = to the A D E P. But the A A B C, A G C having the fame altttudle. 5. The AABC:AAGC = CB:CG. P. t.JLS 6. ConfequentlTtthe A ABC: ADEP=sCB: CG. P. y.Bc. Butfince CB: FE = FE:CG. (Prep), 7. C B : C G in the dofMcate ratio of CB to FE, or as CB« : F E^^ I>.ia&$- 8* Wherefore, the A AB C : ADEP in the duplicate ratio ofC »co FE, orasCB»:FE*« P.ii.^- Which was to be deoMUifirated. FCOROLLART. ROM ibis it it wutmfefi^ thai if three lines (C B, F E, C G) fc pfpmtiea^ ms thefirfi is n the thirds Jeisetnjt^ upen the firfi to aJtmHstr^ If fimilmfy drfcHki A upen thefecond. ^ See Of. t. of tbefolkmng propofitim. Book VI, caesB Of E U C )L I D, %Zi S PROPOSITION XX. ^HKO^EAf XIF. IMILAR polygons ( M & N) may be divided by the dtigonait (A C, A D; F H, r D <n<P the fiiine iMiinl>er of fimibr triangles (ABC, ACD, AD^ & FGH, FHI, FIK) having the fame ratio to one another, chat the poiysens (M & N) have ; an^ the polygons (M & N) have to one another the duplicate ratio of that which their hoii|ologou$ lides (AB^ FO ; or BC, CH &c) have. Thelu* /. Tbofi p9fy»m ma^ h JiviJed f>/f the Janu numier ofjimilar A. //. Wbtrtofy each t9 each Bm ikifome rmfff ^icb tbfp^gfnt banJe. III. Ami the pdfg.M : p^fyg:^^ in the duplir fate ratip oftbt Ihuimwus fides A B, FGj VtffAB* :PG»* ^' Preparation. Draw AC, P H, likewifc AD, F I. P(/.i. B.y BDemonstrajiok. EECAUSEVB=VG&AB:BC=FG:GH(ffr/.WD,i.B.6), Hypothefis^ TbepefygMisfimHar f thpofyg.^i fi that /*# VA,B,C,J^f . are = te the VF,G,H,6rc.#tfrfr» each H the Jides AB,FG| flrBQGH,^^. I. The A A B D is eqiiianrnlar to the A FGH. ^ Wherefore ihofe A are Smilar, & V m s= V «• But the whole "^m-^n is = to the whole V « + f (ffyP)' ^, Conieqnently, V m is =r to V ^« Since then hy the fimil. of the A A B C & F G H (Arp.z)^ A O • R f* *■" P H • f H ^bythe&niLofthepolyg.M&N, BC : CD = GH : HI. ' ^. It follows, £x iEquo» that AC:CD=:FH:HI. That is, the fides about the equal ^i nUevt^ proportionala. $. Therefore the A A C D is equiangular to the A FH I. Arid coniequently is fin>ilar to it. > For the fame reaton, kll the other A ADE, FIIC, &c. are fimilar. r. Therefore, fimilar polygons may be divided into the fame number of fiooUar A- Which was to be dbmoaitrated^ i. * See (V. a* •[ this fnf^thn. B,6. Ax.^.B.x^ h 1.^6. P. 6. A6. SP.A.BJ6. ICir: ^^^ The ELEMENTS Book VL ■^^ C j^m^mm ■MiH ^^a^BB ^^"^^^^ V •v B ^- ^: A E F""*" K Ltkewiic, fwcMfe the A A B C, F G H are fimibr (Arg, %), & The AABC: AFGH = AC» : F H» .♦ 7 Aadthe AACD: AFHI =AC*:FHV» \ 7. Therefore, the A A EC : A FG Hi= A A C D: A FHL It may be demonftrated after the fame manner* that 8. The AADErAFIK =AACD: AFHI. 9. Wherefore, AABC : AFGH= A ACD : AFHI = AADE : AFiK. loTherefore, comtwring the fum of the anteced. to that of the confeq. AABC+AACD, &c. : AFGH+AFHI,arc=AABC:ArGH,&c. That is, the polrf. M : polyg. N=: AABC : AFGHsr AACP : AFHI,fcc. Which was to be demonftrated. 1 1 . Since then the A A B C ; A F G H = A B» : F G»» (F.xi^ J.6). li.Tlie polyg M : polyg. N = A B» : P G»» Which was to be denonftrated. 111. COROLLA Kr I. x\ «^ /^fi Dmnnftratipn may it ^pplitJ t9 fuadriiaierai J^res, ^ tbtfium trwA has already begm frmnd in iriangLes (Pig)* <' '^ tvident umitfnfalfy^ ttmt fioBar le^iliiieal figures are to one another in the duplicate ratio of their hooiologtms fides. Whtref^re^ if t9 KY^^ FG t^wooftbe homohg9US fides a ibird pr9^timtai X be taken ; becaufe A B is io X in the duplicate ratio of ABiFG ; ^ tbm a rtdi^ neml figure M is to anoiber fimilar redilineal figure N, in tbe duplicate rats^ ef tiiv famehdes A B : F G ; it fillawSf that if three (Iraight lines be proportionals, as the nrft is tt> the third, fo is any re£lilioeal figure defcribed upon the nrft to a fio»- kr & iimilarly defcribed rcQilineal figure upon the iecond. (P.f 1. ^.5). Rt^ BA P.tt. »S An. *5 P.iz. Js P. 7. *j P.I I. *$ A- ^ CO R a L L A RT If. *LL fyuares being fimilar figures (D. 30 .B. I. W I>. 1. B. 6), fimilar nffiB- meal figures M (^ H^ are te one anetber as tbe ffuaret of tbeir hum elagH u fidms A B, C D [ntpreged thus A B* : C D*) fer tbefe figures mr9 in iif duplicate 1 f/ tbefe fame fides. kx>k VL Of E U C L I D. az3 PROPOSITION XXI. THEOREM W. XV ECTILINEAL figtiFes (A, C) which are funikr to the (kme re£UUneal figOre (JB), are alio fimibr to one another. 7h€ reSihnea) figures A E^ C , • 7he reSilhual figurr A isfimilar art fimiUr to the figure B. to the redilineoTfipitf C B Demonstration. : figures will be alfo equiangular to the. fisore B» ic fides about the equal V, proportional to the fides of ECAUSE each (tf the figures' A &C is fimilar to the figures I . Each of thofe I will have the ! the figure B. . ' ^' i-B.S- ^ Confequently, thofe figures A & C will be alfo equiangular to one C ^x.i. ^.i. another* and their fides about the equal y» will be proportional ([i^. 1 1. B.^, J. Conftquently, the figiu-ea A & C are finilar. Z>. i,B,6^ Which was to be demonftrated. 324 The ELEMENTS Boole VL I PROPOSITION XXU. THEOREM M -F fourftraightlific$(AB, CD, EF, GH) l>c proportwrfh *«&* reailineal figures & fimiUirly defcrtbed upon them r M, N, ^^> ^ 'o «lfo be proportiontls. And if the fimtlar redilineal bgaret ( M» N, ^ P) QJ fimilarly defcribcd upon four ftraight liaet be pcoportionab^ thdfeftraigliciae* fliall be proportional. Hvpotheds. /. AB : CD = EF : GH. //. 7be figures M& N defcHMup$H AB, CD. ^/fi ibefiguret P & Qjfe/criMu^B £F,GH. mnfim&r, if fimilarfy fituaiea. Preparation. To the h'nes A B, C D take a III proportional Z. To the lines £ F, G H take arlll proportianal X. Demonstratiok. i?4i.Bi B U4i.i'n PrepS^P.ixB.^)- ECAUSEAB,:CD = EF:GH. AB:Z =EF:X. r«*T But the figures M,N, & P.Q^being fimilar & fimilartj ddcribed apoo the ftraight lines A B, C D, & E F, G H (Hjp. %). , ,, - - ~ .. »» <?.»■*» I Or. I & y. Whereftre, AB:Z =M EP:X =P M > N = P N 5t ^(Arg.x\ 'f.wM Book.VI. Of EUCLID; a^ II. Hypothefis. */. M : N = P : (^ _ //. Thofe fifrures arefimilar Ifftmilartj difcribtd uf9H ibeftraigbi lines AB.CD W EF,GH. TheTis. Afl:CD = EF:GH. PreparatioH. I. To AB, CD, £F take a IVth proportional KL. Paz. SS > X. Upon K L defcribe the redil fisure R» (imilar to the reciil. figures P or Q^ Mmilarly fituated.' * >* i8. B.6^ B t)i MONSTRATION. EGA tJSfi AB : CD^EF: KL fPref. ij, & uppn thofe ftraight lines have been iimilarlv defcribed the ngures M, hf» .& P» R> fimilar eacli to each (Hj^^. z- & Prep, zj* J. R {Ifi. part of this prop^/itUn.J M; N ±= P But M : N = P . ^ ,-..-. ^. Confequently, P : R = P : Q^ P.ii.A.J: 3. Wherefore. R = Q. ^- 9- ^J- Moreover, thole figures being finular & fimilarlj defcribtd upon the! ftraight lines G H, K L (Prep. 2). f P.zo. BS. 4. Q : R = DofGH:DofKL. ' I Or. 2. And , Q>>ng=R (Jrm.^J. - yAi6. «•$. 4,TheDofGHis =totheDofKL. t^#r; 5. Copfequently, G H = KL. {ajf*"* Sinct. then A B : C D i±: E F : KLCPrep.i\ & GH z±KL(Arjt.O' 6. AB;CD=EF:GH. P. 7,^5. WUch waa to be demonftrated. Ff 2A$ The ELEMENTS BoekVi. 1 J Hi m m t).. p ■ ^--^sfcj ^lCf=TT= ^^^^ Cj '•jjt PROPOSITION XXIII. THEOREM XFU. Hr QUI ANGULAR parallelograms (M & N) have to one snotfaer te ratio which is oofnfouncM oT the ratios of theb fidos (AC, CD ft £ C, CG] about the equal angles. Hypothefis. Thc6a 7h€ pgrs. M W N iffT emafmtht^, /Jfr. M : J^^T^ ac AC. CD : ECCG /o/^a/ VACD = VECG. Preparatkn. 1. Place A C & C G in the fame ftnright line A G i therefore EC & C D are alfo in a flsaight line £ D. P. 14. f.i- 2. Complete the pgr. P. Wyf,i.B.t. B, Demonstratiok. ECAUSE the pgrs. M, P, N form a feties of three magutnde*^ .,, M : MP = N : N.P. D. r.^^. 2. And akernando. M : N = M.P : N.P. P.t^ Ej^ 7, Confeoieiuly the ratio of the faft M to tkt iaft N^ compowided of the ratios M:P&P:N. D. c.RS But fince AC:CG5=M:P 7. .. & DC:CE=rP :N. ^^.iM^ 4. The ratio of the fides AC : CG is the fame as that of the pgrs. M : P } & the ratio of the fides D C .- C £, the fame as that of the pgrs. P : N. Since then the ratio of M : N is Goapowided of the ratios M : P» & P : N (Jrgi I J. r. This fame ratio- n oompounded of their cf^atAs ; the ratios A C : C G & C D : E C of tfaa fides about the eoual \rACD, ECG. 6. Confeqnently, M : N =s: AC. CD : ECCG. 2>. ^^BS Which was to be demonftrated. Cor. The fame truth is appiichhh t9 tb* trimmgles ( A C D, E C G) bmwt^ mi m^ (AC D) equal /« an angle (B C G)\ for tir diagonals (A D, EG) Jm£ the f^ into two equal parts C^34- J^« >/ Book VL Of E U C LI D. aa? ^ f /.A- 7 ^ /" A -^ p T JPJRQPOSITION XXIV. THEOREM Will. _ H E parallelograms (FH, IG) about the dis^nal (AC) of any para11do» gram (BD), are fimilar to the whole, and to one another. Hypothefe. Thefis. /. B D w ijygr. /. "the pgrs, AFEH, EtCG ar0 ft FH,T6 are pgrs about the fimilar to the pgr. ABCD. • Mag^AQ. IL And fimilar 49 mft mmther. BDEMCm^T&ATION. E C A U S E FE wpUe. to BC (Hyg. ufef a. W-P 30. f. i). a- The £iAFE is equiang. to the A ABC in the order of the letters. 1 In like aaoaer, becaufe H£ ds pUe. Ko DC. S Pzt^ B. i . |t- Thetf^AHE is equnng. to the AADC, in the ^derof ihe lestctsl j 3* Therefore the pgi. AFEH is 4Ub eqaiai^giilar to the pgr. ABCD» in the order of the letters. And l)ecfl.ufe in the A AH£,A&C« the V AHE a^ D are equal (Jr^.z)^ ais alfo M t^e A AF£,-ABC, the V AFE <& B {Arv. i/. ^: AH : HE = AD : PC & AF : EP 3= AB : C«. • P. 4. SS. Moreover, becaufe the V AEH,ACDj alfo FEA,BCA arc equal (Arp. 1.W2). 5. HE:AE = DC-AC*AE:EFj=AC : CB. P. 4. JL6. 6. Therefore, ex«quo, HE : EF == DC : CB. P.aa. Aj. And becaufe the V E A H, £ F A are cammon to the two A AH E, ^ DC & AFE, ABC. ^. HA : EAssDA : CA &EA : AFs CA : AB. P. 4. B.6. S. Therefore, i^m arcyao, HA:AFc=DA:AB. P.ia* A 5. 9. Wherefore the pgrs. AFEH, ABCD have .their n^les equal, each to each in the Order of the letters (Arg. 3^ ; & the Bdes abput the equal anglu, proportionals (Arg. 4. 6. 8.^. lO'CoGrfequently, thofe pers.arei!iniltar. D, i.B.6- 1 1 .It dnay be d«vnonflrated after the faoie ipaiMier tli^t the pgrs. |E I C 0» ABCD^re^anhLC . Which was to be demonftratcd. i. 1 a.Confequently, th« pgrs. AFEH, EIQG are alfo fimilar to one another. P,zi * Ba. Which was to be demonihated II. 328 The £ L E M E N T S Boo): VL Tc PROPOSITION XXV. PROBLEMVIl _ O dcicribe a refiiltiietl figure (I^^which Qiall b# iimilar to t given rtft- lineal figure (L), and equal to another (M). Given. Sougl|t. /. Tht reMiUntml /gun L. The nail, fgurf N» fmiUr it thifM. IL The nSUifualjigirf hL fgwn L» (^= /• the ndiifprtli Refolution. |. Upon the ftrtiffht line AC, defcribe the pgr.AH =:;«to the Stven redilineal figure L. t^. ^-i- knd on the ftraight line CH t pgr. CK =: to the giren redi- liheal figure M, liaving an V « ^= to the V *- '-4$* '*' J. Confequentlf, the fides AC, C I, ft G H, H K will be in Pii^ i^ a ftraight line. &34- A'- 4. Between A C» & C I find a mean proportional D F. t.\\- ^^ 5. Upon thii ftraight line DT, defcribe thc««il. figuie N, funilarly Hi fimihiLf to the redilineal figure L. P iS- ^^ Demonstration. ECAUSRthe pgrs. AH.CK have the fame altitude (Rqf^z.tixy I. pgr. AH : pgr.CK=: AC : CI. f » J^^^ But the pgr. AH;;? rcail. L, & the pgr.CK=s reail.M fRe/AJ^z)- %. Confequenily, L : M = AC : CI. P.wM But AC:DF=DP:CI (Ref. a,), fr upon the ftraight lines AC, DP have been fimilarly delcribed the fimilar figures L & N, (Rtf. 5/ -r 3. Confcquentlj, L ; N = A C : C I. . C Paa W 4. Hence, L : N cr L ; M (Arg, 2/ (Or. 5. Wherefore, N = M <?,ih^ p. Therefore, there has been dcfcribcd a reailineal figure N, fimilar (?> if ^5 t9 the redtilineal figure L (Ref^J, & equal to the redilinetl figure M (Arf. 5/ Which was to be done. B Of E U C L I Dt J PROPOSITION XXVI. THEOREM XIX. X F two rimitar parallelograim '(A C, F G) have % pommon angl^ (F B G), and be fimiltrly fituated, they pre ebpvt the btpe diagonal (B D). .. Hypothefis. Thefia. /. AC isMpgr. £^ B D Us diafnai ^hipgr. F G is fbesJ ahui tbf Ih YGisa pgr.Jimlar to AC l3 having ihf diagonai BD ^ ibe pgr. AQ^ M y FBG fMMMirwfVA AC ^ PfiMONSTRAttOV, If qoti let another line BHD different from BEB be the dfa- {pnol of the pgr. A C, catting the fide G E in the point H. Preparation. Thro* the point H dra^ H I pile, to C B or D A. P.^ \, B,i^ H E pgrs. A C, { Q bein^ about the fame dia^al B H D» & - - - "^ Pg^- A C it limilar to tne pgr. . _^ , 2. Coniequcntly, C 1 : BA == GB : BI. - D. i. 5.6- V F B G being common to the two pg^«. (Sup, ^FrepJ, The per. A C IS fimilar to the p?r. I G. P^M' 5.^. 3ut the pgrs. A C & F G being alfo fimilar, & y B com|no9 to the two pgrs. (Hyp. 2). It follows, that CB : BA = GB : BF, Z>. i. 5.6. 4. Confcquently, GB : B I = GB ; BF. P.ii. 5.5. 5. Therefore, B I ss B F. P.14. ^.5. 6. .Which is impofiible. ^x.g. B \ . 7. Hence, a line BHD. different from the line BED Is not the diagonal of the pgr. P^Q. 8. Conlequentlj, the line B E D is the diagonal, & the pgr. F G i^ placed about it. Which was to be demonftrated. i30 The ELEMENTS Book VI, PROPOSITION KXVII. Ttf^QAEMZX \J P an par«iIlelogranis (A G) applied to thf^ fume ftraight fine (A B), mi deficient by parallclografra (N 1) (imSlar and fimMarlf fiitmed to that (PD) which is defcfiM upon the half (I B) of the line (A B) ; thirt (A E) whkh b Applied to ih^ other lialf (AP), and is (imilar ^ its d«feft<FD)« k the groMcfi. Hypotneni. Vocln. /. A Eisafgr. applied to the hei^ 'A B li the p^mtf^ ff M th^ fp. A F oftbeftraigbt line A B. >aft « A G, appUetHe A B, M //. I^/Vi^ «f /m/Zw* & /imehe^ kave their defeSs fuch «r N I. /it Mated $$ its ditfeB the fg^FD, fmilar i^ fimiUarlj fiiuated f the de/cnieden the pther hafr^. p^. f D, iSr/^y A B, dtJcrM upenY%fbe hdf ^f A B, PrtfBtramn. I. Draw the dia^^al B E. P^.v.Bx. i. Thro' any point G, taken in B £, draw I H, MN pile, to BA, A C 1**11 *»• In order to bave a pgr. A G. «p^i«d to A fi« deficient bj It pgr. N I, fimilar to the pgr. F D & fimiiarly iitiiated. P.a6w Bi%. | . . ■ • n DEMONSTftATIOH. CASE I. Wheo the point N &ls in the half F B. O^E C A U S E the pgr. G D is =s to the pgr. G F ^P.43. ^.1) ; ad- ding the common pgr. NT. i« The pgr. N O wSl be ss; to the pgr. F I. w^.t. J^t. But becaufe the pgr. A K is alfo :;? to the pgr. F L (F. 56. ^. 1 ). 2. The pgr. N O is 5= to the pgr. A K. Ax.u fiLt, And aoding to both fides the pgr. F G. I ' 3. The gnomon «^ r is =; to the f^r. A Q. vfx.2. i3.i. 4. Confequeptly, the whole pgr. FD> or its equal the pgr. A£ (Hyp.%)^ »•> pgr. AG. ifjr.8.iS.i. Which was to be demonftrated. Book VL 6f EUCLID. 831 H.. G ... .. " -1 ID H ctaam KttsaftiHii ' CASE JL \nen the point N &Ui la the half A P. The Mr. N E being =£ to the fgr. I E (P. 43. A. 1), if the commOA MC F D be added to beck fides. 1, tW fg^. N D will be:t= to the pgr. JF I. , Ax.%.B.u Bui becanle the ogt. A K ia alib:=r to the pgr- pi /'P. 56. J?. 1). 2. Thepgn NDwiffbeistothe ngr. AK AxaB.u Therefore the common pgr. F Nlbeing taken away from both fi^eSi ^. The remah^ng per. F D is =: to the gnomon a i c. ' ' Ax.yM.i* But the pgr. F if is =i to the pjr. AE. P.36. B.i. 4.. ^ Wbcremrer the pgr. A E is rf to the gnomon a ^c. Ax.x. B.u ^. Cboftqtieatly the pgf. A E is > the pgr. A G. Ax.Z, B.i. Which was to be dembnftrited. 23a The fiLEMENTS Book VI. PROPOSITION XXVIU. PROBLEM Fill O a given ftrtighc line (A B) to tpply a parallelogram (A G) equal to • given reailioeai figure (V), and deficient by a paraHelogram (M I), (imiiar la A given parallelogram (T) ; but the given redilineal figure (V) muft not be grei|ter than the parallelogram (A F) appfied to half of the given fine, faaviag hs defed (E D) fimilar to the given parallelogram (T). Given. Sought. /. The firaifht lint A B, &f tbi pgr. T. Xbe cwflruBion •[ a pgr. AG, mHUd 11. The rediUneal figureV , not > ^^r.ED, /© AB, 'wbiib wun ^ == /t V» & Jf- JlmHar tt T, applied /• AE, half of AB, ficieni by a pgr. M I fiwuUer /• T. Refolution. 1. Divide A B into two equal p^rts in £. P.iq. ^.i. 2. Upon E B defcribe a pgr. E D, fimilar to the pg^r. T, & fimilarly fituated. P.tg. f^ 3. Complete the pgr. A D^ ^ P.ii. Au The pcrr. A F will be either =: or > V ; fince it cannot be < V, by the determiQatifln. C A 8 E I. rf AFbe = V. There has been applied to AB, a Dgr. AF = to the re£tiUneal V, & daficient by a pgr. E D fimili^r to tne pgr. T. CASE H. If A F be > V. & confequendy E D > V, A F being s:: E D. P.36. B\. 4. Defcribe a pgr. X (imiiar to the pgr. T ^or to the pgr. ED) (Ref^ 2^» & fimilarly fituated, & equal to the exce(s of E D, or its equal AF, above V (i. c. mal^e X = ED— V), & let R S, F D & R P, P E be the homologous fides. ^.45. iTi- And becaufe X is fii^aU. to ED & < ED>(E D being=:V+ X). The fides R S. R P are < their homologous fides F D, F E, ^. Make then F N =R S, & F K = R P. F. y. Ba. i. And complete the pgr. N K. 7^.3 1. B*i- Book VI. Of E U C L I D. zss Demonstration. HE pgr. K N, being equal Sc fimilar to the pgr. X (ReMy%Xi6)i which IS itfelf fimjlar td the pgr. ED (Rtf^ 4), ''""'" o the pgr E D. peTsKN, ED Draw this diagonal F G 6» & complete the defeription of the figure. I. The pgr. KN is fimilar to the pgr E D. P.%\. B.6^ a. Wherefore thofe two pgrs. K N, ED, are about the fame diagonal. P.26. B.6. Since then the pgr. M I, is atfo about the fame diagonal F B. 3. It isfimiUr to the pgr. ED. P.24. A6. 4. Confequently fimilar to the pgr. T (Rtf. %), P.zi- BjS- But the pgr. D G being = to the pgr. E G Y^. 43. B, i)» >f Ac common pgr. M I be added on both fides, 5. The pgr MD will be r= to the pgr. EI. Jx.z.B.i, But tne pgr. A K being alfo ±^ to the pgr. E I (P. 36. B* 1). 6. The pgr. M D is ^ to the pgr. A K. AxiBa, And adding to both fides the common pgr. E G. 7. The gnomon ah c will be isr to the pgr. A G. Ax,i, B,i: But the pgr. £ D being = to the figures V & X taken together (Rifj^.), ortoV&KN, finceXis = KN f/Jr/ 5. &f 6) 5 iTKN be taken away from both fides. S- The remaining gnomon ah cz::iV. Ar.3. Au 9, Confequently, the pgr. A G is == to V (Arg. j). But pgr. A G has for defedl prr. M I, fimilar to pgr. T {Arg. a). io.Therefore, there has been applied to A B a pgr. A G rs V, deficient by a pgr. M I, fimOar to the pgr. T. D. 8. B.6. Which was to be done. SR E M A R K. EVER A L Editors •/ New Elements of Euclid have left out this propofition li the following^ as ufelefs ; hecaufe ther were ignorant of their ufe. Thej^ are not' nvithftanding ahfolutely neceffary for the anafyfis of the ancients^ correfponding to the analitic refolution of equations of the fecond degree. This XXFIlIth propofition corref ponds to the cafe^ 'where the laft term of the egum-^ tion is pofitive. For reducing the given f pace V to an equiangular pgr. T i letW isinli the ratio of the fides QJP, P R of the pgr.XiorT), m : n ; AB = a, A M = x W M B ::^a — x. Confequently, fince the defeS M I, fiould h^ fimilar to the pgr. T or to the per. X. C^P * PR = BM : Wfh (D.i.B.6). m : n :=z a—x : — {a — x). m And hecaufe the pgr. GA ( = MA. MG) /bould he equal to the given fpaceV (= » /), there r^its the following equation (P.z^. J5.6). — (tf— Af) X "^^V or n I. m Which is reduced to - xx—H^a * + V = ». m m Kfrfuhfiituting for V its value, Of multiplying hy mU dividing hy n. xx-^ a X -^ m I zn 0, Gg ^34 The ELEMENTS Book VI T PROPOSITION XXIX. PROBLEM IX. O a given ftraight line (AB), to apply a parallelogram (A G), eqoal to a«iveR reailineal figure (V), exceeding by a parallelogram (MI), rimilaito another given (T). Given. 7. 7he firatpht line A B, W tbe fp. T ;/. Jbtreathneal figure V. Sought. 7be confiruQion «/« pgr- A G, a^fUedn A B, equal to tbe reGtUtneal figm V, H bteuingfor ixcefs fl/jgr. lAi^^aaUer m T- Resolution. 1. Divide AB Into two equal parts in E. P.io. B.i. 2. Upon E B, defcribe a pgr. E D, fimilar to the pgr. T, &T 3. Defcribe a pgr. X (w P S) s= V 4- E D. fimilar & fimi. \ ^•'^- ^'' lafflyfituatcd to the pgr. T; Aconfequeliriy fimilar to tbe J pgr. ED {Jief^,P^\ .^.6)1 h let the (Mes tlS, FD, RP, FE be homologove. 4. Since X. (as i=: V+E D), is > E D; the fide R Sis > FD. U the fide R P > F E ; wherefore having ptodiioed FD & FE» makeFN=RSfirFK = RP;ftcompletetbc |i^. FKG N, which wiU be equal ^fimikr lo the pgr.X. P,^\.Mx. TDCMONSTRAT^OK. H'E pgr. K N bekig eoual and fimilar 10 -che pgr. X> whidi ia itfelf fimilar to the pgr. E D [JM. 3). I. The pgr. KN lefimtlar to the ftr. ED. a. Wherefore thofe two pgrs K N, E D are aboiit the fame diagonal. Draw this diagonal F B G, & complete the defcriptioo of the figure. Since X is±= to V -|. ED ; & Xrir pgr.KN ^*/.3. W4). 3. The pgr. KN = V + ED. Therefore taking away from both fides the common pgr. ED. 4. The remaining gnomon « ^ r is ^ tO ^e re^lineal figuce V. But becaufe A E = E B (/?#/! 1). The pgr. A K = the per. E I. Confequently, this 4)gr. A K it 3: 10 the pgr. NB. P.21.W P.a6. Mh^ I Bock VI. Of E U C L I D. ^35 TThcrefon! addinf* to both fulcs the common pt^r. M K .7. There will tefyli the pgr. A O ■— to the g.iioinon ^2 h c. yJx,2. B.i. But the gnomon nb c i*3=: 10 tliii* twCtUiacal ligurc V u^*-^. 4). I. Confequentiv, the pgr. A G ii rq to ihe rrailincal h'j;ure V. Ama.Bi. Since then this pgr. A G hag lor e;ie<<rs the p^r. M t, hmilar to the ^ pffr. ED (/*. 24. -6.6) ; & coDiif^iiently liruilar to the pgr. T (Jkef. 2. P. 21. ^.6). ^ There has been applied to A B> * pgr. A G = to the reailincal /igurc y, haying fur c xctfi^ a pgr. ^^ 1, fimilai to the pgr. T. Which WM t» U dMMk LR E M 4 n K. as in tbiffinping cafe \^ he made =r a, the giwti fquare V (reiut^d tp « jter. fjWM»guAjr /« //'^ ^gr.T) = ni} the ratio of tbefide$ O P^ P R (/ tfr«^ *«r, X ('twWfA M th^fame as that of the fides of the pgr, T; iw : « ft^ A M = jr, 3),. fequentiyy M B == x — ^. there nvill refult an equation of the fame kind. ForfimeihedtfeaUlfhould he /miUsr tf th ^. T 0r X, wv^H/ have at before the folmoitig proportion- Q>:PR = MB :}AG (D.UB.6J. m : n = ;r— a : Il(M^a). Andhecaufe /*^ /r- A G (= A M. M G)%<wW fc ||jm/ /f I& Ww^ (^z n ij, there refults the fottowing equation, ^ * #r^ * ^(x^aj xz:zV fR2^.B.6f which is reduced i9 i^xjr— «^«;r — V=:9 mm Jndfuhfiitutingfmr V its value n k ihen muitipljing h «-W dmiSsig hy n. xx'^ax^-mi^zo. From whence it appears that the XXlXth Prop, cprefpwds to the (M, in nuhifh fpa lajt farm of the equation is negative. ^ i 'i 436 r" The ELEMENTS Book VI. B A- E -J — "B T PROPOSITION XXX. PROBLEM X. O cut • given ftraight line (A B) in extreme and niean ratio ^ E). Given. Sought. Tbeftraighi line A B. ^^' /o^ii/ E, fucb tba ^ * BA:AE = AE:BL Re/olution. I. Upon the ftrtighl line A B defcribe a fquarc B C. ^46- ^-J- a. Appiv to the fide C A, a pgr. C D ^ to the fquarc B C. P.29^ Bb- whofc excefe A D is (imilar to B C, which will coofe- quently be a fquare. B Demonstration. ECAUSE BC^CD (lUf. z) i by taking away the common rgle. C E from each. The remainder B F = A D. B ^ ^. • AxyB.\. But B F is alfo fcquiangular with A D (P, i^. B. i). Therefore their fides F E, E B, ED, A E about the eoual angles, are reciprocally proportional, that is F E : ED = A E : E B. P.14. 5-6. ButFEi8=sCA(^P.34. -B.i;, or=io BA, &ED=:AE. D.30.B1 Wherefore, BA:AE = AE:EB. But becaufe B A is > A E (Ax. 8. B.\). The ftraight line A E is > E B. Confequently, the ftraight line A B is cut in extreme h mean ratio in E. Which was to be done. Otherwife. Divide B A in E, fo that the reQ. A B. B E be = to the D of A E. Demonstration. ECAUSE BA.BEiszrto the D of A E (Ref.)> BA:AE =AE:BE. Piy, BS And becaufe B A is > A E (Ax.SB.iJ. The ftraight line A E is > B E. P.14. B. Confequently» the ftraight line AB is cut in Extreme & mean ratio in E. />. 3 Which was to be dene. P.14. As- P.I I. B a- u J 3ook VI. Of EUCLID. ^37 PROPOSITION XXXI. THEOREM XXL X N eveiy right anded triangle (A B C), the redilineai figure (E) delcribed upon the hypothenufe (A C) b equal to the Turn of the (imilar and fimilarly defcribed figures (G & H), upon the (Ides (A Bj, B C) containing the right angle. Hypothefis, Thcfis. /. A B C /j « rsU, AinB. fig. E ::;=/^.Q+H. //. ^befiff. E is defcribed upon ibe fypoib. A C cfibis A. ///. And the figures G &f H are fimiiar to E, &f fimilmrfy defcribed upon ibe two otber fides A B, B C. B Demonstration^ CP20.-B4 \ Cor. 2. ^ E C A U S E the figures E, G, H are fimiiar, & fimilarly defcribed upon the homologous fides A C, A B, B C CHvp. %)* G : E = DofAB : nofAC.7 And H : E = DofBC : DofACj Conftquently, G + H : E =: D of A B + Q of BC : D of AC. P.24. S.5. Buthccaufe the A A B C is rglc. in B (Hyp, 1). TheDof AB + Dof BC i8 = to the D of A C. • P.47. ^.i. Therefore, the figure E is := to the figures G + H. C P.16. ^.c, XCor, Which was to be demonflrated. y The E L E M E N T S BooleVL I PROPOSITION XXXn. THEOREM XXIL _ F two trungles (A B Q C D E), which have two Odes (A B^ B C) of die one, proportion^ to two fides (C D» D E) of the othtr^ be joined at one aag^ (C), To u to have their homologous fides (A B, C D, B Q D E) paraDd lo one another, ihe remaining fides (A C» C E) fliall be in a ftiaiRht line. Hjpotheis. Th^. /. AB : BC s CD : DE. IheremMmngUMsK^^Q^^fA^Ck Ih Tbe AABCXDEy^rtjmneJinC. art in a firMi lint XY>. III. StihatABu^Ut.itCD, hf BC/*. /«DE. DSMOHSTRATIOK. B ECAUSE theplles^AB, CDarecntfay theftnught UaeBC, & the piles. B C, D E by the ftraijht line DC (ffyf, zj. I. The V B is :;= to VBCD * VD i8 = to VBCD. Rzg- B.u a. CoafeqneDtlv, V B is =toVD. i£».i.&i. Andbcfidcs AB : BC=:CD : DE(Hjp.i). 3. The A A B C, C D £ are equiangular. P. 6. BS 4. Therefore, V A is == to VDC £» being oppofite to the homologous fides B C, D E. Addioff then to both fides VB, or its ;? VB C P (^rg.i), together with the common V B C A. 5. The V A + B 4- B C A will be 5=: to the V DCE+BCD+BCA. Ax^z. Ai. ^ But the V A + B + BC A are =; to a L (P.^z- B. i). e. Confequentlv the V D C E + B C D + B C A are aUb=: to 2 U. Ax.i.B.i. 7. Wherdfbre the fbraight lines AC» C E are in the lame ftraight line AE. P.14.I.1. Which wu to be demonftratcd. J Book VI. Of EUCLID. »39 I - A • C I PROPOSITION XXXIH. THEO REM XXIIL N eqaal circles (A.I B C, EK F G), angle?, wether at the centres or cir-* cumferences (A M C, E N G or A I C, E K G), as atfo the fedora (A M Cin^ £ N G ff) have the (ame ratio with the arches (A iM C, E » Q) on wt&ich they ftanc^ have to one another. Hypothe(]9- Thefis. /. 7bt.®MY^Qytk?Gare^to one another. I. VAMC : VENG =:: AmC : EnG. //. The "rfat the centers AMQyE^G^ the i/. VA I C : VEKG zs, A«C : E«G. V at the OAIC, EKG flanJ upon the HI. Se&.AMCmi&ed.ENGfe±AmC:En(i. arches A ot C, E a G. I¥eparatw$L 1. Join Ae t&ords A C, E G. 2. In the O A I B<:, draw the chordsC^ b B &c, «ach := to AC« f& m tlK QiE K PGm paieil number of cosiit GH, HP &c, each=:toEG. 3. Diaw MD, M£ iBc, alibl»} H, N)F &e. B'DEMOHBTRATtOK. E C A U S E on one fide the cords A C, C D> DB, ^ on the tidtet the cords E'G', <5 H, H P are =: to one another ("fVi?^. a). 1 . The arches A^'C^C-o D, D B are all equal on the one fide, artbe arches E /»*G, G H, KF are on the other. 2- Confequcntly, the V AW C, CM D, DMB ftc, ft EN G, G N H, H N F *c, are ttlfo= to one another, on one fide ft the other. ^. Wherefore, Vhe V AM B ft the arch A C D B,areequimiilt. of the VAMC ft of the 'arch A »• C 4. Ukcwifc, V E N F &i:he archEGHF are eqoumilt. of V'E N (?, ft of the arch E «f 6. But becwffe the ® A»I B C, E K F G are equal (tfyp. i). Accordhig^*the-tfch AC D B is >, s= or < the archEGHF; V A MB Is ilfo >, = or < V EN P. 5. Wherefore, V A WC : VEN 6 = A w C : E « G. Which was to ije demoifftrated. i. Moreorer, V AM<:-i)eh^ donble^f V A IC, ft V ENG double of V E*GfP. ao.'B. 3;. ^6. tt fcWowf that V AMC : V E W3 = V A I C : V E K G. P.i 5. B.t. 7. Corfcqwmly, V AlC; YEKG= AsiC : EifG. P,\u b\ Which was to be demonftrated. 11. fof.l.SA P. I. J?.4^ F4fi.B.u P.i2. B43. P.27. ir.j. D. 5.^.5. 240 The ELEMENTS Book VL Prep. 4, in thearchea A C, C D, take the poinw « £#•, & join A i«, C «i j C P, D tf &c. P^f^i. Since then the two fides A M, M C are == to the two fides C M> M D (D, \<,B.\), & the V A M C. C M D are equal (Arg, %), «. Thebafe AC isss: to the bafe CD, & the AAMC =to the ACMD. P. 4. Moreover, becaufe the arch A m C is :=r to the arch C« D (Arg. i). 9. The complement A I B D C of the firft is = to the complement C A I B D oF the fecond. Ax.y 10. Wherefore V A « C is = to V C D. P.27. I f .Therefore the (egment A ot C is fimilar to the foment C • D. Ajc.z, P.a4. Ax.%. Beftdes they are nibtended bv equal cords (Arg, SJ» ii.Confequenuy, the fegment A ot C is = to the fegment C « D. But fincc the A A M C is alfo = to the A C M D (Arg. SJ. ij.The fedor A M C m is £= «o the fedor C M D c. Likewife, the fedlor D M B is equal to each of the two foTegoaiMF AMCm, CMDo. 1 4.Therefore the fedors A M C, C M D, D M B are = to one another. 1 5.1t is demonftrated after the fiime manner, that the fedors £ N G» G N H, H N F are s to one another. 16. Wherefore, the feO. A M B D C, & the arch A C D B are equimult. of the fed. AMC M» & of thesrch A mC, the fed. ENPHG, fltthe arch £ G H F are equimult. of the fed. £ N G a, & of the arch £ js G. But becaufe the A 1 B C, £K F G are equal (H^^. 1). If the arch ACD B be = to the arch £GHF,.the fedt. A MB DC is al(b= to the fedt. £ N F H G, as is proved bj the reafoning em- ployed in this third part of the demonftration to are. la inclunVely. And, if the arch ACDB be > the arch £ G H F, the (cO. A MB D C is alfo > the fed. £N FHG, & if le(s, leis. Since then there are four magnitudes, the two arches A »C»^ EitG» & the two fea. A M C OT, £ N G n. And of the arch A as C» & fea. A M C01, the arch AC D B & fed. AMBDC are any equi- mult. whatever ; & of the arch £ it G, & fedor £ N G if, the arch £ G H F, & the fed. £ N F H G are any equimult. whatever. And it has been proved that, if arch ACDBbe>,=ror< the arch £ GH F, fea A M B D C is alfo >, = or < the fea. £ NFH G. 17.It follows, that fca* A MC : fed. £N G=r A m C : £iiG« D. Which was to be denonftratcd. 111. B.U E,t. J? I. A3. B.3. ^3 5*1- J Book XL Of E tf C L 1 f). M* COROLLARY L _ H £ angle ft the center, is to four right angles* as the Urch upon which it ftandr, is to the circnoiference. For (F^. I J, V B C p : L = B I^ : to a quadrant of the O. Whexerore, quadrupling the confequents. VB.CD : 4L = BD : O. Pa^.S,^. C O R A L L A R Y II. ^ H £ arches £ P, B D of unequal circles, are GmiJar, if they fiib* tend equal angles C & C, either at their centers, or at their O (Fig. %}, For EF:OE»F==VECF:4L. 7 ./. . , But V B C D or V E C F : A L = B D : O B « D. j f ^- « v Confequently, E F : O E « F =3 B D : O B « D. P.u. -ff. j. Therefore, the arches £ F, B D are iimilar* C O R A L L A R Y III. W O rays C B, C D cut off from concentric circumfereqces fimilar arches F, BD (Fig. 2). REMARK, \7 is in conftqutnce o/ the ^portiohality ijiablijbed in Cor. i. that an arch tf a ircU (B D) it called the measure of its cwrefpondent angU (B C D.)j that is of the mtU at the center y jubtended by this areh ; tie circumference of a circle being the m^ curve, lohofe arches, increafe or diminipf in the ratio of the correfpondent an^ ies, about the fame point. The nvhoU cirde is conceived to be divided into 360 equal parts, which are alied DEGREES ^ and each of thefe degrees into 60 equal parts, called f INUTBS ; and each minute into 60 equal parts, called seconds (gc. in confe* nence of this hypothefis, y the correfpondence ejlablijhed betiveen the arches, W the mghs, nve are obliged to conceive all the angles about a point in the fame plane (that s the fiiin of 4 L» ;» ai divided into 360 equal parts, in fuch a manner, that the mgle of a degree is no more than the 360/^ part of j^ jjn^, or thecptb of a^,li con" rquentfy, of an amplitude t$ befuitended by the 2fi6tb part of the circumference. H h X l> 'TT' '^•q^! ' ?Y^^|?'~^pBM M^ H* BdokXL DEFINITIONS. SOLID is that which bath lengthy breadth and thfckoeTs. n. ^at wbkb bounds a Solid is a fuperficies. in. A ftraigbt line (A B) // perpendicular to a plane (P L) {Fig. i), if it be per. pendkular to all the lines (C D & F £), meeting it in this plane ; tbat if, The line (A B) will be perfrmdicular to the plane (P L), if it be perpendicular t9 the lines (C D & F E) wbicb being drawn in tbe plane (PL) pafs tbrougb tbe point (B), fo tbat tbe angles ( A B C, A B D, A B E & A B F) jr^ rigbt angles. m IV. A plane (A B) (Fig. z) is perpendicular t$ a ptme (P L), if the lines (D E & F G) drawn in one of the planes (as in A B) perpendicularly to the common fedion (A N) of the planes, are alio perpendicular to the other plane J'be common JeHion of two planes is tbe line wbicb is in tbe two planes : as tbe line (A N), wbicb is not only in tbe plane ( A B), but alfo in tbe plane (P L); therefore if the lines D E & F G drawn perpendicular to A ti in tbe plane A B are alfo perpendicular to tbe plane PL; tbe plane A B will be perpendi* €ular to tbe plane PL-' V. TS< inclination of afiraigbt line (A B) to a plane^ {Fig. 3.) is the acute angle (A'B E), contained by the (Iraight line (A B), and another (B E) drawn from the point (B), in which A B meets the plane (P L), to the point (E) in which a perpendicular (A E) to the plane (P L) drawn from any point (A) of the line (A B) above the plane^ meets the fame plane. ^44 The ELEMENTS Book XI n DEFINITIONS. VL HE incUttdtM af a pUm (A B) (Fig. t) to u plane (PU ; » the i angle (D E F) conltined by two ftraifpit Knet (E D & E F) drawn in each of the planes, (that is D E in the plane A B & E F in the plane PL) frocn s fame pobt (E), perpendicular to their cpmmon fedion (A I^ VIL Two planes an f aid tq bavi fbejame or a like incUnatian to one another, wM two other phnes bav^, when tbejr angles of inclination are equal. VIII. Parallel planee are fuch which do not meet one another tho' produced. IX. • Similar folid figurer are thofe which are contained by the fame nombcr of JTurfaice^y fimilar and homologous. : X. Equal ii Jimilar Solids are thofe which are contained by the fame number of equal, fimilar and honriologous furfaces. XI. A folid Angle (A) is that which is made by the meeting of more than two plaae angles (B A C, C AD & B A D), which are not in the fame plane, b one point (A). XII. A Pyramid (E B A D F) [Fig. %) is a folid contained by more than two iriangolar planes (B AD, B A E &c) having the fame vertex (A), and whofe bafes (viz. the lines E B, B D &c) are in the fame plane (£ B D F). j Book XL Of EUCLID. H5 A PEFINITIONS, xni- Prifm It a Told figure (A HE) (Fig. \.) conttined by plane figures, of which two that are oppofite (viz. GHIKF&BCDA^ are equal fimi- lar, and parallel to one another; and the other (kies (as G A« AK, KD.&c) are parallelograins. If the of>pofit€ parallel flafies hi friangks^ the prifm is ealled a frianguhr wn^ (and it is only of thoTe prirms that Euclid treats in the Xlth and Xllth Book)^ if the 9ppefiie planes are polygons, ibef an ealled polygon prifms. XIV. if Sphere is a Tolid figure ( A E B D) (Fig. x) irhofe fiirf^ce is d^fcribed by the revolution of a feiAicircle {AE B) about its diameter, which remabs un^ moved. XV. Tbe Axis of a Sphere b the fixed dianieter (A B) about which the fefolcircle revolves wbilft it defcribes the fuperficies of the fphere. XVI. Tbe Center of a Sphere is the fame with that of the femicirde which deicribed its fuperficies. XVIL Tbe Diameter of a Sphere is ai^y ftraight line which padfes thro' the center^i and is terminated both ways by the fuperfides of the fphere. 246 The ELEMENTS Book XI P5FINITIONS. XVIII, /\ Com is a fcJid figarc (A BCD) ("/%>. i, 2, Csf 3) dcfcribfd by the revolution of t fight angled triangle (A BE), about one of the fides (B E) Contaifting the tight anglCy which iide remains fixed. If the fixed fide (B E) of the triangle (A B E) (Fig. 2.) be equal to the other fide (A E) ccfntaining the right angle, the cone is called a right «ngled cone ; if (B E) be lets than (A E) (Fig.sJ an obtufe angled, and if (B E) be greiK- er than (A £) (Pig. \.) on acute angled cone. XIX. The Axli of a Cone is the fixed flraight line.(B E) about which the triao^ (A B E) revolved whilft it defcribed the (iiperficies of the cone. XX. rbe h§fi 0/ a Cof» is the circle (AG C D) (Fig. i J defcribed by that fide (B E) containing the right angle, which revolves. Book XI. Of EUCLID. 247 se= Fig.i B ^ig-2 \ZSA DEFINITIONS. XXI. J\ Cylinder is a folid figure (EBD F) CPig. i.) dcfcribcd by the rcvolatn (A N M C) about one of its fides (A Q on of a right angled parallelogram which renvains fixed. XXIL The Axis of a Cylinder is the fixed (Iraight line (A C) about which the paraU lelogram revolved^ whiUl it defcrilied the fuperficies of the cylinder. XXIIL- • The Bafts of a Cylinder (vvl. E N B, & F M D) are the circles defcribed by the two oppofite fides (N i^ M C) of the parailelognuDj revolving about the points A & C XXIV. Similar Cones and Cylinders are thofe which have their axes and the diamefen of their Bafey proportionals. XXV. JtVuie or Bptabedron (Pig, 2.) is a folid figure contained by fix equal fquares. XXVI. . jl Tetrahedron is a pyramid (BDCA) (Pig'S-J contained by four equal and equilateral triangles (^x. A B D Q, B AD, A D C & B A C). 248 The ELEMENTS BoclkXL DEFINITIONS. ' - XXVIL x\ N Oaabedron (Fig. I J is a folid figure conUiped bj eight cqaul «od equilateral triangles. XXVIIL A DoJUcbabedron (Fig.%,) u a (olid figure oontained b; twelve equal pcntagooi vhich are equilatenJ and equiangular. XXDC An lafabedron (Pig. 3 J is a folid figure contained by twenty equal and equi- lateral triangles. XXX. A Parallelipiped is a Iblid figure contaiiied by fix qtiadrilateral figures wbeit- of every oppofite two are parallel. XXXI. A Solid is faid to be in/cribed in a Solid, when all the angles of tt^ iaicribed folid touch the an^es, the fides, or the planes of the folid in which it b iufcribed. \ XXXII. A Solid is faid to bi drcumfcribed about a Solid, when the an^ksj^ the fides, or the planes of the drcutafchbcd folid touch all the aqglesof fbe inicribed iblafc TBmmmam EXPLICATION of the SIGNS. CO ..«•••• Similar. ..»•«.. Parallelepiped. Book XL Of E U d L I D. 849 ^ •G 1^ r y A" B y Z^ ZSf PkOPdSITIO^ I- THEOREM I. V/NE part (A B) of a ftraight line cannot be in a plane (2 X) ; and. another part above It. Hypothdis. Thefis. Ah is a pdrt ifaftraigbt line Another pfrt pf this ftrai^t^bne (as B C) /ituatid intbt plane \ nnill he in thefm Dbmonstratiok, ^plmntZH. 6 If not It will be above the plane as B D Is. Prfparation. I. At the point-B in A B erea in the plane Z ^ the ± G B. 1 t> a. At the point B in B G tred in the plane ZX the J. B C j ^'"* in. ECAUSEVABG isaL.» likewife V G B C, & they meet in the fame point B. I . The lines A B & B C are in the fame ftraight line A C. P. 14. B.i. But the line B D is a part of the ftraight lipe above the plane C^up.J* %. Therefore the lines B D & B C have a common fegment A B. 3. Confequenily, V D B G = V G B' A = G B C, /Atf/ «, the part =3 to the whole. Ax. \o.Ba. ^. Which IS impoilible. Ax. 6.B4U 5. Therefore, B D cannot be a part of the ftraight line A B (Arg. 1/ And as the fame demonftration may be applied to ahv otheiupart oFBC. It follows, that all the parts of a ftraight line are m the tame plane^ Which was to be demonftratfd« I i 2SO The ELEMENTS Book XL 1 / __-X ^c^ -B 7 / ^->^E^--^ / A. y ^ ^ '••••••••'•••••..f,.JIi3!r^ :£/ 2^ ^__ ^BB J^ PRaPOSITION U. THEOREM 11. W O firaight lines which cut one another in ^)ian ia one jifot (ZX> three ftraight lines which conftituie a triangle (EAD) are io thefiune plane (Z X). Hfpothcfit. Thcfik U.EADtsmA. II. TitvMt ^E AD i* in thfkm ZX. Demons TEATiov. If not. The lines ABfrCDarenotinthe fiune ^aoe, Likewife apattofthe^EAD, atAFGDi B. Draw G F. JPl^epariUktk. JECAUSE the partAPGD 9f tlie A EAD iswiwoM nlaae (Z X) with E FG (Sm^.J. I. ftfollow4» tiktttii«.ptmGD»CGof ikeliieCDaieui pknei, & the pftrt»AF,FBoftheikaig^lill• ABytreia planes, u alfo AE G D & F E G; 2* WJbick b impoflible. 3. SiKe tlien the pvu of the two Knea flr of the A am not be hi planea. 4. Thej Auft^Gonfaviiailf be ia the fiuae phuie* Which ^m to be dmoqftmcd I. ft tiw Jt I. I»ii. Book XI. Of EUCLID. 251 K^ A P L ^^X _ X* """y B I PROPOSITION III. THEOREM III. F two planes (R S >& P L) cut one toother, their common fedion is % ftraight line (A B). Hypothefis. Thefis. KS&?Lartiwoplams 9'bei^ commM feSim A Bi tvlficb cut M# oMtbtr. if a firm^i Hnt. DjBMONSTRATIOK. B. If it be not. The feaion will be two ftraight lines. As A X B for the plane RS $ & A T B for the pkne P L. ^E C A U 8 E the ftrtight Ones A X B & AT B hare the fame ex«> tremities A & B. 1. Thofe two fhaight lines AX B fr A T B inClode a ipace A X B T. 2. Which is impoffible. AK»iZf B,i. 3. Confequentljr, the io£Hon of the pUmet PLfrRScannotbe two fbaightlinesAXB&AYB. 4. Therefore their coounoo kOionf is a fbaight line A B« Which waa to be demonfirated as^ The ELEMENTS Book XL "1 PROPOSITION IV. THEOREM B^. X F two ftrtight linei (AB & C D) bterfed each other, and at the poial (E) of their interfedioii a perpendicuUr (E F) be ereded upon tfaofe Goes (A B & CI^ : It will be alfo perpeulicular to the plane (P L) which paflb through thofe Imet ( A B & C D). HyBOtheiia. Thdk. /. AB e^ CD art ftrnifbt finef EF is 1. i9 the jJam F L fituated in the plane P L //. 7bey interject each ether in E. ///. £F // J. /0 thaje lines at the feint B. I^-eparation. 1. Take EC at wiU» & makeEB, ED& AE eadi equal to EC 2. Joio the points A & G, alfe fi & D. 3. Thro* the point E in the feme plane P L, draw the ftraight line G H, terminated by the ftraight lines A C 8b B D, at the points G & H. 4. DrawAF, GF,CF, DF, HF *BF. TDemonstratioh. H E A AEF, CEF, BEF, * DEF have the fide E F common. ^The fides AE, CE, BE, &PE equal fPrep, \) & the adjacent VAEF, CEF, BEF. &DEF cqtial (Hyp, ^J. I. Confequentlj the bafes A F, C F, B F, & DF are equal. ^. 4. Ba. Inthe AAEC &DEB, the fides A £» CE, ED&EBare = (Prep. I.; ft the V A E C & t> E B alfo equal. P.ie. Ba. Therefore, AC = BD. \p Bi And V E A C = V E B D. ThcAGAE&EBH haye VAEGfcVHEB. yEAG=VEBH (Arg, 3.J & A E = EB (Prep. 1/ P.ij. Ai. J Book XI, Of E U G L I Dl 253 4. Confeqyentlv, the fides G A & G C are = to the fides HB Ir EH. P.26. B.t. In the A AF C & F D B, the three fides AF, FC & AC of the firft are = to the three fides F B, F D & p B^ of ^he fecond 5. Therefore> the three V of the A AF C are= to the three V of the AFDBeachtoeach,/i&ii/i/ VFAO= VFBH, &c P. ft. ^.i. The AGAF&HBF have xhp two ijdes A F & A G = to the two fides FB&BH fJirg,\.lij^). Moreover, VFAG = VF3H (Arg. ^. 6. Therefore, G F =r F H. A 4. J. i r Infine^ in the*ih G P F & F£ H, thefidc«GP» G£, & F£aK* ' .= to thefidesFH^EH, &EF (Arp^A.1^6). 7. Confequeatlv, the three V of ihe A GF £ ara = td tPke thm V of the A F E H, each to each, /An/ w V F E G = V F E H, «pc. 6ut'thofe V F^E G & F E H are formed I7 tfje ftraight line EF falling upon G H (becaufe G E & £ H are in the fame ftraight linel S. Therefore, thofe V F EG & F E H arc L» & F E J- upoa GH. ♦ ^'3- ^i- But H G is in the (ame plane, with ^hc lines A B & PQ (Frf^. 3/ I ^•'^- ^•'• And £ F is JL upon thofe lines (Hyp, 3/ 9. Confequently, E F is X ppon the fame plane P L. D. 3, 9 «t Which was to be demonftcatoil. P. 8. B.u »54 The ELEMENTS Boftk XL PROPOSITI'OK v. THEOREM V. 1 F thpee ^riiigbt \\x^ O C, B 0, 4r B £) nmt #11 in ont point (B), And m ftraight lifie (A B) U pirp«ndif:iiiitr lo inch of thesi in thtt point ; rhde three ftr^ght lifi^ (B Q BD, & B E) are in one and th^ fame plane (Z X). nypotbefi?. Thefis. /. BC, BD, WBE«ff/f«B.' BC, BD, W BE^# i« /i# fl. AB is 1, to thoftkmt. /ante fUn$ Z X. • PSMONaTEATlOK. |f not. One of tbofe three as B E is in a difierent plane. B, Let a plane T P pa(^ thro' the X A B & the line B E. _IECAUS£ TP&ZX ar^MTerent planea which meet in B. |. They will cut each other when producedt fr tipeir common fedion will be a ftraight line B P, common to the two planes. P. 3. i?.ii. ButABisXtoBD&BC (Hff.ii). a- Confequfsntljy A B will be alfo X to the plane Z X» in which thofe lines are. P. 4. 2?. 11. 3, Therefore, AP is X to B P & V A B P a U (^g^ O- But V ABE is a L (Hyp.w). And B £ is in the fame plane with A B & 6 P (frep, & Arg, i). 4. Confequentlj, V A B E = V A B P, lAo/ it, the part = the whole. $. Which is impoifible. Jx.S.B, i. 0. Therefore, B £ is not in a difiereat plane from that in which BD&BCare. 7.^Confequentl7y thofe three lines are in the fame plane Z X. A Which was to be demonftrattd. 1 Books. Of EUCLID. iS5 iPWH — X A h^-.. -■::>.. ••••."•'•.. •*,_ c -X „ ••:: ' X ; D / r [ PROPOSITION VL THEOREM VI F two ftfjight lines (A B * GD) be pcrpendlciihr to t plane (Z X), they (hall be parallet to one inoiHtr.^ Hmthefia. Theft. \Bi^ CD m^XutlmpLimZX. ABtfCD^/WiUL I. Join the points B & D in the plane Z X; a. At the point D in B D in this tame plane, ereft the X I>B. P.i i. J. i. ^. MakeDE=AB. 4. Draw AD, A£, &B£. P. 3* A I. B Demonstration. ECAUSE in the A ABD&BDE, tbeikje DEit=sAB (Prtp. 3.^, B D it common to the two A» & th^ V A B D 4c B D E are L (HyPfref. a. &f D. 3. n J The (kte A D is =: B £. P. 4. B. u In the AAB'E a«ADE, thofide A E is cooilDon^ AB issDE, & B E =r A D (Prtp. 3. W ^r^. l.^ Coafe^^encly, V A B E is=: V A D E. P. Z. B. u Bttt VAB&iiaL.. D. 3.^.11. Therefore^ V A D E is alio a. L.» ^jv. i. IT. i. But V C D E is a L. Z>. 3. J9.ii. Confeoaentiff Q E ivX to^C D, D A A DB /ifi^A /9v/k a. & ilr^ O^ Therefore, thofe lines C D, D A & D B are in the fame plane, tbat «r C D is in the plane which pafles thro' D A & D B. P. ;. B.i u Ltkewife AB is alfo in the lame plane which paiTes vbro' DA & DB. P. a- B.i u Therefore, AB & CD are in the fame plane. But the interior VABD&BDCareU (Hyp.) Confei^uently* A B is paraUel to CD. P.28. B. i. \¥ht(h waa to he demonftrateil. « 256 The ELEMENTS Book XL PROPOSITION VU. THEOREM ril. If two points (A & B) in two ptralleb (DC & FE) lie joined by m llnughf imc (A B) $ it will be in the &m€ plane ^P LJ with the ptrallels. Hypothefis. Thefis. /. A a B ar§ iwt points iakett ai win AB is inibefiute^ltmtTL, iniBtparaUelsEF a CD. VHtbthefUes. CD fiEF. II, A B is aftrasgbt line vsbicb j^ins dffe faints, Dbmonstratiok, If not, It will be in t difierent plane A G» as the line A X B is. X5 ECAUSE AXBisintiie plane A G» difibreni from the pfauie P L, & its extremities A & B arc in the lines C D & EP» fitvated in the plane P L. t. The line A X B will be common to the two planes* tkai if» A X B is the common fedion of the two planes A G & P L. i^ 3. ^.i i . But A B is alio a ftraight line having the fidne eitreaiities A & B (Hyf. ii). 3. Which is impoffible. £st.\z,M,i^ 4. Wherefore, the ftrsught line (A B) which joins the points A 1^ B, . is not in a plane A G different from that in which the parallels C D & £ F are. Therefore, A B is in the &ffle plane P L with the piles. C D & E F. Which was to be demonftrated« $ Book XI. Of EUCLID. 257 L "v ■». — .>J — agBaaaaaaaaaa ' aaaaaaagaaaggisaaaaa I PROPOSITION VIII. THEOREM Fill. F two (Iraight lines (A B & C D) be parallel, and one of them (as A B) is perpendicular to the plane (Z X) ; the other C D (hall be perpendicular to the fame plane. Hvpothefis. Thefis. / AB WCD ure plies, QT^isJLu the platu ZX. //. A B ij -L to the plane 7i X. Preparation. Join the points B & D in the plane Z X. Pof.i. S.t- At the point D in B I>, eredt m the plane ZX the XD E. Paz- B.t. MakeDE = AB. P. 3.^.4. Draw A D, A E, & B E. Pof.u B,u B Demonstration. £ C A U S E B D is in the plane X Z, & A B is ± to this plane fffyp. 11). I. VABD is a L. D, 3. A4. a. Confeqnentlv, V B D C is aifo a L. -P.29 ~ But V B D E is a L, D E is = A B (Pnp, 4. W 3.; & B D being common to the two AABD&BDE. 3. The bafc A D is = to the bafe BE. />. 4 In the two AADE&ABE, ABis^DE fPrep.sJ A D i= B E (^f"g: 3 J & AE common. 4. ConTcqnently, VABE=VADE. P. But V A B E is a U. D. Therefore, V A D E is alfo a L. Axi. B. i Confequently, DEisXtoBD&AD {Prep, z- tf Arg. <). Wherefore, D E is alfo -L to the plane pafling thro* thole lines B D &AD. P. 4. S.i, But A D joins two points A & D taken in A B & C D which are parallel {Hyp. i). Therefore CD is in the fame plane with A B & A D. i'. 7. .^.i i. 9. Confequently, D E is _L to D C, or D C is _L to D E» D. 3. B. 1 1 . Since then CD is ± to D B & ED (Arg. %, 13 9;. 10.C D will be alfo J. to the plane pafling thro' thole lines (that is) to the plane Z X P. 4. B.i i. K k I. 2. 3- 4- B.u 4. B.u 8. Ai. 3.-B.1. I 8. m r f t f t r I i 25t Tbe ELEMENTS Book XI XI. I r H -H / ' G ' W R c - - K u PROPOSITION K. THEOREM LIT. jPhE fan (ABI^ CD) whlA ve cxh cf than pmBd to the iknt ftr«f:htEne (£ F) ikiogh finnied b di&tctt {ikMi (SF & R F) aic pt- ralkl to occ another. Hvpocbegs. Tkefis. /. ABumf2«^X<wSF, &CD AUm f&.i9Ctk im tbt p!ane R F. B I. From rhe point H of die liae A B ia the plue F S let fidl a X H G upon E F. a. From tbe poiac 6 in the plane RF let fallthaXGK upon CD. Demonstratioh. P.u.#. I- _ECAUSEEGorEFisitoOH*GK (frtp.i.ft %). I. E G will be X to the plane which pi^es thro* thofe lines. P. 4. Alt. But A B is pile, to E F (Ny^ 2). a. Therefore, A B is X to the plane which pafb th^ thofe Ikes HG&GK. P.t.Eih 3« In like manner, C D is alfb X to this fiune plane. Therefore, the lines A B & C D being X to die frme plane 4. They are pile, to i f another. Which w%i t» be demonftrattd. P. 6.»it Book XI. Of EUCLID. 259 I PROPOSITION X. THEOREM X. _ P two ftraight lines (A B fr B C) which meet one another (in B) be pa^ rallel to two others (D E & E F) which meet one another in (E) ; and are not in the fame plane with the 6rft two; the firft two and the other two fl^all contain eqaal angles (A B C & D E F). Hypothefe. . Thciis. A B If CD men one amnier in B, in a V ABC fx =^ V D EF. Aifferent plane from thmt hi nahkh D E (^ » E F «r^» wiicD alfe meet §ne mnetber in E. I Pffparafiott. Cat ofFat will from the ftraight Hoes A B & B C the parts B 6 fr B I. a. MakeHE=BG, &EK = BI. 3. Join the points B £, G H, G I, H K & I K. Demonstratiok. P. i.B. I. Pof.uB. I. H E line B G being x=: & pile. Jo H E CPrep. 2. W Rjp). "G H win be = & pile to B E. P,^y B, 1 In like maxwcr, I K is = & pUe. to B E. C P. 9. 5.i i Confequtntly, GH is = & pile, tp I K \ Ax.i. B. i Thcretorc, G I ia = & plku to K H. P.33. B. i And becaufe in the A G B I & H E K the three iides B G, B T, & G I of the firft, are = to the three fideA H E, E K, & H K of the laft, each to each, (Prep, 2- if Arg. 4^. V G B I or A B C is = to V H EK or D EF. P.%,B.i Which was to be demonftrated -1 ■A 1^* z6o The ELEMENTS Book XI 1 A / B. X '•• / ""H / DV- 7^^- "-^ .G>^ PROPOSITION XL PROBLEM L O draw a (Irtight linp (A H) perpendicular to a plane (Z X} from a given point (A) above it. Given. Sought. /. rbi plane Z X. Tbeftraigbt line A H UtfaUfrm 11, A point A thrve ii, the point A, J- to the plame X, X. Re/olulion. 1. In the Diane Z X draw at will the ftraight line B C: 2. From tne point A let fall upon B C the -L A D. P.12. B, r J. At the point D in the plane Z X eredt upon B C the X DG. P.ii.B.i. 4. From the point A let fall upon DG the J- A H. P. 12. B. i. Preparation. Thro' the point H draw the ftraight line F £ pile, to B C. Demonstratiok. P.31.A I. 13 E C A U S E the ftraight fine B C is X to D A & D G (Re/.z.^^). 1. It wtll be J. to the plane which paiTes thro' thofe lines. P. 4. £.11. But F E is pile, to B C (Prep), 2. Therefore, F £ is alfo X to this fame plane which paifes thro' D G &DA. P8.A11. But A H is in the fame plane with D A&D G (P. 2B,iiJ& meets F £ in H (Re/. 4. U Prep). 1. Therefore, V F H A is a L- />. 3. B.iu And becaufe V A H D is a L (Ref. 4/ 4. A H is X to the two lines F E & D G fituated in the plane Z X which interfe£t each other in H. J. Therefore, A H is X to the plane ZX. ' P. 4. JB.it- Which was to be done. BopkXI. Of EUCLID. %6i fi D H X 1 \ •'• c \ PROPOSITION Xn. PROBLEM II 1/ R O M a given point (A) in a pline (X Z) to ered a perpendicular BA). Given. Sought. 1 ^oint A in the fUru XZf A ftraight ling B A dravonfhm th^ point XJLto the plane X Z, Refolution. 1. Take at will a point D above the plane X Z. 2. From this point D ; let M upon this plsMie the J. D C. P.ii. B.ii. 3. Join the points A & C. Pof.i.B. r. 4. From the point A draw A B pile, to DC. P.31. J?, i. Demonstration. [JE C A U S E the line A B is pile, to D C (Ref. 4;. And that DC is X to the plane X Z (Re/, 2). . A B will be alfo ± to the lame plane X Z. P. 8. Bai Which was to be done. a6a The ELEMENTS Book XI. I ^^ ^^^^^^ ' p O "^ i A .. c V B ^ ^ PROPOSITION Xlir. THEOREM XL if ROM tht fimie point (B) in a ^«tn plane (ZX) there csiniot be dn«B on the fame fide of it more than one perpendicular (A B). Hypotheto. Thrfia. A B il X 47/ ibifint B, n It is impoffihh f dram frwm iie the plant X Z. /•/»/ B antbir J. /• tbt pUm XZ^nthe fame /JethmiABh^ Demonstration. If not, There maj be drawn from the point B another X. B Pfepdration. From the point B ereft a J. B C difFertnt from A B. E C A U S E the lines AB & BC meet at the point B. They arc in the fame plane P O. Bnt they are each JL to the plane X Z (Sup), a. Conrequendy* the V « + ^« & ^ are each iL. 3. Therefore, V « + * = V ^> ^^at w, the whole = to the part. A, Which is impoilible. But A B is X to the plane X Z fffyp). 5. Therefore, B C is not _L to X Z. 16. Confequently, it is impoffible to draw from a point B any. other line on the fame fide as A B, that wilt be X to the plane X Z. Which waa to be demonftrated. DacB. f. Ax.9.B. I. Book XL Of E U jC L I a aSj * D ...."•■• ...■••■;.■' c" "•••■•:::',v. "B T 1^ X Y ::J PROPOSITION XIV. THEOREM XIL Jl LANES (ZX&TY) to which the ^me flraight line (A B) is per* pendicular ; are par»llel to one another. Hypothefis. Thefo. AB is X to tbeflan€$ XZ tif TY. n* plane XZ is pile, iotbi plane TY. V not, DEMONSTRATION. The planet X Z & T Y produced will meet one another. 2>. 8. B.iu Prtfar^tkm. 1 . Produce thephnes X Z fr T Y untM they meet in D C. 2. Take a point E in the fe^ion D C. 3. Draw£A&£B. DECAUSE AB is X to theplioeTY (Hyp.) k EBisin this plane (Prep, 3/ JP. 3. Ai t« . V ABE isaL.. ^. Likewife V B A E is a L. ^ Genfequently, the A B A E has two U- ^ Which impof&hlt. J^.lj.B. u ;. Frojn whence it follows that the lines A S & E B do not aieet one another, no mare than ike plaiws T Y & X Z. P. t. B.iu ;. Theuftve, thofe planes are pile. 2>. 6. ii^.ii. Which was to be dcmt»nftrated. 264 The ELEMENTS Book XI. • ^f^^ A -s::::^- — - ^G''^--.,,^^ ^^"^c r '''^-^ F ' L B ••••. i "'-, s E^^-.J. i iz PROPOSITION XV. THEOREM XUL J, F rwo ftrtight lines (A B & A C) fuuated in the fame plane (A X), and meeting one another (in A)» fas parallel, to two ftraight lines (D £ & D F) meeting one another, and *fituated tn another plane (D Z) ; thofe planes (A X & D Z) will be parallel. Hypothefis. Thcfis. A B 5^ A C fitunted in the plane A X The plane A X in ^vhich are the Una & meeting each other in A, are pile, to A B tf A C » pSe. to the plarn D Z D £ C^ E F meeting each other in D, tS in which arethe limt D £ (^ D F. fituated in the plane D Z. ' Preparation. I. From the point A let fall upon the plane DZ the X AG. P.I I. All. a. DrawGHpUc.toDE, &GLpUe.toDF. P.31. A i. B Demonstration. E C A U S £ the lines G H & G L are pile, to D E & D F (frep, %). They will be alfo pile, to A B & A C. And G L being pile, to A C. The V C A G 4. A G L arc rr 2 L. But V A G L 18 a L (f'rep, \). Confeqnently, V C A G is alfo a L* It may be demonftrated after die (ame manner that V B AG is a L. P. 9. Hi I. P.29. A I. , Therefore, G A is ± to the plane A X. But G A is alfa JL to the plane D Z (?rep. \). 6. Wherefore^ the plane A X is pile, to the plane D Z. P. A. B.IK Which was to be demonftrated. P.14.S.11 Book Xf. Of EUCLID. s65 K K X ^ \y 6 , ■ , ? D _^ PROPOSITION XVI. THEOREM XIF. 1 F two parallel planes (i& X & Y P) be cut by another plane (A B D C), the common feSiond with it (C D & A B) are parallels. Hvpothefis. Thefis. /. The planes ZX&FY are pile. The common fedions C D W A B //. Tbejf are cut hy the plane A B C D. are pile > If not, B Demonstration. The lines A B & C D being produced will meet fomewhere^ Preparation*, Produce them until they meet in F. foft%, B. t. E C A U S E the ftraight lines B* A F & D C F meet in F* I . TTie planes P Y & Z X in which thofe lines are, will alfo meet one another : (B A F being entirely in the plane P Y, & D C F entirely in the plane Z X). P. I. All* a. Which is impoffible (Hyp. i). 3 . Wherefore, A B & C D do not meet one another 4. Therefore, A B & C D are pile* 2>.35. B. i* Which was to be demonftratecf. "% V zB6 The ELEMENT^ fiookXI. 1 I PROPOSITION XVII. THEOREM JCF. F two fii^ight lines (AC «r BD) be cut by parallei pUnee (XZ^PY ArQM): they (hall be cut in the fame ratio, (tbsi h^ A£:EF = BF:FH &c). Hypothefis. TTicfis. /. A C W B D «r/ Miw firaigbi lines. If. Cut by tbi pile, planes X Z, P Y » QM. Preparation AE: EGsBP: FK t. Join the points A & B, alfo G &H H 2. braw A ft which will pafs thro* the plane P Y in the \p^. p , point I. r V- ' ' ' braw El & IF. J B Demoksthation. £ C A U S £ the pile, planet Z X & P Y are cut by ihc plane cyf the A A B H. I. A B is pile, to I F. P.lS. *ii. 2' Likewile, E I is pile, to G H. 3. Confcquently, A I : I H = B F : F H. 4. And, AI : IH = AE : EG. «;. Therefore, AE : EG = B F : F H. P.ii, M S- Which was to be demonftratei). P. a. A 6- Bookjq. Of EUCLID. a67 / ^^B / i ^ C E ] A* ^ PROPOSITION XVffl. rBEOREMXFI. X F a flraight line (A B) is perpendicular to a plane (Z X): every plane (as QJ£) which pafics thra' this line (A B) (hall be perpendicular to this plane (Z X). Hypothtfis. Theiis. A B » X /« /&# ^i^« Z X. £«;m /i^irf ^«j Q^E) 'which pa^ei tbr^ fbe ±AE is ± t0 tke piafK ZUL Preparation^ I. Let a plane 0£ pais thio' A B, which will cut the plane . ZX inEF. P. 3.5. f. 2» Take in this ftraiffht line E F, a point D at wilL J. From this point D, draw in the plane Q E, the line D C pile, to A B. P.31. B. I. Demokstratiok. XJeCAUSE the ftraight line a B is ± to the plane Z X, &DC wpUe. to A B rHyp. ^Prep. i). 1 . The line D C is X to the plane Z X: 2. Coniequently^ CD is alio X to the common feftion fi F. 3- Therefore, the plane E (^in which the lines A B & C D are, is X P, 8- JJ.ii. D. 3.5.11. ;quenf cfore, to the plane Z X. D. 4. J?.i i And as the fame demonftration may be applied to any other plane which paiTes thro' the X A B, we may conclude, 4. That every plane which paifes thro' this line is X to the plane Z X. , Which' was to be dcmonftrated. 268 The ELEMENTS Book XI. z^ D / 7'^ / C 3"^^^. F / ^1= PROPOSITION XIX, THEOREM XVIL JL F two planes (C D & E F) cutting one another be each of ibem perpn- dicular to a third plane (Z X); their common fedion (AB) (hall be perpen- dicular to the fame plane (Z X). Hvpothcfis. Theiis. /. Tbeplamt C U & E F tfrr X UtlmpkmZX, Tie C9mmmi feaim A B £f J. //. Tbfy cut ««# an^tbif in A B, /• ibe plane 2 X. B Demonstration. ^.11. E C A U S E CB, the common fedton of the plane C D with the plane X Z is alfo in the plane X Z. P, There maj be erected at the point B in C B a X ^P. 1 1. If. 1 1.) which will be in the plane C D (Mfp. tj P.i8. Bai. And becaufe the line F B the common feflion of the planes P E & X Z is alfo in the plane X Z. P. 3. ^.11. There m^j^ be ereded at the fame point B & at the fame fide with the foregoing another X which will fall in the plane F E. P. 18. B.ii. But from the point B only one X can be raifed. P. 13. ^11. Confequentlj, thofe X muft coincide* thai Ut thofe two lines muft form but one which is <ommon to the two planes. But thofe planes have only the line A B in common (Hjff, zj 'f'herefore A B is X to the plane X Z. . Which was to be demonftrated. Book XL Of EUCLID. ^69 PROPOSITION XX. THEOREM XFIII. J[ F three olane angles (CAB, BAD&DAC) form a folid angle A : any two of thofe angles (as B A D S^ C A B) ^re greater than the third (C A D). Hypotheiis. Thefis. /onw fifilid V A. Pemonstration. B CASE I. When the three angles C A B, </, W r -f- i are equal ECAUSE the V C A B, ^ Wf + *are equal. I. It follows that y C AB -f- ^will be > V ^ + ^- CASE 11. When of the three angles CAB, i/ &f r -j- ^ two as C A B C^ d are equals & the third c -|- ^ is lefs than either of them. Because vcABis>Vf + *. iT^V C A B + V ^ will be much > V c + ^. Which was to bp dcmonftrated jIx/^.M, u Ax.^.B. I, t ,1 i snjo The JELEMENTS IbokXl ,^ B ^. v*^ E ^ i/ ___ CASE III. When the three angles are uneqiial, & ^ -{-^ is > C A B or ^. R^eparaiion. 1. At the point A in A C make V A =c V C A B in the phrtte^ CAD. 2. Make A E ;:^ A B. 3. From the point C draw thro' E the (Iraight line C E D, 4. From the points C & D draw C B & ft D. P. y B.i Ba. JP*/,, HE ABCA&CAE hare the fides AB & A E equal (Pm^^J. "The fide C A common &V*=VCAB (Prep. i). Confequentiv, the fide B C is =: to the fide C B. P, 4. B.i Butinthe ACBD the fidca C B + B D are > C D. P^o.B.1. Therefore, if from C B + B D be uken away the part C B, & from C D a part = to C E. The remainder B D will be > E D. jixc.B. i. In the A B A D & £ AD, the fides AB & A £ are ^ (Prfp. %). & A D common. But the bafe B D is > the bafe £ D {Arg, 2). Therefore, \f d\%>'\f c. f^^^^ Ba. If therefore, V C A B be added one fide* & ita equal V^ on tl|e other. VCAB+//wiUbe> V* + f orCAD. Ax.^B,\, Which was to be demonftrat^d. Suok Xh Of E U C L I a a7i ^ PROPOSITION XJti. TtiEO REM XIX. /\l L the plane angles (B A C, C A D & D A B) which form a folic! mglc (A) ; are lefs than four right angles. Hypothefis. Thclis. The VBAC, CAD.WDAB Tir /Amr V B AC + CAD + DAB ^9rm a folid, V A are < ^\^. Preparation. 1 . In the (ides B A, A C, & A D lake the three ^ints B, C, t). 2. Draw B C, B D & C D. ?^i. S. i. 3. Let a plane BCD pafs thro' thoie lines, Which will form with'the planes B' A C, C A D & B At), three folid V ; viz. the fetid V B, formed by the plafle V C B A, A*B D & C B D; the folid V C, formed hf the plane VB C A, ACD&BCD, & infine, the folid V D^ formed hy the plane V CD A, A D B & B D C. 2).ii. An. BDEMONSfRXTtON. E C A U S E the folid V D, is formed by the plane V C D A, ADB &BDC. 1. the VCD A+ ADBare> VBDC •) 2. Likcwife, VABD + ABCare>VDBC. i P.20. B.u 3. And V A C B -f A C D are > V^ C D. 3 ^. Heftce.thefinplane VCDA + ADB-f ABD + ABC+ACB + A CD are > the three pflane VBDC + DBC + BCD. Bnt thofc three plane VBDC + DBC + BCDare = 2L. P.32. B.i. Therefore, the fix plane VCDA-+-ADB+ABD + &C. are > a L {Arg. 4.) But the nine V of the A B C A, C A D & D A B viz. the fiXAkea- d^ menlioned {Arg, ^.) & the three remaining VBACyCAD& D A B Hre together = to 6 L.. P.^z, Ai. If therefore the fix V C D A+ ADB + A B D+ A BC+ AC B + A C D which are together > 2 L hi taken away. 5. The remaining plane VBAC + CAt)+DAB will be < 4L. But thofe plane' V B A C, C AD & D A B form a folid V A. 7. Confequf ntly, the plane V whidh form a folid V A are < ^ L.. Which was to be dcmonftrated. 5 1 ^7^ The ELEMENTS SookXL I I ■^ I PROPOSITION XXII. THEOREM XX. _ F every two of three plane angles be greater than the third, and if die (Iriight lines which contain them be all equal ; a triangle may be made of the ftraight lines (D F» G I & A C) which fubtend thofe angles. Hypothcfw. "^ TTicfis. . /. Aftf two of the three given \f OjbyC^ A A majf he made cf the fira^ are > the third, ash^a^Cyva^ Umi G I, D F & AC^nJnck^ c>h,orh + c>a. tendthrfky, II. The/desKG, HI,DE, EF, AB& fi C tvhich cotitain thofe V, are eqiud. Demonstration. The three given ^ a^h^ti c are either eqaaU or unequal. CASE I If the V «, ^» (^ f be equaL X5 E C A U S E the fides which contain the V, are equal (Hj^^. i) 1. The ADEF» GHI&ABCare equal. P. 4. B. 1. 2. Therefore DF=GI = AC. 3. Confcqucntly, DF + AC>GI. Ax^ B. \. 4. Wherefore a A may be made of thofe ftraight lines DF» AC St G I. P.aa- B. i. CASE. IL If the given \f a,h,licht unequal Preparation. \, At the vertex of one oftheV as B, make VABL = Vii. Pzx.B. i. 2. Make B L = D E. P. 3. B. i. 3. DrawLC&LA. P^.iB. 1. B Demonstration. ECAUSE thetwo V« + farc> V*r^. iJ&LB=:HG = .B C = H I (Prep, 2. W H^p. 2.) 1. The bafe L C will be > G I. P.24. A t. ButLC< LA+AC. P.aaA 1. 2. Much more then 6li8<LA-|-AC. But L A i=D F (Prep. i. fcf P. 4. -B. 1). 3. Therefore G I is < D F + A C Ax.\. B. u 4. Confequeatly, a A may be made of the ftraight lines D F> A C & G I. Which was to be demoaftrated. Book XL Of EUCLID. 273 PROPOSITION XXm. PROBLEM III. X O make a fetid angle (P), which Ihall be contained by three given plane angles (ABC, D £ F & G H I), any two of tliem being greater than the third, arid all three together (V A BC+ V DEF+ V G H I) lefs than four right angles. Given. /. rAr^f VABC, DEF&GHT, tf»^M*»^ 'which art greater than the tbirdy tff V B -{- E> VH, VB+H> VE,& VE + H >VB. //. VB + E + H<4L. Sought. A fdid V P» antaimd hy tbi three fiam V B, E & H. Refolution. Take A B at will, & make the fides B C, D E, EF, GH & HI equal to one another & to A B. P. 3. j9. i« Draw the bafes AC, DF, & GI. Pof.i.B, i. With thofc three bafes A C, D F & G I make a A L M N fo C P.27. B. i. that NMbe = GI, NL = AC, &LM=:DF. X P.aa. Ba i. Infcribe the A L M N in a © L M N. P. 5. S. 4. From the center O, to the V L> M & N, draw the ftraight lines L O, O N & O M. At the point O, erea the X O P to the plane of the © L M N. P.ia, J?.i|. Cut O P fo that the D of L O+the D of P O be = to the Dbf A B. Draw the ftraight lines LP, PN & PM. M m i. i a 274 The ELEMENTS Book XL r Demonstration. X>EC AUSE POis J. to the plane of the Q LMVi (Ref.6,J I. The A P O L will be riVht angled in O (Rif, 5. W 8J a. Coniequently. the D of P O + the □ of O L is = to the D of LP. P.47, R i. But the D of P O + the D of O L = D A B, (Ref. y.) CJx.t. 3. Therefore the D of A B is == to the D of L P, & A B =z L P. < P.40. B. 1. 4. Likewifc P N & P M arc each = to A B. iCm: 7. BuiNMi8=toGI» NL = AC, &LM = DF, (Ref, ^). 5. Confequcntly, A N M P is = to the A G H I, A N P L =1 AABC, ALPM = ADEF, VNPM=VH, VLPnCp. S. i. i. i=VB, &VLPM=VE. j But thofc three VNPM. LPN&LPM form a folid V P- 6. Therefore a foiid V P has been made» contained by the three given plane V B, E & H. Which was to be done. i I. Book XI. Of EUCLID. 275 E A -.... '••- -v^l A •'.**••• '••. *'•»**. j^ c '■■-•• • ,^ ^/ **-.^ y^ B • G H PROPOSITION XXIV. THEOREM XXL In every paralWepped (AH); the oppofite planes (BD&CF;BE& F G ; A F & B H) are fimitar & equal parallelograms. Hypothefif. TheCs. In the eiwn O B F, /A* plan* B D « The ifpofite plants B D, C F, B E irJS4c^F,B£/.FG&AF/.BH. & FG, AF &BH«r. = &OB Preparation. Draw the oppofite diagonals EH & AG, alfo AC & D H. BDem6nstration. BECAUSE the pile, planes B & C F are cut by the plane A R C £ The line B A is pile, to E C P.i6. B.\\- Likewife C H i« pile, to G B. And the fame pile, planes B D & C F being alfo cut by the plane DGHF. . „„ , The line PG will be pile, to FH. I Uke^ifeAEisplle.toBC&DFplle.toGH. ■* " And becaufe thofe pile, planes (Ai^. i . a. &f 4.) are the oppofite ficles of the qnadrtlateraT Egures AECB&DFHG. Thofe quadrilateral figures AECB&DFHG, are pys. />-3S. fl. |. 2.' t ikewife the other oppofite planes BD&CF; AF& B H are pgrs. And fince A B & B G are pile, to E C & C H, each to each {,Arg. 1 .»»). ^ V ABGi» = toVECH. •P.io.Bn ^' ButABis = toEC&BG=CH. P.34. B. , HirLflf-.,-.!.. A ABGis=&OStothe AECH iP a ^ - o Thcreforcthc AABGis=&cutotheAECH. Rut the pgr. B D is double of the A A B G. 7 ,p ^ And the p|r. C F is double of the A E C H j ^^•^'' ^•'•) Tint thofe pgrs. have each an V common with the equiangular A. ronSluentTy, the pgrs. BD & ^ ^' It may be demonftrated after the fame manner that the * ^'== & CO to the pgr. C F, & pgr. A F Js=: & cu to the pgr .BD H. B. I. J. 6. ^. 1.5. 6. I Therefore the oppofite planes of a Q are = & co pgrs. W^^ch was to be dcmonftratcd. »76 The ELEMENTS Book XL 7\ s T " p' UJI O ;::.? ^ J M D ..J :::'-'X ^^^ G 7^ H Jv" ,. p! A F 1 L /^ ^ y •f "w ,B . K C I PROPOSITION XXV. THEOREM XXIL F a parallelepiped (B E D C) be cut by a plane (KIML) paraNclto the oppofite planes (A E F B & C G D H) ; it divides the whole iato tw) parallelepipeds ("v/z. the O BEMK & KMDC), which fliall be to ooc another as their bafes (B F L K & K L H C). Thcfis. Hypothefis. Tbi Ql B E D C w (iMded into i*wo Q! B M & M C, ^.y a plane K M, pile, to the oppofite planes B E & C D. rAfOBM:OMC = haJel^Q. Preparation. B 1. Produce B C botii ways, as alfo F H. A/2. *• i- 2. In B C produced take any number of lines z= to BK & CK: asBO&TOcach = toBK&CW=:KC P. 3 i^- > 3. Thro' thofe points T, O & W, draw the ftraieht lines TU, O P & W X pile, to B F or C H, until they meet the oth^r pile, produced in the points U, P & X. ?\\X i. 4. Thro' the lines TU, OP & W}{ let the planes TR, 00 & W Y pals, pile, to the planes BE & CQ, which will meet the plane AEDG inSR, NQ^& VY. Demonstration. E C A U S E the lines B O & TO, are each = to B K & C W = KC (?rep.z)h\\i^ lines OP, TU & WX pile, to B F or C H. wect F H produced, in the points, P, U & X (Frep. 3). J Book XL Of EUCLID. 277 The pgra. T P & B P arc = tojhe pgr. B Lj & pgr. C X = pp. K H. P.3 5. J?, i . N P pile, pile, to the lines ST or F U The folid OQ^EB will be a 0= & CO to the O B E M K. DAo.Bi\: It may be dembnftrated after the fame manner that the (olid TRQjD is = & CO to S) B E M K; alfo the folid CDYWi8 = &COto OKMDC. But there are as many equal QJ O QJl B, &c. as there are equal pgrs. OF, T P, &c. & thofe O together compofe the QI T E : more* over there are as many equal pgrs. OF, &c. as there has been taken ftraight lines, each = to B K, which together are = to T B. Conlequently, the O T E is the fame multiple of the Q) B E M K that the parts (T O, O B) of the line T B taken together, are multiples of the line B K. Like wife the O C D Y W is the fame multiple of the O KM DC that the line W C is of the line K C. Therefore according as the O T R E B is >, = or < the O B E M K, the line T B will be >, = or < the line B K And according as the QJ C D Y W is >, =or < O K M D C, the line C W will \)e >, = or < the line K C. Confequently, the O B E M K : O KMD C =B K : KC. D. e. B. 5. But B K : KC = bafe B L : bafe K H. P. i. B. 6. ThercforeOBEMK:ejKMDC;=bafeBL:bafeKH. P.ii. B, 5. Which was to be demonftrated. I 278 The ELEMENTS Book XI. t. PROPOSITION XXVI. PROBLEM IV. Jl\ T a given point (A) in a given ftraight line (AB), to make a UA angle equal to a given folid angle (F). Given. 7. A point A in aftraigbt lin$ A B. //. AfdidangkT. Souipht. At the fmm A, ajdidof^ = «i fir fM angU F. Refolution. Pit.Bu 1 . Prom anv point I in one of the feftions abotit the folid V F» let fall a X I L upon the oppofite plane G F H. 2- Draw L F, L G, L H, H I & G I in the planes which form the folid V. PWIi. B. I. 3. In the given ftraight line A B, take A M = F G. -P. 3. ^. i. 4. At the point A, make aplanc V M A D 3= the plane V G F H. P 2? 5 i 5. Cut off A D = F H. p[ r. b[ {, 6. In the fame plane MAD, make a plane V M A E = m the plane 7. Cut off A E = F L. P. I B, u 8. At the point E, in the plane MAD ere& the -L E C. P.it Bai. 9. MakeEC=LI, "^ P. ,. ^. 1. loDraw AC. P^/^i.B. 1. Preparaiion. Draw M E, E D, C D & C M in the planes, M AD, C AD & MAC. Book XI. Of E U C L I D. 279 B Demonstration. . E C A U S E in the A G F H & M A D, the fides F G & F H arc — to the fides A M & A D, each to each, (Ref. 3. &f 5.) & V G F H is :=t to V M A D, (Ref. 4). I. G H will be =i to M D. 1 P . » . i. Likewife in the AGFL & AME, GLisiirtoME. ]^' ^' ^' "* Therefore if G L be taken from GH & M E from M D. 3. L H will be =: to ED. Jx.^. B. i. And fmcc in the A L H I & E UC E D ii = to L H, L I =i E C & the V D E C & H L I, are U (^rg. 3. Ref.^ HD.y B. 11). 4. I H wiU be =: td C D, P. 4. B. i. Likewife in the A F L I & A E C, L I is — to E C, & L F = AE, bcfidcs VFLI & V AEC, arcU, r^^/.y.g. WD.3.J5.11). 5. Therefore F I = A C. P. 4. A 1. 6. It may be demonftrated *after the fame manner that G I is r::: UiQ. Since then the three fides H I, F I & F H of the A I F H are = to the three fides D C, A C & A D, of the A C A D (Arg,\, W 5). 7. V I F H will be = to V C A D. P. 8. B, i. 8. Likewife A G F I is = to the AM A C & V GF I =2 V M A C. Therefore the plane V G F H being =: to the plane V M A D^ The plane V l ^ rt = to tlic plane V ^ A Li (Ai^, And the plane V G F I = to the plane V M A C, (Arg, 8). lie plane V I F H = to the plane V C A D {At^. 7). Befides the plane VGFH, IFH &GFI, form a folid V F. And the plane VMAD, CAD&MAC, fimilarly fituated ai thefe already mentioned, form the folid V A. It follows that the folid V A is = to the folid V F. D. 9. JB.ti; Which waf to be done. \ 280 The ELEMENTS Book XL L H ( =s^= C A ^ f"™ ? 1 / ^y^ ^^^ D M ^ / /^ ^■^sssss^^ I B S=SSB 1 PROPOSITION XXVIL PROBLEM V. X O defcribe from a given (Iraight line (A B)^ a parallelepiped fimiUr, & fimilarly fituated to one given (H N). Given. Sought. /. A flraight line A B. Frtim A B /• defcrihe a QJ A F, tO //. T:he O H N. l^ fimilarly fituated U « Q|HN, \ejolution. I. At the point A in the line A B make a folid V C A D B> = to the (blid V H, or L H M I. 2. Cut A C fo that HI:HL=AB:AC 3. Alfo A D fo that H L : HM=r A C : A 4. Complete the pgrs. A E, B D & B C. 5. Complete the O A F. Demonstration. E:l Pad All. P.J I. A I. HE three pgrs. A E, B D & B C being C\J*& (tmilarly fituated with the three pgrs. H G, M I ft L I of the O H N, each to each (Ref, 1.2. 3.^4.^1). 1.S.6;. As alfo their oppofite ones. P.24. An. Confequently, the i\x planes or pgrs. which form the Si A F, arc C0> & fimilarly fituated to the fix planes or pgrs. which form t|(e given ^3 H N. Therefore the O A F defcribed from A B, is fimilar & fimilarly fituated to the given QJ H N. v 1>. 9. An. Which was to be dane. Book XL Of EUCLID. iU B H ( G A \ ^^-"-^^ r 1 D \ ^^^'"^X I i; I PROPOSITION XXVIIi. rUEORtU XXIIL ^ F • parallelepiped (A B) be cut by a plane (F C D E) paffing thro' tte dt^onals (FC & ED) of the oppoiite planes (BG tz AH): it (hall btf cot into two j^qual parts. Hypotheiis. Thefis. W# QJ A B is cut by a flaw F D ^he plane P D cuts the S3 A. B inti pafingjhrt^ the diagnnals F C & £ D /^o equal parts, •fthe eppojite pldnes B G & AH. B Demonstratiok. E C A U S E the plane F A is a pgr. f. 3- 4- 6' The fides E J & G A are = & pile. > ^24. Ukcwife CD & G A are 3= & pile. J KF. 9. Confequcntly, E F is = & pile, to C D. ( Ax.%. Therefere E D is = & pile, to F C. P.33. From whence it follows that P C D E is a pgr. Z>.3^. But the pgr. B C G F is = & pile, to the pgr. H D A E P.24. Confedvently* the A BCF & FGC are =:: & CO to the A HDE f P.^. &EDiA, t^-4' Moreover,' the pgrs. F E A G & G A D C, are = & CO to the pgrs. B H D C & B HE F, each to each. P.14. Therefore all the planes which form the prifm B F D are := & CO to all the planes which form the prifm D F G. Therefore the prifm B F D or B H E D C F is = & CO to the prirm DFGotDEFCGA. D.io. Confequently, the (Jane F C D E, cuts the O A B into two equal parts. Which was to be demonihated. All. B. u B. f. B, I. B.ii* B, t. B. I. B.tt4 Bff. N n 282 The ELEMENTS Book XL H M PROPOSITION XXIX. THEOREM XXffT. PARALLELEFIP£DS(HBfcKB)upQiiihefiuiielide(BD),ifld o^ ihe fame altitude (A E)» the infiftrag ftraight lines of which (A £» A I; BF^BL; DH, DK; CG» C M) are terminated in the fame ftraighi Ikm (I F» G K) in the plane oppofite the bale, are equal to one another. Hjpothdis. Thciis. //. They ba*ve the fame aitituJe A E. ///. The inftfting lines A E, A I, tic. oftvbkb, are terminated in tbe lines- I F» G K. B Dbmonstration. I. ECAUSE the pgraKC or KMCD, & HC orHGCD. have the fame bafe D C, & their oppofiie fides K D, M Ct 1^ D H» C Gy are terminated in K G which » pile, to D C (Ifyp. 3/ The par. K C is =: to the pgr. H C ^35, Tbererore if from thole equal pgrs. be taken away the coainoa ttapezium H M C D. The remaiodcrs, tnn. the A K H D ft M G C will be equal Jjt.y Likiewife A I £ A is =: to the A L F B. The p^r. K E or K H £ K is alfo ss to the ptr. M P or M G F L. BecauK they are each = to the pgr. D C B A, Ie& the pgr. H M L £, ^Z>.3o fef^.ax. A 11;. But ihe plane G B or C F is = to the plane H A or D E, & the Diane M B or L C is =: to the plaae K A or I D. P.z^. Coaiequently, the prifm H A K D is =: to the prifm G B M C O.ic Therefore h to thoie equal priOas the part HMCBLEAD be added. The prifm HAKD + part HMCBLEAD is = prifm GBMC + ptft HMCBLEAD. ButprifmHAKD + partHMCBLEAD = SIKB. And prifm G B M C + part HMCBLEAD = Q|HB. Therefore the O K B is = O H B. Ax.i. Which was to be demonftrated Ax.u B. I. A 1. B, I. i Book XL Of EUCLID. *«3 ok K O L yd Q/L.^ w~ lyMS^I /b %j/^ A D mMBBBB PROPOSITION XXX. THEOREM XXr. Parallelepipeds (FG HEDCB A &IMLKBC A) npon the fame bafe (A B C D) »d of the (kme altitude, the infifHng finright lines of which ( A F, A I ; D E, D M ; B G« B K; C H, C L), are not terminated in the iaiiie ftraight lines in the plane oppofite rhie bafe^ are equal to one anOr ther. Hypotbefis* Thefis. /. 7be^Hkhl.\ariup9n$h€jamih4if€KQ. SIFHC »s=OlLCA. //. nej bavi tbtfamt altitude* III. The infifiingftra^bt lints A F» A I, lie. ttre not terminateain tbtfame fthiigbt linet. Preparalion. I. Produce L K & F G until ther meet in P. z. Produce I M until it meets F G in Q^ a. And E H to O. 4. Draw QA, PB, DC & ND. 1 Prf.2. M. I. P0/.1.9, 1. B Dbmoksthation. ECAUSE the OFHCA&QOCA have the fame bafc A B C D, & their infifting ftraight lines A F, A Q^; D E, D N ; B G> B P ; & C H, C O are terminated in the lines F P & E O. I. ThcOFHCA is= totheOQ^OCA. P.29. ^.11. ^. Ukewife the O OO CA is = to Q! I L C A. 3. Therefore the Ql F H C A is = to the QJ I L C A. Axx.B. i. Which was to be demonftrated. s94 The ELEMENTS BDdkXn t. If* r. G A I ^ \ \ J \t \ L K C V q W\V \ » \ \ >• ■^I r N ^ E N p PROPOSITION XXXI. THEOREM XXFL Parallelepipeds (ki&nz) which m upon equal btfe H & N q )y And of the fame altitude, are equal to one another. Hypoihefis. Theiia. /. The e\^l V^Z^ have tbiir TArQlKI u =:/«/^0NZ. hafes K H £^ N q ecual- //. They have the fame altitude. Demonstration. CASE I. If the infiftiDg lines A G» &c. of the S5 K Ij i the mfiftint h'nes C M, &c. of the O N Z, are ± to their bafes i or if the inclinations of the infiiUng ftraight lines A G & M C are the fiune. Preparation* ll RODUCE NF, &makeF<i.== AH \p: i. B, u a. At the point F in F Q, make the plane V QF R = plane VHAK. P.23. H. i. 3. Make F R =: A K. 4. Complete the pgr. F QJSR. P.31. A 1. 5. Complete likewife with the lines F Qjk F Dj F R & F D, the pgrs Wi.JL.. Q^TDF&DFR 6. Compleip the O D S. 7. Produce the ftraiffht lines F q & R S until they meet in V S. Thro' the point Q, draw X 6Y, pile, to V a. 9. Produce C q« until it meets X x , in the point Y. ip.Complctc the O Z Q^& V D T X. P.31.S. I. P#/:a. A B ECAUSE the lines F Q^& FR are = to AH & AK. (Prep.ul^th And the V Q F R is =r to the V H AK (Prep. z). ^. The p^r. F S is = & CO to the pgr. K H '^. Z- It may be demonftrated after the lame manner that F T fc O R aie = & (0 to the pgrs. A I, & A L. the lP.iS.B. ID. i.B I. 6- pgr$. Bcxik XI. Of EUCLID. aSs Therefore, fincc the tfiice pgrs. F S, TT, & D R, of the S) D S are = & (0 to the three pgrs. A E, A I, & A L, of the K T, Aadthe remaining pgra. of the Q3 D S, likewife thofc of the QJ K I are =3 & CI9 to thofe already mentioned ; each to each. P.24. B.i i« 3. The D S, will be = & CO to the SI K I. Z>.io. B.i 1. The D X & O S, have the fame bafe D Q^ & their inClling lines F V &. F R, &c. are in the fame pile, direftions V S, &c. 4. Confequentljr, Ql D S ia = to the O ^ X. F.z<^. Bm. But the SI D S is 1= to the O K I (Arg. 3;. 5 . Therefore the S D X is alfo = to the OKI. Axa. B. i. The O M O is cut hj the plane F Z, pile, to the plane M N. 6. Confcquently, the bafe N a : bafe a Qj=: Qj M F : O Z Q. P-a^. J.'i i. The ^ Z )( is Gift by the plane D Q* pQe. to the plane Z T. 7. Confequently, the bafe F X : bafe q 0^=0 D X : O Z Q^ A25. ^.i i. But the pgr. F X is = to the pgr. PS. PjS- B. i. And the pgr. F S is = to the pgr. H K. (Arg. \). 8. Coniequentiy, the pgr. F X b =z to the pgr. HK. Ax.\. B, t, But the bafe H K is = to the bafe q N (Hyp. i). g. Hence the bafe qN = to the bafe F X, But the bafe qN : bafe q Q^=ei MF ; SI Z Q^r^r^.6/ And the bafe q O : bafe F X = O Z O : O D X. (Cnnv.Arg'j). icHence the bafe qN : bafe F X = QJ M F : Ql D }(. P.zz. B, 5. But the bafe q N is = to the bafe F X (Are. oj. ^ 1 1 .Confequently, the O M F >» == to the S) D X. P,i^. B. 5. But the O D X & K I are equal (Arg. %l 1 2.Therefbrc the O MF b = tothe SKI. Ax.i,B. i. Whicl^ w^ t9 be demonftrated. CASE II. u If the angles of inclination of the infiding ftraight lines, A G &c. of SI K I are not equal to the angles of inclinatioQ of the infifting ftraight lines C M, &c. of the SI M F. P O N the bafe K I, make a S^ baring its infixing ftraight lines, either J- : or equally inclined with the infifting ftraight lines of the Q5 MP, & in the fame direction as thofe of SI It I. And confequently, which will be equal to it (P. 30. B.ii). The remainder of the conftrudlion, & of the demonftration, arc the fame as in the foregoing cafe. E COROLLARV. SlU A L parMtUp'tfeds tvbicb have tbt fame aliituJe, bave equal iafeh 286 The ELEMENTS Book XL 1 PROPOSITION XXXIl THEOREMXmi. J: ARALLELEPIPEDS (BDAEP) wfudi hm cqol dduds (B C & F O), are to OK another at their bafes (A K & E G). Hypothefis. TttJUuuks B C {^ F O. »ftie 0BO»EP. « B Thcfis. e:BD:eEP=ii5/>AK:A«j&EG JVeparalion. u PftKSucc E P to M P-/i.^» 2. Upon F G with F M, make ihc pgr. F L =r prr. K A, which will be in the tame diredion with the pgr. b 6. So that the pgn. £ G & F L together, form the pgr. EL f .44 ^ ' 3. Complete the QJ F I. Demonstration. ECAUSE thebtfeFLoftheOFI. is ^tothebafe AK oftheOBD (Prep. i). I. The O F I ii = to the O B D. ?,3iJ.n a. CoDfeqoemlj, eiFI:eJEP = SIBD:eiEP. ?. 7.i5 But, eiFI:.eiEP = balcFL:bafcEG. f^jlu. And the bafe F L is = to the bale A K (Prtp. 2), 1. Therefore, BD : OEP = bafe A K : bafeEG. (P.itJil' u Which was to be demonftnted. 5- Book XL Of EUCLID. PROPOSITION XXXIII. tBEOkEM XXFUL 1^ I M I L A R parallelepipeds (E B 5r F H) are to one another in the triplicate ratio of their homologous Tides (A B & O H). Hvpothefis. Thefis. Ti&* O E B & F H izrf ftj, Cff /i» Tbe^EB is to the^ FKift tbi fijfs A B y G H /ir^ bomohgous. tri^icaU ratio of A B /d O H, or of A/z. AB^;GH^.* Preparation, Produce A B & make A R :i= G H. |p. From A R defcribe the O R L = & CU to the S)F H, fb that the lines A C & A I ; D A & A K be in the fame ftraight Une. P.27. Complete the O A O, fo as to form with O ^ L the OOK. Complete likewife the QJ A F, fb as to form with |3 O A, the 00 C, & with, tte O E B the S P B. Ba u B Demonstration. 2)- 2. 3- 4- AL AK. AK. i;. 9. E C A U S E the (SI E B & R L, are CO (Prep, The pgr. A M is f\j to tHe pgr/ C B. Confequently, A B : A C = A R : A T. Aodalternando AB : A R = A€ : Likewife AR; AD = AR; Andalternando AB : A R = A D ; And (ince A R is = to G H The three ratios A B : A R, AC: A I, & A D to one another & equal to the ratio of A B to G H. But the ^ P B IS cut by the plane A E (Prep. 4/ Confequemlv, the bafc C B : bafe Q^A = O B E And the bafe C B : bafe Q A = A B : A R. Therefore AB:AR=:OBE:eiAP. Which was to be demonftrated. • Zee Cor, z. of this propofition. D. D. :P.i D. P. i.B.^. 16. B. 5. i.B. g. 16. B. 9. AK, are ( tqual :OAP. P.. P. P. z$. B.it. I. B. 6. ii.B. 5. a88 The ELEMENTS Book XL 1 K \ZZJZ 5 ;^H The Q) O C w cut by the plane R D (Pr#/. 4). 9. Confcqucntlv, the hafe R C : bafc A M = Q) A P : O O A. P.a^. ^ i r . And,th«baieRC:bareAM=:AC: AI. P. 1.5,6. loThcrcforc, AC : A I = Q) A P : Ql O A. P.ii. A ^ Infinc, the O O K being cut by the plane A M (Pr«/. 4). ir.It may be demonftrated after the lame manner. That AD:AK = OAO:iSIAN. But the three ratios AB:AR»AC :AI»&AD : AKare=r to the ratio A B : G H {Arv. 6). i2.Confequent!v, the fourO BE, A P, A O, & A N form a (cries of ma^ituaes in the fame ratio (A B : G H). P. it. £. ^• f 3. Therefore, they arc proportionala. ^. 6. 5. 5- 1 4.Con(equently, the S) B £ is to the A N in the triplicate ratio of AB toGH. D.ii.B 5. But the B E is to the F H in the triplicate ratio -df A B • toGH, (or as A B» toGH»).* COROLLARY I. J} RO M ibis it is maniftfty thai if four ftraigbt lines he continual pr9fertiuuk^ ms the Jirft is to the four thy fo is the paralleiepifed Jefcrihed from fheprfi to the fimilar Ij Jimilarly dejcrihed parallelepiped from the f' — ^ ^ '"- -^- ^^ " ' ^' ^^ ROM this it is manijj ms theftfji is to the fourth y ^ ^ ,^ _, ^ --/v- - •>- fimilar li fimilarly dejcrihed parallelepiped from thefecond i hecaufe the prfi fire^St line has to the four thy the triplicate ratio of thai which it has to the fecomd. A ^COROLLARY IL ^ ^LL cubes being fimilar parallelepipeds (D. IX Sc XXX. B. 11), fimnUr fa- railelepipedf ( A B C? F H) «r/ /• one another as the cubes of their homoiegoms Jtia (A B.e^ GH) (exprejfcd thus A B« : G HV 1 t^caufe they are in the tr^Aate ratio of thofe fame fides. Book XI. DF E U C L I Dj 289 J i ^ TJ ■ 1 -1 ■ ■ II 1 ■ V / /^ N r 1 D K P Q 1 s /^ o / G H M [_ C Z^ / A BBB 1 K . J] PROPOSITION XXXIV. THEOREM XXIX. _ H E bafes, (pgrs. A C & I L) and altitudes (G B & I R) of equal pa- rallelepipedsy (AD & IV) are reciprocally proportional ; and if the bates, (pgrs. AC ^IL) and altitudes (GB &*iR} be reciprocally proportional, the parallelepipeds are equal. Hypothefis. Thefis. [S).AD ii = /o ^I V. BafihQ : i^/r IL=:fl//.IR :«//.GB. I. Demonstration. The given parallelepipeds may be either. C A S E I . Of the fame altitude ") ^ . .«„«i|« ;nri;««,i «« ^v^w k«r-* CASE z. OfdifFerentaliitudes j and equally mclmed on their hafes. CASE 3. Having different inclinations : as if one was J. to the bafe^ and the other oblique. B. CASE I. When the SJ have the fame altitude, that w, I R =i G B. >E C A U S E the given (3 are equal, & have the fame altitude. 1 . Their bafe^ are equal (Cor. of P. ^1. B. 11 J. %, Therefore, the bafe A C : bafe I L ==: altitude I R : altitude G B. Z>. 6. B. 5. CASE II. When I R 13 > G B. O cf 290 The ELEMENTS BbokXI. 1 MH ■■ "■ T " V r/ /" I \ D K P 9 s 6 ^» i/ / C N r r G H T M y V y / A B 1 £= saa B /. PreparMtitm. 1 . From the ah. R I, cut ofF the part P I = to the alt. B G. 2. Thro' th« point P» p«6 the pkne PO NQ^ pile to the bafelL. 7 1. B E C A U S £ the parallelepipeds A D & I N have the fame ahi- tudc (I Prep. v). ThcO A D : O I N = bafc A C : bafe I L. P.^x. E.ix. fif^irheO'ADi»=tx>^thcQJI V (Hyp). Therefore. OAD:e)IN = 0IV:OIN P. Confcqucntly, O I V : (3 I N r=l>aie AC : WS I L. Pa The O I V is cut by the plane P O N O (L Prep, %). Therefore, eJPV:(l3lN=:UePS: baftltP. Therefore, coraponendo e)IV:OIN = bafeKR: faiA K F. /lii. A c. But the bafe K R : bafe R P =: R 1 : P F. P. i. A 6. Wherefow, OlV:eriN = KJ:PI. F.u*. A But, ej I V : (31 1 N = bafe AC : bafe I L (An, 3/ And PI = GB (I. Prep, 1). Confcqucntly, bafe A C : bafe LL == I R : B G. />u. IT. ^. CASE III. When iht 1 V hoea difierent.inclination fiom tlltf O A D. IL Preparation. Dcicribe a S5 of the fame altitude with the f^ IT, hav- ing the fame Inclination as the ^3 A D. ^.11. 5- E C A U S E the defcribed O. has the fame baie & the fame altitude with the O A D (lU Prep)'. This O will be = to the given (3 IV. P 31. ^.11. But this defcribed ^i is in the reciprocal ratio' of ite'tsie, Aro£ its altitude with the O A D (Cafe II). Therefore, the E3 1 V will be alio in reciprocal ratio with the O.AD. P 7.S. 5. Which was to be den)ODftnite<L Book XI. Of E U C L I D. 291 Hypothcfw. Thcfis. Bafe iLilafe AC = aJi.QB: ait. ^K. O A D w =: Q) I V, U. Demonstration* Tie ^€^arati§H is thtftune m$ f^r the faregnng cafin B. ^ BECAUSE the0IN&ADhavethertiiietltittKfer/./>iv^.i). 1. TbeQIIN:OAD = bafeIL:l»fcAC. -P.t2.A11. But the bafe IL : baie A C c= alt. G B : alt. I R. (HyO- i. Therefore ej I N : Q) A D = alt. G B : alt. I R. P.\\,B. 5. And as P 1 is = B G. //. Prtp, i). 3. The O I N : ei A D = alt. P I : alt I R. P. 7. B. c, ButPI:IR = pgr.PK:pgr.KR. P. i.B. 6. And pgr. KP : pgr. KR = O I N : QII V. P.32. B,ii. 4. Therefore theQIIN : QIAD = e)IN : QJIV. P.ii.iP. 5. But the (3 I N is the firft & third terms of the proportion* 5. Confequently, the Q) A D is = to the O I V. P.14. B. 5. Which was to be demonftrated. *rbi demonftrations of the firft and tbird cafes in this bypotbefisy art tbe fame^ far ^bicb reafon tve bave omitted fbem. JVf R IB U A R K L HA T bos been demenftraied in tbt fropofiiwHt 25, 29» 30. 31, 32, ^^ W 34, concerning parmtUlipi^ds^ is mlfo true wtb rtffeB to triangular prifms ; becaufe fucb aprifm is tbe half of its fiorattelepiped i (P. »8. B. 1 \,) from ^whence we maf conclude, \, If a triangular trifm he tut hy a plant pUt* to ibt opptfite planes \ tbe ttvo prifms refulting from thence ^ tvill it to one anmbtras tbe parts of tbe pgr. y bafe of tbe lobole prifm, II. Triangular prifms wbicb ba^e tbt famty or tqual bafts, W bave equal altit tudesy are equal. IH. Triangular prifms ivhicb ba<ve tbe fame altitude, art to ont anotbtr as tbeir bafes, IV. Similar triangular prifmsy are to one anotbtr in tbe triplicate ratio, of tbeir bomologous fiius. V. Equal triangular prifms^ bave tbeir bafes and altitudes reciprocally proportional^ If triangular prifms ^vcbofe bafes and altitudes, are reciprocally proportional^ art tqual. 292 The ELEMENTS Book XI. IV. REMARK II. I T H the fume przp^rtia /•-/''v are endued^ ^bufe cff^fite f lanes en r-JHJ. Since ii his be^n demonrtratcd, (P. 20. B. 6 ) that ihofc oppofilc * ilar polvgoni mar be divIJcd inio the fame number of (imilar triai^lcsj ilicrt- fore if thro* the homologous diagonals which form thole triangles, planes, be pafTed : thofe planes will divide the polygon prifms, into as maoj trtasgultr prifms as th^re arc triangles in their oppoliie it pile, planes. But what has been obierved in the foregoing remark, is applicable totbofe rrlangular prifms Confequenily, we may conclude (P. 12. B. 5 J that ftljg>* p'-ifais are endued ^vitb the fawu f'roperties. kwkXI. Of EUCLID. 293 [ PROPOSITION XXXV. THEOREM XXX. F froxp the vertices (A & H) of two equal plane angles (B A C & I H L), here be drawn two ftraight lines (A D & H K) above the planes in which he angles are, and containing equal angles (VQAD==; VIHK "& / D A C :^ V K H L), with the rcfpeaive fides of thofe angles, (viz. A D vith A B & A C ; H K with I H & H L), and from any two points (D & K) n thofe lines, (A D & H K), above the planes, there be let fall the perpcn- liculars (D E & KM), on the planes of the firft nanied angles (B A C & \ H L), and from the points (E & M), in which the perpendiculars meet hofe planes, the ftraight lines (A.E&HM), be drawn to the vertices A & H), of the angles firft nanied : thofe ftraight line? (A E & H M), ftiall rontain equal angles (D A E & K H M), with the ftraight lines (A D & H K) vhich are above the planes of the angles. Hypothefis. Thefis. /. Abwji the planes of the equal >f^ AC^ mUli from VDAE=pVKHM. their vertices A ^ H, there has been dra'wn A D W H K, containing VB AD l^ DAC= VIHK W KHL, each to each. 11. From the t*wo points D y K, in AD W HM, there has been let fall the JL'D 6 y.K M, (?« the planes B A C W I H L. 7/ From the points E y M, ivhere the -L meet thofe planes, there has ieen dra^n A E y M H, /o the ^vertices A ^ H. Preparalion. 1. Make A F i^: HK. 2. Draw FG, pile, to D E, until it meets the plane BAC in G. 3. From the point G, in the plane BAC, draw C G, ± jo A C ; & G B, JL to A B. 4. From the point K in the plane I H L, draw I M, X to HI J &ML, jLtoHL. 5. Draw BF, BC&FCilK, IL&LK. P.31. B. P.12. B. I. Paz,B. Pofi, B. The ELEMENTS F. S.JLil. ^■47 Ax.i. 7^.48. B, I. Dc MOIISTR ATIOir. i5ECAUSE FGisplle. to DEwkickkJulodieplucBAC The line G F is Ju to tlie tuue pkae B AC. And the V FG B. F G A ft F G C ve U 2. Confcqvenly, the O of A F is = to Q of FG + O of G A. But theD of AG 13= to a of AB -f D of BG. fPrtp.^), » 3. Therefore, theDof AF is = to C FG + D AB + D BG. But the D G B + D F G arc = to the D B F (Prep,^). 4. Confequentlj, the D A F is alfo =: to the O B F -f O A B. 5. Therefore, V A B F, is a U 6. It may be deoionftTated after the (ame manner that V F ^ A> is a L.. 7. That atfo the V K I H ft K L H, are U In the A F C A & K L H ; the line H K is = to A F (Prep. 1.) the VACF &KLH, areL. r^r#. 6- W 7), & the V F AC = VKHL, ^/^^i). P-26^^. Therefore the fhJes AC & CF are= to the fides HL& LR, each to each. 9. Likewife A B is = to H I & B F = I R. joConfcquentlj, in the A B A C & 1 H L ; the bafes B C & I L are equal and the V ACB k A B C =: to the VHLI ft HIL, each to each. P. 4. B. Therefore if thofe equal V> be taken from the four L A C Gp ABG, HLM &HIM. II. The remaining V will be equal, vi». VBCG= Vl^M & VCBG=VLIM, Jx.^,B. Since then the A G BC & IML have their bales B C & I L equal 1.1 1. J. I. J. I. B, I. ir. I. 8. (Arg. 10). And the V at thofc bafes are equal, each to each, {Jff. 11). W i2.Thc fides B G & C G will be = to the fides I M & M L. In the A B A G & H I M, A B is = to H I (/frg. o.) BG = I M» fAri^. I2.)^the V ABG& HIMareL. r^rr/. 3. £# 4). P26. B. I Book XL Of EUCLID. ^95 ft.ConfmiemJyy A G = H M P. 4* B. i. Bui tht D df A f (= D A G 4 D G F) C^rg. a.) is = to the D of HK (£sDHM+ DKM)r//jr/. i. W P. 47. B, i.) be- caufe A F is ^ M K. rPr$^. i}. If therefore ff&m the D A F be taken the D G A, A from the D HK, the D HM = D G A, fJrf^, I3.WP.46 B. i.Or 3). f4.The remaittdel, <tAsi. the D of G F will be = to the D of K M. Jx.y B, i- I^.Confequeniiy, G F = K M (Cor. 3. •/'P. 46. B. i). Infine, becaiUe i« Oi* iw» AAGF A HKN4, the fides AP, AG &FG are Wiethe iide»HK,HM& K M» etch ta each, (Pn^. t. W Arg. i> fg i€). i6.The V F A G or D A E IS =: to the V K H M. P. 8. -ff. i. Which wa5 to be demonftrated. COROLLARr. X P A'"' '^^ vertices A Sc H of fvuo eqnal tlane angles B A C & I H L, there be elevated t'w$ e^tml ftraight lines A F & H K$ containhig with the refpeQive fides^ /^rVBAF&FAC equd t$ /i^# V I H K & K H L; esch t$ each, ^ there ie let fall from thofe points F & K (of thofe elevaied ftraight lines) the perpendi- culars Y G tt ¥iU on the planes % AQ & IHL: thofel^YG & KM tsfm ht efual, (Arg, 1 ^J. The ELEMENTS Book XI. 1 I PROPOSITION XXXVI. THEOREM XXXI F three ftra'ght lines (A, B, C) be proportionals, the parallelepiped (D N), defer! bed from thefe three lines as Us fides, is equal to the equiangular piralie- lepiped (£ I), defcril>ed from the mean proportional (B). Hypothecs. /. 7heftraight lines A, B, &f C are fropprthnais, that is, A : B = B : C. //. The SJDN, is defcrihedfrom thofe three /i»«,that is, DK = A, MK=B, fef KL = C. ///. 7he equiangular S) E '» " defcrihed from the mean / 1 oj ortional B, that is, EF:=FGz=:FH=B. Thefis. B Demonstration. ECAUSE DK:EF = EForFH:K.L ("Hyp. 2). And the plane V E F H is = to the plane V D K L fHyp. 3). 1. The pgr. D L, bafe of Ql DN ii =: to the pgr. EH. bafe ofQ) EI P. 14. B. 6. Moi cover, the plane VGFE&GFH contained by the elevated line F G, & the fides E F & F H. being = to the plane V M K D, & M K L, contained by the elevated Tine KM, & D K> & K L, each to each, (Hyp. 3.;,,& F G :i= KM, {Hyp, 2. ^ V- - 2. The X let fall from the point G, on the bale E H, will be = to the -L let fall fiom the point M on the bafe D L. CCor. of P. s^. S, 11;. 3. Confcquentlv, S) E I has.the fame altitude with (he D N. D, 4. B, 6. But the bafe E H of Q) E I is = to the bafe-D L of O I> N, f^rg. i;. 4. Therefore, O E I is = to the D N. P.31. B,ii- Whkh was to be dertionftrated. Book XI. Of EUCLID. 897 I PROPOSITION XXXVII. T H EO REM XXXU. F four ftraight lines ( A, B, C, & D ) be proportionals, (that is, if, A : B = C : D) : the fimilar and fimilarly defcribed parallelepipeds, from the two firft (A & B), will be proportional to the fimilar and (imilarly defcrib* cd parallelepipeds, from tbe two laft (C & D) ; end if the two fimilar and Similarly defcribed parallelepipeds, from the two lines (A &r B) ; be propor- tional to the two other fimilar and fimilarly defcribed parallelepipeds, from the two other ftraight lines (C & D) ; the homologous fides of the firft (A & B)^ will be proportional to the homolcgous fides (C & D) of the iaft. Hypothefis. Thefis. /. A : B = C : D. O A ; OB :^ QJC : 0D. //. From A W B there has been defcribed CQ Q). ///. Alfifntm C W D. B Demonstration. E C A U S E the Ql A is to to the Q) B (Hjf. ^J. 1. The OA : QIB = A» : B». 2. Likewife, the O C : QD . C» : D». But the ratio of A to B being = to the ratio of C to D fHjp. ij. It follows, that three times the ratio of A to B is =z to three times XCtr. Bit. the ratio of C to D, that it. A* : B» = C» : D». Confequently, the QI A : O B = QJ C : Q D. jtx.6. B. P.ti.B. PP r "i^i The ELEMENTS Book Xi, ' / Tif S A I CO rj /^f 5^ i?- ^//, r^f ^ A e 9 = S" c e' D II Demosstratiow, li E C A L S E The e' A 1? 03 10 the S- ^ (K^f O A ; B = C . Di ^j 7 * I. Whkh wa? It) bt ti^MUonft rated ^ Z: M A R K ( /^ 1^ /? ) I >♦ iV /^fUzi^i I w'jth - /^ K^ '^^'' thtjumt truth h ^p^ltt^ihU (9 7 h tfuijr i'*- M^ff *ti^hiti :a ftmtlur /v/>i;ofi /rifwif \ hfau/f t^fj may he JHa^^ .- J Book XI Of EUCLID. 899 I PROPOSITION XXXVin. THEOREM XXXIIl _ F two planet (A Z & AK) be perpendicultr to one another ; and a flr«ight line (C D) be drawn from the point (C) in one of the planes (A Z) perp^mjipilar to the other (A X) : this Araight i^ie fliaU (all on the common kdaon (A B) of the planes. Hypothefia. Thtfis. Tfbi plaM h7fi4 I't^tbi plane A X. Tht lint C D dra^nfrom the point C, fituatedin the plane AZ> X to the plane AXf falls on the common feSion A B. Demonstration. If not, There may Ise drawn a J. as C E> which will not fall op^ the common CeAion A B. Prtfaration* Prom the point C> let fall on A B> in the plaae A Z» a±CD. ?A%.B. u X5 E C A U S E C D is ± to the common fcftlon A B (PrepY 1. C D will be X to the plaae AX. i>. 4: jf.i i. But £ C i> X to the fame plane. (Sup,). 2. Therefore, from the fiime point C» there has bepn drawn to the plane A X, two X E C fr C D. 3, Which J8 impofltble. P.13. A.ii. 4, Confequentlj, the X C D let fall from the point C, of the plane A Z9 to the plane A X (which is perpendicular to h) psiTes thro' their conunon fedUon A B. Which was to be demonfirated. fOO The ELEMENTS Book XI* PROPOSITION XXXIX- THEO REM XXXIV. In « p«T»nelcpi|wl (A E) ifihcMesfGD, AB; G F, A H; FE,HC; ED, &BC) of the oppofiie planes, (FA &EB; FC&GB) bedmded CKh ioto r«« equal fMfts, the coounon fe£bon (M S) oF the planes (I P & LR), piffjig thfo' the points of fedion (K, P, O, I & L, Q, R, N) and the dtameicr (P B) of the paraUeiepipcd (A £) cut each other into two eqni puts io the print (T). HTpotkcfi^. Z /■ /A* S AE, b^ecimg fw dunm FB ; tht JUts D G, A B, l^c. mrt hiftSed in the fmmts K, P, Ifc. Jl Tl^^UmefKOlfLK.lHn;eletm^J tM" tktpmmts^ K, P, 0, 1, er L, Q^R, N. Tbefis. The c%mm9nfe3nn M S •fthtfe^nut & the diam. F B^cut emcb 9tber imn iiv9 eqwul parts in the pmni T. Pteparaiion, DiawSB, SH, FM, & M D. Demonstration. /Vi. IT I. H E fides H Q^& SQ^bcing = to the fides B R& S R (ffypi). P. 34. B. id the V H Q^S = V S R B. i* 29- B. And .... , - ^- The bafe HS of the A H SQ^will be = to the bafe S B of the ABSR, h V HSQ^a: V RSB. P. 4- A i. But the V RS H & H S Q^ together, are = a L. P.rj. B. u Confequentlj, V R S H + V R S B = a U Jx.u B, u Wheretore, V H S B is a ftraight line. -P.14. B. i. It xokj be demonitrated after the (ame numner, that F D b a ftraieht line. Moreover* B D being = & pile, to A G & A G = & pUc. to F H. P.34. B, i. The line B D will be = & pile, to F H. C P. 9. B.i i. lAxA.B. I. J Book XL Of EUCLID. mgaBoaaamtamm 301 m 6. And, confequently, F D is =s & pUe. to H E. f.33. B. i. 7. From whence it follows, that r B & M S are in the (ame plane FPBH. P. 7. An. Bat in the A F M T, & T S B, the fides F M & S B are equal, (becaaretheAFMTi8=&co to the AHSO, HS=:SB), f^rg. I J. MoreovuT, V S T B =: V F T M, & V F M T = C P.ij. -ff. 1. VTSB. lF.zg.B. 1, j8. Therefore, M T =1 T S, & F T 2= T B ('P. 26. B.i.J thai is, the common fe^on M S of the planes K O & L R, & the diameter F B of the jparallelepiped) cut each other into two equal parts, in the point T. Which was to be d^monftr^tec). 3Pi The ELEMENTS BtekXlt I PROPOSITION XL. THEOREM XXXF. F two triangular prifms (F L & £ C) have the Tame altitude (L I & AE], and the bare of one (at C L) is a parallelogram (F I), and the bale of tbe other (E C) a triangle (A B C) : if the parallelogram be double of the triai^, the firft prifm (L F) will be equal to the fecond (E C). Thcfis. 7be frijm F L w =: /• tbe ^fm EC Hypothefls. /. In tbeprifms F L W E C | /*f a//. L I is = $9 iii alt, A E. //. 7be bafe •/ tbe pHfm L F « * fgr. F I, IS tbehafe9ftbefrifm EC« A ABC. ///. Ibtpgr, F its double Qftbe A A B C Preparation. Complete the N I « B p. Demonstration. 15 E C A U S E the pgr. F I, bafe of the pff fvi F L, U double of the A A B C, bafe otihe prifm E C (Ilyp a- W j/ And the |^. B A alto double of the A A B C. 1. The pgr. FI is =r to the pgr. B O. Moreover, the altitude L I being = to the altitude A F (Hjp, i >, 2. ThcOBDi5 = tothcSlNI. The givea prifm L F i« the half of the O N D. ) And the prifm E C is the half of the O B D. J 3. Confequently, the prifm F L is sz to the prifm E C Which was to be demonftrated. ^41. B. I. P.3,. Bau B. I. Book X». . Of E U C L J D. 50J PROPOSITION!. THEOREM 1. Similar polygnm (ABCDE ^ FGHIK), mrcnbed in ciccles are to one another aa the Tquares of their dianieters (E L ^ G M). Hypotbefis. Thefia. /. W#^/^^«ABCDE&fFGHlK. />fl^g. : ACE: a%. F IHzrr^f D are C\J. •ftbediam, E L : D ofthediam, G M, //. Wfjr an infcribedin circUs* ^ as diam. EL* : diam, G M*. Prtparation. 1. In the © A C D, draw A L, & B E, alfo dlara E L. 1. In the ® F M H* draw the homologous lines P M & ^ 7^1. ^. r G K I alfo the dfameter G M. BDemonstratiok^ E C A O SE the polygons ABCD E &GFKIHarero r^ U- Anddie VAorEAB i«ss to VGFK»&AE : AB=eFG:FK ri). I. A6;. 1. The A A B E is cquianguter with the A F G K- A 6. E. 6. a. Wherefore, A A B E is CO to A G F K, & V « = V *, alfo V ^ = V ^. But VELAi8=2 VEB A, ora, & VGMF=i:VGKFor*. Pa i 6. r Confcquenily, V E L A is =^0 V G M F. Likcwifc, VEAL = VGFM. And, bccaufe, in the two A A L E & G F M, the two V E L A * E A L of the tirft are ^ to the two V G M F & G F M of the fecond (Arg, 3. W 4^. The third V A E L of the A E A L will be = to the third VFGMofthe AFMG. Therefore, EL:AE = GM;GF. And altcrnando EL:GM=±AE:GF. But A E & G F are homologous tides of the polygons ABD & PHK. Beiides, £ L & G M are the diameters. of the © in which chofis Ax,\. B. P.iu B. P. A. PaI polygons are inlcribed. 8. Wherefore, polyg. ABCDE; polyg. P K I H G EL*:GM*. P.22. Which was to be demonftrated* B. 304 The E L £ M E N T S Book XII. |. LEMMA. Jl F From the greater (A B), of two unequal magnitudes (A B & Q, there be taken more than its half (viz. A H), and from the remainder (H B) more than its half (viz. H K), and fo on : there (hall at length remain a magoi- fiide (K B)^ le& than the lead (Q, of the propofed magnitudes. Preparation. i. Take a multiple E t of the teaft C» which may furpafi AB, &bc >aC. P#/i.ir. 5. 2. From A B, take a part H A > the half of A B. Pof.2. B. 5. 3. From the remainder H B, take H K > the half of H B. 4. Continue to take more than the half from thofe fuccef- five remainders, Until the number of times, be equal to the number of times, that C is contained in its multiple £ I. Prf.z. B. 5. B. Demonstrxttion. I E C A U S E the magnitude E T is a multiple greater than twice the leaft magnitude C (Prep. i). If there be taken from it a magiiftude G I ^ C. i. The remainder E G will be > the half of E I. But EI is > AB (t^rep, \). 2' Confcquently, the half of E I is > the half of A B. P. 19. B. 5. 3. Therefore, G E will be much > the half of A B. But H B is < the half of A B (Prep. 2). 4. Much more then G E is > H B. 5. Therefore, E P, the half of E G, is > the half of H B. And K B is < the half of H B (Prtp. 3/ 6. Confequentlj, E F is > K B. And as the fame reafonisg may be cdntinued until a part (E F) of the multiple of the magnitude C be attained, which will be equal to C (Prep. 4/ 7. It follows, that the magnitude C will be > the remaininir part (K B) of the greater A^. * ^ Which was to be demonftrated* BookXU. Of EUCLID. 305 C PROPOSITION II. THEOREM IL I R C L E S (A F D & I L P), are to one another as the Tquarct of their diameters (A E & I N). Hypothefis. . Thefis* In tit circles AF D » IL? there Ifos © AFD: © I L P = AE« : IN*, hen drawn the diawuters A E & I N. Demonstration. If not. A £* is to I N* as the © A F D is to a fpace T (which is < or > the © I L P). Z Suppofition. Let T be < © ILP by the fpace V. ibai //, T + V =r © I L P. 7. Preparation. 1. In the © L I P defcribe the O I L N P. P. 6. B. 4. 2. Divide the arches I L, L N, N P, & P I into two equal parts in the pohits K» M, 0» & Q. P.ia B. t. 3. Drawthe lincsIK, KL, LM, MN, NO, OP, PO &Q^I. Pof.uB. I. 4. Thio' the point K, draw S R pile, to LI. P.31. i^. i. c. Produce NL&PItoR&S| which will form the rgle. SRIL. 6. .Infcrtbe in the © A D F a polygon €0 to the polygon of the © I L P. CLq jofi The ELEMENTS BookXIL E C A^US £ the □ defcrlM ak>ar Ae I L P ir itftlf. 1 . The Kair of this □ will be > the half of the© I L P. But the infcribed O I L N P is =: to fadf of thr ckcitmicribcd Q gbe fide of the ciicemfcritted O being :^ to the diftmeter* & the od the dimmeter 3 D Ll + □ L N =: a D L I}. 2. Therefore, the D L I P N is > the half of the I L P. The rgle. S I is > the fegment L K I (Prep, }• & Ax: 81 Bl r). p Confequently, the half of the rg|e. S I is > the half of the (egment Lj Iv !• The A L K I is =s to half of the rgle. S I. 4. Therefore, the A L K F is > the haif<oF the ftgMem t » L 5. It may be proved after the fame manner, thaft all'the'A LM I!^» NOP, &c. are each > the half of the fegment in which it it placed. 6. Wherefore, the fum of all thofe triangles will be > the (iim of the half of ai^' thefe fegiMMIi Continuing to divide the fegments K I, I L, &c. as alA the fig- ments arriung from thofe divifion»< It will be proved after the fame manner. ■y. Thtt the triangles forroed^b^ the ffhiight JiflcrdMwn in thole feg- ments, die togoihcr >r the half-of the^'fegpoeou in wBdh thofe triangKs are placed. Therefore, if.ftodi tbev0rLP ht taketimoie than its Iiali; . vis. theDl L N P, & from the remaining fegments (L it I, I Q;P,te.) be talcen more than the htflf) &^ fe oic 8. There wtOatf iennthi recMdii fesdKfttr whicktOMtlKr, will, be <V. But the ®..Mi P is^ =s T 4- V Ja: ^./ Therefore, taking thofe fegments L K I» Src. from the 0k Ml IC And the fpace V, from T + V (which is > thofe fegments). 9. The remainder, viz. the polygon I K LM N O P Q^will be > T. But the polyg. ADFK : polyg. 1 ll Ot^^zr Dof AE : D of IN. Jx.S-Bi r. P.47. A I. ^.IQ. B- c. /^4i. B. I. P. 19. B. 5. lem.BAz. ^.j. B. 1. P. 1.B.12. SookXH. Of EUCLID. ^ |0} And the D of A E : D of I N sc ® A C EG : T. fSupJ. io.Thcrefore, the polyj. ADFH ; polyjj. ILOCl= 9 ACEG : T. P.n. B, c. But the polrgon ADFHi«<i9AC£G. ^x.S. B. i. ii.ConfequentW, the polygon I L O Q « < T, P.14. A c- Bat the poJygon ILQQ^ii > TT (Arg.^), i4.Thercfbre, T wtfl he > & < the polyg. I LDQ^ (Arg.<^ W 11/ 13. Which is impoiEbJe. i4.Therefore, T i« not < ® I L P. 1 ^.From whence it follows, that the O of the diameter (A E) of a (A C E G)» is not to the D o{tbe dian>eter (I N) of another (I L P), as the firft ® ( A C E G) to a fpace < the fecond ® (I L P). • //. Sufpo/um. Ut t\A (puce T be > the ckdm I L P. B // Preparation. TaU a fpace V, fuch that T : ® ACEG = ® ILP : V, eCAUSE theaofAE:DofIia=®ACEG i6.InTCrtcndo T : ® A C E G =?: D of I N : D of A E. <i*. 4.. *. <. But T : ® AC E G = ® 1 LP : V. (IL Fr$p.). \Ckr. Moreover, T is > ® I L P. (IL Sup J, l7.CQi>iequentIy, the ® AC EG is alfo > V. P.14. 5. j. Befides T: ®ACEG=DofIN;DofAE fAm, 16). Apd T : ©ACEG = ® ILP : V. (IL Ptep.J. i«.Therefore, thcD of IN : D of A E= ® I LP ; V. i^.ii.^. c. But V < ® A C E G (Art. 17/ And it has b^en demonftrated (Arg. 1 ;^, that the D of the diameter (I N) of a ® (I L P), is not to the D of the diameter of another ® ( A C E G) I as the fira ® (I L P) to a fpace < the fccond ©(ACEG). f 9.Confeqaently, V is not < the ® ILP. ao.Thcrcfore, T is not > the ©ILP. Therefore, the fpace T being neither < nor > the ® I L P, {Arg, 14. &f 19^. ^i.Twiil be = to this ©ILP. a2.CoufequcntIy, the ©ACEG: ©ILP=n of AE : D of I N. P. 7. B. t, Which was to be demonftrated. COROLLART. \^ IRCLES gre to 9ne amtber as th toljgmns infcribeJ in them (P. |. B.12. ftp. II. B. 5). 3o8 The ELEMENTS BookXlL 1 E PROPOSITION III. THEOREM III. __ V E R Y pyramid (A B C D) haffng a triangular bafe (A C D), miy be divided into two equal and finnilar prifms, (IDEFLG&GLFHCE}, and into two equal and fimilar pyramids, (L G I A & L F H B), which are (imilar to the whole pyramid ; and the two prifms together are greater thio half of the whole pyramid (ABC D). Hypoihefis A B C D it a pyramid whofe hafi KDQ is a /I. Thcfis. /. 7*e^«rrIDEFLGwii^iyW = tf toi» /^f/flr/GLFECH. //. 7 be part A L G I m « pyrawdd = & CO '• tht part B L F H. ///. «o/f/>yrfl«f^ALGIWBLFH*ifrCO /o the pyramid A B C D. IV, TAf^ri/injIDEFLGesfGLFCH.ri'rt- gtther > than the balf§f the pyr. A B C D. /. Preparation. 1. Cut all tbe fijcs of the pyranud A B C D into t^o equal parts, in the points L, F, H, E, Q> & 1. 2. Draw the hncs L F, F H, F E, G E, G I & I L, alfo L G, & L H. B Demonstration* EC A U S E in the A B C D thp fides B D & B C are dirided into two equal parts in the points F & H (Prep, \), BH:HC = BF:DF. Confequently, F H is pile, to D C. > Likcwife, F E is pile, to B C. ] • Therefore, F E C H is a pgr. It may be proved after the fame manner, that LFEG&LGCH are pgrs. And fincc F H & H L are pile, to E C & G C (Arg, a. W 5/ 6. The planes paffing thro' L F H & E C G will be pile. 7. Therefore, L G E C H F will be a prifm. > 8. Likcwife, L f £ D I G will be alfo a prifm, J P.ic E. I. Pfl/i.J?. I. D.3S. E. 1. Book XH. Of EUCLID. 309 But thoft two prifms haTe the (ame altitude LG» & the pgr.GIDE which is the bafe of the prifm L D is double of the Z^ C £ G, bafe of the prifm L C P.41 . 5. i . 9. Thererore, the prifm L D is =1 to the prifm L C. ^.40. ^.i i. Which was to be demonftrated. i. B. I E C A U S E the iide B D is c^t into two equal parts in F, that F £ & D£ are pile, to &C & F H, each to each, (Prefi. 1. V Jrg.2.V 3). (P.26. B. r, loThe A F D E is z= & CO to A B F H. ( /». 7. -». 6. ii.The AF ED &ILGarealfo equal Da^.B.w. i2.Therefore, ABFH=:ALIG. Ax.i. B. 1. And fince the other fides of the pyramid A B C D are divided into two equal parts. It may be eaiily proved that, mA B L F is =s to the a L a I, a B L H =: a a G L, & ALFH=5 AAGI. i4pFrom whence it follows* that thofe parts BLHF&ALGI are equal & CO pyramids. D. lo.B.i 1. ^1^ Which was to be demonfbated. 1 1. X H E line F H, is pile, to D C. (ylrg. 2). 1 5.1Jerefore, A B F H is C^ A B D C. R z^B. 6 Likawife, all the triangles ^hich form the pyramids BLHF & ALGI are CO to all the triangles of the whole pyramid A B C D. i6.Tlierefore, the pyramids B L H F & A L G I, are CO to the py- ramid A B c a Which was to be demonftrated. HI. //. Preparation, Draw G H & E H. T HE line BH being = to HC (I. Prep, uj FHcrEC {Arg.^J &>fECH = >/FHB (P.2^.B.i). 1 7.ConTequently, the A E C H is = to the A B F H. P. 4. 5. \. i8.Alfothc AHGC & GEC arc = & co to the ABLH&CP. a. B, 1. LHF. \D,x^.Bau 1 9.Therefore, the pyramid L F H B is = to the pyramid H G EG. Z>. 10^.11. But the pyramid E C H G is only a part of the prifm ECHFLG. ao-Therefore, the prifm E C H F L G is > the pyramid E C H G. i^x.8. B, 1 . ^i.Confequently, this prifm ECHFLG is airo> the pyramid LFHB. P. i. B. 5. TheprifmLG ECHF is = lb the prifm EF LG I D, & the pyramid L F H B :;= to the pyramid A I G L (Arg. 9. i^ i±J. xa.Therefore, the prifm E F L G I D is atfo > the pyramid A I G L. :sfc3. Therefore, the two prifms E C H F L G & EFLGID together, will be > the two ovramids BLFH&LAIG together. jix^j^. tf. i. a4-From whence it follows, that the two prifms ECHFLG & EFLGID together, are > the half of the given pyr. A B C D. Which was to be demonftrated. 1 v. 3IO The ELEMENTS. BopkXH. PROPOSITION IV. THMQREM IV. \ F there be two pyramids (A B C D & ^ F O H) ^ ^hl <«fn^ sltM^ upon triangular jiafies (A B C & £ F G), and Mch pf clvm be 4iviiU ifito two equal pyramids fimihr to the whole pyramid^ (viz. tilt pyramid A B C D into the pyramids D L K M fc AN I L, and the pyramid EPOH ifito the pyramids H R QJS & R E P T); and »|fo into two equal prifms, (viz. the pyramid A B CD into the prifm^ L B & LC^ and the pyr^fnid EF Q H into the prifqos R F & R G) ; and if each of th^fe pvramid? (C) L« K M» A N I L, H R <ig, «f p. E P T) »>e divided in the ffurie manner ^ the firfl twoj and fo on. The bafe (A B C), of qne of the firft twp pyramids (ABC D), js to ?hc bafe (E F G) qf f}ie orher pyrs»mid (E F G H), f|s all the prifmf cpn- tained in the fird pyramid (ABC D), is to all the prifni^ contained in the fecond (E F G H)« that are produced by the fame number of divifions. Hypothefis, Thefts. /. 7he triangular pyramids A B C D & EFGH, Th^J[um nf^tlhh pr\^c9ntaintii have the fame altitude. ' .. . .-^-v . - //. Each of thfm arf cut into itvo equal prifms LB W LC; fl/>RFfcrRG, yin^c/w eqj^al pyramids fimilar to the vhole pyramid, fll. Each^tho/epyramidsLDMK^lNlAATfE l3 R QS H, are fuppojed to he divided in the f(^ manner a\ thefirfi |«im, i^fo on. in the pyramid A B C D w t% the fum of thofe contained in the fp-a^ mid EFGH, heing equal in jftf«* her ; as the ha/e A B C, •/ th$ pj- ramid ABCD u /4 thf bafo EFQ, ofthepyramid^YGW. B Demonstration. ,. E C A U 8 £ the pyramids ABCD&EFGH have equal al- titudes, & the prifins LB, LC,RF&RG have each the half of this altitude, (Hyp. i. W P. 3. -B. i2>. Thofe ptifins L B, L C, R P & R G h^ve the fame altitude. Ax.-j- The Knes B C & F G are cut iato two equat parts in the poiats O & V. ' P. 3. B. I. J. IS. j Book XII. Of E U C L I D. ^u 1. Therefore, CB:CO = GF:GV. 1 Aift B. c. 1,. Confcqucntly, AABC:AiaC=±AEFG:ATVG; P.22- B. 6. I And altcrnando A A B C : A E FG = A I O C : A T VG. P.i6. -ff. 5- c. Moreover, bafe I O C : bafc T V G = prtfnr L K M G O I : J C*r.3. Rem. t prifm R O S G V T. I ofP.ss.B.iu 6. AndprifmLKOBNI: prifm LKMC61 ^ priTm RQVFPT: ■^ prifm R Q^S G V T (having the fame altitude (Arg. \.) & being y equal taken two by two (Hyp. w). ^ ^* 1* ^' S* ""f y, Confequently, prifm L B + prifm L C : prifm L C =± prifm R F + prifm R G : prifm R G. . P.i8. B. j. 9. And altcrnando, prifm L B + prifm L C : piifm R F + prifm RG ^^ =: prifin L G : priftn R G. r , . . ^•^^' ^' $• ^ Butpri(tnLC:prifmRG=:=bafeIOC : bafe tVG (jfrg. ^). * And bafel O G : biifeT VG ±Sb|efd^AB G : l?aft EFG (Arg, 4)* 0. Q. Therefore, the prifm L B + pr. L G : pr. R F -|^ pr. R G == bafe • ^ ABC:bafeEPG. Pau B. i. ^ If the remaining pyramids LKMD&LINAi alfoR(iSH& ^ £ P T Rj be di^d after the fame manner as the pyramids A B G D ^ S^l^fGn: it vAstf b^pmired aft«t the fi^ih^ manner. il* lO.That tk^ fbtfr |rftainids' rcfuUiilg from the firft pyramid^ LKMD ^ & A N I L, will have the fame ratio to the fou^ prifms refulting I from the laft R OS H & E P T R, that the bafes L KM & AN I have to the bafes R Q^S fr ft P'T fffyfi. rit . W Arg, gj. And it has been demonftrated. that the bafes L K M & A N I9 are each = I6G| a4f<>R<ij6 ft E P t, each scT ¥& Moreover, AABG:ATEFG=:AI0G:ATVG fA^g^A). I I .Wherefore, the fum of all the ptifms contamed in the pyramid A B G is to the fum of all the prifms contained in the pyramid E F G H,' ai tiie4>a6 A-&G H to^kcf bde B F G. P.ii. B. j. Which ^ai tfr be' dcnfenftrtted. ?» 3i» The ELEMENTS Book XB. 1 D A \ H F / ■ / z vv X C^ ^ A/ >^G^-v^ B PROPOSITION V. THEOREM V. XYRAMIDS(ABCD&EFGH)oftfae ikme aldtude, which have triangular bafes (ABC&EFG):are foooe anoihcr as iheir bafisL (A B C & E F G). HTpothefis. Thcfis. /. TbifyramUshJ^Q'DtiZTGmMnfifmr iyM. A BCD: /^m£FGH= hafet Mf A ABC & E F G. &;/# A B C : A«> E F G. //. They have tbt fawie aliiiUiU. Demonstration^ If not, Pyramid A B C D : pyramid EFGH >l>afe ABC : bafc E F G. PrefafAtion. Take a folid X which may be > the pTtamid A B C D» 2. B fo that X : pjram. £ F GH = bafe AbC : bafe EFG. Divide the pyramids ABCD&EFGHas direded m P. 3. B. 12. E C A U S E the two prifms reftdting from the firft diviiioii> axe > the half of the pyramid A B C D; & the four following, refitt- ing from the fecond diTifion, are > than the halves of the pyramids relulting from the firft divifion, & fb on. It is evident, that the fum of all the prifms contained in the pyra- mid A B C D, will be > the folid X> which was fappofed to be < the pyramid A B C D. P. 3. S.12. lex. Ais< B6ok XU. Of EUCLID. wm 3^S warn But all the prifms . contained in the pyramid A B C D» are to all the prifms contained in the pyramid E F G H» as the bafe ABC is to the bafe £ F G. P. 4. Bm. And the folid X : pyramid EFGHz^boTeABC: bafeEFG (Frtt. X), Qlequently, all the priims contained in the pyramid i to all the prifms contained in the pyramid E F G H> as the folid 2. Confequently, all the prifms contained in the pyramid A B C D are X is to the pyramid E F G H. P.i 1. A $. But all the prifms contained in the pyramid A B C 1>, are > the folid X. (Arg. ij. 3. Therefore^ all the prifms contained in the pyramid E F G H, are > the pyramid E F G H itfelf. P. 1 4. B. 5. 4. Which is impoilible. Ax.S.B, i. J. Confequently^ a folid (as X) which is < the pyramid A B C D, cannot have the fiune ratio to the p ' * "^ ^ bafe A B C» has to the bale E F G. pyramid FGH, which the And as the fiune demonftration holds for any other folid greater than the pyramid A B C D. 6. It follows, that the pyramid A B C D : pyramid E F G H = bafe A B C : bafe E F G. COROLLART I. X T'RAMIDS of the fame altitude^ ii 'which have equal triangles fur their iafes : are equal, (r. 14. & i6« B« 5^). E COROLLART II. f^UAL fyramids 'which have equal triangles for their hafes : have the fame altitude. 1 1 R r The ELEMENTS fiookXn. _ PROPOSITION VI. THEOREM FL r YRAMIDS (FGLIM 8r ABCDE) of the (uat ahitinK \vhich have polygoiu (PGHLI, & ABCD) for their bafet : are io one another as their bafes. Hypothefia. Tliefia. /. The pyramids TGHLlliUBQiy, Prr^iif. MP G HU : #rnw ABODE bavi polygfMi fmr thiir hafts. sAt/^FILMG : ^i/'rABCO. //. 7b^ bavt tbe fame altitude, PtipMraium. 1. Divide the Ufes FILHG & ABCD bto triaifles* hf drawiag the Knes G I,P H ; ft D B. 2% Let planes be pafled thro' tho6 lues & the Tertices of the pTramidsy which will divide each of chofe pyramids ' into as many pyramids as each bafe contains triangles. BDSMQNSTEATM^. ECAUSE the triangular prramidsl L HM &ABDE have the fame altitude. (Hyp, ii. K^ Prep. z). I. The pyramid I^H L M : pyr. A B D E = bafe HIL : bafe ABD. ) Likewife, pyr. G I H M : pyr. A B D E = bafe HIG : bale ABD. IHLM P, 5. B.I 2. + p; ba& Xe GIHM BD. pyr. ABDE = P.24 B. 5- 5- 3, ConfequentlT. pyr. bafe H I L + bafe H ] G 4. Moreover, pjr. F I G M r p*r. A BD E =3 ^fe f I G : bale ABD, P. c. B.1I 5 Therefore, pyr. 1HLM+ pyr. GIHM + »«. FIGM : pyr. A B D E= bafe H I L + bale HIG + hafePIG : bafe ABD. P.X4. B. But pyr. IHLM-f pyr. GIHM -f pft. PIGMai« = toC the pyr. M F G HLI, & the bafe HIL+ bdf HIG + bafe^^j. A a. FIG = bafeFILHG. i 6. ConfequeDtly, pyr. MFG HIL: pyr. ABE>e=: bafe FILHG ' bafe ABD. F. 7. M. $■ It may be prored after the fame manner, that 7. Pyr. M F G H L I : pyr. B D C E = bafe FILHG : bafe BDC. 8. Therefore, pyr. M F G H U : pyr. A BCDE = bde FI LHG : bafe A D C B. P.a$ Wbich was to be denonftrated *5- Book XII. Of EUCLID. 3*5 J I p ^. % A ••••... • / \ , •••.. / <*^E F E PROPOSITION VIL THEOREM ru. f VERY triiogalar priffn (A D E) : may be divided (by planes pafllng through the A B C F & BDF) into three pyramids (AC BF, BDEF & D C B F) that have triangular bifes^ and are equal to one another. Hypothefis. Thcfc. The given frijfm \JilL bos a The prifm A D E wuy he JtinUUd in/d triangular baje. {^1^^ .f^^f '_ iT^^Si^^L _ ^««'''^» three equal triangular ACBF, BDEF,1dCBF. :i.A I. Preparation. 1. In the pgr. D A draw any diagonal C F. \ 2. From the point F m the pgr. A E» draw the diag. B F. V Pof: J. From the point B in the pgr. C £» draw the diag, B D. 3 4. Letaplanebepafledthro'CF&BF,airothro'BF&BD. ' Demonstration. XJeCAUSE a D is a pgr. cut by the diagonal CF. (Prep.i). I. The AACFbafeofthcpyramidABCFft = totheACFD, bafe of the pyramid B C F D. P.J4. B. i . But thofe pyramids ABCF&BCFDi have their vertices at the a.- ^erefore, the pyramid A B C F is = to the oyramid B C F D. I qJ\ ' ^' Likewifc, the pgr. E C is cut by iu diagonal B D. (Pref, 3/ C *-«^- »• Therefore, the A C B D» bafe of the pyraoiid B C F D is = to IDEFB. i>.34.i?. I. the A B D E, bafe of the 1 , And thofe pyramids B C F D,' &c. have their vertices at the pointF. Confequentfy, the pyramid B C D P is ^ to the pyramid B D E F. C P. ;. A 1 2. But the pyramid A B C F is alio s to the pyramid B C D F. ( O. i. (Arg. zj. Therefore^ the pyramids ABCF.BCDFi&BDEFare equal. Ax.t.S. 1 . 3i6 The ELEMENTS BookXIL J c D ^E / 3^ *'••••. ::..< A F 6. Confeqaentlft the triangular prifm (A D £) maj |)e divided into three triangular pyramids. Which was to be denx>iiftrmted. CO RO L L A RT L }i^ RO M this it it wui^fejf^ that every jjyramiJ wbid bos a triatmdm' hetfe^ uiie il^rJ part ef a prifm which pas tbefawee hafe^ U it ef an efual aUitu£ wtb il. C O RO L L A Rr II. J2j yE R T pyramJ which has a pefygm far hafe^ is the third ^ ef a pnfm mhiA has the fame hafe^ H is efan efiu^ altitude with it ; fince it eaof he diifukd smf peifms halving triangular hafes. COROLLARr III X R IS M S ^ ^f^^ altitudes are f em atnther as their hafa^ htcmfe pjra^ mids upea the fame hafes^ H of the fame altitudty are f me amther as iheir hafes. (P. 6, B. iz). Book XH. Of EUCLID. 3^1 PROPOSITION VJII. THEOREM Fill O I M I L A jft pyramids (ABCD&EFGH) having triangular bafcf (B D C & F G H) : are to onp another in the triplicate ratiq of that of their homologous fides. HYpothefis. ^^ 40 tyramdi ABCt>&EFGH havt triafifftlar iafes D B C & G F H, nvhofe Bo- mola^ JiJis are BD ^ ? Gftjfc. Thefis. ne fyrandd ABCD isf the fyramd EFGH, inthetripUcaieeaiU%f%\> t% F G, that is, oi D B* : PG?. B Preparation. 1. Produce the planes of the A B D C, A B D & A DC ; complete the mprs. D R, D Q & D P. 2. Draw P O & O Q^pUe. to A Q^& A P, & produce them ' too. • p,, 3. Join the points O & R ; & O C will be a SI which will have the fame altitude with the pyramid ABCD. 4. After the fame manner defcribe tne ^I M H. 5. Infinc, Join the points Q^& P, alfo M & N, homolosoua to the points B & C j alfo F & H. Demonstration. E C A U S E the pyramids A B CD&EFGHarccc (Ufp.y P.31, B. ,. B. ! tnan^lar planes which forpi the pyramid ABCD are CO the triangular planes which form the pyramid EFGH, 2 3- 4- All the to all the each to each Confequently, A D : B D = E G : G F, &q. * And the plane V A D B is rr to the plane V E G F, Therefore the pgr. D Q^is eo to the ppr. M G Likewiie, the pgr. oppofhe ones A < DR&GI; DP, EL;QR, MI. G N are CO i as alfo their D. 9- 5.1 1. D. i.B. 6. P. 5- -»• 6. D. 1. B. 6. ^.24. 1^.11. 3i8 The ELEMENTS ae Book XII. 6. Coniequently, A R & £ I Are CC 01 />. 9. ^.i 1. 7. Therefore, QJ A R : O E I = D B« : F G«. Pljj. i^.n. And fince the lines QJP & B C ; M N & F H, are diaM|nals fimi- larly drawn in the equal & pUe. pgrs. OA&RD^EL&IG. e parts BQ^APCD&FMENHGwaibeCO prifins : & CD. 9. B.ii. each equal to the half of its Q. \ P. z8. B.ii. (P.i^. B. 5. A. Confequootly, the prifm B P Q C : prifm FNMH = BD» : FG«. < P.34. An. C^«w. I. But the pyramid A B DC is the third part of the prifm B OP C, f P. 7. B.iz. & the pyramid E F G H is the third part of the prifm F NIN H. ( Car, i . laThercfore, the pyramid ABCD : pyraniid.EFGHafc BD» : FG». P. 15. A 5. Which was to be dembnftrated. COROLLART. Xs RO M this it is e^dgnt^ that fimlar pyramids <whick have ptfygms fir their iafesf are to we another in the ttipiicate ratio of their homol^ous fides ^ (hecaufe they mof he Jivi4itl into triof^lar fyramds ; nxmd) itre Jitfdlar^ taken tvt9 hy tw. Book XII. Of EUCLID. Bmmmm PROPOSITION IX. THEOREM IX. _ HE hafeft (ABC arEFG), md altiioda (BD&FH), of equal pyramids, (ABCD&EFG H), having triangular bafes, are reciprocally pioporiionat, {fiat h, the bafe A B C : bafe E F G ^ altitude F H : alti. tQde B D\ and triangular pyramids (A B C D &r E F G H)» of which the bafes (A B C & E F G), and ahrtodes (B D &r F H), are reciprocally pro- portional : are raiial to one another. /. fhet(r^. ABCDUEFGH are iriiwmJar. Adk ABC : taf, EFG t^ altkuJe II Tit ^4im. ABCD is = to the fyram. EFGH. F H : abitudt B D. Preparation. Complete the BO & F K haviiw the fame altimde with the pyramids ABCD & E F G H j as alfb the prifms BAPNC&FELIG. I. Demonstration. B EC AUSE the prifms FNB & LIF, have the fame bafe k altitude with the given pyramids ABCD&EFGH. (Prep). Each p^iihi will be triple of its pyramid, (that isf the prifm P N B triple of the pyramid A B C D, & the priiiii LIF triple of the C P. 7. pyramid EFGH).. , {(kr. Aia, Confeq.uently, the prifm FNB is =s to rhe prifm^L I F. Jx.& B. i, But the (SI B O is double of the prifm P N B, & theSTFK double of the prifm LIF. P.28. ^.11. Therefore, the Ql B O is =; to the SI F K. AfcS. B. 1. But the equal O) (B O & F K) have their bafes and altitudes re- ciprocally proportional (ihat is^ nafe B Q : bafe F M = altitude F H : ahitude B D). And thoft ^5 are each fextuple of their pyramids, (tbatist t<Te (^ B O is =fix pyramids A B CD, & the KF =: fix pyramids E FGH. Jlrguiiil 3«o The ELEMENTS HtxkU 1 I '*V^ ••Vi:£;5a^N Moreover, the bde of the pjrraiiud A B C D Is the htlf of the btfe') of thee) BO. Lp., J?i ' And the hafe of the pmmid EFGH is the half of the baft r*^'* oftheOFK. J 4. CoAfequentlj, bije A B C : bafe £ EG = alt F H : alt B D. (F.i^B. $. Which was to be demonftiated Hypothefis. Thefr. / TbefiprmmJsABCDVEFGliartiriaMilar. TbetrmnfftlargpmmuiASO>is:^ U. Bafi ABC : hafi EFG = «i^. FH : ab. l&D. t$ ibt iriamguIarlifnuddUG^ B. II. Demonstration. BECAUSE theAABC:AEFGr=FH:BD. (l^. %h And the pirr. B O is double of the A ABC» the pgr. F M double ofiheAEFG f^^'\'' ii It follows, that the per. BQ^: pgr. F M = F H : B D. Pis-i*- $• But F? B O has for bafe the per. B Q, & for alt. B D. 7 .pu^ , And O F K has for bafe the pgr. F M, & for alt. F H. ) « ^^-'^ z. Confcquenily, the (Ql B O is = to the O F K. -Pjf *"• But the B O & F K are each double of the prifms P N B & LIF. . i'lS.B". And thofe prifnis P N B & L I F are each triple of their pvrainids C f 7- **** ABCD&EFGH. \Cm.i. 3. Therefore, the triangular pjramid A B C D is = to the triangular pj^ram^d E F G H. Ax.^.^^- Which was to be demonftrated E COROLLART, M-^ SIV A L polygon pyramids have their hafis and ahitudts reciprocal^ ff^ iienai j W polygon pyramids ^hofe hafes ^ altitudes are reciprocally preperttee^- are equal- BookXU. Of EUCLID. $n H P| : s G • F ;>:M ••■•K E PROPOSITION X. THEOREM X. ^VERY cone (BRC) i$ the third part of the cyCnder (HGPE ABDC) which his the fame bafe^ (BDCA) and the ftme altitude (BH) with it. Hypothetia. ^ Thciw, ne r0if#B RC, W the cyismierHF A D C, Tte €9neBRC is epkti to thi third ^ bave the fame At/e B D C A, (^ the fame part rf the finder HFC ABD. mltituJetH. Demonstration; If not. The cone will be < or > the third part of the cjlioder^ bjr apart =:Z. , Z Suppofiti$n. Let the third part 6i the cylinder H C be r=: cone BRC /. Preparation, I. A N the bafe ABDC of the cone k cylinder, defaibc the Q ABDC. P. & A 4. ^ About the fame ba^ defaibe the D P O QjS- il 7. it 4. 3. Upon thofe Tqnarea eted Iwo 0» the firft^S P H B C« ujpon the jnlcribed D, & the (econd, on the circumrcrtbcd D, which will touch the iiiperbr bafe with iu pile, planet, in the poiau H> G, F> J^ £» * having the faaar dtithide with the cylindery & the coae. S f ^ lVeb4tviemttidapart9ftbipr^firatieHinthef^gitm9*09Hd€mfi^9i^ ja« The ELEMENTS BookXII. B 4. Bi&a the arches ATC, CJD.ViB, kBaA/inT^^y&a. c. Draw AT, & T C, &c. 6. Thro' the point T, draw the tangent ITK, which will cut BA & D C produced, in the points I & K & complete the pgr. A K. 7. Upon the pgr. AK, credt theSI ALFK, & upon the A AIT, T AC, & TCK theprifmsETI,ETF, & TFK, hating aH the (aine altitude with the cylinder & cone. ti. Do the iame with refped to the other (egments A a B, 9 ^ B, &c. Pof.i. P.17. A 3^ B, I. A3. EC A USE the D POQ^S IS defcribed about, & the D B D C A defcribed in the 0. i^Prep. i.l3 %). I. The D P O Q^S is double of the D B D C A. And the 0! defcribed upon thofe fquares having the (ame altitude, a. Therefore, the SI upon POQS is double .of the O upon BDCA. But the ^3 upon POQS is > the given cylinder. 3. Thercfore> the ^ upon BDCA is > the half of the dune cylinder. And fince the A T A C is the half of the pgr. A K. 4. The prifm E T F, defcribed upon this A TAC, will be the f half of the jS) upon the pgr. A K. < The 0)defcribea upon the pgr. A K is > the element of the C cylinder, which has for bafe the iegment A T C. 5. Confequcntly, prifm E T F defcribed upon A T A C is > half of . the element of the cyltiKler which has for bafe (egment ^ T C. 6. Likewife, all the other prifms defcribed after the tone manner, will ' be > the half of the correfponding parts or elements of the cylinder. Tlierefore, there may be taken from the whole cylinder more than the half, (viz. the O upon the€D BDCA), & from thofe remain- ing elements (viz. C F £ A T, &c.) more than the half » (viz. the prtfbuKTF, &c), ftfaon. i^47. (Prep ^.32. P.19. P.41. P.34. Rem,i B.J. J.I I. B, I. B s. B, I. B.iu B.iu Cor.^. B. 1. P.ig. M. J. Bodk Xn. Of E U C L I D. saj % Until there remaiiift feveral eleaoenu of the cylinder which together w01be<Z. Lm.B.i%. But the cylinder is =: to three times the cone B R C + Z. (Sup.). Therefore, if firom the whole cylinder he taken thofe elements (Arg. 7./ And from three times the cone B R C -j- Z> the magnitude Z. t. The remainins prifm (viz. that whicn has for bate the polygon ka B ^ D i/ C T) will be > the triple ^the cone. Ax.^. B. 1, But this prifm is the triple of the pyramid of the fame bafe & alti- ( P. 7. B.\%. tijde (vi«. of the pyramid T A n B * D i/C T R). (CV. 2. 9. pon&quentlyy the pyramid A B iDC R is > the given cone. Ax.'j. B^ |« But the bafe of tbf cone is the in which this polygon A B D C is iofcribed, (ft wtiich is confequently > this polygon), ^ this cone has the fame altitude' with the oyramid. ip.Therefore, the part is > tl^^ lyV^c. ii.Which isimpoflible. AK'Z.B. \. l2.Confequently9 the cone is not < the third part of the cylinder. IL Suppojitim. Let the cone be > the third part of the cylinder b/the mgu Z9 that isf the cone = the third part qt the cylinder Hh ^ //. Preparation. Divide the given ^one into pyramid^, in th(^ fyiu w^nfr. that tbt cytinJer nvas diviJedh^ tbf.firjt fupf^tiof^^ X F from the given cone be taken the pyramid which has for bafe the O A B D C, (which is greater than the half of the whole bait o^ the gijen cone, being the half of the circumicribed D, Arg. i. & this U being > the oaie of the cone> Ax. 8. B. i.), & from the lemaining (egments, the pyramids correfponding to thofc fegments^ (as bos been dont in tie cylinder Arg, y.V- f J. There will remain feveral elements of the cone which together will be <Z. tem,Bu. Therefore, if from the cone thole elements be taken which are < Z, & from the cylinder + Z, the magnitude Z. i4.The remainder, viz. thb pyramid A a B ^ D ^.C TR is == to the third part of the cylinder. i£v.^. B. 1. But the pyr. Atf B*l)^CTRi8 = to the third part of the prifm, C P. 7. Bax. which has for bafe the &mepoly^. AaBADyCT, &the(amealt. \Cor. 2.. i5.Thcrefore, the given cylinder, is =: to this prifm. Ax.6, B. 1. But the bafe of the given cvlinder is > the baie of the prifin fioce this (econd is inforibed in tne firft. (L Prep, 4. (^ 5/ i&Therefore, the part is = to the whole. ly.Which » impoflible. Ax.S.B. i. 16. Therefore, the third part of the cylinder is not < the cone. And it has been demonilrated (Arg. i%.)^ that the third part of the cylinder is not > the cone. 19. Therefore, the cone is the third part of the cylinder of the ftme haLk ft altitude. . Which was to be demonftrated; The ELEMENTS C PROPOSITION XL THEOREM XL ONES (E A B D F & H G K I M), and cylinders (CLR BE* $ T K H) of the fume ftUiuide, afc to one inothef tf their bafes. Hyi^thefis. ^be cones EABDF U HGKIM, cr lih- nvife the cylim^rs QRBE W STKH have the faw^ 0kiiM4f' I- 2. 3- 4- B TheGs. /. CWEFB : <!«*fUMKs:fo/€£ABD liafeHGVih II CylimUr QRBE : rr/<Wrr STKH = ^> E ABD : kajk HGKI. Demonstratiok. If not. The cone E F B : Z (which is < or > the coae HMK) szbafeEABD : baieHOKI. / Suppnfition. Let Z be < the cone H M Iv by a magnitude X, that is^ let the cone H M K = Z + X. J/. Preparation^ N ® GHI K bafe of cone H M K ; de£cribe O G H I K. ivide the cone into pyramids (as in 11. Sy^.efP. lo-J. In the bafes of the conet EFB & HMK, draw Jmm. EB^ HK. In the © E A B D bafe of the cone £ F B« defcribe a polyg. CO to the polyg;H i&G^KLIiH»& divide it as the cooe IlM K. P. 6. A E C A U S E the cone HMK has been divided into pyramids. {Fre/. tj- If thoib pyramids be taken from the cone (as was done in the fore* going propofltion, jir^. 13.). The fum of the remainmg elements will be < X. Lem Bai. Therefore* if thofe elemenu be taken ftpm the cone H M K, & the anagnitttde X from Z + X* BbokXa Of EUCLID. 3ts : ft. The remaining pyramid H^^G^KLIfM will be > Z. But ihofe polygons infciibed in the © E A B D & H G KI are tO- fPrep. 4./. , 3. Therefore, ©AEDB:©GHIK= polyg. Qdea\ polyg. C P. 2. A.i*. ibgh. \Cor. But, ©AEDB:®GHIK = cone EF B : Z. (Sup.). And the pyramid 9 J E f A « B C P : pyramid Hi& G^ K LI i M := polygon Qdtai po]ygon ' <^ ^L. P. 6. i^. 1 1^ 4. CooTequendy, pyram.Di/E#A«BCP : pyram. Hi& GjrKLf iM =:coneEFB:Z. f.ii.jR. 5. But the pyramid D^EfA«BCFia< cone E F B. ^x.8. B. i, r . Therefore, the pyramid (I iS G ^ K L I i M is < Z. P. 1 4. i?^ (^ D. But this pyramid is > Z. T^^* 2.^ i. Therefore, it vrill be > & < ZT (Arg. z. y 6/ . Wi^ich is impoflible. 9. Therefore, the fuppofition of Z < the cone H M K is falft. 10. Wherefore, the bale of the cone E F B is not to the bafe of the cone H M K (the cones having the 6jne altitude) as tbe cone f F B %o a magnitude Z < the (:one H M K. //. Suppofition. Let Z be > the cone H^I K. 11. Priparaii$n. Take a magnitude X fi^ch that % : cone E F B as con^t H M K : X. j3eCAUS£ Zis> theconeHMS^ (L Sup.), 1 1. The cone E F B is > X. P.14. B. c. But the cone EF B : Z = bafe E A B D : baft H GKI fSupJ. C P. 4. B. g. la.Therefofe, bafe H G K I : bafe E A B D = Z : cone E P B. \ Cor. r3.Coiifeouently,bareGH!K:bafe AEBDsrcorieHMKtX. -P.it. f. j. But it nas beendemonftrated^>/r^. 10.^, that the bafe of a cone is * Bot to the bafe of another cone, baying the fame altitude, as the Itrft cone is to a magnitude < the fecond. J4.Therefore, X is not < the cone E F B. But X is < the cone E F B. (Arg. 10 J. I ^.Confequently, X will be < ft not < thit ^nc EFB. (Arg.i 1.^14;. 16. Which is tmpoiiible. 17. From whence it fellows, tkat the ibppofitiim of Z > tht codf H M K is falfe. • ^ ThereTore, the magnitude Z being neither < nor > the cone HMK. (Arg.g,^iY.). I g.It will be :t= €0 the cone HMK. iQ,H ence cone EFB : cone HMK = bafe S A BD : bafeHGKI: P. 7. B. j |> Which was to be tieoiofilbated. i. JD E E C A U S E the cone EFB is the third part of the cyhn.QRBE 7 ^ « And the cone H M K is the third part 6f the cyiin. H S T K. j ^- « o. -fi. 1 2. ao.Thccylia.QJlBE:cyl.HSTK = bafcEABD:bafeHGKI. P.15. ^. 5. Which was to be demoiiftratcd. 11. 3Z^ The ELEMENTS BookXn. ^^ ^ PROPOSITION XIL THEOREM JTU. Similar cones (BFE&LOM), and cylinders (B«»Efr htdM) have to one another the triplicate ratio of that which the ' P fti Ktr n (CD & IH) of their bafes (BYDEP & L T H M R), have. Thefi*. I fbtttrnVFEisftbt amlX>Uimat trifliaut talk tf CD t» Itii mr m CD«: IH». //. rhttfi.BatEhftb*€,Ll,tiyi^it tb* tnfUea* mi$ tfCDflHt "» CD«:IH»: Hjrpothefii. VUtmu BFE e^ LOM. Hktwft tht pli^dtrtBakEffLeJM, »to^. Dbmokstration. If not* The cone B F E is to a mamitnde Z (whkh is < or > the coneLOM> aiCD* : IH*. /. Suppojition. , Let Z be < the cone L O M far thi mamitvde X, that i*, ^ecooeLOM = Z + X. I BookM 6f fetJdtiA jif B. /. Preparatitmi Divide the L O M into pyrflmids* as in the foregoing propoiition. In the bafe of the cone B F £ defcribe i polygon 0a to the poiygofl of the bafe of the cone L O M. In the two dones draw the homologous dianittera I H ft C D j alfo the rays L N & B A. J£ C A U S E the cone L O M has been divided into pjnunids. If thoie pyramids be taken from this cone (id the iam^ maimer as in the foregoing propofition. Arg. i .). 1. The fum of the remaining elements will be < X. Item, Aia« Therefore* if thoie elements be taken from the cone LO M9 & the Mtrt X from the magnitude 2 4- X. 2. The remainder, viz. the pyramid LTGHM SRI O' will be > Z. Ate.^. B. li Bat the 01 cones have their axes & the diameters of their baies proportional. D.aA. M.tu And the cones B F E & L O M are fU. (fhp.). 3. Confequently, CD:HI = FA:ON, But, CD:HI = CA:IN. Pac. B. t. 4. Therefore, CA:IN = FA:ON. P.ii. B, L 5. And altemando CA:FA = IN:ON. P.16. B. C4 TheAFAC&IONhavetheVCAFfcioVINO. ^?rf^.3;. ' And the fides C A, A F ; IN, ON about thofe equal aiq|lcs pro- x portional. (Arg,^.), 6^ Wherefore, the A F ACisOl to the AI OR Z). i.^. 6. 7. Confequently, C F : C A =; I O : I N. ^ P. 4. A 6. 8. Likewife, the A B C A is CO to the A L I N. (V B A C beinc = VLNI). (iV./.3.). o. Therefore, CA:BC=:IN:IL. Aa-A^ But, CF : C A = I O : IN. (^.7.). io.Confeqttently, CF:BC=:IO:IL. P.aa. A <^ In the A C A F ft B A F, the fide C A is = to B A (Z>. 1$. A 1.) A F is common, &VCAF=:VBAF. {Frtf. 3.). ii.Therefore, the bafe B Fis =:to thebafeCF. A 4. B. i. 12. In like manner, L O is =: to O I. But, C F : B C = 01 : I L. {.Arg. 10). i3.Tberefore, B F : B C = LO : I L. P. 7. A e. 14 Andinvertendo/BC : B F = I L : OL. CP. 4. A cl 1 c.Confequently, the three fides of the A B F C are proportional to > C^r. thethreefidesof the ALOI. ^ ^ i6.From whence it follows, that thofe A B F C & I O L are CC. P. 5, i?« ^ S7.IC may be demonftrated after the famemanner, that all the tri- angles which form the pyramid B D Q F are CC to all the triaoglee wkidi form the pyramid L H S 0» each to each. gut The ELBMBNTS Bookm ^ ^ ssst And as the bales of thofe pyramida are CO doIwom. (A^ 2.}. i&The pTramid B D QF is CO to the pyramid L H S O. But thofe pTiaiDNts being 03. tgTht fjnTsoitDKlF '. ppuM LHiO = CB* : IL«. D. 9.1.11. But, CA ae-HkertAirc invert- B C 2 1 .And ahernaiM}i», B C ' az-Confequently, B C : BC = IN CA = IL LI ±CA LI =CD XL. IN. IN. IH. {Arg. 9.), $P. 8. J.M. aj.Tbeipferey three times the ratio of B C to L I is = to three times ( P.i i. J?, s- the ratio of C D to I H. ihmt is, B €• ; LI» =rCD* : I H»: BttI C B* : IL* r= pyramid B D QJ : pyramid LHSO. (^rp.19). :24.ConfcquenUr, pyiainfd BDQF : pyramid LHSO = CD* : IHK F.\ u JBL 5. ButtheconeBFE: Z=rCD«; IH». {Suf^^). a^Tkercfore, the py ram. BDQf : pyran. LHSO = cone BFE : Z. JP. 1 1 . #. ^. But the pyramid BDQF being < cone BEE. JxJ^ A u a6.The pyramid LHSO will be alfo < Z. i> 14. M. 1. Bat the pyramid L H S O is > 2. {Arj^ 2.)- STConftqutmly, the pyram. LHSO will be< & > Z. (ifn^.a. & 26). aS.Whicbiatiiipoiible. ao.Therefore, the fuppofition of Z < the cone L O M or L T O HUS9.lOi»Mft. Book xn. - 6f £ U C L I D. S29 3o.From whence it follows, that the cone B F £ u not to a magni- tude lefi than the cone L O M» b the triplicate ratio of the diamc* tcr C D to the diameter I H. IL Suppofition. Let Z be > the cone L O M // Preparation* Take a magnitude X> fuch that Z : cone B F E r: cone lomTx. JlSeCAUSE Zi8> than the cone LO M. (IL Sup). «. The cone B F E will be > X. Pa±. A «•' ButCD*:IH«3=:coneBFE:Z. (Sup.). rp p aaTherefbrc, invert. I H» : C D» =t Z : cone B F E. I ^'4- ^- S* But Z : cone B F E = coneLOM : X. (IL Frftp,). ^^•^• jj.Confequently, I H» : C D» = cone L O M : X. P.i I. A jj And. it has been demonftrated (Arg, 30.^, that a cone is not to a magnitude lefs than another cone in the triplicate ratio of the dia- meters of their baies. 34.Tberefore, X is not < the cone B F E. But X is < the fame cone. (^^K* 3'*/ 3;.From whence it Mows, that X wul be < the cone* h will not \m < at the fame time. 36.Which is impoilibie. 37.Therefore, the fuppofition of Z being > the cone L O Nf, is falfe. Therefore, the magnitude Z bebg neither < nor > the cone L O M. (Arg, zK^li 37.;. 38.It will be equal to it. 39.Confequentl7, the cone B F E : cone L O M = C D* : I H». A 7. A 5; Which was to be demonftrated. i. The cylinder B « * E, being triple of the cone B F E. ) « „ , And the cylinder LcJM, the triple of the cone L O M. t ^•*o- A12. 40.Tl|e cylinder B «* E : cylinder Lc ^M =3 CD* : I H». P.15. B. 5. Which was to be demonftrated. j i. Tt i39 Tho I L 9 M E NT S »>ok XA 1 E. Y Q F \/ X '■ ■■ 2..* *- r-!- • A& i PROPOSITION Xm. THEOREM JCIII. 1 P a cylinder (A B D C) be cut by a plane (H G) ptralid to iu oppofite planes (H A & D C) : It divides the cylioder into two cytiwfers ( A B H G ar G H D C), which are to one another as their axes^ (EK & KF) {iktti^^ the cylinder A B H O : cylinder G H D C =r axis E K : axis K F). Hypothcfo. Thcfis. Thecylin.A V is cui Iji a pLuteHG, CjltM. A I) : c^/m. HCs^xtf EE: pUe, to the 9pf>ofiii plants A B £tf D C axis F K. Prepatatim. 1. Prodace tfaeana EF of iha cjUadir ABDC both «q« towards N & M. JV^a. ^ '• a. In the axis N M produced, take fereral parts sstaEK &FK;as£N = £K, arFX,&c.eachssF8: /. >J|. i. 3. Tfasa' thofe point). N, X & M paA the pfamesSR, T Y & V Q, pile, to the oppoiite planes B A & D C 4. From the points N, X & M» defcribe on thofe plants dK ©SR»TY&VQea€hr=ltoikropp«fite^6A.&IX:. t^v% ^ 5. Cojnpleic the cylinders S A, C Y & T Q^^ DlMONSTRiVTION, B RCAUSE the axes FX^rXNf of the cylinders DP &^T<^ are equal to. the axis F K« of the cyljuider O D. (?rtp, %). Thofe cylmders D T, T QJk G D will be to one another as their bafes. FauBax, But thofe bafes are equal, (^rep, ^)\ 2. Therefore, thofe cylinders T D, TO 4 But there are as many equal cylinders C Y» T Q^&c. whicfi tose* 2. Therefore, thofe cylinders TD, T O 4 G D are alfo equal. P.\^^ A 5. ther are equal to the cylinder G C^as there are paro F X, X M, Sc each equal to the axis K F» which together are eqtsal to M IC f BookXn. Of EUCLID. J. Confequently, th^ cylinder G Q^or G H QV is ttie fiime knultipk of the cylinder G H D C, that the axis K M is of the a«s K P. ^ It may be demonftrated after the fame manner, that the cyliodtr R S H G is the fame multiple of the cylinder A B H G, thut the axis N K is of the axis E K. c. Therefore, according as the cylinder G H QV is >, =, or < the orlinder G H D C, the axis K M will be >, =, or < theaxts P K. And according as the cylinder R S H G is >, =, or < the cylinder A B H G, the ana N K will be >, =, or < than the axis £ K. ^. Confeouently, cylinder ^ B H G : cylinder GHDC:^azii£K 83* •'♦* A 5. B. 5. S3« The ] ELEMENTS BookXn. N 1 L <I.. X M I 0A K1 K L B /a A G H 1 PROPOSITION XIV. THEOREM XIF. Cylinders (noab & iKHG),aiidcoii»(BEA & gfh) upon equal bares (B A & G H) : are lo one another as thdr atatuds (C E & D F). HTDOthefis. Thefis. tdfo tbi €mut BEA & GFH* iavt znalt.QE: alt.DF. fmudhafei. II Qmf BE A : our GFH s=ii2f. C£ Preparation. 1. In the axis of the greater cylinder A O N B» taift a part PC =: to the altimde of the cylinder G I KH. a. Thro' the point P« pafs a plane L M» oUe. to the.bafe BA» which will divide the CTiinder A O N B into twt> t/iin- ders, viz. BAML&LMON. Demonstratiok. B E C A U S E the cylinder B N O A is cut by « plane pile, to its bafe, (Prep. %). 1. The cylinder N O M L : cvlinder L M A B = P E ; PC. P.ij. Baz. 2. Confequently, cylinder NOML + LMAB : cylinder L M A B —"PE-I-PC:PC. P i8* J5- c- But the cylinder N O ML + L M A B is = to the cylin.BNO A, PE + PC = EC. ifx.i.S. I- Moreover, the cylinder L M A B is =: to cylinder I G H K, &' P C = DF. (Prep. t. J. 3. Therefore, the cylinder B N O A : cylinder IGHK = alLEC : alt. D F. jp. y. B, i. Which was to be demonftrated. i. The cone B E A is the third part of the cylinder B N O A. 7 ^ » And the cone GFH the third part of the cylinder G I K H, j ^"^^ ^'^^ 4. Confequently, the cone B £ A : cone G F H = alt. £ C : alt. D F. Pa j. M. (. Which was to be demonftrated. 1 1, PookXU. Of EUCLID. 131 PROPOSITION Xy. THEOREM Xf^. _ HE bafes (AE & GK), and altitudes (C F & O L), of the equal cylinders (A B D E & G H I K), and cones (A C E & G O K) : arc re- ciprocally proportional, (that //, the bafc A E : bafc G K = alt. L O : alt. C F). And the cylinders and cones whore bafes and altitudes are reciprocady proportional : are eoual to one another. Hypothefis. Thefis. /• rhtcjlindtrsk^h^UGYilYiareequa^, Bafe AE : h/eGK = alt. LQ \ //. Thf C9nes AECWGOK^e equal. ^.CE. Preparation* , O9 cut off the 2. Thro' the point N, pafi a plane P M pile, to the oppofite planes of the cylinder H I K G. I. From the greater L O, cut off the altitude L N := the altitude CF. P. 3. Jf, ^, B I. Demonstratiok. ^ E C A U S E the cylinder GHIK & PMKG have the fame hafe. The cylinder GHIK: cylinder P M K G = alt. L O : alt L N. But the cylinders ABDE&GHIKare equal. (Hj^. i J. Confcquently, the cylinder A B D E : cylinder PMKG = alt. LO:alt.LN. Moreover^ the cylinders ABDE&PMKG have the fame al- titude. (Pn^- I./ Therefore, the cylinder A B D E : cylinder P M K G = bafe A E : bafe G K. But the cylinder A B D E : cylinder P M K G = alt. L O : alt. LN. r^rg.z.). And the alt. L N is = to the alt. G F. (Prep. i.). From whence it follows, that bafe A E : bafe G K = alt. L O f P.i i. J5 c •»lt. CF. IP.7B.I Which was to be demonftrated. P.14. -B.12. P. 7. B. s. Pii.B.12. ^ ■ I 4 '4 -iv ■t-Si - -^1 vJ SSA The ELEMENTS BookXn. 1 Hrpotiicfii* B«fi C K : Im/e AE ssait.CT. alt. LO. Thefts. I.CfLABDEiszsfcjLGHlt. U, Th«ctmACEu=zi,iAecmtC(X. B |L Demonstration. ECAUSE the cjUoders GPMK& ABD E, have tiebam altitude, fPr^- %). I. The cjUnder GPMK : cjiinder ABD £ 3 ht(e GK : bale A £. F\\. I.11. Bat the bafc G K : bafe A E= all. C F : alt. LO, (Hyp), " %. CoDfeqaemlv, thecyl. GPMK : cjl. ABD£ = alt.CF : alt. LO. P.ii. B. $. Moreo?er, the cjlinders GPMKftHIKG hare the bmt baie. 3. Therefore, the cvL G P M K : cyl H I K G salt. LN : ah. IXX P.14. AI^ But tht altitude L N is zs, to the altitude C F» (Prtp. i). A. From wheoce it follows that the cyUoder GPMK : crlioder G H I K = altitude C F : altitude L O. P. 7. B. j. But the cyliader GPMK: cjlinder A B D E = alt. CF : alt. LO. fjirg. zj. t. Therefore the cylinder GPMK: cylinder A B D E = cylinder GPMK:cylinderGHIK. P.ii.&|. 6. Confcqaemly, the cylinder ABDE is z= to the cylinder GHIK. Ri4^ B. ^. Which was to be demonftraied i. The cones A C E & G O K being each the third part of the cytin- dersABDE&GHIK. Aio. Aia. And thofe cylinders being equal C^rg. 6J. 1. The cone A C £ is f: to the cone G O K. Ax.j. A i. Which was to be demonftrated. 1 1. BookXil PROPOSITION XVI PROBLEM L WO uo c yi l ctfdM (ABCI& DEF) being given having the fame center (G) ; to defcribe ii> the greater ( A B C I) a p^jgon of ad even nam^ ber of equal fides, that ihall not meet the lefler circle (D E F)» Given. ftnM untquA^^K BX C^ I>E F haknng tbffamt cinor G. Sought. T^dkfcrUt in thegrtaiir A B I, « pdygtn nf an even nuwhtr tf tpiat fiJu^iiuu/baUnu Ul^ (s)bD&F. Refolution. 1 . Draw* the diameter A C ia the greater © A B I which wilt c«rt the O ef the D P ta the point & 2. Thro' the point E, draw the tangent HEI to the f- ^ „ <SF DEF& prodoee it until it meeu the O of the -J «l^ «* ^• © A B I in the ooints H & I. \Prf2. B. i. 3. Cut the feml 0. A B C into two eqtnl pana in the point B. P.30. A 3. 4. Divide the femi arch B C into two* eqoial parta» & To on wrta chr areh R C be < thearch H C Lem. B.12. 5- Draw the chord KC 9e apply it uami in th« O of f i>. i. A a. theOABCL iJ^^.Jil* S3« The ELEMENTS BooklD. B PreparaiiofL From tbe point K, let fall the i. K M vpon tlic dianetcr ( Pn. 1 1* A C> & produce it until it meeu tlie O in L. \h[h^ i* Demonstration. ECAUSE the rem! O ABC, 18 divided into two equal pans at the point B. (Ref, 3.). • And the divHions have been continued until the arch K C has been attained. (Ref. 4../ 1. It follows, that this arch KC will meafiire the Oi an even onmber of times without a remainder, (becaufe it sreafores the (emi O' a. Confequtntlv, the Ihie KC (chord of the arch KQ] will be the fide of a polygon, having an even number of equal fides infoibed 19 the 0. Moreover, the two VHEM& KME being two L. (Rif.i. W Pr^). 3. The line K M or K L is pile, to H E or HI. But tbe line H I is a tangeni of the D E P in E. (Ref. %,f 4. Confequently, K L does not meet the D E F. . But jCC is < K L (P. 1$. B. 3J becaufe KC is remoier fiom . the center than K t* (Prtp,)^ 5.' Much mo^e then KC will not meet the D E P.. And fince the other (ides of the polygon infcribed in the A B C I are each r= to K C. (Rtf, 5.;. 6- It may be demonftratcd after the fame nuuiner, that they do not meet the D £ F. 7. Confequently, there has been defcribed in the AB CI, a poly« gon having an even number of equal fides, which does not meet the © D E F. Which viras to be done. BookXn. Of EUCLID. oBOBammmmmi 337 COROLLjfRr. M. HE UtuKL, ivbicb uJutotbe diam*tir A C, C^ jtins tit two fidet ^Q \i LC, rf tbif4fm<uAitbmttt4Utbti>itrtimty^thisfamJittmtttr: dta ntt mtt tbt kjftr tircb, (Arg. 4.). Uu 33* The ELEMENTS BookXH PROPOSITION XVII. PROBLEM n. WO fphcrw (KON «r G F E H) having the fiwiic center (1) bojS given : to defcrtbe in the greater (K O N) a i»Iyhcdron (KCSPTQ V R O &c ), the riiperficies oF which fliall not meet the lefier (^kat. Given. Sought. TwconciMtric/^biffsKOVl^GTEH. I. A pO^dnm KPTRVO &c J^ . in the £F9ater Mere KOK . //. The fi^fices of^iMcB f^M^,^ mfmcitheUfirfpienGFlH. RefoJution. I. Cut the fpheres by a plane K B N D paifing thro' their center, a. In the A B C D, draw the diameters A C & B D, interfeaiog f P<»/i- » < each other at rijrht angles. C^.'*- * '' 3. In this greater © ABCD, defcribe the polygon C KLMD &c. fo as not to meet the leflcr © G F E H. P-^^ * '** Draw the diameter KIN. From the center I, ere6t on the plane of © A B C D, the ±1 0, C i^ia- 1 '^ & produce it to the furface of the greater fphere in O, XN-^^ *' Thro' I O, & the diameters A C» B D, & K N, pais the planes AOC, BOD, &KON. Divide the arches A O C & K OIQ into an even number of parts in the points P, (^, R, S, T, & V, &c £> that each of thoie parts be equal to C K. Draw the ftraight lines S P, T Q^ V R. 4- 5- BookXIL Of EUCLID. S39 /. Preparation. I. From the points P fr S» let ftll the X P X & ST upon the plane of the © ABCD. ^12. B.it. a. DrawYX. BDemonstratiok. E C A U S E the planes K O N & C O A pafs thro' I O. (ReK). And that I O is X to the plane of the © A B C D. (Ref, 5./. I- Thofe planes K O N & C O A> are X to the plane t)f this ®« P.18.B.11. But the poinu P & S are in thofe planes C O A & K O N. And from thofe points have been Jet fall the X PX & SY. (I Prtp). *. Confcquently, the points Y & X are in the lines K N & C A. P.38. Bax-. In the ACXP&KYS, VPXCis= VSYK. (LPrep i). Moreover, V PCX = V SKY. ^^.27.^3 ;, & CP = KS, (Rm/i). 3. Therefore, the fides P X & X C are sr to the fides S Y & Y K. P.a6. -B. i . But the rays K I k C I are equal. D.i 5. J?, i. Therefore, if the e(|uals X C & Y K be taken from them. 4. The remainders, vix. IX & YI will be equal. Ax.yB, i. 5. Confequently, I X : X C = I Y : Y K. P. 7. B. c. 6. From whence it follows, that X Y is pile, to K C. P. a. B. 6, But P X which is :=: to S Y (Arg. 3.) is alfo X on the ftm^ phuM withSY. (LPnp.i.J. 7. Therefore, P X is alfo pile, to Y S. P. 6. B.i u 8. Ukewile, S P is =& pile, to X Y. P.33. J». 1, But XYis pile, to KC. (Arg. 6). 9. Therefore, S P is aMb pile, to K C. P. 9. B.i u lo.Confequtfntly, the fides of the quadrilateral figure K.SPC are In the fame plane. f, 7. J|*M« 1 1 .It may be demonftrated after the iame manner, that the fides of the quadrilateral figures TQ^P S, V R Q^F, & of the A R O V, are each in the lame plane. i2.And as it may be demonftrated in this manner, that the whole fphero is incompaffed with fuch like quadrilateral figures and triangles. 1 11. Confequently, there has been deicribed in the greater fphere a po* lyhcdronRPCKTVO, &C. Which was to be demonftrated i. //. Preparation. 1. From the center I, let finll on the plane K S P C, the X I Z. P.i i. ^.1 1. 2. Join the points Z P, Z C, Z S, & Z K ; S I & P L Pof.i. B. i. 3. From the point K, & in the plane ABCD, let fidl the X K f on the diameter C A. ^ P-ia- B. 1. B _ E C A U S E in the A KCI, the line YX is pile, to KC. CArv.6). 14. IC:CK = IX:XY. P. 2. B. 6. But IC is> IX. Ax,B.B. I. ic.Therefore, CK > X Y. P.,.. B. c. But PSis=:toXY. (Arp.S.l B6.From whence it follows, that C K is alfo > P S. P, 7. B. 5. 1 7.1t may be demonftrated after the tuac manner, that S P is > T Q, & T<i^> VR. ^ £ L E M EN T S BookSC T^^lZ^ ^ 3 .*4- V..^e?i«. 1 Z ^ c^^ .^ -;: i^e ^ 1 Z F.I ZC» IZK. &IZS. f^47 « Z? = ZC = ZR = ZS. ^?.4(v«.i Cf X— t»cc frca: ihe crnter Z, at the difttncc (Cr. 3* Z P. m J -if? t^-o* tbe rc'rt? R« S & C, ft the qimdrilatenl f r-e R S> C w .' V^ cv c: Sec Is a 0. ^. Fu'- ii< rcu: Sc« cf "be c::w::?*reil tipire were equal ; the arches % h ch lubter^i tlii^ra wi- be lb a Jb» * will be each a quadrant of t'-eO ^F zS B 3\ Eul K SX K &C P, are equi! [Rff - ^ ftCKis > SP. (yfrg.i6.y icFrom whence it is ncanit^d, that ihe three fides K S, CK, frC P, lubtcDd moi-e :han the tb:ee quadrants of the Q ; &. coofcquentlj, C K {which i* = to K S & C P) fubtends more' than ^ quadrant. 2i.Conf€Juemlr» the V CZ K at the center is > L. 22.Htrnce it follows, that the D of K C is > D of Z C + D of Z K. P.ii- ^■ But the D of Z C is = to the G of Z K. (P. 46- B. i. Ctr.y): Becaufe. Z C i< = to Z K. {Jfg 18 ). Z7 'I hercforc, ihe D of KC is > the double of the D of ZC. The V A I K is > L. (being = VAlb.+ VDlK,&VDIA being a L.. /?c/- ^ )• Moreover, Y A 1 K is == V I CK + V r K C. ai.Confcquently, V I C K + V I K C are > l«. . . But V iCKi8 = to V CKI (P.v^.iOt>ccaufeKIi5=:toCI. ^c.Thcrcforc, 2 V I C K are > a l«, & V I C K > half ot a,L.. a6. Wherefore, in A C K, the V C K a ia < half a X^. . R^yB, P31. B. 1 Aje.y.B. BxJkXH; Of..EU.C.:LiJJ>' ^ 34i: But V I C K is > hilf a l«. (/#«• asO- -: "' a^^Erom whence it follows, tl«|t ih'thc' 4^C'» K, ibc{ fide K o» oppofiie to'tfic V RCoor Kt i is >'thcfideCa; oppofiteJotie.yCKV P,i8. * i. 28.Conrcqiicnt!y, the tO of KC (which ia =5c to the D of K'» + tb? DotCa. P.^7.B.i.)h<2DofKo. • And it has been demonftrated {jlrg. 23) that the D of K C ia > the double of the D Z C. ao-Wherefore, 2 D of K « will be > 2 □ of Z C. 30.Hence, theDofKfli8> ihcDofZC. But theDofICis = totheDofIZ+ thcDofZC.l And the □ of I K ( = to the D of I C D. ic. ^. i. & V P.47. B. i, P. 46. B, |. Cor. 3.) is =?to the n of I0+ ^^^ D of K •• ) ji.Therefori, D of 1 Z + D of Z C arc = to D of I •+ Dof K •. ^^.i. B. f. Therefore, if from one iidc be taken. the D of Z C, & from the other the D Ko, (which are unequal, jirg. 30). 32.The remainder, viz. the Q of I Z wiU be > the □ of I ». Jx.^. B. i, 33.Con{equently, I Z is > I o, ^ But the line K ^, (which is X tq the diameter AC: //. Prep.^- is without the fphere.E F G H, & cannot ppeqt i^. f .i6. ^.i*. M^.)i ibat if, I « is > 1 G. ' And I « is < I Z. (^rg. 23.). 34.Much more then I Z, (whfch is much > lO) does not meet the furface of the fphere E F G H. 3 {.Wherefore the plane K S P C, in which Z, is the point neareft the center I, does not touch this fphere £ F G H. 36-It may be demonftrated after the fame manner, that all the other planes which form the polyhedron do not meet the fphere £ F G H. ^^.Conreaueatiy, there has been defcribed in the greater fphere KON a polyhedron K P T R V 0> &c. whofe planes do not meet the leiier fphere. Which was to be demonftrated- u. I COROLLART, F in tnmfpberes (here be defcribed twojtmlar pofybedrom ; tbofe pofybedrons tviH b^ to one anotber in tbe trij^icate ratio of tbe diameters of tbf fpberes in njobicb tbey art defcribed : For tbofe pofybedrons being ftmilnr^ are bounded bj the fame numb^ of planes finnlar eacb to eacby (D. 9. B. 11.); confe^entlj each polyhedron may be dit/ided into, pyramids y bn^ng all their ^vertices at the center of the fphere ^ £jf for bafts tbe planes if tbe pofyhedron, befidet all the pyramids contained in tbe firft polyhedron are fimtlar to all the pyramids contained in the fecond polyhedron^ eacb to each i conpquently, tbey are to one another^ {viz. the pyramids of .the fir fl polyhedron to tbe pyramids of the fecond) in the (riplitate ratio of their bommgous fides ; that is, of tbe femi diameters of their fpheres. (Cor. .P. 8. B- iz.) From ^whence it follows, (P. 12. B.5.) that all the pyramids compofing the firfl polybedrony are to all tbe pyramids compofing tbe fecond polyhtdron in the triplicate ratio of the femi diameters of tbeir fpheres j ^ (P. 1 1. & 1 1;. B. «;.) that thefirji polyhedron is to the fecond in the triplicate ratio of the diameters of tbeir fpheres. 34* The ELEMENTS BookXQ. 1 PROPOSITION XVIII. THEOREM XVI PPHERES(ABCD& HILK) have to one another the triplicitt ratio of that which their diameten (AC & K I) have. Hypothefii. Thcfis. A C is the dmmetefftbifpbm A B C D, Sphift A B C D : fihm^Wl'l^ tfKlih€dmmiierrf$l>e/fhiftHlL¥i AC* : K P. Pemonstration. If not, A Sphere < or > the fphere A B C D will be to a fpbere HltK = AC« : KR / Suppojition. Let the fpbere V R T be < the fphere A B C D, (o thlt ihefphereVRT : fphere HILK = AC* : KI». /. Preparation. |. Place the fphere V R T (b as to have the fiune ceater with the fphere ABCD, aaEFG (which is 3= to the fphere V R T). a. In the greater fphere ABCD defcribe a poljhedron the (iipcrficies of wnich does not meet the leilcr fphere EFG. i^.i?- ^-'^ 3. In the fphere HILK defcribe a polyhedron Oj to that in the fphere ABCD. E C A U S E the polyhedrons ABCD&KHILareCO- (I iV«/.i. WaJ. |. ThepolyhedronABCD : polyhedron K H I L = AC» : KI». f Pi?-^**- {Or. B Book XIl Of E tJ C L I t). Hi CR 4. B, t \Car iP. 7. An. And fincc the fphcrc VRT : fpherc HIKL s=t AC^ : KP. {I Sup.) Moreover, the fptere VRT is = to the. fphcrc EFG. fPrf^). a. It follows, (inveitendo) that the fphere H I L K : fphere EFG =:KI* : ACV 2. From whence it follows, that the fphere H I L K : fphere EFG = po^yg- K H I L : poljg. A B C D. Pu. B. 5. But the fphere H I L K (3 > the polyhedron K H I L. Jx.S. B. 1 . 4. Therefore, the fphere EFG (or its equal VRT) is al(b > the polyhedron A B C D. P. 14. A $« Mi ^ ^^« EFG ts contained in the polyhedron ABCD (Ptefz)^ ^. Confequently, the part will be > the whole. 6. Which U impoffible. 7. Confequently, the Cube of tlie diameter (AC) of a fphere (ABCD1 is not to the cube of the diameter (KlVof another mhejsl (HHK) as a fphere V H T» Jfis diaa ibi^ 6ift fyhfinc (A B QD)p is to this fecoad fphere HI LK. IL Su^Jttion. Let the fphere ZX Y be > the fphere A B C D, fo that <ihefp4iereZXT:4|«ri«HILK=sAe«: KH. // Preparation. Take a fphere VRT, fuch that the fphere ABCD: fphere VRT=:AC«:K.P.. ' B E C A U S E the fphere X Z Y : fphere HILK = AC*:KI«. (IL Sup.}. And the fphere ABCD: fphere VRT = AC« : KI«. (ILPrep). 8. The fphere X Z Y : fphere H I LK =: fphere ABCD : fphere VRT. P.lt.B.e: But the fphere X Z Y is > the fphere ABCD. (IL Sup.), 9. Confequently, the fphere H I LK ta alfo > the fphere VRT- P.14. B. <. But it has been demonfhated {Ai^. 7.), that the cube of the dia- meter (A C) of a fphere (ABCD) is not to the cube of the dia- meter (K I) of anoiiier fphere (H I L K), as a fphere A B C D if to a fphere lefs than H I L K. jcThcrcfore, the fphere VRT is not < tbe fphere HILK (as has been proved, Arg. 9.). 11. Confequently, the fphere XZ Y is opt > .the tf^bmt A B CD, (as has been fuppofcd)* 344 The ELEMENTS BookXU. Ai Theiefixe, as die fiippofed %hae canaot be ddier < or > tbe fyhere A B C D. i2.1t will be equal to it It Jrom whence it follows, that the fthetcABCD : %haeHILK = A C» : K I«. i^Y B. y COROLLARY. iJpHERES mumamaaMtatJmUrtt^lMrmitfcriM 'm Otm (Off. P. 17. B. la. ft P. II. B-s-) FINIS.