/ THE ART of Sound Building,, Demonftrated in Geometrical PROBLEMS: SHEWING Geometrical LINES For all Kinds of Arches, Niches, Groins, and Twijled Rails, both Regular and Irregular. With feveral other Draughts of Buildings and Staircafes. ALL Curioufly Engraven on Copper Plates, Wherein are laid down (fuited to every Capacity) eafy Pra&ical Methods for Carpenters , Joiners , Mafons , or Bricklayers , to work by. By W I L L I A M H ALFPENNY, Architect and Carpenter, LONDON; Printed by Sam. A r is for the Author , in Exeter -Change in the Strand l Benjamin Cole Engraver, in London-Houfe-Yard , St. Paul's Church-Yard ; Tho. Taylor Printfeller, at the GoldenLion in Fleet-Street ; Bowen Whitledge Bookfeller, at the Red Bible in Ave-Mary-Lane-, Tho. Bowles Printfeller, next the Chapter- Floufe in St. Raufs Church-Yard ; Tho. Wright Mathematical In- ftrument-Maker, at the Orrery and Globe in Fleet-Street m , John Senex, over-againft St. SDunJlan's Church in Fleet-Street ; Francis FayraM, at the Royal Exchange in Cornhill\ Tho. Worrali,, over-againft St. Dun flan’s Church in Fleet- Street ; and John Walthoe at Richmond. M.DCC.XXV. Digitized by the Internet Archive in 2015 https://archive.org/details/artofsoundbuildiOOhalf To the Honourable Sir ANDREW FOUNTAIN, Knt. SIR, 1 S with Pleafure I lay the follow- ing Piece at your Feet, as being convinc’d I do it to one who is an abfolute judge of the Merits thereof. You, Sir, are perfectly fenfible of the Neceflity and Ufefulnefs of a Work of this Kind, and will readily perceive the In- duftry and Pains ufed in the Compiling of this. Your Tafte, Sir, will ealily point out to you what is good in it, and what deficient ; and your Humanity will lead you to counte- nance and recommend the one, as your Love to the Art will move you to corredt the other. In Effedl, Sir, if I make my Addrefs to you, kis lefs as a Patron than as a Judge: Your [b] Cenfure DEDICATION, . Cenfure I ftand in more Awe of than that of the Publick ; as I reckon his that will deter- mine this. An uncommon Penetration, an exquifite judgment, a delicate Tafte, and a thorough Acquaintance with the Subject of a Book, are Qualities much to be feared by a Dedicator: And I will add, they are Qualities it had been fo eafy to have found a Patron without, that it ought to be a Preemption of fome Merit in the Work, that I dare to dedicate it to Sir Andrew Fountain . lam , SIR , With entire SubmiJJion , Four Humble Servant ? William Halfpenny, THE Reafons that fir ft induced me lijh this Work, was the daily r JM Errors that I jaw Workmen cm- mit in framing their Works for Buildings , on account of their W ant of the Knowledge of the Proportions contain'd in this Booh, being the only Thing . , that 1 know of that is wanting to make the Art of Building corn- pleat. Here / would not be thought to be fo vain, as to teach our Architects ; neither do I believe they are infenfible of the Ufefulnefs of this , or much greater Works ; but rather blame them for keeping fo advantageous a Work from the Eyes of the World . I t is certainly every Mans Duty to reveal whatever bethinks maybe of Service to the Pub- lick ; and fo I have fhewn the Nature of all Kinds of Arches in this Work, and laid down eafy and practicable Ways of drawing and working of them , fo The P R E E A C E. fo that any Workman, with a little Trains, may underfland the Nature , true Butments , and In- terjections of all Kinds of Arches , from whence he may Jlrengthen and very much beautify them , efpeciaUy Irregular Groins , which have been made very ill , for want of knowing, when the Arch of either Spand being given , what muft be the Arch of the other , fo that the Interjection of them (hall beget the Groin to jiand perpendicularly over its Safe. Therefore I earnejlly defire all Work- men to lay ajide erroneous Methods , and perufe this Jhort T rack of mine , or other, that fo both they and their Art may get Reputation thereby . I h a v e thought fit to fay thus much by way of Preface : and as for the reft let the Work an- Jwer. William Halfpenny. THE Subscribers, a. /A. FAAAZAz zzz/Az Fzz/zzzzo/z/Ajcy/! If. ZG/l/GA/GA, O/H VZtAz '// Aty/zztz: AA. r Ic P/i "GAAzztA/zzzvzz^ / y Y/zc * 'zort 'tv; AA/tzna/ A A/A/yAzz H/A/Z JAGAAFUUziCmyrz: B . AA. FZzZzZjZ/Z '/( VI o/'A/ZU/ //)////'/ Z7Zyj( 7? /'. . AAAF< y /’t 7 VZA A vz; ofR/v/imt vzA/ry/m. «. A A 7zFz 'tzvFa? vz/J, of ' PzcAzzzzzuitA Guy?. Z/Avzz HzpcAuAxizzA/p'FzYiZzzztyYAA AA[ 1/17 7l& I) 7 1771 71, O/'AAAz/G ZZ 7 AA . AAAHzFzJJdd’At// Gazyjf/zzGz; ' C. AA . Ad777('/ (AzA/Fz/OZZlVC/ \ SAZdnVtU'd ( A/7777W/ 1(77/77 t? " . *sM. r Henry ( bv/i,gfPz/Z7i/z/ GzzptziAzr. AHZzzzztJ Gee Ai/f/}//z?c?/ Y/zrpzezzZer. All //?( v??(U GnyAalAz; GzzyvzvzGz *. 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Price, mrrJz/leef. , IPA iinPaniPz iddapez : mPPadeertdezzlzzee// . ta/perzten * l/t l/zc/zaei f/zptuzc, tarpl , tfP/7/ertZaz c/iinfa/Jlierzn 7 di Carp! ^JtlDf77i.Pttfzeil0pldmo/ dzPzic/dt mt'.Sam, C7mzenpc/f/d&v. Bz id/ : ' ’ . t/ilfiil. L taper: faPicivzi'fpnpze/: It. dcAzzLiteApeld, cfPzcfim f f aimer .III It id . Pa/'/c (iff 1 u/zl A J n ' la /pen ft If. 7 'Ac. PacAcr; oft /erzzen tJ In zz llttz/r: IPPclt flallctP/cAz/zcnd ta/penlez: ttlTAcmicrrit, faTAndiz/mJci/zier. r THE ART of Sound Building, SECTION I, Of the Defer ipt ion of Arches by the Interfe 8 ion of Lines , &c. PROBLEM 1 How to ereff a Perpendicular from the Middle of a Right-Line given. PLATE L FIGURE L The given Right-Line is A B, and C is the middle Point there - of ; Upon which it is required to ere Si the Perpendicular . I R S T open your CompalTes at Pleafure to any Diftance, greater than half the Line, and Petting one Foot in the Point A, with the other fweep the Arch e e * this being done, with the fame Opening of your CompalTes Pet one Foot in B, and with the other Pweep the Arch d d. Then from the Point where thePe two Arches cut one another, draw a Right- Line to the Point C, and that Line will be perpendicular to the given Line A B, P R O- 2 The ART of Sound Building- i PROBLEM II. How to erefl a Perpendicular from the End of a Right-Line given . FIGURE II. Let the given Right-Line he A B, and B the End from which it is required to ereCt a Perpendicular. F IRST open your Compafles to any convenient Diftance, and Betting one Foot in B, mark that Diftance five Times in the Line A B from B to C, fo that B G be divided into five equal Parts. Then taking four of thole Divifions in your Compafles, as from D to B, let one Foot upon the End of the Line B, and ftrike the Arch dd ; and afterwards opening the Compafles to the Difiance of five of thofe Parts, or the Length C B, fet one Foot in E, the End of the third of the equal Parts from B, and with the other firike the Arch e e. This being done, if a Line be drawn from the Point where the two Arches cut one ano* ther to the Point B, that will be a perpendicular to A B. PROBLEM III. How to defer i he a Scheme- Arch, whenthe Length of the Bafe and Perpendicular are given . FIGURE III. Let the Length of the Bafe A B be five Foot , and let the Per- pendicular G E be one Foot : The Arch AGB is required to be drawn . F IRST draw the Bafe A B five Foot, and halve it at E, and from E raife the Perpendicular E C of one Foot in Height; then lay a Freight Rule on the two Points G and E, and draw a Line at Pleafure from E The A RT of Sound B uilding. g E towards G. This being done, draw the Hypothenufal Line C B, and halve it at D ; and taking the Length of this Line CB 3 either with a Line, or Beam Compafs, or com- mon Compafs, (if they will open wide enough,) from the Point B fweep the Arch i /, and from the Point C, the Arch b h; then from D, thro’ the Interfe£lion of the two Arches, draw a Line to cut the Line C G at F, and F will be the Center of the Arch; to be drawn about which, you may defcribe the fought Arch A C B with the Diftance F C. PROBLEM IV. How to draw a Scheme-Arch , without finding the Center thereof the Baje and Perpendicu- lar being given . FIGURE IV. Let the Baje KB be five Foot , and the Perpendicular ED be one Foot , the Arch A D B is required . F IRST draw the Bafe A B five Foot in Length, and halve it at E, and on E raife the Perpendicular ED, equal to one Foot, the Height of the Arch, and continue it to C, fo that G D be likewife one Foot ; and from the Point C to the Points A and B, draw the Lines C A and CB. This being done, divide each of them into an equal Number of equal Parts at Pleafure, (the greater the Number is, the exafler will the Work be,) one of which, in this Example may be about two Inches ; then if ftraight Lines are drawn from the Points of Divilion, i, 2, 3, iFc. of the Line A C to the Correfpondent Points of Divilion 1,2, 3, &c. ,of the Line C B, the Points wherein every two of thefe Correfpondent Lines cut one another, will be in the Arch required ; and fo the Arch A D B will be made by drawing thefe Lines, equal to the Arch AC B, of Figure hi. PRO- The ART of Sound Building. PROBLEM V. How to draw a ranipant-Scheme Arch. FIGURE V. F IRST draw the prick’d Line A B, to reprefent the Width of the Door, or Window, upon which the Arch is to ftand, and halve it at G, and from the Point B raife a Perpendicular B C, equal to the Ramp of the Arch, and draw the Bafe Line AC; then from G raife a Perpendicular to A B, which will cut the Bafe A G in half in F, and continue it to D, fo that F D be equal to twice the Heighth the Arch is to rife, that is, twice F E. Having done this, from the Point D to the Points A and C, draw the Lines D A and D C, each of which divide into an equal Number of equal Parts at Pleafure, as i, 2, 3,4, &c. i, 2, 3,4, fcfc. which may be each about two Inches: If A B be four Foot, and F E one Foot; then if ftreight Lines are drawn from the Points of Divilion, 1, 2, 3,4, tisfc. of the Line A D, to the correfpondent Points 1, 2, 3,4 9 istc. of Divilion of the Line C D, the Points wherein every two of thefe correfpondent Lines cut one another, will be in the Arch required ; and fo the Rampant Scheme- Arch A E C will be made, which was required. Note , When the Point C is the fame as B, or when it comes down to B, then will the Arch AEG be equal, and like to thofe of Figure in. and Figure iv. PROBLEM YI. How to defer the a Semicircular Arch by the Inter ~ feftion of Str eight Lines. FIGURE VI. F IRST draw the Bafe Line A B, and halve it at C, and at the Point C raife the Perpendicular C G, equal in Length to C A, or C B | then divide the Semidia- meter The ART c/SoundBuildik(J. 5 meter AC into feven equal Parts, and continue it out to D ; fo that AD be two of thofe Parts. This being done, take CD in your Compafles, and Petting one Foot in C, with the other llrike the Arch D F, of a Length at Plea- fure ; then with your CompalTes, open’d to any convenient Diflance, from A and G fweep the Arches k and e y and through their Interfe£tion, and the Point C draw a ilraight Line, cutting the Arch DF in the Point E, and draw the Lines A E and EG. Again, take A E, or E G, in your Com- palfes, and from the Points B and G llrike the Arches i and/?, and from the Interfe£lion of thefe two Arches draw Lines to G and B ; and if the four equal Lines lail drawn be each divided into the fame Number of equal Parts, and correfpondent interfering Lines are drawn, according to the Directions of the two laft Problems, they will defcribe the femicircular Scheme-Arch A G B required. PROBLEM VII. How to draw a Rampant Semicircular Arch. FIGURE VII. F IRST draw the Lines AG and G I, and 1/ and B /, all equal to one another, after the very lame Manner as the Lines A E and E G, and G h and B L all equal to one another, are drawn in Figure vi. and alfo draw the Arch E H at Pleafure here, as you did the Arch D F there, and the Line G D here, as the Line E C there. Then from the Point B raiie the Perpendicular BC, repre- fencing the Heighth of the Ramp, and draw the Bafe Line AC, which will be halved by the Perpendicular DI in the Point F. This being done, continue the Perpendi- cular D I to K, fo that FK be equal to DI, that is, take DI in your CompalTes, and fet it from F to K, and draw the Lines A G and G K ; then take G K in your CompalTes ; and fetting one Foot in K, with the other llrike the Arch m ; alfo take the Length A G between vour CompalTes, and with one Foot in C fweep the Arch and from the Point of InterfeClion of thefe two Arches C draw 6 The ART o/Sound Building. draw Lines to K and C, and divide the four out Lines laid drawn, each into the fame Number of equal Parts, and draw the interfering Lines according to the Di- rections above giv$n> and they will delcribe the fought Arch AKC* PROBLEM VIII. How to draw an EUiptick Arch to any JHidth and Height h, when the Heighth is greater than the JVidtJg by the Interfeclion of Jiraight Lines. FIGURE VIIL T~^ I R S T draw the Bafe Line A B, and raife the Lines jH AC and B D, Perpendicular to the fame, and each equal to your deligned Heighth, and draw the Line C D, which halve in the Point E. This being done, divide the Lines AC and EC, and ED and DB, each unto the fame Number of equal Parts, and draw interfering Lines accord- ing to the Direrions above given, and you will have the Arch required A E B. The End of Plate I. P R O B L E M IX. How to describe an EUiptick Arch to any THidth and Heighth , when the Heighth is lefs than the JHidtlg by the Interfeclion of Lines . FIGURE IX. F IRST draw the Bafe Line A B, and from the Ends, A and B, thereof raife the Perpendicular AC and B E, each equal to your defigned Heighth, and draw the Line The ART of Sound Building. 7 Line CE, which halve in the Point Q This being done, divide the Lines AC and C D, and DE and EB, each into the fame Number of equal Parts, and draw the interfeft- ing Lines according to the Direftions above given, and you will have the Arch A D B required. problem x. How to draw an FJliptick Arch by Means of a Tram el. FIGURE X. F IRST make the Tramel A B C i in the fame Form as it is reprefented in the Figure, where the Leg C i is at Right- Angles to the Head A B, both of which have Grooves in the Middles of them for the Pins e and /, which are fattened to the Rule or Lath DH, of a Length greater than i B, the half of the Bafe of the Arch to Hide in. Then if the wooden Pins e and f are fix’d in the Lath at fuch a Diftance from one another, that when a Pencil, or any Thing elfe proper to make a Mark with, is put through a Hole g 9 made in the fame with a fmall Gimblet, the Length eg is equal to i B, the Half of the Bafe Line of the Arch, and the Length fg equal to the Heighth the Arch is to arife, and the Tramel be fix’d faft in the Place, you Delign to ftrike the Arch upon, and the Pencil g be put in the Point A, and the Pins/ and e in the Grooves AB and ic 9 and with one Hand you move the Pencil and with the other guide the Pins e and / in their refpe&ive Grooves, when the Pencil is come to A, it will defcribe the Elliptick Arch AHB required. PRO- 8 The AR T of Sound Building, PROBLEM XI. T o draw a Rampant Elliptick Arch by the Jn - terfeflion of ftraight Lines. FIGURE XI. TC" 1 1 R S T draw the prick’d Right-Line A B, and from B •*- ere£t the Perpendicular BE equal in Length to the Heighth of the Ramp, and continue it out to H, fo that E H be equal to the Heighth of the Arch, and draw the Line A E. Alfo at the Point A draw A F Perpendicu- lar to A B, and equal to E H, or the Heighth of the Arch, and from the Points F and H draw the Line F H, which halve in G. This being done, divide the four Out-lines AF and FG, and GH and H E, each into the fame Num- ber of equal Parts, and draw the interfering Lines as above diredled, and you will have the Arch AGE, which was required to be drawn. PROBLEM XII. To draw a Cothick Arch , or Oxf when the Heighth is greater than the TTidth, by the Interfeciion of ftraight Lines . FIGURE XII. TU I R S T draw the Bafe Line A B, and halve it at C, from which eredl a Perpendicular of a Length at Pleafure, and in it fet of the Heighth of the Arch CF, and at the Points A and B raife the Perpendiculars A D and BE, each equal in Length to half the Heighth CF, and draw the Lines F D and F E. This being done, di- vide the four Out-lines AD and DF, and FE and E B, each into the fame Number of equal Parts, and draw the interfering Lines as before direred, and they will make the Arch A F B, which was required to be drawn. PRO- The ART of Sound Building, ti PROBLEM XIII, To draw a Gothic k Arch , or Oxi, when the Heighth is lefs than the Width by the Inter- jection of Straight Lines. FIGURE XIII. I R S T draw tlie Bafe Line A B, and halve it at C, from which ere£l a Perpendicular of a Length at Pleafure, and in it fet off the Heighth of the Arch C and at the Points A and B raife the Perpendiculars AD and B F, each equal in T .ength to half the Heighth C E, and draw the Lines D E and F E. This being done di- vide thefe four Out-Lines A D and DE, and E F and F B, each into the fame Number of equal Parts, and draw the In- terfering Lines as before dire&ed; and they will form or make the Arch A E B, which was Acquired to be drawn. Note, If the Arch is required to be quicker or flatter on the Hanfe, it is but lengthening, or fliortening the Per- pendicular Lines A D and B F* PROBLEM XIV, , _ • To draw a Rampant Gothick Arch, or Oxi 7 ly the Interjection of Lines . FIGURE XlVo F IRST draw the prick’d Line A B, and halve it at C, and from B raife the Perpendicular BE, equal in Length to the Heighth of the Ramp, and draw the Line A E, and from G draw C G perpendicular to A B„ in which take D G equal to the Heighth of the Arch n Then at the Point A raife the Line A F perpendicular to A B, and equal to one Half of the Heighth of the Arch, and continue BE to H, fo that EH be alio equal D to io The ART of § ound Building. to half that Heighth, and draw the Lines F G and G H. This being done, divide thefe four Lines AF and FG, and G H and H E, each into the fame Number of equal Parts, and draw the Interfering Lines as before directed, and they will form or make the Arch A E B, which was required to be drawn. P R O B L E M XV. To fir ike or draw Gothic k Arches by Means of other Arches. FIGURE XV. F IRST draw the Bale Line A B, and fet off the Thicknefs of the .Arch from A to C, and from B to O, and divide G O into three equal Parts in the Points D and E, about wheih deferibe two Half-Circles C i E and D»0. Then taking Cjf, or DO, each of which is two Thirds of C O, between your Compaffes, and fet ting one Foot in C, with the other fweep the Arch E F, and the Compaffes remaining open’d to the fame Diftance, fet one Foot in E, and fweep the Arch C F. And in like Manner fweep the Arches D G and O G, from the Centres O and D, and from the Interfe£lion of the two Arches at F, through the Centre E, draw the ftraight Line FM, and from the Interfedlion of the two Arches at G, thro’ the Centre D, draw the ftraight Line GH. The Centres D, E, F, and G, being thus found, take A D in your Com- paffes, and fetting one Foot in D, with the other ftrike the Arch A H, in like Manner, with the fame Diftance, and one Foot in E, draw the Arch B M. This being done, take F n between your Compaffes, and fetting one Foot in G, ftrike the Arches i K, and H L, and alfo with the fame Diftance, and one Foot in F, deferibe the Arches »K and L M, and the required Arch C AL B O is drawn. PR O- P w. The ART of Sound Building. i i P ROB L E M XVI 'To defer ibe or draw a Gothick Arch o f another Kind by Means of Circular Arches . FIGURE XVI. |_p I R S T draw the Bafe Line A B, and fet off the Thick* - a - nefs of the Arch from A to C, and from B to F, and divide C F into five equal Parts, upon which eleferibe four equal Half-Circles, D and E being the Centers of the outermolf ones. This being done, extend your Compaffes from F to D, or from C to E, and fetting one Foot in F, with the other flrike the Arch DO. The Compaffes re- maining thus open’d, fet one Foot in D, and with the other draw the Arch O F. In like Manner, upon the Cen- ters F and E with the fame Diftance, draw the two Arches C N and D O. Then from the Interfeffions N and O of thefe two Arches, through the Centers E and D of the outermoft Half-Circles, draw the ftraight Lines N L and OGof Lengths at Pleafure, and on the Centers E and D, with the Diftance E B, or A D, between your Compaffes, flrike the Arches B L and A G. Moreover, extend your Compaffes from O to O h, or from N to m, and with one Foot in O, draw the Arch hi, and with one Foot in N, the Arch m I. Laftly, take the Diftance O G, or N L, in your Compaffes, and on the Centers O and N draw the Arches GK andKL, and the Arch CAKBF will be finifhede The End of Plate II. PRO. 12 The ART of Sound Building, PROBLEM XVII To draw a Shipwright* s Arch , by the Interjection of Jtraight Lines . FIGURE XVII. 1C 1 1 R S T draw A B, the Bafe Line, and from A and B, the “*• Ends thereof, erefl the Perpendiculars A G and B E, whofe Difference in Length or Heighth muft anfwer to the Rake of the Arch, or Ceiling of the Cabin, and draw the Line C E, which halve in the Point D. This being done, divide AC and CD, and BE and ED, each into the fame Number of equal Parts, and draw the Interfering Lines as is taught in the foregoing Problems, and they will form or make the required Arch A D B, PROBLEM XVIII. T 0 draw a Bow- Arch by the Interjection of Jtraight Lines , FIGURE XVIII. ' At F IRST draw the Bafe Line A B, and halve it C: Then draw the Line GH of a Length at Pleafure, parallel toAB, or perpendicular to CE, (which is a Perpendicu* lar to A B, drawn from the Middle Point C,) and at a Dif- tance from it, equal to the Heighth of the Arch, and draw the Lines A D and B F. Then take C E in your Compafles, and fet it off from E to D, and from E to F, and draw the Lines ADandBF. This being done, divide the Lines AD and DE, and BF and EF, each into the fame Num- ber of equal Parts, and draw the Interfering Lines as is taught in foregoing Problems, and they will form or make the required Arch A EEL Note* The ART of Sound Building* 13 Note , The aforefaid Arch is very neceflary to be ufed in. wide Spands, or Spaces, where there is not Conveniency to raife a Semicircular Arch, as in Vaults under Streets, or Arches of Bridges , where great Strength is required, becaufe the Butments of this Bow- Arch are much llronger than thofe of Gothick Arches, Elliptick ones, or any Scheme ^Semicircular Arch|§$ PROBLEM XIX To draw the two different Edges of a T wilted Schofeet . FIGURE XIX. HP H E nineteenth Figure reprefents the inward and out- ward Edges of a twilling Schofeet of a Semicircular Door, or Window, whofe Jaums, from the Beginning to the Springing of the Arch, fplays more or lefs, according to the Humour of the Builder, and whofe Crown lies level without fplaying, the outward Arch C F D being even with the Head of the Window-Cafe. Now the Queflion is, how to find the inward Edge AFB, fo that it lhall dimi- niih gradually from nothing at the Crown F, to the Splay of the Jaums at the Springing AG and DB. First draw the Bafe Line A B equal to the Width of the Window and Splays of both Jaums, and halve it at H, and from it raife the Perpendicular HF to AB, and let the Splays inwardly from A to C, and from B to D. This being done, extend your Compalfes from H to D, or from H to C ; and fetting one Foot in H, with the other llrike the Semicircle CFD, and draw the Line IK of a Length at Pleafure, parallel to the Bafe Line, and at a Dillance from it, equal to the Heighth of the Arch CFD ; and take the Height H F between your Compalfes, and lay off that Heighth from F to E, and from F to G in the Line I K, and draw the Lines A E and B G. Lallly, divide the Lines AE and EF, and BG and FG, each into the fame Num- ber of equal Parts, and draw the Interfering Lines as is E taught 14 The ART of Sound Building. taught the foregoing Problems, and they will form or make the Arch AF B, which will fplay gradually from no- thing at F to AG and DB, which is what was required to be done. SECTION II. — Of GROINS. PROBLEM. XX. To find the Angle, or Mitre-Bracket of a Cove. 9 % FIGURE XX. IG 1 1 R S T draw the Bafe A B of the regular Bracket, and from A draw A D, perpendicular and equal to it, and draw the Line D B, and continue the Line D A to C, fo that A G be alfo equal to A B 5 then extending your Com- pares from A to B, and fetting one Foot in A, with the other defcribe the Arch, or Quarter of a Circle C B, and from the Point D draw D F, perpendicular to D B, and equal to DA, or AC, and another as BE from B, likewife equal to D A, and draw the Line F E, which will be parallel to D B. This being done, divide A B into a Number of equal Parts, not exceeding two Inches and an half, and thro’ the Divilions of them draw Lines parallel to A C, to touch the Arch C B, which continue out to the Line D B, and this Line will be divided likewife into the fame Num- ber of equal Parts, as A B is. Laftly, from the Divilions of the Line D B, draw Lines Parallel to D F, and in each of them, fromDB, lay off its Refpedlion Parallel, (from A B to the Arch BC,) and at the Points whereat they end flick fmali Nails, or Pins, and take a thin Lath, and bend it The ART of Sound Building* 15 it round the Nails, or Pins, obferving that it touches them all, and with a Pencil, or any Thing elfe, proper to make a Mark, defcribe the Arch F B round the Edges of the Lath ; and this is the Arch for the Angle, or Mitre** Bracket. PROBLEM XXL To find the Angle , or Mitre- Arch of a Regular Groin , when the Interfering Arches are Semi- circles. FIGURE XXL F IRST draw the Line A B, and from the Ends A and B thereof, let fall the Perpendiculars AG and BD to it, and each of the fame Length as it, and draw the Line CD; then* halve the Line A B in e, and with the Diftance A e, or e B, about the Center e, defcribe the Semicircle A F B, and draw the Line A D. This being done, at the Points A and D raife the Lines A G and DJ-v perpendicular to A EJ each equal in Length to the Heighth of the Semicircle e F, and draw the Line GH. Laftly, divide the Line AB into any Number of equal Parts, thro' the Divifions of which draw Lines to touch the Serhicircle A F B, parallel to B D, and to divide the ftraight Line AB; then if Parallels to AG are drawn from the Divifions of A D, and in each of them from A D, is fet off its Correfpondent Parallel, (from A B to the Arch A F and at the Points whereat they end be iluck in Nails, and a thin Rule be bent round them, fo as to touch them all, a Pencil, or other Thing proper to make a Mark, being mov’d round the Edge of the Rule, will defcribe the Arch A i D, which is the Angle, or Mitre- Arch of a Regular Groin, PRO 16 The ART of So und Building. P R O B L E M XXII. To find the Angle , or Mitre- Arch of a Regular - Groin ? when the Interfering Arches are Go- thick ones , viz. fuch as are jhewn how to be drawn in Figure xv. FIGURE XXII. F IRST draw the Lines AB, BD, DC, AC, and A D,in the Manner as was fhewn in the laft Problem, and upon AB and BD defcribe the Gothick Arches A FB and BLD, according to the Directions laid down in Problem xv. Then divide the Line A B into any Number of equal Parts, and from the Divihons thereof draw Parallels to BD to touch or come to the Arch A FB, and divide the Line AD, and at the Points A and D raife the Perpendiculars A G and 0H, each equal to F e, and draw the Line GFL Then draw -Parallels to A G from the Divihons of A D, and in each of them from A D fet off its Correfpondent parallel, (from A B to the Arch AFB,) and at the Points whereat they end, flick in Nails. This being done, if a thin Rule be bent round them, fo as to touch them all, and a Pencil, or other Thing proper to mark with, be moved round the Edge of the Rule, the Mitre- Arch AID will by this Means be deferibed. Note , In this Figure, and all others, whofe Hanfes rife quick, after you have made the equal Divihons, and drawn the Parallels according to the foregoing Rules, before you defcribe the Arch fought, divide the hrft and fecond Divi- lion next to the fpringing Foot of every Arch, each into two equal Parts, from whence draw other Parallels as is be- fore taught, and fet off their Heighths or Lengths in the Other Parallels ; and by this Means your Hanfe will have its true Bearing, and not be crippled. PRO- The AR T of Sou nd Building?. 7 PROBLEM XXIII If the lejfer Arch of an Irregular Groin he a given Semicircle , it is required to form a larger one, (not a Semicircle,) fo that the Interfeciion of thofe two Arches [hall beget , or make the Arch- Line of the Angle to hang pependicular over its Safe ; as alfo to draw that Arch-Line of the Angle, FIGURE XXIII. f '' I R S T draw the Lines A B and G D, to repre- fent the Walls from whence the Arches ipring, and draw the Line C B, and on the Line A G defcribe the Se- micircle AEG, and divide AC inro any Number of equal Parts, from whence draw Parallel Lines to CD, to touch or come to the Arch AEG, and if thofe Parallels are con- tinued out to the 1 ine CB, they will divide it into the fame Number of equal Parts, as A C is; and if from each of the Divilions of this laft Line Parallels to AG are drawn, they will divide the Line A B into the fame Num- ber of equal Parts, as AG, or CB, is divided into. This being done, continue A G to I, fo that A I be equal to E f 9 and continue D B to K, fo that K B be likewife equal to E f , or A I, and draw the Line I K. Moreover, at the Points C and B raife the Perpendiculars C N and B O to G B, each of the fame Length as E f 9 or A I, or BK, and draw the Line N O. Laftly, from the Divilions of A B, draw Parallels to A I, (that is, continue the Parallels drawn from theDivb lions of the Line C B to the Line I K,) and from the Divilions of C B Parallels to C N. Then fet of the Heighths or Lengths of each of the Parallels in the Semicircle AEG, upon the Correspondent Parallels to A I and CN, and Hick in Nails whereat they terminate ; and if a Lath be bent round them* fo as to touch them all, and a Pencil be moved round the Edge of it, the Arches AHB and BMI will be found; which was required to be done* F Note^ ! 8 The ART of Sound Building, Note x The Prick’d Lines in this, and all other Examples of this Kind, fhew that one Parallel Line has a Relation with the other. For Example: The Lines / E, gh , and l M, are all equal to one another; fo that if the three Arches A FIB, AEG, and CMB, were railed perpendicu- larly upon the Lines AB, A.C, and CB, and a Line drawn from H to M, and another from M to E ; then would the Line HM be parallel to, and dire£Hy over the prick’d Line / g. In like Manner, the Line EM would be parallel to, and direUly over the prick’d Line ft. Under- lland the fame of the other Parallels and prick’d Lines in this Figure, and any others of the like Nature. PROBLEM XXIV. If the lejjer Arch of an Irregular Groin he a Scheme- Arch , it is required to form the greater , ( which will not he Circular ^ ) fo that the Interfeclion of thefe two Arches fid ail heget the Arch-Line of the Angle to hang perpendicularly over its Bqje 7 and to draw this Arch-Line of the Angle . FIGURE XXIV. draw the Lines AB and CD, to reprefent the Walls from whence the Arches Ipring, and upon BD defcribe the given Scheme- Arch BHD, and divide BD into any Number of equal Parts, thro’ the Divifions of which you mull draw Parallels to the Arch, as above ; and when the Line AD is drawn, continue thofe Parallels to it, and they will divide it into the fame Number of equal Parts as A B is ; and from each of theie laid Divifions draw Paral- lels to divide the Line AB into the fame Number of equal Parts as BD, or AD, is divided into. This being done, at the Point A draw the Lines AL and AN perpendicular to A B and A D, each of the fame Length as g H ; and from the Points B and D, the Lines BM and DO, and draw the Lines L M and N O. Laftly, if Parallels are drawn from the ■' - ■/' ■ t The ART of Sou nd Building. 19 the Divisions of A B and A D, after the very fame Way as in the laid Problem, and you lay off upon them the Paral- lels in the Scheme-Arch B H D, and proceed according to the Directions above, you will form the Arches A E B and A K D, required. Note, You need not make ufe of the Diagonal Arch AI{ D, in the making of the Centres for Bricklayers or Mafons to turn their Arches upon ; becaufe the two Arches A E B and BHD do interfeCt each other, and make the Angle or Edge of the Groin hang perpendicularly over its Bafe; and therefore the Ufe thereof is only in the fra- ming of Ceilings, or the like, being in the Nature of a Hip or Valley. The End o/Plate III. PROBLEM XXV. Having one Centre given for an unequal-fided Groin , to form the other , fo that the Interfec- tion thereof Jhall produce the Angle 7 or Mitre- Arch 7 to hang perpendicularly over its Bafe: And \ moreover , to draw the Curve thereof FIGURE XXV. D R A W the Lines A B and B D , and D C and CA, each equal to one another, to reprefent the Walls from whence the Arches fpring, and on the Line AB defcribe the given Arch AFB. This being done, divide the Line A B into any Number of equal Parts, from whence raife Perpendiculars to A B to touch the Arch A F B, and draw jthe Diagonal Lines A D and C B. Then take the Line EF, and fet it perpendicular to the Lines AC, AD, CD, C B, B D, from A to O, from A to I, from C to P, from C to S, from C to L, from D to K, from D to T, from D to V, and from B to M, and from B to Z, and draw the fcraight Lines OP, I K 3 ST $ LM, and Z V. Now 20 The ART of So un d Building. \ Now divide the Bafe Lines BD, DC, CA, AD, and BC, each into the fame Number of equal Parts as A B is di- vided into, and from the Points of Divilion draw Parallel Lines to touch the Lines OP, ST, VZ, LM, and I K. Then take the Lengths of the Perpendiculars to A B, drawn to touch the given Arch AFB, and fet them off in the correfpondent Parallels, drawn from the Points of Divilion of the feveral Bafes upwards, and the Arches ByD, Di;C, CqA, AbD , and CnR, will be defcribed as in the fore- going Examples, ( Figure xxm, & xxiv.) whofe Heighths x y y rq, gh, and gn, are each equal to EF, as like- wife all the other correfpondent Heighths, from the Bafes, to the Curves that are formed. PROBLEM XXVI. The Arch-Line of a large Ceiling , or Vault, fup- pofed to he Semicircular being given : How to form the Curve of a leffer Arch , that fhall interfecl the Side thereof \ to give way for Doors or Windows , fo that their Jnterfeclion fhall produce the Groin to hang perpendicularly over its Bafe ) as alfo to form the Curve -Line thereof FIGURE XXVL F IRST draw the Lines A B, BD, DC, and CA, to reprefent the Walls from whence the Arches fpring, and defcribe the two given Semicircular Arches OAB, CLD, and in the Line BD fet off the Spand of the Inter- fering Arch from v to t . This being done, ^fet off the Heigh th you delign to rife the leffer Arch from g in the Line A O B, perpendicularly to touch the Arch in b , and from v to R, and t to and draw the Line R w, which halve in the Point and draw the Line ^jy, parallel to x>R, or tv, Then drain a Line, or lay a ffraight Rule from The ART of Sound Building. it from b thro’ g 7 towards x m , as alfo from z thro 5 y 9 towards X, and thefe two Lines will cut one another at x 9 from whence to the Points v and t draw the lines x v and x t. Now fet off gh perpendicular to xt from x to w, and from ttosy and draw the Line sw, and divide gB into any Number of equal Parts at pleafure, from the Bivifons of which draw Perpendiculars to gB, to touch the Arch BOA between the Points B and h 9 and divide vy and yt 9 the Halves of the Bafe vt 9 each into the fame Number of equal Parts, as ^B is divided into ; as likewife the Bafe xt; and from the Points of Divifion draw parallel Lines to touch the Lines &R and sw. This being done, take the Lengths of the Lines that were drawn from the Points of Divifon of B^*, perpendicularly to touch the Part Bh of the Arch BOA, and fet them off in the cor* refpondent Parallels from y v to ^R, and from yt to zu? as likewife from xt to ws. Then, if at the Extent of each Line as you fet it off in the Parallels, you flick in Nails, as in the foregoing Examples, and bend a thin Ruler about them, you will defcribe the fought Arches vzt and w t 9 whereof is the true Interfering Arch, and wt the Curve Line of the Groin that is correfpondent thereto. After the very fame Manner the Arches k m z and kp are drawn. PROBLEM XXVII. The Arch-Line of a Ceiling , or Vault , fuppofed to he an Ellipjis , being given , how to form the Curve of a lejjer Arch , to give way for Doors ? or LVindows , that Jhall interfect the Side thereof fo, that their Interfeclion Jhall produce the Groin , or Mitre- Arch, to hang perpendicu- larly ; as alfo to form the Arch- Line thereof FIGURE XXVIL F IRST defcribe the Lines A B, BD, DC, andCA, reprefenting the Walls from whence the Arches G fpring, 22 The ART of Sound Building. fpring, and in the Line AC fet the Spand of the Arch k and on the Lines A B and C D defcribe the Semi-Ellipfes AOB, C L D, reprefenting the given Arches. Then take the Keighth you deiign to rife the lefler Arch, and fet it perpendicularly from the Line AB, as at e, to touch the Arch AB at f: Alio fet the fame Keighth from k to /, and from a: to q , and draw the Line i q. This being done, draw a Right-Line at pleafure from f thro 5 e, towards a, and ano- ther from m thro’ n, cutting it in the Point o, from which draw the Lines ok, o and take mn, the Keighth of the lelfer Arch, and fet it perpendicularly from o to p, and from k to /, and draw the Line pi Then divide Ae into any Number of equal Parts at pleafure, from the Divifions of which draw Perpendiculars to AB, to touch the Arch AOB, and divide the Lines ft, n k, and ok, as in the laid Problem you did the Lines vy,yt, and xt, and draw Paral- lels as there. Now, if you take the Lengths of the Lines that were drawn from the Points of Divilion of A e, per- pendicularly to touch the Part A/ of the Curve AOB, and fet them off in the correfpondent Parallels, from k^ towards 2 q, and from 0 k towards Ip, and at the Extent of each Line ftick in Nails, as in the foregoing Examples, you may de- fcribe the fought Arches k m^, and kp . After the very fame Manner may the Arches and tw be drawn. PROBLEM XXVIII. T he Arch of a round T ower . or any other Circular Buildings being given, wherein a femicircular Window is to Jtandy how to form a Centre fo, that the Mafon , or Bricklayer, (hall twin their Arches thereon without crippling them, FIGURE XXVIII. F IRST draw the Arch A F B from the Centre E, to reprefent the Arch-Line of the Wall, and fet the Width of the Window from C to D, which halve at H, and The ART of Sound Building. 2 a and draw the Line LM, which halve at N ; from whence defcribe the Semicircle L O M. This being done, divide the Semidiameter LN into any Number of equal Parts, from the Divilions of which draw parallel Lines to OL, the Arch of the Quadrant, which Parallels continue out to di- vide the Arch FG into the fame Number of Parts as LN is, and from the Points of Divifion in the Arch.FG draw Perpendiculars to the Parallels, each equal in Length to the correfpondent Parallel of the Quadrant LO; and from the Points of the Divilions of the Line H C, (made by continuing out of each of the aforefaid Parallels,) draw Right-Lines to the extreme Points of the aforefaid Perpen- diculars, as from G to H. This being done, if the Line G FI be laid oft in the Parallel O N continu’d out from H to I, and the reft of thefe Lines laft drawn be laid off in the refpedlive Continuations of the Parallels, the extreme Points of thefe Lines being joined, will form the Curve G I, which, when fet in its due Polition will hang perpendicu- larly over the Arch CF, having its Points co-inciding with the Extremities of the Perpendiculars drawn fom the Ex- tremeties of the Perpendiculars drawn from the Divilions of the Arch G F. SECTION III Of the Formation of Niches . PROBLEM XIX. How to form a Semicircular Nich with Ribs, as is ufual when it is to be plafteTd, FIGURE XXIX. F IRST defefibe the Semicircular Plate ACB, as alfo the Semicircular Front-Rib AD B, equal to it, and fix the Plate ACB level in the Place where it is to continue, and 24 The ART of Sound Building. and upon it fet the Front-Rib ADB perpendicular on AB. This being done, defcribe the Quandrantal Ribs DC, DE, DF, DG, and DH, each equal to AD or BD, and place them about eight Inches and a half from one another on the Plate A C B, as at C, E, F, G, and H, fo as to meet in one Point at D on the Crown of the Front-Rib ADB; and, thus is one half of the Work finifhed. And after the fame Manner may the other be done. PROBLEM XXX. How to form a Semicircular Nich by the Thick - nejfes of Boards , or Planks, and to find the Be- vel of each Thicknefs . FIGURE XXX. “ffp I R S T defcribe the Semicircle on the Front of the A Nich ADB, and divide the Heighth eD into equal Parts, according to the Thicknefs, of the Board or Plank, of which you deiign to make the Nich. Then defcribe the Thicknelfes from whence the Bevels are taken, and draw Lines equal to the prick’d Lines in the Example. This being done, take the prick’d Line i z in your Compalfes, and on the under Side of the Board, or Plank, of which you deiign to make the fir ft Thicknefs, defcribe a Semicir- cle from i equal to ADB, the Semidiameter being equal to the prick’d Line i 2. Then lfrike a fquare Stroke on the Edge from 1 , to find the Centre for the Semicircle on the upper Side of the firft Thicknefs, as at 3, and take the prick’d Line 34 in your Compalfes, fetting one Foot m3, with the other draw a Semicircle on the upper Side of the firft Thicknefs, whofe Semidiameter is equal to the prick’d Line 3 4. Having an Arch defcribed on each Side of tile firft Thicknefs, with a narrow Turning-Saw cut dire&ly thro’ the Arch-Line on each Side the Board, or Plank, and fo you will have the true Bevel and Curve thereof. Now, to make the fecond Thicknefs, defcribe the Semi- circle laft drawn on the under Side thereof, as you did on the The ART of Sound Building. 25 the upper Side of the firft Thicknefs, 34 being alio the Semidiameter equal to it. Then Hr ike a fquare Stroke from 3 on the Edge of the Board, or Plank, to find the Cen- tre for the Semicircle on the upper Side of this fecond Thicknefs, and take the prick'd Line 5 6 in your Com- pall es, and fetting one Foot in 5, with the other defcribe a Semicircle on the upper Side of the fecond Thicknefs, ha- ving its Semidiameter equal to 5 6. Then with a Turning- Saw; cut thro’ the two Arches in the firfl Thicknefs, and the Arch-Line and Bevel of the fecond Thicknefs will be fnifhed. To find the; Arch-Line and Bevel of the third Thicknefs* you are to proceed after the fame Manner as in the firfl and fecond Thicknefs, and fo of the others. Having your Thickneffes all ready, according to their true Arches and Bevels, fet them in good and well-made Glue, letting it If and till it be quite dry, and with a Compafs Smoothing-Plane, a little quicker than the Arch of the Work, plane the Infide thereof till it be fit for the Pur- pofe deiign’d. PROBLE M XXXI How to form an Elliptical Nich ? with Ribs for Elajlering . FIGURE XXXII, & XXXIII. F IRST defcribe the two Figures 32 and 33 , k nm be- ing a Semi-Ellipfis, reprefenting the Plate whereon the Ribs Hand, and being equal to ADB or A^B. The prick’d Lines In, lo , Ip, Iq, l R, and lm reprefen t the Bafe • Lines of the Ribs eD, fD, ^D, b D, iD, and BD ; fo like- wife do the Lines st, su , sw, sx, and sy, and the Perpen- diculars at, bu, cv, drp, ex, and fy, do reprefent the Ri- ling of the Ribs eD, fD, gD, h D, i D, and bD, which is equal in Length to CD; obferving that within thofe Lines the different Arch of each Rib is to be deferibed, vi*. the Arch s a is a Quadrant of a Circle, having t for its Cenrre* H and < 2 $ The ART of Sound Building. and is equal to the Arch of the Rib e D. The Lines u I, equal to gb, bn are the Semi-tranverle and Conjugate Axes of a Semi-Ell ipfis, whofe Arch sb is equal to the Arch of the Rib/ D, which may be {truck either with a Tramel, or by the Interfedtion of Lines. Moreover, the Lines ix, sv, equal to vc, c^, are the Semi-tranverfe and Conjugate Axes of a Semi-Ellipfis, whofe Arch is equal to the Arch of the Rib g D, and fo of the reft. Now having the Ribs all ready, fet the Front-Rib A D B perpendicular on the Plate Ac B, as at AB, and fix the Feet of the fhort Ribs on the Plate AcB, as at e, f, g, /;, i, which correfpond with the Points n, o, p, q, r, and their Points a, b, c, d , e, to the Crown of the Front-Rib at D; and thus may you finifti your Work. P R O B L E M XXXII. Blow to form an Elliptical Nich by the Thick- nefs of Boards , or Blanks. (See Fig. xxxiv.) F I G U R E XXXIV. F I R S T on a Drawing-Board, or Floor, deferibe (. Figure xxxiv, xxxv, xxxvi, and xxxvn,) the Arch ABC and fgh being Semi-Fdlipfes equal to one another. The Arch In is a Quadrant of a Circle, and the Arch op is a Quadrant of an Ellipfls, being the two mod different Arches of the Nich. The Arch fgh reprefents the firft Thickftefs, and is equal to A CD, and the perpendicular Lines mn and gp are equal to eB, and the Bafe Line l m is equal to ig. Moreover, the Bafe Line og is equal to i k, whofe Arches In, op, with their Bevels, do ftand per- pendicularly over ig and ik. Now take the Board, or Plank, of which you deftgn to make the firft Thicknefs, and on the under Side thereof deferibe a Semi-Ellipfis equal to ADC, or fgh, whofe Se- mi-tranverfe Axis is equal to the prick’d Line 12, and Semi-conjugate to 13: Then at 1 ftrike a fquare Stroke on the Edge of the Board, or Plank, to find the Middle of the SjeT?;* .® The AR T of Sound Building, 27 the Bafe to the Elliptick Arch on the upper Side of the firft Thicknefs at 4, whofe Semi-tranverfe is equal to the prick'd Line 4 5, and Semi-conjugate equal to the prick'd Line 4 <5, by Means of which defcribe an Elliptick Arch on the upper Side of the firft Thicknefs. Then by Means of thefe two Elliptick Arches, defcribed upon the upper and under Side of the Piece, with a Turning-Saw, law out the Curve and Bevels of the firft Thicknefs. To find the Arch and Bevels of the fecond Thicknefs on the under Side of the Board, err Plank, of which you delign to make it, defcribe an Elliptick Arch equal to that on the upper Side of the firfh Thicknefs, whole Semi-tranverfe and Semi-conjugate Axes are alfo equal to the prick’d Lines 4 and 4 6 . Then from 4 ftrike a fquare Stroke on the Edge, to find the Middle of the Bale Line to the Arch on the upper Side oi the fecond Thicknefs, whole Semi-tram verie is equal to the prick’d Line 78, and Semi-conjugate equal to the prick’d Line 79, and with a Turning-Saw, as before, faw out the Arch and Bevels thereof; and fa of the reft. SECTION IV, Of the Formation of Twi filed Rails . PROBLE M. XXXIII How to find the Raking Arch , or Mold , for the Hand-Rail to a Circular Pair of Stairs, in fuc'h Manner that it fihall ftand perpendicularly over its Bafe , or Arch of the Well-Hole. FIGURE XXXVIII. F IRST defcribe a Circle equal to the Breadth of the Well-Hole, whofe Diameter is UW; as alfo ano- ther from the fame Centre, whofe Diameter is AG, to repre- 28 The ART of Sound Building, reprefent the Plan of the Rail, and divide the Circumfe- rence of the greater Circle into the fame Number of equal Parts as you would have Steps once round the Circle. This being done, take the Rack, or Rake, of the Bracket equal to CF in your Compaffes, and fetting one Foot in A, with the other finite the Arch/;: Alfo take the Heighth of one Step, as AC, Figure xl, and fetting one Foot in B, with the other ftrike the Arch i ; and when this is done, take the Diftance from A to /; in your Compaffes, and letting one Foot in /;, with the other ftrike the Arch A, and take the Heighth of two Steps, and with one Foot in C draw the Arch /, to interfefh the Arch A, and fo on. The Interfering Points of the Arches hi, and A/, and no, and r s, and tu, are all at the fame Diftance from one another, and the Lines B /;, C A, D», Ep, F r, Gt being the Riftngs or Heighths of the Steps in Figure lx, B b being the Heighth of one Step, C A of two, D» of three, E p of four, Fr of five, Gt of fix. Now if thefe Lines are raifed up perpendicular on the Circle ADG, it is evident that the Point of XnterfecTion of the Arches h and i will Hand per- pendicularly over the Point B ; of the Arches A, / over C; of the Arches p and oj, over J£; of the Arches p and g, over E; of the Arches r and s over F ; and of the Arches t and u, over G. Now if Nails be ftruck into the interfering Points of the faid Arches, and a thin Rule be bent round them, you may defcribe the Arch Ahknprt by the Edge thereof, being the Mold, to ftrike the Arch of the Rail with. PROBLEM XXXIV. The Arch or Mold of the Rail being found , as above , how to prepare the Stuff of which the Rail is to be made , and work the T wift there- of without fetting it up in its due Tofetion. \ FIGURE XXXIX. F IRST ftrike two Circles, whole Diameters are equal to U\V and AG, in Figure xxxviii. and next confi- der The A R T Sound Building, 25? der into how many Pieces you glue the Rail, which in the Semicircle let be fix, as in the Example. Now divide the Semicircle into fix equal Parts, as EF, FM, MS, SL, LD, and DR, from each of thefe Points of Divilion, draw Lines to the Centre A, as A E, A F, AM, AS, AL, AD, AR. Then from F raife F G, perpendicu- lar to A F, and equal to the Heighth of one Step : Alfo at the Point M raife MN, perpendicular to AM, equal to the Heighth of two Steps ; and in like Manner at the Points S, L, D, and R, raife the Perpendiculars ST, LY, DE, and RL, refpeflively equal in Length to the Heighth of three, four, five, and fix Steps. Then draw a Line from G to R, parallel and equal to A F ; as alfo another from N to y, parallel and equal to AM; another from T to W, parallel and equal S A ; another from Y to B, parallel and equal to LA; another from E to H, parallel and equal to D A ; and another from L to P, parallel and equal to R A. From the Point A draw the Line A B, perpendicular to AE, and equal to the Heighth of one Step; alfo at the Points R,jy, W, B, H, P, draw the Lines RL, YZ, \VX, BC, HI, PO, all equal to the Heighth of one Step, and refpeflively perpendicular to RG, y N, TW, YB, EFT, LP, and draw the Hypothenufes EB, LG, Z N, TX* YC, El, LO. This being done, fet oft the Width of the Rail from Et od i G to ?, N to 0 , T to u, Y to a, E to /, and L to m ; and fet the Stem of a Square on the Line E B, till the Blade thereof touches the Point d , and draw the Line c d. More- over, fet a Square on the Line G L, and where it cuts the Line RG, as in the Point /, draw the Line hi ; and in like Manner draw the Lines po, un, gf, an dnm. Then the Angles E dc, G i h, Np o, iFc. and the reft of the little black Spaces, as you fee in the Figure, do reprefent the Twilling of each Piece, and what muft be taken oft from the Back at the lower End, to make the Twill of the Rails. The Lines being drawn, you are next to coniider after what Manner they are to be applied in the working of the Rail. Take the Piece of Timber, of which you defign to make the firil Length, which is reprefented in Figure xli, and plane one Side thereof ftraight, and cut it to its Bevels I ac ? •20 The ART of Sound Building. ac, b d, anfwering to DR A and RDA, Figure xxxix, and both Ends thereof being alfo cut to the Raking joint of the Rail, proceed thus : Take that Part of the Raking Arch in Figure xxxvm, which anfwers to the ftrft Length of the Rail, as A/s in the Arch A^, and lay it on the up- per Side of Figure xli, from / to /;, and ftrike the Arch lb; then cake E c, equal to Gb or N p, in Figure xxxix, and fet it on the Line b d from h to m, ( Figure xli,) and ftrike a fquare Stroke at Pieafure from m tog; alfo take coequal to/;/, or po , IFc. and fet it on the Line from m to g, and draw the Line bg , which reprefents the Back of the Rail when it is work’d, and is equal to E d, or G /, or N o , Ffc. This being done, reprefent the lower End of the Rail bgki , at Right- Angles to hg ; as alfo the upper End l c 0 n at Right-Angles to Ic , and bafte out the inward Arch c m fquare from the up- per Side abed , as mg; and take a thin Lath, and bend it clofe to the Side thereof from c to g, whereon ftrike a Line along the Edge of the Lath, and fo the Lines lb and eg are your Guides in backing the Rail: Which, when done, turn the Piece upftde down, and with the Mold ftrike an Arch equal to lb from a to A, and bafte out the Side to the Lines lb and ok. Then you have one Side, and the Back fquar’d, which is the greateft Difficulty in the Formation of a Twift- ed Rail, becaufe the two other Sides are found by gauging from them. Note, If the Triangles in Figure xxxix, and Lines whereon they ftand be fuppofed to be raifed up perpendi- cularly, then will the Lines AB, K L, jyZ, WX, BC, HI, PO, join to each other, and produce one Line perpen- dicularly over A, equal to feven Rilings or Heighths of the Steps. But in working a Rail of this Kind, you have need of but one Triangle ABcEr/, becaufe they are all equal, and of but one Effect in working, they being drawn only to fatisfy the Curious in the Nature of the Thing. PRO- The ART of Sound Building. gr / PROBLEM XXXY. Flow to frame the Arch ? or Mold for a Hand - to an Oval Fair of Stairs , fo that the fame Jhall ft and perpendicularly over the Fro- file , or Arch of the JHelFHole . FIGURE XLH. HP HE Arch Akmoqs wy is found after the fame Way as the Arch A h k npr t is found in Figure x xxvm ; and Figure xliii bears the fame Relation to Figure xlii, as Fi- gure xxxix does to Figure xxxvm, and is made thus: a, b, c, C, are the Centres* by means of which the Oval is made; a being the Centre for linking of the Part Go of the Oral, from whence the Lines aG, ah , afi are drawn ; b the Centre for ftriking the Arch from whence are drawn the Lines b^, bm , and b n ; and C is the Centre for drawing the Arch^zq from whence the Lines eg, c A, c u are drawn, which Lines fhew where the Rail mull anfwer fquare. Now to ah raife a Perpendicular from h to I, equal to the Riling or Heighth of one Step of the Stairs, and from 0 to aO raiie a Perpendicular to P; raife another to cu 7 from u to V ; another from to c A, from A to Z ; another to eg, from g to h ; another to C n, from n to o ; another to b t, from t to u ; and another to a b , from % to i . Then from I draw the Line I m, equal and parallel to ah; from P the Line P S, equal and parallel to ao; from V the Line V Z, equal and parallel to uc; from B the Line BF, equal and parallel to CA; from H the LineHL, equal and pa- rallel to gc ; from o the Line oz, equal and parallel to ac; from u the Line u X, equal and parallel to tb ; and from 1 the Line 14, equal and parallel to a b. This being done, at the Points a, m, S, Z, F, L, x, 4, to the Lines ab,Im, PS, V Z, BF, HL, 0^, ux, 14, raife the Perpendiculars aD, mn, ST, ZY, EF, ML, S y x, 45, each equal in Length to the Heighth of one Step, and draw the Hypo- thenufal opi The ART o/Sound Building. thenufal Lines DG, »I, TP, Y Y, EB, MH, os, u y, i 5, and fet oft the Width of the Rail from G to e, I to /, P to V to ,v, B to H to k, 0 to p, w to u, and 1 to 3 , and fet the Stem of a Square on the Line GD till the Blade cuts the Point e,and draw the Line ef. After the fame Manner fet the Stem of a Square on the Line nl, till the Blade touches /, and draw the Line k /, and fo draw the other Lines q TP x, cd , i k, qp , zpv, 23, as in the laft Problem. Note , If the Triangles in this Figure were raifed up per- pendicularly, then would aY), mn , ST, ftand perpendicu- larly over a , and ^ Y, FE, ML, perpendicularly over the Point C; and s^s-yx, 45, perpendicularly over the Point b ; lo that in this Figure you will have Occafion for two different Triangles, becaufe there are two different Sweeps that are the Caufe of two different Twifts in the Rail ; and fo the the Triangles a G D, VZY, are enough for fquaring of this Rail; and always obferve, that as many different Sweeps as are contain’d in the Ground Work of the Rail, there are fo many different Twifts, and confequently fo many different Triangles, becaufe the Twift is found by Means of them. PROBLEM XXXVI. How to form the Arch or Mold to the Hand- Rail of a Tair of Stairs that fweep tvjo Steps , fo as to ftand perpendicularly over its Ground , and the Manner of fquaring the fame , without jetting it up in its To fit ion, FIGURE XLIV, XLV, XLVI, XLVII. F IRST draw Figure xliv, to reprefent the Ground- Work of the Rail, whofe Arch G G confifts of two different Arches, one whereof is a Quarter of a Circle, and the other a Quarter of an Oval. AB (equal to AC, equal to CD, equal to BD) is equal to one Third of a Step, and D is the Centre to the Arch CB: Alfo BF is equal to two Thirds of a Step, and FG is equal to one Step and two Thirds. ... _ . 6 . \ i V — The ART of Sound Building. gg Thirds, by Means of which, and BF, is the Arch GB de- fcribed. G K reprefents the ftraight Part of the Rail to one Step, and the Arch HD is drawn by gauging from the Arch GC, that is, it is drawn parallel to it ; and the ftraight Part I H is found by gauging from K G, or is drawn parallel to it. Figure xlv. {hews the Manner of drawing the Rake or Arch of the Rail, which is done thus: Draw IL equal to GK of Figure xliv. andreprefent the Tread of the Steps, as before, by prick’d Lines. Then divide that Part of the Ground-Work of the Rail that belongs to each Step into any Number of equal Parts, as AF into 5, and FK into 4. This being done, draw AB, BC, CD, in Figure xlvi. to reprefent the Riling and Tread of the Steps; and con- tinue out the Line C B, at Pleafure, towards T, in which fet the five Divifions on the Ground of the Rail to the firft Step, FE, of Figure xlv. being equal to G I, of Figure xlvi. alfoED equal to ik, DC to A/, C B to lu , and BA to u T. Then will the Line CT, in Figure xlvi. be equal to the Arch AF of Figure xlv. draw the Line DT. Then is the Triangle CDT the Bracket to the firft Step, according to the Sweep of the Rail ; and as T G is the Length of the Ground to the firft Step, fo is TD the Length of the Rail anfvvering to it. Then from the Points i, A, /, «, raife the Perpendiculars i P, AQ^, / Z, u S, to CT, and take the four Divifions on the fecond Step, and fet them in the Line CT, from C to B, and draw the Line BD ; and then is B G the Length of the Ground to the fecond Step, and B D the Length of the Rail anfwering to it. Draw Lines through thefe Divifions, as from F to m, G to n 9 and H to 0, per- pendicular to C B ; and fo your Perpendiculars are found, according to the Compafs-Brackets of each Seep, and may be pieced thus. In Figure xlvi. take T S in your Compaftes, and with that Diftance, fetting one Foot in A, in Figure xlv. ftrike the Arch w, and take S u between your Compaftes, and with one Foot in B ftrike another Arch to interfedl the Arch m. Again, take SZ, or ST, in your Compaftes, and with one Foot in the Interfeflion of the Arch m, and this latter Arch, deferibe the Arch n ; and take / Z in your Com* pafleSj and with one Foot in C, deferibe an Arch to inter- K fed 34 The ART of Sound Building. fe£t the Arch and thus proceed on, fo that ^q be equal to n o, QJP to op, P D to p q, ^ to Bo, to o», s t to hm, and tu to mD- as alio &Q_to Do, i P to E p, CD to q F, H .- '-J F IRST draw the Line A u , and to A u raife a Perpen- dicular from the Point u to E, equal in Length to the Ramp of the Arch; and draw the Line AE, and halve it at C, and from C raife a Perpendicular towards F. This being done, defcribe the Arch BGD, according to the Di- rections laid down in Promblem xi, and defcribe the Arch AIE thus: Take BA, or DE, and fet it perpendicularly from G to I, and draw the Lines A I, EI. Then take Cl in your Compaffes, and letting one Foot in A, with the other ftrike the Arch O : Alfo with one Foot in E, ftrike the Arch s with the other, and take AC, or CE, in your Compaffes, and fetting one Foot in I, with the other ftrike the Arches p and t, and from the InterfeClion of the Arches op, and st, draw Lines to h and /, cutting the Lines A I, I E, at the Points q and m, perpendicular to them. Now take the Line b q in your Compaffes, and fet it from B towards A, reaching to the Dot in A B, and take the Remain- der from the' Dot fet A in your Compaffes, and fet it twice in the Crofs-Line from q . , towards the Interfering Arches op, to which draw Lines from A and I. Alfo take l m in your Compaffes, and let it from D to the Dot in the Line DE, and take the Remainder from E to the Dot, and fet The ART of Sound Building. it twice in the Crofs-Line from m towards the Interfering Arches st to k , and draw the Lines I A, Ek, each of which divide into the fame Number of equal Parts ; as likewife the Lines drawn from the Interfe(Tion of the Arches op to the Points A and I: Then il Interfering Lines are drawn, they will beget the Arch-Line A IE parallel to the Arch- Line BGD. This being done, divide the Arch-Line AIE into the Courfes of Brick, or Stone ; and from the Divifions thereof draw the ftraight Joints by a Lath to the Line CF, as in the foregoing Example, Fig. lx. and mark the Crofs-Joints pa- rallel to the Arch-Line BGD, and the Work is finiffuh PROBLEM XLVI. How to work a Gothick Arch in Stone . FIGURE LXII. F IRST defcribe the Arch BEC, according to the Di- rerions laid down in Problem xn. and fet off the Width of the Arch A B, CD, and draw the Out-Lines DH, HF, parallel to the Lines CG, GE, wherein fet the equal Divifions, and draw the Interfering Lines, which will generate the Arch-Line FD parallel to CE. Then by by Problem iii. find a Centre anfwering as near as poifible to the Arch CE, which fuppofe to be where the two Arches / and m meet , and this will be that towards which the Joints mull point ; and in like manner, the Interferion of the two Arches i,k, is nearly the Centre of the Arch-Lines A F, B E. This being done, fet the Courfes on the Arch- Line AFD, and then draw the Joints pointing to the In- terferions of the Arches ki and Im . PRO- 44 The ART of Sound Building, PROBLEM XL VII. How to work a Gothick Arch in Brick ; or Stone . FIGURE LXIII. F IRST draw the Arch BFD, according to the Di- regions laid down in Figure xm,and fetoff the Width of the fiat Surface of the Arch AB, DE. Then draw the Out-Lines parallel to each other, as in the foregoing Exam- ple, and defcribe the Arch AGE by the InterfeClions of Lines, which will be parallel to the Arch-Line BFD ; upon which fet the Courfes, and draw the ftraight Joints pointing to C ; and the Crofs-Joints muft be parallel to the Arches AGE, BFD, and the Work is finifh’d. PROBLEM XL' VIII. The Centre whereon the Arch of a Bow-Window is turn’d being given, howto find another Centre that Jhall anfwer parallel to it, according to the upper Edge of the Surface of the Arch . FIGURE LXIV. F IRST defcribe the Arch BKC, according to the Di- rections laid down in 'Problem xxviii. and fet the Width of the flat Surface of the Arch from B to A, and from C to D ; and draw the Lines AD, BC, and halve them at F and E, from whence draw a Perpendicular of a Length at pleafure to H. Then in any convenient Place ( Figure lxv.) draw a Line at pleafure, as from A to G ? and from A draw to A G the Perpendicular A F. Then take E I, in ( Figure lxi v, ) and fet it from A to B, ( Figure lxv.) and E I Vrom A to C. This being done, take the Semi- Diameter B E, or EC, (Figure lxiv.; and fet it from A to The ART oj Sound Building. 45 to D, ( Figure lxv.) Alfo take A B, or C D, and fet it from D to E, and draw the Line EC, which fet in the Line EH from E to g. Again, take the Width of the flat Surface of the Arch A B, or C D, and fet it in the Line ELI, from K to 7, and divide the Remainder from 7 to 9 into feven equal Parts. Alfo divide the Arch B K into feven equal Parts. Then take K 1, in the Line EH, between your Compafles, and fetting one Foot in 1, with the other ftrike the Arch 1 at pleafure : Then take K 2, and ftrike the Arch 2 : Alfo take K 3, K 4, K 5, and K 5 , feverally, and ftrike the Arches 3,4, 5, and 6. When this is done, open your Com« paOes, and divide from A to g, keeping the Points of them on thofe Arches, till you have gotten feven equal Diftances from A to^; at the Points of which, if Nails be ftuck in, and a thin Rule be bent round them, from A to g, along the Edge thereof the Arch A^ may be drawn. And in like manner may the Arch D g be drawn. PROBLEM XLIX. How to work a Nich in Brick . FIGURE LXVI, LXYII. F IRST defcribe the Semicircular Arches AIG, and B H F, and draw the Courfes pointing to the Cen- tre, and from the Courfes in the Arch-Line BHF, draw Lines to the Centre, to reprefent the Joints of the Nich ; and defcribe the Crown c e according to the Thicknefs of the narroweft Part of the Courfes. Then take a Piece of Plank, as is reprefented in Figure lxvii, and upon it draw a Quadrant bd, equal to BH or FA, and cut the End ah off, to anfwer the End AB in Fig. lxvi, and fet the Thicknefs and Summering of the Brick, mark’d 2, on the End thereof ; fo that 2 and 2 are equal to each other. This being done, plane the Piece to its Thicknefs, at the End a b, gradually to nothing at the Point d ; and take c D, Figure lxvi, which is half of the Diameter of the Crown, and fet it from d to e, and lay a ftraight Rule from c through e to /, N and a6 The ART of Sound Building. and cut the Piece off at ef. Then does the Mould abfe anfwer to every Courfe in the Nich. And mark the Grofs Joints upon them, Petting one Foot of your Compaffes at D, according to their Brackets on the Edge of the Mould, whereon you have all the Bevels, Lengths, and Thick- neffes. PROBLEM L. How to work a Level Arch in a Circular IV all. FIGURE LXVIII. F IRST ftrike the Sweep AD, reprefenting the Wall wherein the Arch Hands, and thereon fet B C, the Width of the Door or Window. Then draw the Lines BI, Cl, and fet the Width of the Surface from B to /, and from Cto m, and ftrike the Arch Im from I. This being done, ftrike the Arches E F, G H, from the fame Centre, after the fame Manner as the Arches BC, lm, only letting them be fo much the longer, as are the two Skew- Backs of the Arch ; then halve B C at N, and E F at O, and fet the Mould E F G FI, perpendicularly over the Mould B C, Im , and fo much above it as you delign the Surface of the Arch fhall rife, the Point O being perpendicular over N. Then on the Centre I raife a Perpendicular to K, letting I K be equal to N O. This being done, fet the Courfes on the Arch E F, ac- cording to the Thicknefs of the Bricks, or Stones, you work with. Alfo divide the Arch BC into fo many equal Parts, as EF, which give the Thicknefs of each Courfe at the Bottom of the Arch. Then ftrain a Line, or lay a ftraight Rule from the Perpendicular at K to the Divi- ftons or Joints of each Courfe on the Arch E F, beginning at E or F, and draw Lines to the Arch GH, fhewing the Summering at the Top of each Courfe, according to the Sweep of the Wall. Again, draw Lines from the Divi- fions of the Arch BC, to the Arch Im , pointing to I, which Ihews the Summering of each Courfe at the Surface ; and fo you c The ART of Sound Bu I L DING. 47 you have the Width and Summering of the Mould given for the Surface of eachCourie on the two Arches EF, BC. But the Crofs-Joints are found thus : Take a Bevel, whofe Stem is long enough to reach from C to F, and fet thereon, placing the Blade to the Arch E F towards O ; and this gives you the Crofs-Joints for the firffc Courfe. Alfo fet your Bevel on the Next Divisions to C and F, and place it after the fame Manner, for finding the Crofs-Joints in the fecond Courfe ; and fo of the reft. PROBLEM LL How to work an Arch for a Bow-Window in Brick or Stone , which is a Semi-Circle , and f wells a Scheme . FIGURE LXIX. F IRST defcribe the Sweep A F, to reprefent the Wall wherein the Door or Window ftands, and fet on the Width CD, BE. Then ere£t a Perpendicular Ot, on the Centre O of the Arch A F, and ftrike the two Central- Arches BrE, C^D, to anfwer the Out-Lines of the Tops and Surface, by the Rule laid down mProblemLXiY. Which when done, put them into their due Pofition, letting % and r be perpendicularly over q. Then will Qa be equal to the Semi-Diameter of the Window, and divide the Courfes on the Arch BrE, according as they are Brick or Stone, and divide the Arch CzD into the fame Number of Parts as BrE. This being done, take a fmall Line, and ftrain it from the Centre O to C, bringing it round on the Surface to B, and from B to O again. Then take theHeighth from the Line C D, to the firft Divifion on the Arch C ^ D, at i , and fet it up perpendicularly from O towards t ; from whence ftrain the Line to i, bringing it round on the Sur- face to 2, from whence ftrain it level to the Perpendicular again. Then take the Heighth from the Line CD to 3, and fet it up perpendicularly from O towards t; from whence ftrain a Line to 3 , and bring it round on the Surface to 4 ; from 4 s The A RT o/Sound Building. From whence drain it level to the Perpendicular again, and fo on. As may be feen by the Courfes m p, and g n, let- ting the Lines remain on the Centres and Perpendiculars, that give the Summering, Twidings, and Bevels, of every Courfe, from whence you are to make the Moulds, and take the Bevels. Note , The Point O mud be level with BCD E ; and alfo the Lines that are drain’d from the Perpendicular to the Centre. PROBLEM LIL , Flow to work a Regular or Irregular Groin in Brick or Stone. FIGURE LXX. "DEPRESENTS the Centre for an Irregular Groin, where KL is ten Feet, and OH eight Feet. Divide the Backs or Circumferences O QJd, KFL, into any equal Number of Parts, according to the Thicknefs and Quan- tity of Courfes contain’d on the larger Centre K F C. Then will the Dividons or Thickneffes of the Courfes on the lefler Centre O QFI be fomething thinner, and your Bricks or Stones mud be provided accordingly, and fo you will pre- vent dropping of Courfes. This being done, on thole Di- vidons drike Chalk-Lines, which are reprefented in the Example by Prick’d Lines ; and theie are the true Guides by which the Courfes mud be work’d level, and alfo to take the Levels of the Groin. Having done this, begin to work the drd Groin-Brick after the following Manner: Take the Bevel i on the Cen- tre, and fet it on the Edge of the Brick R, and from the Bevel drike a fquare Stroke on each flat Side of the Brick, to which cut the Levels of the drd Brick. For the lecond Brick, fet the Stem of the Level 2 on the drd Line, and the Blade thereof to the Arch or Curve of the Centre, and this Bevel fet on the Edge of the Brick D : Then take the Bevel 4, and fet the Stem to the fecond Line 3, on the fame \ The ART of Sound B u ILDING, fame Centre, turning up the Blade on the other Centre-, to anfwer the Summering of the fecond Courfe, which Bevel fet on the upper Side of the Brick D from the Bevel 2 ; to which cut the Bevels of the fecond Brick , and place it in its Pofition, and you will find that it does not anfwer the Summering of the Courfe on the reverie Cen- tre ; to do which, make a Mark on the Brick where it touches the fecond Line on the reverfe Centre, and fet back the Width of a Courfe from that Mark to the upper Edge of the Brick where it lhall fo fall out, and from thence flrike a Line to the firft Mark at the fecond Line, as alfo to the upper Corner of the Brick at the Curve of the Centre, to which cut the fecond Brick, ( according as they appear in the Examples C and E,) which anfwers the Summering of both Courfes; and obferve that as much as you take off from the upper Side to anfwer the reverfe Courfe, fo much you muff add to the under Side by a Clofier, the Figure whereof appears in the Example F ; which, if iimmer’d to the Brick* will add much to the Strength and Beauty of the Work. Now find the Bevels and Summering of the third Brick thus : Set the Stem of the Bevel 3 and the Blade to the Curve of the Centre, and then fet the Bevel on the Edge of the Brick B, which done, take the Bevel 4, and fet it on the third Line, on the fame Centre, turning up the Blade thereof on the reverfe Centre, to anfwer the Sum- mering of the Courfe, which let on the flat Side of the Brick B, from the Bevel 3 ; to which Bevels cut the third Brick, and place it in its Pofition ; then make it to anfwer the Summering of the reverfe Centre, as has been before taught, by marking the Brick where it touches the Level- Line on the Centre, and cutting back the Width of the Courfe, and cut the Clofier accordingly. And thus proceed on till you have gone fo far with your Groin, that the Summering turns up the Brick its whole Thicknefs above the Level-Line ; and then the Remainder of the Groin mull be all Headers and Clofiers ; but the Bevels are taken the fame. Only obferve that inftead of placing the Stem of the firft Bevel on the Edge of the Brick, you mull place it on the End, as in the Example on the Brick N Ihews, and fo on ; of- fering that as your Clofier increafes in Thicknefs, from no- O thing 50 The ART of S ound Building. thing at the Springing, fo it muft decreafe to nothing at the Crown. And thus any Regular or Irregular Groin may- be work’d. Figure lxxi. fhews the Infide of the Groin after it is hnilh’d ; where you may fee by the Clofier the Summering of each Courfe, and the Bevel of every Brick. PROBLEM LUX. How to diminijh a Column , or Tillajier. FIGURE LXXII. W HEN the Heigbth and two different Diameters of the Column are given, proceed thus : Divide the whole Heighth into three equal Parts, and draw the firft third Part from A to B, perpendicular to the Bafe ; and from B draw a Crois-Line parallel to the Bafe; upon which defcribe a Semicircle, having its Diameter equal to the Diameter of the Column below. Then draw a Line down the Middle of the Column, or Pillafler, as D E perpendi- cular to the Bafe. This being done, take half of the Diameter of the Column above, and fet it both Ways parallel to the Bafe, from the Middle Line ED to the Arch of the Semicircle before defcribed, where it fhall fo happen, as at 66; and divide that Part of the Circle from the Springing of the Arch to the Points 66 into any Number of equal Parts, as fix, and draw Lines parallel to the Bafe, to the cor- refpondent Divilions on each Side the Circle as i i, 22, 33, 44, 5 5, 66. Then divide the upper Part of the Column, from B to the under Side of the Affragal on the Neck of the Column into the fame Number of Parts as you did that Part of the Arch between the Springing and the Point 6, which is 6, and at each Divifion draw Lines a-crofs at Right- Angles to the Middle-Line ED, and take half the Line 1 1 in the Circle, that is, from the Middle to 1 , ei- ther Way, and fet it both Ways on the Line 1 1 , upon the up- per Part of the Column. Moreover, take half the Line 22, on The ART of Sound Building. 51 on the upper Part of the Column, and fo on; and the fix Lines in the Circle are equal to the fix Lines on the Part of the Column; and in the extreme Points flick Nails, and bend a thin Lath round them, fo that it touches each of them ; then with a Pencil, or any other Marker, defcribe the fwelling Part of the Column that is generated by the Circle, which, when drawn upon a Board, is a Mould to work by. PROBLEM LIV. How to work a Diminijhing Tillafier in Brick FIGURE LXXIII. F IRST make a diminifhing Rule to fit the whole Side of the Column, nr Pillafter ; and if it be large, fo that a Board will not reach the whole Length, you muft make it federal Lengths, always obferving to keep the Rule in its proper Place, according to what Part of the Co- lumn it belongs, then begin to work. The Example F is the firft Courfe, and the Example G the fecond, and thofe continue perpendicular all of the fame Bignefs to one third Part of the Heighth of the whole Column, that is, from the Bafe to the Courfe AB, and from thence it diminifhes fomething to the Collar of the Capital; the Out-lines being drawn according to the Rule in the foregoing Problem, divide your Courfes all of one Thicknels, and divide the breaking Joints, fo that they be parallel to the Out-lines as in the Example; as the Courfe AB confifts of two Streachers and one Header, which is the half of a Streacher, and muft be divided into five equal Parts, being two to each Streacher, and one to the Header, and fo of the Courfes ef. \ i A, pq , and tu ) they bearing the fame Proportion to each other, tho’ they are fhorter, by reafon of the Diminilhing. The Courfe c d confifts of two Headers, one Streacher, and two Clofiers, and muft be divided into ten equal Parts, becaufe two Clo- fiers are equal to one Header, and four are equal to a Streacher, Underftand the fame of the Courfes cd, Im , no y and 52 The ART (//Sound Building. and rs , they all bearing a Proportion to each other, after the like Manner. Then upon thofe Divihons, where- on the Joints are to come, as may be feen by the Dots in the Example, Pick in fmall Nails* and, as in the laft Pro- blem, bend a thin Rule about them, and with a Marker defcribe the Crofs-Joints, which will diminifh parallel to the Out-lines. PROBLEM LV. How to work a Diminijhing Column in Brick FIGURE LXXIV* F IRST defcribe the Out-lines of the Column, and the Thicknefs of the Joints, as in the foregoing Example, and draw the Plane of each Courfe, as bi, from whence the Arch, Dhninilhing, Length, Width, and Bevel, of each Brick is found. One Third of the Column from ab to ab is perpendicular, and from b to h y it is fomewhat fwel- ling, according to Figure lxxii. therefore you muft work thus : Rub the Bricks ail to one Thicknefs, and take a Block juft the Thicknefs of the Bricks you work, and about thres Inches fquare, and fet the Centre thereof juft over the Cen- tre of each Plane, and nail Imall Fillets about it, on the Board upon which the Plane is drawn, to confine it in its Place without nailing it down, becaufe hereafter you will have occafion to take it pff, and fet it on at every Brick you work, the Ufe thereof being to raife equal to the Heighth of the Brick ; fo that you fhall ftrike the Arch and Bevels thereof without Moulds, which would be troublefome in the diminiiliing Part of the Column ; but in the perpendi- cular or upright Parts, you may work with Moulds taken from the Planes, becaufe one ferves for all of that kind. But I fhall proceed thus : Take the firft Brick, and lay it upon the Plane of the Brick 12, in the Plane h ; then take half the Diameter of the Pillar, or Column, in your Compaftes, and fet one Foot in the The ART o/'SotTNDBlJILDlNG, ^ the Centre on the Block, and with the other ftrike an Arch on the flat Side ol the Brick to anfwer the Outiide of the Column. This being done, take the Diftance i 2, 23, or 34, and fet it upon the Brick or the Arch, and on the extreme Points lay a ftraight Rule to the Centre h, and along the Side thereof mark the Joint or Summering of the Brick ; then take it up, and from thefe Joint-Strokes fquare over the Edges thereof to End the Joint-Strokes on the other Side of the Brick, upon which draw the Joints or Summering ; as alio an Arch equal to the former, by Betting the former one firft to fit on the Arch of the Plane. Then, with your Compares opened to the fame Diftance as before. Petting one Foot in the Centre on the Block, with the other ftrike an Arch on the other Side of the Brick. Then you will have the Joints, Arch, and Summering of the Bricks, and fo of the reft, is? c. Now work the diminishing Part of the Column thus: Firft take off the Block, and then take half of the Diame- ter of the Column at duo Joint 1 in ymir Compaftes, and f et ting one Foot in the Centre b, with the other ftrike a Circle within the Circle of the Plane, which Circle Mark 1 : Alfo take half the Diameter of the Column at the Joint 2, and fetting one Foot in the Centre i in the Plane z, with the other ftrike a Circle equal thereto. Moreover, take half the Diameter at the Joint 3 in your Compaftes, and fet- ting one Foot in the Point /;, in the Plane /;, with the other defcribe a Circle. Again take half the Diameter of the Joint 4 in your Compaftes, and fetting one Foot in z, with the other defcribe a Circle, and fo on ; alfo defcribe a Cir- cle, whofe Diameter Shall be equal to 5 in the Plane /?, and a Circle whofe Diameter ftiall be equal to the Joint 6 in the Plaue z, and fo of 7 and 8, and the reft to the Capi- tal of the Column ; and always obferve to draw the Circle that reprefents the under Side of each Courfe in the Plane to which it belongs, that is, if it be all Streachers to b ) if it be Headers, Streachers, and Clofters to the Plane z. Having the Circles all drawn to reprefent the under Side of each Courle, fet the Block in its Place, and begin to work the diminish’d Courfes mark’d thus : Take a Brick, and lay it upon the Plane, as is taught before; then take half of the Diameter of the Circle 1 , in the Plane or the Joint 1 , on the Column, in your Compaftes, ap.d fetting P one £a The ART o/Sound Building. one Foot in the Centre /;, on the Plane /;, with the other ftrike an Arch on the Brick, to reprefent the Outfide of the Column of the Joint i . This being done, lay a ftraight Lath from the Centre h, on the Block, to the Point, let- ting it lie perpendicular over the Line i h, and along the Edge thereof ftrike the Joint or Summering of the Brick; alfo lay a ftraight Lath from the Centre h to 2, perpendi- cularly over h 2, and turn the Brick upfide down, and fet the Arch and Crofs-Joints juft on the Arch and Crofs- Joints of the Arch 1 in the Plane/;; then take half the Diameter of the Arch 2 in the Plane /, or half the Dia- meter of the Column at the Joint 2, and fetting one Foot in b, with the other ftrike an Arch on the other Side of the Brick, and mark the Joints or Summering of the Brick as before, by laying a ftraight Rule from the Centre h to 1 and 2. This being done, cut the Arch on the Brick to re- prefent the Outfide of the Column; which, when done, draw the Crofs-Joints on the outward Edge of the Brick from the Lines on each fiat side thereof, which cut to thefe Lines, and you will find the Brick and Joints to an- fwer the true Diminiftiing of the Column. Underftand the fame for the reft of this fort of Courfes. To work the Courfes mark'd 2, take a Brick, and lay it end-ways upon the Arch 2 in the Plane z‘, at 4 and 5, and keeping the Compafles at the fame Diftance as when they ftruck the Arch on the upper Side of the Brick mark’d 1 , in the Plane /;, which is half the Diameter of the Circle 2, or Semidiameter of the Column at the Joint mark’d 2, and fetting one Foot in the Centre z, with the other ftrike an Arch on the upper Side of the Brick ; then lay a ftraight Rule from the Centre i to 5, and ftrike the Summering of the Brick. Moreover, as before taught, turn the Brick upfide down, and fet the Arch and Summering Stokes or joints of the Brick to the Arch 2, and the Crofs-Lines 4, 5. This being done, take the Semidiameter of the Circle 3 , that is, the Semidiameter of the Column at the Joint 3, and retting one Foot in the Centre z, with the other ftrike an Arch upon the upper Side of the Brick equal thereto. Then, as before, lay a ftraight Rule from the Centre i to 4 and 5, and upon the Edge thereof fet on the Summering Joint, upon the upper Side of the Brick. When you have done this, cut the Arch on the End of the Brick, as alfo the - - . • -• . The ART ofSouND Building. £,5 the Summering Joint, which, when done, is a Header to the Courfe mark'd 2, and anfwers to the diminilhing and true Joints of the Work, the Streachers and Cloliers being mark'd after the fame Manner ; and fo of the reft of the Courfes which when done, the Work is finiih’d. N. B. The Circles, mentioned in the foregoing Problem, are not jinferted, becaufe the Scale being fo fmall would not contain them ; and obferve to number them as the Joints on the diminilhing Part of the Column, becaufe they are equal to each other. PLATE XIV. R Eprefents twoRaking-Collonadoes on the Side of a Hill, wherein Walks are cut; and they are here inferted only to Ihew the Ufe of Raking-Arches, fuch as may be feen on the Side of Ricbmond-Hill , leading to the old Wells. PLATE XV Reprefents three Flights of a Stair-Cafe with the Ceiling under the Gallery, or Landing-Place, under which the Point of Sight is taken ; and they are here reprelented to Ihew the Ufe of the following Machine in Plate xvr. which is to turn Raking-Mouldings, Ballifters, or any other Raking-Work of that kind, which, in my Opinion, would be very beautiful in this kind of Stairs, to have the upper Mouldings next to the Hand-Rale, to rake equal thereunto, but the lower Mouldings next to the Step to be fquare an- fwerably. FIGURE LXXVXIL I S the Machine, the Nature of which may be feen at one View, by lifting up the Handle of the Bow wherein the Swivle of the Crane turns, which will caufe the Socket thereof to turn out of a Perpendicular, the Swivle to which the Ballifter is fatten'd, being let therein mull: confequently draw too and fro, fo much as the Crane turns out of a Perpendicular : Therefore fo much as the different Mouldings rake, you mull lift the Handle higher or tj6 The ART of Sovnd Building. or lower accordingly ; and if you have occafion to turn Square -Work, move the Handle down to the Bottom of the Mortife, and then your Work will be level, and turn perpendicular. The different Parts of this Machine are put together in the following Manner: The two Hangings of the Bow A are let into two level Pieces on each Side the Puppet -Head, fo that the Eyes in which the Spindle of the Crane turns are level with the Spindle to which the Ballifcer is faften’d, and the Bottom of the Mor- tife in the Puppet-Head muff be level with the Gaugings of the Bow. The Eyes BB frame into the Plate of the Bow A, and CC are the Keys to faften them. On the under Side D is the Swivel which goes thro’ the two middle Pup- pet-Heads ; the one End fits to the Socket of the Crane, and the other to the Box that fallens the Balliller. E is the Box which fcrevvs on to the End of D ; f is the Nut to fallen it on; and gg are the Screws that fallen the Ballillers into the Box E. H is the Mandrel that goes thro’ the Puppet-Head, whereon the End of the BalliHer turns, by boring a Hole in the End of the Balliller fome- thing deeper than the Rake of the Mouldings, becaufe of its Hiding to and fro. I is the Spindle of the Crane that is let into the two Eyes B B; and A is a round Nut that fallens the Spindle I into the two Eyes B B. L is the Draught of the Balliller which all Turners mull have be- fore them, to find the Rake of each Moulding. M is the Socket in which the End of D goes; the upper Hole thereof mull be a very little bigger than the End of the Spindle, and the lower End mull be fo large, that when the Handle of the Bow is lifted up to the Top of the Mor- tife, the Inlide of the Hole Hiall be perpendicular* FIGURE LXXIX Is the Plane of the Lath ; and N, O, P ? Q^, are the Mor- tiles wherein the Puppet-Heads Hand. FINIS. ; ■ ' ‘ X / ■ - r>..w \ P/rtfc 17 . Th e ELEVAT I ON , & SECTI ON.of a HOUSE ofmvlNVENTION. withy PLAN S,Sc Sec T ION. as in rhe following- Plate : A - >■■■ .-V-GT.-i ■" ■ •- U'-r v.\ \ \ • ......... i" ’ !'■ - 1 ■' •• j I i tuts*# ' ,-x t ■•«■«. I r . ' > > ' > • ..v ■ ERRATA. P AGE Figure VII. Line 4. read as are drawn in Figure VI ; p. 7. 1. 1. r. Line CE, which halve in the Point D ; p. 10. Fig. XV. 1. 5-. r. Then taking CE; p. 12. Fig. XVIII. 1. r. r. halve it at C ; ibid. 1. p, 8c 6. dele the Words and draw the Lines A D and BP; p. i j. 1. 7. 1 . Scheme under a Semicircular Arch-, p. 7 r. Fig, XXI. 1. 7. r. Lines AG, and D'H -perpendicular to AD; p. 16. Fi^.XXII. 1.8. r. AG and DH; p. 18. 1. 9. r. lg; ibid. 1. 10. r. Line f 1 ; p.19. Fig. XXV. 1.2. for equal r. unequal-, p.20. L4. r. Lines OP, ST, V X, L M, and IK; ibid. Pig.XXVI. 1.6. r. Arch vzt; p.i2. 1. if. r.Lineszn, nk, and ok; p.23. in the two lad: Lines but one of Fig. XXVIII. dele drawn from the Extremities of the Ferpendiculars ; p. 28, 1. 22. r. Arches p and <3, over E; p. 29. I.4. r. DR forDIC; p. 30. ip. r. as AH; ibid, lip. for bafte r. bofle-, ibid. I121. r. Ih from o to k; id. ibid, for bafle r. bofle-, p 31. Fig. XLIL 1. 7. r. Lines aG, ah, aO, are drawn-, p. 3 5". Fig. XLVIII, ,XLIX, L, LI. r. uv to ml; p.48, Fig.LXX.l. 18. for Bevels r. Levels. ADVERTISEMENTS, A L L Gentlemen that have occafion for fuch Mathematical In- ftruments as are ufedin Architecture, I recommend them to Mr .Thomas Wright, (Mathematical Inftrument-Maker to his Royal Highnefs G E O RG E Prince of Wales ,) at the Orrery and Globe next the Globe-Tavern in Fleet-Strep , in Juftice to his great Care in: making them very accurately, and for his great Choice of Ma- thematical Inftruments in general. William Halfpenny. f * r S * Bowen Whit ledge, Son of Mr. Robert Whitledge Deceas'd, at the Red Bible in Ave-Mary-Lane, Ludgate- Street, near St. Paul's, London, S ELLS all Sorts of Large Bibles and Common Prayers for the Ufe of Churches, Chapels, &c. and all other Sorts and Sizes of Bibles and Common Prayers, with or without line Cuts, ruPd or unrul’d, in all Sorts of Bindings, as right Shaggreen or Turky, with or without Silver Work; Common Prayers in French , and in French and Engtifh , and French Teftaments, new eft Edi- tions; ordinary Bibles, Common Prayers, Teftaments, andPfalters; Duty of Man, and other School Books; the two laft Volumes of the Statutes; the new Edition of the Articles and Canons, Books of Devotion, and on the Sacrament; Wholefale and Retale, at rea- fonable Rates. By the faid Bowen Whitledge is Rubli fil’d and fold , A New, Neat, and Curious Set of Cuts ; containing fifty feven in Number; among which are the Heads of King James I. CharlesI. Charles II. and his prefent Majefty King George: Alfo four new Defigns never done in any Set yet publifh’d, viz. Rub lick Baptifm , Matrimony , the Vi ft at ion of the Sick , Burial of the SDead. All the Defigns entirely new, and finely engraven by B. Cole, G. Rick ham, and other the beft Mafters. k & J 'V .. .. : . .