wwfismifi/tm ANCIENT MASONRY- BOTH IN THE THEORY and PRACTICE, Demonftrating the USEFUL R U L E S O F ArithmeticGeometry , and Architecture, I N T H E PROPORTIONS and ORDERS of the Moft Eminent Masters of All Nations, v i z. VITRUVIUS, BRAMANTE, JULIO ROMANO, MICHAEL ANGELO, CARLO CESARE OSIO, ANDREA PALLADIO, VINCENT SC A MO ZZ I, M. J. B AROZZIO of Vignola, SEBASTIAN S E R LIO, DANIEL BARBARO, L. B. A L B E R T I, P. C A T A N E O, P. d e LOR M E, VIOLA, J. BULL ANT, JULIAN MAU-CLERC, J. BE RAIN, SEBASTIAN leCLERC, CLAUDE PER AULT, INIGO JONES, Sir CHRISTOPH. WREN, &c. &c. &c. And alfo of the CAR I At ID ES, T E RSI A NS, FRENCH, SPANISH, and. ENGLISH. Together with Their MOST VALUABLE DESIGNS Temples, Triumphal Arches, Portico’s, Colonades, Piazza’s, Arcades, Frontispieces, Gates and Doors, Windows, Niches, Entablatures, Pediments, Capitals, Festoons, Trophies, Ba llusters. Balconies, Ba llustrades, ClELING-PIECES, Chimney-pieces, Floors, Pavements, Arches, Groins, Stair-cases, Roofs, OBELISqjJES, Ornaments, See. The Whole interfperfed with Critical Remarks and Observations on each Master, Illuftrated by above Three Thou [arid Examples, Engraved on Four Hundred and Ninety four large Folio COPPER PLATES, w I T H A D tcT ion aria l INDEX, explaining the T b r m ; of A R T ufed herein. By B. L A N G L ET LONDON: Printed for, and fold by the AUTHOR, at Parliament-flairs, near Old Palace-yard, Weftminfter; J Milan-, mer-againfi the Admiralty-office; and J. Huggonson, Printer, in Chancery-lane. MDCC XXXVI. ■Sift -A'% * §pM* ■3 ift^ IsY The Moft Noble His Royal Highnefs Francis Duke of Loraine, \Charles Duke of Richmond, G. M. Anno 172c j Charles Duke of Marlbrough, Francis D uke of Buccleuch, G M. Anno 1724 James Duke of Athol, ^ John Duke of Montagu, G. M. Anno 1722 Charles Duke of §peensberry and Dover \Thomas Duke of Newcajlle, Ihe Right Honourable Henry Earl of Pembroke, Daniil Earl of Winchelfea and Nottingham, Philiy Earl of Cbejlerfield, William Earl of Ejjex, Will,am Earl of Albemarle, JohnEsirl of Craufurd, G. M. Anno ‘' 734 ; John Earl of Loudoun, G. M. An¬ no 1736, Thomas Earl of Strathmore, William Earl of luch/quin, G. M. Anno 1727, Alexander Earl of Hume, William Earl of Kellmarnock, James Earl of Wemys, Charles Earl of Port more, James Earl of Abercorne, G. M. Anno 1726, Earl of Ankram, Earl of Horne, Anthony Lord Vifcount Montaru G. M. Anno 1732, Thomas Lord Vifcount Weymouth, G M. Anno 1733, Simon Lord Vifcount Harcourt y Jchn Lord Vifcount Tyrcornel. Lord Vifcount Ershine. Lord Vifcount Vane, Lord Vifcount Bruce, George Lord Carpenter, I John Lord Delawar, Thomas Lord Lovel, G. M. Anno I73E Thomas Lord Southwell, Henry Lord Colerane, G. M. An¬ no 1728, James^ Lord Kmgjlone, of Ireland, G. M. Anno 1729, Charles Lord Cathcart, laird Carmichael, John Lord Hobard, Lord Belhaven, And to all others The Masters Right Honourable and Right Worfhipful of M A S O N R Y My Lords and Brethren, HE, Principles and Practice of Ancient Masonry. icing the Sub,eel of the following Sheets, to whom can *° JU1 7 ,nfcnbe as to Your Moft Noble, Right Honourable, and Right Worjhipjul Selves : not only with regard to your being Masters thereof, but to your great LcZ rage- The DEDICATIO A. ra°ement given, and Honour done to the Art, as well as your mold affectionate Refpecd manifolded to every Brother of the pratemit}, an illuftrious Example to all other Nations. The falfe Conjectures of Masonry, and its great Ufc in fur- nilhing the Fellow Craft , and other Artificers in general, with the ufeful Rules, Proportions, and Examples of the mold eminent tArcbi- tefts, that have lived in all Nations, have been not^the only ° nves of my compiling this Laborious, Extenlive, and Mold Ufeful Work : But as it is the Duty of every Man to communicate fuch Know.edge as he is blefs’d with, that will be of fervice to the Publick, 1 thuTore thought this my Duty ; and which, being happy under your _.ord- flips’Protections, cannot fail of a kind Reception among the § ood- natur’d and judicious Part of Mankind. The Induftry, Care, and Pains that I have taken herein, for the Service of my Country, your Lordflips will readily perceive; and that the cenforious Part of Mankind may thereby not only learn the Principles of this Most Noble Art, but how to correcd their Miftakes for the future. That your Lordlhips may long continue the Encouragers of Arts and Sciences, is the fincere Prayer of My LORDS, Tour Mojl Obedient, Humble Servant, and Affectionate Brother, B. LA NG LET. THE. THE INTRODUCE IO N. By X- X HE Number of Books daily publiili’d on Architecture, and the little Ule they are of to Workmen in their Pra&ice, for want of a juft Knowledge of thofe Sciences on which Architecture immediately depend, gives fo fenfible an Idea of the Advantage a Work of the following Nature would be to all concerned in Building, that a Society (who daily experience and praciije the feveral Arts which lead to a general Knowledge of ArchiteBure) have concluded to demonftrate the fame, in a plain and eafy manner, throughout all its various Parts, for the Ufe of Arti¬ ficers in general. And, as every thing which is necefiary to make compleat Workmen, and good ArchiteSls, will be carefully and ingenioufly handled in the following Work, it will contain more Knowledge than all the other Books of Archi¬ tecture. And for its Utility to thole concerned in Building, as well as other Employments, we hope nothing can be more advantageous; becaule every Branch fhall be carefully and methodically handled, and every Page will be perus’d and modell’d by the whole Society. And as fo voluminous a Work is the Labour of a Society, it will contain the Knowledge of many, and more particularly fuch who daily experience the Advantages of it in the feveral Parts of Building. Vitruvius 4 - INTRODUCTI O N. r Vitruvius reckons no lefs than Twelve necefiary Accomplllhments to make a compleat Architect:, viz. i. He muft be donl and ingenious, i. He mud: be lit irate. 3. Skilful in defining and drawing. 4. In Geometry * w Op ticks. 6. Arithmetic^. 7. Hi [Joy. 8. Philofophy. 9. Mtifick. 10. Medicine. 11. Z.tf'w. And lit Ajlrology. But ii the Law, Phyfick, end Aftrology were left out, I think we fhal 1 treat of all the reft, Mufick not excepted, which will appear by our harmonick Proportions. I t would be of no Advantage to fuch who are pleas’d to en¬ courage this Work, il we were only to colled! the Labours of other Men, and place them in our Magazine oj Architecture, as many have done, and particularly ONE lately publifhed under that Title, -which is wholly compofed of other Mens Works, that were before in different Books extant: therefore we deiign to offer nothing but what is entirely new, and of our own Production ; or at leaft handled in a manner fo different, and fo much more advantageous than what has been hitherto attempted, that every one will find his Expence fufficiently compenfated. Wr. are confident, that not only thofe who pradtife the Art, but likewife Gentlemen may find an agreeable Amufement in the Perulaf of this Under¬ taking, becaufe there is nothing in the OEconomy of Art, but in fome meafure av ill have a Place in it • there will be feveral agreeable and enter- taining Propoftions folv’d, as they occur in their Places, and many ufejtil De- figns, both in the Plan and Elevation, which will add much to the Advan¬ tage and Satisfaction of the Subfcribers. As nothing makes a Man more valuable than Learning, and thofe Accom- / plilhments which are attendant on it; io on the other Hand, nothing ap¬ pears fo deipis’d as the Ignorant and Illiterate, or at leaft ought to be more pittied, for being depriv’d of the happy Advantage which others enjoy. Learning, in all its Branches and Characters, may be properly laid to diftinguifh us from one another, as well as from the reft of the Animal Creation, more diftinetly than Speech : Birds acquire the Faculty of fpeaking, but it is for the moft part milplaced ; it is only the retaining Words and Accents, not knowing how to apply them : In fhort, they may be compar’d to an Engine juftly performing its Rotation, and not knowing its own Ufe : And thus it is with many Workmen, who by Habit and Cuftom perform divers Operations, but know not the Caufe or Reafons thereof. The natural Embellilhments of the Mind, by Convention and Society cultivated and improved, and by a Propenfity to Knowledge rectified and regulated, foon acquires the Faculty of Learning • and in whatever Branch of Learning the Mind makes its Progrefs, it attains its defired End. Learning is a jewel of ineftimable Value, and he who is pofiefied of it, po fie lies all Things. The Gocds of Fortune, by Multitudes of Cafual- ties, perifii and are deftroy’d : Earthquakes, Innundations and Tempefts ruin and impoverifh many Countries ■ but no Misfortunes foock the Soul of the Phihfopher • in Prolperity and Adverfity he is ftili the lame • his Wil- dom, by making Excursions into the Channels of Fortune, make every Stage of Life equal : Knowledge is acquired by Study and Afiiduity, and by culti¬ vating thofe Seeds of Wifdom which Nature has implanted in us. We ihould look into ourfelves, but more particularly our Children, and endea¬ vour introduction. vour to difcover which way Nature has intended to direct the Channel of their Genius; if to the Mathematicks, fuch Branches of Learning w hich lead to thole Arts, ihould be carefully taught them, and not ftop the Current by throwing in Lumber of Lana, Hifiory, &c. which are contrary to Nature’s Defign, and ruins the Artift : Nor muft he who defigns to be a found Workman or Architett, load his Mind with Politicks - he will find Matter enough herein, to employ his whole Study, to become’ Mafter thereof. No Art is fo narrow and confin’d, but it will take up much Time to be acquainted with, and it’s better to know one Thing well, than superficially to know many. There are many Branches of Learning in every Ait, and thofe Paths which lead to them muft be carefully trod • Circum- Ipection and Diligence are requir’d to finilh all our Undertakings. ’ Learning is a lopick which leads us from one Labyrinth of Pleafure to another ■ it is as extenfive as the Univerle ; it confifteth of infinite Di- vifions ; if we trace it from one Channel to another, it never lofes its Beauty ; it, Luftre is always apparent; and whatever Shape you view it in, it always charms you. The Notaries to Learning are 'very few, and the giddy Multitude, blinded with fugacious Pleafures and Follies, have not Good-nature enough to refpect them. I cannot pafs over the Advantages Mankind have reaped by the Wifdom and Unity of our Royal Society ; they have planted Knowledge in the Minds of the Unknowing ; they have cul¬ tivated the natural Genius of many : in ihort, like Orpheus with his Lyre, they have made Stocks and Stones attentive to their Mufick. As to the natural Genius, we fee how many lively Inftances of it have appear d in the Hiftory oi all Ages. The Alan in whom the Seeds of Knowledge are fown, in fpight of all the Obftacles of Fortune, will be ftill the lame * the Ideas which Nature originally ftamps on the Mind, cannot be worn out by Poverty, Want of Education, proper Oppor¬ tunities in Societies, Books, Inftruction, fgrV. I fay, in fpight of all thele Impediments, the bright Ideas will lhine ■ they will appear beauti¬ fully thro all the little Clouds of Fortune, and, like the Sun-Beams on the Surface of the Water, reflect their benevolent Rays on the Eye of the Beholder. How happy is the Fate of that Man, whom Nature, in fpight of all Obftiuctions, has fupply d with beautitul Embellilhments of the Mind, yet wants the nice Correction and Care of Art, to cultivate and im¬ prove them ; to draw them gradually from the little Errors of ignorant and ill-digefted Opinions, imbib’d in Minority, thro’ the Want of a juft and regular Education. Acquired Knowledge flows from the improving and refining the natural Genius. The Seeds of Learning, when firft fown, are like a little Embrio which gradually increales in every Degree of Time, ’till it hath attain d its Maturity. It is firft improv’d by proper Principles inftill’d, iuitable to the Nature of the Genius which is to be refin’d • it takes Root, and fpreads itfeli llowly into Form, which, like a young Fruit-tree, by pru¬ ning and regularly dilpofing, keeps from ihooting into fuperfluous Branches. As Thorns bring not forth Thiftles, fo no Art or Means can make the Man who is naturally horn a Mathematician, to be otherwile ■ and the great Painter and Architect are fu by Nature as well as Art ■ and probably there are many fine Men, now buried in Oblivion, who, if they had the Happi- nefs of a Royal Encouragement, might become Newtons in Philofophy, Raphaels in Painting, and Palladios in Architecture. B Super- t 6 INTRODUCTION. Superficial Learning is a fort of middle Path between natural and acquired Knowledge ; it is as it were the Shell or a gilded Outfide without Value, a Shadow without Subftance, a Multiplicity of Ideas with¬ out Order, a Medium between fomething and nothing ; in Ihort, a Body without a Soul : Such a one grafps at every thing, and can retain nothing ; of which Mr. Pope finely obferves, One Science only, will one Genius Jit ; So vajl is Art, fo narrow human Wit. Pope on Criticifm. I have thus far ventured to define natural, acquired, and JuperJictal Know¬ ledge. I propofe now to fliew you the Ufes of Learning, as far as it re¬ lates to Mankind in general, and Societies in particular.^ In general it is fubfervient to all, in all the Stages and Stations of Life : Our walking, fitting, lying down, rifing, &c. are performed by mechanick Powers ; and tho’ the Ideot, the Illiterate, and Unthinking Part of Mankind, cannot dif- cern it, yet every Mathematician can very evidently demonfirate it ; every Action ii a mechanick Operation that is perform d by the Laws of Mecha- nifm. The Motion and hidden Velocity of our Bodies, are the Effects of a Mathematical Power, and the Contemplation of it elevates us a Degree above the reft of our Species. Learning is necefiary for the hire Direction of Affairs in all Parts of human Life, but more particularly in making Laws, diftributing Juftice, Trade, Traffick, and Commerce 5 in difcerning the Motion of the heavenly Bodies ; in Weights, Meafures, Travel; in Ihort, in every thing w hich concerns Society to be acquainted with. W ithout Reafon and W lldom, Laws were not firft form’d, model I’d, and dehgn’d • nor could Juftice be impartially diftributed. W ithout Navigation and Geography, Traffick and Commerce could not be *, nor could w e judge of, nor defcribe the . Mo¬ tion of the heavenly Bodies, without Aftronomy \ by Geometry, Weights, Meafures, and the Power of Lines are preferved j and indeed we find no one Branch of Learning but is uleful in lome Part or other of the OEco- nomy of human Nature. Besides all this, the Pleafures which the thinking Mind takes in a Purfuit after Knowledge, are inexpreffible. The Ajlronomer can foar from one Planet to another, and from one Region to another, ’till the Mind is loft in an Eternity ojSpace. The Geographer can travel from one Country to another, thro’ the various Climates over Sea and Land, and encompafs the whole Earth in his Imagination, and yet be only retired to his Clofet, or contemplating in the Field. The Painter can lee Groups oj Figures, and lively Land/tips, to divert the Ideas of his Mind. The Architect raifes in his Idea Numbers of pleafing StruB tires, the Orders growing like tall Cedars, beautiful and proportioned, with a regular Symmetry, and juft Exactnels. The Poet reprclents to himfelf beautiful Hills , and Lawns of pleafing Tal¬ lies and circling Rivulets, the Harmony of Numbers and Nature. The Mechanick ideally fees Multitudes of various Machines for Conveyance of Timber, Stone, Water, &c. all perfect and pleafing to his Imagination. The .Mathematician has his Globes, Triangles, Cubes, Prijms, Glajjes of RefleSlion and Refraclion, Lights, Colours, Shades, &c. All thefe I fiy, by a little Expanlion of the Mind, are fecn as natural, as the Statuary views in a Block of Marble, a beautiful Statue, which only requires his nice Hand to take away the grofs Particles that enclofe it, whereby others may view it with equal Plealare as himfelf. I MUST INTRODUCTION. 7 1 must obferve to you, as Buildings are requifite for private Houfes, fo publick Tran&ctions require much Knowledge in the right Application of our Ideas: And to adapt them juftly to the Wants and Conveniencies, is (tho’ feldom regarded) what appears mod requifite in our Tafte and Execu¬ tions. Courts, Pain cos, Seats of Pleafures, &o lhould be gay and airy ; on the contrary, Tribunals, Courts of judicature, Temples, &c. lhould be grave and folemn, to ftrike the Beholder with a majeftick Awe of the So¬ lidity and ferious Purpofes of the Defign. Seminaries of Literature lhould have a little more V lvacity and Gaity. Much Temper is required in Pro¬ ductions of this Kind, and much Study necclfitry, to know where to apply rightly the different Ideas intended to be executed. A Seat continually expofed to the Violence of the extream Seafons or on a Hill unguarded by W oods, lhould have a Mixture of Solidity and Gaity at the lame Inftant. The Tallies require more Chearfulnefs, while the lhady Rivulet, with the encircling Warmth of chearful Greens, and all the Pleafures of a' rural Prolpect, lhould be pleafing and ferious, adorn’d with Sculptures and Feftoons of the various Products of the Seafons, with an enlivening Variety and Regularity. 1 he lilent doom of 11oods , and the dcjart and unfrequented Places, among winding Meanders, and a lliort bounded Prolpect, requires the mod: airy and beautiful Productions, mixing with the folemn Gravity of Nature, the mold exalted Beauties and Gaiety of Art ■, fuch are the different Products which different Situations require, and fuch are the Neceffities which lhould induce us to apply ourfelves to the Study of a Science the moft pleafing and various that the Art of human Invention could produce thro’ every Age fince the Beginning of Time. As I believe it will be granted that the Ancients had this juft Tafte of Building peculiar to themlelves, fo our Endeavours lhould be ufed for the attaining thofe Rules of which the Ancients were pofleffed ; whofe Build¬ ings, as well internal as external, fo charm’d the Mind, and ravifh’d the Eye, that the Arc la lefts themlelves were, by the Vulgar, often thought to be divinely infpired ; when, in Fact, the Beauty and the Pie afire their Works gave, were only the Effects of a mell-chofen Symmetry, connected to¬ gether accotding to the hai montck Laves of Proportion, which of neceffty naturally produce that Effect upon the Mind thro’ the Eye, as the Cords or Difcords of Mufick, pleafe or difpleafe the Soul thro’ the Ear. Their Decorum was always juft in every Reprefentation, whether feri¬ ous, jovial, or charming ; for this End they eftablilhed a certain Modus to be obferved in the Ufe and Application of the feveral Orders. The Dorick Order was always apply’d to Things majeftick, grave, and ferious, and call’d the Dorian Modus. The Ionick to riant Ufes, fuch as to Temples of Bac¬ chus, &c. call d the Ionick Modus. The Corinthian Order was ufed in Pa¬ laces, Triumphal Arches, and Houfes of PleaJ'ure, and call’d the Lydian Mo¬ dus. By thefe Rules they always ke'pt Pace with Nature, and by a ftrict Obfervance of them, they produc’d the various Effects they were intended for. Thus much by way ol Introduction. There remains now* nothing more than to iheiv you what Branches of Learning are to be attain’d before we have a juft Knowledge of Building, which is the Subject and Defign of the following Work. I am, Si . Martin ' s - Lane , Jan , 15. 1732. Tour Humble Servant) x N. At a Meeting of this Society, this 21ft Day of December 1732, the following Manufcript of Vulgar Arith¬ metics was read, and order’d, that the fame be Printed for the Firft Part of The Principles of Ancient Mafonry, tkc. The Members then prefent, A A JB-- B C- C D- D- G—- G- S- S- H H 0 .— Q- R R S S—*— T-- T X- X -- Z z &c.- 6 cc.- i THE PRINCIPLES O F Ancient MASONRY: 0 8, A GENERAL SY ST E M O F BUILDING COMPLEATED. P A R T I. Of ARITH METIC K. By A- A-. ERHAPS it may be objected by fome Perfons, who are already skill’d in Arithmetick, that to begin this Work with that Art is unnecellary and ufelefs; and more elpecially as every Bookfeller in London is fur- nilh’d with the Works of Arithmeticians, that may be purchaled at an eafy Rate : But in Confideration, that amongft Mankind there are many who don’t already underftand Arithmetick, and being able to read, would gladly learn, could they but meet with an Author that was plain, Wpiori r, and injlruffive ; whole Stile of Writing was adapted to their Under- ftanding • and the various Examples applied to Pra&ice in immediate Buli- nels, incident to their lev eral Profeffions, I have therefore, for their Sakes, begun this Work with Arithmetick. 1 \v ould not be undcrftood herein, that I propofe a new Syftem of Arithmetick ; but it I explain the Principles and common Rules thereof, 10 The \Principles of Arithmetic k. fT In a more cafy and injlrutthe Marnier than has been yet done by any, I have reafon to believe that this fir ft Part will be acceptable, and even to them who are the moft learned in this Science : for altho’ for their own Ufe they may not require fuch plain Reafonings , as are herein contain’d, yet if they have, or hope to have, Children hereafter, they cannot but be pleafed in being furnilhed with a Series of Arts, fo well digefted, and made ready to be imbibed by them in their firft advance to Learning. I AM far from finding Fault with the Labours of any one, and more par¬ ticularly of fuch Gentlemen that have been fo good-natur’d to Mankind, as to communicate to the World their feveral Arts and Difcoveries , for every ones Improvement that pleafes to read them. But when I lerioufiy confider, how flenderly the Principles of Arithmetick have, with refpeft to Buli- nefi, been handled, even by the beft of Authors, I can’t help thinking, but that, had they been more copioufiy explain’d, their Labours would have been of much greater Ufe to the unknowing Part of Mankind, for whofe Information fuch Works were defign’d, than at prefent they are found to be. It was this Motive that induced me to the compiling of this ift Part - which I have endeavoured to render intelligible to the meaneft Capacity * and I hope will prove entertaining and inftru&ive to all my Readers that defire to be expert Accomptants. 1 must defire the young Student, for whofe Sake this Work is made publick, to confider, that unlejs a good Foundation be laid , there can be but little Hopes of railing a found Building. And, therefore, fince the Foundation is to be firft, and well confidered, before any Thought need be had about compleating the Structure ; fo we are firft to confider the Principles and Rules of thofe Arts that are the Foundation of the Science which we defire to be Mafters of, before we enter upon the Art itfelf : For by being too rafh and impatient herein, is the Reafon that many who defire to attain a juft Knowledge of an Art, are deceived, and can never arrive to the Perfection thereof. I shall perform this Part by Way of Dialogue, between a Mafer and Pupil, as being the moft familiar and eafy Way of teaching. LECTURE I. Oj Numeration, or the Manner of expreffing Numbers and Quantities by Characlerificks, and to pronounce their Value. M. \ S my Defign is to inftrutt you fully in all the various Mathematical Tx. y 4' rts that are neceftary to make a compleat Architect, therefore I mull, in the firft Place, acquaint you with Arithmetick • that is, the Art of numbering well, it being the Bafs or Foundation of all other Arts. Wherefore I fhall take fome Pains to explain the Principles thereof, that you may underftand the Reafons of all your future Studies, and be enabled to demonftrate your feveral Operations, as they occur in Pra&ife. For, as I before obferved, unlefs a good Foundation be laid, we can have but fmall Hopes of railing a fubftantial Edifice thereon. I P. Tis The Principles of Arithmetic k. P. Tis reefomtble to believe fo: Therefore prey proceed. Tots fry , Sir, That Arithmtick is the Art of numbering well ; prey how ere Numbers ex~ prefer!. M. Numbers are generally exprefs’d by the common Characfters fol¬ lowing, viz. i, 2, 3, 4, 5, 6, y, 8, 9, o ; or by Roman Capital Letters thus, I. lignifies One ; II. Two ; III. Three ; IIII. Four; or thus, IV. V. Five • V,', 5 , VIL Seven 5 VI11 - Eight; IX. Nine ; X. Ten ; XI. Eleven ; XII. Twelve ; XIII. Thirteen ; XIV. Fourteen ; XV. Fifteen ; XVI. Six¬ teen ; XVII. Seventeen ; XVIII. Eighteen ; XIX. Ninteen ; XX. Twenty ; XXI. Twenty-one; XXII. Twenty-two; XXX. Thirty; XL. Forty- L. Fifty ; LX. Sixty ; LXX. Seventy ; LXXX. Eighty ; XC. Ninety ; C One hundred ; CC. Two hundred ; CCC. Three hundred ; CCCC. Four hundred ; D. Five hundred ; DC. Six hundred ; DCC. Seven hundred ; DCCC. Eight hundred ; DCCCC. Nine hundred ; M. a thouland ; alfo thus, ClD ; and io the prefent Year 1733, is exprels’d, MDCCXXXIII ; or, CI 3 DCCXXXIII. and here cbferve, that as Five is lignified by V. and Six with V and I placed on the Right Hand thereof; {0 on the contrary, when the I is placed on the Left Hand of V, as in the Number Four 1 thus, IV. the I lellens the Value of the V once, and thereby makes it'hour; whereas in the Number Six, the I increafeth its Value one time ; and fo in like Manner when an I is placed after X, as thus, XI. it adds one to its Value, and makes it Eleven ; but if it is placed before the X, as thus,, IX. it takes One from the Ten, and both together lignifies but Nine. The Number Twenty is exprefs'd by XX, but if between the X’s an I is placed, as thus, XIX, the laft X is thereby lefTened one, and the whole three Letters fignify but Nineteen : But had the I been placed on the Right Hand, as thus, XXI, then thole Letters would have lignified Twenty-one. You may alfo fee, that Fifty is reprefented by L, and Sixty by LX ; but Forty is exprels’d by XL : In which laft, the the X is placed on the Left, whereby the L, fifty, is made lels by Ten, and both together fiord- fy but Forty ; but in the former, where the X is placed on the Right Hand, the L, Fifty, is encrealed Ten, and both together becomes Sixty? Again' If on the Left Hand Side of One hundred, C, be placed an X, as thus, XC, the C or Hundred is thereby made lefs by Ten, and lignifies but Ninety ; whereas, if X had been placed on the Right Hand, as thus, CX, it would have encreafed the C, and made it One hundred and Ten. P* f understand yon very well, Sir ; bet prey, how do you exprefs the fame Numbers by the Figures you jirft mentioned to me p M To exprefs them, and all other Numbers or Quantities, by thofe Ten Chara&erifticks, viz. 1,2, 3,4, 5, 6,7, 8 , 9, o, is a moll excellent Invention ; and herein yon are to oblerve, that the firft Nine are called fignificant Fi¬ gures, and the laft, a Cypher, which by ltfelf lignifies nothing. P. Pray whet s the Ufe of the Cypher, Jince that of itfelfjignijies nothing ? M. Its Ufe is to augment a Number according to its Place, as thus, 10, where it being placed on the Right Hand of the Figure l, it makes it Ten , and lo in like manner 20 lignifies Twenty, 30 Thirty, 40 Forty, 50 Fifty, 60 Sixty, 70 Seventy, 80 Eighty, 90 Ninety. 12 The Principles of Arithmetics. Again, If to the Figure io, be annex d, or added one other Cypher, as thus, ioo, then the Value of the to is augmented ten times, and is become One hundred ; and lb in like manner zoo figmfies Two hundred, 500 Three hundred, 400 Four hundred, 500 Five hundred, 600 Six hun¬ dred, 700 Seven hundred, 800 Eight hundred, 900 Nine hundred, 1000 One thoufand, 2000 Two thouland, iezC. P. Very 'well, Sir, I underjland it plainly ; I fee that every Cypher en- creafes the Humber to ■which 'tis annexed ten times ; that is, if to a Figure 1 X add one Cypher, it makes it Ten, and if two Cyphers, it makes it a Hundred, or Ten limes Ten ; and fo on, I fippofe, with all other Figures. Pray pro¬ ceed to the next necefary btflruaion. M. The next Thing in Order, is to Ibew you by the following Table, how to numerate and exprejs any Humber tv hen written ; which is called, Numeration. i One 12 Twelve 123 One hundred Twenty-three *234 One thoufand Two hundred and Thirty-four 12345 Twelve thoufand Three hundred and Forty-five 125456 One hundred and Twenty-three thoufand. Four hundred and Fifty-fix. 1234.367 One Million Two hundred and Thirty-four thoufand Five hundred and Sixty-feven. 12345678 Twelve Million Three hundred and Forty-five thoufand Six hundred and Seventy-eight. 1: :45678c) One hundred and Twenty-three million, Four hundred and Fifty-fix thoufand, Seven hun- [dred and Eighty-nine. t 2 34 767890 One thoufand Two hundred and Thirty-four million, Five hundred and Sixty-feven thou- p i ; gjf dd a [fand. Eight hundred and Ninety. P. Pray is not to numerate and exprefs Numbers the fame Thing ? M. No: To numerate Numbers is one thing, and to exprefs or read them is another. P. Pray fhew me the Difference and Manner of both. Me I WILE : But frft, you are to obferve, that the Figures in the firft Place or Column denoted by a, do each lignify fo many Units, or One’s. -idly. The Figures in the fecond Place or Column, denoted by the Letter b, do" each lignify fo many Ten’s ; that is, the Figure 1 at the Top fignifies Ten, the Figure 2 Twenty, the Figure 3 Thirty ; and fo on with all the others down to the Figure 9, which fignifies Ninety. And if to thefe Fi¬ gures in the fecond Column, you add the Lbiits or Ones in the firft, then the One at the Head of the Place of Units, fignifies but One ; the Figure One in the fecond Place, with the Figure a in the firft Place, fignifies Twelve ; the 2 in the fecond, and the 3 in the firft, fignifies Twenty-three ; the 3 and the 4, Thirty-four ; the 4 and the 5, Forty-five; and fo on with the others. In the third Column every Figure fignifies fo many Hundreds, as being ten times greater than thofe in the fecond Place : fo the Unit at the Top fignifies One hundred ; the Number z, Two hundred ; the Number 3, Three hundred ; and fo on of the reft. Now if to thefe Numbers you add the Numbers in the lecond and firft Places, then the firft Number 1, with the Numbers z and 3 in the fecond and firft Places, makes One hundred and Twenty-three ; the next under them, Two hundred The Principles of Arithmetics hundred and Thirty-four ; the next under them, Three hundred and Forty- five ; and lo on with the others to the Bottom. $dly. A s the Figures in the fecond Column exceed the Figures in the firft ten times, and thole in the third Places ten times more than thole in the fecond, fo likewife do thofe in the fourth Place exceed them of the third Place ten times, and therefore is the Place of Thoufands, and each Figure therein fignifies lo many Thoufands • fo the Figure 1 at the Head iignifies One thoilland ■ and if to this you add the oppoiite Figures in the third, fecond and firft Places, viz. 2,3,4, then them taken together with the Number 1, as thus, 1, 2,3,4, fignifies One thouiand Two hundred and Thirty-four. Again, The Figure 2 in the Place of Thoulands, Iignifies Two thoufand, and if to that you add the oppoiite Figures in the other three Places, as before, viz. 3,4,5, then them taken together with the Number 2, as thus, 2345, fignifies Two thouiand Three hundred and Forty- five • and fo on to the Bottom with the others. 4 thly. The fifth Place is the Place of Tens of Thoulands, fo called from its Figures exceeding thole ol Thoulands in the fourth Place ten times ; and therefore the Figure 1 at the Head thereof, fignifies Ten thoufand, the Figure 2 under it, Twenty thoufand, the Figure 3, Thirty thoufand • and fo on with the others. Now if to the Figure 1 in the Head of this Column, you add the Figures againft it in the other four Places, viz. 2,3,4,55 then them taken together, as thus, 12345, fignifies Twelve thoufand Three hundred and Forty-five 5 fo in like manner the next Figures under them, viz. 23456, fignifies Twenty-three thoufand Four hundred and Fifty-fix ; and the Figures 34567 under them, Iignifies Thirty-four thoufand Five hun¬ dred and Sixty-feven ; and fo on in like manner with the others to the Bottom. 5 thly. The Figures in the next, or lixth Place, do alfo exceed them of the fifth ten times, as before, and indeed fo in the fame manner in all other Places, be they ever fo many : Therefore thefe of the ftxth Place become Hundreds of thoulands, as thole in the third Place became Hundreds of Units 3 and the firft Figure 1 at the Top fignifies One hundred thoufand, the Figure 2 under it, Two hundred thoufand, the Figure 3 under that, Three hundred thoufand, and fo on with the others to the Bottom. Now if to thefe three firft Figures you add refpecHvely all thofe in the other Places, then the Number 123456, will fignify One hundred Twenty-three thoufand Four hundred and Fifty-fix • and the Number 234567 under them, will fignify Two hundred Thirty-four thoufand Five hundred and Sixty-feven ; and the Figures 3456.78 under the laft, Three hundred Forty-five thoufand Six hundred and Seventy-eight. 6 thly. The feventh Place of Figures is called Millions, and of the fame Nature as the fourth Place of Thoufands ; for as the fourth Place contains Thoufands of Units, fo this feventh Place contains Thoufands of Hundreds • and therefore it, is that a Thoufand Hundreds, or Ten hundred thoufand, is a Million. The Figure 1 at the Head of the Column, denoted by the Letter g , fignifies One Million ; the Figure 2 under it, Two Million ; the Figure 3 Three Million, and fo on ; and if to the Figure 1 you add the other Figures of the other Places, viz. 234567, then them taken together, that D is, 14- The Principles of Arithmetic k. is, i 234567, will fignify One Million Two hundred and Thirty-four thou- fand Five hundred and Sixty-feven ; and fo in like manner, if to the Fi¬ gure 2 under 1, you add the Figures in the other Places againft them, viz. 345678, then them taken together, viz. 2345678, will fignify Two Million Three hundred Forty-live thoufand Six hundred and Seventy- eight j and fo in like manner with all others to the Bottom. 7 My . If you conlider the Figures in the Place of Millions, as in the Place of Units, fuppoling a Million to be but One, then you may proceed to the Value of all other Places before it, in the lame manner ; and as the firft Place from Units, is Tens of Units, fo likewife the firft Place from Millions, is Tens of Millions, the fecond Place Hundreds of Millions, the third Place Thoufands of Millions, and fo on without End • as you may lee by the following Line, which confifts of Thirty Figures. CT\ • • • • 00 Millions of Millions of Millions of Millions. • ■ • cs Millions of Millions of Millions. on ' '-r Million of Millions ->. - Hundred thouland of Millions | +- Ten thoufand Millions ^ Thoufands of Millions S Millions. on Hundreds of Millions oj Tens of Millions ■ • 4- Millions J u, Hundreds of Thoufands _ '-i Tens of Thoufands 7 Thoflfands. tj T houlands j - Hundreds -"i Tens 7 Hundreds, vo Units j P. I think I underfand yon herein • that is, in the Table, the Figure 1 denotes in the firft Place One, in the fecond Place Ten, in the third Place a Hundred , and fo on in the ref of the Places , when ’ tis con - fiderd with the following Figures that are again ft'it in the other Places. But fnppof that thoje other Figures were taken away, and the Figures of 1 only lejt in their rejpe clive Places, how fall I know their Values at fuch Times - that is, fuppoje the Figure 1 in the Place Thoujands, which has the Figures /landing againft it, had thoje Figures 2,3,4, taken away from it, how rnufl I underfand that Figure 1 to fignify One thoufand ? M. \ou muft always, in fuch Cafes, fupply the Places of thofe Figures fo taken away, with as many Cyphers, as thus, 1000 ; for, as I told you. before, the Ufe of Cyphers is to encreafe the Value of a Figure, according to the Number of Places, as in the following Tables. Io Te n The 'Principles of Ar iThmetxck. io Ten. 100 One hundred. iooo One thoufand. ioooo Ten thoufand, 100000 Hundred Thoufand. 1000000 One million. 20 Twenty. 200 Two hundred. 2000 Two thouland. 20000 Twenty thoufand. 200000 Two hundred Thoufand. 2000000 Two million. 30. Thirty. 300 Three hundred. 3000 Three thoufand. 30000 Thirty thoufand. 300000 Three hundred Thoufand. 3000000 Three million. pF' 1 J’ ou ™ r y well > thank you for this InJlruSlion . iZntZ's idTTT’ pr0Ceed r 1 inform me how t0 nme ° r ***** quantities, and to read or exprefs them, as you before promifed. M I will: To numerate Quantities you muft begin at the Riaht Hand or Place of Units, and number them backward unto the Left H ind calling the feveral Figures to be numbered, by their refpeftive Names in Srteu " fol ' Exam P le > to number the following tour Figures 1234, I begin with the Figure 4 in the Place of Units and lo proceed on to the 1, laying, Units, Tens, Hundreds, Thoufands, and then exprefs them thus, One thonfind Two hundred and Thirty-four ■ and fo in ^dxpTeld* 7 “ " “ the Line 0/Thirty Figures When Quantities confift of fewer Figures or Places than Ten as ,,, °r 123456, or 123456789, then ’tls belt to point every third Figure from the Place of Units, as thus, 1234 in four Figures; thus, 123456 in iix Figures; and thus in nine Figures, 113456789; and then you may readily number and exprefs them ; becaufe the firft three are in the Place of Hun- “f 5 1 h , e | eC0 l nd th . ree , t0 the lec °nd Point, in the Place of Thoufands • and the laft three in the Place of Millions ; which you read thus %%ndred and Twenty-three Millions ; the 7 fecond thme’ C hundred and Fifty-fox Thoufand ; and the laft three, Seven hundred and Eighty-nine , and fo m like manner all other Numbers. ^ WHEN Numbets are very'long, as in the preceeding Line of Thirty Figures, tis beft to 1 point out the Places of MillL, as you fee pointed u,/ derneath the laid Line, which Line may then be thus read ? vZTlZ l 7 d MB^ 0rty ~f\r^ 10 “fi n r S ", hundred Se ™»ty-eight Millions of Million, oj Millions of Millions , Five hundred Forty-three thouland Tn- n l 7 ZLn?tr M ’f m a ' 0 f MiUims Millions, oJe hunled tZI- three Thoufand Five hundred and Sixty-fven Millions of Millions, One hun¬ dred Forty-two thoufand Six hundred and Thirty-four Million, Five hundred Seventy-two thoufand, One hundred and Seveniy-foine. *" P- I thank you, Sir ; / am now able quantity .- Pray what's my next InftruBion ? exprefs and number any M'F'r* neXt T hm @ In 0rder Addition, or the manner of colle&ine; But S before S I t0gethe d or Nu mbers into one total Sunil wL ht " J pt° ce ed thereto, I muft acquaint you with the Meafures, nalstployed er “ ln " ^ ^ ° f Mate ' mS af "' ? ; H ; Ee fereraI Parts of Lands and Buildings are C r J are , x * Meafures Length, called Running Meafures . 2. Mea- * Len § th and Bf eadth, called Square or Superficial Meafures . 3. Mea¬ fures 1 6 The 'Principles of Arithmetics. fores of Length, Breadth, Depth, or Thicknefs, called Solid or Cubical Meafure. These various Meafures I will exhibit in the followingT abl e s, which, by Infpe&ion, lhews you their Magnitudes and Proportions to each other. Table I. Of Meafures of Length. Bark) Corns in Lew/b. 11)0060 Feet ;o4y26000 Ik 1454976000 ; foSoo 121246000 2 1 Fathoms 1760 105600 38016000I 2640 f 2800 19008000 Pole , Rod, or Perch Chain or Acres Breadth 6796000 1 o 1 Rood , Furlong , or Acres Length Ti Mile League 48001 4801 0o 2 Pounds. l j 600 r i 400 Inch thick Plank, is 3 °° 1 3 1 100 14 J 150 Feet long, then 120 to the Bundle ; which fhould be Inch and half broad, and half an Inch thick. I having thus fhewn you the Numeration, and Kinds of divers Quanti¬ ties, I fhall now proceed to fhew you how to colled! them together into one Sum Total, when before divided into divers Parts or Parcels ; which is call’d Addition, and the Second Rule of Arithmetick. LECTURE II. Of Addition. P. T is Addition ? M. Addition is the gathering together of divers Numbers and Quantities into one Sum or Body, which is called the Total Sum. P. Pray, is Addition divided into maty Kinds ? M. Yes; there are many Kinds of Addition ; as Addition of Integers, Money, Materials, &c. as you’ll fee in this. Ledture. P. Pray what do you mean by Integers ? M. An Integer is a whole Number ; as 1, or 2, or 5, or 20, without any broken Parts belonging to it, as *, or i,-which are called broken or fractional Parts : And when fuch fractional Numbers, are annex’d to Integers, or whole Numbers; as 1 £, 2 +, 3 then fuch Numbers are called mix’d Numbers. N°. II. E But, 8 The Principles of Arithmetic k. 1 l But, however, let the Nature of your Numbers to be added, be as they will, you mull always oblerve the following RULE. Take care to place each Kind m their true Places } Units under Units, Tens under Tens, Hundreds under Hundreds, c. and then in the Addition oj Integers, for every Ten that you find in each Place op Figures, carry one to the next Place. Example. 7654311 1 4 1 o t 5 3 4 5 6 7 3 2 1 5 1 7 3 1 3 Total 7 9 3 4 3 1 3 To perform this Example, begin at the Bottom of the Place of Units, and fay, 1 and 3 is 4, and 3 is 7, and 5 is n, and 7 is 19, and 3 is 22, and 1 is 13 : Then becaufe that in 23 there are two Tens, therefore fet down under the Line the odd 3, and carry 2 to the next Place of Tens ; faying, 2 that 1 carry, and 2 is 4, and 7 is n, and 1 is 12, and 6 is 18, and 1 is 19, and 1 is n ; fet down 1, and carry 2 to the next Place of Hundreds ; then 2 I carry, and x is 3, and 2 is 5, and 5 is 10, and 3 is 13 ; fet down 3, and carry 1, becaufe you have Ten but once in 13 : Then fay, 1 I carry, and 3 is 4, and 4 is 8, and 2 is 10, and 4 is 14; fet down 4, and carry 1, and fay, 1 I carry, and 3 is 4, and 4 is 8, and 5 is 13 3 let down 3, and carry 1 ; then fry, 1 I carry, and 2 is 3, and 6 is 9 ; fet down 9, becaufe you have not Ten therein : Laftly, 7 is 7 ; therefore place the 7 before the 9, and your Work is done. Now, from this Example ’tis plain, that in Addition, all the Numbers taken together, are equal to the Sum. I will add the following Examples for your 12 Pra&ice. 123 1234 12345 12345* 1234567 21 213 2143 21354 214365 2135476 12 231 2413 2 3 r 45 24235* 23245*7 2 1 321 4*3 1 3 - 4 r 5 423165 3241657 12 312 4321 34251 432615 3426275 21 132 34 I 1 43521 34*251 4362715 12 123 3 1 4 - 45312 364521 4637251 — — 1324 54132 635411 6473521 1342 51423 *53242 *745312 15243 22534 5*1324 52*342 253*24 135642 7*54132 75*1423 5716243 5172634 15273*4 1253746 io The Principles of Arithmetic! P. Sir, I am greatly obliged to you for thefe various Examples given* me fue Praliice ; but before I proceed thereto,pray anfuier me the following i^nejlion: Whether or no, there is any other thing to be regarded in the placing of'Num¬ bers to be added, more than to place their feveral Units, Tens, &c. in their re- fpeblive Places P M. No ; that is all, and therefore yon need not regard with which of any given Numbers, you fet down at firft in the middle or Lift: That is, fuppole the following Numbers were given to be added together, viz. to, 501, 7235, 40, 90, you may place them 90 7 Z 35 501 10 40 90 10 501 Thus 7235 or thus, 40 or thusj 7235 or thus, 7235 501 10 90 40 10 501 40 90 7876 7876 7876 7876 Here you fee, that tho every one differ in their manner of placing, yet the total Sum is the fame to them all. 1 SHALL now (hew you how to add up Sums'of Money. The Addition of Money, is called Addition of Numbers of divers Denominations, as of Farthings, Pence, Shillings, and Pounds ; wherein you are to obierve the following R U L E. Observe that every Denomination be placed under its correfpondent Deno¬ mination , that is, as in the Addition of Integers, you mere taught to place Units ovei Units, Tens over Tens, &c. So in this, you muf place Pounds over Pounds, Shillings over Shillings, Pence over Pence, and Farthings over Farthings. Then for every 4 contain’d in the Place of Farthings, carry 1 to the next Place of Pence, becauie 4 Farthings make 1 Penny ; alfo, for every 12 con¬ tain’d in the Place of Pence, carry 1 to the next Place of Shillings, becaufe 12 Pence make 1 Shilling ; likewife, for every 20, contain’d in the Place of Shillings, carry 1 to the Place of Pouhds, becaufe 20 Shillings make one Pound Sterling: Laftly, add up the Pounds as you was before taught of the Integers, becaufe here in this Cafe the Integer is a Pound, and the Shillings are Parts thereof, which I before told you are called ffaSional Parts : But more of them in their Place. L Example. s. d. q- 123 1 1 6 2 I 11 2 I 1276 5 OO 3 800 19 II 1 3° 11 IO 2 5 8 2 1 1555 7 8 3 3794 01 05 3 To 20 7 The !'Principles of Arithmetics. To add thele Sums together, you itinft begin with the Column of Far¬ things ; and fay, 3 and 1 is 4, and 2 is 6, and 1 is 7, and 3 is 10, and 1 is 11 : Now bccaule in 11 Farthings you have 4 twice, and 3 remains, there¬ fore place 3 at Bottom, and carry the 2 to the Place of Pence, and lay, 2 that I carry, and 8 is 10, and 2 is 12, and 10 is 22, and 11 is 33, and 2 is 35, and 6 is 41 : Now fince that in 41, you have 12 three times, which is 36, and 5 is remaining, therefore fet down the 5 under the Pence, and carry the 3 Shillings to the Place of Units in the Place of Shillings, and lay, 3 that I carry, and 7 is 10, and 8 is 18, and 1 is 19, and 9 is 28, and 5 is 33, and 1 is 34, and 7 is 41 ; fet down 1, and carry 4 to the Place of Tens, in the Shillings, and lay 4 I carry, and 1 is 5, and 1 is 6, and 1 is 7, and 1 is 8, which is 80 Shillings, wherein you have 20, four times, and o remains, therefore fet down o under the Tens of the Shillings, and carry 4 to the Place of Pounds, which add together, as before taught, of the whole Num¬ bers, or Integers, and the total Sum will be 3794 Pounds, 01 Shilling, 05 Pence, and 3 Farthings. I shall now give you fome Examples for Practice. Exampl I. Example 11. Example III. j Example IV. ]. s. d. q- 1. s. d. 1. s. 1. s. d. I 7 1 123 1 9 11 1 -43 10 12 0 5 19 8 I 7 10 2 7963 11 7 0 2 10 I I 3 8 3 5 12765 8 11 0 3 16 8 2 127 11 2 742 1 9 16 0 10 00 7 2 6 4 8 2222 6 15 0 9 00 OO 3 188 7 *7 0 10 5 8 IO 0 11 6 IO 8 0 9 1 -345 11 In thele Four Examples you have fome Variety: The firft conlifts of Shil¬ lings, Pence and Farthings ; the fecond, of Pounds, Shillings, and Pence ; the third of Pounds and Shillings, and the fourth of Pounds and Pence. To perform the firft Example, you begin with the Farthings, and for every 4, carry 1 Penny to the Place of Pence, and for every 1 2 therein, carry 1 to the Place of Shillings, and for every 20 therein, carry 1 to the Place of Pounds, and place them down in their refpcCtivc Places ; altho’ in this Ex¬ ample there’s no Pounds added, but fuch as arile by the Addition of the Shillings : Then proceed in like manner to the working of the other Examples. But feeing that when you have call up your Column of Pence, the Sum thereof may lometimes be very large, and you may not readily know how many Shillings are contained therein, you muft, before you proceed any further, learn the two following Tables by Heart. Table 21 The Principles of Arithmetic k. Table I. Table II. d. Ito 3 ° 4 ° 5 ° 6o 7°l So 9 ° IOO no I zo is 4 1 z 3 4 5 5 6 7 8 9 xo d. 8 6 4 oo IO 8 6 4 oo >1S < “4 36 48 60 72. 84 96 108 In the firft of thefe Tables the Numbers on the Left Hand aie Pence, and thole on the Right; the firft are Shillings, and the lecond Pence, and are thus to be read ; tod. is 1 r. and 8 d. 3° *'• £ A 4 ° A. » 3 '• 4 A. 50 d. is 4 r. a d. 60 d. is 5 0 o A &c. 1 his Table when learn d will ibew you readily how to divide out your Shillings in the adding up of your Pence. The fecond Table is alfo very ufeful, in fhewing you the Number of Pence contain’d in any Number of Shillings, under Ten ; and which, being once learn’d, will be very ready and uieful to you. Before I proceed any further, you muft here obferye, that altho’in the two firft Examples hereof, I have placed the Farthings in a diftin is < TOO 120 J 5 16 > is J 7 140 17 8 l6o 18 9 l8o l 9 10 _ 200 20 1 F 220 24O 260 280 300 320 34 ° 360 3 8 ° 400 P. I The Principles of Arithmetics, 22 P. I PERFECTLY underfund tbcj'c Tables, and have them by Heart ; therefore be pleajed to proceed. J\f. \\ hen Sums are very long, as m the following Example, you may divide them into Parts ■ and the Parts added together, will he equal to the iVhole. But otherwile to perform it at one Operation, this is The RULE. Fi rst cafl up the Farthings, as before taught, and carry the Pence (ifany) to the Pence in the Units Place of Pence, which number up to the Top, and then yon may come back again down in the Place of Tens as fall as you can Ip?™’ a ” A b the preceding Pence Table, ca/Hy tell how maty Shillings fj any) are contain'd ,u the Pence ■ and place the odd remaining Pence ywhen any) under the Line, and carry the Shillings to the Place of Shillings : Jus done, number up the Shillings in the Units Place, and back again down the Place of Tens ; and then finding bow often zo is contain d therein fet down the a maws [if any) and carry the Pounds to the Place of Pounds, which add np ns before taught. OPERATION. First, I begin with the Farthings, and My, i and l is 5, and + is 6, and i is 8, and + is 9, and 4 is 1:, and A is 13 Farthings in all, w hich is I hree-pence wherefore I put down the I under the Line, and carry 5 to the Place of Pence, and fay, 3 that I carry,' and 8 is 11, and 6 is 17, and 2 is 19, and 3 is 22, and 1 is ~ 3 > and 9 is 32, and 5 is 37, and 2 is 39, and 8 is 47, and 1 is 48, and 6 is 54, and 2 is 56, 59 ? an d r is 60, and 4 is 64, and 2 i, nd, and S is 74, and 2 is 76 .* So J am now come to the Top or uppermolt Figure, from whence I go again downwards in the Place of Tens, laying, 76 and 10 at a is 86, and 10 is od, and io is 106, and 10 is ijf6, and 10 is 126 : Now by the preceding Pence Table, 100 Pence is 8 s. 4 d. and a6 Pence is 2 s. 2 d. making in the whole 10 s. 6 d. Secondly, Set the 6 Pence under the Fence, and carry the 10 Shillings to the Place ol Shil lings, and then beginning with the Units thereof only, number them to the Top, laying, 1 o that I carry and 4 is 14, and 1 is 15, and 6 7 is 33, and 7 is 40, and 1 is 41, and 9 L Example. s. d. IO 1 IS 3 81 I I b 19 2 no tS a 10 106 15 4 111 17 11 16 8 7 6 IS 14 6 27 13 “I 14 16 IO 1 7 9 8 222 19 IO » III 11 18 *7 l6 7 9 * 7 5 1 9 6 3 i 11 2 27 14 6 26 IO 8 i 91 * 9 5 1 is 50 , 1, and 5 and 9 is 26, and 59, and 6 A 65, and .5 is 68, and 4 is y'z, and 6 is 78, and 8 is 86 ,’ and 7 A 98 , and 5 is 98, and 8 is 106, and 9 is 115, and 3 is 118, and 1 is 1 if) : Now iince that 119 Shillings is 11 times ten Shillings, and 9 over, let down the 9 at the Bottom of the Units, then I defeend down the Place of lens, beginning at b, faying, n I carry from the Units of Shillings, and The P rinciples of Arithmetic k . and 1 “ lS > and and To, and WTi if’ ^ ^ '?* Plfcf of t dme ^ H ^ tHe f» i ": h '« *■»£ BSEEfta S‘i;:s- lings and Six-pence i Farthing. 93 ~ 1 ounds 9 ohil- whmhi^e^ -d Memory may be too much affefted, I therefo^ recommend the fearing ^ R U L E. Ho w to perform Addition, without the Trouble of carry!,,? One for eve™ Stte : t Tr- lm v s - befire yjzAzz gnat Ljc tu the Addition of large Sums. J J A s The Manner thereof is as follows w g;".t Sd“'3”ir p1 ''' “ bp <*• L «“ •*«*/ PRACTICE. First add up the Column of Units denoted by /i as v i , before taught, laying 2 and 7 is 9 , and 7 is 16, and i X ‘ W T k L is 17 and 5 is u, and 2 is 24, and 7 is 31, and 6 is Unir Ct 37 Undemeath ’ Wlth the 7 under th e Place of Secondly, Begin again with the fecond Column or Place of Tens, denoted bye, which add up into one Sum, and place it under the lame as before, always ob lerving, that the Units thereof are placed under the Line ot Figures added, and the Tens under the next Place • lo here, the Sum of this fecond Column e being 51, there¬ fore place the 1 under the liime, and the 3 under the next 1 lace d. Proceed ,n like manner, with the remaining lour Columns, whofe_ Sums will be found to be 36 39 , 19, 57 . which being added together, their Sum will be the total Sum required. Now I muft obferve to you, that by this Method of Addition, you will be lefs liable to Errors, than in the ■foregoing where you carry the Tens forward, becaufe here the Mind is difeharged every time at the catting up of each, hngle Column. ” p 7 2 9 2 6 1 1 2 8 5 9 3 8 2 9 2 f 6 7 5 1 7 7 3 7 3 6 9 57 9 4 7 5 8 5 9 3 9 4 7 And again, if an Error Should at any time happen, its much eafier dif- Method wh m ’- > eXaIP T SColumn than in the preceding Opemttn t generally happens, that we muft pafs through the w hole cha ™ b V° mC f th , e /ruth, and even then, if the Memory is over- rtn Mefhfd oHddV ^ 1 f’- nnd ™ fs ° f the trae Sum required 7 : Befides, , >,eth ™ ™ Addition admits of beginning in any Part thereof, without emg confined to the Place of Units ; Tvhich'inftea/of being the fk£ bus added, may be laft, as I ihall illuftrate in Example II. following. First The ‘Principles of A r it h metic k. Example II. a b c d * / / l 4 5 3 6 9 2- 3 4 2 7 6 I 7 5 9 2 1 2 5 7 3 5 8 5 4 3 2 I 9 3 2. 5 4 7 8 2 9 3 5 7 9 1 5 4 3 2 5 7 9 9 6 1 7 I 3 3 3 3 5 9 3 2 9 4 7 First add up the Column a, whofe Sum is 57,which place under the feme. Secondly, add up the Column b, whofe Sum is 19, which place under the iame letting the 9 under the Column b, and the 1 under the 7. Thirdly, add up the Column c, whofe Sum is 3 9, which place un¬ der the lame, l'etting the 9 under the Column, and the 3 under the 9 of the laft ProduS 19. Fourthly, pro¬ ceed to place the Sums of the other three Columns in like manner, and adding them together, after the com¬ mon Manner, their Sum will be the T otal required. Thus you fee, that by this Method, you may begin and end your Addition at Pleafure, either forwards, back- wards, or in the middle, and with one and the lame Trouble • which I muft defire you to well obferve. P I will, Sir ; and with a great deal oj P/e pure alfo. But pray. Sir, cannot Sums of divers Kinds, as Money, &c. be added together by this new Method. At Yfs ' Money, Materials, &c. may be thus added, the Method is univerfal: But that you may well underftand the fame, I wil give you an Example thereof. Let the Sums in Example III be given to be added together according to this Method. As it is the mod: orderly Manner to begin the Addition with the Farthings, 1 will for Order fake, begin - h them and end with the Pounds; but . i. -wife, I might have begun w ith the Pounds, and ended with the Shillings, or Pence, at Plealure. 1. 7-5 125 57 8 9"-4 187 732 Example III s. d 19 14 15 ! 7 9 J 7 First fay, 3 and 1 is 9, and a is 11 r is 5, and 3 is 8, and id 3 is 14 : Now, be- caulf "14 Farthings are equal to 3 Pence 2 Farthings, therefore fet a Farthings under the Place of Farthings, and 3 under the Place ot Pence. 4 3 1 24 3 ° 273 15 Secondly, Begin with Units of the Pence, and (ay, 5 and 1 is 6, and 0 iff, and I is 16, (and coming down the Place of the Tens,m the Pence) £y 16 and ,o is 26, and 10 is 36, and 10 is 46, and 10 is 56 Now becaufe that 36 Pence is equal to 4 Shillings and 8 Pence therefore fet down 8 under the Place of Pence, and 4 under the 1 lace of Shilln g . Thirdly, Begin with the Units of the Place of Shillings , and fay and 9 is r6, and S 7 is 23, and 5 is 28 and 4 . 32 , «nd 9 “ 4 the coming down the Place of Tens In the Shillings fay, 4 >^nd 10 is 5 U 10 is 61 and 10 is 71, and 10 is Si, and 10 is 91- Now becaufe that 1 Shillings are equal to 4 Pounds and n Shillings, there ore e °wa he 11 under the Place of Shillings, and the 4 under the Place of I ounds. 9 the Fourthly, in e 29 The Principles of Arithmetics Fourthly, add together the Sums of the three Columns of Pounds, as before taught, and then their feveral Sums added together, will be the total Sum required. Thus have I made Addition in all its Varieties, very plain, which, I hope, you will carefully remember. P. It is true Sir. But , pray , how am I to depend upon the Totals , when I have cajl them up ; that is , how fhall I know whether they be true or ftlfe ? M. What you now ask, I will readily inform you ; this is the Proof of Addition , and may be performed after two different Manners. As for Ex¬ ample • let the five following Sums be added together. In the Operation of this Example, firft, add to¬ gether from the Bottom upwards, into one Total, the five upper Sums, which appears to be 4345 I. 4 s. 1 d. as at A. Secondly, inftead of adding up each Place of Pence, Shillings, and Pounds, by begin¬ ning at the Bottom of each Column as before, be¬ gin at the Top of each Column, and add them downwards, and if then the Total is the fame as before, you are right, elfe not: So here the Sum at B, produced by the Addition downwards, is the lame as the Sum at A produced by the Addition made upwards as ufual. This is one Way to prove Addition, which may be alfo done as follow¬ ing. Draw a Line underneath the upper Row of Figures, as E E ; and then add together the four under Sums into one Sum, as C, to which add the uppermoft Number before cut off; and if their To¬ tal D is equal with the other Total A, you may affure yourfelf of the Ope¬ ration being truly performed ; becaule the tThole is always equal to all its Parts taken together. 1. 1274 s. 16 d. 1243 I 9 2 1800 I I 9 18 *7 6 5 9 4 Sum 4343 4 1 A Proof 4343 4 1 B 3°68 7 9 C Proof 4343 4 1 D P. I UNDERSTAND this moft reafonable Proof of Addition. Pray be pleafed to give me divers Qneflions that relate to Meajures , &c. M. I will : But I am apprehenfive, that amongft the Variety following, you will meet with fome that you’ll not be Mafter of until you have learned Subtraction. However, be not dilhearten’d thereat, becaufe I fhall foon learn you to fubtradt, and then they will become ealy. Example I. Of Integers. Inches. There are five Windows that have Mold- 1 f firft: contains 227 ings about them. | Second 2,50 >The^ Third 1500 I demand how many Inches of Moldings 1 Fourth 1200 there are in the Whole ? ] f Fifth 800 P. Anfwer 3977 Example G 3 ° The Principles of Arithmetick. Example II. Of Integers. M. There are lix Brick-walls that are covered with Stone copping. I demand Jjqw many Feet of Copping are in the Whole ? >The< Inches. firft is in Length 5^-5 Second 3 ° o Third 272. Fourth 1500 Fifth 1872 -Sixth 2.21 P. Anfvver 4690 Example III. Of A I- A Painter hath painted the Cornices of five Houfes, which are all of the lame Girt, and are to be paid for rnmng Aleafure. I demand how many Feet, rutting Aleafure , in the IF hole ? Note, Here, for every 1 2 contained in the Place of Inches, you muft carry 1 to the Place of Feet, and then pro¬ ceed as Integers. Feet and Inches. Feet Inches. j r Firft Houfe girts 210 11 Second 504 9 The< Third 601 10 | Fourth 100 4 [Fifth 150 9 P. Anfvver 1668 07 Example IV. Of Tards, Feet and Inches. Yards A Joyner has hung ten Safh' Windows, and ufed Line for the lame, as follows. I demand how many Tards he has ufed in the IVhole ? Note , That for every 12 Inches, you carry 1 Foot, and for every 3 Feet, you carry 1 Yard, and then add up the Yards as Integers. >To the< Firft Salh Second Third Fourth Filth Sixth Seventh Eighth Ninth Tenth P. Anfvver 6 6 4 8 3 7 6 8 7 64 Feet Inches. 7 11 10 5 9 11 JO 2, 10 05 For your ready finding how many Feet are contain’d in any Number of Inches under 120, or how many Yards in any Number of Feet under 30, you lhould get the following Tables by Heart; and until you are Matter thereof, you may by Infpe&ion have your Demands anfwer’d. Table The Principles of Arithm.etick. Table I. Table II. Inch Feet. Feet Y ard. 1 Z 1 3 -4 z 6 36 3 9 48 4 12 60 5 15 7 >l > is < 18 > is < 84 7 21 9 ? 8 24 108 9 2 7 120 . 10 3 ° , , sis pur Example. In the laft Example, the Column of Inches amounted unto 77. Look in Tabic I, and in the firft Column under the Word Inch, find the neareft Number to 77, which is 72, againft which ftands 6, the Number of Twelves contain’d therein ; then fet down the remaining 5, and carry the 6 to the Place of Feet, and add them together, which amounts to 20 : This done look in Table II, under the Word Feet, and find the neareft Number to 10' which is it, againft which ftands 6, the Number of Yards contain’d theie- in, which carry to the Place of Yards, and let down the remainder two, un¬ der the Column of Feet; then add up the Yards as Integers. P. These Tables I apprehend will be very ufeful in cajling up large Sums, and therefore I think it advifeable to continue them to greater Numbers, and for that Purpoje, dejtre that you'llfhew me your Rule for making them. M. The Rule for making thefe and all other fuch Tables is, firft, to dou¬ ble the Number you begin with, and to that Sum, add the lame Number again, and to the next, and every Sum after, add the firft or uppermoft Number, and againft each Number fo added, fet the Number of Times in the lecond Column. y4s for Example. I would make Table I. for dividing Inches into Feet. Firft write down i z the Number of Inches in one Foot, and againft it write i, then double the 12, and it makes 24, which fet under the 12, and againft it fet 2 under the 1 aforefaid ; to this 24 add the 1 2 over it, and it makes 36, which write under 24, and fet 3 againft it, fignifying three times 12 ; to 36 add the up¬ per or firft Number 12, and it makes 48, and againft it place the Number 4 in the fecond Column. Proceed on in this Manner, and you may continue your Table to any Length you defire. P. I understand you ; pray proceed'to other Examples. M. I will ; and as Occafion requires, ftiall make you the like Tables to each Example. Example 02 The Principles of Arithmetic k. Example V. Of Fathoms, Tards, Feet and Inches. Fath. Yds. Feet. Inches. A Labourer' was employ’d to dig down a Well, and was fix Days in do- ing it. I DEMAND h(W). deep he dug down in that time ? The A Carpenter hath fet up five Lengths of Pailling. I demand how many Rods there are in the W hole ? he went down z 1 a 11 y 5 0 1 9 4 1 1 7 >y 3 1 2 6 3 1 X 10 4 0 5 Anfwer 25 0 1 OO Yards, as in the laft Example and Fathoms, and add them as Integ ers. f Rods and Feet. Rod Feet. 1 Firft there is 2.0 2, | Second 43 1 f In the Third 70 1 Fourth 110 2 i j Fifth 65 1 i p. Anfwer 308 8i M. For your ready reducing your Column of Feet into Poles, I will give you the following Table, Ihewing the Number of Feet in any Number of Poles not exceeding Ten. Table. Pole. Feet. there is In there P. I thank you, Sir ; I fee by this Table , that I mufl carry i to the Co¬ lumn of Pules for every 16 v found in the Column of Feet. M. You underfiand me rightly, therefore I fhall proceed to Example VII. Of Chains and Links. A Labourer, to enclofe di-' vers Lands, hath digged and planted five Fences. I DEMAND how many Chains Length he hath done in the Whole ? ^The< Fsote, For every ioo in the Column of Links, carry i to the Column of Chains, and add the Chains as Integers. Chains. Links. Firft is in Length. 221 9 1 Second 113 74 Third 37 6 Fourth 179 99 Fifth 242 89 p. Anfwer 1144 75 Table A- A The Principles of Arithmetic k. Chains. Links. In I 100 4' 2 200 8 3 3 °° 12 4 400 16 5 500 20 6 600 24 7 > there is < 700 > equal to < 28 8 800 3 2 9 900 3 6 10 1000 40 A 4 75 3 i 5 ° t- ^5 I Rods length Example VIII. Of Tuns A SmITH hath made five Parcels of Iron-work. Hundreds , Quartet's and Pounds. L f: Tun Hund. qrs ib. Firft weighs 2 98 3 27 Second 1 9 2 18 Third 27 9 1 25 Fourth 25 ! 7 2 11 , Fifth 4 10 3 24 Anfwer 68 l6 2 21 I DEMAND the total JVeight of the IF hole ? Note } That for every 28 Pounds contained in the P. firft Column of Pounds,car- - ry i to the next Column of Quarters of a Hundred. idly, For every 4 con¬ tained in the Column of Quarters, carry 1 to the Column of Hundreds, and for every 20 in the Column of Hundreds, carry 1 to the Column of Tuns, and then add them up as Integers. Table I. Shewing the Number of Pounds in every Quarter of an Hundred unto three Hundred. Qm liters of a Hund. Pounds. Quarters of a Hund. In < Table II. Shewing the Number of Quarters in 1 o Hund. Avoir dupo'ize.- fl 1 ' 28 7 2 i 56 8 1 -3 >there is < 84 112 In < 9 10 there is* 5 140 11 6 168 12 Hundred. 1 * 2 3 4 6 >there is< 7 8 9 IO Quarters. 4 12 16 20 24 28 3 2 3 6 40 Pounds. 196 224 252 280 3 ° 8 33 6 Table III. Shewing the Number of Hundreds in io Tuns. Tun. Hund. weight. 40 60 80 In T 2 3 4 6 there is< 7 8 9 10 j 12,0 140 l60 l80 2,00 B i: H 34 The Principles of Arithmetics By thefe three Tables you may readily call: up the Contents of this and all other fuch Sums. Table I. is for reducing the Column of Pounds into Quarters of Hundreds ; Table II. reduces the Quarters of Hundreds into whole Hundreds, and Table III. reduces the Hundreds into Tuns. Example IX. Of Timber. There are fix Pieces of Timber, number’d i, 2, 3, 4, 5, 6. , I demand how much Timber is in nil I the fix Pieces. >Number< Note, For every fifty Feet carry one to j the Column of Loads. J Loads Feet. 1 contains 2 49 P. Anfwer 18 A Table jhewing the Number of Feet in ten Loads of Timber. Load Feet. Load Feet. 500 I 1 50 r 6 f 2. 100 1 7 f 3 there is< 150 Inc 8 •there is < 4 j 200 1 9 1 5 J 250 l IO 1 Example X. Of Bricks. A Brickmaker fent me in fix odd Parcels of Bricks. I demand how many Loads he fent in the Whole ? Note, For every Five hun¬ dred in the Column of Bricks, carry i to the Loads. > In the < Load Bricks. Fir ft Parcel he fent 1 52 Second 2 472 Third 1 450 Fourth 2 270 Fifth 3 499 Sixth 1 A - P. Anfwe Example XI. Of Lime. 14 172 A Lime-man fent me in five Parcels of Lime, at different times. I demand how much in the Whole ? •At the< Note, For every 25 in the Column o r Hundreds. Hund. Bags. r Firft time he fent me 3 20 Second 1 j 8 Third 4 24 Fourth 2 19 Fifth 4 jy P. Anfwer 17 « 0 s, carry one to the Place of A The Principles of Arithmetics A Table Jheomng the Number of Baggs, or Bujhels in Ten hundred of Lime. Hund. Baggs. Hund. Faggs r *5 6 1 f 150 50 7 | . | 175 75 In 8 >there is<; ;oo 100 9 I I zz 5 . 125 , [10J U JO I have receiv’d feven Parcels of Sand. I demand how mkoh I have receiv'd in all ? Example XII. Of Sand . Load Bufhels. Firft time I receiv’d 21 Second Third 'At thes Fourth Fifth Sixth .Seven F. Anfwer 86 Note, For every iS in the Column of Bulhels, carry i to the Column of Loads. A Table Jhevsing the Number of Bujhels in io Loads of Sand. Load Bufhels. I ^ l. Example XIII. Of Land. AGentleman has five Parcels of Land to let out on Leafe for to build on. >In the Immand bow much Land is contain'd in all the Jive Parcels ? Note That for every contain’d in the Column of Feet, you carry r to the Column of Poles; and for every 40,, in the Column of Poles, you car¬ ry 1 to the Column of Rods ; and for every 4 in the Column of Rods, you carry 1 to the Column of Acres, which you add up as Integers. The ‘Principles of A r it h m e t i c k. Table I. Shewing the Number of fquare Feet hi ten fquare Rods or Poles of Land. In < Rod. Feet. Rod. Feet. f r " r i 6 ' " i6 33 i a 1 5 44 * 7 i 9 o 5 ; 1 3 ■there islnthethere is< 3 °° In. 8 >there is< U 400 9 L 5 J ( .500 10 Feet. 600 . 700 800 900 1000 Example The Principles of Arithmetick. A Painter has Example XV. Of Gilding. gilded over five fquare Firft there is contain’d 5 J 4 3 Pieces of Work. i Second 9 7a >In the< Third 6 l6 I D E M A N D the j Fourth 4 131 Number of fquare Feet j Fifth 5 141 contain'd in all the Jive ' Pieces of Gilding ? p. Anfiver 3 1 O7I In < r 1 1 ft 44 r 6 f 1 a | z88 7 i 3 > there isthere is < 1 4 I 57 6 9 t 5 J [720 IO A PainTer has colour’d over five Pieces of Wainfcoting. Example XVI. Of Painting. ei' j I demand how many Tards in the IVhole ? Firft Piece there is Second In rhc-i Third Fourth L Fifth P. Anfiver Note, That for every 9 in the Column of Feet, you carry of Yards. A Table Jhewing the Number of fquare Feet in ten fquart Tards Yards. Feet. Yards. fi' f 9 r 6 1 \ . 1 18 7 .1 j 3 >there is< 27 In < 8 i>there is< 4 I 36 9 j [5. 145 l 10 J l Feet. 54 6 3 81 Example XVII. Of Solid Tards. A Labourer agreed to dig and carry out Earth required to make a Cellar, which he per¬ formed in fix Days. I demand the whole Quan¬ tity taken out of the Cellar ? In the Firft Day he dug out Second Third Fourth Fifth Sixth P. Anfiver 19 V Squ.Feet. Squ.Inches. Note, For every 144 in the Column of fquare Inches, carry 1 to the Co¬ lumn of Iquare Feet. A Table Jhewing the Number of fquare Inches in ten fquare Feet. Squ.Feet. Squ.Inch. Squ.Feet. Squ.Inch. S64 * 44 ° Yards. Feet, ay 8 18 7 14 8 16 3 15 5 93 4 to the Column Cub.Y. Cub.F. 3 26 a ai 4 *7 Note, That for every 17 in the Column of Cubical Feet, you carry one to the Column of Cubical Yards. N°. III. I A /C- O 3 S The Principles of Arithmetick. A Table fhewing the Number of Cubical Feet in ten Cubical Tards, Yards. Feet. In < Yards. 1 1 ' t -7 6 .» 54 7 3 >there is< 81 In < 8 >there is< 4 108 9 L 5 J 435 . 10 . Example XVIII. Feet. 162 189 216 2 43 270 On the Tenth Day of March , and Eleventh Day of September , the Sun is in the AEquhioSUal Circle , which is 360 Degrees in Circumference, and pades 15 Degrees in every Hour. Ko\y the Queftion is ; Suppole that from Six in the Morn ing, being the Time of her Riling, unto 7, there pafles And from 7 to f after 8 - And from thence to 9 And from thence to i an Hour after 9 - And from thence to 10 And from thence to 7 an Hour after 11 And from thence to 12 at Noon - And from thence to 3 in the Afternoon And from thence to i after 4 And from thence to 6 at Night, the Time of Setting Deg. %6 3 7 7 7 45 26 18 I demand how many pajjed m the Time ? Degrees are } P. AnlWer 180 Min. o 15 45 3 ° 3 ° 3 ° 3 ° 00 15 45 00 Note, That for every 60 in the Column of Minutes, carry one to the Degrees. A Table /hewing the Number of Minutes in io Degrees. Min. In Deg. Min. Deg. I * f 60 r 6 n r i . 1 110 7 3 ■there is< 180 In. 8 there is< 4 I -40 9 j L 5 L 3 °° 10 . l 4z° j 4 8o I 54 ° 1 600 Note, That 60 Minutes, or Miles, make i Degree ofMeafure,as 60 Minutes make 1 Hour of Time . Thus have I given you a great Variety of Examples for PraCtice; and here I muft oblerve to you, that if you find any among them too hard to perform^for want of knowing Subtraction, pals them over, and when you have learn d the following Rule of Subtraction, return back to them again, and you’ll perform them with Piealure. Note, I muft aljo objerve to y o,< p that as all the foregoing Tables are generally ufeful for your expeditions c aft ing up the Contents of all Kinds of Quantities, I would advife you to com- poje them in a handftme Manner on a Sheet of Paper pa/led on Paft-board, and place them in your Study or Accompting-houje, for Ufe , when required . LECTURE The Principles of Arithmetic k. 39 LECTURE III. Of Subtraction. P .HAT is to be underjiood by Subtraction ? M. Subtraction is the third Rule of Arithmetick, and teaches to find the Difference of any two Numbers, by taking or drawing the Leffer from the Greater, whereby the Difference will appear } which Difference is called the Excefs, or Remainder. P. Prat what is to be obferyed particularly herein ? M. That the leffer Number be always fet down under the greater, as in Addition, taking Care to place Units under Units , Tens under Tens , &c. and then draw a Line underneath. As for Example : Let it be required to fubtrad! 54 from 88. Then place them as in the Margin. To perform this Queftion, the Numbers being Example I. placed as aforefaid, begin at the Right Hand, and fay, 4 from 8, and there remains 4, and 5 from 88 the greateft 8, and there remains 3 ■ fo will the remains be 34 ; 54 the imalleft and draw a Line thus: 34 remains. Example II. Begin at the Right Hand, and fay, 3 from 7, and there remains 45 and 5 from 8, there remains 3 • and 4 from 9, there remains 5 : So will the remains be 534. From Take 987 4 53 Remains 534 2 from 8, there remains 6 5 and 4 from 9, there re- ‘ " “ ff 2 55 mains 5 : So will the remains be 5621. 5621 In this Example, the upper Line confifting of Cy- Example IV. pliers only, the firff Figure excepted, we muff bor- From - - 100000 row 10, or fuppofe that 10 be in the Place of the firff Take - - 12345 12 345 87 6 55 Cypher, and then fay, 5 from o I cannot, but 5 from 10, and there remains 5 ; then go on, faying, 1 I borrow’d, and 4 is 5, from 10, and there remains 5 • then again, 1 I bor¬ row’d, and 3 is 4, from 10, and there remains 6 ; then 1 I borrow’d, and 2 is 3, from 10, refts 7 : Laftly, 1 I borrow’d, and 1 is 2, from 10, and re¬ mains 8 : So will the remains be 87655. Now, by this laft Example you fee, that whenever you cannot find your lower Figure in the higher, that then you are obliged to borrow Ten from the next Figure, which you confider but as 1, when you repay it or carry it forward again, and the Reafon why 10 fo borrowed is accounted but 1, is this : That whereas 10 in the Place of Units is but equal to 1 in the Place of Tens, therefore every fuch 10 fo borrowed, is confidered but as 1. I WILL The ‘Principles of A r i t h m e t i c k. 4 ° will make this yet worepafy by another .Example, ' Example V. From - 9227452 Take - 345564 Reft 888188S Begin at the Right Hand, and fay, 4 from 2 I cannot, but borrowing 10 arid adding to it, makes the 2, 12 5 then I fry, 4 from 12, reft 8 ; then 1 I borrow’d, and 6 is 7, from 5 I cannot, but (borrow¬ ing 10) 7-from 15, reft 8 : Again, 1 I borrow’d, and 5 is 6, from 4 I cannot, but (borrowing 10) 6 from 14, refts 8 : Again, 1 1 borrowed, and 5 is 6, from 7 reft 1 : Again, 4 from 2 I cannot, but (borrowing 10) 4 from 12, refts 8: Again, 1 I borrowed, and 3 is 4, from 2 I cannot,but (borrowing 10) 4 from 1 2, refts 8. Laftly, the laft 1 borrowed being taken from 9, refts 8. So will the Remains be 8881888. P. I understand you very well : But how {hall I prove my JVork y that I way know when I aw right. M. I ha ve already told you, that the Whole is equal to all its Parts taken together. 1 herefore, to prove your Subtrattion , add the Sum to be fubtradled ; (that is, the Idler or lowermoft) Sum to the remainder • and if the Total is equal to the greateft or upper Number from which you were to lubtracf, your Work is true, othenvife ’tis falfe. As for Example. Add together the Sum to be fubtradfed, mark’d B, and Remainder C, and if their Total D be equal to the greateft Number A, from which Subtraction is made, the Work is true. 27452 45564 A B 8 1 8-8 8 C 9 2 2 7 4 5 2 D F. I apprehend you plainly , T fee ''tis very reafonahle and eafy. Pray grve me {owe Examples for further Pratt ice. M. I WILL. Example I. I borrowed at divers Times - * 227243 Bricks. I have paid in Part by feveral Payments - _ 115362 Remains due - 111881 Example II. I have lent at divers Times - I have received in Part thereof - Remains yet due to me 997 2 43829ii Ten Foot Deals. 8754342 954° 12180953371 From - Take - Remains Proof _ - - - 101 Example III. 9999988888777776666655554422233322557119 8989898989878787876767656543221275463007 ’""0089898898988789887897879012047094112 - 9999988888777776666655554422233322557119 i Example The Principles of Arithmetick. 41 Example IV. I have lent a Friend divers Parcels of Money; 1 . At one time I lent him 743 217 at another time - - 954321 at another - - 74312 at another - - - - 612 at another 81c Lent in all - *• 1773272 My Friend has paid me ; 1 . at one time - - 621325 at another time - 743216 at another - - - 3 210 at another - 520 Received in all - 1368271 In this Example, I firft add all the Sums lent into one Sum, and alfo what Was paid. I. Then the whole Money lent is - - - - 1773272 And the whole Money paid me in Part is - 1368271 So there remains to pay - 0405001 These Examples are fufficient for Numbers or Quantities of one Deno¬ mination ; and therefore I lhall now proceed to fhew you The Subtraction of Numbers of divers Denominations. Subtraction of Money. Example I. Example II. Example III. l s. d. q. 1. s. d. q. 1. s. d. Lent 7963 3 21 Lent - 572 11 8 Lent^ - 2327 !9 II I Paid 6272 19 10 3 Paid - 431 19 I I in Part N 49 11 IO O Due 1690 3 32 Due - 140 11 9 Due - 1078 8 I I Proof 7963 3 2 1 Proof - - 5 7 2 11 8 M. The Subtradlion of Money is very little different from whole Num¬ bers or Integers, and the Manner of placing Sums is exa&ly the fame as in Addition ; that is, you place Pounds under Pounds, Shillings under Shil¬ lings, Pence under Pence, and Farthings under Farthings, taking Care, that the greateft Number of the two, be the uppermoft ; as you fee in the three' foregoing Examples^ Now Observe, That when the Number , out of which you. are to fubtraCf, is leffer than the Number to be fubtracted ; then, infead of borrowing 10, as you did in the preceding Examples, you mufi borrow Jo many as make an [‘nit of the next Denomination, and add to it, and then fubtratl from that Sum, and place the Remainder under the Line, as in the other Examples. I will illuffrate this by the three preceding Examples. 1. To work the firft Example, I begin at the Column of Farthings, a d fay, o from 1 Farthing, and there remains 1 Farthing, which I let down underneath the Farthings; then 10 Pence from 11 Pence, refts 1 Penny, which K I fet 42 The Principles of Arithmkti c k. I fet down under Pence 5 then 11 Shillings from 19 Shillings, refts 8 Shil¬ lings, which I put down under Shillings ; then 9 Pounds from 7 I cannot, but 9 from 17 (the Pounds being the lull: Denomination, I therefore borrow 10) refts 8, and 5 from 1:, refts 7, and 3 from 3 refts o, and 1 from 2, refts 1 : So the remains is 1078 1 . 8 s. id. 1 q. 2. To work Example II, I begin at the Column of Farthings, and lay, 3 Farthings from 1 Farthing, 1 cannot, but borrowing 4 Units from the Pence, which are equal to 1 Penny, and adding it to the 1 Farthing, makes it 5 Farthings ; then I lay, 3 Farthings from 5 Farthings, refts 2 Farthings, which 2 Farthings I write down under Farthings ; then proceed¬ ing to the Pence, I lay, 1 that I borrowed, and 10 is 11 Pence, from 2 Pence I cannot, but borrowing 1 Shilling, or 12 Pence, from the Place ot Shillings, and adding them to the 2 Pence, makes 14 Pence ; then I lay, 11 Pence from 14 Pence, refts 3 Pence, which I write down under Pence; then proceeding to the Shillings, I lay, 1 that I borrowed, and 1 9 is 20, from 3 I cannot, but borrowing 20 from the Pounds, which makes one Unit thereof, and adding it to the 3, makes 23 ; then I fay, 20 from 23, refts 3, which I fet down under Shillings; then proceeding to the Pounds, I fay, 1 I borrow¬ ed, and 2 is 3, from 3 refts o, and 7 from 16 refts 9, and 3 from 9 refts 6, and 6 from 7 reft: 1 : So that the Remains is 1690 1 . 3 s. 3d. 2 q. 3. To work Example III, I begin at the Pence, and fiy, 11 Pence from 8 Pence I cannot, but borrowing 1 Shilling, and adding to the 8 Pence, makes it 20 Pence ; then 11 from 20, refts 9, which I place under Pence ; then I proceed to the Shillings, and lay, 1 I borrowed, and 19 is 20, from 11 I cannot, but borrowing 20, and adding it to the 11, makes it 31 ; then I lay, 20 from 31, refts 11, which I write down under Shil¬ lings ; and proceeding to the Pounds, lay, 1 I borrowed, and 1 is 2, from 2 reft o, and 3 from 7 refts 4, and 4 from 5 reft 1 : So that the Remains is 140 1 . ns. 9 d. And fo in like Manner all others. P. Pray do you prove thefe Subtractions as you did the foregoing whole Numbers ? M. Yes : By adding the fmalleft Number and Remainder together, as you lee done in the Examples. I shall now proceed to the Subtra&ion of various Things for your further PradHce. I. Subtraction of Feet and Inches. Feet. Inch. Feet. Inch. Feet. Inch. From 123 11 From 7278 3 From 1-435 O Take 102 7 Take 6142 11 Take 7214 IO Reft 21 4 Reft 1135 4 Reft 5220 2 Proof 123 11 Proof 7278 3 Proof I2 435 O In Works of this Nature, you borrow 12, when required from the Feet, and carry 1 for it; as in the Second and Third Examples. II. Sub- The Principles of Arithmetic k, 43 II. Subtraction of Yards, Feet, and Inches. Yards. Feet.Inch. Yards. Feet.Inch. From 107 Take 97 2 7 11 From 2354 Take 1243 0 4 5 From 2721 0 Take 123 2 9 10 Reft 9 2 8 Reft 1110 0 11 Reft 2597 0 11 Proof 107 2 7 Proof 2354 0 4 Proof 2721 0 9 Here, when the Yards. 0 p have Occalion, you borrow 12. from the Feet, and 3 from Yards. Feet.Inch. III. Subtraction of Fathoms, Yards, Feet, and Inches. Fath. Yards. Feet. Inches. Fath. Yards. Feet. Inches. From 272 0 2 10 From 5432 2 2 1 Take 261 0 2 11 Take 4132 2 i 10 Reft 10 1 2 11 Reft 1300 0 0 3 Proof 272 0 2 10 Proof 5432 2 2 1 Here you borrow 12 from the Feet, 3 from the Yards, and 2 from the Fathoms. P. Pray IFhy do you borrow but two from the Fathoms, fince that a Fa - thorn is fix Feet. M. Because that 2 Yards make 1 Fathom, as 3 Feet make 1 Yard. IV. Subtraction of Fathoms and Feet. Fath. From 1234 Take 723 Feet. 5 4 Fath. From 2345 Take 379 Feet. 1 5 From Take Fath. 75 7 2 Feet. 0 5 Reft 511 I Reft 1965 2 Reft 2 1 Proof 1134 5 Proof 2345 1 Proof 75 0 Here you borrow 6 from the Fathoms, two laft Examples. when there is Occalion • as in V. Su btraction of Rods (or Poles) and Feet. Rods. From 299 Take 172 Feet. 1 *4 Rods. From 221 Take 12t Feet. 7 8 From Take Rods. 427 333 Feet. 11 12 Reft 126 3* Reft 99 i 5 i Reft 93 151 Proof 299 1 Proof 221 7 Proof 427 11 Here you borrow 16 and i, becaufe in 1 Rod there is 16 Feet and The Principles of Arithmetics. VI. SOBTR ACT I 0 N of Chains and Links. Chains Links. Chains. Links. Chains. Links. From 91 From 77 - 10 From 6272 is Take 179 95 Take 345 99 Take 5472 75 Reft 7 - 96 Reft 426 11 Reft 839 43 Proof 252 91 Proof 77 - 10 Proof 6272 18 Here you borrow ioo, becaufe in i Chain there is io Links. VII. Subtraction of s 4 voirdupbize JJVight. Tons.Hund.Quart.Pounds. Tons.Hund.Quart.Pounds. From 701 11 2 16 From 882 3 2 14 Take 602 19 3 27 Take 777 18 3 20 Reft 98 11 2 17 Reft 104 62 22 Proof 701 11 a 16 Proof 882 5 2 14 Here you borow 2.8 from the Quarters, 4 from the Hundreds, and 20 from the Tons, becaufe that 28 Pounds is 1 Quarter of an Hundred, 4 Quar¬ ters is 1 Hundred, and 20 Hundred is 1 Ton. VIII. Subtraction of Timber. Loads. Feet. Loads. Feet. Loads. Feet. From 5-7 ■15 From 723 15 From 222 3 - Take 3-7 49 Take 527 39 Take 197 35 Reft 199 16 Reft 195 26 Reft -4 47 Proof 5 2 7 2 5 Proof 723 15 Proof 122 3 2 Here you borrow 50 from the Loads, becaufe that in 1 Load of Timber there is 5 c Feet. IX. Subtraction of Bricks. Loads. Bricks. Loads. Bricks. Loads. Bricks. From 221 472 From 325 28 From 772 101 Take 137 499 Take 279 328 Take 555 425 Reft 8 3 473 Reft 45 200 Reft 216 176 Proof 22i 472 Proof 325 28 Proof . 7 . 7 '-. 101 Here you borrow 500 from the Loads becaufe 500 Bricks is 1 Load. X. SpBTlACTIO.N of Lme. Hund. Baggs. Hund. Baggs. H und. Baggs. F ram 272 18 From 777 <5 From I I Take 192 24 Take 555 20 Take 724 2 3 11 eft 79 19 Reft 221 20 Reft IOO 13 Proof 272 18 Proof 777 1 5 Proof 82s I I Here you borrow 25 from the Hundreds, becaufe 25 Baggs make One Hundred of Lime. XI. Sr- . The Principles of Arithmetick. 45 XI. Subtraction of Sand. Loads. Buftiels. Loads. Bulhels. Loads. Buftiel From 505 8 From 207 11 From 221 9 Take 407 17 Take 118 16 Take 125 11 Reft 97 9 Reft 88 i3 Reft 95 16 Proof 505 8 Proof 207 11 Proof 221 9 Here you borrow 18 from the Loads, becaufe 18 Baihels make i Load. XII. Subtraction of Lamt-Meafnre. Acres. Roods. Poles. Feet. Acres. Roods. Poles. Feet. From 1728 2 37 15 From 2237 I 27 19 Take 507 3 38 207 lake 14 3 2 3 35 201 Reft 1220 2 38 8oj Reft 804 1 3 1 9 °i Proof 1728 2 37 15 Proof 2 -37 1 27 19 Here you borrow 17 z i from the Poles, 40 from the Roods, and 4 from the Acres, becaufe that 171 Feet and j is 1 Pole, 40 Pole is 1 Rood, and 4 Roods are 1 Acre. XIII. Subtraction of Flooring. From Squares. 729 Feet. 75 From Squares. 55 2 4 Feet. 7 From Squares. 33 2 Feet. 18 Take 34 1 - 80 Take 4444 95 Take 271 19 Reft 386 95 Reft 1079 12 Reft 60 99 Proof 729 75 Proof 5514 7 Proof 232 18 Here you borrow 100, becaufe that 100 lquare Feet make 1 lquare of Flooring. XIV. Subtract; On of Gilding. Squ. Feet. Squ. Inches. From 2 79 14° Take 139 J 43 From Take Squ. Feet. Squ. 339 222 Inches. 49 97 Reft 139 141 Reft 116 96 Proof 279 140 Proof 339 49 Here you borrow 144 from the Feet, becaufe that in 1 fquare Foot there are 144 fquare Inches. X 1 6 77t Principles of Arithmetics XVI. Subtraction of Solid Yards. Solid Ys. Feet. Solid Ys. Feet. Solid Ys. Feet. From 527 18 From 666 21 From 726 16 Take 471 19 Take 198 26 Take 517 24 Reft 54 26 Reft 467 22 Reft 208 19 Proof 52 7 18 Proof 666 21 Proof 726 16 Here you borrow 27 from the Yards, becaufe that 27 Solid Feet make 1 Solid Yard. XVII. Subtraction of Degrees and Minutes. From Deg. 926 Min. 45 From Deg. 365 Min. 20 From Deg. 24 Min. 18 Take 728 50 Take 321 59 Take 18 54 Reft r 9 7 55 Reft 43 21 Reft 5 24 Proof 926 45 Proof 365 20 Proof 24 18 Here you borrow 60 from the Degrees, becaufe that 60 Minutes make 1 Degree. XVIII. Subtraction of Plank 1 Inch thick. iSote , 600 Feet make 1 Load. From Take Loads. 72 36 Feet. 527 582 From Take Loads. 672 492 Feet. *37 272 From Take Loads. 291 172 Feet. 11 272 Reft 35 545 Reft 1 79 465 Reft ns 339 Proof 72 527 Proof 672 J 37 Proof 291 11 Here you borrow 600, becaufe 600 fquare Feet make 1 Load. XIX. Subtraction of Plank 1 Inch and ; thick. Note , 400 Feet make 1 Load. From Take Loads. 437 327 Feet. 271 3 * 1 From Take Loads. 521 328 ■ Feet. 11 3°7 From Take Loads. 60 57 Feet. 332 399 Reft IO9 360 Reft 192 104 Reft 333 Proof 437 271 Proof 521 11 Proof 60 332 Here you borrow 400, becaufe 400 fquare Feet make 1 Load. XX. Subtraction of Plank 2 Inches thick. Note, 3 °° Feet make i Load. Loads. Feet. Loads Feet. Loads. Feet. From 792 18 From 54° 207 From 327 256 lake 672 27 lake 432 299 Take 291 270 Reft 119 491 Reft 107 208 Reft 35 286 Proof 792 18 Proof 54° 207 Proof 327 256 Here you borrow 300, becaule 300 lquare Feet make 1 Load. XXI. Sub- 1 47 The Trinciples of Arithmetic^ XXI. Subtraction of Plank three Inches thick. Note, That 200 Feet make 1 Load. Loads. Feet. Loads. Feet. Loads. Feet. From 53 1 105 From 123 27 From 520 79 lake 427 199 lake 102 170 Take 472 170 Reft 103 106 Reft 20 57 Reft 47 109 Proof 53 1 105 Proof I2 3 ■ 27 Proof 520 79 Here you borrow 200, becaufe 200 Feet make 1 Load. XXII. Subtraction of Plank four Inches thick. Note , That 150 Feet make 1 Load. From Take Loads. 291 172 Feet. 140 1 4 1 From Take Loads. in 104 Feet. 6 9 149 From Take Loads. 222 215 Feet. 87 13 1 Reft 118 149 Reft 6 70 Reft 6 106 Proof 291 140 Proof III 69 Proof 222 87 Here you borrow 150, becaufe 150 fquare Feet make 1 Load. XXII. Subtraction of Time. I was Born in 1696. How many Years is! The prefent Year 1732 my Age to this prefent Year 1732? Y The Year Born in 1696 Anfvver - 36 Thus have I given Varieties of Queftions that are of very great Ufe as well as entertaining in their different Natures. I {hall now proceed to the Fourth Rule of Arithmetick, called Multiplication. LECTURE IV. Of Multiplication. P. T AT is Multiplication? M. Multiplication is no more than a concife Method of -adding Numbers together, and therefore may be juftly called Short Addition. P. Prat explain this to me. iz I will. Suppofe you were to add together the following Num- 12 bers, viz. 6 times iz ; then, according to the Rule of Addition, you 12 let them, as in the Margin, and adding them up, they make 72 : ia But to find their total Sum by Multiplication, you muft write once 12 i z only, with 6, the Number of Times under it, as following : 12 Then The Trinciples of Arithmetick. 48 Then beginning with the 6, fay,6times a is 12 ; let down the z iz underneath the 6, and carry 1 in your Mind for the 10, and then fay, 6 6 times 1 is 6, and 1 you carried is 7 ; then write down 7 under the 1, ~ and it makes the 2, 72 ; which is equal to the total Sum of the 6 ' Twelves that you added together. Is’ow from this you fee, that s 4 dditton and Multiplication are in effeff the lame Things, and differ only in the Manner of their Operations, or Working. You are flow to obferve , That in Multiplication there are three Numbers that are diftinguiihed by their particular Names, which you muff well take Notice of ; that is to lay, Fuff, The Multiplicand. Secondly, 1 he Multiplier. And laftly, the Product. P. Pray explain them feparately , as I may underfundyou rightly. M. I WILL. In the Example before-mention’d, to iz Multiplicand. multiply 12 by 6, the Number 12 is the Multiplicand ; 6 Multiplier. the Number 6 the Multiplier ; and the Number 72 is 72 Produff. the Produff ; as in the Margin lignified. P. Pray why is the Multiplicandfo called ? M. Because it is the Number that is to be multiplied, or added to itfelf, as many times as there are Units in the Multiplier. P. Pray why is the Multiplier fo called ? M. Because it is that Number which increafes or multiplies the Multi¬ plicand as many times greater than itfelf, as the Number ol Units contain’d therein ; and as the laft Number or Produff is produced by the Multiplication of the Multiplier into the Multiplicand, it is therefore call’d Produff. P. Pray, am I to have the fame Regard to the placing of the Figures in the Multiplicand and Multiplier , as I have in Addition and Subt raff ion ? M. Yes; you muff here, as before, obferve to place Units under Units , Fens under Tens , &c. but you are not obliged to place the greateft Num¬ ber uppermoft, as you were direffedto do in Subtraction ; for it matters not which of the Numbers is made the Multiplicand, or which the Multi¬ plier, for the Produff is in both Cafes the lame, 12 times 6 being 72, as well as 6 times 12 is 72. But however, its more convenient to make the lefs Number the Multiplier. That you may have a per- feff Idea of this Rule, do you fuppole that each of the little Squares mark’d a, in the Dia¬ gram, to be a fquare Foot, and let there be 12 of them ranged clofe together in a ftrait Line, a A 12 B : Now if you were to to multiply the fiid 12 Feet by 2, Multiplicand <5 5 1 a a a | a a a a a a a a • 1 1> b if b * b b b b b b c c c c c ’ c c c c c c c 1 d d ~d d if d d d d e e e 1 e e e e e e e e L ,/ Al A I J JlL } ./ t * Product. the The Principles of Arithmetick. the Produdt would be 24 ; that is, to the 11 Squares mark’d a a, Ac. there would be the Addition of the iz other Squares b b, &c. making 24 Squares in the whole: Again, if you multiply the laid iz Squares by three, the Pro- du£f will be 36 ; that is, then to the 12 Squares aforefiid, there would be the Addition of 24 other Squares, which are the Squares b b, &c. and c c, Ac. making 3 6 Squares in the whole. Now here you fee, occularly demonftrated, that this Multiplication is no more than the Multiplicand 12, added as oftentimes as the Multipliers 2 and 3 confided of Units; and fo in like Manner, 12 multiplied by 4, pro¬ duces the little Squares a a, Ac. b b, Ac. c c, Ac. and d d, Ac. making in the whole 48 Squares : And 12 multiplied by 5, produces the little Squares a a, Ac. b b, Ac. c c, Ac. dd, Ac. and e e, Ac. making in the whole 60 Squares : Alfo 12 multiplied by 6, produces the little Squares a a, Act h b, Ac. i c, Ac, dd, Ac. ee, Ac. and f f Ac. making in the whole 72 Squares, which all are comprized m the oblong lquare Figure marked A B C II. Now you are to obferve, That in this lad Multiplication, the Side A b is the Multiplicand, the End A C, the Multiplier, and all the 72 Squares taken together, make the Product. P. Vert well, Sir : But what does the Lines B D, and C D reprefent. M. The Lines BD and C D, are but boundary Lines, compleating the Squares, or they may be confidered as A B and A C were ; that is, you may make C D the Multiplicand, as we did A B ; and B D may be made the Multiplier; and then in fuch Cafe, the Lines A B and A C do become boundary Lines compleating the Squares or Product in their reipcctivc Places. P. I UNDERSTAND you very right, and from hence T think it appears, that the Multiplicand and Multiplier, in all %nefiions of Multiplication, are to be confidered as two Lines, the one as the Length of an oblong. Square, and the other as the Breadth ■ and the ProduSt the Space included within the Lines or Bounds of the Square thereof. Mud if I am right herein, then when the Multiplicand and the Multiplier are equal to each other ; that is,, when thofe two Lines are of the fame Length, and the boundary Lines oppofte to each, be the fame, they together muft make a ProduSt, or Figure that is exaSlly jquare. M. ’Tis true ; you have a right Underdanding thereof, it is the Reafon of Multiplication, and now the whole will become very eafy. But before you proceed any farther, get the following Table by Heart. M Multi- The "Principles of Aeit it metick, 50 P. I will. But really, at pntfent, I dont know how to begin. M. I will make it eafy. Firft you fee, that on the Head of the Table are pla¬ ced the Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,and the like down the Left-hand Side ; and the Ufe of this Table is to multiply thofe at the Head, into thofe at the Side, and tell their Pro¬ duces. As for Example : I would multiply 2 by 2, or know how much 2 times 2 make. Multiplication Table. I 2 _ S 4 5 6 j 7 3 9 10 1 1 2 4 6 6 9 _S I 2 10 L 5 irt '4 16 . ... ai 2 7 _20 SO ay 33 24 S 6 4 8 1 2 i 6 20 2 4 3 2 36 40 44 48 5 IO 15 20 2 5 30 35 40 45 5 0 4 5 60 6 I 2 18 -4 3 ° 36 4 : 45 44 60 66] 72 I 'oofa £4 16 2 J -4 28 .r- iiP 4048 49 56 56 64 ^3 7 2 go 8p 77 1 84 SB j 96 9 _ 10 1 h 2 C sO f 6 40 45 50 54 60 65 70 I M | O ' 1 W cn 81 90 90 r oc 99 108 110120 r 1 1 2 24 76 id id 4S|6o 66 7 2 7 7 54 88 96 99 108 11 c 1 20 1 2 1 I 3 2 r ^ 2 777 First, I find z in the Side of the Table, and alio z in the Head of the I able, then in the little Square that is under the z at the Top, and ag.unft z in the Side, Hands 4, which is the Produift of 2 multiplied by 2. . Again, I would multiply 6 by 4, or know how much 6 times-4 make. Firft I find 6 m the Side, and under 4 at the Top, Hands 24, which is the Producf of 6 multiplied by 4 ; and lo in like Manner all other Numbers, as following : Twice < 4 5 6 7 i> is < " | Three I ^i > times 10 ! 2 4 3 4 5 6 7 > 8 9 10 11 Five times ihi 9 10 11 12 > six J I times I 4 5 6 7 > is 8 9 10 6 1 r 8’ 9 \ 3 [ I 2 r 2 4 16 15 | 5 20 r 8 1 < zi 2 4 1 Four times 6 8 > is 2 4 :=f 3 - 2 7 9 3 6 3 ° 10 40 33 11 44 J 6 . 12 48 L J ■ I , ■ ‘4 18 3 ZI 2 4 4 z8 . 30 5 35 36 6 4 2 1 |S Seven 1 times 1 7 8 is < 49 !> 5 b 54 9 63 60 IO 7 ° . 66 n 77 : l 72 12 84 J. Eight The Principles of Arithmetic k. 51 . p - S IR, f learn d this Table perfeBly by Heart, ami can readily mul- tiply any Numbers therein. M. Very well : Now I will proceed to make you perfedt in this Rule, fo as to work any Sum that lhall be Rated you. Say, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, and 2 times 7 is 14 ; So the Product is 14868. By 1 Figure J Multiply 7432 IBy 2 Product 14864 If. Figure C Multiply 2By Produdt 5947 2 7 Example Say, 3 times 7 is 21, fet down 1, n and carry 2, for the 20, and lay 3 times 2 is 6, and 2 I carry is 8, which let n ~ . . down under the 2 ; then fry, 3 times r0 uc 4 1 7 is 21, fet down the 1, and carry 2, and fay 3 times 4 is 12, and 2 I carre¬ ls 14, let down 4 and carry 1 for the Ten, and lay, 3 times 9 is 27, and 1 I carry is 28, let down the 8 and carry 2 • then 3 times 5 is 15, and 2 I carry is 17, which being the laft, fet it down under the 5 : So will the Pro¬ duct be 1784181. Example III. First fay, 7 times 4 is 28, let down 8 and carry 2 • then 7 times 5 1S 35 > and 2 is 37, let down 7 and carry 3 ; then 7 times 3 is 21, and 3 is 24, fet down 4 and carry 2 ■ then 7 times 4 is 28, and 2 I carry it 3 °, let down o, and carry 3 ; By 2 Figures ^Multiply 77-4354 _97 54070478 69519186 ProduS 749262338 then $ 2 The Trinciples of Arithmetick. then 7 times a is 14, and 3 I carry is 17, fet dow n 7 and carry 1 ; then 7 times 7 is 49, and 1 I carry is 50, let down o and carry 5 ; then 7 times 7 is 49, and '5 I carry is 54, which fet down, and fo have you finilh’d the Multiplication of the firft Figure. Secondly, lay 9 times 4 is 36, fet down the 6 under the the 9 and carry 3 • then 9 times 5 is 45, and 3 I carry is 48, fet down 8 and carry 4 5 then 9 times 3 is 27, and 4 I carry is 31, fet down 1 and carry 3 ; then 9 times 4 is 36, and 3 1 carry is 39, let down 9 and carry 3 ; then 9 times z is 18, and 3 I carry is 2,1, fet down 1 and carry 2 • then 9 times 7 is 63, and 2 I carry is 65, fet doum 5 and carry 6 • then 9 times 7 is 63, and 6 I carry is 69. Thirdly, add together the two Sums produced by the tw r o Multipliers, and their Total 749162338 is the Product required. Example IV. 9 » 5 “ p ' r 3 31275 Produce of the Multiplier 7 378600 Produce of the fecond Multiplier 8 425925 Produce of the third Multiplier 9 Product 46709775 Total Sum of all the Produffs. First fay, 7 times 5 is 35, fet down 5 under the 7 and carry 3 ; then fay 7 times 2 is 14, and 3 I carry is 17, fet down 7 and carry 1 • then 7 times 3 is 21, and 1 I carry is 22, fet down 2 and carry 2 • then 7 times 7 is 49, and 2 I carry is 51, fet down 1 and carry 5 ; then fiy 7 times 4 is 28, and 5 I carry is 33 : fo will the Produce of the firft Multiplier be 331 275 as above. Secondly, Begin with the fecond Multiplier 8, and fay 8 times 5 is 40, fet down o under the Multiplier 8, and carry 4; then 8 times 2 is 16, and 4 I carry is 20, let down o and carry 2 ; then 8 times 3 is 24, and 2 I carry is 26, fet down 6 and carry 2 • then 8 times 7 is 56, and 2 1 carry is 58, fet dowm 8 and carry 5 ; then 8 times 4 is 32, and 5 I carry is 37 : fo will the Produce of the lecond Multiplier be 378600 as above. Thirdly, Begin with the third Multiplier 9, and fay, 9 times 5 is 45, fet down 5 under the 9, and carry 4 * then 9 times 2 is 18, and 4 I carry is 22, let down 2, and carry 2 ; then 9 times 3 is 27, and 2 I carry is 29, fet down 9, and cary 2 ; then 9 times 7 is 63, and 2 I carry is 65, fet down 5, and carry 6 ; then lay, 9 times 4 is 36, and 6 is 42 : lo will the Produce ol the third Multiplier be 425925 as above. Fourthly, Add together thefe three Produtfts, as they ftand in their refpeclive Places, and their total Sum 46709775 is the Produdf required. Example V. First fay, once 2 is 2, once 4 is Multiply 50734: 4, once 3 is 3, once 7 is 7, once 0 is 0, and once 5 is 5. By 400 I 50734 - Secondly, As the next tw r o 202936800 Multipliers are Cyphers, therefore firft let them down under themfelves, and ProduQ 102987534: thea The Trinciples of Arithmetick. 53 then proceed to the Multiplier 4, and fay 4 times a is 8, which fet down under the 4 ; and 4 times 4 is 16, fet down 6 and carry 1 ; then 4 times 3 is 12, and 1 I carry is 13, fet down 3 and carry 1 ; then 4 times 7 is 28 and 1 I carry is 29, fet down 9 and carry 2 • and becaufe 4 times o is o’ therefore fet down the 2 you carried ; laftly, 4 times 5 is 20. Now thefe’ two Produ&s being added as they ftand, their Total will be the Produfl required. P. Pray what is the Re af F Thirdly, Multiply 27, the Multiplicand, " J ^ crols ways into 10 Inches of the Multiplier, ^ * laying 27 Halves, or Six’s, is 13 Feet 6 432 o 2 Inches, which let down • and 27 Quarters, or - Three’s, is 6 Feet 9 Inches • and 27 One’s is 2 Feet 3 Inches, which let down as at D E F. Fourthly, Multiply the Inches into themfolves, flying 10 times 11 is 110, containing Twelve 9 times, and 2 remaining, which fot down as at G. Lastly, Add up the Inches, carrying 1 for every 12 to the Feet, and add up the Feet as Integers • their Total will be the Produdf required. First, That Feet multiplied into Quarters of Inches, each one of the fquare Foot. The Realons for which hereafter in its Place. Secondly, Inches multiplied into Quarters, each 12 is 1 Quarter, and every 3 is a Quarter of a Quarter, or One lixteenth Part of an Inch. Thirdly, Quarters multiplied into Quarters, each 12 is a Quarter of a Quarter, or One lixteenth Part of an Inch. The Principles of Arithhstick. 6 This I will make plain to you b■ t by pie. the Operation of the prefent Examp, First, Multiply the Feet into themlelves, and let down their Produdh Secondly, Multiply 29, the Mul¬ tiplier, crols ways into 7 Inches of the Multiplicand, laying 29 Halves is 14 Feet 6 Inches, which let down under Feet and Inches, and 29 ones is 2 Feet 5 Inches • fet thefe down as before. Multiply By Feet. Inch. Quart. 37 7 3 X X 333 74 18 9 *3 Thirdly, Multiply 37 Feet in the Multiplicand, into 11 Inches in the Multiplier, laying 37 Halves is 18 Feet 6 Inches, and 37 Quarters is 9 Feet 3 Inches, and 3 7 two Inches, equal to 74 Inches, equal to 6 Feet 2- Inches ; let all thefe down under Feet and Inches, either feverally as ProdudI 1061 3 A 2 B 9 C 2 D 6 E 10 multiply them, or in one Sum, adding them all three together. In the Example I have let them down feverally, but in Praftice you may do as you pleale. Fourthly, Multiply 11 Inches, the Multiplier, into 7 Inches in the Multiplicand, and their Product is 77 Inches, containing Twelve 6 times, and 5 remaining : Therefore fet down 6 Inches 5 Parts under Inches and Parts. Fifthly, Multiply 29, the Multiplier, into 3, the Quarters, and their Product is 87, which, as I told you before, are each Quarters ol an Inch fquare ; therefore every 4 of them is but 1 lquare Inch, and in 87 there is 2i of thole Inches, and 3 remaining. Now, as I told you, that 12 of thofe fquare Inches made but 1 Inch in the Column of Inches ; therefore take once 12 out of the 21, and let that down under the Inches, with the Remainder 9 under the Parts, and the firft remaining 3 fquare Quarters of an Inch in another Column next the Right-hand, as at A. Sixthly, In the fame Manner multiply crofs ways the 37 of the Multi¬ plicand, into the 2 Quarters of the Multiplier, faying 37 times 2 is 74, containing Four 18 times, in which 12 is once, and reft 6 : Therefore let dow n 1 under the Inches, 6 under the Parts, and 2 in the laft Column, as at B. Se venthly, Multiply 11 Inches, the Multiplier, crols ways into 3 Quar¬ ters in the Multiplicand, and they produce 33 ; and, as I told you, that each 12 is a quarter ol an Inch ; therefore twice 12 from thence refts 9, fet 2 under the Quarters, and 9 in the next Column, as at C. Eighthly, Multiply 7 Inches, the -Multiplicand, crols ways into 2 Quar¬ ters in the Multiplier, and they produce 14, which is 1 Quarter 2 Parts, as at D. Lastly, Multiply 2 Quarters into 3 Quarters, equal to 6, which place down in the laft Column, as at E ; and then adding up every Column, carry P 1 for 62 The Principles of Arithmetick, i for every I z from the Quarters to the Inches, and from the Inches to the Feet, their Total will be the Produdl required. I SHALL now give you divers Queftions for your further Prafiice ; as following : Question I. In 57-34 Pence, how many Far¬ things ? Rule. Becaufe that 4 Farthings make 1 Penny, you mult therefore multiply the given Number by 4, as in the Margin. 57-34 _ 4 2.-8936 Q. II. Inf 9-435 Shillings, how many Pence? Rule. Multiply the given Number by 12 5 be- caufe 1 2 Pence is 1 Shilling. n ' V 79-435 9509220 To multiply Numbers of this Kind, where the Multiplier conjijls but of two Figures, tis befi to male the ufp r Number the Multiplier, and the lower the Multiplicand, as follows ; Jay 5 times 12 is 60, Jet down o, and carry 6 ; then 3 times 1 2 is 36, and 6 I carry is 42 ,Jet down 2, and carry 4 • then 4 times 12 is 48, and 4 1 L iny is 52, Jet down 2 and carry 5 ; then 2 times 12 is 24, and 5 I carry is 29, Jet down 9 and carry 2 ; then 9 times 12 is 10S, and 2 I carry is no, Jit down o, and carry 11 5 then 7 times 1 2 is 84, and 11 I carry is 95, which Jet down; And Jo will the Produbl be 9509220, the true Produbl required. Q. III. In 755367 Pounds, how many Shillings? 7553^7 20 Rule. Becaufe 20 Shilling's is 1 Pound, therefore r ' 1 • 1 1 x- 1 1 Anfw. 15107340 multiply the given Number by 20. " Q. IV. In 5274 Feet, how many Inches ? 5 2 74 12 Rule. Becaufe 12 Inches make 1 Foot, therefore . r : , • , . • XT 1 1 Anlw. 62288 multiply the given Number by 12. _ Q. V. In 72545 Yards, how many Feet ? Rule. Becaufe 3 Feet make 1 Yard, therefore multiply the given Number by 3. Anfw. 7 2 345 3 217035 Q. VI. In 57243 Fathoms, how many Feet ? Rule. Becaufe 6 Feet make 1 Fathom, therefore multiply the given Number by 6. Anfvv. 57 2 43 6 343453 Q. VII. In 57928 Rods or Poles, how many Feet? Rule- Becaufe 16 Feet and i make 1 Rod or Pole, therefore multiply the given Number by 16 *. 57928 i6t 926848 28964 Anfw. 955812 Note, The Trincipies of Arithmetick. 63 Tl ?: lt to . I f uIt . i P 1 y the Multiplicand into the j, you need only halve the Multiplicand, beginning with thefirft Figure next the Left Hand viz. 5. and lay half 5 is 1, which fet down under the 5, and carry on the odd ’1 to the 7 and lay half 17 is 8 fet down 8, and carry on the odd 1 to the 9, and fay halt 19 is 9, fet 9 under the 9, and carry on the odd 1 to the 2, and fay half under the 8 d ° Wn 6 Undei ' ^ and then half 8 is 4 > lvhich fet down Thus will you have finilhed the Multiplication of the 1 or 6 Inches and their Total being added, will be the Produft required. CT. V III. In 5749 1 Chains , how many Links ? Rule. Becaufe 100 Links is contained in 1 Chain, therefore multiply the given Number by 100. Q. IX. In 792927 Hundreds Avoirdupois (Height, how many Pounds ? Rule. Becaufe 112 Pounds make one Hundred, therefore multiply the given Number by n 2. hf' ■ Tt - 7543 “ Tons (Height, how maty Hundred"? Rule. Becaufe 20 Hundreds make 1 Ton, there¬ fore multiply the given Number by 20. Q; XI. In 72597 Loads of Timber, how many Solid Feet ? J Rule. Becaufe 50 Solid Feet make 1 Load of Timber, therefore multiply the given Number by 50. (X XII. In 75924 Loads of Bricks, how tnaty Bricks ? Rule. Becaufe that 500 Bricks are 1 Load, there¬ fore multiply the given Number by 500. Q.. XIII. In 72431 Hundred of Lime, how many Pages ? Rule. Becaufe that 1 Hundred of Lime contains - 5 Baggs, therefore multiply the given Number by 25. Qs XIV. In 51243 Loads of Sand , how many Bujhels ? Rule. Becaufe that 18 Bulhels make 1 Load of Sand, therefore multiply the given Number by 18. 57492 . 100 Anfiv. 5749200 791927 112 ,9515114 792917 Anfw. 88807824 1 7543 2 Anfw. 5508640 72-597 5 ° Anfw. 3619850 75924 500 Anfiv. 37962000 72431 _25 Anfw. 1810775 51245 18 Anfw. 922374 The ‘Principles of Arithmetics. T 99"-5 144 319700 1128950 113214700 Q.XV. I// 799-5 fquare Feet, how many fqu,we Inches? Rule. Becaufe that 144 fquare Inches make 1 fquare Foot, therefore multiply the given Number by 14+ Kote, That whereas it may be too bird for you to multiply the 2 firft Figures of the Multiplier 44, at once, therefore, firft multiply by the firft Figure 4, and afterwards the other two, viz. 14 by the Multiplicand, as directed in Queftion II ; then add them together, and the Total will be the Product required. O. XVI. J» 99713 fquare Tards, how many fquare Feet ? Rule. Becaufe that 9 fquare Feet make 1 fquare Yard, therefore multiply the given Number by 9. q. XVII. /h 57259 Squares of Tiling, how many fquare Feet ? Rule. Becaufe that 100 fquare Feet make one Square of Work, therefore multiply the given Num¬ ber by 100. q. XVIII. In 84327 fquare Rods or Poles, how many fquare Feet ? Rule. Becaufe that 1 fquare Rod contains 172 Feet and;, therefore multiply the given Number by 997 1 Anfw. 897507 57*59 100 Anfw. 5725900 84327 *7*i 168654 590289 168654 21081J To multiply the Multiplicand 84317 into the ; of a Anfw. 229580281- Foot, you mull firft write down on a Piece of wafte Pa- - per the Multiplicand; as at A in the Margin, then halve it, as taught in queftion \ II, which will be the Num¬ ber 42165;, as at B ; then halve this Number again, as at C, which will be a quarter Part, and is the Product of 84327 multiplied by j, which you muft add to the other Products, and the Total will be the Produdt re¬ quired. A 84317 B 42163; C 21081I q. XIX. Feet 3 In 72431 Jhlid Tards, how maty Jolid Rule. Becaufe that 1 folid Yard contains 27 fo- lid Feet, therefore multiply the given Number by 27. q XX. In 360 Degrees, how maty Minutes ? Rule. Becaufe that in 1 Degree there is 60 Mi¬ nutes, therefore multiply the given N umber by 60. 72432 Anfw. 1955664 360 60 Anfw. 11600 q XXI. The Principles of Arithmetick. 65 Q; XXI. In 5732 Loads of 1 Inch Plank boon many Feet ? J ' J Rule. Becaufe that 600 Feet of 1 Inch Plank Anfiv. make 1 Load, therefore multiply the given Number by 600. Q. XXII. In 79425 Loads of Plank 1 Inch and j thick, how many Feet ? 57 V- 600 5459200 794*5 4 00 Rule. Becaule that 400 Feet of t Inch and 1 thick ^ n ^ v - 5 * 77 ° 00 ° Plank make 1 Load, therefore multiply the given Num¬ ber by 400. Or XXIII. In 5542 Loads of z Inch Plank, how many Feet ? Rule. Becaule that 500 Feet of 2 Inch Plank Anfvv. make 1 Load, therefore multiply the given Number by 500. Qt XXIV. In 7725 Loads of 5 Inch Plank, how many Feet ? Rule. Becaufe that 200 Feet of 3 Inch Plank AnAv * make 1 Load, therefore multiply the given Number by 200. fb XXV . In 9972 Lunds of 4 Inch Plank , ho-w many Feet ? Rule. Becaufe that 150 Feet of 4 Inch Plank Anfw. make 1 Load, therefore multiply the given Number by 150. 55 4 * 3° o 1602600 77 2 3 200 1544600 997 2 I50 149586 Thus have I given you a large Variety of Examples and Queftions, which being well underftood, we may proceed to Division.- lecture v. Of Division. P ' W HAT is Division? Division is in effeQ: no more than Subtradlion, by which we difcover hoiv often one Number is contained in another : For was we to lubtract one Number out of another as often times as we can find it therein we mould perform that Work which is called Division. As for Example : ouppole I was to find how often 5 is contain’d in 15 by Subtradlion ; then 0, I place 0 66 The Principles of Ahithmetick. 1 place my Numbers Us in the Margin, and proceed as fol- 15 lowing : Saying, 3 from 13 reft 1 a, which is one time 3 i 3 once then 3 from 1: reft 9, which is twice 3 , then 3 from 9 reft 6, which is three times 3 ; then 3 from 6 reft 3 which is four times 3, and the remaining 3 is 5 times 3 ; to bom hence it appears, that 3 is contain’d 5 times in 15, and o remains/ 3 twice 9 3 thrice But feeing that this Way by Sukra£Hon requires, much ^ fourth Time and Trouble in the feveral Subtractions, therefore a more concife Manner of working the fame EfTeS, has been , fi ve times invented, which is called Dtvijmi , wherein there are three principal Parts to be obferved • that is to lay, First, The given Number that is to be divided by fome other Number, which is therefore called the Dividend. Secondly, The Number by which we divide the Dividend, or feek how often it is contain’d therein, which is therefore called the DrjiJor. And, Thirdly, The Number exprefting- how often, the Number of rime 3 that the Divifor is contain’d in the Dividend, is called the Quotient. l o which we may add a fourth Number, which fometimes happens when Divi- fion is made, that is always lefs than the Divilor ; and therefore called the Remainder. I will illuftfate this by an eafy Example: Suppofe tis required to di¬ vide 10 by 3 ? a Dividend n Divifor 3) 10 (3 Quotient 9 n 1 Remainder a ) F1R s T, I place the Numbers 3 and 1 o, as in the Margin, feparating them from each other by the crooked Line a <7, and alio making another crooked Line on the Right Hand of the Divifor, as iin , to feparate the Dividend from the Quotient. Secondly, I fay, how often is 3, the Divilor, in 10, which is 3 times 5 then 1 let down 3 on the Right Hand of the Dividend, and multiply it by the Divifor, faying 3 times 3 is 9, which I let down under the 10. Thirdly, I fubtra& 9 from 10, and there reft 1, which I place under the 9, fo is 1 the Remainder. And thus have you a View of the Divifor, Dividend, Quotient, and Remainder in their refpe&rve Places. F, Very well. Sir : Pray proceed, for I believe I /hall foon underfiani Divijion, face that there is no more to do, than frfi to plate the Divijor and Dividend in their Places, and then fading how often the Divijor is contain d in the Dividend, Jet down the Ja/ue in the Quotient ; after which multiplying the Quotient by the Divijor, and Jetting the Prodult under the Dividend, and f'.btr adding it therefrom , gives the Remainder , which I Jee plainly muft be lejs than the Divijor, othemije the Divijor is not taken as open in the Dividend as it might have been. M. ’Tis 6 ? The Trinciples of Arithmetic k. M. ’Tis true, you obferve rightly, and your Obfervation on the Rule for Working, is very juft ; but as in farther Praftice, you will find it more difficult than you are now appreheniive of; I muft therefore endeavour to in¬ troduce you in as plain and eafy a.Manner as I can. And in the firft Place you muft take Notice, that Divifion is either Single or Compound. P. Prat explain them federally, and give me Examples therein. I will. Single Divifion is when the Divifor is but one fino-le Figure and the Dividend but two at moll, as in the foregoing Example. This kind of Divifion is very eafy, as you'll fee by the following Examples. Divide io by 3 , 11 by 4 , n by 5 , 13 by 6, 14 by 7, 15 by 8 , 16 by 9 , 17 by a, 18 by 3 , and 19 by 4 . Example I. 3>°(3 9 1 rem. Example II. 4)11(2 8 3 rem. Example III. IO 2, rem. Example IV. 6 ) i i(i 12 1 rem. Example V. ■ 7)14(2- r 4 0 rem. Example VI. 8 )t 5 (l 8 7 rem. Example VII. 9)r6(r 9 7 rem. ExampleW III. 2)17(1 l6 i rem. Example IX. 3)18(6 18 0 rem. Example X. 4)19(4 16 3 rem.- I n thefe Examples you are to obferve, That in Example I. 10 divided by 3, the Quotient is 3, and 1 remains. That in Example II. n divided by 4, the Quotient is 2, and 3 remains. That in Example III. 12 divided by 5 , the Quotient is 2, and 2 remains. That in Example IV. 13 divided by 6, the Quotient is 2, and 1 remains. And fo in like manner obferve the Quotient and Remains of all the other Examples, which-being fo very- plain, needs no further Account. P. ’Tis true, Sir, I fee this kind of Divifion very perfectly, and is what 1 believe I can perform by the Table of Multiplication ; for Juppofe I /eek my Dhijor at the Top of the Table, and run down the fame Column until I find the Dividend, or the uearejl Number to it, then over-agninft it in the firji Co¬ lumn funds the Quotient repined. As for Example ; To divide 10 by 3. Firf Ifind 3, the Divifor, at the Head of'the Table, and run down it to find ro ; but there being no fetch Number in the third Column, therefore I take the nearejl Number to 10 , which is 9, againft which flands 3 for the Quotient-, and as 9 is 1 lefs than 10 , therefore 1 is the Remainder : And fo in like manner any other Numbers. M. I MUST affine you I am highly pleafed to fee that you fo well uiv derftand as you proceed. Here you have in a manner conne&ed Multipli¬ cation and Divifion together, which I could not have expefied fo early. I ihall now proceed to Compound Divifion. P- Prat what is to be under food by the K r ords Compound Divifion< 68 The ‘principles of Arithmetick, M. Compound Division is when the Dividend confifts, or is com¬ pounded of more Figures than two, and the Divifor of one or moic 5 and when a Queftion of Compound Divifion is propofed, it mall be performed by the following RULE, First, Write down your Dividend, with crooked Lines at either Ends thereof, as before taught; that On the Left-hand to contain the Divifor, and that on the Right-hand for the Quotient. Secondly, Diftinguilh with a Point, fo many Places of your Dividend towards the Left-hand, as are equal, or next exceeding your Divifor. Thirdly, Ask how often your Divifor is contain’d in the ftid Number, or Figures lo pointed, and place the Number of Times in your Quotient on the Right-hand the Dividend. Fourthly, Multiply the Divifor by the Figure laft placed in the Quotient, and let the Produft underneath the pointed Figures. Fifthly, Draw a Line under the Produft laft fet down, and fubtraft that Produft’from the Figures of the Dividend pointed out, and to the Re¬ mainder bring down your next Figure of your Dividend, with w hich pro¬ ceed as you did with your firft pointed Number, and fo on 'till you have pointed and brought down all the Figures of the Dividend. Sot:, 1 F it Ihould fo happiin, that at any time, when you have pointed and brought down a Figure to a Remainder, you cannot find your Divifor one timeriierem; then, at every fuch time, you muft place, or add a Cypher to the other Figure or Figures in the Quotient, and point and bring down another Figure from the Dividend, and then proceed as before. You may alfo here obferve,- That as many Points as you have made in your Dividend, 1 b many Figures will be in the Quotient; and therefore from hence you lee the Neccihtv ol placing a Cypher in the Quotient ..t all fuch Times, when your Divifor cannot be found once in your Remain¬ ders with the next pointed Figure brought down as aforelaid. This Rule I will illujlrate by the following Examples . Example 1 . By one Figure'. Divide 79143 by 6. 6)79x43(13107 6 • • • • 19 18 11 12 ° 43 ' 4 2 1 Remains Quotient. Fprs-t, Place your Dividend and Divifor, as in the Margin ; then feeing that you can have your Divifor once" in the firft Figure 7 of the Dividend, fay the 6’s in 7 once, let down 1 in the Quotient, and fry, once 6 is 6, which fet under the 7, and fubtraft the 6 from 7, reft 1, which fet under the 6. Secondly, Make a Point under the next Figure of the-Dividend 9, and bring down the 9- to the 1 re¬ maining, which will then be 19 } then fay, how often f-he Divifor 6 is in 19 P Anfwer 3 ; fet down 3 in the The Principles of Arithmetic]*. 69 Quotient, and multiply the Divifor by it, faying, 3 times rt is 18, which fet down under 19, and fubtradt it from 19, refts 1. Thirdly, Make a Point under the next Figure of the Dividend 2, and bring down the 2 to the Remainder 1, which will then be 12 ; then fay how often is the Divifor 6 in n ? Anfwer, Twice ; then fet down a in the Quotient, and multiply the Divifor by it, frying, a times 6 is 11, which fet down under 11, and fubtraft it from 1 2, reft o. Fourthly, Make a Point under the next Figure of the Dividend 4, and bring down the 4 to the Remainer o ; and fince that in 4 you cannot have the Divifor 6 once, therefore fet a Cypher in the Quotient, and point and bring down the next Figure of the Dividend 3, which will make the 4 43 ; then fay the 6’s in 43 p Anfwer, 7 times, write down 7 in the Quo¬ tient, and multiply the Divilor by it, frying 7 times 6 is 42, which f-t under 43, and fubtradl it from thence, and 1 remains. Thus will yon ran finilhed your Sum, whole Quotient is 13207. Example II. By Two Figures. Divide 9547-43 by 47. 47)9547243(203132 First, Your Divifor and Dividend being placed as before taught, begin the Divifion, and lay, how often is 47 in 95, the firft two Figures of the Divi¬ dend, which will be found 2 times, fet down 2 in the Quotient, and fry 2 times 47 is 94, which fet under 95, and fubtraft it from thence, reft 1, which let under the 4 of the 94. Secondly, Make a Point under the next Figure of the Dividend 4, and bring down the 4 to the 1 remaining, which will make it 14 ; and becaule that 47 cannot be had in 14, therefore place a Cypher in the Quotient on the Right-hand of the 2, and then point the next Figure 7, and bring it down to the 14, which then will become 147 ; then fay the 47’s in 147 ? Anfwer 3 times, rvhich write down in the Quotient, and multiply the Divifor by it, flying 3 times 47 is 141, rvhich fet under 147, and fubtraift it from thence, refts 6. Thirdly, ! oing and bring down the 2 to the 6 remaining, making it 61 ; then lay the 47’s in 62 one time, fet down 1 in the Quotient, and once 47 under 62, and fubtraift it from thence, refts 15. Fourthly, To the 5_remaining, bring down the 4, making the 15, r 54 ( then fay, the 47s in 154? Anfwer, 3 times, fet down 3 in the Quotient, and fubtrafl: 141 from 154, refts 13. Fifthly, To the 13 remaining bring down the 3, making the 15, 133 : then fry the 47 s 111 153 ? Anfwer, 2 times, fet down 2 in the Quotient, and lay twice 47 is 94,_ which fet under 133, and fubtrading it, there refts 39, which is the Remainer, and the Quotient is 203132. 94 " • • 147 141 62 47 J 54 141 *33 94 39 N° V. R P. Sir, 7° The Principles of Arithmetics P. Sir, I underftandyour Method very rightly , and I fee that every Part of it is eafy , excepting that of finding hove often the Divijor is containd in the Numbers made by the fever at Figures of the Dividend brought down , which I mu ft oven is Jo?ne thing difficult to me, and I apprehend will be yet more diffi¬ cult, when that the Divifion confjls of 3, 4, 5, or more Figures. M. That Piece of Difficulty I will remove, and make it eafy and de¬ lightful to you, without charging your Memory in the leaft. P. Pray proceed , for herein Ifjail have abundance of Pkafure. M. That you may readily find how often your Divifor will go in any Number propofed, you muft firft (after your Divifor and Dividend are truly placed) make a Table of Divifors , which is no more than your Divifor multi¬ plied into the 9 Figures, as fol¬ lowing ; Suppofe lam to divide 53)7 1495 1 792423, by 53, then I firft place Table of Divifiors. 53 ** * * the Divifor and Dividend in their Places, and make a Table of A 53 1 262 Divifors, as in the Margin. This B 106 1 212 Table, and all others of the like C *59 3 5 ° 4 kind, are made moft eafy, as fol¬ D 212 4 ?-Times 477 lows : E 265 5 First, Write down your Di¬ F G 318 37 1 6 7 27Z *65 vifor 53, as againft A, and on the H 414 S 1 z Right-hand Side of it, write down I 477 9 . j 53 20 the Number 1, fignifying it once, or one Time. This done, double it, or multiply it by 2, laying, twice 3 is 6, and twice 5 is 10, making 106, againft which fet Number 2, fignifying 2 times. Secondly, Add the 2 Numbers together, and they make 159, againft which place Number 3, fignifying 3 times • then to this laft Number 159, add the upper one 53, and againft the Total fet Number 4, fignifying 4 times ; to this laft Number add the firft, as before, until you have fo added the Divifor 9 times 5 which being done, your Divifion will become very eafy, as following : First, Say the 53^ in 79 is once, I write 1 in the Quotient, and 53 under 79, and fubtrating it from thence, refts 26. Secondly, To 26 I bring down the Figure 2, making the 26, 262 ♦ then I fay how often 53 in 262, and looking in the Table of Divifors for the neareft leaft Number to 252, I find 212, againft which ftands 4 times, then I fet 4 in the Quotient, and 212 under 262, and Subtra&ion being made, refts 50. Thirdly, Bring down the 4 to the 50, making it 504 • then fiiy how often 53 in 504, and looking in the Table of Divifors for the neareft leaft Number to 504, I find 477, againft which ftands 9 times; then I fet 9 in the Quotient, and 477 under 504, and Subtraction being made, refts 27. The Principles of Arithmetic k. 71 Fourthly, Bring down the next Figure 2 of the Dividend, to the 27 making it 272 ; then fay how often 53 in 272, looking in the Table for the next lead: Number thereto, I find 265, againft which (hands 5 times- then I let 5 ,n the Quotient, and 265 under 272, and SubtraSion beins made, retts 7. Fifthly, To the 7 remaining, bring down the 3 of the Dividend, making the 7, 73 ; then faying the 53’s in 73, is 1 time, fet down 1 in the Quotient, and 53 under 73, and Subtradlion being made, refts 20 for a Remainer, and the Quotient will be 14951. By this way of finding your Divifor, you will attain to a good Knowledge ol Divifion, and be enabled to work eafily without fuch a Table. I will give you fome other Examples for Pradhice. Example I. Divide 998877 by 543. Table. 543 ) 9 9 8877 ( 54? 1 543 ‘ 1086 2 455 8 1629 3 4344 2172 4 2 7 J 5 5 2I47 5 2 5 8 6 1629 3 8° 1 7 5187 4544 8 4887 4S87 9 300 rem. Example II. Divide 5432729 by 4532 Table. 45 3 1 1 9064 2 1 3 S 96 3 18128 4 22660 5 27192 6 4 7 36256 8 40788 9 453i)54ii7-9(”p8 45 3 2 ' - ' 9OO7 4532 4475 * 407S8 39649 36256 3393 rem. I. Sir, I thank you ■ I fee that by making Tables of Divifors, Divifon is very eajy ; but pray tell me how I mlift know to fold the Ealue of the Remainer, which in the lajl Example was 3395. J M. The Remainer, when any, after Divifion is ended, is the Numera¬ tor of a Fradhion, and the Divifor is a Denominator thereto, and are gene¬ rally annex’d to the Quotient; as in the firft Example the Quotient is 1839, and 3I00 remaining, which muft be thus written, 1839 f“ And in the lalh Example, to the Quotient 1198, fhould be annex’d the Remains 3393, as thus, 1198 if ji : In both of which Examples, the Remainer is fet over the Divifor, leparated by a Line, as you fee here, and the Divifor is always to be placed the undermoft, as being the greateft of the two. P. Pray what is to be underfood by the PPPrd Fruition? and why is the Remainer called the Numerator, and the Divifor the Denominator of a Frac¬ tion ? J El - -h Fraction is a Broken Number, and always lels than Unity, as f, or i, or reprefents Three quarters, One half, or One quarter of any thing, 01 Unity : And if the Divilor to a Sum of Divifion be confidered as an Integer or Unity, then the Remains, after Divifion is made, being lels, is therefore called a Fraction, being but a Part thereof: And as the Divifor expreiles- The Trineiples of Arithmetick. expreiles the Number of Parts jnto which tis divided, it is therefore called the Denominator j and fo in like manner, as the Remnmer expreiles, or numerates how many of thofe Parts are remaining, it s therefore called the Numerator to the FraAion ; and FraSions of this kind, are called Vulgar Fractions. More of which I thall inftrudl: you hereafter in its proper Place. Having thus fhewn you how to exprefs and write down your Remainer, and to annex it to your Quotient as a fraction, I {hull in the next Place lhew you how to find the Value thereof. Suppose that the left Example was 5431719!. to be equally divided between 4531 Men, where you fee that the Quotient is 1198 1 . to each Man, and 3393 remaining, whofe Value is found by the following RULE. First, Multiply the Remainer by the Number of Parts into which an Integer of the Dividend is divided ; as here an Integer of the Dividend, which is one Pound, is divided into 10 s. therefore multiply the Remainer 3393 by 10, and dividing the Product by the fame Divifor 4531, the Quo¬ tient is Shillings. Secondly, Multiply the Remainer of this laft Divifion, by the Num¬ ber of Parts into which a Shilling is divided, viz. 12 Pence, and dividing the Product by the aforefiid Divifor, the Quotient will be Pence. Lastly, Multiply the Remains, if any, by 4, the Farthings in a Penny, and dividing the Produft by the Divifor aforefaid, the Quotient will be Farthings. Example. The Remains of the laft Sum was 3393 Multiply by Divide by jo the Shillngs in a Pound. 4531)67860(14 Shillings. 453 -' 11540 18128 Multiply by Divide by 4412 remains. 12 the Pence in a Shilling. 45 3 944 (i 1 Pence. Multiply by Divide by 3092 remains. 4 4532)12368(2, Farthings. 9064 3304 So The Principles of Arithmetick. 73 S o the true Quotient, or each Man’s Part, is 1198 1 . 14 s. 11 d. i : And here note, That the fame Table of Divifors ferves for dividing the feveral Produ&s hereof, as you made for the Divifion of the given Number. P. Sir ; I fee your Method very plainly • hut how /hall I know when my JVork is right or wrong , for I don't remember that you have yet taught me how to prove Divifwn. M. ’T 1 s true, I have not yet taught you how to prove your Work, which I will now do, as follows. RULE. Multiply your Quotient by your Divifor , arid the Produtf will be equal to the Dividend . Example. Table of Divijori, 721 1 1442 2 2163 3 2884 4 3 6 °5 5 43 i6 6 5047 7 5768 8 6489 9 Divide 55431, ty 7 2I < Multiply 711 the Divifor 711)55431(76 By 76 the Quotient 5047 • 4961 43 2 6 636 re. C - - - To which add E - - - 4316 5047 54796 Produdl. 636 the Remains. 55431 Total, equal to the Dividend, which is the Proof required. T o prove this Divifion, you fee here, that 711, the Divifor, is multi¬ plied by 76, the Quotient, and the Produdt is 54796, as at C. To this add 636, the Remainer, and the Total will be 55431, as at E, equal to the Dividend given. P. Sir, Tour Manner of proving Divifion is very demonftrable ; for fnee that the Quotient is no more than the Number of times which the Divifor is contain'd in the Dividend ; therefore if the Divijor be multiplied by the Quo¬ tient, it mufi produce a Number equal to the Dividend at all Times when the Divifion is ended and nothing remaining : But when there is a Remainer, then that being added to their Product, makes up the Total equal to the Dividend ■ for, as you have taught ms to know, the INhole is equal to all its Parts taken together in one Sum. M. I AM pleas’d to obferve that you fo juftly lee into the Reafons of your Operations, and therefore I fhall, in the next Place, proceed further to ihew you fome Rules for contraQing your Works in many Cafes, which I call Contractions in DIVISION. First, When your Divifor is an Unit, with any Cyphers annex’d to the Riuht-hand, cut off from your Dividend the fame Number of Figures in their refpe&ive Places, the Remainer is the Quotient, and the Figures cut off, are the Numerators of a Decimal Frafition : So if I was to divide 1719 by 10, I cut off the laft Figure 9, as in the io ) i 72|9( Margin, becaufe that there is one Cypher in the Divifor ; then 74 The ‘Principles of A RITHMETICK, then will the other Figures 171, be the Quotient, and the 9 (truck off is a Remainer rl. t|oo>7i|-5( In Again, Todivide 2.7325 by 100, I ftrike off 2 Fi¬ gures towards the Right-hand, as in the Margin, becaufe in the Divifor 100, there are Cyphers • then will the remain¬ ing Figures 273 be the Quotient, and 25 remaining, which is t like manner 9762543, divided by 1000, the Quotient is 9762 riH. Secondly, When your Divifor and Dividend conlift of Cyphers to the Right-hand, cut or point off from both, an equal Number, and then proceed with the remaining Figures, as by the Rules before given. As for Example. Divide 47325000, by 12000. Here T cut off three Cyphers in the Divifor, and as many in the Dividend, and then 12 is be¬ come the Divifor, and 47325 the Dividend. Thirdly, When your Divifor has Cyphers annex’d, they may be omitted, and fo many of the laft Figures in your Dividend cut off, and then proceed as before. As for 12(000)473 25)000(3 943 36 * * * Divide 6327495 by 13000. Example. But after the Divifion is made, the Cyphers are to be reftored to the Divifor, and the Figures cut from the Dividend added to the Remainer for a Numerator. i 3|°oo)6327|495( 4 86 51- ■ • I I 2 I04 So here the Figures cut off, 495, muff be added to the Remainer 9 for a Numerator, which 9 will be 504, and the Denominator is the Divifor 13000; wherefore the Fraftion is ,Co*, and the Quotient 486 : These are rhe moft material Contraftions in Divifion, which beino- well underftood, with the preceding Rules, you will eafily divide any Number or Integers required. J fo/Practice th ‘ S Leiflure with giv ' in S >' ou the foIlo «’ ; ng Queftions Q. I. \ The Principles of Arithmetic! 75 t>-I. In 719543 Inches in Length, how many 11)719543(60795 Feet "' 7 ^“ Rule. As 11 Inches make 1 Foot, therefore °95 divide the given Number 12. 84 Answer. 60795 Feet 5 Inches. Q: I* 9543-7 Feet, how many Tards ? Rule. Becaufe that 3 Feet make 1 Yard, therefore divide the given Number by 3. 3 ) 9543 2 7 ( 3 1 S109 Yards. 114 108 6 3 60 3 Inches. Note, That to divide by 3, you need only write down the Quotient as you run through the Dividend ; faying, the 3’s in 9, 3 times, fet down 3 in e Q uotlerlt ; tllen the 3 s in 5 once, let down 1 in the Quotient, and carry 2 to the next Figure 4 ; and fay, the 3’s in 14 is 8 tinms, fet down 8 in the Quotient; then the 3’s in 3 is once, fet down the 1 in the Quo- tient then the 3 s in 2 not once, therefore place a Cypher in the Quotient. tuny, the 3 s in 27 is 9 times, fet down 9 in the Quotient, and then the Quotient will be 318109 Yards, the Anfwer required Q. III. 7299367 Tards , how many Fathoms ? Rule. Becaufe that 2 Yards qjake 1 Fathom, therefore divide by 2, 1)7299367(3649683 i Note, That to divide by 2, you need only take half the Dividend, and fe td" h f 7iS 3 . fet 3 in the Quotient/then Aen ’ f* thC ^ U f ,ent i then h:llf 9 ^ 4 , fet 4 i^the Quotient: then half 19 ,s.9, fet 9 in the Quotient; then half 13 is 6, fet 6 in the Quotient, then half 16 is 8, fet 8 in the Quotient ■ then half 7 is 3 j • So the Quotient is 3649683 i. ' 6 Q IV. In 7253476 Feet, how many Rods or Poles ? Rule. Becaufe that 16 Feet and i make 1 Pole or Rod, therefore divide the given Num¬ ber by 16 t. Note, When you have a Fraction at the End of your Di- vilor, as in this Example, you mull multiply both the Integers of the Divifor and Dividend, by the Denominator of the Frac¬ tion, and then proceed as be¬ fore. So in this Example I multiply 16, the Divifor, by 2, the Denominator of the Frac- Multiply by j6 ‘) 7 2 5347 2 ( 33 ) 1 4506944(439604 Rods. 2 3 2 . 130 99 316 297 1 99 198 I 44 111 1 2 remains, equal to 6 Feet. non 76 The ‘Principles ef Arithmkt i c k. tion, which makes 32, and the Numerator 1 of the Ftaaion, added to makes 33 which is my new Divifor; then multiplying the Dividend 71 5347^ by a, the Produft is 14506944 , 'y hich being divided by 33, the and 11 hut 6 Feet. And fo in like Q. V. In 7294327 Tards, how many Rods or Poles ? Rule. Becaufe that 5 Yards and i make 1 Rod or Pole, therefore divide the given Num¬ ber by 5 as directed in the laft Queftion. 5 *) 7 1 943 1 7 ( Multiply by - - - 1 11)14588654(1326241 ix. 35 11 28 Q. VI. In 5729257 Links, how many Chains ? Rule. Becaufe that in one Chain there is i oo Links, there¬ fore divide the given Number by 100. • i|°°)57292|57( Here you need only cut off the two laft Figures in the Di¬ vidend, the 5 other remaining Figures is the Quotient, or An- iiver, and the Figures cut oft, are Remainers. 68 66 Is 22 45 44 _ 14 11 3 remains, equal to ii The Reafon of this is, becaufe that In the Divifor there are 2 Cyphers, and that the 1 doth not divide any more than it multiplies, wherefore the aforelilid Part of the Dividend is equal to the Quotient. Q. VII. In 72 54379 Pounds Avoirdupoize, how ninny Hundred weight at 112 lb. to the Hundred, which is called the Great Hundred ? Rule. Becaufe that in one Hundred there is 112 Pounds, therefore divide the given Number by 112. 112 1 224 2 33 6 3 448 4 560 5 672 6 784 7 896 8 1008 9 Note, For your ready performing the Operation, firft make a Table of Di- vifors, as you fee here prefix’d. 112)7254379(64771 Hundreds. 672— • 534 448 863 7 8 4 797 784 139 I 12 27 Pounds remains. q. VIII 77 The Trinciples of Arithmetic*:. 0 ; VIII. In 97154978 Hundreds 10)97154978(4861748 Tons. Avotrdupoize, how many Tons ? 80 * • * • • • Rule. Becaufe that 20 Hundred weight make one Ton weight, there¬ fore divide the given Number by 20. 0 / IX. In 67432543 Hundreds weight oj Lead , how many Fodder ? Rule. Becaufe that 19 Hundred and t make one Fodder of Lead, therefore divide the given Number by 19 {. Note, Here you muft firft mul¬ tiply your Dividend by 2 the Deno¬ minator of the Fradfion i ; alfo the Divifor the fame, adding to it the Numerator 1 • and then your Divi¬ for and Dividend conlifts of fe many i Hundreds, which being divided by the foregoing Rules, the Quotient will be Fodders, and the Remains after Divifion is ended, are i Hun¬ dreds, as exprefs’d in the Margin. Q. X. In 7954379 Joltd Feet of Timber , how many Loads ? Rule. Becaufe that 50 Feet make 1 Load, therefore divide the given Number by 50. 172 t6o 125 120 54 4 o *49 140 97 80 178 160 18 Hundred remains. 197)67451545 59)134865086(5458079 Fodders. ffr . 178 156 12.6 195 3>5 3™ 308 356 351 5 remains, equal to 1 Hundred and i. 50)7954379(159087 Loads, 59- .... 295 250 454 450 4 37 400 379 350 29 Feet remains. Q. XL T ?3 The Principles of Arithmetics. Q. XI. In 9543217 Jolid Feet oj 4°(9543 2I 7( 2 :>S58o * onv - Timber, how many Tons ? 80. Rule. Becaufe that 40 folid Feet make 1 Tun of Timber, therefore divide the given Number by 4c. 232. 200 154 120 343 2ZO 3 2 ° 1 7 Feet remains. Q. XII. In 93274359 Bricks , how 500(93274359(186548 Loads. many Loads? 500. Rule. Becaufe that 500 Bricks make one Load, therefore divide the given Number by 500. -743 2500 -435 2000 43 2 7 4000 3 2 74 3000 4359 4000 359 Bricks remains. Q. XIII. In 793274 Baggs of -5)793-7 Lime , how many Hundreds ? 75 •• • Rule. Becaufe that 25 Baggs make 43 one Hundred of Lime, therefore divide 2 5 by 25. 1 8i 175 77 7£__ M Kags remains. Q. XIV. The Principles of Arithmtick. 79 Q. XIV. In 743175 Bujhels of Sand, horn many Loads ? Rule. Becaufe that 18 Buthels of Sand make one Load, therefore divide the given Number by 18. 18)743275(41293 Lo: «k. Ill'" 23 18 5 1 3 6 167 162 55 54 1 Bufhels remains. Q.XV.///9S7654311 Square Inches, how many Square Feet ? Rule. Becaufe that 144 Square Inches make one Square Foot, there¬ fore divide the given Number by 144- Dwifort 144 1 2.88 2 43 - 3 57 1 4 72'cr 5 864 6 1008 7 1152 8 1296 9 144)987654321(6858710 Square Feet. 864. 1236 1152 845 720 1254 1152 1023 1008 152 144 81 Square Inches remains. Q. XVI. In 5432176 Square Feet , htm many Square Lards ? 9)5432176(603575 SquareYards. 54-•• •• Rule. Becaufe that 9 Square Feet make one Square Yard, therefore di- __ vide the given Number by 9. 51 45 67 6 — 46 45 1 Square Foot remains. Q. XVII 8o The 1 Principles of Arithmetics, Q.XVII. In 97154517 Square Feet of Brick-work, how viany Square Rods ? Rule. Becaule that 2 7 2 lquarc Feet and quar¬ ter make one fquare Rod, therefore divide the given Number by 272 i. Note, That as the Divifor confifts of a Frac- Denominator is 4, you muft therefore multiply the Dividend and Divifor by 4, to which idd the Numera¬ tor 1, and then they will be tranlpoled into Quar¬ ters. This done, divide the one by the other, and the Quotient will be the Anfvver required. _ 1 Poles. Feet. 1089)389017308(357224 118 3-- 6 7. 6231 5445 7867 7623 2 445 217S 2650 2178 4728 4356 Divifor s 1089 1 2178 2 3167 4356 5445 5 6534 6 7623 7 8711 8 9791 9 472 remains, which muft be di¬ vided by 4, they being but Quarters of Feet, caufed by the multiplying of the Divi- , dend and Div i lur by 4, and are equal to 118 Square Feet. Q. XVIII./» 8832574 Square Rods of Lam], bow many Acres. ’ Rule. Becaufe that 160 (quare Rod make 1 Acre, therefore divide the given Number by 160. 160 1 310 2 480 3 640 4 800 5 960 6 1120 7 1280 8 I 44° 9 160 ) 8832574 ( 55203 Acres. 800 800 3 2 5 320 574 480 94 Rods rem. r W, XIX /? F" 11,ae / La> ‘ d is 15 Rod in Breadth, bow many Rod in Ungth muft I go to meajure out an cxaB Acre. 7 a^iviue 100, the Number of R01 an Acre by the given Breadth 15, and the tient will be the Length that muft be taken to 1 an exact Acre. For as many 15 Rods as are 1 had ill 160 Rods, f 0 many times 1 muft be t in Length. I5)i6o( 10 Rods y 15 10 Anfwer. io Rods f? equal to y. Q: XX. fo T* The Principles of Arithmetick. CK XXVII. In 55317145 Feet of Inch and half thick Plank> how many Loads ? Rule. Becaufe that 400 fquare Feet of Inch and half Plank make 1 Load, therefore divide the given Number by 400. CL XXVIII. In 77243257 Feet of Two Inch thick Plank , how many 'Loads ? Rule. Becaule that 300 Feet make one Load of Plank two Inch thick, therefore divide the given Number by 300. CL XXIX. In 543 9762 Feet of three Inch Plank, how many Loads ? Rule. Becaule that 200 Feet of three Inch Plank make one Load, therefore divide the given Number by 200. 4 0 °)553 2 7M5(id83i8 Loads, 400••••• * 53 * 1200 33 2 7 3200 1272 1200 724 400 3 2 45 3200 45 Feet remains* 300)77243257(257477 Loads, 600. 1724 1500 2243 2100 143 2 1200 2 3 2 5 2100 22 57 2100 157 Feet remains; 200)5439762(27198 Loads, 400••* • *439 1400 397 200 1976 1800 1762 1600 162 Feet, remains* Q. XXX. The Principles of Arithmetick. 89 Q. XXXIV. In 7325479 Weeks, 52)7325479(14.0874X6315. how many Tears ? . Rule. Becaufe that 52 Weeks 211 make 1 Year, therefore divide the given Number by S2. 1 > 4 5 4 416 387 364 239 ao8 51 Weeks remain. Q_ XXXV. How many Paving Bricks, each 9 Inches tong, and 4 Inches i wide i -will pave a Cellar that contains 120 fquarc Feet ? Rule. Firft find the Quantity of fquare Inches contain’d in one Brick; which is done by multiplying the Length, 9 Inches, by the Breadth, 4 Inches and the Product is 40 Inches and ; This you are to referve for your Divifor. Secondly, Find the Quantity of fquare Inches in 120 fquare Feet; which is done by multiplying 120 by 144, and the ProduQ is 17280,; which is your Dividend. Thirdly, Divide 17280 by 40 [, and the Quotient will be 426, the Number of the Bricks required, and 27 fquare Inches remaining. 4 I 3 6 4 1 40 ; the fquare Inches of I Brick. '44 120 2880 ] 44 17280 the fquare Inches in 120 fquare Feet. 40 ;)I728o( 81)34560(426 Bricks. 216 162 54 which is equal to 27 Inches. X Q. XXXVI, 86 The Principles of Arithmetics XXXVI. How many Paving Tiles , each i o Inches fquare , will pave a Kitchen that contains ioooo fquare Feet? Rule. Firft find the Quantity of fquare Inches contained in one Tile; which is done by multiplying io Inches, the Length, by io Inches, the Breadth, and the Product is ioo Inches. This you are toreferve for a Divifor. Secondly, Find the Quantity of fquare Inches contained in ioooo fquare Feet; which is done as before in the laft Queftion, by multiplying ioooo by 14.4, and the Product will be i 440000, which is your Dividend. Thirdly, Divide 1440000 by 100, 10 and the Quotient is 14400, which is 1 o the Number of Tyles required. ito T [ le fquare Inches in aTyle. ioo)i44oo]oo(i440o Tyles. ioooo *44 1440000 The fquare Inches in the ioooo Feet. Thus have I exemplified the principal Rules of Vulgar Arithmetick; which being well underftood, you will be enabled to engage with the Solutions of any Queftion comprifed by them. It is ufual for all Mafters of Arithmetick to impofe on the young Stu¬ dent a fixth Rule, which they call Reduction ; when in Fa£t it is no more than the Application of Multiplication and Divifton, according to the Na¬ ture of the Propolition to be folved. That is, if we are to change Money, Weight, Meafures, fefc. out of one Denomination into another, we have only this to confider ; that is to fay, if the propofed Quantity be to be changed into another of a lefs Denomina¬ tion, fuch as Shillings into Pence, or Feet into Inches, then we muft con¬ fider how many Pence are contain’d in a Shilling, or Inches in a Foot, and mul¬ tiply the Number propofed by the lame : That is, if I am to change 10 Shillings into Pence, then I muft multiply 10 by ia, the Number of Pence contained in one Shilling, and the Product is the Anlwer : And if I am to change Yards into Feet, then I multiply the Yards by 3, the Number of Feet in a Yard; and fo in like manner all other Quantities, as has been al¬ ready very largely handled in Multiplication. On the contrary, if the propofed Quantity be to be changed into another of a greater Denomination, fuch as Pence into Shillings, Feet into Yards, iffc. then we muft confider how many of the Number propofed will make one of the Denomination intended, and then we muft divide the Number propofed by the fame : That is, if I am to change Pence into Shillings, I muft divide by 1 2, the Number of Pence in a Shilling: But if I am to change Pence into Pounds, then I muft divide by 240. the Number of Pence in a Pound ; and fo in like manner any other Quantity, as has been already very largely ex¬ emplified in this Ledture of Divifion. I shall in the next Place fhew you the Application of the Rules hitherto taught, to Practice in the Rule of Proportion, vulgarly called the Golden Rule, or Rule of Three. LECTURE The Principles of Arimhmetick. 87 LECTURE VI. Of the Rule of Proportion, or Golden Rule. p. Pra r why is this Rule called the Golden Rule ? M. For the excellent Ufe thereof; which will be demonftrated in this Le&ure. P. Avd is it therefore alfo called the Rule of Three ? M No ' It is called the Rule of Three, becaufe three is always three Num¬ bers given to find a fourth, which muft bear fuch Proportion to the thud as the fecond doth to the firft. That is, if you divide the iecond by the hrit, and the fourth by the third, and the two Quotients are equal, then thole four Numbers are faid to be proportional. As for Example : Let the Num¬ bers 8 : 1 6; 3a : 64 be given. Firft, Divide 16 by 8, and the Quotient is 2. 8)16(2 Secondly, Divide 64 by go, and the Quotient is 2 alfo. Now, I fay, thefe Numbers are Proportionals. For as 8 is to 16, fo is 01 to 64. And when four Numbers are thus proportional to the Product of the Means (that is to fay, the fecond and the third) is equal to the Product of the Extreams, which are the firft and laft : For g 2 multiply’d by 1 6 (which, as I faid before, are the two Means) produce 512 ; and fo in like manner 64 multiply’d by 8 (which are the two Extreams) produce 512 likewiie. Therefore, if the Produfl of the two Means (that is to fay, 512) be divided by the firft Number 8, the Quotient will be equal to the fourth or laft Num¬ ber 64. It is from this that the Knowledge of a fourth Number arifeth, which flrall have fuch Proportion to any one Number given, as the two Numbers given have to one another. This Rule is performed either iimple, by one Operation; or compound, by two Operations; and thofe both direfl and indirect. The Golden Rule DireB, is, if the fecond Number be greater than the firft, the fourth Number fhall likewife be greater than the third ; and fo in like manner, if the fecond Num¬ ber be lefs than the firft, the fourth Number fhall likewife be letter. There¬ fore obferve, that when in any Queftion, more require more, or lefs require lefs, it is to be folved by the Golden Rule Direct: As if 10 loot of Oak ihould colt 20J. then 15 Foot muft needs coft more than 70s. that is, it will coft gor. which is, more Things require more Money, i*. The next Thing which you are to obferve, ' is the Manner of placing the Terms or Numbers in their true Pofitions, which perform as following: After th« 83 The Principles o^Arithme tick. the firft Term, place two Points; after the fecond Term, four Points • and after the third Term, two Points.: As thus, i : 6 : : 4 ’ l2 which muft be thus read. As a is to 6, fo is 4 to 12. And here note,' That the firli and third Terms are always of the fame Denomination or Name ; as alio are the fecond and fourth ; that is, If 6 Men perform a Piece of Work in 10 Days, then 1 a Men could have done the fame in 5 Days; which is thus !tated : M,*. Days. M,j. Days. 6 : 10 : : 12: 5 Wherein you fee, that the firft and third Terms do both denominate Men, ancl tne lecona and fourth denominate Days. Th e greateft Difficulty in this Rule, is the Stating of your Queftions tru- ly , and that you may be Pure thereof, obferve as followings : 1. That whereas there are always three Numbers given to find out a Four h, you muft diftmgutlh thofe Numbers, the one from the other as follow mg I That is the firft two, I would diftinguilh and call by the Name of Stated Numbers; and the third Number I would call the Demand- mg IN umber. P. Pray 1 thy do you thus diftwguijh them ? rhe followin g Feafons. Firft, As the Proportion of the Fourth to firft Te‘ rd 18 Z er 1 ° be a f the SeCOnd is CO theFirft ; therefore in the two firit Ierms, or Numbers, the Proportion of the fourth Term is ftated ; and fhe t ft K r ° n ,’ Z ‘ T fe Terms ’ the Stated N ™bers. Secondly, as the the third Number demands the Fourth, in Proportion to itfelf, as the Se¬ th ? a cl f’ thcreforeIc311 >t the Demanding Number; and as it theffiirdPhce Cthree ^ Demand ’ muft therefore be always placed in Now, the only Difficulty remaining, is to know which of the two Stated folWs 111 firftP,aCe ' which you may readily deteremine as Consimi of what Denomination your Demanding Number is, and make that of your Stated Numbers, which is of the fame Denomination, the firft the Solution ^ ^ " tMr trUe teady for To folve all Queflions in the Golden Rule Dired ; This is the Rule. bv ffie L ffift L T the feC ,° nd and , third Term, together, and divide the Produfl by firft Term ; the Quotient is the fourth Term, or Number required which is of the fame Denomination with the fecond Number. tSS If an Unite be in the firft Place, the fourth Term is obtain’d by the Mul- no P !'S 0 Alffi, Dd and thrd TCrmS J « Unity, dot! notV^hJ* r tIle reC r°r nd “a Mrd PIaCes ’ whereh y Multiplication can- - ade, becaufe 1, or Unity, doth not multiply, then the fourth Term required The Principles of Arithmetick. 89 required is obtain’d by only dividing the fecond or third Term by the firit, EXAMPLE I. If 25 Men are paid 251 /. for fix Months Work, how much will 72 Men be paid for the fame Time, and at the fame Rate of Payment ? In this Example 72 is the demand¬ ing Number, which I place for the third Term ; and it being Men , there¬ fore I make 25 my firlt Term, which is Men likewife, and then is my Queftion truly fated ; for when the firft and third Terms are known, the fecond is known alfo, being given; and the fourth likewife, when difi covered by the Rule aforefaid. Men. 1. Men. 1. 25 i 252 : : 72 : 7*5 5 JL 5°4 25) 18144(725 J 75- • 64 5° 144 El '9 EXAMPLE II. If 1 Leaf of Gold will cover 16 fquare Inches, how much will 100 Leaves cover ? Here the Solution is made by Mul- Leave. tiplication only; becaufe the firft i Term is anUnite. Inches. Leaves. Inches. 16 :: too ; 1 6oo 16 1600 EXAMPLE III. If i i Square of Flooring take up 120 Deals, how many are required for I Square ? Here the Solution is made by Di- Square. Deals. Square. Deals. vifiononly; bccaufe the third Term n : 120 : : i ; io " is an Unite. , ii) iao(io ii 11 io y E X A M P L E IV. ■'SC. V 90 The 'Principles of Arithmetick. EXAMPLE IV. If 19 Yards of Wainfcoting coft 57 s. what will 37 Yards coll? Turds. Shill. Yards. Shill. >9 ■' 57 : : 17 ■■ hi 17 399 1 7 ‘ 19) 2109 (111 I 9" 20 i_ 9 _ J 9 example v. If 7 Rod of Brick-Work require 31500 Bricks, how many will e 2 R 0 d require ? J Hair. Bricks. Boab. BnVir. 7 : 31500 : : 52 : 234000 _ 5 ^ 63000 1575 Bricks. 7) 1638000(234000 theAnfwer. !_+■:: 2 3 2 I "28 28 o EXAMPLE VI. Limec 5 om“ ed ^ ^ 42 *' 6 * what will 25 Hundred of Before this Queftion can be „ , lolved, I muft reduce the 'll. is. 1 . 6i into Pence ; whereby the Num- — ' bers will be of one Denomination, as 4° r. in the Margin ; where 2 /. 2 j. 6 d. Add 2 is reduced to 510 A — Sum 42 s. Multiply by lid. 504 6 d. 5 io or ( ourth Number, is 2550 Pence; which being di- v,Jed by 12, as at A, is equal to 212 Shillings and 6 Pence remaining. 2.1 Divide 212 Shillings by 20, as at B, and the Quotient is 10 Pounds theASw«r^& Ioi ^ EXAMPLE VII. If -7 Feet of Marble coft 4 1 . 141-. 6 d. what will 75 Feet and ■ coft? fc " d N — *'■ ■** “ - * -.-vi 27 l s. d. 4 75 4 '4 6 ~ 4 20 Ioa Quarters. — — 300 80 Add the Numerator 1 14 94 *• 12 1128 6 1134 A- Now 301 Quarters. 92 The Principles of Arithmetics Now Jlate and work the Queflion as following : Quarters. ‘Pence. Quarters. Pence. 1 . s. d. 108 : 1134 : : 301 : 3160. £, equal to 13 : 3 : 4. 301 A 11 34 12) 3160(263 s. 3402 24" B i° 8 ) 34 i 334 ( 3 i«o 76 20)263(13 l. 324-■• ?! 20* 1 73 4 ° 6 1 I08 3 6 60 6 53 4 d. 3 r. 64.8 54 1. ) Here the Anfwer is 3 x 60 Pence; which divide at 12, as at A, and the Quotient is 263 Shillings and 4 Pence remaining. 2. ) 262 Shillings divided by 20, as at B, the Quotient is 13 Pounds 3 Shillings remaining; which, with the 4 Pence remaining at A, make 13/. 3 s. 4 d. the Anfwer required, EXAMPLE VIII. If 50/. will purchafe 25 Loads of Timber, how much Timber will 750 /. purchafe ? Pounds. Loads. Pounds: Loads. 50 : 25 •' •' 75° : 375 the Anfwer. -5 50) 18750(375 1 5 ° 375 • 35 ° 250 25O Thus far with Refpefl to the Golden Rule Dirett; which you fee is per¬ formed moil: eafily by the Help of Multiplication and Divifion only. I fhall in the next Place give you fome Examples in The Golden Rule Indirect. This Rule is called the Indirect Rule of Proportion, w ith Regard to its in¬ verting the Practice of the Direct Rule : For as in the Direct Rule the fecond and third Numbers were multiplied together, and their Product divided by the The Principles of Arimhmetick. 93 the fir ft Number; fo on the contrary, m this Rl jJ e t!,e arK J e ° n . Numbers are multiplied together, and their Produft divided by the third. And further, as in the Direft Rule, it was noted, that more required more, or lefs required lefs; fo on the contrary, in this Rule, more requires le s, or lefs requires more. That is, if 20 Men in 3 Days eat 20 Quartern-Loaves of Bread 9 40 Men mull needs eat the 20 Loaves in leffer Tune, that is, m hal the Time, becaufe double in Number. So here more Men require lels 1 ime to do the fame Thing. &ND contrary, if 10 Men have 20 Loaves to eat, in manner as aforefaid, they will require more Time, 6 Days, becaufe they are lefs, but have the firll Number: So here lefs Men requires more Time to do the lame Thing. EXAMPLE I. If 15 Men can perform a Piece of Work in 12 Days, how many Days will 12 Men be in performing the fame Work ? Men. Days. Men. Days. 15 : 12 : : 12 : 15 12) 180(15 60 EXAMPLE II. IF 11 Men build a Wall in 21 Days, how long mull 7 Men be to do the fame Work ? Men. Days. Men. Days. 11 : 21 : : 7 : 35 11 7 )’ 8 ! (33 11 * 11 11 z example 94 _ The Principles of Arithmetick. EXAMPLE III. thjfame Work F ? erform “ W ° rk17 Da 7 s > how Ion g will 25 Men be doing MfK. Days. Men. Days. 33 : l ? ■ ■■ 25 : 32 ” li 25)561 (22 50. 61 _ 5 ° 11 EXAMPLE IV. If 72 Men begin and finifh a Houfe in doing fach another Work ? 45 Days, how long will 52 Men be Men. Days. Men. Days. 7 - ■' 45 •' •' 3 ? •' 202 ; 64- ■ 080 16 remain, or ^equal to equal tot, equal to;, equal to In all thefe Examples, lefs Men have required more Time • following Examples, more Men will require lefs Time. and in the EXAMPLE V. If 21 Labourers can empty out 517 cubical Yards of now loon will 30 Labourers do the fame Work ? Earth in 9 Days, Men. Days. Men. Days. 21 ■ 9 •• : jo : 6 _9 8°) 189(6 180 9 remains, equal to s s „, or or EXAMPLE The Principles of A rithmetick. 95 example VI. 1F T Carpenters can frame and raife five Roofs in 11 Days, how long will 40 Carpenters be doing the fame. ° Men. Days. Men. Days. 32 : 11 : : 40 : 8 *. 11 40)353 (8 320 3 2 remains equal to or L 5 , or or ♦ These Examples being well underftood, I need mention no more : And therefore I Ihall juft give you two or three Examples in the Golden Rule Com¬ pound, and fo conclude this Leflure. The Golden Rule Compound. The Golden Rule Compound confifts of five Numbers given, to find out a fixth in Proportion to them; wherein you mull obferve, That the three firft Numbers may contain a Suppofition, and the two laft a Demand. Now, that you may place them right, obferve, That the Firft and Fourth ^ Second and Fifth>Terms, be of the fame Denomination. Third and Sixth J SurrosE that the following was a Queftion propofed, vitc If 16 Men for iS Weeks Servitude, are paid 136 /. what mult 18 Men for 25 Weeks Servi¬ tude be paid ? Place the Terms as following : /. Men. Weeks. >36 : : 18 : 25, and then work by the following Men. Weeks. 16 : 18 RULE. Multitlx the firft and fecondNumbers into themfelves; alfo the fourth and fifth Numbers, and note their ProduQs feverally. This done, you may folve the Queftion by the Angle Golden Rule, making the Produft of the firft two Terms the firft, the third Term the fecond, and the Produft of the fourth and fifth Terms the third Term: Then will the fourth Number pro¬ portional to them, be alfo proportional to the five given Numbers, and the An- lwer required. As The Principles of Arithmetick. 96 As for Example. The firft Number 16 The fecond Number 18 Produfl: 1188; which isthe firft Term. The third Term 136 is the fecond Term. The fourth Number 18 The fifth Number 25 Produft 2250; which is the third Ter m - These Numbers thus atttained, ftate your Queftion, as follows : Table. 1188 : : 136:: 2250 : 257 1188 I 136 2 MS 00 35 6 + 8 675° 47 5 a 4 2250 59 ts 7,28 8516 5 6 1188)306000 (257 l 2376.. 95°4 10672 8 9 684O 5940 09000 OS^I^ 684 Here the fourth Number produced, is 257 : 775* ; which is the Anfwer required; and is to be placed in your original Queftion, as follows : Men. Weeks. Pounds. Men: Weeks. Pounds. If 16 : iS : ] 36 : : 18 : 25 : 257 P. 1 thank you , Sir, for the Trouble you have been at. 1 fee that the Golden Rule, in all its Varieties, depend principally upon the true Stating of the Queftion ; in rvhich I ft j all be very careful. But, Pray Sir, how {ball I know when I have done right or wrong ? That is, how mu ft I prove my Work ? M. In the very Beginning of this Rule, I told you; and proved to you alfo, that the four Numbers are Proportionals. And therefore, To prove the Golden Rule , The Square of the Means (that is, the Produ£lof the fecond and third Num¬ bers multiplied into each other) is equal to the Square of the Extreams, that is, the Product of the firft and fourth Numbers multiplied into them¬ selves. The Principles of Arithmetick. 97 As for Example. In this laft Queftion, The third Number is The fecond Number is 2250 7 which are 136 ^ the Means. I 35°° « 75 ° 2250 Product 306000 Square of theMeans. The firft Number 1188 The fourth Number 2 57 8316 Product 305316 To whi^h add 68qtheRemainerafcerDivifion. And the Total 306000 is equal to the Square of the Extreams, and to the Square of the Means before found; which is a Proof that your Work is true. But you mull here obferve, that this Manner of Proof holds good for no other kind of Queftions, but fuch as are direfl Proportionals, vi%. when of four Numbers, the firft is to the fecond, as the third is to the fourth : Therefore when Numbers happen indirect, that is to fay, when the third Number is lefs than the firft, and require more; or more, and require lefs; then the Product of your firft and fecond Numbers will be equal to the Product of the third and fourth. EXAMPLE. If 24 Men build a Column in 16 Days, in how many Days will 48 Men do the fame Work ? M. D. M. D. 24 : 16 : : 48 : 8 16 48)384(8 Anfwer. 584 Now, here you multiply the firft Number 24 By the fecond Number 16 Produfl 384 Alfo the third Number 48 By the fourth Number 8 384 Here you fee that the Produfl of the firft by the fecond, is equal to the Produ£t of the third and fourth ; which proves the Operation to he true. A a Having f The Principles of Arithmetick. Having thus inftrufled you in the Principles of Vulgar Arithmetic upon which Foundanon our SuperltruScre is to be railed," I (hall, in the riext Lecture, give you fome QueftionS of Arithmetical Proportions, for your further Exerc.ie : And after them, proceed to Geometry ; and when that you have acquired a competent Knowledge therein, then I will inltruft you i n the Do&me of Vulgar Fractions, Decimal Arithmetick, PxtraElion of the Sat,are and Cube Roots , Geometrical Progreffion, and Logarithmetical Arithmetick • For as Geometry is the Real Eafis of all Arts, fo it is impoffible to well underhand any one, without being firft acquainted with the Principles thereof. And indeed, though you may at firft believe that Geometry is a Digrelliun from Arithmetick, yet you will find in the Practice thereof, Chat the^ one hath a very great Affinity to the other, or, if I may be permitted to lay, they are oth the fame Thing: For, as Arithmetick expreffes Numbers and Quantities by Charaflerifticks or Figures, fo likewife doth Geometry the lame by Lines Figures, and Bodies; and whatever is exprefs’d by Arithme.. tide the very fame are either Geometrical Lines, Figures, or Bodies; as will evidently appear in tne following Sheets. Wherefore ’tis plain that Arithme- ttek ts Geometry or at leaft a Branch or Part thereof, and not a Science or Ait abiolute of ltlelf as many fuppofe it to be. \M LECTURE VII On Arithmetical Proportion. jU A RITHMETICAL Pr0 P 0rtl0n is by f°me called Progreffton, as be- inga continued Progreflion or Series of Numbers, increafing or de- creahng by equal Differences, or the continual Addition or Subtraflion of iome equal Number. So 5 > g > 7 - 9 , is a Rank of Numbers increafing by rinn r . ^ Q „ / _ U / ' ' > ’ I J a i , i ? the continual Addition of I ; and 9, 8 of Numbers decreafing by the continual Subtraflion of 7 ’ b, 5, 4, q, 2, 1, is a Rank ALSO, -> 4 ) > > I0 > '7 Hi 1 1 S, 20, are a Series of Numbers increafing by 2 added to each preceding Number; and 20, 18, 16, 14, J3 IO ’ a u’ are Serles ^creEling by 2, being fubtra&ed from each preceding Number. LIKEWISE 7 he 'Principles of Arithmetics. 99 likewise, S, 15, 22, 29, 36, 45, 50, is 3Series or Rank of Numbers increafing by the continual Addition of 7 ; and 100, 90, So, 70, 60, 50, 40, 10, a°, io, is a Series or Rank of Numbers decreafmg by the continual Sub¬ traction ot 10. f Ear well, Sir-, I fee by thefe Examples, that in e-uery Series of Num¬ bers, each Number is greater or lejfer than the following, according to the Difference affgned them, be it 1, 2, 4, 10, &c. Pray what enfues ? M. Five Things, or rather fo many Confiderations, 1tig,. Firfl, The firft Term, (which is generally the Ieaft.) Secondly, The Iaft Term, (commonly the greateft.) Thirdly , The Number of Terms, or Places. Fourthly, The equal Difference, (called the common Excefs.) Fifthly, The Sum of all the Terms taken together in total Aggregate. P. Prat explain all tbefe Confiderations more fully, M. I will : Suppofe the Series of Numbers following be given, vig, 1,5, y, 1^, 17, 21, 25, 29, 54, 47, 41. Then the Number 1 is the firft Term,’.and” the ieaft alfo;) and the Number 41 is the laid Term, and greateft. Again, the Number of Terms is 11, and the equal Difference is 4 5 becaufe I and 4 make 5, and 4 make 9, and 4 make 14, and 4 make 17, Isle. Laftly, The Sum of all the Terms taken together, is the total Aggregate; which in this Series of Numbers is 241, as in the Margin, where all the Numbers are added together, accord¬ ing to the common Method of Addition : But herein I fhall ftiew you how to find the Total of any fuch Series of Numbers in a more concife Manner; and for the well underftanding thereof, I Ihall, in the firft Place, acquaint you with fome neceffary Theo¬ rems, for the better underftanding of the following Problems. I’. Prat what are Theorems and Problems? 5 9 1 7 21 -5 29 33 37 41 24! M. A Theorem is a Propolition, wherein the Truth is confider’d only, without defending to the Praflice thereof. But when we defend to the Practice, that is, when fomething is propofed to be done or made, then fuch a Propolition is called a Problem. P. Thank you. Sir: Pray proceed to the Theorems you juft mentioned. M. I will, as follows. THEOREM I. ASy Term of an Arithmetical Progreffion is equal to the firft Term added to the Produfl produced, by the Number of Places preceding it multiplied in¬ to to the common Excefs: Or, if the common Excefs be multiplied by a Number that wants one of being equal to the Number of Places in the Progreflion, and to the Produfl thereof be added the lealf Term, the Sum is equal to the greateft Term, Arithmetical Progreflion; then 1 fay, any Number thereof, luppofe 19, is equal to the firft Term, 3 being added to 16, the Produfl of the preceding Number of Places, S multiplied by 2 the common Excefs. P. Pra t achat do you mean by the Number of Places preceding ; and which are they ? M. The Number of Places preceding any Number given, is the Number of Places that are before it. So in the firft three Numbers of the above Se¬ ries, 3, 5, 7. if 5 be a given Number, then the 3 preceeds it, that is, it is before it, and confilfs but of one Place. And if 7 be a given Number, then the 3 and the 5 preceds it, and are two Places before it. And if the 9 was the given Figure, then there would be three Places preceding it, the 3, the 5, the 7; and fo in like Manner to any other Term. Therefore, In the above Series the Term 19 is equal to S, the Number of Places preceding it, multiplied by 2 the common Excefs, added to the firft Term. If three Numbers are in Arithmetical Progreflion, the Double of the Mean, or middle Number, is equal to the Sum of the Extreams, being add¬ ed together. Then I fay, that the Mean or Middle Term 20, being doubled, is equal to 40; and if the firft Term to be added to the laft Term 30, the Sum is equal to 40 alfo : And fo the like of any other three Numbers in Arithme¬ tical Progreftion. Hence it follows. That in any Arithmetical Progreflion, any Term doubled is equal to the Sum of any other two Terms, equally diftant on each Side from it. Then I fay, that any Term thereof, fuppofe (the 7th Term, which is) 43, being doubled, (equal to 86) is equal to any other two Terms equally diftant on each Side from it, being added together ; fo 36 and 50, added toge¬ ther, are equal to 86, the Double of 43. And in like manner 29 and 57, The Principles of Arithmetics. ioi which are at equal Diftances from 45, being added together, are equal to 86 alfo. And 15, which is four Places before 43, being added to 81, which is four Places after 43, makes 86 likewife : And fo of all other Numbers in Arithmetical Progreffion, that are equally diftant on each Side of the Num¬ ber given. THEOREM. III. In any Series of Numbers, that are in Arithmetical Progreffion, the Sum of any two Terms, taken in any Part thereof, is equal to the Sum of any other two Terms of equal Diftance from them. DEMONSTRATION. Let 7, 14, si, (28,) 35, 42, (49,) 56, 63, 70, be a given Series. Then, I fay, that the Sum of any two Terms, fuppofe 28 and 49, which added together, make 77, is equal to the Sum of any other two Terms of equal Diftance from them. Suppofe, the Terms 35 and 42, which ate equally between them, be added togther, they make 77 ; or 21 and 56, the two outward Numbers next them, make 77 J or 14 the next two outward, make 77 ; and fo 7 and 70, which are the firlf and laft, at three Places diftant on each Side, make 77 alfo. AGAIN, SurposE, 42 and 49 be the two Terms taken together, making 91 ; then 35 and 56, the two next to them, make 91 alfo; as likewife doth 28 and 63, or 21 and 70. Now from this Theorem, ’tis plain, that if four Numbers are, in Arith¬ metical Progreffion, the Sum of the two middle Numbers, or two Means are equal to the Sum of the two Extreams For if 35, 42, 49, and 56, be given Numbers, whofe common Excefs is 7, then the Sum of the two Means, 42 and 49, equal to 91, ate equal to the Sum of the Extreams, 35 added to 56, equal to 91 alfo. THEOREM IV. 1. In any Series of Numbers in Arithmetical Progreffion, if the greateft and leaft Terms be added together, and their Sum multiplied by the Num¬ ber of Terms, one Half of the Produfl is equal to the Sum of all the Terms. E b Numb. VII. DEMONSTRATION. Tiipn I fay, if 34, the greateft, be added to 2, the leaft Term, and their Sam 3 A be multiply’d by 9, the Number of Terms, one half of the Produfl 324, which is 162, is equal to the Sum all the Terms taken together. I J . How {ball I be Jure that the A Greateft Term 34 I Produtl is equal to the Number of 2 Leaft Term 2 Terms taken together. £ — IO Sum 3 6 AT. Place them one under another, 14 Number of Places 9 as in the Margin at A, and add them iS together according to the common 22 One half W ay of Addition, and their Total 26 Will be equal to the I Produfl, as 3° The Sum of the greateft and leaft Term Halt the Number of Places Produa equal to the Sum of all the Terms taken together 163 The !Principles of Arithmetick. 103 D E M O N S R A T I O N. Half of the greateftand leaft Terms is 18 The Number of Places 9 Produdl ! Laflly, ^ hen the Series confilf of an odd Number, as that before-going, then multiply the middle Number thereof, (which there is 18) by the Num¬ ber ot i erms, 9, and the Product is equal to the Sum of all the Terms required. The middle Number 18 The Number of Terms 9 16a Thus have I fhewn you four feveral Ways of tolving this Theorem, that are entertaining and ufeful, to prove the Truth by. THEOREM V. In aSeries of Natural Numbers, as 1, 2, 4, 4, 5, 6, 7, 8, 9, isfc. if the laft Term be multiplied by the next greater, one half of the Produfl is equal to the Sum of the whole Series taken together. DEMONSTRATION. Let J, a, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 14, 15, 16, 17, 18, 19, 20, be given to find, the Sum of the Whole taken together. Then, I fay, if 20, the laft Term be multipliedby 21, which is the next greater, one Half of the produft 420, which is 210, is the Sum of the whole Series required. Here, for Proof, I add together 1 11 the given Series, divided, for Conve- 2 12 niency’s fake into two Parts ; whofe 4 14 Totals added together, are equal to 4 14 21 o, the half Sum before produced. 5 15 6 16 9 ! 9 10 20 55 '55 55 210 The laft Term 20 The next greateft 21 Produfl: 420 One Half 21 o theorem VI. The Principles of Arithmetics 104 THEOREM VI. Ifj a Series of natural odd Numbers, 1, 3* 5, 7 » 9 > Ir > ^‘■ the Sum of the Whole is equal to the Square of the Number of Terms. DEMONSTRATION. Let 1, 3, 5; 7, 9> ir > ! 3> ] 5> '7> J 9> *'• -f 5> confflinr of 1 3 Places, or Number of Terms: Then I lay, that Multiplied by be the given Series, 13 13 ProduS is 1 ^9 Which is the Sum of the whole Se- 1 15 ries taken together ; as appears by the 3 '7 fame added together in the Margin 5 r 9 at B. 7 31 9 B. 23 11 25 13 — .— 120 49 JV 169 THEOREM VII. In a Series of natural even Numbers, the Sum of the Whole taken to¬ gether, is equal to the ProduA of the Number of Terms, multiplied by the lame Number, more one. DEMONSTRATION. Let 2, 4, 8, 10, 12, 14, if, 13 2 l6 18, 20, 22, 24, 26, be the given 14 4 18 Series, confijling of 1 3 Places, or Num- her of Terms : Then I fay, that 13 6 20 8 22 multiplied by 14, which is the fame 10 C. H Number, more one, the ProduA is 12 26 equal to 1S2 equal to the Sum of 14 the whole Series-taken together, as — j 26 is proved by their Addition at C in 5 6 5 6 the Margin ; where, for Conveniency’s 182 fake, I have divided them into two Columns, and added their Totals in¬ to one Sum ; which you fee is equal to 182, as before. T H E O R E M VIII. In any Series of Arithmetical Progreffional Numbers whatfoever, if the lead Term be fubtraAed from the greateft, and the Remainder divided by the common Excefs, the Quotient having Unity added to it, will be equal to the Number of Terms contained in the whole Series. DEMONSTRATION. The Principles of Arithmetic?;. DEMONSTRATION. i r 1 ,C ’ I ^’ 72, ' 2 J< ? r > 44 , is the given Series. Then , la £ d fr ° m 54 . the greateft Term, be fubtrafled 4, the leaft Term, and the Remainder 40 , divided by 4, the common Excefs, the Quotient is 10, to " Hen add Unity, and it makes 11 ; which is equal to the Number of 1 laces, or Terms required THEOREM IX. In any Series of Arithmetical Progreffional Numbers whatfoever, if from the leait Icrm, the firft Term be fubtrafied '(as before in the laid Theorem,) and the Remainder divided by the Number of Terms, ieis by one, the Quo- tient will be equal to the common Excefs. DEMON STRAT ION. en" Ct r’ 2r ’ l 1 ’ 36, 41, 46, 51, be the given Series, con- Jijting of ten Places. Then I lay, if from 51, the grealeft Term, you fub- n ,„ 6 ’ tbe ,e f Term > the Remainder will be 45 ; which divided by 9, Number of Places, or Terms, lefs one, (there being ten Places in the 10 e,) the Quotient is 5 ; which is equal to the common Excefs required. From 51 the greateft Term. Subtradf 6 the leaft. 9 ) 45(5 the common Excefs. ^AALt. now give you a few Problems for Praiftice ; and fo conclude this firft Part. PROBLEM I. Hove many Strokes doth a Clock Jlrike, in finking all the 12 Hours? RULE. ©y Theorem V. th , 1 p 7L J I A c the Nl ' raber ’ b 7 tbe next greateft Number 14, and half the Product thereof is the Anfwer required, 1 1 the laft Number. 14 the next greateft Number. Produfl 156 * The Half 78, the Number of Strokes in 1 a Hours. This Problem may be alfo perform’d by Theorem IV. as follows : RULE. Add the firft and laft Strokes together, which is i and p and thev make 14; which multiply by half the Number of Terms, which here is l and the Produft is the Anfwer required. The firft Number is The laft Number is Sura Half the Number of Terms 1 11 6 Produfl Cc 78 as before. PROBLEM II. V 1 ■!!;•• 0 .': ■ 6060 2 /O/OO 106 The Principles of Arithmetics. PROBLEM II. A Hundred Stones are laid in a Right Line at i Yard afunder , which are to.be collected together into a Basket by one at a time , how many Yards muft a Man travel to gather them together. This Problem is to be folved by the laft Rule of the foregoing Problem. That is to fay , Ado the firft Stone fetched in, (which the Man travels two Yards in do¬ ing, vi%. one Yard from the Basket to the Stone, and one Yard back again,) unto the laft Stone fetched in, (which the Man travels two Hundred Yards to perform, vi^_. one hundred Yards from the Basket to the Stone, and one hundred Yards back again,) and the Sum is 202 ; which being multiplied by half the Number of Terms 50, the Product is die Anfwer required. The firft Stone 2 Yards. The laft Stone 200 Sum ao2 Which being multiplied by 50, half the Number of Stones. Product 101 co Yards; which is the Anfwer. This Problem may be folved by Theorem IV. as follows : To the firft Stone 2 Yards. Add the laft Stone 200 Their Sum 202 Which multiplied by theN° of Stones 100 Product 20200, of which take one Half. Which is 10100, the Anfwer as before. If we divide 10100 by 1760 the Number of Yards in a Mile, the Quo¬ tient will ftiew the Number of Miles contained in 10100'Yards. 1760) 10100 (5 Miles. 8800 1300 Yards, which is three Quarters of a Mile, all but 20 Yards. I will illuftrate this by another Example : Suppofe 200 Stones are laid at one Yard diftance from each other in a Right Line, as the former, to be colle&ed together by one at a time; how many Yards muft a Man travel to perform that Work ? The firft Stone is 2 Yards. The laft Stone is 400 Their Sum 402 Which multiply by 100, the half Number of Stones. Produfl 4 roo, the Number of Yards required. Now % 'Z do / a / o 0 2 The Principles of Arithmetick. Now 40200 being divided by 1760, (as before.) the Quotient is 22 Miles 5 . and 160 Yards. 1760) 40200 (22 Miles. 35 2 °' 5000 85 ^° 1480 Yards remaining. Subftra& 1320 Yards, which is \ of a Mile. And 160 Yards remain. P R O B L E M III. An Architect traveled from London towards Rome : His fir ft Day's Travel was 'en Miles , his laft Day s Travel was 65 Miles-, he increafed his Journey every Day five Miles : I demand how many Days did he travel , and the Number of Miles travelled ? RULE I. lo find the Number of Days, (which is the Number of Terms,) Prom laft Days Journey 65 Miles, fubtrafl the firft Day’s Journey ro Miles, the Remainder 55 Miles, divided by 5, the Number of Miles he in¬ creafed his Journey every Day, which is the common Excefs; the Quotient, more one, is equal to the Number of Terms, or Days, he travelled, vi%. to J 2j as in the following Operation. Prom die laft Day’s Journey 65 Miles, Take the fir ft Day’s Journey 10 Which divide by 5) 55 (11 To which add, more one, 1 The Sum is equal to 12, the Number of Days Travel. "Ihf Number of Miles travelled is found by the Rules of the preceding Problem, as follows : To the Lift Day’s Journey 65 Miles, Add the firft Day’s Journey 10 The Sum is 75 Which multiply by half the Days 6 And the Anfwer is 450 Miles; which is the Number of Miles he travelled in the Whole. PROBLEM IV. I have paid twelve Sums of Money: The firft Payment I made was 5 l. and the laft was 3 8 1 . which Payments increafed in an Arithmetical Progreffion : I de¬ mand, What was the common Difference of my Payments , and how much Money I have paid in the Whole ? i A // / X /4 A '3 t.c 'Ll 1C -a ti •zf 7 / iz 73 ?; ? * •V / 4 4 3 4 4 < *v 4 S- 4 t> sc d & y tf f * t / if L *7 a 4 * <■ as T'C a a 4 5 ft f f yt- it a A? RULE. io8 The Principles of Arithmetick. RULE. From the greateft Sum paid 38 /. fubtrafl the firfl Sum paid 5 /. the Re¬ mainder 33, divided by 12, the Number of Payments, lefs one, which is 11, the Quotient is 3, which was the common Difference of Payment. From 38 Take 5 Divide by 11) 33 (3 c Naw if to 38 /. the greateft Payment, you add 5, the firfl: and leaft Pay¬ ment, th£^n\s is 43 /. which multiplied by 6, the haft Number of Pay¬ ments, the Product is 258 /. as appears in the Operation following: The firfl: Payment 5 / The laft Payment 58 Sum 43 Multiplied by 6, the half Number of Payments. Produifl 258, which is the Total Sum paid in the Whole. PROBLEM V. 1 bought 20 Blocks of Marble in Arithmetical Progrefjion: For the firfl I paid 24 1 . and for the lafl I paid 120I. What did the Whole amount to ? RULE. Add 24/. the Value of the firfl Block, to 120/. the Value of the laft Block, and their Sum 144, multiply by 10, the half Number of Blocks, and the Product 1440 is the Anfwer required. To 120 /. Add 24 Their Sum 144 Multiply by 10, the half Number of Blocks. The Product 1440 is the Total Sum paid in the Whole. PROBLEM VL 1 am to receive 987 1 . at 1 4 Payments , each Payment to exceed the former by 7 1 . I demand the firfl Payment ? RULE. Divide 987, the Total Sum to be received, by 14, the Number of Terms, or Times of Payment, and from the Quotient 70 1 fubtra£l half the Pro- du£t produced by the Number of Terms, or Times of Receiving, lefs one, vi^. 1 3 multiplied The Principles of Arithmetick. 109 13 multiplied by 7, the common Excefs, the Remainer is 25 /. which is the firft Payment required. In the firft Place, divide 987 by 14. 1 4) 987 (70 4 , or l, the Quotient. * °7 In the 2d Place, multiply, 1 3 the Number of Terms, lefs one, they being 14, By 7 the common Excefs. Product is 91 And the Half is 45 [; which being fubtraQed from the Quotient 70;, There remains 15 which is the Number of Pounds paid at the firft Payment. From 70 1 the Quotient, Take 45 1 the half Product of 14 multiplied by 7, 25:0 remains, the firft Payment required. PROBLEM VII. 1 A' 8 Days Time I travelled 480 Miles', every Days Journey teas greater than the Day before by 4 Miles ; and my lafi Day’s Travel teas 79 Miles, I demand how many Miles 1 travelled the firft Day ? RULE. Multiply 4, the common Excefs, or Difference of each Day’s Journey, by the Number of Days, lefs one, 7, and fiibtrafi the Product 08 from 79, the Number of Miles travelled the laft Day, being the greateft Term, the Re¬ mainder 5 j is the Number of Miles travelled in the firft Day. Multiply 4 From 79 By 7 Take 28 28 51 remains, the N° of Miles travelled in the firft Day. Now have I inftrufied you in the firft Part of Arithmetick with Abun¬ dance of Satisfaction; and that you may, with equal Pleafure and Eafe, pafs through Vulgar FraSlions, Decimal Arithmetick, the ExtraBion of Roots, Geome¬ trical Progrejfion, and the Nature and Ufes of the Table of Logarithms ; I mult, in the next Place, make you a little acquainted with PraBical Geometry, that you may well underftand the Reafon of every Operation contained therein ■ which, for want thereof, will be very difficult. To which I proceed. Dd THE O F Ancient M A S O N R Y : O R, A GENE R A L SYS T E M O F COMPLEATED. PART II. Of G EO MET RT. By A- A- S Numbers are the Subject of Arithmetick, fo Lines, Angles, Superfices, and Solids, are the Subject of Geo¬ metry : All which will be herein confidered, fo far as they relate to Practice, in the various Operations in¬ cident to the feveral Arts contained in this Work. Geometry is the very Balls of all Arts; and by a clofe Application, is foon and eafily acquired. To become a Proficient herein, Thought is required in our Reafonings and Reflections, on the various Effects of the feveral Schemes that we are to con fid er. P. Prat The Principles of Geometry. P. Prat what do you mean by Thought, or Thinking , which I fuppofe to be the Jame Thing, and in nh.it Manner is it performed ? M. This is a Qiieftion that few can anfwer; fince there is not many in the World who give themfelves the Trouble of Thinking : But, however as thinking is abfolutety neceifary herein, I will therefore define to you what Tc is to think, and how ’tis perform’d. The Manner of Thinking defined. Since that we have no innate Idea’s, therefore all Objefls, or Materials for Thinking, mull be firft let in upon the Mind through the Organs of Senfe • That when they are fo communicated, we may then reflea or reafun upon them. That is, when the Aflion of exterior Bodies flrike open us, they at that Inftant caufe a fecond Aflion internally, which is continued in infinitum and differently named, as it aftefts the different Parts of our Bodies: that is, when it a 11 cels the Eye, tis called Seeing ; when the Ear, Hearing', when the Palate Taflmg', and when the Nofe, Smelling : All which are no more than fo many different Kinds of Feeling. 1 A s foon as either of thefe feveral Parts are thus affeQed by the Obiefl, the Motion is continued farther, and inftantly communicated to the Brain, where it caufes that Eftefl, which is called Thinking; after which proceeds our Con- fiderations, Determinations, Expreflions, and Afiions analogous thereto. Thus it is by Thinking that the Beauties of Geometry axe to be acquired; and [“/a 31 ; End ’ the Definition of Li " es > Angles, Superfices, and Solids, mull be iirit let in upon the Mind, as Materials or ObjeSs of Thought; which being well underllood, will enable ustoreflefl and reafon on their Properties when they come to aft on one another, in their various and numberlefs Theorems and Problems , whofe EffeQs are called Dimonstation ; an Art that will herein be fully illuftrated, to which many have pretended, and but few underftood. I shall herein pnrfue the fame Method that Euclid obferved in his Me- thod of Teaching, as being the moft familiar, concife, and inffruftive. LECTURE I. Geometrical Definitions. M. r | TIE Method of teaching Geometry, according to Euclid, was, Firft, JL To define the moft ordinary Terms. Stcondly, To exhibit certain Suppositions; and then proceed to Propofitions, wherein he treated of Lines, and the feveral Angles made by their Interfeflions, or Meetings; together with their Properties: All which he proved and demonftrated, lb as to con¬ vince any Perlon,^ who will confent to nothing but what he (hall be obliged to acknowledge, fo which I lhall annex, their various Ufes in the feveral Art- whole Foundations they are. P. Br The Principles of Geometry. p Hr thejc I imderftand, that the Princicipks and P rallice of Geometry, are comprised within three Terms that is to Jay, Definitions, Suppofitions, and Propo. fitions : Pray explain the Definitions of the mofi ordinary Terms a M. I will : And you are to obferve, that the feveral Quantities of which Geometry treateth, are Points, Lines, Superficies, and Solids. P. Wha t is a Point ? DEFINITION I. jyj. A Poin t, according to Euclid, is that which hath no Parts, by \\ Inch you are to underftand, that what is but one, cannot be made two. There¬ fore any Quantity whatfoever, as an Inch, Foot, Yard, i*. may be con- iidered a Point, provided you do not fubdivide the fame into Parts. But with regard to the fmalleft Quantity, and the Divifibility thereof, the moft mi¬ nute Atom has its Dimeniions, and by the Mind may be fubdivided into an infi¬ nite Number of Parts, which may be feverally fubdivided again ad infinitum: So that to determine theBignefs or Magnitude of the leaft material Point, is as difficult a Talk, as to conceive Bounds to the Univerfe. Hence ’tis plain, that if a Point be materially confider’d, either infinitely fmall, or determi- nately large, as a Foot, £*. it doth confift of fomething, and therefore con¬ tains Matter ; wherefore a material Point muft be either a Superficies, or a Solid. In the Praflice of Geometry, a Point is the leaft fuperficial Appear¬ ance that can be made by the Point of a Pen, Pin, Pencil, ire. as the Point A Fig. I. Plate I. P. Wha t is a Line ? DEFINITION 11. JVI. A Line is a Length without Breadth or r I hicknefs, which I thus prove: Suppofe two Figures, as A and B, Fig. II. be differently colour’d, viz- that of A with red, and that of B with white Colour; then I fay, that the Place d c, where the Edge of each Colour meet, or clofe one another,_ is a Line, or Length without Breadth or Thicknefs: And whereas in Praflice we cannot draw a real Line, which hath not a determinate Breadth, we muft therefore, to conlider it ftriflly, confider one Side thereof only, that is, the very Edge or Meeting of the Colour of the Line on either Side thereof with the Colour of the Paper, ilfc. on which tis drawn. Hence it follows, thac all Lines in Practice, though drawn never fo fine and narrow, have Breadth, and confe- quently’fuperficial Quantity; for otherwife they could not be vlfible to the Eye. But however, a Line is generally confider’d in Praftice to be a Length, without making any Refle&ion on its Breadth, and as fuch ’tis to be un¬ derflood. p. How many Kind of Lines are taught in Geometry ? M. Three, viz - Right Lines, Circular Lines, and Curved Linesi P, What is a Right Line', and how is it generated ? DEFINITION The Tvinciples cf Geometry. I 12 DEFINITION III. M. A Right Line, is a ftraight Line, and the lhorteft of ail the Lines that can be drawn between two Points; fo the Line i c, Fig. II. is a Right Line, being the neareft Diftance between the two Points d and c. Hence it follows t that the two Ends of a Line are Points ; and if the Point d be moved directly to the Point c , it will, by its Motion, trace or generate the right Line dc. P. I nnderjland yon very well: Pray now explain to me what is a circular Line ; and how *tis generated. DEFINITION IV. M. A Circular Line, is an arched Line, as he. Jig. Ill. and is gene¬ rated by the Motion of one End of a right Line, having the other End at the fame Time fix’d : As for Example ; Suppofe the right Line ha have its End at a fix’d, and the other End b moveable; then if the End h be mo¬ ved to each will 5 reprefent l 6 J j 5 l. Minutes. When the Circumference is thus divided, draw right Lines from every De¬ gree in the fame unto the Center e, whereon deferibe two Circles at Pleafure, as n 0, ih, Zip km; and obferve, that by the feveral Lines drawn from the Center c, to the Degrees in the Circumference, the Circumferences of the two inward Circles are divided into the fame Number of Degrees, as the outward Circle, wherein the firlt Divifion was made. And again, if the fame Lines were to be continu’d out farther from beyond the outward Circle, their Number would not be increafed, although they’d grow larger. Therefore you fee, and mull always well remember, that the leaft Circle as can be imagin’d, contains the fame Number of Degrees in its Circumference, as the very 'largeft. This being underflood, I will now explain to you the Manner of determining the Quantity of Angles. P. I fall be very thankful for the fame ; and hope foon to underfund it, face I ca: divide the Circumference of any Circle into q6o Degrees, and each Degree into 60 Minutes. definition xii. M. The Meafure of Angle;, in general, are determined by the Quantity of Degrees and Minutes contain’d in the Arches which meafure the fame, that is the Meafure of the Angle made by the right Lines ae, and e c, is’deter¬ mined by the Quantity of Degrees and Minutes contain’d in the Arch a fc, or in the Arch 0 x i, orintheArch s r /:, which, in general,’contain the lame Number of Degrees, as before obferved : So in like Manner, the Meafure of the The Principles of Geometry. ll 7 — the Angle madeby the Lines / e and b e is determined by the Degrees and Minutes contain’d in the Arch l b, or Arch gn, or Arch s p. . Now here obferve, that if the Lines s e and p c beconfider’d as an Angle, vhofe Meafure is the Arch s p, and that the Side e s be continu’d out to /, and Side e p out to b, yet the Angle is not increafed thereby; becaufe that the Arch / b contains but the fame Number of Degrees as the Arch r p. Therefore you fee, that you are miftaken in thinking an Angle to be in¬ creas’d by the Continuation of its Sides. P. I underfiend you very rightly ; end fee plainly that the Continuation of the Sides of an Angle doth not increafe it, as I once imagin’d it to do ; and that every Circle contains the fame Number of Degrees in its Circumference, that are leffer or larger, according to the Magnitude thereof. I think, Sir, 1 have heard you Jay, that Angles are denominated according to the Quantity of Degrees they contain ; pray, will you be pleajed to inform me thereof ? M. I will : Angles are denominated, as you obferve, according to the Number of Degrees they contain, and are of three Kinds, viz.. Acute-angled, Right-angled, and Obtufe-angled. P. What is an acute Angle ? DEFINITION XIII. AI. Any Angle that contains lefs than 90 Degrees, as the Angle made by the Lines f e and e c. P. What is a right Angle ? DEFINITION XIV. Al. An Angle that contains juft a Quadrant, or 90 Degrees ; as the An¬ gle made by the Lines a e and e c, P. What is an obtufe Angle ? DEFINITION XV. Any Angle that contains more than 90 Degrees; as the Angle made by the Lines b e and e f. And here note, that an Angle is always expreffed by three Letters of which, the fecond or middlemoft denotes the angular Point, as the Angle b e f is the Angle which the Lines b e, and e f form at the Point t. DEFINITION XVI. aeb* W. a. Deing given, me , • t » , caufe both thofe Angles being taken together, make the right Angle a Again, 118 The Principles of Geometry. Again, If the obtufe Angle b e d was given, then the Angle b e a would he the Complement thereof; becaufe both taken together, compleat the Semi¬ circle acd. Hence you are to obferve, that the Complement of an acute Angle, is fo much as it is lels than a right Angle; and the Complement of an obtufe Angle, is io much as it is lefs than a Semicircle. Note, The Complements of Degrees in a Quadrant and Semicircle, are the fame. P. P ray do Geometricians fignify the Kinds of Angles by any particular Cha - raders ? Yes: An acute Angle may be exprelfed thus IN, aright Angle thus [_, and an obtufe Angle thus _/; fo likewife the Word Angle is exprelfed thus A , and Angles thus A A ■ P. I thank you for this full and plain recount of the Names and Nature of Angles. Notv I mufl beg Leave to remind you of what I have often heard you fpeak of, relating to a perpendicular Line, which I define you will define. DEFINITION XVII. Whin a right Line falling on another right Line, maketh the Angles on each Side equal, thofe Angles are right Angles, and the Line fo falling, is called a Perpendicular : As for Example, Fig. X. IF the right Line c e, falling on the right Line a d, make the Angles a e c, and c ed, equal, that is, if having on e , as a Center deferibed a Semicircle acd, and the Arches ac, c d, are then found equal, the Angles aec and ced are called right Angles, and the Line ce a Perpendicular : And becaufe that the Arches cd ar.d ac do each contain Degrees, therefore the Semicircle acd doth contain I So Degrees, which is half 360. Haying now fully explained the Names and Kinds of Lines and Angles, I (hall in the next Place, proceed to ftiew the like of the various plain Geometrical Figures that are form’d thereby. P. Pray bow many plain regular Geometrical Figures are form’d by Lines ? M. Four: tig. The Circle, the Equilateral Triangle, the Geometrical Square, and the Polygon. P. And are thofe all the Figures that can be formed by Lines ? M- No: Befides thefe, there are various Kinds of Ellipfis’s and Or ah, divers Kinds of Triangles and Trapeziums ; alfo the Parallelogram, or Oblong, the Rhombus, and Rhomboides ; alfo all the compound Figures that Invention can form: But of all thefe laft, none are regular in their Sides and Angles, as the former are; fo that they are deem’d irregular Figures, although fome of their Parts are correfpode'ntiy regular. P. Pray ex: lain them (everally ; and be pleafed to begin with the regular Fi¬ gures ; and firfl, of a Circle. DEFINITITION ii 9 The 'Principles of Geometry. definition xviii. M. I HAVE already Ihewn you, in the Generation of curved Lines, how by the Revolution of a right Line, a Circle is generated ; and therefore I need now only add, to refreih your Memory, that a Circle is a plain Figure, whofe Bounds are made by the winding or turning of a Line, which is called Circumference, (as before obferved,) and which is equally dillant from the middle Point, that is called the Center. DEFINITION XIX. Secondly, That the Diameter of a Circle is any Line vhatfoever which palleth through the Center, and which ends at the Circumference, cutting the (ante into two equal Parts. DEFINITION XX. Thirdly, That half the Diameter is called the Semidiameter, or Radius. DEFINITION XXI. Fourthly, That a Semicircle is a Figure terminated by the Diameter, and half the Circumference ; and a Quadrant, one half Part thereof. P. 1 thank you, Sir, for reminding me of a Circle, and the Lines thereof: But fnppofe that a Circle have a Part divided in it, as the Part e c d. Fig. XL which is lejs than a Quadrant; pray what is the Name of fuch a Figure ? DEFINITION XXII. M. The Name of fuch a Part of a Circle is called a SeBor of a Circle ; and is terminated under the two right Lines ce and cd, and the Arch or Curve ed; fo likewife is bxaec, a Sector alfo, terminated under the two right Lines be and. c e, and Arch bxae. P. Very well. Sir; fo far I underfland you. Butfuppofe that a Part of a Circle fhould be divided ojf, as the Part I gh, Pray what’s the Name of fuch a Figure ? DEFINITION XXIII. M. When a Circle is divided into two unequal Parts by a right Line, the Parts fo divided, are called Segments of the Circle ; fo the Part fgh is the letfer Segment of che Circle badh, and fxeg the greater Segment. P. And is the right Line f g called a Diameter, as the Line b d is ? DEFINITION XXIV. III. No : It is called a Chord Line: None are Diameters,as before obferv’d, blit fuch that pafs through the Center. P. 1 ask Pardon for my Forgetfulnefs. Pray proceed ; and explain to me the next regular Figure in Order, which I think you Jaid was the Equilateral Tri¬ angle. M. I 20 The Principles of Geo metry. • • i definition XXV. M Yes; the equilateral Triangle is the next plain Figure that is com¬ prized under the next feweft Lines; that is, an equilateral Triangle is a plain Geometrical Figure, bounded by three right Lines, which are each equal to one another, as a, b, c, Fig. XIII. P. Pray, is there any particular Lines belonging to an equilat: at Triangle, more than the Sides thereof f M. Yes; perpendicular Lines, which may be drawn from any Angle to the Side oppollte to it, as the Lines bd, ce, and ah. P. Pray, are all Triangles to be made equal in their Sides ? A/i No; they may have two Sides equal, and the third unequal; as i,l,k , or every Side unequal, as m, n, o ; but neither of thefe two laid are regular Figures. P. That I know ; and therefore beg Pardon and Leave for this DigreJJion from our Difcourje on regular Figures ; which I am induced to defile, that in this Phut, where you are fpsaking of Triangles, l may be fully informed in all the various Kinds thereof, and in what Manner they are feverally diftinguijhed. ' M. I will explain them feverally to you. And, firft, Euclid diftinguiiheth Triangles after two different Manners, by their Sides, and by their Angles. DEFINITION XXVI. Firft, If a Triangle have all its Sides equal, ’tis therefore called an equila¬ teral Triangle, as before obferved. DEFINITION XXVII. Secondly, IF two Sides are equal, and the third unequal, (either longer or Ihorter, as l,i,k,) it is called an lfofecles Triangle. DEFINITION XXVIII. And, Thirdly, If every Side is unequal, as m,n,o, it is called a Scalcnwm Triangle. These are the Diftinflions of Triangles, with rcfpefl to their Sides. Now, with regard to their Angles. DEFINITION XXIX. Firft, IF a Triangle have one right Angle, as the Triangle rpq, it is there¬ fore called a right-angled plain Triangle. DEFINITION XXX. Secondly, If a Triangle have one obtufe Angle, as the Triangle ts u, whofe Angle at s is obtufe angled, it is called an Amblygonium Triangle. DEFINITION The Principles of Geometry. I 2 I DEFINITION XXXI. Thirdly , If a Triangle have all its Angles acute, as the Triangles a,b,c, or as /, i, k, it is called an 0 xygonium Triangle. And thefe are the Diftin&ions of Triangles, with refpe& to their Angles. P. Pray are there any other Particulars to be known relating to the Sides and Angles of Triangles ? DEFINITION XXXII. M. Yes; you are alfo to obferve, Firfl, that in a right-angled plain Tri¬ angle, thofe two Sides that form the right Angle, are called the Legs, as rp, and p q ; and fometimes one of them is called the Bafe, and the other the Perpendicular. P. Pray which of the two Legs is generally taken for the Bafe ? M. The longeft, as pq. P. 9 Tis very reafonable that the Bafe fhould be the largefl : Pray have you any Name for the other Line rq ? M. Yes : That is called the Hypothenufe ; and fo the three Lines that form a right-angled plain Triangle, are the Bale, Perpendicular, and Hypothenufe. P. And are the Sides of all other Triangles thus denominated ? M. No; they are only diftinguifhed by calling any one Side (but generally the longeft, as aforefaid) the Bafe, and the other two Sides are always called the Sides oppofite to the Bafe, either greater, equal, or lefter, according to their Proportions, the one to the other. So in the Triangle m no, if the Side wo be made the Bafe, then the Sides mn, no, are the Sides oppofite to the Bafe. These are the Particulars to be obferved, with refpeCt to the Names of the Triangles. The next relating to the Nature, Quantities, and Affetftions of their Angles, I mnft refer unto Plain Trigonometry, wherein they are applied to immediate Practice. P. I thank you, Sir. Pray return to your Difcourfe on the regular Figures, and excufe this DigreJJion. DEFINITION XXXIII. AL The next regular Figure is the Geometrical Square, alfo called a Rect¬ angle; which is a Figure bounded with four equal Sides, and all its Angles right, as a, b, c, d. Fig. XIV. P. Pray is there any particular Lines incident to the Geometrical Square ? ill. Yes, there are two Kinds, v'vg. the Diagonals, and the Diameters . P. What are the Diagonal Lines of a Geometrical Square ? Gg definition "■■I ill > 122 The Principles of Geometry. DEFINITION XXXIV. Af. Those right Lines that are drawn from one Angle to the other, as ad and be, and the Point i, where the two Diagonals interfe£t each other, is called the Center of the Square. P. Wlmt are the Diameters of a Geometrical Square ? DEFINITION XXXV. Af. Right Lines drawn through the Center from one Side to the other, making right Angles at meeting, as gh and e f. Euclid calls the Lines ad, b c , Diameters, inftead of Diagonals; but, I think, very improperly. The next regular Figures, are the various forts of Polygons, namely, the Penta¬ gon, Hexagon, Sept agon, or Heptagon, OB agon, Non agon, Decagon, dec. P. Pray what is a Pentagon ? definition XXXVI. Af. A Pentagon is a regular Figure bounded by five equal Lines, which conftitute as many equal Angles, as A, Fig. 15. So in like manner a Hexa¬ gon confifts of fix Sides, as B; a Septagon, or Heptagon, of feven Sines, as C; an OBagon , of eight Sides, as D; a Nonagon, of nine Sides, as E; a Decagon , of ten Sides, as F; an Undecagon, of eleven Sides, as G; a Duodecagon, of twelve Sides, as H ; y the Arch « m 0, and likewife deferibe the Arch re over the given Point b. e. Lav a Ruler from o to d , and it will interfeci the Arch ee m a. v. D a* hb, and ’tie the Perpendicular required. RULE III. Fig- X. Let am be the given Line , and m the given Point. Fja.hc i . Draw a right Line at pleafure, as A B, a - Tjl- e three Inches from the Scale of Inches in your Compailhs; and 1 tit : , ■ oo, , n the given P« int m, ith the other deferibe an Arc t < ove the lame, as oo. . - an d fet them on the Line a m, from the given Point m t0 c- alfo taki Infcl es, and fetting one Foot of your Compaffes in c with in erfctt the Arch op, in the Point i. Lathy, draw the ..gut Line im, and ’twill be the Perpendicular required. t> l Pall demonftrate this Problem hereafter, in my LeBure on the Transforma¬ tion and. Equality of Geometrical figures, when the Manner of extrattmg the jqo.a.re Root is known to you. NoTr, This Method of railing a Perpendicular, is very ufeful in ferring out’the Foundations of Buildings : For being provided but with three gods t ig- the one of j Feet, the fecond of 4 P eet > and the third or 5 Feet in Length, you may readily form a fquare Angle at one Opera¬ tion, as following : SittoSe CF, Fig- XI. to be Part of a Foundation,^- and ’tis required to raife a Square or Perpendicular from the Point F. First, Lay your 4 Foot Rod.from F, along the Line FC towards C; then taketl other two I ds, and apply their Ends to the Ends thereof, as at D .and F, o 1 ■ in that the , 1 01 1 Rod be always at the given Point r their th< E Is in; brot ht t ether at E, the 3 Foot Rod will then Perpendicular required. NoTr, That as a 10 Foot Rod is the mofl general in PraBice, you way, m- pead of the Numbers 3, 4 and 5, make uje of theje Numbers dou:n. , -^z. 6 , b, and 1:, and tv ith them proceed as before. By this Method, you indy not only very readily fee out a fijuare Angle 01 a Building, &c. but you may all'o examine Angles that are already built, it they are truly fijuare, or not. Perpendiculars The 'Principles of Geometry. 129 Perpendiculars may be raifed inftrumentaUy two Ways, vig, by the Scale of Chords, and by the Help of a Protraflor. P. Pray what is the Scale of Chords ? M. The Scale of Chords, is the Degrees contained in a Quadrant, or quar¬ ter Parc of the Circumference of a Circle, transferred unto a right Line, as you may fee in Fig. XII. where the Degrees in the Quadrant cd are trans¬ ferred unto the Diameter ad, as following : Your Quadrant cd being divided into 90 Degrees, as before taught in he firft Leflore hereof fet one Foot of your Compaffes in d, and extend the other on the Arch to 10 Degrees. Then with that' '•pening, remove the faid Point away, until it fall upon the Diameter at b ; then will bd of the Diame¬ ter Signify to Degrees, as the Parr oi the Quadrant d 10 doth. Again, let¬ ting one Foot of'your Co.mpslles in d, as before, extend the other on the Arch to 10 Degrees ; which Opening or Extent fet on t! e Diameter, front d to to, as before ; thenwill d 20 on the Diameter, figuify 2c Degrees, in the fame manner as d 20 on the Arch doth : Proceed on in like manner with every Degree in tfe Arch, and fo will you have transferred the 90 Degrees in the Arch cd of the Quadrant ecd, unto the right Line, or Diameter ad, as re- prefented in Fig. XIII. to- And here obferve, That 60 Degrees taken on the Arch c d, is exactly equal to the Radius, or Semidiameter e d. P. And are 60 Degrees of any Circle always equal to the Radius or Semidiameter thereof? M. Yes ; and therefore, before your Perpendicular can be raifed, or any other Work done, you mult always firft take 60 Degrees from your Line ot Chords, and with that Diftance deferibe an Arch on the given Point. As for Example ; I would raife the Perpendicular dc on the right Line ab, from the Point c. PRACTICE. Fig. XIV. 1. Take 60 Degrees in your Compaffes from your Line of Chords, and on the given Point r, deferibe an Arch at Pleafure up from the given Line, as edf. 2. Take 90 Degrees in your Compaffes, and fet them from e to d, and then drawing dc, it will be the Perpendicular required. P. 1 thank you. Sir ; ’tis very eafy. Pray proceed to Jhew me, how to perform the fame by the Inftrument, which you call a Protractor : But in the firft Place, pray what is a Ptotr actor ? At. A Protractor is a Semicircle made of Ivory, or more generally ot Brafs ; as Fig. XV. whofe Circumference is divided into Degrees, and half Degrees, and fometimes to Quarters, or 15 Minutes, by Help of which In- ftrument, you may molt readily raife a Perpendicular from any given Point, at one Operation, as following : Ii To I co « The Principles of Geometry. To raij-e a Perpendicular by Help of a Protraclor, Fig. XVII. I.et / be a given Point in the right Line eg, and ’tis required to raife a Perpendicular therefrom. Practice. Apply the Center of the Protrafloril to the givenPoint/ with the inward Edge ik, exactly to the given Line eg ; and, at the fame Time, with a Pin, Pencil, &c. make a Point on your Paper, clofe by the Edge of your Protraflor, cxaflly at 90 Degrees Then removing away your Pro- trudlor, draw a right Line from that Point unto the given Point /, and it will be the Perpendicular required. Now you fee, that thefe two laft Methods are much the fame, excepting that here in this laft, you have no need of a Pair of Compaffes to defcrib'e Arch, becaufe the Protraclor is the Arch itfelf. But by the way, I mult remind you, that on all Lines of Chords there are. or fhould be, two Brafs Studs bxd in the Line, the one at the Beginning thereof, and the other at 60 Degrees ; which are there fix’d for your better applying your Compaffes to the Line, when you want to take off the Radius, or any other Number of Degrees and Minutes required, without flicking them into the Rule, and thereby deface and fpoil the Divifions. These Inftruments are generally made and fold by Mathematical Inflru- ment-Makers, and are commonly included amongil thofe that are made up into Cafes, for the Ufe of Mathematicians, and all others who delight in Drawing, Defigning, Meafuring, iSr. P. Pray, what are thofe Inflruments that are made into Cafes : and of whom can I pure baft them ? M. rHE Inftruments that are proper for your Purpofe, are two Pair of Compares, about fix Inches in I ength ; of which one Pair hath one of its Feet made to take out, and in its Place ferew in another, either with a Black-lead Pencil in it, or a Drawing-Pen ; vvhofe L'fcs are to deferibe Arches or Circles, m Black-lead or Infc Sometimes there is a third Point added, with a fmall A heel, todefcribe pricked or dotted Lines, and is called the Wheel-Point : But as Ink is fubject to receive Hairs, and other Impediments, that oftentimes caules it to run into an entire Line, and thereby fpoil or deface many times an elaborate Drawing ; therefore I advife you not to make any Ufe there¬ of, and in its Head take Time, and draw your prick’d or dotted Lines as near as you can with your Hand Drawing-Pen : Which is another Inflrument of t he Calc : whofe Ufe is to draw right Lines of any Breadth or Finenefs required, which you regulate by a Screw that’s fix’d in the Chops of the Pen for that Purpofe. Befides thefe, there are, firft, a plain Scale, which is made either of Box, Ivory, Brafs, or Silver, on which is graduated Varieties of Scales of equal Parts, and generally two different Lives of Chords. Laftly, a Seftor, which opens to a Foot; on which is deferibed tile Lines of Sines, Tan¬ gents, Secants, Humbert, Polygons, &Tc. whofe feveral Ules I lliall hereafter make known unto you m their proper Places. Thefe with good Black-lead Pencils Ins, Paper, Draw mg-Board, Te-Square, and two or three Rulers of different Lengths, as a Foot, eighteen Inches, two Foot, isfe. are fully fufficieut for your preient Purpofe. Thefe, and all other Mathematical Inftruments, are molt accurately made and loll, at very reafonable Rates,, by Mr. Tho. Wright, Inllrument-Maker S _ Tbe Principles of Geometry. _ > 3 ' S*5t‘fe,54 h ' 0m *" 1 “ ,l Mr - DmBL Sem ’' "■ ‘ >*-**-' - «<“'= JW. A Drawing-Board, is a fquarc Board,' made of Mohogony or Warnf cot, about 15 Inches wide, and to Inches in Length, or anfsiL that the argenefs of the Paper you make ufe of requires. To this 7 is belonging Square, as », F,g XVI. Plate III. ma de in the Form of the Letter T and ma e dfa re r ,S t Ab ° UC the Ed ? e <* Dratlng-Wd t made a Groove, wherein the Stock m of the Te-Square Hides when in ufe w.th’s I-" 316 l ' fe ° f i th , e B ° ard 3nd S and thereon, with any Opening greater than ci, defcribe Arches- as nh, and kk, lnterieituig each other in/. a. Dra w fd, and ’tis the Perpendicular required. PROBLEM X. Fig. XIX. To erett a Perpendicular upon a concave Line, from a given Point, in or near tbe Middle thereof, as df on adb. Practice. Set oft re as in the laft Problem, and likewife defcribe the required* ^ m f' Draw J< ’tis the Perpendicular PROBLEM XI. Fig. XX. To creB a Perpendicular upon a concave Line, from a given Point, at the End thereoj, as ab on fb. Perpendiculars ■ • . i The Principles of Geometry. Practice i. Affurae 5 Points in the concave Line at pleafure as the Points I) c n and draw the right Lines he, en ; which Lines, by Iroblei VI. hereof, bifefl by the Lines ag and ae, and continue them until they meet in a. 1. Draw the right Line ab, and ’tis the Perpendicular required. PROBLEM XII. Fig. XXII. To rare a Perpendicular on the Angle of an Equilateral or Ifofecks Triangle, as he, on the Angle acb of the Triangle cab. Practice, i. On the given Point c, with any Opening of your Com- paffes, deferibe an Arch as if, interfefling the Sides at :and cb in the Pom s l .. on which, as two Centers, with any Opening of your ComnaiL-, de fcf.be Arches over the Angle, as ik, and Im, mterfefling each other ,n h. 3i Draw the Line he, and ’tis the Perpendicular required. PROBLEM XIII. Fig. XXX. To let fall a Perpendicular from a given Point, upon a right Line afigned ; as lk, from the Point l, on the right Line iib. Practice. On the given Point /, with any Opening of your Compares greater than the nearett Pittance of the given Point from the Line de¬ feribe an Arch cd, cutting the given Line in cd; on which, as two Centers with any Diftance greater than ck, deferibe Arches below the given Line ; as ee, hh, interfefling each other in 1. Lay a'Ruler from the given Point / to t, and draw the Perpendicular re- quired. Which will bijeEi cd in k. D E M ONSTRATION. Draw the Lines Ic, and Id; which, being drawn from I, the Center of the Arch cd, are therefore equal. 2 The ri< J htLine cd, being bifefled by the Perpendicular, therefore ck and and’i id are equal : And as the Perpendicular lk is common to the Triangles lek, and Ikd, therefore the Angles Ike, and Ikd, are equal, and the Line Ik perpendicular to cd. Which was to be proved. PROBLEM XIV. Fig. XXIII. To let fall a Perpendicular from a given Point, that is filiated nearly over the End of a given right Line, mxb from the Point h, on the right Line ab. Practice, i . Lay a Ruler from the given Point h, unco any Part of the right Line, as unto^; and draw a right Line, as gh. - a. Bisect gh inf, and on/, with the Radius fg, deferibe the Semicircle hbg, interfefling the given Line in b. Draw hb, and it will be the Perpendicular required. P R O E L E M The Principles of Geometry. 1 33 PROBLEM XV. Fig. XXIV. To let fall a Perpendicular, from a given Point, upon a concave circular Line-, as a h on the concave Line c i. Practice, r. Affume three Points in any Part of the curve Line, as at c,e,g, and draw the right Lines ce and eg; which bife<£t in d and f, by the Lines dk and fm, and continue them until they meet in b. Winch is the Cen¬ ter of the Carve ci. 2. Lay a Ruler from b unto the given Point a, and draw ah’, which is the Perpendicular required. PROBLEM XVI. Fig. XXV. To let fall a perpendicular Line, from a given Point, upon a convex circular Line, as hd, on, i k. Practice, i . Open you Compaffes to any Diftance greater than the Di¬ ftance of the given Point from the Curve, and deferibe an Arch, as //, in¬ terfering the given Curve in two Places, as at e,g, on which Points, with any Diftance greater than half eg, deferibe Arches, as a a, c c, interfering each other in b. 2. Lay a Ruler from b to the given Point h, and draw hd, the Perpen¬ dicular required. PROBLEM XVIL Fig. XXVI. To make an Angle, equal to an Angle given, in any Point of a given right Line. At the Point k in the right Line cn, make the Angle Ikn , equal to the given Angle rpq. Practice. On the Points p k, with any Opening of your Compaffes, de¬ feribe equal Arches at Pleafure, as bo and t r ; then take the Diftance t s be¬ tween your Compaffes, and fet it on the Arch no, from n to m ; and from the given Point k, through m, draw the right Line k l ; then will the Angle Ikn be equal to the Angle rpq. DEMONSTRATION. Draw the Lines ro, and mn, and the Triangles spt and mkn will have the Sides ps, pt , equal to the Sides k m, kn ; becaufe the Arches nm and st, were defcribecl with the fame Opening of the Compaffes ; and their Baffes st and mn being alfo equal ; therefore the Angle Ikn is equal to the given An¬ gle rpq; which was to be proved. Kk PROBLEM S The Principles of Geometry. i 34- PROBLEM XVI 1 L. Fig. XXVII. To make an Angle, even to a Jolid Angle given, in any Point oj a given right Line, own, equal to bac. i. Because that the given. Angle is folid, and therefore cannot deferibe an Arch therein, as nr the foregoing Problem ; therefore continue out the Sides thereof, as the Side ba, towards /' and the Side ca, towards u v. With any Opening of the CompalTes on the Points a and v deferibe the Arches kg anAt&g making.^equal to hg; then through the Point y draw the right L.itie viv, and io will the Angle n v .v be equal to the folid Angle bac ; becaufe that the oppofite Angle iaj is equal to the given An¬ gle bac. This Problem is very ufeful in taking folid Angles of Buildings, as will hereafter appear, when I come to (hew you how to take the Plans of Build¬ ings, isle. p. hltfuppofe that a jolid Angle be Jo punted by a Rncr-fdc, ash ac, by the Rivry A B, which prevents my continuing on the hide ba, towards t : Pray horn rrmji 1 proceed at Juch Times to find the Quantity oj an Angle ? M As following : Continue on the Side ca towards h, and then will you have formed the Angle bah, wbofe Complement to a Semicircle, or I So De¬ grees. is (by Def. XVI. Page 117.) the Quantity of the Angle required ; for the Angle ba c, added to the Angle bab, being taken together, are a Semicircle. P. Very rvcll. Sir ; I mderfland you : But fuppofe that 1 have an Angle to take, as the Angle bah, that has a Chimney (landing in the Angle, as at the pricked Line a. 1 r. 1 4. g. which prevents my roc a] wring into the Angle at a; Pray how mufi I proceed to know the (hi. entry ? M. As follows : (1.) Alfume two Points in each Side of the Angle in any Parts thereof, as at the Points 5 and i, in the Side ah, and the Points I and a, in the Side ba; from all which raile Perpendiculars, continu’d at pleafure, as i- v , a.10, 2 -n, i 11. (a.) Set off on each Perpendicular any Length Irom the Sides of the Angle, as the Diftance i 5, 7-4, 2-7, J-S; and through the Points 4, 5, S, 7, draw the right Lines 1 5-7, and 11-ia, which iorm the lame Angle as rhe Lines ba and ah. (4) Whofe Quantity may be found, as before taught. PROBLEM XIX. Fig. XXVIII. To draw a right Line parallel to another right Line, at any affigned Di/lancc, as c parallel to n f, at the Difiance of a b. P. Pray what are parallel Lines ? I don’t know that I ever heard of them before. M. Parallel Lines are right Lines equidifiant in every Place, and therefore are juch, that being in the fame Superficies, if infinitely produced, would never meet, *s the right Lines c and nf, which are delineated as following • Practice. The Principles of Geometry. 6 3$ Practice. Take the affigned Diftance ab in your Compares, and to¬ wards each End of the given Line nf, affign two Points, as m and b, on which,, as two Centers, deferibe Arches, as ee and gg. a. Lay a Ruler unto their Convexities, until you can but juft difeern them ; and then draw the right Line c d, which will be parallel to nf. at the albgned Diilance ab, as required. PROBLEM XX. Fig. XXIX. To draw a right Line parallel to another right Lins that Jhall pap through a given Point, as the Line oo, parallelto p p, puffing through tbs given Point's. Practice, i . .From the given Point r draw a right Line at Pleafure unto any Part of the given Linepp, as unto r. * I-Ai.e the Length of the Line sr in your CornpalTes, and, on the Points sr deferibe Arches, as s q and' tr, and make rt equal to sq. T La y a Ruler from j to t, and draw the Line oo, which will be parallel to pp, and pafs through the given Point s, as required. DEMONSTRATION. Because that the Angle srq is equal to the Angle tsr ; therefore the Lines oo, pp, are parallel. «■ Note, That the Angles sr g and tsr, are called Alternate Angies. PROBLEM XXI. Fig. XXX. At a Point given, to make a right Line equal to a right Line given ; as at the given Point c, to make the right Line ed, equal to to the given right Line ab. Practice, i. Draw a right Line from the given Point c, to one End of the given Line, as to b ; and on the Line c b, erefl: the equilateral Triangle c c b. 2. On the Point b, with the Radius ab, deferibe the Circle a f b, and continue eb to f, and ec at pleafure. .>. On the Point e, with the Radius ef, deferibe the Arch fg, interfeflinu the Line ec continued in d ; then will ci be a right Line equal to ab, as required. demonstration. c d and ef are equal, and eb and ec are equal; wherefore cd, bf, and ab, are equal. Which was to be done. PROBLEM XXII. Fig. XXXI. To divide a right Line into any Number of equal Parts: Sttppofe a b into feven Parts. Practice. From either End of the given Line, as at a, draw a right Line as ac, making any Angle at pleafure; alfo from the other End of the given Line w The Principles of G e ometr v. i ? 6 I.inaf, draw the right Line /-//parallel to tic, an 1 continue it out at plea- Lire. Then opening your Cumpafles to any fraall Pittance, let oft on each Line, from the Ends ot the given line, the lame Number of pittances, lefi one, as the Number of Parts, into which the Line is to be divided, which is 6, becaufe that 7, lefs 1, is 6, as at the Points 1, 2, -J, 4, 5, 6. j. La-sing a Ruler from (, 5, 4, 3, a, 1, in the Line continuing them until they meet in tfe Point m, which is the Center of the given Arch, on which you may com¬ plete the Circle as required. DEMONSTRATION. Draw the Lines ab and be, and they will be divided equally; and the Lines / o } and f h, will be perpendicular thereto. PROBLEM XXVI. Fig. XXXV. To find the Center of a Circle. ’Tis required to find the Center c of the Circle abge. Practice, i. Draw a Line a-crofs the given Circle in any Part thereof, as b lo that it it intei fe£t the Circumference on each Side, as at b and e; which h hi in/, and thereon raife the Perpendicular fa, and continue it from / to g. 1. Bisect ag in c\ which is the Center required. DEMONSTRATION. I f the Point c is denied to be the Center, let any other Point be faid to be it; fuppofe die Point d: hen I lay, if d is the Center, the Lines db and de , muff be equal, and the Triangle db f mult be equal to the Triangle dfe, and confequently df mult be perpendicular to be, and not j a, which would be contrary to the Hypothelis : Therefore ’tis evident, that the Center can¬ not be out of the Line aj. Furthermore, fince that the Point c divides the Line ag into two equal Parts, and c b, c a, c e, eg, being equal, therefore c Numb. X. Ll is The Principles of G ko mmtr y. i ; - the Center ; otherwife thofe Lines cb, ca, ce eg, drawn from it to the Circumference, would not be equal. Which was to be proved- - c plain, that the Center of a Circle is in a Line which div'.deth another Line i » the Middle, and that at Sight Angles : So c is in the Line a f which di¬ vides b e in the Middle at f at Right Angles, becaufe that at is perpendicular to b e. p R O B L E M XXVII. Fig. XXXVI. To divide an Arch Line into two equal Parts, as the Arche dx. PitACTicE. Draw a right Line from one Extreme of the Arch unto the other, as ex, which bifeft in », and thereon raile the Perpendicular gn, and continue it to the Arch at d ; then will ed be equal to dx, as required. demonstration. The Center h is in the Line dg and g%, being perpendicular to ex; therefore ex cuts£*L at Right Angles, and eh mull be equal to hx ; alio ed mull be equal to dx; otherwile the Lines dig and ex carrot cut erh other at Right Angles; wherefore the Arch ed is equal to the Arch dx. Which was to be proved. PROBLEM XXVIII. Fig. XXXVII. To divide the Circumference of a Circle into thirty-two equal Parts; or otherwife, To deferibe the thirty-two Points of the Compafs. Let AitcB be a given Circle, to have its Circumference divided as aforefaid. Practice, (i.) Through the Center D draw a right Line, as Ac; aifo draw a B at Right Angles thereto, and then will the Circumference be divided into four equal Parts, at the Points A, a, e, B, which may reprefent the four Cardinal Points of the Horizon, vig. North, Kaft, South, and Welt, as therein prefTe-d. (: ; By the Lift Problem, divide the Arch ae into two equal Parts by the Line i c, drawn from r, the Interfeclion of the Arches a a and b e, unto the Center D. In the fame Manner divide the other three Arches e B BA, and A a; and then will thofe four Diviiions reprefent the North-Eaft, South-Eaft, South Weft, and North-Weft Points, and the Circle will then be divided into eight equal Parts. (->.) Divide the Arches ai and i e in like manner, by the Arches a 6 , 4/- an d ir, it; and from 5 and q, the two Points of Interfeflion, unto the Center D, draw the two right Lines 5 g and l q, then will the Quadrant or Arch a e, be divided into four equal Parts at the Points g, i, l. (4.) Is the fame Manner divide theArches ag in /, gi in h, i l in k, and le iu m ; and then will the Arch or Quadrant ae, be divided into eight equal Parts. (5.1 Complete the other three Quadrants in like manner, and the Cir¬ cumference of the Circle A a. e B, will be divided into 4 - Parts, as required. PROBLEM The Principles of Geometry. 1 39 PROBLEM XXIX. Fig. XXXVIII. A right Line being given, tc find, a Point in a dire ft Pofition thereto ; jor the Continuation thereof, Let hd be the given Lire, and ’ris required to find a Point in a diredl Pofition thereto, that the laid Line being continued, fhall pafs through the fame. Practice, (i.) On the End d, with any Opening of your Compafies, deferibe an Arch as a a, interfering the given Line in z. (a.) Set off any Diftance from i to c, and the fame from i to b; on which, as two Centers, with any t Opening greater than b d , deferibe Arches, as ee and ffi interfering each other in g, which is the Point required, through which the right Line h d will pafs, being continued. PROBLEM XXX. Fig. XXXIX. Trvo Points being given, to find two other Points direftly interpofed■ Or to draw a right Line between two given Points with a Ruler, whoje Length is equal but to Half the given Diftance contain d between them. Let A and B be the given Points. Practice, (i.) Upon the Points AB, as Centers, with any Opening of Compafies greater than Half the Diftance of A B, deferibe Arches, as d d and cc , interfering in ef (7.) Upon the Points ef, as Centers, with any Opening greater than Half ef, deferibe Arches, as gh, and hg interfering in the Points nm which are direriy interpofed between the given Points A and B; by Help of which, with your lhort Ruler, draw the right Line A B, as required. PROBLEM XXXI. Fig. XL. To deferibe a Segment of a Circle , in which any Angle being drawn from the Ex¬ tremes of the Chord Line, fhall be each equal to an Angle given. Let EC be the given Chord Line, and A the given Angle. Practice, (i.) Make the Angle cB/j equal to the Angle A, and on B erer the Perpendicular B/, continuing it out at pleafure. (a.) Bisect BC In e, and thereon raife the Perpendicular ed, until it meet B/ in d, on which, as a Center, deferibe the Arch B«C; then will every Angle form’d in the Arch Bgn fC, whofe Sides pafs through the Ends of the Chord Line BC, be equal to the given Angle A. So the Angles B^C, B»C, and B/C, are each equal to the given Angle A. DEMONSTRATION. [ 4-0 The Principles of Geometry. DEMONSTRATION. The Angles liCB, and C B d, being equal, the Lines d C and dB are equal; becaufe d is the Center, on which their Ends B C are terminated by the Arch. Now the Angle bBf being right, the Line h B toucheth the Circle in the Point B; therefore the Angle, which the Segment EgnfC com- prehendetb, as the Angle B/C is equal to the Angle hBC, or to the given Angle A. The well underftanding of this Problem will be of admirable Ufe to all Workmen that work by a Square; for thereby they may molt readily make, or prove them. P. Fray bora can that lie ? I don't fee that in this Scheme or Diagram there be one Square or Right Angle, more than thofe made by the Line d e on the given Chord Line BC, which 1 believe is of [mail Ufe for that Purpofe, more than what has been already delivered: Therefore pray explain it to me- jVl. I w ILL : Through the Center d draw a Diameter, as k o, and let k i be drawn at Right Angles to ko, as fuppofing the given Angle to be a Right Angle; then I fay, for the fame Reafon, as the Angles B^C B#C, andD/C, were each equal to the Angle CBi; fo wiil every Angle form’d on the Extremes of the Diameter ko, in the Semicircle klo, be a right Angle alfo; as klo kmo, or any other two Lines whatiuever. Therefore to prove if a Square be truly made, deferibe a Semicin le. and therein apply the Square, and if the two Sides thereof pals by the fnds of the Diameter, and the angular Point of the Square touch theCircumference of the Circle in any Part at the fame Time, the Square is truly made; otherwife ’tis falfe. P R O B L E M XXXIL Fig. XLI. A Circle being given to cut off a Segment that frail contain an Angle given-, Let bade be the given Circle, and h the given Angle. Practice, (i.) Draw the Semidiameter ce and eg at right Angles thereto. (c.) Make the Angle gef equal to the given Angle h, and continue ef to d. (q.) All the Angles made upon e d, in the Segment ebad, will be each equal to the given Angle h, as required. PROBLEM XXXIII. Fig. XLII. From a given Point, to draw a Chord Line in a given Circle that Jhall bt equal to a given Line ; Let D be the given Point, A B the given Line, and gIf the given Circle. Practice, (i.) Take the given Line A B in your Compafles, and fet on any Part of the Circumference, as from f to g, and draw the Line f k in¬ finitely through the Points gf. (a.) From The ‘Principles of Geometry. .(?■), Fr ° s 5 . the S' ven Point D . draw the Line D e; and on the Center e, with the Radius cD, defcnbe the Arch Dr, cutting fk in h. (^.) 1 ake h J in your Compaffes, and fetting one Foot in the given Point , with the other interfeft the Circle in » by the Arch mm, and draw the me D »; then will the Chord Line In, be equal to the given Line AB, as required. ° P R O B L E M XXXIV. Fig. XI .m To cut off, om a Line any Part required. It is required, to cut off two ninth Parts of the Line xy. Practice, (i.) Draw two right Lines, making any Angle at Pleafure as h 4 > 5 > 6 > 7 > ^3 9 , and let ad be equal to feven of the fame, and ac be equal to xy. (a.) Draw the Line c«, and db parallel thereto; then will ab be equal to leven .Ninths or ac, or xy. ■* DEMONS'!' RATION. In the Triangle cae, bd being parallel fon of ad to d e, as of it A to Ac; and as therefore ab fhall contain feven Ninths of to ce, there will be the fame Rea- a d contains feven Ninths of a e, a c, or xy. Which was to be proved. LECTURE III. By Z-Z— Plate IV. On the Generation of Regular Geometrical Figures. M. HP he next Part of Geometry, with which you are to be acquainted, JL is the Generation of regular fuperficial Figures, wherein you will be agreeably entertain J, and enabled to pafs through the Conffruflions there¬ of in the next Ledlure with equal Pleafure. The Figures to be here con- iidered, are the various Rinds of Triangles, the Square, the Parallelogram, the Khombus, the Rhomboides, and the Polygons. P. Arc there not other regular Geometrical Figures ? M. Yes: There are alfo the Ellipfis, Parabola, and Hyperbola. But I ihall explain them hereafter, in my Le&ure upon Conick Se&ions. P. How is an equilateral Triangle generated ? ww\ BY u three -M qUa ‘ "S ht Unes > h;lvin S their Ends applied to each other, whereby they will conftitute or generate an equilateral Triangle. And fo in like manner if three unequal Lines, of which any two, being taken together are greater than the third, have their Ends applied together,' they Ihall gene- rate aScalenum Triangle. And alio, if of three Lines there be two equal and the third unequal, they will generate an Ifufeles Triangle. Mi P. 7 14 ' T. I uKdi-Jlamt yodperfeSIf **%> Pry proceed to tie Square. ?,i. I will. The Geometrical Square is generated as follows : I F r the Lines at. ag, Fig. XLIV. form aright Angle, and „r Ire equal to a '■ ti en if the Line.it b he removed forward unto c, lu as for its End a to be always on the Line ag, and allies Parts move equally at the fame time it Willi i Mo n, have paffed through a fuperficial Space equal to ah it, and fince that at i- equal to at, and all the Parts of the Line at moved equal¬ ly in the lame time, therefore at, in the Place of cd, will remain perpend.- ailar to ag, as at firtt; and the'Superficies defenbed by its Motion, will be a geometrical Square. Now from hence you lee, that as a Line is generated by the Motion of a Point, on a plain Surface or Superficies ; fo in like manner tms fquare Super- ficies is generated by the Motion of a Line. The Parallelogram atgb, is alfo generated in the fame manner ; for ,f the Line at, after ’tis moved to cd, paf on with equal Motion m all its Parts further to e I, and thence to g b, it will then have paffed through a Space, whofe Length is ag, and Breadth at, and generate the Oblong, or Parallelo- gram abgh. The Rhombus, Fig.'S. LV. is alfo generated in the fame manner; for if the Line at, making an Angle of 45 De S ref * wh the Line tj, move to¬ wards f with equal Motion in all its Parts, and the End b always in the Line b I it will generate the Rhombus adtc, when the Point b, be arrived at c ; fuppofing t c equal to ab . And if ab, when moved to dc, move on to ef, it will deferibe the Rhomboides a, e, b,f As three equal right Lines, having their Ends applied together geperate an equilateral I nungle, fo fix equilateral Triangles having their Sides applied regularly together, generate a Hexagon ; that is, if the equilateral Triangles a 7 >, c, it, e, f. Fig. XLVI. be applied together, as g, b, i, k, /, m, they will com¬ plete the Hexagon n, o, p,q,r, s. The other Polygons, namely, the Pentagon, Septagon, OQagon,Nonagon, Decagon, Vc. are generated by the Applications of Five, Seven, Eight, Nine, Ten, itfe. Ifofceles Triangles, as in Fig. XLVII. having their Sides propor¬ tioned, as will be hereafter taught. These being feverally well underftood, will render the following Leflure very eafy. LECTURE i 4 3 The ‘Principles of Geometry. L ECTU RE IV. By X-X- Plate IV. On the Geometrical Conjlruclion of Superficial Figur es. PROBLEM I. Fig. XLYIIt. To defcribe an Ifofceles Triangle, that fhall have two Sides each-equal to the given Line a a, and the third Side equal to the given Line bb. Practice Make fg equal to bb\ and with the Length of aa in your CotnpaffeSj on f defcribe the Arch cc; and with the fame Opening on £, the Arch dd, interfefling cc in e. Draw the right Lines ef, eg, and they complete the Ifolceles Triangle required. PROBLEM II. Fig. XLIX. To detcribe a ScalemmTriangle, wbofe three Sides /hall be equal to three given L, net, of which any two taken together he greater than the third , as a a, bb, cc. Practice. Make gh equal to a a, and on g, with the Length of bb, de¬ fcribe the Arch ff; and on h, with the Length cc, defcribe the Arch cc, in, terfefling ff in d. Draw the right Lines dg, db, and they complete the Scalenum Triangle required. PROBLEM III. Fig. L. To defcribe a Geometrical Square, wbofe Sides {ball be refpettively equal to a given Line, as a a. Pr iCTicE. Firfl, Make fg equal to a a, and on g raife the Perpendicular ,C which make equal to a a. Secondly, With the Length a a, on e deicribe the Arch bb, and on /', with the fame Opening, defcribe the Arch cc, m- terfeffino bb in d. Draw the right Lines de, df, and they complete the Geometrical Square de fg, whole Sides are feverally equal to the given Line a a, as required. PROBLEM IV. Fig. LI. To defcribe a Parallelogram, wbofe Length fhall be equal to a a, and Breadth to bb. Practice. Firfl, Make gh equal to aa, and on h raife the Perpendicular hf- which make equal to bb. Secondly, On g, with the Length it, defcribe the’Arch tid; and on f, with the Length a a, defcribe the Arch cc, inter- fefting dd in e. Thirdly, Draw the right Lines ef, eg, and they complete the Parallelogram required. PROBLEM V. Fig. LII. To defcribe a Rhombus, wbofe Sides fhall be feverally equal to a given Line a a. PRACTICE. Make bf equal to aa, and on f with the Length bf defcribe the Arch bcde, and fet b] from b to c, and from thence to d Lajtly, Draw the right Lines be, cd, and df, and they will complete the Rhombus required. PROBLEM '44 The Principles of Geometry. PROBLEM VI. Fig. LIII. To deferibe a Rhomboides, whofe tbngefl Sides Jbatt be each equal to the given Line a a, the fhorteft to bb; and acute Angles each equal to the given Angle h. Practice. Firfl, Make kg equal to a a, and at k make an Angle equal to the given Angle B, and make k h equal to b b. Secondly , On g, with the Length kh, deferibe the Arch ff; and on h, with the Length kg, deferibe the Arch ee, interfering ff in i. Thirdly, Draw the right Lines hi, ig, and they complete the Rhomboides required. PROBLEM VII. Fig. LIV. To deferibe a Trapezium, wboje four Sides J.ball be equal to four given Lines , (as a a, b b, cc, dd,) with the Angle made by the Sides a a and cc, equal to an Angle given, as the Angle e. Practice. Firfl, Make// equal to the greateft Line dd-, and at f make an Angle equal to the given Angle e, alfo make f'g equal to cc. Secondly On g. with the Length b b, deferibe the Arch kk-, and on /, with the Length a a, deferibe the Arch hh, interfering kk in/. Thirdly , Draw the right Lines A 4 ai -d i l, and they will complete the Traperzium required. PROBLEM VIII. Fig. LV. To make a Geometrical Square, having the Difference between its Side and the Diagonal given. Let an be the Difference given. Practice. Firfl On n erefl the Perpendicular nm , which make equal to an-, and by the Points a, m, draw the Line ax of any Length. Secondly, Upon the Point m, with the Radius mn, deferibe the Arch nx, cutting the’ Line ax in x; then will ax be the Side of the Square required. Thirdly, On x, erecl the Perpendicular xo equal to ax, and draw' a n o, which is a’ Diagonal; of which, no is equal to the Side xo, or ax; and an, the Re- mainer, is the Difference that was given. Fourthly, Bifefl a o in ^ and on s; as a Center, with the Radius a'r, deferibe the Circle, and complete the’ Square a x o r therein, as required. PROBLEM IX. Fig. LVI. To deferibe an Oval of any njfignd Length. Let the given Length be ax. Practice. Firfl , Divide the Length ax into three equal Part?, at e and f; upon which, as two Centers, with the Radius ae , deferibe the Circles abejng, and ecdxbn, interfering each other in c and n. Secondly, From the Points e and n, through the Points e and f, draw the Lines nfd, neb, t\h, ceg; then on the Centers cn, with the Radius nb, deferibe the Arches bdstnigh, which will complete the Oval required. The ‘Principles of Geometry. Note, This, «d .< /- »*■**» * TX’XVtZ, contained, in it, ns will be hereafter demonjlrated in us Place. PROBLEM X. Fig. LVII. To defcribe an Oval of any given Length, without RefpeB being had to its Vreadth, 1 after a different Manner from the foregoing. Let Ik be the given Length. Trial Z,lfa„ h, through (, ». right Anglo f ““ */ *■»“ *» complete the Oval as required. PROBLEM XI. Fig. LVIII. An Oval being given, to find its Center and two Diameters', Let an eh be the given Oval. v - n r>aw at pleafure two parallel Lines in any Part of the 2* Vlstli r if rS3 h ft s:; ■s «o d ": “S"S” d£.«i 4 i- .in r Lve difcovered the Center p, and two Diameters an and nh, as reqmred. PROBLEM XII. Fig. L 1 X. Plate V. To defcribe an Oval of any Length and Breadth required. Let «, be the given Length, and bb the given Breadth. PRACTICE Firfl, Make cd equal to a a, which1 bifefl: in g by the Per- sss it. V”i i .id« »/. «.a d» ...».,ci. r i T-g'o n ‘"“i Breadth will be equal to the given Lines a a and bb, q N n Numb. XI. c * Note, 1 4<5 The Principles of Geometry. 'Note, Since that Ovals are compofed by Arches of Circles, therefore you may on the fame Centers defcribe other Ovals concentrick to the frit as the Oval vxmg, which is defcribed on the Centers h, i, k, l, at the Dildance of jy ; and fo m like manner any other. P. But, Sir, fuppofe I am to defcribe an Oval in a Place that is not broader than the given Breadth of the Oval, and confeqttently cannot have an Opportunity of find- fittmoZZ? k ’ ’ ’ liewithoM th ' sivenL,mils; Pr . ayh ™ muJl 1 ^ crihe M. A s following : By three Methods. METHOD I. Fig. LX. Let the given Length be the right Line a, and given Breadth the right Line b. Practice. Firft, Make cn equal to the right Line *, and bifefl it in g, and draw ef at right Angles thereto, and equal to the given Breadth b. Se¬ condly Take Half the Jongeft Diameter g„ in your Compafl'es, and on / i n - terfeff cn m dh ; which Points are called the Focus Points of the Oval or El- hpfis. In which fix two Pins, Nails or Stakes, if the Ellipfis be large, and he fln n eStended at itS Bendl "g’ 11,311 exa< % reach to a Bkck Te fp fl me r r 4 at f °r " hichIaI,e & fixed, fhall, with rhe Ov I PM en r ’ ° e ' 1 ‘? p T d P er P endlcuIar, y thereto, trace or defcribe tile Oval, or Ellipfis, as required. N°te, That the Diameters of an Oval, or Ellipfis, are diflinguijbed from each o et by the Fames of Conjugate and Tranfvcrfe, that is, the longefl is called the Conjugate, and the Jhortefi the Tranfvcrfe Diameter. S "METHOD II. Fig. LX. Let the given Length and Breadth be the fame , and dram the treo Diameters at right Angles as before. Practice. Make a Ruler as p q, whofe Length muft be equal to the greater Semldiamer eg or gn; upon which fet off the leffer Semidiameter eg tl/iy f ZZ/ “ m ' i / 1S f one ’, a PP‘y Ruler in fuch a Manner upon he Diameters cn and ef, that the Point » paffing along the Diameter cn, the End q may always be in the Diameter ef; then moving the End p to e thence to c, thence to f thence to n, and thence to p, the Point from whence’ you moved firft, the End p will have defcribed the true Curvature of the Elljplis, as required. METHOD III. Fig. LXI. j r 'i^‘ Lines f, g, be the given Diameters. Practice Firft, Make a Parallelogram, as need, whofe Length cb fhall ue equal to/, and Breadth ed equal to g, and bifeft the Sides and Ends there¬ of m the Points x,h, nm Secondly, Divide k and cn, each into any Num¬ ber of eqnal Parts as 6, 8, to la, , 4> ,6,^. the more the better. In this Example, I have divided each into eight equal Parts, as k at the Points ’ ’’ 4 > 5 » 6 > 75 and at the Points 7, 6, 5, 4, j, a, ,, Thirdly , Draw the right Lines *7; 1,6; a,5; 5,4; 4, 3 i 5 > 2 i °»1; -jn Which Lines will The 'Principles of Geometry. *47 will form one quarter Part of the Oval. Laflly, The other three Parts being formed in the like Manner, will complete the Oval, as required. P. This is a very eajy and delightful Method, being perform’d without any Re¬ gard to its Center , or Focus Points, as in the loregoing ; and I imagine muft be of very great Vfe at fome 1 imes , when an Elliptical or Ovallar Wall may be re¬ quired to be built, to inclojing a Wood, &c. where there’s no Pofjibility of making ufe of Lines, &c. as in the ^ preceding Methods. .Vf. ’Tts true; you obferve very juftly : In iuch Cafes, 'cis the only Way yet made publick. P. Pray, Sir, is it an Invention of your own ? M. No : It was fir ft invented by Mr. William Hafpeny , alias Hoare, lately of Richmond in Surry, Carpenter. P. Pray is this Method applicable to any other curved Figures ? M. Yes : Circles may be alfo thus deferibed, as Fig. LXV\ Which is done by firll making a geometrical Square, as abfh, whofe bides ihall be each equal to the Diameter of the given Circle; and then each being bifefted in deeg, and divided into any Number of equal Parts, as before in the Parallelogram, and Lines drawn from and to the refpeftive Points of Divifion, they will form the Curvature or Circumference of the Circle, as required. But you are to conlider, that the Formation of the Circle depends upon the Angles of a geometrical Square, being truly right-angled ; for were the Angles not to be right-angled, the Figure form’d would be an Oval, and not a Circle, as you may obferve in Fig. LX 1 I. where the Sides of the Rhombus aefb. are equal to tile Sides of the geometrical Square abjh, and are befefl- ed, divided, and Lines drawn, in the very fame Manner; but the Angles there¬ of not being right-angled, the Lines do therefore form an Oval, and not a Circle, as before noted. The like is to be obferved in the Rhomboides acjg, Fig. LXIII. whofe Length ac, and End eg are equal to ae and ed, the Length and Breadth of the Parallelogram aced. Fig. LXI. For by the Obliquity of the Angles of the Rhomboides, the Length xx of the Oval therein deferibed becomes greater, and Breadth Idler than xm and bn of the Oval, F’ig.LXI. Hence ’tis plain, that unlefs the Angles of a Parallelogram are truly right- augled, the Oval within it cannot be made equal to the given Length and Breadth, as required. P. I fee it plainly : Pray, Sir, proceed to other Works that are capable of being perform’d by this Method. M. I will, in fome few more; and then refer the Remainder unto my Leflore on t he Generation of Arches and Groins, wherein it will be more largely handled. What I fhall here further take notice of for the prefent, is, Fiift, The Manner of deferibing Ovals, that are broader at one End than at the other, as Fig. LXVI, commonly called Egg-Ovals. And Laflly, The Manner of deferibing a regulaisCurve within an irregular Trapezium; as bd, ej, with¬ in the Trapezium aegh. Fig. LXIV. w % PROBLEM The Principles of Geometry. 148 PROBLEM XIII. Fig. LXVI. To defcribe an Egg-Oval to any afjignd Length. Let xx be the given Length. Practice. Firfl, Divide xx into three equal Parts at w . and make the Parallelogram ache, whofe Length (hall be equal to xx, and Breadth to xw. Secondly , Make ab, hf, each equal to xz., and bifeft ah in g, and ce in d-, then dividing ab, and be, each into the fame Number of equal Parts; alio cd, de, ef, fh, hg, and ga, in like manner; and from the refpeftive Points of Divifion, draw right Lines as before taught, you will form the Egg-Oval as required. PROBLEM XIV. Fig. LXIV. To inferibe a regular irregular Curve within an irregular Trapezium, as ebdf within aegh. Pr act ICE. Bifefl ac in b, ch in d, gl) in /, and ag in e; and divide ab, be, cd, dh, hf, fg, ge, and ea, each into any and the fame Number of equal Parts ; and then drawing right I ines from the refpefiive Points, they will form the Curve required. I shall now proceed to fhew you, How to determine or find a fufficient Number of Points, through which you may trace the Circumference of any Circle or Curve of any EHipfis required, without any Regard being had to their Centers, or Focus Points, for deferibing the fame. PROBLEM XV. Fig. LXXIII. The Diameter of a Circle being given, to find a fufficient Number of Points, through which the Circumference thereof may be traced. '.V Let the right Line fh be the given Diameter. Practice. Firfl, Make ac equal to fh, and b 1 le fit it in e. Secondly , Draw bd through e, at right Angles to ac, and make eb, ed, each equal to half ac. Thirdly, Draw a right Line at Pleafure, as ac, Fig. LXVIII. of any Length ; which bifefit in d, and thereon raife the perpendicular Line db, which make equal to dc, and on d deferibe the Semidiameter a b c, whofe Circumference you are to divide into 1 So Degrees, that is, each Qua¬ drant thereof into 90 Degrees, as has been already taught. In this Example, I fhall only divide to every 10 Degrees, as exhibited in the Figure. The Circumference being thus divided, draw the chord Lines 10, to; co, ao; 30,50; 40,40; 50,50; 60,60; 70,70; and 80,80; which will icterfeft and divide the Semidiameter b d unequally in the Points xxx, isle. Fourthly, by Problem XXIV. Lefiture II. divide the Semidiameters ae, be, ec, cd, each in the fame Proportion, as bd, Fig. LX\ III. as at the Points OxS., CT. through which draw right Lines parallel to ac, and unto bd, that will interfefl each other in nnn, £sV. which arc the Points through which a Line being traced, will be the Circumference of the Circle required. N. B. The Semidiameter of any Circle being fo divided, is called a Line of Sines, whofe Ufes will be largely handled in plain Trigonometry. J ' P. Pray, The ‘Principles of Geometry. i 49 P. Pray, Sir, can an Oval or Ellipfis be thus traced. ? It. Yes : As follows. PROBLEM XVI. Fig. LXVII. The conjugate and tranjverje Diameters of an Ellipfis being given, to find a fttffi- cicnt Number of Points, through which the Curve thereof will pafs. Let ah and ce be the Diameters given, interfering each other at Right An- gles m n) and let cn be equal to ne, and an equal to nb. Practice. Divide the Semidiameters cn and ne, alfo the Semidiameters a n and n b, each in the lame Proportion as b d. Fig. LXVIII. and through thofe Diviftons draw right Lines parallel to ce and ab, they will intericct each other in the Points xxx, i$c. through which a Line being traced, will be the Circumference or Curve of the EllipGs required. In the fame Manner, you may defcribe the curve Line of an Ellipfis within a Rhombus, ebjh within acgi, as Pig. LXIX. where the Semidiame¬ ters cd, db, df, dh, being divided into fuch Proportion as bd, Fig. LXVIII. and right Lines being drawn through the fame, parallel to themfelves, (as in the Figure,) the Interfeclions thereof will produce Points, through which the Curve of the Ellipfis will pafs, that will be equal to the Curve of the Ellipfis dbeg form’d in the Rhombus acfh, Fig. LXII. Note, Half the Ellipfis, Fig. LXII. that is dbe, may be confidered as a Ram¬ pant Arch ; as aljo may e bf, in Fig. LXIX. But more of this in its proper Place. Thus have I largely explain’d to you the various Methods to defcribe an Oval or Ellipfis: I lhall now proceed to the Manner of deferibing regular Poly¬ gons. PROBLEM XVII. Fig. LXX. A Circle being given, to find the Side of an Equilateral Triangle, Geometrical Squat e, Pentagon , Hexagon, Septagon, OCtagon, Nonagon, and Decagon, that may be made in its Circumference. Let a dig be the given Circle, and h its Center. Practice. Firjl, Draw the Diameter ai, and making ac and am each equal to ah, draw cm, which is the Side of an equilateral Triangle that may be deferibed therein; therefore on c, Fig. LXXI. with the Radius ah, defcribe the Circle abd; and then alfuming a Point in any Part of its Circumference, as at d-, from ther.ce fet off the Length cm to a, and from d to b, and draw the Lines da, db, they will complete the equilateral Triangle within the Cir¬ cle, as required. Secondly , Draw dg through the Center h at right Angles to ai, and draw ad, which is the Side of a geometrical Square; therefore on e. Fig. LXII. with the Radius ah, defcribe the Circle bacd-, and then alfuming a Point in any Part of its Circumference, as at d, from thence let off the Length ad to b, thence to a, and thence to c ; and drawing the Lines db, dc, ab, ac, they will complete the Geometrical Square, as required. Thirdly, On f, in the Line a i- with the Radius f d, defcribe the Arch d e ; and draw the Line d e, which is the Side of a Pentagon; therefore on/, Fig.LXXIV. with the Radius ah, O o defcribe The 'Principles of G k o m e t r y 150 defcribe the Arch abdee ; and then alTuming a Point in any Part of its Cir¬ cumference, as at e from thenoe fet off de (in Fig. LXX.) to c, thence to a, thence to b, and thence to d ; and drawing the right Lines ce, c a, ab, b d, and de, they will complete the Pentagon, as required. Fourthly, The Semi- diameter ah, is the Side of a Hexagon ; therefore on g. Fig. LXXV. with the Radius ah, defcribe the Circle abode f; and then affuming a Point in any Part of its Circumference, as at f, from thence fet off ah to e, thence to c, thence to a, thence to b, and thence to d; and then drawing the right Lines J e, e c, c a, ab, df, they will complete the Hexagon, as re¬ quired. Fifthly, Half cm that is cf, is the Side of a Septagon; therefore on h, Fig. LXXVI. with the Radius ah, defcribe the Circle cabedgf; and then affuming a Point in any Part of its Circumference, asar^, from thence fet off cf to d, thence to c, thence to a, thence to b, theDce to e, and thence to /; and then drawing the Lines gd, dc, ca, ab, be, ef, and fg, they will complete the Septagon, as required. Sixthly, Divide the Arch abd into two equal Parts at b, and draw ab, which is the Side of an Oflagon; theref re on i, Fig.LXXVII. with the Radius ah b feribe the Circle abciefgh ; and then afluming a Point in any Part of its Circumference, as at g, from thence "let off ab to b, thence to a, thence to b, thence to c, thence to d, thence to e, thence to /: and then drawing the right Lines gh, ha, ab, be, cd, de, ef, and fg, they will complete the Oflagon, as required. Seventhly, Divide the Arch cam into three equal Parts, then will one third Part thereof, as xm, he the Side of a Nonage n ; therefore on l, Fig. LXXVIJI. with the Radius ah. defcribe the Circle a be defghi; and then alTuming a Point in any Part of its Circumference, as at a, from thence fet off xm to h, thence to i, thence to a, thence to b, thence to c. thence to d, thence to c, thence to /, and thence to g ; and then drawing the right Lines gh, hi, i a, ab, be, cd, de, ef and Jg, they will complete the Nonagon, as required. Eighthly, The Diftance he, or Half de, is the Side of a Decagon; therefore on/. Fig. LXXIX. with the Radius ah, defcribe the Circle a be defg hi k; and then affirming a Point in any Part of its Circumference, as at i, from thence fet off he to k, thence to a, thence to b, thence to c, thence to d, thence to e, thence to /, thence to go and thence to h; and then drawing the right Lines ik, ka, ab, be, cd, de, ef, jg, and gh, they will complete the Deca¬ gon required. Thus have I ffhewn you how to make any Polygon within the Circumfe¬ rence of a given Circle : 1 Thall now proceed to fhew you how to make them feverally, having a Side only given. PROBLEM XVIII. Fig. LXXX. To make a Regular Pentagon, whofe Sides fball be each equal to a given Line, as g £ Practice. Fivfl, On the Points g and /, with the Radius gf, defcribe the Arches ge and jd, interfeQing in n. Secondly, Bifetff.f/' in x, and draw xn. Thirdly, Divide the Arch nf into two equal Parts at k-, alfo divide nk into three equal Parts at the Points h, i ; then making ng equal to nh, ^ will be the Center of the Pentagon. Fourthly, On sc, with the Radius f, defcribe the Circle fgabc, and fet gf from g to a from a to b, and from b to c; and then drawing the right Lines ga, ab, be, and ef, they will complete the Pentagon, as required. PROBLEM The Principles of Geometry. I 5 I PROBLEM XIX. Fig. LXXXI. To make a Regular Hexagon, rvhofe Sides Jhall be each equal to a giver. Line, as hg. Practice. Firfl, On the Points hg, with the Radius hg, deferibe the Arches bf and eg, interfering each other in «; which is the Center of the Hexagon. Secondly, On n, with the Radius ng, deferibe the Circle abedhg, and fet hg from h to a, from a to b, from b to c, from c to d, and from d co g; and then drawing the right Lines ha, ab, be, cd, and dg, they will complete the Hexagon, as required. PROBLEM XX. Fig. LXXXII. To make a Regular Septagon, rvhofe Sides {hall be each equal to a given Line, as y £ Practice. Firfl, On the Points yf, with the Radius yf, deferibe the Arches y h and fg, interfering each other in s. Secondly, Bife£G/ in z, and draw k z through s. Thirdly, Divide the Arch sf into two equal Parts at o; alfo divide so into three equal Parts at the Points nm, and make sx equal torn; then will the Point x be the Center of the Septagon, Fourthly, On x, with the Radius xf, deferibe the Circle abedefy; and let yf from y to a, from a to b, from b to c, from c to cl and from d to t; and then drawing the right Lines y a, ab, be, cd, dc, and cj, they will complete the Septa¬ gon required. PROBLEM XXI. Fig. LXXX1II. To make a Regular Octagon, rvhofe Sides {hall be each equal to a given Line, as p 1. Practice. Firfl, On the Points pi, with the Radius pi, deferibe the Arches ph and li, interfering each other in Secondly BifefG / in o, and draw of through z< Thirdly, Divide the Arch zl into two equal Parts in x ; alfo xz into three equalParts at the Points mn, and make zk equal to z»; then will k be the Center of the Oflagon. Fourthly, On k, with the Radius kl, deferibe the Circle Ipabcdeg, and fet pi from p to a, from a to b, from b to c, from c to d, from d to e, from e to g, and from g to /; and then drawing the right Dines pa, ab, be, cd, de, eg, and gl, they will complete the Octagon, as required. PROBLEM XXII. Fig. LXXXIV. To make a Regular Nonagon, rvhofe Sides fljall be each equal to a given Line, as e f. Practice. Firfl, On the Points ef, with the Radius ef, deferibe the Ar¬ ches ec and If, interfering each other in a. Secondly, Bifer ef in h, and draw bh through a. Thirdly, Divide the Arch af into two equal Parts at z, and make ad equal to az~, then will d be the Center of the Nonagon. Fourthly, On d, with the Radius i if, deferibe the Circle fegiklmnop, and fet ef from c to g, from g to i, from i to k, from k to I, from / to m, from m to n, from n to o, and from o to/; and drawing the right Lines eg, gi, ik, kl, Im, mn, no, and of, they will complete the Nonagon, as required. PROBLEM ) I i -2 The Principles of Geometry. PROBLEM XXIII. Fig. LXXXV. Jo make a Regular Decagon ,. n'bofe Sides Jhall be each equal to a given *• -Line, as ep. Practice. Firfl, On the Points ep, with the Radius defcribe the Ar¬ ches cb and cp, interfering in a. Secondly, Bifefl ep in d , and draw f d through a. Thirdly, Divide the Arch ap in ^; alfo divide zip into three equal Parts and make a e equal to ax, (which is azi and l of zip, equal to J of ap’,) then will e be the Center of the Decagon. Fourthly, One, with the Radius ep, defcribe the Circle pcnghiklmo, and fet up ep from e to n, from n to g, from g to b, from b to /, from i to k, from k to l, from / to m, from m to o , and from o to p •, and drawing the Lines en, ng, gb, hi, ik, kl,lm, mo, and op, they will complete the Decagon, as required. PROBLH M XXIV. Fig. LXXXVI. To make a Regular Undecagon, rrhofe Sides JJjall be each equal to a given Line, as d e. Practice. Firfl, On the Points de, with the Radius de, defcribe the Ar¬ ches db and ce, interfering each other in a. Secondly, Bifeft de in x, and through the Point a draw the Line fx. Thirdly, Divide the Arch ae in h; al¬ fo he into three equal Parts at the Points no; and then making ag equal to a o, the Foint g will be the Center of the Undecagon. Fourthly, On g, with the Radius ge, defcribe the Circle edikl mpqrst; and fet de from d to i, from i to k, from k to l, from / to m, from m to o, and from o top ; and drawing the Lines i d, ik, kl, l m, mp, pq, qr, r s, st, and te, they will complete the Undecagon, as required. PROBLEM XXV. Fig. LXXXVII. To make a Regular Duodecagon, ivbofe Sides fJoall be each equal to a given Line , as g f. Practice. Firfl, On the Points g,f, with the Radius gf, defcribe the Arches gb and cf, interfering each other in a. Secondly, Bifin x, and through e draw the right Line ex. Thirdly, Make ad equal to af; then will d be the Center of the Duodecagon. Fourthly, On d, with the Radius dj, defcribe the Circle fghiklmnopqr; and fet gj from g to h, from h to i, from i to k, from k to /, from / to m, from m to n , from n to o, from o to p, from p to q, from q to r, and from r to f; and then the right Lines hg, hi, i k, kl, Im, mn, no,op,pq, qr, and rf, being drawn, will complete the Duo¬ decagon required. PROBLEM XXVI. Fig. LXXXVIII. To defcribe all Manner of Polygons, rphofe Sides are required to be equal unto a given Line, as a e. ''x the Extremes of the given Line, as a and e, with the Length thereof, defcribe Arches, as af, and be, interfering in 6. Bife£l the given Line in n, and through £ draw the Perpendicular nd at Pleafurc, and divide the Arch g e into The Principles of Geometry. [t ?3 into fix equal Parts at the Points 1,2, 3,4., 5. This being done, you may de- fcribe any Polygon as follows. Firft, Make the Diftance 6,5, on the Line nd, equal to ;of the Arch 6 e; then will the Point 5 be the Center of a Pentagon. Secondly , The Points a, 6, e, being equi-diftant, therefore the Point 6 is the Center of a Hexagon. Thirdly, Make the Divifions on the Line nd, from 6 to 7, 8, 9, 10, 1 t,'i 2, Isfc. each equal to , of the Arch 6e; and they will be the Centers of fo many Polygons, whofe Number of Sides will be always equal to the Number of Divifions that the Center thereof is from the Point e. So the Center of a 'Pentagon r 5" Hexagon 6 Septagon 7 O&agon 8 i Nonagon is at the Point < Decagon IO j Undecagon I I ^Duodecagon y 1 u And Polygons of 'G 14 '4 '5 ' 5 | 16 .6 >7 *7 ) 18 Sides, have their Centers 18 [ s *9 at the Points * 9 * 20 20 21 2 I 22 22 7 S I24J chJ ■ on the Line n d. Now if on thefe Centers you defcribe Circles, whofe Radius’s are equal to the Diftances contained between them, and either of the Ends of the given Line a or e, you may, as before taught, fee off therein, the Sides of the Polygon, or Polygons, required. If you confider the Operations of the eight foregoing Problems, you can’t help feeing the Reafon thereof, which is very plain and eafy to under¬ hand. PROBLEM XXVII. Fig. LXXXIX. To defcribe any regular Polygon by Help of the Scale of Chords. Before we can proceed in thele Operations, we muff firfl difeover, how many Degrees and Minutes are contained in the Side of the Polygon we de- defign to delcribe. As for Example, I would defcribe a Pentagon ; which confining of five Sides, muff there¬ fore therefore divide 360, the Number of Degrees in the Circumference of a Circle, by 5, the Number of Sides in the Pentagon, and the Quotient 72, is the Number of Degrees that are contained in each Side thereof. h P Now, S 1 54 The Principles of Geometry. Now to defcribe the Pentagon Fig. XG. With 6c Degree? of your Scale of Chords, defcribe a Circle, as cdbef, and in its Circumference alTiime a Point, as at /; then taking 7 2 Degrees in your Compares, and fetting them from / to b, from b to c, from c to d , from d to e, and from e to /, and drawing the Lines bj, be, cd, de, and ef , they will compleat a regular Pen¬ tagon ns required. Nowfince that we mufl: firft find the Number of Degrees that are contained in the Sides of thole Polygons we defign to reprefent, therefore obferve, 3 Triangle ico 4 Geom. Square yo 5 Pentagon 77 6 Hexagon 60 7 the Number Septagon the Quo¬ 5 1 8 ofSides in a O&agon tient is 45 9 Nonagon 40 10 Decagon 11 Undecagon 3 a T ^ V. 1 - . Duodecagon 30 Which are the Number of Degrees contained in each of their refpe&ive Sides. By this Method you may inftantly deferibeany Polygon in any Circle having its Diameter given, although you have but one Scale of Chords, as follows: Sui’Tose I am to defcribe a Pentagon in the Circle hi klg, whofe Diame¬ ter is much lels than the Diameter of the Circle bedej, whofe Radius I fup- pofe to be equal to the Radius of Scale of my Chords, by which I work. Practice. Firfl , With 60 Degrees of my Scale of Chords, I defcribe the Circle bedef, and therein compleat the Pentagon bedef Secondly , Draw the Lines ab, ac, ad,, a e, and af and on a, with the Radius of the given Circle, defcribe the Circle hiklg, which will interfefl the Lines ab in h, ac in i, ad in k, ae in I, af in g; and then drawing the Liner hi, ik, kl, l g, gh, they will compleat the Pentagon, as required. P. Very well. Sir, I fee the Reafons thereof; and can after that Manner per¬ form any Polygon: But, fuppofe that the given Diameter of the Circle, in which I am to make my Polygon, be greater than the Circle bedef, that's made by the Radius oj my Scale of Chords ; Pray how mufl I proceed at fuch Times ? M. As following : Suppofe the Radius of the fmall Circle hiklg to be equal to the Radius of your Scale of Chords, and that you were required to make a Pentagon in the great Circle b c def■ then having firfl: compleated the Pentagon in the Circle bedef, as before-taught, continue out the Lines ab, ac, ad, ae, af, at Pleafure; and on a, with the Radius of the given Circle, as af, defcribe a Circle, which will interfefl the Lines a 6, ac, a d, and af in the Points bedef, and then drawing the Lines be, cd, de, ef, fb, they will compleat the Pentagon as required. And fo in like manner may all other Kinds of Polygons be deferibed. P. This Method, I apprehend, will do, when their Diameters are given ; but when their Sides only are given , this Method or Rule will not do. Pray how mufl I proceed at fuch Times, to defcribe Polygons with the Scale of Chords. M. Before The 'Principles of Geometry. 155 M. Before you can enter upon the defcribing ofPoiygons by this Method, you mull firlt obferve, and always remember, that the three Angles of every Kind of Triangle, are exadtly equal to 180 Degrees, (the Reafon of which will be hereafter fliewn in its Place;) and therefore, if from a Triangle, as a/e, you fubtrafl. the Angle, e af, which contains 72 Degrees, (the Quan¬ tity of the Arch fe,) the Remains are 108; which are the Number of De¬ grees contained in the other two Angles a/e, and fea: And whereas the Sides n c and a f being equal, therefore the Angles a] e, and a ef are equal, each containing 54 Degrees, the half Part of 108. And fince that every of the five Triangles afe, afb, a be, acd , ad e, are all equal to one another, therefore the Angle baj is equal tothe Angle bfe; that is, each 54. Degrees, and both being taken together, to 10S Degrees, which is the Quantity of the Angle bfe. Now to the Purpofe: Let a b. Fig. XCI. be the given Side of a Pentagon. Practice (i.) On each End of ab, with 60 Degrees of your Scale of Chords, defcribe Arches, as fg, hi, and therein from the Points g and h, fet off 108 Degrees, from g to /, and from j h to i, and through the Points f and i draw the.right Lines a e, and be, each equal to a b. (2.) At the Points e and c, perform the fame Operations as at a b, and you will compleat the Pentagon as required ; or otherwife, on the Points e and c, with the Diftance a 6 , defcribe Arches, as 00, m m, interfering in d, then drawing e d, dc, the Pentagon will be compleated as before. P. J obferve you fay I muft fet 108 Degrees on the Arches f g, and h i : Pray 'note tan that be done, fince that my Scale of Chords contains but 90 Degrees ? Af. Firft fet off 90 Degrees on each Arch from g to k, and from h to n, and afterwards 18 Degrees from k to/, and from n to i, and then will fg and h i be each equal to 108 Degrees, the Angle of the Pentagon. This being underftood, you may defcribe any Polygon at Pleafure, whofe Angles will be found to contain as following, mg. ' Hexagon 120 Septagon 128) O&agon G 5 «j Nonagon . 1 4 ° Decagon *44 Undecagon >47 A k Duodecagon > 5 ° - Degrees PROBLEM XXVIII. Fig. XCII. A Right Line being given, to find the Semidiameter of a Circle that Jhall be capable to circumfcribe any given Polygon, whofe Sides Jhall be equal to the given Line. Let A B be the given Line. Practice. Firft, Bifeft AB in K, and on K raife the Perpendicular Km. Secondly, With the Radius A B, on A and B defcribe the Arches A k and h B, interfering in I; on which defcribe the Arch A 1 2 q 4 5 B, which divide into fix Parts. Thirdly, Set one Foot of your Compaffes on A, and extend the other unto the Point i,and defcribe the Arch 1 a. Fourthly, Make In, no, Ip, pq, qr. 156 The Principles of Geometry. qr, rs, st, tv, cTc. each equal to Arf; and then drawing the right Lines oh, *B, pte, 1 B, r B, jB, t B, tB, they will be the Semidiameters of Cir¬ cles, wherein may be defcribed any Polygon, as required, whofe Sides fiiall be each equal ro the given Line AB. hor the Line 0 B is the Diagonal of a Square. And the Line <( C ‘ 'n B' IB p B VB • the Semidiameter of a < s B I t B U B „ v. Hexagon. Sepragon. O&agon. Konagon. Decagon. Undecagon. Duodecagon. P R O B L E M XXIX. Fig. XCIV. To deferibe a Spiral Line at any given Diftance, as the Right Line n x. Practice. Draw a right Line at Pleafure, as ov, and in any Part thereof aflign a Point, as r, and thereon, with half x.x, deferibe a Circle, as qs. This done, on q deferibe sp ; on r deferibe ptr; on q the Arch iro; and fo on thofe two Centers q, r, to as many Circumvolutions as you pleafe. If when on q you have defcribed the Arch sp, you make s your next Center, and thereon deferibe the Arch ft, the fpiral Diftance will be doubled, and admit of an. other Spiral to pafs by it at equal Diftance. For if on s you deferibe the Arch qw, and on q the Arch wo, you will form the fame Spiral from the Points, rsprefeuted by dotted Lines; as the Spiral from the Point r, which may both be revolved on the Centers q, s, at Pleafure. When fpiral Lines are continually open’d in their Revolutions, as Fig. XCI 1 I. they are called Scrolls, (I fuppofe from their Manner of winding or turning themfelves up,) and may be fo defcribed, as following. Firft, Draw a right Line at Pleafure, as bn, and therein alfume a Point, as g. through which draw another right Line, as am, making an Angle of 45 Degrees. Secondly , Set off from g ro e, from e ro d, from g to h, and from h to x, any fmall Diftance, and let the Points deghx reprefent fo many Cen¬ ters. Thirdly, On g deferibe the Arch he- on h the Arch ei; on e the Arch ib; on x the Arch hi; and on d the Arch lb-, and fo on with other Centers added at the fame Diftance at Pleafure, you will complete the Spiral, as re- q 111 red. ™te. It on the v .enter e you deienbe the Arch he, alio on * the Arch ck, and on d the Arch ky, you will have produced a Lift or Fillet that may be continued about, whole Breadth is equal to twice the Diftance of each Center : There¬ fore, whenever the Breadth ol your Lift is determin’d, the Diftance of one C-enter from another will be exactly equal to half the Breadth of the given Lift. 6 LECTURE The ‘Principles of Geometry. '57 LECTURE V. By S— S— Plates VI. VII. On the Jnfcribing and Circumfcribing Geometrical Figures. P. Jf/HAT do you mean by Jnfcribing and Circumfcribing Geometrical r ' Figures ? M Figures are faid to be infcribed within one another when the Sides or Angles of the one, touch the Stdes of the other; fo a right-lined Figure is infcribed in a Circle, when all its Angles are in the Circumference of the lame Circle: Or a Circle is infcribed in a right-lined Figure, when all the Sides of the Figure touch the Circumference of the Circle. As for Example, Fig. XCY I. The Geometrical Square fgik, is infcribed in the Circle fgt ek, becaufe all ics Angles are in the Circumference thereof. A Ho THf Circle nlme, is infcribed in the Geometrical Square nlme becaufe :al its Sides touch the Circumference thereof: and fince that the Geometrical Square bead contains the Circle nlme within its Terms, therefore it is findo circumfcribe the Circle nlme. And fo likew.fe the Circle nlme, is alfo faid to circumfcribe the Geometrical Square j'gik* P. I perfeBly underftand rehat it is to infcribe or circumfcribe one Figure within or about another : Pray proceed to /hew me how to perform the fame. M. I will; in the following Problems. PROBLEM I. Fig* XCV. To infcribe a Circle in any kind of right-lined Triangle, as bco in the Equilateral Triangle a h f. Practice. Firfl, Bv Problem V. Lefture II. divide any two Angles, as abf and afh, into two equal Parts by right Lines, as he and £/ interfer¬ ing each other in g. Secondly , On g, with the Radius gc, ox gb, defenbe the Circle bco, as required. DEMONSTRATION. The Triangles gbh, gho, have the Angles at b and o right-angled, and therefore equal; the Angles gbh and gho are alfo equal, becaufe the Angle bho is divided by the Line he equally : And fince that the Line hg is com¬ mon to the Triangles bbg and gho, therefore the Sides bg and go lhal be equal In the fame Manner we may prove that gc is equal to go. Now feeing that gb, gc, and go, are equal; therefore, if on g, with the Radius^, we deferibe a Circle, it fliall pafs through the Points bco: And becaufe that that the Angles b,c,o, are right Angles, the Sides of the Triangle ah, aj, hf do touch the Circumference of the Circle in the Points b, c, o, only; and therefore the Circle tea is infcribed in the Triangle abf. Q E. D. In the very fame Manner Circles are infcribed in all manner of lioleles Scalenum Triangles, as the Ifofeles Triangle Hi, and Scalenum pqr. and ! Numb. XII. o.q PROELHM J 7 'he Principles of Geometry. x u d l t M II. Fig. XCVI. iorfnle a Circle within a Geometric! Square, as nlme within bead. trough f ° irde ft"* iore is inferibed, as required. " r lc s< 2 uare j anc * there- PROBLEM III. Fig. XCVJI. 7 0 ' V " & * Clrck in an y *X*r Polygon, as in the Pentagon a beef. divide sny two Angles, » , will meet in the Point / from whence let fab b'"t "’“f b( j' n S com,nued > if. On rf, with the Radius dr, defoibe a Grc ^“ d ? k \ “ other Sides of the Polygon in s Itc t0Uch al1 the W,N t iight Angles •‘and^therefoe if 'the'aS^l ^ thofe Perpendiculars, it will touch eterv Sid b* * tbrou S h tllc Enc L of fta Rv the d fami n M bEd ^ ^ P °‘4 n ' ™ tTlZ ° ^ Septagon A, or any othe/reguW Pd/got' as ^uired!"" 1111 ^ ^ PROBLEM IV. Fig. XCVIII. Plate VII. To mferibe a Geometrical Square within any Triangle, as efdx within abc. e V aal to the Ba kt. thS’p“ d f: a '; d dra ^ thc Li '« hg. Thirdly, Draw ef parallel to be PROBLEM v. Fig. XCIX. .6 inf cube an Equilateral Triangle in a Geometrical Square, as cbe in cadg. JaSVtth oL'SKStTtfr 5”?“ “ “* die Arch hnf, inte.feaing the Circle i„ ilird ji 7/rL^ tw’drfl'P* ch and cf, cutting the Square in the Points /,„/ a /u a ™ L,nes b '’ tke Tr ' ang!e ° bc Wi “ bc the Triangle infcri|ft S 4S* PROBLEM •59 P R O B L E M VI. Fig. cm. Scribe an Equilateral Triage tn a Pentagon, as beg «, abd.hi. 0 * 35 , defcribe t b, xv- Secondly, Divide the Arche/ “ h Kadl11 ! de^ribe the Arc! the Points m0 , through which from d/p /*/ Int °, nV ° e< R' al Pa “ s ir alfo the Line eg; thfn wilu’ b^r A ' the LinrS b c and quired. * C e ec l UI, ateral Triangle mferibed, as re- PROBLEM VII. Fig. CIV. To inscribe a Regular Pentagon in an Equilateral Triangle, as dehfk within a, v, i. V M tbe Per P cndicu!ar m and on into five equal Parts at thrPe , ’ ltsr lP°m and divide the Arch ip •>« :".vt mk ’ •> •**» m. -i and on v, with the Radius v l defeibe the/ ^ tT° * 7 a ^ Parts in ; > unto /. Fourthly, Make vg equal to I^ ^ ^ Line Pijthly, On the Points k/;*, with the Radius i/j H CUttln 8 40 111 and ke, and make the Arches kd hr i \ e ^ cri ^ e ^ le Ar ches bn, kd, ing the Lines ^ / / t/“i-fTh ^ ^ then dra - as required. to * J ' J " 1 be mferibed in the Triangle problem. Vlil. Flg . CVI inferibe a Geometrical Square within a Pentagon, *r befh within eadgn. which make equa/ro /^/ncT dmw /’ i”* 1 Iet , faiI the Perpendicular ek. Pentagon in/ w/ Saw/Tn n h™ “ K wHch wiil ^rfed the * «efl the two Perpendicuia // ,fd"/ “St “? fr ° m the Po-ts/and * and r, and be aIfif each equal tofh / Z t """ ^ the PeI W™ » Square itfb w iU be the S[]uare i„/ib ed , ttquhed. ^ ^ *'* “ d the PROBLEM IX. Fig. cv. re Aj/ir* * Penta-Deeagon, or Regular Polygon, coning t f ,j ^ ^ , given urcle, as baenfi T P ang,e within the c;rcle > as Angles meet in the Point* • then wiiu/b^ /'Tr f ° that the »r is one fifth of the Circl’e therefore W ^ ^ An DEMONSTRATION. the Circle, equal to fiwe Rfenths^and the a// /j an§le ’ or one Third of fiiall contain three “* ^ Pa “’ PROBLEM The Principles of Geometry. PROBLEM X. Fig. CX. To circumfcribe a Circle about a Geometrical Square, as a bee about the Square a bee. Practice Draw the Diagonals be and ^ interfering in d, which is the Center; on which, with the Radius db, deferibe the Circle, as require . PROBLEM XI. Fig. CIX. To circumfcribe a Geometrical Square about a Circle, as abed about the Circle e g f i. Pr sctice. Firft, Draw a right Line, as gi, through the Center fr alfo abed, and complete the circumfcnbing Square, as required. PROBLEM XII. Fig. CVIII. To circumfcribe a Pentagon about a Circle , as abode about the Circle hwxgf. Practice. Firfl, Infcribe a Pentagon within the Circle, as hwxgf, and a,b,c,d,e, and form the circumfcnbing Pentagon, as required. PROBLEM XIII. Fig. CVIII. To circumfcribe a Circle about a Pentagon. Practice Bifeft any two Sides thereof, as ed in g, and at m f; and deferibe the circumfcnbing Circle, as required. PROBLEM XIV. Fig. CVII. To circumfcribe any Regular Polygon about another Polygon of the fame fort, as the Hexagon lacegi about the Hexagon K,m,b, Qtn. Practice. Firfl, Draw the Lines m f, bl>, kd, and through the Points m l d f,h,k, draw the Lines ac, ce, eg, gi, lb If, at right Angles o e Lines hh,'dk, fm; which will form the circumfcnbing Hexagon, as required. PROBLEM The Principles of Geometry. 7 i PROBLEM XV. Fig. CXI. To circumjcribe a Pentagon about an Equilateral Triangle, as aorv f about upk. Practice Firft, With any Opening of your CompafTes, on the Points defcribe Arches., as bde, Iki, mnq Secondly, Divide the Arch ^Tnto five equal Parts; and on r, with the Radius equal to four of thofe Parts defcribe the Arch nxb, and from a, through b draw the .right Line ’ Thirdly, Make the Arch qrn equal to the Arch gb, and from fcthroug m draw the right Line ompr. which will cut the Line oo in o; “ifo make ’ i to ao Fourthly Make the Arch h equal to the Arch mq, andhorn belle fiU making kf equal to op, and An equal to J’,. . then drawing the right Lines af and rt,, they will complete the circum¬ feribing Pentagon, as required. PROBLEM XVI. Fig.CXII. To circumscribe a Pentagon about a Geometrical Square, «non about 5 lw v. Practice. Firft, Bifefl 5 / in i, and continue the Side x 5 unto d, mi- , p “ ll to Vi, and defcribe the Arch defghi ,. divide into five klng l 5 Pa,M at the Pmnts e,fg,h. Secondly , On i ereft the Perpendicular t k, throlf the find Kvlfion in the Arch tbe blde e f. 18 e T' al t0 Side bn, and the Side fg equal to the Side nk, and the Angle efg equal to the Angle but; then Iffy ?hat the Side eg is al& equal to the Side hk. ’ y ’ tnaC demonstration. i lF . thePomt b bei applied on the Point e, and the right Line bn placed on thejight Line ef the Point »fhall fall on the Point ], becaufe bn is equal Also the right Line nk fhall fall on the right Line fg, becaufe nk is equal to fg. Lifcewife the Point * will fall on the Ftfnt* and the fth Line hk on the right Line eg, becaufe hk is equal to eg. g No w feeing that all the Sides of each Triangle agree with the Correfpon- f, " / a fetJU « nt 7 f hC rnang 6 ejg ' S 6< 5 Ual t0 the Triangle hnk. Which ivas to be demonstrated. 5 tilde t0 What V t t thiS r T , he0rem afp,yd ? For at » /«<* to be of little Vfe, os many other Things I have learned in the preceding Leblures. 1 vo ,f „ lTS f eat ; as alI ° ever y Proble nr I have hitherto taught you as will be prefently f ee n. But the Ufe of this Theorem I will now dluftrate : Suppofe ab Fig. CXV. to be the Bafe Line of a Hill, or Diametef of a Concave, or large Pit or Pond of Water, whofe Breadth is required but cannot meafure direffly from 4 to b, to obtain the fame. S ’ Pi!AcT,c£. Fir ]i Align any Point on the Ground (being level) where you can fee and meafure to the Points 4 and b, as at i ^ Se ln 7 ly, Off any D,dance m ftreight L.nes fVom d, towards the Points 4 and *■ as to die The ‘Principles of Geomet R V. 163 38&SWK A S'* — o.k, =«l Ground, n»k, , Tri.ng'fc, , , *“« c ’y v ' f“ T Triangle,, a* and continue out the Sidesc/Va r / e ^ al /° ths Pleafure. T&>^, Make + 4 equal to a /, „ A 4 ’ r ? rds « and b, at the Triangle abc, be equal to the Triangle aH*\ ^ t!;en wiil equal to the Angle .A; and conStlv A TV^ A A " gle *' U ’ is is equal to the Lin ea b of Fig CXIY J ' I ? lftani ? of ab, Fig, CXV. . Now that you may nter bfataSmnd^V cion Point, only obferve, That if you can but I n ° W I '™ t0 your Sta " treams of the Line, whofe Length required A' " ** ACCe , fi t0 the Ex ' if tnftead of the Point being chofen a A had h “* Certainly r, g ht > for. Fig. CX VIII. would have been equal ^ -rfi, consequently the Diftance of 4 b would be fonnd as Wbre § ' ^ “ d Diftance Z/t takln ol ^ ^ngth or Cm?® s as f a 1Win£; JKffSJSSs 74 S“ rr 3 *- . a r: vnriisj™' pi?* THEOREM II. That Triangle which hath Fig. CXV. ~ **- »*/-./ DEMONSTRATION. I he Triangles efg, ink, have theSides nk, fa, eg, ik equal fbv W fit.on,) and the Angle egf, is alfo fuppofed equal to he Ing e id T from thence the Triangles efg, J, ftj be equal t tety Re^ and the Angles, ink, and hnk, Ihould be equal alfo - that rh ’ bv th A ft bC T a ‘ “ A Angle bn hi whiah Cam,ot U becaufe itisHefi by the Angle bni; but according to our Hypothefis the Analp - ■ 1 i SrED°'d‘ b “ J “ r Vr "“I" 1 •»*!. wl ctsir refpefKvs Angles are equal. Sill Jr S T trUe ' Sir ’ by the frying Theorem, you have proved, that when the Sides Triangle, are equal, the Triangles themfelves are alfo eaual. Pray to ivbat Ufe may this Theorem he apply d ? J *1 s-> M. To meafure inacceffible Diftances, which frequently happen in the Praa.ce of taking Plans of Buildings, Lands, tfc where you arAnterrupt! ed by Water, or other Perions Lands, on which you muft not enter, but I y The Principles of G k o m k t rJL at the fame Tim. muft comply «* wftl t ss-^'ra.tsrsts• •» « - fure. , . Rrpa d,h of the River MX, which cannot bs 1 *“ ^ Ceed t0J ‘ termlneitS BrCadth ’ lowing: v .n R , nee ^ on the Ground a right Line as xf, and PkActice. F"ft' ' § e;A> „f the River, affign a Point, asato, and in any Part thereof, near the S Length, at Pleafure ; wherein ah thereon raife the Perpendicula , Y 'secondly. Sight or range in your fo aifign a Point ,n any Par ^ ^ the Point /, and the Point Five or Ten Foot Rod, m a rig cuts on the farther bide of the *». f wl ’ erc the ^'jX^Feet or°any other known Diftance from the Point 1 , as Riverat 5, i°> & • ’ Diftance of ll, on the Perpendicular 0 Z, from at the Point 7 • Alf “ fe “'? l g T \ nr dly, Make the Angle 8 1 % equal to / to 8, and meafure the Diftance / »■ f . then wi U the Tri- the Angle 7/8, and le »/„; and if you make a 5 equal to Dfc’“ ”f 5 f. «U b= W*' » - '• Br ““ ” f R "“ "*** theorem 111. Fig. CXV1U. Aff f.-,a the ritht Angle of any right-angled plain if a Perpendtcit m » ft ffftf i' /t w m aivide the fame into nvo Triangles, Triangle on the h p to. which will he alike tbeuto. r n • Ft Andie IM be drawn the Perpendicular t r, to the It from the r.^ht A * right-angled Triangle qts into.two Tri- Ml b- Jk 0. 'Rtonsto, ,0 the Triangle DEMONSTRATION. ■ »».«■* *mss Station, as following: T/J required to determine the Diftance of r s. Length. Second, Being furmlh d with a (quare Jot ^ ^ ; rpen , i; J ular ly P “ 3 , Turn oneSide of the ^re about, until 5 ort I me r q ; which is done by Sighting in an upright Staff placed q, continuing on the Side tc, until it meets the Line r q. Now j The ‘Principles of Geometry. Now ’tis evident, that there is the fame Reafon or Proportion of t r to rg , as of tr to rs; therefore if t r be multiply’d into itfelf, and the Produfi divided by r q, the Quotient will be r s, the Diltance required. EXAMPLE. Suttose tr be 12 Feet, and r q, 6 Feet. Now from the Confideration of thefe Fbeorems, you may readily de¬ termine all inacceftible Diftances, as they occur or happen in Practice; which no Learner could imagine at his firft: Reading of the Theorems only : And it is the very fame Thing with every other Problem of the pre¬ ceding Lectures, as will now be de¬ clared : So that my witty Readers, who have faid, many of thefe Problems were ufelefs, will be convinced that they have found Fault with what they have not underftood ; and which indeed, 1 believe, has been the only Caule of their ill-natur’d Cenfure : For none but the Stubborn, the Conceited, and the Ignorant, will condemn the Labours or good Intentions of other-, or pretend to be Judges of a Knowledge, to which they are entire Stran¬ gers : Whilft the judicious and thinking Man maturely conliders, without Prejudice, the Reafons and Caufes of every Thing prefented to his Confide¬ ration ; and to fuch only do I dedicate this Work. PROBLEM I. Fig. CXIII. To take the Quantity of an Angle in a Building , and delineate the fame on Paper, Rlc. by the Help of a Two-Foot Rule, or Ten-Foot Rod only. Suttose I am to take the Angle ace, and reprefent the fame on Paper. Practice. Firft, From the angular Point c, meafure off any Diilance, as 10 Feet towards a, as to b ; and from c towards e fet oft the lame, or any other Diftance, as to d, which is 15 Feet from c , becaufe of the Opening at A. Secondly, Carefully meafure the Diftance contain’d between the Points b and d, which note down on a Piece of Paper, and fuppofe to contain 20 Feet and ten Inches. This do, e, draw a right Line on Paper, at Pleafure, as / B Fig GXVII and make/ st equal to c d, 15 Feet, of any Scale of Feet. Thirdly, Take in your Compaffes 20 Feet 10 Inches; and on the Point d, de- fcribe the Arch‘d; alfo on/’ with the Diftance of 10 Feet, defcribe the Arch a a, interfering b b in d, and draw the right Line jc through the Point d , then will the Angle cf^ } be equal to the Angles cd; becaufe the Triangle dj^, is fimilar to the Triangle bed. In the fame Manner, the Angles, hik and v xy are taken as follow¬ ing. Firft , Set off from the angular Point i, to h and k , any Number of Feet; fuppofe three Feet each, and meafure the Diftance hk , which let be four Feet and four Inches. This done, let the Line gy. Fig. CXVII. be drawn parallel to/sc, at the fame Diftance from as gy. Fig. CXVII. is from c e, and make them both equal. Secondly , Make gk equal to gi; and on the Point k, with the Diftance of three Feet, defcribe the Arch hi. Thirdly , Make hi equal to four Feet four Inches, the Length of hk, and from k. Fig. CXVII. through/, draw the Line ki, and the Angle hki , will be equal to the Angle h i k, Numb. XIII. " Sf Fig. Then 12 Multiply’d by 1 2 Divide by 6) j( 24. 12 H H o 1 66 The Principles of Geometry. Fig. CXIII. Proceed in the fame Manner with the Angle wxy, Fig. CXIII. and Angle xiy, Fig. CXVII. which will be equal to each other alfo. Now you are to obferve, by this Method, that all manner of inward or internal Angles may be very correfldy taken, provided that the Diflances you fet off from the angular Point, be as great as poifible, and not very fhort; becaufe in long Lengths you will be more liable to Errors. P. Pray what do you mean by internal Angles ! . 11 . All fuch Angle:, whofe Quantitles are each lefs than I So Degrees, as the Angle ace. Fig. CXIII. whofe Meafure ?s the Arch DI, which is lefs than I he Degrees, by the Arch PD ; becaufe PD and DI, taken together, are but a Semicir,le, or i 8o Degrees. Of this Kind are the Angles At, I, 6, 9, 7, of Fig. CXIV. and K, a, 6, vsn of Fig. CXVI. If the Arch D I'^I be confider’d without Side of the Angle DC I, it will be the Meafure of the Complement of the internal Angle, and is called an ex¬ ternal Angle, as being greater than a Semicircle, or 1 80 Degrees, by the Quan¬ tity of the Arch P D. Of this Kind are the Angles at k, n, 9, and v, in Fig. CXIII. and at m, i, d, b, s, and y, in Fig. CXIV". as alfo are the Angles «, k, a, 6, v, and s, in Fig. CXVI. Now the Manner of taking thefe external Angles, are exaflly the fame as before for internal Angles; which will appear by the following EXAMPLE. 'Tis required to take the Quantity of the external Angle ikn. Fig. CXIII. and to delineate the fame. Practice. Firft, Apply your Ten-Foot Rod from the angular Point k, towards m, fo that the End thereof reft in a ftraight Line with the Side i k, at rn, at which Place make a Mark. Secondly, Remove the End of the Rod at m, along the Side k n, from the angular Point k unto /, at which Place make a Mark alfo; and then meafure the Diftance ml. This done, apply to your Paper-Drawing, and continue out k i. Fig. CXV II. towards /, and on the Point i, with a Radius of 10 Feet, equal to the Length of your Ten- F'eet Rod, deferibe the Arch 1 m, and make Im thereof equal to lm of Fig. CXIII. and then drawing the right Line in through the Point m, and equal in Length to £ n, you will have deferibed the external Angle kin, which will be equal to the Angle i kn, as required. After the very fame Manner are all the other Angles in Fig. CXIII. taken and delineated in Fig. CXVII as alfo are the external Angles of Fig. CXIV. that are delineated in Fig. CXVI. which being fo very plainly ex¬ plain’d by the Lines themfelves, needs no farther Explanation. Before I conclude this Problem, I muft obferve to you, That when in¬ ternal Angles are very large, as to contain about 160, 170, or 178 ,isfc. Degrees, there is fome Difficulty to determine the real Points of Interfeflion. As for Example: I would take the internal Angles 7, 9, 6 , and at 1 of Fig. CXIV. Now if I proceed, as before deliver’d, and fet off equal Diftances on each Side of each angular Point, as to v and y, at the Angle t, and to Z X, at the Angle 9 ; then the Subtendent Lines will be vy and Z X, which laft is not vaftly lhorter than Z 9 and 9 X taken together, and therefore very liable to Error; which may be prevented by the following Methods : METHOD i 6 j The Principles of Geometry. method I. Firft furniih'd with two Bods of equal Length, (the longer the better,' fuppofe each ten beet; meafure off one of their Lengths from the angular Point a unto c; at which Point apply one End of the other Rod, anti bring both their other Ends to meet at the Point ir, at which Place make a t ^nfrhTp H iSfc - Secon ty'-‘ Keeptng yet the Rod at the angular Point ’ and the End of the other Rod being placed at/, bring both their Ends TriarHe ' wh f 't" W ‘ U y ° U haVe defcribed r '™ equilateral Triangle., whole Sides are feverally equal unto to Feet. Thirdly 1 A Mark being made at a, remove away the Rod, and place it from t to/and mea- fure the Diftance y x : And thus have you taken the Angle in three Parts, which is eafily defcribed on Paper as following : Practice. Firfl, Suppofe A t was a Line drawn at Pleafure on Paper, and made equal mLength to At, by your Scale ofFeet. Secondly, Take ic Feet irom your Scale of Feet, and therewith complete the two equilateral Triangles tvrr, and twj and on t, with the fame Radius of 10 Feet, deferibe the Arch xn Thirdly, On *, with the Radius xy, deferibe the Arch pp inter- ietting the former in y; and then drawing the Line M through the Point y you will very correftly lay down the Angle Am, as required. METHOD If. Obtuse Angles of this Kind may be taken at twice, and very iufly alio. As for Example : I would take the Angle k a 6, of Fig. CXVI. Practice. Firfl, Compleat one equilateral Triangle as before; as caf, and make ag equal to./; alfo meafure/f. This done, let the fame Line k a represent a given Line, and at the End a, compleat the equilateral Tri¬ angle caf. Secondly Take 10 Feet, i*. equal to the Side af and on . de- knoe the Arch nn alfo on/ with a Radius equal to the mealured Diftance of Jg delcribe the Arch m m, interfiling the former in g. Thirdly, Through the Point g, draw the Line ag-, which will compleat the Angle, as requir’d. METHOD III. Supiose the obtufe Angle nsv, Fig. CXVI. is to be taken ; which may be ealily done as follows: J Practice. Firfl, Meafure in a right Line from v towards n, and when you are come at x, direflly oppolite to the Angle r, there flop, and note your Length meafur d; which we will fuppofe to be 42 Feet Secondly Irom the angular Point r, let fall the Perpendicular r a-, on the Line xv and meafure its Length, which fuppofe to be two Feet and fix Inches • after which meafure the Reiidue of the Length nx, which fuppofe to be / Feet This done, delineate the Angle as following: * Firfl, Draw a right Line as n-u, which make equal to 67 Feet, from your Scale of Feet, which is the Length of n x and xv taken together, and make v x equal to 42 Feet. Secondly, on w, ereft the Perpendicular a r which make equal to two Feet and a half; and then drawing the Lines n s and 1 v, they will compleat the obtufe Angle nsv, as required. W PROBLEM 168 The Principles of Geometry. p R O B L EMI. Fig. CXXV. The Out-Line of a side of an irregular Building (as abeknos) bang given, to delineate a Plan thereof , as tvbhnol. Practicf. Firfl, Draw a right Line at Pleafure, as tx and make tv enm^ to ah I 2 Feet, (by your Scale of Feet and Inches,) and continue ah to- wards c. Secondly, Make the Angle bvs equal to the AngL ebe as taught in Prob XVII. I ea. II. and in the Ufe of Theorem III. Lett. VI- alio make ^ equal to eh, . 6 Feet. Thirdly, Make the Angle hbv i equal to the Ang e k e h and the Side b h equal to the Side ek, 15 Feet. Fourthly, Make thei Angle b h f equal to the Angle ekg, and continue) h towards k, making hn equal to „/ fs Feet.° Fifthly, Make the Angle hum equal to the Angle k m o, and make 5 «V equal to .o: S. Sixthly, The Side m, being continued to¬ wards a and the Side no towards m, make the Angle -. low equal to th e Angle Toa and draw lo equal tot,,,. Feet a Inches; and then the Thicknefs < 4 , being drawn parallel thereto, the Plan will be completed, as required. PROBLEM II. Fig. CXXI. and CXXIII. The Out Line of an irregular Building, as fa bed c, Fig.CXXL being given, to delineate a Plan thereof, as m g n i k I, hig. LAXlli. Practici. Firfl, Draw a right Line at Pleafure, as nlj and at one End thereof as at k, make the Angle »ki equal to the Angle ; Alfo make Ik audw^TarFeetioInclies 6 and ki equal to dc , a. Feet. Secondly, Make the Angle kih equal to the Angle deb, and make ih equal to c , jjo ee;. Thirdly, Make the Angle thg equal to the Angle eba and make jh equaGo ab 3a Feet Fourthly, Make the Angle hgm equal to the Anje /, „’ke Ym equal to J*6 Feet 6 Inches. Fifthly, Make the Angle#*/ equal “the Angle,/,; which, if your former Work be truly perform d, will be done by joining ml, and ml will be alfo equal to fe, i ? Feet and q Inches, and complete the Plan, as required. PROBLEM III- Fig- CXXII. To take the Plan of the Out-Line of any irregular Building, without meafuring any Side or taking any Angle thereof-, and inaccefjible alfo, as the irregular la fbcdefenhiklmopq, which is inviron’d with Water, and therefore none of if Sr Angle) can lemeafured ; and yet a true Plan with the juft Meafures hereof Zfl be made, and that alfo with no other Jnflrument than a common Z °lr Ten-Foot Rod, as bang always to be had in any Place, and an fftrumem, MAfpa is not Jo aflonifhing to vulgar Eyes, as the Plain Table, Theodilite, W Circumferentor; nhofeVjes ifhallvery carefully explain, and compare with the Ten-Foot Rod, &c. in the Fifth Part hereof. This Plan may be moft exaffly taken by the three different Methods fol- l0Wmg ' METHOD I. Practicp. Firfl, Walk about the Building, and obferve at how many Stations you can be capable of feeing all the Angles containd therein; for I The ‘Principles of Geometry. 169 perpendicularly. Secondly, Being furniihed with an AftiiLnt, who muft alfo be furniihed with lhort Station-Staves, or ftrait Rods, about three Fee: in Length, place yourfelf at one of the Stations, as C, and direct him in right Lines, between the Station C, and the feveral Angles p,o,m,l.!,i,h,n,g, to erefl thole Staves or Rods at the Brink of the Water, as at the Points x, x,x, &c. Thirdly, Meafure in a right Line from C, towards the Angle p, unto the Rod at ,y, which Length fet down on a Piece of wafte Paper; and then find the Length of xp, as before taught in Theorem II. and 1 L. hereof. This done, order your Alliftant to Rand in a right Line with yourfelf at .v, and Station C, as at or about the Point 9, and meafure in a right Line from C unto him, until you have meafnred firft the Diftance C 22 equal to G,v, and afterwards from 22 unto 9, the Diftance equal to px; then will the Diftance C9 be equal to the Diftance C p. Fourthly , Meafure from G, towards the Angle a, unto the Rod at x, which Length let down, and find the Length xo, which add unto Cx; then direft your Alliftant to Hand in a right Line between x and C, as at or about the Point 8, and in a right Line from C unto him, meafure off the Diftance oC, as at the Point 8; then will the Diftance of the Points 8 and 9 be equal unto the Diftance'of the Angles p and 0. Fifthly , Meafure from C, towards the Angle m, unto the Rod at x , which Length fet down, as before; as alfo the Length mx, which you mull find as aforefaid, and add unto the other : Then direeft your Alliftant to ftand in a right Line between x and C, as at or about the Point 7, and in a right Line from C unto him, meafure oft the ! iftance m C , as at the Point 7 ; then will the Diftance of the Points 7 and 8, be equal to the Diftance of the Angles Dand m; and the Angle l 8 9 is equal to the Angle pom, but is reverfed. Sixthly , After the fame Manner, make C6 equal to /C; alfo C5 equal to kC; alfo Cq equal to i C; alfo C 3 equal to hC ; alfo C 2 equal to nC; alfo G 1 equal to^C ; and draw the right Lines 6,7; 5, 6 • 4, 5; 3, 4 ; 2, 3 ; and 1,2; which will be equal to the Sides of the Building ml, /(, hi, ih , hn, and ng ; but are reverfed, as before obierved. Now, to reprefent them in their true Politions, Ive mull reverfe them again ; which is done as following. Assign a Point in any Part of the Area before them, as at F; through which, from the feveral Angles before found, as at 1,2, 5,4, 5, 6,7, 8. 9, draw- right Lines at Pleafurc, and make F 18 equal to 1 F; alfo F 17 equal to 2F; alio F l 6 equal to 3 F; alfo F 1 5 equal to 4 F ; alfo F 1 q equal to 5 F; alfo F 13 equal to 6F; alfo F 12 equal to 7F; alio F 11 equal to 8 F; andlaftly, F 10 equal to 9 F : And then drawing the right Lines 10, j j ; 11,12; 12,13; 13,14; 14,15; 15,16; 16,17; and 17, 18; they will be exactly equal to the Sides of the Building po , om, ml, Ik, ki, ih, hn, and ng; and the feveral Angles at 10, 11, 12, 13, 14, 15, 16, 17, and 18, will be the very fame as thofe of the Building a t pom l k i h n g This being underftood, which is very eafy to do, you may at the Station D, in the fame manner, find the Sides ha, aq, qp, and po; which laft Side, tho’ taken before at Station C, muit be now again taken to find the Angle qpo , otherwiie we could not truly join theSides ha, aq, and qp, unto the others before taken in their true Politions. Which Laid Sides, by the firft Reverlion, will be 1,2; a, 3; 3,4; and 4,5 : And then affigning at Pleafure the Point 6, by reveriing them again through the faid Point, as before taught, the Lines GH, HI, IK, will be equal to the Sides of the Building ha, aq, qp, and po; as alfo will the Angles H, I, K, be equal to the Angles a, q,p, and in a right Pofition alio. T c In I N the fame Manner, P, O, N, M, L, will be found equal to the Sides q, a, b,c,d; and R, S, T, V, W, equal to c,d,e,f,g ; and fo will you have taken all the Sides and Angles of the given Building, without once meafuring any Part thereof. Practice. Firft, By Pkob. I. hereof, make b,c, d,e,J g,b,i,k, equal to to, it, it, i 3, 14, 1 5 16, 17, ofFig.CXXU. and becaufe that the Sides KQ_, and 10, 11, are the fame, therefore on b. Fig. CXX 1 V. make the Angle abc equal to the Angle IKQg and then make baqp equal unto KIHG. Secondly, GH and N O being the fame Side, (equal unto b,i of the Building.) therefore make the Angle pqa equal unto the Angle GHI, which is alfo equal to the Angle NOP; and then make pan equal to NML. Thirdly, ML and RS being the fame Side equal unto cd of the Building, therefore make the Angle on m equal unto the Angle RST, and make nmlk equal to STVW; and^f that you have truly perform d, the Line V\V, which in your Drawing is the Line kl, will clofe up your Plan at k, as required. As this Alethod of meafuring Plans is intirely new, I make no doubt but that the carping Critick will make his Cblervations and Reflections thereon, and Objections againft it. But that he may not have all the Trouble thereof, 1 will, for his Eafe, make the frit Object.un, and then refer the Remainders, if any, for him to difeover. To difeover the real Length of the Sides, and the Quantity of the Angles of Buildings by this Method, much Space or Room about it is required for the Operation, and the Ground to be very nearly fmooth or level, other wife ’tis not practicable; and this is the only Objection that I know of can be made againlt it.. For where there is Room fufficient, and the Ground nearly level" it is infallible, if Care be taken in the Operation. N.B. And fince that oftentimes we may not have Room fufficient for the Practice of this Method, I (hall therefore Ihew how to perform the fame Work in lefs Space with a Ten-Foot Rod only, as aforefaid. But however let not the foregoing Method be rejeted, fince that on fome Occafions it may be of very great Ufe to you, as I in Practice have often experienced. Fir/ 1 , Affign the proper Stations, as A,B,D,C, and then begin at any one thereof, as at C, as follows. Secondly, Let your Affiftant fix up two fmali Station-Rods, or Staves, at the Points X and 10, at any equal Number of Feet dillance from C, as 10, 20, ist’e. and in right Lines between the Stations DC and BC. Thirdly Being provided with a Ten-Foot Rod, and a Line long enough for the Purpofe, apply one End of the Line unto the Rod at X and ftram it by the other Rod at 10, as nearly level as you can. Fourthly o’n two Sticks fet lip, as ab, bd. Fig. CXXVI. hang a Plum-Line as be, fj that the Bob c hang exactly over the Station-Point C; and your Affiftant havim* an¬ other Plum.Line alfo, caufe him to hold it up by the Side of. the Line mov¬ ing it backwards and forwards, until you have direted him, in a ri»bt-lined The TrincipJes of Geometry. i 71 the Station C; at every of which Time, let him exa£Hy mark the Side of the LineXlo, and meafure the Diflance from X, as at the Points I, a, q, q, e, 6, 7, S, ), jo. Fifthly , Meafure the feveral Di I lances from the Station at C, unto the Angles po, Imk, ibng, and enter them down in a Book for that Purpofe, as follows- f ’ Feet. Inch. The Diflance from the the Station C, unto the Angle r p 7 1 0 0 57 6 m 47 3 l 37 9 And Quantity of An¬ i k 26 0 gles on the Line, < i 3 2 7 from X to b 47 0 n G 4 S 68 Feet. Inch. 4 4 4 5 9 ‘I 16 it) 3 * 10 6 Sixthly, Proceed in the like Manner at the other Stations D, A, B, and enter them accordingly. Which being done, you may delineate from your Book a true Plan thereof, as following. Firfl, Conftder to what Magnitude you would make your Drawing, and accordingly proportion the Size of your Scale. Which being done, draw a right Line at Plcafure, as b>x, Fig. CXX 1 V. which make equal to the mea- fured Diflance of your Stations D and C, 8) Feet; and on r, with theRadius of 26 Feet, equal to XC, deferibe an Arch as ry, and thereon fet 24 Feet from r to 10, which is equal to X to, the Meafure of the Angle DCB, and draw the Line r 10. Secondly, Make the Divilion n, ri, rj, r r 5^ r6 r 7 > ’; g > r 9 . equal to the Quantity of the Angles obferved and meafured by the Side of the Line, and enter’d in the lail Column of the Table; that is, make r 1 equal to X 1, and draw h x equal to 71 Feet, the Diflance of tile Angle p from the Station C; alfo make r 2 equal to Xi, and draw cot equal to 57 Feet 6 Inches, the Diflance of the Angle 0 from the Station C, and fo ill like manner the others : Which, when done, you will have produced the Points b,c,d,e,fg,h,i,k; and then the right Lines be, cd, de, ef, fg, gh, hi and in, being drawn, will be the Sides po, om, ml, Ik, hi, ih, bn, ng, which are Part of the given Plan. Seventhly , Begin again at the next Station D, and proceed in every Refpeft as at C; after which, fet off the fame by your Scale of feet at IV, Fig. CXXIV. In like manner proceed at the Stations A and B, and fet off the fame at v and t, and you will have com, leted the Plan within the Limits of A, B, D, C, as required. Note, When Lines of Diflance from tile Station.Points falls very obliquely on the Sides of the Building, as pG and 0C on tile Side po, ’twill be bell to make Oft fets, as go and bp, from the Line DC, wliofe Lengths may be found by Theorem II. and III. and will determine the angular PoInts . p and 0 with Certainty, that is not fo eaftly done by tile foregoing Dire&ions. The like is to be^oblerved at the Angles of the Building ab, cd, which are in general determm d in this Manner, in '"Big. CXXIV. as appears by the Ifule- cles Iriangles at the Angles cb , qp , on. The whole being fo very plainly ex¬ hibited by the Lines of the Diagrams, it is therefore needlefs to fay further thereof Irregular The Principles of Geometry. 172 Irreegular Inacceffible Buildings may be alfo pland as following. METHOD III. Fig.CXX. Let abcdefghiklmno be the Out-Line of a Building, environ d nitb Hater, and ’tis required to make an exact Plan thereof. Practice. Pirfl , Affign about the fame a fufficient Number of Stations, aS rjrst, vp, at which Places fix down Stakes, &c. and with your Ten-Foot Rod meafure the Diftance between each■ and aifo take the Angle at c\ery Station, as is made by the right Lines contain’d between them. Secondly, By p non. II. hereof, lav down a Plan of the Stations and Lines between them, (which we will fuppole to be the Plan qrs, pvt.) Thirdly, Being pro\ jded with the fame fquare Board, or Joint-Stool, as direfled for the Ufe of finding Inacceffible Diftances in Theor. II. and III. apply oneSide thereof unto the Line pv, and move it along the fame, until by the other Side you fee the Angle n; then by its Side, oppofite to the Angle, draw the Line 1,15. In the lame Manner, perform again at every other Angle as at the Points 2, y, 5,6, 7, S, 9, jo, 11, 12, 13, 14; and then will the Lines n 1, 0 2, aj, b 4, £-5, dP, o'], f 8, £9, h 10, ill, k 17, 1 13, m 14, be fo many Off-lets, whofe feveral Dillances on every relpeffive ifalionary Line, being truly placed according to their Meafures found, and Length difeover’d by Theorem II. and III. being made correfpendently equal ; then Lines being drawn unto the fame, will complete the Plan, as required, as in the Figure is mold plainly leen, by the feveral Lines that conftruci the lame. THEOREM IV. Fig. CXVIII. Equiangular Triangles have their Sides proportional. If the Triangles wxy, 1 yz-, are equiangular, that is to fay, that the An¬ gles wxy, 1 jz; xwy, y I z, be equal, there will be the fame Realon of xm to xy, as of y 1 toyz- In like manner, the Reafon of xw to wy, lhall be the fame with that of y 1 to 1 g. Continue the Sides xw, and ri, until they meet in v; and becaufe the Angles wyx, and 1 zy, are equal, therefore the Sides wy and tug are parallel, as alfo are the Sides v x and 1 y- DEMONSTRATION. In the Triangle vxz> nay is parallel to the Hypotheneufe vg, and therefore there ffiall be the lame Reafon of xw to it, as of xy to yz- alfo there lhall be the fame Reafon of wx to xjy, as of \y to yz- In like manner, 1 y being parallel to the Perpendicuar vx, there ffiall be the fame Reafon of vi to 1 r, as of xy to yz; and alfo of wy to xy, as of 1 r to yz- Hence ’tis evident, that the Parts of Equiangular Triangles are refpeflive ly proportional to one another. This being underftood, we will now proceed further, on divers other Me¬ thods for Taking and Delineating of Plans, as they may varioufiy occur unto us in our Praflace. PROBLEM The ‘Principles of Geometry. 173 PROBLEM IV. Fig. CXXVIII. To take the Plan of an Irregular Curved Line, as AE. BrroRE you begin to take the Plan of any Lands or Buildings, walk over the fame, and make, as you go, a rough Draught of the Lime at Guefs, as near y true as you can, on a Piece of Paper, dignifying therein every Side and Angle, Without any Regard being had to the Exaftnels thereof; which youare to call an Eye-Draught, (as being made by the Eye only, without any Mea- furement for forming of the fame,) whofe Ufe is, for to receive on its feveral Sides an Account of their Lengths, as alfo of their reipeftive Angles: From which you are enabled to delineate an ex-aft Plan of the Premifes meafured; which is called by Artizans, The Taking of a Plan. Practice in the Field: Firfl, affign two Points, as as, on whi.h ereft two Sticks; and then fuppofe a right Line to be drawn between them, which alfo reprefent on your Paper. Secondly, In the Linear, and againft every re¬ markable Turning in the Curve, as at the Points b,c,d„ef g,h,i,k,l,m n fix down fmall Sticks, and meafure from each of them, as nearly at; right Angles as you can, from the Line a s, unto the) Curve ; and to each of them, in your Eye-Draught, affix its true Length, as alfo its trueDiftance from thePoint a :Or othenvife, beginning at a, meafure towards r,until youcomeati- which, fuppofe to be 11 Feet, which let down in your Eye-Draught; and then meafure theOff-fet it, (as nearly fquarefrem ab as you can,) which fuppofe to be 1 5 Feet, which fet down thereto in your Eye-Draught, as in the Figure. This done, proceed towards s, until you come to c, where you fup¬ pofe ’tis neceffary to take a fecond Offifet; whofe Diffimce from a, fet dov/n thereto in your Eye-Draught, and then meafure the perpendicular Off fet v,c, fuppofe to be 50 Feet, which fer down alfo as inthe Figure. Proceed in like Manner, to take all the Reiidue of the differs, as ffiall be judged neceffary for the Purpofe, (the more, the better,) and then you may delineate the lame truly, as following: Practice on the Paper: Draw a right Line at Pleafure, as a s , and with any Scale of Feet, as you ffiall make choice of. Feet. Feet. r ab • li¬ rbt ' r 15 ac as c V 3° ad 3 2 drv 34 ae 43 ex 3 1 *f 5> fy 3 2 a g (yi « 38 ab 74 h 1 39 : 6 a \ ak >equal to- 85 98 _ and the Oft-Set i2 > equal to ^ 37 31 a l n 4 4 eS am iaa m 5 3 1 an 4 2 n 6 37 a 0 14 a °7 38 ap I 5 I pH 35 aq '59 19 -1 K ar j i 6 5> no j U u Numb. XV. and 174 - The Principles cf G eomet ry. And though the Extreams thereof, f, v, rr, x,y, ^ i, 2, q, 4, 5, 6, 7, 8, 9 10, draw, or trace the curved Line required. PROBLEM V. Fig. CXXIX. To make the Plan of a Piece of Land bounded by an h regular curved Lin :, as b, d, e, f, h, 1 , p, which may be Jeeh from one Station in or near the Middle thereof. Practice in the Field: Firft , Make an Eye*Draught thereof, and in its Sides a IF gn four Points, as b, e 3 h 9 k; in which ereft four Station-Staffs, and reprefent the fame in your Eye Draught, and therein alfo draw Lines repre- fenting b e, e /;, h k, and bk , whofe Lengths being feverally meafured, and the Of-Sets that are necell'ary for the taking of the Curve, being taken, as before taught in the lull Problem, you may proceed to the Delineating a true Plan thereof, as following : Practice on Paper. By Prob. II. of Lect. IV. make the Triangles eh k and ebb, that their refpe£live Sides fhall be equal to the Meafures thereof taken and exprelfed in your Eye-Draught; that is to fay, Feet . In the Triange c h k, the Siae< e k c k h C 9 1 to be equal to < 89 And the Triangle b e k } the Side < b c Wherein obferve, That the Side e k is common to both [the Triangles be b and ehky Lailly, Meafure and let off every Oft-let, as they have occurr’d, and through their Extreams, draw or trace the curved Boundary, as, required. PROBLE M VI. Fig. CXXX. To take the Plan of a Piece of Land , bounded by divers unequal Sides , whofe Angles cannot be all Jecn from any one Point taken within the fame ; as a,b, c, d, e, f, g, h, i, k, n, 1, m. Note, When the Out-Lines of Lands or Buildings cannot be all feen from one Station, we mull have Recourfe unto two, or more Stations, in Manner following: Practice in the Field. Fir(l , Go about the Out-Line, and make an Eye- Draught thereof, and draw Lines from Angle to Angle, to divide the fame into Triangles, as in the Figure. Secondly, Meafure every Side, and note it down in your Eye-Draught on every Side, as in the Figure; and then proceed to delineate the lame, as following: Practice on Paper. Firft, Draw a right Line, to reprefent a c, equal to ya Feet io Inches, by your Scale of Feet, ar.d by Problem II. Ltd. IV. compleat the Triangle a be, making nb equal to qo Feet, and be equal to 47 Feet : Or othenrife, If at 0 you had taken the Off-let Ac, and made it equal to 24 Feet, 6 Inches, at qa Feet diftance from 4, and drawn the Lines bn. The Principles of Geometry. 17s ■ ’ \ they . WOuld a '{° have ™mpleated the Triangle* bac, as before • and whirl, isfometimes neceflary to he fo perform’d, whin, by Water or’o he m - t0 *■ ?■ 4 *■ 4 re equal to 7 r Feet, and Side « e equal to 75 Feet 6 Inches • Alfi, nn re at q, 4I Feet D.ftance from the Angle c, fet off the Off-frt qd - 1 Feet- 6 and compleat: the Fr,angle r, d, e. Fourthly, n k, being fuppofe/in vowEve! Draught to be continue! to r , and equal to ->H Feer tlilf ^ , f^ 6 " ”4°*; sr&ft? f 4 t , equal to 63 Feet to Inches, and Sidee h J 77 Feet' ,o"n£’vfi! 8 the me e h compleat the Triangle A, making the Side eg equal 'to 42 leet and Side g h equal to 53 Feet. Laftly, on the Line ,, a^ the Point ? 31 Feet from the Angle e, fet off the Off fit ff equfl to ro F t’ ;li" •! -/« By this Method, if Care is taken in meafuring truly from one Angle to thac l;Vre l ;3. m COrrC% ^ ^ a » 7 &ch irregular^ pr oblem vii. Fig.cxxxr. A Piece of Land [intendedto be built on .) which is fo verv irretrular /u f r ullits Angles, under lejs than three’Stations as 7 Vf?t f 7 F nVr ' 5 - ’, + ’ i h *7 IO - 9 , being given, to th a Plan th’ereof by Off-Jets., taken from flanonary Lines, divided through any Part thereof l Pleafure, unto the fevsral Angles thereof. & J r tnereoj at Practice in the Field. Firfl , Walkabout the Out-Lines or Bounds and make an Eye-Draught thereof Secondly, Errfl a Station-Staff in any of the End-Angles, as at A; alfo another in any Part of the Field as at^B- a lf„ WLcfTf A s U ; p eWlfe , an ° ther ’ as ac E i anJ , ' a %, another, as at F - Whicn faid Sta.ion-Points do you represent in your Eye-Draught, and draw right Lines from one to the other, as in the Fi^urp r ri- l : c A r from A towards B, and againft the Angle 9, take & the Off-fet, ^'’e^reffing its Length Io Feet, and D,fiance from A 20 Feet Then IrrZld f P r g . 8 - ,h, 4 i.»:« p ,TSd r ;rs B and If 1 t A 1 A , * 5 F =«- Again, go forwards towards from A 4 1 Feet A 1 f? % “ *’a * ? e b Feet ’ andDiftance ,nd TTIltl t ' w S “ for "' ardf > and at d, take the Off-Set d 11, 36 Feet Off fet e a ^ 6°Fe ’ 'l fya' C “ ntlnue A B towards C, and at e, take the let e 3 .6 Feet, and Diltance from A 93 Feet. Thirdly, At any Dilfance from B. fuppofe 16 Feet, make a Mark in the Line A Q at b Ld« 4" fame Dilfance from B, fight 111 a Station-Staff g, and meafure the Line oh - cry e«£Hy, which fuppofe to be thirteen Feet : All which enfer down m your Eye-Draught on their refeeflive Places. FouZfy, meafure I m The Principles of Gfio\i etry. iff Meafure from B towards D, and againft: the Angle 4^ at /, take toe Off f’t fa 51 Feet, and Diftance from B 28 Feet. Again, go on towards D. and'againft the Angle 5, at re, take the Off-fet W 5 1+ Feet, and iMance from P» 4S Feet. Alfo go on towards D, and againft the Angle 12, at take the Off-let i I a 42 Feet, and Diftance from B 47 Feet. L.ikewile meaiure forwards to l, aril there take the O.fffct *6 .5 Feet, and 6a Feet from B. I -lftly Meafure towardsD, and at n take the Off-fet» 1 3 28 Feet, and Diftanc- from B 88 Feet. Fifthly, Set back 16, or any other Number of Feet from D to l and at the fame Diftance from D, fight in a Station-Staff at m, and meafure Im very exaQIy, which fuppofe to be 22 Feet All which Particu¬ lars do you enter down in your Eye-Draught, on their refpedhve I-laceo Sixthly, Meafure from D towards E, and againft the Angle 1 4, at the Point 0, take the Off-fet <114 10 Feet, and Diftance from D 2 Feet. Go on towards E, and againft the Angle 7, at p, take the Off-fet p 7 20 Feet, and Diftance from D ti Feet. Likewise meafure on towards E, and againft the Angle 15, at b, take the Off-fet h 15 15 Feet, and Diftance from D 60 Feet Laftly, Meafure home to E, whole Length from D let be 84 Feet. This done, let 1 6 Feet back from E towards D, on the Line DE at q, and at the lame D.l- tance from E towards F, at s, fight in a Station-Staff and meafure the Diftance sq very exaflly: All which carefully enter down in your Eye Draught Seventhly, Meafure from E towards the laft Station F, and againit the Angle y, take the Off-fet ry 21 Feet, and Diftance from E 5 Feet. Alio meafure on, and againft the Angle b, take the Off-fet 18 21 Feet, and Diftance from E 36 Feet. This done, meafure home to the Station Point r, 62 Feet and fet any Number of Feet back from F to v, fuppofe 11 Feet, and meafure rev, which let be 21 Feet; alio ux, which let be 17 Feet. Laftly Meafure re F 18 Feet, and Fx 15 Feet. Which Dimenfions being carefully entered down in the Eye-Draught, you may mold exaffiy del.neate the Plan thereof as following. Practice on the Paper. Firjl, Draw a right Line at Pleafure, as AB ^’ making A B equal to 95 Feet. Secondly, Set 1 k Feet from B to h, and on B<> complete the Triangle Bhg, making equal to 1 6 Feet, and g h equal to 12 Feet Thirdly, Continue B,j unto D, making BD equal to 99 Feet, and fet 16 Feet back from D to l. Fourthly, On the Line / D, complete the Triangle ImD, making the Side In equal to 22 Feet, and Dm to 16 Feet; and continue Dm unto E, making DE equal to 84 Feet, and fet back 16 Feet from E to q. Fifthly, On the Line q E, complete the Triangle qs E, making the Side qs equal to 27 Feet, and the Side rE equal to 17 Feet; and continue Er unto F, making EF equal to 62 Feet: And thus will you have laid down all your Station-Lines ready for fettmg on the ieveral Off-iets as following: Maked ' Feet. Feet. f' 7 l 'b 9 -I 20 a 1 '5 j 35 1 4» and the Off-fet • CIO *2 equal to. H ! 8 > , 6 7 d 11 i .* 3 26 ) Secondly, The Principles of Geometry - . 177 Secondly, On the Line BD, Feet. 4 S (Rf) B» I 1/ +1 ! rv 5 I Feet. fm Alake -< B i j- equal to-j 48 j- and the Off-fet equal to-i 45 j> l 88 j I B k | v B b J Thirdly, On the Line DE, i* 6 | V.»XJ J ■5 Us; :d»: Feet. C a T Make -^DpS- equal to -JjiVand theCff-fet Jp 7k equal to ■ 'Dh '60 '* r 5- F‘et. ‘to 7 }‘°r . 1 5 J Fourthly, On the Line E F, Feet. Feet. Make^^| equal to and the Off-fet ^ ^ equal to Laflly, Set 11 Feet back from F to v, and on the Line Fv make the two Triangles wvx, and Fiw; fo that rff v ~. r 2 The Sides

at the Points 3:3 ; and the right Line j; 2 being llrain’d to pals through the Points y, 1 , it will cut the Line rx in and the Angle rx7 will be equal to the Angle ghi: And io in like manner all the remaining Angles may be found, as required. And if we are required to make a Plan of the fame alfo, then we mult proceed as following : Fir ft ., Having ranged Lines parallel unto every Side of the Field,.as afore- faid, for thereby to dilcover the Quantity of each Angle, as the Lines 29,3c ; 50, 17; 1 7, 1 1 ; 12,8; 8, 4. ; 4. x ; x s ; s 31; and 3 1, 29. Secondly , Make an Eye-Draught, exprefnng every Side, and every Angle thereof ; alfo the parallel Off lets to each Side. Thirdly, Meafure every Side, and every parallel OfT-fet from each Side, and place down their feveral Meafures on their refpeflive Lines; alio take the Meafure of every Angle, as before has been taught, which alfo place down accordingly. Fourthly , On Paper draw a right Line at Pleafure, as 29, 3T; and therein affume a Point as at 29, and from thence draw the right Line 29, 30, making an Angle equal to the Angle there meafured, and make the Length of the Line 29, 30, equal to the Length meafured on the Ground, as expreffed in the Eye-Draught. Fifthly, Make the parallel Off-fets 22,24.; 21,3°; alfo 25,26; and 27, 28; each equal to their refpeflive Lengths meafured; and through the Points 25,27, draw the Line af ; and through the Points 22, 21, draw the Line ah, both at Pleafure; and then will the Angl ej ah be equal to the Angle 31, 2?, 30. : . . ; L Sixthly , Make the Line 29, 31, equal to the meafured Length expreffed in the Eye-Draught, and at the Point 31, make the Agle 29, 31, p, equal to the Angle meafured, and draw 31 p, at Pleafure. Seventhly , From any two Points in the Line 31 p , fet off the two parallel Off-fets, which make equal to their meafured Lengths, and through their Extremes k and n, draw the Line fg, which will cut af in J; and then will af be the true Length of that Side, and the Angle afg will be equal to the 29, 31, p. Proceed in like Manner to lay down every other Side, Angle, and parallel Off-fets; and then rightLines being drawn through their refpeftive Ex¬ tremes, will interfeft each other, and form all the remainingSides and Angles, as required; and as truly, as if you had free Accefs to mealure into every Angle without Obllru&ion. Note, If when the feveral parallel Lines for determining the Angles had been ranged, you had from any one of thofe Angles (as from the Angle 5) ranged out a right Line at Pleafure, as A 5, for a Stationary Line, and from thence taken Off-fets into every new Angle, as has been already taught in the Iaft Problem, and expreffed the fame in your Fye Draught, you would very readily from thence have laid down the fame by your Scale, with the parallel Off-fe:s alfo ; through which, Lines being drawn, would have expreffed the Plan of the Whole, as required. PROELEM The !'Principles of G eo met r r. r 79 PROBLEM IX. Kg. CXXXIII. If ST 0f T i rr,£u!ar ° f Und ' in tht Uld fl thereof there is a os a b'c d e f Jhfk 1 ,heJOreg0,nS ** P‘“ » »**ice, drawlhffA ■“ ‘f /,f/ l T irfl ’ Make an Ey^ Draugttt thereof, and therein AH" 8 fr0m Ang ' e t0 Ar, g Ie > ( as as nray be,) as in the Fiat,re fpcflfve LKe fUre T^Ar and 0ff - fet i whofe Meafures place on the re-' hems L a , d A f ° many Angles thereof as are necefliry, which ’ A '• ‘ f “ r “t «i«« * makeA'Af-™ ^ ^ :r ‘ Pl, f’ Draw a ri ght Line to reprefent K», which make equal to no Feet, the meal'ured Length; and at ai Feet S1 Fee and 7+ Feet from m, fet off the Off-fets A„ t r Feet 6 Inches; C, 4 Feet’ Get ^ Chord K °V A? ^ AngIe ab m > with a RaditAf to ^and Chord L me of 13 Feet 6Inches, and make ah equal to ,8 Feet. to hf M , eS f P>P °’ ° n ’ and and make *= Angle cbm equal Gne oftf/eeV P ”’ A/-’ A ^ ^ ° f 20 ^ ^ ° ff * cIS ft , r ? eer, as exprelTed m the Eye-Draught, and make be equal to 21 Get 6 Inches. Thirdly, On c , with the Radius of io Feet, fet from the Line be the Chord Line of , a Feet, as unto .v; and draw *e out at Pleafure unto GeGfeVoiT 13 F nu ° n * with the Rad ^ of ,6 throucih F f d V rP" 6 -', I 1 ' h01 ' d Une ' 7 FeeC ’ 33 untoF I and f™ra d, ° ’ r i W * le . r, 8 ht Line d & which make equal to 72 Feet, and ereon at 5 eet, and 41 Feet Diltance from g as at the Points 5 and G fet off the Off-fets 5/ equal to 5 Feet, and G ( equal to I 7 Feet; and then draw the Sides de, of and fg. Fifthly, Continue dg unto h, making e b Goff M ’ki ee V ° n /; ’ WUh / he Radius of lS F “t, from ‘he ujgh let off the Chord Line a5 Feet 6 Inches, as unto H; and draw hH out at ea tire as unto/ making hi equal unto 76 Feet, and thereon, at iq Feet and 5 I Feet 6 Inches, as at the Points * and I, fet off the Off-fets & and Ik making y equal to 1 q Feet 6 Inches, and Ik equal to 18 Feet; and draw the Len&th ‘’l if the W 1 AA’ A “ Z ’ f hkh wil1 com P lete the PIa n, whofe Length (if the Work be truly taken, and truly laid down,) will be found to “ 55 Feet, and the Chord Line of the Angle mlh ,\ith the Radius of 17 Feet 6 Inches, will be equal to 26 Feet,as exhibited in the Eye-Draught. PROBLEM X. Fig. CXXXIV. T ° Uk ‘ the PUa °f an irre £“ !ar Pim of Water fnuated reithin an irregular Field.' Practice.~ Fir ft, Make an Eye-Draught, and meafure all the Sides and fiofe aTth % 7 t iaftPr ° blem ’ “/hmgyout Stations as few as poffiblc; ,0 10 8 S a’ 'l’/ 7 ’/’ 7 ’ and 8; and dra "' the Lines H uro’ner Offf r 4 75 7 k, AA’ ^ 7)8 ; From a H which take the FCH wh Vr nt ° K c? 0i the WaKr > as alfo from the sid “ of the Field Which delineate, as before taught in Problem IV. hereof, and you will complete the Plan, as required. 5 y PROBLEM i Bo The Principles tf Geometry. PROBLEM XI. Fig. CXXXIV To take the Plan of an irregular Piece of Land, by going about the fame Witbout-fide, in a Lane; and to dejcribe the Lane aljo. Pbacticf. Firfl, Make an Eye-Draught by going about the fame, as alio of the Sides of the Lane, as they happen; and then beg.n at any one Ang e thereof, fuppofe at the Angle 14, and meafure the Side 14. I 3 to be 48 Feet. Secondly, To take the Angle 14 1 I> 12 , continue on the Side 13.1 a towa. d^ 47, and make .3. 47,and 13.48,each equal to toFeet and meafure the Chord Line 4*^.4^ F eet : Proceedin like Manner to meafure all the other Sides and Angles, whofe Quantities note down in your Eye-Draught, and afterwards delineate them, as taught in the foregoing Problems. r , f Note, If, as you go round the Field, you take the proper Off-lets from the Side thereof, into every of the Angles of the Lane as are exprelled in the Figure, you may truly delineate the Sides thereof alfo, and compleat the Whole, as required. PROBLEM XI. Fig. CXXXV. To make the Plan of a Serpentine River, Brook, See. Practice. Firfl, Make an Eye-Draught, and affign proper Stations along the Side of the River, as at *, h, e, d, e, f g, h i, k, I, m. Secondly Begin¬ ning at 4, meafure to b, and as you go forward, take the proper Ofi-fets from the Line a b unto the Edge of die Water, whofe Diftances from a, and Lengths from the flationary Line unto the W ater, be careful to place down truly in your Eye Draught. Secondly, Meafure ,n like Manner from* to e, and note down the fame; alfo meafure back from c to a, and then you have the three Sides at, b, e, and r, si, which firming the-Triangle abe, is delineated by Prob. II. Lett- IV. hereof, and truly forms the Angle ab e. Thirdly, Meafure from r to d, alfo from d to 4, and on ae compleat the Triangle aed, obferving, as you meafure the Side rtf, to take the proper Off- lets therefrom unto the Water’s Edge. Fourthly, Continue e d tom and take the Angle m in; alfo meafure the Line d e , and take the proper Off-lets from the fame. Fifthly, Repeat the like Operation, at every of the other Angles, as you fee exprefly in tile Figure, and your Eye-Draught will be compleated. 4 fter which, make your Plan therefrom, by laying down the itationary Lines with their Angles and Off-fets, as before taught in the foregoing Ex- amples, and you will compleat the Whole, ts required. PROBLEM XIII. Fig. CXXXVI. Plate X. To make a Plan of any Town, City, life. The Figure prefented, for this Example, is an imaginary Part of the City of London, began at Aldgate, and continu’d unto the Royal Exchange. Practice. Firft, Range a right Line, as far as can be, through the prin¬ cipal Street as A B, from Aldgate to Leadenhall Street and Cornhill; from which take Off-fets unto the feveral Angles of each Street, which come into the fame on the Right and Left-hand Sides, as from d to k, from p to 0, from r to I The ‘Principles of Geometry. iBt to t, from s to 0, from x to y, from n> to 5; from I to 6, from 5 to 5;, from 8 to 7, from 12 to 64, from 14 to 14, from 18 to 17, from 19 to 20, from 22 to 27, from 25 to 26, from 29 40 28, from 72 to 77, from 75 to 65, from 76 to 77, from 40 to 41, and to 79,, from 44 to 45, from 46 to 66, from 48 to 50, from 51 to 52, from 57 to 67, from 56 to 68, from 57 ro 5-> from 60 to 61, from A to 67, and 62; obferving, as you go for¬ ward, to mealure and fet down on your Eye-Draught the ieveral Didances between the Off.fets; and then will you have taken the true Dimenfions of Cornhill, and Leadenhall-Jlreet ; together with the Enterances in Exchange-Alley, Royal-Exchange, Smthin’s-Alley, Bircbin-Lane, Finch-Lane, Grace-church-Street, Hijbopjgate-flrcet, into Leadenhall-Market, Lyme-Street, St Mary-Axe, Biliter- Lane, Greyhound-Alley, Fencburch-flreet, Crouchet-Friars, and Duh’s-Place. ' This done, draw a Line at Pleafure, and make it equal to the raeafured Dillance from A unto B, and thereon fet off from your Scale of Feet the feveral Dif- tances of the Ofl-fets, and the Length of every Off-fet alio ; and draw the fe¬ veral Lines from one Off fet unto the other, which will form the Sides of Cornhill and Lcadenhall-Street. Secondly , Begin with the next principal Street that comes into the laft taken, as Bijbopfgate-Street; ar.d in the Line A B, as at the Point 71, allign a llationary Point; as alfo another at the upper End of Bijbopfgate-Street, as at 1 ), at which Place ereft a Station-Staff. Thirdly, Sight in another Station-Staff at 70, between the Points 71 and D, at 10 or coFeet from the Point 71; alio fet the fame Dillance from 71 to 27, and meafure the Chord Line 27, 30, the Quantity of the Angle, which note down on your Eye-Draught. ’Phis done, meafure from 71 towards D, and take the Off-fets to each Side at the Angles of the Streets coming into the fame, as at 64, the Off-fet to 65 the End of Threadneedle-Street; alfo at 69 and 73, the Enterance into Crosby-Square ; alfo at .75 and 76, theEnterance of the Paffage but of Bijbopfgate-Street into Broad-Sttreet; alfo at 79 and 80, the Enterance in¬ to Great St. Helens. All which being plan’d according to their refpeclive Dif- tances and Lengths,.and Lines being drawn from one Off-fet unto the other, will form the feveral Sides of Bijbopjgate-Street. In the fame Manner proceed with all the other Streets from one to the other, obferving to plan every Street fo loon as your Eye-Draught thereof" is completed, before you take the next. In fhort, this Method is fo very plain and intelligible, by the Lines drawn in the feveral Streets of the Plan, that it will be but Tautology to fay any more hereof. PROBLEM XIV. Fig. CXXXVII. Plate XI. To take the Plan of a Fault, or Cellar, that is groined over, as bade. Practice. Firft, Make an Elye-Draught thereof, exprefling the Thicknefs of the Out-Walls, and ProjeQion of the Pillallers againft the Sides, and in the Angles thereof, as at FCBA, El, »iF, GH; alfo the Piers at K and L. This done; meafure the true Length of every particular Side and Part there¬ of, and place them to their refpeflive Parts in your Eye-Draughr. It will be alio neceflary to take one of the Angles in a Plan of four Sides, as this Example; becaule fome Buildings are not truly fquare, or right-angled; and when fuch Buildings happen, and you fuppofe them to be fquare or right- angled, and make the Plan accordingly, it will be falfe : Therefore, to be always fure of the Truth, take one Angle; and that being truly laid down, with the Sides proportion'd thereunto, the Plan cannot but be truly deferibed, in Manner as following. Numb. XVII. Yy Practice The Principles of Geometry. i 8*2 Practice II. To draw the Geometrical Plan. Firfl ) Make a Scale of Feet of fuch a Size as will belt fuic your Purpofe. Secondly , Draw a right line, as dc, which make equal to 45 Feet; and on the Ends d and c, ered the Perpendiculars db and ca each equal to 28 Feet, the Dimenlions noted on the Eye-Draught, and draw ba , which will be alfo equal to 45 Feet; becaufs ’cis parallel unto dc, and the Angles at a and b arc equal to the Angles at c and d, and confequently bade is a Para- lellogram. Thirdly , The Thicknefs of the Wall being three Feet, as fignified by the Dimenlion between gn, therefore at three Feet, within the Out-Line bade , draw Lines parallel thereto, as AD, Dp oy , and 1 H, which will re- prefent the Thicknefs of the Foundation. Fourthly , Becaufe the Enterance is in the Middle of dc, therefore divide dc into two equal Parts; and be¬ caufe the Breadth of the laid Enterance is 4 Feet 10 Inches, therefore make hg bp each equal to 2 Feet 5 Inches, and draw fe and^iz parallel unto 4c; and then will you have exprefled the Enterance in its true Situation. Fifthly, Since that the Pillafters have all the fame Projection from the Wall, vi%. each 9 Inches, as fignified at the Pillafter f, therefore at 9 Inches, within the Lines DA, Dp, 1 H, 0 H, draw Lines parallel thereto, as 15, 2, ijm,py, and 2^; which limits the Projections of all the Pillafters againft the Sides at CBEI, FG, and forms thofe in the Angles at I)A 0 H. Sixthly , Becaufe the Diftance of the Pillafter B is 10 Feet 2 Inches from A, therefore fee toFeet 2 Inches from A to 5; alfo from 2 to 7, and draw the Line 5, 7, the Side of the Pillafter. Again, becaufe the Breadth of the Pillafter is 3 Feet, as fignified in the Eye- Draught at B, therefore fet 3 Feet from 5 to 6, alfo from 7 to 8, and draw the Line 6, 8, and lo will you truly have reprefented the Pillafter B in its true Situation. Proceed in like manner to fet off’ the Diftance of the next Pillafter G from B, according to its Dimenlions found, which is 10 Feet 4 In¬ ches and Pillafter 3 Feet, as before; and fo in like Manner all the Remainers at E, F,G, I. Seventhly , When all your Pillafters are truly placed, draw right Fines from the Sides of every one unto its oppolite, as the Lines 7, 25; 8,33; alio 9,2^; 12,22; and likewife 15,27; 17,30; which will interbed each other in the Points 24, 23, 26, 25, and.20, 19, 22, 2^; and form the Balis of the Piers at L and K. Laflly , if right Lines be drawn from the Angle of every Pillafter and Pier unto its dired oppolite, as from 2 to 23, 7 to 27, 8 to 19, 9 to 24, itfe. they will reprelent the Bafis of the feveral Groins of the Arches, over which they ftand perpendicularly, and complete the Plan as required. Note-, When the Lines of your Plan are drawn, fill up the folid Parts thereof with a faint Walk of Indian Ink, fo diftinguifh the folid from the open Parts of the Whole. PROBLEM XV. Fig. CXXXV 1 II. Plate XI. To take the Plan of the Ground-Floor of a Dwelling-H life t as bade. Practice. Firfi, Make an Eye-Draught thereof on a Piece of wafte Paper, and therein reprefent the Out-Walls, with the Windows and Doors, diftinguifhing each from the folid Brick-work; by making the folid Parts black wich your Bla.k-Lead Pencil, and leaving the Doors and Windows white. The Principles of Geometry. _ ]* 1 _ The Doors muft alfo be difllnguifhed from the Windows, by their Sides being drawn parallel to each other, as Imon; and the Windows, with their pides from the Window-Frames, to fall back or open themfelves immediately mto the Rooms, io as to admit of the Light’s faffing freely therein, as the e’ d r f 4 ’, 6 ’ a p 1 5 , 9 , °f the f Wm - d ° w h 6 , 9 ; which is called th eShrr-B.uh, lo dre Skew-Backs of the aforefaid Winders are the Pittances 7,6, and 3 9 being fo much back from the Sides of the Frame, if the fame had been con’ tinned m right Lines from 4 to 7, and from 5 to 8. The Quantity of the Angle neceflary for the Skew Backs of Windows, will be declared in the Lee- ture on the Kinds and Proportions of Windows. LV hi^ in your Eye-Draught you have reprefented the feveral Windows and Doors, then proceed ro divide the Infide thereof into its feveral Parts By I artittons, Walls, istc. expreffing the Doors, or Enterances of each Room m their proper Places; as alio the Chimneys, Clofets, <57. Together with the Stair-Cafe, or Stair-Cafes, when more than one is in a Houfe 3 ; Al¬ io the 1 ortico dc, 89 ,3 J I with the Plans of the Bafe of each'Column as they appear unto your Eye. it’s not material whether yourEye-Draught be truly he e pT t0 f b" r” g ’ ,' Vh ° ie Plan , y0l - lare t0 mike i fur ™ you to make the I lan of a Building that was truly iquare, that is, every of its Sides equal and the Sides of your Eye-Draught were each unequal, it doth not avail any Thing provided that to each Side yen place the juft Dimenfion or Meafure thereof, from whence you lay down in your Plan the Length of each with Exaflnefs. Therefore obferve, That in your Eye-Draught, if you do but ex- prels every Part with its true Dimeniion, and be careful not to confufe vour Dimenfions together, you may with great Pleafure delineate you r Plan as re- quireu, in manner following. Practice Your Eye-Draught being made, and all the feveral Dimenfions taken, and placed to their refpedlive Parts; proceed to draw the Geometrical 1 Ian, as follows : ( 1 .) Having made a Scale of Feet proper for the Size of your Plan draw a right Line dc equal to 45 Feet 6 Inches; and on the Points dc erect the Perpendiculars ca and db, and make each equal to 45 Feet 6 Inches the Dunenlions taken and expreffed in your Eye-Draught, and draw ab- then \\ ill b a dc be the Out-Line of your Plan. Feet. Inch. (a.) Make r ,l 2 ' '7 8 ' 5 7 , I 6 8 I t "'equal toe 5 6 b t IO 6 8 10,1 1 5 7 M b j '7 89 . “wuwa 1 1, and -i with the Door 1 1 alfo. fedat ^ H k ThiCkne E ° f the bebg 3 Feet and + Inches, as expref- ied at the Door 1 t between t jt; therefore at the Dilfance of a Feet and 4 Inches within the Out-Lines, draw the right Lines fc, fb, hr and er which determines the Thicknefs of the Out-walf, (4.) Fmm tl/plts t t l?’! 1 ’ dra ) V "S ht L>nes a, 7; 3,8; ,y; tv; 10jIS; ’ dicular to b a , and parallel to bd; which will determine the Front of each Y tndovv, and Breadth of the Door. Fifthly, Since the inward Face of tfe Window-Frames I The Principles of Geometry. Window-Frames ftands I Foot and 2 Inches inwards from the Face of the Front, therefore make 2,4; 3,5; 10,12; and II, 1}; each equal unto I Foot and a Inches; and draw the Lines 13, 12; and 4,5; reprefenting the fame. Sixthly, Becaufe the Skew-Backs 9, 8; 7,6; are each equal to 1 Foot 1 Inches ; therefore fet I Foot 2 Inches from 7, and from 8 to 7 ; and draw the Lines 4, 6; 5,9; which are the Shew-Backs to the Window 7,2, 5, 4. Proceed to finilh in like Manner all the other Windows and Doors, ac¬ cording to their federal Dimenfions, and then begin with the internal Parts, follows: Firft, Bifefl the Breadth of each Door in the Points r and k, and draw the right Line rk through the Middle of the Plan. Secondly, Becaufe that the Breadth of the Patfag e pq is 8 Feet and 6 Inches, therefore draw 4.5 and xp each at the parallel Diftance of half pq , vi 4, 4 Feet and 7 Inches, making px equal to 21 Feet. Thirdly , Since that the Partition-Wall As; qG is 9 Inches in Tbicknels, therefore draw AG at the parallel Diftance of 9 Inches from ^ q . In like manner draw' the other Partition-Wall xp, and in both ex- ptefs the Doors at PQ, (which the Engraver by miitake has filled up ) RS, and TV, according to their refpeftive Dimenfions. Fourthly, Make GE equal to 17 Feet and 6 Inches; and at E ereft the Perpendicu¬ lar F..v equal to 8 Feet 1 1 Inches. Alfa make ^F equal to 11 Feet 6 Inches, and draw Fx, in which fet oft' the Dimenfions of the Chimney, and then that Room is completed. Fijtbly , Make AD equal to ] 7 Feet and 2 Inches, and on D ereft the Perpendicular CD equal to 8 Feet 9 Inches ; al¬ fo make e B equal to 1 1 Feet 6 Inches, and join BC, in which fet off the Chimney according to its Dimenfions, and fo will that Room be completed alfo. Sixthly, Divide CD and xE, each in the Middle by the Partition, as in the Plan, and thereby each Room is accommodated with a convenient Clo- fet as X and W 1 Seventhly, In the fame Manner, complete the other Room IH, according to the refpeflive Dimenfions taken. Eighth, Draw *24 parallel to kH, which completes the folid Back of the Chimney. Ninthly, As the Stair-Cafe » >f, x-24, is the next in Order, therefore draw the right .Lines 22; 18, parallel to wf 0,21, parallel to 24X; and <1:5, parallel to flip, each at 7 Feet Diftance, as expreffed by the F imenfions. Tenthly, Divide 1 ra, 18, 19, each into as many equal Parts as there are Steps contain’d, (which here is fuppofed to be but 5,) and draw right Lines through each Divifion to reprefent the Steps. In the fame manner divide the Lines 21,20, and 27X, and draw them Steps alio. Eleventh, On the Points 19 and 20, with any Opening of your Compaffes, deferibe two Quadrants, and divide the Arches into 4 Parts, when the Stairs are not very large, or into 5, or more Parts, when they are large as in this Example ; and through the Divifions thereof draw the feveral winding Steps, as exhibited in your Eye-Draught. Twelfthly, Divide 19, 20, and 22,0, in c c, and draw the Line cc; which will complete the Stair Cafe. Thirteenthly, Continue on the Sides hd to 79, and a c to 77, making d 77, and c 72, each equal to 6 Feet; alfo make 77 79, and 72 77, each equal to 7 Feet and 7 Inches, and draw the right Lines 7 7 8 L and 79 88 ' Eourteentbly, Becaufe that the Diftance between the two middle Coloumns is 9 Feet, therefore fet 4 Feet and a half from A to 41, and from A to 44; and draw 41, 47, and 44, 46, each perpendicular to the Line 72, 77. Laflly, Make 72, 74; 77, 75 ; 41,40; 47,42; 44.45; 4 * 1 2 * 4 * 6 > 47 .; 87- 8 6 i and 79, 78; each equal to 7 Feet and 7 Inches; and draw the Lines 74,75; 40,42; 45,47; and 76, 78; they The ‘Principles of Geometry, i Sc; they wiH complete the Plan of the Portico c,d, and then filling up the federal folid Parts with Indian Ink, not too black, which looks rather too hard for the Eye, the Whole will be completed, as required. Note, As this Figure is only laid down for Example fake, the Proportion of its Parts has not been confidered or regarded, that being the Work of the bixch Part. ° These Problems being well underftood, there can no Difficulty arifi in taking and delineating the Plan of any Building whatever: Therefore I ffiall conclude this Part of the Ledure with Plate XII. which I have given for •** * fa " *■" k >' * on.. The next Part of this Leftm-e, is the Manner of delineating the Geometri¬ cal Elevation of Buildings in general, wherein will be comprized every 'filing that is uleful and curious, and which herein will be more extenfively handled than has been yet done by all the Authors who have wrote on the Architecftu- ra Art For as Geometrical Elevations conlift of Windows, Doors, and Inter- cm n fWo “ d ’ Brick or Stone, which are oftentimes enrich’d with Columns, I, afters, Cornices, Entablatures, Rufticks, Key-Stones, and divers other Em- belliffiments; I am therefore under a Neceffity in this Place, to explain and teach their various Conftruaions and Proportions, before I make any further Advance to the delineating Geometrical Elevations : For unlefs the Manner of Drawing the Orders be well underftood, ’cis impoffible ro delineate the Geo- metrical Elevation of Buildings, wherein any one or more Orders are intro¬ duced. And whereas, fince the Time of the antient Archirefls, many Methods and Rules for proportioning and drawing the Orders have been invented by many Atchitefts; in Confideration thereof, and of the various Opinions of leop e, who io differ among themfelves, that every Architefl has his Admirer I ihall therefore, m hopes of giving a general Satisfaflion, prefent the World’ with ail the various Methods and Proportions that have hitherto been pra&ifed • wherein, as I proceed, Ihall give fome general Remarks and Obfervations on the Whole. Before I proceed to this mod curious, and mod delightful Subiefl, I mud advertife my Reader, Firft, That the Conftruflion of the five Orders of Ar¬ chitecture, in this Geometrical Part, is, as before has been obferved, abfolute- lynece lary, as being the Bufinefs of Geometry to teach, and which mud be firit well underftood, before we arrive unto the fixth Part hereof, which will confilt only of the Manner of applying the Orders to various Ufes in the For¬ mation of Defigns for Buildings in general- Secondly, That after the £ve Or¬ ders have been herein univerfally ilfuftrated, I will then alfo illuftrate the Pro¬ portions of Doors, Windows, Chimney-Pieces, and Niches by one General Rule • by which all Kinds will be as eafily performed; as to delineate th eTufcan Bafe when the Meafures of the Heights and Projeftions of its Members are given. And whereas Mr. Gibes has, contrary to the Ufe of Workmen, given a very great Va¬ riety of good Defigns for Doors, Windows, and Chimney-Pieces, without their Mealures being affix d thereto, I ffiall, in Honour to that ingenious Gentleman, and with an affectionate Refpeft to Workmenin general, comprise all thofe his feveral Defigns with general Meafures affix’d to each : By Help of which, every Work¬ man wul be enabled to execute them mold readily, to any Size required, hav- ing the Breadth of a Door, Window, Chimney, or Nich only given To Numb. XVIII, Zz the f e 186 The Principles of Geometry. thefe will be prefix’d the Defigns of Doors, Windows, and Chimney-Pieces by Inigo Jones, in like manner; as likewife of all other Architeils that are worth the Regard or Notice of Workmen. When that I lhall have thus illuftrated the Proportions of thofe Orna¬ ments which enrich the feveral Fronts of Buildings in general, I fhall then proceed to fhew Geometrical Rules entirely new, perfectly eafy, and more ufeful than has been yet publiihed, for framing all manner ■ of regular and irregular Roofs, and twitted Rails to Stair-Calcs; which will conclude the Geometrical Architecture of this LeClure on the various Conftru&ions of Plans, and Geometrical Elevations of Buildings, in general. The Refidue of this Geometrical Part will confift of Le&ures on the following Subje&s, to'h- On the Proportion of a Circle, and its Parts; on the Ratio, Redu&ion, Transformation, and Equality of Geometrical Figures; on the Divifion of Lands; on the Power of Lines, wherein the Reafon of Menfuration is clearly demonftrated; on the Divifion and Proportion of Lines, wherein the Ex¬ tractions of the fquare and cube Roots are demonftrated ; on Arithmetick geometrically performed, in a very eafy Manner by Lines only, to greater Ex- a&nefs than can be done by Figures; on FraQions proper and improper; on Decimal and Duodecimal Arithmetick ; on the Generation of Solids ; on the Menfuration of the Superficies and Solidities of folid Bodies; and laftly, on the various SeQions of circular and elliptical Cylinders and Cones. All which will be moft concifely, fully, and familiarly handled, to the Underftanding of the meaneft Capacity, and both advantageous and delightful to every Lover of Architecture. Now to the Purpofe ; wherein I defire, that every one will confider every Paragraph without Prejudice or Conceit, and not make any Ob je&ions, except that he can prove them to be juft. As the Antients well underftood the beautiful Proportions of the Or¬ ders, I fhall therefore introduce each Order, with their Methods of de- feribing the Five Orders of Archite&ure geometrically, without any Re- fpeft or Regard being had to Models, Minutes, or Parts, as invented by latter Archite&s. And although the Doric was the firft Order of the Greeks, who invented it a long Time before the Tufcan Order was in¬ vented by the Latins ; yet as the Tufcan is the moft maffy, ftrong, and ro buft, and Cuftom has prevailed to place it before the “Doric, fol fhall alfo place it at the Head of this Difcourfe, in the Manner as following : I. Of the TUSCAN ORDER of the Antients. To whom, of the antient Architects, the Honour is due, for the geometri¬ cal Rules of dividing and proportioning the Five Orders of Columns in Ar¬ chite&ure, I believe to be unknown, fince that neither Vitruvius, Palladio, Sca- fnoszi, or Vignola, has taken any Notice thereof, although the Proportions of the Orders of Vignola are nearly the fame, as may be feen by comparing them tovether. But however, as I am pofleffed of their valuable Rules, which are hath delightful and ufeful, I lhall therefore communicate them for the Publick Good, in Manner following: But before I enter upon the Orders of Archite&ure in particular, I think it is very reafonable, that I fhould in the firft Place explain them in gene¬ ral ; and afterwards diffe& them feparately in their Turns. And as I am certain all the Ev p? of Mankind arc on me, to behold the D C &ion of this The Trinciples of Geometry. 1 87 this Part, I fhall therefore, without Favour or Affeftion to any, difplay the Beauties and ImperfefKons of thofe Authors, who are the Subjeft of this Difcourfe. Of an ORDER, and its Parts. An entire Order of Architecture, be it Tufcan, Doric , Ionic , Corinthian , or Compofte, confifts of three principal Parts: AsGHI, Plate XIX, vi%. I the Pedeftal, H the Column, and G the Entablature ; which are feverally di ¬ vided into three principal Parts alfo, as in Plate XXI. where the Pedeftal V N W is divided into its Bafe W, its Die or Cube N, and its Cornice or Capital V. The Column RST into its Bafe T, its Shaft or Fuft, S and its Capital R: And the Etablature O PQ_ into its Architrave Q^, its Frize, P, and its Cornice O : All which are compofed of divers Parts, varioufly divided, called Members or Moldings, that are either right-lin’d or curved. The right-lin’d Members or Moldings are called Plinths; as 7; and B, Plate XX. or Fillet, as A ; or Lift, asS; or Abacus, as kl; or Architrave, as i h; or Frize, as g ; or Corona, as d; or Plat-Band, as B, Plate XIX : Which in general differ in their Names, according to their Situations. The curved Moldings are of two Kinds, -viz- Single and Compound; in each of which, there are two Varieties, viz- Firft, In the fingle curved Moldings, they are Convex, as tbo Ovolo E, Plate XVII. and Concave, as the Caveto G under¬ neath it. And in the compound curved Moldings there are Cima’s, or Ogee’s of two Kinds; as B, Plate XVII. which is called Cima Re£la, or the Fore-Ogee- and as f Plate XX. which is called Cimafium, or Cima Reverfq, or the Back- Ogee. And whereas the feveral rig'nt-lin’d Members are, in their Angles, truly fquare ; therefore from thence ’tis evident, that of Moldings, there be but three Kinds that are abfolute, that is to fay, in common Terms, the Square, the Hollow, and the Round ; and he who underftands how to apply them well together, may juftly be efteem’d a good Judge of the Orders in Architeflure. Before that we proceed to the Application of thefe Parts together, we flrould confider how to give Dimenfions to each, proper to the Ufes for which they are defign’d; as that of being more or lefs ftrong, and capable to fuflain a great Weight ; or more or lefs capable of receiving thofe Embel- lifhments, that are requifite for the Ufe, and agreeable to the Situation of the Building. The Proportions of Columns have their Differences in their Heights, as be¬ ing higher or lower, and of equal Diameters. Thus to the Tujcan Column, Palladio gives feven Diameters; And, as Perrault obferves, the Form of their particular Members, proper to their Proportion, takes its Differences from the Plainnefs or Richnefs of the Ornaments of their feveral Parts. It was from hence, that the three Orders of the Antients, namely, the Do¬ ric, Ionic, and Corinthian, were confider’d : For we find that the Doric, which is the fhorcelt of the three, and the raoft maffy, has in all its Parts a plain, but noble 188 The Principles of Geometry. noble Afpe£t; although its Capital has neither Volutes, nor Leaves. And on the contrary, the Corinthian, which of the three is the higheft, has in its Capital, Enrichments of Leaves, and Volutes, with Modillions in its Cor¬ nice, adorn’d with Leaves alfo ; which, with its fluted Column, continuing the Side a 19 towards rv, and af, towards x, and ma¬ king ah and an, each equal unto the given Height bn; then the Triangle ahn will be equilateral alio; and right Lines being drawn from a, through the fe- veral Points at 1,2, 5, 4, 5, 6, 7, 8, &c. in the Line 19/, will divide the given Height bn into 19 Parts alfo; of which, give 4 to tn n, the Height of the Pedeflal, three to bg, the Height of the Entablature, and 12 to gtn, the Height of the Column. Or otherivife, Let bn be the the given Height as before. Practice. Firfl , Bifleifl h n in i, and on the Point i, ere£I the Perpendicu¬ lar i k, of any Length at Pleafure. Secondly , Make the Angle i n k equal to 50 Degrees, and draw n k , cutting i k in k. Thirdly , From k draw km, ma¬ king an Angle of 45 Degrees with the perpendicular Line i k ; then will m n be the Height of the Pedeflal, and b m, the Height of the Column and Entabla¬ ture, which divide into live equal Parts, by Problem XXII. of Lett. II. here¬ of? and giving the upper one to the Height of the Entablature, the remain¬ ing four will be for the Height of the Column. Now feeing that the Entablature bg is one fifth Part of hm , it is al¬ fo equal to one quarter of the Height of the Column. Upon fome Occa- fions, the Antients abated the Height of the Pedeflal, making it equal unto The 'Principles of Geometry. 189 one fourth Part of the Entablature and Column taken together; and there¬ fore ’tis equal to one fifth Part of the Pedeftal, Column, and Entablature, taken together; which, in my humble Opinion of the two Methods, is the moll: preferable, as being readily perform’d, by dividing the Height of the en¬ tire Order given into five equal Parts, of which the lowermoft one is the Height Oi me Pedeftal; and then the remaining four Parts being again di¬ vided & b C 5 ? equal Parts, give theC Tujcan and Doric, C J upper one to the {ionic, Corinthian, and Compofite[ ■ Entablatures, f Tufcan I Doric and then*< Ionic the Compofitc I Entabla- ! , r | of the Height ^turewillJt 1 of the Column, Compofitc J U] C0 two and Tone ) Diame¬ r 45 cwo ter or 00 < one y Mo- ■ 4 * j two | dule, 00 ItwoJ and co . -Minutes. These are the Proportions which the Antients p. ftifed, and Palladio ob- ferved, and which Mr. Gibbs recommends in his New Rules ier Drawing. Folio 4. J a Having thus eftabhfhed the ancient Rules for dividing an entire Order into its Pedeftal, Column, and Entablature, according to an/Order, I Ihall in the next Place proceed to the Manner of dividing Tedeftals into their The-< Columns into their vEntablatures into their C Babes, , vGeometrical Lines. W VM b3 7Equal Parts. A a a Plate The Principles of Gkomktry. 190 Plate XIII. Fig. CXXXI Let k y F be the given Height. Practice. Fit ft. Draw the Lines ao, LE, through the Ends or Points k, F, parallel to each other, and at right Angles unto the given Line. Se¬ condly, Draw a F, whole Angle a F k be equal to thirty Degrees; and from F draw the right Line Fir, making the Angle kF rv equal to the Angle aFh; alfo from the Point A, draw kn> perpendicular unto Ftp, and draw a a? cutting AF in m. Thirdly , From k draw k ^ parallel unto af , and from m let fall the Perpendicular ml. Laftly, Through the Point /, draw the right Line gv, which is the lower Part of the Capital or Cornice to the Pedeftal ; and then drawing the Line H 1 8 parallel unto LE, at the fame Diftance from LE as g t is from the upper Line b 0, you will have divided the given Height into the Bafe, Die, and Cornice, as required. Note, By this Rule the Ancients made the Height of t fi e Bafe and of the Cornice equal, and which in fome Cafes might have an agreeable Elfe£l; but I think far fhort of that Beauty which is feen in the ; r other more general Method, which is to divide the given Height of the Pedeftal in¬ to four equal Parts; of which give half of one Part to the Height of the Capital, and one whole Part, and one third Part thereof, unto the Height of the Bafe, as exhibited in Plate XX V. This Manner of di¬ viding the Height of the Cornice and Bafe is obferved by Mr. Gibes, and which is very reafonable fhould be fo, fince that as the Bafe is the Foun¬ dation and Support of all the reft, it fhould therefore confiftof Strength fuperior to the Cornice, which is no more than a Covering to the Whole. The Bafe having one Part, and a Third, for its Height, is divided as fol¬ lowing : To the Height of the Plinth one Part; and the one third Part re¬ maining, being dn ided into fix equal Parts, give one to each Fillet, above and below the Ogee, and the remaining four Parts is the Ogee. Thf Cornice being one half Part of a fourth Part, or an Eighth of the whole Height of the Pedeftal, is divided into fix equal Parts; of which give one to the fillet, two to the Cima Reverfa. and the remaining three to the Platband. But more of this hereafter, when I fhali fpeak of the Order in general. PROBLE M XVIII. Plate XIII. Fig. CXXXI. The Heght of the Die of the Tufcan Pedeftal being given , to find its Diameter or Breadth geometrically. Let NISI be the given Height. Practice. Firft , Bifedl NM in y, and draw xi 7 through the Point y, making the Angle xyk equal to 45 Degrees, and continue the Line H 18, un¬ til it meet the Line x 1 7 in the Point j 7. Secondly , Bifedt M 17 in the Point 1 5, from which let) fall the Perpendicular 1 5 O on the Line x if. Thirdly. Through the Point O draw the Line «D parallel unto ayf‘ } alfo on the other Side, draw the Line iG at the fame parallel Diftance; and thenar, be¬ ing drawn parallel unto H 18, the Line hv, or GD, will be the Diamecer or Breadth of the Die, as required. PROBLEM The Principles of Geometry. PROBLEM XIX. Plate XIII. Fig. CXXXI. The Height of the Bafe to the Tufcan Pedeflal being given, to divide it into the Billet and Phntl). Let xD be the given Height. Practice. Firfl, Make the Angle CD* equal to qo Degrees, and draw the Line CD; which bifefl in B, and frumB raile the Perpendicular B A. Se¬ condly, Bifefl * A in the Point V; then will * I 6 be the Height of the Fil¬ let, and 1 6 D the Height of the Plinth, as required. . ° r otherwife. Divide *D, the given Height, into fix equal Parts, and giving one to the Billet, the remaining five fhall be the Heighc of the Plinth, as required. PROBLEM XX. Plate XIII. Fig. CXXXI. Vie Height oj the Bafe to the Tufcan Pedeflal being given, to find its Projettiou before the Upright of its Die geometrically. Let * D be the given Height. Practice. Firfl, Make the Angle CD*equal to qo Degrees, and draw CD, and bifefl; it in B ; alfo raile the Perpendicular BA, as before done in the lall Problem. Secondly, On D, with the Radius D A defcribe the Arch A E; then will DE be the Projeflion of the Plinth, as required. Thirdly, The Bafe being before divided into its Fillet and Plinth, therefore from the Point E draw the Line E ao parallel unto 16 D, and then bifefling 16, ao, in 19, the Point 19 will be the Projeflion of the Fillet. Fourthly, Continue the Height of the Fillet 19, 18, unto 11, making 18, 1 1, equal to x 1 8 ; al¬ fo make * 14- equal to 11, 18.; and on the Point 11, with the Radius 1 1 10, defcribe the Hollow of the Die : And thus will you compleat tile Pro- jefture and Forms of the Members, to the Bafe, as required. PROBLEM XXI Plate XIII. Fig. CXXXI. To divide the the Height of the Cornice of the Tufcan Pedeflal into its Cima- Reverla and Fillet. Let i h be the given Height. Practice. Make the Angle b hi equal to qo Degrees, and draw the Line bh, which biffefl in e and on e, erefl the Perpendicular ef; which will di¬ vide the given Height i h inf; and then will if be the Height of the Fillet, and and / h the Height of the Cima lleverfa, as required. P R O B L E M XXII. Plate XIII. Fig. CXXXI. The Height of the Cornice to the Tufcan Pedeflal divided into its Fillet and Ci¬ ma being given , to find the Projeflion of the Fillet, and to defcribe the Face oj the Ogee, or Cima Reverfa. Let the Line a 5,4, be the given Height of the Fillet and Cima Reverfa, di¬ vided at 1 ; and let it alfo reprelent the Faceor Upright of the Die. Practice. I 92 The Principles of Geometry. Practice. Make the Angle 21, 4, 25, equal to 30 Degrees, and bifc£l the Line a 1, 4, in 22, from whence raife the Perpendicular 22, 24, which will cut the given Height 25, 4, in the Point 1, at the Divilion of the Cima and Fillet, and the upper Part of the Fillet 25, 24, in the Point 29, which faid Line 25, 24, is fuppofed to be drawn beforehand, at right Angles, to the Upright of the Die 25, 4; as alfo the Line 1, 10, from the Point r, paral¬ lel unto 25, 24, and of any Length at Pleafure. Secondly., ’Tvaia fix d Rule of the Antients, for to give to their Members a Projection equal to their Height ; which is undeniably the mofl jufl and beautiful , and 1 here/ ore to be always obferv d. This being underilood, make 25, 24, and 1, 10, each equal to 1, 4, and draw 24, 10, which determines the Projection of the Platbai.d, or fillet. Thirdly , From the Point 23, draw the Line 23, y, parallel 10 24. 1 , which will cut the Line 1, 10, in the Pointy, which is tne PiojeCtiou of the Cima Reverfa , and draw the Line 10. 4, which bifeCl in the Point b. F unh'y. Bi- feCl the Height of the Cima 1, 4, in the Point 5, and draw the Line A A pa¬ rallel unto 1, 10 alfo draw the Line 4, 3, out at Pleafure, and parallel to the Line 5, 8 . Fifthly , Draw the Line 5, 6, making an Angle of 30 Degree-, with the L : ne 1,4; alfo draw the Line 8, 7, making an A gle of -}o Degrees, with the Line 5, 8, which will mterfeCl the Line 5, 6, in the Point 2, from which draw the Line 2, 3, parallel to the Line 1, 4, and will cut the Line 4, 3, in the Point 3 ; and there limits the lower Part of the Cima. Lafi/y, Draw the Line y, 3, and on each half Part thereof, erect equilateral Points, and deferibe the Face of the Cima Reverfa, as required. PROBLEM XX.IF Plate XIII. Fig. CXXXI. A Height being given, to proportion and compleat a Tufcan Psdeflal thereto. Lit k F be the given Height. P. ACTICI. f By Problem. < XVII. Divide the given Height into its Bafe , Die , and Cornice. XVIII. Find the Diameter or Bre dth of the Die. XIX Divide the Bafe into its fillet and Plinth. XX Find the Projection of the Bale and Fillet. XXI. Divide the Height of the Cornice into its Platband, and Cima Reverfa. XXII. Find the Projection of the Platband or Fillet, and deferibe the Face of the Cima; which be¬ ing done, will compleat the Pedeftal, as re¬ quired. PROBLEM XXIV. Plate XIV. Fig. CXXXII. The Height of the Tufcan Column being given, to divide it into its Bafe, Shaft, and Capital geometrically , Let bo be the given Height. Practice. Firfl, Bife£t bo ini, and from i draw the Lines ai, and di, each making an Angle of 50 Degrees, with the Line b i. Secondly , From the Point b, let fall the Perpendicular be, on the Line di, and draw the Line/?*?, which will cut the Line bo, in c , then will be be the Height of the Capital, and equal unto one fourteenth Part of the whole Height. And whereas the, ancient Rule for making the Height of the Bafe to the Pedeilal, equal to its Cor- The ‘Principles of G eo metrt. 193 nice or Capital, fo likewife they made the Height of the Bafe to the Column equal to the Height of the Capital thereof; and therefore make no, the Height of the Bale, equal to be, the Height of the Capital; and then will the Column be divided into its principal Parts, as required. PROBLEM XXV. Plate XIV. Fig. CXXXIII. The Height of the Tufcan Bafe being given, to divide it into its Plinth, Vents, and Cinttut e, geometrically. Let ar be the given Height. Practice. Firfl, Bifecl arm n, then {hall nr be the Height of the Plinth. Secondly, From » draw the Line xn. making an Angle of 50 De¬ grees with the given Line an. Thirdly, ‘ raw xb through the Point a and at right Angles to ar, which will cut the Line xn in a-. Fourthly Bifid xa in and from s; draw gc, making an Angle of yo Degrees with .1 Line s:<*, which Line will cut the given Line in the Point c; then will at be the Height of the Cin&ure, and cn the Height of the Tor us, as required. PROBLEM XXVI. Plate XIV. Fig. CXXXIII. The Height of the Tufcan Bafe divided into its Plinth, Torus, and Cincture, being given, to find the Projection of the Plinth before the Upright of the Column ; as alfo the Center of the Torus, and Projection of the Cinilure. Let ar be the given Height, divided into its Members at 11 and c. Practice. Firfl, Draw the Line or, making an Angle of jo De¬ grees, out at Pleasure. Secondly, From the Point n, let fall the Perpendi¬ cular np on the Line or; and from the Point p, let fall the Perpendicu- _ lar pq on the given Line ar, which will cur it in the Pointy. Thirdly, On the Point r, with the Radius rq, deferibe the Arch qs, which will cut the bottom Line of the Plinth in s, Projection; and if from the Point s a right Line be drawn parallel to a r, alfo from the Point n draw the right Line nm parallel to rs; they will interfeCl each other in the Point m, and complete the Plinth. Fourthly, From the Point a draw the right Line at, making an Angle of 30 Degrees with the Line nr, and bifeiffc 1 ;; in g ; from whence draw the Linear parallel unto nm, which will cut rhe Line «/ in the Point i, the Center of the Torus. ' Fifthly, Let full a Perpendicular from the Point g on the Line a I, which will interfeCl it in/. Sixthly, Through the Point / draw the right Line bh parallel to ar, which will determine the ProjeCture of the Cincture at bd. And thus you fee with what Beauty and Accuracy Geometry has divided and completed the Bafe of the Tufcan Column. PROBLEM' XXVII. Plate XIV. Fig. CXXXVI. To deferibe the hollow or concave Part of a Column next above the Cincture, as cl e. Let eigb be the CinCture to the Shaft of a Column, and let ah be the Upright of the Column. Practice. Firfl, Continue ge to c, and make fd and ec each equal to the Projection ef. Secondly, On the Point c, with the Radius c e, deferibe the Arch or Quadrant ed', which will complete the Concave, as required. Nome. XIX. Bbb PROBLEM j i94 The Principles of Geometry. : ■ 5 •: PROBLEM XXVIII. Plate XIV. Fig. CXXXV. To proportion, divide, and defcribe an Aftragal about the upper Part of the Shaft of a Column . Note, The Antients divided the Semidiameter of the Column at its Bafe into fix equal Parts, and made theHeight of its Aftragal equal to one of thofe Parts, •»«;. five Min. or one twelfth Part of the whole Diameter. Let am reprefent the Upright of a Column, and equal unto one Twelfth of its Diameter, to be divided into its Aftragal and Fillet, or Lift. Practice. Firft , Draw the Line ac perpendicular unto am, and make nb equal to am. Secondly , Draw bd parallel to a m, and md parallel to ad ■ alfo draw the diagonal Lines ad and bm; then will bd be the Limits of the Projection, which is equal to the Height, becaufe that abmd is a geometrical Square. Thirdly, From the Point m draw the Line mk, making an Angle of qo Degrees with the Line mb, which will cut the Diagonal ad in n ; on which Point, to the Line mk, raife the Perpendicular ni, which will cut the other Diagonal in h, the Center of the Aftragal. Fourthly, Set the Di- ftance from o, the Interfeflion of the two Diagonals, unto h; from o towards m, on the Semi-Diagonal om, as unto p; and draw the Line If parallel unto md , which will divide the Lift from the Aftragal. Laftly, Draw oje parallel to am, which will inrerfe <3 If in f and terminate the Pro¬ jection of the Lift; then on the Point h defcribe the Face of the Aftragal, as required. To defcribe the Concave or Holloiv under the Li/ 1 . ContiisLe down fe unto g 0 making eg equal to m e • and then on f, wuh the Radius g e, defcribe the Hollow, as required. PROBLEM XXIX. Plate XV. Fig. CXL. The Height of a Tufcan Column being given, to find its Diameter, and dimimjh its Shaft geometrically. Let di 8 be the given Height. Practice. Firft, By Problem XXIV. hereof, divide off the Height of the Bafe 22, 28, and Capital bd. Secondly, Since that the Height of the Capital Is equal to one fourteenth Part of the whole Column, and fince the Height of the Bafe 22, 28, is equal to the Height of the Capital, therefore the Heights of the Capital and Bafe taken together, are equal unto one Seventh of the whole Height ofthe Column, and consequently the remaining Part d 22 is equal unto fix Sevenths of the whole Height; therefore divide the Shaft into iix equal Parts,. as at the Points 18, 1 5, 10, a, rj and then bifefling each of them, their Points of Bi lection will be the Centers of fix Circles of equal Ra¬ dius, which defcribe as in the Figure : Where alfo in the Places of the Bafe and Capital are Semicircles, which denote, by Inlpeflion, that their refpeflive Heights, are each equal unto half of one of the fix Circles, or a fixth Part of the Shaft. Thirdly, If from the Extremes of the two Semicircles at a c, and - 7 1 y 9 , the right Lines at 7 and c 2y be drawn, they will touch every Cir¬ cle in its Circumference, but will not interfeiftit, and thereby form the Body The \Principles of Geometry. >95 of the Column without Diminutionand then ’tis plain, that the Diameter of either of the Circles, or Semicircles, is equal to the Diameter of the Co- lumn required. From Problem XXIV hereof, with this, is feen the Reafon why the Tiijiari Column is made of feven Diameters complete. For it is by Problem XXIV. tnat one Seventh of the whole Height- is divided off geometrically to the Bale and Capital; and by this Problem the remaining lix'are divided into its own remaining even fix Parts, of which one is the Diameter of the Co- lumn required. Thus much by way of Reafon, why the Tujcan Column hath been efta- bldhed and made of feven D.ameters high, which no one of our modern An- thors have accounted for, they having been contented with telling their Readers asMr. Gibes has done in his New Rules oj Drawing, Page 5. wherein, fpeaking of the general Proport,ons of the Tujcan Order, he fays, after having de- ter nun d the Heights of the Pedeftal and Entablature, the remaining (half “ be ‘ T H J ei § ht °J r tbe 0>lum including hs Baje end Capital, and this Height being ■ a,ruled into [even Parts, one jhall be the Diameter or Thicknefs of the Column" And further adds, ‘ The Bafe and Capital are each in Height one Semidameter of - the Com But gives no reafon why they muft be fo, nor can their Heights be triarchy known, before the Thicknefs or Diameter of the Column is known. It has been by this imperfed Wanner of Writing, that young Stu¬ dents m Architecture are difabled of giving a Angle Reafon for any one Onera- non they have performed t My next Buflnefs is, to fliew the Manner of diminilhing the Shaft of the Column; which mail exhibit in two Manners, and begin with the moft antient, that to Vignola was well known, and which he has recommended in Ins Treatije on the live Orders oj Columns in Architecture. I cA icx° T find that the Antients did ever aflign any other Reafon, that ar¬ tificial or made Columns fhould be diminifhed, than that to diminiih them IS to bring them as near unto natural Columns as is po/Eble. By natural Columns, I mean the Bodies of ftrait well-grown Trees, which ’tis natural to believe were hr ft ufed by the Antients, for the Support of their Buildings before the Ufe of Stone was known : And as Trees in their Growth are naturally diminifhed, by the upper Parts being further and further remote from the Roots, and thereby receive leffer and letter Nourilhment, according to the Diftances of the Parts from the Roots; therefore, in Imitation of Nature, it was, that afterwards, when Columns came to be form d by the Hands of the Artificer, he then diminifh’d their upper Parts accordingly. I find by the Tujcan Works of the Antients, that they had not any efta- bhihed Quantity of Diminution; for in fome of their Buildings they dimi¬ nished a fourth Part of the Diameter at Bottom; in others a fifth, a fixth, and in fome a ninth Parr, as in the Trajan Column at Home ; which Diffe¬ rence, I fuppofe, arifed from the Difference in tile Heights of their Buildings, for to very high Buildings, as that of the Trajan Column, whofe Diminu¬ tion is but one ninth Part of its Diameter at Bottom, its own Height caufed it to appear diminifh’d much more than it really is; becaufe the Parts there¬ of are feen under letter and letter Angles, as they arife higher and higher above the Eye; and therefore feems to be diminiih’d, as I laid before, more than The principles of Geometry. tl n in Fafl it doth. The Reafon vhf CAumv.s do tints appear 2, 7 8 8 9 3 4 4 5 1 y x equal to. X I I w TV H r 1 l l r v» 4 l k D.J and then, through the Points n p ty 37 11, 16, draw the Contour or out Line, which completes the Shaft di- minifli'd as tequired. PROBLEM XXX. Plate XV. Fig. CXXIX. To Diminifh the Shaft of the Tufcan column after the ufital Method of Mo¬ dern Arch/tecis. Let b 20, he the Height of the given Column. Prafticc, Firft, Draw the Line 19, 2t, through the Point 20, and at right Angles to the Line 20, d. Secondly , Make 18, 20. and bd each equal to one Fourteenth Part of the given height b 20, and draw 11, 12, and e f Parallel to the Bafe 19, 21; and then will you have divided off, the Heights of the Bafe and Capital. Thirdly , Make C 18, equal to one Third part of d 18, and through C draw y 7, parallel to 19, 20. Fourth¬ ly, make 19, 20 ; 20, 2 1 ; y, C; C 7 ; each equal to 1 8, 20 ; and draw the Lines y 19, and 7. 21. Fifthly, Forafmuch as the Tufcan Column is diminilh’d at its Top, one quarter part of ics Diameter at the Bafe,as hath been already obferved, therefore, make h i, and i 4 , each equal to Three fourths of 19, 20 ; and then will h 4 - be the Diameter of the Column at the Top, under the Aftragal. Sixthly , on the Line y 7 make D C and C E, each equal to b i, the Semidiameter at the Top, and draw the Lines h D, and 4 E ; alfo on the Point C, with the radius y C, deferibe the Semi-circle y w 7, cutting the Lines h D, and 4 E in the points 2 and 3. Seventhly, Divide the centeral Line i C, into any Number of equal parts Suppofe, four, as at the points, n q v, through which draw right Lines at Pleafure parallel to y 7. Eiglotly, Divide the Arches y 2, and 3, 7, each into as many equal parts as you divided the centeral Line d C ; which in this Example are four, as at the points xi 1, and 4, 5, 6, and then draw the Lines, 23, 1, 4 ; 2. 5 ; x 6 ; y 7; Ninthly, Make v t, vs. each equal to B 6 ; alfo q r, p y, each equal to A 5 ; alfo no, in n ; each equal to 9, 4; Tenthly, Through the Points h m p s y, and k^o r t 7, draw the contour or Out-line of the Shaft, as required. PROBLEM XXXI. Plate XIV. Fig. CXXXIV. The Height of the Titjcan Capital being given , to divide it into its feveral Moldings. Let a m be the given Height. PraSiee, Firfi, Divide a rv into three equal parts at m t, and draw in ot v and y, at right Angles to am. Secondly, Make the Angle wax, equal to 30 Degrees, and draw a x, Secondly, from x to t, draw x t, con¬ tinued at Pleafure towards p. Thirdly, Make the Angle bat, equal to 20 Degrees, alfo the Angle a m i, and from i, through the point m, draw NUMB. XXIV. C c c the The Vriniiples of Geometry. 198 the Line i q, curcing the Line p x in q. Fourthly, Make w 0, equal ro rut, and draw the Line 0 t. Fifthly, From the point q, draw the Line q j\ cutting the Line 0 t in s. From t draw t>v, parallel to rs , and from a, draw s v, parallel to r t, and then will the Lift or Annulet of the Ca¬ pital be form’d. Again, from the point/, draw the line if, out at Pleafure. parallel to tv y, aifo the line a h , parallel thereto. Sixthly , from the point 0, draw the line c 0, parallel to zr, cutting i c in d, and in c. Make c b, and each equal to c cl , and draw b e, then is the Upper fillet of the Abacus Form’d. Seventhly, Continue b e to f, making e f equal to b 0, and on f with the radius e f, deferibe the Arch e g, which completes the Hollow to the Fafcia of the Abacus. Tightly, Bife& a A in h^, and from thence raife the Perpendicular / lg, cutting a tv in /, which is the Center of the Gvolo, on which with the radius / j, deferibe the curve, and then will the Capital be compleated as required. PROBLEM XXXII. Plate XV. Fig. CXLI. The Height of the Titfcan I ntablature being given, to divide it , into its Archi¬ trave Freeze and Cornice. Let a s, be the given Height. Practice, Firfi , Through the point s, draw the line t s 7 0, at right Angles to the given line, and from the point a , draw the right line a 0, making an Angle of 30 Degrees, with the line a s , untili it meet the line t 0, in 0. Secondly , Bife& a 0, in /, and at /, raile the Perpendicular / m t: cutting a s , in m, and of any length at Pleafure. Thirdly, Draw j P making an Angle of do Degrees, with the Line a s, which continue untill it meet the Line l1 in P. Fourthly , from nr draw the Line m n , pa¬ rallel to s 0, and from P to draw the Line P cutting a j, in q. Fifthly, Bile# m q, in p, then is p s, the Height of the Architrave ; Fa fly, Bift& a q , in /, then pi , is the Height of the Freeze, and i a , the Height of the Cornice, as required. PROBLEM XXXIII. Plate XV. Fig. CXLI. To Divide the Tiifcan Architrave, into its Tenia and Fafcia ancl Deferibe the Fro fie thereof. Let X 7. be the given Height. Pra&ice, Firfl, Thro’ the point x, draw the line p y , at Pvight Angles thereto, and of any length at pleafure. Secondly, From the point 7, draw the line <7 7, making an Angle of 30 Deurces, with the linex 7, which continue untill it meet the line p x in q. Thirdly, Bifc£f q 7 in 4, and thereon raife the Perpendicular 4 3 ; cutting the line, x 7, in the point 3. Fourthly, Bifcft x 3 in and through the Point a,, draw the line q a,, at pleafure. Fifthly, Make xy , and z. 2, each equal to x^and draw y 2, which terminates the Face of the Tenia or Lift. Fafily, Deferibe the Hollow or Arch 3, 2 * as deferibed for The Principles of Geometry. 1 99 for the Hollow of theCin&ure in Fig. CXL.II. and the Architrave will be completed as required. This is likewife exhibited at large in the Architrave on Plate XVI. PROBLEM XXXIV. Plate XV. Fig. CXLI. To Divide the Tuscan Cornice, into its Cima, Corona, and Ovolo. Let a z, be the given Height. Pra&ice, Firfi, Make the Angles b i a , and h a i, each equal to 30 Dc grees, alio make the Angles c a i, and g i a, each equal to 60 Degrees, and draw the lines forming the faid Angles, untill they Interfcft each o- ther in the points e and b m , from whence draw the light lines c e, and h f, at right Angles thereto, which will divide the given Height at the points e and f-, and then will f z, be the Height of the Cima ; f e the Height oi the Corona j and e a , the Height of the Ovolo , as required. Note, As the Triangle a c i, and a b z, are equal by having their Angles corcfpondently equal, therefore their Perpendiculars c e , and h f, which di¬ vide the given Height into its Cima , Corona, and Ovolo , doth alfo divide off equally the Parts a c , and f i, each being equal to one quarter of the given Height-, and hence it is, that the Cima and Ovolo , are equal, and which be¬ ing taken together, ace equal to the Corona, that is compriz'd between them. PROBLEM XXXV. Plate XVI. Fig. CXLIII. To Describe tbe Profile of the Tic can Cornice. Let b 8. be the given Height. Praflice, Firft, Divide off b x, for the Height of the Cornice, and t 8, for the Height ot the Cima, as by the laft Problem, and draw out the line b e and x A. at Pleafure. Secondly , Bifefi x a. in 2, and make x t , equal to one fourth of x 2, and draw t o , Parallel to b e , at pleafure alio. Secondly, Make the Angle i x x, equal to 6o Degrees, and draw x x, untill it cut the line t o, in x. Thirdly, Bifefl x x in y, complete the equilateral Tiiangley I x, and through the Point r, draw the line v p, at Plralure ; then will you have determin’d the Height of thcaftragal and Fillet. To Determine the Trojechire. Firjl, From m, draw the line w ly A, making an Angle of 6o Degrees with the line b w, which continue untill it meet the under pai t of the Ovolo at A. Secondly , From the Pointy, of the equilateral Triangley x I, draw the right line J'y h, out at pleafure, and from the point A, let fall the Perpen¬ dicular A b,on the line / h, then will the Point 6, be the Center of the allra- gal or BaqHdte, as Term’d by SebeJUan le Clerc. Thirdly, 200 The “Principles of Geometry. Thirdly, Make h i, equal to f v, and from the Point i, draw the line i y 20, parallel to the line x which will determine the Proje&ion of the Corona. Fourthly, Draw i ;//, parallel to x w, and make \n equal to : Alfo make l^o, and ni p, each equal to m n , and draw op, the Face of the Fillet. Fifthly, Continue i m, to 20, making m 10, equal to v rt>, and draw the right Line rv 20. Sixthly , make 2 a , equal to / 2 and from the point a, draw the Line a 18, parallel to rv 20. Seventhly, Make 15 18, equal to twice 18, 20, and from the point Z^draw the Line h^g. Making the An^le 20, A, 9, equal to 30 Degrees. Lightly , Bifefr 10, 15, in I and on the point 13, raile the Perpendicular 13, 14., Ninthly, on the point 1 with the Radius 1 2, 15, defcribe the Quadrant 15, 14.; alfo on the point 11, with the Radius 12 , 13, deferibe the Quadrant 1 3, 11: Likewife from the point 9, raife the Perpendicular 9, 10, and then will the Throat or Drip of the Corona be Completed. Tenthly, Make a 3, equal to a 8, and draw the Line 8, 3. Draw the Line d, 2, Parallel to a n\ which &ifc&, and from the point of Interfe£H- on, draw rhe Line 6, 1, Parallel to 8, 3, cutting the Line tp 20, in the point 1, the under part of the Cima, being continued in the point 6 -, Lajlly, Bifc£f the Line 6, 1, in 7, and deferibe the Arches 8, 7,. and 7, 1. Which will complete the profile of the Cornice as required. Thus have I gone through the Geometrical Conffruclion of every part of the Tufcan Order of the Antients. Which tho’ perhaps, may feern to be more Tedious, than fome of the following Rules j yet as ’tis what I may venture to call New to this Age (altho the mod antient of all, and in itfelf very Eafy and Demondrable) I am perfwaded rhat to every Lover and Judge of Arts, It will be very acceptable : For it was not without Rea- fon the Antients thought the Rules of Geometry to be the beft, by which the Orders of Archite&ure could be proportion’d ; and more efpecially, becaufe Geometry itfelf had its Rife from humane Bodies, which Nature has fo made, as to fit all the various purpoics in Life ; and therefore thofc for Labour are made Robud and Strong ; thofe for a&ivity and addrefs of a more (lender and genteel Body, and the Man for Bufinefs, a mean between thofe Extreams. It was from the Confideration hereof, that the antient Greeks conftituted th c three Orders of Columns, of which the Doricl^v>ns made the mod: Mafly and Strong j the Corinthian the mod: (lender and deli* cate, and the lonick^ ^ a mean between both. The general forms of the antient Orders being thus ordain’d, they were then under a Ncceffity of fubftituting Rules by the lame Art viru. Geome¬ trical Rules, by which the parts of each Order were proportion’d, fo as to be agreeable to the Charafler it was madeto R.eprefcnt, of which thofe of the Tulcan \ I have now declared, and the others will follow here¬ after in their Places. Fray Sir , in what manner had Geometry it's Rife from human Bodies. From The Principles of Geometry. 20c; M From a well made Man , extending the Extrcams of bis Body, as follows, Firft , If a good proportion’d Man be laid on his Back, and extend his Armes and Legs, as in Fig. I. Plate XLV 1 I, and parallel right lines, be drawn to touch the Extrcams of his Head, Fingers and Feet, they will at their angles of meeting generate a Geometrical Square. And the Diagonal Lines thereof being drawn, will Interfeft each other, on his privies, as in the figure. Secondly, It the Body be extended, as in Fig II. Then a Line being extended from n, the Navel of his Body unto the end of his longed Fin¬ ger, fhall be the radius of a Circle, that being deferibed, will pal's by and touch the other extreams of his Body; and all lines, drawn from one part of the circumference unto the other, that doth pafs through his Navel as n a and n m , will be equal to one another. Hence you fee, that the Circle and Geometrical Square might with very great reafon be fir ft taken or difeovered from a well made Humane Body fo extended ; and from them all the various Properties and affeftions, that now form that moft Noble Science, Geometry; have arifed ; by which all affairs are Govern’d and Determin’d, a i ; will be herealter fully explain’d, when I come to demonftrate and fhew the Ufe of Mcchanick Powers. Since the firft Inftitution of the Antient Orders, which originally were no more than the Doric\fonic\ and Corinthian, as hath been before oliferv- ed in Fol 188; The People of Tufcuny^ (a confiderable part of Italy) Invented another kind of Column, which they made as Vitruvius Obferves, the plained: and moft Simple of all the orders, and from their Name was called Tufcan Order-, and l think with great reafon alfo, altho’ Ferrattlt and other Modem Architects will argue, that it is no other than the Dotick order made Stronger, by Shortening the Shaft; And more Simple and plain, by the finall Number of its Moldings, and largenefs of them than at its firft Inftitution. As Vitruvius is the only Antient Writer on this Subject, whofe Works have been perferved; and who has not given us any part of the Compofite Order, I am therefore apt to believe that it has beenCompofed, fince his time; and was wholly unknown to the Antients before Him. Otherwife, in his drift Search amongft their works, he would very probably have meet with fome which he might thought worthy of communicating to poftcrity with his o- ther works. As 1 have thus taken a Slight Survey of the Orders in general, So far, as is worthy of the Workmans Notice ; I lhall now proceed to the Explanation of all the Several Mafters herein,who have affign’d Proportions to the Orders, and given Examples forPraftice ; which with no finall pains, and Expence, I have Collefted, and Exhibited in the following Plates, and which contains fo great a Variety of ufefull Examples, that, ’tis impoffible, that any Defign can be wanted, but that, if the very thing itfelf is not here found, there are fuch that will fo furnifh the Mind with Invention ; that the meaneft Ca¬ pacity will be fully enabled to adapt and perfeft the Defign required in an Inftant of Time To which I proceed, I. Of NUMB. XXV. XXVI. Dd d The Trinit pies of Geometry. 20 6 I. Of the Tufcan Order , by VITRUVIUS- Th? T ufcan Order of Vitruvius is Exhibited in Plate XVII. and hath its Meafure determin’d by Modules and Minutes. A Module is the Diameter of the Column, at its Bafe divided into 60 equal parts, called Minutes, as the Line 4 60. which being divided but into 6 equal parts at the Points 10. 23. ">o 40. 50. do therefore, each repreient ten Minutes. In Pra£iicc, ’tis {Efficient to Subdivide the firft ten Minutes only, and the others to re¬ main as they arc, for by the Divifion of the firft Ten, you may take from thence with your Compares, any Number of Minutes required in the fame manner, as any Number of Feet, &c. are taken from a Scale of that kind. It is by this Module ,or Scale of Minutes that the Heights and Projecfures of every part of an entire Order are regulated, and Determin’d ^ and iheieiorc before an Order can be began, the Diameter thereof muft be fir ft given or found. When the Diameter of a Column is given, the Scales of Mi¬ nutes is moft readily made by dividing the lame into 60 equal parts, as be¬ fore obferved j but when the Height of the Order is given, either with, or without its Pedeftal, then we muft find the Diameter thereof before we can begin to make the Scale. In order thereto, we muft add into one Sum, the Number of minutes contained inthe Height of the order ; and divide the given Height, into the lame Number of equal Parts: Sixty, of which will be the Diameter of the Column, and Module or Scale of Minutes required. As for E X A M P L E. Mod. Min. The Height of the Cornice, f o 35 Freeze g 027 Architrave h^q 0, o 45 Capital q s o 30 Shaft s b 6 00 Bafe of the Column am o 30 Sum 8 47 Since 60 Minutes are equal to 1 Module, therefore 8 Modules are equal to 480 minutes, and the 47 min, make 517 minutes in the whole, which are the Number of Minutes contain’d in the Height of the Column and Entab¬ lature. If to theColumn its Pedeftal, or rather Sub-Bafe, is required, which is the fame, as that to the T ufcan Order of Palladio , who Height is one Module ; then to the prececding Sum 527 muft be added 60 more, and the Sum of the Height of the entire Order, will be 587 minutes. Now to proportion this Order, to any given Height we muft proceed as follows. J. Admit the given Height to be 20 Fret, and thereto , we are to Proportion , the Column with its Entablat ure, exclufive of its Pedeftal or Sttb-baje. f By the foregoing it appears, that the height of the Column with its Entablature only, contains 527, Minutes. In 20 Feet height, there are, 240 Inches, and if we fuppofe each Inch, to be Sub-divided into 100 equal Parts; The Principles of Geometry. 207 Parts; then the 240 Inches contain 34000 Hundred parts of an Inch.- Which being divided by 527 the Number of Minutes contain’d in the give:. Height, the quotient will be 45 the Number of too Parts, contain’d in each Minute. (j_) Multiply the quotient 44 the Number ot Hundred Pairs of an Inch contain’d in one Minute, by 60 the Number of Minures in one Module; and the produfl being divided by 100, the quotient will be the Diameter or Module required, in Inches. EXAMPLE. The given Height in Feet 20 Multiplied by the Inches in a Foot 1 2 Produfl: 240 Inches Which Multiply again by 100 And the Height will be reduced into 24000, Hundred parts of an Inch, which divide by 427, as following, and the Quotient will be the Number of too Parts of an Inch, contain’d in one Minute. 427 ^24000/45. Hundred parts of an Inch 72108 \ in one Minute 2920 2645 284 Remains, which is fomething more than , a Hundred part, or 264! the i of 427. But as the over and above remains, are more than the 4 of the Hundred part, in Practice is of no Notice, therefore I lhall State the Quotient at 45 Hundred parts and 4 , which 1 multiply by the Number of minutes in a Module. 45 i 60 Produft 2730 Hundred parts of an Inch in the Diameter or Module, which I Divide by too, the Number of Parts into which the Inch was Sub-divided. Inch 100^)2730(27 ,Vo) equal to of an Inch, which is the Diameter or Module required. P. Pray Sir , How do you prove that 27 Inches' and £ is the Module required. Feet Inch M. As follows, Multiply the Module found,•z/ix. 2 3 ^ by the Num¬ ber of Modules contain’d in the Height of the Order which arc 8, and to the Produft, add £ of 27 Inches., lor the 47 Minutes, which cho’ to lit¬ tle, becaufe 4^ Min. are equal to £ of a Module, yet ’ti.s near enough lor Pra&ice ; and their Sum will be nearly equal to the given Height. EXAMPLE, 2o8 The Principles of Geometry. EXAMPLE. Feet. Inch. i 3 , x 0 the Module produced 8 The Number of Modules in the given Height 16 o 2 o 2 4 i 8 2 X the 3 quarters of 27 Sum 19 1 o l * which is within 1 Inch and of an Inch, of the given Height of 20 Feet. And if this Inch and , : 3 be given to the Neck ot the Capital, ’twill be very well difpenfed with •, ior by the Proje&ion ol the Aftragal, in large and high Columns, part of the Height ot the Necks of Capitals, are take off thereby *, and therefore, to make a Capital appear in good Proportion, fuch an allowance ought to be made, lo as to Suit the Difbnce, which the Building is to be viewed. I he Manner ot finding out fuch allowances will be Explain’d, in My Dilcourfeon the manner of placeing Columns, over Columns. 2. li to the Column and Entablature, the Pcdeftal or Sub-Bafe be added, then the Height (hereof (which is one module or 6c minutes) mult be added to 8 modules 47 min the Height ot the Entablature and Column alone, and the Height ot the entire Order will be 9 modules, 47 min. or 587 minutes, which is the Divifor, by which you are to Divide, 24000, the Number of Hundred parts of an Inch in the given Height. 587 24000.(40. 0° The Fra&ion being reduced, is equal to ^ for rejecting the Ia(T Figure 7 in both the Numerator and Denominator, the Remains of the Fra&ion will be — equal to equal to ' 4 \ Therefore the Quotient is 40 ^ which are the Number of Hundred Parcs of an Inch contain’d in one minute of the Orders Height. 60 and then, 40-iT 15 Multiplied by 60 the Min. in a Module 14) 780 ($ 5- 2400 A 70 55"t4 80 produces 2455-}; equal to * the Num- 70 her of 100 parts of an Inch in a module which - being divided by too, as before the quotient 10 remains, is 24 Inches equal to it, equal to two Feet and ii parts of an Inch, the Diameter, or Module required. P. I dont conceive, How yon have performed the fore-going Multiplication , that ij, In what manner did yon produce the 55* M. By Multiplying the Multiplier 60, into 1 3 the Numerator of the Frac¬ tion, and dividing that Product by the Denominator the quotient is 55. X x the 209 The Tr inciples of Geometry. the Sum added to the Product. See this lad Operation in the Margin, at A, which in Praftice, is perform’d on a waft Paper by it belt. 1 his being well underftood, (and which muft be, before you proceed fur'her) we may now advance to the Delineating of the Order, according to the Meafures affix’d to each Member. Wherein Obferve; F/rff, 1 hat the Height of every Member, is fignifieJ at the centeral line bh, by the figures, thac are there placed to be read upwards, and which fig- nifie as many Minutes; So the number t 5, placed between the two lo-w r- moft lines, fignific 1 5 minutes; the Height of that Member which is called the Plinth, from the Greek Plinthos, a fquare Brick. And the number 11 placed between the Wand (ford lines, Signifie 12 minutes and half, the Height of that Member, which is called Torus from the Greek word lorn a Cable, which its Swelling fomething refembles, or rather from the Latin Torus abed, and the number 2 i placed between the next two lines, Sig¬ nifie the Height of that Member which is called the Cincture, Fillet, or l ift, f om the Italian Uftello a Girdle ; and fo in the like manner, the fame is to be underftood of all the other like Numbers fo placed on each Member. The ProicSions of moldings or Members, are exprefted by the figures placed level with the Eye, either againfteach Member, or otherwife between the Upright line of the Column, (from which their projeflion arc generally accounted) and their F.xttearns. So in fuclimanner,on the / limb there hands the Number 10, on theCinflurc theNumber ^figr.ifingthar theffo;h projects to minutes, and the Cinfhtre 4 minutes, before the line 8 e, which is the upright line of the column at its Bate, and which is equal to the proj.cftmn of the Capital. And as the Capitals projeflion is exaflly equal to the mod¬ ule or Diameter of the Column, therefore its projeSion is equal to f the Diminution of the fhaft at the aftragal, which is 7 minutes and i, for a 5 minute-rhe Diameter of the (haft at the aftragal, being taken from 60 the Diameter of the column at its Bafe, the remains is t 5, one halt of which is 7 minutes and t, To the projeSlion of the Capital, the Engraver hat plac'd the figure 8 inflead of 7 which is the true projection, of which you are to talp Notice and correct. Hence tis’ plain, That to Delineate an Order is no more, than (when your module is prepared; That is, determin'd and divided into minutes as has been already expreffed ) Firft for to draw a cenreral line as b Is, and on each fide thereof draw two right lines, parallel thereto, at 30 minutes d.fiance, asq r and’60 /; Secondly, To draw a right line thro’ any point thereof, as at h. at right angles thereto, for the Ground or Bafe line of the PI,nth, or lowcr- moft Member .Thirdly, To draw right lines parallel thereto, and at inch d'lfancc from each other,as are expreffed by theNumber of minutes there placed for thac purpole. Fourthly to make the length of every fuch line, before the Upright line of the column,equal to theNumber ofVlinutes placed there for tlut purpole and then Clofing their excreams, with the our line of the feveral moldings exhibited,the profile of the Order will be completed as requtred.Thts is fo very plain and eafy,needs no more co be laid on this Head, and therefore as I proceed to theOrders of otherMafters whoDetermme their meafurts in the fame manner by the Diameter, or module divided into 60 minutes, the lame is to be therein underftood, as herein hath been deltverd, and therefore wffl Numb.XXVI I. E c c. 2 io The 'Principles of Geometrt. be needlefi to repeat there again. I (hall now proceed to my Obfervations and Kcmirk^t on the whole. And Firft, Oi the ESafe whole Height iiequal to minutes or J the Diameter ot the column at its Bale, including the Cincture, which in faff is a part of the (haft, and not of the Bafe, and therefore,ought not to be made a part ot theBafe,any more herein,than in all the otter Orders; Where every of them Excludes it from their Bafe and ma te it a part of the Shaft, as being Originally placed there to otrength- cn the Bottom thereof, from whom, when, or where this Error had its Rife, I can't determine: But as Vitruvius pals'd it over in 'ilence, rhe Archite&s of Succeeding ages have continued it, and every one his unde the Cinfture a part of the I uf- can Bale, ot which, with fome other miftakes, ! (hall rake further notice when I hive pa U d rhrough the works of all the Matters, herein propoftd. The Plinth bring 15 Minutes high, is equal to half the height ot the whole Bafe, and therefore tnaffy and ftrong It was the Praftice ot Vitruvius to take away its four Corners, and make n : ' rffed in the tfes to the Tufcan Ituercohimnation in flats XXXI fig I. But notwirhttanding, tli.t Vitruvius was fo great a Mail, I mutt needs Join wi h Vcrrmtlt, who do not think it ought >w’d, becaufe the angles of the Bafe, anfwertng thole of the Ca¬ pital, the Bafe would appear difabled, maim’d, and Incapable to fupport when depriv d of them: I mutt own, that when Columns are placed wirh- in Halls, or other Publick Buildings 'tis very convenient, tho’ vciydifa- grecable to takeaway the corners ot their Plinths, for thereby Pcrfons are not Interrupted fo much in their walking, as when they are trucly Square, and there! ire it is, that at fome times. Beauty and order niuft fub- mic to convcnicncy which may in many Cafes be helped very greatly, as herein ; If the Capitals have their Abacus round as the Plinth, or rather to have them Octagons, which is a mean, between the ex reams ofa Circle and a Square; then the Order would leem to be perferved, and the convcnicncy the lame as before. Thus far, with refpefl to the Bafe, the next in order,-is the Shaft, whofc thicknefs at the Bale is continued up f part ot rhe Height of tin Shaft, and from thence is Diminifhed unto the Aftragal, where its Diameter is but 45 min. as therein exprelfcd. The Capital is very particular, as having a plain Abacus without any Ogee at its top, and equal to to min. or t part of the whole height of the Capital. The Ovolo under the Abacus is alfo the fame height, and underneath is placed an aftragal in manner of rhe Irajans Column at Rome Plate XLII. from whence i believe lie borrowed the Hint, and which Scamoisn.i and Fer- ratill only has followed ; all other Architeffs having put only a fillet there. The Proportion and Character ot the Entablature is very different from all the following Mafters; the Architrave being not only larger than the freeze, but even than the Cornice, which 'tis very reafonable to believe lie fo ordain’d for the fake of ftrengih to fupport it felf in its Bearings, where the Intercolumnations are very great. The Projeflion of his Architrave, feems to be unreafonable, but when we confider, that had he placed it to be ranged with the upright of the Column,as it is all in one part, then that, and the Freeze would have appeared as one Member, and therefore to avoid that, 1 do fuppofe he gave it the projeflion as exhibited in the Figure Ic is my humble opinion, that if Vitruvius, had Divided the Height of his Architrave The ‘Principles of Geometrt. 21 I Architrave and Freeze into 7 equal parts , as at the points b il n a p, and given four to the Architrave, three to the Freeze, and Divided his Ar¬ chitrave into two talcia s with a tenia, as exprefsed by the pricked line.*; //, t r;r, and v tv, the whole entablature would be ftrong enough and more a- grcablc ro the Eye than at prefenc it appears to be. The Height ot the Entablature being 107 minutes, is 1 minutes more than i c ^ e length ot the Column. For 7 times 60, is 420, and the quar¬ ter 105, 1 I I Of the Tufcan Order, by A. PALLADIO. The Tufcan Orders given by Palladio, are exhibited in Plates XVII and XVIII sV herein their Members areDctermin’d bymodulcsand minutes,as before done by Vitruvius. In theXIV Chapter of theFirfiBook of Palladio concerning thcT„fcanOrdcr,Hc fays, That the Column with 'its Bale and Capital mulF be y modules in length, and its Diminution a fourth part of its Bignefs(T fuppofe he means its Diameter next the Safe and afterwards goes on to deferibe the parts of Vitru- 1’iHr Tulcan column and Architrave; But has quite forgot the Freeze and Cornice thereof, as well as to fpeak a Angle word concerning the Two Tufcan columns taken from the arena’s of Verona and Pola , which he but juft mentions to have taken the profiles oft’,which that in Plate X ' III. is alter the manner of Inigo Jones , in the portico of St. Pauls Covent Carden as exprelid in PlateXXVI, XXVII, and XLV This Order I have Delcribed at Length in Figure, CXLIX. Plate XVIU to which I have added the ogee B to cover the Joifts ( whole projeflion are a fourth part of the length of the Column as at itt. Pauls Covent garden Am to that of Palladio's , there is not any Ogee cornice on the Joifts, as herein let forth ; which I Judge to be an Omiltion or miftake. The Members that eompofe his Tufcan Order Plate XVII. are as follows. A C F, Fillets or Lifts. B. S II Cima reSla's or Fore Ogee’s, from the Greek word Kymation a wave, called by fome the Throat, Cola, Genic or Doncine. D Coiona, by fome called Supercilium or rather Slillicidium the Drip; The under part hereof is called by the Italians Soffito-, by the French, Pllncerr, both fignifying no more than the Cielling. Which laft Term is ’molt com¬ monly tiled by Workmen to exprels thac part of the Corona. E Ovolo, or Echinos, a Greek word, Signifying the Shell of a Chefnut which many Workmen called quarter round. G N Caveto’i from the Latin, Cavils a Hollow. H The Freeze from the Latin Phrygiom Embroiderer, or from the Italian Freggio a Fringed or Embroider’d Belt; the Greeks call it Zophoms, %- nifyingrheZodiack .and it in the Freeze of a Rotunda, or round Temple, the it Signs were depifled, it could not have an ill efFeft. I The Tenia allb called Diadema, a fmall Fillet with which they ufed to bind their Heads. K L Fafcia’s of the Architrave, called by fome Paftes, from the Latin word Fajcia, a large Turban ; they are alfo by fome called Swathes or bands. I 212 The 'Principles of Geometry. \ } Architrave from the Greek E piftilim, the lower-moft principal part of ^ f the Entablature. M ^ t he Abacus from the Greek word, Abax, fignifyiflg a Square Table N or 1 rencher. Ch Fafcia of the Abacus. R T. Fillets. W C A(irZTcalled by the French Tabu, or Heel by the Greeks AJhagah, the Bone of the heel; The Italians call ir T budmo, as being like s Tor us. Some call it Hypotracbehum : But I chink very wrongly, becaule Hy- potrachelium denotes the Neck of the Capital, called by fomt, Lolla- rino , the Collar. a i V. Fillet of the Aftragal Cinffure, Fillet or Lift. X Y 7 Torus. 1 I fillet. IV Plinth VI Pedeftal, Socle or fub-Rafe. , . . „ r It Seems that the Architefl of this Column, has trod in the fteps of .. . . ot otherwife, Vitruvius in his with refj-efl to the Architrave, which has a near affinity to that of Vitruvius,*, being much Larger than r andalfoproieasforwardovertheCapital.il lute manner. But however as the height of the Cornice is greater, and in n,y opinion more proportionable to the whole, than that oiVitruv.us, the plamcf. thereof nnv bedifpenfed with; but were it to be divided into the Terns i I. and two falcia’s K L with the fame projeflion and Height, a. Fxpre fs d by the pricked 1 ines and numbers placed between them, it would in moft Calcs, it not in all have a much better Effeft .• But this is Submitted to the Confide ra- Lon of thole whom it may concern ; as alio is die Compofition ot the Cornice, which in my eye, has a very agreeable efted The next in oider is th .Capital, whole Abacus d <> crown d with a Caveto and Fillet which in Vitruvius is quite plain, and inftead of he Ovob there next under it, here is an Concluded between two fillets; all "hi -h feems to be too great a number of parts,as not being in the leaft agreeable o h plainefs of the Architrave, which 1 fuppofc Palladio alio obfemd, Becaule 1 find he ha, given another Tufcan Capital of the fame bind which is Plain and more agreeable as the Capital Exhibited at Figure, rXlV. which Confifts ot a plain Abacus, with an Ogee between its billets in rhe Place where Vitruvius has the Ovulo-, But tiro this Capital is Letter than the other, yet 1 think the compofition not fo proper for a Capital as that of tbrntedHr or that in Plate XVI I I . ... , . Th-B tle hereof,at firft view ;feems to be broken into many parts,and inftead of _ torus 1 an Ogee is introduced, which perhaps may be an agreeable Mem¬ ber tho’ 1 think not fo proper as a Torus, whofe roundnefs eeins to be caufed by the weight prefling thereon, and therefore is a natural and proper Member in that Place. III. Of The Principles of G e o m t t r y. 2‘3 Mod. Parts Mod. Min. C Column and Entablature 717 6 1 f 8 4,7 (< 5 ) The Height of the .^Pedeftal and Column si8 s(or< 9 zo (.Pedeftal, Column and Entablature 52 1 xj ! 11 y (7) The Intercolumnations and Arcades, with their Imports of this Order, are reprei'ented in Plates XXXIV. XXXV. and XXXVI. (8) The Proffc of this Order is alfo reprefented in Plate F, to follow Plate XX. where its Projec- tures are accounted from the central Line, according to Mr. Evelyn (9) As the Parts of this Order are very grand, it is belt to be employed in magnificent Buildings, that are to be feen at fome confiderable Diftance. Plate F, to follow Plate XX. Besides the Tufcan Profiles of Talladio, Scamozzi and Vignola, here are the Profiles of Serlio, and that famous Column of the Emperor Trajan , now Handing a^t Rome, whofe Meafures are accounted from the central Line, accord¬ ing to Mr. Evelyn. In Plate XLII. you'll find the Bale and Capital of this Column exprerted more largely by Sebaflian Serlio, a famous Collector of the Remains of the Antients, where Fig. I. reprefents the Pedeftal and Bafe, Fig. II. the Capital and Plan of the Shaft, with the Stairs and Newel, and laftly, Fig. III. reprefents the whole Column entire, whofe Altitude is as follows, viz. The Pedeftal is one Diameter of the Column and 30 min. including the Zocolo or Plinth, whereon refts the Eagles and Feftoons. The Column, including its Bafe and Capital, is 8 diam. and 30 min. The Capital is 10 min. and the intermediate Members are as exprerted a- gainft the Profile in Plate F. But more of this you'll find in the Ex¬ planation of Plate XLII. Plates G, H, I, following Plate XX. The Tufcan Order by Julian Man-clerc. These three plates reprefent th eTufcan Order of this Mafter; as firft, its Pedeftal and Bafe of the Column in Plate G ; fecondly, the Elevation and Plan of its Capital in Plate H ; and laftly, its Entablature in Plate I. The entire Order is reprefented on the Left Hand of Plate LXXX 1 X. where, on its Right, you fee, ft) That the whole Height being divided into 9 equal Parts, the Height of the Pedeftal is equal to two of thofe Parts. (1) That the remaining Height of the Column and Entablature being divided into 15 Parts, there are li to the Column, and 3 to the Entablature. (3) The Height of the Column divided into 7 equal Parts, one of thofe Parts is the Diameter. The Height of the Bafe and of the Capital are each ; a Diameter, and the Column is diminifh- ed one Quarter of its Diameter. Now tofub-divide thefe principal Parts, proceed as follows, viz. (1) The Pedeftal, Plate G, whofe given Height being divided into 6 equal Parts, as on the Right Hand, give the upper one to the Cornice, the other to the Bale, and the remaining 4 to the Dado or Die. The Sub-divifions of the Bafe and Cornice you have on the Left, the Bafe of the Column having its Height found as before, divide it, as on the Right Hand is exprerted. (i) The Capital in Plate H, its Height being found as before, divide out its Parts, as on the Right Hand exprerted. The feveral dotted Circles, iilfcribed within the dotted Squares, reprefent the Plan of the Bafe and Capital. As the Ovolo*' of this Capital is carved into Eggs and Darts, this Mafter thought it necelliiry to inftrufl us. How to deferibe the Egg-Oval. Let 3 7 be the given Breadth, which divide into 4 equal Parts; alfo di¬ vide the given Height into 3 equal Parts; or rather, which is better, firft G g g divide 2i4 77>e ‘Principles of Geometry. divide the given Height into 5 equal Parts, and make the Breadth equal to two of thole Parfs^’aud'dravv the Line A B at Right Angles to 5 E ; alfo on the Point 5 deferibe the Circle EG, and draw CD parallel to AB. (2) Make 7B apd 3 A each equal to j of the Diameter 3 ", and on the Points A andB , with theRadius'AD, deferibe the Arches DE and CE. (3) Bifedl the Arch ED, and from the Point of Bifeclipn lay a Ruler to A, and it will cut the Line j-E in the Point I, on which, with the Radius taken from the Point I, to the laid Point of Bifedlion, deferibe the Arch H, which will complete the Whole as re¬ quired. (4) The Entablature Plate I. its Height being found as before, divide it into 3, giving 1 to the Architrave, 1 to the Freeze, and 1 to the Cornice ; and then fubdividing the Architrave and Cornice, as on the right Hand.of the Profile is exprefs'd, the Whole will be completed as required. Note, That this Mailer makes the' Projections of his Members always equal-to their Height. REMARKS. ( 1 ) The Block Rultics, in the Bafe and Cornice of the Pedcftal, are rather Nuifanccs than Ornaments, becaufe they break the Courfe ol the Mouldings without any Reafon. (i) The Mouldings on the Left of the Bafc to the Pedel’- tal would be very good for an lonick Pedcftal, but not for the Tufcan, as be¬ ing too delicate, and too fmall for the “Die they fupport: A plain Plinth would have been better. (3) The Cornice is fomething out oi the common Way, as being crowned with an Aftragal, inftead of a Regula, which would, I believe, have been more mafterly, and more agreeable to the Bafe of the Shaft, that is placed thereon, whole Torus is miftakenly carved, as is its Capital in Plate-H, which have no Affinity with the plain Entablature in Plate 1 . whole upper Member (the Ovolo) is much too fmall for the Corona under it, and ought to have been made a Cymatium , as before obferved. Plate XXL. The Firjl Example of the Tufcan Order by Sebastian’ L e C l e r c. *. In this Plate we have two Examples of Entablatures and Capitals, which differ very little Irani one another, as may be feen by comparing the reflec¬ tive Members of each Part, and their Meafures, wherein no material Fault can be found, excepting his having crown’d both the Entablatures with Ovo- lo’s, inftead of Cymatiums , as they ought to have been ; and that the Dado or Die of the Pedcftal would have been more agreeable to the Tufcan Mode, had its Height been made fio inftead of 80 min. Note, As by this time it is Suppos'd, that the Reader is well acquainted the with Names of each Member in this Order, 1 [hall therefore forbear to repeat them again. The alphabetical L.etters, placed on each Member, are to diftinguilh one Member from another ; as for Exa- ple, If I would remark to you any thing particularly relating to the Ovolo A, as being in a wrong Situation, &c. then I fav, the Ovolo A is wrongly placed ; but were that and the other Members not denoted by Letters, then I mult fay, the Ovolo of the Cornice is wrongly fituated, to diftinguilh it from I, the Ovolo in the Capital, which is in its true Situation. R E M A R K S. 1 he Meafures, by which thefe Examples are determined, is the Diameter divided into 6 o min. as by ’ Vitruvius, Palladio , &c. The Column contains 7 dram, in Height, and is diminiflfd Thefe Entablatures do each contain y diam. The Trinciples of Geometry. 2. , diam. and 40 min. in Height, which is- y min. lefs than,! of the Column’s Length. " . • Diam. Min. CColumn and Entablature 8 4.0 The Height of the thereof. ^ flic Pro¬ jection of the Pedeftal s Bale is equal to that of the Caping, and that of the Dado, 01 Die, to.h C. lo- The Profile of this Order is ; nted .... .1 foil w Plate XX. where its Members are detc mi ules. I . iccounted from the central Line, according to J . — elyn. Plate XXVI. XXVII. The Tufcan Order by Mr. Stone in the Portico of St. PaulT, Covent-garden. This Order is built in the Portico of the Church aforefad, whofe fnr.pie Grandeur excels all other Buildings in this City. The Conttruction ot this Portico is as follows: Let GL be the given Breadth, and through its Ex- treams The 'Principles of Geomet r y. 219 tr earns GL, draw the Right-lines A B and M(/, v hi-jh fhall be the central ] anas ol the exl'ieam Columns ; divide G f, into 6 equal Parts, then, with the Radius GH, which is jot GL, thereon defcribe the 4 Circles, cutting GL m H I k : With the fame Radius defcribe two Circles on the Line G B, touching each other at FD; perform the fame on the Line M (/, then will the Points BQ. be the under Part ol the Plinth to the Columns, and the Line L/a drawn thro the Points FO, is the upper Part of the Cima reBa, v h >le Projection js equal to the Radius HE, and confequentlv to one fourth Part of I' B, the Height of the Order. If 1 M is made equal to GA, and the Lines i\ M and MZ be drawn, then will the Triangle N Z, KM, and M Z, exhi¬ bit the Pitch of its Pediment, whole perpendicular Height is equal to half tiie Height of the Order that fuftains it, as is evidently demonftrated by the large Circles which meafure the fame. To divide the principal Tarts of the Order. (1) Divide PO into 9 equal Parts at/, W, e, X, d, Y, h, Z, (/, then Z.(/is the Height ol the Bafe, which doubled is the Diameter of the Column. (1) Divide the Height OZ into 3 equal Parts, or rather into 6 , and then OS, the upper 6th Part, will be the Height of the Entablature. If SZ be divided into 7 equal Parts, one of thofe Parts is the Diameter of the Column, and will be equal to twice ZQ_, the Height of the Bale before found. (3) Make tlie Height of the Sub-plinth (/equal to the Height of the Bafe. Tiih Entablature, Fig I. having its Height as divided into 6 ' equal Parts at x > /> 1 > r > the lower 3 is the Height of the Architrave, and the upper 1 the Cymatium and Aftragal included, and the next one is the Height of the Modilhon, whofe Projection is equal to the Fleight of the Entablature. The Projection of the Plinth is a fixth Part of the Diameter at the Bafe, and the Projection of the Cindture is half thereof The Abacus of the Capital has the lame Projection as the Cincture, and the Diminution of the Shaft is } of the Diameter at the Bale. The Height of the Capital is a twelfth Part more than the Semidiameter at the Bafe, which 1 fuppofe is allowed with refpect to the Projetlure ol the Aftragal, which, when we are near the Building, eclipfes a Part ol its Height, and therefore this extraordinary Height is justifiable. Plate XXVlll. Fig. II. exhibits the Tufcan Order of Inigo Jones, as ’ris executed in the Frontifpiece at Tork-flairs, whole entire Fleight a y being di¬ vided into 5- equal Parts, at b, g, p, y, the upper one a b is the Heigh Entablature, and the lower 4 being divided into 7 equal Parts, at e, h, l, r, iv, 2, one of thole Parts is equal to the Diameter of the Column. The Dia- meter of thefe Columns is precifely two Feet, and therefore the Diameter, Fig. I. is divided into 24 Inches, and is the Scale by which the Whole is delineated. This Compofition is not unlike that cf Barozzio of Vignola, in every refpect, the Cincture (/excepted, which differs from all P ever fee. The feveral Figures affixed hereto, to denote the Heights and Projaitures of each Member, are Inches and Parts of Inches, as their feveral fractional Num¬ bers expreis. Piute XXVIII. The Tufcan Order by Inigo Jones, at York- Stairs. I his fo much celebrated Mafter would have made a fine Corfipofitiori here, had he placed a Cymatium on the Corona, inftcad of an Ovolo, which I have already obferved to be ablurd. The remaining parts ol the Order are very noble, lave its Bale, where I think its Torus S is too fin: 11 , and Cinfture R too large: Had the Torus been made y Inches inftcad of 4 Inches, and the Fillet 1 Inch inflead of 2 Indus, they would have been better proportii ned than r The 'principles of Geometry. than they are, to the great Parts they fuftain. The Cavetto QJs very abfurd, as is the Manner of the Shaft fitting on it: In fhort, the Cincture R and Cavetto Q_ ihouid have been a Part of the Shaft, and not a Part of the Bate, as they are made tube. The Height of the Column is 7 Diameters, the Height of the Entablature i of the Column's Length ; the feveral Members are determined by Inches and Parts of Inches, both in their Heights and Projedturcs. To ru/iicale this Column, divide the Length of the Shaft- c + into 17 equal Parts at c, d, f, efye. then give two to each Ruftic, and the like to each Interval, except the upper Interval, which mult be but 1, as c d. The Projections of the Rallies are determined by Lines drawn from the Projections of the Cincture F and G, unto the Projection of the Aftragal 7 and 8. P. Tray, Sir, why were Columns fir ft ru/Heated ? M. To ftrengthen their Shafts, by binding their Parts clofely together, lb as not to fuller them to built or fplit by very great Weights, that were then impofed on them. Therefore to rulticate Columns, that do not Main great Weights, is a great Ablurdity ; of which the molt ridiculous Exam¬ ples 1 have feen are the Columns in an Ionick Frontifpiece to the Entrance of the Court-yard, before the Lord Burlington's old Houle at Chejwicb, and thofe to the Frontifpiece of the New Play-houfe, in the Piazza of Cogent- Garden. Nor can I indeed very juftly commend Mr. Inigo Jones in this Ex¬ ample : But as they are done in a Grotefque Manner, upon the Brink of the River Thames, intended more for Ornaments than Strength, they are there¬ fore not to be condemned, nor very greatly commended: For a Rural Grotefque Frontifpiece, like unto the Entrance into a Cave or Grotto, would have been much, more fuitable to that Place, than a regular Piece of Architecture. Thefe lead me to obferve to you, the fame Abfurditics in the rulticated Co¬ lumns to the Fxontifpiece of Lord Burlington's Gate in Piccadilly, and thofe againll the Meufe Stables, Chartng-crojs ; where in both thefe Buildings, the Columns are not only rulticated, without any impofed Weight, five that of their broken Entablatures, but their Surfaces allude to'different Situati¬ ons from thofe they poflefs: For thofe lficles in Burlington- Gate, would have better becomed the Situation of TorLStairs, or the Entrance to a Grot¬ to, than that to a Nobleman’s Palace in a common high Road. And thofe defaced Ruitics of the Meufe are but a Monkey Affectation (ot Antiquity, as if worn fo by Time, or caufed fo by Nature,) being no wife fnnilar to the other Parts of the Building, or agreeable to the Age in which die Date of the Year placed over them denotes their being erected. Plate XXIX. The Tufcan Order of Sir Christopher Wren,' in the Frontifpiece of St. Antholine’r Church in Watling-Street, London. The Compofition of this Order is the fame as the foregoing of Inigo Jones, excepting in the Bafe to the Column, wherein this Mailer has very wifely ex¬ cluded the Cavetto above the CinCture, and given a better Proportion to the Torus and Plinth. To proportion this Order to any Height is a Work of lome Trouble, as that the Height of the Entablature, which is but t diam. 1, bears no good Proportion to the Height of its Pilafter, which is of - Diameters ; and indeed 1 am furprifed to find lb fmall an Entablature, by fo great a Maf- ter ; it being actually 27 min. too low, to be equal to a quarter Part of tiie Pilafter's Height, which it ought to have been. To proportion this Order, divide the given Height into ay equal Parts, or rather into y Parts; and then dividing the upper one into y, give the upper four to the Entablature, and the Relidue of the Whole to the Column. The Height of the remaining 21 Parts being divided into 7, take 1 for the Diameter. To divide the Bafe into its The Principles of Geometry. 221 its Mouldings, divide xa (which is equal to half the Diameter) into 19 Parts, give 10 to the Height ot the Plinth, 8 to the Torus, and z to the Cincture. To divide the Members of the Capital, divide ao, which is equal to ap the Semidiameter, into 3 Farts at df ; divide f 0 into 8 Parts, and make no equal to ; of / 0, and draw u ID for the upper Part of the Aftragal: Again, dividers into 6 Parts, give the lower z to the Neck 6 8, half the next 1 to the Annu¬ let, the next half to the Ovolo, the next 1 to the Abacus, which fubdivide into 3, ar.d give 1 to its Regula. To divide the Entablature into its Archi¬ trave, Freeze, and Cornice, divide ap its Height into z~j Parts, give 8 to the Architrave, 8 to the Freeze, and 11 to the Cornice. To divide the Tenia of the Architrave, divide bp into 9 Parts, and give z to the Tenia. The Cor¬ nice being divided into 11 Parts, the upper 3 is the Height of the Ovolo, the next 1 of the Aftragal, and half the next 1 of the Fillet. Divide dx into z equal Parts, and the lower 1 of thole 1 Parts into 3 Parts, then give 1 of thole Parts to the Cima never/a t x, whofe Fillet is \ thereof! The 'Projection of the Cornice fg is equal to its Height f 10. The 'Projecti¬ on of the Capital is equal to j of the Height from the under Part of the Fillet to the Aftragal, unto the upper Part of the Abacus. The Projection of the Bafe is (fomething very odd, being) z thirteenth Parts of the Diameter 3 O, which is equal to the Diameter, being divided into 13 Parts, at B, C, D, E, &c. 8 A3, and OP y, are each equal to 7] of 3 O. The fever at Figures placed againfl the Members fignify no more, than References to thofe Parts, when we need to mention any of them particularly. To rujiicate this Column or 'Pila/ler, draw klpq parallel to the Cincture ms, and at'fuch Diftance from it, as to be juft clear of the Curve; divide the upper Part from 0, to the under Part of the Fillet to the Aftragal, into 10 equal Parts, and each of thofe Parts into 6 ; then give y to the Ruftick, and 1 to the Rabbit, Groove, or Chan¬ nel, as fignified by the fmall Circles between e and 0, Fig. I. The Projecture of the Rulticks is equal to that of the Cincture. Plate XXX. Tufcan Intercolumuations , Arches, and Impojls, accord¬ ing to the Ancients. F i g. I. An Import at large. Fig II. Intercolumuations for Portico’s or Co- lonades. Fig. ill. Arches, or Arcades on Sub-plinths. F’ig. IV. Arches, or Arcades on Pedeftals, vvhofe Imports ZZ and SS are reprefented by Fig. I. The Proportions of the principal Parts are exhibited bv the large Semicircles and Circles, and thofe of the particular Parts by the Idler Circles; all which, being very plain by lnfpedtion, need no further Explanation. Plate XXXI. Tufcan Portico’s to Temples by Vitru vius. T hese three Figures do not only exhibit, by the proportional Circles, the proper Intercolumuations of this Order, but the Proportion of Portico’s to Tuf¬ can Temples alfo, which Infpedtion will better explain, than Words can do. Plate XXXII. Tufcail Intercolumnations , Arches , and Imposts, by A. Palladio. As the Meafures and proportional Circles demonftrate the Magnitudes and Proportions of their feveral Parts, I need not fay any thing thereof; and there¬ fore I fliall only note, that if the Semicircle O hgv be divided into 11 Parts, the Key-ftone will be one of thofe Parts. 11; Plate 222 The ‘Principles of Geometry. Plate XXXIII. Tufcan Intercolumnations by Vincent Sc amo 7.7.i. A s all the feveral Members of thefe 'Portico's and Arches have their reflec¬ tive Meafures affix'd to them, as likewile are to Fig. V. VI Vii. which repre- Jent his various Imports, Architraves, and Entablatures, no more need be laid relating thereto. Plates XXXIV. XXXV. XXXVI. Tufcan Intercolumnations, Arches, and Impojls, by BarozZio of Vignola, and Sebastian Le Clerc. The Intercolumnations for Porticos and Colonades by Barozzio are repre- fented by Plate XXXIV. and thofe for Arches or Arcades to Piazza's by Plates XXXV. XXXVI. where the firft reprefents the Tufcan Arch without Pedeftals, and the laft, with Pedeftals. The Intercolumnations for Colonades and Arches by Le Clerc are reprefented by Fig. II. and III. at the bottoms of Plates XXXV. and XLII. alfo by F'ig. I. II. III. IV. V. VI. of Plate XLIII. and his various Imports and Architraves for thofe Arcades are exhibited at the bottoms of Plates XXXIV. and XXX VI. The Meafures to Barozzio' s are Mo¬ dules and Parts, (the Semidiameter being the Module divided into n Parts, as before obferved) and thofe of Le Clerc s are Modules and Minutes, wherein the Semidiameter of the Column is accounted the Module divided into 30 Minutes. Plates XXXVII. XXXVIII. XXXIX. XL. XLI. Tufcan Trium¬ phal Gates , Arcades / Intercolumnations to Colonades, Niches, Doors, and Windovos, by Sebastian Serlio, and Julius Romanus. To delineate the Tufcan Triumphal Cate or Entrance , Fig. I. Plate XXXVII. (1) Divide the given Breadth into n equal Parts, as demonftrated by the ii Circles E, D, C, &c. then will one of thofe Parts be the Diameter of the Pi- lafters. (il Make the middle Opening equal to 3 Parts, and the fide Openings each equal to 1, and then complete the Order, throughout its Height, by the Diameter before found. (3) Make d f equal to half the Height f r, then the whole Height is in 3 equal Parts. (4.) Divide d / into 7 Parts, then the upper i is the Height of the Pediment; the out Lines of the middle Pilafters, being continued to K and G, terminate the Extent of the Nttick Pilafters KI and HG. (y) Divide the Height of the Pilafters into 15 Parts, which terminate with curved Lines, reprefenting their rultick Swellings, making the 9th Ruf- tick from the Bafe the Import of the Arch. (6) Divide the femicircular Arch into 19 Parts, that the middle 011c (h) may Hand perpendicular to the Diame¬ ter of the Arch, which will complete the Whole. Note , The Divifion h 0 n m l k 2, and the Semicircle p d q are ufelefs in this CunftruiStion. R E MA RKS on Fig. I. and Fig. II. Plate XXXVII. (1) The Breaking off the Architrave and Freeze, for the fake of enlarging the Rufticks of the Arch, is abfurd, and deftroys that noble Look of continu¬ ed Entablature, as is feen in Fig. II underneath : It is alfo a weakening to the Whole, and therefore not to be pradtifed. (a) If the Number of his Ruf- ticks were fewer, they would be more grand and noble, the Whole being (If 1 am not miftaken) broken into too many Parts. Fig. II, is another Gate of this Mafter, which is not fo much to be con¬ demned, but indeed may be received as a good Example ; but I would recom¬ mend, that circular Recedes, for the Reception of Bufto’s, be made in the Places The Principles of Geometry. 22 Places of the fquare Apertures 6 and y, which Apertures, appearing as Win¬ dows, have not fo noble an Eff'e£t. As the proportional Circles E, D, C, B, &c. demonftrate the given Breadth is divided into iy Parts, of which i Part is the Diameter of the Column, the Height of the Order and its Parts is from thence determined, as before has been taught at large ; by which you may complete the Whole as required. Plate XXXVIII. Fig. I. is a Tufcan Frontifpiece oi Serlio's, of very un¬ common Proportion, viz. Its Intercolumnation is but 3 diam. and its Pilaf- ters but 6 diam. in Height, and, what is yet more odd, the Height of the Bafc is but i ol the Diameter, and Height of the Capital but f, which, tho' Prefidents. are not to be followed ; therefore 1 recommend, that when any luch Delign as this be required, make the Height' of the Bales and Capitals each equal to half the Diameter of the Pilafter or Column, for otherwife they will appear too finall for the Shaft they belong to. Fig. II. is an Arcade for Piazzas of Columns in Pairs, to be ufed where a great Weight js to be fup- ported over them in the fecond Story. Plate XXXIX. Fig. II. reprefents a Tufcan Arcade for Piazza's alfo, to be ufed where the Weight above is not fo great as the aforefaid, and when much Light is required below. Fig. I. is a Tufcan Gate rullicated after the Manner ol Julius Rcmanus, wherein tis to be oblerved, (t) That he makes the lower- molt Ruftick the Bale of the Column : (xf That his Columns are dirninifhed immediately from their Bottoms unto their Capitals : And laftly. That he has abfurdly broken the Architrave and Freeze, as Serlio has done before him in Plate XXXVII. Plate XL. Fig. I. reprefents Tufcan Niches, which are of good Compoli- tion lor Grotefque Works ; as alfo is the Tufcan Door, Fig. II. But the double Arch to Fig. Ill I cannot commend, for if thellreight Arch aa, &c. have good Butments, there's no occalion for the Semicircle A over it, todilcharge off the Weight, the other being llrong enough of itfelf: Indeed, in fome Cafes, it is required to let in Light at A ; but, when l’uch Demand is made, pray why may not the llreight Arch be taken away, and its Height turn'd into an Im- polt, and Itopt over each Side of the Window, with an agreeable Projedtion, thereby giving to the Whole a noble Afpedt ? From whence Serlio got this abfurd Method I know not, but fure I am, that many modern Pretenders to zirchite&ure have copy’d and pradtifed it, not only in the Doors of the War- ojfice, by the Hotje-guards, at IVbite hall, and to the Doors of His Majefty s new Stables in the Meufe, but many other Buildings in and about London, as not being able to judge of the Abufe. Plate XLI. exhibits another rullicated Arcade for Piazza's, with a ruftick Gate not unworthy of our Regard, where Strength and fnnple Beauty are to be expreffed at the fame Time. Plate XLII. The Trajan’.! Column at Rome, by Sebastian Serlio. Fig. III. reprefents the Column eredled by the Senate and 'People of Rome, in Recognition of thole great Services that the Emperor Trajan had rendered his Country, and indeed it is one of the molt fuperb Remainders of the Ro¬ man Magnificence to be now lecn Handing; and, that it might more immor¬ talize him, than all the Pens of Hiltorians could do, they cauled this Column to be made of Marble, on whole Shaft they engraved, in Bas-relievo, all his memorable Adis, by the molt exquifitc Hands, without Limit of Expence therein, to perpetuate his Memory to all fucceeding Ages, and to continue as long as the very Empire itfelf; and which, for their Excellency ofWorkman- Ihip, are juftly admired by the whole World. That noble Encourager of Arts and Sciences, Lewis XIV. of France cauled 70 of the Bas-reliefs of this Column to be moulded, fome of which were call in Brafs, and the others employ'd to incruftate and embellifh the arch’d Cieling to the great Gallery in the Louvre. As 22 i, The Trinciples of Geometry. As many of our Englifh Nobility have taken pleafare to credl in [date Co¬ lumns, in their Gardens and Parks, to the Memories of their Friends • and which are not only very great Marks of Gratitude and affectionate Love, but Ornaments alfo ; i mult therefore deftre my young Readers, not to pafs over the Conlideration hereof in a flight Manner, but confider well their Nature and Ufe ; fo that, when it may be required to raife a Column for any fuch Purpofe, they may readily know, how to complete the Compofition agreeable to the Sullied its made to perpetuate. Now, to make this more plain, 1 will defcribe this Column in the Words of Mr. Evelyn, as follows: “ The firft, and as it were the Foundation of the “ whole Structure, is the Pedeftal, which is here no left nccefiary, than is the “ Cornice to the Columns of the other Orders ; and its Proportion, though “ fquare and folid, requires an Enrichment of handfomc Modenatures, and of “ all other Sorts of Ornaments at the ‘Plinth and Cymatium ; but above all in “ its four Faces or Sides of the Die, which are as it were Tables of Renown, “ where fhe paints the Victories of thole Heroes to whom fhe erects fuch zlori- “ ov.s Trophies (See Plate F, following Plate XX. where the principal Parts of “ this Column, with its Meafures in Minutes, are exhibited, with their Fm- “ beililhments, dec.) It is there that we behold all the military Spoils of the t; Vanqmlhed, their Arms, the Machines they made tile of in Fight, their “ Enfigns, Shields, Scimeters, the Harneft of their Horf.s, and of their Cha- “ riots, their Habiliments ot War, the Marks of their Religion, and, in a “ Word, whatever could contribute to the Pomp and Magnificence of a Tri- " umph. Upon this glorious Booty, as on a Throne, is eredted, and revetted “ with the moll rich and fplendid Apparel which Art can invent, and indeed, “ provided the Architect be a judicious Perfon, it cannot be too glorious. I “ repeat it again, that this ought in no fort to alter, or in the leait confound, “ the Proportions and Tufcan Profiles of the Bale and Capital, res bcinv the “ very Keys of the Concert and Harmony of the whole Order. The la ft, but “ principal Thing, becaule it lets the Crown upon the whole Work, is the “ Statue of the Perfon to whom we credl this fuperb and magnificent Struc- “ tore : This hath an Urn under his Feet, as intimating a Rcnafcency from “ his own Allies, like the Phoenix, and that the Virtue ofgreitMen tri- “ umphs over 'De/liny, which has a Power only over the \ ulgar." Fig. I. ex¬ hibits a Profile of the Pedeftal and Bale of the Column at larges as Fig. II. doth of the Capital, and Plan of the Stairs within the Shaft. The Circles B and A taken together denote the Thickneis of the Shaft at its Bale, the Circle A the Thicknefs of the Shaft next the Aftragal, the Circle C the Going of the Stairs and the Circle D tlie Well-hole or Newel. Serlio meafured the Parts of this Column with the old Roman Palm (which I think is equal to 9 of our Inches) divided into iz equal Parts, called Fingers and each Finger (being again) fubdivided into 4 equal Parts, were called Mi¬ nutes ; fo that in 1 Palm there are 48 min. The Height of the Pedeftal, with the Sub-plinth, on which refts the four Eagles, he lays is zj palms 8 min the Column, including its Bafe and Capital, 148 Palms' ;z min. and the Urn on the Capital, with its Crown, 14 Palms 14 min. which together make 184 Palms 1 6 min. equal to 138 Feet 3 inch. The Breadth of the Peddial's Die is Z4 Palms 6 min. on vvhofe Faces are cut two Compartments of many Tro¬ phies, with the following Infcription : s. P. Q. R. IMP. CAESARI DIVI NERVAE. F. NERVAE. TRAIANO AVG. GERMANIC. DACICO PONT. MAX. TRIB. POT. XVII. COS. VI. PR AD DECLARANDVM QVANTAE ALTITV- DINIS MONS ET LOCVS SIT EGESTVS. The The Principles of Geometry. __ 2*5 The Diameter of the Column at its Bafe 1 6 Palms, and r + Palms at its Af tragal; fo that its Diminution was but r* or i. The Height of the Fife i jireeifeiy 8 Palms, and the Height of the Capital f thereof! The Height of the Column, including its Bafe and Capital, is 8 diam. and a half- ft that we are here to obleiwe, that, altho’ its Members are chiefly Tufcan, yet its Altitude is really Tonck, as indeed are its Flutings, which are without Fillets Plates XL1I. and XLIII. Tufcan Intercolumnations by Sebastian Le Clerc. Plate XLII. Fig. V. reprefents the Intercolumnations of this Matter, for CoUimns in Colonades, winch are determined by Modules and Minutes as affixed in their l laces ; as alio are the Intercolumnations of the Arcade Fig IV Hate XLIII reprefents fix Examples of Intercolumnations to Arcades for l>i- azza s ; of which Fig. I. and Fig. II are with Pedeftals, to be ufed either with w n th ‘i ee f q V a i te , r Columns ’ °r Columns in Pairs, detached free from the IMill, with ledeftals behind them, as their Plans exprefs, as the Nature of the Cafe may require. Figures V. and VI. are alfo with Pedeftals : but thefe our Matter has g‘ven, as Examples to be pradtif'ed, when Pilafters are to be Hied lnftead of Columns. Figures III. and IV. arc arcaded Intercolumnations, without Pedeftals who e Meafures being hgnifled by Modules and Minutes, need no further Explanation. Plate XLIV. The Intercolumnation of the Portico of St. PaulV, Co vent-garden. This Building, I have already noted, was built by Mr. Stone, and not by Mr. Inigo Jones, as moft do imagine, and indeed as I did myfeif, when 1 mea- lurtd the Columns in the Portico ; but happening, loon after ray Meafurement to be fpcaking of it to my worthy Friend, Sir James Thornhill dcceat'ed’ he did allure me, that Mr. Stone was the Architect of that Building • but whether it was his own Defign, or Mr. Jones' s, I will not undertake to fay To proportion this Portico, (i) divide the given Breadth into 4 equal Parts at g, h i ■ alfo divide the r outer Parts, d g, and i f, each into r equal Parts at l and m, which arc the Places, where the two outer Columns are to Hand' (i) Divide Im or pg into equal Parts ; then give 4 to p r, and the like to v q, lo from the Points r, v, mutt the central Lines of the inward Columns be railed. The Altitude of the Cornice h is equal to d h, the half Breadth- and a I>, the Altitude, or Pitch of the Pediment, is equal to half 1 > d, as is plain by the proportional Semicircles, which exhibit the whole by Infpeflion The Manner of dividing the particular Members is fhewn in Plates XX VI XXVII. Fig. II. and Ill. are Examples of Tufcan Frontifpieces for Doors, of which Fig. If is a double Example, having a Baluftrade on the right Hand, and a Pediment on the left ; which laft Example exhibits the Effeft, that’a Pedi¬ ment hath with a femicircular-headed Door; and Fig. Hi. fhews the Efte® with a fquare-headed Door. The Intercolumnation f h. Fig. II. is 6 diam and the Aperture 4, diam. the Height g h divided into 3 Parts, the upper 1 at i is the Center ot the Arch: So likewife in Fig. 11. the Intercolumnation p q is 6 diam. and the Aperture 4,; therefore, the Diameter of a Door being giv en, divide it into 4, and take 1 for the Diameter of the Column, and then proportion the Order according to any Matter, as required. Note, 111 making circular-headed Doors, that the Impoft be always above the common Height of a Man, viz. 6 Feet. Note alfo, that the Height of Doors and Gates be not lefs than twice their Breadth, nor more thanVvice and one fixth Part. K k k Plate 2 2 6 The “Principles of Geometry. Plate XLV. Divers Compofilms of Block Cornices examined, with ^ the Manner of pro portioning them to the Height of any givenBuilding: (JIVork entirely new.) Alfo, Mr. Gibb* erroneous Method oj pla¬ cing Cornices over Ruilick Quoins of Buildings detccled. Before we can make Block Cornices fit for Buildings, we muft know how to find their proportionable Heights, which Mr. Gibbs, nor any other Mailer has Jet thought neceflary to teach. To effect tins; (.) biv.de the given Height into five equal Parts, then will the upper one be a Height inade fit to receive a Tv.(can Entablature, (xl Divide the upper one into 7 equal 1 arts, .. n cl tike the upper four for the Height of the Cornice required. Pig. I- IP :,nd I ! ! are various Defigns (which differ in their lower Parts only, the Q- v ,,urri and Corona in every of them being the lame) by Mr. Gibbs of which the firft is worthy of Regard ; but the two laft being of iTteanei Defign, on Account of the fmall trifling Members, on which hi. Blocks 01 Truffes are placed - and efpecially that of Fig. III. which are two numerous they arc therefore to'be re'icdlcd. As the Aftragal qf, Fig. I. cannot properly be faid to be a Part of the Cornice, it being to the Wall, the fame as it is to the Shaft of a Column; therefore divide the Height of the Cornice ^/ mto 6 p. u - t , at a b c, cL e, x, and give to each Member its Height as the proportional Semicircles exprefs. Make/A equal to ! of the Cornice s Height and at that Height let the upper Ruftick fimlh. Make hi, 1 k &-C each equal to half a h, which is the Heights of the Rufticks. 1 he Breadth of each he. d- Ru fti c k from the Upright of the Wall (not from its own, viz. from 0 > to mll |t be equal to ! of the Height of the Cornice, (that is, to a c) and the Breadth of the ltrctching Ruftick equal to the whole Heigh *[ h ° ,d (that is to af) over which mutt Hand the out Line of the id Block 01 Trulls of the Cornice, whole Breadth muft he equal to t of the Cornice Height The out Line of the lit Block, or Trufs to the Cornice, muft Hand prcalclv over the Upright of the Wall, and thereby hath a true Bearing, and its Breadth the fame as aforelaid : But when they are placed without the Perpendicular of the Wall, as in the lower Example, they have a talle Bear- mg on the lower Members. Divide the Height of any Ruftick, 38 W /f ’ into 8 equal Parts, and take 1 for the Height of their champheied Edges. As there Cornices are generally tiled without Architrave and Freeze, they are therefore made with Ids Projection than is equal to their Height as othei Cornices generally arc. As I have now done with the Divifion of the C01- mcc and Rulticks, I muft now proceed to take Notice oi a moft fmpitting Error committed bv Mr. Gibbs, and many other Perlons, in placing Cornices over ruft,cared Walls, in Manner as follows, viz. Inftcad of iprmging from the Uptight of the Wall n 0, Fig. i they make the Height n 0 into a Sort of Fafcia or Band, carrying it out equal to the Upright of the Ruftic > and from thence /ring the Cornice, as is reprelented by the dotted ^loftle ot the Cornice 7 v x y z q 11, which brings the whole Cornice too loiw d, .vires fa lie Bearings to the Truffcs, and a clumfy Afpcft to the Whole. It is alfo ve,-v common, in the rufticating of P,Liters and Columns to.make the Face of the Ruftick the Upright of the Shaft and cut the Clianigumgs into the Body of the Shaft. This is a monftrous and ftupultclity • for here, under a Pretence of ttrengthemng the Shaft as Rufticks i d , when thev embrace it, it is weakened as much as the Depth of ther tres o’- Champhers are cut in. Of this Kind oi Rufticks are thole ruitica Brick Frontilpieces on each Side the Church of St. Paul, C^arden which, i think, were not long fince beautified, either undei the D refti n, or at the Expence of the prefent Earl of Burlington : But luiely, foul h. Lord 111 in well examin'd their Architecture, he could not have avoidec the feeing of this fo vilible an Abfurdity ; and inftcad of giving to them a The 'Principles of Geometry. 227 new Cloathing of Plafter in that maimed Condition, would have taken Pity, and order d thofc barbarous Wounds to have been healed up, as Mr .Dodding- ton did thole monftrous rubricated Columns built at Gunmll in DorfetJlAre, under the Direction ol Sir John Vanbrugh. The Ruftick Piers before the Houle of the prelent Lord Chancellor in Lincoln s-inn-ftelds, built under the Direction of Mr. Jones, Clerk of his Majefty’s Works at Kenfmgton Palace, is alfo another horrid Example of this Kind. There are alfo many other wretched Things of this Naure in and about London, which at prefent I omit. Figures IV. and V. reprefent four Defigns for Ruftick Doors, every Side being a different Defign : Thole of Fig. IV. are tolerable good for grotefque Buildings, but thofe of Fig. Y. are both monftrous: For this great Mailer (as he’s thought by foine to be) has not been contented with breaking the natural Courle ol the Architrave, but of the Freeze allb, which of themfelves, with¬ out the Rufticks, would have made a good Frontifpiece; befides, the breaking of the Architrave into fo many Parts, is a weakening (not a ftrengthening) to the Whole, and therefore abiurd. Plate XLVI. Two Rujlick Doors. Fig. I. is a Defign of Vignolas,, but not to be commended, he having de- ftroyed the natural Collide of the Architrave and Freeze, lor the Sake of making the Key-Hones monftroufly high and narrow. Fig. II. is a Profile of the Entablature to this Frontifpiece, which for grotefque Edifices is very good. Fig. III. is another double Delign for two Ruftick Doors, by Mr. Gibbs, where you fee lie has now crept with his Key Hone into the Bed-mould of the Cornice, which is furprifingly monftrous. As to the Rufticks on each Side, they are indeed of very pretty Invention, and would better become a Frontif¬ piece to a Paftrycook's Shop, than they do the Window's and Doors to the Church of St. Martin in the Fields, as being analagous to the rufticated Edges of Patties and Pies. Plate XLVII. The Geometrical Conjlrucl'wn of the principal Parts of the Dorick Pedejlal, and of the Mouldings of its Bafe, by Carlo Ce¬ sar e O sio. I. Lei a i. Fig IV. be the green Height of the Pede/lal. Practice. (1) Draw i 0 of any Length at Right Angles to a i. (1) Draw a 0, making an Angle of 30 deg. with the Line a i, interfefting io in 0. (3) From the Point i , let fall the Perpendicular tk on the Line a 0, and from k draw k n parallel to a i. (4) From n draw n m parallel to ik, and from m draw hm parallel to is. (y) Bileft e k in f, and through / draw gl parallel to h d. ( 6 ) From l draw l b parallel to a 0, cutting a i in b ; then is g i the Height of the Bale, g b the Height of the Die, and a b the Height of the Cor¬ nice, as required. II. Let m c, Fig. III. be the given Height of the Tedeflal's Bafe. Practice, (x) Divide me into 5 equal Parts at the Points y, 6 , a 7, and give the lowermoft 1 to the Height of the Sub-plinth, or Zoccolo, abeg. (1) Draw g c at Right Angles to me, and b a parallel thereto, both of Length atPleafure. (3) Make cf equal to ac, alfo draw the Lines ch and f h, mak¬ ing the Angle fc h equal to 30 deg. and the Angle cfh equal to 60 deg. and continue h f out at Pleaiure towards e. (4) From h let fall the Perpen¬ dicular h d, and on d, with the Radius d h deicribe the Arch h e, cutting h f in e. (y) Through the Point e draw the Line geb parallel to ac, which will determine the Projefture of the Zoccolo. (6) Thro’ the Fo:nt in draw k n At 22 0 The Trinciples of Geometry. at Pleafure parallel to g c, and from the Point a draw ah, making the Angle k a waRqual to 30 deg. (7) Biiecf km in l, and draw Is, making the Angle vi Is equal to 30 deg. (8) From m let fall the Perpendicular m t on the Line ha, cutting the Line Is in 0, from which draw or parallel to kn, lor the Height of the Fillet. (9) From the Point of Jnterfcdlion, made by the Lines /s and m c, draw a Line to n, making an Angle of 45- deg. and draw n r pa¬ rallel to me, which will determine the Projecture of the Fillet mn. (10) From t draw t tv parallel to k n, cutting me in the Point 9, then will 8 9 be the Height ol the Torus. (11) Bifcdt 10 a in 4, and draw 4 s perpendicular to k a, cutting Ip m s, from whence draw sr parallel to 8 9. (n) Bifcfl: r s in q, the Center of the Torus. (13) Make 9 vj and six each equal to 9 v, and draw the pricked Line wx. (14) Make .vs equal to w x, and draw the Line w z. which bii'ect in /, and then deferihe the Cima ay a. (ijj Make a z equal to si z, and bii'ect zb in the Point 3. (16) Make z 1 equal to z 3, and draw the Line 1 3, which completes the Projections of the Members in the Bale as required. Fig. I and II. demonftrate, how from a Man's Body the Circle and Square were firlt taken, of which I have already taken .Notice. Plate XLVIl. The Geometrical Conjlruchon of the Die and Cornice of the Dorick Pedeflal, as alfo of the Dorick and Attack Bafe, by Car¬ lo Cesare Oslo. I . The Height of the 'Die to the Dorick 'Pedeflal, Fig. I. being given, to find its Diameter : Together with the Height and Projection of its Ft tie t at Bottom. Let the given Height be i s. Practice. (1) Bifcft is in n, and on n, with the Radius ni, deferibe the Circle fmqr. (z) Draw op through n at Right Angles to is. (3) Divide the Quadrants 0 i in /, ip in m, ps 111 r, and so in q. (4) Thro' the Points i s draw the Lines k h and t v parallel to op\ alfo, thro’ the Points Iq and mr, draw the Line- it and k t parallel to is, interiefting the former in the Points 0, k, v, t, thereby completing the Body of the Die, as required. T0 deferibe the Fillet xy at its Bottom. (1) Continue tv to z, making js equal to ns. (r) Make the Angle wsz equal to 30 deg. and the Angle wz s equal to do deg, and from w, the Point of InterfeCtion, draw w y parallel to h v, then will xv be the Projection of the Fillet. (Let df reprefent v x and c e. Part of hv, which being larger, the Manner ot the Operation is more intelligible.) (3) Make/g equal to half fd, and drawge parallel to fd, alio join fg, and then is the Fillet completed, with its Height and Projection required, (i) Continue// to a, making fa equal to f d, and then on f, with the Radius f d, deferibe the Hollow fb, which completes the Whole, as required. II. The Height of the Cornice to the Dorick Pedeflal, Fig. II. being given, to divide it into its feveral Mouldings. Let a n be the given Height. (1) Thro' a draw cb at Right Angles to an, as allb nk through the Point n. (a) Make the Angles cn a and kan each equal to 30 deg. (3) Bi- fect c f in d, from whence draw dz parallel to be, alfo bifeft fn in if and from k, thro’ i, draw hi, continuing it till it meet an in x, (4) From x draw xg, and from i draw i t parallel to a b, and of any Length at Pleafure. (JO Fromc, thro' h, draw c w, and from d draw de parallel to an, then from Tlx Principles of Geometry. 229 from e draw the Line e 4, and from 7 v, the Line w 1, and thus are the Heights of every Moulding determin’d. To determine the Trojeclitres. (1) Make ah equal to an, and from h, draw b 4 parallel to an, which will determine the Fillet. (2) Make 4 3 equal to J b 4; alio 3 6 equal to ed, and on tlie Point 6 , deferibe the Ovolo 3 z. (3) Continue 6 1 to 1, then is the Fillet under the Ovolo done. (4) Make qg equal to .V9, and thro’ the Point g draw the Line 7 t parallel to a n ; then continuing z 1 to/, making 1 y e- qual to 1 z, on /, with the Radius/ 1, deferibe the Arch 1 8, which com¬ pletes the Face of the Corona, (7) Bifedt 9 g in 0, make qt equal to 0g, and draw oq ; alfo bifedt qt in j, and on q deferibe the Arch 0 s, which com¬ pletes the Drip. ( 6 ) Make x l equal to p 10, and draw Ip the lower Fillet. (7) Make nm equal to half tp, and deferibe the Lima reverfa, which completes the whole Cornice as required. HI. The Height of the Dorick Safe, Fig. IV. being given, to divide it into its Jeveral Mouldings. Let b y be the given Height. Practice, (i) Bifedt by in u, and draw xw and ys at Right Angles to by, and of any Length at Pleafure ; then will xy be the Height of the Plinth. (1) Draw a c parallel to w x ; alfo draw a x, making the Angle axb equal to 30 Degrees. (3) On a, with any Radius, deferibe an Arch, as 3 1, which divide into z equal Parts at z, and draw the Line a z 4, from whence draw the Line 4?/ parallel to w x, then will 4 x be the Height of the Torus. (4) Bifedt 4 x in 0, and from 0 draw the Line ot, parallel to wx, ot any Length at Pleafure. (7) From 6 let fall the Perpendicular b d on the Line a 4, and from d draw the Line d i, cutting by in/; make ef equal to f 4, and from e draw the Line eg, parallel to 4 n : And thus are the Heights of every Moulding determin'd. To determine their Trojedures. 1) Draw the Line / c, making an Angle of 47 Degrees with the Line f b, which will cut be in e, then drawing eg parallel to be, the Cindture or Fillet is completed, (2) Continue eg to q, bifedt oq in p, and make q r equal to p q, and draw Ir parallel to 4 0. (3) Make kh equal to k l, and draw hi, cutting / i in i, the Center of the Aftragal. (4) Make x w equal to e x, and draw e w, which will cut 0 t in s, the Center of the Torus. (7) From w draw w z parallel to xy, until it meet zy in z, and then is the Bale completed as required. IV. The Height of the Attick Safe being given, Fig. III. to divide it into its Mouldings, with their TrojeBures. Let a 6 be the given Height. (1) Bisect a 6 in t, and from thence draw the Line t c, making the Angle ate equal to 30 deg. (2) Draw a c, and 7 6 , at Right Angles to a 6 , and of any Length at Pleafure. (3) Bifedt t c in g, and on g raife a Predendicular, which will cut a 6 in in ; then will a m be the Height of the upper Torus. (4) From m draw m e parallel to a 6 , alfo, thro’ the Point g, draw the Line by pa¬ rallel to a 6 , which will cut me in e. (7) Bifedt b e in cl, the Center of the upper Torus; alfo, bifedt g e in f, and, making mo equal to e f\ draw of, the Fillet under the upper Torus. ( 6 ) From the Point m draw tire Line ms parallel to t c, and at t let fall the Perpendicular ts on the Line m s, and from the angular Point r draw the Line 18 1, which is the upper Part of the lower Torus. (7) Bifedt in 8 in p, and from p draw p q, making the Angle qp 8 L 1 1 equal The 'Principles of Geometry. 230 eqi-al to do deg. which Line continue until it meet ms in q. (8) Draw the Line qy cutting a 6 in r, through which Point draw r z parallel to 8 1, which is the Height of the lower Fillet. (9) Make the Angle 8ty equal to rto deg. then will ,r 7 cut a 6 in 7, then 8 7 is the Height of the Torus, and 7 6 the Height of the Plinth. (10) Bifeft 8 7 in v, and draw v a the central Line o! the Torus. (11) Make 4 7 equal to my, and draw the Line m 4 cut¬ ting o a in a., the Center ot the lower Torus. (la) Ahike 7 d equal to 4 7 and join 4 7, which completes the Plinth. (13) From / draw f z parallel to >n 4, cutting r z in z, and me in h, and draw s 1 parallel to a 6, which com¬ pletes the Proiecture of the Fillet. (14) On the Point a, with the Radius a 3, dc'tribe the loner Torus. (15) Draw h w parallel to ey, and bifeft hz 111 k, Irom which Point draw the Line nk parallel to r z. (id) Make ni equal to l n. then the Point i is the Center of the Arch f n, and the Point k is the Cen¬ ter of the Arch ;/ x, which together form and complete the Scotia : And thus are all the Mouldings completed as required. Plate XLIX. The Geometrical Conjlruclion of the Attick Safe, accord¬ ing to L. B. Alberti; and the Cincture and Ajlragal to the Dorick Shaft , by C ar lo Ces are Osio. I. To deferibe the Attick Bafe according to Leoni Baptifta Alberti. Let b p he the given Height. Practice. (1) Through the Extreams bp draw the Lines gb and zp at Right Angles, and of Length at plcaiiire. (a) Draw b m until it meet zp, being continued in m, making the Angle ni bp equal to 30 deg (3) Divide the Angle bmp into a equal Parts by the Line mn, which continue until it meet bp in n, then is np the Height of the Plinth, and therefore from n draw ny of any Length parallel to p z. (4) From n draw a n, making the Angle • '■ equal to 30 deg. alfo draw a b, making the Angie a bn equal to 60 deg. and from a, their Point of meeting, draw a d 1 parallel to bg, then will bd be the Height of the upper Torus. (7) Bifeft b d in c, and from c draw c f pa¬ rallel to bg, which is the central Line of the Torus ; alfo bileft t n in 0, and from 0 draw 0 v, the central Line of the lower Torus. (6) Bileft d t in 1 , make bg equal cob l, and draw Ig ; alfo make ov equal to b g, and draw gv. (7) Bileft tg in i, and through i draw e k parallel to b p, cutting cj in f, the Center of the upper Torus. (8) From the Point h, where g l cuts i d, draw hq parallel to a n, interfefting e k in k ; from whence draw 1 k parallel to di, and then is that Fillet completed. (9) Make Is equal to It., and draw J q inter¬ fefting k q in q, and joining q r, that Fillet is alfo completed. To deferihe the Scotia. (1) Make ii equal to i k, and from h draw h 7 parallel to a s, which divide into 4 equal Parts, at the Points 3, 6, 7. (a) From the Point 3 draw the Line 31a parallel to bg ; alfo, from the Point q draw the I.ine q 2 parallel to bp, interlocking the Line 3 a in the Point a ; then is the Point 1 the Center of the Arch k 3, and the Point a the Center of the Arch 3 q, on which deferibe the Scotia required. (Laftlv) On the Center /) with the Radius e f, deferibe the upper Torus ; and on the Center v, with the Radius v w, deferibe the lower Torus ; and then will the whole Bafe be completed as required. II. The Height of the Dorick Column, Fig. I. being given, to divide of} the Heights of the Baje and Capital. Let agbe the given Height. Practice, (i) Bifeft ag in c, from whence draw c d, making an Angle of 30deg. alfo from g draw g d, making an Angle ol 60 deg. with the Line ag, which The 'Principles of Geometry. 3 " hich continue until it meet c d in d. (2) Biledt g d in c. from whence draw e f, perpendicular 10 ag. (3) Make ah equal to /g, then a h will be the Height ot the Capital, f g the Height of the Bafe, and bfthe Height of the Shaft. JIT. To proportion the CinBure to the Shaft of the Dorick Column. Let e d, Fig. IV. he the Semtdiameter of a Column, andbd the central Line. . Practice, (i) Draw the Line dp of any Length, making the Angle p do equal to 30 deg. alfo draw the Line cp, making the Angl epca equal to 30 deg. interfering each other in the Point p ; from which let fall the Perpendicular p 0 on the Line d c, being continued to 0. (2) From 0 draw 0 k, making the Angle hoc equal to 30 deg. which continue until it meet pc in k, from whence draw / / parallel to c 0, alfo draw k m parallel to a c. (Laftlv) On k as a Center, with the Radius k l, deferibe the Arch / h, and from h draw hq parallel to dm, which will complete the Cincture as required. Note, I hat the molt ready Way ofdelcribing an Angle of 30 deg. is to open the Compalles to any Diftance, as d e, and deferibe an Arch, as f e, on the Point d: This done, let up the fame Opening from e to f, on which Points, with the lame or any other Opening, deferibe Arches interfedling in g; then a Line drawn from d thro' g will make an Angle of 30 deg. with the Line dm. IV. To proportion the A(iragal to the Dorick Shaft, the 'Projection being given. Let a c. Fig 111 . he equal to the TrojeBion given. Practice, (i) Make ag equal to a c, and draw gc, which bifedt in e, from whence draw e l parallel to a k of Length at pleafure. (2) Divide g c in¬ to 3 equal Parts at f d, and through f draw m n parallel to a c ; alfo from g draw g i parallel to m n, cutting e l in h and i, and completing the Fillet. (3) lire Point d is the Center of the Aftragal, therefore on d, with the Radius h d, deferibe the Aftragal h 0 n. (Laltlv) Continue h i to l, and make i l e- qual to g i ; then on the Point /, with the Radius 1 l, deferibe the Hollow or Arch i k under the Fillet, which completes the Aftragal required. Note, In this Example the Height and Projedture are equal ; but in the following the Height is but i of the Projedtion. V. To proportion the Dorick Aftragal, Fig. V. with its Fillet, fo that its Height fall he hut 3 fourths of its ‘Projection. Let ai he equal to the given TrojeBion. Practice, (i) Make i m equal to ai, and draw am, which bifect in h, through which draw kd parallel to a i ; alfo bifedt: a h in h, the Center of the Aftragal, from whence draw eg parallel to i n. (2) Biicdt k m in l, and from l draw If parallel to k d, cutting eg in/. (3) On b, with the Radius bd, de¬ feribe the Aftragal 0 p d, and making /g equal to f l, on g, with the Radius / g, deferibe the Arch f n, which completes the Whole as required. VI. To proportion the Aftragal, Fig. VI. with lefj'er Height and TrojeBion than the former. Let eh be the given Height. Practice, (i) Draw ke at Right Angles to e b, and divide e b into 3 e- qual Parts at de, make ba equal to cb, and draw the Line ha, which bifedl in h, and from thence draw h c parallel to k e. (2) From b draw f b parallel to he ; alfo from b draw/ i parallel to a k, then biledl e c in d, and from d draw di parallel to k e, cutting b i in z, the Center of the Aftragal. (3) From g draw' g f, the Face of the P'lllet, parallel to ae, and on z, with the Radius ih, deferibe the Semicircle Ih, which completes the Aftragal as Required. 2 32 The Trinciples of Geometry. Plate L. The Geometrical Conjlruclion of the Dorick Capital and Enta¬ blature, by Carlo CksareOsio. I. The Height of the Dorick Capital, Fig. IV. being, given, to defcribe its Mouldings. Let i - he the given' Height. Practice, (i) Divide i 7 into 3 equal Parts at pv, and draw f p and 2 v at Right Angles to i 7 of Length at pleafure. (2) Make pf equal to p v. and draw 7 v ; bilect p v in q, q v in s, q s in r, and s v in t, and from the Points r, s, i, draw the Lines wr, xs,yt, at pleafure, parallel tofp. (3) From the Points w, z, 1, draw the Lines w x, zy, 12, parallel to r v, which will complete the Annulets. Continue 1 2 to 3, the Center of the Hollow 14,. (4.) Draw 0 r, making the Angle 0 rp equal to 30 deg. meeting f p in 0 ; ailb draw on and np, making the Angles nop and np 0 each equal to 30 deg. in- terlecting each other in n, the Center of the Ovolo, on which, with the Radi¬ us nw, defcribe the Ovolo wg. (y) From 0 draw 0 h parallel to i p, make p m equal to p r, and front m draw rn d at pleature ; make h a, the Projection of the Abacus, equal to h 0, and draw the Diagonal a 0 cutting dm in e, from whence draw e f parallel to m p. (6) Divide 1 m into 3 equal Parts at k /, and from k draw kb, parallel to a i, at pleafure. (7) Draw ab parallel to He, to complete the Fillet. (Laftly) Make tic and de each equal to half ab, and draw the Line c d, on which defcribe the Cima Reverfa : And thus arc all the Parts of the Capital determined. T0 delineate the Aflragal. (1) Bisect v - in 5, make 7 6 and 7 18 each equal to 7 7, an ' draw the Diagonal 6 18, which bifeft in 10, through which draw 9 12. (2) llifeift 6 xo in 8, the Center of the Altragal ; alio 8 10 in 11, from whence draw 11 14, parallel to7 17; alfo bilect 1 2 18 in 1 6, and from 16 draw 1613 parallel to 12 9, cutting 10 14 in 13, and thereby completes the Fillet. (Continue 10 1; to 14, making 13 14 equal to 13 id.) (Lnltly) On 8, with the Radius 8 9, de- Icribe the Aflragal; and on the Point 14, with the Radius 16 13, defcribe the Arch 13 17, which completes the Whole as icquired. II. To defcribe the Dorick Entablature, Fig. J. II. HI. Let aX be the given Height equal to 1 fourth Tart of the Column's Height. Practice, (i) Draw the Line a Z, making the Angle X a Z equal to 30 deg. and the Line XZ, making the Angle aX 7 . equal to 60 deg. interfecting a Z in Z, from whence draw ZB perpendicular to aX, then will B X be the Height of the Architrave. (2) Bifedt a B in b, then will b B be the Height of the Freeze, and ab the Height of the Cornice. The principal farts of the Entablature being thus found, 1 will now proceed to the Divuions of their Parts, of which the Architrave is the firlt in Order. Let a X reprefent the central Line of the Column, and b X is the given Height of the Architrave. Practice, (i) Draw the Line 13X, making the Angle J; X 13 equal to 30 deg. cutting B 10 in 13. (2) lliledt B 13 111 a, and from a draw a C, making the Angle B a C equal to 30 deg. and cutting BX in C; then is 13 C the Height of the Tenia or Band, and therefore from C draw C 7 at pleafure parallel to B 9 (3) Bilect X 13 in z, from whence raife the Perpendicular zy, cutting BX. in y, then is Cy the Height of the Drops with their Fillet. (4) Make C7 and X 13 each equal to CX, and draw 13 7 continued until it meet B 9 in 8 ; make the Projeftion of the Tenia 9 8 equal to 3 fourths of its Height The Principles of Geometry. 2 33 Height 8 7, and draw the Line 9 y for the Face of the Tenia. (7) Make the Projections of the Gotta 1 11 11 equal to half the Height of the Tenia. ( 6 ) Make 4 3 equal to ; of 8 7 for the Height of the Fillet : And thus are the Parts of the Architrave determined. The Gutta’s or Drops /, m, p, q, t, w,y, altlio belonging to the Architrave, will be better deferibed with the Triglyph in the Freeze, which now comes next in Order. Let b 15 be the Height of the Freeze. Practice, (i) Divide b B into 3 equal Parts at c d, and make B 9 0 equal to c B, and draw 90 91, which will be the Bounds or Limits of the Serni- triglyph. (a) Divide B 90 into 6 equal Parts at the Points g, f, e, d, c, and from thence draw the Lines g 1, f z, e 3, d 3, and c y, parallel to b B. (3) Make 91 11 equal to y 91, or f Part of the Semi-triglyph's Breadth, and draw the Line 11 7 parallel to 91 b ; alfo make n iz and 8 9 each equal to 10 11, or i Part, and draw the Lines 6 9, 13 9, and 10 it, which will com¬ plete the Semi-triglyph as required. (4) The Line c 90 being drawn, and in- terleclcd in the Point K by the Line z /, from thence draw K L parallel to B 90, cutting 91 90 in the Point L. (y) Make 90 8 equal to 90 L, and from the Point 8 draw tile Face or Upright of the Freeze, which is alfo in the lame Plane with the Upright of the Face of the Architrave. Note, The fame being performed, on the other Side of the central Line, will complete a Triglyph entire. Under the Tenia of the Architrave are placed 6 Gattds or Drops in fuch man¬ ner, as if they hadflowed from the Channels of theTriglyph thro’ the Tenia and Fiilct under it, therefore feem to be an Ornament belonging to the Triglvph; and indeed, when the Triglyphs are omitted, as oftentimes they are, thefe Gutta’s are omitted alfo ; wherefore ’tis evident, that tho’ they are placed in the Architrave, yet are a Part belonging to the Triglyph only. Their Forms are twofold, being made by fome as the Fruftums of a Setni-pyramis; and by others, as the Fruftum of a Semi-cone, cut from their Vertexes, down their Axifes, perpendicularly unto their Bafes. The Manner of deferibing them is as follow s. Practice. (1) Continue down the Lines 91 90, y c, 4 d, 3 e, zf, and 1 9, towards X 13. (1) The Line X 13 being before drawn, make the Angle B X 13 equal to 30 deg which bifeft in z, and front thence raife the Perpen¬ dicular zy, cutting BX in y, from whence draw / ix, parallel to X 13, for the Depth of the Gutta’s, interfering the aforefaid continued Lines in the Points /, m., p, q, t, w. (3) Draw the Diagonals y h , C t, h p, t i, i l, k p, in¬ terfering D 3 in the Points n, 0, r, s, v, x. (Laftly) Draw the Lines ,r y, v t, s t, rp, 0 p, n l, and then u t xy, r p s t, and 0 p n l, will be tire Profiles or Sections of the Gutta’s to the Semi-triglyph, as required. This Ornament, with the Triglyph over it, has a Very agreeble Effect; and therefore Workmen are fond of introducing it, without confidering what it re- prefents, or whether 'tis agreeable to the Frontifpiece they make, which is entirely wrong: For, as they were firlt ufed in the “Delphic Temple, to repre- lent an Antique Lyre, which Inftrumcnt Mpollo had been the Inventor of; they cannot therefore be a proper Ornament to every Building, any more than the Ox-skulls placed in the fquare Intervals, or Metops between them, which th Antients introduced, alluding to their Sacrifices, fyc. with which we are en¬ tirely unacquainted : And therefore, inftead of fuch Ornaments, we fhould in¬ troduce fuch as allude to the Situation of the Place, or Perfon, to whom the Building belongs. Having thus gone through the Architrave and Freeze, there is now the Cornice only remaining. Let a b, Fig. I. be the given Height. Practice, (i) Bifett a b in c, alfo c b in x, make 1 d equal to ; of 1 b, M m m and The 'principles of Geomf.tr y. 23.1 and from the points c, z, d, draw Lines parallel to each other, and at Right .Angles to ah, of Length at plea lure, (i) Make the Angle dbh equal to : :■ .... hi . • . in c, and draw kc at Right Angles to bh, cutting db in k, tiom whence draw k •v at pleafure. (3) Make k v equal to i of kb, and from -e draw ■■;3 3 T TT J IS u 18 1*7 3 ° z 9 3i 34 33 _ 3 Y then will the lanes a 24, b 17, c z 6 , da, e zj, f 18, g 19, h 30, i 31, k 31, l 33, and m 34, form the Profile of the Guilds, in the Tlancere of the Corona. (13) Draw 0 36 at Right Angles to d 7 ; alfo divide 0 4 in 37, and delcribe the fmall Semicircle for the Drip. (14) From the Point 4 draw the Line 4 41, parallel to the central Line, lor the Face of the Mu- tile (17) Divide 40 43 into 3 equal Parts, and make 41 39 equal to one of thole Parts. (16) Make 41 a equal to 41 41, draw a 39, and delcribe the Cima reverfa ; alfo make a 40 equal to 41 39, and draw the Line 40 49 ibr the Face of the Corona. (17) Make the Projeffions of the Cima reverja and Cima retda each equal to their Heights, and that will complete the Whole as required. R E M A R K. This Geometrical Conftruclion of the Orders may probably have been the very firH Method uied, as Carlo Cesare Osio doth affirm, and in which there is a great deal ot Pleafure, in confidering with what great Difficulty thole Rules have been acquired, and how; furprilingly, in many Cafes, the Proporti¬ ons The Principles of Geometry. 235 ons of one Member to the other is produced : Vet upon the Whole, I cannot think it to be io good a Method for a young Beginner, as many others cent prifed in this Work ; and therefore to fujrh, it will be better to read this Matter, after they have well acquainted theml'elves with others more eafy. But, becaufe 1 give this early Admonition, don’t let it be an Excufe not to read this Matter at all: For it that Jliould be the Cafe, they will deprive themfelves of many beautiful Proportions, and fine Methods of Working, not to be feen in any other Mailer. The Manner of fluting ‘Pilajlers and Columns. Ihe Flutings of Columns do particularly affeft the [onick Order {rarely the Dorick faith Evelyn) uti Stolarum Ruga, in Imitation of the Plaits of Women’s Robes, as Vitruvius faith. The hr ft Order that the Anti- ents fluted was the Ionick, in that renowned Temple of Diana, built at E- phefus (as many think) by the Amazons, which employed above two hun¬ dred hears to finifli, at the Expence of all Ajia, whole Columns were of Marble 70 Feet in Height; of which more hereafter, when I come to fpeak of the Ionick Order. As this Temple was built after that immemoriable magni- liccnt Temple, erected to the Goddefs Juno, in the famous City of Argos, by Dorus, Prince of Achaia, and Sovereign of : Teloponnefas , it therefore J'eems, as if the Original Dorick Columns were not fluted, but made plain : Xot but the Allufion may be made as well to the Goddefs Juno, as to the Goddefs Diana. But however, we find many lnftances of fluted Columns among the antient Dorick Buildings of Rome, as in the Theatre of Marcelius, Plate LI. the Bath of Dwclefian, Plate LVII. &c. and therefore thofe Ex¬ amines may be pleaded as Authority to do the fame when required. The antient Manner of fluting the Dorick Shaft was, to divide its Cir¬ cumference into io equal Parts, (as Fig. I. which is a Semicircle divided into io Parts.) The Depth of each Flute was equal either to the Segment of a Circle on the Side of a Geometrical Square, as/Jio, where the Breadth of the Flute is the Side of the Square; or othcrwile, to a Segment of a Circle on the Side of an Equilateral Triangle, as n A 1, on the Side of the Equilate¬ ral Triangle no 1; which Flutings had 110 Divifions of Fillets, (as are in the other Orders) but were worked to fharp Edges or Angles, without any Spaces of Fillets between. I will not undertake to aflign the Reafon, why the An- tients did thus flute their Dorick Shafts ; but fore 1 am, that they are abun¬ dantly more liable to Abufe and early Decay, than they would bej were each Flute divided by a Fillet, which would be a Strengthening, and an Orna¬ ment to them alfo. To diftinguifh the Dorick Fillets from the Ionick Fil¬ lets, divide the Breadth of each aoth Part into y equal Parts, give 1 to each Fillet, and 4. to each Flute. To flute the Ionick, Corinthian, or Compofite Columns. (t) Divide the Bafe of each Column into 14. equal Parts, and divide each Part into 4 ; give 1 to each Fillet, and 3 to each Flute, as in Fig. III. (1) if from the feveral Divifions of the Flutes and Fillets, you draw Right Lines at Right Angles to the Diameter, their parallel Diftances will reprefent the Breadth, that every Flute and Fillet will be icen to diminifh, from the Mid¬ dle to both Sides of the Column. To divide the Flutes and Fillets on the Shaft of a plain Column. (1) Draw a Right Line, as a b (in the lowermoft Figure of the Plate) of Length at pleafure, and therein afl’ume a Point, as b. (1) Open your Compafles to any fmall Diltance, fo that 14 of thofe Diftances, Pet along the Line a b, ill 11 be lei's than the Girt of the Column at its Aftragal. This done, divide - Opening of the Compafles into 4 equal Parts, and take 3 of them into your ompalles ; alfo take the other 1 into another Pair of Compafles, and th n will 20,6 The Principles of Geome t r y. will the one be the Breadth of a Flute, and the other the Breadth of a Fillet. With thele Oj'c nines, prick along the Line ah 24 Flutes, and as many Fillets; and from thole Points draw Right Lines, parallel to each other, and at Right Angles to ah. (3) Take the exa& Girt of your Column with a lira it Piece of Parchment, &c. which fuppofe to be dc ; and then laying its Ends, fo as to touch the two Out-lmes e a and f b, the feveral parallel Lines will divide its Edge into its proper Flutes and Fillets, as at the Points 1, 2, 3, 4, y, < 5 , yb-'c. (4.) Apply one lend of your Parchment, thus divided, unto a Right Line drawn on the Shalt, from its Aftragal to its Cindture, and therewith girt the Shaft at its Bottom, and from the feveral Points on its Edge, prick off the Breadth of every Flute and Fillet, as required, (7) Take the Girt of the Co¬ lumn at its Aftragal, and apply it to the Out lines, as before, which fuppofe to be e /, then the Edge will be divided, with its Flutes and Fillets proporti¬ on tblv diminilhed ; and if one End be placed to the aforefaid Right Line, drawn on the Surface of the Column, (not under the Aftragal) and the Breadths of every Flute and Fillet be pricked oft" from the Papers Edge ; when 'tis girded about the Neck of the Column, as before at the Bottom, proceed to draw Right Lines, from the Divilion of the Flutes and Fillets above, to thole below, and vou will divide the Superficies of the Shalt ready for working, as required. Note, by the fame Rule you may find the Breadth of the Flutes and Fillets, in any Part of the Shaft you are pleafed to girt it at. To flute Pilafters. Some Architects divide Pilafters into 9 Flutes and 10 Fillets, and others but into 7 Flutes and 8 Fillets, of w hich the lalt is moll generally pradtifed. As the Breadth of a Fillet is one third Part of aF'lute, therefore divide your Pilafter, if for 9 Flutes, into 37 equal Parts, but if for 7 Flutes, into 29 e- qual Parts, and then give 1 to each Fillet, and 3 to each Flute. Sometimes Workmen place a Bead at the Angle of Pilafters, as in Fig. II. and then the Breadth nluft be divided into 31 equal Parts ; give 3 to each Flute, 1 to each Fillet, and 1 to the quarter Round, or Bead, at each Angle, as in the Figure exprellcd. But this lad Method is not to be commended, bccaufe the break¬ ing of the Angles by the Beads is a feeming Diminution of their Breadth, and indeed it makes the Angles look weak, by being divided into fmall Parts, which othenvile would be more malfy, and confequently much itronger, and of grander AfpeCL Note, That Elutings arc called by lome S/riges, and Fillets Stria, Rates, or Lifts. Note alfo, that the Flutings of Columns and Pilafters are generally filled up with a Swelling, a third Part from the Bale, called Staves, or Cablings. P. Pray, wherein doth a Pilafter differ from a Column ? M. A Pilafter hath no other Difference from a Column, than that the Shaft of a Column is round, and diminilhed from a third Part ol its Height unto the Aftragal; and that of a Pilafter is fquare, and Ihould never be diminilhed (as erroneoufly is done by Inigo Jones, at the Banquetting-houlc at Whitehall ) but when it Hands behind a Column: When a Pilafter Hands alone, tis called by the Greeks Paraftate, and by the Italians, Membretti. The Projection ol Pilafters, from the Wall they Hand in, is foinetiines a fourth, or fifth, or fixth Part of their Diameter, as Occafion may require; and the re¬ maining Part of the fquare Body is always fuppofed to be Handing within the Wall. From their Projedture, or Coming forward, they are alio called slnte, or Wnta, as having been placed before the Walls of antient Temples, and at their Quoins, for Security and Strength. In the Ufe ol Pilafters tis to be ob- ferved, that tho’ they have a noble Afpect in large Buildings, yet in fmall Fronts they make but a very poor Figure, and therefore in fucli Buildings Ihould be avoided. Plates The ‘Principles of Geometry. Plates LI. LII. Two Examples of the Dorick Order in the Theatre of Marcellu s at Rome. Altiio many Examples of the Dorick Order are given us, without any Lale, as the id Example hereof, and that of the Bath of Diocleftan , Plate L.VIf. yet here we lee, in this ill Example, the slltick Bale introduced, and I think not improperly : For if Columns wtre antiently uied without Bales, they were not io beautiful, nor had fo good a Foundation, as thofe with Bales. P. Tray what is the Attick Bafe ? -d/. The flttick Bafe (or, as fome call it, the dttick Curgi) conlifts of a 1 linth, two Torus s, and a Scotia, with its Fillets, as the Bale to this ift Ex¬ ample, and which exceeds the Tujcan Bafe by the Scotia y, and the upper Torus x x. ‘ r P. Tray why is the Scotia fo called ? M. From the Greek Scotos, Darknefs, or from its Obfcurity, pro¬ ceeding from the Shade of its Hollowncls, but more vulgarly (faith Evelyn) they call it Cafement ; though, 1 mult confefs, 1 never heard of this Name before. It is alfo called by ibme Trochile, T e%" or a Rundle or Tally - wheel, which it refembles. The Italians call it Baftone. By the feveral Divisions of equal Parts, and their Subdivifions, you fee by Inlpedtion, how the principal and particular Parts are determined, and which, to conlider and find out, I have placed here as Examples for Practice ; though indeed there is but very little in them, they being made fo very plain, as to be underftood at firft Sight by thofe who have made thcmfelves Matters’of the foregoing, and which any one may foon do with good Attention. 1 lhall now proceed to give you my Remarks on thole Examples: And frfl, I cannot be¬ lieve the Annulets, in cither of the Capitals, to be proportionable to the Ovo- lo, and Abacus next above them, they having a poor and weak Look, by be¬ ing many and final]. This I cannot help calling "an Error, (notwithftanding that Vitruvius, 'Palladio , and Ibme other Matters have followed it) and which is evident, if we do but compare thefe, or any other of the aforelaid Matters with Scamozzi, Plate LXII. and with Mr. Gibbs, Plates C. and Cl. where the firft hath a Cima Reverfa, and the latter a Cavetto, either of which have a much grander Look. P. Tray which are the Annulets, you [peak of ? M. Those three final! Fillets or Rings immediately under the Ovolo of each Capital, comprifed between HK in the firft Example, and hk in the fe- c°nd. the Capitals oi thefe Examples confift of the lame Members, but dif¬ fer in their Magnitudes ; the Ovolo of Example l ft being greater than that of the 2d, and the Abacus of the ad greater than that of"the ift. The Archi¬ traves of both confift of one Fafcia, and the Cnttds or Drops in botli have a noble Aipect. Uiefe Guttas are made either as Sections of Fruftums of Tyramifes, or of Cones, but fquare at their Bales ; they are alfo under the 'Planton or Tlancere of the Corona, as are exprefled in the ad Example, and contain iS in Number, placed exaftly over the Triglyphs in the Freeze.’ P. Tray what doth a Triglyph reprefent ? M S\ N Antique Lyre, firft ufed in the Delphic Temple, of which Inftru- ment, tis laid, Apollo was the Inventor. The Word, in Greek figni- fies a three-fculptur d Piece, Quafi tres habens G/yphos. The Italians call them Tlanetti, l’mall Plains. Their Breadth lliould always be equal to the Semidiameter of the Column at its Bale; but fome Matters make it equal to the Semidiameter at the Altragal. N n n The 2 3 s The 'principles of Geometrv. The broad Lift BC, next above the Freeze, is called the Capital to the TrEvph From wlienee arifes the Cornice. In Competitions of thefe Cornices there .snot a little Variety ; and, 1 think, they are the very firft that I have , to finilli w ith Cavetto's. They have both a lofty and noble Air, and tiicrcforc are belt for the Oatfidc of magnificent Buildings. Plates LLS. LIN The Dorick Order of Vitruvius, lumnations for Portico’s to Temples, &c. I with literco- rt IG I is a Profile of this antient Mailer, whole Members are determined b' ccual Parts as Inlpeftion makes plain. Fig. II. is the Bale ot the Column more at lame/ Fig. HI. is a Plan of the Plancere of the Corona Fig V I. exhibits th' Manner of placing the Triglyphs, fo as to preferve perfect lquaie Motors between them ; that is, the Triglyph C being placed directly over t ] ie central Line o 1 the Column, the Interval E, which is ca.ied Metop, mult, bv'the Dillance of the next Triglyph B, be made a perfect Square There- l 0 ,- e if Tnglvphs arc not placed exactly over a Column, as A, or ir thc Me- too’be a Paraileltogram, inftead of a geometrical Square, both are ablurd. An Lxamplc of the lait may befeen in that monftrous Frontilpiece to the South- Sec. Houle in Threadneedle-Jtreet. The Word Me top is from the Greek Mela, and Ope, between three, and was antientlv enriched with Oxes Skulls, _ Lillies, Targets, Battle-axe-, Thun- derbolts &c. nay, even to this Day thefe Ornaments are ufed, but .oi the Oe- Rcrai.ty* wich great Impropriety, as being inconliltent with the Buildings and Satiations, of which I have already taken Notice. Fig. VIII. is a Plan of a Triglyph, exhibiting BA, the Depth, i f the t.ha- nellings and G F F, the Intervals between. Fig. V. reprefen - . tuviuss anner of fluting his Dorick Column ; where, you fee, that his Flutes are de¬ ft ribed on the Center of a Square, whole Side is equal to the Breacun ot a Fluie as before taught. The Ancients did foraetimes cut their Columns into. Cants’inftead of Flirtings, as reprefented by/, g, h, i, k. Figures IV. VII. and X. are Portico's after fhe antient Dorick Manner, where, you lee, there are nut any Bales to the Columns, as there are to the Portico, Fig. X. The Intercoiumnation of this Temple is called yJr,eo/iyle , from the Greek Ar- mod. 4 par. which is equal to z diam.40 min. Nor can 1 think the Height of his Members in the Pedeftals, Cornice and Bafe, to be of fufficicnt Height and Strength, proportionable to the Greatnels and noble Ai’peft of thof. Members in the Entablature they belong to. Diam. Min. r Pedeftal and Column 9 40 The Height of the ^Column and Entablature 10 00 ^Pedeftal, Column and Entablature iz 40 Plate LXVI. Dorick Intercolumnations, by Barozzio. Tims Plate reprefents the Dorick Intercolunmation for Colonades by Mi¬ chael James Barozzio of Vignola, whole Meafures arc determined by Mo¬ dules and Parts, as in the preceding. Note, This Plate, by Miftakc of the Engraver, is numbered LXVII. Plate LXVII. A Dorick Arcade without Pedejlals, by Barozzio. This Plate exhibits the Intercolunmation for a Derick Arcade, where no Pedeftals are required, and which would have had a much more noble Effect, had this Matter made the Pilafters under the Imports equal to the Semidi¬ ameter of the Column, inftead of one quarter or half a Module, which has a thin and mean AfpeCl, no wife harmonious with the Diameter of the Co¬ lumns. The Parts are determined by Modules and Parts, as in the fore¬ going. Plate LXYIII. A Dorick Arcade with Pedeftals, by Barozzio: Alfo, Dorick Intercolumnations, for Colonades mid Arcades, without Pedejlals, by Sebastian le Clerc. Here this great Matter has fell into an Extream again, by making the Pi¬ lafters as much too broad, as the foregoing are too narrow; theie being 1 mod. and half, equal to 4s- min. are, in my humble Opinion, 15- min. too much ; and which any indifferent Eye may difeover, if the Diameter, and Height of the Column be compared with the Diameter and Height of tire Pilafters, which laft have a greater Diameter for their Altitude, than the for¬ mer. The Error of the preceding, in Plate I, XVII. is judicioufly corrected by Seba/lian le Clerc, in his Dorick Arcade at the Bottom of this Plate, where his Pilafters have their Diameters equal to half the Diameter of the Column; and the exceflive Error in the Pilafters of this Plate is, in like Manner, cor¬ rected by the fame Matter in Fig. III. Plate LXXX 11 I. where they are all'o equal to the Semidiameter of the Column, as before. The Intercolumnati¬ ons for Colonades by Le Clerc, on the left Hand of the Bottom of this Plate, are regulated by the Number of Triglyphs between each Column, as in¬ deed are the Intercolumnations in the aforefaid Arcades, which InlpeCtion doth fully demonftrate. O o o Plate 242 The Principles of Geometry. Plate LX IX. Two Dorick rujlicatecl Gatts, by Barozzio. These two Defigns, at firft View, pleafe a common Eye very well, but, when they are critically examined, they will be found to confilt of as many Abfurdities, as Beauties. In the firft Place, I will not contend, about the Pro¬ priety of rufticating this Order, fince that this great Maftcr has done it: But however, I do again affirm, that the Multitude of finall Members in thePedef- tal and Bafc of Columns and Impolts of Fig. II. have no Affinity with the Ruf- ticks, and that the Breaking of the Entablatures, in both Examples, is mon- ltrouflv ablurd ; for tho the Kcy-ftones to both Arches have a grand Look, by being made large, yet that Grandeur is infinitely lefs, than the Continuation of the Entablatures : Befides, as I have already obferved, the Breaking of the principal Parts makes the Whole appear defective and Weak. Figures A and B are the Profiles of thefe Examples. Plate LXX. The Frontifpiece to the principal Entrance into the Farne- fean Palace at Rome, This Frontifpiece is one of the 1110ft limply grand Compofitions of the Do- rick Order, that is to be feen in the World. Here, in its native Lines, free from all Manner of Embellifhments and Ornaments, you behold all the folemn Greatnefs and Magnificency that can be defired, and therefore I mult recom¬ mend it, as an Example worthy of Imitation. Plate LXXI. A Triumphal Arch, by M. J. Barozzio. This is a magnificent Defign for the Entrance to a Nobleman's Palace, pro¬ vided that the Entablature be not broken, as is done here, which is ablurd. The Pilaltcrs under the jhnpolts are here too broad, as before in PI. LXV 11 I. which mult be obferved to be made of lefs Diameter, (as before noted) when a Defign of this Nature may be required. As to the other Parts, they arc in general of good Compofition, the oblong Windows, between the outer Co¬ lumns on each Side, over the continued Impoit, excepted, which ought not to be there. The great Pannel in the Parapet over the Arch may be ufed as a Table to contain an Infcription when required. Plate LXXII. The Donck Gate of Cardinal Farnese, at Caprarola. Here \vc have another rufticated Example of the Dorick Order, but with this Difference, that as the preceding Examples conlilted of champhered or mitred Ru/hcks, thefe are fquare or rabbit Ru/licks ; and as thefe Rufticks project beyond the Upright of the Columns, they do therefore, by fuch Em¬ bracing, ftrengthen them very greatly, according to their Defign. The En¬ tablature of this Frontifpiece being entire is noble and grand, as indeed are all the Parts ot the Whole, and therefore 1 recommend this Defign, as another Example worthy of Imitation and Regard. Plate LXXIII. The Dorick Order, by Sebastian Serlio. This Mailer prefents here an Entablature, crowned with a prodigious Cy- matium, placed on a final] Cima reda, or Cavetto, which bear no Proportion to each other; nor indeed doth the Regula, or Fillet on the Cymatium, which is too low and thin for fo great a Member as that to which it belongs. The Cima recta under the Corona is rather too low and finall for the Corona and Cymatium-, but the Height ol the Freeze and Architrave, with the Tri- glyphs The Principles of Geometry. 243 glyphs and Gutta's, are very good. The Capital could not be condemned were the Annulets excluded, it being of a tolerable Compofition. The Bale of the Column is nearly Nttick, wherein he has placed a Fillet between the Plinth and lower Torus ; and which is intirely right, when the-Bafe is elevated con- fiderably above the Eye; for thereby the Torus is feen more diftinCtly, than it could be, were it to fit immediately on the Plinth, whofe Projection would eclipfe a great Part of its Height, and thereby caufe it to have an ill EfteCt. This being confidered, ’tis evident, that, when the Bafe is placed beneath the Eye, this Fillet mult be excluded, and its Height given to the Plinth and To¬ rus, viz. 1 min. to the Plinth, and the half min. to the Torus. The Pedef- tal is of a Compofition very particular, having a Cima reverfa for its Cornice, fitting on an Aftragal, and crowned with a large Fillet or Regula. The Bale of the Pedeftal leems to have been taken from the Bale of the Dorick Co¬ lumn by Barozzio, being an Aftragal placed oil a Torus and Plinth, as that Bafe is. Thefe Mouldings are divided as follows, (1) The Height of the Pc- deftal is equal to 3 diam. of the Column, which being divided into 7 equal Parts, give 1 to the Height of the Cornice, and 1 to the Height of the Bafe. (1) Divide a e, the Height of the Cornice, into 4,, at a, b, c, d, e, and fubdivide d e into 3, then the lower x is the Fillet, the next 2 the Aftra¬ gal ; and then giving a b to the Regula, the Remainder b d will be the Height of the Cima reverfa. (3) BileCt z m , the Height of the Bale, in l , then / m is the Height of the Plinth ; bifefit i l in k, then k l is the Height of the To¬ rus: Divide z k into 3 Parts, then the upper 1 is the Fillet or CiriCture, and the lower 1 the Aftragal. The Projection ol the Die is always equal to the Projection of the Plinth to the Bafe of the Column. Divide % h, which is equal to the Projection of the Die, into 6 Parts, and make h w, and 3 2, each equal to 1 of thefe Parts, for the Projection of the Bale to the Pedeftal. The Projection of the Cornice to the Pedeftal is a little more than that of the Bafe, that thereby the Bafe may be cleared from the Perpendicular Drip of its Cornice. The Height of the Column is but 7 diam. and the Height of the Entablature 1 diam. j 2 min. which is 7 min. more than one fourth Part of the Column's Height. If Serlio had made his Column 7 diam. and a half in Height, then the Height of his Entablature would have been within 2 min. of one quarter Part of the Column’s Height, p’ig. II. reprefents the intire Order, and Fig. III. the fluting of the Column. Diam. Min. IO GO is s 8 ya (11 yx Note, The Meafures of the Bafe and Capital of the Column, and of the Entablature, are determined by Modules and Minutes, of which the Projections are accounted from the central Line. Plates LXXIV. LXXV. Dorick Intercolumnations for Portico’s with Front ifpieces, by S. Serlio. FiG.LandFig.il. reprefent Intercolumnations for Portico's to Temples, d fc. which are both of the fame Kind, as appears by the Triglyphs over each Interval. That of Fig. I. confifts of fix Columns, and that of Fig. II. of four only, each having three Triglyphs in the middle Interval, and two only in every of the other, exclufive of thofe over each Column. Fig. 111 . is another rufticated Dorick Example, which, in a Grotefque Building, would have a fine Effect, was but the Entablature whole, inftead of being broken in fo barbarous a Manner by the rufticated projecting Key-ltones, which do not only fill up the Place of the Architrave, Freeze, and Bed-moulding, but the Tympanum of the Pedeftal alfo. The lower Rufticks of the Columns being placed in the Stead of their Bafes, have a good EffeCt, and do better become thofe' The ‘Principles of Geometry. 244 tholl' Places, than the common Bale to the Column would have done. Figures IV end V. are two Examples of Doors from the Ancients not unworthy of our Notice. Plate LXXVI. Dorick Intercolumnations for Colonades and ylrcades , by S. Serlio. Figure I. represents the Intercolumnations proper for a 'Dorick Colonadc, wherein the Columns are placed nearly in Pairs, and which in fome 1 laces have a very noble EffeCt Fig. II. is an Arcade to a Piazza, whole Arches fpnng from Pilaltcrs, or rather from fquare Columns, placed to fupport the Impolls, which here is turned into Capitals that feem to be fimilar to the Ca¬ pitals of the Columns, that fuftain the Entablature over them. I muft own, that tins is the licit Example oi the Kind 1 have yet ieen, and which is very well worth our Notice, as that in many Cafes it may better (hit our Purpoles, than any other we may think on. Fig. 111 . is another Kind of Arcade m the Venetian Manner, which is alio ol good Invention, and of greater Strength, than the preceding. Plate LXXVII. yl Dorick Temple , by Bramanie. This Temple is a Defign of that famous Architect Bramante, who defigned St. ‘Peter s at Rome, and is of a noble Talte. By the dotted Circles you fee, that the whole Height is divided into two Parts, the lower one ex¬ tending from the Bale of the Columns, to the lop of the Balnllrade, and tlie other from thence, to the Vertex of the Dome. The Corndore of Columns which environ the Building, and which fupport the Gallery above, have a noble Alpect, as well as afford good Shelter from the Weather below, and a pleafant View from the Gallery above, when a Building ol this Kind is creel¬ ed on a pleafant Situation 111 a Garden, Park, &c. Plates LXXVIII. LXXIX. Dejigns, by S. Serlio. Fig. 1 is a Defign for a Gate or Door, where the Corona is fupported by Trufles placed in the Freeze ; which Trufles have their Face s channeled in Manner of the Triglyphs, and their Gutta's under them in the Architrave. I mull own, 1 think the whole has a good A 1 peel; but whether the Conver- fion ot the Triglvphs into Trufles (called, by fome Mailers, Mutates) be war¬ rantable I will not undertake to determine. Fig. II. is the Delign ot an yiltar- piece of good Invention. Fig. HI. is the Defign of a Triumphal Arch of good Invention alfo ; wherein you fee, that between the two middle Columns, there are five Triglyphs, and between the outerColumns but two. Fig. IV. is a Dorick Arcade in a very grand Talte, and where there are but lour Tri- glyphs, between thole over the Columns, initead of five, as in the preceding. Plate LXXX. The Dorick Order, by S. le Clf. rc. In this Plate we have two Varieties of Entablatures, viz. A and B, which are both finifhed w ith Cavetto’s, in Manner of fome of the foregoing Mailers, as indeed are three other Entablatures in Plate LXXXI. The Meaiure, by which this Mailer determines his Members, is the Semidiam. ot the Column divided into 30 min. as before taught. As to the Difference of each E ntabla¬ ture, from each other, that is better feen by comparing them together, than deferibing them by Words, to which I refer you. The Height of the Column is id mod. or Semidiameters of the Column; the Entablature 3 mod. min. and the Pedeltal y mod. 10 min. The The ‘Principles of Geometry. 245 Diam. Min. rPcdeftal and Column 10 40 The Height of the ^Column and Entablature 9 7(3 ^Pedeftal, Column and Entablature ix 38 Plate LXXXT. Dorick Entablatures and Sofito’s, by S. le Clerc. The three Entablatures exhibited on this Plate (as I before obferved) being compared together, confill of the fame Members, and differ only in their Heights and Prqjettures, as exprelled by the Me^fures affixed. Eig. III. V. VI. and VII. reprefent various Ways of turning the Sofito of the Corona at an Angle, and of dividing and placing the Mutules over each Triglyph in the Freeze, which a little Inl'peCtion will make more plain, than Words can do. Plate LXXXII. Other Examples of the Tivifions of Dorick Sofito 1 s for Practice, by S. le Clerc. Plate LXXXIII. Dorick Intercolumnat ions for Arcades, See. with their Imposts, by S. le Clerc. Fig. I. reprefents a Dorick Arcade without Pedeftals, to be made either with tingle Columns, as on the Right, or with Columns in Pairs, as on the Left. Fig. II. and III. are Arcades with Pedeftals, to be tiled as Occafions require. Figures A, 11 , C, D, are four Varieties of Imports, of which Choice may be made at pleafure. Fig. IV. is an Intercolumnation proper lor a Colonade, to be ufed either with a continued Pedeftal, as here reprefented, or without, as the Nature of the Building may demand. Plate LXXX1V. Dorick Examples for Practice, by S. le Clerc. This great Matter having given ns his various Entablatures, Sofito’s, Ar¬ cades, gfre. he now finilhes this Order with an Example of placing the Dorick Order on the Tufcan, as exhibited on the right Hand of this Plate, wherein tis to be noted, (1) That the Diameter of the Dorick Order at its Bale is equal to the Diameter of the Tufcan at its Aftragal. (x) That the central Lines of the lower Order and of the upper Order be the fame, or one continued Line, lo that the upper Column may Hand exactly over the under. (3) That the Pi- lafters of the upper Arcade itand exactly over thole of the lower, whereby the Solid will Hand over the Solid, which is a general Rule to be obferved in all Parts of Buildings in general. The other two Examples for Gates, and the Rotunda, or round Temple, are added for the Exercife of the young Prac¬ titioner. Plate J ,XXX V . The Dorick Order, by Claud e P f. r a u lt. Ip this Mailer had omitted the Annulets in the Capital, and introduced an Aftragal in its Head, the Whole would have been a tine Competition. To proportion this Order to tuiy given Height. Divide the given Height into 37 equal Parts, give 7 to the Height of the Pedeft.il*, x+ to the Height of the Column, and 6 to the Height of the Enta¬ blature. The Diameter of the Column is equal to 3 of the aforelaid Parts. To divide the Tedeflal into its Cornice, Die and Baje, divide the given Height into 8 Parts, give 1 to the Cornice, 7 to the Die, and x to the Bafe. To di¬ vide the Mouldings of the Cornice of the Tedeflal, divide the Height by into <3 equal Parts, give x to the Regula, 7 to the Falcia, 1 to the under Fillet, P p p and j i ;ind z to the Cavetto. To divide the Mouldings of the Bafe of the Pedeflal, divide the Height g k into 3 equal Parts ; give z to the Plinth h k, and the nj-ptr 1 being divided into z, give the lower 1 to the Torus; and then, this hfr upper I being divided into 3, give the lower 1 to the Fillet, and the up¬ per a to tlie Cavetto. To divide the Mouldings of the Bafe to the Column ; 1) the entire Height being equal to half the Diameter, divide it into 3 equal Parts, and give the lower 1 to the Plinth, (z) Divide the remaining z into 4. Parts, and give the upper 1 to the upper Torus. (3) Divide the Remainder into v, and give the lower 1 to the lower Torus. (4.) Divide the laft 1 re¬ maining into 6, give the.uppcr 1 to the Fillet under the upper Torus, the lower 1 to the Fillet on the lower Torus, and the remaining 4 to the Scotia. This is the true Attick Bafe. To divide the Capital into its Tarts, divide the given Height into 3 equal Parts, give 1 to the Abacus, 1 to the Ovolo and its Annulets, and the lower 1 to the Keck. The Aftragal, with its Fillet, is equal to halt the Height of the Keck, and the Fillet is a third of that Height. The Height of the Ovolo and its Annulets being divided into 3, give the lower 1 to the Annulets, which fubdivide into 3 alio. The Height of the Abacus be¬ ing divided into 3, give the upper 1 to its Cima and Fillet ; and that being l'ubdivided again into 3, give the upper 1 to the Fillet, and the lower z to the Cima reverfa. To divide the Entablature into its Architrave, freeze, and Cornice, divide the Height into 14 equal Parts, give 6 to the Architrave, 10 to the Freeze, including the Capital of the Triglyph, and the upper 8 to the Cornice. To divide the Architrave , divide the Height into 7 equal Parts, and give the upper 1 to the Tenia, and the next 1 and a third to the Depth of the Guttds. To divide the Cornice ; its Height being divided into 8 equal Parts, as aforefaid, give the upper 2, and a fourth of the next 1, to the Height of the Cymatium, and the remaining three lourths oi the third Part, to the Height of the Cima reverfa. The next 1 and half is the Height of the Coro¬ na, the next half to the Cima reverfa on the Mutule, and the next 1 and half is the Height of the Mutule itfelf; the remaining 1 and half is the Height of the Cavetto, which finilhes the Whole. To determine the Projeciures. (1) Divide the Diameter of the Column into 14 equal Parts, and make the Projeflion of the Plinth, to the Column's Bafe, equal to 3 of thofe Parts. The middle 1 of the 3 Parts in the Plinth being divided into 3, and Lines drawn from thence perpendicular to the Bafe (as the dotted Lines running up) to the Capital) terminate the Projections of the upper Torus and Fillets to the Scotia. The Projection of the Plinth limits the Projedtion of the Die. One, with the Radius e f, deferibe the Semicircle J d z\ then is e d the Projedtion of the Plinth to tire Pcdeftal. The Projedtion of the Abacus of the Capital is equal to that of the upper Torus ; that of the Ovolo to that of the Cindlure, and that of the Aftragal to the Upright of the Column. The Diminution of tire Shaft at the Aftragal is one ieventh Part of the Diameter at its Bafe. To find the Projection of the Cornice, (1) Divide the Height of the Cornice, in¬ cluding the Capital of the Triglyph, into 11 equal Parts, {2) Continue the upright Line of the Architrave and Freeze through the Cornice, until it cut the upper Line ol the Regula on the Top of the upper Cima of the Cornice. This done, on the upper Line of the Fillet, fet along 16 equal Divifions, each equal to 1 of the 12 found in the Height of the Cornice; then will the 16th be the Proiecture of the Cymatium, the 14th of the Cima reverfa , the 13th of the Corona, the nth of the Mutule or Modillion, the yth of the lower Cima reverfa, the 4th of the Cavetto, and the id of the Triglyph in the Freeze; and thus is the Whole completed, as required. Figure Kreprefcnts the Sofito of the Corona, and Figure M the geometrical Rule for deferibing the Cima redta and reverfa. That ol the Cima redta is no more than z equilateral Triangles, whofe Sides are The Principles of Geometry. 247 are each equal to half a b ; that of the Cinia revcrfa hath its Projection divided into 6 Parts, of which 1 is given to the Projection of its Foot, and the other 1 to the Proje£lion of its Fillet, then the Curve is defcribed by two equilate¬ ral Triangles, as before. Plate LXXXVI. This Plate was number’d Plate LXXXV 1 I. by Miftake of the Engraver, and printed oft before difcovered, therefore the 'Dorick Orders of / iola, Al¬ berti , de Tonne, and Bullant, which were to have been Plate LXXXVI. are now become Plate LXXXV11. The Dorick Orders o/LeoniBaptisTA Viola Leoni Baptisti Alberti, Philip de Lorme, and Job:. Bullant. These Profiles are from Mr. Evelyn, and have their Members determined by Minutes, of which the Projections are from the Central Line. As Infpec- tion demonftrates the Difference of each Matter, I need not enlarge thereon; and therefore I fhaU only oblerve, that the Capital of Alberti is monftroufly high, being 43 min. which is 13 min. more than by any other Matter, and of a poor Projection. The Ovolo in his Bed-moulding, being without the Cavet- to under it, looks very heavy and dull, and unworthy of Imitation. In ttiort, the whole Entablature is a bad Compolition, whilft every of the other three are worth our Confideration. Plate LXXXVIH. The Dorick Order , bj the Reverend Daniel Barbaro, and Cataneo. These Matters arc alfo from Mr. Evelyn , and have their Parts determined by Minutes, as the preceding ; and their Variations in each Member are alfo demonftrablc by InfpeCtion, as the former. The Dorick Temple A is an Ex¬ ample for Practice, by way of Digreffion from the Courfe of the Order. Plate K to follow Plate LXXXVIH. The Dorick Pedefial, hj ] u- LIAN MAU-CLERC. To find the Height of the Pedeftal to an entire Order, as Fig. A, Plate LXXX 1 X. divide the given Height into 8 equal Parts, and give the lower z to the Height of the Pedeftal. The Height of the Pedeftal being given, divide it (as in Plate K) into 7 equal Parts ; give 1 to the Height ol the Bale, and r to the Height of the Cornice. To divide the Mouldings of its Bafe, divide its Height into x Parts, and give the lower 1 to the Height of the Plinth; alfo divide the upper 1 into x, and give the lower 1 to the Height of the To¬ rus ; alfo divide this lalt upper 1 into 3, give the upper 1 to the Fillet, and the lower 1 to the Altragal : and thus is the Bale completed, as exhibited on the Right-hand Side. On the Left-hand Side, the Bafe is divided in a different Manner, as follows; viz. The Plinth is equal to half its Height, the Torus to two Thirds of the Remainder, and the Fillet to a Sixth of the Whole. The Mouldings of the Cornice are alfo divided in x different Ways ; as firff that on the Right, the Height being divided into 4 Parts, give the upper 1 to the Regula, the lower 1 to the Aftragal, and the middle z to the Cima reverfa ; the Height of the Aftragal being divided into 3, give x to the Aftragal, and 1 to the Fillet. Secondly , that on the Left; divide the Height into y Paits, give the lower 1 to the Aftragal, fubdivided into 3, as before ; the next z to 248 The 'Principles of Geometry. the (tma reverfa, the next 1 and half to the Fafcia, and the half of the upper 1 fubdivided into 3 to the Regttla and its Chna. If the Diameter of the Die of the Pedeftal be given to find its Altitude, complete a geometrical Square whole Side is equal to the Diameter of the Die, and make the Altitude of the Die e- qual to the Diagonal of the Square, as is very plainly demonftrated in the Fi¬ gure. This done, divide the Altitude of the Die into y Paits, and then giving x to the Cornice, and x to the Bafe, proceed to divide their Parts as belore. Plate LXXXIX. The Dorick Order entire, by Julian Mau-clerc. The Manner of finding the Height of the Pedeftal being explain'd in the laft Plate, we will now proceed to find the Height and Diameter of the Column, as alio the Height of the Entablature. Divide the Height ot the Pedeftal, Column and Entablature into 11 equal Parts, and give the upper 2 to the Entablature; the Diameter of the Column is equal to 1 of thofe ix Parts; the Column is diminilhed ' of its Diameter, and the Height of the Capital is equal to a Semidiameter of the Column. Plate XC. The Tufcan Order at large, by J. Mau-clerc. This Plate exhibits by Infpeftion the Divifions of the Members in the prin¬ cipal Parts of this Order, of which I have already fpoken, in the Explanation ol the Tn/cati Order by this Matter; and as the Divifions of the Members are very plain and eafy, they need no further Explanation. Plate XCI. The Dorick Order at large, by J. Mau-clerc. The Manner of dividing the Pedeftal into its Parts being demonftrated by Plate K, and the Manner of finding the Height of the Pedeftal, Column and Entablature being exhibited by Fig. A, Plate LXXXIX. 1 fhall now explain the Manner of Dividing the Bale and Capital of the Column and the Enta¬ blature into their refpedtive Members. I. To divide the Members in the Bafe to the Column. The Height being equal to the Semidiameter of the Column, divide it in¬ to 3, the lov er 1 is the Plinth ; the remaining Height divided into 4, the upper 1 is the upper Torus ; the Remains divided into 2, the lower 1 is the lower Torus ; the upper 1 divided into 7, the upper and lower ones are the Fillets and the Remains between the Scotia. This is the Attick Bafe, as be¬ fore delivered in \Perault on the Dorick Order. The Projection of the Plinth is equal to one fourth Part of the Diameter of the Column. This is alfo exhi¬ bited in Plate E to follow Plate XCI. II. To divide the Members in the Capital, The Height being equal to half the Diameter, divide it into 3 Parts, give r to the Neck, 1 to the Ovolo with its Annulets, and 1 to the Abacus; then lubdivide them, as in the Figure for their Parts. III. To divide the Entablature into its Architrave, Freeze and Cornice. Divide its Height into 7 equal Parts, give 2 to the Architrave, 3 to the f reeze, and 2 to the Cornice; then lubdivide the Members, as exhibited by the Divifions againit them. Here are two Varieties of Cornices, and 1 think nei¬ ther of them goed ; that on the Right being finiihed with a monltrous Ctma recta, on a decripid Cima reverfa under it ; and that on the Left being much too high for its Freeze, and conlifts of the two Falcia's, next over the Capi¬ tals of the Triglyph, more than it ought to have, they making a treble Repeti¬ tion ol the fame Member, which is abl'urd. Plate The 'Principles of Geometry, 249 Plate E to follow Plate XCT. T ms Plate exhibits one of the ancient Manners of enriching th <1 At tick Bale, when uled with the "Derick Order, as all’o ot the Capital, which are giv¬ en here as Examples for Imitation, or Help to Invention. Plate O and Plate P following Plate E after Plate XCI. These Plates reprefent the Entablatures'of the preceding Dorick Exam¬ ples more at large than in Plate XCI. wherein the aforefaid Errors are more obvious. Plate P following Plate O after Plate XCT. The Dorick Order, by A. Palladio, V. Scamozzi, and M. J. Barozzio of Vig¬ nola, according to Mx . Evelyn. Note, The feveral Members of thefe Profiles are Minutes accounted from the central Line. Plate XCII. The Dorick Order, by Inigo Jones. This Dorick Order is executed in the Screen to the Royal Chapel in Somerfet- Houfe, and is one of the molt beautiful Performances I everfaw. The Heights and Projections of the feveral Members are determined by Inches and Parts. The Diameter at the Bafe is 17 inch, and {, and 14, inch, at the Aftragal, be¬ ing' diminifhed ;. Here we fee, that this Mailer has at once kick'd away the Triglyphs, and introduced Leaves in their Stead, which a very noble Al’pedt, as well in Profile as in Front ; and that thefe Leaves might not be thought ufelefs, he has brought forward the Sofito of his Bed-moulding, for them th fupport. In brief, the Compofition is grand, and the Enrichments are very noble, and Worthy of Imitation. Plate XCIII. Dorick Intercolumnations, by I. Jones. This Plate reprefents the Plan and Elevation of the lower Part of the Screen in the Royal Ghappel alorelaid, where the Intercolumnations are (de¬ noted by Feet and Ihches, and) very grand. Plate XC1V. The Dorick Order, by Sir Christopher W R e n. Th is Profile is an exadt Reprefentation of the Dorick Order in the Frontif- piece of the Steeple of St. Mary-le-bow, in Cbeapfide, London , vvhofe Entabla¬ ture is of a very extraordinary Compofition, and has a very good Effedt. The Height and Projedtures of the Members are denoted by P’eet and Inches, as al¬ to are thole of the Impoit and Intereoliunnation in the Frontifpiece itfelf, ex¬ hibited in Plate XCVIL Plates XCV. XCVL The great 'Pillar, or Monument of London, begun in the Year 1671, and finifhed in 1677, according to the Defigns, and under the Condudt of Sir Chri- /lopher IVren, Knt. Surveyor General of the Royal Works, and of the Cathe¬ dral of St. 'Paul, and all the Parochial Churches and publick Buildings o(Lon¬ don, after the Conflagration of the City, 1666. 0.9 q This' 2KO The ‘Principles of Geometry. This fuperb Column nevpr having appeared in Print, with its Meafures af¬ fixed tliereto, 1 have therefore, with great Pains and Care, meafured every Part theieof. As 1 am ignorant of the Manner how this great Matter deter¬ mined the Height and Projefture of its Parts, I have been obliged to form a Method different from what is generally pradtifed, in order to come at a right Knowledge of its Dimcnlions, which Method is lb eafy, that Infpedlion only will make it underftood by the Judicious. But as this is a capital Building in the Capital City oh Great Britain, (and which I believe the whole World cannot equal for Magnitude) 1 am willing that thole, who have as yet but a flendcr Knowledge of MrchiteQure, lhould be thoroughly acquainted with the Conftrudtion of its Parts, therefore 1 lhall give a full Explanation as follows. The general Proportions of the 'whole Monument , as exhibited in Fig. 1 . The Height ol the Column with its Capital and Bafe. but wuhout its Sub¬ pit is B tunes its Diameter at the Safe, which is ry Feet. the Height of the Pedeftai (without the Sub-plinth of*the Column) is one tin d part of ihe Column s Height ; and the Height of the Cippus (or circular Pedeftai) on the top of the Column, together with the Vaje , (or Fire pot) excluding the Flame, is equal to the Height of the 'Pedeftai. To determine the Heights of the particular Parts. The Semidiameter of the Column , Fig. II is equal to the Height of its Bafe , viz. the Plinth and Turns. Divide the Height of the Baft into 7 equal Parts, which Parts will ferve as general Meafures throughout the Whole, give 4 Parts t0 the Plinth, 3 to the Torus, and 1 to the Fillet of the Cin&nre. The Pede/tal is equal to 36 parts, give y to the Bafe, 11 and 1 thirds to the 'Die, and 7 and 1 third to the Capital. To divide off the Members of the Bafe of the Pedeftai, give 3 parts and half to the Sub-plmth , 1 and halt to the Plinth, 1 to the Torus, z to the lima recta and two Billets ; divide 1 part into 4., and make each Fillet equal to one of thole 4 parts, give one whole part to the up¬ per Torus and Fillet ; the Fillet is equal to one fourth, as before. The Frame of the Punnet in the Die is equal to x parts, give 1 to the plain Margin, and the other to the carv d Cima recta and Fillet ; divide this laft into 8 parts, and give 1 to the Fillet. To di vide the Members of the Capital of the Pedeftai. Give i part to the C.avetto, Fillet, and dftragal, viz. one half to the Si f- trcgal, and the other half being divided into 3, give 1 to the Fillet, and 1 to the Cave/to ; give 2 whole Parts to the Cima retlta, one and half to the Faj¬ ita 01 Corona and the Billet under it, which is one fixth ot a part, half a Part to the Cima reverfa, and one third to the Regnla ; give the remaining z thirds, and 6 more whole Parts, to the Sub-plinth ol the Column The Height of the Capital is 8 parts, fo that it is 1 part higher than the Bafe, which is very uncommon ; doubtlefs the Reafbh is its great Altitude, which in Km meafilre contracts, or fore-lhortens the perpendicular Lints, and makes the V\ hole appear not lb high, as it really is, therefore this lhould never be pradtiled in other Kinds of Buildings. To divide ihe Members of the Capital. Divide 1 part into 3, and give 1 to the Regnla, and x to the Cima rever- Ja ; give z whole parts to the Mb ins , and z to the Ovolo; divide 1 part into 3, and give z thirds to the Hftragul , and 1 third to the Fillet ; give x whole 1 arts to the “Seek ot the Capital, 1 to the sljlragal ot the Column, and half one to the Fillet. To The Principles of Geometry. 251 To divide the Bafe of the Cipptis into its Members. Give a Parts and half to the Sub-plinth, 1 and half to the Plinth, 3 fourths to the Torus, and the remaining 1 fourth to the Billet of the Cincture, which is Part ol the Pie • The Height of the Pie is 3 times the Height of the Bate viz. the Sub-plinth, 'Plinth , and Torus. To divide the Capital of the Cipptis into its Members. G ive 1 fourth of a Part to the Regnla, half a Part to the Fafcia, half a Part to the Ovolo, and 1 fourth to the Aftragpl and Fillet ; divide this fourth into 3, and give r to the Aflragctl, and 1 to the Fillet. The Height of the great Cima retta, or the Crown of the Gppus, is 4 whole Parts; the Ajlra- gal and Fillet together are 3 fourths ; divide the lovvelt fourth Part into 5, and give 1 to the Fillet ; the Height of the great Fillet above the si [hagai, or the 7 hath, to the T^aje (or pire-pot) is equal to the Height of the sijh ,/' al. To determine the Projeclures. The ProjeHure of the Fillet of the CinBme is equal to its Height, viz. 1 Part from the Upright of the Column, and that of the ‘plinth is 3 Parts, le Swelling of the Toms being perpendicular to the Out-line of the Plinth, e : /iue the Height of the Plinth into 3 equal Parts, one of which is the P10- . ure of the Sub-plinth^ the ProjeCture of the Pie of the Pedejlal is the lame with that of the Sub-plinth' of the Column. 1 0 determine the ProjedureS of the Members of the Capital of the Pedejlal. The Projection of the Fillet over the Cavetto is equal to the Height of the Cavetto a” Fillet together; the Bottom of the Cavetto projects as much as the H ■ a the Fillet above it ; from the Perpendicular of the Projection of Fitlet, delcribethe Out-line of the A/lragal, which determihes its Probation ; the ProjeCture of the Fillet over the Cima reda is equal to the Height of the Cima and the Aftragal without the Fillet under it ; and that of the Fafcia to the Height of all the Members below it; the Projection of the Regnla, from the Upright of the Fafcia, is equal to the Height of the Cima reverfa. To determine the Projeclures of the Members of the Bafe of the Pedeflal. The ProjeCture of the upper Torus is equal to its Height, and the ProjeC¬ ture of the z upper Fillets is half that of the Torus ; the Projeflure of the lower Fillet is equal to the Height of the Cima reda, and both the Fillets ; • the Torus and Plinth project one half Part from the lowelt Fillet, and the Sub-plinth twice as much, or one whole Part. The Semidiameter of the Co¬ lumn is diminiflied, from 7 Parts at .the Baje, to ■; and one third at the Af¬ tragal. To determine the Projeclures of the Members of the Capital. The ProjeCture of the AJhagal of the Capital is equal to its own Height, and that of the Fillet under it is equal to its own Height alfo; the ProjeCture and Height of the Ovolo are likewife equal, and the ProjeCture of the Abacus is equal to the Height of the Ovolo and slftragal together; the Regnla pro¬ jects, before the Upright of the Abacus, as much as the Height of the Cima reverja. The fillet under the Aftragal of the Column projects equal to that of the fijtragal of the Capital, and the Swelling ol the Astragal beyond the Fillet is half its own Height. The Projection of the Fillet and AJtragal of the Column is very final! in Proportion to their Height; the Reafon of which I take to be this ; had they projected more, at fo great a Height, the Aflra¬ gal would have hid a great Part ot the Neck of the Capital, which would have had an ill EffeCt. To The Principles of Geometry. 248 To determine the ProjeBures of the Members of the Bafe of the Cippus. T h f. Body ot' the Cippus is equal to the Column of its Mragal ; the Torus of its Bafe projects as much as its own Height, the Plinth equal to the Torus, and the Fillet of the Cincture half as much ; the Sub plinth projefts, beyond the 'Plinth, 1 fourth of the Height of the Plinth. To determine the TrojeBures of the Members of the Capital of the Cippus. The Fillet, Astragal and Ovolo do each project equal to its own Height ; the Projctture of the Regu/a is the fame with that of the Torus and Plinth ; and that of the Fa/cia is half the Diftance between the Projefture of the Ovolo and that of the Regula. The Projefturc of the Fillet under the upper ASira- nal is ; Parts from the central Line, the Swelling of the Astragal half its Height beyond it ; the great Cima, and the Fillet above, are 1 lixth part backward. Having thus fhown how this noble Column and its particular Members may be divided by Parts, I fhall in the next Place film up the Number of Parts contained in the whole Height, as thus. Bafe of the Pedeftal 9 Die of the Pedeftal n r! or ; Capital of the Pedeftal y ri or t Sub-plinth of the Column 6 r! or * Bafe of the Column 7 Shaft of the Column 93 Capital of the Column 8 Bafe of the Cippus 4 if or i Body of the Cippus 14 ri or £ Capital of the Cippus 1 ri or 1 Great Cima 4 - Aftragal with thefmall Fillet? ^ Qr . and Plinth of the Vafe f 11 1 Yafe n _ ► 188 ri or \ Thus it appears there are 188 parts and half in the whole Height, let us now reduce thefe Parts into Feet, and fee what the Height of the 188 i Whole is by that Meafure. You may remember I divided ri the Semidiameter of the Column into 7 Parts, fo the Whole , is 14 parts and ly Feet; but before I ftate the Queftion, I ’ ^ \ mull reduce the Parts into Half-parts, becaufe there is an-— odd Half: So the Queftion Hands thus. If 18 Half-parts give 377 18 iy Feet, what do 377 Half-parts give ? 377 z$)s 6 sf{roi IT yd i88y ooyy 377 18 jG r 17 Bv this Operation it appears, that the whole Height is ioi Feet, and 17 twenty-eighths of a Foot, which is fo near ioi, that we need not fcruple to fay it ’is fo by this Method of working. This is the Height which one of the Inscriptions upon it afligns it; and it was that Diftance Eaftward from it, that the Fire of London began, which was the Reafon of its being that Height. Fig. The “Principles of Geometry. 253 Fig. III. reprefents the Plan of the Column at its Bafe, which being ip Feet is divided into y equal parts of 3 Feet each, the firft is iblid Wall, the fecond is the Length of the Stairs, the third is the Newell or Well-hole from top to bot¬ tom, the fourth is the fame with the fecond, and the fifth as the firft, the Cir¬ cle of Stairs is divided into 24., which fhows there are lo many in one Round. F1 g. IV. V. exhibit the Column of London , and that of Trajan at Rome by the fame Scale, which lbows the latter is but 3 fourths of the Height of the former. Plate XCVIf. A Dorick Intercolumnation , by Sir C. W ren. For the Explanation of this Frontifpiece vid. the Explanation of Plate XCIV. Plate XCVIII. XCIX. The Dorick Order, by Mr. Gibbs. (1) The firfl Figure on the left Hand, reprefents the Dorick Order entire, according to this Mafter, whole entire Height being divided into five parts, give the lower 1 to the Height of the Pedefhil, and the upper 4. to the Height of tile Column and Entablature ; which laid Height being divided into y, give the upper 1 to the Height of the Entablature, and the lower 4. to the Height c the Column, (2) Divide the Height of the Column into 8 Paris, and tube i : r its Diameter. The Height of the Bale (which is si!tick) is equal to the Semidiameter of the Column, as alfo is the Height of the Capital; and the Shaft is diminifhecl, from one third of its Height unto its Aftragal, one fixth Part of its Diameter ; and thus are the general Parts of this Order determin¬ ed. The next Figure reprefents the Pedeftal and Bale of the Column at large, whole Parts arc divided as follow, viz. (1) The Height of the "Pedeftal being given, to divide 1! into its Bafe, "Die, and Cornice , divide the given Height into 4 Parts; give the lower 1 to the Height of the Plinth, and \ of the next 1 to the Height of the Mouldings. The Height of the Cornice is equal to one eighth Part of the whole Height, or half of the upper 1. (2) To divide the Mouldings of the Bafe , divide the Height of the Mouldings into 8, give the upper 2 to the Cavetto, the next t to the Fillet, the lower 1 to the lower Fillet, and the remaining 4 to the Lima recla. The Projection of the Bale of the Pedeftal, from the Upright of the Die, is equal to its Height. The Upright of the Die is equal to the Projection of the Plinth to the Column's Bafe ; and the Projection of that Plinth, from the Upright of the Column, is equal to 011c fixth Part of its Diameter. If the Projection of the Plinth of tile Pedeftal, from the Upright of the Die, be divided into 8 Parts, (as under its Mouldings) one half of the firft 1 is the Projection of the upper part of the Cavetto, the next 2 of the Fillet, and the 7th of the lower Fillet on the Plinth. (5) To divide the Mouldings of the Cornice , divide the Height into 4, give the lower 1 to the Cavetto with its Fillet, the next 1 to the Gvolo, half the upper 1 to the Regula, and the Remainder to the Platband or FaJ'cia. (4) To divide the Mouldings of the Bafe to the Column , divide the Height in¬ to 3, give the lower 1 to the Plinth ; divide the middle 1 into 4, and give 3 to the lower Torus, and half the upper 1 to the Fillet; divide the upper 1 into 4, give the upper 2 and half to the upper Torus, and the other half to the billet, the Remainder is the Height of the Scotia. Thefe Mouldings ot the Bafe and Cornice of the Pedeftal, and of the Bafe to the Column, are dc- feribed at large by Fig. V. and VI. 111 Plates C. and Cl. (y) To divide the Mouldings of the Capital , Fig. I. Plate C. divide the Fleight into 3, give the lower 1 to the Neck A, the next 1 to theOvolo with its Cavetto, and the upper 1 to the Miacus ; then fub-divide them as is repreiented by the Sub-divihons. The Projection of the Abacus is equal to one quarter of the Column's Diame¬ ter at its Aftragal, which being fub-divided into 4 parts, at 2, 3 ar.d 4, the R r r Point 254 The (principles of Geo m e t r y. Point z is equal to the Projection of tlie Fafcia C, and the Point 4 to the Pro¬ jection of the Fillet under the Ovolo\ the Projection of the Fafcia C, over the Ovoh B, is equal to j of the Part a 3 ; lajily, the outer Divilion being lub- divided into 6 , the fir ft and laft ones determine the Projection of the Cima revei jit, which compleats the Capital. (6) To divide the Entablature into its Architrave, f reeze and Cornice, divide the given Height into 8 Parts, give z to the Architrave, 3 to the Freeze, and-as many to the Cornice. The Breadth of the Triglyph is equal to the Semidiameter of the Column at its Bale, and the Diltance of the Triglyphs, which is the Metops, is equal to the Height of the Freeze. I mult here take the Liberty to obferve, that this and all the preceding Mailers on the 'Dorick Order, are entirely wrong in making their , Metops truly fquare, inftcad of making them to appear lb, which they can't do, if they are made truly fquare. This is indeed a Paradox ; but the Truth is, if the Architrave be above the Eye, the Projection of the Tenia will eclipfe a Part of the Height of the Freeze, and confequently the Metops will appear Parallelograms, of greater Length than Height, and that more and more as you approach the Building: There¬ fore, to make the Metops appear as Geometrical Squares at any aflignd Di- itance, there mult be an extraordinary Height given to the Freeze, as lhall be equal to that Part of the F'reeze, that may be cclipfed by the Projection of the Tenia, (9) To divide the yjrchitrave into its Tenia and Gutta s, Fig. IV. Plates C. and Cl. divide the given Height into 6 Parts, give the upper 1 to the Tenia ; divide the next z each into 4, give the upper one to the Fil¬ let of the Gutta, and the next 4 to the Depth of the Gutta. The Projection of the Tenia is equal to its Height, and the Projection of the Gutta to two Thirds of the Tenia. (8) The Height of the Cornice being given, to divide it into its Members , Fig. 11 . Plate Cl. divide the Height into 9 Parts, give the lower 1 to the Capital of the Triglyph, the next 1 arid * to the Height of the Ovolo, the upper z to the Regula and Cima recta, the next z to the Corona with its Fillet, and the remaining z and ; to the Modillion, with its Cima and Fillet. The Projection of the Cornice is equal to the Height, and one third Part of the Height of the Freeze. The Projection of the Cornice from the Upright of the Freeze being divided into 4 Parts, and thofe lub- divided again, as againft the Tenia is clone, thole Sub-divifions terminate th? Projections of the other Members, as alfo of the Tlancere, with its Bells or Drops, ot one ol the Mutules or Modiiiions a, b, d, d. Plates C. Cl. The principal Parts of the Derick Order at large, by Mr. Gibbs, The Figures 1 . II. IV. V. VI. being already explained in the laft Plates, need not be repeated here again. Figures III. and VII. are two 'Dorick Fron- tifpieces given by this Mailer for Practice ; of which Fig. 111 . having its Height divided into 11 Parts, give 1 to the Sub-plinth, z to the Entablature, and the Refidue to the Column. The Diameter of the Column is equal to the Height of the Sub-plinth: The other Figures denote the Breadth and Height in Diameters' and Parts. Figure VII. hath its Height divided into 1.3 ; of which the upper z go to the Height of the Baluftrade, the next z to the En¬ tablature, and the Remainder to the Column and its Bale. To find the'Pitch of the 'Pediment in F’ig. III. make FI D equal to E A ; then make D F equal to D A, fo will the Angle CF'A be the angular Pitch of the Pediment required. Plate CII. Dorick Arcades , by Mr. Gibbs. Here are two Varieties of Arches, the one without Pedeftals, the other with Pedeftals. That without Pedeftals contains 4 Triglyphs between the Co¬ lumns, and that with Pedeftals contains y. To proportion the Arches without The Principles of Geometry. 2 55 without Te deft ah to any Height, divide the Height into n Parts, give i to the Sub-plinth, 4 to the Entablature, and the Refidue to the Column. To .find the Height of the Top of the lmpoft , divide the Height of the Column and Sub-plinth into 3 Parts, and the id is the Height required, as on the Left- hand is divided. To proportion the Arches ninth Tedeftals to any given Height , divide tlie Height into y, give the lower 1 to the Pedeftal, (as has been alrea¬ dy taught) and the Remainder being divided into y, give the upper 1 to the Entablature. This done, divide the Height of the Column and Pedeltal into 3, as on the Left Hand, then will the fecond Divifion from the Safe be the Height of the Top of the lmpoft. The lmpoft and Architrave to thefe Arcades or Arches are reprefented by A B in Plate CHI. where you'll alfo fee, that the Height of the lmpoft (Which is always equal to the Diameter of its Pilafter it Hands on) is divided into 3, of which I is given to the Neck, the middle 1 to the Ovolo with its Aftragal, and the upper 1 to the Fafcia and Fillet. The Aftragal under the Neck is equal in Height to 1 the Neck. The Architrave B is generally made equal to i the Diameter of the Column, which Breadth being divided into 3, give three fourths of the firft to the fmall Fafcia, the uppet 1 to the Fillet and Cavetto, and the Refidue to the great Fafcia. The Projection of the'lmpoft is equal to one third of its Pilafter’s Diameter, fub-divided as in the Figure. The Projedtion of the Architrave is equal to five eighths of the Height of its Fillet and Cavetto, as at a is demonftrated. The Meafures of the other Parts being Dignified by the Figures affixed to each, there needs no further Explanation. Plate CI1I. Dorick Intercolumnations for Triumphal Arches and Colo * nades, by Mr. Gibbs. The uppermoft Figure is a Triumphal Arch of the Ttorick Order, vvhofe Height being divided into 6 , the lower 1 is given to the Pedeltal, and the upper 1 to the Parapet; then the Remainder being divided into 5, give the upper 1 to the Entablature, and the lower 4 to the Column. The Diftance of each Column is determined by the Figures affixed, and Height of the Import, aS in the preceding. Note, The Diameters of the Dies to the Pedeftals in the Parapet, rnuft be equal to the Diameter of the Column at its Aftragal. The Intcrcolumnatioiis for Portico's, or Colohades, have their Diftances ex- prefied by the Figures affixed. Plate CIV. The Ionick Pedeltal, by Carlo Cesare Osio, Geo¬ metrically deferibed. Bisect XL. the given Height in Y, and make the Angle X Y c equal to 30 deg. From X draw X c perpendicular to Y c, and thro’ the Points X and c draw Right Lines at Right Angles to the central Line X L ; then will X a be the Height of the Pedeftal's Cornice, and equal to one eighth Part of the Whole. Make Z L equal to X a, for the Height of the Bafe. T0 divide F A, the Height of the Bafe , into its Mouldings. (1) D e aw the Out-lines of the Die parallel to the central Line, at the Di¬ ftance of d z, or 1 of the whole Height of the Pedeftal. If the Pedeftal be ufed fingly without its Column, or otherwife, at the Diftance equal to the Projection of the Column's Bafe. (i) Make the Angle F’A B equal to 30 deg. and divide A B into 3 Parts at D C. From D raife the Perpendicular D c to cut F A in c, then c A is the Height of the Plinth. (3) Divide FA into 3 Parts at b E, draw b D, which divide into 3 Parts, then through/, which is the upper third Part, draw the Fillet. Bifedt b B in H, and throftgh IT draw the The Trinciples of Geometry. the upper Part of the Cima reEla ; the Remainder is the Aftragal, which com- pleats the Heights of the feveral Members. To find their Troje&ures. (!) Make F n equal to F b, and draw nm parallel to F A, for the I rejec¬ tion of the Aftragal. (i) Draw nb, and make bn perpendicular thereto in- terlefting the upper Line of the Plinth in h, which determines the 1 rejection of the Plinth. Make h i equal to q r, and that determines the Projeciure ot the Fillet. The Center a of the Aftragal, is let back half the Height of the Aftragal and the Beginning of the Cima recta is from that Point as is per¬ pendicular under it. The Height of Z, the CMure to the Die , is equal to half the Height of the Aftragal, and it ne be made equal to the Height ot the Aftragal, the Remainder F is the Projeftion of the CMure. To divide the Height of the Cornice into its Members. (i) Make the Angle MFz equal to 50 min. and divide F~ into 5 Parts at A B. (») From z draw zG, dividing the Angle D z K into % equal Parts, and through the Point B draw the Line Hg parallel to M F, cutting G z ing, through which draw bg E, the lower Line of the Cima reverja. Diviae *.1 h in 3 and the upper 1 is the Height of the Fillet, and tne lower x the Height of the Cima. (?) Make the Angle M F FI eqfial to 60 deg. then will the Line FH cutgH in H, thro' which draw the Line c H, the under I ait of the Ovolo. Bileft gH, and give the upper Half to the Fajcta and the lower Half to the Ovolo, and thus are the Heights of every Member deter¬ mined. Make h, the Prelection of the Ovolo, equal to the Height of all the Members above the Ovolo, and draw, the I .ine h M ; alfo thro the Point h draw the Line p r at Right Angles, to h M, by whole Interjections m the Points p, d, c, r, the Projections of the Members are determined. The Drip of the Corona 6 y is equal to J ot the Corona s Height. The Height of t ‘ 1 '- Fillet F, under the Aftragal, is equal to half the Height of the Aftragal, and its Projection is determined by the Line p r. Plate C V. The Ionick Bcife and J^oluta, by Carlo Cesare Osio. Altho' the Ancients were people of great Invention, and whofc Examples are in molt Cafes worthv of the greateft Regard, yet by the Compoution of this Bale Fig. 1 . ’tis evident, that they were miftaken 111 feme Things ; for lurcly nothing can be fo monftrous, as to lee fo many fmall Members placed between a monftrous 'Plinth and an overgrown Torus , and indeed, leemingly, as if it was intended to prefs them to pieces, mltend ot thc-ir being made Members capable to lupport the incumbent Weight. This horrid Competi¬ tion (for no other can 1 call it) is alfo followed by Vitruvius, Barozzto, Ser- lio, Cataneo, Barbara, Viola, De Lorme, Bidlant , ‘Perunit, and Julian Mau- Clerc w hi lit 'Palladio, Scamozzi, and the other Matters, have abhorred it. Ps Height is equal to the Semi-diameter of the Column, and its Members are divided as follow, (1) The Height of the Plinth is one thud part of the Height of the Bale, and its Projection is determined by the Line aw, mak¬ ing the Angle w az equal to 30 deg. (i) Make fg, the Height of the Sco¬ tia’s and Attragals taken together, equal to wg, the Projection of the Plinth, then will a / be the Height of the Torus. Make a c equal to aj , and draw c e parallel to a z ; make cb and e l each equal to half the Height of the To¬ rus, and bileft it in d, the Center, on which defcribe the Semi circle. (3 ) Bifeft/g in 4, the Divifion of the two Aftragals, and draw the Line w /, which interfeft in the point p, by the Line/) 9, making the Angle p 9 w e- qual to 30 deg. then thro’ the point/) draw the Line ps, the under part ol the lower Aftragal. (4) The Fillets y 6 and 8 9 are each one eighth ot y 9. (y) Make * 4, the Height of the upper Aftragal, equal to 4 y, the Height The ‘Principles of Geometry. 2 57 ot the lower Aftragal, and the Fillets fg and i z, equal to j off z. The Projec- tton of the Fillet under the Torus is determined by h l continued, the Fillet m r by the Interledlion of the Lme« w in the Point m\ the two Aftragals by the Torus c e continued, the Fillet p y by the Interfeaion of the Line fw in the Point/). (6) Divide the Height of each Scotia into 3 equal parts, as h and and diaw the Lines 1 h, and r 7, on which will be found the Centers for to dc- ienbe tne Curves of the Scotia's as follows, viz. Continue Ik to n, then is n tlic Center of the lelTer Quadrant; then taking the Height h 1 111 theCompaf- les, let that Durance on the Line i h, from the Point where the lefler Qua¬ drant meets it, unto i, then on i dclcribe the greater Quadrant, which will compleat the upper Scotia ; and it in like Manner you form the lower Scotia, the whole Bale will be contpleated as required. To dejeribe the Ionick Capital , Plate CV. V Dl , V1D , E J he „ gI ^ n Hc, S ht 1 1 ’ ( whic}l is always equal to two thirds of the Height or the Bale) into 13 equal Parts, give 1 to the upper Fillet, a to the Cuna reverfa 1 to the Lift of the Volute, 3 to the Fafcia, and the refi- due y to the Ovolo; continue on downwards, and make the Aftragal equal to t., and ils l iilct to i of thofe Parts, and fo arc the Heights of all the Mem- hers determined. To determine their TrojeBures before the Line et (which being continued downwards is the Upright of the Column) make e h equal to i ?, and cb equal to i i, then from b draw b h parallel to e t, which termi¬ nates the fillet; draw the Line cf\ cutting the Lift of the Volute in x and l, and then making g h equal to a fourth Part of h h, draw x r, and deferibe the Lima reverja. Thro' the Point /, where the Line c f cuts the Line y / draw the Line d l A F w, &c. continued on at Pleafure, interfering the up-’ per Line ol the Aftragal in w, the Center of the Volute; from - draw the T.ine 7 m parallel to « t q, which will cut the Line d lA, &c. in the Center o the Quadrant Im and then m q being drawn parallel to et , the Mouldings ol the Capital will be conrpleated as required. To defer the the Valuta. (1) Draw i 3 V Mg the upper Line of the Aftragal, out at Pleafure, as on to ’ a 0 cdhtinue downwards the Line dhw, as unto 9 ;, fyc. alfb draw the Lines B w 10 f, and M » 8 t’, cutting each other at Right Angles, and iW' dc§ ft, i ' W f ance from the Lines A w and W 7//, which continue out both Ways at lkafure: This done, apply to your ty equal Divifions, by which \ou divided out the Heights of the Members of the Capital, and fub-divide them into 3 Parts, of which every z will be equal to rf Part of the Diameter ot the Column, and equal to 1 min. (1) Of thefe Minutes make 7 v M equal to 11 A and w N equal to 10 7;; alfo make w P equal to 11 min. A, and wO to 9 min. U: This being done you have the three Points x, M, P, given to uelcribe the Arches x, M, P, and the three Points /, N, O, given to dclcribe the Aich / K (). Proceed in like Manner with every other Quadrant, fetting oft the Diltances from the Center, as ftgnified by the Figures affixed, and Arches being defcribed to pafs through every 3 of them, will compleat the \ olute, as required. This Volute hath the molt eafy and gradual Di¬ minution of any that I have yet leen, and the Rule being the 1110ft ealy, I do therefore recommend it before all others that follow, which in general are infinitely deficient of the graceful Curvature and eafv Diminution of the Lilt. 1 late 7. to follow Plate CV. Ionick Volutes of various Kinds. Method I. The Figures C and D reprefent an ancient Method of deferib- ing uie Ionick Volute, which Barozzio of Vignola and many others have fol¬ lowed, altho the Lift is far from having an eafy Diminution S l'f To 2g8 The Trinciples oj Geometry. To underftand this Method, 'twill be belt to have Recourfe alio to Plates CXXVI. CXXVI 1 . where this Volute is deferib’d more largely in Fig. [. there you lee, that where the perpendicular I.ine HA 6 (which is called the Cathetus) mterletts the upper Line ot the Altragal, that is the Center o! the Eye of the Volute. To find the Radius or Diameter of the Eye of the J'olute, divide the upper Part of the Cathetus, from the Center of the Eye, to the under Part of the Cnna reverfa, into 9 equal Parts, and make the lower 1 the Radius of the F.vc, with which defenbe the Circle. To find the Depth of the Volute on the Cathetus below the Eye, make that Part equal to 6 of the Parts above; this done let the Line 1 y, in Fig. D, Plate Z, re- prefent the Cathetus, and the Line 3 7 the upper Line of the Altragal, and let their Point of Intellection be the Center ot the Eye of the Volute, oil which let the Eve be deferibed ; alfo draw the Lines z 6 and 4 8 through the Center of the Volute at Right Angles to each other, and at + y deg. Pittance from the Cathetus 1 y, and horizontal Line 3 7, on which, and on the Ca¬ thetus and horizontal Lines, the Limits of the Volute mult be determined as follows, viz. (1) Draw he in Fig. C Ion the Left Hand) at Pleafure, and at its End h creft the Perpendicular h a, of Length at Pleafure. (1) Take the Radius of the Eye of your Volute in your Compalfes, and on the angular Point h de¬ fenbe a Circle. Make the Height e a equal to 4 times the Diameter of the Eye; alfo make he equal to 3 Diameters and half, and draw the Hypothenufe a e. (4) On e, with the Radius c b, delcribe the Arch bed, and draw the Line e c (y) Divide the Arch e d into 6 Parts, and lub-divide each Part into 4 Parts, then w ill the Arch e d be divided into 14 equal Parts. (6) Lay a Ruler Irom c to every of thofe Parts, and it will divide the Line a e into 14 une¬ qual Parts, by which the Curve of the Volute is determined as follows, viz. The Pittance from the Center of the Volute to 1, the upper Point ot the C ithetus, in Fig. D, being equal to b a or 6 1 in Eig. C ; therefore let y ,1 [V &c. thro’ which the Contour, or b ;\ 3 Curve of the Volute mult pals, b q 'CLr. in Eig. C, from the Center of 4 as following, viz. (P Set one b y(thc Eve ol the Volute in Fig. D, ky > Foot of the Compalfes in the Ccn- b 6 unto the Point d| ter ot the Volute, 'and extend the y y 7 other to the Point 1, then with y 3 8j that: Radius on the Points 1 and 1, delcribe Arches interfe&ing each other, on which Point of Interjection dc- lcribe the Arch 1 z. (z) Set one Foot of the Compalfes in the Center of the Volute, and extend the other to the Point z, then, with that Radius, on the Points 2 and 3, deferibe Arches interfefting each other, on which point of I11- terfeftion deferibe the Arch 23: In like Manner proceed with all the remain¬ ing Points, 3, 4, y, 6, 7, 8, 9, 10, 11, lz, till the Whole is compleated. The inward Line of the Lift of this Fillet is carried, for the greateft Part, at a parallel Pittance, which has a very ill Efted; and therefore to diminilh it gradually is a Work which none of thefe Matters have taught, and may be ealily effected as follows: Divide the Height of the Lift at 1 into 14 equal Parts, and from the Point r - *1 , 3 1 u D IT [ 4 21 1 7 20 1 6 *9 I 1 7 fet towards the Center !8 > 8 17 9 16 10 T 7 11 14 .& c * [&c.\ (•Parts; and The Principles of Geometr y. 2 3 9 and then, thro" thefe laft produced points, dcfcribe the inward Curve, by the fame Rule as you did the outward Curve, and that will compleat the Voluta, diminilhcd in an eafy agreeable Manner, as it ought to be done. Meth od II. 7 *o defcribe the lonick Toluti % by Means of 24 Centers , differ¬ ently from the preceding. Fig. H, Plate Z. (1) Draw the Cathetus p 4, and let the Height p 4 be the given Height of the Voluta. (2) Divide the given Height into 8 equal Parts, as by the dotted Lines 7 7, 6 6, y y, efyc. is done, and let the fifth Divifion 4 3 be bife&ed in g, from whence draw the Line g d, and its Point of lnteri'ertion, with the Cathetus, is the Center of the Eye of the Volute : The Diameter of the Lye is equal to the fifth Divifion, or ; of the whole Height: Within the Circle of the Eye inferibe a Geometrical Square, as before in the laft Example, drawing its Diameters through the Center, and dividing them each into (> equal Parts. This done number the feveral divided Points in the Diameters of the inferib- ed Square, as is done in Fig. K, which reprefe’nls the Eye of the Volute more at large, as 1, 2, 3, 4, y, 6, 7, 8, 9, 10, 11, 12; thefe 12 Points arc the Centers on which the outward Line of the Volute is deferibed, as follows, •viz. (1) From the Point 1 draw the Line 1 q parallel to the Cathe¬ tus. (2) Continue 1 r to k, and on 1, with the Radius 1 q, defcribe the Qua¬ drant q k. (3) Thro' the Points 2 and 3 draw the Line 2 3 c parallel to the Cathetus, and on the Point 2, with the Radius 2 k, defcribe the Quadrant k c. (4) Through the Points 3 4 draw the horizontal Line 3 l, and on the Point 3, with the Radius 3 c, defcribe the Quadrant cl. (y) Through the Points 4 $ draw the oblique Line 4 j u, and on the Point 4, with the Radius 4 /, dcfcribe the Arch lytt. { 6) Thro' the Points y 6 draw the Line y 6 r, and on the Point y, with the Radius y u, defcribe the Arch u r. (7) Through the Points 6 7 draw the Line 67 r, and on the Point 6, with the Radius 6 r, dcfcribe the Arch r y. (8) Through the Points 7 8 draw the Line 7 8 A, and on the Point 7, with the Radius 7 r, defcribe the Arch r A. (9) Through the Points 8 9 draw the oblique Line 8 9 w, and on the Point 8, with the Radius 8 A, defcribe the Arch A tv. (to) Thro' the Points 9 10 draw the Line 9 10 x, and on the Point 9, with the Radius 9 tv, defcribe the Arch ■w x. (1 1) Thro’ the Points to 11 draw the Line to 11 i, and on the Point jo, with the Radius 10 x, defcribe the Arch x 1. (12) Thro’ the Points 1 1 11 draw the Line it ix&, and on the Point 11 defcribe the Arch i; lajl- ly, on the Point 12, with the Radius 12 &, defcribe the Arch Sc h, which compleats the Contour as required. Note, The inward Line mult be dirni- nifhed, as in the laft Example, but with this Difference, that whereas in that vou divided the Height of the Fillet into 24 equal parts, it having 8 Lines that terminated the Arches, fo here, where in Effedl there are but 4 Lines which terminate the Arches, the Height mult be divided into 11 equal Parts, and therefore 1 Part mult be abated at the Termination of each Arch. Tlie Centers to thefe inward Arches may be found by Interfedions, as in the preceding Example. It is to be noted, that the nearer the Centers are together the more circu¬ lar the Volute is, as in Fig. D and H, and the more diftant they are, as iri Fig. F, G, 1 , the more elliptical they*are. Fig. 11 reprefents at large the Manner of placing the Centers of Fig. G. Fig. L is no more than a Repeti¬ tion of Fig. K, inlerted by Miftake of the Engraver. Fig. N reprefents the Volute Fig. H, but as if it were prefled clliptically, it having the fame rcfpecl to the Parallelograms made by the dottcdLines, as the Volute Fig. H hath to the dotted geometrical Squares. 7 he Manner of deferibing thefe elliptical Vo¬ lutes is as follows, Suppofe the given Height be e a and Breadth b a ; (1) It is to be obferved, that the Height and Breadth of the circular Volute, Fig. H, are to one another as 8 is to 7, that is, 8 Squares in Height by 7 Squares in Breadth; this being underltood, divide e a, the given Height of your ellipti¬ cal Volute Fig. N, into 8 equal Parts, and its Breadth into 7 equal Parts, and compleat ■z6o The ‘Principles of Geometry. com pleat the dotted Parallelograms, (x) Compleat a circular Volute, as Fig. H, making its Height equal to the Height of the elliptical Volute. (3) Observe to trace the elliptical Curve through the Parallelograms, in the very lame Manner as the circular Volute paffeth through the geometrical Squares, and that will compleat the elliptical Volute, as required. F1 g. F_, on the plead of the Plate, reprefents the Centers at large of Fig. 1 . which being nearer together than Fig. B, the Volute Fig. I, is lei's elliptical than the Volute Fig. G. The Manner of deferibing thefe Volutes are the very fame as the other before delivered, that is, the Volute Fig. F is by the fame Rule as Fig. D, and the Volutes G I the fame as Fig. H. Fi g. M is the Manner of deferibing the Ere of the Volute, according to the Invention of Mr. Nicholas Goldman, as follows. ( 1) Let the Circle g x 4, reprefent the Eye of the Volute, let the Line p 4, be the Cathetus, and g d ci (x) L.et the Points I, 4., < and con,’. ; sently tire 1 (3I Compleat the gc ■ ..... 1 4 ‘ 9 10 at o 6 equal 11 ! i j 87 pai alk Sides of the Eye, as in the pre. minates the Arches, intlead o; at Right Angles in the Center of the F.ye. ido the two Semidiameters into x equal Parts, ■ be equal to the Semidiameter ot the Eye. . e 1 x 4, 3, and from the angular Points .ic Eye. (4.) Divide the Side of the Square mts y, 9, 11, 8, and draw the Lines 5 6 , l.iaegd. (y) Draw occult Lines in three .mg, which with the Cathetus p 4 (that tcr- ie Lines q n w, as in Fig. K) will terminate every Arch contained in the Vonite, which are to be deferibed on the Centers 1, 1, 3, 4, efic. as thole in Fig. H. P. Tray from whence is the Word Voluta derived? M- The Word Voluta, or Scrowl, is from the Latin, Voluta d vohendo, for that it Items to be rolled upon an Axis or Staff: Alberti calls them Snail- Shells from their fpiral Turn ; it is the principal and only appropriate Mem¬ ber of the Ionick Capital, which lias four, in Imitation of a Female Orna¬ ment. The Face is called the l i ons, or Forehead, a little hollowed between the Edge or I .ill; and the Return, which the next Plate exhibits, is called Tul- vin or Pillow, betwixt the Abacus and Echinus or Ovolo, refembling the fide- plaitcd Treffcs of Women’s Hair, to defend as it were the Ovolo from the Weight of the Abacus (over which the Voluta is placed.) The Eye of the Vo¬ lute is, by lbmc, from the Latin, called Oculus. P. Tray why is this Order called Ionick ? M- Because it was invented by Ion, when he was Tent from Athens with a Colony into that Part of Greece bearing his Name (and where he erected a Temple to 'Diana the Goddels, whole Columns were ol ftupendous Magni¬ tude, but made of greater Altitude, with refpeft te their Diameter, than the Dorick, as being more fimilar to feminine Slendernefs ; not like a light Houfewife, as Vitruvius compares it, but in a decent Drefs, hath much of the Matron; and to make it ftill the more feminine, its Shaft was enriched with its 14 Flutes, and as many Fillets, alluding to the Folds and Plaits of Gar¬ ments, of which I have already taken Notice. Plate CVI. The Profile or Pillow of the Ionick Volute. (1) The Abacus rp m v being firft drawn, as before taught, together with the Ovolo w, and Aftragal x’zr, admit the Line easy to be the. central Line of the Capital, (x) Make a xx equal to a c, and then a xx divided into 3 Parts, the outer 011c on the Right is the Fillet, and the other x the Band. (3) Make k l equal to I k, and draw l 19 for the upper Line of the Pillow ; allb divide z x, the Height of the Fillet to the Aftragal, into y equal Parts, and The Principles of Geometry: 2 6 1 and from the lower x draw the Line i s 7 13 for the lower Line of the Pillow. The Depth of the Volute m 3 is determined from the Volute itfelf, and the Breadth of the Lift tk is equal to k l. (4) Make r q equal to rp, bifeft rp in 0, and from q, through 0, draw the Line qh is, cutting the Line 1 19 in the Point h, and the Bottom of the Ovolo in the Point if: On k h, with the Diftance k h, del'cribe the Curve k h. (jj From the Point 13 draw the Line 13 10, making the Angle 10 13 it equal to 30 deg. cutting the Line x iz in 10; from the Point 10 draw the Line 10 zi parallel to the Line zz 18, cutting the Line /19 in the Point at, at which Point the Curve begins; the other Point, where the fame Curve ends in the Line zz 18, is the fame Diftance jiom 19 as / is from zz ; the Arch is defcnbcd cqudaterally, as the former. (6) Make 13 7 equal to 10 iz, and fet i of 13-011 the Line zz 13, from 13 towards iz, in which Point the Arch will terminate, and which is to be de¬ fended equilaterally as the former ; laftly, the Curve s + is del'cribed on a Cen¬ ter, whole Diftance from the Point 4 is equal to the Diltance 1 4, on the Line 3 4, continued. Plate CVII. The. Ionick Entablature geometrically deferibed according to the Ancients, by C arlo Cesare Osio. This Entablature is rather of tco rich a Compolition for the Ionick Order, and particularly its Architrave, which would have been better had it been broke into two Fal'cia's inftead of three, w hich is more proper for the Corin¬ thian or t ompo/i.te Order. To divide tine Height of the Entablature as into its Architrave s C, Freeze C r, and Cornice r a. (1) From the Points a and s draw the Iines a A and A s, each making an Angle of 30 deg. with the Line a s, and interdicting each other in the Point A. (r) Divide A s into 3 equal parts at D F., and from the Point D draw the Line D P at Right Angles to a s. (3) Divide F t into 6 equal Parts, and give the lower fC s to the Height of the Architrave. (4) Bifeft a C, the Height of the Freeze and Cornice, in the Point r, then C r is the Height of the Freeze, and r a the Height of the Cornice. To divide the drehitrave-into its Tenia and three Fafcia's. (1) Draw a s, making the Angle C s a equal to 30 deg. alio from the Point a diaw the Line a rn, making the Angle C a rn equal to to deg. then will m C be the Height of the TeniaDivide C m into 3 equal Parts, and give the upper X to the Fillet, and lower a to the. Lima reverja. (z) Bilcct / s in q, then is m 9 the Height of the upper Fafcia, and one fourth Part of rn s is the Height of the lower Faicia 3 a. To determine the Projecittre of the Architrave, bilcct a s in a, on which erect the Perpendicular z n ; from n draw n i parallel to a s, cutting m a in 1 ; from i draw 1 k parallel to n z, cutting C i in k: On m, with the Radius m k, deferibe the Quadrant k g : Divide g m into o equal Parts, let h p project 3 and half of thole Parts, and 0 r one half Part thereof : The Projection of the Cima reverfa is always equal to its Height. To find the Swelling of the Freeze, make r C the Radius, on C deferibe the Arch r B, and on r the Arch C B, then is the Point B the Center, on which, with the fame Radius, dderibe the Swelling of the Freeze required. T0 divide the Cornice into its Mouldings. (1) Divide the Height a r into 3 equal Parts, as is done by the 3 dotted Semicircles, of which the lower one determines the Height of the Dentils and Cima reverja under them, (z) From the Point 3 draw the Lines h 3 and i 5, the firlt making the Angle h 3 a equal to 30 deg. and the latter making the Angle i 3 a equal to 60 deg. Bifeft b 3 in f, and from / draw f g at Right .Angles to h 3, cutting i 3 in the i’oint g, thro which draw the Line 17 s for the Bottom of the Fillet under the great Cima. (3) Divide ab into s equal Parts, and give the upper 1 to the Regula, or up¬ per Fillet. (4) Make b e equal to b d, then is be the Height of the frnail Cima, and the Height of the Fillet over it is equal to one fourth Part of its own Height, (s) Bifeft 6 iz in 8, and draw 8 m 14 for the lower Part of the Fillet to the Ovolo. (6) Divide e m into 6 equal Parts, then the lower 1, / m, is the T t t Height 262 The ‘Principles of Geometry. Height of the Fillet aforefaid. (7) Make l n, the Ovolo, with its Fillet in- eluded, equal to l e, the Height of the Corona, and draw the Line 1 n 9. (8) Draw the Line i r, making the Angle i r a equal to 30 deg. which bifect in k, and then dividing k r into 4. Parts, the lower 1, that is 14 r, determines the- Height of the lower Cima reverja ; alio thro’ the upper 1, at the Point 10 draw the Line z 10 lor the central Line of the Aftragal under the Ovolo! (9) From the Point k draw the Line ky parallel to the Line a r, which will cut the Line z 10 in the Point z, the Center of the Aftragal; divide the Di- ftance from the central Line of the Aftragal to m into 7 equal Parts, and ffive the lower 1 to the upper Half of the Aftragal, alfo make the under Half of the Aftragal, and the Fillet under it, each equal thereto ; laflly , Make the Height of the Fillet under the Dentils equal to one fixth Part of their Height, and thus are the Heights of every Member determined. To determine t%eir Projechtres, (1) Make 64 7 equal to 64 r, and draw 74), the Face of the Fil¬ let under the Dentils ; make 64 q equal to 64 y, and from the Point y draw the Side of the Dentil parallel to r a, until it meets its own Fillet; make 7 equal to - y, and draw w x ; bifeCt the firlt Dentil w 7 in the Point 8, and make the Diftance between every Dentil equal to nv 8, that is, every Inter¬ val mult be equal to half the Breadth of a Tent 'd, and every Dentil equal to w 7. (a) The Line zy being before drawn from k, let it be continued to meet the Line 17 8 in the Point iy, which is the Center of the Ovolo, whole Fillet 18 17 hath its Projeftion equal to its Height. (3) Make a 31, the Pro- jeftion of the Cornice, equal to a r its Height, and draw 31 30, the Face of the Regnla: From the Point i draw the Line i 19 parallel to a r, and make the Projection of the Fillet between the two Cima’s, before the Line i 19, e- qual to its own Height, deducting the upper Cima and its Regula only. (4) Draw the Line 30 19, cutting the upper Part of the Corona in the Point ay, from whence draw the Face of the Corona z6 z+; bifeft 14 19 in zo, and on zo, with the Radius 10 zz, deferibe the Quadrant 11 n ; laflly, de- feribe the two Cima s, and the whole Entablature is compleated as re- quired. Plate CVIII. lonick Orders taken from the Theatre of Marcellus, and Temple of Mania Fortune in Rome, by Peter Ti¬ ll U RTINE. F I g. I. reprefents the lonick Order in the Theatre of Marcellas at Rome, whole entire Height A C being divided into 13 equal Parts, the upper 3 is given to the Height of the Entablature, and the lower 10 to the Column, with its Bale and Capital. To find the Diameter of the Column is a Work of fotne little Difficulty, feeing that the Height of the Column is 8 Diameters and f, which compels us to divide the Height ofthe Column into 43 Parts, and take y of them lor the Diameter. Fig. II. reprefents the lonick Order in the Tem¬ ple of Manly For tune m Rome, whole entire Height AB being divided into 8 equal I aits, as on the Right Hand, the Entablature pollellcs the upper 1 and ofthe ad, as a b, and half b c; the Height of the Column is 8 Diameters and ;, and to find the Diameter we mult divide its Height into 3 3 equal Parts, and take 4 ol them for the Diameter. I t is here to be obferved, that the Ancients did not confine themlelves to ceitain Rules in their Orders, but added to them or diminilhed them, as the Situation and Place required, but then it was always with the utmoit Circurn- lpection and greateft Judgment imaginable, which mult be always obferved when we depart from eltablilhed Rules. Plate The Principles of Geometry. 2 63 Plate CIX. The Bafc, Capital and Entablature at large of the Theatre of Marcell us at Rome. The feveral Parts of this Order are determined by equal Parts, as (1) The Bale, whole Height is equal to the Semidiameter of the Column, is divided into 3, of which the lower 1 is the Height of the Plinth, the remaining a be¬ ing divided into 3, the lower 1 is the Height ol the Torus, the middle 1 the Scotia and its lower Fillet, and the upper 1 the upper Torus and under Fil¬ let ; the Height of the Fillets is i of the Scotia; the Bale of the Column being divided into 11, the Out-lines of the upper Part of the Shaft, Handing over the Points 0 p, lhews that its Diminution is i; the Height of the Entablature being divided into 10 Parts, as a b, give 3 to the Architrave, as many to the Freeze, and the upper 4 to the Cornice, then lub-divide each as the Divifions exhibit. Plate CX. The Bafc, Capital and Enatblature at large of the Ionick Order, in the Temple of Manly Fortune at Rome. The feveral Parts of this Order are alio determined by equal Parts, as thc_ preceding Example. The Height of the Baje is equal to the Semidiameter oi the Column, and that of the Capital unto two third Parts thereof. The Height of the Entablature is two Diameters and ri of a. Diameter, fbe Height of the Architrave is equal to the Semidiameter of the Column, as alio is the Freeze; but the Height of the Cornice is equal to both their Heights, and a fourteenth Part more, as before obferved. The principal Parts being thus divided, the particular Members have their Heights determined by the Sub-divifions, which a little lnlpection will make plain. REMARKS. Tho’ thefe two Examples arc of the Ancients, yet lure it is, that no¬ thing can be l'o monftrous as the upper Parts of both thefe Entablatures. In that of Marcellas there is a very l'mall and poor Cymatium let upon a large and noble Corona , and that of Manly fortune has a monftrous Cima recta on a fmall Cima reverfa, and thole placed on a poor and low decrepid Corona , fcandalous to behold. The Bed-mouldings to both thefe Examples are not amifs, nor is the Freeze of Mar cel Ins fo liable to Cenfiire as that of Manly Fortune , which Items to be much too low for the Cornice over it, and more el'pecially as that the gieat Pro¬ jection of the Tenia to the Architrave doth caufe it to appear much lower than it really is. Plates CXI. CXH. Profiles of the Ionick Orders in the Theatre of Mar- cellus, and in the Temple oj Manly FoRTune in Rome, by Mr. Evelyn. The Profile on the Left is that of the Theatre of Marcellus (not oi the Temple of Manly fortune , as it is by the Engraver miftakenly exprefled.) This and the other Profile, which is of the Temple ol Manly Fortune 3 differ veiy greatly in their Coronas and Cymatiums 3 from the preceding Examples, and which I believe have been altered by Mr. Evelyn himlelf; loi in that ol Marcellus there is a poor trifling Corona, which in the preceding Example hath a very large and grand one; and in that ol Manly Fortune theic is a Corona of a noble Afped, which in the other Example is very low and difpre- portioned: On the Right Hand of this Plate is the Portico to this Temple, which 264 Tlx ‘Principles of Geometr y. w hich is now called the Temple of St. Mary the Egyptian, and is one of the molt beautiful Portico's of the Kind that the Ancients ever erected; the Parts , both thcic Profiles arc determined by Minutes, .accounted from the central J ,inc. Plate CXI1I. The Ionick Order in the Bath of Dioclesian hi Rome. Hr re this Order is reprefented, as well perfpeftively as in its geometrical Klevation, where its Members are determined by Minutes: The Temples 'Di¬ li/iyle and Syjlyle are according to Vitruvius , whole Ionick Orders come next in couri'c. Plates CXIV. CXV. Two Examples of the Ionick Order, by Vi- x r u vi us. Pig. I. is the firft Example, whofc Column and Entablature being divided into 8 Parts, the Height of the Entablature is I Part, and T of a Part, and the remaining 6 parts and P is the Height of the Column, which being divi¬ ded into 8 parts, as on the Left, one of thofc 8 parts is the Diameter of the Column. Fig. III. reprclents the parts of this Order at large; the Bale is of that horrid Compolition which I have already taken Notice of whole Height is equal to the Semidiameter of the Column, and whole Members are fub-divi- ded, as bv the Divilions appear. T H e Height of the Capital is equal to i of i of the Column and Entablature; the upper eighth part of the entire Height in Fig. I. being divided into 8 parts, give the upper 4 to the Height of the Cornice, and the next 3 to the Height of the Freeze, then fub-divide the Architrave firft in - lor to find the Tenia, and afterwards in it to divide theFafcias: The Height of the Cornice iilb-divide into 9, and give each Member its Height, as exprefled in Fig. III. Fig. If is his fecond Example, which is much better regulated than the o- thcr ; in this the entire Height is divided into y parts, of which the Enta¬ blature is one, which is directly the 'Derick Proportion, the lower 4 being the Height of the Column divide it into 8 parts, and take 1 for the Diameter: The Height of the Bale to this Order is equal to the Semidiameter of the Column, and it is compofed of the fame Members as the other Example ; but the Torus of this is not quite fo monftrous, nor are the Aftragals fo very fmall as thofe of the other. We may alfo obferve, that in this Exam¬ ple "Vitrmius has divided the Semidiameter of the Column into it parts, and confequently the Whole into 14; of thclc parts he makes the Height of the Architrave equal to 13, and Height of the Freeze to n J, which being taken out of two Diameters (which is the Pleight of the Entablature) the Remainder is the Height of the Freeze: The Height of the Capital is equal to 7, exclu- live of tlie Altragal, which is equal to z ; the particular Members are Rib-di¬ vided according to the Divifions annexed. REMARK S. I f we cotifider the very fmall Chna reverfa under the great Lima retta 111 Fig. 111 . and the fmall Cymatium over the Corona in Fig. V. it cannot be laid that either of thefe Cornices are good, altho' they are the Compolition of lb great a Matter. The ‘Principles of Geometry. 265 Plate CXYI. The Ionick Voluta , Ay Vitruvius. This Voluta of Vitruvius is defcribed by Method 11 . in Plate Z, after Plate CV. whofe Height is equal to 8 times the Diameter of its Eye, as is l'een on the Cathetus P D, or rather of the Height of the Aftragal, to which the Eye of the Volute is made equal, and on whofe central Line its Center is placed, as at O. The Fig. H reprefents the Eye of the Volute with an infcribed Square, whofe Diameters are each divided, as I have already fhewn, into 6 Parts, which gives the ix Centers for deferibing the Out-line of the Volute, and then each of thole 6 parts being fub-divided into 4 parts, the upper 1 of each part will be n other Centers, on which you may deferibe the inward Line of the Volute with a graceful Diminution. The 14 Centers of this Eye are each numbered as they are to be employed. Fig. K is the Eye of a Volute, for turning about the Contour by 6 Centers, which are found by dividing its Diameter into 6 equal parts at the Points 3, y, 6 , 4; the other Points 7, 9, 11, n, 10, 8, are 6 other Centers for deferib¬ ing the inward Line, caclt being one fourth Part of the firft: The Cathetus 1 x, is the only Line at which every Arch terminates, and therefore they are all Semicircles. The Volute thus produced is very agreeable, its Diminution very ealy, and the Method very plain. Fig. W is a Duplicate of Fig. Id, inferted by the Engraver's Miftake. Fig. V is the Manner of dividing the Eye of the Volute into its Centers, accord¬ ing to Mr. Goldmans, Invention, Fig. M, Plate Z, after Plate CV. But as Mr. Goldman only gave us the ix Centers for the Contour or Out-line, here are 14, viz. ix for the Out-line and as many for the Inward-line, which laft are found by dividing each of the 6 parts of 1 4 into 4 parts, as in the Figure is exhibited, and the laft one from the Center of every fuch Sub-divifton, as the Points 13, 17, xi, X4, 10, id, are the Points from which you are to draw Lines parallel to the former, until they cut the oblique Lines in the Points 14, x8, xx, iy, 19, 23, which, with the aforefaid, are the Centers re¬ quired for the Inward-line. Fig. X is a Side-view of the Volute, with its Pil¬ low enriched. Plates CXVII. CXVIII. Divers Ionick Portico’s , by Vitruvius. This Plate contains the geometrical Plans and Elevations of the three diffe¬ rent Kinds of Ionick Portico’s, as firft, the Temple Euflile of four ; fecondly , Fig. B of fix ; and laftly. Fig. A of eight, which laft is the Temple of "Diana at Ephefns, and the firft Example of this Order. On the Left Hand is an Ionick Door by Vitruvius, which with the Portico's is given here by Way of Example for the Practice of young Students. Plate CXIX. Ionick Temples, by Vitruvius. The Plans of the Portico's to thefe two Temples are exhibited in the laft Plate, which, together with thefe Elevations, are given as further Examples lor Practice. Plates CXX. CXXI. The Ionick Order, by A. Palladio. This Plate comprifeth all the Ionick Meafures of the Pedeftal, Column, Entablature, Impofts, Arcade and Intercolumnation, which in general are de¬ termined by Modules and Minutes; Fig. I. reprefents the Entablature, which is a fine Compofition of Mouldings, wherein the Dentils are excluded, and plain Modillions introduced. The Freeze is fwelling, and feerns to be eaufed by Bun the The 'Principles of Geometry. •i65 the Weight of its Cornice in manner of a Cufhion preffcd. The Architrave is no wile "lei's noble than is the Capital; though I muft again obferve, lean- net think but that the Architrave in this Order would be more noble, did it conlilt but of two Falcia’s, and more el'pecially when tis placed over the Do- i;ck, and the Architrave of the Dorick confifts but of one Fajcia, excluiive of theTetiiu. The Altragal under the Cindhire is a pretty additional Member, making the sJttick Bale under it, fomething more rich, than before in the 'Dorick. Figure 11 . is the Pedeftal, wherein there are two Examples for the Mouldings of its Cornice and Bale, which are very good. Figure III. is the Volute of the Capital, which is deferibed by Method II. Plate Z. Figure IV. i, a Plan of t ne quarter Part of the Capital, with the Manner of dividing the Flutes and Fillets of the Keck of the Shaft, and the Swelling of each Cabling. I'be Manner of dividing the Shaft is, to divide its Circumference into 24. equal Parts, and fub-dividing each Part into 4, give 1 to a Fillet, and 3 to a Flute. To find their Depth, divide eacli Flute into a, and deferibe Semicircles for their Depths. To find the Swelling of the Cablings, take the Breadth of a Flute in vour Compafles, as h c, and make the Section a, on which, as a Cen¬ ter, deferibe the Curve e b, which is the Swelling of the Cabling required. Fi¬ gure V. is a molt beautiful Impolt and Architrave lor the Arch. Figure VI. is another Impolt of good Compofition alfo. Figure VII. is his Intercolum- nation for Colonades or Portico's. The Height of the Pedeftal is 1 diam. 37 min. the Height of the Column 9 diam. and the Height of tile Entablature 1 diam. 70 min. Diam. Mm. .Pedeftal and Column The Height of the)Colnmn and Entablature Column and Entablature 11 37 10 70 13 27 Plate CXXH. The Dorick Order, by V. Scamozzi. The Meafures, by which the Members of this Order are determined, are Modules and Minutes, as exprelled in the Plate. The Compofition of the whole Order taken together is, I think, too rich, excepting theInfide Works, where the Eye cannot be removed very tar from it. The Contrivance of pla¬ cing the Volutes in a Diagonal View, as exprelled by Figure D in Plates CXXI 1 I. CXXIV. is much preferable to the ancient Method of placing them in a direift View ; and their being made of an elliptical Form, is a Means of preventing their forelhortening too much, when viewed very near. The other Parts being obvious, I need to fay no more as to their Particulars; and therefore 1 lhall proceed to the general Parts, wherein ’tis to be obferved, (1) That the Module, bv which they are meafured, is the Diameter of the Bafe of the Column, (a) That the Height of the Pedeftal is 1 mod. 30 min. (3) The Height of the Bafe to the Column ;• mod. the Height of the Column, in¬ cluding its Bafe, Shaft and Capital, 8 mod, 47 min. the Height ol the Capital 18 min. i; the Height of the Entablature 1 mod. 47 min. the Diminution of the Shaft is 10 min. or ; of the Diameter. Mod. Min. 11 zy 10 30 13 CO (Pedeftal and Column The Height of thetColumn and Entablature (Pedeftal, Column and Entablature Plates CXXI 1 I. CXXIV. Two Ionick Frontifpieces, with Impojis, by V. Scamozzi. F1 g use A is an Ionick Frontifpiece, with a femicircular-headed Door, whofe Impolt and Architrave may be as Figure B, or Figure G. Figure C is a good •Entablature, with a fwelling Freeze, to be ufed over Windows, or Doors, as Occafion The Principles of Geometry, 267 Occafion requires. Figure F is another Frontifpiece for a Door or Window, crowned with a circular Pediment. This and the other Frontifpiece are given by Scamozzi, as Examples for Pradice. Figure D reprefents the under View', or Sqfito, of tile Voluta’s and Abacus of the Capital, as alfo the Pofitions of the Volutes, in their Diagonal Situations, whereby the Capital has the fame Appearance in Profile, as in Front. Plate C XXV. Ionick Intercolumnatwns, by V. Scamozzi. The two Examples next the Right-hand are Intercolumnations for Temples or Colonades, viz the upper 1, without Pedeftals, for Temples ; the other be¬ low without Pedeftals, for a continued Colonade. The two other Figures, on the Lett, are Intercolumnations for Arcades, wherein the upper 1 is without Pedeftals, and the lower one with Pedeftals. Their teveral Diftances are de¬ noted by Modules and Minutes, or Parts. Plates CXXVI. CXXVII. The Ionick Order, by M. ]. Barozzio of Vignola. The Module, by which the Members of this Order are meafured, is the Semidiameter of the Column, divided into 18 parts, ot which every Member contains as the Figures to each exprefs. . As to the Compofition ot this Order in general, no material Fault could have been found, had not the Pedeftal been made lb very high in its Die, and that monitions Torus introduced in the Bale to the Column. The Capital is really as good, as any of the ancient Capitals, and the Entablature is a grand Compofition. The Manner of defcribing the Volute has been already taught in Method I. Plate Z, after Plate CV. Figure 111 . is a View ot abide ot the Capital, where the Pillow fills up all the Space between the Lifts ot the fore and back Volutes. Fig. IV. is a View of the Members of the Capital, with their Meafures diverted of the Volutes. Fig. IX. is the Sofito, or View of the under Part of the Capital, which projects beyond the Aftragal ot the Column. The Height of the Pedeftal is 6 mod. of which its Bate is f mod. its Die y mod and Cornice j mod. Vide Fig. IF and VII. The Height of the Column is 16 mod. £, of which its Bale is 1 mod. its Capital 11 parts, and Aftragal 3 parts. Tile Height of its Entablature is 4. mod. of which the Architrave is x mod. i, the Freeze 1 mod. i, and the Cornice 1 mod. j. The Shaft is di- minifhed 1 part of its Diameter at the Bafe. x Mod, CPcdeftal and Column i The Height of thePColumn and Entablature 11 >■ ['Pedeftal, Column and Entablature 17 J Plate CXXVIII. /In Ionick Arcade without Pedeftals, by M. J. Ba¬ rozzio of Vignola. The Diftance of the Columns in this Arcade would have been better, had he placed them 1 mod. more diftant, to have admitted of the Pilafters being made 1 mod. in their Diameter, inftead of half a Module, which is half a Mo¬ dule too narrow, and which, being compared with the Diameter of the Co¬ lumns, have a very bad EfTed ; the Architrave of the Arch would have been alfo of double the Breadth, and confequently more noble. Therefore I advde, whofoever follow this Matter, to make the Intercolumnation 1 mod. greater than here reprefented, which give to the Pilafters equally, and encreafe the Architrave the fame alfo. Plate The 'Principles of Geometr y. 268 Plate CXXIX. An Ionick Arcade with Pedejlals, by M. J. BaroZ- zio of Vignola. The Pilafters under the Impofts of this Arcade being equal to 1 mod. or Semidiameter of the Column, have a much more noble Afpedt, than thofe of the lait Plate; and may be taken as a good Example for the Practice of the young Student. Plate C-XXX. Ionick Intercolmmations for a Colonade, by M. J. Ba- rozzio of Vignola. The three upper Columns reprefent the Intercolmnation for Columns in a 'Portico , or Colonade, whole Diftances are denoted in Modules and Parts. In the lower Part of the Plate, on the left Hand, are the Intercolumnations for Colonades, or 'Portico s, by Sebajlian le Clerc ; and on the Right arc two Kinds of Arcades, the one with tingle Columns, the other with Columns in Pairs, by the fame Mailer, whole feveral Meafures are fignified by the Module or Dia¬ meter of the Column, divided into do Parts or Minutes, as in his preceding Orders. Imufl here advertife , that thefe Intercolumnations of Le Clerc are brought on before their Place, which was occafioned by that Part of the Plate being void, and thought convenient to be filled therewith, and more efpecially, as that the Ionick Order of this Matter is reprefented in Plates CXXXV 1 I 1 . CXXX 1 X. CXL. CXLI. CXLII. which are very near unto it. Plate CXXXI. The Ionick Order, l>y S. Serlio. This Matter gives us two Examples, as Figures A and B, the one having a flat Freeze, with Dentils only; the other with a fuelling Freeze, with both Dentils and Modillions ; which laft, as Vitruvius obferves, is abfurd, as that Dentils are peculiar to the Ionick , and Modillions to the Corinthian. As to the Entablature with Dentils only, which is exprellcd on the left Hand at large, there is not any Thing 111 it that is very good, the upper Cima recta being much too large for the Cima reverfa under it, as indeed are the Den¬ tils to the Cima recta under that: And, as the Tenia of the Architrave hath a confiderable Projection, the Height of the Freeze (which in itfelf, I think, is much too low) is made to look very low. The Capital is truly ancient, as its Bale is truly ugly, with its great Torus and fmall Aftragals. The Height of the entire Order, divided into 9 equal parts, the lower a is the Height of the Pedeftal ; the remaining 7 parts divided into 5- parts, the upper 1 is the Height of the Entablature, the lower 4, the Height of the Column ; the Height of the Column divided into 8 equal parts, take 1 for the Diameter of tlic Column ; the Height of the Pedeftal divided into 8 equal Parts, (as on the left Hand) give one to the Cornice, 1 to the Bale, and 6 to the Die. The Members of the Bale and Capital of the Column being determined by Mo¬ dules and Minutes, I refer you to them for the fame. Plate CXXXI I. 7 he Bafe, Capital, and Entablature of the Ionick Or¬ der at large, determined by equal Parts, by S. Serlio. The Bafe, Capital, and Entablature of this Matter, reprefented in the laft Plate, having their Parts determined by Modules and Minutes, according to Mr Evelyn , 1 lhn.ll here explain Serlio' s own Method of determining their Heights and Projectures by equal Parts. The Bafe, whole Height is equal to the Senndiameter of the Column, is divided into 3 parts, the lower 1 of which, as The Principles of Geometry. 269 as cb, is the Height of the Plinth; the other 1 parts divided into 3 parts the *W er ? dethc Hei S ht of the Torus, and lower x, the two Scotia s and Aftragals. Draw kg, which is the Height of the upper Scotia and upper Af- tiagal, divide it into 4, the lower 1 is the Height of the Aftragal and the upper 3 being divided into 4, give the upper 1 to the Fillet under the Torus and the reft to the Scotia. Draw r s, and being divided into 4, give the un- th 1 ' the T f ft r Sa ’ , a " d thc lower ? t0 the Scotia, and that will complete the Bale. The Capital being defcribed in the next Plate, I come next to the Fntablature, whofe Height is equal to 1 mod. or Diameter, and A m^n of which the Aichitrave is 30 min. the Freeze iz and the Cornice 30 1 The Tenia of the Architrave is a 7 th part of the Whole ; the remaining 4 di¬ vided into ix give y to the upper Fafcia, 4 to the middle, and 3 to the low¬ er Make the Cum ed equal to a 7 th of the Freeze, the Height of the Den¬ tils equal to the middle Falcia of the Architrave, its Ctma reveda to a yth of itlell ; the Height of the Corona, and thc Cima r ever [a over itTare alfo equal ^ afCla ’ and r which bc '”g divided into 3, the upper 1 is the Height of the Lima reverfa ; the remaining Height, which is alfo equal to he Height of the middle Fafcia, and ; part thereof, is thc Height of the great Ctma reel a and its Regula. The Projedtion of the Cima rever la over the riovin'? 1 t0 ’ ts He, ght, as alfo is the Projeflion of the Dentils before nlnfl h r I" ST’ cqual tothe Height. Divide the Projeftion of the ^efwe the Fdlet into 6 equal parts, then take the outer 3 for the B eadth of a Dentil, and the next z for an Interval between ; by which fet out all others. The Projeftion of the Corona is equal to the Height of the nfrheVr r C T™, Clma 'i° 7 c,: k t0 their ™n Heights. The Diminution of the Column (which is of 8 diam. in Height) is a 6 th of its Diameter at the C Pedeftal Thc Height of thc)S edeftal and Column ) Column and Entablature (Pedeftal, Column and Entablature Plate CXXXIII. The Ionick Capital, by S. Se Diam. RLIO. I hi A olute of this Capital js defcribed by 6 Centers, as reprefented in thc F.ye ; the Height of the Eye is equal to the Height of the Aftragal, and thc Cathetus is drawn (to mtcrfea it) from the Upright of the Column. The Height of the Capital is equal to i of the Diameter, or xo min. which divid¬ ed into 11 parts, the upper 1 is the Regula or upper Fillet, the next z the Cima the next 4 the Volute, the next 4 thc Ovolo. The Aftragal and its I diet is equal to x parts. The Height of the Lift of the Volute is a yth of its Height. J Plate CXXXIV. An Ionick Portico, and a Colonade, by S. Seri, 10. The upper Figure reprefents a Portico to a Temple, and is very good • the lowei is an arcaded Colonade, whofe Intercolumnation is, I think, rather too Examples ^ ^ 3 VCry grand Thefe aJ ' e » llateCXXXV. CXXXVI. Three Ionick Frontijpieces, by S. Serlio. 1 ig. A is a nifticated Frontifpiece, which would have made a noble Figure had not the natural courfe of the Architrave and Freeze been broken by the Key-Hones of the Arch over the Door, and had another Bafe to the Columns o fewer and larger Members, been introduced, inftead of that monftrous X x x Bale, The ‘principles oj Geometry. 270 I?.. c which he has copied from dtrii'dius, and which the l aigenefs of the ,. caiiic to appear tnnch worfe, than were tire Columns not 1 u ft icatcd. Pi'juie 11 is a good Frontilpiece for a Window, as Fig. C would have been for a Boor, had a~better Bafe to the Columns been introduced. Plate CXWYll. The lonick on the Dorick Order, as in the Theatre of Marcellus^ Rome. i'H is Plate reptefents the Manner of placing one Order over another, where¬ in the Intereoiumn.itions of both Orders are regulated by the intercolumnati- , m proper to the lower Order ; becaufe the central Lines of tire Columns in the upper Order mull Hand perpendicular over the central Lines of the lower Or¬ der And as the Diameter of the Columns in the upper Order mult be made, at the if Bales, equal to the Diameters of the lower Columns at their Altra- '■ais, whereby’the Law of placing Solid over Solid is maintained ; therefore the’Arcades above will either have their Pilafters under their Arches, or the Diameters of the Arches, of greater Diameter than thole under them in the lower Order, excepting when Pilafters without Diminution are tiled inftead of Columns. But of this much more will be demonftrated in the following Plates. Plate CXXXVIII. The lonick Order, by S. le Clerc. This Plate reprefents the entire Order, the Pedeftal entire with the Bafe to the Column, (which is jlttick) tiie Cornice and Bale to the Pedeftal at large, and the Capital with its Volute, which is delcribed according to Method 11 . in Plate Z, following Plate CV. The Module, by which the Members are determined, is the Semidiameter ot the Column divided into 30 min. Plate CXXXIX. lonick Capitals and Entablatures, by S. le Clerc. That we may be the better able how to judge of the different Effects of the antient and modern Capitals, tins Mallei has here placed them together, at Fi gurcs M and N with different Entablatures; Fig. G is yet a more modern Ca¬ rdial than that to the Entablature N, and which, being elevated lomething more than the other, above the Aftragal of the Column, has not a bad Effect, but, I think, is a Grace to the Capital, and gives it a more noble Afpeft. Fig: H is the Seiko of the Ovolo, and Aftragal of the under part of the Capital. Fi¬ gures K and L. are Sofito's of the Dentils, exhibiting two different Ways tore- turn them at a Light Angle. Pig- O and P reprefent the Kev-ftone of the lo nick Arch, that of P in Front, and that of O in Profile. Fig. ACDBEF re- prcprclents the Manner ol dividing the Flutes and Fillets in the Shaft ot the Column, wherein the Jladius AC is divided into 7 equal parts, and the Fillet C D is made equal to + of thole parts. Plate CXL. lonick Capitals , by S. i.e Clerc. This plate exhibits the various parts of the ancient and modern Capitals, in different Views, whole Members are in general determined by Modules and Minutes. Plate CXLT. lonick Entablatures and Impofls, by S. l e C l e r c. This Plate reprefents four Varieties of Entablatures, and fix Varieties ol I m- pofts and Architraves to Arches, all whofc Members are determined by Mo¬ dules and Minutes, as in the preceding. Plate The Principles of Geometry. 271 Plate CXLII. Ionick Arcades, with the Ionick on the Dorick Order, by S. le Clerc. The two uppermoft Figures are two fine Examples of Arcades, as indeed arc the two Examples below, for Galleries, where that on the Left is arcaded with Balufters between the upper Columns, and that on the Right a Colonade with a continued Pedeftal. I11 both thefe Examples you fee the central Lines of both Orders make but one Right Line, as before obferved. Plate CXLIII. Ionick Arcades , by S. le Clerc. T h is Plate reprefents 4 Figures of Arcades, as alfo a Plan of the Sofito of the Cornice, wherein is exhibited the Manner of its Return as well at an internal, as at an external Angle. The Intercolumnations are here regulated as well by the Number of Dentils, between the central Lines of each pair of Columns, as by the Modules and Minutes affixed thereto. Plates CXLIV. CXLV. The Ionick Order, by Cataneo, D. Bar. baro, Viola, L. B. Alberti, P. de Lorme, and J. Bul- lant, according to Mr. Evelyn. Asa little Infpettion will make plain the Difference between thefe Mailers, who in general affeft the monftrous Bale of Vitruvius, I need only add, that the Meafure, by which all their Parts are determined, is the Diameter divided into 60 min. and that their Projections are all accounted front their central Lines. Plate L K, to follow Plate CXXVII. The Ionick Order, by Pall a. dio, Scamoyzi, and V1 gnola, according to Mr. Evelyn. This Plate having by Miflake efcapcd its proper place, I introduce it here oh account of the Members being determined by Modules and Minutes, and the Projections let off from the central Line, according to Mr. Evelyn, as the lix preceding. The Bale of Vignola's is the fame with that of Vitruvius, but Pal¬ ladio has taken the AttickS, ale, and added an Aftragal above the upper Torus. All the foregoing Matters ufed the ancient Capital, but Scamozzi invented a new one, whole Volutes rowl out at the Angles, and make all the foul hides of the Capital appear alike ; this, being better approved of by fucceeding Matters, is called the modern Capital, and is more in Ufe than the ancient. The Bale of Scamozzi has the fame Members -ns Palladio's, but different Proportions, as expreffed by the Figures. Plate CXLVI. The Ionick Order, by C. Perault. Had this Matter been fo happy, as to have omitted the Bafe of Vitru¬ vius to the Column, and introduced lome other of fewer and more propor¬ tionate Members, than thofe poor trifling Altragals, and monftrous Torus, the Compolition of this entire Order would have been very good, and more efpecially, had he made but two Fafcia’s, inftead of three to the Architrave. The Freeze and Cornice are very noble, as the Architrave would be, weie it divided as aforefaid. To proportion this Order to a given Height. (1) Jf for the Order entire, divide the given Height into 40 equal parts, c i. e the lower 8 to the Height of the Pedeftal, the upper 6 to the Height of ' the The 'Principles of Geom e t r y. 2 the Entablature, and the remaining ab to the Height of the Column. The Diameter of the Column is equal to 3 of thole 40 par. (a) If for the Column and entablature only , divide the given Height into 31 parts, give the upper 6 to the Entablature, and lower ad to the Column. (3) If for theVedeftal and (Column only , divide the given Height into 34 parts, give the lower 8 to the pedeftal, and the upper ad to the Column. (4) The Height of the Tedeflal being green, to divide it into its Bafe, Hie, and Cornice , divide the given Height into 8 parts, give the lower a to the Bafe, the upper 1 to the Cornice, and the remaining y to the Die. To divide the Mouldings of the Bafe to the Tedeflal, divide the given Height into 3 parts, give the lower a to the Height of the Plinth, and the upper 1 to the Height of the Mouldings, which fub-di- vidc into 8, and give the upper a to the Cavetto, the next i to the Fillet, the next 4 to the Cima reffa, and the lower 1 to the Fillet. To divide the Height of the Cornice into its Mouldings, divide the given Height into 10, as on the Left, give the upper 1 to the Regala, the next 1 to the Cima reverjd , the next 4 to the T/at hand, the next 1 to the Fillet, and the lower a to the Ca¬ vetto. The Troje&ion of the Cornice to the Tedeflal is thus found, divide the Diameter of the Column into iy equal parts, give 3 of thole parts to the Pro¬ jection of the Plinth to the Bale of the Column, (under which Hands the Up¬ right of the Die) and 4 to the Projection of the Cornice beyond the Die; and if, from every of thole 4 divilional parts, as 1, a, 3, 4, Right Lines be drawn at Right Angles to the Members, they will determine the Projections of the leveral Members, as well in the Bafe as in the C mice, (j) The Height of the Bafe to the Column being given, divide it into 5 equal parts, give the low¬ er one to the Tlinth, the other a being divided into 7, give the upper 3 to the Torus, the next a to the upper Scotia and upper Aftragal, the Remainder to the lower djlragal and lower Scotia , which lub-divide as in the Figure. The Height of the Capital is 1 third part of the Column's Diameter at its Bafe, viz. from the Top of the Aftragal to the Top of the Abacus, which being divided into 11 parts, give the upper 3 to the Abacus, the next 4 to the Folate, the next 4 to the Ovolo. The liune parts being continued, make the Aitragal with its Fillet equal to 3, and the lower part of the A'olutc equal to y more. The Vo¬ lute is deferibed according to Mcth. II. PI. Z, after PI. CV. (6) To divide the Height of the Entablature into its Architrave, Freeze and Cornice, divide the Height into ao parts, give 6 to (he Architrave, as many to the Freeze, and the upper 8 to the Cornice. To divide the Architrave into itsTenia and Fajcia’s, divide the given Height into y, and, giving the upper 1 to the Tenia, divide the lower 4 into la parts, give 3 to the lower Fajcia, 4 to the middle, and y to the upper. To divide the Cornice into Us Members, divide its Height into 8 parts, give the lower 1 to the Cima reverja, the next a to the Dentils and their Aftragal , (which lub-divide into 8) the next 1 to the Ovolo, the next 3 to the Corona with the Cima reverfa, and the upper a to the great Cima recta and its Regula, u hole Projection is equal to the Height of the Cornice. Di¬ vide the Projection into ia parts, and determine the Projection of the Dentils from the ad and 3d, the Ovolo from the yth, &c. Plate CXLVIL The Ionick Order entire, by J. Mau-clerc. Hf.re arc two Examples, the one with a Pedeftal, the other without; fo likewiie the one with a dwelling Freeze, the other with a plain Freeze. The Height of the entile Order being divided into 14 parts, the lower 3 is the Height ui the Pedeftal, and the remaining 11 parts being divided into 10, toe upper a is the Height of the Entablature, and the lower 8 the Height of l g L ', hunn 5 the Diameter of the Column is one 8th of its Height. The o- ther Example on the Lit Hand, without a Pedeftal, the given Height being divided into 7, the Height of the Entablature is equal to 1 part and one 6th; the remaining Height divided into 8 parts, take 1 for the Diameter of the Column. Plate The Principles of Geometr 273 Plate CXLV1IL The Ionick Bafe, Capital, and Entablature at larre by J. Mau-clerc. 6 «* s^T2/as™£“«~ ™ - &£z 5»s,ri ki:” “ ■? “™™ Members, » „„ „,e id Hand „ esh.bS 'S d“S T&&? Plate CXLIX. The Bafe and Cornice of the Ionick Pedeflal together zL’Lit: abm ■ mi >“ w- °/ En- but the Proportion of its Members are not the fame nor are the n ln f eCedl " S ’ fame, that being by Diviiions of a - , ’,1 u Divifions the »■ ^ Capital /this ^Tolc f°T ofZEnJlattZul YoTheArcS Plate CL. Th Ionick , I. J. M r, . 1 he Height being given, divide it into 8 conal mrts rri\ 7 T> 1 B de^u/f'r and thC UPF , er 1 to . theCor ^; £ beii fubdivide°the Height^f^he Bate and Cornice into their refpeftiye Members, as the Divifions exhibrt Plate CLI. The Bafe to the Ionick Column, its Volute, and Impofi by J. Mau-clerc. j i 1 H A ^ efi ^ n of this Bafe bein § Plcwn here at large, is to fhew the . e ‘S Ch .! n ?,‘ tS ^ eulbers » whlch I^eed is very ext, Manner of , u i ------ 0 y~ vviuui inueea is verv extravnenn,-, i are to beoniv ufed m Altar-pieces, or other infide Works. The Volute E A deftnbed accord,r,g to Method 11. Plate Z, after Plate CV. FLureF tr£ Impoit enriched alio. ri & ure p is the Plate CLIP The Bafe and Capital of the Ionick Column enriched by j. Mau-clerc. j iWSMissm&sm merits to thole Member^which ^tobSTjb^S ^ Y y y Plates 274 - The Principles oj Geometry. pi c j |( f CLIV. CLV- CLVI. The Ionick Capital and Enta¬ blatures at large , by J. Mau-ci.erc. These Plates reprefent the various Ornaments, with which this Matter \ niJ the Members of his Capital, and of his three Varieties of Enta- cmbelliihed the Membas M r P ^ magnificcnt Xafte . The Capital in Phte'cL III is very noble and rich, and its extraordinary Height is a very 1 f Addition to its Beautv, and therefore worthy of Imitation. The Enta- BJ e tA . pi., fe CLIV is'alfo very grand, and its Enrichments are truly elegant, provided that ihe Freeze be not enriched according to the fmall Deli"n placed againft it, which would croud the W hole. Plate CLV exhibits a very grand Sofito to the Corona of the Cornice, and not a bad Entablature, altho the great Qmatm n on the Cornice i= want- mT and which, I do really think, was the firit Method of fin.Aung of Cor- n,«s by the Ancients; for, if we rightly conl.der ,he Cnna reverfa on the Corona it is as a Band, or Finifliiug to it; and as the Corona is lo called, which fignifies the Crown , tis very reafonablc to believe, that it was the up- neniioft part of the Cornice, as the Crown of the Head is the uppettnoft part of the Body: Betides, when Pediments are tiled, the Corona is diiengaged from the Cnna refta, that being a part of th. Pediment, and not of the continued Cornice. But however, be that Conjefture of nunc as it will, certain it is, tint the Ornaments of thefe Members are of very great Help to Intent- n. Plate CLVL is an Entablature of very great Majefty, and fine Defign, to be ufed in the Infides of Hills, Saloons, State-rooms , die. Plate CLVII. CLVIII. The Ionick Order , by I. Jones, in the Royal Chapel at Whitehall. The Banquetting-hmle at Whitehall, now made one of his Majefty's Roy¬ al Chanels was built by this Mafter in the Reign of James I. Its Height withinfide, as well as without fide, is divided into i Heights of Orders, that s to fay the lower Order Ionick , and the upper Order Compofile. The lo- nick is here reprefented very accurately in Plate CLv lb and its Intercolum- nition in Plate CLVIII. wherein the Heights and Projeaions ofeveiy Mem¬ ber are expretPed by Feet, Inches and parts. Plates CLIX. CLX. The Ionick Order, by I. Jones, in the Royal Chapel at Somerfet-houfe. As 1 have already exhibited in Plates XCII. and XCIII. the Donck Order of the Skreen, in the Royal Chapel at Somerjet-hmje, I fliall he ^ re ptefent the Ionick Osier in the Altar-piece, and the Altar-piece alfo. llate CLL . contains the Profile of the Order, and Plate CLX. the Altar-piece; wherein the Members of the Order, and its Intercolumnation, aie determined, and ineafured by Feet, Inches and parts. Plates CLXI. CLX1I. The Ionick Order, by Sir C. Wr EN This Matter of immortal Memory has given us a fine Example of this /- omck Order in the Church of St. Mamies, at the Foot of London ■bridge whole Profile is reprelentcd in Plate CLXI and Intercolumnation in Plate CLXII. wherein the Members of the Order, &c. are meafured by Feet, indies and parts, as the Figures affixed thereto exprefs. Plate The Principles of Geometry. 275 Plate CLXIII. The Ionick Order by Mr. Gibbs. This Matter has very judicioufly excluded the monftrous Bafe of / ilrtraus which almolt every of the foregoing Mafters have followed and has divided his Architrave into two Fafcia’s only, which is more agreeable to tins Oidei, than three, of which I have already taken notice. 1 he At tick Bale hcie in¬ troduced 1 cannot think to be fo proper to this Order as it is to the Corinthi¬ an, and therefore I would recommend, m its Head, a Bafe confiding-of a Plinth a Scotia, and a Torus, with their proper Fillets for Separation. The Height of the Plinth to be one third of the Height of the Bafe, that is, of the Senn- diameter ; the Height of the Scotia, and of the Torus to be each the fame, abating the two Fillets, one out of each, whofe Height mil!! be one fixth pait of then' Height. This Bafe would I fubftitute for the lomck Bale, and em¬ ploy the At tick Bafe to the Corinthian Order only. To proportion this 0 / del entire to a given Height 3 divide the Height into y equal paits, an to i\e tic lower 1 to the Height of the Pedeftal; and the other 4 parts being divided into s parts, give the upper 1 to the Height of the Entablature, and the low¬ er 4. to the Height of the Column ; which being divided into 9 c< l ual P arts > take 1 for the Diameter of the Column, whole Bale hath its Height equal to the Semidiameter of the Column, and Capital to one third of its Diameter. To divide the Height of the Tede’ftal into its Bafe, Die and Cornice, divide the given Height into 4 equal parts, give half the upper 1 to the Hug t the Cornice, the lower 1 and one third of the next to the Height of the Bale, and the Remainder to the Die. To divide the Height of the Baje into its Mouldings (1) The Height of the Plinth is one fourth part of the whole Height of the. Pedeftal, and the Remainder is the Height of its Mouldings. ( 1 ) To divide the Height of the Mouldings, as Fig. H Plate CLXIX divide the Height into z parts, and each part into 4 ; give the pwer Fillet fitting on the Plinth 1, the Lima rebta +, the Aftragal 1, its Fillet halt ot one and the upper 1 and half to the Cavetto. To divide the Cornice into its Members, divide the Height into 3, and each into 4 ; give the uppei 1 to the Regula, , to the Chna reverfa, 3 to the Plat-band, and as many to the Ovolo, the vr , to the Altra'gal, and the Remainder to the Fillet and Cavetto. fo divide the Ba/e of the Column into its Mouldings, Fig. DACFEB, Plate CLXIX divide the Height into 3, the lower 1 is the rteight of the Krnth then the remaining z being divided into 9, give the lowermoft 3 to the Height of the lower Torus, the upper z ; to the upper 2 orus, and the others to the tl S : pijSn of the Bafe to the Pedeftal is equal to the Height of its Mouldings on the Plinth, and the Projection of the Cornice is the lame. To determine the TrojeBions of their Members , divide the whole Projection into 8 parts, as between the Figures G, H, and from thence determine each Mem¬ ber as there exprefled. The Projeftion of the Die of the Pedeftal is equal to tb it of the Plinth to the Bafe of the Column, which is equal to two thirds of the Diameter ot the Column, whofe Height is 9 diam. and Diminution one fixth part ot its Diameter. Plate CLXIV. The Ionick Capital, by Mr. Gibbs. This Plate exhibits three Figures, as firjl, the upper one, which reprefents a direct View of the Capital ■ Jecondly, the middle one, which is a l lan, re- prelcnting , quarter part of a Column on the left bide, and a quarter part of a Pilalter, or fquare Column, on the right Side Between t ide z Figures is placed a Scale of 1 diam. of the Column at its Bafe, which hath 1 half pait divided into 6 parts; and which, being continued both ways equal to 3 parts on each Side, determines the Projection of the Capital. To dCj tribe the Capf 276 The Principles of Geometry. f 1 ’, dra ' v , its cclltral Li,le > and another, at Right Angles [crofs-wife as Mr. Gibbs, in his low I .anguage, terms it) for the upper part of its Abacus. The Altitude ot the Capital from A to B, the lower part of the Volute, is equal to the Semidiameter of the Column, which divide into 5 parts, and give the" up¬ per t to the Abacus, which divide into z, and the upper 1 into 4 of which give the upper 3 to the Ovo/o, and lower i to the Fillet. Divide the remain¬ ing part C !i into 8 parts, give the upper a to the Fafcia of the Abacus the next i to the Ovolo, the next 1 to the Aftragal, and the next ; to its Fillet. 1 no A olute, being the next Work, is defenbed at large in Plate CLXV to which 1 refer you. ‘ ‘ Hm in,; thus done with the Capital in its direft View, I mall now proceed to flicw how to defenhe the Tlan of a Capital. (1) As the Shaft of the Co iumn at its Aftragal is dimimfhed one fixth part of the Diameter at its Bife therefore (if your Column be circular) rieferibe a Circle, whofe Diameter is equal to of the Diameter at the Bafe ; or, if a Pilafter, deferibe a geometri- C ^m. <1Ua1 / ° f ^ he , la !? e P lameter - (2) Take the Projections of the Fillet the sl/lraga( y and r he Oyo/o, and their refpective Diftances from the Upright of tic Column ; fet oft from the Circle, or Square, reprefenting the Planof thc Head of the Shaft; and, through thofe Points, defenbe Circles, if to a circu p a ians°SZfrMe q mbers. ‘‘ t0 * ^ Column ’ and they " lU the The Number of Flutes are H , which deferibe as follows: Divide the Cir- cunifeience of the Shaft into 14 equal parts, which will be the Centers of each Flute tins done, divide each j part into 4. parts, and on the Centersaforefaid with the Radius of 3 of thofe 4 parts, deferibe the femicircular Flutes which wil! leave z 4 .Fillets between them, whofe Breadth will be equal to - of a Flute thSnivn ,rtS Th th w )VOl °?r ° f thc fame Nuniber > and flwuld anfwer the fame Divii ons. The Flutes of lquare Columns, or Pilafters, are the fame as m round Columns. J 3 ' '£'??n ex P lal ?, l , hls > f° r 1 uevef could he informed hr any, why the Side of a 1 ilafler mu/l have 7 Flutes, and no more. Mr. Gibbs fin his New Rules of Di a Wing, Fol. 17, Jays, The Flutes of Tilesfters, or fquare Columns mull he the fame as in round Columns , 1 which will make 7 in Number divided from the middle, with a Remainder of * the Corner But real£Sir leant understand hts language , his Terms being very uncommon: He lays they mu/l be Jo , but why they mujt be Jo, and why they are Jo, Imu/l defire you to demon tZr hed ° tbn °‘ meantheAn S^ when he Jays, thc M. I 'vi el make all thefe cafv to your Underftanding, and his Terms alfo • you muftnot cnt'cahy read this Mafter, not be angry if his Terms be a £e as he no! “ ' n 1 ^ ^ mally Pla “ S of his New R ^ of Drawing as he only can call them ; therefore, when he fays, you mult draw a : for the middle of the Capital, and another crofs-wife for the upper Tart of it drawn r / + ’ “ T bc £ailly ie^ufe o„fu„e may diaun aojs-wtfe to another Line, as well at Oblique, as at Right Angles vet uu are always to underftand, that, when he lavs 'crofs-wife, he means ’ (t ho’ peakn:e°of e a n c en0USh l t0 P,y) at R 'S h t Angles : So like wife, when L is fpeakmgof a Bead. “““ *" ^ and aftragal, when he is of N °pLf 0 H e P r rpoft ' 35 the Breadth ° f ever y Fillet is equal to one third . lime, therefore we may account every Flute with its Fillet as j parts- • id as thcie arc 14 Flutes, and as many Fillets, therefore they togethe/make M l-iits, in tne whole Circumference of a circular Column,' which are the Sth^oTe ;^^ ld Vr e r Uft C ° nCeiVe thc Circumference ol eve?y find h w l v „f S' T 1S l,ndcrft00d > we “* in the next Place f N man > ot fuc!l parts are contained in the Diameter • which mav h ■ found near enough for our Purpofe by this ’ umch n,ay bj A N A- The Principles of Geometry. 2 77 4 N A L 0 GT. As 22 is to 7, fo is 96, the parts into which the Circumference is fuppofed to be divided, to a fourth Number, which is the Number of parts 111 the Dia¬ meter required. E X A M P L E. ir : 7 :: 96 ■ 3 ° rt 11)672(30 66 11 remains, equal to T ; This fourth Number, 30 rf, being the Number of parts in the Diameter, We muff now conceive this Circle to be circumfcribed by a geometrical Square, whole Diameter is equal to that of the Circle, and its Side to the Diameter; and as one Side of this Square contains but 30 it parts, and as 1 Flute, with its Fillet, contains but 4 parts, ’tis plain; that if 30 T f be divided by 4, the Quotient is 7, and i 7? remains. Now, as 7 Flutes mult be included’between 8.Fillets, therefore take 1 of the parts from the 1 parts and T ; remaining, and give it to the 7 Flutes and 7 Fillets, and then there is but 1 part and ,1 re¬ mains, which is to be divided into 2 parts, and each half placed at the Ex- trearns, or Angles, to be worked into 1 three-quarter perpendicular Cylinders, commonly called Beads : And thus have I demonftrated to you, why a Pi- lafter muff be divided into 7 Flutes and 8 Fillets, as you required. Now I'll return to finifh the Plan of the Capital, whole central Line being the fame with the central Line of the Profile, compleat a geometrical Square, fo that its Sides be fo far dilfant from the Center of the Plan, as the Projedtion of the Abacus is from the central Line of the Capital. Divide each Side of the Square into 11 parts, and the 10 inward Parts will be the Extent of the Arch of the Abacus, whole Center D is at an equilateral Diftance. Draw GG pa¬ rallel to F F, fo that the Diagonal Line of the Capital may bifedl G G, and make G G equal to half F F, and drawing the Lines F G, F G, bifedl them to find the Divifion of the 2 Members of the Abacus. The greateft Projection of the Volute L, Fig. 3, [Jailsplumb, faith this Mailer, with the lower part of the Abacus, that is,) having the fame Projection, as the lower part of the Abacus, is therefore perpendicularly under it. Plate CI .XV. Mr. GibbsT Rule for dra wing the fpiral Lines of the Scotch Volute to the Ionick Capital. (1) Divide the given Height A B into 8 parts, and make the Breadth C 3 A equal to 7 of thole parts. The Height of the 4th part, in the Line B A, is the Height of the Eye, which divide into 2 parts, with the Line m 12 continued out at pleafure. (2) From 4, the 3d part in the Line CA, eredl a Perpendicular, which will cut the Line m 11 in the Center of the Eye, on which deferibe a Circle, making its Diameter equal to the Height 3 4; wherein infcribe a Square, whole Diagonals (hall be in the two Lines aforelaid, whofe Interfedlion is the Center of the Eye. Draw the Diameters of the Square through its Center; and divide each Semidiameter into 3 parts, as are expreffed more at large in the lower Figure, where the Centers are marked i, 2-, 3, &c. From thele Centers draw Lines parallel to the Diagonals of.the Square extended on each Side ; then, on the Center 1, with the Radius l i, which this Mailer calls Length, according to the Scotch Mode of Z Z Z Speech, 273 The ’Principles oj Geometry. Speech, deferibe the Arch n; alfo on the Center a, with the Radius ax, del'cnbe the Arch a 3 ; and on the Center 3, with the Radius 3 3, deicribe the Arch 3 4. Proceed in like Manner at the remaining Centers, which will com- ,,l ca t the fpiral Line at the upper Point of the Eye. The inner ipiral Line is parallel to the outer from 1 to 3, and therefore is defenbed on the fame Centers; the other Centers are exprefted by the refpective Letters as b, c d f e h &c. The Breadth of this Fillet is equal to one 16th part of the w hole PTeight cf the Volute. Thus have I given you this Rule for to deicribe the Scotch Volute, which hath a very difagreeable and clumfy Diminution, not to be prabtifed by any. Plate CLXVI. The Ionick Entablature by Mr. Gibbs. The Altitude of the Entablature being found, divide it into 10 parts, give the lower 3 to the Architrave, the next 3 to the Fieeze, and the upper 4 to the Cornice whole Projection is equal to its Height, and divided into 4 parts, from whence’the Projection of the Corona, &c. is determined. The particular Members of the Architrave aud Cornice are deferibed more at large m the iol- lowing Plates. Plates CLXYII. CLXVI1I. CLXIX. Ionick Cornices, Arcades and Impofis, by Mr. Gibbs. Thf. Figure A IB, Plate CLXIX. reprefents the Architrave at large, divid¬ ed into 3 parts, of which the lower 1 is the Height of the lower Fafcia, and its Lima > ever fa is a fourth part thereof. The Tenia is three fourths of the up¬ per 1 divided into 3 ; the upper 1 is the Fillet. 1 lie Projection of the lema is equal to its Height; which Projection being divided into 3, the firit 1 gives the Projection of the upper Fafcia. T h f lower Figure in Plate CLXVII. reprefents an Ionick Modillion Cornice, whole Height is divided into 4 principal parts, and thoie lubdivided again for the Divilion of the fmaller Members. It is to be obferved, that, when this Cornice is ufed with Columns, it mult have its Modillions 1 of the Diameter of the Column, and the Interval between them :, or of the Diameter, that is, the Diitance from the eentral Line of 1 Modillion to the other mult be equal to the Semidiameter of the Column ; and that being divided into 6, give the outer ones to half oi each Modillion, and the Interval will be the middle 4. The Breadth of each Modillion in Front mult be made equal to x of thole 6 parts, and the Length of a Modillion in Profile mult be equal to 3, or 1 fourth part of the Column s Diameter. The prick d Line A B is the central Line of the Column, over which the xd Modillion muff Hand. The Line I D fhews the diminifhed part of the Column, which toucheth the Side of a Modillion. The Contour, or Out-line of the Modillion in Profile is deicribed by 3 Centcis, thus ; its Projection being divided into 6 parts, erect Pcrpcndiculais el x and 5- • the firit Center will be at a, with the Radius x 1 ; the fecund, 1 and a half below it, and the third x and a half above the Point y. The Projection of the Cnna severja over it, which is the Cap of the Modillion, is exprdled by a dot¬ ted Square at the End of the Modillion. To divide the [queu e Tannels in the Sofi'to 0] the Cornice , divide the Space between the Caps of the Modillions into 6 parts, as they are numbered; take 1 on each Side for the Border, and the other 4. will remain for the Panncl. Divide out the fame Divifions on the Profile of the Corona at E, and bilect the remain¬ ing part for the Drip. The w hole Projection being divided into 4, as in the Scale underneath, lubdivide, and determine the Projections of the Members, as there reprefented. . , The upper Figure, next over the Modillion Cornice, is an ionick Den¬ til Cornice, whole Height being divided into 4 principal parts, as before, and then The Principles of Geometry. 279 then fubdivided, the Height of every Member is moft eafily determined. It is to be obferved, that when this Cornice is to be ufed upon Columns or Pilafters, the Dentils inuft be exadtly divided by the Semidiameter of the Column as followeth. Suppofe the Line A B to be the central Line of a Column, divide the Semidiameter of the Column into iz parts, then make the Breadth of each Dentil equal to two of thofe parts, and the Diftance between each 'Dentil equal to one of thofe parts. The central Line of tire Column BA, rauft pafs dircdtly through the midft of a 'Dentil. The Line CD reprefents the Upright of the Column continued, from whence the Pro¬ jection ol the Cornice is accounted, and its Members are determined by its Scale, whofe Length is equal to its whole Projection divided into 4 Parts, and fubdivided again as in the Figure exprelfed. It is alfo to be further obferved, that in cafe this Cornice is to be ufed without Columns in Rooms, or over Doors, Windows, gfre. to find the Magnitude of the Z dentil, divide j of the Scale of the Projection of the Cornice into 6 Parts, then take the Length of 7 fuch parts, and divide it into y parts, then take z of them for the Breadth of a Dentil, and 1 for an Interval, as may be feen by the Scale CC. The other two Figures on the 1 ight Hand Side are lonick Arcades, the upper one without Pedeltals, the lower one with Pedeltals; whofe Intercolumnations are determined by Diameters and parts. The Import to thefe Arcades, is re- prel'ented by Figure A, and the Architrave by Figure B, in Plate CLXIX". on whofe upper part, is an Iimick triumphal Arch, or Gate, of very good Defign. Plate CLXX. lonick Intercolumnations , with lonick Frontifpieces , and the lonick Order, on the Dorick, by Mr. Gibbs.. Figure B reprefents the Intercolumnations for Portico’s, or Colonades, and thole of A and C, for Frontifpieces to Doors or Windows ; whofe feveral Diltances are determined by Diameters and parts. Figures D, E, F, reprefent the lonick on the Dorick Order, and the Dorick on a indicated Baiement, wherein the only Thing to be regarded is, that the fame central Lines be common to the Columns in both the Orders, that void be over void, and lolid over folid : To which 1 mult alfo add, that 'tis abfolutely neceflary to place a Sub-plinth under the Plintlt of the lonick Pedeltals, to raife them higher, lo that they might be feen at a tolerable near Diftance from the Building, which they cannot be as they are now' placed ; becaufe the Projedtion of the Cornice will eclipfe the Plinth. The Like lhould be alfo obferved with the Plinths to the Bales of the lonick Columns, whofe Heights are in great part eclipfed by the Projection of the Cornice to the Pedeltals, The little rufticated Window F, placed in the rufticated Arch, makes a very poor Figure, and its Architrave being broken by the Rultick Blocks, feems to be more of the Invention of a Blockhead, than of a real Artift. Plate CLXXI. Venetian Windows , by Mr. Gibbs. This Plate reprefents to our View two Deiigns of Windows after the Vene¬ tian Manner, the lowermoft of the Dorick, the uppermoft of the lonick Order, xvhofe Intercolumnations are exprelfed by Diameters and parts. I inuft here beg leave to obferve, that when thefe Kind of Windows are let in Fronts, where greater Columns are made ufe of, as in the Chancel End of St. Martin s Church in the Fields, nothing can be fo Blocking; lor the large Co¬ lumns and their Entablature, being feen at the fame Time with thefe irnall ones, they caufe them to appear much fmaller than they really are, and con- i’equently they cannot lail of having a very poor Effect. To remedy this, I would recommend, that an Import only be ufed, inftead of an Entablature, placing 2 g Q The ‘Principles of Geometry. niacin" Pilafters without Capitals under them; nor would I give them any Bile Yavc that of a Plinth on the Window-itool. To proportion a M in clow of this Kind, I would divide the given Breadth into iz parts, ol which Id give r to each of the Pilaltcrs, a to each of the lide Apertures, and 4 to the middle Aperture The Height of the middle Aperture, unto the Topol the Import:, from whence the Arch fprings, I'd make equal to twice its own Diameter ; and that Height being divided into y parts, I'd take the upper 1 (which now is made anVntablature) and divide it into an Import ol Dontk Compobtion, and then the Pilafters would conlift of 8 diarn. in Height. A Window thus com- Poled would be verv light, and airy, and an Ornament to a Front, inrtead ol an Eve-lore, where large Columns are introduced ; lor then there would be no Comparifons made of a fmall Kntabiature and its Pilafters with the greater. Plate CLXXII. An lonick Dentil Cornice , fupported by TruJJes, fir Doors , Windows, &c. by Mr. Gibbs, When we arc to place Cornices over Doors or W indows, to be lupportcd bv Tallies only, and to make inch Cornices of this Order, we mult iirlt nnd the proper Height lor Inch Cornices, before that we can proceed to the Divih- on of their Members. Now, to do this, we mult divide the Height ol the Window into y parts, and let up 1 of thole parts above the Height ol the W in- dow, which will give us the Limit, or Top ol the Cornice. 7 "0 .find the Depth of the Cornice, divide the Height between the Head ol the Window and lop of the Cornice into 10 equal parts; of which take the upper 4 for the Height of the Cornice, and the other 6 being equally divided, give one hall to the Freeze, and the other half to the Architrave. The Proceedings thus made, is the fame, as if Columns were to be placed under the Entablature, inftcad ol Trlilies ; fo that the next Thing to be found is the Depth and Projection of the Tails, which is as follows: (1) The Height ol the upper Volute is equal to the Height of the Freeze, which divide into 7 parts, (i) f he Face ol the Cornice being deferibed, let the Height of the Freeze from 1, the under part of the lower Cima reverja, to 8 r-~, and lrom the Point 8 e-, draw the Line 8 t- DO o, which is to be taken for the Upright, or Face ol the Limiting, a ". gainll which the Cornice is placed, and the Diftance 1 8 t- is tnc Projection or the Cima r ever fa , under which the Trufles are placed. I he Diftance, from the under part of the upper Volute, to the upper part of the under Volute, is equal to 4 parts ; the Height of the Cornice, and the Depth of the lower Vo¬ lute, is equal to z parts, or hall the Height ol the Cornice ; as alio is the Depth of the Leaf underneath it. The Heights of the principal parts being now determined, proceed to deferibe the Volutes, as follows: (1) ihe Height of the Five of the upper Volute is equal to 1 feventh of the Height ol the Freeze, whole Center is found by an horizontal Line, drawn from D, the 3d Divilion, on the upright Line, and another cutting it at Right Angles, drawn lrom the middle of the 4th part, under the Cima r ever fa. (z) Continue out the Bottom of the Cima reverfa equal to 1 z, and from thence draw a pricked Line, paral¬ lel to the Upright of the Building, for to terminate the levelling, or project¬ ing part of the Volute. This being done, delcribe the P.vc ol the V olutc, where¬ in inferibea Square, and divide its Diameters, each into 4 parts, as reprcfcnted at the Bottom ol the Plate, where the Eye is fhewn at large, for the better L'nderftanding of its Diviiions, drawing from every Center the leveral horizon¬ tal and perpendicular Lines, as are to determine the Quantity oi each Arch ol the Volute; and on the refpettive Centers del’cribing the Arches r z, z ;, 34, fyc. until the Volute is compleated. The Height of the lower Volute being before determined, divide its Height into ~ parts, and make its Projection A B equal to 8 of thole parts. The Height of its Eye is equal to 1 ol thole parts, and Hands over the 4th horizontal Divilion, with its Center exactly again!! the 4th perpendicular Point. The Centers of this Eye are found in the very fame Manner, as thole of the other. The Diameters of the Circles, in which The Trinciples of Geometry. 28 1 the Rofes are comprifed, are each equal to t of each Volute’s Height. To find the Curve of Communication of the two Volutes, draw the Line CA from 1 part below the Line G G, and make A C equal to A B, and draw the Line D C, which bilea ; then, on the Points C and D, with the Radius i D or CG, de- feribe the two Arches, and their Concentricks, which will complete both Vo¬ lutes. The Breadth of the Front or Face of the Trufs mu ft be equal to 7 pths ol the Height of the Freeze, and, it being divided into - parts, give 1 to each ol the Fillets L, 1 to the Aftragal and its Fillets H, and 2 to each of the Ci- ma recta s on its Sides, file fide Projechire of each Rofe, between the Plain of the Trufs, is equal to the Breadth of a Fillet. Plate CLXXIII. The Safe and Cornice to the Corinthian 1 Pedestal, geo¬ metrically deferibed, by C. C. Os 10. Before we can proceed to deferibe the Mouldings in the Bale and Cornice of this Pedeftal, we mult find their Heights, and, in order thereto, we muft have the Height ol the Pedeftal given, which well fuppofe to be AD, Fig. II. To find CD the Height of the Bafe, and AB the Height of the Capital, (1) Draw the Lines e a and / k, through the Points A and D, at Right Angles to AD. (2) Make the Angle f DA equal to 50 deg. bileft fD in g, raife the Perpendicular g f, cutting A I) in f ; draw the Line a f I, lb that the Angle Af a be equal to 30 deg. from the Point h raife the Perpendicular h k, to cut l k in k ; divide the Angle h k l into 2 equal parts, bv the Line 1 k cut¬ ting AD in C; then is CDthe Height of the Bale: Allb divide the Angle Aa f into 2 equal Parts, by the Line a d B cutting A Din B; then AB is the Height of the Cornice. The Height of the Bafe b A, Figure 1. being given, to divide it into its Mouldings . Continue outthe Bafe towards r, make the Angle A hr equal to 30 deg. bifeft b r inq, raile the Perpendicular q 2, then is 1A the Height of the Plinth. Make 2 0 and A p, each equal to 2 b, and draw 0 p the Face of the Plinth. Draw bo, and divide b 2 into 8 parts, at z, y, x, w, v, t, s; then is .v 2 the Height ol the Torus, w x the Fillet, s tv the Lima recta, and b j- the Aftragal. Bifedt/ 1 in z, draw z m parallel to 0 2, cutting b 0 111 m the Cen¬ ter oi the Iorus ; on / ereft the Perpendicular f h, cutting the central Line of the Aftragal h e in h, the Center of tire Aftragal ; from h, draw h k parallel to b 0, cutting k x in k ; make i tv equal to k x , and draw i k the Face of the Fillet; draw t g, and thereon deferibe the Cima recta ; on the Center h, with the Radius hg, deferibe the Aftragal ; make a b equal to be, and draw ca e- quat to twice a b ; draw c d, and the Whole is completed. 7 he Height of the Cornice a 2, Figure III. being given, to divide it into its Mouldings. Dr aw e a at lit. Angles to a 2 ; make the Angle e 2 a equal to 30 deg. and draw e z, cutting ae in e, which is the Projection of the upper Lift, or Regu- la- Make a v equal to a e, and draw e v; make ek equal to f of e 2 ; draw b k and k 1 continued to a 2; make b d equal to } of b k, and thro’ d draw / d to the Line a z, e f for the Face ol the Regula ; allb bifett k 1 in h, draw g h, and thereon deferibe the Cima reverfa. Bilett k 2 in 1, on 1 raife the Perpendi¬ cular 1 s, bilea sv in t, make sr equal to j t, from r draw l r, and from i draw 1 L the Face ol the Plat-band ; from t draw t z cutting e v in w, from whence draw w 4 parallel to a z ; through x (where the Perpendicular 11 cuts the Line ev) di.aw qz parallel to a z, which bifect iny the Center of the Aftra¬ gal ; from the Point 2 draw the Line 2 7 at Right Angles to a z, cut¬ ting the Line tv 4 in y; make 6 y equal to y 2, and through the Point 6 diaw the Line 8 6 parallel to 72 ; make 7 y and 8 6 equal to 4 3, then bilect 4 A the a8'2 The ’Principles oj Geometry. the Diagonal 7 6 for the Center of the Aftragal; the Height of the Fillet y 4 - is equal to half 6 y, which completes the Cornice as required. Plate CLXXIV. The Corinthian Pedefial entire, according to the An¬ cients, by C. C. Osio. The Figures of the lalt Plate having taught the Manner of dividing out the Members in the Cornice and Bafe of the Pedeftal, after that their Heights were found) the Manner ol which being allb given theie, we are now to find the Diameter of the Die of the Pedeftal, fuppofmg that ’tis to be ufed alone for the Support of a Statue, and the Projection of the Plinth to the Bafe of its own Column is unknown. Now it is evident, by the Semicircle deferibed on the Center b , in one of its Sides, that the whole Altitude of the Die, is equal to twice its Diameter; therefore having divided the Height of the Die, on its central Line, into 4. equal parts, fet off 1 of thofe parts on each fide of the cen¬ tral Line, and draw the Out-lines of the Die, as required. Plate CLXXV. The Bafe of the Corinthian Column, according to the Ancients , geometrically deferibed, by C. C. Oslo. According to this Mailer, we are to underftand, that the Ancients gave but 9 diam. and half to the Height of their Corinthian Column, of which they gave the i to the Height of its Bafe, as yz in Fig. 1 . The Members, into which they divided this Bafe, were the upper Torus B, the upper Scotia D with its Fillet C, the upper Aftragal F with its Fillet E, the lower Aftragal G with its Fillet H, the lower Scotia 1 with its Fillet K, the lower Torus L, and the Plinth M: Wherein 'tis to be obferved, that, excepting the lower Torus, all the reft is no more than the Ionick Bafe. The Cindture A is a part of the Co¬ lumn, not of the Bafe, as I have long fince declared. To divide d 15, the Height of the Baje, into its Mouldings, proceed thus: (1) Draw the Lines gd and zy z3 at Right Angles to d 23. (1) Make the Angle zy d z3 equal to 30 deg. drawrfzy, which bile ft in 18, on which Point ereft the Perpendicular 18 zi, and from zi draw zr zz, cutting d zy in the Point zz ; make zq zg equal to zz zi, and draw zz zq, the Face of the Plinth. (3) Divide d zi into 3 equal parts at r 13, then is 13 zi the Height of the lower Torus; alfo make g d equal to d r, and draw g k, &c. parallel to d Z3, to terminate or limit the Projections of the upper Torus and Jljlragals. (q) Divide d r into q parts, and / will be equal to 3 of them, which is the Height of the upper Tonis ; biledt dl in h. and from h draw h k, cutting g r in t, the Center ol the Torus, on which deferibe the Semicircle f k n. (y) Make r t equal to r l, and draw t y of Length at pleafure; make / m and s t each equal to j of It, and draw the Parallels m 0 and ti s of Length at pleafure. (, Q-c. are each equal to half a Dentil. Make 14 a, the Height of the Atlrag.il, equal to 1 fifth of the Dentil a c whole Fillet op is one third Part thereof The Projection ol the Fillet, before The ‘Principles of Geometry. 28 before the Dentil is equal to the Height of the Fillet, and the Center of the Altiagal is perpendicular over it. Make the Height of the Ovolo 11 24 equal to 1 third of to 14, and draw the Line 23 1 at Pleafure, and parallel too#- continue the Face oi the Fillet to the Altragal op to 1;, which is the Center of thc i raaac 10 11 ec ! lla l to 1 fifth of to 13, and from 21 draw the Line V ’ 0 Length at Pleafure ; divide b g e; make e f equal to 1 filth of e e A the Hetght of the Corona; make to equal to half ef, and thro’c diau l he I auc z c h, then is c / the Ph ight of the Lima reverfa with its Fillet • divide be .the Height of the great Cpiiatium, into 6 parts, and give the upper 1 to the Fillet; make b E equal to b 7, then is b E the Projection of the Cor- mce ; make the Breadth of every Modiiiion equal to their Height, including then 6/7/.V/ reverfa. As the Projections,of the two Cana's are equal to their own Heights, defer ibe their Out-ii; cs, and Irani the lower one draw the Face of the Corona, with thc Cima neverja j the Mcdillions underneath it. To find the Height of the Centers of the Eyes o; the two Volutes to the Modillions make 10 1 equal to the Height of the Modiiiion without the Lima reverfa and from the Point 10, draw the Line 10 9 parallel to the Line 8 1, and draw the Diagonals 9 11 1, and 10 11, interlecting each other in 11, which is the Center of the great Volute; draw thc Line u 12 at 1 third of the Modillions Height, and making u t equal to s u, the Point t is the Center of the l'maller Lye This Mailer proceeds no farther to Ihew how to deferibe the Volutes which will be taught hereafter, by other Malters, in the following Sheets Fi« I. is thc Order entire. Plate CLXXVIII. Corinthian Profiles, taken from the Temple of Jeru- falem, and the Portico of the Rotunda, according to Mr. K v e 1 , y y. f he fiift of thefe is certainly the moil beautiful Order that was ever in¬ vented, its Bafe excepted, wherein there is the fame Abfurditv of fraall Mem¬ bers, as I have already obierved in many of our Mailers on the lonick Order- therefore, to make this Order the Order of Orders , as Mr. Evelyn calls it, we mult give to it the /ittick Bafe, which indeed ought to be uied with the Corin¬ thian Order only. I he other Profile of the Portico of the Rotunda hath the fame kind of Bale, but their Capital and Entablature, which are both very good, are very different in their Leaves, thole of the Temple of ferufatem be¬ ing like unto Feathers, having their Volutes enriched with Palm-branches and the other of Acanthus, with plain Volutes. If we confider the Triglyph in the Freeze of the firft, which is a part of the Portck Enrichment, we are then Ihewn, that in this Order there are all the 0- thcr Greek Orders comprifed ; for, befides thc Triglyph, there are alio the Io- nick Dentils comprifed in its Bed-moulding, and therefore it may be laid to be a Porick, lonick , and Corinthian Compoiition. As to the placing of the Modil¬ lions in pairs over each Column, lean fay but little in its Praife, for, where Joifts are lo placed in a Building, their Strength oi lupporting is unequally di¬ vided. The Height of the Column in both Profiles is 10 diam. and their Mem¬ bers in general are determined by Minutes. As to the Heights of their Archi¬ traves and Cornices, they are both equal, but their Freezes are different, that of the Temple of Jerujalem being yq min. and that of the Rotunda but 43. I mult alio remark, that I think the great Cymatimn of the Rotunda is really Tufcan, as that it hath no Cima reverfa under it. Plate CLXXIX. An Altar in the Rotunda. This Frontifpiece reprefents a Corinthian Altar in the Rotunda, with a circular Pediment, which, being one of the molt finiple and grand Compofiti- ons I ever law, is given here as an Example for Help to Invention. 4 B Plate I ^ •2 86 Tlx ‘Principles of Geometry. Plate CLXXX. The Door of the Rotmula at Rome. The Entablature marked P N, on the left Hand, is the Members at large of the Door, reprefented on the right Hand, and whicn, being a very gland Compofition, is given here as a further Help to Invention. The Bale, placed over the Door, which confifts of 3 Torus’s, 1 Scotia's, a Plinth, and their Fil¬ lets, is from S'erho, and taken from the Rotunda ; but from which part he makes no Mention, and which, 1 believe, is a very proper Bale to an Order, v hen placed very much above the Eye, in all which Cafes the Heights of Mem¬ bers are very much fore-lhortened. Plate CLXXX1. The Corinthian Bafe, Capital, ami Entablature of the Portico to the Rotunda at Rome, deferibed by equal Parts. As many Perlons delight in working by various Methods, this Example is given here for their Entertainment; which is very eal'y to underhand by a very little Infpection, the fcveral Divifions being very plain to the meaneft Capacity, and el'pecially to luch, who have wifely examined all the preceding Plates. Plate CLXXX1I. The Corinthian Order 'within the Rotunda at Rome. This Plate reprefents a Corinthian Column with its Entablature, taken from the lnlidc of the Rotunda, whofe Height being divided into 9 parts, the Entablature poflefles the upper 1, and 6 ninths, or z thirds of the next 1. The Height of the Architrave is T ninths of the zd part; the Height of the Cornice is equal to the Diameter of the Column at its Aitragal, which isdimi- liifhed i of its Diameter at the Bafe; the Remainder is the Height of the Freeze. The Height of the Column is 7 ninths of the whole Height, and one third of one yth part, which is the part remaining below the Entablature. The Height of the Capital is equal to eight yths of 1 of the 9 parts contained in the whole Height; and then, if 7 of the 9 parts of the whole Height, and four yths of the 8th part be divided into 9 parts, 1 of thole parts will be equal to the Di¬ ameter of the Column. Fig. 1 . is the Plan, and Fig. II. a Seftion of the Capi¬ tal at large, divided by equal parts, as therein are exprellcxl. Plate CLXXXIII. Two Corinthian Profiles, taken from the Frdntifpieces of the Bath oj Dioclesian, and of El ero, at Rome, according to Mr. Evelyn. Here we arc again prefented with two of the moll noble and rich Entabla¬ tures, that perhaps have been yet invented, that of ‘Diodefian being enrich’d to Excels, and the other not much Short of it. We are here to obferve, that, altho’ many of the Ancients run into that grofs Error of placing double Altragals between the z Torus’s of the Bafe, yet tis plain, by thele two Examples, that it was not generally, obferved; for here, in the Bafe of Diode'fans, there is but a Angle Aitragal, which, tho' bad, is yet much better, than when the lame Height is divided into two. In the Bafe to the Column of Nero, we find both of them banilhed, it being direftly the Nttick Bafe, which I have all along re¬ commended, and which has an Affinity with the Grandeur and Magnificency of its noble Entablature, whofe Greatneis of Parts excels in Majefty all others, that I have yet l’een. And as its Architrave confifts but of two Fafcia’s, and its Cymatium of one great Cima retia only, with an O’eolo under it, crown its Co¬ rona ; it mult therefore be never ufed, but to the Out-lides of Palaces, whilft the more delicate and rich oi Diode fian is received \vithin-iide, where its beau¬ tiful parts are near to the Eye, which without-fide would not only be loft at diftant The Principles of Geometry. 287 diflant Views, but the tender parts of the Enrichments more liable to early Decay by the Injuries of Weather. The Cima reverja, ufually placed under the great Cymatium of the Cornice, is alfo excluded in the-Cornice of Z Ytodefi- an , as I before obferved it to be in the Columns of the Portico to the Rotunda at Rome. The little geometrical Profile contains the Meafures of thofe of Ne¬ ro , whole Parts, as alfo thofe of Diocleftan s, are determined by Minutes, and their Projections are accounted fi om their central Lines. Plate CLXXXIV. The firfi Example of the Corinthian Order, by Vitruvius. Altho' this great Mailer was happy in the Knowledge of many noble Ex¬ amples of this Order, built by the antient Creeks and Romans, yet he as mad- ly runs into the fame Abfurdity in the Bale of this Order, as he has done in tiie Bafc of his lonick. To proportion this Order to any given Height, divide the Height into 9 parts, give the upper 1 and r 9ths of the fecond to the Height of the Entablature, and then the Remainder being divided into 9 parts (as on the left Hand) 1 of them is equal to the Diameter of the Column, and to the Height of the Capital alfo. The Height of the Cornice is equal to 1 18th part of the entire Height of the Column and Entablature, or to half of the up- permolt 9th part. The Height of the Architrave is equal to the Semidiameter of the Column, and the remaining part of the Height of the Entablature is the Height of the Freeze. The general parts being thus divided, divide the particular parts as the Divilions exprefs. Plate CLXXXV. A fecond Example of the Corinthian Order, by Vitruvius. As the Entablature of the laft Plate doth not contain any Modillions, we have here an Entablature With Modillions; the general parts of which Order are found as follow : (1) Divide the given Height into f parts, and give the upper 1 to the Entablature, (which I think is too much, as being the fame Height as is generally given to the Tufcan and Dorick Entablatures .) (a) Di¬ vide the under 4 parts into 9, give the upper 1 to the Height of the Capital, and half the lower 1 to the Height of the Bale. The Diameter is equal to one 9th of the Column’s Height. (3) Divide the Height of the Entablature into 10 parts, give the lower 3 to the Architrave, as many to the Freeze, and the upper 4 to the Cornice. The Figures C and E. reprefent two Corinthian Capi¬ tals ; that of C having that kind of Leaves which are called Tatjley-leaves, and the other of the Acanthus , or Branca Urfina, Bear’s Foot. This Capital, its faid, was invented by Callimachus, an ingenious Statuary of Athens, by the following Accident: A young L.ady of Corinth being buried, it happened that on her grave grew a Root or Plant of Bears Foot, on which her Nurfe placed a Basket covered with a Tyle, containing the feveral Toys with which foe uled to be delighted ; the Basket happening to be placed direftly on the Plant, its Weight prevented the perpendicular Growth, and compelled it to creep tinder the Basket, (which ’tis reafonable to believe was not very heavy) until its Leaves had got to the outfide, when they changed their horizontal Growth to that of perpendicular, quite about the Basket, as represented by Figure F ; and when the leading Branches of this Plant had grown fo high as to be again obftrucTed, by the covering Tyle, they were then compelled to gently turn about their Extreams, which, together with the Weight of thofe parts, formed themfelves into an eafy free Curve, in Imitation ot which Cal¬ limachus made the Scrolls or Volutes'of this Capital, dreffingthe lower parts with two Heights of Leaves about a Vale, refembling thofe about the Basket, which were of different Heights, and gave it its Abacus, in Imitation of the Tyle, Figure B reprefents the Projeaure of the Leaves, Volute, and Abacus before 288 The 'Principles of Geometry. before the Upright of the Column. Figure D is a Plan of the Capital, exhi¬ biting tlie Curvature of its Abacus. Plate CLXXXVT Yl third Example of the Corinthian Order, by Vitruvius. This thiiil I- sample . winch the Engraver has mift.tkenly called the id) the Bale excepud, c a tine Compofition ; its Capital is open and free, and contains i di.im. in I leiglit cxclulive of its Abacus, which is - min and half more. The Height of the Abacus above the Aftragal being divided into 4. parts, the Heights of the I .caves and Volute are from thence determined, as exprefled in the Figiu c To divide the Entablature into its Architrave, Freeze and Con/ice, divide the Diameter of the Column into 16 parts, of which give 1 1 and thiee q.ths to the Architrave, 11 to the Freeze, and 1 diam. 3 parts and hair to the Cornice, which together make x diain.and 11 parts for the total Height of the Entablature, which is half a Diameter greater than the quarter part of 9 diam. the Height of the Column. The particular Members of the Architrave and Cornice arc fubdivided as exprefled by the equal Divifi- ons. This Entablature, as f before obferved, is of a fine Compofition, and efpecially for the Outfide of Buildings, as that its Members are of grand Di- menfions, and the extraordinary Height given to it, 1 believe, is with refpeft to its Height being much forclhorten’d, when ufed in very lofty Buildings, when Allowances of this kind fliould be always given. I mult own lam in fome Doubt, whether this Cornice is an Invention of Vitruvius, who often protefts againfl the introducing of Modillions and Dentils together, as here is done : But however, let who will lay claim to the Invention, certain it is, that the Compofition is very grand ; and here we are alfo to obferve, that an Ovo- lo is placed under the Cymatium of the Cornice, as in the Example of the Por¬ tico to the Rotunda at Rome. Plate CLXXXVII. Corinthian Intercolmmations, by Vitruvius. In this Plate is reprefented, the Intercolunations of Columns for two Rinds of Temples, the upper one confuting of fix Columns, called 'Pycno/tyte ; the other of eight, whole Plan is reprefented above by a lefler Scale, which are both given as Examples for Praftice. Plate CLX.XXY III. The Temple of | UPITER, by VIT R U V I u s. This Plate reprefents a Plan and Elevation of the Temple of Jupiter, whofe Porticos confift of double Rows of Columns, each ten in Front, which, being one of the moft magnificent Defigns of the Ancients, is given here for a Help to Invention. Plates CLXXXIX. CXC. CXCI. Yl Corinthian Temple, and a Ro¬ tunda, by Vitruvius. These three Plates exhibit two different Defigns, as ,firfi, that of Plate Cl .XXXIX, which is the Delign of a Temple, with a Coridore about it, whole Plan is reprefented in the upper part of Plate CXC. And lajlly, that of Plate CXCI. which is the Delign of a Rotunda, whofe Plan is reprefented in the lower part of PI. CXC. Thefe Defigns being both very magnificent, and fit to adorn the Gardens of the greatejl Prince, are given as Examples for further Help to Invention. Plate The Principles of Geometry. 289 Plates CXCII. CXCIII. The Corinthian Order, by A. Palladio. This Plate reprefents, (1) The Corinthian Pedeltal and Bafe of the Column, in which laft we find two ylftragals between the two Torus's , but not placed together, as in many of the preceding Examples; and indeed I mult confefs, if we mult be obliged to introduce them, that this is much the belt Way. The Altragal, placed on the upper Torus, is, I think, not arnifs neither, as being a gradual beginning of the impreffed parts of the Bafe. (i) The Capi¬ tal and Entablature are, beyond all Difpute, of noble Compofition, and ef- pecially for Infides of Buildings ; but fomething againit the Precept of Vi¬ truvius, as that its Cornice comprifeth as well ‘Dentils as Modillions. The Intercolumnation for Colonades, and of his Arcade with Pedeitals, is very grand, as is the Compofition of the Members in his Impolt and Architrave un¬ derneath it. The Height of the Pedeltal is 7. diam. a 7 min. of the Bale to the Column 30 min. ot the Column 9 diam. and halt, whofe Diminution is one 7th of its Diameter ; of the Capital 1 diam. 10 min. and of the Entablature 1 diam. 5-4. min. Diam. Min. Pedeltal and Column II J 7 Column and Entablature I I 14 Pedeltal, Column and Entablature 13 LI Plates CXC1V. CXCV. The Corinthian Order, by V. Scamozzi. T h e Corinthian Order of this Maftor, who is with many the next belt after 1 Palladio , is reprefented in thisPlate with all its Embellilhments. Fig. 1 . repre¬ fents his Pedeltal and Bafe of the Column, whole Plinth is made curved, for the better difeharging the Rains from the Cornice of the Pedeltal, which, tho' convenient, looks clurafy, and leems to overload the Cornice. The Members of which his Bafe confilts are the fame as thole of Palladio , but vary fome finall matter in their Dimenfions. The Capital is of the fame Compofition as ‘Palladio s, the Ornament of the Abacus excepted, which here is a Sun-flower, and that in 'Palladio'!, the Tail of a Filh. The Entablature, Fig. II. is very different from that of Palladio ; for, where Palladio has a Cvma reverfa in the Tenia of the Architrave, here is a (a-velto, and Modillions in the Cornice with¬ out Dentils, as are in Palladio. On the Side of the Capital Hands a part of its Section, wherein the Heights of the Leaves, and their Curvatures, &c. are exprelfed by Minutes. The Figures HI. and IV. reprefent his greater and lefs- er Impofts for Arcades, the greater to be ufed in Arcades without Pedeftals, the Idler in Arcades with Pedeftals. Fig. V. is a Profile of the Entablature at large, the Ule of which is. To find /he 'Projection of the Cornice oner the Mi¬ ter of a Right single, as follows: (1) From any point in the Line ah draw the Right Line hi, to make an Angle of 45- deg. with the Upright of the Freeze, then will that Line be equal to the Bale of the projedting Miter at the Angle ; alfo draw the Line l x, &c. at Right Angles to b l, and from the point l let oft' the Heights of every Member in the Cornice, that is, make l m e- qual to p q ; alio in n equal to q r ; alfo n 0 equal tori; alfo 0 tv equal to st; alfo w x equal to t n ; and fo in like manner all the other Members of the Cornice, and from the points /, m, n, 0, 70, x, &c. draw Right Lines of Length at plealiire. (1) From the points a, d, g, 1, &c. in the Profile, draw Lines parallel to the Line a h, until they cut the Line b l in the points, b, e,g, &c. then Right Lines being drawn from the points b, e, g, &c. they will ter¬ minate the leveral Members at the points c, f, g, h, k, &c. which will be the very points or Extreams in which the tw r o fide Cornices will meet in the Mi¬ ter line of the Angle ; and fo in like manner the fame is to be underftood of the Architrave, as alfo of the Mouldings of the Bafe to the Column, and of the Bafe and Cornice to the Pedeltal, as exhibited in Plates CXGVI. CXCV 1 I. 4 C Fig. 290 The ‘Principles of Geometry. Fig, VI. reprcients the entire Order, whofe principal parts are the following Heights, via. Tire Height of the Pedeftal 5 diam. or mod. and xo min. of the Column, including Bale and Capital, xo diam. of which the Bale contains ->o min. and the Capital 1 diam. 10 min. and of the Entablature z diam. \i hich is equal to part of the Column. The particular Members here, as they are alio m : Palladios, are determined by Modules and Minutes. The Diminution of the Shaft is begun at x diam. 46 min. above the Bafe, and tile Diameter of the Shaft at its Aftragal is equal to yx min. and a half. Diam. Min, ( Pedeftal and Column 13 xo The Height of the ^Column and Entablature ix 00 (Pedeftal, Column and Entablature iy 10 Plates CXCVI. CXCVII. Corinthian Frontifpieces, by V. Sca- moZZi. As in the laft Plate I have explained the Manner of finding the Miter Bracket of the Cornice and Architrave, and have there obferved that the fame Rule is to be followed for finding of the Miters of the Bafe to the Column, when made fquare, and of the Bafe and Cornice to the Pedeftal, 1 have no need to lay any Thing further thereon ; but lliall proceed to the other parts of this Plate, which conftft chiefly but of two Defigns of Frontifpieces lor Doors or Windows, the one marked D, with a ftreight Head, part ot which is Ihewn at large over it; and the other marked E, with a femicircular Head, whole feveral Members being exprefied by Modules and Minutes, need no fur¬ ther Explanation. Plate CXCViiL Corinthian lntercolumnations for Portico’s to Temples, Arcades, &c. by V. Scamozzi. The two upper Figures reprelent the Profile and Front ol a magnificent Temple, that on the Right being the Portico, the other on the Lelt its Pro¬ file, which is arcaded in a very grand Manner. As thele two Figures repre- fent unto us the lntercolumnations of this Order without Pcdcftals, fb thole at the Bottom reprelent the proper lntercolumnations in a Colonade and vlrcadc with Pedeftats ; all which have their lntercolumnations determined by Mo¬ dules and Minutes. Plates CXCIX. CC. The Corinthian Order, by M. J. Barozzio, of Vignola. This Malter, like many others, has his Faults as vifible as his Beauties, and which I mull own are very furprifing ; for who but himlelf, after having compofed fo grand, fo noble, and fo magnificent an Entablature and Capital, would place them with their Shaft on fo monitions a Bafe, and that on fo very flender a Pedeftal, whofe Altitude I think is much too high, and which would yet appear higher had he not made a Kecking under the Capital, by placing an Aftragal there, which makes its flender Height appear fomething lei's than it really is. The Module by which the Parts of this Order are de¬ termined, is the Semidiametcr of the Column divided into 18 parts. Before 1 proceed to the Meafures of the principal parts, I mult beg Leave to ob- ierve, that was the Pedeftal of Palladio, with the ylttick Bafe, given to this Column and Entablature, I much doubt if it could be exceeded by any other Compolition that mortal Man is able to compofe. The Height of the Pedeftal is 3 diam. and i ; that of the Column 10 diam. including its Bafe, which is equal to the Semidiameter; and the Capital, whofe Height is 1 diam. The ‘Principles of Geometry. 291 and one ( 5 th ; the Diminution is one 6th of its Diameter at the Bafe, and t'ne Height of the Entablature z diam. and half; Diam. CPedeftal and Column 15 t The Height of the tiree +ths to each Scotia. To find theTrojettion of the !Plinth, before the Un¬ right of the Column, divide the Diameter of the Column into 16 parts and make the Projection equal to 3 of thofe parts; the Height of the Capital bein'* divided into 7 parts, give the upper r to the Abacus, the lower 6 to the Leal es and } olutcs. The Projection of the abacus is nearly equal to the Piojedtion of the 1 inth. The Height of the Architrave is equal to the Semi- i th n e C ° lumn ’ Wh0le Te p ]S a 7th of its Height; the remaining Height divided into 12, give 3 to the lower, 4 to the Middle, and y to the uppei Pafcia s; the remaining part of the Entablature being divided into 2 P aitb * § 1VC tIlc lowcr 1 t0 the Freeze, and the other to the Cornice. To divide tlje Members of the Cornice. (1) Make the Height of the Cima reverfa equal to an 8th of the whole Height, and the fillet a 3d of the Cima. (2) Make the Height of the Den- tns equal to a yth of the whole Height, as alfo the Ovolo, with the Fillet of the Dentils included, which Fillet is equal to a ( 5 th of the Height oftheDen- tils j tlie remaimng Height being divided into 17 parts, give the upper 1 to the Height of the Regain, and the next 8 to the Height of the Cima retta ■ then the Height of the remaining 8 being divided into 3, give the upper 1 to the Cura reverfa and lower 2 to the Corona ; laflly, The Projection is equal to its Height, and thus is the whole Entablature completed. Plate CCXXIX. The Corinthian Bafe (at large, enriched) and Capital, ty J. Mau-clerc. Tbe Bafe here reprefented confifts of the fame Members as the preceding and is divided 111 the lame manner; fo that the only Reafon of repreifentin- it here again at large, is for nothing elfe but to fliew, how to enrich Inch Members with carved Ornaments, when required. The Capital reprefent- cd in the upper part of the Plate, is one 14th part greater Altitude than the pieccding, and one of the belt I have leen. Fig. S reprefents a Front View of one Angie of the Abacus, with its Volutes. llate CCXXX. yl perspective JTiew of another Kind of Corinthian Bafe and Capital, with Corinthian lmpofls, by J. Mau-clf.rc. 1 he Members of the Bale reprefented here have fonie Difference in their Heights from thole of the foregoing Bale, but the Kinds are the lame; indeed we have here an Aitragal placed on the upper Torus, which is not in the o- thei. 1 0 divide this Ba/e into its Mouldings, divide the Height into o parts give the lower 3 to the ‘Plinth, the next 2 to the Torus, the upper 1 and a 3d of the next to the upper Tow, and then the Remainder being divided into ° l Mrls > 6 lve to each ol the other Members fuch of thofe parts as thole Divi- hon.s exprcls. As the Members I have now divided are thofe which make the bale, exclusive of the Aitragal and Cincture, which this Matter makes a part of the Shaft, 1 think it neceflary to add, that the Height of the Aitragal and ■incline taken together are equal to a 9th of the Bale, and which being divided into y parts, give x to the Aitragal, and 3 to the Cinfture. The Capital here reprefented is another grand and elegant Compofition, and the twining together the two Helices, or finall Volutes, is admirable good. The Enrichment of the Abacus, with the fmgie Pink in its Middle, is 4 E ■ very The ‘Principles of Geometry. 298 verv nobie and rich; but fuch carved Ornaments m an Abacus are only to be ufed in intide Works, as that the Angles of fuch Carvings, lo openl) e\- pofed, are the mod liable to an early Decay. , , 1 1 'hk Impofts are both very good, but metlunks they Teem to have been taken from ‘Palladio and Barozzio . Plates CCXXXf. CCXXXIL The principal Members of the Pedejlal, Column and Entablature , by J. Mau-clerc. These principal parts arc the fame as thofc reprefented before in the entire Order of Plate CCXXVII. which are feverally divided as following . Plate CCXXXIL divide the Height of the Pedeftal into 9 equal baits gne. e lower . to the Bale, the upper 1 to the Cornice, and the imt.imdiate 7 to the Die To divide the Height of the Bafe into its Members, on ide the Height into S , give the lower a to the Plinth, and the remaining 3 being S into I give the lower 1 to the Torus, and upper 1 to the Altragal and Fillet, which divide into 3, give the upper 1 to theFillet and the lower % to the Altragal; the Remainder being divided into r give the Ion u 1 to the Fillet on the Torus, and the other 4 to tire Cima re&a. Divide the Height of the 'Die into 5-parts, and make its Diameter equal to 3, >> t " h to d equal parts and give . to the Projection of the Bafe, before the Up¬ right or Face of the Die. To divide the Height of the Cornice into its Mem- fters divide the Height into x parts, give the upper 1 to the Falcia 01 Hat¬ band, with the Cima reverfa and its Regula, which mate equal t .. i thereof- the lower 1 being divided into 4, give the lower 1 to the height of the lower Cima reverfa, and the remaining 3 being divided into a., give the upper 1 to the Ovo/o, and the lower 1 to the Cavetto and its 1 diet. On Right of the Pedettal this Matter has varied the Members in the C01 nice, having placed an Altragal on the Top, which on the left Side is Lima re- verla The Bafe to the Column, Plate CCXXX 1 . is the fame asm the pre¬ ceding Plates, as alfo is the Capital; but the two Samples of Cornices are different. To divide the Height of the Entablature into its Architrave, Freeze and Cornice, divide the Height into 10 parts, give, to the Arch- trave as many to the Freeze, and 4 to the Cornice. The Tema of the Ar¬ chitrave is a -th of its Height, and the Remainder being divided into n ™ rts o-ive 3 to the firft, 4 to the fecund, and ? to the third Fafcta. To V divide g the Cornice on the left Hand divide the Height into1 10 pairs an fub-divide them as therein expreffed ; lo in like mannei. To divide the Cor nice on the right Hand, divide the Height into y parts and iub-diMt e as the Divifions^ exhibit. The Projections of both are equal to their Altitudes. The Diminution of the Shaft is a 6th of its Diameter. 1 Plates CCXXXIIL CCXXXIV. CCXXXV. Three Corinthian Enta¬ blatures enriched , by J. Mau-clerc. These Entablatures are the three laft deferibed more at large, with many sf their Members very finely enriched, which are given here as further Helps to the inventing of Ornaments for enriching fuch Members, when ic quired. Plates CCXXXVI. CCXXXVII. The Corinthian Order, by Inigo Tones, in the Front of Somerfet-houfe, next the River Thames. The Principles of Geometry. 299 Plates CCXXXVIII. CCXXXIX. The Corinthian Order of Sir C. Wren, in the Portico of St. Paul’r Cathedral in London. The other Orders of this great Matter having their parts exprefted by Feet, Inches and Parts, 1 have alio reprefented this Order and its Intercolumna- tions in like manner, which were taken by me with the greatelt ExaCtncls and Care. Plate CXL. The Corinthian Order entire, with the Pedejlal and Bafe to the Column at large, by Air. Gibbs. To proportion this Order entire, (1) Divide the Height into y equal parts, give the lower 1 to the Height of the Pedeltal, and the remaining Height being divided into 6 parts, give the upper 1 to the Height of the Entabla¬ ture, and the low er y to the Height of the Column, including its Bafe and Capi¬ tal. (x) Divide the Height of the Column into 10 parts, and take 1 for the Diameter ; the Diminution of the Column is a 6th of its Diameter, and the Height of the Capital is 1 diarn and a 6th. To divide the Height of the Pedefal into its Bafe, Die and Cornice. (1) Divide the given Height into 4 . parts, give the lower 1 to the Height of the Plinth, a 3d of the next 1 to the Height of its Mouldings, and half the upper 1 to the Height of the Cornice, (a) Divide the Height of the Mouldings into 4, parts, give the lower 1 to the Torus , and a 3d of the next 1 to its Fillet ; give half the upper 1 to the Cavetto, the other half being divided into 3, give the upper 1 to the Fillet, and the lower x to the Si frugal ; the remaining 1 and two 3ds is the Height of the Cima reffa. To divide the Mouldings of the Cornice, divide the Height into 6 parts, give half the low¬ er 1 to the Cavetto, the other halt being divided into 3, give the lower 1 to the Fillet, and the upper x to the H/lragaF, give the next two 6th parts to the Height of the tuna recta and its Fillet, whole Fillet is a 6th ; divide the upper 1 into 6 parts, give the upper 1 and a 3d to the Regnla, the next x and two gds to the Cima reverja, and the next 1 to the f/lragal: Thele Members have not the Divilions of their parts reprefented here, as being ihewn at large by Fig. Q. in Plate CCXL 1 II. The Height of the Baje is equal to the Semidiameter ol the Column, which is alfo reprefented at large by Fig. P, Plate CCXL 1 I 1 . To divide the Moulding of the Bafe. (1) Divide the Height into 3 parts, give the lower 1 to the Plinth. (1) Divide the other 1 into 10 parts, give 3 to the Torus, 1 to its Aftragal and Fillet, of which the Fillet is a 3d, 3 to the Scotia, including the Fillet of the upper Jjtragal (which Fillet is a 3d of 1 part) and the upper x and a 3d to the upper Torus. The Projection of the 'Plinth is equal to its Height, which divide into y parts, make the Projection of the Fillet to the upper Aftragal equal to two 3ds, and of the lower to three yths. The Projections of the Aftragals and Torus’s, beyond the Fillets, ate equal to their own Semidiame- ters. To deferibe the Scotia, continue down the Face of the Fillet to the upper Aftragal, unto the Fillet of the lower Aftragal, which divide into 7 parts ; from the 3 Diviftons draw a Line parallel to the Plinth, cutting the Conti¬ nuation of the Face of the Fillet to the lower Aftragal, which Line continue upwards, equal to the Difference between the Projections of thofe Fillets, and then a Line drawn from thence, thro’ the third Point, will determine the Quantities of the two Arches which compofe the Scotia, the firft of which is deferibed on the third Point, and the laft on the Extream of the Line afore; goo Tlx Trim:i pies of Geometr y. fr.id. The Shaft of this Column is divided into 14. Flutes, and as many Fil¬ lets as in the lomck. The Aftragal and Cincture are equal to two jds of the upper Torus, which being divided into 3, give i to the Aftragal, and 1 to the Cincture. The Diminution of the Column is a 6th of the Diameter. Plate CCXLI. Tlx Corinthian Capital, by Mr. Gibb s. The Height of this Capital is 1 diarn. and a 6th, which laft is given to the Height of the Abacus, and the remaining Height being divided into 3, and thole fubdividcd into 4, the Heights of the Leaves are from thence de¬ termined. The Aftragal, with its Fillet, is equal to a 4th of the lower Di¬ vifion of the Leaves, which divided into 3, give 2 to the Aftragal and 1 to the Fillet. The Scale under the Capital, numbered from 1 to 6, towards the leit Hand, is equal to theSemidiameter of the Column, and which being cut in the fifth Divifion by the Upright of the Column, fhews that its Diminution is a 6th as aforefaid. The next two parts from 6 to 8, fhews the utmoft Pro¬ jection of the Abacus-, and the next Divifion to 9 is the angular Point from the central Line, equal to half the Side of a geometrical Square, in which you may deferibe the Plan of the Capital, as here reprefented. Fig. 3 is a Profile and direct View of one Angle of the Volutes of the Capital. Pl a(:es CCXLII. CCXL 11 I. The Corinthian Entablature, by Mr. Gibbs. To divide this Entablature into its Architrave, Freeze and Cornice, divide the Height into 10 parts, give 3 to the Architrave, as many to the ’Freeze, and the upper 4 to the Cornice. To divide the Architrave, Fig. R, Plate CCXLIII. divide the Height into j-parts, give the lower 1 to the iirft ’Fafcia with the Bead, which is a 4th thereof, the fecond 1 to the fecond Fafcia, tile next 2 to the third Fafcia, including the Chna reverfa, which is a 6th, and the Bead under the Tenia, which is an 8th ; the upper 1 being divided into 4 parts, give the upper part and a 3d to the Regula of the Tenia, and the Remainder to its Chna reverfa. The Projection of the Architrave is e- qual to the Height of the Tenia with its Bead, which divided into 5 equal parts, the Projections of the Cima in the Tenia, the Fafcia's, &c. are deter¬ mined. To divide the Mouldings of the Cornice, divide the Height into 5- parts, and fubdivide each part as is exhibited. To find the'Projections of the Members, divide the whole Projedtion into 4 parts, as is done againlt the Freeze in Plate CCXLIII. and Subdivide each Part, as therein expretied, from which Divifioiis give to each Member its Projection, as exhibited. The Scale under the Entablature, in Plate CCXLII. is equal to the Diameter of the Column, which is divided into 12 parts, by which the Modillions and their Intervals in the Plan of the Cornice are determined. Note, The Breadth of every Member 111 fuch a Plan is equal to its Pro¬ jection. Plate CCXLIY. Corinthian Intercolumnations for Portico’s and Colo¬ nies, with the Impojl and Architrave for Arcades ; alfo the Corin¬ thian on the Ionick Order, by Mr. Gibbs. 1 he Figure C reprefents the various Intercolumnations of this Order pro- pei fin Porticos and Colonades, whole Meafures are expreiled by Diameters, i iKl- igtues A and B reprefent the Imports and Architrave of the Arcade, whole Members are determined by the Height of the Import (which is equal to the belliidiameter of the Column) divided into 3 parts, and fubdivided as re- quned. lhe Aftragal is equal to half one 30 part of the Height of the Im¬ port. The Principles of Geometry. 301 port. The Projedtion of the Import is equal to a 3d of its Height, and that of the Architrave to eight ()ths tf the lame, whole Fai'cia’s are divided, by dividing its Height or Breadth (which is always equal to the Semidia¬ meter of the Column) into 3 parts, and iubdividing them as exhibited. Figure DEF rcprefents the Ionick and Corinthian Orders on a Rurtick Bafement, whole Height may be coniidered as Titfcan or ‘Dortch. It's to be here obferv- cd, where we place one entire Order over another, either arcaded or otherwife, that in fuch Cafes the Heights of Pedeltals mult be lefs than a yth of the whole Height ; otherwife, the Stools of the Windows would be too high ; and especially when the Columns are of a large Diameter. In thcfe Examples, the Height of each order is divided into 7 Parts, of which the Pedeftal con¬ tains the firft, and the Entablature the lalt. On the left Side of Plate GCXLV 1 I. is the fame Order on a Rurtick Bafement, where the Intercolum- nations are fomething different; the other being arcaded, and this not: But the Heights of the Pedeltals are regulated here as in the other. The Inter- columnations on the right Hand of this Plate are the Ionick and Corinthian on the Dorick, which lalt gives the Rule to the other two, becaufe ol its Metops nnATrigtyphs, whole Intercolumnations are expreiled by Diameters and parts. Plates CCXLV. CCXLVI- Corinthian Doors, and Arcades by Mr. Gibbs. In Plate CCXLV. we have fix Defigns lor Doors, of which the upper tw'o are fquare-headed with Pediments; the others below are arcaded, either with Pediments or Ballultrades, wherein ’tis to be obferved, that the Height of the Balluftrade is equal to the Height of the Entablature. Thcfe Defigns are in general very good, and worthy of Imitation. The Arcades in Plate CCXLVI. are of two Kinds, the upper 011c being without Pedeltals, the other with Pedeltals, whole Intercolumnations are expreiled by Diameters and parts. Plate CCXLVII. The Ionick and Corinthian Orders, on the Dorick Order, by Mr. Gibbs. This Plate is explained in the Explanation of Plate CCXLIV. Plate CCXLVI H. The Compolite Pedejlal , Geometrically deferibed , according to the Ancients , by C. C. Oslo. The Height AE being given, bifedt it in C, and make the Angle AC a equal to 30 deg. Draw w E and A a at Right Angles to AE, and each of Length at Fieaiure; then will C a cut A a in a, which is the utmort Projec¬ tion of the Cornice. Bifedl A a in h , make the Angle A be equal to 30 deg. cutting A FI in c, on c erect the Perpendicular c e, of Length at Pleafure, and from A draw Ac, making the Angle e Ac equal to 30 deg. on e A eredl the Perpendicular e d, cutting A E in d, then is Ad the Height of the Cornice. Make FID equal to A d, then is DE the Height of the Bale. Divide BE into 4 Parts, let 1 of thole parts on each Side the central Line for the Projection of the Die. To di vide the Height of the Bafe into its Mouldings, draw the ] .incs aq and au, making an Angle of 30 deg. biiect au 111 l ; make the Angle a l k equal to 30 deg. then will Ik cut aq in ki ; bilecT kl in z, on z raile the Perpendicular z p, cutting a q in p ; divide kp and a k , each into 4, then is qp the Plinth, po the To-rus, 0 k its Fillet, k m thcCima inverfa, and m a theAltragal. The Projection of the Plinth is equal to the Height of all the Mouldings above it. The Mouldings to its Bafe may be alio divided very eafily, as follows; divide the Height into iz parts, give 4 to the Plinth, 3 to the Torus, 1 to its Fillet, 3 to its Cima inverfa, and x to the MJlragal. To divide the Mouldings of the Cornice. The upper Cima inverfa, with its Fillet, is a < 5 th of the Height, of which the Fillet is a 3d ; the Height of the Aftragal is a The ‘Principles of Geometry. 302 mh of the Height, exclufive of its Fillet, which is equal to a 3d of the Af- t .‘,; a i xhe Height of the Neck n z is equal to a 3d of the Height. Eiil-ft f ., n j the upper half is the Height of the Fafcia, the lower halt being di- v'h-d' into i, the upper 1 is the Ovolo, the lower > is the Cavctto with its Fillet, which is a third part. Make the Angle i Z 6 equal to 30 deg. and drew the J ,ine z /, cutting z A a in z, then will i A be equal to ha, the Pro¬ jection oi the Cornice being found. p l cbfer-jeyou call the Cinia, or the upper Member of this Tedejlal, forme- tunes rcvcrla, and at other times inverfa, pray why is it called by different Names, when the Moulding is thejame ? M- The Moulding is indeed the fame; ’tis call'd reverfa when ulcd in any of the upper parts of an Order, and is always placed with the Fillet uppennolt, but inner la when 'tis inverted or turned uplide down, which can never happen any where but in Bales. Plate CCXLIX. The Ba[e to the Compolite Column, with a Seclion of its Capital, by C. C. Oslo. The Height of the Bafe is (as of all other Mailers) equal to the Semidia¬ meter of the Column, and its Members are thus divided. Fig. I. The Height of the Plinth IC is a 3d of the whole Height. The Height of the Torus IH is a 3d of the Remainder. Divide AH into 8 parts, and'give the upper 3 to the Height of the Torus, the next x to the Scotia with both its Fillets, each of which is an 8th of its Height; the next three 4-thsto the Aitragal, including its Fillet, which is a 3d, and the remaining x one 4th to the Scotia and Fil¬ let, which is an 8th of its Height. The Projection of the upper Torus is e- qual to AD, and the Projection of the A/lragal is the lame. Draw the Line AB, making the Angle CAB equal to 30 deg. cutting I .v, tire Height of the Plinth, in x, then is lx the Projection of the Plinth. The Projections of the Fillets’to the Torus's and Aihagals arc perpendicular to their Centers. To defcribe the lower Scotia , draw w s r ; to divide the Height of the Scotia in¬ to two parts, cutting A B in r, draw po perpendicular to wr, then are the Points 0 and r the Centers, on which defcribe the Archpj and s u. the upper Scotia is a Semicircle, which completes the Whole. To defcribe the Capital, Figure II. (1) Divide the Height AC (which is equal to the Diameter and a 6th) into 7 parts at the Points n, i, kj, L, b, A ; make Ad equal to a 3d of A b, and divide A d into 4 parrs ; alio divide b L into 3 parts at c e ; make L M equal to a 3d of I./, and h M a 4th thereof; bill'd: fk in g, and make il equal to a 4th of i n ; thefe being done, from the Points A, a, d, b, c, e, L, h, M, /, g, k, z, l, n, C, draw Right Lines out at Plealurc at Right Angles to AC, then will the Height of the feveral Members be divided, (x) Make B A and 80 b each equal to b C, then is BA the utmoft Projection of the Cornice. Make 80 7 equal to 4 80, and draw the Line 6 7 parallel to 4 80, for the D plight of the Fafcia; draw the Diagonal 4 7, which billet in f, the Center of the Curve x 8, and the Line j- x 1 terminates the Fillet. (3) Make r H equal to half M /, and complete the geometrical Square w rZH, then the Side t w, being continued, terminates the Fillet of the Aitragal, whole Projection be¬ fore it is trail its own Height. Through the Points s, r, draw the Line sr q, and defcribe the Ovolo equilaterally. (4) Make the Height ol the Aitragal, including its Fillet to the Neck of the Column, equal to half n C, of which give a 4t h to the Fillet; make the Projection of the Fillet equal to its Height, and the Projection of the Aitragal before the Fillet equal to its own Semidia- nieter, and the Height ot the Fillet alfo as the Line DC, then will both the Altragals have equal Projections, and thus are all the Mouldings deicribed. (j-) Draw the Line BD, cutting the Line G / in G; from whence draw the Line GF parallel to AC, for the Cathetus of the Volute, cutting the Lines drawn The Principles of Geometry. 3°3 drawn from the Points e and L, in the Points 12, 13, which is the Height of the Eve of the Volute; bil'cct 12, 13 m T, the Center of the Eye; divide the Diameter of the Eye into 6 parts, on which deferibethe Volute, as before done in the Ionick Volutes of Vitruvius and Serlio. Plate CCL. The Compolite Entablature of the Ancients , according to C. C. Oslo. The Height A Q_ being divided into 3 parts, give the lower 1 to the Ar¬ chitrave, the next 1 to the Freeze, and the upper 1 to the Cornice. To divide the Architrave, make the Tenia n pcqual to two 9ths of n t, its whole Height, of which make the Regula n 0 one yth ; make the Beadp q equal to n 0 , and bilect q t in r ; alfo make r s equal to one yth of r t, then from the points k, 0, p, q, r, J, t , draw Right Lines at Right Angles to the Line A Q_, which w'ilGeprefent the Height of each Member in the Architrave. To find their ; Projections , make b M equal to nq, the Tenia with the Aftragal, and make d e equal to op the Height of the Lima reda, exclufive of its Regula and A- Jlragal, make k r equal to h p, and draw g k ; divide k R into 4 parts, and deferibe the Cima rtverfa, which completes the Architrave. 7 0 divide the Mouldings of the Cornice, (1) BifeftAD in G, and draw the Line ay G of Length at Pleafure ; make Ac equal to two cjths of A D, of which make A a two 9ths, and b c one 4th of the Remainder ; make c e equal to cne 9th of c fq, and c n equal to one 3d of 1 D; alfo make n l equal to one 4th of e n, make G m equal to one 4th of G D, of which P m is one 6th ; alfo make the Height of the Ovolo p 17 equal to one 4th of G D ; lajlly, make sr equal to one 6th of tv r, and then from the Points A, a, h, c, d, e, l, n, G, F, m, 0, p, draw Right Lines of Length at Pleafure, and right angled to the Line AD. (j) Make A B equal to A D, then is A B the utmoft Projeftion of the Regula of the Cornice ; make Dy, the Aftragal of the Freeze, equal to one 18th of the whole Height of the Freeze ; alfo make 17 D equal to Dy, and draw the Line 28 p, then is p the Center of the Ovolo q 17 ; draw qt perpendicu¬ lar to B D, and s q perpendicular to qt, then thro’ the Point s draw the Line rsw parallel to A Q_; from the Point 10 draw the Line 10 26 parallel and equal to w s ; on s, with the Radius s tv, deferibe the Arch y tv 11, mak¬ ing the Arch tv it equal to the Arch y tv, and thro the Point u diavv the Line z u 1 ; make the Intervals zm and 1 2 each equal to half tv z, and m 42 3 equal to tv z s 1, and fo in like manner divide all the other Dentils, dhc Point 12 terminates the Fillet to the lower Cima reverfa, and as the upper Cimds have their Projections equal to their Heights, complete them accoid- ingly with tile Head and Corona under them, and thus is this Older completed, which finiilies the Works of Carlo Cejare Ofw. Plate CCLI. The Compolite Order entire, by J. Mau-clerc, not Vitruvius, as miftakenly inferted by the Engraver. I know not bv what Accident this Plate was placed here, it being the en¬ tire Compolite Order of J. Mau-clerc, which fliould have immediately preced¬ ed Plates CCLXXXIV. CCLXXXV. &c. wherein the particular parts of this Order are exprefl'ed at large ; but as tis too late now to correct this Miftake, 1 will here give the Manner of dividing the principal parts of this Order, and then refer you to its particular parts, as they are expreffed in the 1 lates a- forefaid. . rT . 7 To proportion this entire Order to any given Height , divide the given Height into 81 parts, that is, into 1 6 parts and one 4th, accounting every 4 equal to 1 ; this done give the upper r with the odd one 4th, as alio three 8ths of the third part unto the Height of the Entablature, and the lowermoit ? and two gds of the fourth unto the Height of the Pedeftal, then the re¬ maining Height being divided into 10, take 1 for the Diameter of the Column- 3°4- The Principles of Geometry. The He gh; -if the Capital is i diam. and one 6th, and the Diminution of the os..ft one 6th of its Diameter, as by its Divifions is demonflrated. Plate CCL1I. The Bafe, Capital and Entablature of the Compoiite Or¬ der, in the Arch oj Titus at Rome. This Compoiition is one of the fineft that is to be feen in the Remains of the Ancients; its Bafe is equal to its Senndiameter, which being divided into 11 parts, or the Diameter into u, abate i on each Side, and take the 20 remaining for the Diameter of the Column at its Aftragal, fo that the Dimi¬ nution of the Column is nth of its Diameter, which is very inconfidera- ble. The Height of the Capital is equal to i diam. and a quarter. To find the Height of the Architrave, Freeze and Cornice, divide the Diameter of the Column at its AAragal into -8 parts, make the Height of the Architrave equal to 6 and three jths, the Height of the Freeze equal to 7, and the Height of the Cornice to the Diameter of the Column at its Bale ; the parti¬ cular Members are determined by the Subdivifions of each part, which are very plain to Infpection. In Plate CCLV. is a perfpeefive View of the parts of this Order very finely enriched, according to Mr. Evelyn. Plate CCLIII. The Geometrical Elan and Elevation of the Triumphal ylrch of Titus at Rome. This Plate reprefents the Plan and Elevation of the Arch of Titus , which was one of the fineft Works of the Kind that the Ancients ever performed, and which will fo appear, after having deliberately confidcred the Magnifi- cency and Richnefs of its parts, which are expreffed in the two following Plates. In the fquare Pannel Z was an Infcription, which not being material to our prefent Purpofe is omitted : By the Plan underneath it is evident, rliat the Centers of the Plans of the Columns are in a Right Line, and therefore the Entablature muff have been entire throughout, and not broken forwards as is here reprefented in the Elevation, which I believe to be a Miflake of Ser- lids, from whom I had both Plan and Elevation. Plate CCLIV. The Members at large of the Arch of Titus at Rome. The feveral Capital Letters refer to the like Letters in the laft Plate wherein we fee tire Proportion of each Member and its Enrichments more di- ftindUy, from whence home ufeful Hints may be taken for the ornamentin' 1 of other Works. 0 Plate CCLV. A perjpeclive View of the Bafe, Capital and Entablature (enriched) of the /Itch of Titus at Rome; as aljo a Geometrical Profile of the Compoiite Order in the Cafile of Lions at Verona by Mr. Evelyn. The Entablature, fyc. of the Arch of Titus having been already taken Notice ot in Plate CCL 1 I. I fhall therefore proceed to my Remarks on the Profile of the Order in the Caftle of Lions at Verona, whole Members are determined by Minutes, and their Projections accounted from the central 1 .ine. By the great Height of the Plinth, it feems as if this Order flood very high, on a Pedcftal whole Cornice eclipfcd a part of its Height. The Mem¬ bers placed on it are of the At tick Compofition, making the total Height 57 mm. and a half The Column hath 8 min. Diminution, and the Height of its Capital is equal to 70 min. its Architrave 49 min. its Freeze y 6 min. and its Cornice (which is very remarkable, in having the Dentils placed immedi¬ ately The Principles of G EOMETir. 3°5 ntely on the Freeze, without any Oma, Ovolo or Cavetlo under them) is m mm. which is equal to the Architrave. ' Plates CCLVT. CCLVII. The CompofiteOrder, by A. Palladio. 1 he Meafure by which this Mailer determines the parts of this O.dcr is the Diameter of the Column divided into do min. as in his other Orders rig. A lcpiefents the Pedeftal and Bale to the Column, of which the firft is good, but the laft, namely the Bife to the Column, is horrid, he havin^in- ihlcreetly introduced the fmall Aftragals together, which in the Ccriit’hian dale he had wifely feparated ; this lliews a Poverty of Invention in this Ma¬ her, who is lo much celebrated by the Ignorant. Fig. B reprefents the Capi¬ tal and Entablature, of which the firft is nothing more than the l nick Vo¬ lutes Ovolo and djtragal, crowned with the Corinthian Abacus, and ihufefet on the Vale or Bell of the Corinthian Capital. The Architrave, u hich Ihould liavc con lilted of three Fafcias, to have been elegant and airy, is here but in two, and thole very clumfy, crowned (as it were) with a double Tenia con- hlhng ol a Bead, a Cimareverfa, a Cavetto, and Regula, which together al- tbo enriched, have a heavy, dull, Tufcan Countenance. The Freeze he has taken from the Iomck, and which is not amifs; but his Cornice is of little better Compofition than that of his Architrave, the Members being all very lajge and heavy, inftead ot having a delicate Elegancy fuperior to that of the Corinthian. Big. C reprefents his Intercolumnation in an Arcade, with l'ede- ftals Fig. E his Impolt, which is very good, and Fig. D his Intercolumnati- ons foi Columns in Colonades, whofe Meaibres are determined by Modules or Diameters and Parts ; the Height of the Pedeftal is 3 diam. and 10 min! the Height of the Column 10 diam. whole Bale is equal to the Semidiameter and its Diminution to 7 min. and a half, or one 8th of the Diameter ; the Height or the Capital is 1 diam. and one yth, the Architrave is 40 min. the rieeze 30 min. and Cornice 70 min. CPedeftal and Column The Height of the !S . we11 confulered, in cale that the Cornice of the Pcdeftal is mu’cli above the Lye , foi was the Plinth under the Torus to ffand on the Pcdeftal the Projection ot us Cornice would totally ecliple it. The Bale to the Column is ot the AUidi Compofition, the Aftragal on the upper Torus only excepted and is very good. It is fomething very uncommon to li e Modillions in an Import, as are in C, which is the Import to the middle Arch ; but if wc con- hdei, that at that Height there might have been a Floor within, whofe foifts were laid out, to ftrengthen the upper Members, they were therefore verv piopei. Inc Import G, which belongs to the Side Arches, is a horrid Com¬ pofition, but the Architrave, Freeze and Cornice, B, is very good: And hcie again is another Example of the Corona , finifhing with a Fillet or Revda without any Cymatium whatfoever. 6 ’ Plates CCLXXIII. CCI ,XX l\ . The Triumphal Arch at Benevento, by S. Serlio. B y the Plan of this Arch it is evident, that all the Centers of the Columns arc m one Right Line, and therefore 1 am lurpriled that the Kntablature was not mad. entire as the Pedertals are; tis true, that the bringing the F.ntabla- turc and Parapet over the Arch, before the Sides, doth give fonie Majcfty to that part;_but then to break over the two extream Columns, and bring them J ! ' 1 ■ ai *°> ’ 5 vc, y Poor and trifling, not to be praclifed by any that would Ic ciieemcd a Judge of Architecture. To have made the middle part truly fr t.wi, thole Columns ihould have advanced quite clear of the Wall, with ?■- i.dteis behind them to fupport the returned Entablature on each Side In i..: - CM, XXIV. the feveral Members of this Arch arc exhibited at large O. winch the Entablature C, and the Bafe and Cornice to the Parapet A B arc very pretty Competitions. 1 ’ Plate The Principles of Geometry. 3 °9 Plate CCLXXV. The Gallery at Belvedere hy B ram a NT. I cannot tell what bewitched fo many of the ancient Architefts, to break the Entablatures over their Columns as here is done, in this, and many other of their Examples; fince that the only Way to make all Build¬ ings truely grand, is to avoid Abfurdities, and fmall parts; for as the Archi¬ trave is the Bails of the Entablature, it ought not to be weaken d, by being broken into many parts, thereby having a regular Bearing in fome parts, and not any in other parts, which is abfurd. The Gallery here reprefented would have been very grand, had not its Entablature been broken over every Pair of Columns, on which the Architrave has a regular Bearing ; but the Archi¬ traves over the Arches have no Bearings at all, fave that in the Wall over which they lye, and therefore are abfurd; as are alfo the little parts, into which the Entablature is broken. The principal Members of this Arch are exhi¬ bited at large in the lower part of this Plate. Plates CCLXXVI. CCLXXVII. The triumphal Arch of Con¬ stantine, by S. Serlio. As the Columns of this Arch do advance clear of the Building, and have Pilafters behind them, to have made this Entablature in one continued Piece, would have been furprifingly grand and noble; but to break it over every Co¬ lumn is monftroufly lhocking, and ieems to intimate, that they were not Mailers fufficient to carry its Architrave in Stone from one Column to the other. In Plate CCLXXVII. is exhibited the principal Members of this Arch at large, wherein are many fine Compofitions, well worth our Conlide- ration. Plates CCLXXVIII. CCLXXIX. The Compofite Order, by A. Pal¬ ladio, V. ScamoZZi, M. J. Barozzio, of Vignola, and S. Serlio, by Mr. Evelyn. In Complaifance to many, who delight in Mr. Evelyns Method of deferibing the Orders by Modules and Minutes, with their Projeftions accounted from the central Line, I have therefore, in thefe two Plates, reprefented the Corn- pofite Order according to the above Malters in that Manner, wherein we may obferve the Entablature of Serlio, whole Cornice is really Tajcan, to be (be¬ yond all Manner of Doubt or Difpute) the very worft Compofition of Mem¬ bers that have been yet placed over the Compofite Capital, with which they have no Kind of ConneHion, or Affinity of Parts, and is therefore unworthy of our Notice. Plate CCLXXX. The Compofite Order entire , with the principal Parts of the Pedefial at large , by S. le Clerc. The Module, by which this Mailer proportions the parts of this Order, is the Semidiameter of the Column, divided into 30 min. To proportion this entire Order to any Height, divide the given Height into 30 mod. and 7.0 min. that is, divide the Whole into 91 parts, and each part will be equal to ao min. and confequently the Diameter of the Column is equal to 3 oi thofe parts. The Height of the Pedeltal is equal to 6 mod. iy min. not 1 6 min. as ex- preiled in the Plate. The Height of the Column, including its Bale and Ca¬ pital, is 19 mod. and yo min. (not 70 min. as exprefled in the Plate) and the Height of the Entablature to 4 mod. and ry min. The Height of the Bale 4 H to The ‘Principles of Geometry. 3 io to the Pedeftal is 34 min. and one half, and its Cornice to 1 6 min. The Height of the Bafe to the Column, is equal to its Semidiameter, and of its Capital, to a mod. and 10 min. The Diminution of the Shaft is 8 min. The Height ol each Member is exprelled by Minutes, and their Projections are accounted from the Upright of the Die, and of the Column. The Mouldings, that compofe the Bale and Cornice to the Pedeftal, and the Bale to the Column, are in general very noble, and worthy of Imitation. On the Right-hand of the lower part of the Plate is the Vale of the Capital, under which is its Plan ; and on the Left of the following Plate are the orna¬ mental parts of the Capital at large. Plate CCLXXXI. Compolite Entablatures, •with Impofis, and the or¬ namental Parts of the Capital at large, by S. le Clerc. The Height of the Entablature being 4 mod. and 18 min. give 40 min. to the Architrave, 44 min. to the Freeze, and 3-4 to the Cornice. Both thefe Entablatures have the Uprights of their firlt Fafcia’s of their Architraves of 30 min. or 1 mod. in Projection from the central Line, as being made for Pi- lalters without Diminution. The Imports are both compofed of the lame Mem¬ bers, but differ in their Heights,' the under Import having 34 min. Altitude, and the upper but 30. The Members that colnpole the Entablatures are well choicn, and have a very good Fifed. The Volute on the Right-hand is Uc- icribed by Method II. 111 Plate Z, alter Plate CV. The leveral Leaves, &c. ol the Capital arc here Ihewn at large, to be oftentimes copied lingly, that thereby the young Student may be well acquainted with each particularly, and with Kale and Delight reprefent them molt exactly in Concert in their proper Places of the Capital. Plate CCLXXXlf. Compolite Entablatures , Impofis and Intercolum- nations , ivith the Compolite on the Ionick Order, by S. le Clerc. Here are two other Entablatures rcprdhnted by this Mailer, the one for a Column, where its Architrave has but r 6 min. Projection; and the other for a Pilafltr, where it has 30 min. Projection; the lalt of which has a perpendi¬ cular Freeze, and the former a convex, or fuelling Freeze. But the Mem¬ bers in both thefe Entablatures are of the lame Kind with thole in the lalt Plate, and differ only in their Heights ; tor, as the Entablature of the lalt Plate confuted otq mod. and 18 min. thefe here are, one of 4 mod. and 10 min. and the other ot 4 mod. and ir min. in Height. Here are alio three Imports, whole Members are the fame as thole in the lalt Plate; but their Height are all various, and their Architraves arc different; therefore their Choice is at Difcretion, as the Nature of the Place, wherein they arc to be ufed, doth require. This Plate doth alfo reprelent the Compofite Order on the Ionick, where the Columns arc placed in Pairs, and wherein the Intercolumnation ol the Compofite is regulated by that of the Ionick , becaufc the central Line or the upper Order mult be exactly perpendicular over the central Line of the lower Order, as has been already obferved. We have here alfo the In- tercolumnations of this Matter lor this Order in Portico’s, or Colonades, and underneath is an Arcade of lingle Columns without Pedeftals. Plate CCLXXXIII. Compofite Arcades, by S. le Clerc. The uppermoft Figure on the Lcft-fide of this Plate is another Arcade without Pedeftals, which is to be made with lingle Columns, or Columns in J ans; the other tour Examples are all with Pedeftals, and very grand Deligns; and elpecially thole crowned with the Balluftrades, Statues, and Trophies of War. the Intercolumnatio'ns of each are exprelled by Modules and Minutes. Plate The ‘Principles o/ Geometry. 311 Plate D, to follow Plate CCLXXXIII. Compofite Sofito’s, See. by S. LE ClEIIC. We have here another Entablature of 4 mod. and 15- min. in Height, whole Architrave, projecting but ad min. is therefore calculated for a Column. Here is alio another Import of 34 min. in Height, with a View of the Key- itone of the Arch, as well in Profile, as in Front, witli its particular Meafures of Height and Projefture. The Quad rant, number’d 1, l, 3, 4, S, 6 , repre¬ sents one 4 .th part of the Plan of the Bafe of the Shaft, divided into fquare Flutes and Fillets, inftead of circular Flutes. The two upper Figures repre¬ sent the Plans of the Sofito's of the Cornice, and Manner of returning it, as well at an internal, as an external Angle. Plate CCL’XXXIV. The Compofite Pedejlal by J. Mau-clerc. The principal parts of this Order being demonftrated in Plate CCLI. I fhall therefore proceed to the Divifion of the parts of the Pedefhil, whole Height being divided into 10 parts, give the upper 1 to the Height of the Cornice, and the lower 1 to the Height of the Bale ; then divide the Height of each into 7 parts, and fubdivide them again, as on the Right-hand is exploded ; the remaining 8 parts is the Height of the Die, whole Diameter is equal to halt its Altitude. Plate CCLXXXV. Two Examples of Compofite Entablatures, with the Capital, Bafe, See. by J. Mau-clerc. Alt ho’ this Mailer has made choice of two very bad Compofitions of Members for Freezes and Cornices to this Order, which he has here exhibited in the fame Figure, yet his Architrave and Capital are pretty tolerable, and his Bafe to the Column is very good. The Height of the Bafe to the Column is equal to its Semidiameter, and the Diminution of the Shaft is one 6th of its Diameter. The Members of the Bale are found by the Divifions and Subdivi- fions on its Left-hand, and the Projection of its Plinth is equal to one 4th part oi the Diameter, and that of the Pedelcal is the fame before the Upright of the Die: Or otherwil'e, divide the Breadth of the Die into 6 parts, and give 1 on each Side to the Projection of the Plinth. By the three Circles over the Capital, to which the two upright Lines of the Shaft are Tangents, tis evi¬ dent, that the Height of the Entablature is equal to three times the Diameter of the Column at its Aftragal, and that its Architrave, Freeze and Cornice are equal to each other. Plate CCLXXXVI. A perspective View of the Compofite Capital, by J. Mau-clerc. This Bafe and Capital is very different from thole in the laft Plate, that Bafe being very good, and this monftroufly bad, here being thole wretched, little Aftragals placed between th e great Torufes, of which I have often com¬ plained. The Leaves of the Capital are of the Parfley Kind; and indeed I think the whole Capital to he an extraordinary Competition. Plates CCLXXXVII. CCLXXXVIII. The aforefaid Compofite En¬ tablatures at large, with their Enrichments, by j. Mau-clerc. As the Compofition of the Members in the two upper parts of thefe Cor¬ nices 312 The Tnnciples of Geometry. nices are not good, it may be a Matter of Surprize, why I exhibit them again here at large. To this I anfwer, That, altho' the Members themfelves are not well choien for the Places they are employed in, yet their Ornaments are very grand and noble, and teach us how to enrich thofe Members in a very g ran d Manner, in their proper Places, when required. Plate CCLXXXIX. The Compofite Order by C. Perault. To proportion this Order to any Height, divide the given Height into ±6 parts; give 10 to the Pedeftal, 30 to the Column, including its Bale and Ca¬ pital, and 6 to the Entablature. Thefe parts this Matter calls Modules of which three are equal to the Diameter of the Column. To divide the Pede- ftalinto its Bafe, T)ie and Cornice, divide the Height into 4. parts of which gi ve the lower 1 to the Height of its Bafe, and half the upper 1 to the Height of the Cornice. To divide the Mouldings of the Bafe to the Pede/lal divide its Height into 3 parts, of which give the lower x to the Height of the Plinth and the upper 1 to its Members, which fubdivide into 10, and give to each as there exhibited. To divide the Mouldings of the Cornice to the Pede/lal "d i¬ vide its Height into 11 parts, and give to each, as there exhibited. The Height of the Bafe of the Column is equal to its Semidiameter, and its Height bein^ divided into 4., give the lower 1 to the Plinth-, the remaining 3 parts divid¬ ed mt0 5 P arts = S‘ ve th e upper 1 to the upper Torus ; the remaining 4 divided into 3 paits, give the lower 1 to the lower Torus; the remaining x parts di¬ vided into 3 parts, give 1 to each of the Scotia's, with their Fillets and the other 1 to the two Aftragals. The Height of the CinBure is equal to two ads of the upper Torus. To determine theProjeffure of the Plinth to the Bafe of the Column, divide the Diameter of the Column into iy parts, and give a to each Side for the Projeftion of the Plinth. To determine the Broiettion of the Bafe and Cornice of the Pedeflal, make p 0 equal to one 4th of the Diame¬ ter of the Column, or 4 parts, for the Projection of its Bafe, and ca to . parts and a half, for the Projection of the Cornice. The other Members are deter mined from thole equal parts, as by Infpeftion is very plain. The Diminution of the Shaft is two iyths of the Diameter, and the Height ot the Capital 3 mod. and three yths, which being divided into 7 <>i ve upper 3 to the Height of the Volutes and Abacus, and the lower 4 to & the two Heights of Leaves; which being divided into 6 parts, and the others above into 8 parts, from thence all the Members and Leaves are determined. To divide the Entablature into its Architrave, Freeze and Cortiice, divide its Height in¬ to 10 parts, give 6 to the Architrave, as many to the Freeze, and the unuer 8 to the Cornice ; then divide the Architrave into 18, and the Cornice into 10, and give to each Member, as Iiifpection doth exhibit. Laflly The Proicc non of the Capital is equal to the Projeflion of the Plinth of the Bale to the Column, wanting 1 part; and the Projeftions of the Tenia of the Architrave and of the Cornice, are both equal to their own Heights, Thus much for this Matter, who would have finilhed his Orders tolerably well, had he not ( French m ?n- like) introduced the fmall Aftragals in the Bafe of his Column and crowd- cd into the Tenia of the Architrave one Moulding too much, which make the firlt look little and trilling, and the other heavy and difproportionate. Plate A A, to follow Plate CCLXXXIX. I. Jones. The Compofite Order, by This Plate exhibits the uppermoll Order in the Royal Chapel (commonly called the Banquettmg-houfe) at White-hall, in which the molt remarkable rhrng is, that the Capital has but one Row ofLeaves; there are likewife two Things in this Buildings, which, I cannot but think, are very great Abfurdi- ties, notwithftanding it was defigned by fo great a Matter, and is, by many" efteemed The Principles of Geometry. 3 X 3 efteemcd a perfedt Piece of Architedlure : (i) The Pilalters are diminifhed, which has a very ill Effedl at the Angle of the Building, making the Wall ap¬ pear not to Hand upright upon its Balls, (a) The Entablatures, both of this, and the lower Order, are broke back over every Column and Pilafter, the Ab- furdity of which I have before fhewn, and therefore I fliall not enlarge upon it here. The parts of this Order are determined by Feet, Inches, and Parts, as are all the other Orders of this Mailer. Plate B B, to follow Plate A A, after Plate CCLXXXIX. The Com- polite Order, by Sir C. W ren. This Example is taken from the Infide of St. Swithiris Church in Cannon- Jlreet , which is one of the many, built by this Mailer after the Conflagrati¬ on of the City in 1666 ; the Members are determined by Feet, Inches, and Parts, as arc his other Orders. Plates CCXC. CCXCI. The Compofite Order entire, with its Pedejlal and Capital at large, by Mr. Gibbs. To proportion this Order to any Height, Fig. I. divide the Height into y parts, and give the lower 1 to the Pedeftal ; the other 4 parts divided into 6 parts, give the upper 1 to the Height of the Entablature, and the lower y to the Height of the Column, which divide into 10 parts, and take 1 for the Di¬ ameter of the Column, whofe Diminution is one 6th of the Diameter. To divide the Tedejial into its Bafe, 'Die and Cornice, Fig. II. divide its Fleight into 4 parts, give the lower 1 and one 3d of the next to the Height of the Bale, and half the upper t to the Height of the Cornice. To divide the Members of the Bafe, Fig. IV. Plates CCXCII. CCXCI 1 I. divide its Height into 3 parts, give the lower 1 to the Height of the Plinth, and the upper 1 to its Mouldings, which divide into 4 parts, and fubdivide the ad and upper 1 into 3 parts, and give to each Member, as there is exhibited. To divide the Members oj the Cornice, divide the Height into 6 parts, which fubdivide, as exprelled, and determine each Member, as is exhibited. The ‘Projection of the ‘Plinth to the Bafe is equal to the Height of its Members, as alfo is the Projection of the Cornice, beyond the Plinth of the Bafe to the Column. 'I he Height of the Bafe to the Column is equal to its Semidiameter, which being divided into 3 parts, as in Fig. III. Plates CCXCII. CCXCIII. give the lower 1 to the Plinth, whofe Projedfion is equal to one 6th part of the Dia¬ meter ; the other a parts being each divided into y parts, give the lower 3 to the lower Torus, the upper a and a half to the upper Torus, and to the other Members, as Inlpedtion will inform you. The Height of the Capital, Fig. III. PL CCXC. CCXCI. is 1 diam. and one 6th, which being divided into 7 parts, or 3 parts and a half, and the lower a being each mbdivided into 4 parts,froin thence youlce, that the Heights ofits Leaves, their Foldings, Volutes, andAbacus, are determined. Underneath the Upright of the Capital are Plans of one quarter part of a Column, and of a Pilafter, with the Projedfion of their Capitals ; and under thefe are Views of the Volutes of the Capital, both at its Angle, and in Profile, wherein "tis to be noted, that the Volute is the fame, as in the lonick Capital. Plates CCXCII. CCXCIII. The Compofite Entablature, by Air. Gibbs. The Height and Projedfion of this Entablature is the fame as the Corinthi¬ an ; and its Height being divided into 10 parts, give 3 to the Architrave, as many to the Freeze, including the Aftragal, (which is a part of the Freeze) and 4 to the Cornice. The Height of the Aftragal, with its F'illet, is equal 4 I to The ‘Principles of Geometry. 3 [ 4 - to one 16th of the Height of the Cornice, or one 4th of a fourth part, as in Fig. VI. the Cornice at large is exprefled. The Height of the Cornice being fubdivided into 16 parts, give to each Member as exhibited. Fig. VIJ. is the Architrave at large, whofe Height divided into 4 parts, and fubdivi¬ ded, give to each Member its parts, as exprefled. The Projection of the Cor¬ nice is divided into 4 parts, and tliofe lubdivided again into 6 parts (as again ft the lower part of the Cornice at large is exhibited) from which Divilionsthe feveral Members of the Cornice are determined. This Cornice is very plain, and cafv to be delineated; tis -taken from the /omck Modilhon Cor¬ nice, which differs from this only in the Kind of the Modillion. The Fine A B, in Fig. VI. repreferits the central Line of the Column, and the Line CD, the upright Line over the diminifhed part of the Column, from whence the Projection of the Cornice is accounted The Scale divided into 4 parts, and thole fubdivided into 6 parts, for determining the Projections of the Members in the Cornice, is alfo the Scale bv which the Modillions are pro¬ portioned ; therefore make the Breadth of each Modillion equal to y parts of that Scale, which divide again in 4, and let off 1 on each fide for the Fro- jedtion of the Modillion s Ovolo, or Capping. Thefe being done, the Diftance, from the central i ,ine of one Modillion to the central Line of the next, will be equal to the Scmidiameter of the Column precifelv. Fig. V. reprefents the Import and Architrave to this Order, for the proportioning of which (as alfo the Imports of all the other Orders) take this GENERAL RULE. Make the Height of Imports to Arches equal to 011c 8th of their Opening; which divide into three parts, give the lower 1 to the Height of the Keek or Freeze of the Import, and the upper % to the Mouldings, which divide as the per end cular Scale direfts. The Height of the Aftragal, at the Bottom of the Freeze of the Import is equal to one 6th of the Imports Height, which divide into 3, give a to the Aftragal, and 1 to the Lift. The Breadth of Pi- lafters under Imports Ihould always be equal to the Scmidiameter ol the Co¬ lumn, againft which they Hand, and the Breadth of the arched Architrave Handing on the Import, mult always be equal thereto. In the 7 it/can and 'Donck, divide the Breadth of the Architrave into 3 parts, and give one to the Projection of thole Imports; but in the Ionick , Corinthian , and Cotapo- Jite, divide the Architrave's Breadth into 11. parts, and make the Projection equal to y of thole parts. The Architrave or this, and of the other Orders of this Mailer, hath the Members divided by the horizontal Scales, which are lb very plain and ealy, as to be underltood at the firlt Infpection. Plate CCXCIV. Intercolumnations for Arcades and Colonades, See. by Mr. Gibbs. T h ts l’iate reprefents the Intercolumnations ol Columns in Arcades and 1 Terifyliams, or Colonades, or Portico’s; of which the uppermoft Figure is an Arcade w ith Pcdeitals, the middle Figure an Arcade without Pedeftals, and the lowennoft the Manner of placing Columns, iingle or in Pairs, in Colonades, Galleries, Porticos, &c. whole Mealures are in general denoted by Diameters and Parts. Plate CCXCV. Compofite Doors with Exotick Pedeftals, by Mr. Gibbs. The Figures A and B reprefent unto us three Defigns for Frontifpieces to Doors, and every one very good: That of A is a fquare-headed Door, to be made either with three quarter Columns, or with Pilafters only, as in its Plan The 'Principles of Geometry. is exhibited. The Figure marked B may be finilhed either with a Pedi¬ ment or Balluftrade, and therefore it may be underftood as two Defigns, which, as Figure A may be executed either with three quarter Columns, 01 Pilafters only, as may be alio feen by the Plan; or inftead of Pilafters only, they may have infular Columns placed before the Pilafters, which laft is much the molt grand Manner. Figures C, D, Exotick Pedejlals. B y Exotick Pcdcftals, we are not to underhand Pedeftals natural to ano¬ ther Country, and prelerved here by an artful Imitation of the Clime, as Gardeners do, to prelerve and continue the Growth of Plants brought from foreign parts, which they call Exotick Plants ; but, lays he, / mean they are [uch°as have their Mouldings otherwife formed, and adorned, than the re¬ gular Pedeftals that belong to each Order, which arc ufed generally for lup- porting of Statues or Yales in Gardens, <&c. not but thole Ornaments may be lupported by regular Pedeftals. If we nbferve the Mouldings of the Cornices and Bafts to there Pedeftals, we find, lit. That the Cornice on the Right is colilpofed of a Regain, a Plat-land, or Fafcia, with an Ovalo and a Cornetto under it ; and the Cornice on the Left, we lee, is compofed of a JRegula and Plat-hand, or Fafcia, with a Lima recta, and Cavetto under it. idly, The Bafe on the Right-hand is Compofed of a Cavetto, an ytflragal, a Cinta red a inverted, and a Plinth ; and that on the Left of a Cavetto , a Cima ieBa in¬ verted, a Torus, and a Plinth-, all which are Members not otherwife formed, as this’ Mailer (abfurdly) lays they are, than the regular Pedeftals; but are only differently compoied, with the fame Forms, as in the regular Pedeftals of the Orders ; nor are the Enrichments new; for in many ol the Mailers aflembled in this Work, we find the fame Members, as here fhewn in thefe Pedeftals, enriched with the lame Ornaments; and indeed, tis my Opinion, that from thole Mailers thefe Ornaments have been taken, and given us here as Rarities of Northern Growth. The Height ol thefe Pedeftals, and Manner of dividing them into their Bale, Dado, or Die, and Cornice, this Matter has forgot to l'peak of fo that I am under a Nccclhty of eftablifhing A GENERAL RULE for dividing an Exotick Pedeftal into its Bafe , Die, and Cornice. Divide the given Height into 9 parts, give 3 to the Bafe, 4. to the Die, and a to the Cornice; make the Breadth of the ‘Die equal to its Height, and every Member of common Projection. To divide the Bafes and Cornices into their Mouldings, divide the Height of the Bale into 9, and the Cornice into 6 parts, which fubdivide again, and give to each Member fuch parts as are exhibited in each. Thus much by Leave of this Matter, until he affigns a new and better Proportion, than I have done, for completing thefe Kinds ol Pedeftals. Plate CCXCVI. A geometrical Elevation, Sofito, and Profile of the Corinthian Modillion, by Mr. Gibbs. To deferibe the fpiral Lift of the Volutes to this Modillion, (1) Divide its Height into 8 parts, and from the Angle 7, fet feven of thole parts towards the Left- hand, and then proceed to deferibe the Volute, as lor the Jonick Capital. (1) Divide the Perpendicular, that limits the Projection of the frnall Volute, into the fame Number of parts, and equal to thole, that limit the great Volute on the Right. (3) Divide the upper 4 parts each into i parts, and let thefe 4 parts be confidered as 8 parts; this done, let oft 7 ol thele 8 parts, from the laid Perpendicular Line, towards the Right-hand, on the low- 31 6 The ‘Principles of Geometr y. er Line of the Cima reverfa, from whence draw a Line parallel to the Perpen¬ dicular, to bound the Right-hand Side of the Volute ; then will the Height of this Volute be equal to 8, and its Breadth to 7 parts, which is juft the lame Proportion, as the great Volute hath ; and by the fame Rule defcribe this Vo¬ lute alfo. Now, as the fmall Volute ends at the Point F, and the great Vo¬ lute at the Point L, the next Thing is. How to join thole two Volutes, fo that their Arches may meet each other at Right Angles: To effecT this, draw the 1 fine F L, which bifeCl in D ; divide F D and D L into two equal parts by the two Perdendiculars, E H and B I, which laid perpendicular Lines continue, until they meet the Cathetus of both Volutes, which are the two Centers, on which defcribe the Arch F D and D L, which will join together the two Vo¬ lutes, as required. Note, the Out-line of the Lift is defcribed on the fame Centers, it being concentrick to the firft. Note alfo, that the dotted Curve is tlie true Curve ; the other full-lin'd is the Curve of this Mafter, which, not meeting the two Volutes at Right Angles, is therefore falfe, and his Method of delcnbing it rejedled. Plate CCXCYII. Three Entablatures fir Doors , Windows, or Niches , by Mr. Gibbs. These Entablatures arc of three Kinds, viz. Fig. III. of Tufcan, Fig. II. of Doi/lIc, and Fig. 1 . of Iomck , and which are in general very good. To proportion thefe Entablatures to the Height of any Window, Door, or Nich, divide the Height into 4. parts for the Tufcan and Domck, and into 5 for the Iomck: Make the Height of the Entablature equal to 1 of thole parts, which divide into their Architraves, Freezes, and Cornices, by firft dividing the Height into 3 parts, of which the lower 1 is the Architrave, three + ths of the next 1 is the Freeze, and the remaining 1 and one 4th the Cornice; lubdivide thefe again, that is, the Architraves each into 4 parts, and the Cor¬ nices into 6 , y, and 7 parts, which Infpeftion doth demonftrate. I here mull obferve, that 1 think the fuelling Freeze to Figure II. whofe Cornice is a Do- nek Compoiition, fhould have been upright, and the fwelling Freeze given to Figure 1 . which is of Iomck Compoiition, and therefore are more properly to¬ gether. Piute CCXCVI1I. Profiles of Block Cornices, by Mr. Gibbs. The Manner of proportioning thefe Kinds of Cornices to Buildings, being already declared in Plate XLV. I fhall therefore only add, that thefe Profiles are here given for further Examples, not only to flievv their various Compofi- tions, but how this Mafter perfifts in his erroneous Method of placing his Rulticks under them, which fhould have been 111 the Places of the dotted Ruf- ticks, as I have before obferved. Fig. G' H is an Architrave, Freeze and Cor¬ nice, for a Door, Window, or Nich, by 'Palladio, whofe Heights are thus found divide the Height of the Entablature into 27 parts, give 8 to the Architrave 7 to the F’reeze, and 10 to the Cornice. Plate CCXCIX. Three Entablatures fir Doors, Windows, or Niches, by A. Palladio. These Entablatures have each a fwelling Freeze, but their Cornices and Aichitiaves have their Differences, as thole of Mr. Gibbs, and therefore I can¬ not think, that the fame kind of Freeze is proper to all of them. To find the Heights of the two upper Architraves, Freezes, and Cornices, divide the given Height into 12 parts; give 4 to the Architrave, 3 to the Freeze, and y to the Cornice, in each Example : But, to find the Heights of the Architrave, Freeze, The Principles of Geometry. 317 Freeze and Cornice, in the loweft Example FE, divide the given Height into 8y parts; give 31 to the Architrave, 13 to the Freeze, and 40 to the Cor¬ nice, whole Members divide as the Subdivifions exprefs. Plate CCC. Mouldings for fmall Pannels, b y Mr. Gibbs. Here are five Varieties of Mouldings reprefented, of which the firft is a final) O-volo, over a Cavetto, on an Aflragal ; the fecond, a Cametto, over a Cima reverfa, on an Aflragal ; the third, a Cima reverfa only on an Aftra- gal ; the fourth, a Cima recta on mCavetto-, and the lalt, anOw/s only on an Aflragal, and that on a Cametto. Thefe Mouldings arc divided in their Heights by the perpendicular Divifions on the Right-hand Side, and their Projections by the horizontal Divifions on the upper part ol each : They are in general very good, and their Ornaments are well confidered and very rich. Plate CCCI. Large Mouldings for Pannels or Picture Frames, by Mr. Gibbs. These Mouldings are alfo very rich and well proportioned, and wherein is a good Variety ; the upper one conlifts ol an Ovolo and Cavetto, the le- cond of a Cima reverfa and an Altragal, and the lower of an Ovolo and finall Cametto over the Freeze. The Freeze ol the upper ij enriched with a plain Fret, that of the Middle with a Vitrumian Scroll, (but not the very belt that I have leen) and the lower with circular L.aceings and Roles : The iower or inward Mouldings are alfo different, and varioufly enriched with proper Ornaments, which have a noble Effect. The perpendicular Scales, denote the parts by which their Heights are determined, and the horizontal Scales, which are each equal to one 3d of their Heights, their Projettures. Plate CCCII. The Spamfli Order entire, by S. le Clerc. As this, and the French Order which follows, arc no other than compofed Orders, 1 have therefore placed them to fucceed the Compofite Orders of the preceding Mailers. The great Difference of this Order from other Mailers, confilts chiefly in its Capital and Ornaments of the Freeze, which they en¬ rich with the terreftrial Globe, embraced with Cornucopia's, alluding to the many parts of the Earth fubjetl to them, and which, tis faid, are fo nume¬ rous, fo large, and fo fituated, that the Sun is always fhining upon forne one or more of their Dominions. The Capital is of the fame Proportions as the Corinthian, and its Volutes alfo ; but the Leaves are of another Kind, charg¬ ed with a Kind of Husks, which have no dilagreeable Kft'ea. The Module, by which the parts of this Order are determined, is the Semidiameter of the Bale divided into 30 min. To proportion this Order, divide the given Heights into 30 mod. or parts, ol which, give 6 mod. and 10 min. to the Pedeftal, 19 mod. and ly min. to the Column, including its Bafe and Capital, and. 4 mod. and iy min. to the Entablature. To divide the ‘Pedeftal into its Bap, Die, and Cornice, give 3y min. to the Bafe, 16 to the Cornice, and the Remainder to the Dado or Die. The Height of the Bafe is equal to the Se- midiater of the Column, and the Capital to t mod. and 10 min. and fome- times to 1 mod. and 1 y min. as the Capital at the Bottom of this Plate on the Right-hand is; where the Heights of the Husks, Leaves, Volutes, and Abacus, are each l'lgnified by Minutes. The Heights and Projeaures of the feveral Members in the Bale and Cornice of the Pedeftal, and Bafe and Ca¬ pital of the Column, and of its Entablature, are determined by Minutes, as before in the other Orders. 4 K Plate 3 i 8 The ‘Principles of Geometry. Plate CCCI1I. Five Entablatures of the Spanillt Order, with its Im- pojls , by S. le Clerc. The Entablatures of this Plate are not of the word Compofitions, and therefore T recommend them to the Confideration of thole who delight in new Inventions. Plates CCCIY. CCCV. Intercolumnations of the Spanilh Order, with the Sofito of its Cornice ; alfo the Corinthian Order on the Spa¬ nilh, the Spanifh on the Roman or Compolite, and the Corinthian on the Spanilh and Roman, by S. le Clerc. In Plate CCC 1 V. we have five Examples of Intercolumnations, of which the firft are Intercolumnations proper for Portico’s and Colonades; the next two for Arcades without Pedeftals, and the lower two for Arcades with Pe- deftals, and thofe either with Columns in Pairs, or fingle Columns. The Meafures of every Example are Modules and Minutes. The feveral Figures, and equal Divifions in the Entablature of every Example, denote the Num¬ ber and central Lines of each Modillion, between every two Columns in each Intercolumnation. On the Left-hand Side is a Plan of the Entablature, exhi¬ biting the Manner of placing the Modillions, and returning the Sofito, as well at an external, as an internal Angle. Thefe being the feveral Interco¬ lumnations of this Order, I fliall now proceed to Plate CCCV. wherein we have three Examples of placing one or more Orders over another ; of which the firft is the Corinthian upon the Spani/h , and wherein he is obliged to fet the Corinthian on a Plinth, bccaufe the Height of the Windows cannot ad¬ mit of a Pedeftal, whole Cornice would be too high for their lower parts. It’s to be here obferved, that this Mailer pays the greateft Refpefl to the Corin¬ thian Order, by placing it above, as being (as it really is) the molt noble and elegant of all the Orders that have been yet, or may be invented. In his fecond Example he prefers the Spaui/h above the Compofite , and indeed I think very juftly ; as that the Spanifj Capital is not much unlike the Corin¬ thian, and much more light and airy than the heavy Compofite ; and indeed, had Sir Chri/lopher Wren at St. Tatil's, placed his Compofite Order in the firft Story of that Building, it would have been much more agreeable to true ArchiteCdurc, and to his Reputation alfo. On the Right-hand, at the Bottom, is his third Example, where he has placed the Corinthian trium¬ phant over the Spauifj and Compofite, and which being leen together in one Front, and well executed, mult appear very rich and magnificent. Plates CCC\ I. CCCVI1. The French Order, by S. le Clerc. I n the firft ot thefe Plates we have this Order entire, with the Pedeftal at large, and two Varieties of Capitals very prettily compofed, and its En¬ tablature with a very rich and elegant Compofition. This Order is meafured by Modules and Minutes, as the Spanifh and other Orders are. To propor¬ tion this Order is a Work of fome Trouble ; as that its total Height is equal to 31 mod. and ir min. which mult be reduced into Minutes, which are equal to 94a. Now fuppofe the given Height be 10 Feet, we mult re¬ duce them into Inches, which are equal to 140, and then fay. As 941, the Minutes contained in the whole Order, Is to 140, the Inches contained in the whole Height, So is 3 o, the Minutes in 1 Module, To 7 Inches which is fomething more than 7 Inches and f The The 'Principles oj Geometry. 31 9 The Module being thus obtained, and being divided into 30 min. give 6 mod. %r min. to the Height of the Pedeftal, zo mod. y min. to the Height of the Column, including its Bafe and Capital, and 4 mod. ty min. to the Height of the Entablature. To divide the Tedejlai into its Bafe, 'Die and Cornice. Give 36 min. to the Bafe, 17 to the Cornice, and the Remainder to the Die. The Height of the Bafe to the Column is equal to its Semidiameter, and the Height of the Capital to 1 Diameter and one 6th. The next Part ot^ this Order is its Entablature, of which we ha\*e two Examples in Plate CCCVII. which arc of different Heights, the one on the Right-hand being 4 mod. iy min. and the other 4 mod. and 10 min. in Height, of which the firft has 39 min. to its Architrave, 4Z to its Freeze, and y4 to its Cornice, and the latter hath 38 to its Architrave, 40 to its Freeze, and yz to its Cornice. The Projec¬ tions are allb different, the one having yz and the other yy mm. The Dimi¬ nution of the Shaft is 8 min. We have alfo two Kinds of Imports and Archi¬ traves, with a direCt or front View, and Profile or fide View of the Confole, or Key-Itone, enriched in a very elegant Manner, which, tho' not proper to be ftriftly copied, may be a good Help to Invention. Plate CCCVIII. The Grotefque Order for Entrances in Grotto’s, Her¬ mitages, &c. of my own Invention. Before a Frontifpiece of this Kind can be made, the Diameter of the Door inuft be given, which divide into 4 parts, or one half into z parts, as in this Example. This being done, (1) draw a central Line and a Bale Line at Right Angles to it; on this Bafe Line fet on each Side the central Line the given Breadth, or Diameter of the Entrance, and divide each lialf into two parts, (z) Set on the Bafe Line one +tli part of the Diameter of the Entrance; from the Sides of the Entrance outwards both Ways, as to l i, and from thofe Points draw Right Lines, as l m (on the Right-hand, and the like on the Left) parallel to the central Line, which Lines are the central Lines of the two in¬ ward Columns. (3) Make li equal to half the Diameter of the Entrance, and draw i k parallel to m l, for the central Line of the outer Column. Per- form the like on the other Side, and then will all the central Lines of the four Columns be deferibed. (4) The Diameter of the Entrance being divided into 4 parts, take z parts and a half, and let it up on the central Line of the Fin- trance, y Times as to Z, or rather on one Side, as on the Right-hand is done, where’ the outward Scale, which is equal to the whole Height, is divided into y parts, (y) Give half the lower 1 to the Height of the Sub-plinth, the upper 1 to the Height of the Pediment, above the Entablature, and the re¬ mainin'* 3 and a half to the Height of the Column and Entablature, which di¬ vide into y parts, (as in the lecond Scale is done) give the upper 1 to the Enta¬ blature, and the lower 4 to the Column. The Height of the Entablature be- in" divided into 3 parts, give 1 to the Architrave, 1 to the Freeze, and 1 to the Cornice. (6) Divide the lower 1 of the outer Scale into y parts, and make the Height of the Bafe r t, and Capital z8 Z7 of the Columns, equal to 1 of thofe parts. (7) Divide the Height of the Shaft into 7 parts, and give the zd, 4th and 6tii to the Rulticks. The Height of the CinCture is one 6th of the firft part of the Column, contained between its Bafe and the lowed Ru- itick, as likewife is the Fillet under the Capital. The Height of the principal parts being thus determined, and the Diameter of the Columns being equal to one 7th of their Height, and Diminution to one 4th of their Diameter, com¬ plete their Shafts, giving Projections to the CinCtures and Fillets equal to their own Heights; make the Projections of the Bales and Capitals, from the Cinc¬ tures and Fillets, equal to the fame. The Projection of the Sub-plinth tv is equal to the Diagonal Diltance y z, or q r, and the Projections of the Rulticks are 320 The 'Principles of Geometr y. are terminated by Lines drawn from the Cinctures to the Fillets, parallel to the Shafts of the Columns. To determine the Impoft, divide OP, the' fecond Ruftick, into + parts, and give the middle 2 to the Height of’the Impoft The Projection ol the Impoft is equal to one 4th of its Height. The Center of the Arch being placed on the Line F O, the upper part of the Ruftick con¬ tinued, the Diftance f At is a Right Line, and which is fo made to elevate the Arch; lb that the Projcftion of the Impoft may not take off or echpfc a part of it, and caufe it to appear Ids than a Semicircle, which it ou°ht not to do. 0 To proportion the ftp/-[lone lo the sir oh. Divide GF into 7 parts, and give 1 to one half of the Key-ftone which T divide mto 2 and give 1 for the Delcent or Drop of the Key-ftone, below the Son to of thcAich. The next Work is to proportion the Architrave and Freeze, io that from a certain given Point of View they fhall appear of equal Height. To do this, (1) Let the Face of the Architrave y 29 be drawn whofe Projection from the perpendicular Line of the Column is equal to that of the Fillet. (2) Draw Right Lines from the Top of the Freeze, and Bottom of the Architrave unto the given Point, and the Angle made there, by the meet¬ ing of thofe two Lines, being divided into 2 equal parts by a third Line drawn from the Angle until it meet the Architrave, will cut the Architrave and Freeze into 2 Parts, as at the Points 3 and y, then will y 29 be the Height of the Architrave, and 3 8 the Height of the Freeze, which will appear equal to one another, becaule they are feen under equal Angles, and the Part of the Freeze 3 4 will be hid, and have no more Effect on the Eye than ifitcvas not there, tlieieioie tis evident, that to make the Freeze appear equal to the Ar¬ chitrave at a given Point, the Height of the Freeze inuft be greater than the Architrave, as much as is equal to the Height 3 4. ft is fbr want -of knowing this Rule that the Metops in the Donck Order do appear to be Pa¬ rallelograms of Ids Height than F.xtcnt, inftead of Geometrical Squares as they ought to do; and the fame is alio to be leen in Htlick Windows, which though made precifely fquare do not appear fo, bccaul'e the Projection of the 1 ema of the Architrave ,11 the firft, and of the Wmdow-itools in the laft do echpfe a part ol their Height in every Situation and View, when the Heftht of the Eye is below the horizontal 1 ones of their Heights: Therefore, before a Work is erected, a Coni,deration Ibould be firft made, of the molt common 1 Jace at which the Building will be viewed, and then to give to everv part next above every projecting Member above the Eye, Inch Allowances in Hei-ht as mail preferve the Symetry and true Proportion of the Whole, in the very lame Manner as if the Eye was placed at Right Angles againlt every of its parts at the fame Time. The next Work to be done is to give the Corona its 1 rejection, which is equal to its Height; and its Height bang divided into 4, parts give 1 to its Regain. ThcOw/s on the Corona, 1 am iatisfied, was ne¬ wt tiled above the Corona, or any other Member in its ftead, by the Anci¬ ents (and of whom 1 have already produced fonie luch of their Examples in the preceding Plates) indeed where Pediments are uied, then there Items to be a xxeceflity for a greater I rojebhon to keep oft the Weather, becaule Pediments aie never to be tiled but over Doors, Windows, or other fuch Apertures, which r %‘^r° bC IS ter ^ .'J 011 -- from Rains than any other parts of the Outfides 0 Buildings. The Height of the Ovolo is equal to one yth of the Pediments 1 Light 01 1 itch, and as its Center is perpendicular over the Extreme of the (a rena, and beinga Quadrant, its Projection is therefore equal to its Height As it is luppolcd, 111 this Defign, that the two middle Columns which carry the 1 ednnent, come forward from the Line of the Columns, at the Extremes 1 t lam. and a halt clear of the Pilafters behind them; therefore the Enta- blaturc is broke back from the Outlide of the middle Column, and the fame I rofile completed up to f, as is over the extreme Column up to W; laflly draw the Lines 7 , 1 , A V, &c. and complete the Pediment ; alfo divide the’ Diftance Tlx Principles of Geometry. 32 r Diftance EC into it parts, and make C D equal to 1 of thofe parts, then give x of thofe parts to each Trufs, and x to each Interval, and thus will the whole Frontifpiece be completed. N. B. Perhaps it may be asked; why I have placed an Quote to crown this Entablature, when 1 have declared fo much againlt it in the Tuj- can Order ? To this I anfwer. That whereas thofe Orders are luppoled to be worked very accurately in Stone, the looner their upper parts are decayed by Time, the lei's beautiful they appear; and their bringing the Water down upon the Face of the Corona makes them ufelefs, becaule the Corona is as well able to do that Work of itielf, as when the Ovolo is on it: But as this Order is calculated for Entrances into Grotto's, Subterranean PaJ/ages, Her¬ mitages, &c. the more and fooner its parts are fractured and defaced, the more it is agreeable to thofe Works ; and its introducing the Ram Water upon the Face of the Corona, thereby caufing it to become ol a green Colour, is rather a Beauty than a Deleft; for the more ruinous and antient thofe Buildings feem to be, the nearer they approach the Tafte of the prefent Age. Plates CCCIX. CCCX. The Englilli Order of my own Compofition, geometrically divided hy Air. R. West. I make no doubt but that many will fay, I have given myfelf fomc unne- cellary Trouble in the compoling of this Order, fince that there are already io o-rcat a Variety among the feveral Mafters here affembled : Indeed I mult own, that here is the greateft Collection that hath been yet (or perhaps ever may be) feen in one Work : But then, to the immortal Shame of ourlelves, they are all either the Inventions of Foreigners, or Monkey Imitations of them ; nor has any one Engli/hman, that I know of, ever vet attempted to compofe an Odar 'in Honour to his Country, as the Greeks, Latins, Romans, French and Spaniards have done, and it is therefore that I have taken the Liberty to annex this Order to the foregoing Collodion, which every one is to receive or reieft at his Pleafure, and whole parts are divided as follows: Let A H (m the uppermoft Figure of Plate CCCIX.) reprefent the given Height of an en¬ tire Older To find the Height of its Pedeftal, Column and Entablature, divide A H in the Middle, and thereon, with a Radius of half its Length, defenbe a Circle, as FMAIL, &c. whofe Circumference divide into ix equal parts as at 1 LRQ.H, &c. which is eafily done, thus; fet one Foot of the Compailes in the Point A, and with the fame Opening that turned the Circle, mark the Points P I; in like manner fet one Foot in H, and mark O K, which divides the Circle into 6 parts, bileft each of thofe, and the Whole is divi¬ ded into ix, as required. Draw the Right Line MI, cutting AH in B, then is A B the Height of the Entablature; alfo draw the Right Line N I,, cutting the Line A H in D, then is D FI the Height of the Pedeftal, and B D the Hei°ht of the Column, including its Safe and Capital. To divide the Height of the Pedeftal into its Rafe, Pie and Cornice, divide D H in the Middle, and thereon, with a Radius equal to half its Height, defciibe a Cir¬ cle as q D pH &c. whofe Circumference divide into ix parts, as before, at the Points 0, p, u, w, H, &c. draw the Right Line rp, and it will cut D H in E then is E D the Height of the Cornice to the Pedeftal; alfo draw the RRht fine t u, cutting D H in F, then is H F the Height of the Bale to the Pedeftal; draw the Right Lines .r w and p H, interfediug each other in the Point G, from whence draw a Right Line, to cut the Line F FI at Right An¬ gles, then will the lower Segment of F H be the Height of the Plinth, and the upper the Height ol its Mouldings. To divide the Height of the Column into its Bafe, Shaft and Capital, (1) Divide BD in the Middle, and thereon, with a Radius equal to half its Height, deferibe a Circle as Y W TBS, &c. which divide into .ix equal ° 4 L parts. 322 The Principles of Geometry. parts, as before, draw the Right Lines Y D and Z k (to the firft part on the Left of D) interfering in k ; from k draw k i at Right Angles to B D, bifect / D in m, then is m D the Height of the Bafe, and / D equal to the Diameter of the Column, (x) Draw the Right Line W R, cutting B D in Y; bii'ecl B V in C, then is B C the Height of the Capital. The Diminution of the Shaft is one 6th of its Bale. To divide the Entablature into its Architrave, Freeze and Cornice, Divide A B in the Middle, and thereon, with a Radius equal to half its Height, defcribe a Circle; divide the Circumference into ix equal parts, as before; (the Points are not all marked in the Figure, as not being all want¬ ed) draw the Right Lines f a and b d, cutting each other in c, from c draw a Right Line to A B, and at Right Angles thereto, then will the Diftance from that Point to A be the Height of the Cornice; draw the Right Lines f d and a B, interfering each other in e, from whence draw a Right Line at Right Angles thereto, then will the Diftance from that Point to B be the Height of the Architrave, and the other Intermediate between that and the Cornice is the Height of the Freeze. As I have thus ihewn the Manner of dividing out the principal parts of this Order, I ftiall now proceed to the Divilion of its particular Members ; and fir ft, thofe in the Bafe of the Pedeftal, which are reprefented at large at the Bottom of this Plate. To divide the Mouldings of the Bafe to tbe'Pede- flal, make r s equal to one jd of the Height, and draw the Line noi both ways at Pleafure, and in any part thereof, as at i, with the Radius r s, de¬ fcribe a Quadrant, as x 10, and on the Point 10, with the fame Radius/ the Quadrant 4. 1, interfering the former in 3; then on the Point 3 take the neareft Diftance to the Line s 10 1, as to g, and with that Radius interfeCl the two Quadrants on each Side, and through thofe InterfeCtions draw the Fillet to the Cavetto. Note, this Method of taking out a Fillet from a given Breadth, as from r s, will be often done in the dividing of the other Members; and as I (hall not again explain the Manner of doing it, you mull well underftand it (which is very eafy to do) before you proceed further, and which being done on anv Point in the Line P 13, as at 1 3, with the Radius s u, defcribe a Quadrant, as 1 ix, and on ix, with the lame Radius, defcribe the Arch 13 11, then thro’ the Point 11 draw the lower part of the Aftragal. The Depth of the Cima ieda inverfed is half the Remainder, and its Fillet is one 4th of the other half, the other three qths is the Torus. The Projection of the Plinth is equal to the Height of the Mouldings, and the other Members finifli in common couife. Secondly, To divide the Mouldings in the Cornice of theTedeJlal, with the Radius A B defcribe the Circle D A G, fyc. and on the Points A and C, with the fame Radius, defcribe the Arches F B H and DBG, interfeCting the Circle in D G F H ; through D G draw the Head of the Flat-band or Fa- Jcia, through B draw the Head of the Ovolo, and through F H draw the Head of the Cavetto, then will thefe Members divide the Height into 4 equal parts ; in the lower fourth part make a geometrical Square, as n Ip q, draw the Diagonals n q and Ip, then take half the Diagonal, as 1 0, and let it down towards q, which is the Height of the Cavetto, the Remainder is the A- ftragal. Out of the 3 upper parts take the 3 Fillets, as taught befofe, and the Remainder will be the Ovolo, the Fafcia, and the Cima reverfa, with its Regula included, which Regula is one 4th part thereof As the Projection ol the Mouldings of both the Bafe and Cornice depend on the Diameter of the Die of the Pedeftal, and as the Die is regulated by the Projection of the Bale to the Column, we mult therefore, in tile next Place, proceed to the Divifion of the Mouldings of the Bafe and their Projections; make A» equal to tlie Semidiameter of the Column, and complete the Geometrical Square b a A u , on a, with the Radius a b, defcribe the Quadrant b mu, and draw the Diagonal a A, interfeCting the Quadrant in m ; Divide the Arch b m into 4equal The Trinciples of Geometry. 323 4 equal parts, at the Points F, e, E, through which, from the Point a, draw Right Lines to cut the central Line, through which the Height of the Plinth, Torus, and upper part of the Scotia rnuft pais, draw both Ways at Pleafure. Out of the upper part, take a Fillet at F, the Operation of which is on Left-hand at A BCD; take another at e, out of the third part, the Opera¬ tion of which is on the Right-Hand, and the Remainder is the Scotia. Make ivn equal to nu, then wu is equal to the Diameter of the Column. Di¬ vide wn into 3 parts, and make xw equal to one of thofe parts, then will x w be tlie Projection of the Plinth, before the Upright of the Column. Draw x Q_ parallel to the central Line, which will determine the Projeclion of the Die. The Projection of the Plinth to the Bale of the Pedeltal, is equal to the Height of its Mouldings ; and that of the Cornice to the Pe- deftal, to its own Height. The Projection of the upper Torus in the Bale to the Column, is equal to the Center of the lower Torus. To deferibe the Scotia, the Line 4 32;/ being drawn through the Center of the great Torus, draw the diagonal Line 3 z, which bile cl in the Point 1, which is the Center of the upper Curve, and the Point 3 is the Center of the lower Curve, U'hich together form the Scotia. The Projection of the CinCture is equal to the Center of the upper Torus, as alfo is the Fillet under it; and thus are the parts of the Pedeftal and Bafe deferibed. The next in order is the Capital and Entablature, which are reprefented in Plate CCCX. and divided as fol¬ lows : Divide the Height of the Capital into 3 parts, the lower z determines the Height of the Oak Leaves, the others, the Volutes and Abacus, of which the Abacus is equal unto two yths, as on the Left-hand is exhibited. To find the Projection of the Capital, divide the Semidiameter of the Co¬ lumn at its Aftragal into y parts, and makeg/a, the Projection of the Aba¬ cus, equal to 3, and the extream Projection of the Ovolo 3 and two 3ds. V.-Pray, Sir, JVhat do the Oak Leaves of this Capital reprejent ? M. The Blefting and Strength of the Nation : By the Blelfing, I mean his prefent Majefty, who could not have been our Sovereign Lord, had not a glorious Oak preferved the facred Perfon of Charles II. from the Fury of his Enemies, fyc. By the Strength of the Nation, I mean our naval Forces and Trade, which are both dependant on the Oak, and which no Nation in Europe can parallel for Strength and Duration. P. / fee that you have introduced Palm Branches, afeending out of the Cornucopia s, in the Place of the Corinthian Volutes, pray what do they re¬ prejent ? M. Peace and Plenty, which we have always enjoy’d fince the happy Re- ftauration ; but more particularly at this Juncture, when almoft every Na¬ tion in Europe, befides our own, are feeling the Miferies of bloody Wars, wherein many Thoufands have been llain; we, by the prudent Management and great Care of his moft facred Majefty George II. live in Plenty, fleep in Peace, arife at our Pleafure, enjoy our Liberties, and keep, or difpofe of our Properties, according to our own free Wills ; BleJJings that no other Peoptb in the World enjoy. 1 ’. / alfo objerve, that in the Abacus, there is a Star and Garter, injlead of the common Ornament of the Eifh Tail, &c. Pray what does that al¬ lude to ? M. Honour, an Ornament more peculiar to this Nation, than to many others in the Univerfe, and yet is known but to very few of its People. I fhall now proceed to the Divifion of the Members in the Entablature. (1) To divide the Architrave into its Members, divide its Height into tw'o equal parts at N, make CL equal to CN, and draw LM parallel to CN; through the Point N, draw the Line NO, both Ways, of Length at Plea¬ fure, 3 H The Principles of Geometry. fare, cutting LM in O ; draw the diagonal Lines CO, NL, and NM, Of; on N, with the Radius Na, defcribe the Semicircle abce, and, with the fame Opening, on e defcribe the Arch Ci d, interfering the Arch eb in the Point c ; through the Points f, a, b, draw Right Lines parallel to the A- bacus of the Capital. Now, as the Line / includes the Height of the fir It Falcia, and firft Bead, take out the Height of the Bead, by the fir It part of the Method for taking out a Fillet. The Like is alfo to be taken from the fecond Falcia, as the interfering pricked curved Lines on the Left of the cen¬ tral Line reprefent. The Height of the Regula is one gd of the Tenia ; the Projections of the parts of the Architrave are found thus; divide the Height of the Tenia, viz. the Regula, Cima, and Bead, into y parts, and let them off on the horizontal Line, the middle Falcia projedls I, the upper Fafcia z, the upper Bead 3, the Cima y, and the Regula 6 parts. Alfo, (z) To divide the Members of the Cornice, divide the Height AB into z equal parts, and on its Middle, as a Center, defcribe a Circle ; with the fame Ra¬ dius on B, intellect the Circle, and from its Center, through the InterfeCtion, draw a Right-Line that fhall cut the Line IK in S: This Line IK muff be drawn parallel to A B, at the diftance of one half AB, or the Seinidiameter of the Circle ; make SR, RQ, and Q_P, each equal to SK ; and through the Points R, Q_, P, draw the Right Lines / R, rQ, and o P, at Right Angles to A B. Divide TB into 3 parts at VW, and through the Points V, \V, B, draw Right Lines at Pleafure, parallel to / R; alfo, through the Point A, draw the Line HA parallel to the former. Divide IP into 4 parts, give the upper 1 to the Regula, and the other 3 to the Cima reCla. Divide PQ_ into 3 parts, give the upper 1 to the Cima reverfa, including the F'illet, (which take out, as before taught) and the lower z to the Corona. Divide QR into 4 parts, give the upper one to the Cima re-ver/a over the Modilli- ons (out of which take the Fillet) and the lower 3 to the Height of the Mo- dillion. Out of VT take a Fillet, and the Remainder is the Ovolo; take the fame alfo out of \ T W and W B, and the Remainder is the Dentils and Cavetto; the Breadth of each Dentil is two 3ds, and the Interval one 3d of its Height. The Projection of the Cornice is equal to its Height : The Freeze is compofed of z Cima s, or may be made vertical or upright at pleafure. Plates CCCXI. CCCXII. Fractional Architecture, by Mr. Edward Hoppus. These two Plates contain an Attempt to proportionate the five Orders in Architecture, which this Author publilhed not long iince, under the mil- taken Title of 'Proportional MrchiteClnre bp equal parts, inllcad of Difpro- portional yIrchitecfure by Fractional parts, which in Truth it is, as will ap¬ pear by the following. J11 the firft place, before we can proceed to form an en¬ tire Order, we mull aflign a certain Number of parts, into which its Height is to be divided, whereby we may proportion the Whole, and which, ac¬ cording to this Author, muff be found as follows. To proportion the T ttjean CDorick lonick Corinthian Compofite C‘° i) Order, to any given.Niz 1/ and take 1 for the Height, divide the^i; }i Diameter of the given Height into j)iq Column. Here, in his very firft fetting out, he perplexes the young Beginner with fractional parts. To proportion the Tufcan Order, the Height muft be divided into 10 parts and three 4ths, which is not eafy to be done by a young Beginner ; and as he has not been l'o kind, as to lay down a Pro¬ blem, How to divide a Right Line into Integers and fractional parts , I very much doubt, if he is a Matter of fo little Geometry, as to know it The Principles of Geometry. 325 it himfelf. To have made the Manner of dividing the Heights of the Or¬ ders familiar to a mean Capacity, he ihould have faid, Divide the Height of the And of thefe Parts take 4 for the Diameter of the Column in the Tufcan and Corinthian Orders, (becaufe the Denominator of the Fractions is 4) and 3 for the Diameter of the Derick, Ionick, and Compojite , for the fame Realon. The next Thing to be considered is, the Proportion of the Orders with relpeft to each other, wherein we fhall alfo fee fomething very blocking and difproporti- onate. But, before I proceed thereto, I mull reduce all his fradtional Altitudes to one Denomination, that thereby the meaneft Capacity may be able to judge of the great Dijcord, that is here offered to the World under the Name of 'Pro¬ portional Architecture by equal Parts. The Parts afiign- cd for the Tu[can Dorick Ionick Corinthian Compofite 4th Parts. are equal to 43 )49 t |f 4 f 5-8 , C are indeed Dorick, the Leaves only excepted, which makes them compoled Capitals, and of which that marked B is the molt beautiful of all that i have vet l'cen. The Bales D and G are both very noble, that ol D is the Attick Bale, and is Ihewn here only on account of its Ornaments; but that of G is a compoled Bale for the Compofite Order, and one ol the very belt I have leen anv where. In Plate CCCXIV. there are three very curious Capitals and three Bales of which that marked C is always to be reieftcd, on account of thole fmall’Aftrasrals between the two Torufes, which have not only an ill Effect in their being very lmall, but indeed, when fuch a Bale Hands very much above the Eye, there are no other Members vifible but a continued Line ol repeat¬ ed round Mouldings, leemingly fitting immediately on each other, without Separation. The Bale D having the /ingle Ajlragal next under the upper 7 o- ms and the double BJlragal immediately on the lower Torus, has fomething in it very elegant, and-worthy of our Attention. The Attick Baje H is from Barozzio and is given here only to Ihcw the uncommon Manner ol under¬ cutting the upper Torus, making the Face of the Fillet under it reclining, in- Head of perpendicular, as is ufually done. Plates CCCXV. CCCXVI. CCCXVII. Compofed Capitals, by John B E R AIN. In thefe three Plates are contained twenty-four different Capitals of very great Invention, and different from all that we have been yet lpeaking of, and which are of fuch various Heights and Kinds, that without any great Difficulty there may be five chofen, which together are much fupenor (in In. vcntion) to thole commonly ulcd to the five Orders by all the foregoing Ma-_ llcrs. To find the T report ions of thoje Capitals, divide the Diameter of their Shafts next under their Aftragals (they being defigned chiefly for Pila- llers) into 6o min. and therewith mealure and number the Height and 1 ro- iedturc of every part at your Pleafure. The many Ornaments with which thcv are enriched are very helpful to Invention, not only for compofing of new Capitals, which every Man that can has a Right to do ; but many other noble Ornaments which naturally follow in the Courfe of Study. Plate CCCXVUI. and Plate S, to follow Plate CCCXVI ii. The Orders of the Perfians and Canatides, by Air. Evelyn, S. i.e Clerc, and J. Gouion. The Terfian Order is no other than the 'Dorick, but inftead of Columns its Entablature is fupported by aged Men ; and that o! the Cariatides, is the lo- nkk Entablature fupported by Women. I muff own I think them both un¬ natural altho' in great Efteem by the Ancients; and indeed it doth reprelent very ftrongly a natural Cruelty and tyrannizing Inhumanity in the Inventors, and whereloever they have been executed they could not be but very blocking to every judicious Eyd. The Tribunal in Plate S, done by John Gouion in the Swifs Guard-chamber, in the Louvre at Taris, may have been pleahng to a French Tyrant, but could never be fo to a Lover of Humanity : But, beiides this ridiculous Cuftom, of turning Mankind into Columns, there have been others The ‘Principles of Geometry. 327 others who have ufed Angels with no left Severity, as reprefented by I e Clerc in Plate CCCXVIII. which is yet more fhocking than the former; therefore be it underftood, that I reprefent thefe Orders not as Examples for Practice, but to fhew, that the Ancients had their Follies, in many Cafes to as great a Degree as the Moderns. Plate T, to follow Plate S, after Plate CCCXVIII. The Manner of defer thing wreathed Columns, by Andrea Pozzo. This Mailer gives us three Methods for wreathing of Columns ; they?/;/? Method is reprefented on the Right-hand by Number I. and is wreathed as follows, viz. (1) Defcribe the Z )orick Shaft, which is here reprefented by the dotted Lines, that go up from the Cinfture B to the Aftragal A. (x) From A draw a horizontal Line, of Length at Pleafure, and therein, from the Point A, let off 9 Diameters of the Column, from which Point draw a Line to B, and on which, with any Radius, defcribe an Arch, as AP. (3) Divide the Arch AP into ix equal parts, thro’ which, from the Center, draw Lines to meet the Outfide of the Column, as at h g, &c. and from thefe Points draw Right Lines parallel to the Cinfture, as h /, g k, &c. (4) The whole Height of the Shaft being divided into ix Parallelograms, defcribe Equilate¬ ral Curves on every of their Sides, as the Curve h g on t, and l k on m, &c. and they will compleat the Contour, or Out line of the Wreath, as required. The fecond Method is reprefented by Fig. II. as follows. Divide the Height of the Shaft into 3 parts, and fet on the Line from the Bale 1 part from the Upright of the Shaft (which is fuppofed to be firft delineated) to C, then on the Points C and D, with the Radius C D, lnterfedl in E, on which, as a Cen¬ ter, defcribe the Arch D C, which divide into ix parts, and from thence draw Right Lines parallel to the Bafe, cutting the Out-line of the Column in the Points b, a, &c. then will the Shaft be divided into ix Parallelograms, as b , e, a, d , &c. This done, divide the Side of every Parallelogram into 4 parts, and with 3 of thofe parts defcribe Ifofceles Triangles , as b c a and efd, &c. on whole angular Points, as c and /, defcribe the Curves nccefl'ary to com¬ plete the Shaft, as required. The third Method is reprefented in Fig. III. wherein tis fuppolcd, that the Out-line of the Shaft is delineated as before. (1) Draw G F, and make L. P’ and I H equal to the Diameter F H, and draw the Line L I, which will be divided into x unequal parts by the Line G P’. (x) Make I N equal to the greateft Segment or Part of L I, and draw M N parallel to L I. Now it is to be obferved, that the Line F G, as it afeends, ihortens the greateft Segment of every horizontal parallel Line in a gradual Manner ; and it is from thofe Segments that the Heights of all the Paralle¬ lograms are determined, that is to Jay , the Length of the greater Segment ot every horizontal Line is the Height of the next Parallelogram above it ; and their Diagonal Lines being drawn out until they interfedl each other, as in the Plate is reprefented, their Angles of meeting are the feveral Centers, on which the wreathed Curves may be defcribed, as required. Plates V, T andW, to follow Plate CCCXVIII. Divers Defigns of Obelifjues, by S. Serlio, compofed with Dejtgns of modern Ar¬ chitects. In the firft of thefe Plates are four Defigns for Obelifques, R, Q_, P, O, by S. Serlio , wherein is a Ample Grandeur and Majefty, not to be found in the other ; on the Left, or in thofe of Plate W, which are poor Inventions of modern Pretenders, and whole Pedeltals with fmall Mouldings have no Affinity to the plain folid Body they fupport. The proper Bafement for an Obelilque is a plain Cube, whofe Diameter muft be one 3d greater than the Obelifque at its Bafe, which may be ufed alone, or placed on a 'Plinth The Principles of Geomf.tr y. of half its Height, either with or without a fquare, plain Capping (of one 4th its Height if with a Capping) on the Pedeftal mult be placed a fquare 'Plinth, equal in Diameter to the Die of the Pedeftal, and in Height equal to the Se midiameter of the Obelifque, on which the Obelifque mult be placed ; but if without a Capping, with a Plinth only, then the Cintture of the Obelifque mult Hand immediately on the Cube. The Height of the Obelifque (to be grand) mult be 11 Diameters, diminifhed one 4th, and finilhed with a Right Angular Vertex. An Obelifque thus compofed will appear noble in every View, which thofe whole Pedeltals are made with final! Members (as in Plate W) cannot do, they being inconfiltent with the Majejly and Grandeur which thefe monumental Pillars Ihould reprefent, and are only lit for Lamp-pofts in Streets. Plate X W, to follow Plate W, after Plate CCCXYIll. The Man¬ ner of building Pilajlers of Stone againjl Brick brails, by S. Serlio. The three Figures, reprefented on this Plate, exhibit the ancient Ways or Methods for facing or incruftrating Brick Walls with Stone,’ and placing one Order over another; wherein is feen, that the Projeftion of the Pedeftal, to the upper Order, is equal to the Upright of the Freeze of the lower Order, and which is abfolutely right, and good Architecture, altho' it doth require the lower Wall to be very fubftantial. To incruftrate Brick Walls, we Ihould firft complete the Brick-work, and let it be entirely fettled before we begin to face it with Stone, otherwife the Work cannot be found, nor will the Walls ftand perpendicular, altho’ built lb with the grcatelf Care; and this is often feen in many Buildings incruftrated with Stone, which have been carried up with the Brick-work, wherein the inward Side of Brick, having perhaps 8, 10, 11, or more Joints of Mortar, to one of the Stone Outfide, doth, in its dry¬ ing, fettle 8, 10, or more Times as much as the Stone; and as the Stone Facing muft, at proper Places, have heading Pieces laid into or a-crofs the Wall to bind in the outer parts, if thofe Pieces do not break by the Weight of Bricks they fuftain, they muft change the horizontal Pofition, they were firft laid in, into an inclining one, or the Brick-work beneath them muft fe- parate and fettle from them ; and let either of thefe be the Cafe, the Wall cannot be found. If I miftake not fomething of this Kind may be now feen in the Walls of St. Giles’s in the Fields, lately rebuilt, which had the Stone and Brick carried up together, that caufed lbme Irregularities by their differ¬ ent Settlements. Plates CCCXIX. CCCXX. Ba/lujlrades, by S. le C1,erc. In the firft of thefe Plates this Mailer gives us the Balluftrades proper to each Order, and the Manner of placing them with Pedellals, from which the ingenious Student may compofc others equally as good; thofe at B and A, (jyc. are of different Kinds, not particularly adapted to any Order, but to be u- fed difcretionallv. On the Right-hand, at the Bottom of the Plate, is an anci¬ ent Ornament ufed inftead of Balluftrades, called Circular Interlacing, which has a very good Effett, as alfo have the other fix Kinds on the Left of Plate CCCXX. where are other Examples of Balluftrades, as well raking in the attending Range of a Stair-cafe, as level in their Landing 'Places, Balconies, &c. as B A, B A C. The Balconies F, E, D, are Deligns for Iron-work, to be uled where the fiighteft Balluftrade would be too maflive. Plate The Trinciples of Geometry. 329 Plates CCCX.XI. CCCXXII. Ballufirades, Balconies, and their Truf- fes, by Mr. Gibbs. In the firft of thefe Plates we have five Examples of Ballufters, which are proportioned to three Kinds of Balluftradcs, and may be applied to the Dorick, Ioftick and Corinthian Orders; the lowermoft Figure may be applied to the Dorick. The Height of the Pedeftal, 111 this and the other Orders, mult be equal to the Height of the Entablature it Hands over; or, for want of an Entablature, which fometimes may happen, we mult fuppofe an Enta¬ blature of the Order we approve belt of, to be placed on its Column only, under the Balluftrade, and make the Height of the Ballultrade equal to it. As the Height of the Ballufters is equal to the Height of the Dado or Die of the Pedeftal, divide its Height into 8 parts for the Dorick, 9 or 10 (as in the middle Example) for the lonick, and into ix (as above) for the Corinthi¬ an. The Diameter of the Bale of each Ballufter is equal to r parts of its Height, and their Diltance to 1 part; the Thicknefs of each at its Neck, and at the (everal Scotia’s, is equal to 1 part, as may be feen by the half Ballufters deferibed at large on the Sides of the Plate. By the feveral Subdi- vilions the parts of each of their feveral Members are determined. The Cen¬ ters, for deferibing the Out-lines of the curved parts of each Ballufter, are the Points where the feveral Diagonal Lines meet on the central Line of each Ballufter. Plate CCCXXII. contains divers Defigns for Balconies, which are reprefented as well in Profile as in Front, that thereby we may the better judge of their different Truffes made for their Support. Plates CCCXXII I. CCCXXIV. The ancient Fret Ornament, Vitruvian Scrolls, Interlacings, Eggs and Darts, by S. S e r l i o, Mr. Evelyn, and Mr. Gibbs. The firft of thefe Plates contains a very great Variety of the ancient Fret Ornaments , by Mr. Evelyn, and Mr. Gibbs, with the Manner of turning them at an Angle, which indeed is the only Difficulty in making this Ornament. I11 Plate CCCXXIV. Fig. G is another Fret by Mr. Gibbs , and in Plate CCCXLj. are two others, and both returned at an Angle, of which that marked A is of my own Invention, and that marked B of Mr. Edward Stephens , an inge¬ nious Cabinet-maker in Condon. It is to be obferved, that as the Breadth of the Fillets (which are the unfhadowed parts) are equal in Breadth to the fhad- ed Diftances between them, therefore, to make either of thefe Frets in any given Margin, Sofito, or other Breadth, divide it into as many equal parts as the fhadow'd and unfhadow'd parts contain ; fo the upper Fret of Plate CCCXXIII. is divided into 7 parts, and the lowermoft into 9 parts, &c. The Figures A, D, F, Plate CCCXXIV. are divers Kinds of an Ornament, called the Vitruvian Scroll, perhaps from its being invented by Vitruvius. That marked A contains three Varieties by Mr. Gibbs, the other two are by S. Serlio. The other Figures marked B, C, E, H, I, are marginal Ornaments as the Frets, being Interlacings of various Kinds, of which thofe marked B, C, F, are by Serlio, and thole marked H, I, by Mr. Gibbs, as likewife are the Eggs and Darts Fig. K, which are thus deferibed ; divide the Height of the Ovolo into 9 equal parts, and fet 7 of them on each Side for the central Line of each Dart. The fmall Stars on the Left-hand are the Centers of the Curves that form the Egg, and the dotted Lines fhew the Limits of each Curve. The Darts are formed by the dotted Lines which interfeft each other, the one from the Top of the central Line of the Egg to the Bottom of the Dart, the other from the fecond Divilion, up the Dart, to the Top of the Dart on the other Side. Fig. L reprefents Husks and Leaves in the place of Darts, which are often ufed in many Works for Variety. + N Plates The “Principles of Geometry. Plates CCCXXV. CCCXXVL Two Frontifpieces for Gates, by M. Angelo. These two Frontifpieces are compofed of all the Abufes in Architecture that this Mailer could polfibly invent; and altho’ they are lb vaftly ltrangc, and of Inch uncommon parts, yet there is a Grandeur in the firlt,- and a No- blenefs in the l'econd not undeierving our Regard. To view and confider the monllrous Bale of the Pilalters in the firlt, whole Height is equal to their Diameters (without Precedent) its long necked Capital, clumfy Impoit, low and finall broken Architrave, furprifing high Freeze, and that broken over the Pilalters, its aukward Cornice, open Pediment, with its little infcriptional Table, and fuper Pediment above that, one would think that its Inventor had never feen or heard of regular Architecture, and yet the whole taken toge¬ ther, without confidering its parts, makes a verv grand Appearance, if we can believe its upper part is not too mallive for the lower. Fig. A, in Plate CCCXXVI. is a Profile of this Dcfign, wherein we fee, that the open Pedi¬ ment has a very great Projection before the fuper ' Pediment , which is alfo very confiderable. The l'econd Defign, in Plate CCCXXVI. being taken to Pieces, is as full of Abfurdities as the firlt ; lor here are mallive Rulticks en¬ vironing the Shafts of the Columns, feparated by Altragals let very dole toge¬ ther in the lower parts, as if they had forne mighty Weight to lultain : The Entablature broken over each Column, the Bed-moulding cut in two by the Key-Hones in the Freeze, the Cornice crowned with an open Pediment, with a Table juft fit to contain the Name of its Architect, behind which rifes a kind of Parapet, finifhed with Heads, like thole on Temple-bar : But the whole being taken together, and not critically reviewed, makes a noble and grand Ap¬ pearance, w hole Profile is reprelented on the Right-hand. Plate CCCXXVII. Iron Gates with Tufcan Piers, after the French Manner. The Piers of this Gate, having two three quarter Columns in Front, are very grand, and the Entablature being continued over the Iron-work, fupport- cd by the Side Pilalters, is very Itrong and fecure. The Iron-work is very light and airy, and if liable to any Confute it is tor its Work, which is rather too rich for the Tufcan Order. Plates CCCXXVI1I. CCCXXIX. CCCXXX. Three Examples of Iron Gates for Dorick Piers. The firlt of thefe Deiigns reprefents a Pair of curious Gates, between two Piers, compofed of “Dorick Pilalters, wherein tis to be obferved, that if their Bodies be made fquare, as in the Plan, they will cut into one another, and are therefore abfurd. This I mention here in order to prevent fuch Abufes for the future, and which at this Time almolt every one who pretends to Architec¬ ture is fond of running into, altho’ nothing can be fo abominable; for two Bodies cannot potlels the lame Place at one Time; therefore I recommend the placing ol entire Pilalters, either tingle or in Pairs, and not to fplit one Pilaltcr into r parts, and place i on each Side of the whole Pilalter, as here is done. The carrying of the Entablature over the Gate is a very Itrong Way of building, but is not fo elegant and airy as Plate CCCXXIX. and CCCXXX. where the Entablature is broke all round its broken Pilalters, of which that ol Plate CCCXXIX. hath exactly the fame Abufe in its Pilalters as that in Plate CCCXXVill. and which is more vifible here, by feeing half Triglyphs on each Side the projecting Freeze of the entire Pilalter, which are very block¬ ing to coniidcr; whereas if the two Pilalters had been made entire, and the Enta- The Principles of Geometry. 33 r Entablature continued front the one to the other, then the Triglyphs would have been entire, and the Whole of grand and noble parts. Before I proceed any further, 1 mult beg Leave to advertife, that the Defign of thefe, and the nine following Plates isto reprefent the many curious Defigns for Iron Gates, of which wc have none lb noble yet executed in England. ; and lajlly, to expofe the breaking of their Piers into fo many fntall parts, which makes them look not only very mean and poor, but con¬ temptible in the Eye of every Judge, and which, to the eternal Shame of our prelent greateft Architcfts, is daily done by them, as well in their capital, as in their fmall Buildings. Plates CCCXXXI. CCCXXXII. CCCXXXIII. CCCXXXIV. Four Examples of Iron Gates for Ionick Piers. P as sing by the Defign for the Iron Work, which is very rich, we come to the Piers of Plate CCCXXXI. which are rufticated Pillars, wherein the Co¬ lumns are inferted, and crowned with the Entablature of their Order, and which may be pleafing to many, and in fome Degree juftified, as that the Vo¬ lutes of the Capital of the Column are not continued over the Rufticks; yet I do infill, that if a Pair of Columns were placed on each Side, with a continued Entablature, they would be more grand, and far exceed all that is afforded by the mean Variety of fmall Rufticks on each Side the lingle Columns. The fecond Example in Plate CCCXXXII. hath rufticated Pilafters, with half Pi- lalters on their Extreams, with their Entablatures alfo, which is another Kind of Abuie to be avoided ; therefore in all Inch Defigns omit the half Capitals on each Side, as alfo the Freeze and Cornice over them, and then the Column or Pilafter, with its Entablature, will be entire. The Mouldings of its Bale ought not to be continued and returned, it being a Diminution of the Beauty, that fhould be only feen in the Bafe of the Pilafter. The Iron Work of this Gate is very magnificent and rich. I now proceed to Plate CCCXXXIII. whole Iron Work is of great Invention, but the Piers are, without Exception, the very worft I have ever feen, the Front Pilafter being backed by no lefs than two half Pilafters on each Side; and, to play as many Monkey Tricks as was poflible, we have an efcallop'd Shell ftuck up in the Stead of an entire Capital, with a Cove on each Side. It is my real Opinion, that fuch abfurd Compofi- tions, as this now before us, arc the Delight and Study of many who profefs themfelvcs very great Judges ; but if we do but confider the great Number of Angles and parts, into which it is divided, and the poor Effects, that their tall and fiender Dimenfions have, wc may foon be affined, that the Abilities of their Defigners are no more capacious than themfelves ; and of this Sort I believe the celebrated Mr. Archer to be the capital in this Kingdom, who, in lus feveral Buildings, has expofed to publick View more Abfurdities, than all the Architects, antient and modern, did before. The laft Example of this Order is Plate CCCXXXIV. which is a very grand and noble Defign, and wherein all the aforef.iid Errors are excluded. The Columns, which ftand de¬ tached from the Walls, have a noble EfFedl, as likewife have the others, that are inferted in the Walls. Plates CCCXXXV. CCCXXXVI. Two Examples of Iron Gates for Corinthian Piers. fu e Piers to both thefe Defigns are Parallelopipedons, the one having a Pilafter, the other a Column inferted in them, with their Entablatures conti¬ nued, and which is to be juftified, becaufe the parts of the Parallelopipedons cn each Side of the Pilafters and Columns, tho’ crowned with their Entabla¬ tures, do not appear as half Pilafters ; and as in the Bafes to both Defigns, the Torus and Scotia of each are maintained no where, but under the Tilajlers The 'Principles of Geometry. 332 , nH Columns ■ the upper Torus and Tlinths therefore may be allowed to conti¬ nue throughout the whole Walling on each Side in both Deligns. Plates CCCXXXV1I. CCCXXXVIII. Two Examples of Iron Gates for Compolite Piers. In the firft of tliefe Examples the Columns hand clear from the Pier, but in the laft it is inferted. Both thefe Defigns are very good and grand, the Modillions in the laft only excepted, which are entirely falie ; tor as they re- prefent the Ends of foitts, how can they be cxpecled here, where the Ln- tablature is broken all round, and where no Joiits can be employed . Thciclore to place Modillions in the Entablature of a Pier is abfurd. Plate CCCXXXIX. A Door with Compolite Pilajlers, by Vitru- I have already r • refented a Door of this Kind by Vitruvius, which was of lefs Diameter at the fop than at the Bottom, and generally tiled to 1 cm- ples for the Sake of Ihutting themfelves. The Height of this Door is 1 Dia¬ meters and one +th, and its Diminution two ijths of its Diameter at the Bottom. The Pilafters on each Side are compofed ot the Dorick and Corin¬ thian Orders, the Abacus being Dorick and the Leaves Corinthian , and which together form a very agreeable Capital, worthy of our Notice. Plate CCCXL. A femicircular and circular Window, with a Door, by Vitruvius and S. Serlio. The uppermoft Figure reprefents a femicircular Window divided into three T.if'hts bv two Munions, which molt ot the Architects ot this Age call a Pal- ladian Window, as if "Palladio had been its Inventor: But, as in the Works 0 f Vitruvius, this Window is reprefented, tis plain that 7 cilladio was not its Inventor, and therefore I do aienbe its Invention to Vitruvius and call it a Vitrmian Window. As to the Merit ot its Invention, I think tis not worth contending for; for if we confider the Strufture of a Semicircle, it will appear, that to place Munions for its Support is really needlefs, and indeed ridicu- Ions becaufe the Strength of the Arch is fuch as not to ftand in Need of any Support. But however, as the modern Tafte feems to countenance Abfurdi- tics more than real Beauties and naked Truths, 1 have therefore given the Proportions for this Window. The Diameter being given divide it into 6 parts let i and one 4 th on each Side the Center, and give half of I to the Thicknefs of each Munion , whole central Lines will be at do deg. Diitance from the Diameter. The Breadth of the Architrave is equal to one 6th ot the Diameter of the Window. Thete Windows are only proper to be placed in the Tympanums of Pediments, or to illuminate Stables, as thole of his Maje- fty’s Meufe at Charing-crojs ; but to place them in the Fronts of Noblemen s Houfes, as is done by Mix Ftitcrojt in the Front of his Grace the Duke of Montague's, Houle, in the 'Privy-garden, Whitehall, nothing can be more ab- furd. The middle Figure is S. Serlio s Method for proportioning a Door within a Geometrical Square, whole Aperture hath its Breadth determined by the Interfeftions of the Diagonals, and Lines drawn from the Middle of the upper Side to the two lower Angles, and its Heights by the Right Lines drawn perpendicular to its Baft. The Breadth of the Architrave is equal to one 6 th of the Aperture, and the open Pilafters are equal to the lame, ihe Free7e, Cornice and Pediment are made proportional to the Architrave, ac¬ cording to any Order, and after any Mailer at Pleafure. The lower Figure reprefents, the Manner of proportioning a circular Wmdow in an ub ‘™& The ‘Principles of Geometry. 333 'Parallelogram, whofe Breadth is equal to twice its Altitude. The Diagonals being drawn, as alfo Right Lines from the Middle of the upper Side to the two lower Angles, let fall Perpendiculars from their Interfedlions to the Bale • then taking the neareft Dillance from the Center, to either of thofe perpen¬ dicular Lines, deferibe a Circle, which is the Window required, whofe Ar¬ chitrave is equal to one 6th of its Diameter. Note, The aforefaid Door is reprelentcd by Mr. Hoppus in Plate CCCXII. as of his own Invention, alter that it had been publick by Sebajhan Serlio for many Ages paft. Plate CCCXLI. Rufiicated Piers for Gates, Ay Inigo Jones, and the Earl of Burlington. The upper Pair of Piers are the Invention of Inigo Jones, and the lower of the prelent Earl ol Burlington-, in which fall 'tis oblervable, (i) That the Piet Ornament, next above the Rufticks, doth not end as it begins, wherefore I am of Opinion, that if this noble ArchiteCT had thought-proper to have be¬ gun the Fret from the central Line of the Pier, inftead of one End, and made each equidiftant part the fame; then the Termination of the Fret on each Side, would have been equal, and which it ought to have been, as being both ieen and conlidered at one View, (i) The upper part, againlt which are ftuck Fe/loans of Drapery (which are more proper to adorn the Drapers Shops at Charing Crofs, than the Entrance into a Nobleman’s Palace) have a Projection equal to the Rufticks underneath; which, if I miftake not, is falfe Archi¬ tecture-, for here the lower part of the Tier, which ought to have been the molt mafly and ftrong, is made the weakeft, and is therefore an unpardonable Abfurdity. The Frets A and B are dclcribed with Plate CCCXXIII. Plates CCCXLII. CCCXLIII. iVindows and Niches, by Mr. Gibbs. Here are reprefented three Dcligns for Windows, each confifting of two Diameters in Height: The firlt with a circular 'Pediment, the Middle one with a raking 'Pediment, and the lalt, without either, being finilhed with s fuel¬ ling Freeze and plain Cornice. If we are to make thele into Niches, as in Plate* CCCXLIII. then the Diameter of each 'Parallelogram, which "before was made into Windows, being divided into io parts, give 8 to the Diameter ol the Nich. I he Breadth of the Architrave to each is equal to one 6th of the Diameter. Plate CCCXLIV. The ancient Manner of proportioning and placing Win- dovis between Ptlaflers or Columns, by S. 8eri,io. Alt ho’ Serlio gives us this Example, which I believe he took from one of the Altars in the Rotunda at Rome-, and altho’ Sir Chriftopher Wren has ornamented the upper Windows of the Cathedral of St. Pauls London, in this very Manner with Pediments, fupported by Columns on Pcdeftals ; yet I mult here obferve, that the Method of placing fmall Columns, or Pilafters, in any Front, where there is a large Order ieen with them at the fame Time, is ablolutely a Got hick Mode, and to be avoided by every one that delights in good Architecture- For, as all Things are (aid to be fmall, when com¬ pared with larger of the fame Kind ; fo thele fmall Columns appear diminu¬ tive and poor, when feen and compared with greater at the fame Time. Therefore, as Pediments and Entablatures are neceflliry Ornaments over Win¬ dows and Doors, they cannot be better applied in any Manner, than is done by Mr. G i bbs in Plate CCCXLIII. wherein there are three Varieties, which are all very good. 3 O Plates The 'principles oj Geometry. Plates CCCXLV. CCCXLVI. Two Tufcan Frontispieces , by An¬ drea Bosse. In the firftof thefe Plates, is reprefented, a very good Tufcan Frontifpiece, with a raking Pediment, whofe parts are adiufted by Feet, Inches, and Lines, according to the French Meafure; that is, by Feet, Inches, and nth parts of an Inch, thev calling every twelfth part of an Inch, a Line. At the bottom of this Plate is a Door after the Talladian Manner, from which 1 believe Mr Gibbs learned to make his rufticated Doors: But lurely, nothing can be fo mon¬ strous as this, where the whole Entablature is entirely cut through,’for the lake of introducing a heap of Key Stones, that have not any Bufinefs there, any more than the Rollicks, and mangle the Architrave on the Sides into many pieces, which ought to be entire. In Plate CCCXLIX. is another wretched Door by Tallidio, of the Ionick Order, whofe Entablature is cut quite through, in the lame Manner ; which othenvife (as this we have been now {peaking of' would have been a good Defign, that is, either with the Rufticks only, or having them entirely excluded) The Frontifpiece repefented in Plate CCCXI A 1 . is a very n luck Defign, v hole parts arc exprclled by Modules and parts, as well as by Feet and Inches. Plates CCCXLV1I. CCCXLYIil. Two Dorick Frontispieces, by A . Bosse. Heri. is alio in the firlt of thefe Plates a very grand Defign, with a raking Pediment, together with its profile on the Right, bv which its Projection is the better underftood. The parts hereof are mcafured by Feet and Inches, and contain Inch as the Figures exprefs. In the lower part of this Plate is another Talladian Door, which having ei'eaped the Havock as the other underwent, is therefore a tolerable good Defign, tho not the very belt I have feen of the Kind ; and which would vet be better, il lire Mouldings of its hnpojls were not fo numerous, that, by its being more limply plain, it might be more agree¬ able to the Plainnefs of the Rulticks. The arcaded Frontifpiece in Plate CCCXLV III. is a very elegant Defign, whole parts arc determin’d as well by Feet and Inches, as by Modules and Parts. Plates CCCXLIX. CCCL. Two Ionick Frontifpieces, by A. Bosse. Both thefe Defigns are fmilhed with circular (commonly called by Workmen lompafs) Pediments, the lirlt on a Sub-plinth, which may be ufcd as Occafions require; the other with Pedejtals detached from the Pilajlers, as exhibited in the Plan: The parts ol both thefe Defigns are mcafured as well by Feet and Inches, as by Modules and Parts. Plates CCCI.T. CCCL1I. Two Corinthian Frontifpieces, by A. Bosse. The firlt of thefe Defigns hath a Scheme Arch, for the Head of its Door, which has a good Effect, as like-wile hath its Pilafters, Flntablature, and Pedi¬ ment, whofe Pitch or Height of its Vertex above the Horizontal Line of the Re- gula on the Cornice, is equal to one fourth part ot the Extent of the Cornice ; and which 1 mult own, I think has a more noble and majeftick Appearance, than the low Pediment-pitch of 'Palladios, whole Height is but two (jths of its Extent. As this, and the following Defigns are enrich’d with Modiliions, I mull beg leave tooblerve, that as Modiliions in level Cornices were originally made by the ends of Joiits in large Buildings, which were left longer than ordinary for the Support of the Corona ; fo, for the fame Reafon, the Purloyns at Gable-ends of Buildings were continued out, to help fupport the raking Cornice to Gable-cnjs, from which Pediments were firlt taken: Now if we confider, that The Principles of Geometry. 355 the Modillions in a Frontifpiece of a Door cannot be confider’d as the Ends of Joilts,and of Purloynsas aforefaid, 'tis therefore evident, that they ought not to be imployed there, as being direftly abfurd ; but however, as Cultom has made them common, and as they are an Ornament and Strengthening to the Corona , (tho not lo great as Joifts are) wc will therefore confent to their being tiled herein, provided they are not carried out in the Cornice, further than the Up¬ right of the Freeze, where they cannot have any pretence to a folid Bearing, as this Mailer, and I think all others, very erroneoufly have done. The other Delign in Plate CCCLI 1 . is an Arcaded Door with Pedeflals, and a raking Pedi ment alio; and which is a very good Delign, and would make a much finer Appearnce, had not the Engraver, by Miftake, fhorten'd the Projection of the Cornice, which makes it appear contracted, in its natural Extent. The Scales by which thefe Frontifpieces are deferibed are Feet, Inches, and Lines, and Modules and Parts, as in the preceding. Plate CCCLIII. A Compofite Frontifpiece, by A. Bosse. This Frontifpiece reprefents the Cotnpojile Order complete, in its Pedcftal, Column, Import, Arch, Capital, Architrave, Freeze, Cornice, and Balluftrade ; and had not here the Engraver miltakenly fhorten’d the Projection of the Cornice, the whole would have made a very magnificent Figure. The Scales for this Frontifpiece are Modules and Feet, as in the foregoing. Plate CCC.LIV. The Manner of inferting Columns in a hTall, by A Bosse. As I have given a very great Variety of Defigns for Doors, Windows, and other parts of Buildings, wherein Columns are employed, I fliall now give you this Matter's Method of inferting them in Walls, which take as follows. The three Figures here reprefented are thePcdeltal and Column of the Tufian, 'Do- rick, and fonick Orders, where the Circles, or Plans of their Shafts are each divided into S parts, of which 3 are inferted, and 5- project from the Upright of the Wall. The Thicknefs of each Wall, in which they are inferted, is equal to 1 diam. and a half or to 3 mod. or Feet, as the Figures exprefs. Plate CCCLV. The Manner of finding the Sk ew-backs fund dividing all the various Kinds) of Streight and Scheme Arches, both regular and rampant. In order to underftand well the Magnitudes of Windows, I have reduced them here to three Kinds, mz. to three different Magnitudes, of which the firft is the geometrical Square, of 011c Diameter in Height ; the fecond, the Parallelogram of 1 diam. or the double Square ; the third, the Paral¬ lelogram, whole Height is equal to the Diagonal of the firft Kind, that is, equal to the Diagonal of a Square, whofe Side is equal to its own Diameter. Thefe Heights are, in each Kind of Window, to be confidered feparately from the Height of the Arch on their Heads, be they either Scheme , Semicircular, Elliptical, &c. Before wc can proceed to the Divifion of the Courfes for a Brick Arch, wc muff confider and meafure the Thicknefs of our Bricks, and the Size of every Divifion in the upper part of the Arch mult be fomething lei's, than a Brick s Thicknefs, that a fhiall Allowance may be made for its Diminution in rubbing, this being done, we mult then proceed to find out the Height of the Arch, that will be proportionable to the Diameter of the Window, which is done by this GENERAL RULE. Draw the Diagonals of a Square, as in Figure A ; from the Center f raife the Perpendicular fb ; on /, with the Radius f d, deferibe a Circle, cutting the 3 The ‘Principles of Geometry. the Perpendicular in h, through which draw the Line a c ior the Height of the ttrei „ht \rch ; and if a fcheme Arch be required, then, on the Center /, with the Radius fe, deferibe the Arch dh e, and, with the Radius f c, the Arch a b c. The skew Back of this Arch is the Diagonals of the Square, and which of all other ltreigbt Arches is the itrangeft, as requiring the lead Butments for j, s Support. In the dividing out the Collides, always oblerve to divide them odd for thereby the odd one will Hand perpendicular over the Center, and the others on each Side will be correfpondently equal. As to the Term, /height Arch , it is very abliird, becaufe nothing can be llreight, that is arched , blit, iiuv.e rer, as the Courft s in theft Kinds ol Arches, as they are called, lia\ e re 1 - fpecc to the Center of an Arch ; as the Courfes of the ./height Arch etc d e, Fi " A, are no other, than the Courfes ol the fcheme Arc'll a be, a h e conti¬ nued towards the Center/, they may therefore be called /freight Arches, j bein'* in fome Degree affected by divilionary Lines of a real Arch. To divide the Courfes ns flreight Arches, there arc two Methods ; the one is, to divide the circular Arch, as abc, into an odd Number of equal parts, and draw Right Lines from thence to the Center, which will divide the ureight Aich into the fame Number of parts, but unequally ; and the other Method is, to divide the upper Line of the itreight Arch into an odd Number of equal parts, without anv regard to the circular Arch, as Figure C, and which 1 think is piefeiable to the former. The Figures B and F. reprofent another kind of skew Back, whole Center is at g and /, the Difiance of i diam. from the Lop of the Arch u ; and it is here to be oblcrved, that the Ids the skew Back is, the lefs is the Height of the Arch, and which is milled bv the gicatci oi leliei Diftance of the Center, from the Head of the Window. Now as the skew Back of the Windows A C D, which are all the fame in the gieateft Extream, and the Windows B E are of the leaft Extream. L have therefore introduced two Means, viz. the Windows F I, whole Centers ol their skew Backs are at equilateral Diftances; and the Windows GK M, whole Centers are in the Center of their Bales, and which laft 1 think to be the molt graceful of all the others The Arch to the Window M is defer,bed by the fnterkchon of Right Lines, as following; the Height of the llreight Arch being firft found, as a kirn let up the Point b, on the central Line ol the Window, lo that its Height above the Line a k be equal to one +th part thereof; and draw the Lines a b, and b k, which divide into any Number ol parts, each the fame; and then drawing the Right Lines it, i d, 3 e, \ j, t&c. they will form the arched Line required, which divide into Courfes, as beiore duectcu. lne Windows H I, are called rampant Windows, the one having a fcheme Arch rampant, the other a llreight Arch rampant. The fcheme Ant.') is thus de¬ fer ibed, Let g n be the Difference ot Height above the Level, draw the Line g rn and the central Line bops alfo, becaule on the ccntial Line the Cen¬ ter for the Courfes is to be found ; make o s equal to 3 times 0 m, then the Point s is the Center for the skew Backs and Courfes, to the level Arch, and on m with the Radius m s, and from the Point s draw the Lines s as l at pleafure • make l m equal to the skew Back, delcribe the Arch s t ; chaw the Line m r through q the level Bottom of the Window, cutting the Arch si in r which is the Center of the rampant Arches a l and gm. T h fse Arches may be deferibed by the Intcrfecfion of Right Lilies, in Man¬ ner 0 f the Arch to the Window M, as follows: Draw a right Line from a to 1 and fet up the Point b, from thence on the central Line, equal to one 4-tit of? m and draw the Lines ah, hi, which divide into equal parts, &c. as 111 the Window M. The rampant llreight Arch, Figure I., hath the lame skew Backs as that of Figure H, and its Limits a d are found as a l, in Figure M. Plate Plate CCCI/t I. The Manner of deferring all the Varieties of regular Semicircular, Semielliptical, and Gothick Arches, in Brick-work on Windows of the firfi Magnitude. Let S.fh be the Diameter, make the Breadth of the Arch h c equal to one 4th of the Diameter, or half f h ; on/ deferibe the Semicircle ah c, and con- Number ot equal parts, each of which to be within the Thicknefs of a Brick, as before obferved ; and from thence draw Right Lines to the Center, which will be the Divifion of the Arch required. circular Arch, and which is dclcribed, as follows: The Diameter do, and Height / 71 being given, and placed, as in the Figure, at Right Angles to each other, deferibe the Semiellipfis dn 0 ; make 0 c and bn, each equal to one 4th ot d 0 the Diameter, and deferibe the Semiellipfis a eb m c. To dtp crib e an Ellipfis hath been already taught by divers Methods ; but, left they fltould have efcaped the Memory, and it being troublefomc to turn back to thole Rules, I will here repeat fo much of them, as concerns our prefent Purpofe : Make d b equal to n i, make g h equal to one 3d of h i, make 1 k equal to g i, and g l and k l each equal to g k ; from the Point /, thro’ the Points g and k, draw the Lines / g e, and l km-, on the Points g and k, with the Radius & a > delcribe the Hunches, ae and me ; and on the Point /, with the Radius le, deferibe the Scheme e b m, which will complete the outer Curve aebme. O11 the fame Centers deferibe the inward Curve a n 0. To divide the Courfes in a femielliptical Arch there are two Ways, and both of different Ft Lets : The firft is, to divide the outer Curve, as b m c, into equal parts, agreeable to in which every fuch Divifion happens ; fo all the Courfes in the half Scheme b m arc drawn to the Center /, and all the Courfes in the Hanch m c are drawn to the Center k. Now, as thefc Arches are of different Curvatures, and both divided equally in their outer Curves, it therefore follows, that the inward Curve illuft be unequally divided, and the Thicknefs of the Courfes within the Hand) mult be much Ids, .than the Courfes within the Scheme. The fecond Way is, to divide the inw ard Curve, as cIf n, into the fame Number of parts, as you divide the outer Curve a e b\ and then, drawing Right Lines from one to the other, they will be the Courfes required, it is by iliefe two Methods, that all other Arches, which confift of more than one Curve, are divided, ex¬ cepting the femielliptical Arch, Fig. C, which is the next Example in courfe. E X A M P L E III. Of a femielliptical yirch on the Jhortelt Diameter , Fig. C. These Arches are oftentimes ufed ill particular Places, where the Height of a Semicircle of equal Diameter would be too low. To delcribe this yprch, the Diameter fo, and Height m h being given, and placed at Right Angles to each other, as in the Figure, find the Centers k, l, n, p, as in the laft Example, and on the Centers n , l, k, deferibe the Curves f g, gh i , and i 0; and on the fame Centers, at any affigned Diftance, fuppofe 0 e, neceffary for the Breadth pf the Arch, deferibe the Curves ab, bed, de, which divide into an odd Num¬ ber of equal parts, agreeable to the Thicknefs of a Brick, (as before obferved) as at i, i, 3, 4, (tpc. This done, take in your Compafles the Height of the Arch c m, and with that Diftance fet one Foot in the leveral Points of the out¬ ward Curve 1, i, 3, 4, & c . and turn the other Foot, to fall on the Diameter The ^Principles oj Geometry. 338 f 0 and at every fuch Time lay a Ruler, from the laid Points 1, i, 3, 4, &c. to the Points in'the Diameter f 0, on which the Foot of the Compaffes tall, and drawing Right Lines, they will be the Courfes required. EXAMPLE IV. Of the firfl Kind of Gothick Arches, Fig. F. The Diameter fg, and Breadth of the Arch gh, being given, with the Ra¬ dius fg, on the Points g and /, deferibe the Arches a c, f e, c h, and e g; divide a b m&d h into any-Number of equal parts, agreeable to the Thicknefs of a Brick and draw Lines from thence to the Centers g and /, which will be the Courfes required. This Arch may have its Courfes divided, as in Fig. D, where the inward Arches, he and e g, are each divided into the fame Number of e- qual parts, as the outward Arches, a h and be; and Lines drawn from the one to the other, will be the Courfes required. E X A M P L E V. Of the fecond Kind of Gothick Arches , Fig. E. Th e Diameter g m, and Breadth of the Arch a g being given, divide g m into 3 parts, at h n ; make g p and m g, each equal to g n, and draw the Lines p n k]iL nd g h b, alfo q n d; with the Radius n m, on the Points h, n, deferibe the Arches g / and i m ; alfo, on the fame Centers, the Arches b a and k l; with the Radius q /, on the Points q and p, deferibe the Arches f e and e i, an d on the fame Centers, the Arches be and ch Thefe Arches have their Courfes divided, either with refpeef to their Centers, as the Side a b d, or by both Curves being divided into the fame Number of parts, as chi. EXAM P L, E VI. Of the third Kind of Gothick Arches , Fig. G. The Diameter / n, and Breadth of the Arch n e, being given, divide / n into ’ equal parts, at i k; with the Radius in, on the Points f and k, deferibe the Arches k in, f m, intorlccting m m ; alio, on 1 and n, deiciibe the Aichcs 1 i and n /, intcrfefiling in l ; draw the Lines in h d and lib at pieafure ; with the Radius’ h n, on the Points i and k, deferibe the Semicircles fg k and ton; on thcle Centers alfo deferibe the Arches ab and d e; on the Points m and l, v. ith the Radius Ig, deferibe the Arches g b and h 0, alfo the _ Arches b c and cd; then divide their Courfes, with relpecT to the Centers of the Curves, as the’Side c de, or both Arches equally, as a be, fg h. E X A M P L E VII. Of the fourth Kind of Gothick Arches, Fig. H. The Diameter/ i, and Breadth of the Arch a f, being given, divide ft into 3 equal parts,' at g h; with the Radius/ h, on the Points g and h, deferibe the Arches fe and e 1, alfo a c and c k; then divide the Courfes, with refpett to both Centers, as in the Figure, with a Key-ftone on the Head of the Arch ; or divide the inward and outward Arches on each Side equally, as in the afore- liiid Examples. E X A M P L E VIII. Of the fifth Kind of Gothick Arch, Fig. I. The Diameter d g, and Breadth of the Arch g k ., being given, divide dg into y parts, at /, in, n,o; with the Radius of 3 of thofe y parts, as dn, on the Points V, n, deferibe the Arches dp and 0 p ; alfo, on the Points in, 0, deferibe the Arches g q and / q ; draw the Lines p 0 1 and q lb ; with the Radius 0 g, on the Points l, 0, deferibe the Arches /a gaud de, alfo the Arches i k and ab; on the Interfeftions p, g, with the Radius p h, deferibe the Arches h f and e f, alfo b c and c 1 ; then divide the Courfes, with refpect to the Centers of the Curves, as on the Right, or both Arches equally, as on the Left. Plate CCCLYII. The Manner of rujiicating Semicircular, Elliptical, and Gothick Arches. T 0 divide the Rufticks of thefe Arches over Windows or Doors, whofe Di¬ ameters do not exceed fix Feet, take this 1 GENE- The Principles of Geometry. 339 GENERAL RULE. Fig. A. Make the Length of the ftretching Ruftick fg equal to one + th of the Diameter k /; delcribe the Semicircles k, e,f, and 2, b, h, g ; make the in wu thc s J™ e m tile upper Semicircle equal to one" 9th part of the Whole, and the Side Stones each equal to 011c 4th of its Breadth • di vide b a the Breadth of one of the fide Key Stones, into 2 parts, and Vet , up for the Height of the middle Key Stone ; its Drop below the under Arch is equal to one dth of its Breadth in the lower Semicircle. The Kev Stones being thus proportion d, divide each Side, as h g, into 4 parts ; alfo divide fg into 4 parts, and make the heading Rollicks equal to three qths of ft in this manner make the Key Stones to all the other Arches equal to one 9th of the \\ hole, their Side Stones to one 4th of themfelves, and the Re¬ mainders on each Side divide into 4 parts, either with refpe Plate CCCIA III. The fir ft Magnitude of Windows ruflicated. These Rufticks are divided by this RULE. Fig. D, Plate CCCLIX. 1 m Vs 1DE . Semidiameter of the Window into fix parts, give 1 to the half Breadth of tile Key Stone, 1 to the Side Stone, and z to each Side Ruftick The Height of the Side Rufticks are the fame, as in the ftrait Ar¬ ches of Brick-work, and the Height of the Key Stone above the ftrait Arch is equal to nm , the Breadth of a lide Key Stone at its Top. To make the Ruf- ticks^on the Sides, Fig. D, Plate CCCLVIII. divide the Height into 6 parts • the Projection of the ftretching Rufticks is equal to the Skew back, and that of the heading Rufticks to one 3 d of the Diameter ; which is alfb the Height of the Window Stool. Phte CCCLIX. The geometrical Confiruclion of Semicircular , Ellipti¬ cal and Gothick rampant Arches in Brick-work. EXAMPLE I. Of a rampant femicircular Arch, Figure A. Let a l be the Diameter of the Window, and ac the Height of the Ramp • dunv c /; make e g, c b and z l, equal to ah > he, and cn, each into any Number of equal parts and defenbe the Curves by Intellections, as has been already taught EXAMPLE 340 The ‘Principles cf Geomet r y. ]'X AMPLE III. Of a rampant Gothick Arch , Pig-C. t oh he .he triven Ramp, f? the central Line, and gd the Height of , T . . b ,. , ct ; n m through which draw the Line ./ m t, parallel to the Aich bduSg , wl t0 m „ an d draw the Lines f d, and df, egh, make f/, and *z, ea^h equ. at thc Diftance of ai'o drawee LmesiG *nl ck, parallel ^ My Numb of df diMdci/, / > ’ A interfeftrons, they will defcribe the Arch re- c(]iia! parts, and «:Jn erlcttion ^ Courfes drawn , will qu.rcd ; Which being yvidedwitnin ana J . Win dows of the fecond Slt^^rSck^^IL as thofeof the firif Mag¬ nitude. Plates CCCLX. CCCLXI. CCCL.X1I CCCLXI1I. CCCLXIV. Rufticated lVindows in all their Variettas. ... th ,.fc five Plates, 1 have compelled all the Varieties of rufticated Win- Delight. Plates CCCLXV. CCCLXVI. Windows, by S. le Clerc, and Mr . Gibbs. In the Bottoms of the laft two Plates are feme Defigns for Windows by Sebaftian !e Clerc ; and in thefe two Plates - many others by thefime Milter which are given here lor Examples; and as he hath alio extiiDitea v “"if fheir feveral parts at large, both in Front and m Profile, I cannot thmk" but that they will be helpful to Invention. In the upper part of Plate CCCI’VI are y Defigns for Windows by Mr. Gibbs, whole Proportions are cx- preflbd by the'dottef Circles. The Figures CCCLVI « Plans, Elevations, and a Sedlion tor a Niche, by h. le Uerc. Plate CCCLXY1I. Venetian Windows, Tajle. according to the modern IN this Plate is contained nine Defigns for Windows, of which thofe mark- i j) V H are called Venetian. The upper Window B in the Lyes of the V . ’ 1. .. -w olcifimr a Figure, as thc patched Bawd does m the Har¬ lot's'progrefs and indeed,°not much unlike her, as being patched m the lame Manner with thefe miierable little Block Rufticks, which cut and mangle the . i ' r ,! Vl ,- . inc » Entablature, in a molt barbarous Mannei. I he \\ into > 3 Aichitiat es an j ntion . tbr hcrej not contented with defacing the Ar¬ chitrave ° there are half Pilafters fet againft the Columns, to make the Whole ib'ufivc as poffible; whereas, had the Architraves been omitted, and he lg Rulticks introduced in their Head, or the Rufticks wholly excluded and the Archittaves'remain'd entire; thc Defigns would have been tolerable good. A chitiaxes lc Apertures too narrow lor the middle Se -StStn.Sh “m, ,n A. «h, Tr.fr, « W*+ S' ik i. *,k= but , very ,x„r The tfher >re ,» «- nera'l very good, if rightly applied, that is, ll the \\ indows A, C, , imploved in Fronts, where there are large Columns; and the others D, G, I, where there are not any feen with them at the lame Time. Plates The Principles of Geometry. 34 1 Plates CCCLXVIII. CCCLXTX. The geometrical Omjlruclioms of fe- micircular , femielliptical. Scheme-headed Windows in circular ellipti¬ cal Walls. 1 Before thefe Kinds of Arches are made, we muft form tlic Centers where¬ on they are to be turnd ; and in order thereto, we muft fir ft make the Ribs as follow. Let the dotted curved Lines c a b d, and °) /> &C. tiace the Curve x 0 0 0 0 f t , whole Ordnates n 0 , 71 0 Sec be¬ ing equal ill Height to the relpecftive Ordnates g b, Sec. of the Seini- ciides; and the Diftance of the Ordnates n 0, being proportional to the the Line x t, as the Ordnates g h are to the Diameter of the Semicircle a b ; therefore it the Semicircle a e b be railed perpendicularly over its Diameter a b, and the Curve x f t over the Line ,r t, they will be the Ribs required If a third Rib is required (fupnofe at 1 k, then by laying a Ruler from the Center l to the Points g g. See. you will divide the Line i A into the fame Number of Parts as are 111 a b , and in the lame Proportion alfo, as you did the I.ine x t ; and ii from the Divifions in the Line 1 k, you draw Lines pa¬ rallel to l e, and make every one of them correfpondently equal to the Ord¬ nates g h, in the Semicircle ac b, and through their Terminations trace the Curve i z h, it will be the third Rib required; and fo, in like Manner any other at Pleafure. J 1 The Ribs being thus prepared and fixed in their Places, let them be cover¬ ed with Hit Deal, or rather Pantile Laths, which irmft projeft outwards fomething beyond the Upright of the Wall ; and as the Covering of the Ribs muft be confidered before the Ribs are made ; therefore, in their making an Allowance for its Thicknefs muft be made; fo that the Whole together fliall exabtly fit the Arch required. The Center being thus made and "fixed the femicircular Rib a e b, Piate CCCLXIX. will be over the I.ine I D H* Plate CCCLXVIII. Fig. I.jmd as the Line 1 D H in Plate CCCLXVIII. ’ equal to a r b, in F’ig. I. the Semicircle a e b, over the Line 1 H, it will be in the Place of the femicircular Rib; this be¬ ing well underttood, nnke a Chalk Line about the boarded Center Rom the Points 1 and FI, lo that it may be exactly perpendicularly over the Diameterl H and let this Line be reprelented by the Semicircle I C H, which well fuppoib to Hand perpendicularly over the Line I H ; on a large Door or Floor, "lav down a Triangle, equal to the Triangle I H (L., and on CL with the Radius I O , delcribe the Arch 1 H; alio on D deferibe the Semicircle I C H ; which will be equal to your Semicircle, deferibed by the Chalk Line oil the Center • di¬ vide 1 H into any Number of equal or unequal Parts, as at the Points y - 1 y and from thence draw Right Lines at Right Angles to I H, until they meet 4 Q. the * X. «.iu ao J .mc i ri ill J.'iatc C.C,L.J-,A\ ill. is :g. 1. Plate CCCLXIX. the Semicircle I C H is equal to L and if the Semicircle 1 C H be railed perpendicularly Tlx ‘Principles of Geometry. 342 the Semicircle in the Point- C, x,W, t>, q; this done, transfer the feverall in , „ “ .v C, &c. in the Semicircle on the Floor, to thatdeienbed in Chalk on die Center and from thole Points draw outwards Right lanes at Right An from C ..Wards A, > D K, t x equal to 6 y, r w equal to 4 3, r u equal to 1 1, P ? <- - v n . ~ 10 > 9 ’ .. d which 'has no Difference in the Operation from the former. The like isalfo to be und rftoodh Sche re A < , Fig. V. at the other End of the Plan, ca i Ar , g. III. and VII. on the Sides; and which s of each gt re 11 a e 5 plain at the firft View. - de, andpre to receive the Arch of B. Stone k, Wc mu ft now proceed to that Work as follows : (0 Let G I, - LateCCULAV . 1 be e iven Breadth of the Arch or Architrave, that is to Sow .S i equal to G I, and draw G F, which 1 1 L, on w hich, h’the ribe the Semicircle G B F nude G E a " d B F l " mv Number of equal or unequal Parts, as at the Points „ «, 9 > 1 ■> ' l. Z an 1 ■ ence draw Right 1 dues to the Sem.c.rcleG I F, at Plight 5 Angles to it'elf, which will alio cut the Semicircle 1 C H in the oints 5 «wx&c ' e the Right Line A E, Fig. II. equal to the Length of tiheCurveG IKH 1 in Fig. I. alio make A B, and D E in Fig. H. equal to T i tt r-. ■ y r j /.\ Divide A E in Fis. Li- in inch Proportion as GF in G I and H 1- in Fig. 1 . (3 ^ Vldc A ^ m , c &c . and from thofe L-io- I ; s divided, us at the Points g, /, e, u, cl, c, 0, a, c, c-l. a Points draw Right Lines, right angled to itfelf of Length at 1 leafure; this.don , fg h If l e k dl Make 'c m f in Fig. II. equal to b n a o CG &c and thn igh the its E, h, 8 c 9 / 10 g in l 11 l 13 n E B &c. &c. deferibe the Setniellipfis i, k, 1 , m, n, o, G, —. . V G A. In the fame Manner, and on the fame Lines, from the 1 oints d, c, if a c /y C . fet up the Ordnates of the teller Senucime yg, i «, 3 - > f > ; Y f fi fwd af fffp'uuU Vfhe jfc lx*c*Ur Arch re- : s'-h- fssssa ' , r YYT asirom i to i, x - g ;> 1 3 > © & t hisdone, transfer theDivifions of the SSIcSi^SS^ WoS and fix it perpendicularly in the Center O (of FL 1 Plate CCCLXVIII.) fo that the Beginning of the Divi- fionf be^e.\aftly°'level with the Diameter of the Arch; then divide the The 'Principles of Geometry. 343 Curve I t r s p H cn the Center, in Fig. I. into the finne Number of Parts, as you have the Semiellipfis BFD into Comics, and which will be qual to each other; and then a Chalk Line being applied from each of thofe Divilions on the Center, to the refpective Divilions, in the Standard Piece, erected at Q_, that is, from the firiton the Center, to the firlton the Standard, and from the ad cm the Center, to the ad on the Standard, &c. linking the Line at each Time, you will truly let out on the Center every Courfc as required. Lallly, apply one Side of a Bevel to each Line itrudt on the Center, w ith its Angle to the Curve on which the Arch is to be placed, and open the other Side, until it meet the Perpendicular of the aforel'aid Curve; and the Angle lb taken, will be the true Angle, that the Sofito cl every Courfe mult make with the Perpendicular of its Front, The next and bill Thing to be done is, How to find the Curvature of eve¬ ry Courfe contained in the Arch ; which may be found as follows, Plate CCCLXX. Fig. IV. and V. Admit Fig. IV. to be a femicircular-headed Win¬ dow in a cylindrical Wall, whole Arch is to be rulficated with Brick or Stone, and tis required to make the proper Templets, for the forming of the Curva¬ ture of each Rollick ; let i l be the Diameter of the Cylinder, and Window the Point k ; deferibe the Semicircle, and divide every of the Rufticks, as be¬ fore taught; continue the Sides of each Rultick both Ways, until the Lines meet the outiide of the Cylinder, as h g, to x and /«, / e, to w n, &c. in a- ny Part of the central Line k q, draw a Line at Right Angles to it, as the Line x- r , interfering it in k; on k, with the Radius k r, deferibe the Circle rqp', divide one of its Semidiameters, as p q, into any Number of equal parts, (liippofe eight) as at the Points x, r, 3, 4, y, 6 , 7, 8, and from them draw Ordnates parallel to x r ; take one half the Line x m , Fig. IV. viz. xk or k m, in your Compafles, and letting 1 Foot in the Point q, turn the o- ther to fall on i, in the Line x k. Fig. If and through the Point i draw the Line q i m, making z m equal to i q\ through the Point i draw the Line n 0, at Right Angles to the Line in q\ divide m 1 and z q into 8 equal parts at the Points 1, 1, 3, &c. thro’which draw Ordnates, which make equal to the refpec¬ tive Ordnates of the Circle r q p, and through their F.xtreams deferibe the Ellipfis m 0 qn ; take 1 half the Line w n in your Compafles, viz. k n , Fig. IV. and letting one Foot in q, with the other touch the Line x k, Fig. V. in /a, through which, from q, draw the Line q d equal to w n , and through the Point h draw the Line/'/a l, which make equal to the Diameter of the Circle p q ; divide d h and h q into 8 equal parts, and draw the Ordnates parallel to fh /, and equal to the refpective Ordnates of the Circle, and through their Extreams deferibe the Fdliplis d l q f In like Manner make g q. Fig. V. equal to k 0, Fig. IV. and through g, from q, draw the Line q g b, making g h equal to g q ; through the Point g draw thc.Lme e c equal to the Diameter of the Cylinder, and at Right Angles to b q, and then, dividing the Lines bg andg q each into 8 parts, draw the Ordnates equal to thofe of the Circle, and through their Ex¬ treams deferibe the Ellipfis b c q e. Laltly, make q x. Fig. V. equal to k s. Fig. IV. and through * draw the Line q xt, making x t equal to q x; through the Point x draw the Line w a equal to the Diameter of the Cylinder, and at Right Angles to t x q; divide t x and x q eacli into 8 parts, and, drawing the Ordnates equal to the refpe&ive ones of the Circle; deferibe the E llipfis t « q W. Note, If the Ordnates of the Circle be continued out to meet the Line t q, they will divide the longeft Diameter of every Ellipfis, in the fame Pro¬ portion its pk the Diameter of the Circle is divided ; therefore, tho’ I did before direbt the dividing of the Diameter of the Circle into equal Parts, as being eafier underllood at firft, yet ’tis plain, that, if it is unequally divided, tis the fame Thing; therefore, if enough Ordnates are drawn, fufficient to deferibe the Curve of the Ellipfis, 'tis all that is required. Thefe feveral Ellipfifes are to be confidered as the Outlines of lo many Sections of the Cylinder, which, in ge¬ neral, pais thro’ k the Center of the Window; and therefore the left Hand, 01- lower Ends, of the fliorteft Diameter of every Ellipfis, as the Points w, e, f, and 344 The ’Principles of Geometry. and n will all fall together in the Side of the Cylinder at k ; wherefore making the parts of cadi Ellipfis next above the fhortelt Diameters, as w 30, e ’a. f , , an d n 36, each equal to g k Fig. IV. the Semidiameter ot the Arch ; and the nest parts of each Ellipfis, viz. 30. 19; 3a, 31; 3 +> cac ij equal to h g, Fig IV. the Freight ot the Rufticks, they will be the Curvatures of thole Rufticks ; that is to fay, (1 The Templets lor the tirft lluftick i h ", are the Templets marked D, found on the Ellipfis moq n, which is for’ihe upper part h g; and a Templet made to the Curve of the Cylinder's Circle, is for its lower Edge at i. (1) The Templet C is for the Oirve at fe, and the Templet B for d c ; the Templet A is for ha, and the central Line of the Key Stone is perpendicular. The fame Templets in tire relpcclive Parts on the other Side, work the fame Lftect. And thus is the Arch complete. „ . , , , . r Now from this tis plain, that every Cotirfe m fuch an Arch being conh- dered as a Section, palling through k the Center; the Templet required for every fuch Cotirfe, is that Part ot the Ellipfis of every fuch Section that is at the fame D,fiance from the End of its fhortelt Diameter, as the Cotirfe or Luftick 1 he ( nter of the Window. _ As I have thus dc tra ed the Nature ot thefe curved Courfes of Baick or Stone, it now remains, How to find on a plain Superficies, as on Taper , a l-'/oor Sc- /he singles made by every Conr/e contained in fuch an Arch, which continued through the Center k, unto both Sides of the Cylinder, as erved, are die longeft Diameters of the Ellipfifes of fuch Seftions, whof fhortelt Diameters are ail equal to the Diameter of the Cylinder ; -Make the Rivht-line DB, Fie. II. Pi. CCCLXIX. equal in length to the curved Line 1 k 1 r. gig. i. pi. CCCLXVUI. and make CF. in Fig. II. afdrefaid, equal to gk, the Semidiameter of the femicircular-headed Window in Fig. IV. IT CCCLXX. and deferibe the Semiellipiis DFB, which divide into fuch Number of equal v>rts as are confiftcnt with theThicknefs of the Bricks ; this done, draw Right- s frt m the Center C, through the aforefaid Divifions, and they will he the Angles of the feveral Courfes required: For i! the Ellipfis be thusdefcribed on Paper, and applied again!! the Side of the Cylinder, fo that the LincPC is per¬ pendicular ; then the Diftance from the Center C, to every part of the Semi* ellipfis DFB, will be equal, (and the Points D, B will Hand perpendicularly over the 1 onus g, w hich are equal to the Chord-line, or real Diameter of the Window) and therefore will be lcmicircular, and the feveral Linespafling fiom its Center through its Circumference, are the Angles required. Now, feeing that the Plan in Plate CCCLXVUI is an Ellipfis, or rather an Oval which is compofed of Segments of Circles; to find the Ieimunitions of the long Diameters of the Arch, Fig. I. we mult imagine that the Segment in which it Bands, be the Segment of a Cylinder, whole Bale is a Circle, and whole Semidiameter is equal to IQj Wherefore drawing the Lines F E, and j\ c t i„ l If plate CCCLXIX. at the Diftance of IQ. (Fig. I. Plate CCCLXVUI ) on each Side F C they will reprefent the Sides of the Cylinder, at which all the longeft Diameters of the feveral Ellipl'des muff terminate, and thofe- Angles transferred to the Building, will be the Collides lequiied. Ac I have not yet taken any Notice of the Divilion of Courfes in Arches, with , - mfelves, [ muft now obferve, that each other Courfe 111 an Arch, whole Height is 14 Inches, confifts of two Stretchers, as n n in Pi,r ji pi.r C CCCLXIX. which are Inches each in Length; and the other Courfes between thefe of three Headers, as a,a,a\ each of three Inches and one half, and two Gofers, as c, c, each of 1 Inch and three 4ths, which are continued alte nately the Whole. Haying thus explained all the parts belonging to a Window of this Kind, wherein I have been very copious, and which ) hope is lb intelligible as to be underliood by Perfons of mean Capacities, for whole Sakes I have compiled tills whole Work. I muft 111 the next Place obferve, that the very fame Methods and Lines are to be uled in the Formation of the Centers and Arches ol Elliptical and Scheme-head- The Principles of Geometry. 345 ed Windows in fuch circular Walls, and which is very evident if we pleale to confider the feveral Lines in Figures III. IV. V VI VII VII! mt compare them with Fig. I. and II. Plate CCCLXVIII. as alfo Figures V VI and \ II. with Fig. I. Plate CCCLXIX. 8 ' ' To make a Jireight Arch in a circular Wall, Fig. III. Plate CCCLXIX ■ L u T , the C ! rcIe C j E D re P refent the plan ° f a Cylinder, or round Tower m winch is to be made a Window, as between C and D, whole Head is level’ W "if, w ‘ AlXh ° f Brick 1 the Right Line g f pg’ Jh e to M f T t C T e r L ", le C D ’ Kig - I1L which divide into Courffs Jet p.l m h 9 nJd Ckn ° f I i£ BnCkS ; make ? e > in Fig- IV. equal to G A in Fit,. Ill- and draw Lines from e through all the Divifions in the Line e f which are the Angles of each Courfe, that mult be deferibed on the Point A and continued through botli Ways, as n n,lo, k p, &c. Fig. HI for if Fig- IV was horizontally applied again!! the Cylinder, with the Point q on life Pofot G, , t , £ " T the lomtsj and / would cap about, and Hand over the Points C D and the Lines * k, &c. drawn through the Center, and continued upwards a, d downwards to meet the Sides of the Cylinder, will be the long Xneter of thofe Eilipfifes m whole Curves the Templets of each Courfe may be found as before taught. The ieveral Divifions in the Plan of the Window between Tl Uft i* 6 eq u al c t0 u ° fc ° f g f’ Flg ' IV ' and a Plane or Board fo divided and fet level with the Head of the Window, and over its Plan are the Lines to which every Courfe of the Arch mull be fet. spore I proceed any further, 1 mull beg Leave fora Digreflion to oh ferve, that altho ,n the Conftruftion of Geometrical Figures in the Peal,- '. of our Geometry, I have taught divers Methods of deferibing Ovals and FI? ipfifes and have alfo here, in the Conftruftion of femicircular-headed Win dows, hewed another Method of deferibing an Ellipfis to any Length and Bicadth, by tiansfeiring the Ordnates of a Circle, whole Diameter is eou-il to the fooiteft Duimeter of the Ellipfis, &c. which indeed are Efficient‘for . Kinds of Bufiriels, where they are required; but yet, as there are three o hei Methods, that have fince occurred to me, the Knowledge of winch may be enteitaming to the Curious, I will, for their Safes, here infert them. To describe an Oval or Ellipfis to any Length and Breadth given, Plate A, to follow Plate CCCLXIX. Fig. II. E' 1and a d be the given Diameters ; make c q equal to a d make A s and A t each equal to two ?ds of qb on / s comnlorc equilateral Triangles t y s, i. and'draw o ? ut the Lmes /l ^ , : 011/ and s ’ w Vh the Radius c t, deferibe the Arches iv c v and h 1 ’. alio on/ and w, with the Radius , n, deferibe the Arches * * andl // which completes the Oval required. “ a ’ METHOD II. Fig. III. EE : a l b e ’ b u the §iven Dlamctcrs interfering in a, on which with the Radius a a, deferibe the Circle a dh c, alfo wnh the Radii s Suai p et r e ® enlici ; cle Q f/^ ; divide the Quadrant /^into ny Numberof SlM *■ k ‘ 1 ‘ &C ' and from Whence draw Ki 0 nt Lines parallel to the Diameter h e, of Lemrth at Pleafure ■ aid, i;„ i the Qpadrant f e into the fame Number’of equal parts as you did' Xe S al f \r’ at thC loints ■/>/>/> &c. from whence draw Right Lines paiMlel to the Diameter af, cutting the aforefaid in the Points *, w ) Tt fTl [h^wtl'e as .eqmkd ^ ° f the Ellipfis > and Manned This Method is an Invention of Selajiian Serlio. 4 R ME- H 6 The 'Principles of Geometry. METHOD III. Fig. II. By Means of a Tramel. t r t r Z> and a d be the given Diameters, and e f g h the Hamel fixed with Heads or Sides to fix them atP^afure , let^be a^hxu^ ^ ^ J ^ '"to any of die Extreams of the Diameters, and turning it about, the Point ' ta’-'-S itnlsmavTiScxpeditioufly defcribed, as in Figures TV, - ■ ' viherein T is a Semicircle; W, a Semielliphs; and 1, a Eheme the Elevation of the rampant Diameters to the Figures V>X,Z» V leafure - and, if therein be placed Ordnates equal to thofe m the Fi- W, Y, as reprefented, the Curves traced through their Extreams H^SdShTwXS'being the next in Order, I fliall Hmce.d A belbre 1 can make any Beginning therewith, I nnift hilt fiiew How to cover the Superficies of a Cone , Plate CCCLXX. t t r.» VI renrefent a given Cone,and’tis required to cover its Superficies [,et r i r . > • } & with the Radius i h, on the Side ot the ; : ' ; ;,s Hg’m' in whole Circumferenc > - P< p, » Conc A P X ni a ' crence of the Bafe of the Cone, and let it m the Circle, c; meal me the Uieui -cientc > j (Quadrant aec , vu r tY . m f to a, and draw the .Lines e a, ec, t ..,rJ' andequaltol kX Srde of the Cone; and from a to c, ig equal.hunt 1 ’ p yn is equal to the Circumlerence How to hue With thin Stuff, &c. the Sofito of aChrular Window, as Fig. VIII. Plate A, following Hate CCUAIA. T rT ac P a rcprcic.taSea.onof the Wall, in which Hands the Window, Lkt acpq icyi , Height or Diameter of the Window, ff 'i 111 - V ;^o"n„titv of tSe Spl^-lSr and * and «g the Splay-backs angle b 1 .» flffp j y-g £ u fiippofe the.Splay-back of the W mdow ’f take n E the S.dcol the w ^ ^ ^ C£nt . r w> whercin a fl,gn a Point ( as - ^ I .i - - r SST; Arssnsss SSK S-f-i whole Splay, „ fZttS cover the whole Cone, as before is demonitrated, therefore/ a the Splay, which is a part. §U.E. V. j x The Principles of Geometry. 347 [ T is on this very Principle of covering a Cone, that all circular Mouldings laid on arc worked ; A s for EXAMl’L E, Let it be required to /lick the Mouldings to the Baje, or Cornice of a circular Pede/tal, as Fig. III. Plate CCCI.XX. (i) Make a Section, on Paper, of the Pedeftal with its Mouldings, as in tlie Figure ; continue the Face ol their Projections inwards, until they meet on the central Line, l'o Imaad on, being continued inwards, meet in the Point k : Now, it you conlider the Line l o as the Diameter, and the Lines k l and k o as the Sides of a Cone, the Mouldings are eafily found, as follows ; With the Radius ko, or kl, on a Point, as e. Fig. II. defcribe the Arch of a Circle, as hkl, which make equal in Length to the Circumference of the Bafeof the Cone, and draw the Lines el, ch; make gl equal to ln.i, or n o, and defcribe the Arch g dbf\ then is the Piece h k If ig the Form and true Magnitude of the Piece, in which you are to work the Mouldings for the Bale, and which will exactly iimili, as required. Ilic like is alio to be oblcrved m working of the Mouldings for the Cornice, as is evident by the Lines a d and g f bein'* continued to /, and whole Piece is deferibed by Fig. I. To cut out the Lining to the Sofito of a femicircular-headed Door , or Win¬ dow., flanding in a circular Wall, Fig. VI. Plate A, following' Plate CCCLXIX. Let DBzC reprefent the Plan of part of a circular Building, in which is placed a femicircular-headed Window with the Splay-back, as B r, or ^ i, the Infide of which is to be lined with flit Deal, <&c. (i) Make a Plan of the Win¬ dow, as in the Figure ; continue on the Splays, until they meet on the Center Line, as at a; draw II a and r r, on F defcribe the Semicircle i 2 3 ; divide the Quadrant r 1 into any Number of equal parts, (fuppofc 4) as at the Points 4, y, 6 , and from thence draw Right lanes parallel to the central Line A 2. until they meet the inward Line ol the Building in the Points t, on, x, from whence draw Right Lines, parallel to B unto the Splay-back B r/as\o the Points jAd.' Now, if you fuppofc the Semicircle to be railed perpendicularly on its Diameter, and Lines drawn from every part of its Circumference to the Point a, they will form half a Cone; and if we l'uppole the Lines B y, / x, Aw, and lit, be railed perpendicular to the Plane of the Paper, then the Line B r is equal to the Length of that part of the Cone out of the Wall, that is perpen¬ dicularly over the Line y F ; and lb, in like manner, the Line iris the Length ovci the Line .vG, A r over w H, and v r over t I; this being done, on a, with the Radius ar, defcribe an Arch, as hr, which make equal to the Circum¬ ference of the Semicircle r 21, and divide it into double the Number of equal parts, as the Quadrant r 2 is divided into, viz. eight, as at the Pointsc,- e, g, k, m 0, p, from whence draw Right Lines to the Center a ; make k i, cn the mid¬ dle Divilion, equal toBr; make g h and m l each equal tort; make e f and 0 n each equal to A r ; make c d and p q each equal to v r, and thro’ the Points d, f, h, i, l, n, q, trace the curved Line b dfh i l n q r ; make b 7, d 7 , f-j, l J 7 ) tj, lq, n 7 : q 7, each equal to B r, and through the Points 7, 7, 7 ’&c. trace that Curve, then will the Area, included between thefe two Curves, and the Lines 7 b and B r be the Sofito, or Lining of the Splay required. Fig. VI. will be explained, when we come to fpeak of finding the Centers of Pediments' Of the Formation of Niches , Plate CCCLXIX. Having now done with Doors and Windows, the next in Order is the For¬ mation ol Niches, quafi Nidi, or Nefls, of old Concha, which are a Kind of 1 'Pluteus The principles °f G e o m F. t r y. 348 Tluteus, or final) Tribunals, and are fo called by the Italians to this Day, wherein Statues are placed to proteft them from the downright Injuries of the Weather. The magnificent Throne of Solomon, mentioned in 1 Kings io. was undoubtedly a Niche, as the Words exprefs. The Throne had (l Steps, and the Top of the Throne was round behind., &c. wherefore, as Mr. Evelyn obl'erves, it feems to have been fuch an ample Niche , in which a pr incipal Perfon might fit as it were, half canopyd over, within the Thicknefs of the Wall. Niches have the n Recedes, not only femicircular, as Figures if and e, Plate CCCLXVI. but arc fometimes made jemielliptkal\ or fquare, which loft have their Depth equal to one third of their Diameter. They are in general ufed as Or¬ nament' to enrich very large Tiers between Windows, Columns, &c. and fometimes to fupply the Places of Windows. The femicircular Niches are the bell for Statues, whole Height fliould be eqi to the Center of the Import, or Center of the Head. If we would enrich their Heads, nothing is more natural and becoming than the Efculop, whether printed or carved, or being divided into fquar e, hexangular, or oSiangt lar Pannels, proceeding from their Center, with Roles of divers Kinds, as represented in the Infides oi Cupola's, Figures I, ii, C, Plate CCCLXVIL But indeed Niches without Doors, fhew ! eft when : lain, and therefore it is to be noted, that fuch Enrichments arc only proper for Niches within Doors, as at the End of a Gallery, ; Church, &c. Elliptical and fjttare Niches are molt proper for Bullos or Va- . , not being fo deep as the femicircular, and which may be placed on a fmall Keck or Pedeftal, on a Truj's or Mutule, either plain or enriched with St. U. , Leave , &c ; is the Occafion requires.) When Niches are placed in the ju ft Story, they muft have Pedcftals under them, as in Plate CCCLXIII. and Fig. iand h 111 Plate CCCLXVI; but when they are placed in a Range with Window's, they muft be conformable to their Ornaments. Niches are made in lour different Ways, as. Fir ft, with plain Brick¬ work, rendered, and white-vvaihed or painted within, cl otherwife fin idled with Ornaments of i’lai Her, called by the Italians Siit.cc-',an k. Sc. airily, With Stone, either plain or enriched. 1 Thirdly , With \Vood,as Wainlcot or Deal glued toge¬ ther painted : And la/lly , with wooden Quartering, lathed and plaillered. A le- mici: cul-’-r or fcmielliptical Niche iscompofed of two parts ; the lower one being one half of a Concave Cylinder, whofe Bale is a Semicircle, or Semieilipfis, and the other, 011c Quarter of a concave Sphere, or Spheroid, or Semidome. To make the cylindrical parts of both Kinds, is to do no more, than to raile them j erpendicuiar over the curved Line of their Plans; but to form their upper parts is a Work lomething more curious, as will appear by the following. To make the Head of a Semicircular , or Semielliptical Niche, in rough Brick or Stone. Before thefe Niches can be made, there muft be Centers made to turn them on, which, to do well, is a curious Piece of Workmanfhip, and may be done as follows. I. To make the Center for a femicircular-headed Niche. ( 1 1 M are a femicircular Plate, anfwerable to the Diameter of the Cylin¬ der, on which the Feet of vour Ribs muft Hand, like fo many Rafters, as B, C, A, Fig. XV. Plate CCCLX 1 X. make the Front B D A exactly equal to the Plate B C A, which fix on B and A, at Right Angles to B C A ; this done, cut out as many llibs as are necelfary, all which have the lame Curve or Arch, as D A ; then fix one on C, equal to B, or D A, (the Thickncis of the Front at D excepted), and which will Hand at Right Angles to B D A. Proceed in like Manner, to fit in all the others on each Side, making an Allowance for fuch as muft be cut away in their meeting together at D. Having thus prepared the Ribs, we muft now proceed to the covering of them with flit Deal, which per¬ form form as follows. Fig. I. Plate CCCXCVII. Suppofe the upper Semicircle reprefent BDA, the Front of the Center in Plate CCCLXIX. on its Center eieCl a Perpendicular, terminating in a, which divide into any Number of e- qual parts, through which draw Right-lines parallel to the Diameter, as e / fS T r tpnilinntimr o <- tlm C , ... I. f ’ 1 .1 Tl , . *' rallel Lines cut the Perpendicular a, deferibe Semicircles to terminate in the Extremes of thole Lines, as in the Figure is reprefented: This being done conhder how many Pieces, and their Breadths, will belt fuit to cover the feve- ral Ribs, (fuppol'e four) then divide one Quadrant of every Semicircle, as lb ■VM O n Illtn 1-10 I f or- __ . 1 T-> ajb\ let on the Line l c, from l to k, &c. the leveralDiftances, that the Ex- equal to Ik, &c. and on the Point c, with the Radiufes c /, c k, &c. deferibe the Arches hid, i k g, &c. and then making Id, lb each equal to half lo, alfo kg, ki each equal to halt m n, and. lb in like manner the other Arches between k and c equal to the Halves of all the other Arches of the Semicircles, being fo transferred, and lines being traced through all the leveral Points, as h i, &c. c, on the one Side, and dg, &c. c , on the other Side ; then the Fi¬ gure h dc will cover one quarter ot the Head of the Niche, and, confequently iom- of them will compleat the Whole; lor, as the Length Ic is equal to the Convexity a b, and as the fevcral Curves hid, i kg, &c. are equal to one 4th ot the Semicircles, taken at their refpeftive parallel Heights, therefore the Fi¬ gure hdc will cover one 4th of the Whole. E. D. But this Demonftration may be yet plainer underftood, if wc imagine eve- ty of the Semicircles to be fo turned up on the Paper, their Diameters remain¬ ing fixed, as_to cut it at Right angles, 1. a rallel to the Diameter b; for then to¬ gether they form one 4th ot a Spheie •.Inch is the Body, that the Center is to iepiefent. Fig.II. is another Exon ;-;e ot this Kind, for the Covering of a jpherotdical Dome, either 011 the cor;., ate Diameter db, or on its tranfverfe a e\ and altho this Rule is given for covering of Semidomes, or Niches, and entire 'Domes , either J'pherkal or fpheroidical, yet it is alfo to be underftood lor the Covering of the Irtjide of A idles, or Domes, with thin Fiueers &c By this Rule Plumbers may cut out their Lead for to cover Inch Roofs'with Certainty ; but more of this will be laid, when 1 come to explain the Remain¬ der of Plate CCCXCVII. In Hate CCCCV1I. Figures M, N, Q_, is a Method propofed for to cover the 1 lead of a femicircular Niche, by Air. Francis 'Price, in his Treatife of Carpentry, wherein tis to be oblerved, that as the Ordnates k 1 is s in rhe. I in.' h /. 171 tr ,\/T o,-,. iHnkt _ .1 • , 3 all the fevcral parts in thole Right-lines cannot be equidiftant from the Points a a, that is to lay, if the Line aj. Fig. M, which ftands over the Middle of 8 0, was to be continued, until it meet the Line 8 9, its Length would be lei's than the Diftance taken from a to 8, or to 9, which fhouldnot be • for as the Point a reprelents the Vertex of a Dome, or Niche, and the Line 8 9 is fup- poied to be a part ot its Bale, ’tis certain, that every part thetvof fhould be equidiftant from a, but they not being fo, this Method is falfe ; as alfo is that oi Mr. Michael Hoar, in his Builders Pocket Companion, whole Method is lepiefented alfo in Plate CCCCVli, Fig. V, T, wherein T is a fiippofed 1 Piece to cover part of the Whole : But here, as in Mr. Price s, the Ordnates fe, ba, nm, &c. are all Right-lines, and parallel, which ought to have been circulai, and concentric!, wherefore the Points/ and e are at greater Diftance from a, than w is, which ought not to be, and therefore is abfurd alfo. The Realm why I have inferted their Figures is, for the fake of demonftrating then Fallities, which are very evident, and therefore to be avoided • and which indeed, 1 believe both thefe Perfons were jealous of, as not havinggiven ■any rational Account of their Method, to be underftood by any Reader. 35 ° The Trinciples of Geometry. As the Covering of the Ribs is thus taught, and fuppofed to be done, and the Cente 1 ' fixed in its place, proceed on with the Brick-work about it in circu¬ lar courfes which is belt done with all Headers, breaking joint over each o- ther and which being placed with their true Somenng, and key d m well be¬ hind with broken Tyles, &c. the Whole will be completed as required If it is required to make an entire Dome in Brick-work, it is to be oblerved, that every Courfe will kev itfelfi in, and therefore there never need be any Center made to turn them on, as mult be done for a Niche ; tor if a Nail be hxedin the Center and a Line ftrain'd from it, it will Ihew the true Somenng of every courfe ; and a Lath being cut to the Semidiameter, and applied to the Center, will give the Diftance of all its parts. 1 ). To male the Center for a [emiellipt ical-headed Niche. (i) Make fit a Plate to the Plan of the Niche, as Fig. XII. PI. CCCLX 1 X. on which the leveral Ribs are to hand, and thereon aflign their Places, as at „ p o n , &c. (i) Cut on the front Curve, as B DA, Fig. XI. which mult be equal to the Plate, and fix it at Right-angles, as in Fig. XI. the Rib Dee is a Semicircle; but the other Ribs, as z, h, g,f, and the like on the other Side arc Curves generated by fuppofed Sections ol the Spheroid, cut thio the j fines rl q l, p l, o l, n l, Fig. XII. and which are all Quarter-parts of lo many Ellipfifcs, that have the Lines r l, q l, p l, and o l, in Fig. XII. for one Half¬ part of their tranfverfe Diameters, to every of which the hall conjugate Dia¬ meter nl is common, and which may be deferibed with a Irammel, as ano¬ ther Ellipfis, or geometrically, as in Fig. XIX. which contains a Demonftia- tion of their being fo many Ellipfifes. let 6y t he the 'Plan of a femiellipt ical Niche, and 'tis required to find the Curvature of a Rib, flanding over the Line a q, which is taken at plea Jure. Divide aq into any Number of equal parts, (fuppofe 6 ) as at the Points n , m tec- through which draw Lines, fir[l, parallel to qy, fecondly, parallel to’ Cal and. Wily, at Right-angles to itfelf, of length at plealure. Now, if we fuppofe that Seftions are made, through the Quarter-fpheroid, at the Lines z p 7 i, 8 a, 9 3, io 4, and qy, tis evident that every fuch Section would be a Quadrant, as z up, whole Arch cuts the Perpendiculai c b in the Point b. In like manner the Line 7 1, being made Radius, and turned upon the Center 7, it will cut its Perpendicular e h in h ; fo likewife will The Line) 9 ?>being made Radius, cut its Perpendicular^ m /kin the Point U ' v (nr j (r through the Points r, s, t,v, w, f, trace the emrve, which is the Rib required ; For as the Points h, h, k, l , r, are the real Interfedtions of the Semi¬ circles on every Seftion, and the Lines c h, e h, i k, m l, and nr, cr, e s, 1 1 , m v, n w, are real Perpen- Rib required, ana 10 m iiKe inaiiuei any uuiti. Now it is to be here oblerved, that as all Sections of a Sphere , that pals • :: „h its Center, arc Circles, fo alio all Sedions of a Spheroid made thro ; ts Center, arc Ellipfifes ; and therefore Ellipfifes may be generated three ditte- iet Wavs. viz. bv the Section of a Cone, a Cylinder, and a Spheroid. The Nature and Manner of delcribing of the Ribs being thus made eafy, as alfo their Covering, the Making of Centers for Brick or Stone-work is from thence taught, as alfo is the Making of Niches with Lath and Pbufter; for when the Ribs are cut out, and fixed up, they are very eaiily lathed and pku- itered. The leveral Curves V, W, X, \, Z, Fig. IX. Hate CCCLXIX. are the Ribs propel- to Hand over the Lines wg, mg, Ig, kg, and ig, in the 1 lan T The molt curious Performances, that are made in the Heads ol Niches, m re. (10 4- c r e s c b\ eh\ Make • i t m v n iv »equalto< i *1 mV. n r ! -iJ - circles over the Line a q. The Principles of Geometry. _________3$i are to make them in rubbed and gaged Brick-work, an Example of which (and the veiy belt I have ken) is a Niche in the Gardens of the Right Honourable :h & Earl of Strafford, at Twickenham, ill the County of Middlejex which contains ten feet in Diameter, made by the ingenious John Gregory deceased to whore(Memory this Niche, and feme curious Brick-work in the Chancel-aid of that 1 anfli-church, will be lafting Monuments. The belt Way of performing thefe Works is, (i) to have the Center made wit i gieat Exacaicls ; (2) to ftrikc out thereon every Courfe, from its Front to itsCeiitci behind, with every Joint, which mutt be concentrick thbreto If the Head isfpherical, then making Templets, or Moulds to the firit two Courfe wi l be ftifficient for the Whole, the Curvature being equal • but if h is elliptical, then there mutt be Templets made to every Courlc contained in one hall part, becaufe the Curvature of every Courfe therein is different ■ This done, and the Somering of every Courfe being truly kept, the Whole may be completed with Beauty and Delight. y low f HE next and laft Method is by theThickneffes of Boards or Planks, as fol- 1 To make a femidrcular-headed Niche, by Thicknefs of Boards Plank &c. Fig. XIV. Plate CCCLXIX. • In L E , T t 1 he J Sc .“ lcircle A P ® reprefent the Face of the Niche, whofeHeieht of th: b'^'I 1 dlV pi e U t t0 T , ecLual P ;lrts > as wiil be agreeable to the Thicknefs ^ the n, vi d n t°- lla ?k, and thr-mm thole parts draw Lines parallel to BA, nte vou firft K 1! mV n pf rh ‘^ bein S P^red * Board, or Wall take \oui flirt Boaid or Plank, and on a Center chofen on its Edge, with the Radius 1 2, defcribe a Semicircle, which will be equal to the Front B D A • ap- finTh t0 H hC CC ' 1 | Cr, aad dKUV a Llne onthe Edge to the other Side to circlf tbT " lth thC M “3 4 . defcribe another Semi- cuc.e , then, with a turning Saw, cut through from the under to the urmer Semicircle, and your hrtt Thicknefs is made. (2) Take the Piece intended fj dmlc on,' 1 '’‘cl IK i‘ S i> a ? J 011 ‘ t3 ^ gC f' gn a Center ’ Miereon defcribe a Semi- cude cqiiai L o the .aft (becaufe the under part of this fecond Thicknefs mutt T th n UPPC ‘ P;U , °T he and therefore mutt be equal to it) then liniare f f k f Csntc ‘ ' ! !l • a.:e to the upper Side, whereon, with the Radius yd uclcnbc another Semicircle, and, with a turning Saw, cut through all round to tncothci ... miuiclc, as before, and thus is your fecond Thicknefs done Pioceed 011 111 .ike manner with all the others, always obfervine to make the under .amende ol every Piece equal to the upper one of the Piece next under MLg T w r hlch slcwcd to§ether ’ and dr “ S , Vr £ fmoo thing Plane, whofe Curve is part of a Circle of a iomething Ids Diameter, than that of the Niche. 1 01 I T. To make a Jemielliptical-headed Niche, by Thicknefs of Plank , or Boards. I ? pyfo™ 1 * hl . s thcrc ls no Difference from the foregoing; for, as there vou divided theHe.ght into equal parts, agreeable to the Thicknefs of the Boards or 1 lank, and, duo thofe Divifions, drew Right-lines parallel to the Bafe or Dia- nietci, whole Lengths determined the Radius of every Semicircle on each Pi,>-. 10 here, the Height being divided m like manner, Jd Lines d^Z from t S 1a1.lL 1 cunts as well parallel to its femiconjugate, as femitranfverfe Diameters to find the oem,diameters of both Kinds, Ellipfifes are defcribed on each Piece L VS r,’ vin™ ?S V ai ! d "u 0fe Diamcters arc foMd “ fellows tt- vuru A ’ 1 P’ ‘Y ,;-i c l 1 ’ lc Clit thc Optical Niche required ; make s e 0 Plg ' rV {- ' iy# Ua l_ t0 dadCBA ’ that IS, to 1 BA; alfo make n m /, Ffo XVII* equal to thc Depth and Height of CBA, which, in plain Terms, is itsSeS' And as urn. Fig. XVII. is equal to p g, Fig. XVlil. they both representing the The Principles of Geometry. 352 , , x- hr therefore divide each of them into the fame Number the Height of the\ hofe parts draw Lines parallel to ml and go; then (: : equal paits, a,.d l fcniitranlverie Diameters, and the Parallels '''uv^XVII will b^le'twmnjugate Diameters of the feveral Ellipfilis to be de- »? ^ 1§ , X ;;; h Thicknefs as‘the Semicircles aforellud, and which being done «».p.W.h=ad»l Midis, » Pl Te CCCLXXL The Butments of Arches to Boors, Windows, and Niches demonflrated. ,. , Arc h C s to the 1 leads of Dotrs, Windows, or Niches, are either Simple ‘ j c :.,, 0 . - , 3 thofe which contain a Semicircle, .i.ri^. B, or compound Single Arches which laft is called a fcheme Arch. It a or a part of a Semicir , i , al artSj t he two outer parts are called '.■jir.icircular Aich 1><- gi\ - + i* t-^cthor arc called its Scheme ; arid and the two middle ) < ££ /the Scheme , ill meet’each =s drawn from the Center tothe Termrn ^ ^ ^ p )ints 0 f the I L .i.r m an An 0 lt r 9 S' tl ,„ j nw ard and outward Arches, Right-lines j ,c ‘Y-- V ^Va' t Ri"ht-an o les to thofe I.incs, and continued, until they meet the liL J “' V 1 mirin-le (being continued out on both Sides) they will be Duimetei <1 ,hc Senncir , ( S t h c Thruft of the Scheme on both ^an any acute : -";r r cr C ^ trSe Bu . e t required. And if the Mater,- or obtule Anye, lstnu^ro , ,, , » i c g a f c or Foundation, on there cannot be qnv which inch an a rc:n 1 1 , it down. If wc cncreafc the x •c Sn th■ &nfeTas is reprefented by the lower dotted 1 anes, us Scheme, and luU .11 ti - , in the former, as being nearer to its but as thc^'cheme is encreafed’in its Curve, 'tis weakened alio, as being ’ b lL J , r de« Di I lance from the Middle of its Key-done, where die Idt it uime as many daily do, to the Shame of their Builders, and Lois oi many Lw ts. at Rfoht-angles to thc AlC % 1 '' the 0 renter w the Thruft, and the higher, the left, agamft its Butments, is made, the D reatu s fo c w here the firft is a bcmiclliphs, ' V i llA ^;' Cr -.foSviiv Sgand°ow, it; Butments, at Right-angles to its who.e Scheme bemgt ei) wllcr£as the f eCO nd, which is a com- Extreams, req y § e( j hkk ; s very high, and hath its Butments pound Aiun and mei h tenninate thc Hanfes, and from which aSentfroccS at Right-angle, are drawn from the Center of each Arch, - f dc - D twfthmfca mftmJlVand^nd Arches, which in general r»us much with e pect to utments at Right-angles to their Schemes, ' forid:, £ iW-wjfi; on th. Ham. , -■ - a ii the Butments, that can be had. Now, Scheme,„c— The Trinciples of Geometry. 35 3 away, and the Arch will Hand ; but if the refilling Weight of the Ranches be lefs, than the thralling Weight of the Scheme, then the Scheme will inevita¬ bly fall, and which is the Calc very often, when obftinate and ignorant Brick¬ layers will ftrike the Centers of femicircular Arches, before they have fo brought up the Work over the Handles, that theWeight thereof be luperior to the Thrall of the Scheme. Plate CCCLXXII. The Manner of framing Naked Flooring. Before that a Floor can be began to be framed, there mull be a Plan of the Building made, that thereby we may form a Judgment, where to place the Girders in the moil fubllantial Mariner ; and indeed this Should be done, before the Brick-work is railed high enough to receive them, that not only the Lin¬ tels may be well placed over Doors and Windows, (which ought never to be lefs, than five by feven Inches) but in thole Places, on which the Ends of the Gndeis aie to lelt , which Lintels, or Bearing-pieces, being made equal in Length to the Diitance that is contained between Girder and Girder, will com¬ municate the Weight equally on the whole Wall, and which is much better, than when the Bearing is on the parts juft underneath them only, as is the Cafe, when Lintels are not ulcd fo; and befides, when Lintels of Inch Lengths are lo laid, and contain 4., y, 6, or 7 Inches 111 Thicknefs, according to the Thick¬ ly the Walls, they are very great Strengthenings, and tic thofe parts very firmly together, wherefore they are called Bond 1 imbers ; but left Bond Tim¬ bers be not perfectly underftood, I mult aifo-obferve, that Bond Timber is to be laid in Walls wherein no Girders are, as in End-walls, Crofs-walls, and which, being laid throughout all 1 'uch Walls, at every 6 or 7 Feet in Hefoht, and being dove-tail’d, or cogg'd together at every outward Angle of the Build¬ ing, as PO, and at every Party-wall, as ST, or V W, in Plate CCCLXXV. will molt firmly bind the Whole together, fo that, even if a Foundation be unfirm, they will oblige the Settlement to be regular, and prevent Cracks and P radii res, that would be, if inch were neglected. But before I proceed any further, 1 mult beg leave to cbierve, that neither the double Dove-tail at S or treble Dove-tail at V, are fo itrong, as the Angle Dove-tail at O. Thus much by Leave of this Mailer. It is a Matter of Difpute among Workmen, which is the belt Way of pla¬ cing the Ends of Girdeis, on ftrong Lintels, or Bread: Sommers, over the Heads of Windows, or 111 the Piers between them; for certain it is, that when the End of a Girder is decayed, that is placed over a Window, if the Head of the Window be fliored up, whilft the Lintel is taken out, a new Girder put and the Lintel put in and fixed in its Place, no part of the Brick-work is injur¬ ed ; whereas, when a Girder laid into a Pier of Brick-work is decayed, to take it out, and put another in, doth generally make a very great Frafture and where the Pier is very narrow, doth very greatly endanger it. The turning of (mail Arches over the Ends of Girders is an excellent Method, becaufe then they may be taken out, and put in again at pleafure without Prejudice to the Building. The Figure ON M is the Plan of a fuppofed Building, wherein the dotted lanes exprefs the Situations of the Girders, which have all Jolid Bearings on the Walls, and which are fo laid, (and are always to be obferved) as that the Boards lie all one Way through the Middle of the Building, as from O to M, becaufe the whole Breadth may be fecit at one Time, either front the Point O, or the Point M; and if the Joints of the Floor of one Room were not ranged the fame Way, as the Joints of the other, they would have a very ill .xfieCl. The Figures VR, TQ_, and SP, reprefent the Joifts framed into the Girders aforeiaid, of which more will be faid in its Place. The Situation of Girders being determined 111 the Plan, we are thereby taught to find their Lengths, their Number, and their Diftance, which fhould never exceed iz Beet in any Building whatfoever ; nor ihould Joifts exceed that Length. It is 4 T alfo 354 The ‘Principles of Geomf.tr y. alfo to be obferved in the placing of Girders, always to lay them the fliorteft wav and that their Ends have at leaft 14 Inches Bearing in the Wall, except¬ ing thofe in very fmall Buildings, where Walls are of thin Dimenfion, then their Bearing may be reduced to 10 Inches. As nothing is a greater Enemy to Timber than Lime, 'tisbeftto lay the Ends of Girders in Loam ; and Lintellmg, and other Bond Timber (efpecially that of Fir) is belt preferved by being anoint¬ ed over with melted Pitch and Greale, of which the laft to be a fifth part, viz. to four Pounds of Pitch put one Pound of Greafe. I t bein cr neceffary that Scantlings of Girders fhould be of fufficient Strength, according to their Lengths, I lhall therefore deliver them, ijl , as appointed by Parliament, for the Rebuilding of the City of London, juft after its being reduced to Allies ; and, idly, according to the prefent Praftice 111 our modern Buildings. Feet 1 y (1) If the Length of )^( t0 VV a Girder be from^^ ^ " ythen its Scantling mull be by Inches But as Timber is very different in its Strength, it is not poffible to aflign certain Scantlings, and therefore they are to be varied at the Difcretion of the Carpenter. It is alfo to be obferved, that altho’ Oak is much ftronger than Fir, yet as it is of a greater fpecifick Gravity, it mult therefore have larger Scantlings for the fame Purpofes, than Fir, which is weaker. The laft Scant¬ lings are given by Mr. Smith, and the following by Mr. Trice, wherein you'll fee, at one View, their Differences. fi) If a Girder of Fir iiA lf 7 in Length, then its ' ,, ,, , ...... <_•-,i:„„ k- a fmall Building be Scantling mult be But if of Oak, then the Scantling rnuft beby I'd) Now, fince of all the Scantlings given here, not any of them arc perleftly fquare therefore, in every of them, let the fmalleft Dimenfions be under- ftood, as the Horizontal Surfaces, and the greateft, the upright Sides. For by Experience ’tis evident, that that Weight which will juft break down a ten Foot Deal, bearing on its Ends only, with its Breadth parallel to the Ho¬ rizon, cannot bend it, when it's placed with its Breadth perpendicular to the Horizon. As Girders are fubjecl to the Weight they fupport, as well as to The Principles of Geometry. 355 to their own Weight, they do therefore in Time give way to fuch Force, and bend downwards (called Saving) if not prevented; wherefore ’tis ufual to cut them Camber, that is, to cut them with an Angle in the Midft of their under Surface, as Fig. E, Plate CCCLXXV. and Fig. z , Plate CCCLXXXIf. which Angle mull rife above the Horizontal Line, i half an Inch for every to Feet that the Girder is in Length. But as this Expedient will not do in Girders ol great Lengths, they are alfo trufs’d up within-fide in Manner following. The Manner of truffing Girders. Plates CCCLXXV. and CCCLXXXII. The Figures A, C, D, E, F, G, H, I, in Plate CCCLXXV. are Methods propofed by Ms. Smith; and thofe of Plate CCCLXXXII. by Mr. 'Price, which ate perform d as follows, (i) Saw the Girder down the Middle the deepeft Way, and having prepared two Pieces of dry Oak, about four by three Inches, or fix by four Inches, as the Strength of the Girder may require ; let half of the Thicknefs of one Piece into one of the inward Sides, as in g, Plate CCCLXXXII. at m l, as tight as its poflible to drive them in ; and having cut out the Cavities in the Piece h, fitting to receive the other half parts of the Tiufles, drive it on as tight as is poffible, and then, the two Pieces of the Girder being well bolted together, 'tis made fit for Ufe. The Girder rqpo hath its Trulles, each equal to one jd of the Girder's Length ; and if at q and p you mortife down through both Fletches, and therein drive Wedges you may when the Building is cover’d in, tighten the Trufies at Fleafure! The Ti uls Figure G, Plate CCCLXXV. is alfo tighten’d by a Wedge driven through at cd, having beforehand cut out, within-fide of the Fletches, a free Panagc foi the V edge; both thefe laft Methods, I efleem to be the very heft of all the various Ways, propofed in the feveral Figures of thcl’e two Plates. Leon Baptifta s'llboti taught the flitting of Girders, and reverfmg of the Fletches (being well dryed) without any truffing, and fo bolted the Top of one Fletch againft the Butt of the other, as Fig. k, Plate CCCLXXXII • and which in many Cafes I believe to be an excellent Method. In the Choice of Timber for Girders, great Care fhould be taken to have it as free from Knots as is poflible; becaufe thofe parts are more fubiefl to bi eak, and to early decay, than any other. The Figure nyw reprefents how Timber may be tiled, when two Ihort of itfelf, for the Purpofe required ; that is, fuppofe the Piece iv s, fhould have extended, or reached further than its Lnd ,v, and no longer Piece can be had; in any part towards the End, mor¬ ale and tenon in the Piece t at w, alfo at x mortife on the Piece v, but obliquely, not at Right Angles, which alfo tenon into the Piece t at y ■ 'then will the Ends of the two Pieces u and t become Bearings for the Piece s at a greater Length than itfelf could extend to. Having thus largely explained the Situation, Scantlings, and Truffing of Girders, we may now proceed to the Joifts, which are of various Kinds viz Common foijis. Trimming JoiJls, Binding JoiJls, Bridging Joills, and’ Ciel- ing Joifts. (i) Common Joifts, are thofe reprefented in Fig. A, Plate CCCLXXII. and at SP and VR in Plate CCCLXXXII. which are framed flulh (that is, level) with the upper Surface of the Girders, and which fometimes are all of equal Depth, but lefs than the Depth of the Girder, whereby the Girder becomes lower than the Cieling: But the moll genteel Way is, to have every third or fourth Joift, equal in Depth with the Girder, whilit the other intermediate Joifts are of lefts Depths, and between thofe deep Joifts fix finall Joifts to car¬ ry the Cieling, which then will conceal the under Surfaces of the Girders that otherwife have an ill Effect, ’ (2) Trim - 2$6 The !Principles of Geometry. fz) Trimming Joifts are fuch as are framed into other Joifts, for other Toifts to be framed mto them, which are againft a Chimney, as a in fig. A. ' H ite CCCLXXII. or to give Way for a Stair-cafe, as reprefented on the left. Now as thefe Joifts are weakened by receiving many Mortifes, and being to fuftain the Weight of many Joifts bearing in them, they are therefore made of greater Scantlings than common Joifts. The Scantlings of Common Joifts and Trimming Joifts framed, as Fig. A. may be as follows. If the Length of the Common Joift be 1 7 ^ Inches I O / _ I 4 . o the Breadth muft be * ; fniall Buildings have lings muft be [ 6 But if of Oak, then the Scantlings muft be by 3 3 1 h e Figures K and L, in Plate CCCLXXXII. is the Manner of proportion¬ ing of their Tenons and Mortifes, by Mr. 'Price , and which I look upon to tea much better Kind of Tenon, than any of the eight Kinds, Ihewed by Mr. Smith, at the bottom of Plate CCCLXXil. Note, That all binding'joifts ought to be half as thick again as common Joifts ; therefore, if common Joifts are three Inches thick, a binding Joift lhould be four Inches and a halfj altho - the fame Depth. (4) Bridgings, or Bridging Joifts are reprefented by thole marked mm, &c. in Plate CCCLXXIL lying on the binding Joifts, as b, and which are alfo lepidented in Fig. I, (f , Plate CCCLXXXII. wherein a and b I’eprelents the Sections of two binding Joifts, and which, to be eaiily underftood, mult be lo turned, as to invert the Letters bottom upwards, the Figure being wrongly placed by the Engraver ; and which being done, then cd (hews the Situation of a bridging Joift on thole binding Joifts, and f, which is a Cieling Joift, exhibits the Manner of their Reception by the binding Joift. The Section over V ex¬ hibits the Manner of fixing Cieling Joifts between deep Joifts, where lhallow ones, as />, 0, n, are framed in between them, as I obferved, to be the molt genteel Way ol framing Common Joifts. The Diltance of Bridgings is generally about 12 or 14 Inches, and their Scantlings about 3 by 4 Inches, or 3 and a half by y Inches. Their Bearing is never more than the Intervals of the binding Joifts, which is from 3 Feet to 10 Feet, and which are laid even, orflulh with the Girders, (as aforefaid) to receive the Bi arding. (4) C 1 e 1,i n g Joifts, the molt flendcr of all other Kinds of Joifts, as having H the leaf! Weight to lupport, are made about 2 Inches by 3 Inches, or 3 by 4 Inches, according to the Strength of the Building. Theie are reprefented Hi in Plate CCCLXXil. by thole marked n n, &c. whole Diftanccs arc generally at 1201 14 Inches. Ihele Joifts (as I oblcrved before) arc tenon'd into the binding Joifts, as reprefented by the Section kih, over S, in PI. CCCLXXXII. where h rcprclents a linglc Mtirtife, made on the one Side of a binding Joift, and i and k two double Mortifes, called 'PuUy Mdrtijes, in the Side of a parallel binding Joilt, to receive the other End of the Cieling Joift. Thefe Cieling Joifts and Bridgings are feldom fixed, until the Houle is covered in, when the laft are pinned down to the binding Joifts. Thefe Kinds of Floors are called Bridnw ploors, and are the belt Kind of Carcafe Flooring. N. B. The upper Figure in Plate CCCLXXIII. reprefents a pretty De¬ vice, by Seba/han Serlio, How to frame a Floor by the Help of ‘very fort Pieces, which I have given here for the Amulcmcnt of the young Student. Plate CCCLXXIII. The Manner of framing Timber Partitions. The Examples given for this Kind of Work are three by Mr. Smith, in Plate CCCLXXIII. and [even by Mr. Price, in Plate CCCLXXXIII. (1) Thole by Mr. Smith, of which the upper one is framed in the common Way, and " hciein 1 think are more Mortiles and Tenons, than need to be; for were the Braces to be let into the principal Polls, fo as to butt againft Shoulders of a- bout half an Inch 111 Depth, and nailed in, they would do the fame Office in a much more able Manner, than being tenoned in,'(as here reprefented) and would be done in lefs than halt the Time; and as the Quarters are only to fuftaiu the 5 8 The ‘Principles of Geometry. Lath and Plaifter, &c. the Weight of the Roof, &c. being carried by the Polls and Plates, they have no Need of being framed into the upper and under Plates, which takes up much Time, and will not en¬ dure longer, than when they are cut and nailed in only, and which is done with very little Time and Expcnce. The middle Example A A, its Au¬ thor fuppofes to be for a Warchouic, &c. where Girders, or home other Weight are to reft on the King-pelts A, A, &c. but Purely he means that, when under the Polls A, A, there are Windows, or bad Foundation, that cannot i'ultain the Weight laid on A, A, and therefore mult be difcharged bv the Struts to bear on the bottoms of the intermediate Polls, and which, on Inch Occafions, is an ex¬ cellent Method; but where the Weight may be imprellcd equally throughout the whole Length, then this Kind of framing is to be avoided, as being very expenfive in Waite of Timber to cut cut the Joggles of the King-polts, much Time to its framing, and at laft to no other Purpofe, than the Work of the up¬ per Example will perform, which is full as ltrong, and much cheaper, both in Timber and Time. T.h e Example, Fig. E, is propofed to rife the Height of two Stories, the lower of 13 Feet, the upper of 11 Feet, or otherwife in one Height only, as the Side of an Out-houfe, Barn, &c. more properly than of a Hall, or Salloon, as our Author fuppofes. Now it is to be obl'erved, that as [oilts are fuppofed to lye on the Inter-ties (which is the middle horizontal Piece againlt D, and) which is framed into the King-poll and outward principal Polls, the whole Weight, at each End, mull depend on the Strength ot the Tenons, excepting fuch Aid, as is given to it by the under Quarters ; wherefore lam of Opinion, that the lower Braces are placed exactly the wrong Way, becaufe they now fupport the Inter-tie near to where they are fupported by the middle Polls, and where they do not Hand in any Need ; whereas, were their Ends turned, to aftilt next to the bearing Tenons, the Inter-ties would be made very fecure, and they would, at the fame Time, perform their Office of bracing, &c. As to the King-poll at E, on which he luppol'esa Girder, or Beam of the Roof to be placed, I mult own, I cannot ice any Occafion for fuch Waftc of Timber in cut¬ ting out the Joggles; for if fuch Polls were made a fmall matter more in Breadth, and their Struts let into them with fmall Shoulders, commonly call¬ ed by Workmen Birds-mouths, they would be as ltrong and lecure, as they can be done this way. The Examples given by Mr. Trice, Plate CCCLXXXIII. come next in Or¬ der, of which Figure V is the firft, which he fuppofes to be a Partition between two Rooms, wherein Doors are required next the Extreams, and therefore lias placed a King-poll in the Middle, with Prick-polls between it and the Doors. It is here to be noted. That the middle horizontal Piece, which is called the Inter- jie, is halved, not only into the Prick-polls, but even into the King-poll alfo, which is a great Weakening to it, therefore abfurd; nor indeed is there any Need for any Inter-tie at all, it the Height is intended but for one Story : And, ad¬ mit there was an Occafion for it, would not its being llightly tenoned into the King-poll have been a Ids Weakening to it, and, to have given it a ltrong Bear¬ ing, to have turned the lower Struts the contrary way? The halving of Timbers together 1 know to be a common Method, but ihould never be done blit with very great Judgment, and always avoided in Braces and Struts. Fig. D is his fecond Example, and which is a very ltrong Method, where¬ in three Doors are fuppofed, and wherein the two King-polls, and Inter-tie are halved as before. F 1 g. E is his third Example with a Door in the middle, and which is a good Defign, as alfo is his fourth Example, Fig. X. wherein the Inter-tie is halved into the Polls as aforefaid. F1 g. C is alfo a very ltrong Method of framing, but is very expenfive. F 1 g. B is a Partition fuppofed to fuftain a Gutter, common to two Roofs, with their Beams and Rafters, or to carry a Wall of Brick or Stone. Lastly, The Figure A reprefents the Manner of a Timber Front, with an The Principles of Geometry. 359 .in Arcade in its lower Story, whole Polls being placed on reverfcd Arches of Brickwork, caufes the Weight on thofe Polls to be imprelfed as well on the Foundation between, as direftly under them. Note, All thefe feven Exam¬ ples, arc, by the Miftake ol the Engraver, inverted, therefore, to rightly view them, you mult turn the Book upfide down. T he proper Scantlings of principal Polls, given by Mr .Trice, are as fol¬ lows, viz. Fir Polls of Oak Polls of I. For [mall Buildings, Feet in Height to be Feet in Height to be Inches fquare. Inches fquare. Fir Oak Polls of Polls of II. For large Buildings. \ V k i 8x Feet in Height to be Feet in Height to be rj 8> Inches fquare. 103 81 iri Inches fquare. ,6 J N ow, though all the Scantlings given here are fquare, yet it is to be under- ftood, that we arc not compelled to make them lo, though, perhaps 'tis ne- cefliiry, we lliould keep up to their Strength ; and therefore, as Mr. Trice ob- lerves, if convenience will not admit us to make a Poll, for Example, 6 inch, fquare, whole Bale is equal to 36 inch, yet 'tis likely we can make one 9 inch, by 4 inch. &c. whofe Bafe is equal to 36, as the other, and its Strength I believe equal all’o, and lo in like manner all others. Plate CCCLXXIV. The ancient Manner of Framing the Timber Fronts of Buddings. We may very reafonably believe, that, when Buildings were framed as re- prefented by -the lower Figure, there was a greater Plenty of Timber in Eng¬ land, than is at prefent; and, as I have in the preceding demonllrated the modern Manner ot Framing, I thought it not arnifs to compare it with the ancient Manner, and to give my Readers an Account ot the Names of the fe- veral parts, which are number’d, and whofe Names in the Plate Hand againlt the relpedlive Figures'. Plate CCCLXXV. Divers Methods for trujjing Girders, and fcarfing or piecing of Timber together. The Manner of Trading Girders being explained in Plate CCCLXX 1 I. it need not be repeated. The Manner ol fcarfing Timber together is reprefent- ed by Fig. K, L, M, N, as alio in Plate CCCLXXXII. where the Joints a and b are for Tlates, Lintels, or Timber for Ties, and, if for Beams to a Roof, add the Bolts and Screws, as reprelented in the Figures. In Works where much Strength is required, the other Fig. c, d, e, f, may be confulted; and when it happens that the Length ol Timbers cannot be abated, then Fig. f mult be the Joint, and that ol e lor an extraordinary Ufe, for, by its being in two Thicknefies, it is made as ltrong as though it were an entire Piece. Note, In forming of Ground, or railing Tlates to Timber Fronts, that their Joints be fo contrived, as never to be in the Breadths afligned for Doors or Windows, hut always in the intermediate parts between them. As qi 5 o The Principles of Geometry. As I am now come to the Formation of Roofs, of which the aforefaid Plates nrc a part, as being the Bale on w hich the Beams lie, and fmall Rafters Hand, we mult (after having formed it to the Plan of the Building, and leaned its Angles, in the Manner reprefented by Fig. OPQ., Plate CCCLXXV.) confider of die proper Diftances and Places to lay the Beams on, wherein obferve, (i) To avoid the Joints of the 'Plate, (i) That their Diftance be not too great, left you arc obliged to have large deling Joifts, and large Purlins, which are but a Load to a Building, and therefore fhould not exceed to Feet. (5) That they lie over, or nearly over, the Heads of the principal Pofts in Timber Buildings, and on Piers ol Brick or Stone Buildings. The m Situations and Lengths being determined, their under Suffice at each p.nd, being equal to the breadth of the Plate, is dove-tail’d an Inch and half or 1’Inches in depth, according to their Strength, and which are let into both Plates, as the Pieces T and W into the Plates S, Y, Plate CCCLXXV. but, as 1 have obferved already, with one fingle Dove-tail, not double, or treble, as arc here represented as abfurd Rarities by Mr, Smith. If the breadth of the Beam be divided into 3 parts, give the middle one to the breadth ot the nar¬ row part of the Dove-tail, which opens to the whole breadth at the End of the Beam. When Beams are thus dove-tail'd into the Plates, they are then laid (by Workmen) to be cogg’d down, and ready to receive their Ceiling Joifts, and principal Rafters. But before the principal Rafters can be framed, the . Light of the Pitch, and their Length muft be determined. The Pitch of every Roof muft have refpect to 1 Cov ring, whether it be of Lead, Pan tile, plain Tile, or Slate, and which Heather-hoarding, Shingles, and Thatch) l think, are all the Covering . have in England. The ufual Pitches made ufc of arc Pediment latch. Common Pitch, (ge¬ nerally called True Pitch) and Gothick Pitch. Pediment Pitch is, when the perpendicular Height is equal to two yths of the Breadth of the Building. Common Pitch is making the Length of the Ralters equal to three +ths of the Breadth of the Building, if it ipan the Build¬ ing at once ; and Gothick Pitch is making the Length of the Rafters equal to the whole Breadth of the Building, and therefore is equilateral. Mr. Smith re¬ commends the Breadth of the Building within to be divided into - parts, of which he gives z to the perpendicular Height, and makes the Length of the Rafter equal to 4 of thole parts, and which is very good for Lead or Pan-tiling, Examples of which he has given in Plates CCCLXXV! and CCCLXXVII. and whole Beams he proportions as follow's : Feet raeh. TiJ 6 ' 8 J 16 6i 8 ;’ TOI 6i 9 If the Bearing of the Beam in the Clear be* 1+1 aS s its Scantling muft be r-l x 8 •by- 91 9i 10 36 8 5 IOf 4°) 8 * 11 14+ J .9 in , Inches. Principal Rafters, at their Feet, fhould be nearly as thick as the breadth of the Beam, and at their top one yth or one 6th lefs. King 'Pofts fhould be as thick as the top of the principal Rafter, otherwife thev will not be able to receive it ; and their breadth of Efficient Strength to receive the Struts, that are defigned to be framed into them, their middle parts being left fomething broader than the Thicknefs. Struts fhould alfo diminiffi one 6th, as Rafters. It is to he noted. That as the lower parts of principal Rafters are the ftrongeft, the Purlins, Collar- beams, and Struts fhould be placed fomething higher than juft in the middle of The Principles of Geometry. of the Rafter, that the Bearings may be proportional to the Strength, and not in equal parts, as Mr. Smith recommends. Purlins mull have the fame Thicknefs as the principal Rafter hath, in that part which lies on the Purlin, and the Breadth of Purlins fhould be to their Thicknefs, as 3 is to 4 ; therefore, if the Thicknefs is 6 inch, its Breadth muft be 8 inch. Purlins are generally framed into the principal Rafters, but fometimes are laid in the Collar-beams. When they are framed into the prin¬ cipal Rafters, their Lengths are equal to the Diftance of the Principals, exclu- livc of their Tenons ; but when laid in the Collar-beams, they are of twice, thrice, &c. that length, as the length of Stuff will admit of Small Rafters may be in their Scantlings 4 inch, by z inch, and half, or 4 inch, and half by 3 inch, and half, or finch, by 3 inch, and half according to the Nature and Strength of the Principals, and whofe Length, in a purlin'd Roof, fhould not exceed 7 feet; and when the length of the principal Rafter exceeds ly feet, ’tis belt to frame in z feet of Purlins, as reprefented by A A, on the upper Side of the lower Figure, in Plate CCCLXXXL which reprefents the Plan of a Roof : But I cannot recommend the Method of framing the Pur¬ lins in a Right-line, becaufe, when the Mortiies in the principal Rafters are a- gainft one another, the Rafter is not only weaken'd very greatly in thole parts, but vou lole the Pinning alfo, and therefore they fhould be framed as in Fig. BD,’ Plate CCCLXXXVI. The Proportions ol Pitches, for the various Coverings, as afligned by Mr. Price, are as follow : (1) For Coverhig with Lead, Fig. A, Plate CCCLXXXVI. the perpendicu¬ lar Height is equal to one 4th of the breadth. (z) For Covering with Pan-tiles, Fig. B, Plate CCCLXXXV. the perpendi¬ cular Height is three 8ths of the breadth; or as Fig. D, where the Per¬ pendicular is cut by an Arch, whofe Radius is equal to two gds of the breadth. (3) For Covering with plain Tiles, Fig. C, the perpendicular Height is e- qual to half the breadth; or as Fig. F, whofe length of Rafter is equal to three qths of the breadth, which laft is called True Pitch. (4) For Covering with Slates, Fig. E, divide the breadth into 7 parts, and, with the Radius of y parts, interfeft the Perpendicular. The necefiary Scantlings, affigned by Mr. Trice, for Beams and Rafters, are as follow: I. For Beams or Ties. (1) For finall Buildings. Feet If the Length of a Beam of Fir be 6 by 7 7 by 8 >its Scantling mull be< 9 8 ;>butif of Oak<10 10 11 a 13 iy ii 11 (z) For large Buildings. muft be but if of Oak Inches 3<>2 The ‘Principles of Geometry. II. For Principal Rafters. (i) For fmall Buildings. I,lch '> Inches and at bottom . (7 by 8) (. 8 by 9 But if of Oak, at tops8 9>and at bottoms 9 lot 9 1 o 1 Go li (a) For large Buildings. F '" Inches , Inches If the Rafter be of 5 2 tuts Scantling atS 2 G ®) ^ 8by 9 Fir, audits Length)G) top moll be} 8 9 - anc * at bottom < 9 10; l 4b y r C 9 io-> Cio IX \ Sty 9) C 9 by 10 But if of Oak, at top< 9 io)and at bottom) 10 ’ ix Cto ii-J in 14 bby 7 8 ' 10 10 IX If the Rafter be of its Scantling atG, G „ Fir, and its Length)^) top muft be)g ^ III. For Small Rafters. (1) For fmall Buildings. feel Inches Inehts It the Rafter be of) °)then its Scan-)’ ’ G 1 *) , .. r (4 ■ by 3 Fir, and its Bearing) tling muft be)^ f ) ■ > but il of Oak i 3 (1) For large Buildings. I’ect Inches If the Rafter be of) 8 ?thcn its Scan -) 4 - i G G , .. r ^ , Fir, and its Bearing) IO ( tling mult beH ; 3 ) but if of Oak (n) & C <5 i 3 j Inches £ S' t by ; V 3 /9 3 Having thus demonftrated the various Pitches of Roofs, and the feveral Scantlings of their parts, the next in Order is their different Kinds of Trulfes. Plates CCCLXXVI. CCCLXX Y J1. Divers Kinds of Trujjes for Roofs. The upper two Figures in Plate CCCLXXV 1 . arc Seftions of Roofs by Vi¬ truvius, which are lomething higher than Gothick Pitch ; the other five are of Pediment Pitch, as alfo arc the next five, in Plate CCC] .XXVII. and therefore niult be coveied either with Lead or Pantiles. The Alctliod of each Trulling is diffeient, feme being Itronger than others, as the Nature of Buildings requires. The lower Figure hath a Valley in it, to take off the Barn-roof Afpetf, which it would have, were the principal Rafters continued up to an Angle. In this Roof is Space for two Lodging-rooms, as being framed with a Collar Beam and middle Poll, which laft mult befupported by a Wall, or a Partition, otherwife the Roof cannot be depended on as very fecure. Plates CCCLXXVIII. CCCLXXIX. Di vers Kinds of Truffes for Roofs. i hi. fine two Figures of Plate CCCLXXVIII. are framed for the Conveni¬ ent) of making two Lodging-rooms in each, and which are lomething Itronger Tiri} 1 " 11 * 1Jn ?' n §’ .than the former, but yet not to be depended on, without the AtiUtance of Partitions or Walls under their middle Polts. The other two Sec¬ tions The Principles of Geometry. 363 tions in this Plate, and the four others in the next are Defigns for Roofs over fpacious Rooms of conliderable Breadths, as Halls, <&c. whole Cielings are lup- pofed to be arched, or covered ; but I mult own, that if fome Provifion be not made to itrengthen or help the firlt in Plate CCCLXXVIII. as alfo the third and fourth in Plate CCCLXXIX. at the Meeting of the Collar Beams with the Rafters, I Ihould not be willing to trull any conliderable Weight of covering on fuch Bearings, left the Principals break at, or juft under the Tenons of the Collar Beams. The lalt Scdtion of Plate CCCLXXVIII. and the two firlt of Plate CCCLXXIX. are helped very greatly by the lower Struts, as the others mult be, to be pratlifed with Safety. Plate CCCLXXX. Divers Kinds of Trujfes for large Poofs to Church¬ es, &c. The two upper Figures reprefent two Sections of Roofs fitting for Churches, &c. the firlt, marked BAB, mult be fupported within by Columns, which are luppofed to divide the Breadth of the Church into three parts, or I files, whole Cielings are circular. As 1 am now lpeaking of the l'upporting of Roofs by in- lide Columns, as is done in the Church of St. Martin in the Fields, 1 multad- vertife, that nothing is fo uionltrous and abl'urd, as to break the Entablature about fuch Columns, making them as Capitals on Capitals ; nay, even to place an Entablature on them, and to continue it is an Abfurdity; for, as within Churches no Rains can fall, why mult Cornices be introduced there, lince that their Buiinefs is to carry off the Rains from the Freeze, Architrave, and Co¬ lumn only, excepting when they finilh the Height againft a flat Cieling, and indeed then they may be confidered, as an ornamental Way of finilhing, and a Strengthening to the Cieling alfo. The other Section CDC hath a femicir- cular Cieling in the Middle, with flat Cielings on the Sides, and where they reft on the Architrave, from whence the fcmicircular part Ihould have fprung, (and not from the Cornice, which has no Buiinefs there) as is judicioufly done by Sir Chriftopher IFren within the Church of St. Mndrew, Holbourn. The principal Polls of this Section have their Support on Columns at EE, as the aforefaid. The two lowcrmoft Sections are after the Gothick Manner, and of great Strength ; the Principals of the lower one are mortified to receive the King-poll into them at its Top, and both have Demy King-pofils in their Quar¬ ter-parts, which, if ftrapped with Iron, would contribute very greatly to the Strength of the Whole. Plate CCCLXXX I. The Plan of a regular Roof with the Manner of finding the Length of its Hip-rafters. The Ufe of fuch a Plan is, to determine the Situation and Number of prin¬ cipal Rafters, final] Rafters, and Jack Rafters. The principal Rafters are thofe 4, that lie thro’ the Body ol the Plan, with Tenons reprelented in their Middles ; the fmall Rafters are thofe between them, marked B ; and the Jack Rafters are thofe Ihorter ones, whofe Tops bear againft the Hip-rafters, as thofe marked C and 1 ). Hip-rafters are thofe, that Hand over the Diagonal Lines, drawn from the Angles, whofe Lengths, being longer, than the principal Raf¬ ters, arc thus found: Suppofe, in the upper Figure, that B y reprelent the Breadth of a Building, and B 3 5 a Pair of principal Rafters ofcommon Pitch; continue out the Backs of both Rafters at pleafurc, and make 3 F and 3 E each equal to 3 M, the Height of the Pitch, and then, drawing the Lines F y and E B, they will be the Lengths of two Hips, as required; for, as the Lines 3 B and 3 y are the Diagonals, or Bafes, over which they mult Hand, and as the Lines 3 F and 3 E are perpendicular to thofe Diagonals, and equal to the Height of the Pitch ; therefore, if the Lines F y and E B are raifed with the Perpendiculars 3 F and 3 E, to Hand perpendicularly over the Diagonals 3 y and 3 B, the Points E and F will meet in the perpendi¬ cular 3RAw a Right Line at Right Angles to the Bafe of the Hip, through any T 41; ' ,:i ‘ft as the Line hn ; then letting one Foot of your Compafles on the 1 din. ot interjection, take the neareft Diltance to the Hip g a, and turn that Foot on the Bafe, as to n, from whence draw Right Lines to the Points l, m , where the Line cuts the Out-lines of the Angle of the Plan, and the Angle l n m The Principles of Geometry. 367 IrimvnW be the Angle of the Back of the Hip ga required. In the like Man¬ ner the Angle hei is the Back of the Hipg c ; and fo of all others. The Fi¬ gures S, R, T, relate to finding the Angles of Purlins againft Hip-rafters, an Ac¬ count of which you'll find in the Index , under the Word Purlin. Plate CCCLXXXV. Divers Truffes for Roofs. Here are eight Examples of Roofs, of which thole marked I, E, G, H, D are framed open, fo as to contain Garrets within them; the others are tor larger Buildings, where fuch Room is not required. Plate CCCLXXXV I. The principal, and Hip-rafters of a regular, and an irregular, or bevel Building exhibited. The upper Plan Q_ is a Parallelogram, wherein are reprefented its Plates, Beams, and Mortil'es, to receive the principal and Hip-rafters. The Figure S ihews one of its Hip-rafters, as when handing in its place, and by which you fee the Quantity of its Angles at the Head and Foot. Fig. R reprefents one pair of the principal Rafters fixed upon its Beam, with its King-poll and Struts. Figures W, T, reprefent the Hip-rafters of one Find, together with one prin¬ cipal Rafter, and the Purlins framed into it, between them. Fig. V reprefents the Principals contained in one Side, with their Purlins framed into the Hips, and wherein the Purlin joint is reprefented. The irregular Plan A is a Trape¬ zia, wherein tis fuppofed, that the Beams are required to lie bevel to the Sides. The dotted Lines, drawn from the Plan A to the Skirt B, how much each prin¬ cipal Rafter mull lie bevel; and which is juft as much as half the Beam doth, that the Rafter is to Hand on. The Side of each principal Rafter, and the pricked Line is the true bevel of each, as is evident by the Skirts, reprefented here in Ledgment, being confidered with the Obliquity, or bevel of the Plan. The Trufs'A is the molt plain and Ample of all contained herein, and the bell for all kinds of Roofs, that are not of very great Extent. Plates CCCLXXXVII. CCCLXXXV1II. Various Trujfes for Roofs of a large Extent. In the fir ft of thefe Plates arc nine different Truffes, of which thofe mark¬ ed s, q, arc fit for Churches, etc. of which that marked q fpans beyond the Walls, as that of St. 1 'Patti, Cogent garden, where they are fupported by Co¬ lumns within, which is a great Help to the Walls, and is a very firm Way. The Truffes p, 0, are for thofe Kinds of Roofs, that are called M-roofs, hav¬ ing Gutters on the King-polls in their Middle, and which are often ufed to a- bate the Height; that of 0 is two thirds, and that of p is three quarters of the Height. The others, marked n, L, m, K, are for Rooms with arched Cielings, and of very lining Compofition. The'Piece marked bb is called a Collar-beam, and thofe marked d d are called Hammer-beams. In Plate CCCLXXXVI 1 I. are eight Defigns for Truffes of great Span ; that marked A is fit fora Building, from whole top fine Views may be feen ; that of B is called a Cirb-roof, and much eileemed 011 account of its giving much Space within-iide. The Trulfes Y X, W V, and Z, and G Z are different in their Sid*^ ; thofe of the left, being as fome particular Roofs, in or near Lon¬ don, are framed, and not being of the very belt Compofition, the right-hand Sides are to (hew, how they might have been framed with a great deal lefs Timber, and a great deal more Strength. As tis oftentimes neceffary to for¬ tify the Meeting of Timbers with Iron Straps and Bolts, 'tis good to obferve (as in Fig. D) to turn up the End of the Straps fquare, and to bolt on the Straps with fquare Bolts, which cannot turn within the Holes at the time of ferewing a68 The ‘Principles of Geometry. fcrewing on the Nuts, which round Bolts will do, and therefore cannot be fcrerv'd fo fait as they ought to be. Plate CCCLXXXIX. The Proportion and Manner of framing Spires on Steeples or Towers. The three Spires A, B, C, have their Heights proportioned as follows, (i) Make a regular Octagon, as cegd, Fig. D, whofe oppolite Sides are equi- diftant, equally to the Sides of the Tower, on which the Spire is to hand, (a) Make the Height of the Spire A, equal to 8 times the Side of the Octa¬ gon, the Spire B to nine, and the Spire C to ten. Make the Height of A equal to four of its own Diameters, that of B to 4. Diameters and 1 half, and that of C to y Diameters. To find the Height of Weather Cocks with their Ornaments. Make a regular Octagon, as cegd. Fig. D, whofc oppofite Sides are equi- dillant, equally to the Sides of the Tower, on which the Spire is to Hand. Set up 8 times the Side of the Octagon for the Height that the Hip-rafters are to rife in Fig. A, 9 times in B, and 10 times in C ; and the Remainder of the Height is the Height of the Vane, whofe Length is equal to two gds of one Side of the OCtagon, divided into three parts, viz. one for the Pointer, and two for its Plate. Altho' Towers are generally built fquare, yet Spires are commonly made oftangular at their Bafe ; as is that furprizing Spire at Salisbury, which Hands on a Tower of 200 Feet high, and its felt rifes near 210 Feet more. The Manner of forming their octangular Bales is reprefented in Fig. D, wherein caefghdh is the Bafe of the Spire, which is tyed in a very ltrong Man¬ ner, by the interfeCting Squares that are halved together and framed into it, and by the fliort angular Beams n, 0 , x, q, r, z, s, t, being cogg’d down, on which the Hips Hand, and framed into. A Frame being thus prepared, and bolted down upon the Heads of eight Standard Oaken Pofts, worked up in the Body of the Walls of the Tower, wherein, at proper Diftances, crols Pieces are let in, and worked up within the Walls, will Hand to the End of Time, could the Materials fo long endure; for by its being fo tyed down by the Weight of the Tower, if tis made with good feafon d Timber, and well framed together, it can never rack, fliake, heave, or fall down, except the Tower and that are overfet together. As each Side of fuch Spires are re¬ clining and contracted at their Tops, they do therefore trufs tip each other, as in the Figures is demonllrated. As there is more Difficulty to frame a Spire with a Lanthorn under it, as the Spire G, this Mafter has given us Fig. 1 . which reprefents the Manner of the Timber framing, embracing the Top of the Tower at c d ; alfo the Man¬ ner of framing the Lanthorn, as Fig. L, and the Cirb to its Head at ef as Fig. K, which two laft Figures are reprefented more at large, than the others are, for the better underftanding of them. The Plan H, reprefents a pro¬ per Frame to be placed at ha, under the Spire F, whofe Timbers are very well connected together. As to the Defigns of both the Towers, placed un der thefe two laft Spires, I mult own, would have been better, had this Mafter omitted them, they being a manifeft Proof of a Barrennefs of in¬ vention. Plate CCCXC. The Manner of framing curved Roofs to cylindrical Buildings. fur great Difficulty in thefe Kinds of Roofs is the Plan, which 11111ft be :o contrived, that the Prefliire of the Trulles may not thruft out the Plates. I he belt Way to eliedt this, is to frame and cogg down an octagon Plate on the The ‘Principles of Geometry. 369 the circular Plate, fo that the Middle of its inward Side be flulh with the Upright of the Wall within, and its Angles being braced, with Tyes cogg'd down, it will be very fecure agninft all the Thrult that can be made by any Roof The upper Trufs L is a Ca-veto , the next two, KI, are Cima redds in¬ verted, and the lower one an (hole, which laft is very ftrong, and of my own compofing. Plate CCCXCf. The Manner of framing Hemifperical Roofs, com¬ monly called Domes. This Plate exhibits the Plans and Sedlions of three Kinds of Domes, which I fhall deferibe as follows. Firft, the Semiplan B, by which the whole Plan or Circle is to be under- ftood, (whole Settion is Fig. A.) the outer Circle of the Plan marked b b, reprefents the Plate, the fmall Circle cc the Kerb, on which Hands a Lan- thorn (as in the Seflion,) the I.ines a a the Bales of the principal Rafters or Ribs, and g, h, i, the Purlins. This Dome being half a Sphere, its principal Ratters would be all Quadrants, were they not fhortened by the Lanthorn on its -Vertex ; and as their Height is equal to the Semidiameter of the Plate, they have therefore the fame Curvature, and are cut by the fame Radius, or Mould. As the Plate and principal Rafter cannot be made too fecure, 'tis therefore belt to make them in two Thicknefies, well pinn'd (and bolted, if in large Roofs) together, and, if pollible, to cut them out of Engliflj Oak, whofe natural Curve is nearly the fame with that of the Plate, &c. The Diftances of the Purlins e, d, in the Seflion, are equal, each being at one 3d of the Raf¬ ters length ; and if from e d Perpendiculars be let fall on the Diameter of the Plan, as / and g, to pals thro' thofe Points, where the Perpendiculars cut the Diameter, they will be the Moulds, by which both the Purlins are to be cut out, in order to be worked up (not fquared, as Mr. Trice calls it) for Ufe : In doing which. Care mull be taken to make their Backs agreeable to the Curva¬ ture of the Principals, and that their upper and under Surfaces have a true Somering to the Center, that thereby the Angles of the fmall Rafters on the under and upper Side ol the Purlins may be equal, and at Right-angles to them, as being the ftrongeft, and belt Manner of framing. To find the Somering and C urvatures in the Section of the ’Purlins. Set up on the Back of a principal Rafter, from its Foot, the Height of the Purlin from the Piate, and the Height of the Purlin alio ; from which draw Right-lines towards theCenter of the Principal, until they meet its under Side ; then thofe Right-lines, taken with the outward and inward Curves of the prin¬ cipal Rafter, will be a Section of the Purlin, from whence the feveral Bevels, or Angles being taken, and transferred upon the rough Purlins, and the Sur¬ plus cut away, the Remainder will be the Purlins, with their true Curvatures and Somerings, as required. N. II. it is abfolutely neceflary, that the Curvature and Sedtion of thefe Pur¬ lins be well underftood, before the Sawyers go to work, and wherein there is no Difficulty ; for if Care be taken herein, the Work and Time required to fi- nifii them will not he very conliderable. The middle Figures F, K, reprefent the Plan and Sedtion of the Domes in the Cathedral of St. 'Paul, Loudon ; the interior Dome, whofe Painting is a Monument to the Memory of Sir Janies Thornhill , deceafed, is exprefled by the Segments oi Circles on each Side ee, which is of Bricks, made 1 feet in length for that Purpofe. The middle part e e reprefents a circular Newel, or Opening in the Vertex of the Dome, thro' which from below you fee up to the Windows b bb$ &c. which give Light from the upper part of the external Dome. From the Bale of the internal Dome aforefaid riles the Fruftum of a Cone, nude of Bricks 18 inch, in Thicknefs, whofe Ufe is to carry the Cupola 4 7 . Handing The Trinciples of Geo m e t r y. 370 ftandin 0 ' on the external Dome, at G. This Fruftum (which Mr. 'Price mif- takcnlv calls a Cone ) having a very confiderable Tliruft againft the Walls that earn- it, I fuppofe, were one Motive of inducing Sir Chri/lopher to ftrengthen that part with a Corridorc on the Outfide, which is very grand, and beautiful alfo ; and, indeed, if we confider, that the Cupola Handing at G on the laid Fruftum is built with Portland Stone, and near do feet high, it is a very great and mafterly Performance, and an undeniable Proof of its Ar- chite&'s molt exquifite Judgment, and extenlive Knowledge in Geometry. The external Dome H is fpheroidical, and hath forne Dependence on the Fruftum aforefaid, its horizontal, or Hammer-beams e e, &c. having their Ends dependent on the Stones c c, &c. where they are curioufly tied together with Iron Cramps, that are run with Lead into the Stones, and then bolted through the Hammer-beams. As the Manner of framing the Timbers of this Trufs is made plain by the Seftion, nothing more need be added, but that the Number of thole lingle- trufs d principal Ribs is thirty-two, as in the Plan I is demonftrated, where they are obltrufted at a a, &c. to admit Light to the Windows b b, See. as afore¬ said It is to be obferved, that in this great Dome there arc not any Purlins, but it has horizontal Ribs inftead, by means of which the Covering-boards, that are nailed thereon, have very little Curvature. Within this Framing is a Stair-cafe, not here reprefented, which leads to the Balcony dd, from whence extenlive Views may be feen, when the Air is ferene and clear. The Figures C and D reprefent the Plan and Seftion ofa third Dome, which is alfo fpheroidical, and hath an internal, fpherical Dome, as that of St. Paul's. This Dome is made to fit a Temple of about So feet Diameter, and the Walls to an 8th part of the Opening ; but the Lanthorn, placed on its Vertex, mult not be made of 'Portland Stone, as that at St. Pauli, becaufe here is no Provi- fion made to carry fuch a Weight. The Manner of framing the Kerb to the Opening of the Lanthorn at C, as reprefented by the interfering Timber-fquares halved together, as at D, is very ftrong, and ties that part of the Roof well together, but there is not a- ny Provifion made to prevent the Weight ol the Cupola from thrufting out the Handles ol both Domes, excepting the lower Brace on each Side, where¬ fore I can't but recommend the placing of Struts from the Bafe of the outer Dome up unto the Bottoms of the two upper King-polls, which, together with the Struts, that go from the Bottoms of thofe King-polls up to the Side of the Opening, will be capable of fupporting a Lanthorn of a very great Weight, which, without them, would never Hand- Thus much by Leave ol this Ma¬ tter, whofe Works, as well as thole of Mr. Smith, I have (if 1 miftake not) explained to the Underftanding of young Students fomething more largely, and plainer, than they themfelves (or their Scribes) have done, and that, 1 hope, without Offence, as being done for the publick Improvement of the no¬ ble Art we are now treating of. Note, This Dome is made to confift of fixteen principal Rafters, or Ribs, which is a mean Proportion between the former two, the one of eight, the o- ther of thirty-two, and which may be framed with Purlins, as the firlt, or with horizontal Ribs, as that of St. Pauli. Note alfo, That if to a Dome there are but 11 principal Rafters required, then, inftead of making the Kerb of its Opening with two geometrical Squares, as at D. you mult apply two equilateral Triangles together in the fame Man¬ ner, which will produce fix external, and as many internal Angles, in the fame Manner as the interfering Squares produce eight internal, and as many exter¬ nal Angles, for the Reception of the lixteen Ribs. Having thus explained the Formation of circular Roofs, I lhall only add, that the Feet of all principal Rafters to Domes lliould extend 110 farther, than the Upright of the Wall, but thofe to Roofs, where their Forms are Cavetto's, or Lima's, as Figures I,, K, 1 , Plate CCCXC. may extend to the Extremity of the Cornice Plate Plate CCCXCII. Triangular, fquare, and oblong Roofs demonjlrated. Jt d. To form this Roof , divide each Angle into two equal parts by the Lines i ds, b d, and a d, which are the Bales, over which the Hips mult Hand ; on the Line c s, at the Point d, eredt the Perpendicular d k, which make equal to the given Height of the Pitch, and draw kc for the length of one Hip, and he for the length of one principal Rafter. The length of the Hip k c being thus found, complete the Triangles c f b, hr a, and c p a, making each of their Sides, as cf, fb, &c. equal to the length of the Hip kc, and then will the Extreams Continue the Sides cf and c p out at plealure, and from the Points b and The Plan capo, Fig. II. is a geometrical Square, and the Plan bayz. Fig. III. is a Parallelogram, whofe requifites arc found, according to any given Pitch, in the fame Manner, which their Lines do very plainly demonftrate. Plate CCCXCIII. Oblique-angled (commonly called bevel) Roofs demon- The Plan acbd. Fig. I. isaRhomboid.es, in which its principal Rafters are placed at Right-angles to its Sides, and the Plan bodi, Fig. II. is the fame Figure, with its principal Raltej-s -placed parallel to its Ends. The Plan edab. Fig. 111 . is a Rhombus, whofe Hips are ci and dg, &c. and Principals n'f and o h, ike. the length of which, as all'o of the other two Examples, being found by’the aforefaid Rule, and which their Lines very plainly demonftrate, need no farther Explanation. Plate CCCXCIV. fin irregular Roof demonjlrated, 'with a General Rule for backing of principal Rafters. T h e Plan cb da. Fig. I. is a Trapezium, whole Sides in general are une¬ qual, and confcquently all the Angles arc the fame. To frame this Roof, lb as to make the Ridge level, is a Work of lome Difficulty, and the Method of performing it is as follows. Suppofe the Side a b to be the Front, to which the Ridge mu ft be level ; divide the Angles dab and cb a each into two equal parts, by the 1 .ines 11 a and g b, and let the Line 11 g be the Bale of the Ridge ; 'all'o let n i+ be the given Height of the Pitch; from the Point n draw tlic Line n iz at Right-angles to ri a, and 1113 at Right-angles to nd-, make 11 iz and n 13 each equal to 11 14 the Height of the Pitch, and draw’the Lines iz a and 13 d for the length of thole Hip-rafters: In the fame Manner, on g, draw g h at Right-angles to g b, and g 1 at Right-angles to g c, each of which make equal to the Height of the Pitch, and draw ci and hb for the lengths of thd'e two Hips. This done, aftign the Places for the Beams, and where the Beams cut the Line g 11, there raile Perpendiculars to them equal to the Height of the Pitch, and from thence draw Right-lines to the End ol each relpcftivc Beam, and they will be the lengths of the feveral principal Rafters ; in the fame Manner the lengths of every pair of final! Rafters are to be found ; the backs of each Hip are found by my Rule before given, which the Lines within the Angles at c and d demonftrate. The Triangles d 1 a and c eb are the Flip-ends, and the Trapeziums itc 10 d and by a z are the two Sides of the 3?2 The 'Principles of Geometry. Roof laid out, and which, being turned up to meet each other, will fliew the whole Figure of the Roof: As the Side cd doth not cut the Feet of the Raf¬ ters at Right-angles, they mull therefore be all backed, which may be done as fellows : • _ To find the Backs of Rafters in a bevel Roof. I. ct the parallel Fines t reprelent the Plate, the Line r the Bafe of the R Fge, and the Lines k x and i w the Bafe of a Rafter, over which 'tis to Hand when up in its place, and let h i reprelent the breadth of the Foot of the Raf¬ ter : From the Points o and n draw the Lines o q and nm, at Right-angles to tiie Lines k o and i u and make each equal to the Height of the Pitch, and draw the Lines qh and mi, which are the lengths of each Side of the Rafter, alii, draw the Lines a l and pf for the Depth of the Rafter ; from a draw a c at Right-angles to z m , alfo from f draw fh at Right-angles toiq; from c draw i li at Right-angles to i tit, and from h draw hg at Right-angles to k x; from g draw ge, and lfom b draw bd, at Right-angles to" the Plate; on a, v. ith the Radius a c, cut the Line b d in d ; and on f, with the Radius / h, cut tne Line g e in e ; then drawing the Lines ef, e d, and da, the Trapezium ed fa, will be a Section of the Rafter cut through at Right-angles, and then ns ieveral Angles are the Angles of the Back required. For if the Sides of the Rafter qp kf, and l m i a, w ere turn'd up to Hand over the Lines ko and z n, then the Points q and m would be perpendicular over the Points o and ?z; and it the Trapezium e df a was turn'd up on the Line f a, to meet the Sides cl the Rafter, the Point d would be at the Point c, and the Point e at the 1 oint h , and as the I .ines f h and a c, are at Right-angles to the Sides of the Rafter, therefore the Trapezium edf a will cut the Rafter at Right- angles, and be the Section required, if E. D. I late CCCXC\ . Other oblique for bevel) Roofs demonjlrated. The two Plans reprefented here are both the fame, but their Manner of Framing are different. That of Fig. I. hath a level Ridge all round it, with a Flat or \ alley f e h g in the midit, and which, in fuch Calcs, is the belt and handfomeft Manner of Working, rhat of Fig. If hath its Ridge level; and as every Pair of its Rafters arc of different Lengths, the Sides of the Roof will be curved, or rather twilled, not flat as in other Roofs, and which has not only a veiy ill Effect, but is very troubleiomc in the working. The Manner of finding the Lengths and Backs of every Rafter is the fame, as afoieiaid, and which the 1 .ines demonitrate ; or as Air. Trice commonly lays, (in his 1 reatije of Carpentry') due Infection will make plain. Plate CCCXCVI. The Reajons and Manner of Backing Circular and Convex Hip-rafters. Admit BI)P to be the Angle of a Building, over which is to Hand the convex Hip ■rafter KL io, whole principal Rafters are Quadrants or Ovo- los, as the Arch i a, which is fuppofed to ltand over the Line i E. To find the Curve of the Tip-rafter. Let FE reprelent the Bafe of the Hip, and i E the Bafe of the Principal, ■ s aii relaid. Divide z L into any Number of equal parts, luppofe four, as at ogf, irem whence draw the Ordnates h d, g c, and fb\ divide EE into the i.w.c Number of equal parts, as at ay, 24, 23, and draw the Ordnates zy, 18, • l t> - 7 , -F -h, and E io, equal relpedtively to the Ordnates h d, gc, f b, aid h.a; and through the Points io, z 6 , 27, 28, M, trace the elliptical Uuve, which is the Curve of the Hip required, whole Depth or Thicknefs is iuppoled to be LX. To The ‘Principles of Geometry. 373 To find the Bach of an elliptical Hip. Strike a Chalk-line down the middle of the Back of the Hip, and to its Foot fix a Mould of the Angle equal to that of the Plan, fo that the middle Line Hand exabtly at the Angle, as at F ; and if we l'uppofe the Breadth of the Back to be equal to ON, then the Parallellogram ON AIL will be the Plan of the Foot of the Hip, which projects beyond the two Sides of the Angle, equal to the Triangles FM*, andFL*; this done, draw Right-lines on each Side of the Hip-rafter, at any Diilance from each other, parallel to its Bafc ; and on each, from the outward Angles of the Hip, let off the Diilance L*, as P*, Q_*, rc. in which Points, fix Email Nails, and with a thin Lath apply’d to every of them trace a Curve on each Side. Laftly, cut away all the Timber contain’d between thele Curves and the central Chalk-line, and the Angle made thereby, will be the Back of the Hip required. DEMONSTRATION. Draw F. e at Right-angles to Ff, and continue it to the other Side at B; divide BE and E e into 4 equal parts (as F) at the Points m, iy, 14, 15, n, 11, and from thence draw the Ordnates wn, iyai, 1410, EG, 13 18, it 17, 11 1 6 , which make equal to the Ordnates of the Principal or Hip, that is, make 14 10, and 13 18, each equal to 13 id; allb the Ordnates iy ii, and n 17, each equal to the Ordnate 14 17 ; and laftly, the Ord¬ nates m n, and 11 id, each equal to ly 18, and from the Point e, through the Points id, 17, 18, 19, 10, ir, 11, to B, trace the Semiellipfis B 19 e: Now, forafmuch as the Ordnates of the Hip are refpetlively equal to the Ord¬ nates of the Semiellipfis, therefore, il the Ordnates of the Hip) be erefted perpendicular on its Bafc, and thole of the Semiellipfis on its Diameter BE e, their refpeftive Heights will be equal; and the Semiellipfis B 19 e will be the Section of the Root, cut through at Right-angles to F FI, and conl'equently that part of its Curve Handing over E, is that part of the Hip's Back. To find the Angles of the Back Handing over any given Place, fuppole over the Point 14, draw the Line 14 e, or 14 B, and at the End 14, erect the Perpen¬ dicular 14 30, equal to the Height of the Pitch, and draw the Line B 30, which transfer to I, and draw BI, which divide into 4 equal parts, from whence draw Ordnates of Length at plcafure. Now, forafmuch as the Ordnate eg, of the Principal i m a , is the Height of the given Point in the Hip, which is allb equal to the Ordnate a.4 17; therefore draw the Line c 2, which divide into the fame Number of parts as B I, and from thence draw Ordnates to the Arch 2 c ; make the Ordnates of BI equal to thole of the Arch i c, and thro’ their Extreams trace the Curve B a I, which is 1 half of the Angle of the Back, Handing over the given Point 14; make the Curve le equal to the Curve BI, and the Angle Bl e is the Angle required. In the fame Manner the curved Angle at any other Point may be found, as by the Lines is de¬ mon ft rated. To find the Hip-rafter, where the principal Rafter is a Cavetto, ay Fig. I and the single of its Back. Admit C.g a to be the Angle of a Building, and the Curve 8 6 y a principal Rafter, Handing over the Line ty, and tis required to find the Curvature of the Hip dg ; draw the Line l g for the Bale of the Hip, which divide into the fame Number of equal parts, as you do the Bale of the Principal ty; fuppole 4, as at sp 0 ; from whence draw Ordnates, equal to thole drawn in the Bafe of the Principal, and making s h equal to pu, p k equal to w, and m equal to x ; from the Point d, through the Points h, k, m, to g, trace the Curve d b k m g, which is the Curvature of the Hip required. Note, The Method by which the foregoing Hip was back'd, will alfo back this, or indeed, any other Kind whatfoever, as Cima's, &c, T A To' 374 The Principles of Geometry. To dejcribe the Angle made on the Back in any ajjigned part of it, fnppofc at the Point of h. Make g e at Right-angles tog/; and from e, through the Point h, draw the lime eh i, meeting gt continued in i. Through the Point i draw the Line i si a, cutting each Side of the Angle in i y a ; make i h equal to ih, and draw the Right-lines h if, b a. Now, forafmuch as the Ordnate 7 u is equal to the Ordnate s h, they are, when in their Places of equal Altitude; and as h is the given Point, and the Point is equal to it, therefore draw the’ Line 7/, which divide into Ordnates, as 1, a, 3, 4, j-,6; divide i iy into the fame Number of equal parts, as 7 y, and make its Ordnates equal; thro’ whole Extreams trace the Curve 1713 11 10 h\ make the Curve h \6 a equal to the aforefaid, and the curved Angle 17 b a is the Angle of the Back of the Hip, over the Point h, as required. In like Manner, the curved Angle z r f is the Back over the Point k, and that at l over the Point m. It is to be ob- ferved, that the Angle of the Back of this Hip increafes from its Foot, even from a real Point, and opens as it afeends to the Top, where it becomes an Angle, equal to that of the Building over which it Hands; and on the con¬ trary, the Angle of the Back in the aforefaid Hip, where the principal Rafter is an Ovolo, there the Angle at the Foot of the Hip is equal to the Angle of the Building, and decreafes as it afeends, unto a very Point at the Top. Thus much for Roofing, which, being underftood, will enable anyPerfon to eafily perform all Worksof this Kind, that may happen to be done, and which J have been the more copious in, as being one of the mofl eminent parts of a Building, and never before made plain to mean Capacities, which here I have endeavoured to do. Plates CCCXCVII. CCCXCVIII. Coverings for curved Roofs. I n Plate CCCXCVII. are reprefented fix Figures, that of Fig. I. being a fphe- rical, and Fig.II. a fpheroidical Dome, whole Manner of Covering has been already explained in my Explanation of Niches: Vide the Word Riche in the Index. The Plans, Fig. III. IV. V. VI. are all Hexagons, but the principal Rafters of each are different, that of Fig. III. being a Cima reCla , Fig. V. a Cima reverfa, Fig. IV. an Ovolo inverted, and Fig. VI. a Cavetto inverted. To find the Curvature of their Hip-rafters. R U L E. Divide the Bafe of the Principal, and of the Hip, each into any Number of equal parts, as a 9 and 9 1, Fig. III. and draw equal Ordnates in each, which will give the Points p, q, r, s, /, u, w, e, through which a Curve being traced, is the Curvature of the Hip required. In the fame Manner all the other Hips to the other Figures are iound, as the Lines of each demon- ftrate. Their Manner of Covering. Continue out the Bafe of a principal Rafter, as 9 1, to 18, making its length 1 18 equal to the length of the Curvature of the principal Rafter 1 17; divide the Bale of the Principal 9 1 into any Number of equal parts, as at the Points x, 3, 4, y, 6 , 7, 8, through which draw Right-lines, at Right-angles to 9 1, cutting the principal Rafter in the Points jo, 11, ix, 13, 14, iy, 16 , and the Bafe of the Hip in the Points h, i, k, /, m, n, 0. Make 1 xy equal to the Curve 1 10, alfo 1 14 equal to the Curve 1 ix, alfo 1 13 equal to the Ciuve 1 ix, ef'e. and through the Points xy, X4, X3, xx, xx, xo, 19, draw Right-lines, at Right-angles to 1 18 ; make xy 31 equal to h 1, 14 31 equal to 3 i, X3 30 equal to yk, xx 19 equal to y /, &c. and through the Points 31, 3 L 3 °> 2 9 > " i8 » V> R', from a to 18, trace a Curve, and make the Curve 18 h equal theieto ; then will a 18 b cover one Side of the hexangular Roof; for. 11, iz, &c. in the Principal, and as the Points zf, 14, 15, <&c. will lye pei- pendicular over the Points z, 3, 4, &c. and as the Lines 32 if, 31 24, 3 ° 2 3 > are equal to the Lines h z, i 3, £4, &c. therefore 18 b will exactly co¬ ver the part of the Roof over the Triangle a 9 b, and confequently fix of fuch Pieces will cover the Whole. The other Rools, figures IV. \ . AT. are co¬ vered by the fame Rule, as is plainly feen by the ieveral Lines in each Plan, but more particularly in Fig. V. where j e iy is the Covering-piece to the Side f b e. * . In Plate CCCXCVII 1 . are nine Figures, which are as follow; Fig. 1 . repre- fents a conical Roof on a Cylinder, and Fig. If one 8th part of its Covering ; Fig. III. reprefents a Bottle-roof, and Fig. IV. one 8th part ot its Covering; Fig. V. reprefents a Bell-roof, and Fig. VIII. one 8th of its Covering ; Fig. AT. R U L E. Divide the perpendicular Height into any Number of equal parts, and through every of thole Points draw Right-lines parallel to the Bale, which conlider as Diameters of fo many Semicircles, which deferibe, as i r c, n q l, &c. Fig. I. make u b , Fig. II. equal to the Side ol the Cone d c, and divide ab in the fame Manner, as dc is divided by the Semicircles ; divide rcinto 2 equal parts, and draw the Line dm, which will divide every ol the Quadrants into two equal parts alfo; then make If, Fig. II. equal to half r c in Fig. I. alfo h i equal to qm, &c. and the Figure b If will be one 8th of the Cover¬ ing required. It is here to be noted, that by this Method all other circular Roofs may be covered, it being a General Rule, and which I fhewed here in the Example of covering the Cone, altho’ I have already demonftrated that Covering in a different Manner in Plate CCCLXX. Fig. VII. demonftrates the Seftions of a Cone by common Ordnates, of which fee Conick Sections, in the Index. Plates CCCXCIX. CCCC. Dcmonjlrating the Proportions of Rooms. The geometrical Figures, made ufe of for the Generality of Rooms, are, (1) The Circle, asg; (2) The Square, as f, Plate CCCC. (3) The Octagon, as e, and (4.) The Parallelogram, as dbjca, &c. Rooms, that are Parallelograms, are of divers Proportions, viz. (1) Their Length equal to the Diagonal of their Breadth, as d b ; (i) To the Breadth and half, as /, and Fig. b, Plate CCCC. (3) To the Breadth and one yth, as d, Plate CCCC. (4) To the Breadth and two 3ds, as c; (?) To the Breadth and three qths, as k, Plate CCCC. ( 6 ) To the double Square, or twice the Breadth, as a, and as h, Plate CCCC. When the Lengths of Rooms exceed twice their Breadths, they become Galleries, which may have their Lengths three, four, or five (but not more) times their Breadth, as. Fig. m 0, Plate CCCC. As Rooms are differently ciel'd, forne being flat, and others arched, their Heights are therefore different. Grand Rooms with flat Ciclings fhould have their Heights equal to their Breadths, but where Grandeur is not to be ftri&ly obferved, a lefs Altitude may be given, provided that the Breadth of fuch Rooms be not lefs, than 16 Feet, when three qths thereof, viz. ai Feet, may be taken for its Height. The Height of Covings to Rooms, whole Height is 15 Feet, or lefs, is one 4th of the total Height; but of Rooms, whofe Height exceeds ry Feet, as 30, t&c. the Height of the Cove fhould be one 3d of the whole Height. Hale the Sum of the Length and Breadth of all Rooms, not exceeding the double Square, maybe an eitablifh’dRule for the Height of cov d Rooms in general; and when their Lengths exceed the double Square, their Heights 376 The Principles of Geometry. are to be confidered as the double Square; fo a Room zo by zo Feet its Height mult be zo, being one half of 40 ; and a Room zo by zy, whole Sum is 4f, gives zz and a half for its Height; alio a Room 50 by 20, whole Sum is yo, gives zy, its half, for the Height ; and a Room zo by 40, whofe Sum is 60, and its half 30 is the Height. This lait being the double Square, its Height is the fame, as that of the Gallery m 0, Plate CCCC. whofe Length is 100, and Breadth 20. Palladio makes the Heights of his Rooms to be a mean Proportion be¬ tween their Lengths and Breadths, as in Fig. B, Plate CCCXCIX. where hk being the Length, k m equal to the Breadth, the Line hi is a mean Propor¬ tion between them, and the Height affigned, which is fomething leis than the half Sum of the Length and Breadth, as aforefaid. Plates CCCCI. CCCCII. CCCCIII. CCCCIV. CCCCV. Acute, Right, and Ohtufe-angled Brackets demonfirated. As I have fo largely explained the Formation of divers Curves by Ordnates by Which all thefe angled Brackets are found, it feems to be almoft need left to fay any thing hereon, the Whole being very plain by Infpedtion ; but, that the young Student may not blame me, 1 will explain one Example, by which all the others are to be underflood. Let e a b, Fig. III. Plate CCCCI. reprejent a Front-bracket, whofe Curve e />, is Juppofed to ft and perpendicular over the Line e c, and let the Line d e reprejent the Baje of the Angle-bracket. Divide ec into any Number of parts, no matter whether equal or une¬ qual, as at the Points p, q, r, s, t, u, w, and through them draw Right-lines parallel to e B, cutting the Curve of the Bracket in the Points 0, n, m l k, h, 1 , and the Bale ol the Angle-bracket in the Points 1, z, 3, 4, y, f from whence draw Right - lines at plealure at Right-angles to the Line e d; make the Ordnates 18, z 9, 310, &c. equal to the Ordnates p 0, q n r m, &c. and through the Points 8, 9, .0, 11, 12, 13, 14, from e to ? trace’ the Curve of the Bracket required. The hill thiee of thefe Plates exhibit all the various Mouldings, Acute Right, and Obtufe-angled, which in general have the Mould of their Brack¬ ets found by the foregoing Rule; as is alfo the Tufcan Cornice, in Plate CCCCIV. and CCCCV. The Figures B, C, are a Front and Angle-bracket, ac¬ cording to Mr. 'Price's Method, which is as follows. The Fig. C being a Front-bracket, whofe Height is n 0 , and Projection n m, draw the Line rft 0, and parallel to n 0, the feveral Lines a 1, bz, c 3, ^4’ &c. dividing the Line m 0 in the Points a, b , c, d, &c. meeting the Curve in the Points 1 i, 3, 4, &c. The Angle-bracket is reprefented by Phg. B, where o p is the Height (equal to n 0, the Height of Fig;. C) and 0 n, its Projeftion; diaw ?ip ) and divide it into the fame Number of parts, and in the fame Pro- portion as m 0 in Fig. C, from whence draw Lines parallel to 0 p, and each 1 cfpeclivcly equal to a i, b z, c 3, d 4, &c. ot Fig. C, and through their Ex¬ tremes r, z, 3, 4, &c. trace the Curve », 11, 10, Scc.p, which is the Mould of the Angle bracket required. F : g. 1) is another Example olthe fame kind. The Fig. E and F are Brackets to make lath d and planter d Cornices on. That of Fig. E is a Front-bracket, made fit to a propoled Cornice, and F is' its Angle-bracket, which is made as follows ; I ne Front-bracket being prepared, as in the Figure, and the Projec- foii of ihe Angle-bracket being given or known, which here is bg, divide it in the lame proportion, as m f in Fig. E, as at the Points c, d, e,f, which a ! e at T ., iame proportionable Diltances from each other with refpeft to r , lc , as rhe P°’ nts L <5,7, 8, in the Line mf are with refpect to r.iat \\ dole ; make the dotted Lines parallel to ba in Fig. F, whofe Heads The 'Principles of Geometry. 37 7 marked with c, d, e, f, equal to si, 6 c, 7 d, 8 e, in Fig. E, and through their Extreams draw the Out-line of the Angle bracket as required. Plate CCCCVI. The Formation of Centers for turning the Vaults of Arches on, in Brick or Stone. E X AMPLE J. Fig. A. The the Saliant Angles a, b, c, d, being work’d up with the Walls to the Height from whence the Arch is to fpring, and the Curve of the Arch beinp determined (fuppol'e) a Semicircle, as reprelented in the Section B, begin at dec, and center it through as a common Vault, and board it. To make the Groins, let Centers, as from a to c, and from b to d; divide the Curve dec into + equal parts, as at g and /; then is g e f a Mould for fmall Centers which will be wanted to nail on the Centers firti boarded, whole Bale is at h Ihel'e fmall Centers are to he put in at Difcretion, as the bearing of the Boards may require. To make the Groih /height, over its Bafc, at Ionic little Height over the Centers, ftrain a Line, as from b to c, or from d to a, from which, with a Plumb-line, drop Perpendiculars on the Boarding, (which is fuppofed to be firlt fixed) at as many places as necclfary, and therein itrike Nails, to which apply a Itreight and pliable Ruler or Lath to touch them, and, with a Pencil or Chalk, delcribe the Curve, which will be a Semiellipfis, to which brin" the Boards to be nailed on the aforefaid little Centers, and their Joints will form a Itreight Groin. To cut the Angles of Boards, to cover any Center required, Fie* VIII . Plate CCCCVH1 The Plan dacb is to be vaulted, with a lemicircular Arch, from a tod and an elliptical Arch from b to a. To cut the Boards to cover this Center, let n be the Center of the 'Plan : On one End, as d c, delcribe the Semicircle dgc\ continue out c d both Ways to l and m, fo that l m be equal to the Cir¬ cumference of the Semicircle egrt, and draw the Lines mnmdln-, divide ml in inch manner as the breadths of the Boarding will allow, as the dotted Lines reprefent, and cutting their Ends to the Angles or Bevels of the Lines mn and n l, and their Lengths anfwerable to the Line ml, they will exactly Cover that End of the Center, as required. In the like mannermake k i equal to the Girt, or Circumference of the Semiellipfis cfb, and drawing the Lines kn and a i, and dividing k i according to the Breadths of the Boards, as was done with m l, the Angles or Bevels made by the Lines k n and n i arc the Angles required for that Side, and fo in like Manner all others. EXAMPLE II. Fig. C, Plate CCCCVI. As this Plan is of greater Extent than the former, and if the Weight on it be great, it mult not only have its Angles fa'liant, as the other, but project¬ ing Piers, as c e, &c. and others entire in the middle, as / l alfo. The Section D ihews, that the Length contains three Arches, and thole lemicircular, where¬ fore their Groins, as ce and a/) &c. are femielliptical; by this Method the Arches will fuftain a very great Weight, with fmall Abutments ; but if thole middle Piers /and / are found inconvenient, and the Abutments can be made fecure, then the aforefaid Piers may be rejected, and elliptical Vaults turned both IV ays, as the dotted elliptical Curves exprefs in D and E, which is equal¬ ly as ltrong as the former, and much more fpacious, if the Abutments are made fecure. EXAMPLE III. Fig. G, Plate CCCCVI. I he Length of this Plan hath an elliptical Vault, as denionftrated by the S B Section cr. The ‘Principles of Geometry. Sc. Hon I. whole Height is regulated to the Height of the three fcmicircular Vault: that pals through its Breadth reprefented by Section II. which iiinng from the laliant Piers h e, a d, &c. in the Sides and Angles. To ' -:d the Groins , perpendicular and freight ever their Ba[es. Draw Ordnates in the Semiellipfis of Seftion I. at any Diftances at Plea- j'ure, which continue to meet the firft Groin in G; make n c the Breadth of the ParaUeloaram L equal to the,Circumference of the Semicircle e f b, in Fig. H ; and ne its Length, equal to the Circumference of the Semiellipfis f, and draw h e, and m 9 in Fig. L ; divide m 9 in Fig: L, in fuch Proportion, as 9 m 111 Fi°. G is divided by the Ordnates being continued, and through thole Points draw Right-lines parallel to c n both Ways at Plealure; divide e 9 and 9 6 each in the fame Proportion, as m 9, and through thofe Divifions draw Right-lines parallel to n e, which will cut the others in Points, through which the curved Lines of the Groins mult be traced, and thus will the Figure I. reprefent the fioiito or Area of one of the fcmicircular Arches, prefled down on a flat Su- pirficics. The Figure K reprefents the fame of the whole being conlidered in Breadth only, fuppofing the Piers that fultain the Arches on each Side, were laid down at the Kxtreams of the Extent of the femielliptical Vault, when, prefled down to a flat Superficies. E X A M P L E VI. Fig. M, Plate CCCCVI. The Figure M reprefents the Plan of a Cieling, and Fig. O a Sedlion of the Room it belongs to, whole Cove is one 4th part of the Height. As the Pro¬ jections of the Coves are equal to their Heights, the Diftance between their Projections, as g h , is the Breadth of the Panned g h, in Fig M, and the fame, being taken out of the Length, leaves h k for the Length of the Pannel; and as the Angle-brackets and Aftragals are found by the preceding Rules, 1 don't fee what more is to be done, or what Mr. 'Price means by Fig. P, or to what Ufe 'tis to be applied, which he calls, the Face of O, as /Fetched, or extend¬ ed out , on which any thing , propofed to he deferihed therein, may he truly performed. EXAMPLE V. Fig. III. Plate CCCCV 1 II. The Example, given here by this Mailer, is in order to prove the necefia- ry Abutments to large Arches, that thrult againft final] Arches, as the Arch L againft the Side-arches M and N, and which he endeavours to proportion by the Height of the Curves of thofe Side-arches, as follows; viz. Make the /.'eight oj the (mail Arches fimilar to that of the great Arch ; (which, fays he, are the proper Curves, according to the Laws of Strength.) But herein 1 mull ask his Pardon ; for if the refilling Weight, laid on the Hanches and Piers, be not fuperior to the Thrult of the Scheme-part of the Arch, it cannot Hand; therefore, let the Curvature of the finall Arches be what it will, if the Weight of the Scheme of the great Arch be fuperior, they will rife up in their Tops, and permit the great Arch to fall down, as many have fo done; and (if I mil- take not) fuch was the Cafe in the vaulting of the new Church in Spittle-fields. E XAMPLE VI. Fig. I. Plate CCCCVIII- The Plan O reprefents the Plan of a Cellar in a Dwelling-houfe, which is given to Ihevv the Variety of Groins, whole Bales are exprelled by the dotted Lines, and a Section of tile Whole by Figure P. EXAMPLE VII. Fig. C C, D D, &c. Plate CCCCVII. Tms Figure reprefents the Manner of forming curved Groins, an Example of which may be feen in St. Clement e, Danes, London, and other Buildings of the like Nature, and (as Mr. Trice obferves) is a Work worthy of our Regard. To find the Bafe of thefe Groins, (x) adhc being the Plan, continue ah and The Principles of Geometry. 379 £;3?5ass=5i§a mrt - , s a], viz tell, as at the Points I, %, 3, 4 -. Lc. .V 0 " 1 , ,. , St-linesto the Center which will interim the atorclaid Arches m Points, r c and have their Curves thus formed ; make ad, in Fig. BB, equal to L ’ nlfo make it Fie EE, equal to the fmall C.iuve be, anddniUL Curve a$ cl, alio nun 1 i b . ^ 1 1 divided fnl a fo^arB C complete TeVemfcirclc'^lV > ts Slxtr^ms trace their Curves, and then BB and E E being bent, lo as to ftand on the curved Lines of the Plan ad and and the . ^ and D D being fet up over the Lines a b and dc , they ‘ ^ make r ,F ? .V eqm,ltothe ( Sr • %/ ,i iKn • nlfn make £ Fie A A, equal to the Cuive ec in tne nan, 'immmissm gSl “mt fc .rue Groin, Ming ferprndicnl.rt, o»„ „ M. » required. The other for b e d is found in the fame Mann Plate CCCCVII. Various Centers for Vaulting demonflratel The Plan 6 adc being a geometrical Square, and the Arches ^oA.Vau!£ - -a* and which is defenbed by Ordnates, as the Figure expielles. EXAMPLE II. Fig. P. Th , s Plan acid, being a Prallelogram and the Arch of the Lengg, of whole HeLht is equal to the former, and Length to the Diagonal b c and are both defenbed by the fame Ordnates taken from the Semicircle, as the Lines deinonftrate. EXAMPLE III. Fig. R. This Plan badc'm geometrical Square, and the Arch of the Vaulting 1 <1 Wavs is a Gothicl Arch to find the Curvatures of the Groins, draw da, both W ays is a L iOiijick ./ . 7 mil-p the Ordnates others from b to c. EXAMPLE IV. Fig. X. I N this Plan a c b d, the Arch of the Length of the Vault bhd is a Seg- sa from thofe of b h d, as the Lines deinonftrate. E X A M- q8o The ‘Principles of Geometry. )' X A M 1 'LE V. Fig. Z. T h is Plan dhac is a Rhombus, and the Arches of the Vaults both Wav are Semicircles. To find the Curves of the Groins, draw the Diagonals b e and a d, divide each in luch Proportion as be a, draw Ordnates in each e- qual to rhofe of tlie Semicircle b f a, and through their Extreams trace he femielliptical Curves b n c, and a hd, which are the Groins required. EXAMPLE VI. Fig. W. In this Plan fade there is but one Vault, and that in length, whofe Arch is a Semicircle bhca. It is fuppoled that in its Sides there are to be fmall Arches made, over the Heads of two Doors or Windows, which will Theie/fjtW'T / f ; hC Va r k at R >gEt angles. To find the Curves of - the/e [mail T emits, which they make with their meeting of the great Vault whVtfhT a ° WS: f £t ' \ re P refentthe Breadth of the Window, of fmall Vault’ Inch bileOin/; diaw the Ordnate hg, in the Semicircle, equal to y t- alfo make , i and u t equal to / t, and draw r u ; draw « / at Right-angles to b d continue hg tow or to meet zy continued in .v, then drawing ,\ v and tx he Tnangle i xt will be the Plan of the fmall Vault; make xw at Right- angles to v /, and equal to ; divide bg into any Number of parts, and draw k * t0 th n Ar< i h bb \ dlVldc EE/G i V .rV, each in the fame Propor¬ tion as bg, and making their Ordnates equal to thofe of bhg, throught their Extieams trace the Curves z,zt, and t w, which are the Curves of the layover [T V , , f re . t l uircd i for lf the Arch r zt was raifed perpendicu- l . , 1 lC '™ llld bc the Stand Rib of the fmall Arch, and if the Curve ' eat Vault rl a Cn h ' ^ ^^ k wouId the Groin meeting the S- , } . t; a3 havin § e< i uaI Ordnates; the Triangle 2. 0 k, on the Right-hand mde of this Figure, is a Idler Arch than that deferibed, which hath its Groins A Vh f it 1 the 0th vr ’ , the llke 13 ^ to be obferved m Fig. S, where the Aich o the g>eat Vault is a Semiellipfis, which is interfered in its Sides by A 1 B CD vt ar r 1 * t’ a " d 1 ° k - F(,r the Defcription of Figures I’. , i D ’ u b - ^’„ G ’ &C ’/ ee the Word faults of Brickwork in the In - de.\ "here then Manner of working is deferibed at large. p„t HE Fisl ", es I , V U V ; VI VIL demonftrate the Pitch, and Mitre-joints of ediments, ol which lee the Word Tediment in the Index. Plates CCCCIX. CCCCX. CCCCXI. Defigns for Geling-pieces after the ancient Manner. In thefe three Plates are contained eleven Defigns for Cieling-pieces bv Se ff“* r :^' vheran , ar ? i ? 1 ” e ’ 5 hich ,node ™ Architects have not e.xceed- Se ven Defigns for Cieling- ed, and which are very helpful to Invention. Plates CCCCXII. CCCCXIII. CCCCXIV pieces, by I. Jones. l , HF \ F lEfigns, being of grand parts, and thofe not crowded with Orna- K,1 d ’andMurh 316 ^ S T I ud S ment of this Mailer in Decorations of this ration. “ 6eneril1 ' are worth >' of our greatell Regard and Confide- ‘ -as CCCCXV. CCCCXVI. Modern Defigns for Cielings. Pla\es 'cCQdIX r and 1 CCCCX? e vi^: n ar' n th^ e Fdates Wlth thofe of itolen from V ' sVrL t L J we J aie r then informed, that thefe Defigns were v ho, and indeed, if we compare thofe on Plate CCCCXVI. with The Principles of Geometry. 38 1 with them of Inigo Jones in the preceding Plates, we are alfo informed, that his Manner of dividing out the Pannels were borrowed from that Matter, but not his Ornaments, which I think are too numerous, and therefore lets noble. Indeed, I mult own, that the Proportions and Profiles of thofe in Plate CCCCXV. are very inftruaive to the Workman, and which are Improvements on Serlio worthy of our Thanks. These Defigns may be executed either on flat or curved Cielings, but in both Kinds it is to be obferved, that the nearer the Cieling is to the Fiye, the lefs the Mouldings and Ornaments mu ft project: For it the Mouldings and Ornaments of a Cieling iz Feet high, were made as prominent as thofe to a Cieling of zo Feet high, they would appear, as Weights, almoft infupporta- ble, and give Offence, inftead of Pleafure, to the Eye ; therefore, herein, Difcretion is abfolutely neceflary. Plate CCCCXVII. Demonjlrating the Ornaments of Cupola's, and Cir¬ cular Sojito's. Admit the Line D in Fig. I. whofe Length is divided into 14, parts, re prelent the Breadth of one Side of an odtangular Cupola, whofe Centre is C ; from whence draw Right-lines to its Extreams, and the Iloceles Triangle form¬ ed thereby, will reprefent the Plan of one 8th part ot the Cupola. Make AB, Fig. II. equal to CD, and on A, with the Radius A B, delcribe the Qua¬ drant CB; divide the upper half of the Line D into 7 equal parts, as at the Points 1, 1, 3, 4, s, 6, 7, D, and from the Points 1 and 4, draw Right-lines to the Centre C ; -make B 1, in Fig. II. equal to 4 of the 7 parts into which the half of D is divided ; and from the Point 1, draw the Line iG parallel to AEB, and alfo the Line 1 z z parallel to BB ; make the Dillance 1 z in the Arch CB, Fig. II. equal to tire Dillance z z in Fig. I. and from the Point z, draw the Lines zG and z 3 3, parallel to the Lines AB and BB as before; alfo the Dillance z 3 on the Arch C B, Fig. If equal to 3 3 in Fig. 1 . and from thence draw the Lines 3 G and 3 4 4, parallel to A B and B B : And fo in like Manner proceed to let up on the Arch CB, the Diltances 34, 4 y, y6, 6 7, and 7 8 ; from whence draw Right-lines to meet DC, the central Line of Fig. 111 . which is the Upright of this 8th part of the Cupola , whole Breadth BA, is equal to the Line D in Fig. I. and Height to AC Fig. II. and whofe Ordnates, through which its Curve pafleth, are taken from, and are equal to thofe dotted ordnate Lines in Fig. I. which arc parallel to the Line D, and included between the Sides of that Triangle, which proceeds from the Centre C. The horizontal Diltances of the Angles of each Pannel are alfo determin'd by Curves, trac'd through Points found in this Manner, from the inner Divifions ot the Plan. The Figures H, I, are two Sides of the fame Cupola, but differently adorned, they being divided into Hexagons and Octagons, and this into geometrical Squares. The Figures K, L, M, re¬ prefent the Sofito’s of Arches divided into Pannels, wherein 'tis always to be obferved, that they confift ot an odd Number, that thereby one may be di- reftly in the Vertex, and the others equally on each Side. The Border mull be not more than one 6th, nor lefs than one 7th of the whole Breadth. The concentrick Arch F E in the Profile Fig. II. on which the Pannels of this femicircular Sofito are divided, will be fully fufficient to demonftrate them. The two femicircular Sofito’s O, N, are of a greater Breadth, and given as Examples to ihew how they may be adorn’d. Plate CCCCXVIII. Ornaments for coved Cielings. Here is reprefented fix different Ways of adorning coved Cielings, under which are their Platforms enrich’d ; the upper two are with Groins, which have a very pieafing and beautiful Appearance. Thefe Coyes are gene- S C rally The Principles of Geometry. 382 rally made to be a 4th part of a Circle, and adorned with Fret-work or Paint¬ ing, and oftentimes with both. Plate CCCCXIX. Ornaments for the Infides of Cupola's, and Circular Sofito's. Here are eight good Defigns of Ornaments for the Infides of Cupolas, vhofe Heights and Breadths are found by the foregoing Rule ; as alio feven Defigns for Circular Sojitos, with 5 Defigns of Roles for Pannels. From Plate CCCCXX. to Plate CCCCXXVIII. inclufive. Defigns for Chimney Pieces. In thefe nine Plates are contained twenty two Defigns for Chimney-pieces, of which the largeft in every Plate are by Serho ; the four Defigns on the left of Plate CCCCXX 1 II. and thole of the Bottoms ofPlates CCCCXXV. and CCCCXXVI. are by Inigo Jones, and which in general are good Defigns. The other Defigns at the Bottoms of Plates CCCCXXI. CCCCXXII. CCCCXXIV. CCCCXXVII. CCCCXXVIII. CCCCXXX. and CCCCXXXI. are Defigns by Mr. Kent, to which he has made fmall raking Pediments, which are not only improper Members to fuch Ornaments, but, by their extraordinary pro- ieCtures, have fiilfe Bearings, renders the Mantles ufelefs, and deftroys the Magnificency of the Entablatures, which ever ought to be entire. As Peui- dents are Ornaments adapted to carry off Rains from Portico’s, &c. 'tis ah- iurd to introduce them where Rains do not come. If we compare thefe Kpntijh Defigns with thole of Set lio's, which are truly grand and magnificent, we fee immediately, that they are nothing more than lo many taftlels Whims of poor Invention. From Plate CCCCXXIX to CCCCXXXIV inclufive. Defigns for Chimney Pieces, by other Alajlers. The firft of thefe Plates reprefents four Defigns for Chimney Pieces by Vincent Scamozzt, of which I cannot recommend the upper and lower ones; that above, having the Range of its Entablature broken by a projecting Ta¬ ble, as il thereon an Infcription was intended; and the other for its Pediment, which, like thole of Mr. Kent's, has a poor nigard Look, and feems to have been taken from the Form of an old Woman’s Forehead-cloth. The other two half Defigns arc very good, as likcwile would the upper one in Plate CCCCXXXI. be, which is by M. J. Baroz zio of Vignola, was but that Gothick Table in its Freeze removed, lb that its Freeze might be entire. I11 Plates CCCCXXXII. CCCCXXX 11 I. and CCCCXXXIV. are twelve Defigns for Chim¬ ney Pieces, by Mr. Gibbs, ol good Invention. The firft three Chimney Pieces in Plate CCCCXXXil. have Pannels over them with broken Pediments, and which are much more proper for inlide Ornaments, where no Weather comes, than abroad, where, by their being open, they are ufelefs. The next three Defigns in Plate CCCCXXX 1 II. have alfo Pannels over them with raking Pedi¬ ments, and tho they cannot be laid to be perfectly properly introduced, tor the Reafons aforefaid, yet, as they crown the Whole, and in fome Meafure protect the lower parts from the perpendicular Fall of Duft ; and as they fpan each Defign, and are of grand parts (not poor and little, as thole of Kents) they are worthy of Eiteem. In the laft of thefe Plates are fix De- figns, by the lame Mailer, which are in general very good. The upper three are fquare, and their Architraves one 6th of their Aperture ; the Apertures of the other three vary, as the Divifions exhibit. To each Defign is a Scale, by which the Proportions of their Heights and paits may be verv accurately determined. Plates The Trinciples of Geometry. 383 Plates CCCCXXXXV. CCCCXXXVI. CCCCXXXVII. Exhibit¬ ing Elans and Sections of fir ait and cylindrical Stair-cafes The upper Figures i s, e 0, ah, reprefent a Seblion railed from the Plan m k a c, which is the Plan of a ftrait, but double Stair-cafe, whofe Extreams are at a and k, its half Paces at lb, and landing Places at m and c. The lower Figures c c and d d, reprefent the Plans and Sections of two cylindrical Stair-cafes, whofe Difference conlifts chiefly in the Form of their Steps; that of c being ftrait from the Centre, and the other circular, which laft has not only a Beauty, but a greater Length than thofe of c. Thefe Kinds of Stairs are made, either to wind about a folid Cylinder, (which fome ignorantly call a Column) as d c in Plate CCCCXXXVI. or with an open Newel, as herein is reprefented, and which in large Stair-cafes is very convenient, as well as beau¬ tiful, to admit Light from above to the lower parts. The Diameter of the Cylinder, about which the Stairs wind, mult not be lets than one 6th, nor more than three 7ths of the Diameter of the Stair-cafe. But I think the moll beautiful Proportion is, to divide the Diameter of the Stair-cafe into 4 equal parts, give two to the Column, or Well-hole, and the other two to the Steps. The uppermolt Figure in Plate CCCCXXXVI. is a Plan and Scdtion of a beautiful and grand cylindrical Stair-cafe, made by Order of Francis I. King of France, at Chamber, a Palace eroded in a delightful Wood. This Stair-cafe is quadruple, having four Entrances, as at a IJ q, which afeend the one over the other in fuch a Manner, that, being made in the Centre of the Building, they lead to four Apartments, each at 90 deg. Diftance; fo that the Inhabitants of one Stair-caie, need not go down thole of the other; and, as it is open in the Middle, they may all lee each other afeend and defeend, without being in the leaf! incommoded. The Figures e e and ff, Plate CCCCXXXVII. are the Plans and Sections of elliptical cylindrical Stair-cafes, which are proportioned in the fame Manner as the aforefaid ; the other two Figures are cylindrical Stairs alfo, of which the lowcrmoft is a double Stair- calc, winding about a Cylinder in fuch Manner, that two Perfons, afeending together in equal Times, will never fee or meet each other, and yet be al¬ ways of equal Height above Ground. A Stair-caie thus made would lead to two feparate Apartments, in as private a Manner as two diftinbt Stair-Cafes could do. Circular Stair-cafes are tiled cither for Grandeur or Conveniency; when they are tiled to exprefs Grandeur, they mult be ipacious, as in the foregoing Example in Plate CCCCXXXVI. but when for Conveniency only of going up in a little Space, they mult be made much narrower. In the Formation of fuch fmall Stair-cal'es it is to be obferved, that the Breadth of the Treader of each Step, at about ro Inches or % Feet Diltance from the Middle of the Rail, be not lefs than 9 Inches, nor more than ij- Inches as aforefaid ; becaufe, as in going up and down, the Hand being generally on the Rail, the Feet travel at about that Diftance from the Middle of the Rail’s Bafe. It is alio abi'o- lutely neceffary to make Quarter Paces, for Eafe in going up, as well in thefe, as in other Kinds of Stair-cafes, which, if placed at proper Dillances, fo as not to obftrubl the Head-way under them, will be found very ufeful. The Plans C, B, A, ID, H, in Plate CCCCXLIL have the Breadths of their Treaders pro¬ portioned as aforefaid ; but the Plan F, which is an Ellipfis, is Varied a little, the Curvature being much quicker towards the Ends, than in the Side. Note, Thofe ol A and B may be lighted from above; thofe of CFD by fide Lights, &c. The Plans E, I, G, are recommended for Buildings, wherein are half Stories, called Mizzano’ s. The Plan GH feems to be of pretty In¬ vention, having the fmall Stair-cafe H for the Ufe of Servants in its Vacuity. The Seftion T, Fig. II. is taken from the Plan D, exhibiting the Meeting of the Steps and String-board, wherein at dc, eh, is fhewn the ill EffefI of plac- The Principles of Geometry. 384 ; n g circular taper Steps with parallel Steps, which the fudden Turn at their meeting occalions, and which caulbs the Figures of the String and Kail to be not the molt agreeable, if not worked by a judicious Hand, that can humour their meeting with Difcretion. The Section G is taken from the Plan C, which conlilts of two Quadrants, on which the Steps are equally divided ; the Right-lines a, a, a, a , are each equal to the Curves of the Quadrants in the Plan; l'o that being bent to the Curvature of the Plan, they become cir¬ cular and twilled every Way. The Seftion H K 1 is taken from another Plan of the fame Kind, as the Plan C, wherein at M the Doors and Windows in the Wall being exprefs'd, and the Profile of the Steps, as well for the fecund as the firlt Floor, you are thereby Ihewn the Space for Head-way, which is a material Point, as this Matter oblerves, and greatly afliftant to tliofe of lfnall Experience. Plates CCCCXXXVIII. CCCCXXXIX. Exhibiting Plans of mixt Stair-cafes. Mixt Stair cafes are Inch as be partly itrait, and partly circular, as b, Plate CCCCXXXVIII. as D, Plate CCCCXLU. and A, Plate CCCCXLIII. The upper Figure a 5, in Plate CCCCXXXVIII. is a Section of the half Plan a 1. The Figure 3 b is a Section of the half Plan b, of a grand Stair-cafe and cir¬ cular Portico; the upper part of this Plan, marked r b, is the Plan of the upper part over the Plan b. The Figure c is the half Plan of another mixt Stair- calc ; the part on its left, marked x c, is the Plan of the upper part of the fame. Figure 3 c is its Section length-wife, and Figure 4 c its Section breadth- wile. Thele Examples being given for Practice, the voting Student mult ob- lcrve in performing them, to make each complete (not in half) that thereby he may be the better able to judge of their Effects. In Plate CCCCXXXIX. are fifteen Figures. In that marked A, the part a reprei'ents the ground Floor, b the Alcent of the Stairs, wherein 'tis fuppofed the Strings and Steps of Stairs are contained; and c, a horizontal half Pace. In Fig. B, the parts a, b, c, reprefent the Floor, Afcent and half Pace, as in Fig. A; the parts d, d, the Space for an Architrave; the parts e , e, e, the Bale to the Ballufters ; f, Newel-Pofts; g , g, the Hand-rail, under which are placed the Ballufters. The Fig. C is a Reprelentation of Figures A and B, as when completed with its Mouldings, &c. To deferibe the “Plan of a Stair-cafe, Fig. Z. (1) Let CDAB be the internal Angles of a Stair-cafe, the Length of whofe Steps (which fliould never be lefs than three feet) arc equal to one 4th part of the Breadth, and whofe Bounds arc the Parallellogram HIGH, within which, deferibe the Breadth of the Ornaments, viz. the Ballultrade, Hand-rail, &c. as higf. (i) Confider the Height of the Story, and the Humber of Steps necefiary to attend its Height ; wherein obferve, that the Height of Steps fhould never be lei's than y Inches, nor more than - Inches; and that their Breadths ihould never be lefs than 9 Inches, nor more than iy Inches (feme lav 18 Inches, blit 1 think 'tis 3 Inches too much.) (3) Suppofe the Height of this Story be 9 Feet 4 Inches, then 16 Steps of - Inches each will rife to that Height. (4) Set out the Breadth of the Steps, and conlider how many Steps can be had in the Length GH for the firlt Flight, which l'uppofe to be fix ; thefe fix divide into Halves, which call twelve, and letting 1 of thofe Halves off:' from the Angles G and H, with the other ten Halves deferibe five Steps, which draw through the Plan, lb as to divide the Side IF into the fame Number of Steps. . (y) Proceed in the fame Manner to deferibe the Steps at the Return H I, which fuppofe to admit of four Steps, and let off the Breadth of half a Step from eacli Angle, and deferibe three Steps, which produce to the Line C D, and thus is the plan completed. Note , ’Tis beft to divide the Heights The Principles of Geometry. 3«5 Heights of the Steps exactly on a Rod ; and that the Height or Rife, and the Tread or Breadth of a Step, is called a Pitch-board, whole Ufe will be hereaf- ter explained. To raife the SeSion or Upright of each Flight. (1) Continue out every Step, and make e b equal to the Height of the firft five Steps; divide e b into j- equal parts, from whence draw Lines parallel to C B, and they will determine the Height of every Rifer, and Length of every Treader in that Flight, (a) Draw d h parallel to C D, at the Dillancc of e b, and from h fet up the Rifers and Treaders of the Flight from H to I. Conti¬ nue C D to d, making D d equal to D d, in the L,inc A D continued ; and from m, fet up the Rifers and Treaders of the Flight from I to F ; and thus will the Upright of each Flight from the Ground be completed. The Figure D is an irregular Stair-cafe, whole Plan is a Trapezium bcdci, wherein obfcrvc, that if from the Angles Perpendiculars are drawn, as ic, ch, and dg, df, alfo kb and a e, and the Steps being divided as before, leaving the Dillancc of half a Step from each Perpendicular, the Whole may be com¬ pleted in the fame Manner. Fig. E reprefents the ufual Method, where the Quarter Paces are made fquare to the Angle of the Newel, which occafions the Hand rail of the firft Flight to drop below the Rail of the fecond by the Height of three Steps, and fo the fame in all other Flights. Fig. F exhibits the Stair fet to the Middle of the Newel, which drops its Rail the Height of two Steps below the Rail next above it; and fo in like Manner that of G, where the Stair is placed to the Outfide of the Newel, the Drop of the Rail is but one Step; and laftly, that of H, having its Stair fet half a Step clear without Side the Newel, brings the Rails to meet, as in Fi¬ gures B and C. To the Figures I and K are large Mouldings, as a a, and to preferve the Regularity here, as in the lalf, fet the Step the Breadth of half a Step on the Outlidc or the Moulding. It is alfo to be obferved, that a half Ballufter is of¬ ten joined to the Newel, and whenever it happens that the Interval or Space is too great for a half Ballufter, then the Newel may be augmented, as at b b in Fig. K. Fig. L l'eprelents the regular Method, and Fig M the irregular Method, ol joining Rails and Ballufters, which laft, tho’ done in the new Stair-cafes at the Welt End of the Parifh Church of St. Martins in the Fields, by Direc¬ tion or Permiflion of Mr. Gibbs, yet it is a Practice to be abominated by every Artift, and what none would do or fufter to be done that knows how to do better. The Figure N exhibits the Manner of continuing Lines from a regu¬ lar Ballufter, for the dividing of the Parts of a raking Ballufter. Lastly, Fig. O is a Plan of a Stair-cafe of five Flights, fitting for a very lofty Story, whole middle Flight is made larger than the others, as being more convenient and grand. Plates CCCCXL. CCCCXLI. CCCCXLIT. CCCCXLIII. Exhibit- ing Plans of Right-angled Stair-cafes, with the Manner oj kneeling, ramping and jqmrmg twifl Rails, fluting Newels , &c. The Plan A, Fig. III. Plate CCCCXLI. confifts but of two Flights, and therefore is cal!?d 'Dog-legged, wherein bed reprefents Door-ways, and e a Window lor Illumination. Right-angled Stair-cafes are either Geometrical Squares, as Figures G, H, Plate CCCCXLIII. or Parallelograms, as A B in Plate CCCCXLI. The Plan B, Plate CCCCXLI. hath an open Newel (as the Space B) which conlilts ol three Flights, wherein fnmlk are Door-ways, and g h i a Venetian Window to illuminate it. It is to be obferved, that when Stair cafes are made with open Newels, as in this Example, they may be illu- f D initiated The ‘Principles of Geometr t. 38 6 minuted from above by Means of Lanthorns, Cupola s, &c. when the Sides will not admit Light, as in this Example; and therefore if a Stair-cafe be but fpacious, and conveniently iituated, we need not regard whether it is illumi¬ nated from its Sides or from its upper Parts. Quadrangular or geometrical fquareStatr-cafes *) teCC< have their Steps divided into three or tour Flights, as the Space and He ht of the Story requires. The Length of the Steps, as I before oblerved, ihould not be lefs than three Feet, and is molt beautiful when each is equal to one 4th of the Breadth of the Whole. In the Figure G, it is funpofed that there is a Wall within Side, from whence it receives the Light, but that of 11 hath an open Newel. The Section B, Plate CCCCXLII 1 . reprefents a Stair-cafe of the 'Dog-legged Kind, whole Steps are on Strings of Wood, which are eaied underneath to re- prelent folid Steps: In Stair-cafes of this Kind 'tis fometimes neceflary to put Steps in the Quarter Paces, which ought not to exceed four in Number, un- lefs the Stairs are very large, viz. where the Length of the Step is four Feet divide it into four Steps ; where five Feet into live Steps, and where eight or nine Feet into twelve Steps; but indeed in large Stair-cafes it mult be avoid¬ ed, if podiblc, bccaufc fuch telling Places are not only ufeful, but ..H alfo. Thus much with refpeft to the Formation and Difpofition ui Stair-caies, the next in Order is their Ornaments. To find the Kpieeling and Ramp of Rails, Plate CCCCXL 1 . At C in Fig. 11 is exhibited a Ihort Flight of four Steps, and part of a half Pace ; the Height a h is the Height of the firlt Step, on which Hands the Newel h c and the firlt Ballufter 0 ; the Heights and Breadths of the other Steps are exprelfed by the dotted Lines, on each of which are placed but two Ballulters. The Height of the Newel may be from two Feet and four Inches to two Feet and fix Inches, &c. and c d , the Thicknefs of the Rail, is at Pleafure. The horizontal part of the Hand-rail c 0 is called the Kneel, whole Joint is at 0, the Middle of the firlt Ballufter. The Height of the Plinths to the Ballulters is equal to the Height of the Steps, and their Breadth to one 4th part of the Breadth of a Step ; as alfo is fe the Plinth of (he Column on the half Pace, whole Column /g is equal to the Column h c on the firlt Step. The Height of the Hand rail at hg is alio equal to itsHeight at dc , and the Length of tlie horizontal Hand rail 1 h is generally equal to the under part of the Kneel c 0. To find the Center of the Ramp. The Arch n i is called the Ramp, whole Center k is thus found : Continue h 1 towards k, at pleafure, from which find, with your Compafles, the neareft Diftance to the upper part of the llrait Rail, that, when turn'd up, ftiall meet the Line h i in i. Now it is to be here oblerved, that as the raking Line of the llrait Rail, (but not the Point n therein) and the Point z, with the Line h i continued, are given, to find the Points k and n. I mull own, 1 do not know any Propofition in Geometry, that will determine thole two Points; and therefore, the Point k mult be found by making divers Eflays, by moving the Point of the Compafles either backwards or forwards on the Line h i k, un¬ til the other Point extended to the ltreight Rail, which will, when turn’d up, fall on the Point i. When k the Center is lo found, let fall frflin thence a Per¬ pendicular to the raking Rail, as the Line k n, then will the Point n be the Point from which the Ramp afeends. I f 'tis required to have three Ballulters on each Step, then the Kneeling at t p and q s Ihould come to the backlide of the firlt and lalt Ballulters, as at p and q. To The Principles of Geometry. 387 To find the Height of the Ramp. ■ Let « reprefent the Hand-rail, whofe bottom is continued out both ways to « and w, make w x equal to u t , and xy equal to the Height of a Step ; then is y the Height of the lower part of that Kneel, and qy is equal to the Inter¬ val between the Plinth of the Ballufter, and the Riler next to it. Having pro¬ ceeded thus far, you muft draw r s parallel to qy, at the Diftance affigned for the Height of the Rail; alfo, draw the upper part of the raking Rail parallel to p w. Now, to deferibe the Ramp, and thereby find the Point r, continue out the Line r r at pleafure towards the left, and front thence take the neareft Diftance to the Line p w, fo that the extended Point of the Compafles, when turned up, lhall fall in the Point q; then will the other Point, in the Line .r r continued, be the Center of the Ramp, and which will determine the Point r ; alfo, by the upper Arch of the Ramp, being deferibed from the upper part of the raking Rail. The Figure D reprefents the manner of fluting Newels and Ballufters for Stairs. The Semicircle marked * hath fix Flutes, and is lor Newels the other Semicircle hath but four, and is for Ballufters : But where Newels and Ballufi ters arc any thing large, inftead of giving twelve Flutes to the Newels, and eight to the Ballufters, the Newels may have iixteen, and the Ballufters twelve, the Whole being always at the Difcrction of the Archil eft. The Manner of deferring Scrolls, for the Plans of tvjified Rails Fia' I Plate CCCCXL1. First Sketch out with Chalk, &c. a Scroll proportionable to the Place in which it is to Hand, and determine on the Blends of the Stuff to be uied, and the Kind of Mouldings on the Side of the Rail; which, for Example's fake, we ll fuppofe to be as reprelented in Fig. C. Secondly, On the Center of the F.ye of your chalk'd Volute deferibe a Circle, whole Diameter make equal to gf in Fig. C, and, concentrick thereto, deferibe another, as h, lb that hg be equal to half g e in Fig. C ; then will this laft Circle be the Bignefs of the Eye in the Scroll. Thirdly, Set one Foot of your Compafles in the Center of the Eye, and extend the other to k, the Infide of the Rail, and, with that Radius, deferibe the Circle k l w, which divide into 8 equal Parts, as the dotted Lines exprefs; draw the Line h l, and on the Point k deferibe the Arch 11345-678, from the Point where the Line In cuts the out Circle of the Eye; which Arch divide into 8 equal parts, as at the Points 1,1, 3, 4. &c. and from the Point £ draw the Lines 11,11,33,44, &c. which divide the Line l n at the Points m, n, o,p, q,r,s. Fourthly, Take the Diftance / m, and let it on the Line q m ; from the out dotted Circle to the Point m. In like manner fet the Diftance In on the Line i n, from the out dotted Circle to the Point n ; alfo fet the Diftance lo, on the Line os, from the out-dotted Circle to the Point 0; and fo in like manner, fet off the Diftances from the out-dotted Circle on the Lines pk, qm. See. Diftances equal to Ip, lq, lr, and l a ; and then will tire feveral Points m, n, o,p, q, r, s, t, be the Points, through which the Out-line of the Scroll mult pals, and which is deferibed by eight Centers, as follow : Take the Di¬ ftance from k to the Center of the Eye, and with that Diftance, on the Points m and k deferibe a Sedtion of two Arches, whofe Point ol Interfedlion is the Cen¬ ter of the Arch mk. This done, take the Diftance from the Point m (in the Line q m ) to the Center of the Eye, and, with that Diftance, on n and »z deferibe a Sedtion ol two Arches, as before, whofe Point of Interfedlion is the Center of the Arch nrn ; proceed in like manner to find the other fixCenters, on which deferibe feveral Mouldings concentrick to the Out-lines, whofe Sections do not terminate, or meet each other at the eight dotted Lines, as the Out-line did, but at thole Lines, that are drawn from the Points m, n, 0, p, &c. to the Centers of the Arches h m, m n , n 0, &c. As 1 have thus explained the Conftrudtion of lliis • 388 The 'Principles of Geomktry. this Scroll, which this Matter has omitted, I may venture to refer the Reader to the Ini'pection ot the other Scroll, Figure E, as being cleferibed by the limie Rule, altho'it confifts of two Revolutions, or Turnings about, and is there¬ fore made from a Divifion of iG, as tire other was from a Divifion of 8. To Jquare a Pwifled Rail, Plate CCCCXL. ft) The Out-lines of F, Figure V. are the fame of Figure D in Plnte CCCCXL 1 . whole Centers are the Points i, r, 5, 4, 5-, 6 , 7, 8, that form a-. Circle in the Eye of the Scroll; and as the Center 2 is the Center of the Arch hr, therefore from the Center 1 to h draw the Line dh. (r) From the Plan deb a trace the Mould K, Fig. IV. whole Curves fhall Hand perpendicularly over thofe in the Plan F, when applied on the Rake, which is traced as follows, •viz. i/i , Oblervc, that as the twitted part of the Scroll begins at a, and ends at n , therefore, in F'ig. M, make n a equal’fo the Curve an in Fig. F; in like manner, make oc, Fig. N, equal to the inward Curve cd, &v. in Fig. F. I have already l'poken of a Pitch-board, which is nothing more, than a Right- angled plain Triangle, as I, whole Ball- is equal to the breadth, and its Per¬ pendicular to the Height of a Step. Make g e, in the Step-mould, equal to ae, in Fig. F, and draw e i in the Pitch-board parallel to its Perpendicular; on the Points a and c, in the Figures N and M, ereft Perpendiculars, each e- quul to e i in the Pitch-board ; alio make a e in Fig. M, and c d in Fig. N, e- qual t oge in the Pitch-board, and draw the Line er p in Fig. M, and d sq in Fig. N; divide tie and er in Fig. M, and 0 d and ds 111 Fig. N, each into 8 equal parts, and draw the interlefting Lines in both, by which their Curves are generated. The Curve in M, at ht, Ihews how much Wood is wanting- on the back of the Rail, which let from e to a in Fig. L, and there deferibe the bignefs of the Rail; the other part of the Twill is cut out of a parallel Piece, as Fig. O. It is alfo to be noted, that the under part of the Rail will be deficient of Wood, as at g h. The aforefaid Wood being made good on the Top and under part of the Rail, make k 1 in K equal to g i in the Pitch-board, and k l in K equal to eh in F* and draw the Line i /; make g cl in the Pitch-board equal to dh in F, and draw d in parallel to i e \ make lm parallel to h i, and equal to gm in the Pitch-board, and make mp, in K, parallel to k l, and equal to df, in F; draw Ordnates at plcafure, either equidittant, or otherwife, in F, from the Lines c f and dj to the Curve d c, and from the Lines ae and to the Curve bra', divide the Lines ik and k l, in K, in the fame Proportion, as ae and be, in F, and making the Ordnates equal, trace the Curve il in K; and then, dividing pm and op, in K, in the fame Proportion, as c f and d /in F, make thofe Ordnates in K equal to thofe in F, and trace the Curve 0 m, which • compleats the raking Mould K, whole Curves (when in their Places) will Hand perpendicular over the Curves in the Plan F, as required. Take the Raking Mould K, and fet the Point 1 to the Point / (in L) and there ttrike it: Wherein obferve, that the Angle / (in L) mutt be made equal to the Angle m (in the Pitch-board) or applying the Angle m of the Pitch- board to the Point f (in L) with the Hyporhenule g m of the Pitch-board, to the L,inc f e (in L) draw the Line / bv the Perpendicular of the Pitch-board. At the Bottom of the Rail apply the Mould K, fet 1 to the prick'd Line, and there deferibe it with your Pencil; laftly, cut that Wood away, alfo cut the remaining part ol the Scroll out of the Biock (as) O, then glue thefe toge¬ ther, and binding both the Moulds M and N round the Rail, ttrike them and cut away the Wood ; lo will the Back of the Rail be fquared, as re¬ quired. To 389 The Principles of Geometry. ~~ '—-- To fet out the Diftances of the Balluflers on the twifted Rail, Fig. VI. Plate J CCCCXL. The Scroll U is of the fame Magnitude as that ot F, and QP reprefents the firft two Steps in this Figure, as H G doth in P. The Pitch-boa. d R is alfo e q U al to the Pitch-board I. Before the Diftances of Balluflers can be fet out, their Bignefs muft be determined, which, for Example fake, we ll fuppofe to be a, b, c, d, e,f\ for the more exaft Divifion of the Balluflers, 'tis beft to defcribe a Line through the midft of the Rail, and thereon fet out their Diftances at plealure; this middle Line will terminate at the Circle g, under which muft Hand the Newel, and the Extreams of the Plans of the Balluflers on the inward Side will be p q, r s', t v, u w, xy, z, at which laft the twifted part terminates, and from thence to the Eye is horizontal. To find the Lengths of the Balluflers and Kernels. (1) Draw a Right-line at pleafure, as zp. Fig. II. Plate CCCCXL. and therein affiime a point, as at l. (z) In Fig. VI. oblerve where the Scroll be¬ gins, as at l, againft which fuppofe a point in the Line r p; take the Diftances from this point to r and to s, and fet them from the point l, in Fig. II. to r and r; alfo make the Diftances of r q, qp, in Fig. II. equal to r q, qp, in Fig. VI. as likewife the Diftances s t, t v, vu, u w, w x, xy, z, in Fig. II. equal to the Diftances s t, t v, v u, n , w x, xy, z. in Fig. VI. Take from the Plan, Fig. VI. the Diftance from l to m, and make h n in the Pitch-board equal thereto, and draw n 0 in the Pitch-board parallel to i k ; make the Triangle Ih 0, Fig. II. equal to the Triangle h n 0 in the Pitch- board R, then will the Angle & h p. Fig. II. be the Angle of the Rail, and the Line h 0 & will be its Slope. Divide h 0 and h z. Fig. 11. each into any Number of equal parts, and draw the interfecting Lines to generate the Curve z 0 ; from the Points r, q, p, in Fig. II. draw Right-lines at Right-angles to the Line z p, and each equal to the Length of the fixed Balluflers, as b, a, and defcribe the Step S, below which fet the Step T, whofe Height is equal to i k of the Step-mould R. Draw Right-lines from the Points t, v, u, w, x, y, z, at Right-angles to z p. Fig. II. which continue up to the Curve, and down to the Steps, which are the true Lengths of the Balluflers a, b, c, d, e, f, in Fig. VI. Tht Length of the Newel is equal to the Length of the Ballufter at s, becaufe there the Twill ends, and are both on the lame Level. ’ filote , The Plan of the firft, or Curtal-ftep P, in Fig. VI. is formed in the fame Manner as the Plan of the Rail; and what is here faid with refpebt to the firft two Steps of a Stair-cafe, the fame is to be underflood as if a whole Flight had been underftood. To Jquare a Rail that ramps on a circular Bafe. The Plan Fig. VIII. Plate CCCCXL. is of a Stair-cafe, at whofe landing is a Quadrant of a Circle; to make this familiar to the Underftanding, the upper three Steps are reprefented at large in X, Fig. IX. with a Seblion of the Rail, with its Ramp and Kneel, which are defcribed by the fame Method as thole of c. Fig. II. Plate CCCCXLI. In Fig. Y is the Plan of the Rail, from which trace a Raking mould, as before taught in Fig. IV. that will be agreeable to the Angle made by the Raking-rail m l, and the Line 0 k. In this Operation there will be a conflderable Thicknefs required on the Back of the Rail, as appears by Fig. III. The next Work is to fquare the Rail, as before taught, which being done, make 0p. Fig. I. equal to 0 c. Fig. IX. and compleat the geometrical Square caop\ make p g. Fig. I. equal to that part of a p 111 Fig. IX. as is contained between p and the upper Part of the Rail, and draw g i. Fig. I. parallel to 0 p. Make a b. Fig. I. equal to the Curve of the outer Quadrant g h in Y ; and as 0 p, the Bra^tjr of the upper Step, is equal to de the Radius of the Arch h g, therefore make ab \\\T, equal to the Curve h g, and then the Point b in Z will reprefent the Point b in Fig. IX. S E Now The 'Principles of Geometry. 290 Xo», to find the Curve bg in Z, divide b a in Fig. IX. into any Number of parts, and divide b a in Z in the fame Proportion, and then drawing e- (|iul Ordnates in each, you may defcribe the Curve b g in Z ; and as the lower raking part in Z is the fame as that in X, therefore the Mould Z will bend about with the Rail, becaufe a b in Z is equal to the Girt of the Curve h g in Y, and becaule the Height of the Curve b g, over its Ordnates, is equal to the Curve of the Ramp in Fig. IX. Make p k, Fig. III. equal to p k .in X, alio make p e equal to p g in Fig. I. and draw e /"parallel to 0 p h; make e a equal to g a in Fig. I. and draw a c parallel to op-, make a b equal to e f the inner Quadrant in V. Divide b a in fuch proportion, as before you divided b a in Fig. IX, and draw equal Ordnates, through which trace the Curve b e, which i? the Curve of the inner Mould, Note, The Heights 0 c, in Figures I. and III. are each equal to the Height «in Fig. IX. Now if you bend both thele Moulds round the Rail, they will form an exact Iquare Back, bv drawing by their upper Edges Lines with a Pencil, and cutting away the fuperfluous Wood. The Figure M exhibits a Method to have the Newel under the Twift the fame length as the roll, by which Means the Rail twills no farther, than the firft Quarter, and therefore the remaining part may be cut out of a Plank the Thicknels ot the Rail without Twilling, wherein If is the Thicknefs of Wood wanted on the Back of the Rail. Other Methods for Jquaring twijled Rails. This Mailer propofes three other Methods to perform this Operation, but doth not heartily recommend them, as that, when they arc done, they will not have that agreeable Turn in their twilled part, as they would have by the Rule aforefaid. That of P, in Plate CCCCXL.II 1 . is the raking Mould, taken from K in Plate CCCCXL. the Triangle Q_in Plate CCCCXLIII. is the Pitch-board taken from I in Plate CCCCXL. which gives the Rake, or Reclination of the Rail. In R, Plate CCCCXLIII. is fhewn, How to fquare a Rail without bending a Templet about the twijledpart, and which is done by making the Back your Guide, as follows ; defcribe the bignefs of the Stuff to be tiled, as the Paralle¬ logram ab h i, which fhews how much Wood will be wanted at bottom, fup- polingthat S is the Side ol the Rail ; and in confideration that tfie Grain of the Wood fliould be agreeable to the falling of the Twill, therefore conlider how many Thicknefles of Stuff will make the Body required, to cut the Twill out of, which, 111 this Example, are three, therefore (as in S) continue the Line ab to c ; on a, with the Radius a c, defcribe the Arch dc, which divide into 4 equal parts, as at 1, 1, 3, becaule the Rail S mult be always reckoned as one. This (liiith this Mailer) by Infpeclion Ihews how the Grain of the Wood is to be managed, as the Forms of the Pieces T, U, W, exhibit, which will be belt, if cut fo by the Pitch-board, before they are glued together. In X, Plate CCCCXLIII. is fhewn. How to fquare the twifledpart, mak¬ ing the bottom your Guide, whole Seftion Ihews how much Stuff is wanted on the back. In Y is fhewn. How to Iquare the twijled part, making a middle Line on the back your Guide, whole Seftion Ihews the Stuff wanting 011 the back, and at the bottom. That ofZ may be cut out of a parallel Piece, of the Thicknels of the intended Rail, which, when it is glued to the twilled part, will want very little (if any) Amendment. Thus much for the Works of this molt perplexed Mailer, who, not having been able to exprels his own Meaning, has given me much Trouble to make his Rules practicable and eafy to the young Student. Plate The Principles of Geometry. 39 J Plate CCCCXLIV. Cylindrical Stairs, hy J. Vredeman On the Left Side of this Plate is a Plan and Sedtion of a Cylindrical Stair cafe, whole Newel is very fmall, .as generally practifed in Stair-rales to Clur ch Steeples and is given here as an Example of that Kind. The Figure on the Right Side is a perfpe&ve Sedhon of fuch a Stair-cafe, whole Top or upper Part being viewed, under a much Idler Angle than, its Bale, doth theXe appear damn,(hed in fuch a Manner, that if its Sides were continued they would meet in a Point ; and as the Heights of every Step from the Bottom is feen under leffer and efler Angles, therefore their Heights do appear Id KSfiSKf*" mK,,[ ™ - wm Plate CCCCXLV. A Plan and Seclion of a Stair-cafe, by I. f 0 nes. J; is “ 1S '!° W fta, , ldmg in a Houfe adjoining to the Cloyfters of Wejlminjier-abbey, wherein the Right Hon. the Earl of Hfibamham lately W Stair-cafe his Lordfhip did inform me was built by Mr Webb, a Thic^le A bugo jmes, not by Inigo Jones himfelf, tho’ perhaps he Defign might have been made by Inigo Jones, and executed bv Mr JVebb Eftal '‘X laft 1 3 f P her ° ld ‘Ml Dome, fupported by fmall Columns on Pe- deftals, between which are Ballufters, and, if 1 miftake not, a Gallery within them. The Whole is not large, and of the lomck Order, and which would have a better Effeft than it now hath, was it of greater Dimenfions; and in deed if the upper Order of Columns, that fuftain the Dome, had been made of the Corinthian Older, it would have been more mafterly, and better Archi teaure than it now is, where the lomck on the lomck feems to be abfurd Plate CCCCXLVI. The Formation of twijled Rails, by Mr. W Halfpenny. PROPOSITION I. Figures N and K. To find the Curvature, or Rahng-arch of a Hand-rail to a cylindrical Stair cafe, whofe Plan ts a Circle, as Fig. N, or an EUipfis , as Fig. O First, make Fig K, reprefenting the Height and Tread of the Steps, which laft inuft be adjufted from the Plan N, wherein the Number of Steps about the Cymder being before determined, (fuppofe twelve) the Diltances no op &c. will be the Length of the Tread of each Step. The Height of each Step mull be filch, as that their under Parts may admit of a free Head-way and therefore fhould not be Ids than 7 Inches rife ; for n times 7 is but 7 Feet nor fhould their He,ght be much greater, left they are found too laborious in attending Take the Back or Rake of a Step, as c fi and on «, i„ p°™N defenbe the Aich m\ alio on o, with the Height of one Step, as a c, dettnbe the Arch /, and from the Point of their Interfedion to the Point * trace a « U w fi ti" w C - XT ’ w ‘ th the Radius nm > delcribe the Arch i • and on p with the Height of two Steps, defenbe the Seftion k; alio on the Seftion k with the Radius n m defenbe the Arch h ■ and on q, with the HeHn of l ucc Steps make the lntcrleiftion g h. Proceed in like Manner with °the o thers and thro their Interfedions, with a thin Rule, trace the Curve re’ qmred. Fig. O ,s the Plan of an elliptical Stair-cafe, whofe Rail hath its Curve found m the very fame Manner as its Lines do exprefs. PRO- The ‘Principles oj Geometry. PROPOSITION II. Figure I. To prepare their Rails, and work their Twijis. . VT A K , the Circles i, i,+ equal to / n w n. Fig. N, and confute how ma- nv^Pieces you’ll give to the Rail, which, in this Example wc fuppofe to be fix- divide the Semicircle i 1 3 + into 6 equal parts, atthelomts 8, 9, 13. v and from thence draw Right-lines to the Center 3 ; on the 1 omts 8, 9, ,, v , mile Perpendiculars, as 8, 11, equal to (c a in Figure k) the Hei D ht V ’’ .,if 0 „ z c q V1 al to the Height of two Steps, 13 » to the Height of S° e sS S /to the Height of fom Steps, W * to the Height of five Steps, mnrto the Height of fix Steps; at the Ends of all thefe Perpendiculars e- reCt other perpendicular Lines, as 11 ", zx, #f,M H, and eh each equal 0 1 the Radius of the Cylinder; alfo on thele laft Perpendicula s ereft o- thers asTi,/g, lm, qr,xv, 1 r 7 , &c. making each equal to the Height of one Step, and then compleat the Triangles a b e, /g *, m ip, &c lfo make - 6 equal to the Height of one Step, and compleat the Triangle 1 3 6. S E t offthe Width of the Rail from 1 to 2, from n to 12, from Z to y, from u to r, from p to a, from* to K and from c to d, fromthe Poins 2,10, ,6 sure let foil Perpendiculars on the Lines 1 6 , n i), zv, ur, pm, I f and lea as 2 y, 10 ia, 1 6y,st, n 0, y h, and dc ; then will the Triangles dee, hik’onp , &c. reprefent the parts, that mult be taken oft from the hack at the lower End, to form the Twilt of the Ran. , . , ‘ To apply thefe Lines to Praftice, take the Piece eft Timber, of which you defion to 1 make the firft length, which fuppofe to be Figure M , plane one Side foaC and cut it to its Bevels « c, f l, anlwering to 4 1 * 3 > J- and then both its Ends being cut to the raking Joint of the Rail, proceed as follows ; take that part of the raking Arch in Figure L, which is equal to the firft length of the Rail, as nm, and lay it on the uppei Side of Figure M, from b to *, and deferibe the Arch b k ; make ki equal to 11 12 in Fig. I. from the Point 1 draw ig, at Right-angles to 1 k, and equal to \(>y m Fig. I and draw the Line g i, which is equal to 1 2 in Fig. I. and repreients the buck of the Rafik when worked. This being done, reprefent the lower End of the Rail gho?n at Right-angles to kg, alfo the upper End a be d, at Right-angles t0 a h and emboft out the inward Arch a 1 fquare from the uppei fide c da f as i « • take a thin Lath, and bend it clofe to the Side from a to g, and by its “d 4 delcribe a Line ; then will the Lines * g and h * be your Guide to back the Rail, and which being done, turn the Piece uplide down, and with the Upirribo an Arch from d to m equal to bk, and emboli out the Side to the Lilies b k and dm, and then will one Side and the Back be lquared, which is the greatell Difficulty in the forming of a twifted Rail, and which is a Gage fo V T h fo°Sbe t Zen?d in Fig. I. that if the Tnaingfr» 1 3 ^ £?£* **£ V s p and'the Center 3, then th/lines’-’ 6 yiy, «, fh lm,gf and ba will generate one Right-line, md whofeWfe will be the Point 3, and their Heights, being taken together ','11 1 . , ,, 1 ro ifiven Rifings, or Steps; but as in the working of a Hand-rail ^orc lie '1 e but < ne ofthefe Triangles made to work from, they being all e- the fame Effebt in working, therefore its to be noted, that the repre 'enting of l'o many is for no other Purpofe, but to make the Whole more intelligible to the young Student’s;Underftanding. To pei-tbrm the like Operation in an elliptical, or oval Stair-cafe, theres no ether Difference, than the following, that ts to Jay, whereas, in Fig. I. the Safes of the feven right-angled Triangles, as be, gh. Ip, qn, xz, 711, and 1 Z ■ ■ equal t< the Radius of the Cylinder, as 1 3, or 3 4, &c. fo ere the Bales of thefe right-angled Triangles in'Fig.L, as bd,gk, and mp, mutt 393 The Principles of Geometry. each equal to the Lines 13 14, 10 19, and 6 3, becaufe the Point 6 is ti c Center of the Arch i 18, as alfo is the Point 3 the Center of the Arch 7 n ■ and the Bales 9 w, 1+17, of the three lefler right-angled Triangles nnift each be equal to the Radius of the lefler Arch at the End of the Oval ■ the o- ther Particulars being in every Refpcft the fame, 'tis evident, that the whole Difference confifts in the two Triangles only, and which, being applied to its proper Curve, will form the two different Twills in the Rail, as required Now from hence tis plain, that as many different Arches as are in the Babe of a Rail, fo many different Twifts, and as many different Triangles will be made, becaufe it is by thofe Triangles, that the Twifts are found. PROPOSITION Ill Pig. D. To form, the Curvature or Mould of a Hand-rail that /lands on two Steps. ['} Le T the dotted Lines k, g, l, e, reprefent ihe Breadths of three Steps and let k 1, gh, l , deb a, reprefent the Plan of a Rail, whole curved part hands over the lower two. It is alfo fuppofed that the Curve g lb is one wth part of an Elliplis, and the Curve be one 4th part of a Circle. The Outlide of the Newel c is fixed from the Line k /, at the Diftance of a Step's Breadth and which being divided into 3 equal parts, gives ba equal to one 3d for the Newels Diameter. Make a c and b d each equal to b a, and drawer the Diameter of the Newel. It gf, whofe Length is equal to 1 Step and two -fos be coiifidered as one half of the longeft Diameter of an Elliplis, and fb as one half of its fhorteft Diameter, we may deferibe the Curve ? lb and on the I oint d. with the Radius db, deferibe the Arch b c. Affign 11 for the Thick- riels of the Rail, and draw the Line ih parallel to the Line kg, and the Curve h d concentrick to the Curve gib-, and thus is the Plan completed. (i) Draw kl m Fig. Pi, equal to kg in Fig. D, to reprefent the Tie.id of the Steps, as before, by dotted Lines Divide that part of the Plan of the Rail, which belongs to each Step, into any Number of equal parts, as a f mto 7, and fk into 4. ' J ’ On a Piece of vvaftc Paper make a Parallelogram, as bcae. Fig. H ma¬ king b a, c e, each equal to the Rile of a Step, and a e, be, to the Tread of a Step ; alfo continue e c to d, making c d equal to e c ; alfo continue c b out at 1 icafure. Make et equal to the Curve /a in Fig. E, and divide t c into y equal parts at the Points u, /, k, 1, from whence draw Right-lines parallel to d c, as s u, z l, ok, and p i ; draw alfo the Lines / d and b d, and the Ti .angle / d c is the Bracket of the Si ft Step, according to the Curve of the Rail. As t c is the Length of the Ground to the firlt Step, fo is t d the Length of its Rail. Make c b. Fig. H, equal to the Curve k f. Fig E • and as this Curve is divided into 4 equal parts, therefore divide the Line c b Fig H, into 4 equal parts, at the Points h, g, f, and from thence draw Right- lines parallel to e c. Ihe Line c b is the Ground-line of the fecund Step and db the Length of its Rail. 1 ? ) In Fig. H, with the Radius t s, on the Point a, in Fig. E, deferibe the Arch m ; with the Radius s u, Fig. H, on b. Fig. E, interfetl in m ■ on m with the Radius a m , deferibe the Arch n; and on c, with the Radius z i FN II, interfecl in n : In the like Manner on n, with the Radius n m deferibe the Arch 0 ; and on d, with the Radius 0 k. Fig. H, interfea in 0; and fo , n like Manner find the other Points, p, q, z, s, t, u, w, wherein flick Nails /cy C to which bend a thin Lath, fo as to touch every of them, and by its Side de- fenbe the Curve required. To Jquare this Rail, Fig. A. Describe the Curve fe a equal to the Curve l k a. Fig. E, and exprefs the Centers of the different Arches, as the Points h and g, from whence draw dotted Lines to the Places where you defign to join the Rail, as from g to b, f ^ and 394 The ‘Principles of Geometry. and c and from h to « and d\ and becaufe the firft Step is to be joined m j equal'pieccs, you muff take one 3d of the Riling or Height of the Step and let it from b to i at Right Angles to bg, and draw the Line mi parallel and equal to t b. Draw m n at Right-angles to m i, to rile lo much as the Rail rakes over which is one 3d of the Height (or Riling) of the Step, becaufe that irl rt of the Rail is one ;d of the Length on the firft Step, and draw the Line m then will nm i be the firft Triangle. From the Point c draw the Line c a at Right-angle to eg, and equal to two 3cls of the Height of one Step, alfo draw the 1 ine q z equal and parallel to eg ; on qz draw z s, at Right-angles toys, and equal to the Height of one 3d of one Step, and then, drawing the Line "r q the l'econd Triangle is compleated. On the Point d ereft the Line dt at Right-angles to dh, and equal to the Height of one Step; draw t w equal and parallel to dh, and, on w, erect x w at Right-angle, and equal to the Height of one Step, and draw the Line xt, which will compleat the third Triangle. Make ik, q 0 and tv each equal to the Width of the Rail, and, from the Points k, 0, v, let fall Perpendiculars on the Lines oi, s q and x t, then will thole fmall Triangles reprefent the parts, that mult be taken off from the lower End of each Piece, to bring the Rail to its Iwift. PROPOSITION IV. Figures B, F, C, G. To find the Curvature of a Hand rail , whofe Bafe /lands on two Steps, as before, but hath a quicker Curvature. Let FR. B reprefent the Plan of a Rail, wherein e a is equal to the Height of a Step, which bifeft in c ; compleat the geometrical Square cadb, and draw e d, which bifect alfo for the Center of the Newel; on the Center d defcribe the Arch f c b ; continue a b tog, making b g equal to fix 7ths of the Width of a Step and make h g equal to one Step and two 3ds : This being done, defcribe one 4th part of an Ellipfis, as the Curve bb, and thus is the Plan compleated. The Figure F reprefents the Out-line of the Plan, and raking Curve of the Rail, which Curve is thus found ; divide the Curve, that ftands on the lower Step from g to a, into fix equal parts, as at the 1 oints g, f, e, d, c n- alfo divide the other part of the Curve Ig into 4 equal parts, at the Points k, 1, h. In Fig. G, make the Parallelogram hcae equal to the Height and Tread of a Step ; continue c b to q, making q c equal to the length of the Curve e a in Fig F, and divide c q into fix equal parts at the Points/), o,b, k, 1 ■ make c d, in Fig. G, equal to c e, and draw dq, alfo, from the Points p, ol k, i, draw Right-lines parallel to dc, cutting dg in j , t, v, u. Now, to find the raking Curve x w q a. Fig. F, on a, with the Radius qy, Fig. G, de- fenbe the Arch b, and on n, with the Radius v p, Fig. G, interfeft in b ; on b, with the Radius ab, defcribe the Arch 0, and on c, with the Radius no, Fig. G interfedtin 0. Proceed on in like manner to find the othei Points/), q. r, s, J ’v it, w, x, (as before in Figures H and E) through which, with a thin Rule, trace (or draw) the raking Curve x q a required. The part of the Curve g a, Figure F, is equal to the Line qc, in Figure G; and the Line qd, Fig. G, anJ the part of the Curve sr qp ob a. Fig. E, are alfo equal to each other. n Figure C, the Triangles elk, t s e, and guv reprefent the fuperfluous Stuff, that mult be taken away from the lower End of each Piece, to make the Twift required. Plate CCCCXLVII. The Curvatures of twifted Rails demonfirated. I f we confider, that the Area of the Outfide of a circular, or elliptical Cy¬ linder 1, nothing more, than a right-angled Parallelogram, whole Height is equal to the Height of the Cylinder, and Breadth to the Cylinder s Circumfe¬ rence, bent about the Cylinder, tis very eal’y to underftand, that if on inch a Parallelogram we defcribe the Rifes and Treadcrs ol a fhaight Stair-cale, with 395 The Principles of Geometry. its Rail, &c. in one continued Flight, or othenvife, if required, with half Paces, Ramps, Rails, eppc. and bend the fame about the Convexity of the Cy¬ linder, then will the Curvature, made by the ftraight Rail fo bent, be the twifted Rail required. Hence tis plain, that a twilled Rail about a Cylinder is nothing more, than a Right-line placed at an Angle equal to the Afccnt of the Steps, and bent about the Cylinder, which will always make the lame Angle with the Horizon. DEMO NSTRATIO N. Let the Circle cha, Fig. E, reprefent the Plan of a circular Newel, or of a circular Cylinder, whole Altitude is t a: (i) Divide the Semicircle cha into any Number of equal parts, that will be fpacious enough for the fmall Ends of the Steps, (fuppole fix) as at the Points h, g, h , f\ e, and draw the central Lines h d, gd, h d, f d, and e d', alio, from the Points h,g,f,e, draw the Lines hm, g l, f k, and e i, which continue out at pleafure. (a) Continue out the Diameter sc at pleafure ; draw cn parallel to h m, and equal to the Height of a Step; let y n be the length of a 'H eader, and y a the Height of a Riier. In the fame Proportion compleat all the other Steps, (as in the Figure) and continue the Tread of every Step, until they meet the Lines m b, lg, d b, k /, and / e continued, and the Line a i in the Points o, p, q, r, r, t, thro’ which trace the feftional Line, or Curve ; make o x equal to the Ordnate m h, alfo piv equal to the Ordnate l g, and q u equal to the Ordnate dh, and, through the Points it, iv, x, n, trace the Curve n it. Now, fmee that the Points o, p, q, r, r, t are of the fame Altitude above the Bale c a, as the levcral 'Headers ;»,!*, 31, r 4> 7 < 5 , 9 d, and io ; and as the Lines o x, p w and q it are equal to the Ordnates mb, Ig and dh. therefore the Curve n*nv u is one 4th part of the Curve, that will encompafs the Cylinder in one Revolution ; for if the Lines c n, om, p l, and q d were to be railed perpendicularly over the Line c a, and the Lints 0 x, ivp, and u q were to be fix’d at Right-angles to them, then their Points n, x, w, u would be perpendicular over the Points c, b, g, b in the Circumference, and equal in Altitude to the Headers n, z, z, 4, and confequently the Curve paffing through the Extreams n, x , iv, u , is a quar¬ ter part of the Curve required. Z). The Figure F is a Plan of an elliptical Newel, whole Section and quarter part of its curved Rail is found in the very lame Manner as the aforefaid which the Lines exprefs. Now, from all that’s delivered tis evident, that every fingle Flight of Stairs (whether itraight or circular) may be conlidered as a right-angled Trian¬ gle, as BCA, whole Bale, CA, is equal to the Tread of the leveral Steps, its Perpendicular B C to the Heights of the leveral Steps, and BC, its Hypothe- nule, to their String ; and therefore, if we divide B C and C A each into fuch Number of parts, as we lhallaflign for the Number of Steps, aud from thole Divilions draw Lines parallel to B C and C A, their Interieftions will deter¬ mine each Step in its Rile and Tread, as required. Plate CCCCXLVIII. is a perfpective View of a cylindrical Stair-cafe, whole Rail is continued through many Revolutions, and whole Curve is no other, than that of Figure E (Plate CCCCXLVII.) continued. Plates CCCCXLIX. CCCCL. CCCCLI, CCCCLII. CCCCLIII. Divers Kinds of Pavements. At Stunsfield, near Woodjlocb in Oxford/hire,iome few Years lince, was found underground a Pavement, l’uppofed to have been made by the ancient Romans, of Mofaick W 01k, whole Plan is reprefented by Plate CCCCXLIX. The leveral Devices exprelfed in this Plan are all made with little Cubes, whole Sides are faid to be lels than an Inch, fet together in a ftrong Cement (the Knowledge of which, I believe, is known but to very few); the Cubes thcmfelves The Principles oj Geome t r. y. 396 thcmfclvcs are of divers Colours, and many of Glai's Compolition, and which, harm" their Surfaces placed truly level, and polifhed, do make a molt beauti¬ ful Appearance. The two Sides and End of the outer Margin ggg arc enrich¬ ed with ifnall Cubes, as the End /, and the l : ret is the fame all round, as the angular part above on the right-hand Side, which is compoied ot very linall white ai.d black Cubes, as the little Squares in the Plan exprefs. The Paralle¬ lograms cl, d, d have the lame Interlacing as the Parallelogram c, and the inner Margin next within the Fret is interlaced all round, as at the right-hand Angle above. In the Diftribution ol the other parts there are fome Things very edd and remarkable, as fir ft, the Manner of dividing them, and laftly, their Enrich- mcuts. In the upper part we fee a geometrical Iquare Margin, whole Sides are each adorned with Parallelograms, and Rhombuies inferibed, as b bb, a, which, in general, are enriched with the Fret, as that at a, and the levcral angular pans have the fame Enrichments, as the Angle contained between a and b on rhe right Hand ; the Spandrels h, l are enriched as their Oppofites; the next inward Circle is enriched with Interlacings, as the Specimen exprefles ; and the inferibed Square, and its parts, with its circular Spandrels, as in the Plan. The remaining part, being a Parallelogram, is very oddly divided with re- fpedl to the Uniformity of the Whole ; for, inftead of placing the lower geo¬ metrical Square to the Extreams of the hither End, (as the other is to the Ex- treams of the upper End) here is a Parallelogram above and below it, which (had they been placed together in the Middle, and the geometrical Square brought down to the End) would have prefervd a beautiful Regularity of parts, which, I think, is now wanting. A s to the many Devices herein exprefled, and their Allufions, I refer to the Antiquarian, and therefore fhall only add, that as herein is a great deal , tr(-,. C (j Invention, that may be helpful to the young Student, it was there¬ fore that I inierted this Plan. Plate CCCCL. In this Plate is reprefented a very great Variety of Pavements, as well of the molt common and ordinary, as the molt rich and expenlive Kinds: The Fi¬ gures F, N reprelent a rough Kind of fquare Pavement fitting tor Roads, as that before the Privv-gardcns at Whitehall ; that of Z is of common Pebbles, as in the Streets of London-, and that of h of irregular Purbeck, a flat Kind of Pavement of great Strength and Duration, belt for Foot-ways, Pavements ol Kitchens, Brewhoufes, &c. and when laid in lquarc Work, makes a handfome Pavement. The Figures d and e reprefent two Kinds of Pavements made with Dutch Clinkarts, that of d is called a Jiraight-joint Pavement, and that 0 fe a Herring-bone Pavement. The Figures / and g are of paving Bricks, of which that ot f is Brick in Breadth, and that of g Brick on Edge, which is much the flrongeft of the two; thcle two Kinds of Pavements arc named as thofe of d c. The Figure p is another Kind ol Brick Pavement, that is very handfome, when the iquare Spots (which arc nothing but half Bricks laid in Breadth , are in their Colour, a good Oppolite to the Stretchers that enclofe them. The Figures i,x.y may be made, either of fquare paving Tyles, or of Stone, or of Stone and Marble, or of Marble only ; wherein ’tis to bo oblerved, that as the Angles of every two Squares (in Fig. 2) come againft the Side of a Square, that Kind of Pavement is therefore much ftronger, than thofe of */, where ilie Angles of four Squares come together in one Point. The Figures 3 and ^ may be" made either with paving Tyles made in hexangular Moulds, or with Stone, inch as 'Portland, Newcaftle, &c. That of Fig. 3 is compofed of regular Hexagons, whole Sides are equal; and that of Fig. a of oblong Hexa¬ gons, whofe oppolite Sides only are equal. The Figure 1 is compofed of regu¬ lar Ottagons and geometrical Squares, or Dots ; the Octagons are generally made of "'Portland Stone, and the Dots of black Marble ; but as the 'Portland Stone The Principles of Geometry. 397 Stone doth wear away much fafter than the Marble Squares, 'tis much better to lay the Whole with white and black Marble, or with Tortland, &c only, as Fig. 3. The Figures t, v, are compofed of triangular parts; thole of t by the Interfe&ions of the Diagonals, and Ihorteft Diameters of Parallelograms; and thofe of v by the InterfeCtions of the Diagonals of geometrical Squares. The Fig. w is aCompofition of geometrical Squares and Parallelograms, which in black and white Marble has a pretty EffeCt; as alfo have the Figures A, B and C. The Fig. E is an Invention of my own, and which being made with White, Black and Dove colour'd Marble, reprefents fo many Tetraedrons , or Pyraments, with their vertical Angles, feemingly perpendicular to their Bafes; as alfo doth the Angles of the Cubes in Figures q and r, which in the Dusk of an Evening appears as fo many folid Bodies not to be walked on. In Plate CCCCLI. I have exprefled the Pavement of Cubes lying on their An¬ gles more at large, as alfo another Kind, as Fig. B, where the Cubes are lying on their Bales, and appear as Steps, which makes this Difception very agree¬ able. Alfo, the other Figures of Plate CCCCL. are divers Varieties of beautiful Compartments and Bordures to Pavements, for Halls, Cabinets, &c. Fig. D, Plate CCCCLI. as alfo Figures A and B, Plate CCCCLII. and Figures A, B, C, D, E, F, G, are divers Pavements fit for Temples, &c. in Gar¬ dens, Which I have given here for the lVadticc of young Students. o Plate CCCCLIV. Proportional Lines, and the Similarity of Figures demonjirat ed. For the better underllanding of the preceding Plates, and for the Enter¬ tainment of the curious and induftrious Student, I fliall conclude the Trinci- ples of Geometry with the geometrical Proportions of Lines, the Similarity, Reduction, Transformation, Multiplication and Divifion of geometrical Fi¬ gures ; and the geometrical ConftruCtion of the twenty-four Letters of the Alphabet, by means of which the Magnitude and Proportions of large Capi¬ tal Letters, for Infcriptions, Motto's, &c. againft Buildings of coniiderable Heights, are very exactly and eafily determined. I. Of proportional Lines. PROBLEM I. Fig. A. Between two Lines given to find a mean Troportion. Let the Lines ah and c d be two given Lines, which together are equal to k l. PRACTICE. Make 2 e equal to h /, which bifect in /, on which, with the Radius if, delcribe the Semicircle 1 h e ; make 1 g equal to a b, and e- reCt the Perpendicular g h, cutting the Semicircle in h, then is g h the mean Proportion required ; for as e g is to g h, fo isg h to g i. COROLLARY. Hence a Right-line drawn in a Circle, from any Point of the Diameter perpendicularly, and continued to the Circumfer¬ ence, is a mean Proportion betwixt the two Segments of the Diameter. PROBLEM II. Fig. B. To cut a given finite Line in extream and mean Troportion. Let ah be the Line that is to be cut, fo that the Reftangle of the whole Line a h, and one of the parts e b (which is the Parallelogram g h a b) may be equal to the Square [dfa e) of the other. PRACTICE. Erect the Perpendicular a d, which continue towards c; make a c equal to half a b ; upon the Point c , with the Radius c b, deferibe the Arch b d\ upon the Point A, with the Radius a d, deferibe the Arch 2/?, which will cut a b in e in the Proportion required; for if you complete the Pa- y G rallelo- 39$ The ‘Principles of Geometry. rallelogram a gh b, of the Whole ah, and part be, it will be equal to the Square df ae. For if you complete the Parallelogram adib, and draw the Diagonal a i, it will interfea the Lines/ e and g h in the fame Point, and di¬ vide the Parallelogram into two equal Triangles, which are tab and dai. The Parallelograms g e and//j are alfo each divided into two equal Triangles by the Diagonal, in the fame Manner. Now, feeing that the Triangles of thefe two Parallelograms are refpeftivcly equal, and as the Triangles da i and tab are both equal, therefore the Parallelogram g f is equal to the geo¬ metrical Square eh, and confequently the two Parallelograms eg and^/, which together make the geometrical Square df a c, arc equal to the Paralle¬ logram g e, and Square e h taken together, becaufe the Parallelogram g e is common to them both, and that the Parallelogram gf is equal to the Square e h. Tfi E. T> ? II O B L E M III. Fig. C. Two Lines being given {as a b and c d) to find out a third in 'Proportion to them. P P A C T I C E. Draw two Right-lines, forming any Angle, at pleafure, as h e and k e ; make / e equal to c d, and / e equal to ab, and draw i f • make h I equal to/e, and draw k g parallel to i fi then is the Line fig the third Proportional ; for, as e i is to ik, fo is e f to fig, which was to be done. FROBLE M IV. Fig. D. Three Right-lines being given, {as a be) to find out a fourth Proportional, f h. PRACTICE. Draw a right-lined Angle at pleafure, as in the preced¬ ing Problem ; make / e equal to the Line c, alfo d f equal to the Line b, and e g equal to the Line a ; draw ef, andg h parallel to e f, then is h f the fourth proportional Line required; for as d e is to e g, fo is dfxafih which ivas to be done. PROBLEM V. Fig. E. To divide a Right-line given {as a b) into two parts, in Proportion one to the other, according to two given Lines {as d, c). Draw a Right-line, at pleafure, from one End of the given Line, as a e making any Angle ; make a f equal to the Linec, and/ e equal to the Lined- draw eb, alfo/ g parallel to eb, dividing the given Line ab in g; then as a f is to f e, lo is a g to g b, which was to be done. PROBLEM VI. Fig. F. The greater Segment of a Line divided by ext ream and mean Proportion being given, to find the whole Line. Let be be the greater Segment given. Continue ia towards c at pleafure; mak cbd perpendicular to b a, and equal to hall b a, and draw the Line da ; make d fi equal to db, and a c e- qual to a f, then is b c the Length of the whole Line, as required. PRO- The Principles of Geometry. PROBLEM VII. Fig. G The Sum of the Extreams, and the mean Proportional being given, to find the Means. Let b a be the Sum of the Extreams, or two given Magnitudes connected without any Diftindtion, and g h the mean Proportional, by means of which the Point f, where the Extreams join, is to be diftinguifhed. PRACTICE. Bifedt the Line ha, and deferibe the Semicircle b e a ; erelt the Perpendicular h k equal to the mean Proportional g b, and draw k i parallel to b a, which will cut the Semicircle in e; from e draw e f parallel to k b, then will the Point/ be the Point required, and e f will be a mean Proportional between b f and / a. PROBLEM VIII. Fig. H. To divide a Right-line in any Ratio propojed. Suppofe ab to be divided according to the Ratio’s of the Lines a i, hr, c 3, d+. PRACTICE. From the Point a draw the Line am at pleafure, with- out regard to the Quantity of the Angle mab\ make a h equal to the Ratio ai, make h i equal to the Ratio h z, make i 3 equal to c 3, and 3 m equal to d + 1 diaw t ^ le Line m h, alfo the Lines 3 n, i 0, and k p parallel to m b, cut¬ ting the Line ab in the Points p, o, n, and divide it according to the Ratio demanded. PROBLEM IX Fig. I, To find a mean Proportional between two given Right-lines. Suppose the Lines g h and n m be the two given Lines, between which a mean Proportional is to be found. PRACTICE. Draw the Right-line f e at pleafure ; make / b equal to n m, and b e equal to g h ; bifedl/e in d; on d, with the Radius d c, deferibe the Semicircle/ 0 e ; on b, raife the Perpendicular bo, which is the mean Pro¬ portional required. P. Pray, IP hat is to be under flood by a mean proportional Tine ? M A mean proportional Line, as 0 b is a Line, which being multiplied in¬ to ltlell, produces a geometrical Square, whole Area is equal to the Area of a Parallelogram, oi Length and Breadth, equal to the two given Extreams fb and b e. PROBLEM X. Fig. K. A Right-line being given, to cut off a Part, that /hall be a mean Proportional between what remains, and another given Right-line. Suppose ab be the Line, of which a part is to be cut off, that fliall be a mean Proportional between what remains, and de the Line propofed. P R A Cl ICE. Draw the Line If at pleafure ; make li equal to de, and if equal to a b ; deferibe the Semicircle Ini f; from i, eredt the Perpen¬ dicular i m ; bifedt li in A j draw m k, and on k, with the Radius km, deferibe the Arch mg, cutting If in g ; then is ig equal to a c, the part demanded. P R O- 400 The Principles of Geometr y. PROBLEM XL Fig. L. Two Right-lines being given, to cut each into two parts, fo as that the four Segments may he proportional. Suppose ha and de be the two given Right-lines. PRACTICE. Draw g n and o n; erett the Perpendicular n e, both of Length at pleafure; make ng equal to ha, and n e equal to d c, and draw the Hypothenufe eg; defcribe the Semicircle g bn; from the Point h, draw the Line h i, parallel to e n, alfo h f parallel to z n ; then g n being cut in i, and e n in f, the part g i will be to i h, as i h is to hj, and i h is to hf, as h f is to / e, which was to he done. PROBLEM XII. Fig. M. To find out Right-lines, having fuch Proportion one to the other, as two geo¬ metrical Squares given. Suppose the Lines a and h are the Sides of two Squares. Make the Right- angle e dc at pleafure; make dc equal to the Line a, and de equal to the Line b ; draw the Hypothenufe ec, and let fall the dotted Perpendicular d thereon, which will divide ec into two parts, the leller equal to 9, the greater to 1 6, and which are in proportion to one another, as the Square of a %o, is to the Square of b 1 y. For as aay, the Square of the Line h ly, is to 400, the Square of the Line a vo, fo is 9, the Idler Segment of fr, to 1 6, the greater Segment. II. Of the Similarity of Figures. Similar Figures are fuch which are equi-angled, or have the Angles of the one feverally equal to the Angles of the other; having alfo the Sides a- bout thofe equal Angles proportional, as in the two Triangles N and T, where¬ in the Angles p and b are common; that is, the Angle p is common to the whole Triangle N, and to the fmall Triangle op q; and as the Line 0 q is parallel to the Hypothenufe of the Triangle N, therefore the Angles of the lmall Triangle opq, are equal to the Angles of the Triangle N, and confe- quentlv its Sides are proportional alio. It the L langle min be made equal to the Triangle opq, then the Triangle min will be iimilar to the Triangle X ■ and fo in like Manner, the fame is to be underftood of the Triangle V, equal to the Triangle ah c, to be fimilar to the Triangle T, and the Triangle i k h to the Triangle S. As the Angles of all geometrical Squares are Right- angles, and all the Sides of every Square are equal, therefore all geometrical Squares are fimilar to each other ; fo in Fig. R, the Square t sw v is fimilar to the Square q p s r, and all equilateral Triangles, as V and T are fimilar lor the fame Realon alfo If oblique Triangles, as X and \, have their re- fpe&ive Angles equal, they will be fimilar to one another, and their Sides will be proportionable alio. Ihe lame is alio to be underftood of T*apezia s. Pentagons, &c. which are fo made as followeth. PROBLEM I. Figures Q, W. A Right-line being given (as m a) to make a Trapezia ( iema ) fimilar to (. xypg, Fig- QJ a Trapezia given. PRACTICE. Draw the Diagonal xg ; make the Angle i m a equal to the Angle x pg ; make the Angle m a i equal to the Angle p g x, then will the Triangle i m a be fimilar to the Triangle X gp. In the lame Manner, make the Triangle z e a fimilar to the Triangle xy g ; that is, make the An¬ gle _ Th e Principles »/ Geometry. 401 ?*/-‘£n q t a he T° the An?le WXS ' a " d thc An S^cab equal to the Angle tofhe ^ S,V£n Lme *» bc P R O B L E M II. Figures O. P. ^ Right-line being given [as 1 * Fig. P.) to defer the an irregular Tension ( 1 y x w z) fimilar to an irregular ‘Pentagon [as Fig. O.) * *«se s - ssmsat; aw. rW £ - J-, / », ■> * ud 7, * which will complete ,h. infl/r Peno® Plate CCCCLV. rhe Reduction, Transformation, and Equality oj too- metrical Figures demonjlrated. I. Of the Reduaion of geometrical Figures. B y Reduction of geometrical Figures, is meant the Manner of red, ' or transforming any given Figure, as a Triangle, &c mto another f " S aXfe S |r metriCal SqU ‘^ ^ whofc *« ** be equaTto'thc 1 ^ P R O B L E M I. Fig. A. To reduce a Triangle (as do a) into another Trianrle (at h c a) bavin* one Side (as c a) common to botlj. ° Since that by the 37th Propofition of the firft Bool; of Fn Pd T / /landing upon the fame Bafe, and between the lame Paril.bh‘ Tua, f le 1 s one to the other ; therefore from the Pnim d \ ™ rallel P at e equal the .i,c m, 0 . ..hid, * „ v:z,2fz„ x?,r.te' 'zi/esr^r aflign any loint, as at b> and complete the Trian«de c*h a wi ' i i inc ^ f for its Bafe, and being between the Parallels cl f and r , UC 1 iaVjf} § ca the Triangle do a, as required. 1 ° ‘ S therefoie e< i ua > P R O B L E M II. Fig. B. Sin Single being given, as e b c, and from a Point, as e in the T ; drawn a Line by Cbance of any Lemth as e f nitt'nw ^ Gy 7 e b c the Triangle g . Nol fs retired.from TTdrt/Tn t ^ P Zgle°L ’ aStheUnecf > toenclojea Triable as h, equal totl/Tn thf Point e drawn!the L fnef e -^andlrom th ^ °Z ^ £ Bgle 6e - * 4 P-allel to e c, cuttingVhi Se ef f X draX V- raW / he L '"< will enclofe the Triangle h equal to the Triangle/'as required. I” 6 / ^ Whld y H P R O- • 4 02 The \Principles of Geometry. PROBLEM hi. Fig. c. To reduce any Triangle [as cal) into a geometrical Square [as e df c). “s a mean iroport.on between « and cl; and the btde of a bquare [e df is equal to the given Triangle, as required. P R O B L E M IV. Fig. D. n Triangle being given [as cl a) to make another Triangle e da equal thereto, 1 whoje Perpendicular e e is limited to the Length oj toe Line j g. . At the Diftanceof the Line fg, draw a Line parallel to b a as the dotted I ineV cutting the Side c * in * ; draw eh, alio cd parallel to eh, cutt.ng the Ba'fe a h, being continued in d; and draw ed, then is Ole T ™ n S£ eaual to the Triangle eba; for as the Inangles ted, and dbt, a, , it will divide the Rhomboides into two equal parts, and be alio equal to hah the Triangle c b a, becaufe it Hands on half its Baft, and is of the fame perpenuicu- The ‘Principles of Geometry. 4.03 lar Height ; and, as the Triangle g fa is equal to gae, therefore the Rhombus get a is equal to the Triangle c b a, as required. PROBLEM VIII. Fig. H. To reduce a Parallelogram {as dah e) into a geometrical Square {as c h hf). Continue a Side, as e h , from h to i , making h i equal to h d ; bifedl i e in g ; deferibe the Semicircle icb e ; continue h d to c ; then is h c a mean Pro¬ portional, and the Side of a Square {cl hf) equal to the Parallelogram, as re¬ quired. PROBLEM IX. Fig. I. To reduce a geometrical Square {as deb a) into a Parallelogram, (as m l x h) whofe Height is limited equal to the given Line ef. Continue alto x at pleafure, and make h l equal toe/; make d l e- qual to la , and draw dl, which bilcdt in /a; on h erect the Perpendicular h i, cutting x a in z ; on 1, with the Radius i a, deferibe the Semicircle Id x ; on x ereft the Perpendicular * m equal to b l, and join m l, which completes the Parallelogram mlxh , equal to the Square deb a, as required. PROBLEM XII. Fig. K. To reduce a geometrical Square (as c db a) into a Parallelogram, (as je a l) whofe Length is limited equal to the given Line z.x. Continue b a to/; make al equal toz*; draw the Line d /, which bilect in 0, whereon raife the Perpendicular 0 k , cutting h l in k, on which, with the Radius k l, deferibe the Semicircle Idm ; make a / equal to a m , alio e l equal and parallel to zz /; draw / e, and the Parallelogram f e a l will be equal to the Square x d b a, as required. P R O B L E M XIII. Fig. L. To reduce a given Parallelogram (as d cab) into another Parallelogram (as f eh l) whoje Breadth is limited, equal to x z. Continue a b towards /, at pleafure, alfoadtog, making d g equal to x Z • from g, through the Point c, draw the Line del, cutting the Line ab l in /,’then will hi be the Length of the Parallelogram ; make b f equal to*z the'given Breadth, and then, making le equal and parallel to b f, draw f e, which will compleat the Parallelogram fell, as required. PROBLEM XIV. Fig. M. To reduce a given Parallelogram ( asebda) into another Parallelogram, (as h gf a) whofe Length is limited, equal to e e. Continue ad towards f, at pleafure, and make*/ equal to e e: Now fay. As e e, the Length limited, is to c d, the given Breadth, fo is a d, the given Length, to h /, the Breadth required. O P E R A T I O N, Fig. O. Make the Angle 0 i pat pleafure ; make z n equal to e e, and having c d in your Compafles, fet one Foot in >1, the other will fall on /, and draw ln \ make i m The "Principles of Geometr y. 404 i m equal to a d, and from m draw m k parallel to l n, which will be equal to h /, the Breadth required. PROBLE M XV. Fie P O' To reduce a Rhombus (as c db a) i/do a geometrical Square (as nlh e). Draw the Perpendicular de, and extend it up towards /; make eg equal to e d\ and make e h equal to a b ; bifect h g, and delcnbe the Semicircle hlg, which cuts the Perpendicular e d in l, then is le the Side of tile Square required. PROBLEM XVI. Fig. Q. T0 make an equilateral T'riangle (as iff) ■ ual to a geometrical Square, whofe Side is eqi . to a a). Make fg equal to twice a «; at the End f make the Angle hfg equal to 60 deg. draw ee parallel to / c, at the Diitance of a a, which will cut fh in /;; make / b equal to f h\ find a mean Proportion between fb and fc, which is f g, and draw the Line ig parallel to h A; then is the equilateral Triangle ifg equal to a geometrical Square, whole Side isecual to a a, as required. P R O B L E M XVII. Fie. R O T0 reduce a Trapezium given (as w a cl 0) into a right-angled Parallelogram [as nek i) or right-angled Triangle (as h k i). Draw a Diagonal, as w d, and thro’ the Points a and 0 draw Right-lines parallel to w cl, of Length at pleafure ; alio, thro' the Points w and d, draw Right-lines, at Right-angles to the former, which will compleat the Paralle¬ logram h gk 1, and circumferibe the Trapezia w ado, whole Area is equal to twice the Area ol the Trapezia; for if, from the angular points a and 0, we iiippole Perpendiculars to be drawn to the 1 fine 7 u d, then each oppofite Tri¬ angle will be equal, and thole included within the Trapezium will be equal in Number and Area to thofe without it, that makes good the Refidue of the Parallelogram. Now, feeing that the Parallelogram hg k i is equal to twice the Trapezium w ado, therefore bifedt h k in n, and gi inf, and draw ne which will divide the Parallelogram hg k i into two equal Parallelograms, viz. hgu e and nek i, which are each equal to the Trapezia w a do, as alfo is the right-angled Triangle hki, which is half the Parallelogram hg k 1. P R OBLE M XVIII. Fig. S. To reduce a Trapezia (as chad) into a Triangle, as c a e. Continue the Bafc a d towards e at pleafure ; draw the Diagonal c d and he parallel thereto ; alio draw the Line c e ; then is the Triangle c ae equal to the Trapezia chad-, for as the Triangle c de and c h d do both Hand on the Line cd, and being between the Parallels be and c d, thev are therefore equal; and as the Triangle c h d is common to both the aforefaid, therefore the Triangle c a e is equal to the Trapezium chad. PROBLEM XIX. Fig. T. o To reduce an irregular 'Pentagon (as c d e ah) into a Triangle. Draw the Lines ce and c a from the Angle b ; draw hg parallel to c a, and from d draw elf, parallel tocc; draw the Lines c f and eg, then will the The Principles of Geometry. 405 the Triangle c f g. be equal to the irregular Pentagon c de a b, for the lame Reafon as delivered in the laft Problem PROBLEM XX Fig. V. To reduce a T rapezia {as d c a b) into aTriangle i k l, whofe Bafe k l, and one Angle k, is limited. (1) Draw db, continue out a b; draw c e parallel to db, cutting the Bafe continued in e, and then drawing d e, the Triangle dae will be equal to the Trapezia dc ab. (z) Draw k l equal to the affigned Length of the Bafe, and on the End k make the Angle ikl equal to the given Angle h. Now, to find the angular Point i, lav. As k l, the Bafe limited, is to ae, the Bale of the Triangle d a e, fo is df, the perpendicular Height of the T riangle dae, to the perpendicular Height of the Point /, above the Line k l. Plate CCCCLVI. Demonjlrating the Equality oj geometrical Figures. P KOBLEM I. Fig. A. There is a Triangle, as acb, and from a , is drawn a Right-line, as a e d, wherein is a Point, as e. given at Plea fare, from whence 'tis required to draw a Line thro' a c {as e f) in Inch Manner, that the Triangle eg f cut off, be equal to the Triangle e ga taken in. Thro the Point c draw the Line d c parallel and equal to ab, to cut the Line ae d in d; join da, and draw the dotted Line e b, and from the Point d draw the Line df parallel to eh, cutting cb in f, and draw the Line e f ; then will the Triangle eg/ be equal to the Triangle e ga, which was to be done. P R OBLE M II. Fig. B. There is a Triangle, as b a c, and from the End a, there runs a Right-line, as a de, given ; to find how from a given 'Point, as e, to draw a Line to forrtS 'Part of the Bafe c a, {as e f) that the Triangle e f a may be equal to the Trapezia bgc f. Draw the Line ec, alfo b d, parallel to c a, cutting a e in d ; from d. draw df parallel to e c, and draw e f, then is the Triangle c f a equal to the Trapezia bgef, as required. PROBLEM III. Fig. C. There is a Triangle, as be a, from a, is drawn a Line at Pleajure, as ad, and tis required, from the Angle c to draw a Line to fame Part of a d {as c e) in fitch Manner, that the Triangle eca may have ProOortion to the given Triangle b a c, as the Line 4, to the Line y. Bv the Rule of Proportion fay. As the Line 1 y is to the Line K 4, (o is b a to ga; then thro’ g draw the Line g e parallel to c a, and from the Point e draw the Line e c ; then will the Triangle eca be to the Triangle b c a, as the Line K 4 is to the Line i y, as required. S I PRO 40 6 The ‘Principles of Geometry. PROBLEM IV. Figures D, E, F. Two Figures being given, and a Right dine, to find another Line in Proportion to the given Line, as the one Figure to the other. The Trapezia be da, Fig. E, the Triangle k h f, and tile Line m 15-, are given to find the Line n 40. (1) Reduce the Trapezia bed a into the Trian¬ gle b e a, and draw its Perpendicular ho. (2) Reduce the Triangle khf Fig. D, into another Triangle, whole Bale is equal to a e, as the Triangle g i f\ Fig. D, and let fall its Perpendicular gp, then will the Triangle g if, or h k f, (which are equal to one another) be in Proportion to the Trapezia, as the Perpendicular gp is to the Perpendicular b 0 ; now if gp were equal to the Line m 1 s, then b 0 would be the Line required ; but as it is not, therefore, as the Perpendicular gp is to the Perpendicular bo, lb is the Line m ly to the Line n 40. OPERATION, Fig. F. Draw the Angle b 14 at-Plcafure ; make 1 2 equal to the Perpendicular g p, and 1 4 to the Perpendicular b 0 ; lake the given Line m 1 y in yourCompalies, and fetting one Foot on the Point r, the other will fall on the Point 3 ; from Point 4 draw the Line 4 y parallel to the Line 1 31 and it will be the Line required. P R O B L E M V. Fig. G. To divide a Line given into two fitch Tarts as another given Line (■ wbofe Length is not more than half the fir ft) may be a mean ‘Proportional between the ‘Parts. Suppose hf lie the given Line to be divided, and ah a mean Proportional between the Parts to be found. O P E R A T I O N. * Bisect hf in ft, and deferibe the Semicircle h ef\ draw dc parallel to k /, at tin: Diftance of .the given Line a b, cutting the Semicircle in e; draw e 1 at Right-angles to h f, dividing h f in the Parts h i and if, the parts re¬ quired. P R O B L E M VI. Figures H and I. O To make a Triangle (as h i a ) equal in //tea to a given Trapezia (as b c d a) and ftmilar to a Tr tangle given (as e / g). Reduce the Trapezia beda into the Triangle b h a, and at the End a make the Angle hai equal to the Angle egf; at the End h make the Angle a h i equal to the Angle g e f, and draw the Line h 1, to cut the Line a i in the Point i, then will the Triangle h i a be iimilar to the Triangle e fg ; but as 'tis not equal to the Trapezia b c d a, therefore from b draw b k parallel to h a, to cut a i in k, then will the Triangle a k h be equal to the Triangle b h a (as being 011 the fame Bale h a, and between the fame Parallels kb and ha) but as the Angle at h is not equal to the Angle feg, therefore from the Point k draw the Line k l parallel to 1 h, to cut ha in this being done, find a mean Proportional between a t and ah, which is a m, and from the Point m draw the Line m n, to cut the Line a 1 in n, then will the Triangle a n m be equal to the Trapezia b c d a, and iimilar to the Triangle e f g, as required. P R O- The Principles of Geometry. 407 PROBLEM VII. Fig. A. To reduce a Parallelogram (as cb da) into a Tarallelogram (as nmdl ) whofe Breadth /hall have Proportion to its Length, as the Line e e n is to the Line iv 72. Find a mean Proportional between the given Lines ee 31 and K 71, which i_ s & & 4 8 i a™ a mean Proportional between the Breadth ba and the Length cb, winch is k k 71; then if gg 48 give k k 71, what will K 71 ? Anfwer //I° b ; lor as 40 1S to 71, lo is 71 to 108. And again, if? ? 48 give k kvr what will , e 3x ? Anfwer the Line L +8 ; for as 48 is to 71, fois 3 * to 48. Con¬ tinue da to l, making d l equal to ff 108 ; alfo make dn equal to L 48 • draw nm parallel to dl, and Im parallel to nd, then the Parallelogram nmd’l , be equal to the Parallelogram cbda, having its Breadth to its Length, as the Line ee 31 to the Line K 71, as required. PROBLEM VIII. Fig N To increafe or decreafe a Plan given, according to any Proportion ajjigned. S dppos , e , c f de a he a 1>lan S ,vei1 to be decreafed in Proportion as the Lines k t0 {■ Add the Lengths ot the Lines / and k into oneSum, and then fay. As the Sum of thofe two Lines is to any one of them, (luppofe l the (hort one) fo is the Length of any given Side of the Plan, (fuppofo b a, to b f) the Quantity ot its Decreafe; or as the Sum of thefe two Lines is to k, the longeft, fo is the Length of any given Side of the Plan (as aforefaid) to a f, the Length of the Side of the dimmifhed Plan: From a, the End of the Side fa, draw Light-lines into the Angles c and d ; from the Point/draw ^/parallel" to cb- from g draw g h parallel to dc, and from h draw hi parallel to de, then will the i lan g f h 1 a be dmnnifhed in Proportion to the given Plan cb d e a as the Line k is to the Line l, as required. If the Plan fg hi a had been given for to be enlarged, as the Line / is to the Line k, then the Point b mult have been found as following, viz As the Line k is to the Line /, fo is a f the given Side of the Plan, to fb its Increafe and which being found, and the Lines ag, ah, and ai continued out at Pleaiure, from the Point b draw the Line b c parallel to yf, alfo c d parallel to g h, and de parallel to h i , and then will the increafed Plan cb d e a be to the given Plan g fh ia, as the Line l is to the Line k, as required. ’ P ROBLEM IX. Figures S, T. si Trapezia being given (as ebda. Fig. T) to make two other Figures equal to it, but each of thofe two Jtmilar to the Parallelogram O another Fi- gm c given-, and yet the lejfer to have Proportion to the greater as the Line E 9 to the Line F 16. (1) Reduce the Trapezia ebda into the Triangle b h a, and that Triangle into a geometrical Square, whofe Side will be equal to the Line P R, and Area F 05 , 1 ^ ed:lce [ he Parallelogram O alfo into a geometrical Square, whofe Side will be equal to the Line Q^ and Area to no. (3) Reduce the Trapezia ebda into a Parallelogram, fimilar to the Parallelogram O, as following viz. As the Line E is to the Line F, lb is the Length of the Parallelogram O, to the Length of the greatelt Parallelogram in Fig. S ; and as the Line E is to the Line P, fo is the Breadth of the Parallelogram O, to the Breadth of t^e greateft Parallelogram in Fig. S; and which Parallelogram is equal to the Irapezia ebda , and fimilar to the given Parallelogram" O (4,) Di¬ vide the Length of the greateft Parallelogram, Fig. S, into two parts, fo that 4.0 8 The ‘Principles oj Geometr y. that thofe parts may have the fame Proportion to each other as the Lines R and F have to each other, and raife the dotted Perpendicular, to cut the Arch of a Semicircle deferibed on the Side of the Parallelogram, from which Point draw Right-lines to the Extreams of the Parallelogram's upper Side, thereby forming a right angled plain Triangle, whole Sides that contain the Right-angle are the Lengths of the two Figures re- q uired. 1 The Breadths of thefe upper two Parallelograms are thus to he found, vi~. as the Length of the given Parallelogram O, is to its Breadth, lb is the Length of either °of thele two Parallelograms to the Breadth required. As the Breadths of thele Parallelograms are found by the Ratio of the Parallelogram O they are therefore fimilar to it; and if both their Areas are added together, the Sum will be equal to theTrapezia e h d a, as required. PROBLE M X. Fig. V. To reduce a regular Polygon (as the Hexagon a / c e d) into a geometrical Square. Thro’ its Center g draw a Right-line mb at Right-angles to a Side ; make ? m equal to half its Circumference ; on g raile the Perpendicular g r, of length fufficient to cut thd Semicircle mr/j, delcribed on the Line mb in the Point r, then is the Line rg the Side of the Square required, and a mean proportional between m g, the Semiperimeter , or half Circumference, and g h the Semidiameter. PROB L E M XI. Figures W, X. To reduce an irregular 'Plan (as c ah gf e il) into a geometrical Square. (i) Draw the Lines dh ancle g, Fig. W, which will divide the Whole into two Trapezia's and one Triangle, (i) Draw the Diagonals of the Trapezia's ad and b e, and let fall Perpendiculars thereon from the Angles c, h, d, g, f. (-) Reduce every Triangle into a geometrical Square, and note their leveral Sides, as i, *, 1 , &c. (4-) Add the leveral geometrical Squares together in Manner following, viz. Take the Sides of any two Squares and find a mean Proportional to them, which will be the Side ol a Square equal to both of them ' proceed on in like Manner to find mean Proportionals until there be not any Squares remaining, and the lall Proportional will be the Side ol a Square" equal to the irregular Plan cahgf e d, as required. PROBLE M XII. Fig. Z To reduce a Circle into a geometrical Square. \s the fquaring of the Circle has been a Work attempted by many more able Pens than that of mine, which in general have failed of their delired F.x- aelnels, 1 ihall therefore in this Place content myfelf with delivering a Ride, that will come fufficiently near for all Kinds of Bufinefs, relating to Buildings in general, without troubling my Readers with a long Series of Calculations for that Puipoie, as many have done, more for the Sake thereof than any real Ufc in Bufinefs. Let ofgabea Circle given to be reduced into a Geometrical Square (as ch da. PRACTICE. Continue the Diameter g o towards h at pleafure, on the Center a erect the erpendicular ab of Length at pleafure ; divide the Diameter og into 7 eqi al parts, and make a j equal to three Diameters and The Principles of Geometry, _ 4° 9 one 7 th of the Diameter. From / draw fk parallel and equal to h a and Circle ’lahlv I 5 ,* 6 . Parall « Io g^m kfba equal to the Area of ,h Circle . Laftly, bifedf hg in o, whereon, with the Radius o h, deferibe the Semicrcle h bg continue aftoh, now, as ag is equal to a f therefore ff 1Sa P ro Portional between the Length and Breadth of the ParaI- required* 3 a “ d the S,dc ot a Square equal to the Area of the Circle, as 1 he Circle aforefaid may be reduced into a right-angled plain TriaiHc by continuing ,hk to e, making ke equal to kb, and drawing the Line ea’ foi bj the firft Proportion of the eleventh Book of Euclid, every Circle ,s equal to that right-angled Triangle, of whofc containing Sides the onefas hi) Diameter 13 Che Circumference, and the other eh to twice fa, equal to the PROBLEM XIII. Fig V T° reduce an irregular Elan {aslchagfe) of many Sides, into a right-End Triangle, as a p o. (!) Draw the Line g e alfo fk parallel to g e, and draw the Line g k M Draw the Line a k and from g, the Line g o parallel to a k, and draw the Line a o, cutting the Line l e continued in o. to) Draw the Line hi an I from the Point* the Line cd parallel to*/, ind draw the Line ^ ? + ) Diaw the Line a d continue out e d towards p at Pleafure ; from the Point fj'T- Lme If? 1161 t0 « d > Cl 'tting k d \n p, and Tawmg Z JequiTed 56 aP ° W ' £<1Ual t0 thE 11Tegular ^ lan ^bagfe, as PROBLEM XIV. Figures g, j, &c. To reduce a Geometrical Square [as h a c d) into the Figure of a Lunula as g ef a. Continue ha to e, making he equal to ha-, on h, with the Radius !h’A de , Ulbe th f Semicircle e g a ; alio one, with the Radius e c deferibe required' 6 ' 1C * S the Lunula & efa > e< l ual to the Square b’a c d, as Plate CCCCLVII. Arithmetick geometrically demonjlrated. I. numeration. As Right-lines arc compofed of Points, fo are Numbers'of Units A Unit figmfies one as a Man, a Tree, a Line, a Square, a Circle, &c fo the Right-line a b. Fig. iI. is a Unit, as not being divided into two parts as c h The Qrcle/i/^/ Rg. If the Square efgh, and Parallelogram bode being conhdered as undivided Figures, are alio Units, or One's, thd of different t p * rB - ,hc p,rB ™ ciiw « T K 4 IC The ‘Principles of Geometry. So in< Fig. 1.1 r c b 1 eb\ h b x ' 9 4 l I one half 1 one third one fourth r f 1 * .mb s equal one fifth 1 1 and are | J | r b is di¬ 6 parts. one fixth '1 s b 11 b vided 7 which one feventh -thus ex- b j > into 8 are one eighth prefled j 1[ \n x b 9 10 11 called one ninth one tenth one eleventh j ; ;| \yb . 1Z one twelfth LTij part of the Line a b, which is equal to each of the Lines c b, e b, &c. wherein thofe fractional parts are reprefented. If the Line hd be drawn thro’ Hence 'tis plain, that a part is a Magni g rrv tSTcSr 1 S"n S- S’ ^ » a » d ^^ LaeS b^L^^ quite through the Center, from Side to Side, at Right-angles, will divide each Semicircle into a Quadrants, each equal to one +t ifhke manner, the Arch c l being one 6th of the whole Circumference the Seftor “ b is one 6th of the Unit; fo likewile e ad is one 8th, alfo d a c one rrth, and / « e oneifthof th.eUmt divided as in the p.gure. The seometncal bqaaie e j g u, 1 '&■ ilJ - » ■ r_ tIle snuares a b and c are each one 4 th part of the Unit e f g h lo likewile the Squares a and h taken together are i half; the Squares a, b and c taken to¬ gether are three 4 ths ; and the Parallelogram c or '■« is three laths, 01 one 4th, &c. 11. ADD! T I O N. \s hv the 4-th Propofition of the firft Book of Euclid the Sum of the Squares; ,Se + on the llgs of a right-angled ^ square made on the Hypothenufe the Addition of the Ama s of geometnea^ i; s , s verv eafv; lor it the Areas to be added aie hilt reduced m So in res is before taught, this Propofition will add them together into one Sun ' T m b th“ Proton to be underftood to the meaneft Capacity, Sum. 10 make 1 n 1 P , „. „ , its Legs k m and m l equal; Let the right-angled mangle k m l, i ig. \ . nave a con ipleat The ‘Principles of Geometry. 41 i compleat the Squares qkpm, min 0, and skr l, continue q k to r, 0 l to s, and draw the Diagonal qmo. Now, ’tis plain that the Triangle h is equal to the given Triangle e, becaufe it Hands on the fame Bale m l, and is between the fame Parallels k l and q 0 ; and as n 0 is equal and parallel to m l, and m n to lo , therefore the Triangle i is equal to the Triangle h, and to the Triangle e all'o ; for the fame Reafons the Triangles / and g are each equal to the Tri¬ angle e. Again, as k t is equal and parallel to m l, and t L to k m, therefore the Triangle c is equal to the Triangle e ; and as s t is equal to 1 1 , and h s to k l, and as k l is common to both, therefore the Triangles a and c .are c- qual, as likewife are the Triangles b and d, becaufe t r is equal to kt , s r to j- k, and r l to / k. Now, feeing that all the Triangles a, b, c, d, /, g, h, 1, are equal to each other, therefore the Triangles a, b, c, d, which compofe the Square skr l, are equal to the Triangles/ g, h i, which compofe the Squares made on the Legs of the Triangle, as was to be proved. This is alf'o proved arithmetically, by the Squares of the Numbers 3, 4, and y, as in Fig. VI. Suppofe a right-angled Triangle, as b c a, whofe Side l c is 3, that of c a is 4, and the Hypothenufe b a is 5, compleat their Squares, dividing each Side, and draw the Lines in each as in the Figure. Now, ’tis plain that the Square e b d c, whofe Side is 3, contains 9, and that the Square cafjg, whofe Side is 4, contains 16, which added together is equal to ay, and which are equal to the Square h iba ay, whofe Side is y. Hence tis evident, that making the Sides of any two Squares given, the Legs of a right-angled Triangle, the Hypothenufe ’is the Side of a Square, whofe Area is equal to the two Squares given. EXAMPLE Fig. VII. Let the Geometrical Squares a , b, c, d, e, be given, to be added into one Geo¬ metrical Square. (1) Make the Legs of the right-angled Triangle fg i equal to the Sides of any two of the given Squares, fuppofe of the Squares a and b, then the Square hg f i, on the Hypothenufe, is equal to the Squares a and b. (a) Make the Legs of the Triangle m n 0 equal to f i, the Side of the Square k. and to one Side of the Square c, then will the Square m q p 0, made on the Hypothenufe m 0, be equal to the Squares a, b, and c. (3) Make the Legs of the Triangle t s r equal to the Side ol the Square q p ?n 0, and Side 'of the Square d, then is the Square tvrw equal to the Squares a , b, c, d- (4) Make the Legs of the Triangle 3 1 a equal to t r, the Side of the Square .r, and to the Side of the Square ied in any ate of its Sides. Suppose dfhl be the given Trapezia, the Point e in the Side df, the al¬ igned Point, to cut on .1 pan equal to a Square, Wiiole owe is equal to the Line The Principles of Geometry. 419 Line b. Reduce the Trapezia into the Triangle eg m, alfo reduce the Square, whole Side is equal to b, into a Triangle, whole perpendicular Altitude be equal to the Perpendicular of the Triangle e g m, from e upon the Bafe gm\ then will the Bafe of that Triangle be equal to the Line a, which fet from m to k, and draw e k, and the Trapezia e f hi cut off will be equal to a Square, whole Side is equal to the Line b, as required. PROBLEM VII. Fig. IV. To di vide a Trapezia into two parts, by a Line parallel to one of its Sides, in Inch 'Proportion the one to the other, as two given Right-lines ; and to lay the part required towards an Angle or Side afflgned. Suppose s qm p be the given Trapezia to be divided into two parts, in fuch Proportion to one another, as the Line h has to the Line 1, by a Right-line drawn parallel to the Sidef/>, and to lay the greater part next the Side s m. (1) Confider through which two Sides the Line of Partition mull pals, which mult always be continued until they meet, as qs and pm, which being conti¬ nued meet in k. (i) Reduce the Trapezia sqmp into the Triangle q Ip, whole Bale is Ip. (3) Divide Ip in n, ib that In be to n p, as the Line b to the Line c, the lefler part being placed next to p. (4.) Find a mean Proportion between kn and k p, as ko ; from the Point 0 draw the Line or parallel to qp ; then will r qo p be the lefler part next to q, which has the fame Pro¬ portion to the part sr mo, as the Line i to the Line h, as required. But if the part required to be cut off had been with a Parallel to m p, then the Line of Partition would have palled through the Sides q p and j m, which mull have been increafed till they meet; and then proceed in all Points, as before. PROBLEM VIII. Fig. IV. To cut from a Trapezia a Part equal to a Figure given, by a Line parallel to one of the Sides, ana to lay the Part out off towards a Place appointed. Suppose the Parallelogram a c d g be given to cut off from the Trapezia sqmp a part equal thereto, by a Right-line drawn parallel to the Side qp. (,) Confider through which two Sides the Line of Partition mult pafs, which here are r^and mp, which continue, until they meet in b. (a) Reduce the Parallelogram ac dg into the Triangle ace, and that into the Triangle a b f, whole perpendicular Altitude be equal to the Perpendicular from q on the Bafe k p, and Bafe to a b. Now, feeing that the part to be cut off is to be laid next to’ p, therefore make p n equal to a b, and find a mean Proportional be¬ tween k n and k p, which is k 0, (3) From the Point 0 draw the Line 0 r pa¬ rallel to the Side qp-, then will the part r q op be equal to the Parallelogram aedg, as required. Note, If the part cut off, equal to the Parallelogram, ihould have been laid next to the Angle m, then the Point l mull have been firfl found, and a b mull have been found, and placed from m towards p, and then proceeded, as before. PROBLEM IX. Fig. V. To divide a Geometrical Square {as d e f g) into two equal Parts, by a Right-line {as b c) drawn thro a given Point {as b) within the Square. D e aw the Diagonals dg and e f interfeaing in h the Center; from the given Point b, through the Center h, draw the Right-line a c, which will di¬ vide the Square defg into two equal parts, as required. Note } 4.20 The 'principles of Geometry. Note, The fame is to be underftood of the Parallelogram, Rhombus and Rhomboides. PROBLE M X. Fig-. XL To divide a Trapezia into two 'Parts, according to Proportion between two Lines, with a Right-line parallel to a Right-line given; and to lay the Tart required towards a 'Place affigned. Suppose d f g i be the given Trapezia to be divided into two parts, in Pro¬ portion one to the other, as the Line a to the Line b, by a Right-line paral¬ lel to the Line hn, and to lay the greater part next tog. (i) Consider thro' which Sides the Line of Partition mult pafs, which here are g i and d f, which continue until they meet in the Angle c; from/draw the Right-line / k pa¬ rallel to the given Line l m, to meet the Bale g i continued in k." Suppofe a 1 Jne be drawn from/to g, and another parallel thereto from dt o the Bale c k, cutting the Bale in t; and then fuppofmg a Line drawn from /'tor, theTriang e will be equal to the Trapezia dfg i. (i) Divide t i. the Bale of the aforefiid Triangle, into two parts, at the Point v, id that the pai f iv be to the part vt, as the Line a to the Line b, oblerving to lay the greateft part next to g, as is required. (5! Find a mean Proportional between cv and c k, which is c h, and from the Point h draw the Right-line h e parallel to the Line Im ■ then . ill the Trapezia dfg i be divided into two parts, according to the Pro¬ portion affigned, with the greateft part next to g, as required. PROBLEM XL Fig VI o To cut from a Trapezia a Part equal to a Figure given, by a Right-line drawn parallel to a Right-line drawn by Usance, and to lay the Part cut of towards a 1 Place ajjigned. Suppose the Trapezia dfg i be given, to cut off an Area equal to the A- reaof the Trapezia no r q, by a Right-line parallel to the Chance-line Im and to lay the part cut off n xt to the Angle i. (1) Confider thro' which of the Sides the Line of Partition e h mult pals, which h -re are g i and d /, which continue until they meet in the Angle c. (1) Reduce the Trapezia nor q into the Triangle npr, and that again into the Triangle n 0 s, whole perpendicular Altitude may be equal to the Perpendicular from /upon g i continued, then will the Bale of this new Triangle be n 0. (3) Set n 0, horn ? > towal ■ on he L ; . .> the 1 oint ®, and f Right-line p oallcl to the Chance-line l m, ro cut the Bale g i continued in k (+) Find a mean Proportion between cv and c k, which is ch, and from h draw h e parallel to Im, then will the part e f h i, next to the Angle i, be equal to nor q, the Trapezia given, as required. Note, Thai 1 ■ been hit-o aid conce ingti Dit ifion of Trai e : the fame is to be underftood of the Geometrical Square, Parallelogram, Rhom¬ bus and Rhomboides. PROBLE M XII. Fig. VII. To divide an irregular Plan into two Parts (according to any poffible Proportion given) by a Line drawn through a given Point will n the Jame. Suppose the Plan In q s w be given, and let the Point 2, where the Line m t cuts the Line op, be the given Point, and tis required to divide the Plan into two parts, that ihall be to one another, as the Line a to the 1 .me b. (1) Confider thro which Sides the Line of Partition mt mult pals, which here are The Principles of Geometry. 42 1 are the Sides n q and sin, and which continue until they meet in x (a) Re¬ duce the whole Plan into a Triangle, as before taught, and that Triangle into another, whofe Perpendicular is equal to a Perpendicular from 71, upon the X in C r x ■ from n let fall a Perpendicular on the Bafe sx, and on each Side thereof on the Line r x, fet off the Segments of the Bafe on each Side the Perpendicular of the laft produced Triangle, which fuppofe to termi¬ nate in y and r (3) Divide /r in v, fo that the part / v may be to the part v r as the Line b is to the Line a. (+) Take the neareft Diftance from the given Point z to the Bafe r x, and lay. As that Length is to the Length of the Perpendicular from n to the Line r x, fo is the Half of x v to vw which fet from a- to w. (y) From the Point w draw the Line w 0 pa¬ rallel to «*; alfo from the given Point z the Line op parallel to the Bafe r x cutting each other in the Point 0. (6) Take the Square of 2 tp from the Square of 0 z, and theSide of a Square, equal to the Remainder, will be equal to wt, which fet from w to t. Laftly, Draw the Line turn, which will di¬ vide the Plan into two parts, which are in Proportion to each other, as the ] jne a to the Line b, and the Line of Partition drawn thro’ the given Point z, as required. P R OELE M XIII. Fig. VII. To cut f rom an irregular Plan given, a Pari equal to a given Figure {-whofe Strea is lefs than the irregular Plan given) by a Right-line puffing thro a ‘point given within the Plan. Suppose In q s, &c. be the given Plan, and the Triangle rig i a Figure given, and z the given Point within the Plan. (1) Conhder which Sides the Line of Partition, drawn through z, limit pals, which here are tne Sides n q and s w, which continue, until they meet in the Point x. (1) Reduce the given Triangle dg i into the Triangle e g k, whofe Perpendicular kg may be equal to a Perpendicular from n on the Bale s .v. (3) Find the loints r, /, as 111 the Lift Problem, and let eg (the Bafe of the Triangle e g k) from / to v, and fay a ‘ s before ; As the neareft Diitan.ce, from the given Point z to the Bale a *, is to the perpendicular Height of the Point n above the Bale s r, fo is the half of a- v to x ii’. (4) Frc m if draw the Line w 0, and then proceed, as before in the laft Problem, and draw the Line t ztn , then will the Lfier part, next the Angle q, be equal co dgi, the Triangle given, as required. P R O B L E M XIV. Fig. VIII. To divide a Trapezia, according to any Proportion given, by a Right-line drawn from a given Point without the Trapezia, and to lay the Part cut off towards any Place ajfigned. Suppose o r s t to be a given Trapezia, and let v be the given Point, from which is to be drawn a Right-line, as v q, that fhall divide the parts of the Trapezia, as the Line a is to the Line b, and to lay the Lifer part next to t. (i) Since that the Line of Partition vq mult be drawn from the Point v, it muit pais through the Sides s t and o r, which continue, until they meet in the Angle m ; and, from the Point v, draw a Parallel to the Line in t, to meet >• in continued in n. (2) Suppofe a Line be drawn from r to s, and, parallel thereto, another from 0 to the Bafe-line m t, cutting it in w ; then is wt, the Bale of^ the Triangle w r t, equal to the Trapezia 0 r s t. (3) Divide t w in x, fo that t x may be to x w, as the Line b is to the Line a. (4.) Now fay. As n v is to m x, fo is m r to mf. (/) Find a mean Proportion between")* m and m p, as m */and bileft mf in /, and draw the Line k l ; make 0 q equal to k /, and, from the Point q to the given Point v, draw the Right-line q v, which will divide the Trapezia, whofe parts will be to each other, as the Line a to the Line b, as was required. y N P R O- The 'Principles of Geometry. 422 PROBLEM XV. Fig. VIII. T0 cut from a Trapezia a Tart equal to a Figure given, by a Line drawn from a given'Point without, and to lay the Part cut off towards a Place appointed. Suppose 0 r s t be a given Trapezia, and v a given Point, from whence muft be drawn a Line to cut off a part towards the Angle t, that fhali be equal to the Parallelogram cdhi. (1) Tis evident, that the Line ot Partition, from v, the given Point, muft pal's through the Sides s t and or; therefore conti¬ nue thole Sides, until they meet in the Angie rn (i) From the given J oint v draw the Right-line v n parallel to m t, until it meet mr continued in u. (5) Reduce the Parallelogram c d h i into the Triangle ceh, and that again into the Triangle cfg, whole Perpendicular c f may be equal to the Perpen¬ dicular from r upon the Bale fu t. (4) Make / * equal to eg, and then lay As n v is to m x, fo is mr to mj>. (?) Find the Pointy, as in the laft Pro¬ blem, and draw the Line q v ; then will the part q t be equal to the given Parallelogram, and be next to the Angle t, as required. PROBLE M XVI. Fig. IX. To divide a Plan given, according to any Proportion afftgned, with a Line dr anon from a given Angle, and to lay the Part required towards a Place appointed. Suppose the Plan n ik cp be given, to be divided into two Parts fas the Line h, is to the Line a) by a Right Line drawn from the Angle k, and to lay the Idler Part next to n. (1) Conlider on which Side the Line of Partition muft fill, which in this Example will be n p , which continue both Wavs towards m and q at pleafure (1) Reduce n 1 k c p, the Plan given, into a Triangle, as k m q, whofe perpen¬ dicular Height above the Bale m q be equal to the Height of k above m q. (3) Now beckile the Idler Part is to be laid next to n, therefore divide m q in 0, fo that m 0 may be to 0 q as the Line b, to the Line a-, and then drawing the Line k 0, the Plan wi.l be divided as required. P R OBLEM XVII. Fig. X. To divide a Trapezia into two Parts (according to any poffible Proportion given) with a Line drawn through a given Point within the Trapezia. Suppose / k n p be a Trapezia given, and let q be a given Point, and 'tis required to divide the Trapezia into two Parts, in Proportion < ne to the other as the Line a is to the Line b, by a Right Line drawn through the given Point q. { 1) Conlider through which Sides the Line of Patition s q r 'mult pals, which in this Example are the Sides i k and n p, which continue to meet in 0. (i) Find the Bale of a Triangle equal to the given Trapezia, w hofe Per¬ pendicular may be equ.d to the Perpendicular from k upon 0 p, as the Bale v p, which divide in w lo that v w may be to wp, as the Line a , is to the Line b. (3) Through the Point q draw the Line g l, parallel to 0 p; and, taking the nearelt Diftance from the Point q to the Bafe 0 p , fay. As that ncareft Dif tance, is to the ncareft Diftance from the Point k. to tlx Bafe op-, fo is the halt ol 0 w, to 0 m, wh,ch et from 0 to m, and from m draw m l parallel to ok, cutting g l in l. (4) Take the Square of h q, Irom the Square ot q / (which in the tught-angled Triangle d e /, is d e, which Jet from m to r, and from tile Point > through the 1 oint q, draw the Line ol Partition r q s, which will divide the Trapezia into luch two Parts as required. V R 0- The Principles of Geometry. 423 PROBLE M XVIII. Kg. XI. T0 di vide a Plan according to any Proportion given, with a Line drawn from a 'Point qffigned in any of the Sides, and to lay the Part required towards a Place appointed. Suppose# iking be a Plan given, and m a Point afligncd in the Side In, from whence a Line is to be drawn, to divide the Plan into Parts in Proportion one to the other, as the Line a to the Line h, and to lay the greater Part next to h. (1) Confider to which Side the Line of Partition muff be drawn, which in this Example is h g, which increafe both Ways towards r and s. (2) Find s r, equal to the Bale of a Triangle, equal to the given Plan, whole Perpendi¬ cular is the nearcft Diftance from the Point m, on that Bale. (3) Divide that Bafe in p, as the Line a to the Line h ; and becaufe the greatelt part mull be laid towards h, therefore fetthe greater Segment of the Bafe from r top, and draw the Line m p, which willdivide the Plan as required. PROBLEM XIX. Fig. XII. To divide a Plan according to any Proportion a/fignedbetween two Sides, with a Parallel to a Line given, and to lay the Part required towards a place ap¬ pointed. S u r p o s e the Plan hg c d f n i be given to be divided into two parts, in proportion one to the other, as the Line 0, to the line p, with a Parallel to the Line ah, and to lay the lefier part next to n. (1) Confider through which Sides the Line of Partition mult pafs, which in this Example are the Sides in and df, which continue until they meet in *. (2I Find qx, the Bafe of a Triangle, equal to the given Plan, whole Perpendicular is equal to the near- eft Diftance from the Point d, to the Bafe in. (3) Divide that Bafe q x in m. fb that as m be to m q , as the Line p to the line 0; and fet the Idler Segment, from q to m, next to n (becaufe it is fo required). (4) From the Point d draw the Line d k, parallel to the given Line a b, to cut the Bafe i n in k. (y) Find a mean Proportion between x m, and x k, which is xl. ((,) From the Point l draw the Line L e parallel to k d, which will divide thegiven Plan into two parts with the Idler next to n, as required. PROBLE M XX. Fig. XIII. A Triangle as h i e being given with the Triangle om e adjoining to it, whofe Side e rn is juppojed to be continued out at pie a jure towards l, to find a Point in the Side h c, as g, from whence a Parallel drawn to h i, as g k, and another to 0 rn, as g n, injuch fort, that the Part k i h g cut off, may be equal to the 'Part gn 0 m taken in ; that is, that the Trapezia k g n e may be equal to the two Triangles h i e, and 0 m e, taken together. (1) F ind a mean Proportional between h e, and a Perpendicular on h e from the Angle i, as be, alio a mean Proportional between 0 e, and a Perpendicular on 0 e, from the Angle m, as a c. (2) Add the Squares of thei'e mean Pro¬ portionals into one Square, whole Side is b a\ divide the Side he in Power, as e 0, to its Perpendicular from m\ fo that, as the Perpendicular from m, upon his Bafe 0 e, is to 0 e, lo is the Idler Part of the Power of h e, to the Power ol he, being fo divided; which Idler Part in Power is the Line c d, whole Square being added to the Square of b c, their Sum is equal to a Square, whole Side is b d. (3) If b d give b a, what will h e (which is equal to b / )? Anfwei, lz\ which take and fet, from e tog, upon the Side h e. (4) From the Point g draw g k parallel to h i, and g n parallel to 0 m ; then will the Trape- The (Principles of Geometry. zia h ik g cut off, be equal to the Trapezia g n o m taken in ; and confequent- I v the Trapezia k gn e will be equal to the Triangle h t e, and the Triangle o m e, as required. P R O B L E M XXL Fig. XIV. There is a given Plan, as dh chi, and in the Side h n is a ’Point, a< c, from whence 'its required to draw a Line, as c f to cut off two Triangles b dc, and h g f, and to lake in the Triangle d eg equal to them bulb. So that the Trapezia c n f l may be equal to the given Plan d b c h l. (i) Reduce the given Plan into the Triangle a n /equal thereto; and from the given Point c draw the Line c f, foas to make the Triangle k f l equal to the Triangle ack, then will the Triangle d eg taken in, be equal to the two Triangles bed, and hgf left out, and confequently the Trapezia cjnl will be equal to the given Plan required. Plate CCCCLX. The Divifion oj Geometrical Figures , by Vincent W IMG, with 'Decimal Arithmetick lineally demoufirated. P R O B L E M I. Fig. I. To divide a Right-line (as c) given into two fuch parts, that Jhall have fucb Propa lion, as the Line b to the Line a. (i ) Make an Angle as f dg at pleafure; make dg equal to the given Line c, alio d e equal to the Line and c/to the Line/;, (r) \o\n f g, and from thePomt e draw the Line e h parallel to / g, cutting dg in h; then as d e is to ef, lo is d h to h g, which are the Parts divided as required. PROBLE M II. Fig. H. To divide a Right line (as d f) in Power , according to any Proportion given : Suppoje as the Line a is to the Line b. (l) Divide the given Line d /'in e, fo that the Idler Segment d e may be to the greater e /, as the Line a is to the Line b. (v) Bilecf d f and on it de- feribe the Semicircle d c f, and irora the Point e rate the Perpendicular ec; alfo draw the Lines dc, c f, which two Lines together are equal in Power to the given Line, and the Power of c f is in fuch Proportion to the Power of c d, as the Line b, to the Line a, which was required. PROBLEM III Fig. III. To divide any Triangle given (,use dj)according to any Proportion required (as the Line a, is to the Line b) by a Line drawn from d, an jingle affgned. Divide the Safe cj in e, fo that c e may be to e /, as the Line a , is to the Line b, and then drawing the Line de, the Triangle is divided as required. PROBLEM IV. Fig. IV From a givenTriangle as m h k, to cut off any part , by a Line drawn, from eit her o] its Angles. S u p p o s e the Area of the Tf angle be equal to 371 Poles three q-ths, and the Sided/, 41, and t is required to cut off 90 Foies with a Line proceed¬ ing from the Angle d. To p eilorm this, fay. It 37 % three 4ths, the Area or the The Principles of Geometry. 425 the whole 1 riangle, give 42 for its Bafe, how much of that Bafe, in the lame Parallel, will 90 require ? Anfwer, io T ;t; for as 372 three 4ths is to 41, fo is 90 to 10 Tit, which fet from k towards m, as to l, and draw the Line/a/; then will the Triangle hlk be equal to 90 Poles, as required. PROBLEM V. Fig. V. T0 divide a Triangle, according to any Troportion given, by a Line drawn parallel to any of the Sides affigned. Suppose a s f be a Triangle given, and 'tis required to cut off three j-ths by a Line parallel to a g. (1) On the Line zz/defcribe the Semicircle ahf , and divide a f into five equal parts, according to the greater Term. (2) On c, which is the third part from f, let fall the Perpendicular ch, and draw the Lines h f; make/# equal to fh, and from the Point# draw a Right-line pa¬ rallel to a g, as # z; then will the Triangle # if contain three yths of the Tri¬ angle a g /, as required. PROBLEM VI. Fig. IV. To divide a geometrical Square, (as *Agf) according to any Troportion af- pgried, by a Line drawn parallel to one of its tides. Suppose the Square contain 67 6 fquare Poles, and 'tis required to cut offioS by the Line be. (1) Find the Side of the given Square, which here is id, and then liiy. If 676 requires/ Length of id, what doth 208 require ? Anfwer 8- for, as 67b is to id, fo is 208 to 8; therefore fet 8 from a to #, and from d toe, and draw be ; then will the Parallelogram abdebe equal’to 208 Poles, as required. PROBLEM VII. Fig. V. From an irregular Tlan (as pmikn l q) to cut off any parts required. 'Tis required to cut off 420 fquare Feet by a Line drawn from the Angle / to cut the oppofite Side. (1) Meafure the Area of the Trapezia mlpq which contains 314 Feet; alfo meafure the Area of the Triangle i l m, whole Area 137, added to the former, is equal to 479, which being 39 Feet more than the part to be cut off therefore, from the Triangle 1 1 m, cut off 39 Feet, according to the 4th Problem hereof which is the Triangle i In, and the Remainder n m l, with the Trapezia m Ip q, is the parts cut off as re¬ quired. Note, This and the following Problem will be better underftood, when The Mensuration of Superficial Figures, &c. is learned, which will be comprifed in V ol. II. PROBLEM VIII. Fig. V. To divide a given irregular Tlan into any equal parts required. Suppose s 0 t v w be given to be divided into two equal parts, the Area of the Whole being 707 fquare Feet, whofe half is 372, 1 half. (1) Meafure the Triangle a s w, and its Area will be found to be 290, which taken from 072 1 half the Remainder is dz, 1 half. (2) Take 62, 1 half fi 0111 the Triangle 0 t iv, which is the Triangle 0 r w, which with the Triangle 0 s w is one half of the Whole, as required. By the fame Rule any geometrical Figure, be it ever fo irregular, may be divided into as many parts, and of any Quantity, at pleafure S O As The i Principles of Geometry. 426 As l have now fimlhed thefe geometrical Problems, which afloat a great Variety 0!' Practice and Pieaiure to fuch as delight in this Science, as well as of verv aeat Ufe in fetting out, and dividing of Lards, that are intricate, and vlnch very often happens in Practice, i ihall now proceed to demonllrale the Nature of Decimal Anthmetick. and alcerwaros the Geometrical Con.lruct.on ot the Alphabet, with which I fhall dole this hrlc \ online o. this molt extern live and laborious Undertaking. Decimal Arithmetics demon fir at ed. I. N U M ERATION of Decimals, This Kind of Arithmetic!, fuppofes the Integer to be always divided into ten equal parts, and fometimes each ot thole parts are ..u pmod to he lubdivid- „ into ten equal partsalfo, which thereby divide the nteger mto tea parts Suppofe the Line a l. Fig- VI be an Integer, ..ed mto .0 equal parts at the Points b, c, d, e,f,g,h,i,f the ... is < ei h, act w. tohs \ d three xoths, &c. ot the whole Line al\ but U each . . tl.c.c ic.n fonts is divided or luppofed to be divided intoiof as m n, which is one icth of m *, then m * is thereby divided into < 1 e hundred pa its, and m „ w hich is equal to one 10th of a l , doth now represent ten ot the ico pans, i"“fo — L f ; of”,hire * ... g to their Places, .. lot knifes 7 -i, the firlt and . . 1 parts-.*, the fir t, fe « tl ■ ; and : •,* hke manner all the thefe If each of thefe 100 parts are again lubdi- vide d into 10 equal ; ■ then the Integer will ho divided int , and mn , which before reprefented rjj, will now represent riL, al.o mo 1,,,, 4 we place -’ , 7:*; thus, 'tis evident, that c ; is but one 10th of the Whole, as al .j ^ one roth ot 1000 is too and one tenthof 100 is 10 ; anc there re, if the hundredth part of one Integer ,e c qual to another Integer divided into 1000 parts, then -it of the hdt is equal to - of the lad and each but one 10th ol the Integers. Now irom this tis nla'in” that the Addition of Cyphers doth not encreale the Quantityof the f,... -y : n • for - 1 ,1 t ’ 1 ; ! . are no more, than tS, -,i, as is evident, it from 1 ich Fraftiony u cut off towards the Right-hand the laft three Cyphers, when the Remainders to the Lett will be ri, rl, as ahndaid. As the De- nomi Decimal Fraftion is here Ihewnto bean Integer divided into ™ 1 P ar ts. it is never written 1 ice, but alway s ~Y d to fi as many Cyphers foil, v mg the l mt, as arc Figures and Cypt fi - - « r; mi is exprefibd by the F« re 1, witha Comma 1 before hus ,1 ; and wd thus ,01; lo likewife r. is expreffed thus ,a ; and thus ,16; alfo ttIS thus ,01 .6. Now, bv the , leading it appears, that the prefixing of Cyphers to the Nu merit, r'ofa Fraction doth dimimlh it ten times; for 1:1 the atorclaid Inac¬ tion of s, whicl I nified thus ,x, and figniEes < k tent 1, by having a Cvuher prefix'd i it, as thus ,01, doth now hgmiy but one hundredth .... , and if to it two Cyphers had been prefix'd, as thus ,00 1 , it won d then have bonified but one thoufandth part, becaufe (as I laid before) the Denominator nuilt be underlfood to contain as many Cyphers following the unit, as are Cyphers and Figures contained in the Numerator. II. ADDITION®/ Decimals. The Addition of Decimals is the fame, as of whole Numbers in Vulgar Anthmetick, carrying one for every ten, as is evident by the following Lx- ample. E X A M- The Trinciples of Geometry._427 EXAMPLE I. Fig. VIII. (ab equal to 1,4 ) vv hieh is equal to the Line ik. Add together the Lines J C * becaufe « * » equal to ab, m „ to b ) e J LC l UJi to 1,71 c > } m t0 e / ( and , / to g /a. ( g h equal to 1,9 ) and their Sum is 8,5- \ab\ C i )9 In Fig. VII. the Line£ >,/ Ac of Decimals, which is common in almolt every Author of Arith .net,cal Hafts, and of very little, or no Ufe m Menfuration, when known! Plates CCCCLXI. CCCCLXII. CCCCLXI IT CCCCLXIV. The \ r lC i Con fi ruat °l °i the Twenty-four Letters oj the Alphabet, by Mr. Robert West. FERHAPf it may be thought, that to teach the Horn-book at the Concluli on ot Geometry may be ablurd and ncedlcfs ; but if we confider the great Dif- thar V ’V VVol ^“ £ u are ^ t0 - t0 P ru P° rt 'on Letters of Infcriptions Me that Hand much above the Level of the Eve, it will appear to bTa uieful liiftiuctiun. The Methods hitherto pracihed have been to make them eithei at a \ enture, or by Trials ot Letters of divers Magnitudes firft m ,de and then placed up agaimt the Building, where the Infcnption, &c is requir¬ ed, which they have either mcreafed or decreafed in Magnitude,Is their rudi¬ ment directed them. An Inftance of which was done, for to find the Height and Proportion ol the Letters of his Majefty’s Name and Date of the Year a f‘ nft Ac Front,fpiece to the Meufe-Stables at Charing Crops , and (if 1 ‘ mif . take not Vi- y the Dl :, c « on *hc Board of Works, or at leaf! of their Sur- !, eyo ' ( 1 mr.rolk r, who, tis reafbnable to believe, knew not how to effeft th . 1 ™ : Ifay ’ 11 this ’ and fuch Blc lull an ccs be confdered of vmch l cotud enumerate many, to the eternal Shame of thole who have molt hem h l ' n R rt K 1 \ V Dircchon ol JVorks) it muft be allowed, that to teach them then Hom-book by geometrical Rules, how at once to make thofe Let te« proport,enable to any Height, will be doing them, and fuch part of the 1 ubuck, as may be demons to know their Conitruftions, a Service worthy of thanks ; to which 1 proceed. y I. Of the Letter A, Plate CCCCLXI. (i! Make a geometrical Square, of Diameter equal to the given Height of the Letter, as a d r A, and draw the Diameters ct and fg: draw cr¬ isis SC c z equa to c t, and from 2 draw 2 r perpendicular to r A; alfo from z Tim P a ;. a lcl t0 ’ A > thal 15 1 5 r the Thicknefs of the full Stroke, 13 y the I hicknefs 0l the fine Stroke of every Letter, and yr the Projeftion of each Grace to The Principles of Geomktrt. 4^9 to every Letter, whether right or oblique-angled. The Height of each Grace is equal to the fine Stroke, and which is lieceflary to be fo, when Letters are much elevated above the Lye; therefore draw the Line h i parallel to the Bale r A. Now to form the Letter draw cs ; make t v equal to s t and c v; draw i i w at the parallel Diftance of 13 r, alfo up parallel tocr, at the Di’ftance of the fine Stroke / , make the ciois Stroke 11 equal in f hicknels to the fine Stroke, and wholly below the Diameter fg ; and thus is the Letter form'd, the Giaces only excepted, which are thus perform’d. The Projection of tile Grace to every Letter, as aforefaid, is equal to the Line y r , thcrcloie make q h t q n, p 0, p ?n, 10 8, toy, y 1 , y 4, each c- qual toy r, and from thole feveral Points raife Perpendiculars, as h k, n k, ol, m /, 8 6 , 7 6 , and 4' 5, 1 3, interfcfting in the Points k , l , parallel to the Bale ; make c d and f g equal to f e, and draw / d, which terminates the Extreams of the Letter ; make i h and k l, the Depth of the Graces, equal to the Breadth of the full Stroke, and turn their Curves as before; draw gc cutting a bin m, through which Point draw n o parallel to the full Stroke ; the Thicknefs of the Tongue is c- (iual to the fine Stroke, placed equally above and below the Line a b ; the Graces are of the fame Dunenlions, as thole of the lop and Bottom of the Letter. VI. Of the Letter F. This Letter is the fame as Letter E, the Bottom excepted, which hath its Graces as before. VII. Of the Letter G, Plate CCCCLXI1. The upper part of this Letter, ab c de, is the fame as Letter C, and the Remainder is thus defended : The Diagonal / g , and the Perpendi¬ cular b b being drawn, make i k equal to the full Stroke; from l draw Im parallel to the Bafe, and no, the Thicknefs of the fine Stroke, paral¬ lel to the former ; then defenbe the Graces as in Letter B ; with the Radius lp , on l, defenbe the Arch p q, which compleats the Letter. VIII. Of the Letter H. The Diftance of the inward Lines a b of the two full Strokes are equal to half the Height of the Letter, as c d ; the crofs Bar equal to the fine Stroke, let equally above and below the middle Line d e, and the Giaces as in Let¬ ter B. IX. Of the Letter I. This Letter is onlv the full Stroke with its Graces, as the Letter B. X. Of the Letter K. Th e firft part is the fame as the preceding Letter I, whole Height is bileft- ed in c ■ make a d equal to a c, and draw the Line c d, and make ef parallel thereto’ at the Diftance of the fine Stroke; make bg equal to be, and draw the Line eg ; make h i parallel to eg, at the Diftance of the full Stroke ; fet off the Graces from k l and m n, and compleat their Curves, as in the oblique Graces in Letter A. XI. Of the Letter L. Make the full Stroke as the Letter I, and the Projedion of its Bottom equal to half its Height, with its Grace the fame as Letter E. XII. Of the Letter M. Make a geometrical Square, as abed, equal to the Height, draw the fine Stroke u e r/' on the Left, and the full Stroke g b h d on the Right ; hiked The Vrinciples of Geometry. 43 1 / h in i, and draw the Line mi, and make no parallel to mi , at theDiftance of the full Stroke ; draw li, and go parallel thereto, at the Diftance of the fmall Stroke, and making the Graces as the Letter B, the W hole is com- pleated. XIII. Of the Letter N, Plate CCCCLXIlI. Divide the Perpendicular a b into 4. equal parts; make if equal to b c, and draw fg parallel to a b; make albk and hgif each equal to' the fine Stroke, and draw the Lines l k and h 1 ; draw the Line m f, and parallel thei e- to the Line n 0, at the Diftance of the lull Stroke, and then, making the Graces as the Letter B, the Whole is done. XIV. Of the Letter O. To-form this Letter, make the firft part of the Letter C right and re. verfed. XV. Of the Letter P. Make the full Stroke with its Graces, as in Letter B ; bifed its Height in b, and draw b c parallel to its Bale ; fet the Breadth of the fine Stroke equally above and below the Line b c ; make a d equal to a e, and / /a equal f g, and draw the Perpendiculars he and d 1 ; bifed h c -in k, and draw the Ll "® k l parallel to the Bafe ; make k m equal to h k, and p n equal to p 0 on the Point m, with the Radius mk, defenbe the Arch q l: r, and on «, with the Radius np, deferibe the Arch sp t, which edmpleats the Whole, XVI. Of the Letter Q. First make the Letter O, as before direfted ; Infect ab me, and draw the Line cd\ bifeift the Arch dc in/, and draw f to the Center g ; on the Point d, with the Radius di, deferibe the Arch hk, and on the 1 oint c, with the Radius e k, deferibe the Arch k /; with the Radius b m on the Point h, defenbe the Arch m n, and on « deferibe the Arch h n interlea.ng mn m n on the Point n, with the Radius nh, deferibe the Arch b m ; on the 1 oint 0 (found before) with the Radius om, deferibe the Arch ml, which compleats the Whole. XVII. Of the Letter R. Make the Top and left Side as the Letter P; bifeft ab in c, anddraw cd parallel to the Bafe ; from the Points e g (found before) draw ef and g h per¬ pendicular to the Bafb, cutting c d in / and h ; on h, with the Radius g 1, de¬ feribe the Arch Ik, and on/, with the Radius em, defenbe the Aid no 5 make , p equal to' h k, and 7 c d equal to fo ; on the Pointy with the Radius po, deferibe the Arch oq, and on d, with the Radius k d, deferibe the Aich k q, which compleats the Whole- XVIII. Of the Letter S. Draw a geometrical Square, as abed, with its Diagonals d b and do alfo the Diameter e f; make g b equal to the fine Stroke, and h i equal to the full Stroke • make e k and If each equal to the fine Stroke ; bilect e h in m, and k i in n ; on m, with the Radius m e, deferibe the Arch p e h, andl on n, with the Radius n k, the Arch k q i ; on m, with the Radius m k, defenbe the Aich 43 2 The ‘Principles of Geometb y. kr; draw the Line r s from the Point r perpendicular to the Rife i i kt parallel to the Bale; draw the Line s m, and on v with the R , defcribe the Arch vj /, which finithes the upper part ’Biktt • /Lt f’, ■V i ° n *• " ith th ^ Radius 4 defcribe the Arch i % an y s nh , ' W ' yh, defcribe the Arch h i /; on .v, with the Radius x l, deli u c the Arch /’ US from the Point x draw the Line i , perpendicular to the Bale draw 1 then on the Point J-, with the Radius r 6, delcnbe th ■ Arch 6 • / compleats the Whole. x ami which XIX. Of the Letter T, Plate CCCCLX1V. Compleat a geometrical Square as ahrA- ,t »i,„ ta- fet the Breadth of the full Stroke equally on each Side in e f‘ , and g/aand it; make / m parallel tct a b. at the D Lr1 t,e Lln « W«, „« Top £ Letter Ji, SS&SSSZSS*? XX. Of the Conjunant Letter V. Tins Letter is no more, than the Letter A inverted, without the crofs Bar. XXI. Of the Vowel u. A 1 ake the full Stroke abed nnrl HitXrJr. • . , equal to a g, and draw the Line hi perpendicular to the^Pir’^H 5 mA * ah ralhd to h i, at the D,fence of the fine Strokeb^ L"' k/ V the Radius m c, defcribe the Arch r i ■ ri ’ i 1 o'” m ’ and 011 m ’ "Ml lei to the Bale; on *, with the Radius * l d°efcriteth ^'‘a ^ ° P paral - d l ,11 q, and draw q r perpendicular to theBafe • on r tti U the Tl V defcribe the Arch s v /, and then, complcating the Graces, the Whole 'is d«L XXII. Of the Letter W. Make the Confonant V, as Aha- hlfcA „ z. - , fonant V, whofe firft Line e f (hall interfeft the I l) mnke anothcr c °n- plcats the Whole. J eCt the Line nb m c, which com- XXIII. Of the Letter X. Jir&sozgtzs?. ittis ft: Di “ * f ebih; one, with the Radius e,6 deferiho th„ «. i L bfl j :lrcs ae *Z and both perpendicular to theBafe; draw Ip and „ o'whidb ire tit"' Tt-^ of the Stiokcs of the Letter- let the fnii i ’ , 1 ‘ c antral Lines ip, and draw the Lines Jr ’and x t S tl fineT f “o' 1 S,dc thc ^ the Line n o, and draw the Lines -ow and ,v ,, wlm'h whhthe ^ C pleated as ,n Letter A, compleats the Whole Glaces con> XXIV Of the Letter Y. Af A k r a geometrical Square, as abed, and draw the D .- f 7 on e, with the Radius e h, de'icribe the Arch hi?, hIP ' ; the Radius e t, defcribe the Arch okp- H rnv r i,Vr 111 L on e > VVlth in the Point m, and iq touching the Arch‘ the l Z / Zl The Arch ton/, at the Dillauce of the full Srw,L-^ , ““ :nC rs parallel ftan.ee of the fine Stroke - Ll L L r Z A. * lel t0 ^> at ' - the central Line.,/, T/v ' and 1 l " e ,Uli ‘ ^ 'W on each Side Graces as in Letter A, and thole at Bottom Letter ^R * ^ Upp£r XXV. The ‘Principles of Geometry. 433 XXV. Of the Letter Z, Plates CCCCLXV. CCCCLXVI. Complete a geometical Square, as a, b, c, d, and make the Top and Bot¬ tom Strokes equal to the line Stroke ; let the Fulnefs of the full Stroke from b l° e, and c toy, and draw the Lines e c, and bf\ then completing the Graces, as in Letter E, the Horn-book is completed as required- Plates CCCCLXV. CCCCLXVI. Of the Geometrical Conjlruclions of Hour-Figures for Clocks and Sun-dials, by Mr. Robert West The uppermoft Figure of this Plate contains three Examples of Hour- Circles for Clock-Dials : I he firft is the Manner of proportioning and divid* mg the Marginal Circle into twenty four Hours, with a Circle of fixty Mi¬ nutes on its Limb, as follows. 1 I. To proportion the Breadth of the Margins. (i) T he Diameter a n being given, divide the Radius a m into four equal Parts, of which the outermoft {a / ) is the Breadth of the Margin. (z) Sub- div'de a f into four equal Parts, of which the outermoft ( a b) is the Breadth of the Mniute Circle, (f) Divide df into two equal Parts in e, then is e f the Breadth of the Circle tor the Quarters of the Hours, and the remaining Part be is the Circle for the Capital Figures. The Thicknefs of the Divifio- nal Lines ot the Circles a b and e f are each equal to one 4th of e f to be fet oft from a towards b, and from b towards a ; alfo e towards t\ and from f to- wards e. (4) On the Point m , with the Radius m a, m b, m e, and m f deicribe the Circles a b, e f, with their Thicknefles, as aforeiaid. H. To divide the Hours and Minutes. (1) F rom the Point / divide the Circle / into twenty four equal Parts and from the Center m draw Right lines through each of the twenty four Fo nts continued through the Marginal Circle b e, which are the central Lines of each Figuie. (i) Subdivide each of the twenty four equal Parts in the Circle e into four equal Parts for the (Quarters of.each Hour, and make the Divifionary Strokes in Thicknefs equal to the circular Lines. (5) Divide the outer Circle a into twelve equal Parts, which are the Places for r, io i-r zo eGr Minutes, and which being each fubdtvided into y equal Parts, gives do Minutes in the whole ’ ° “ The fecond Example is the Manner of dividing a Circle of i» Hours with¬ out Minutes, as follows. I. To proportion the Breadth of the Marvin The Diameter f o being given, divide the Radius / m into three equal Parts, and / h the outer Part, will be the Breadth of the Margin : Divide / /j into eight equal 1 arts and gb, the innermoft, is the Breadth of the Circle for the Quarters and the Breadth of the circular Lines is one 4 th of g h, to.be fet oft from b towards g, from g towards b^ and from / towards g. T Q. II. To m The ‘principles o] Geometry. II. To divide the Hours. Divide the Circle f into twelve Parts, and draw Lines to the Center m, which are the central Lines of the Figures; and then lubdividing each Part into four equal Parts, completes all the Divilions required. The third Example is the Manner of dividing a Circle ol twelve Hours, with a Circle of 60 Minutes on its Limb, as follows. 1. To proportion the Breadth oj the Margins, <5cc. (i) The Diameter h p being given, bifeft the Radius h m iiW; make h i equal to one 4th of h l , and k l equal to x half h i ; then tile Remainder i k is the Margin for the Hour Figures, k l for the Quarters, and h 1 for the Minutes. (a) Set off the Breadth of each circular Line, and divide the Quarters and Minutes, as in the fuff Example. K. B The Diftance between the Extreams of each Figure, and the circular Lines of their Margins, in every Example, mult be equal to the Thicknefs of thole circular Lines. The Letters that exprefs the Elours, arc the Capitals I, V, and X, which are thus formed ; (1) The Letter I, hath its Breadth equal to one 8th of its Height, and the Projeftion of its Graces equal to one half of its Breadth, (i) To form the Letter V , divide its Height into eight equal Parts, and diaw the central Line c d, with the Line h'f through the Head ol the Figure at Right Angles to the central Line ; make cf and c h each equal to one one 4th of the Height, and from the Points h and /' draw Lines to the Point d ; draw g 1 parallel to h d at the Diftance of the full Stroke, which is one 8th of the Height; alfo draw i t parallel to d f, at the Diftance of one 4th of the full Stroke. The Projedtion of the Graces are the lame, as thole of Figure I, and are de- feribed as the oblique Graces in the Alphabet. (3) The Letter X, for thefe Ufes is thus formed, viz. Divide the Height into eight equal Parts, as be¬ fore, and draw the central Line a b ; make a l, b m, each equal to one 4th of the Height ; on the Points / m , with the Radius equal to the Breadth of the full Stroke, defct'ibe the Arches n and 0, and draw the Lines n m , and 1 0 ; Bifebt l n in p, and 0 m in q, and draw p q the central Line of the Stroke ; make a r and b s each equal to a p and b q , and draw the Line r s ; fet the Thicknefs of the fine Stroke on each Side the Line r s. Laftly, Set off, and deferibe the Graces as before. Af B. It muff be obferved, to place the Figure to each Hour, equally on each Side the central Line of the Hour ; as Fig. VI in Example If. is on each Side ot the Line f g m. V. S. To find the Height of Letters at any Height above the Fye, vide the Word Letter in the Index. 1 FINIS. THE CONTENTS. I. Arithmetical Lectures, Lect. "Lect. X. N Numeration, or the Manner IV. On Multiplication P. 47 L/ 0 f expreffing Numbers and V. On Dieijion by Quantities by CharaQertJUcks , and VI. On the Rule oj Troportion or to pronounce their Value Page to Golden Rule 87 II On Addition 17 VII. On Arithmetical Troportion or HI. On Subtraction 39 TrogreJJton 98 II. Geometrical Lectures. Lect. I. Geometrical Definitions, Plate 1. Pag. s 111 >113 >114 Pkob. i. Of a Point I. Of a Line 3. Of a Right-line 4. Of a Circular Line y. Of the Circumference of a Circle 6. Of a Curve Line 7. Of Curved Lines 8. Of Right-lin’d Angle 9. Of a Spherical and a ( mixt Angle ) 10. Of the Increale and - * Quantity of An¬ gles fny II. Of Degrees and Mi- | notes J 1%. Of the Meafure of An-1 gles 13. Of an Acute Angle 14. Of a Right Angle 5-117 iy. Of an Obtufe Angle 1 6. Of the Complement of an Angle 17. Of a Perpendicular 18. OfaCircleanditsCenter 19. Of the Diameter of a Circle 10. Of theSemidiameteror 5-119 Radius 11. Of a Semicircle li. Of the Sector of a Circle _ Prop. *1- 14. Pag. Of the Segment of a ) „ Circie .>,19 Of the Chord Line of ^ ' an Arch Of the Equilateral Triangle Of an Ifoceles Triangle Of aScalenum Triangle 19. Ot a Right angled Tri¬ angle Of an Amblygonium I Triangle Of an Oxygonium Tri- I angle J Of the Legs, Bafe, Per¬ pendicular and Hypothenufe of a Right angled f Triangle 31. Of the geoniet. Square j Of the Diagonals of a Square Of the Diameter and Center of a Square Of a Pentagon, Hexa¬ gon, Septa^on, M Odtagon, Nona- I gon. Decagon, | Undecagon, Duo- ! decagon 3 6 . Of a Superficies if- 1 < 5 . 17 18. 3 °. 3 !- 3 i- 33 ' 34 - if- >110 37 - Of The CONTENTS. Pkob. 37. Of a Trapezia, Parallelogram, Oblong or long Square, P.hom ? bus, Rhomboides, and of the Altitude of Figures V : Lecture 11 . Geometrical 'Problems , Plate a, and 3. 1. To make a Scale of equal Parts reprefent:ngluches,Fect,Yards.^r. 123 1. To draw Right-lines of determinate Lengths, to reprefent any] Number of Inches required, and to mealure Right-lines by the Scale of Inches ' ; 3. To make a Scale of Feet and Inches, and to draw Right-lines j reprefenting any Length required 4,. To deferibe an Equilateral Triangle 5-. To divide a Right-lined Angle into two equal Parts 6 . To bil’ecT a Right line 7. To erect a Perpendicular inor near theMiddle of aLine 8. To erect a Perpendicular at the End of a Right-line by di¬ vers Methods 9. To erect a Perpendicular upon a Convex Line 10. To ereCtaPerpendicularupon a Concave Line,in or near ifsMiddlcC ■> 1 11. To erect a Perpendicular on the End of a Concave Line 3 it. To erect a Perpendicular on the Angle of an equilat.or IfocelesTriang. s 13. To let fall a Perpendicular upon a Right line,on or near its Middlef 131 14. To let fall a Perpendicular upon a Right-line,near one of its Lnds ' 35-. To let fall a Perpendicular on a Concave Line a \ 6 . To let tail a Perpendicular on a Convex Line ML 17. To make an Angle, equal to a given Angle 3 18. To make an Angle equal to a lolid Angle given ^ ^'fTodraw Right-lines parallel to Right-lines ii. Tomake a Right-line equal to a Right-line given v _ • To divide a Right-line into any Number of equal Parts To divide a Right-line into any Number of unequal Parts in the! fame Proportion as another Line is divided j ' 3 “ To compleat a Circle, having a Part given id. To find the Center of a Circle 17. To divide an Arch-Line into two equal Parts 18. Todivide the Circumference of a Circle into 31 equal Parts To continue a Right-line given To draw a Right-line between two Points with a Ruler, whofeiij9 Length is equal but to half their Dtftance 3 31. To deferibe a Segment of a Circle, to contain an Angle given ) 31. A Circle being given to cut from it a Segment, that fhall con-( 140 tain an Angle given ( 3 3. To draw a Chord Line in a Circle, equal to a given Line ) 34. To cut from a Line, any Part required 143 Lect. III. On the Generation of regular geometrical Figures , Plate 4. 141 Lect. IV. On the geometrical Conjh uchon of fnperficial Figures, PI. y, C. Prob. ^ 3 - 14. if- T9- 30. } fe Xi: 12 8 To deferibe an Ilbceles Triangle To deferibe a Scalenum Triangle ? To deferibe a geometrical Square /I43 To deferibe a Parallelogram ( To deferibe a Rhombus 3 To deferibe a Rhomboides 144 To The CONTENT S. in PrOB. 7. 8 . To defcribe a Trapezium 7 To defcribe a geometrical Square having the Difference be-Sp. 144 tween its Side, and the Diagonal given J To defcribe Ovals and Ellipfis by divers Methods to >14 T , L 14-b, (1+8 18. 19. 2 .0. 'Ll. ’LL. il- 14 - ry. t.6. lH. z8. To defcribe a regular Curve within an irregular Trapezium ( To trace the Circumference of a Circle through Points, without/ having any recourfe to its Center V To trace the Circumfererence of an Ellipfis in like manner lo find the Side of equilateral Triangle, geometrical Square, Pen¬ tagon, Hexagon, Septagon, Octagon, Nonagon, and De- cagon, that may be infcribed within the Circumference of a given Circle. To make a regular Pentagon To make a regular Hexagon 7 To make a regular Septagon To make a regular Octagon To make a regular Decagon To make a regular Undecagon To make a regular Duodecagoil To defcribe all manner of Polygons \ To defcribe any regular-Polygon by the Scale of Chords -7 iy , A Right-line given, to find the Semidiameter of a Circle, ca-Ciy/ pable to inferibe any given Polygon, whofe Sides fliall bcCitT each equal to the given Line J To defcribe a Spiral Line >49 iyo Ur 1 iyx Lect. V. On the inferring , and dr cum [cubing geometrical Figures, Plates C Prob. 1. To inferibe a Circle in any Triangle 1. To inferibe a Circle within a geometrical Square 3. To inferibe a Circle within any regular Polygon 4- To inferibe a geometrical Square, within any Triangle / 5 - To infciibe an equilateral Triangle within a geometrical Square 6 . To inferibe an equilateral Triangle within a Pentagon 7. To inferibe a regular Pentagon within an equilateral Triangle 8. To inferibe a geometrical Square within a Pentagon 9. To inferibe a Penta-decagon (of iy Sides) within a Circle 10. To circumfcribe a Circle about a geometrical Square 11. To circumfcribe a geometrical Square about a Circle iz. To circumfcribe a Pentagon about a Circle 13. To circumfcribe a Circle about a Pentagon 14. To circumfcribe any regular Polygon, about another Polygon of the fame fort 17. To circumfcribe a Pentagon about an equilateral Triangle 16. To circumfcribe a Pentagon about a geometrical Square 17. To circumfcribe a geometrical Square about anyScalenum or Ifo -1 celes Triangle fid 1 18. To circumlcribe a geometrical Square about an equilateral Tri¬ angle 177 U-8 H 9 >160 b Le ct. IV The CONTENTS. Lect. VI. Methods for taking and drawinggeometrical Plans of Lands, and Elevations of Buildings, ike. Plates 7, 8, 9. P. jiiz Prob, 1. 1. x. 3 - 4. 7 - IX. I 3 ' 14. 17. 1 6. 17. 18. 19. xo. To take the Quantity of an Angle in a Building, by a two-foots Rule, <&-c. and to delineate the lame on Paper f"* To delineate the Out-line of the Sides of an irregular, acceffible? Building ( l68 To take the Plan of the Out-line of any irregular, inacceffible? Building, without meafuring anv Side or Angle thereof '-168 03 14. X 7 - z6. 17. x8. 19. 3 0 . To make the Plan of an irregular curved Line To make a Plan of a Piece of Land, bounded by an irregular? curved Line, by one Station To make the Plan of a Piece of Land, bounded by divers un- ? equal Sides, by two Stations $ WT To make the Plan of an irregular piece of Land, bounded by map nv Sides, by means of three Stations S I ~ 5 ’ To make the Plan ot a piece of Land, whofe Angles are inacceflible 177 To make the Plan of an irregular piece of Land, in the midft of) which is a Lake of Water ( To make the Plan of an irregular piece of Water in an irregular ( 1 ^ Field \ To make the Plan of an irregular piece of Land by going about ) it without-iide, as in a Lane, &c. C ^ To make the Plan of a Serpentine River f lu0 To make the Plan of any Town, City, &c. Plate 10. ) To take the Plan of a Vault or Cellar with groin’d Arches, Plates? lgl 11, 11. S To take a Plan of the Ground floor of a Dwelling-houfe i8x To divide a given Height into the principal parts of an Order,p viz. the Pedeftal, Column and Entablature, geometrical-^ 188 ly, Plates 13, 14. 3 The Height of the Tufcan Pedeftal being given, to divide it? Q into Bale, Die and Cornice geometrically l Ii5 9 The Height of the Die to the Tn/can Pedeftal being given, to? find its Diameter or Breadth geometrically £ 1 9 ° The Height of the Bale to the Tufcan Pedeftal being given, to"! divide it into its Fillet and Plinth geometrically The Height ofthe Bale to the Tufcan Pedeftal bein find its projection geometrically To divide the Height of the Cornice of the Tufcan Pedeftal 191 into its Cima Reverfa and Fillet The Height of the Cornice of the Tufcan Pedeftal being giv¬ en, to find the Projedtions of its Members, and to de- lcribe them geometrically A Height being given, to proportion and compleat a Tufcan Pedeftal thereto geometrically The Height ofthe t ufcan Column being given, to divide it in-{^ r to its Bale, Shaft and Capital geometrically \ The Height of the Tnjcan Bale being given, to divide it into’! its Plinth, Torus and Cindlure geometrically To find the Projection of the Members to the Bafe of the Tuf \ raw Column geometrically f 1 93 To deferibe the Hollow above the Cincture (called Mpophyes) I geometrically To proportion, divide and deferibe theAftragal to the Top of ) the Tufcan Column geometrically ( ^To dimimlh the Shaft of a Column geometrically, Plate 17 r given, to I 31. To The CONTENTS. v Prob.ji. To divide the Height of the Tufcan Capital into its Mem¬ bers geometrically ) 32. To divide the Height of the Tufcan Entablature into its Av-) chitrave, Freeze and Cornice geometrically ( 33. To divide the Tufcan Architrave into its Tenia and Fafcia geoT metrically ) 34. To divide the Tufcan Cornice into its Cima, Corona and Ovo-- lo geometrically 35. Todefcribe the Profile of the Tufcan Cornice geometrically J Plate 17. and Plate A, to proportion an Order to any given Height,? 202to and make the Module thereto J 207 The Tufcan Order ol^Vilruvius and Talladio 207 Plate B to follow Plate A after Plate 17. The Manner of making Pro-x traffing-fcales for delineating the Five Orders of Columns in Architec-/ ture, according to the proportions of any given Mailer, by Mr. R.We/tfzo 6 Vide the Word Trolrahing-fcale under the Letter P, in the didtiona-C rial Index following, where its Ufe is explained. ) Plate C, to follow Plate B, after Plate 17. To defcribe the Cima refta,/ Cima inverfa, Ovolo and Cavetto, by divers Methods 5 * Remarks on the Tufcan Order of Vitruvius 208 Plate 18. Other Examples of the Tufcan Order by si. Talladio, taken from the Arena's of Verona and Tola y ~ 7 Remarks on the Tufcan Order of si. Talladio ? Plate 19. The Tufcan Order of V. Scamozzi , with Remarks thereon y Plate ic. The Tufcan Order by Barozzio of Vignola, with Remarks thereon 212 Plate F, following Plate ao. Tufcan Profiles of Talladio, Scamozzi, and) Barozzio ; alio of the Column of Trajan according to Mr. Evelyn S213 Plates G, H, I, following PI. zo. The Tufcan Order by J. Man clerc j Remarks thereon x Plate 11. The firlt Example of the Tufcan Order by S. le Clerc, withCn4 Remarks thereon y Plate zz. Two other Examples of the Tufcan Order by S. le Clerc, with? Remarks thereon Cry Plate 13. The Tufcan Order of Claude Terault y Plate 24. The Tufcan Order of the Antients, as pratlifed by Mr. Gibbs? ^ ^ and others, with Improvements £ Plate 27. The Tufcan Order by S. Serlio 1 Plates z 6 , 27. The Tufcan Order by Mr .Stone, in the Portico of St. >218 Tauls, Covent-garden Plate 28. The Tufcan Order by Inigo Jones at Tork Stairs Plate 19. The Tufcan Order by Sir Chrijlopher Wren, in the Frontif- piece of St. Antholms Church, in Walling flreet, London Plate 30. Tufcan Intercolumnations, Arches and Imports, according tc the Antients ^ V Plate 31. Tufcan Portico’s to Temples, by Vitruvius Izzi Plate 31. Tufcan Intercolumnations, Arches and Imports, by A. Talladio) Plate 33. Tufcan Intercolumnations by Vincent Scamozzi 1 Plates 34, 37, 3 6. Tufcan Intercolumnations, Arches and Imports, by Barozzio of Vignola, and 5 . le Clerc Plates 37, 38, 39, 40, 41. Tufcan Triumphal Gates, Arcades, lnterco- | 11 lumnations to Coionades, Niches, Doors and Windows, by Seka/iian\ Serlio and Julius Romano J Plate 41. The Column ol Trajan at Rome, by S. Serlio Plates 42, 43. Tufcan intercolumnations by S. le Clerc j> Plate 44. Thelntercolumnation of the Portico of St. Taul's, Covent -gar deny x * Plate 45. Dive, s Compoiitions of Block Cornices examined, gfyc. Plate 46. Two Ruftick Doors 22 6 zzt Plate VI The CONTENTS. 227, 2.28 Plates 47, 48. The geometrical Conftruftion of the Dorick Pedeftal ?P and Sit lick Bale, by Carlo Cefare Ofio ’c Plate 49. The geometrical Conltruction of the Attick Bale, bv £, ]] sllberti, and of the Cincture and Aitragal to the Dorick Shalt bvii ( ark Cefare Ofio ’ Plate 5-0. The geometrical Conftruftion of the Dorick Capital and EntaV Mature, by C. C. OJio Plate R, following lhate 70. Divers Methods for fluting Pilafters and Coo iumns. Vide the Word Fluting under Letter F, where the Ufe of this/, „ Plate is explained Plates y 1, 72. Two Profiles of the Dorick Order in the Theatre of Mar- / ceHits at Rome rig 7 Plates j-j, yc, 77, y 6 . The Dorick Order of Vitrmiits, with Interco-V lumirations for Portico’s to Temples, &c. £2.38 Plates 77, y8. Divers Dorick Kxampl.es from the Anticnts 1 Plates 79, Ho. The Dorick Order by Andrea 'Palladio >2 3 9 Plates 61, 62, 63. Che Dorick Order by V. Scamozzi, with Intcrcolum-s nations for Arcades and Portico's < l’lates 64, 67. The-Dorick Order by Barozzio of Vignola ) " r Plates 66 , 6 7, 68. Dorick Intercolumnations for Colonades and Arcades? with or without Pedeftals, by Barozzio and ,V. le Clerc ’£141 Plate 69. Two Dorick Gates rufticated by Barozzio , Plates 70, 71, -2. Ihe Dorick Frontifpiece to the principal Entrance in-/ to the palace of Famefe at Rome, a Triumphal Arch, and Dorick GatefH’- at Caparolit, by Barozzio \ Plates 73, 74, 77, 76. The Dorick Order of Sebaflian Serlio , with,041 Intercolumnations for portico's, Frontifpieces, Colonades, and Ar-C,,' cades " JH4 Plates 77, 78, 79. A Dorick Temple by Bramante ; alfo a Gate, an Al- Triumphal Arch, and an Arcade, by S. Serlio "l 144 2 44 > 147 Hr tar-piece Plates 80, 81, 81, 85, 84. The Dorick Order of S le Clerc, with its va lions Entablatures, Sofito's, Impolts, and Intercolumnations Plate 8 7. The Dorick Order by Claude Perault Plates S7, 88 The Dorick Orders of Leom Bapti/la Viola , Leoui Baps tijla Alberti,'Philip de Lonne and John Bullant, Daniel Barbara and/ Cataneo, according to Mr. Evelyn Plate K, following Plate 88. The Dorick pedeftal at large, by JuliatA 1+7 Mau-clerc " ' j Plate P, following Plate K (not Plate O) after Plate 88. Dorick Enta- -l r ^147 ^48 ,bv^i 48 249 Mature at large, by J. Matt clerc Plate 89. The Dorick Order entire, by J. Mau-clerc Plate 90. The Ttjcau Order of J. Man clerc Plate 91, allb Plates E and O. A focond Example of the Doric k Order J. Mau-clerc ’ Plate P, following Plate O, after Plate 91. The Dorick Order bv A. Pal'- ladio, V. Scamozzi, and Barozzio, by Mr. Evelyn Plates 92, 93. The Dorick Order by I. Jones, in the Roval Chapel at Somerjet-houle Plate 94. The Dorick Order of Sir ChriJlopherlVren, at St. Mary le Bow, in Cheapjide, London ’ \ Plates 97, 9 6. The great pillar, at Monument, on Fifb-ftreet-biU, Lon-\ don, built by Sir C. Wren I Plate 97. A Dorick Frontifpiece by Sir C. Wren Plates 98, 99, too, 101, 102, 105. Gibbs, with its imeicoiumnations. The Dorick Order by Impolts, Arcades, &c. Mr. ^2 Hr, Th 174 > 2-77 Plates The CONTENTS. ■vn i6z, t -61 Plates 104, ioy. The Ionick Pedeftal, Bafc of the Column,and Volute; P.ij-y, of the Capital geometrically deferibed i ty 6 Plate Z, following Plate toy. Ionick Volutes of various Kinds 257 Plates 106, 107. The Profile of the Ionick Volute and Entablature; geo^ado, metrically deferibed by C. C. Ofio S261 Plates 108, 109, no, 111, in. Ionick Orders, taken from the The¬ atre of’Marcellas, and Temple ol Manly Fortune, in Rome , by Te ter Tiburtine and Mr. Evelyn Plate 113. The Ionick Order in the Bath of pioclefian at Rome. s , Plates 114., 2iy. Two Examples of the Ionick Order by Vitruvius y Plates 116, 117,118, 119. The Ionick Volute, alio Ionick Portico's and ) Temples, by Vitruvius ( 6 Plate 120, X2i. The Ionick Order by A. 'Palladio, -with its Intercolum-f nations. Arcades, Imports, &c. ) Plates in, 113, 114, ny. The Ionick Order by V. Scamozzi, with its Intercolumnations, Arcades, Imports, Doors, Temples, &c. Plates 116, 117- The Ionick Order by M. J. Barozzio of Vignola ) Plate L K, following Plate 117. Ionick Profiles of 'Palladio, Scamozzi and Barozzio, according to Mr. Evelyn Plates 118, 119, 140. Ionick A rcades with and without Pedeftals, alfoy z 6 ‘ Intercolumnations for Colonades, by M. J. Barozzio Plates 131, 131, 133- The Ionick Order bv S. Serlio Plates 134, 135, 136. An Ionick Portico, Colonade, and three Frontif pieces by S. Serlio Plate 1 37. The Ionick on the Dorick Order, as in the Theatre of Mar-^ cel/us at Rome Plates 138, 159, 140, 141, 141, 143. The Ionick Order by S. le Clcrc, with his various Entablatures, Imports, Arcades, &c. Plates 144, 14;'. The Ionick Order by Cataneo, Daniel Barbaro, Viola, L. B. Alberti, P. de Lor me, and J. Bullant, according to Mr. E- velyn Plate 146. The Ionick Order by Claude Perault Plates 147 to 1 56 inclufive. The Ionick Order at large, with its) 171, 173 Enrichments, &c. by Julian Mau-clerc 1 174 Plates 15-7, 178. The Ionick Order by I. Jones in the Royal Chapel at Whitehall Plates 179, 160. The Ionick Order by I. Jones in the Royal Chapel at Somer fet-houfe Plates 161, i 6 z. The Ionick Order by Sir Chriflopher Wren, againft the Church of St. Magnet, at the Foot of London-bridge Plate 1 (it, to 170 inclufive. The Ionick Order of the Ancients, with its Imports, Intercolumnations, &c. as prabtifed by Mr. Gibbs, and the Manner of delcribing the Scotch Volute Plate 171. Ionick Venetian Windows Plate 172- An Ionick Dentil Cornice, fupported by Trufles, for Doors, Windows, &c. according to Mr. Gibbs and others Plate 17; to 177 inclufive. The Corinthian Order geometrically vj.8i, to deferibed by Carlo Cejare Ofio $ *84 Plate 188. Corinthian Profiles, taken from the Temple of Jerttfalem,-. and the Portico of the Rotunda at Rome, according to Mr. Evelyn C Plate 179. An Altar in the Rotunda at Rome 5 Plates 180, 182. The Corinthian Bafc, Capital, and Entablature of the ) Portico to the Rotunda ( Plate 183. Two Corinthian Profiles, taken from the Frontifpiece of thef Bath of Z lioclefian and of Nero at Rome, according to Mr. Evelyn ) Plate 184 to 187 inclufive. Three Examples of the Corinthian Order byl.287, Vitruvius, with its Intercolumnations JxBB c Plate 274 286 • V viii The CONTENT S. Plate 188 to 191 inclufive. The Temple of Jupiter, a Corinthian Temple, and Rotunda by Vitruvius Plates 191, 193. The Corinthian Order by Andrea 'Palladio, with its? Q Arcades, Imports, and Intercolumnations ( Plate 19+ to 198 inclufive. The Corinthian Order by V. Scamozzi, with? its Intercolumnations, Arcades, Imports, Portico's for Temples, &c. £ Plate 199 to 104 inclufive. The Corinthian Order by M. "j. Barozzio,; 29 >P. i 83 190 S 2-91 9 * of Vignola, with its Intercolumnations, Arcades, Imports, &e Plates ioy, 106. The Corinthian Order by Seba/lian Serlio Plates 107 to 113 inclufive. Triumphal Arches of the Romans, by Se- bajlian Serlio Plates 114, ny. The Corinthian Order by 'Phlladio, Scamozzi, Baroz zio, Serlio, and the Rev. Daniel Barbara, according to Mr. Evelyn Plates ii 6 , 117. The Corinthian Order by Viola, Alberti, Cataneo, de\ Lorme, and Bullant £ 1 9 5 Plate 118 to 114 inclufive. The Corinthian Order by S. lcClerc, with its 9 various Entablatures, Sofitos, Arcades, Imports, Intercolumnations,/ 1 ^’ &c. alfo Enrichments for Cablings to fluted Columns, and for Cnna's,? 1 ^’ Ovolo’s, Cavetto's, &c. Plate 117. Corinthian Frontifpieces by Seba/lian le Clerc ? Plate ii( 5 . The Corinthian Order by Claude Perault S 1 9 I' Plate 117 to 135-, Divers Examples of the Corinthian Order fineli 719(5, 197, enriched, by Julian Mau-clerc f Plates 136, 137. The Corinthian Order, by Inigo Jones, in the Front of? , Somerfet-houfe next the Thames ^- 9 ° Plates 138, 139. The Corinthian Order of Sir Chrijlopher Wren, in the? portico of St. Pauls Cathedral in London 5 a 99 Plate 140 to 147 inclufive. The Corinthian Order entire, and in parts, at ) large, with its Intercolumnations, Arcades, Imports, &c. alfo the Io-( nick and Corinthian Orders on the Dorick, according to the Ancients/^ 00 and as practifed by Mr. Gibbs, and other modern Architects ^ Plate 148 to iyo inclufive. The Compolite Order of the Ancients?3oi, 301, g@ometrically deferibed, by Carlo Cejare Ofio s 303 Plate !)'i. The Compofice Order entire, by J. Mau-clerc, not Vitruvius,! as milhikenlv infected by the Engraver .1 3°3 Plate iyi to 1 ST inclufive. The Compolite Order in the Arch of TitusZ at Rome, alfo in the Gallic ol Lions at Verona, by Mr. Evelyn S 3 °+ Plates iy( 5 , 177. The Compolite Order of Andrea 'Palladio, with its? ts? Intercolumnations, Arcades, Imports, &c. Plate iyS to 160 inclufive. The Compolite Order by Vincent Scamozzi} ;cy, with its Intercolumnations,Arcades, Imports, portico's for Temples, &C.5306 Plate ib 1 to 164 inclufive. The Compolite Order of M. J. Barozzio of) Vignola, with its Sofito, Intercolumnations, Arcades, Imports, &c.(3o<5, alfo a Comport'd Entablature with Rollick Quoins, and a Compolite(307 Door in a rullicated Wall ^ Plate 16y. A Door by M. Angelo in Capiaeglio -■ Plates 1 66 , 167. A Compolite Frontifpiece near St. George Belabro, built/ in the Time of Lucius Septimus, by S. Serlio \ 507 Plate i (58 to 170. The Triumphal Arch at the Gate Dei Leom in J^eroK na, by Seba/lian Serlio ) Plates 171, i _ i. The Triumphal Arch of L. S. Severus, by S. Serlio 7 Plates 273, 174- The Triumphal Arch at Beuevento, by the fame Hand 5 Plate 177. The Gallery at Belvedere by Bramante 9 Plates 17b, 177. The Triumphal Arch of Conflantme Plates 178, 179, The Compofite Order by zt. Palladio, V. Scamoi M. J. Barozzio, and S. Serlio, according to Mr. Evelyn ■■Zl,\ >309 Plate The CONTENTS. IX Plate 280 to 183 inclulive; alfo Plate D, following Plate 183. The Com pofite Order of Sebajiian le Clerc, with various Entablatures, Sofito’s,| Arcades, Impofts, Intercolumnations, &c. , Plate 184 to 288 inclulive. The Compofite Order of Julian Mau-clerc\ with various Capitals and Entablatures finely enriched J Plate 289. The Compolite Order by Claude ‘Perault 7 Plate A A, following Plate 289. The Compolite Order by I. Jones, ink the Royal Chapel at Whitehall y Plate B B, following Plate A A. The Compofite Order of Sir C. Wren, inr St. Swithiris Church in Cannon-Jlreet, London £ Plate 290 to 294 inclulive. The Compofite Order of the Ancients, as) praetifed by Mr. Gibbs and other modern Architects, with its Intcrco-C lumnations. Arcades, Imports, &c. y Plate 295- to 301 inclulive. Compolite Doors, Exotick Pedeftals, Corin-'\ thian Modillion deferibed at large. Entablatures for Doors, Windows/ and Niches, Block Cornices, Mouldings for panned:, piCturc-frames,/ &c. from the Ancients, as praftiied by Mr. Gibbs and other modern \ Architects l Plate 302 to 305- inclulive. The Spani[h Order by S. le Clerc with its I Sofito, Intercolumnations, &c. the Corinthian on the Spanijo, theftiy, Spamjh on the Compolite, and the Corinthian on the Spanijb and(318 Compolite y Plates 30b, 307. The French Order by Sebajiian le Clerc 3x8 Plate 308. The GV-ote/ync Order for Entrances intoGrotto's,Hermitages, &c. 319 Pag. 312 313 . 314 3 IT, 3 id, 317 321 * 32+ 32 b Plates 309, 310. The Engh/h Order geometrically deferibed Plates 311, 312. Fractional Architedture by Mr. Edward Hoppus Plates 313, 314. Compofite Bales, and Capitals oi the Ancients, by Ba-~) rozzio and Serlio / Plate 315- to 317 inclulive. Compofite Capitals of very fine Inventions,? by John Berain y Plate 318, and Plate S, following Plate 318. The Orders of the r Perfians\ and Cariatides, by Mr. Evelyn, S. le Clerc, and J. Gordon I Plate T to follow Plate S. The Manner of deferibing wreathed Columns? Plates V, T, and Plate W following PI. T. Divers Deligns of Obelifques,? by Sebajiian Serlio, compared with Deligns of modern Architects S Plate X W following Plate W. The Manner of building piiaiters of? Stone again!! Brick Walls, by S. Serlio S Plate 319 to 322 inclulive. Various Kinds of Balullrades, Balconies, and their Trades, according to S. le Clerc and Mr. Gibbs Plates 323, 324. Great Varieties of the ancient Fret Ornaments, Vitru-i Scrolls, Interlacings, Eggs and Darts, by Serlio, Evelyn and Gtbbs\ Plates 327, 326. Two Frontifpieces for Gates by Michael Angelo Plate 327. An Iron Gate after the French Manner, with Tufcan Piers ( Plate 328 to 330 inclulive. Three Examples oflron Gates for Dorick Piers) Plate 331 to 334 inclufive. Four Examples of Iron Gates for lonick Piers? Plates 337, 33d. Two Examples oflron Gates for Corinthian Piers 1 Plates 337, 338. Two Examples of Iron Gates for Compofite Piers ) Plate 339. A Door with Compofite Piiaiters, by Vitruvius C Plate 340. The Vitruvian Window by S. Serlio y Plate 341. Rufticated piers for Gates by 1 . Jones, and E. of Burlington A Plates 342, 343. Windows and Niches of the Ancients, as praetifed by/ Mr. Gibbs, &c. Plate 344. The ancient Manner of proportioning and placing Window between piiaiters and Columns Plates 347, 34b. Two Tufcan Frontifpieces by A. Bojfe 1 Plates 347, 348. Two ‘Dorick Frontifpieces by A. Bojfe f Plates 340, 370. Two lonick Frontifpieces 7 , r , . ? 33 + Plates 371,372. Two Corinthian Frontifpieces^ ^ e ame ^- ut hor ^ ’ 328 ! 328, 329 33c 331 4 ^ 13 Plates X The CONTENTS. Plates 373, 374. A Compofite Frontifpiece, alfo the Manner of inferring Columns in Walls, bv st. BoJJe ( Plate 3 77, The Manner of finding the Skew-backs, and dividing all the, various Kinds of ftraight and ftheme Arches, both regular and' rampant Plate 3rd. The Manner ofdefcribing all the Varieties of regular, femi- circular, femielliptical, and Gothick Arches in Brick-work, on Windows of the firlt Magnitude Plates 35-7, 35-8. The Manner of rufticating femicircular, elliptical, and' Gothick Arches Plate 35-9. The geometrical Conftriuftion of femicircular, elliptical, and Gothick rampant Arches in Brick-work Plate 360 to 364 incluiive. llufticated Windows in all their Varieties Plates 367, 36 6. Windows according to S. leClerc and Mr. Gibbs Plate 367. Venetian Windows according to modern Tafte ^ Plates 368, 369, Plate A following Plate 369, and Plate 370. The geo-^ metrical Conftruftions of femicircular, femielliptical, and fchcme / headed Windows in circular or elliptical Walls ; alio to find the Tern plets of their Courfes, and Figures of their Sofito's ; alfb the Forma tion of circular and elliptical Niches Plate 371- The Abutments of Arches to Doors, Windows, and Niches dc monftrated Plate 371. The Manner of framing naked Floors The Manner o! framing Timber partitions The antient Manner of framing the Timber Fronts of Build ! ? 7 337 III 38 , piec- 3!^i3@0*saB* A Didionarial INDEX O F T H E 'Principal Matters contained in the following W O R K, explain¬ ing all the Terms of Art, in Theory and Practice, as ufed by Artizans and Workmen herein. A B A U S E D by the Ancients numerically, fig- 7 nified 500 ; with a Dafh thus, a, 5000. ABACUS, what \ the upper part of a Capital, as E, Fig. 47, and M N Q_, tfif. Fig. 46, Plate 17 ; alio vide p. 187, 210. ABBREUVOIRS, the Joints in Stone-work, as eg, di, f k, fg, &Cc. Plate 42. ABUTMENTS of Arches, vide p. 352, PI. 371. ACANTHUS-Leaf, the firft kind of Leaves ufed to adorn the Corinthian Capitals, vide p. 287, Fig. E, Plate 185. ACCESSIBLE, that Objeft that may be come to for any Purpofe required. ACRE of Land, its Length 40 Poles. -its Breadth 4 Poles. -its Quantity 160 Poles. ACRE'S Length in a Mile, League, Degree, or Circumference of the Earth, vide Table 1, p. 16. ACROTERES, little Pedeftals on Pediments, as thofe marked 45, Fig. 3, Plate 33. ACROTERIA, Pinnacles ranged on a Tower, with a Baluftrade, or Battlements between them. ACUTE Angle, what, vide Dcf. 13, p. 117 - ADDITION, what, p. 17. _geometrically demonftrated, p. 410, PI. 45 J- --.the Kinds, p. 17. --how performed, p. 17. -how proved, p. 29. -of Integers, p. iS. --- of Money, p. 19,—2.2. -by a new Method, p. 23. .- of Feet and Inches 7 , _of Yards, Feet and Inches $ -- of Fathoms, Yds, Feet and Inch.p --of Rods and Feet >p‘ 3 2 * -of Chains and Links, 3 -of Tons, Hundreds, Quarters and Pounds, p. 33- _of Timber, Bricks and Lime, p. 34. -of Sand, p. 3 5. —-of Land, p. 35. -of Flooring, p. 36. -of Gilding, 7 ---— of Painting, S --offolid Yards, p. 37. -of Time, p. 38. ADJACENT-Angle, vide Angles. A D ADJUST, to fettle, or Bate an Account of Dimcnfions taken, or to conclude a Difference. ADMEASUREMENT, Dimenfions taken of Lands, Buildings, &c. AGGREGATE, in Arithmetick the total Sum of divers Numbers added together, p. 99. ALCOVE, a Recefs within a Chamber for a Bed of State, which fhould be afeended to by three Steps, and divided off from the Chamber by a magnificent Arch, or Colonade. ALIQUOT-parts, luch as are contained in a given Quantity any Number of times without a Rerhainder, as 3 is contained in 12, 4 times ; 4 in 20, 5 times, &c. not the 3's in 13, becaufe there 1 remains, Sfir. ALPHABET, how deferibed geometrically, p. 426. ALTARS, from Altus , high, a Place facred for divine Worfhip in Churches, which arc ge¬ nerally enriched with ALTAR-Pieccs, Compofitions of Architec¬ ture, Painting, ffcfc. as that by Serlio, p. 244, Plate 89. ALTERNATE-Angles, what , p. 123. ALT 1 METRIA, the Art of meafuring the Heights of Buildings, ££)f. ALTITUDE, Height , as the Altitude of a Figure, &C. is the Height, or neareft DiftancC from its Bafe to its Top, p. 123. AMBIT of a geometrical Figure, is the Sum of all the Circumference ; lb the Ambit of a Circle is all its Circumference ; of a Triangle, all its three Sides ; and of a Square, all its four Sides, AMBLIGONIUM-Triangle, vide Def. 30. p. 120. AMPHIPROSTYLE, a Portico of a Temple, confifling of four Columns, as Fig. B, Plate 119. ANALOGY, the Relation which divers Things, as Numbers, bear to each other; that is, as 5 is to 12, lb 10 is to 24, Sfc. ANCHORS, or rather Darts, an Ornament introduced between Eggs, carved in an Ovolo, as Fig. K, Plate 324. ANCONES, the fquare Returns at the Angle of a Door or Window, as thofe in Fig. A, B, C, D, Plate 366. ANCIENT Orders, firft us’d, were the Dorick, lonick, and Corinthian, p. 201. ANGLE, A Dictionarial Index. A N ANGLE, what, Def. 8. p. 11". - Tlain, Def. 8. ') - Spherical , 1 c s P- 11 4* — Mix,, j Drf j - S - Adjacent , or Contiguous , fuch as have one Leg common to both Angles, as the Angle a b e is contiguous, or adjacent to the Angle b c a, and the Leg a c common to both Angles. -- Acute, * - Right, J»Def. 13, 14, 15. p. 117. - Obtufe, S - Oblique, thofe that are not right-angled - Oppojite, thofe that arc made by the In- terlettion of two Right-lines, as by the Lines ef and ab, Fig. 3,Plate 2, where the Angle eh b is oppolite to the Angle ah f t lo likcwil'e the Angle a b e isop- pofite to the Angle b h f, &Cc. -how exprels’d by three Letters, p. 118. -its Quantity, how taken by a two-foot Rule, and delineated, p. 163, 166. -how meafured, p. 116. -Right-lin'd, how divided, p.126. -their Complements, what, p. 117. -to make equal to an Angle given, p. 113. -to make equal to a folid Angle given, 134. -Internal and External, what, p. 166. -Solid is made by the Meeting of three or more Angles in a Point -Alternate, thofe which are refpe&ivcly e- qual to one another. ANGULAR Point, that Poin where the two Sides, or Legs of an Angle meet, ast the Point a, Fig. S, Plate 1. ANNULET, what, p. 237, Plates 31,32. ANTZE, or ANTES, p. 036. ANTICK Compofition, a Confufion of Men, Birds, Bealls, Flowers, Fruits, Fillies, Z$c. re- prelented by Painting or Sculpture. ANIICUM, Latin , a Porch before a Door. ANTIPAGMENT, Latin, the Jaumbs, or Architrave to a Door, Window, or Chimney, ei¬ ther plain or carved. ANTIQUE Statues, Bafts, Medals, &c. fuch that were made by the ancient Creeks and Romans. APERTURES, or APERTIONS, from the Latin Aperio , an Opening, Arc. Doors, Win¬ dows and Chimneys. APEX, the uppermofl: Point of a Cone, An¬ gle, &c. APIARY, a Place wherein Bees arc kept. APOPHYGE, what, and how deferibed, p. 207. AREOSTYLE, what, p. 23$. Plates 33, 34. ARCADE, an Arch, as Pig. 3, Pi. 30, p. 221. ARCH, I.aim, Arcus, a Bow. ARCHES, the kinds, arc /freight, as A B C, FLte 3.55 i o r / chenie , as D E F, Plate 333- or femicircular , as A, Plate 336 • or / emtcircular , as B and C, Piate 336 ; or GothicI:, as D, E, F, G, H, I, all which may be made rampant, as A, B, C, Plate 363. Arches, how divided, J -their Skew-backs, A F 35 - --how rufticated, p.338. -how made rampant, p. 33(7 Arch-line , how divided, p. 138. ARCH-BOUT ANT, an arched Buttrefs, as thole againft Hen. VII’s Chappel at IVefiminfler. ARCHITECTONICK, of, or belonging to a Building. A R ARCHITECTUS Ingenio , a Perfon well skill¬ ed in Arithmetic!-:, Geometry, and all other Arts by which Buildings in general are performed. ARCHI PECTUS Sumptnarius , a Builder well Bored with Money, ARCHITECT, a Perfon skilful in the The¬ ory and Pradice of Buildings in general. ARCHITECTURE, the Art of Building ARCHITRAVE, Aide p. 210. ARCHIVAULT, from the French Archivolte the Quoin, or Face of an arched Vault. AREA, the fuperficial Content of any geome¬ trical Figure ■ or ’tisthe Number ol lquare Inch¬ es, Feet; x$c. contained on the Surlace of any given Figure, ARITHMETICS the Art of Numbers, which herein is performed vulgarly, decimally, duodecimally. J ART, of Ars, Latin, that which is performed by help ol Arithmetick, Geometry, Grammar , k, l in PI. ^BEVEL-angled, that which is not right-an¬ gled, therefore every acute and obtule Angle is a Bevel Angle. BINDING-JOISTS, vide p. 3 °J- To BISECT a Right-line, vide p. 126. BLOCK-CORNICE, what, page 226, Plates 45 BLOCK-RUSTICKS, what, page 214. BODY, a Magnitude, that is contain d under three Dimenfions, viz. Length, Breadth, and Depth, or Thicknefs. BOND-TIMBERS, what, p. 353 - BOULTIN, a Moulding, the lame as Ovolo. C A ^ | ' H E Letter C numerically fignifies ioo. 1 CABLINGS, what, how deferibed and enriched, p. add, 2514. CAMBER-BEAMS, Beams cut with an Ob- tufe-angle in their Middle to help prevent their fagging, as Fig. B, plate 375. CAMERATED, Vaulted, arched, deled, $$c. CANALICUL/E, the angular Hollows cut in the Dorick Triglyph, as in Fig. 8, Plate 53, 54, which is a Set!ion of a Triglyph, where B A afe the CanalicuU , and D C the SemicdnalicuU. CANT, the Side of a regular Polygon, as of a Pentagon, Hexagon, &c. CANTALIVER, a carved Modillion fuftain- ing a Cornice, as Fig. 12, Plate 263. CANTALIVER-Cornice, a Cornice wherein Cantalivers arc uled, as in Plate 263. CAPITAL, what, p. 210.. _of various Compofitions, p. 3 26. CARACOL, a Stair-cafe of a lpiral Form, as in Plate 448. CARIATYDES Order , inftituted by the an¬ cient Greets , in Memory of their having fubdued the rebellious Cariatydes , a People of Caria , re¬ prefenting their captive Wives placed in the Read of Columns, as a Symbol of their Obedience, Ser¬ vitude and Slavery. In the lame Manner came the Terjian Order, by the Berfuxns being van- quifhed by the Lacedemonians at ‘Plattfa, p. J CASCADE, a Water-fall, or Cataraft ma'c by Art. CARDINAL POINTS, of the Horizon, p. 138. CARTOOSES, CartOUZes , or Cartouches , a kind of Bracket, or Trills for the fupport of Cornicesover Doors, Windows, or Windovv-ftools, as O P, plate 365. CATARACT, a great Fall of Water made by Nature. CATENARIA, the Curve Line, into which a Chain forms itfelf, when it hangs freely between two Points of Sulpenfion. CATHETA, a Perpendicular let fall from the Abacus of the lonick Capital, palling through A DlCT IONA RIAL INDEX. C A the Center of the Eye of the Volute, as A C, Plate 13S, which Sebafliau le Clerc calls the Axis. CATHETUS, the Perpendicular of a right- angled plain Triangle, as r p, Fig. 14, Plate j. CAVUS, what, p. 209. CAVETTO, what, p. 1S7, 209. -how deicribed, p. 208. .-how enriched, p. 294. CAUKING, Cogging, or Cocking of Beams, &c. down on Plates, the letting of a Dove-tail, cut in the under Face of the End of a Beam, into a Dove-tail concave made in a Plate, as Fig. S T, Plate 375, whereby they cannot draw afunder. CAU.LICOLT, the Volutes of the Corinthian Capital, as U, Fig. 1, Plate 201. CAULICOLES, lmall Volutes of the Compofiie Capital, finifhing with a Role, as reprelented at large in Plate 281. CIELING-JOISTS, p. 355. CIELINGS, coved, p.381. -their Ornaments, ibid. CIELING-PIECES, p. 380. CELERITY, the lame as Velocity,or Swiftnefs. CENTER of a Circle, what, p. 119. -how found, 137. Center of Gravity, a Point, upon which if a Body was l'ufpended, all its parts would be in ML- quilibrio. Center of an EHipfis, a Point where the two Diameters interleft, asg, Fig. ip, Plate 5. Center of Motion of a Body, a Point, about which a Body being fattened may, or doth move, as the Center of a Pulley, or the Middle of a Ba¬ lance is the Center of its Motion. Center of a Sphere, or Globe , a Point, from which all Line, drawn to its Surface are equal. Center of a Square or ^Parallelogram , a Point where its Diagonals interfeft, as /, Fig. 14, PI. 1. CENTR AL-Line of a Column, a Right-line patting directly through the Midi! of the Column, as the Line d 28,Fig. 140, Plate 15, from whence the Projection of the Members is accounted, by Mr. Evelyn. CENTER, for turning Arches of Brick or Stone, how made, p. 577, 579. CENTER, of a Column, or round Tower, ffk. how to find without-iide. ‘Practice, Pig A,‘Plate A, following Plate 369. (1) Apply the limit Edge of a Board, whole length is known, thereto, as g n, and meafure the Diftances, g h and n l. (2) On Paper, with a Scale of equal parts, draw a Right-line reprefent- ing the lengtli of the Board, and at each End let off Perpendiculars equal to the Off-lets gh and n l ; then, by Prob. p. 137, find the Center of a Circle, that (hail pal’s through the Points £/, k , /, as the Point e. CELLAR, how pl.mu’J, p. iSr. CHANNEL, of the lonick Capital, is that part of the Capital that is next above the Ovolo, as 3, Fig. 1, Plate 1 26. CHAPITER, or Chapter , from the French, Cbapiteau, the Crown or Capital of a Pillar or Column, of which there are two Kinds, viz. thole with Mouldings, which have no Ornaments, as the Tnfcan and Dorick Capitals • and thole with fculptur’d Ornaments, as the lonick , Corinthian and Compofite , ol which, in the tollowing Work, is a very great Variety. CHAPLET, vide BagueVe. CHAPTRELS, Carved bnpofs , as of the lonick , C H Corinthian and Compofite Orders, as Fig. V. ££1. Plate 126. CHIMNEY-PIECES, various Defigns, p. 382. CHORD-LINE, what, p. up. CHAIN, its Length, vide Table I. p. 16. Chain fquare, vide Table II. p. 16. Chain's Length in a Rood, Furlong, or A- cre’s length, vide Table I. p. 16. - -in a Mile, •-in a League, -in a Degree, -in the Circumference of the Earth. CIMA RECTA, what, p. 1S7, 400. --how deicribed, p. 207. CIMA INVERSA, or Reverja, what, p. 1S7. 20 5 >- -howdeferibed, p.207. CIMACIO, Italian, the lame as Cymativ.v:. CINCTURE, what, p. 206. CYPHER, its Ufe, p. 11. CIPPUS, a little Column, or rather a Cylin¬ der, creftcd on a Column in Memory of lome- thing remarkable, as the Cippus or Cylinder A, P'ig. 1, plates 95, 96, on B, the monumental Co¬ lumn of London , in Memory of the Conflagra¬ tion, &c. CIRB-ROOF, what, p. 367. CIRCLE, what, p.112. -irom whence originally taken, p. 2co. -how generated, s -how completed, a part being given,' p. 113. -its Center, how found, i -its Circumference, how traced, p. 1481, CIRCLE INSCRIBED lan Equilateral, p. 157. within Va geometrical Square, /p. 113. )a regular Polygon, j CIRCLE CIRCUMSCRIBED about a Pentagon, ^ . about a geometrical Square, S °‘ CIRCULAR LINE, what, p. 113. CIRCUMFERENCE of a Circle, what, p. 113. ■-how divided into Degrees, p. 113. -of the Earth, what, p. 16. CIRCUMVOLUTION, in Architecture, a Eurning-about, as the turning, or delcribing of a Volurc. Vide Volute. CIRCUMSCRIBING FIGURES, what, p. 137 - CIRCUS, a fpacious Circular 'Theatre for ex¬ hibiting of Spcftacles to the Populace j Baths were all’o called Cinufes. Vide Baths. CITY, how plann’d, p. 1S0. COFFERS, Iquare Concaves in the Sofito of the Corinthian Order between the Modillions, which are generally enriched with Rofes of vari¬ ous kinds, as reprelented in the Sofito, Plate 155. COLLAR I NO, Italian, the Aftragal at the top of the Shaft of a Column, (net the Freeze of the Capital next above it in the 'Tnfcan and Do¬ rick Orders, as many underftand it to be.) COLLAR-BEAM, a fmall Beam framed into, or near the midft cl a pair ot principal Rafters, to (Lengthen them, (and lomctimes to help fup- port the 1 Purlins) as thole marked 2, in PI. 374. COLONADE, a Range, or Ranges of Co¬ lumns placed before, on the Tides, or entirely a- bout, or within a Building, at an afligned Diflance, as in the Plan of tie Temple of Jupiter, Plate 118. Note, When a ■ ’^nade cannot be feen at one View, ’tis called a 'Polyfyle Colonade , as the Colo- A Diction irial Index. c o Colonade aforefaid. COLOSSUS, any Statue or Column of an e- normous lize, as the famous Statue of Rhodes , de¬ dicate! to the Sun, which was 70 Cubits high, and coft about 44,000 /. fterling . It was placed at the entrance of the Harbour, the right Foot on one fide of the Land, and the left on the other • the Height was lb great, that the tailed Ships could fail between the Legs • the Magnitude of the lit¬ tle Finger was fuch, that few Men could encom- pals it with both their Arms. It was thrown down by an Earth-quake, and the Brais of which it was made, loaded 900 Camels ■ this wonderful Work, was made by Chares , who completed it in 1 2 Years. COLUMNS, what, p. 187. -their Proportions, ibid. -how diminifhed, p. 194, a to. -how canted, p. 2.38. -how rufticated, p. 220. ■ -how fluted, p. 235, ■ -how wreathed, p. 327. The Kinds are five, vis. Tuff an, Doric!:, Ionic k , Corinthian and Compofite, which are differently proportioned according to the ievcral Mailers contained in this Work. 9 COLUMNS PARTICULAR are fuch that are diftinguifhed according to their various Situa¬ tions and Ufes, viz. 1. Angular , a Column inferted into the An¬ gle of a Building, as thole in the Angles of the Tower of St. Bride’ s in Fleet-flreet. 2. Doubled, that is, where the Shafts of two Columns penetrate each other, with about a 3d of their Diameters, as is done (tho’ not to be com¬ mended ) by Sir Chriflopher Wren, within the Church of St. Bride aforelaid. 3. Coupled , as the extream 'Columns in PI. 93. 4. Carolttick , a Column enriched with Foliages, or Branches twilled fpirally around the Shaft. .5. ' Triumphal , a Column creeled in Honour of a Hero, as that of the late Duke of Marlborough at Blenheim. d. Chronological , fuch as bear an hillorical Ac¬ count of Fadls, digelled according to the Order of Time. 7. Hiflorical , one whofe Shaft is adorned with Bajfo-relievo’s in a fpiral Manner, as the Trajan Column at Rome, vide plates 42, 112. 5. t Colojfal , one of an enormous fize, too large to be employed in any Ordinance of Architecture, as the Monumental Column of London. 9. Hermetick, a fort of Pilafter made in man¬ ner of a Terminus , with the Bud, or Head of a Man for its Capital, as reprelented on the Right- hand of plate S, following plate 318. 10. Rolygonous , the fame as a canted Column, W'hofe Bale is an OCtagon, &c. 11. Elliptical, fuch whofe Bafe of the Shaft is an Ellipfis, which is to be uled, where the Thicknefs in Depth of a circular Column would be too thick, and where a Column of Diameter equal to the given Thicknefs would be too fhort and weak. 12. Funeral , a Column crown’d with an Urn, wherein is luppofed to be the Alhes of the De- ceafed, with its Shaft enriched with Tears and Flames , Symbols of Sorrow and Mortality. 13. Inferted , fuch as are attached to a Wall, C O a third or fourth of their Diameter, as reprefent- cd in Plate 354. 14. Infulate, one that Hands free on all fides, detached from all other Bodies, as the Monumen¬ tal Column on FiJh-Jheet-hill , London. 15. Grouped , fuch as Hand by 3, 4, &c. on one Pedeftal. i( 5 . Sepulchral , one ereCled on a Tomb or Se¬ pulchre, with an Inlcription on its Bale. 17. Statuary , a Column which lupports a Sta¬ tue, as that ot Trajan in Plates 95, p6. COMMON EXCESS, what, page op. COMMOM DIVISOR, a Number which ex¬ actly divides any two (as the Numerator and De¬ nominator of a FraClion) without leaving a Re¬ mainder. COMMENSURABLE <% antities, fuch as will meafure one another precifely, as 4 will mea- fure 8, 12, 16, 20, without a Remainder; fo likewile 3 will equally divided, 9, 12, 1 c, 18, ®c. COMPLEMENT of an Angle , what, p. 171. *- of Degrees in a Quadrant or Semicircle, p. 118. COMPOUN D FIGURES, what, p. 122. COMPOSITE ORDER, -by Carlo Cejarc OJio, p.301, 302, 303. -by Julian Mau-clerc , p. 303, 304. -in the Arch of Titus 1 -in the Caftleof Lions j at R ° m ’ P ' 3 ° + -by Ralladio, } - Scamozzi , J> 3 ° 9 - —Baro -. 'to, p. 306, 309. - Serlio, 7 - LedercM^ 9 ’ 3 ‘ 0 ' - J. Mau-clerc , p. 311. -Sir Chriflopher Wren , 7 -Mr. John Gilts. l p ' 313 ’ CoNCAMER ATE, to arch, or cove the Ciel- ing of a Chamber. CONCATENATE, to chain the Out-walls of a Tower, £yr. together. CONCAVE, the inward Superficies of a hol¬ low Veffel. CONCAVITY, the Ipace contained by the fupcrficies of a Concave. CONCENTRICK Circles, tlicfe that have one and the lame Center, as the Circles a, 0 , z, Fig. p, pi. 1. Concentric/: Ovals are thofe that are deferibed on the fame Centers, as in Fig. 59, plate 5. Conccntrick Rolygons, are fuch whofe fides are parallel to each other, and have but one Center common to both, as Fig. po, plate 6 . CONCURRING Figures , fuch that are in e- very relpcdt equal to one another. CONF, a geometrical Solid, generated by the Revolution of a right-angled plain Triangle, whole Perpendicular, or Cathetus , is made the Axis of its Motion. — its Superficies, how covered with Lead, &c. page 34(5. CONJUGATE Diameter, of an Oval, or El¬ lipfis, the fhortefb, as e f. Fig. do, plate 5. CONNOISSEUR, French , a Perlon (curious and) skilful in Arts. CONSECTARY, a confequent Truth arifing from a Demonflration ; the fame as Confequencc, or Corollary. CONSEQUENT, the laft Term in any Set of Pro- A Diction a rial Index. c o Proportionals, fuppoll- the following ; As j is to j 2 lb is 10 to 04 i Now here the lalt Term 04 is ’the Consequent , or Confequetice ol multiplying (the Means) ra by 10, and dividing their Pro¬ duct by 5, which gives 04 for the Quotient, and is a fourth Proportional. CONSOLE, the Key-ftone of an Arch, (from the French , Confolider , to dole up) as B, Plate 108, whole Profile at large is B, pi. 209, alio in the bottom of Plate 307. CONSTRUCTION, of geometrical figures, is the Formation, or making of them. CONTACT, ‘Point of ContaCf, the Point where a Tangent, or Right-line touches the Arch of a Circle, as in Fig. 95, Plate 6, the Eight- line ah touches the Circle obc in b, its Point of Contafr. CONTENT, the A ' gure, or Solidity of any lolid Body, which is ac¬ counted in fquare or cubical hicbelect, laias, 6Cc. viz. If a l'quare Superficies have each fide equal to 3 Feet, its Content is 9 fquare Feet; and if a Cube have each fide equal to 3 Feet, its iohd Content is 27 cubical Feet, Igc. CONTINUED Proportion is, where the 3d Term is to the 4 th, as the ill is to the 2d ; and confequently the firft and third Terms are of one Denomination, and the lecond and fourth of ano¬ ther. Continued Proportion is the Golden pule Dirctt, vide p. 87, and its Charafteriftick, uled by Geometricians, is as thus, — CONTOUR, the Out-line of any Member or Ornament. CONVEX, the outer Superficies of a Sphere or Globe, as Concave is the inward Superficies. CONVEXITY, the outward Curvature of a (I be, f oid, or any < her curved Body. CORBELS, coved TrulTes made for the Sup¬ port of Statues, Bufto’s, &c. The Holes left in the Walls of ancient Churches, ffjtr. for Statues to ftand in, are by l'ome called Corbels , but (I think) very improperly. CORINTHIAN ORDER, -by Carlo Cefare Ojio , p. 2.81, to 2S4. --in the Temple of Jerusalem, \ -in an Altar of the Rotunda, S ^ --in the Poitico of the Rotunda , > -in the Bath of Dioclejian , ( P- -in the Frontifpiece of Piero , P V ^ - - - Barozzio , p. 290, 292. *- Serlio , p. 29 r, 292. - Bar bar o, p. 292. -- Viola , - Alberti , - Cataneo , - de Lome ^ v - Bullant, \ - Le CUrc , J - Perault , p- 295. -- Julian Mau-clerc , p. z$ 6 , nyj, ayS. - Inigo Jones , p. 29S. -Sir Cbrifiopber Wren , p. 299. -Mr. John Giibs , ibid. CORINTHIAN CAPITAL, from whence taken, and by whom firft made, p. 287. CORNICE, what, p. 209. -of a Pedeftal, what, p. 1 87. --of an Entablature, what, ibid. --for Doors and Windows, pi. 298, 299. C O CORONA, what, p. 187, 209. COROLLARY, an ufcful Conlcquencc drawn from a geometrical Propcfition. CORRIDOR, a circular or fquare Colonade, or a Gallery about a Building, which leads to all iti lcveral Appartments. CROSETTE, the lame as Ancones, which is alfo called Protbyi sdcs. CROSS-MULTIPLICATION, p. 60. CUBE, a lolid Body generated by the Motion of a geometrical Square, erected perpendicularly on a Plane, and moved regularly along a Right¬ line, whole Length is equal to the fide of the Square: Its Surface is terminated by fix equal Squares, and its Figure is a Dye truly made. CUBE-ROOT is the fide of a Cube, whole Solidity is equal to a Quantity given. When a Number reprelenting a Quantity is given to find its Cube-Root, that Number or Quantity niuft be conceived to Le a Cube, containing as many cubical Inches, Feet, &c. as the Numbers ex- prels; and to find the Root of fuch Number is no more, than to find the Length of a Side of fuch a Cube, which is called Extracting its Root. Vide Extractions of Roots. To CUB£ a Number is to multiply it into itfelf, and the Pro :uc: for ] ample, to cube the Number 3 ; firft, 3 multipli¬ ed into 3, the Product is 9 \ and then 9 multipli¬ ed into 3 again, the Product is 27, which is the Number cubed. CUBIT, anciently was a Length from the El¬ bow to the Finger’s end, but now fixed at iS Inches. CUPOLA, a hemifpherical concave Roof, or Covering for a magnificent Building, generally called a 7 , which ant i ntly a Imitted the Light at the Top, or Zenith-point only, without any Lanthern, as in Plate 191, and which is yet to be iecn in that incomparable Piece of the Pan¬ theon at Rome ; bur. indeed I muft own, as this Climate is more fubjcct to uniettlc-d and wet Wea¬ ther, ’tis better to admit the Light otherwiie, and finifh their Vertex, cither with a Husk, Pine¬ apple, Ball, Vale, &c. as in Plate 189, or with a Lanthern and Ball, Efr. as in Plates 1S8, 391. Cupola’s , how framed, p. 369. -how ornamented, p. 381. CURVATURE, the Bending of a Line, or Surface. CURVES, crooked or arched Lines, of which there arc many kinds, but thole which relate to r Pur pole are, Spiral, 1 Serpentine, S 1 14 - CURVILINI AT. Figures, are tbofe whofc Spaces are bounded by a curved Line, as-a Circle, Ellipfis, tit- . CYLINDER, a lolid Body, whole Bale is a Circle, and is g nerat d b] the Revolution of a right-angled Pal d it one of.its Sides, of by the Motion of Circle, crefted at Right, angles to the Surface of a Plane, moved regularly thereon in a Right-line, ft na| iven Point, to a Dihance equal to the required Length of the Cy- linder. Ellipfis. CYMATIUM, or Cymaife , what, p. 1 17, 209. -how A Dictionarial Index* D A -how enriched, p. 2514. D A D I N Latin, {lands for 500. 7 DADO, what, p. an. DARTS, p. 325). DECAGON, p. 122* DECASTYLK, a 1 Portico , that hath ten Columns, as the Temple of "Jupiter , PI. 18S. DECIMAL Aritbmetick geometrically dc- monftratcd, p. 426. DECORATIONS, the Ornaments, or En¬ richments that adorn a Building. DEFINITION, a full and plain Defcription of a Point, a Line, a Superficies, f$c. DEGREE, what, p. 16, 115. Degrees in the Circumference of the Earth, p. 16. DELINEATE, to draw, or reprcfent by Lines any Plan, Elevation, DEMONSTRATE, to prove, to {hew, to make plain, by evident Proofs, the Truth of a Propofition, &c. DEMONSTRATION, a clear and convinc¬ ing Proof. DEMY King-poft, what, p. 353. DENOMINATOR, ofaFra&ion,what,p. 71. DENTICI.ES, or Dentil-cornice , what, p. 280. DEPENCILLED, dejigned (or drawn) with a Pencil. DESCRIBE, to draw, or reprcfent by Lines a geometrical Figure, &c. DESCRIBENT, that Line, which by its Motion generates a Superficies } or that Superfici¬ es, which by its Motion generates a folid Body. Vide Generation of geometrical figures. DESIGN, the Dittribution and Proportion of The Parts, into which a Plan, or Elevation for a Building is divided. DIADEMA, the Tenia of the Doric/: Order, as the Member I, Fig. I, Plate 60. DIAGONAL of a geometrical Square, what, p. 122. DIAGRAM, a geometrical Figure, confiding of divers Lines drawn for its Demonftration, &c. DIAMETER, of a Circle, what, p. up. -how found, p. 205. --of a Square, what, p. 112. DIAMETRICALLY, that which pafleth diredly through the Center of a Circle, &c. from fide to fide. DIAMOND-Pavcment, vide Plates 450, SCc. DIAPHANOUS, that which is tranl'parent, as Glais, DIASTYLE, an Intercolumnation of three, and fometimes four Diameters, p. 238. DIATHESIS, the fame as DifpoJitioH. DIGLYPH, an imperfect Triglyph, or rather a Confole with two Channels only, as thofe in the F'reeze of the Entablature, Plate 263. DILAPIDATION, a Building, or Buildings in Ruin, for want of having been timely repair'd. DIMINISHING of Columns, how, vide p. 104, 210. " DIMINUTION of Columns, from whence taken, p. 195. -—the Quantity of Diminution in each ally is as follows; gener The Shaft of the < D I Tufcart ' ^ , . 1 7-.' • 2. / Column is \ 7 / )w (“-)!( \ Corinthian C . rt aClts . \ f 7 - Compoftte \ Aftragay, ) of its Di- ’ameter. DIPTERE, a Temple, ^r. environed with two Ranges of Columns, as Plate 188. Line oj DIRECTION, the Line made by the Motion of a natural Body, in its Afccnt, De- l'ccnt, according to the Power impreffed upon ir. DIRIGENT, the Line or Path, along which the Defcribent Line, or Superficies is carried, in the Genefis or Generation of a Superficies, or of a Solid. DISPOSITION, or Dijlribut'ton of Parts, the wclhdiipofing of the fcvcral Parts and Members, into which a Plan, or Elevation of a Building is divided according to their proper Places, Ufes, &c. DISTANCE, to determine, p. 174. DITRIGLYPH, the fame as Metope, the Space between two Triglyphs in the Dorick Freeze. DIVIDEND, a Number given to be divided, p. 66. DIVISION, what, p. 65. -Bow performed, p. 66. -Single, p. 67. - - —Compound, p. 68 to 71. -how proved, p. 73. -how contracted, p. 72. Dtvifion of geometrical Figures, p. 414, to 417- DIVISOR, what, p. 66 . DODECAGNO, or Duodecagon , a regular Polygon of 12 equal fides, and as many equal Angles, vide p. 122, 142, 152. DODECAHEDRON, one of the regular Pla~ tonick Bodies, bounded by 12 equal and equilate¬ ral Pentagons. DOME, vide Cupola. -how framed, p. 363). -of St. 'Paul's, London, ibid. -its Plan, Section, and Manner of fram¬ ing. P- 370- DOOR of the Rotunda at Rome , p. 286. Doors Tufcan , Plates 40, 44. —— Dorick, Plates 7c, 74, 78, 100, 101. - Iotiick, Plates 123, 124. - Corinthian , p. 301. • - Compojite , Plate 25)5. --rufticared, Plates 43, 46, 69, 74. DORICK Order, -by C. C Ofio, p. 227, to 234, Plates 47. 48, 4 pi 50. -of Marcellus at Rome, p. 237, Plates --of Vitruvius , p. 238, Plates 54. -of Dioclejiau at Rome , p. 239. • -of A Palladio, p. 239, Plates 59, 60. -of S camozzi, p. 240, Plates 61,62. -of Barozzio, p. 240, Plates 64, 65. -of Serlio, p. 24a, Plate 73. -of S. le Clerc , p. 244, plate So. --of C. Perault , p. 245, plate 85. --of Viola, p -of Alberti , p. 247, plate 87. -of P. de Lorme ,) Order Dorick A Dictionarial Index. d o Dor 'tck Order of Bullant, p. 247, plate S7. --of Barbara. ? . , „ _ of Catatteo, 5 plate SS. --of J. Mau-clerc, p. 247, plates P, K, 89, 90, pi. -of Inigo Jones , p. 24p, plates pa, 93. -of Sir Cbrijlopher Wren , p. 249, pi. 94, 95, 9 6 , 97■ ■-of Mr. John Gibbs, p. 253, 254, pi. p8, pp, 100, 101. Dorick on Tujcan Order, p. 245, plate 84. DOVE-TAIL, what, p. 353. DOUCINE, vide Cima inverfa. DRAMS in an Ounce, Pound, &c. p. 1 6. DYE, of a pedeftal, the lame as Dado. DYPTERE, the fame as Diytere. E Numerically fignifies 230. ECHINUS, the lame Member as Ovolo, ECPHORAS, the Quantity of Projection, that any Member of a Capital or Entablature hath before the upright of a Column. EGGS of an Ovolo, how deicribcd, &c. p. 14S, 213, 329. ELEMENTS, the Principles of an Art or Science. ELEVATION, or upright of a Building, &c. a geometrical Draught of a Eront, (fir. exprel- fing by a Scale of equal parts, the Meafure of rt. Elevati ns, how made, p. 162. ELIPSIS, a geometrical Eigure made by an oblique Section of a Cone, in whole Curve there is not any part of a Circle, and therefore cannot be an Oval, which is compoled of four Arches of two Circles 3 as fome Writers of Dictionaries have laid it to be. -how traced, p. i4p, 345. EMBRASL t R.E 7 the Splay-back of a Window, or Door, within-lide, made to give more Light, than when tlie Piers or Jaumbs are made fquare. ENGLISH Older, p. 321. ENNEAGON, the lame as Nonagon , a re¬ gular Polygon of nine lides, p. 122. ENRICHMENTS of Mouldings, (fL. their Carvings, of which we have here a very great Variety, byalmolt every one of the Mailers 3 but more particularly by Julian Mauclerc. ENTABLATURE, ox Entablement, p. 210. Entablatures for Doors and Windows, p. 316. ENTRESOLE, or Enterfole , the fame as Mcz-aninc , a little Story between two grand Stories, as in the Keza Treasury, Whitehall. ENTASIS, what, p. 210. EPICTHEATES, r. 209. EP 1 STYI.UM, the lame as Architrave. EQU ALITY of geometrical Figures, p. 405. EQUAL parts, how divided, p.135, 136. EQUIANGULAR, when in a geometrical Fi¬ gure, two or more of its Angles are equal to each other. EQUICRURAL, a right-angled plain Tri¬ angle, whole Legs are equal. EQUILATERAL, equal Tided j hence it is, that a Triangle of three equal Sides is called an equilateral Triangle 3 hence alfo, a geometrical Square, and all regular, equal-iided Polygons, are equilateral Figures. Equilateral Triangle, p. 120, 141. --to inferibe in a Square, 158. in a Pen¬ tagon, 139. EQUILIBRIO, the exact equality of Weight, in a Ballance. E S ESTRADE, the eminent part of an Alcove in a Bed-chamber. EURYTHMIA, the harmonious Proportions of Rooms, into which the Limits of a Plan is divided. EUSTYLE, what, p. 238. EXERGUM, the Space without the Figure of a Bafle Relievo, of a Medal, ^r. wherein the Name, Infcription or Date is placed. EXTRACTION of Roots, is the finding of Numbers equally equal, that being multiplied into themfelves, once, twice, (£>r. their Product fnall be equal to a Number given. This will be underllood by the two following Examples. I. To ext raid the Square Root. Let 43,1584 be a iquare Number given to find its Root, or if 'tis fuppofed to be a geometri¬ cal Square, as a eg i. Fig. B, Plate 466 con¬ taining as many Feet; to extract its Root, is to find the Length of a fide, as a e , (1) Make a Table of Squares, with their Ge¬ nitive equal Numbers, as far as the nine Digits, as follows. Hoots. 7 8 cR By this Table you find by lnfpection, the Square in the 3d Column, and Root in the 4th of any fingle Number, compofed or made up of any one of the Digits. (2) Point every other Place of the given Number beginning with that of Units j and as many fuch Points that the given Number contains, fo many Figures will 'll H 4 3 3 6 14 4 16 whofe 5 >into 5 its Square 2 S >Square-( 6 6 ' is- 36 Root 1 7 7 4 9 is 1 8 . 8 *4 19. 9 , .Si, the Root confiff of, and on the Right- hand fide make a Crotchet as a b , as is done in Divifion. This done, have rccourfe to the firlf Pun&ation, which is 43, and in the 3d Column of the above Table find the neareft lei's Number thereto, which is 36, whofe Root is 6, as in the 4th Column. Place 3b under 45, and its Root 6, in the Quo¬ tient, as under A, and then fubtracling 36 from 43, the Remains is 9. This is your firil Work, and is no more to be repeated. (3) For Plainnels fake place the 9 a Line lower, as at D, and to it bring down the next Pun&ation 15, making it 915, which is the firft Rel'olyend . . .aABC 431,384(672 36- ■ 6 9 remains 12,7^889^ E— 26 remain reiolvend. F 134,2)26,84 (2d refol- 26,84 vend. 0000 remains A Dictionarial Index. E X or Dividend, and make a Crotchet on the Left- hand fide as in Divifion. (4) Double the Root 6, it makes 12, which is a Divifor ; rejed 5, the laft Figure of the Re- i'olvend, and then, dividing the other Figures (cji) by ia, the Quotient is 7, which place in the Quotient under B, and to follow the Divifor 12 alio, making it 12,7: This done, multiply 127 the Divifor encrealed by 7 (the Root placed under B in the Quotient) and placing the Produd (88p) under the Dividend (5)15), fubtract it from thence, and the Remainder will be 26, as at E. (5) For Plainnefs fake, place the Remainder 26 a Line lower, (as at F) and to it bring down the next and laft pundation 84, making it 26,84, which is a lecond Reiolvend, and make a Crotch¬ et on the left Hand, as before. (6) Double 67 the Root hitherto found, which makes 134 for a new Divifor; reject 4 the laft Figure of the fecond Reiolvend, and then, divid¬ ing the other Figures (268) by 134, the Quotient will be 2, which place in the Quotient, as under C, and aflfo to follow the Divifor 134, making it 1342. This done multiply 1342, the Divifor encreafed, by 2, the Root laft placed in the Quo¬ tient under C, and placing the produd 2684 under the Dividend 2684, fubtrad it from thence, and nothing will remain, which ends the Opera¬ tion, and gives 672 for the fquare Root required. Note , It fometimes happens, that when an Ex¬ traction is thus ended, there are Figures remaining, which are called Irrational Surds, and the given lquare Numbers, from whence they come, are called Irrational Numbers, or Squares whole Roots or fides cannot be exprelfed numerically, neither by whole Numbers nor Fractions, there being al¬ ways fomething remaining; and fuch Numbers are 3, 7, i156 plied, mull be abated, 6^/156 until the Produd pro- —- duc’d thereby be equal 000 remains, to, or lefs than .the Rcfolvend. In this Example the firft pundation is 2, and the neareft lefs Square thereto is 1, which I place under 2, and its Root 1 in the Quotient; then, fubtrading 1 from 2, there re¬ mains 1, which I remove a Line lower to prevent Confufion, and to it, bring down the next punc- tation, 56, making it 156 for a Refolvend : This done, double the Quotient, which makes 2 for a Divifor, and rejeding the laft Figure of the Re¬ folvend, 6, I find the Remainder 15 to contain the Divifor 7 times; therefore I place 7 after the Divifor 2, making it 27, and multiply it by 7, the Produd is iSp,, which is placed under the Reiolvend for a Subtrahend; but as this Subtra¬ hend is greater than the Reiolvend, a new Sub¬ trahend mult be found, that will be equal to, or lefs than the Refolvend, as aforefaid. To find this new Subtrahend, we mull take the Divifor 2 but 6 times (inltead of 7 times) in 15, and place 6 after the Divifor 2, as at A,mak- E X ing it 26, which being multiplied by 6, the Pro¬ dud is 156, which being equal to the Refolvend, nothing remains. And in cafe that this new Sub¬ trahend had yet been greater than the Refolvend, the Divifor 2 mull have been taken but 5 or 4,^. times in 15 ; and fuch Number of times as is found to produce a proper Subtrahend, fuch Num¬ ber mull be placed in the Quotient, as herein, 6 is placed for the Quotient, which completes the Root as required. 1T. To extract the Ciibe Root. Let 146,363,183 be a cubical Number given, to find its fide or Root : (1) Make a Table of Cubes, with their Ge¬ nitive equal Numbers, as far as the nine Digits, as follows. Roots. Squares. Cubes. r '1 t r n 8 1*1 M 3 4 5 f < 5 K S_ ,1 *5 Ij 3 4 5 C p 27 ■2? H 5 f 6 6 36 jj 6 d- zi6 .2 6 7 7 1 49 7 .5 343 j- 7 * ,9J 3 L9J 6 4 80 f 8 19j 512 1729 1 19 J (2) Set a Point over the place of Units in the Cube Number given; omit two, and point every third, and as many fuch Points that the given Number contains, lb many Figures will the Root confift of. . ad 1 (3) In the above Table of 146,363,183(52.7 Cubes find the greateft Cube 125 that is neareft to 146, the firft- pundation, which is 125, whofe 21 Root is 5, as Hands in the firft Column of the Table againft c, firft Refolvend. 125; place the Root 5 in the 21,363 Quotient under A, as is done b in Divifion,and its Cube Num- 75) 2 ! 3( 2 ber 125 under(146)the firft 150 pundation, and then lubtrad- - ing 125 from 146, the Re- 63 remains, mainder is 21 ; this is your firft Work, and is no more to E 150 . be repeated. F 6° (4) To 21 the Remainder G 8 annex, or bring down 363, the -• fecond pundation, which to- 15,608 gether make 21,363 for the --- firft Refolvend, as at C. 5755 remains. (5) Square the Quotient 5, and triple its Produd 25, the H. fecond Refolvend. Produd is 75, which is a Di- 5 i 755» i8 3 vifor, by which the Refolv-- end (its two laft Figures 6 3 exedpted) is to be divided ; K that is to fay, the Figures 8112)57,551(7 213, as at B, wherein the 56,784 Divifor 75 being found twice, -* and 63 remains, therefore 7^7 place 2 to follow 5, as under D. l (6) Treble the Root 5, it 5 6 >7 8 4 • makes 15, which multiplied O 7^44 ■ by 4, the Square of the Quo- P 343 tient B, makes 60 ; alfo cube- D . 5>755y lS i the Quotient 2, which is 8 ; bring down 150, the produd of the Divifor 75 multiplied A Diction a rial Index. e x multiplied into the Quotient 2, as at E, and under it place the 60 and the 8 laft found, each a place backwardcr from each other, as at F and G 3 thefe three Numbers added together make a Sub¬ trahend 15,608, which muft be fubtrafted from the Rcfolvend C, 21,563, and the Remainder will be 5755 - (7) To thefe Remains 5755 annex or bring down 183, the next (and lall) pun&ation of the given cubed Number, which together make 5,755,r83, as at H, and isa l'econd, or new Rc¬ folvend with which proceed, as with C the firlt Rclolvcnd, in manner following, viz. AB (1) Square the Quotient 52, it mak.es 2704, which trebled, or multiplied by 3, makes 8112, which is a Divifor, by which the l'econd Rel'olv- end 5,755,183 its lall two Figures S 3 except¬ ed) is to be divided,‘y/ir.the Figures 57,551,as at K. ( 2 ) Divide 57.551 by 8112 (as at K) the QUO¬ AD tient is 7, which alfo place following 52, as un¬ der I, and 767 will remain. AB (3) Treble the Quotient 52, it makes 156, which, multiplied by (45)) the Square of 7 the Quotient K, makes 7644. Alio cube the Quo¬ tient 7, it makes 343; bringdown 56,784,- the Product of the Divifor 8112 multiplied into the Quotient 7, as at L, and under it place the 7644, and the 343 laft found, each a place backwardcr towards the right Hand, as at Q and P. Thefe three Numbers added together make a Subtra¬ hend 5,755,183, which being lubtracted from H the 2d Rclolvcnd, 5,755*183, nothing remains5 ADI wherefore the Figures 527 are the Cube Root required. Note, I. As many Periods as you have, except the ift, fo often this laft Work is to be repeated. II. That in all Extractions, when a Divilor cannot be found fo often as once in its Dividend, or if it can be found, and yet there fhould arile a Subtrahend greater than the Relolvend, in both thefe Cafes a Cypher mu ft be put in the Quotient, and annexed to the laft Divilor alio, for a new Di¬ vifor, and then the next Punctation being brought down and added to the laft Relolvend for a new Rcfolvend, proceed in every particular as aforefaid. III. When Numbers remain after the laft Sub¬ traction is made, which oftentimes happen, fuch are called Irrational or Surd Numbers, bccaufe their Roots cannot be exactly expreffed by Nume¬ rical Figures, altho’ by adding of Cyphers, we can come very near the Truth. EXTERNAL ANGLES, p. 166. EXCESS, what, p. pp. EXOTICK Pedeftal, PI. 2p 5 . EXTREAM and Mean ‘Proportion, is, when a Line is fo divided, that the whole is to the greater Segment, as the greater Segment is to the lei's. EXTREAMS, what, p. 87. EYE-DRAUGHT, what, p. 173. --how made, p. 173. F. F I N Latin Numerals, Itands for 40, with a 9 Dalh thus, 1? 4000 FACIA, Fafcia , or Fafce, vide p. 210. F A FASTIGIUM, the upper angular Point of a Pediment. FATHOM, what, p. 16. FESTOON, from Fejlus, Inrichments of Wreaths, that were anciently made of Fruits, Flowers, t£ir. on Feftival Occafions 3 and which are now made in carved Wood, Stone, &c. for the Embellifhments of Buildings, as in the Freeze of the Portico of St. Mary the ^Egyptian, Pl.i 12. FEET in Length in a Yard, -Fathom, * --Statue Pole or Perch, I -Chains Length orAcres Breadth, -Rood,Furlong, or Acres Length, Ip. 16. -Mile, ] -League, -Degree, -Circumference ol the Earth, I ^In a fquare Yard, "\ rc .Square of 10 Feet, p. 16. -fquare. , - Vn a lquarc Rod, r r fin an Acre of Land, 3 E1GLRE, geometrical, a Superficies termi¬ nated by one or more Lines, as a Circle by one a Semicircle by two, a Triangle by three, &c. Figures right-lined, or reSihneal, are thofe Whole Limits ccnfift of Right-lines only, as plain Inangles geometrical Squares, Parallelograms which are called plain Figures. Curvihneal Figures, are Fuch that are bounded by Curved-lines, as Circles, Ovals, Ellipfcs, g.-. Mixed Figures, are Fuch that are bounded with Right-lines and Curved-lines alfo. Regular Figures, are fuch whole oppofite fides and oppofite Angles are equal. I) regular Figures, arc thofe whole fidcs and Angles are unequal. . FINAL, what, p. 187, 206. _ A FINAL in Sculptuie, an Inrichment or.a lomb or Euneral Monument, reprelenting the end of Mortal Life, as a Lamp extinguilhed 1 or a Boy holding in his Hand an extinguilhed I orch, fixed on a Death’s Head at his Feet. FINISHING of a Building, to cover the Out- walls with Lime and Drift-land, fo as to relemble or imitate Pori land-(lone, which by fome is call¬ ed rough cajt. FLOORING, bow framed, p. 353. FLUTES, or Flutings of Pilalters, and Co¬ lumns. —By Fitr'rshs, p. JjS . pi. J3 , J4 . “- Palladio , p. 2 3 p. pi. 5p, 60. 10 reprclent on Paper the geometrical View of fluted Columns and Pilaltcrs, wVfc PI. R, following Plate 50, wherein Infpc&ion fhews, that having divided the Semi-plan ot the Column into its Flutes only, as Fig. 1. or into its Flutes and Fi¬ lets, as in Fig. III. and Right-lines drawn from the leveral Divifions in each Bafe, at right Angles to the Diameters, they thereby determine the Breadth of the Appearance of each Flute and Fillet, i he Pilalter, Fig. II. having its Diameter divided into 31 equal parts, give one to the Bead at each Angle, three to each Flute, and one to each Fillet. A Pilafter thus fluted, with Beads at the Angles, is molt proper for the Corinthian Order. Draw A Dictionarial Index. f o To divide at once the Flutes and Fillets on the Shaft of an Ionick or Corinthian Column, this is the Rule: - Draw a right Line on Paper, &c. as a b, in the lower Figure, and therein prick off with two pair of Compaffes (the one being opened to one 3d of the other) 24 Flutes, and as many Fillets, of any Size at Pleafure ; lo that the whole 24 of each, from a to b, be fomething lei's than the Girt of the Column, in that part where you want to divide'out the Flutes and Fillets. This done, from the feveral Divifions in the Line ( z b, draw right Lines at right Angles to a b , of Length at Pleafure ; and then having taken, the exact Girt of the Shaft with a piece of Parchment, &c. that hath one flrcight Edge, apply the Ends thereof fo as to touch the two outer Lines, as the piece d e ; then will the leveral parallel Lines cup the Edge thereof, in the Points 1, 2, 3, 4, &c. which are the true Breadths of every Flute and Fillet required ; and then the Parchment being applied about the Shaft, the Breadth of each Flute and Fillet may be moft readily let off with great Exa&nefs. In the fame Manner the Breadths of the Flutes and Fillets may be determined in every other part of the Column's Height at Pleafure. FOCUS-'?0/w?r of an Ellipfis, two points in the longeft Diameter, whofe Diltance from the Ends of the fhorteft Diameter, are always equal to half the longeft Diameter, as d h, (Fig. 60, Plate 5.) on which every Ellipfis may be dcicribed. Vide Ell/pfis. FODDER of Lead, what, p. 17. FOLIAGE, Inrichments of Branches, Leaves, and Fruits, rcprclented by Carving, Painting, or-Plaftering, as in Plate 4id. FOOT in Length 12 Inches \goide Tables I, - (qua) c 144 lq. Inches, S II, p. id. - cubical or (olid, 1728 cubical Inches. FRACTION, what, p. 71. FraCl tonal 'Parts, what, p. 17. - - Architecture , p. 324. FRAMING ol Partitions, p. 357. -of Floors, p. 353. ■-of Roofs, p. 3d2. FREEZ-E, Frize, or Friefe , by the Italians called Fregio . vide p. 187, 2op. FRENCH Order, p. 318. FRETT, Ornament of the Ancients, p. 325. FRIGERATORY, a Place wherein the Air is alwavs very cool, as a Grotto , Cave, £tc. FRONTISPIECE, what, p. 242. ( Tufcan , 7 \Doricb, S p - 3 ^ 4 ‘ Frontifpieces(Ionic k, p. 2dd, 275), 334. j Corinthian, p. 2p. 271. -‘ Terault , j - -J. Mau-clerc , p. 272. - Inigo J one Si 7 -Sir Chrijiopber Wren, .v ‘ 74 ’ -Mr. John Gibbs 1 p. 275. IRON p. 330. IRREGULAR Buildings 1 how plann’d,p. 168. IRREGULAR Curves, how plann’d, ibid. ISOCELES Triangle, what, p. 120. .-how generated, p. 141. -'how dclcribed, p. 143. K K Numerically reprefents 250, and ancient- 7 ly, with a Dalh, (thus, kT) it fignified 150,000. KEY-bVfltte of an Arch, the uppermoft Vouffoir, as s ii Fig. C, Plate 357. KING-? 3 *?/?, what, p. 356, 360. KNEE of a Hand-rail to a Stair-cafe, the Angle made in its bottom, by the level part meet¬ ing the raking part, as the Angle at k , Fig. 3, pi. 440. L L I N Latin Numbers fignifies 50, with a 7 Dafh (thus, l) 50,000. LACUNAR, an arched Cieling. LANTHERN, ox Turret 1 an Ornament plac¬ ed on a Dome, as that on Fig. A, pi. 3pi, which is nothing more, than a fmall Dome lup- ported by little Columns, either abfolute of them- lelves, or with Imports and Arches between them, and which are made entirely open, or glaz'd, as Occadons require. LARMIER, the fame as Corona. LATERAL, of, or belonging to the fides of a Figure ; hence a Triangle, whofe fides are all equi, that is to fay, equal, is called equilateral. LANDS Irregular, how plann'd, p. 174, to 1 79 - LATHS, their Length, and Number to a Bundle, p. 17. LEAGUE, what, ^ Leagues in a Mile, £p. 16. ■ -in Circumference of the Earth, S LEGS of a right-angled Triangle, what, p. 121. LEMMA, a Propofition preparatory for the Demonftration of a Theorem , or Conftruttion of a Troblctn. LENGTH, the firft kind of Dimenfions of Superficies and Solids. LETTERS, Capitals for Infcriptions ; their Height mull be regulated according to their Num¬ ber, which is required to be in a given Length, and which may be thus found, viz. Number the Letters in the Words, and thereto add as many ones lefs one, as are Words in a Line j divide the Length into as many equal parts, as the Sum of the Letters and the ones lefs one, and make the Height of the Letters equal to one of thole equal parts j then each Letter will poffels a geometrical Square, and the ones added, as aforelaid, will give the Interval between each Word equal to a geo¬ metrical Square alfo: When the Spaces for the Letters are thus determined, proceed to make them, as taught in PI. 461, &c. If ’tis required to proportion the Height of a Letter Handing high, as at g, Fig. A, pi. 466, to L I appear at a equal in Height to another Letter, as e d, which is much lower, proceed as follows: From a, the given Point of View, draw the Lines a d, a e and ag, and on a , with any Radius, de- feribe an Arch, as i h k l, make k l equal to ib , and through / draw the Line a If, cutting the upright Liner/ in f, then will the Height fg be the Height of the Letter required • for as the Angle l a k is equal to the Angle h ai, therefore they are both feen under equal Angles, and con- fequently their Heights will appear equal alfo. In the fame manner, the Height of Statues requir¬ ed to ftand on Buildings, may be found to appear equal to the Height ot a Man on the Ground, from any given Point of View. LINE, what, 7 -the Kinds, } P* 112 ‘ LIST, Liflello, what, p. 187, 20 6. LINKS in a Chain, p. 33. LINTELS, their Scantlings, 7 -their Length, 5 LOAD of Timber,A -— Earth, / -Bricks, ( —--Sand, FP* -Lime, . —Plank, LUTHERN -Windows, the fame as Dormant , or Dormer-windows, fuch as Hand on the Raf¬ ters of a Roof 1 they are called Dormer , or Dor¬ mant-windows from the Word Dormitory , a Sleeping-place, to which Ule the Rooms enlight- ned by thele Windows are generally applied. • 353* M M Numerically fignifies 1000, and with a 7 Dafh (thus, m) 1,000,000. MAGNITUDE, a certain Quantity of Mat¬ ter or Space, polfelfed by a geometrical Figure, or Folia Body , that is, the Bignefs of every geo¬ metrical Figure or Superficies, as alio every lolid Body, is called Magnitude , and therefore equal Things are of equal Magnitudes- but when two Things of the lame Kind are, one greater, or Idler than the other, as a Man and a Boy, then they are laid to be of different Magnitudes, that is, the Man is of greater Magnitude than the Boy, and the Boy of lelfer Magnitude than the Man. MANNERS of Building are three, viz. the Solid, the Mean , and the Delicate , which are well expreffed by the Doric!, Ionic!, and Corin¬ thian Orders. MASONRY, the Art of preparing, forming and putting together the various Materials of which Buildings are made, fuch as Stone, Bricks, Timber, Tiles, Lead, Iron, which, to well perform, requires an extenfive Knowledge in A- ritbmetic !, Geometry , and Architecture , and there¬ fore no Perfon can be a perfect good Malon, who is unacquainted therewith. Hence 'tis plain, that the Art of Malonry is not the Art of hewing and fquaring of Stones only, which is the Bufinels of a Stone-cutter, but comprizes the Manuaris , or Work of the Carpenter, of the Brick¬ layer, of the Joiner, and, in Ihort, of every other Artificer employed in the raifing and finilhing of Buildings in general; for without the Carpenter’s, Bricklayer’s, Joiner’s, t : £c. Works, that of the Stone-cutter is ufelefs, and therefore thefe laft arc as much Malons as the firft, bccaufd all of their Works A Dictionarial Index. 16. M E Works depend on each other, and together do but compleat a Building. MEANS, or middle Terns, what, p. 87. MEASURES of Length, what, p. 16. MEMBERS, what, p. 187. MEMBRETTI, p. 236. MENTUM, vide Corona . METOCHE, the Interval, or Diftance be¬ tween two Dentils, as the Intervals 26 3, 8 11, in PI. 167. METOPE, what, p. 238. MEZZANINE, vulgarly called Mtzzana, p. 383. MILE, v.hat, p. to. Miles in a League, ■-Degree, --Circumference of the Earth, MINUTES in a Degree, p. 113. -in a Module, p. 202. MITRE of an Angle, a Right-line dividing a folid Angle into two equal parts. ; . by t- 3 PI. I- Mix’d Number, what, p. 17. . ■ ■■ made of Wood, or any other Material in Mini¬ ature by a fmall Scale, wherein every part is lo proportion’d, as the Building is to be which tis made to reprelent, and which gives a much better Idea of large Buildings, to many Perfons, than geometrical Plans and Elevations can do. MODULE, or Modulus, of a Column, what, p. 202. --how found, p. 203, 205. MODILLION, from the Italian , Modigliani , a plain Support to the Corona of the C orintbian and ComPofite Cornices, as C, Plate 166. In La¬ tin a Modillicn is called Mutuli, from whence the fquarc Modillior.s in the Doriik Order, whole Sofits are enriched with Bells or Drops, arc called Mutules, and which are always of much lcls Depth than the Moddlions in the Corinthian Or¬ der. When the Modillions of the Corinthian Order arc eniichcd with Scrolls carved on their Tides, and Leaves on their Sofits, they are call- ¬ are there (ore called Cant a liver Cornices, as in PI. 177. MONOPTERE, a Dome fupported by Co¬ lumns inflcad of Walls, as in Plate 191. MONUMENT of London , p. 249. -a Column, Statue, Urn, ereeled to perpetuate the Memory of a Perfon or Ac.: n. MOSAICK ‘ Pavement, or rather Mnjaick Work, from the M:C.i\i of the Creels, a kind of Pavement compoicd of very invall Pieces < f Mar¬ ble, Stone, GET, igc. which are To placed, as to eprefent I .- Plate 449. MOULDINGS, (ingle and compound, p. 187. - , p. 3 17. MORTISI S, their Proportions, p. 36$. MULTANGULAR, a geometrical Figure or Body of many Angles. MULTILATERAL, a geometrical Figure of many Tides, MULTINOMINAL, that has many Names. MULTIPLICATION, what, p. 47. -—geometrically demonitrated, p. 412. --Table, p. 50. -of Integers, p. 51, to 54. M U Multiplication by a new Method, p. 36. -of Integers and Fractions, p. 412. -of Pence and Farthings, p. 57. ■ -of Pounds, Shillings and Pence, p. 38. ■ -of Feet and Inches, p. 59. -of Feet, Inches and Parts, p. 61, 413. MULTIPLICAND,) . MULTIPLIER, jvvhat, p. 47. MUNIONS, Plant ins, or Montans, upright Polls, that divide the lcvcral Lights in a Win¬ dow-frame, as thole two in thew^/f nun an Win¬ dow, pi. 340. MUTULE, vide Modillion. N IN Latin Numbers fignifies 900, with a Dafti (thus, n' «;-oo. NAKED, of a C, lumn or Wall, the upright, or out-lide Pace thereof - j . eticulation, its Rd'emblance of a ' l Wal.s built with iquare Stor.es, whole D; :g< i -Is a;e, one par.dk-1, and the other perpendicular to the Horizon. NEWE L-'Foji, the u iight Poll, about which ir, or elli{ p. 3S4, jSy. NICHES, what, p. 333. -— their Formation, p. 347. --lemicircular, 7 -lemiclliptical, S -’* t .-to make their Centers, p. 350. -how formed cut of Thicfcncflcs of Planks, p. 331. NON -how generated, p. 142. --how deferibed, p. 131. NUDITIES, tl> he parts of a human Figure not covered with Drapery. NUMBER 0} Tlaces, what, p. ico. NUMBERS, how expreffed, p. 11. NUMERATION, what, p. 12. -geometrically demonftrated, p. 409. -Tables, p. 12, 14. NUMERATOR of a Traci ion, what, p. 71. NUMERICAL Figures lor Clocks, PI. 467. O O Numerically was ufed by the Ancients to 7 de note 11, and with a I 1,OOC,OCO. OBELISK, or ObCnque, a Memorial Pillar without Capitals, p. 317. OBL! QU h'-Angle, any right-lined Angle, that is not right-angled. OBLONG, what, p. 123. OB FUSE- Angle, what, ••. 11-. OCCULT Lines , dotted Line , as the Terpen¬ tine Line C, Plate 1. OCTAGON, what, p. 122. OCTAHEDRON, one of the five regular Bodies, confiding of c , ,inod erat demonflrandum , that is to fay, Which was to be demonftrated. QUADRANGLE, or gucldrate, the fame as geometrical Square . a u QUADRANT of a Circle, what, p. n<5. QUADRILATERAL Figures , thofe that are of four fides, either regular or irregular. QUART ER-Pace of a Stair-calc, what, p 0 385. QUIN-Z)(?cr3goff, the fame as P endec agon, a polygon of 15 fides. QUINQUE -Angled, five angled. QUOIN, a lolid Angle, as of a Building, &c which are often rullicatcd, as in pi. 263. QUOTIENT, what, p. 66 . R R Signified among the Ancients 80, and with •y a Dalh (thus, r ) 80,000 RADIUS, what, p. 1 id, 119. -equal to do Degrees, p. 129. RAFTERS, how framed, p. 364. -their Scantlings, p. 3da. •-their Kinds arc three, viz. Principal Rafters , Hip-rafters, and Jack-rafters. RAILS to Stair-cafes , how knee'd and ramp- edj p. 3Sd. - Twifted, how fquared, p. 388, to 393. -on a circular Bafe, p. 389. RAMP of a Rail to a Stair-cafe, its Height how found, p. 387. •-its Center and point of Contact may be found as follows: Suppole 0 n. Fig. 1, pi. A, fol¬ lowing pi. 3d9, to be the raking part of a Hand¬ rail, and wi its Ramp, to find its Center/’, and point of Contact w. On any point, as q, taken at plealurc, in the line h i continued delcribe the Arch ir O’, let fall the perpendicular^;* on ow, and continued tor; draw* the line ir cutting the Raking-rail in w, from whence draw w k paral¬ lel to q r ; then is k the Center of the Ramp, and w the point of Contaft, from whence the Arch Or Ramp proceeds.-For this general Method I am obliged to Mr. Henry Maxted of Canterbury , who was fo kind as to communicate it to me, for the publick Good of Workmen that are employed in fuch Works. RAlSING-Y’/tf/f’j, how Icarfed together, p, 3JP- 3<>°. ' RAKING-Ctfrw/Vtf of a Pediment, pi. 345. RANK of Numbers , what, p. 98. RATIO, the Rate, or proportion which feve- ral Quantities have to one another, that is, as 3 is to 7, lb is 12 to 28 ; now here the Ratio , or proportion of 12 to 28 is as 3 is to 7 ;- and fo the like of any other Quantities. RECTANGLE, the fame as a Right-angle. RECTANGLED, the fame as Right-angled. RECTILINEAL, the fame as Right-lin’d. REDUCTION, a Rule in Arithmetick, teach¬ ing how to reduce Money, Weights, Meafures &c. into the fame Value in other Denominations. ReduBion is two-fold, viz. Afcending or De¬ fending. Reduction Amending, is to reduce a lower De¬ nomination into a higher, as Farthings into pence, Inches into Feet, &c. vide p. 6 2, to 64. ReduBion Defending , is to reduce a higher Denomination into a lower, as pounds into Shil¬ lings, Feet into Inches, ^r. vide p. 75, to 85. ReduBion of geometrical Figures , p. 401. REGULA, the uppermoft Fillet, that finifties a Cornice, or Capital, as A R E, Fig. 2, pi. 24. REGLET, or Rigid , the flat, fquare Lift, or Fillet, with which the ancient Fret Ornament, &c. A Dictionarial Index. r E is made. REGULAR Body, vide Platonick Bodies. RELIEVO, Italian , the Projecture of em- boffed Work, as Figures of Men, Beafts, above the Plane on which they are reprelented, fuch as the Heads on Money, Medals, i$c. which rile above the Surface. There are three Kinds of Relievo, viz. i. Baffo-relievo , low Relief, as of Money, Medals, &c. 2. Mezz-rclievo, called by l'ome Demi-relievo, that is to lay, Half-relief, when one half of the natural Thicknefs - is pro¬ minent. 3. Alto-relievo , High-relief, when the Figures are more than half prominent, and not quite clear’from the Plane. REMARKS on Vitruvius , p. 208. --on Palladio and Scamozzi, p. an* -on Barozzio and Mau-clere , p. 212, 214. ■ -on Le'Clerc, p. 214, 215. RESOLVEND, a Number in the Extra&ion of the iquare and Cube Root, arifing from the Remainder being encrealed by the next Pun&ati- on being brought down and annexed thereto. Vide Extraction of Roots. RHOMBUS, what, p. 123. -how generated, p. 142. — -how delcribed, p. 144. RHOMBOIDES, what, p. 123. ---how generated, p. 142. -how delcribed, p. 144. IWGHT-Angle, what, p. 117. RIGHT-Angled Triangle, what, p. 120. R\GHT-Line, what, p. 113. Right-lines , how continued, p. 135?. ROD, in length 16 Feet, 6 Inches, the fame as Statute-pole. - fquare, 272 Iquare Feet \ Rods fquare in a Iquare Chain, f ^ — --in an Acre of Land, .3 P‘ ROOD, in length, what, ^ Roods in a Mile, -in a League, -in the Circumf. of theEarth,^ - fuf crficial, one 4th of an Acre, or 40 fquare Rods, ibid. ROOF S, Reguk r ,| how delineatcdj p flbid. 36 . 5 . Irregular, A ‘ — Curved, how framed, p. 368. -—'Hemilpherical, how framed, p. 365). —Triangular,"? —Bevel, > demonftrated, p.371. —Irregular, S —Curved, how covered with Lead, bsr. P- 374- ROOMS, their Proportions, p. 375. ROTUNDA, or Mono ft ere, by Vitruvius, p. 288. ROOT fquare, a Number, which being mul¬ tiplied into itlelf is equal to a Iquare Number gi¬ ven ; or ’tis the Side of a geometrical Square, whole Area is given : So likewife the Cube Root is the Side of a Cube, whole Solidity is given. Vide Extract ion of Roots. ROTATION, the Circumvolution of a Sur¬ face round an immoveable line, as a Semicircle on its Diameter, which generates a Sphere 3 a Semi- ellipfis about its longeft Diameter, which gene¬ rates a Spheroid 3 a right-angled Triangle about one of its Sides, which generates a Cone 3 a Paral¬ lelogram about one of its Sides, which generates a Cylinder, (sf. R U RUDIMENTS, the firft Principles of an Art or Science. RULE of Proportion, or Rule of Three, p. 87. RUS11CKS, large lquared Blocks of Stone, which the Ancients employed in the Quoins of their Brick Buildings, lor the more effectual bind¬ ing thole parts (which are as Nerves) together, as in pi. 263 3 w herein oblerve, that the longeft are called Stretching Rujiicks , and the fhorteft Heading Rujiicks. The ancient Way of forming the Edges of Ruiticks was, to chamfer them off 45 Degrees back, as in Fig. 1, 2, pi. 45, where¬ by their Angles became obtulc and ftrong, l'o as not to be injured at their letting, (sr. which a fquare Angle is more liable to" It was all'o a Pradice among the Ancients to build their molt beautiful Phones rufticated in this Manner, leav¬ ing the Out-faces of each Stone rough, as it came out of the Quarry, until the whole Height was railed, when they began at the Top, and worked down a fmcoth Face, cutting away all tire pro¬ minent parts ot the ruftjck Stones in fo exact and neat a Manner, that the Joints were fcarcely per¬ ceptible. Rabbit, or fquare Rujiicks, as in pi. 263, is a modern Mode contrary to their original Ule, their Angles being Iquare, and liable to In¬ juries, which the others are nor. RUSTICATED Gates, p.242. -Columns, p. 220. •-Doors, p. 46, 227. -Quoins, p. 226. S S Among the Ancients denoted the Num- 7 ber 7. SAFFITA, vide Sofito. SCABELLUM, in ancient Architecture, a Pc- deltal to fupport a Bujlo or Relievo, the lame as Piedovche. SC A L E N U M -Triangle , what, p. 120. SCAMILl Iwpa/cs, Zocco’s, or fquare Blocks, placed under an Order, or Statue, to elevate it to a proper Height; in fhort, they are nothing more, than two or more Plinths on one another in manner of Steps, oi greater Proje&ions, than the Plinth of the Order or Statue they fupport. The levcral lquared diminifhed Zocco’s, that form the pyramidical Top of St. George , Bloomsbury, on which the Statue of K. Geokge I. is placed, are Sc ami l i imp ares. SCALE of equal Parts , how made, 1 --its Ule, jP- I2 4- --of Feet and Inches, ibid. -its Ule, p. 125. -of Chords, how made, p. 120. SC A PUS, vide Shaft. To SCARF Timber together, p. 353). SCENOGRAPHY, the Art of reprefenting pcrlpedive Appearances of Plans, Elevaions , &c. SCHEME-^,r.b, Fig. D, E, F, pi. 353. Scheme of a femicircular Arch,„ what, p. 352. SCHOLIUM, a Remark, or Comment on a Propofition before demonllratcd. SCIAGRAPHY, the Art of draw ing the ele¬ vated parts within a Building, (fuppofing the Whole to be cut down thro’ its length or Breadth) fhewing the Infide of every Room, Thicknels of Floors, Walls, (ffc. which is therefore commonly called the SeCtiou. SCIMATUM, vide Cyma, SCOTIA, p. 237. Scot in. A Dictionarial Index. s c Scotia, how defcribed by Alberti, p. 4p, 230. SCROLLS for twilled Rails, how delcribed, p. 3S7. SECTION of a Building, vide Sciagraphy. SECTOR of a Circle, what, p. 119. SEGMENT, Latin, a Piece. -of a Circle, vide Def. 23, p. 119. SEMI, Latin , Half. SEMIDIAMETER of a Circle, what, p. 1 16, 119. SEPT AGON, a Polygon of feven Sides, the lame as Heptagon. -how generated, p. 142. --how delcribed, p. 151. SERPENTINE River, how plann’d, p. 180. SHAFT, what, p. 1S7, 210. SIMILAR, of a like Nature. Similar Figures , p. 400. SKETCH, a rough Draught of aDefign, made in a flight Manner, by Hand, without Rule or •Compafles. SKEW •Sack of Arches over Windows, f£c. the Quantity that the upper Angle (as a. Fig. C, pi. 355) recedes, or falls from the Upright of c, that is, if the Line of the Side of a Window be continued up to the upper .Line of its Arch, the Diltance from that Line to the outer Angle, as to a , is the Quantity of the Skew-back, and which is made more or lefs, according to the pla¬ cing of the Center of the Arch, which the feve- ral Examples in plate 355 demor.llratc. SOCLE, or Sub-bafe, p. 211. SOFFIT A, Soffito, Saffita , Soffit , Soph eta, or Sophete, from Subfixum, a kind of Cieling, by which the Cielings of Windows are underftood. Soffila of a circular Window, how ornamented, p.381,382. --of circular Windows, how round, p. 3-A 347- SOLID Angle , the Meeting of three or more Planes in a point. SOLIDITY, the lolid Content of a Body. SOLUTION, an Anfwer to a Problem or Quell ion. SOMMERING Courfes of an Arch in Brick or Stone, thole which point dire&ly to its Center. SPANISH Order, p. 317. SPHERE, a Globe or Ball, vide Rotation. SPHEROID, vide Rotation. SPIRAL Line , how defcribed, p.ijd. SPIRES, their Proportions and Framing, p. 368. SQUARE- Root, vide Extraction of Roots. - Angle , an Angle containing 90 Degrees. - Meajure , ) - Foot, ( - Tard, V , -of 10 Feet,f 1 -Rod, S T STEEPLE, a Building eretted at the Weft- end of a Church for the Conveniency of hanging Bells therein, and for Ornament alio • when 'tis finifhed with a Pyramis, ’tis called a Spire , and when without, as with Battlements, &c. 'tis called a "Tower. STEREOBATE, or Stylobates, vide Rede- jlaly. p. 211. STRETCHERS, vide Pleaders. STRIAE, or Singes, vide Flutings. STRUTS, their Scantlings, p. 36c. -how framed, p. 364. SUBTRACTION, what, p. 39. -geometrically depronftrated, p. 411. -How proved, p. 40. -how performed, p. 39, to 46. SUBTRAHEND, the lefler Number in Sub trabtion , that is fubtra&ed out of the greater. SUMMIT, the Top, Vertex, or upper Point of a Triangle, Pyrament, or Faltigium of a Pe¬ diment. SUPFRCTLTUM, vide Corona. SUPERFICIES, the Surface of a Solid, which is confidered ro have Length and Breadth (as a Shadow) without Thickneb. SUPERFICIAL Figure, what, p. 122. SUPERSTRUCT, Latin, to build one Thing upon another, as a SUPERSTRUCTURE, the upper parts of a Building railed on the lower. SUPPLEMENT of an Arch, the Number of Degrees that it is lefs than 1S0, which is fometimes called its Complement to 180 Degree 0 . SYMMETRY, the Harmony, Proportion, or Uniformity of parts, that runs between the parts of a Building and the Whole. SYSTYLE, or Syftylos, p. 238. - Chain , d ■ - Inches in a Iquare Foot, p. 37. - Feet in a fquare Rod, - Roles in a fquare Rod, >p. 36. — -- Roods in an Acre, j STAIRS circular, p. 383. STAIR -Cafe, cylindrical , p. 39 u - circular , p. 383. - mixt, p. 384. -a Plan, how defcribed, ibid. -a Se&ion, how raifed, '? - Dog-legg’d, and right-angled , T 385. T Anciently denoted 160, and with a Dalh __ y (thus t) ido,coo. TABLES for Inf r tiovs are fquare or oblong planes, placed againft Walls, alio in the Archi¬ trave and Freeze of an Entablature. TALLIOR, French , a plain, fquare Abacus to the "Tufcan Capital, as E, pi. 17, Fig. 147. TALON, vide Afiragal. TAMBOUR, the iafe, Drum , or Bell of a Corinthian or Compofite Capital. TAXIS, with the Ancients was the fame as Ordonauce is with the Moderns. TEMPLE, Dorick by Bramante, p. 77, 244- (> Barozzio , ? - Ionick by . Serlio, S 1 y C Vitruvius , p. 2 65. -of Jupiter, p. 288. - Corinthian by Vitruvius, ibid. TFNIA, p. 210, TENONS of principal Rafters, p. 365. TERMS geometrical , are Points, Lines, and Superficies 3 viz. (1) Points are the Terms or Ends of Lines; (2) Lines are the Terms or Bounds of Superficies 3 and (3) Superficies are the Terms or Limits of lolid Bodies. T E-Square, p. 131. TESSELATED Ravemeni , a rich pavement of fmall fquares, in manner of pi. 44 9 - TETRAHEDRON, one of the five regular Bodies, comprehended under four equal and equi¬ lateral Triangles. i theorem. A Dictionarial Index. p. 144. T H THEOREM, p. 99. THEORY, the Study of an Art or Science, exclufive of the Practice. THERMAE, a Hot-bath. . THESIS, a Subjeft to be difeourfed upon : Alio a Sentence, or Propofition advanced to be cleared by Demonltration. TIMBERS in a Building, how to be fituated, p. 5 6<5 - TONDINO, vide Aflragab TON Weight, what, p. 1 6. TORUS, what, p. 206. TOWER, vide Steeple. 1'OWN, how plann’d, p. 180. TRABF.ATION, the lime as Entablature. TRABS, Latin, a Beam. TRACERIX, vide C atanaria. TRAJAN’s Column , p. 213, 215, 224. TRAMMEL, what, p. 346. TRANSFORMATION of geometrical Fi¬ gures, p. 40 r. TRAPEZIA, what, p. 122. -How delcribed, TRIANGLE equilateral, -Ifoceles, ? -Scalenum, f • ' 1 -right-angled, j -Ambligonium, 7 -Oxygonium, jP‘ 111 ' TRIGLYPH, what, 237. TRIMMERS, or Trimming Toy Its, vide Fie. A, Plate 372, p. 356. TRISECT, to divide a Line into three equal Parts. TRIL T MPHAL ArxheSj p. 307, to 309. TROCHILUS, what, p. 23-. TROPHIES, Reprefentations of Drums, Pikes, Halberds, and other Inftruments of War. TRUNK or Tige, the lame as Shaft . TRUSSES for Roofs, p. 362, to 367. - Ionick, p. 2So, 329. TUN, a Meafure of Capacity in Liquids, equal to 2 52 Gallons. TUSCAN-(lWt7 by the Ancients,p. iSS,to 199. --by Vitruvius, p. 202. -by Talladio, p. 209. -by Scamozzr, p- 21 j. -by Barozzi , p.212. -by Julian Mau-clerc , p. 213. ■ -by Sebaflian leClerc , p. 214. - by Claude Terault , p. 215. ■—-by Mr. John Gibbs, p. 216. ■ -b y S. Serlio, 1 R -by Mr. Stone, ,fP* " --by Sir Cbriflopher Wren, p. 220. Tufcan Portico, p. 221. TUSK, a Bevel Shoulder made on the Tnon of a Toy ft to Hrengthen its bearing. TWISTED Rail, how fquared, p. 38S, 393. -their Formation, p. 391, 392. TWIST ED Columns , vide Wreathed Columns. TYMPAN of a Pediment, the Inward Tri¬ angular upright Part, that Hands perpendicular over the Freeze. , and \ T In Latin Numbers Hands for Five, 9 with a Dafh (thus v) for 5000. VAGINA, Latin , a Sheath, the lower part of a Terminus , which riles as out of a Sheath. V A VALLEY of a Roof, the external Concave Angle, made by the meeting of two Roofs. VASES, Vcflels that were ufed by the An¬ cients in their Sacrifices, in Imitation of which lolid Ornaments are made in form of Flower-pots’ with ornamental Covers, enriched with curious Mouldings, and fometimes the Convexity of the Vale with Bafs-reliefs • for adorning Piers to Gates, Parapet-walls, &c. VANES of Weather-cocks, their Heights and Lengths, how found, p. 368. VAULT with Groins, how plann’d p. 18;. VELOCITY, the Degree of Swifn.els, that a Body goes with, in palling through a ce:tain Space, in a certain Time. VENTALF, vide Corona. VENETIAN Windows, p. 279. VERTEX, the top or uppermoit Point. VERT UOSO, Italian, a curious and ingenious Perlon, delighting in collcfting Rarities in Ait and Nature. VISTA, or V1JI0, Italian , a Hreight Open¬ ing made through a Hill, Wood, &c. to admit a diHant View, being Teen from a Houle, fsE. VIVO, vide Shaft. Y ITRUVIAN Window , p. 332. Vitruvian Scroll, p. 329. UNDECAGON, what, p. 122. -how delcribed, p. rj 2 . UNGULA, a Section, or a Segment of an Ellipfis, whole outer ordinate cuts the longelt Diameter at Right angles. UNITE, what, p. 12. VOLUTE, or Valuta , vide p. 256, to 260. VOUSSOIR, Trench a Stone of an Arch, whole Sides have a dired Sommcring to its Center and its Bottom a part of its Cuivc. UPRIGHT, geometrical of a Building, the fame as the geometrical Elevation. URN, Latin , Urna, a Veflel wherein the Ancients depofited the Afhes of their deceased Friends. It is no other than a covered Vale ie- prefenting Flame ifluing out at its Vertex,’ and is generally uled as a crowning or finifhing of a Monument, (jf. W W EATITER-CV&, their Heights and Or¬ naments, p. 368. WINDOWS, p. 333. -RuHicated, 7^ - Venetian , j P’ -Scheme-headed,7 ■-Semicircular. > 540- in circular Wall? P- -Semi-elliptical, S ^ I " Windows in a Diane, circular or elliptical, are called Lucar Windows , and lbmelimcs La- cun ars, as thole are in an arched Roof, or coved Ceiling. Vide Lacunar. WREATHED Columns , p. 327. X X I N Numbers fignifies 10, and with a 9 Dalh XYSTOS, a Grecian Portico of mere than common Length, which were fometimes un¬ covered or open, wherein the Athleta exercifed themlelves in Racing , WreJUing , 5tc. The Word is Greek , fignilying to polilh, or make fmooth; it being their CuHom to anoint their Bod : es P R Bodies with Oil before their Encounters, to prevent their Antagonifts from laying faft hold of their Fleih. The Romans had their Xyftos alfo, which was a very long Portico arched over, with Plantations of Trees on each Side, forming an agreeable Place to walk in, in very hot or wet Weather. Y ARDS in a Fathom, - -in a Statute Pole or Perch ,C --in a Chain’s Length, C -in a Acre’s Breadth, J Tards in a Rood, Furlong or Acre’s \ Length, S Z WAS : ? aooo; Times 0.000. ZOCCOLO, ZOPHORU; Advertifement to the R E A D E R. T H E Reader is deiired to obferve, that this AY o r k was pro- pofed to the Publick for to be performed by a Socie t y, (or Set of People) as let forth in die Introduction , who for that Purpofe entered into Articles with me, and undertook to do very great Things herein ; but upon future Examination, when thofe Parts were wanted, that they undertook to perform, in theCourfeof the Wor k, none of them produced any Thing: So that (had*not {, of myfelf, been able to carry on and finifh the Whole, as well as the Arithmetick, and Part of the Geometry to that Time) I ihould have been a very- great Sufferer in the Expences, that I had then been at, for Paper and Printing, and the World difappointed of the Performance alfo. Parliament-Stairs, 10 Sept. 1736 . B. LANG LET. mm ■ ■ X ,225 ZXZ 'X" 'Xy rtryyv Tfr t - Young NOBLEMEN and GENTLE ME N Taught to Draw the Five Orders of Columns in Architecture, To Deftgn geometrical Plans and Elevations for Temples, Hermitages, Caves, Grottos, Calcades, Theatres, and other Ornamental Buildings of Delight, To Lay-out, Tlant, and Improve Parks, and Gardens, By the A U T H 0 R. By whom Buildings in general are Dejigned, Surveyed, and Tier formed in the molt Malterly Manner, Artificers Works Meajured and Valued, And Engines made, for raifing Water for the Service of Towns, Cities, &c.