THii 
 
 Carpenter’s New Guide: 
 
 BEING A 
 
 COMPLETE BOOK OF LINES 
 
 FOR 
 
 CARPENTRY and JOINERY. 
 
 TREATING FULLY ON 
 
 Pradtical Geometry, Soffits, Brick and Plaifter Groins, Niches of every Deferip- 
 tion. Sky-lights, Lines for Roofs and Domes, with a great Variety of Defigns 
 for Roofs, Truffed Girders, Floors, Domes, Bridges, &c. ; — Stair-cafes and 
 Hand-Rails of various Conftrudions ; Angle Bars for Shop Fronts, &c. ; 
 afld Raking Mouldings ; with many other Things entirely new. 
 
 The whole founded on true Geometrical Principles ; the Theory and Pradice 
 well explained, and fully exemplified 
 
 ON SEVENTY. EIGHT COPPER-PLATES, 
 
 CORRECTLY ENGRAVED BY THE AUTHOR. ^ 
 
 INCLUDING 
 
 SOME OBSERVATIONS AND CALCULATIONS ON THE 
 STRENGTH OF TIMBER. 
 
 B Y 
 
 PETER NICHOLSON. 
 
 L O N D O N; 
 
 PRINTED FOR I. AND J. TAYLOR, 
 
 AT THE ARCHITECTURAL LIBRARY, No. 56, HIGH HOLBORN, 
 
 MOCCxeilU 
 
 
PREFACE. 
 
 T O a book intended merely for the ufe of Praaical Mechanics, much Preface 
 is not neceflfary : — it is proper, however^ to fay, that whatever rules by 
 previous authors have on examination proved to be true and well explained, thele 
 have been feleaed and adopted, with fuch alterations as a very clofe attention 
 has warranted for the more eafily comprehending them, for their greater accuracy 
 or facility of application •, added to thefe, are many examples which are entirely 
 of my own invention, and fuch as will, I am perfuaded, conduce very much to 
 facilitate the workman, and to the accuracy of the work. 
 
 In this Second Edition the arrangement is gradual and regular, fuch as a ftii- 
 dent Ihould purfue who wifhes to attain a thorough knowledge of his pro- 
 feflion ; and as it is Geometry that lays down all the firft principles of building, 
 meafures, lines, angles, and folids, and gives rules for defcribing'the various kinds 
 of figures ufed in buildings ♦, therefore, as a neceffary introduftion to the art 
 treated of, I have firft laid down, and explained in the terms of workmen, fuch 
 problems of Geometry as are abfolutely requifite to the well underftanding and 
 putting in pra<5tice the neceffary lines for Carpentry. Thefe problems, duly con- 
 fidered, and their refults well underftood, the learner may proceed to the theoretical 
 part of the fubjeft, in which Soffits claim a very particular attention ; for by a 
 thorough knowledge of thefe, the ftudent will be enabled to lay down arches 
 which fhall ftand exadly perpendicular over their plan, whatever form the plan 
 may be : on this depends the well executing all groins, arches, niches, &c. con- 
 ftru(fted in circular walls, or which ftand upon irregular bafes ; wherefore the 
 importance of rightly underftanding thefe I cannot fuffidently infift on, their 
 conftruition being fo various and intricate, and their ufes fo frequently required. 
 
 The next fubjed which regularly prefents itfelf is Groins-, for the conftrudion 
 of which there will be found many methods entirely new, and befides the common 
 figures, I have fhewn many which are difficult of conftrudion, and not to' be found 
 in any other author, I have difplayed a large affortm.ent of niches of each kind ; 
 
 A tliefe 
 
VI 
 
 PREFACE. 
 
 thefe are frequently wanted, thofe of the elliptic form only have yet been cit- 
 plained : in addition to thefe, here will be found fchcmcs for globular ones,, 
 which occur frequently in praftice. 
 
 Among the various methods for finding the Lines for Roofs, I have given an 
 entire new one for finding the down and fide bevels of purlines, fo that they 
 lhall exadlly fit againfl the hip rafter i and by the fame method the jack rafter 
 will be made to fit. 
 
 Of Domes and Polygons, I have (hewn an entirely new method for finding 
 their covering, within the fpace of the board, thereby avoiding the tedious and in- 
 commodious method of finding the lines on the dome itfelf, as has been always 
 pradtifed heretofore j alfo a method for finding the form of the boards near the 
 bottom, when a dome is to be covered horizontally.^ Of Dome-lights over ftair- 
 cafes, or in the centre of groins, a rule upon true principles is given, for finding 
 their proper curve againft the wall, and the curve of the ribs. This has never 
 before been made public. 
 
 Having gone thus far in the Science of Carpentry (viz. through the theoreti- 
 cal part), it is necefiary for me to fay, by way of caution and guard to the ardent 
 theorift, that there are on fome furfaces curve lines which cannot be found 
 abfolutely true to one another; fuch as fpherical or fpheroidical domes, where 
 their coverings cannot be found by any other means than by fuppofing them 
 to become polygonal ; in which cafe, they may be performed upon true prin- 
 ciples, as may be demonftrated. — Let us fuppofe a polygonal dome infcribed in 
 a fpherical one ; then, the greater the number of fides of the polygonal dome, 
 the nearer it will coincide with its circUmfcribing fpherical one. — Again, let 
 us fuppofe that this polygon has an infinite number of fides j then, its fur- 
 face will exadly coincide with the fpherical dome, and therefore in any thing 
 which we lhall have occafion to praftife, this method will be fufficiently near ; 
 as for example, in a dome of one hundred fides, of a foot each,, the rule for 
 finding fuch a covering will give the pradice fo very near, that the variation 
 from abfolute truth could not be perceived.. 
 
 The Theory of Carpentry being now gone through, and I hope well ftudied 
 and attended to j for the young ftudent mult not exped to be perfed mailer of 
 
 this 
 
P R E A e E. tU 
 
 this intricate and important fubje£t without feme pains and application, not- 
 withftanding the plain and dear manner in which I have endeavoured to lay- 
 down and explain thofe parts of his profdTion, which, on account of their im- 
 portance, claim his utmoft attention j and obferve, that, in order to combine 
 theory and pradtice, as well as to vary and enliven the fubje£t, I have uniformly, 
 with the theory, given examples of the pradice, yet keeping them diftind: 
 and feparate : , 
 
 We now proceed to the Pradical Part of Carpentry, in which is given ,a great 
 variety of examples for floors, trufles, girders, roofs, domes, and partitions, on 
 the neweft and beft principles. 
 
 In that nice and elegant branch of the Building Art, called Joinery, Stairs and 
 Hand-rails take the lead; and notwithftanding the great importance of this 
 fubjedb, I am forry to find it has been treated, by authors in general, in a very 
 clumfy and fiovenly manner. For Stair-cafes, in general, I have laid down right 
 methods, on principles entirely new, and which, fince the publication of the 
 former edition of this work, I have the fatisfadion to fay, have been put in 
 pradice, and found to anfwer well. 
 
 Various methods for diminifhing Columns are fhewn, together with two 
 new ones, which I flatter myfelf are more eafily adapted to pradice ; among 
 other things of inferior note, is a method for finding the Lines of a circular 
 Safh in a circular wall ; alfo, a method, to the fame purpofe, for Architraves in 
 a circular wall ; neither of which have before been given, or explained. For 
 mitring raking mouldings, I have, with fome pains, confirmed a true method, 
 not merely in theory, but by models, which I have by me, and am willing to 
 fhew, at convenient feafons, to any enquirer. 
 
 1 muft not here omit to obferve, for though laft not leafl, that my fpecula- 
 tions and calculations on the ftrength of timber, will, I hope, be found parti- 
 cularly ufcful ; and not merely fo, but may alfo tend to induce others to confider 
 this fubjed, whofe leifure and abilities may lead to more important difeoveries : 
 I beg leave to add, that, to confirm the mathematical calculations, I have tried 
 feveral of the queftions by experiments. He who is a perfed matter of this 
 branch, may err in decoration, but never can in ftrength and proportion. 
 
 A 2 I beg 
 
viii 
 
 PREFACE. 
 
 I beg leave to fay of the Conclufion, it is intended to guard the young and 
 incautious ftudent againft error*, for wrong maxims are with more difficulty 
 obliterated from the mind, than originally obtained. 
 
 To conclude*, as I pretend not to infallibility, I hope to be judged with 
 candour, being always open to convidion, from a knowledge of the difficulty 
 and intricacy of fcience*, yet I hope that my labours may be of fome ufe to others, 
 in fhortening the road, and fmoothing the path through which, for feveral years, 
 I have been a perfevering traveller for knowledge : I lhall then be fatisfied, 
 and not deem time milpent, if my labours tend to the public good, 
 
 P, NICHOLSON. 
 
 ' P. S. In this Second Edition the arrangement of the fubjec^s is progreffive 
 and regular*, and befides eighteen additional plates, many of the others have been 
 re-engraved, the fubje^ls, in fome, made more intelligible, and, in others, multi- 
 plied ; So that this edition may be confidered as a New Work. 
 
 C O N- 
 
c o 
 
 N T E N T S 
 
 OF THE 
 
 PLATES, 
 
 GEOMETRY. 
 
 Definitions 
 
 PerpfndicularSf bifediing angleSy ^c. ’ ' 
 
 Polygons^ tangents, ^c. • ~ 
 
 To defcribe figures whofe dmenfions are given, or to make one figure equal and fimilar to 
 
 another, ^ c. ' ^ ^ ^ ^ 
 
 To draw an arch of a circle to any given length und height, to defcribe an ellipfis to any given 
 
 length and breadth, with compajfes, or by ordinates — ^ 
 
 To defcribe an ellipfis with a firing, or trammel j to find the centre, and the two axes of an 
 ellipfis to proportionate an ellipfis about a parallelogram, ^c, — ' 
 
 To defcribe the eonic fediions from a given cone ~ 
 
 To defcribe elliptic arches, fegments, ^c. by interfe6ling lines 
 
 To defcribe the feaions of a cylinder, or any fegment of a cylinder, who/e plane is parallel to 
 
 its axis, to any given angle whatever ■ _ 
 
 To defcribe the jeaions of a globe, or any other folid generated by the revolution of any irre^ 
 
 gular figure about an axis — " ‘ 
 
 Plate 
 
 1 
 
 2 
 
 3 
 
 JO 
 
 iito IS 
 
 CARPENTRY. 
 
 Offofiits parallel, fining, or winding, in firaight or circular walls -- ■ 
 
 iGiven the form of an arch, to defcribe any other arch of the fame height, of any given width 
 jBuhatever — ■ ■ '' '■ ' 
 
 To 
 
X CONTENTS. 
 
 To defcrlhe the curb when a or elliptic window cuts above the celling line j to 
 
 back and defcrlbe elliptic ribs with compajfes ■ - 
 
 BRICK GROINS.- 
 
 'Centring for groins when level^ the ftdes cutting through each other at right angles — — 
 
 Ditto for inclined groins.^ when the interfediion of the angles are given upon the plan 
 Centring for inclined groins when the fide arches are given — " • - ■ — . 
 
 PLAISTER GROINS. 
 
 Various methods to defcrlbe the ribs of a groin when parallel to the horizon — — — 
 
 *To defcrlbe the ribs when inclined to the horizon — — — — — 
 
 To defcrlbe the arches of a groin parallel to the horizon, when the fide arches cut under the 
 body range, fo as to have the angles Jlraight upon the plan — — ' 
 
 To defcrlbe the interfeSling ribs of a TV eljh groin parallel to the horizon, the fde arches being 
 
 given, and cutting acrofs each other at right angles - “ 
 
 To defcrlbe the interfedling ribs of a TVelJh groin parallel to the horizon, Jlanding upon a 
 bevel plan '• •• — — — — — . ' 
 
 To defcrlbe the interfeSiing ribs of an odlagon groin parallel to the horizon, the fide arches 
 being given — 
 
 To find the interfediing body ribs of a circular groin parallel to the horizon, when the fide 
 arches are given, and the interfediion of the angles Jlraight upon the plan 
 Having the inclination of a circular groin, and one of the body ribs, to find the interfedling 
 and fide ribs - ■ ^ " ' " 
 
 NICHES. 
 
 The plan of a globular niche being a given fegment, and the front rib a femlcircle, to find the 
 fweep of the back ribs — ' ■ " ' 
 
 To find the fweep of the back ribs of a globular niche when the plan and front rib are feg- 
 ments of the fame width, but of different breadths ' ' 
 
 The plan of a femicircular niche being given, Jlanding in a circular wall, to find the front 
 
 and back ribs ■ ' — — ' ' ' 
 
 The plan or elevation of an elliptic niche being given, to deferibe the back ribs 
 
 SKY-LIGHTS. 
 
 To find the mitres or bevels of the files ofJky4igbts to any plan given — — 
 
 Plate 
 
 1 6 
 
 17 
 
 i3 
 
 19 
 
 2« 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 3* 
 
 LINES 
 
CONTENTS. 
 
 XI 
 
 LINES IN ROOFING. 
 
 Plate 
 
 To find the length and hackings of hip rafters^ and the form of the ends of a purline fiandlng 
 againjl them j alfo the proper forms of the ends of the jack rafters^ where they come againji 
 
 ^he hips ' , ■■ " »■■■ '■ ■ I.,. — ^2 
 
 To lay a fquare roof in ledgmertt ■ ■ • ■ ■ ■ 33 
 
 To lay a winding roof in ledgment . 1 ^< >^ ■ ■ ■ - — 3^ 
 
 To find the hips^ and to cover all manner of polygon roofs^ without drawing the covering 
 from the plan — — — — 1 . — — . — 3^ 
 
 To cover a dome with horizontal boards — _ 36, 
 
 To draw the ribs of an elliptic dome^^ and to find the form of the covering • — 37 
 
 To find the fehiions of domes placed over a fiair-cafey or when it is placed between the inter- 
 
 feSiions of groins 1.- — . . 3^^ 
 
 To find the fe^ions of an elliptic dome^ placed in the fame manner^ and to defcribe the ribs 39. 
 
 PRACTICAL CARPENTRY. 
 
 Trujfmg of girders u...— — 
 
 Roofs fuited for various purpofes . ■ — 
 
 Story pojisyand the manner of framing and piling of a bridge 
 
 40 . 
 
 41 to 49 
 50 
 
 JOINERY. 
 
 OF STAIRS AND HAND-RAILS. 
 
 To find the mitre cap of a raily thefeblion of the rail being given ■- — — 
 
 J dog-leg Jiair-cafe, with a half [pace 
 
 A dog-leg fiair-cafey with winders all round — -- 
 
 To defcribe the fcroll of a hand-rail — to find the face mould and the falling mould to any 
 
 given pitch whatever - . - 
 
 T 7 find a face mould for getting a fcroll out of a folid piece of wood 
 
 5I' 
 
 52 
 
 53 
 
 54 
 
 55 
 
 The plan and feSlion of a geometrical Jiair-cafe circular on the plan^ upon a level landing 56 . 
 To find the moulds of ditto when the opening is fmall - . . „ 
 
 To find the moulds of a Jiair-cafe circular on the plan^ with a quarter fpace in it sq 
 
 To find the moulds of dittOy with winders all round 60 to 61 
 
 8 The: 
 
XU 
 
 CONTENTS. 
 
 Plate 
 
 The plan and feSllon of an elliptical Jlair~cafe ■ ' " — — 62 
 
 To find the moulds of ditto ■ — — — — 63 
 
 The plan of a fair- cafe being given^ to diminijh the ends of the Jleps at the rail regularly ^ 
 
 Jo that the hannijiers Jhall he all regular ■ ' ■■■■■■■ ■ ■ ' ■ 64 
 
 Ditto when there is a half [pace on theftde - " ' - *"■ — ■ ■ — — 65 
 
 To cap an iron rail — — — ' ' 66 
 
 To glue a rail up in thlcknefs, and to trace the brackets round the winders — 67 
 
 To fit the Jkirtlng to any kind of flairs Jlraight or circular — 6* 
 
 Various methods of diminijhing columns — — — — — 69 
 
 How to find the bars for a circular circular fajh 7 ® 
 
 How to find an archivolt to a circular circular window — — — 71 
 
 To find the angle bars of Jhop fronts^ and raking mouldings for pediments — 7* 
 
 Ditto for flair-cafes — — — - ' ' 73 
 
 To diminijh or enlarge a cornice^the method of finding the joint of a jib door 74 
 
 To find the proper curve of a board to be bent round a cylinder^ fo that it Jhall f and to any 
 
 angle given^ and be parallel to the horizon when if*nt round » — 7 S 
 
 CONCLUSION — ’ 76 7 * 
 
 f 
 
 OF 
 
PRACTICAL GEOMETRY 
 
 G eometry is the fdence of extenfion and magnitude ; It teaches the conftrudlon of 
 all right lined and curvilineal figures, and is divided into Theoretical and Pra£lical. 
 The Theoretical partis founded upon reafon and felf-evidence ; it demonftrates the con- 
 firudion of varioufly formed figures, and evinces the truth, or detcds the falfehood on 
 v^ich they are made. This is the foundation of the Pra£lical part ; and without a know- 
 ledge of the Theory, no invention to any degree certain can be made in the Pradlice. 
 The ufes of Geometry are not confined to Carpentry and Architedure, but, in the various 
 branches of the Mathematics, it opens and difeovers to us their fecrets. It teaches us to con- 
 template truths, to trace the chain of them, fubtle and almoft imperceptible as it frequently 
 is, and to follow them to the utmoft extent. 
 
 Its ufes are great and neceflary in Aftronomy and Geography. The fcience of Perfpec- 
 tlve is entirely dependent upon its principles. To enumerate its many ufes is beyond 
 my power. Thofe who defire to become thoroughly acquainted with Geometry, will do 
 well to ftudy attentively the Elements of Euclid. 
 
 As my labours are not intended for the abftrufe Mathematician, but for the inflrudion 
 of the Praiiual Carpenter^ I fhall omit all fpeculative demonftraiions, the fe(Tions of Cylin- 
 <3ers and Globes excepted (which are not to, be found in Euclid), and confine myfelf to 
 the ufeful part of the fcience, viz. Practical Geometry. 
 
 PLATE I. 
 
 Definitions. 
 
 A. POIN^ hath neither parts nor magnitude, 
 
 2, A line is length.^ without breadth or thicknefs. 
 
 B 
 
 3. A fuperficies 
 
2 
 
 PRACTICAL GEOMETRY. 
 
 3. AJuperJicies hath length and breadth only. 
 
 4. A folid is a figure of three dhnenfions^ having lengthy breadth^ and thicknejs. 
 
 Hence furfaces are the extremities of folids^ and lines the extremities of furfaces^ and points 
 the extremities of lines, 
 
 5. Lines are either right, curved, or mixed of thefe two. 
 
 6. A right or Jiraight line lies all in the fame direction between its extremities, and is the 
 jhorteji difiance between two points, as A. 
 
 7. A curve continually changes its dir colons between its extreme points, as C. 
 
 8 . Lines are either parallel, oblique, perpendicular, or tangential. 
 
 g. P arallel lines are always at the fame difiance, and will never meet, though ever fo far pro- 
 duced, as D and E. 
 
 10. Oblique right lines change their dfiance, and would meet, if produced, as F. 
 
 ■ 1 1. One line is perpendicular to another when it inclines no more to one fide than another, as G. 
 
 1 2. One line is a tangent to another when it touches it without cutting, when both are produced, 
 asU. 
 
 .13* angle is the inclination of two lines towards one another in the fame plane, meeting in a 
 point, as I. 
 
 14. Angles are either right, acute, or oblique, as K. 
 
 1^. A right angle is that which is made by one line perpendicular to another, or when the 
 angles on each fide are equal, as G. 
 
 16. An acute angle is lefs than a right angle, as K. 
 
 17. An obtife angle is greater than a right angle, as L, 
 
 18. A fuperficies is either plane or curved. 
 
 ig, yf plane, or plane furf ace, is that to which a right line will every way coincide \ — hut if 
 not, it is curved. 
 
 20 . Plane figures are bounded either by right lines or curves. 
 
 21. A folid is fald to be cut by a plane when it is cut through in any particular place, and the 
 place that is cut is called the feSiion of the folid. 
 
 22 . Plane figures, bounded by right lines, have names according to the number of their fides, 
 or of their angles, for they have as many fides as angles — the leajl number is three. 
 
 23. An equilateral triangle is that whofe three fides are equal, as M. 
 
 24. An 
 
 * 
 
I'’ late 1 
 
 A 
 
 B 
 
 C 
 
 D 
 
 
 F 
 
 G 
 
 H 
 
 1ST 
 
 k 
 
 T 
 
 IT 
 
 Y 
 
 , 1)4 
 
 
 («■ 
 
PRACTICAL GEOMETRY. 
 
 3 
 
 S4. An ifofceles triangle has only two fides equals as N. 
 
 25. A fcalene triangle has all fides unequal.^ as O. 
 
 26. A right angled triangle has one right angle^ as P. 
 
 27. Other triangles are oblique angled, and are either ohtufe or acute. 
 
 28. An acute angled triangle has all its angles acute, as M or N. 
 
 29. An obtufe angled triangle has one obtufe angle, as O. 
 
 30. A figure of four fides and angles is called a quadrangle^ or quadrilateral, as Q_, R, S, T, U 
 and V. 
 
 31. A parallelogram is a quadrilateral, which has both pairs of its oppofite fides parallel, as Qj 
 R, U and V •, and takes the following particular names, 
 
 32. A rectangle is a parallelogram having all its angles right ones, as Q^nd R. 
 
 33. A fquare is an equilateral redlangle, having all its fides equal, and all its angles right oneSy 
 as Qi 
 
 34. A rhombus is an equilateral parallelogram, whofe angles are oblique, as U. 
 
 35. A rhomboid is an oblique angled parallelogram, as V. 
 
 36. A trapezium is a quadrilateral which has neither pair of its fides parallel, as T. 
 
 37. A trapezoid hath only one pair of its oppofite fides parallel, as S. 
 
 38. Plane figures having more than four fides, are in general called polygons, and receive 
 other particular names according to the number of their fides or angles. 
 
 3Q. A pentagon is a polygon of five fides, a hexagon hath fix fides, a heptagon feven, an oSiagon 
 eight, a nonagon nine, a decagon ten, an undecagon eleven, and a dodecagon twelve fides. 
 
 40. A regular polygon hath all its fides and its angles equal; and if they are not equal, the 
 polygon is irregular, 
 
 41. An equilateral triangle is alfo a regular figure of three fides, and a fquare is one of four ; 
 the former being called a trigon, and the latter a tetragon. 
 
 42. A circle is a plane figure bounded by a curve line called the circumference, which is every^ 
 where equidifiant from a certain point within, called its centre. 
 
 43. The radius of a circle is a right line drawn from the centre to the circumference, as a b 
 at W. 
 
 44. A diameter of a circle is a right line drawn through the centre, terminating on both 
 fides of the circumference, as c A at W. 
 
 B 2 4^. An 
 
 V- 
 
4 
 
 PRACTICAL tJEOMETRY, 
 
 4'. Ai arch cff a arcle is a> y part of the circt mference, 
 
 46. A chord is a right line joining the extremities of an archy as a b at X. 
 
 47. A fegment is any part of a circle bounded by an arch and its chords as X. 
 
 48. A fern' circle is half the circle^ or a fegment cut off by the diameter, as Y. 
 
 49. A fedior is any part of a circle bounded by an arch and two radii,, drawn to its extremd- 
 tits, as X, 
 
 50. A quadrant, or quarter of a circle, ts a feblor having a quarter of the circumference f9r 
 its arch, and the two- radii are perpendicular to each other, as K j.. 
 
 51. The height or altitude of any figure is a perpendicular let fall from an angle, or its vertex,, 
 to the oppofue fide, called the bafe, a b ^ 7 / B 2^ ’ 
 
 52. When an angle is denoted by three letters, the middle one is the place of the angle,, andi 
 the other two denote the fides containing that angle r thus let ?ih c be the angle at C b is the 
 angular point, and a b and b c are the two. fides containing that angle^ 
 
 S 3 ’ meafure of any right lined angle is an arch of ' any circle contained between the tJUB 
 lines which form the angle, and the angular point being in the centre,as D 4». 
 
 PLATE II. 
 
 P R O B L E M S.- 
 
 Figure r. To draw a perpendicular to. a given point in a line,. 
 
 ^ 5 is a line,^ and c a given point ; take a and' b, two equal diftances on each fide of Cy 
 and with your Gompaj(fes in a and ^ mak.e an interredion V,, and dfaw which is the 
 perpendicular. 
 
 Fig. 2. To make cr perpendicular with a ten foot rod. 
 
 Let a b be fix feet, then take eight feet and make an arch at c in the point h, and in the 
 point a with the diftance ten feet crofs at c, then draw c b, which is the perpendicular. 
 
 Fig. 3. To let fall a perpendicular from a given point to a line. 
 
 In the given point c make an arch to crofs the line in a and b, and in a and h make an 
 interfetSHon at d, and draw c d, the perpendicular. 
 
 Fig. 4. To draw a perpendicular upon the end of a line 
 
 Take any point at pleafurc above the line, and with the diftance make an arch ah c, 
 and draw a line a d crofs it at c, and draw c b the perpendicular. 
 
 Fio, 5, 
 
Plate ^ . 
 
5 
 
 PRACTICAL GEOMETRY. 
 
 ^ Fig. 5. T'o divide a line in two parts by a perpendicular. 
 
 In the points a and b defcribe two arches to interfe^Sl at c and d^ and draw e dy whick 
 divides the line in two. 
 
 Fig. 6. To divide any given angle into two equal angles.. 
 
 Take two equal diftances b and c on each fide of the angular point a^ and with the fame 
 opening of the compafs on any other place the foot of your compafs in b and c, make an* 
 interfe£Uon at dy, and draw d Uy which will divide the angle into two equal parts. 
 
 Fig. 7 and 8. An angle being giveuy, to make another equal to iiyfrom a given pointy in a 
 
 right line. 
 
 Let b a c be the angle given, and c a right line, c the given point ; on a make an 
 arch b c with any radius, and on c with the fame radius defcribe an arch d Cy take the 
 opening of b r, fet i-t from d to ey and draw- e C'y. then the angle e c. d will be equal to c a b,. 
 
 F L A T E im. 
 
 E re., l.. Upon a right line to make an equilateral triangle. 
 
 Take a,b the given fide with your compafs, jjdace one foot in and by make an interlec- 
 tion at Cy and draw c a and c b. 
 
 Fig. 2. Upon a right line to make a geometrical fquare. 
 
 With the given fide a by and in the points a by defcribe two arches to interfe£f at r, divide 
 b e into two- eq_ual parts at y, make e d and e c each equal to e fy draw a dy d c, and c b. 
 
 Fig. 3. PROPOSITION. 
 
 If through the point c in the extreme of the diameter c d, is dra wn a - tangent line e c 8, and 
 if on this point c with any radius a femicircb be defer ib^d and divided into equal partSy and from 
 the centre c lines be drawn through theje points to terminate in the circumferenccy it will be di- 
 vided into equal parts, 
 
 F I G. 4, 5) 6. The fide of any polygon being giveUy to defcribe' the polygon to any number of 
 
 fide s whatever. 
 
 On one extreme of the given fide make a femicircle of any radius, but it will be moll 
 convenient to make it equal to the fide of the p.)Iygonj then divide the femicircle into the 
 lame number of equal parts as you would have fides in the polygon, and draw lines from 
 
 the 
 
PRACTICAL GEOMETRY. 
 
 the centre through the equal dlvifions In the femicircle, always omitting the two laft, 
 and run the given fide round each way upon thefe lines, join each fide, and it will be com- 
 pleted. 
 
 Example in a pentagon^ Fig. 4. 
 
 Let ^ be the given fide, and continue it out to c ; on as the centre and the given 
 fide, defcribe a femicircle, divide it into five equal parts, through 2, 3, 4, draw a a d.^ a 
 make b e equal to a 2 d equal to 2 a or a b, join 2 d, d and e b. In the fame manner 
 may any other polygon be defcribed. 
 
 Fig. 7. Through a given point a to draw a tangent to a given circle. 
 
 From a draw a 0 to the centre, then through a draw b c perpendicular to a 0, it will be 
 the tangent. 
 
 Fig. 8. yi tangent line being given.^ to find the point where it touches the circle. 
 
 From any point a in the tangent line b c, draw a line to the centre and divide a 0 into 
 two equal parts at w, and with a radius m a-^ ot m Oy defcribe an arch, cutting the given 
 circle in «, which is the point required. 
 
 Fig. 9. Two right lines being given^ to find a mean proportion. 
 
 Join a h and b c \n one ftraight line, divide it into two equal parts at the point r, with 
 the radius 0 a ox 0 c defcribe a femicircle, and ere£t the perpendicular b dy then \s a b to b d 
 as ^ r/ is to b c. 
 
 Fig. 10. Through any three points to defcribe the circumference of a circle. 
 
 From the middle point b draw chords b a and b c to the two other points a and c, divide 
 the chords a b and^ c into two equal parts by perpendiculars meeting at Oy which will be the 
 centre. 
 
 Fig. II. To find a right line equal to any given arch of a circle. 
 
 Divide the chord a b into four equal parts, fet one part b c on the arch from a to dy 
 and draw d r, which will be nearly equal to half the arch. 
 
 Note. This method fliould not be ufed above a quarter of a circle ; fo that if you would 
 find the circumference of a whole circle by this method, the fourth part mull only be ufed, 
 which will give one eighth part of the whole exceedingly near. 
 
 PLATE IV. 
 
 Fig. I. Any three lines being glveny to make a triangle. 
 
 Take one of the given fides a by and make it the bafe of the triangle 3 take the other fide 
 
7 
 
 PRACTICAL GEOMETRY. 
 
 « f, and put the foot of the compafs in and make an arch at c ; then take the third fide 
 h c, and put the foot of the compafs in and crofs ibe other arch at r j jom the tides. 
 
 Note : That any two lines muft be greater than a third. 
 
 Fig. 2} 3 . T'o make a quadrangle equal to a given quadrangle. 
 
 Divide the given quadrangle, jig. 2, in two triangles j make the triangle e g f equal to 
 ah C’i and e g h equal to « c and it is done. 
 
 Fig. 4) 5 . Any irregular polygon being given.y to make another of the fame dimenfons . 
 
 Divide the given polygon into triangles, and in fig. 5, make triangles in the fame po- 
 fition, refpedlively equal to thofe in fig. 4» then will the irregular pol^'gon f g'i hy /, be 
 equal and fimilar to a b c d e : in the fame manner may any other polygon be made equal 
 and fimilar to another. 
 
 Fig. 6. To make a reBangle equal to a given triangle. 
 
 Draw a perpendicular c d^ divide it into two equal parts at through e Axzvj f gy parallel 
 to the bafe a b^ draw af b g^ perpendicular ; then will the redangle ab gf he equal to 
 the triangle a b e. 
 
 F IG. 7. To make a fquare equal to a given reBangle. 
 
 Let a b c dhe the given parallelogram ; continue one of its fides a b out to make 
 b e equal to the other fide b r, divide ^ in two equal parts at /, with the radius i e ox i a 
 make a femicircle a f e^ and draw b f perpendicular to a by make the fquare bfg hy which is- 
 equal to the parallelogram abed. 
 
 Fig. 8. To make a fquare equal to two given fquares. 
 
 Make the perpendicular fides a c and a b ol the right angled triangle ah c equal to the 
 fides of the given fquares A and jS, draw the hypotenufe c by which is the fide of the fquare 
 C, equal to the two fquares A znA B. In the fame-manner may a femicircle be made equal 
 to two given femicircles, or any fimilar figures whatever. 
 
 Fig. g. To make a fquare equal to three given fquares. 
 
 Let A B C he the three fquares j make a b equal to the fide of By a c equal to the fide 
 of Ay at right angles to a b join b Cy then make a d equal to b r,.make a e equal to the fide 
 of Cy join d 6y which will be the fide of the fquare D equal to the fquares ABC, 
 
 PLATE 
 
8 
 
 PRACTICAL GEOMETRY. 
 
 P L A T E V. 
 
 Fig. I. To draw a fegment of a circle to any length and height, 
 
 A b is the length, t h the height ; divide the length a b into two parts by a perpendicular, 
 divide a h hy the fame method, then their meeting at g will be the centre ; fix the foot 
 of the compafles in g, extend the other leg to make the arch a h which is the fegment. 
 
 Fig. 2. To draiu a fegment by rodsy to any length and height. 
 
 Get two rods c e and c f each equal to a b the opening ; place them to the height at r, 
 and to the ends a by put a piece acrofs them to keep them tight, then move your laths 
 round the points a by and it will defcribe the fegment at the point c. 
 
 Fig. 3. To defcribe a fegment of a circle at twicey upon true principleSy by a fiat triangle. 
 
 Let the extent of the fegment be a by its height c dy from the extreme b to the top d 
 draw b dy through the point d draw e d parallel to the bafe a by equal in length to d by 
 defcribe one half, as you fee at G \ then move your nail, or pin, out of tf, ftick it in the 
 point by and defcribe the other half. 
 
 F ig. 4. The tranfverfe and conjugate axis of an ellipfis being giveUy to draw its reprefentation. 
 
 Draw a d parallel and equal to n Cy bifedf it in e j draw e c and d g cutting each other at «z, 
 join m Cy bifeft it by a perpendicular meeting c gy produced at h ; draw h dy cutting b a ztky 
 and make n i equal to « i j n I equal to « /; ; through the points /, /, ky hy draw the lines 
 h iy k ly and i /, h ky then defcribe the four fedlors by help of the centres i I k hy and it will 
 be the reprefentation required. 
 
 Fig. 5. To defcribe an ellipfis by ordinates. 
 
 Make a femicircle on the length a by divide it into any number of equal parts, as 16, on 
 the end at <7, make a 8 perpendicular, equal to half the width, and draw the ordinates 
 through all the points in the femicircle, draw the line 8 i to the centre, then a 1 S will 
 be a fcale to fet your oval off ; take i i from your fcale, and fet it from i to i in your oval 
 both ways at each end ; then take 1 2 in your fcale, and fet it to i 2 in your oval, and find 
 all the other points in the fame manner j a curve being traced through thefe points will be 
 the true ellipfis. 
 
 P A T E VI. 
 
 Fig. I. To make an ellipfis with a firing. 
 
 Take half the longeft diameter a by that is, a gy with that diftance fix the foot of the 
 1 ' compaft 
 
Plate 
 
PRACTICAL GEOMETRY. 
 
 9 
 
 compafs in f, crofs a b zX. e ftick in two nails or brads, then lay a firing at ef to come 
 out to r, fix a pencil at r, and move your hand round, keeping the firing tight, will de- 
 fcribe the ellipfis. 
 
 Fig. 2. Ttf defer ibe an elUpfs by a tratmnel. 
 
 I 2 3 is a trammel rod, at i is a nut with a hole to hold a pencil j at 2 and 3 are two 
 other Aiding nuts j make the diftance of 2 from i, half the fhorteft diameter of your ellipfis, 
 and from the nut 1 to 3 equal to half the longeft, the points 2 and 3 being put into the 
 grooves of the fame fize, then move your pencil round at i, and the pencil at i will deferibe 
 the true curve of an ellipfis. 
 
 Fig, 3. An ellipfs being given, to find the centre and two axifeu 
 
 Draw any two parallel lines a b and c d zt pleafure, divide each of them in two equal 
 parts at the points e znd f, and through e f draw the line k /, divide k I into two equal parts at 
 the pointy, place the foot of the compafs in g, with the other foot make two crofles h and 
 /, on the circumference draw a line h i, through g draw m n parallel to h /, alfo through 
 g draw 0 p zt right angles to m n\ then 0 p\% the tranfverfe axis, and mn the conjugate, and 
 g the centre of the ellipfis. 
 
 Fig. 4. How to proportionate one ellipfs within another ; that is, to give it the fame length ire 
 proportion to its width, as the length of the other has to its width. 
 
 Let the given ellipfis be a db c, make the parallelogram e h gf, to touch the Tides and 
 ends of the ellipfis, draw the diagonals e f and g h, of the parallelogram, let r ^ be the 
 width of the lefTer ellipfis given, through the points q, or r, draw I 0, ox m n, parallel to the 
 tranfverfe axis, at the points m and n, where it cuts the diagonal, draw m I and n 0, parallel 
 to the conjugate axis, will alfo fhew its length. 
 
 Fig. 5. How to deferibe an ellipfs about a parallelogram, to have the fame length in proportion 
 to. its width, as the length of the parallelogram has to its width. 
 
 Let the given parallelogram he a b c a’,\ct the diagonals a c, and b d, be drawn from the 
 centre i ; draw the quarter of a circle, 2 i i, to half the width of the parallelogram ; divide 
 the quadrant into two equal parts at i ; through the point i, draw the line / 3, parallel to- 
 the tranfverfe axis, to cut the diagonal b d m the point 3 ; then draw the lines 3 2, and 
 3 4 : again, draw/ d, parallel to 2 3, then z/will behalf the width, and d e parallel to 3 4.; 
 and / e will be half the length of the ellipfis : make i h equal to i e, and i g equal to if, 
 which will give the four points through which the ellipfis muft pafs j deferibe the curve, 
 and the thing will be done. 
 
 Fig. 6 . To divide a line in the fame proportion as another is divided, 
 
 D a, is a line given already divided, and ^ ^ is a line to be divided in the fame proportion, 
 
 ^ making 
 
10 
 
 PRACTICAL GEOMETRY. 
 
 making any angle at a e, draw If and eg, parallel to « ^ ; then d is^ divided at 
 
 f and g, a d is at b and c. 
 
 F I cr 7^ To do the fame by an equilateral triangle, 
 
 \ 
 
 J b\% the given divided line, take c d the length as you would have divided d e\s the 
 ■fame length, and is divided in the fame manner as a b. 
 
 f Fig. 8. To make an odiagon the neareft way in a fquare. 
 
 Draw the diagonal of the fquare to crofs at Cy fix the foot of your compafs in r, and make 
 an 2sd\fe g j then fet your gauge io df ox b gy which will gauge off each angle. 
 
 PLATE VII. 
 
 CONIC SECTIONS BT INTERSECTING LINES. 
 Definitions. 
 
 1. A cone is a figure funding upon a circular bafcy and dminijblng to a point at the top. 
 
 2. If a cone is cut by a plane pajfmg through its fidesy then the figure fo cut is an ellipfis. 
 
 3. If a cone is cut by a plane parallel to one of its fideSy then the figure is a parabola, 
 
 4. If a cone is cut by any plane paffing through the oppofite conCy then the figure is. an hyperbola. 
 
 To deferibe the ellipfis from the cone. 
 
 Figure A. Let B be half the circle of the bafe of the cone, n the vertex at the top ; 
 then n a and n d are two fides j let the cone be cut by a plane paffing through g h j bifedl 
 g h at the point k, and through k draw r q, parallel to the bafe a d ; alfo bifedl r q mmy 
 deferibe the femicircle r p q, draw k p zt right angles to q r y then \s g h the length of the 
 ellipfis, and h k half its width ; then the figure may be deferibed at Cy which is explained in 
 the next plate. 
 
 To deferibe the parabola from the cone. 
 
 Figure A. Let i e be the axis of the parabola, parallel to the other fide n d oi the cone, 
 and through e draw e c zt right angles to the bafe ; then will ^ r be half the width of the 
 parabola, and e i its height ; then the figure will be deferibed, as at D, by interfering lines 
 upon each ordinate, up to the crown, from the equal divifions on each fide. 
 
 T 0 deferibe the hyperbola from the cone. 
 
 Figure A. Let the axis of the hyperbola.be i f cut by a plane paffing through/'and /, 
 till it cut the oppofite cone at / 5 diaw fb at right angles to a by then is f i the height of 
 
 an 
 
to ^ ^ to >0 K 
 

PRACTICAL GEOMETRY. 
 
 ir 
 
 an hyperbola, and f b half the width of the bafe, and i I its tranfverfe axis j then make f i 
 at E equal to f i in figure make i I'm E equal to / / in figure A^h b \w E equal to twice 
 f b in figure A j let the bafe ^ ^ in A be divided into ten equal parts, as at o i 2345, that 
 isj into five equal parts on each fide from the centre, and draw lines to the point / through 
 thefe points j likewife divide the height into five each way, and draw lines to the crown 
 at t i this will ihew the points through which the curve muft pafs, 
 
 PLATE VIII. 
 
 Hozv to draw any fiefni-ellipfis upon the tranfverfie^ or conjugate axis, or even a fiemie'irel: itfiefi 
 
 by a new method of interfieSling lines. 
 
 Figures A and 13 . Let the given axis be a b, and let it be divided into any number 
 of parts, as 10 ; alfo let the height be divided into half the number of parts ; make e d equal 
 e f, that isy to the height of the arch ; then, from the point d, draw lines through the equal 
 divifions of the axis a b ; likewife, through the points 12345, height a /, draw 
 
 lines tending to the crown at r, which will interfed at the points h i k l\ and lines being 
 drawn through the divifions of ^ to r, at the crown, in the fame manner, will give the 
 points n 0 p q \ a curve being traced through thefe points, will ftiew the true curve of an 
 ellipfis.* 
 
 The femicircle,/^Kr^ C, is drawn in the fame manner, by making a f equal to one half 
 
 a b. • 
 
 How to draw the true fegment ofi a circle, by the method of interfieSiing Vines. 
 
 Figured. Let ^7 ^ be the length of the fegment, and c r its height, and draw the 
 chord b c for one half of the fegment, and draw b m2X right angles to Z> r; and from the 
 centre at 0, divide the diameter a b, each way, into five equal parts ; alfo from c, at the 
 crown, in the centre of the line m n, divide c rn, and c n, each into five equal parts j and 
 draw I i» 2 2 , 3 3, 4 4, 5 5, on each fide, through the divifions 12345 on as, and 
 j 2345 on b r \ draw lines to the crown at r, which will interfed the other lines at 
 the points d e f g, and h i k I : the curve being traced, the thing is done. 
 
 Hoiu to draw a fiat fegment of a circle nearly true. 
 
 Divide the length of the fegment Into equal parts each way, from the centre d, as before, 
 and draw the lines i i, 2 2, 3 3, 4 4, 5 5, all at right angles, to the length a b j lines 
 being drawn to the crown at c, from the divifions at each end, will fhew the points which 
 the fegment muft pafs through; the curve being traced, the thing is done. 
 
 Remark. Although this laft method is not the true fegment of a circle, but a para- 
 bolic curve, yet it will be found ufeful in pradice, in tracing, any fegment whofe height is 
 not more than one tenth part of its length ; and If the centre of the fegment is found, and 
 drawn with a compafs, the difference will hardly be feen, and the flatter the fegment is, 
 this difference will become the more imperceptible ; but if the height exceeds one tenth of 
 
 C 2 its 
 
12 
 
 PRACTICAL GEOMETRY. 
 
 its length, the diftercnce will be vifible ; for then the arch will be quicker at the crown, 
 and get fatter and flatter towards each extreme. 
 
 In the fame manner may all kinds of rampant ellipfes be defcribed, or any fegment of 
 theJin, as at F and G, alfo a rampant parabola in the fame manner as at H. 
 
 ^ P L A T E IX. 
 
 THE SECTIONS OF A CYLINDER. 
 
 Definition. 
 
 A cylinder is a figure generated by the revolution of a right angled parallelogram about one of its 
 fules ; co'nfequently tloe ends of the cylinder are equal circles^ and the line paffing through the 
 centre of the cylinder^ is called the axis. 
 
 The feSlio?i of a cylinder^ cut by any plane., is an eUlpfis. This is evident to the meaneji con- 
 ception y but for a farther fatisfacliony it is proved by the writers of conic fe£lions. 
 
 To find the feSiion of a femi-cylindery by ordinateSy when it is cut at right angles to the planey 
 pajfing through its axis, in the dirediion a b. Fig. i. * 
 
 Let the circle of the bafe be divided into equal parts at and drawn parallel up the 
 cylinder to the line a by at the points o i 2345, &c. and from thefe points draw lines at 
 right angles to a b, then B being pricked from Ay as the figures diredt, B will be the 
 fedlion of the cylinder. 
 
 DEMONSTRATION. 
 
 Conceive the circle A at the bafe to be turned at right angles to the pbne, alfo the ellipfis B at right angles 
 to the fame plane ; then will the oi'dinatcs of B be. parallel and perpendicular over the ordinates of A, and 
 every correfponding point in the circumference of B will fall perpendicular to the fame correfponding points 
 in A : therefore B is the true feftion of the cylinder, cut in this pofition. 
 
 To cut a cylinder in the diredlion a b, upon a plancy pajfing through its axisy to make an acute 
 
 angle with that plane. Fig. 2. 
 
 Let Cy at No. i, be the given angle, whi^h the fedllon at B is to make with the 
 plane of the cylinder; take « ^ in figure 2, that is, the radius of the bafe, and fet it from 
 b Cy at No. I, perpendicular to i b draw c c parallel to i by alfo from c draw c e perpen- 
 dicular to i b ‘y then take the diflance c i, fet it from i to fy in figure 2, zt B y likewife 
 take i e from No. i, and fet it from i to e in figure 2, at d? ; draw e d parallel to m in the 
 bafe, to cut the rake in d, and join d f ; then is df the bevel of the firfl: ordinate of the 
 fedlion B. And draw the lines e c and d a parallel to the axis ; join a c zt A 'y then will 
 <7 c be the bevel of the firfl: ordinate of the bafe. Then draw all the other ordinates of A 
 parallel to a r, and at the points 1234, &c. in m n, draw lines parallel to the axis of the 
 cylinder, to cut the raking line 12345, &c. From thefe points, let lines be drawn parallel 
 6 to 
 

13 
 
 PRACTICAL GEOMETRY. 
 
 to (1 /; then the ordinates of being pricked from the fame correfponding ordinates of the 
 bafe at will give the fe6lion of the cylinder. 
 
 Note. The point/will fall beyond the fweep at the feffion B. 
 
 DEMONSTRATION. 
 
 Since / /is equal to i c, let the plane B be conceived to be turned round the line a ^ to make an angle 
 at /, with the line / e, equal to e i c, at No. i ; then the point / will be perpendicular to the point e, and the 
 line joining e and/, will be equal to e c, at No. i. But e c \st equal to G c at A upon the bafe j therefore tlic 
 point/, when oppofite to e, will be perpendicular over the point c, in the bafe : confequently the line ctf, will 
 be perpendicular, and parallel over the line joining a c-, becaufe all the ordinates in B are drawn parallel to 
 df, and perpendicular to every cori efponding ordinate in the bafe, which are parallel to a c-, and as all the 
 ordinates of B are equal to their correfponding ordinates in A, fo they are alfo parallel and perpendicular to 
 them ; confequently every point in the circumference of B, will be over the fame correfponding points in the 
 bafe; therefore B is the true fedtion of the cylinder, cut in this pofition, which was to be demonftrated. 
 
 *To cut a fegment of a cylindeVy in the dircSiion a b, to make an obtufe angle with the plane of the 
 
 fegment. Fig. 3. 
 
 Let No. I be the angle given, which the fedlion 5 is to make, with the plane of the 
 fegment ; from /in No. i, draw f g at right angles to f c, and g e alfo perpendicular, to 
 make the right angled triangle e gf. And in figure 3, at B, draw gf at right angles to 
 a by and make g e equal to ^ ^ at No. i. Alfo, make gfztB equal to e/at No. i. 
 Draw e at 5 , parallel to m nzt A, the bafe, and at the point where it interfe£ls the line a by 
 join df; then dfh one of the ordinates. From e and dy draw the two parallel lines e e, and 
 d 3, join c 3 ; then c 3 will alfo be an ordinate of the bafe. Draw parallel lines at diferetion 
 to c 3, for the other ordinates of the bafe j and from their interfe6lion upon m w, draw lines 
 parallel upon the cylinder, to cut a bm 1234, &c. and from thefe points, draw parallel 
 lines to d fy which are the ordinates of B : thefe, being pricked from the bafe as the 
 figures direct, will give the points through which the curve muft pafs, which being traced, 
 will be the true feition of the fegment of the cylinder. 
 
 DEMONSTRATION. 
 
 Let the feftion B be turned round the line a b, to make an angle with the plane of the fegment equal to 
 No, I ; then, g f at B being equal to e f at No. i, and e g atB equ^ to eg at No. i, therefore a line joining 
 e and /at B is equal tof g,or e h at No. i, that is, equal to z c, the width of the bafe A ; but efh alfo 
 parallel and perpendicular over 2 c, therefore the point /will be perpendicular to the point c in the bafe : but 
 the point d is level with the point e, that is to fay, e d is parallel to m n ; and the point /becomes alfo level 
 with the point e, when turned round ; therefore the line joining/ and d, will be parallel to the bafe, and per- 
 pendicular over 3 c : for d is perpendicular to 3, and/ is perpendicular over c; confequently d /is an ordi- 
 nate of the feftion, and 3 c an ordinate of the bafe : but all the ordinates parallel to d f, are refpedlively equal 
 and perpendicular over thofe of A, which are parallel to 3 £■ ; therefore they are in the true curve of the feftion 
 B, which was to be demonftrated. 
 
 That the Reader may perceive this more clear, the bed way is to draw thofe lines on pafteboard, the 
 feftion and the end being made to turn round, in their proper pofition ; then the demonftration will be clearly 
 ften. 
 
 Figure 4, is to be laid down and demonftrated in tlie fame manner as Figure z. 
 
 
 Remark. 
 
14 
 
 PRACTICAL GEOMETRY. 
 
 Remark. Upon thefe figures depend the whole principles of hand rails for flairs : the Reader ought to 
 underftand how to form the fe£lion of a cylinder, in any cafe whatever; for the face or raking mould of a 
 hand rail is nothmg but the double feftioa of a cylinder, as m figure 4, at B, where the double cii cle upon 
 the bafe A reprefents the plan of a rail, and the bevel at No. figure 4, reprcfents the fpring of the plank, 
 and « b the pitch of the rail ; therefore, it is very necelTary that the Reader ought to have a knowledge of 
 thefe figures and their dcmonftrations ; and not to be fatisfied with only doing of it, but to read thefe demon- 
 ftrations, and confidcr them with attention ; then he will be able to fee the reafon why every line is drawn iti 
 the manner it is. 
 
 PLATE X. 
 
 THE SECTIONS OF J GLOBE, OR ANT OTHER FIGURE STANDING 
 UPON A CIRCULAR BASE^, ALSO, THE SECTION OF ANT FIGURE 
 STANDING ON AN IRREGULAR BASE. 
 
 Definition. 
 
 A globe h a figure generated by the revolution of a femiclrcle round its diameter, which becomes 
 
 the axis of the globe. 
 
 AXIOMS; OR, SELF-EVIDENT TRUTHS. 
 
 ift. From this definition it appears, that every plane JeSHon pajfiwg through the centre, is e^ual 
 
 to one another » 
 
 2d. Every fieaion of a globe, cut by a plane, is a circle \ for the generating circle may be made 
 to revolve round any line, as an axis-, and therefore every point in it will generate a circle, 
 whofie diameter muji be twice the radius of that circle, diji ant from the axis of the globe. 
 
 3d. If afiemi-globe is cut by a plane at right angles to the plane of its bafe, the fiaion will he a 
 
 femicircle. 
 
 To find thefiaion of afiemi-globe cut by a plane at right angles to the plane of its bafe. Fig. i. 
 
 It appears from the laft axiom, that there is no tracing required r for, let the fedion be 
 cut acrols a b, figure i j divide a b m two equal parts at the point c ; and on c, as a centre 
 with the radius, c a, ox c b, defcribe the femicircle A, which is the true fedlion required. 
 
 The fame by ordinates. Fig. a. 
 
 Draw any line d e through its centre, and let a b ho the place of the feaion upon the baije, 
 as before j place the foot of your compafs in the centre of the globe at f and, with a radius 
 f c, draw an arch from c, round to in the diameter d e -, the foot of your compafs re- 
 maining ftill in/, draw the concentric dotted circles from c b tofe, and at the interfering 
 
 points 
 
*5 
 
 PRACTICAL GEOMETRY. 
 
 points 12345 likewife in c by ere£l perpendiculars to thofe lines ; then A being- 
 
 pricked from Cy as the figures direft, will give the points through which this femicircle 
 
 muft pafs. 
 
 DEMONSTRATION. 
 
 Conceive the femicircle C to ftand at right angles upon d e, alfo the fcaion A to be at right angles to 
 fl d-, now it is evident, if^- / is the height of the globe over the pointy in the bafe, i, which is equal to 
 g I muft alfo be the height of the feftion, becaufe the points c and g ftand at an equal diftance from the 
 cenw; and therefore the point i over r, is in the furface of the globe. In the fame manner it may be 
 proved, that any other points carried round by the dotted lines are in the fame furface; but the feftion that 
 ftands upon a b, in Ay is a femicircle ; and confequently the method of tracing is alfo a femicircle. 
 
 Cbfa-vation. Hence appears the erroneous principle of tracing ufed by a late writer upon this fubjeft, as 
 you may fee zX figure z, where A is the feftion of a globe, and the bracket at D is tire feaion acrofs the dia- 
 meter. A is truly traced from D, becaufe the ordinates are carried round in circles ; but by his method of 
 tracing, as you fee at C, upon the other fide, the point of the bracket C falls within the fweep of the circle, 
 by reafon of the ordinates of C being carried ftraight through between the two bafes, which I have proved to 
 be falfe. And this he has applied in bracketing up the angles in the fquare well-hole of a ftaircafe, to the 
 circular curb of a Ikylight, which, if truly done, is nothing elfe but upon the fame principle as the fedions of 
 
 a globe. 
 
 Figure 3, is done upon the fame principle as figure i. ^is the fe£Hon traced from Cy 
 and wants no other demonftration than what has been given \n figure i. 
 
 Figure 4, is an ogee fe£lion, (landing upon a circular bafe acrofs the diameter; and A 
 is the fedlion traced from it, upon the fame principle zs figure i. 
 
 From thefe examples it is clear that this method of tracing does not depend on the 
 form of the top, but entirely upon the bafe. Thefe figures are fuppofed to be generated 
 round an axis ; and, as every circle is carried round at an equal diftance from the axis, 
 the perpendicular height of the figure, upon any circle, muft be the fame height in every 
 point throughout that circle ; which proves itfelf to be the only method for any thing of 
 this kind. 
 
 A femi-glohe being cut by a cylindrical furface perpendicular to the plane of its bafcy to find 
 the form of a veneer that will bend round it. Fig. 5. 
 
 Let deho, drawn through the centre / ; and place the foot of your com pafs in f the 
 centre ; and from the points ^1234, which are equally divided from the centre at b in 
 the circular furface, draw the concentric dotted lines round to the diameter i at o i 2 3 4, 
 and at thefe points raife the perpendiculars 00, i i, 2 2, 3 3, 4 4. Take the ftretchout 
 round ^12345, which is one half; and lay it upon the bafe of No. i each way, from 
 o I 2 3 4, &c. and No. i being pricked from Ay figure 5, as the figures dired, will give 
 the points through which the curve muft- pafs for the veneer. 
 
 DEMON- 
 
i6 
 
 PRACTICAL GE 
 
 O 
 
 E T R y. 
 
 DEMONSTRATION. 
 
 For, fince the feaion ftanding upon deh a femicircle, which is equal to the femicircle upon the bafe • 
 and as the points i 2 3 4 in the circular furface, ftand at tlte fame diftance from the centre /, as o i 2 3 4, ia 
 d e ; now if the point 0 at No. i, is made to coincide with the point b \nfgure 5, then the height 0 ftand- 
 mg over the point b, will be equal to the height 0 A thefe points are at an equal diftance from the 
 centre, therefore the top of each ordinate will be in the furface of the globe. In the fame manner every other 
 point may be proved, when bent round and elevated, to be of the fame height, and at an equal diftance from 
 the centre with thofe of A ^ and therefore No. i is the true form of the veneer. 
 
 the ribs of a gothic niche, being the plan, and No. 1 the front elevation. 
 
 Fig. 6 . 
 
 Take the length of each bafe upon the plan, and make them the bafes of No. 2, No. 3, 
 No. 4, and No. 5 ; divide each bafe into five equal parts ; alfo divide the half of nJ. i 
 into fix parts, and draw the ordinates from the equal divifions, perpendicular to each 
 bafe ; then prick each from No. i, as the figures direa, will give the form of each rib. 
 This wants no demonfiration. 
 
ppiiilllHI III 
 
F/ate :/7 
 
 cn7fin^ cF/iane 
 
THE 
 
 THEORY AND PRACTICE 
 
 O F 
 
 CAR P E N T R Y. 
 
 PLATE XI. 
 
 Definition. 
 
 in the Theory of Carpentry^ JigniJies the covering of any furface whatever^ fpread out on » 
 
 plane ifpojftble. 
 
 How to Jlretch out a foffit, when a window or door^ having a femicircular head^ cuts into a Jlraight 
 
 wally in an oblique direSiion. 
 
 Let C be the plan or opening of the window, in fig. and let the bafe of the femicircle B 
 be drawn at right angles to the jambs, or fides of the plan C\ divide the femicircle into 
 any number of equal parts, as ten, and draw the ordinates acrofs the plan C, then ftretch the 
 divillons round 5, along the foffit, in the fame ftraight line with the bafe of B ; under fig. y/, 
 the ordinates being drawn acrofs, and traced off from the plan C, as the figures and letters di- 
 redl, the foffit will then be completed. 
 
 If you would make a cylinder to be only the thicknefs of the wall, D fliews the end of it, 
 which is to be traced from the femicircle B. 
 
 How to draw a foffit when the top is a femicircle.^ cutting right into a circular wall. 
 
 Fig. E. This and the other below are performed the fame as that above, with this dif- 
 ference, that you are to prick from the circular plan, infiead of the ftraight plan. 
 
 Fig. I . ftiews the method when a circular headed window cuts oblique into a circular wall. 
 
 Note. In all kind of foffits, when the ’two jambs are parallel, the ftraight line, which 
 the foffit is pricked from, muft be drawn at right angles to the jambs, as is ftiev/n in this 
 plate •, for want of this confideration they are fhewn in books upon wrong principles. 
 
 But in the following foffits, where the jambs are not parallel, they muft be continued 
 till they meet in a point, and the lire which the foffit is to be pricked from muft be made 
 to form ^in ifocalas triangle with the jambs. 
 
 D 
 
 PLATE 
 
i8 
 
 THE THEORY AND PRACTICE 
 
 PLATE Xir. 
 
 To draw a foffit in a Jiraight wall^ jiuing equally all round with a circular head. 
 
 In jig. continue the Tides of the plan that is « t and b d^ to meet at e ; then about 
 the centre e^ and from the points a and c, defcribe the foffit C, and ftretch the femicircle B 
 along the outline of the foffit t?, it will be completed. 
 
 To drotu a foffit in a circular wall^ Jiuing equally all round with a circular head. 
 
 Fig. B. The ftretch out of this foffit is managed the fame as in the laft ; draw the 
 ordinates of the femicircle B, from thence continue them to f the centre of the flue, and at 
 the points a b c d e^ where they interfedf the plan, draw the parallel lines a b f c g^ &c. 
 and from the points e f g h and./, circle lines to abed and e round the centre f which will 
 give the half of one edge of the foffit, the other half being pricked from it ; the other edge 
 is found in the fame manner. 
 
 Note. This cannot be pricked from the plan as the others are, as the lines round the 
 flue are not level with the plan, and will be longer than thofe on the plan. 
 
 DEMONSTRATION of Fig. A 
 
 Conceive the femicircle B to be turned at right angles to the plan A, then every point in the circumference 
 of the femicircle B will be at an equal diftance from the point e, but the foffit C is deferibed with the fame 
 radius ; therefore the edge of the foffit C, that is the arch line a f, will exaftly coincide with the ardt of the 
 femicircle B, which was to be proved. 
 
 DEMONSTRATION of Fig. .P. 
 
 It is eafy to conceive from the laft demonftration, that if the femicircle B is turned up, and the foffit at C 
 bent round it, the points i 2 3 4 5 at C will coincide with the equal divifions in the femicircle B, and the points 
 abed, &c. at C, will fall perpendicularly over the points abed, &c. in the plan A j for the arches a e, bf, 
 c g, dh, and e i at C will fall over the parallel ftraight lines e a, f b, g c, h d, i e, in the plan A, which was 
 to be demonftrated. 
 
 The learner is advifed to cut thefe and the following foffits out of pafteboard, and their 
 demonftrations will be more clearly feen. 
 
 PLATE 
 

 
OF CARPENTRY. 
 
 19 
 
 PLATE XIII. 
 
 How to draw a foffit in a Jlraight wall^ fining from the jamhs^ and level at the crown. 
 
 ^is the plan of the wall, 5 is a femicircle on the outfide, and ^ is an ellipfis in the in- 
 fide, traced from B, or got by a trammel ; draw the lines a bf.^ and h /;, all perpendicular 
 X.0 a h the fide of the flue of A\ then take half the compafs of jB, and lay it on bf'm D \ 
 likewife take half the compafs of C, and lay it from a to c in Z), and through the points e andy* 
 draw a line to cut the line h h he. in and continue it to' g ; put your compafs in the 
 centre of the flue at and with the other extreme point deferibe the quadrant 12345, 
 which is divided into five equal parts, and draw ordinates 5 4 3 he. then take one 
 
 of the divifions of the femicircle 5 , and fet the foot in b, and make the fmall arch at r, and 
 take h 1 from the centre of the flue in A \ then put the foot of your compafs in /;, in the 
 foffit, and with the other foot crofs the fmall arch at i ; and with the aforefaid divifion of 5 , 
 fet the foot of your compafs in i in the foffit Z>, and deferibe the fmall arch at 2 ; and take 
 h 2 from the centre of the flue, and fet that to its correfpondent 2 in Z>, and by this 
 means you will get one half of your foffit; put the foot of your compafs in /;, and with- 
 the other extrem^^ draw a circle h g k, and put the foot of your compafs in and with 
 the diftance g that is from g to h the centre of the flue, deferibe an arch from h round 
 to and draw the line h k where thefe two arches interfedl ; then fet the divifions oi h h 
 he. on h k, and deferibe the other half in the fame manner, and fo the out-line of the foffit 
 will be completed. 
 
 The infide line is got by pricking it from the plan according to the figures. 
 
 PLATE XIV. 
 
 Yi? draw a fojfit in a circular wall^ fining firojn the jambs and level at the crown. 
 
 Proceed as in the lafl: plate and get the line b df h i.^ then prick the foffit D from the 
 plan yf, according to the letters. 
 
 In the fame manner may any other foffit, fluing from the jambs and level at the crown, be 
 drawn, let the form of the wall be what it will, by getting the line b d f h i firft ; then the 
 foffit may be pricked from the plan whatever may be its form. 
 
 D 2 
 
 PLATE 
 
/ 
 
 THEORY AND PRACTICE 
 P L A T E XV. 
 
 cutting right in a wall which does not Jland perpendicular to the 
 ground^ to a level bafe. Fig. J. 
 
 Let a e zt D be the level of the ground, a I the inclination of the v/all, equal to the radius 
 of the cylinder, let fall the perpendicular from / to c, in the bottom line a e make the lemi- 
 circle in Jig. A \ to the width of the cylinder, or the double of / at D, take the diltance a c 
 at D, and make a b equal to it in Jig. yf, and defcribe a femi-ellipfis to the length of the 
 femicircles d d, and to a b its v/idth ; lay che equal divifions round the fcmicircle in Jig. v/, along 
 the line d d double at C, then take the parts e d.^ d c b., b from the plan j 5 , and lay them 
 at D refpedlively from e towards deb., and from / draw I e to make a right angle with / <7, 
 and at the points abed eredl perpendiculars to r? ^ to cut I e 2X f gh and f, take the diftances 
 e z, i h^ h ^,,.and g /, and lay them on the foffit at C refpeclivtly, from i z/, 2 c, 3 4 zz, each 
 
 way, then will the ftraight line d d in the foffit, when bent round, be perpendicular over the 
 elliptic line in the plan and the curve line d d c b a., &c. d will fall over the points d c b am 
 the plan: in the fame manner the edge of the foffit may be brought to anfwer any curve line 
 propofed. 
 
 r 
 
 To draw the arches of groins by a new method, whether right or rampant, fo that their arches 
 Jhall interfebi or mitre truly together, from a given arch of any form. 
 
 Let fig. E be the given arch of a gothic form, draw the chord a c for one half the arch, 
 divide it into any number of parts, as 4, and through the equal divifions draw lines from the 
 centre e to terminate in the circumference zth g I, draw lines from c through h g I to cut 
 the perpendicular a d b, c, d and if No. 2 is required to be wider, but the fame height 
 as fig. E, draw the two chords a c and c b for each fide of the arch, divide each into four 
 equal parts, as before, and fet the divifions a, b, c, d, perpendicular on each end of a b z.t 
 No. 2, and from thefe divifions draw lines to the crown at c, then trace the curve through 
 the points h gl. Sic. fo the arch at No. 2. will truly mitre into fig. E ; in the fame manner 
 the rampant curve at No. 3, will be brought to correfpond with fig. E and No. 2 
 
 * Nothing can be more ready than this in praftice, becaufe a chalk line will foon ftrike all the radial linei, 
 having only to move it but once from the point f up to c at the crown ; Agy F thews the common method by 
 dividing the bafis of each into a like number of parts, and transferring the height, as the figures explain, at 
 No. I, and No. 2; nothing is more tedious in praftice than raifing a number of large perpendiculars, and going 
 •ontinually from one curve to get the height of another. 
 
 20 
 
 THE 
 
 To draw a cylindrical foffit. 
 
 PLATE 
 

OF CARPENTRY, 
 
 21 
 
 PLATE XVI. 
 
 As It happens fometlmes in church work^ that windows go higher than the ceiling line^ therefore- 
 
 the ceiling ’ivants to be hollowed out^ fo that the light tnoy be thrown down into the body of 
 
 the church j / /hall, in this place, foew the method of making a curb for that purpofe. 
 
 To find the form of the curb. 
 
 Let k b Ihe the head of the window, figure A, and let it come as high as a b, above 
 the ceiling * ; and let a b zt No. i be the fame height, and b c the diredlion of the light, 
 and a c will be the length of the curb. Make « c at No, 2, equal to 0 at No. i, and divide 
 it into fix equal parts ; alfo divide a b, in figure A, into fix equal parts, and let the ordinates 
 be drawn as is explained in the figure, a curve being traced round the points of interfedtion, 
 will give the form of the curb. 
 
 The ceiling is here fuppofed to 'be level, which is feldoni the cafe in a church} bvtt the method will 
 be nothing different if the ceiling line a e was to incline to the horizon in any angle whate\er. 
 
 Figures B and C fliew the method of drawing and backing any elliptic rib with a com- 
 pafs, which is exceedingly handy in drawing, and will be near enough for the reprefenta- 
 tion of an elliptic rib on paper, as no other method will be fo clean when done j but 
 for pradlice, nothing is more handy than a trammel, or interfedling lines. 
 
 To draw and back ribs by this method. 
 
 In B, let c hhe the height, and r ^ the width ; divide, the difference into three equal 
 parts, and fet four fuch parts on each fide of c, tO ' d and d, and make an interfedlion with 
 the diftance d d zt e, and draw a line through e and d out to /, then d and e are the centres 
 for the ribs. And fuppofe the rib is to be backed as much as a b upon the bottom, fet 
 a b from d to f and from e to g, parallel to the bafe ; and draw a line through g f out 
 to k\ then ^ andyare the centres for deferibing the backing. 
 
 The rib E is traced from D, and a b is fet all round on the parallel lines, fhews the t 
 backing is alfo ufed for drawing on paper. 
 
 The method of drawing the rib C is only the reverfe of the other at B, and therefore 
 wants no other defcripiion. 
 
 Note. The word backing fignifies the beveling of any rib, fo as to range in a ftraight or 
 regular curve line with any other number of ribs already fixed \ nothing is more difficult to 
 un ift ind in the pradice, where groins or other arches meeting together do not forma 
 ftraight line upon the plan, then the ftiifting of a mould cannot be applied to a curve furface 
 as It is to a ftraight one j in this cafe we muft have recourfe to another method. 
 
 
 PLATE 
 
22 
 
 THE THEORY AND PRACTICE 
 
 PLATE XVII. 
 
 Definition. 
 
 Groins are the InterfeSlion of arches or vaults cutting acrofs one another^ meeting on their diagonal 
 
 fedlions. 
 
 BRICK G R O I N S. Description. 
 
 J, -7, a^ a^ See. is the plan of the piers which the vault is to Hand upon, a h is the end 
 opening, which is a given femicircle ; in this b c is the opening of the fide arch, which is 
 to come to the fame height as the end arch a b : fix your centres over the body range, 
 fg. A.^ as fliewn in the feiSIion at C, then board them over. In fig. A is the manner of 
 fixing the jack ribs upon the boards, which likewife fhews at C. 
 
 To find the mould for the jack ribs. 
 
 Take the openings of your arches in fig. 4 that is a b and b r, and lay them down in 
 
 fig. D, zt a b and b to make a right angle. Divide one half of the given femicircle E 
 
 into five parts, and fquare them acrofs i i i, &c. to cut d b and d r, the diagonals, in 2 2 2, 
 
 he. and through the points 2 2 2, he. draw lines parallel to i i i, he. the bafe of E 
 
 both ways towards F and G; flick in nails at 12345 in T, and bend a thin flip of 
 wood round them, which mark with a pencil at every nail ; this flip of wood being 
 flretched out from <i, i 2 3 4 5, and fquared over to G, will interfecl the other lines in 
 fmall fquares : a curve being traced through the diagonals of each fquare, wdll give a 
 mould for to fet the jack ribs. 
 
 How to fix the jack ribs. 
 
 Bend your mould G from to the crown at e, in fig. A-, that will give the edges of 
 your boards ; then fix a temporary piece of wood, level upon the crown, in the diredion of 
 f f and let it come the thicknefs of your boards lower than the crown, then it will give the 
 height of your jack ribs, which is a very fure method of placing them. 
 
 To find a mould to cut the ends of the boards. 
 
 The rib F is traced to the height of T, or got by a trammel, which will be fully exem- 
 plified in the following plates. Take the parts round T, and lay them out to i 2 3 4 5 ; 
 then H will be got in the fame manner as G, which will be a mould to cut the ends of 
 your boards that goes upon the jack ribs againfl the body range. 
 
 Fig. I. Is an eafy method of getting the moulds when both arches are the fame opening. 
 
 Take half the opening of the arches, whatever they are, and draw a quarter circle, and 
 divide It into fix; bend a flip round it to take its parts, then ftretch it out upon the bafe 
 from I to 6, and fquare over your points i, 2, 3, he. Through the points in the arch 
 draw the lines on both fides the other way, and being traced as before, gives both moulds, 
 being the fame in this, 
 
 1 he curve F may be drawn in praftice with a trammel, independent of the other, and the two 
 moulds F and G may be drawn feparate, without any connexion of lines, as lhall be (hewn hereafter. 
 
 ^ - PLATE 
 

 
OF CARPENTRY. 
 
 23 
 
 PLATE XVIir. 
 
 Defin ition. 
 
 Groins are faid to he ajcending or dejc ending when they are not built upon level grounds 
 
 CENTERING FOR ASCENDING OR DESCENDING GROINS. 
 
 The plan^ and afcent or defcent^ of any groin being given^ and one of the body ribs., as B, al/o the 
 place of the angles upon the plan, to find the form of the fide ribs, fo that the interfe^ion of both 
 arches will be perpendicular over the plan. 
 
 Divide half the circumference of the given rib B, into any number of equal parts, and 
 draw them to^'in ter feel the angles ; and from thence let them be returned up to the rib C, 
 upon the fide ; then C being pricked from the given rib at B, as the letters direft, will give 
 the form of thefide centre. 'Fhe fame is (hewn at F, by the method of interfedling lines. 
 
 To find the two moulds D and E for placing the jack ribs, to bend over the angles in the body 
 range, when boarded in, fo that they may be perpendicular over the angles upon the plan. 
 
 At C, draw lines from the points a b c d e f g. See. where the ordinates of C interfedl the 
 top of the arch, and perpendicular to the rake, and draw the femi-ellipfis A, to the width of 
 the body range ; and to a h, the height of the fide centres, perpendicular to the rake ; and 
 continue the ordinates of B, up to A, to interfedl at i 2 3 4 5 Bend a flip round thefe 
 points, and mark them oppofite to every point, and flretch it out along i 2 3 4 5 6, 
 between D and E, and draw lines through thefe points, at right angles to k 6, to interfedl 
 with the perpendiculars. Begin at 6, and trace a curve both ways, will give the edges of 
 the two moulds for placing the jack ribs. 
 
 To cut the jack ribs to the rake of the groins. 
 
 Set the number of the jack ribs upon the arch iS, at their proper diftances, anc^take their 
 feveral Heights, that is, h i, k I, and m n, and fet them upon the arch G, from a to b, and from 
 a to c, and from a to d draw lines through thefe points parallel to the rake, which will 
 * fhew how the jack ribs are to be cut, fo that they (hall range properly with the other raking 
 centres. 
 
 Note. All the^dy ribs muft be backed according to the rake of the groin; to do this exaftly the under 
 edges of all the l%s muft be beveled according to the rake ; tlien make a mould as B, or one of the body 
 ribs themfelves will anfwer inftead of a mould, which being applied to each fide of any other rib, keeping the 
 bottom fair with the under edge upon each fide, and drawing the curves by the ether, it will give the backing 
 in this cafe. 
 
 PLATE 
 
THE THEORY AND PRACTICE 
 
 2 + 
 
 PLATE XIX. 
 
 Given the two fide arches of any groin^ and the afcent j to find the interfediion of the angles upon 
 
 the plan. 
 
 Divide half of the body rib By into equal parts, and draw parallel lines tob c d e and fy 
 and on the point a^ as a centre, draw the concentric dotted circles round^to g h i k l» 
 then draw parallel lines to the rake, to cut the centre C, at 12345, and abode on the 
 other fide ; and from thefe points let lines be drawn perpendicular througii the plan. And 
 on the centre of the rib Cy at^, fquare a line up to 5, the top of the arch C\ and from 5 
 draw a line perpendicular through the plan. Alfo through the points 12345, at By 
 draw perpendicular lines to the plan the other way ; begin at hy and trace through the 
 angles both w.ays, will give the place of the angles tipon the plan. 
 
 The moulds for bending over the angles are found in the fame manner as in the laft 
 plate, by taking the ftretch out round Jy and laying it between D and E. 
 
 The Reader may fee fuch groins executed under the Adelphi Buildings in the Strand, 
 London, where the defcent is very rapid in going down to the river. 
 
 The jack ribs of the groin are cut in the fame manner as directed in the laft plate, and 
 in the pradice there will be no occafion for tracing the angles, as the two moulds D and 
 E are done independent of them : the Reader v/ill farther obferve, that the arch B muft 
 not be ufed inftead of the arch Ay which would produce a very great error in the moulds 
 D and E, as it muft be evident to every one, that the feiftion upon the fquare of the 
 cylinder, or body range, muft be lefs in the height than the perpendicular or plumb fedlion B 
 which in this cafe is oblique ; if thefe things are properly underftood, there will occur 
 nothing in brick groins but what may be eafily furmounted. 
 
 In all kinds of brick groins the centres or body ribs muft be fixed firft in the fame 
 manner as if there were no fide-arches cutting acrofs them ; then the centres muft be boarded 
 over ; then to find the place of the angles upon the boards, that is, the proper interfedlion 
 of the fide-arches upon the plan, the moulds D and E muft be both bent round the boards 
 _at one time, by keeping the points / and e of the moulds D and E upon the tops of the 
 p-ers at (3 and ; then keep the top points together, and bend them round, keeping them 
 ftiil togedier, then the point at^ 5, will fall perpendicular over h in the plan ; round the 
 inner edges of the moulds draw a curve upon the boards, which will be the proper inter- 
 fecUon of the fide-arch. The jack ribs are cut in the fame manner as diredted in the laft. 
 
 PLATE 
 
CARPENTRY. 
 
 O F 
 
 as 
 
 PLATE XX. 
 
 PLJISTER GROINS.- 
 
 The angles or diagonals of any plaijier groin^ which are Jlraight upon the plan^ arid one of the 
 fide arches being given^ to find the other fide arch and angle rib. 
 
 Fig. I. Case i. If the given rib is a femicircle femi-elHpfis they may be defcribed as 
 in fig. 2, plate 6, with a trammel, which is by far the readieft method; bat if a proper trammel 
 is not to be got, a temporary one may eafily be made, which will anfwer equally as well, 
 by fixing two pieces of wood in the form of a fquare, that is to make a right angle ; each 
 leg mull be as long as the difference between the femi-tranfverfe and femi-conjugate axis, 
 and inftead of the Aiding nuts in the rod, two brad awls will anfwer the purpofe, being put 
 through any ftraight flip of wood, and by moving this round either the exterior or interior 
 angles of the fquare, keeping the pins or brad awls clofe to each leg, it will defcribe one- 
 quarter of an ellipfis at one timci 
 
 To find the length of the jack ribs. 
 
 Lay down the plan of the ribs, as at and draw a rib upon each opening ; then draw 
 perpendicular lines from the plan of each opening, at the extremities a c e^io cut its cor- 
 refponding rib at b df\ then the diftance from b to b fhews the length of the firfl: jack rib,, 
 from dtod the length of the fecond, and from f to / the third. 
 
 How to back or bevel the angle ribs^fo that they Jh all range with each opening of the groin. 
 
 Firfl: get the ribs out in two halves or thickneffes, as at E and F, then draw the plan of 
 your angle rib which is placed between E and F, will fhew the true bevel upon the bottom 
 of the rib; then fhift your hip mould parallel upon the bale of E and F, will fhew how much 
 wood there is to be bevelled off; then nail the two halves together, and it will be completed. 
 
 METHOD I. 
 
 Fig. 2. Case 2. When the given rib is a fegment of a circle, or any other curve what- 
 ever, the ribs will be defcribed as in plate i Sr fig' ^r are fhewn at B, E, and F. 
 
 METHOD II. 
 
 When the given arch is a fegment of a circle as at y/, take its height b c, and place it from 
 ^ to r at Cand D ; then take the whole diameter of the arch J, that is, twice the radius a Cy 
 and place it from the crown of the other arches perpendicular to their bafes from r to at C, 
 and from c to d ^t D\ then the arch may be drawn as in plate 8, by interfcdting lines ; the 
 backing of the ribs is done in the fame manner as in the laft groin. 
 
 Either of thefe two methods are much readier in pradUce than tracing the ribs through 
 ordinates. 
 
 E 
 
 PLATE 
 
20 
 
 THE THEORY AND PRACTICE 
 
 PLATE XXL 
 
 Given one of the body ribs, and the angles Jiralght upon the plan^ and jhe afcent of a groin not 
 Jlanding upon level ground^ to find the form of the afcending arches^ and the angle ribs. 
 
 Let b a c zt B he the angle of the afcent, from the point b make b c perpendicular to a 
 and deferibe the rampant curve 5 , as in plate 15, at No. 3, in fig, E ; then draw the dia- 
 gonal a b zt E, and make b c perpendicular to it, and equal to b c zt B \ then draw the hy- 
 potenufe a c, and deferibe the angle rib E, in the fame manner as that of B. 
 
 To find the length of the jack rlbs^fo that they Jhall fit to the rake of the groin. 
 
 Draw lines up from the plan to the arch, as at A In the fame manner as explained in the 
 laft plate ; then the arch from a to a is the firft jaca. rib, from b to b the fecond, and from c 
 to c the third, &c. 
 
 How to back the angle ribs for fuch fort of groins^ fo that they Jhall range ea:h way. 
 
 Get the ribs out in two halves, as in the laft plate, then the bottom of the ribs muft be 
 bevelled agreeable to the afcent of the groin, and the plan of it muft be drawn upon the level, 
 and from thence they may be drawn perpendicular from the plan to the rake of the ribj then 
 take a mould to the form of the rib, or the rib itfelf, and Aide this agreeable to the rake to the 
 diftance that is marked upon the bottom to be backed ofF, will fhew how much the rib is to 
 bevel all round. 
 
 PLATE 
 
Plate 22 . 
 
OF CARPENTRY. 
 
 27 
 
 PLATE XXII. 
 
 OF GROINS CUTTING UNDER PITCH, 
 Definition. 
 
 fF'hen the fJe arches of a groin are lower than the body archy then they are called under pitch groins-. 
 
 Given one of the body ribs B, and the height f g, of a door or window^ &'e. at D, and ifs width 
 m 1, to find the fide and angle rltrs D and ILyfo that the interfediion of the fide arch D, with 
 the body rib B, Jhall be Jlraight upon the plan. 
 
 Draw c e perpendicular to c the bafe of Py and equal to the height of the window at 
 Dy that is, equal to f g \ through e draw e a parallel to c by cutting the arch B in a, let fall 
 the perpendicular a b to c b, and continue it fo as to cut the line f g produced to ky and draw 
 k m and k /, which is the place of the angles upon the plan, or the bafe of the angle ribs ; 
 then the ribs D and E may be deferibed from the given rib T, as direded in plate 15 , fig. Ej 
 from a centre j or they may be deferibed as at fig. F of the fame plate, as you fee on 
 the other fide at ^ and C by ordinates ^ but th-e firft is by far the eafieft method for 
 pradice, for rf you flick a pin or brad awl in gy at D, and lay a chalk line to it, you may 
 ftrike all the radial lines g ly g 2, g 2r S much lefs time than the parallel lines 
 
 in ^and Ccan be drawn, and with much greater accuracy j and the divifions upon c n of 
 the arch T, may be marked upon a rod, and readily transferred to the arches D and Ey on 
 m py and fg'- then move your brad awl out of gy and fticlc it in the crown atfy and 
 ■flrike lines from the divifions of m p to erofs the other lines,- will give the points through 
 which the arch muff pafs ; but the Reader muff recoiled that four or five points will not be 
 fufficient in the pradice for tracing the curve with accuracy, and therefore a greater number 
 muft be found. At the other end of the groin is fhewn the manner in which it maybe 
 fixed,, fuiftciently intelligible for a. workman. 
 
 - E 2 
 
 PLATE 
 
THE THEORY AND PRACTICE 
 
 ^8 
 
 PLATE XXIir. 
 
 ^ TF 'E, L C H GROIN. 
 Definition. 
 
 JV dch groin is an under pitch groln^ whofe ftde and body arches are both given femlclrcJes^ or they 
 may be fmillar fegments of circles cutting through one another, whofe interfeSilons do not meet 
 in a plane furface, that is, the place of the ribs will not be Jiraight upon the plan, but will ge- 
 nerate a curve line. 
 
 Given the body rib A, and the fide rib B, of a Welch groin, to find a mould for the interfedling ribs. 
 
 Divide half the arch B into any number of equal parts i, 2, 3, d, or they may be taken 
 at difcretion, and from thefe points let fall perpendiculars to a b, its bafe; produce them at 
 pleafure j alfo from the fame points i, 2, 3, d, draw lines parallel to a b, the ^afe of B, to 
 interfedt the perpendicular line e /; transfer the divifions from efioeg\ then from the di- 
 vifion of ^ ^ draw lines parallel to p q, to interfea the body rib J at the points h uw y, 
 from thefe points draw perpendiculars to p q, its bafe, and continue them to interfedf with 
 the perpendiculars from B, at the points k, I, m, n, between C and Z) j then trace a curve 
 through thefe points, which will be the place of the interfering ribs upon the plan j then 
 draw two other curve lines on each fide of h, I, m, n. See. to make the thicknefs of the rib 
 upon the plan j on the infide of the curve draw two chords for each half to their extremi- 
 ties, draw two other lines parallel to them to touch the outfide curve, then the diftance 
 between thofe two ftraight lines will fhew what thicknefs of fluff it will take to make the 
 interfering rib j through the points ^ / zw «, &c. draw perpendicular lines to the chords, 
 ina’ce the heights c <5^, 3 3, 2 2, i i, kc. at D, equal to their correfponding heights at 5 j 
 then D is the mould for the interfering rib ; C is the fame as D. 
 
 To bevel the ribs, fo that they will fland perpendicular over the plan. 
 
 At the points x v t i, draw the parallel dotted lines to the ordinates of Cand D, and 
 make their correfponding heights equal to thefe of the arch B ox A \ draw the dotted curve 
 line h u w y zt C, and it will fhew how much is to be bevelled off on that fide of the rib • 
 in like manner the other fide D is bevelled, as is fhewn by the dotted curve line. 
 
 To find a mould to bend under the interfecling ribs, fo that it Jhall give the place of the angle 
 
 truly upon the plan. 
 
 Take the ftretch round the under fide of the rib D at the dots, by bending a thin flip of 
 wood round it, mark it at each dot, and flretch it out along the flraight line ^ r at E, 
 draw the ordinates acrofs, and prick them from the plan that lies between D and C, then E 
 agreeable to the letters will be the mould required. 
 
 Note, The ftraight edge of the mould muft be kept exaftly to the face of the rib; when it is bent round, 
 then draw a curve round the under fide of the rib, by the other edge of the mould, will give the true place of 
 the angle. 
 
 7 
 
 PLATE 
 
mmmmmamrn 
 
OF CARPENTRY. 
 
 *9 
 
 PLATE XXIV. 
 
 There will be no occafion for explaining the lines of this groin, as they are of the fame 
 nature as thofe in the laft plate; but it will be proper to take notice, as this is a bevel groin, 
 the ribs muft lie in the fame diredion as the plan of the groin, which will make them longer 
 than their correfponding given arches at the top, but of the fame height ; they are con- 
 fequently ellipfes, being the fedlions of cylinders, therefore to make a rib over / m acrofs 
 the two piers take the extent of the bafe I and the height of the given arch n #, and d-e- 
 (bribe an ellipfis ; and to deferibe the fide arches between any two piers, as from a to 
 take the extents and the height of the given arch p gy at and deferibe an ellipfis, it will 
 give the proper form of the rib to ftand over a b ; the interfering ribs will require two 
 moulds C and Z), owing to the groins being bevel upon the plan. 
 
 Note. The letters are marked the fame upon D and C as they are upon E, to fhew 
 they are traced from it. 
 
 PLATE XXV. 
 
 Ti? deferibe the Interfediing or angle ribs of a groin Jlandtng upon an oSlagon plany the tide and 
 body ribs being given both to the fame height. 
 
 Fig. I. E is a given body rib which may be either a femicircle or a femi-elllpfis, and 
 is a fide rib given of the fame height ; Z) is a rib acrofs the angles, trace from E, the 
 bafis of both being divided into a like number of equal parts, divide the bafe of the given 
 rib Ay into the famejtumber of parts; from thefe points draw lines acrofs the groin to its 
 centre at w, and from the divifions of the bafe of the other rib Z), draw lines parallel to the 
 fide of the groin, then trace the angle lines through thefe fquares, will be the place of the 
 interfering ribs, draw the chords a by and b r, then prick the moulds B and C from E or Z), 
 but take care not to prick them from the crooked line at the bafe, but from the ftraight 
 chords a by and b c. 
 
 To deferibe and back the angle ribs of a groin circular upon the plany the fide and body arches 
 
 being giveUy as in the lafi groin. 
 
 The ribs are deferibed in the fame manner as in the laft example for the oflagon groin, 
 or in the fame manner as the Welch groin, plate 23, and the backing or bevelling is found 
 in the fame manner as is deferibed in that plate. 
 
 Note, E and F are the fame moulds as are fhewn at B and D. 
 
 PLATE 
 
THE THEORY AND PRACTICE 
 
 PLATE .ICXVL 
 
 The fide rih A, and the angles being given Jlralghi upon the plan^ to find the angle rib C, 
 
 and the body rib C. 
 
 Fig. I. The rib A\s fuppofed to be placed over the ftralght line a and its bafe divi- 
 ded into any number of equal parts, as 8 j from the divifions draw lines to the centre of the 
 groin, to interfecl the angles at the points abode f g \ place the foot of your compafs in the 
 centre of the groin, and from the points a b c d^ &c. draw lines to the bafe of C, and make 
 the ordinates of Cequal to thofe of then C is the body rib ; draw lines at right angles 
 from the points ab c dy &c. and prick the moulds G and B from Ay will be the angle ribs 
 required ; this wants no mould to bend under the angle ribj as in the others that are crooked 
 upon the plan. 
 
 How to deferibe the ribs of a groin over flairs upon a circular plan^ the body rib being given-. 
 
 Fig. 2. Take the tread of as many fteps as you pleafe, fuppofe nine, from Ey and the 
 heights correfponding to them, which lay down F -y draw the plan of the angles as in the^ 
 other groins, and take the ftretch round the middle of the fteps at Ey and lay it from a to b 
 at F-y make d e perpendicular to h e at By equal to d e at Fy draw the hypotenufe e r,.draw per- 
 pendiculars from d c to By and prick B from Ay as the figures diredf, then B is the mould 
 to ftand over ah’, draw the chords a 4, and 4 m at the angles, make 9, 4 perpendicular to 
 them, each equal to half the height d Cy at 5 or Fy draw the hypotenufe g 4, and h my 
 draw the perpendicular ordinates from the chords through the interfe£lion of the other 
 lines that meet at the angles, then trace the moulds D and G, from the given rib Ay will 
 form the moulds for the angle or interfe6ftng ribs. 
 
 Note.. The reafcn that the angle ribs D and Gare laid contrary ways, is only to avoid 
 confufioHi 
 
 PLATE 
 
OF CARPENTRY. 
 
 31 
 
 PLATE XXVII. 
 
 A$ the fur face of a globe is every where of the fame curvature^ confquent^y the furface of ary 
 fegment of a globe^ or part^ niuji Jlill retain the fame curve as before it was cut ; and for 
 this reafon^ it appears that the curves of the back ribs of a niche mitjl be the fame fweep as 
 the ground plan itfelf \ and the front rib is a femi circle. See Axiom the yU page 14, and 
 figure i, plate 10. If a femi- globe is cut by a plane at right angles to the plane of its hafe^ 
 then the feSlion is a femicircle. The practice of this is eafy. 
 
 To get out the ribs for the head. 
 
 From the centre c draw the ground plan of the ribs as at figure and fet out as many 
 rih^ upon the plan as you intend to have in the head of the niche, and draw them all 
 out towards the centre at c. Place the foot of your Compafs in the centre r, and from 
 the ends of each rib, at e and c^ draw the fmall concentric dotted circles round to the 
 
 centre rib at m and n j then draw m g and n /, parallel to r the face of the wall ; then 
 
 from q round to e upon the plan, is the length and fweep of the centre rib, to ftand over a b\ 
 and from i round to <?, the length and fweep of the rib that ftands from rto d upon the 
 plan ; and from ^ round to e is the fweep of the ftiorteft rib, that ftands from e to f upon 
 the plan. 
 
 How to bevel the ends of the hack ribs againjl the front rib. 
 
 The back ribs are laid down diftinft by themfelves at C Z> and from the plan. 
 
 Take c i, in figure A^ and fet it from c to i in JD, will give the bevel of the top of the 
 
 x\h D. And it oxx\ figure A^ take from ^ to 2 upon the plan, and fet it from ^ to 2 in the 
 rib Ey will give the bevel of the top. 
 
 To find the places of the back ribs where they are fixed upon the front. 
 
 From the points a c and at the ends of the ribs, in the plan,^^«r^ draw the dot- 
 ted lines up to the front rib, to d f and w, which will fhew where they are to be fixed 
 upon the front rib. The double circle upon the front rib fhews the backing. 4 
 
 « 
 
 o 
 
 PLATE 
 
32 
 
 THE THEORY AND PRACTICE 
 
 PLATE XXVIII. 
 
 To find the curve of the ribs of a globular niche, the plan and elevation being given fegmenit 
 
 of circles. 
 
 fiS" elevation of the niche, being the legment of a circle whofe centre is t, at B 
 
 is the plan of the fame width, and may be made to any depth, according to the place it is in- 
 tended for, and its centre is r ; on the plan B, lay out as many ribs as you think it will 
 take, draw them all tending to the centre at c, they will cut the plan of the front rib 
 \ngfe i/j through the centre r, draw the line m n, parallel to a b, the plan of the front 
 rib ; put the foot of your compafs in the centre at r, draw the circular lines from g,f, e, d, 
 to the line m «, and make c s equal to u /, that is, make the diftance from the middle of 
 the chord line m n to r, the centre of the arch at C, equal to the diftance from the middle 
 of the chord of the top at fig. A, to its centre at t-y then place the foot of your compafs in r, 
 as a centre, and from the extremities m or «, deferibe the arch at C, with the fame centre 
 draw another line parallel to it, to any breadth as you intend your ribs Ihail be, then C is the 
 true fweep of all the back ribs in the niche. 
 
 Note. The points I k 1 hy fhew what length of each rib will be fufficient from the point nr^ 
 from h to m\% the rib that will ftand over d Xy from / to m is the rib that will ftand over ^ Uy 
 from k tom over/?;, and from I to m over gw'y the other half is the fame. ’ 
 
 The truth of this is eafy to be conceived by thofe who have previoufly ftudied the Sedions 
 of a Globe, in plate lo of this book. 
 
 Through the centre /, draw d e, parallel to a b, complete the fweep of the top, g f, 
 to the line d e, then d e is the diameter y through n draw n a parallel to u d, in the 
 centred; with the diftance r a deferibe. another femicircle, whofe diameter is A; then 
 Will the femicircle c f g a e, be equal to a vertical feaion of the globe, ftanding on i k, 
 pairing through its centre at r, which is the fame fweep as the rib at C, becaufe u a is 
 equal to r k, and r r; bifeding m n 2X right angles, is . equal to t Uy bifeaing e a at 
 right angles j therefore the hypotenufe / a, that is, the radius of the circle b a g E c, 
 is equal to s the radius of the circle or rib at G 
 
 PLATE 
 
OF CARPENTRY. 
 
 33 
 
 PLATE XXIX. 
 
 The plan of a niche being glven^fandlng In a circular wall^ to find the front rib. 
 
 B is the plan given, which is a femicircle whofe diameter Is a and /, /, /j», the 
 
 front of the circular wall ; fuppofe the femicircle B to be turned round its diameter a b^ fo 
 that the point v may ftand perpendicular over h in the front of the wall, the feat of the 
 femicircle (landing in this pofition upon the plan will be an ellipfis ; therefore divide half the 
 arch B upon the plan into any number of equal parts, as 5 ; draw the perpendiculars i d^ 
 3, 3 4 5 upon the centre c with the radius c h, defcribe the quadrant of a fmaller 
 
 circle, which divide into the fame number of equal parts as are round B j through the points 
 Jj 2, 3, 4, 5, draw parallel lines to a b^ to interfe£l the others at the points </, h ; 
 
 through thefe points draw a curve, it will be an ellipfis ; then take the ftretch out of the 
 rib 5, round i, 2, 3, 4, 5, and lay the divifions double at i^, ftretched out; take the fame 
 diftances d /, e k^fl^g from the plan, and at F make d /, e k^fl^ equal to them, which 
 will give a mould to bend under the front rib, fo that, the edge of the front rib will be 
 perpendicular a^ r, ky /, m. 
 
 Note, The fweep of the front rib is a femicircle, the fame as the ground plan, and the 
 back ribs at C D and E are llkcwife of the fame fweep. 
 
 The reafon of this is eafy, the niche being part of a globe, the curvature mull be every 
 where of the fame fweep, and confequently the ribs fit upon that curvature. 
 
 Note. The curve cf the mould F will not be exa£lly true, as the difiances d /, e kyf ly See. 
 are rather too ftiort for the fame correfponding difiances upon the foffit at F, but in pradlicc 
 it will be fufficiently near for plaifier work ; but thofe who would wifti to fee a method 
 more exa<^, may examine plate iStfig, A where C is the exad foffit that will bend over its 
 plan at B, 
 
 In applying the mould F when bent round the under edge of the front rib, the ftraight 
 fide of the mould F muft be kept clofe to the back edge of the front rib, and the rib being 
 drawn by the other edge of the mould, will give its place over the plan. 
 
 F 
 
 PLATE 
 
34 
 
 THE THEORY AND PRACTICE 
 
 PLATE XXX. 
 
 *The plan and eleuatlon of an elliptic niche being given^ to find the fwecp <f the ribf^ 
 
 Fig. a. Defcribe every rib with a trammel, by taking the extent of each bafe from the 
 plan whereon the ribs ftand to its centre, and the height of each rib to the height of the top 
 'of the niche, it will give the true fweep of each rib. 
 
 To back the ribs of the niche. 
 
 There will be no occadon for making any moulds for thefe ribs, but make the ribs them- 
 felves j then there will be two ribs of each kind ; take the frnall diftances i e, i d, from the 
 plan at S, and put it to the bottom of the ribs D and E, from d to 2 , and ^ to i ; then 
 the backing may be drawn ofF by the other correfponding rib j or the backing may be drawn 
 ofF with the trammel, as for example at the rib E, by moving the centre of the trammel 
 towards upon the line ^ c, from the centre c, equal to the diftance i e, the trammel rod 
 remaining the fame as when the infide of the curve was ftruck. 
 
 Given one of the common ribs of a cove bracket ^ to find the angle bracket for a fquare 
 or redi angular room. Fig. H, 
 
 Let H be the common bracket, b c its bafe ; draw b a perpendicular to b r, and equal to 
 it, draw the hypotenufe a r, which will be the place of the mitre j take any number of ordi- 
 nates in perpendicular to b r, its bale, and continue them to meet the mitre line'^ that 
 is, the bafe of the bracket at 1 ; draw the ordinates of / at right angles to its bafe j then the 
 bracket at /, being pricked from Hy as may be feen by the figifres, will be the form of the 
 angle rib required. 
 
 Note, The angle rib mufl: be backed either externally or interpally, according to the 
 angle of the room. 
 
 Having given a common bracketlLJee fig, G^for plaljiery to find the mitre bracket L* 
 
 Proceed as in the laft example, and you w^ill have the bracket required. 
 
 PLATE 
 
35 
 
 OF CARPENTRY. 
 
 PLATE XXXI. 
 
 SKT^LIGHTS. 
 
 'To find the length of the hips of a Jky -light funding upon a fquare plan, the height being given. 
 
 In fg. A, draw the diagonals a h, and c d ; they will bifeifl each other at right angles at e ; 
 take a e kx the bafe of any hip ; from e make e /equal to the height of the fky-light j from 
 atof draw a line a f it will be the length of the hip required. 
 
 To find the backing of the hip. 
 
 Draw any line k /, at right angles to a e, the bafe of the hip rafter, to cut it in any point A, 
 put the foot of your compafs in h, as a centre, and with the other deferibe a circle to touch af^ 
 the hip rafter, to cut the bafe line a e, 2 X g\ then draw g i and g k ; then the angle k gi 
 will be the backing of the hip, as is fliewn by the bevel at B ; but the beft way to v/ork the 
 hips is to apply a bevel to the parallel fides of the hips, as is fhewn at C, by making the 
 other fide of the bevel parallel to a e, the bafe of the hip. 
 
 Note. The fame lines will extend to any fky-light, whatever may be the form of its plan ; 
 if it be any polygon, to find the length of the hip rafter, draw a line through any point in 
 its bafe at right angles to it, fo as to cut the two contiguous fides to that bafe, and on the 
 faid point as a centre deferibe a circle to touch the hip rafter from the point where this circle 
 cuts the bafe line, draw two lines to meet the ends of the perpendicular line at the fides of 
 the polygon ; then the angle formed by thefe two lines will be the backing required ; but 
 perhaps a few more examples will make it plainer than many words can. 
 
 Fig. 5 is a fky-light, ftanding upon a redangular bafe, having a ridge in the middle ; 
 make c d upon the ridge line equal to half the width o^a b\ then the angle b d a will be 
 a right angle } every other requifite is the fame as direded for fig. A\ if thefe hips are to be 
 mitred, the bevel at C (hews the mitre. 
 
 Fig. C is another fky-light, ftanding alfo upon a redlangular bafe ; but the hips all meet 
 over the centre of the plan at r, and confequently the diagonals do not bifedb each other at 
 right angles ; therefore take any bafe line as a e^ or eg, and make ef perpendicular to a e, from r, 
 and equal to the height of the fky-light ; and draw / a, ox f g, for the length of the hip, by 
 drawing the line I m zt right angles to a e. The backing will be found in the fame 
 manner as the others above. This fky-light will require two different bevels D and E, to 
 be applied to the parallel fides of the hip, which are both found from the backing by drawing 
 the ftocks of the bevel parallel to a e, the bafe of the hip. 
 
 But if the hips are to be mitred together, F and G fhew the two bevels for the mitring 
 each half, fo that -^vhen put together (hall form the proper backing. 
 
 Fig. X) is a fky-light ftanding upon an odangular plan, as is deferibed In fig. 8, plate 6, 
 of the Geometry ; the lengths of the hips and backing of the angle are found in the fame 
 manner as diredfed for others. 
 
 Fig. F is a fky-light whofe plan is a trapaziod; upon each end as a diameter deferibe 
 a femicircle to cut the ridge-line^ from thefe points draw lines to the extremities of their 
 refpedfive diameters, which will form a right angle for the bafe of the hips to ftand upon the 
 backing or mitring of the hips, will be found as is deferibed in fig. A and B. 
 
 F2 
 
 PLATE 
 
36 
 
 THE THEORY AND PRACTICE 
 
 PLATE XXXII. 
 
 LINES FOR ROOFING. 
 
 The lines for roofing are found in the fame manner as the fky-lights in the laft plate \ 
 the length and backing of the hip muft be found in the fame manner as direfled for the Iky- 
 lights i and if it is required to find the end of a horizontal bar, fo that it fhall fit againft 
 the hip, it will be found in the fame manner as finding the form of the end of a purline, fo 
 as to fit againfl: the fide of a hip rafter, which method will be defcribed below ; the fame may 
 be faid of the jack bars to be fitted againft the hip of a Iky-light as in a roof. 
 
 In this plate one end of the roof is {hewn in order to (hew two cafes : the firft is when 
 the purline lies level, or having two fides parallel to the horizon j the fquare at and 
 the bevel at C, will lliew how to draw the end of the purline in this eafy cafe; but the 
 following method is univerfal in all pofitions of the purline. 
 
 Note. There will be no occafion to draw this at large ; as the bevels will be the fame 
 if done to ever fo fmall a fcale, and the fides may be meafured from a fcale. 
 
 Let a b he the width of a fquare roof, make b fox a e one half of the width, and make c d 
 perpendicular in the middle of ef the height of the roof, which is here one third, and draw 
 d and df which are each the length of a common rafter. 
 
 To find the bevels tf a purline againfl the hip rafter. 
 
 Let the purline be in any place of the rafter, as at /, and in its moft common pofition, 
 that is, to {land fquare, or at right angles to the rafter ; and from the point as a centre 
 with any radius, defcribe a circle. Draw two lines q, /, and p «, to touch the circle in 
 ^ p and parallel to f b \ and at the points s and r, where the circle and two fides of the 
 
 purline interfcd, draw two parallel lines to the former, to cut the diagonal in m and and 
 draw m n and k I perpendicular to / m and r and join the points n i and k i ; then G is 
 the down bevel, and F the fide bevel of a purline : thefe two bevels, when applied to the end 
 of the purline, and when cut by them, will exadly fit the fide hip rafter. 
 
 To find the bevels of a jack rafter againfl the hip. 
 
 By turning the ftock of the fide bevel of the purline, at T, from <7, round to the line i z, 
 will give the fide bevel of the jack rafter. And the bevel at that is, the top of a 
 common rafter, is the down bevel of the jack rafter. 
 
 At the bottom is (hewn the manner of cocking down the tie beam upon the wall plate j 
 the proper fize of the cocking is figured at a. 
 
 PLATE 
 
"/.///• .'32 
 
 J‘,a >-yiij,2^jV}VAo/.}cu 
 
I^/c(^c‘ 34 
 
 tft,cAcf- c/ircrt}Ao\'yi 
 
OF CARPENTRY. 
 
 37 
 
 PLATE XXXm. 
 
 This plate fliews the manner of framing a roof in ledgmentj but as roofs are feldom exe- 
 cuted in this manner, I (hall not be very partitular in defcribing its lines. The following de- 
 fcription for winding will ferve for any. 
 
 PLATE XXXIV. 
 
 Hoxv to lay out an irregular roof in ledgmentj with all its beams lying bevel upon the plan^ fo 
 that the ridge may he level when tinijhed ; the plan and height of the room being given. 
 
 The lengths of the common and hip rafters are found as ufual. From each fide in the broad- 
 eft end of the roof, through c and d^ draw two lines parallel to the ridge-line; draw lines from 
 the centres and ends of the beams perpendicular to the ridge-line, and lay out the two fideS' 
 of the roof D and F, by making e d&t E equal to x n'm the length of the longeft common 
 rafter, and c am E equal to « v at and fo on with all the other rafters. 
 
 71? find the winding of this roof. 
 
 Take y -y, half the bafe of the iborteft rafter ; and apply this to the bafe of the longeft 
 rafter, from z to i ; then the diftance from i to 2 ftiews the quantity of winding. 
 
 How to lay the fides in winding. 
 
 Lay a ftraight beam along the top ends of the rafters at E, that is, from c to and lay 
 another beam along the line a b^ parallel to it, to take the ends of the hip rafters at m and 4 
 and the beams to be made out pf winding at firft. Raife the beam that lies from a to at 
 the point to the diflance i 2 above the level ; which beam, being thus rai fed, will raife all 
 the ends of the rafters gradually, the (anic as they would be when in their places. 
 
 The lame is to be underftood ot the other iide D ; the ends are laid down in the fame 
 manner as making a triangle of any three dimenfions. 
 
 To fatisfy the curious, I have given the lines of this roof; but in pra<*ilice there is not the 
 leaft occafion for framing the lide: in winding, for inftead of the ridge-line, the top is made 
 level at the wideft end of the ,ool, from tne narroweft end, which begins at a point; and by 
 this n.eans the itdes may be mcd quite out of winding, which will have a much better 
 effetSl than any winding roof can have. 
 
 PLATE 
 
3 « 
 
 THE THEORY AND PRACTICE 
 
 PLATE XXXV. 
 
 P O L r G O N ROOFS. 
 
 The methods of conftrudling regular polygons upon any given fide, are fbewn at/^. 4,-5, 
 and 6, in the Geometry. 
 
 The plan of a polygon roof being given., and one of the common ribs Jlanding upon that plan., to find' 
 the angle rib., and the form of the boards that will cover it when the ribs are fitted up. 
 
 In fig. A let B be the given rib ; divide the curve into any number of equal parts, as four, 
 and lay them at D from a to 4, which bifedts b the fide of the polygon, at right angles ; 
 through thefe points draw lines parallel to the fide b b the polygon ; at B and D make i c 
 at D equal to c c, between B and C l d equal to d dy and 3 e equal to e e^ &c. and through 
 the points b, c, d, <?, f draw a curve line, which will be the form of the boarding ; from the 
 points gy f e, d, c, draw lines at right angles to g by the bafe of the angle rib, and p. ick the 
 rib C from D as they are marked by the letters, which is plain. 
 
 Note. The more parts there is in this operation, the truer it will be, or any other of this 
 nature. 
 
 In the fame manner may the covering and angle ribs of any other polygon be found, 
 whatever may be the form of the ribs, as is fhewn at figures B and C. 
 
 To find the covering of a fpherical dome. 
 
 Fig. D. Make a circle i c kf of the plan of the dome, and if it is a femi-globe take the 
 flretch out of one q-uarter for the length of a board ; make the length of K from <7 to 4 equal 
 to it, and let t ie w idth c c, at the bottom, be any thing that the board will admit of yon the 
 bafe r r, as a diameter, make a femicircle ; divide half the arch-line into any number of equal 
 parts ; draw the little lines i i, 2 2, 3 3, parallel to r c the bafe of the board, and divide the 
 height into the fame number of equal parts ; draw the ordinates acrofs ; make i i, 2 2, 3 3, 
 upon thefe ordinates, equal to i i, 2 2, 3 3, in tHe femicircle at the bottom ; a curve being 
 drawn through thefe points, will be the mould K for the covering. 
 
 ' To cover a fpherical dome when the top does not rife fo high as a femicirchy but only a fegment, 
 
 Suppofe I dto\)Q the height of the dome at F, and the width c/ of the dome as before, 
 upon the chord c f\ with the perpendicular height / d deferibe a fegment, which will be the 
 fame as a vertical fedtion Handing upon c f\ here is only one half of the fegment, which is 
 fufficient : draw the chord c d ; take c a equal to half the width of a board, whatever it will 
 admit of ; draw a b perpendicular to cut the chord c d 2Xb y take the ftretch or circumfe- 
 rence of the arch c dy and make the length of 1 from « to 4 equal to it ; take the double of 
 a Cy at Fy and make it the bafe of the board at /; take a b from T, and fet it upon the bafe 
 of /, updn the middle of c c, from a to b\ and with the chord c c, and the height a by deferibe 
 a fegment upon the bottom of the board at /; divide one half into any number of equal 
 parts ; likewife divide the height of the board 1 into the fame number of equal parts ; draw 
 ordinates in both, and the board / will be completed, as in the fame manner is that of H, 
 deferibed before. 
 
 PLATE 
 
JJ. 
 
OF CARPENTRY. 
 
 39 
 
 plate XXXVI. 
 
 DOMES. 
 
 _ As the common method of finding the centres for defcribing the boards to cover a ho 
 nzontal dome will be found m praaice very inconvenient/for thofe boards which come 
 near to the bottom j I fhail in this place fhew how to remedy that inconvenience. 
 
 To find thefweep of the boards at the top. Fig. A. 
 
 Divide round the circumference of the dome into equal parts at i, 2, 2, 4, c 6 kc e?ch 
 divifion to the width of a board, making proper allowance for the camber of each board • 
 draw a line through the points 1, 2, to meet the axis of the dome at ; on as a centre’ 
 with the radii x- 1 and at 2 defenbe the two concentric circles, it will form the board G ■ in 
 the fame manner continue a line through the points 2 and 3 at C, to meet the axis in re : ’then 
 w IS the centre for the board C; proceed in the fame manner for the boards D, E, and F. 
 
 Now fuppofe Fto be the laft board that you can conveniently find a centre, for want of 
 room ; on t its centre, and the radius t 5, make from t on the axis of the dome t a 
 eyial to < 5 ; through the points 5 arrd a dr.w the dotted line 5 to cut the other 
 
 j. circumference of the dome at ; from the points 6, 7,8, q, 10 ij draw- 
 
 radial lines tob, to cut the axis of the dome at i, h, m, n, 0 •, alfo thrLgh’ the’ points 
 
 1 I’ f;.?’ parallels b c, y d, S e, &c. then will each of thefe parallel lines 
 
 be half the length of a chord-line for each board ; then take c 6 from fg. J, which transfer 
 to No. I from r to 6 and 6 ; make the height r at No. I, equal to rf at fig. A-, and drL 
 the chords i 6 and / 6 ; then upon either point 6, as a centre with any radius, deferibe an arch 
 
 two equal parts at r, and through the points 6 and idraw6tf- 
 bifea tb m;>; draw perpendicular; then r 6 is the length, and pq the height of the 
 board G wh,y may be deferihed .s in fig. 4, plate 5, of the GeL^try. 'Ae reader 
 muft obferve, that the length of a board is of no confequence fo as the true fween is mV 
 which IS all that IS required. Proceed in the fame manner with No. 2, by taking Av from 
 
 /g 4 and place It at No. 2, on each fide of at 7 and 7, and take d hi Uoxn fig.A and 
 
 make d k at No. 2, equal to it; draw the chords k^ and f 7, and bifea f 7 at n ; draw n a 
 perpendicular ; upon jhe other extremity at 7, as a centre, deferibe an arch o i 2, and bifedf 
 It at r, and through tliejjoints 7 and I draw the line 7 a, to cut the perpendicular » a at a • 
 but if ine alliance k 7 is too long for the length of a board, bifed the arch o 1 at « ■ throinrh 
 
 7 and i draw 7 t. and draw the little chord a 7, and bifed it at t; draw / u perpendicular 
 
 to mterfed 7 4 at a ; and with the chord 7 a and the height ( u, deferibe the fegment H 
 In the fame manner may the next board / be found, and by this means you may brine 
 the fweep of your board into the fmallefl: compafs, without having any recourfe to the centre. 
 
 Suppof it were required to draw a tangent from 8 at No. 3, without having recourfe to the centre. 
 
 Bifea the arch 8 / 8 at / on 8, as a centre; with a radius 8 ly deferibe an arch e 1 make It 
 equal to / ^ ; draw the tangent t 8. ' 
 
 Given three points in the circumference of a circle, to find any number of equUdifant points be^ 
 yond thofe that will be in the Jame circumference, 
 
 . ^^’PPO^e.the three points b, c, to be given to one of the extreme points <7- 
 
 join the other two points b and c by the lines « b, and f? c ^ with, a radius a b, and the centre / 
 defenbe the arch of a circle i 2 3 ; then take b i and fet it from i to 2, and from 2 to z • 
 through the points 2 and 3, draw « d and f? then take b c, put the foot of your com- 
 pafs in r, and with the other foot crofs the ad at d-, with the fame extent put the 
 foot of your compafs in d, and with the other foot crofs the line ae ate-, in the fame 
 manner you may proceed for any number of points whatever. 
 
 7 ■ 
 
 PLATE 
 
4 ® 
 
 THE THEORY AND PRACTICE 
 
 PLATE XXXVII. 
 
 Fig. a is the plan of an elliptical dome; B is the longeft fedlion, Cthe ftiorteft fc6Iion ; 
 ztaa \n 5, and b b \n C, Ihews how to fquare the purlines, fo that one fide may be fair 
 with the furface of the dome ; the dotted*lines from a a m B, and b b \n C, fliews how 
 to get the length and width of the purline \nfig. but if the fides of the purline were made 
 to Hand perpendicular over the plan, the fweep of it would be found m the fame manner 
 as before; then it would require no more than half the fluff that the other would, and 
 take only half the time in doing> which is a confiderable advantage. 
 
 How to proportionate the infide curb for the JkyAight, fo as it Jhall anfwer to the fuface of 
 
 the dome. 
 
 Draw the diagonal i / and i m in fig. A, and let A r or jr/ibe the width, then h g 
 or ef will be the true length of the curb ; becaufe every fedion parallel to the bafe will 
 be proportional to the bafe of the dome. 
 
 T 1 find the ribs for this dome. 
 
 The ribs in this are got in the fame manner as the ribs for a niche, as direded in page 34, 
 and if the reader uuderftands that, he mull know this. 
 
 To find the form of a heard to ft and in an, place of this dome, to he bent up to the crown. 
 
 Suppofe you would find a board over u i c in the plan ; divide D into three parts round 
 a,h,c,d, and draw u c, i c, r r, and d r, to the centre ate; then take the triangle u Jc 
 in D, and lay it down .tabeXuG; then draw the line c . . ., &c. at rignt angles to u h 
 and deferibe a rib Gto the height of the dome, and the length to the perpendicular ol the trungle 
 u b c, and divide it into five equal parts i lay them along the line 1 i l, &c. ^ 
 
 prick the mould H from the triangle u « r, as the letters are marked. The board A will 
 
 be found in the fame manner. 
 
 Note. In the praftice, you are to divide one quarter of this dome into as many parts 
 as you think the breadth of a board will contain ; and the boards, when got out by this 
 method, will fit to a very great exadnefs ; this is only into three, that the parts may 
 
 be clearly feen to learners. , i .u!,. 
 
 If the boards are got out for one quarter of this dome to the lines here laid down, the 
 
 boards that are in the other three quarters will not require any other lines, for every boar 
 
 in the firft quarter will be a mould for three more boards. 
 
 PLATE 
 
Plate 
 
 14 
 
 iw 
 
 : O 
 
 ;'k 
 
 
 
 /• 
 
 c 1 
 

plate XXXVIII. 
 
 OF DOMES, WHEN PLACED OVER THE OPENINGS OF STAIR- CASES, 
 
 One of the ribs of a dome being given, and the plan of the opening of a fair-cafe which is fquare, 
 and an odfagon curb at the top for a fy -light ; to fnd the ribs and the curve on each ftde of the 
 opening of the fair -cafe, where the foot of the ribs comes, fo that part of the dome fall be an 
 
 Fig. a. Let B be the given rib ; take any number of perpendicular ordinates to its bafe 
 at pleafure; from the points a, c, e, g, i, I, where they interfedl its bafe, draw parallel lines to 
 the fides of the curb, returning round each diagonal, if there is more than one, till it cut the 
 bttfe of the angle rib D ; at the points a, c, e^ g, i, I, draw the ordinates of D, and prick it 
 from B, will be the angle rib ; and at the points g, i, I, at C, upon the fide of .the opening 
 of the flair-cafe, draw the perpendicular ordinates and prick C from B, agreeable to the 
 letters j then the curve C will be the true place for the foot of the ribs upon the fide of the 
 ftair-cafe, and the part that lays in the middle is a flraight line parallel to the horizon. 
 
 The vertical feaion of a femictrcular dome through its centre being given, the opening of the fair- 
 cafe being afquare as before, to find the curve D on the fide of the fair-cafe for the foot of the 
 ribs,fio that it fall finif to a circular curb at the top. 
 
 On the fide of the flair-cafe I m, as a diameter, deferibe a femicircle ; D will be the true 
 place for the foot of the ribs ; this is evident, for every fedion of a femi- globe, at right 
 angles to its bafe, is a femicircle, and this is the fame thing if truly confidered. 
 
 Note. All the ribs of this dome are cut by the rib at C, as explained by the perpendicular 
 lines ; draw round the centre from the points of each bracket, c d e f to the points k i 
 
 h g ; from thefe points draw perpendicular dotted lines, and thefe will fliew what length 
 each bracket muft have according to its place. 
 
 The vertical feaion ofafegmentdomepajfing through its centre being given, the plan of the opening 
 of the fair-cafe being fill a fquare, as before, to find the feaion upon each fide of the f air- cafe 
 for the foot of the ribs, to finif to a circular curb at the top. 
 
 Let the fedion D acrofs the angle be given, whofe centre is K and the diflance of fb^ 
 
 oaagon finif, agreeable to the curb. 
 
 angle rib D ; all this is evident from the fcdlions of 
 the Geometry. 
 
 Fig. Z) is of the fame nature as the others, havino- 
 from E. ^ 
 
 an ogee top ; the fedlion F is traced 
 

 THE THEORY AND PRACTICE, &c. 
 
 PLATE XXXIX. 
 
 Fig. a is the plan of an elliptical domical fky-light over a ftair-cafe; B and C are the 
 fedtioiis, vi'hich fhews how to place your ribs. 
 
 H:w to proportionate the length of the infide curb to any width given. 
 
 Proceed as dire£led in page 40 for an elliptical dome, that will determine the true length 
 to the width. 
 
 Hoiv to proportionate the circimifcribing ellipfis^ to pafs through the angles at a, b, c, and d, to 
 have the fame proportion as a b, and b c, the ftdes of the fair-cafe. 
 
 Proceed as dire£led in fig. 5, plate 6, in the Geometry. 
 
 To deferibe the ribs. 
 
 The rib over n to the centre of the trammel in fig. A., is a given quarter of a circle, as is 
 (hewn at A, and of courfe all the other ribs muft come to the fame height with it, Suppofe 
 it was required to find a rib over d p.^ you muft take the full extent from d to the centre, and 
 deferibe the quarter of an cllipfis D ; then the part over d p will be as much of it as is wanted : 
 in the fame manner E will be deferibed, and the part over i 0 is what is wanted of this rib ; 
 the fame letters are marked upon the bafes of D and as they are in the plan,/^. A. Every 
 other rib is deferibed in the fame manner. 
 
 To find the feSlions on each fide of the fiair-cafe for the foot of the ribs to fiand upon. 
 
 Deferibe the fcmicircle C, to c b^ the width of the opening of the ftair-cafe, which will 
 give the bottom of the ribs on that fide; and deferibe a quarter of the ellipfis for the 
 bottom of the ribs on the other fide, to the fame height as C. 
 
 This method depends on this principle, that all the parallel fedlions of a fpheroid are 
 fimilar figures ; therefore a vertical fe£tion ftanding upon a b^ will be fimilar to a vertical 
 fe6Hon pafiing through its centre ; both will be fimilar ellipfes ; but a b is an ordinate to 
 the conjugate axis, and ^ r is an ordinate to the tranfverfe of the circumferibing ellipfis; by 
 conftrufftion half the length of the parallelograijr is to half the length of the ellipfis, as 
 half the width of the parallelogram is to half the width of the ellipfis, and a fpheroid may be 
 fuppofed to be generated by the revolution of a femi-ellipfis about its axis; hence it follows, 
 that all fe£lions of a fpheroid parallel to the axis are fimilar figures, confequently the 
 fedlion B is fimilar to the circumferibing ellipfis of the ground plan. 
 
 / 
 
 I NTRO- 
 
INTRODUCTION 
 
 T O 
 
 PRACTICAL CARPENTRY. 
 
 ’ — — — i— ■ -i 
 
 OF THE COMPARATIVE STRENGTH OF TIMBER. 
 
 PROPOSITION I. 
 
 The ftrengths of the different pieces of timber, each of the fame length and thlcknefs, are 
 in proportion to the fquare of the depth ; but if the thicknefs is to be confidered along 
 with the depths, then the ftrengths will be in proportion to the fquare of the depth, mul- 
 tiplied into the thicknefs ; and if all the three are taken jointly, then the weights that will 
 break each will be in proportion to the fquare of the depth, multiplied into the thicknefs, and 
 divided by the length : this is proved by the do£lrine of mechanics. Hence a true rule 
 will appear, for proportioning the ftrength of timbers to one another. 
 
 RULE. 
 
 Multiply the fquare of the depth of each piece, into its thicknefs ; and each product being divided 
 by their refpeSiive lengths, will give the proportional Jirength of each, 
 
 EXAMPLE. 
 
 Suppofe three pieces of timber, of the following diraenfions : 
 
 The firft, 6 inches deep, 3 inches thick, and 12 feet long. 
 
 The fecond, 5 inches deep, 4 inches thick, and 8 feet long. 
 
 The third, 9 inches deep, 8 inches thick, and 15 feet long. The comparative weight 
 that will break each piece is required. 
 
 G 2 
 
 OPERA- 
 
44 
 
 INTRODUCTION TO 
 
 
 OPERATIONS. 
 
 
 Firft. 
 
 Second. 
 
 Third. ■ 
 
 6 deep 
 6 
 
 5 deep 
 5 
 
 9 deep 
 9 
 
 36 
 
 25 
 
 81 
 
 3 thick 
 
 4 thick 
 
 8 thick 
 
 15)648(43 and a fifth. 
 
 60 
 
 48 
 
 45 ^ 
 
 3 
 
 Therefore the weights that will break each are nearly in proportion to the numbers 9, 12 
 and 43, leaving out the fradlions, in which you will obferve, that the number 43 is almoft 
 5 times the number 9 ; therefore the third piece of timber will almoft bear 5 times as much 
 weight as the firft ; and the fecond piece nearly once and a third the weight jof the firft 
 
 piece ; becaufe the number 1 2 is once and a third greater than the number 9. 
 
 The timber is fuppofed to be every where of the fame texture, otherwife thefe calcula- 
 tions cannot hold true. 
 
 PROPOSITION II. 
 
 Given the length, breadth, and depth of a piece of timber ; to find the depth of another 
 piece whofe length and breadth are given, fo that it fhall bear the fame weight as the firft 
 piece, or any number of times more. 
 
 RULE. 
 
 Multiply the fquare of the depth of the firft piece into its breadth, and divide that pro- 
 duct by its length : multiply the quotient by the number of times as you would have the 
 
 other piece to carry more weight than the firft, and multiply that by the length of the laft 
 
 piece, and divide it by its width ; out of this laft quotient extradl the fquare root, which is 
 the depth required. 
 
 EXAMPLE I. 
 
 Suppofe a piece of timber 12 feet long, 6 inches deep, 4 inches thick j another piece 20 feet 
 long, 5 inches thick j requireth its depth fo that it fhall bear twice the weight of the firft piece. 
 
 Length 12)108 
 
 Length 8)100 
 
 Length 
 
 1 2 and a half 
 
 6 deep 
 
PRACTICAL CARPENTRY. 
 
 45 
 
 6 deep 
 
 6 
 
 36 
 
 4 
 
 12)144 
 
 12 
 
 2 times 
 24 
 
 20 length 
 
 Proof, 
 
 9-7 
 
 9.7 
 
 67 9 
 
 873 
 
 94.09 
 
 1. 9 1 remainder added 
 
 96.00 
 
 5 
 
 5)480 
 
 96 (9 . 7> or 9 . 8, nearly for the depth 
 I 81 
 
 187)1500 
 
 1309 
 
 20)480 
 
 24 
 
 191 
 
 EXAMPLE II. 
 
 SUppofe a piece of timber 14 feet long, 8 inches deep, 3 inches thick; requireth the depth 
 of another piece 18 feet long, 4 inches thick, fo that the laft piece fhall bear five times as 
 much weight as the firft. 
 
 8 
 
 8 
 
 64 
 
 3 
 
 As the length of both pieces of timber is 
 divifible by the number 2, therefore half 
 the length of each is ufed inftead of the 
 whole; the anfwer will be the fame. 
 
 half 7)192 
 
 27.4, &c. 
 
 5 times 
 
 J37 
 
 9 half the length 
 
 4)1233 
 
 308.25(17.5 the depth nearly 
 
 27)2 o 8( 
 
 189 
 
 345) . 19251 
 
 PROPO^ 
 
46 
 
 INTRODUCTION TO 
 
 PROPOSITION III. 
 
 Given the length, breadth, and depth of a piece of timber; to find the breadth of another 
 piece whofe length and depth is given, fo that the Lil piece liiall bear tr.e fame weight 
 as the frA: piece, or any nuftiber of times more. 
 
 RULE. 
 
 Multiply the fquare of the depth of the firft piece into its thicknefs ; that divided bv its 
 length, multiply the quotient by the number of times as you would h.ive the lart piece 
 bear more than the firft; that being multiplied by the length of the laft piece, and divided 
 
 by the fquare of its depth, this laft quotient will be the breadth required. 
 
 EXAMPLE I. 
 
 Given a piece of timber 12 feet long, 6 inches deep, 4 inches thick; and another piece 16 
 feet long, 8 inches deep ; requireth the thicknefs, fo that it ihall bear twice as much weight 
 as the firft piece. 
 
 Or this, at full length, 
 
 6 6 depth of the firft piece 
 
 6 6 
 
 4 thicknefs of the firft piece 
 3)144 Length 12)144 
 
 12 
 
 2 by the number of times ftronger 
 24 
 
 1 6 length of the laft piece 
 144 
 
 24 
 
 8 ) 3«4 
 
 8) 48 
 
 6 thicknefs. 
 
 8)384 
 
 8) 48 
 
 6 thicknefs. 
 
 48 
 
 96 
 
 4 
 
 36 
 
 4 
 
 EXAMPLE 
 
PRACTICAL CARPENTRY, 
 
 47 
 
 EXAMPLE II. 
 
 Given a piece of timber 12 feet long, 5 inches deep, 3 inches thick ; and another piece 
 14 feet long, 6 inches deep j requireth the thicknefs, fo that the laft piece may bear four times 
 as much weight as the firft piece, 
 
 5 
 
 5 
 
 25 
 
 3 
 
 J2)75 
 
 6.25 
 
 4 
 
 25.00 
 
 14 
 
 100 
 
 25 
 
 6)350 
 6) 58.266 
 9.7II8 
 
 PROPOSITION IV. 
 
 If the ftrefs does not lay in the middle of the timber, but nearer to one end than the other, 
 the ftrength in the middle will be to the ftrength in any other part of the timber, as i di- 
 vided by the fquare of half the length is to i divided by the redangle of the two fegments, 
 which are parted by the weight. 
 
 EXAMPLE I. 
 
 Suppofe a piece of timber 20 feet long, the depth and width is immaterial ; fuppofe the 
 ftrefs or weight to lay five feet diftant from one of its ends, confequently from the other 
 
 end 15 feet, then the above proportion will be — - — ~ -L. : — I — — as the ftrength 
 
 10 X lo 100 5X15 75 
 
 at 5 feet from the end is to the ftrength at the middle, or 10 feet, or as — ~ i I — zz i 
 
 100 75. 3 
 
 Hence it appears, that a piece of timber 20 feet long is one third ftronger at 5 feet dif- 
 tance from the bearing, than it is in the middle, which is 10 feet, when cut in the above 
 proportion. 
 
 EXAMPLE 
 
48 
 
 INTRODUCTION TO 
 
 EXAMPLE II. 
 
 Suppofe a piece of timber 30 feet long ; let the weight be applied 4 feet diflant from one 
 end, or more properly from the place where it takes its bearing, then from the other end 
 
 it will be 26 feet, and the middle is 15 feet; then — - — “ — or as 
 
 I5XJ5 2,^5 4X26 104 225 
 
 .221; 17 1 I 
 
 ~ I ; — zz « — or nearly 2 
 
 104 104 b 
 
 Hence it appears, that a piece of timber 30 feet long will bear double the weight, and one 
 fixth more, at 4 feet diftant from one end, than it will do in the middle, which is 15 feet 
 diftant. 
 
 EXAMPLE III. 
 
 Allowing that 266 pounds will break a beam 26 inches long, requireth the weight that will 
 break the fame beam when it lays at 5 inches from either end j then the diftance to the other 
 end is 21 inches; 21 X 5 = 105, the half of 26 inches is 13 . 13 X 1 3 “ 169; therefore the 
 ftrength at the middle of the piece is to the ftrength at 5 inches from the end, as 
 
 ~ ; : — or as i I — the proportion is ftated thus : 
 
 169 105 105 
 
 ft. 
 
 I : : 266 : to the weight required, 
 
 169 
 
 2394 
 
 1596 
 
 266 
 
 105)44954(428 
 
 420 
 
 29s 
 
 270 
 
 854 
 
 840 
 
 .14 
 
 From this calculation it appears, that rather more than 428 pounds will break the beam 
 at 5 inches diftant from one of its ends, if 256 pounds will break the fame beam in the 
 middle. 
 
 By fimilar propofitions the fcantlings of any timber may be computed, fo that they 
 {ball fuftain any given weight ; for if the weight one piece will fuftain be known, with 
 
PRACTICAL CARPENTRY. 4^ 
 
 its dimenfions, the weight that another piece will fuftain, of any given dimenfions, may alfo 
 be iromputed. The reader muft obferve, that although the foregoing rules are mathemati- 
 cally true, yet it is impoffible to account for knots, crofs-grained wood, &c. fuch pieces 
 being not fo ftrong as thofe which are ftraight in the grain ; and if care is not taken in 
 choofing the timber for a building, fo that the grain of the timbers runs nearly equal to 
 one another, all rules which can be laid down will be baffled, and confequently all rules 
 for juft proportion will be ufelefs in refpe^ to its ftrength. It will be impoffible, however, to 
 eftimate the ftrength of timber fit for any building, or to have any true knowledge of its 
 proportions, without fame rule; as without a rule every thing muft be done by mere conjedurc. 
 
 Timber is much weakened by its own weight, except it ftand perpendicular, which will 
 be fhewn in the following problems ; if a mortice is to be cut in the fide of a piece of 
 timber, it will be muich lefs weakened when cut near the top, than it will be if cut at 
 the bottom, provided >the tenent is drove hard in to fill up the mortice. 
 
 The bending of tiimber will be nearly in proportion to the weight that is laid on it; 
 no 1 earn ought to be trufted for any long time with above one third or one fourth part 
 -of the we ght it will abfolutely carry ; for experiment proves, that a far lefs weight will 
 break a piece of timber when hung to it for any confiderable time, than what is fufficient 
 to break it when firft applied. 
 
 PROBLEM I. 
 
 Having the length and weight of a beam that can juft fupport a given weight, to find 
 tlie length of another beam of the fame fcantling, that ftiall juft break with its own weight; 
 Let / zz the length of the firft beam, 
 
 L zz the length of the fecond ; 
 a zz the weight of the firft beam, 
 w = the additional weight that will break it. 
 
 And becaufe the weights that will break beams of the fame fcantling are reciprocally as their 
 lengths. 
 
 It , * 
 
 therefore J t ^ ^ 4* - I ^ = the weight that will break the greater beam ; 
 
 becaufe w + j is the whole weight that will break the lefier beam. 
 
 But the weights of beams of the fame fcantling are to one another as their lengths : 
 
 Whence, / • A ; : - j the weight of the greater beam. 
 
 Now the beam cannot break by its own weight, unlefs the weight of the beam be equal 
 -to the weight that will break it : 
 
 La w + f X / 
 
 Wherefore, — = : 
 
 2 / Li 
 
 iiu a X / 
 
 TL 
 
 a \ iw-\- a :: /* : U, confequently y/'Z* = Z = the length of the beam that 
 can juft fuftain its own weight. 
 
 H 
 
 PROBLEM 
 
5 ® 
 
 INTRODUCTION TO 
 
 PROBLEM II. 
 
 Having the weight of a beam that can juft fupport a given weight In the middle j to 
 find the depth of another beam fimilar to the former, fo that it (hall juft fupport its own weight. 
 
 Let d = the depth of the firft beam j 
 X = the depth of the fecond beam ; 
 a — the weight of the firft beam ; 
 
 w = the additional weight that will break the firft beam : 
 then will w -f - or - = the whole weight that will break the lefler bea m; 
 
 2 i 
 
 And becaufe the weights that will break fimilar beams are as the fquares of their lengths. 
 
 ^ 
 
 weights of fimilar beams are as the cubes of their correfpoading fides ; 
 
 . Ct Cl JC ^ 
 
 Hence • • 7 • 77 ~s 
 
 ax^ _ ix'^w-]rx'^a 
 
 •’TTs ~ 
 
 a \ a + OiVU : d X =. the. depth required. 
 
 As the weight of the lefler beam is to the weight of the lelTer beam, together with the 
 additional weight ; fo is the depth of the lefler beam, to the depth of the greater beam. 
 
 Note. Any other correfponding fides will anfwer the fame purpofe, for they are all pra- 
 portional to one another. 
 
 example. 
 
 Suppofe a beam whofe weight is one pound, and its length lO feet, to carry a weight of 
 399.5 pounds, requireth the length of a beam fimilar to the former, of the fame matter, 
 fo that it fhall break with its own weight : 
 
 then a = I 
 and a + 21V =. 800 
 ' </ = 10 
 
 Then by the laft problem it will be 
 I : 800 10 
 
 10 
 
 8000 ~ 9 ( for the length of a beam that will break by its own weight. 
 
 PROBLEM 
 
PRACTICAL carpentry. 
 
 51 
 
 PROBLEM HI. 
 
 The weight and length of a piece of timber being given, and the additional weight that 
 will break it, to find the length of a piece of timber fimilar to the former, fo that this lalb 
 piece of timber ftiall be the ftrongeft pofiible : 
 
 Put / = the length of the piece given, 
 w = half its weight, 
 
 the vveight that will break it ; 
 
 A- = the length required. 
 
 then, becaufe the weights that will break fimilar pieces of timber are in proportion to the 
 fquares of their lengths, 
 
 : A * : ; W + w : — = the whole weight that breaks the beam ; 
 
 and becaufe that the weights ot fimilar beams are as the cubes of their lengths, or any other 
 correfponding fides, 
 
 then :: zu : the weight of the beam ; 
 
 JV X ^ “1“ *VJ Y ^ .it* ^ • 
 
 confequently — — i is the w’eight that breaks the beam = a 
 
 loaximum j therefore its fluxion is nothing, 
 
 that is, 2 W X X + 2ZV X X — nothing. 
 
 rrr % 1U X 
 
 2 rr + 2 TO = — 
 
 , , Z IF 2 VJ 
 
 hence, x — i X 
 
 3 w 
 
 Hence It appears from the foregoing problems, that large timber is weakened in a much 
 greater proportion than fmall timber, even in fimilar pieces, therefore a proper allowance 
 muft be made for the weight of the pieces, as I fhall here fltew by an example. 
 
 Suppofe a beam 12 feet long, and a foot fquare, whofe weight is 3 hundred weight, to 
 be capable of fupporting 20 hundred weight, what weight will a beam 20 feet long, 15 
 inches deep, and 12 thick, be able to fupport? 
 
 12 inches fquare 
 
 
 12 
 
 15 
 
 144 
 
 75 
 
 12 
 
 15 
 
 12)1728 
 
 225 
 
 ■ — 
 
 12 
 
 144 
 
 2.0)270.0 
 
 H2 
 
 135 
 
 But 
 
52 
 
 I N T R O D U C T I O 
 
 But the weight of both beams are as their folid 
 
 therefore, 1 2 inches fquare 
 12 
 
 *■44. 
 
 144 inches = 12 feet long 
 
 contents ; , 
 
 15 deep-' 
 
 12 wide - 
 
 180 
 
 240 length in inches'*. 
 
 576 7200 
 
 576 360 
 
 144 •- 
 
 . 432CO folid contents of the 2d beam ? 
 
 20736' folid contents of the- ift beam. 
 
 20736:43200 : : 3 144 - ^3S •• 21.S by prop, u 
 
 3 21. 5 
 
 » ■ — - cwt. lb. ■ ■ — ~ 
 
 207'36) 129600(6... 28'= the weight of 67 5. 
 
 124416 the 2d beam 135 
 
 270* 
 
 ..5184 
 
 1 12 12)2902.5' 
 
 10368 
 
 5184 
 
 5i«4 
 
 20736)580608(28 
 
 41472 
 
 165888 
 
 165888 
 
 12) 241.875 
 
 20.15625 
 
 112 
 
 3I25P 
 
 15625 
 
 15625 
 
 17,50000' 
 
 16 
 
 30 
 
 5 
 
 8,0 
 
 21 cwt. 561b. is the weight that will break the firft Beam, and 20 cwt. 17 lb. 8 oz. 
 the weight that will break the fecond beam j dedud out of thefc half their own weight, 
 
 20. . 17. .8 
 
 3 . . 1 4 . . o half 
 
 17. . .3. .8 
 
 Now 20 cwt. is the additional weight that will break the firft beam ; and 17 cwt. 3 lb. 
 8 oz. the weight that will break the fecond: In which the Reader will obferve, that 17. .3. .8 
 has a much lefs proportion to 20, than 20 cwt. 17 lb. 8 oz. has to 2i . .56. From thefe ex- 
 amples the Reader may fee that a proper allowance ought to be made for all horizontal beams ; 
 that is, half the weights of beams ought to be deduced out of the whole weight that they 
 will carry, and that will give the weight that each piece will bear. 
 
 4 
 
 If 
 
PRACTICAL CARPENTRY. 
 
 53 
 
 IF feveral pieces of timber of the fame fcantling and length are applied one above another, 
 and fupported by props at. each end, they w\\\ be no ftronger than if they were laid fide by fidej 
 or this, which is the fame thing, the pieces that are applied one above another are no ftronger 
 than one Tingle piece whofe width is the width of the feveral pieces colleded into one,- 
 and its depth the depth of one of the pieces j it is therefore ufelefs to cut a piece of timber 
 lengthways, and apply the pieces fo cut-one above another, for- thefe pieces are not fo ftrong 
 as before, even if bolted^. 
 
 EXAMPLE. 
 
 Suppofe a girder i6 inches deep, 12 inches thick, the length is immaterial, and let the 
 depth be cut lengthways in two equal pieces; then will each piece be 8 inches deep, and 
 12 inches thick. Now, according to the rule of proportioning' timber, the fquare of i6' 
 inches, that is, the depth before it was cut, is 256, and the fquare of 8 inches is 64 ; but^ 
 twice 64 is only. 128, therefore it appears that the two pieces applied one above another 
 is but half the ftrength of the folid piece, becaufe 256 ^ double 128. • 
 
 If a gifder be cut lengthways in a perpendicular direction, the'ends turned contrary, and 
 then bolted together, it will be but very little ftronger than before it was cut j for although 
 the ends being turned give to the girder an equal 'ftrength throughout, yet wherever a 
 bolt is, there the girder will- be weaker, and I am very doubtful whether it will be any 
 ftronger for this procefs of fawing and bolting ; and I fay this from experience. 
 
 If there are two pieces of timber of an equal fcantling (Ph 52,/^-. B\ the one lying hori- 
 zontal and the other inclined, the horizontal piece being fupjjorted at the points e andy^ and 
 the inclined piece at r and y, perpendicularly over ^ and/, according to the principles of 
 mechanics, thefe pieces will be equally ftrong. But, to reafon a little on this matter, let 
 it be confidered that although the inclined piece D is longer, yet the weight has lefs effeef 
 upon it when placed- in the middle, than the weight' at has upon the horizontal piece C, 
 the weights being the fame; it is therefore reafonable to conclude, that in thefe pofitions- 
 the one will bear equal to the other. 
 
 The foregoing rules will be found of excellent ule when timber is wanted to fupport a 
 great weight; for, by knowing the fuperincumbent weight, the ftrength may be computed to 
 a great degree of exaeftnefs, fo that it ftiall be able to fupport the weight required. The 
 confequence is as bad when there is too much timber, as when there is too little, for nothing' 
 is more requiftte than a juft proportion throughout the whole building, fo that the ftrength 
 of every part ftiall always be in proportion to the ftrefs ; for when there is more ftrength 
 given to fome pieces than others, it encumbers the building, and conlequcntly the foundations' 
 are lefs capable of fupporting the fuperftrudiure. 
 
 No judicious perfon, who has the care of conduaing buildings, ftiould rely on tables of 
 fcantlings, fuch as are commonly in books; for example, in ftory pofts the fcantlings, ac-' 
 cording to feveral authors, are as follows ; 
 
 For 9 feet high 6 inches fquare 
 
 12 8 
 
 15 10 
 
 i 8 12 
 
 Now, 
 
5 ^ 
 
 INTRODUCTION, &c. 
 
 Now, according to this table, the fcantlings are increafed in proportion to the height; but 
 there is no propriety in this, for each of thefe will bear weight in proportion to the numbers 
 9, 16, 25, and 36, that is, in proportion to the fquare of their heights, 36 being 4 times 9 ; 
 therefore the piece that is 18 feet long, will bear four times as much weight as that piece 
 which is 9 feet long ; but the 9 feet piece may have a much greater weight to carry than 
 an eighteen feet piece, fuppofe double, in this cafe it muft be near 12 inches fquare inftead 
 of 6. The fame is alfo to be obferved in breaft*fummers, and in floors where they are 
 wanted to fupport a great weight ; but in common buildings, where there is only cuflomary 
 weights to fupport, the common tables for floors will be near enough for pradlice. 
 
 To conclude the fubjedf, it maybe proper to notice the following obfervations which 
 feveral authors have judicioufly made, viz. that in all timber there is moiflure, wherefore 
 all bearing timber ought to have a moderate camber or roundnefs on the upper fide, for till 
 that moiflure is dried out the timber will fwag with its own weight. 
 
 But then obferve, that it is beft to trufs girders when they are frefh fawn out, for by their 
 drying and ftirinking, the trufles become more and more tight. 
 
 That all beams or ties be cut, or in framing forced to a roundnefs fuch as an inch in 
 twenty feet length, and that principal rafters alfo be cut or forced in framing, as before ; 
 becaufe all joifls, though ever fo well framed, by the flirinking of the timber and weight of 
 the covering will fwag, fometimes fo much as not only to be vifible, but to offend the eye ; 
 by this precaution the trufs will always appear well. 
 
 Likewife obferve, that all cafe bays either in floors or roofs do not exceed twelve feet if 
 poflible, that is, do not let your joifls in floors exceed twelve feet, nor your purlines in roofs, 
 &c. but rather let their bearing be eight, nine, or ten feet j this fliould be regarded in form- 
 ing the plan. 
 
 Alfo, in bridging floors, do not place your binding or flrong joifl above three, four, or five 
 feet apart, and that your bridging or common joifls are not above ten or twelve inches 
 apart, that is, between one joifl and another. 
 
 Alfo, in fitting down tie beams upon the wall plates, never to make your cocking too 
 large, nor yet too near the outfide of the wall plate, for the grain of the wood being cut 
 acrofs in the tie beam, the piece that remains upon its end will be apt to fplit off^ but 
 keeping it near the infide will tend to fecure it. See plate 32, at the bottom, where the 
 dimenfions are figured. 
 
 Likewife obferve, never to make double tenents for bearing ufes, fuch as binding joifls, 
 common joifls, or purlines; for, in the firfl place, it very much weakens whatever you frame 
 it into, and in the fecond place it is a rarity to have a draught in both tenents, that is, to 
 draw both joints clofe, for the pin in palling through both tenents, if there is a draught in 
 each, will bend fo much that, without it is as tough as wire, it muft needs break in driving, 
 and confequently do more hurt than good. 
 
 Roofs will be much flronger if the purlines are notched above the principal rafters, than 
 if they are framed into the fide of the principals ; for by this means, when any weight is 
 applied in the middle of the purline it cannot bend, being confined by the other rafter;, 
 and if it do, the Tides of the other rafters muft needs bend along with it, confequently it 
 has the flrength of all the other rafters fide-ways added to it. 
 
 7 
 
 PRACTICAL 
 
Plate 4 - 0 . 
 
 
PRACTICAL CARPENTRY 
 
 PLATE XL. 
 
 A and B fhews the method of trailing girders, as is ufed by the greateft mailers at this time. 
 
 C is a horizontal fedtion of B. 
 
 D is a fedlion of the butment, by cutting acrofs a h'm A, 
 
 E and F Ihews the two fides of the king-bolt, at c in A^ which is made with a wedge-way 
 upon the top, fo that it may force out the trulTes upon the butments. 
 
 To tighten the girders. 
 
 Your trulTes ought to be let clofe in to the fides of the girder, about an inch and a half on 
 each fide, and the- head of your king-bolt ought to be greafed, fo that it may Hide freely 
 paft the ends of the trulTes ; firft fcrew your girder clofe fideways, then proceed to turn 
 the nut of the king-bolt, and another perfon to hit the head at c of the king, with a mallet, 
 which will make it ftart every time it is hit, and give frelh eafe at every turning of the nut, 
 fo that you may camber the girder to any degree that you lhall have occafion for, but gene- 
 rally not above an inch in twenty feet. 
 
 Note. The fedions X), X, and T, are to one eighth part of the real fize. 
 
 PLATE XLI. 
 
 This contains the moll hmple conllru£lion of roofs. 
 
 Fig. a is calculated for a fmall building; at one end of the colar-beam is the Car- 
 penters’ boall, what they term a dove-tail tenent ; but I think rather a rule joint, as it is 
 worked out to a centre. This roof will do for an extent of 20 or 25 feet. 
 
 Fig B is a trufs for a roof, the purlines to be notched upon the principal rafters, as 
 will be defcribed in the following plates of ledgment roofs : this roof may be well calculated 
 for an extent of 30 or 35 feet, the height one fourth of the fpaii for Hate coverings. 
 
 Fig. Cis a fimple conftruflion of a roof, for the fegment finilh of a dome, which will 
 anfwer to any of the above extents. 
 
 PLATE 
 
56 
 
 PRACTICAL CARPENTRY. 
 
 T L A T E XLII. 
 
 Fig. is a roof for the’ purlines to be framed in, and the common rafters to come fair 
 ■•with the principals. 
 
 Fig. B is a roof calculated for a greater extent than any of the foregoing roofs, and may 
 •-.well extend 50 or 60 feet. Here likewife is fhewn the connection of the roof in the walls. 
 
 Fig. is a roof fupported by two queens, inftead of a king, to give room for a paflagc 
 or, any other conveniency in the roof. 
 
 PLATE XLIII. 
 
 This roof is calculated for a fpan of feventy or eighty feet. 
 
 'You will obferve in this, and the foregoing roofs, that the trufles are the fame in number 
 as the purlines which they have to fupport ; for how abfurd it is to give a roof more ftrength 
 than neceffary ; but, on the other hand, the confequence will be dangerous if too w’eak. 
 
 Fig. yf is a defign of a roof for a theatre, which may extend from 80 to 90 feet. 
 
 As it happens frequently in buildings that walls run acrofs the roof, in fuch cafes there 
 will be little occafion for truffing the roof j then the purlines may be trufled, which will (avc 
 one or two pair of principals, which is a confiderable advantage. 
 
 Fig. B is a roof of this kind, which fhews-the end3 of the purlines, and C fhews howt* 
 .trufs the purline. 
 
 S>y Ey and F, are the methods of fcarfing timber. 
 
 ,P L A T E X'LIV. is explained on the Plate. 
 
 PLATE XLY. 
 
 Fig ^is a curb roof, with a door in middle of Che .partition.; the beam a b to run quite 
 .acrofs the pole plate, to be tenented into the beam 2X a and the ftory poll: a rand h ^like- 
 wife to be let in with a fmall tenent to the beam, as it ftiould proje£l about an inch on each 
 fide of the beam, to take the flioulder of the pole plate, which will difcharge the weight 
 from the tenent. 
 
 >)F IG. B is a roof calculated for two rooms. 
 
 Fig. Cfhe.ws the method of framing a bridge floor.. 
 
 ' plate 
 
 / 
 

folate z^,3. 
 
 
IVa/^:44^. 
 
 A.(ieoi(^n of a ?^ccf to yLuis/i niNi f/)<' fiarafu’t ?r/u'7t r<n'rrrft m j/to rr/r//t fi'iui/ 
 
 30 O) '1-0 f/’tl ^ 
 
 l^//3 ,x^ t/ic Act hrcit^t Oct^'J ?.,7~^y2.ijyJ^.A\t/n:l-on 
 
JPu^. as/Ac.A^t' . 
 
-P/atf 46 
 

PRACTICAL CARPENTRY, 
 
 57 
 
 PLATE XLVI. 
 
 Fig. y/ is a defign of an M roof> which is ufeful in fome cafes where the fpan is great, 
 and no wall between, and the roof is required not to appear of a great height ; but this 
 feldom happens in pra<Sl;ice, for if there is any wall between the external walls, the roofs are 
 in general made double, as are fhewn zX figures Cy 5, and C. 
 
 PLATE XLVII. 
 
 Fio. //is a defign for a church roof, the extent marked on the plate. 
 
 F iG. 2? is a defign of the fame kind, but may be applied to an extent much greater. Thefe 
 two roofs, when finiftied, wHl be the fame in every refpe£l as the Welch groin deferibed in 
 plate 23 ; as the manner there ihewn of fixing the ribs will not be different in this, I refer 
 the reader to the defeription of that plate. 
 
 Fig. Cis another defign for a church roof, where the ceiling over the galleries is to finilU 
 level. 
 
 PLATE XLVIir. 
 
 Fig. >/Is a defign for a domical roof j B fhews the manner of framing the curb for it to 
 ftand upon, the fefUon of the curb being alfo fhewn upon the bottom oifig, 
 
 PLATE XLIX. 
 
 Fig. a is another defign for a domical roof j the bottom of it is made into a very narro’WT 
 compafs in order to gain room within the dome. 
 
 Figures B and C are defigns for circular and, elliptical trufles for bridges, &c. Thefe trulles 
 may alfo be applied to roofs where there is no cavity wanting within. 
 
 PLATE L. 
 
 Fig. y/is.a defign for a ftory poft and breft-fummers. 
 
 F iG. jB is a defign for a bridge. G is a fedlion acrofs. D Is part of the plan, which 
 alfo fhews the manner of fixing the piles, R fhews half the plan of the bridgings. 
 
 I PART 
 
PART II 
 
 Of the Theory and Pra£lice of Joinery, 
 
 PRINCIPLES of HAND-RAILS for STAIR-CASES. 
 
 I AM now going to enter upon a fubje£t, which wants more particularly to be laid dowa 
 by new methods, than any of the others which I have before touched upon ; as the methods 
 laid down by all authors on the fubje6l are grounded upon erroneous principles, without 
 any proper foundation, arwl not confidering it truly according to its nature. For it is evident^ 
 that if a cylinder is any how cut, but not parallel to- the bafe, that fedfion will be an ellipfts ; and 
 if the cylinder is perpendicular, the fedlion will alfo be perpendicular to the bafe, or plurrib 
 from every point in it to every correfponding point in the bafe or plan : and lilcewife, if you 
 fuppofe any other fedioato be cut under the former, and parallel to it, then the elllpfis under 
 will be the fame as that above ; and therefore I fay, if a mould is made to this ellipfis, let It 
 Jbe drawn upon the upper fide of a plain piece of wood, of a parallel thicknels j then if the 
 bevel where the ellipfis cuts the cylinder be applied to either of the extremes upon the edge, 
 and the fame mould being applied from the rake of that line to the under fide, then let 
 the plank be cut out between the rake of the upper and under lines; and if this is taken into 
 confideration, it will appear that hand-rails are the fedions of cylinders, and confequently 
 the rules for drawing them will be the fame as thofe for finding the fedion of a cylinder, 
 which has been explained in the Geometry, fee plate 9 and its explanation ; or if they are 
 ^^tpade in feveral different pieces, they will ftill be fome portion of a cylinder, which are all 
 explained in the plate before mentioned,. 
 
 PLATE LL 
 
 draw the form of a hand-rail. 
 
 A make an equilateral triangle v w t upon its width, and divide it into five equal 
 parts, and from one part on each fide draw z s and y w, then t g and m are the centres, 
 / m being made equal to / ^ ; the centres are found the fame for the upper fide. 
 
 Tl'he form of the rail being given.^ to draw the mitre cap. 
 
 Let the projedion of the cap be three inches and a half, and make the di fiance of the 
 infide circle from the outfide circle the projedion of the nofe on each fide of the rail, and 
 draw the mitre n 0 and p 0 ; then continue parallel lines down to the mitre p 0, put the foot 
 of your compafs in the centre of the cap, and circle the parallel lines round to a c e g and /, 
 and draw the ordinates a b, c d, e f See. and then prick the cap to the rail according to the 
 letters. 
 
 How to draw the form of the cap for the mitre to come to the centre. 
 
 It is only drawing the parallel lines from the rail to the mitre wherever it is, and circling 
 them round to the ordinates, and fo pricked from the rail, and the thing is done. 
 
 I PLATE 
 
I^/af€52. 
 
 Pu/z^oa ^/te Act dir^cAi Octr22 j.J^'2 ^PAtcAo/jofc 
 

Pla/c‘ o3. 
 
 A d(^/y 
 
 Pul/ , cut Hi^A.ct olc?^€d^PovT(^l^J-t^cc/ioL/to'>i' 
 
Plati34. 
 
 Outside falling iMbuld- 
 
 Pitc/v Board 
 
 
 
 
 1 1 
 1 1 
 
 
 
 
 
 
 
 
 2 
 
 
 
 1 
 
 
 
 — 
 
 - 
 
 E 
 
 c 
 
 
 
 
 ' Z 
 
 
 
 
 
 
 
 Scale 
 
 Inches. 
 
 O.S t/teA-cC di-rectJAu^*^. i^jn^PjS2e/i<riie'' 
 
 Z % Irv. 
 
‘THE THEORY AND PRACTICE OF JOINERY. 59 
 
 PLATE LIV. 
 
 To draw thefcroU of a hand-rail, 
 
 In fig. A make a circle three inches and a half diameter, divide the diameter in‘o three 
 «qual parts, and make a fquare in the centre of the eye to one of thofe parts, and divide each 
 fide of the fquare into fix equal parts ; this fquare is {hewn in at the bottom, in full fize 
 for pra£lice, and laid in the fame pofition as the little fquare above, fo that the centres 
 may be more readily found, which are all marked in a regular pofition ; the centre at i 
 draws from i round to X*, the centre at 2 draws from k to /, and the centre at 3 draws 
 from / to m, &c. which will complete the outfide revolution at <?, with the centre c ; then 
 fet the thicknefs of the rail from a to f and go the reverfe way to draw the infide ; then the 
 
 fcroil will be completed. ^ ' / 
 
 To draw the curtail fiep^ 
 
 Set the ballifters in their proper places on ea?ih quarter of the fcroil in fig. A ; the firfl 
 ballifter (hews the return of the nofing round the ftep, the fecond ballifter is placed at the 
 beginning of the twift, and the third ballifter a quarter diftant, and ftraight with the front of 
 the laft rifer : then fet the projedlion of your nofing wuthout, and draw it all round equal 
 diftant from the fcroil, which will give the form of the curtail. 
 
 To draw the face mould for fquarlng the twiji part of the fcroil. 
 
 You will obferve here, that the joint is made at 3, 6, juft to clear the fide of the fcroil ; 
 draw ordinates acrofs the fcroil at diferetion, to cut the line db^ a h c being the pitch-board ; 
 take notice that lines be drawn from 3 and 6 to meet d by fo that you may have the faid 
 points exa£l at 3 and 6 in your face mould, then take the line d by and mark the places of 
 the ordinates upon a rod, and transfer the divifions to d b m By then trace By from fig. Ay 
 according to the marks. 
 
 To find the falling mould C. 
 
 In C, ^ ^ c is the pitch-board; the height is divided into fix parts, to give the level of the 
 fcroil, the diftanceiT d, is from the face of the rifer to the beginning of the twift ; and the 
 diftance from d to k in C, is the ftretch out from ay the beginning of the twift round to h 
 in fig. A ‘y each beihg any point taken at diferetion, more than the firft quarter ; divide the 
 level of the fcroil, and the rake of the pitch-board, into a like number of parts, and complete 
 the top edge of the mould by interfedling lines, and the under edge parallel to it to the depth 
 of the rail. 
 
 How to find the parallel thicknefs of fluff for the twifi and fcroil. 
 
 Take the compafs round abcdcytobyin fig. Ay and ftretch it out upon the bafe of the 
 pitch-l'oard from dto g\ draw g h perpendicular to interfe<5l with the top of the mould; 
 then draw the dotted line h f parallel to the level of the fcroil both ways ; then take the 
 diftance 6 i, in fig Ay that is, the length of the plan, for the twift part, and fet it from d to ^ 
 in Cy and draw e f perpendicular, to cut the parallel f h ; then draw a dotted line through 7 ^ 
 parallel to c the longeft fide of the pitch-board, which gives the thicknefs of fluff for the 
 twift, about three inches and a half ; and the parallel line from / to the bale, Ihews the 
 thick, efs of the fcroil. 
 
 Note. The falling mould X>, for the outfide, is found in the fame manner as the other 
 falling mould C, 
 
 I2 PLATE 
 
6o THE THEORY AND PRACTICE OF JOINERY, 
 
 PLATE LV, 
 
 As the method of getting a fcroll out of a folid piece of wood, having the grain of the 
 wood to run in the fame dire£lion with the rail, is far preferable to any of the other methods, 
 with joints in them, being much ftronger than any other fcroll with one or two joints, 
 and much more beautiful when executed, as no joint can be feen, and confequently no dif- 
 ference in the grain of the wood at the fame place. I fhall here give you a fpecimen, the 
 method for defcribing a fcroll being already given in the laft plate } and likewife the falling 
 mould. 
 
 How to find the raking or fiace mould. 
 
 Place your pitch-board, a h c^\n fig. Z), as in the laft plate ; then draw ordinates acrofs 
 the fcroll at dlfcretion, and take the length of the line d with its divifions on the longeft 
 fide of the pitch-board, and lay it on d b \n E \ then the ordinates being drawn in it 
 will be traced from fig. D, as the letters diredL 
 
 How to find the parallel thicknefs of fluff'. 
 
 Let ^ £• be the pitch-board, in F, and let the level of the fcroll rife one fixth, as in the 
 laft plate j and from the end of the pitch-board at 6, fet from b to d half the thicknefs of the 
 ballifter, to the infide ; then fet from dtoe half the width of the rail, and draw the form 
 of the rail on the end at e^ the point b being where the front of the rifer comes, then 
 the point e will be the projedlion of the rail before it ; then draw a dotted line to touch 
 the nofe of the fcroll, parallel with c i, the longeft fide of the pitch -board, then will the 
 diftance between this dotted line and the under tip of the fcroll fliew the true thicknefs 
 of ftufF, which is nearly five inches and a half : but there is no occafion for the thicknefs to 
 come quite to the under fide ; if it comes to the under fide of the hollow, it will be quite 
 fufficient, as a little bit glued under the hollow could not be difcernible, and can be no 
 hurt to the fcroll, therefore a piece about four inches and a half will do. 
 
 Fig. ^is a fcroll of a fmaller fize, drawn in the fame manner and with the fame centres 
 as the others are, but with a centre lefs. The method of finding the raking mould and 
 thicknefs of fluff are the fame as fig, D. 
 
 PLATE 
 
2^bi-S Shows' /jo'w a, Sc^'o// ts to l>o ^ot oaf oy- ' t/^o So/^( 7 ^ 
 
 l^hickyi^s ^ Stu^^or Fu^ T> 
 
 '■cyT^.j^y2 ^FJVti^v/jon-. 
 

 
THE THEORY AND PRAQJICE OF JOINERY. 6i 
 
 PLATE LVII. 
 
 To find the face mould of a fiair-cafe^ fo that when fet to its proper rake it tvlll be perpendicular 
 to the plan tvhereon it Jlandsfor a level landings as is Jhewn in the lajl plate. 
 
 In fig. A draw the central line b parallel to the fides of the rail ; on the right line b 
 apply the pitch-board of a common ftep ; from ^ to a draw ordinates n d, m If k g, and 
 i k, at difcretion, taking care that one of the lines, as k g, touch the infide of the rail at 
 the point gy fo that you may obtain the fame point exadly in the face mould j then take 
 the divifions q h g f e d, from y, and apply them at B from qh gf e d\ from thefe points 
 draw the ordinates of and prick them from the plan, as the letters explain j then B will 
 be the mould required. 
 
 To find the falling mould. 
 
 Divide the radius of the circle into four equal parts, and fet three of thefe parts from ^ 
 to b \ through n and v^ the extremities of the diameter of the rail, draw b n and b to cut 
 the tangent line at the points c and g ; then will c d\s& the circumference of the rail, which 
 is applied from r to d, at C, as a bafe line j make c ^ the height of a ftep j draw the hypo- 
 tenufe e ri, at the point e and d\ apply the pitch-board of a common ftep at each end of 
 their bafes, parallel to c d^ make equal to d e^ if it will admit of it, and by thefe lengths 
 cafe off the corners by the common method of interfe£ting lines j then draw a line parallel 
 to it, for the upper edge of the mould. 
 
 To find the parallel thicknefs of ftufi\ 
 
 The fame falling mould is again ftiewn diftinaiy at D ; bife£l the line r, at ; divide 
 c c into any number of parts, as 6 ; on r, as a centre, deferibe a quarter of a circle to the 
 radius of the rail ; divide the arch alfo into fix equal parts; from the points c g i I n, draw 
 the parallel lines e f g h^ i k^ I ?n, n o from the equal divifions in the arch, draw the per- 
 pendiculars of 1 hy 2 ky 3 w, 4 <j, and 5 y, to interfe<a the parallel lines at f h km 0 q\ 
 through the points draw a curve line j draw a right line r my parallel to //, to touch the 
 curve ; then is the diftance from r to r the parallel thicknefs of fluff. 
 
 PLATE 
 
U THE THEORY AND-PRACTICE OF JOINERY. 
 
 PLATE LVHI. 
 
 To a face mould of a rad for a large opening on a level landing, 
 
 'Let fg. J be the plan of the rail ; through the centre C draw the diameter s z, and produce 
 k to A ; alfo produce the fide of the rail out to 2 i then take the diameter z r, put the foot 
 of your compafs in y, and crofs the line a z at a ; through a andy draw the line J 2, 
 cutting the fide of the rail produced at 2 ; then the diftance from z to 2 is half the arch 
 line of the r^il; take the diliance z 2, and place it on the right line v v 2 lX. G, on each 
 fide of to V and v ; draw v B and v c, each perpendicular to the right line v Vj and 
 equal to the height of a flep; make c e and b d parallel to v z', each equal to the tread of a 
 flep ; draw the hypotenufe v r, and the common pitch-boards v '& D and c E F, at each end; 
 make v h equal to v d, and c g equal to c f ; and eafe off the angles g c F and d H by 
 the common method of interfedfing lines, which will give the curve of the under edge of 
 the falling mould; draw a line parallel to it , equal to the thicknefs of the rail, will give 
 the upper edge ; produce* the line v v out toy, from the middle w of the line v v, at G \ 
 make w y equal to wy at the plan fig. D ; y being the place of the joint upon the plan, 
 draw the liney 2 i perpendicular to v cutting the upper fide of the falling mould at 2, 
 and the under fide at i ; from i draw the line i 6, parallel to v v, cutting the line 8 zu, 
 produced to 6; draw a tangent line m l ; parallel to the chord a at the plan S, to the 
 chord a b draw any number of indefinite perpendicular*;, obferving to draw a perpendicular 
 through every joint, as from the joint ^ ^ and ^y; then take the diftance i 2 from your 
 falling mould at G, and fet it from m to o of the plan at 5 ; alfo from L make l n o 
 equal to 6 7 8 at G ; then the ftiaded parts at n o and m o are fedlions of the rail ; then 
 draw a line 0 b, to touch the corners of the fedfions at 0 and b\ at the points <7, a^ d^ e,fi 
 g^ /), and draw perpendiculars to 0 b\ then G being pricked. from the plan at as the 
 letters diredl, will be the true face mould. 
 
 ,Fig. /) is a plan of the fame fize, (hewing the face mould at F, when fprung, which will 
 be a very great faving of ftuff ; and not much more trouble in laying it down when properly 
 undeiftood. This metliod will be clearly explained in the following pages. 
 
 PLATE 
 
THE THEORY AND PRACTICE OF JOINERY. 63 
 
 PLATE LIX. 
 
 T } draw the falling mould of a rail having a quarter fpace in it j thence to find the face moulds of" 
 
 the circular part. 
 
 At the plan fig. Jy a c \s. the ftretch out of half the circular part of the rail, found by 
 the fame method as in the foregoing platesj or it may be found more exadUy thus : Divide 
 the radius into four equal parts, and fet three of the divifions out to 3, and draw a line from 
 3 to by cutting the fide of the rail produced zt a* y from the point f in the right line h g 
 at By make / hy and f gy each equal to the ftretch out of half the rail, that is, equal to a r, 
 fig. A-y draw the perpendiculars h Oyf /, and ^ r ; at 5 apply the pitch hoard of a common 
 ftep at F'y through the point ty draw t ky parallel to g hy cutting the line //, at i j from k to / 
 fet up the height of the four winders ; through / draw / «, parallel to g hy cutting 
 the line h Oy at « ; from n make n Oy equal to the height of a ftep, for the quarter fpace 
 upon the landing which only rifes one ftep j draw the hypotenufe Icy again, draw 
 0 p parallel to g hy and p q ; perpendicular to o py draw y o ; then 0 p q \% the pitch-board of 
 another common ftep above the winders j then thefe angles being eafed olF by the 
 method of interfering lines, the falling mould will be completed, as in the laft plate ; 
 make f u and f Vy from / equal to a dy fig. Ay that is, the ftretch out from the middle of 
 the arch at by to the joint; draw v Xy and u z parallel to//; then take the heights from i 
 toy and z , and fet them from a to b and c, will give the fecftion b C ; then take m I from 
 the falling mould, and from d make d e equal to it, will give the fedlion d e ; then take 
 w X from By and make f g, at Ey equal to it;'^from w, draw zu r, parallel to g hy cutting 
 f m2Xr 'y from r take the heights from m and /, and fet up thefe heights from H, to i knd K 
 at Ey it will give the fedlion i k ; then the face moulds upon D and E will be traced as di- 
 redled in the laft plate. 
 
 '* The line a c nearly equal to the femicircumferenoe, and is the moft exad of any that ever has yet been 
 fhevvn by a geometrical method ; it may be depended on in praftice ; it is abfolutely impoffible to find a right 
 line exaftly equal to the circumjference of a circle ; this has exercifed the attention of the greateft mathema- 
 ticians in every age. . 
 
 PLATE LX. wants no explanation* 
 
 FL ATE 
 
4 THE THEORY AND PRACTICE OT JOINERY. 
 
 / 
 
 P L A T E LXI. 
 
 To draw a falVirg mould for a rail having a winder all round the circular part-^ as is Jhcwn in 
 the lajl plate^ thence to find the face mould. 
 
 To defcribe every particular in this," would altnofl: be repeating what has been already 
 defcribed in the laft plate ; the heights are marked the fame upon the falling mould at Z>, 
 as they are at the face mould, which will give the heights of the fesffion^^ of the rail ; and 
 the face mould at C, is traced from the plan according to the letters j In plate 6o is 
 fhewn the fame thing, only with this difference, that the fate mould is partly flraight at 
 one end, becaufe the joint muft have been weaker, had it been made where the cir- 
 cular part of the rail boigins, as. is fhewn in this plate ; but the method of tracing this is 
 nothing different from the other in the laft plate ; Only i would have the reader to obfervc, 
 in plate 6o, that ordinates are drawn through the pLces where the circular part begins, 
 which will give the fame points on the face mould ; for, by this means, you will be able to 
 determine what part of the face mould is exadly ftraight, and where the crooked place of 
 your mould begins. I hope the reader will underftand the fame thing in the following 
 plates, without being told a fecond time. G (hews the application of the mould to tha 
 plank ; take the bevel at /f, and apply it to the edge of the plank at D, and draw the 
 line b c\ then apply your mould to the top of the plank, keeping one corner of it to the 
 point and the other corner clofe to the fame edge of the plank ; then draw the top 
 face of the plank by your mould j then take your mould, and apply it to the under fide at f, 
 in the fame manner. 
 
 TLATE 
 
lied. 
 
Pl/ilz 65. 
 
THE THEORY AND PRACTICE OF JOINERY: 
 
 PLATE LXIII. 
 
 Hsiu to draw the face moulds of an elliptic fair. 
 
 The plan and feiliion being laid down as in plate 62, the reader will obferve, that the 
 ends of the fteps are equally divided at each end ; that is, they are equally divided round the 
 elliptic wall, and alfo at the rail. In this plate, the rail is laid down to a larger fize than 
 that in the laft plate \ the plan of this rail muft be divided round, into as many equal 
 parts as there are fteps j then take the treads of as many fteps as you pleafe, fuppofe 8, and 
 let h hi at fg. H, be the tread of 8 fteps from i/j on the perpendicular h w, fet up theheight 
 of as many fteps, that is, 8 ; and draw the hypotenufe m h^ which will give the under edge 
 of the falling mould. The reader will obferve, that this falling mould will be a ftraight 
 line, excepting a little turn at the landing and at the fcroll, where the rail muft have a little 
 bend at thefe places, in order to bring it level to the landing and to the fcroll ; then mark 
 the plan of your rail in as many places as you would have pieces in your rail (in this plan 
 are three) ; then draw a chord line for each piece to the joints j alfo draw lines parallel to 
 the chords, to touch the convex fide of the plan of the rail; from every joint draw per- 
 pendiculars to their refpe<ftive chords. Now the tread of the middle piece at C being juft 
 8 fteps, the height of the fedfion from h to m h 8 fteps ; and the feilion m n h the 
 fame as m n on the falling mould, and the feflion h i is the fame height as h t upon the 
 falling mould ; then draw a line to touch the fedllons, and complete your face mould 
 in the foregoing plates, each of the other pieces at F and G, the treads, being fix fteps ; 
 therefore, from your falling mould fet the ftretch of 6 fteps ; from h to n draw H /, parallel 
 to h «, then h k I will give the heights of the fedlions at D and E ; every thing elfe agreeable 
 to the letters. 
 
 Kote. By the ftretch out of 8 fteps, or any other number, is not reckoned on the chord ; 
 but it is the ftretch out round the convex fide of the rail, or what fome people call the infide. 
 
THE THEORY AND PRACTICE OF JOINERYo 
 
 44 
 
 PLATE LXIV. 
 
 7 'he tread of a lulnding fair being givth^ round the middle and the plan of the rail ; to diminifn 
 the ends of the feps at the rail^ fo that the ballifers Jhall be all regular^ or of an eqnal height^ 
 when fin'ifoed. 
 
 Let the firfl winder begin about the firft ftep before the circte of the rail, at Z) ; from 
 a to in the plan fig f is the ftretch out of half the circular part of the rail ; the method 
 of finding it has already been explained in the foregoing plates ; from e draw e H, perpen- 
 dicular to the fide of the rail; by reckoning round the dotted line, from 5 to lO, you 
 will find there are five treads, or five winders ; therefore from ^ to 5 fet up the height 
 of 5 fteps ; produce the lon:2:efi: fide a b of the pitch-board- D, to r ; bifeil r, at 2 j draw 
 a line from 2 to 5 ; then divide 2 b and 2 5, each into the fame number of equal parts i and 
 interfedt the angle by the common method of interfedliing lines, will give the under edge 
 of the falling mould ; then a line being drawn parallel to it, to the thicknefs of the rail, 
 will give the upper edge, which is the falling mould for half the rail; draw the lines i /, 2 ky 
 3 /, and 4 hy parallel to 6 gy to intcrfedl; the falling mould at the points hy iy ky I ; from thefe 
 points draw the parallel dotted lines to q H, down to the rail Tutftvu) from c draw c s, c ty 
 c Vy and c w, cutting the arch line of the rail w, «, 0, py will give the ends of the fteps at 
 the rail ; then draw lines from w, ?ij o, py through 6, 7, 8, 9, will be the plan of the fteps. 
 
 To find the face mould of a rally fo that it may he got out of the leaf thicknefs of fluff pojfible. 
 
 Lay down the plan of the rail at any convenient place, as No. 3 ; draw the chords of 
 the rail N o and o p, from the centre K draw K E, perpendicular to the chord N o, cutting the 
 outfide of the rail at c ; in the fame manner draw the chord w x, at the plan, 7%. yf ; from 
 the centre gy draw gf perpendicular to it, cutting the outfide of the rail at f\ from c draw a 
 line c fy to cut the tangent line at v ; draw a line v E, parallel to ^ h ; from the joint of the 
 rail at h draw h B, alfo parallel to q H, interfc61:ing the under fide of the rail at A, and the 
 top fide at B ; draw from A a line A F parallel b q from Fy at No. 3; make F G H equal 
 to F c H, at No. I ; from C, at No. 3, make c d e equal to c d e, at No. i ; and make 
 J By at No. 3, from Jy equal to yf By at No. i ; draw a line B Ry for the chord of the 
 mould to touch the fhaded feeftions, perpendicular to N o and A F ; from E draw E M, per- 
 pendicular to B R ; make c iy at No. 2, equal to i c, at No. 3 ; make i e perpendicular to 
 £ iy equal to e l, at No. 3 ; make l m equal to c at No. 2 ; from e draw e t, parallel 
 to A F, cutting the chord line b r at t ; from the points t and m, draw the line t m, 
 then T M will be one of the ordinates; all the other ordinates are drawn at diferetion, parallel 
 to it, and completed in the fame manner as is fhewn in plate 9, for the fedions of a 
 
 cylinder. v 1 i. y 
 
 The reader will take notice, before the face mould at No. 3 can be applied, the edge of 
 the plank muft be firft bevelled according to No. 2 ; then the plumb-line will be drawn on 
 the bevelled edge of the plank, by the bevel that is drawn at No. 3. 
 
 Note. By this method of proceeding, a three inch plank will almoft be fufficient for 
 any rail of this kind however it ramp ; whereas in many cafe.s, by the common method, it 
 may require a plank five or fix inches thick. Many other advantages will attend the manner 
 of fetting out this plan ; 1 (hall mention one or two: in fixing the bannillers, they will be 
 all regular, and the firing board will be as eafy as the rail itfelf ; the fkirting will alfo be 
 quite regular, for the ends of the fteps are wider and wider as they go rounu to the middle 
 if the femicircle ; laftly, a blackfmith may put up an iron rail with very little trouble, the 
 bannifiefs being all regular j whereas no other plan will adroit of it, unlefs it is fet out in 
 
 this manner. PLATE 
 
THE THEORY AND PRACTICE OF JOINERY. 67 
 
 PLATE LXV. 
 
 To dlminljh the Jlep of a fair winding round one of the quarters to a lev:l landing. 
 
 Find the ftretch out round half the circular part of the rail, as direfled for the foregoing 
 plates, and complete the falling mould as dirc£led in the laft plate, for the winding part of 
 the rail, which is fix fteps from / to in order to bring the rail with an eafy turn round t® 
 the landing, fet off the height of another ftep from « to 7, and let the under edge of the 
 rail be half the height of a ftep above that to e } or it may be more, according to the dif- 
 cretion of the workman ; then the rail will be half the height of a ftep more upon the land- 
 ing than it is upon the winders ; through c draw cf parallel to the bafc, and continue the 
 line 2 », that forms the interfeiSHon below for the winders up to d, and eafe ofF the angle 
 K D f by interfeifling lines, will give the under edge of the mould turning up to the landing: 
 in order that the laft ftep beyond the quarter fhould alfo follow the mould, draw a line 
 through 7, the height of the laft ftep, parallel to k or c r, cutting the under fide of the 
 falling mould at a ; through A draw A B, parallel to c / j then « B is the tread of the laft 
 ilep at the rail, which is fet from g to e. The face moulds at D and F are completed in 
 the fame manner as dire£led in the laft plate, and the mould in plate 58, at fg. D, is alfo 
 laid down by the fame method, the height of the fedions being taken from the falling 
 mould that correfponds to that place of the rail which the face mould is made for ; and the 
 bevels that are laid down above each face mould will fhew how much you muft bevel the 
 edge of your plank, before you can apply the face moulds to the plank ; then draw the plumb 
 of your rail, upon the bevelled edge, by the other bevels tliat are (hewn at the fe(ftionsj 
 then apply your mould to each fide of the plank, keeping it fair with the bevelled edge, 
 the fame as in other cafes before mentioned. 
 
 Ka 
 
 PLATE 
 
6B the theory and practice of joinery< 
 
 P L A T E LXVI. 
 
 This plate (hews the method of capping an Iron rail, upon much the fame principles as the 
 others, but with lefs- trouble. 
 
 How to find the thicknefs of fluff for capping of an iron rail. 
 
 Lay a thin broad piece of wood, as b, upon the top of the iron, upon the place that is to 
 be capped, and turn it round upon the iron, till you fee the greatefl: fpace between the iron 
 and wood to be as little as poflible j then the open fpace will fhew what thicknefs mpfl: be 
 added to the thicknefs of the rail. 
 
 How to find the plumb of the piecc^for the application of your mould. 
 
 After haying found what thicknefs of fluff will do, apply the folid piece itfelf, b^ to its 
 place, then let one of the ends, as d e, be cut plumb. 
 
 How to trace the under fide of the plank to the iron^ inflead of the face mould. 
 
 Make a pricker, as r, with a fleel point in the upper end, and let it be notched out, fb 
 that when it is applied on each fide of the ballifter, it may juft leave the thicknefs of the rail 
 between each point ; then take your pricker r, and prick your piece b, at every ballifter j 
 always keeping your pricker clofe to every ballifter. This being done both outfide and 
 infide, if you infpedl C, it will make it plain, where you fee the fides. of the plank flatted out, 
 which Ihews upon the under fide at the black dots ; ftrike a plumb-line d ey upon the end 
 of your plank b, and this plumb-line will fhew how the top is to be pricked off from the 
 bottom, which you fee at Cj the under fide is fquared over to the edge, from your pricked 
 points, and from thence drawn acrofs the edge to the rake, which is formed by the plumb 
 upon the edge, then fquared over the top fide, and then it is to be pricked off from a line 
 drawn from the point dy in the end feiStion D, which the plumb-line gives upon the end, 
 along the top face of the plank, parallel to the edge, and not from the fquare edge of the 
 planko 
 
 PLATE 
 
r/a'/,' 0-6. 
 
the theory and practice of joinery. 69 
 
 PLATE LXVII. 
 
 This plate fliews the method of gluing a rail in thicknefles ; but if I muft give my 
 ODinion, a rail got out of the folid is much more preferable, although you are obliged to 
 hLe more end joints in it; but if your joints are well fcrewed together, a fol.d rad has 
 a more beautiful appearance than a rail glued up, having fo many different thicknefles of 
 elue which makes it have a black and nafty look with it ; if a perfon is ever fo careful, the 
 foints will ftill fhew, and this rail in itfelf having a natural tendency to fpring, the lealt 
 dampnefs will make it give way in time ; but as this is held by fome a very valuable ac- 
 quifltion, I lhall proceed to lay down the moulds for it. 
 
 ro (heuo the application of the outfide and hfide falling moulds to the upper and under faces of the 
 
 planky to give it the form of the twiji, 
 
 A b is the ftretch out of the greateft circle in B, and a c the height of the fteps ; again, 
 d e is the compafs of the leffer circle, fet in the middle between a b and df the height of 
 the fteps the fame with a c : therefore the triangle a b c \s the pitch-board of the inflde 
 falling mould ; and ^ w 0 at the bottom, and i h c the top, are the pitch-boards of two 
 common fteps ; which lines, when interfeaed, will give the under line of the inlide falling, 
 mould. In the fame manner dfe, with the two common fteps ^ at the top, and ^ « 
 
 at the bottom, will give the under line of the outfide falling mould. The top lines are 
 only drawn parallel to the under lines to the thicknefs of the rail. 
 
 How to apply thefe moulds to the plank. 
 
 Draw a line t p, to touch the moulds fo laid down in two places, and as both moulds in- 
 terfea toc^ether at then draw a fquare line p q, upon the top of your plank, at the fame 
 diftance zs p q is from the bottom ends of your moulds, and this line being fquared acrofs 
 the edge, and from thence acrofs the under fide ; then fet the diftance p q on both fides of 
 Tour .plank, from the fame edge, and likewife fquare over t s r, zt the diltance p t on your 
 plank on both ftdes, then fet the diftance of r from t upon the top fide of your plank, and 
 fet the diftance of j t upon the under fide ; you will obfei ve to mark the point q upon both 
 your moulds, then apply your outfide falling mould to the top of your plank, making the 
 point q to coincide with the fame point q in the plank, and make the front of the falling 
 mould to come to r, and with this mould, placed in this pofition, draw the upper face of your 
 plank with it, and in like manner apply your infide falling mould, that is, by applying the 
 point q to the fame point q in the plank, on the under ftde, and let the front edge be to r in 
 the plank ; then draw the under fide ; your plank being fuppofed of a fufficient thicknefs, 
 making allowance for the faw cuts, and plaining up your veneers, this plank, when cut out 
 twifted to thofe lines will be the true form of your veneers : the piece being thus formedj 
 you are to cut your veneers the. other way into thickneffes as you think they will bend eafy, 
 and fo I fhall leave you to complete the rail. 
 
 ig one thing that I would have you take notice of here, the general cuftom among 
 workmen to keep the hand rail higheft upon the winding part of the flairs, on the fuppo- 
 fltion of a perfon coming down flairs being liable to fall over the rail, when the delcent is 
 very rapid ; and therefore to remedy that inconvenience, I have all along made the under 
 fide of the falling mould, or the under fide of the rail, which is the fame thing, to be equi- 
 diftant upon the rake from the face of the rifers, which will caufe the upper fide of the rail 
 to run higher on the rake, than the height of the rail above the common fteps, and the quicker 
 the afeent is, the difference will be greater. In laying down this, the diftance v w (hews 
 how much the winding part rifes above the common fteps, which is about five inches and a 
 half; this is done by continuing the top line of the rail upon the winding part to w, and by 
 continuing the top of the ftraight part to meet at then the diftance oi v w will always be 
 according to the pitch of the winders, 
 
 ^ J* L A T E 
 
70 THE THEORY AND PRACTICE OF JOINERY. 
 
 PLATE LXVIII. 
 
 This plate fhews the method of fitting down the fkirting, upon any fort of a ftair-cafe 
 whatever ; whether fl:raighr, circular, regular, or irregular j if the treads are ever fo crooked, 
 and the rifers out of an upright. 
 
 InJ/f. is fliewn a bevel, made to the rake of the Ikirting, and the other perpendicular 
 to the ftair, and a Aiding piece to be applied to the perpendicular fide of the bevel, with a 
 hooked point of iron or fteel, to fiand forward at the bottom fo much, that the Aiding piece 
 may clear the nofing of the Aep, I ftiali proceed to fliew its application. 
 
 How to fit down the Jkirting, 
 
 Lay the fkirting over the top of the fteps, and let a very fine notch be made on the front 
 edge of your Aiding piece, to the height of a ftep, or rather higher ; then apply the point of 
 the Aiding piece to the internal corner of a Aep, and prick your Atirting in the notch at 
 the bevel being fiippofed to be brought clofe to the Aider : again, fuppofing you want to 
 take a point at the nofing, where you fee the bevel applied under, apply the point of your 
 Aiding piece to the nofing at c j then prick your fkirting In the not'h at that will give 
 the point d^ which is to co lefpond with r, &c. and by this means you may taktf'as many 
 pricks as will be fufficient, till the whole is completed, 
 
 COROLLARY. Hence it is evident by the fame method, that one thing may be fitted 
 into another, whether confidered as a Aair-cafe or not, Aanding either raking, horizontal, 
 or perpendicular. 
 
 If the Aeps of a Aair-cafe be very true, two pricks from each rifer and tread will be fuffi- 
 cient, as it is only joining thefe pricks by lines, which will form the rife and tread of each 
 Aep, and three pricks in each nofe, becaufe a circle may be eafily drawn through the three 
 points. 
 
 If the nofings are all exa£V, let a mould be made to fit one of them, and your nofings on 
 the fkirtings be drawn by this mould, which will likewife be exadl. 
 
 Fig. £■ fhews the method of laying down a raking fided Aair, which is clear in itfelf, the 
 height of the Aeps being the fame on each fide. 
 
 C and D fhews the method of tracing one bracket from another, in a Aair-cafe; C being 
 the bracket for the common Aep, D a bracket for one of the winders. 
 
 There is fhewn a method in the laA plate, at E and Z), for doing the fame thing by means 
 of a triangle, which is performed thus : let the other bracket at fig. Z>be given, whofe length 
 is A B ; and it you w'ant a bracket for the winders, whofe length is b C, draw B c, making 
 any angle at the point B join a c ; take as many ordinates as you pleafe, to touch - all the 
 principal lines of the given bracket; then draw lines parallel to A c, from thefe ori i lates j 
 and complete the other bracket as you fee by the letters. 
 
 PLATE 
 
Flaf^e 
 
 cAreoSiAii^f^.iJ^zfyP2Kck olaon. . 
 
 V 
 
THE THEORY AND PRACTICE OF JOINERY. 
 
 72 
 
 PLATE LXIX. 
 
 How to cUmlniJh the Jhaft of a column^ by the ancient method, 
 
 In/^. A, defcribe a femidrcle at the bottom ; let a line be drawn through the diameter, at 
 the top, parallel with the axis of the column, till it interfeds the femicircle at 1, at the bot- 
 tom ; then i i at the bottom will be equal to i i at the top ; divide the arch into four equal 
 parts, and through thefe points draw lines parallel to the bafe, the height of the column 
 being alfo d vided into the fame number of parts, and lines drawn parallel to the bafe, then 
 the column is to be traced from the femicircle, according to the figurer. 
 
 How to dhnimjh the column by lines drawn from a centre at a dijlance. 
 
 Fig. C. Take the diameter a b, at the bottom, fet the foot of your compafs in c at the 
 top, and crofs the axis in the point a\ continue c at the top, and a ^at the bottom, to 
 meet at ^ ; then draw from e as many lines acrofs the column as you pleafe, and take the 
 diameter a b at the bottom, and prick each line upon the axis equal to b a, which will give 
 the fweli of the column. 
 
 *To diminif) a column by laths upon the fame principle, 
 
 In^^. D, the point e being found, as in fig Cy take and plow a rod d b, and lay the grove 
 upon the axis of the column, and plow the deferibing rod upon the under fide, and lay 
 the grove upon a pin fixed at e^ and fix a pin at gy to run in the grove upon the axis of the 
 column, and the diftance of the pencil atfy equal to b ay then move the pencil at fy it will 
 deferibe the diminifhing. 
 
 How to deferibe the column by another method. 
 
 Take the diameter ^ at bottom, and fet the foot of your compafs in the top, at f, and 
 crofs the axis -at 8, and draw the line a 8 on the outfide, parallel to ^ 8 on the axis, and di- 
 vide each of thefe lines into eight equal parts, and fet the diameter ^ ^ at the bottom along 
 the flant lines 1 i, 2 2, 3 3, &c. from the axis ; this will alfo give the diminiftiing of the 
 column. 
 
 How to make a diminijhing rule. 
 
 Divide the height of your rule Into any number of equal parts, as 6 ; draw lines at right 
 angles from thefe points acrofs the rule, and divide the projedtion of the rule at the top ; that 
 is, half of what the column diminifhes i into the fame number of equal parts put a pin or 
 brad-awl ; lay a ruler from « to 5 ; mark the crofs line at f\ then lay a ruler from 4 to Oy 
 and mark the next crofs line at e j then lay the ruler from 3 to ay mark the next at dy and 
 fo on to the bottom ; bend a flip round thefe points, and draw the curve by it, will give a 
 proper curve for the fide of the column. 
 
 Note. This is the readieft method, and it gives the beft curve of any that I have tried. 
 
 PLATE 
 
72 
 
 THE THEORY AND PRACTICE OF JOINERY. 
 
 PLATE LXX, 
 
 The plan and elevation of a circular fajh^ in a circular wally being given j to find the mould 
 for the radial bars^fo that they Jhall be perpendicular to the plan. 
 
 Draw perpendiculars from the points i i i i, &c. at A and 5 , in the radial bars, either 
 equally divided, or taken at difcretion, down upon the plan, to i 2 3 4 5 6 7, at C and D ; 
 and draw a line from the firft divifion upon the backfide parallel to the bafe j then draw 
 ordinates from i i i i, &c. at right angles to the radial bars, z.t A and which being 
 pricked from the plans at D and C, will give a mould for each bar j and the bevels upon 
 the end will fliew the application of the moulds. 
 
 To find the veneer of the circular bar. 
 
 To avoid confufion, I have laid down the plan and elevation for the head of the fafh under. 
 The ftretch out of the veneer is got round i 2 3 4 5 6, on the -circular bar, w'hich being 
 pricked from the fmall diftances on the plan at My will give the veneer above, at E, 
 
 To find the face tnould for th e fajh-head. 
 
 Divide the fafti-head round, into any number of equal parts, at G, and draw them per- 
 pendicular to the bafe at H ; draw the chord of one half of the plan at and draw a 
 line parallel to it to touch the plan upon -the back fide ; then the diftance between thefe 
 lines at Hy will fhew what thicknefs of Huff the head is to be made out of ; and from the 
 interfe6ting points on the back fide, draw perpendiculars from the bafe of the face mould,, 
 which being pric-ked from the elevation, as the figures direft, will give the face mould. 
 
 To find the moulds for giving the form of the heady perpendicular to the plan. 
 
 The bafe of L is got round the arch i 2 3 4 5 6, at and the bafe of K is got round 
 abed efgy alfo at Fy and the heights of the ordinates of each are pricked either from H 
 or /, which will give both moulds. 
 
 By the fame method, a circular architrave, in a circular wall, may be got out of the folid. 
 
 Note. The face mould at G, muft be applied in the fame manner as in groins ; fo that 
 the fafh-head muft be bevelled by fhifting the mould G, on each fide, before you can apply 
 the moulds K and £; the black lines at K and L are pricked from the plan at H\ thefe 
 ^ black lines will exa<5lly coincide with the front of the rib when bent round; a line being 
 drawn by the other edge of the moulds, will be perpendicular over its plan, and the thicknefs 
 Hof the fafh frame towards the infide will be found near enough by gauging from the outfide. 
 
 PLATE 
 
Fcice 
 
P/cUe 71 
 
 iln c^7' ^lj'c/icc'x?/^i>r ix (i?'Cif/a/' Tl^hc/ofi/ lyji ct Ct?'cie/ar all 
 
 I'/ie v-enecr to 1/^ out dut o£ t/z^ tTo/t ot ■ — ^ 
 
 AroAitz'oiue ..Jv ^ar cui /i z oizidk^f' on eotcA dide.to ^puv 
 !To /t/ie JUi (A/A’^teecc t/iat /oi^ l-ctzvocfi /i.v an </ ?L 
 
 JdrepiA A/io ^otnt fVi//i tAe zzczct ncfu^er tAatto Izetzveen / 4,6 
 
 a fid nt ft .07' 6 .i. Izc 
 
 PiilP a.dt/veAci ddcctP2tbvt<^.ij^’i^ lp/PNic/toL><m 
 

A/^tt?-e of a Oo//toc/c 
 
 Tlatc . 
 
 Tbe 2 Tof/jcd of oai a?2y/o laJ's fi,' S'/jofb F?-o 22^-2 . 
 

THE THEORY AND PRACTICE OF JOINERY. 73 
 
 PLATE LXXII. 
 
 To defcrihe the angle bars for Jbop fronts, 
 
 \nfig, A'y B is a common bar, and C is the angle bar of the fame thicknefs ; take the 
 raking projeftion i i, in C, and fet the foot of your compafs in i at B, and crofs the middle 
 of the bar at the other i j then draw the lines 2 2, 3 3, &c. parallel to i i ; then prick your 
 bar at C from the ordinates fo drawn at B, which being traced will give the angle bar. 
 
 How to draw the mitre angle of a commode front for a fioop. 
 
 In E divide the projeaion each way in a like number of equal parts, then the parallel 
 lines continued each way will give the mitre. 
 
 How to find the raking mouldings of a pediment. 
 
 In fig. B, let the cimiredfa on the under fide be the given moulding, and let lines be 
 drawn upon the rake at difcretion} but if you pleafe, let them be equally divided upon the 
 ‘ cimireda, and then drawn parallel to the rake \ then the mould at the middle being 
 pricked off from thefe level lines at the bottom, will give the form of the face. The return 
 moulding at the top muft be pricked upon the rake, according to the letters. 
 
 The cavetto,/^. C", is drawn in the fame manner. 
 
 N. B. If the middle moulding, fig. B, is given, perpendiculars muft be drawn to the top 
 of the middle moulding ; then horizontal lines muft be drawn over the mouldings at each 
 end, with the fame divifions as are over the middle moulding ; and lines being drawn per- 
 pendicularly down, as above, will ftiew how to trace the end mouldings. 
 
 PLATE LXXIII. 
 
 Figures B and A ftiew how to trace bafe mouldings for (kirtening ftairs, upon the fame 
 principle as ftiewn in the laft plate ; at the bottom are g^iven two methods of mitring 
 mouldings of different proje<ftions together. 
 
 PLATE 
 
74 THE THEORY AND PRACTICE OF JOINERY, 
 
 PLATE LXXIV. 
 
 Given the form of a cornice to draiv it to a greater proportion. 
 
 In fig. let th€ given height of the cornice he ab \ fet one foot of your compafs in <r, 
 and crofs the under fide at b with that height,^ and from the point c draw the line c d at 
 right angles to a b-, then the height of ali your mouldings will be on a b^ and the projedions^ 
 on c d\n proportion to a b. 
 
 NoUy af (hews another height, c e its projection in proportion to that height. 
 
 How to dimtnip a cornice in the proportion of a greater. 
 
 Defcribe equilateral triangles on the bafe and projection as at D, and make i f and i g 
 equal to the intended height, and draw the line/^ acrofs the triangle, which will give the 
 heights in proportion to a b\ put the foot of your compafs in b as a centre, and circle b c 
 round to b h^ and draw the dotted line h /, cuttingy^ inky then fet ofF i e and i dy each 
 equal to ^ ^ ; draw d e ; then take the divifions oi e dy and fet them from f to m\ in the fame 
 orders draw perpendiculars : it will give the diminifhed cornice at D. 
 
 Another method. 
 
 At Ey let the given height be a by and draw the hypotenufe a gy and lines being fquared 
 up to a by from the divifions of a gy will give the heights ; and if you draw the line g d zt 
 a right angle with a gy then dcw'iW give the projection in proportion, when returned upon d c. 
 
 Fig. C /r the method for hanging a jib door. 
 
 Let a c he the projection of the furbafe or bafe moulding, and c the centre of the hinge j 
 make a b equal to a c, and in the centre at c defcribe the arch b d e-y then the arch b d e will 
 be the proper joint for the furbafe to work in. 
 
 The joint of the furbafe or the bafe may alfo be ftraight, as you fee by the dotted line 
 touching the circle at the point by as a tangent to it. 
 
 PLATE LXXV. 
 
 To find the fiweep of a moulding to be bent upon the fpring round a circular cylinder. 
 
 In fig. Ay which ftands upon a femicircular plan, fet a c to the height of your moulding, 
 and make a b the projection ; draw the form of your moulding, and draw a dotted line to 
 touch the face of your moulding ; then draw the line e d to meet in the centre of the body 
 at dy io a,s to keep your moulding to a fufficient parallel thicknefs j then in your centre d 
 •leferibe your moulding. 
 
 How to find the fiweep of your moulding when the plan is a fegmenU, 
 
 Complete the femicircle as in plate ly fig. i, then proceed as deferibed in fig. 4. 
 
 Figures Cand D (hew the method for bending a moulding round the infide, which is 
 performed the fame as above. 
 
 The demonftration may eafily be conceivfd from the covering of a cone. 
 
 CON. 
 

CONCLUSION: 
 
 In which are examined, by way of preventive Caution to the Student, 
 feveral prevailing Methods, which are founded on wrong Principles, 
 and better ones here propofed. 
 
 Of the Ellipjis, Plate 76. 
 
 The old manner of interfe£l:ing all kinds of lines, applied to gothic and elliptical - figures, 
 to this day is exceedingly ufeful in forming the ramps of flairs, or eafing off any angle, as 
 at Gi but when this is applied to elliptical figures, it is far from forming a true ellipfis, being 
 too full at the ends, as at fig. D and E : and this is no certain rule for drawing an ellipfis, 
 for the more divifions there are, the worfe is the ellipfis j as for example,/^. F is divided 
 into double the number of parts as £) j it is plain that neither D nor F is an agreeable 
 ellipfis, and F is much worfe than A which is contrary to general opinion ; for 1 have 
 been frequently told, the more parts it is divided into, the truer it is ; but by this it appears 
 the more parts it is divided into, the worfe it is ; if this is doubted, try it. Fig. is an 
 ellipfis drawn on true principles, as laid down in Plate 7, at C, of this book, and is here . 
 repeated to be compared with the others. Figures B,, Cy and Ej are ellipfes drawn with a 
 compafs : I may call them reprefentations, as it is impoffible to draw them true with a 
 compafs j there is no part of the curvature of an ellipfis that will exaaiy agree with any 
 part of a circle, for in every quarter of an ellipfis, from the extremity of the tranfverfe, . 
 the curvature in every fucceeding part is continually flatter towards the extremity of the 
 conjugate axis; but yet there is a method to reprefent an ellipfis, which will differ very 
 little from the truth, as is fliewn at figures B and Ey which are both drawn by the fame 
 method ; fig. E ox B is the nearefl to the ftiape of fig. A'y fig. C, the method ufed by almoft 
 every author who has written upon the fubjea, is full at the ends, but not in fo great a 
 degree as fig, D and E. 
 
 Of raking Mouldings. Plate 77.' 
 
 In Plate 77, fig. 4 let the moulding at the bottom be given, and let the perpendicular 
 height be divided into any number of equal parts, as fix ; likewife divide the perpendicular 
 height of the top moulding into fix, and the face moulding into fix, at right angles to the 
 rake ; and let the ordinates of each be drawn through the equal divifions of each refpeaive 
 perpendicular, and pricked from the bottom, as the figures dire^. It is evident, that if the 
 under moulding is compofed of two quarters of a circle, the upper mouldings will be com- 
 pofed of two quarters of an ellipfis j confequently the returtl moulding at the top will be too 
 quick upon the round, and likewife in the hollow. 
 
 But 
 
76 
 
 CONCLUSION. 
 
 But if this demonftration fhould not be fufficient, let a dotted line be continued from r, 
 in the given moulding, parallel to the rake ; then it will be evidently feen, that this dotted 
 line correfponds with neither the face nor the return moulding ; for in the face moulding 
 it falls between the points i and 2 ; and in the top moulding it falls almoft at the point 2 ; 
 whereas it ftiould only come to the point i in each; but the horizontal projedlion from 2 to 2, 
 at the top, ought to be equal to i 1 at the bottom; but it is much greater; therefore this 
 method is falfe, as they will not mitre together. 
 
 I fiiall alfo notice another method ufed by fame authors, figure i 5 , at D and Ey 
 where they are pricked perpendicular to their chords, in the middle, which is alfo falfe ; 
 but if they are pricked as at B and C, on the rake, will be exceedingly near, if deferibed 
 with a compafs through three points. 
 
 Of dhninijhing of Columns. Plate 78. 
 
 The method for drawing a column, deferibed by fome authors, and which is properly 
 called a conchoid column, is not only very inconvenient on account of the cumberfomc 
 inftrument which is neceflary to find the curve for pra<Sfice, but alfo the appearance is, 
 1 think, lefs graceful than when produced by other methods. The conchoid curve, or 
 column, is concave towards the bottom, and convex towards the top; and if this curve was 
 infinitely extended, it would never meet the axis ; which fliews it to be different from the 
 elliptic curve or column, as fome have called it. 
 
 The column called hyperbolic, plate 78, is not fo named from the general properties of 
 the conic hyperbola (becaufe there may be an infinite number of hyperbolas Handing upon 
 the fame bafe, having one common vertex, which will all be contained between a triangle 
 and a parabola, according as its axis is longer or fhorter), but becaufe it will nearly coincide 
 with fome of thefe hyperbolas. This curve has been known among workmen, and by 
 them has been miftaken for an elliptic curve ; to refute which I have, on the fame plate, 
 (hewn a true elliptic column for comparifon ; the lines of their curvature are continued 
 only to fhew their true figure ; either of thefe is a more commodious method than the 
 conchoid. 
 
 The method which I recommend as eafieft in practice for diminifhing of columns is al- 
 ready deferibed on plate 69, by means of a diminifhing rule, which is infinitely more con- 
 venient than the trammel, and which, to my eye, alfo produces a pleafanter contour ; but 
 as this will depend on the fancy of the architefl:, the workman will find fome ©f the methods 
 fhewn will anfwer his purpofe for any' curve. The conchoid is flatteft at the top, the 
 hyperbolic is a little quicker, the parabolic is Hill more fo, and the elliptic is4he moft quick. 
 
 F I N I 
 
 S. 
 

 z/z'z'r.Ai .iV/z- 'J^y7y2^<'2^uYMsozi. 
 
 df 
 
A CATALOGUE of Modern Books on Archite^uie, 
 
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 Bridges, &c. btair-Calesand Hand-Rails of various conftruaions, Angle-Bars for Shop Fronts, and Raikinc Moulding 
 with many other thinp entirely new. The whole founded on true Geometrical Principles ; The Theory and Praafei 
 
 If ^Tgreat“ varil Doors, Windows, Chimneys, Cornicesi 
 
 underftoocl. Circular circular Sotn.,, Hewing and winding i‘n ftraight and cbcular VVauloranrA^ngfe BWAt rchf^ 
 
 the, r circular Bars, Shop Fronts, Ac: ’ b/T^ ■ 
 
 Pam's Britijh Palladio, or the Builder's General Affflant ; demonftrating, in the moft eafy and praaical method all th. • • ' 
 
 rules of Architeaure, from the ground plan to the ornamental finiffi. Illuflrated with feveral new nnH 
 houfes, with their plans, elevations, and feaions. Alfo clear and ample inftruaions annexed to each 1 r® & 
 
 with a lift of prices for materials and labour, and labour only. This work will be univerfalh uCeful to all T 
 mafons, joiners, plaiferers, and others, concerned in the feveral branches of building, &c com'pr^{ndin<^ the fnU ' 
 viz. Plans, elevations, and fedions of gentlemen’s houfes. Defigns^for doors chirn^vsan^^^^^^^^ 
 embelliffiments, in the moft modern tafte. A great variety of mouldings, for b^e and ftubafe t chi raves h iDoftrne?e< 
 and cornices, with their proper ornaments for pradice, drawn to half fize • to which are pHrleH flilc i 
 leffenmg m pleafure, if required Alfo great variety of ftair-cafes ; (hewing the praaicd tnethod f etcuting 
 cafe required, -y/z. groins, angle-brackets, circular circular fle wing and winding ffiffits domes fkv Ss 11 
 plain and eafy to the meaneft capacity. The proportion of windows for the light to mnm/ p ^ ^ j 
 
 tlie proportion of chimneys to rooms, and fedions of flews. The principal timbers nronerlv ;;/ ^oundaUasss 
 
 the manner of framing the roofs, and finding the length and backiJ^g of hips, either fqnLe or bevel SWind ^ 
 timbers, figured in proportion to their bearing. The method of truffing girders fc^fina ^ ^ ^ i 
 
 articles, particularly ufeful to all perfons in the building profeffion. Thf vvhole coirebfR 
 
 copper- pl.ates, from the original defigns of William and James Pain, bound, i6s ^ ^ ^ ffirty-tvv9 fob 
 
 The Pralticat Heufe Carpenter-, or Youth', InJhuBor: containing a great Variety of ufeful Defigns in Carpentry and Architeaur 
 
 as Centeng for Grotns, Ntches, &c. Examples for Roofs, Sky-Lights, &c. The Five “Orders laid down by ThC sc, 
 Mouldm^.c, &c. at large, with their Knnrhmf.nfc P liSOC In .M... .] O— d ! /*rt r ^ ^ 
 
 ....... v^iiim icy-ricccs, onop-rronts, Uoor^Lafes. Section of a Dinincr-Room and 
 
 brary. Variety of Statr-Cafes w.tl, many otlter important Articles, and ufeful Embelliffiments. To whi?h is Sded a J 
 o! Prices for Matc/ials and Labour, Labour only, and Day Prices. The whole illuftr.ated, and made perfcaiy eafv by , 
 
 Copper- Plates, with Explanations to each By miUam Pain, Author of the Ptaftical Builder, and Britiflt Fa'lalo 
 Fourth lulition, with large Additions. Price 15s. bound. .oniiin ra.iae.o. 
 
 N. B. This is pain’s laft work. 
 
Tlr Framed Builder, or Workman’s General Affiftant; (hewing the moft approved and eafy methods for drawing and working 
 the whole or feparate part of any building ; as, the ufe of the tramel for groins, angle brackets, niches, &c. femiciicu ar 
 arches on flewing jambs, the preparing and making their fofhts ; rules of carpentry, to find the length and backing oi 
 ftraioht or curved hips, truffes for roofs, domes, &c. Truffing of girders, fc6fion of floors, &c. The ^oportion of the 
 five orders in their general and particular parts; gluing of columns; fiair-pfes, with their ramp and twilter. rads, fixing 
 their carriages, newels, &c. Front! fpieccs, chimney-pieces, ceilings, cornices, architraves, &c. in the ne weft talte ; wiifi 
 plans and elevations of gentlemen’s and farm houfes, barns, &e. By JV. Fain, Architeft and Joiner. Engraved on eighty- 
 three quarto plates, bound, I2S. A new edition, with improvements by the Author. 
 
 The Carpenter'' s Pocket Diretiory ; containing the beft methods of framing timbers of all figures and dlmenfions,^ with their 
 feveral parts; as floors, roofs in ledgments, their length and backings; trufled roofs, fpires, and domes; truffing girders, 
 oariitions and bridges, with abutments; centering for arches, vaults, &c. cutting ftone ceilings, groins, &c. with iheir 
 moulds : centres for drawing Gothic arches, ellipfes, he. With the plan and fe£lions of a barn. Engraved on twenty-tour 
 plates, with explanations. By JV. Fain, Architedl and Carpenter, bound, 4s. 
 
 The Builder's Complete AManf, or, a Library of A ts and Sciences, abfolutely necefTary to be underftood by Builders and 
 Workmen in rrencral, viz, i. Arithmetick, vulgar and decimal, in whole numbers and fractions. 2. Geometry, lineal, 
 funerficial and folid. 3. Architeaure, univerfal. 4. Menfuration. 5. Plain trigonometry. 6. Surveying of land, &c. 
 
 -7 ^Mechanic powers. 8. Hvdroftaticks. llluftrated by above thifteen hundred examples of lines, fuperficies, folid*, 
 mouldings pedeftals, columns, pilafters, entablatures, pediments, impofts, block cornices, ruftic quoins, frontifpieces, 
 arcades porticos, he. proportioned by modules and minutes, according to Andrea Palladio and by equal parts. Like- 
 wife great varieties of trufled roofs, timber bridges, centerings, arches, groins, twifted rails, compartments, obelifks, vafes, 
 oedefHs for buftos, fun-dials, fonts,’ he. and methods for raifing heavy bodies by the force of levers, pulleys, axes in peri- 
 ' trochio ferews, and wedges; as alio water by the common pump, crane, &c. wherein the properties and prefTure of the 
 air on water, he. are explained. The whole exemplified on 77 large quarto copper-plates, by Batty Langley. The fourth 
 edition, 2 vol. royal o6lavo, bound, 12s. 
 
 n Cwns in Architetlure' confifting of plans, elevations, and feftions for temples, baths, caflines, pavilions; garden feats, obelifks, 
 ^^and other buildings : for decorating pleafure-grounds, parks, forefts, he. he. By John Soane. Engraved on thirty-eight 
 copper* plates, imperial o£tavo, fewed, 6s. 
 
 ArchiieTure, or Rural Amufement ; confifting of plans, elevations, and fedions, for huts, fummer and winter hermit- 
 ages retreats terminaries, Chinefe, Gothic, and natural grottos, cafeades, ruftic feats, harps, mofques, morefque pavi- 
 vFr.nl arotefnue fcats, green-houfes, he. many of which may be executed with flints, irregular ftones rude branches 
 and loSro?tLe3rconta.ning twenry-eight Lw def.gns, with fcalcs to each. By ff. Wrigh,, Archueft. Oaavo, 
 fewed, 4s. 6d. 
 
 Ideas for Ruftic Furniture, proper for Garden Chairs, Summer Houfes, Hermitages, Cottages, &c. engraved on 25 Plates, 
 Odavo. Price 4s. 
 
 The Temple Builder's moft ujeful Companion: containing original defigns in the Greek, Roman, and Gothic tafte. By C. T. Overton^ 
 Engtaved on fifty copper-plates, odtavo, fewed, -7s. 
 
 The Carbenter'sTreafure- a colkaion of defigns for temples, with their plans ; gates, doors, rails, and bridges, in the Gothic 
 
 -2 toe t.ap (-entres at large for ftriking Cjothic curves and mouldings, and fome fpecimens of rails m the Chinefe tafte, 
 
 f^mmcT a complete fyftem for rural decorations. By N. JVallis, Architeft. Sixteen plates, oftavo, fewed, 2S. 6d. 
 
 Cnthic Architeaure improved by Rules and Proportions in many grand defigns of columns, doors, windows, chimney-pieces, 
 arcades colonnades, porticos, umbrellas, temples, pavilions, &c. with plans, elevations and profiles, geometrically exem- 
 plified * By B. ^ ‘Z- Langley. To which is added, an^Hiftorical Difeourfe on Gothic Architeddure. On 64 plates, quarto, 
 
 bound, 15s. 
 
 The Modern Joiner • or, a Colleftion of original Defigns, in the prefent tafte, for chimney-pieces and door-cafes, with their 
 mouldings and’enrichments at large; frizes, tablets, ornaments for pilafters, bafes, lub-ba(es and cornices for rooms, he. 
 with a Table, fhewing the proportion of chimneys, with the entablatures, to rooms of any fize. By N. JVallts, 
 
 Architedl, quarto,. 8s. 
 
 Outlines of Defigns for Shof Fronts and Door Cafes, with the Mouldings at large, and Enrichments to each Defign. Engraved 
 on 24 Plates, quarto, 5s. 
 
 Currus Civilis, or Genteel Defigns for coaches, chariots, poft-chaifes, vis-a-vis, road and park phaetons, whifkies. Angle horfe 
 ''' difes, he. Engraved on thirty plates, quarto, fewed, los. 6d. 
 
 An intolerable a nuifonce ; with a Table to proportion ctoti^f ^to’^he lize o 7 lu. 
 
 ‘"1 il"d with A new edition. By Rohr, Clavorhg, Builder, fewed, as. 6d. 
 
 Obfervations on Smoky Chimneys, their Caufes and Cure, with Confiderations on Fuel and Stoves, llluftrated with proper Figures, 
 
 Mh-k dift 
 
 ‘•'i* f“bjea. oaavo, fewed, as. 
 
 n, Builder, 'p«k,t Treufure, in which not only the Theory but the Praaical Part of Architeaure are carefully explamed and cor- 
 reaiy engraved on fifty-five Copper-plates, with printed Explanations to each, by fPilliam Pain. O&uvo, bound 6s. 
 
 Tandevs Builder's Direaor, or Bench Mate; being a pocket treafury of the Grecian, Roman, and Gothic orders of Archi- 
 ^tefture, made eafy to the meaneft capacity, by near 500 examples, engraved on 184 copper- plates, i2mo. bound, 4s. 
 
 Langleys Builder s bound, 4s. 6cl. 
 
 Uawney's Complete Meafurer, 3 ®* . 
 
 Moppus's Meafurer. Tables ready call, 2S. 6d. ^ 
 
 Plate Glafs Book, 4s. 
 
 Everv Man a Complete Builder-, or eafy Rules and Proportions for drawing and working the ftvcral parts of Architeaure. In 
 which are given a plan, elevation, and feaion of the curious trufled carpenters work ereaed to fupport the centre arch ot 
 Black-Friars Bridge, from an exaa meafurement. Compiled hy Edward Oakley. Oaavo, fewed, 4s. 6d. 
 
 The Joiner and Cabinet Maker's Darling ; containing fixty different defigns for all forts of frets, friezes, &c. fewed, 3s. 
 
 The Carpenter's Companion ; containing thirty-three defigns for all forts of Chinefe railing and gates. Oaavo, fewed, 2S. 
 
 The Carpenter's Complete Guide to the whole Syftem of Gbthic Railing; containing thirty-two new defigns, with fcales to each, 
 oaavo, fewed, 2s. 
 
 The Carpenter's and Joiner's Vade Mecum, By Robert Clavering and Company, fewed 2s. 
 
 A Geometrical View of the Five Orders of Columns in Architeaure, adjufted by aliquot parts ; whereby the meaneft capacity, by m- 
 fpeaion, may delineate and work an entire order, or any part, of any magnitude required. On a large meet, is. 
 
[ 4 3 
 
 Elevation of the New Bridge at Blaci-FriarSi with plan of the foundation and fuperftru£fure. By R, Baldwin. 12 inches by 
 48 inches, 5s. 
 
 Plans, Elevations and SeSiions of the Machines and Centering ufed in cre£Hng Black-Friars Bridge; drawn and engraved by 
 R. Baldwin, clerk of the work; on feven large plates, with explanations lOs. 6d. or with the elevation 15s. 
 
 Elevation of the Stone Bridge built over the Severn, at Shrewjburyi with plan of the foundation and fuperftrudure, elegantly 
 engraved hy Rooker. is. 6d. 
 
 ATreatife on Building in Water. By G. Semple. Quarto, with 63 plates, fewed, I2s. 
 
 Plans Elevations, and Se£Hon of the Gaol, Bridewell, and Sheriff's Ward, lately built at Bodmin, in the county of Coi-nwall/ 
 by fohn Call, Efq. upon the plan recommended by John. Howard, Elfq. On a large (heet, 2s. 6d. 
 
 London and Wejlminjier improved. Illuftrated by plans. To which is prefixed, a Difeourfe on public Magnificence; with obferva-. 
 tions on the ftate of arts and artifts in this kingdom ; wherein the ftudy of the polite arts is recommended as neceflary 
 to a liberal education: concluded by fome propofals relative to places not laid down in the plans. By John Gwynn, 
 Architedf. Boards, 5s. 
 
 Plans Elevations, and Sedlions, prefented to the corporation of Bath, for the improvement of the baths in that city; intending to 
 make the whole one grand, uniform, elegant, and convenient ftrufture of the Ionic order. By the late R. Dingley, Efq. 
 Engraved on nine folio plates, by Rooker, See. fewed, 6s. 
 
 Encaujlic, or Count Caylus’s Method of painting in the manner of the Ancients. By J. H. Muntz. Odiavo, bound, 5s, 
 
 The Young Draftfman's Guide to the true Outline of the Human Figure; or a great variety of eafy examples of the Human 
 Body; calculated to encourage young beginners, and thereby lead to the habit of drawing with accuracy and facility 
 on true principles, By an eminent Artiji, deceafed. Engraved on eighteen copper-plates, folio, fewed, 5s. 
 
 BOOKS OF ORNAMENTS, &c. 
 
 Ornaments Difplayed, on a full fize for working, proper for all Carvers, Painters, &c. containing a variety of accurate Examples 
 of Foliage and Friezes, elegantly engraved in the manner of Chalks, on 33 large folio plates, fewed, 15s. 
 
 JNew Book of Ornaments y containing a variety of elegant defigns for modern pannels, commonly executed in ftucco, wood, 
 or painting, and ufed in decorating ptincipal rooms. Drawn and etched by P. Columbani. Quarto, fewed, 7s. 6d. 
 
 A Variety of Capitals, Frizes, and Cornices how to increafe or decreafe them, ftill retaining the fame proportion as the original. 
 
 Likewife, twelve defigns for chimney-pieces, drawn an inch and a half to a foot. On twelve plates, drawn and etched by : 
 P. Columbani. Folio, fewed, 6s. ^ f 
 
 The Principles of Drawing Ornaments made eafy, by proper examples of leaves for mouldings, capitals, fcrolls, hulks, foliage, &c. V 
 engrawed m imitation of drawings, on fixteen plates. With infiruaions for learning without a mafter. Particularly ufeful I 
 to carvers, cabinet-makers, ft uccO- workers, painters, fmiths, and every one concerned in ornamental decorations. V 
 By an Artijl. Quarto, fewed, 4s. 6d. , f 
 
 Ornamental Iren Work', or defigns in the prefent tafte, for fan-lights, ftalr-cafe railing, window guard Irons, lamp irons, palifades ' 
 and gates. With a fcheme for adjufting defigns with facility and accuracy to any Hope. Engraved on 21 plates, (juarto^ 
 fewed, 6s. 
 
 'A new Book of Ornaments, by S. Aiken. On fix plates, fewed, 2S. 6d. 
 
 Twelve new Defigns of Frames for Looking-GlafTes, Piaures, &c. by S. H. carver, fewed, 2s. 
 
 A Book of Tablets, done to the full fize commonly ufed for chimney-pieces. Defigned and etched by J. P ether, on fix plates, 
 fewed, 3s. 6d. 
 
 Ornaments in the Palmyrene Tajle\ engraved on twelve quarto plates. By N. Wallis, fewed, 4s. 6d. 
 
 Law\ new Book of Ornaments, fewed, 2s. 
 
 A Book of Vafes, by T. Law, fewed, 2s. 
 
 A Book of Vafes, by P. Columbani, fewed, 2s. 
 
 A new Book of Eighteen Vafes, Modern and Antique, 2S. , 
 
 A Book of Vafes from the Antique, on twelve plates, fewed, 2s. 
 
 Book of Foliage, fewed, 2s. 
 
 A fmall Book of Ornaments, on fix leaves, by G. Edwards, is. 
 
 A new Book of Defigns for Girandoles and Glafs Frames. By B. Pajlorini, on ten plates, fewed, 4s. ^ J 
 
 An Interior View of Durham Cathedral, and a View of the elegant Gothic Shrine in the fame. Elegantly engraved on twt > 
 large Sheets, fize 19 and a half by 22 and a half; the pair, 12s. 
 
 An Exterior and Interior View of St. Giles's Church in the Fields, elegantly engraved by Walker, fize 18 inches by 15 ; the pair, 5s. 
 
 A Plan and Elevation of the King of Portugal’s Palace at Mafra, on two large fheets, 6s. % 
 
 A north-weft View of Greenwich Church, is. 
 
 An elegant engraved View of Shoreditch Church, 2 feet 4 inches by i foot 8 inches, 3s. 
 
 An elegant engraved View of the Monument sX London, with the parts geometrically; fize 2i by 33 inches; from an ori- 
 ginal by Sir C. Wren; and an Hiftorical Account in letter prefs, 7s. 6d. 
 
 Sir Chrijlopher Wren's Plan for rebuilding the City of London after the great Fire 1666, is. 
 
 Plan and Seftions of a curious Sailing Machine, neatly coloured, 5s. 
 
 The Art of Pradical Meafuring by the Sliding Rule ; (hewing how to meafure timber, ftonc, board, glafs, painting See 
 alfo gauging, &c. By H. Coggefhall. A new edition, by J. Ham, bound, is. 
 
 The Building Act of the i\th Geo. III. With plates, (hewing the proper thicknefs of party walls, external walls, and chimne; 
 
 A complete index, lift of furveyors and their refidence. Sec. In a fmall pocket fize, fewed, 2s. 6d. ^ 
 
 N. B. The notice and certificate required by the above a6l, may be had printed with blank fpaces for filling up, price 2c 
 each, or 1 3 for 2S. 
 
 Animals drawn from Nature, and engraved in Aquatinta, by Charles Catton, on 36 plates folio, 2I. 17s. in boards. 
 
 Smeaton's Experiments on Under-fliot and Over-(hot Water Wheek, &c. Oftavo, with 5 Plates, boards 4s. 6d. 
 
 A General Hiflory of Inland Navigation, Foreign and Domejiic ; containing a complete Account of the Canals already executed in E 
 
 ■ land ; with Confederations on thofe proje^ed. To which are added, Pradiical Ohfervations. Illuftrated with four Plates ef Lc 
 Bridges, is’e. and a large Map of England coloured, Jljewing the Lines of the Canals executed, thofe propofed, and the Navii 
 Rivers. A new Edition, with Two Addendas which complete the Hiflory to 1795. Boards, il. 8 s. * j 
 
 N. B, The Acldendas may be had feparate, by former purchafers of the work. Alfo, I 
 
 The Map may be had feparate, price 5s, coloured. / 
 
 I 
 
-;v, V' 
 
 
 SPECi^u 85-6 
 
 150(55 
 
 GETTY CENTER LIBRARY