YALE UNIVERSITY LIBRARY A39QO2005913752B j@xlrrvERif§r YALE UNIVERSITY LIBRARY I J" 70AART. — A Treatise of Perspective, by Bernard fiftrvr ' r Lamy, illustrated ivith above 50 figures in ' ' copper, post 8vo, old calf, 2s 6d 1702 "" "¦ "-Jt^-- A TREATISE 0 F Perfpective. O R, Thc Art of Reprefenting all manner of Obje£ts> as they appear to thc Eye in all Situations. CONTAINING Thc Elements of Defigning and fainting. 3lfluftcateti uut& aboue 5°Jf teutetf in Coppec Written originally in French, By Bernard Lamj Prieft of the Oratory, and Faithfully ¥ ranflated into Englijb, by an Officer of His Majeftics Ordnance. LONDON Printed, and Sold by moft Bookfellers, 1702. TO THE , RIGHT HONOURABLE- Sir Hairy Goodrich, Kt. & BaA . Lieorenant-Geheral of the Ordnance; and one of 'His Ma/efty's moft Honourable Privy-Council. And to the HONOURABLE John Charlton Chrtftopher Mufgravec James Lowther i^fq; John PuJteney and William Boulter Commiffiohers of His lyiajefty's Ordnance. ¦ May it fleafe your Honours, Since the Nobleft parts of the Mathe- maticks are not only ufeful but necef- fary, to fuch as are concerned in an Artillery, any Branch of that Noble Science may claim your Honours Patronage, as being intrufted with the Management of fuch a weighty Concern as the Ordnance of the A Kingdom. the Epiftle Hedicatory. I ^dom. Your arduous Application, and , ri, .^ .dentCdtiduft, are fuffieient Evidcnces,that His Majefty has difchveredin your Honours, thofe excellent Qualifications fo proper fou# executing his OFderS ; and that he may ex pect an inviolable fidelity , from fpcb as neither Pleafure^ nor Ambition, can Corrupt, or divert from the meaneft things t.that con- , cern his Service. * ' | I am proud of the Opportunity I ndW hage, to make a Juft Acknowledgment of the Fa* vours I ha%receiVed$ fince I had the Honour^ to ferve in the Artillery ; andi humbly beg ' your Honours Protection for this fmall Trca- tife, which the. Author's Character and my own Curiofity nrnVinvited me to read, and which his Plainnels and Succinctriefs encou raged me to tranflate-mto £»£/$&. I hopeyoui! Honours will pardori my Prefumption„ which I was prompted to by the Zeal 1 have fofc' your Service^ andthe Profound Reipect wi# which I am *> Yours Honours, #hft Obligek&nd ntofi Devoted Servant A. F. THE THE PREFACE- THE Mathematicks are fo ufeful in all other parts of Knowledge, the Affinity of their noble Parts fo con fiderable, and the Enquiry Which hath been made after them fo curious, that they have both contributed wonderfully to the Advance ment and Perfection of- Arts, and have oc casioned feveral furprifing Difcoveries. "¦The Science of the Mathematicks (to ufe the Author's words) ci is as Profitable as ic it is Spacious and Difficult : It compre- " hends the Principles of all Arts ; and there " is fcarce a Science that ftands not in need " of its affiftance. Thofe therefore who " underftand the Mathematicks, are the beft " qualified for all manner of Employments ; " they are more capable either for praftifing " Arts or for governing and dke&ing thofe << who'praaife them. Being accuftomed to " clear Idea's and coherent deafenings, they " have a greater Capacity for all the Sciences, A3 " and The PREFACE. li and are entitled to fuch Penetration and " Exaftnefs, as renders them Ingenious in all " Profeflions. " In former Times the Mathematicks " dwelt, as 'twere, in a Sanfruary, where " every one was not allow 'd to approach : " They were Myfteries known only to a u few. Their Obfcurity proceeded not from " any Artifice of thofe who improv'd them, " to make them feem more admirable, the „ Truths they contain are'fimple and clear; " but cannot be perceiv'd without Labour and Attention, and a patient Study of their long Connexion ; for fuch a Truth, or fuch a Suppofition , can be Only clear to him who has already unfolded a hundred others, of which this is the Confequence. " Few are capable of that painful Atten tion that muft be given to the Mathema ticks, we have a general Averfion to Dili gence and Application, it was therefore the Work of feveral Ages to render them perfect; a long Series of time was re quired to -invent and collect the Truths they contain, before they could be dif- poled in that natural oi der, which makes it eafier now to comprehend their greateft " Difficulties in fix Months, than formerly in " fix Years. a d u u cc a u t>. ci u it u The "PREFACE. The Example of feveral Perfons of Piety and Judgment, who applied themfelves to render the Mathematicks more eafie, fore- feeing what Ad vantages might be drawn from them, encourag'd our Author to publili his Elements of Geometry ; and he would ha /e given us a compleat courfe, if the purfuit of fuch Studies as are proper for one of his Character had not diverted him. His Defcrip tion of the Temple of Jerufalem was t1 ; oc cafion of his recollefting what he had many Years before ftudied of Perfpe&ive ; and the Benefit he»found by it in that great Defign, was his chief Motive for publifhing this fol lowing Treatife, which ought not to be defpifed for its fhortnefs, fince the Excel lency of a Book, is to be fhort and plain. Attention is always painful, and what lafts too long cloys us. 'Tis not the enlarging on a Mathematical Demonftration that renders iteafier, but the doing itfo, as that the Brevity does not fink the Perfpieuity. Here is nothing laid down without Demonftration 5 • and no Rules are propofed without proof 5 and it contains as much Theory as any Book of Perfpe&iv e can pretend'to : It was not huddled together of a fudden, but after its being copied feveral times, and paffing through the hands of feveral of his Friends ; he examin'd it carefully, explain'd fome A j things The PREFACE. things more clearly, turn'd others more ad- vantageoufly, adding in fome places, and re trenching in others, till he gave it the Form that it now bears. The Subject is one of the moft noble parts of the Mathematicks, 'tis a Science that con- fiders the fight, and explains by Principles of Phyficks and Geometry, the Reafons of the different Appearances of the fame Objefta or to give it a plainer Definition, 'tis the Art of reprefenting Objects in a Picture, fb as they would appear if the Pifture were tranf- parent. From this Art of reprefenting or imitating of things feen with the Eye, pro ceeded the Art of Designing, and from thence the Arts of Painting, Limning, Carving, &c. all which, Painting efpecially, have luch de- pendance on Perfpeftive, that it was the Opir nion oi Pamphilus the Matter of Jppe ffe /* that without the knowledge of Arithmetick, Geo metry and the Opticks, that Art could never have been brought to Perfection. °Twas thought proper to gratifie the En- glifh V/orld with a Translation of this Ex cellent Piece. »The Author's Charafter is fufficiently known from his other Perfor mances: And thofe who take thc pains to read this, will find 'tis not unworthy of him. I know no Book upon this Subject, in any Language, that contains fo full and fo plain Tbe PREFACE. a Syfteme of Perfpe&ive, as this I now pu- blifti : And how much we in England are at a lofs, for want of a Performance of this Na ture, is fufficiently known to all Lovers of Painting. I have taken care to render the Tranflation as |uft and Exaft as poffibly I could, choefinf rather to keep clofe to my Author, than to purfue the Embellifhments of Language, which things of this Nature eannot well admit of. TABLE O F T H E CHAPTERS AND Principal Articles. CHAP. I.nP//£ Excellency of Painting, • JL Perfptttive U its foundation. page i CHAP. II. An Explication *f the ufual Terms, and the Principles fuppifed in Per? PpeBive. 1 9 An Explication of fome Terms not belonging to Perfpeffive, yet often ufed in [peaking of this Science. 24 Suppojitions or Peftulata. * 2? CHAP. III. Of the Properties of the SeUions of two or more Plans, which meet or cut bnt another. 40 CHAP, Thc CONTENTS. CHAP. IV. Of the Properties of the Sections of a Picture, and the Plan made by the Rajs that come to the Eye from the vifible Ob ject. 45 CHAP. V. Of the Difpofition oftheObjegs defigned to be reprefented ; and of the fituattm of the Picture, or Perfpective Plan, with re lation to thofe Objects, And to tbe point from whence it ought to be feen, 6$ I. Of the Difpofition of Objects on the Geome trical Plan* 65 II. The fituation andfize of the Picture. 7 J III. Of the fituation of the Eye, or of the point of fight. So IV. Eftablifhing the Neceffity of-PerfpeSive, and anfwering the Difficulties which art framed againft this neceffity. 26 CHAP. VI. How to find the Perfpective of any Object whatever ', its fituation being given in regard to the Picture and the Eye. 95 CHAP. VII. Of Pictures which are not per pendicular on the Geometrical Plan\ which are inclined or parallel 5 which are Jloping in refpetf of the Eye ; and laftly, of thofe which reft on an unequal and irregular ground. I 26 I. Of Pictures that are inclining or leaning 129 II. Of Pictures parallel to the Horizon. 1 j 1 ill. Of Pictures that ftand fideways itf.reJpecT' of , the Eye, *_34 IV. Of The CONTENTS. IV. Of Pictures ut>oti Concave and Convex bo dies, or in fuch as have Cavities and Emi- nencies, 135 V. Of Pictures and Statues made to be plated in high and eminent places. 140 CHAP. VIII. Shewing that the moft import tant Rules for laying on of Colours, proceeds from Perfpective. 1 50 CHAP. IX. A general Obfervation on the pro- ¦ portion of Shadows. 160 CHAP. X. A Conference of ^Socrates with Parrhafus the Famous Painter , and' with Cliton the ingeniom Carver. \ 69 A N A N INDEX ACcidentalPoint ,what 22. 51.96. how to I be found 51. Action, the moment of an Atiion can only be re' prefented 7. 74. Aerial Perfpe&ive , -wJatf 35- J52- J53- Alcamenes the Carver 141. ^//ey of Trees to put in Ferfpeftive 123. Anatomy, Painters ought to know the Anatomy of a Humane Body 15. Angle, things feen under equal Angles,are not al' ways equal 36. 37. Aintonine Column, a fine Monument at Rome 175 foot high 142. appearance, things Con cave pr Convex, appear flat, if feen at a diftance 3.3. what is high ap pears low, and what is tow appears high at a diftance 33. things at a diftance feejn fmaller 34 B. Bafe of the PiiJure, what Beauty of Painting con* fifts in Imitation 91 . 92 C. Curving , its difference ^ from Painting 7, Ceiling appears lower at the farther end of a long Gallery 33, Center of the Pi&ure, what 21. Cilindrick body, howrepre- fented 133. Circle, how to put it in Perfpeclive 112. Cliton the Carver 1 69. Colouring An INDEX. Colouring a confiderable part of Painting t8. Colours, their natural caufe 4. Rules for right ap plying them 1 $0. Colours of Figures change accor ding to the different points of fight 151.154, Columns, how put in Per fpeclive 1 1 9. Concave, things Concave appear fiat, if feen at a diftance 33. Piclures on Concave bodies ought to be feen at a diftance 135.148. Concourfe, Point of Con courfe 48. Concurrence, Point of Con currence, what 21. Convex, things Convex appear flat at a diftance 33. Pi&ures on Convex bodies, ought to be feen at a diftance 135. 148. D. Degradation of the Piclure \vbat 54. how found 70. Difficulties againft the ne ceffity of Perfpeclive an- fmered 86. Difpofition cf Objecls on the Geometrical Plan 6 5 Diftance, Points of diftance whet 34. 98. Diftance of the Point of fight 84. dftcnw cf the Eye, how founc ct. 85- DiVif.v' f a Line, .. ho» found in Perfpeclive 1 06 E. EItvationtvbat 25 Explication of Terms be longing to PerfpcS'm 1 9- 2.4. Eyeyhowit fees,! and 3. is a Point 28. Its fitm. tion 80. What is exatllj oppofite to it, is moft fen fible Si . ta regulate its diftance 85. F. Figuri, to put any Figm in PerfpeQive m. Fleeing Lines, what 2 \ . Fieeing Scales, what 48. how made 115. Fortifications reprefentti Geometrically 27. Frame for finding tk largenefs of a Statue in a high place 144. G. Gallery adorrid with Pih- fters, how put in Pet- fpeilive t2i. Geometrical Plan,what 19,1 what is called Geomt-l trie A1 An INDEX. tricel by Artificers 26. Geometrical reprefenta tion differs from Per- fpeOive 26. Geometrical Scalds, what 48. . Ground Line, what 1 9. H. Height of the Eycjwbat 20. Height of tbe Point of fight is tbe natural height of the Eye 81. Hollow body, bow fut in PerfpeSive 1 24. Horizontal Line , what 19-21. I. Icbnograpby, what 25. Imitating, fainting is tbe Art of Imitating 1 1. Imitation pkafes in Paint ing pi. Incidence, point of Inci dence, what 23. Inclining or Leaning Pi clures 129. L. Leaning or Inclining Pi clures 1 29. Light, wherein it confifts 3. 4. 32. Light of tbe Sun different from that of a Lamp 165. tine, Ground line, what 23 Fleeing-line, what 21. Radial Lines, what 21 - Line of Station, what 20.21. Lineal PerfpcBive, what 3S- 153- IS4- ' Lines, to find their Perfpe ctive, 1 04. 105. toS. 114. Luminous body 160. 161. M. Magnitude offhadows \6\ Mathematicks necejfary t» a Painter 6. Mathe maticks alone notfuff- cient to make a goad Pcrfpeclivefy. Military Perfpeclive,what 27. Minerva'j Statue at A- thens 141. Model of a Picture, how U make it 1 29. Moment, a Painter cm only reprefent tbe mo ment of an Action 8.74. W. Nature tbeObject of Paint ing and Sculpture 1 42. Nature corrected by Pain ters 87. Neceffity of Perfpective efiablifhed 86. Nolle PerfpcQivt, what O.Ob- An J ObjeSs at at a diftance ap pear fmatter 34. Difpo fition of Objects to be painted 65. Oclaedre, how to put it in Perfpective 68. Opaque body 160. 161. Orthography, what 27. P. Parrhafus thePaiyster \ 69. Painters can only reprefent the moment of an Aili- on 7. 74. they have lefs liberty than Ppets 17. 73,74. they correct Na ture 87. Faintingfits de(tgn 6.- its difference from Carving 7. its Perfection, ypbat 1 1 . 1 3. what it is effen- tiatty 10. .11. it differs from Sculpture 75. • Parallelogram,, to find its Perfpective ic8. Pavement appears higher at the further end of a long Gallery 33. Perfpetlive9 what is taught thereby 5 . what it is 1 8. the neceffity of it efta- blifhed %6Jtis the Foun dation of painting 1 8. Perfpetlive of a Point, E wba% 31. of a right lmt\ '; 31. of the furface of an Object' 32. 1 .Perfpective Aerial, what 3^. 152.U53. Lineal, ifbat ibid, at a Birds fight 82. , Perfective 'Scales , what 115. hbw made ibid. Picture, eveiry Picture is a Perfpective 6. "' "' Picture,whdt 1 9: howiitu- afed on tbe Geometrici\ .,_ Plaii'2,o.j3.'ifsfiz.e73i conftBcnd as a Window 78. 79. 80. ' it may ftil be reduced to a. vertisd . fitwition 1 27. Pictures itklining or lean ing j 29. of Pictures pa- rattel to tbe Horizjar^ 1 3,1. of Pictures to be placed on eminent places 140. Pictures on Con cave or Convex bodies^ ought to be feen at a di-i fiance 135. 148. Pilafters, how put in Per fpective 1 1 6. Pillars, how "put in Per- , fpective 116. Point, Point of fight, what 20. its juft diftanceZo. ¦ 84. Point of Concur rence j An INDEX;. ,rence,wbat 21. Point of Incidence 23. Points of diftance 24. 98. to find the Perfpective of a ¦Point 97. 101, 11 3. Polygon, bow to reprefent 133. Principal Point, what 2 1 . Not always in the mid dle of tbe Picture 80. fometimes without the Picture 81. moft com monly in tbe Center of the Picture 81. fome times above, and fome times fce/ow8i.82. Principal Ray, what 20. Profile, what 25. 26. Projection, what 25. Proportion of Shadows, how found 160. 161. R. Radial Lines, what 2 1 . Ray, Principal Ray, what 20. every Rap is a firaight Line 28. Rays, what 29. Rays by which we perceive a Line make a Plan 30. Retina of the Eye, what 2.3. Row or AUey of Trees, to put in Perfpeclive 123. Scales Geometrical, what 48. Fleeing Scales, what ibid. Perfpective Scales, what, and how made 115 Scenography, what 26. Sculpture differs from Painting 75. Seat oj Objects, what 23. Point of SCat, what 23. 68. Semiteint, what 151* Shadows, general Obferva tions on their Propor tion 160. 161. Sight,how it is a. 3. Point of fight, what 20. Situation of Ob)ects, how determined 2 1 . Situa tion and Size of the Pi cture 73. Situation of the Eye,or Point of fight 80. Situation of a Pi cture may be reduced ta Vertical 127. Secrates bis Conference With Parrhafus andCli- ton, 169. Spiritual part of Painting, what 66. Spirit may be reprefented 171. 172. Station, line of Station, what 20. 2 1 . Statues An INDEX. Statues to put in Perfpe- fpective 1 23. T. Tacquet'i Propofition falfe $6. 37. and 142. Teint, what is meant by it Terms of Perfpective ex plained 19. Trajan'iCo/«>»w at Rome 143 foot high 142. W. Windows may be confi der 'd as Luminous bo dies 162. Workmen profit by tbe Con verfation of tbe Learned 170. 174. Explanation of the Notes ufed in this Treatife. Equality "7 C-4- I/tors'? This^ fignifies rThis E fignifies Proportion- ERRATA. PAge 10. Line 17. for feen : 'tit tad feen, 'tis, p, 14. 1. 3. for ; thefe an reprefent read etn reprefent. p. 86. 1. 2. for twi read mi. p. 112. 1. 12. for Flan read VUin. p. 124* 1.1.2. for high read height, p. 141. 1. 21. for Alconunei read Ala- menes. p. 14^. 1. 22. and p. 146. j. 5. for vtuh read Arch. p.146. la & 2. for Belie read £//>*. p.J47. 1.13,19,2, & 26.^ for Vault read Arch. p. 148. 1. 12. for Helie nudEliuh. l 13. for her read him. TREATISE O F Perfpective. CHAP. I. The Excellency of Painting; Per- fpetlive is its Foundation. T 1 0 reprefent updrt Cloath, what is not there, as Cavities and Eminences, where all is Flat; and. Diftances when every thing is Near; is a Performance that merits Admiration. 'Tis an Effect, and at the fame time, a Proof of what the Eye (to fpeak Philofophically) doth not fee, but only the Soul which forms to it feif different .Images of Objects, according to the different B Impreffions 1 A Treat if e of Perfpective. Impreffions of the reflected Light on the Eyes. Nothing is more difficult to be ex- prefTed', than the Nature .of thefe Images ; whither it be that the Soul forms them out of its own Subftance, and fo fees it feif, as it Were, transform'd into all things 3 or if it fees thefe Images in a Subftance above it, which being the Principle of all Beings, can reprefent all. By this Advance, I mean only to difcover a Difficulty, on which it would be requifite to make ferious Reflections ; for it concerns not the Subject I intend to treat of, and therefore it's fufficient at prefent to confider, that the Operations of Nature being Simple and Conftant, the like Impreffions in the Or gans of Senfe, ought to be followed with the fame Sentiments 5 fo that as often- as the Eyes are ftruck after the fame manner, the Soul muft have in its view the fame Images, what ever be their Nature and Origine. If (I fay) the Rayes, by which we fee a Picture, pierce the Eye ;n the fame order as if they came from the Objects themfelves, tho' we fee but the Painting ; and if the fmall luminous Bo dies which com pofe thefe Rayes, fhake and move after the fame manner the Retina, that is, the fmall fixings of the Optick Nerve, which line the bottom of the Eye, then the Picture muft have the fame Effect as the Ob jects ATreatifeofTerfpeftive. 3 jecls themfelves : For the Optick Nerve fur- nifhes the bottom of the Eye with a vaft number of fmall Filaments into which it divides it feif, and makes what we call the Retina : 'Tis there that the Rayes do in fome meafure paint the Features of the Objects from which they are reflected ; as when a Chamber is fhut fo clofe, that there is no paflage left for the Light, but through a Per- fpective-glafs, the Rayes paint the Objects that are without, on a 'piece of white Paper, if it be oppofed to the Glafs. This Cham ber reprefents the Eye, and tbe Paper the Retina. The new Philofophy fuppofes the World full of fmall Bodies, and that it is their Action or Preflure, that makes us fenfible of the Light. Thefe fmall Bodies, in removing themfelves from that which caufes their Re flection, prefs thofe that oppofe their Motion, and thefe in like manner prefs thofe that fol low, by a Communication of Motion made in a direct Line from the Object to the Eye. 'Tis this Motion (according to our Philofo- phers) that informs the Soul of the Figure of the Object, as the Staff does a Blind-man of the Nature of things, by the Impreffion it makes in his Hand, as it is thruft forward or pulled back. So the Impreffion of the Luminous Bodies on the Retina, occafions B 2 the 4 ATreatife of Perfpective. the Soul's having the Idea of the Object, thai: caufed the Impreffion. For, we muft know, that the Picture made by the Rayes, is only the Motion they impart to the fmall Strings of the Optick Nerve: Now, fince the Pref lure of the Matter which refts on the Optick Nerve, is the Natural Caufe of the Senfe of Light and Colours, becaufe Red appears ftill Red, both with a ftronger and weaker Light ; we may conclude the different Celerity of Shakings or Vibrations of the Matter which prefles the Eye, to be the only caufe of the Variety of Colours ^ different Colours arc in the fame cafe as different Sounds, for the Sounds change not becaufe of the variety of the force whereby the Air is agitated in the Motion, (for inftance) of the Strings of a Lute , but becaufe of the diverfity of Readi- nefs and Celerity in the Vibrations, the Pa rallel holds in all Points, bating that the Action of the Air conveys the Sound, and that of a Matter yet more Subtile, the Colours. s Thus 'tis the difference of the Shakings or Vibrations of the Matter which preffeth the Eye, that adds the Colour of the Objects, to the Features of the Image, which the Rayes paint in the bottom of the Eve ; that is to lay, that 'tis the difference of' the Motions, that this matter takes on the Surface of the Objects, A Treatife of P erf pe clive. 5 Objects, from which it reflects it feif, that ' caufes different Senfations ; or 'tis the occa fion of thefe Sentiments that we call Colours; Juft as the Soul is fenfible of different Tafts, according te the Variety of the Food that affects the Fibres of the Tongue. However it be in this New Philofophy, it is agreed upon, that a Picture, when feen from a certain Point, reflecting the Light in the fame mar.nsr as the Object it felt' would do ; ought to have the fame Effect as the Object whofe Features and Colours it repre sents. That is to fay, When it fends back the Luminous Rays in the fame Order and Difpofition, and with thofe very Motions that give the true Sentiments of each Colour, the feme will the Effect be. 'Tj&iis is what's taught by that part of the Mathematicks call'd Perfpective, which I defign to treat of. A Picture may be conlider'd as an open Window or tranfparent Glafs, through which the Eye which is fuppofed to be at a certain Point, may fee the Objects reprefented by the Picture. Now by the help of the AUtk- maticks, the paffage of the Rayes which ren der the Object Vifible, may be trac'd in the Picture or tranfparent Glafs. This paffage being marked with fuitable Colours, the Picture reprefents the Features of thefe Ob jects, their Form, their Colour, and in a B 3 word, 6 A Treatife of Perfpeclive. word, all their Appearances. And fince it makes the fame Impreffion, the Soul muft have the fame Images in its view, and be thought to fee the fame things. Mathematicians draw only Lines, they can not finifh a Picture ; And on the other hand, Painteis cannot begin it, without a regard to the Rules taught by the Mathematicians. Every Picture, is a Perfpeclive, fo that what is taught in this part of the Mathematicks, is the Foundation of Painting, which ought to be well adjufted, for all Painters do not agree to it. The Defign of Painting, is to reprefent on a flat Body, as Paper, Cloath, or a Wall, whatever is defired. This can never be done, if the view of the Pill are makes not the fame Impreffion on the Eyes, as if they faw the things themfelves: And this is what Perfpective does exactly. Painters that are ignorant of this, can never fucceed but by chance ; for in Painting by the Eye in Imi tation of Nature, as they do, 'tis impoffible to form their Features fo juft, or range them fo exactly in their true places, as the Rays of things fuppofed beyond the Picture, would do other Features in piercing the Picture if it were tranfparent. For a clearer Conception of thefe things, Let us confider, that there is a great diffe rence A Treatife of P erf pe clive. 7 rence between Carving and Painting. A Statue that ftands by it feif may bd feen on all fides, it fhews all its parts. As for Ex ample, the Statue of Hercules in the Palace of Far neff, reprefents the Body of Her elites intire ; ip may be feen all round, and viewed from different parts ; it is not the fame with a Painted Pifture, which is terminated by a fingle Stroke, reprefenting only and precifely the Circumference in which the thing that is Painted appeared to the Painter that defign'd it, and in which he defigns it fhall appear. So that this Circumference is different ac cording to the different points of fight ; and cannot be proper for reprefenting the fame Object feen from another fide. This is the reafon why all tlie Draughts of the Hercules of the Palace of Farnefe are not alike, becaufe this Statue was defigned by different Per- fons, who did not all behold it from the fame fide. Let us here confider, that Stones and other Inanimate Matters, can keep the fame Situa tion a long time, whereas all that hath Life, is continually changing,and in a perpetual Mo tion. The moft ingenious Painter cannot reprefent thefe Changes ; all he can do, is to paint the Moment of an Action ; that is to fay, the Situation of every thing, the Motions, the Poftures proper to every Aftor, and the B 4 Character 8 A Treatife of Perfpeclive. Character of the Paffion with which he was animated in the Moment of the reprefented Action. Nor can he thus Paint feveral Actions in one Picture. ' In Drawing the Picture of a Perfon fup- pofed to be alone, it fuffices to obferve in his Vifage, and in his Countenance, the Cha racter of his Genius, and of his ordinary In clinations, his Phifiognomy, or the Strokes of his Face, which are peculiar to himfelf; but in reprefenting an Action of Confe- quence, to which many contribute as Actors, or Witneffes, every one, according to the part he undertakes, fhould make appear by his Eyes, and by his Pofture, what he is, thinking that Moment. This is the Moment a Painter can reprefent \ this is the Point where all his Work tends ; his niceft Point, I fay, is, that having placed at a certain Sta tion, him that is to confider his Picture, he fees the fame thing as if the Cloath becoming in that Moment Tranfparent, he faw the Aaion it feif, which is the fubjed of the • Picture. This being well confidered, it is eafie to eftablifh the Neceffity of Mathematical Per* fpeSlive. It is impoffible to fee preeifely the fame things, from two different Stations or Points of Sight : The Eye being placed in a certain Point, from whence it fees at once a whole A Treat if e of Perfpeclive. 9 whole Action, perceives nothing but what is oppofite to it. If it fee the Front of a Perfon, his Back is hid ; it cannot at the fame time fee above and below the fame thing. The Line which terminates what the Eye difco- vers, is fo peculiar to what it fees in the Situ ation it is in, that there would be a neceffity of drawing a new the out-lines of a Figure, if the Tfye were moved to another place ; for it is evident, that the Figures of things alter, as they are feen doping, fidewife, or in Front j they become likewife fmaller or bigger, as they are more or lefs diftant from the bafe of the Picture ; fo that a Piflurc can be only made for one fingle Point or Station. 'Tis impoffible to find exactly by chance, all the out-lines of a Figure, and the bignefs fuitable to it in the place where it is defigned to be ; This cannot be done true by Juft and Infallible Rules, without the Mathematicks : But after all, one cannot err in thefe things without miftaking grofly •-, for, once more, can a Figure be feen behind and befdre at the fame time ? W hat is feen at a diftance, hath it the fame Appearance as if it were near? Is it reafonable, to give a Figure almoft the whole height of a Column, which, in the Object: reprefented, is 20 or 40 Feet, when at the fame time the Natural Heighth of this Figure io A Treatife of Perfpeclive. Figure is not above 5 or 6 ? Thefe are ne- yerthelefs Faults very ordinary to Painters, particularly to thofe who Copy the Works' of fome great Matters, not taking notice that the Painter, whofe Works they fteal, hath given a Circumference to his Picture, which will not be convenient fos it, in the place whither he carries it. Many imagine Perfpective to be ufeful only in reprefenting Walks, Trees, or Architecture; becaufe they diftinguifh it only by a Con- courfe of Lines to one fingle Point. But fince a Picture cannot have its Effect:, if the Rayes, which it reflects, come not to the Eye, in the fame order as if the Cloath let the Light pafs through, or that by the opening of the Picture,the things themfelves were feen : 'Tis the paffage of the Rayes that is fearched for in the Painting, as well as in the Perfpective, that ought not to be fo diftinguifhed. To in- able us to judge the better, let us fee what Painting is effentially. We may fay of Painting as of Eloquence : There are general Rules for Writing and Speaking Judicioufly and Nobly ; but as be ing Mafter of thefe Rules, is not fufficient for Speaking and Writing on all forts of Sub jects, on Philofophy, on the Mathematicks, or on Theology ; and what Eloquence foever one may have, he can never fpeak: reafonably on a A Treatife of Perfpeclive. 1 1 a Subject he knows little of. So a Painter cannot reprefent but what he knows, tho' he fearch to the bottom of his Art. For Ex ample, He cannot reprefent a Battel well, if he is ignorant of the Method of Drawing up an Army; nor a Sea-fight, if he be no Sea man. But this does not imply, that a good Painter ought to be both Soldier and Sailer. 'Tis true, there are fome Subjects that Pain ters fo commonly treat of, that it feems effen- tial to their Art, not to be ignorant of them. Could a Painter be Excellent, if he knew not a Man ? I mean the outfide of a Humane Body, and what may appear on that out-- fide; the Veins, the Mufclcs, and the Ten* dons ; he ought therefore to be perfectly well acquainted with the Anatomy of the out fide of a Humane Body. Thofe who apply themfelves to the Painting of Beafts, ought to raake the fame Inquiry. To paint a Horfe well, 'tis convenient to know the Anatomy, and moft efteemed Proportion of his parts. In fine, Painting is not effentially limited to reprefent any particular Subject. It is in general the Art of Imitating; and its Per fection is, that the Imitation is fo Natural, that the Picture makes the fame Impreffion, as the Object it feif that the Painter would imitate. There lies the Beauty of his Art: 'tis the Addrefs with which he imitates what he 12 A Treatife of Perfpeclive. he would reprefent,that makes him efteemed ; for we are fometimes charm'd to fee that in a Picture, which would be. frightful if we few it really. A Serpent cauies Fear, but its Picture, if weH done, is charming : So that 'tis the Skill of the painter that pleafes. Now the Imitation is not perfect, if it have not the fame Effect as the thing it feif; that fo the Eyes may be agreeably deceived. Therefore fince a Picture can have only one point of fight, and fince each Figure expos'd tq view, hath a certain Circumference pecu liar to it feif, in relation to the point from whence 'tis fuppofed to be feen ; and a cer tain bignefs, which depends on the diftance in which it is reprefented, we muft of ne ceffity have recourfe to the Mathematicks, without which it cannot be done to the ut- moft Precifenefs. It may be faid, there are Pictures that pleafe without that Nicety. I own it ; but whom do they pleafe ? None but fuch as do not compare them with the things the Painter would have them- repre fent. 'Tis the Refemblance of Truth that pleafes in painting, as hath been faid. How can this Refemblance exift in a Picture, when every thing confutes it, when the Ground he reprefents is too large or too fmall for the Actions that are fuppofed to be done there ? When what ought to be feparate, are hud dled ATreatife of Perfpeclive. i% died together ; and what fhould be joyned, are remote? When all is too great, or too little, and nothing hath its juft Meafure? Painters, after having made their Figures, do generally adorn the bottom of their Pifture with a piece of Architecture, rich and fine in appearance, for if we examine the Plan in order to find, for Example, the foot of a Column, we fhall find it refts on the head of fome Figure. Can fuch Pictures pleafe indifferently ? We fhall now examine its Perfection, and the Enquiry is neither Vain nor Ufelefs ; for the Rules of Perfpective are as eafie as they are fure. Let us then conclude Perfpective and Paint ing to be the fame thing ; fave only that Perfpective is made toconfift, in finding Geo metrically, ( as I do in this Treatife ) the Points, at leaft the Principal, through which the Rayes pafs, that would fhew the Object that is painted, if the Picture were Tranfc parent. I fay, the Principal Points, for it would be too tedious to fearch for all with Rule and Compafs. The Eye alone hath an infinite Number of Features proper to it feif; and befides fhat, two Men have not their Eyes intirely alike, the fame Eye can change as many ways, as the Soul can have different Motions. Only they who have ftudy'd Nature, and by a fearch after all its Cha- 14- ATreatife of Perfpeclive. Characters, have acquired that Facility of imitating what they fee, or what they con ceive ; thefe can reprefent things as they are. All a Geometrician can do, is to determine the Greatnefs and Situation of Figures in a Picture. That's what belongs to Perfpective, the reft is the Work of the Painter, or of one who is wont to imitate or defign what is feen ; efpecially to mark well the Out lines. 'Tis in this they exercife themfelves in Academies, who defign after a Copy, or in particular, after a Relief-work. A Skilful Painter having once, by the help of Perfpective, found the Pofition of fome Points in the Circumference, which he endeavours to compafs , finiflies it eafily ; which is impoffible to thofe that cannot de fign. There are a thoufand fine Strokes, of which one may find feveral Points, and yet not be able to finifh them. Every Motion hath a proper Pofture ; every Paftjon hath a Character in the Vifage ; every Age, every Sex, and every Condition, a certain Air , which he muft be a Judge of, and know how to exprefs 5 otherwife, what is done, reprefents nothing that hath Life, all is dead ; for the Air and Features of a Body full of Life, are very different from thefe of a dead Body, However it may refemble fomething that had once Life, there are ftill the fame Features A Treatife of Perfpeclive. 1 5 Features but much changed. Nothing, ,b,ut a long Exercife, and an extraordinary Ge nius, a Curiofity and a Nicety more th^n ordinary, can make one fenfible of the dif ference. . , Let us add, that tho' the Art of painting copfifts in Imitating, yet a Painter that can only Imitate what he fees, is no Artift. , What he reprefents fhould be Beautiful and Fine, and there is no Beauty but what is Imperfect- He muft therefore imagine what is not, and form a Refemblance finer than the fineft things can be found ; for when he forms to himfelf an Idea, it may happen that nothing can be found intirely refembliug it. He ought therefore to dive into what things may hap pen, in all the Conditions in which they can be apprehended. To reprefent the Pofture of a Body that one fees before him, 'tis not requifite to be an Anatomift ; but without the Knowledge of Anatomy, 'tis impoffible to reprefent correctly a Pofture, which he con ceives but does not fee. The Science of a Painter ought to be infi nite, if he undertakes to meddle with all Sub jects s but he undertakes a large Task, when he confines himfelf to Man ; that alone is fufficient to imploy him. He cannot paint all the Interior Motions that are hid from him; £ut fince thefe Motions have their Signs 1 6 ATredtife of Perfpeclive. Signs on the Vifage, he may ftill reprefent them. Thofe who have writ of Painting, have done this at large '. They have examined the Proportions of a Humane Body, as to Age and Sex, and as to the Condition to which Nature renders them moft proper. For it is evident that there is a certain Difpo fition of Body neceflary to make a Wreftler, which is not always found amongft thofe that are proper for Government --, fo that we muft judge of a good Proportion, with rela tion to the Quality of the Subject. This Search after fuitable proportions, was the Study of the great Matters of Antiquity : A fingle Statue, or the Painting one Figure in a Picture, imployed a part of their Lives : They were not fatisfied to imitate only what they faw ; if they painted a Wrefiler, they formed the moft noble and perfect Idea that was poffible, of a robuft well-made Body. To this end they confidered all the Wrejllers, meafured them exactly, and taking from each what appeared to be perfecteft, in re lation to their Defign, they formed that per fect Idea of an Agile, Robuft, and Well-pro portioned Body. If they made a Statue, or Picture, of Venus, that is to fay, of a fine Woman, they made the like Search on all Bodies, where they perceived the Features of a%are Beauty. Since A Treatife qf Perfpeclive. 1 7 . Since I fpeak of P.iintirg more as a Ma thematician than as a Painter, 'tis not. my bufinefs to advertife Painters of Hiftory, that they ought (as Poets in a Comedy) to be careful of the Unity of Action, Time and Place, and for that reafon to paint but one Action in one Picture, with what relates to i^ and is neceffary to denote it. A multitude of Perfons caufes Confufion,this may be avoided by making none appear but who are ne ceffary towards the execution of that Action, in their convenient Poftures : All ought to be attentive Witneffes, and fhew in their Coun tenances the Motions with which they may ¦ and ought, to' be animated, in relation to their State, Sex, Age, Quality, and to their part in the Action,which is thefubject of the Picture. Above all things* Likelihood and Decency muft be obferved ; Painters have much lefs liberty than Poets, for a Poet may allow 24 hours to the Action he reprefents, but a Pain ter, can only reprefent the Inftant of an Aclion,and what is feen at one fingle look. This would need a more ample Explanation, if I defigned a compleatTreatife of Painting 5 I fpeak but a^s a Mathematician, and therefore cannot treat of Colours,of the matter of which they are compofed, nor of the manner of mixing them, fo as to imitate the natural Colour' of the Objects to be painted. There € ate 1 8 ATreatife of Perfpeclive. are Secrets for preferving Paintings always clear and lively. Colouring is a confiderable part of Painting. The Mathematicks make Abftraction of the fenfible Qualities. Per fpective then, which is but a part, can be no other than an Application of Geometry , to find the paffage of the Luminous Rayes, that will make the things themfelves appear, which are fuppofed to be behind the Picture, and are to be there reprefented. perfpe&ive, I fay, is the Foundation of Painting, but it is not fuScient to make an accomplifh'd Painter, I am far from pretending it : The Idea which I have given of Painting, (hews that I have other thoughts ; but after all, what I have faid, will ferve to prove, that Perfpective is ufeful to a Painter, that 'tis that which regulates his Defigns ; that without it he works but at random, and cannot keep up to the nicety of juft Meafures. In the firft place, we will fee which are the Terms ufed in Perfpective, arid which are its Prin ciples. CHAP. A Treatife of Perfpeclive. 1 9 CHAP. II. generally it is all can be feen at a fingle view, fuppofing the Eye infinitely diftant, or greater than the Object. Orthography is marked with perpen dicular Lines, reprefenting the Heights pf an Edifice. This is what's underftood by the Word Orthos, which fignifies Right. Orthography ov Elevation, is fometimes con founded with the Profile. Profile is an Eleva tion, but this word implies particularly the Cut of a Building, which fhews the infides and 16 ATreatife of Perfpeclive. and outfides that are behind the Plan, which makes the Cut; and at the fame time all that ¦is cut by the Plan, as the thicknefs of Walls, Timbers, and all other Objects fo cut, the Eye being ftill fuppofed infinitely diftant. X is the Elevation or Orthography of part of a Pilafter with its Cornifh , and Z is the Pro file or Cut of the fame part, which fhews the thicknefs and heighth of its parts. It is particularly the Line which makes this Cut or Section that's called the Profile. Ste nography is likewife the fame thing with Orthography , 'tis the reprefentation of an Object elevated on the Geometrical Plan, with its fhadows as it appears to the Eye. (See Plate 2. Figure 1 &; 2.) Artificers call Geometrical whatever keeps its proper Meafures, but Figures put in Per fpective, and feen from a certain determined diftance, change, and have no more the fame Meafures. In the Perfpective of a long Gallery, the laft Squares are the narroweft, and the Columns fmalleft and fhorteft ; fo they diftinguifh betwixt what is Perfpective, and what is Geometrical, or betwixt a Per fpective and Geometrical Reprefentation. As for Example (fee Plate 3. Figure 1, 2.) the fame Column reprefented two ways, X Geo* metrically, and Z in Perfpective. You fee that A the Geometrical hhnography, keeps both its Meafures late Paye uS The Proftui anj OkcrHOGB^pmE of 'the, Cortiifli cmoL C apitail of co 'late 3 ,";V> JS^ hQe^Ke^twe of whicW 34 A Treatife of Perfpeclive. which is then in E, appears at the point F, and therefore higher than it is ; as that which is in B, appears in C, and confequently lower. The Figure fhews likewife, that the equal parts A B and D E, ought to appear unequal ; and that the moft diftant have the leaft appearance : Becaufe the Rayes of the fartheft objects prefTing the Eye lefs, it judges not the object to be in the place where it is 5 B G appears H C, and therefore lefs than it is. It is the fame with E L, which appears as F K, lo that at a diftance equal parts ap pear unequal, and th$ fartheft appear the fmalleft. This is what Painters cannot be ignorant of. therefore in Painting of things, as they fee them, they never fail to reprefent the molt rugged bodies fmooth and even, and all their parts confufed, if they fuppofe the Eye at a great diftance. They notice nothing di- itinctly in reprefenting a very remote Figure. They raife whatever is below the Eye, ac cording to the diftance, and decline all that is abo\ e it. They reprefent, I fay, all that is remote, lefs than Nature. This is what they call Degradation of a Picture. We have fbewn, that it is the manner after which the Luminous Bodies ftrike the Optick Nerve, that makes the different Sen timents of Colour; fo that as the action of thefe A^40;i^^::~~-~~ — <7? X ..„->;S'"*-'"" itj.3. A Treatife of Perfpeclive. 3 5 thefe bodies, keeps not it feif all cntir^when they remove from the bodies from whence they are refletted,fo'they ought not to appear fo lively coloured as if the Eye were near* Remote bodies appear without colour; there fore they -ought tb have none in a Picture, or at leaft it ought to be weakened ; and in adjufting this faintnefs, lyes the Myftery of Paintings for tho' -an object be coloured with the fame colour, as to its kind, yet its parts are diftinguifh'-d by the faintnefs or fullnefs of the fame colour, according to their fitua tion, as we fhall fhew more particularly in the laft Chapter of this Treatife. Painters call this Diminution of the Tincture or Col ours, Aerial Perfpective ; and the Diminution of Lines which reprefent others remote from the Picture, they call Lineal Perfpective. It is this only I have undertaken to treat of. Ne- verthelefs I fhall have occafion to reduce Aerial Perfpective in fome meafure under this Head, and touch lightly upon it. Many advance for an Axiom, a Propofi^ tion which on feveral occafions is falfe, and capable of caufing us to commit greatMiftakes in the Practice of Perfpective. They pretend, that things feen under equal Angles, have the fame appearances, or appear of an equal fize % from whence Father Tacauet concludes*. that if a Statue,or right Line,appearing equal D 2 to 3 6 A Treatife of Perfpeclive. to B C, (fee Plate 6. Fig. t.) were erected on the Column B D, then a Line drawn from D to A, and the Arch e f taken equal to b d, the line A E drawn through/, would give the line D E, which in appearance would be equal to B C, becaufe feen under an equal Angle. This Propofition holds not always true, it holds when the objects are near ; but at a di ftance things of the fame bignefs have diffe rent appearances according to the variety of the Eftimate we make of their diftance. For fince we fee in a right Line, objects feen un der the fame Angle muft appear fo much greater, as they are judged to be diftant- We may be faid to have within us, a kind of Na tural Trigonometry $ for when we view an object, there is a Triangle form'd by the Rayes of that object, and the diftance of our Eyes$ which being known with the two Angles of the fituation of the Eyes, deter mines the fenfation of the remotenefs of the object. But the Interval of our Eyes is too fmall a bafe, to judge of the diftance of re mote bodies; becaufe the Eyes change not fenfibly their fituation, to fee an object at a thoufand Paces,or to fee it at 100000. Where- tore we cannot judge of the diftance of ob jects by Natural Trigonometry, but by the ap parent bignefs of interpofed bodies. There- ATreatife of Perfpeclive. 3 7 Therefore the Sun and Moon appear much lefs, and nearer to us, when we fee them above our Heads, than when we fee them on the edge of the Horizon. I fhall add no thing more here, referring the Reader to what Father Malbranche hath faid in his Search after Truth. To demonftrate the falfehood of this Pro- pofition, which Father Tacquet takes for an Axiom, That what is feen under the fame Angle, appears equal. Suppofe the Eye to be in A (fee Plate 6. Fig. i.) and a Statue to be erected on the Column BD, that accord ing to him fhould appear equal to B D. If the Angle b A d were of 45 deg. it behooved, according to the pretended Axipm, that the Angle DAE were equal to the Angle b A d, to the end that D E might have the fame appearance as D B , but 'tis not poffible, in this cafe, that thefe Angles can be equal, tho' DE were prolong'd ad infinitum j forA^ being fuppofed parallel to BE, the Angle DAE muft be always lefs than the Angle B A D, which is fuppofed of 4 5 degrees. But allowing it to differ a little, even then D E would be almoft infinite. Neverthelefs, ac cording to this Axiom, it fhould appear lefs thanBD, which is contrary to experience. We fhall fee in the fequel, what may be done in placing an objeft in a high place, D W and 3 8 A Treatife of Perfpeclive. and preferving the appearance of its natural bignefs, Before we finifh this Chapter, we muft examine a Difficulty that fome make againft this Suppofition \ That the place from whence the Picture ought to be feen, is but a Mathe matical Point. They arc deceived, fay they, who confider the vifual Point as a real Point ; becaufe the two Eyes that fee, are not one Point, What fhall we anfwer to this Diffi culty ? Why, in the firft place, thofe that examine a Picture nicely, ufe but one Eye. In the fecond place, The fmalllnterval that is betwixt the two Eyes, may be taken for a fingle Point, in refpect to the remotenefs at which the Picture ought to be feen. Others anfwer, That they fee but with one Eye ; That the two Eyes act only alter natively ; That one alone fees ; and tliat fo the Ocular Point, or place from whence the Pifture is feen , may be fuppofed a fingle Point. They go upon Experiments, which if tryed, fhew the quite contrary. For let X be a black Ball, (fee Plate 6. Fig. 2.) and 7. a white Table, if the Eye B be fhut, and the Eye A ufed, it fhall fee X in D, as B will fee it in C, if it look to X alone -, , now the two Eyes A and B being open, they fee X in £ ; which I have tryed.feveral times. •. "'it JD Qricrrir Paye38 >tg ATreatife of Perfpeclive. 39 It is not then true, that one Eye only acts ; or that we fee only with one. j. But as I have faid, this hinders not, but that we may fup- pofe, the Point of fight, to be truly a fingle Point; that is to fay, that the two Eyes of him who is in a place, from whence the Pain ter would have him confider his Picture, may be taken for a fingle Point. A Picture feen too near, hath not the Effects it ought to have; and at a diftance, the Interval of the two Eyes is as nothing in refpect of the Picture ; as experience fhews, in what we have been faying ; for if the Ball X be at a great diftance from the Eyes, and near the body Z, which ever Eye is fhut, or kept open, whither the object be beheld with one or both Eyes, the right or the left, X fhall ftill be feen in the fame point of the Table Z, without any fenfible difference, D 4 CHAP. 4

and fo the Angle D G I, or F G I, is equal to C B L, according to what hath been demonftrated in Prop. 1 1 . Chap.y. COROLLARY. The Perfpectives of lines equally inclining on the Geometrical Plan, are parallel. Thefe lines are all fuppofed parallel to the picture. By the preceding Demonftrations they make the fame Angle on the ground line, and con fequently they are parallel, according to what Jiath been fhewn in the Elements of Geometry. E 3 THEOR. 54 A Treatife of Perfpeclive. THEOR. X. Two or more equal lines being perpendicular, or equally inclined on the fame fide, and on the fame line perpendicular to the Picture, their Perfpectives are betwixt two lines, which ter~ minate at the Principal Point. DEMONSTRATION. Let there be on the fame line H K, the equal lines L H, I M, and N K perpendicular or equally inclin'd, the Perfpective of H K, is in the line D A, which terminates at the principal point A (Theor. 4.) then the Per fpectives of the points H I K, common to thefe lines, and to the line H K, are in A D, at the points D, C, B ; and becaufe H L, I M, and KN are equal, perpendicular,/ or equally inclin'd, on the fame fide, their tops' muft be in one line perpendicular to the Picture, to wit in L N, whofe Perfpective is in G A, which terminates at the principal point A (as by the fame Theor.) and confe quently L, M, N; the tops of thefe lines and common ppints to tfic line L N, "are in the line G A, to wit, at the points G, F- E ; which was to be demonftrated, (fee Plate 10^ fig. 2.) -.. . a* [Mcuifo ^~Pcje Sj. ($fg: a 1 ATreatife of Perfpeclive. 55 COROLLARY. Several lines being given, equal and perpendicular on the fame line as on HK,after having found the Perfpeaive of the firft line H L, and the top or bafe of the others, the reft is eafily found. For the Perfpectives of thefe lines are pa rallel (Theor. j & 8.) then by drawing pa rallels to the line D G,the Perfpective of H L, by the points Dr C, B, or G, F, E, between A G and A D, the Perfpective of H L, I M, K N, are found. THEOR. XI. Tbe Perfpective of a line being a line ; Ifay^ the parts of the Perfpective of a line parallel to the Picture, are proportionable to thofe of the line, whofe Perfpective it is. Suppofe B D parallel to the picture X, and5 divided in the point C (Plate n. Fig. i.) its Perfpective is E F ; fo E is the Perfpective of B, F of D, and G of C. Now E F is parallel toBD; thereforeasDCistoCB, foisFG to G E. COROLLARY. If the parts B C^and C D of the line B D are equal, the parts EG and GF of the Per- Jpective E F, are likewife equal. This is evident. E 4 THEOR. $d" ATreatife of Perfpeaive. THEOR. XII. The Perfpective of a Figure, parallel to the Plan of the Picture, is fimilar tp that Figure. ..-'¦¦ DEMONSTRATION I. The Angles comprehended between the fides of the Figure, ^are equal to thofe of the Perfpective, becaufe all the lines of che Per-. fpective make the fame Angles with the ground line, as thofe made by the Lines of the figure, (Theor.- 9.) So the Figure of the Per fpective is fimilar to the vifible Figure. DEMONSTRATION II. - The Rayes that come from this Figure to the Eye, make a Pyramid, of which this Fi gure is the bafe. The Picture is fuppofed pa rallel to this Figure ; therefore its Section with the Pyramid is fikewife parallel to this Figure, and confequently fimilar, as is dempnftratecl injny Elements, Now this Section is the per fpective of the faid Figure, therefore the per fpective of a Figure parallel to trie Picture, i$ fimilar to that Figure. COROLLA RY I. The Perfpective of a Square or a Circle, &c *^ rallel to the Picture, if a Square or a Circle. This is a neceffary Confequence, becaufe in diis cafe the Figureof the perfpeaive js fimjlar * CO. A Treatife ofPeffpeffive. t,j COROLLARY II. The. Perfpeaive of the parts of the- Front of * piece of Architecture, that are on the fame line, keep the Proportions of the parts ef the faid Front, COROLLA.RY III. 4»ypart of the Perfpective of a piece of Archi tecture, or its Diminution, being known, the re ft. i$ eafily. found. Make only a Scale^f Diminution. If (far in stance; the Architecture is diminifhed half in the Perfpective, then take half of each part, THEOR. XIII. The Perfpectives of equal parts of a Line, per pendicular to the Picture, are unequal, v Suppofe B E a line perpendicular to the Picture, and divided equally by the points C and D (Plate i 1. Fig. 2.) the Perfpective of this line is F G; which we muft prove not to be equally divided. DEMONSTRATION. If B E and F G were equally divided, then they would be parallel 5 but fince E B is per pendicular to the Picture, in the Plan of Which is F G, they cannot be parallel ; therefore not being divided according to the fame Propor- tion,the parts of the Perfpective F G,eannot be equal, as thofe of B E are : Which was, &c. *¦ L E M- 5 8 A Treatife of ^Perfpeclive. LEMMA. The bafe WWW of the Triangle ABE, being di vided into equal parts ; I fay Firft, of the Angles of which the equal parts are theBafes, the higheft are the leaft : Secondly, in the Trir angle FAG, the -bafe F H of the fuperior Angle, is lefs than H I the bafkofthe inferior. DEMONSTRATION. Firft, The lines A fc, A C, &c. {Plate^i i. Fig. 2.) become longer as they are more "di ftant from the perpendicular A M. So A E fhall be greater than A C. Then if the angle; E A C were equally divided, Euclide, Book 6. Prop. ?. A E, A C : : ED, DC; then E D Would be greater than D C ; but being fup pofed equal, the fuperior angle E A D ought to-be fmaller than D A C, or F A, H than. H A I, and fo of the reft. Secondly, All lines, as A F, A H, &c. as, they are more diftant from the perpendicular' A P, become longer 5 fo A F fhall be lefs than A I. If then in the triangle F A I, the angle A be parted equally, Euclide, Ubi 6. Prop 3. F H, H I : : FA, A I. .Then FTI would be lefs^han H I, for that reafon, and becaufe the angks F A H is lefs than H A I ¦ which, we have demonstrated (Plate 1 1 .Fig.2.) THEOR. PcujeSd lLate~'ii ATreatife of Perfpeclive. 59 THEOR. XIV. The Perfpectives of the parts of the line B E (the fame Figure, and the fame Cafe, as in the foregoing Theorem) that are remotest, are fmalleil. This has been proved in the foregoing Lemma. V- THEOR. XV. The farther Objects are from the Picture, their Perfpectives are the fmaller. If the line B E were infinitely prolong'd, and ftill divided in equal parts (Plate ii. Fig. 2.) the laft of thefe parts would have the leaft Perfpective. So the Perfpective of that object which is the bafe of the fame angle in the Picture, is leffer. It is evident (fame Figure) that if the Eye A draw back from the Picture X, or if B E be placed at a greater diftance ; as the Rayes A B and A E grow longer, the angle B A E becomes lefs 3 and the fecond bafe F G Perfpective of B E becomes fhorter. Which was, &c. CO- 60 A Treatife of Perfpeclive. COROLLARY. In removing the Eye too far from the Picture^ or in reprefenting Objects at too great a di ftance on the other fide the Picture, aS muft be confufed.For inftance, in the Perfpective of a too long row of Columns, all the Columns muft be confounded. This depends particularly on the fituation of the Eye, in regard of the Objects. The Eye and the Objects being determined in their fituation , the Picture may be nearer or farther from the Eye, When iris nearer the Eye, the Perfpective is really lefs ; and the contrary when it is more remote, tho' the Object appear ftill of the fame bignefs, and under the fame angle \ For 'tis thp diftance from the Object to the Eye, that makes the different Senfation, and not the Picture which changes not the Sen fation, be it near or far from the Eye. THEOR. XVI. The Perfpective of all Objects placed lower than the Eye, is below the Horizontal Line ; but above the Horizontal Line, if the Objects be placed higher than the Eye. DEMONSTRATION. Suppofe B an Object below the Eye A {Plate 12. Fig. i.) feen by the Ray A B, the Horizontal A Treatife of Perfpeclive. 6 1 Horizontal line is E F, and the' principal Ray AD, parallel to the Geometrical Plan. .¦> If B were drawn back infinitely, the Ray A I> could never reach the Horizontal line E F -5 becaufe it ought firft to be parallel to, AD, and then B would not be below" E F, or A D, as it is fuppofed to be fituated. IJt > Let C be a vifible point above A D or E F, the line A C fhall never reach E F for the fame reafon ; and confequently the Perfpective C, fhall not be found below E F* (Which was, &c.) THEOR. XVII. The Perfpectives of Objects that are below the Eye, are higher, the remoter the Objects be ; and thofe of Objects above the Eye, ar& lower, the remoter they be. DEMONSTRATION. It is evident, that the farther B is removed, the Ray A B draws nearer the line A D, which is the principal Ray, (Plate 12. Fig.i.) and fo cutting the Picture nearer the Hori zontal line E F, the Perfpective of* B be comes higher : So that the more C is re moved, the Ray A C drawing near or def cending to A D, cuts the Picture nearer E F ; and confequently the Perfpective defcends nearer 6 whofe Perfpective we look for : Suppofe like- wife F E to be the Perfpective of A H, which Perfpective is parallel to the bafe of the Picture, . Theor. 3. Now A is in the parallel A H, and in the perpendicular A B, fo that the Perfpective ot A, which is in the Per fpective of thefe two lines, is certainly in the common Section of B G, and E F, the Per fpectives of A B and AH: Which was, &c COROL- 64 ATreatife of Perfpeclive. COROLLA RY. Having thus found the Perfpective of the Line parallel to the Picture, where A the vifible . Point is ; draw from its point of Incidence, a Line to the Principal Point , and the Per fpective of A is found. For the Point of Incidence of A, according to the Definition (Chap. 2. Def. 6.) is the Point where the Perpendicular falls, that is drawn from the vifible Point to the Picture : Now the Perfpective of the vifible Point, is in the Perfpective of that Perpendicular ; and fo in the common Section of that Perfpective, with that of the parallel Line that paffes through the vifible Point A, in the Geome-s tricalPlan. CHAR I ATreatife of Perfpeclive. d$ CHAP. V. Of the Difpofition of the Objecls defign id to be reprefented', and of the Situation of the Piclwe, or Perfpeclive Plan, with re* lation to thofe Objecls, and to the Point from whence it ought to b&feen. THE Art We here treat of, corififts iii finding in a Picture, the paffage of the Rayes, which Would fhew the Objects that are behind it, if it were trarifparent. Before We fearch for this paffage, we muft confider which is the fituation of the Picture, with re lation to the Objects we would reprefent, and to the point of fight, from whence it ought to be feen; This will make three Heads, Whichfhall be the fubject of this Fifth Chapter. I. Of the Difpofition of the Objects on the Geome trical Plan. What the ordinary Painters do by chance, znc\ M random, he that is a Geometrician cap: F 4& 66 A Treatife of Perfpeclive. do artfully and preeifely. We have already faid, that every Picture is fuppofed to be on a Geometrical Plan, and that it is there, Verti cally or Perpendicularly, "if 'it be not mentioned to be in another fituation. We have likewife fuppofed the Geometrical Plan to be Hori zontal, that is to fay, parallel to the Horizon. 'Tis on a regular Plan, that we ought always to conceive the Objects, whofe Perfpective we fearch after, to determine and meafure their juft Situation, in regard to the Picture and point of fight. Altho' the fubject be given, and that it doth not depend on the Painters choice, he may imagine it under the fineft Form it is capable of, in which he fhews his Ingenuity. The Features and Colours are the Materials, which can only exprefs the body of the fub ject j the Difpofition we fpeak of, paints what the Senfes cannot perceive ; and this is called the Spiritual part of Painting. But 'tis not my bufinefs to explain this Difpofition, becaufe I talk of Painting, in fo far as it bor rows the help of the Mathematicks, to deter mine in the Picture, the place where each thing fhould appear. When the Subject is great, and contains feveral things, the Imagination is not ftrong enough to comprehend them all, in the Dif pofition in which it is defired they fhould ap pears ATreatife of Perfpeclive. 6f pear. He rriuft at leaft have fome help to fix and fupport his Imagination, and to keep it ftill lively, for fuch a time as is neceffary for £ Painter to exprefs with his Pencil ana Colours^ what it reprefents to him. There* fore it is, that thofe who defire to fueceedj make ufe of this Artifice : They difpofe on a fort of Theater all they would reprefent 5 aftei* Which they take their Meafures, their height, their diftance from the Picture, and Jrom the Eye ' of the Spectator. They marlc their- Situation on the Geometrical Plan, with relation to the bafe of the Picture, and to that Vertical Line which is the Section of the Geometrical Plan, and of a Vertical Plan, which paffes by the Principal Point of the Picture. The thing is neither fo long, nor fo diffi cult, as it appears ; it may be made eafie* To that end we muft imagine the Geome trical Plan to be a Square divided into feveral Other fmall Squares, by the help of which, the place of each Object, and the Ichnography br Plan of the whole fubject is determined. This Plan cannot be made, without fearching the point of Seat of all that is above the Geo metrical Plan That is to fay, it mult be found in What point of this Plan, or in which of thefe little Squares that divide it, a per pendicular line falls, from the point of the F a Object 68 A Treatife of Perfpeclive. Object which is in the Air. 'Tis this Per pendicular that meafures the height of that point. If (for inftance) it were defired to pot a Statue in perfpective, according to the Rules j to find the perfpective of its principal points, we muft confider which is the Situation of its Foot, and after that of the upper parts, by letting fall a Plummet, which gives their point of Seat. Let us make this more intelligible by fome eafie Example. If the Octaedre X were to be put in Ferfpective (fs&Plate 1 5. Fig. 1 & 2.) that is to fay, the body X having eight fides, it muft be examined where the Perpendicular lines B a, Ch, Df, ~Ed, AF do fall, which meafure the height of its angles. The foot of thefe perpendiculars makes the Figure abed, which muft be put in Perfpective 5 then fearching the Perfpective of all thefe perpendiculars, the points A, B, C, D, E, F^ are found, which being joyned by ftraighfc lines, forms the body Z fimilar to X. This is neither difficult, nor very long. Now when the Ichnography of what you would reprefent, or at leaft of the principal things is found, all is eafie, and you may work fafely, for the Perfpeclive of this Ichnography is eafily found, in dividing (as fiath been faid) the GeometricalPlan in fmall Squares; Plate i3 P A Treatife of Perfpeclive. 69 Squares. The whole Plan is conceived as a Sjquare. Suppofe then Z or the Square abed to be the Geometrical Plan, (fee Plate 14. Fig. 1 .) Divide the fide a b in equal parts, by drawing right parallel lines by the points efg. Divide likewife the fide a d and draw parallels: Thus the whole Square will be divided into feveral fmall Squares. The Per fpeclive will be found by the Operations which we fhall fhew in the next Chapter. This Perfpective is B F 0 p ; which being found, it is eafie to find the Perfpective of the Ichnography of the fubject that is on the Geometrical Plap. This may be done with an eafie practice, without any fenfible error, by marking in the Perfpective of each fmall Square, whatfoever is in that Square. The fight of this figure alone makes it eafily com prehended, (fee Plate 14. Figure 2.) X or ABCD is the Geometrical Plan^ on which we imagine a fortification. This Plan hath been divided in feveral fmall Squares, which contains the Geometrical Ichnography of this Fortificationt I fuppofe Z to be the Per fpective of this Geometrical Plan, and of the other Squares into which it is divided. By tranfporting then what is in the Squares of the Geometrical Plan, into the Perfpectives of the faid Squares , the Perfpective of the Geometrical Ichnography which was defir'd, is found. F j This jo A Treatife of Perfpeclive. This is an admirable ready way to find what Painters call the Degradation of the Picture. The firft Objects which are con* ceived behind the Picture, are thofe which are the firft beyond the Ground line, or bafe of the picture. According as they are more diftant, their Perfpective is raifed, and at the fame time gradually diminifhed. This may be obferved without Perfpective. Generally whatever is feen at a diftaace appears fmaller. So a Figure, which on the bafe of the Picture ought to be five foot, may be placed at fuch a diftance, that it can have but four, or lefs if what it reprefents ought ftill to be imagined more remote beyond the Picture, This is what I have been calling the Degradation of the Picture; which is known as foon as we have found the Perfpective of all the Geome trical Plan ; and of the Squares into which it is divided. We know Geometrically the Diminution of each Object, as it is placed in fuch or fuch a parallel of the Geometrical Plan ; for inftance, How much a body of five foot, which is in the fecond parallel line of the Geometrical Plan, fhould appear di- minifh'd in the Picture ; and how much in the third ¦-, and confequently what is the De gradation of the Picture, or by how mucfi muft the natural fize of the Object be di minifhed, according p the place where its Per? A Treatife ofPerfpeclive. 7 1 Perfpective is put. This being as eafie as it is neceffary, Painters that neglect it are much to blame. And can do nothing well but by chance. The Geometrical Plan, on which we ima gine the Objects we would reprefent, may comprehend a great Country. • But every thing may be reduced from great to fmall ; and we may fuppofe a Board or Table of five or fix foot fquare, to be a large Country. Neither is it neceffary to make effectually any Geometrical Plan, it is enough to imagine it* and without marking the lines which are the meafures of the Objects which are to be painted, we may exprefs them by number, thefe meafures "making only a Scheme of the Picture which is to be painted, as the Archi tects, who without drawing any line, calcu late what they defign to make, and make it known to the Workmen that are to execute it, marking by numbers all the meafures of their Work, the fize of its Plan, the length, and breadth of all the Apartments, the height of the Stories, and the largenefs of the Win dows. A Painter may likewife prepare his fubject in his fancy, and write it down ; He may mark it on Paper by numbers, deter mining the Situation of fuch and fuch a part, how many feet it is diftant from the bale of the Picture, how much it is elevated above F 4 the 72 A Treatife of Perfpeclive. the Geometrical Plan, or depreffed. under it, and fo make an exact defign. Painters have a great liberty ; All is allow'd them, providing they clafh not in any thing with livelihood and decency. They may therefore imbellifh and difpofe things to tfie; beft advantage. I believe likewife they may take fome liberty in the Defigns of Per fpective, which they make upon 'Paper to reprefent an Edifice, particularly when the bufinefs is only tp give an Idea ; and pot to finifh it, but only to fhew which way it is done. If the bufinefs be to make a general view, they need not confine themfelves to a true Plan. For inftance, if in this Plan there be found great Courts before* the Entry, thej Perfpective pf thefe Courts which will be upon the bafe of the Picture, fhall appear top great ; whereas that of the Edifice which is to be reprefented, will appear too fmall and confufed. I believe that upon fuch an occa fion they may imagine another Plan than the true one •¦> that is to fay,they may fuppofe one, fuch, that the parts which they would render moft fenfible, may appear in the defign, as, obvious as is neceffary. They muft fuppofe theCouit of eqtiy leffer •-, and if there be any Buildings on the Wings that they woulq make, appear, they may detach them more £han' they are; v tliat is to fay, they may fup pofe A Treatife of Perfpeclive. 7 g pofe the Court that feparates thefe Buildings to be greater; to the end that in the Per* fpective thefe Buildings may not be confufed. This can run them into no error, for the true Geometrical Plan, reforms and makes under^ flood what is requifite to be known. There may be many things obferved,touch* ing the Difpofition of the Objects on the Geo metrical Plan ; but as I have faid before, it does not concern me. I fhall only add, that a Painter ought to fuppofe nothing there, but what the Eye .can comprehend at a fingle view. We are now going to fhew, that the Eye comprehends a greater or leffer number of Objects, as it moves farther off, or draws nearer, XL Of the Situation and Size of the Picture. A Picture can reprefent but one fingle Action, or Actions that relate to one thing, and can be feen wholly at one view. The liberty of Painters is therefore not fo great as that of Poets •¦> thefe are indeed obliged to unity of Action, Time, and Place, whither jt be in a Comedy, Tragedy or Heroick Poem. in Drammatick Poejy, or Pieces of Theatre, the place is the Theatre it feif, on which all paffes, pr is there related, if it has been acted elfe- where. 74 ATreatife of Perfpeclive. where. In Heroick Paefy the place is all the Provinces Which the Heroes have run oyer in the time in which the principal Action was done, which is the fubject of the Poem. The time of this Action may be a whole Year. TlW Pieces of Theatre are allowed at moft but twenty four hours. Painters are yet more ftraitned, for they can only paint the Moment of one Action, the Situation where the things were in that Moment •-, the Tigure, and the Character of the Paffion with which every Perfonage was animated at that time. The Figure of all the Bodies, the Poftures, the Vifage according to the Mo tions of the Spirit, which being active, is con tinually changing all thefe, which is impoffi ble to be mark'd with a fingle ftroke of the Pencil ; for the fame Features cannot ferve for quite different things. 'Tis then only the moment of an Action that hath its proper Features which can be painted. So the time of the Action^ that is the fubject of the Picture, is but one inftant, becaufe the next inftant will requite other Features; the things having no logger the fame Situation, the fame Difpofition, nor the fame Character. Painters are likewife more ftraitned than Poets in the unity of the Action : Every Comedy has five Acts ; and every Act hath different Scenes,in which are feen different Ornaments, changings A Treatife of Perfpeclive. 75 changings of the Scenes, and ftill fomething that is new* Jn Heroick Poems they give Battel, and befjege Towns. 'Tis noLthe fame in Painting} as the Eye can fee nothing diftinctly, but what is before it, and that its overture is limited and narrow, it cannot re ceive but a fmall number of Objects at once. Befides that the Rayes that enter obliquely, cannot unite in the Retina, to form there the Xmage of the Objects from whence they come. Wherefore a thing is not diftinctly feen, but when the Rayes by which it is perceived,* fall directly upon the Eye, or mifs little pf doing fo. In changing place, or turning the Eye, things are difcovered that were not feen before •-, but a Painter cannot reprefent exa&ly what is not feen at a fingle view. There is great difference between Sculpture and Paint ing : A Statue that ftands by it feif, can be feen on all fides, and by parts ; and every point from whence it is feen, hath a parti cular Circumference. But in a Picture there cannot be given to the fame Figure different Circumferences, to terminate it by two diffe rent ftrokes. So that if, in corifidering an Action, the place be chang'd, or the Eye turn'd, it is feen in a different manner, and then the fame Figure can no more ferve j for its Circumference will not be proper, for it, but when it is feen in its firft Situation. If j6 ATreatife of Perfpeclive. If we would reprefent different Actions to be feen at feveral times, we muft make feve/al Pictures. But finally, what Rule muft we obferve for the Grandure of the Action that is the fubject of the Picture ; oi^ what fize muft we allow the Picture ? To find this Rule, let us reflect on what we have been faying, that the Eye fees not di ftinctly but what is before it -, and that it is but little affected with what enters it flanting. Likewife the Eye rowls in the Head, and Iturns it feif to what it would fee 5 it draws near, or retires to fee better; for as this Fi gure makes plain, while the line Z is betwixt D and E more diftant from the Eye,the Rayes by which it is feen, finite the Eye more di rectly, than when it is in B and C. So the Rule we look for, is, if the Picture be large, and the Figures in their natural fize, and in great number, we muft fuppofe what it re prefents to be at a great diftance ; for other- ways ATreatife of Perfpeclive. 77 ways 'tis impoffible for the Eye to receive fo many things at a fingle.. view. For in- flances Let us confider the great Front of the Louvre. If we are near it, we can fee but a fmall number of Columns, which Will ftrike the Eye direftly : Whereas in retiring to a reafonable diftance, the Eye, tho' it continue fixed, comprehends eafily all this great Front, which becomes fmaller, and contracts it feif, to be proportionable to the Capacity of the Eye. But if the diftance be too great, the Objects become too fmall 5 they appear no more, and the Picture is all confufed. We fpeak here of a Picture, in which we would have things appear in their natural greatnefs ; that is to fay, that they make, as near as poffible, the fame Impreffion upon the Eyes, as they would do in their natural State. To determine the greatnefs of this Picture, we muft have a regard to what we would have appear there. But above all things, we muft remember, that it ought not to be exceffive large ; that it ought to be pro portioned to the Capacity of the Eye, which ought to fee it wholly at one view. It may be done by taking a great diftance, even tho' the Picture were very large ; but then it. would in a manner difappear 3 all the fmall parts would not be feen. Now a Picture is net made to dazle the Eye with colours, nor w 78 ATreatife of Perfpeclive. to fhew ftrokes that cannot be diftinguifhed at too great a diftance. Neverthelefs there is no Action, whatever multitude of Actors it may have, but may be expreffed in one Picture of a fize propor* tionable to the Capacity of the Eye. There is a certain convenient diftance, which hin ders not from feeing what the Painter would reprefent. Battels and all forts of Actions which fuppofe a great Multitude of Perfons, may be reprefented, as is feen in a Picture^ reprefenting a Basket of young Chickens. But Which is that convenient diftance ? We muft acknowledge it a little difficult to determine juftly. To fpeak nothing here at random^ let us call to mind what we have faid feveral times ; that a Picture ought to be confidered as a Window. According to the largenefs of the Overture of a Window, and the nearnefs of the Eye, we perceive more or fewer Ob jects. If we be near, and the Objects touch the Window, it is evident that what we fee, is no larger than its Overture ¦¦> which gives me reafon to fay, that the Rule which Pain ters ought to follow, is, that if the fubject they treat of, ought to be imagined on the forepart of the Picture, they can place no thing there but What the largenefs of the Picture allows to be reprefented in its natural bignefs j fo that if they be great Hiftories, which ATreatife of Perfpeclive. 7.jg> which require a great number of Figures, a large place, or a Country, as the Eye cannot fee fo many things through a Window, if they be not at a convenient diftance, fo they ought not to be on the forepart of the Picture, but in the Degradation, and of a fize much lefs than the natural ; becaufe every thing fhould diminifh and decreafe in proportion to its diftance ; and grow likewife-more con futed, and dark. It is true, that when we are at the Window, 'tis not its Overture which limits the extent of the Objects which we may fee. But as the Eye ought not to touch the Picture, that it ought to be at fome di^ fiance, its largenefs which reprefents the Overture of a Window, determines the great nefs and multitude of the Objects which this Overture would let us fee. According as we are near or far, it lets us fee more or fewer Objects. So to refolve the Queftions that may be made concerning the Difpofition of Objects on the Geometrical Plan, and the fituation and fize of the Picture, we muft confider which is the fituation of the Eye, and at what diftance it is from the picture. Whiehfhall.be the fubject of the next Head. 111.0/ 80 ATreatife of Perfpeclive. in. Of the Situation of the Eye, or of the PoiM of Sight. The Situation of the Eye, in regard of the Picture, would feem to be left to our choice, becaufe the Rules that are given for Per. fpective, are for all the different fituations of the Eye. There is neverthelefs a Rule* which is, to place it where naturally it ought to be, to comprehend the whole Object that can be feen through a Window, for as fuch we now confider the Picture, Sometimes it is placed in the Center j fo generally fpeaking, the principal point, that is to fay, the point on which the principal Ray falls directly, ought to be in the middle of the Picture. But as we difcover different Objects, according as we prefent our felves at a Window ; that in looking fideways we difcover what we did not fee when we look'd ftraight forward^ fo we muft place our Eye differently, accord ing to the different fituations of the parts we would fee. For inftance, being placed oppofite to the Gate of a Church, I difcover its front ; but if I would fee the fides or wings of this Church, I muft change my ftation. So in reprefenting the fides or wings of this Church, the point of fight ought not to be in the middle A Treatife of Perfpeclive. 81 middle of the Picture, it may bt fometirhes without the Picture ; for it may happen, that to fee the fides and wings of a Church, whole Gate is exactly oppofite to the Window from whence it is feen, we may be obliged to take a ftation fidewife, fo as the principal Ray fhall be without the overture of the Window and fall aTide. This Is not ordinary ; it is moft natural to be placed oppofite to the Center of the Picture, whereby confequence the principal point fhohld be. It is on this Center we look : What is directly oppofite to the Eyes, is moft fenfible, and is beft feen. So it is at the principal point, or near it, that the beft Painters place the chief Perfon of the Hiftory they reprefent ; and who ought to attract the Eyes of the Spectators moft. The natural height of the point of fight, is the natural height of the Eye. We ought then to fuppofe the point of fight, and con- quently the principal point, which are both on a level, to be at the ordinary height of bur Eye. But as there are Objects which cannot be feen, but from low to high, or from high to low, that we cannot fee the Wainfcot of a Ceiling if the Eye be not be low, nor the infide of a Court furrPunded With Buildings, if the Eye be riot elevated higher than thefe Buildings ; fo there are Pictures j in which the principal point is above, 6 and 82 ATreatifeof Perfpeclive. and there are others, in which it is below the Picture. When we would fee a great Edifice, we take our flation in the front, to the right, to the left, on the ground, or in eminent places, according as the Edifice is fituated, and ac cording to the parts we would difcover. Such Perfpectives, whofe point of fight is placed in fuch mannei', as that we fuppofe the Spectator elevated in the Air, as if he were a Bird, are called, Perfpectives at a Birds fight. But the Eye without being raifed above the Earth, can fee things that are more ele vated, and fee them directly ; for we may turn our head backward without incom moding us. Befides the Eyes can turn them felves upwards, fo as the fight of things much above their natural height, fhall finite them in a direct line. This happens in Pictures which are placed above the Eye. But as We can fee nothing through a Win- dow which is above the Eye, but what is in that Window ; and that we cannot per ceive what is beyond it, unlefs it be in the Air, higher than the overture of the Win dow ; foin a Picture reprefenting thatWindow, all ought to be forwards; and we ought to make the head appear, or the higheft parts of what we conceive beyond tb.atWin.dow. Pictures A Treatife of 'perfpeclive. 8 3 „ Pictures that are placed in high places, are often made to. incline, fo as we muft raile pur Heads, and turn our Eyes upwards, tp fectbem, to the end the principal Ray may be perpendicular to the Picture. In this cafe the principal point may be in the Picture, tho' it be placed above the Eye \ but this is not natural, for the Perfons appear in a fitu ation where they cannot be. A body in clining riiuft faG, if it be not fupported; fo if the Picture be truly inclin'd, it muft re; prefent the Figures with fo much Art, that they appear upright or ftanding; , We fhall fpeak of this Art in the feventh Chapter. Ordinary Pictures ought to be upright; if they be inclined, it is becaufe the Dull may flick lefs to thern. ; , We have already faid, that in a Picture, Whatever is above the principal point, is feen lower -, and that what is placed lower, is feen higher, and feems to mount ; whereas what is placed above this point feems to defcend. 'Tis evident that the Circumferences of Fi gures alter as they draw nearer, or remove farther from that point. SomC fhould be feen in front, when they are at the principal point, and directly oppofite to the point of fight; others are feen only in Profile -, and fuch a Figure as being placed on the left, fhews the Stomack, in another fituation would prefent. G i the $4 -A Treatife of Perfpeclive. the Shoulders ; this is the reafon that thofe who in their Defigns make ufe of Figures bor rowed and copied from the Draughts of diverfe Matters, or from their own proper Studies in the Academy, ought to be careful to place them as they fhould be, in relation to the point of fight, uader which they were firft defigned. We have already faid, that a Figure, what ever it be, being once fet on a Plan, can ne ver have the fame appearance in any other place of that Plan, to which it may be tranf ported, if the point of fight continue fixed. So that without the help of Perfpective, it is not eafie for thofe who lteal any part of the Works of another Painter, to place it as it ought to be in a new Compofition ; as is ob- ferved by Rowland Freardof Ch ante Ion of Cam bray, in that excellent Idea he hath given of the Perfection of Painting. It remains now to fpeak of the juft diftance that is to be given to the point of fight ; this depends on thc largenefs of the Picture .- What is large cannot be feen at once', if it be not at a diftance ; fo that it is the fize of what we would reprefent, that regulates this diftance. When the Eye is too near, the Plan on which the Figures are placed, appears elevated in Talus, and the Diminution of the Figures is too fudden, without a proportion to thofe that are- A Treatife qfPerfpeclive. 8 $ are on the forepart of the Picture. On the con* trary, if the point of fight be more remote than it ought, the things are confufed and hud dled together. This point ought therefore to be at a moderate diftance, which will be eafier found by Experience,than by Arguments; for before any thing be concluded on, we fee very near what effect the things will have, ac cording as the point of fight is at a diftance, or near. What we would have appear diftinctly, ought to be feen very near,for what is feen at a great diftance,we perceive confufedly. Where fore when we would reprefent diftinctly any Object,the diftance of the Eye ought not to be great. As we cannot difcover a great Building it a fingle view, unlefs we retire to a great diftance, fo the Perfpective of a large Building s never very diftinct. Likewife if we would lave our Defigns clear, we muft reprefent in sarticular every one of its parts, and make lifferent Defigns. In a word, to regulate the liftance of the Eye, it muft be fuch as to com- )rehend every thing entire without trouble ; md to fee and diftinguifh plainly all the fmall trokes. We fhould not here have regard to uch as have bad Eye§, becaufe the Mufician loes not fing to the Deaf. G I IV.Efta- 8*5 ATreaiife if Perfpeclive* IV. Eftiiblfhrrg tire Neceffity of Perfpeclive, arvd ::~fvering the Difficulties which are framed against th:. Neceffity. Before we begin the next Chapter, which contains the practical part of Perfpeclive, let us confider what feveral of the moft indiffe.-. rent or lazy Painters are ufed to object ; that if we fhould confine our felves to the rigorous Rules' ol Perfpective, there are fome cafes, in which/ inftead of Figures well proportioned, we fhould reprefent theni monftrous, as hap pens m Reprefentations of Architecture, when we follow the Rules of Perfpeaive. They can not fhun (fay they) meeting with fome things. extreamly troublefotne, fuch as the exceffive lengthening of the Cornifhes, of the Capitals, and of the -Bafes,their fudden falkand their dif- order -, which happens when any one would put them in . Perfpective accordingto the Rules of Art*, for as we have demonftrated, the Perfpectives of lines perpendicular to the Picture tending all to the fame point, which is the principal point 5 if it be Columns that are to be put in Perfpeaive, and if the point of fight be elevated, the Bafes of the Columns which ough,t to be Horizontal, muft riever- thelefs mount exceffively.;' and on the contrary, A Treatife of Perfpeclive. 8 7 if the point of fight be placed low, the Capi tals feem to overturn, and fall toward the place where the point of view is. To this I anfwer firft, If the bufinefs of Painting were only to reprefent things to the. Life, it would be no fault if they appealed maimed and deformed,if they muft effectively appear fo,in the Situation they are fuppofed to liave, and in which they are painted. But as Painting is only made ufe of to pleafe, fo no thing ought to be painted but in a ftate where the Eye is fatisfied. Thus Painters correct Nature it feif, or chufe only what is the fineft and biggeft ; to this end they never chufe a Situation, which, by following the Rules, will infallibly make their Pictures maim'd and monftrous ; that is to fay, if they muft appear fo from the point from, whence they ought to be feen. They place therefore their point of fight, according to the pofture they would have their Work appear in ; and if they mix any Architecture with it, they order it fo, as nothing appears offenfive. Secondly^, there is any thing difagreeable in fuch Perfpectives as are made according to the rigour of theRules,it is occafioned by want of Art : The Mathematicks alone fufBce not to make a good Perfpective, they only fhew us how to find certain Points, and to draw ne ceffary Lines, but this is not enough; the G 4 prin- 88 A Treatife of Perfpeclive. principal thing is, the Light and Shadows, 'tis the Colours that makes us judge of the di ftance of things, and of their Difpofition and Situation. So (for inftance) whereas in the Draught of the Bafe of a Column, all its parts, tho' Horizontal, feem tp elevate and mount. Painting can preferve its natural appearance,in, reprefenting the parts moft diftant from the Eye as fading,and as they would appear if they were feen really without the Picture : In which cafe they feem to mount and elevate on thq Horizon.lt is the fame with thefe Poflures we call maimed,, which never appear but either too lively, or not faint enough. The weaken ing and diminution of Colours, make the principal Performances of Painting. For the farther defence of this Art, whiclj we treat of as Mathematicians, let us add, that Pictures or Perfpectives cannot have the de fired Effect,but when they are feen from a cer tain point. But we place them yery feldorn in this point, in order to be examined ; neither is it poffible to do it, when the Pictures are fmall, and their Figures not of a natural bignefs. The Bafis of the Picture ought to be the ground line; that is to fay, common to the Picture, and the Planbn which we fuppofe the Specta tor, and the Perfons reprefented in the Picture; but this is, never done. For thefe Pictures are placed on a Table much higher than the Flap we A Treatife of Perfpeclive. $9 we ftand on tp confider them $ and as jthey have fome fine ftrokes, and fome fmall parts, which cannot be feen but near, fo we cannot fee them at a neceffary diftance 5 neverthelefs becaufe they have not their natural bignefs, they ought not to be feen but at a great di ftance. So if any thing in Perfpective appear deformecj, that Deformity happens, becaufe the Rules of Art are notobferved 3 for in Per fpectives, where the Bafes. of Columns are feen to mount exceffively, and the Capitals to overturn, the Point of Sight is too near ; if it were at a good diftance, the Perfpectives of the Bafes and Columns would be fo fmall, that the great difference bet wixt the advanced part, and the remote part of the fame Column, would not be obfervable. When we reprefent Architecture, or would fhew a great Front with very high Columns, we are obliged to take our point pf Sight at a confiderable di ftance ; for it is impoffible for us to fee a very lofty Building, and of a great extent, at one view, if we do not ftand at a confiderable diftance. Therefore the fuppofing the Point of Sight too ne,aiy for difcovering at one view what we reprefent, is a fault which ought not to be imputed to the Art. It may be anfwered, That we are fome«, times obliged to it, efpecially when we would fhe^ tfie fmalleft parts of a piece of Archi tecture; p6 ATreatife of Perfpeclive. iectiire ; tha* in that cafe the Eye muft be near, becaufe diftance renders every thing confufed. I anfwer, That this Conftraint is but Imaginary, and that we ought not to de fire any thing but what is reafonable ; that- if we fuppofe a confiderable diftance, the fmall parts ought not to appear diftinctly ; and if the Edifice we reprefent, is very high, and near the Picture, we muft fhew only the foot of it, becaufe we could fee no more, if we few it effectually through the overture of the Picture, fuppofing it to be tranfparent or open. Architects, in the Defigns they make of great Works, ufe Perfpective only for the general view of an entire Edifice. They make only Geometrical Reprefentations of the parts, For, once more, we cannot fee an intire Building of a great height, and large extent, at one view. I do not deny but there are many fine Pictures, in which Perfpective is not obferved ; it is not obferved in fmall Pictures, as I have faid. This ConfeffiOn does not deftroy what I have eftablifhed, tliat perfpective is the Foundation of Painting, and that a Picture cannot have the fame effect as the things themfelves, if all the Rules be not obferved, We muft confider the fmall Pictures but as Imitations, wtiereas a -true Picture is a Repre fentation, which makes the fame Impreffion on the Eyes, as the thing reprefented would do, ATreatife of Perfpeclive. 91 do, if it were feen ; which ought to appear- in the. Picture in its natural bignefs, according to the fhape it hath, and all the meafures oi that fhape. To make an Imitation perfect, what is done in fmall, ought in that imallnefs to have the fame Proportions. Confequently1- a fmall Picture, which is a Reduction, that is to fay, which is reduced from great to fmall/ tho' it be not feen from the point from whence it ought abfolutely to be feen, i. e. from the true point of Sight} it ought to have a point to which all bears ; fo that if this Picture were reduced from fmall to great, all the Figures would appear in their natural Size, and the point pf Sight would be,, that where the Eye being placed, the Sight of this Picture would make the fame Impreffions on it, as the things themfelves would do if they were prefent. 'Tis the Imitation which pleafes,and which we admire in Painting as well as in Poefy. A Poet reprefents upon a Theatre in a fhort fpace of time and place, an Action, which (fo fpeaking) hath taken up a great deal of ground, and a confiderable time. He is careful of the Likelihood, and all he fays, may be done in the manner he fays it, and in the place and time of the Reprefentation. It ought to be the fame With Painting. A Picture is the Re prefentation of an action, vvhich we fuppofe ; . ' - to 92 A Treatife af Perfpeclive. to be acted in -the prefence of him who con? fiders the Picture. He ought tp fee nothing that belyes this Reprefentation ; it .is not fo much the Features end Colours that appear ad mirable, as the perfect Imitation which fuf> fifts always/. whether the Picture be of 'a na tural Size or not ;, as we take pleafure in readr ing a Comedy, which can be read in lefs than: an hour. Every Picture is an Imitation, but a fmall Picture': is, theilmitation of an Imitation, A fmall picture cannot make the fame im? . preffion on the Senfes, but it may upon the Mind, which'is charm'd with the Ingenuity of a Painter, that can ingenioufly imitate in fmall, what may be done in great. Now this Imitation would be imperfect, . and this little Picture would not fatisfie the Mind, if Per fpective were not oblerved in it. For as we have often faid, we can only paint the inftant of an action feen from a certain point ; in re fpect of which, the Actors have a certain de- termin'd Circumference. Therefore Likelihood is not oblerved in a Picture great or fmall, if all be not ordered with relation to a certain point ; tho' that be not the fame in a fmall (icture, where the Eye that confiders it might be placed ; but, the Soul finds this point, and it is with relation tp this point that it judges of the Picture. We may likewife fay, that in a fmall Picture all is intirely reduced from great to -A Treatife of Perfpeclive. 9^ to fmall, and the Spectator himfelf becomes. fd fmall, that his height doth not e'xceed that of the principal point above the bafe or '.the1 Picture. 'Tis the Ingenuity of him that imi tates, which pleafes ; what we fee is nPt real; and Falfehood cannot be agreeable; there fore 'tis the Likelihood, or the 'art of repre fenting Truth , which renders Painting fo' Charming; and Painting "doth not pleafe, but fo far as it favours of this Art. We fhould take no pleafure in feeing a Picture, in which every thing were reprefented in its natural bignefs ; that is to fay, if it were a perfect Perfpective made according to all the Rules •¦> and if Wfe were fo deceived, as not to perceive our Eiior,: or not to be fenfible of its being only an In genious Imitation. Inftead of confidering fmall Pictures as a Reduction of greater, we may confider them as the things themfelves reduced. 'Tis fo we ought to confider the defigns of Architecture, which are contained on a leaf of Paper : We may fuppofe the Defigner did not intend to reprefent the Edifice fuch as it is ; but that it is reduced to fmaller, that he hath made a Model of it, and that 'tis only this Model he defigned to reprefent. Having proved Perfpective to be the Foun dation of Painting, before I come to the Pra ctice of the. Art I teach here, I will firft re- m©W 94 A Treatife ofl^effpeftive. move a thought which fome may entertain, as if I pretended that my Work were fufficient to make an excellent Painter 5 and thereby undertake to teach What I do not know. I repeat once more, that the Science I explain,- fuppofes a perfect knowledge of the defign. By that Word is underftood the juft Meafures^ the Proportions, and the external form that the Objects Ought to have, which we repre- fent in Imitation after Nature, and that with relation to the point of Sight. For according as an arm (for inftance) is feen, fo it appears to have a certain meafure. 'Tis not its real meafure which muft be reprefented, but the meafure it affumes by being feen from a cer tain point -, this is what the Painters learn,' who make their Exercifes in Academies, de- figning after Relief-work, or from a living Model. There are Books which explain the Proportions of a Humane Body, and difcover which are the fineft, for Painters fhould not exprefs but what is rare and fine. For my part, I only treat of Perfpective as a Mathematician, tho' at the fame time I alledge, that it is the Foundation of Painting. ATrtatife of Perfpeclive, 9^ CHAP. vi. :i How to find the Perfpeclive $ anyObjecl whatever, its Situ ation being given in regard U the Piclure and the Eye. o. THE Practical part of the Art of whicti we treat, confifts in finding the Per fpective of a given point in the Geometrical Plan, or elevated above that Plan, whofe Situ ation is known, in regard to the Picturei and the Eye that is to fee it. For all vifible Ob jects whatever, are but a heap of Points $ and thefe Points may be innumerable. It may thenbeask'd, Whether it be not an infinite labour to fearch for them all ? But the thing is not fo difficult as it at firft appears. To comprehend this, obferve First, That a Line is known when its Extremities are known : So that to find the Perfpective of a Line, we muft fearch for the Perfpective of its two Ex tremities ; for as we have already demon* ftrated, the Perfpective of a Line is a Line. Secondly, Since to draw a Parallel to a given line, 'tis fuffkient to know one fingle point through which it muft pafs ; fo to find the per- $6 A Treatife of Perfpeclive. Perfpective of a line parallel to the Picture, wo need only' fearch for the Perfpective of one point of that parallel ; for we have already tlemonftrated, that the Perfpective of a line, which is in the Geometrical Plan, and is pa rallel to the Bafe of the Picture, will in the Picture be parallel to the fame bafe. Thirdly, Having the Perfpective of one point of a perpendicular on the Geometrical Plan, we have the Perfpective of the whole perpendicular ; this Perfpective, as hath been proved, being perpendicular on the bafe of the Picture ; and the Pofition of a perpendi cular depending only upon a fingle point. Fourthly, Since the Perfpectives of Figures parallel to the Picture, ar£ fimilar to thefe Figures, the Perfpective of one of their fides being found, it ferves as a common Meafure, or Scale of Reduction, with which it is eafy to finifh the fame Figure iri the Picture. Fifthly, The Perfpective of a line in the Geometrical Plan being known, to find one parallel to it, 'tis fufficient to know one of its points ; for if thefe parallel lines be perpendi cular to the Picture, then their Perfpectives being prolong'd, pafs through the principal point : If they be not perpendicular to the Picture, t\\€vc Perfpectives being continued, cut the Horizontal Line in the fame point, which is called the Accidental point ¦; So" A Treatife of Perfpeclive.' 97 So there needs no extraoidinary pains to nhd the Perfpectives Pf all the points of an Object, becaufe the knowing of a few gives the reft. In the following Problems,we muft fuppofe thefe things known, to wit, the Situa tion of the Picture ia regard of the Eye, and confequently the point of Sight, or principal point, the diftance of.theEye from that point, the. remotenefs of the Object -behind the . Picture, and all that concerns its Situation, in regard to the Picture. PROBLEM I. To find the 'Perfpective of a point which is in the Vertical Line of the Geometrical Plan. A * B D is the "Vertical Line, A the Eye, and £ the Object, whereof C is the Perfpective. The diftance between the Object arid the Picture is D E ; the Line A B or H D is the diftance of the eye from the Picture j thefe distances are known. A H the height of the eye istikeWife known; fo is the Triangle AHE; railing then upon the poiat D a per-* pendicular, terminating at the line A E, the point C Perfpective of ti is found. Which, was to be done (Plate 1 5. Fig. j .) This is eafily done; by trarifportirig the diftance of the eye A B, on the Horizontal H Line* 98 ATreatife of Perfpeclive. Line. Take B F equal to A B, and on the ground line mark the diftance of the Object E, by taking D G equal to D E 5- then draw a right line from the point G to the point F, (thefe are the points which in the fecond Chapter we have called the points of diftance) which xilie F G cuts the Vertical line B D at the point C Perfpective pf E. This is evident •-, for the triangle F K G is equal: to the Triangle AHE, and the Triangle D C G is equafto DCE ; fo H D, DC :: K D, D C. *a'- Therefore the point C may be found by Calculation ; and as all the following Pro blems are reduced to this, when the Situa- tion of the Objects, in relation to the Picture, and to the point of Sight, is known, it will be eafy to make a Tablaof Numbers, which fhall give the meafures of a whole Per fpettive. LEMMA I. Let P and Q^ be two parallels, between which A B and M N are perpendiculars. A D=FG and B C=I H ; I fay, that D C cuts A B at the fame height . that G I cuts the perpen dicular M N, drawn through the point K, the Section ofYHbyG \.mdthat BE=NK **f..AE=MK. D E- A Treatife of Perfpeclive. ,99 DEMONSTRATION. The Triangles A E D and B E Care fimi lar, andfo are F K G and I K H ; X K H So BC, A D : : B E, A E, and as I H, F G : : NK, MK, -no%aC=I H and A D=F G ; therefore B E, A E : : N K, MK. / ; . Et Componendo. BE-FAF/(orAB) AE :: NK^MK (or M N) M K. SoAB, AE :: MN, MK, ->andAB, MN : : A E, MK. r. ¦ - . f - - Now thefe two perpendiculars A B and M N are equal, being between the fame pa rallels -¦> therefore the lines A E and " M K are likewife equal , and confequently B E is equal to N K 5 which was to be demon- ftrated. H z LEM- ioo A Treatife 'of Perfpeclive. LEMMA II. Dividing the di fiances F G and I H ''according to the fame proportion, a line drawn. through the points of this divifion, will cut F H and Gl at the fame height a? BE/ Whereas we have fuppofed A D equal to F G, and B C equal to HI; fuppofe now A D is not equal to F G", nor B C to M I ; but only that there is the fame psopPrtioa INH between thefe two lines ; that is to fay,"! BC, AD : : IH, F G, in that cafe the fame will happen as in the former Lemma ; that is to fay, A B and M N will be cut at the fame height. DEMONSTRATION. Becaufe the Triangles A E D and B EC are fimilar; as likewife F KG and I K H ; therefoie B C, A D : : B E, AE, and IH, F G : : NK, M K ; fo the reafon of BC ATreaiife of Perfpeclive. 101 B C to A D; is the fame as that of IH to FG ; therefore two reafons being equal to a third, they are all.equal. BE, A E : : N K, M K. Confequently Componendo. BErAE (or AB) AE :: NKtMK (or-MN)MK. SoAB, AE :: MN,MK. Then Permutanio. A B, M N : : A E, M K. Now A B=M N ; therefore A E=M K, then A B-AE=MN-MK. But A B- AE=BE, andMN-MK=NK, There fore B E=N K. Which was, &c. P R O B. II. To find the Perfpective of a point, in any place ' what foever of the Geometrical Plan a Suppofe Y a point given in the Geome trical Plan, and its Situation known. Its In cidence on C D the bafe of the Picture X, is B, the point of fight is A ; the line A O is the Section of the Vertical Plan. The diftance of Y from the Picture* is equal to B E, which I mark on thc ground line, prolong'd at dif- cretion. The diftance of the Eye is equal to A F,% which I have alfo marked on the Hori- H i zontal 1 02 A Treatife of Perfpeclive. zontal Line G H, prolong'd as far as requi- fite: So E and F are the points of diftance^ through which I draw the line E F, which cuts A B in a point, which is the Perfpective of Y 5 which was to be demonftrated, (Plate 1 5. Fig. 2.) First, The Perfpective of Y, ought to be in the line B A, Chap. 4. Theor./*.. Secondly, If the point Y were in the Ver tical line O P, then A B would be a perpen dicular. Thirdly, Let Y be elfewhere than in the Vertical line. By the Theor. 1 9. Chap. 4. its Perfpective is in a parallel to the bafe of the Picture, to wit, in the Perfpective of Y P, which is parallel to the fame bafe : There fore whither Y be in the Vertical line or not, its Perfpective is in the fame parallel, and confequently at the fame height. /Now the diftance of the Eye, and the diftance of Y, is in this cafe the fame, confequently ac cording to the firft preceding Lemma, FE cuts A B, whither A B be a perpendicular or an oblique line : Therefore the Perfpective of Y, which is in A B, and in F E, will ne- ceffarily be in the Section of thofe two lines. AD VER- Qlatc. Kraqel/acL, A Treatife of Perfpeclive. 103 ADVERTISEMENT. FirB, The point of Incidence B being known, there is no occafion for drawing the line A B; for haviag marked AN equal to A F, and B M equal to B E, the Perfpective of Y is in F E and in M N, and confequently in their Section. Secondly, If the Picture be too narrow, and k be difficult to prolong the ground line, and the Horizontal line (which may be done ne verthelefs by applying long Rulers) we need only to diminifh proportionably the diftance of the eye, and of the vifible point ; taking only a third or a fourth of one and the pther diftance $ for then, according to the fecond Lemma, the line A B fhall be cut in the fame point by FE. Thirdly, To facilitate the Operations, they tye a fmall firing to the point of fight, which changes not ; and moving it to the points of Incidence of all the points whofe Perfpective is fought, this firing reprefents A B- They tye likewife a firing to the diftance of the eye which is always in the fame point, and moving it to the points of diftance of the vifi ble object, the place where it cuts the firft firing, is the Perfpective look'd for. Inftead of a firing, a Ruler may be applied to the H 4 point 1 04 A Treatife of Perfpeclive. point of Sight, and point of Diftance, and the effect will be the fame. PROB. III. To find the Perfpective of a line which it in the Geometrical Plan. I find by the preceding Problem, the Per fpectives of the two Extremities ; between which I draw a line^ which is the Perfpective of the line propofed ; and this Perfpective is a line, according to what hath been demon- ftrated, Chap. 4. Theor. 2.. PROB. IV. To find the Perfpective of a line which falls per pendicularly on the Picture. Find as in the foregoing Problem, the two- points, which are the two Extremities of the line propofed, and draw a' right line betwixt them ; this will be the Perfpective fought for. If it were not neceffary to determine its great nefs, it would be fufficient to find the Per fpective of one of its points ; for the line which contains its whole Pertpective, muft pafs through the principal point, if prolong'd, as hath been demonftrated, Chap. 4. Theor. 4. PROB. ^Treatife of Perfpeclive. i o 5 PROB. V. To find the Perfpective of feveral Lines parallel ' amongst them/elves, but not to the Picture. CASE I. Thefe Lines which are parallel amongft themfelves, may fall perpendicularly on the Picture; and in this cafe their Perfpectives are found by the preceding Problem. CASE II. If thefe L;nes parallel amongft themfelves, are not pe-per-tci iar'to the Picture, the Per fpective of one of them muft be found by the third preceding Problem, which being pro long'd to the Horizontal line, which it will cut, as hath been proved by Chap. 4. Theory. it gives the Accidental point, in which ail the Perfpectives of thefe parallel Lines do meet, becaufe they all cut the Horizontal line in the fame point by Chap.t\. Theor. 6. So it fuffices to find the Perfpective of one finglei point of thefe lines, for that gives the Perfpective of every, one of them. But to determine the length of thefe Perfpectives, the Perfpective of the Extremities of each of thefe lines muft be found, if it be not found already. PROB. jo6 A Treatife of * Perfpeclive. P R O B. ' VI. To find the Perfpective of feveral lines parallel {0 fhe Pictute. Since the Perfpective of thefe lines is pa rallel to the bafe of the Picture, Chap. 4. Theor. 3. then by finding the Perfpective of one of their points, you have that of the lines. For to draw a line parallel to a right line given, it's enough to have, a point through which it paffes, as is fhewn in the Elements of Geometry. PROB. VII. To find the Perspective of the Divifion of aLine% Let C D be a line divided into three parts (fee Plate 16. Fig. 1.) C H. H G, and G Dj the ground line is M E, the principal point A, the point of diftance of the eye is B, the points of Incidence ofC, H, G,D areL,I,F, E. I firft enquire after the Perfpectives of the Extremities C and D, taking L M equal to LC, and drawing from M to the point B,, the line MB, which cuts the radial A L in N, which is the Perfpective of C.T take like- wife EK equal to DE, and draw i from K to B a right line which cuts A E in O }'" fo that A Treatife. of Perfpeclive. 107 th&pQ is the Perfpective of D $ confequently N 0 is the Perfpective of C D. Now having drawn radial lines from the points of Incidence L, I, F, E of the parts of C D, to the prin cipal point A, thefe Radials fhall give the divifion of N O ; for the Perfpectives of the points H and G are in the line N O ; and likewife in the Radials A I, A F, and confe quently in the points/ and q common to N O, and to thefe Radials. If we were in queft of the divifions of a line perpendicular to the Picture, as if L C were divided into three parts, then L M muft be divided into three equal parts, and a line drawn from each point to the point B, which will give the divifion required, for the line L N is the Perfpective of C L, which will be cut by thofe lines in the points defired. ADVERTISEMENT. Tho' tfie principal point be unknown, yet the points of diftance may be known, as in this other Figure where D and F are the points of diftance. The point of which K is the Perfpective, is diftant from the Picture the length of the line AE, (fee Plate i6.Fig.2.) and the point H is the Perfpective of A, which by this is on the bafe of the Picture. To io8 ATreatife of Perfpeclive. To have the eight Divifions of A K, divide A E in eight equal parts, and from each of thefe parts draw a line to F, whidh gives the divifions required ¦, for thefe lines, as well as the Radials, ought to cut /.. K in the points which are the Perfp'^lves of thefe points, the diftance of Which is rnark'd .by the divifions of A E. The fame muft be done to find the divifions of A I, dividing A C in eight equal parts ; and from each of thofe parts drawing right lines to the point D. PROB. VIII. To- find the Perfpective of a Parallelogram divided in feveral Parallelograms, on the Geometrical Plan. This Problem is eafily refolved by the pre ceding, Problems, and thus abridged. Let the Parallelogram B C D E be propofed, and di vided in feveral other fmall Parallelograms, (fee Plate 17. Figi 1.) I fuppofe it on a Geo metrical Plan behind the picture. : It may he fo placed, tharall its lines fhall be fome pf them parallel, and fome of them perpendi cular to tlfe Picture, or elfe they fhall all be oblique. This makes two Cafes. . . CASE ^v Trcige jur Vo acg: i D T : % A Treatife of Perfpeclive. 1 09 ,1.1 C A S Ej \. Let BfDE be a Parallelogram. I fup pofe B Gj)« the ground line or bafe of the Picture .Crbe principal point A, to which I draw lines from each part of tlie.divifipri of BCi J apply the Ruler to the points Hand L wfiich I have found, and to the Accidental point ISf^arkrdraw betwixt' IT L and IK right lines': ' I do the fame on the divifions of K L, with relation to the Accident al point M, which gives me the figure H,L K, L, Per fpective of A, B, C, D, arid of all the Paral lelograms w.hjch it inclofes. CASErJl .r-;n ¦ ¦: ¦ Suppofe A B C D a Parallelogram con-? taining feveratathers ; she- principal: poind is E, the pO'i*its,Tof diftance iaTerF andG-; I .fearch first by the* common Rules the Per fpective of the four points A, B,i C, D, which IfindtobeH,I,K,L. Secondly, I apply the Ruler to H I, or on IK, and marks the point N, which fhall be one of the Accidental points, where all the Perfpective lines of the parallels to A D and to B C fhall terminate. I apply likewife the Ruler to H L or I K, and I find the Acci dental point M. Thirdly, 1 1 o A Treatife ofPerJpective. Thirdly, I fearch by the foregoing Problem, the Perfpective of the divifions of A D and of D C : I rapply the Ruler .to the points of H and L which T have found, and to the Accidental point N, and draws ftfaight lines between H L arid j K : I do the fame ort the divifions of K h,, "with relation to the Acci dental point :M, ; -Which gives me the figure nj. K L, Perfpective of A B '£ D, and of all ¦the Parallelograms which it inclofes which Waft 6 tie done. ADVERTISEMENT. This Practice is of important^, ' for by it we perform, what, without its affiftance, would be very difficult. In the firft place it gives Painters the Degradation of 'a Picture: They imagine the Geometrical Plan to be a Parallelogram comprehending, feveral leffer Parallelograms. Then they tifaee firft the. Perfpective of all thefe Parallelograms, which is eafily found by the preceding Operation. After which, as I have already obferved, they have the Degradation of the Picture ; and know what Diminution to give their Figures. '-'^ This Practice furnifhes a fhort.way of copying all forts of Figures, and putting them iri1 Perfpeaive. To copy a Figure, in- clofe it in a fquare Frame, divided in feveral < fmaller p«syn lLy 1 ^ v"I / /'¦:. .•-¦'. \ / '¦'"' '--..\ \ / •*** \"'--.\ ' ATreatife $ Perfpeclive. - 1 1 1 fmaller Squares. Then if the Copy be defired 'of the fame bignefs, as the Original, 'rikflte another Square equal to the fofrrier, or' greater dr fmaller in 'proportion to What yo'it defire your Copy to be* after that carry what is contained in each Square of the Orlgifialj into ; the Square that, answers it in the Gopji This is eafily comprerienclefd.lIf tfidS^ttafce^m^dfed for the CPpy, ise^ual tothat^nichlrreto'fe the Original, the Figure that is copied, will be preeifely alike, and equal to the Original. If it be only a third part, the Copy will be ftill like, but three times lefs.. Now 'tis eafy to pu; this Figure in Perfpective, whe- 'fcher it be an Original or a Copy, as the fol lowing Problem will deinonftrate. PROB. IX. To put any Plan or Figure whatever in Per- f fpective. The Figure rriuft be put in a Frame, or Parallelogram, divided into leffer Parallelo grams, as was faid in the beginning : Then the Perfpective of that Parallelogram muft be found by the former Problem j and the Fi gure in it, placed in the Perfpective of each of the fmall Parallelograms. As for example, it is defired to put the Plan of the Cittadel X in 1 1 2 A Treatife of Perfpeclive. in Perfpective, fee Plate i 8. Make the Square A B C D, which comprehends feveral fmaller Squares, thefe Squares contain the parts of the Plan of this Cittadel. Then find E F G H, Perfpective of the Square A B C Dj and of all its parts. -This done, tranfport into the parts of the Perfpective Plan Z, all the parts. of. the Geometrical Plan X, that, anfwer to it ; This may .be conceived eafily by the Figure/ PROB. X. To put a Circle in Perfpective. Generally in putting any Plan Figure in Perfj :.ive> whither it confift of right or crooked lines, the Practice muft be as we have faid. So to find the Perfpective of the CirC'<= "X (fcp. Plate ig. Fig.i.) after having iac'oied if m ae Square B C D E,and divided the Square as you fee, feaixii for -Z Per- fpecoveof this Figure; which being found, draw the crooked line Z through the points where the Perfpectives of the right lines that cut the Circle X cut one another ; and this Z is the Perfpective of X which was PROB, Page. 711, A Treatife tif 'Perfpeclive. 1 1 ^ PROB. XI. To find the Perfpective of a point in the Air over the Geometrical Plan, fuch as Y. Firft, Find its point of Seat, and the point of Incidence of this Seat ; I fuppofe the point of Sent "to be in the line K B, of which B is the point of Incidence, (fee PUte ig. Fig. 2.) I erect on B the perpendicular B C of the height of Y. > After which, having drawn a line from C to the point of fight a, I fay the Per fpective of Y muft be in the line AC; for having drawn a perdendicular trom Y to the Picture X, its point of Incidence is C } now the Perfpective of the line Y C is in the line A C ; therefore the Perfpective of Y is fome point of the line A C, fuch as^. Secondly, To find which point of A C is the Perfpective of Y, find by the fecond Pro blem the Perfpective of K the foot of Y K ; I fuppofe it to be d 5 then I erect on this point' a perpendicular which fhall cut AC^ in g the Perfpective of Y ; for it is in AC as we have feen, and that Perfpective muft be in dg according to Chap. 4. Theor. -j. PROB. ii4 ATreatife of Perfpeclive, PROB. XII. To find the Perfpective of Y K a perpendicular en the Geometrical Plan. This Problem is the fame with the for* mer ; for to find the point Y, its Seat K muft be found, and confequently the perpen dicular Y K, of which d g is the Perfpective, (fame Fig.j PROB. XIII. To find the Perfpective of feveral perpendicular Lines. The Vertical or perpendicular B C being found, it remains only to find the points of Seat of the perpendiculars given : I fuppofe the points to be H, L, K, their Pferfpectives are a, b, d ; from thefe points draw parallel lines between AB and AC, as hath been fhewn, Chap. 4. Theor. 10. (feeFlate ig.Fig.2iJ ADVERTISEMENT It is eafy to find by Calculation all thefe perpendicular Perfpectives, or the Perfpective greatnefs of lines which are perpendicular en the Geometrical Plan ; for in the Tri* angl Qlabs ig ^Se f7*\ W'V A Treatife of Perfpeclive. I 1 £ angle ABC as B C, a e, hf, dg, are pa ralleled they are proportional ; fo that know ing B C to be equal to Y K, you have the length of ae for > A B, BC : : A a, a e, (fame Fig.) It is as eafie to make a Scale for the Per fpective as for the Geometrical Reprefenta- tions ; which Scale gives all the Meafures the parts of a Picture ought to have. Per fpective Scales are called Fleeing, as we have faid, becaufe that whereas all parts of a Geo metrical Scale are equal, thofe of Perfpective decreafe gradually. To have all their Divi fions, make a common Scale in a Geome trical Plan ; that is to fay, divide into equal parts one of the lines of that Plan, which makes a right angle with the Picture, or is perpendicular on it : Then find the Per fpective of that line and its divifions, and that Perfpective will be the Scale defired ; as if B K were a Geometrical Scale, the line B A, Which is its Perfpective, will be the Fleeing Scale. We have feen that the Perfpective lines of lines parallel tp the Picture are divided in the fame proportion as the lines of Which they are Perfpectives. A Scale in frorit, is the line which marks this proportion : 'Tis eafy to find this proportion, and confequently the Scale is eafily made. I z, TVv* 1 1 6 A Treatife of Perfpeclive. This done, 'tis eafy to make a Scheme of the whole Compofition of a Picture .- We may find by Calculation, as we have faid before, the fize of all the lines which meafure the Objects that are to be reprefented, and confequently the meafure of all its parts. They may be exprefs'd by numbers, as an Architect does of a Building, of which he is forming the Defign. PROB. XIV. To put in Perfpective the Pilafler X and Z, parallel to the Picture. Put firft in Perfpective, the Geometrical Plan of the bafes of thefe Pilafters : So A be ing the Geometrical Flan of each bafe, their Perfpectives (fee Plate, 2©>) abed, efg h, i n op, muft be found. The Operation is fhort, for according to Chap. 4. Theor. 3. all that is in the Geometrical Plan upon the fariie parallel with the Picture, ought to be in the fame parallel in the Picture : Having thus found the Perfpective of abed, the other per fpectives of the bafes are' found, by drawing parallels to the bafe of the Picture, and lines to the principal point, as the Figure it feif fhewsi To A Treatife of Perfpeclive. 1 1 7 To finifh the Perspective, we muft raife perpendiculars On all thofe points which are the Perfpectives of the points in the Geome trical Plan, on which are raifed the fides of the Pilafiers. ADVERTISEMENT. Obferve tliat all thefe Perfpective bafes are equal, that abed is equal to efg b, for the Triangles abe and e f g being betwixt the fame parallels, and on equal bafes, to wit, ab and ef, are equal. It is the fame with the Triangles a.c d and eg h, which are likewife betvvixt the fame parallels, and on equal bafes d c and* g h ; for according to Chap. 4. Theor. n. the Ferfpectives of the equal parts of a line parallel to the Picture, are equal. The fides 'of thefe bafes which arc in the fame line parallel to. the Picture, as o n, ef,ab, are equal; but the fides which are in the Radials, or in the lines which terminate at the principal point, are unequal ; the line ni is .longer than eh, and eh is longer than a d. So that as a. Pi Lifer is remote from the principal point, the fide of its bafe augments, ¦which is the reafon that the face of the i ilajhr, which is raifed" on that fide, augments and I 3 becomes 1 1 8 A Treatife of Perfpeclive. becomes larger, and infinite, if its diftance were infinite. This is what makes a Per fpective deformed, when a great number of Pilafters are reprefented on the fame line pa rallel to the Picture; for the Pilafters which are at the extremity, fhould have their faces on thofe fides which are in the Radials ; and much larger. In the two Pilafters X and Z,(fee Platevo.) the faces of the front M and N are equal j but the face Qpf the cilafter Z which is the re moter* , is greater than the face P of the Pilafter X which is nearer the Eye. This inequality is ofienfive ; but to avoid it, we fhould not put too great a number of Pi lafters on the fame line parallel to the Picture. It is likewife impoffible that the Eye can dif cover at one view, an intire row of Pilafters all parallel, if it be not at a great diftance ; and -then all thefe Pilafters are found' to be very near the principal point ; and become fo confufed, that their inequality is not leeri. Al Pictures ought to be limited, as we b.ave Uid beioi ?.: becaufe they ought t , be feeriat one view. It would be impoffible to fee thofe parts which are too remote from the principal point ; for the Rayes that fhould tome from thence to the Eye, could not (enter it.' Such as take right Methods', never bake Pictures, or Perfpectives, in which any t>--> <¦ • .-, ......... i j :., . , . ,..- j.^ Pit^i, JjS otati e %o :z^i ¦ ATreatife of Perfpeclive. 119 thing fhall appear monftrous or irregular. If the fubject be vafl, they fuppofe it at a great diftance, and then all its parts come near enough to the principal pioint ; and become fo faint, that we are not fenfible of the irre gularity we fpeak of. Befides that, according to the Rules of Painting, as what is directly oppofite to the Eye. ftrikes it more lively, fo we generally colour it more, and by confe-' quence we darken what is remoteft from the principal point. Now this darkening regu lates the bignefs, and is the reafon why the irregularity is not fo perceptible. When a long Hiftory is to be painted to adorn a Gallery, as Poets divide their matter in different Books, fo Painters fhould diftribute this Hiftory in different Pictures, having each of them their point of fight. PROB. XV. To put in Perfpective Columns which are parallel to the Picture. Suppofe X to be the Plan of each of thefe Columns : We muft put in Perfpective all their bafes, that is to fay, their Geometrical Plan, as the figure fhews, (Plate u.) The difficulty is, to determine on what points of thefe bafes we fhould raife perpendiculars to inclofe the vifible parts of thefe Columns. I 4 Firft, 120 A Treatife of Perfpeclive. Fir It, For the Column A which is oppofite to the Eye, it is evident, that having drawn, a b the Diameter parallel to the Picture, .it is on the points a and b that we muft raife thefe perpendiculars. So the vifible pa. t of the Column will fall upon the portion a b c. Secondly, In -^Column we can only fee what is betwixt the vifual Rayes which touch it, 'that is to fay, the Rayes which areas Tan gents to the Column. We fee different parts according as we. change our fituation ; or as the Perfpective becomes more diftant from the principal point. So that we fhould be deceiv'd, if on the bafe of the Column E, we raifed the fides of the faid Column upon the extremities 7 and g of the diameter /^ par rallel to the Picture ; as if in that fituation, the fame vifible part of E were alike to that of A. Thirdly, What muft then be done? Ifwc raife a Pa'afitr on D, we fhall fee. two faces raifed on ik and'kl. Now a Column isa Pilafter whole angles are cut off} fo it is upon the points d and o where the Diagonal cuts the Circle (or Perfpective of the Circle) which is the bafe of the Column, that we muft raife fhe perpendiculars, and we fhall fee, that by this' Rule we reprefent oply that part of the fofimn which is vifible. ADVER- <*fC T2.B P Late 3-J A Treatife of Perfpeclive. 121 ADVERTISEMENT. According" to what we have faid, Columns are Pthfters 'whofe angles are taken away. We mav then difttnguifh in each vifible part of a Column, two parts tliat anfwer to the two faces "Of a Pilafler, as in the Columns B and C, F and H anfwers to the face which is in front, and E and G to that which is on the fide. The parts F and H are always equal, but E and G unequal. as we have fhewn in Pi lafters. This is the reafon why Columns parallel to the Picture, Wwtii usey are remote from the principal point, ought to appeai more grofs, which makes a bad profpect , becaufe if their diftance were infinite, the-) inuit Oc infinitely grofs; but this never happ-jns, for the Eye, continuing fixed on a certain point of fight, cannot fee to the right and to the left, but in a limited extent, unlefs the Eye be infinitely di ftant ; and then all xhtColumm being confufed, the excefs of the one half above the other in their Perfpective, cannot be fenfible. PROB. XVI. r~ To put in- Perfpective a Gallery adorned with . Pilafters and Columns. The firft thing to be done, is to put the fchnography pr Plan of this; Gallery in Per fpective j ancl if they be Pilafters tp raife per pendiculars 122 A Treatife of Perfpeclive. pendiculars on the Plans of their bafes, as we did in the i jth Problem.But if they be Columns, we muft mark the Squares which inclofes their bafes as here CDEF; and having drawn the Diagonal C E, raife two perpendiculars on the points G and H, where this Diagonal cuts the crooked line, which is the Perfpective of the foot of the Column ; as we have told in the foregoing Problem j and as the Figure will give you to underftand. ¦'A. ADVERTISEMENT. I do not believe it is poffible to find furer and exafter Methods of Practice 5 I fay more exact, becaufe I allow this not to be altoge ther fuch. It fuppofes the Eye to fee always the 4 Treatife of Perfpeclive. i % ^ the half of *a Colump, which is npt true. If the Column be large, the two Rayes which touch the crooked line G H Q in the points G and H, cannot reunite in the fame point, to wit, the Eye. Neither is it true that the two perpendi culars which inclofe the Column, ought to be raifed on the points of the bafe, where a Dia meter parallel to C D, fuch as M N, cuts this bafe ; for it is evident, that 'tis not that part of the Co'umn which is feen, when the point of fight is in A. PROB. XVII. To put a fucceffion of Pillars, or St at ties, in Per- fpe&ive, or a Row or Aflty of Trees. The Plan muft firft be put in Perfpective. I fall thefe Trees and Terms are equal, and the Statues alike, aU we have to dq, is to di- rninifh them in proportion to their diftance. Now we have already fhewn, that it is eafie to find the Perfpective bignefs of an Object ; fo when a Painter knows,that where he places a certain Figure,it ought (for tnftance) to be a fourth part lefs,than if he placed it on the fore part of the Picture ; there is no doubt but he muft make it a fourth part lefs. It is the fame with a Term or a Tree. All Books of Per fpective are full of Figures, fhewing how to pradice the Problems which are here propofed. PROB.- 124 ATreatife of Perfpeclive. PROB. XVIII. * To />«£ * hollow body in Perfpective. Every body,of whatever Figure, is inclofed by the lines which meafure it, and certain principal points which terminate its Figure. So to put in Perfpective a body of any fort of Fi gure, we need only to find the Perfpectives of its principal lines and points. I fuppofe its fitu ation, in refpect to the Picture, to be known : To put the hollow body X in Perfpective ; this, body is comprehended between eight faces, fuch as A B C D ; its hight is A B ; the figure of its bottom is A E F.G H D ; that of its overture is B C I K alike and fimilar (fee Plate 22. Fig.i.) To put then this hollow body in Perfpective, we muft' First find the Per fpective of A EFGH D, then the Perfpective of the perpendiculars A b and D C, and of the others which meafure the height of X, and by joynirig their- tops, we have the perfpective of X, as is evident ¦> but that the Perfpective may have its effect, and that X may appear hollow, it muft be fhadowed as-youfee. The Figure alone is fuffitient tq make us comprehend how to put the body Z in Perfpective (fee Plate 22. Fig. 2.) We muft find the perfpective of its Plan, and of its principal . lines , and then fhadow the parts which are.not expofed to the light, as we fee in the Figure. PRO B. A Treatife ofPerfpetlive. 125 P R O B. XIX. How to put aS forts of Objects in Perfpective1. Perfpective, as we have already faid, teach eth not how to Defign, how to paint Men, Beafts, Trees, and Architecture; all we ought to expect from a Treatife of Perfbective, is to know how to put the principal points of an Object, andthe principal lines which fhew its Dimenfions, in Perfpective. The reft is the, bufinefs of a. Painter, or of one who pan Defign. So that if it be a piece of Architecture which we would put in Perfpective, after having agreed on the place it is to have in the picture, and on its largenefs, if we underftand not Architecture, we muft employ one that knows how to defign it ; or if it be a Figure, we muft imploy a Painter who knows the defign. GHAK. i2<5 ATreatife of Perfpeclive* CHAP. VII. QfPiclures which are not per- pendkular on the Geometrical Plan ; which are inclined or parallel to the Horizon ; which are floping in refpecl of the Eye ; and laflly, of thofe which reft on an unequal and irregu lar ground. Hitherto We have fuppofed the; ground of the Picture level and plain ; and placed Vertically, the- 1 is to fay, perpendicular ori the Horizon, and feen in front. Let us fee what is to be obferved in all the cither fitua tions it may be fuppofed to have ; whether it be ori a Flan, or on a place Concave or Convex, in a Ceiling, or in a Vault, on a Wall in which there are Cavities, and Sally ing or Re-entring Angles^ or on an unequal and irregular Ground; AH this requires no other Rules than thofe we have already pro pofed ; but it is neceffary to be well un- deiftoodi White ATreatife of Perfpeclive. isy Whatever be the fituation of a Picture, or whatever its ground be, 'tis eafy to make appear there, whatever is poffible to be re- preferited in a common Picture, in following itill the fame Rules. The fituation of a Picture, whatever it be, may be eafily re duced to that of a Vertical Picture. To that end we muft fuppofe the difference to be in the fituation of the Spectator. As if (for in ftance) the Picture be parallel to the Horizon, we need only fuppofe the Spectator, not to be ftanding, but lying upon the ground, and confequently parallel to the Horizon; and then the Picture hath its ordinary fituation with refpect to the Spectator. We have fhewn, that bodies Concave, Convex, or Rugged, if feen from afar, appear flat and even 5 that the greateft parts appear fmall ; andfo confequently thofe that are unequal, may have the fame appearance as if they were all equal. We may then divide an un even ground of any figure, fo as to make it appear even, and compofed of equal parts,. Confequently if we there paint things ac cording to the extent of the place, that is to fay, making that to be greater that is in a greater place ; if we paint two different parts of a Man, that are equal, as (for inftance) the two Eyes in two places of a different bignefs, this Painting, which if feen near, would ap pear 12B ATreatife $ Perfpeclive. pear deformed and without proportion, yet if feen from a-far, may have an agreeable ap pearance ; becaufe what is unequal appears equal 5 and fo thefe two Eyes unequal in the place where they are painted, fhall have an equal appearance. A general expedient, for fucceeding in all the forts of Painting we fpeak of in this Chapter, and for doing things eafily, is to make a Picture on an even ground perpendi cular to the Horizon, and fuch as we would have that appear which is painted on a ground that is not fb, or that hath hot an ordinary fituation. There is a neceffity of making this firft Picture, even then when we under take to work on a level ground, which hath neverthelefs inconvenient fituations^, in which it is difficult to take all its meafures. I do not except even the Vertical Pictures ; if it ought to be placed much above the Eye, ap- ply'd to a Wall, and feen at a diftance, jn which it will not be neceffary to raife the Eyes too much, nor to turn the Head back wards to confider it , for then the point of fight would be much lower than the f icture ; and fo the performance would be very trou- blefom; Now to avoid what difficulty may happen in this cafe, make a rude draught or model in fmall, of the great Pictnre which is to be work'd. This A Trtatife of Perfpeclive. 1 2 9 This may be eafily underftood ; for to make this Model, give the Cloath the fame meafures as you give to the great Picture, only reduced ; that is to fay, fuppofing (for inftance) that the true Picture ought to be of 2fPpt} and that the principal point fhould' be as many feet below the bafe of the Picture^ take the Cloath of 20 Inches, and place; the principal point 20 Inches below this Cloath. There we paint the rude draught or model of the. great Pitture : We- divide this rude draught into fmall Squares, and likewife the Cloath of the great Picture j then there is no difficulty, but to reprefent in each of thefe Squares, what anfwers fo it in fuch or fuch a Square of the Model. This being done, the great Picture hath the defir'd effect. I. Of Pictures that are Inclining or leanings This fituation requires no new Rules. We muft do the fame as if the Picture were perpendicular to the Horizon. Xisa Picture inclin'd on the Horizon Z, the point of ftation is B, and B <*'the height of the Eye. Suppofe the line B A parallel to the Picture X, and! equal to B a ; that is to fay, we mnft ima gine that B being ftill the pointof ftation, the Spectator inclines himfelf and becomes pa rallel to the Picture. Then A C is the height K of 1 5 o A Treatife of Perfpeclive. of the Eye, which is fitiated at the point A. I fuppofe A D parallel to the Horizon,- or to B C ; fo D is the principal point, or point of fight, and C D is the Vertical line , (fee Plate 25.) Thefe things being fuppofed, if E be the vifible point, 'tis evident that its Per fpective will be F. If the Pifture X were not fix'd, that fo we could fet it upright, and that the line AB became perpendicular on the Plan Z : Then in this new fituation the point A would be the fame as a, and D the fame as the point d. It remains to prove, that the Per fpective of E will, after this change, be in th& fame point of the PictureX. That is to fay,that as C D would be the fame line as C d, fo C / would be the fame line as C F. A D and a d are parallel to B E, and a d—A D. So the Triangles af d and C/'E are fimilar. Therefore ad, CE : : df, f C. ' The Triangles A D F and C F E are likewife fimilar. Q A D, C E : : D F, F C, b0 a a, CE:: df / C. Now a d,=A D, and C E=C E : Then AD, CE :: 4tHK-D F, F G. Confequently df, fC : : DF, F C. So Jf+fC, /C : : D F-f-F C, F C. But df+f C=D F-h F C. And confe- * quently F C=/C $ vhichwas, &c. Thus. A Treatife of Perfpeclive. I g r Thus we fee what is to be done when the Picture is inclin'd^ in regard of the Eye. If the Picture proprjfed is X, and the Eye that muft fee it be in A, we muft imagine a pa rallel to the Vertical fine C D, which parallel paffes through A the place of the Eye. This parallel is A B. On the point B where this parallel cuts the Horizon Z, we muft raife the perpendicular B a equal to B A. Then there will ftill be the fame point of fight, which will be D or d : So we muft draw on this Picture the Perfpectives of the Objects propofed, Recording to the Rules which are given for fuch Pictures as are perpendicular on the Geometrical 'flan. Then giving to the Picture X its firft Inclination, and placing the Eye again at A, the Perfpective will an fwer our expectation ; that is to fay, the Picture will have the appearance which it ought to have in that fituation, the Eye being placed at the point A, and the vifible Objects having the fame difpofition in regard to the point A. II. Of Pictures parallel to the Horizon. 'Tis the fame thing whether thefe Pictures be feen from low to high, as when they are in the Ceiling of a Hall 3 or from high to low, K 2 as 132 ATreatije ofperjpenive* as if they were in the Pavement of a Church, and were feen from fome Gallery. There is nothing particular for fuch Pictures; the Rules are the fame. Suppofe Z the Horizon, and X a Picture parallel to it,placed in the Ceiling, A is the Eye, and fo B ' is the point of fight, fuppofing the Eye placed under the Center of the Picture. But becaufe in fuch a fituation' we are obliged tp turn our heads quite back, Which is uneafy, it is more natural to retire from under the Center of the Picture, and then Bis no more the point of fight. If the Eye be placed in I, the point of fight will be in G. Suppofe then m to be the vifible Object, the line D M may be- cgnfidered as the Geo metrical Plan on which is the line D m. The height of the Eye above the Horizon is I L; but its 'height above the Geometrical Plan MD, is I F, and its diftance from the picture X, is IG. This Picture X is perpendicular on LB; ; and fo this fituation, as we con ceive it differs nothing from the ordinary fitu ation, there is therefore no occafion for new Rules to work upon a Picture parallel to the Horizon. It is evident that the Perfpective of m is n,. which is found after the fame man ner, as 'if the Picture were perpendicular on the Horizon. When thefe Pictures, which are fituated parallel to the Horizon, are round, or when they ilMHIIlut^' ill 1 1 w T •si A Treatife of Perfpeclive. 123 they reprefent a Cylindrick body, or Polygon, if the point of fight be not at the Center of the Picture, we muft chufe almoft as many different points of fight, as the Polygon, hath fides ; becaufe in fuch a cafe the Eye changes its place, to confider the different faces which are raifed on each of the fides of the Polygon. For 'tis evident, that to the end the Eye may fee an Object which fhould be in the line G H, it muft be placed in F, and then L H repre fents the Geometrical Plan, F is the fituation of the Eye-, and the letter C marks the point of fight. In the Ceiling of a Room, the point of fight ought to be in a place which may be , eafily feen, and confequently near the moft confiderable Door, to the end that the Paint ing in the Ceiling may be feen from the entry, inviting, as it were, to enter farther, and find out the place from whence it may be fem with moft advantage ; that is to fay, from whence the Figures appear in an agreeable proportion. If the Ceiling were not even, but like the top of a Coach, we may then reprefent in it feveral Objects, or different Complica tions of Pictures, having each their point of fight ; for it is not poffible that the Eye, in whatsoever place it may be fituated, can at a view comprehend the whole Painting of K ? a 134 A Treatife of Perfpeclive, a Cuiling. Therefore it muft change place ; and confequently we ought to fuppofe feveral points of fight ; but all thefe pieces of Paint ing ought to have a relatipn, and form by a certain union one fubject by which the whole Painting is joyn'd. When a Ceiling is only painted with Ornaments, there is no occafion for Perfpective, Geometry is fufBcient ; that is to fay, we muft draw each Ornament in its juft Meafures, as if the Eye were exactly oppofite to it. III. Of Pictures that ftand fideways in refpect of the Bye. -, A Picture is faid to ftand fideways in re fpect: of the Eye, when it is not oppofite in front. We maj confider it in this fituation as if it weFe inclin'd ^fee Plate 25.) The Picture X ii fideways in refpect of the Eye A ; it is incline on the wail Y, which we may con fider a:- the G :ometrical Plan. So that this fitu ation is the fame with that of which we fpoke in the firft Article of this Chapter, and needs po other Rule. IV- Of A Treatife of Perfpeclive. 1 3 5 IV- Of Pictures upon Concave and Convex bodies, or on fuch as have Cavities and Eminences. Such Pictures ought to be feen only at a great diftance, to the end that the Concavity or Convexity of the ground on which they are painted, and the Eminencies and Cavities which are there may not be fenfible, becaufe of the great diftance ; and confequently that the ground may appear equal and flat : And then the parts of fuch a Picture feen at a di ftance, appear not what they really are : Thofe that are great appear fmall ; therefore what is here painted, ought to have quite different. Features, from what is drawn in a level Picture feen at a fmall diftance. The fame happens in all forts of Pictures, which are feen fideways and at a confider able diftance. Suppofe (for inftance) the wall Y to be a Picture (fee Plate 2 5.; The Eye is in the point A at a diftance, which I could not here reprefent fo great as it ought to be imagined. The Picture Y hath unequal parts ; a is lefs than b, b is lefs than c, and c is lefs than d. Yet thefe parts feen from A may appear equal, and have the fame appearance as the Picture X. So that inftead of repre fenting a Head in the Picture X, where all its K 4 parts 136 A TreatiJ.e of Perfpeclive. parts muft be done in a natural proportion, as (for inftance) the Eye muft be equal : In -the Picture Y, the Eye which is in 6, ought to be fomuch greater than the other Eye, as this other is in a place whofe appearance is leffer. The Figure gives a fufBcient Idea of this matter here. In this fituation, and generally in all others where Painting is to be done on irregular bodies, and which ought to be feen from far, we muft make a Model of what we woulcj reprefent; that is to fay, we muft paint upon a Picture that is level and even, as we have already faid, whatever we would reprefent on another Picture that is not fo. We divide the level Picture in feveral fmall Squares ^ then make a frame of the fame bignefs, which we divide in equal fquares with the help pf feveral Threads. This frame we put in the place from whence the uneven Picture, whofe parts are unequal, hath the appearance of the Picture which is fmooth, and whofe parts are equal. As heie the Picture M hath the ap pearance pf the picture X . This frame being prepared ai.:! placed, we put a Candle in thp point A, where the Spectator is fuppofed tp be placed. The fhadow of the Threads pf the frame A, will mark Figures on Y, to re prefent the Squares of "X ; fp we muft tranf portwhat is jn each Square of X, into the ' cor- ra.00 szy ~* r-<— t— Jill J. A Trmtife ofPerfpeclive. 137 correfpondent Figures of Y. The Figure is fuf- ficient to demonftratfe this Artifice. For a clearer light into this matter, confider the Figure reprefenting a proportioned head in the* Square Z. If the Figure X were fo fituated, that irom a certain point from whence it might be feen, it would have the appearance of the: Square Z; then by ob- ferving what parts of '.r. correfpond to thofe ofX, and painting in hefelaft, what is in thofe of Z; that head inX, which at a near view, appears monftipus, will, if feen from afar, appear proportioned, and every way, like that in Z. We ought to follow the fanie method in painting Figures on a Concave place, which fhould appear upright and perpendicular, as that of our Saviour fix'd to the Crofs. A Con vexity or Concavity feen from afar, appears flat 5 fo that to paint a Crucifix qn that fort of Surfaces, it muft not be dpne in its natural proportion, but in fuch, as if feen from afar, whatever be the figure or bignefs of its parts, they may appear in their juft proportion. Suth Pictures are not made for a near view. A Painter ought therefore to confider whicb, is the place, where a Convex or Concave picture, hath the appearance of one ftnootb, and even ; it is in this place where he ought fp put the frame. The light of the Candle placed 138 ATreatife ofPerfpeclive.. placed in-the point which reprefents the Eye, will. mark in the place that is Concave or Convex, the parts to which every pare of the frame anfwers- So in Painting the fame things according to the proportion that is -betwixt the Squares of the Concave or -Convex picture, and the Squares of the frame,, she ap pearance cught to be the fame. ; J& '*> To do this fort of Perfpectives after an eafy Method,,-fome pretend to prick all the fea tures of the Modefj and place it as the Picture is here, (fee Plate 27.) for the light of the Flamboy A which paffes through thefe 'little holes, will delineate all the features of this Perfpective. But let us confider well, that the light which paffes through thefe fmall holes,as it fpreads it weakens ; fo that it can never denote at a great diftance the features of tbe Model that is prick'd, through which it paffes. Whatever Method is ufed,fuch Perfpectives may be rhade to furprize thofe that fee them near and at a diftance. If {for inftance) we would paint St. John the Evangelift on a wall, with a green Habit and a blew Mantle ; we may paint in thePerfpective,Meadows,Fields, Forrefts and Seas. If his Girdle be white,there we may paint Streams of Water i likewife Lakes in the place where hisGofpel is opened, and fhews the Leaves white and "extended. But all fuch Figures as are different from the natural A Treatife of Perfpeclive. 1 3 9 natural Features pf the principal Figure, muft be fo fmall, tjjat from the point from whence the Perfpective is to be? feen, they have no • npearance ; and their colours muft agree with thofe of the principal Figure, and pro mote its beft appearance. Much after the fame manner we may. reprefent in. this Per fpective, fmall Creatures, Fifhers, Ships, Birds, and a thoufand other things, which at a di ftance will not appear, their Features lofing themfelves becaufe pf their fmallnefs, and their colours beingconfounded by that pf the Habits of St. Jebn, which is the fame. Neverthelefs all thefe little Figures, if fees near at hand,. make the Perfpective fo different from a St. John, that one would not think it could ever reprefent him, be it feen from what place it will. I have already faid, that this fort of Per fpectives fhew beft in a long Gallery. To ren der them more furprifing, they are fhewed through a hole at the door, to which is ap plied a Profpective with two Glaffes •¦> fo that tho' the Image of St. John be turned up-fide down, this Image appears upright, which makes the Perfpective yet more con fufed, and St. John lefs eafy to be known ; for we can conceive nothing in its proper Features, nor can we perceive any relation to *hefe fmall Figures, which might have been made 1 40 A Treatife of Perfpeclive. made upright. This only happens when we fee the Perfpective at a near view. V. Of Pictures and Statues made to be placed in high and eminent places. After what we have faid, it will be need- lefs to acquaint you again, that in Pictures which are upon flat and fmooth grounds, but yet are placed much above the Eye, things ought not to be reprefentedt in their natural proportion ; this muft be underftood of Sculpture as well as Painting. In fuch a cafe, if we would have things appear what they are, we muft paint them otherwife than they are. ( For inftance, ) In making a Picture of our Saviour for a very high place, to the end that his Head may not appear too fmall in refpect to the reft of his Body, it muft fee made greater. The higheft parts which are feen under greateft Angles, appearing fmaller than they are, ought to have a more than natural fize, to the end they may feem to be of a natural fize. That is to fay, in the Ex ample propofed, that the Head of Chrift ought to be bigger, to bear proportion to the other parts of his Body, which being lower, and nearer the fight, preferve better the ap pearance of their true fize. Tzetzes A Treatife ofPerfpeclive. 1 4 1 Tzetzes relates a Hiftory which fuits to Our purpofe: He fays, that the Athenians having refolved to place the Statue of Minerva on a high Column, gave orders to Phidias and Alcamenes to make each of them a Statue of this Goddefs, defigning to dedicate to her the fineft of the two ; Alcamenes made the Figure of Minerva, very fine, with an agree able Vifage ; there could be nothing finer, when feen near ; every one went to his Work- houie to fee it with Admiration. Phidias, on the contrary, madp her Figure quite other- wife ; he placed her Lips in a wide diftance, her Mouth open, and her Noftrils large and great. Thefe great parts had no beautiful Afpect in his Work-houfe, for which reafon he was to be fton'd by the People. But when the two Statues were put in their places, that of Phidias appeared with all the Beauty that could be defiled, and was much efteemed by the Athenians : And that of Alcomenes loft all its Grace, and appeared ugly and ridi culous.. There is likewife a Rule, that in all large Gigantick Statues, no part ought to be made fine ; it is enough that it appears as if it were hammered out } becaufe, befides its being needlefs to make it otherwife, what is .rough appears beft at a diftance, as being more palpable. • Painting 1 4<2 A Treatife ofPerfpetlivL Painting and Sculpture have nature for their Object; they try to reprefent it ; \i ought therefore to appear in their Works what it really is. Nothing muft appear too fmall, or too great, in their Mixture. Thus a great Figure, which exceeds the natural fize, does not well, unlefs it be fo remote, as to appear in its natural magnitude. This was obferved iri the two fineft Monuments now remaining of ancient Rome, namely, the Trajan and Antonine Columns. This laft is 175 foot high, and thaf of Trajan is 140. The proportion of all the Figures of the Relieves that cover thefe Columns, anfwers to their fituation ; for their parts enlarge as they mount farther from the Eye ; fo that thofe which are on the top of the Column, are as well . feen, as thofe which are below ; and all is fo equal, (as M. Raguenet, who has lately given us a Defcription of thefe Monuments obferves) that the Soul deceived by the Eyes, does not think of the difference of the fituation of Objects, which muft of confequence fink the difference of their Gran- diire. There is no certain Rule for judging what ought to be the grandure of a Statue placed in atiigh place, to the end it may appear in its naturaj fize. Father Tacquefs Rule agrees not with Experience; it does not fatisfie the Eye* ATreatife of Perfpeclive. 1 43 Eye. This is what obliges Painters, Carvers, and Architects, in-Works -of confequence, to make Experiments to. know what fize a Statue oughjt tp haye in a place, where it is to be put, to the end it may appear in a rea fonable bignefs from the place from whence ' it is to be feen. We have already obferved, that it is not the diflartce alone, the remote- nefs, or the height, that can pr'efcribe a cer tain Rule, trhe Difpofitipn of the place, and the Interpofition of bodies,* fupporting the fight, makes us judge an Object to be greater or leffer according as we perceive more things interpofed.Vi'It js t|us that we take the Sun and Moon to he fmaller, when they are above our Heads. We muft therefore on ?!! occafions have recourfe to Experience. We have fhewn that this may be done., by applying- a Frame to the place appointed. I fhalfhene add, That to know the true fize that an Object ought to have in that place, to the end it may ap pear proportionable, the Lines Or Rulers bf this Frame which are parallel, ought to be moveable, like thofe of the frame Z. Con fider one of thefe Rulers, the Figure G fhews how they may move and remain parallel. This frame Z muft be put in the fame place where the Picture, > Statue, or any other piece of Work is to be> Remove or approach the Lines 44 ATreatife of Perfpeclive. Lines of this Frame, till they appear equally diftant one from another ; that is to fay, till the Intervals A, B, C, D, E, F, appear equaL There is a neceffity for their being fo, and 'tis the real inequality of thefe Intervals, which fhews how much we muft augment the different parts of a Statue, according to the appearance We would have it make. It fuffices fometimes to place a courfe grofs body, in the place where the Statue is to ftand, and to diminifh and augment it till it appears in the natural fize. of what we would have A Treatife of Perfpeclive. 1 4 £ nave the Statue reprefent. Thefe Experi ences are very neceffary. The other Rules Which are given, are falfe, becaufepthey fup pofe ; tliat things feen under 'equal Angles, appear; equal. I do not pretend to advije Painters w,hat Subjects* they ought to. chufe Jfot their Pictures ; but in fpeaking of fuch as are parallel to the Horizon, or inclined, and in general pf all thofe which have a fitu- atlpn^uncoinmon, I cannot' forbear faying, that they ought never to reprefent, but what js convenient for the place where the Picture is to be puu It is not proper to paint any thing in a Ceiling, but what may appear in the Air, and be above oiir Heads, So that it would be rediculbos.. to paint Ships, and Ship- wracks in fuch a place. -,.r We muft neverthelefs acknowledge, that the Deft Painters have not obferved this Rule : Dominick Zampietri, called commonly the Dominican^ Native of Bologna in Italy, painted in the Vault of the Church of St. Andrew of tie Valley, a Ficture of our Saviour, who from the edge of the, Lake of Genefareth , difcovering Simon and Andrew in a Bark, calls them to him to be two of his Difeiples. This Painting appeared fo excellent to M. Raguenet, that he allotted it the firft place in his De fcription of the Monuments of Rome. L I may i*\.6 ATreatife ofPerfpeclive. I may be allowed to fay, that Helie fnatched away in a flying Chariot, the Rap ture of St. Paul, or the Affumption of the Virgin Mary, would be more agreeable to a Vault : But in thefe Paintings, as in all others, the Pencil of a Painter does more than the Calculation of a Mathematician. The prh> cipal thing in it, is, That it depends upon a certain Pofture, which makes (for inftance) the Blefled Virgin, in her Affumption, mount in 6ur view, that the more we behold* the Chariot of Helie, ths, more we believe it really to fly. The Ganimedes carried away by Jupiter in the form of an Eagle, defigned by Michael Angelo, Is much admired. M. Raguenet, in defcribing this piece, fays, the Pofture that this Famous Painter hath given to thefe two Figures, namely, the Ganimede and the Eagle, is wonderful ; for the Bird fo intangles him, with his Neck and one of his Talons, that fie holds him with an invincible force, neither can he hinder him from taking his flight. One of his Talons alone, with which he fur- rounds the Leg of Ganimedes, and his Fjead end Neck with which he incompajfes th* Body of that young Man, commands him. fo much, that he has the motion Qf his Wing? £ree to fly, without fuffering bis Prey ip efcape. I make, tjjsfc Remark^ tq 0e end A Treatife of Perfpeclive. \ 47 none may imagine, I pretend that a Mathe matician, without being a Painter, can per form any thing in Perfpective. . We have faid, that nothing pught to be reprefented in a Ceiling, but what may rea- fonably be imagined to be there. Yet if we paint there a Cupola with Niches, we may reprefent Figures, which may appear Hand ing. 'Tis in fuch occafions that Perfpective is wonderful. Amongft the Monuments of Rvme is reckoned, the Perfpective which the Father' Mathew Zacolino T heat in made in the Vault of the Church of St. Silveftre a Monte Cavallo. He has reprefented a Dome in the Quire of that Church with fuch Art, that the moft cucious Eyes are deceived ; neither can Reafon correct the Error of the Eyes. It can- npt be imagined,, but that there is a Cavity in the Vault, at the place where this Dome is pointed, and yet all is fmooth and even. Neat this Dome is feen a fmall Angel painted in trie Mould of an Arch, which commences the Vault of the Quire ; and never did any piece of Painting appear fo truly ernboffed as that is. This Angel feems to be clear of the Vault, not .touching it but with his Head. M. R&guenet fays, 'Tis impoffible to carry Impofture farther. The knowledge of the Rules we have given, war ^ceffary to him $iat painted this Perfpective j But he is ftill L ,2 more 148 ATreatife of Perfpeclive. more obliged ko Painting than to the Mathe maticks. / ¦:.".. Before we finifh this Article, let us fay once more, that a rational Soul can never be fatisfied with what is not likely. We cannot but be offended to fee a weighty thing bear falfe ; I fay, a heavy thing ; for if it had any levity, fuch as the Air could fupport, and the Winds carry, it might then be repre fented without any fupport, in the Air, like Birds that have Wings. There is fomething miraculous in the Rape of Helie, fo that it is not offenfive tp fee her in a flying Chariot ; but When , a Man has not an extraordinary Talent, he fhould not be allowed to reprefent any thing inclining, if he do not at the fame time allow- it a fupport. It is likewife a palpable fault, to place a Picture near the Eyes,- when it cannot per form its effect but at a diftance. Such are all the Pictures which have the fituation, treated of in this Chapter. A Ceiling fhould be lofty 5 if Concavity or a Convexity are not proper for Reprefentations, if they be not at a great _diftance from the Eye of the Spe&ator. To conclude, I'll fay freely, tho' I con- troul the Practice of moft Painters, that their performance in Pictures to be placed much above the Eye, is not tolerable. They fup pofe the. point of fight above the groUnd line, • and A Treatife of Perfpeaive. 1 49 and therefore they reprefent there all the Geo metrical Plan. , If they paint a Chamber, we fee the Pavement : Now this cart never be, for a Picture being as, a Window, can we fee a Horizontal Plan of an equal height with that Window, if this Window were, much above the Eye.. On fuch occafions a wife Painter fhould fuppofe his point of fight where it ought fo be, above the ground line, and fo neither fuffer his Geometrical flan, nor the feet of his figures which are not on the ground line, or too near, fp appear; urilefswe may fuppofe them to be in fome high place, or on a Hill, that theTictufe would fuffer to appear if it were a Window, tho' the Spectator 'were in a Jaw place.1 t L 3 C H A P^ I $d \A Treatife of Perfpeclivh CHAP. VIII) Shewing that the Mofi important #< Rules for laying on of * Colours , proceed from Perfpeclive. * P Aiming, by the meansof Light and 5ha? dpw , fhews . Cavities and Eminencns where all is flat. It can reprefent the Colour of every thing in particular, in all the dfgrees of Strength or Faintnefs j. and according; to the Changes which the Colours' of neigh bouring Bodies make on its proper colour. For diftance does not only weaken and de face the Colours, but the oppofition pf thofe that are near,, alters and changes them.. It is a different thing tp fee a Flqwer alone, and to fee it in a Garden With others : This variety of Colours occafiorls a new one, which unites all the reft. So that "when a Painter is not ingenious in reprefenting this vfnion, all his Colours contradict and fpoil ea£h other. The diverfity of Colours, and how they enliven and weaken by infinite degrees, is a thing to be adrriired. An Arm of admirable whitenefs, exactly ^heifame in all its parts^ is not equally white in refpect to the Eyes of A Treatife ofPerfpeclive. 151 of thofe that fee it, its whitenefs differs ac cording as it is expofed to the light, as its parts are higher or lower, nea* or diftant from the Eye, and 'tis by this we judge it to be round. Its whitenefs cannot then be reprefented, but by the different degrees of whitenefs which we give it in Painting. 'Tis not enough to fhadow fome pai ts of it, and to ftrengthen thefe fhadovvs, it ought ftill to appear what it is,- truly white: So that it is only by a feafonable leffening or augment ing its whitenefs, that we make it appear round, without changing its colour. Painters call an artificial Colour Te'mt ; and Semiteint implies , the diverfe Colours as they are clearer or darker, more lively or more dull. 'Tis the Light and Dark nefs, the Black and the White, that caufe the appearance of a Relieve or embofs'd Work. 'Tis not my bufinefs to fpeak of Colours ; yet fince the fubject requires it, I -fhall copy what is faid by tfiofe who have writ of Painting. Good, Colour, fay they, is that which hath the neareft refemblance to Na ture; when the Objects painted have the fame Colour, and the fame Teint, as the Natural : When the Carnations (by which Painters mean the naked parts of a Figure) appear like true Flefh, whether it be in the Tight L 4 or 1^*2 A Treatife^ Perfpeclive. or fhadow : When the Drapery refembles Stuffs of Silk, Wool, or any other matter 5 and when thefe Colours keep their natural Luftre, which is called Frefhnefs. Finally, Colouring well, is, when the Colours have nothing that is too ftrong to fix and retain the fight more in one place than another 5 or when there is not in the Work any one Colour too pniverfally fpread over the whole ; when, neither the Black nor the White, the Yellow nor the Red, nor any other Colour reigns, but an agreeable Variety of feveral Teints agreeing together, both in the Light and Shadow. Painters do always fuppofe a principal Light falling on the middle of the Picture, where they place what is molt remarkable 5 and as the Colour reflects all round, what is in thc middle is moft enlightned, and the reft appears by reflection. All the Colours are confufed at a diftance ; fo that the Ob jects which are at the Extremity, ought to come near to the colour of the Sky, or of the ground of the Picture. Painters, under the name of Aerial Perfpective, comprehend the Science that is neceffary to make Obje£ts feem to fbun pr to approach us, by a fuc- ceflive order of Teints. This belongs not to fhe Mathematicks ; for Colours are different things from Lines and Features : Yet the Art which A Treatife of Perfpeclive. 153 which we have fhewn, will be a great help for the mdre judicious application of Colours ; and \is that Which remains to be explain'd in this Treatife. To grind and mix the Colours, is the bufi nefs of a Mechanick. It belongs to an Artifan to know tfie Effects of Blew and Red, and what may make thofe Colours more lively or more dull, more clear or more obfcure. It is likeWife the Experience which is got in working, that makes us obferve the different Effects of Colours according to the fituation they are in, and the Changes occafioned by a great Light, or according as an Object Is fuppofed to be expofed to a greater Light. Thefe things I meddle not with- Yet a Philo- fopher who knows perfectly the Nature of Colours, (which is yet a Secret) might give good Inftructions to Painters ; but this does not any way concern what I am to fay here, concerning the Perfection of the Art I treat of. That which concerns the Diminution of Lines in the Plan of the Picture, according as the, Lines are reprefented diftant or more near, is called Lineal Perfpective ; and the Diminution of the Teints and Colours, as we have already faid, is called Aerial Perfpective-. This laft may be afiifted by the firft •¦> for what does a -Painter in applying his Colour*, does i'54 ATreatife of Perfpeclive. does he not, in fome manner, draw the Lines With his Pencil ? Now he cannot do this ex actly, if he does not draw thefe Lines ac cording to the Rules which I have propofeoV To explain this, There needs no great Art in reprefenting a Body which is flat and fmooth, one colour is fufficient $ neither need we weaken or for- tifie this colour in any of its parts, becaufe we fuppofe it entirely alike without any dif ference. It is not the fame iri Relief work. A Statue, tho' all made of the fame Metal, and coloured with the fame fort of Colour, cannot be reprefented but by varying of its colour ; for as all its parts are not turned after the fame manner j and fome advance or fall back, fo they are not equally expofed to •the light ; and confequently there is a dif ference in their colour. Their being diffe rently expofed to the light, caufes a change, and occafions that amongft its different parts, fome have the fame colour clearer than others, Thefe Alterations follow the dif ferent turnings or Features of tlje Objects. Therefore the Science of Shadows, or of what is called the Clear and the Obfcure, depends on Lineal Perfpective, which is only capable of finding thefe Lines by Art., To A Treatife of Perfpeclive. 1 5 5 To conceive this more clearly, and to con vince our felves of the ufefulnefs of Per fpective in applying of Colours, let us give attention to Experience , viz. That there is nothing but what may be; imitated and reprefented with one Colour, with Ink, or with red or black Paftil. We may give Life to a Defign, only by drawing the Out lines. Now Perfpective is neceffary in the drawing of thofe Lines, which areas guides to the Pencil of a Painter, as we have faid. Let us imagine a Statue made of Plans of an equal thicknefs, placed parallel the one above the other, fo as if the Extremities of thefe Plans fhould appear, the Statue would feem to be covered all over with parallel Lines or Strokes. Suppofe this Statue is to be imitated or reprefented. It is certain, that if we find the Perfpectives of the pa rallel ftrokes of this Figure, we fhall make a perfect Image. It would be an infinite trouble if we were obliged to find all thefe ftrokes by the Rules ; but it may be done exactly enough, by the Eye, provided we ob ferve what hath been demonftrated , that what is parallel in the Object, is not always fo in the Reprefentation j and that except what is level with the Eye, and exactly 'oppofite I $6 ATreatife ofPerfpeclive. - oppofite to it , nothing continues parallel ; that fo thofe of the parallel Strokes or Lines, which are above the Eye, in tf?s parts which are moft djftant, feem to defcend 5 and that •on the contrary, the ftrokes feem to moiric \n thofe which are below the Eye, and at -s. diftance, (fee Plate z%.) This may be obferved in the Statue Z, which I fuppofe ¦made of parallel Plans, as I have faid, tho' .they could not be all mark'd : Excepting the Plan which is exactly oppofite , and level with the Eye A, the extremities of the others mount 01- defcend according as they are higher or lower than the Eye A. We fee likewife the effects of the Opticks in the two Globes B and C, whofe ftrokes are diffe rently turned, becaufe thofe of the Globe C, feen from high to low, ought not to have the fame appearance as thofe pf the Globe B, which is feen from below to above by the Eye A. This Figure expreffes my meaning but very imperfectly; but a fmall attention will eafily fupply what neither my words nor the Engravers Art could explain. We may eafily conjecture what ought to be the appearance of thefe parallel ftrokes of this Statue. Now Experience fhews, that no thing can be more capable of giving Life to a Picture, than an exact pbfervance of thefe ATreatife ofPerfpeclive. 157 thefe Lines. This Experience we have in the Prints where Engravers reprefent threads of Drapery; all thefe threads are parallel amongft themfelves in their ordinary Difpo fition , when the fluff is extended in the Loom; but ths Paralleling changes, and each line turns after they are .cut, or made in a Habit ; and that the Stuff is differently folded. When Engravers kriOw how to imitate this Defcription^ when they exprefs the turnings of thofe threads as Perfpective requires j then the Image of what they re prefent, hath a roundnefs equal to that of the beft Pictures ; for it is not the Colours, alone that makes the Relief, but the Strokes of the Pencil, when they are done by ti|e • Rules of Perfpective. We may likewife fuppofe the Statue Z to be made of Plans placed parailefbut ver tically; that is to fay, that this Statue is made fo, that the Plans of which it ;s com- pofed are vertical. It muft be covered with Lines which we may imagine in all forts of Objects. Thefe Lines will be imaginary ; but there are fome that are real , which determine the out-ftrokes of an Object 5 and thefe are the Lines which a Painter ought to follow, and by which he ought. to i $8 ATreatife ofPerfpeclive. to regulate fiis Teint : It is impoffible to mention every thing we ought to ob ferve. In Reprefentations of Architecture , each Stroke or Feature ought to be fuit able to the things ; fo to reprefent a thing that is perpendicular, the Lines which make the Shadows fhould crofs each other per pendicularly. If we fuppofe an Architecture covered with Lines, we muft of neceffity fuppofe a great number, which determine the parts of this Architecture, and mark out its meafures. If I fay, we fuppofe an Ar- fhitetture compofed of Lines, thefe Lines muft be reprefented as is taught by Per fpective 5 that is to fay, as they ought to appear. Thofe which are the Perfpectives of Lines parallel to the Picture , ought to continue parallel to each other; thofe which are perpendicular to it, fhould lofe and confufe themfelves in the principal Point. What I have faid is fufficient : There remains nothing more to be done before I finifh this Treatife, but to fpeak of the Shadows. I fhall fay no more of Colours that are fhadowed, that is to fay, of thofe which are made fainter, or more obfcure, ac'con&ng as the Object they reprefent is mere ATreatife ofPerfpeclive. 1 59 more diftant, or lefs expofed to the Light : My Intention is, to fpeak of Shadows caft. upon adjacent Bodies, by that which robs them of the Light, they beiqg placed be hind it. CHAR 1 66 ATreatife of Perfpeclive. CHAP. IX. A General Obfervation onnthe^ Proportion of Shadows. THIS Subject might be enlarged upon, but 'tis fuflicient to take notice of what we may remark every day, that when the Luminous and Opacpue Bodies are of an equal bignefs, the Shadow made by the Opacpue -Body, ought to be contained between two parallel Lines ; that if the Luminous Body be the leffer, the Shadow grows and aug ments infinitely ; and oh the contrary , if the Opaque Body be the leffer, the Shadow diminifheth, and terminates in a Point. AThjs is the Rule for the Magnitude of "Shadows; their Figures are different, ac cording to the difference of the Opaque Bo dies that occafion them. Experience fhews the effect of the Light, when a Body is ex pofed to it ; and what happens to that Body, when it is deprived of it by another. All 4bis is eafily comprehended : Yet becaufe I would leave nothing to conjecture, I have made thefe following Figures ; X reprefents « Luminous Body, and Z a dark Globe lefs than *9 J^OAje j 6j -ytTi'eatife'of Perfpeclive. \6\ than the L uminous Bod y ( fee Plate 2 9. Fig. 1 .) it. is evident that the fliadow of X ought to terminate at a Point, and form a Cone, whofe top or point is A. If the Opaque Globe Z were equal to the Luminous Body X , the fhadow A of the Opaque Body Z, would be contain'd between two. parallels, "(fee Plate 29. Fig. 3. J When the Luminous Body Y is lefs than the"0^*e Body Z; the contrary ought to happen from what we have obferved, when the Opaque Body is the fmalleft; to wit, that its fhadow diriiinifb.es ; for here* the fhadow A grows larger, (fee Plate 29. Fig. 2.) 'Tis likewife to be obferved , that the fhadow is longer or fhorter as the Luminous Body is more or lefs elevated. In" the Morn- . ing when the Sun begins to appear, on the Horizon; or in the Evening, when he is going down, the Shadows are infinite , as he elevates, they grow fhorter : At Noon, when he is above our heads, if he were di rectly fo, we fhould have no fhadow s ; but this happens only ta fuch as are under the Torrid Zone. What fhadow he makes in our Climate, is always fhorteft at that hour. For the clearer Conception of this, fee Plate jo. Fig. 1 . where X reprefents the Sun, and Y and Z two Opaque Bodies of Cubical Figures, M as 1 62 ATreMifeofPerjpe&ive* asGasm^gffe: Ms t%e Sim is more <$p~ v%m$. ia vegntfl to Z, fe ks fhadow A h fkmmx ibm dae todow B -of the bodv Y, feHq^ritowfeidi!, the Sum hadi a fcls%fe- vfttiea. Sks&m is a privation <&f Light i fo drat wfaam skb Qf&jpee body hinders the Communi cation, etas melt bea Ifbadow, AH Windows may fee camfidemed as Luminous bodies ; f© O&^Ss uiskfi cawsot be enEghtned becaiife of tfae feffisarpditbra of fome Ofatfke body, agsgeas: dbfeasdL There are different de gress u IfoaiSows, tftey saay be ftronger or iaifseer; fiir as the light may come from federal piasKs, the fame Object may have the foetsefo of one fight, acid be deprived of anotlier. If it weceives no light at all , then it mv& be covered with a thicker fhadow. The Sigfot is fometimes refkfled ; and there are bodies which do not receive it directly, and confequently ought not to appear in a lively Brightnefs =, but at the fame time they are not quite obfeare, becaufe in the (ituation in which they ate, the Light is reflected upon therm from the neighbouring bodies. Pewters make a diflinction betwixt a Principal Light, aad what is called a Reflected Light. They chufe their Light as they pleafe: They ought, to be careful in the obfervanceof their Syftem 5 PlatD30 ^aq^jfa, ¦lj:i m If''! £ A z. A Treatife of Perfpeclive. J and that it hides the light from the parts behind it. We have endeavoured to defcribe all the Effects of thefe two Lamps, to the end that upon oc cafion, nothing may efcape the Diligence of a Painter, and that he may obferve where the Snadows of Opaque bodies fhould be, {'fee Plate g r .J But ATreatife of Perfpeclive. 165 But after all, there is a great difference between the "Effect of a Lamp, and that of the Sun : Colours feen by an Artificial Light, appear very different from what they are, when feen in full day light. The Light which is fpread over all the Air when the Sun fhines, and furrounds the bodies, lights them on all fides, which a Lamp cannot do : Therefore it is only good for reprefenting the time which we then ufe; that is to fay, a night enlightened by Art, while the Natural Light is abfent. 'Tis needlefs for me to flay longer on this Subjed, or to be particular upon all the dif ferent fhapes of Shadows, with refpect to the bodies that occafion them, .and to the Luminous bodies of which they hinder the Effects. That is eafy ; therefore I fhall only fay, that after having well obferved the Figure, and the Meafure of the Shadows of Objects on the Geometrical Plan ; we rtiuft find the Perfpective of this Figure, and the Perfpective bignefs of that meafure. , This is the true way of working exactly ; but Painters are not at fo great pains, their work will not allow, it ; they are forced to make feveral Pictures in a Year for an honeft M $ Lively- 1 66 ATreatife of Perfpeclive, Livelyhood. The Illuftrious Painters in an cient times, imployed whole Years in making one Picture: What did I fay, in making? They were whole Years in forming the Defign ; which cofts the , leaft trouble to moft Painters now. The moft Laborious and the moft Exact , defign by the Eye what they would imitate : So that we feldom fee Pieces worthy of Admira tion. This Treatife of Perfpective cannot be ac ceptable, but to fuch Painters as aim at Perfection : Thofe of the Indifferent fort, look upon it as ufelefs, becaufe they are fen fible of their own Weaknefs in. making ufe of it : I hope fuch as are in a Condition to examine it, will judge otherways. Ik- fides, I pretend not that every Picture ought tq be a a exact Perfpective ; or that none can be fine, it the Objects there repiefented, and which we fuppofe would be feen, if it were tranfparent, appear not in their na tural bignefs, and in their true diftance. This is to be wifhed : Yet I acknowledge there are' fome very fine Pictures , tfiat have not thefe Advantages : But we can not from thence conclude , that Perfpective, is only good for painting the Schens of a Tbeattr, ATmtife$Pfrjpe&faje» i6j If Beater, or iotm Archite&ms m ' ^m{Qs0ai " c©# of a GarTery. A PieTure, t 'fay, may fee fecettenr, a8$r* k have not the Efiect of aus cip6f F&$f- ifrvt~, which is knpoffibk, wfeeai the JFi* guns are aH much fnaalter tfeaa the ?*tamraL and fa cannot make the feme IsEfedlBws as t|* things themfelvss wfeiefe we repre fent. In fine, Painting €3»bh>£ pleafe if k be not reafonable: And this fccaneot h&. if the Imitation be e©t perfect; tliat is to fey ». if it do not imitate the tx^h* W-c are fenfible, that what h acted oa a Theater^ i$ but a Eictiori; the Theater ss toofroaf]; al! is there ?PQ much contracted, both) the time and the place, tot conceal its beisg only a Representation ; bust fho* the troth be wanting, we find the Likelyhood, xtixh- put which, tbe piece woujd appear ridi culous. Every thing mull; likewife, in a Picture , have the refemblance of truth. Haying therefore fuppofed the Action which, is to. be reprefented , to be feen from a certain Point, 'tis to that Point that al) oughj to be directed. A Picture may be great or fmall. We may fuppofe Objefls ^ch as we would have them ; but there fliuft be a point of fight, to which all tends ; M 4 and 1 68 ATreatife of Perfp$i%e.. and we cannot draw oneftrokeof a Pencil, without having a regard to this' Point.; Which is impoffible to be done without Perfpective 5 and this,. proves Perfpective to be the Foun dation of Painting- CH A PV . : imenfiohs of a' Body propofed to be put: in Perfpective }1 which belongs t;o Geom epy. But, fay tfey," you go beyond jtne extent of perfpective, in meddlipg to; give your Sen timent of Pdintfn'g in general^" I muft give a reafon for this, and tjbfjtiat ^ejid^fhali". re late, a Conference beiWe&foSocraffi and two, famous 170 jTtmife*fPtir)^fak lameus Workmen, one of which wasa Pam» Per, and the other a Carver _5_ wlhich will be Efficient to demonftrate, that we may con tribute to the Perfection of Arts, wkhoW be fog MechalMs, -* Z.enophm % in his third . Book pf the memorable things of Socrates , relates a Conference tn$t State haAifkh Pifrka^ fus the e^eflent Painter} and with Cfoo» (he ingenious Carver. The Phtlofopher in fracts then* in what might; render their Works rhore perfect 5 This he does after Ms- ordinary mariner, interrogating tbem in fpch order, that in, their anfwers they fpeak a> if they knew b^bre-nanci whstt pe asked w&n, arRFwhat they wanted to be is- Ibwcted in; ^Tfiey ' eafily J^knOWTettge the' truths he. difeoyers- 'This was his metnbdi ik ill the Inductions ne gave, it being, act- jnfrapty tyett caituMwi for Instruction. ¦$& fWererice was' ib% s^go tranftajted'rmtft W*cb {fori ffie Greek, fey atf&aient £olTe- gift*. Ahd-itist^us: Zwo^^kiagof Socrates. , * He \(is adifiirable in all "Cohverfatiori, * and if he met e>en with a MechanJcV * hfe fpoke always fbinething that might he ** fervlceable to\huii, -' ; rt Being once in %H Wotlc-no^e of Pari Z tirf* «** Painter, lie confer^ with hTm Rafter. 4 Treatife of Perfpeclive. 171 •' after this manner. Is not Painting a Re? u prefentation of all that we fee ? For with •' a little Colour, you reprefent on Cloath, " Mountains and Caves, Light and Obfcu- *f rity : You make a difference between -¦ what is hard, and what is foft } between " what is fmooth , and what is fugged ; " You give to Bodies Youth and Old Age ; *' and when you would reprefent a perfect " Beauty, fuch as i§ impoffible to be found, " without fome fault or other; you ufe " ro confider a great many; and taking " from each what pleafes you, you make one *f every way accomplifh'd. " You are in the right, anfwered Par* " rhafus. " Can you reprefent, faid Socrates, what xl is yet more Charming, and more ami* f able in a Perfon, I mean the Inclina* " tion ? " How would you, anfwered ParrhafiuJ .* that I fhould paint what cannot be ex- f prejs'd by any Proportion, nor with any " Colour, and what hath nothing common " with all thofe things you have named, li which are imitable by the Pencil; in a '' word, what cannot be feen ? " Do not Men, replied Socrates, make " Hatred and Friendfhip appear in their " Countenances? 1 7 2 ATreatife^of Perfpeclive. " Yes, I think fo, faid Parrhafus. t^ Then it is poffible to obferve Hatred or '•' Frieadfhip in the Eyes? xi I acknowledge? "Dolyou likewife believe, purfued So- tl crates, that in the Adverfity and Profpe- K rity qf Friends, thofe who are interefted '} Lcep the fame Countenance as thofe who $. are not concern'd ? "Noways/ faid he, for* in the time *5 of our Friends Profperity, our Vifage is "Gay, and full of Joy ; whereas in their "... Adverfkyy we are Dull and.Melancholly. " Ll This then can belikewife painted. -;*" 'Tis certain. ¦ .;¦( ; " Farther, faid Socrates, Magnificence, ¦friGenerofity, Bafenefs, Cowardice, Mo- iSrdefty,- Prudence, Irifolence, and Rufti- **<¦ city •-, all thefe appear in the Vifage or " in the Pofture of a Man, whither fitting or f ftanding. . -sP You fay right. . ; ." Then this can be imitated by the Pencil ? and this is the* tf reafon that they feem to be likewife alive. " 'Tis fo, faid Clitou. lC Certainly then, replied Socrates, you " muft obferve very exactly amongft the dif^ " ferent Poftures of Bodies, which are the " moft natural Difpofitions of all the parts j *' for when fome bow down, others are lifted il up; when fome are preffed, others are ex- u tended ¦¦> when fome are ftrongly "bound, " others are relaxed 5 and in imitating all this, •< you make your Statues approach very near *' to Nature. « >Th 174 ATreatife ofPerfpeclive. " 'Tis true, faid Ctittn. *l Is it not likewife true, continues So- *' rr^ftfj, tliat 'tis a great Satisfaction to the ** Spectators, when all the Paffions of a Man " in Action, are well exprefied > So in the '•'Statue of a Champion 'fighting, thatfuri- +' ous look with which he threatens his