EGERTON LEIGH, ESQ #? 
 
Dr. BROOK TATLOR's 
 
 Method of 
 
 PERSP ECTIVE 
 
 Made Eafy, 
 Both in Theory and Practice, 
 In TWO BOOKS, 
 
 BEING 
 
 An Attempt to make the Art of Perspective eafy and familiar i 
 
 T o 
 
 Adapt it intirely to the Arts of Design 5 
 
 AND 
 
 To make it an entertaining Study to any Gentleman who fhall 
 
 chufe fo polite an Amufement. 
 
 By JOSHUA KIRBY, Painter. 
 
 Illuflrated with Fifty Copper Plates; moil of which are Engrav'd 
 
 by the Author. 
 
 The Practice [of Painting] ought always to be built on a rational 
 Theory > of which Perspective is both the Guide and the Gate, and, 
 without which, it is impojjible to fucceed, either in Defgning y or in any of 
 the Arts depending thereon. 
 
 Leonardo da Vinci upon Painting, p. 36. 
 
 The Second Edition* 
 
 — — — ~ ■ — — ^ ' ■ - 1 ■ ~i ' ' • — 
 
 IP S W 1 C H : Printed by W, CraiChton, for the A V T h g r. 
 Sold by the Author, at the Golden Eagle in Great Queen's -Street, Lincoln s -fan Fields-, London ; and alfo 
 by J. and P. Knafton, on Ludgate Hill; T. Osborn and Co. in Grafs-Inn; T. and T. Lokcman, 
 in Pater --nofier Ro<w ; R. and J. Dodsl*y, in Pall- Mall; W. MeadoWs, in Corn bill >; W. Ow^en, at 
 Temple-Bar ; J. Swan, near Northumberland- lloufe in the Strand; F.Noble, in King's -Street, Convent- 
 Garden; and J. Noble, in St. Martins Court. At Cambridge, by W. ThurlbourK; at Oxford, 
 by J. Gresn, Engraver ; at Norwich, by J. Glebh ; and at I?swigh, by W. Craighton. 
 
 Mcccir, 
 
T O 
 
 Mr. HOGARTH. 
 
 S I R, 
 
 IF your extenfive Knowledge and Genius in 
 the Art of Painting did not entitle you 
 to a Dedication of the following Theory 
 of Pe rsp EC t i v E, the great Obligations which 
 I am under for your Friendfhip and Favour, 
 would claim not only this, but every other 
 Token of my Gratitude and AfFe&ion. But 
 this Work in a peculiar Manner has a Right to 
 your Patronage and Protection, as it was You 
 who firft encouraged me to write upon the Sub- 
 ject: And if it has any Merit, the Publick, in 
 a great Meafure, are obliged to you for it. 
 
 I fhall not follow the common Method of 
 Dedicators, by attempting a Panegyrick upon 
 
 your 
 
DEDICATION. 
 
 your amiable Qualifications j which might ap- 
 pear like Flattery, and offend your Modefty : 
 I fhall only beg Leave to fay, that your own 
 inimitable Performances are greater Inftances of 
 your Genius in the Arts of Defign, your Know- 
 ledge of the Human Paffions, and your Con- 
 tempt of Vice and Folly, than it is in my Power 
 to exprefs ; and that, 
 
 Jam, SIR, 
 
 With the greateft Efteern and Gratitude, 
 Tour mofi obliged. 
 
 And obedient Servant, 
 
 JOSHUA KIRBY. 
 
A LIST of Subscribers. 
 
 N. B. Thofe ivhofe Names are 
 Academy of Painting, Sculptur 
 
 A 
 
 QIR Thomas Allen, Bart: 
 Nathaniel Acton, Efq; 
 Mr. George Adams, Mathematical- 
 
 Iriftrument- Maker 
 Mr. David Adkinfon 
 Mr. William Aldrich 
 *Mr. Allen, Painter 
 Mr. Richard Allick, Painter 
 Mr. Henry Anderfon 
 Mr. Thomas Ailing 
 Mr. H. Atkinfon 
 Mr. James Audley. 
 
 B 
 
 Right Hon. Earl of Bute 
 John Bagnall, Efq; 
 William Baintun, Efq; 
 Mr. Andrew Baldrey, Painter 
 Mr. John Banks 
 Mr. T. Bardwell, Painter 
 Miles Barne, Efq;. 
 Mr. John Barnard, Ship-Builder 
 Mr. Chriftopher Barry 
 *Mr. James Bafire, Engraver 
 Mr. R. Bateman, Painter 
 Mr. W. Bathurft 
 John Battie, Efq; 
 Mr. John Bayley, Engraver 
 The Rev. Mr. Charles Beaumont 
 Mr. James Bencraft 
 Mr. John Bennet, Mathematical- Ih- 
 
 ftrument- Maker 
 The Rev. Mr. Ralph Blois 
 Mr. William Blois 
 
 *Mr. Jacob Bonneau, Drawing- Matter, 
 
 6 Books- 
 Boteldale Book-Club 
 Mr. Thomas Bowell 
 Mr. George Bowling, Mafon 
 Mr. Edward Bowman 
 The Rev. P. Bracebridge, D. D. 
 Mr. Bernard Broadbelt 
 Mr. Richard Brompton,, Painter 
 Philip Broke, Efqj 
 
 ma^k'd thus * are Members of the 
 e, Architecture, &c. in London. 
 
 Rev. Mr. John Broke 
 Mr. Wm. Browne, Architect 
 Mr. John Browne 
 Mr. Jofeph Browne 
 Mr. Daniel Brown a 
 Mr. Nathaniel Bucke 
 Mr. John Burrel 
 James Burrough, Efq; F. R. S. 
 Mr. William Butcher, Engraver 
 Robert Butler, Efq; 
 The Rev. Mr. Robert Buxton 
 Mrs. Ann Buxton 
 C 
 
 Turner Calvert,. Efq; 
 Rev. Mr. Richard Canning 
 Mr.. William Caftleton 
 The Rev. Mr. Philip Carter 
 The Rev. Mr. Samuel Carter 
 *Mr. Thomas Carter, Statuary 
 *Mr. Charles Catton, Painter 
 *Mr. Wm. Gazal, Drawing-Matter 
 Richard Charlton, Efq; 
 *Mr. Charles Cheere 
 The Rev. Mr. Chriftian 
 The Rev. Mr. Church 
 Mr. George Church, Bricklayer 
 John Clarke,. Efq; 
 Mr. William Clarke 
 Mr. William Clarke 
 Mr. Samuel Cleverly 
 The Rev. Mr. Henry Clofc 
 Mr. Robert Clowes 
 Mr. Butler Clowes, Engraver 
 The Rev. Mr. John Clubb 
 *Mr. Francis Coates, Painter 
 Thomas Coggefhall, Efq; 
 *Mr. William Collins, Statuary 
 Mr. William Collingwood 
 Mr. John Cooke 
 Mr. Jofeph Cooper, jun. Painter 
 Mr. John Cooper, jun. 
 Mr. T. Coopley 
 The Rev. Mr. Daniel Copland 
 Mr, Henry Copland, Engraver 
 
Mr. John Cornifh, Painter 
 Mr. William Craighton 
 Mr. -Edward Crane, Painter 
 Mr. Richard Crawley 
 Philip Crefpigny, Efq; 
 
 D 
 
 The Right Hon. Earl of Dyfart 
 
 John Dade, Efq; 
 
 Dr. William Dade 
 
 *Mr. Richard Dalton, Painter 
 
 Sir Francis Dafliwood, Bart. 
 
 Mr. Eleazer Davy 
 
 James Dawkins, Efq; 
 
 Henry Dawfon, L. L. D. 
 
 Mr. Richard Dawfon, Painter 
 
 Mr. R. Deard 
 
 Peter Del me, Efq-, 
 
 Robert Del ton, Efq; 
 
 Mr. Arthur Devis, Painter 
 
 Mr Robert Dingley 
 
 Mr. P Dixon 
 
 *Mr. John Donowell, Architect 
 Mr. F. Drummond 
 
 E 
 
 Theodore Eccleftone, Efq; 
 
 *Mr. John Eccardt, Painter 
 
 Milefon Edgar, Eiq; 
 
 Robert Edgar, Efq; 
 
 Mr. G. L. Edwards 
 
 Mr. George Elliott 
 
 Mr. Robert Ellifon 
 
 Mr. William Elfden, Carver 
 
 Mr. J. Etty 
 
 Mr. T. Evans. 
 
 F 
 
 Sir Gordel Firebrace, Bart. 
 
 Mr. Andrew Fergulon 
 
 Mr. Jofeph Finley 
 
 Mr. Thomas Finley, Architect 
 
 Mr. Zachariah Fifke, Painter 
 
 Mr. William Fokard 
 
 Mr. Ingham Fofter 
 
 The Rev. Mr. Richard Fowller 
 
 Mr. George Fowley 
 
 Mr. Richard Francklin 
 
 Mr. J. Francklin 
 
 Mr. William Franks 
 
 Mr. M. Freak« 
 
 ( a ) 
 
 Cha. Frederick, Efq; Surveyor General 
 of his Majefty's Board of Ordnance. 
 G 
 
 Sir Thomas Gooch, Bart. 
 *Mr. Thomas Gainfborough, Painter 
 Mr. Thomas Gardner 
 Mr. James Gillingham 
 Mr. G. Glynn 
 Mr. John Gorham 
 Mr. John Gravenor 
 The Hon. Col. Gray 
 Mr. James Green, Engraver 
 *Mr. Charles Grignion, Engraver 
 Jofeph Grove, Efq*, 
 *Mr. John Gwyn, Architect. 
 H 
 
 Mr. Hailftone 
 
 Mr. Gavin Hamilton, Painter 
 Mr. Hamilton, Architect 
 Mr. William Hammond 
 Mr. William Hammond 
 Mr. Hammond, Painter 
 Mr. John Hammond 
 Mr. Hannan, Painter 
 Robert Harland, Efq; 
 Mr. Jofiah Harris 
 Mr. Can ham Hart 
 Mr. William Havard 
 *Mr. Franeis Hay man, Painter 
 *Mr. Richard Hayward, Statuary- 
 Mr. Nathaniel Hedges 
 Mr. Thomas Henderly 
 The Hon. Nicholas Herbert, Efq; 
 The Rev. Mr. Thomas Hewett 
 Mr. Jofhua Highmore, Painter 
 Mr. Edward High moor 
 Mr. John Hingefton 
 The Rev. Mr. Robert Hingefton 
 The Rev. Mr. Peter Hingefton 
 Benjamin Hoadley, M. D. 
 Mr. William Hoare, Painter 
 *Mr. Hogarth, Painter, 6 Books 
 Rowland Holt, Efq; 
 *Mr. Nathaniel Hone, Painter 
 Mr. Peter Houltum, Painter 
 The Rev. Mr. Henry Hubbard, FeWow 
 of Emmanuel College, Cambridge* 
 Mr. Thomas Hudfon, Painter 
 Mr, Hunt, 
 
( ifl ) 
 
 Mr. Samuel Jacomb 
 Mr. Jeffreys 
 Mr. R. Ingham 
 Mr. Thomas Jolly 
 Mr. Owen Jones 
 Henry Ifaacs, Efq; 
 Mr. Richard Jupp, jun. 
 Mr. Bartholomew Juftinier 
 K 
 
 Sir John Kemp, Bart. 
 
 *Mr. Wm. Keable, Painter, 3 Books 
 
 Samuel Kent, Efq; 
 
 Mr. Samuel Kilderbee 
 
 Mr. Dover Kilderbee 
 
 Mr. William Kirby 
 
 Mr. Knapton, Painter. 
 
 L 
 
 Sir James Lowther, Bart. 
 
 Sir Richard Lloyd, Knt. 
 
 Mr. George Lambert, Painter 
 
 Mr. J. P. Lamborn 
 
 Mrs. Lancafter 
 
 Mr. Robert Lane 
 
 Mr. Robert Larwood 
 
 Robert Lcman, Efq; 
 
 Offley Lewis, Efq; 
 
 Mr. William Leflie 
 
 Library of Catherine Hall, Cambridge 
 
 _ CorpusChrifti College, ditto 
 
 ■ Emmanuel College, ditto 
 
 ■ Jefus College, ditto 
 
 — St. Peter's College, ditto 
 
 Trinity College, ditto 
 
 Mr. Littleton 
 
 Richard Savage Lloyd, Efq; 
 
 The Rev. Roger Long, D. D. Matter 
 of Pembroke Hall,Camb. Lowndes's 
 Profefifor of Aftronomy, and F.R.S. 
 
 Mr. J. Lovatt 
 
 William Lynch, Efq; 
 
 M 
 
 Right Hon. Lord Vifcount Middleton 
 *Mr. Robert Maberly, Painter 
 *Mr. James M'Ardell, Engraver, 
 
 2 Books 
 *Mr. Thomas Major, 2 Books 
 Mr. Jonathan Mallet 
 Mr. G. Manning 
 
 Mr. John Manning 
 Mr. Marquand, Architect 
 Mr. Marquois 
 Mr. Marfden 
 
 Mr. Thomas Martin, F. R. S. 
 
 Mr. Jofeph Martyr 
 
 The Rev. Charles Mafon,D.D. Senior 
 Fellow of Trinity Coll. and Wood- 
 wardian Lecturer of the Univerfity 
 of Cambridge 
 
 The Rev. Mr. Mafterman 
 
 Mr. John Meadowe 
 
 John Middlemore, Efq; 
 
 Mr. Charles Middleton, Chafer 
 
 *Mr. J. S. Miiller, Engraver 
 
 Mr. T. Miller 
 
 Robert Milner, Efq; 
 
 Montague, Efq; 
 
 Mr. T. Moore. 
 
 Mr. James Morris 
 
 Mr. Benjamin Morris 
 
 *Mr. G. M. Mofer, Chafer, 2 Books 
 
 Mr. Thomas Mulliner 
 
 Hutchenfon Mure, Efq; 
 N 
 
 Right Rev. Lord Bilhop of Norwich 
 Hon. Richard Savage NafTau, Efq; 
 *Mr. F. M. Newton, Painter 
 Mr. Nichols 
 Mr. Edward Nickfon 
 Mr. Robert Stiles Norman 
 O 
 
 Robert Onebye, Efq; 
 Mr. John Oldfield, Writing-Matter 
 Mr. Thomas Olborne, Bookfcller. 
 P 
 
 Mr. Thomas Paddington 
 
 *Mr. James Paine, Architect 
 
 Mr. John Palgrave 
 
 Mr. Thomas Palmer, Plaitterer 
 
 Mr. John Parlons 
 
 Mr. Robert Parfons, Carver 
 
 Mr. Francis Patton, Engraver 
 
 Mr. Thomas Patty, Carver 
 
 Mr. Pennie, Painter 
 
 Mr. R. Pennifon 
 
 Mr. Francis Perry 
 
 Mr. F. Peters 
 
 Mr, A. Pond, F.R.S. 
 
Mr. Peter Pullyn 
 
 Mr. Robert Pyle, Painter 
 
 His Grace the Duke of Qaeenfbury 
 R 
 
 His Grace the Duke of Rutland 
 Sir John Rous, Bart. 
 *Mr. Benjamin Radeliffe 
 Humphry Rant, Efq; 
 Mr. John Redgrave, Architect 
 The Rev. Mr. William Reeve 
 Mr. E. Reeves, Carver 
 *Mr. Jofhua Renolds, Painter 
 Mr. Nicholas Revett 
 Mr. William Richards 
 Mr. Richards, Painter 
 *Mr. Peter Roberts, Chafer 
 *Mr. Edward Rooker, Engpaver 
 Mr J. Rofe, PlaiuW 
 *Mr. F. L. Roubiliac, Statuary 
 Capt. Robert Rowning 
 The Rev. Mr. Tobias Ruftat. 
 S 
 
 Mr. Thomas Sanby, Draughtfman to 
 his Royal Highnefs the Duke of" 
 Cumberland 
 
 Mr. P. Sanby 
 
 *M ,i . Samuel Scolt, Painter 
 
 Mr. Scullard 
 
 Mr. Seldon 
 
 Mr. Seton, Engraver 
 
 *Mr. Seton, jun. 
 
 *John Shackclton,Efq; Principal Pain- 
 ter to his Majelty 
 John Sheppard, Efq; 
 Mr. Robert Sherman 
 Mr. George Silverfide 
 Mr. John Simmons, Painter 
 Mr. John Simpfon 
 Mr. Thomas Singleton, Carver 
 Mr. John Slarke, Painter 
 Mr. Henry Smart 
 The Rev. Mr. Juftice Smith 
 Robert Sparrow, Efq; 
 Thomas Staunton, Elqj 
 Mr. Stuart, Painter 
 Mr, Stevens 
 
 Mr. Stevens ^ 
 Mr. Luke Sullivan, Engraver, 4 Booki 
 Mr. John Swan, Bookfeller, 6 Books. 
 T 
 
 The Right Hon. Lord Talbot 
 Maj®r Taylor 
 
 The Hon. Horace Townfhend 
 Philip Thicknqfs,- Efq; 
 Michael Turkic, Efq; 
 Mr. R. Trevor 
 Mr. Samuel Trcw 1 - 
 Mr. N. Tucker, Painter 
 
 : v ; 
 
 The Right Hon. Lord Villieres 
 
 The Hon. Ed ward Vernon, Elq; 
 
 E. Venn, M. D. 
 
 The Rev. Mr. Edward Ventris 
 
 Mrs. Sarah Un'ett c 
 
 Mr. Urell ' 
 
 Mr. T. Urlihg 
 
 W 
 
 •Mr. Sam. Wale, Painter, 4 Books 
 
 Mr. Charles Walf'ord 
 
 Mr. William Walford 
 
 *Mr. A. Walker, Engraver 
 
 John Warburton, Efq; Somerfet H?- 
 
 rald and F.R.'S. 
 Mr. Francis Warden, Plaifterer 
 Juftly Watfon, E(q; Engineer 
 Mr. Weecks 
 
 Weftern, Efq; 
 
 Mr. Wieland, Chafer 
 
 *JVIr. Benjamin Wilfon, Painter, and 
 
 F. R. S. 4 Books 
 Robert Wilfon, Efq-, Gray's-Inn 
 Mr. R. Wilfon 
 *The Rev. Mr. James Wills 
 Mr. Wolfe, Architect 
 Mr. Jofeph Wood, Engraver 
 Mr. John Woods 
 
 Mr. James Woridale, M after Painter 
 to his Majefty's Board of Ordnance 
 Mr. Thomas Wright 
 Wiiliam Wyndham, Efq; 
 Y 
 
 *Mr. Yeo, Engraver, 2 Books 
 Mr. R. Young. 
 
PREFACE. 
 
 * k HE many Treatifes already puhlijhed upon Perspective, 
 Ji may make it appear needlefs to augment the Number ; it is there- 
 fore necejfary to give the Reafons that induced me to undertake 
 fuch a JVork. 
 
 The Defign of the following Treatife, is, by exhibiting a New System 
 of Practical Perspective, to make this hitherto intricate, but 
 ufeful Art, eafy and familiar to every Capacity : And to drefs it in the 
 mofl fimple Garb ; that its Parts may be clearly feen^ and its whole De- 
 jtgn (f° f ar as it relates to Painting, &c.) eqfily under flood. For cer- 
 tain it is, that no Subject hath been treated in a worfe Manner than this, 
 notwithstanding the many Volumes which have been wrote upon it ; fome 
 purely Mathematical, and therefore unfit for the generality of Perjons 
 who are concerned in the Arts of Defign , and others wholly Mechani- 
 cal, made up of incoherent Schemes, unapplicable Examples, and fuch a 
 Confufion of unnecejfary Lines, as tend only to puzzle and difcourage 
 the Learner. My Intention, therefore, is, to fleer between the abflrufe 
 mathematical Reafoning of fome, and the tedious andfalfe Explications 
 of others and from thence to produce a Syftem of Perjpettive upon cer- 
 tain and fimple Principles, eafy to be underflood and applied to Prac- 
 tice. 
 
 This is a general Account of the following Work; which is the Pro- 
 duel of feveral Tears Study and Application : And how I have fucceeded 
 in the Attempt, is fubmitted to the Candour of every ingenuous Reader. 
 But that fuch an Undertaking was necejfary, is fufificiently teftified by 
 the many eminent Painters, and other curious Artifls, who perfuaded 
 me to projecute my Defign, and have generoufly encouraged the Publi- 
 cation of this Work. 
 
 I have intitled this Treatife Dr. Brook Taylor's Perspective, 
 &c. out of Gratitude to that ingenious Author, for furnijhing me with 
 Principles to build upon and becaufe his, though a very Jmall Pam- 
 
 A phlet, 
 
PREFACE. 
 
 phlet, is thought the mofi correct, concife, and comprehenjive Book upon 
 the Subject, i have not proceeded exactly in his Method ; for that was 
 not agreeable to my Plan : Nor have I explained his Proportions, 
 Theorems, &c. in a regular Manner Jince that alfo was inconfiflent 
 with the Order of my Work : But I have had Regard to his Principles 
 in general, fo as to make his Meaning more intelligible, and that kind 
 of Perfpective of more univerfal Ufe. This Book of Dr. Taylor's 
 was firfi publifhed in the 7 ear 1715, and was intitled Linear Per- 
 spective ; and in the Tear iyig he publifioed another fmall Tract, 
 which he called, New Principles of Linkar Pfrsplctive j and 
 which he intended as an Explanation of his firfi Treatife. But, not- 
 withfianding both thefe Treatifes are fo curious and ufeful, few have 
 been able to underfiand his Schemes j and when they have underfiood 
 them, have been as much puzzled in applying them to Practice. And 
 in the Tear 1738, Mr. Hamilton favoured the World with two 
 Volumes in Folio, intitled Stereography, which he has explained in 
 the Manner of Dr. Taylor, and which, though a very curious Work, 
 and worthy the Perufal of every good Mathematician, yet, I may ven- 
 ture to affirm, that but very few of thofe Perfons who are Students in 
 the Arts of Defign can comprehend it j and were they qualified with a 
 fufficient Stock of Mathematical Knowledge, it would take up more Time 
 than they either could, or would chuje to fpare. However, I mujl 
 frankly acknowledge, that I think it the befi Syftem of Mathematical 
 Projection * that ever was, or, perhaps, ever will be, made public j 
 and I Jhould be very ungenerous in not conf effing that it has been of great 
 Service to me in fever al Parts of my Work j and that I am indebted to 
 it for fome Things which 1 Jhould never have thought of had not 
 that ingenious Gentleman pointed them out to me ; and I hope, that this 
 publick Acknowledgment will prevent the Imputation of Plagiarijm, 
 and be a fufficient Satisfaction for the Liberties 'which I have taken 
 with his Work. 
 
 * Mathematical Projection comprehends all kinds of Projection whatfoever ; fuch 
 as the Projection of the Sphere, the Cylinder and its Sections, Conic Section* and the 
 
PREFACE. 
 
 iii 
 
 The Plan which I have proceeded upon in the Profecution of my 
 Dcfgn, is this : 1 have divided the whole Work into two Books j the 
 Jirft I have called A compleat System of Perspective, which 
 contains the Theory and its Application to Practice and the fecond, 
 The Practice of Perspective, which contains the practical Part 
 only. This Method of treating the Subject feemed to me the moft eligible ; 
 becaufe there are fome who do not like to take Things for granted, but 
 chooje to be convinced by Demonftration, and to have the Reafon of 
 Things explained upon certain Principles. For fuch, I intend the firjl 
 Lock ; and for others ;, who either want Time or Capacity to go regularly 
 through the theoretical Part, I have wrote the fecond Book ; that fuch 
 Per/ens may be enabled to determine the Appearance of all Kinds of 
 Objecls upon the Picture with the greatejl Eafe and Expedition : So 
 that the whole together (if I am fo happy as to have fucceeded in my 
 Attempt ) may be called a compleat Syjlem of PerJpecJive, fo far as it 
 relates to the Art of Paintings ($c. 
 
 In the firjl and fecond Chapters of the fir ft Book, I fuppofe my "Reader 
 a mere Novice , not enly in Perfpeclive, but in every Thing which it is 
 necejjary he Jhould know as previous thereto j and therefore I have begun 
 with an Explanation of Mathematical Inftr uments, and have Jloewn their 
 different Ujes ; after which I have explained a few Geometrical Defini- 
 tions and Propofitions, and from thence I have proceeded to /hew how to 
 defcribe (in a mechanical Manner) fuch Geometrical Figures as may 
 occur in the following Work. And becaufe PerJpecJive is an optical 
 Science, 1 have given fome Jhort Abjlracts from the moft eminent Wri- 
 ters upon Opticks j by which Means the unlearned Reader will have a 
 general Notion of tbe Eye and the Nature of Vifion, the Refection and 
 Refraction of the Rays of Light, and of the Caufe of Colours. I fay 
 the unlearned Reader, becaufe 1 do not prefume to give InftrutJions to 
 Perfons who are well acquainted with the Mathematicks or PhiloJophy 9 
 but only to fuch as are ignorant in thefe Matters, but are neverthelefs 
 defirous of feeing the Foundations upon which Perfpe&ive is built ; and 
 therefore, all that hath been hitherto advanced, may be omitted by the 
 learned Part of my Readers, as an imperfect Abfiraft of what they 
 ere infinitely better acquainted with than myfelf 
 
 A 2 
 
PREFACE. 
 
 The Third Chapter of the fame Book begins with an Introduction to 
 Perfpective, which 1 have endeavoured to explain in a familiar Manner ; 
 by fuch Objects as we are every Day converfant with ; and then 1 have 
 proceeded to the Theory >, which I have ranged under the following Heads, 
 viz, it Of Objects which are in Planes perpendicular to the Pic- 
 ture ; 2. Of Objects which are in Planes perpendicular to the 
 Ground ; And 3. Of Objects which are in Planes inclined to the 
 Ground ; becaufe there are no other Situations in which Objects can be 
 difpofed ; that is, they mujl be either perpendicular ; parallel, or inclined: 
 And every Example, which I have produced, is immediately applied to 
 Practice and by that means the Theory and Practice are fo clofely con- 
 nected, that they ferve to explain each other, and to fix both very Jlrongly 
 in the Memory. 
 
 And thus having explained the Theory and general Practice of Per- 
 fpettive, I have in the fourth Chapter of the fame Book, confidered t'e 
 different Kinds of Perfpective, viz. when the Picture is either perpen- 
 dicular, parallel, or inclined. The perpendicular PiSlure is what is 
 commonly made ufe of, and is placed perpendicular to the Ground ; the 
 parallel Picture is placed parallel to the Ground, (fuch as Ceilings, and 
 the like -,) and the inclined Picture, is fuppofed to be inclined to the 
 Ground, and is rather more curious than ufeful. I have next given an eafy 
 Method of determining the Reprefentation of any Objects upon Domes, 
 vaulted Roofs, and other uneven Surfaces, by means of Reticulation or 
 "Net-Work ; which is all that is contained in this Chapter. The fifth 
 Chapter of the fame Book contains the Perfpective of Shadows, both in 
 Theory and Practice and this I have treated in the very jame Manner 
 as I have done the Perfpective of Objects : So that what is contained in 
 this Chapter, is deducible from what hath been already advanced in the 
 third Chapter, and follows like fo many Corollaries from thofe general 
 Proportions . The fixth Chapter of the fame Book contains fame general 
 Inftruttions for choofing a proper Diflance for the Eye, &c. and the bad 
 Effects of viewing a Picture from any other than the true Point of Sight -> 
 and the feventh and lafl Chapter, is principally copied from Mr. Hamil- 
 ton, and contains an Explanation of Aerial Perfpective, the Chiara 
 Ofcura, and Keeping in Pictures.— And thus having gone through the The- 
 oretical 
 
PREFACE. 
 
 oretlcal Party I have proceeded to the fecond Book, which contains the 
 practical Part only ; in which 1 have obfcrved much the fame Order and 
 Method as in the Theory, and therefore that needs no farther Explanation. 
 
 The Figures I have made choice of to demonflrate and explain this 
 Syjlem of Perfpeftive, are not fet off with Ornaments, to attract the 
 Eye, but are done with Simplicity, to inform the Under/landing ; and 
 are fuch as every common Mechanick has clear and determinate Ideas of 
 and confequently of the mofl univerfal ZJfe : For the Square, the 
 Triangle, and the Circle, are not only the Foundation of mofl 
 geometrical Figures, but are aljo the fimple Materials of Shapes in 
 general, and of which regular Buildings, in particular, are always 
 compofed. And I may, without the leaft Arrogance, affirm, that, had 
 the fever al Writers upon PerfpecJive Jhewn how to find the Reprefenta- 
 tions of thofe three Figures only, upon different Planes, and in various 
 Situations, their Works would have been more intelligible, and of 
 much more Service, than they now are with fuch a Multiplicity of orna- 
 mental Schemes and unapplicable Examples. In jhort, the Principles of 
 Perfpeflive are few and fimple, and therefore to explain them by a vaji 
 Number of ornamental Figures, would ferve only to divert the Eye and 
 mijlead the Judgment, and to make that appear obfeure and difficulty 
 which in its own Nature is extremely clear and eafy. 
 
 And here it may not be improper to obferve, that the Learner is de- 
 fired to draw out every Figure as he proceeds, which will ferve to fix 
 them in his Memory, and to make their Explanations more eafy and fa- 
 miliar to him. It is a Method I have always praBifed myfelfi with 
 Succefs ; and therefore think it may be of Service to others : However, 
 thofe who by an extraordinary Capacity, can carry on a long Train of 
 Ideas together, and can recollect, compare, and combine them as they 
 pleafe, need not give themf elves this Trouble. 
 
 This is a general Account of the following Work : But before I quit 
 the Subjecl, I JJjall beg leave to fay Jbmething upon the Ujefulnefs of Per- 
 Jpeclive to every Per/on that is any. ways concerned in the Arts of Defign, 
 and to recommend the Study of it in particular to every Tyro in the Art 
 of Painting-, which I could wijk might put a Stop to that Ridicule and 
 
PREFACE. 
 
 Contempt with which it has been treated by a fort of People* who art 
 too ready to condemn a Branch of Science, which they have negleBed to 
 gain a jkfficient Knowledge of. Thefe Per fans bring to my Mind a Story 
 of Leonardo da Vinci, a famous Italian Painter who fjourijhed in the 
 latter End of the fourteenth Century *. He tells us, that a Friend of 
 his, named Boticello, had a peculiar Pique againfi Landfkips, thought 
 them much beneath his Application, and looked upon them in a mojl con- 
 temptible Light : But, fays Leonardo, the Reafon was, becaufe he 
 was a very forry Landfkip Painter : And our Author udds, that for 
 this Reafon his Merit in other Matters was the lefs regarded. 
 
 That PerfpeBive is an efential Requifite in a good Painter, is aitefled 
 by all our mojl eminent Artijis, and is moreover confirmed by almoft every 
 Author -f* who his wrote upon Painting ; my, the very c lerm Painting 
 implies PerfpeBive. For to draw a good PiBure is to draw the Repre- 
 fentation of Nature, as it appears to the R e ; and to dr ^w the Per- 
 fpeBive Reprcfentation of any ObjeB, is to draw the Reprefent Jtion of 
 that ObjeB as it appears to the Eye : Therefore the Terms Painting and 
 PerfpeBive feem to be fynonymous, though I know there is a critical Dif- 
 ference between the Words. Tet this will Jerve at leafl to jhew the near 
 Alliance between P inting and PerfpeBive ; that if the one doth not com- 
 prebend the other, PerfpeBive, howtver, may be faid to be the Bafis upon 
 which Painting is built ; and therefore he who attempts to paint a Pic- 
 ture without having a general Knowledge of it, will always wander in 
 the Mazes of Uncertainty, be JubjeB to the greatejl Errors, and his 
 Works, like thofe of Boticello, will be the lefs regarded. And what is 
 fi. id of the Ufefulnefs of PerfpeBive to Painters in particular, may be 
 applied to Artijis in general ; fucb as Engravers, A, chiteBs, Sta- 
 tuaries, Ch Jers, Carvers, &c. It will alfo be an entertaining Study to 
 any Gentlem. «, who has either a Tajle for Drawing, or is a Lover of 
 Painting ; as it will enable him to draw cut the Reprefent at ion of any 
 Building or \ roJptB, and to form a tolerable judgment of a PiBure 
 without any other AJiJlance. 1 would not be underjiood to mean, that a 
 
 * Vid. Leonardo da Vinci upon Painting, p. 31. 
 
 i Ibid. p. 29. Frefnoyt Poem upoa jramting, p. 19, v. 115. And Du Piht ttpon Paint, 
 ifig, Cb.ap. 18. 
 
 Perfm 
 
 \ 
 
PREFACE. 
 
 Perfon is always to follow the rigid Rules of Perffiecthe, for there are 
 fome Cafes in which it may be necejfary to deviate from them j but then 
 he muji do it with Mode fly, and for fome good Reafon, as we have 
 Jhewn in fome Parts of this Work. Nor would I be thought to defire 
 the Artijl to make life of Scales or Compafes upon all Occafions, and to 
 draw out every Line and Point to a Mathematical Exaclnefs ; no, the 
 Defgn of this Work is quite the Reverfe ; it is to teach the general 
 Rules of Perfpective, and to enforce the Practice of it by eafy and 
 felf-evident Principles ; to afjifl the Judgment, and to direSl the Hand, 
 and not to perplex, either by unnecejjary Lines or dry Theorems. Upon 
 the whole : He that has a true Genius, and will take Pains to learn the 
 Principles delivered in this Treatife, will be taught to See Objects with 
 fuch Exaclnefs, and his Judgment will be founded upon fuch folid Prin- 
 ciples, that he will be enabled to draw out any Reprefentation with more 
 Eafe, and with much more Correclnefs, than the greatejl Genius who is 
 ignorant of Perfpettive, or he who defpifes the Rules of fuch a necejfary 
 Art. 
 
 Con- 
 
CONTENTS. 
 
 Compleat System of Perspective. 
 
 book jr. 
 
 Chap. I. 
 Containing Matters previous to Perfpe&ive. 
 
 SECT. I. 
 
 <y Inftruments ufed in Drawing, ■ ■ , ■ Page i 
 
 SECT. II. 
 
 Geometrical Definitions and Proportions, > — 2 
 
 SECT. III. 
 Practical Geometry, - . - . 
 
 Chap, II. 
 Some Optical Obfervations. 
 
 SECT. I. 
 
 Of the Eye and the Nature Vifion, - 8 
 
 SECT. II. 
 
 Of the Reflection and Refra&ion of the Rays /Light, — 15 
 
 SECT. III. 
 Of the Caufe of Colours, . r 8 
 
 [ a ] Chap. 
 
CONTENTS. 
 
 Chap. III. 
 The Theory of Perfpe&ive. 
 
 SECT. I. 
 
 Introduction to ^Theory c/Terfpective, .. Page 20 
 
 SECT. II. 
 
 Definitions, Theorems, &c. » ■ 2 6 
 
 SECT. III. 
 
 Of Objects that are in Planes perpendicular to the Picture, 30 
 
 SECT. IV. 
 
 Of Objects that are in Planes perpendicular to the Ground, 33 
 
 SECT. V. 
 
 Of Objects that are in Planes inclined to the Ground, ■ 37 
 
 Chap. IV. 
 Of parallel inclined Pictures. 
 sect. 1. 
 
 Of the Parallel Picture, . ■ 43 
 
 SECT. II. 
 
 Of the Inclined Picture, ■ ■ 47 
 
 SECT. III. 
 Of Vaulted Roofs, Domes, &c. ; 51 
 
 Chap. 
 
CONTENTS. 
 
 Chap. V. 
 The Perfpe&ive of Shadows. » 
 
 Introduction to the Theory of Shadows, ■ Page 56 
 
 SECT. I. 
 
 Of Shadows projected by the Sun, — , ■ 
 
 General Rules applied to Practice, ■ „■ 63 
 
 SECT. II. 
 
 Of Shadows projected by the Candle, &c. 6 5 
 
 Chap. VL 
 Containing feveral eflential Requifites. 
 
 1. Of the Diftance of the Eye, ■ __. , 66 
 
 2. Of the Height of the Eye, ■ , 67 
 
 3. The Confequence of viewing a Picture from any other 1 
 than the true Point of Sight, t ° 
 
 4. Of the Size of the Picture, — ibid. 
 
 5. Some Confiderations upon the Appearance of round or cy- 7 , 
 lindrical Objects upon the Picture, J 9 
 
 Chap. VIL 
 
 _ Of Aerial Perfpective, the Chiara Ofcuro, and Keeping in I 
 Pictures, . * 74 
 
 [ a 2 ] 
 
 BOOK 
 
CONTENTS. 
 
 BOOK II. 
 The Practice of Perspective. 
 
 Chap- L 
 
 Introduction to the Practice of Perfpective, ■ Page 5 
 
 Chap. II. 
 
 SECT. L 
 To Prepare the Picture, GV. 
 r. Of the Size of the Picture, — — 
 
 2. For the Height of the Eye, — — ■ ibid. 
 
 3. For the Diflance of the Eye, — « . 10 
 
 SECT. II. 
 
 To find the Reprefentation of Objefls which He flat upon the Ground, 
 fuch as the S qua r e, the Tr i a n g l e, and the Circle, &c 3 
 
 1 . Of the Projection of a Point ; — by two Methods ; 1 1 
 
 2. Of a Line that is perpendicular to the Picture j by two Methods, 1 2 
 
 3. Of a Line that is parallel to the Picture; — by two Methods , ib. 
 
 4. Of a Line that is oblique with the Picture j — -by two Methods, 13 
 
 5. Of an Equilateral Triangle, when one of its Sides is pa- I 
 rallel to the Picture ;— by three Methods, — — ' 
 
 6. Of an Equilateral Triangle, when all its Sides are \ 
 oblique with the Picture ; — by three Methods, . : ' 1 $ 
 
 7. Of a Geometrical Square, having fome of its Sides parallel I 
 
 to the Picture j — by three Methods, * lb * 
 
 8. Of a Geometrical Square, having fome of its Sides oblique ? , 
 with the Picture; — by three Methods, ■ • « * 
 
 9. To find the Reprefe?jtation of a Square of any determinate \ 
 Width, * l 7 
 
 10. Of a regular Hexagon ; — by two Methods, 18 
 
 11. Of Circles — by three Methods, ■ ■ ■ ■ 19 
 
 SECT. 
 
CONTENTS. 
 
 xiii 
 
 SECT. III. 
 
 To find the Reprefentation of objects that are perpendicular to the 
 
 Ground. 
 
 1. Of a Plane perpendicular to the Picture, ■ Page 21 
 
 2. Of Planes parallel to the Picture, 22 
 
 3. Of Planes oblique with the Picture, — 23 
 
 4. 5, Of T riangular folid Figures, 24 
 
 6, 7. Of Cubes and Parallelopipeds, 1 25 
 
 8. Of Hexangular Figures, 27 
 
 9. Of Octangular Figures, ■ ib. 
 
 10. (^Cylindrical, or round Objects, > 28 
 
 SECT. IV. 
 
 To determine the Reprefentation of Objects that are inclined to the 
 
 Ground. 
 
 1. Of a Square; — by three Methods, ■ ■ 29 
 
 2. To find the vanifhing Lines and vaniftring Points of a Roof, 3 1 
 
 3. To find the Appearance of a Square by the vanijhing Points^ ^ 
 of its Diagonals ; — by three Methods. — 
 
 4. General Rules for Roofs, and other inclined Objects, 1 H> 
 
 5. General Rules for the Projection of the Regular Solids, &c. 36 
 
 viz, 
 
 — of Cubes refting upon one Edge only, ■ *b* 
 
 of a Cube /landing upon one Corner only, 37 
 
 »■ of a regular T ?trahedron y 39 
 
 ■of a Canted Cube, 4° 
 
 6. Of the Double Crofs, 4 1 
 
 Chap. III. 
 
 The Practice ofPerfpe&ive abbreviated. 
 
 sect. 1. , 
 
 General Rules, &c. — 4 2 
 
 1. When any vanijhing Point is out of the Picture, to draw^ ^ 
 a Line that jhall tend to that Points 1 : 
 
 2. To 
 
CONTENTS. 
 
 2. To make one Line equal to another Line given. Page 43 
 
 3. To cut off a Line in any given Proportion, _1 , ^ 
 
 4. To prove if an ObjeB be properly di mint/bed, /: 
 
 5. To fid the Length of any Reprefentation by Calculation only, 4 
 
 6. To find the Diftance of any Picture by means of the va-l . 
 nifiing Points of a Square, ■ , - 3 
 
 7. To find the vanifhing Lines and vanifhing Points of in- 1 
 clined Planes, together with their Centers and Diftances, — _ * ib - 
 
 £ See alio 
 
 8. General Rules for all Kinds of vanifhing Lines, * ° P * 11 
 
 9. A Method taken from Marolois, ufeful in fome Cafes, - 47 
 
 SECT. IL 
 
 The foregoing Rules more particularly applied to common Practice 
 
 1. Inputting a Tufcan Pedeftal into PerfpeBive, .3 
 
 ' . 2 ; -Circular Mouldings, the Tufcan, Bafe, WCa- ? 
 
 pital, and the Corinthian Capital,. . . j 51 
 
 3- Columns in general, 
 
 4- ~ Stairs in general, , 
 
 5. An Arch and Pediment, . H 
 
 6' Houfes in general, . . r, 
 
 7. The infide of a Room, ~~ * 
 
 Chap. IV. 
 Horizontal Perlpe&ive. 
 
 ri 1 \j t °Jj aW r a ^ ece P tion u P° n * Cieling, which Jhatt appear I 
 like the Sides of a Room continued upward?, 1 PP i 
 
 2. To put two Cylinders into PerfpeBive, . , 
 
 3. O/Pilafters, J lb - 
 
 60 
 
 4. Of oblique Objects, ^ 
 
 5- Of Arches, lb * 
 
 6. Of Windows, m . lb * 
 
 7. Of Mouldings, 
 
 62 
 ib. 
 
 8 > 9> Of 
 
CONTENTS. 
 
 8, 9. QfBafes and Capitals, 1 ■ Page 62 
 
 10. Of the Human Figure, • 63 
 
 1 1. Of the Reprefent 'attorn of Domes upon flat Cielings, &c. — 65 
 
 Chap- V- 
 
 The Perfpe£tive of Shadows. 
 sect. 1. 
 
 Of Shadows projected by the Sun. — — — 66 
 
 1. When the Light comes in Planes parallel to the Picture, 67 
 
 2. When the Light conies from behind the PiBure, 68 
 
 3 . When the Light comes from before the Picture, ib. 
 
 From whence are deduced the Jollowing general Examples, viz. 
 
 To find the Shadow of a perpendicular Object, when it is l , 
 cafl upon the Ground only, ■ * " 
 
 To find the Shadow of an oblique Object, when it is caft I 
 upon the Ground only, 
 
 To find the Shadow of a perpendicular Object, when it is 
 caft upon an inclined Plane, \ ' 
 
 To find the Shadow of an inclined Object upon a perpendi- j 
 cular Plane, 
 
 To find the Shadows of perpendicular Objects, when they are l ib. 
 cafl upon Planes that are every Way oblique with the Picture, — * j l 
 
 Of Shadows projected by the Candle, &c. ib, 
 
 SECT. II. 
 
 Some Confiderations upon the Degree and Colour of Shadows, ? 
 &c. and of Reflections in Water, — — — — 3 - 
 
 1. Oj the Colours and Degrees of Shadows. — ib. 
 
 2. Of Reflections in Water, 74 
 
 Chap- VI. 
 
 Of ScENOGRAPHY J Or S C E N E-P AINTING, 
 
 76 
 
 CHAP. 
 
xvi 
 
 CONTENTS. 
 
 Chap. VII. 
 
 Several Methods of Perfpe&ive, by the moft 
 
 eminent Authors- 
 
 1. Vignola* Method, — 
 
 2. Marolois'j Method, - 
 
 3 . Vredeman Friefe'j Method, 
 
 4. 'The Jefuit'j Method, 
 
 5. Andrea Pozzo'j Method, 
 
 APPENDIX. 
 
 Page 82 
 
 ib. 
 
 83 
 
 ib. 
 
 ib. 
 
 A Com- 
 
Compleat SYSTEM of PERSPECTIVE. 
 
 BOOK I. CHAP. I. 
 
 ♦ ♦ » * » ♦ ■> * * 1' * * * * * * * * * * * * * * * » ***** 
 
 Of Inftruments ufed in Drawingy Geometrical 
 Definitions and Propofitions 9 and Practical 
 Geometry. 
 
 S E C T. I. 
 
 Of Instruments ufed in Drawing. 
 
 THE Inftruments necenary in Drawing are as follows, 
 viz* 
 i. A Tee-Square. Fig. i. 
 Mi A Parallel Ruler. Fig. 2. 
 
 3. A Drawing-Board ; which is a fmooth Board made 
 
 exactly fquare at the Corners. 
 
 4. A Sector. Fig. 3. 
 
 5. A Pair of Compares and a Drawing-Pen. 
 
 6. A Semi-Circle, or Protractor. Fig. 4. This Instru- 
 
 ment is half a Circle, divided into 180 equal 
 Parts, which are called Degrees. 
 Thefe are all the necenary Inftruments in Drawing, and may be 
 had at any Mathematical-Inftrument Maker's *. 
 
 In regard to their different Ufes : They are almoft univerfal ; but 
 I fhall only confider them as applied to particular Purpofes ; and 
 firft, of the Tee-Square and Drawing-Board. 
 
 1! Let it be required to draw one Line CD, perpendicular to Fig. 5; 
 
 another Line AB. . 
 
 After having fixed a Piece of Paper upon the Drawing-Board, 
 apply the fquare Arm ED of the Tee-Square to the Side of the 
 Board, and draw the Line AB; then lay it in the fame Manner 
 asainft the Top or Bottom cf the Board, and draw a Line touch- 
 ing the other Line in the given Point. Thus, let D be the given 
 
 * Mr. John Bennet, at the Globe in Crown Court, between St. Ann's and Golden Square, 
 London, has had particular Directions from the Author, for making a very firaplc and uieful 
 Cafe of Inftruments, fit for the above Purpofe. © • <. 
 
 Book I. A r°int 5 
 
i 
 
 Of Instruments, 
 
 Point; then draw CD, and the Line CD will be perpendicular to 
 the Line AB. — And if an oblique Line AC k wanted; lay the 
 Arm AB of the Square, which turns upon a Screw C„ againft 
 the Edge of the Board, and move the Ruler backwards and for- 
 wards, 'till you have got it to the Inclination you want. 
 Tig, 2. 2. The Parallel Ruler is to be ufed when we would work 
 upon a loofe Paper, without ufing a Drawing-Board : Thus, let 
 it be required to draw one Line CD, Fig. 8, parallel to a given 
 Line AB ; and let C, or D, be the Diftance it is to be from AB : 
 Lay the Edge AB of the Ruler, to the given Line AB, and keep 
 the Limb ab, fixed -> then move the other Limb cd, to the Diftance 
 propofed, and draw a Line, as CD, which will be parallel to AB. 
 So that having given only one Line, and erected a Perpendicular 
 thereon, we may draw any Number of parallel Lines, or Perpen- 
 diculars to them ; only obferving to fet off the exact Diftance of 
 every Line by a Prick of the Compares, like C or D. 
 
 3. The Sector is made of two brafs Rulers, AB, AC, artifici- 
 ally fixed upon a Center A: This Inftrument is ufually filled 
 with a great Number of different Scales, which, tho' very ufeful 
 in many Parts of the Mathematicks, are neverthelefs foreign to 
 our Purpofe; and therefore, I fhall confider it only as having 
 what is called a Line of Lines on one Side, and a Line of Poly- 
 gons on the other ; which different Scales are exprefled upon the 
 Sector, by the Letters LL and PP, as in the Figure; LL ftands 
 for the Line of Lines, and PP for that of Polygons. The Line 
 of Lines ferves as an univerfal Scale for dividing any Line into 
 equal Parts, or into any given Proportion ; for inftanee, divide 
 F fr H- the Line AB into fix equal Parts : Take the Length of the Line 
 in your Compafles, and fet one Leg of them in the Point 6, 
 upon the Line of Lines ; then open the Sector 'till the other Leg of 
 the Companies coincides with the Point 6, which is on the Line 
 of Lines upon the other Limb of the Inftrument ; in this Pofition 
 keep the Sector fixed, till you have taken the Diftance from 1 
 to 1 ; which Diftance will be one fixth Part of the Line given. 
 And in the fame Manner, a Line may be divided into any Number 
 of equal Parts, even though they mould exceed the Numbers upon 
 the Sector; fuppofe, for inftanee, it was required to divide a Line 
 into 24 equal Parts; then fet the Length of the Line from 12 to 
 12 and divide the i-i2th Part into two Parts, which will anfwer 
 the Purpofe. 
 
 The 
 
Of Instruments, * 
 
 The Line of Polygons is called fo from its Ufe ; which is, to di- 
 vide a Circle into any Number of Parts, as in Fig. 27. which Fi- 
 gures are called by the general Name of Polygons ; and the Method 
 of ufmg this Scale is extremely eafy. For having firft defcribed a 
 Circle, take the Radius (that is half its Diameter) and fet it upon 
 thefe Lines from 6 to 6 in this Pofition keep the Sector fixed, 
 and you will have a Scale for dividing any Circle of that Radius 
 into any Number of equal Parts ; for if you want a feven-fided 
 Figure, (or Heptagon) take the Diftance from 7 to 7 ; if an eight- 
 iided Figure, (or Octagon) take the Diftance from 8 to 8, and fo on. 
 
 4. The laft Inftrument is the Semi-Circle or Protractor, Fig. 4; 
 which is ufed in drawing all kinds of given Angles, and in the 
 following Manner. 
 
 Let it be required to make a right Angle* CAB, from the Fig. m 
 
 Point A, upon the given right Line AB. Lay the lower Side 
 
 BC of the Inftrument, exactly even with the Line AB, and in fuch 
 a manner, that the Point, or Center A, will coincide exactly with 
 the Point A upon the Line AB ; then make a prick at 90, ana* 
 draw AC, and the thing propofed is done. — And after the fame 
 Manner any other given Angle may be drawn, which a little Ex- 
 perience will make much more eafy than Words can do. 
 
 SECT. II. 
 
 {Jeo metric At Definitions and Propositions, princU 
 pally from SimpfonV md Pardie'j Geometry. 
 
 t. A N Angle is the Inclination of two right, or ftraight Lines, Fig. $; 
 ± \ AD, CD, meeting each other in a Point, as D ; and the 
 middle Letter D always denotes the Angle. 
 
 z. When one right Line CD, falling upon another AB, makes 
 the Angles on both Sides equal, thofe Angles are called right An- 
 gles, and the Line CD is faid to be perpendicular to AB ; and if 
 any Line AC be drawn from a Point A in one Line, to any Point 
 C in the other, the Line fo drawn is called the Hypothenufe. 
 
 3. An acute Angle BDE, is lefs than a right Angle BDC 
 
 4. An obtufe Angle ADE is that which is greater than a right 
 Angle ADC, 
 
 5. Parallel Lines are fuch as are equally diftant from each Fig. $ 
 qther, as AB, CD, 
 
 * For right Angle, fet Geometrical Definitions m tte next Seftion. 
 
 A 2 & A 
 
g Geometrical Definition's, Gff. 
 
 6 A plane Figure is that which lies evenly between its Bounds 
 or Extremes ; thus any fmooth Surface is a plane Surface, and is 
 therefore called a Plane. . 
 
 Fig. 9 , io. 7. All plane Figures bounded by three right Lines, AB, AC, BC, 
 are called Triangles. . . r _ , c . , 
 
 Kg.,0. 8. An equilateral Triangle ABC, is that whofe Bounds or Sides 
 
 Fie 11 are o a Lerypiane Figure ABCD, bounded by four right Lines, is 
 called a Quadrilateral) and if its Sides and Angles are equal, it is 
 
 called a Square. „ . 
 
 10. Any quadrilateral Figure, whofe oppofite Sides are parallel, 
 but not equal, is called a Parallelogram-. 
 
 Fig. 12. 1 1 A rieht Line is faid to be perpendicular to a Plane when it 
 flands on it at right Angles y thus the right Line EF is perpendi- 
 cular to the Plane ABCD, when it ftands like a Pillar upon the 
 Ground, and is inclined no more to any one Side of the Plane* 
 than to the other. , , , - 
 
 Fig. ,5. 12. One Plane ABCD, is right and perpendicular to another 
 EF, when, like a well-made Wall, it inclines and leans on one Side 
 no more than it does on the other. 
 
 Fig. 9 ,!o. 13. Two. right Lines, if they meet fo as to cut or crofs each 
 other, are in the fame Plane > wherefore all the Angles, A B, C, 
 and Sides AB, BC, CA, of every Triangle, are in the fame Plane. 
 
 Fig. 14, 14. If two Planes ABC, EFGH, cut or mterfedl one another 
 they mall do it in a right Line EF, which Line is caned their 
 
 C Tc m if a'ShtLine FG, be perpendicular to two right Lines 
 FD, FE, which are in the fame Plane ABC, that Line is alio per- 
 penchcular to ^hat Phme^ ^ kr three right Lines, 
 
 FI FE, and FD; thofe three Lines are all in the fame Plane, ABC. 
 
 t 7 . If two Lines FG, EH, are perpendicular to the fame Plane 
 ABC, they will be parallel to one another. 
 
 Fig. 15. ,8. Two Lines EG, FH, perpendicular to the fame Plane 
 ABCD, cannot be drawn through the fame Point 
 
 Fi g . 16. 10 If two parallel Planes. ABCD, EFGH are cut by a third 
 Plane IKLM, the common Seaions, OP, QR, are parallel. 
 
 FJg 17 20. If the Lines GM and HN, are divided by paralle Planes, 
 
 F £ ' 7 ' then GI will have the fame Proportion toIM, as HL has to LN, 
 and the Seclion MN, IK, of any plane Triangle MGN two 
 parallel Planes, is always in a given Ratio; that is, IK is m tne 
 fame Proportion to IG, as MN is to MG. 
 
Geometrical Definitions-, &c. S 
 *, A folid Angle E, is made by the meeting of three or more Fi g . ,8, 
 l^SWiniig m a Point; like the Pomt of almond, 
 
 W a f ^^I^EB, fixt above in the Point E *> 
 
 UnC>h ^vTt^ Tbef 0 v e , move round a Circle, AB, it Fig. .9 
 
 m1 3 hX^ F gTe tha is call'd. a Cone s and the Circle is its 
 
 r1 and a lL dfawn from the Vertex C, to D, is called its Axis. 
 
 t\ If a Lbe A d7 move uniformly about two angular Figures, Fig. *. 
 
 ABC DEF which are every Way equal having their Sides 
 
 id Andes mutually parallel "and correfponding exactly to one 
 and Angles mmu j f then that Lme by lts 
 
 ^^Vdefame^f it\ath three Sides, a Prifrn; if four, a 
 
 C t Tf P aT!!S° P m P ov d e- uniformly round two equal and parallel Fig., 
 
 Circles it lha"l defcribe or generate a Cylinder } and the Lme 
 Circles, it miu & ^ lkd lt£ , Axls> 
 
 ,0 T S I^ ^ 'foUd Ldy is kid with one Face upon another Plane, Fig. i 
 the Space whS that y Face takes up is called its Seat ; thus *e 
 Gube CG refts with its Face CDEH, upon the Plane IK ■ there- 
 fore CDEH is the Seat of the Cube on that Plane , and thus the 
 Points C and D, are the Seats of the Lines C A, DB; , as are alio 
 GD DE, the Seats of the Lines AB and BF; and fo hkewrfe, the 
 Seat of the oblique Line DF, is the Line DE. 
 
 SECT. III. 
 Of Practical Geometry. 
 
 To ereSt a Perpendicular CD, from D, near the Middle of a right Fig 
 
 Line AB. 
 
 T^ROM D.fet off on the Line AB, any Diftance DA, DB, 
 F equal to each other-, then from A, defcribe at Pleafure the 
 Ire cd! and with the fame Diftance from B, defcribe the Aic ab , 
 and then from the Point C, where they cut each other, draw CD; 
 fo will CD be perpendicular to AB. 
 
 to let fall a Perpendicular CD upon a Line AB, from a Point C, Fig. 
 
 •without the Line. 
 From the given Point C, defcribe the Arc AEB at pleafure 
 an! frl The Points A and B, defcribe two other Arcs cutting each 
 etltrTn F i draw CF, then are CD and FD perpendicular to AB 
 
1 Geometrical Definitions, & c . 
 
 m 7- Toereff a ?erpendicukr AC, upon the Extremity A, of a Line AS 
 With any Diftance, .defcribe the Arc e fa- ft™ Xf A B * 
 A, and fet off the fame Diftance upon A c ll ?7\ P ° in * 
 fronvf to g f then from the Points^, f, defcrihc fell if* *!* 
 and A from the, Sedion C, draw C £ which fift JjJ^ffl 
 
 & «« CD, P„W d, ^ | „ ^ 
 
 Draw, as you think proper, the obliaue Line A n t 
 |iven Point D cutting An in ? A, and £guj m^A 5g 5* 
 
 Sis steals ir^S 
 
 ^aUo B.D{ ana^^^te^Df^^ 
 Or it may be done yet eafier, by defcribine; two Arc, r n , 
 
 % 9- make. «ny Triangle, as ABC, Lines, AF, fi E CD 
 
 Xi?eT F a Xn^ 21 3nd make AB to the gS 
 
 Lme A * \ tben from the Point B, with the Radim RP Lr -l 
 
 m A f the Radius CD, fiXSST3Sri£S? 
 
 tther Arc ab, cutting the former Arc in C • ther Tfrct , c a 
 Lines to A and B ; and then will ABC be a Triani Thofe tT 
 are refpediively equal to the given Lines AF, BE, f nd CD? 
 
 * iS " l °' ~rll 7f e T eq i lat T al * rian Z k I*"* a Li ™ S™n, AR 
 
 Pobts A a nH R Cn | r $ AB in I™ Co^ffe,: and from the 
 
 Triangle. C '* n ^ u "ateral 
 
 fffc-u 7i make* Geometrical Square ABCD, «, jfo- ab 
 
 From the Point B erea the Peroendicular rp r D . , 
 Jta Radius AB, defcribe IfjJjKSfeffi Serpent 
 -cular ui C, from A and C, (with the fame Radius) defcribe tw» 
 gore Arcs cutting each other in D } then draw DA,' DC, 5 the 
 Figure prqpofed is compleated. ' ' rne 
 
 *%■ tfi *^^-§ifc Line AB, ^ *, ^ Angle jg. 
 
 © ^ 1 ?° mt o '7 lth Radius > defcri ^ the Arc cdf from 
 M, mth the iame Radius, 4efcribe the Arc a b, take the Length 
 
Geometrical Definitions, &c. y 
 
 of a b, and transfer it from c to d, and through d draw a Line 
 to A j then is the Angle BAC equal to the given Angle aDb. 
 
 To bifecJ, or divide an Angle A intd two equal Parts. Fig. 2 $; 
 From the Point A, with any Radius, defcribe the Arc BG3 di- No * * % 
 vide BC into two equal Parts, and draw AD. 
 
 To divide a right Line AB into any Number of equal Parts. Fig. 24; 
 
 From the Point A draw at pleafure the Line AC, and make BD 
 parallel thereto j then carry as many equal Parts along the Line 
 AC, from the Point A, and along the Line BD, from B, as you 
 would divide the Line AB into (for infrance, fix Parts) and draw 
 the tranfverfe Lines, which will divide the propofed Line as 
 was required. 
 
 Or, it may be done by drawing a Line AB, parallel to a given Fig. 2$, 
 Line CDj then by fetting as many equal Parts upon the Line 
 AB as CD mould be divided into, and by drawing Lines from 
 thence to a Point, as E, from every Divifion, and in fuch a Man- 
 ner, that the outward Lines AE, BE, mail touch the Ends of 
 the Line CD, as in the Figure. I fay then, the Line CD will 
 be divided into fix equal Parts. 
 
 To infcribe a Circle within a Square ABCD. Fig. 26. 
 
 Draw the Diagonals AD, B C, and where they crofs each other 
 will be the Center of the Square, which confequently is the Center 
 of the Circle alfo. 
 
 To infcribe a Square in a given Circle. Fig. 26. 
 
 Draw the Diameter AB, from A and B defcribe the Arcs a, b, No ' 2 ' 
 and draw DE ; from A, D, B, E, draw Lines as in the Figure, 
 which will be the Square required. 
 
 As to the Geometrical Contraction of Polygons, I mall not Fig. 27. 
 take up the Reader's Time about them ; for they may be defcribed 
 very eafily by Means of the Scale upon the Sector for that Pur- 
 pofe, as has been obferved before, under the Word Sector. 
 
 CHAP. 
 
ISii w ; li 
 
 CHAP, II. 
 
 Of the Eye and the Nature of Vifion y the 
 Reflection and RefraBion of the Rays of 
 Lights and the Caufe of Colours. 
 
 S E C T. I. 
 
 Of the E y b and the Nature of V i s i o n. 
 
 THE Defign of this Chapter is, to explain to the unlearned 
 Reader the Conftruction of the Human Eye, and to give 
 him a general Idea of the Nature and Caufe of Vifion ; 
 and not to proceed in a regular Manner upon Opticks, but only 
 to take Notice of fome particular Parts of it, by which he will be 
 enabled to fee more clearly -the Nature of Perfpeclive. In order 
 to which, I mail take Quotations from the moil eminent Writers 
 upon that Subject, and not prefume to give him much of my own, 
 as ^nothing which I can offer will be new, or fo much to the 
 Purpofe. 
 
 " /Every vifible Body emits or reflects inconceivably fmall Par- 
 cc tides of Matter from each Point ; of its Surface, which iffue from 
 <c it .continually (not unlike Sparks from a Coal) in ftrait Lines 
 *} <and in all Directions. Thefe Particles entering the Eye, and 
 cCC ftriking upon the Retina (a Nerve expanded on the back Part of 
 <c the Eye to receive their Impulfes) excite in our Minds the Idea 
 « of JLight, and as they differ in Magnitude, they produce in us 
 t{ the Ideas of different Colours. 
 
 cc That the Particles which conftitute Light, are exceedingly 
 " fmall, appears from hence, lifys. that if a Hole be made through 
 <c a Piece of Paper with a Needle, all the Rays of Light which 
 <c proceed at the fame Time from all the Objects on one Side of it, 
 t£ .are capable of paffing through it at once without the leaft Con- 
 x< fufion; for any one of thofe Objects may as clearly be feen 
 " through it, as if no Rays paffed through it from any of the 
 <c reft. Further, if a Candle is lighted, and there be no Obftacle 
 <c in the Way to obftruct the Progrefs of its Rays, it will fill all 
 * c the Space within two Miles of it every Way with luminous 
 ** Particles, before it has loft the leaft fenfible Part of its Subftance 
 I s thereby. 
 
 « That 
 
Of the Eve and the Nature of Vision, 
 
 " That thefe Particles proceed from every Point of the Surface? 
 " of a vifible Body,, and in alL Directions, is clear from hence, 
 «« viz. becaufe where-ever a Spectator is placed with regard to the 
 " Body, every Point of that Part of the Surface which is turned 
 " towards him, is vifible to him. That they proceed from the 
 « Body in right Lines, we are aflured, becaufe juft fo many and 
 " no more will be intercepted in their Paifage to any Place, by an 
 «« interpofed Object, as that Object ought to intercept, fuppofing 
 " them to come in fuch Lines. The Velocity (or Swiftnefs) with 
 «« which they proceed from the Surface of the vifible Body, is no 
 " lefs furprifing than their Minutenefs : For by the Calculation 
 " of the moft accurate Philofophers, they are no more than about 
 " feven Minutes in pairing over a Space equal to the Diftance be- 
 " tween the Sun and us, which is about eighty-one Millions of 
 u Miles, and is- confiderably more than a Million Times greater 
 " than the Velocity of a Cannon Ball. 
 
 " A Stream of thefe Particles iffuing from the Surface of a 
 " vifible Body in one and the fame Direction, is called a Ray of 
 " Light. J 
 
 " As Rays proceed from a vifible Body in all Directions, they 
 M neceflarily become thinner and thinner, continually fpreading 
 " themfelves as they pafs along, into a larger Space, and that in 
 •5 proportion to the Squares of their Distances from the Body* 
 " that is, at the Diftance of two Spaces, they are four Times 
 " thinner than they are at one > at the Diftance of three Spaces, 
 " nine Times thinner, and fo on : The Reafon of which is, be- 
 " caufe they fpread themfelves in a twofold Manner, viz. upwards 
 
 V and downwards, as well as fide-ways."* This may be the Fig. 
 
 more clearly comprehended by the following Experiment. 
 
 " Let the Light which flows from a Point A, and paffes through 
 " a fquare Hole bcde, be received upon a Plane, BCDE, parallel 
 " to the Plane of the Hole j or, if you pleafe, let the Figure BD, 
 " be the Shadow of the Plane bd; and when the Diftance AB is 
 " double of Ab, the Length and Breadth of the Shadow BD will 
 u each be double the Length and Breadth of the Plane abj and 
 " treble, when AB is treble of Ab, and fo on; which may be 
 " eafily examined by the Light of a Candle placed at A. 
 
 " Therefore the Surface of the Shadow BD, at the Diftance 
 -** double of Ab, is divifible into four Squares, and at a treble 
 
 • Vid« Rowning't Ojpt. Part 3, p. 4, 
 
 B 
 
 •« Diftance 
 
Of the Eye and the Nature of Vision. 
 
 ■« Diftance into nine Squares, feverally equal to the Square bd, as 
 « reprefented in the Figure. The Light then which falls upon 
 » the Plane bd, being fuffered to pafs to a double Diftance, will 
 be uniformly fpread over four times the Space, and confequently 
 " will be four times thinner in every Part of that Space, and at 
 " a treble Diftance will be nine times thinner, and at a quadruple 
 6 < Diftance fixteen times thinner than it was at firft; and fo on 
 c< according to the Increafe of the fquare Surfaces bcde, BCDE, 
 « or of the fquare Surfaces Abf g, ABFG, &c. built upon 
 « the Diftance Ab, AB, &c. Confequently the Quantities of this 
 " rarified Light, received upon a Surface of any given Size or 
 ££ Shape whatever, removed fucceffively to thofe feveral Diftances, 
 " will be but one quarter, one ninth, one fixteenth, of the whole 
 « Quantity received by it at the firft Diftance Ab. Or in general 
 <« Words, the Denfities or Quantities of Light, received upen any 
 " given Plane, are diminifhed in the fame Proportion as the 
 « Squares of the Diftances of that Plane from the luminous Body 
 ** are increafed; and on the contrary* are increafed in the fame 
 " Proportion as thofe Squares are diminifhed. For the Lights of 
 <1 the feveral Points of the Body, which feverally follow this Rule, 
 e< will compofe a Light which will ftill follow the fame Rule."* 
 
 Having thus far explained what we are to underftand by the 
 Rays of Light, we will now proceed to a Defcription of the Hu- 
 man Eye, and confider the Nature of Vifion. < . 
 5 « ATYC is the Heprefentation of an Human Eye, differed 
 through its Axis f , all the Parts being twice- as big as the Life. 
 " Here the tranfparent Coat, called the Cornea, is ABC; the 
 « Remainder ATYC being opake, and a Portion of a larger 
 J£< Sphere. Within this outward Coat Anatomifts diftinguiih two 
 " others; the innermoft of which is called the Retina, being like 
 « a fine Net, compofed of the Fibres of the Optick Nerve YVT 
 «" woven together, and is white about the Parts p, q, r, at the 
 C{ Bottom of the Eye. The Cavity of the Eye is not filled with 
 " one Liquor, but with three different Sorts. That contained in 
 ■« the outward Space ABCOEGFD is called the Aqueous Hu- 
 « mour, being perfeftly fluid, like Water,; the other, contained 
 
 * Vide Smith's Opt. P. 17, Art. 57, 58. 
 
 t The Axis of the Eye is a Line drawn through the Middle of the Pupil and of the Cry- 
 stalline Humour, and confequently falls upon the Middle of the Retina. And the Axes of 
 iboth Eyes produced, are sailed. the Optick Axes .; which will be better underftoed after the 
 ^Qefcription of the Eye. 
 
 *" in 
 
Of the Eye. and the Nature <?/ Vision. 
 
 ^ in the inward Space EpqrDFG is a little thicker than the 
 « White of an Egg, and is called the Vitreous Humour j the third 
 « Humour, FG, is fhaped like a Lens* of unequal Convexities, 
 « lvine between the two former, and fixed to the fide Coats by 
 « Filaments or Threads extended all round it, and is called the 
 « Crvftalline Humour, being hard like the White of an Egg 
 « boiled but as clear as the other two, and differs from them in 
 «< a greater Degree of Refraftivef Power; whereby the Rays that 
 «c came from the Points P, Q, R, having, received a Degree of 
 « Convergence % by the Refraaion of the Cornea ABC are 
 « made to converge a little more by other Refradions at the Sur- 
 « faces of the Cryftalline FG; fo that uniting in as many Points 
 c« pj q , r , upon the Retina, they reprefent the. Points of the Ob- 
 
 « fe'a pjQi r > from whence tlie y came -" II ,,1, 
 
 The PicluTe of an Objea upon the Retina being; produced much 
 in the fame Manner as a Piaure by a Lens, viz. in both Caies by 
 Means of the Refraaion of the Rays of Light, we will therefore, 
 firft (hew how by the Paffage of thofe Rays through a Lens a 
 Piaure maybe produced; as this will be one considerable Step 
 Towards plaining the Nature of Vifion : For which Purpofe I fhau 
 ouote an Experiment from the incomparable Sir Ifaac Newton. } 
 
 « Let PR reprefent an Objea without- doors, and AB a Lens Fig. * 9 , 
 « placed at a Hole in the Window-mutter of a dark Chamber, 
 « whereby the Rays that come from any Point CL of that Object, 
 « are made to converge and meet again in the Point q; and it a 
 « Sheet of white Paper be held at q, for the Light there to faU 
 « upon it, the Piaure of that Objea PR will appear upon the 
 « Paper in its proper Shape and Colours. For, as the Light 
 «« which comes from the Point goes to the Point : q . fo i he 
 « Light which comes from other Points P and R of the Object, 
 « will go to fo many other correfpondent Points p and r> io 
 « that every Point of the Objea mail illuminate a,^^^^ 
 « Point of the Piaure, and thereby make a Piaurehke the Object 
 
 * By a Lens in this Place is meant a Glafs which collet the Rays of Light into a Point, 
 
 "I W™ay rf Lfgnl paffes cut of one Medium into another of a different 1 Den%, it 
 wil be bent near y the Su?fac« of thofe Medi.ms, which bending » called R^*"' . 
 
 1 If feveral Rays approach each other fo as to meet in a Point, they are faid ^ converge 
 J Tf they proceed from a Point and go further off continually, they are then faid to diverge. 
 
 (j Vide Smith's Opt. p. 26 
 § Vide Newton's Opt. p. 11. 
 
 B 2 
 
 in 
 
Of the Eye and the Nature of Visiosr, 
 
 f in Shape and Colour, this only excepted, that the Picture fhaB 
 " be inverted. And this is the Reafon of that vulgar Experiment 
 cc of catting the Species of Objects from abroad upon a Wall, or 
 <c Sheet of white Paper in a dark Room.* 
 28. " In like Manner, when a Man views any Object PQR, the 
 " Light which comes from the feverai Points of the Object is fo 
 " refracted by the tranfparent Skins and Humours of the Eye, 
 " (that is, by the Cornea ABC, and by the Cryftaliine Humour 
 " FG ) as to converge and meet again in fo many Points in the 
 et Bottom of the Eye, and there to paint the Picture of the Ob- 
 " ject upon the Retina. And thefe Pictures, propagated by Mo- 
 " tion along the Fibres of the Optick Nerves into the Brain, are 
 ct the Caufe of Vifion. For accordingly as thefe Pictures are per- 
 <f feet or imperfect, the Object is feen perfectly or imperfectly. 
 * l If the Eye -be tinged with any Colour ( as in the Difeafe of 
 ** the Jaundice) fo as to tinge the Pictures in the Bottom of the 
 « Eye with that Colour, then all Objects appear tinged with the 
 " fame Colour. If the Humours of the Eye by old Age decay, 
 Si fo as by fhrinkrng to make the Cornea and Coat of the Cry- 
 Xi ftalline Humour grow flatter than before, the Light will not be 
 " refracted enough, and for. want of a fufficient Refraction will 
 " not .converge to the Bottom of the Eye, but to fome Place be- 
 " yond itj and l>y .confequence paint in the Bottom of the Eye a 
 M confufed Picture.—This is the Reafon of the Decay of Sight in 
 " old Men, and fhews why their Sight is mended by Spectacles. 
 xc For thefe .Convex GlalTes (or Lenfes ) fupply the Defect of 
 £C Plumpnefs in the Eye, and by increafmg the Refraction make 
 the Rays converge fooner, £o as to convene diftinctly at the 
 et Bottom of the Eye, if the Glafs has a due Degree of Convexity. 
 " And the Contrary happens in ihort-fighted Men, whofe Eyes 
 rc are too plump. For the Refraction being now too great, the 
 " Rays converge and convene in the Eyes before they -come at the 
 Bottom ; and therefore the Picture made in the Bottom, and the 
 «< Vifion caufed thereby, will not be diftinft, unlefs the Object be 
 brought fo near the Eye as that the Place where the converging 
 Rays convene may oe removed to the Bottom, or that the 
 * f Plumpnefs of the Eye be taken off, and the Reflations dimi- 
 
 * A Perfon may eafiiy fathfy himfelf of the Truth of this, by only talcing a common Butb- 
 sng-GJafs in one Hand, and a Piece of white Paper in the other, and let him hold the Glafr 
 before any Objeft, and the Paper on the oppofite Side of the Gkfs ; then by moving tfieGlafs 
 or Paper backwards and forwards 'till he gets the Rays to their proper Focus, he will fee the 
 > ittu.re.of x\e Objea.upon the Paper, but is will not be fo diftintt as in the dark Chamber. 
 
 " niih'd 
 
Of the Eye and the Nature ef Vigrotf. f j 
 
 rt mfhed by a Concave-Glafs of a due Degree of Concavity ; or 
 *' laftly, that by Age the Eye grows flatter, 'till it comes to a due 
 <e Figure : For ftiort-fighted Men fee remote Objects beft in Old 
 fC Age; and therefore they are accounted to have the moft laft- 
 *' ing Eyes."' * 
 
 " As to th« Diftinctnefs of Vifion in a perfect Eye, that evi- 
 <f dently depends upon the Refraction of the Rays; and it is then 
 <£ as diftinct as poffible, when the Refraction is fo made, as that 
 <e all the Rays which come from one and the fame Point of the 
 " Object, meet together exactly in one and the fame Point of the 
 " Bottom of the Eye : But this is never precifely fo, but in thofe 
 
 Rays which come from that Point of the Object which is at the 
 *-« Extremity of theoptick Axis Q^q-; for it is evident, that thofe Fig. 2g< 
 " Rays which come from the other Points, are reunited fo much 
 " the lefs exactly one than the other, as they are more diftant 
 <c from this Axis ; wherefore we cannot have at the fame time, 
 
 the moft diftinct Scnfation but in this Place alone, and the reft 
 " will be more confufed." -f- 
 
 The farther diftant the Eye is from an Object, fo much lefs Fl v. 
 will the Picture of that Object be upon the Retina: For let E be 
 an Eye viewing the feveral Objefts AB, CD, EF, at the Diftance 
 OQ, OR, OS. — Having drawn the feveral Rays A a, Bb, Cc, 
 D g/TE e, F f, through the Pupil O, it will be manifeft, that the 
 Picture of the neareft Object AB, will be painted at the Bottom 
 of the Eye in the Space ab, the Object CD in the Space eg, and 
 the fartheft Object, EF, in the Space ef; therefore, as the Space 
 a b, is larger than the other two, eg or ef, the Picture of AB 
 will be larger than the Picture of the other two Objects CD or 
 EF, which are at a greater Diftance from the Eye; and thefe Pic- 
 tures will be to each other, as the feveral Diftances OQ^ OR, 
 OS, are to each other, t From hence then we may eafily con- 
 ceive, that the Eye may be fo far removed from the ObjeSf, that at 
 laft the Image of that Object will totally difappear. || 
 
 " But the Degree of Brightnefs of the Picture of an Object 
 <£ painted upon the Retina continues the fame, at all Diftances be- 
 <( tween the Eye and the Object, provided aone of the Rays be 
 
 * Vide Newton's Opt. p. 12. 
 
 f Vide Clarke's Rohault, vol. 1. p. 249. 
 
 t Ibid , . p. 243. 
 
 II The Rays of Light OA and OB, which come from the extreme Points of the Obje£l 
 •to the Eye O, form an Angle AO B, which is sailed the Optick or Vifual Angle; and the 
 3Uy« O A and 0£ aie called Vifual Rays. 
 
 ** ftopt 
 
Of the Eye and the Nature of Vision.. 
 
 " ftopt by the Way, and that the Pupil does not alter its Aperture.*. 
 " For inftance, when the Eye approaches as near again to th« 
 " Object, the Picture upon the Retina becomes double in Length 
 " and double in Breadth, and confequently quadruple in Surface j 
 " for the Surface would be double, if its Length alone or Breadth 
 " alone was double. The Quantity of Rays received through the 
 " fame Aperture of the Pupil, at half the Diftance from the Object, 
 " is alfo quadruple, and being equally fpread over four times the 
 " Quantity of Surface of the Retina, they are juft as denfe as 
 " before, when the Object was at twice that Diftance. 
 
 " It follows then that the faint Appearance of remote Objects, 
 is owing to the Opacity of the Atmofphere, which hinders Part of 
 «< their Light from coming to the Eye. Accordingly we find that 
 " the Sun, Moon, and Stars, appear very faint when near the 
 <c Horizon, and brighter continually as they rife higher; becaufe 
 w the Tract of Vapours which lies in the Way of the Rays, is 
 " longeft and thicker!: near the Horizon ; and becomes thinner and 
 " fhorter as the Objects rife higher, and confequently does lefs 
 " obftruct the PafTage of the Rays.* 
 
 " Parallel Lines feen obliquely, as ABC, DEF, appear to con^ 
 " verge more and more as they are farther extended from the Eye, 
 " Becaufe the apparent Magnitudes-)- of their perpendicular Inter- 
 " vals AD, BE, CF, &c. are perpetually diminifhed. And for 
 " the fame Reafon they appear to converge towards an imaginary 
 t£ Line, OG, drawn from the Eye parallel to them. 
 
 " This is the Reafon that the remoter Parts of a Walk or Floor 
 " appear to afcend gradually, and the Ceiling to defcend towards 
 << the Horizontal Line OG : And that the Surface of the Sea, feen 
 " from an Eminence, appears to afcend gradually in going from 
 " the Shore ; and that the upper Parts of very high Buildings feem 
 <c to lean forward over the Eye below, becaufe they feem to ap- 
 <l proach towards a vertical Line OG. 
 2 " The apparent Magnitude of a given Line, AB, feen very ob- 
 " liquely at a given Diftance, OA, increafes and decreafes in pro- 
 " portion to the Increafe and Decreafe of OP, the perpendicular 
 " Diftance of the Eye, from the Line AB produced; provided 
 " the Line AO be very large in comparifon to AB. For let the 
 " Ray BO cut a Line AC perpendicular to AB in C; and while 
 
 * Vide Smith's Opt. p. 29. 
 
 t By apparent Magnitude is here meant the Bignefs of the Pi&ure upon the Retina. 
 
 * 6 the 
 
Of the Eye and the Nature of Vision. t$ 
 
 « e the Eye is raifed or deprefTed in the Perpendicular OP, the Line 
 «* AC will increafe and decreafe as OP does, and fo will the 
 " Angle AOC fubtended by AC, and this Angle meafures the 
 " apparent Magnitude of AB. 
 
 « Hence the apparent Magnitudes of equal Parts AB, ab, of 
 " a Line PAb, feen very obliquely at great Diftances from the 
 " Eye, are reciprocally in a duplicate Proportion of thofe Dif- 
 « tances. For Example, let Ob be double of OB, and the 
 " Angle OBP will be double of ObP, and accordingly fince AB, 
 te ab, are equal, the Perpendicular AC will be double of ac, and 
 " being ieen twice as near as ac, will appear four times bigger 
 " than ac. Again, if Ob be treble of OB, the Line AC will be 
 " treble of ac, and being feen three times nearer than ac, will 
 " appear nine times bigger than ac, and fo on. 
 
 " Hence the apparent Magnitude between a Row of Columns 
 < c are diminiilied in a greater Proportion than their Heights. 
 
 " The quick Diminution of the apparent Magnitudes of the 
 « c remoter Parts of long Lines or Diftances, is the Caufe of great 
 £C Difficulty and Uncertainty in our Eftimate of their Quantities, 
 « For be the Differences of feveral Diftances or Heights never fo. 
 €C great in themfelves, they will become invifible at laft by reafon 
 « of the Smallnefs of the Angles they fubtend at the Eye, occa- 
 < ( fioned by their Obliquity ; and then thofe unequal Heights and 
 *< Diftances will appear equal." * 
 
 SECT. II. 
 
 Of the Reflection and Refraction of the Rays of 
 
 LlGH T* 
 
 cc IT* THEN a Ray of Light falls obliquely upon a fmooth po- Fig. 54 , 
 < 4 VV Hfhed Surface, it is turn'd out of its Way either by Re- 
 <£ flection or Refradion in the following Manner. Imagine the 
 « Paper upon which this Figure is drawn to be perpendicular to 
 « the Surface of ftagnating Water, and to cut it in the Line RS, 
 cc and that a Ray of Light, coming in the Ak along the Line 
 « AC, falls upon RS at the Point C. Then fuppofing the Line 
 ■** PC Q^jto be perpendicular to the Surface of the Water, if the 
 *' Ray be reflected, or turn'd back at C into the Air again, it will 
 
 .* Vide Smith's Opt p. 5$. 
 
 tx defcribe 
 
s6 Of Reflection and Refraction. 
 
 " defcribe a ftraight Line CB, inclin'd to the perpendicular CP 
 u at an Angle PCB exactly equal to the Angle PC A, and there~ 
 " fore the Angle of Reflection is always equal to the Angle o£ 
 M Incidence. 
 
 Tig. 35. " But if the Ray that came along A C goes into the Water at 
 " C, it will not proceed ftraight forward, but being refracted or 
 « bent at C, it will defcribe another ftrait Line C E, inclined to 
 « f the Perpendicular CQ^ at a lelfer Angle ECQ^ than the 
 «< Angle ACP 5 and the Line CE^will always be fo fituated, 
 * that when any Circle, defcribed about the Center C, cuts the 
 " Line C A in A, and C E in E, the Perpendiculars A D, and 
 " EF, drawn from A and E to the Line PQ, fhall always bear 
 *' the fame Proportion to each other, whatever be the Magnitude 
 " of the Angle ACP. In Water the Line EF is always three- 
 4( quarters of AD. 
 
 " In both thefe Cafes the Line AC is called the incident Ray, 
 " CB the refle&ed Ray, CE the refracted Ray, C the Point of 
 " Incidence, PCQ^the Perpendicular (at the Point) of Incidence, 
 M the Angle ACP the Angle of Incidence, BCP the Angle of 
 " Reflection, E C Qjhe Angle of Refraction j the Line A D the 
 « Sine of Incidence, that is, of the Angle of Incidence ; and 
 «' EF the Sine of Refraction, that is, of the Angle of Refraction. 
 
 " As Rays of Light are inceflantly thrown out and dif- 
 4t perfed in all poffible Directions from every Point of a luminous 
 >l Body; fo when they illuminate other Bodies, on which they 
 4< fall, they are alfo inceflantly thrown back from every Point of 
 " thofe Bodies. Far the Points of opake Bodies fo enlightened, 
 " are vifible to the Eye at any Point of Space and in any Point of 
 «' Time, as well as the Points of the luminous Body that en- 
 •« lightened them. The numberlefs Rays which flow fre-m all 
 '* yifible Bodies, called Objects, may be methodically diftributed 
 " in this Manner. The Surface of the Object is confidered as 
 " confifting of Phyfical Lines, and thefe Lines as confirming of 
 «« Phyfical Points, and thefe Points are conceived to radiate all 
 *' manner of Ways. It is ufual to make ufe of nothing elfe fox 
 *« an Object hut a Phyfical Line. For by how much that Line is 
 w increased or dirninifhed in apparent Magnitude, or Brightnefs, 
 *' or Diftinctncfs, fo much the Diameter, or Length, of any Ob- 
 " ject, in its Place, would be increafed or diminished. 
 
 Fig. 36. " The Point from which Rays diverge, or towards which 
 ** they converge (being made to go back towards the fame Point, 
 
 " though 
 
Of Reflection and Refraction, 
 
 ** though they may never meet at it) is called their Focus. And 
 ** in both Cafes any Parcel of thefe Rays, as QJB C,, or QB A, 
 fft confidered apart from the reft, is called a Pencil of Rays. 
 C{ This Figure reprefents the Manner in which the Rays of a Pen- 
 " ciL, Q^A B y diverging from any Point of an Object Q, and 
 " falling upon a ftrait Line ABC, or upon a polifhed Plane re- 
 ** prefented by it, do all diverge after Reflection as if they came 
 •« from another Point q. The Ray Q^C, which falls perpendicu- 
 " larly upon the Plane AB, is reflected back again along the 
 <c fame Line C but all the reft falling upon it with greater 
 • c and greater Degrees of Obliquity, as the Points of Incidence lye 
 " farther and farther from C, are alfo reflected with Degrees of 
 " Obliquity refpectively greater. It will feem reafonable therefore, 
 ** efpecially by attending to the Figure, that the reflected Rays, 
 •* produced backwards, {hould meet the Perpendicular QC, pro- 
 " duced in a Point q, fituated as far from the reflecting Plane on 
 «5 one Side, as Q^is on the other: And confequently that all the 
 «« Rays flowing from a fmgle Point will after Reflection di~ 
 ** verge from a fingle Point q,, at an equal Diftance on the other 
 " Side of the reflecting Plane.. 
 
 C£ On the contrary, if q be a Focus to which the incident Rays 
 " are made to converge, the Point Q^_will be their Focus after 
 " Reflection from the Surface ACB. 
 
 " What has been faid of the Point is applicable to every Tig. ^7 
 '* other Point of an Object PQJR.; namely, that as the Focufes o>% f <2>$ 
 " Q^q, lie at equal Diftances on each Side of the reflecting Plane, 
 fC fo the Focufes P, p lye on each Side at other equal Diftances, 
 " and R, r at other equal Diftances, in Lines P p, R r, drawn 
 ** perpendicularly through the Plane AB. Hence it is eafy to 
 " underftand by Infpection of the Figures, that thefe Focufes 
 " P» <!> r > w ^ tn innumerable others, lying all in the fame Order as 
 <£ the correfponding Points P,QvR, compofe an imaginary Line of 
 *' the fame Length and Shape as the Line P Q^R ; and that the 
 " Situation of the Line p q r, with refpect to the back fide of the 
 * f reflecting Plane, is the very fame as that of PQ^R with refpect 
 " to the fore fide of it. This Line pqr is called an Image or 
 " Picture of the Objeft PQR."* 
 
 This may fuffice to fhew the Nature of the reflected Images of 
 Objects from polihYd Planes , the Knowledge of which is abfo- 
 
 * Vide Smith's Opt. p. 7, 
 
 C lutcly 
 
1 8 Of R ET LECTION R EPR AC T I O M- 
 
 lutely necelfary in feveral Parts of Painting, efpecially in Landfkips* 
 where Water is often introduced; the Tranfparency of which, 
 depends upon giving the Reprefentation of that Fluid its true or 
 local Colour, and in giving the Reflections their proper Depths 
 and Appearances. — Proceed we now to a farther Consideration of 
 ithe Refraction of the Rays of Light, as introductory to the Caufe 
 of Colours. 
 
 In the 35th Figure we obferved, that if a Ray of Light went 
 out of Air into Water, it would not proceed ftrait forward, but 
 foe bent and turned out Of its direct Courfe at the Point of Inci- 
 dence C; and that the Reafon of this Refraction, or bending of 
 the Ray, was owing to its palling out of a rarer or thinner, into 
 a denfer or thicker Medium ; and in Proportion as this Medium 
 into which the Light enters, is more or lefs denfe, the Ray will be 
 snore or lefs refracted. Now what is faid of one Ray, will hold 
 .equally true as to any Number of Rays : But fince the Rays of 
 Light arc not alike, but diffimilar, fome greater and others lefs, 
 they will be differently refracted at their Exit out of one Medium 
 into another Medium ; and being thus feparated, each Species of 
 Rays will exhibit a Colour peculiar to itfelf ; which is the Subject 
 of the next Section. 
 
 SECT. III. 
 
 Of the Cause of Colours. 
 
 <ee ^TpHE Sun's Rays are not homogeneous (that is alike) but of 
 V JL different Kinds, and each Sort has a different Degree of 
 " Refrangibilityj that is, in paffing through a denfe Medium they 
 « f are differently difpofed to be refracted, being bent or turn'd 
 6< out of their firfr. Courfe to different Diftances from the Perpen- 
 *' dicular; and thefe feveral Sorts of Rays have each a peculiar 
 *\ Colour, viz. thofe which are leaf! refrangible, are Red ; the 
 <c fecond Sort, Orange; the third Sort, Yellow; the fourth Sort, 
 " Green; the fifth Sort, Blue; the fixth Sort, Indigo; and the fe- 
 £C venth Sort, Violet, which laft are mofl refrangible, or refracted 
 * c to the greater!: Diftance from the Perpendicular. 
 %. 4o. ** To illuflrate this Matter, let GF reprefent a Parcel of the 
 " Solar Rays entering through the Hole H of a Window-Shutter, 
 •* into a darkened Room, and there let them fall on the Prifrrt 
 <c ABC, in the Point F: In paMing through the Prifm they will be 
 ** feverally refracted in a different Degree, and thus feparated from 
 
 <c each 
 
Cause of Co io xj r s* 
 
 «* eacli other, fo that at their Exit on the other Side at E, they 
 * will proceed at different Diftances from the Perpendicular EP 
 « to the other Side of the Room, where they will make a long 
 « and various-coloured Image of the Sun XY ; which is, perhaps, 
 ** one of the raoft furprizing and agreeable Spectacles in Nature.' 
 
 " The feveral Sorts of Rays, after they are refracted, appear in 
 " their own proper Colours, in Order as follows, viz. Thofe 
 " which are leaft refracted, or fall nearer!: the Perpendicular EP, 
 <c are Red, and make the red Part of the Spectrum at Rj the 
 <c next are the Orange at O, the Yellow at Y, the Green at G, 
 " the Blue at B, the Indigo at I, and the Violet at V: And thefe 
 " feven are all the original limple Colours in Nature; and of 
 " which,, by various Mixtures, all others are compounded, in the 
 " common Refractions and Reflections from natural Bodies. * 
 
 " From hence then we may conceive, that Colour is a Senfation 
 " produced in the Mind, by the Impreflion made in the Eye, by 
 " certain Kinds or Sorts of Rays of Light, feparated from others 
 " by means of their different Refrangibility and Reflcxibility, 
 " whereby they are divided into feveral Parcels, each endowed with 
 sc its own diftinct colour-making Power. And Bodies, whofeSur- 
 " faces are difpofed to reflect one kind of thefe Rays more copioufly 
 <£ than any others, exhibit, and are faid to be of that Colour 
 " which is peculiar to the Rays they mofr. copioufly reflect; and 
 " the infinite Diverfity of Bodies, and the different Mixtures and 
 " Modifications of different colour-making Rays thereby occa- 
 " fioned, muff therefore produce that infinite Variety of Colours 
 u which beautifies the Face of Nature."^ 
 
 * The Truth of all this any Perfon may convince himfelf of by making the Experiment, 
 •r by only holding a Prifm between his Eyes and the Sun ; then by turning it round, he will 
 fee the feveral Colours in their proper Order, as above defcribed. 
 
 t Martin's Philof. vol. II. p. 156..— —Hamilton's Perf. p. 1. 
 
 CHAP. 
 
» 
 
 P 
 
 C HAP- IIL 
 Theory of Perspective. 
 
 SECT. I. 
 
 An Explanatory Part, by Way c/ Introduction, 
 
 iE.r spec tive is the Art of drawing upon any Surface the 
 Reprefentation of Obje&s as they appear to the Eye : In 
 order to which, it is neceflary to fuppofe the Light fhould 
 come from every Part of the Reprefentation in the very fame 
 Manner, and with the very fame Strength of Colour, as it would 
 .do from the real Objects themfelves, were they put in the Place of 
 the Picture j becaufe then, the Eye will not be able to judge, whe- 
 ther what it fees be a few Colours artificially laid upon a Canvas, 
 or the real Objects themfelves in the fame Situation. 
 
 This is a general Definition of that kind of Perfpeftive I am 
 going to explain ; which, is only what relates to the Arts of Paint- 
 ing and Defigning; but not to any of the Mathematical Arts, 
 which are too abftrufe for my Speculations, and would be of no 
 real Service to thofe for whofe Ufe this Work is chiefly intended: 
 And although Perfpeaive Reprefentations may be drawn upon 
 any Surfaces, be they ever fo irregular, yet I (hall firft confine 
 myfelf to fmooth even Planes, fuch as a Canvas, Wall, Cielmg, or 
 the like. — This being premifed, 
 ffig.41. Let E be the Eye, HE its Height from the Ground OP, and 
 TOSX a fquare Object laid flat upon the Ground. Now it is evi- 
 dent, from what was faid in thelaft Chapter, Sett. 1. that the Eye 
 will fee the' Object TOSX, by means of the Rays of Light which 
 come from every Part of the Objea to the Eye. Let us therefore 
 fuppofe a tranfparent Plane GLPP, like a Glafs- Window, to be 
 fixed perpendicularly upon the Ground OP, between the Spectator 
 HE, and the Object TOSX; and it will be as evident, that the 
 Rays TE, OE, SE, and XE, will be cut by the tranfparent Plane 
 GLPP, in the Points tosx; which Points are called the Projec- 
 tion, or in other Words, the Perfpeaive Reprefentation of the 
 correfponding Points TOSX, of the original Objea.* And if 
 Lines are drawn from the feveral Points tosx, fo as to join each 
 mother, the Figure fo defcribed, will be the Projeaion, or Perfpec- 
 
 * See Definition 6, S«a. II. of this Chapter. 
 
 tive 
 
Introduction to the Theory ^Perspective. 
 
 21 
 
 live Reprefentation, of the whole original Figure TOSX, upon 
 the Picture. 
 
 In like Manner, fuppofe TOSX to be raifed perpendicular to Fig. 42. 
 the Ground OP, and parallel to the Picture, but every thing elfe 
 remaining in the fame Situation as in the former Figure ; then will 
 tosx be the Reprefentation of TOSX: For it is the Section of 
 the Picture with the Rays TE, OE, SE, and XE, which come 
 from the original Object to the Eye. And here let us obferve, 
 that when the original Object is parallel to the Picture, its Repre- 
 fentation, tosx, will not only be parallel to the Original, but 
 exactly like it, though fmaller in Proportion as the original Object 
 is farther from the Picture ; and if the Original be brought to G, 
 fo as to coincide, or touch the Picture, then the Reprefentation 
 will be equal to the Original : But on the contrary, the Original 
 may be fuppofed fo far removed from the Picture, that the Angles, 
 which the Rays fubtend at the Eye, growing fmaller and fmaller 
 continually, it will at laft totally difappear, and confequently its 
 Reprefentation upon the Picture will difappear alfo. Again, when 
 the Original is brought to coincide with the Picture, the Repre- 
 fentation of TX will not only be equal to the Bottom of the Ori- 
 ginal, but will be at the Bottom of the Picture, in the Line GL, 
 which is its Section with the Ground Plane OP : But as the Ori- 
 ginal is removed farther and farther from the Picture, the Repre- 
 fentation will rife higher and higher, 'till at laft, the Original 
 being fuppofed at an infinite Diftance, its Reprefentation will va- 
 vifh into an imaginary Point C, exactly as high above the Bottom 
 of the Picture as the Eye is above the Ground, or original Plane 
 OP, upon which the Spectator, the Picture, and the original Ob- 
 ject are now fuppofed to ftand. And fo alfo in regard to Objects 
 that lie flat upon the Ground; when their Sides are parallel, then 
 the Reprefentations of thofe Sides will be parallel alfo : Thus the 
 Reprefentation tx of TX, and os of OS, are parallel to their Fig> 4 
 Originals, but feverally diminifhed in proportion to their Diftance 
 from the Picture; and therefore the Reprefentation of their oblique 
 Sides TO, XS, which muft join tx, os, to compleat the Repre- 
 fentation of the whole original Figure, cannot be parallel to their 
 Originals, but will be oblique in the Picture, and would, if con- 
 tinued towards the Top of the Picture, converge into an imaginary 
 Point C, exactly as high above the Bottom of the Picture, as the 
 Eye is above the original Plane OP. Now thefe Points, into 
 which we fuppofe the Reprefentations of the Sides of Objects do 
 
 vanifh 
 
2 2 Introduction to the ^Theory ^Perspective.. 
 
 vanifh upon the Picture, are called by the general Name of Va 
 mining Points. 
 
 From hence then, we may form an Idea of the Nature of thet 
 Perfpeclive Plane or Piclure, and of Perfpeclive Reprefentations • 
 which Reprefentations are nothing more than the Seclion which 
 the Mure makes with the Rays of Light in their Paffage from 
 original Objefts to our Eyes; and that the whole of this Art 
 depends upon finding the exact Section, or true Shape, which that 
 cutting of the Rays makes upon the Picture in all kinds of Situ- 
 ations, and in giving them their proper Force and Colour 
 
 But to dluftrate this by a very familiar Inftance. Suppofe a 
 Spectator to be looking at a Profpeft without Doors, from within, 
 through a Glafs-Wmdow ; he will perceive not only the vaft Ex- 
 tent which fo fmall an Aperture will admit to be feen by his Eve 
 but the Shape Size, and Situation, of every Objeft upon the- 
 G ais: If the Ob;ecT:s are near the Window, the Spaces which thev 
 take upon the Glafs will be proportionably larger than when thev 
 are at a greater Diltance; if they are parallel to the Window, then 
 their Shapes upon the Glafs will be parallel alfo; but if they are 
 oblique, then their Shapes will be oblique, and fo on. And he will 
 always perceive, that as he alters the Situation of his Eye the Si- 
 tuation of the Objecls upon the Window will be altered alio: If he 
 raifes his Eye ever fo high, the Objecls will feem to keep pace with 
 his Eye, and rife higher upon the Window ; and the contrary, if he 
 places it ever fo low. And fo in every Situation of the Eye, the 
 Objecls upon the Window will feem to rife higher or lower: and 
 confequently, the Depth of the whole Profpeft will be proportion 
 ably greater or lefs, as the Eye is elevated or deprelTed; and the 
 Horizon will, in every Situation of the Eye, be upon a Level with 
 it: Inat is, the Horizontal Line, or that imaginary Line which 
 appears to part the Earth and Sky, will feem to be raifed as far 
 above the Ground upon which the Spectator ftands, as his Eye is 
 removed from the fame Place. 7 
 Fig. 43- Let us now fuppofe two Planes ABab, CDcd, of the fame 
 Height, and parallel to each other, one to pafs through the Eye E 
 the other through any Point as e, and both to be perpendicular to 
 the Ground ABCD; and let us imagine another Plane, abed, to 
 be laid upon thefe two Planes, ABab, CDcd, as in the Figure, 
 
 GrluL W ^ abcd is P^l to* the 
 
 Oround A BCD becaufe it lies upon two Planes ABab, CDcd 
 
 of the lame Height, Now if we fuppofe this Plane, abed, to be con - 
 
 tinued 
 
Introduction to the Theory, ^Perspective. 
 
 tinned at an infinite Diflance, and the Line cd to reprefent a Part 
 of the real Horizon, and then imagine a Picture GLPP, to be 
 placed between the Eye E, and the Horizon cd then its Section 
 HL, with the horizontal Plane abed, will be the indefinite Re- 
 prefentation of the Horizon cd, upon the Picture; and this Re- 
 prefentation is called the Horizontal Line. Now fince all Objects 
 which lye flat upon the Ground, or are parallel to it, ieem to 
 vanifh into the real Horizon, therefore the Reprefentation of all 
 fuch Objects upon the Picture, mufr. vanifh into this Horizontal 
 Line j becaufe it is the perfpective Reprefentation of the real Hori- 
 zon : And for the fame Reafon, the Ground, or whole Extent be- 
 tween the Eye and the real Horizon, will not appear to lye flat, 
 but to rife upwards. For let E be the Eye, ABCD the Ground, f 
 and HI the utmoft Extent which the Eye can diftinguifh ; now, 
 I fay, the Ground will not appear to lie flat, as ABCD, but to 
 rife upwards, like A Bed, till it cuts the Plane abed, which is 
 drawn through the Eye E, parallel to the original Plane ABCD j 
 and the Section c d, which the Planes A B c d and abed make 
 with each other, will reprefent the real Horizon. And, as before, 
 if we fuppofe a Picture, GLPP, to be fixed between the Eye and 
 the faid Horizon ; then the Section H L, which the Picture makes 
 with the parallel Plane abed, will be the indefinite Reprefentation 
 of the Horizontal Line upon the Picture j becaufe the Rays of 
 Light, in their Paflage from the Section cd, or real Horizon* 
 would cut the Picture in the Line HL. 
 
 From hence then, we may fee, the grand Principle on which 
 Perfpective depends ; namely, on finding thofe Lines and Points 
 into which Objects feem to vanifli upon the Picture. And whoever 
 will give himfelf the Trouble to underftand the following fhort 
 Theory, will have maftered all the Difficulty in Perfpective : For it 
 only requires to have a clear Idea of the Nature and Property of 
 vanifhing Lines and vanifhing Points, and a few other Requisites 
 as previous thereto ; which he may partly conceive by what has 
 been faid already, and by confidering, that as the Horizontal Line 
 H L, is produced by means of the Plane abed, which panes 
 through the Eye parallel to the Ground, or original Plane ; fo, in 
 the very fame Manner, all other vanifhing Lines are determined 5 
 namely, by imagining a Plane to pafs through the Eye, parallel to 
 thofe Planes whofe Reprefentations are required upon the Picture* 
 — Again, in regard to vanifhing Points ; they are determined by 
 drawing Lines from the Eye, parallel to the original Lines, till 
 
Introduction to the theory o/*Perspectiv?, 
 
 they cut the Picture ; in order to which, we muft alvays fuppof© 
 thefe Lines to lie in fome Plane, and then, having found the va- 
 nifhing Line of that Plane, the vanifhing Point of any Line, in. 
 that Plane, may be found alfo. And from hence we inay obferve,. 
 that the Horizontal Line is of the fame Nature with any other va- 
 nifhing Lines, and differs from them only in being nore ufeful, 
 becaule, many more Objects are perpendicular and parallel to the 
 Picture, than oblique with it : And therefore, the great Strefs which 
 hath been laid upon this Line by mo ft Writers, is not fo very 
 fignificant as they apprehended j for, in fome Cafes, it is of no 
 ufe at all in a Picture. For let us confider a little. If vanifhing 
 Lines upon the Picture, are always to be produced by Planes- 
 paffing through the Eye, parallel to original Figures, then no- 
 original Plane can have its vanifhing Line in th« Horizontal 
 Line, unlefs it is parallel to the Ground j but, if any Object be 
 obliquely fituated with regard to the Ground, ther., the Plane 
 which is to pafs through the Eye, parallel to the Original, in order 
 to determine its vanifhing Line, will be oblique with the Ground 
 alfo; and therefore it cannot pafs through the Horizon:al Line, but 
 will be either above, below, perpendicular to it, or crofs it in an 
 oblique manner : All which may be conceived by infpecting the 
 following Figures. In Fig. 45, the original Object, TOSX, lies 
 upon the Ground ; therefore, the Plane, abed, which pafTes 
 through the Eye E, parallel to the Ground, cuts the Picture in 
 the Horizontal Line HL. In Fig. 46, the Original, TOSX, is* 
 fuppofed perpendicular to the Ground, and to be perpendicular to 
 the Picture alfo ; therefore, the Plane ABPD, which paries through 
 the Eye E, parallel to the faid Plane, will be perpendicular to the 
 Ground and perpendicular to the Picture ; and therefore will pafs 
 through the Center C of the Picture, and produce the vanifhing 
 Line PD, which will be perpendicular to the Horizontal Line HC. 
 But, if the original Object is perpendicular to the Ground, and 
 oblique with the Picture, as in Fig. 47, then its vanifhing Line 
 PD, will be perpendicular to the Horizontal Line HL, but, will 
 not pafs through the Center or Middle of the Picture, but will be 
 on one Side of it. Again, if the fquare Object A B T S, Fig. 48, 
 (which is, inclined to the Ground, at the Angle ATO, but reclined 
 to the Picture) have two Sides A B, T S, parallel to the Picture ; 
 then the Plane OPVL, which paffes through the Eye E, parallel to 
 the original ABTS, will produce a vanifhing Line V L, above the 
 Horizontal Line H C 3 and exactly parallel to it. But if the fame 
 
 Objeft, 
 
Introduction to the Theory of Perspective. 
 
 Object, (Fig. 50.) be turned fo as to have all its Sides oblique 
 with the Picture, then the Plane EPL V, which panes through the 
 Eye E, parallel to the original ABTS, will produce a vaniftiing 
 Line VL, which will be aflant the Horizontal Line HL. Again; 
 if the Object ABTS, (Fig. 49.) be inclined both to the Ground 
 and the Picture, but have its Sides AS, BT, parallel to the Pic- 
 ture, (as in Fig. 48.) then its vaniftiing Line, VL, will be parallel 
 to the Horizontal Line HL, but below it. And To in regard to 
 the vaniftiing Points of any original Lines : As thefe Lines are 
 fuppofed to lie in fome Planes, therefore, having found the vanifh- 
 ing Lines of thofe Planes, as above, the vaniftiing Point of any 
 Line in thofe Planes may be eafily found alfo; viz. by drawing 
 Lines through the Eye, parallel to fuch Lines, 'till they cut the 
 Piaure: Thus, in Fig. 45, EL is drawn from the Eye E, pa- 
 rallel to the Original el, and therefore L is the vaniftiing Point of 
 el upon the Picture. And fo again in Fig. 47, Es, EL, and Eo, 
 are parallel to the Originals ST, SX, OT and OX, and therefore 
 will produce the correfponding vaniftiing Points ; viz. s for the 
 Line ST, L f or the Lines SX and OT, and O for the Line XO. 
 In like Manner the Points L, in Fig. 48, 49, 50, are determined* 
 viz. "by drawing the Lines EL, from the Eye, parallel to the Ori- 
 ginals el and SB. — From hence, then, we may perceive, that the 
 various Situations of Objects may be reduced under three general 
 LIeads * viz. 
 
 1. When they are perpendicular to the Picture, or parallel to 
 the Ground. 
 
 2. When they are parallel to the Picture, or perpendicular to 
 the Ground. 
 
 3. When they are obliquely iituated, both as to the Picture and 
 the Ground, or any other Plane upon which we fuppofe them: 
 All which I ftiall now endeavour to explain in their feveral Orders, 
 and apply them to Practice. 
 
 SECT. 
 
Definition «• 
 
 SECT. II. 
 DEFINITION S. 
 
 I. ryiHE Point of Sight, is that Point where the Spectator V Eye 
 A is placed to look at the Picture. Thus the Point E, of 
 all the Figures in Plate 6, is the Point of Sight, or Place of the Eye. 
 45« 2. If from the Point of Sight E, a Line, EC, be drawn per- 
 pendicular to the Picture GLHL, the Point C, where that Line 
 cuts the Picture, is called the Center of the Picture* 
 
 |. The Dijlance of the Picture, is the Length of the Line EC 
 which is drawn from the Eye, perpendicular to the Picture. 
 F'g- 4«, 4. If from the Point of Sight E, a Line EP be drawn perpen- 
 49, S 0 ' dicular to any vaniming Line VL, the Point P, where that Line 
 cuts the vanifliing Line, is called the Center of that vanijhing Line. 
 
 5. The Dijlance of a vanijhing Line, is the Length of the Line 
 BP, which is drawn from the Eye perpendicular to the faid Line. 
 
 6. By Original Object, is meant the real Object whofe Repre- 
 fentation is fought, whether it be a Line, Point, or plane Fi- 
 gure : And by Original Plane, is meant that Plane upon which the 
 
 Fig. 45. real Object is fituatedj thus the Ground OP, is the Original 
 Plane, and TOSX the Original Object. 
 
 7. The Line GL, where an original Plane OP cuts the Pic- 
 ture GLHL, is called the Section of the Original Plane, or the 
 Ground Line. 
 
 8. If any Original Line OT, be continued fo as to cut the 
 Picture; the Point G, where it cuts the Picture, is called the Inter- 
 fection of that Original Line. 
 
 9. The Vanijlring Line of any Original Plane,. &c. is that Line,, 
 where a Plane drawn through the Eye, parallel to that Original 
 Plane, cuts the Picture : Thus HL in this Figure, and VL in Fig.48, 
 49, 50, are the vanifliing Lines of their feveral Original Planes, 
 TOSX and ABTS. 
 
 10. The Vanijhing Point of any Original Line, is that Point where 
 a Line drawn from the Eye, parallel to that Original Line, cuts 
 
 Pig. 48. the Picture : Thus EL, being parallel to the Original el, produces 
 the vanifhing Point L ; and fo on. 
 
 Theorem i. 
 
 % 5 t. If two or more Planes, ABCD, EFGH, are parallel to each 
 other, they will have the fame vanifliing Line HL. 
 
 For 
 
Theorems, &c* vj 
 
 For let GHLL be the Picture, E the Spectator's Eye, and 
 ABCD an original Object. 
 
 Imagine the Plane HIKL to pafs through the Eye E, parallel 
 to the original Object ABCD, and it will cut the Picture in the 
 Line HL, which will be the vanifhing Line of the original Plane 
 ABCD : And fmce the other original Plane EFGH, is parallel to 
 ABCD, therefore the Plane HIKL is parallel to that alfo ; and 
 confequently HL is the vanifhing Line of the Plane EFGH, and 
 of every other Plane which is parallel to ABCD, 
 
 Theorem 2. 
 
 The vanifhing Points, H and L, of Lines AC, BD, in any ori- 
 ginal Plane ABCD, are in the vanifhing Line HL, of that Plane. 
 
 From the Eye E, draw EH, EL, parallel to BD and AC; 
 then becaufe the original Plane ABCD, and the Plane HIKL, 
 are parallel ; therefore the Lines EH, EL, that are drawn from 
 the Eye E, parallel to the original Lines BD, AC, will be in the 
 Plane HIKL; and confequently muft, cut the Horizontal, or va- 
 nifhing Line HL, in the Points H, L, and thereby produce the 
 proper vanifhing Points of the original Lines B D, A C. 
 
 Theorem 3. 
 
 If the original Plane ABCD, is parallel to the Picture GHLL, fig. $z< 
 it can have no vanifhing Line upon it, and therefore its Repre- 
 fentation will be parallel, as in Fig. 42. becaufe its parallel Plane 
 abed, which panes through the Eye E, can never cut the Picture, 
 and confequently, will not produce a vanifhing Line upon it. 
 And fo in regard to the Line BD : It can have no vanifhing Point 
 upon the Picture, but its Reprefentation will be parallel to the 
 Original, as o s, t x, in the above Figure. 
 
 Theorem 4. 
 
 The Reprefentation ab, of a Line AB, is a Part of the Line Fig. 53, 
 GC, which panes through the interfering Point G, and the va^ 54- 
 nifhing Point C, of the original Line AB. 
 
 For imagine the Plane AHEF, to pafs through the Eye E, 
 and the original Line AB, and it will pafs through both the in- 
 terfecting Point G and the vanifhing Point C, and cut the Picture 
 in the Line GC: And if the vifual Rays AE, BE, are drawn 
 from the Object to the Eye, they muft be in the Plane AHEF, 
 
 E a and 
 
a g Theorems, &c. 
 
 and confequently, their Section a b with the Picture, will be m 
 the Section GC of that Plane with the Picture j therefore, ab„ 
 which is a Part of the Line GC, is the Reprefentation of the 
 Line A B, 
 
 C O I O L l r 
 
 When the Original is perpendicular, as AB, Fig. 53, then its 
 vanifhing Point will be in die Center C of the Picture becaufe 
 a Line drawn from the Eye perpendicular to the Picture, deter- 
 mines its Center j and therefore, fince AB is perpendicular to the 
 Picture, EC is parallel to it, and confequently will produce the 
 Center G, for the vanifhing Point of A B» 
 
 CoR O L. 2. 
 
 If the Original A B is in a Plane OPB, perpendicular to the 
 Picture, but lies obliquely in that Plane in regard to the Picture, 
 *ig. 54- as AB ; then its vanifhing Point L, will be in the Horizontal Line 
 HL> but on one Side of the Center C : And fo whatever be the Si- 
 tuation of any original Line, its Reprefentation upon the Picture 
 will always be in that Line which is drawn through its Interfectioa 
 and vanifhing Point, 
 
 C O R O L, 3> 
 
 yi s- jrs» For let AB be inclined to the original Plane OP, at the 
 Angle ABD. 
 
 Continue AB till it cuts the Picture in G, and from the Eye E x 
 draw EF parallel to it* which will cut the Picture in the vanifhing 
 Point F ; then draw FG, and the vifual Rays AE, BE, cutting 
 FG, in a and bj, then will the Line ah be the Reprefentation of 
 the Original AB, and is a Part of the Line FG, which pafles 
 through the interfering Point G, and the vanifhing Point F, of 
 the Original AB. This, from what was, obferved above, is felf- 
 evidentj becaufe the Rays AE, BE, are in the Plane AFEG> 
 which pafles through the Eye and the original Object, and there- 
 fore muft cut the Picture in the Section FG, 
 
 COXOL 4. 
 
 From hence it follows, that all Lines which are parallel to each 
 other, but not parallel to the Picture, will have the fame vanifhing 
 Point j becaufe a Line which panes through the Eye, being parallel 
 to one, is parallel to all the reft; and therefore can produce but 
 
 one- 
 
Theorems, &c. 
 
 29 
 
 one vanifhing Point, let the Number of parallel Lines be ever fa 
 many This I have explained by Paper Planes, where OPHL is 5 6 - 
 the Picture, TPE a Plane which pafles through the Eye parallel 
 to the Picture, and AB, CD, EF, three original Lines parallel to 
 each other. Now if we raife the Picture OPHL, and the Plane 
 TPE, 'till they are perpendicular to the original Plane A E K I» 
 and then turn the other Planes, which pafs through the ori- 
 ginal Objeas AB, CD, EF, 'till they coincide with the Eye at E ; 
 they will all meet upon the Picture in the Point C, which is the 
 common vaniftiing Point of all the original Lines AB, CD and EF. 
 And by obferving the vifual Rays, which are drawn from the 
 Extremities of every original ObjecT: to the Eye, at E, we may 
 perceive that the Reprefentation of the Line AB, will be ab 
 upon the Piaure j of CD, cd> and of EF, ef : All which Repre- 
 fentations will tend to the Point C, as a common Center, and there 
 vanifti into the Pi&ure. And we may moreover obferve, that 
 fmce the original Lines AB, CD, EF, are not only equal and pa- 
 rallel to each other, but at equal Diftances from the Pidure or 
 Section GLj that therefore their Reprefentations will be at the 
 fame Diftance from the Section, GL, of the original Plane, and 
 between the fame parallel Lines a e, b f. 
 
 This laft Theorem, and the Corollaries deduced from it, are the 
 principal Foundation of all the Practice of Perfpeaive; and there- 
 fore the Reader will do well to make it very familiar to him : And 
 to help his Reflations upon it, I have annexed the laft Figure. 
 But although I have confined myfelf in this Figure to an original 
 Plane which is perpendicular to the Piaure, yet the fame Rules 
 will ferve for any other original Planes, be they ever fo obliquely 
 fituated in regard to the Piaure; provided they are parallel amongft 
 themfelves: As muft appear extremely obvious, by a Httle Attention 
 in examining the Figure. 
 
 Theorem 5. 
 The Reprefentation ab, of any Line AB, that is parallel to the Fig. 56, 
 Piaure, is to its original Line AB, as the Diftance EC of the No. 2. 
 Reprefentation ab is to the Diftance ED of the original Figure. 
 For let the original Figure AB be two Parts, and the Diftance 
 ED (or which is the fame Thing, AH) five Parts; and the Dif- 
 tance EC, (or HG) of the Reprefentation ab, two Parts ; then 
 will AB be to its Diftance ED as five to two. For if we divide 
 
 the 
 
3<> Theorems, ®c. 
 
 the Diftance CE of the Reprefentation ab, into five Parts, then 
 the Reprefentation ab will be equal to two of thofe Parts; that is, 
 as five is to two. Again, the Diftance Ca, between the vanifhing 
 Point C, of a Line AO, and any Point a in its Reprefentation 
 O a ; is to the Diftance CO, between the vanifhing Point C and 
 the Interferon of that Line, as the Diftance EC (or HO) of 
 the Eye, is to the Diftance HA of the original Point. For let HA 
 be five Parts, and HO two Parts; divide OC into five Parts; and 
 the Diftance C a, between the Reprefentation a of the Point A, 
 will be two of thofe Parts; therefore, Ca is to CO, as HO is 
 to HA ; that is, as two is to five : As is evident by mfpe&ins: 
 the Figure. J r 6 
 
 From hence, then, we mayobferve, that the perfpe&ive Repre- 
 fentations of Obje&s are diminifhed upon the Picture in an har- 
 monkal Proportion; and that, if the Length of any original Ob- 
 jed, its Diftance, together with the Diftance and Height of the 
 Eye, are known, that then the Appearance of thofe Objects upon 
 the Picture may be found by Calculation ; which will be exempli- 
 tied in the pra&ical Part. Proceed we, therefore, in our propofed 
 Order*, to determine the Reprefentations of Objects which are 
 in Planes varioufly fituated in regard to the Picture, 
 
 SECT. IIL 
 
 Of Objects which are in Pkncs perpendicular to the PiBure. f 
 
 1-Ji Plane OGLP and let E be the Eye, and EC its Diftance. 
 *rom what has been -faid already it is manifeft, that abed it 
 
 & v-rfp ntat prL ABCDj for the P <^ts a,b,c,d, are where 
 the vifual Rays BE, flfr. are cut by the Pitfure, as was obferved in 
 
 i 1 !' 4 *' 42 ' the Reprefentations ab, cd, arc Parts of the Lines 
 1 bt, which are drawn from the interfe&ing Points T and S 
 and the vammmg Point C, of the original Lines AB, CD ; as was" 
 fhewn in Fig. 53 f + . and confequently ad, be, are the Reprefen- 
 tations of then- Originals AD, B C. r 
 
 • Vid« Page 25. 
 
 Now 
 
 Kg- 57- 
 
Of Perpendicular Objects. 
 
 Now let us fuppofe the original Plane OGLP to be turned 
 upon its Section GL ; and the parallel Plane HIKL to be turned 
 alfo upon the vanifhing Line HL, 'till thofe Planes and the Picture 
 become one ftrait Plane, like flD^llt j then it is manifeft that the 
 Eye E, will be tranfpofed into the Point €, and €C will be equal 
 to its Diftance. And if we moreover fuppofe the original Figure 
 ABCD, to be drawn upon the under Side of the Plane OGLP^ 
 and exactly in the fame Situation as in the Plane JJDGLp } 
 
 then, I fay, if Lines are drawn from the feveral Points in 
 this tranfpofed Plane, to the tranfpofed Place of the Eye, that 
 their Sections a, b, c, d, with the Lines TC, SC, will be in the 
 •irery fame Points, in which thofe Lines are cut by the Rays, which 
 go from the original Points A,B,C,D, in the Plane OGLP, to the 
 Eye E : Thus the Ray BE cuts the Line TC in b; and if a Line 
 is drawn from % to <fc 9 it will cut TC in the fame Point b; and fo 
 of the reft. From whence it follows, that the Reprefentation 
 abed, may be as- exactly determined by thus tranfpofing the Planes, 
 as by thofe imaginary Rays of Light which go from the real Ob- 
 ject to the Eye. 
 
 That the Senfe of this Figure may be the more clearly compre- 
 hended, in Fig. 57, No. 2, are all the above Planes laid flat upon 
 the Paper ; and may eafily be diftinguifhed by the Letters which 
 denominate each Plane. Thus OPLG is the original Plane, ABCD 
 the original Object, T and S the Section of the Sides AB, CD, 
 with the Picture GLHL: The parallel Plane is HIKL; and HL 
 the vanifhing Line of the original Object. C the Center of the 
 Picture ; E the Eye 5 and EC its Diftance. — Thefe Things being 
 premifed, let us apply them to Practice by drawing the above 
 Reprefentation. 
 
 From T and S draw TC, SC, and from the feveral Points Fig 
 A, B, C, D, draw Lines to the Eye at E, which will cut TC, SC, N< 
 in the Points a, b, c, d; then draw ad, b e, parallel to HL, and 
 the Reprefentation is compleated. 
 
 From hence, then, it follows, that if the Situation, or Seat of 
 an original Object, together with the Place of the Picture, and the 
 Diftance of the Eye, are known, that then the Reprefentation of 
 that Object may be eafily determined : For let us now, without 
 any. Regard to the former Figure, call OPGL the Ground, ABCD 
 an original Object, GLHL the Picture, HL the Horizontal Line, 
 C the Center of the Picture, and CE the Diftance of the Eye. 
 
 Front 
 
i 32 Of P £ R J? E N D I C U L A R 0 EJECT;. 
 
 . From the Eye E draw EC, parallel to the Sides AB, CD of 
 the Original, which will cut the vanifhing Line HL, in C, the 
 Center -of the Picture ; becaufe AB and CD are perpendicular to 
 the Picture, that is, perpendicular to the Section G L j therefore 
 C is the vanifhing Point of AB and CD. — Continue the Sides AB, 
 CD, 'till they cut the Section GL in T and S. From T and S 
 draw Lines to C ; then from the feveral Points A, B, C, D, draw 
 Lines to E, which will cut TC, SC, in the Points a, b, c, d : 
 Finally, draw the Lines a d, b c, which will give the Represen- 
 tation required. 
 
 This Reprefentation may alfo be determined without drawing 
 Lines from the original Points A,B, C, D, to the Eye E, by means 
 
 of the Diagonal AC continued, and its parallel EN. For 
 
 Continue the vanifhing Line HL, and the Section G L, at 
 pleafure ; continue alfo the Diagonal AC, till it outs the Section 
 m M : From E, draw EN, parallel to AC; and from N, where 
 EN cuts the vanifhing Line, draw NM, cutting TC, SC, in the 
 Points a and c; then is a the Reprefentation of A, and c the 
 Reprefentation of C; therefore from a and c, draw ad, be, 
 parallel to HL, and the Thing propofed is done. 
 s 8 ' For let ABCD be an original Square, and AC, BD, Diagonals 
 drawn in it; and let A Bed be its Reprefentation upon the Pic- 
 ture. — C is the Center of the Picture, and CE its Diflance. 
 
 Through E, draw EL and EH, parallel to the Diagonals AC, 
 B D, cutting the vanifhing Line in L and H ; then are L and H 
 the vanifhing Points of thofe Diagonals j for there the Picture 
 is cut by Lines which are drawn from the Eye parallel to the Ori- 
 ginals AC, B D, And for the fame Reafon, (as we have obferved 
 before) C is the vanifhing Point of AD, BC ; and therefore, if 
 Lines are drawn from the Sections A, B, to the vanifhing Points 
 H, C, L, their mutual Interactions c, d, with AC, and BC, will 
 determine their feveral Reprefentations : Thus Ad is the Repre- 
 fentation of AD, Be of BC, Ac of AC, and Bd of BD ; and 
 by drawing cd (which will be parallel to the Horizontal Line) 
 the Reprefentation of the whole Square will be compleated. 
 
 The practical Part is reprefented by the 59th Figure ; where all 
 the Planes are laid down, as before, with correfponding Letters to 
 diftinguifh them. 
 
 From hence, then, we may obferve, that any plane Figure 
 may eafily be drawn upon the Picture by refolving the whole into 
 Triangles. 
 
 For 
 
Of P E R- P E N D I C U L A R OBJECTS. 3 £ 
 
 For let A BCD be a Square refol-ved into four Triangles,, as Fig 59 » 
 AND, ANB, CND, CNB. Then, by means of the three va- 
 niihing Points H, C,L, which are found by drawing EC, EH, EL, 
 parallel to AD, BC, AC, BD, the Reprefentations of thofe Tri- 
 angles may be found j as in the Figure. And fo likewife in Fig. 
 60, the Reprefentation of the Parallelogram ABCD, by means 
 of the Points H, L ; or the Triangles ABC, ADC, by means of 
 the Points P, H, L, may be determined. 
 
 Thefe two laft Figures, though fo very fimple, contain the 
 greater!: Part of Practical Perfpective : For, however original 
 Planes are fituated, or however any Lines are drawn upon them, 
 their Reprefentations may always be determined upon the Picture, 
 by continuing the original Lines 'till they cut the Picture, and 
 by drawing Line* through the Eye parallel to them. All the Dif- 
 ficulty lies in being careful to, draw the Lines from the right 
 interfering and vanifhing Points ; which a little Practice will 
 make extremely eafy : And, therefore, here the Learner will do 
 well to exercife himfelf with the Examples under this Head in. 
 Book II. Sea. a. 
 
 SECT. IV. 
 Of Objects which are in Planes perpendicular to the Ground- 
 
 HEre T OSX is a fquare Plane which fliands upon its Side TO, Fj g . $ %t . 
 perpendicular to the Ground Plane OP, and is alfo perpen- 
 dicular to the Picture. 
 
 Now let E be the Eye, C the Center of the Picture, and CE its 
 Diftance. — From the Eye E, draw EH parallel to TX or OS, 
 and EC parallel to OT: Then beeaufe EC is parallel to TO 
 and S X, therefore C is the vanifhing Point of thole Lines ; and 
 therefore, from C draw CL, cutting the Section GL in L ; and 
 then from L draw LH parallel to CE, which will compleat a. 
 perpendicular Plane CEHL, that panes through the Eye parallel . 
 to the original Object TOSX; and therefore CL, ks Section with 
 the Picture, is the vanifhing Line of that original Plane. And 
 fence CE is by Conftruction perpendicular to CL, therefore C is 
 the Center of the vanifhing Line, and alfo the Center of the Pic- 
 ture, and CE is its Diftance. 
 
 Again, continue CL at pleafure; and from the EyeE, draw EA, 
 EB } parallel to the Diagonals OX, TS, which will cut the vanifhing. 
 
 E Line 
 
34 0/ Parallel Objects. 
 
 Line A B in the Points A and B; therefore A and B are the va- 
 nishing Points of thofe Diagonals, by means of which the whole 
 Reprefentation may be determined. Thus G is the Section of 
 the Side OT, and C its vanifhing Point, therefore draw GC; 
 then from T and O draw Lines to E, which will give the Ap- 
 pearance to of TO; and from t and o draw the Lines tx, os 
 parallel to the vanifhing Line AB (that is, perpendicular to the 
 Ground Plane) and continue them at pleafure : Finally, from A 
 draw a Line through o, cutting tx in x, and from B draw a 
 Line to t, which will cut o s in s, then draw sx to its vanifhing 
 Point C, which fmifhes the Figure. 
 62 • But to apply this to Practice. — The Planes being fuppofed to 
 be laid flat, as in Fig. 57. No. 2, 
 
 Then OT reprefents the Seat, or Plan, of the original Plane 
 TOSX, in the laft Figure, TG its Diftance from the Picture, 
 AEB the parallel Plane, E the tranfpofed Place of the Eye, and 
 CE its Diftance. J 
 
 From the Extremities O, T, of the Seat OT, draw T 1, &% at 
 pleafure, but parallel to each other, cutting the Section in 1 
 and 2 ; make CB equal to the Diftance CE of the Picture, and 
 from B draw BH, parallel to T 1, O 2, cutting the horizontal 
 Line in Hz Then is H the vanifhing Point of the Lines T 1, O 2 ; 
 therefore draw H i, H 2, and from G draw G C, which will be cut 
 by the above Lines in the Points t, o ; and thereby give t o for the 
 Jteprefentation of TO. Again, from t and o, draw the Lines t x, 
 o s, at pleafure, but parallel to the vanifhing Line AB; then from 
 A draw a Line through o, cutting tx in x; and from B draw a 
 Line to t, which cutting os in s, will determine the laft Angle of 
 the Square; and therefore, by drawing sx to its vanifhing Point 
 C, the whole Reprefentation will be compleated. — I have made 
 ufe of both the vanifhing Points A, B, to exercife the Learner, but 
 one Point will do; thus, Ax determines the Side tx; therefore 
 draw xC, which will cut o«, and give the other Side o s. 
 pfcti. Here let us obferve, that when the Seat OT, of any Plane, is 
 perpendicular to the Picture, the vanifhing Line of that Plane will 
 pafs through the Center of the Picture, and be perpendicular to 
 the horizontal Line; But, if the Seat OT, Fig. 63, of any per- 
 pendicular Plane, TOSX, be oblique with the Picture, then its 
 vanifhing Line, A B, will not pafs through the Center of the Pic- 
 turej but on one Side of it ; neverthelefs, it will always be perpen- 
 dicular to the horizontal Line, and will pafs through the vanifh- 
 ing Point L, of its Seat OT, For 
 
Of Parallel Objects. 3 1 
 
 For, draw EL, parallel to OT, and it will cut the horizontal Fig. 63. 
 Line in L: From E and L, draw EH, LL, parallel to TX or 
 OS j and from L, where LL cuts the Section GL, draw LH pa- 
 rallel to EL - 3 then is the Plane LLHE parallel to the original 
 Plane TOSX, and confequently perpendicular to the Ground y 
 and therefore LL, its Section with the Picture, is the vanifhing 
 Line of that original Plane, and is perpendicular to the horizontal 
 Line : And fince the vanifhing Point L is in the Section L L, 
 therefore LL continued will pafs through that Point, and confe- 
 quently AB is the vanifhing Line of the Plane TOSX* Again j 
 mice EL is perpendicular to the vanifhing Line AB, therefore L 
 is the Center of that vanifhing Line, and EL its Diflance 5 and 
 therefore, from E draw EA, parallel to the Diagonal OX, and 
 EB parallel to the Diagonal TS, cutting the vanifhing Line in A 
 and B then are A and B the vanifhing Points of thofe Diagonals j 
 from whence the Reprefentation may be compleated, as in the 
 former Figure.. 
 
 But to apply this to Practice, Let the feveral Planes be fup- 
 pofed to be laid down as before. 
 
 Then T O is the Seat of the original Object, L its vanifhing Fig. 64; 
 Point, C the Center of the Picture, EC its Diflance, L the Center 
 of the vanifhing Line AB, and EL its Diflance. 
 
 From the Section G> draw GL to its vanifhing Point, and from 
 the Extremities T, O, of the Seat TO, draw two parallel Lines 
 at pleafure, cutting the Section GL in 1 and 2 ; from E, draw 
 E3, parallel to T 1 and O2, cutting the horizontal Line in 35 
 then draw 13, 23, which will give the Reprefentation ot ; again, 
 from t and o, draw the Lines tx, os, parallel to the vanifhing 
 Line AB : And then, by means of the vanifhing Points A and B, 
 the whole Reprefentation may be compleated, as in Fig. 62. 
 
 This Figure alio deferves the Learner's particular Attention ; for 
 if he obferves, in Fig. 62, the vaniming Line AB pafTes through 
 the Center of the Picture, and therefore the Diflance CE of that 
 vanifhing Line, is equal to the Diflance of the Eye, or principal 
 Diflance : But in this lafl Figure, fince the vanifhing Line does 
 not pafs through the Center of the Picture, therefore, the Diflance 
 E L, of that vanifhing Line, is greater than the principal Diflance 
 CE, and will be proportion ably greater and greater, as the vanifh- 
 ing Line is removed farther and farther from the Center of the 
 Picture. For the principal Diflance EC, is one Side of a right- Fig. 64. 
 angle Triangle ECL j but EL, the Diflance of the vanifhing Line 
 
 E 2 AB, 
 
Of Parallel Objects, 
 
 AB, is the Hypothenu-fe of that Angle, and therefore greater than 
 either of the Sides EC or C L : From whence it follows, that if a 
 line CL be drawn from the Center of the Picture, perpendicular 
 to any vaniming Line AB, the Point L, where that Line cuts the 
 Picture, will determine the Center of that vaniming Line j and 
 if a Line be drawn from the Eye to that Point, as E L, it will 
 determine its Diftance *. 
 
 Let us now, without any Regard to the Theory, find the Ap- 
 pearance of a fquare Object fituated like TOSX, in Fig. 63. 
 *%• 64. Let T O be the Seat of the Object propofed, FI L the horizontal 
 Line, C the Center of the Picture, CE the principal Diftance, 
 and GL the Section of the Ground Plane with the Picture. Con- 
 tinue the Seat O T, 'till it cuts the Section in G, and parallel to 
 OT, draw EL from the Eye, cutting the horizontal Line in the 
 vaniming Point L; then draw GL: Finally, draw Ti, O 2, and 
 and alfo their Parallel E 3 ; by which means the Heprefentation ot, 
 may be found. Again, through the vaniming Point L, draw B A, 
 perpendicular to HL, and continue the horizontal Line towards 
 <fe, at pleafure 3 then, becaufe C L is perpendicular to the vanim- 
 ing Line AB, therefore L is the Center of that vaniming Line, 
 and confequently, EL is its Diftance : Therefore continue the Per- 
 pendicular CL, at pleafure, beyond the vaniming Line A B, and 
 from L, with the Radius LE, defcribe an Arc ACBE, cutting the 
 vaniming Line in A and B, and C L continued in <t , then are A 
 and B the vaniming Points of the Diagonals o x, and t s, and <t 
 is the proper Diftance of the Eye : Therefore by drawing Perpen- 
 diculars from t and o, and Lines from A and B, through the 
 fame Points t and o, they will cut the Perpendiculars t x, o s, in 
 x and s, and thereby give the Height of the Square 3 from whence, 
 by drawing x L, it will be compleated. 
 
 From hence, then, it is manifeft, that the Method for finding 
 the Reprefentation of an upright Plane, is exactly the fame as that 
 for determining the Appearance of a Plane which lies flat upon the 
 Ground, only the Situation of the vaniming Line is different , but 
 the Operation in both Cafes is the very fame , which may be con- 
 ceived by turning the Figure, and imagining A B to be the hori- 
 zontal Line, L the Center, and LC the Diftance of the Picture : 
 For then this Figure will be like Fig. 59. But that the Learner 
 
 * Sec Definition 2, Se&. 2. Chap. 1 — alfo Definitions 4, 5, Sell. 2, of this Chapter. 
 
 may. 
 
Of Inclined Objects. 37 
 
 may underftand the Meaning of this more perfectly, he is tlefrred, 
 before he proceeds any farther, to exercife himfelf with fome Ex- 
 amples of this Kind ; which he will find in Book II. Seel 3. 
 
 SECT. V. 
 
 Of Objects which are inclined to the Ground, 
 
 fipHE Objects which come next under Confideration, are fuch 
 JL as are neither perpendicular nor parallel to the Ground, but 
 inclined to it ; like the Roofs of Houfes, Pediments, and the like ; 
 the vaniming Lines of which cannot be the horizontal Line, nor 
 any Line that is perpendicular to it. 
 
 For let T O S X Y Z be the original ObjecT:, having one Side Fig. 6 f . 
 TOSX upon the Ground, and one Side TOYZ inclined to it, 
 at the Angle YTX; and let the other Sides be perpendicular to 
 the Ground.-- -E is the Eye, C the Center of the Picture, and 
 C E its Diftance. 
 
 From E draw EC, parallel to XT or SO; then is C their 
 vaniming Point : And becaufe TX is perpendicular to the Pifture, 
 therefore its vaniming Point C is the Center of the Pi&ure. And 
 fmce the Plane TXY, is perpendicular to the Ground, therefore its 
 vaniming Line LD, is perpendicular to the horizontal Line ; and 
 therefore, through the vaniming Point C, drawLD, which con- 
 tinue at pleafure, then from E draw E D, parallel to T Y or O Z, 
 which will cut the vaniming Line LD, and give D for the vanifh- 
 ing Point of the inclined Sides TY, OZ. And if a Line, VL, be 
 drawn through D, parallel to the horizontal Line H C, it will be 
 the vaniming Line of the inclined Plane, TYZO; becaufe, if a 
 Plane was to pafs through the Eye, parallel to T Y Z O, it would 
 cut the Pi&ure in the Line VL. And fince ED is perpendicular 
 to the vaniming Line VL, therefore D is the Center of that 
 vaniming Line, and E D its Diftance. 
 
 To apply this to Practice. Let us fuppofe the Planes to be laid 
 down as m the former Figures ; only for Convenience, we have 
 removed the Seat TOSX, farther from the Middle of the Pifture. 
 
 Here TOSX is the Seat of the original Object, H L the hon- 
 
 zontal Line, E the Eye, C the Center of the Picture, and C E its 
 Diftance. 
 
 Find the Reprefentation of TOSX, as before direfted, by means 
 of the Lines O 1, S 2, and their parallel DH s Then, parallel to 
 
 the 
 
g8 Of Inclined Objects, 
 
 the horizontal Line HE, draw a Line ab, Fig. z, at pleafure, an<f 
 through C draw the vanifhing Line DL, perpendicular to the 
 horizontal Line, at pleafure alfo : With the Line ab, and at the 
 Point b, make an Angle a be, equal to XTY, the Angle of Incli- 
 *Fig. 65. nation of the original Figure j * then from E, the Diftance of the 
 Eye or principal Diftance, draw ED parallel to be, cutting the 
 vanifhing Line in D 5 finally, from D draw Dt, Do, and from s 
 drawsz, parallel to DL, which will cut o D in z ; therefore* 
 from z, draw z y parallel to t o, or H E, and the Thing propofed 
 is done* 
 
 Or the vanifhing Point D may be determined without the Fi- 
 gure z, by making an Angle at E, the Diftance of the Eye,, with 
 the horizontal Line HE, equal to the Angle of Inclination, and 
 then drawing ED. 
 
 In Fig. 65, the Plane TO ZY is inclined to the Ground Plane* 
 but reclined in refpect to the Picture, and therefore its vanifhing 
 Line V L will be above the horizontal Line : But in Fig. 67, the 
 inclined Plane T OZY is inclined to the Ground and to the Picture 
 alfo ; for which Reafon, its vanifhing Line VD will be below the 
 horizontal Line.-— The 68th Fig. reprefents the laft Figure applied 
 to Practice, the Operations of which are the very fame with thofe 
 in Fig. 66; only the Seat TOSX, and the vanifhing Point D, are 
 inverted ; that is, are below, inftead of above the horizontal Line. 
 Fig. 65, From hence, then, it is evident, that D is the vanifhing Point 
 66, 67, &q£ a u Lines which are parallel to the Sides oz, and ty; and 
 therefore, when the Figure confifts only of parallel Sides, as o z 
 and t y, there will be no Occafion for drawing the vanifhing 
 Fig. 65. Line VL or YD; fince the vanifhing Point D of thofe Sides 
 « — 6 7- is only wanted. But if any other Lines are fuppofed to be drawn 
 upon the inclined Plane, as in Fig. 69, then thofe vanifhing 
 Lines become necefTary ; becaufe the vanifhing Points of thofe 
 Lines will be fomewhere in them. Which comes next under 
 Confideration. 
 
 Fig 69. Let t o z y be the Reprefcntation of one inclined Plane, whofe 
 vanifhing Point isD; and cdef another inclined Plane, whofe 
 vanifhing Point is D ; and let V D L be their vanifhing Lines. 
 — E is fuppofed the Eye, C the Center of the Picture, and C E 
 its Diftance.— Continue the vanifhing Line DD at pleafure : Then,, 
 becaufe CD is drawn from the Center of the Picture, perpendi- 
 cular to the vanifhing Lines VL, VL, therefore D, D, are the 
 Centers of thofe vanifhing Lines, and DJE, DE, their Diftance from 
 
 the 
 
Of Inclined Objects. 
 
 the Eye; confequently if DI, DI, be made equal to DE, DE, then 
 I, I, will reprefent the tranfpofed Places of the Eye; and therefore 
 if Lines are drawn from the Points I, I, parallel to any original 
 Lines, they will cut the vanifhing Lines VL, VL, and give the 
 vaniming Points of fuch Lines. Thus, let it be required to find 
 the vanifhing Points of the Diagonals of a Square, t o 1 2, one of 
 whofe Sides to is given.*-- Any where apart draw a Square, as X, 
 at pleafure, but in fuch a Manner that its Sides, a b, c d, are pa- 
 rallel to the vanifhing Line V L ; and likewife draw its Diagonals, 
 — Firft for the Figure t o s z y. 
 
 From I, draw IL, IV, parallel to ac, bd; which will cut the 
 vanifhing Line in V and L; and from t draw tL, cutting o D in 
 2 ; from o, draw o V, cutting t D in 1 ; then draw 1 2 parallel 
 with t o, and then is t o 1 2 the Reprefentation of a Square 
 upon the inclined Plane tozy ; and 1 2, 01, are the Reprefenta- 
 tions of its Diagonals. And were it demanded to make the Length 
 of the inclined Plane equal to feveral Times its Width, as in this 
 Figure, we may do it by means of the Points V and L ; becaufe 
 having determined one Square, all the reft are to be found in the 
 fame Manner. 
 
 Here let us take Notice, that if one vaniming Point of any 
 Plane is determined, all the other vaniming Points of Lines which 
 can be drawn any how in that Plane, will be fomewhere in a Line 
 which is drawn through that Point. Thus C is the vanifhing 
 Point of the Side o s, which lies upon the Ground, and the hori- 
 zontal Line HE panes through that Point: Again, C is the vanifh- 
 ing Point of o s, which is one Side of the perpendicular Plane 
 osz; therefore DCD, the vanifhing Line of that perpendicular 
 Plane, panes through the Point C : And fo again, D is the vanim- 
 ing Point of the inclined Planes, and therefore VL, VL, their 
 feveral vanifhing Lines, will pafs through the Points D, D ; and 
 confequently, all the Lines which can be drawn in either Plane, 
 will have their vanifhing Points fomewhere in the vanifhing Lines 
 of thofe Planes. AH which is explained by various Examples in 
 the fecond Book. 
 
 Hitherto I have confidered the inclined Planes, as having one or 
 more of their Sides parallel to the Piclure, for which Reafon the 
 vanifhmg Lines of thofe Planes are parallel to the horizontal Line. 
 Let us now fuppofe the Plane to be fituated in fuch a Manner as 
 to have all its Sides oblique with the Picture, as in Fig. 70. 
 
 Here 
 
Of Inclined Object f* 
 
 Fig. 70. Here TOZY, is a fquare Plane every way oblique with the* 
 P*5ture; TOSX, its Seat on the Ground; YTX, its Angle of 
 Inclination; E the Eye; C the Center of the Picture, and CB 
 its Diftance.~r.Draw the Horizontal Line HC, and continue it at 
 pleafure; then parallel to TX, or OS, draw EH, cutting the Hori- 
 zontal Line in H ; and then is H the vanishing Point of the Lines 
 TX, OS. Again, parallel to TO, or SX, draw EL, cutting the 
 Horizontal Line in L; then is L the vanifhing Point of the 
 Lines TO, S X ; from whence the Reprefentation of its Seat may 
 be found. Now fince the Plane TYX is perpendicular to the 
 Ground, its vanifhing Line HV will be perpendicular to the 
 Horizontal Line; therefore from the vanifhing Point H, draw HV 
 parallel to XY, and EV parallel to TY, cutting HV in V ; then 
 is V the vanifhing Point of the parallel Sides TY, OZ; and fince 
 L is the vanifhing Point of TO, it is alfo the vanifhing Point of 
 its parallel Side YZ, and therefore, a Line drawn through V and 
 L, will be the vanifhing Line, (as VL) of the inclined Plane 
 TOZ Y.. Here let us obferve again, that if a Line, ED, be drawn 
 from the Eye E, perpendicular to the vanifhing Line VL, then D 
 is its Center, and D E its Diftance. 
 
 To apply this to Practice.— Imagine the feveral Planes to be 
 laid down as before. 
 Fig. 71. Then, HL is the Horizontal Line, E the Eye, C the Center 
 of the Picture, CE its Diftance, HV the vanifhing Line of the 
 perpendicular Plane tyx; VL, the vanifhing Line of the oblique 
 Plane toz y, C its Center, C <£ its Diftance, and H, L, V, the 
 vanifhing Points of the feveral Planes ; or, if you pleafe, of fhe 
 feveral Sides of fuch a Figure. 
 
 Let ot be given for the neareft Side. Continue ot, 'till it cuts 
 the vanifhing Line HL in its proper vanifhing Point L : From L 
 draw L € 3 and from t and o, draw Lines to the vanifhing Point 
 V, and draw V (J : Then is V€L a right Angle ; which bifecl, and 
 draw E£>, cutting the vanifhing Line VL, in SD; then is SD the 
 vanifhing Point of the Diagonal of a Square : Therefore (fince the 
 inclined Plane was fuppofed to be a Square) draw BDt, cutting 
 oV in z; from L, through the Point z, draw Lzy, cutting tV 
 in y ; then draw y z, parallel to H V, which will compleat the 
 whole Reprefentation, not only of the inclined fquare Plane, but 
 the whole Appearance of a Figure like 6 5, 67, but in a different 
 Situation. 
 
 Since 
 
Of Inclined Objects. 4* 
 
 Since this Figure is as difficult in regard to the Practice of Per- 
 spective, as any I can think of, I have annexed the Paper Planes 
 in the 72d Figure, to help the Reader's Reflections upon it ; and 
 to affift him ftill further, we will now find the Reprefentation of 
 fuch an Object without any Regard to the Theory. 
 
 Let E be the Eye, C the Center of the Picture, CE its Dif- Fig. 7 i, 
 tance, HL the horizontal Line, and t o one Side given of the 
 inclined Face. . 
 
 Any where apart draw AB, Fig. X. parallel to the horizontal 
 Line HL, and draw CB perpendicular to AB; then make an 
 Angle at A, equal to the Angle of Inclination (as TYX m 
 Fig. 70) and draw AC. — Continue ot to its vanifhing Point L, 
 and from L draw LE to the Eye ; then at E make a right Angle 
 with the Line LE, and then, becaufe the Side whith lies upon the 
 Ground is fquare at the Corners, therefore H is the vanilhing 
 Point of the two Sides tx and o s, and L is the vanifhing Point 
 of the other two Sides to and sx.— From the vanifhing Point H, 
 draw HV perpendicular to the horizontal Line, and continue the 
 horizontal Line towards 1. From H fet off HI, equal to the 
 Diftance HE of the vanifhing Line H V ; then from 1 draw IV, 
 parallel to AC in Fig. X ; which will cut HV in V, and give 
 HV for the vanifhing Line of the perpendicular Plane t y x ; and 
 by drawing a Line through the Points V and L, we fhall have VL 
 for the vanifhing Line of the inclined Plane t o z y : Therefore 
 from C the Center of the Picture, draw CC, perpendicular to the 
 vanifhing Line VL, and continue it at pleafure; then is C the 
 Center of that vanifhing Line. Again, from C the Center of 
 the Ficture, draw CI perpendicular to CC, and make CI equal 
 toCE the principal Diftance, and then draw IC, which is the 
 Diftance of the vaniming Line VL; therefore, make €<£ equal to 
 Cl and from the vanifhing Points V and L, draw V<B, L<fc, which 
 will be a right Angle : Bifeft the Angle (£, and draw (CSD, cutting 
 the vaniihing Line in 2D; then, as before, 3D is the vanifhing Point 
 of the Diagonal of a Square tozy, from whence the whole Re- 
 prefentation may be compleated. Here alfo the Learner is referred 
 for Examples to Book II. Chap. 2. Sect. 4. 
 
 Thus have I endeavoured to explain the Theory of Perfpective, 
 and to apply it to Practice by the moft familiar and ufeful Ex- 
 amples, and in all the Variety of Inftances which can come with- 
 in the general Practice of Painting, &c. As for other Matters, < 
 * which are out of the common Road, and which ferve rather to 
 
 p perplex 
 
Of Inclined Objects. 
 
 perplex than benefit a Learner, I have purpofely avoided them * 
 and believe, I may venture to affirm, that whoever has attended 
 to what has been faid, and exercifed himfelf regularly with the 
 Examples to which he was referred in the Practical Part, Book the 
 Second, will find no kind of Difficulty in determining the Appear- 
 ances of any Objects upon an upright Picture, let them be of. ever 
 fo irregular a Figure., or howfoever they are fituated 
 
 But thus far I have confined myfelf to the Appearance of Ob- 
 jects upon an upright Picture only, fuch as are generally made 
 choice of for Perfpective Reprefentations : But as there are fome 
 Cafes in which the Situation of the Picture is different, fuch as 
 Ceilings, inclined Walls, or the like, I fhall now proceed to the 
 Confideration thereof, and fhew, that the Reprefentation of Objects 
 upon fuch kind of Surfaces, is deducible from the fame Principles, 
 and confequently, is to be determined after the fame Manner ; 
 which is the Subject of the next Chapter, 
 
 CHAP. 
 
CHAP. IV. 
 
 Of Parallel and Inclined Pictures. 
 
 sect, % 
 
 Of the Parallel Picture. 
 
 WHEN the Picture is perpendicular to the Ground, or any 
 other Plane upon which the Spectator ftands, I call it a 
 perpendicular Picture ; when it is parallel to the Ground, 
 I call it a parallel Picture ; and when it is inclined to the Ground, 
 I call it an inclined Picture. The firft of thefe Situations I have 
 already confidered at large, as being the moft ufefuk Proceed we 
 therefore to the Second, which principally relates to Ceilings or 
 immoveable Pictures. 
 
 Now, whoever has attended to what hath been faid upon the 
 upright Picture, will (I apprehend) find no fort of Intricacy in 
 this, becaufe, on either Picture, the Projection of Objects is deter- 
 mined in the very fame Manner. But if there mould appear any 
 Difficulty, it cannot be in the Operation, but in considering what 
 Objects are proper and what not for fuch kind of Pictures j and 
 the Situation of thofe Objects. For inftance, to reprefent a Land- 
 flap or any Objects which are fuppofed to be upon the Ground, is 
 extremely improper for a Ceiling ; for fmce the Picture is always 
 fuppofed parallel to the Ground, and the Eye is placed between the 
 original Object and the Picture, therefore the Rays of Light in 
 their Paflage from original Objects to the Eye, will not be cut by 
 the Picture, and confequently fuch Objects can have no Projections 
 upon the Picture ; for which Reafon they ought not to be repre- 
 sented. But any Objects which may rcafonably be fuppofed to exift 
 in the Air, or any Story which can be fupported either by Hiftory 
 er Fable, may be reprefented with the greateft Propriety j as may 
 likewife feveral Parts of Architecture, which may ferve either for 
 Ornament, or be ufeful as to the main Defign. And in regard to 
 the Situation of Objects, they are generally fuppofed to be erect, 
 and therefore I mall principally confider them in that Situation s 
 which will be fufficient for our Purpofe, as it will give the Reader 
 a very clear Idea of all that can be faid upon the Subject ; and 
 which, together with the Examples under this Head in the Second 
 
 F 2 Book, 
 
Of the Parallel Picture. 
 
 Book, will enable him to find the Reprefentation of all Objects 
 upon a Ceiling with the fame Facility as he can determine thofe 
 upon an upright Picture. 
 
 Let KM NO be the Ground, E the Eye, EH its Height, DGLP 
 the Picture, (which we will fuppofe a Ceiling) C the Center of the 
 Picture, CE its Diftance, and GACL a Plane perpendicular to 
 the Picture whofe Reprefentation is required. 
 
 From the Section GL, draw GC, LC; then from A and C 
 draw AE, CE, cutting GC, LC, in a and c; then draw a c, 
 which will compleat G a c L, the Reprefentation of the original 
 GACL ; which will, to an Eye placed in E, appear to be erect:. 
 
 Let us now turn the Figure in fuch a Manner that the Picture 
 may become an upright one; then ACMK is the Ground Plane, 
 E the Eye, EC its Diftance, EI its Height, C the Center of 
 the Picture, VL the horizontal Line, and GacL the Reprefen- 
 tation of GACL, which lies upon the Ground. From hence 
 then it is evident, that in order to determine the Appearance of 
 any perpendicular Plane upon the parallel Picture, we muft pro- 
 ceed in the very fame Manner as in finding the Reprefentation of 
 an Object which lies flat upon the Ground in the perpendicular 
 Picture; for in both Cafes, the original Plane ACLG is perpen- 
 dicular to the Picture, only the Situation of the Picture is different 
 in regard to the Eye, and therefore the Reprefentation in both 
 Cafes will be the fame, as is manifeft by infpecting the Figure. 
 
 But fuppofe the original Plane be parallel to the Pidure > then 
 the Reprefentation will be like the Original, and muft be found 
 by the fame Rules as Objects thus fituated are determined upon 
 the perpendicular Picture. 
 
 Thus, let LGPO be the Picture, E the Eye, C the Center, 
 and CE its Diftance, and ABCD the original Plane parallel to 
 the Picture. 
 
 From A, B, C, D, draw Lines perpendicular to the Picture, in- 
 terfering it in the Points G, L, P, Oj then from thofe Points 
 draw Lines to the Center of the Picture; and from the Points 
 A, B, C, D, draw Lines to E, which will interfect GC, LC, PC, 
 and OC, in a, b,c, d, and thereby determine the Reprefentation 
 required. 
 
 Now let us turn this Figure alfo, and call ABMK the Ground 
 Plane ; then this Picture is an upright one, and the Reprefentation 
 a, b,c, d, of the parallel Plane ABCD, in either Situation of the 
 Picture is the fame ; and confequently the Reprefentation of all 
 
 parallel 
 
Of the Parallel Picture. 45 
 
 parallel Objects are to be determined after the fame Manner as in 
 the upright Pifture. ' 
 
 Now, fince the Rules for drawing the Appearance of Objects 
 upon the parallel Picture, are exaaiy the fame as thofe for drawing 
 the Appearance of Objects upon the perpendicular Picture, it fol- 
 lows, that the fame Rules will do in both Cafes, and therefore the 
 Artift has nothing more to remember than this, viz. thofe Objects 
 which in the parallel Pidure are to be reprefented as erect, muft 
 be determined as thofe which lie flat upon the Ground in the per- 
 pendicular Picture ; thofe which are parallel in one Picture, as 
 thofe which are parallel in the other ; and thofe which are oblique, 
 after the fame Manner : Or in other Words, however original Planes 
 are fituated, the Reprefentations of them muft always be deter- 
 mined by imagining a Plane to pafs through the Eye parallel to 
 thofe Planes, which will give their feveral vanifhing Lines, from 
 which the whole Reprefentation may be compleated. Thus, 
 the Plane FGVL, which panes through the Eye E, parallel to the Fig. ni 
 original Plane ACLG, produces the vanifhing Line VL of that 
 Plane; and therefore having the Diftance EC of that vanifhing 
 Line, the Reprefentation of any Lines which can be drawn m the 
 original Plane are eafily found alfo. 
 
 And here we may obferve, that if the original Plane ACLG 
 were infinitely extended, the Triangle GLC would be its indefinite 
 Reprefentation, and confequently the Appearance of all Lines which 
 can be drawn in that original Plane, will be fomewhere within 
 that Triangle. And fo like wife, if perpendicular Planes are erected 
 on the other Sides LP, PD, DG, of the Picture, their indefinite 
 Reprefentations will be the feveral Triangles LCP, PCD, and 
 DCG, and the Center C will be their common vanifhing Point. — 
 For draw the original Plane ACLG upon the Side LG of the Fig. 7 # 
 Pi&ure, and let every thing elfe remain as in the former Figure. 
 —Through E draw the Plane FHGLV, parallel to the Plane 
 ACLG, which will cut the Picture in VL; then is VL the va- 
 nifhing Line of that Plane. Again, from E draw EC, perpendi- 
 cular to V L ; then is C the Center of the Picture. And fince 
 EC is parallel to AG, BS and CL, therefore C is the vanifhing 
 Point of thofe Lines ; and therefore, from C, the Center of the 
 Piaure, draw Lines to G, S, L ; and from A, B, C, draw Lines to 
 E, which will cut the former Lines in the Points a, b, c ; then is 
 a G the Reprefentation of A G, bS of BS, and cL of CL ; and 
 G a c L is the whole Reprefentation of the original Plane ACLG. 
 
 And 
 
Vf the Parallel Pictur f ; 
 
 And after the fame Manner any other Lines, as xz, may be found 
 iiipon the Picture. 
 
 And from hence alfo, we may obferve, that if perpendicular 
 Planes are fet on each Side of the Picture, the Reprefentation of 
 thofe Planes will appear like the Sides of a Room continued up- 
 wards j from whence it follows, that by fuch Deceptions as this, 
 & Room may be made to appear of any Height, by drawing a 
 Reprefentation of this Kind upon a Ceiling with Accuracy and 
 Judgment,' and viewing it from the proper Point. One Example 
 of which I mail give in this Place, by way of Practice, and then 
 refer the Reader again to the fecond Book for more Examples of 
 this Sort. 
 
 76. Let GLPO be a Ceiling, E the Eye, EC its Diftance, and 
 C the Center of the Picture. 
 
 Through the Center C draw Lines parallel to LP, LG, and 
 continue them at pleafure ; then with the Diftance CE defcribe a 
 Circle, cutting thofe Lines in D, F, H: Then DCH is the vanifh- 
 ing Line for the original Planes, which ftand upon the Sides GL 
 :and OP; and ECF is the vanifhing Line of the Planes which 
 ftand upon the Sides GO and LPj and the feveral Lines EC, 
 DC, F C and H C, are the Diftance of the Eye from thofe Lines. 
 Having fettled the vanifhing Lines of the four Sides, their Center 
 and Diftance, it matters not upon which Side we begin to work ; 
 for upon any Side, as G L, draw out one of the original Planes, 
 as ACLG, and upon it draw the Lines XZ, BS, which will make 
 it like the Plane ACLG, Fig. 74. From the feveral Sections 
 G, S, L, draw L ines to Cj and from A, B, C, draw l ines to E, 
 cutting GC, SC, LC, in the Points a, b, c; then from a to c 
 draw a£, and then will Ga be the Reprefentation of GA, S b of 
 SB, and Lc of LC: Therefore, GacL is the Reprefentation of 
 the whole original Plane GACL, and the Triangle GCL is the 
 Reprefentation of that Plane infinitely extended. — In like Manner 
 x z is the Reprefentation of its Original X 
 
 Or the Operation may be fhortned thus. From the extreme 
 Point B of any Perpendicular in the original Plane, draw a Line, 
 EI, at pleafure, cutting the Section in I; then from E draw EK 
 parallel thereto, cutting the vanifhing Line DH in K j from the 
 Sections, of the Perpendicular SB, draw SC; and from the Sec- 
 tion I draw IK, cutting SB in b : Then is bS the Depth of the 
 Reprefentation ; therefore, by drawing GC, LC, and by drawing 
 a Line through b, parallel to GL, the Thing propofed is done. 
 
 Now, 
 
Of the Para an Picture. 47 
 
 Now, in order to transfer this Reprefentation unto all the other 
 Sides, proceed thus. 
 
 From O and P draw Lines to the Center C ; then will the re- 
 maining Part of the Ceiling be divided into three Triangles, GCO, 
 OCP„ PCL; which Triangles may reprefent three Planes perpen- 
 dicular to the Ceiling, infinitely extended, and at right-Angles with 
 each other ; and GC, O C, PC, and LC, reprefent the joining of 
 thofe Planes : For GC and LC are the Representations of GA and 
 LC infinitely extended ; and therefore, having found the Depth 
 (as G a) of the Reprefentation of any given Plane, as above, from 
 the Point a, which determines that Depth, draw a Line, as a e, 
 parallel toOG; and from e, where a e cuts O C, draw another 
 Line e d parallel to OP; and from d, where e d cuts P C, draw 
 dc, which will cut L Cine ; then will GaeO, OedP, and 
 P d c L, be the Reprefentations of three perpendicular Planes of the 
 fame Height as ACLG, and fituated in the fame Manner ; that is, 
 upon the feveral Sides GO, OP, and PL ; and confequently, to an 
 Eye placed at E, and at the Diftance EC, the Sides of a Room 
 will appear to be continued above the Ceiling by the Length of the 
 Perpendicular GA, $. e. the Height of the original Plane ACLG, 
 
 SECT. II. 
 Of the Inclined Picture. 
 
 I Have before obferved, that by an inclined Picture, I would be 
 underftood to mean when the Perfpe&ive Plane is neither per- 
 pendicular nor parallel to the Ground, but inclined to it. Indeed, 
 this Situation of the Picture is very feldom made ufe of , yet as 
 there are fome Cafes which may require the Knowledge of this 
 kind of Perfpe&ive, I have therefore given it a Place in this Work, 
 
 Let O P H be the Ground or original Plane, H L G L the Pic- Fig 77 , 
 ture, inclin'd to the Ground Plane at the Angle PLL; and let E 
 be the Eye, EH its Height, and H its Seat upon the Ground. 
 
 Continue the Picture HLGL downwards at pleafure, as GLFO. 
 From the Seat H of the Eye draw HS perpendicular to the Section 
 GL, cutting GL in S; then through S draw SD, perpendicular to 
 GL alfo, and continue it at pleafure towards FO ; and then from E 
 draw E D, parallel to H S, cutting the Picture in D, and continue 
 EH 'till it cuts DS in V; then from V draw VI, parallel to ED, 
 and from D draw D I, parallel to EV : And then will EDIV be a 
 
 r Plane 
 
48 Of the Inclined Picture. 
 
 Plane which panes through the Eye perpendicular to the Ground 
 Plane OPH, interfering the Picture in the Line DV; and there- 
 fore the Section DV will be the vanishing Line of all Planes that 
 are perpendicular to the Ground Plane and parallel to the Plane 
 EDIVj and for the fame Reafon, V will be the vanifhing Point 
 of all Lines that are perpendicular to the Ground Plane OPH, be- 
 caufe EV which is drawn through the Eye parallel to thofe Lines, 
 will cut the Picture in the Point V: For as in the upright, or pa- 
 rallel Picture, lb alfo in this, the vanifhing Line of any original 
 Plane muft be determined, by imagining a Plane to pafs thro' the 
 Eye parallel to that original Plane 'till it cuts the Picture. And fo 
 alfo in regard to the Center and Diftance of the Picture, or the 
 Center and Diftance of a vanifhing Line j the firft is found by 
 drawing a Line from the Eye, as EC, perpendicular to the Picture, 
 and the latter, by drawing a Line from the Eye, as ED, perpen- 
 dicular to that vanifhing Line : The Method for doing either is as 
 follows. 
 
 1. For the Center and Diftance of the Picture. 
 Having continued the Picture downwards as above directed, and 
 drawn the vertical Plane EDIVj from E, draw EC, perpendicular 
 to the Section DV; then will C be the Center of the Picture, and 
 CE its Diftance: For fince the vertical Plane cuts the Picture at 
 right Angles, andfince EC is in that Plane, and perpendicular to the 
 Section D V, therefore EC is perpendicular to the Picture alfo, and 
 confequently C is the Center of the Picture, and CE its Diftance. 
 
 2. For the Center and Diftance of a vani/hing Lme. 
 Let the Plane ABHL pafs through the Eye E, parallel to the 
 Ground Plane OPH, and it will cut the Picture in H L, which 
 Line HL is the vanifhing Line of the original Plane OPH; and 
 if from E, a Line, as ED, be drawn perpendicular to HL, then D, 
 where it cuts HL, is the Center of that vanifhing Line, and DE is 
 its Diftance. 
 
 Fig. 78. Now, let it be required to find the Reprefentation of the ori- 
 ginal Plane ABGL upon the inclined Picture GLHL; and let E 
 be the Eye, H its Seat upon the original Plane, E C its Diftance, 
 and C the Center of the Picture. 
 
 From H, the Seat of the Eye, draw HS, perpendicular to the 
 Section GL; from S, draw SD perpendicular to GL, and con- 
 tinue it at pleafure; then from the Eye E, draw ED parallel to 
 H S, cutting S D in D finally, through D, draw H L, parallel to 
 
 GL, 
 
Of the Inclined Picturs. 49 
 
 BL, then is HL the vanning Line : of the original Plane : ABGL, , * 
 and D Is the vanhhing Point of the Sides AG, B L , therefore, 
 from G and L draw GD, LD, and from A and B draw AE EE, 
 cutting GD, LD, in the Points a and b 5 then is GabL the Re- 
 
 ^S^iilS^i^M be the Pianre laid F % .,o. 
 fiat as fn fome of the preceding Fignres.-Bifea the Bottom GL, 
 and draw cD perpendicular thereto, and continue it atpleafure: 
 Then from the 78th Figure take SC, CD, and transfer them turto 
 c D in this Fisure, beginning at the Point c ; draw HL ; then is 
 C the toter of the Pifture, cD the Height of the vamftiing Line, 
 and D its Center. Again, make D* equal to the Diftance of the 
 Eye and AG equal tl the Length of the original Plane, (tha . , 
 eaual to AG Fig. 78.) then from G and L draw GD, LD, and 
 fr q om A draw A't, cutting GD in a; finally, from a draw a b, 
 parallel to GL; winch will compleat a Reprefentation GabL, 
 exaaiy like GabL Fig, 78. pj„ w rF nualld 
 
 Or it may be done thus.— From the Center C draw CE, parallel 
 to GL and make CE equal to the Diftance of the Picture and 
 ED equal to^the Diftance of the vanning Line HL ; then from 
 D whh the Radius DE, defcribe the Arc EL«H 5 and from G, 
 with the Radius GA, defcribe the Arc Ac; and ther jfrom c and 
 H draw He, which wilkcut GD 111 a, and give the Depth of the 
 "Renrefentation ; from whence the whole may be compleated. 
 
 Tnhke Manner, let it be demanded to find the Proctor , of a F*. 7 ,. 
 Line AB, which ftands perpendicular to the Ground I Plane OPH. 
 
 From B, the Seat of the Line AB, draw a Line BH to the Seat 
 of the. Eye H ; and from V draw Vd, through the Seftion c, and 
 Continue it at pleafure; then from A and B draw Lmes to he 
 F^eT cu tinged in a and b; and then is ab the Reprefentation 
 ol the Original AB. For fince EV is parallel to the Original AB, 
 the efore tie Point V, where it cuts the Pifture, is the vanilhmg 
 Point of AB, and of all other Lines which are parallel to AB . 
 1 And if we imagine a Plane ABHE to pafs through the original 
 AB, anX the Lme HE, it will cut the Figure m cai«fc ^there- 
 fore, fince the Rays AE and BE are in that Plane, the Section 
 n h will be the Reprefentation of AB. ,•■,„.„ 
 
 To app y this to Praaice.-Let MNGL be the Piaure, laid F,,. ... 
 flat as before. Then C is its Center, CE its Diftance V the va- 
 nifliing Point of Lines perpendicular to t^ Ground P ane HL 
 the vamfliing Line of Planes parallel to the Ground Plane, D the 
 
 G 
 
t>f the I *M C L I MED f'lCTUEE, 
 
 Center of that vanishing Line, and DE its Diftance. Now, fet it 
 be required to find the Reprefentation of a fquare Plane which; 
 ftands perpendicular to the Ground Plane, having one Side, a b,, 
 of the Reprefentation given. 
 
 From D, the Center of the vanifhing Line HL, and with the 
 Diftance DE, defcribe an Arc ELFH, cutting the vanifhing Line 
 in H and L ; then is H the vanifhing Point of the Sides ad, be: 
 Therefore, draw aH, bH, and from H draw HV; fo will HV 
 be the vanifhing Line of a Plane perpendicular to the Ground* 
 and by finding A ( the vanifhing Point of the Diagonal of a 
 Square) the whole Reprefentation may be determined.. 
 
 The 8 2d Figure reprefents a Cube upon the inclined Picture : 
 For having determined the Appearance of one Face abed, as in 
 the laft Figure, the whole Reprefentation may be compleated, by 
 means of the vanifhing Lines HL, HV, and LV, and the vanifhl 
 ing Points of the Diagonals, B, D, G. 
 
 I have hitherto confidered the Picture as reclined from the Eye; 
 let us now fuppofe it to be inclined to the Eye, as in Fig. 83* 
 where E is the Eye, LV the Picture, C its Center, CE its 'Dis- 
 tance, V the vanifhing Point of Lines perpendicular to the Ground,, 
 and DL the vanifhing Line of Planes parallel to the Ground.--— 
 In the 84th Figure the Picture is laid flat, and the Reprefentation 
 of one Face of a Cube is determined: ~And in the 85th Figure, 
 the Proje&ion of the whole Cube is compleated. — -Thefe Figures 
 need no Explanation, being only as it were the Reverfe of the 
 others j and therefore a little Attention mufr. render them extreme- 
 ly ohvious. 
 
 From hence then it follows, that the Method of determining 
 the Reprefentation of a Cube upon an inclined Picture, is exactly 
 the fame as in finding the Appearance of a Cube any ways in- 
 clined to the Ground ; and therefore the Rules which ferve for the 
 one will ferve for the other alfo : For Which Reafon the Learner 
 is defired to compare this with what has been faid in Sect, c. 
 Chap. j. r" 
 
 iECf, 
 
Qf Vaulted Roq**, 0om*su 
 
 * S E C T. HI, 
 
 Of Vaulted Roofs, Domes, &c. 
 
 TO draw Perfpective Reprcfentations upon vaulted Roofs, 
 Domes, or any other uneven Surfaces, has always been 
 efteemed a Work of great Difficulty ; and among all the Methods 
 which have been given us for this Purpofe by different Authors, 
 none feems fo practicable as that by Mr. Hamilton, in his ingenious, 
 Treatife intitled Stereography; where he directs us to *Re- 
 ticulate the propofed Surface, in fuch a Manner as may be beft 
 fuited to its Shape, and can with the moft Eafe be done ; then to 
 draw out, on a Plane properly chofen, a Picture of the intended 
 Defign, by way of Model; after which, to draw on this Model f 
 the Image of that Reticulation, by the common Rules of Perfpec- 
 tive ; which will divide the Defign on the Model, into fuch Parts, 
 as are proper to be transferred into each correfponding Cell of the 
 original Reticulation ; and finally, by means of this Reticulation, 
 to transfer the Work unto the Dome or Roof, in the fame Manner 
 as one Picture is copied from another, by the common Methods of 
 Reticulation. 
 
 Thus, fuppofe it was required to paint fome perfpective Repre- 
 fentation upon a vaulted Roof, HOIKPG. 
 
 Let this Figure be a Model drawn out upon Paper, of a vaulted Fig. 86, 
 Roof ; and let G H I K reprefent a Plane, which is fuppoled to 
 pafs through the Foot of the Arch, parallel to the Horizon. 
 
 Now, if we fuppofe the Spectator's Eye to be placed directly 
 under the Middle of it at E, and then imagine a Plane ABLQM 
 to pafs through the Eye, perpendicular to the Ground Plane AB, 
 it will cut the Picture in ML; and therefore, by drawing EC 
 perpendicular to the Section ML, we (hall have C for the Center 
 
 of a parallel Picture, and CE for its Diftance. Let us next 
 
 divide the Roof into any Number of Squares, or Parallelograms, 
 as in the Figure ; and then imagine a Line to be drawn from the 
 Angte of every Square to the Eye E ; and it muft appear extremely 
 evident, that the Sections of thefe Lines with the Plane, or parallel 
 Picture GHIK, will be the Projection of thofe Points upon the 
 Picture ; and it muft alfo appear as obvious, that, when the Pro- 
 jection of the Angle of every Square is determined upon the Pic- 
 
 ■* To Reticulate any Surface, is to divide it into Scares like Net-Work 
 
 G 2 ture, 
 
Of V a ti l t e b R o o f s, Domes, 
 
 ture, the whole Reprefentation of thofe Squares may eafily he 
 compleated. But farther, fmce the perpendicular Plane ABLQM 
 pafles through the Eye, and cuts the Pi&ure in a ftrait Line;, 
 therefore the Projeaion MCL, of the Arch MQL, will be a ftrait 
 Line upon the Piaure ; but the Projeaion of all the other Arches, 
 1R4 HOI, &c. will be curve Lines. Again, fince the tranfverfd 
 ftrait Lines 7 5, PO, 6 8, are parallel amongft themfelves, and 
 are alfo parallel to the Piaure ; therefore the Reprefentation of 
 thofe Lines upon the Piaure, will be ftrait Lines, and parallel 
 
 to each other. • 
 
 Thefe Things being premifed, let us now fuppofe this Figure 
 removed to the 87th Figure.— About the Arch HOI, defcribe the 
 Parallelogram HIPN; and through the Points 5, O, 6, draw the 
 Lines 17 O 8, 29, perpendicular to the Pidhire, and cutting the 
 Piaure in the Points 7, 8, 9; then through 5, 6, draw QR parallel 
 to HI, and from the feveral Seftions H, 7, 8, 9, 1, draw Lines to 
 C, and from N, O, P, Q, R, draw Lines to the Eye E which, wilt 
 determine the Projeaion of the Parallelograms by which means 
 the Reprefentation HaocI, of the Arch HOI, may be compleated; 
 After the fame Manner, the Projeaion of all the other Arches 
 may be found; but as one is fufficient for our Purpofe, we will 
 now fuppofe this parallel Piaure to be laid down flat in the 89th 
 Figure/where C is the Center, CE the Diftance of the Piaure, 
 and H, 7, 8, 9, 1, the Seaions of the Perpendiculars NH, 1 7, &c 
 
 m Continue IH (Fig. 89,) atpleafure, towards N, and maRe HQN 
 in this Figure equal to HQN in the 87th Figure; then from 
 H, 7, 8, 9, 1, draw Lines to C, and from N and O^draw Lines to 
 E, which will cut HC in n and t, and thereby give the Depth of 
 the Parallelogram Hnpl j by which Means the Points H, a, o, c I, 
 will be determined : Which being fo many Points m the Reprefen- 
 tation of the Curve, they will be a fufficient Guide for drawing it, 
 as in the Figure. After the fame Manner, the Reprefentaiion of 
 the other Front Arch is to be found : From whence it follows that 
 the Projeaion of the whole curved Roof upon this parallel l ie- 
 ture, will be contained within the two curved Lines Hoi, GgK, 
 and the two ftrait Lines GH and IK; and therefore GHoIKg is 
 the whole Space allotted for the Defign. Now having determined 
 this Space, let us next find the Projeaion of the feveral Squares 
 which were fuppofed to be drawn upon the original Root. 
 
 From 
 
Of^ Vaulted R o o f s, Doom s> &c. 
 
 , from a, o, c, draw Lines parallel to the Side GH, or IK ; then 
 will a f og, cd, be the Projeftions of the tranfverfe Divifions (or 
 ftrait Lines) which are parallel to the Piaure • and by dividing the 
 feveral Lines HG, af, og, cd, and IK, into four equal Parts, we 
 fhall have the Points given, through which the other Curve Lines 
 are to be drawn, as in Fig. 883 by which Means the whole Repre- 
 fentation may be compleated. 
 
 If it be required to paint any Perfpective Reprefentation upon a 
 Dome, that alfo may be done after the fame Manner, viz. by ima- 
 gining' the Dome to be divided into feveral perpendicular Seftions, 
 drawn at equal Diftances from the Bafe, through the Center of the 
 Dome- and by fuppofmg thofe Seaions to be cut by other Seaions-, 
 which are made by Planes that are fuppofed to pafs through the 
 Dome parallel to the Horizon : Then by making a Model upon 
 Paper in a given Proportion, and taking the Diftance of the Eye 
 accordingly, we may find the Projeaion of thofe Se&ions upon the 
 parallel Plane, as in the former Figures : For then we mail have a 
 parallel Piaure, which we fuppofe paffes under the Bottom of the 
 Dome properly reticulated, and by that Means, whatever is drawn 
 upon it, may be transferr'd unto the real Dome or Cupola. 
 
 Thus, let ABDE, Fig. 90, reprefent the circular Plane (or pa- 
 rallel Piaure) which we fuppofe to lie under the Bottom of the 
 Dome; and let AacegfdbB, Fig. 91, reprefent one of the perpen- 
 dicular Seaions above-mention'd ; and let us imagine the Dome to 
 be divided perpendicularly by four of thefe Planes, and horizon- 
 tally by four Planes, the Seaions of which horizontal Planes are 
 expreffed by AB, ab, c d, e f : Then let us divide the Circumfe- 
 rence of the Plane, or Piaure, ABDE (Fig. 90) into eight equal 
 Parts and from each Part draw Lines through the Center C ; and 
 then Will thefe ftrait Lines be the Projeaions of the perpendicular 
 Seaions upon the Piaure. And in order to find the ^Projections of 
 the parallel Seaions j from C, the Center of the Pifture, draw CE F.g 
 perpendicular to AB, and equal to the Diftance of the Eye j then 
 from ab, cd, ef, and g, draw Lines to the Eye E which cutting 
 the Piaure, will give 1 6 for the Projeaion of ab, 2 5 for that 
 of cd 7 4 for that of ef, and C the Center of the Piaure for 
 g the Center of the Dome; therefore, from the Line AB transfer 
 the feveral Divifions Ai, 1 2, &c. unto the Line AB in the 90th 
 Fi-ure i and from the Point C, defcribe the feveral concentric Cir- 
 clet through the Points 1, 2, .3; and fo will the whole Mure be 
 properly divided for the Work : For each Reticulation upon the 
 
tif Vaulted Hoof s, BoM£s, tM 
 
 Picture, is the exact Projection of its correfponding and original 
 Reticulation upon the Dome ; and therefore, all that now remains 
 is, only to divide the Picture into an agreeable Number of Parts, 
 and to confider each Part as a parallel Picture, whofe perpendicular 
 Sides will vanifh into the Center of the Picture ; and to be always 
 careful to take the Center of the Model perpendicular to the fup- 
 pofed Place of the Eye s and the Diftance to be work'd with, muft be 
 the fame as that between the Eye and the Plane AB, Fig. 91, as 
 well for describing the Model itfelf, as for the Reticulation. * ! 
 
 We have hitherto confidered the Eye as placed under the Center 
 of the Dome, in which Cafe the Reticulation upon the parallel 
 Picture is done with great Eafe 1 But if it were placed obliquely, 
 the Reticulation would become a little more troublefome ; in re- 
 gard that in fuch a Polition of the Eye, the perpendicular Sections 
 of the Dome would not form {trait Lines upon the parallel 
 Plane, but Curves. 
 ^ Thus, let A g B, be a perpendicular Section of the Dome, and 
 a b, c d, e f, its Sections with the horizontal Circles, as before^ 
 and let E be the Place of the Eye. 
 
 Then Lines drawn from E, to the Vertex g of the Dome, and 
 to the Centers and either Extremity b, d, f, of the horizontal Dia- 
 meters, will cut the Bafe of the Dome AB in correfponding Points; 
 which being transferred by Perpendiculars to the Diameter A B of 
 the parallel Plane, will give the apparent Vortex C, and the Cen- 
 ters and Radii of the Images of the horizontal Circles, on the 
 parallel Plane ; and thefe being drawn, and each divided into the 
 fame Number of equal Parts, as the Bafe of the Dome is fuppofed 
 <to be, Curve Lines drawn through the correfponding Divifions of 
 thefe Circles, will give the Projections of the feveral perpendicular 
 Sections of the Dome, as in the Figure. 
 
 For as the horizontal Circles are all fuppofed parallel to the cir- 
 cular Plane AD BE, it is evident their Projections will frill remain 
 ^Circles, and their Subdivifions will be equal, like thofe of their 
 ^Originals. 
 
 ,AncLhere all the perpendicular Sections of the Dome form 
 Curves upon the parallel Plane, except the Section A gB, which 
 Jls projected into the (trait Line AB, the Eye being fuppofed to lye in 
 the Plane of -that Section. But in Painting on curvilinear Grounds, 
 the raoft,direel Situation of the Eye ought always to be chofen* 
 ythat the IDefign, when painted, may appear the more agreeably j 
 ,^nd indeed, in all fuch Works, the Defign ought, as. much. as pof- 
 
xmi. 
 
Of VAPtf ed Roofs, Domes, &c . 
 
 fible, to be fuited to the Shape of the Surfaces, and to confift 
 principally of ornamental Architecture fitted to it, (putting the 
 Hiflorical Part into fmall Compartments, to be difpofed in proper 
 Places) or elfe of fome Aerial View, where the Sky and Clouds*, 
 with other Objects proper for that Situation, may be defcribed; in 
 which Cafe, the principal Objects not being confined to regular 
 Figures, there will be lefs Danger of their appearing diftorted by 
 the Shape of the Surface painted upon. 
 
 But when a Cupola, Dome, or Vault, is to be defcribed on a 
 flat Ground, there may be a greater Liberty taken in placing the 
 Eye j which may have either a direct or oblique Pofition, as the 
 Artift judges beft for the View he intends to reprefent, and will 
 not be liable to thofe Inconvenjencics which attend Painting upoifc 
 an uneven Ground, 
 
 JCHAf* 
 
C H A P. V. 
 
 "The Perspective of Shadows. 
 
 T HE Meaning of the Word Shadow is too obvious to need 
 any Explanation ; and therefore I mall not trouble the 
 Reader about its Etymology, nor mall I confider that in- 
 finite Variety of Shadows which may be projected by different 
 Planes j but proceed to mew, that the Perfpective of Shadows upon 
 the Picture, is to be determined after the fame Manner as the Per- 
 fpective of Objects, being founded upon the fame Principles, and 
 deducible from the fame Rules: It is therefore very furprizing, 
 that almoif. every Author who has handled this Part of Perfpective, 
 mould have committed fuch egregious Miftakes, in giving fuch 
 Rules as are falfe in Theory, and in Practice the moft abfurd. 
 
 But to proceed. All Shadows are produced by the Interposition 
 of fome opake Objects, which flop the Progrefs of the Rays of 
 Light in their direct Courfe from any luminous Body or Point. 
 And fince the Rays of Light do always proceed in ftrait Lines, 
 therefore, when they pafs over the Extremities of an Object, they 
 leave a Space unilluminated, which Space is called, the ProjeSfion 
 ef the Shadow of that ObjeB : And 'tis the Bufmefs of this Part of 
 Perfpective, to determine the Appearance of that Projection upon 
 the Picture. In order to do which, we muft firft confider whether 
 the Light be fuppofed to come from the Sun, a Candle, or any 
 other luminous Point : If from the Sun, then, from its immenfe 
 Diftance with refpect to us, the Rays may be confidered as pa- 
 rallel ; but if from a Candle, as flowing from a Point in a di- 
 verging Manner. 
 
 And in regard to the Theory of the Perfpective of Shadows ; 
 there needs but little more to be faid than what has been already 
 advanced upon the Perfpective of Objects : For fince every Ray of 
 Light is to be confidered as a ftrait Line, that Line may be con- 
 ceived to lie in fome Plane ; and therefore, if the Reprefentation 
 of that Plane can be eafily found upon the Picture, the Reprefen- 
 tation of a Line which is in that Plane, may be eafily found 
 alfo. 
 
 And here let us obferve, that the Planes in which I mall fuppofe 
 $he Rays of Light to be^ will always be coniider'd as perpendicular 
 
 to 
 
Of Shadows by the Sun. 57 
 
 to the Horizon, as that will be fitter for our Purpofe, and render 
 the Thing more intelligible. 
 
 If the Rays of Light come, from the Sun in Planes parallel to 
 the Picture, they then can have no vanifhing Point ; in this Cafe, 
 therefore, the Shadows will be parallel in the Picture : But if they 
 come in Planes not parallel to the Picture, then becaufe a Line 
 drawn from the Eye parallel to thofe Rays will cut the Picture in 
 fome one Point, therefore they will have a vanifhing Point upon 
 the Picture, which will be the common vanifhing Point for all the 
 Rays in that Direction, whether they be all in one Plane or in any 
 Number of Planes, provided thofe Planes are parallel to one 
 another. 
 
 SECT. I. 
 Of Shadows projeBed by the S u n. 
 
 LET HP reprefent a Plane parallel to the Horizon, or, if Fig. 93. 
 you pleafe, call it the Ground Plane, and let ABCD repre- 
 fent a Plane of parallel Rays, as EL, Dd, &c, each Ray making 
 an Angle, RLA, with the Ground Plane, 
 
 Now, in order to find the Perfpective of any Shadow upon the 
 Picture, two Things are neceffary to be given;* viz. the Incli- 
 nation, or Angle, which any Syftem of Rays makes with the 
 Ground Plane, and the Situation of the Plane (in refpect to the 
 Picture) in which thofe Rays are fuppofed to be. As to the Angle 
 of Inclination, that may be given by a fingle Ray only ; for fmce 
 the Rays RL, Dd, &c. are all fuppofed parallel amongfr. them- 
 felves, therefore the Angle RLA, which any fingle Ray RL makes 
 with the Plane H P, is common to all the reft : And as to the Si- 
 tuation of the Plane of Rays, that is to be chofen at the Difcretion 
 of the Artift, fo as to be moft productive of Effect as to his main 
 Defign. 
 
 Lemma 1 . 
 
 If the Rays of Light come from behind the Picture towards the 
 Spe£tator's Eye, then the vanifhing Point of thofe Rays will be 
 above the horizontal Line. 
 
 Let PQj>e the Ground Plane, GO the Picture, E the Eye, EC Fig. 9J ; 
 the Diflance of the Picture, C the Center of the Picture, HC the 
 
 * What is here faid to be given, is exclufive of the Diftauce of the Eye, the Center of th« 
 4?i£turc, &c. which, it U prefumed, will be taken for granted, without mentioning them. 
 
 H hori- 
 
Of Shadows by the Sun.. 
 
 horizontal Line, ABOD a Plane of parallel Rays interfecling trie 
 Picture at right Angles in the Line BO 3 and let RL be a Ray of 
 Light, and RLA the Angle of Inclination which the Rays make 
 with the Ground Plane. 
 
 Through the Eye E, draw E F parallel to any of the Rays, as 
 RL, cutting the Picture in F; then is F the vanifhing Point of 
 all the Rays of Light j for EF being parallel to one Ray, is parallel 
 to all the reft. — Now, fince the Plane of Rays ABOD is perpendi- 
 cular to the Picture, and panes through C the Center of the Pic- 
 ture, therefore the Plane BFEH, which paries through the Eye, 
 will be perpendicular to the Picture, and will pafs thro' its Center 
 alfo; and therefore BO, the common Section of thefe two Planes, 
 will be the indefinite Reprefentation of the Plane ABOD; and 
 confequently, F, where EF cuts the Picture, will be the vanifhing 
 Point of the Rays RL, &c. 
 
 C o r o l. 1. 
 
 ' Since the Plane of Rays panes through the Center of the Pic- 
 ture, the vanifhing Point of the Rays will be in a Line drawn 
 from the Center of the Picture perpendicular to the horizontal 
 Line- \ 
 
 C o r o l. 2. 
 
 From hence alfo we may perceive, that C may be the Repre- 
 fentation of the Seat of the luminous Point ; for the Seat of the 
 real Luminary is fuppofed to be in a Plane parallel to the Plane of 
 the Horizon ; and therefore, if we confider A, the Seat of R, as 
 at an immenfe Diftance, and fuppofe R a real luminous Point, 
 then will C be the Reprefentation of the Seat Rj that is, the Re- 
 prefentation of the Seat of a luminous Point upon the Picture, 
 which is fuppofed to be at an immenfe Diftance from it : Or, in 
 other Words, fince C is the vanifhing Point of LA infinitely ex- 
 tended, therefore, it is alfo the vanifhing Point of any Point in 
 that Line at an immenfe Diftance. 
 
 C O R O L 3. 
 
 And here likewife we may obferve, that in order to find the va- 
 nifhing Point of a Ray of Light, or of any Number of parallel 
 Rays, we need only have the Angle of Inclination given ; then by 
 fetting off the Diftance of the Picture upon the horizontal Line, 
 and making an Angle at that Point of Diftance with the horizontal 
 Line, equal to the given Angle of Inclination, we may determine 
 
Of Shadows by the S u n» 
 
 the vamftiing Point of thofe Rays; as is ihewn in the 98th Figure 5 
 which will be more fully explained hereafter. 
 
 Lemma 2. 
 
 In the laft Figure wc confider'd the Rays as coming in a Plane 
 perpendicular to the Picture ; we will now fuppofe them to come 
 in a Plane oblique with the Picture. 
 
 Let ABOD be a Plane of Rays which cuts the Picture obliquely Fig. 96; 
 in the Line OB, every thing elfe remaining as in the former 
 Figure. 
 
 Through the Eye E draw the Plane HLLE parallel to the Plane 
 of Rays, cutting the Picture in LL; then continue LL upwards 
 beyond F, at pleafure, and from E draw EF parallel to the Ray 
 of Light RL; then is F the vaniming Point of that Ray, Be. 
 
 C O R O L. I. 
 
 From hence it follows, that when the Light comes from behind 
 the Picture, the Shadows of Objects will be thrown towards the 
 Bottom of the Picture. 
 
 Lemma 3. 
 
 When the Rays come from behind the Spectator's Eye towards 
 the Picture, (that is, when the Spectator is between the real Lu- 
 minary and the Picture) then the vaniihing Point of thofe Rays 
 will be below the horizontal Line. 
 
 Let FHIL be the Picture, E the Eye, C the Center of the Pic- Fig. 97. 
 ture, and EC its Diftance ; and let ABOD be a Plane of parallel 
 Rays whofe Seat upon the Ground Plane is in the right Line LH 
 continued : Or in other Words, fuppofe the Plane of Rays was 
 continued towards the Picture in the Line BL, it would pafs thro* 
 the Eye E, and would cut the Picture in the Line LI. 
 
 Through the Eye E, and its Seat H, draw EC, HL, parallel 
 to AB or CD; and from L, where HL cuts the Section GL, 
 draw LC parallel to AD ; then is EHLC a perpendicular Plane 
 which paffes through the Eye parallel to the Plane of Rays ABOD, 
 cutting the Picture in LC ; therefore LC continued will be the in- 
 definite Reprefentation of the Plane ABOD, and it will alfo be the 
 vanifhing Line of all the Rays which can come in that Plane ; 
 and if EF be drawn parallel to RL, then is F the vanifhing Point 
 of that Ray, and C the Reprefentation of its Seat upon the hori- 
 zontal Line H L* 
 
 H 2 For 
 
Of Shadows by the' Su n. 
 
 For fuppofe the Plane ABOD to be tranfpofed into the Line' 
 XZ, then it will be like the Plane ABOD in the 95th Figure, 
 with this Difference only, that the Rays coming in a contrary 
 Direction, will have their vanifhing Point upon the Picture below 
 the horizontal Line. 
 
 Corollary. 
 
 When the Light comes from behind the Spectator's Eye towards 
 the Picture, the Shadows of Objects upon the Picture will be 1 
 thrown towards the horizontal Line ; and fmce the Light is gene- 
 rally fuppofed to come upon the Front of the Picture, and not 
 from behind it, therefore thefe Kind of Shadows are more gene- 
 rally ufed. 
 
 I fhould have been more particular in the Explanation of this 
 Figure, if there appeared the leaft Difficulty to me in understand- 
 ing it : Indeed, as the Eye is fuppofed to be between the Picture 
 and the original Object, it may feem to contradict our general 
 Definition of Perfpective, in which- we have always eonfider'd the 
 Picture as placed between the Eye and the original Object ; and 
 therefore this Lemma may appear not to be fo aptly drawn from 
 the preceding Theorems* as it really ought to be : Yet, fmce the 
 Method for determining a vanifhing Line, or Point, is the fame in 
 either Cafe, viz. by imagining a Plane to pafs through the Eye, 
 parallel to the original Plane, 'till it cuts the Picture, &c. I have, 
 therefore, only explained that lingle Article, and endeavoured to 
 make myfelf underffood, in the raofl familiar Manner; not much 
 regarding ftrict mathematical Demonftration, nor yet that Order 
 or Method which would be necefFary were this Treatife purely 
 Mathematical. 
 
 Lemma 4. 
 
 When the Rays of Light come in Planes parallel to the Picture, 
 they can have no vanifhing Point ; becaufe a Plane which paifes 
 through the Eye parallel to thofe Planes, and which in other Cafes 
 would cut the Picture, and thereby produce a vanifhing Line, in 
 this Cafe can never cut the Picture, and therefore cannot produce 
 any vanifhing Line : From whence it follows, that when the Rays 
 come in this Direction, the Appearance of their Shadows upon the 
 Picture will be parallel, for the very fame Reafon that the Repre- 
 
 * Chap. 3, Seft. 2, of this Book. 
 
 fentation 
 
Of Shadows by the Sun, 6i 
 
 fentation of any original Plane which is parallel to the Pidure, is 
 exa&ly like its Original.. 
 
 We will now give fome general Rules for applying to Practice 
 what has been faid upon this Head. In order to do which, let 
 AB reprefent a Piclure laid flat, as in the preceding Examples ; Fig. 9 t. 
 and let HL be the horizontal Line, G the Center of the Pi&ure, 
 and CE its Diftance. 
 
 Method i. 
 
 To find the vanifhing Point of a Ray of Light, when it is fup- 
 pofed to come from behind the Piclure towards the Spe&ator's 
 Eye, in a Plane like ABCD, Fig. 95, which cuts the Pifture in 
 its Center * 
 
 Any where apart, draw NP parallel to the horizontal Line HL, $>*• 
 and draw NO, at pleafure, for the Ray of Light; then is ONP 
 the Angle of Inclination.— Through C the Center of the Picture, 
 draw EK perpendicular to HL, and continue it at pleafure ; then 
 make CH equal to the Diftance EC, and from H draw HD pa- 
 rallel to the Ray NO, cutting CE in D; and then is D the 
 vanifhing Point of the Rays, of Light. For fince EK is the ya- 
 nifhine Line of the Plane of Rays, C the Center of that vanifhing 
 Line and CH equal to its Diftance, therefore H may be con- 
 fidered as the Eye ; and confequently, fince HD is drawn from 
 that Point parallel to the original Line NO, the Point D, where 
 it cuts the vanifhing Line ED, is the vanifhing Point of that 
 original Line. 
 
 Method 2. 
 
 When the Rays come from before the Picture, as in Fig. 97; 
 
 Every Thing remaining as before, -Let TW be a Ray of Light, Fig. 98, 
 and VTW its Angle of Inclination. — From H draw HK, parallel 
 to the Ray TW, cutting, the vanifhing Line EK in Kj then is 
 K the vanifhing Point required. 
 
 Method 3. 
 
 When a Ray of Light comes from # behind the Ficlure in a 
 Plane oblique with the Picture, as in Fig. 96 ; 
 
 Let IG be the vanifhing Line of a Plane of Rays, RS a Ray Fig. 98. 
 of Light, and RSQjts Angle of Inclination.-- Continue the hori- 
 zontal Line beyond L at pleafure, and from F, the Center of the 
 vanifhing Line, draw FE 5 then is FE the Diftance of that va- 
 
62 
 
 Of Shadows by the Sun, 
 
 nifhing Line; therefore by making FL equal to the Diftanee FE, 
 and by drawing LI parallel to the Ray R S, we fhall have I for 
 the vanifhing Point of that Ray. 
 
 Method 4. 
 
 When a Ray comes from behind the Spectator's Eye towards 
 the Picture, in a Plane oblique with the Picture. 
 -Fig 98. Let IG be the vanifhing Line of that Plane, ZY a Ray of 
 Light, and XYZ its Angle of Inclination. — From L, the tranf- 
 pofed Place of the Eye, draw LG parallel to the Ray ZY, which 
 will give G for its vanifhing Point. 
 
 Corollary. 
 
 From hence let us remember, that the Center C, or F, of a 
 vanifhing Line EK, IG, of a Plane of Rays, will be the vanifh- 
 ing Point of all Shadows which are cart by perpendicular Objects 
 upon the Ground 3 becaufe that Point * muft be in the horizontal 
 Line, and alfo in the vanifhing Line, of the Plane of Rays 3 fuch 
 are the Points C and F, 
 
 1o find the Shadow ef an Object which is fuppofed to fiand perpendi* 
 cular to the Ground^ when the Rays come in Planes parallel to 
 the Picture. 
 
 ^'g- 99- Let FG be the Picture, AB the Reprefentation of a perpen- 
 dicular Object whofe Shadow is fought j and let HL be a Ray of 
 Light, whofe Inclination with the Ground is equal to the Angle 
 HLC. — Through B, the Seat of the Object, draw Ea at pleafure, 
 but parallel to the horizontal Line ; and through A draw R a 
 parallel to the Ray HL, cutting Ea in a; then is Ba the Sha- 
 dow of BA. 
 
 "3Tig. too. Again, Let abed be a perpendicular Plane, whofe vanifhing 
 Point is C the Center of the Picture, and let HL be a Ray of 
 Light. — Through the Seats a, b, of the Perpendiculars ad, be, 
 draw a f, be, parallel to the horizontal Line, and through d, and 
 c, draw Lf, Re, parallel to the Ray HL, cutting af, be, in f and 
 €; finally, from f and e, draw fe, then is abef the Shadow of 
 the Plane abed, and f e continued will vanifh into C, the vanifh- 
 ing Point of ab, and c d. 
 
 * See the Appendix to Shadows, Book II. 
 
 Now, 
 
Of Shadows by the Su N* 63 
 
 Now, when the Shadow of any perpendicular Object is produced 
 by Rays which are fuppofed to come in Planes parallel to the Pic- 
 ture, that Shadow may be found by Calculation : Thus, when the 
 Angle of Inclination is 45 Degrees, then the Shadow will be equal 
 to the Height of the Object, as in the two laft Figures ; therefore, 
 by putting Unity for the Height of the Object, we may have the 
 following Proportions, viz* 
 
 Angle of Inclination. Length of the Shadow. 
 Deg. Min. 
 
 'No Shadow. 
 i-5th Part of the Object. 
 2~5ths ditto, 
 b-ioths ditto, 
 is; The Height of ditto. 
 
 1 2-4th?. once the Length of ditto and Half* 
 
 2 4~9ths. twice the Len. of ditto and 4-Qths. 
 5 Times the Length of ditto. 
 Infinite. 
 
 90 00 
 
 78 45 
 
 67 30 
 
 5 6 15 
 
 45 00 
 
 33 45 
 
 22 30 
 
 11 15 
 
 00 00 
 
 From hence, then, we fee the Reafon why the Shadows produced 
 by the Sun are very long in a Morning and Evening, and why 
 they grow fhorter and fhorter the nearer the Sun approaches to the 
 Meridian. 
 
 The foregoing Rules applied to PracJice. 
 
 To find the Shadow of an Obje6r, when the Light comes from 
 behind the Spectator towards the Picture. 
 
 Let AB be the Picture, C its Center, CE its Diftance, IK a Fig. iot. 
 Ray of Light, D the vanifhing Point of the Rays of Light, abed 
 a perpendicular Plane whofe Shadow is fought, and L the vanifh- 
 ing Point of the Shadow which is call: upon the Ground by the 
 perpendicular Sides ad, be. 
 
 From a and b, the Seats of the Perpendiculars ad, be, draw 
 Lines to L, the vanifhing Point of the Shadow ; and from d and 
 c, the Extremities of ad, be, draw Lines to D, the vanifhing Point 
 of the Rays 5 then from where they cut aL and b L, draw ef, and 
 then is abfe the Shadow of abed ; which if continued will vanifh 
 into C, the vanifhing Point of ab, cd. 
 
 To 
 
Of Shad o w £ by the S u n .' 
 
 To find the Shadow of a perpendicular Object when the Light 
 comes from behind the Picture. 
 
 In this Figure, H is given for the vanifhing Point of the Shadow, 
 KI for a Ray of Light.— —Draw FD parallel to IK, which will 
 give D for the vanifhing Point of the Rays of Light j then from 
 the Point H of the Shadow, draw i ines through all the lower 
 Corners, a, e, f, of the Object, and continue them at pleafure ; 
 then from D, the vanifhing Point of the Rays of Light, draw 
 Lines through the upper Corners b, c, d ; which will give the Points 
 g, h, i, from whence the Shadow a g h i f, may be compleated. 
 
 In the two laft Figures, I have drawn out every Line and Point 
 which is neceffary in the Work, and have alfo added the Angles 
 IGK, IKL, for the Inclinations of the Rays, to make the Thing 
 more intelligible. 
 
 Here let us obferve, that as the Shadow of every perpendicular 
 Line, will vanifh into the vanifhing Point of the Shadow ; fo alfo 
 the Shadow of every oblique Line, will vanifh into the vanifhing 
 Point of that Line : Thus ag is the Shadow of the Perpendicular 
 ab, and gh of the oblique Line cbj and gh, c b, will both 
 vanifh into G : For fmce the Shadow is caft upon a Plane perpen- 
 dicular to the Object which projects it, therefore the Shadow hg, 
 and the Edge c b, are to be confidered as parallel, and confe- 
 quently will tend to the fame vanifhing Point. 
 
 I have hitherto confidered Shadows as projected upon the 
 Ground, and the Planes which project them as perpendicular to 
 it; but by the fame Rules any other Shadows are to be deter- 
 mined, whether the Planes upon which they are caft are perpen- 
 dicular, parallel, or oblique, or however the original Objects are 
 lituated : And therefore, thus much might have fufficed to explain 
 the Theory and Practice of Shadows, fo far as is generally necef- 
 fary in a Picture ; but that this Part of Perfpective may be made 
 as familiar as poffible, I have added feveral ufeful Examples in the 
 •Practical Treatife, Book the Second, 
 
 S E C T, 
 
< 
 
Of Shadows by the Ca ndle, £fo 
 SECT. II. 
 
 Of Shadows frojeBed by the Cakdli, Lamp, &c. 
 
 THE Rays of Light from a Candle may be conceived to flow 
 from a Angle Point, like the feveral Radii of a Circle from its 
 Center. The 94th Figure reprefents a Plane of thefe Rays, which 
 is fuppofed to ftand perpendicular to the Ground Plane HP; where 
 L is the luminous Point, and S its Seat upon the Ground. 
 
 Now lince thefe Kinds of Light are but feldom chofen for a Pic- 
 ture ; and fince the Method for determining Shadows projected in 
 this Manner, is extremely eafy; there needs but very little to be 
 faid upon it ; I mall therefore treat this Section with the utmoft 
 Brevity. 
 
 The firft Thing neceilary in order to determine the Shadow by a 
 Candle, is, to give a luminous Point, and its Seat in the Picture ; 
 then by drawing Lines from thofe Points through the Extremities 
 of any Object, their mutual Interferons with each other will 
 give the Appearance of that Shadow. 
 
 Thus, let L be a luminous Point, S its Seat, and abed the Fig. 
 Reprefentation of a fquare Plane : From S and L draw Lines 
 through the Extremities a, b, c, d, and their Interfections at e and 
 f will give the Shadow a e f b. 
 
 Again, let L be a luminous Point, S its Seat, and ab, cd, ef, gh, Fig. 
 be the Reprefentations of feveral perpendicular Objects whofe 
 Shadows are fought : 
 
 From S and L draw Lines through the Extremity of each Line, 
 and the Points where they cut each other, will mew the Length of 
 the Shadows, as in the Figure. 
 
 I 
 
 CHAP 
 
CHAP. VI. 
 
 Of the Diftance and Height of the Eye, of 
 the Size of the PiSlure^ and of the true Point 
 of Sights QyV. with fome Confederations upon 
 the Appearance of circular Objects upon the 
 PiSlure. 
 
 i. Of the Distance of the Eye. 
 
 THE choofmg a proper Diftance for the Eye is fb effential 
 in all Perfpective Reprefentations, that without a nice 
 Obfervance thereof every Object will appear unnatural and 
 prepofterous, be the Rules by which it was drawn ever fo true in 
 Theory, or fo exactly obferved in Practice. And the Reafon of 
 this will appear extremely obvious, if we confider that there is a 
 certain Diftance at which the Eye can fee an Object with more 
 Diftinctnefs than in any other Point of View. Now, That Dif- 
 tance may be called the true Point , of Sight in refpect to That 
 Ob j eel: j and what is faid of one Object will hold equally true of 
 any Number of Objects : And therefore, as it is the Buhnefs of 
 Perfpective to draw the Reprefentations of Objects, as they appear 
 to the Eye, under the moft agreeable Shape, it follows, that the 
 Diftance to be work'd with upon the Picture, fhould be chofen in 
 fuch a Manner that each Reprefentation fhall make the fame agree- 
 able Figure to the Eye, as the Originals themfelves would do were 
 they feen under the fame Angle, 
 ffig. 105. To explain the Senfe of this more fully, let ADFK be a Pic- 
 ture, C its Center, NL the horizontal Line, AB one Side of a 
 geometrical Square parallel to the Picture ; and let it be required 
 to find the Reprefentation of that Square as feen at the feveral 
 Diftances CG, CI, CE. 
 
 From A and B draw Lines to C, the vanifhing Point of the 
 oblique Sides ; and from C fet off the feveral Diftances of the Eye 
 upon the horizontal Line in the Points O, N, H ; then from thefe 
 Points draw Lines to B, cutting AC in the Points a, d, ej and 
 from a, d, e, draw Lines parallel to A B : Then fhall we have the 
 Reprefentation ABab as feen at the Diftance CE, the Reprefenta- 
 tion ABcd as feen at the Diftance CI, and the Reprefentation 
 ABfe as feen at the Diftance CG. 
 
 Now, 
 
Of the Distance of the Eye. 67 
 
 Now, by infpecting the Figure, we fhall find that the apparent 
 Depth Ae, of the Reprefentation Abf e, which ought to be fore- 
 fhorten'd, is longer than the parallel Side AB, fo that the Figure 
 which fhould reprefent a Square, is a Parallelogram 5 and therefore 
 this Reprefentation will not appear to be true : And if the Diftance 
 be at I, then the Depth will be longer than it ought, becaufe the 
 Figure AB c d will ftill look like a Parallelogram : But if the Dif- 
 tance be taken at E, then the Reprefentation will appear of a 
 more proper Depth ; and therefore the Diftance CE is properer 
 for a Picture of this Dimenfion. And if Lines are drawn from 
 the feveral Points of View G, I, E, to the Extremities N and L of 
 the Picture, then thefe Lines will mow the Angles under which the 
 Picture is feen at thofe Diftances j viz. that at G will be an obtufe 
 one, that at I a right one, and that at E an acute one : And 
 therefore from hence we may conclude, that the Angle under which 
 any Picture is to be feen, ought never to be fo great as a right one 
 and, by making an Experiment, we mail find, that if it is much 
 lefs than an Angle of 50 Degrees, the apparent Depths of fquare 
 Objects will be too much foremorten'd, by which Means thofe Ob- 
 jects which mould reprefent fquare Bodies, will appear like fo many 
 Parallelograms : However, in fome Cafes, fuch as in painting De- 
 ceptions for Gardens, or for Pictures with curvilinear Objects, the 
 Diftance lhould be taken as great as poffible ; which is left to the 
 Difcretion of the Artift. 
 
 There are feveral other Reafons to be given for choofing a proper 
 Diftance for the Eye ; but as one Example is fufncient to fhew the 
 Abfurdity and Inconveniency of difregarding, or not knowing, this 
 eftentialPart of Perfpective, it becomes needlefs to produce any others. 
 
 2. Of the Height of the Eye. 
 > r "pIS the Height of the Eye that determines the Height of the 
 JL horizontal Line from the Bottom of the Picture 5 and there- 
 fore, it is that which gives the whole Space for the Reprefentation 
 of the Ground. And in taking the Height of the Eye, we muft be 
 careful not to let it be fo great as. the Diftance of the Eye ; fince the 
 fame bad Confequences will follow from thence as in choofing an 
 improper Diftance. For let PP be a perpendicular Section of the 
 Picture, AP an original Line perpendicular to it, and HE the Fig. tel. 
 Height of the Eye. — Draw AE, and then will Pc be the Reprefen- 
 tation of PA : But if the Eye be placed at I, fo as to be equal to 
 its Diftance IP, then will the Reprefentation PC be too long in 
 
 I a the 
 
68 
 
 Of the Height of the Eye, &c. 
 
 the Picture ; and the nearer the Eye is brought to P, (fuppofe atQ) 
 that is, the more the Height QR exceeds the Diftanc« QP, the more 
 prepofterous will the Reprefentation P a appear. Indeed if any 
 Fig. j 07. original Object, as AB, be parallel to the Picture, the Height of the 
 Eye will have no Effect upon this kind of Reprefentations, pro- 
 vided the Eye moves in the fame perpendicular HI 5 for the Repre- 
 fentation ab, is equal to the Reprefentation CP. 
 
 3 . The Conferences of viewing PiBures from any ether than the true 
 
 Point of Sight. 
 
 FROM what has been faid upon the Diftance and Height of 
 the Eye, it mull be manifeft, that no Perfpective Reprefenta- 
 tions will appear fo natural as when viewed from the true Point of 
 Sight 5 becaufe, at that Point, all the Rays which are fuppofed to 
 come from the original Objects, and produce their feveral Projec- 
 tions upon the Picture, will concur at the Eye in their proper Point, 
 and thereby exhibit a Picture upon the Retina exactly fimilar to 
 that of their Originals. 
 
 Sut again, If the Eye is not placed in the true Point of Sight, 
 the Projection of all Objects which are not parallel to the Picture, 
 will not feem to tend to their proper vanifhing Points ; and for that 
 Reafon fuch Reprefentations will feem to Hart .out of their proper 
 Places, will lofe their juft Proportions, and confequently, will con- 
 vey a jumble of confufed Appearances to the Eye : And to this we 
 may add alfo, the mocking Effect it will have upon the horizontal 
 Line in particular, which is always governed by the Place of the Eye. 
 
 What has been faid upon this Head, relates principally to Pic- 
 tures painted upon uneven Grounds, fuch as Domes, vaulted Roofs, 
 irregular Walls, &c. where the leaft Variation from the true Point 
 of Sight, will be productive of the above, and other bad Confe- 
 quences : For as to flat Pictures, the Fancy will be ready to give 
 fome Afliftance towards correcting what is not ftrictly right in 
 them i and therefore, a little Variation of the Eye from the true 
 Point of Sight, is allowable in fuch Cafes : For no great Inconve- 
 niency will appear, fo long as the Eye keeps upon a Level with the 
 liorizontal Line. 
 
 4. Of the S 1 z e of the Pj ct u r e. 
 
 THE Size of the Picture is to be governed by the Diftance and 
 Height of the Eye. Thus, let CE be the Diftance of the 
 
 iEye, and CP its Height, 
 
 With 
 
t)f the Size ^ ^ Pict.ur e. 69 
 
 With the Liftance CE defcribe the Circle FKAD, and make 
 CI equal to CP; through P and I draw AK, DF, parallel to the 
 horizontal Li.ie, cutting the Circle in A,K,D, F; from which 
 Points draw tie Lines AD, FK : Then mall we have a Square, 
 which will give the uttnoft Size a Piclure mould be of if feen from 
 no greater Dilance than CE. But if the Height of the Eye be 
 lefs than CP, then the Piclure will be a Parallelogram, which is 
 the moil general Shape given to Piclures. — This Method of limit- 
 ing the Size of the Piclure to the Diftance and Height of the Eye, 
 will be of great Ufe in feveral Operations. 
 
 |. Some Conjiderathns upon the Appearance of round Objects upon 
 
 the Fiffiure. 
 
 Y?ROM what has been faid upon the Diftance of the Eye, &c* 
 I. it may feem very improbable that any perfpeclive Representa- 
 tion mould have a difagreeable Effecl, if the Rules we have laid 
 down be nicely obferved : Yet there are fome Cafes, perhaps, in 
 which the Artift will think it better to be guided by his own Judg- 
 ment, than to follow the ftricl Rules of Perfpeclive. This feems 
 to have been the Opinion of Monfieur Frefnoy : For in his excellent 
 Poem upon Painting, tranflated by Mv.'Dryden, he fays, « Though 
 " Perfpeclive cannot be called a perfect Rule for Defigning, yet it 
 s: is a great Succour to Art, and facilitates the Difpatch of the 
 " Work; tho, frequently falling into Error, it makes us behold 
 ■«« Things under a falfe Afpecl ; for Bodies arc not always rcpre- 
 " fented according to the Geometrical Plane, but fuch as they ap- 
 *' pear to the Sight." But as there are different Opinions upon 
 this Subjecl, I fhall beg Leave to offer my Thoughts upon it. 
 
 Snppofe it was required to draw the Reprefentation of a Range 
 of Columns parallel to the Piclure ; if they are drawn according to 
 the {Irici Rules of Perfpeclive, then that Column which is in the 
 Center of the Piclure will be the leafl, and confequently, thofe on 
 each Side of it will be larger and larger continually, the farther they 
 are removed from the Center of the Piclure. But to explain this 
 more fully: Let KLMN be a Plane which panes through the Eye Fig. 10& 
 parallel to the Ground ; then will PP be the horizontal Line, and 
 C the Center of the Piclure : And let AB, H, g, be three Co- 
 lumns cut by this Plane ; then let Lines be drawn from the Extre- 
 mity of each Circle to the Eye; and the Seclions ab, cd, ef, with 
 
 the 
 
70 
 
 Some neeejfary Obfervations^ &c. 
 
 the Picture, are the Projections of thofe Circles up>n the Picture % 
 and by meafuring the feveral Reprefentations we fh;ll find, that cd, 
 and ef, are much longer than ab. From whence v e m ay conceive, 
 that the farther any Column is removed from tte Center of the 
 Picture, the larger will be its Reprefentation j anc we may more- 
 over conceive, that this Increafe of the apparent Magnitude of the 
 Columns, is owing to the Obliquity of the Line? g h, ik, with 
 the Picture, which Lines meafure their apparent Widths.. Now the 
 Queftion is, Whether Columns fituated in this Minner are to be 
 thus reprefented upon the Picture, or not ? 
 
 The Definition I have given of the Word Perfpective, is this; viz. 
 To draw the Reprefentations of Objects as they appear to the Eye, 
 &c. and I have avoided the more general Definition, viz, of drawing 
 the Reprefentation of Objects by the Rules of Geometry, &c* as the 
 former appeared to be more fignificant of what I intended to ex- 
 prefs by the Term Perfpective. For lince the Fallaces of Vifion are 
 ib many and great*, and finee we form our commor. Judgment and 
 Eftimation of the Appearance of Objects from Cutom and Expe- 
 rience -fy and not from mathematical Reafoning ; therefore it feems 
 
 reafon- 
 
 * The ingenious Dr. Smith, in his Treatife upon Opticks, has given us feveral Inftances of 
 the Fallacies in Vifion, amongft which, he fays, " We are frequently deceived in our Eftt- 
 " mates of Diftance by any extraordinary Magnitude of Objects feen at the End of it : As in 
 ** travelling towards a large City or a Caftle, or a Cathedral Church, or a Mountain larger 
 " than ordinary, we think they are much nearer than we find them to be upon Trial. For 
 ** fince by Experience the Ideas of certain Quantities of known Diftances are ufually annexed 
 " to the apparent Magnitudes of known Obje&s of a common Size ; and fince the apparent 
 " Magnitudes of thofc larger Obje&s at a greater Diftance are the fame as of the fmaller at 
 " a fmaller Diftance, it is no Wonder they fuggeft the ufual Idea of fmaller Diftance annext 
 " to more common Objetts. This is further evident, becaufe we are ignorant of the Coua- 
 ** try, and of the Inequalities in the Ground interpofed." Again, he obferves, " the Part of 
 " the Monument extant above the Tops of the adjoining Houfes, I am told, is five times 
 *' longer than the Height of the Houfes, and yet from below that Part appears but two or 
 *' three times longer at moil ; becaufe of its unufual Magnitude and Obliquity to the Sight." 
 And the fame curious Gentleman adds, " I remember a red Coat of Arms, upon the Top of 
 " an Iron Gate at the End of a Walk, was taken for a Brick Houfe in the Fields beyond 
 " it". Vide Smith's Of ticks, Book I. /. 6 1, 62. 
 
 f In regard to Perception, that acute and judicious Reafoner Mr. Locke, obferves, ** We 
 " are to confider concerning Perception, that the Ideas we receive by Senfation are often in 
 " grown People alter'd by the Judgment, without our taking Notice of it. When we fet 
 " before our Eyes a round Globe, of any uniform Colour, v.g. Gold, Alabafter , or Jet, 'tis- 
 " certain, that the Idea thereby imprinted in our Mind, is of a flat Circle varioufly fhadow'd, 
 " with feveral Degrees of Light and Brightnefs coming to our Eyes. But we having by ufe 
 " been accultom'd to perceive, what kind of Appearances convex Bodies are wont to make in 
 " us ; what Alterations are made in the Reflections of Light, by the Difference of the fenfible 
 «< Figures of Bodies, the Judgment prefently, by an habitual Cuftom, alters the Appearances 
 " into their Caufes : So that from that, which truly is Variety of Shadow or Colour, collecT:- 
 w ing the Figure, it makes it pafs for a Mark of Figure, and frames to itfelf the Perception, 
 i " oi 
 
Some necajfary Obftrvatiom, ®C« 
 
 reafonable not to comply with the ftricl: Rules of Mathematical 
 Perfpective in fome particular Cafes (as in this before us) but to 
 draw the Representation of Objects as they appear to the Eye; and 
 therefore, I pre.ume, a Painter mould represent thofe few Objects 
 which are an Exception to the General Rules of Perfpe&ive, in 
 fuch a Manner as may not offend the Eye of any common Spec- 
 tator. For if the above Columns are to be reprefented according 
 to the ftricl Rules of this Art j then the Columns as they recede 
 from the Center of the Picture will grow thick and clumfy, their 
 Intercolumnations will be continually growing lefs and lefs, and 
 the whole Beauty of the Building will be intirely deftroyed. * 
 
 " of a convex Figure, and an uniform Colour; when the Idea we receive from thence, is 
 *« only a Plane variouily colour'd, as is evident in Painting. To which Purpofe I (hall here 
 " infert a Problem of that very ingenious and ftudious Promoter of real Knowledge, the lear- 
 " ned and worthy Mr. Molineuxi and it is this : Suppofe a Man bom blind, and row adult, 
 " and taught by his Touch to diftinguifh between a Cube and a Sphere of the fame Metal, 
 *' and nearly of the fame Bignefs, f@ as to tell when he felt one and t'other, which is the 
 *' Cube, which the Sphere. Suppofe then the Cube and Sphere placed on a Table, and the 
 " blind Man be made to fee : Quaere, Whether by his Sight, before he touched them, he 
 " could now diftinguifh, and tell, which was the Globe, which the Cube. To which the 
 *' acute and judicious Propofer xnfwers : Not. For though he has obtained the Experience 
 *« of, how a Globe, how a Cube, affe&s his Touch ; yet he has not yet attained the Ex- 
 44 perience, that what affedts his Touch fo or fo, mult affeft his Sight fo or fo : Or that a 
 *' protuberant Angle in the Cube, that preffed his Hand unequally, mail appear to his Eye as 
 *» it does in the Cube. I agree with this thinking Gentleman, whom I am proud to call my 
 *« Friend, in his Anfwer to this his Problem ; and am of Opinion, that the blind Man, at 
 «' firft Sight, would not be able with Certainty to fay, which was the Globe, which the Cube, 
 ** whilft he only faw them, though he could unerringly name them by his Touch, and cer- 
 '* tainly diftinguifh them by the Difference of their Figures felt. This I have fet down, and 
 «« leave with my Reader, as an Occafton for him to confider how much he may be beholden 
 '« to Experience, Improvement, and acquired Notions, where, he thinks, he has net the 
 '* leaft Ufe of, or Help from them." Vide Locke's EJJay upon Human Underjianding, Vol. I, Ch. 9. 
 
 But this is no new Opinion; for fo old an Author as Lucretius, takes particular Notice of it ; 
 for he, after having given innumerable Inftances of the Errors in our Judgment, in regard 
 to Sight, fums them up in the following Lines. 
 
 tCeetera de genere hoc mirando tnulta uidetnus, 
 
 ££uce <vio/are fdem quaji Senjibus omnia quarunt: 
 
 Nequicquam. ^uoniam pars horum maxima fallit 
 
 Propter opinatus Animi, quos addimus ipfi, 
 
 Pro njifis ut Jint, qua non Junt fen/ibu' <vi/a. LvCRET. Lib 4. 
 
 Or, as Mr. Creech has tranflated it: 
 " Ten thoufand fuch appear, ten thoufand Foes 
 " To Certainty of Senfe, and all oppofe : 
 *' In vain, 'tis Judgment, not the Senfe miftakes, 
 
 *' Which fancy 'd Things for real Gbjeds takes." Vide Creech's Lucre/. #4. 
 
 * Of this we have feveral Inftances in Pozzo's firft Book upon Perfpeftive, particularly in 
 Fig. 45, 46, 50, and 51 ; in which, thofe Columns that are fartheft from the Point of Sight, 
 are fo prodigioufly increafed in their apparent Widths, as to lofe very near one Diameter in 
 Height; and, I think, the Difproportion is too viable to be difputed, efpecially in the 45th 
 and 46th Figures. 
 
 What 
 
Some neceffary Obfervations> &c*- 
 
 What has been faid upon this Subject, relates principally to rounds 
 or cylindrical Bodies, fuch as Globes, Columns, ©r the like ; but as to? 
 angular ones, (efpecially thofe that are Square) fince their apparent 
 Widths are perpetually increafed the more diagonally they are feen 
 by the Eye, therefore, the Reprefentations of fuch Objects upon 
 the Picture mould continually grow larger and larger in Width the 
 more they are removed from the Center of the Picture. Thus the 
 Reprefentation of the Square which is feen only in Front, 
 cannot appear fo large as the Reprefentation of the Square R, which 
 is viewed as a Triangle. I fay, that the apparent Magnitude of 
 Objects that are Square or Triangular, will be greater when view'd 
 Angle-wife, than when feen in Front : But the apparent Magnitude 
 of Columns, or any other round Objects, will always be the fame 
 at the fame Diftance ; becaufe, in the firft Cafe, the Diagonal of a 
 Square (which in fome Views meafures its apparent Width) is 
 longer than its Sides j but in the latter Cafe, the Diameter of a 
 Circle (which constantly meafures its apparent Width) is always of 
 the fame Length ; and therefore to reprefent Columns, &c. larger 
 and larger, when they are at a greater and greater Diftance, is, I 
 prefume, falfe in Theory, (I mean in an optical Senfe only) and 
 cannot be true in Practice. To this it may be faid ; Why then 
 fhould they not be reprefented lefs and lefs in proportion to their 
 feveral Diftances, fmce in fact they are fo ? To which I anfwer 
 again, that by a Habit of judging, and from the prevailing force 
 of Experience, we are taught to think, they are all of the fame 
 Size, becaufe they are upon the fame Parallel with the Eye. Thus, 
 for Inftance ; when we ftand before the Middle of a Building of 
 any considerable Length, we apprehend the Ends to appear exactly 
 as high as the Middle of it, though in fact they cannot, becaufe 
 the Angle fubtended at the Eye from the Middle, is greater than 
 thofe fubtended at the Corners. Again, fuppofe it was required to 
 draw the Reprefentation of round Balls, or Globes, which are fup- 
 pofed to be at the fame Diftance from the Picture, according to 
 the ftrict Rules of Mathematical Projection : Then the Projection 
 of that Ball only which is in the Eyes Axis vs ill be a Circle, and, 
 being properly fhaded, will appear like a Globe j but all the other 
 Projections, which are not in the Eye's Axis, will be Elliptical, 
 and, made them how you will, they can never appear like Globes 
 to any common Spectator : I fay to any common Spectator, becaufe 
 fuch Appearances contradict the common Idea which Men in ge- 
 neral have form'd to themfelves of Rotundity. In fhort, Perfpective, 
 
Sme necejjary Ofyervattons, &C0 
 
 In a ftrictly Mathematical or Optical Senfe, is one Thing 5 and 
 Perfpe&ive, according to the Acceptation of that Word among 
 Painters, is another : The Firft teaches how to defcribe on a Plane, 
 to a mathematical Exactnefs, the Projections of any Objects ; but 
 the Second, like a modeft and judicious Matter, teaches the moil 
 limple and general Principles of Arts and inftead of leading us 
 into the Mazes of Lines and Angles, and lofmg us in the Laby- 
 rinths of mathematical Rcafoning, directs us only to the Study of 
 Simplicity, which is the Foundation of Grace and Beauty. 
 
 I know it may be faid, that if we make choice of a proper 
 Pittance, all Inconveniencies of this Kind may be avoided: But 
 let the Diftance be ever fo proper, yet ftill the Projections of Co- 
 lumns, &c. as they are removed farther and farther from the 
 Center of the Picture, will grow larger and larger continually ; 
 which furely ought not to be admitted. 
 
 Thefe are the Reafons which induced me to confider this Sub- 
 ject in a particular Manner ; but whether they are fufficient, or 
 not, to anfwer the intended Purpofe, is fubmitted to the Candour 
 gf every ingenuous Reader, 
 
 C H A P, 
 
CHAP. VII. 
 
 Of Aerial Perspective, Chiara Qscuro,. 
 and Keeping in Piffiures. 
 
 JFrom Mr. Hamilton. 
 
 tc ^ Aerial Perspective is meant, the Art giving a 
 " Hi ^ ue Diminution or Degradation to the Strengh of Light, 
 «. JL-# Shade, and Colours of Objects, according to their difFe- 
 " rent Diftances, the Quantity of Light which falls ipon them, 
 " and the Medium through which they are feen. 
 
 " The Chiara Oscuro confifts more particularly in* ex- 
 C£ prefiing the different Degrees of Light, Shade, ani Colour of 
 €< Bodies, arifing from their own Shape, and the Ponton of their. 
 " Parts with refpecl: to the Eye and neighbouring Objt&s, where- 
 Ki by their Light or Colours are affected. 
 
 " And Keeping, is the Obfervance of a due Proportion in 
 <£ the general Light and Colouring of the whole Picture ; that no 
 C£ Light or Colour in one Part, may be too bright 01 ftrong for 
 €£ another ; but that a proper Harmony amongft them all together 
 t£ may be preferved. 
 
 « All thefe are neceflary Requifites to a good Picture, and may 
 " be properly enough included within the general Name of Aerial 
 " Perfpective, as they all relate to the different Degrees of Strength 
 « of the Light and Colouring, according to the Circiimftances of 
 « the Shape and Pofition of the Objects with regard to each other, 
 <£ the Eye, and the Light which illuminates them. 
 
 ££ The Eye does not judge of the Diflance of Objeds barely by 
 " their apparent Size, but alfo by their Strength of Colour and 
 <« Difrinction of Parts j it is not, therefore, fufficient to give an 
 <£ Object its due apparent Bulk, according to the Rules of Perfpec- 
 " tive, unlefs at the fame Time it be expreffed with that proper 
 t£ Faintnefs and Degradation of Colour which that Diflance 
 <{ requires. 
 
 ££ Thus, if the Figure of a Man at a Diflance were painted of 
 « a due Size for the Place, but with too great a Diftinclion of 
 
 * But it is the Opinion of fome very eminent Painters, that the Words Ckiara O/curo, mor« 
 gfoperly iignify a Clearuefs of Shadow. 
 
 « Parts, 
 
Of AejrulPbr.spective, ChiaraOscuro, &c» 
 
 " Parts, or too flrong Colours, it will appear to ftand forward, 
 " and feerm proportionably lefs, fo as toreprefent a Dwarf fituated 
 " nearer the Eye, and out of the Plane on which the Painter in- 
 " tended hie (hould ftand. 
 
 " By the Original Colour of an Object, is meant that Colour. 
 <c which it exhibits to the Eye when directly expofed to it in a full, 
 " open, uniform Light, and at fuch a moderate fmall Diftance as 
 5* to be clearly and diftinctly feen. 
 
 " This Colour receives an Alteration from many Caufes, the 
 <c principal of which are thefe : 
 
 " i. From the Object's being removed to a greater Diftance from 
 " the Eye, whereby the Rays of Light which it reflects are lefs vivid, 
 " and the Colour becomes more diluted, and tinged in fome mea- 
 «> fure with the faint blueifh Cart, or with the Dimnefs or Hazi- 
 * c nefs of the Body of Air through which the Rays pafs. 
 
 " 2. From the greater or lefs Degree of Light with which the 
 « Object is enlightened ; The fame Original Colour having a dif- 
 " ferent Appearance in the Shade from what it has in the Light, 
 *' although at an equal Diftance from the Eye, and fo in Propor- 
 ** tion as the Light or Shade is rtronger. 
 
 " 3. From the Colour of the Light itfelf which falls upon it, 
 " whether it be by the Reflection of coloured Light from any neigh- 
 " bouring Object, or by its PalTage through a coloured Medium ; 
 ie which will exhibit a Colour compounded of the Original Colour 
 " of the Object, and the other accidental Colours which the Light 
 f { brings with it. 
 
 " 4. From the Pofition of the Surface of the Object, or of its 
 li feveral Parts with refpect to the Eye j fuch Parts of it as are 
 " directly expofed to the Eye appearing more lively and dirtinct 
 f than thofe which are feen flanting. 
 
 " 5. From the Clofenefs or Opennefs of the Place where the Ob^ 
 * c ject is fituated, the Light being much more varioufly directed 
 *' and reflected within a Room, than abroad in the open Air; 
 *' every Aperture in a Room giving an Inlet to a different Stream 
 " of Light with its own peculiar Direction, whereby Bodies in fuch 
 " a Situation will be very differently affected with refpect to their 
 " Light, Shade, and Colours, from what they would be in an 
 f * open Place. 
 
 " 6. Some Original Colours naturally reflect Light in a greater 
 11 Proportion than others, though equally expofed to the fame 
 
 K. 2 *« Degrees 
 
Of Aerial Pirspective, Chiara Oscuro 3 ^. 
 
 ,£ Degrees of it; whereby their Degradation at fevera ]D>iftance$ 
 cf will be different from that of other Colours which reflect lefs 
 «« Light. 
 
 " From thefe feveral Caufes it arifes, that the Col.oms of Ob- 
 " jeCts are feldom feen pure and unmixed, but general^ arrive at 
 <* the Eye broken and foftened by each other; and tlerefore, in 
 " Painting, where the natural Appearances of Object: are to be 
 « defcribed, all hard or fharp Colouring ought to be avoided. 
 
 " A Painter, therefore, who would fucceed in Aeral Perfpec- 
 « tive, ought carefully to ftudy the Effects which liftance, or 
 <* different Degrees or Colours of Light, have on eacl particular 
 « Original Colour, to know how its Hew or Strengthis changed 
 < £ in the feveral Circumftances above-mentioned, and to reprefent 
 CK it accordingly; fo that in a Picture of various-coloucdl Objects^ 
 " he may be able to give each Original Colour its own proper Di- 
 « minution or Degradation according to its Place. 
 
 " Now, as all Objeftsin a Picture take their Meafuiss in Pro~ 
 " portion to thofe placed in the Front, fo, in Aerial Perfpective, 
 « x the Strength of Light, and the Brightnefs of the Colours of Ob- 
 < £ jects clofe to the Picture, muft ferve as a Meafure, vith refpect 
 «« to which, all the fame Colours at feveral Diflances, mufl: have 
 <l a proportional Degradation in like Circumftances. But, as in 
 « Mufick, it is not necefTary to the Harmony, that the Inftru- 
 « ments mould be tuned to the Concert Pitch, but tiey may be 
 x< fet above or below it, fo long as they are in tune, to each other; 
 « fo in Painting, it is not requifite that the Meafures on the in- 
 « terfecting Line of the Picture, or the Brightnefs of the Light 
 « there, mould be equal to the Life ; but they may be taken 
 f< greater or lefs, fo long as every Thing elfe in the Picture bears 
 «« a true Proportion to that which is chofen as the firft Standard, 
 
 " Hence, almoft any Degree of Light may be taksn for the 
 * greater Light in a Picture, when the leffer Degrees of Light are 
 « exprerled with darker or weaker Colours ; for any Degree of 
 « Light may either reprefent a Light in refpect of a darker, or it 
 " may ferve as a Shade to a lighter ; and it matters not in Point 
 « of Keeping how light or how dark a Picture is in general, fo 
 " that its feveral Parts have proportionable Degrees of Light and 
 " Shade given them. 
 
 " In order, therefore, to the giving any Colour its due Dimi- 
 « nution in Proportion to its Diftance, it ought tc be known, what 
 
 " the 
 
O^Aexial Perspective* Ckiara Oscuro, &c* 
 
 * the Appearance of that Colour would be, were it clofe to the 
 «< Picture Regard being had to that Degree of Light which is 
 <c c hofen as tie principal Light of the Pi&ure 5 as in order to the 
 « giving amy Object its due apparent Size, its true Size muft be 
 «« reduced to the fame Scale with the Meafure on the Bottom of 
 
 « the Picture ; • • , . ; l ■ 
 
 " For if any Colour mould be made too bright for another, or 
 « for the general Colours employed in the reft of the Picture, it 
 « will appear too glaring, and feem to ftart out of its Place, and 
 *< throw a Flatnefs and Damp on the reft of the Work; or, as the 
 « Painters exprefs it, the Brightness of that Colour will kill the 
 f< reft. 
 
 « No Painting can exprefs the dazzling Brightnefs of the Sun, 
 « c or even its reflected Light coming from poliftied Metals, with 
 " that fparkling Vivacity as it appears in the Camera Obfcura, in 
 « the Images of polifhed Surfaces on which the Sun ftiines ; or if 
 " it could in fome Sort be imitated in a Picture, by the Afliftance 
 « { of Gilding, it would not have a good Effect with regard to the 
 « ( other Colours, which it would too much outfhine; and thereby 
 « c hurt the Keeping; And this is one Defect which the Reprefen- 
 « c tation of Objects in the Camera Obfcura is liable to ; for by 
 « reafon of the Refraftion of the Rays by the Glafs, thofe Obje&s 
 « which naturally reflect lefs Light, lofe a greater Proportion of 
 " it than thofe which reflea Light more plentifully j whereby the 
 < £ due Keeping m the whole, is not fo exactly preferred as in 
 «« direct Vifion, the Lights and Shades appearing generally too 
 « f ftrong for each other." 
 
 Thus far Mr. Hamilton. And I have thought proper to add the 
 following Figure, with a Defign of fixing what he has faid upon 
 the Subject more ftrongly in the Memory. 
 
 Let E be the Eye, PP the Picture, RS, AB, CD, EF, and 
 HL, the fame Object feen by the Eye at different Diftances ; now, 
 the farther they are removed from the Eye, the larger will be 
 the Space of Air through which they are feen, and the more they 
 will be tinged with its Hew therefore, in Proportion as the Re- 
 prefentations Ps, Pb, &c. are more and more diminifhed upon 
 the Picture, fo likewife, in the fame Proportion, muft the Original 
 Colour of thofe Objects be more and more broken and diluted 
 with the Colour of the Air. Thus, fuppofe the Reprefen tation P s 
 of RS, to be painted of the Original Colour; then, becaufe the 
 Reprefentations Pb, Pd, Pf, and PI, are perpetually diminifhed in 
 
 propor- 
 
Of AERIAtPERSPt CTI VI, ChIARA OsGURd, Bei 
 
 proportion to the Diftance of their refpe&ive Originals j therefore 
 in colouring thofe feveral Reprefentations, Care muft be taken t& 
 diminifti that alfo in the fame Proportion. 
 
 There may be fome Exceptions made to this general Rule by a 
 nice Obferver of Nature, for there are fome Incidents will happen 
 which feem to contradict it y fuch as a white Houfe, or any other 
 very light Object: directly oppofed to the Sun at a great Diftance y 
 yet notwithstanding, what has been advanced will be of great Ser- 
 vice in fixing right Ideas of thefe eflential Requifities : And I may 
 venture to affirm, that without a general Obfervance thereof, every 
 Picture will be at beft but a flat and lifelefs Performance. 
 
 I might now proceed to the Confideration of fome other Things 
 relative to Perfpective but fmce they are not at all eflential in the 
 Theoretical Part, and as I muft take Notice of them in another 
 Place, I ftiall therefore put an End to. this Book, 
 
 End of the First Book. 
 
THE 
 
 Practice 
 
 Of 
 
 PERSPECTIVE: 
 
 Being 
 
 The SECOND BOOK 
 
 or 
 
 Dr. BROOK TAYLOR's Method 
 c/* Perspective made eafy> &c. 
 
 By JOSHUA KIRBY, Painter. 
 
 %he PraBice [ of Painting ] ought always to be built on a rational 
 Theory, of which Perspective is both the Guide and the Gate y 
 end without which it is impojjible to fucceed either in Defigning, or 
 in any of the jirts depending thereon. 
 
 Leonardo da Vinci upon Painting, p. 36 
 
 , . , . * 
 
 The SECOND EDITION. 
 
 IPSWICH: 
 Printed by W. Craightqn. Mdcclv* 
 
TO THE 
 
 ACADEMY 
 
 O F 
 
 PAINTING, SCULPTURE, ARCHI- 
 TECTURE, Sec. in London. 
 
 Gentlemen, 
 
 AS Perspective is abfolutely neceflary 
 to a Just Desjgn, give me Leave to 
 dedicate this Book to You on the Subjedl, It 
 is the Product of many Years Application and 
 Study, and wrote with an Intention to render 
 that hitherto perplexed, but ufeful Art, eafy 
 and familiar. How I have fucceeded in the At- 
 tempt, is fubmitted to your Candour and Judg- 
 ment 5 and I hope that this Dedication will be 
 received as an Inftance of my Gratitude, for the 
 Favour of that Encouragement and Recommen- 
 dation, which you have been pleafed to give to 
 the Work. 
 
 I do not prefume to offer any Thing new to 
 the Principal Members of the Society 5 for I am 
 Book II. A not 
 
DEDICATION. 
 
 not fo vain as to think I can give any Inftruc- 
 tions to Perfons of fuch fuperior Abilities : But 
 if I can contribute a little towards inftru&ing 
 the Pupils in the firft Rudiments of Defign, it 
 may fpare fome Time and Trouble, and I hope 
 will be accepted as a Token of my Regard for 
 
 You, and AfFe£tion for thofe Arts. 1 mall 
 
 only add, that it is my fincereft Wifh, that 
 every Encouragement may be given to your in- 
 defatigable Endeavours, in promoting the Arts 
 of Painting, Sculpture, Architecture, 
 $PV. That the Pupils may do Honour to their 
 feveral Masters, and become Ornaments to 
 their Country; and that every other Advantage 
 may concur to raife the iGlory of the English 
 Academy to the higheft Pitch. 
 
 / am, 
 
 Gentlemen, 
 
 Tour moft Obliged, 
 Humble Servant, 
 
 JOSHUA KIRBY. 
 
PREFACE. 
 
 IN this PraBical Book upon Perspective! Jhall endeavour 
 to give fome general Methods for finding the Reprefentations of 
 all Kinds of Objects, however they are ftuated in regard to the 
 Eye or the Pifiure, or however irregular they are amongft themfelves. 
 And find great Care has been taken to adapt every Example in this 
 Part to the Theory, the Reader may be fatisfied that every Figure 
 is ftriclly true, and capable of a Mathematical Demonjlration ; fo 
 that thoje whom Curiofty will not invite, or Leifare permit, to go 
 regularly through the preceeding Theory, need not trouble themfelves 
 about it, becaufe what follows will be fufficient for their Purpofe. 
 But let them confder, that it is in this as in all other Studies, with 
 which, if a Perfon defres either to be thoroughly acquainted, or to 
 profit by his Study, he mujl read with Attention, draw out every Fi- 
 gure as he proceeds, and be well acquainted with one Example 
 before he begins with another* 
 
 And flnce it is prefumed that every Example in the following Work, 
 may be as eqfily underjlood and applied to Practice by every Student in 
 the Arts of Defign, as are the common Principles of Arithmetick by every 
 ordinary Mechanick j therefore it is hoped that Perspective will be 
 no longer thought an abjlrufe and difficult Study, nor be difregarded as 
 trifling and infignijjcant but that the young Tyroes in the above 
 Arts will firfi mak/P erspective familiar to them, and treat her 
 •with the RefpeSi which Jhe deferves, as the Parent of the noble Art 
 of Painting ; and upon whofe general, though not rigid Precepts, 
 every Dejign mufl be regulated, if the Arttfl intends it Jhall appear a 
 true Reprefentation, of Nature. 
 
 A aU 
 
 THE 
 
THE 
 
 RACTICE of PERSPECTIVE 
 made Easy, &c 
 
 BOOK II. CHAP. I. 
 
 « t if ft t< ft - ft ft ' 1' * '* 1 1 1 
 
 An Introduction to the PRACTICE 
 
 of PerfpeSiive. 
 
 TN order to convey a general Idea of PBW«^ 
 I much Eafe as the Nature of the Thing wil admit, let ABCD F* . 
 J. reprefent a fquare Board ftanding perpendicularly upon the 
 Ground, which is reprefented by H M, and fuppofe the Figure 
 EH to be looking at It , then it will be e Vl dent that he muft fee 
 it by means of In infinite Number of Rays of Light which are 
 continually reffeaed from every Point of the laid Objea to his 
 Eve But fince the Rays which come from the four Corners only 
 will be fufficient for our Purpofe, we will fuppofe that he fees it 
 by means of the four Lines AE, BE, CE and DE which repre- 
 fent thofe four Rays of Light : And, if we fuppofe a tranfparent 
 Plane like a large Piece of Glafs, to be placed between the Objeft 
 ABCD and the Skater's Eye, it mull be obvious, that this Plane 
 will cut the Rays of Light in their Paflage to his Eye. Now, the 
 Shape abed, which that cutting of the Pifture makes with he 
 Rays, is called the ProjtMon of the real ObjeSl ABCD, upon the 
 Glafs. And if inftead of this tranfparent Plane, we fuppofe GLOl 
 to be a Canvas, and the above Projeftion to be drawn upon it in 
 the very fame Manner as it was projeaed upon the Glals, then the 
 Fieure fo defcribed is called the PerfpeBiw Appearance of the real 
 ObieB ABCD ; for the Rays of Light coming to the Eye from the 
 Points a. b, c, d, which are drawn upon the Piaure, m the very 
 fame Manner as they do from the correfponding original .oints 
 A, B, C, D, of the real Figure ; therefore, if they are painted with 
 
Introduction to the PraBice ofVt, rspectme. 
 
 the fame Strength of Colour, &c. they will to the Spt&ator EH, 
 appear like that original Object : So that the whole \rt of Per- 
 fpective confifts, in determining thefe, and the like Appearances, 
 upon the Picture, and in giving them their proper Force and 
 Colour. 
 
 Now let us ohferve, Firft, If the original Figure be parallel to 
 the Picture, then its Reprefentation will be exactly like it thus, 
 ABCD is parallel to the Picture, therefore, abed, its Reprefenta- 
 tion, is exactly like it. For which Reafon, the Reprefentations of 
 the Sides of all Objects that are parallel to the Picture, will not 
 tend to any Points upon the Picture, but will be parallel amongft 
 themfclves, and only proportionally diminished as their Diftance 
 from the Eye is greater or lefs : But the Reprefentations of the 
 Sides of all Objects that are not parallel to the Picture, will vanifh 
 into various Points upon the Picture, which, are therefore called, 
 the mani firing Points of fuch ObjeBs. 
 
 Secondly. The Projections A B c d and e f g h, of the Squares 
 ABCD, E F G H, which lie fiat upon the Ground, HM, will to 
 the Spectator, EH, be the perfpective Appearance of thofe Objects : 
 And the Reprefentations of the Sides AB, cd, e h and f g, will be 
 parallel to the Bottom of the Picture, but will be federally dimi- 
 nifhed in proportion to their Diftance from the Picture. Thus AB 
 is even with the Bottom of the Picture, and therefore its Repre- 
 fentation is the fame as the original Line A B, and confequently 
 equal to it : But the Reprefentations cd, eh, and f g, will be per- 
 petually diminilhed in the Degree of Diftance they are from the 
 Picture ; as is evident by infpecting the Figure. For which Reafon, 
 the Reprefentations Ad, Be, ef, and gh, of Lines AD, BC, 
 E F, and G H, which lie flat upon the Ground, and are parallel 
 amongft themfelves, but not parallel to the Picture, will conti- 
 nually approach towards each other, till they vanifh into a 
 Point C, exactly as high above the Bottom GL of the Picture, as 
 the Eye is removed above the Ground HM. Thus, FC is equal 
 to the Height of the Eye E H, and the Sides Ad, Be, @fi will va- 
 nifh into the Point C. From hence then, we fee the Reafon why 
 the Reprefentations of Objects are more and more diminilhed upon 
 the Picture, the farther thofe Objects are fuppofed to be from it. 
 
 Thirdly, We have obferved that the Reprefentations Ad and Be, 
 of the oblique Sides AD and BC of the real Object, will vanifh 
 into the Point C upon the Picture and therefore the Point C may 
 very properly be called the vanifhing Point of the Lines A D and 
 
 BC, 
 
IntroduSiion to the FraSlice ^Perspective, f 
 
 BC— — Now, in order to determine the vanifhing Point of any 
 Line, we muft always draw a Line from the Eye parallel to that 
 Line: Thu.< EC is parallel to AD, or BC, and therefore, C, 
 where EC cats the Picture, is the vanifhing Point of AB, or BC. 
 
 And in like Manner, E J being drawn parallel to the Line BK, Fig. i. 
 which is oblique with the Ground, will give J for its vanifhing 
 Point upon :he Picture. For, from K draw the Ray KE to the 
 Eye, which will cut the Picture in k ; then is k the Reprefentation 
 or K, and b is the Reprefentation of B j therefore, k b is the Re- 
 prefentation of KB: And, if kb was continued upwards upon the 
 Picture, it would cut E J in the Point J, and therefore J is its 
 vanifhing Point. 
 
 Fourthly, If thro* the vanifhing Point C, a Line HL be drawn 
 parallel to the Bottom of the Picture, then that Line will be the 
 vanifhing Line of all Objects that lie flat upon the Ground, or ar& 
 parallel to it. Now this Line, HL, hath always been called the 
 Horizontal Line, and therefore, I fhall call it by that Name in the 
 following Work. Indeed, it is the moft ufeful of all vanifliing 
 Lines ; but neverthelefs, too much Strefs hath been laid upon it by 
 almoft all Writers upon this Subject ; who have paid no Sort of 
 Regard to any other vanifhing Lines. But had they confider'd, that 
 there are feveral Objects whofe Reprefentations cannot be correctly 
 determined upon the Picture, without a general Knowledge of all 
 Kinds of vanifhing Lines and vanifhing Points, they would not 
 have confined themfelves to the Horizontal Line only; and had they 
 built their feveral Syftems of Perfpective upon as folid Principles 
 as Dr. Taylor or Mr. Hamilton, their Works would not have been 
 crouded with fuch a Confufion of Lines, nor with fuch a Number 
 of ufelefs Examples ; but they would have been more true, Jimple, 
 and of more general Ufe. But to return from this Digreffion. 
 
 Fifthly, Since the Horizontal Line is level with the Eye and pa- 
 rallel to the Ground, and, for that Reafon, the vanifhing Line of 
 all Objects which lie flat upon the Ground, or are parallel to it; fo, 
 for the very fame Reafon, the vanifhing Line of any other Object 
 will be parallel to that Object. Thus, fuppofe ABK to be a tri- Fig. t. 
 angular Plane, which flands upon the Edge AK, perpendicular to 
 the Ground : Then the vanifhing Line of this perpendicular Plane 
 will be perpendicular to the Horizontal Line and if this Plane be 
 perpendicular to the Picture alfb, then its vanifhing Line will pafs 
 through the Center of the Picture 3 thus JE is the vanifhing 
 Line of ABK. 
 
 Thus 
 
3 ' Definitions, & 
 
 Thus much, I prefume, may fuffice to give the unlearied Reader 
 a tolerable Idea of Perfpedive.— We will now give an Explanation 
 of a few Terms made ufe of in the following Work ; and then 
 proceed to the Mechanical Part of Perfpeclive. 
 
 DEFINITIONS. 
 
 i Hp HE Point of Sight, is that Point where the: Spectator's Eye 
 Fig< I( ' 1 is placed to look at the Pi&ure. — Thus E is the Point 
 
 0f 2 Sl ^f t from the Point of Sight E, a Line EC is drawn from the 
 Eve perpendicular to the Picture, then the Point C, where that 
 Line cuts the Picture, is called the Center of the PiSlure. 
 
 7 The Dijlance of the PiSlure, is the Length of the Line E C, 
 which is drawn from the Eye perpendicular to the Piclare. 
 
 4. If from the Point of Sight E, a Line EC be dravn perpen- 
 dicular to any vanifhing Line : HI, , or JF, then tb Point C 
 where that Line cuts the vanifhing Line, is called the Center of 
 
 that vanifhing Line. . « » c U, T . 
 
 c The Dijlance of a vanijhing Ltne, is the Length cf the Line 
 EC which is drawn from the Eye perpendicular to the faid Line- 
 And if PO was a vanifhing Line, then EJ will be the Diftance 
 
 of that Line. ' . . . ^ c r - 
 
 6 The Diftance of a vanifhing Point, is the Length or a Line 
 drawn from the Eye to that Point: Thus, EC is the Diflancc of the 
 vanifhing Point C, and EJ is the Diftance of the vanning Point Jo 
 
 7 By Original Objett, is meant the real Object whofe Repreien- 
 tation is fought : And by Original Plane, is meant that Plane upon 
 which the real Objeft is fituated: Thus, the Ground HM is the 
 Original Plane of A B CD, &c. Fig. i, 2, 
 
 AXIOM S. 
 
 Fis> 20 i The Reprefentations of all Lines that are parallel to each 
 other, but not parallel to the Picture, will have the fame vanifhing 
 Point: Thus, the Reprefentations Ad, Be, of the parallel Lines 
 AD, BC, have the fame vanifhing Point C. 
 
 2 . The Reprefentations of all Lines that are parallel to each 
 o&er, and parallel to the Picture alfo, will not vamfh into any 
 Point upon the Piaure,. but will be parallel to each other : Thus, 
 
 i AB, cd, eh, and f g, are parallel to each other, becaufe the Ori- 
 ginals AB, CD, EH, and FG, of thofe Lmes, are parallel to 
 each other. CHAP* 
 
r, CHAP. IL 
 
 Practical Perspective 
 sect. I. 
 
 To prepare the Picture. 1. Of the SrzE of the Picture. 
 
 2. Height of the Eye.. 3. Of the Distance of the 
 
 Eye. 
 
 I. r ■ i H E Size of the Picture muff be adapted. ta the Diftance 
 of the Eye, if it be an immoveable Picture, like the 
 Side of a Room, Ceiling, or the like ; which may very 
 eafily be done by means of Frames, or other Compartments : But, 
 if the Picture be an Eafel-Piece, * then the Bignefs of it may be 
 left to the Artift's Difcretion. The Height of the Eye, rauft always 
 govern the Height of the Horizontal Line from the Bottom of the 
 Picture ; and particular Care muff be taken not to let it be fo great 
 as the Diftance of the Eye, fince it will be productive of very bad 
 Confequences. And great Regard muft be had to the choohng a 
 proper Diftance to be worked with; for Other wife, every Perfpective 
 Reprefentation will have a very bad Effect. 
 
 2. Of the Height of the E y e. 
 
 SuppofeGLTP to be a Canvas, reprefenting the Size of the. Fig, 
 
 Picture.' Divide the Bottom of it, GL, into two equal Parts in 
 
 P, and draw FI perpendicular to the Bottom GL ; then from F 
 fet off FC, equal to the Height of the Eye from the Ground, and 
 draw HL, through the Point C, parallel to the Bottom GL : And 
 then will HL be the Horizontal Line, and C the Center of the Pic- 
 ture. Now, though the Height of the Eye in Eafel-Pictures is 
 left intirely to the Difcretion of the Artift ; yet, in general, low 
 Horizons have a much better Effect than high ones j for which 
 Reafon, the Height of the Horizontal Line mould never exceed 
 one half of the Height of the Picture; and, I believe, a little Ex- 
 perience will teach any one, that if it is made equal only to one 
 third Part of the Height of the Picture, it will be the moft proper 
 Height of any : I mean only in regard to Eafel-Pieces ; for if the 
 
 * The Inflrumcnt upon which a Picture is placed to be painted, is called an Ea/el ; and 
 therefore, every Pi£«re which is moveable, is called zn Eafel-Piete. 
 
 Book II. B Pidure 
 
"The D i s T a n c e of the E r 
 
 Picture be a fixed one, then the Height of the Horizontal Line muft 
 he exactly level with the Spectator's Eye. 
 
 3.. Of the Distance of the Ey e. 
 
 The choofing a proper Diflance for the Eye is fuc3i an effential 
 Requifite, that without a nice Obfervance thereof, every Perfpec- 
 tive Reprefentation will appear a fhocking Deformity ; therefore, 
 we fhall be the more particular in fettling a proper Diflance for 
 the Eye; the Neceffity of which will appear by the following 
 Example. , ■ 
 
 In the 4th Figure I fuppofe A B E t> a real Square upon the 
 Ground, and CE one Diflance of the Eye, and CI another. Now, 
 by putting this Square into Perfpeclive, agreeable to thofe different 
 Diflances, we fhall have a b E D for the Reprefentation of the 
 Square as feen at the Diflance CE, and EDcd for the Reprefen- 
 tation of the Square as feen at the Diflance C I ; and by infpecting 
 the Figure, we may perceive, that the Reprefentation EDcd, which 
 is feen by the Eye at the Diflance CI, does not appear like a 
 Square, but looks much longer than 'tis wide, and therefore, it is 
 a falfe Reprefentation ; but the Reprefentation a b E D, which is 
 feen by the Eye at the Diflance CE, has a more agreeable Appear- 
 ance, and looks like a Square feen in Perfpeclive, and therefore 
 is a more jufl Reprefentation. Now, that this Difference between 
 the two Reprefentations of the fame Object is wholly owing to the 
 different Diflances of the Eye, is apparent from the Figure ; and 
 therefore, this one Inflance, out of many, may fuffice to fhew 
 the Neceffity of choofing a proper Diflance to be worked with ; 
 In order to do which, the following Method feems the mofl eafy, 
 and the mofl ufeful, of any I can think of, 
 
 Having drawn the horizontal Line HL, and fixed the Center C, 
 of the Picture ; draw a Line (as CP) from the Center C, to one 
 of the farthefl Corners (as P) of the Picture; drawalfo the Per- 
 pendicular C D, and continue it at pleafure ; then from C, fet off 
 the Length CP, upon CE, and call CE the leafl Diflance : Again, 
 from C, fet off C D, upon the Line C D, equal to the longefl Di- 
 menfions of the Picture, and call CD the greatefl Diflance. That 
 is, never let the Diflance you work with be greater than C D, nor 
 lefs than C P $ beeaufe, as was obferved before, if the Diflance be 
 lefs than C E, the Reprefentations will be too deep $ and if it be 
 more than CD, the Reprefentations will not be deep enough and, 
 1 think, if a Medium between thofe two Diflances be taken as a 
 
 general 
 
TO? Dis tance of the Eye. 
 
 ri 
 
 general Rutlr, it will produce the moft agreeable Shape of any Dis- 
 tance whatfrever. Thus, the Reprefentation ED mn, (Fig. 4.) 
 is determined by fuch a Diftance. 
 
 In this Place it may not be improper to take Notice, that the 
 Diftance of the Picture is fometimes placed upon a Line as CD, Fig. 3; 
 perpendicular to the horizontal Line, and fometimes upon the 
 horizontal Line itfelf; as the Nature of the Work may require. 
 And I will alfo obferve, that the Diftance generally made ufe of 
 in this Work, is the lean: Diftance, for the Conveniency of having 
 as many Figures upon each Plate as was pofllble. And the Reader 
 is defired ta remember, that the Letters CE will always ft and for 
 the Diftance of the Eye, E for the Eye, or Point of Sight, C for 
 the Center of the Picture, HL for the horizontal Line, and GL for 
 the Ground Line, or Bottom of the Picture. For, to avoid Pro- 
 lixity, I Ib.aU not mention either of thofe Terms but upon fome 
 particular Occafion. And that he may fix them the eafier in his 
 Memory, I have made every Letter as fignificant as porlible : Thus 
 E is the Eye, C the Center, HL the horizontal Line, &c. And I 
 fhall, moreover, always fuppofe a Picture, as GLTP, (which may 
 be confidered as a large Picture in Miniature) to be laid flat, and 
 that we are actually at work upon it, in determining the Repre- 
 fentation of the following Figures. 
 
 SECT. II. 
 
 Of 0 ( B jects which lie fat upon the Ground, or that are 
 in Planes perpendicular to the Piclure, 
 
 I. To find the Reprefentation of a Point upon the Piclure, after 
 having prepared the Pifiure as above directed. 
 Method i. By one tanijhing Point only, 
 
 LE T A be the Point upon the Ground. From A draw any Fig. 5 
 Line at Pleafure, as A 1 , cutting the Bottom of the Picture 
 in 1 j and from the Eye E, draw EL parallel to A 1, cutting the 
 horizontal Line in L then is L the vanifhing Point of A 1 j there- 
 fore, draw the Line L 1, then from the Point A, draw a Line to 
 E, cutting Li, in a; and then is a, the Reprefentation of the 
 original Point A. 
 
 Method 2. By two vanifing Points, 
 
 Draw A 1, A 2, at pleafure, cutting the Bottom of the Picture 
 in 1 and 2 -„ and from the Eye E, draw EL parallel to Ai, and 
 
 B 2 EH 
 
12 Vf O b j e t t s ugen the Grou *d>; 
 
 EH parallel to A2; then draw L 1 and Ha, whxfo will cut each 
 other in a, and fo give a, for thue Reprefentation of A. 
 
 $lg. IL ¥0 find the Reprefentation of a Line KB, ivhid its perpendicular 
 
 to the Bottom of the Picture. 
 
 Met h o d 1 . By one vanijloing Po'mtt. 
 Let AB be the real Line upon the -Ground. — !So>w (ince AB is 
 iperpendicular to the Bottom of the Picture, therefore EC is pa- 
 rallel to it j and therefore C, where E C cuts the horizontal Lme, 
 is the vanishing Point of AB. — Draw AC, and fronn B draw JBe[ 
 cutting AC in b ; and then is Ab the Reprefentation of AB. 
 
 Method 2. By two vaniflring Pdnts. 
 
 From B draw B 1 at pleafure, cutting the Bottom of the Picture 
 in 1 ; and from the Eye E, draw EH parallel to B 1, cutting the 
 horizontal Line in H; then is H the vanifhing Point of Bi ; there- 
 fore draw 1 H, cutting AC in b which will determtine the Repre- 
 fentation propofed. 
 
 In like manner, the Reprefentation Fd, of FD, which lies -direct- 
 ly againft the Middle of the Picture, is to be determined. For C 
 is the vanifhing Point of FD, and H is the vanifhing Point of D2. 
 
 From hence then we may conceive, that if there were ever fo 
 many Lines parallel to AB, they would all vanifh into the Center 
 of the Picture ; and that the Reprefentation Fd, of any Line that 
 ties directly againft the Middle of the Picture, will be perpendicular 
 to the Bottom of the Picture j that is, will be Part of the Perpen- 
 dicular FC, which is drawn from F to the Center C but in pro- 
 portion as any other perpendicular Lines (as AB) are more and 
 more removed from the Middle F, the Reprefentation s Ab of fuch 
 Lines will be more and more oblique with the Bottom of the 
 Picture, 
 
 III. Of a Une parallel to the Bottom of the PiBure. 
 Method i. By one vanifhing Point. 
 XFig. 7; Let AB be the Original Line. — Draw Ai, B3, perpendicular 
 to the Bottom of the Picture; then is C their vanifhing Point $ 
 therefore draw iC, 3C; and from the Extremities of the Line AB 
 draw Lines (as AE) to the Eye, cutting iC, 3C, in a and b; 
 then draw a.b, which will be the Reprefentation of AB. — Or it 
 may be done by finding one End only of the Reprefentation (as a) 
 .and then drawing ai> parallel to the horizontal Line, 'till it cuts 
 
 Met hop. 
 
tf Objects upon the G r o xj n d; 
 
 Method 2. By two vani/hing Points, 
 Draw A . 2, \ 4, at pleafure, (but parallel to each other) cutting 
 the Bottom of the Picture, as before: Then draw EH parallel to 
 ft 2 5 and then is H the vanifhing Point of A 2 and B4 ; therefore 
 draw 2H, 4H cutting iC and 3C in a and b; finally, draw* 
 the Line ah, vhich is the Reprefentation propofed. — Or, finding 
 one Point orily (as a) and then drawing ab parallel to the hori- 
 zontal Line, ai before, will be fufficient. 
 
 IV,. Of a Line AB oblique <with the Bottom of fbe Pitture. 
 Method j. By one vani/hing Point. 
 Continue AB to the Bottom of the Picture, and draw EH pa- 
 rallel thereto and from 3 draw 3H ; then from the Extremities 
 A and B draw Lines to E, which will cut 3 H in a and b ; and 
 then is ab the Reprefentation of AB. 
 
 Mithod 2. By two vanijhing Points. 
 
 From the Extremities AB draw Ai, B2, parallel to each other; 
 and from E draw EL, parallel to A i, Bs ; then draw i L, 2L* 
 cutting 3H in a and&; and then is ab the Reprefentation of AB. 
 
 Here let us obferve, that fince Lines muft be either perpendicular 
 to the Picture, parallel to the Picture, or oblique with the Picture, 
 the three laft Examples may ferve as univerfal Rules for the Situ- 
 ation of all Objects that are fuppofed to lie upon the Ground 1 
 which is fully explained in the following Figure. 
 
 V. Of an equilateral triangle* one of whofe Sides is parallel to the 
 
 Pifture. 
 
 Me th od I. By halving the Original Figure ABO drawn out 
 
 upon the Ground* 
 Continue BA and BD to the Bottom of the Picture, and draw 
 EL parallel to AB, and EH parallel to BDj then draw 1 L and 
 4H, cutting each other in b; and then is b the Reprefentation of 
 the Angle B. Again, from D draw D3 parallel to B i ; then is L 
 its vanifhing Point; therefore draw 3L, cutting 4H in the Point 
 d ; and then is d the Reprefentation of the Angle D : And fince 
 AD is parallel to the Bottom of the Pi&ure, therefore, if from the 
 Point d, the Line ad be drawn parallel to the Bottom of the 
 Ticture, it will compleat the Reprefentation propofed. — Or, it may 
 
 * Aa Equilateral Triangle is that whofe Sides and Angles arc all equal. 
 
 be 
 
Of Objects upon the G r o tx n w 
 
 be done by drawing A 2 perpendicular to GL, and then drawing 
 2C, cutting iL in the Point a. 
 
 Method 2. By making an Angle at the Eye E, equal to the given 
 
 jingle ABD.. 
 
 Let ab be one Side of the Reprefentation given upon the Pic- 
 ture Continue ab to the horizontal Line j then is L its vamhV 
 
 inz Point: From L draw LE to the Eye ; and any where a-crofs. 
 the Line CE, draw ef parallel to the horizontal Line, and then 
 make ef equal to fE: Again, draw EH through the Point e, 
 and then is H the vanilhing Point of the Side bd; therefore, thro 
 b draw bd, and from a, draw ad parallel to the horizontal Line* 
 which will compleat the Reprefentation.— Or it may be done thus : 
 Having; continued ab to its vanifhing Point L, and having drawn 
 LE, make an Angle at the Eye E equal to 60 Degrees*, . and draw 
 EH, which will cut the horizontal Line in the vanilhing Point ot 
 the other Side bd. 
 
 Method 3. Without having any Original Figure drawn ouf y . 
 hut by having one Side given only. 
 
 Let ab be the Side given.— Continue ab to ^^S^™* 
 L, and draw LE 5 then at L, with the pittance LE, defer ibe the 
 Arc EH, cutting the horizontal Line m Hfi then is H the vamfh. 
 ing Point of the other Side bd; and by drawing ad parallel to 
 the horizontal Line, the Reprefentation will be compleated. 
 
 Here let the Reader obferve again, that if the Durance LE, ot 
 any vanifhing Point L, be transferred unto the horizontal Line as 
 LH, it will cut off one Line equal to another Line given. Tfcus, 
 let ba be a given Line.— From a, draw ad parallel to the hori- 
 zontal Line 5 and from H (the Diftance of L from the Eye E) draw 
 Hd through the Point b, cutting ad in d; then 
 ab .j for they are both the Reprefentations of two Sides AB, AD 
 of a Triangle ABC, whofe Sides are all equal.~ : In like Manner, 
 if ad was a Line given, and L the vanifhing Pomt of a Line ab, 
 which is required to be cut off equal to ad : Then make lh 
 equal to LE; and from d, draw dH, cutting a L m b; and then 
 is ab equal to ad.-f 
 
 * This Angle may be fet off with a* Inftrument called a Protraftor, which is a Semi«i 
 Circle divided into 1 80 equal Parts, called Degrees. ^nditnous Ufe. 
 
 " t The Learner cannot make this Figure too familiar too him, as Us of prodigious uw. 
 
 VL Of 
 
'Of O b j e c ¥ 6 vpdn the G rotj n i>7 fef 
 
 yi, Equilateral Triangle AB C, whofe Sides are all oblique fig. »®i 
 
 /itf Picture. 
 
 Method i. By having the Original Figure drawn out upon the Ground, 
 Continue the Sides of the Triangle to the Bottom of the Pic- 
 ture, as i, 2, 3, and draw EI parallel to AC, EF parallel to AB, 
 and EK parallel to B C, which will feverally cut the horizontal 
 Line in the vanilhing Points <of AB, AC, and BC s therefore, from 
 thofe vanifhing Points draw Lines to i, 2, 3, and their mutual In- 
 terferons a, b, c, with each other, will give the Reprefentation 
 ab c of the original Triangle ABC. 
 
 Method j, By making Triangles at the Eye, as before. 
 
 Let a b be one Side of the Reprefentation given. Continue it 
 
 to its vanifhing Point F, and draw F E ; then at E, upon the 
 Line EF, make the equilateral Triangles MEN, MEO ; continue 
 EN and EO 'till they cut the horizontal Line, which will give the 
 vanifhing Points required ; therefore from a, draw a I, and thro' 
 b draw K c, which will compleat the Reprefentation a b c. 
 
 Method 3. By giving one Corner a, of the Triangle ; and from 
 thence finding the whole Reprefentation a be. 
 
 From a, draw a F, and call F one vanifhing Point ; then from 
 F draw FE, and at E make an Angle of 60 Degrees, and draw 
 E I j then draw la, and through a, draw f e, parallel to the hori- 
 zontal Line, at pleafure, and make af, ae, each equal to one Side 
 of the fuppofed Reprefentation ; then from the vanifhing Point I* 
 fet off the Diftance IE to D ; and from e, draw eD cutting a I in c> 
 then is ac equal to ae. Again, from F fet off the Diftance FE, 
 (as FP) and draw f P, cutting aFinb; then is a b -equal to af; 
 finally, draw be, which will compleat the Reprefentation a be. 
 
 VII. Of a Geometrical Square A B C D, having one Side A B parallel Fig. iu 
 
 to the Picture, 
 
 Method 1. By a Plan ,; that is, by having the Original Square 
 
 drawn out upon the Ground. 
 Draw AC, BC, to the vanifhing Point C, of the perpendicular 
 Sides AD, BC; and from the JEye E, draw EH and EL parallel 
 to the Diagonals BD and AC ; then from A and B draw Lines to 
 L and H, cutting AC in d, and BC in c; then draw dc, which 
 
 compleats the Reprefentation. Having found the Reprefentation 
 
 of 
 
i6 Of Objects upm the Gro ttn eC 
 
 of one Square, any other Square, as ik, may be found alfo. For 
 let L ik be one Side of the Reprefentation given. — -From i and k 
 draw i C and kC then from i draw i L, and from k draw kH 5 
 
 which will give the Depth of the Square, as in the Figure. Or* 
 
 one Diagonal only will be fufficient. Thus, A L cuts B C in cl 
 therefore draw c d parallel to the horizontal Line. 
 
 From hence we may obferve, that when original Squares are 
 thus fituated, the vanifhing Points H and L of their Diagonals, 
 are exactly as far from the Center of the Picture, as the Eye is 
 from the Center of the Picture. Thus HC and LC are each equal 
 to the Diftance C E ; and therefore, by fetting off CH, or C L > 
 equal to C E, the Lines EH and E L may be omitted. 
 
 Method 2. By having only the Depth FI of the Square 
 
 F I L given. 
 
 Set off IL and 1 1, equal to the Depth FL From I; and L> 
 
 draw Lines to C, and make C L equal to CE 5 then draw 1 L, 
 cutting 1 C in f j then is If cutoff equal to Il ; therefore draw 
 f e parallel to the horizontal Line* and the Reprefentation will be 
 compleated* 
 
 Hethod 3, By having only one Side, as G K, given. 
 From G and K, draw Lines to C j continue G K, and make 
 K 2 equal to GK ; then make CH equal to C E, and draw H 2, 
 cutting KC in b ; finally from b, draw ab parallel to the hori- 
 zontal Line, and the Thing "propofed is done. In like manner 
 any other Square, m n o, may be found. 
 
 VIII. Of a Geometrical Square, when its Sides are oblique with the 
 
 Picture* 
 
 1*. Method i. By a Plan ABCD. 
 
 Parallel to the Sides A B, CD, AD, BC, draw E L and E H 5 
 then are L and H the vanifhing Points of thofe Sides ; for con- 
 tinue the Sides of the Square 'till they cut the Bottom of the Pic- 
 ture in 1, 2, 3, 4 ; then from 1 and 2, draw Lines toH, and from 
 3 and 4, draw Lines to L, and their mutual Interfe&ions a, b, c, d, 
 
 will g lve the Reprefentation propofed. Or the original Square 
 
 may be made at the Eye, as in the Figure. 
 
 Me tho» 
 
Of Objects upon the Groitn d; 
 
 Method 2. By having only one Side, as G i, given upon the 
 
 FiSiure* 
 
 Continue G i till it cuts the horizontal Line in H, and from H 
 <lraw HE ; then at E, made a right Angle * with the Line HE, 
 and draw EL ; then is H the vanifhing Point of the Sides G i, lk, 
 and L is the vaniihing Point of the Sides G 1 and i k ; therefore, 
 from G and i draw GL and i L, then from G draw GC, cutting Li 
 in k; finally, from H draw a Line through k, cutting GL in 1, 
 fcnd then is G i k 1 the Reprefentation propofed. 
 
 M E T no d 3 . By having only the Length of the Diagonal e L given 
 upon the Bottom of the Piffure, as LP. 
 
 From L draw LCj then from P draw PH, cutting LC in e; 
 then is Le the Reprefentation of the Line L P : Again, from L 
 draw L L, cutting P H in f, and from L draw L H, then thro' 
 € draw L h, cutting L H in h ; and then fliall we have the Repre- 
 fentation L f e h. 
 
 IX, Tofnd the Reprefentation o f a Square, abed, of any determinate 
 
 Width ; fuppofe three Feet. 
 
 Let a be one Corner given, and K the vanifhing Point of the 
 Side ad. — Make KD equal to KE, and from D draw a Line thro' 
 a, cutting the Bottom of the Picture in f j then from f, on the 
 Side of a d, fet off three Feet upon the Bottom of the Picture, and 
 from e draw eD, cutting aK in d ; then is a d equal to three Feet i 
 Again, from K draw KE, and make a right Angle at E, then 
 •draw EL; and then is L the vanifhing Point of the Side a b ; 
 therefore draw aL, and bifecl f the Angle KEL, and draw E B 
 cutting the horizontal Line in B; then is B the vanifhing Point of 
 the Diagonal of the Square; by which means the whole Figure may 
 be compleated. For draw aB and dL, cutting it in c, then draw 
 K. b through the Point c ; and then is ab c d the Square propofed. 
 
 * A right Angle is one Corner of a Square, or 90 Degrees. 
 
 f Bifeft, is to divide any thing into two equal Parts : Thus, on E defcribe any Arc, 
 
 * V»? l V -ri l , hat Arc ' ia ^ im two e< l aal Parts » and dKW fiB » and then i» the 
 angle KEL biiedted. 
 
 Book II. 
 
 C 
 
 X. Of 
 
Of Objects upon the Ground? 
 
 X. Of a regular Hexagon *, having one of its Sides parallel to the 
 
 Fifture. 
 
 fig. 14. Method i. By a Plan ABCDEF. 
 
 Continue the feveral oblique Sides 'till they cut the Bottom of 
 the Picture, and draw EH and EL parallel thereto 5 then will L 
 be the vanifhing Point of AB and DE, and H will be the vanifh- 
 ing Point of BC and EF: Therefore, through the Corners A,D, and 
 F, C, drawn D 2, C 5, which being parallel to the Sides AB, EF, 
 will have H and L for their vanifhing Points ; and therefore, froni 
 1 draw 1 H, and from 3 draw 3 L, cutting each other in b then 
 is b the Reprefentation of the Corner B. Again, from 2 draw 
 2 H, cutting 3 Lin a; then is a the Reprefentation of the Corner 
 A, and ab is the Reprefentation of the Side Ab; therefore, draw 
 4H and 5L, and then is f the Reprefentation of F; then draw 
 a f, which will be. the Reprefentation of A F ; finally, draw 6 L, 
 which will cut 4H in e, and give the Reprefentation of E F ; and 
 fo on. Here the Learner may take Notice, that this whole Repre- 
 fentation is found in the fame Manner as the fingle Point A, 
 Fig. 5 ; only the Operation in this Figure is repeated fix times, 
 becaufe here are fix Points, which reprefent fix Corners, inftead 
 of one. 
 
 Method 2. By having one Side a b, in the Reprefentation given^ 
 Through the Corner a, draw a Line f h, parallel to the hori- 
 zontal Line, and continue ab to its vanifhing Point L 5 and from 
 L draw LE, and make L H equal to LE; then from H draw a 
 Line through b, cutting ah in hj then is ah equal to ab;. there- 
 fore make af equal to ah; and then is af the Reprefentation of* 
 the parallel SideAF: From f draw fL, and from a, draw aC, 
 cutting f L in c; then is c another Corner; therefore draw be, 
 which reprefents another Side ; then from c draw c d, parallel to 
 the horizontal Line, and draw f C cutting it in d ; then is d ano- 
 ther Corner, and cd another Side ; finally , through d drawLe, 
 and from f draw f H, which compleats the Reprefentation. 
 Fig. 15- In like Manner the Reprefentation of an Oaagon (or eight-fided 
 Figure) AB C D E F G H, is to be determined. 1 have put every 
 
 * A Hexagon is a fix-fided Figure j and when its Sides are all equal, it It called a regular 
 
Of Obj'ec T s upon the Groun n» 
 
 Line and Point ufed in the Operation, which, it is prefumed, is 
 now fufficient, without any further Explanation. 
 
 XL To find the Reprefentation of the Circle ABC. Fig. 
 
 Method i. By finding the Reprefentation of feveral Points, as 
 
 A, B, C, &c. 
 
 From thefe Points draw any Lines at pleafure, but parallel to 
 each other, as C i, C2, B3, B4, &c. cutting the Bottom of the 
 Picture; then find their vanifhing Points, as H,L, and from each 
 original Section at the Bottom of the Picture, draw Lines to their 
 refpe&ive vanifhing Points, and their feveral Interferons will give 
 the Reprefentations of the original Points ; from whence the Re- 
 prefentation of the Circle may be drawn by hand : Thus a, b, c, 
 are the Reprefentations of A, B, C, &c. This alfo is a Repetition 
 of Fig. 5. 
 
 Method 2. Or the Reprefentation of a Circle may he found by means 
 of a Geometrical Square ■; which is the moft ufeful Method of any. 
 Thus, let G be a Circle, and ABCD a Square defcribed about Fig. 
 it, as in the Figure. — Find the Reprefentation of that Square ; 
 which will be a fufficient Guide for drawing the Reprefentation of 
 the Circle, to any one who has but the learl: Notion of Drawing. 
 But if the Circle is very large, or if this Method mould not be 
 thought correct enough, then divide the Circle into any Number 
 of Parts, and draw Lines through thofe Divifions parallel to the 
 feveral Sides of the Square, as in the Figure ; then by finding the 
 Reprefentation of thofe Lines, the Appearance of the Circle may 
 be determined with great Exactnefs. 
 
 Method 3. To find the Reprefentation of a Circle by having only 
 the Diameter given upon the Picture. 
 
 Let e f be the Diameter given. Divide e f into two equal 
 
 Parts ; then is n the Center of that Circle. From C, and through 
 the Points e and f, draw Cg and Ch, at pleafure, and make CL 
 equal to the Dillance CE; then through n draw Lg, cutting gC 
 and hC in g and k; finally, from the Points g and k draw two 
 Lines, gh, ki, parallel to the horizontal Line ; and then is ghik 
 the Reprefentation of a Square equal to the Diameter of the pro- 
 pofed Circle; and confequently, will be a fufficient Guide for 
 drawing its Reprefentation, 
 
 C 2 From 
 
Of Oejects upon the GRouNiy, 
 
 From hence then it is evident, that the Reprefentation s of Circles 
 are as eafily to be determined upon the Picture, as any other Re- 
 prefentations whatfoever ; and, that after having fixed upon the 
 Diameter of any Circle, and the Place that Circle is to poffefs upon 
 the Picture, then fuch a Reprefentation may be determined with 
 the greateft Exactnefs, without the tedious Method of Plans, and 
 that Infinity of occult Lines, which have hitherto been made ufe of. 
 
 What hath been faid in regard to finding the Reprefentations 
 of Circles without a Plan, or having the original Object drawn 
 upon the Ground, is equally applicable to any of the preceding 
 Figures, as I have fhewn in the Courfe of this Work ; and there- 
 fore, though I have put the Plans at the Bottom of each Figure, 
 it was for no other Reafcn than to explain the Truth of the Ope- 
 ration , and therefore the Reader will do well to exercife himfelf 
 with feveral Examples of the like Nature, before he proceeds to 
 the next Section. 
 
 And here let us take Notice, that the Figures I have been put- 
 ting into Perspective, though few in Number, and the moft fimple 
 in Nature, yet they are fuch as comprehend Forms in general * I 
 fay, the Forms or Shapes of Objects in general are compounded of 
 fuch Figures as I have been reducing into Perfpective ; that is, they 
 muft be either Square, Triangular, or Circular, or elfe compounded 
 of fome, or all of thefe put together. Thus, a Cube is compofed 
 of fix Squares joined together at rightAngles; a Pyramid, of feveral 
 Triangles meeting in a Point; and a Column, of a Number of round 
 Superficies laid upon each other exactly even, and perpendicular to 
 the Ground. Thefe, and the like, may therefore be called fimple 
 Objects; but when they are joined together, fo as to make but one 
 Object, then that Object may be called a Compound one : Thus a 
 Building may be called a Compound Object ; the Body of which 
 is either a Cube or Parallelopiped ; the Roof and Pediments feveral 
 Triangles, and the Arches, Domes, Columns, &c. are nothing, 
 elfe but Circles, or Parts of Circles, put together. And therefore, 
 it follows from hence, that whoever is able to put a Square, 
 a Triangle, or a Circle, rightly into Perfpective, has got all the 
 Materials that are neceflary for drawing the Reprefentation 
 
 * I mean fuch Forms only as are proper Subjects for Perfpective; for as to Objects which 
 are compofed of an infinite Variety of Curve-Lines, I will not pretend to give any Rules for 
 determining their Appearances according to the ftrict Rules of this Art ; and was I able to do 
 it, I mould think it unneceffary ; fmce a good Eye, in fuch Cafes, will direct the Hand with 
 more Safe, if not with as much Certainty, as any Rules wkatfoever } efpecially if the Pexfon 
 ha« a general Notion of Perfpective. 
 
 Of 
 
n. 
 
 / J 
 
 
 i 
 
 L L/' f ( 
 
 §: \ 
 
 \ \ IK**'"'' 
 
 /I 
 
 G 
 
 
 I 
 
 ht ' 
 
 7- 
 
 L < 
 
 % \ 
 
 " ' ** \ ^'''/ 
 
 ' \ y f: \ 
 
 ^-'a 
 
 — f i- i — ^ 
 
 4 \ 3 
 
 2\ ". <I 
 
 
 \ IO. 
 
 1 
 
 L C |F D H 
 
 
 
IV. 
 
 
 
 
 12 . 
 
 < 
 
 
 r'-' 4? l \ p 3w 
 
 
 
 
 
 
 11. 
 
 o 
 
 13. v 
 
 
 
 / 
 
 
 
 
 ( 
 
 \B h 
 
(^Objects perpendicular to the Groun dv 
 
 of any Object upon the Pi&urc. I mall therefore follow this 
 fimple Method throughout this Work, and will now proceed to 
 fhcw, how to determine the Reprefentation of Obje&s when they 
 fland perpendicular to the Ground ; which is the Subject, of the- 
 next Seclioa. 
 
 SECT. III. 
 
 Of Ob j e c t s that are perpendicular to ^Groukd. 
 
 I* Wo find the Reprefentation of Planes when their Bafes are perpen- 
 dicular to the Bottom of tfje Picture, like KB, Fig. 6,. 
 
 Case r. When one Corner A,, of the Plane ABCD, is at the Bottom 
 
 of the Piclure : And let it be required to draw the Reprefentation 
 
 of a Plane fix Feet high and four Feet wide, 
 
 FRom A, upon the Bottom of the Picture, make a Scale of Feet* 
 at pleafure, as in the Figure ; and from the Point A draw 
 AD, perpendicular to the Bottom of the PiBure,, and con- 
 tinue it at pleafure ; then upon the Line AD fet fix Feet from A, 
 and draw DC to the Center C, and from A draw AC; then make 
 CH equal to C E, and draw H 4 cutting AC in B ; and then is 
 AB equal to four Feet ; therefore, from B draw B C, parallel to 
 AD; and fa will ABCD reprefent a Plane four Feet wide and' fix 
 Feet high. 
 
 Case 2. Let it be required to draw the Reprefentation of a Plane 
 GHIK, three Feet Square ; and let the neareft Corner G, be one Foot 
 from the Bottom of the PiSlure. 
 
 Draw any Line at pleafure, as C 9, and make CL equal to CE; 
 then fet off one Foot from 9 to 8, and draw L 8 cutting C 9 in 
 G ; then is G one Foot from the Bottdm. Again, take three Feet 
 more, as 8 5, and draw 5 L cutting G CinK; then is G K equal 
 to three Feet; therefore, draw GH and IK perpendicular to the 
 Bottom of the Pi&ure, and continue them at pleafure ; and from 
 G draw G F, parallel to the Bottom of the Picture ; then is G F 
 equal to three Feet ; finally, make G H equal to G F, and draw 
 HC ; which will compleat the Reprefentation propofed.^-Or it may 
 be done thus : From 9, draw 9 N perpendicular to the Bottom of 
 
 * What I here mean by a Scale of Feet, is not to make a Scare of fo many Feet long, but 
 only to divide the Bottom of the Piclure into fuch a Number of eoual Farts, which are to be 
 eoniidered as fo many Feet : A Thing very common amongfl Workmen. 
 
 the 
 
22 Of Object s perpendicular to the Groun b» 
 
 '.the Picture, and make it equal to three Feet; then draw oC and 
 NC, and from 8 draw 8L, which will give the Point, or Corner, 
 G; then draw 5L, which will give three Feet for GK; therefore, 
 by drawing GH and IK parallel to 9 N, the Thing propofed is 
 done. — In like Manner, fuppofe a Plane abed, four Feet fquare, 
 was removed five Feet into the Picture, and fuppofe the lower 
 Edge, or Plan, to be fomewhere in the Line AC. — From A, where 
 it cuts the Bottom of the Picture, fet off five Feet for its Diftance, 
 (as A 5) and four Feet for its Width (as 5 9 ; ) then draw H $ . 
 and H9, cutting AC in a and b; then is ab the Reprefentation 
 -of its Depth ; therefore, draw AD, ad, be, perpendicular to the 
 Bottom of the Picture, and make AO equal to four Feet ; and 
 then draw OC, cutting ad and c b, which will compleat the 
 propofed Reprefentation. 
 
 II. T 0 find the Reprefentations of Planes that are parallel to the Picture* 
 
 S%. i0. Case .1. For a Plane ABDE, four Feet fquare, which, we fup- 
 pofe, is removed two Feet Jrom the Bottom of the Picture. 
 
 Divide the Bottom of the Picture into any Number of Parts, 
 which call fo many Feet; then from G and 4, draw GC and 4C,. 
 and make CL equal to CE, and draw 6L cutting 4C in E; then 
 is 4E equal to 4 6, that is, equal to two Feet; therefore, draw 
 EA parallel to the Bottom of the Picture, winch will cut GC in A, 
 and give the Length of one Side, upon which make the Square 
 ABDE, which will be the Reprefentation propofed. 
 
 Case 2. For a Plane fix Feet high and three Feet wide, which is 
 to be jive Feet from the Bottom of the Picture. 
 
 Any where, at pleafure, draw C4, and fet off three Feet, (as 
 47) then draw 7C, and from 4 fet off five Feet, (as 49) and 
 draw 9L, which will cut 4C in e; then will e 4 be equal to 
 4 9, that is, to five Feet. Again, from e, draw ef parallel to 
 the horizontal Line, which will give e f equal to three Feet ; that 
 is, equal to 4 7 ; therefore, continue e f at pleafure, and call a, 
 one Corner of the intended Plane ; make a b equal to e £ and 
 draw ad, be, perpendicular thereto ; then make a d equal to 
 twice ab, and draw dc parallel to the horizontal Line, and then 
 will the Reprefentation be compleated. 
 
 IIL Ts 
 
Of Objects perpendicular to the Ground. 
 
 III. To find the Reprefentatisn of Planes, when their Plans or Bafes 
 ' are oblique with the Bottom of the Picture, like AB, Fig. 8. 
 
 Let ABDF be the Reprefentation fought, which let be fix Feet Fig. 20 
 high, four Feet and a Half wide, and one Foot and a Half from 
 the Bottom of the Picture j and let L be its vanifhing Point, and 
 LG the Line in which the Plane is to ftand. — Draw GI perpen- 
 dicular to the Bottom of the Picture, and fet fix Feet upon it (as 
 in the Figure) and draw IL; then from G fet off Ga equal to one 
 Foot and a Half, and draw a-H cutting GL in Aj then is GA 
 equal to Ga j that is, equal to one Foot and a Half. Again, from 
 a, fet off four Feet and a Half, (as a 6) and draw 6H cutting GL 
 in B j then is AB equal to a 6, that is, equal to four Feet and a 
 Half> therefore by drawing AF and BD perpendicular to the 
 Bottom of the Picture, we mail have the Reprefentation of a 
 Plane, fix Feet high, and four Feet and a Half wide. 
 
 If the vanifhing Point L is out of the Picture, the Figure may Fig. 21 
 be draw thus. — Let BA be the Reprefentation of one Side given, 
 and AD its Height. — Continue AB at pleafure, and anywhere 
 upon it draw ab perpendicular to the Bottom of the Picture ; then 
 make cb to ca, as FD is to FA, and draw Db; which will give 
 the Length of BC ; for if DC be continued it will vanifh into L*. 
 
 From hence then it is evident, that the Reprefentation of any 
 perpendicular Plane may be immediately determined upon the Pic- 
 ture, without having Recourfe to the tedious Methods of Plans, 
 Elevations, &c. and but very few Lines are required, even when 
 the Reprefentation is to be of any given Dimenfion, or however it 
 is to be fituated upon the Picture: But, if the Reprefentation is not 
 to be of any particular Dimenfion, being left to the Difcretion of 
 the Artift, then nothing can be more fimple than the Operation. 
 For let AB be one Side given, and AD its Height 5 then from A Fig. it 
 and B draw AD, B C, perpendicular to the Bottom of the Picture 5 
 and from C, draw CD; which will compleat the Figure. 
 
 Here let us obferve again, that the vanishing Point C of the 
 Line AB, is the vanifhing Point of every Line DC, NM, &c. that 
 is parallel to AB; agreeable to the fecond Axiom. 
 
 Having fhewn how to find the Reprefentation of Square Planes 
 perpendicular to the Ground ; let us now proceed to join them 
 together, which begins the Perspective of folid Figures. 
 
 * Suppofe FD is equal to FA, then cb muft be made equal to ea. 
 
 IV. <To 
 
g| Of Objects perpendicular to the G R o u *n>.* 
 
 IV. To find the Reprefentation of Triangular Pieces of Woody &c. 
 //vy <i& ^ below the Horizontal Line. 
 
 Tig. -ii. Case i. JVhen they are below the Horizontal Line % and have one 
 
 Side, abed, parallel to the Pi Bur e. 
 Find the vanifhing Points H and L, of the oblique Sides^ as 
 taught in Figure 9; then from b and d, 4raw bL, dH, cutting 
 each other in c; then is bed the Top; which corapleats the Fi- 
 crure. In this Figure the Front Side ab-de, is parallel to the Pic- 
 ture, but in the Figure nopqrm, the back Side npmq is parallel 
 to the Pi&ure; therefore let us find the Appearance of that alfo: — 
 Here let mqpn.be the parallel Side given.— Through the Points 
 n and m draw Lines from H, and through the Points p and q 
 draw Lines from L, which will cut the Lines Hm, Hn, continued 
 in o and r ; therefore draw or, and the Thing propofed is done.-- 
 -Gr it may be found by giving one Edge rar,-- Continue mr to 
 its vanifhing Point H, and draw rL; then draw mq parallel to 
 .the horizontal Line, cutting rL in q; then is rq the other Side; 
 .therefore draw mn, ro, pq, each perpendicular to the horizontal 
 line, and make any of them the propofed Height, (fuppofe ro) 
 then .draw ©H, oL, cutting mn, and qp, inn and p, and then 
 cdraw .np ; which finifhes the Figure. 
 
 Case 2. When they are above the Horizontal Line. 
 Let f g be one of the Bottom Edges given.— Draw fi perpen- 
 dicular to the horizontal Line, and make it equal to the propofed 
 Height; from f draw f L, then draw gl parallel to the horizontal 
 Line, cutting fL ih i; and then is fl the Deptk of the other 
 Side ; therefore, draw g h and lk parallel to f i, and from 1 draw 
 iH and iL; which compleats the Reprefentation* 
 
 V. To find the Reprefentation of any triangular Figure, when all ih 
 Sides are oblique with the PiBure* 
 
 % 23. Case 1. When it is below the Horizontal Line. 
 
 Let AD be one Edge given, whofe vanifhing Point is out of 
 the Piaure, and let L be the vanifhing Point of the other Edge 
 AG. — Make AG equal to AD, by Figure 10; and from A, V,^, 
 draw Lines perpendicular to the Ground; then make AB equal to 
 the Height, and draw BL ; which will compleat the Side AUt u . 
 Again, continue AB and DE to the horizontal Line, and make 
 
Of Objects perpendicular totheGnovxD* 25 
 
 l>E to MF, as AB is to AR* ; then draw BE,, which compleafs 
 the Side AB»ED; finally, draw EF, which finiflies the Reprefen- 
 tation. For, if AD and BE were continued, they would both, 
 vanifh into the fame Point in the horizontal Line 1 as was obferved; 
 in Fig. 21. 
 
 Case 2. When it is above- the Horizontal Line. 
 
 Let HO be one Edge gjvsn,, whofe vanifhing Point is out of 
 
 the Picture, and let L be the vanifhing Point of the Edge HN. 
 
 Make HN ejual to- HO, as before; and from N, H; 0> draw 
 Lines perpendicular to the Ground , then make FFK equal to the 
 propofed Height, and drawKL, which will give one Side : Again, 
 make PQ.J the Part above the horizontal Line) to PO- (£he Part 
 below the horizontal Line) as IX is to* IH (which- in 'this Fi- 
 gure is as z is to, 3), and draw K% which will compl eat the Re- 
 prefentation propofed. For if HO. and K Qjvere continued, they 
 would meet in a Point upon the horizontal Line^. 
 
 This Method of determining the Appearance of any Line, when 
 its vanifhing Point is out of the Picture,. is> extremely ufeful ; and 
 therefore, the Reader cannot make, it too familiar to him; the 
 general Method for which I have, farther explainednn Figure 4 u 
 
 VI.- To find the Reprejentativns of Cubes^ both above and below the 
 Horizontal Line, when feme oj their Sides are parallel to the PiBure. 
 
 Case. Iv When below the Horizontal Line. 
 
 Let abeg Be one Side given. Draw bC, eC, and gC, and Fig. H: 
 
 find the vanifhing Point H of the Diagonal e f, by Figure 1 1 ; 
 and draw eH cutting Cginf; then from f, draw fd parallel to 
 the horizontal Line, cutting eC in d ; and then draw d c parallel 
 to ebi and the Reprefentation is compleated. And for the Cube 
 E-Make the front-fide like the Cube A, and draw Lines from the 
 upper Corners to the Center V C ; then by continuing eg and fd 
 we may compleat the other alfo ; as in the Figure. And therefore^, 
 having got one Reprefentation, That will be fufficient for any, 
 Number of thefameKind, provided theyftand all in the fameLine, 
 a n j that is, at the fame Diftance from the Bottom of the Picture. 
 
 * Suppofc A B fa four Parts, and BR one Parr- then divide JfcH int« five Parts ; . and thea 
 a nh to £ D as R B to B A ; that is, as one to four. 
 
 Book II. D Cass 
 
z& Of O ejects perkndieular ftp the Q'k^vu 3*. 
 
 Case 2. When above tke Horizontal Line, as B*md D. 
 
 Here let the Reader obferve, that the Rule in either Cafe is tne 
 fame and therefore he is to proceed in the fame Manner in firtd* 
 ing the Heprefentation of a Cube above the Eye, as we have done 
 in determining the Appearance of a Cube below the Eye j which 
 is fufficiently explained by the Figures.— And fo likewife for the 
 Depth m o, of the Parallelopiped F ; wfcich is found by drawing a 
 Line from the Corner n, to the vanishing Point of the Diagonal H. 
 
 VII. Of a Cube and Parallelopiped, whofe Sides &re all oblique with 
 
 the Picture, 
 
 Case i» Of .the Cube. 
 if ig. 24. Let a b be given, whofe vanishing Point is L; and let H be the 
 
 vanifhing Point of the other Side ag. From a, draw a 1 parallel 
 
 to the horizontal Line, and make L A equal to the Diitance LE; 
 tthen through b, draw A 14 cutting A 1 in 1 ; and then is a 1 
 equal to a b ; and from a and b draw Lines perpendicular to the 
 .horizontal Line, and make a e equal to a 1 j then from e, draw 
 «L, cutting be in c; and then we fliall have one Side : Again, 
 from c and e, draw Lines to H, and from e, draw a Line to C, 
 <the vanifhing Point of the Diagonal) which will cut c H in d ; 
 then from L draw a Line through d, cutting eH in f* finally, 
 from f, draw f g parallel to- a e * and then will the Reprefentatioa 
 be compleated. 
 
 Cass 2. Of a- Parallelopiped, or oblong Piece of Wood:, rejling upon 
 
 one of its longejl Faces. 
 Let it be required to make it three Feet long, one Foot thick, 
 and one Foot high z And let o be the neareft Corner, andH, L, the 
 
 vanifhing Points of the Sides. Through o, draw 2 o 3, parallel 
 
 to -the horizontal Line, and fet off three Feet upon it; then draw* 
 oH and oL, and make HB equal to HE, and draw B2 cutting 
 oH in i; then is oi equal to three Feet : Again, make L A equal 
 to LE, and from o, fet off 03, equal to one Foot; then draw 
 3 A, cutting oL in n; and then is on equal to 03 : Again, 
 from i, o, n, draw Lines perpendicular to the horizontal Line, and 
 make om equal to 03; finally, from m, draw mH and mL, 
 cutting ih in h, and nl in 1 ; then from 1 draw a Line to H> 
 and from h draw a Line to L, which will cut each other in k, and 
 fo finifh the Reprefentation ; which will be three Feet long, one 
 Foot thick, and one Foot high. 
 
 VIIL ?p 
 
@f Q b J e*c T s perpendicular to the G r o u n d. 
 
 2 
 
 ^HT.. To Reprefentation of an Hexangular Figure \ both above 
 
 and below the Eye. 
 
 Case i .. When below the Eye. 
 
 Let ab be one Side given, and; let H and L be the vanifliing Fig. 
 Points of the other Sides. — Continue ab on cither Side, at plea- 
 fore, and make ai, b 2, equal to ah ; then cut off a e and bf>, 
 equal to 1 a and 2b; and from e, a, b, f, draw Perpendiculars to 
 ab ; then make ad equal to the propofed Height, and draw c d. 
 parallel to ab; then from c and d, draw Lines to L and H, cut- 
 ting f h and eg in h and. g; and from g and h, draw Lines to 
 L and H, and from c and d, draw Lines to C, cutting them in ju 
 and k.j; finally,, draw ik, which will be parallel, to cd, and: will, 
 fee the Reprefentation propofed. 
 
 Case 2.. Whm abme the Eye,. 
 
 Let m n be one Side given, and H, L, the vanimihgPoihts of 
 the other Sides, as before.—- Continue mn, at. pleafure, as 3 4, 
 from whence the other Sides may be found, , and confequently, the 
 whole Reprefentation; as is evident, by the Figure:: In which I. 
 have put every Line in the Operation, to make it eafy to be under- 
 stood without any further Explanation* 
 
 Erom hence then it follows, , that the Appearance of any Objects 
 may be as eafily determined ■ above, the horizontal Line as below it;: 
 fmce one Rule ferves in both Cafes;, and therefore it matters not 
 whether, we begin our Work at the Bottom or at the Top of the 
 Pifture.. Now, this Method of finding the Reprefentation of, Ob- 
 jects is of prodigious Ufe. . For fuppofe it was required to draw 
 the Reprefentation of the Top of any Building; we need notfketch. 
 out any more of it than is to appear upon the Picture ; but we 
 may begin in the very Place where that Top is to be, without, 
 undergoing the tedious Tafk of beginning at the Bottom of fuch . 
 & Building, and afterwards rubbing out what is not to appear.. 
 
 IX. To put an Off angular Building into Perfpe5live. 
 Let ah be one Side given, and let H, C, L, be the vaniftiing Fig. 
 Points of the feveral Sides. — Find the Reprefentations of the Sides, 
 as ah, be, and cd, which are vifible to the Eye (by Fig. 15.) and 
 from the feveral Points h, a, b, c, d, draw Perpendiculars to the Bot- 
 tom of the Picture; then make ak equal to the propofed Height, 
 and draw kg parallel to ab, which compleats one Side; then from k 
 
 D z draw 
 
£f Of Ob ject s 'perpendicular to the <J ro tr'N fc. 
 
 draw kH, -and 'From g draw gL, which finiihes two Sides more % 
 finally, from f draw f C, which will compleat the whole Repre- 
 sentation. 
 
 X. To find the Representation of Cylindrical, or round Obje&s^ fucb 
 
 as Columns, and the like, 
 
 in the 1 6th and 17th Figures we have fhewn feveral Methods of 
 finding the Appearance of' Circles upon the Picture, by which 
 ■means the Reprefentations of Circles of any Dimenfions may be 
 determined with great Exactnefs ; -and fince a Circle is tfhe Bale of 
 a cylindrical Object, therefore, by finding the Reprefentation of 
 two Circles at any determinate Diftance, the Appearance of that 
 Object may be determined alfo. 
 
 ^ig. 28. it bi required to find the Reprefentation of a round Objecl like D. 
 
 Case x. When it ftands upon one End, . 
 
 Let ef be the Diameter given, and n the Center of the Circle. 
 —Draw nb perpendicular to the Bottom of the Picture, and 
 make it equal to the Height you intend for the Reprefentation \ 
 then, by Method 3, Fig. 17, find the Reprefentation of the Square 
 g h f k i e ; which will be a fu'fncient Guide for drawing the Ap- 
 pearance of the Circle, as in the Figure. Again, through b, draw 
 cd parallel to ef, and, by the fame Method, find the Reprefenta- 
 tion of the Square al, which will be a Guide for the upper Circle ; 
 finally, from the Extremities of both Circles, draw ec, f d, parallel 
 to n b, which will exhibit the vifible Appearance of the round 
 Objecl:, as in Figure D. 
 
 From hence then it is manifeft, that any Number of round Ob- 
 jects, (fuch as Columns, &c.) may be found upon the Picture, by 
 having only their Diameters and perpendicular Heights, as we 
 have further fhewn in Figure 63, Sfc. 
 
 Kg. 29° Case 2, When a Cylinder -lies upon the Ground oblique with ife 
 
 FiSlure, 
 
 This alfo may be done with the grcatefr. Eafe, by finding the 
 Appearance of two geometrical Squares. Thus, let A be the Cor- 
 ner of the Square for the neareft End of the Cylinder, AB its 
 Diameter, and AC its given Length ; and let H be the vanifhing 
 
 Point of the End, and L the vanifhing Point of the Sides. Cut 
 
 off AE equal to AB, and from, the Points AE draw Perpendi- 
 culars, and compleat the Square a ; then draw its Diagonals, 65V, 
 
 and 
 
Of Objects perpendicular to the Groun d. 
 
 &nd then the Appearance of the Circle, as in the Figure. And 
 for the Length, cut off AD equal to AC, and from the Point D. 
 make another Square for the farther! End, in which draw the Ap- 
 pearance of another Circle ; then draw Lines from the Extremities 
 of the Circle a, to L, which will cut the Circle in the Square at 
 D, and thereby cofnpleat the Reprefentation as required. 
 
 If there mould not be Room enough for the whole Draught 
 below the horizontal Line, it may be done above it, as in the Fi- 
 gure, taking great Care to make them both at the fame Diftance 
 from it.-— Thus, cn is equal to cA. 
 
 From hence it appears, that this Figure is determined after the 
 feme Manner as the oblong Piece of Wood, Fig. 25; <mly, one is 
 fquare and the other circular. 
 
 Thefe Examples are fufficient to (hew the whole Practice of 
 Per'fpeftive, fo far as it Telates to Objects which lie Hat upon the 
 Ground, or are perpendicular to it : For, as I obferved before, the 
 immediate Obj efts of Perfpective, are a Triangle, a Square, and a 
 Circle ; and therefore were we to multiply Objects to infinity, they 
 would be compounded of fome or all thefe put together ; and con- 
 fequently, what has been faid already, is fufficient for our Piarpofcv 
 
 SECT. IV. 
 
 Of Objects which are inclined to the G Rous D r fich as Pedi~ 
 ments, Roofs $f Houfes, and the like. 
 
 f T 1 H I S Part of Perfpec~tive, neither the Jefait> nor Pozzo, not 
 JL many others, feem to have had the leaft Knowledge of ; for 
 they have confined themfelvcs wholly to the horizontal Line, with* 
 out confidcring any other vanifhing Line ; and therefore, when 
 they have fhewn how to find the Appearances of inclined Objects, 
 they did it by means of Plans, Elevations, &c. which is not only a 
 tedious, but an uncertain Method. But, Dr. Taylor has fhewn 
 us, that inclined Objects have their proper vanifhing Lines and 
 Points, as well as thofe Objects which lie flat upon the Ground, or 
 are perpendicular to it ; and that the Method for determining the 
 Appearance of Objects in either Cafe, is exactly the fame. Which 
 wc are now going to demonftrate. 
 
 I n 
 
Of Objects inclined to the Q R p u p; 
 
 fig. 3©- I, fo fnd the Representation if a Square, by means of its. Diagonals 
 only, when it is fituated like G H I K, Fig. 18. 
 Method i . Let a b be the lower Edge. 
 Continue ab to its vanifhing Point C, and through C, draw EV 
 perpendicular to HL, which will be the vanifhing Line ef the 
 propofed Plane ; then make C€ equal to the Diftance of the Eye, 
 and draw AB parallel to the horizontal Line , upon which make 
 the Square ABC D of any convenient Bignefs, and draw its Dia- 
 gonals AC, BD-, then from draw <t£ and #V parallel to the 
 Diagonals AC, BD$ and where they cut the vanifhing Line EV,, 
 will be the vanifhing Points, fought : Thus, E is the vanifhing 
 Point of the Diagonal ac, and Y i* the vanifhing Point of tha 
 Diagonal bd. 
 
 Method 2, By making a gfaen Angle at the Eye. 
 The Angle of a Square is a right Angle, and contains 90 De- 
 grees, the half of which is 45 Degrees; therefore, at <t, with 
 the horizontal Line, make two Angles, C€E^ and CdpV, each 
 equal to 45 Degrees, then draw v£E, d£V, cutting the vanifhing. 
 kine in E and V; which will be the vanifhing Points, as before. 
 
 Method 3. By the Dijlance of the Eye only*.. 
 
 Through the vanifhing Point C, draw the vanifhing Line EV, 
 and make CE, CV, each equal to the Diftance CE of the Eye, 
 which gives the vanifhing Points propofed. 
 
 From hence then it appears, that Planes which are perpendicular 
 to the Picture, and to the Ground ako, wilj have their vanifhing 
 Lines pafs through the Center of the Picture, perpendicular 
 to the horizontal Line ; and that all the oblique Lines which can 
 
 S>e drawn within thofe Squares, w ill vanifh into this Line, for the 
 ame Reafon that all the oblique Lines which can be drawn within 
 a Square that lies upon the Ground, will vanifh into the horizontal 
 Line. And from hence alfo we may conceive, why Roofs, Pedi- 
 ments, &c. will have their proper vanifhing Points as well as any 
 other Objects. For let a b e be the End of a Roof or Pediment, 
 then is E the vanifhing Point of the Side a e which is next the Eye, 
 and V is the vanifhing Point of the other Side b e ; and if Lines are 
 drawn through E and V, parallel to the horizontal Line, then 
 thefe Lines will be the vanifhing Lines of the Sides of the Roof, 
 for the fame Reafon that E V is the vanifhing Line of its Ends : 
 As is. evident from the next Figure. 
 
 11. r$ 
 
Of Objects Inclined to the Ground. 
 
 tl, find the vani/hing Lines and vanifhing Points of a Roof, when Fig. 31. 
 the End of the Building is fituated like abed in the laft Figure, 
 
 Draw the vanifhing Line I J, as before taught, and make C€ 
 equal to the Diftance of the Picture. — -Parallel to the horizontal 
 One, (or if you pleafe upon the horizontal Line) draw AB, upon 
 which, draw the End of the Roof, as ABC ; tlien from (£, draw 
 (£V, <& L, parallel to AC y BG, cutting the vanifhing Line in V and 
 *L j then are V and L the vanifhing Points of the inclined Edges 
 ac, b e. Again, through V and L draw Wlfy and v j parallel to 
 the horizontal Line, and then will WL % be the vanifhing Line of 
 the inclined Side aecd, and vl will be the vanifhing Line of the 
 inclined Side bed. 
 
 Now in order to find any vanifhing Point upon either of the 
 vanifhing; Lines oft %, or vl, we muft proceed exactly in the fame 
 ■manner as in finding any vanifhing Point upon the horizontal 
 Line; namely, by fetting off* the Diflance of the vanifhing Line,' 
 and then drawing Lines from thence parallel to any original Linea 
 whofe vanifhing Points are required. Thus, let it be required to 
 find the vanifhing Points of the Diagonals of a Square, whofe 
 Sides vanifh into the Center V, of the vanifhing Line cEX > like 
 ab, dc, Figure 32.— Make VI equal to V<£, and LJ equal to L<£j 
 then at I and J, with the Lines VI, LJ, make Angles of 45 De- 
 grees each, as in the Figure, and draw the Lines IU, l% y Jv and 
 J„l ; which will give the vanifhing Points propofed. — Or it may be 
 done by making BV, HV, ^r. equal to the Diflance Vdc^ which 
 comes to the fame Thing. For fuppofe the Picliire removed into 
 the Place of the gal Figure; where U% is the vanifliing Line of Fig. & 
 the Square abfg, and WL and % the vanifliing Points of its Diago- 
 nals; and let ab be one Edge of the Square, which ftands upon 
 the Ground. — FrOm a and b draw "Lines to V, which is the vanifli- 
 ing Point of the Sides a e, bd; and then from b draw a Line to 511, 
 and from a, draw a Line to U, cutting aV and bV in f and g* 
 finally, draw gf; and then will abfg be the Reprefentation of a 
 Square inclined to the Ground, like the Line A C, Fig. 31. And 
 in like Manner, if another Square was required, as fged, it may 
 be found by repeating the laft Operation ; that is, by means of the 
 Diagonals, as is evident by infpe&ing the Figure : Or any Number 
 of Squares may be found by the fame Method. From whence it 
 is manifeft, that the Reprefentation of any inclined Object may 
 'very eafily be determined, and made of any given Proportion. 
 
 And 
 
32 (yO-BjrcTs inclined to the Gro^n-d» 
 
 And what has been faid about the inclined Side abect, is equally 
 applicable to the oppofite inclined Side ;, fmce the only Difference 
 confifts in working below the horizontal Line, inftead of -above it i 
 For vl is its vanifhing Line, and' v and 1 the vanifhing Points of the 
 Diagonals, &c.~ I have added the Figure A, which reprefenti, as it 
 were, the Frame-work of the other and will ferve to explain the 
 Thing more fully. 
 
 The principal Difficulty in determining the Reprefcntation of 
 any inclined Planes, lies in finding the Center and Diftance of their* 
 peculiar vanifhing Lines j therefore, before we proceed any further, 
 we will give fome general Rules for that Purpofe, as is moreover 
 explained by the 50th Figure. 
 
 I. To, find the Center of a vanifhing Line. 
 Pig^ M> Let fl&L be a vanifhing Line given. — From C the Center of? the 
 Picture, draw C H perpendicular to the vanifhing Line L, and? 
 then is H the Center of that vanifhing Line. Again, lelSML be d 
 vanifhing Line given. — From C the Center of the Picture, draw 
 CO perpendicular to WL% y and then is O the Center of that vanifh- 
 ing Line.. 
 
 sr. To find the DiHance of a vamping Line. 
 Continue the Perpendicular C H, at pleafure, towards <fc j and 
 from the Center of the vanifhing Line draw H E to the Eye j then 
 is RE the Diftance of the vanifhing Line ULj therefore, fet off 
 H<£ equal to* the Diftance HE, and then is <£ the Diftance to be 
 work'd with. Again, for the vanifhing Line % — Continue the 
 Perpendicular CO towards I, at pleafure, and from O fet off O J 
 upon the vanifhing Line, equal to CE, the real Diftance of the 
 Eye, and draw CJ j then is C} the Diftance of the vanifhing Line 
 5flU- and by making OI equal to C J, we fhall have I for the 
 Point of Diftance of the vanifhing Line WL%.. From hence then* 
 it will appear, that C is the Center of the horizontal Line in the 
 30th Figure, and it is alfo the Center of the vanifhing Line E V; 
 that C E is the Diftance of the Picture, (that is, of the horizontal 
 Line) and C <£ is the Diftance of the vanifhing Line E V : And 
 becaufe the vanifhing Line EV panes through the Center of the 
 Picture, therefore the Diftances CE and CC muft be equal. 
 Again, V and L, of the 3 ift Figure, are the Centers of the vanifh- 
 ing Lines *&% and v i s and V I, LJ, are their Diftances, and fo 
 on. Thus much is fufficient for our prefent Purpofe but in the 
 
 50th 
 
Of Objects inclined to the G ft o v a d. 33 
 
 50th Figure I have given one general Rule, not only for deter- 
 mining the Center and Diftance of each vanishing Line, but for 
 finding the vanifhing l^ne of any Plane, let its Inclination be 
 what it will : All which mould be well remembered. 
 
 Having determined the Center ^nd Diftance of any vanifhing 
 Line, we are then to proceed with our Work in the very fame 
 Manner as in drawing the Reprefentations of Planes that lie flat 
 upon the Ground ; and, by turning the Figures, we may conceive 
 every vanifhing Line to be a horizontal Line, &c. 
 
 III. To find the Reprefintation of a Square a b e d, by the "vanijhing Fig. jj,' 
 Points of in Diagonals, when it fands perpendicular to the Ground 
 
 but oblique with the Piclure* like ABDF, Fig. 20* 
 
 ^Method r. By drawing out 4 Square. 
 Let ab, be the under Side given. — Continue a b to its vanifhing 
 Point O, and through O, draw I J perpendicular to the horizontal 
 Line; then is I J the vanifhing Line of the Square. Again, from 
 E draw EO, which is the Diftance of the vanifhing Line IT; 
 therefore, fct off 0(t equal to OE, and parallel to the horizontal 
 Line draw a Line, as AB, atpleafure; upon which, make a Square 
 ABCD, and draw its Diagonals; then from <fc draw Lines pa- 
 rallel to thofe Diagonals, which will cut the vanifhing Line in I 
 and J; and then are I and J -the vanifhing Points of the Diago- 
 nals ac and bd. 
 
 Method 2. 
 
 Make at fc, on each Side of the horizontal Line, an Angle of 
 45 Degrees, and draw CI, <£J, which will produce the vanifhing 
 Points propofed. 
 
 Method 3, 
 
 Or, the vanifhing Points may be determined in this Example by 
 making OI and O J equal to 6 tfc 
 
 IV. Suppofe a be to reprefent the End of a Roof as %efore- y then the 
 Sides of that Roof will be oblique with the Eye, like a We, 
 Fig. 34; therefore, let us next find the Reprcfentation of a Plane 
 fituated in this Manner. 
 
 Method*. Let at be the lower Edge of the Roof which let us ni g . 34% 
 fuppofe to be a Square that refts upon the Edge ae. 
 
 Continue the Side ae to its vanifhing Point %, and draw L<£ ; 
 then at E make a right Angle, and draw EH; and then is % the 
 
 B o o x II. E va . 
 
Of, Objects inclined to the G r o u: n dy 
 
 vaniihing Point o£ one Side of a Square which lies upon the 
 Ground, and H the vaniihing Point of another Side of the fame 
 Square; therefore, from the Corner a draw aH, and through H 
 draw Wih perpendicular to the horizontal Line j then is H the 
 Center of the vaniihing Line 3lXL,. and HE its Diflance. Again, 
 makeHC equal to EH, an4 make AC parallel to it; then upon 
 AC draw the End of the Roof, (that is, the Angles of its Incli- 
 nation) and from C draw <tL, parallel to A B, CB, which 
 will cut the vanifhing Line <0L, in the vaniihing Points <H and L$ 
 and fo will flft be the vaniihing Point of the Edges a b, e d, and % 
 will be the vaniihing Point of the Edges ae, bd, and L will be the 
 vaniihing Point of the Edge c b : And if a Line be drawn from WL 
 to H, it will be the vaniihing Line of the inclined Face aebdj; 
 from whence it is evident, that after we have found two vanillic 
 ing Points of any inclined Plane, if a Line be drawn through thofs 
 Points, it will be the vaniihing Line of that Plane- But to com- 
 pleat the Reprefentation propoled : Find the Center of the vaniih- 
 ing Line B%, and fet off its Diftanee upon the Perpendicular OI 5 
 then draw Lines from the vaniihing Points WL and H, which wilt 
 make a right Angle at the Eye ; then bifeel: that Angle by the Line- 
 ID, which will give D for the vaniihing Point of the Diagonal of 
 a Square ; by which means the Square aeb d may be compleated. 
 
 Method 2. By having the Width, as ac, of the Roof given. 
 Let ac be the Width, or what is generally called the Span, of 
 the Roof ; and let WL and L be the vaniihing Points of the Roof. 
 
 From a, draw aH, and through c draw L b, cutting aH in 
 
 b; then is abc the Reprefentation of one End. Again, from H> 
 draw HE, with which, at E, make a right Angle, and draw EH; 
 then is % the vaniihing Point of the Side ae which rells upon the 
 Ground ; therefore, draw aH and b % ; then find the vaniihing 
 Point of the Diagonal of a Square, whofe vaniihing Points are H 
 and %, and from a, draw aD, cutting b% in d; then from WL 
 draw a Line through d, which will cut aH in e, and thereby com- 
 pleat the Reprefentation propofed. 
 
 M e t hod 3* By cutting off one Line equal to another Line given. 
 
 From the Corner a draw af parallel to the vaniihing Line 
 Wi% 3 and make af equal to one Side of the intended Square 
 then fet off UF equal to 5111, and draw Ff, cutting 
 aflfl in b; and then is ab equal to af. Again,, from a and b 
 
 draw 
 
■t}f Objects Inclined to the Ground; 
 
 draw alt, b% y and from a draw ag parallel to the horizontal 
 Line, and make it equal to af; then fet off %G equal to %E f 
 and draw Gg, cutting aH in e ; then is ae equal to ag, that is, 
 equal to af* therefore, draw e0, which compleats the Squarp 
 gibed ; finally, draw aH and bL, which will finifh the whole Figure, 
 
 Method 4. By having the vani/hing Line <&% given, at pkafure* 
 
 From the Center of the Picture draw CO perpendicular to the 
 vanifhing Line WL %, and fet off the Diftance of the vanifhing Line 
 -fiom O to I, and let ab be one Side given,— Continue ab 'till 
 it cuts the vanifhing Line in its proper vanifhing Point iti, and 
 from WL draw 3131; then at I make a right Angle, and draw IH; 
 and then is % the vanifhing Point of the Sides ae, bd; and by 
 finding the Point D, the Square may be compleated, as before. 
 Again, for the upright End j— Continue the horizontal Line at 
 pleafure, and make IX € equal to <H I, cutting the horizontal Line 
 continued in <B ; then is 4 H the Diftance of the vanifliing Line 
 WLL-, by which means the vanifhing Point L of the Side b e may 
 
 be determined : Or the vanifhing Points of any Lines may be 
 
 found upon WL H, by inferibing a Figure at the Eye, like the Ori- 
 ginal of our propofed Reprefentation ; as the Square I. Now what 
 is faid of a Square, will ferve for any other Figure ; which, it is 
 prefumed, is now fb evident as to need no farther Explanation ; 
 efpecially, fince a little Practice will make all that has been ad- 
 -vanced in this Section very eafy and familiar. 
 
 Here let us obferve, that when Objects are parallel to the 
 Ground, they will have their feveral vanifhing Points in the hori- 
 zontal Line ; when they are perpendicular to the Ground, they 
 -will vanifh into a Line perpendicular to the horizontal Line, like 
 Fig. 30, 3*> 32, 33; when they are inclined to the Ground, 
 "but have fome of their Edges parallel to the Picture, like a b, ed, 
 Fig. 32. they will then vanifh into Lines parallel to the horizontal 
 Line; and will be above the horizontal Line when the Plane leans 
 from the Eye, and below the horizontal Line when the Plane leans 
 towards the Eye; but when the inclined Planes are every way 
 oblique with the Picture, the Eye, and the Ground, like Fig. 34, 
 then the vanifhing Points of thek feveral inclined Sides will vanifh 
 into Lines aflant the horizontal Line, like (E %. Now, thefe 
 being all the Variety of vanifliing Lines which can ever happen in 
 common Practice, it were needlefs to produce any other Examples 
 
 E a o£ 
 
Of O b J e c T s inclined to the G r o u n d; 
 
 of this Kind : But to affifl the Curious in determining the Repre- 
 fentations of Regular Solids, * or fuch-like complicated Bodies, I 
 have added the fix following Figures ; which may be omitted by 
 the Generality of my Readers, as Things more curious than ufeful; 
 and which are not in the leaft eflential to common Practice ; and 
 therefore, they are. now referred to the next Chapter. 
 
 3*. V. Tq fni the Reprefentation of a Cube that refy upon one of 
 
 its Edges a b. 
 
 Example i . When fame of its Edges > as a b, c d, f e, are parallel ' 
 
 to the Pi&ure*. 
 
 Let ab be one Edge given-, which let us fuppofe refts upon thc„ 
 Ground. Now, becaufe the Edges ab, &c. are parallel. to the Pic- 
 ture, therefore the End adf g will be perpendicular to the Picture;, 
 and confequently, the vanifhing Line VL of that. End will pafs 
 through the Center of the Picture,, and will be perpendicular to 
 the horizontal Line l And. if we fuppofe the Diagonal af to be. 
 perpendicular to the Ground, then the vanifhing Point of the other 
 Diagonal dg will be the Center of the Picture, becaufe it is pa- 
 rallel to the Ground. Therefore, through C draw the vanifhing; 
 Line VL, and make C<£ equal to theDiftance of the Eye 5 then 
 at C make a Square, in fuch a manner that its Diagonal 1 2 may 
 be parallel to the vanifhing Line VL; or, which is the fame thing) 
 make the other Diagonal a Part of the horizontal Line then 
 draw <£ V and <t L parallel to the feveral Sides of the Square,, 
 which will produce V and L for the vanifhing Points of thofc 
 Sides. This being premifed, let us now compleat the Represen- 
 tation, from the Side ab, which is given. — Through L, draw vl 
 parallel to the horizontal Line, and make LI equal to the Diftance 
 LJ of the vanifhing Line vl > then from L draw Lines through a 
 and b, and continue them at pleafurej and from 1, (which is the 
 vanifhing Point of the Diagonal) draw a Line through b, cutting 
 Ld in d ; then from d, draw dc parallel to ab, which will com- 
 pleat the Face abed. Again, from a and d draw Lines to V, and 
 from d draw another Line to C, cutting a V in g 3 and from L 
 
 * Regular Solids, are Bodies terminated by regular Planes, and are five in Number, viz. 
 h the Tetrahedron ; 2. the Hexahedron, or Cube; 3. the O&ahedron ; 4. the Dodecahe- 
 dron j and y. the leofahedron : The firft of which is compofed of four equal and equilateral 
 Triangles; thefecond, of fix geometrical Squares ; the third, of eight equal and equilateral 
 Triangles ; the fourth, of 12 regular Pentagoas ; and the fifth, of a* eq«al and equilateral 
 Triangles. * * 
 
 draw 
 
Of Ob j fi c r s inclined to the O r © u n d. 
 
 <fraw a Line through g, cutting d V in f ; which finifhes another 
 Face : Finally, from c draw cV, and from f draw a Line parallel 
 to dc, which will cut cV ia e, and thereby compleat the whole 
 &eprefentation. ■' 
 
 Again^ When the End adfg ftands in fuch a Manner that the 
 Diagonal af is perpendicular to the Ground; in this Cafe, the 
 Angles at <£, made by the Sides of the Square with the horizontal 
 Line, will be each equal to 45 Degrees; and therefore C is the va- 
 nilhing Point of the Diagonal of the Square : But if the End be 
 fituated like the Square B, then draw a Square, repofing upon one 
 of its Corners on the horizontal Lines, in the fame Manner as^you 
 fuppofe the End of the real Cube to be fituated' upon the;Ground ; 
 after which, draw Lines from <£, parallel to* the- Sides of that 
 Square, and then its Diagonals will cut the- vanilhing Line V L, 
 continued, in the proper vanilhing Points of thofe Lines ; thus, F 
 and H are the vanilhing Points of the Sides of the Square, and G 
 is the vanilhing Point of one of its Diagonals;. 
 
 Example 2. When the Cube reft s upon one Edge ab, that is F 
 
 oblique with the. Pitfure. 
 Continue ab to its vanilhing Point U, and from % draw a Line 
 to the Eye E, with which make: a right Angle ; then draw E H, 
 and through H draw 211 E, perpendicular to the horizontal Line, 
 and then is WL L the vanilhing Line of the End a d e f . Again , let 
 H be the vanilhing Point of the Diagonal ed; then, by making 
 HfllX and HL equalto the Diftance HE, we lhall have the vanilh- 
 ing Points of the Edges ad, df, &c. and if from 211 and % we draw 
 511 %, L%, we lhall have the vanilhing Lines of the Faces abed, 
 cdfg; Again, find the vanilhing Point D of the Diagonal bd, 
 and from D draw a Line through b, then from L draw Lines 
 through a and b, and then will L d cut D d in d, and thereby give 
 a d for the Edge a d ; therefore, from d draw dlt, which compleats 
 one Face. In like manner, from a and d draw Lines to %. 
 and from d draw dH, cutting a ill in e ; then from L draw a 
 Line through e, cutting d<E in f ; and then we lhall have com- 
 pleated another Face ade f : Finally, from f draw a Line to %, 
 and from c draw a Line to 2L*, which finifhes the whole Figure. 
 
 Example 3. When the Cube ftands upon one Corner ■, as a* ' F 
 Let us fuppofe the Cube to ftand in fuch a manner, that a Line 
 pafling through the upper and under Corners will be perpen- 
 * dicular 
 
Of Objects -inclined to the o u n i*; 
 
 dicular to the Ground; in which Cafe, -a Plane a c e h, that 
 •pafles through thofe Corners, will.be perpendicular to the Ground 
 alfo, and confequently, its vanifhing Line will be perpendicular to 
 the horizontal Line. And let us, moreover, fuppofe/ that this va~ 
 
 nifhing Line, as KI, panes through the Center of the Picture. , 
 
 Any where, at pleafure, draw a Square z and its Diagonals; then 
 .upon the horizontal Line, draw a Perpendicular AE, at pleafure 
 ;alfo ? ; and at the Point A, with -the horizontal Line, make an Angle 
 W 54 Degrees, and draw AH; then at A make another Anile 
 CAD of 36 Degrees, and draw AC j and make AH equal to the 
 Diagonal of the Square z, and A C equal to one of its Sides • then 
 draw CE <parallel to API, and EH parallel to AC : So mall we 
 have a Plane AC EH, which may.be confidered as the original of 
 the Reprefentation aeeh; whofe longeft Side ah is inclined to the 
 'Ground at an Angle of 54 Degrees, and whofe fhorteft Side is in- 
 clined to the Ground at an Angle of ^Degrees; which together 
 make a right Angle ; that is, one Angle of a Square. Now 
 having made thefe neccflary Preparations, .let a be the Corner upon 
 
 which it ftands. Set off the Diflance of the Eye CE, and draw 
 
 KI through C; thenfrom E, draw EK parallel to AH, (Fie X) 
 .-and EI parallel to AC, (Fig.X) cutting KI in I and K ; then 
 are I and K the vanishing Points of die Plane aceh. Again 
 through K draw 211 % parallel to CE, which will give the vanifh- 
 mgLineof the .upper Eaee c d c f ; and fince K is the vanifhing; 
 Point of its Diagonal ce, therefore, by making KWL t and K% 
 *each equal to the Diftance EK, we mall have the vanifhing Points 
 of the Sides cf, cd, i&c. by which means that Face may be corn- 
 pleated. Then let ce be the Diagonal given—From c draw cM, 
 <xt, and from III and % draw Lines through e, which will pro- 
 duce the Face propofed. Again, having two vanifhing Points WL 
 and I, of the Face abed.— Draw Utt, which is its vanifhing Line - 
 and by finding the Center O, and its Diflance OC, together with 
 its vanifhing Point H of the Diagonal be, the Face abed may 
 be compleated. The fame may be faid of the other Face aefg. 
 The Figure K is drawn in fuch a manner, as tosfhew all the Faces 
 *>f a Cube in the above Situation. 
 
 Let us now, without any Regard to a particular Situation of the 
 fube, iuppofc ab one Edge-given, 0 its vanifhing Point, .and .01 
 its vanifhing Line 5 and let C be the Center of the Picture, and CE 
 its Diflance.— Find the Center and Diflance of the vanifhing Line 
 SI I, and draw^C; then, at * make a right Angle, and draw (tU 
 
 and 
 
Of Ob j e c t s inclined to the Grounc, 39 
 
 and then will I be the vanifhing Point of the Edge ac; and by 
 finding tlie vanifhing Point H of the Diagonal be, the Face abed 
 may be compleated. For,, from the- vanifhing Point !, draw a Line 
 through the Center of the Picture, and continue it at pleafure, (as 
 IKj) then from C draw a Line perpendicular to IK, and make CE 
 equal to the Diftance of the Eye; then draw IE, and at E make a 
 right Angle ; then draw E K , cutting IK in K 3 finally through. K 
 draw a Line parallel to CE, which will pafs through the vanifhing 
 Point P, and produce the vanifhing Line fill % of the Face c d e f. 
 Again, from % and I draw %I, which will give the vanifhing Line 
 of the other Side ac f g.. 
 
 I have dwelt the longer upon this laft Figure, as it is a very 
 curious Example, and, as it were,, opens theJWay to the Projection 
 ©f all the regular Solids. 
 
 Example 4, To find the Representation of a regular Tetrahedron, Big. 38, 
 repofmg upon one of its Faces. 
 
 This alfomay be done eafieft by finding a perpendicular Plane 
 which is fuppofed to pafs through the Middle of the Body, as ade.-- 
 Now in order to find the Inclination of the Sides of this perpendi- 
 cular Plane, draw an Equilateral Triangle A G F, and divide the 
 Side G F into two equal Parts, and draw A E ;. then at E, with the 
 Diftance EA, defcribe an Arc; and at A, with the Diftance AG, 
 defcribe: another Are, cutting the former Arc in D; then draw AD* 
 ED; and then will AD be the Inclination of the Edge ab, and is 
 55 Degrees; and ED is the Inclination of the Edge ed, and is 70 
 Degrees. Having thus got the Inclination of the above Edges, the 
 next thing is to find the Reprefentation of the Face a b c, the va- 
 nifhing Points of whofe oblique Sides are H, L. Bifect the Angle 
 
 HEL, and draw E C ; then is C the vanifhing Point of a Line that 
 will divide the Side b c into two equal Parts ; and therefore C is the 
 vanifhing Point of ae, that is, of the Bottom of the perpendicular 
 Plane a de. Again, through C draw WL D perpendicular to the 
 horizontal Line, and continue it at pleafure ; and then is OX D the 
 vanifhing Line of the Plane ader Then at E, the Diftance of the 
 vanifhing Line 5L1 D, make an Angle with the Line CE equal to 
 55 Degrees, and draw E 511 ; and then is 0 the vanifhing Point of 
 the Edge ad. Again, make another Angle at E of 70 Degrees, 
 and draw ED; and then is D the vanifhing Point of the Side de, 
 by which means the Plane a d e may be compleated ; and by draw- 
 ing bd, and cd 3 the whole Figure will be finifhed. — Or it may be 
 
 done 
 
4® Vf Objtects incfined to the Ground; 
 
 <done by making the Figure ADE in fuch a manner that 4E may be 
 parallel to the horizontal Line ; for then, by drawing El, and ED, 
 parallel to AD andDE, the vanifhing Points WL and b will be 
 produced. — Or this Figure may be found by having onl>. the Incli- 
 nation of the Edge ad, which, we obferved before, was 4a Angle of 
 55 Degrees. Thus, make an Angle of 55 Degrees at E, and draw 
 E5U ; then fmce H is the vanifhing Point of the Edge sb, and 3ft 
 is the vanifhing Point of the Edge ad, therefore by draving 211 H, 
 and continuing it at pleafure, we (hall have the vanifhing Line U% \ 
 and by rinding the Center and Diftance of that vanifhing Line, and 
 making two Angles of 60 Degrees each at C, we mall have the va- 
 nifhing Points of all the Edges of the Side abd; and confequently, 
 by joining d c, the Figure will be compleated. — What is faid of the 
 vanifhing Line 311 % } may be faid alfo of the other vanifhing Line 
 
 Exa.mple 5. to put a Canted Cube \ into PerfpeBive, refting 
 upon one of its Square Faces, 
 T,et a c be one Edge of its under Face, A its vanifhing Point, and 
 H the vanifliing Point of another Edge of the under Face ; that is, 
 let A and H .be .the vanifhing Points of a Square that lies flat upon 
 the Ground. — Through H draw F I perpendicular to the horizontal 
 Line, and make H<£ equal to the Diftance of the vanifliing Line 
 FX Then at <E, on each Side of the horizontal Line, make an 
 Angle of 45 Degrees, and drawCF, then from F and I, 
 draw FA, IA; and then is I A the vanifhing Line of the Face 
 ahed; therefore, by rinding the Center and Diftance of this va- 
 nifhing Line, and by that means the vanifliing Point K of the 
 
 Diagonal be, the Square abed may be compleated. In like 
 
 manner, it is eafy to (hew, that FI is the vanifhing Line of the 
 Face g; GM the vanifhiag Line of the Face h; UN the vanifli- 
 ing Line of the Face b ; BD the vanifhing Line of the Face f ; 
 FA the vanifhing Line of the Face k ; F N the vanifhing Line of 
 the Face i ; HH the vanifhing Line of the Face m j and N the 
 vanifliing Line of the Face e. 
 
 * Thefe two Examples are fufficient to point out the Method for determining the Rerre- 
 fentatiori of all the regular Solids ; for having got the Inclinations of their fevcral Planes &c. 
 their peculiar vanifhing Lines and Points may be found with the greateft Eafe, 
 
 f A Canted Cube, is a folid Body comprehended under li Geometrical Squares and eieht 
 et|ualarvd equilateral Tnangl.i. * 6 
 
 E XAM P IE 
 
Of the Double Cro s s. 
 
 Ex amp le 6. To put a double Crofs into Perjfre&iw. 
 Let C be the Center of the Pi&ure, CL the horizontal Line, Ffc. 
 and L C the Diftance of the Eye ; and let L be the vaniftiing Point 
 of the Diagonal of a Square whofe Side AB is parallel to the Pic- 
 ture. — Now fuppofe AB to be the neareft Edge of the Bottom of 
 the Crofs, — Continue AB, and make BD equal to it; then from B 
 draw BC, and from D draw DL, cutting BC in a; and from the 
 Points A, B, a, draw the Perpendiculars An, Bra, ak, and make 
 Bm equal to thrice AB ; then draw nm parallel to AB, and finifli 
 the Top nk; after which, draw a 3 and s 1 parallel to AB, and 
 continue 2 3 and s 1 on both Sides, at pleafure ; then make 3 4 
 and 2 5 equal to 3 2, and draw 4 q and 5 6 parallel to n A ; then 
 from 4, 2, 5, draw Lines to C; and through i draw p 8 parallel 
 to 4 5, and from 6 draw 6 C, then from 8 draw 8 7, parallel to 
 5 6 5 which fmifhes a fmgle Crofs. Again, from C draw Lines 
 through 1, 2, 3, at pleafure ; and from L draw a Line through 3, 
 cutting cC in c ; then draw c e parallel to 3 2, and from e and 
 c draw ef, cd, parallel to 12; then from d draw df parallel to 
 e c j finally, from the Corner 8 draw a Line to L, which will cut 
 iC in b, and by that means give the fartheil Corner b; from 
 whence the whole Figure may be compleated. 
 
 I have added this laft Figure, to fhew the vaft Eafe and Expe- 
 ditioufnefs of this Method, preferably to any yet made publick ; 
 and I fhall have a further Ufe for it in another Place : However, I 
 will iuft obferve, that there are no more than four Lines in the 
 whole Operation but what are a Part of the Reprefentation itfelf. 
 
 Book U, 
 
 F 
 
 CHAP* 
 
CHAP. III. 
 
 The PraStice of Perfpe&ive abbreviated. 
 
 SECT. L 
 General Rules, &c» 
 
 IN the laft Chapter I have given fame general Rules, and have 
 explained them by the moft ufeful Examples, fo that the whole 
 Practice of Perfpective might be deduced therefrom by any 
 one who will confider them with a proper Attention : But left their 
 Application, in general, fhould not appear fo eafy as could be 
 wilhed, (and to fpare the Learner as much Trouble as poflible) I 
 fhall, in the firil Section of this Chapter, collect all the Rules toge- 
 ther in their proper Order » and then, in the fecond Section of this 
 fame Chapter, I mail apply them more particularly to common. 
 Practice. 
 
 fig. 41. I. Having one Line AB given,, whofe vanifhing Point is out of tha 
 Picture, from thence to draw another Line CD, which Jhall tend ta 
 the fame Point.. 
 
 From the Point A draw A C at pleafure, and at any convenient 
 Diftance, (the farther the better) draw B D parallel to A C. — Now 
 let A C reprefent the Corner of any Building,, and C the Top of 
 it s and let it be required to draw a Line, as CD, which is fup-* 
 
 pofed parallel to AB. Make B 6, which is below the horizontal 
 
 Line, in the fame Proportion to 6 D, which is above the horizon- 
 tal Line, as A 2 is to 2 C 3 that is, fugpofe 2 A two, equal Parts, 
 and 2 C three equal Parts ; then divide the Space 6 B into two 
 equal Parts, and make 6 D equal to three fuch Parts ; and then 
 draw CD, which will, if continued, vanifh into the Point E. And 
 by proceeding in the fame Ratio, any other Line may be drawn, 
 either above or below the horizontal Line. 
 
 Iv When r tis below the Horizontal Line, to draw a Line, as 1 £ 
 which tends to an inaccejjible Point E. 
 
 Let AB be the given Line. Draw A 2, B6, parallel to each? 
 
 other, and divide them in the fame Proportion j thus, let A 2 be 
 divided into tvvo equal Parts, then divide B 6 into two equal Parts: 
 alfo, and a Line, as 1 5, which is drawn through the Points 1, 5, 
 will vanifh into E, 
 
 2-. When 
 
General Rules, &c. 
 
 2. When it is above the Horizontal Line*, as 4 & 
 Here C D is the Line given, and 'tis required to draw 48 in 
 fuch a manner that it fhall cut off one-third of the Plane C 2 6 D. 
 Divide 2 C and 6 D each into three Parts, and draw 4 8 j which 
 is the Thing propofed. — Thefe Rules are applied to Practice in the 
 13th, 2 1 ft, and 23d Figures. 
 
 II. To make one Line eqmt to another Line given. 
 
 This may be done by giving a Line, as a c, parallel to the hori- Fig. 4e, 
 zontal Line, — For let cH be an indefinite Reprefentation, Part of 
 which is to be cut off equal to ac. From H, fet off HI equal to 
 
 H E, and draw a I, which will cut off b c equal to a c. And 
 
 fuppofe it were required to find the Length of any Line which is in 
 an inclined Plane, then the very fame Method is to be ufed. Thus, 
 in Fig. 44, let V be the vanishing Point of ab, and VL the va- 
 niftiing Line of an inclined Plane, in which ab is fuppofed to lye. 
 — Find the Center C of that vanifliing Line, and its Diftance CE; 
 then make V 2 equal to the Diftance E V of the vanifliing Point V, 
 and from a, draw ac parallel to the vanifliing Line VL, and 
 make it equal To the propofed Length, and then draw 2 c, cutting 
 aV in b; which will give ab equal to ac. Again, to make one Fig. 4* 
 Line, df, equal to another Line cb, which is given. Let H be the 
 vanifliing Point of c b, and C the vanifliing Point of d f. — From 
 the Points H and C, defcribe the Arcs IE, LE; from c draw ac 
 parallel to the horizontal Line ; and from I draw a Line through b, 
 cutting ac in a; then is ac equal to cb : For continue ac 'till 
 it cuts Cd in d, and make ed equal to ac ; then draw eL, which 
 will cut off d f equal to c b. * 
 
 III. To cut off a Line in any given Proportion. 
 Let a Line be drawn parallel to the horizontal Line, and conti- Fig. 43, 
 nued at pleafure ; and let it be required to cut off ac equal to three 
 Feet. — Thro' one End, as a, draw a Line 3 4, cutting the Bottom 
 of the Picture in 4, and the horizontal Line in 3 ; then fet off 3 1 
 equal to the Diftance 3 E ; and from 1 draw a Line through s, 
 cutting the Bottom of the Picture in e ; then from e fet off three 
 Feet upon the Bottom of the Picture, (as eh) and draw h 1, cut- 
 ting ac in c; fo will ac be equal to three Feet. — Again, To cut 
 oft an oblique Line, as a b, equal to three Feet. — Set off the Dif- 
 
 * Thefe Methods are applied to praftice in feveral of the preceding Figures. 
 
 F 2 tance 
 
44 General Rules, 6f<?. 
 
 tance^i, and from e and h draw ml hi, cutting ai in a and? 
 bj then is ab equal to three Feet.— Or this may be done without 
 taking the whole Diftance 3 E : Thus, take half the Diftance, as 
 3 2, and divide the given Line ac in the fame Proportion, that is 
 into two equal Parts, and draw fa, which will cut off ab equal to 
 ac. After the fame Manner, a Line may be divided into any 
 Number of Parts, or be made of any given Length j for by fetting 
 the real Proportions upon the Bottom of the Piclure, it may ferve 
 as a general Scale for regulating the apparent Size of any Perspec- 
 tive Reprefentation : Thus, ab, or ac, may be divided into three 
 Parts each, or three Feet, by dividing e h, and then drawing Lines 
 to i, in the above Manner.— Thefe Rules are particularly applied 
 to Practice in the i8th, 19th, and 20th Figures. 
 
 What is here faid m regard to the cutting off Lines in any 
 given Proportion, when thole Lines vanifh into the horizontal 
 Line, is equally applicable to Lines in all Kinds of inclined Planes, 
 "'Jf* VL bc a vanifhing Line of an inclined Plane, and V the 
 *»g. 4i. vanifhing Point of a Line ab in that Plane.— Continue Va to the 
 Bottom of the Pi&ure in B, and from B draw AB parallel to the 
 vamlhing Line VL, and continue it at pleafure ; then, upon this 
 Line AB fet off the feveral. Meafures, as if it were the Bottom of 
 the Picture, and confider V L as the horizontal Line, C as the 
 Center of the Pi&ure, and CE as its Diftance; and then the Ope- 
 ration will be the fame as in the laft Figure— This alfo is applied 
 to Practice in the 34th Figure. 
 
 IV. Having Jketched-in the propofed Size for an ObjeB upon the Pic- 
 ture, to prove whether it be di minified in Proportion to its Diftance. 
 
 %• 45- Let a b reprefent the Height of an Objc&. From an y Point, 
 
 as H, in the horizontal Line, draw Lines through the Extremities 
 a and b of the Figure, and from C, where the loweil Line cuts 
 the Bottom of the Picture, draw the Perpendicular CB ; then make 
 this Line a Scale, the fame as if it was the Bottom of the Picture v 
 and that will fhew whether the Figure be in Proportion for the 
 Place it porTefles in the Pi&ure, or not : Thus, fuppofe the Height is 
 20 Feet ; then g f is too much, and c d too little. Thus again, fup- 
 
 Jig. 46. P ofe . the °bjea to be a Houfe, I fay, its Height ab may be pro- 
 portioned to its Diftance by the above Rule. Thus, continue the 
 ^dge ab upwards at pleafurc, and from the vanifhing Point C, of 
 the End, draw C b through the Top of the Roof, cutting a b in b ; 
 then is ab the whole Height of the Houfe ; therefore from a and b 
 
 draw 
 
Gene ra t Rules, &c. 45 
 
 draw two Lines parallel to the horizontal Line, and continue them 
 at pleafure ; then any where from the Bottom of the Picture draw 
 a Perpendicular, as 1 2, and to any Point 5 in the horizontal Line 
 draw 1 5 cutting 3 a in 3, and from 3 draw the Line 3 4 parallel 
 to 1 2, cutting b 4 in 4 ; finally, from 5 draw a Line through the 
 Point 4, cutting 1 2 in 2 ; and then is 1 2 the real Height of the 
 Houfe, which being meafured by a Scale of Feet, will, ftiew whe- 
 ther the Houfe be in proportion, or not, to its Diftance.*- — This is 
 likewife applied to Practice in abed, Fig. 18. 
 
 V* To find the Length of any Reprefentation by Calculation only. 
 
 Let AB be a real Line whofe Reprefentation is fought, and vjg.47, 
 CE the Diftance of the Eye i which, in this Cafe, is parallel to 
 
 AB. Make 6 a in the fame Proportion to Ca, as AB (the real 
 
 Line) is to CE, the Diftance of the Eye ; Thus, let AB be two 
 Parts, and CE three Parts (or, if you pleafe, fo many Feet;) then 
 divide AC into five equal Parts, and the Reprefentation Ba of 
 BA, will be two of thole Parts;- that is, as 2 is to 3 1 For draw 
 AE, which will determine the Reprefentation of A B , as in the jjth 
 Figure. In like Manner, ec is to cC, as 2 is to 3 ; or* if you 
 pleafe, ec is to the whole Line eC, as 2 is to 5. And fo alfo for 
 any oblique Line, as DF 1 For Dd is to dL, as DF is to LE. 
 Or this may be determined, without taking the whole Diftance, 
 by Analogy. Thus, half the Diftance of CE, as C 1, and half 
 the Length of AB, as A 2, will come to the fame thing ; for draw 
 2-1, and it will pafs through the Point a. — The next Figure is a 
 farther Explanation of the fame thing, but by a different Method. 
 
 VI. To find the Diftance of the Picture from having two vanishing 
 
 Points of a Square given. 
 Let V L be the horizontal Line, C the Center of the Picture, and Fig. 49 
 V, L, the vani&ing Points of the Square. — Divide VL into two 
 equal Parts, in A, and with the Diftance A V defcribe the Semi- 
 circle VEL; then from C draw the Perpendicular CE, cutting the 
 Semi-circle in E; and then is CE the Diftance of the Picture. For 
 draw VE, LE, and they will make a right Angle at E. 
 
 VII. To find the vanijhing Lines of any inclined Plane, and their proper 
 vanijlring Points, together with the Center and Diftance of thofe va- 
 nijhing Lines. 
 
 1. When the Plane is inclined to the Ground, kit has Jome oj its Sides 
 parallel to the Picture, like a c d e. Fig. 31. 
 In this Cafe, the Rules are laid down and fully explained by that 
 Figure. 2. When 
 
46 
 
 •General Rules, &c. 
 
 a. When the Plane is not only inclined to the Ground, but has alliti 
 Si-des oblique with the PicJure, like a b d e. Pig-. 34., 
 In this Figure alfo the Rules are fully explained. And thefe 
 ■are all the Rules which are necefiary in common Practice, as I 
 have obferved^ before ; neverthelefs, to affift the Curious, I have 
 added the foffowing Figure, 
 
 'g. 50. VIII. Having given one Side of any inclined Plane, at pleafure, toge- 
 ther with its vanijhing Psint, and the vanijhing Li?te of that Plane 
 thence to determine the whole Reprefentation. 
 
 LetVU be a vanifhing Line given, and fuppofe V the vanifh- 
 ing Point of one Edge of a Square, (as ae, Fig. 36) and let C be 
 
 the Center of the Picture, and C E its Diftance. From V, the 
 
 given vanifhing Point, draw a Line through the Center of the 
 Picture, as V D, and continue it at pleafure ; then from C draw 
 CI perpendicular to VD, and make CI equal to the Diftance CE, 
 and from V draw VI, then from I draw Ic perpendicular to VI* 
 •cutting V D in c; and parallel to IC draw LU, cutting VU in 
 U ; then is UL the vanifhing Line of a Plane perpendicular to 
 that Plane, whofe vanifhing Line is VU; that is, VU and LU, in 
 this Figure, are the fame as Wl L and ILL in the 36th Figure, 
 where the Plane abed is perpendicular to the Plane adfe! 
 Again, Suppofe the Angle which another Plane made with the 
 Plane whofe vanifhing Line is V U, was a different Angle (fup- 
 pofe 60 Degrees) and it was required to find its vanifhing Line. 
 
 Then, as before, draw a Line V D through the Center C of the 
 Picture, and find the vanifhing Line LU of a Plane perpendicular 
 to that Plane, whofe vanifhing Line is VU; after which, continue 
 cD at pleafure, and make cD equal to the Diftance Ic of the 
 vanifhing Line LU; then draw DU, with which make an Angle 
 at D equal to 60 Degrees, and draw DU; finally, draw VU; and 
 then will V?L be the vanifhing Line propofed. — Again, to find the 
 Center and Diftance of a vanifhing Line. — From the Center of the 
 Pidure, draw a Line perpendicular to any vanifhing Line, which 
 will give the Center of thai vanifhing Line : Thus, CH and Cc 
 are perpendicular to VU and L U, and therefore H and c are the 
 
 Centei s of thofe Lines. Again, for the Diftance of a vanifhing 
 
 Line. Upon the Perpendicular Cc, and at the Center C, draw 
 
 another Perpendicular C I, and make CI equal to the real Diftance 
 CE; then draw Ic 3 which will be the Piftance of the vanifhin* 
 
 Line 
 
General Rules, &c. 
 
 Line LU, which being transferred into Cc continued* ascD, 
 will give the proper Diftance to be work'd with. 
 
 IX. The following is a Method for finding the 'Reprefentation of the Fig. 5 
 Plan of any Building, &c. when the Difiance to he work'd with is 
 not greater than jrom the Center to one Corner of the Pifture, as CE„ 
 
 Draw C E j perpendicular to which, draw K through C ; which 
 eonfider as the Bottom of the Picture. Under K draw out the 
 real Figure in its proper Situation, as ABC; then from the tranf- 
 pofed Place of the Eye, draw Lines parallel to the feveral Sides of 
 the Figure, which will give H and 1 for the vanifhing Points of 
 thofe Sides, and which are to be tranfpofed into the horizontal 
 Line, as H and L; after which, draw the Perpendicular CM, and 
 from M fet off the feveral Diftances CF, CG, &c. upon the Bot> 
 torn of the Picture ; and then, by drawing Lines to the proper va- 
 nifhing Point of each Line, as in the Figure,, the whole Reprefen- 
 tation may be compleated, exactly in the fame Manner as if the 
 original Figure had been drawn out under the Picture. 
 
 In the next Place, I fhall lhew how to determine the Appear- 
 ance of thofe Sorts of Objects which moft frequently occur in 
 common Practice ; for this will explain more fully the Ufe of the 
 preceding Rules, and at the fame Time, will mew the Shortnels 
 and Expeditioufnefs of this Method of Perfpective. And as I have by 
 former Examples, fo I mall likewife, in the next Section, make ufe 
 of fuch Objects as are fimple in their Parts, and of the moft gene- 
 ral Ufe. To explain my felf more fully. A Pedeftal, for inftance, 
 is but one Part of an Order in Architecture, and the Idea we have 
 of it is, of its being the Bafe, or Support, of a Column; but by 
 enlarging the Idea of a Pedeftal to that of a large fquare Building, 
 enriched with Mouldings, &c. we may then eonfider it as fuch a 
 Building y and therefore, we may conceive, that the fame Rules by 
 which the Appearance of a Pedeftal is determined upon the Pic- 
 ture, will ferve for finding the Reprefentation of any Building 
 which is fimilar to it. In like manner, as to the Situation of Ob- 
 jects which are perpendicular to the Ground, (fuch as the Walls 
 of Buildings, and the like) they muft be either perpendicular to 
 the Picture, parallel to the Picture, or oblique with it ; as we have 
 (hewn before : And therefore, one Example in each Situation, 
 adapted in a general Manner, will be of much more Service than 
 4en thoufand. different Schemes by way of Examples ; for the one 
 
 fixes 
 
48 General Rules, &c, 
 
 fixes our Attention to a particular Set of ufeful and general Ideas 
 but the other diffracts the Mind with Confufion and Obfcurity. * 
 
 The fame Arguments will appear equally true, if we apply them 
 to the particular Parts of any Building, fuch as Columns, Mould- 
 ings, and the like. For, nrft, in regard to Columns; by this 
 Method, we have no Regard to Plans, Elevations, &c. and there- 
 fore, it matters not where we begin the Operation, whether at the 
 Top, the Bottom, or at the Middle of it ; fb that one Rule alfo in 
 this Cafe will appear to beuniverfal: And in refpect to Mouldings, 
 they mull be either plain or curvilinear, either above or below the 
 Eye ; and therefore, one Rule in either Cafe will be fufficient for 
 our Purpofe. The fame may be faid of every other Example in 
 this Section ; but what has been faid already, will, I hope, be 
 fufficient to explain the Senfe of the following Figures, and to 
 filence any Objections which may be made againft my not having 
 fwell'd my Work with more ornamental Schemes, or, as they are 
 generally called, Curious Examples, 
 
 The firft Example which I fhall produce, is the Tuscan 
 Pkbestal, in order to fliew how to find the Representation of 
 ftrait Mouldings, when they are either parallel, perpendicular, or 
 oblique with the Picture, or when they are either above or below 
 the Eye. In the 5 2d Figure one Side is parallel to the Picture, 
 the other perpendicular to it ; and in the 53d Figure, both Sidet 
 are oblique with the Picture. 
 
 SECT. II. 
 
 The foregoing Rules of PerfpeSfhe more particularly applied to 
 
 common Practice. 
 
 I. fc 5 Tuscan Pedestal into Perfpettive. 
 
 1. When one of its Sides is parallel to the Picture, then the other Side 
 will he perpendicular to it*, fo that one Rule will do in both Cajes. 
 
 Tig. 52. T ET AB reprefent the Bottom of the Plinth in Front.— -Now, 
 1 j from this one Line AB, the Appearance of the whole Pedeftal 
 may be found: For continue AB at pleafure, and draw a i ine 
 I K perpendicular thereto, and make I K equal to the Height of 
 the Pedeftal ; then, upon IK, draw the Capital and Bafe iif their 
 proper Proportions ; This being done, continue Lines from each* 
 
 Moulding, 
 
General Rules applied to common FraBice. 
 
 Moulding, which will form feveral Rectangles, and thereby -divide 
 the Planes i 2 34, 56 78, into a Sort of Net- Work ; then, by- 
 putting thefe Planes into Perfpeclive (as iia the Figure) we mail 
 have fufficient Guides for drawing all the Mouldings. But to be 
 
 more particular in the Operation. Make AD equal to AB, and 
 
 cut off AE equal to AD ; from whence the Plinth may be com- 
 pleated. — In like Manner, for the Die. — Draw the Diagonals upon 
 the Top of the Plinth, and any where upon the Edge hg, fet off" 
 the Projection of the Plinth, as 1 9 ; then draw one Line from 1 
 to C, and another Line from 9 to L, which will give the Projec- 
 tion 1 2; then, if you draw a Line through 2 parallel to hg, it 
 will cut the Diagonals ah, bg, and give the Corners of the Front 
 Side; and if you draw a Line from b to C, it will cut the Diagonal 
 dc, and give the further Corners ; therefore, by drawing Perpen- 
 diculars from a, b, c, we mail have all the Edges of the Die which 
 can appear in this Situation. — As to the Height of the Die, or the 
 Height of the feveral Mouldings, they may be found by drawing 
 a Perpendicular from the neareft Corner of the Reprefentation, as 
 B H, and transferring thereon the feveral Heights from I K, as in 
 the Figure ; then by drawing Lines from the feveral Points upon 
 B H (which meafures the Height of each Moulding) to the vanifh- 
 ing Point of the Diagonal ah, we ftiall have thofe Heights tranf- 
 ferred unto the Edge am of the Die; thus ai and km are the 
 Heights of the Bafe and Cornice ; fo that by finding the Appear- 
 ance of the Planes 1234, 5678, and drawing the Mouldings 
 therein, and by drawing the triangular Planes at the Corners, we 
 may finifh the whole Reprefentation with the utmoft Eafe'and 
 Expedition. 
 
 But before we begin to draw out any Object in Pcripeclive, we 
 muft firft confider, whether the Whole, or only a Part of it, is to 
 appear ; and muft fketch out the Size we intend it fliall be of, or, 
 at leaft, give one Line for its greateft Dimenfion. Thus, if the 
 whole Pedeftal is to appear, then give AB, which is neareft the 
 Eye, and call it the utmoft Length of the Plinth : But if only the 
 Top is to appear, then give H O, and call it the utmoft Extent of 
 the Cornice ; then, by cutting off O r equal to O M, that is, equal 
 to O H, we ftiall have the Depth O r of the Cornice, &e. from 
 whence, and with the Affiftance of the Plane 5678, (which is 
 found exactly in the fame Manner as the Plane 1 2 3 4) we may 
 compleat the whole that is wanted. 
 
 Book II, 
 
 G 
 
 Now 
 
£0 General Rules applied to common Practice. 
 
 Now, in order to do all this, it is neceflary that the Artift fhould 
 (as was obferved before) be able to apply: the preceding Rules with, 
 the greater!: Readinefs particularly That which teaches how to, 
 cut off one Line equal to another Line given. 
 
 2. When both Sides are oblique with the Picture. 
 
 Fig- 53. In this Figure, let A be the neareft Corner of the Plinth, AC, 
 AB, the Length of two Sides A G, AF, and Ak the Height of 
 the whole Pedeftal properly divided ; (that is, like BH in the laft 
 Figure.} — Cut off AF, AG, equal to AB, AC, and draw the 
 Plinth and the Diagonals upon the upper Square ; then draw i b 
 parallel to A B, and make it equal to the Projection of the Plinth 
 then cut off i c equal to i b, and from c draw Lines to H, and 
 from i draw a Line to C, (the vanifhing Point of the Diagonal 
 I 2) which cutting each other in 2, will give the Edge 2 7; and 
 by drawing Lines from 2 to their proper vanifhing Points L and 
 H, they will cut the Diagonal k g, and thereby give the other 
 
 Corners of the Die, as in the Figure. For the Height of the 
 
 Mouldings ; draw Lines from 4 and 5 to C, which will cut 2 7 in 
 the Points required; by which means the triangular Planes aik> 
 f b g, &c. may be found, and from thence the Mouldings may be 
 compleated. 
 
 Here alfo, if we want only the upper Part, we may begin at the 
 Point 8, making 8E, 8D, each equal to the Length of the Cap, 
 &c. then, by finding the Plane 5678, as in the former Cafe, we 
 jhall have funicient Guides for compleating the Figure. 
 
 Here let us obferve, that when the Pedeftal has one Side parallel 
 to the Picture, then the Plane 1234, (Fig. 52) which is a Guide 
 for the Moulding, may be begun any where upon the Edge a b ; 
 But when it is oblique with the Picture, then we mufl begin from 
 the neareft Corner, as a ; and by attending to the Figures, we may 
 conceive, that in the firft Cafe, the Mouldings in the Directing 
 Plane, are like the Ends of Mouldings cut off fquare ; but in the 
 latter Cafe, they are like Mouldings cut off at what is called the 
 Mitre Joint. And from hence we may alfo obferve, that all the 
 Difficulty in putting Mouldings intoPerfpective, lies in finding the 
 little Planes 1234, &c. and therefore the Reader fhould confider 
 them attentively before he proceeds any farther. 
 
 II. Of 
 
General Rules applied to common PraBice* 
 
 II. Of Circular Mouldings, &c* 
 
 The Method for determining the Appearance of Circular 
 Mouldings, is much the fame as that for finding the Reprefentation 
 of ftraight Mouldings, viz. by imagining a Plane to pafs through 
 the Mouldings in a perpendicular Manner, and then putting that 
 Plane into Perfpeclive : As in the two laft Figures. 
 
 i . To put a Tuscan Base into Perjpetfive. 
 
 Give one Line for the Width of the Plinth, and draw out the Fig. 54, 
 proper Projection of the Mouldings, and the Plane ABCD ; then cut 
 off the oblique Side equal to the Front, and compleat the Plinth ; 
 after which, draw the Diameters and the Diagonals upon the Top 
 of the Plinth, as in the Figure ; and then draw the Reprefenta- 
 tion of a Circle for the Seat of the lower Torus. Again, for the 
 Bottom of the Shaft of the Column 5 from the Center H of the 
 Column, draw the Perpendicular HL, and from a, where the 
 Diameter ae cuts the Plinth, draw ad parallel to H; then 
 make ad equal to the Height of the Mouldings AD, and from 
 d draw a Line to C, which will cut HL in I ; then will I be the 
 Center of the Square for the Bottom of the Column : Therefore, 
 upon the upper Edge of the Plinth, and from the Point a, make 
 a 1 equal to the whole Projection of the Mouldings j then cut off 
 ab equal to a 1, and draw be parallel to ad, which will give 
 the little Plane for the Mouldings j within which draw the Mould- 
 ings; and then we mall perceive that c is the Middle of the 
 neareft Edge of the upper Square : Therefore, through c draw a 
 Line EF parallel to the Edge B 1 of the Plinth ; then from H draw 
 a Line through the Center 1, cutting EF in E, and then is cE 
 half the Width of the Square : Therefore make cF equal to cE, and 
 from thence compleat the Square, and within it draw the Repre- 
 sentation of the Circle, as in the Figure : Finally, from the Extre- 
 mity of each Circle draw the two oblique Lines 2, 3, which toge- 
 ther with the little Plane for the Mouldings, &c, will be fufficient 
 Guides for compleating the whole Bafe, as was propofed ; which is 
 evident by infpecting the Figures 57 and 59. 
 
 As for making Columns, &c . of any given Proportion, or at 
 any Diftance ; the Rule for cutting of a Line in any given Propor- 
 tion, in the 43 d Figure, is fufficient for that Purpofe* 
 
 2. TV 
 
General' Rules applied to common Practice. 
 
 2. To put # Tuscan Capital into Perfpe&ive. 
 Fig. $$. Let K be the Center of the Square for the Bottom of the Capi- 
 tal. — Through K draw a Line AB parallel to the horizontal Line 
 at pleafure, and from f and e of the Bafe, draw Lines parallel to- 
 the Axis HI of the Column, cutting the above Line in E and F^ 
 then is EF the Diameter of the Column : Therefore diminifh it ia 
 its proper Propertion, as 3 4 ; then is 3 4 the Diameter of the Neck 
 of the Column : Therefore with the Line 3 4 draw the Appearance 
 of a Square, and in that Square draw the Reprefentation of a 
 Circle as before directed ; fo mall we have a Guide for the under 
 Part of the Capital. Again, make KI equal to KF, (that is, equal 
 to Half the Diameter of the Column) and through I draw.GH 
 parallel to AB, then make IG, IH, each equal to KA, or KB, 
 (that is, equal to Half the Diameter of the Top of the Abacus) 
 and then with the Line GH draw the Appearance of another Square, 
 which will reprefent the Top of the Abacus : Finally,, from C draw 
 a Line through I, cutting the Edge 5 6 in a, and from a fet off a 1 
 for the Projection of the Capital, and draw the little Plane abed 
 for the Mouldings, as before : From whence the remaining Part of 
 the Capital may be compleated, as in the 56th and 59th Figures*.. 
 
 I mall juft mention a Method for finding the Point where the. 
 Diagonal of a Square will be cut by a Circle inferibed in that 
 Square ; which may be of ufe in this and fome other Cafes. It is 
 Fig. 54. this: Divide the Edge BG into feven equal Parts; then fet one 
 Part from each Corner, as B4, G5, and draw Lines to C, which 
 will cut the Diagonals in the Points required. I do not fay, this 
 is mathematically exact, but, I prefume, it is near enough for the 
 intended Purpofe. 
 
 Thefe are the moft limple, as well as the moft general Methods 
 I can think of for mix'd Mouldings ; and I believe any Perfon 
 who is but tolerably fkilled in Drawing, will find them fufficient 
 for his Purpofe, upon all Occafions. 
 
 3. To find the Reprefentation of a Corinthian Capital. 1 
 Fig. 62. Let AB be the Diameter of the under Part of the Capital, and 
 let ca be the Center, or Axis, of the Capital, properly divided for 
 the Height of its Leaves, Volutes, and Abacus. — From the Line 
 
 *' In this Cafe tne Projection of the Capital, (according to Mr. Gibbs, from whofe Book I 
 have taken my Proportions) is one fixth Part cf its Length, and the Projections of any other 
 Mouldings may be determined in the fame ctfy Manner by a Scale and Compafles.S 
 
 AB, 
 
Qeneral Rules applied to common Practice* 53. 
 
 AB„ which is given, find the Appearance of a Square, in which,, 
 draw the Diameters and Diagonals, and then the Reprefentation of 
 the Circle ; which will determine the Places for the Stalks of the 
 great Leaves, as reprefented by the Dots : Again, through o draw 
 bd parallel to AB, and make ob, od, each equal to half the un- 
 der Part of the Abacus with which Line b d, draw the Appear- 
 ance of another Square, and divide it like the Plan 1234, Fig. Z. 
 of the under Part of the Abacus, and then draw the Reprefenta- 
 tion of it, as in the Figure : Again, through a, draw another 
 Line parallel to bd, and make it equal to the upper Part of the 
 Abacus 5 then, by finding the Reprefentation of the Square a b c d, 
 Fig. Z, we may draw the Appearance of the upper Part of the 
 Abacus, and from thence compleat the Abacus, as in the 60th Fi- 
 gure: Finally, find the Middle of each Face of the Abacus, as 
 n, e, and draw Lines n 1, &c. to the correfponding Points at the 
 Bottom of the Capital; then find the Height of the Leaves by 
 drawing Lines from C through the Dots in c a, 'till they cut 1 n 
 in 2 and 3 s after which draw the Bafket; then, by a nice Eye, 
 compleat the Capital beginning as is exemplified in the 60th 
 Figure. — The Lines drawn from the Corners of the upper and 
 under Square, will ferve as Guides to prevent our giving the 
 Leaves too much Projection. In Fig. 61, the Capital is compleated, 
 and the Figures X and Z are added to explain the Thing more 
 fully ; one of which is the Plan, and the other Half the Profile of 
 a Capital. 
 
 Here it is necefTary to take Notice, that upon Account of bring- 
 ing the Diftance of the Pifture within the Compafs of each Plate, 
 and to make the Figures as large as poflible, feme of them have 
 not that agreeable Shape which could be wifhedi but if the Reader 
 will choofe a greater Diftance, and follow thefe Rules, he will find 
 every Objection of this kind, that may arife, immediately vaniih, 
 
 III. Of Columns parallel and oblique 'with the Eye. 
 
 I. Let it be required to find the Appearance of two Columns in Front , Fig. 63. 
 and let a b. be the Diameter of the under Part of the Plinth », and 
 € the Center of the Column^ 
 
 Continue ab at pleafure, and any where upon it, as at A, 
 draw a Line AB perpendicular to Ab; upon which Line fet the 
 feveral Heights for the Bafe, Capital, Entablature, &c. then from 
 c>, the Center of the Column, draw a Line cd parallel to AB, 
 
 and 
 
'General Hu7es applied to common TraSfiee. 
 
 and from tlie feveral Divifions upon AB, draw Lines parallel to 
 the horizontal Line, which will cut cd, and give the Heights of 
 the Bafe, Capital, and Entablature. Now, having got the ieveral 
 Heights, we are to confider cd as the Axis of the Column, (that is, 
 a Line which paries through the Middle of it) and then at every 
 Dot make a Square, equal to the Diameter of that Part of the 
 Column, &c. which that Dot ftands for : Thus ab is the under 
 Part of the Plinth, and by means of H, and the Diagonal i 2, 
 we may compleat the firft Square. So alio, r is the Square for the 
 Bottom of the Shaft, p for where the Column begins to di mi- 
 nim, t for the Top of the Shaft, and v for the Top of the Abacus ; 
 and therefore, having got thefe feveral Squares, we mall have 
 fufficient Guides for compleating a Column of any Order. Again, 
 for the other parallel Column ex. — From c to e, and upon the 
 Line AB continued, fet off the Diftance which the Center of one 
 Column is from the other, and draw ex* upon which, fet off 
 the feveral Heights, as before, from the Line AB, and then find 
 the feveral Squares, as before directed, 
 
 2. For oblique Columns. 
 
 Upon AB continued, fet off the Diftance which the Centers of 
 thofe Columns are from the Center of the Corner Column, and 
 draw a Line from e to C; then cut off e m, en, equal to ef, eg, 
 and from the oblique Sides of the Square e, draw Lines to C; 
 from whence the other Squares m and n may be compleated, as 
 before. — For their Heights,-- -Draw mo and nq, parallel to ex, 
 and from x draw xC, which will give their feveral Heights. 
 
 Now, if we would put an Entablature over the Columns ; then, 
 the Height of the Architrave, Freeze, and Cornice, may be drawn 
 from their refpective Divifions, upon the Line AB ; and the Ap- 
 pearance of the Mouldings peculiar to each Part, may be found by 
 the Rules already laid down for that Purpofe. Or, they may be 
 found thus,— Let agbf, in the Figure %, re£refent the upper Part 
 <of the Capital x belonging to the comer Column; and we want to 
 find the Corner of its Entablature; that is, the Corner of the 
 lower Facia, or Freeze:— Set off the Projection of the Capital from 
 n to e, and draw a Line to C, which will out the Diagonal ab, and 
 give the Corner of the Facia ; fo that by drawing the Line 1 from 
 the Dot in the Diagonal, we fhall have the Corner propofed. In 
 like Manner the Projection of the Cornice may be found. Con- 
 tinue 7 © ( which meafures its Height ) at pleafure, and make 
 
 oh 
 
General Rules applied to common. PraSlict. $ : $ 
 
 © h equal to the real Predion of the Cornice ; then through K 
 draw a Line from C, and from H draw a Line through o ; and 
 where it cuts Ch continued in i, will be the Corner of the 
 Cornice &c» 
 
 From hence then it is evident, that any Number of Columns 
 may be drawn with the greateft Eafe and Expedition, let their Si- 
 tuations be what they will; and from hence aho we may obferve, 
 that any Part of a Column may be immediately produced upon 
 the Picture in the very Place in which it is intended, without 
 drawing, out the Whole, or any other Part than that which is 
 
 really to appear.. 
 
 Here it may not be improper to take Notice of what we have 
 obferved in Chap. VI. Book I. concerning the Reprefentations of 
 parallel Columns, which, we obferved, would grow bigger and 
 bigger the further they are removed from the Center of the Picture; 
 and to point out a Method to be pra&ifed by thofe who are fatis- 
 fiedwith theReafons upon that Head; and by that Means, to 
 
 give all the Columns an agreeable Shape. It is this : Firft find F lg . 6 3? 
 
 the Reprefentation of that Column which is neareft the Center of 
 the Picture, as ex; then fet off the Diftance for the Centers of the 
 other Columns, and draw the Squares for the Plinth, Capital, &c. 
 and then, upon each Side of the Axis, fet off at the Bottom of 
 each Column Half the Diameter of the corner Column, and at the 
 Top of the Column fet off Half the Width of the Neck of the 
 corner Column : Finally, draw Lines from thence, fo as to diminifh 
 the Column in a proper Manner ; and thereby we may make all the 
 Columns that are parallel to the Eye of the fame Bignefs. As to 
 the great Projc&ion of their feveral Bafes, they will not look at all 
 prepofterous, if they are done by any one who has but a tolerable 
 Eye for Drawing, and is careful in taking a proper Diftance for 
 
 the Eye, ^ o/Stairs parallel and oblique. 
 
 I. For PARALLEL ST AIRS. 
 
 Let AB be the Length of one Step, B i its Height, B4 its Fig. 6 4? 
 Depth, C the Center of the Picture, and CH the Diftance of the 
 Picture.— Find the firft Step by the Height Bi, and its Depth B 4; 
 then make 4O equal to 4 B, and iD equal to 1 B ; by which 
 means the upper Step may be found. Now, by continuing B O, 
 and BD, and by dividing them properly, any Number of Stairs may 
 be determined. Or, if only thofe above the Eye are to be feen, 
 
 then 
 
5 6 General Rules applied to common Pratthe. 
 
 then begin with the Line a be, and proceed in the aforefard 
 Manner. 
 
 2. For o b l i qjt e Stairs. 
 Fig. 6 S . Let AB be the Length, BO the Depth of two Steps, BD the 
 Height of two Steps, and C the Center of the Pi&ure. — By means 
 of the Points H, L, cut off Ba, B F, &c. equal to B A, BO, & f , 
 then by making BD equal to the Height of the Steps, they'may 
 be compleated, as in the Figure : And after the fame Manner the 
 Stair above the Eye is to be found. 
 
 V. Of an Arch and Pediment. 
 
 i. For the Arch. 
 Fig-. 66. Let A be the Corner of the Arch. — Set the feveral Divifions for 
 the Width and Center of the Arch upon A B, and for its Height 
 upon the Edge A i, and draw Lines to C ; by which means the 
 parallelogram cdef may be found, which will be a Guide for 
 drawing the Arch. 
 
 2. For the Pedimen t. 
 Let abD be the Fitch of the Pediment, and CD the Diftance 
 of the Picture.— Find the vanifhing Points V and L, and draw 
 Lines from thence through the Top and Bottom of the Cornice ; 
 which will interfecu each other, and give the Reprefentation of the 
 Pediment. 
 
 VI. Of H o u s es parallel and oblique. 
 
 I. For PARALLEL HOUSES. 
 
 Fi £ . 6 7 . Let AB, and BF, be the Length and Depth of the Houfe, BD 
 
 the Height of the Walls, and C the Center of the Picture. 
 
 Draw the vanifhing Line VL, and find the vanifhing Points V, L, 
 of the Roof j then draw V% parallel to the horizontal Line, which 
 will be the vanifhing Line of that Part of the Roof which fronts 
 the Eye. And the Roof which covers the Pediment b d e is found 
 by drawing the perpendicular ab, and, from the Extremities there- 
 of, Lines to C and V, which will give a Triangle a be, whofe 
 Side cb is the Top of the Roof-- Or the Height of the Roof 34D, 
 may be found by continuing BE, and making DE equal to the 
 propofed Height, 
 
 2. For House s that are o b l i qjj e. 
 %.G8. In "this Figure AB is the Depth, and AC the Length of the 
 Houfe, VI is the vanifhing Line of the End, and V, I, are the va- 
 nifhing 
 
G-eneral Rules 'applied to commo)i TfaBieh 
 
 51 
 
 nlming Points of the Roof j the vanifhing Line of the Pediment, 
 &c. is vl, and its vaniihing Points v and lj the Length of the 
 Roof over the Pediment is found by means of the Triangle a b c, 
 as before. 
 
 VII. To pat the Injide of a Room into Perfpefiive. 
 
 Let ADFE be a Picture upon which the infide of a Room is m%. 6 9 
 to be drawn. — Here C is the Center of the Picture, and AB the 
 Length of the Room. — Set off the feveral Divifions from A to B, 
 and cut off the feveral Spaces upon A 1 equal to thofe upon AD ; 
 then fet off the Heights upon AE, and draw Lines to C j which 
 will be fufficient for our Purpofe. 
 
 If any Reprefentation of this Kind is to be drawn upon a Wall, 
 fo as to make a Deception like the Continuance of a Room, 
 Care muft be taken to choofe a proper Diftance, and to make 
 the Height of the horizontal Line exactly equal to the Height of 
 the Eye, viz. about five Feet fix Inches. If the Wall be too large 
 for any Diftance that can be taken within the Room, then fome 
 other Subject muft be painted upon it, fuch as will admit of its 
 being divided into Frames, Compartments, or the like: And the 
 'fame may be faid in regard to Cielings; for, in either Cafe, if the 
 Diftance be an improper one, all the Reprefentations will have 
 a bad Effect. 
 
 Thefe Examples are the moft general I can think of ; and I 
 flatter myfelf, that they will be found fufficient to anfwer every 
 Dehgn which can be propofed in common Practice: But if there 
 fhould appear any Difficulty in applying the aforefaid Rules upon 
 fome extraordinary Occafions (as when the Defign confifts of many 
 Parts, or when it happens that any of the vanifhing Points are 
 out of the Picture) then the beft Way will be to draw out the 
 whole Defign, by Way of Model, in a fmall Compafs, upon 
 Paper, and from thence, by a proper Scale, or by Net-Work, to 
 transfer the whole unto the real Picture for then the moft diftant 
 vanifhing Points may be very eafily come at. Or in many Cafes> 
 the real Picture may either be laid flat upon a Floor; or elfe have 
 Rules made to fix upon the back Part of the ftraining Frame by 
 Screws, or fome fuch Contrivance, whereby, and with the Aflif- 
 tance of fmall Twine fixed upon Pins at each vanifhing Point, we 
 may produce almoft every Reprefentation which can be defired. 
 
 Book II. H The 
 
General Rules applied to common Tradlice* 
 
 The three following Figures I have not only given as Examples 
 in Perfpefrive, but have attempted to difpofe each Object in 
 fuch a Manner as to produce agreeable Shapes, EfFecT:, &c. — The 
 firft reprefents a Variety of Figures tending to various vanifhing 
 Points in the horizontal Line, below the horizontal Line and 
 above it : amongft which, are the five regular Solids y and the 
 whole together, contains all the Rules and Principles of Perfpec- 
 tive. The next Figure is a View of Framlingbam Caftle in Suffolk \ 
 a Place of great Antiquity, and formerly the Seat of the Howards, 
 Mowbrays, &c. which is produced in this Place as an Example of 
 a Building that tends to feveral vanifhing Points upon the hori- 
 zontal Line only : And the laft Figure is an Example of a Land* 
 ikip, by a very great Genius in that Way. 
 
 CHAP; 
 
B2. 
 
 XV. 
 
CHAP. IV. 
 
 Of the Parallel Picture, fuch as Cielings, 
 or the like $ or what is ufually c ailed > Hori- 
 zontal Perfpe£tive> 
 
 THIS Kind of Perfpective is extremely eafy, becaufe little 
 more is required to be known than what has been already 
 taught in Seel. II. Chap. II. of this Book ; viz. How to 
 find the Appearance of Objects which are fuppofed to lie upon the 
 Ground. For mod Objects which are drawn upon Cielings, are 
 fuppofed to be perpendicular to them ; and therefore, the Rules for 
 determining the Reprefentations of Objects in this Manner, are 
 exactly the fame as thofe for determining the Reprefentations of 
 Objects which lie flat upon the Ground, in the perpendicular Pic- 
 ture j and confequently, the Rules which ferve in one Cafe, will 
 ferve in the other alfo. 
 
 The firft Things to be coniidered in thefe and the like Repre- 
 fentations, are, the Diftance of the Eye, and the Center of the 
 Picture. — As to the Diftance of the Eye, that is unalterable, be- 
 caufe the Picture is fixed therefore, if the Cieling be fo large, or 
 fo low, as to fubtend too great an Angle at the Eye ; that is, if 
 ■ the longeft Dimenfions of the Cieling be much greater than the 
 Diftance at which the Spectator is to look at it; then, in this Cafe, 
 the Cieling fhould be divided into Compartments, which may ferve 
 as Frames for the intended perfpective Reprefentations : And we 
 muftbe always careful, when we take the Height of any Cieling 
 from the Floor, to deduct the Height of the Spectator's Eye there- 
 from, which is ufually about 5 Feet 6 Inches. And in regard to 
 the Center of the Picture, the general (and, I believe, the beft) 
 Method has been, to fix it in the Middle of the Picture, unlefs any 
 Thing prevents the Eye from feeing it conveniently from that 
 Place j becaufe then there will be a Uniformity of the Parts, which 
 will agree with each other, and be more likely to deceive the Eye. 
 
 And it is to be obferved, that in thefe Kinds of fixed, or im- 
 moveable Pictures, the Spectator ftiould always fix his Eye directly 
 againft the Center of the Picture ; for otherwife the Reprefenta- 
 tions will not have their defired Effect, 
 
 H 2 £Tow, 
 
6o Of Horizontal Perspective. 
 
 Now, in order to draw any Piece of Perfpective upon a CieliW 
 the belt Way feems to be this, viz. Take the Dimenfion of the 
 Cielmg, and make an exact Calculation of the Diftance and Height 
 of the Eye j then draw out the intended Defign upon a large Piece 
 of Paper, by Way of Model, and from thence transfer it unto 
 Canvas, with the Addition of Colouring, Effect, &c. and finally, 
 from thence draw it upon the Cieling, by Net- Work. 
 
 I. To draw upon a deling a Deception, which, viewed from a proper 
 Point, Jhall appear like the Sides of the Room continued upwards. 
 
 FlS 74' Let ABFD be a Cieling drawn upon Paper to a certain Scale • 
 ' 4 * and let E be the Eye, EC its Diflance, and C the. Center of the 
 Pi&ure.^ Now, let it be required to make the Cieling appear as 
 if the Sides of the Room were continued upwards equal to the 
 Length A B.— Through the Center of the Picture draw E 1, E 2, 
 parallel to A B, which maybe confidered as the horizontal Line \ 
 then draw Lines from the Corners A, B, F, D, to the Center C y 
 and make CEi equal to CE ; by which means A a may be cut 
 off equal to the given Length AB, and confequently, from thence 
 all the Reprefentations may be compleated, as in the Figures: 
 Thus, ab being drawn parallel to AB, gives the Side ABab, and 
 
 ad being drawn parallel to AD, complcats the Side AD ad, &c. 
 
 In the 73d Figure, the Center lies out of the Middle of the Pic- 
 ture ; but in the 74th Figure, the Eye is directly in the Middle. 
 
 Here we may obferve, that if Lines are drawn thro' the Center 
 of the Picture parallel to the Sides of a Room, then thofe Lines 
 may be confidered as fo many horizontal Lines, and may be made 
 ufe of accordingly. Thus, E1CE2 will ferve as a horizontal, or 
 vamming Line, for all Obje&s which can lie upon the Planes 
 ABab, DFdfi and ECE3 will ferve as a horizontal Line for 
 
 any Objects which can lie upon the Planes ADad, BFbf. By 
 
 turning the Figures we may conceive this very clearly 5 but in the 
 next Figure it is more fully explained. 
 
 In this Figure I mall mew how to find the Reprefentation of 
 fuch Objects only as may occur in common Practice, fuch as Co- 
 lumns, Pilafters, Arches, and Windows. And fir!* of Columns. 
 
 2. To find the Appearance of Two Cylinders upon a Cieling, 
 75- Let the Circles H, I, reprefent the Ends of two Cylinders, and 
 let E2CE4 be the vanifliing Line of the Plane ABab, CE3 the 
 Diftance, and C the Center of the Picture.— About each Circle 
 
 H, I, 
 
Of Horizontal Perspective. 
 
 H, I, defcribe a Square, and make CE4 equal to C E 3 ; then draw 
 a Line from each Corner of the Square to Cj and then, by means 
 of the Point E4, a Parallelopiped may be made of any Length, 
 which will be a Guide for compleating the Cylinder 5 as is fhewn 
 in Figure 29 of this Book. Now, by the fame Method, the Ap- 
 pearance of Columns may be determined, with this Difference only, 
 that three Squares muft be found as Guides inflead of two 5 that 
 is, one for the Bottom of the Column, another where it begins ta 
 diminifh, and the Third at the Neck of the Column. 
 
 3 . To fnd the Reprefentation of Two Pilasters.. 
 
 Let F and G be the Ends of the Pilafters. — From each Corner 
 draw Lines to C, and, by means of the Point E 2, cut off each 
 Pilafter to its proper Length. 
 
 4, To determine the Appearance of a Square Object which lies 
 
 oblique with the Picture. 
 
 Let 5 be one Corner, 5 7 the given Length, and E 2, E4, 
 
 the vanifhing Points of the Sides. From the Corner 5 draw 
 
 Lines to the above vanifhing Points, and cut off 5 6 equal to 5 7 j 
 from whence the Figure may be compleated. 
 : On the oppofite Side DEde, I have finifhed thefe Reprefenta- 
 tions with Shadows, &c. 
 
 5. To put an Arch into PerfpeBive. 
 
 Let KM be the Width of the Arch, Mh the Height to where 
 the Arch fprings, and hi the Height of the circular Part 3 and 
 let E 1CE3 be the vanifhing Line, C E 2 the Diftanee of the Pic- 
 ture, and C its Center.— From K and M draw Lines to C, and 
 cut off Mn, no, equal to Mk, ki, then draw the Parallelogram 
 nopq, which will be a Guide for drawing the Arch. Again, for 
 the Depth of the Opening,— From K draw a Perpendicular to 
 D A, and make it equal to the propofed Depth 5 and from its 
 Extremity z draw a Line to C; then from q draw a Line parallel 
 to K z, which will cut Cz, and thereby give the proper Depths 
 do the fame on the other Side; then draw the bottom Curve for 
 the other Side of the Arch, parallel to the upper Curve, as in the 
 figure; and fo will the Reprefentation be compleated. 
 
 6. To 
 
62 Of Horizontal Perspectives 
 
 6. To find the Appearance of # Window, the Top of which we 
 will fuppcfe to be even with the Top of the Arch, and to be tw9 
 Diameters in Height, 
 
 Set off the real Width L f, and its Height L g, and from the 
 Points L, f, g, draw Lines to Cj then continue the Line op to r, 
 which will give the Top rs, and from E i draw a Line through 
 the Corner r, cutting g t in w ; then from w, draw w v parallel to 
 rt, which compleats the Window rs ox. The Depth is found in 
 the fame Manner as the Depth of the Arch, viz. by the Perpen- 
 dicular L. 
 
 On the oppofite Side to this alfo, are the above Figures wholly 
 compleated. 
 
 7. To put a Cornice into Perfpefiive. 
 
 Draw out the Projection, &c. of the Cornice, about which de- 
 Fig. 76. f cr - be the p] ane a BCD; then put that Plane into Perfpective, as 
 FGHI; from whence all the Mouldings may be determined, as in 
 the Figure, 
 
 S. To put a Base and Capital into Perjpetfive. 
 
 Fig. 77 . For the Bafe, Altho' nothing more than the Plinth can in 
 
 general be feen by the Eye, yet I have here given a Method for 
 determining the whole Projection. — Let AB be the Diameter of 
 the Plinth, and B F the Height of the Bafe ; make a Square with 
 AB, and from B draw a Line to C, and cut off BD equal to BF, 
 and from D draw D C parallel to A B, and with D C make ano- 
 ther Square ; then divide A B into eight equal Parts, and one of 
 thofe Parts (according to Gibbs) is the Projection of the Mouldings : 
 Therefore, make B 1 equal to one of thofe Parts, and from 1 draw 
 a Line to C, cutting the Edge of the fartheft Square in 3 ; then 
 from 3 draw a Line parallel to C D, and fet off the Diflance 3 D 
 upon the other three Sides of the parallel Square; then, by drawing 
 Lines through thofe Points, we fhall have a Square equal to the 
 Diameter of the Column ; and from thefe two Squares the whole 
 
 Appearance is to be compleated. The Figure G reprefents it as 
 
 nnifhed, 
 
 9. To put a Capital into Perjpeffive. 
 Let AB be the Diameter of the Bottom of the Capital, 1 2 the 
 Diameter of the Abacus, 1 3 the Height of the Capital, and B 2 
 the Projection of the Capital. — Set off B b equal to B 2, and make 
 
 the 
 
Of Horizontal Perspective. 6;J 
 
 the Square abed; then draw iC, 2C, and cut off iD equal to 1 35 
 then draw DF parallel to AB, and with DF make a Square ; 
 finally, from the Corner of one Square draw Lines to the corref- 
 ponding Corners of the other, as in K j then mail we have fufficient 
 Guides for compleating the Capital. 
 
 10. To put the Human Figure into PerfpeBive, 
 
 Having made a Defign of the Figure, defcribe the Frame about Fig.. 73 
 it, as AB C D, and then reticulate it in a proper Manner; after 
 which, put that Frame and the Reticulation into Perfpective, as 
 abed, which will give all the Forefhortnings, as in the Figure. 
 
 I am very fenfible, that 'tis impoffible to give Rules for putting, 
 the Human Figure correctly into Perfpective, and that the greateft 
 Part muft be left to the Judgment of the Artift 5 yet the above 
 Hint may be of fome Service in defigning Figures for the above 
 Purpofes. So likewife, as to the Size of Figures which are to be 
 feen at a confiderable Diftance; I know of no Rules by which they 
 can be correctly determined ; and therefore, in fuch Cafes, the beft 
 Way is, to fketch out feveral Figures of different Sizes upon the 
 intended Picture; then, by furveying them from the Point of View 3 
 the Eye will immediately inform the Artift which is of a proper 
 Proportion. * 
 
 In Book I. Chap. IV. Sect. 3, we have given fome general 
 Rules from Mr. Hamilton, for drawing any Perfpective Reprefen- 
 tations upon vaulted Roofs, Domes, or other uneven Surfaces y 
 and therefore, if the curious Reader would inform himfelf of that 
 Kind of Perfpective, he mult refer to thofe Figures, where this 
 Article is conficlered at large ; which will fufficiently explain the 
 78th and 79th Figures, fince they are thofe Rules applied to Prac- 
 tice. But as I have faid very little upon drawing a Dome, &c. 
 \iDon a flat Cieling, and as the Operation is quite Mechanical, I 
 fhall therefore introduce it in this Place. 
 
 In order to find the Reprefentation of Domes, &c. it is neceffary 
 to draw out the Plan, and Half the Elevation, of the Defign which 
 we intend to reprefent, to a proper Scale, upon Paper: Thus, let 
 the 80th Figure be the Section, or Half the Elevation, of the in- 
 tended Defign ; and let the two outward Circles,* and the fmall *Fig. t 
 Squares and Circles within them, reprefent the Plan of it : From 
 which two Figures we may perceive, that the Defign confifts of* 
 eight Columns upon Pedeftals, with an Entablature, in the Co- 
 rinthian Order j that thofe Columns are fuppofed to Hand againft 
 
 a per- 
 
Of Domes. 
 
 a perpendicular Wall A e, Fig. 80 j and that the Dome Is a Semi- 
 circle, and begins to fpring from the Top of the Cornice. Now- 
 having drawn out the Plan and Elevation, as above directed the 
 Representation of any Defign may very eafily be determined in 
 the following Manner. 
 
 ^ 11. To draw upon aflat Cieling the Reprefentativn of a Dame * 
 
 Having given the Elevation and Plan, choofe the Center and 
 Dinance of the Picture. Thus EC, Fig. 81, is the Diftance of 
 the Mure , and C is its Center j that is, EC is the Diffance at 
 winch the Eye is to view the Dome when painted, and dire&ly 
 under C is the Point from whence the Eye fhould be placed to 
 1 ° JL Pi # ure - Now, this Point C being taken out of 
 the Pifture, will give a greater Length for the Columns, Gfr. 
 and will prevent fome Confufion, which would be occafioned by 
 placing the Center within the Picture. Thefe neceffary Points 
 being iettied, let us next defcribe the Parallelogram ABCD about 
 the Elevation, Fig. 80, and then draw Lines parallel to AD, from 
 the feveral Heights g, f, e, &c. as in tlie Figure : After which 
 from the Center of the Picture C, (Fig. 81) draw CD perpendi- 
 cular to EC, and from EdrawEA, parallel to C D ; then from 
 A draw AD parallel to EC, and continue it beyond D at pleafure- 
 and then will EA be a Line for the Plan, and AB a Line for the 
 Elevations: Therefore, from the Point A fet off the feveral Meafures 
 from AD, Fig. 80, which are the Meafures of the "Plan- and 
 from AB of the fame Figure, fet off the feveral Diftances, which 
 are the feveral Meafures for the Elevations: Thus Ad Fij? 81 is 
 the Width of the Plan AD, and AB the Height of the whole De- 
 fign, properly divided for the Height of the feveral Members, which 
 may eafily be conceived by comparing the two Figures 80 and 81. 
 Having proceeded thus far, the next Thing is to put the Elevation 
 into Perfpedive, as the 8 iff Figure; where Abed, &c. is the 
 Representation of ABCD, Fig. 81. This is done by drawing Lines 
 from A and d to the vanifhing Point C, and then drawing other 
 Lines from the feveral Divifions upon A B, which cutting A C in 
 correfponding Points, will give the apparent Depth of each Part ; 
 
 JlTiT M 5 thod / 0 v r find j n g *e Reprefentatian of a Don* upon a flat Cieling, is princi- 
 pally taken from Andrta Po ZZO \ Firft Book upon Petfpeffive, publilhed by MujJnSturt 
 
 SSTni 11 l 7 7L an V he , refore > if I am going to advance upon the Sub/eft fhouTd 
 appear not to be fufhciently dear, the Reader is referred to the above JSook. 
 
 from 
 
Of Domes. 
 
 from whence the whole Elevation may Be reduced into Perfpe&ive^ 
 as in the Figure. Having proceeded thus far, the next Thing: 
 (and indeed the Principal of all) is* to. defcribe feveral Circles* 
 each from a different Center, and each of a different Diameter 
 which is done thus : From the feveral Divifions, as n, m, k, draw- 
 Lines parallel to AE, cutting d C in r, p, o j then from d, r, p, o y 
 draw Lines parallel to AB, cutting the Perpendicular CD in 
 h 2 > 1> 4 > tnen * s r tne Center of the outward Circle, and i D- 
 (which is equal to Ad) is its Radius therefore, defcribe the 
 outward Circle,, and from the fame Center defcribe the fecond; 
 Circle, and then, within thofe two Circles, draw the Plan, as in 
 the Figure.. Again, for the Height of the Pedeftal 5 draw 56, r2^ 
 paralleL to AB, cutting CD in 2 and 6 ; then is 2 the Center, and 
 2 6 the Radius of that Circle which governs the Heights of the: 
 Pedeftals. In like Manner, 3 is the Center of the Circle which 
 limits the Length to where each Column begins to diminifh, and ; 
 4 is the Center of the Circle for the Nofe of the Cornice 5, and fo - 
 of the reft : All which may be made very familiar by drawing out 
 the Figure. As to the Returns of the Pedeftals and Mouldings,, 
 they all vaniftr into the feveral Centers of thofe Circles which, 
 determine their Heights : Thus 2 is the Center of the Circle for 
 the Height of the Pedeftals 5 and therefore, the oblique Sides of 
 thofe Pedeftals terminate in that Point. And as to the Ornaments 
 which may be drawn upon the Dome, they alfo are to be deter- 
 mined in the fame Manner s as will be evident by a very little 
 Attention to the Figure, and by applying thefe Rules to Practice 
 
 in a larger Scale than this upon the Plate. The 80th Figure is 
 
 the Reprefentation more nearly compleated, and between each Co- 
 lumn I have introduced a Pannel to fill up the Vacancy, and to 
 give a Hint how to introduce Ornaments proper for this Kind of, 
 Reprefentations j for whether Figures, Feftoons, or any other kind 
 of carved Ornaments, are intended by infcribing Squares about 
 each, and by dividing them into fmaller Squares, we may reticu- 
 late each Cell, which will be fufficient for forefhortning all Kinds 
 of Ornaments. 
 
 Book II. 
 
 1 
 
 CHAP. 
 
CHAR sr. 
 
 173fe Perspective / Shadows, ^ 
 
 SECT. I. 
 
 THIS Part of Perfpectivc has been very little attended to by 
 moft Writers upon the Subject, and yet it is very neceflary 
 to be known, and very eafy to be underftood ; for it is 
 built upon the fame Principles as the Perspective of Objects, and, 
 therefore, is deducible from the fame Rules. But I would not be 
 underftood to mean, that the Shadow of every particular Object 
 upon the Picture is to be determined in the following Manner ; 
 z\o; my Intention is, only to give forae general Principles, in order 
 to explain the Reafon and Nature of fuch Shadows as are neceflary 
 in the Arts of Defign ; by which means the Artift will form a 
 general Idea of the Perfpective of Shadows, and will be the better 
 qualified to difpofe them in a Picture. 
 
 Shadows are either projected by the Sun, or elfe by a Candle* 
 Torch, or fome fuch luminous Point. But fince thofe produced 
 by a Candle, &c. are but feldom wanted, I fhall therefore princi^ 
 pally have regard to fuch Shadows only as are projected by the 
 Sun : Which may be reduced under the following Heads. 
 
 i. When the Light comes in parallel with the Picture. 
 
 Z. When the Light comes from behind the Picture towards the 
 Spectator. 
 
 3. When the Light comes from before the Picture. 
 
 In the firft Cafe, the Shadows will be parallel to the Bottom of 
 the Picture ; but in the fecond and third Cafes., fince the Light 
 comes in oblique with the Picture; therefore, both the Rays of 
 Light^ and the Shadows projected by them, will have their proper 
 vaniming Points ; and confequently the Shadows produced thereby 
 will be oblique with the Bottom of the Picture. The vanishing 
 Point of the Rays of Light will be either above the horizontal 
 Line or below it ; and thofe Points will always be in Lines drawn 
 perpendicular to the horizontal Line * : And we may moreover ob- 
 ferve, that when the Light comes from behind the Picture, then 
 the vaniming Point of the Rays of Light will be above the horizon- 
 
 * See the Additions apcra this Head in the Appendix, p, «* 
 
 tat 
 
The Per spe cti.vs of Shadows. tj 
 
 tat Line ; but when the Light comes from before the Picture, then 
 the vanifhing Point of the Rays of Light will be below the hori- 
 zontal Line :. All which is exemplified in the following Figures. 
 For in Figure 83, the Light is fuppofed parallel to the Picture y 
 therefore the Shadows are parallel: In Figure 84, the Light is 
 fuppofed to come from behind the Picture, and S is taken at plea- 
 fure for the vanifhing Point of the Shadow of the perpendicular 
 Sides, and L for the vanifhing Point of the Rays of Light : In the 
 85th Figure, the Light is fuppofed to come from before the Pic- 
 ture ; and here S is the vanifhing Point of the Shadow, and L the 
 vanifhing Point of the Rays of Light*, which are both taken at 
 pleafure. 
 
 From hence then, and from the following Examples, it will be 
 obvious, that after having drawn out any Perfpective Reprefen- 
 tation, the Shadow of it may be very eafily determined upon the 
 Picture 5 therefore let us now apply what nas been faid to Practice. 
 
 Case t». When the Light comes in parallel to the PicJure, 
 
 To find the Shadows of the Objefls A and B, which are fuppofed to be Fig. $3; 
 
 cafi upon the Ground, 
 
 Through all the Corners of the Bottoms of the Objects draw 
 Lines parallel to the horizontal Line, and through every Corner 
 of the Top of the Objects draw Lines parallel to each other for 
 the Rays of Light ; and their Intersections with the loweft parallel 
 Lines will determine the Appearance of the Shadows, as in the 
 Figure : Thus a is the Shadow of A, and b of B. 
 
 From hence we may obferve, that fince EB is confidered as a 
 Ray of Light, therefore EBD is its Angle of Inclination with the 
 Ground ; or, in other Words, with the Plane of the Horizon. 
 And wc may alfo obferve, that in Proportion as this Angle of 
 Inclination of the Rays is greater or lefs, the Shadows will be 
 longer or fhorter ; which accounts for the Reafon why the Shadows 
 of Objects are longer in a Morning and Evening, than when the 
 Sun is at any confiderable Height above the Horizon : All which 
 may be clearly apprehended by attencjing to the Figure ; or by 
 drawing out other Figures, and then giving different Inclinations 
 to the Rays of Light. 
 
 I 2 
 
 Case 2. 
 
The r sTic tive c/ Shadows; 
 
 Casx 2. When the Light comes in from behind the Picture. 
 
 Tig. «4« To find the Shadows of the Objects A and a, which are fuppofed to be 
 
 caji upon the Ground. 
 Take S at plcafure in the horizontal Line, for the vanifhing 
 Point of the Shadows which the perpendicular Edges call: upon 
 the Ground (for as the Shadow lies upon the Ground, it muft va- 
 nifh into the horizontal Line;) and from this,Point S, draw a Line 
 SL perpendicular to the horizontal Line: Then will SL be the 
 vanifhing Line of the Rays of Light, and, confequently, fome- 
 where in this Line will be the vanifning Point of thofe Rays. 
 Now, in this Cafe, the vanifhing Point of the Rays is above the 
 horizontal Line j therefore, take L at pleafure for that vanifhing 
 Point, and from thence draw Lines thro' all the upper Corners of 
 the Figures j then from the vanifhing Point S of the Shadow, draw 
 Lines through all the Bottom Corners ; and their Sections with 
 each other will be fufficient Guides for compleating the Shadows, 
 as in the Figure : Thus, L3 being drawn through 1, and S 3 be- 
 ing drawn through 2, will give the Point 3 for the Shadow of the 
 Point 1, and 2 3 for the Shadow of the Edge 1 2, Gfc. 
 
 Here let us obferve, that in order to determine any Shadow, 
 nothing more is required than to find the Places of a certain 
 Number of Points upon the Picture, which Points are to repre- 
 fent the Shadows of all the upper Corners of any given Objects : 
 Thus, 3 is the Shadow of 1, and 4 is the Shadow of A; there- 
 fore, draw 3 4, which is the Shadow of the upper Edge 2 A ; and 
 fo of the refr. 
 
 C A s t 3. When the Light comes from before the Picture. 
 
 Fig. «$, To find the Shadow of the Object A, which is fuppojed to be caji upon 
 
 the Ground. 
 
 Here S is given for the vanifhing Point of the Shadow, L S for 
 the vanifhing Line of the Rays of Light, and L for their vanifh- 
 ing Point i which, in this Cafe, is below the horizontal Line. 
 
 From S draw Lines through all the lower Corners of the Object, 
 and from L draw Lines through all the upper Corners of the 
 Object^ as in the Figure ; and then their feveral Sections with each 
 other will be fufficient for compleating the Shadow, as before. 
 
 From 
 
The Perspective of Shadows^ 69 
 
 Prom thefe two laft Figures alfo, we may obferve, that the far- 
 ther the vanifhing Point of the Rays is taken from the horizontal 
 Line, the fhorter will be the Projection of the Shadows; and the 
 contrary, the nearer it is placed to the horizontal Line : That is, 
 the nearer it is to the horizontal Line, the 1efs is the Angle of In- 
 clination which the Rays make with the Ground ; and the contrary, 
 the farther it is from it. Again, by infpecting the two laft Fi- 
 gures, we may perceive, that when the Light comes from behind 
 the Picture, the Shadows will be caft towards the Bottom of the 
 Picture, and grow wider and wider continually, and the Front of 
 every Obj eel: will be in Shadow ; but in the laft Figure, the Sha- 
 dows will be caft towards the horizontal Line, and will grow nar- 
 rower and narrower continually, and the Front of every Object 
 will be enlightened ; and therefore, thefe Kind of Shadows are the 
 moft proper for a Picture, and confequently, deferve the mofl 
 Attention: For which Reafon, I fhall henceforth fuppofe the Light 
 to come in this Direction only; and fhall now proceed to mew how 
 to determine the Appearance of Shadows as they are projected by 
 ^different Planes, &£. 
 
 In the laft Figure the front Side is parallel to the Picture, and 
 the Method for finding the Shadow has been fhewn already; there- 
 fore proceed we to a Figure whofe Sides aie oblique, though the 
 fame Rule is ufed in both Cafes. 
 
 To find the Shadow of the ObjeB A, which is fuppofed to be caft upon 
 
 the Ground. 
 
 From the vanifhing Point S of the Shadow, draw Lines thro* 
 the Bottom Corners, and from the vanifhing Point L of the Rays 
 of Light, draw Lines thro' the Top Corners, which (as before) will 
 cut each other, and thereby give feveral Points, as Guides for com- 
 pleating the Shadow.— If only the Shadow of the Top was required, 
 then the Seats of each Corner muft be found upon the Picture ; 
 and from thence the Appearance of the Shadow may be deter- 
 mined: Thus 2 is the Seat of i, and 3 is its Shadow ; and a is the 
 Shadow compleated. 
 
 "To find the Shadow of an oblique Qbje&, which is fuppofed to be caft 
 
 upon the Ground. 
 
 Here A is the oblique Side, S the vanifhing Point of the Shadow, 8 7 . 
 and L the vanifhing Point of the Rays of Light.— From d draw dS> 
 and from a draw aL ; then will dc be the Shadow of the Perpen- 
 dicular da j therefore by drawing be, the Shadow will be corn- 
 pleated. <Tq 
 
7 0i Perspective ^Shadows, 
 
 To find the Shadows of Objecls when caft upon different Planer* 
 
 Fig. 88. I. To find the Shadow oj a perpendicular Objett A, when it is cafc 
 upon a Plane inclined to the Ground, but has Tome of its Edges 
 as i 2, parallel to the Pitture. 
 
 From the lower Corners of the Objeft A draw Lines to S, and 
 from the upper Corners draw Lines to L, which would determine 
 the Shadow of A, upon the Ground j but this Shadow being cut 
 by the Bottom i 2 of the inclined Plane, therefore Part of the 
 Shadow will be caft upon it. Now, to find this Shadow, from b, 
 (where the Line which is drawn from the loweft Corner of the 
 Object A cuts the Edge 2n) draw be perpendicular to the hori- 
 zontal Line, cutting the inclined Edge 2e, in « ; then from b and 
 c draw Lines parallel to the horizontal Line, and from a (where 
 ba cuts the Line drawn from the other Corner of the Object A 
 toS) draw ad, which compleats the Parallelogram abed ; finally, 
 from where the Edge 1 2 cuts the Ground Shadow, draw Lines' 
 through c and d, which will cut the Lines drawn from the upper 
 Corners of A to L, and thereby determine the Length of the Sha- 
 dow upon the inclined Face of the Object, as in the Figure. 
 
 2. To find the Reprefentation of a Shadow when it is cafi by an inclined 
 ObjecJ upon a perpendicidar Plane enrs. 
 From n and e draw Lines to S and L, which will give m for 
 the Shadow of e, and 2mn for the whole Shadow of the Side 
 2ne ; but fmce 2m is cut by the Edge nr of the Plane enrs, 
 therefore. Part of the Shadow will be caft upon it; which Shadow 
 is determined by drawing a Line from o (where 2m is cut by nr) 
 to e ; thus, noe is the Shadow which is caft upon the perpendicu- 
 lar Plane, and 2 on is the Shadow that is caft upon the Ground. 
 
 ^ig.89- 3. To find the Shadow of a perpendicidar ObjeSi A, when it is caft 
 upon a Plane B, that is every Way oblique with the Picture, but is 
 neverthelefs fituated in fuch a Manner as to have the vanijhing Ling 
 LP, of the perpendicular Side a be, pafs through the vanifhing 
 Point of the Shadow. 
 
 From the upper and under Corners of A, draw Lines to S and 
 L, as before ; then, from where the Ground Shadow is cut by the 
 Edge a 1, draw Lines to the vanifhing Point P of the oblique 
 Side B; which will cut the Lines drawn from the upper Corners of 
 A, and thereby determine the Length of the Shadow. 
 
 4. To 
 
The Perspective of Shadows. 
 
 4. To find the Projection of the Shadow of a perpendicular ObjeB A, Fig. < 
 when it is ca/i upon an inclined Plane ' that is every Way oblique 
 with the Piblure, 
 
 Draw Lines from the tipper and under Corners of A, to S and L, 
 then, from where 3S cuts the loweft Edge 1 a of the farther 
 Side of the inclined. Object, draw ab perpendicular to the hori- 
 zontal Line, and continue the inclined Edge ib 'till it cuts ab; 
 then through a and b, draw Lines from the vani filing Point of 
 the Edge 1 2, which will cut 4S in c; then, from c draw cd 
 parallel to ab, which will compleat the perpendicular Plane abed ; 
 finally, from where the Ground Shadow is cut by 1 2, draw Lines 
 to b and d, which will cut 5L and 6L, and thereby give the 
 Depth of the Shadow, as in the Figure. 
 
 Here let us take Notice, that as the Shadows of all Objects that 
 are caft upon the Ground will vanifli into the horizontal Line, fo, 
 for the very fame Reafon, the vanifhing Points of all Shadows 
 which are caft upon any inclined, or other Plane, will be fome- 
 where in the vanifhing Line of that Plane, as was obferved in 
 Figure 89. 
 
 The 91ft Figure is an Example of the Shadow of a cylindrical 
 Object A, caft both upon the Ground and the Object B ; and of 
 the fquare Object C, which is caft upon the Ground and the Ob- 
 ject D ; which, it is prefumed, wants no Explanation. 
 
 Before I conclude with the Shadows projected by the Sun, I mall' 
 juft obferve, that altho* I have taken the vanifhing Points of the 
 Shadows always within the Picture, for the Conveniency of Room 
 in each Plate ; yet, it is to be obferved, that, in general, the far- 
 ther it is taken from the Picture the better.* 
 
 Of Shadows projeBed by the Candle \ &-c. Fig. 
 In Shadows of this Kind, nothing more is required than to have 
 the Luminous Point, and its Seat upon the Ground for by draw- 
 ing Lines from thofe Points through the upper and under Corners 
 of each particular Object, the Shadow of that Object may be found, 
 as in the former Figures; and only the Shape of the Shadows will 
 be different ; that is, they will grow wider and wider continually, 
 the farther they are projected. I have given feveral Examples in 
 this Figure, and have put every Line and Point that is neceffary in 
 «ach Operation j which, it is prefumed, is fufficient for the Purpofe. 
 
 t Here the Reader i& referred again to the Additions upon Shadows in the Appendix. 
 
 SECT* 
 
Sbe FiRspECTive f Shadows, "VJ ~» 
 S ECT, n. 
 
 TTAving (hewn how to determine the Appearance of Shadows, 
 XXI might now proceed to the Confutation of Aerial Pcrfpec- 
 *m>§c~ but as that is handled at large in the laft Chapter of the- 
 nrft Book, the Reader is now referred to that: However* by way 
 of Supplement to what is there advanced upon the Subjecl, I mall 
 beg leave to make the following Obfervations, For, fince various 
 have been the Opinions about the Colour of Shadows, and as va- 
 rious the Methods purfued by Painters and other Artifts, I mall 
 therefore only offer a few Hints taken from Nature, which perhaps 
 may be of Service to the young Tyros m the Arts of Defign. 
 
 By Shadow then, in this Place, I mean the Colour of that Part 
 only of an Objea, which is either turned from the Light, or is 
 94- wholly in the Shade. Suppofe, for Inftancc, the Pillar W to be 
 placed near this Side of the Wall b, and fuppofe alfo, that the Rays 
 ot Light came from the other Side of the Wall ; then, it will be 
 evident, that Part of this Objea will be enlightened, and Part will 
 be wholly m Shadow. Now, that Part which is wholly in. 
 Shadow, is of the fame Colour as the whole Objea would be of 
 were the Sun not to mine upon it; or, in other Words, 'tis of the 
 fame Colour which the whole Objea would be of in common 
 Light. From whence I infer, Firft, (allowing for the different Ac- 
 cidents of the Sun's Light, the Air, &e.) that the Shadow of the 
 Pillar W, is the real Colour of that Objea in common Light, but 
 being oppofed to a fuperior Light, is, in comparifon of that fupc- 
 nor Light, a Shadow. Secondly, That therefore the Colour of all 
 Shadows muft be proportionably lighter or darker, as that Objea 
 to which it is a Shadow, is of a lighter or darker Colour. This 
 I have explained in the following Manner. The Objeas W W 
 I fuppofe to be White; the Objea Y, to be Yellow; the Object G* 
 Green; the Objea R, Red; the Objea B, Blue ; and the Objea 
 25 Black. — Here the fhadowed Parts of each particular Objea are 
 made darker and darker, in proportion as the Colours of the 
 feveral Objeas proceed from White to Black; which is evident by 
 the Figure. Thirdly, Since then the Shadows of all the above 
 Objeas are nothing more than the EfFeas of common Light com- 
 pared with the EfFeas of the fuperior Brightnefs of the Sun j and 
 fince Objeas are as diftinaiy feen by a common uniform Light, as 
 they are in the Sun-mine; therefore thofe Objeas which are in 
 Shadow, mould be as highly finiflied, and their Parts as well made 
 
 out, 
 
P E R 3 P EIJTIVE Of SHADOWS. 
 
 ©lit, in the Picture, as the Parts of the neighbouring Objects, 
 which are in the highert Light. And fourthly, from hence it fol- 
 lows, that the Shadow of every Object muff partake of the real 
 Colour of that Object; and therefore, Black can never be the. 
 Shadow of White, nor of any other Colour than that of Black. 
 
 By thus ranging, the Colours in their proper Orders, we may 
 eafily conceive the Degree of Darknefs which is peculiar to the 
 Shadow of each Colour. And if any one would moreover fatisfy 
 himfelf of the Truth of this, let him have a Number of fquare 
 Pieces of Wood, painted of different Colours ; then, by oppofing 
 one Side to the Light, the Degree of Shadow will be very vifible. 
 
 And from hence alfo we may obferve, that the jftronger the Light 
 fhines upon any Object, the darker will be its Shadow ; for in 
 Proportion as the Sun fhines ftronger or fainter upon an Ob- 
 jeci:, the Oppofition of the Light and Shadow will be greater or 
 lefs; and confequently, the more perceptible will be the Shadow. 
 And this accounts for the Blacknefs of Shadows by Candle or 
 Torch-Light ; becaufe the violent Oppofition between real Light 
 and total Darknefs, together with the Faintnefs of the Reflections 
 from the Smallnefs of the Luminary, muff produce that Effect. 
 
 From thefe Obfervations then it appears, that the Colour and 
 Degree of Darknefs to each Shadow, is abfolutely neceffary to be 
 known, and ought to be well underftood, in order to produce a 
 good Effect in a Picture, or to reprefent any Object as it appears 
 in Nature. It is in this, and in a proper Diftribution of the Lights 
 and Shadows in a Picture, that the Chiara Obfcuro confifls ; and it 
 is this, and this only, which can give a Clearnefs to any Shadowy 
 whether in a Painting, Print, or Drawing. 
 
 I mall juft offer a few Hints for determining the Appearances 
 of the Reflections of Objects in Water, &c. and fo put an End to 
 this Chapter. 
 
 The Reflections of Objects in Water, or any other transparent 
 Medium, may be conlidered, Firft, as to their Colour 3 and, 
 Secondly, as to the Length of their Reflections. As to their Co- 
 lour, If the Medium be very clear and tranfparent, the Colour of 
 the Reflections is very near the Colour of the Objects ; but in a 
 thick or dirty Medium, the Reflection of an Object very fen- 
 fibly changes its Colour, and partakes more and more of the Co* 
 lour of that Medium in, proportion as it is more denfe and muddy, 
 
 Book II. K till 
 
 I 
 
74 O/* Reflections in W ate r; 
 
 'till, at laft, the Reflection will entirely difappear. And in order 
 to make Water appear tranfparent (which is done principally by 
 means of Reflections ) the Reflections fhould be as perfect as 
 poflible* 
 
 To determine the Reflection of any Dbjedi in Water. 
 
 Fig. 93. Let 1 be an Object ftanding upon a Hill, and a its Bottom ; 
 then comtinue the Sides of the Object downwards, at pleafure (as 
 the prickt Lines in the Figure) and fuppofe c is where the even 
 Ground cuts the Bottom of the Hill; then fet off cd equal to c 1, 
 which will give the Length of the Reflection. Again, for the 
 Object 3, which ftands upon the flat Ground; make the Length 
 of the Reflection f equal to that Object. The Object 4 is too far 
 from the Water to be reflected by it ; and the Reflections of the 
 Objects o, n, g, which are floating upon the Water, are each 
 equal to the Height of its peculiar Object. So alfo as to the 
 inclined Object 2 ; the Reflection of that muit have the fame An- 
 gle of Inclination with the real Object, and be of the fame 
 Length, as in the Figure. From which it appears, that all Kinds 
 of Reflections are very eafily determined ; fince nothing more is 
 required, than to fet off the perpendicular Height of each Object, 
 downwards, upon the Water, &c. 
 
 What has been advanced upon Reflections, relates only to a 
 ftagnating Medium ; that is, a ftill or fmooth Water, or the like 5 
 which is the fitted for an Explanation of this Matter, and will be 
 fuflicient for giving the Learner a general Idea of Reflections : But 
 when either the Objects, or the Water, or both together, are in 
 Motion, then, though the Reflections will be wavering and uncer- 
 tain, yet the above Rules will be of great Service in fuch Cafes ; 
 and efpecially, if they are joined to the 5tudy of Nature. 
 
 I cannot conclude this Head without the following Quotation 
 from Mr. Pope s Second Paftoral ; which, to me, feems an inimi~ 
 table Picture of Nature, and much to our prefent Purpofe, 
 
 " A Shepherd's Boy (he feeks no better Name) 
 " Led forth his Flocks along the filver Tbame^ 
 " Where dancing Sun-Beams on the Waters play'd, 
 " And verdant Alders form'd a quiv'ring Shade." 
 
 I have 
 
€^f Reflections in Water; 
 
 I have now gone through with all I intended to advance upon^ 
 the Subject of Perfpective, and wifli the Work may anfwer the 
 Expectations, of my worthy Friends and generous Subfcribers ^ 
 and that the great Pains, Labour, and Expence it hath coft me, 
 may not prove in vain. I fay, that here I intended to have put an 
 End to my Subject but, by the Defire of fome particular Friends, 
 I ihall take a Tranfcript from Pozzo and Mr. Hamilton, in 
 relation to S c e n e-P aintingj and then mall add the different 
 Methods of the moft confiderable Authors upon Perfpe6tive ; : 
 which may either ferve to divert or inflrucT: the Reader ; and, at: 
 tiie fame Time,, will mew him, which are the preferable Methods* 
 mine a or theirs, cither as to Eafe or Expedition, 
 
 CHAPj 
 
CHAP, VI 
 
 Of Scenography,; or Scene-Painting, 
 
 SCENOGRAPHY is the Art of Painting upon feveral 
 Planes, or Scenes, at different Diftances, and in various 
 Pofitions with refpect to the Eye, in fuch a Manner that 
 <c all -thofe different Scenes, when feen from one certain determi- 
 <l nate Point, may correfpond with each other, and reprefent one 
 " entire View of the Defign without Breaks or Confufion, as if 
 « it were one continued Pi&ure." This is Mr, Hamilton* Expla- 
 nation of Scenography 5 who has handled this Subject in a very- 
 clear and comprehenfive Manner, both in Theory and Practice ; 
 and therefore, what I intend to offer upon it myfelf, mall be 
 principally an Abftract from him. For, fince 'tis impoffible for me 
 -to treat it in a better Manner than he has done before me, I mail 
 therefore refer my Reader to his and Pozzos* Books, if what I 
 ihall offer be not fufficient for his Purpofe and in order to be 
 as concife as poffible, I mall fuppofe him to be acquainted with the 
 Nature and Contraction of Theatres in general, and that he only 
 wants to know, how to draw fuch Reprefentations as are proper 
 for fuch Places. 
 
 The Defign of Scene-Painting, is not only to decorate the 
 Theatre, but to make that Part of it which lies beyond the Stage, 
 appear much longer than it really is. This is effected by raifing 
 the Floor to a certain Angle, by Hoping the Cieling, and by raifing 
 the Scenes in fuch a Manner, that both Floor, Cieling, and Scenes, 
 ihall be a Part of a hollow Pyramid, like LlOoNnMm, which, 
 if continued, would meet in the Point Tj and after having dimi- 
 Fig. ^. nifhed each Scene in its due Proportion, then by drawing there- 
 upon the intended Defign, by the common Rules of Perfpeclive, 
 fo that every Scene, when put in its proper Place, mall appear as a 
 Part of the general Defign. 
 
 As to the Inclination of the Floor and Cieling, and alfo the 
 Ranging, and the Space between each Scene j thefe, as I obferved 
 before, I mail fuppofe my Reader acquainted with, and therefore, 
 mall have Regard only to the drawing Perfpe&ive Reprefentations 
 upon Planes fituated in the above Manner, 
 
 fam ^S^ a ^ 6ZX9> * a ^°°k s u P on Perfpe&ive, has alfo been very copious upon the 
 
 v Here 
 
Of S C E N O G R A P H Y. 
 
 Here APFD is that Part of the Theatre which is allotted for Fi & 9I« 
 the Spectators, KGLM the Profcene, LMON the Curtain, abed 
 the Aperture in the Curtain through which the Scenery is feen, 
 ML ml the Floor upon which the Scenery is placed, PQRS the 
 farther End of the Theatre, E the Eye, EH its Height above the 
 Floor AB-DF, h its Seat upon the horizontal Plane ef QJP, and 
 T the Center of Contraction for the Scenes, Floor, &c. 
 
 Now let ABCD be a Plan of the Theatre, ee the Seat of the %- 9 6 » 
 Curtain, ML the Opening of the Curtain, bb, the Grooves 
 for the Scenes to Aide in, H the Eye, and T the Point of Con- 
 traction. — Here the Diftance C H of the Eye from the Curtain, 
 is not taken fo great as in the laft Figure j for, was the Point of 
 Sight placed at one End of the Houfe, then the moft ordinary Part 
 of the Company would have the beft View of the Scenery; and 
 therefore, about the Middle of that Part of the Houfe which is 
 allotted for the Spectators, is thought the moft proper Place for the 
 Eye as in the Figure. 
 
 And having determined the Plan for the Scenery, and fixed the 
 Point of Sight and Center of Contraction, let us next determine 
 the Height of the Eye, and the Height of the feveral Scenes. 
 
 Let ABCD be a perpendicular Section of the Houfe in the Line %-97- 
 OT (Fig. 96.) — Draw Lines perpendicular to HT from the Points 
 H and T, till they cut the Line CS of the 97th Figure in the 
 Points h and S ; then is h the Seat of the Eye, and S the Seat of 
 the Point of Contraction : Again, continue the parallel Lines 
 through the Seats of the Scenes till they cut the Line CS in 
 I, 1, 2, 3, 4, which will give the Diftance between each Scene ; and 
 from the Point I draw I e, for the Inclination of the Stage, and 
 continue it beyond T, at pleafbre : Then, for the Height of the 
 Eye and Point of Contraction, make E h in this Figure equal to 
 E h in Fig. 95 * and draw E T parallel to C S, cutting I T in T; 
 then is Eh the Height of the Eye, and T the Point of Contraction. 
 
 From hence it is evident, that fince the Floor I c is fixed, the 
 Point of Contraction muft be governed by the Height of the Eye. 
 For let xh be the Height of the Eye; draw &e parallel to hS, 
 and then is e the Point of Contraction ; and in Proportion as the 
 Height of the Eye is greater or lefs, the Point of Contraction will 
 be nearer or farther off. By this Method of varying the Height of 
 
 * The Reafon why Eh is taken for the Height of the Eye, and not E H, is, becaufe Fig. 95* 
 *e"F QP is confidcred as the Ground Plane Upon which the Picture is fuppofed to itand. 
 
 the 
 
78 
 
 €>/ SCENOG R APf Y. 
 
 the Eye, great Variety of Scenery may be introduced ; but bow, 
 far 'tis allowable to alter the Height of the Eye in Scenes for the 
 fame Entertainment, muft be left to Experience to decide : How* 
 ever, this we may obferve, that it ought never to be above the 
 Middle of the Opening of the Curtain, (that is, above s in Fig. 95) 
 nor much below the Face of an Actor upon the Stage. And in. 
 regard to the Point of Contraction ; it is not neceflary to have it 
 upon the End of the Wall, at t, but it may, and ought in general, 
 to be placed beyond it : For when it is placed at the End of the 
 -Houfe, then the Scenes will be too fuddenly diminifhed, and will 
 have a difagreeable Effect, befides other Inconveniencies. 
 
 Fig. 97. Again, for the Height of the Scenes.— The Line r s is the per- 
 pendicular Section with the Curtain ; and the Curtain being con- 
 sidered as a Picture, therefore C is the Center of the Picture ; and: 
 therefore, upon the Line r s fet off the feveral Diftances from s» 
 for the hanging Scenes and Tops of the fide Scenes; thus c is for 
 the Tops of the Scenes, and c a for the Widths of the hanging 
 Scenes; therefore from s, e, a, draw Lines toT, which will give 
 the Height of each Scene, &c. as in the Figure. 
 
 Fig. 96. The Side Scenes are made to project beyond the Line M m, &a 
 which meafures the Opening of the Curtain ; and they mould be 
 brought fo forward upon the Stage, that a Line Hb, drawn from 
 the Seat of the Eye thro' the Corner of the fir&. Scene a, may meet; 
 the fucceeding Scene in the Point b, where it is cut by Mm : For, 
 by this Means, the Spaces between the Scenes will not be vifible to. 
 many of the Spectators ; but the whole together will appear lite 
 
 Fig 97> one continued Picture. In like Manner, Lines drawn from the 
 Top Corner of each Scene, as b of the Scene f b, to the Eye, 
 whlrgive c a for the Width of the hanging Scenes. 
 
 Fig. 96. Again, if Lines are drawn through the Points f, g, h, i, till; 
 they cut the Line eD, then thefe Points f, g, h, i, will be the Pro* 
 jections of the Points i„ 2, 3, D upon the Floor of the Stage; and 
 confequently, the oblique Line e i, will, to the Spectators at H, 
 appear to be equal to the Line eD ; fo that the Back Scene in the 
 Line ik, will appear to be as far from the Eye as the End of the 
 Houfe CD ; and, by that means, the Depth of the Theatre will 
 appear to be much greater than it really is. 
 
 Having made thefe neceflary Preparations, we will now proceed 
 to (hew how to draw the Reprefentations upon each Pair of Scenes, 
 fo that the whole, when viewed from a proper Point, mall appear 
 as one continued Picture, 
 
 To 
 
Of Sgenographv. 
 
 To prepare a Pair of Side Scenes for Painting. 
 Draw a a at pleafure, which call the Line of Interfection that 'Fig* 9* 
 the Scenes make with the Floor of the Stage ; then from any Point 
 c, erect the Perpendicular cE, and from the Plan (Fig. 96) take 
 the Diftances dx, dx, which the Scenes xb, xb, are from HT, 
 and transfer them from c to b ; take alfo the Width of each Scene, 
 and transfer it from b to a, as in the Figure; then continue Cc 
 downwards, and make eg equal to f 1, (Fig. 97) and draw^ f h 
 parallel to a a : Then are ab, da, the Seats of the fecond Pair of 
 Scenes, and gc their Height from the horizontal Plane efQJP, 
 
 (Fig- 95.) And fo alfo for the Height of the Scenes; From 
 
 a, b, d, a, draw Lines parallel to gE, then take the Height f b of 
 the fecond Scene, (Fig. 97) and fet it from b to i; which gives the 
 proper Height. Again, for the horizontal Line and Center of the 
 Picture ; — Take 1 d (Fig. 97) and fet it from g to C, and through 
 C, draw HL parallel to a then is HL the horizontal Line, and 
 C the Center of the Picture. In like Manner, the Diftance of the 
 Eye for each Pair of Scenes is to be determined j— Thus Ed (Fig. 97) 
 is the Diftance of the Eye from the Scene f b ; therefore, fet off 
 CE in this Figure equal to Ed in the 97th Figure : And having 
 got the proper Diftance of the Eye for one Pair of Scenes, &c. we 
 are to proceed with our Work in the very fame Manner as if it was 
 an upright Picture. And the fame Methods are to be taken for 
 all the other Side Scenes, the Back Scene, &c. taking their Breadths 
 from the Plan, and their Heights from the Elevation : All which 
 may be very eafily done by drawing a fmall Model, according to 
 the above Rules, and then transferring the feveral Parts unto each 
 •Scene, &c. 
 
 Or this may be done by confidering the Curtain as a Picture 
 which is to reprefent the whole Defign, and upon which are drawn 
 the feveral Parts proper for each Scene ; then by reticulating the 
 whole, as in the ioift Figure, we may transfer the Part peculiar 
 to each Scene, in the fame Manner as one Picture is copied from 
 another by the common Method of Net- Work : But we muft take 
 great Care to divide each Scene exactly in the fame Manner as that 
 Scene is divided by the Reticulation upon the Curtain. 
 
 And here it is neceffary alfo to obferve, that fince the Space I f, 
 {Fig. 97) which is the Diftance between the Scenes In and f b, 
 repi efents the whole Space from I to o ; therefore, no £art of the 
 Diftance Io fhould be drawn \ipon the Scene fb; but all that 
 
 comes 
 
Of Scenography; 
 
 comes within that Diftance, fhould be painted upon the Scene Tm 
 And fo of the reft. 
 
 Again, we muft take Care to give each Scene fuch a Proje&ion, 
 that a Line drawn from the Eye through the Edge of one Scene' 
 may cut its fueceeding Scene in a proper Manner ; as was obferved 
 
 before : For which Purpofe we may ufe the following Method. 
 
 m- $9- Set off the feveral Widths for the Opening of the Curtain, and 
 Width of the Scenes, from the 96th Figure, upon the Line a f, 
 (which I here fuppofe the Bottom of the Model ; ) draw alfo the 
 horizontal Line, &c. then, from the Points a, b, c, d, e, f, draw 
 Lines to C, and make ag equal to In (Fig. 97 5) then draw gm 
 parallel to af, and fet off the feveral Divifions gh, hi, &c. from 
 g towards m j then draw Lines from all thofe Points to C, as in 
 the Figure. Thus again, fuppofe n o the Seat of the firft Scene i 
 then draw np, cutting Ck in p ; and then is np the Height 
 of the firft Scene. Again, from the Point 2, where the Edge of 
 the firft Scene cuts eC, draw 1 2, which will cut Cd in i 5 then 
 is i 2 the apparent Breadth of the fccond Scene : And fo of the 
 reft. — In the 100th Figure is a Set of Scenes complcatedi where. 
 C is the Back Scene, which parts in the Middle; r, 2, 3, 4, the Side 
 Scenes and the prickt Lines ab; &c> are the Hanging Scenes. 
 
 CHAP; 
 
CHAP. VII. 
 
 An AbfiraB of feveral Methods of PerfpeBive ; tran- 
 fcribed from the mofi eminent Authors. 
 
 T 
 
 H E oldeft Books which I have met with upon the Subject of 
 Perfpective, are, one by Vignola *, and another by Ma- 
 
 rolois -f*. And thefe two Authors feem to me, to have laid 
 the Plan for every Syftem of Perfpective (except Dr. Taylors and 
 Mr. Hamilton s) fince their Times though few of the Authors 
 who have built upon their Principles, have been fo generous as to 
 acknowledge their Obligations to them but, on the contrary, have 
 fet off their Books with pompous Titles, to allure the Public, 
 and to raife in them an Expectation of finding fomething new 
 and curious. This, though a Practice too common among Authors, 
 is, in my Opinion, an unpardonable Tranfgreflion of the Rules 
 of Modefty and Plain dealing j and therefore, to avoid any Impu- 
 tation of this Kind, I have conftantly acknowledged my Obligations 
 to every Author who has lent me any Affiftance. It was for this 
 Reafon, principally, that I gave my Book the Title of Dr. Brook 
 Taylor's Perspective, &c. But though I muft acknowledge 
 my Work to be generally built upon the Principles of that inge- 
 nious Author, I hope, I may at the fame Time affert, that who- 
 ever will compare my Schemes with thofe that have been before 
 made publick, will find very few but what are intirely of my own 
 Invention. 
 
 The following Examples, which are taken from Vignola, 
 Marolois, Vredeman Friese, thejFsuiT, andPozzo, 
 will be fufficient to mew how one Author has copied from another, 
 and the various Methods which have been published. I mail begin 
 with Vignola's. 
 
 * Vignola was a famous Italian Architeft, who flourimed in the Beginning of the 15th Cen- 
 tury : He wrote a Treatife upon Perfpeaive, which was publimed in 1644, by Ftlippo de Roffi, 
 with Annotations by Ignatius Danti. It was printed in Folio at Rome, and is in the Italian 
 
 k^'I his Work was printed in Folio at the Hague, is in Latin, and was engraved and pub- 
 lifted by Henry Hondius in 1 615 j and though tedious in its Operations, is neverthelefs a very 
 curious Performance. 
 
 L I. VIG- 
 
?2 Different Methods of PerfpeBive. 
 
 I. VIGNOLA's METHOD. 
 
 To put a Cube into PerfpeSfive. 
 
 Fi 102 Here AC is a perpendicular Se&ion of the Picture, AB is the 
 JS ' Bottom of the Picture, and C the Center of the Picture, E the 
 Eye, and ES its Height, D is the Elevation of the Cube, and F 
 its Plan upon the Ground. Now, having fettled the above Requi- 
 lites, draw Lines from every Corner of the Elevation D, to the 
 Eye E, and from the Plan F draw Lines from every Corner to the 
 Seat of the Eye at S j and their feveral Interferons upon the Line 
 BC, will give the proper Meafures for the Height and Depth of 
 the propofed Reprefentation. Thus, from the Points i, 2, 3, 4, 
 on the Line of Elevation AC, draw Lines parallel to the horizon- 
 tal Line j then from the Line AB of the Plan, take A b, b a, and 
 fet from 1 to a, and take Ac, cd, and fet from 2 todj which 
 will give the proper Heights and Depths, as in the Figure. Or, 
 by fetting off A 5, 5 6, equal to 7 8, 8 9, and drawing Lines to 
 C, we may get the Depth of the Plan abed. 
 
 By this Method, we are taught how to make a perfpective Scale 
 for any Reprefentation : For having drawn the Elevations and Plans 
 of the propofed Objects, the Line AB may be conlidered as a Scale 
 for the Plans, and the Line AC as a Scale for the Elevations. 
 
 II. MAROLOIS'sMETHOD, 
 
 To put *z Double Cross into Perfpeflive. 
 
 Fig- 103 Here ce is the Ground Line, DC the horizontal Line, C the 
 
 Center of the Picture, and CD the Diftance of the Eye. Draw 
 
 out the Plan of the Crofs, as A, and put it into Perfpective, as in 
 the Figure ; then, at any convenient Diftance c, raife a Perpendi- 
 cular cd upon the Ground Line, and fet the Elevations a, 1, 2, b, 
 upon it; then from c, 3, 4, d, draw Lines to any Point H in the 
 horizontal Line j after which, draw Lines through every Angle of 
 the Plan, parallel to the horizontal Line, which will cut the Line 
 c H, and thereby give the Points by which the Perfpective B of the 
 Elevation may be compleated j finally, from every Angle of the Plan 
 draw Lines perpendicular to the horizontal Line, and from every 
 Angle of the Elevation draw Lines parallel to the horizontal Line ; 
 and then, their mutual Interferons with each other, will produce 
 
 the propofed Reprefentation, as in the Figure, The Reader is 
 
 dented to compare this with my Method in the 40th Figure- 
 
 III. JAN 
 
Different Methods of Perfpective, 83 
 III. JAN VREDEMAN FRIESE's METHOD. * 
 
 T ? put jCube into Perfpective. 
 
 Make the Bottom BP of the Picture a Scale of Feet, from whence Fi S- I0 4« 
 find the Reprefentation of any Number of Geometrical Squares, as 
 
 in the Figure. Now let it be required to find the Appearance of 
 
 a Cube abed, equal to two Feet in Diameter, and let it be one . 
 Foot from the Bottom of the Picture.— Make the Front Face abed 
 two Squares wide and two Squares high, then give two Squares for 
 the Depth, and from thence compleat the Figure. 
 
 IV. The JESUIT'S METHOD, f 
 To put a Cube into Perfpective. 
 Draw the Plan A BCD, which put into Perfpective, asabcdjFig. 105; 
 from thence draw another Plan e f g h, then, by Maroloiss Method, 
 find the Elevation, and from thence compleat the Figure. 
 And this fame Method is taught by Kircher, in his Work, entituled, 
 Jlrs magna Lucis et Umbra, Chap. 3. 
 
 V. ANDREA POZZO's METHODS. || 
 
 1. To put a Parallelopiped into Perfpective, 
 
 Draw the Elevation A, and from thence the Plan B ; then put Fig. 10& 
 the . Plan into Perfpective, as a g f d j from the Corner a of the 
 Plan, erect the Perpendicular ab, and continue the Top of the 
 Elevation A 'till it cuts a b in c; from whence the Perfpective Ele- 
 vation may be compleated by Maroloiss Method : And having got 
 the Depth of one Plan, and the Height of the Elevations, the 
 whole Reprefentation may be compleated by Fignolds Method. 
 
 2. To put a Parallelopiped into PerfpeSfive, which will ex- 
 plain Pozzo'i other Method. 
 
 Here in Conformity to Vignolds Perfpective Scale, A C is the Fig. ioy. 
 Section of the Picture, AB the Ground Line, D the Eelevation of 
 ,an Object, and F, H, the Plans of two Objects parallel to the Pic- 
 
 * This Book is a Folio, in Trench, was printed at the Hague [in 1619. It was corre&ed 
 ; by Marolois, and engrav'd by Henry Hondius. 
 
 t This Book is in Quarto, was wrote originally in French by a Jefuit at Paris, was tran- 
 slated into Englijh by E. Chambers, and was printed at London in 1726. 
 
 j| His firft Book was publifhed in Latin and Englijb by John Sturt, Engraver, in 1707; and 
 >is fecond Book was publiihed by himfelf, in Latin, in 1700 j and are both in Folio. 
 
 h* ture "5 
 
#4 Different Methods of Perfpeftive. 
 
 ture 5 fo likewife, E is the Eye, E S its Height, and I is confidered 
 as the Seat of the Eye. From the feveral Angles of the Ele- 
 vation draw Lines to the Eye E, and from the feveral Angles of rhe 
 Plans draw Lines to I, which will cut BC, and thereby give the Ele- 
 vations and the Depths of the Plans j from whence the 108th 
 Figure may be compleated. Thus, A i, A 2, A3, A4, are each 
 equal to their correfponding Divifions, A 1, A2, A3, A 4, upon the 
 Line of Elevation A C, Fig. 107 j and 10, it, &c. are equal to 
 K o, K t, &c. of the fame Figure ; and A C is equal to the Height 
 of the Eye S E. But I have put every Line and Point, to explain 
 the Thing the better. 
 
 There are feveral other fmaller Treatifes upon Perfpective, and 
 particularly one by Bernard Lamy, entitled, Perfpefiive made Eajy> 
 which, as it contains fome curious Obfervations upon Painting, &c. 
 is worthy of Notice. 
 
 FINIS. 
 
 * 
 
APPENDIX. 
 
 TH E favourable Recteption of the firft Impreffion of this 
 Work had been a flifficient Inducement for publifhing a 
 fecond Edition, if the Number of my Subfcribers had not 
 made it abfolutely neceffary. 
 When I firft engaged in this Undertaking, I much dreaded the 
 Difficulties which prefented themfelves, both from my own Inca- 
 pacity, and from the Nature of the Subject : For although I had 
 made Perfpe&ive my particular Study for feveral Years, and was fatif- 
 fied in my own private Opinion as to the fhortnefs and clearnefs of 
 the preeeeding Method ; yet to make it intelligible to others, and 
 ufeful in general, were Things not to be accomplifh'd without much 
 Study, Labour, and Expence. I therefore determined to proceed 
 very cautioufly, to view every Article in various Lights, and not 
 to print any thing without having it firft approved of by competent 
 Judges. 
 
 As this feem'd the mo/t likely Means to prevent my publiming any 
 ufelefs or undigefted Figiures, fo I thought it alfo a very likely way 
 of avoiding the little ill-rnatured Criticifms, which are fo often made 
 upon the Works of a yomng Author:. And I muft confefs (with the 
 utmoft Gratitude and Thanks) that my Succefs hath abundantly ex- 
 ceeded, my utmoft Expectations ; for I. have been fo fortunate as to 
 have the Work approved of in general, and recommended in fuch a 
 peculiar Manner, oy Gentlemen of great Genius and Knowledge, 
 that I now begin to think it fecure^from public Cenfure, under their 
 kind, an.d powerful Protection. 
 
 But it may be neceffary to inform my Reader of the Additions which, 
 he may expert to find in this Appendix : And, in the firft place, I have 
 more largely and more fully con£dered the Perfpective of Shadows 5 I 
 have alfo given one Figure to mew why a Down-hill (if merely con- 
 fidered as fuch) cannot be reprefented upon the Picture ; then I have 
 added another Figure, to explain the nature of what is called a Bird's- 
 eye-view, a fort of Perfpective ufed in drawing Fortifications, and the 
 like. I have alfo (hewn the Ufe of an Inftrument of my own Invention,, 
 
 EST which: 
 
APPENDIX. 
 
 which may be of Service in Drawing extenfive Views, large Build- 
 ings, &c. and, lafUy, I have given the Conftru&ion of a fmall Pocket 
 
 Camera Obfcura. To begin therefore with the Additions to the 
 
 Perfpective of Shadows. 
 
 In both the Theory and Practice of Shadows, I have frequently^ 
 made Ufe of this Expreffion, viz. " The vanifhing Point of the Sha- 
 " dow -," which, poffibly may require fome farther Explanation : Be- 
 caufe the Shadows of any Objects which are compofed of perpendi- 
 cular or parallel Planes, will, when put into Perfpective, vanifh into- 
 various Points upon the horizontal Line ; and therefore this Article 
 may not feem fo very fignificant, as in fact it is* 
 
 By the vanifhing Point then of the Shadow, is meant the vanifhing 
 Point of fuch Shadows only, as are fuppos'd to be carl: upon the 
 Ground Plane (or upon a Plane parallel to it) by the perpendicular 
 Edges of Objects. For fince thefe Species of Shadows will always 
 vanifh into the Center of the vanifhing Line of the Plane of Rays, 
 therefore this particular vanifhing Point will be found to be more ufe- 
 ful than any other; as will appear by the following Examples. 
 
 And, as I found it neceffary to make fome confiderable Additions 
 to this Part of Perfpective, fo I have made Choice of fuch Figures 
 as might contain the moil general Rules, and have given fome of the 
 moft curious and difficult Examples which can be propofed : And 
 that they may be the more clearly comprehended, we will range what 
 we have farther to advance, under the following Heads, viz. 
 
 •I. When the Shadow is cafi upon the Ground, or upon a Plane paral- 
 lel to it. ■ 
 
 II. When the Shadow is cafi upon a perpendicular Plane, 
 
 III. When the Shadow is cafi upon an oblique Plane. 
 
 C A S E I. 
 
 When the Shadow is caft upon the Ground, &c. — • — -Firfl, by a 
 perpendicular Object ; fecondly, by a parallel Object ; and tliirdly, 
 by an inclined Object. 
 
 Example I. When it is cafi by a perpendicular Objedl. 
 
 , _ * Here R S is given for the vanifhing Line of the Rays of Light, 
 ' R for the vanifhing Point of the Rays, and A B for the perpendicular 
 Object, whofe Shadow is fought. 
 
 * The Line R S, whether continued or not, will always fignify the vanifhing Line of the 
 Plane of Rays, and R the vanifhing Point of the Rays of Light. 
 
 From 
 
APPENDIX. 
 
 From A and B draw" A S, and B R, cuting each other in b ; then 
 is A b the Shadow of A B. -j- 
 
 Example II. When it is caft by a parallel Objetf, as the 
 
 Plane 1234. 
 
 Find the Seats of the four Corners upon the Ground, as a, e, n, f j Fl S- < 
 then from thofe Seats draw Lines to S, and from 1, 2, 3, 4, draw 
 Lines to R, which will interfed each other, and thereby give the 
 Appearance of the Shadow m. — Now becaufe S and H are the va- 
 nilhing Points of the Plane 1234, therefore the Sides of the 
 Shadow will vanifh into thofe Points. 
 
 Example III. When it is cafi by an inclined ObjeSt. 
 
 Let 1 2 3 4 be a Pyramid, whofe Shadow is required.— From Fig- 7- 
 n, the Seat of its Apex or Top 3, draw a Line to S, and from 3 draw 
 a Line to R, cutting 118mm; then from 1 and 2 draw Lines to m, 
 which will compleat the Shadow.-- And in the fame Manner the 
 Shadow of the inclined Edge A B, of the perpendicular Plane A B D, 
 is to be determined. 
 
 CASE II. 
 
 When the Shadow is caft. upon a perpendicular Plane.-— FirhS 
 when it is caft by a perpendicular Objetf: ; fecondly, when it is caft 
 by a parallel Object and thirdly, when it is caft by an inclined 
 Object. 
 
 Example h When it is caft by a perpendicular Objetf. 
 
 Let ab be an Object perpendicular to the Plane A B C D, and let Fig. 4. 
 it be required to find the Shadow of the ObjecT: a b upon this Plane. 
 
 From a and b draw the Perpendiculars a f, b g, and through f, 
 
 where af cuts the Bottom AB, draw Cg, cutting bgingj then 
 
 + Now that S, die Center of the vanilhing Line R S, is likewife the vanifhing Point of the 
 Shadow of A B, may be thus demonftrated. The Lines A B and R S being parallel there- 
 fore the Plane ABRS will pafs through them both ; and fince the Shadow of AB is cait upon 
 the Ground, it muft vanifh into the horizontal Line : And becaufe RS is the vanilhing Line 
 of the Plane of Rays which projefts the Shadow, the Point S muft likewile be in that Line, 
 and therefore S, the common Sedtion of the vanifhing Line RS, with L S the vanilhing Line 
 of the Plane upon which the Shadow is caft, muft be the vanifhing Point of the Shadow upon 
 
 that Plane. And in the fame Manner all the foregoing Figures upon this Head may be 
 
 demonftrated, r 
 
 JST 2 fr° m 
 
APPENDIX. 
 
 from g draw g S cutting A B in h, and from h draw a Line at plea- 
 fure, but parallel to a f, then from b draw a Line to R cutting h e in 
 e ; finally, draw a Line from e to a, which will be a Guide for corn- 
 pleating the Shadow, as in the Figure. 
 
 Example II. When it is caft by a parallel Objecl. 
 
 Fig- u Here A B is the Objecl:, and i 2 3, the Plane upon which the 
 bnadow is caft.— From A draw A S cutting the lower Edge 1 2, in 
 a, and from B draw BR, and from a, draw ag parallel to A B • 
 then will af be the Shadow propofed : And was the Plane 1 2 % 
 'continued as high as g, then would ag be the Shadow of A B.~- 
 The fame may be faid of the Shadows a, c, in Figures 2, 3. 
 
 Example III.- 
 
 -When it is caft by an inclined Objecl. 
 
 h * S ' j In thls figure 1345, is an inclined Obje£t, which cafts a Sha- 
 dow upon the Planes A, C— F«d the Shadow of the perpendicular 
 Plane 1 2 3, upon the Ground, which will cut the lower Edge of 
 the Plane A in a j continue a a till it cuts the horizontal Line in f • 
 then is f the vanifhing Point of the Shadow of the Edge 1 3, there- 
 fore from R draw a Line through f, and continue it at pleafure, 
 which will pafs through C, (the vanifhing Point of the inclined Face 
 J 3 4 5) then from H, the vanifhing Points of the perpendicular 
 Planes A, C, draw H V perpendicular to the horizontal Line, which 
 will cut R V in V, and thereby give V for the vanifhing Point of the 
 Shadows ab and cdj therefore from a draw aV, and from b draw 
 b S, from c draw cV, and finally, from 3 draw 3 R, which will give 
 the Point d for the Shadow of the Corner 3.— And in order to find 
 the vanifhing Point of the Shadow which is caft upon the Plane G 
 by the Top 34 of the inclined Plane, continue V S below the hori- 
 zontal Line, and draw a Line from R parallel to the horizontal Line 
 which will cut V S in S, and thereby give S for the vanifhing Point 
 of that Part of the Shadow ; as is evident by the Figure. 
 
 CASE III. 
 
 When the Shadow is caft upon an inclined Plane.— Firft, by an 
 Objea perpendicular to the Ground ; and fecondly by an Obied in- 
 clined to the Ground. J 
 
 EXAMP. 
 
APPENDIX. 
 
 Example I. When it is cafi by a perpendicular Objedi ; which 
 
 will admit of great Variety. 
 
 ift If the vanifliing Point S of the Sides i 5, 3 4-*%^ vanifh-Fig. u 
 W Point of the Shadow upon the Ground, then will S be alfo the 
 vanifliing Point of the Shadow upon the inclined Plane -Thus the 
 Shadow cd which is call: by A B upon the inclined Plane will va- 
 *fl into S 
 
 111 2dlv° If the vanifliing Point S of the Shadow be taken within Fig . t , 
 the vanifliing Point H of the Edges 15, 34, then the vanifliing Points 
 of the Shadow cd, will be above the horizontal Line ; and may 
 always be found in this Manner 5 viz, find the Shadows A a, ac, 
 and continue the vanifliing Line R S above the horizontal Line at 
 Pleafure 5 then from the vanifliing Points H, V, of thr inclined Plane 
 draw HV, cutting Rs in s, and then is s the vanifliing Point or 
 
 Cd ^dlv If the vanifliing Point S of the Shadow be taken without Fig. 3^ 
 the vanifliing Point H of the Edges 15, 3 4> then the vanifliing 
 Point 1 of the Shadow c d will be below the horizontal Line : And 
 this is found by drawing a Line through the vanifliing Points V and 
 H of the Edges 13, 15 of the inclined Plane, till it cuts the vanifli- 
 ing; Line R S in 1. , c 
 
 Now the Reafon of all this muft appear extremely evident, it we 
 confider, firft, that however a Shadow is caft upon any Plane, it 
 muft vanifli into a Point or Points in the vanifliing Line of that Plane ; 
 becaufe the Boundaries or Out-lines of every Shadow, are confidered 
 only as Lines drawn upon a Plane. And fecondly, becaufe the va- 
 nhW Points of the Sides of any Shadows, and the vanifliing Point 
 of the* Plane of Rays which projects thofe Shadows, muft always be 
 in the fame Plane : But, as we obferved before, this will more fully 
 
 aP ^tSy^ y To e fiXhe Shadow of A BCD upon the inclined Plane Fig. 4 
 1 74.— Here VL continued will be the vanifliing Line of the 
 inclined Plane : And to find the Shadow cd, firft determine the Sha- 
 dow of A B C D upon the Ground, which will cut the lower Edge or 
 the inclined Plane ; then continue R S till it cuts V L in V and from 
 R draw R L parallel to the horizontal Line which will cut VLmL; 
 and then are L and V the vanifliing Points of the Shadow, as m the 
 
 r ig 
 
APPENDIX. 
 
 Figure.-- — From this Figure we may obferve, that the Line V L 
 paffes through the vanishing Point H of the Edges i, u anc j 
 therefore the vanifhing Point V may be found by drawing L V 
 through H. ■ 6 j*- * 
 
 Fig. 6. 5thly. To determine the Shadow of the Pillar A, when it is can: 
 
 upon two inclined Planes. Here HF and FL are the vaninW 
 
 Lines of the inclined Planes 124, 23 4. Find the Shadow of 
 
 the Pillar upon the Ground which will cut the lower Edge of the 
 Plane 124 in a b ; then continue HF till it cuts R S continued in V 
 and then is V the vanifhing Point of the Shadow abed, and fo alfo 
 s where the vaniming Line F L cuts R V; is the vanifhing Point of 
 the Shadow c d e f. to 
 
 Example II.- 
 
 -When the Shadow is cqft by an inclined Line 
 AB 3 upon an inclined Plane 123. 
 
 Fi ? . 7. . ^ a . vin g fo " nd the Shadow of A B upon the Ground, continue 
 it till it cuts the horizontal Line in s j then from R draw a Line thro' 
 s cutting the vanifhing Line H V of the Plane 1 2 3, in f, and then 
 is f the vanifhing Point of the Shadow a b ; And if L f be continued 
 it will cut the vanifhing Line N U in U. 
 8. To find the Shadow of a perpendicular ObjecT: when it is cafl upon 
 
 a Tetrahedron. Determine the Shadow A b of A B upon the 
 
 Ground, and draw the Seat a e n f of the Plane 12345 then from 
 where the Shadow cuts a f , draw Lines parallel to A B cuttino- the 
 Edge 1 4, which will be a Guide for drawing the Shadow upon the 
 upper Face, to S. And for the Shadow on the Face 1 4 D continue 
 the vanifhing Line HM of 14D, and S R the vanifhirV Line of 
 the Plane of Rays, and their Interferon with each other will be the 
 
 vanifhing Point of that Shadow. As to the Shadows which are 
 
 cart upon the Ground by the above Objects, it is prefumed, that they 
 want no farther Explanation. • 
 Thefe are a few of the many Examples which might be produced 
 as a farther Illuflration of the Perfpedive of Shadows j for this Part 
 of Perfpedive might be extended to Infinity : However thefe Figures 
 contain fome of the moft general Principles that I can think of, and 
 are abundantly fufficient to fhew how the Appearances of any Sha- 
 dows are to be exadly determined, upon all forts of Planes, and in 
 £he molt difhcult Situations, 
 
 Borne 
 
APPENDIX. 
 
 Some Confiderations upon drawing the Reprefentation of an inclined 
 Plane going from the Eye, or what is ufually called a Down-hill. 
 
 To reprefent a Down-hill hath always appeared a Matter of great 
 Difficulty to Painters, and thits will ever remain impracticable, iince, 
 in the Nature of the Thing, :it is impoffible to be done. 
 
 For let HL be the horizontal Line, and let Fi, F2, F3, F4,Fig. 
 and F H, reprefent the fevenal Angles, or Inclinations of five differ- 
 ent Hills : Then we may conceive thefe Hills to be like fo many 
 inclined Planes. And if theyr are fuppofed to vanim into Lines paral- 
 lel to the horizontal Line, th<en a a is thevanifliing Line of the Hill 
 F 2, b b of the Hill F 3, c c of the Hill F4, d d of the Hill F H, and 
 the' horizontal Line H L is the vaniming Line of the even Ground 
 F K ; for the feveral Lines L C, L d, &c. are parallel to the ori- 
 ginal' Lines F K, F H, &c. Prom whence we may obferve that the 
 even Ground (fuppofe a Roadl ABC) will feem to rife upwards, and 
 vanifh into the horizontal Lime H L ; and the leaft inclined Plane wilt 
 vanifh below the horivontal Line, but will take up the Space D d 
 upon the Picture the next imclined Plane will take up the Space Dc 5 - 
 the next D b, and the next: Da; and when the Inclination of the 
 Hill is fo great (as F 1) that its vaniming Line will fall below the 
 Bottom of the Pi&ure, then tthat inclined Plane will totally difappear, 
 and therefore can have no Place upon the Picture. So that from 
 hence appears the ImpoffibiLity of reprefenting a Down-hill, (fmgly 
 as fuch) by the Rules of JPerfpective 5 becaufe, what we actually 
 knowtto go down-hill in Nature, will, if ever fo correctly drawn 
 upon the Picture, appear to rife upwards ; which is another ftrong. 
 Inftance of the Inefficiency of Perfpedive upon fome particu- 
 lar Occafions. For in ordeir to reprefent fuch an inclined Plane, 
 we muft have Recourfe to E xperienee, which will teach us to dif- 
 pofe particular Kinds of Obje&s in fuch a Manner, as mall convey 
 to the Mind the Idea of a Down-hill ; fuch as mewing Part of a 
 Figure, or making the Tops of lofty Objects full below the horizon- 
 tal Line, &c. As to the Manner of reprefenting Hills when 
 
 they are fideways with the Picture, that is fo very eafy,. as not to be 
 worthy Notice in this Place. 
 
APPENDIX, 
 
 Of a Bird's Eye View, and how to put a Fortification, .&c. into 
 
 Ferfpeftive, 
 
 Although this Part of Perfpedive is eafily deducible from our 
 general Rules, yet I have here added the following Figure, which is 
 fufficient to explain the whole of this Matter. And I have made 
 the Figure very fimple, with upright Walls only, and without Baf- 
 tions, or any the leaft common Parts of Fortifications. 
 
 In drawing the Reprefentations of Fortifications, it is neceflary 
 not only to fhew one View as feen upon the Ground, but to exhi- 
 bit alfo fo much of the feveral Buildings as the Eye can poffibly take 
 in at one Time from any Situation. And in order to do this we muft 
 fuppofe the Eye to be removed to a confiderable Height above the 
 Ground, and to be placed as it were in the Air, fo as to look down 
 into the Building, like a Bird that is flying. 
 Fig. io. ; Suppofe therefore M, N, O, to be the Walls of three Fortifica- 
 tions, the loweft (O) of which, is furrounded with a Ditch. Now 
 
 to draw thefe feveral Reprefentations, we muft firft choofe a proper 
 Height for the horizontal Line, and then proceed exactly in the fame 
 Manner, as if we were drawing any Objects by the common Rules ; 
 only obferving to let the Diftance we work with, be fomewhat 
 greater than the Space between the Bottom of the Picture and the 
 horizontal Line.. And, if it were required, to draw the Appear- 
 ance of a Ditch, or the like ; then from the Surface of the Ground, 
 as IK, fet off I 2 equal to the fuppofed Depth, and draw a Line to 
 the vanifhing Point of the top Edge of the Ditch ; and fo %ewife 
 for the Surface of the Water, which we will fuppofe to be at the 
 Diftance I 1 from the Top of the Ditch. Set this Diftance from I 
 to 1, which will be & fufHcient Guide for the above Purpofes. 
 
 It is eafy to conceive that the higher the horizontal Line is placed, 
 the more of the Fortification will be feen, and the contrary the lower 
 it is placed. * 
 
 4 Defcription of an Infirument that may be ufeful in taking exten- 
 
 five Views, &c. 
 
 The Ruler A B is 19 Inches long, and is graduated into 19 equal 
 Parts ; upon the upper Edge of it is a douftail Groove to receive the 
 perpendicular Ruler G, which has one End fitted to it, fo as to Aide 
 very eafily ; this Ruler is 1 5 Inches long, and is divided into 1 5 equal 
 
 Parts 
 
APPENDIX. 
 
 Parts, and upon the Back- fide of it (reprefented by F) is a Line 
 drawn exactly in the Middle, and upon this Line is fixed a piece of 
 Barber's Silk, with a little Plummet at the End. The Ruler A B is 
 fixed by two Screws a c, to two pieces of thin Brafs j and thefe 
 pieces of Brafs are fixed at the other Ends by two Screws d e to a 
 ftronger piece of Brafs b f - y this Brafs b f goes clofe to the Ruler 
 A B, and has a Joint at x which turns upon a Screw j below this 
 Joint is a piece of round Brafs about fix Inches long, which goes into 
 a Hole made in the Top of the Staff, and may be raifed higher or lower 
 like a Barber's Block by means of the Screw f 5 Part of this Staff is 
 C D E, and the whole Length is ab-out 3 Feet, and at the Bottom is, 
 what we call a rank Screw made of Iron and is fixed to the Staff. H 
 I is a Wire 22 Inches long, with a Screw at h to go into the Hole b; 
 the piece of brafs Wire, bent into the Form i k is fixed to the Wire 
 H I by the Screw k ; and the Part i goes into the Hole f, in the brafs 
 Piece b f. The fmall Wire K L. is about 12 Inches long and is flat- 
 ted at K, at which Place is a little Hole about i-8th of an Inch in 
 Diameter ; this Wire K L is fitted to the Holes 1, m, n, o, which 
 are made in the larger Wire H I, and it may be placed higher or low- 
 er, by means of a fmall Screw. This is a Description of the feveral 
 
 Parts of the Inftrument; we will next fhew its Ufe. 
 
 Fix a Paper upon a Drawing-board, as in Fig. 1 2, and divide the 
 Paper length- ways into 19 equal Parts, and Perpendicularly into 15 
 equal Parts ; and in Proportion as you intend the Drawing to be larger 
 or fmaller, make thefe Divifions greater or lefs. Then take the Staff 
 and fix it ftrongly in the Ground, by means of the Screw at the Bottom, 
 and at a convenient Diftrance from the Profpect which you intend to 
 take. After this, put the Inftrument together as in Fig. 13; and 
 fix the Ruler A B, exactly Horizontal by means of the Plummet on 
 the perpendicular Ruler and the Brafs Joint x; then fix the Wire K L. 
 fo as to have the Eye-hole exactly level with the Horizon, that is 
 equal to the Height of the Eye, and take care alfo to have the 
 greater!: Diftance of the Eye-hole from the Ruler, equal to the whole 
 Length of the longer!: Ruler A B, and never lefs than the Diftance 
 
 h 1. -Having thus fixed the Inftrument, place yourfelf on a Seat, 
 
 and proceed to make your Drawing in the following Manner. 
 
 Look through the Eye-hole, and then move the perpendicular Ruler 
 in the Groove, till you get one Edge exactly againfl fome principal 
 Object: j then will the Parts upon the Ruler fhew how high the Ob- 
 ject is from the Bottom of the Ruler (that is from the Bottom of the 
 
 O Picture) 
 
APPENDIX. 
 
 Picture) and you will alfo have its apparent Height ; therefore trans- 
 fer this unto the Paper in thefe Squares which correfpond with the 
 Divifions upon the Rulers. So alfo for the Breadths of Objects ; move 
 the perpendicular Ruler fo as to be even with the Sides of an Object j 
 and the Divifions upon the lower Ruler will (hew their apparent 
 Breadths. And after the fame manner, get the Places and apparent 
 Sizes of as many principal Objects as are neceflary for affifting you 
 in compleating the whole Drawing ; which may be done by this Me- 
 thod with great Exactnefs. And having finiilied the Drawing, 
 
 the Inftrument may be taken to Pieces and put into a Box, which 
 may ferve as a Drawing-board ; the Top M may be fere wed into 
 the tarT which will ferve as a Walking-ftick, and the Stool to fet on 
 may be made very Portable, fo that every Part of this Apparatus may 
 be carried by one Perfon with great Eafe. * 
 
 I mall juft Obferve that the Inftruments which have been Published 
 of this Kind, have no Diftance limited for the Eye hole, which make 
 all the Reprefentations that are drawn by an improper Diftance 
 moft egregioufly Falfe j as is Demonftrated in what we have faid 
 concerning the Diftance of the Eye in Chap. 6. B, i. and Chap. 2, 
 B, 2. 
 
 The 14th Figure is a fmall pocket Camera Obfcura. The lower 
 Part of this inftrument is a fquare Box, 4 Inches in Diameter with a 
 Looking-glafs E fixed at an Angle of 45 Degrees. In the Middle of 
 the Side B C is a fmall Hole 2 Inches in Diameter, in which goes a 
 Tube to Slide 2 Inches long, and in that a Lens for the Object - 
 Glafs. The Top part F of this Box is a Piece of ground Glafs to 
 receive the Image from the Looking-glafs E. But as the Picture 
 will be very fmall, and confequently the Object's too much dimi- 
 nifhed ; therefore, on the Top of the Box CD a b, is another Tube 
 G, with a Lens of a large magnifying Power ; which being raifed 
 higher or lower will fo increafe the Size of the Picture as to make the 
 whole View very diftinct. The Fore-part which is left open, may 
 
 * But although this Inftrument is very Simple, and ufeful to any Perfon who is tollerably 
 ikilled in Drawing; yet I muft be fo ingenuous as to acquaint my Readers, that there is an- 
 other kind of Inftrument, made by Mr. Adams, at Tycho Brake's Head, the corner of Raquet- 
 Ccurt, in Fleet Street London : which is fo conftructed, that any Perfon, without either the 
 knowledge of Perfpe&ive, or the leaft Notion of Drawing, may take the Perfpe&ive View 
 of any Building, Profpe£r., or Figure, with the greateft Accuracy. This Inftrument is newly 
 invented, (but upon a Plan of Sr. Chrijlopber Wrens) by a Clergyman in the Country, whofe 
 Name I am no: Commifiioned to mention ; however in Juftice to fo ingenious and ufeful a 
 Production, I have taken the Liberty of Recommending it in this Place. 
 
APPENDIX, 
 
 be either made like two Doors to move upon Hinges, or may Aide 
 in Grooves for that Purpofc : One of which is absolutely neceflary 
 
 on account of cleaning the Glaffes &c. This with the other In- 
 
 ftrument may be had of the fame Perfon, mentioned in Page I 
 Book I and of Mr. Adams in Fleet-Street. 
 
 FINIS. 
 
1HE6ETTY CENTER 
 
 LIBRARY