5 \ # r jr~' Jf L ; , - ij ~ fi'jz $r gtAJh'uJ £%czed 1 A«, i'/j ft*-* Z T H E H E O O F R Y PERSPECTIVE DEMONSTRATED; IN A METHOD ENTIRELY NEW. By which the feveral PLANES, LINES, AND POINTS, USED IN THIS ART, ARE SHEWN BY MOVEABLE SCHEMES, In the true Portions in which they are to be confidered. Invented, and now publifhed for the Ufe of The ROYAL ACADEMY at WOOLWICH. BY JOHN LODGE COWLEY, PROFESSOR OF MATHEMATICKS. LONDON, Printed for T. Payne, at the Mufe-Gate ; J. Dodsley, Pall-mall; J. IVIillan, Charing - Crofs ; B. White, in Fleet - Street j Webley, Holborn ; T. Longman, in Pater-nofter Row; R. Hors field, in Ludgate-Street ; S. Hooper, in the Strand ; and J. Bennet, in Crown-Court, St. Anne’s, Soho. MDCCLXVI. Digitized by the Internet Archive in 2016 with funding from Getty Research Institute https ://arch i ve .org/detai Is/theoryof perspectOOcowl TO THE MOST HONOURABLE JOHN MANNERS, MAR Q_U I S OF GRANBY, One of His Majefty’s Mod: Honourable Privy Council, Lord Lieutenant Cuftos Rotulorum of the County of Derby, Colonel of the Royal Regiment of Horfe Guards, Lieutenant General of His Majefty’s Forces, and Matter General of the Ordnance, &c. &c. &c. My Lord, The great Honour Your Lordfhip has condefcended to bettow upon me in Your acceptance of my endeavours to facilitate the theory or true idea of PERSPECTIVE, warms me to the mod; faithful difcharge of this, and every other duty of my flation ; well knowing it to be my bed: recommendation to Your Lordfhip’s future favour and pro- te&ion. I am, with perfect edeem. My Lord, Your Lordfhip’s mod: obedient. Royal Academy, Woolwich, May i, 1765. And mod devoted humble fervant, John Lodge Cowley . rx, o Vj -/ ^ L • cUrT/^ k P R E F A C E. I Shall not here point out the ufefulnefs of PERSPEC- TIVE, or the fuperior excellence of the principles I have endeavoured to explain, fince the numerous writings of very eminent perfoiis ; , employed on this fubjedt, are fufficient teftimonles of the former ; and it is hoped the enfuing Treatife will fully evidence the latter. On the other hand, I frankly own, that my performance allows me not any reafonable claim or pretence to an enlargement of fcience, by extending its dominions, though this is confehedly the chief merit of an author. Inftrudting youth in certain branches of mathematical knowledge, planned and improved by others, is the pro- vince to which I am devoted ; and to convey the fenfe of my Ledtures to ftudents, in the moll eafy and perfpicuous manner, is the main point I afpire to. The quick fale of the whole imprelTion of my Appen- dix to Euclid, and the repeated demands for that treatife, are indubitable marks of publick approbation ; and have determined me to expedite a fecond edition of PREFACE. it, or rather a new and more comprehenfive work, which I believe will prove a .confiderable help to fuch as engage in that part of mathematical learning ; no lefs perfuaded that this my prefent attempt, to explain and facilitate the dobtrine of Planes as applicable to Perfpebtive, will have a fimilar effebt, efpecially as it has been approved of by The honourable Sir CHARLES FREDERICK, sur- veyor GENERAL OF HIS MAJESTY^ , ORDNANCE, who Was pleafed to confider it in manufcript ; on which foundation I, with the more confidence, fubmit it to the publick, J. L. C, ji: n.: A SUCCINCT HISTORY OF PERSPECTIVE. f^ERSPECTIVE, according to the accounts of hiftorians, is derived from painting, particularly as employed in theatrical decora- tions, but was very little known until about the beginning of the fixteenth century. The defire of reprefenting on a plane, or flat furface, fuch figures as fhould produce, on the eyes of the fpedta- tors, effects fimilar to thofe which would be occafioned by their being feen in relief, and at different diftances, engaged the artifls of thofe times to confider the apparent diminution of magnitude and alterations of pofition, which fuch different objedts exhibit to the eye, according as they are nearer to or farther from the fame. In this refearch it is, at leaf!:, very probable, that a due obfervance of natural appearances furnifhed the firfl: hints to intelligent painters how to proceed in this part of their art ; for, by looking at a range of objedts placed on lines parallel to each other, as rows of trees, &c. they could not but fee them as converging together, and ap- pearing nearer and nearer to each other in proportion to their re- motenefs : the ground, although really level, would feem as rifing ■upwards with a gentle afcent ; on the other hand, in viewing a level plane elevated on high, as a flat cieling, &c. they would fee it as appearing to decline or fink in going off from the eye. Thefe, and fuch like notices, may reafonably be fuppofed the primary or principal guides by which artifls werediredted to make the imitations which they firfl produced of thefe kinds of appearances; and that illu- sion's of this fort being publickly exhibited, induced geometricians, a A SUCCINCT HISTORY u who are not to be fatisfied with any thing lefs than rigorous exadtnels,. to examine ftridtly into the caufes of thefe effects, and the means of accurately inveftigating perfect imitations of thefe illufive appear- ances, and thereby Perfpedtive is become a fyftem of mathematical rules, the true guide and firm fupport of the imitative arts. The ancients made it a general and leading principle to confider the objects, which they would reprefent, as being beyond a tranf- parent plane, fo placed between the eye and the objedts, that all rays ifiuing from them to the eye fhould pafs through the fame, and imprefs thereon fuch images as would have the fame efifedt to an eye placed in the proper point for viewing them, as the real objedts themfelves would produce when feen in their natural Hate ; and the moderns have afiented to this pofition of the ancients ; for, feeing that the image or reprefentation, thus imprinted on that plane, fends to the eye the fame rays as the real objedt itfelf would do, it is ma- nifelt, from all that has been difcovered in optics, that no other fenfation can be produced, but that fimilar effedts will be wrought upon the fpedlator and that,, if the variations of magnitude and po- 1 * fition be duly afiilted by the art of colouring, the illufion will be compleat : whence the art of Perfpedtive confifls abfolutely in a ge* ometrical determination of the points in which the feveral rays cut the faid tranfparent plane in proceeding from each point of the given objedts to the eye.. In fhort, a perfpedtive reprefentation is no other than a projedtion of objedts, due regard, being had to the pofition or place of the eye. This fundamental principle of the ancients may be yet more ge- nerally extended ; for it is not neceflary to imagine the objedts be- yond a tranfparent plane, nor even that the plane fhould be tranf- parent ; for the objedts may be confidered and treated as being between the eye and the plane on which they are to be reprefented* OF PERSPECTIVE, Hi and the vifual rays, by which the objedts are perceived, may be pro- duced till they meet the plane which is to exhibit their images, and thereby mark out other points analogous to thofe of the original objedts from whence they proceeded, which feveral rays will form on that plane, when thus pofited, correfpondent images, which will, in like manner, fliew the perfpedtive reprefentations of the given ob- jedts i but the other way of applying this principle is mod; generally ufed, and is therefore particularly alluded to in def. I.. page 17. ne- verthelefs the geometrician is at liberty to have recourfe to either, as fhall be mod; convenient. Vitruvius, in his Architedture, book VII. chapter I, has pre- ferved feme of the Perfpedtive of the ancient Greeks ; he obferves, that one Agatarchus, having been indrudted by Eschylus how to draw theatrical decorations,, was the fird who wrote upon the fubjedt ; that Agatarchus taught'. his* art to Democritus and Anax- agoras, and that thofe two, who were geometers, had alfo wrote about it. Vitruvius adds, that they Ihewed how to draw lines from a point in a certain place,, fo as' fliall afluredly reprefent buildings in a decoration, in fuch manner, that fome of them fhall appear to pro- jedt or advance towards the eye, while others at the fame time feem to fall back and retire. In this manner does the Roman arehitedt explain the performances of Democritus and Anaxagoras y we need only be barely initiated in the rudiments of optics to enable us for didinguidiing the refem- blance which the principles of the ancient Perfpedtive have to thole of our own y for the point in a certain place, is that we call the point of fight, or place of the eye, which determines the pofition of almod all the lineaments of the objedt. This is all we have left concerning the Perfpedtive of the ancients, *s far as I can find y fo that we may, at lead, fairly account the mo- A SUCCINCT HISTORY av derns, the fecond inventors of this art, which took its rife among us in thofe glorious days of painting, the latter end of the fifteenth, or the beginning of the fixteenth century. Two artifts, who were alfo good geometricians, Albert Durer in Germany, and Pietro del Borgo San Stephano in Italy, gave rules for putting objeCts in perfpeCtive. Durer did it mecha- nically by the help of a machine, whofe conftruCtion and ufe are founded upon the ancient principle before-mentioned. Borgo, who was a little before Durer, wrote three books upon the fubjeCt, which Ignatio Dante highly extols, but they are loft to us. After this author Daniel Barbaro, patriarch of Aquileia, pub- lifhed a treatife on this art, in 1579. Balthazar Peruzzi, of Sienna, was happy in clearing PerfpeCtive of many incumbrances which it laboured under in Italy, and gave it a degree of elegance, by introducing the ufe of points different from thofe before made ufe of, as known to us, which are called points of distance : this author has been exaCtly followed, even to a degree of fervility, by Vignola, the famous Italian archi- tect, in his Treatife of PerfpeCtive. Sirigatti made ufe of Vig- nola in the fame manner j and Andrea Pozzo, fo lately as the year 1700, indulged himfelf in a free ufe of the fame liberty. Guido Ubaldi Marquis de Monte, in a folio treatife, printed at Pefaro in 1600, confidered PerfpeCtive in a more fcientific view than any of the former he appears as being the firft whofe ideas tended towards rendering the principles of PerfpeCtive univerfal, and greatly improved the art, by advancing this very prolific prin- ciple, viz. that all lines, parallel to one another and to the horizon, although inclined to the table, or picture as it is now called, would conftantly converge towards a point in the horizontal line, and that OF PERSPECTIVE. the faid point is that where the faid horizontal line is interfered by the line drawn from the eye parallel to the afore-mentioned lines. Ubaldi might indeed have made his principle more general, by fhewing, that all lines parallel to one another, though not parallel to the horizon, do meet in one certain point of the picture, namely, in that where it is cut by that particular parallel ray which is drawn from the eye ; and there are conditions which require this augmen- tation to what he advanced. But what Ubaldi has given us may be made to fuffice in the ordi- nary cafes of Perfpeftive, where fuch objects mod; generally prefent themfelves as are terminated by lines perpendicular or parallel to the horizon ; for it is plain, that the apparent concourfe of all lines per- pendicular to the plane of the picture, being in the principal point, is only a particular cafe of Ubaldi's principle j for the faid principal point is no other than that in which the pi&ure is interfered by the perpendicular drawn to the eye. In like manner, lines inclined 45 degrees to the plane of the pidture, meet in that point of the hori- zontal line in which it is cut by a line, drawn from the eye under an angle of 45 degrees. ruAll-q>arallels, inclined 30 degrees to the plane of the pidture, will appear to concur in that point thereof, in which a line, drawn from the eye under the like angle, meets the fame ; and the like of others. So that it w r ould be no difficult mat- ter to folve the general and fundamental problem of all Perfpedtive after 25 different ways, as Ubaldi has done, but the fame may be performed by other methods innumerable. As for the reft of Ubaldi’s work, it has the fault common to others of that time of day, and what he has there expanded, through a multitude of proportions, might have been elegantly comprehended in the compafs of a few pages. b VI A SUCCINCT HISTORY With refpeCt to what has been tranfmitted to us by Marolois, the Jefuit, and others, whom we call the writers on the old PerfpeCtive, in contradiflinCtion to the fyftem we here endeavour to explain, it feems fufficiently manifest, that the works of Peruzzi and Ubaldi have been the general ftore-houfe to which thofe feveral writers have had recourfe for the principles they make ufe of we Shall there- fore pafs over, for tire prefent, the numerous writings which have appeared on this part of the mathematicks, and not make any far- ther excurfions on a fubjeCt, whofe greateft difficulties are within the reach of a moderate geometrician, but proceed to ffiew what were the next advances made towards bringing it to a Rate of perfection, and into general ufe among artifls, and others defirous of being ac- quainted with this ufeful and pleafing art of deception. The celebrated geometrician. Dr. Brook Taylor, F. R. S. ob- ferving how confined, intricate and ungeometrical, the principles of PerfpeCtive were in his time, how few, fimple and univerfal they in reality might be, defirous of eftablifhing new principles of fimple eonftruCtion, univerfal in application, and fupported by geometrical demonflrations, condescended to write on this fubjeCt, and pub- lished a hnall traCt, under the title of linear Perspective, in the year 17 1 5, In this little piece the doCtcr made prodigious advances towards bringing this art to its ultimate degree of perfection. He juflly ob- ferved, that all planes, confidered asfuch, are alike in geometry, and Should in like manner be applied to PerfpeCtive ; that there are no exclufive honours due to the ground plane, nor any particular magic in the horizontal plane, nor confequently in their correfpondent lines or points. He relieved us from the contracted limits in which our conceptions of this fubjeCt were inferibed, enlarged and extended cur ideas to univerfality itfelf, taught us the true ufe of vanishing OF PERSPECTIVE. vii planes, lines and points, in all fituations in which they can poflibly be conceived, whether parallel, perpendicular, or any how inclining to the pidture, or to the original objedts, &c. But finding that many objedted again# it on account of its not being fufficiently eafy to be generally underftood ; he therefore, in the year 1719, publilhed a fecond fmall treatife, called New Prin- ciples of linear Perspective. But this likewife, on account of the mathematical drefs and bre- vity of expreflion in which it was delivered, was not fo generally ca- reffed as it deferved to be, nor the principles it contained fo much examined as he wilhed ; wherefore he defigned the publick another treatife, propofing therein to fet forth thofe principles in another light, fuch that their preheminence above all others then in ufe might be more readily perceived, and the whole better adapted to the conceptions of young artills applying themfelves to Perfpedtive for works of defign ; but his death happening before the completion of thofe intentions, the world was thereby deprived of the advantage of fuch a perfedt piece as might reafonably be expedted from his great abilities, and this method of Perfpedtive remained a knowledge enjoyed by few, and thofe chiefly fuch in whofe hands it was not of the greate# utility to the publick, through their not being of em- ployments which required fldll in thofe principles, for making the proper applications of them required in the arts of defign. In the year 1738, John Hamilton, Efq; F. R. S. publilhed two volumes, folio, under the title of Stereography, or a compleat body of Perfpedtive in all its branches, the projedtions of fhadows, reflexions by polifhed planes, &c. in feven books. This learned and ingenious author hath very copioufly treated the fubjedt of Per- fpedtive in a flzridt mathematical way, with the afliftance of Dr. Tay- lor’s principles, whom he acknowledges to have made, in a few page A SUCCINCT HISTORY viii only, more real advances towards perfecting the fcience of Perfpec- tive, than all the writers who went before him. In this treatife, and alfo in thofe wrote by Moxon, Pozzo, Marolois, Ley- sour n, and many others, he may fee his miftake, who aflerts that it hath not been fhewn how to find the perfpeCtives upon concave or convex furfaces, nor upon a figure of feveral faces, much lefs by reflexion or refraCtion, &c. * But this treafury of mathematical pro- jection is little attended to, as being too difficult and diffufive for ge- neral ufe among artifts, notwithftanding it is worthy of being con- fulted occafionally in fome extraordinary cafes that may chance to occur, and has been confidered in what is here offered the reader. In the year 1754, Mr. Kirby, defigner in perfpeCtive to their pre- fent majesties, published a treatife in quarto, titled Dr. Brook Taylor’s Method of PerfpeCtive made eafy, &c. and in the year 1755, a fecond edition thereof; which D. Fournier thought worthy for him to make free ufe of in his publication in the year 1764. In 1761, he publiflied the PerfpeCtive of Architecture, a large and elegant work in folio, containing two rules of univerfal application ; improvements in the doCtrine of fhadows, the defcription and ufe of a new and very ufeful inflrument, called the architectonic feCtor; and the prefent year 1765 produces the third edition of that firft publiflied in 1754, This ingenious author, by attentively examining and applying Dr. Taylor’s new principles of PerfpeCtive to practice, was gradually led to a difcovery of their generality and facility in operation, faw how preferable and excellent they were in practical applications, how * See.Elements of Mathematicks, &c. together with a new Treatife of Perfpedtive, for the ufe of the Royal Academy at Woolwich. Printed for and fold by J. Milan, Bookfeller, oppofite the Admiralty, Charing-crofs, 1765. OF PERSPECTIVE. IX fimple and extenfive their conftruCtions, what a vaftconfufion of un- neceflary lines were thereby avoided, and how beneficial they would be if generally known to artifts concerned in works of defign ; pof- fefled with thefe and fuch like confiderations, he employed himfelf zealoufly to retrieve them from that ftate of darknefs in which their author’s brevity of expreflion and manner of writing had concealed them, and became the firft among artifts, who appeared in publick, to explain their true nature and ufe in adapting them fuitably to the arts of defign. The encouragement he met with at his firft communicating this de- fign, howjoyfully our artifts in general embraced it, the fuccefs which attended and overcame all the various oppositions he had to encounter in the profecution of it, would come now to be explained, were I to continue the thread of my hiftorical narrative beyond this period of time : but as thefe are things which would lead me to tranfmit an impartial account of fome of this author’s inventions, yet in ma- nufcript, as alfo to recite the exploded fyftems again revived, the controverfal writings produced thereupon, and other fuch like at- tempts, made for continuing thefe new principles in their former obfcurity, which, being matters that I think not altogether proper to be explicitly handled in this co-temporary publication, I referve them to a future feafon, and pafs over this part of its hiftory in filence, for the fake of preventing any undue fuggefiions which my perfonal acquaintance with Mr. Kirby might, perhaps, be the means of bringing upon the moft fair and juft account that can be given, fliall therefore confine myfelf to only one general and well-attefted faCt, publickly declared to all, and which fhould alfo, for the very fame reafon, have been here omitted, had it not appeared to me, that the juftice due to the memory of thofe ingenious authors, who have contributed to improve and exalt this art to perfection and ge- c A SUCCINCT HISTORY neral utility, required my endeavours to refcue them from the un- deferved cenfure lately palled upon them, of not having made the leafl improvement*; as well as to vindicate ourfelves, by Shewing that what we have advanced in favour of our author is. not our own bold aSfertion only, but has the united fuffrage of a body of artiSls, well qualified to judge decifively in this matter, by whofe order the following paragraph was inferred in the fevera.l publick papers of the year 1754, viz. Academy of Painting and Sculpture, in St. Martin’s-Lane. Jan. 24, 1754. Mr. Kirby, author of a work, intituled. Dr. Brook Taylor’s Method of Perspective, made easy, &c. has read three lec- tures (being the fubftance of his intended work) to the gentlemen of this fociety, which appeared to them fo clear, fimple and exten- five, that, in order to do juftice to fo excellent a performance, they have unanimously given this their public approbation, and declared the ingenious author an honorary member of their body. By order, F. M. Newton, Secretary. To conclude, I Shall now only mention a few more of the many writings we have on this fubjedt, viz., Hondius, his perfpedtive institutions were formerly held in great efteem. Alleaume’s deferves to be more known than it is, being well adapted to the purpofes of artifls. Father de Chales’s is remarkably neat.. S'Gravesande’s Essai de perspective, published at Leiden in 1711, contains many new things, and is recommendable for practice. See page 31 1 of the Treatife before alluded to. OF PERSPECTIVE. xi Father Lamy’s contains fome proper notices on the fubjedt of painting. M. DELACAiLLE’sperformance deferves to be noticed with refpedh The new Treatise, referred to in the preceeding note, is diftinguiihable for its extreme brevity and peculiar Angularities, &c. As to what I have here done, my whole defign has been to lay- before learners the method I have ufed in my private courfe of com- municating a general knowledge of the theory on which the art of perfpedtive is grounded, fo far as concerns one of my profeflion. I do not pretend to a new treatife on perfpedtive, my only view is to illuftrate and render the art eafy to be underftood, by a new manner of {hewing the rationale of it, to facilitate the fludy, point out the true principles that ought to be applied by thofe who would arrive at perfection in it, and knojv how to diftinguifh them fiotrt falfe ones, in an art where deception lofes all its beauty when fo grofsly handled as not to bear a correfponding refemblance to reality. Page ii fhould begin thus, Raife up the plane D Y, pafs the plane S H through the fame, and make the plane B M pafs through the plane S H. Whenever lines are mentioned, ftraight lines are to be underftood. The references allude to the Elements of Euclid publifhed by Pro- feffor Simfon of Glafgow, Odtavo Edition 1762, thus (4. 1.) denotes the fourth proportion of the firft book, &c. And the dodtrine of planes is principally taken from my predeceffor’s Elements of Geo- metry, Second Edition. - THE THEORY OF PERSPECTIVE DEMONSTRATED. PART I. OF THE DOCTRINE OF PLANES, DEFINITIONS. A Line is perpendicular, or at right angles to a plane, when it makes right angles with every line meeting it in that plane, ILLUSTRATION. Plate I. Fig. i. Make F in the plane F B coincide with F in the plane F N. Then, if the angles EDF,EDK, EDN, &c. are each a right angle, the line E D is perpendicular to the plane F N. II. A plane is perpendicular to a plane, when all lines drawn in one of the planes, perpendicularly to the common feftion of the two planes, are perpendicular to the other plane. B 2 OF THE DOCTRINE OF PLANES. SAME FIGURE. The lines BA, ED, GF, &c. being drawn in the plane B F per-i pendicularly to A F, the common fedtion of the two planes B F, F N, if they are alfo perpendicular to the plane F N, the plane B F palling through thofe lines, is perpendicular to the plane F N. III. The inclination of a line to a plane is the acute angle formed by that line, and another line drawn from the point, in which the firft line meets the plane, to the point in which a perpendicular to the plane, drawn from any point of the firft-mentioned line above the plane, meets the. fame plane. SAME FIGURE. Thus the acute angle F D G is the inclination of the line D G to the plane F N. •' IV. The inclination of a plane to a plane is the acute angle formed by two lines drawn from any the fame point of their common fedtion at right angles to it, one upon one plane, and the other upon the other plane. SAME FIGURE. Raife up the planes D Y, Z X, making W Y coincide with W y. Then the line D W, being the common fedtion of the planes D Y, D Z, and R T perpendicular thereto, and R S alfo perpendicular thereto, at the fame point R, the acute angle S R T is the inclina- tion of the plane D Y to the plane D Z. V. Two planes are faid to have the fame or a like inclination one to the other, which two other planes have to each other, when the faid angles of inclination are equal to one another. OF THE DOCTRINE OF PLANES. T wo planes, which, being either way produced, do not meet each other, are faid to be parallel one to the other. VII. A pyramid is a folid figure contained by planes that are conftituted betwixt One plane and one point out of that plane, all meeting in that one point. ILLUSTRATION. Plate II. Fig. 2. Make B in the triangle LAB coincide with B in the plane L B, alfo make C coincide with C, and L with L, and B in the triangle B A E with B in the plane L B, then will the Figure thus formed reprefent a pyramid. VIIL The point A is the vertex of the pyramid. The plane L B the bafe thereof. The planes LAB, B A C, &c. the Sides thereof. And the line AH the perpendicular altitude or height thereof above the plane of its bafe L B. THEOREM I. One part of a line cannot be in a plane and another part thereof above it. PLATE I. Fig. 1. Make F in the plane F B coincide with F in the plane F N. DEMONSTRATION. For, if it be poflible, let A D r part of the line A D G, be in the plane F N, and the part D G above the fame, then A D being in the plane F N, it can be produced in that plane fuppofe to F ; now the points D and G are in the plane BF, which pafies through the 4 OF THE DOCTRINE OF PLANES, line AF, whence the line D G is in the plane B F, (7 def. 11.) wherefore the two lines A D G, A D F, are in the fame plane, and have a common fegment A D, which is impofftble (cor.//. 1.) ^ E. D, THEOREM II. Two lines, which cut one another, are in one plane, and three lines, which meet one another, are in one plane. SAME FIGURE. Make F coincide with F, and the planes B M, S H pafs through each other in the line H L, and let p k, r H be two lines cutting one another in C ; I fay thofe two lines, as alfo the three lines C H, C k, H k, meeting each other, are in one plane. DEMONSTRATION. For neither can p C, part of p k, nor r C, part of r H, be in one plane, and the other parts C H, C k, out of that plane by the pre- ceedent ; wherefore the lines p k, r H are in one plane. And if you fay that C P x, part of the triangle C H k, is in one plane, and the other part P H k x in another plane, then muft C P be in one plane and P H in another, the fame likewife of C x and x k, which is im- poffible } therefore the triangle C H k is in one plane. ^ E. D. THEOREM III. If two planes interfedt each other, their common fedtion is a ftraight line. SAME FIGURE. Let H and L be the extremes of the common fedtion of the two planes B M, S H, join thofe extremes by drawing the line H L. OF THE DOCTRINE OF PLANES. 5 DEMONSTRATION. Then is the line H L in both the faid planes ; that line is therefore their common fedtion by conllrudtion, for if you fay it is not in both the planes, then may two lines be drawn between the fame ■extremes, one from L to H in the plane S H, and another from L to H in the plane B M ; thus would two lines, bounded between the fame extremes, include a fpace, which is impoffible, (axiom io. i.) wherefore H L mull necelfarily be in both the planes S H, B M, which is therefore their common fedtion. E. D. THEOREM IV. If one line be perpendicular to the common fedtion of two other lines which interfedl each other, it is alfo perpendicular to the plane which palfes through the faid two interfedting lines. ILLUSTRATION. Plate II. Fig. 3. Make}£ in the plane EBC to coincide withEJ in the plane CDEF, and pafs the plane F B D through the plane EBC, alfo make the plane G B H pafs through both the faid planes, making H coincide with H. DEMONSTRATION. Let there be taken AC — AD = AE = AF, and draw the lines CD, C F, FE, ED alfo through A in the plane CDEF, draw the line G H any how, meeting the lines C F, D £, in G and H, and let A B be perpendicular to the two lines C E, D F at^ their com- mon fedtion A, and draw the lines B C, B F, BE, B H^and B D. Now by conltrudtion AC, AD, A E and A F are equal to each other, and the angle CAF— DAE, (15. 1.) whence CF = DE (4. 1.) and the angle FCA orGCA = DEA or H E A, fo like- C 6 OF THE DOCTRINE OF PLANES. wife G A C=H A E, and A C = A E y therefore is A G = A H and G C=H E (26. 1.) Again, becaufe A B is perpendicular to the plane C D E F by hy- pothecs, the triangles CAB, DAB, E A B, FAB, are each right angled at A, having their bafes equal each to each, and the perpen- dicular A B common to them all, whence their hypothenufes will alfo be equal each to each, B C = BD — BE = BF; wherefore the triangles C B F, D B E, being mutually equilateral, the angle F C B or G C B = D E B or H E B, (8. 1.) and becaufe G C = H E and B C=B E, therefore is B G = B H. But it was proved above, that AG = AH, and A B is common, whence the angle G A B = H A B, confequently A B is perpendicu- lar to the line^G H (def. 10. 1.) and in the fame manner it may be proved, that A B is perpendicular alfo to C E, F D> and all other lines whatever that can be drawn in the plane C D E F through the point A, on which it infills 3 therefore A B is perpendicular to the plane palling through the lines C E, D F, &c. E. D. COROLLARY. Hence it follows, that when a line, as A B, is at right angle? to feveral lines, as A F, A E, &c. which it meets in the fame point, as A, the lines which it fo meets, are all in the fame plane 3 for a line, as A K, drawn from A out of the plane C D E F, cannot be perpendicular to A B, becaufe the angle B A K will be lels or greater than a right angle, or BAH, according as A K is drawn above or below the faid plane CDEF. THEOREM V. If in a given plane a line be drawn through any point thereof, and two other lines perpendicular thereto be alfo drawn from the faid 7 OF THE DOCTRINE OF PLANES. point, one of them in the given plane, and the other in any other plane, palling through the firft mentioned line j I fay, that if from the faid point a line be drawn, in the plane palling through thofe two perpendiculars, at right angles, to the firlt of them, it will be perpendicular to the given plane at the faid point. PLATE I. FIG. 1. Raife up the plane D Y, pafs the plane S H through the fame, and the plane B M through the plane S H. Let now B M be the given plane, C a point, and L H a line drawn through the fame. C E, C R lines perpendicular to L H, the firlt in the given plane B M, and the other in another plane, as S H, palling through the firlt mentioned line L H. Alfo let D Y be a plane palling through both the perpendiculars C E, C R, and conceived as being produced. Now let C I be a line drawn from C, in the produced part of the plane D Y, at right angles to the firlt perpendicular C E then will C I be alfo perpendicular to the given plane B M at the faid pbint C. DEMONSTRATION. Becaufe C H is perpendicular both to C E and C R, by hypothecs, it will alfo be perpendicular to C I, by the preceedent, whence C I being perpendicular both to C E and C H at the point of meeting, it is therefore perpendicular to the given plane B M, containing the faid lines, by the preceedent. E. D. THEOREM VI. Two lines, which are perpendicular to one and the fame plane, are parallel to each other. SAME FIGURE. Make K in the triangle G F K coincide with K in the plane F N, 8 OF THE DOCTRINE OF PLANES. and let E D and G F be each perpendicular to the fame plane F N, and in that plane draw the line F D, and perpendicular thereto draw D K=F G, alfo let there be drawn the lines FK, G K and G D. DEMONSTRATION. The triangles G F D, F D K, having G F=D K, by conflruftion, the fide F D common, and the angle G F D=F D K, each being a right angle, will alfo have GD=F K, (4. 1.) whence the triangles G D I(, G F K being mutually equilateral, the angle GDK or GFK is a right angle, (8. 1.) but the line E D is alfo perpendicular to D K, as well as the lines G D, F D ; it is therefore in the fame plane with them E D F G, confequently the angles G F D, E D F being right angles, E D is therefore alfo parallel to F G. E. D. COROLLARY, Hence it appears, that there cannot be drawn more than one per- pendicular from the fame point to one and the fame plane, for all lines perpendicular to the fame plane are parallel to each other, which lines, drawn from one and the fame point, can not be. THEOREM VII. If there be two parallel lines, one of which is perpendicular to a plane, the other will alfo be perpendicular to that fame plane. SAME FIGURE. The angles F D K, GDK being right angles, and G D in the fame plane with the propofed parallels G F, ED, the angle E D K is alfo a right angle, (4. 11.) as is alfo the angle EDF; therefore is E D perpendicular to the plane FD K. E. D. OF THE DOCTRINE OF PLANES. 9 THEOREM VIII. If a line be drawn from any point in one of two parallel lines to any point in the other, that line will be in the fame plane with the faid parallels. SAME FIGURE. Let B X, E QJbe two parallel lines in the plane B M, suid take any point H in one of them, and any point C in the other, then will the line which joins the points H and C, be in the plane of the pro- pofed parallels. DEMONSTRATION. If you fay it is not, fuppofe it above that plane as the line C u H, and in the plane of the parallels draw C H, then will the lines C u H, C H, be terminated by the fame extremes, and include a fpace, which is impoflible, (ax. io. i.) whence the line joining the points C and H can not be above the plane of the propofed parallels, in like manner it will appear that it can not be below the fame; therefore it mud: be in the fame plane with them. ^ E . D. THEOREM IX. If a line be perpendicular to a plane, any plane palling through that line, will be at right angles to the plane whereto the faid line is perpendicular. SAME FIGURE. Let ED be^ line perpendicular to the plane F N, and B F a plane palling through that line, and it will alfo be perpendicular to the faid plane F N, DEMONSTRATION. From any point, as B, taken at pleafure in the plane B F, draw D 10 OF THE DOCTRINE OF PLANES. B A perpendicular to A F, the common fedtian of the two planes B F, F N, then is the angle FAB=FDE, equal a right angle, (def. 3. 11.) whence B A is parallel to E D, and is therefore per- pendicular to the plane F N, by the foregoing theorem 7. In like manner it may be proved, that every line which can be drawn in the plane B,F perpendicularly to A F, the common fedtion of the planes,, will be alfo perpendicular to the plane F N, therefore the plane BF, paffing through thofe lines, will itfelf be alfo perpendicular to the plane F N. i^. E. D. COROLLARY I. Hence it follows, that the plane F N, according to the fenfe of the definition, is perpendicular to the plane B F j for let D K be a line drawn in the plane F N, perpendicular to A F, the common fedtioi* of both planes, at the point D, and it will alfo be perpendicular to E D ; and therefore the plane F N, in which, it was fo drawn, is alfo perpendicular to the plane B F, paffing through the faid line ED. COROLLARY II. Hence it alfo follows, that a line, drawn at right angles to one ot two perpendicular planes from any point of their common fedtion, will be in that other plane. For let the two perpendicular planes be F N, BF, their common fedtion A F, and D a point in the fame $ alfo let D E be at right angles to the plane F N, then muft D E be in the plane B F, for no other line can be drawn from the point D at right angles to the plane F N, as appears by the corollary to the foregoing theorem 6. THEOREM X. Planes, to which one and the fame line is perpendicular, are pa- rallel to each other. OF THE DOCTRINE OF PLANES. n SAME FIGURE. ^ Let E C be perpendicular to the plane B F, and alfo to the plane S H, then are thofe planes parallel. DEMONSTRATION. From any point, as D, in the plane B F, draw D R parallel to E C, and it will alfo be perpendicular to both the planes B F, S H, by theorem 7. Let the lines ED, C R be drawn, now feeing the angles at E, C, R and D, are each right ones, (def. 1.) the figure E C R D is a redtangled parallelogram, by reafon its fides E D, C R, are in the fame plane D Y, with the parallels EC, DR, confe- quently EC — DR, the fame way it can be proved, that all other perpendiculars, terminated by thofe two planes B F, S H, are alfo equal to each other > therefore the faid planes are parallel. 5 ^. E. D* COROLLARY. Hence it appears, that all lines, which are perpendicular to one of two parallel planes, are alfo perpendicular to the other. SCHOLIUM. From the two laft theorems, the fenfe and propriety of the two definitions of perpendicular and parallel planes appear manifelL THEOREM XI. Lines which are parallel to one and the fame line, though not in the fame plane with them, are parallel to each other. SAME FIGURE. Pafs the plane S H through the plane D Y, alfo pafs the plane B M is OF THE DOCTRINE OF PLANES. through the planes S H and Z X, and let B G, X M, be each parallel to Z Y, then are they alfo parallel to each other. DEMONSTRATION. Draw E W, W Q^, each perpendicular to Z % then will W 12 L- be perpendicular to the plane paffing through E W and E B, QJ£, will alfo be perpendicular to the fame plane, whence they are alfo parallel to each other ; therefore the lines B G, X M, are pa- rallel one to the other. £*. E. D, THEOREM XII. If two lines, meeting each other in one plane, be refpeflively parallel to two other lines alfo meeting each other in fome other plane, then will the angle, formed by the two lines meeting in the firft-mentioned plane, be equal to that formed by the meeting of the other two lines in the other plane. SAME FIGURE. Let the lines L C, C R, in the plane S H, be refpe&ively parallel to the lines G E, E D, in the plane B F, then will the angle L C R be equal to the angle GED, DEMONSTRATION. Take CL, C R, G E, E D, all equal to one another, and let C E, R D, D G, L R, G L, be drawn, then C R and D E, being equal and parallel to each other, as alfo L C equal and parallel to G E by hypothefis and conftrudtion, G L and R D are both equal and parallel to C E, confequently equal and parallel to each other, (33. 1.) whence DG = LR, and the triangles L C R, G E D, be- OF THE DOCTRINE OF PLANES. r 3 Ing mutually equilateral, the angle L C R is therefore equal to the angle GED (8. i.) E. D. THEOREM XIII. If two lines, meeting each other in one plane, be refpe&ively pa* rallel to two other lines, meeting each other in another plane, then the two planes, palling through thofe lines, will alfo be parallel to each other. SAME FIGURE. Let the lines ED, EG, in the plane B F, be refpe&ively parallel to the lines U c, U O, in the plane S H, then are the planes B F* S H, palling through thofe lines, parallel to each other. DEMONSTRATION. Draw E C perpendicular to the plane B F, meeting the plane S H in C, in which plane let there be alfo drawn C r, r f, refpe&ively parallel to U O, U c, then are C r, r f, refpedtively parallel to E G, E D, by theorem 1 1, and becaufe C E G, C E D, are right angles by conftrudtion, E C L, E C R, are likewife right angles ; whence E C, being perpendicular to the plane S H, by theorem 4, and alfo to the plane B F, by conltru&ion, the two planes B F, S H, are therefore parallel one to the other, by theorem 10. E. D. THEOREM XIV. If two parallel planes are cut by a third plane, the fe&ions made thereby are parallel to each other. SAME FIGURE. Let the parallel planes B F, S H, be cut by the plane D Y, then are the fe&ions ED, C R, parallel one to the other. E i 4 OF THE DOCTRINE OF PLANES. DEMONSTRATION. Draw E L and D S parallel to each other, alfo draw EC, DR, perpendicular to the plane S H, and draw CL, SR, then E L be- ing parallel to D S, by conflru&ion, and E C parallel to D R, by theorem 6 , the angle C L E = R S D, by theorem 12, but the angle ECL = DRS, as being each a right angle,, and EC = DRj whence E L is both equal and parallel to D S, therefore E D is alfo parallel to C R. ^ E. ZL COROLLARY. Hence it appears, that lines parallel to each other, and terminated by the fame parallel planes, are alfo equal to each other. THEOREM XV. If, from the extremes of a line palling through a given plane, be drawn two lines, each meeting the faid plane at right angles, the line, which joins the two points where thofe two perpendicular lines meet the faid plane, will be cut by the firfl-mentioned line, in that point where it pafles through the faid plane, and be thereby di- vided into two fuch parts, as bear the fame ratio to each other, which the two perpendiculars, meeting the plane, have one to the other. SAME FIGURE. Let n m be the line cutting the given plane B M in the point C, and from n and m, the extremes thereof, draw the perpendiculars nr, m H, meeting the plane S H in the points r and H, and draw the line H r , then will rC:CH::rn:Hm. DEMONSTRATION. Produce n r as here to f, then feeing nf, m H, are each perpen- dicular to the plane B M, by hypothecs, they are parallel to each OF THE DOCTRINE OF PLANES. *5 other by theorem 6, whence the lines n m. Hr, being both in the fame plane with thofe parallels, and terminated by them, as they are not parallel to each other, they will meet or interfedt each other, and thereby make the alternate angles equal one to the other, that is, r n C = H m C, as alfo rCn = HCmj whence the triangles n r C, m H C, being fimilar, it will therefore be r C : C H : : n r ; mH. %.E.D. COROLLARY. Hence it follows, that if, in the plane B M, there be drawn the lines r f, H X, parallel to each other, and in them be taken rp=nr, and Hk = Hm, then will the line p k, joining the points p and k, interfedt H r in the felf fame point in which it was before interfedted by the line n m. For if h be conceived as the interfedtion point of p k with H r, the triangles p r h, H h k, will be equiangular ; whence we have rh:hH::rp:Hk, and nr:Hm::rC:CH, butrp = nr, and H k = H m, wherefore nr:Hm::rC:CHj therefore, feeing r H is divided into one and the fame ratio by both the points C and h, thofe two points muft neceffarily coincide together, or vanifh into one point only. THEOREM XYI. If two planes, interfedting each other, be both perpendicular to a third plane, the common fedtion of thofe two planes will be alfo per- pendicular to the faid third plane. SAME FIGURE. Let the planes D Y, S H, be each perpendicular to the plane A Z, interfedting each other in the line C R, then is their common fec- tion C R perpendicular to the plane A Z, i6 OF THE DOCTRINE OF PLANES. DEMONSTRATION. From R, one extreme of the common fedtion, draw a line, as R C, perpendicular to the plane A Z, then this perpendicular, being in both the planes D Y, S H, by cor. 2 to theorem 9, it muft necefia- rily be their common fedtion, which is therefore perpendicular to the plane A Z, by conftrudtion, E. D, PART II. OF THE DOCTRINE OF PLANES APPLIED TO THE TRUE PRINCIPLES OF PERSPECTIVE. fu DEFINITIONS. T) I. ' s "' - X ERSPECTIVE is that part of mathematical projection which gives rules for defcribing., upon any given plane, the reprefentations of any given objeCts, fo as to exhibit thereon the exaCt forms, mag- nitudes and pofitions thereof, fuch as they would be feen to have upon that plane, by an eye fixed in one certain point, and viewing them from thence through the faid given plane, fuppofing it to be perfectly tranfparent, or as glafs, &c. ILLUSTRATION. Plate III. Fig. 4. Raife up the plane D Y, and pafs the plane S H through the fame, in the line C P, then bring R, in the plane F G, to coincide with R in the plane F M, raife up the plane M X, and pafs the plane G X through both the planes S H and M X. Suppofe now the plane S H tranfparent, and A B a line in the plane F M, feen by an eye fixed at E, and viewing it through the plane S H, then will A and B, the extremes of that line, appear on the plane S H, in the points a and b, where the lines A E, BE, drawn from, or ifluing thence to the eye at E, cut or interfeCt the faid plane S H ; and all other lines which can be drawn, or con- ceived to ifliie from any other of the points in the line A B to the eye at E, will, by their interfeCtions with the plane S H, mark out other intermediate points between a and b, and the whole line A B F j8 OF THE DOCTRINE OF PLANES will be thereby depidted, on the plane S H, by the line a b, which is therefore the true defcription or reprefentation thereof. SCHOLIUM. Hence it appears, that the given plane, on which the reprefenta- tions of the given objedts are to be drawn, mult be fo lituated un. rejf&ct of the given objedts and the point in which the eye is placed, that all lines, which can be drawn or conceived to ifTue from them to the eye, may pafs through the faid plane ; for, feeing the repre- fentations thereof are determined upon that plane by the interfedlions made thereon, by lines palling through it in proceeding from them to the eye, it is plain there can not be any reprefentations formed on that plane of objedts which are fo pofited, that the lines ilfuing from them to the eye, do not cut the plane on which they are re- quired to be defcribed ; whence the neceflity of the above-mentioned pofition of the faid given plane appears manifelt. II. The plane S H, upon which the given objedts are reprefented, is, in mathematical terms, called the plane of projedtion, but here we term it the picture. r III. The plane, which contains the objedts given to be defcribed on the pidture, is called the original plane. Thus the plane F M is here the original plane, IV. The point, in which the eye is fixed for viewing the given ob- jedts, or their reprefentations on the pidture, is called the place of the eye, or point of sight. Thus E is the point of fight. APPLIED TO PERSPECTIVE. *9 SCHOLIUM. The place of the eye, or point of fight, ought to be juftly afcer- tained, by having due regard to the fize, fituation. See. of the pic- ture and original plane ; for an injudicious determination in this re- fpeCt, produces ill effeCts in the reprefentations formed on the picture, as appears from a recent inftance hereof in a late production, in which the author has made ample difplay of inconfiftent reprefen- tations, formed by tranfgrefling the rules requifite to be obferved on this occafion. How faults of this kind are to be avoided, will be fhewn in the annotations or general comment hereto annexed. V. A line drawn from the eye, perpendicular to the picture, and meeting the fame, is called the distance, or axis of the eye, or diftance of the picture. Thus E C is the diftance or axis of the eye, or diftance of the picture. VI. The point, in which the axis of the eye meets the picture, is the CENTER OF THE PICTURE. Thus C is the center of the picture S H. VII. The deferiptions made upon the picture of any original objeCts, whether they be points, lines, furfaces or folids, are called the per- fpeCtive reprefentations or images thereof. Thus the line a b is the perfpeCtive reprefentation or image of the given line A B. VIII. A plane, paffing through the point of fight, parallel to any ori- ginal plane, is called the vanishing plane of that original plane. 20 OF THE DOCTRINE OF PLANES Thus the plane G X, being parallel to the original plane F M, and paffing through E, the point of fight, is the vanilhing plane of the faid plane F M. IX. A plane, palling through the point of light, parallel to the' pic- ture, is called the directing plane. Thus the plane F G, being parallel to the picture S H, and palling through E, the point of fight, is the directing plane. X. A plane, palling through the axis of the eye at right angles to the original plane, is the vertical plane of the faid original plane. Thus D Y is the vertical plane of the original plane F M. XI. The line, in which the directing plane cuts the original plane, is called the directing line of that original plane. Thus F R is the directing line of the original plane F M. XII. The line, in which a vanilhing plane cuts the pidture, is called the vanishing line of the original plane correfponding to that vanilhing plane. Thus H L is the vanilhing line of the original plane F M. XIII. The line, in which a vanilhing plane cuts the directing plane, is called the eye’s parallel. Thus I G is the parallel of the eye when placed in E. XIV. The line, in wlfirh the vertical plane interfedts the directing plane, is called the director, or height of the eye. Thus E D is the diredtor or height of the eye when fixed in E. APPLIED TO PERSPECTIVE. 21 XV. The line, in which the picture interfedls the original plane, is called the intersecting line of that plane. Thus S P is the interfering line of the original plane F M. XVI. The line, in which the picture interfe S P, H L and F R, are parallel to each other. ^ E. D, 24 OF THE DOCTRINE OF PLANES COROLLARY. Hence it follows, that any one of thofe lines, and any point in ano- ther of them, being given, that other line may thence be determined ; for a line, drawn through that point, parallel to the given line, will be the line required. But it does not follow, as an axiom, that thefe two lines will be in the fame plane, for, before that can be rightly afferted, the interpofition of another plane, diftindt from the former, mud: be introduced. THEOREM II. The vertical plane is perpendicular to the picture, the vanifhing, directing and original planes, and alfo to the parallel of the eye, the interfering, vanishing, and directing lines of that fame original plane. SAME FIGURE. Let D Y be the vertical plane, and the other mentioned planes and lines the fame as in the preceedent. DEMONSTRATION. The line E C is perpendicular to the pidlure, (def. V.) and the vertical plane D Y palfes through that line, (def. X.) the vertical plane is therefore alfo perpendicular to the picture, (theorem IX. part I.) and like wife to the directing plane, which is parallel thereto, (def. IX.) the faid vertical plane is alfo perpendicular to the original plane, (def. X.) whence it is alfo perpendicular to the vanifhing plane, that being parallel thereto ; (def. VIII.) wherefore the faid vertical plane, being perpendicular to each of the four above-men- tioned planes, it is therefore perpendicular to the common fe&ions IG, HL, SP, FR. %.E.D. APPLIED TO PERSPECTIVE. *5 THEOREM III. The director is perpendicular both to the direring line and the parallel of the eye. And the vertical line is perpendicular both to the interfering and vaniihing lines. SAME FIGURE. Let E D be the director, C P the vertical line, and the reft as before. DEMONSTRATION. The lines I G, H L, S P, F R, heing all perpendicular to the plane D Y by the preceedent, they are therefore perpendicular to all lines meeting them in that plane, (def. I. part I.) Now, feeing EC, CP, P D, D E, are in the plane D Y, therefore is E D perpendicular to F R and I G ; alfo C P perpendicular to H L and S P, confequently it is as the theorem enounces. E. D. THEOREM IV. Original planes, parallel to the pidture, have neither vaniihing, interfering, nor direring lines, neither have the lines, fituated in thofe planes, any of thofe correfponding points. SAME FIGURE. Let M X be an original plane parallel to the pidture S H, and Z X an original line fituated in that plane j I fay, the plane M X hath neither vaniihing, interfering, or direring lines, nor hath the line Z X any of them points. DEMONSTRATION. For the original plane MX, being parallel to the pirure, can not cut it to determine the interfering line as is required, (def. XV.) the H 26 OF THE DOCTRINE OF PLANES vanifhing plane F G being alfo, in this cafe, parallel to the pidlure S H, can not determine a vanifhing line to the original plane M X, (def. XII.) for the like reafon there can be no directing line, the directing plane F G being here parallel to the original plane M X, as well as to the picture S H. Again, feeing the plane G X drawn through the eye at E and the original line Z X, cuts the picture in H L, and the directing plane in I G, thofe two lines are parallel, (theorem XIII. part I.) they are alfo parallel to Z X, fince a plane M X may be drawn through that line parallel to the plane S H whence I G is the line which ought to produce the vanifhing point of the line Z X, (def. XIX.) but, being parallel to the picture, it can not. The original line Z X, being alfo parallel both to the picture and directing plane, can cut neither of them •, therefore the plane M X and the line Z X, are deftitute of either vanifhing, interfering, or diredling lines, or points. E.D . COROLLARY I. If an original plane M X, parallel to the picture, cut any other plane whatever, as F M, their common fedtion A M will be parallel to the vanifhing, interfering, and direring lines of the faid plane FM. For the original plane M X, being parallel to the pirure and di- rering plane F G, the ferions A M, S P, F R, made by thofe three planes, with the plane F M, are parallel, (theorem XIII. part I.) Now S P is the interfering line, F R the direring line of that plane, H L the vanifhing line thereof, which is alfo parallel, thereto ; therefore A M is likewife parallel to that vanifhing line, (theorem X. part I.) APPLIED TO PERSPECTIVE. 27 COROLLARY II. Hence it appears, that the image of an original line, parallel to the picture, is parallel to the faid original. SAME FIGURE. Let Z X be an original line parallel to the pidture S H, and let the plane G X pafs through the point of light, and the original line Z X, cutting the pidture S H, in the line H L, draw the lines X E, Z E, interfering the pidture in the points W and V ; then is the line W V the image of Z X, and they are parallel to each other, by what was proved above. COROLLARY III. All parallel original lines, which are parallel to the pidture, have parallel images, for thofe images are parallel to their correfponding originals, by the preceedent corollary. COROLLARY IV. Hence it follows, that whatever angle is formed by the meeting of two original lines, which are parallel to the pidture, an angle equal thereto will be formed on the pidture by the meeting of their correfpondent images. SAME FIGURE. Make the planes D Y, D Z, both pafs through the pidture S H in the lines CP, VS; raife up the plane M X, fo that Y coincide with Y, and make the plane G X form one continued plane with the plane F E, and let A Y, Y M, be two original lines parallel to the pidture, making the angle A Y M, then is the angle a C m, formed on the pidture by their correfpondent images Ca , C m, equal thereto* 28 OF THE DOCTRINE OF PLANES For thofe images are refpedtively parallel to their originals, (by the preceedent corollary) therefore the angle aCm=AYM. (the- orem XII. part I.) And, univerfally, all lines in an original plane, which is parallel to the pidture, are themfelves parallel to the pic- ture, and alfo to their refpedtive images, THEOREM V. The image of an original line, which is not parallel to the pidture, produced both ways, will pafs through both the interfedting and va- nifhing points of the faid original line. SAME FIGURE. Let now the plane D Y form one continued plane with the plane D M, pafs the plane D Z through the pidture S H in the line V S, making D coincide with D and M with M, and pafs the plane G X through the pidture. Let now M N be an original line in the ori- ginal plane FM; S the interfedting, and V the vanifhing point thereof. DEMONSTRATION. Becaufe the lines E V, M N, are parallel, (def. XIX.) all lines that can be drawn from E to the line M N, will be in the fame plane as the lines EV and MN are in. (7. 11.) Now, feeing the points S and V are in this plane, and alfo in the pidture, the line V S is the common fedtion of thofe two planes, but n m, the image of N M, is a part of that fedtion ; therefore n m, produced both ways, will pafs through S and V, the interfedting and vanifhing points of the original line. E. D. C O R O L L A R Y I. The diredting point D can have no image on the pidture, by rea- fon the line E D, which ought to cut the pidture in order to form APPLIED TO PERSPECTIVE. 2.9 an image, is in the directing plane FE j it is therefore parallel to the pi&ure, confequently can not interfedt it. COROLLARY II. The images of all the points, that can be conceived between the interfering point S and the extreme point M, how far foever it be produced on from thence, will fall on the picture between the inter- fering point S and the vanifhing point V ; for, wherever the point M be fo taken, the angle E M S will be equal to the angle M E V, as being the alternate angles of two parallel lines ; wherefore the points m and V can never coincide, nor can the point N be fo taken, but that its image will always be between S and V, the angle E N S being, for the like reafon, always equal to the angle N E V. Now, when N comes into S, its image n then alfo comes into the inter- fering point S j in like manner, if M be fuppofed at an infinite diftance, fo that the angle M E V entirely vanifh, and the line E M coincide with E V, then will its image m vanifh into the point V. Hence appears the propriety of calling this a vanishing point, as alfo why the line H L, pafling through that point, is termed a va- nifhing line, and the plane G X, pafiing through that line, a va- nifhing plane, &c. COROLLARY III. If the original line itfelf pafles through its own vanifhing point, its image is then contrared into that point, and the line, in this cafe, may be faid to Vanifh. Thus, fuppofe V the vanifhing point of N M, and let N M be conceived as moving parallel to itfelf along the lines N K, M Z, till, being in the pofition or line Z K, its pro- duced part N S or K V pafles through the point V, then is that point only the image of the line N M when in that pofition, the images n, m, both now coinciding in V. I 3 o OF THE DOCTRINE OF PLANES COROLLARY IV. Hence it follows, that if an original line be produced and pafs through the point of fight, the vanilhing and interfering points thereof will coincide, and its directing point will be the point of fight j whence any point in the pidure may be the image of any point in the original line palling through the eye and that point of the pidure, or may be taken as the vanifhing point of that line, or as the image of its interfedion with all planes or lines whatfoever, which it cuts j wherefore, a point being given on the pidure, its original can not be thence determined, unlefs the pofition of that original, with refped to fome other known point, line or plane, be alfo given. THEOREM VI. Original lines, which are parallel to each other, but not to the pidure, have the fame vanifhing point, and their images produced all meet in that fame point. SAME FIGURE. Raife up the plane D Y, and pafs the pidure S H through the fame ; bring R to coincide with R, and make the plane G X pafs through the pidure. Let now O Q^be parallel to A B, then is C their vanilhing point, and their images a b, o q, produced, will pafs through the fame. DEMONSTRATION. Becaufe E C is parallel to A B, it will alfo be parallel to O Qj (theorem XI.) wherefore C is their common vanilhing point. And, by the preceedent, o q, the image of O Q^, will, when produced, pafs through the vanilhing point, and a b, the image of A B, when produced, alfo palles through the fame ; therefore the images a.b* APPLIED TO PERSPECTIVE. 3i o q, of the two original parallel lines A B, O Q^, when produced* meet in that fame vanifhing point C. ^ E. D . COROLLARY I. The fame vanifhing point can not belong to any two original lines which are not parallel to each other, for the fame line, drawn from the point of fight to meet the pidture, can not be parallel to more than one of them. COROLLARY II. The center of the pidture is the vanifhing point of all lines which are perpendicular to the pidture, becaufe the axis of the eye is pa- rallel to all fuch lines, (def. V. part I, and theorem VII, part II.) THEOREM VII. Original lines, whofe diredting points fall in the fame diredtor, have parallel images, and the angle made at the point of fight, by the diredtors of any two original lines, is equal to that formed on the pidture by their images, SAME FIGURE. Bring D to coincide with D, raife up the plane D Y, and make the pidture pafs through both the planes D Y and E M ; now let N M and O QJie two original lines, having their diredting points D and T each in the fame diredtor E D ; then are the images n m, o q, of thofe two lines, parallel to each other. DEMONSTRATION. For the lines CP, VS, are each parallel to the diredtor E D, whence the images n m, o q, being in thofe lines, are alfo each pa- rallel to the fame diredtor E D } they are therefore parallel to each 32 OFT HE DOCTRINE OF PLANES other ; and when there are different directors, as all directors meet in the point of fight, they will alfo be parallel to the images. E. D. THEOREM VIII. If two original lines meet each other, their images will alfo meet each other in that point of the picture, which is the image of that point in which the originals meet one another. PLATE IV. Fig. 5. Form the figure according to the directions annexed to definition XXVI. DEMONSTRATION. Let B G, AG, be two original lines meeting in G, then are b g and a g the images thereof. Now, feeing G is a point common to both the lines AG, B G, the image g of that point will alfo be common to both the images a g, b g j thofe images therefore meet in the point, which is the image of that point in which the ori- ginals met each other. §>* E. D, COROLLARY. The images of all parallel lines whatever are either parallel, or meet in fome one point ; for, when the originals are parallel to the picture, their images are alfo parallel, and when not parallel to the picture, their images meet in one common vanifhing point, as was Ihewn above. THEOREM IX. All lines in an original plane have their vanifhing, interfering and directing points, in the vanifhing, interfering and directing lines of that plane. APPLIED TO PERSPECTIVE. 33 PLATE III. Fig. 4. Let the pidture now pafs through the plane D Y only, and the plane G X pafs through the picture, and let the plane F M be the original plane, and AB, M N, two given lines in that plane. DEMONSTRATION. The lines S P, F R, in which the pidture and directing plane are cut by the original plane F M, are the only lines which can be cut by lines drawn in the original plane, as M N, A B, See. wherefore P and S are the interfedting points of thofa lines, and D the diredt- ing point thereof ; for thefe lines meeting, in this cafe, the diredting line in the fame point D, have their diredting point common. Now E C, E V, being drawn parallel to A B and M N, determine their vanifhing points C and V, but thefe lines are alfo parallel to the ori- ginal plane ; wherefore the plane G X, drawn through thofe lines, will alfo be parallel to the plane containing the lines A B, M N ; and as no other plane can be drawn through E parallel thereto, it will be the vanifhing plane of the original plane, and the line H L the vanifhing line thereof ; but C and V, the vanifhing points of A B and M N are in that line ; therefore the proportion is manifeft. %>E.D. COROLLARY I. The image of any point, line or figure, fituated in that part of an original plane, which is beyond the interfedting line, muft fall on the pidture between the interfedting and the vanifhing line, as is evi- dent by what was premifed above in cor. II. to theorem V* SCHOLIUM. From the above it appears, that when the ground is confidered as the original plane, the vanifhing line thereof will be the ultimate K 34 OF THE DOCTRINE OF PLANES extent to which the image of any part of the ground, efleemed as level, can extend; this line therefore, in that cafe, determines the apparent boundary of the horizon, whence, being taken in this fenfe, it is called the horizontal line ; on the fame fuppofition, the interfering line is called the ground line, as being then the inter- fedion of the pidure with the ground. But if we flop here, at this imperfed and confined way of confidering and applying planes, as adapted by authors of this kind of perfpedive, the art will be in- fufficient in many cafes, and in general intricate and complex ; which a due application of the improvements, furnifhed by the principles here treated of, will render perfed, and make the art of drawing perfpedive reprefentations fimple, eafy and univerfal. COROLLARY II. Any two original lines, in the fame plane, which are not parallel to the pidure, having their images parallel to each other, have a common direding point ; for only one diredor can be drawn parallel to both their images, and that diredor can cut the direding line of the original plane only in one point ; that point is therefore the com- mon direding point of the two original lines. And when any two lines in an original plane, cut the direding line of that plane in the fame point, their images on the pidure will be parallel. COROLLARY. III. Lines, drawn through any two vanifhing points, any two inter- feding points, and any two direding points, of lines in an original plane, are the vanifhing, interfeding and direding lines of that plane ; for any two points, in a line being given, the whole line is thereby determined. APPLIED TO PERSPECTIVE. 35 COROLLARY IV. All original lines, parallel to an original plane, have their vanifh- ing points in the vanifhing line of that plane ; for lines may be drawn in the original plane parallel to the propofed lines, and all parallel lines, having the fame vanishing points, (theorem VI. part II.) the vanifhing points of the propofed lines are therefore in the vanifhing line of that original plane to which they are parallel, (theorem IX. part II.) THEOREM X. The vanifhing line of a plane, perpendicular to the pidure, paffes through the center of the picture, which it can not do when that plane is in any other pofition. DEMONSTRATION. For the vanifhing plane will then pafs through the axis of the eye, which is perpendicular to the picture, (def. V. part II.) and all planes, which pafs through the axis of the eye, are perpendicular to the picture, wherefore thofe planes can not be parallel to planes which are not at right angles to the picture ; therefore the vanifhing planes thereof do not pafs through the axis of the eye, confe- quently the vanifhing lines of fuch planes do not pafs through the center of the picture. E. D. THEOREM XI. All parallel original planes have the fame vanifhing line, center and axis of the eye, and their interfering and directing lines are parallel to each other. PLATE VI. Fig. 7. Make S, in the plane S L, coincide with S in the plane A K j 36 OF THE DOCTRINE OF PLANES make A coincide with A, and the plane G P pafs through the plane S L, in the line R T ; raife up the plane K L, fo that the plane G P may pafs through the line U O, and the two lines K P coincide with each other; turn the plane K L about, till K coincides with K, then bring D, in the plane D Y, to coincide with D in the plane A K, and make the plane Y G pafs through the plane S L in the line WX, and pafs the plane F M through the plane K M in the line M V, and pafs the plane V li through the plane S L in the line H L. Now, let the plane S L reprefent the picture, K M the directing plane, E the point of light, A K one original plane, and Y G ano- ther parallel thereto, then are the interfering and directing lines SI, K P, parallel to each other, and both thofe planes have the fame vanilhing line F N, the fame center c, and axis of the eye Ec. DEMONSTRATION. For the plane F M, palling through E, the point of fight, parallel to either of the given planes, will alfo be parallel to the other ; it is therefore the vanilhing plane of both the faid given planes, (def. VIII. part II.) Now the fame vanilhing plane can produce but one vanilhing line F N, and E c is the axis of the eye belonging thereto, and c the center thereof, and the interfering and direring lines SI, K P, being parallel to the fame vanilhing line FN; they are therefore pa- rallel to each other, (theorem XI. part I.) i^. E. D. COROLLARY I. Planes, which are not parallel one to another, can not have the fame vanilhing line j for the fame vanilhing plane can, in fuch cafes, be parallel only to one of them. APPLIED TO PERSPECTIVE. 37 COROLLARY II. All parallel original planes have the fame vertical line, vertical plane, and parallel of the eye ; for there can be but one line drawn through the center of the picture, perpendicular to the fame vanifh- ing line, nor but one line through the point of fight, parallel to the faid vanifhing line ; neither can two different vertical planes pafs through the fame point of fight and fame vertical line. THEOREM XII. If two original planes cut each other in a line parallel to the pic- ture, their vanifhing lines will be parallel to each other, as alfo their interfering and directing lines, if neither of thefe laft coincide. SAME FIGURE. Let the planes G P, Y G, be two original planes interfering each other in the line B G, parallel to the pirure S L, and let the planes M F, V H, pafs through the point of fight E, parallel to the faid original planes, and then will the vanifhing lines H L, F N, be pa- rallel to each other, as alfo the direring lines K P, QY, and inter- fering lines WX,T??L DEMONSTRATION. Let the plane ABG pafs through the line BG parallel to the pirure and direring plane K M, then will the feffions B G, X W, Y made by the plane Y G, with the three planes ABG, S L, K M, be parallel to each other, (theorem XIV. part I.) and for the fame re a - fon the ferions B G, R T, K P, made by the plane G P, interfer- ing the fame three planes, will alfo be parallel one to another, con- fequently X W, R T, the interfering lines of the two given original planes, will alfo be parallel, and their direring lines QJf , K P, will L 38 OF THE DOCTRINE OF PLANES alfo be parallel, (theorem I. part II.) Now the vanifhing lines H L, F N, of thofe planes are paralleFto their interfering lines, by the aforefaid theorem, and the original planes, not being themfelves pa- rallel, they can not have the fame vanishing line ; (cor. I. to theo- rem VI. part II.) therefore the vanifhing lines H L, F N, are diftinr and parallel to each other. E. £). COROLLARY I. All planes, whofe vanifhing lines are parallel, have the fame ver- tical plane, vertical line, and parallel of the eye ; for C n, which is at right angles to F N, is alfo perpendicular to H L, and M E,. being parallel to F N, is alfo parallel to H L. COROLLARY IL If the interfeclion B G of the given planes were in the pirure* it would them be their common interfering line, and if it were in the direring plane, it would be their common direring line ; but, im either cafe, their other lines would be parallel one to another. THEOREM XIPT If the vaniihing lines of two original planes be parallel* their com- mon interferion will be parallel to the pirure. SAME FIGURE. Let the two original planes be the fame as in the preceedent. DEMONSTRATION. Then (by cor. L to the laft theorem) the original planes, having the fame vertical plane perpendicular to them both, it is alfo per- pendicular to their common ferion B G but the vertical plane is APPLIED TO PERSPECTIVE. 3-9 perpendicular to the vanifhing lines of the original planes ; (theo- rem II. part II.) therefore B G, the common interfedlion of thofe planes, is parallel to their vanifhing lines H L and F N, and confe- quently parallel to the picture. ^ E. D. COROLLARY I. If an original line B G be parallel to the picture, it will be parallel to the vanifhing, interfering and directing lines of all original planes, which can pafs through the faid original line. For all planes whatfoever, which pafs through the line B G, cut the plane A G in that fame line ; confequently the vanifhing, inter- fering and directing lines of all fuch planes are parallel to BG. (16 . ii.) COROLLARY II. The original of the image of any line in an original plane, pa- rallel to the vanifhing line of that plane, is parallel to the picture. For a line in an original plane, parallel to the picture, being pa- rallel to the vanifhing line of that plane, by the corollary above, its image mufl be parallel to the fame line ; (cor II. theorem IV. part II. and theorem XI. part I.) therefore a line, as B G, parallel to the pRture, may be found in the original plane, which will produce the given image ; but no two different lines in the fame plane can pro- duce the fame image ; therefore, if the given image be the image of a line in the original plane, it muil be the image of BG, a line in that plane parallel to the picture. THEOREM XIV. If an original plane, being produced, pafs through the point of fight, its vanifhing and interfering lines will coincide, and its di- rering line will be the fame with the parallel of the eye. 40 OF THE DOCTRINE OF PLANES ILLUSTRATION. Plate IV. Fig. 5. Let B A Z R be the original plane, which, being produced to the line P L, pafles through the eye at E, cutting the direding plane N Y in the line P L. DEMONSTRATION. Now, feeing that L B is the only plane which can pafs through L P parallel to the original plane B A Z R, it follows that the ori- ginal and vanishing plane coincide, and form one continued plane j therefore L P is both the direding line and the parallel of the eye, belonging to the faid original plane, and x y, the interfedion of the pidure O P, with the original plane produced, is both its inter- feding and vanifhing line. E. D. COROLLARY I. The image of an original plane, palling through the point of light, is a line only, and in that line the images of all points, lines and figures, fituated in the original plane, mull be formed ; where- fore the original plane, in this cafe, hath no depth. For a line, drawn from E, the point of fight, to any point in the plane BAZR, will cut the pidure O P in fome one of the points in the line xy; therefore the image of that point mull be in that line x y. COROLLARY II. All original lines, fituated in the plane BAZR, have the fame line P L for their diredor. For no line in the original plane can cut the direding plane N Y, but only in the line P L. COROLLARY III. Any line, drawn in the pidure, may be the image of an original APPLIED TO PERSPECTIVE. 4 * line in a plane paffing through the eye, and the line given in the pidture ; or it may be taken as the vanishing line of that plane, or as the image of its interfedtion with all other planes whatfoever which it interfedts, as is evident from the above, (cor. I.) COROLLARY IV. The original of a line given in the picture can not be determined, unlefs two points in that line be known, one of which at lead mull be an original point, but the other may be the vanishing point of the faid original line required. For if the vanifhing point alone be given, the direction only of the original line is thence determinable, as being parallel to the line producing that vanifhing point, but the original line itfelf may be any line parallel to that faid line, and being in a plane paffing through the eye and the line given in the pidture ; but when the direction of the original line and an original point in that line are known, the original line itfelf is then determined, by reafon there can not be two different lines drawn through a given point parallel to the fame line. THEOREM XV. The original of any figure in the pidture, may be any objedt which is bounded by the fame pyramid of rays indefinitely produced. SAME FIGURE. Let O P be the pidture, E the point of fight, A G B F the ori- ginal figure, N L the directing plane, and a g b f the image of the original on the pidture O P. DEMONSTRATION. Now g c may be the image of any original line in the plane GEC, M 42 OF THE DOCTRINE OF PLANES terminated by the lines EG, EC, indefinitely produced, and b c may be the image of any original line in the plane C E B, bounded by the lines EC, EB; in like manner produced (cor. III.) to the preceedent, it is the fame with the other fides of the given figure ; therefore a g b f may be the image of any figure whatfoever, bounded by the fame pyramid of rays E F B G A, indefinitely produced. ^E.D. COROLLARY. If the image of any plane figure be given in the pidture, its ori- ginal can not be determined, unlefs three points in that figure be known, one of which at leaf! muft be an original point, but the other two may be vanifhing points. For the two given vanifhing points will determine the vanifhing line of the plane containing the original figure } (cor. III. to the- orem IX. part II.) confequently the direction of that plane is known,, feeing it muft be parallel to the vanifhing plane which produces that vanifhing line ; (def. VIII. part II.) but, notwithflanding this, the original plane is yet undetermined, for it may be any plane parallel to that vanifhing plane ; (theorem XI. part II.) whence fome one point in the original plane is necefiary to be known, in order to af- certain that plane ; which point being given, the original plane will be thereby determined, for two different planes can not pafs through the fame point parallel to the fame vanifhing plane. Now the ori- ginal plane, containing the original figure, being thus found, the original of the given image on the pidture is truly afcertained. THEOREM XVI. Any line in the pidture, parallel to the vanifhing line of an original plane, if it be the image of an original line, mufl be either the image APPLIED TO PERSPECTIVE. 43 of a line parallel to the picture, or of one whofe directing point is fome where in the parallel of the eye belonging to that plane. DEMONSTRATION. For, if the original line be parallel to its image, it mult alfo be parallel to the picture y but, if it be not parallel to its image, that image mult then be parallel to the director of the original line ; that director, being therefore parallel to the propofed vanilhing line, (theorem XI. part I.) it mult be the parallel of the eye belonging to that vanilhing line, (theorem I. part II.) feeing there can not be drawn two different lines through the point of fight parallel to the fame vanilhing line. ^ E. D. COROLLARY I. Any line in the picture, parallel to the vertical line of an original plane, if it be the image of an original line, mult be either the image of a line parallel to the picture, or of one, whofe directing point is in the director of that plane. For, if the original line be not parallel to the picture, its image mult be parallel to its director ; which director, being therefore parallel to the propofed vertical line, it mult be the director belonging, to the original plane. COROLLARY II. Any two parallel lines in the pidture, if they be the images of any two original lines, they mult be either the images of lines parallel to each other and to the pidture, or of fuch original lines as have the fame diredtor. For, if the original lines be parallel to their images, they mult have the fame diredtor, by reafon there can be but one diredtor drawn parallel to the given images. 44 OF THE DOCTRINE OF PLANES THEOREM XVII. If two original planes cut each other in a line not parallel to the picture, their vanilhing, interfering and direring lines will alfo cut each other ; and the interferion of the vanilhing lines will be the vanilhing point, the interferion of the interfering lines will be the interfering point, and the interferion of the direring lines will be the direring point of the common interferion of thofe planes. ILLUSTRATION. Plate VII. Fig. 8. Bring in the plane K C, to coincide with Q^in the plane "B M, bring D, in the plane N D, to coincide with D in the plane B M, and make T, in the plane D V, to coincide with T in the plane B M, and make the plane L R pafs through both the planes D V, K C, in the lines V T, C Qj bring D, in the plane D W, to coincide with D in the plane B M, and pafs the plane W H through the plane L R, in the line H L. Now, let the planes K C, B M, be two original planes, cutting each other in the line K which is not parallel to the pirure L R, let O be now the point of light, Z B the direring plane, W H the -vanilhing plane of B M, and D V that of the plane KC ; then are the lines H L, V T, the vanilhing, T R, C Q^, the interfering, and P B, P N, the direring lines of the faid original planes ; I fay, thofe feveral lines will cut each other, and the reft be as the theorem declares. DEMONSTRATION. Becaufe the line K Qjs not parallel to the pirure, it muft have a vanilhing, interfering and direring point. Now K Q^, being a Jine common to both the original planes, its vanilhing point V muft be in both their vanilhing lines ; (theorem IX. part II.) whence H L, the vanilhing line of the original plane B M, muft crofs S T, APPLIED TO PERSPECTIVE. 45 the vanifhing line of the other original plane K C, in the point V, feeing thofe lines can not coincide, by reafon the propofed planes are not parallel to each other j (cor. I. theorem XL part II.) therefore V is their common vanifhing point. For the fame reafons the interfering lines T R, C Q^, and the di- recting lines P B, P N, or P Z, muft cut each other, as here at and P, thofe lines being parallel to their refpeCtive vanifhing lines ; therefore Q^is the interfering, and P the direring point of the faid original planes. r E, D. COROLLARY I. Hence it follows, that, if two original planes be both perpendi- cular to the pirure, their vanifhing lines will make, with each other, an angle equal to the angle of inclination, which the ori- ginal planes have one to the other ; and their common intcrferion will be in the center of the pirure, which is alfo the common cen- ter of thofe faid vanifhing lines. SAME FIGURE. Raife up the plane D V, and make the plane L R pafs through the fame in the line V T, bring D, in the plane D W, to coincide with D in the plane B M, -and pafs the plane W H through the plane L R, in the line H L ; make the line B W, in the plane D V, pafs through and coincide with the line B W in the plane D W, and revolve the plane E X about the line E I, fo as to pafs through the plane L R, in the line X C. A £e/i #<>,,/ Now, let the planes B M, D V, be the two original planes, per- pendicular to the pirure L R, then will the interfering lines T R, V T, of thofe planes with the pirure, determine their angle of in- clination, which is, in this cafe, a right angle, they being here per- pendicular to each other, as well as to the pirure ; and, as the in* N 46 OF THE DOCTRINE OF PLANES terfedting lines determine that angle, the vanifhing lines H L, X C, or X muft alfo do the fame, (10. ji.) and the line G T, which is the common fedtion of the original planes, being alfo perpendi- cular to the pidture, (19. 11.) its vanifhing point is in C, which is the center of the pidture ; (def. V. part II.) therefore C is alfo the common center of the vanifhing lines of the propofed planes. COROLLARY II. Hence it alfo follows, that when two original planes D V, B M, are perpendicular to each other, as well as to the pidture, their va- nifhing lines will be at right angles to each other, and the vanifhing line of either plane will be the vertical line of the other plane ; for the vanishing lines H L, X Q^, are perpendicular to each other, becaufe they make with each other the fame angle with that of the propofed planes, by the preceedent corollary ; and the vertical line of every plane, palling through the center of the pidture, is perpendicular to the vanifhing line of that plane 3 (theorem II. part II.) wherefore the vanifhing lines H L, X pafling through C, muft be reci— procally the vertical line one of the other. COROLLARY III. If two original planes D V, B M, be perpendicular to each other, and only one of them, fuppofe B M, perpendicular to the pidture, their vanifhing lines will ftill be at right angles to each other, and the vanifhing line H L, of that plane B M, which is perpendicular to the pidture, will be the vertical line of the other plane D V, which is not perpendicular to the pidture, but the vanifhing line of the laft- mentioned plane D V will not be the vertical line of the firft-men- tioned plane B M, but only parallel thereto ; for the plane B M, being perpendicular to the pidture, all lines, perpendicular to that APPLIED TO PERSPECTIVE. 48 plane, are parallel to its vertical line X Q^, and alfo to the pidture ; and feeing the planes B M, D V, are perpendicular to each other, a line may be drawn, in the plane B M, perpendicular to the plane D V, (38. 11.) and confequently parallel to H L, and alfo to the pidture L R ; that line being therefore parallel to the vanifhing lines of all planes which pafs through it, (cor. I. theorem XIII. part II.) it muft be alfo parallel to H L, the vanifhing line of the plane B M ; (theorem XI. part I.) that vanishing line is therefore parallel to the vertical line of the plane D V, and confequently perpendicular to the vanifhing line thereof. Now. the line H L, pafling through C, the center of the picture, at right angles to the vanifhing line of the plane D V, is the vertical line of that plane ; but, as the plane D V is, by hypothecs, not perpendicular to the pidture, it's vanifhing line can not pafs through C, and confequently H L can not, in this cafe,, be the vertical line of the plane D V, but parallel thereto only. THEOREM XVIII. If an original line be given in a plane, parallel to the picture, it will be in the fame proportion to its image on the picture, as the diflance between the eye and original plane ; is to the diflance between the eye and the pidture.. PLATE IV. Fig. 5. Let O P be the pidture, c its center, E the point of fight, A B a line given in the plane S T parallel to the pidture, E c the axis of the eye produced till it meets the original plane in C ■, then, as the original line A B is to its image a b, fo is E C, the diflance of the eye from the original plane, to E c, the diflance of the eye from the pidture. 8 OF THE DOCTRINE OF PLANES DEMONSTRATION. Becaufe A B is in a plane parallel to the picture, it is itfelf pa- rallel to the pidture, and alfo to its image a b ; (cor. II. theorem IV. part II.) wherefore the triangles E A B, E a b, are fimilar ; and, for the fame reafon, the triangles E C B, E c b, are alfo fimilar ; whence, in the fimilar triangles E A B, E a b, we have A B : a b : : E B : E b, and, jn the fimilar triangles E C B, E c b, it is E B : E b : : E C : E c, confequently AB : ab :: EC : Ec. ^E. D. COROLLARY I. If the original line A B be any wife divided intofeveral parts, the images of thofe parts will have the fame ratio one to the other, as the original parts bear one to another : becaufe each original part is to its image as the diftance between the point of fight, and original plane, is to the diftance of the eye from the pidture, which two di- ftances remain invariable j and, for the fame reafons, the images of the ftdes of any figure A B G F, in a plane parallel to the pidture, have the fame proportion to each other, as the correfponding fides of the original figure have one to the other. COROLLARY II. The image of any figure, in a plane parallel to the pidture, is fimilar to its original ; becaufe the angles, made by the fides of the image, are equal, and its fides proportional to the correfponding an- gles and fides of the original figure, by cor. IV. theorem IV. part II. and the preceedent. APPLIED TO PERSPECTIVE. 49 COROLLARY. III. The image of a given line, in a plane parallel to the picture, will be of the fame length, in whatever point of the directing plane the eye be fixed ; becaufe the diftance between the eye, the picture, and the original plane, undergoes no alteration, let the point of fight be taken in the directing plane where it will j therefore the propor-* tion of the image to its original remains the fame. COROLLARY IV. If the pidture and original plane are both on the fame fide of the eye, the greater the diftance is between the eye and original plane, the nearer does the image of any given line in that plane approach to equality with its original. ILLUSTRATION. Plate V. Fig. 6. Raife up the planes P Z, D K, and make the planes M N, S Q^, pafs through the plane D K ; bring R to coincide with R, and make the plane K A pafs through the planes M N, S Q^, P Z. Now let A D be a given line in the original plane P Z, and S the pidture, a d its image when feen by the eye at E ; fuppofe now the eye to be removed farther from the original plane, as at K, then, drawing the lines D K, A K, the line F G will be the image of AD, when viewed by the eye placed in K ; whence, feeing the original line A D is to its image feen at E, as E L is to E C, by the precee- dent theorem, the fame line A D will be to its image F G, feen at K, as K L is to K C ; but K E, being greater with refpedt to E C, than it is with refpedt to E L, (8.5.) KE + EC = KC, will be greater in refpedt of KE-f EL = KL, than EC is to E L 5 and confequeptly F G, the image of A D, when feen from K, will be greater iil proportion to AD, then it will be when feen from E. O 5 o OF THE DOCTRINE OF PLANES Now, feeing the image of A D is always lets than its original when thus fituated with refpedt to the eye and the pidture, and, as the image becomes greater in proportion as the diftance . SCHOLIUM. It having been fhewn, in cor. I. to theorem V. part II. that the directing point of an original line can have no image on the pidture, it follows, that all points whatever of an original line, which can be reprefented on the pidture, mull be at fome diftance from the o rt— ^^-|f?l^plane. Now the line on the pidture, bounded by the image of any point in an original line, and the vanifhing point of the faid line, is called the indefinite image of that faid original line. And the image of any determinate part thereof, bounded by any two points given therein, is the perspective image of that de- terminate line fo bounded. And the remainder of the indefinite image is called the complement of the image. And that part of the original line, contained between its diredting point and that point of the original line which is nearefi the diredting point, is called the complement of the original line. PLATE VIII. Fig. 9. Raife up the plane G E, and make the pidture S H pafs through the fame, make D coincide with D, and the plane M H pafs through the pidture S H, in the line H L. Now, let AG be an original line, C its vanifhing point, A the nearefi: point thereof to its diredting point D ; then is a C the inde- finite image of A G, If G had been the nearefi point of the line re- APPLIED TO PERSPECTIVE. 5 1 prefented on the picture, g C would then have been the indefinite image of that line. If both the points A and G be reprefented on the pidture, as here in the points a and g, the line a g is the perfpec- tive image of that determinate original line A G. Now a g being the perfpedtive image, g C is the complement of that image, and AD is the complement of the original line A G. THEOREM XIX. If an original line be divided into any two parts, the redtangle contained between the extremes of the indefinite image thereof, will be to the redtangle contained between the mean part thereof and the whole indefinite image, as that part of the original line, which is neared: to the directing point, is to the other more diftant part thereof. SAME FIGURE. Let A G be the original line, any how divided in the point K, into the two parts A K, KG, and the indefinite image thereof, a C, into the three parts a k, k g, g C ; I fay, the redtangle under the ex- tremes g C, and a k, is to the redtangle under the mean part k g and the whole indefinite image a C, as A K, the nearer part of the original line A G, is to K G, the other part thereof. DEMONSTRATION. Through a, the image of the point A, draw the line T V parallel to the original line A G, cutting E G and E K in the points T and X, then are the triangles E C k, k a X, fimilar one to the other ; wherefore aX:EC::ak:kg + gC, and the triangles E Cg, g a T, are alfo limilar ; whence we have E C : aX+XT :: gC : a k + k g, OF THE DOCTRINE OF PLANES 5 2 and, by multiplying thefe two proportions together, the product will be aXxEC:ECxaX-f ECxXT ::akxgC:kgxak-f kg xkg + gCxak + gCxkg. But, by (i. 2.) aCxkg = kgxak + kgxkg-fgCxkgj whence aX x EC : EC xaX + E C x XT :: ak xgC : gC xak + a C x k g. Now, fubtradting the antecedents from the confequents, it is aXxEC : EC xXT ::akxgC:aCxkgj that is, aX : XT :: akxgC :aCxkg, but, feeing that A G and T V are parallel to each other by conftruc- tion, it will be aX : XT :: AK:KG; therefore ak xgC : aC x kg :: AK : KG. E. D. SCHOLIUM. Although the line A G is here fuppofed wholly beyond the pic- ture, the propofition is neverthelefs true univerfally, wherever the point A be taken in the line A G, fo long as the line C F is parallel toEDj for any line whatever, that is parallel to C F or E D, cutting the lines EC, E A, E K, and E G produced if neceffary, will be thereby divided into parts, bearing the fame ratio to each other, which the correfpondent parts a k, kg, g C, of the line C F, have one to another. COROLLARY I. The image a k of A K, the nearer part of the original line A G, is to k g the image of the farthermoft part K G, as the redtangle under A K the nearer part, and the whole line G D, or line G A, pro- duced to its diredting point D, is to the redtangle under K G and D A, the extremes of the whole line G D. APPLIED TO PERSPECTIVE. 53 For, through g draw g (^parallel to A G, cutting the lines E A, E K, in the points R and P ; then, if a g be confidered as an ori- ginal line, C as its directing point, E C as its director, and Q^g as its indefinite image, it follows, from the preceedent theorem, that RPxQg : Pgx QR : : a k : k g, and becaufe Qjg and A G are parallel, C^g : Qjl :: DG : D A, alfo RP:Pg::AK:KG; therefore RPx Qjg : P g x QJl :: DGx AK : DAxKGj oonfequently ak : kg :: DGx AK : D A x KG. COROLLARY II. If the parts A K, KG, of the original line, be equal to each other, then a k, the image of the nearer part, will be to k g the image of the remoter part, as the indefinite image a C is to its com- plement g C. For, by the preceedent theorem, akxgC : k g x a C ; : A K : KG. Now, fince by hypothefis, AK = KG, it will be akxgC = kgxaC; whence arifes this analogy, ak : kg :: aC : gC j wherefore the line A C, in this cafe, is divided into harmonical proportion, by the points k and g. COROLLARY III. The hypothefis being the fame as in the preceedent corollary, it will be ak :kg::DG:DA. For, in the fimilar triangles E V T, ECg, we have EV = aC:gC::VT:EC = Va 3 but, by the laft corollary, ak:kg::aC.gC. P 54 OF THE DOCTRINE OF PLANES Now, feeing V T and D G are parallel by hypothecs* V T : V a : : D G : D A 5 therefore ak:kg::DG:DA. COROLLARY IV. If the images a k, k g be equal, then A K, the nearer part of the original line is to K G, the farther part thereof, as D A, the com- plement of the original line, is to D G, the whole line G A, when produced to its directing point D. For, by the above corollary I. ak:kg:: AKxDG:KGxDA. Now, if a k = k g, then is AKxDG=KGxDA; whence arifes this analogy, AK : KG :: DA : DG. COROLLARY V. The hypothecs being the fame as in the preceedent corollary, it will be AK : KG :: gC :aC; for the triangles E V T, ECg, being fimilar, it is E C = V a : VT :: gC : aCf but Va:VT::DA:DG. And, by the preceedent corollary, DA:DG::AK:KG } confequently AK ; KG :: gC : aC. CONCLUSION. Having thus fhewn the feveral pofitions of the planes, vanifhing lines, points, &c. necelfary for conveying to learners a general idea of the nature of perfpedtive repfefentations, when formed flereogra- phically on the plane of the pidture ; we now proceed to confider them in another view, fuch as is required in the practice of drawing the perfpedtive reprefentations of any given original objedts, or for determining the original objedts from having their refpedtive repre- fentations on the pidture given ; and in this we are to confider them. each as being bounded by its own proper dimenfions, and fo placed with relation to each other, as to coincide together, and form one continued plane. PLATE IX. Fig. 1 Or Let there now be given the center of the pidture, the height of the eye, the diftance of the eye from the pidture, and the petition of the original plane with refpedt to the pidture. Thus, fuppofe P M to reprefent the original plane, L M the pidture confidered as being perpendicular thereto, L X the vanifhing plane, E the point of fight $ then, upon the paper or cloth, &c. intended for the pidture which is to contain the required reprefentations, draw at pleafure the line M N, and parallel thereto, at a diftance equal to the given height of the eye, draw the line H L, and perpendicular thereto, through the point of fight E, draw the line E r, interfedting the lines H L, M N, in the points C and r, and through E, parallel to H L, draw the line Z X ; then is M N the interfedting line, H L the vanifhing line, Z X the parallel of the eye, C r the vertical line, E C the diftance of the eye from the pidture, C the center of the pidture, and alfo that of the vanifhing line H L, they both coinciding together in this particular pofition of the pidture. This preparation being made. 56 OF THE DOCTRINE OF PLANES the feveral planes and lines here mentioned are in the proper fitua- tions for the prad:ice of drawing the reprefentations required to be determined by the help of thofe feveral planes. See. As to the parti- cular dimenlions proper to be affigned to thofe planes, we fhall treat thereof in the annotations which are hereto annexed. PROBLEM I. To find the indefinite image of an original line, not parallel to the interfering line, by help of the vanifhing and interfering points thereof. CONSTRUCTION. Let the line N I be an original line, meeting the interfering line M N in the point I ; from E draw the line E V parallel to N I, cutting the vanifhing line H L in the point V, draw the line I V, and it will be the indefinite image required. DEMONSTRATION. The .original and vanifhing planes being parallel to each other, (def. VIII. part II.) and the line E V parallel to N I, by conftruc- tion, the point V is the vanifhing point of the line NI; (def. XIX.) now the interfering line M N being common both^the plane of the pirure and the original plane, J is therefore the interfering point of the given original line N I ; confequently I V is the indefinite image thereof, by the fcholium in page 50. SCHOLIUM. The line E V mull be fo drawn, that the vanifhing point of the given line may be on the fame fide of the vertical line, to which the inclination of the original line tends towards the interfering line 5 thus N I, being the original line, its vanifhing point V falls between APPLIED TO PERSPECTIVE. 57 C and H ; but if D M be the original line, then its vanilhing point V falls between C and L ; and if the original line be perpendicular to the interfering line, as the lines AN, B D, &c* then will the Vanilhing point fall in C, the center of the picture. thence to find the PROBLEM II. The indefinite image of a line being given ; .. „ — — ^ original, by help of the vanilhing plane and interfering point thereof. CONSTRUCTION. . Let I V • be the indefinite image given ; produce the fame both ways when necelfary, fo as to meet its vanilhing and interfering points V and I, and from I draw a line, as I N parallel to E V. DEMONSTRATION. Then the line I N, being drawn through the interfering point I, parallel to the line E V, drawn in the vanilhing pjane .through the point of fight, by conftrurion j it is therefore the* original line re- quired. (def.XIX.) A E. D. PROBLEM III. The common vanilhing point of parallel lines in the original plane being given ; thence to find the vanilhing point of all other lines in that plane, which make a given angle with the lines firftpropofed. CONSTRUCTION. Let TO, S U, be two given original lines parallel to each other, V their common vanilhing point, and let S T, U O, be two other given lines, making any given angle S T O, or U O T, &c. with the former. From the given vanilhing point V draw a line to the point of fight E i then from E, towards that end of the vanilhing line to which Q. 58 OF THE DOCTRINE OF PLANES the lines tend, whofe vanifhing point is fought ; draw the line E y,. making with V E an angle V E y, equal to the given angle, and the point y, in which the line fo drawn cuts the vanifhing line, is the other vanifhing point required. DEMONSTRATION. By fcholium, in page 50, the lines o V, u V, are the indefinite images of the given parallel lines TO, S U, and oy, t y, are thofe of the lines U O, ST, and the angle V E y = T O U, by conftruc- tion ; therefore V and y are the vanifhing points of thofe feveral. lines, (theorem VI. part II.) £. D. COROLLARY I. If the original lines propofed, as A B, N D, be parallel to M N,, the interfering line of the original plane, then the vanifhing point of fuch lines, as AD, B N, which make a given angle with the propofed original lines, may be found thus j through E draw E V, to cut the vanifhing line in V, on the fame fide or fides of C, to which the re- quired originals are fuppofed to tend, making the angle X E V, or its equal E V y, equal to A B N, or D 7YB, the angle propofed ; then will V be the vanifhing point of all fuch lines as B N, and v that of all fuch lines as A D. For the line E V makes the fame an- gle with the eye’s parallel EX, as the originals, whofe vanifhing point is V, make with the interfering line of the original plane, or any line parallel thereto. The fame holds good with refper to the line E v, and the lines whofe vanifhing point is v. COROLLARY II. Hence it follows, that if the original lines propofed incline fo much to the interfering line of the original plane, as to caufe their APPLIED TO PERSPECTIVE. $ 9 vanifhing point to be out of reach ; yet if the angle, which the ori- ginal lines make with the interfering line, be known, the vanifhing point of lines, which make a given angle with the propofed original lines, may be found in manner as- above ; for, drawing E V on that fide of C, to which the original lines are fuppofed to tend* making the angle X E V equal to the angle of inclination which the original lines make with the interfering line y draw another line Ev, cutting the vanifhing line in v, and making the angle V E v equal to the angle propofed, and v will be the vanifhing. point defired,. for the: fame reafons as above- LEMMA, The direror of a line in an- original' plane makes the fame angle ' with the eye’s parallel and direring line of that plane, as the image of the given line makes with the vanifhing and interfering lines of that plane.- Becaufe the direror and the image of the. original , are parallel one to the other, (theorem VII. part II.) COROLLARY L The direror of an original line makes the fame angle with the eye’s direror of that plane, which the image of the original line makes with the vertical line 1 of that plane. Becaufe the eye’s direror and vertical line are parallel, , as is alfo the direror of the original line to the image thereof.. COROLLARY II. The indefinite images of all lines whatfoever, whofe direring - points are any where in the parallel of the eye, relating to an ori- ginal pfane, are parallel to the vanifhing line of that plane. For the images are parallel to their direror, which being the pa* 6o OF THE DOCTRINE OF PLANES rallel of the eye, in this cafe, they are therefore parallel to the va- nishing line of the original plane. COROLLARY III. The indefinite images of all lines whatever, whofe directing points fall any where in the eye’s director, relating to an original plane, are parallel to the vertical line thereof, whether the propofed lines be in or out of that plane. For the director of the eye, being the director of the propofed lines, it is therefore parallel to the vertical line. COROLLARY IV. If any two lines in an original plane cut the directing line of that plane in the fame point, their images will be parallel. For the original lines muft then have the fame dire&or. COROLLARY V. If the images of any two original lines in the fame plane be pa- rallel, the original lines, if they be not parallel to the picture, muft have the fame directing point. For there can be drawn but one director parallel to both the given images, and that director can cut the dire&ing line of the original plane only in one point, which point is therefore the com- mon directing point of the propofed original lines. SCHOLIUM. It is eafy to reverfe this problem, that is, from a directing point given, thence to find another directing point, fuch that the images of all original lines, which have thofe points for their directing points, may make in the picture an angle equal to any angle pro- APPLIED TO PERSPECTIVE. 61 pofed, only by ufing the dire&ing plane and directors in like manner as the vanifhing plane and the lines drawn to the vanifhing line from the point of fight parallel to the given original lines, were ufed in the above, regard being had to the pofition of the directing point fought, which muft fall on the contrary fide of the given directing point to that, whereto the propofed images are intended to incline, the demonftration whereof is deducible from the preceedent lemma and its corollaries. N. B. The angles determined by this problem, when neither of the original lines are fuppofed parallel to the interfering line, are thofe comprehended between the two vanifhing points, or the two interfering points of the images, as V m v, or M m I, or the cor- refponding angles of the originals, and not the angles which the originals, or their images make fide, ways, as V m M, or v m I. DEFINITION. The angles Vmv, Mm I, or any others in a like pofition with their correfponding originals, are here called internal angles to diflinguifh them from the angles V m M, v m I, which are called external angles. PROBLEM IV. A vanifhing point being given ; thence to find two other vanifhing points, fuch that all lines, drawn in the pirure from thofe three points, on the fame fide of the vanifhing line, may, by their mu- tual interferions, form triangles, whofe originals fhall be fimilar to an original triangle given. PLATE X. Fig. ii. Let the fame things be fuppofed as before, and let V be the va- nifhing point given. R 6 2 OF THE DOCTRINE OF PLANES CONSTRUCTION. Draw the line E V, and from E take any part thereof, as E W ; upon the line E W make a triangle E W Z limilar to the original tri- angle propofed, having either of its angular points in E ; then pro- duce E Z till it cut the vanishing line in fome point, as H, from E draw another line, as E K, parallel to W Z, cutting the faid vanishing line in K ; then are H and K the vanishing points required. Let there now be drawn, from the points H, V and K, any lines, as H m, H p, H s V n, V m, V p j K k, Kn, Km; making by their mu- tual interferons any triangles, amp, a b k, a k n, a n m, &c. I fay, the originals of all thofe feveral triangles are fimilar to the ori- ginal triangle propofed. DEMONSTRATION. Becaufe the originals of a p, a m, m p, whofe vanishing points are H, V and K, are refpe&ively parallel to the lines EH, E V, E K, drawn from the point of light to thofe vanilhing points, (def. XIX. part II.) and E K is parallel to W Z by conftrudtion ; therefore the original of in p is alfo parallel to W Z, (theorem XI. part I.) for the fame reafons, the originals of the three fides am, a p, m p, of the triangle amp, being refpedtively parallel to the three tides E W, E Z, W Z, of the triangle E W Z, and their correfponding angles being equal; (theorem XII. part I.) therefore the original of the triangle a m p is limilar to the triangle E W Z, but the triangle E W Z is fimilar to the original triangle given, by conftru&ion ; whence the proportion is manifest. The fame may, in like manner, be proved of each of the other triangles a b k, a k n, &c. E. D. SCHOLIUM. If it happens, in conltru&ing the triangle EWZ, limilar to the APPLIED TO PERSPECTIVE. 63 original triangle given, that the line W Z be parallel to the vanilhing line H L, then the original triangle will have but two vanifhing points, from which drawing any two lines fo as to interfedt each other, and thefe being again interfered by any line drawn parallel to the interfering line M N, a triangle will thereby be formed, whofe original is limilar to the triangle propofed. Suppofe now, in the fame figure, S T to be parallel to H L or X Y, then a line drawn through E, parallel to S T, will coincide with X Y ; whence the fide of the original triangle, correfponding to S T, can have no vanilhing point in this cafe ; (theorem IV. part II.) it will therefore be parallel to the interfering line M N, (cor. I. theorem XIII. part II.) to which its image is alfo parallel; (cor. II. theorem IV. part II.) wherefore any two lines H R, K D, being drawn from the vanifhing points H and K, any where cutting each other, as in m, any lines g f, n p, drawn parallel to the inter- fering line M N, or even the interfering line itfelf, will, by their in- terfedions with the lines H R, K D, form triangles mpn, mgf, D m R, &c. whofe originals will be fimilar to the original triangle given. COROLLARY Hence, if three vanifhing points, or two only, in fuch cafes as the above, of a triangle be given, the fpecies of the original triangle may be thence determined. Thus, fuppofe H, V and K to be given, draw the lines E H, E V, E K ; then, through either two of thofe adjacent lines, draw a line parallel to the third, and it will determine the fpecies of the original triangle; for, drawing W Z through E H, and E V, and the line W Z parallel to E K, the triangle E W Z will be fimilar to the original tri- angle, from what has been premifed, or a line W drawn through E V, and E K, parallel to E H, determines E W Q the fpecies of the 6 * OF THE DOCTRINE OF PLANES original triangle ; for the triangles E W Z, E W Q^, are fo evidently fimilar, as not to need a demonftration thereof ; or if H and V be the only two vanifhing points of the original triangle, it is evident, that by drawing S T parallel to X Y, it will determine the fame by the triangle E S T, which is alfo fimilar to the triangle E H K. SCHOLIUM. In this problem it is limited, that the lines, drawn from the given vanifhing points of the given original triangle, be all on the fame fide of the vanifhing line, in order that the triangle, formed by the interfedtions of thofe lines, may reprefent a triangle fimilar to the original ; for if thofe interfedtions fall fome on one fide, and fome on the other, of the vanifhing line, the original of the tri- angle fo formed will be two diftindt and feparate indeterminate figures. The reverfe of this problem, that is, from a diredting point given, to find two other diredting points, fuch that the images of all original lines, drawn from thofe three diredting points, may, by their mu- tual interfedlions, form on the pidture, triangles fimilar to a given triangle, may be eafily performed, by making ufe of the diredting plane, and the diredtors of the original lines, in the fame manner as was fhewn, of the vanifhing plane and the lines drawn from the eye to the vanifhing points of the given original lines, as was obferved in the fcholium page 60, which is fo evident, that it needs no farther explanation. PROBLEM V. AT o find the image of any given point in an original plane. •Phis is done by finding the indefinite images of any two lines in applied to perspective. 6 5 the original plane which pafs through the given point, for the inter- fedion of thofe images is the image of the point required, (theorem VIII. part II.) EXAMPLE I. PLATE IX. Fig. io. For, let z be the point given, D M, N I, two lines in the original plane, interfering each other in that point, M v, IV, their indefi- nite images on the pidure, interfering each other in the point m ; then is that point m the image or perfpedive reprefentation of the point z, which was to be found.. EXAMPLE II. To find the perfpedive reprefentation of B D, a line perpendicular to the interfering line M N, or bottom of the pidure ; draw from one of the extremes, as D, a line D M, any how cutting the inter- fering line, as here at M ; through E, parallel to D M, draw E v, cutting the vanishing line in v, the vanifhing point of D M, through B, the other end of the given line, draw N I, and parallel thereto draw E V, continue B D till it meet the interfering line as at i ; then drawing M v, I V and i C, the intercepted line b d will be the perfpedive reprefentation of B D required. EXAMPLE III. To find the perfpedive reprefentation of A B, a line parallel to the interfering line or bottom of the pidure. Draw B i, Ajc, each perpendicular to M N, and meeting the fame in the points i and^e ; drawx C, i C, and the other lines as before then is a b the reprefentation required. S 66 OF THE DOCTRINE OF PLANES EXAMPLE IV. To find the reprefentation of a line, as NB, in an oblique pofition, with the interfering line. Produce N B to I, and draw the feveral lines as before directed, then will the line b n be the reprefentation required. SCHOLIUM. Now, feeing that all plane right lined figures are bounded by- lines, which are either perpendicular, parallel or oblique to the in- terfeCtion line, the examples here given fuffice for finding the repre- fentations of any given rectilinear figures, the picture, original and vanifhing planes being in the fituation here defcribed, we might now proceed to the finding any other reprefentations, or to the finding of the originals themfelves, from having their reprefentations given on the picture, in every fituation that can be given to the picture, with refpeCt to the original plane but that would be foreign to the defign of this little traCt, which is not intended to teach the art of drawing perfpeCtive reprefentations in its full extent, but only to remove the difficulties which lay in the way of learners at their firft entrance on this fiudy, through their not being able rea- dily to conceive, with fufficient clearnefs, the aCtual pofitions of the feveral planes and lines neceflary to be duly diftinguiffied and applied in fiudying the rudiments of this, art wherefore, hoping this ffiort Effay will be of fome affiftance to learners in this refpeCt, ffiall now endeavour to fhew them, by a kind of ocular proof, the exaCt agreement which fubfifis between the practical conftruCtions which 'are ufed in this art, and what we have previoufly advanced under a ftereographical confideration. To effeCt which, let us now fuppofe the picture to be raifed at right angles to the original plane, by bringing P to coincide with P, and Q^with Q^; alfo fuppofe the va- APPLIED TO PERSPECTIVE. 67 nifhing plane parallel to the original plane, by making Y coincide with Y, and W with W ; then, fuppofing ABDN, GFKPR, OUST, to be three original figures given in an original plane, and a b d n, g f k p r, o u f t, to be their refpedtive reprefentations on the picture, as inveftigated by the rules of practical perfpedtive, when thofe feveral planes formed one continued plane : let now the filks be drawn through the point of fight, they reprefenting fo many rays of light ifluing from the extremities of the given objects, and all uniting together in the eye or point of fight ; then, by what was delivered in def. I. part II. it appears, that thofe feveral rays will, by their interfedtions with the picture, mark thereon the feveral points requifite for determining the required reprefentations of the given original objedts ; and, feeing thofe rays interfedt the pidture in the very fame points as were before found by practical conftruc- tion, it is plain the refult is the fame in both ; whence the perfedt coincidence between the theory and pradtice of forming perfpedtive reprefentations appears manifelt. And the fame may, in like manner, be thewn, let the pofition of the pidture and the original plane be what it will ; whether confi- dered as parallel to each other, as obliquely fituated one to the other, as inclining to, or as reclining from each other, under any given angle whatever. See. which, it is hoped, will evidently appear from what has been delivered, and is contained in the following annota- tions j wherefore, having thus endeavoured to elucidate the dodtrine of planes as applied to the primary operations ufed in the pradtice of perfpedtive, which is the propofed boundary of this attempt, I now refer to Mr. Kirby’s treatife afore-mentioned, and thofe matters whofe province it is to exhibit the ufes of thofe principles under the - different modifications required in the pradtical applications thereof. 63 OF THE DOCTRINE OF PLANES, &c. leaving to another opportunity the consideration of what may here- 'after appear farther neceflary for rendering this method of explaining the theory more extenlive. V ANNOTATIONS, O R GENERAL COMMENT ON PART II. I. OF MATHEMATICAL PROJECTIONS IN GENERAL. ^^lAthematical projections are of two kinds, viz. geometrical and flereographical ; the former whereof fhould be firfl underflood, as being ufeful and neceffary to the latter. II. OF GEOMETRICAL PROJECTIONS. Geometrical projections are conflruCted by drawing lines, parallel to each other, from the feveral points of the given obje&s, cutting the plane of projection either perpendicularly or obliquely, under any angle whatever. In this kind of projection, the place of the eye is not confidered otherwife than by fuppofing it very remote, or at an infinite diflance from the plane of projection ; whence it can re- prefent only two dimenfions at a time, as length and breadth with- out thicknefs, or length and thicknefs exclufive of breadth, &c. The projected images of objects differ according as they are fituated with refpeCt to the plane of projection : hence- arife three cafes. I. When the objeCts to be projected are in planes parallel to that of projection. 2. When perpendicular thereto. 3. When inclining to the fame. T 7 ° GENERAL COMMENT In the firfl of thefe cafes, the images are exadtly equal and fimilar to their originals ; in the fecond, they become flraight lines ; and in the third, are neither equal nor fimilar. CASE I. ILLUSTRATION. PlateLFig.i. MaLe F coincide with F, and the planes B M, S H, to pals through each other in the line H L. Now, let A B E D reprefen t an objedt to be projedted on a plane S H parallel thereto. Draw AT, DR, B H, EC, parallel to each other, and perpendicular to the plane of projedtion S H, meeting the fame in the points T, R, H, C ; then, drawing the lines R C, C H, H T, R T, the figure HCRT is the geometrical projedtion of AB E D, and exadtly equal and fimilar thereto. DEMONSTRATION. The planes B D, R H, are parallel to each other by hypothefis,, B H parallel to E C, and both perpendicular to the plane R H by conllrudtion ; H C is therefore equal and parallel to B E. In like manner it may be proved, that the lines R C, R T, T H, are refpec- tively equal and parallel to E D, DA, AB; and all the angles of the given figure being equal to the correfpondent ones in the projec- tion, therefore is R C H T equal and fimilar to A B E D. Or if the projedting lines are oblique to the plane of projedtion, thus ; Let A o, D V, BP, Ed, be drawn from the fame points A, D> B, E, any how inclining to the plane R H, and meeting it in the points o, V, P, d, it may be proved as above, that the figure V o P d is alfo equal and fimilar to ABED. E. D, COROLLARY Hence it appears, that the projedtion will always be the fame,. ON PART II. 7 1 whether the plane of projection be before, behind, above, below, nearer to, or farther from the original plane, provided the projecting lines are parallel to each other. CASE II. SAME FIGURE. Suppofe now E f p q, in the plane B M, to be the objet to he projeted on the plane R H, perpendicular thereto ; then, drawing f r, EC, to meet R FI perpendicularly, and they will give the line C r, for the projetion of E f p q, for all the lines that can be drawn) from any points in that figure, to meet the faid plane perpendicu- larly, will fall in the common fetion C r ; therefore the projection, in this cafe, is a flraight line only. CASE III. SAME FIGURE. Make the plane S H pafs through the plane D Y ; let now the plane D Z be the plane of projection,DY an original plane containing a line a o, inclining to the plane D Z. Draw o R, a e, perpendicular to D Z, and they determine the projection e R of the given line ao. SCHOLIUM. In this cafe, when the projecting lines are perpendicular to the plane of projection, each line in the given figure will be, to its image, as radius is to the co-fine of the angle of inclination. For, from the extreme o, draw o b parallel to e R, and it will alfo be equal thereto; whence o b a, being a right-angled triangle, o b = eR, is the co-fine of the angle of inclination ; therefore it is a o : o b *= eR Radius : the co-fine of the angle of inclination. GENERAL COMMENT 7 2 OF THE DIFFERENT WAYS OF APPLYING GEO- METRICAL PROJECTIONS. There are feveral ways of applying this kind of projection, in all which the projecting lines are generally confidered, as being perpen- dicular to the plane of projection whence it is diftinguilhed by dif- ferent names, fuch as Ichnography or plan. Orthography or eleva- tion, SeCtion, Profile, See. Ichnography or plan, when it deferibes the perpendicular feat or place which objeCts have on the ground, not regarding their heights above it. Thus the plan of a town, fort, or any building, is the geometrical defeription of the perpendicular pofitions which their feveral parts have with refpeCt to the ground, that being con- fidered as the plane of projection. Orthography or elevation, when it reprefents the plane or furface of objeCts, according to their breadth and heighth above the ground, excluding the depth or fpace which they occupy thereon. Thus the elevation of a building is the geometrical defeription of fome one front or fide thereof, deferibed upon the plane of projec- tion, as to the height and breadth thereof. And if the building be fuppofed to be cut by a plane paffing through the fame, and the fe- veral parts thereof be deferibed upon that plane, according to the true meafures which the parts have through which that plane palfes ; this is called a section of that building. Profile, when it is applied to works of fortification, for de- feribing the feveral heights. thereof, as they would appear upon a plane pafling through them perpendicular to the ground. Chart or map, when ufed geographically, to deferibe the form of any part of the earth, and the pofitions of places fituated upon the fame, without confidering the fphericity thereof, but looking upon the part fo deferibed as being a plane. ON PART II. 73 In all thefe applications, the plane of projection is conceived as parallel to the plane of the original or thing defcribed j whence the projection fhould, in ftriCtnefs, be equal and fimilar to its original, but, in the practical ufes hereof, it is fnnilar only ; for it is chiefly ufed to reprefent or defcribe large things in a fmaller compafs than they really poffefs, fo as to be in any affigned proportion to the ori- ginal, at the fame time preferving, as much as poflible, the true iorm and pofition of the feveral parts thereof. This kind of projection is alfo ufeful to agronomical purpofes, in projecting the fphere and its feveral circles in piano, in which thofe circles are confidered as being in planes, whereof fome are parallel, fome inclining, and others perpendicular to the plane of projection j and they are accordingly reprefented on the given plane, by circles in the firft cafe, ellipfes in the fecond, and by ftraight lines in the third. OF STEREO GRAPHICAL PROJECTION. Stereographical projection is when the projecting lines,- drawn from the original objeCt to cut the plane of projection, are not pa- rallel to each other, but converge together, and all meet in one point, ObjeCts are here confidered as feen from one certain point, called the point of fight ; whence this kind of projection can reprefent at once, all the three dimenfions, length, breadth and thicknefs j that is, as it were, the folidity of objeCts. Here alfo arife three cafes or varieties proceeding from the different pofitions which the point of fight, plane of projection, and the original plane may have with re- fpeCt to each other. i. When the plane of projection is between the point of fight and the objeCt, U 74 GENERAL COMMENT 2 . When the object is between that point and the plane of pro- jection. 3 . When the faid point is between the objeCt and the plane of projection. The hr ft of thefe cafes is called perfpeCtive, and here the image is always lefs, in the fecond it is always greater, and in the third it may be either lefs, equal, or greater than the given .objeCt. But it is not true, that the parts of a magnitude may be less* EQUAL TO, OR GREATER THAN TEE MAGNITUDE ITSELF It is only the reprefentation of magnitude that is here defcrihed. ILLUSTR AT I O N* Plate V. Fig. 6 . Raife up the plane K D, and make the planes S Q^and M N pals- tli rough the fame ; raife up the plane P Z, bring R to coincide with R, and pafs the plane K A through the planes M N, S Q^and P Z. Now, in the hrft cafe, let E be the point of fight, the plane S Q^. the picture or plane of projection parallel thereto, and P Z the original plane containing an original line AD. Then, drawing the lines-, E A, ED, they will interfeCt the picture or plane S in the points a and d, and thereby determine the line a d .for the perfpeCtive ap- pearance of the original AD. Now it is evident, that a d muft al- ways be lefter than A D, and that if the plane S Q be fitnated nearer the plane P Z, every thing elfe being as before ; the image a d will be greater and greater, as that plane approaches nearer to the plane PZ, and reverts into the original, by the coincidence of thofe two planes. On the other hand, if the plane S QT>e fuppofed to approach, in like manner, the plane M N, the image decreafes, and when thofe two planes coincide together, it will vanifh into the point E, * See page 27. Elements of Mathematics, &c. for the ufe of the Royal Academy at Woolwich, 1757. ON PART II. 75 In the fecond cafe, let now the plane S Q^be confidered as the original plane, a d the original objedt, and the plane P Z to he this pidture or plane of projedtion, and E the point of fight, as before > then producing E a, Ed, till they meet the plane of projedtion, as here at A and D, the line A D will be the image of a d, and greater than it, as is evident from the conflrudtion itfelf, and be- comes Rill greater, as the plane S Q comes nearer to the plane P and will become infinite, or rather be no image at all, when thofe two planes coincide ; for then the lines, which fliould produce the image, will be parallel to each other, as in geometrical projedfions, therefore can not meet in a point to form an image, which flereo- graphical projedtion requires them to do. Laftly, fuppofe B S to be the original objedt in the plane R W, and S Qjhe plane of projedtion parallel thereto, and E the point of fight i then drawing through E the lines B E, S E, producing them till they meet the plane of projedtion, as here in the points d and a, and they will determine the line d a for the image of B Si Now, when KE = E C, it is evident that the image da will be equal to the original B :S, as in the pofition before us ; but when K E is lefs than E C, the image d a will be greater than B S ; but if E K be the greater, then the image will be leffer than the original, and that when E C is nothing, or that the planes M N and S coincide, the image then vanifhes into the point E ; but if E K becomes no- thing by the coincidence of K with E, the image fhould then be infinitely great, and therefore can not be reprefented on the plane of projedtion.. COROLLARY. Hence it appears, that, in all thefe three cafes above-mentioned, the point of fight, plane of projedtion, and the original plane, mult be at fome dillance from each other ; for if either the plane of pro-* GENERAL COMMENT 76 jection, or the original plane, coincides with the eye, no image can then be formed ; and, if the plane of projection and original plane coincide, the image then changes into the original object. OF SPHERICAL PROJECTIONS. In projecting the fphere in piano, the geometrical and ftereogra- phical projections are introduced, and applied fometimes fingly, and fometimes both together ; thus, fuppofe the eye to be placed in one of the poles or extreme points of the axis of the fphere, and the plane of projection to be perpendicular to that axis, and palling through the oppofite pole or other extreme thereof j in this cafe, the whole fphere will be included between the eye and the plane of pro- jection ; and therefore the fecond cafe only is here applied. But the eye continuing in the fame pofition, and the plane of pro- jection fuppofed to pafs through the equator, or any of the lefler cir- cles parallel thereto, as the tropics, polar, or parallels of latitude, &c. then a due application of both the firft and fecond cafes is re- quifite ; for the projections of thofe parts of the fphere, which are between the eye and plane of projection, are derived from cafe II. and thofe which are beyond that plane, from cafe I. Becaufe the eye is here fuppofed to be in one of the poles of the fphere, it is ma- nifeft, from the above, that the faid pole can not be reprefented in the projection, by reafon thofe parts of the fphere, which are neareft the eye, will be projected fartheft out, and recede infinitely from the center of the projection, proportionably to their vicinity to the eye ; whence thefe kind of projections are generally confined to fome par- ticular circle of the fphere, taken at pleafure, according to the parts which the projection is to defcribe, for then none of the parts between that circle and the eye are delineated on the plane of projection. Now, if the plane of projection pafs through that particular circle. ON PART II. 77 the whole projection is perfpedtive, and comes tinder cafe I. In all thefe inftances the projections are fxmilar one to another, and differ only as being one greater or leffer than the other, according as the plane pf projection is fuppofed to be more or lefs diflant from the eye. Having thus defcribed the nature of mathematical projections in general, I now proceed to confider cafe I. more particularly, as be- ing the principal defign of this little tradt, referring the fpherical projections, and what I have prepared for illuftrating and explaining that fubjedt more fully, to another opportunity. OF THE DIFFERENT APPELLATIONS GIVEN TO PERSPECTIVE, Authors have confidered and diftinguifhed Perfpedtive varioufly. Some according to the different portions which the picture may have with refpedt to the ground ; thence denominating it to be di- rect, inclining and horizontal, according as the picture be fuppofed either perpendicular, oblique, or parallel to the ground. Others from the different portions of the picture with refpedt to the eye, as being either diredtly in, above, below, or on one fide of the axis thereof. And others, again, by the different diftances of the eye from the picture. But no advantage has accrued to the art from any of thefe varieties, there being no effential difference in the practice, whether it refult from one or the other of thofe confiderations, for the rules ferve alike in all of them, let what will be the pofition of the eye and the picture, or that of the picture with refpedt to the objeCts. For it is only from the different form which the picture may be of, that any effential difference can arife, as whether it be a plane, concave, or convex furface, &c. Such are the cafes of appearances X GENERAL COMMENT 78 defcribed on cupolas, arched cielings, uneven walls, or theatri- cal fcenes, &c. which laft is commonly called Scenography, and confifts in making the defcription upon feveral different planes, dif- ferently pofited, and at different diftances with refpect to the eye; And fometimes objects are reprefented by reflexion, by fo drawing them on the picture, that although they then appear confufed, and without any veftiges of a regular defign, yet, by placing a polifhed piece of metal or glafs, of a fuitable- form, and in a proper pofition* the feveral parts of the picture will be thereby duly reflected to the eye, in their proper- fituations, and the true image of the original} from whence they were drawn, will be perfectly exhibited to view, however confufed and unintelligible it feemed before. For, whe- ther the rays which compofe an image be refledted, refradted, or any other way disturbed from proceeding in a rectilinear courfe to the eye, if they be fo fubjedted by any contrivance, as that they at laft duly unite in the eye, they will thereby difcover the true imag3 of that objedt ; but thefe are fpeculative matters more for curiofity than real ufe, fhall not therefore digrefs farther on this* head, but leave the reader, who is deflrous to learn this particular fpecies of Perfpedbive, to the directions of a treatife, eompofed by a jefuit, titled, LE CURIEUSE PERSPECTIVE. Some authors have alfo introduced a third kind of projection, called by them military perspective, or geometrical elevation* by which they pretend to reprefent length, breadth and depth to the view all at the fame time, by raifing the fides of the objects from their ichnography or geometrical plan, adhering to the true meafures of thofe fides, regardlefs of their different diflances froro- the eye, but varying the angles of elevation by the laws of flereo- graphical projection. The effects of this unfit mixture of both kinds of projection, is an inconfiflent medley, both unnatural and difa- ON PART II. T 9 greeable to the eye, and can anfwer no purpofe but what may be better attained by true Perfpedive, is therefore unworthy of a place in this trad: j but a more particular account thereof may be feen in the firft volume of the jefuit’s Perfpedive, printed at Paris in 1679,, to which we refer thofe who would learn this kind of projedion, which we here think not worth our while to explain any farther. II. OF THE SEVERAL PLANES NECESSARY TO BE CONSIDERED IN DETERMINING THE APPARENT POSITIONS OF OBJECTS ACCORDING TO THEIR RE- SPECTIVE SITUATIONS. The fundamental principle, upon which all manner of projedions depend, confifts in knowing how to determine or mark out, upon any given plane, the place of any point of a vifible objed, as feen by the eye in any given pofition.. Now the apparent place of any point of the objed will be in that point of the given plane where a line, drawn from the eye to the faid point, interfeds the faid plane, or when the given plane is beyond’ the objed, in that point where that line duly produced meets the faid plane : and the pofitions of any points being obtained, thofe of lines are eafily derived, as being terminated' by points ; Surfaces, as bounded by lines y alfo folids, as being contained under one or more furfaces wherefore the rules of projedion have their origin from hence. But the real pofition of a point in abfolute fpace can not be known other ways than by com- paring it with other objeds, whofe fituations are given y whence feveral planes, differently fituated one to the other, mult be affumed, fo that, by means of one of them, the pofition of the objed may- be compared with refped to its being even with, above, or below the eye, and by help of another, whether it be in the fame right line with the eye, or on either fide thereof. Thus aflronomers fup~ So GENERAL COMMENT pofe a plane, which they call the vifible horizon, to pafs through the eye, and extended every way till it coincide with the heavens ; by this they diftinguifh the pofitions of the moon, itars, &c. either as being in this plane, or as being elevated or deprefted above or below the fame ; alfo another plane, in like manner extended, they fuppofe to pafs through, and at right angles to the former, and thereby dif* tinguifh objects as being to the eaft or weft, or on the right or left hand of the obferver y and many other pofitions relating to the fun, moon, ftars, &c. are afcertained from fuch like imaginary planes, as is made appear in the aftronomical fcience ; fo likewife, in Per- fpedtive, three planes are neceftary for determining the true appear- ances and pofitions of objects. ILLUSTRATION. Plate IV. Fig. 5. Raife up the plane D C, and make the plane O P pafs through the fame, then raife up the plane S T ; bring D, in the plane P D, to coincide with D in the plane D K, and make the plane L R pafs through both the planes OP, ST; again, bring D, in the plane D Y, to coincide with D, and make the plane M N alfo pafs through the faid two planes OP, ST. Now, fuppofe the plane S T to be an original plane, and AGBF to be an objedt feen in that plane by the eye at E, and let L R be a plane pafting through the eye, and cutting the objedt, fo that the line E C, drawn from the eye to the objedt, be perpendicular to the plane containing, that objedt ; let now the plane O P be placed fome- where between the objedt and the eye, and let M F be another plane, interfedting the two other planes at right angles, then will the plane L R £hew what parts of the objedt are above, what below, and what are even with or in the plane of the eye ; the plane M F ftiews what parts thereof are on the right, and what on the left ON PART II. 3i hand iide thereof ; and the plane O P {hews the perfpedive appear- ance or image thereof, when projected by lines drawn through the fame, from the feveral points of the objed to the eye, as feen in that pofition. Now, if LB be confidered as a level or horizontal plane, cutting the picture or plane O P perpendicularly in the line ab or x y, that line is called the horizontal line ; and if the plane M F be confidered as a perpendicular or vertical plane, cutting the fame in the line H I, this line will be called the vertical line; now the vertical and horizontal planes interfed each other in the eye’s axis, or line EC ; whereof that part E c, which is intercepted between the eye and the pidure OP, is called the principal ray, or diftance-of the eye from the pidure. Now, when the three planes O P, L B, M F, are given in pofition, the ho- rizontal and vertical lines are alfo given in petition on the plane of the pidure ; wherefore if E, the place of the eye in the interfedion of the horizontal and vertical planes, be known, that is, if E c, the diftance between the eye and the pidure be given, and the pofition of any point, fuppofe B, be determined from any known diftances thereof from the three before-mentioned planes, its pofition or place on the pidure may from thence be eafily obtained : thus, draw B C perpendicular to the vertical plane M F, and C G perpendicular to the horizontal plane L B, then will thofe lines meafure the diflances of the point B from the faid two planes ; then, drawing BE, C E, G E, the figure B C G E will be a triangular pyramid, having the plane of its bafe perpendicular to the vertical plane, and parallel to the fedion beg, made by the plane of the pidure OP; now, feeing the line C c expreffes the distance of the faid plane of the bafe from the pidure, it is alfo the diftance between the pidure and the point B ; and, becaufe the two pyramids B C G E, b c g E, are fimilar. Y $2 GENERAL COMMENT EC : Ec :: GC : gc, and EC : Ec :: BC : be. COROLLARY. Hence it appears, that as the diftance between the eye and any original objedt is to the diftance of the eye from the pidture, fo is the diftance of the faid objedt, from the horizontal plane, to the diftance of its image on the pidture from the horizontal line. Alfo, as the diftance between the eye and the original objedt is to the diftance between the eye and the pidture, fo is the diftance of that objedt, from the vertical plane, to the diftance of its image on the pidture from the vertical line, &c. And if the original plane be be- tween the eye and the pidture j thus, fuppofe now the plane S T to be the pidture, and O P the original plane, then making the dif- ference between E C and E c, the firft term of the proportions, the 1 reft will be the fame as in the two preceeding analogies. OF THE DIFFERENCE BETWEEN THE TWO WAYS OF CONSIDERING AND APPLYING PLANES, AS USED IN THE OLD AND NEW METHOD OF PERSPECTIVE. Becaufe the pofition, in which the pidture is moft frequently con- fidered by painters, when not concerned with works to be exhibited on cielings, &c. is that of being fituated perpendicular to the ground or plane of the horizon, it has occalioned the horizontal plane or ground itfelf to be confldered as the principal, and oftentimes as the only plane neceftary to be referred to ; thus we fee the horizon- tal plane called the geometrical plane, ground, floor or pavement, &c. and the line, in which the pidture interfedts the faid plane, when thus conftdered, called the ground line ; now, when the ground and pidture are perpendicular to each other, what we call O N P A R T II. 83 the vanilhing line of the ground will then pafs through the center of the pidture, (theorem IX. part II.) and the center of the pidture will, in this cafe, be the center of that vanishing line, (def. XIX.) This vanifhing line the writers on the old perfpedtive call the hori- zontal line, as being the apparent boundary of the vifible horizon confidered as a level plane, as was before obferved in the fcholium to theorem IX. part II. Now, feeing the pofitions of objects, whofe perfpedtive reprefentations are required, are many of them placed on the ground, and elevated perpendicular thereto, as the front and fides of buildings, &c. fuch objedts may therefore be con- ceived, as being in planes perpendicular to the ground ; whence the vanishing lines of thofe planes will, in fuch cafes, be perpendicular to the line thus denominated the horizontal line, and the center of thofe. vanishing lines will alfo be in the faid horizontal line. (cor. I. theorem XVII. part II.) When the pidture, as thus confidered, is fo fituated with refpedt to the original objedts, as that they have one line or lide thereof, as the front of a building, &c. parallel thereto ; and the other lines or lides thereof, as the tide fronts. See. perpen- dicular to the pidture ; then the vanilhing lines of thofe fide fronts, as well as the horizontal line, will pafs through the center of the pidture, which center will be the common center of both thofe vanilhing lines, (cor. II. theorem XVII. part II.) and the vanilhing point of the common interfedtion which fuch planes make with the ground, as well as that of all other lines, in any of thofe planes, parallel to their interfedtion with the ground, will alfo be in the center of the pidture (cor. I. of the fame) ; whence it appears, that the center of the pidture is of confiderable ufe in determining repre- fentations on the pidture, when thus fituated with refpedt to the original plane, and is therefore called the principal point, or point of light ; but, in the method we efpoufe and with to recommend. $4 GENERAL COMMENT the point of fight is that which is defcribed in def. IV. part II, and the axis of the eye being the line which produces the va- nifhing point of all lanes perpendicular to the pidture, it is therefore the line which produces the vanifhing point of all thofe lines which moft frequently occur in this fituation of the pic- ture 3 whence it is called the principal ray, and the length of this line being fet off from the principal point, both ways upon the ho- rizontal line, will give two points, called the points of diftance. Now, in this particular lituation of the pidture, with refpedt to the original objedts, the principal rays being either perpendicular or pa- rallel to the ground or to the pidture 3 they will therefore make the fame angle with the ground line, as the original lines themfelves make with the ground, (cor. IV. theorem IV. part II.) and thofe lines which are parallel to the ground, but not to the pidture, will have their vanifhing points in the horizontal line. (cor. IV. the- orem IX. part II.) Now, when thefe points fall befide the princi- pal point, they call them accidental points in the horizontal lines and when it is required to find the reprefentations of lines, which are neither parallel nor perpendicular to the pidture, nor to the ground, they generally feek their images, by having recourfe to the feats which fuch lines have on the ground, for thereto they moft commonly refer all fuch lines as do not come within the reach of the above defcription, without enquiring after their vanifhing points, or the lines which fhould produce them. But thefe principles are too confined for general ufe, and can ferve only in this particular cir- cumftance, of confidering the pidture as being perpendicular to the ground 3 whereas the practice of this art requires that any other plane, and not the ground only, may be taken as an original plane, and the pidture may have other fituations with refpedt to the ground, or the original objedts, fuch as being parallel thereto, inclining to. ON PART II. or reclining from the fame, 5ec. for the parts of a building, as roofs, &c. are not always either parallel or perpendicular thereto ; and there are alfo cafes which render it neceffary to conlider, in the fame pidture, feveral different original planes, each having its own inter- fering, vanifhing, and the feveral other lines and points peculiar thereto, by means whereof many operations are eafily performed, which would otherwife be extremely tedious and difficult, if not, in fome cafes, impracticable ; but, that we may candidly favour this old fyllem of Perfpedtive, as far as propriety will permit, will allow the horizontal line and ground line to be proper terms for expreffing the vanifhing and interfering lines of the ground, when that is taken for the original plane, but think it abfurd to give thefe appellations to any other plane than the ground ; for the original plane may be fuch, that the vanifhing line thereof fhall interfeCt the picture either in, above or below the horizontal line., or it may be that it crofs or interfeCt the fame, as may be thus exemplified . PLATE VI. Fig. 7 . Bring A to coincide with A, make the planes G P, S L, pafs through each other in the line T R ; make K, in the plane K L, coincide with K in the plane K A or KB, pafs the plane FM through the plane K L, in the line V M, and pafs the plane M H through the plane S L, in the line H L. Now, let the plane G P reprefent an original plane, inclining to the picture S L j let the plane F M be con- fidered as an horizontal plane, paffing .through the eye at E, and in- terfering the pir ure in the line F N, then is the line F N the horizontal line : let now a plane, as M H, pafs through the eye, and be parallel to the faid original plane G P, cutting the pirure in the line H L, then is the plane MH the vanifhing plane of the original G P, (def.VIII. Z 86 GENERAL COMMENT part II.) and H L the vanifhing line of the fame, (def. XI. part II.) confequently the vanifhing line falls above the horizontal line in this cafe ; and if the original plane had declined from the pidture the contrary way, the vanifhing line thereof would then have cut the pidture in a line below the horizontal line, as is eafy to conceive from what has been here fhewn, and when the original plane meets the pidture at right angles, the horizontal and vanifhing lines thereof coincide, or fall together in one and the fame line only ; fuppofing, as is here done, that one fide of the original be parallel to the pic- ture, for, if all the tides thereof are oblique thereto, its vanifhing and horizontal lines will then interfedt each other, as was fhewn in theorem XVII. part II. the vanifhing line H L in that figure be- ing now the horizontal line. The feveral efforts, exclufive of the laft, which have been made in oppofition to the endeavours ufed for conveying and eftablifhing a general knowledge and ufe of this new method of Perfpedtive will,. I hope, fufficiently juftify me in this attempt, to fhew the truth and univerfality thereof j and thus endeavouring to fet learners right in their firft notions of the different methods here alluded to, by warn- ing them of the defedts under which this art will labour, when dis- robed of the improvements added to it, by confidering the pofitions of planes in a general view, and thence making the neceffary pradtical applications fuitable for obtaining the advantages it offers to us. OF THE POSITION OF THE PICTURE WITH RE- SPECT TO THE ORIGINAL OBJECTS. Although the pidture may have any pofition with refpedt to the objedts which are to be reprefented thereon j yet, in order that it may fhew an agreeable and natural reprefentation of the things de- ferred, it fhould be fo placed, that the feveral objedts may appear ON PART II. *7 in their natural lituation, that is, fuch objedts as would really be feen by rays parallel to the horizon, or by rays inclining upwards above the fame, or by others declining downwards below it, may appear on the pidture by rays in fimilar correfpondent pofitions ; for the ground or plane of the horizon being the natural or apparent feat of all vifible objedts, whether terreftrial or celeftial, the mind judging them to be higher or lower, according as they are more or lefs ele- vated above it ; the fituation of all objedts may and ought therefore to be referred to that plane, and have fuch pofitions given them in refpedt thereto, as is agreeable and confiftent with their natural ap- pearance ; wherefore the mod: proper diftindtions, to be made of the different fituations which may be given to the pidture, arife from its polition with refpedt to the ground or plane of the horizon, and are therefore three, perpendicular, parallel, or inclining. OF THE PERPENDICULAR POSITION. The perpendicular fituation of the pidture is heft adapted for re- prefenting the ground itfelf, and the feveral objedts which are eredted thereupon ; and this being a pofition which is parallel to the ufual pofture of the body of the fpedtator wherein the eye is moft ufed to view original objedts, it follows, that the reprefentations hereby formed on the pidture, appear the moft natural and agreeable to the originals themfelves. In this pofition the ichnography of the defign on the ground is confidered as defcribed on a plane perpendicular to the pidture, the vanifhing line whereof paffes through the center of the pidture, (theorem X. part II.) and is parallel to the horizon, and alfo reprefents the fame, is therefore, in this cafe, called the horizontal line ; the elevations of upright objedts are confidered as being in planes perpendicular to the ground, which faid planes may be either perpendicular, parallel or oblique to the pidture, but their 83 GENERAL COMMENT vanilhing lines are always perpendicular to the horizontal line, (cor. II. theorem XVII. part II.) and all lines, which meafure the perpendicular altitudes of objects, thus lituated above the ground, are parallel to the pidture ; but it muft be here obferved, that the ground, defcribed in the pidture, is fuppofed to be level or truly ho- rizontal ; for, if the ground have afcent or defcent, the picture con- tinuing perpendicular to the horizon, the vanilhing line of the ground will not, in either of thofe circumftances, coincide with the horizontal line, but will interfedt the picture either above or below the fame, according as the ground is elevated or deprefled ; but, neverthelefs, the lines, which meafure the perpendicular heights of the original objedts, will yet be parallel to the pidture, as being perpendicular to the horizon, and they, with refpedt to the ground, will reprefent the oblique fupports of the feveral points above that inclining plane. The pofition of the pidture here defcribed is proper to landfcapes, views, buildings, hiltorical pieces, and in general to all pictures where the fpedtator is fuppofed to Hand on the ground, and having the axis of the eye directed parallel thereto. OF THE PARALLEL FOSITION. Here are two cafes, i. When the eye is between the ground and the pidture. 2. When the pidlure is between the eye and the ground. In the firft, the axis of the eye is fuppofed to be turned perpen- dicularly upwards, and confequently the ground, being behind the eye with refpedt to the pidture, can not have any part thereof re- prefented on the pidture ; whence all kinds of terreftrial objedts are here excluded, excepting fnch as may be fuppofed to afcend above the plane of the pidture, fuch as the higher parts of mountains or ON PART II. 89 buildings, or fuch other objedts as may be conceived in the air, but all their uppermoffc furfaces mull neceflarily be concealed from the eye. Pictures or paintings on flat cielings are of this kind, and thofe on cupolas or arched roofs may be reduced hereto, they all agreeing with refpedt to the particular kind of objects proper to be defcribed thereon. A negledt in giving due attention to this particular cir- cumftance, has rendered feveral performances of this fort difagreeable and unnatural ; but a judicious painter will forbear to reprefent on a pidture, in this fituation, the fea, ground, part of the floor of a building, or the upper faces of fteps, &c. notwithflanding examples of fuch inconfiftencies are to be met with. In the fecond cafe, the eye is fuppofed to be at fome height above the pidture, and its axis turned perpendicularly down- wards ; in this circumftance no part of the fky can appear, nor any other thing but what can be fuppofed to lie either upon the ground, or between it and the pidture, fuch as the pavement or floor of a building, the plan of a garden, &c. and thofe parts of the building, or fuch other things which, ftanding upon the ground, do not reach up to the pidture whence this fltuation of the pidture is the moffc confined of any, and feldom ufed but out of curiofity, as on the floor or pavement of a church or dome, which, by an artful difpo- fition of materials differently coloured, may be made to reprefent objedts proper to that fituation, or even the refledted image of the building itfelf, as appearing in a looking glafs, or ftagnant water, to an eye viewing it from fome convenient place, as a gallery, &c. at the top or upper part of the building. In. both the cafes here mentioned, the ichnography of the objedts on the ground is defcribed as on a plane parallel to the pidture, and is therefore fimilar to its original} the planes of the elevations, and A a go GENERAL COMMENT the lines, which meafure the perpendicular altitudes of the objects above the ground, are perpendicular to the picture > wherefore the vanifhing lines of thofe planes pafs through the center of the pic- ture, which center is the vanishing point of all the faid lines of al- titude now, feeing the ground can not interfedt the picture in ei- ther of thofe two cafes, it is not proper to be ufed as the principal original plane, there being no horizontal line in thofe cafes, but any of the upright fides of the building defigned to be reprefented, or any other fubftituted plane perpendicular to the picture, may be ufed for that purpofe, and the objedts required to be reprefented may be thereto referred. OF THE INCLINING POSITION. The various inclinations, under which the pidture may be placed with refpedt to the plane of the horizon, are innumerable, which will have correfponding effedts on the place of the horizontal line and the vanishing point of the perpendiculars to that plane ; as to the kind of objedts proper to be reprefented on a pidture of this fort, they are to be determined from what has been delivered concerning the two pofitions before mentioned, and will differ according as the inclination of the propofed pidture approaches nearer to one or to the other of them } but thefe oblique pofitions are feldom pradtifed, ex- cept neceffity obliges it, from the particular pofition of the wall or place in which the pidture is to be painted or placed, SCHOLIUM. Let the pofition of the pidture be what it will, if it he fuch that the horizontal line can appear therein, it ought, in ffridtnefs, to be fo placed as that the eye, when in the true point of light, may be on a level with that line for then all the reprefentations, formed ON PART II. 91 on the pidture, will appear as being in their true and natural por- tions. This may, however, be difpenfed with, in fome meafure ; for if a picture be drawn, by fuppofing it perpendicular to the ho- rizon, and it Ihould be neceffary to place it fo high, that the axis of the eye can not reach up to a level with the horizontal line, then the true appearance which the pidture ought to exhibit may be pre- ferved, by inclining the pidture forwards, fo that a line may be drawn or conceived to iffue from the eye perpendicularly thereto* and meeting the fame in the center; for, although the ground re- prefented in the pidture does not then appear parallel to the horizon, but as riling upwards, yet the fpedtator, in viewing and confidering the reprefentations according to the relations which they have one to the other, without referring to their true pofitions, or his own real podure, with refpedt to other objedts which are out of the pic- ture, will readily pafs over the defedt of this deviation from the true polition, fo long as the whole pidture is every where confident with itfelf ; and the pidture being in. this fituation will, in fome fort, ex- hibit a refemblance of the appearances which the like objedts- would fhew, if refledted by a well-polilhed plane mirror, or common looking-glafs r placed inclining to the horizon, which makes the refledted ground appear as being elevated or deprefled with refpedt to the real ground; but the refledtions of all other objedts preferve the fame relations with refpedt to the refledted ground, which the real objedts have to the ground itfelf : for, let the fpedtator but only imagine that he dands perpendicular to this refledted ground plane,, and then all will appear as having a natural fituation with refpedt to the ground on which they are fuppofed to be placed. But let it be obferved, that this liberty be not ufed too freely ; it is allowable only to detached pieces, which are to be placed where conveniency fuits ; for, fuch as are exprefsly defigned for a certain 92 GENERAL COMMENT fixed fituation, as a wall, deling. See. do not admit of this variation from the true pofition, by reafon fuch pieces have generally a more immediate relation to the reft of the building itfelf. Suppofe, for example, that a pidure, defigned for a pofition which is perpendi- cular to the horizon, were placed in one that is parallel thereto, as a deling, &c. the fpedator will not then be able to conceive it as exhibiting reprefentations of objects according to their natural ap- pearance, without imagining himfelf to be ftanding ered on a plane which appears perpendicular to the horizon, a conception which re- quires fuch a force of imagination as is not eafily acquired, never- thelefs, we have fome examples which render this extraordinary ef- fort of the mind abfolutely neceflary, before the pidure can pofiibly have its due effed. OF THE DISTANCE OF THE EYE FROM THE PICTURE. From what has been advanced on the nature of ftereographical projedions, it is manifeft that the reprefentations, formed on the pidure, can not appear to the fpedator exadly as they ought to do, if the eye be not in the true point of fight ; whence it is evident that they Ihould, in ftridnefs, be always viewed from that point ; therefore a true determination of the pofition of the eye is fuch an eflential requifite in this art, that no examples of authority whatever fhould induce us to difregard it, or be too carelefs about the choice of it. In a room or place that is bounded on all fides, whatever be the pofition of the pidure, fuch a diftance muft be taken for the eye as is within reach ; thus, fuppofe the pidure were to be on the cieling, the diftance is then determined by the height of the room, dedud- ing therefrom the height of the fpedator’ s eye from the floor ; but. ON PART II. 93 when the place is not thus circumfcribed, the artift is more at liberty to make fuch a choice for this diftanee as fhall be moil fuitable to the grand defign of making the objects appear to the beft advan- tage, and in their natural fituations. Now, when the picture is in a perpendicular pofiti-on, the artift has generally the greateft room for exercifing himfelf in making his choice of a proper diftance to work from ; let us therefore briefly enquire into the effects which are produced from different diftances of a pidture in this pofition, for, what we fhall learn from hence, ffnay be eafily applied to any other position of the pidture whatever. In this enquiry I fuppofe the height of the eye to be given or taken at pleafure, and remaining conftantly the fame. The eye thus abftradtedly confidered, being placed at different diftances from the picture, will produce corref- pondent changes in the ichnography and orthography of the original objects ; for the ichnography thereof will be affedted either with re- fpedt to the quantity of the whole fpace which it contains, or the proportions of that fpace taken up by its different parts, according as they are nearer to or farther from the pidture, and alfo as to the apparent breadths of thofe parts. Now here are two things which ought to be carefully guarded againft : firft, that the apparent de- creafe of equal parts in the ichnography may not be too fudden, as they become more remote. Secondly, that fuch remote parts as are to be diftindtly expreffed, may not approach too near the vanifhing or horizontal line ; for if they do, their images will be too fmall, and too much crouded together, both which contribute to render them indiftindt, and the pidture will be thereby defedtive ; which imperfedtions are avoided by making a right choice for the place of the eye, or the diftance thereof from the pidture. Bb 94 GENERAL COMMENT LEMMA I. The diftance between the image of any point given in an original plane, and the vanifhing line of that plane, is to the vertical line thereof, or to the director, as the line which produces the vanifhing point of a line drawn in the original plane through the faid given point, is to the diftance between the faid point and the directing line. ILLUSTRATION. Plate XI. Fig. 13. Bring D, in the plane DB , to coincide with D in the plane N B 5 make S, in the plane S H, coincide with S in the plane N B ; make the plane QJI pafs through the plane S H in the line F G, and pafs the plane Q Z through the plane S H, in the line line H L. Let A be the point given in the original plane N B ; through the fame draw the line A B parallel to the interfering line SI; let C^Z be the vanifhing plane of the original plane N B j from the di- recting point D to any point, as K, in the line A B, draw the line D K, and parallel thereto, from the point of fight, draw the line E C ; then is C the vanifhing point of the line D K, and E C its diftance, P C the vertical line, E D the director, M N the directing line, &c. draw alfo the lines E A, E B, E K : now A B, being pa- rallel to S I, its image a b is alfo parallel to S I ; and K being a point in the line A B, and k the image thereof, the line a b is that which muft contain the image of A, as has been before fhewn ; but the diftance of a, from the vanifhing line PI L, being equal to k C, and the diftance of A from M N is equal to D K, becaufe E C and P C remain conftantly the fame, it follows, that in what point foever of A B the point A be taken, the diftance between the image of that point and the vanifhing line will be to P C as E C is to the diftance between the given point and the directing line. ON PART II. 95 COROLLARY I. The distance between the image of any point in an original plane and the vanifhing line remains the fame wherever the point of fight be taken in the parallel of the eye ; for A B, a b and X Q^being all in the fame plane, a line drawn from any point in X Qjo any point in A B, muft cut the pidture fomewhere in the line F G, and all points in that line are equally diflant from the vanifhing line H L. COROLLARY II. If the height of the eye be increafed or diminifhed, the eye being in the fame directing plane, the diflance between the image of an original point and the vanifhing line will be increafed or diminifhed in the fame proportion. COROLLARY III. If the height of the eye remain conftantly the fame, and the di- flance thereof be varied, the diflance of the image from the vanifh- ing line will vary accordingly. LEMMA II. The difference between the image of a nearer part of an original line and the image of the part next beyond it, is greater than the difference between this lafl image and that of the next fucceeding part, and fo on of others. PLATE VIII. Fig. 9. For, let O be the point of fight, the parts A K, K G, G Y, being equal, qn:nm::DG:DA, by cor. III. theorem XIX. and for the fame reafon nm:my::DY :DK, alfo q n — n m ; nm :: DG — D A = A G : DA, GENERAL COMM E N T n m — m y : my :: D Y — D K = K Y or A G : D K. But A G is greater in proportion to D A than it is to D K, therefore qn- — n m is alfo greater in proportion to nm than nm — m y is to m y ; and n m being greater than m y, therefore is q n — nm fo much the more greater than n m — ■ m y. COROLLARY I. The more remote that the equal parts of the original line are from the direding point, the nearer do the images of thofe parts approach to an equality, by reafon their differences are accordingly diminilhed. COROLLARY II. The greater the di fiance of the eye, the more nearly equal do the images become of any two adjacent parts of the original line. For, fuppofe the eye removed from O to E, then, becaufe A K and K G are equal, the image of A K is to that of KG from the point of light O, as N G to N A, and, from the point of fight E, they areas DG to DA; but D G is lefs in comparifon to D A, than N G is to N A ; therefore the image of A K is lefs in proportion to that of K G, from the point E, than it is from the point O : but a k, kg, the images of A K and K G, are in the fame ratio one to the other wherever the eye be placed in the fame directing plane ; therefore, whether the eye be placed at E, or any where elfe, in a plane palling through E parallel to the pidure, the images of A K, K G, feen from thence, will be nearer equal to each other than they would be if formed by viewing them from O. COROLLARY III. If, from the interfeding point F of any line FY, feveral dillances, FA, A K, KG, GY, &c. be taken, each equal to N F, the ON PART II. 97 point N being the directing point, q C, n C, m C, y C, the images of thofe parts, will be in proportion to ON, or F C, as the follow- ing fractions, 4> 4> 4* 4, 4, &c. and the images of the parts them- felves will be as 4 , 4 ,Vt, At, -A, 6cc. of F C, the differences of the denominators of each of thefe lafl fra&ions increafing from o in this feries 2, 4 , 6, 8, 10, 6rc. In the fimilar triangles ON A, qFA, it is N A : F A : : ON = FC : Fq, but N A — 2 F A by hypothefis ; whence F C = 2 F q, confequently Fq=qC = 4 FC. Again, in the fimilar triangles O N K, n F K, we have NR:FK::ON=FC:nF, but F K = r N K j whence n F = } F C, confequently n C = 4 F C, and in the fame manner it may be proved, that m C is 4 F C, and y C = 4 F C. Now, if we call FC=i, qC = 4 , nC=i, mC=4 yC== 4 > the image n q, which is the difference between q C and n C, will be 4, nm = TT, my=Ar, 6cc. the difference of the denominators of which fractions increafe from o in the feries 2, 4, 6, 8, &c. Suppofe now the parallelogram S Y to reprefent part of the ichno- graphy of the original defign, fubdivided into fmaller ones as in the figure, N F being chofen for the fpe&ator’s line of ftation, which determines the pofition of the ichnography with refpedt to the pic- ture ; take any height at pleafure, as N O, for the height of the eye, fuppofed now in O, and C O its diftance from the picture ; then, if the images q, n, m, y, of the points or divifions A, K, G, Y, and confequently the diflances of the images of the crofs divifions of the ichnography, pafhng through thofe points, fhall appear too unequal, or fall off too fuddenly 5 or that y, the image of Y, the moff remote Cc 9 S GENERAL COMMENT divifion, appears too far up in the picture, or too near the vanifhing point C, the fault may be removed by inlarging the diflance of the eye, and fixing it in fome other point of the line C O produced, fuppofe at E, &c. Now, when C O, the diflance of the eye, is taken equal to F A, the diflance of the firfl divifion from the pic- ture, the heights of the images q, n, m, y, above the interfering line, will be 4, r> 4, t> &c. of F C, the depth of the original plane, their refpedtive diflances from the vanifhing line being 4 , 4 » 4 -» &c. of that depth, and the images of the parts F A, A K, K G, G Y, &c. will be 4 , 4 , -A, A, &c. of that fame depth j but if, at this diflance of the eye, the parts near the pidture occupy thereon a greater fpace than is agreeable with refpedt to what is pofTefTed by the other more diflant parts, or that the objedts, fituated between G and Y, appear too fmall and crouded, the diflance of the eye may be taken equaL to the two firfl parts F A and A K together, which will bring the images of the divifions, next fucceeding thofe here mentioned, down to the fame points on the pidture which they themfelves were in before ; and thus may there be found fuch a diflance for the eye, as fhall make any point in the ichnography to appear at any propofed height on the pidture, within the limits of the depth of the original plane but, if it were required to reprefent any particular part of the ichnography, fuppofe that between G and Y, fuch as fhall oc- cupy the greatefl fpace in depth pofiible, the diflance of the eye mufl then be a mean proportional between the two parts F A and F Y for, at that diflance, the fpace A Y will occupy a larger field in depth than it would do at any other diflance of the eye in the line C O, produced either nearer to or farther from the pidture. Hence it appears, that the diflance of the eye is that which prin- cipally commands the diflance of the mofl remote ground that can be defcribed with any tolerable diflindlnefs ; for, if the ground be- ON PART II. 99 yond the pidture be divided into fpaces equal to the diftance of the eye, the feats of all fuch objedts as are contained in the ninth fpace from the pidture, can then occupy no more than the one-ninetieth part of the depth of the original plane, and the image of the extre- mity of that fpace being but one-tenth part of that depth diftant from the vanifhing line, that one-tenth part is therefore the whole fpace that is left wherein all fpaces beyond the ninth can poflibly be reprefented on the pidture, as appears from what has been premifed, for there being now nine divifions, the firft fradtions are >,13456 7 8 9 . -a» -rf T> s-t T > T* itf * the fecond r» t> r> t# tV » and 9 x io — 90. EXAMPLE. Suppofe the height of the eye to be 5 feet, and its diftance from the pidture 20 feet. Then the feats of all fuch objedts as are between 160 and 180 feet diftance from the pidture, can poffefs no more fpace than one- ninetieth part of 5 feet, or two-thirds of an inch, and this reach- ing within 6 inches of the vanifhing line, or, which is the fame, the tenth part of 5 feet, it follows, that the feats of all objedts on the ground, from 180 feet to any affignable diftance, muft be con- fined within that 6 inches, fo that, even at fo great a diftance of the eye as 20 feet, the feats of objedts, 20 feet in depth at the diftance of 60 or 70 yards, can be reprefented but imperfedtly, and all be- yond that diftance will be faint and confufed ; and, if the diftance of the eye be leffened, the fpace which can be diftindtly defcribed will be proportion ably decreafed. There is alfo another effedt which the diftance of' the eye produces upon the ichnography j for, as by enlarging the diftance the images of the feveral crofs divifions are 100 GENERAL COMMENT brought lower down towards the interfering line, fo their apparent meafures are proportionally increafed, and confequently the lines, which meafure the breadths of objedts parallel to the pidture, ap- pear longer, at the fame time that thofe, which meafure their depths, become fhorter ; and, as the apparent breadths of objedts are thus encreafed, fo alfo are their apparent heights or elevations, thofe heights being fuppofed parallel to the pidture ; fo that, upon the whole, fuppofing the pidture and original objedts as retaining their pofition, if the diftance of the eye be enlarged, their apparent breadths and heights, or thofe tides thereof which are parallel to the pidture, will be thereby encreafed, but their depths will be di- minifhed. Now, although we may, from what has been faid, ea- iily find fuch a distance for the eye as is fuitable for reprefenting the objedts in the moft advantageous fituation on the pidture, fo as that the remote# parts may not appear advanced too near the horizon, nor their depths decreafe too fuddenly, but that a due and agreeable proportion between their apparent heights, breadths and depths may be preferved in the reprefentations formed on the pidture, yet, let it be obferved, that what has. been advanced on this fubjedt, is to be underftood as relating to a pidture, formed from a fcak, equal to the true meafures of the original objedts, that is, when the objedts adjoining the pidture are reprefented bigas the life ; but, as this fcale may be contradted to any dimenfons at pleafure, it follows, that a pidture may be rendered capable of reprefenting much greater dis- tances than the above-mentioned, and yet the diftance of the true point of fight may be greatly leffened, thus, PLATE XL Fig. i 3 . Make D, in the plane D K coincide with D in the plane N B, and pafs the plane S H through the fame 5 make B, in the plane ON PART II. ioi B R, coincide with B in the plane N B ; then make the plane Q^Z pafs through both the planes S H and BR. Now let A B M N reprefent the ground plane on which the ich- nography of the intended objedts is fuppofed to be formed in its true dimenfions, and let A B be the interfedtion of the ground with the fuppofed pidture A B Z R, and confequently the neared: part of the ■ground propofed to be reprefented on the pidture ; now, if D K or E O be the diftance of the eye neceffary for rendering the reprefen- tations fufficiently diftindt, and this be too great for the place where the pidture is to be fixed, it mufl be drawn from a leffer fcale, which may be thus determined. Between the diredting point D and the interfedting line A B draw in the original plane a line, as S I, parallel to A B, on that line eredt a plane parallel to the plane or pidture A B Z R, and upon the plane thus eredted find the perfpedtive reprefentation of the given plane, the eye being at E, which is here expreffed by abzrj then, if a bz r be taken for the pidture, E C for the diftance, and E d for the height of the eye, and that the true meafures of the objedts to be repre- fented be all reduced to the fame proportion as a b bears to A B, then all objedts, defcribed in the pidture a b z r, with thofe propor- tional meafures with the 1 aft- mentioned diftance and height of the eye, will be exadtly fimilar to a pidture of the fame objedts, drawn on the plane A B Z R, according to their true or real dimenfions, having E O and E D for the diftance and height of the eye, but as much leffer as is the proportion affigned between a b and A B. Thus may the fcale, and confequently the diftance of the eye, be reduced to any convenient meafure to fuit the fituation in which the pidture is to be placed. The fpace between a b and A B is entirely hid in this cafe, no part thereof being to have a place on the pidture ; and, as A B is the true D d 102 GENERAL COMMENT interfedting line of the ground on which the true meafures of the objedts Should be taken, if that were the place of the pidture ; fo a b, being the reprefentation of A B, is therefore to be considered as; the interfedting line of the pidture abzr, on which the proportional meafures muft be ufed according to the diminished fcale. If the pidture a b z r were continued down to S I, its interfedting line with the original plane, the reprefentation of the fpace between S I and a b will fall between S I and a b, and the objedts Situated in that fpace may be defcribed either by fetting off their true meafures on S I as the interfedting line, or by their proportional meafures on a b according to the diminished fcale j for, whether the one or the other of thefe meafures be ufed on thofe refpedtive lines, fueh parts of the reprefentation as come within abzr will, in either cafe, be the fame j and thus may a pidture be formed that Shall reprefent a very large and extenSive profpedt, the eye being at a moderate distance only, by confidering the pidture a b z r as a window in a fecond or third floor, and the fpedtator as viewing the objedts ^through the fame ; for, in proportion to the height of a b above the ground, and the distance E C taken for the eye, the vifible interfedting line A B of the ground, where the defcription begins, may be removed to any required distance, which will have a correfponding effedt on the fpace which is poShble to be reprefented diitindtly. As the pidture abz r is a kind of miniature reprefentation of the pidture ABZR, the neareSt objedts which are reprefented thereon muSt be lefs than the life, they being fuppofed equal to the life at A B : but, if ob- jedts were reprefented according tp- their real dimensions on the plane abzr, and thence projedted upon the plane ABZR, they would there appear bigger than the life, which is proper to be done when the distance of the eye is neceSTarily very great, as in the cafe of pic- tures formed on the cieling of a church, &c. where the height i3 ON PART II. 103 very great, and the pi&ure defigned to exhibit the reprefentations as being nearer than they really are, which it will do, if to this circum- stance be added proper and fufficient Strength of colour, for that alfo ought now to be encreafed beyond that of real nature, in order to pro- duce the effeCt here mentioned. In Short, if the reprefentations be form- ed from a fcale lefs, equal to, or greater than the real dimenfions of the original objects, they will accordingly appear to the Spectator as being farther from, at an equal distance, or nearer than they really are when duly coloured according to thofe feveral different circum- stances : but, notwithstanding this, it is not neceffary, in miniature paintings, to defcribe the figures in that faintnefs of colour with which the originals would really appear when fo far distant as re- duces them apparently to fuch a fize, by means of the angle under which the vifual rays do then exhibit them, for a greater distinction of parts and vivacity of colour may be here allowed, taking care only that a due diminution be obferved amongft the feveral objects repre- sented on the picture with reSpeCt one to the other ; for a picture thus drawn may flill be looked upon as fuch a reprefen tation of the objeCts as would be produced by viewing them through a concave glafs, which leffens their apparent magnitudes, but does not take away the distinction of their parts nor their Strength of colour in Co great a proportion. ' OF THE HEIGHT OF THE EYE. It appears, from what has been Shewn, that the height of the eye determines the depth of the original plane, and is always equal thereto, consequently is that which gives bounds to the Space which muff: contain the ichnography of all objects on the original plane that can be reprefented on the picture ; that the image of a line, in a plane parallel to the picture, is of the fame length wherever the f 104 GENERAL COMMENT eye be placed in the directing plane ; therefore the elevating or de- prefling the point of fight will produce no difference in the apparent heights and breadths of objects, or fuch of their dimenlions as are parallel to the pi&ure, for they remain of the fame length, let the height of the eye be what it will, fo long as its didance from the pidture remains the fame ; alfo, that the images of any determinate parts of an original line, which inclines to the pidture, will have the fame ratio to each other at all different flations of the eye taken in the diredting plane, and therefore the altering its height, without changing its didance, can have no influence on the apparent decreafe of the equal parts of the lines which meafure the depths or didances of the objedts, by reafon they have dill the fame proportion one to another, let the height of the eye be what it will, and are effedted only as to their being greater or lefs in proportion to the height which is given to the eye. DEFINITION. A line or meafure, taken upon any line parallel to the interfering line, having the fame ratio to a given original line as any determi- nate part of that parallel bears to its original, is called the propor- tional measure of the original line on that parallel. LEMMA. If any determinate part of the indefinite image of an original line inclining to the pidture be taken, and a proportional meafure thereof be found upon a line, drawn parallel to the interfering line, through that extreme thereof which is neared the interfering line, it will be as the complement of the propofed image is either equal, greater or lefs than the line producing its vanifliing point, fo will the affumed II. & ON PART i°5 part of that image be accordingly equal, greater, or lefs than its proportional meafure. PLATE X. Fig. n. Let D K be fuppofed the indefinite image of an original line in- clining to the pidture, and fuppofe the line producing its vanishing point to be equal to FI K j let g m be the determinate part, and g f its proportional meafure taken on a line parallel to the interfering, line, and pafiing through g, its nearefi: extreme to that line ; then if m K, the complement of the afiumed part, be equal to, greater, or lefs than H K, g m will be accordingly equal, greater, or lefs than its proportional meafure g f. For feeing the triangles FI Km, mg f, are fimilar, it will be mK:KH::gm:gfj therefore, if m K be equal to K H, or whether it be greater or lefier, the fame will g m be with refpedt to g f. PLATE VIII. Fig. 9. Suppofe now O N the height of the eye, to be taken fo great in refpedt of its diftance O C, that the image q n of any part of an origi- nal and inclining line fhall fall at a greater diflance from its vanishing point, fuppofe C, than the length of the line O C which produces its vanifhing point, then will the image n q be greater than the pro- portional meafure of its correfponding original line, when meafured on a line parallel to the pidture pafiing through its nearefi: extre- mity ; but, as it mufi: feem unnatural that the image of a line, in- clining to the pidture, fhould appear equal to or greater than the image of a line, of the fame length, parallel to the pidture, and op- pofed diredtly to the eye, at a diftance no greater than the nearefi: extremity of the faid inclining line, fuch an appearance mult needs E e 106 GENERAL COMMENT be deformed and difagreeable, if ought therefore to be avoided, which it may, by taking fuch a height for the eye, that the inde- finite image of any line in the original plane jfhall not be longer than the line which produces the vanifhing point of that line, for then,_ whatever part of the original comes to be defcribed within thofe bounds, its image will always be lefs than its proportional meafure j whence it plainly appears, that the height of the eye has an immediate dependance on its diftance, which it ought neither to equal nor exceed, but may be lefs at pleafure, according to the na- ture of the defign, and defire of the artiffc for obtaining more or lefs room for the depths of his objects, fo as fhall procure an agreeable proportion between their ichnography and elevations. Nor is the place of the eye always to be confidered as being the fame with that of a Ipedtator Handing upon the original plane, for the eye may be conceived as placed on 'an eminence, at a confiderable height above the original plane, fuch as an hill, the upper part of a building, &c, as when objects, fituated below, in low grounds, or valleys, 5cc. are to be defcribed. PLATE XI. Fig. 13. Bring D, in the plane D A B, to coincide with D in the plane N B j make B, in the plane A Z, coincide with B in the plane N B ; make the plane S H pafs through the plane D A B in the line F G, and make the plane QJZ pafs through both the planes S H and A Z. Let now A B M N be the original plane containing the objects to be reprefen ted, A B the neareft limits of the ground that is to be defcribed, then may the fpedtator be placed in d, or any other point of the line E D, provided that the picture can be fo placed as that the eye, when thus proportionally railed, may be on a level with the horizontal line Z R. ON PART II. toy OF THE SIZE OF TH.E PICTURE. What has been premifed with refpeCt to limiting the greatefi height of the eye relates equally to the fize of the picture ; for, in both, the principal inconvenience, necefiary to be guarded againfi, is the excefs of the image of any part of an original line above its proportional meafure. PLATE X. Fig. 12. Let C be the center of the picture, E C the difiance of the eye 5 from C, with the radius E C, defcribe a circle, then if that circle, or any rectilinear figure, D F L P infcribed therein, be made the bounds of the picture, the image of no line whatever, which is per- pendicular to the picture, can, within thofe limits, be extended far- ther than the radius, which is the difiance of the eye from the pic- ture, and therefore, when the principal lines of depth contained in the defign are perpendicular to the picture, its fize may be thus de- termined. But, when the principal lines of depth incline to the picture, as buildings, having an oblique pofition, &c. then the vani£hing points of the ichnography of their inclining fides mufi be taken as centers, and the refpeCtive diftances of thofe vanifiiing points as radii, and circles defcribed therewith, which, by their mutual interfeCtions, will mark out a fpace, beyond which no part of the images of thofe inclining fides ought to extend. Thus, fuppofe V, v, were the vanifhing points of the ichnography of the fides of a building, then taking thofe points for centers with the diftance V E or v E, draw the arcs E S e, E T e, interfering each other in E and e, and the fpace included thereby will be that in which the reprefentation of the propofed building ought to be con- fined. 108 GENERAL COMMENT : Thefe rules are to be confidered as more immediately relating to t he reprefen tation of pieces of architecture and animal figures, or fuch other objeCts as are of a certain known and determinate form, fo as not to fuffer their reprefentations to be projected too far diftant from the center of the picture ; for, as to fuch other objeCts as have un- certain, variable and indeterminate forms, as clouds, hills, moun- tains, and fuch like, a greater latitude is allowable : for, although what has been before afferted, concerning the necefiity of viewing a picture from the exaCt point of fight, be true, wherever that point be taken, yet, if the diffance of that point be too finall for the fize of the picture, the images, of objeCts near the fides thereof will be extended to great lengths, and take up more fpace on the picture, than the objeCts themfelves would do if feen direCtly, and when a picture thus drawn is feen from a different point,, the images, of thofe objeCts will appear deformed or difforted, and be difagreeable to the eye ; we fhould therefore adapt the fize of the picture to the diffance of the eye, fo that none of the reprefentations may appear monffrous or unnatural, wherever the eye be placed to view it ^ for, notwith- ffanding that it be from the true point of fight only that a picture can appear exaCtly as it ought to do, yet, when the diffance of the eye is comparatively large with refpeCt to the fize of the picture, fo that the greateff dimenfion of the picture may be feen under an angle of ninety degrees or lefs, any little deviation of the eye from its true place will not have fo fenfible an effeCt on the appearance of the picture, as when the diffance is fmaller, or the picture of a greater extent ; and feeing pictures are frequently, if not always, placed in fuch pofitions, as that they may be viewed from different points $ they ought therefore to be fo drawn that, in any of the points fronting them, they may appear as little difagreeable to the eye as is poffible, and if nothing contained in them appears remarkably de« ON PART II. 109 formed, little variations from the ftriCt appearances which they ought to exhibit, will be readily excufed, and the defeCt fupplied by the imagination of the obferver. OF THE CONSEQUENCES WHICH ATTEND THE VIEWING OF A PICTURE FROM ANY OTHER THAN THE TRUE POINT OF SIGHT. It has been before obferved, that the reprefentations formed on the picture can not appear ftriCtly as they ought to do, if the fpeCtator be not in the true point of fight j and the great perfection of this art confifts in forming the picture fuch, as fhall excite in the mind of the obferver the fame fenfations as would arife therein, by feeing the real objeCts themfelves, which is not indeed to be expeCted barely from geometrical inveftigations ; recourfe mult be had to the pain- ter’s fkill in the art of colouring, for rendering the illufions thus truly perfeCt : but, neverthelefs, it is of the utmofl import- ance to determine rightly the pofitions which the feveral lines and points, terminating the original objeCts, mult have on the picture, fo as to enable them, when duly coloured, to produce the effeCt wilhed for by the artifl ; whence the arts of perfpeCtive and colour- ing fhould go hand in hand, and afford reciprocal afliftance to each other. From whatever point it be that a picture is viewed, the reprefen- tations will appear as the images of objeCts correfponding to that pofition j whence the fame reprefentations may be made to allude, by changing theplace of the eye, tQ different original objeCts, confequently can not be a true exhibition of the objeCts there depicted, whofe real forms, magnitudes and pofitions ought to be invariably preferred, as feen from one certain point, that is to fay, the true point of fight ; now every different pofition of the eye produces correfponding altera- F f 1X0 GENERAL COMMENT iions in the apparent places and other affedions of the originals, and difpofes the mind to judge of them differently from what they, re- ally are in their natural Rate ; for different pofitions of the eye, taken in the line which is the parallel pf the eye, belonging to the plane containing the original objeds, will produce alterations of one fort, while others, taken in the eye’s diredior, occafion changes of another kind ; and the placing the eye nearer to or farther from the pidure than it ought to be, caufes variations different from either, as we fhall now endeavour to make appear, by a few general obfer- vations derived from what has been before delivered. And fhfl, let us fuppofe the eye of the fpedator to be placed fome- where in the parallel of the eye out of the true point of fight, then will the center of the vanifhing line appear to be as far diflan t from the true center as is the fuppofed place of the eye from the true place, and on the fame fi de thereof ; but the vanifhing line of the plane which belongs to that parallel of the eye, as alfo the height of the eye, will remain without alteration, and thofe dimenfibns of objeds, which are parallel to the pidure, will preferve their true magnitudes and pofitions : if the original plane be horizontal, it will flill appear fo ; the elevations of objeds, fituated thereon, will con- tinue to appear perpendicular to it, and the fame judgment will be paffed on all lines in the ichnography which are parallel to the pic- ture, as would be done from the true point of fight ; but the angles, formed by lines which incline, the planes railed upon them, and the apparent meafures of their feveral parts, will be different from what they really lhould be. P'or the true center of the vanishing line is the vanifhing point of the images of all lines in the ichnography which are perpendicular to the interfeding line, and when the ap- parent place thereof is changed, that true center then becomes the vanifhing point of fuch original lines as incline to the line of Ration, ON PART II, in or line which determines the portion of the ichnography, fo as to form therewith an angle, whofe tangent is the diftance between the -real and apparent centers, the diftance of the eye being made radius, and a correfpondent change will enfue in the appearance of all other original lines, whofe images tend to any other vaniftring points, and ■confequently in the apparent inclinations of any elevated planes, whofe vaniftiing lines pafs through thofe points, PLATE IX. Fig. io, Suppofe now the eye to be placed in e, and draw eV perpendicular to the vaniftiing line H L, then will V be the apparent center j for, in whatever point the eye be placed, a line drawn from thence per- pendicular to the picture, the point wherein that line cuts it, will be judged to be its center ; and x C, the whole image of an original line, perpendicular to the interfering line, will appear as inclining to the line of ftation, which is parallel to E C, under the angle Ve C now, if e V or EC be aftumed the radius, then is V C the tangent of the faid angle V e C. And if e V or E C were the vanifti- ing line of a plane palling through x C perpendicular to the picture and original plane, it will appear, from the ftation e, as a plane perpendicular indeed to the original plane, but it will feem as incli- ning to the picture in the angle eC V : hence it is that a plane, which would appear perpendicular to the picture when feen by an eye at E, will, as the eye is removed in the line E X, feem to in- cline more and more towards the picture, and always the contrary way to that in which the eye is tranftated from one place to the other ; for imftance, the plane feems to incline towards Z, as the eye moves from E towards X, and when the eye moves from E towards Z, then i.t feems to incline in like manner towards X. 112 GENERAL COMMENT If any two vanishing points, H and h, be taken, and there be drawn from the true point of fight the lines EH, Eh, they will fubtend the true vifual angle under which thofe two points will ap- pear j alfo drawing the lines e H, eh, they will fubtend the angle under which the fame points will appear from the place or Ration e, and confequently all original lines tending to thofe points will appear at e, under the angle H e h, which will be equal, greater or lefs than the true angle, according where the points H and h happen to meet the vanishing line H L, with refpett to the real and apparent centers C and V. For if, upon the line H h, fuch a fegment of a circle be defcribed on the vanifhing plane, which, palling through E, contains the true vifual angle H E h, (33. 3.) it is plain that, in whatever point of its circumference the eye be placed, the angle fubtended by lines drawn from thence to the points H and h, will appear the fame ; therefore if the eye be at q, the point where it cuts the parallel of the eye, the apparent angle H q h will be equal to the true angle ; but, if the eye be in or out of the circle bounded thereby, the apparent angle will be accordingly greater or lefs than the true angle H E h. Now, as the line x C appears at the point e to incline to the pic- ture, the line e C becomes that which apparently produces its va- nilhing point 3 and therefore, if x C be any how divided, fuppofe in the points n, c, &c. the apparent meafures of the parts x n, n c, &c. will be increafed or diminifhed in the fame proportion which the apparent line e C has to the true line E C, which really produces the vanifhing point of that line ; fo that, inftead of reprefenting their true meafures x I, I J, from the place e, the mind will judge them as being equal to x w, w 1 . And feeing that the apparent line, pro- ducing the vanifhing point, may be equal, greater or lefs than the true one, according to the pofition which any vanilhing point in the ON PART IL ”1 line H L has to the apparent center V, excepting only the point C, for the apparent line belonging thereto can never be lefs than E C» it follows, that the images of the parts of any inclining original line may accordingly reprefent parts equal, greater or lefs than their true originals ; but, notwithllanding this, the perpendicular didance between the pidure and the points n, c, Sec. will appear true, fo long as the eye continues in its parallel X Z. The fame holds good alfo with refped to any elevated plane, whofe vanifhing line is E C, or a line parallel thereto ; for, although its center is not varied while the eye moves in the line X Z, yet the line, which apparently deter- mines its vanifhing point, becomes equal to e C, and as that is greater than the true one, a correfponding effed will be thereby wrought on the apparent lines belonging to all other vanifhing points in that line, as alfo on the apparent angles and didances from the interfering line of the originals of any points in that elevated plane, though not on their perpendicular didances from the pidure. The eye being placed in any point of the diredor of any original plane, thofe dimen lions of objeds, which are parallel to the pidurej appear unvaried both with refped to their magnitude and didance as .above-mentioned ; but the apparent place of the horizon will be changed, and feem as being above or below the true horizontal line, according as the eye is higher or lower than the true point of light ; the original plane will accordingly appear as raifed above or funk below the horizon, and the perpendicular fupports. of all original points appear as oblique fupports ; alfo the apparent line, producing the vanilhing point, will be increafed, by altering the apparent place of the center of the pidure, the originals of the parts of all inclining original lines will be judged proportionably greater than they really are, and the objeds will thereby feem to occupy more fpace in depth, on this inclining plane, than they adually do, though their perpen-? Gg GENERAL COMMENT 114 dicular diflances from the picture are not affected. Now, feeing the. eye is fuppofed always in the fame director, the vertical line of the original plane will remain unaltered, and confequently all elevated planes will appear inclining to the picture under the fame angles as. they would do from the true point of fight, but will not appear per- pendicular to the original plane, but as the planes of the oblique feats of lines on that plane, and thofe lines in the planes of elevation which fhould appear parallel to the horizon, fuch as the ranges of windows, cornices, or other members in buildings of architecture, by the apparent change of their centers, will appear elevated or de- preffed above or below the horizon, according as the height of the eye is greater or lefs than that of the true point of fight. Alfo the apparent angles, formed by inclining lines in thofe planes, and the apparent magnitudes of their parts, will be in like manner affeCted from the apparent changes in the centers, made by varying the lines which apparently produce the vanifhing- points thereof. But fo long as the eye continues in the fame director, the perpendicular diftances betwen the picture and all the points in thofe planes con- tinue to appear as they fhould do, by reafon the director is, in this cafe, the parallel of the eye with refpeCt to the faid elevated planes.. But, if the eye be placed any where in the line producing the va- nifhing point of any original line, farther from or nearer to the pic- ture than the true point of fight, the center of the picture, hori- zontal and vertical lines will retain their true appearance, and all planes of elevation will continue to appear perpendicular to the original plane ; but the diftance of the eye, being thus varied, will have an effeCt upon the apparent magnitudes of the parts of all inclining original lines, and confequently on the apparent diftances of thofe parts from the picture, and likewife on the apparent angles fub- tended by lines joining the vanifhing points of lines in that plane, ON PART II. 1 15 for they will appear greater or lefs than they truly ought, according as the diltance of the eye is lelfened or increafed ; alfo the apparent inclinations of all elevated planes, and lines therein, with refpedt. to each other and the pidture, will be affedted, except only the right angle contained by the parallels and perpendiculars to the interfedting lines of the feveral planes, for they will all continue to appear perpendi- cular to each other, wherever the eye be placed in the line here mentioned, becaufe the center of the pidture continues the fame, the centers of all vanifhing lines in the pidture will remain unvaried ; and the magnitudes of the parts of thofe parallels, and of all other lines which meafure the dimenfions of objedts parallel to the pidture., will be judged the fame. PLATE VIII. Fig. 9. Thus, fuppofe SH to be the pidture, C its center, O C its dis- tance, O being now taken for the true point of light, if that diltance be increafed as to E, the apparent line above-mentioned will then be E C, and the apparent angle L E H, fubtended by the line joining the vanilhing points L and H, will be lefs than the true angle L O H, and would be more fa, were the eye in fome other point of that line between O and C, and the reft, it is plain, is as above de- clared. Now, let Y E be the vertical plane, C the center of the pidture, O C its true diftance, D F the line of ftation let A B be an original line in that plane perpendicular to the original plane, and parallel to the pidture, whofe image, feen from O, the true point of light, is k y or q y. Let now the eye be fuppofed in E, then will K U, which is the apparent original of the image k y, although it appears far- ther diltant from the pidture than its true original A B, be judged* ii6 GENERAL COMMENT by the mind of the fpedtator, to be of the fame fize as ABj that is, A B and K U will be equal to each other. For, in the fimilar triangles E O k, k A K, Ek : kK :: Ok: k A, by compofition Ek-f kK = EK : Ek :: Ok-fkA=OA: Ok. But, in the fimilar triangles E k y, E K U, EK : Ek :: KU : ky, and, in the fimilar triangles O k y, O A B, OA : Ok :: AB : kyj therefore, by a parity of reafoning, KU : ky :: AB : k y, confequently KU = AB. If we fuppofe E the true point of fight, and O the 'point aflumed for the place of the eye, the fame reafoning will fhew, that the dis- tance of the eye being lelfened, the original of k y will be judged nearer the pidture than it really is, but ftill of the fame magnitude. Now, feeing it is as A B to ky, fo are all other lines -in a plane parallel to the pidture paffing through A B to their images fee n from the fame point O, and likewife as K U is to k y, fo are all other lines in a plane parallel to the pidture, paffing through K U to their images feen from E ; therefore, feeing A B and K U are equal, the originals of all lines in the pidture, which meafure the dimenfions of objedts parallel to it, will be judged of the fame length, let the eye be placed where it will in the line O C or E C. What has been advanced relating to the feveral pofitions of the eye as being placed in its parallel, diredtor, or as lafl-mentioned, is eafily applied to any other pofition of the eye whatever ; for, fup- pofing it placed any where in the diredting plane, the confequences will be derived from what has been faid of the parallel and diredtor. O N P A R T II. ll 7 and, if placed any where out of the directing plane, by taking in the above-mentioned confiderations, the effects will readily appear. Upon the whole, we may obferve, that the ill effedls produced on the pidture, by its not being viewed from the true point of fight, are not fo confiderable by its being feen from a point that is too near or too far, fo long as the eye remains on a’ level with the horizontal line, as are thofe which fpring from a pofition of the eye that is higher or lower than it ought to be ; and that a reafonable licence may be allowed for deviating from the true point of fight in viewing fuch pictures as are on a plane furface, without incurring the fault of any other inconfiftencies or mifreprefentations, but what may, in fome fort, be corrected by the imagination of the fpedtator j but, as to thofe pictures which are formed on other kinds of furfaces, as domes, or arched roofs, or other irregular figures, it is there abfolutely' necefiary to pay the ftridtefi: regard to the true point of fight ; for any the leafh variation from it, occafions the moll grofs appearances, and makes the figures feem as being dis- jointed, broken or diftorted, and the unity of the reprefen tation iv thereby deftroyed. Having far exceeded the bounds firfi: prefcribed to this little Tradt, we here conclude, referving the dodtrine of fihadows, the method of applying perfpedtive to fcenographical reprefentations, as now pradtifed in painting the fcenes of theatres, the manner of draw- ing anamorphofes or deformed appearances, &c. for a future dif- quifition. F I N I S. ' Plate HI ■ •• v * * •• . • fe. * . Plate IV. Plate V. ) Plate VI , . - % Plate vnr. * - * « *• v' 4 >* O • . V. ! ¥ ■k; % Fig.io . xN*Vi > £>3 & • , *■ V. ' 0- 5 ■ 0* ■ ' \^ I \V w-xcA A^ ,>J >*"r^