m 
 
PRACTICAL PERSPECTIVE 
 
 BEING 
 
 . A COURSE of LESSONS, 
 
 EXHIBITING 
 
 Eafy and concife RULES for drawing juftly all Sorts 
 
 ' of Objefts. / 
 
 Adapted to the USE of SCHOOLS. 
 
 By H. CLARK E. 
 
 IN TWO VOLUMES. 
 V O L. | 
 
 Sempre la pratica di pittura de<ve ejfere edificata fopra la huona teo- 
 rica, della quale la profpetti<va e guida, e porta: e fenza quella 
 niente Ji fa bene, coji di pittura* come in ogrfaltra profejfione. 
 
 Lion ardo da Vinci della Pittura. Cap. XXIII. 
 
 LONDON: 
 
 Printed for the Author, and fold by Mr. Nourse, in the Strar.d, and 
 Mr. Murray, No. 32, in Fleet-Street* 
 
 M DCC LXXVI. 
 
! 
 
i T O 
 
 CHARLES WHITE, Efq: 
 
 MANCHESTER, 
 
 FELLOW of the ROYAL SOCIETY, 
 
 S your extenfive Knowledge and 
 good Tafte command the Efteem 
 of all that profefs or love the Liberal 
 Arts and Sciences, I have thence pre- 
 fumed to addrefs the following Sheets to 
 ypu ; being well affured, that if they 
 fliall have the good Fortune to meet with 
 your Approbation, I need not be anxious 
 about their Reception elfewhere. 
 
DEDICATION. 
 
 Were your Character, Sir, lefs dif-. 
 tinguifhed, the Obligations I am under 
 to your Goodnefs, would, alone, be Mo- 
 tives fufficient to make me gladly embrace 
 this Opportunity of publicly exprefling 
 my warmed Gratitude, and of teftifying 
 the perfed Efteem with which I am, 
 
 S I R, 
 
 Your mod obliged 
 Humble Servant, 
 
 HENRY CLARKE. 
 
IT may, perhaps, have the Appearance of Va- 
 nity to attempt to write upon a Subject which 
 feems already to have been quite exhaufted. 
 But as the Utility of Perfpective begins now to 
 appear to ' every one, and is no longer looked 
 upon merely as a genteel Accomplimment, but 
 as a moft necefTary Acquisition I prefume, an 
 Attempt to render this agreeable Art more eafy 
 and intelligible to the Learner, will not be thought 
 impertinent. Indeed, I do not pretend to offer 
 any Thing new in Principle ; but the Method I 
 have obferved, I flatter myfelf, will be found to 
 be better adapted to the Capacities of Youth, 
 than what is propofed in any Treatifc that has hi- 
 therto appeared on the Subject. In order to be 
 well informed in this Particular, I have been at 
 the Trouble of collecting all the different Treatifes 
 of Perfpective I could hear of, both in our own 
 and other Languages ^ and from a Comparifon it 
 appears, that while the one Part have written 
 purely for Mathematicians., the other have treat- 
 
 a ed 
 
vi PREFACE. 
 
 ed the Subject in fo limited a Manner, that the 
 Reader could not from thence acquire the leafr. 
 Idea of the Principles of the Art he was endea- 
 vouring to learn. 
 
 The Principle, upon which this Method of 
 Perfpective is founded, is fo obvious to the Un- 
 derftanding, that it is a Matter of Surprize, it 
 has not been univerfally received by the Connoif- 
 feurs. For what can be more eafy to conceive, 
 than, if a Plane of Glafs be fet betwixt our Eye 
 and an Object, that the Rays, in palling from the 
 Object to the Eye, will, by their Interferon with 
 the Glafs, there exhibit an exact Reprefentation 
 of whatever the Eye is directed to. The Bufinefs 
 then of this Method of Perfpective (and it is the 
 only rational one) appears to be nothing more 
 than the finding of thofe Points of Interferon. 
 This is done in LefTon I. in fo plain and fimple 
 a Manner, that the Method of Operation is 
 known by a bare Infpection. Indeed, the De- 
 monftration given to this fundamental Propor- 
 tion, may not be comprehended by every one ; 
 but, perhaps, to fuch it may be thought a fuffi- 
 cient Proof of the Truth of the Operation, to find^ 
 
 that 
 
PREFACE. vii 
 
 that when the Planes are all brought into one, 
 or extended on the Ground Plane, the Thread, 
 or Vifual Ray, ftill interfects the Picture in the 
 fame Point, in a ftraight Line to the Eye. 
 
 When the firft LefTon is clearly underftood, 
 the main Point is accomplifhed. For as Lines, 
 Planes, and Solids, of whatever Figure, may 
 be eafily conceived to confift of Points, the Per- 
 fpective of them may be found by this one ge- 
 neral Rule. This alone I have often found fuf- 
 fkient to feveral grown Perfons, to let them in- 
 to the whole Practice of Perfpeclive. But (as 
 Mr. Locke obferves) fmce we always find the 
 Minds of Youth are opened by Degrees, and that 
 it is Exercife alone which opens them, it is ne- 
 ceflary for them to go through a regular Courfe 
 of Leflbns, in order to fee the Application of this 
 Rule to different Objects. Having therefore 
 mown the Method of finding the Perfpeclive of 
 a Point, I have proceeded to Lines and Planes 
 fituated on, and parallel to, the Ground Plane 
 then have gone to Lines, Planes, and Solids 
 perpendicular to that Plane ; and then mown the 
 Perfpeclive of Lines, Planes, and Solids inclined 
 
 a 2 to 
 
PREFACE. 
 
 to the fame Plane. And in this lafl I have been 
 the more particular, as it is a Part of Perfpective 
 which is feldom underftood. I do not know 
 any that has rendered it practical, but Kirby. 
 I have thence proceeded to Sciagraphic Perfpec- 
 tive, which, as it is a very effcntial Part in Draw- 
 ing, I have confidered every Way, both from the 
 Sun and Candle. The Rules for Catoptric Per- 
 fpective are next given, when the reflecting Sur- 
 face is parallel, perpendicular, and inclined to 
 the Horizon. I have then exhibited the Nature 
 of Theatrical Perfpective ; and alfo mown the 
 Theory of Horizontal Pictures, or Ceiling Pieces. 
 And have concluded the firft Volume, with a 
 a practical Method of taking Perfpective Views 
 without the AfTiftance of any Inftrument. 
 
 In the fecond Volume I have fhown the Ap- 
 plication of the foregoing Rules to a Variety of 
 Subjects j as in Architecture, to the Perfpective 
 Elevations of the Orders, Arches, Doors, 
 Niches, &c. with the Perfpective of various 
 Modes of Buildings, Squares, Streets, Avenues, 
 &c. in Geography, to the Projections of the 
 Sphere, for the Conftrudtion of Maps, Charts, 
 
 &c. 
 
PREFACE. it 
 
 &c. in Aftronomy, to the Conftru&ion of Ce- 
 leftial Planifpheres, Analemmas, Aftrolabas 9 
 Dials, &c. as alfo to the Conftrudtion of Tran- 
 fits, Solar and Lunar Eclipfes, &c. I have 
 alfo tranflated from that excellent Italian Pieee^ 
 Frattato delta Pittura di Lionardo da Vinci, his 
 Obfervations on the Keeping of Mrial Perfpettive* 
 And have concluded the Whole with fome ufe- 
 ful Hints to Painters for the Mixing of Colours,, 
 on the Principle of Sir Ifaac Newton's Theory. 
 The inverfe Method of Perfpedtive I have omit- 
 ted, as being of little or no Ufe ; for any Perfori 
 may find the vanifhing Line of a Perfpeclive 
 View, by producing any two Lines which are 
 perfpedtively parallel, till they interfedt each o- 
 ther ; for that Point of Interferon will always 
 be in the vanifhing Line. Neither have I faid 
 any Thing of the Mechanical Methods of taking 
 Views, copying Prints, &c. as thinking thefe 
 beneath the Attention of the real Artift, who is 
 feldom inclined to have Recourfe to fuch Helps. 
 A Camera Obfcura, indeed, is fometimes ufeful 
 in taking the Perfpeclive View of a Town ; as 
 it is not a very eafy Matter to draw fuch a Num- 
 ber of Objects all in true Proportion, merely by 
 
 the 
 
PREFACE. 
 
 the Eye. But when the View is not crowded, 
 nor very extenfive, the Method I have before 
 laid down mould always be preferred. In the 
 Catoptric Perfpective I might have fhown the 
 Principles of the Anamorphofis, and their Rec- 
 tification ; but as this hath not an immediate Re- 
 lation to the common Practice of Perfpective, I 
 thought it would be rather leading the Learner 
 out of his Way, and have therefore omitted it alfo. 
 
 Such is the Plan of this Performance : Which, 
 however, I do not pretend to fay hath no Im- 
 perfections, for Humanum efi err are. But this I 
 will venture to affirm, that though it is as concife 
 as pofiible, yet it is as complete as neceffary, 
 either for the School or private Study. I have 
 alfo been particularly careful to have it correct 
 in the Printing, having frequently obferved, that 
 when a Work of this Kind appears in the World, 
 the greater Part of thofe who fet up for Judges, 
 are fo bent to depreciate every Thing to which 
 they can frame the lead Pretence of an Objection, 
 that a fmgle typographical Error would be fuf- 
 ficient for them to pafs Sentence on the whole 
 Performance. 
 
 It 
 
P R E F A C E. xi 
 
 It were well that thofe who are ambitious of 
 palling forConnohTeurs in Drawing, would make 
 themfelves perfectly acquainted with the Princi- 
 ples of this Art. And then we fhould not fo 
 often meet with thofe unjuft Criticifms, thefe 
 Gentlemen are fo apt to throw out indifcriminate- 
 ly without regarding the Prejudice they may do 
 a Perfon's Reputation, with thofe who may have 
 an high Opinion of their affumed Knowledge. 
 It is in this, indeed, as in moft other. Sciences, 
 that as foon as fome Perfons have got a Smatter- 
 ing therein, they immediately fet up for profeffed 
 Adepts. But as fuch muft be fenfible of their 
 own Deficiency, (otherwife their Cafe is pitiable 
 indeed) they mould be cautious how they ad- 
 vance, left they mould not be furnilhed with a 
 good Retreat, The Jefuit's Perfpective has made 
 more Pretenders to this Science, than any other 
 Book on the Subject. It is a mere Jumble of 
 Practice without Theory, and full of Abfurdities, 
 and is therefore a very improper Book for 
 Learners : 
 
 pecipit exemplar vitiis imitabik. — — * — 
 
 Nothing 
 
PREFACE. 
 
 Nothing is more common with this Author 
 (and all thofe who have copied from him) than 
 to give the Perfpective of an Houfe, &c. with 
 one Side parallel to the Line of Interferon, and 
 the other diminifhing into the Centre of the Pic- 
 ture. Indeed there is nothing wrong in this Me- 
 thod, if the Views are accounted only partial 
 ones ; but when the View of a Gentleman's Seat, 
 &c. which makes the whole Picture, is exhibited 
 by this Method, there is nothing more abfurd. It 
 is juft as if a Painter mould mow you a fine Por- 
 trait with Half the Face covered, and then leave 
 you to judge of the Likenefs from that. For as 
 the Centre of the Picture is always oppofite to the 
 Spectator's Eye, is it not plain, by this Method, 
 that the Houfe, &c. which fhould be the princi- 
 pal Object of his View, is only a Part of a gene- 
 ral One, and that he directs his Eye to one Ob- 
 ject, in order to look at another ? And is it not 
 equally as evident, that unlefs the Obferver be 
 lituated exactly in the Front of an Houfe, it will 
 be always viewed on the Angle, and muft there- 
 fore have two Points of Diminution ? How ridi- 
 culous then muft a Perfon, who has no better No- 
 tions of linear Drawing than what the Jefuit's Per- 
 fpective 
 
PREFACE, xiii 
 
 fpective affords him, render himfelf to the real 
 Connoiffeur, by calling this bad Perfpeffive, when, 
 in Fact, there is no other Method right. 
 
 I have been naturally led into thefe Refleo 
 tions, by being more than once informed of the 
 ill-natured Critkifms* on a Perfpective View 
 which I lately took of a Gentleman's Houfe near 
 Manchefter* by a Perfon who pafifes for a Con- 
 noiffeur in Drawing and thofe Criticifms made 
 merely becaufe it has two Points of Diminution ! 
 Rifum teneatis ? Whether the Defect lies in this 
 Gentleman's Heart or Eye, I mall not pretend 
 to determine, but will charitably fuppofe the 
 latter- Perhaps the Perforation of the Uvea is 
 not fituated exactly in the Axis of his Eye. If 
 fo, the Rays of Light, after Refraction by the 
 Cryftaline and Vitreous Humours, will fall ob- 
 liquely on the Expanfion of the Optic Nerve in 
 the Fundus, and thereby caufe an Obliquity of 
 Vifion. In fuch a Cafe* his Eye may appear to 
 be directed to one Object, and at the fame Time 
 the Retina be impreffed with the Image of ano- 
 ther ; and therefore his Point of Sight may be 
 out of the Picture, and yet the Central Ray 
 
 thereof 
 
xiv PREFACE. 
 
 thereof coincide with the Axis of his Eye. The 
 Effect is obvious. The moft juft Reprefenta- 
 tion to fuch an Eye, muft appear to be a mere 
 Anamorphofis* 
 
 Another Abfurdity in the common Practice 
 of Perfpective, which it may not be amifs juft to 
 take Notice of, efpecially, fince even Pozzo 
 himfelf has fallen into the Miftake, is the Re- 
 ducing of Objects of equal Magnitudes, which 
 are elevated above the Centre of the Picture, 
 according to the Angle they fubtend at the Spec- 
 tator's Eye. This will immediately appear erro- 
 neous by a bare View of Fig. 4, For, if 1, 2, 
 3, 4, be Points equally diftant from each other 
 on the original Plane O R, and P E be the Per- 
 fpective Plane, and S the Eye ; it is plain, that 
 the Points of Interfection 1, 2, 3, 4, by the 
 vifual Rays 1 S, 2 S. &c. will be at equal Dif- 
 tances from one another, becaufe P E is parallel 
 to O R, and therefore no Reduction is neceflary. 
 
 A great many more Abfurdities might be 
 pointed out in the Practice of Perspective, through 
 proceeding on wrong Principles. But as my In- 
 tention 
 
P R E F A C E. xv 
 
 tendon is not fo much to turn Critic on others^ 
 as to avoid the Lafh of Criticifm myfelf, I fhall 
 fay no more on that Point •, being well allured, 
 that Excellence in Writing upon any Art or 
 Science, depends not on the Expertnefs of de- 
 tecting the Miftakes of others, but on Docility 
 and good Senfe : Which is, in Part, agreeable 
 to the Sentiment of Horace, 
 
 Scribendi rette^ fapere eft et frincifium it fons. 
 
 Commercial and Mathematical School,, 
 Salford, near Mahchefter, June i, 1776. 
 
 1 
 
 PRACTICAL 
 
THE 
 
 O N T E N T S 
 
 Of the FIRST VOLUME. 
 
 Paje 
 
 Definitions i j 
 
 Axioms 26 
 LefTon I. A Point being given on the Ground Plane, to 
 Jind its Place or appearance on the Perfpeclive Plane ; 
 or that Point, where a Line drawn from the Specta- 
 tor's Eye to the given Point, will cut the Picture 2 J 
 LefTon II. A Line being given on the Ground Plane, to 
 
 Jind its Appearance on the Perfpeclive Plane 29 
 LefTon III. To Jind the Perfpeclive of any rectilineal 
 
 Figure 30 
 LefTon IV. To put a Square in Perfpeclive 3 1 
 
 LefTon V. To find the Perspective of the Ordinate Fi- 
 gures, a Pentagon and a Dodecagon 32 
 LefTon VI. To put a Circle in Perfpeclive . 3 3 
 LefTon VII. To throw the Ground Plan of an Heuje ink 
 
 to Perfpeclive 34 
 Leflon VIII. To find the Appearance of a Pole, of which 
 
 the Height and Seat on the Ground Plane are given 3 5 
 LefTon IX. To Jind the Perfpeclive Appearance of a Paral- 
 lelogram, which is perpendicular to the Ground Plane 36 
 LefTon X . To find the Perfpeclive of a Plane Figure 
 
 filiated on an Upright Plane 37 
 LefTon XL T ? put a Cube and Parallelopipedon in Per- 
 
 fpettive 38 
 LefTon XII. To find the Perfpeclive of a Square Pillar 39 
 LefTon XIII. To find the Perfpeclive of a Pyramid 40 
 LefTon XIV. 71? put an Hcufe in Perfpeclive 4 1 
 
 LefTon XV. To find the Perfpeclive of an Houfe with 
 m Hip- Roof, viewed upon an Angle ib. 
 
 LefTon 
 
C O N T E N T S. 
 
 LefTon XVI. To find the Appearance of a Pole inclined 
 to the Horizon ; the Angle cf Inclination, and the Si- 
 tuation on the Ground Plane being given 43 
 
 LefTon XVII. To find the Perfpeblive of a Parallelogram 
 • inclined to the Ground Plane, and parallel to the Line 
 of Interfeciion, having the ufual Data 45 
 
 Leflbn XVIII. To find the Perfpeblive of a Point, on a 
 Plane inclined to the Horizon, and parallel to the Line 
 of Interfeciion 46 
 
 Leflbn XIX. To find the Appearance of a Line fituated 
 on an inclined I lane, which is parallel to the Line of 
 Interfeciion 48 
 
 Leflbn XX. To find the Perfpeblive of a Figure, fituated 
 on a Plane inclined to the Horizon, and parallel to the 
 Line of Interfeciion 50 
 
 Leflbn XXI. To find the Appearance of a Pole inclined 
 to the Horizon, and fituated obliquely to the Line of 
 Interfeciion 5 2 
 
 Leflbn XXII. To find the Perfpeblive cf a Parallelo- 
 gram inclined to the Horizon, and fituated obliquely 
 to the Line of Interfeciion 5 3 
 
 Leflbn XXIII. To find the Appearance of a Point, 
 fituated on an inclined Plane, 'which fiands obliquely 
 to the Line of Interfeciion 5 5 
 
 LefTon XXIV. To find the Perfpeblive of a Line, 
 which is fituated on a Plane inclined to the Horizon, 
 and to the Line of Interfeciion 56 
 
 Leflbn XXV. To find the appearance of a plane figure, 
 fituated on a Plane, inclined to the Horizon, and oblique 
 to the Line of Interfeciion 57 
 
 Leflbn XXVI. To find the PerfpeBivc of the fame Tri- 
 angle, by Vanifhing Points 5^ 
 
 Leflbn XXVII To put an Hexahedron in Perfpeblive, 
 
 ftanding upon an angular Point 0 1 
 
 LefTon 
 
CONTENTS. 
 
 Leffon XXVIII. 7b Appearance of a Dodeca- 
 
 hedron 63 
 
 Leffon XXIX. To find the Perfpeciive of an Icofahe- 
 dron 64 
 
 Leffon XXX. To pat a Sphere or Globs in Perfpeciive 66 
 SCIAGRAPHIC PERSPECTIVE 70 
 
 Leffon XXXI. To find the Shadow of a Pole., which is 
 pependicular to the Ground Plane ; having the Ground 
 Seats % and Heights of the Light and Pole given ib. 
 
 JLefTon XXXIL To find the Shadow cf a plane Gbjetl, 
 when it is perpendicular to the Ground Plane 71 
 
 Ledbn XXXIII. To find the Shadows of Inclined 
 Planes ; fuppofe an inverted Pyramid j 2 
 
 Leffon XXXIV. 71? find the Shadow of a plane circa- ; 
 lar Ring 74 
 
 Leffon XXXV. To find the Shadow of an Hoop ' ib. 
 
 Leffon XXXVI. To find the Shadow cf a Pyramid, 
 when it is intercepted by Objefls Jlanding upon the 
 Ground Plane 76 
 
 Leffon XXXVII. To find the Perfpeciive and Shadow 
 of a Crofs, ccnfijling of two plane Parallelograms in- 
 terfering each other in any Angle 
 
 Leffon XXXVIII. To find the Shadow of an upright 
 Plane, when the Sun is in the Plane of the Pitlure 79 
 
 Leflbn XXXIX. To find the Shadow of a perpendicu- 
 lar Plane , when the Sim is behind the Plane of the 
 Pitlure 80 
 
 Leffon XL, To find the Shadow of an upright Plane, 
 when the Sun is before the Perfpeciive Plane, or 
 Pitlure 81 
 CATOPTRIC PERSPECTIVE 83 
 
 Leffon XLI. To find the Perfpeciive of the Reflexion of 
 an Objetl, fituated on an horizontal refletling Plane, or 
 Surface; as Water, Glafs, <hc. 84 
 
 Leffon 
 
CONTENTS. 
 
 Leflbn X LI L To find the Reflexion of an Object, when 
 fiiuated above the horizontal refletling Surface ; as on 
 a Bank by the Side of a River, <bc, 8 5 
 
 LefTon XLIII. To find the Reflexion of a Pole from a 
 polifhed Plane or Looking-Glafs., when both are perpen- 
 dicular to the Ground Plane 8 7 
 
 Leflbn XL1V. To find the Reflexion of a fqnare Pillar 
 viewed on an Angle, when the refletling Plane is filiated 
 obliquely to the Line of Interfecfion 88 
 
 Leflbn XLV. To find the Perfpeclive of the Reflexion 
 of a perpendicular Pole, when the refletling Surface or 
 Glafs is inclined to the Ground Plane 89 
 
 Leflbn XL VI. To find the Appearance of the Reflexion 
 of a Pyramid, from an inclined Looking-Glafs 9 [ 
 
 Leflbn XLV1I. To take the requifite Meafurements for 
 putting an Objetl in Perfpeclive 92 
 
 Leflbn XLVIII. To find the Diftance of the Pitlure 94 
 THEATRICAL PERSPECTIVE 101 
 
 Leflbn XLIX. To determine the Form and Pofltion of 
 the Stage ; the V anifhing Lines and Centres of the 
 Scenery ; and the Situations and Lengths of the 
 Grooves for the fliding and lateral Scenes 102 
 HORIZONTAL PERSPECTIVE 106 
 
 Leflbn L. To draw a Line upon a flat Ceiling, which 
 Jhall appear, to the Eye of the Spetlator, to be per- 
 pendicular to the Ground Plane, or parallel to the 
 Sides of the Room I o 7 
 
 Leflbn LI. To draw the Reprefentation of a Window 
 upon a flat Ceiling, which Jhall appear to the Spec- 
 tator's Eye, as if it were in the Side of the Room con- 
 tinued upwards 108 
 
 Leflbn LII. Diretlions for taking Perfpeclive Views 
 without any Meafurements 1 1© 
 
PRACTICAL PERSPECTIVE, 
 
 DEFINITIONS, 
 
 D E R I. 
 
 E R S P E C T IV E, is the art of deferr- 
 ing on a plane the exact reprefentation oi 
 any object. 
 
 D E F. If. 
 
 The Original Plane, is that plane on which an 
 object is fituated. If it be parallel to the horizon | 
 and that on which the fpectator ftands 5 it is 
 called the Ground Plane. 
 
 B * D E F. 
 
iS PRACTICAL 
 
 D E F. III. 
 
 The Perfpeftive Plane, or Pifture, is a trans- 
 parent plane, which is imagined to {land perpen- 
 dicular to the ground plane, between the Spec- 
 tator's eye and an object. 
 
 DEF, IV. 
 
 The Point of Sight, is that point where the 
 Spectator's eye is placed to look upon the pic- 
 ture, 
 
 DEF. V. 
 
 The point where the picture is cut by a per- 
 pendicular line drawn from the point of fight, 
 is called the Centre of the Pitture. And the dis- 
 tance between this point and the point of fight, 
 is called the Diftance of the Pifture* 
 
 DEF. 
 
PERSPECTIVE, 
 
 D E F. VI. 
 
 The Interferon of an Original Plane, is the 
 line where that original plane continued cuts the 
 picture. And the Interferon of an Original Line, 
 is the point where that original line continued 
 cuts the pi&ure. 
 
 D E F. VII. 
 
 A plane pafllng through the point of fight 
 parallel to an original plane, is called the Parallel 
 of that plane. And a line paffing through the 
 point of fight parallel to an original line, is Cal- 
 led the Parallel of that line. 
 
 D E F. VIII. 
 
 The Vanifhing Line of a plane, and the Va* 
 nifhing Point of a line, are the interferons of 
 their refpective parallels, with the perfpeclive 
 plane, or picture, 
 
 13 J2 F« 
 
so PRACTICAL 
 
 D E F. IX. 
 
 A line drawn from the point of fight perpen- 
 dicular to a vanifhing line, is called the Biftance 
 of that vanifhing line. And the point where the 
 vanifhing line is cut by the perpendicular, is cal- 
 led the Centre of that vanifhing line. The dis- 
 tance between the vanifhing point of a line and 
 the point of fight, is alfo called the Biftance of 
 that vanifhing point. 
 
 D E F. X. 
 
 The point where a perpendicular, drawn from 
 an original point, meets the line of interferon, 
 is called limply the Seat of that point. And if 
 the original point be fituated on an inclined 
 plane, the point where a perpendicular meets the 
 ground plane, is called the Ground Seat of that 
 point. 
 
 D E F. XI. 
 
 A line drawn from the' feat of a point to the 
 centre of the picture, is called the Bire cling 
 
 Line, 
 
PERSPECTIVE. 21 
 
 Line, or Biretlor of that point. And a line 
 drawn from the interfection of a line to its va- 
 nifliing point is alfo called the DireElor of that 
 line. If the original line be fituated on the 
 ground plane, its directing line is called a Pri- 
 mary Director if on an inclined plane, a Se- 
 condary Director* 
 
 DEF, XIL 
 
 The Vifual Ray, is a ftraight line, proceeding 
 from the object to the fpectator's eye. 
 
 In order that the learner may more readily 
 comprehend the fenfe of thefe definitions, I have 
 given three different moveable fc hemes ; • which 
 will alfo be found of great fervice in illuflrating 
 the different methods of operation in the follow- 
 ing leffons, by an occular demonftration; 
 
 FIG. i, 
 
 Let the plane A I be raifed up perpendicular 
 to the plane G D ; and the plane S C be 
 brought parallel thereto, fo as the point S may 
 
 be 
 
2i PRACTICAL 
 
 be perpendicular to, or exactly over the point 
 T. And let A B be an original line anyhow 
 fituated on the plane G D, and S V a line drawn 
 parallel thereto. Aifo let the thread A a S (be- 
 ing drawn up to S) reprefent a ftraighc. line drawn 
 From A to S, and S C a perpendicular to the 
 line V A. Then will the plane G D reprefent the 
 Ground Plane, def. 2 ; T the place where the 
 fpectator ftands, and S his eye, or the Point of 
 Sight, def. 4 ; A I the Perfpective Plane, or 
 Picture, def. 3 ; and the plane S C the Parallel 
 of the original plane G D, def 7. I N is the Line 
 of Interferon, def. 6 ; V A the Vaniming Line 
 of the ground plane, def 8 and C the Centre 
 of that vanifhing line, def 9. which is, in this 
 cafe, the Centre of the Picture, def 5 ; and S C 
 the Diflance of that vanifhing line, def 9. which 
 is alfo in this cafe the Diftance of the Picture, 
 def 5. S V is the Parallel of the original line 
 A B 3 def 7. V the Vaniming Point of that line, 
 def 8. and SV the Diflance of that vanifhing 
 point, def 9. 5 s are the Seats of the original 
 points A, B, def 10. and y the Interferon of the 
 line A B, def 6. The lines s C, ;C, and y V, 
 are the Directing Lines, or Directors, of A B, 
 
 def 
 
PERSPECTIVE. 23 
 
 def 11. and the lines AS, B S, are the Vifual 
 Rays of the faid line A B, def, 12. 
 
 FIG. 2, 
 
 Bring the plane A I perpendicular to the plane 
 G D, and the point S in the plane C S Z perpen- 
 dicular to the point T. Bring down the plane 
 S Z, fo that the line S Z (which is perpendicular 
 to V A) may coincide with Z S. Elevate the 
 plane O M X U, by bringing the plane Ly M 
 perpendicular to the plane G D, and let the 
 plane O X thereby become parallel to the plane 
 S Z. Alfo let the line S V be parallel to A B, 
 and S C perpendicular to V A. Then will the 
 planes GD, A I; the lines IN, V A, S C 5 
 and the points S, C, y 9 reprefent the fame as in 
 Fig. 1. The plane S Z V will alfo be the Pa- 
 rallel of the original plane OMXU, def. 7. 
 V Z A the Vanifhing Line of that plane, def. 8, 
 and S Z the Diftance of that vanifhing line, def 
 9. The line S V is the Parallel of the original 
 line A B, def 7. V the Vanifhing point of that 
 original line, def 8. and SV the Diftance of 
 
 that 
 
PRACTICAL 
 
 that vanishing point, def. 9. The line y v is 
 the Primary Director, and fV the Secondary 
 Director of the line A B, def 11. Alfo, the 
 points ^, are the Ground Seats of the original 
 points A, B, def, 11. and therefore the line a b y 
 is the Ground Seat of the original line A B. 
 
 FIG. 3. 
 
 Let the plane A I be raifed perpendicular to 
 G D as before, and the points S, S, s> be all 
 brought exactly over the point T, which will 
 then be united in one. Bring likewife the plane 
 L y M perpendicular to G D, and let the planes 
 
 V S F y V s v, thereby become refpectively pa- 
 rallel to the planes O M X U, L y M. Alfo 
 let the line S 2 be parallel to the line A B, S C 
 perpendicular to V A, and S Z perpendicular 
 to V V. In this Scheme as well as the laft, the 
 planes G D, AT, the lines I N, V A, S C, 
 and the points S, C, are the fame as in Fig. 1. 
 The plane V S V is the parallel of the original 
 plane OMXU; the plane Vsv the Parallel 
 of L y M, def 7. and therefore the lines <v V, 
 
 V are the refpective Vanifhing Lines of thofc 
 
 planes 
 
PERSPECTIVE. 25 
 
 planes, def. 8. S Z is the Diftance of the va- 
 nishing line V/ 7 , and Sv the Diftance of the 
 vanifhing line V v 9 def. 9. The line S 2 is the 
 Parallel of the original line A B, def. 7 j the 
 point 2. the Vanifhing Point of that original line, 
 def. 8 and S 2 the Diftance of that vanifhing 
 point, def. 9. Alfo the line y 2 is the Primary 
 Director, f2 the Secondary Director, and ab 
 the Ground Seat of the faid line A B, def. 10, 11, 
 
 Note, As a perfect knowledge of thefe defi- 
 nitions is indifpenfibly neceffary in the reading ot 
 the following ieflbns, the fcholar mould not at- 
 tempt to proceed till he thoroughly underftands 
 them for then he will meet with few or no dif- 
 ficulties through the whole book, and the reafon 
 of every operation will be almoft felf-evident, 
 which, without fuch precognita, might feem to 
 him to be involved in obfcurity, and totally in- 
 explicable. 
 
 C AXIOMS, 
 
26 PRACTICAL 
 
 A X I O M S, 
 
 Or things which are fuppofed to be either pre- 
 vioufly underftood, or evident in themfelves. 
 
 I. 
 
 The interferon of two planes is a ftraight line, 
 
 II. 
 
 Two parallel lines may be in the fame plane. 
 
 III. 
 
 A plane may pafs through a ftraight line and a 
 point, however fituated. 
 
 IV. 
 
 If a ftraight line interfecl: two or more parallel 
 lines, they may be all in the fame plane. 
 
 2V. B. In every leflbn, I fuppofe the Centre 
 and Diftance of the Picture, and the Dimenfions 
 and Situation of the Objecl, to be given. 
 
 LESSON 
 
PERSPECTIVE. 
 
 LESSON L 
 
 A Paint being given on the Ground Plane, to find 
 its Place or Appearance on the Per/peclive Plane 
 or that Point, where a Line drawn from the 
 Spectator's Eye to the given Point, will cut the 
 PiHure P 
 
 Operation. By the Seat. From the given 
 point A, draw A s perpendicular to the line of 
 interferon I N. And from the feat s, draw a 
 ftraight line to the centre of the picture C. 
 Then, from the original point A, to the fpecta- 
 tor*s eye, or point of fight S, draw the vifual 
 ray A S \ the point of interaction a, with the 
 director s C, is the perfpective of the given 
 point, or the place where it appears on the per- 
 fpective plane. 
 
 By the Interjection. In any direction draw 
 A P ; parallel to which draw S V \ and from 
 the point of interfection P, to the vaniming 
 point V, draw the ftraight line P V. Then 
 draw the vifual ray A S, and the point of in* 
 
 C 2 terfectioo 
 
28 PRACTICAL 
 
 terfection with the director P V, is the per- 
 fpective of the original point A, as before. 
 
 Demonftration. Let the plane S V C, (Fig. i.) 
 be brought parallel to the ground plane G D, 
 the point S perpendicular to the point T, and 
 the thread drawn up to the eye at S. Then, 
 becaufe S C is parallel to A s (being both per- 
 pendicular to the parallel lines V A, I N,) and 
 the plane SVC parallel to the plane G D, it 
 is evident, that a plane will pafs through the 
 lines S C, As, (Ax. II.) and intericft the pic- 
 ture in the ftraight line Cj, (Ax. I.; And for 
 the fame reafon it appears, that a plane will pafs 
 through the lines S V, A y, (A P) and interfccl 
 the picture in the line V y 9 (VP). Moreover, 
 it is obvious, that both thefe planes pafs through 
 the points S, a, A, which points muft therefore be 
 in their line of interferon 5 hence the thread, 
 or vifual ray, pafTmg from the original point to 
 the eye, is a ftraight line, (Ax. I.) And what- 
 ever pofition the perfpective plane has in refpect 
 to the ground plane, if the planes G D, S V C, 
 continue parallel, the vifual ray will always be a 
 ftraight line, becaufe the points of interferon 
 
 S, a 9 
 
PERSPECTIVE. 29 
 
 S, a, A, will manifeftly continue the fame. 
 Now let the planes be ftretched out, and laid 
 down on the ground plane GD-, the planes 
 G D, S V, being yet parallel, the thread, or vi- 
 fual ray A S, will evidently be a ftraight line, 
 and inter feci the lines s C, yY i (PV) in the 
 fame point a ; which is therefore the place where 
 the original point A appears to the fpectators 
 eye on the picture. 
 
 LESSON II. 
 
 A Line being given on the Ground Plane, to find 
 its Appearance on the Perfpeclive Plane. 
 
 Operation. By Seats. From j, J, the feats 
 of the points A and B, (being the extremities 
 of the given line) draw the directing lines s C, 
 s C ; and from the original points A, B, draw 
 the vifual rays AS, B S ; which will interfeft 
 the directors in the points a 9 k Join a, and 
 it will be the perfpeclive of the original line A B. 
 
 By 
 
PRACTICAL 
 
 By the Point of Interferon. Produce D E 
 to P, and draw S V parallel to ED; and from 
 the point of interferon P, draw the director 
 P V. And the vifual rays, E S and D S, will 
 interfecl the director in the points e and d. So 
 v/ill e d be the perfpective of the line E D. 
 
 LESSON III. 
 
 To find the Perfpeftive of any rettilineal Figure. 
 
 Operation. By Seats. Find the perfpective 
 of the points A, B, K, by lelfon I. then join the 
 points a, b> and it will be the picture of the 
 given triangle A B K. 
 
 By the Interferon. Draw S i parallel to 
 F H, S 2 parallel to PI E, and S3 parallel to 
 F E. Then from the points of interfection 
 i, 2, 3, draw the directors 1 1, 2 2, 3 3 ; and 
 h e f will be the peripe&ive of the triangle 
 HEF 
 
 LESSON 
 
PERSPECTIVE, 
 
 LESSON IV. 
 
 7*o put a Square in Perfpeflive* 
 
 Operation. Let A B C D be a given fquare, 
 of which a fide is parallel to the line of interfec- 
 tion. Having found the feats, draw the direc- 
 tors to the centre of the picture, and the vifual 
 rays will inter feci; them in the points ab c d\ 
 which being joined will give the perfpeclive of 
 the fquare. 
 
 If the fquare be fituated obliquely to the line 
 of interfection as E F G H, find the points of 
 interferon i, i, 2, 2, and the vanifhing points 
 1, 2 ; then the directors being drawn to their re- 
 fpective vanilhing points, will, by their inter- 
 ferons, form the figure e fg h ; which will be 
 the perfpective of the fquare when viewed on an 
 angle. 
 
 LESSON 
 
32 Practical 
 
 lesson v. 
 
 To find the Perfpeffive of the Ordinate Figures, a 
 Pentagon and a Dodecagon, 
 
 Operation. Let P be the given Pentagon. 
 Find the directing points by producing each fide 
 to the line of interferon. Then find the va- 
 nishing points, i, 2, 3, 4, and draw the directors ; 
 which will form the perfpective of the given 
 pentagon by their interferons. 
 
 When the figure confifts of many fides, the 
 perfpective is moft conveniently found by Seats. 
 So to find the picture of the Dodecagon D ; 
 find the feats of all the angular points ^ from 
 whence draw the directors to the centre. Then 
 draw the vifual rays, and join the points of inter- 
 feron and trie figure made thereby will be a 
 true reprefentation of the given Dodecagon. 
 
 LESSON 
 
PERSPECTIVE, 
 
 LESSON VL 
 
 <Z0 put a Circle in Perfpetlive. 
 
 Operation. By Seats. (Fig. i.) Find the 
 feats of any number of points in the periphery ; 
 from which draw the directors. Then from the 
 arTtimed original points draw the vifual rays, 
 which will interfect their refpective directors in 
 the correfponding parts of the periphery ; thefe 
 points being joined with a fteady hand, will mew 
 the appearance of the circle on the perfpective 
 plane. 
 
 By the Points of Interfeclion. (Fig. 2.) In 
 any angle find the interfe&ions of any number 
 of points in the periphery, from which draw the 
 directors to their common vaniftiing point V, 
 and they will be interfered in the correfponding 
 points by the vifual rays ; and thence the ap- 
 pearance of the circle determined. 
 
 D It 
 
PRACTICAL 
 
 It is oftentimes very convenient to circum- 
 fcribe the given circle with a fquare, as in Fig. 
 3. drawing parallels through the points where 
 the diagonals cut the periphery ; and then pro- 
 ceeding after the manner of feats. 
 
 LESSON VII. 
 
 To throw the Ground- Plan of an Houfe into Per- 
 
 JpeMve. 
 
 Operation. Find the feat of every angle, 
 point, &c. on the ground plane. Then draw 
 the directors and vifual rays, and they will inter- 
 fec~t each other in the correfponding points on the 
 perfpeclive plane. Thefe points being joined, 
 will give the perfpective of the plan. 
 
 In regard to the femicircle ; the perfpe&ive of 
 that may be found by the laft lelfon, by taking 
 a few points in the periphery. 
 
 After 
 
PERSPECTIVE. 
 
 After the fame manner may any horizontal 
 plane figure be drawn ; as flagged areas, walks, 
 avenues, &c. either of the fquare, diamond, or 
 grotefque forms. 
 
 LESSON VIII. 
 
 T o find the Appearance of a Pole, of which the 
 Height and Seat on the Ground Plane are given. 
 
 Operation, Find the perfpeftive of the point 
 A as before, and draw indefinitely B E perpen- 
 dicular to I N, on which take B D equal to the 
 given height of the pole. From the point D 
 draw D C \ and from a draw a d parallel to 
 B D, and it will be the apparent height of the 
 pole. 
 
 For a plane being imagined to pafs through 
 B D and the centre C, will pafs through a d. 
 And becaufe B C and D C have the fame va- 
 niming point C, they are parallel ; therefore 
 
 D 2 ad 
 
3 6 PRACTICAL 
 
 a d is equal to B D, and perpendicular to the 
 ground plane. 
 
 LESSON IX; 
 
 To find the Perfpeclive Appearance of a Parallelo- 
 gram^ which is perpendicular to the Ground 
 Plane. 
 
 Operation. Let A BCD be the given paral- 
 lelogram, and A B its feat on the ground plane. 
 Find the perfpective of the points A, B, as be- 
 fore ; and on the line of elevation take E F 
 equal to A B, from whence draw F C (FV); 
 and raife the perpendiculars a € , b d. So will 
 acdb be the appearance of the given parallelo- 
 gram. 
 
 LESSON 
 
PERSPECTIVE. 
 
 LESSON X. 
 
 To find the Perfpeftive of a Plane Figure fititated 
 on an Upright Plane. 
 
 Operation* Let A B be the feat of the given 
 plane, and D E F G the given figure. Produce 
 the fides to i, 2, 3, 4 ; and make E H parallel 
 to AB. Make the angle CSV equal to 
 H E 4, and the angle equal to H E 2 % 
 
 then will V, and a^ be the vanifhing points of 
 the fides of the figure. On the line of elevation 
 P^, take Pi, P 2, &c. equal to A 1, A 2, 
 &c. and find the points 1, 2, 3, 4, on the 
 edge of the perpendicular plane ; then draw 1 V, 
 2 V, 3 4 and the interfe&ions defg will 
 form the perfpe&ive of the figure DEFG, 
 
 LESSON 
 
38 PRACTICAL 
 
 LESSON XI. 
 
 IT 9 put a Cube and ParaUelopipedon in PerfpeElive. 
 
 Operation. Let the fquare A B C D be a fide 
 of the given cube, and abed the perfpective of it, 
 found by leflbn IV. On the line of elevation 
 take L F equal to AC ; and draw the line F C. 
 Draw a g perpendicular to a e and make c h, 
 parallel and equal to ag, and join g b. Then 
 to the centre C draw b e ; and make d e paral- 
 lel to ch^ and fe parallel te gb- y fo will the 
 picture of the cube be completed. 
 
 Let H I K L reprefent a fide of the given 
 parallelopipedon. Firft find the interfering and 
 vanifliing points thereof, i % 2 *, and the inter- 
 ferons of the directors will form the figure 
 hi k I, which is the perfpective of the fide of the 
 foiid contiguous to the ground plane. On the 
 line of elevation take 2 m equal to I K, and 
 draw the line m 2 \ then raife the perpendiculars 
 ir 9 bp, and to the vanifhing point 1, draw 
 
 r n, 
 
PERSPECTIVE. 39 
 
 r n^p o. Laftly, make k n> to, each parallel to 
 i% and join no, and there will be formed a 
 true reprefentation of the parallelopipedon. If 
 the line n o produced paries through the vanifh- 
 ing point 2, it mew sthat the drawing is accurate,. 
 
 LESSON XIL 
 
 To find the Perfpeflive of a Square Pillar. 
 
 Operation. Let A B C D be the ground 
 plan, or feat of the pillar on the ground plane, 
 of which find the perfpective as before. From 
 the feat of the line D C, raife the line of eleva- 
 tion ; on which take L M equal to the given 
 height of the pillar, and draw the line M C. 
 Then raife the perpendiculars de, a /, and bg ; 
 and draw e f parallel to d a, and fg to the 
 centre C , and it will produce the perfpecYive of 
 the given pillar. 
 
 If it {lands obliquely to the line of interfec- 
 tion, find the interfering and vanilhing points 
 
 1, 2. 
 
4 o PRACTICAL 
 
 i, 2. Then draw the directors ; and the inter- 
 fections will give the appearance of the bafe of 
 the pillar efg h. Make % i the line of eleva- 
 tion, on which take 2 f equal to the given 
 height of the pillar ; and draw the line i 2, to 
 which raife the perpendiculars e k, f L From 
 k to the vanfhing point 1, draw k m, and raife 
 the perpendicular h m ; fo will the true perfpec- 
 tive of the pillar be had. 
 
 The learner will find little or no difficulty in 
 thefe two laft lefibns, if he remembers, That all 
 lines which are parallel to one another in the 
 given object, will always have a common va- 
 niihing point ; for from that the reafon of the 
 operations will be obvious. 
 
 LESSON XIII. 
 
 To find the Perfpefiive of a Pyramid. 
 
 Operation. Throw the ground plan of the 
 pyramid into perfpeclive ; and then imagine a 
 
 pole 
 
PERSPECTIVE. 
 
 41 
 
 pole to pafs from the centre of the bafe to th® 
 vertex. From the centre of the picture refer 
 the point e to the line of interferon, and raife 
 the line of elevation L F, which make equal 
 to the given height of the pyramid then - 
 draw F C ; and the point where it cuts the per- 
 pendicular e b 9 will be the vertex of the pyra- 
 mid. Laftly, draw a b, b b 9 c b, and db ; 
 and the perfpeclive of the pyramid will be com- 
 pleted. 
 
 If the pyramid is inverted or ftands upon its 
 vertex, find the perfpeclive of the ground plan 
 e fg) and the centre 2, by vanifhing points, as 
 before. Then make 1 M the line of elevation 1 
 and draw M 1, and raife the perpendiculars gg y 
 e e* From g draw g 2, and raife the perpen- 
 dicular //. Lailly, join ei 9 gi> and / /; and 
 the perfpective of the inverted pyramid will be 
 had. 
 
 E LESSON 
 
42 PRACTICAL 
 
 LESSON XIV. 
 To put an Houfe in Perfpective. 
 
 Operation. Let A B D E be the ground plan, 
 and the angle BHD the pitch of the roof. 
 Having thrown the ground plan into perfpective 
 as before, raife the perpendiculars g b 9 f k ; 
 and on the line of elevation, take P F equal 
 to the given height of the raifing, and FH 
 equal to F H. Then draw H C and make 
 h k parallel to a b ; and m h> n k, ko will be the 
 pofitions of the rafters. The reft is plain from 
 the figure. 
 
 LESSON XV. 
 
 To find the Perfpeflive of an Houfe with an Hip» 
 Roof viewed upon an Angle. 
 
 Operation. Let A B C D be the plan of the 
 raifing, and F E the height of the pitch. Throw 
 the ground plan into perfpective, and raife the 
 
 indefinite 
 
PERSPECTIVE. 43 
 
 indefinite perpendiculars G H, IK, L O. 
 Then, on the line of elevation, take P M equal 
 to the height of the railing, and M N equal to 
 F E the pitch of the roof. Draw M F, N V 9 
 and through the interferon H draw the inde- 
 finite line v t ; then raife the perpendiculars 
 d r, a /, draw t and raife the perpendicular 
 h X.< Laftly, join r k> t K, and X O ; which 
 will be the perfpeclive of the Hip-Rafters. The 
 reft needs no explanation. 
 
 LESSON XVI. 
 
 T o find the Appearance of a Pole inclined to the 
 Horizon ; the Angle of Inclination^ and the 
 Situation on the Ground Plane being given. 
 
 Operation. Let A D be the pole, A B its 
 feat on the ground plane G D, and the angle 
 BAD the inclination of the pole to the ground 
 plane. Produce C S, and make the angle C s Z 
 equal to the angle BAD: So will Z be the 
 vanilhing point of the line A D. Find the per- 
 
 E 2 fpeclive 
 
44 PRACTICAL 
 
 fpective of the point A by its feat s \ and on 
 the line of elevation take sg equal to B D, and 
 draw gc and A Z. Then will ad be the 
 picture of the inclined pole. 
 
 For if we imagine the plane Z s C to be 
 brought up perpendicular to Z C, and the plane 
 G D turned back into its proper fituation, s C 
 will be parallel to A B, and s Z parallel to AD; 
 becaufe the angles C sZ and BAD are equal. 
 And becaufe the points A B D are in the fame 
 plane, and perpendicular to the ground plane - 3 
 and sg equal to BD, and perpendicular to the 
 line of interferon IN; therefore sg and ad 
 are in the fame plane, which is alfo perpendicu- 
 lar to the ground plane G D. Hence a d is 
 the appearance of the pole A D on the perfpec- 
 tive plane. 
 
 LESSON 
 
PERSPECTIVE. 
 
 LESSON XVII. 
 
 fo find the Perfpeffive of a Parallelogram inclined 
 to the Ground Plane^ and parallel to the line of 
 Interferon, having the ufual Data. 
 
 Operation. Let O M X U be the parallelo- 
 gram, O U its interferon with the ground 
 plane, and L O M its angle of inclination. 
 Make the angle C S Z equal to the angle 
 L O M then will Z be the vanifhing point of 
 the parallelogram. Having found the perfpeo 
 tive of O U as before, take L M on the line 
 of elevation equal to L M $ and at r ? M C, 
 o Z, and u Z. Then make mx parallel to 4 u t 
 and omxu will be the perfpedive of the given 
 parallelogram. 
 
 For when Fig. 2. is put in its proper pofition, 
 the plane S Z will be the parallel of the original 
 plane OMXU, (becaufe the angle C S Z is 
 equal to the angle of inclination L O M) and 
 V Z A its vanifhing line. And fin'Ce S Z is 
 
 parallel 
 
46 PRACTICAL 
 
 parallel to the fides of the inclined plane O M 
 and U X, and interfe&s the perfpective plane in 
 the point Z, it is therefore the vanifhing point 
 of the fides of the inclined plane, def 8. And 
 LM is the altitude of the point M above the 
 ground plane, and o u the perfpedtive of OU; 
 therefore omxu is the perfpeclive of the given 
 plane OMXU. 
 
 LESSON XVIII. 
 
 T o find the Perfpeclive of a Pointy on a Plane in- 
 clined to the Horizon^ and parallel to the Line of 
 Interfeclion* 
 
 Operation. Let OMXU be the original 
 plane ; and omxu the perfpective of it, as be- 
 fore. From B, the given point on the orginal 
 plane, draw B x perpendicular to O M, and 
 xy perpendicular to OL; then will xy be 
 the height of the given point B above the ground 
 plane. Draw B s perpendicular to I N, which will 
 
 in 
 
PERSPECTIVE, 47 
 
 in this cafe give the feat of the point B, becaufe 
 OU is parallel to IN; then draw the primary 
 director s C, and from the point of interferon 
 with the inclined plane, draw the fecondary di- 
 rector k Z. Make s t the line of elevation, on 
 which take s t equal to y x - 9 then draw the 
 line / C, and it will interfect the fecondary di- 
 rector k Z in b which is the perfpective of the 
 given point B on the inclined plane. 
 
 For if the plane O X (Fig. 2.) be brought to 
 its proper elevation above the ground plane, it will 
 be evident that x y is the height of the point B 
 above the faid plane. And fince a plane which 
 is perpendicular to the horizon may be imagined 
 to pafs through B s, the point B will be in the 
 interferon of that plane and the original one 
 OX. (Ax. III.) But kr is the .perfpective of 
 that interferon, and / j is the diftance of the 
 given point from the ground plane ; therefore b 
 is the perfpective of the point B, as required. 
 
 L ESS O N 
 
4 8 PRACTICAL 
 
 LESSON XIX. 
 
 To find the Appearance of a Line fituated on an in* 
 dined Plane, which is parallel to the Line of In* 
 terfeclion. 
 
 Operation. Let om x u be the perfpecYive of 
 the original plane as before, and A B the given 
 line. Having drawn A X perpendicular to 
 O M, and X Y perpendicular to O L, take 
 O i equal to O y ; and draw i b parallel to 
 B x 9 and B b parallel to O x ; then will b be 
 the ground feat of the point B* Produce A B 
 to c 9 and draw b cy, and it will give the ground 
 feat of a perpendicular plane pairing through the 
 given line A B on the inclined plane. 
 
 Find v the vanifhing point of b y 9 and draw 
 the primary directory y v. Take Z S equal to 
 Z S, and make the angle Z S V equal to bB c, 
 and V will be the vanifhing point of the line 
 A B. Then draw the fecondary director /V, 
 and it will produce an indefinite reprefentation 
 
PERSPECTIVE. 49 
 
 of the line A B. From the point of interfecYion 
 j, make jY a line of elevation, on which take 
 y x equal to y x, and y Y equal to Y X, and 
 draw x v 9 Y v ; and thefe lines will interfect the 
 fecondary director in the points b 9 a ; fo will 
 a b be the perfpective of the line A B. 
 
 For let all the planes in Fig. 2. be brought 
 into their proper pofitions, then will h evidently 
 appear to be the ground feat of the point B \ and 
 therefore a plane palling through A c 9 perpen- 
 dicular to the ground plane, will interfe6t that 
 plane in the line by. (Ax, L) And becaufe 
 S V is parallel to the line A B, and interfefe 
 the picture in V, the line /V will be the per- 
 fpeclive of the interfection of the perpendicular 
 and inclined planes. But the line A B is in 
 that interfe&ion, and y x and y Y are the re- 
 fpeclive heights of the points A and B above 
 y v, which is the perfpective of the ground feat 
 of the perpendicular plane Y v y 5 therefore a b 
 is the reprefentation of the given line A B on 
 the inclined plane. 
 
 F. 
 
 LESSON 
 
PRACTICAL 
 
 LESSON XX. 
 
 ST o find the Perfpetlive of a Figure, Jituated on a 
 Plane inclined to the Horizon, and parallel to the 
 Line of Interferon. 
 
 Operation. Let O M X U be the original 
 plane as before, O U the interferon with the 
 ground plane, L O M the angle of inclination, 
 and A B C D the given figure. Make the angle 
 C S Z equal to the angle LOM; and parallel 
 to O U draw V Z A, which will be the va- 
 nifhing line of the original plane O M X U. 
 Make Z S equal to Z S ; and draw S V pa- 
 rallel to B A, and S A parallel to BD; then 
 will V be the vanifhing point of the lines A B, 
 C D \ and A the vanifhing point of the lines 
 B D, AC. Produce A B to r, and C D to 
 c. Find b the feat of B on the ground plane, 
 and draw bey, and parallel thereto draw c y, 
 which will be the feats of two perpendicular 
 planes paffing through A B and C D ; and 
 therefore y> y y will be their points of interfeclion. 
 
 Find 
 
PERSPECTIVE. 
 
 Find v the vanifhing point of thefe lines, and 
 draw the primary directors y v, yv ; and from 
 their points of interfedtion with the inclined 
 plane /, /, draw the fecondary directors / V, 
 /V; which will give an indefinite reprefenta- 
 tion of the lines A B and C D. Produce D B 
 to p, and C A to q\ and draw pp^ qq y each 
 parallel to O L. On the line of elevation take 
 L p equal to L p, and L q equal to L q> and 
 and draw p C, q C ; then from the points of 
 their interfedion with the fide of the inclined 
 plane, draw p A y q A; and abed will be the 
 perfpective of the figure A B C D on the in- 
 clined plane. 
 
 For when the planes VN and MLjy (Fig. 
 2.) are fet perpendicular to the ground plane, 
 S A will appear to be parallel to Bp and there- 
 fore the point A will be the vanifhing point of 
 the fides of the figure B D and AC. And 
 fince L pi L q, are the refpedlive heights of the 
 points />, above the ground plane, the lines 
 p A, q A, will be indefinite reprefentations of the 
 fides of the figure, B D and AC; but by the 
 laft leflbn the lines /V, /V, are alfo indefinite 
 
 F 2 rep re- 
 
52 PRACTICAL 
 
 reprefentations of the fides A B, CD; therefore 
 the interfe&ions of thefe lines will form the per- 
 fpedive of the given figure A B C D. 
 
 LESS'ON XXI. 
 
 T 9 find the Appearance of a Pole inclined to the 
 Horizon, and fituated obliquely to the Line of 
 Interferon. 
 
 Operation. Let O M be the pole, O L its 
 feat on the ground plane, and L O M its angle 
 of inclination. From v the vaniming point 
 of O L, take v s equal to v S and make 
 the angle v s V equal to the angle LOM; 
 then will V be the vaniming point of the 
 pole O M. Draw o V, and on the line of 
 elevation take y M equal to L M, and draw 
 TS/Lv 9 fo will om ht the perfpeclive of the in- 
 clined pole, as required. 
 
 For let the inclined plane O M X U, (Fig; 
 3.) be raifed to its proper angle, and fuppofe the 
 
 fide 
 
PERSPECTIVE. 53 
 
 fide of it MO to be an inclined pole. Then, 
 becaufe S v is parallel to O L, and v s equal 
 to v S, the line v f will coincide with v S, 
 when brought forward to the eye at S, and the 
 plane v sV be parallel to the plane L O M. 
 And fince the angle v s V is equal to the angle 
 LOM, sV will be parallel to O M, and in- 
 terfect the picture in V - 9 which is therefore the 
 vanifhing point of the pole O M. The other 
 part of the operation is plain from the preceding 
 leflbns. 
 
 LESSON XXIL 
 
 To find the Perfpeffive of a Parallelogram inclined 
 to the Horizon^ and fituated obliquely to the Line 
 of Interferon* 
 
 Operation. Let OMXU be the original 
 plane, LOM its angle of inclination, and O U 
 its line of interferon with the ground plane. 
 Produce MO to y 9 X U to y, and U O to 
 w j and find the vanifhing points % V y of the 
 
 lines 
 
54 PRACTICAL 
 
 lines U U w. Take vs equal to yS 9 and 
 make the angle v s V equal to the angle of in- 
 clination L O M j and V will be the vanifhing 
 point of the fides of the inclined plane O M 
 and U X. From o and u the refpective inter- 
 ferons of the primary directors wV 9 y ov % and 
 w V, y u v, draw o V, u V , and on the line 
 of elevation take y M . equal to L M ; then 
 draw M v, and through the interferon x draw 
 Fm 9 and o m x u will be the appearance of the 
 inclined plane. 
 
 For when Fig. 3. is put in a proper pofition, 
 s V will plainly appear to be parallel to O M, 
 and V v will be the vanifhing line of the plane 
 L O M, becaufe that plane is perpendicular to 
 the ground plane, and the angle v sV equal to 
 the angle LOM. Therefore V, v, and V 
 will be the refpecYive vanifhing points of the 
 lines O M, O L, and O U. 
 
 LESSON 
 
PERSPECTIVE. 55 
 
 L E S S O N XXIII: 
 
 To find the Appearance of a Point > Jituated on an 
 inclined Plane, which ftands obliquely to the Line 
 of Interferon. 
 
 Operation. Let OMXU be the original 
 plane, and B the given point. Find the per- 
 fpective of the inclined plane o m x u 9 as before, 
 and draw B y parallel to O y, then from the 
 point of interferon y, draw the primary direc- 
 tor y z\ and from the point of interaction r, 
 draw the fecondary director r V. Make B x 
 parallel to OU, and xy parallel to LM- 3 then 
 will xy be the perpendicular height of the point 
 B above the ground plane. Then on the line 
 of elevation y M, take y x equal to xy 9 and 
 draw x v - 9 and through g 9 the interfection with 
 the fide of the inclined plane, draw V b \ and the 
 point of interfeclion b y with the fecondary di- 
 rector r V, will be the perfpective of the given 
 point on the inclined plane. 
 
 For 
 
$6 PRACTICAL 
 
 For when the original plane, Fig. 3. is fet to 
 its proper elevation, xy will plainly appear to 
 be the perpendicular height of the point B 
 above the ground plane. And fince the point 
 B is fituated in the interferon of the two 
 lines x B and / B, (which are refpectively paral- 
 lel to O U and OM) the perfpeclive interfec- 
 tion of thofe lines will evidently be the perfpec- 
 tive of the given point. For Vb is an inde- 
 finite reprefentation of the former, and r V of 
 the latter ; therefore b, their point of interfec- 
 tion, is the perfpective of the given point, as 
 required. , 
 
 LESSON XXIV. 
 
 T* 0 find the Perfpeclive of a Line, which is fituated 
 on a Plane inclined to the Horizon, and to the 
 Line of Interfeclion. 
 
 Operation. Let A B be the given line on the 
 original plane O M X U. The perfpecYive of 
 the inclined plane being found as before, find 
 
 the 
 
PERSPECTIVE. 57 
 
 the points of interferon y, of the given 
 points A and B. Draw the primary directors 
 y rv, y tv 9 alfo the fecondary directors r V, 
 and t V ; and on the line of elevation take y x 
 equal to y x 9 and y X equal to Y X. Then 
 draw the lines x v, X v ; and from the points 
 of interferon with the fide of the inclined plane 
 draw V gb, Vna, and join by and the line 
 a b will be the appearance of the line A B on 
 the inclined plane. This is evident from the laft 
 ieiTon. 
 
 LESSON XXV. 
 
 To find the Appearance of a plane Figure? fituated 
 on a Plane, inclined to the Horizon^ and oblique 
 to the Line of Interferon. 
 
 Operation. The inclined plane being thrown 
 into perfpective as before, draw the parallels 
 B y, A y 9 and C y ; and the primary directors 
 y rv, y t vi and y ev. Alfo from the points 
 of interferon with the inclined plane, draw the 
 
 G fecondary 
 
58 PRACTICAL 
 
 fecondary directors r V, t V, and e V. Then, 
 on the line of elevation y M, take the perpendi- 
 cular heights of the points of the triangle above 
 the ground plane, y x, T X, and Y X ; and 
 draw the lines xv 9 Xv, and X v. Laftly, 
 through g h the points of interferon with the 
 inclined plane, draw Vg V h c> and V n a. 
 So will a b c be the perfpective of the triangle 
 on the inclined plane. This is alfo evident from 
 kflbn XXIII. 
 
 LESSON XXVL 
 
 To find the Perfpettive of the fame Triangle, by 
 Vanijhing Points* 
 
 Operation. Take Oi equal to Oy% and 
 draw i b parallel to O U, and B / parallel to 
 O M ; then will b be the ground feat of the 
 point B. And after the fame manner will c, 
 the ground feat of the point C, be found. Pro- 
 duce the fides of the triangle to p, z, and d-, and 
 from the points b and c 9 draw the lines b p i, 
 
 bz2, 
 
PERSPECTIVE. 59 
 
 h2, r i 3 ; and they will reprefent the ground 
 feats of three perpendicular planes, paffing 
 through the fides of the given triangle on the in- 
 clined plane. Then find their refpective vanifh- 
 ing points i, 2, 3 ; draw the primary directors 
 1 1 j 22, 3 3 and through the points V, 
 draw a ftraight line, and it will be the vanishing 
 line of the inclined plane. Take V S equal to 
 V S, and V S equal to V s and make the 
 angles 1 S V 9 2 S Z 7 , and 3 5 F, refpeclively 
 equal to the angles Cp U, A 2 U, and A iU; 
 Laftly, from the points of interjection of the 
 primary directors with the inclined plane, /, 
 and /, draw the fecondary directors k 1, * 2, 
 jf 3 •, and they will form, by their interferons 
 on the inclined plane, the triangle a b c ; which 
 is the perfpective of the original one ABC. 
 
 For when the original plane OMXU (Fig. 
 3.) is brought to its proper elevation, the point 
 b will plainly appear to be the ground ieat of 
 the original point B, a the ground feat of A, 
 and the line bey (b z 2) the ground feat of a 
 perpendicular plane interfering the original plane 
 in the line A B. And after the fame manner it 
 
 G 2 will 
 
6o PRACTICAL 
 
 will appear that the lines b i, c 3, are the ground 
 feats of two perpendicular planes, paffing 
 through the lines BC, AC, refpeclively, which 
 are fituated on the inclined plane. Alfo when 
 all the planes are brought to the eye, perpendi- 
 cular to the point T on the ground plane, the 
 line v s will evidently coincide with v S, the 
 line V s with V and the line V S with VS. 
 And becaufe S V is parallel to O U, and s V 
 parallel to O M, the plane V S V is parallel to 
 the inclined plane OMXU, and interfects the 
 picture in the line V V\ which is therefore the 
 van idling line of the original plane OMXU, 
 def. 8. Now, becaufe S V is parallel to O U, 
 and the angle 2 SV equal to the angle A z U, 
 the line S 2 is parallel to the fide of the triangle 
 A B. And for the fame reafon, the lines 6' 1, 
 S 3, are refpeclively parallel to the lines B C, 
 A C and therefore the points 1, 2, 3, in the 
 vanifhing line V F, are the vanifhing points of 
 the fides of the triangle. 
 
 LESSON 
 
PERSPECTIVE. 61 
 
 L E S S O N XXVII. 
 
 SI o put an, Hexahedron in PerfpebJive^ /landing upon 
 an angular Point. 
 
 Operation. By the Interjections^ or vanishing 
 Points. Let HKNQ^bea fide of the given 
 Hexahedron, Take K O equal to the diagonal 
 H N, and complete the parallelogram KOPH, 
 which will then reprefcnt a vertical lection of the 
 Hexahedron ftanding on one of its angular points. 
 Upon the diameter F C (equal to I R) defcribe 
 the regular hexagon ABCDEF: and find 
 V and v % the refpective vanilhing points of 
 A D and B E. Take v s equal to v S ^ and 
 make the angles v s V, VvW^ each equal to 
 the angle KHI; and the angle Vvw equal 
 to the angle P H R. Then will V and W 
 be the vanifhing points of thofe interferons of 
 the fides of the Hexahedron which are not paraU 
 lei to the picture and w the vanifhing point 
 of a diagonal. By the interfection of the line 
 
 AD, 
 
Gi PRACTICAL 
 
 A D, find <2, g, the perfpcctive of the points 
 A, G and on the line of elevation, take L M 
 equal to H O, and L p equal to IK; 
 draw the lines p V 9 M V 9 and raife the perpen- 
 diculars apy gr\ then will p reprefent one an- 
 gular point, and r the vertex of the given Hexa- 
 hedron, on the perfpective plane. Now, fince 
 the interferons of the fides of the original object, 
 which are perpendicular to the lines A B, FC, 
 and E D, are parallel to the perfpective plane, 
 they will have no vaniming point ; therefore, 
 draw indefinitely p u, r /, and g n> each parallel 
 to s V, and complete the Hexahedron by draw- 
 ing the lines rw 9 nh, r r z/, p /, and p g 9 
 as in the figure. 
 
 For if we imagine the planes vsV, V $ v> 
 and ^i/W, to be each turned up on the lines 
 v Y'i V z\ and V\Y respectively, fo that the 
 line i s may coincide with v S, and Vv with 
 FS 9 then will sV plainly appear to be the 
 parallel of the interferons of the fides of the 
 Hexahedron p f 9 u r, and nh \ vW the paral- 
 lel of rfo, u % and pg; and *v*w the parallel 
 of the diagonal r n. 
 
 LESSON 
 
PERSPECTIVE. 63 
 
 LESSON XXVIIL 
 
 To find the Appearance of a Dodecahedron . 
 
 Operation. By Seats. Let a b be a line af- 
 fumed at pleafure. Take a e equal to half of 
 a b 9 from the centre e dcfcribe the arch bf„ 
 and draw the ftraight line b f\ then from the 
 centre a defcribe the arch b J, and draw d c 
 parallel to fb. Defcribe two concentric penta- 
 gons, of which the fides A P, C D, are respec- 
 tively equal to a b, b c ; and complete the 
 ground plan as in the figure. Make F 0, F I, 
 F K, F Lj and F D, refpedtively equal to 
 F0 5 FI, FK, FL, and FD; and raife the 
 perpendiculars F ON, K //, and D E. 
 Then make IG equal to I L, and L E equal 
 to CD; alfo make G N parallel and equal 
 to L £, and EH parallel and equal to IG 0 
 and join NH; then will GILEHN be a 
 vertical fedtion of the Dodecahedron, {landing on 
 a fide. Having found the perfpedtive of the 
 
 ground 
 
64 PRACTICAL 
 
 ground plan, take de equal to D E, dg equal 
 to FG, and dh equal to K H and draw e C, 
 gC, and h C. Refer all the angular points in 
 the perfpeclive ground plan to the line dC, 
 as b i 9 c £, &c. from which raife perpendiculars 
 to the line h C ; then will the interferons of 
 the perpendiculars, drawn from the correlpond- 
 ing points in the perfpeclive plan and feclion, 
 give the appearance of the angular points of the 
 Dodecahedron, as b q 9 r q ; p w, o *», &c. 
 The reft is plain from the figure. 
 
 The mathematical reader, I fuppofe, will 
 readily perceive from the method of operation, 
 that the fides of the concentric pentagons are in 
 the ratio of the fegments cf a line cut in extreme 
 and mean proportion. 
 
 LESSON XXIX. 
 
 To find the Perfpefffce cf an Icofahedron. 
 
 Operation. By vanifloing Points. Find the 
 lines H E and I D as before ; and complete 
 
 the 
 
PERSPECTIVE. 65 
 
 the plan and fcction, as in the figure then find 
 S and », the vanifhing points of L M* and H I. 
 Take S s equal to S S ; and draw s V parallel 
 to A F 9 and s v parallel to B E i alfo take 
 ny equal to S v 0 and n V equal to S V. Then 
 will V be the vanifhing point of the fides of the 
 triangular faces H I, P Qj and v the vanifh- 
 ing point of the perpendiculars of the faces 
 W H, K Alfo V and y will be the re- 
 fpe'ctive vanifhing points of the lines M L s 
 TO, and U L, T Y. And becaufe FA, 
 AB, BE, ED, DC, and C F are all parallel 
 to the perfpeclive plane, they will therefore be 
 parallel in the picture to their refpeclive lines in 
 the fection. Find m d the perfpe&ive of M D, 
 being the fide of the triangular face MDP, on 
 which the body refts ; ui 0 the perfpeclive of 
 the horizontal face A I O ; and the points u, w 9 
 and h y Then from V 9 draw V ml% and from 
 draw yuh) alfo draw V i b 9 and vwb- 9 
 and make d e 9 and b e 9 refpectively parallel to 
 D E 9 and B E. Laftly, join the points a /, i /, 
 I b 7 mb 9 b d> be, i e 9 and 0 e ; and the figure 
 formed thereby, will reprefent fo much of the 
 
 H Icofahe- 
 
66 PRACTICAL 
 
 Icofahedron as will be vifible to the fpectator's 
 eye, at the diftance C S. 
 
 After the fame manner, and by the fame va- 
 nifhing points, may the perfpeftive of the other 
 fides of the body be obtained ; but thefe are 
 here omitted to avoid a confufion of lines. 
 
 LESSON XXX. 
 
 To put a Sphere or Globe in Perfpeffive. 
 
 Operation, Let A B be the dimenfions of 
 the Globe, and A the feat of the centre on the 
 ground plane. Make D R on the line of eleva- 
 tion equal to A B the femidiameter of the Globe; 
 from whence find g the perfpeclive of the cen- 
 tre. Make C S perpendicular to C R, and 
 equal to C S, and draw S g alfo make R q 
 perpendicular to C R, and equal to A D. 
 Take AG equal to qg, and GF equal to 
 g S \ and make the angle F G I equal to the 
 angle CgS. Draw I G K, and KL, HI, 
 
 each 
 
PERSPECTIVE. 67 
 
 each perpendicular to A F ; and take g i equal 
 to G I, and g k equal to G K. Take alio 
 N M, M O, rdpeclively equal toKL, HI; and 
 from the centre V defcribe the femicircle O P N, 
 and draw M P perpendicular to N O. Laftly, 
 bifecl: k i in d, through which draw p m 9 per- 
 pendicular to k 4 and equal to P M ; then will 
 kmip, to the eye of the fpeclator, be four ex- 
 treme points on the furface of the fphere 5 
 through which an eilipfe being defcribed, will 
 give the appearance of the whole fphere on the 
 perfpective plane. 
 
 As it would be in vain to attempt to (how the 
 reafon of this operation to any but the mathe- 
 matical reader, the common learner tnuft there- 
 fore reft fatisfied with the bare operation, and 
 pafs by the following demo nitration. 
 
 Imagine the triangle C S g to be brought 
 forward, and the triangle R qg turned back- 
 ward, fo as to be both perpendicular to the per- 
 fpeclive plane. Then, becaufe the lines D R, 
 R C, are both in the fame plane, R q is per- 
 pendicular to DR; and fince R q is equal to 
 
 H 2 A B 
 
68 PRACTICAL 
 
 A B the diftance of the centre of the globe from 
 the perfpective plane, and g is the perfpective 
 of the centre, Sg, g q, is a ftraight line paffing 
 from the centre of the glebe to the fpectator's 
 eye, and cutting the picture in g. JCow it is 
 evident, *hat the rays proceeding from the ex- 
 tremities of the fphere to the eye will form a 
 cone; and as this is cut obliquely by the per- 
 fpective plane in the angle S g C, the feet ion will 
 be an ellipfe , therefore, fince A F is equal to 
 S g q 9 AG equal to qg 9 and the angle FGI 
 equal to the angle S g C, IK will be the tranf- 
 verfeof the fection. And by the property of the 
 curve L K multiplied into I H is equal to the 
 fquare of the conjugate •, but N M is equal to 
 K L, and M O equal to H I ; therefore M P 
 is a mean proportional between KL and H I, 
 which is equal to the conjugate, or morteft ap- 
 parent diameter of the fphere. Q^E. D. 
 
 Remark. From hence appears the necefiiry 
 of a picture being looked upon from a right 
 point of view, in order to produce a proper 
 effect in the fpectator's eye. For the figure here 
 drawn is a perfect ellipfe, when placed directly 
 
 before 
 
PERSPECTIVE. 69 
 
 before the eye ; but when looked at obliquely at 
 the given diftance, it appears perfectly circular, 
 and is an exact reprefentation of a globe. And 
 the fame is to be obferved in all perfpective draw- 
 ings whatever ; otherwife, when the diftance is 
 taken very near the perfpective plane, or the ob- 
 ject, feen very obliquely, the common rules of 
 perfpective will not exhibit a true reprefentation. 
 But if the eye be brought to the proper point of 
 view, the drawing will no longer appear diftorted, 
 but will produce the fame image in the eye as 
 the object irfelf would, if feen from a propor- 
 tionable height and diftance. 
 
 I think we may from hence obferve, that it 
 would be no bad method, if our capital landfcape 
 painters, and even thofe in portrait too, were to 
 write down on the back of the canvas, the height 
 of the Centre, and the Diftance of the Perfpec- 
 tive plane. For then, the Picture mio-ht be 
 placed to the greateft advantage, fo that not 
 only the contour, but alfo the light and made in 
 the colouring, would appear to the eye, exactly 
 agreeable to the painter's intention ; in which 
 light, it is plain, no drawing can be feen from 
 any other point of view. 
 
 SCIAGRAPHIC 
 
PRACTICAL 
 
 SCIAGRAPHIC PERSPECTIVE. 
 
 Or, the Method of finding the Shadows of Objecls^ 
 on the Perfpeclive Plane. 
 
 Shadowing from a Radiant Point ; 
 as Torch, Candle, &c. 
 
 LESSON XXXI. 
 
 To find the Shadow of a Pole^ which is perpendicu- 
 lar to the Ground Plane ; having the Ground 
 Seats, and Heights of the Light and Pole given. 
 
 Operation. Let L and P be the refpe&ive 
 ground feats of the light and pole ; and 1 H, 
 N K, their refpective heights above the ground 
 plane. Having found the perfpective of the 
 |] *ht and pole, by the lines of elevation I H, 
 
 NKj 
 
PERSPECTIVE. yi , 
 
 N K from /, the feat of the light on the per- 
 fpeclive plane, draw Ip s-, and from b 9 the 
 point of light, draw hk s; then will p s hz the 
 fhadow of the pole. 
 
 For as the pole is fuppofed to be a line only, 
 and the light a pointy a plane may be imagined 
 to pafs through them both (Ax. III.); but this 
 plane will interfect the perfpeclive ground plane 
 in the line Is (Ax. I.) ; therefore p s will evi- 
 dently be the fhadow of the pole p k 9 on the 
 picture. 
 
 LESSON XXXII. 
 
 To find the Shadow of a plane ObjetJ, when it is 
 perpendicular to the Ground Plane. 
 
 Operation. Let ABCD be the original ob- 
 ject, and B C its feat on the ground plane L 
 the original feat of the light, and I H the 
 height of it. Produce C B to N, and make 
 N K equal to ABj and from thence find the 
 
 per- 
 
72 PRACTICAL 
 
 perfpective of the given object, abed. From 
 /, the perfpeetive feat of the light, through the 
 points b r, draw the indefinite lines le, If-, 
 and from draw the rays of light h a e, h d f\ 
 and join e f. Then will befc be the perfpec- 
 tive fhadow of the object abed. 
 
 For the planes h I e> hlf, h e f pafs 
 through the ftraight lines ab, de, ad, and in- 
 terfecl: the ground plane in the lines b e, c f 
 and ef refpeclively (Ax. I. III.) therefore the 
 lines b cf and ef, are the refpeclive fhadows 
 of a by cd, and a d, which are the extremities 
 of the given objeel: ; and confequently befc is 
 the fliadow of the whole figure abed. 
 
 LESSON XXXIII. 
 
 To find the Shadows of Inclined Planes ^ fuppofe 
 an inverted Pyramid. 
 
 Operation. Let ABC be the feat of the py- 
 ramid on the ground plane ; and L the feat of 
 
 the 
 
PERSPECTIVE. 73 
 
 the light. Through L, draw the indefinite line 
 P I perpendicular to the line of interferon y 
 and join A L, B L, CL, and D L ; refpec- 
 tively parallel to which draw D D b, and 
 D c ; and they will reprefent the feveral interfec- 
 tions of the inclined planes, or fides of the pyra- 
 mid produced, with the ground plane. Find d 
 the perfpe&ive of and upon the director 
 I C, find the perfpective of the points a- % 
 from which draw the indefinite lines b d b, c d c 9 
 and a d a. Having found (by the feat L and 
 elevation I H) the point of light h on the pic- 
 ture, draw the rays b a q i h c c, b b b y and 
 join b c , c ay then will d a 9 a c> c b, and b d > 
 be the feveral perfpective interferons of the in- 
 - clined planes, which may be imagined to pafs 
 through the extremities of the pyramid, with the 
 ground plane y and therefore the fpace included 
 d a cb d will be the ihadow of the pyramids 
 
 I 
 
 LESSON 
 
PRACTICAL, 
 
 LESSON XXXIV. 
 s To find the Shadow of a plane circular Ring. 
 
 Operation. Let rpou be the appearance of 
 the Ring on the perfpective plane, and v the va- 
 nifhing point of its ground feat. Draw the lines 
 vry, vpo, vum \ and raife any number of perpen- 
 diculars, as my, ab,ur, &c. Then, from the feat 
 of the light /, draw the indefinite lines Imn, lag, 
 lut, &c. and the interferons of the rays h on, 
 hzg 9 hrt, &c. will give the correfponding 
 points of the fhadow ; which being joined with 
 a fteady hand, will form a true reprefentation 
 of the fhadow of the whole objecl:. 
 
 LESSON XXXV. 
 find the Shadow of an Hoop. 
 
 Operation. Let bqpr be the Hoop in per- 
 fpedtive, and C the vanifhing point of its 
 
 ground 
 
PERSPECTIVE. 75 
 
 ground feat; h the perfpective of the light, and 
 / its feat on the picture. Through x 9 the cen- 
 tre of the circular edge of the Hoop b q e r 9 
 draw the perpendicular r x q ; and from the cen- 
 tre of the picture C, draw the lines C q a 9 
 C x b 9 Crr, and raife the perpendiculars, c a 9 
 f d. Then, from the feat of the light /, draw 
 the indefinite lines / c i y Ir t, If s ; and the rays 
 proceeding from the light at h 9 through the 
 points q, e, will interfect thofe lines in the cor- 
 refponding points of the fhadow /, /, s ; through 
 which and the point r the elliptical figure 
 ritsr being traced, will mow the fhadow of 
 the circular edge of the Hoop b q e r on the 
 perfpective ground plane. In the fame manner 
 proceed with the other circular edge of nig, 
 and through the points mvnum defcribe an 
 ellipfe, and complete the fhadow as in the figure. 
 
 If thefe four points be not thought fufficient 
 to trace the ellipfe through, we may eafily 
 find as many more as we pleafe, by raifing per- 
 pendiculars from the directing line c C ; and 
 then proceeding as before, by drawing lines 
 from / and h, through, the refpedtive feats and 
 
 I 2 inter- 
 
7 6 PRACTICAL 
 
 interferons of the perpendiculars with the edge 
 of the Hoop. 
 
 LESSON XXXVI. 
 
 To find the Shadow of a Pyramid, when it is in- 
 tercepted hy Objetls fianding upon the Ground 
 Plane. 
 
 Operation. Let the intercepting objects be a 
 parallelopipedon and a wall \ which are thrown 
 into perfpective, from the given dimenfions and 
 fituations on the ground plane. Then imagine a 
 perpendicular plane to pafs through the point of 
 light h and the vertex of the Pyramid •, and it 
 will evidently interfecl the ground plane in the 
 line IC ; the parallelopipedon in the lines m m, 
 m 4, 4 q and the wall in the line o r. Now it is 
 plain from the figure, -fince the plane h IC pafles 
 through the perpendicular of the Pyramid, that 
 a C b is the fhadow of it on the ground plane 
 (fuppofing the intercepting objects removed) ; 
 therefore having produced m m to /, from the 
 
 points 
 
PERSPECTIVE, ( 77 
 
 points of interferon x, 2, draw 1 /, 2 ty and 
 refpectively parallel to a C, £C, draw ^4, 23. 
 Lailly from the points of interferon 5, and 6, 
 draw 5 r and 6 r \ and the fhadow of the Pyra- 
 mid will be truly reprefented as it falls upon the 
 given objects. 
 
 LESSON XXXVII. 
 
 To find the Perfpeclive and Shadow of a Crofs, con- 
 Jifting of two plane Parallelograms interfering 
 each other in any angle. 
 
 Operation. Let A B C D be the feat of the 
 Crofs on the ground plane, and A E F D its 
 feclion. Find V and v 9 the vanifhing points 
 of BC and AB; take VS equal to FS; 
 and make the angles V SY^ V S W, refpec- 
 tively equal to D A F, and ADE; then will 
 V be the vanifhing point of the plane A F, 
 and W the vanifhing point of the plane ED. 
 Find ah the perfpective of AB, and raife the 
 perpendiculars a g J bh\ then, on the line of 
 elevation, make Ik equal to A E, draw kV 9 
 
 and 
 
yS PRACTICAL 
 
 and vgb. Having thus determined the points 
 a 9 b 9 g 9 b, draw a V, b V, £ W, h W, and 
 complete the perfpective of the Crofs by means 
 of the vanifhing point v> as in the figure. 
 
 From m the feat of the light, draw the inde- 
 finite lines m b p 9 m a q, mrn 9 m o i ; and from 
 the radiant point b 9 draw the rays b b p, and 
 htn. Then to the vanifhing point v 9 draw p q 9 
 n i ; and join a t 9 b n, r p 9 o q 9 and it will give 
 the perfpective of the fhadow. For the extre- 
 mities of the Crofs g h 9 a t 9 are parallel to the 
 ground plane, and p q, n i 9 are their refpective 
 fhadows on that plane, they will therefore be 
 parallel to each other, and have a common va- 
 nifhing point. 
 
 The fhadow might alfo be determined by 
 drawing one ray of light only, as bp for fince 
 the points h t are in a fbraight line which is pa- 
 rallel to the ground plane, and having its vanifh- 
 ing point in V 9 a line being drawn from p to V 
 will determine the point n ; from whence i and 
 n may be found as before, and thereby the 
 whole fhadcw completed. 
 
 Shadowing 
 
PERSPECTIVE. 79 
 
 Shadowing from the Sun, 
 
 The Magnitude of the Earth being but a 
 Point when compared to the Sun's Di (lance, the 
 Rays of Light which fall thereon, will evidently 
 proceed in a Direction parallel to each other. 
 Therefore to find the Shadow of any Object by 
 the Sun, we need only know the Direction of the 
 Rays with refpect to the Line of Interferon - y 
 and their Inclination to the Ground Plane. 
 
 LESSON XXXVIII. 
 
 T0 find the Shadow of an upright Plane> when 
 the Sun is in the Plane of the Picture. 
 
 Operation, Let A B D F be a parallelogram 
 feen on the perfpe&ive plane ; and ah c the 
 angle of the fun's altitude, or inclination of the 
 rays of light to the ground plane, Draw the 
 
 indefinite 
 
So PRACTICAL 
 
 indefinite lines A G, and F H, each parallel 
 to the line of interfedtion I N ; and LBG, 
 L D H, each parallel to the ray of light a b. 
 Then join G H, and A G H F will be the 
 fliadow of the parallelogram. 
 
 LESSON XXXIX. 
 
 To find the Shadow of a 'perpendicular Plane, 
 when the Sun is behind the Plane of the PicJure. 
 
 Operation. Let A D be the original feat of 
 the parallelogram abed-, p A the direction of 
 the fun's rays ; and p A r the angle of the fun's 
 altitude, or inclination of the rays to the ground 
 plane. Find V the vanifhing point of p A, 
 and take V s equal to V S \ then make the 
 angle V sV equal to the angle p A r, and V 
 will be the vanifhing point of the fun's rays. 
 Laftly, from V draw the lines V c e, Vdf> in- 
 definitely, and the parallel rays of light V b e, 
 V cf, will interfect them in the points /, e 
 therefore a efd is the perfpedcive of the fliadow. 
 
 If 
 
PERSPECTIVE. 81 
 
 If the line e f produced, meets the line of in- 
 terferon in W the vanifhing point of A 
 it mows that the drawing is accurate. 
 
 LESSON XL. 
 
 To find the Shadow of an upright Plane, when the 
 Sun is before the Perfpeftive Plane, or Pifture. 
 
 Operation. Let A B be the ground feat of 
 the parallelogram ac db\ pq the direction of 
 the fun's rays ; and rqp their angle of inclina- 
 tion to the ground plane. Having found V the 
 vaniming point of p q, take Vs equal to FS 9 
 and make the angle FsV equal to the angle 
 r q p \ then will V be the vanifhing point of 
 the fun's rays. Therefore, by drawing the lines 
 a V, bV, c V, d V, and joining the points of 
 interferon / e, the perfpective fhadow of the 
 parallelogram b ef a will be determined. 
 
 The reafon of the method of Operation in 
 thefe two lad leflons will be obvious, by irna- 
 
 K gining 
 
82 PRACTICAL 
 
 gining the plane S C V to be turned up perpen- 
 dicular to the picture, and the plane V s v fa 
 raifed up that the line V s may coincide with 
 VS for then the line s V will evidently be pa- 
 rallel to the rays of light, and interfect the pic- 
 ture in V, which is therefore their common va- 
 nifhing point (def. 8.) And, moreover, becaufe V 
 is the vanifhing point of the direction of the rays 
 on the ground plane, it is plain that the interfec- 
 tions of the lines, drawn through the extremities 
 of the object to thofe vanifhing points, will give 
 the perfpective of the fhadow of the parallelo- 
 gram. If this be clearly underftood, there can 
 no difficulties occur in finding the fhadow of any 
 kind of object whatever, whether the rays pro- 
 ceed in a direction parallel to each other, or di- 
 verge from a radiant point. 
 
 CATOPTRIC 
 
PERSPECTIVE. 
 
 83 
 
 CATOPTRIC PERSPECTIVE. 
 
 Or the Method of finding the Appearance of an Ob- 
 ject on the Perfpeclive Plane, when reflecled 
 from a plane Surface. 
 
 An Object feen by Reflexion, always appears 
 to the Eye of the Spectator, at the fame perpen- 
 dicular Diftance on the other Side the reflecting 
 Surface, as the Object itfelf is on this Side. 
 Therefore to find the Perfpective of an Object 
 feen by Reflexion, we need only fuppofe it to be 
 fimilarly fituated on the other Side the reflecting 
 Plane ; and then proceed according to the fore- 
 going LefTons. 
 
 LESSON 
 
8 4 PRACTICAL 
 
 LESSON XLI. 
 
 To find the Perfpetlive of the Reflexion of an Ob- 
 jeft> fituated on an horizontal reflecting Plane> 
 or Surface ; as Water^ Glafs, &c. 
 
 Operation. Let BR be the brink of the 
 water, and B A a perpendicular pole. Draw 
 Ba perpendicular to BR, and equal to B A; 
 then will B a be the fhadow, or reflexion of 
 the pole on the water. If the object (lands at a 
 diftance from the water (but on the fame level 
 or plane) as at D, make D c perpendicular to 
 BR, and equal to DC; and ce will be fo 
 much of the object as can be feen by reflexion in 
 that fituation of the eye. 
 
 To find the fhadow of the fquare pillar G F, 
 produce GE, HF, and KI, indefinitely; and 
 make E^, F h, each equal to the height of the 
 pillar. Then to the centre C draw h k C, and 
 gib km will be the reflexion of the pillar on the 
 
 water 
 
PERSPECTIVE. 85 
 
 water. In like manner will the reflexion of the 
 cube M be found, by the vanilhing points V 
 and v the object and its fnadow having always 
 a common vanifhing point. 
 
 LE S S O N XLIL 
 
 5T 0 find the Reflexion of an ObjeB^ when fituated 
 above the horizontal reflecting Surface , as on a 
 Bank by the Side of a River \ . &c. 
 
 Operation. Let Mn be the brink of the 
 water, and BmnM. a perpendicular eminence 
 or bank. Firft take M b equal to M B, and 
 draw the irregular line b u for the reflexion of 
 the bank.. Then, if the object be clofe to the 
 edge of the bank as at B, produce A B, and 
 make 3VE a equal to MA; then will b a be 
 the fhade, or reflexion on the water. If the ob- 
 ject be fituated at a diftance from the edge of 
 the bank, as the pole at E ; from the centre 
 C draw C E G, let fall the perpendicular GH, 
 
 and 
 
S6 PRACTICAL 
 
 and drav/ H I C then produce D E indefinite- 
 ly, and the point of interferon I will be the 
 feat of the pole on the horizontal reflecting fur- 
 face continued. Therefore take I d equal to 
 I D, and di will be the reflexion on the water. 
 By the fame method, with the vanifhing points 
 (V, v) of the fides of the triangular prifm K L tf, 
 KL£, find R, the feat of KL; then make 
 R k equai to R K, and draw the lines k v, 
 k V ; and the figure formed by the inte;fec~lions 
 of the perpendicular fides of the prifm produced, 
 will be the perfpective of the fhadow. 
 
 To find the reflexion of the pyramid, firft 
 find o the feat of the centre q 9 as before then 
 make op, px, x /, and x r, refpeclively equal 
 loop, p x, x% and x r and draw pp 9 t p, 
 and it will produce the (hade or reflexion of the 
 pyramid. > 
 
 LESSON" 
 
PERSPECTIVE, 87 
 
 LESSON XLI1I. 
 
 2*0 find the Reflexion of a Pole from a polifhed 
 Plane or Looking-Glafs^ when both are perpendi- 
 cular to the Ground Plane, 
 
 Operation. Let A B and P be the refpec- 
 tive ground feats of the looking-glafs and pole. 
 Draw Pp perpendicular to AB; and make 
 rp equal to rP. Having found the reprefen- 
 tation of the reflecting plane on the picture, 
 a bfg ' y find the perfpective of an upright plane, 
 the ground feat of which is P p, and height af- 
 fumed at pleafure. Then will the perfpective of 
 that fide of the perpendicular plane of which the 
 feat is p y be the reflexion of the pole on the 
 looking-glafs. 
 
 For the planes afg D E e d, are perpen- 
 dicular to the ground plane, and to each other. 
 And becaufe rp is equal to r P ; d t 9 en, are 
 refpe&ively equal to / D 5 n E, and perpendicu- 
 lar 
 
88 PRACTICAL. 
 
 lar to the reflecting plane ; therefore d e is the 
 perfpective reflexion of D E. 
 
 LESSON XLIV. 
 
 To find the Reflexion of a fquare Pillar viewed on 
 an Angle i when the reflecting Plane is filiated 
 obliquely to the Line of Interferon. 
 
 Operation. Let A B be the original feat of 
 the reflecting plane, and D E F G the feat of 
 the pillar. Make i d equal to D i, 2 e equal 
 to E 2, &c. and complete the fquare d efg ; 
 which will then reprefent the bafe of a pillar 
 fimilarly fituated on the other fide the reflecting 
 furface A B. Having determined w, x, the 
 vanifhing, points of E F, E D, the bafe of the 
 given pillar; V, the vanifhing points of d e, 
 dg, the reflexion of the bafe ; and v the vanifh- 
 ing point of D d ; then find the perfpective of 
 the pillar m and nb or the perfpective of defg. 
 Then raife the indefinite perpendiculars ty, n 0, 
 r x > and to v the vanifhing point of lines per- 
 pendicular 
 
VIYY > 
 
PERSPECTIVE. 89 
 
 pendicular to the reflecting plane, draw mv\ 
 and the point of interferon q with the perpen- 
 dicular n 0, will be the correfponding point in 
 the reflexion ; from whence the whole may be 
 completed by the vanifhing points V V, as in 
 the figure. Or, the reprefentation of the re- 
 flexion might be mown without the vanifhing 
 point by finding the perfpective of a pillar 
 of the fame height with the original one, and 
 ground feat d e fg. 
 
 LESSO N XLV. 
 
 Ho find the Perfpetlive of the Reflexion of a per- 
 pendicular Pole, when the reflecting Surface or 
 Glafs is inclined to the Ground Plane. 
 
 Operation. Suppofe A B to be the feat of 
 the glafs, DBE the angle of inclination, and P 
 the feat of the pole. Having thrown the glafs 
 and pole into perfpective by leffons VIII. and 
 XXII. from the centre C, draw C H perpen- 
 dicular to v V the vanifhing line of the re flee- 
 
 L ting 
 
9 o PRACTICAL 
 
 ting plane, and produce it indefinitely towards 
 W. Take CK equal to CS, and perpendicu- 
 lar to H W then draw K H and K W, mak- 
 ing the angle H K W a right one ; fo will W 
 be the vanifhing point of lines perpendicular to 
 the plane of the glafs. Make the angle DBF 
 equal to DBA, and the angle V s Y equal to 
 FBE; then will Y be the vanifhing point of 
 the reflexion of the pole± Draw O W and 
 TW; and OTW will be a plane perpendicu- 
 lar to the glafs, and interfering it in the line 
 r N. Laftly, from N, the point where the pole 
 cuts the reflecting plane continued, draw N Y ; 
 and o t will be the perfpeclive of the reflexion 
 of the whole pole T O, fuppofing the plane of 
 the glafs to be continued downwards ; and o x 
 that part of it which will be vifible to the eye 
 at the diftance C S.- 
 
 Or, find Z the vanifhing point of B H, the 
 angle of the reflexion of P B with the feat of 
 the glafs, to which draw n Z, and it will cut 
 T W in / ; from whence draw / Y, which will 
 give t o, as before, for the reflexion of T O. 
 
 LESSON 
 
PERSPECTIVE. 
 
 LESSON XLVI. 
 
 T o find the Appearance of the Reflexion of a Pyra- 
 mid, from an inclined Looking- Glafs. 
 
 Operation. From the feat of the glafs A B, 
 and angle of inclination to the ground plane 
 DBH, find the vanifhing line of the glafs Vv. 
 From whence find W, the vanifhing point of 
 lines perpendicular to the plane of the glafs ; 
 arid Z the van i Pain g point of lines on the re- 
 flected ground plane, which are perpendicular 
 to the interfedtion of that plane and the glafs. 
 Then proceed as in the laft lelTon to find the re- 
 flexion of the perpendicular or pole of the pyra- 
 mid o d, by its vanifhing point Y ; and from 
 the perfpective of the feat of the pyramid, draw 
 the lines a i, c 2, d 3, b 4, perpendicular to 
 the line of interaction P v. To the vanifhing 
 points Z and W, draw reflectively 1 Z, 2 Z, 
 &c. and c W, d W, &c. and the points of in- 
 terferon a cb will give the reflexion of the 
 
 L 2 bafe 
 
92 PRACTICAL 
 
 bafe of the pyramid. Laftly, join o a, o c, oh, 
 and it will produce the perfpedtive of the re- 
 flexion of the whole pyramid. 
 
 If what has been done before be well under- 
 ftood, and the law of reflexion attended to, the 
 learner will eafily fee the reafon of the whole 
 operation in thefe two laft lelfons. 
 
 LESSON XL VII. 
 
 To take the requifite Mcafurcmcnts for pitting an 
 Objeff in Perfpeffive. 
 
 Operation. Let C be a chair to be put in 
 perfpeclive, and E the fpectator's eye. Then, 
 any where betwixt the eye and the object on the 
 ground plane, fuppofe the line IN to be drawn 
 to reprefent the interfection of the ground plane 
 with the plane of the picture. Let alfo 
 M L T R reprefent the paper on the drawing- 
 board, on which draw IN for the line of inter- 
 feclion. Meafure the height of your eye P E 
 
 from 
 
PERSPECTIVE. 93 
 
 from the ground plane, and make IV equal to 
 it; then draw V A 9 on which take at pleafure 
 the point C for the centre of the picture. Per- 
 pendicular to I N, meafure P B, to which make 
 C S equal, for the diftance of the eye from the 
 perfpective plane. Imagine d i, a 2 to be lines 
 perpendicular to I N \ and meafure B 1, B2, 
 Thefe diltances being fet off from D towards 
 N give the points ^, q ; from which raife per- 
 pendiculars, and make p m equal to d 1, and 
 q n equal to a 2. Laftly, raife qr perpendicu- 
 lar to IN for the line of elevation ; on which 
 take qo equal to b and qr equal to b f. 
 So will every requifite be known to find the per- 
 fpeclive of the object by the preceding leifons. 
 
 Remark. If abed be a right angled figure^ 
 and d 1 equal to a 2, the object will be viewed 
 partially. And if • d 1, a 2 be not equal, the 
 object will be viewed on an angle, and muft 
 therefore have two points of diminution. If B 
 falls at an equal diftance between the points i, 
 and 2, it will produce a central view. 
 
 LESSON 
 
PRACTICAL 
 
 LESSON XLVHl. 
 
 To find the Dijlance cf the Piflure. 
 
 As this is a point of the greater importance 
 in perfpective, it would be well for every one, 
 who would either draw well themfelves, or be 
 able to judge properly of a piece of drawing, to 
 to be particularly attentive to it being equally 
 as neceffary to the fpeculative connoilTeur as to 
 the real artift. For no one can form a proper 
 judgment of the beauties or defects of either por- 
 trait or landfcape, unlefs he knows exactly the 
 centre and diftance of the picture ; as it cannot 
 poffibly appear to the eye of the fpectator in any 
 other point of view, agreeable to the painter's 
 intention, as we have before obferved. 
 
 Irt order then that the learner may have a 
 clear idea of this very effentiai point towards 
 giving a juft representation of any object, it may 
 not be amifs to give a fhort explanation of the 
 ftructure of the eye, whereby he will the more 
 
 readily 
 
PERSPECTIVE. 95 
 
 readily fee the reafon why one diftance of the 
 picture is preferable to another. ' 
 
 In Fig. 5. which reprefents an eye in its juft 
 proportion, A is called the Cornea Tunica ; B 
 the Uvea, in which is a fmall perforation c, 
 called the Pupil, through which the rays of light 
 enter; C the Aqueous Humour ; D the Cry- 
 ft aline Humour, which is nearly of the form of 
 a double convex lens, being fufpended in the 
 middle of the eye by an annulus of mufcular 
 fibres, called the Ligamentum Ciliare, as at a 
 and b \ E the Vitreous Humour ; F the coat 
 called Choroides \ G the coat called Sclerotica, 
 being a part of the Dura Mater continued about 
 the ball of the eye on its inner part ^ H the Op- 
 tic Nerve \ and I the Retina, which is a part of 
 the Optic Nerve, expanded over all the interior 
 part of the eye to a and b. Each of thefe parts 
 have a different refractive power, which all con- 
 cur towards dlftincl vifion, by forming the 
 image of every external object on the retina, 
 which is by nature appointed the immediate or- 
 gan of fight. 
 
 Thus 9 
 
96 PRACTICAL 
 
 Thus, if A B be an object placed at any dif- 
 tance from the eye, a pencil of rays proceeding 
 from any point C will firft fall on the cornea, 
 and then be refracted by the aqueous humour 
 under it to a point M, in the axis of the pencil 
 behind the retina. The rays then falling on the 
 cryftaline humour are again refracted, and 
 meet the axis ftill nearer the retina at N. After 
 which they enter the vitreous humour, where 
 they are again refracted fo as to meet the axis 
 on the retina at Hence it appears, that the 
 rays of light in flowing from every part of an 
 object A B placed at a proper diftance from the 
 eye, will form an image on the retina, which 
 is always inverted, or in a contrary pofition to 
 that of the object, becaufe the rays which come 
 from the extreme parts thereof A, B, crofs each 
 other in the middle of the pupil. 
 
 Now it has been found from repeated meafure- 
 ments of the feveral parts of the eye, that the 
 extremities of the cryftaline humour fubtend an 
 angle at the pupil of about 90 degrees ; confe- 
 quently no object can be distinctly feen by the 
 
 eye 
 
PERSPECTIVE. 97 
 
 eye under an angle which is greater than that. 
 For if we fuppofe the object A B to be brought 
 nearer to the eye as at A B, it is plain that the ex- 
 treme rays A c y B c> in palling through the pupil, 
 will not fall on the cryftaline humour, but on the 
 ligamentum ciliare, and therefore cannot be duly 
 refracted to the retina. And though the rays pro- 
 ceeding from an object fubtending an angle at the 
 eye of ,90 degrees, may be refracted to the fundus, 
 yet it will not be feen diftinctly ; for as the rays 
 have a greater or lefs force as they are lefs or 
 more oblique, it follows, that thofe rays which 
 pafs through the very middle of the cryftaline 
 are of moft efficacy in caufing vifion, as entering 
 the pupil perpendicularly, and failing in the 
 fame direction on the retina. Agreeable to this, 
 it is found from obfervation, that any large ob- 
 ject is beft viewed at an angle of about 60 de- 
 grees, and fmaller objects under any angle be- 
 tween that and 30 degrees; If the object be 
 very fmall it cannot poffibly be feen diftinctly un- 
 der a greater angle than 30 degrees, becaufe the 
 refractive powers of the humours will not admit 
 of a lefs diftance, which in a perfect eye is never 
 lefs than fix or feven inches. Indeed when an 
 
 M object 
 
98 PRACTICAL 
 
 object is not far diftant, there is a power in the 
 eye of lengthening the focal diftance after the 
 laft refraction in the vitreous humour, and there- 
 by adjufting itfelf for diftinct vifion at different 
 diftances but as this is limited to a very narrow 
 fpace, it cannot fenfibly affect the above angle 
 for fmall objects* It has been a matter of dis- 
 pute for fome time among the gentlemen of the 
 faculty, whether this is effected by means of the 
 ligamentum ciliare, in bringing the cryflaline 
 humour a little nearer the cornea or, by a 
 power in the optic nerve to alter the convexity of 
 that humour. 
 
 But to mow the application of this more im- 
 mediately to the practice of perfpective, let 
 A B C D, Fig 6. be a fquare fituated on the 
 ground plane, I N the line of interferon, V A 
 the vanifhing line, &c. as before. Now if the 
 eye of the fpectator was at s, it is plain he could 
 not fee the two ends of the fquare A and B, 
 becaufe the eye could fee at mofl but fo much 
 of it as is comprehended under the right angle 
 c s d. But if the eye was at /, it could fee the 
 
 ends A and B, becaufe the angle afb is a 
 
 right 
 
PERSPECTIVE. 99 
 
 right one. And the eye would fee the ends A 
 and B better at S, and better yet at S, where 
 the angle a S b is 60 degrees. And from fuch 
 a diftance, the objects may be feen in perfection ; 
 the vifual angle being neither too great nor too 
 little. Therefore when the perfpective plane is 
 imagined to be contiguous to the nearer! part of 
 an object,- the diftance C S mud be taken fo 
 large as to make the angle C b S not lefs than 
 45 degrees. But as the perfpective plane may 
 be imagined to ft and any where betwixt the fpec- 
 tator's eye and the object, it is evident that this 
 is not a general rule, though it is given as fuch 
 in moil books of perfpective. For it is plain 
 from Fig. 5. that whether the perfpective plane 
 be placed contiguous to the object, as at PE; 
 or any where nearer to the eye, as at p e , that 
 the object A B will appear to the fpectator's 
 eye under the fame angle A c B. But if we 
 fuppofe the object to be placed at p e, it would 
 fubtend an angle at c greater than a right one 5 
 and therefore could not be diftinctly feen by the 
 eye at that diftance. Hence it appears, that it 
 is not the diftance of the piElure^ but the diftance 
 
 M 2 of 
 
ioo PRACTICAL 
 
 of the otyeft that muft be principally regarded in 
 the practice of perfpective. 
 
 Now let E F D C, Fig. 6. be a parallelogram 
 placed at a diftance from the line of interfection, 
 and we (hall have this general rule for finding 
 the diftance of the picture, when the object is 
 viewed under a given angle. Draw C H perpen- 
 dicular to I N, and make the angle H F Z equal 
 to the complement of half the ariumed angle 
 (which muft not be greater than 60, nor lefs than 
 45 degrees, as hath been obferved before) and 
 Z K will be the true diftance of the picture, for 
 the object to appear to the fpectator's eye under 
 the given angle. 
 
 THEATRICAL 
 
PERSPECTIVE. 
 
 lot 
 
 THEATRICAL PERSPECTIVE, 
 
 As the whole Defign of a Theatre is founded 
 in Perfpective, it ought to be ftrictly regulated 
 by the Rules thereof. For if we imagine a 
 tranfparent Plane to be put in the Place of the 
 Profcene or Curtain, it is evident, that an Ob- 
 ferver would view the whole Affair tranfacted 
 perfpectively and therefore, not only the Stage, 
 but alfo the Sides and Ceiling, Profcenes and 
 Poftfcenes mould be all in regular Perfpective. 
 
 Hence it is, that the whole Houfe, internally, 
 is fo contrived, as to reprefent a perfpective 
 View of an hollow Parallelopipedon, or large 
 Room, about three or four Times as long as it 
 is broad or deep, the Ceiling, Sides, and Floor 5 
 all converging by the Rules of Perfpective to 
 the Centre of the Picture at the fartheft End 5 
 which is the remoteft Part of the Stage. 
 
 LESSON 
 
jo2 PRACTICAL 
 
 LESSON XLIX. 
 
 To determine the Form and Pqfition of the Stage 
 the Vanijloing Lines and Centres of the Scenery ; 
 and the Situations and Lengths of the Grooves 
 for the fliding and lateral Scenes. 
 
 Operation. Let A B D E be an horizontal 
 plane, F'G a line drawn through the middle of 
 it, and F H the height of the eye. Then if 
 H c be drawn parallel to F G, it will cut the 
 profccne or perfpective plane I K L N, in C ; 
 which is therefore the centre of the picture, and 
 and V A the vanifhing line. Now upon this 
 perfpective plane the picture of the ftage will 
 be represented by INC; the lateral fcenes or 
 fides by I K C, N LC; and the aerial fcene or 
 ceiling by KLC; all which Ihculd have a per- 
 fect coincidence in the fpectator's eye with the 
 parts they reprefent. This is effected in the fol- 
 lowing manner: 
 
 In Mead 
 
PERSPECTIVE. 103 
 
 Inftead of the upright plane I L, let another 
 plane I N a v be placed on the fame line I N, 
 and fo far inclined, that its elevation G c may 
 be j uft equal to the height of the eye F H. 
 Then will the end of the inclined plane v a ex- 
 actly coincide to the eye at H with the vanifh- 
 ing line V A, and the perfpeclive ground plane 
 INC on the profcene be projected into the 
 large one INc; which is therefore the true 
 form and elevation of the ftage. The heights 
 of the feveral lateral fcenes, or Aiders, are to be 
 adjufted from the inclination of the lines K 
 I c ; and the diftances for the grooves may be 
 thus found \ Divide the line N C, which is the 
 perfpective of the ground line of the ftage, into 
 the perfpeclive equal parts, 1, 1 ; 2, 2 ; 3, 3, &c. 
 and from H refer them to the fide of the ftage 
 N c \ then will 1, i y 2, 2 , 3, 3, &c. be the 
 true places and diftances of the grooves, for the 
 lateral fcenes to appear in true perfpective to the 
 eye of a fpectator at H. 
 
 The vanilhing line V of any poftfcene 
 iiklmy is beft found by a fed ion of the ftage 
 
 € O G 5 
 
104 PRACTICAL 
 
 fOG; from whence the height of the eye A C, 
 from the bottom of the fcene, is eafily determined 
 to any given diftance from the profcene O P 
 being only the difference betv/een the height of 
 the horizontal plane O C, and the perpendicu- 
 lar height of the ftage A P in that pjace. By 
 obferving this rule, there will be a perfect uni- 
 formity in the fcenery as the vanifhing line of 
 every poftfcene will then be in the fame hori- 
 zontal plane, and the whole will thereby have 
 a very agreeable and natural appearance to 
 the fpeciators. 
 
 It appears from what has been faid, that a 
 perfon will fee a theatrical performance mod 
 juftly reprefented, from the farther! end of the 
 pit, or in the middle boxes ; where every thing 
 will appear to the greateft advantage. For the 
 eye will then be in the centre of the perfpeclivc 
 plane, and at a proper diftance from the pro- 
 fcene, which generally appears there under an 
 angle of about 55 or 60 degrees. The Man- 
 chefter Theatre-Royal is admirably well execut- 
 ed in this refpecl, and does great honour to the 
 
 judgment 
 
PERSPECTIVE. 105 
 
 judgment of the projectors. For to the eye of 
 an obferver fituated at the farther end of the pit, 
 or in the middle boxes, there is a perfect coinci- 
 dence in the perfpective, not only of the poft- 
 fcenes, but alfo of the ftage and lateral fcenes ; 
 all uniting in one common point of view. And 
 the diftance of the front of the middle boxes 
 
 from the orcheftra is fo adjufled, that the pro- 
 
 i 
 
 fcene, or field of view, fubtends an angle at the 
 eye of the fpeclator of 60 degrees ; under which 
 angle we have before mown, any large objects 
 will be feen in the greateft perfection, 
 
 N 
 
 HORIZONTAL 
 
io6 PRACTICAL 
 
 HORIZONTAL PERSPECTIVE. 
 
 This Kind of Perfpective relates chiefly to 
 Ceiling Pieces *, which, when properly executed, 
 will make the Objects appear to the Spectator's 
 Eye, as if they were drawn on the Sides of the 
 Room continued upwards. And in order to 
 produce this Deception, nothing more needs be 
 done, than to make the Centre of the Ceiling the 
 Centre of the Pitture and the Height of it above 
 the Spectator's Eye the Bijiance of the Piffure-, and 
 then proceed according to the foregoing Direc- 
 tions for Objects fituated on the Ground Plane. 
 
 LESSON 
 
PERSPECTIVE. 107 
 
 LESSON L. 
 
 <To draw a Line upon a flat Ceilings which fball 
 appear to' the Eye of the Spectator, to be per- 
 pendicular to the Ground Plane^ or parallel to 
 the Sides of the Room. 
 
 Operation. Let A B C D be the ceiling, E 
 the place where the perpendicular is to be ap- 
 parently raifed from, and E F the given height 
 of it ; and let C be the centre of the picture, and 
 C S (or C S) the given diftance. Then from E, 
 draw EC; and from F, draw F and E/ 
 will be the reprefentation of the perpendicular 
 line, as required. 
 
 After the fame manner may the ceiling be 
 made to appear as the fides of the room con- 
 tinued upwards, as follows : From each corner 
 of the room to the centre of the picture, draw 
 the lines AC, B C, C C, and DC; then draw 
 D S, and from the point of interferon c 7 draw 
 
 N 2 ; 
 
ic8 PRA, CTICAL 
 
 c b, b <?, a d, d c> refpectively parallel to the 
 fides of the ceiling C B, B A, AD, DC. 
 And abed will be the perfpective ceiling, 
 which will appear as if it were elevated higher 
 by C D than before and the parts A a d D, 
 D d c C, Cf^Bj and B b a A will have the 
 fame effect to the eye at S, as if the fides of the 
 room were actually continued upwards to that 
 height. 
 
 LESSON LI. 
 
 o draw the Reprefentation of a Window upon tk 
 a flat Ceilings which jJoall appear to the Specla- 
 tor's Eye, as if it were in the Side of the Room 
 continued upwards. 
 
 Operation. Let A B C D be the ceiling of 
 the room, and abed the perfpective of it. Let 
 alfo EF be the breadth, and E^ the height 
 of the window ; and E e the thicknefs of the 
 wall. To the centre of the picture C, draw 
 EC, eC, and F/$ then draw g S, and make 
 
PERSPECTIVE. 109 
 
 m mfi refpectively parallel to AD, A B ; 
 and it will produce a juft reprefentation of a win- 
 dow in the fide of the room, when it is completed 
 as on the oppofite fide A B. 
 
 If the window be arched at the top, find the 
 points otxp as before \ and take hi equal to 
 the height of the arch, and draw i S. Then 
 make r v parallel to B C, vu parallel to A B, 
 and u s parallel to B C. Laftly, draw the 
 curved lines 0 /, xp, and finidi the whole ac- 
 cording to the figure on the other fide AD $ and 
 it will have the defired effect to an eye oppofite 
 to C, and at the diflance S C. After the fame 
 manner may cylinders, pilafters, arches, doors, 
 cornices, domes, human figures, &c. be drawn 
 on a ceiling, fo as to appear upright, or per- 
 pendicular to the ground plane ; which is fully 
 exemplified in the fecond volume. 
 
 The practical method of drawing upon ceil- 
 ings, is to lay down the dimenfions of the ceiling 
 upon paper, and with the height of the room as 
 the diflance, to finifh the whole drawing, both 
 with regard to the lines and colouring; then 
 
 trans- 
 
no PRACTICAL 
 
 i 
 
 transferring the whole to the ceiling by reticu- 
 lation, and finifhing it agreeable to the model. 
 
 LESSON LII. 
 
 Direclions for taking Perfpeffive Views without 
 any Measurements. 
 
 Operation. Having fixed your paper on the 
 drawing-board, draw a line to reprefent the bafe 
 line of the picture, or the place on which you 
 ffand ; and about the middle of the paper draw 
 another line parallel to it, for the horizon or 
 boundary of your fight. Then, in order to fad* 
 liiate the meafuring of the diftanccs of each part 
 of the view by your eye, draw the neareft object 
 on the left hand, as church, houfe, tree, &c. 
 £s accurately as you can. Having done this, 
 hold the edge of your drawing-board horizon- 
 tally, fo as to cut or lie over the extremities of 
 the next difcant object \ and obferve what part 
 of the object on the fore-ground is there covered 
 by the edge of the board. Then draw, or ima- 
 gine 
 
PERSPECTIVE. in 
 
 gine to be drawn, on the paper, a line parallel 
 to the bafe line and palling through the corres- 
 ponding part of the object already drawn on 
 your paper. Thus proceed till you have got to 
 the moft remote objects. Then do the fame 
 with what is to be feen on your right hand, and 
 you will have every thing laid down pretty near 
 the truth, both with refpect to diftance and pro- 
 portion. When you have finiuhed your rough 
 draft, which may be done in black lead, Italian 
 chalk, &c. transfer the whole to the paper on 
 which you intend to complete your drawing 
 and before you begin to out-line or finifh any 
 part of it, correct the perfpective by obferving 
 the following directions : i. All objects, as the 
 fides of houfes, walls, hedges, walks, avenues, 
 &c. which you obferved to be parallel to your 
 drawing-board (when held in the before-mention- 
 ed pofition J muft alfo be parallel to the line of 
 interferon or bafe line in the picture. 2. What- 
 ever you obferved to be perpendicular to the 
 board, muft diminifh into the centre of the pic- 
 ture. 3. If an object was fituated obliquely to 
 the board, lay the edge of a ruler along any line 
 which is horizontal, as the eaves of an houfe, &c. 
 
 and 
 
H2 PRACTICAL 
 
 and mark the point where it cuts the vanifhing or 
 horizontal line, and that will be the point of di- 
 minution, and alfo the vanifhing point of all 
 other objects which were in the like fituation in 
 rdpe£t to the drawing board. 4. If in produc- 
 ing the flant lines of the fame object, you fhould 
 find that they do not interfect in the vanifhing 
 line, it mows that you have not been accurate in 
 your rough fketch, which muft therefore be cor- 
 rected by the point of interferon being brought 
 into the vanifhing line. For it cannot poffibly 
 happen in that manner, unlefs the object ftands 
 inclined to the horizon. 
 
 When your out-line, is correct, begin with the 
 fhading part : This is bed performed by fhad- 
 ing the whole faintly over firft, and fb proceeding 
 by degrees of made till you have made the whole 
 agreeable to your eye. By this method you will 
 be better able to judge of the propriety of the 
 fhades, than by finifhing it by piece-meal, which 
 a great many practife very injudicioufly. You 
 muft likewife obferve to make all your light and 
 fhades fall one way, and let every thing have its 
 proper motion, as trees fhaken by the wind, 
 
 the 
 
PERSPECTIVE. 03 
 
 the faiali boughs bending more than the large 
 ones ; water agitated by the wind, and dafhing 
 againit mips or boats, or falling from a precipice 
 upon rocks and ftones, and fpurting again into 
 the air. The work mould always imitate the 
 feafon it is intended to reprefent, and every view- 
 have its proper adjuncts ; as the farm-houfe 
 mould be accompanied with herds of cattle, 
 flocks of fheep, mills, woods, &c. But to ac- 
 quire a facility of performing with propriety all 
 the additional graces in a perfpective view, a 
 good copy will be found of far greater fervice 
 than a verbal defcription. I know of none more 
 proper for this purpofe than Collet's landfcapes, 
 
 Plate LII. is added for the learner's exercife 5 
 where C is the centre of the picture C S the 
 diftance ; a b c the falient angle of the central 
 piers, ib their perpendicular projection, and 
 V, V their vanifhing points; and e b the 
 breadth of the bridge. 
 
 . THE END OF THE FIRST VOLUME 
 
I 
 
 r 
 
/ 
 
 / 
 
rtttement 
 
 THE fecond Volume of this Work will be publiftied 
 as foon as ever the Plates can be got ready ; which 
 will occafion fome Delay, as they confift chiefly of finifh- 
 ed Objects, Views, &c. and are executed in a very maf- 
 terly Manner. 
 
 In the Prefs, and will fpeedily be publi/bed, by the 
 fame Author, 
 
 I. The Rationale of Circulating Numbers ; with the 
 Inveftigations of all the Rules and peculiar ProcefTes ufed 
 in that Part of Decimal Arithmetic. To which are 
 added feveral curious Mathematical Que ft ions, with fome 
 ufeful Remarks on adfected Equations, and the Finding 
 of Fluents. Adapted to the Ufe of Schools. Price 
 5 s. in Boards. 3vo. 
 
 In this Treatife the young mathematical Reader will 
 meet with many Things which are not only curious, but 
 will alfo afford him much Information in thofe Parts of 
 the Mathematics which Teem to be rather obfcure. In 
 adfecled Equations, a very clear and concife Rule is given 
 for extracting the Cubic Root of an impoffible Binomial ; 
 whereby Cardan's Theorem is rendered generally ufeful, 
 in finding the Roots of an Equation when they are all 
 real, as well as when there is but one real, and two ima- 
 ginary. The Impoffibility of obtaining general Formula 
 for the Surfolid and other higher Equations is alfo mown, 
 and the Reafon of it. The Method of tabulating literal 
 Equations is likewife fully explained, and illuflrated with 
 
 Examples ; 
 
( ) 
 
 Examples from whence the Reverfion of a Series, 
 however affected with Radicals, is rendered extremely 
 eaiy. In Fluxions, the Principles are fully explained, by 
 avoiding all metaphyfical Confiderations, and rendered 
 clear to the loweft Capacity. The whole Bufinefs of 
 finding Fluxions is reduced to one general Rule ; and 
 the particular Forms of fluxionary Expreffions are fo dif- 
 tinguifhed, that the Learner may almcft immediately de- 
 termine in what Manner the Fluent may be obtained. 
 The Correction of a Fluent is alfo clearly explained, and 
 the Reafon of it. And at the End is given a complete 
 Catalogue of the mod approved Authors in the feveral 
 Branches of Mathematics, Philofophy, and Aftronomy. 
 
 IT. An EfTay on the Ufefuinefs of Mathematical Learn- 
 ing ; wherein is fhown the progreflive Growth of the 
 Mathematics, from their Infancy to theprefent Time; and 
 a Comparifon drawn between the Ancients and Moderns ; 
 proving the high Estimation they were held in by the for- 
 mer, as comprehending Kofid rOL Ma^uaTtf, 
 or the whole Circle of Human Learning. With an Al- 
 phabetical Account of the moll eminent Geometers and 
 Mathematicians, ancient and modern, and the Works 
 they have published. To which is added, A Treatife on 
 Magic Squares, tranflated from the French of Frenicle, 
 as publithed in Les Onv rages de Mathematique par Mef- 
 fieurs de P Academic Roy ale de Sciences, with feveral Ad- 
 ditions and Remarks : A Subject, though not very in- 
 fere/ting in itfelf ; yet, which affords- the Mind a pleafing 
 Satisfaction in ohferving the wonderful Properties of 
 Numbers, Price 6s. in Boards, 8yd, 
 
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 Globes, with their Appendages, 
 »nd other necefDry Definitions of 
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 Derivations. 
 
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 Sphere in RefpecT- to the Horizon, 
 and of the inhabitants of the Globe 
 in Refpcct or their Stuations to one 
 another. 
 
 All the u r efui Problems perform- 
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 Of the ufe of Maps, general and 
 pirticular, and of the constituent 
 Parts of the Terraqueous Globe, as 
 they are delineated thereon. 
 
 A general Defcription of the Ban h 
 and Seas, and a particular Defcrip- 
 tion of Europe, in which is given 
 the Geography of ancient Britain. 
 
 A Defcription of Afia, with the 
 ancient LetTer Afia, and the ancient 
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 the Chaldeans, and Armenia. 
 
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 Romanum Imper'ium ad Acmen evec- 
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 Of the folar Syftem. The Sun 
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 given to the Planets, Signs, and 
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