( e 7 W.).-^oa fiave the vfoiro or^l7K,«8^also the appeodiX' of 1783. The ." loniie" iBltBvM conteia frontispiece and forty-six . plates. If botli volumes are complete they will ' be north 18s. or 20s., as bound. - ■■ i l 5. r i.-s** ; --1 VL."" V. : V ■ < .-."' legd' Reynolds (Sir Joshua) The Great Por trait Painter’s Own Copy, with Signature and Stamp on Title, Brook Taylor’s ' "■ Compleat Treatise on Perspective, edited ■ by Malton, 46 fine plates with diagrams, sm folio, original bds. Unique Copy, £7. 2s y, rr ■X. I ,i' ^ V J-.a 'r ■ 17' ' Ir- It’'' . ’/ %y: h i'V- (a' ' r m !■ : ii.’ . i' "V . I -.Vr tAr / ■ ' ‘i - . ■' .t" '■t" H. t-' ■ ' r'j iV M / IL* Li T^M-T>*^tcr u. Cu^ytJTL , 4 !^ . * ■S: ■f IT / 6 ' » Digitized by the Internet Archive ' in -2017 with funding from Getty Research Institute ‘ • • •V • . i • '.'V . , >■ * • ■ - , T ■ , / https :/7archive.org/details/compleattreatise00malt Aibl^d fiy JTiof^lfalton^lprii /i'/y/.f A COMPLEAT TREATISE ON PERSPECTIVE, I N THEORY AND PRACTICEj ON THE TRUE PRINCIPLES OF DR. BROOK TAYLOR. MADE CLEAR, IN THEORY, BY VARIOUS MOVEABLE SCHEMES, AND DIAGRAMS; AND REDUCED TO PRACTICE, IN THE MOST FAMILIAR AND INTELLIGENT MANNER; SHEWING HOW TO DELINEATE ALL KINDS OF REGULAR OBJECTS, BY RULE. THE THEORY AND PROJECTION OF SHADOWS, BY SUN-SHINE, AND BY CANDLE-LIGHT. THE EFFECTS OF REFLECTED LIGHT, ON OBJECTS; THEIR REFLECTED IMAGES, ON THE SURFACE OF WATER, AND ON POLISHED, PLANE SURFACES, IN ALL POSITIONS. THE WHOLE EXPLICITLY TREATED; AND ILLUSTRATED, IN A GREAT VARIETY OF FAMILIAR EXAMPLES; IN FOUR BOOKS. EMBELLISHEDWITH AN ELEGANT FRONTISPIECE, AND FORTY-EIGHT PLATES. CONTAINING DIAGRAMS, VIEWS, AND ORIGINAL DESIGNS, IN ARCHITECTURE, &c. NEATLY ENGRAVED. ALL ORIGINALS; INVENTED, DELINEATED, AND, GREAT PART, ENGRAVED BY THE AUTHOR, THOMAS M ALTON. THE SECOND EDITION, CORRECTED AND IMPROVED; WITH LARGE ADDITIONS. LONDON: PrlntCQ for the Author; and fold by Meflrs. Robson, In Bond-ftreet ; Becket, Adelphi, Strand; Taylor, near Great Turn-ftile, Holborn; Dilly, In the Poultry; and by the Author, No. 56, Poland-ftreet, Oxford Road, near the Pantheon. MDCCLXXVIII. T O THE PRESIDENT AND MEMBERS, OF THE ROYAL ACADEMY, For PAINTING, SCULPTURE, and ARCHITECTURE; INSTITUTED AT LONDON^ By, and under the Aufpices of, his moft gracious Majesty GEORGE THE THIRD. Gentlemen, I F the unremited labour and aflldiiity, with which I have prole- cuted the ftudy of the Science and Art of Perfpedtive, have thrown any new Light on that moft neceflary branch of the Polite Arts, fo, as to render its Principles clearer and better underftood, in Theory, more ealily applicable, in Pradice, more generally ufcful and fubfer- vient to the Arts, of which, it muft be allowed to be the foundation, ’tis what I have chiefly aimed at ; and prefume, this Work may not be wholly undeferving your Patronage and Encouragement, although it may not merit your entire Approbation. Such as it is. Gentlemen, I fubmit to your Candour, and claim your Protedion. The propriety of dedicating fuch a Work to Gentlemen, who are, undoubtedly, the moft competent Judges of it, will plead an excufe for my prefumption ; if, on an accurate examination of its Contents, it be found, that I have rendered an apparently intricate Science more familiar, and better adapted to the capacities of young Students, in fo eflential a part of their Studies ; the negled of which, amongft the riling Artifts, is much to be lamented. Perfpedive feems to be looked on as an Appendage, only, which may be difpenfed with, inftead of the firft requiflte ; in which, the Student, who would be a Candidate for Fame, fhould be well grounded. It DEDICATION. It is with reludtance I add, but it is too obvious, that, in many fine Pieces, in refped of Defign, Compofition, Drawing, or Colouring, there feems to be a want of a juft Idea of Perspective. Pardon me, Gentlemen, I do not mean to depreciate the Merits r ' fuch experienced Artifts, in their feveral Performances; for, in my opinion, there only wants a thorough knowledge of Perfpedive, to ren- der the prefent Age as famous as any former JEr2L ; which, now, is eftabliflied on the moft permanent and infallible Principles ; whereby, the trouble of projecting Objects, perfpectively, is, in the procefs, greatly abridged and facilitated. I am. Gentlemen, with great refpect, Your moft obedient, and obliged humble Servant, THOMAS MALTON. I N every Age, and in every Climate, where the Polite Arts are cultivated and encouraged. Emulation, and a defire of Fame, infpire the Breafts of the rifing Generations; the ingenious Mechanic catches the ardent and laudable Flame; r'- Avmerce is extended, the People are necefl'arily enriched, and the State becomes ^*ft and formidable. Architedlure rears her ftately Domes and lofty Turrets to thc'ifkies; the fumptuous Edifices, raifed by the hand of the able Mechanic, and embclliflaed by the ingenious Sculptor, with breathing Statues and almoft ani- mated Marble, proclaim abroad the Magnificence of the Founder, and jufily im- mortalize the fkilful Architeil; whofe Works may vie with the moll celebrated relicks of Antiquity, which feem to be of more than human Compofition. Whilll Painting, their Sifter Art, aflifts and unites in decorating the interior Domes, Cielings, &c. or, in well-chofen Pieces and Subjeds, and mafterly Performances, adorns the Stair-cafe, the Gallery, the Drawing-room, and the Cabinet. Perfpedive is allowed by all, who are well acquainted with it, to be the Bafis of all the Polite Arts which have their foundation in Drawing; particularly Paint- ing, or Delineating ; for. Colouring does not come within its Rules. The Sculp- tor and Architedt may receive great afiiftance from Perfpedtive. 1 am fenfible it is not abfolutely neceffary to the Art of Defigning, but it is efl'entially fo to fee the eftedl of the Defign. An Architedl may, doubtlefs, be famous in his Art without Perfpedive, but more fo if he had that accomplifliment ; as he would be better able to judge of the etFedl it would produce, and convey a juft Idea of it to others, before the Defign is executed. Drawings, in Perfpe(ftive, from particular Stations, where the Building will be moft confpicuous, would anfwer in that refpedt, the fame purpofe as a Model, and at a much lefs expence. At a Time when the Arts are in fuch a Degree of perfection, a Treatife on Perfpeclive may be thought an unneceflary Publication; feeing there are, already extant, many valuable Books on the Subject, containing all the Rules, for prac- tice, which are either neceffary, or ufeful ; and confequentiy, unlefs fome new Prin- ciples are propofed, by which, the trouble of projecting ObjeCts, perfpeCtively, may be lefiened, it is ufelefs to increafe the number. For fome, with truth, alledge, that, unlefs PerfpeCtive be comprized in a fmall compafs, it will never be ftudied by thofe, for whofe ufe it is chiefly intended. On PerfpeCtive, as on other Subjects, there are, indeed, a fufiicient number of Authors; and yet, I may venture to affirm, that, no SubjeCt, whatever, has been worfe handled, in general. 1 do not pretend to have found out new Principles, nor do I think, there can or need be any other; thofe given, by Dr. Brook Taylor, being fufficient for any purpofe, whatever; and that, the Principles, on which he has founded his Syftem, are the moft Ample and perfeCt that can poftibly be conceived. Notwithftanding what many, who have not a true Idea of PerfpeCfive, imagine, that there is im- perfection in it; that the Rules, prefcribed, do not always produce a true or pleafing Reprefentation of Nature; I maintain, that all, which can be done by Rule, is performed on the moft perleCt and infallible Principles that can be devifed; that Perfpeclive is, abfolutely, at its ne plus ultra ; where then, it may realonably be afked, is the neceffity for, or ufe of this Treatife? ' There are, generally, two Motives, which induce every Perfon to publifh to the World, his Inventions or Improvemen'.s, or Studies of any kind. The firft is Vanity ; or, to give it a milder Appellation, a laudable thirft of Fame; (I believe, I fliall not err wide if 1 affirm, that, Vanfty is the firft, and chief fprlng of all that is great and laudable) the other is the lucrative Emolument, if not always the Conlequence, at leaft, it may reafonably be expeCled an Attendant on Fame; it is fcarce determinable, vyhether Vanity or Intereft ftimulates moft to aClion. To pretend that we are not aCluated by either of thefe Motives is the height of folly ; lor though our Station, in Life, may fet us above the fordid views of Gain, yet no Station, whatever, is exempt from Vanity; which, moft wifely, for the bell of purpofes (the good of the whole) is made a neceffary ingredient in our Compofition. a Now, Now, although I have not the lead pretence to the Invention of new Principles, yet I am hrmly perfuaded, that I have made ufe of thofe we have to the bed advantage ; that, trom the irregular and imperfect Order, as they are given by Dr. Brcob Taylor, I have digefted it into an ufeful and pradtical Syftem; not involved in a labyrinth ot mathematical Demondraticn, of things uhich are to little pur- po!c in the Art of Delineating; as the voluminous Work of Mr. Hamilton, pub- lithed in the Year 1738 ; which, though a mod elaborate and valuable Production, has not been of the lead ufe, to the Arts it was intended to promote. In the ma- thematical -part, I have not entered further than is really neceflary, to evince and enforce the Priirciples, on which the Practice is founded ; and, in Pradlice, I have Ihewn its immediate and abfolute dcpendance on the Theory. I have by means of an A'japaratus, contrived for the purpofe, united the Theory and Pradtice fo toge- ther, that he mud have very little penetration, and a fhallow Capacity, who does riot in a flrort T. ime, and with little Study, conceive a clear and cornprchenlive idea of the rationale of Perfpedrive. I have, every where, made fuch remarks, and thrown fuch light on the Subjedf, as, I datter myfelf, will make it judly deferve the Title 1 have given it : my Defjgn, throughout, having been to make the dudv and pradlice of Perfpedtive, at the fame time, eafy and entertaining. The Subjedts I have choien to diiplay, and embellifh it with, are fuch as are common and fami- liar ; for, to vrhat purpofe is an intricate and puzling delineation of the Dodecah^E- dron, &c. except, to die w that the Author underdood it? I am of opinion, that the Reader will be much better pleafed, with the defeription of fuch Objedts as are frequently before his Eyes, and I'uch as are fit Subjedds to introduce into a Pidture; or to form a Pidlure on. 1 have not been fatisfied with the common LelTons for plane Figuies, and Solids compofed of or bounded by Planes, and left the Student to decorate his Buildings with the embellilhments of Columns, Entablatures, &c. of the Icveral Orders; but I have, minutely, diewn, how to delineate them, fingly, and to compofe a regular Building of the feveral Materials, deferibed and treated leparateiy of before. 1 here are, at this time, but few Gentlemen who are not pleafed with a fine Pidture, wliic'h truly reprelcnts Nature, on Canvas; the Deception being fo well managed, that, vve can almofi: imagine we fee the real C,)bjedts, themfelves, through the aperture of tl.e Frame. For, it is pofiible, in perfpedtive Delineations, in which there is a true gradation of Light and Shade, and judicious Colouring (on Ciclings, &c, which we cannot coiVie near) to deceive the Eye and Judgment, fo, as to imagine, what vve fee are real, and foiid Exifiences. Although Perfpedtive is not confidcred as a necefiary part of the Education of a Gentleman, it n.ult be looked on as a genteel and polite Accomplifiiment, a Qua- lification tor a Prince, For, has not our moil gracious Monarch cultivated and particularly encouraged it? nav, defeended to learn its Rules, and to delineate, vvitli his own Hand, pieces-cf Architedture ? Is it pofiible, for a Gentleman to fee, as lie oug t, much lefs to judge of a Pidfurc, with true tafie, and with the difeern- inent of a Connoiffeur, without having fome notion of Perfpedtive? it expands the Ideas, and makes us lee Objedts as they really appear, to a judicious Obferver ; it makes a Perfon a judge of the uifpofition, and proportion of the feveral parts of a Pidture, to each other; the Syu;etry and Harmony of the whole are perceived ; if timre be any difcordance, or unfeemly diftortion of the Parts, owing to an inju- dicious choice of the Artilt, either in the Difiance or Situation of the Objedt, or in the Pohtion of the Pidture, they will be quickly feen ; efpecially, when there is intnuluccd regular Architedture, in Buildings of any kind. T. he manner, in which Perfpedtive has hitherto been treated, fcientifically, is too rigidly and mathematically fo, to be entertaining, to a Perlon not converfant With the Elements of Euclid ; vvliich, though a branch of Univerfity Education, has been treated, in thofe jeminaries of Learning, in a manner, which deters num- bers from the fiudy of that molt ufeful and necefi'ary Science, the foundation of all math'“matical refearches. Every Science, and neceffary Art, fhould be fo treated as to allure a Perfon to the purfuit; by making him (in a familiar way) relifli preface. relifli a fludy, for which he has not a natural propenfity. Many Gentlemen ima- gine, if they neither paint nor draw, that, Perfpedive is wholly ufelefs, and un- neceffary to them; they are greatly miflaken who think fo, if they dcfire to be accomplifhed, and would beftow a little time and pains to be fo. Of all the mathematical Sciences, the Trudy of Perfpedlive is perhaps the moll entertaining; the pleafure and fatisfadiion which refult from it, in delineating, can only be felt, not deferibed. Aftronomy (to which Perfyeftive greatly con- duces) is the mofl: fublime of all, and next to it T Ihould rank PerTpedive. The entertainment, refulting from the former, is of a higher kind; we are, as it were, lifted from the Earth, in contemplation of the fuperior Works of the great Cre- ator, in fuch wife, as to look on all below not worthy of our notice. The advantages to Society, which are deduced from this moft exalted Science, are many and great; and, being well known, would be impertinent to enumerate. On the other hand, the ifudy of Perfpedive, fcientifically, is highly entertaining to a rational Mind; to thofe who would purfue it, 1 recommend the Works of Mr. Hamilton. 1 have not cholen to go further into it, than is really ufeful ; and yet, to fome Perfons, 1 Tnall be thought prolix enough ; who want to know it, without the labour of Study; to which, Artifts, in general, have great antipa- thy ; I mean, to the Itudy of any thing mathematical. It is ufual, with rnofl: writers on Perfpedive, to introduce To much of Geometry into their Work, as they judge neceffary for the knowledge of Perfpedivc. There w'ould be, in my opinion, as much propriety in prefacing every Book, on Litera- ture, with Grammar; left the Reader ftiould not be acquainted with Syntax, and the Idiom of the Language. I fairly own it above my Ikill, to draw a line between the neceftary and unneceftary, for all is more or lefs fo. Some Theorems, in Geometry, are only needfary for the attaining of others ; which, if they cannot be obtained without them, are neceftary. In fhort, if the Student be quite unac- quainted with Geometry, he is very unfit to ftudy Perfpedive ; let him firft become intimate with Euclid, and then he may fafely purfue Perfpedive ; the pradical part of one is fufticient for the other. Wherefore, feeing 1 have compiled a Syf- tem of Geometrv, to which this Work is fubfervient, I have wholly omited the Definitions of geometrical Terms, here ufed ; becaufe, it is reafonable to luppofe the Student already acquainted with them ; otherwife, he muft firft ftudy that Grammar, of the Science of Perfpedive. Perfpedive being a branch of the Science of Optics (as it is founded on dired Vilion) it becomes necefi'ary to confider, in the firft place, the ftrudure of the human Eye, and the nature of Vifion ; to conceive (as far as our Inteiieds can trace) an Idea, how that extraordinary and afionilhing Senfe is performed ; which being, in fome degree, underftcod, we firall be well prepared for a clear under- ftanding of Perfpedive, and enabled to diftinguifti between the Reprefentation of an Objed on a Plane, and its real Appearance; a circumftance, which is a great ftumblir.g-block to many, who have not rightly confidered the diflerence. 1 have prefaced this Work with a Sedion on Light and Colour ; and as a more eflential requifite, to Perlpedive, the fecond contains a brief deferipLion of the Eye, and of Vifion. In the next is contained the foundation of Perfpec- tive ; it treats on dired Vifion, comprifed in two Theorems, of univerfil appli- cation ; whicli are the very Eftence of Perfpedive. The two remaining Sedior.s, of the firft Book, are wholly digrefiive ; they not being at all necefi'arv, to the underftanding ot wdiat is contained in the followinc: Work; the Reader mav therefore pals them over, if he be fo difpofed, w'ithout breaking the thread of the Subjed. They contain matter of mere Opinion ; refpeding the materiality of Light, Refiedion, Refradion, &c. Book the fecond contains the whole nfeful Theory of Perfpedive, redllinear and curvilinear ; which is fomewhat copious, by reafon of the Examples given lor illuftration; and Corollaries, deducible from the Theorems. 4 Tlie PREFACE. T he firfi; Sedion is a general Intrcdudlon, containing many preliminaries, necef- fary to be known, previous to what follows. TT,e jecond is alfo introdudory, and contains a full explanation of all the various kinds of Projedion, ichnographic, orthographic and ftereographic ; with a cir- cumftantial and comprehenfive Definition of Perfpedive, and other introdudory matters. The third is more elementary; it contains full, yet brief Definitions of all the Terms made ufe of in the Theory. The fourth Scd;ion contains the whole Theory of right-lined Perfpedive, in fourteen Theorems; from which are deduced feveral ufeful and pradical Lefibns, in Corollaries, Scholia, &c. In the firfi: feven Theorems, is contained all the neceflary knowledge of Perfpec- tive, relative to Interfedions and Vanifhing Lines, in general; the firfi and moft efiential requifite in Pradice. The eighth Theorem, though felf-evident, is intended as a refutation of an ablurd opinion which many entertain, that continued Right Lines, feen dired, cannot be reprefented by right Lines; feeing that, parallel right Lines always appear to incline towards eacli other. The ninth teaches all that relates to Planes and Lines parallel to the Pidure. The tenth fiiews where the Interfeding and Vanifiiing Points of all other Lines are to be found; and the eleventh determines how to find them. The tv/elfth and thirteenth contain all that relates to Lines not p>aral!el to the Pidure, in the inofi concife, pet lull and clear manner that I can conceive. Perhaps the Demonfirations of the lafi may deter thofe, who are not Ceometricians, from examining it witii that attention it requires; let fuch rem.ember, that, in order to pradife Perfpedive, it is not abfolutely neceflary to be a Geometrician ; becaufe, I pradifed it long before 1 underftood Geometry ; excepting a little pradical, and, a fuperficial knowledge of Lines, in general; which are certainly requifite ; as the whole of pradical Perfpedive con fills in it. For which er.d I have feleded, all that is necefiary, not only for Perfpedive, hut, alfo for various mechanical ufes, into one Book (the firfi part of the Roval Road to Geometry) lately publiflied ; perhaps, in the mofi ealy and intelligent manner, ever yet done; to which 1 refer the Reader, for the georn trical Conftruc- tion of all kinds of Figures here ufed ; as well as to the Elements, for Demon- firation, in the Theory. If thofe Readers, who have neither time nor inclination to become acquainted Vv^lth the Elements of Geometry, at leafi acquire fo much, as that Trad contains; I will vouch for the great advantage he will find to his fiudy of Perfpedive. In- deed it is jmpofilhle to underfiand or pradife Perfpedive without it; for, being in itfelf, wholly geometrical, the very Language, and Grammar, of Perfpedive is Geometry; let them take my word, for once, they will not find their time mif- Ipent; bur, on the contrary, fo much knowledge, as it teaches, will be purchafed at a very eafy rate. They may, then, boldly venture on the Theory ; omiting the Demonfiration where they find they cannot comprehend it; which, in general, they will not find enveloped in myfiery ; having treated that part in the mofi familiar manner pofil- ble, by plain reafoning only, without other reference than to the Elements, lor proof of what is advanced. Let them, at leafi, by the mofi attentive perufal 'of the Theorem, endeavour, to conceive the Premiles of it; which, in fome of them, perhaps, may not be perceived, clearly, at the firfi reading; the fecond will open fome Ideas, and the third perhaps compleat it; which, if it require a fourth, fometimes, will not be lofi labour; for, when the Premifts of a Theorem are clearly underfiood, the proof of it is not difficult, in many cafes. However, whether the proof of what is advanced be perceived or not, it is not efiential to the Prac- tice of Perfpedive; the Reader may give me credit for it, I will be refponfible fur the truth of it; he may be fully latisfied that the thing is certainly fo, and make ufe of that knowledge accordingly. X The P R E F A C E. The Theorem being underftood, let him, at leaft, go through one Example in each, as he proceeds ; his Ideas will not only be ftrengthened, but he will, poffibly, fee the truth of the Premifes. Then, the Corollaries, Scholia, &c. are but fo many eafy, ufeful, and pra61Ical Leflbns, deduced from the Theorems, which will be found of the greateft ufe in Pradlice ; and, if the Theorem be clearly conceived, the Corol- laries will be fo too. The fourteenth Theorem has no affinity with, or dependance on the foregoing ; it contains the whole Theory of praftifing without Vanifhing Lines ; an ingenious method, but feldom pradifed ; nor, indeed, is it fo generally applicable to practice. The fifth Sedion contains fo much of the Theory of curvilinear Perfpedive, as is really ufeful in Pradice ; or neceflary to be known, by any Artift, whatever. The fixth is a refutation offeveral capital Errors and abfurd Opinions, which many Perfons entertain of Perfpedive ; and which are, there, clearly and fairly ftated, and fhewn to have no real exiftence. The third Book is a copious Treatife on ufeful pradical Perfpedive, containing Leflons and Examples in every neceflary Cafe, that can occur in Pradice ; fhewing alfo its dependance on the Theory, in brief Denionftrations, where it is neceflary. The Method, in which I have treated this ufeful part of Perfpedive, is fuch, as I prefume will fufficiently recommend it. The Book is divided into twelve Sedions, each treating on a diftind: fubjed from the foregoing ; begining with the moll Ample and firfl Principles or Elements, and going on, from the moll Ample, to the moll complex Objeds, in a regular, progreflive fuccelAon. After the neceflary Preliminaries and Elements, in the three Aril Sedions ; the third Ihewing alfo how to And Vanilhing Lines, and Vanifhing Points, in general, the Aril requifite in Pradice ; in the fourth is contained all the pradical elementary Problems in Brook Taylor’s Eflay, which are, of themfelves, a complete Syftem of pradical Perfpedive. Thefe Elements, of the following Work, being well under- llood, will be found of great Utility ; which has induced me to perfed fuch as the Author of them had left very imperfed; and fome, particularly his 21 ft and 2 2d Figures, are greatly defedive *, we are neverthelefs infinitely obliged to him, for* giving them to the World, fuch as they are. To underftand this Sedion clearly, and be enabled to apply the Problems It con- tains, in all Cafes, pradically (which, it muft be obferved, is the foundation of the whole) a tolerable (hare of geometrical knowledge is requifite ; without which, it Is impoflible to turn them to that general ufe, in Pradice, in which they are fo uni- verfally applicable ; Inafmuch that, it may be truly faid, he who can comprehend them, perfcdly, and apply them in all Cafes, generally, underftands Perfpedive, thoroughly ; but, being fuperficially looked over and not well digefted, they will be found of little ufe. This’eonfideration, perhaps, has induced fome Perfons to imagine, and affirm, that the whole ot Perfpedive may be comprifed in a very fmall Compafs. But, I mull make free to tell them, they are under a miftake ; for, notwdthftanding the Principles are few and general, yet, to know how to apply the Rules, in all Cafes that may occur in Pradice, cannot be fo foon acquired ; if fo, how comes it, that they are no greater Proficients ? Or, can they imagine, that, amongft the number of able Men, who have wrote on the Subjed, none Ihould hit on that much wilhed-for Expe- dient ? Either they muft acknowledge that their own faculties are not fo peute as others (which Is paying a forry Compliment to their Underftandings) or they mull fuppofe that there is more in it than they at firfl: imagined. ’Tis true, the whole of Pradical Perfpedive is comprifed in this and the foregoing Sedions ; yet, how few would be able to apply thofe Problems, though clearly underftood, in delineating a regular piece of Architedure, is but too obvious to enlarge on ; without previous, pradical Leflons, diverfified varioufly, to familiarize them. For which reafon. It is neceflary to fliew their application, in various Examples, which I have not been b fparing P R E F A e E. fparlng of. I have not, as others have done, ftudied the tnoft eafy pofitions of Objefts ; but I have endeavoured to give the moft pleafing, pi6lurefque, and natural Reprefentations, and render the moft difficult Pofitions eafy. Some Perfons know a great deal (in their own opinion) becaufe they know, per- haps, that all Right Lines, which are parallel, have the fame Vaniffiing Point ; who yet, know not how to fix and afcertain any one, except the Center of the Pidure, and that often very abfurdly (indeed they feldom, if ever, ufe any other) nor how to proportion a Line perfpeclively ; and further, to ffiew their confummate knowledge in it, if you ffiew them a piece of Perfpeclive, in which there is not one Line tending to the Center, or Point of View, they will call that Point on the Pidure, in which the principal Lines converge, the Point of View ; fo that, there may, by this crite- rion, be feveral Points of View in the fame Pidure ; a circumftance too palpably abfurd to need a refutation. ^ ^ ^ ^ If this be to know Perfpeftive, it may indeed be very foon acquired ; It may taught to a Child. But, when they have drawn a Line to its proper Vaniffiing Point, how that indefinite Line is to be proportioned. In order to reprefent, certain finite parts of the Original, they are wholly ignorant, though the chief requifite in Pra6fice, and the principal bufinefs of the fourth Sedlion of the third Book. However, not to difcourage them, from attempting to acquire fo neceflary a part of their Studies, they may be affured that a fixed refolutlon and perfeverance, will foon furmount every apparent difficulty. Some are alarmed, and even frighted, at the fight of fo many Lines in a Diagram, for a Leflbn in Perfpedive, imagining they can never comprehend them all. Certainly, they muft, to a perfon wholly ignorant of the ufe and meaning of them all, appear a general confufion ; but when they have fet about it, with a refolutlon to know, and have well digefted the Definitions of them, they will, at one glance, comprehend the ufe of feveral ; and the reft, by ana- lizing them regularly, will foon become familiar to the Eye, and to the Under- ftanding, - ■ This, I believe, has induced feveral Perfons, who have wrote on Perfpeffive, to deal very unfairly in It, in not giving all the neceflary Lines In their Diagrams. I have, in mine, given all, and more than are neceflary to be drawn, at once ; for let it be obferved, that the Radials of Lines, for determining Vaniffiing Points, feldom need be drawn ; and feveral of the operative Lines, or Vifual Rays, which are only ufed for cuting others, need not be drawn, at all ; a Ruler being applied to the two Points (the Eye and the original Point) the indefinite Line may be cut, without drawing the whole Line, which could not anfwer. In a Diagram, fo well as drawing the whole, in order to ffiew the dlredlion of the Line ; befides, in Praftice, one part is effeded and the Lines rubbed out before others are drawn, which cannot be done in a Diagram, all muft remain together. Others, again, there are, who have almoft determined to make an effort, only want to know fo much of the Praftice as to be able to projeft a plain Building, &c. and feem quite afraid of attempting the Theory ; becaufe there Is occafion for Geometry, which, to many, is a terrible Bugbear ; fo that, rather than exert them- felves, they do without it, though the firft eflential, and chief requifite perhaps of ^ their Profeffion. They want to know Perfpe6live, but are afraid to venture on the Theory. They deceive themfelves if they Imagine it an unneceflary Appendage ; can it be unneceflary to know how, and why, before we attempt any thing, pradi- cally ? Is not one, the means of effedfing the other ? I do aver, that the ffiorteft way to acquire Perfpedtive is to underftand it firft, in Theory ; the Pradfice of it will readily follow. How different are the dlfpofitlons of mankind. After I had (from the Jefuits) acquired (as many would fuppofe) a proficiency in Pradlicai Perfpedlive, I was far from being fatisfied with all that could be obtained from that Author ; and although I was not a Geometrician, at that time, and he not giving any Theory, I fet about contriving means to have conviction of the truth of his Rules j by which means, I 2 fell p R E F A C E. fell infenfibly into the method iifed by Vignola and Sirigattl ; than which, nothing can be more convidive ; and I was accordingly. fatisfied, that the Rules, given, might be depended on. But, having acquired fuch a proficiency in Geometry, as to enable me to comprehend Brook Taylor’s Principles, how trifling, how limited appeared all I had learned of it before ; I was even afhamed of my own prodiidions, fome of which had, unde- fervedly, been admired : I found, indeed, I Icarce knew what was meant by Per- fpeftive, till then. The greater progrefs I made in Geometry, the clearer, and more comprehenfive Idea 1 had of his Principles ; which, though founded on common ob- fervation of Objects (fupported by Geometry) are beyond every thing ever thought " on, by any who had wrote on the Subjed j and yet, it is fo Ample, when under- ftood, that ’tis ftrange it fliould not have been brought to light fooner. Having, in the fourth Sedion of the third Book, difplayed the whole Elements of Pradical Perfpedive, the fifth begins the application of them to real ufe ; the whole of which, coiififts in finding the reprefen tation of a Line or Point, in any Plane, any how fituated : for. Points are the extremes of Lines, and Lines of Surfaces ; and. Solids, or Objeds, are bounded by their Surfaces. A Plane, however fituated, it is manifeft, is the fame, in all its properties ; its fituation, in refped to the Horizon, is, therefore, of no confideration, in Perfpedive ; and, to delineate Plane Figures is all that can be done by Rule, with abfolute certainty j the apparent Contours of fome round Objeds are fomewhat difficult to determine, yet they may be done fufficiently corred. In this Sedion, the Conftrudion of all kinds of plane Figures, perfpedively, in Planes any how fituated, is comprifed ; and, in the fixth is fhewn how to conftrud Solids, of Planes only ; by which, any Plane Building may be projeded. The feventh Sedion teaches, fully, how to reprefent Mouldings in general, and to break the fame at the Angles of a Building, &c. internal or external, right-angled or otherwife, with other necelfary, and decorative, parts of Architedure, &c. as Triglyphs, Confoles, Modilions, &c. alfo, how to form a Pediment. The eighth is wholly adapted to circular and round Objeds of all kinds ; as Arches, Columns, Tufcan and Doric Bafe and Capital, round Steps, Wheels, Vafes, and circular Mouldings ; including alfo the Ionic and Corinthian Capital. ^ Being now furnilhed with all the materials of a Building, the ninth Sedion fhews how to compound them, and to form a Building ; from the moft plain and Ample to the moft elegant and rich, decorated with the various Orders ; alfo detached Build- ings, Views, &c. The tenth is for internal Subjeds, as Rooms, infide views of Churches, Arcades, Staircafes, and Cieling-pieces, reprefenting Domes, and Cupolas ; which, though fomewhat particular, is founded on the fame invariable Principles. The eleventh is adapted for houfiiold Furniture ; as Tables, Chairs, Cabinets, Bookcafes, Beds, &c. alfo, for Machines, Coaches, See. The twelfth treats of inclined Pidures and Planes, in general, containing Leflbns in the moft difficult cafes, from known data ; in which, the excellency of the new Principles is exemplified. Book the fourth treats on the Perfpedive of Shadows, which is indeed a copious Subjed, and much more might be faid of it } but, as I am fenfible, that very few take the neceflary pains to projed their Shadows by Rule (general Effeds being all that is ftudied, or regarded) it would therefore be to little purpofe to give Rules, which will feldom, if ever, be followed. Nor is it at all neceifary, or even pradical, to projed every Shadow, mathematically ; it would be attended with great lofs of time, and perhaps, in fome Cafes, produce a bad effed, though truly projeded. In this branch of Perfpedive, licence may therefore be taken, juftifiably, provided they do not run into grofs and palpable abfurdities ; fuch as projeding the Shadows of Objeds, already immerfed in Shade ; or, as 1 have feen in the works of eminent Mafters, Shadows catt both ways, in the fame Pidure, to the right, and to the left. I have PREFACE. 1 have alfo feen, the Shadow of a curve Line reprefented by a Right Line, when the Luminary was not nearly in the Plane of the Curve ; which., with others, lefs ob- vious, ought carefully to be avoided, and guarded againft. Neverthelefs, there are certain general Rules may be giv^en, which ought ftridlly to be adhered to ; and never, on any account, departed from. 1 have, therefore, in this Work, given fo much as I conceive neceffary to be known ; for, in order to be converlant in Shadows, it is abfolutely neceffary to be acquainted with the invariable Law of Nature, in the projection of Shadows; although it may not always be ne- ceflary to follow her didlates, implicitly. The effect of refledted Light on Objects, from other Bodies, in vicinity with them, is likewife treated of in this Book ; alfo the effect of Diftance, ufually underftood by the term Keeping, properly, aerial Perfpedtive ; both which, contribute greatly to the perfedtion of a Pidture ; infomuch that, without due regard being had to both, a Pidture, ever fo well defigned and delineated, or coloured, will be but a flat and fpiritlefs performance. And laftly, the reflected Images of Objedts on the furface of ftill Water, or oii ' polifhed Mirrours, in any Pofition, are, in the laft Sedtion of this Book, treated on. Plaving, now, given a brief account of the Work, I fhall juft make one more obfervation, in refpedt of the notions of fome Artifts ; who being always accuftcmed to fketch by fight, only, and having not the lealt conception of projedling Objedts by any Rule, they vainly imagine that Perfpedtive will teach them how to reprefent Objedts, exadlly as they appear to the Eye, without knowing any thing of their Proportions or Situations, in refpedt of each other, or of the Pidture, which Is im- poffible ; fo that, when they find there is a neceffity for taking and applying their real Meafures, or Proportions, being fo foreign to what they conceived of It, they imagine fo many obftacles to lie in the way, that, they foon defift from the attempt ; choofing rather, to remain in ignorance, and proceed with uncertainty, than take the neceffary pains to acquire, what, being acquired, they would deem invaluable. He, therefore, who can hit on fuch an Expedient, as to teach them how to de- lineate Objedls truly, by Rule, without applying any Rule at all, will be likely to meet with great encouragement from thofe Indolent Artifts. I acknowledge my incapacity for the talTc j and therefore, I fliall abide by the Rules, as they are given in the following Work. s UBSCRIBE RS TO THE COMPLEAT TREATISE on PERSPECTIVE. T THE Earl of Northampton "I'lie Duke of Portland The Duke of Devonlhire The Counteis of Efi'ex I'he Marcjuifs of Rockingham The I'uke of Richmond I'iie Vifcountefs Weymouth Tiie f)uke of Northumberland 1'Iie Earl of Beiborough 1 he Earl of Clanbral'il 'J'he CountcE of Clanbrafil 1 he Tvlarquils of Granby Lord George Cavendilh h.arl 1. irzAvihlam Tiie Earl of Seaforth Lord Le Defpencer Lord Mulgrave Karl of Stafdiope The Earl of Rofeberry Tlie Earl of Stamford The Earl ©f Gainiborough Lady Beaulieu The Duke of Athol Lord Dude The Duke of Grafton Lord Newberry Lord Vifcount TownOiend The Earl of Dartmouth Lord North Lord Clifford The Earl of Oxford The Earl of Avlsford The Marquis of Carmarthen Lord Vifcount Irwin Lord Petre The Duke of Beaufort The Earl of Scat borough Lord Sondes Lord Willoughby De Broke Lord Richard Cavendifii Lord Vifcount Howe Earl Spencer J-ord Robert Spencer Lord Darn ley Lord Craven The Duke of Queenfberry I'he Earl of Lincoln The Earl of Radnor The Duch. Dow. of Newcafll'e TheCountef s Dow.ChefLeriield Lord Romney Earl Ferrers Lord Pohvarth '1 he Earl of Offory The Earl of Nortliington Lord Rivers HE KIN The Duke of Roxborough Lord Robert Manners Sir George Savile, Bart. Sir John Thorold, Bart. Major General Fitz Roy General Parker Sir H arbord HarborJ, Bart. Sir Cha. W. Bamfylde, Bart. Sir Charles Price, Bart. Thomas Dundafs, Efq, - — Hamilton, Efq, F. F. Foljambe, Efq; Frederic Montague, Efq, John Bagnal, Elq; John Offley, Elq; Robert Child, Efq; David Garrick, Efa; Stephen Wright, Efq; W. Kenrick, LL. D. Robert Weflon, Efq; Joleph Hickey, Efq; Lieu. Colonel Campbell John VValton, Efq; R. J. Gbodenough, Efq; Sir Cha. Saunders, Knt. of B Beilby Thompfon, Ffq; The Rev. Dr. Barvvis Hon. Admiral Keppel Sir William Dolben, Bart, Sir Abraham Hume, Bart. Sir William Jones, Bart. Sir Hugh Paiifler, Bart. Thomas Butler, Efq; John Bentinck, Efq; Richard Dalton, Efq; Richard Bennit Lloyd, Efq; Robert Udny, Efq; John Courtenay, -Efq; Sir Francis Vincent, Bart. Henry Srevenfdn, Efq; Thomas Mellifh, Efq; Thomas Thornhill, Efq; The Hon, Mr. Watfon David Hartley, Efq; Sir Herb. Mackworth, Bart. George Dempfler, Efq; Tho. Brand Hollis, Elq; Charles Mellifh, Efq; Cha. Orby Hunter, Efq; Captain Page William Backwell, Efq; John Hewit, Efq; lyieut. Col. Alex. Johnftonc Peter Del me, Efq; John Bowater, Efq; Sir John Smith, Bart. G. The Archbifhop of Armaghi Sir Robert Bcu ker, Bart. James Connell, Efq; Richard x‘\lexander, Efq; IfaacV/all, Efq; Major General Tayler Freeman, Efq; John Ellis, Efq; General Trapaud William Frankland, Efq; Sir Tho. Robinfon, Bart. Jofeph Stacpoole, Efq; Mathew Duane, Efq; Theoph. Foreff, Gent. DoSlor Boucher J. Gideon Loten, Efq; John Lloyd, Efq; Captain Salter John FJill, Efq; Lien. C. Selby, Efq; r Sckell, Gent. R. W, A. Bennit, Efq; Richard Stonhewer, Efq; Edward Stanley, Efq; . Sir Charles Knowles, Bart. John Nicholl, Efq; F. W. Skipwith, Efq; Henry Stevenfon, Efq; George Forbes, Efq; John Sawbridge, Efq; Sir William Draper, K. B. James Hamlin, Efq; Samuel Meek, Efq; Sir Charles Whitworth, Bvirr. Arnold Nefbitt, Efq; Marmaduke Tunflall, Efq; William Confrable, Efq; Mr. Norton, Surgeon H. A. Fellows, Efq; William Flenley, Efq; W. P. Georges, Efq; Ralph Ward, Efq; John Aftley, Efq; Captain George Smith Profeflbr Hutton, JVoohvkh Mr. Burrows, Math.M. Toiver Captain Watfon Richard Tavlor, Gent. — Luntley, Gent. Thomas More, Efq; Rawfon Adlabie, Gent. Roger Blount, Gent. Mr. Broughton, Merchant Jennings, Gent. Dodor Falck Alex. Mapin Bailey, Gent, ( ii ) Mr. Anth. Martinez, Merch. J. Jackfon Malo, Gent. Joleph Gilbert, Gent. Mr. Meredith, jun. Mr. Galini, Gent. Mr. Greathead, jun. Mr. Reynolds Mifs Pelham Mrs. Raby Vane Mifs Keene Mifs Fanfhawe Mifs Winter ROYAL ACADEMICIANS and other ARTISTS, &c. Sir Jofliua Reynolds, Prefident Sir Will. Chambers, Treafurer Mr. Penny, Prof, of Painting Mr.T. Sandby, Prof, of Arch. Mr. Paul Sanby, Painter, R. A. Mr. Stubbs, Painter Mr. D. Martin, Port. Painter Mr. Weft, Hift. Painter, R. A. Mr. Cipriani, Painter, R. A. Mr. Wilton, Statuary, R. A. Mr. Cofvvay, Painter, R. A. Mr. Payne, jun. Sculptor, Mr. Richards, Laiidl.Pr. B. A, Mr. Webb, Portrait Painter Mr. Greenwood, Painter JVlr. Catton, Painter, R. A. Mr. Mever, Min. Paint. R. A. Mr. Romney, Portrait Painter IMr. W'. Pars, Painter, AiTociate Mr. Woolett, Engraver Mr. Burch, Sculptor, R. A. Mr. Hone, Port. Painter, R. A. Mr. Ed wards. Painter, Aflbc. Mr. John Hamilton, Painter M. Dali, Landi'c. Paint. Afto. Mr* Steuart, Painter Mr. Rooker, Engraver, Aflbc. Mr. Elias Martin, Paint. Aflb. Mr. Rigaud, Painter, Aflbciate Mr. Carver, Lanfeape Painter Mr. Hodges, Landfeape Paint. Mr. Bonneau, Drawing Mafter Mr. Barralet, Pr. & Dr. Maft. Mr. V. Green, Mezzo. Scraper Mr. Pars, Drawing Mafter Mr. Gainfborough, Paint. R. A. Mr. J. A. Grefle, Draw. Maft. Mr. James, Portr.Paint. Afloc. Mr. Cotes, Miniature Painter Mr. Vandergucht, Painter Mr. Noble, Drawing Mafter Mr. Cole, Painter Mr. Boydel, Engr. & Printfel. Mr. Mortimer, Painter Mr. Smart, Miniature Painter Mr. Wheatley, Painter Mr. Chamberlain, Port.Pa.R. A Mr. Parfons, Painter Mr. Englehart, Minia. Painter Mr. Hickey, Portrait Painter Mr. Brompton, Portr. Painter i Jr. Davey, Portrait Painter Mr. Scouler, Miniature Painter Mr. Bogle, Miniature Painter Mr. Grignion, Painter Mr. Jolhua Smith, Modeller Mr. Riccard, Miniature Painter Mr. W. A. Baron, Landf. Paint. Mr. Nixon, Miniature Painter Mr. Taflaert, Painter, Mr. Craft, Painter in Enamel Mr. Haywood, Sculptor Mr. Stubble, Painter Mr. Sanders, Painter Mr. Daniel, Painter Mr. Day, I.andfcape Painter Mr. Grim, Painter Mr. Roma, Paintet Mr. Reinagle, Painter Mr. Seaton, min. painter Mr. Paxton, Painter Mr. Burgefs, fen. Painter Mr. Burgefs, jun. Draw, maft. Mr. J. Taylor, Painter Mr. Gilder, Dr. M. Tower Mf. Durand, Painter Mr. Adamfon, Painter Mr. Southgate, Painter Mr. Buckley, Painter Mr. Brooks, Glafs-ftainer ARCHITECTS, &c. Mefl'rs. Adam, Architects Mr. Stuart, Painter & Arch. Mr. Wyatt, Architect Mr. Crundon, Architedt Mr. Harrow, Carpenter Mr. Shakefpear, Carpenter Mr. Carter, Statuary Mr. Grooves, Bricklayer Mr. Weftmacott,'- Sculptor Mr. Allan, Carpenter Mr. Allen, ditto Mr. Howell, ditto Mr. Hoberaft, Carpenter Mr. Byfield, Carpenter Mr. Grinell, Carpenter Mr, Filewood, Carpenter Mr. Babb, Bricklayer Mr. Devall, Mafon Mr. White, Carpenter Mr. Holland, Architect Mr. Gray, Bricklayer Mr. Collins, Plaifterer Mr. Rofe, Plaifterer Mr. Geo. Adams, Builder Mr. Abr. Adams, ditto Mr. Lyfter, Builder Mr. Johnfon, Builder Mr. Wolfe, Architect Mr. Gandon, Architect Mr. Hall, Carpenter Mr. Edwin, Architect Mr. Rogers, Architect Mr. G lantham. Carpenter Mr. Cook, Surveyor Mr. T. Leverton, Architect Mr. F'iftier, Carpenter Mr. Lewis, Architect Mr. Brounton, Surveyor Mr. Goldfmith, Carpenter Mr. G. Steuart, Architect Mr. Oldfield, Architect Mr. Dixon, Carpenter Mr. Peacock, Architect Mr. Gowan, Surveyor Mr. Lewis, Carpenter Mr. Mecluer, Surveyor Mr. Dotchin, Surveyor Mr. Mofs, Surveyor Mr. Gibfbn, Carpenter Mr. Meadows, Carpenter W. Robinfon, Efq; Architect Mr. Coufe, Architect Mr. An fell. Carver, &c. Mr. Newton, Architect Mr. Hillier, Architect Mr. Keene, Architect Mr. Hunter, Carpenter Mr. Harlowe, Carpenter Mr. Saunderfon, Mafon Mr. Cameron, Architect Mr. Prince, Carpenter Mr. Stephens, Architect Mr. Stephens, mafbn Mr. Beatey, Carpenter Mr. Heaftord, Plaifterer Mr. Baker, Surveyor Mr. Jagger, Surveyor Mr. White, Surveyor Mr. Davifon, Surveyor Mr. Hedges, Bricklayer Upholders, Cabinet-makers , &c. Mr. Bradfhaw, Upholder, Mr. Saunders, Upholder Mr. Fell, Upholder, &c. Mr. Belcher, Cabinet-maker Mr. Gray, Cabinet-maker Mr. Chipendale, Upholder, &c. Mr. Jenkin, Upholder, &c. Mr. Linnel, Upholder, &c. Meflrs. Mahew & Ince, Uph. Mr. Mackay, Upholder, &c. Mr. Tate, Upholder &c. I Mr. Reilly, Upholder, &c. Mr. Chowles, Upholder, &c, Mr. Marfh, Upholder, &c. Mr, Phillipfon, Upholder, &c. Mr, Dollond, Optician Mr. Barnard, Coach maker Mr. Fofter, Coach- maker Mr. Gale, Upholder, &c. Mr. V/at!on, Upholder, &c. Mr. Hatchet, Coach-maker Mr, Godlal, Coach-maker Mr. Bradburn, Cabinet maker Mr, Gilroy, Upholder Mr. Seddon, Cabinet-maker Mr. Graham, Upholder Mr. Chapman, Upholder Mr. Mill, Upholder Mr. Fox, Cabinet-maker, &c. Mr. Ilankin, Coach-maker Mr. Gilding, Upholder Mr. Romer, Goldfmith Mr. Arnold, Watch-maker Mr. Aitkin, Cabinet-maker Mr. Allan, Cabinet-maker Mr. Collins, Uph. Grantham Mr. Walfli, Carver and Gilder Mr. Tu rner, Cabinet-maker Mr. Giliows, Cabinet-maker Mr. Robins, Upholder Mr. Kay, Cab-maker & Uph. Mr. Rand, Upholder Mr. Sandell, ditto Mr. Edwards, Uph. Dublin Mr. Newton, Upholder Mr. Fryer, Cabinet-maker Mr. Platt, Carver, &c. Mr. Twaddel, Glafs-grinder Mr. Dodd, Cabinet-maker Mr. Dalrimple, Cabinet-maker Mr. Shade, Cabinet maker Mr. Bubb, Cibinet -maker Mr. Harrifon, Cabinet-maker Mr. Halton, Cabinet-maker Mr. Noone, Upholder. Mr. Chmnery, Writing- mafl. Mr. Harper, Mailer of Acad. Mr. Williams, Wnting-mafl. Mr. Crocker, Phil. 1 iminher Mr. Alhby, Letter Engraver iiubfcribers in Brijlol. The Library Society John Gordon, Efq; Mr. Paty, Architedl. Haac Piguenit, Efq; Thomas Whitehead, Efq; Joi'eph Flarford, Efq; Tho. Tyndall, Efq; Charles Partridge, Efq; James Harford, Efq; Mr. Daniel, Merchant I’he Rev. Mr. Lee Mr. Neufville, jun. merchant ( iii ) Mr. Gilbert Davis, Builder Mr. R. Tombs, Ship-Builder Mr. John Garnett, merchant Mr. Howel, Upholder Mr. Henry Marfh, Carver Mr. Wells, Surgeon Mr. Burgam, Pewterer Mr. Parfit, Cabinet-maker Mr. Clayheld, Tobacconitt Mr. James Moody, jun. Mr. Edward Langdon, jun. Mr. John Rees, Builder Mr. Walldo, merchant Mr. Hague, malon Mr. James Allen, Arch. &Stat. Mr. Tullin, Cabinet-maker Mr. John Stephens, Gent. The Rev. Mr. Thomas Robins Mr. P’rancis Safi res Mj. Richard Webb, Builder Mr. Court, Cabinet-maker Mr. Edward Stock, Builder Mr. Meyler, Broker Mr. Donn, mathematician Mr. Roller, mafon Dodler Samuel P'arr Mr. Shiercliff, minit. painter Mr. John Harris, Carpenter Mr. Miles, Carpenter Mr. W. Shorland, millwright Mr. Richard Perkins, DiRiller Mr. Samuel Atlee, Dilliller Mr. Carpenter, Builder Mr. S. Wilmot, Carpenter Mr. W. Hopkins, Carpenter Mr. Marlh, Cabinet-maker Mr. Rich. Hale, Copper-fmlch Pvlr. Abraham Elton, jun. Pvlr. Giles, Attorney Mr. Mac Taggart, merchant Mr. Daniel, mafon Dodlor John Ford Mr. Welliek, Coach-maker Mr. Mark Davis, jun. merch. Mr. R. Champion, merchant Mr. John Rawlins, Surgeon Mr. T. Shellard, Apothecary Mr. Hill, Plaillerer Subferibers in Baih. The Hon. Mr. Achefon Alexander Champion, Efq; The Hon. Mr. C. Hamilton Col. Fleming Martin, Sir John Stapleton, Bart. R. Dick foil ckrine, El'q; IMr. Hoare, Portrait Painter Mr. Beach, Ditto Mr. .Pine, Hif. and Port. Paint. Mr. Wright, Painter Air Redmond, Minit. Painter Mr. Garvey , Land (cape Painter Air. Davis, Scene Painter Mr. Baldwin, Architcdl Mr. Blrchall, Cabinet -maker Mr. Daniel Brown, Builder Mr. J. Palmer, Builder Mr, J. Dearc, Carver, &c. Mr. 'F. Young, Cabinet-maker Air. T. Cottel, Builder Mr. T. Beale, Builder. Subferibers at Satijhury, James Tobin, Efq; Hen. Pen. Wyndham, Efq; William Benfon E.arl, Efq; William Hufley, Efq; Dodlor Jacob Mr. Lulh, jun. Builder Mr. Eaflon, Bookfeller Mr. Brompton, Painter Mr. Moulton, Builder Subferibers in Southampton. Bdiomas Rumbold, Efq; William Blackwell, Efq; Rd. Vernon Sadleir, Efq; Arthur Atherley, Efq; John Gowan, Efq; Capt. Thomas Sadleir, William Yeates, Gent. Air. G. Hookey, Mafon Air. R. Waight, Carpenter Mr. Thomas Jeans, Surgeon Meflrs Linden and Ward Mr. J. Taylor, Builder Mr. Daman, Attorney Mr. T. Slade, Coach-maker Mr. Hillgrove, Merchant Air. Watts, Builder Subferibers in JVincheJlcr. Henry Penton, Efq; Edward Grace, Efq; The Rev, Dr, Warton, H. M. Air. J. Hawkins, Commoner Mr. William Kernot, Builder Mr. John Barton, Builder Mr. W. Greenville, Bookfeller Mr. W. Cave, Painter Air. W. Gover, Builder Air. Hollavvay, Attorney Subferibers in Portfmouth. Thomas Meik, Ai. D. Mr. Damerum, Builder Mr. Wheeler, Carpenter Air. Mac Ilraith, Secretary Mr. J. Jefi'ery, Dock-A^arJ Mr. J. Gilbert, Dock-Yard Portfmouth Common. Air. Andrews, Carpenter &c. Air. James Hay, Alafon Air. J. Monday, Builder Air. Twynam. Builder Mr. Harding, Watch-maker Mr. Douglafs, Cabinet-maker Subfcrihers in Gofport. Henry Lys, Efq; Henry Corte, Efq; Mefi'rs. Daman and Page Mr. Carver, jun. merchant Mr. Whitcomb, Upholder Mr. John Kent. Builder Mr. Sprint, mafon Suh/enhers in ChicheJieJ\ H. Coote Molefworth, Efq; Edward Long, Elq; Thomas Sanden, M. D. John Bayley, M. D. Mr. W, Clowes, Maft. Acad. Mr. W. Humphry Mr. Andrews, Builder Mr. Weller, jun. Cab. Maker Mr. Garrard, Ditto Mr. Houniom, jun. Merchant Mr. Rand Siibfcrib.^rs in Guiicford. Mr. Higginbottom, jun. Rev. Mr. Cole, Mali. Gra.Sch. Rev. Mr. Anderfon, Ditto Mr. RuiTell, D rawing- inafler Mr. Killick, Cabinet -maker Mr. John Terry, Carpenter Snbjcribers in Birmingham, Mathew Bolton, Eiq; Mr. Jofeph CLeen, Merch. Mr. Fergufon, Merch. 6c Mann. Mr. Steward, Attorney Mr. Price, Button Manuf. Mr. Lewi.<;, Builder Mr. Wheeley, Jun. Coach-M. Mr. Millar, Painter Mr, Clay, Painter and Manuf. Mr. Humphrys, Merchant Mr, Winftanley, M ifon Mr. Higginfton, Builder Mr. Cottrel, Manufadlurer Mr. Samuel Ford, Merch. Mr. Faulconbridge, Merch. Mr. Meredith, Attorney Mr. Horton, Builder Mr. Thomas Gill, Manufa. Mr. .Lee, Manufadurer Mr. Fletcher, Painter &Manu. Mr. Charles Marllin, Manuf. Mr. Joleph Kendrick, Manuf. Dodlor Marriot Mr. Hedlor, Surgeon, Mr. Charles Birch Mr. Hammond, Button-manu. Mr. Tho. Green, Ironmonger Mr. Harvey, Jun, Merch. Mr. Jeremiah Vaux, Surgeon ( ) and j Manufadt. Builders Mr. John Fox, Manuf. Mr. Levi Perry Mr. Pickering, Maft. of Acad. Mr. Bloomer, Attorney Dodtor John A(h Mr. Sampfon Llyod, Jun. John Taylor, Elq; Mr. W. Taylor, a Merchants Mr. Cl. Johnfon, |- Mr. Riccards, ‘ Mr. Charles Cartwright Mr. Starton, Merchant Mr. Palmer, Merchant Mr. John Dearman Mr. Winwood, Merchant Mr. Sam. Cope, J Mr* John Cope, J Mr. Allport, Painter Mr. Eglington, Mafon Mr, Bromley, Mai on Mr. Nixon, Builder IVir. Kennedy, Surgeon Mr. Henry Henn, Merchant Adr. Dickenfon, Mercer Air, Werton, Manufitdlurer Mr. Dudley, Cabinet-maker Mr. Dolphin, Attorney Mr. John Goodall, Merchant Mr. John Scape, Land Suiv. Adr. Eiv'ins, Builder Air. Cooper, Cab-mak. & Uph. Mr. Giles, Painter & Dr. Adr. Doctor W ithe ring Mr. Illownes Mr.W. Burr, Chaf. &Alodeller Mr. Thomas Will more Mr. Hazcldine, Drawing Air. Mr. Rufton, Japaner, &c. Mr. Js. Brown, Japaner, &c. Mr. John Weft wood, Alanuf. Mr. William Bache, Chymift Air. William Dowler, Merch. Air. J. Haywood, Brafs- founder Mr. John Taylor, Builder Adr. Gibbs, AdanufaClurer Air. Hickin, Gun -maker Air. Robinfon, Painter Adr. Horne, Alanufadlurer Mr. Adynd, Buckle- maker Air. P'ord, Scale-maker. Snbjcribers in IFoherhamptofi. George Moleneux, Elq; Mr. Falconner, Attorney The Rev. W. Robertfon, D.D. Air. Jofeph Lane, jun. Fadlor Dodlor Steward Air. Flendcrfon, Cabinet-mak, Air. Randle W’alker, Ditto Air. Brawne, mafon, Kidderm. Mr. John Smith, Builder Air. Jofeph Turner, mafon Air. Jofeph Barrington, Gent. Subferibers in IForceJcr. John Berkeley, Efq; Air. J. G reenwood, Painter Mr. Gwin, Architect; Air. James Rols, Engraver Adr. Leonard Mole, A^afon Air. John Soule, Ironmonger Adr. Graifiger, Drawing-Mad:.- Mr. James Fell, Schooimafter Adr. Stephens, Builder Mr. Berney, Gent. Mr. Cropper, Cutler Snbjcribers in Olocejfer. John Ready, Efq; Rev. Dr. Tucker, Dean of GI. Rev. Adr. John Newton Adr. Thomas Weaver, Gent. Adr. W. Price, Builder Air. J. Ricketts, Statuary Air. Bubb, Builder Adr. Braidwood, Painter ' Air. Bryan, Mafon. Air. W. Bratt, Button-manuf. Air. Will. John Banner Air. Swanwick Air. John Bingham, Alerch. Air. James Rickard, Alanuf. IMr. John Hallen, Manuf. Air, Will. Hawkins, Alerch. Air. Gough, Button-manufac. Mr. Winheld, jun. Iron-manu. Mr. Steuart, Attorney Air. Sanders, Sword-cutler Air. Barker, Attorney Mr. William Collins, Builder Air. Tompfon, Builder Mr. James Bingham, Manuf. Mr. Jofeph Bell, Fainter Kingjion on Hull, Jofeph Sykes, Efq; Ilaac Broadiey, Elq; Thomas Williamlon, Efq; Henry Alaifter, Efq’; Henry Stephenfon, Efq; John Shipman, Gent. Air. John Dixon, Merch. Air. J. Pdetcher, Painter Air. Johnfon, Surveyor J. GrimRon, Efq; Beverley Air. Carr, AtchiteCt, Turk Air. T. Atkin fon. Ditto, York Air. '/ ay lor, Ditto, Tork Air. Hutchinfon, Surv. IVhitby bfhe Library, at Leeds. BOOK I. OPTICS or VISION. SECTION I. Of L I G H T and COLOUR. \ T O treat on any Science, properly, it is necelTary to begin with the Elements of it, or firft Principles ; which is the reafon, as I fuppofe, that fome writers, on Perlpeftive, have prefaced their Works with a chapter on Optics and Vifion, Some of whom, not daring to advance any thing of their own, on the fubjefl, have favoured the World with extracts from Sir Ifaac Newton’s and Smith’s Optics; which, indeed, is very little to the purpofe, as the much greater part is no way fubfervient to Perfpedive ; conlidering it not merely optical, but purely a mathematical Science, iuppotied on the firm Balis of Geometry. The ingenious Mr. Hamilton, in his Stereography (a work which does great credit to the Author) to which I acknowledge, myfelf indebted, has given fome judicious obfervations on the fubjedf of Vifion ; and fome, which arc rather ex- ceptionable. But, the Theory of Colours (if that can be called a Theory, which is lo little known, and nothing in it demonftrable^ he, very wifely, declines en- larging on ; as not being fubfervient to Perfpeclive, nor indeed to Painting ; I mean the Theory of the Prifm, which. Sir liaac Newton and others have fo copioufly expatiated on. I would by no means have it thought that I intend, or have the leaf): defire, to derogate from the charafler of fo great a Man, whofe Name I highly revere ; but I am perfuaded, that the purfuits of the greatefi: Men are, fometimes, in them- felves, trivial and merely amufing ; when the charader of a great Man is efta- blifhed, it gives a fandion of confequence to his moft infignificant amufements ; for, why may not the moff profound reafoners relax from their intenfe ffudies, and amufe themfelves, fometimes, with trifling matters? by which means, things of great utility have been brought to light; as muff have been the cafe in refped of the Prifm : nor do I find that he has perfeded what he had in view, but left his obfervations for others to improve on, having (in his own words) not incli- nation to take it up again, and purfue it further, as he had intended. Had the Theory of Colours, as deduced from the Prifm, been amongd the firf^ and chief of that great Man’s purfuits, I am much in doubt, if the reputation he has acquired had ever been eftablifhed, at lead on that Bafis; things of infinitely more importance to the Community, fixed his credit (mod delervedly) on the highed pinnacle of Fame ; for, what ufeful and neceffary knowledge has been communicated to mankind, by this acquifition to the Science of Optics ? which (with fuch, apparently wondrous, fagacity and penetration) he has explored and given to the \Vorld. A To Book I. OF LIGHT AND CODOUR. To define, with any degree of precifion and perfpicuity, what Light is, is not poffible ; feeing we cannot comprehend, and enter into an accurate difquifition of Fire; which is the only caufe we can conceive of Light. From the different and obftinate refrangibility of the different Colours produced by a Prifm, when applied to the Sun’s Beams, it is concluded, that Light is, in its nature, hetero- geneous; that it is compofed of Particles, of different qualities, which produce ve'ry different effe£ls on Objedls. I cannot, from any thing I have yet feen or read on the Subjed, give my aflent to that opinion; for, I believe, that what we call Light (the vivid glow occalioned by any luminous body) is perfeclly homogenial ; and that it is not compofed of Particles ; which implies that it is material, a Body; for, how elfe can it be a compofition of Particles? which, by means of the Prifm, are luppofed to be dilfevered, and fepavated from each other. But, what realon can be affigned, that a Prifm fhould have this moff: extraor- dinary power of feparating them; owfing to its form only not the matter of which it is compofed : But it is not owfing to either, as it is manifefl by Expe- riment. Allowing the rays of Light to be thus fifted or feparated, what ufe is made of it, or what further knowledge is deduced from it ? ’Tis obvious, that the fame Objed, having its Surfaces differently difpofed, or fituated, exhibits different Colours, according as its Surfaces are fituated to the Light; that is, to that quarter, from which any thing luminous caufes an illu- mination of them ; and thole parts, which are mofl diredfly oppofed to it, are moft intenfe in Colour; whilft the oppofite Faces (the Light being obffruded by the Objed) are almoff: deprived of Colour; which they would be entirely, but for other Objeds in vicinity with and oppofed to them : which other Objeds being flrongly illumined, are faid to refled Light to them. Hence, a conclulion is drawn, that Objeds have, in themfelves, no inherent Colour ; but that, all the fenfe we have of different Colours, in the fame or in different Objeds, is entirely owing to their different qualities and modifications (or, rather, to the conffrudion of their Surfaces, fimply) by which, they are fitted and difpofed to refled different coloured Rays, fome more copioufly than others. But, it does not neceflarily fol- low, that they have no inherent Colour ; for although Objeds, when oppofed to the Sun (that is, when nothing opake intervenes) exhibit very different, and more brilliant colours than otherwife ; yet, if its Rays are obftruded, inftantaneoufly (the Light around them being then luppofed ftagnated), they ftill exhibit Colour, and of the fame kind, or hue, though lefs vivid, and greatly inferior in the degree of it. The chief Argument advanced in favour, and the only reafon, of weight, affigned, in fupport of their Hypothefis, is, that Objeds exhibit no Colour, when they are entirely deprived of Light ; which argument cannot be denied; becaule, no Ob- jed, in fuch Cafe, is perceptible by means of Vifion; without I ight our Optics are of no ufe at all. It is the fame if we clofe our Eyes, when Objeds are fully illumined ; but, certainly, the Colour, as well as the Objed, remains, whether, by fhuting our Eyes, wc perceive it or not. I look on this as a fpeclous, and fubtlc, not to call it a fophiftical way of reafoning ; fince, without Light, there can be no Vifion; and confequently, the fenfe of Colour, more than of the Figure, Magni- tude, or Situation of Objeds, cannot be communicated. But, if Light be con- ffdered as a Medium exifting of itfelf, without the Luminary, how can there be a privation of it, if it fills the whole furrounding Ipace, as Air does? which, feeing that. Air is alfo confidered as a material Body, is fomewhat repugnant to the eftablilhed and univerfally received Maxim, that, two Bodies cannot fill or occupy the fame Space, at the fame Time. And, if Light be a Medium, of w^hat ule is it, without the Luminary ? or, how can there be a perpetual emiffion, from the Luminary, the whole Space being, every where, filled with Light ? and alfo, if it be material, what becomes of it? To fuppofe that it is abforbed, by the Earth and other Planets, would be ridiculous in the hlgheff: degree ; becaufe, the whole Surfaces of all the Planets bear no fenfible proportion to the immenfe Space between • A Prifm, as defined by Euclid (Def. 13. 1 1.) is a Solid, contained by Planes, of which, two oppofite are equal, fimilar, and parallel ; all the rclt are Parallelograms ; fo that, every Parallelopiped is a Prifm, by which, being right- angled, no effeft of Colour is produced. Now, as it depends, folely, on the inclination ot one Plane to another, an acute angled Pyramid (in which all the Plaues are Triangles) or fruftum of a Pyrninid, is the fame, in refpedt of Colour, as a Prifm. them SeO. I. OF LIGUT AND COLOUR. 3 them, ill which there is no obftruclion to the progreffion of Light, flowing from the Luminary in all diredions. That Light fliould exift in darknefs (notwithftanding the Scriptures fay, that Idght was created before the SuiA is not only a direct contradiflion in terms, but to reafon and common lenfe, and the nature of things. Alfo, that Objedts ex- hibit no Colour, when deprived of Light, is certain ; becaufe they are no longer vifible ; but, fince every other quality remains, it is moft probable (Colour being perceptible by fight only) were they objedls of Vifion, in total Darknefs, Colour would remain likewife : For, although Colour cannot be perceived without Light, yet it is not the efficient caufe of different Colours, in Objedls compofed of different materials. In treating on Colours, the learned Boyle gives us a wonderful account of a blind Man’s diftinguifliing Colours by the touch ; which can only be by the afperity or roughnefs of the furface of the coloured Body, and which is con- Ifrued as favouring their Opinion. It may be poffible, that a difference in Colours, artificially laid on the furfaces of Bodies, may be felt; or, that the qualities of Dies or Stains may alfo affedl Bodies fo, as to make them fenfible to the touch. But, will any perfon venture to fay, that if Bodies, ffained or otherwife coloured, were polifhed or made equally fmooth, the difference, in Colour, could be perceived by feeling; or, that the natural Colours of Wood, Metals, or Stones, nearly of the fame qualities, except Colour, when equally poliflied, could be diftinguifhed by that mode of Perception ? I affirm they could not; nor can I give any credit to fuch affertions. Then, fince that. Bodies, equally hard and equally polifhed, exhibit different Colours, I fee no reafon for fuppofing that the caufe of their differing in Colour is owing to the different texture or con- ftruilion of their Surfaces ; or, that it is poffible to diftinguifh them by their afperity ; for, Bodies of equal hardnefs, and being equally polifhed, muff have Surfaces alike, except in Figure and Colour. Light, however propagated, has, 1 prefume, the fame properties and influence ; and, confequently, afls the fame on Bodies. I would afk then, what is the reafon, why Blue and Green are fcarce diflinguifhable by the light of Fire ? If the aflion and reaffion of Light, on the furfaces of Bodies, be uniform, by a certain law, according as they are fitted and difpofed to refledt any particular fpecies of colour- making Kays, whence arifes the difference in this cafe ? I fhould be glad to hear or read an ingenious difquifition of that point; as, I do not doubt, there are Men of fagacity and penetration lufficient to let it in the cleareft light, and make it as evident, to the underftanding, as any common Cafe in any Science whatever. To return to the Prifm. It is certainly true (and it is a curious and entertain- ing Experiment) that, through an acute angled Prifm, we perceive Objedls very differently coloured from what they really are by Nature ; alfo, by applying the Prifm, properly, to the Sun’s Beams, there is exhibited a curious and moft extra- ordinary Phaenomenon, viz. a diverfity of Colours, the moft lively and beautiful that can be conceived; and, the more uncoloured the Surface is, on which they fall, the more vivid they appear. Now it is certain, that here is a perception of Colour, on the furfaces of Bodies, which is not inherent ; and fo is there when the Sun’s Beams, only, fall on them: yet, we cannot from thence conclude, that they have no inherent Colour. The Colours which the Prifm exhibits are Red, Orange, Yellow, Green, Blue, In- dico, and Violet or Purple. But why is there, here, made a diftinition of feven Colours, when, in reality, there are not above three or four Ample Colours in Na- ture? (unlefs Black and White may be called Colours) viz. Red, Yellow, Blue*; and Green ; which may alfo be compounded of Blue and Yellow. Indico is Blue, Orange and Purple are Compounds, or mixed Colours. Now, if any Philofo- pher, or Artift whatever, can, from thefe three or four Ample and brilliant Colours, produce all the variety which we fee in Nature ; or, did the Prifm exhibit di- • It is faid in Emerfon’s Optics (Cor. 5. Prop. 9.), There are as many Ample Colours as there are degrees of Retrangibility ; and therefore an infinite number. And, hence (in Cor. 5.) it is plain, if the Sun’s Light conlifted but of one fort of Rays, there would be but one Colour in the World. In one place, he fays, Objeds have no Colour naturally ; and in another, they become coloured, by refleding the Light of their own Colour, more plenti- fully than others. 3 ftinftly 4 OF LIGHT AND COLOUR. Book I, ftin£lly, all the fimple Colours which are in Nature, only, without mixture, I fhould then be better clifpofed to give credit to their Theory : or, when I can have convidlion, that a mixture of all thefe Colours, together, in any ratio, will produce a perfedl White (as Snow) I may then be a perfedl Profelyte ; till then, 1 am perfuaded that I mull: diflent from their Opinions, I fhall, in the next Place, enquire, how a Prifm, particularly, has the wonderful property of feparating the different coloured Rays of Light. The Prifm is a body of Glafs, or it may be of Cryftal, or other pellucid, uncoloured Stone ; is the property then in Giafs, or Stone? No ; in the Figure that is made of it, only i amazing! that the difpolal of the fame Matter into different Forms fhould produce fuch very different Effects. Glafs, difpoftd into a portion of a convex Sphere, will make Objeds, feen through it, appear magnified ; and, being oppofed to the Sun’s Beams, will collect real Fire; altonilhing indeed! If the Surfaces be concave, Objefls, feen through them, appear lefs than they really are; in the forms of Prifms it has various effeds ; but we do not fee the effed of exhibiting Colours through a right-angled Prifm ; I mean, a four-fided one, whofe oppolite faces are parallel, and the Angles right ones ; neither has it that effect through the right Angle, of a right-angled triangular Prifm, but only through the acute Angles. The caufe, then, of exhibiting Colour is not in the Matter itfelf, or Figure, nor in the Surffees, fimply, but in the inclination of the Surfaces to each other ; for, being parallel, or at right angles, or nearly fo, they do not produce that effed. We may as well alk, why convex Surfaces colled Fire (but, that is well known) rather than plane or other parallel Surfaces, as, why a Prifm, whofe Planes are inclined to each other, fhould have the property of exhibiting the dif- ferent Colours I have mentioned, rather than one that is right-angled ; and I am perfuaded, that, with all the fagacity and penetration Man is endowed with, he will never be able to account, truly, for either. Yet, I do not condemn all enquiries into the caufes of the various effeds which we perceive in Nature, but think the purfuit rational, and truly commendable, when it is founded on certain Data, and real Hypothefis ; and the refult of our refearches produdive of real utility, deducible from it, as in fome other branches of Optics. It is afferted, by lome, that the perfedion of Telefcopes is owing to the Theory of the prifmatic Colours. As 1 am not converfant in the mechanical conftrudion of Lenfes and their application to Telefcopes, 1 cannot affirm that it has not been fubfervient thereto ; but, in my opinion, it has not the leaft apparent tendency to benefit Mankind accruing from it. Certain I am, that it will never Le of ufe to a Painter, to compound his Colours and form a Theory thereof ; by means of which, he may fooner, and with certainty, arrive at perfedion, in his Art; and I muff needs fay, that the attempt made by Brook Taylor, in the Ap- pendix to his fecond part, does not fhew the Dodor’s judgment, in that, to be of a piece with the reft of the Work. The Sun’s Beams paffing through Glafs, whofe Surfaces are neither parallel nor perpendicular to each other, or nearly fo, exhibit the various Colours, fpoken of above, in fome degree, according to the inclination of the oppofite Surfaces, and purity or clearnefs of the Glafs ; through the thick part, near the knob, in the middle of a table of Crown Glafs, I have often obferved the Phaenomenon, in a Window of thofe Squares : but, it is alfo obfervable, that the effed ceafes when the Sun ceafes to fhine on them. It is alfo certain, that Objeds appear coloured other wife than what Nature affigned them, when we look through Mediums denfer than Air ; whofe Surfaces, through which we look, are inclined to each other in certain Angles ; and confequently, there will be the effed of Colour pro- educed, in fome degree, which is not natural to Objeds, in looking through Objed- Glafles, or Lenfes, with which Telefcopes are conftruded; feeing that, their fur- faces are inclined, at the edges. The bufinefs, therefore, of an Optician, in this refped, is, fo to contrive and difpofe his Glaffes, as to diveft the Objed of the borrowed Colours, which do not belong to them, naturally ; and I could almoft affirm, that the perfedion, fpoken of, has been found out by repeated trials ; not from any certain eftabliffied Law, deduced from the Theory of the prifmatic 5 Colours ; Seft. I. OF LIGHT AND COLOUR* 5 Colours: for, if' a certain Theory had been eftablifhed, what hindered the im- mediate perfe(£lion of them, as foon as the caufe of the imperfedlion was known, and certain Laws eftablifhed, whereby the unnatural effed of Colour might be removed from the Field of View ? From what has been obferved, refpe^ling the effe£l of Colours produced by the Prifm (unlefs it can be proved that the particles of Air, through which. Light mud neceffarily pafs, are Prifms) to infer, that Nature has given no inherent Colour to Objects, is bold and afl'uming ; it is alfo groundlefs, feeing that the Colour, which is natural to each, remains when the Sun does not fliine on them, though it dilTers greatly in the degree of it. We may with as much reafon conclude, that Objects are much larger than we perceive them to be with the naked Eye, becaule they appear fo, when view’d through Glafs whofe furfaces are convex ; and, we may as well enquire how Vegetables, as Flowers, Fruit, &c. fpring out of the Earth, and are adorned with all that beautiful variety of Colours which we fee in Nature ; by what means they are continually varying in their Colours, from their firfl forma- tion, in embryo, to maturity ; as why, the Prifm exhibits the various Colours of the Rainbow, to which they are, in a great meafure, fimilar. Colours produced by alkaline Salts, &c. and mixtures of different Fluids, with all fuch like chymical operations, I look on, eq^ually, as entertaining Experiments ; but no way productive of any advantage, accruing from it, to the Theory of Colours neceflary for a Painter to know, in order to reduce it to real ufe, in praCtice. I fhall not, therefore, trefpafs any longer on the Reader’s Time, as it is entirely foreign to the purpofe of PerfpeCtive ; but confine my further obfer.vations to that part of Optics relative to Vifion, only* In refpeCl of direCt Vifion, it is certainly of ufe and fubfervient to PerfpeClive ; and effentially neceffary to give a clear Idea . of the Principles on which it exiffs* both in Theory and PraClice. Colour, whether inherent or incident to Bodies, is not material towards a perfpec- tive Delcription or Delineation ; ’tis the Figure of the Objeft, only, it contemplates. All folid Bodies, whatever, become objeCfs of Perception from their Figure or Colour. Figure is more particularly infeparable from our Ideas of Matter, than Colour; it being impoflible, in the nature of things, to have any notion of Exten- tion, abftraCted from the Idea of Figure, which limits the Space Bodies occupy or fill. It is the fame with or without Light, and may be perceived by the fenfe of Feeling only. From the. known external Figure, an'dfituation of an ObjeCb, may be delineated, by mathematical rules, a true perfpeClive Repreientation, on a Plane ; which, by the help of Light and Shade, will raife a perfect Idea in the Mind, of the figure of the Objeft', and the different forms and pofitions of its Surfaces, in refpedt of each other, without the afliffance of Colour; but, to exihibit a true and natural Pifture of Objects,- and create a perfe6t Idea in every circumflance, Colour is abfolutely necefl'ary ; and is the laft degree of perfeflion, with which the omni- fcient Creator has embelliffied Nature, that can be given to a Picture. Lights confidered as a IMpdium, by which Vifion is convey’d to the Eye, is of too refined a nature for my fpeculations. . But, as the great Author of Nature has given a portion of Reafon’ to every human Being, we have certainly a right to make ufe of that Reafon ; .and, if we will exert it, properly, it is poffible, nay certain, that one Perfon ipay penetrate as deeply into the myfteries of Nature as another, though not blefs’d with quite fo much Learning. We are told by feveral great and profound Philofophers (to be a Mathemati- cian is not neceffary) thi^t vve perceive Objeds, only by means of Rays of Light, refleded frpm every point, in their Surfaces to the Eye; which enter there, and form an Image-, or Pidure of tjie Objeds perceived, on the Retina, or fine Membrane which furrounds all the back part of the Eye, internally. The Retina is faid to proceed diredly from the Optic Nerve, which dilates or fpreads itfelf as before- mentioned ; and, the impreffion being made thereon, we are further told, is con- veyed, by the Optic Nerve, to the Brain, or feat of Perception. B But, 6 OF LIGHT AND COLOUR; Book L Buti why do they flop here? I expeded. and ftiould be glad to be conveyed into the inmoft recefles of the Brain, and be (hewn or told, how the Image of the Objedl, on the Retina, is there perceived ; for I mud: own it is aftonifh- ing ! that, from what is perceived or felt within, we (hould have a true Idea of . the Figure, Colour, and Magnitude, Situation and Dlftaiice of Objeds, which are external, or lituate without the Eye.- NoW< after all this parade, and pompous difplay of great fagaclty and deep pene- tration, what does the Sum total amount to ? Why, that the Objed is perceived, i.e. the Mind is fenfible of the exiftence of inch Objeds as it perceives; and that, the Vlfion of them is conveyed in right or dired Lines from the Objed to the Eye, or from that wondrous Organ to the Objed ; but how, or in what manner, remains as much a myftery as before, So that, after all which has been faid on the Subjed, and, allowing the Image, formed on the Retina (inverted or otherwife) as perfed as they pleafe, what nearer are we ? where is the Perception of that Image or Pidure of the Objed ? How the Mind, or Soul, perceives the Image on the Retina, any more than the real Objed, we are juft as much at a lofs to account for as ever. Jn vain, therefore, does Man torment himfelf, in endeavouring to explore the hid- den myfteries of Nature, which are for ever hid from our refearches. Let us then purfue what is within our reach, and not an airy Phantom, which will ever elude the purluit. In mathematical Sciences we have fome certain Data, whereon to frame Hypothefes ; and although we muft not exped perfedion, except in Theory, yet, we can difeover lo much of Truth as gives us fufficient encouragement to purfue it, and makes ample amends for the imperfedion of human nature. 1 do not intend, nor (hall I attempt to explain the nature of Vifion, or enquire into the true caufe of it; being fully convinced of the infufficiency of our reafoning facul- ties for fuch a dlfquifttion. It is fufficient, for the fcience of Perfpedive, to know, or even to fuppofe that it is conveyed to the Eye,^ the feat of Sight, in Right Lines. But, whether it be by means of Rays* of Light, refleded from all parts of Objeds to the Eye (as is the general Opinion) or whether the Allwife former of the Eye has given a pow'cr to that truly wonderful and amazing Organ, to convey to the Mind, by Vifion, the perception of Objeds by other means, I do not mean to make the fubjed of my enquiry. Yet, 1 muft acknowledge, that I have ftrong objedions againft the general Opinion, as it is now received and almoft univerfally aflented to; V!%. that the perception we have of external Objeds, from Vifion, is by means of Rays of Light, refleded from all parts of their Surfaces to the Eye ; and that thofe Rays are material, or compofed of Matter. But, as it is of no confequence, in Perfpedive, by what means Vifion is performed, fo it be conveyed in Right Lines, which may pafs for an Axiom ; feeing that, the re- fradion of the Rays (if there be any) in the Air which furrounds us, is fo very lit- tle, in Objeds which are at an immenfe diftance beyond the whole Atmofphere, it cannot be lenfible at any diftance we can perceive Objeds, within it, fuppofing the refradion uniform or regularly curved, between the Objed and the Eye. But, I ra- ther fuppofe the refradion of the Rays to be at the common Surface of two Mediums, only ; at entering any other Medium, denferor rarer, than that through which they firft pafs, and thence proceed again in Right Lines ; confequently, there is no Refradion at all, within the Medium ; either in Air or Water, I fuppo(e Vifion to be convey’d in Right Lines, if the Objed and the Eye are both in the fame Medium. At the fame time, I would not have it thought, that 1 fuppofe any Rays to pa(s, by means of Refledion, from opake Bodies ; and in luminous ones, the Light, which proceeds from them, filling a concave Sphere, as far as they can be feen, I do not fuppofe to proceed in Rays, nor in Planes (as a modern Author has fuppofed) ; the Hypothefis is abfurd ; feeing that, the whole Space is illumined in every part, it is impolfible, that from every Point in the furface of the luminous Body, Rays can proceed to an immenfe diftance, filling all the Space between. * I r ake ufc of this Term, though I have not the leaft conception of a Line, or thread of Light, called a Ray ; m ch lefs of a Pencil, or bundle of Rays, Suppolc Se 8 ;. IL OF LIGHT AND COLOUR. Suppofe every Point in the furface of the Sun to emit a Ray of Light, proceed- ing in Right Lines from the Center ; it is evident, that every Ray, at the diftancc of the femi-diameter, will fill a Space four-fold of its firft dimenfions, viz. in a duplicate ratio of the Diftance ; at the diflance of the whole Diameter, it muft fill a Space nine times as large; i. e. it increafes in proportion to the Squares of the Diftance j what, then, will be the magnitude of a lingle Ray of Light, proceeding from the Sun to Jupiter or Saturn, before it reaches thofe Planets*? or how can the whole of the Space it occupies be filled (as it certainly is) with Light, which proceeded from a fingle Point, in the Luminary only? Gan one individual Ray be (battered into thoufands, into millions, lying clofe to each other? what direction do they proceed in, from the Luminary? not in the diredlion of a Right Line from the Center, I prefume. I • , I know' it may and will be alledged that it expands as It proceeds, ftill filling the whole Space. 1 prefume then, it cannot proceed in Rays, or Right Lines from the Center of the Luminary, as it is imagined. Air may be rarified and expanded to a very great degree; alfo, Water and other Fluids are expanded, as in fteam, by Fire, or, in Mifts and Vapours, by Exhalation, till it floats on Air. But can one drop, or the minuteft particle of Water be expanded Infinitely? as a Ray of Light, emited from one fingle Point in the Luminary, muft, in this Cafe,’ neceffarily be; elfe, how could we fee thofe Stars, whofe diftances are, to all fenfe, infinite? Thefe things^ which may amufe and pafs for orthodox with the generality of mankind, cannot be fatisfadtory to a Perfon who dare think for himfelf, and is furnifhed with the means to do fo. Therefore, fince all which can be faid about it is, at beft, but an ingenious conjedlure, and fince it is in no wife fubfervient to Perfpedlive, 1 fhall at prefent take my leave of the Subjedl, to purfue that part or branch of Optics, which is eflentially fo, refpedting diredt Vifion; and after that, in a feparate Chapter, or Sedtion, 1 lhall give my objedtions to the received Opinion of the caufe of Vifion. * I find by calculation (frOm the Tables in Harris’s Ufe of the Globes^ which I fuppofe are as much to be depended on as any other) that the diftance of the Earth, from the Sun’s Center, is 212 femi-diameters of the Sun, nearly. Wherefore^ if, inftead of a Point, we fuppofe a certain quantity of Light to flow from one fquare Inch in the furface of the Sun; each fide of that fquare Inch will, at that diftance, be increafed to 212 inches, or 17 feet 8 inches; the fquare of which is 44944 fquare inches. Confequently, a quadrangular Pyramid, whofe Bafe is 1 7 feet 8 inches fquare, and its altitude, the diftance of the Sun from the Earth, is filled with Light, from one fquaire Inch, only j and confequently, each fingle Ray, proceeding from every Point in its Surface, will, at that diftance, be multiplied, or magnified 44944 times. At the diftance of Saturn (by the farhe Tables) a fquare Inch, and confequently a fingle Ray, will be encreafed or multiplied 4,149,369 times ; its diftance being 2037 femi-diameters of the Sun, nearly ; confequently, a quadrangular Pyramid, whole Bafe is 169 feet 9 inches, on each fide, and its altitude 777 millions of Miles, is filled with Light, flowing from every fquare Inch on the furface of the Sun ; an amazing circumftance indeed ! if it is fuppofed, or confidered to be a material Body. But, how far Light may proceed; from the Sun, beyond the realms of Saturn, is not yet, 1 preiume, determined. SECTION II. Of the ftrudlure of the EYE, its parts deferibed^ and a fhdr8 Introduction to the nature of VISION. T H E conftrudlion of an Eye, that moft wonderful organ of Sight, by means of which, we have intelligence of the Objedls which furround us, is va- rious; yet has nearly the fame Parts, in all the aninial Creation. The Parts are few, and apparently fimple, but aftonifhing in their mechanifm 3 as, indeed, are all the animal Funiftions; but, what is there in Nature which is not fo? The Figure of an entire human F.ye, within its Socket is globular, of fomewhat more than an Inch in Diameter; the Furniture of which are as follow, ■ * ' 7 Let 8 OF LIGHT AND COLOUR. Bock I. Plate I. Fig. 1. * Let H I K L be a vertical Sedion through the center of the Eye, which, in this Figure, is confiderably larger than Life ; in order, that the leveral parts may be delcribed with greater accuracy. - The external part, from H to I, is called the Cornea, or Horny Coat. It Is per- fedlly tranfparent, and is fomewhat more convex than the jnu t which is within the Socket, by means of a fine, clear, aqueous or watry Humour, between the Cornea and the Iris (the coloured Circle within). Immediately behind the iris, (ghik,) through which there is in Aperture, in the. middle, to let in Light (or Vlfion by means of Light) is a whitifh Subfiance (MN) like a flrong Jelly, or cold Glue, of a modeiate confiflence. It is as clear and pellucid as Cryflal ; from which, it is called the Cryflalllne Humour. It is in every refpecl of the Lme nature and ufe as a double convex, microfcopic Lens ; and is faid to be more convex on one Side (the inward)- than the other. Its ufe , is fuppofed to contradl the Rays of i.ight, by which Vi- fion is convey’d, to a Focus, beyond this Humour, in the Center of the Eye, at E ; fromwhence, they become’ diverging, and fall on the bottom or back part of the Eye, at azb. i ‘ The Cryflalline Humour (MN) is faid, by thofe who are acquainted with and accuftomed to difledlions of the Eye, to be furnifhed with mulcuiar Fibres ; by means of which, it is made either more or lefs convex ; or, by contrafling them, is brought forward or otherwife as occafion requires ; in order, to render the Images, or Piflures of Objefls perfefl, in the bottom of the Eye, as the Objects are nearer to or further from the Eye; all whch feeras confonant to Reafon. ..The Aperture or Pupil (h i) is alfo, by fome luch means, expanded or contrafled, as there is occafion for more or lefs Light to enter. I have oblerved, in the Eve of a Child, in a Room juft light enough to dlfcover it, tire Pupii enlarged to three tenths of an Inch in diameter; and immediately, on bringing into a tu-U Light,: it was contriufted to one tenth, or lefs ; amazing Strufture ! and- this is perfoi mcd in- voluntarily, whether we will or no. There can fcarce be any Pferlon. who has noD' made the experiment ; on going into a darkilh Room, he can fcarce, at firff, diftinguifh any thing ; when 'prefently, he will, after the Aperture in the Iris is open’d to a proper dimenfion, difcern Objefts dlflinclly ; and again, on going- into a full Light, he cannot bear to look on any illumined Objefl, till the Pupil' is contrafled ; it is even painful to the Eye, at firfl, going out of a dark Rooin^ into a flrong Light,. efpeciaUy if the Sun fhines out. .. . All the back part of the Eye, within, is lined with a fine Membrane of a mofP curious and delicate texture ; called the Retina, from Its relemblance of net-work.! The large cavity, V V, which is bounded by the Retina, backward, and the^ Cryflalline Humour, before (which is contained within it) is rilled with a glutinous Fluid like the white of an Egg; being perfeflly pellucid and uncoloured, it is- called the vitreous or glafly Humour; in which the Rays converge, at E, the cen- ter of the Eye; and, croffing each other, in that point of contraClinu, they fall diverging on the Retina, and form the Images, of Obje^s therpon ; the fenlation, of which, is' fuppofed to' be* communic;lted, by the Optick Nerve (which is con- nedted with the Retina) to the Senforium, in the Brain. As the rnanner in which the Rays of Light are. rerfi-adled, in paffing through the various humours of the Eye, is entirely conje<51ural ; feeing that, through a final! pin-hole, applie.d clbfe to.the Eye, at q, there may be perfect Virion, of an Obj Cl of any magnitude ; in which Cafe, the whole Syftem of Rays fall on the Cornea in one Point; how, then, are they refradted ? As, this Eflay is not intended as a Treatife on Optics, T flaall not infift on anything; but, fhall only fuppofe tJie Vifiial Rays (when ‘the Lye is naked) to converge and crofs each other in the center of the Eye, at E (without confidering their Refradlions) from whence, 1 fiippofe, they diverge lii equal Angles, falling on the Retina, at. a c b, in, tfie back part of the Eye. DE: 1 Hate I ® ^W.A. J ^ G H ^ t 1 4^1 %• 9 OF V I ^ S I O N. DEFINITIONS. \ 1. The AXE of the EYE is a Right Line pafling through the Center of the Eye and the Center of the Aperture or Pupil. As c E C. The Point g, where the Axe cuts the Retina, is the place where Vifion is perfect j which point is varied as the Eye is moved, or its Axe directed to diferent Objedts, or to different parts of an Objedt, 2. VISUAL RAY. If A B be fuppofed an Objedl of Sight; the Right Lines A E, B E &c. from all parts of the Objedl to the Eye, or to its Center (E) under which, or by the means of which, the Objedl is feen, or fuppofed to be feen, are called Vifual Rays. 3. OPTIC ANGLE. By Optic Angle may be underflood, either a plane or folid Angle. If the Objedl AB be confidered as having no dimenfions but length, only ; i. e. if it be a Right Line, fimply ; the two Vifual Rays E A, E B, from the Eye to each extreme, form a Plane Angle, A E B ; which is the Optic Angle, under which, the Line A B is feen. And, A E D or D E B is the Optic Angle of the Segment A D or D B. 4. CONE, or PYRAMID of RAYS. If the Objedl of Sight be either glo- F'g* 2* bular, or circular; as ABG; the Vifual Rays, EA, EB, &c. from the Eye to every part of the extremes of its Surface, being towards the Eye, form a Figure refembling a Cone, which is called the Optic Cone. But, when the Objedl is either a right lined Solid, or Plane Figure, as CD; the Vifual Rays EF, EC, EH, &c. form a folid Angle, compofed of feveral plane Angles, F E C, C E H, &c. which, being of a pyramidal form, is called a Pyramid of Rays. The, Vifual Rays croffingeach other, in E, and, from thence, become diverging to the Retina, each oppofite pland Angle being equal, -t there is formed a fimilarf 2. i. El. Pyramid, cEd, which is called the oppofite Pyramid ; and E is their common Vertex. The Figure, a b or c d, which is the Bafe of the oppolite Cone or Pyramid of Rays, is the Image or Pidlure of the Objedl, A B or C D, on the Retina. The correfponding Charadlers fhew in what manner the Image is Inverted. Let AB be fuppofed an Objedl, diredl before the Eye, that is, perpendicular to its Axe (EC). It is fuppofed to be feen, by means of the Vifual Rays E A, EB, EC, &c. from every part or point in the Objedl, towards the Eye, whether it be Line, Surface, or Solid ; which, from the point of contradlion (E), proceed diverg- ing, to the back part of the Eye, where they fall on the Retina, and form the Pidlure, az h, of the Objedl A B. It may be alledged, that it is impoffible for the Points A and B to be feen, at the fame time ; feeing, the Rays, from thofe Points, do not enter the Pupil. I grant they cannot ; nor F and G, diflindlly ; nor indeed, little more than a Point, with- out moving the Eye; we have but a confufed intimation of the furrounding Ob- jedls, at a little diftance from that Point, to which the Axe of the Eye is diredled ; of which, every Perfon’s own Experience mull: foon convince him. But, the quick tranfition of the Eye, or the diredtion of its Axe, from one part of an Objedl to another, is fo little attended to, that, many imagine the field of View is much larger than it really is. As for the refradlion of the Rays, through the various Humours of the Eye, I fhall leave for the fpeculation of thofe, who are blels’d with a more fertile imagination than I have any pretenfion to. C The 10 OF VISION. Book I. Plate I. The Image of the extreme point A, by means of the Vifual Ray A E, falls on Fig. I. the Retina at a, and the other extreme, B, is reprefented at l>i andCE, the Axe of the Eye, conveys the Image of the point C to c, in the center of the Retina. From which it is evident, that the Pidure of an Objed, formed on the Retina, is necefl’arily inverted. Nor is there any thing furprizing in this, as fome may imagine. For, fuppofe A the top, and B, the bottom of the Objed. It may be imagined, from the inver- ted pofition of the Image, acb, that the Objed muft iieceflarilly appear upfide down. But, if we confider, that the fenfation of the Point A, being perceived or felt at a, is by the diredion of the Vifual Ray A E, which determines its place in the Objed ; the Mind, by long experience, having acquired a habit of determining that part of an Obejd, perceived at a, to be above, and that at 3, below ; by the fame reafoning it is manifeft, that the points of Objed on the right hand are pic- tured on the left, and thofe to the left, on the right; and which, is no matter of furprize at all ; no more than, that, by the fame experience, we have acquired an Idea of the diftance of any Objed perceived. If a Perfon, who was born blind, could, when grown to maturity, be made to fee, he would have no Idea at all of Diftance, or the fituations of Objeds ; which way was up, or which down ; and would as foon attempt to lay hold of the moft diftant Objed, which he perceived by the fenfe of feeing, only, as others which w’ere near him ; it being impoflible to diftinguifti, merely by Sight, whether the Point perceived at c, on the Retina, be at C or C, or at any diftance beyond Q in the diredion of EC. But, by Experience and a familiarity with Objeds, accord- ing to their known magnitudes or diftindnefs of parts ; or, according to the length of Ground, which we imagine to lie between us and the Objeds, we judge them to be at fuch or fuch a Diftance. Whereas, when we look at the Heavens, one Star appears as far off as an other ; nor can we form the leaft Idea of their diftance from fight, fimply ; for, a Star may as well be ten thoufand millions of Miles off as ten Miles, its Magnitude being in Proportion to its Diftance. (See Fig. 7 . A, B and C.) The Art of Seeing is acquired, regularly and progreffively, as all other Arts or Knowledge whatever. Do not we fee a Child attempt to catch the Moon or other ftriking Objed, though at an immenfe Diftance ? but, growing up in a familiarity with them, by common and frequent experience, they become fo intimately con- neded with our Ideas, that we form a judgement of their Magnitudes and Dif- tances, inftantaneoufly, at firft fight ; and alfo of their Situations, in relped of themlelves and of each other. It may, to fome Perfons, appear ftrange to call Seeing an Art ; but that it is fo is certain, although Nature has furnilhed us with that moft wonderful Organ of Sight, the Eye ; which, being perfed, is not poflible to be open (with fufficient Light for the purpofe) but we immediately fee what is before the Eye. A Child lees as foon as it is born ; but it fees, without judgement, or difeerning what it lees. So would a Perfon born blind, and continued fo, till he was mature in Judgment, in every other kind of Perception. Let us fuppofe fuch a Perfon, immediately made to fee. What Ideas he would have of what he merely faw, at firft opening his Eyes ’tis impoflible to fay ; but, of this I am well alfured, that he could not diftinguifh the moft familiar Objeds, without Experience ; as Figure and Colour are things of which he had formed no conception. There are not many more oppofite, regular Objeds, than a Cube and a Sphere, or Globe; nor any, more eafily dlftinguilhed, by feeling; yet, even thefe could not, with certainty be known, merely by feeing them, which was one and which the other ; becaufe. Experience had not taught, in what manner fuch Objeds af- fed the Eye. In refped of Colour, ’tis impoflible he could have any Ideas, by which he could diftinguifti them; they would be, to him, as if never exifting before. Of Diftance, he could have no Conception ; for being wholly unacquain- ted with the real Magnitudes of Objeds and the efieds which Diftance has on them, their relative and apparent Magnitudes could not be difeerned, fo as to know that one was farther off than another. This /*■ Sea. II. O F V 1 S I O fJ. it , This may appear ftrange, to thofe who have not confidered it with attention j but I maintain it to be a truth, founded on found reafon. For, as he knows no- ( thing of the difference between real Magnitudes, and the apparent Magnitudes of Objefts, according to their feveral Diftances j he could never imagine that a Man, or a Horfe, which he faw at a Diftance, was of the fame kind as thofe he faw near him ; till Experience had (hewn, that the difference in apparent Magnitudes is^ / merely, the effe<£t of Diftance* So, of any other Objedl, as a Houfe, for inftance, into which, as he finds he can freely enter, being near, he could never imagine that to be a Houfe, which by means of its Diftance, appeared not fo big as his Head. Confequently, lince he could not imagine that a Man, or a Houle, at a Diftance, was a Man or a Houfe, of the fame Magnitude as others which were near him, he could form no Idea of Diftance, by fight ; befides, the difference in Effed and Colour, occafioned thereby, are things entirely new ; of which, he had not, till now, any Ideas. Nowi the judgement we form of Diftance, &c. being acquired early, and grown up with us, is fo little attended to, that at firft thought of thefe things, we fuppofe it to be merely from Sight ; than which, no Seilfe is more delufive and uncertian* The deceptions of Vifion are many and frequent ; as, when we look at Objeds through refrading Mediums, i. e. through any tranfparent Medium whatever, fo- lid or fluid ; according to the different denfity of the Medium, or to its Figure* See. we frequently imagine an Objed to be nearer or farther off, larger or fmaller than it really is ; or to be where it really is not j nay, it is poffible, by Light and Shade, judicioufly difpofed, and the affiftatice of Perfpedive, to deceive both the Eye and Judgment. Of all the Senfes, that of Sight is by much the nobleft ; it is, neverthelefs, the moft deceptive. He who can moft deceive the Eye in the art of Delineating and Colouring is efteemed the greateft Genius. It is poffible, to deferibe the reprefen- tations of many Objeds fo very accurately on a plane Surface, that (as I have al- ready obferved) with the affiftance of Colour, ahd a juft diftribution of Light and Shade, they would, in the true Point of view, deceive a judicious obferver, and ap- pear to be real Objeds. As a proof of the deception, in Vifion, in refped of Magnitude, (imply, I (hall give a moft ftriking inftance * of which, though it has fallen within the cognizance of Thoufands, few, or none, that I have conver(ed with on the fubjed, have formed a juft Idea, or right judgment. I have afked feveral Perfons, what were their Ideas of the apparent magnitude of the Moon ? moft have concluded, that it appears, when full, as large as the crown of their Hat ; fome, as a common Plate, or fmall Di(h. I (hould be glad to know, by what ftandard luch a judgment could be formed ? for, unlefs fome Diftance,- of the Objed to be compared with, be determined, no Comparifbn can be made' ; as, it may appear as large as the Dome of St. Paul’s ; but not, I am perfuaded, at any diftance we can fee that Objed, by the Light of that Luminary *. Now if we fuppofe ulie Plate laid on a Table, before us ; or the Hat in one Hand, when we make the coinparifon, ’tis, perhaps, as random a fuppofition as can be conceived ; and, without fome determined Diftance, it is vague and undeterminatc. I know it will furprize many ; but I do affirm, that, at the greateft diftance we can hold any thing to compare with the Moon, it does not appear larger than a Silver Penny. Now, if Seeing be not an Art, to be acquired by Experience, how comes it, that fome Perfons fee with more judgment than others ? Since as I am perfuaded, that, all human Eyes (being perfed) fee alike by Nature ; and confequently, all would judge alike, of what they faw ; but, it is certain we do not ; of which, ftriking * inftances might be given. A Pidure, or an Objed, it is manifeft, muft ftrikq every Eye alike, in the fame Point of view; how is it then, that the moft judi- cious and accurate reprefentation, of a well-known Objed, does not communi- cate, to every beholder, the fame Idea of the Original ? • There was a Moon-Lighr Piece (by a capital Artift) in the Strand Exhibition, lall, in which (if I remcmbci well) the Moon appeared near that Objecl, and almoft of equal Magnitude. ? ■ ft 12 O F VISION. Book I. It would fcarce gain credit, to affirm, that, on alking a Pcrfon, who was admir- ing a large, and well-executed. Print of St. Paul’s Cathedral (one who had lived, fome Years, near that well-known Objed) what it reprelented ? When, after a flridt attention to it, I was anfwered, with an Interrogation, is it Weflminfter Bridge ? ’Tis true, it was a Female ; but, Ihe had Eyes, and could diftinguiffi- between a Cow and a Horfe, at firft fight, as well as any other Perfon. It is, to me, almoft impoffible to account for this ; as I have known many Children, of thre or four Years old, fpeak with certainty of very paltry Reprefen- tations. I think it can arife, only, from their inattention to the Objeds they fee ; as it is impoffible, that a Child can have had more Experience than a Perfon grown to maturity. Neverthelefs, it is certain, that fome have much more Judgment of what they fee, with lefs Experience ; or, at leaf!:, they acquire it much lefs time. That Vifion is conveyed in Right Lines to the Eye, I prefume no peffon will attempt to difpute (I would be underflood to mean, of fuch ObjeiSls as are fituate on or near the Earth; and when no other Medium, but Ai,r, is between the Obje(^ and the Eye). I flaall, therefore, give it as_a general Axipm; which, neverthelefs, may be illuflrated or ocularly demonfirated, by flrainihg a fine filken thread, to a Right Line, from any Point, as F, of an Objetl (DF) to the Eye. Thenj if any other Objefl, as Q R, be interpofed, as foon as it touches the Thread E F, they will appear to be in contact ; and if it be fo interpofed, as to hide the Point F, the Right Line or Threajd, which may reprefent a Vifual Ray, will be refra£led, or broken into two Right Lines, ER, RF forming an Angle, ERF. Hence it is evident; that the Points F and F; alfo, the Point C or C being in the fame Right Line with Cz, will perfedlly coincide with each other, to the Eye, at E. S E C T r , O N III. Containing a brief 'Theory of direct VISION. THEOREM I. O BJECTS appear to have that Proportion to each other, refpec- tively, as the Angles under which they are feen. Let acb repr^ent a feflion of the Retina, a portion of a concave Sphere ; and, let A B be an objedl diredl before the Eye ; divided, in any Ratio, in D and F. Fig. I. Dem. It is evident that the Objefl A B, by means of the. Vifual Rays E A, EB, is feeir “under the Angle AEB; and the feveral Parts of AB, as AF, ED, and D'B, are feen under the feveral Angles A E F, &c. ■•*-Eut, the oppofite Angles ^Ef, fEd and dE^, are, refpedlively equal t6 the Angles A EE, FED, and DEB — — ■ — 2. i. El. And, the Images <3f, fd and d^, on the Retina, of the Originals, A F, F D, and DB, are the meafures of thofe Angles, refpedively (Th. P. Ang.^) the Ptetina being fuppofed a portion of a bphere; each part is, therefore, equally diflant from the Center; * But, the Ark ^ f is to the Angle f, as the Ark fd is to the Angle fEd, and as d 3 to dEA — — — — — 19. 6. El. * (Th. P. Ang.) refers to the Theory of Plane Angles, in my Trcatife of Geometry. I- Therefore, Sea. III. *3 b F D I R E C T V i S I 6 N. Therefore, the Images or Piaures on the Retina, and confequently, the apparent magnitudes of Objeas a!re in the fame ratio, or proportion, as the Angles they fubtend at the Eye. Q. E D. Nor is there pccafion to have recourfe to the Images on the Retina ; but,’ allowing the Vifual Rays EA, EB, &c. to be Right Lines, aiid an ark of a Circle, of any radius, being drawn, curing thofe Rays; the portions of the Ark, 1 m n o, intercepted between the Vilual Rays, as 1 m, m n, &c. meafure the feveral Atigles AEF, FED, &c. refpeaively ; and confequently, the parts A F, F D, &c. of the Objea A B, will appear to the Eye, at E, in proportion to thfe Arks 1 m, m h, &c. which are equally diftant from the Eye. Alt remote Objefls appear equally diftant from the Eye ; as the Planets, Stars, &c. appear as if they were all m the fame concave Sphere ; of which every Eye is the Cenl;er. So likewife, when we ftan'd in the line of dire£lion of two, three, o’r more Objects, they appear to touch each other; nor could we judge, merely from fight, which was fartheft off, or which the largeft. For, fuppofe the Vifual Rays E F, ED, E B, produced to F, D and B any -other Objects, as D F and B D, being in the fame diredllon with F D, and D B, ar.d touching the fame Vifual Rays (E F, E B) will coincide with FD and DB; the lefs will hide the greater from Sight, and appear to be of equal magnitude. 2 . I am fomewhat furprized at what Mr. Hamilton has advanced, in Art. 6, Sect. I. where he fuppoies, that Objedls do not appear In proportion to the Angles they fubtend, but in proportion to the Tangent of half the Angle; and imagines the Retina to be, nearly, a Plane, in the center of the feat of Vifion. How Mr. Hamilton could conceive and endeavour to account for fuch a ftrange and unwarrantable Opinion 1 cannot imagine. If, as he has truly aflerted, the fpace on the Retina, fo far as there is diftindf Vifion, does not exceed an Angle of one or two Degrees, on every fide of the Axe ; which is as much as to fay, it does 'not exceed an Angle of two or four degrees; and I am fully convinced, that it does not exceed one Degree; what difference can it make, whether that fmall portion of the Retina be a Plane or a portion of a Sphere ? he certainly could not fuppofe a large portion of the Retina to affume that Form. Now, if the Retina does affume the form of a Plane, of the fpace ab, then it is certain that the Images, af^ f and db, on that Plane, (of which, the Right Line ab may be fuppofed a Sedlion) are not in proportion to the Angles AEF, FED, and DEB, equal aKf, f'Fd^ and ^E^, refpedlively. Dem. For, the Tangent cf or cd is not in proportion to ca or cb as the Angle cEf is to cEa; becaule, c f is equal fa (AF being equal FC) therefore, the Angle cEf is not equal fEa, or cEa double cEf. — 3 * 6. El. But, ab h parallel to a b ; therefore, af, fc and cb are in the fame Ratio, as a f, f c and c b. — — — — Cor. to 0. 6, El. Thus it is clear, if the Retina does take the form of a Plane, that Objedls do not appear in proportion to the Angles they fubtend ; but, as it is the only inftance in which 1 ever knew it contefted, fo, I am perfuaded, it will never be univerfally adopted, on that Hypothefis. However, as it is merely conjedfural, it feems to me more rational, to luppofe the Retina to retain its original fpherical Form, which the all-wife Creator has given it, than, that it Ihould, continually, at every motion of the Eye, be changing to that of a Plane. Wherefore, on the prefumption that the internal Eye is truely fpherical, it is very reafonable to fuppofe, that, Objedls appear, exa6Hy, in proportion to the Angles under which they are feen ; and not in proportion to the Tangent of half the Angle, as,Mr. Hamilton afferts, and has endeavoured to demonftrate, in 7 and 8 of the lame Seclion ; which cannot be as himfelf has ftated It, but on the uippofi- tion of the Retina ailuming the form of a Plane, which is abfurd to fuppofe. Or D granting 14 OF DIRE QT,. -VISION. Book I. Plate I. Fig. I. F'g- 3- granting It really fo, the portion of the Retina, which the compafs of an Angle of one or two Degrees, at moft, takes up, is fo very fmall, that it is Irapoffible to diftinguifh it, in a Sphere of an inch in Diameter, from a Plane. As the 360th part of the circumference of a Circle, of a fmall radius, Is not diftinguiihable from a Right Line ; confequentlyj there could be no difference in the apparent magni- tude of an Objed, in either Cafe^ which is feen under fo Imall an Angle. As, the fmall portion at d, on the Retinaj fubtending an Angle, at E, of about five Degrees, is not diftinguifhable from a Right Line. 3. It is alfo evident, if the Vifual Rays are refrar^ed In pafling through the aqueous Humour, before they enter the Pupil (unlels that Refraction be rectified again at the Cryftalline Humour or elfewhere, and, by that means, proceed in the lame direction to the point of convergency, in the center of the Eye) that OhjeCts cannot appear exaCtly in proportion to the Angles they fubtend. Nor have I any conception how a Vifual Ray, AE, falling remote on the Cortiea,- can be refracted fo as to pafs in the fame direction through the Aperture, h i, to the point of conver- gency, at E, and from thence to the Retina. For, luppofe a Vifual Ray, A from any Point A (feen very oblique) falls per- pendicularly on the Cornea, at if it pafs at all through the Pupil, h i, whilft the Axe is directed to C, it muft pafs in the direction e g\ where it falls on the Cryf- talline, at and where, it Is pofiible it may again be refraCted, in the direction gh^ to the Retina ; making an equal Angle with the Axe, (EC) as the original Ray, A e ; in which Cafe, the Image of the Point A is feen at /a, in its proper polition. But, it is manifelt, that, whilfl the Axe remains in the direction EC, the point A cannot be feen, at all, nor even the point F, diflinCtly ; the Intimation we iiave of Objects, at a lefs diftance from C, is fo very confufed, that, without moving the Eye, we fcarce perceive their exiflence j of which, every Perfon’s own experience will foon convince him. Unlefs the Axe be direCted to any Point (as E) no per- fect Image of that Point is formed in the Mind, or on the Retina. Confequently, the Rays go dIreCt through the various humours of the Eye; as when vve look through a Pinhole; for, how can it be otherwife? feeing, it is manifeft they mufi: ail pafs through the fame Point ; (q is then their common point of convergency) the Image formed on the Retina is, in every refpeCt, the fame (except in Hue, for want of fufficient Light) and of the fame Magnitude as it appears to the naked Eye. Which lingle circumflance explodes, in my opinion, the notion of RefraClIon at the Cornea and through the various Humours, in the palTage of the Rays to the Retina. For It is manifelt, that, if the Point of convergency be at q, without the Eye, the Image will occupy more fpace on the Retina than at E, or any other Point within the Eye. I fliall therefore fuppofe, that, when the E^ye is naked, the Point of convergency of the Rays is in its Center ; and, on that luppofition, do infer, that ObjeCls do appear in proportion to the Angles under which they are feen. Of which I will give an eafy and familiar Example. ‘ 4. Let AB, CD, and FG be three Globes, at different DIRances from the Eye, at E, and of different Magnitudes ; feen under the Angles AEB, CED, and FEG refpeCtively *. Now, if the Diftance of the Globe CD be equal to twice the Diftance of AB, and FG be double of CD, four times the Diftance of AB ; then w'ill the Angles AEB, CED, and FEG, which they fubtend at the Eye, if the Diameters of the Globes arc in the fame proportion as their Diftances, be equal; and confequently, their apparent Magnitudes, to the Eye, at E, are equal. Draw Pi S, to the Center of the Globe F G ; draw C D perpendicular to E S, and equal to the diameter of the Globe CD ; alfo, draw Byf perpendicular to ES, and equal to the Diameter of the Globe AB. * The full Diameter (FG) of a Globe cannot be feen; for, the Vifual Rays being Tangents to the Globe, from the fame Point E, as E/,' E^, are all equal (Cor. z. i6. 3. El.) and, by real'on of the convexity, towards the E)e, they cannot touch the two extremes of the fame Diameter ; unlefs the Diftance be infinite; and then, the Vifual Ravs are confidered as being parallel ; fg is, therefore, the apparent Diameter of the Glebe FG. Dem. 5 e£t. III. OF DIRECT VISION. *5 Dem. It is evident, that the Vifual Rays FE and GE, from each extreme of the Diameter of the Globe F G, will pafs through the extremes of G D and B A. For, in the llmilar I'riangles FEG, DEC and BE^; FG : SE :: DC (equal CD) : KE, and, as BA (equal A B) : LE. — — 6. 6. El. Then, fiiice they touch the fame Vifual Hays FE and GE ; confequently, they are all leen under the lame Angle, PEG. — — by Theorem And, the Diameters, A B, C D and F G, being in proportion to their central Dlftaiices, are, therefore, all feen under equal Angles, AEB, CED and FEG. Conlcquently, the Images, a b, be, and cd, on the Retina, fubtending equal Angles, are equal ; and therefore, the appearances of the three Globes A B, CD, and F G are equal. Q.E.D. If the Globe C D be fuppofed to be moved to M, its central Diftance, EM, being equal ES; and, being, in Diameter, equal half F G ; it will fubtend an Angle GEIT, i. e. CEl, half CED, equal FEG; as the Arks 1234, which mealure thofe Angles, exemplify. Confequently, the Image, (h c) of the Globle G H, at that Dlftance, is, in mag- nitude, but one fourth part of the Image (c d) of the Globe CD, of the fame Magnitude, and feen at half the Diftance; for, they are in the duplicate ratio of the portions of the Ark 3 o to 3 4. — — Cor. 1. 13. 6. El. 5. Yet it is manifefl, that two or more unequal Objefls, at an equal Diftance from the Eye, and feen dired, do not appear exadly of the proportion of their refpec- tive Magnitudes ; likewife, equal parts 6f the fame Objed, on the fame ftde of that Point where it is cut by the Axe of the Eye, do not appear equal. Let AB be an Objed, of length (imply, bifeded by the Axe EC, to which it is Fig. I* perpendicular, in the Point C ; and fuppofe AC and C B again bifeded, in F and G. Then, the parts AF, F C, &c. are equal, and AB is double F G ; but, they do not appear of that proportion, at E. ' ' Dem. Now becaufe AF is equal to FC, the Angles AEF, FEC, under which they are feen, are unequal ■ — — — — 3* 6. El. For, if the Angle A EC be bifeded, the Right Line EO, blfeding it, will cut A C, in the Point O, in the ratio of E A to E C. But, EA is greater than EC; for, EC is perpendicular to AC. — Cor. to 12. i.El. Wherefore, A O is greater than O C, and the Angles A E O, O E C, under which they are leen, are equal ; confequently, the equal parts A F, F C, are feen under unequal Angles ; therefore, they appear unequal, by the Theorem. And, becaufe the Angle C E F is greater than F E A, the equal part C F will appear greater than F A, After the fame manner it may be proved, that the equal part C G appears greater than GB. Confequently, CG -|-C F, i. e. F G, appears greater than half A B. Q.E.D. Cor. Hence, it is evident, that if AB be produced, and the equal divifions, CG, G B, BP are continued, they will appear continually lefs, being feen under lels Angles. For the Angle B E P is lefs than BEG, which, is le(s than G EC; as the meafures, po, or, and rC, of thofe Angles, fufficiently evince. N. B. The farther any two or more Obje£(s are from the Eye, the nearer they will appear in the pro- portion of their refpeftive Magnitudes. Becaufe they arc leen under lefs Angles ; and, the Tangents of fmall Angles deviate lefs from tlie ratio of the Angles, than the Tangents of greater Angles. 6. As Objeds of different Magnitudes, at equal Diftances, do not appear in the proportion of their refpedive Magnitudes, fo neither do Objeds of equal Magni- tude, feen at different Diftances, appear exadly in the ratio of their feveral Diftances. Let Book 1 . i6 Plate I. Fig. 4. OF DIRECT VISION. Let A B be an Obje(ft at any Diftance (E F) from the Eye, at E ; and, let C D be another Objed:, parallel to A B, of equal Magnitude, as to length limply ; alfo let CD be at twice the diftance of AB (EG double E F). And let them be fo fituated, that, right lines, AC and BD, joining their extremes^ are parallel to EG i. e. perpendicular to AB and CD. By means of the Vifual Rays EA, EB, &c. thofe Objects are leen under the Angles AEB and C ED. I fay, the Angle A E B is not double of tlie Angle C E D. Dem. Becaufe AB is parallel to CD, EF: EG Ec : EC, as cF: GG. 6 . 6 . El. But E F is equal to f E G (Hyp.) wherefore, c F | C G, equal A F ; that is, A F is bifedled, in the point c. Now becaufe A c is equal to c F, the Angle A E c is lefs than c E F 3 . 6 . El. for, if they were equal ;• A c would be to c F as E A to E F. But, EF is perpend, to AB, therefore, EF is lefs than EA. Cor. 3 . 12 . i. El. But A c and c F appear (to the Eye at E) in proportion to the Angle under which they are feen ; i. e. as AEc to cEFj (by Theo.) and cF appears equal to C G, being feen under the fame Angle ; confequently C G appears greater than half AF. By the fame reafoning, GD appears greater than half F B. Therefore, GD appears greater than half A B; i. e. the Angle CE D is greater than half AEB; although CD is equal to AB, and the diftance double. Eet I K be another Obje£l:, equal and parallel to A B. and at three times the Diftance, from the Eye, at E. I fay, that I K will appear greater than one third of A B. Produce E G to H, and IK to L, making K L equal to II K ; and draw E K and EL. Then will AF be trlfeded, in k and 1. Dem. Becaufe A F is parallel HK, kF:KH::EF:EH|‘ . , -p. And, becaufe KL is equal to K H, kl is equal to kF. J But, E F is equal to one third of EH; confequently, Fk and kl are, each, equal to one third of AF, equal K H. Then, becaufe F k, k 1, and 1 A are equal, the Angles F E k, k E 1, and 1 E A are unequal ; for, if the Angle F Ek, was equal to k E 1, F k would be to k 1 as EF to El. — — — 3 * 6 . El. But E F is perpendicular to A B ; wherefore E F is lefs than E k ; confe- quently, F k would be lefs than k 1. And, for the fame reafon, if the Angles k El and 1 E A were equal, k 1 would be lefs than 1 A ; becaufe E k.is lefs than E A. But F k, k 1, and 1 A are equal ; therefore, the Angle F E k is greater than k El, and k E 1 than 1 E A. Therefore, F k, i. e. H K, appears greater than one third of A F. By the fame reafoning, it is manifeft, that H I appears greater than one third of F B, equal H I ; Therefore, I K, appears greater than one third of AB. Q. E. D. Thus, I have made it appear, to Demonftration, that Objects do not appear in that proportion to each other, as their refpedtive Diftances; in which I have been more particular, becaufe, Mr. Kirby has expreflly faid, Page 13 , that thefe Pidlures (i. e. the appearances of Objedfs) will be to each other, as their ieveral Diftances are to each other*. Rohauk fays, nearly fo; Page 243 . • The Appearances, i.e. the Angles under which Objects are feen, being in proportion to their Diftances is, literally, ablurd ; becaufe, as the diftance of an Object is incrcafed, the Angle, under w hich it is feen, decreafes ; which kind of reciprocal Proportion is always to be uudcrftood in this Cafe. 6 Dem. Sea. III. OF DIRECT Vision. *7 7, If the Eye be moved from E to O, making O F equal to F H, and O I, O K be drawn, E B will be cut in the fame Points, c and d, as by the Vifual Rays E C, E D* But, I K will not appear of the fame proportion to AB, at the Point O, as C D, at E, the Objeds AB, CD, and IK being equal. Dem. Becaufe AB, CD, and IK are parallel, the Triangles E c F, ECG, alfo, OcF, and OKH are limilar ; wherefore E F : EG : ; c F : C G, and, OF : O H : : c F : K H. — — — 4. 6. El. But, E F is equal to half E G, and O F is equal to half O H ; confequently, c F is equal to half C G, and alfo to half H K ; equal C G. By the fame reafoning, F d may be proved equal to half G D or HI. Therefore, AB is cut in the fame Points c and d ; as it was affirmed. 2- C D appears to the Eye, at E, in proportion to A B, as the Angle C E D to A E B ; and 1 K appears, at O, as tlie Angle I O K to A O B. Now, the ratio of the Angle C E G to AE G is greater than K OH to AO H. For the Angle A E G is greater than A O F, equal C E G ; wherefore, the ratio of c E f to A E F is greater than c O F to A O F. For, let E m be drawn parallel to O K ; the Angle F E m is equal F O c conleque'ntly, the ratio of F E m to F E c is equal to that, of F O c to F O A, equal F E c. But F c is equal to c A (proved) wherefore, F m is equal to m c. Ahd, equal Parts of a Right Line fubtend lefs Angles, the farther they are from a perpendicular to the Line, from the Eye ; proved. Therefore, the Angle F £ m to F E c, I. e. F OK to F O A, has a lefs ratio than FEc to FEA, i. e. than GEC to G E A, being nearer equality, Confequently, H K appears lefs, in proportion to A F, at O, than C D at E. After the fame manner, H I may be proved to appear lefs, in proportion to • FB, at O, than G D at E. Therefore, I K appears lefs, in proportion to A B, at O, than C D at E. Q.E.D. Hence It Is manifeft, that, notwithftanding the fedions of the Vifual Rays EC and OK, ED and O I, with A B, are the fame ; the Didance of the Eye on one fide of A B, and the Objed on the other being proportional ; yet, the appearances of the Objects, to each other, are varied, confiderably.as the Didance is greater or lefs. For, two Objects of equal height, at a great Didance from each other, and pa- rallel between themfelves, and the Eye at the fame Didance from one of them, and in a Right Line with both, will appear, in proportion to each other, nearly as their Didances; i. e. the farthed off will appear half the height and half the width of the neared, being at twice the Didance, and, one third in height and width at three times the Didance, nearly, i. e. fomewhat more. As I K appears, at O, fomewdiat more than half A B. But, C D (equal I K) ap- pears much greater than half A B, at E. 'Confequently, if the Eye be at a greater didance from AB, than at O, and the Objed: at an equal didance on the other fide, they would appear nearer in proportion to their Didances, reciprocally ; and there- fore, wdien the didance of both is fo great, that the Vifual Rays, O I, OK, ap- proach nearly to parallelilm (which, it is obvious mud be infinite, before they can he perfed:ly fo) they w'ill appear nearly as their Didances. Notwithdanding, on a Picture parallel to the Objeds, they mud ever be reprefented in the ratio of their refpediive Didances, and Magnitudes. E THEOREM OF DIRECT VISION. Book J. iB THEOREM li. . t Parallel Right Lines, however htuated, being produced, appear to ap- proach towards each other ; and, if produced infinitely, they will appear to meet in a Point, at an infinite Diftance Plate I. Let I F and H G be two parallel Lines ; and, E, an Eye, fituated any how be- Fig. 5. tween them. Let K, B, D, and G reprelent various Uihances in the Line HG, and let the Lines I K, AB, &c. be drawn perpendicular to H G, cutlng tlie other Line (alfo perpendicularly^ in the points 1, A, C, and F. Alfo, let the Vifual Rays E A, E B, dec. be drawn. Dem. The fpace between the Lines, IF and KG, at the feveral DIftances ES, E O, &c. appear, to the Eye at E, under the feveral Angles I E K, A E B, &c. , each of which, as the Dilfance is farther from the Eye, is lefs. C. 14. 1. El. For, the Space between the parallel Lines is, every where, equal. Def.^. Geo. The Space, at IK, is feen under the Angle lEK ; and the fpace between them, at A B, under the Angle A E B ; which is lefs than IE K, nearly in pro- portion to the Diflances, E S to E O ; CD fubtends the Angle C E D, and F G the Angle FEG; each of which is lefs than the former; for their Subtenies, A B, C D, &c. are equal; and the Lines EA, EB; EC, ED, &c. forming or containing the Angle, are continually longer; wherefore, the Angle BEG is lefs than CED, which is leE than AEB+, &c. — Cor. to 14. i. El. from which it is manifeR, that at a greater DiRance the Angle will be Rill leE ; confequently, it will, at laR, become infenfible, the interval, between the Lines, is loR to Sight, and, the Lines appear to meet each other. Q. E. D. Hence, the Lines, I F and H G, are faid to vanifli ; and confequently, the fpace between them vanifiies alfo. 2. Let ER be drawn, from the Eye, parallel to I F and HG; thofe Lines will appear to approach, continually, towards ER; and being piod'.'!.ed, infinitely, they w'ill all appear to fneet in the fame Point ; however E R may be nearer to one than the other; for, the farther any parallel Line is from ER fwhich may be called its Radial) the more Ridden is its apparent approach to that Radial, The Space 1 A, of the one, appears to advance from h to /; an equal Space K B of the other, advances from p to 0 \ which is lefs than the other, in propor- tion to the Angle K E B to I E A ; i. e. as the Ark po x.o hi. The Space from A to C appears to advance from i to k\ an equal Space, BD, in the other, from 0 to n. From C to F it advances from k to /, an equal Space, DG, from n to in the proportion of the Arks kl to mn. It is evident, that If thofe Lines w'ere produced longer, they w'ould Rill appear to approach nearer to the point e, as EL, EM, evince. And, being infinitely procluced, they would at laR appear to terminate in the Point e, and be loR to fight. P'or, it is manifeR, that the farther any Points, L and M (confulered as being in the parallel Lines IE" and G H) arediRant, the nearer the Vifual Rays, EL, EM, are to a coincidence with ER; which, it Is evident, muR be infinite before they can coincide perfedlly ; in which cafe, EL, EM, and ER will be the lame ; for they will all appear to unite. * It is faid ill Smith’s Optics (Vol. I. Art. i;6, Page 58) that parallel Lines, feen obliquely, appear to con- verge more and more as they are farther extended irom the Eye; which, will ever be the cafe however the Eye is fituated, in refped of the parallel Lines. For, being ften obliquely, means nothing more tlian when we look to- wards either ot their extremes ; as they are feen direct, only where a Line, from the Eye, cuts them both at Right Angles; on either fide ot the Perpendicular they are iiecefiiirily feen oblique. t When the Angle, AEB or CED, under which any Object, as AB, is feen, does not exceed 20 or 30 De- gres, at the molt, and that Object is removed to the feveral Dlftijnces, from E O, to E P, EQ^ < 5 <:c. the ditfeience between the 'I'angeiits O A, O C, &e. deuates but lltlle from the Arks OCq, OC, &c. and that flill lefs as the Angle is lefs ; as PC, P F 2, &c. 3. Hence Jl 6 • Sea. III. >9 b'F DIRECT VISION. 3. Hence, It Is ealy to account for the appearance of a horizontal Plane, or con- tinued level Surface ; which, being below the Eye^ will gradually appear to afcend ; and, being above the Eye, it appears to defcend ; and, if they were produced in- finitely, they would appear to meet in a Right Line, on a level with the E)*e. Let the Right Line H G be a feaion of a horizontal Plane, below the Eve, at E; and let 1 F be a fection of one above the Eye. alio, let 1 K be a ledtion of a Plane diredt before the Eye* (at E) perpendicular to the other two. Now if the Vifual Rays, E A, E B, EC, See. be drawn, they muft necelTarlly cut the Plane of which i K is a Sedlion ; in a, b, c, &c. It is evident that the Space KB will appear to rile, towards S, on the Plane IK^ from K, its interfedtion, to b, where the Vilual Ray EB cuts that Plane; the fpace K D will rife to d, and KG to g. For the lame realon, the fpace 1 A will appear to defcend from I to a ; 1 C, from I to c ; and 1 F, from 1 to f ; each Place appearing gradually to approach towards S, where a Perpendicular, ES, to that Plane, cuts it. And, if the Planes were produced infinitely, they would at laft vanifh, or be loft to liglit, in a Right Line paffing through the Point S, parallel to the inter- fedtion of the other two Planes, with the Plane IK. And this is evid.ntly the cale however a Plane be fituated ; whether horizontal, vertical, or inclined, it is the fame; for, a Plane is ftill a Plane, in all pofitions, and lias no properties peculiar to any pofition, in refpedt of the Horizon. 'Phnefore, there may be drawn this Conclufion, that every Plane, in which the Eye is not fituate, wfill appear to approach towards, and at length to meet, another Plane, palfing through the Eye, parallel to the former. 4. Let A B, CD, and F G reprefent three Objedls of equal height, at the feveral Dilfances, EG, E P, aind E Q, from the Eye, at E. The Vifual Rays E A, EB, EC, &c. being drawn, thole Objedls will appear in proportion to the Angles AEB, CED, and FEG. For, the portions of an ark of a Circle, whofe Center is E, intercepted between the' Vifual Rays, meafure thole Angles, refpedlively, and determine the apparent proportions Of the Objedls A B, CD, and F G to each other; feeing that, the Angles whicli they fubtend are in the lame Ratio — — 19. 0 El. Wherefore, fince, in Feripedllve, the Reprefentatlon is always on a Plane, and not on a fpherical Surface; luppole I K a ledlion of a Plane, in a diredl pofition; the Spaces, a b, c d, and f g, between the Points where the Vifual Rays cut and pals through that Plane, confidered as a Pidlure, are the proportions of the Repre- lentations of the Objedls, AB, C D, a^id FG ; but their apparent Magnitudes are the portions /o, kn, and /m of the Ark hep. Let HG reprefent a fedlicn of tlie level Ground, or any other horizontal Plane; and A R, CD, and F G, Objedls perpendicular thereto. If I K be fuppoled a fedlion of a vertical Plane, the foot or feat B, of the Ob- jedl A B, will appear, on that Plane, at b; and confequently, Kb will rep reCnt the fpace of Ground, K B, between the Piclure and the Objedl A B ; i. e. it will appear to fife lo high on the Pidlure; K d will reprelent the Space of KD, and Kg of KG So likewiie, a, c, and f Ihew how much the tops of thofe Objedls appear to defcuiid, on the Pidlure. 5. Now, if the pofition of the Pidlure be inclined to the Horizon, as rtK, the Objedls and the Eye remaining as before; the Reprefentations ab, c d, and fg^ on that Plane, are very different from the proportions of ab, cd, &c. on the Plane IK. • By being direft before the F.\e, I would be undeillood to mean, in a vertical Pofition ; and when a Right Line, from the Lye, falling in the middle, beivveen its Bounds, curs the Plane perpendicularly. For, it the Kye be not in the Flaue, or in a continuation of it, the Eye, being confidered as a Point, can have but one pofition, either to a Plane or to a Right Line, being produced ; however the Plane may be fituated, in refpedl ot the Horizon. But, 20 Book I. Plate I. Fig- 5 - OF DIRECT, VISION. But, the apparent proportions, of both, are the fame, atE; viz^ the portions on the Ark which mealure the Angles AEB, CED, and PEG. Nor, is It poffible to be otherwile; feeing that, the Objeds, AB, CD, &c. remain the fame, in refpect of each other; and the htuation of the Eye, is not varied; confequently, the Angles AEB, CED, &c. are not altered, and the Objeds A B, CD, &c. the Subtenfes of thofe Angles, mull, necellarily, appear the fame, reprefented on any Surface whatever, and in any Portion, cuting the Vifual Rays EA, EB, &c. N. B. The Proportions CD, and FG, &c. on AB, are the fame, as c d and f g on IK (AB being parallel to IKj; wherefore, the Triangles AEB, aEb, alfo CED, CEZ), and cEd, &c. are ilmilar; and therefore, as fg ; cd, or to ab, FG ; CD, or to A B. — — 4. 6. El. Confequently, Vifual Rays, being cut by parallel Planes, in any pofit'on, are not only them- felves cut in equal Ratio, but alfo the SeAions, or the Proje£fions on the Planes, by their ledtions with the Rays. — — — — — — 1.8. El. And, their Proportions are as their Diftances E S, EO, EP; i. e. as fg : ES FG : EO as Fi Gi : EP, and, as F G : EQ_; (6. 6. El.) and fo of any other. Hence, it is manifeft, that the Sections of any fyftem of Rays, by parallel Planes, are limllar. PROBLEM.- A Right Line (AB) being obliquely fituatcd in refpetd of the Eye, p. g at E, to determine the Point D, to which if the Axe of the Eye be * direeded, the two Parts, AD and DB, (hall be feen under equal Angles. Without drawing the Vifual Rays, or bifeding the Angle made by the Rays. Make AD to AB, as the dlftance of the Point A, from the Eye (E) is to the dlflance of the fame Point A, added to the diftance of B. — Pr. 32. El. Then, AD : AB::EA lEA-j-EB; and, AD :DB::EA:EB. Dem. The Vifual Rays EA, ED, EB being drawn, the Angle AEB is blfeded by Ed ; wherefore, AD and D B are feen under equal Angles, at the Point E, For, if the Optic Angle AEB be bifeded by the Right Line ED; then is A D : D B : : E A : E B. — — — 5. 6. El. 2. If the Axe of the Eye be direded to C, the middle Point of AB, then will the Point B be more diftindly feen; becaufe its Image falls nearer to the Image of C than the Point A. And, if the Ark cf be made equal c a, and Y.f be drawn, meeting AB produced, in F ; AC will then be to C F as E A to EF' ; and, not- withllanding the difference between AC and C F, their Images on the Retina, and confequently their apparent Proportions, are equal, being feen under equal Angles. Now, if thofe Vifual Rays are cut by a Plane in the politlon I K ; the part a c, the greater portion of a f, will reprefent A C, the leffer Segment of A F ; and c f, the leffer portion, reprefents C F, the greater Segment ; For, fince the Optic Angle, A E F, is bifeded, by E C ; a c : c f ; : E a : E f ; 3. 6. alfo, asAC:CF ::EA:EFj conlequeutly, ac, cf reprefent AC and CF. 3. If the Right Line A B be produced, at the extreme A, cuting the Plane I K, at 1 ; the whole indefinite reprefentation of that Line, on the Plane I K, is the lec- tion I K, of a Plane paffing through the Line A B and the Eye, and terminated by a parallel Line from the Eye (E K) to that Plane. For, becaufe EK is parallel to AB, they will appear to meet in a Point at an t Thso. 2. infinite Dlflance ; f which is reprefented by the Point K, on the Plane IK; for, feeing that the Line EK palfes through the Eye, its whole appearance is but a Point; confequently, the Point K reprefents the whole Line E K, produced infinitely ; and A B, infinitely produced, will appear to meet E K in that Point. I If • Sed. III. OF DIRECT VISION. 21 If the Line A B be produced on the other Side of the Plane IK, and any Point (G) be afllimed, between the Eye and the Plane; its reprefentation, on that Plane, is the Point g, in which the Right Line EG, produced, cuts K I produced ; the part Ig, beyond the Interfedion, I, reprelents IG, the portion of the Line, A B, lying on the fide of the Plane, IK, towards the P>ye ; as I a or I b repre- fents the part I A or IB, on the other Side; all which is I'o very obvious, from infpedion of the Diagram, it is need lei's to fay more about it. 4. After the fame manner as the apparent magnitudes of Objeds are determined, fo are their apparent Diltances, or Bearings, in rcfpcd of each other; viz. by the Degrees on the Ark of a Circle, which meafure the Angles they make at the Eye. Let C, D, and F, be three Objeds, and E the place of the Eye. Fig. 7. Draw the Vifual Rays CE, DE, FE; then, the Angles DEC, CEF are the Optic Angles of their apparent Diftances, or Bearings at the Station E. Notwithhandlng their real Diftances from each other, C F is nearly double that of C D, yet, the Angle CEF, is much Icfs than the Angle DEC; as it is evi- dent, from the portions, c f and cd, of the Ark d f. If y'r/ be fuppofed the fedion of a Plane, the appearances, or places of thofe Ob- jeds thereon, are at y, c, and d. 5. Suppofe C, D, and F to be Stars, in the unbounded Expanfe of the Heavens, at an immenfe diftance from each other and from the Eye, at E. It is impolfible to form an Idea of their real Diltances or Situations, in refped of each other ; for, if the Star C was at B, and F at G, or any where elfe in the di- redion EG, their apparent places are ftill at c and f, in an imaginary i\.rch of the Heavens, as it appears to us, equally diftant in every part. Now, lince the whole Diameter of the Earth’s Orbit is not fufficient to make any fenfible difference in their Bearings, and, confequently, of their apparent Places, in the Arch d f, there cannot be any pofitive Idea formed of their real Diftances; for the portions fc and c d, of the Arch fd, are the meafures of their apparent Diftances, only ; i. e. of the Angles d E c, c E f, or the Originals of thefe Angles DEC, CEF under which they are feen. Hence is conftruded the Celeftial Globe or Sphere, which is a true Pidure of the Heavens, of Stars, &c. divided into various Figures called Conftellations. If dcf be fuppofed a portion of a Sphere ; or, let HI be an entire Globe. It is evident, that from its Center, E, which is alfo the Center of the Arch df (E being fuppofed the Eye of a Spedator) the Star C will appear on the Globe at o, and D at s, &c. in the fame pofition, fituation, and diftance, in proportion to the Radius, EH or Eo, to E c, as in the imaginary Arch of the Heavens, df. So that, w'hether the real Star be at A, B, or C, or any where elfe in the Right Line EC ; its apparent or reprefentative Place, on the Globe H I, will ftill be at o ; and confequently, can make no difference, in its true Place thereon ; but its apparent Magnitude will be in proportion to its Diftance, nearly. 6. Let AB, DF, and GH be three Objeds, promifcuoufly fituated in refped of Fig. 8. the Eye, at E. It is manifeft, that, whilft the Eye remains at E, the three Ob- jeds muft neceflarily appear the fame, upon any Surface whatever, cuting the Vifual Rays EA, EB, &c. Suppofe aha fedion of the Rays, made by a Plane, and E G a Perpendicu- lar from the Eye to the Plane, cuting it at C. Then, the Vifual Rays E A, E B, which are neareft to the Point C, cut that Plane lefs oblique than E F ; and E G, E H are ftill more oblique, as they are more remote from C. Whereas, on a fpherical Surface of which TLdfh may be fuppofed a Sedion, being an ark of a ‘Circle, defcribed on the center E, the Rays EA, EF, EH, &c. are all perpen- F dicular a ,0 F D I R E C T VISION. Book I. S2 Plate I. (^IcuuU- to its Surface, being perpendicular to the Ark Cfh. Wherefore, fince 8. * ^ ‘b F>^^, iSec. are equal, for, they are all Radii; confequently, the furface of * ^ a Sphere is equidiilant from its Center, in every part ; and, the Reprefentations, a b, df^ and ^ h, of the Objedls AB, DF and GH, on its Surface, .are alfo their true Appearances, from the Point of View E. But, on the Plane, or its Sedlion, a h, the Objedl AB, has its Reprefentation, a b, nearly equal to a L- on the Curve ; whereas, d f, it is obvious and demon- llrable, is larger than d f‘, and g h flill larger than ^ /a, as it is more remote from the Point C ; neverthelefs, its true appearance \% g h on the Curve, or ipherical Surface; for, they both reprelent the fame Objedf, from the fame Point of View, which cannot vary in its Appearance. Therefore, the Reprefentations of Objedls, on various Surfaces, and in various Situations, although they may differ greatly in Figure, will, if truly reprefented, appear the fame in the true Point of View. The three Objefts A B, D F and G H, though of various dimenfions, appear equal ; not owing to their Biftances, but to their Pofitions in refpedl of the Eye, • being feen under equal Angles, A E B, D E F and G E H ; as the portions a by dfd and gby of the Ark Cfhy indicate. Whereas, their Reprefentations on the Plane, of which a h is fuppofed a Seflion, are unequal ; for, a b, the reprefentation of A B, being nearefl to the Center (Cj is the leaf ; d f is greater than a b, and • g h than d f. Yet, it is manlfef: that the Reprefentations, a b, d f. and g h, have the fame appearance, to the Eye at E, as their correfpondmg Images, a b, df and^^, on the Ipherical Surface. For, the Eye being in the true point of View (at E) they are both feen by the fame Vlfual Rays as the original Objedls themfelves ; and Gonfequently, under the fame Optic Angles. Wherefore, their Appearances are . the fame on either Surface ; by Theorem ift. From what has been advanced in this Sedlion, it is evident that there is a manlfeR . difference betwen the Reprefentation of an Objedl, on a Plane, and its true Ap- pearance ; which difference is the greater, as the Eye is nearer to the Objeft. For, fince the Vifual Rays mufl necefi'arily cut any Plane, on which they fall, more and more oblique, the farther their Interfedlions are from that Point, in which a perpendicular Line from the Eye would cut the Plane ; confequently, the Repre- lentations of Objeifts, on a Plane, cannot be in proportion to their true Ap- pearances, but deviate continually, more and more, as they fall farther from the Point delcribed. Whereas, on a fpherical Surface, which is every where equally have loft one eighth of the velocity it had, at its return to C ; at which time the calculation is made. Wherefore, the motion of Light, from C to E, has, at raoft, but half the ve- locity it has from S, to C or E, diredtly ; and, it paffes, according to Roemer’s cal- culation, from C to E in 15 Minutes; confequently, it pafles from S to E (with double velocity) in a fourth part of the time, that is in 3 minutes and 3 quarters. But, if the velocity, to and again, be regular and uniform (which I believe is contrary to the laws of motion, of Bodies impelled and reflefted again) being 15 minutes in paffing from C to E, half of which, S E, muft confequently take 7 minutes and a half. For a Satellite, emerging at a, when the Earth is at C, ap- pears 7 minutes and a half Iboner, and at E 7 minutes and a half later, than at D or F where the diftance of Jupiter from the Sun and the Earth is equal, accord- ing to the calculations made by the ableft Men. But fince it is not poflible, or, at Se(J^. IVi Materiality of light. ^7 leaft, probable that the velocity of Light can be uniform and equal, at all Dif- tances from the Sun (if it has any motion at ail) the greatell velocity muft be at its firft emiffion from the Luminary ; and confequently, it takes lefs time in pafling from the Sun to the Earth than in its return from Jupiter. 3. It is certain that no opake Body, I mean, fuch as are not luminous, can be feen at a great Diftance; unlefs its magnitude be fufficient to obftrudl a great quantity of Light, when oppofed to a luminous Body; or, when it is fo htuated, in refped of a luminous Body, that the Light, it receives from the Luminary, is reflefled again * ; of which, the various appearances of the Moon is fufficient proof. Confequently we could not difeern either Jupiter or his Satellites (whofe immenfe Diftance is more than feventeen hundred times the Moon’s Diftance) if they were not illumined by the Sun. Jupiter being a fuperior Planet, that is, he moves in an Orbit beyond, yet con- centric with the Earth’s, he is, confequently, always illumined on that Face to- wards the Earth ; and confequently (being of immenfe magnitude) may always be feen, when above the Horizon, by the naked Eye; except when in near vici- nity with the Sun. But the Satellites are fecondary Planets or Moons (of which Jupiter has four) accompanying the primary ones, as the Moon does the Earth ; and are fo very fmall, in comparifon of their primaries, that they cannot be feen at that diftance without a good Telefcope. Now, if thofe Satellites cannot be feen, after they emerge from behind Jupiter, until the Light (which is always ready, at hand, to illumine them) is reflected, back again, from the Satellite to us ; and, fuppofe this Light to be a material Bo- dy, I am firmly perfuaded that they would never be feen at all, by us ; fuch an immenfe Space, from a to E, above 500 millions of Miles, for Matter to be pro- jefted in about 45 minutes, is beyond the power of my reafoning faculties to find credit for (with God nothing is impoffible.) Befides, they not only retle(ft the Light direftly, but alfo obliquely, in all Diredlions, filling a Hemifphere; which, I fhould fuppofe, is too grofs an improbability (being Matter) to gain credit with a thinking rational Being ; let thofe, who can, find belief and give credit for it. The whole of this Phaenomenon I fuppofe is this ; (for I have never feen the Experiment ; as it muft require extraordinary Telefcopes, which magnify to a great degree, to difeern the emeriion diftindly) either it is impoffible to perfedl Tele- fcopes fufficient to difeern the emerfion, till after the Satellite is confiderably ad- vanced from behind Jupiter; and alfo, from the near vicinity of the Sun, when in Conjundlion with the Earth, at E, whofe fuperior fplendor muft hinder the Sattcllite from being feen, for fome time; together with the fo much greater Dif- tance, than when at C ; which, 1 know from my own obfervation, renders them fcarcely vifible, with a common refledlor ; all which, may concur to render it in- vifible for 1 5 minutes, after it might be vifible from C. For I make not the leaft doubt, that their motions and revolutions are as regular and equal as the motion of the Earth itfelf; which, to us, is the only ftandard meafure of Time. 4. If we fee Objefts, only by means of material Rays of Light paffing from the Objedl to the Eye ; by what means are opake Objedls, which are immerfed in Shade, feen at all ? as they do not receive Light', immediately, from any luminous Body ; nor, perhaps, from any other, oppofed to them, by refledlion. But, with fuch reafoners, who can give what properties they pleafe to Matter, there is no arguing ; feeing that, they can make Light refleft and rebound from one Objedf to another, at pleafure. But, are not thofe ideal properties of Light of their own creating, entirely? That Light is refleded from one opake Objeft to another is beyond a doubt ; but, that real Matter is projedled, to and again, in every diredlion, I cannot acquiefee in. • By Light being refleiSleci, I would not be underftood to mean, that there is any kind of real Matter projedled from the reflecting Body; but only, by being illumined, itfelf, it becomes, in fome degree, luminous, fo as to ifiine with its borrowed luftre, and illumine others. Let. 28 OBJECTIONSTOTHE ' Book- 1; Plate II. ^ ^ reprefent, what is called, a Ray of Light, from fo'me luminous Objed, io falling on. any SuiFace, as CD. * 1 fhall fuppofe this Ray of Light to be a phyfical Right Line, the lead in its di- menfions of breadth and thicknefs, that can be conceived; confequently, this phyfical Ray of Light can ftrike any Surface, on which it falls, in a phyfical Point only, the fmalleft that can be conceived. 1 Now by the laws of Refledlion, according to all the Writers on Optics, when- ever a Ray of Light ftrikes or impinges on any Body, it is refleded again from that Body, making an equal Angle with a Right Line (B F) perpendicular to the Sur- face, at the Point in which it ftrikes the Surface*. Or, if it be a Plane Surface, it makes equal Angles with the Plane (ABI=EBK) both, the Original Ray, A B, which is called the incident Ray, and the refleded Ray, BE, being in a Plane, which is vertical to the other Plane. It is reafonable to conclude, if this Ray of Light, A B, be material, and it is by means of material Rays ftriking the organs of Sight, that Vifion is performed, or the fenfe of Seeing is effeded, that, the Point B could only be feen by an Eye placed, any where, in the diredlon of BE, as at E; BE being confidered as the fame individual Ray, A B, refleded or broken at the Surface, in the Point Bf. It is certain, that if the furface C D be Water, or a polhhed Mirrour, the Image of any Objed, at A, will be feen at B, by. the Eye at E, or fome where in the di- redion B E, only. But experience convinces us, that the Point B, or any other Point in the Sur- face C D, may be feen as perfedly in a thoufand other Diredioiis ; as at F, G, H, &c. Confequently, if Eyes were placed all around, within the compafs of a Hemifphere, of which the Plane C D may be confidered as the Bafe, or being ele- vated but a few Degrees above the Plane C D, the Point B may be feen by them all, at the fame moment of Time. Now, ftnee the Point B can receive but one Ray of Light from the luminous Body, how can that identical Ray be refleded in ten thoufand Diredions, in all Diredions above the Plane C D, and thofe Rays to be material ? at the fame Time, the whole Space is filled with Air in every part. But, the fame Eyes can alfo fee every other phyfical Point in the whole Surface C D, as C, D, &c. confequently, ■ every Point in the Surface muft alfo refled Rays in all Diredions, crofting and cut- ing the former in every Point ; which, unlefs, not only two, but, an infinite num- ber of Bodies may occupy the fame Space, and at the fame Time, cannot poftibly be, on the fuppofition that Light is a material Body. 5 . Again. Since, as I have obferved before, it is aflerted that Vifion is perform- ed by means of thofe material Rays of Light, which enter the Aperture of the Eye, and, impinging on the Retina, affeds the Optic Nerves with the perception * It is advanced by Sir Ifaac Newton, (Prop. 8, Part 3. of the Second Book of his Optics) that Light is refledicd from Bodies before it impinges on their Surfaces. But 1 mull own, that in all the feven reafons he gives for that Opinion, I cannot find one of weight, nor all of them together, fufficient to prove it. Which, Smith, from that au- thority, and from Sir Ifaac’s firll and fourth Quere, endeavours to prove, is not refiecled immediately in an Angle, but makes a regular parabolic kind of curve, at a very fmall dillance from the Body, which he calls the fpace of acti- vity; and which fpace, he fays, is lb extremely fmall, that, confequently, in phylical lixperiments, the incurvation of a Ray of Light may be confidered as performed in a phyfical Point. Emerfon fays (in a Scholium to Prop. i. of his Catoptrics) that the Curve, deferibed by a Ray of Light, is fo extremely fmall, that it may be looked upon as a fingle Point. It is, to me, allonilhing, how any Perfon dare prefume to advance fuch Opinions ! and, for what purpofe they do advance them, except it be to make the World have a high veneration for their extraordinary fagacity and penetra- tion. Can they, ei'her by experiment or otherwife, prove the truth of their Hypothelis? or do they fuppofe man- kind fo credulous as to give them credit for it, Irom the ridiculous experiment of a Hair, or of the Knives, and fringes ot Colours produced by them ? And yet it is certain, that foine (not to be behind them in penetration) either do or pretend to do ; for I have Jieard the faiue thing advanced, verbally, by a Perfon who has a tolerable lhare of mathe- matical knowledge. See Fig. 10. No. 2. A B is fuppofed to be an incident Ray, and C D the reflected Ray ; making a Curve, B E C; before it touches the Surface G 11 ; which Curve is performed in fo linail a Space, that the Points, B and C, are fup- poted to coincide. Quere ; how is the Curve to be afeertained, and determined.' or, how fliall we have convidtioa that it does not touch the Surface, G H ? 3 ot Sea. IV. MATERIALITYOFLIGHT. ^ of the Objeas from which they flow, how fhall we'account for Vifiori through Glafs, or much harder diaphanous Subflances, as Cryftal, or Adamant ? which is impenetrable, by any tool made of the hardeft Steel. Let any objea, as A B, be feen, by an Eye at E, by means of the Vifual Rays AE, BE, &c. and, let any tranfparent uncoloured Stone, as Cryftal, &c. be ,in- terpofed direaiy between the Eye and the Objea, as C D, of any thicknefs ; having ■ its oppofite Surfaces parallel Planes, and well poliflied. The Objea, AB, will be feen through it Inftantaneoufly, as reprefented at a b, with every circumftance of Colour, &c. Now, I fhould be glad to know, how or by what means thofe Rays of Light* tranfmited or refleaed from the Objea (A B) to the Eye (at E) pafs in an inftant through this folid Body of Cryftal or Adamant, if they are material? and firft, I Ihould be glad to know, what is the caufe of Tranfparency ?, or, in what the diffe- rence confifts, between tranfparent and opake Bodies ? The reafons given by Sir Ifaac Newton, in the third part of the fecond Book, and which. Smith, in his Optics (Gh. 8 , P. 95 ) has very wifely copied, word for word, concerning the caufe of Tranfparency, Opacity, and Colours, in Bodies, is but little to the purpofe; it does not convey, to me, the leaft Idea of the caufe of Tranfparency, and how Vifion is conveyed through tranfparent Mediums, in every diredfion, inftantaneoufly. The learned and reverend Dr. Harris, in his Lexicon, tells us (under the article Tranfparency) that, a diaphanous or tranfparent Body is one which, probably, has its Pores all Right or Direif, and nearly perpendicular to its Surface. By the Pores being all right or direft is meant, I prefume, that they go direflly through, in Right Lines, without any obftrudlion ; that they are perpendicular, or nearly fo, to the Surface, 1 mull; needs fay, he was extremely fagacious who firft found out that extraordinary quality. But how flaall w'e interpret the Word, probably ? is it not indeed a tacit acknowledgement, that it is all nothing more than Conjedlure? I cannot fay that it is even a probable Conjedture. I am afraid that the Dodtor was not very happy in his memory; for, .in another place he tells us (under the article Diaphaneity) that the Pores of a diaphanous Body are fo ranged and dilpofed, that the Beams of Light can pafs freely through them, every way ; which plainly contradidts the former aflertion, that they are nearly perpendicular to its Surface ; and which, in reality, is faying nothing, be- caufe, a Surface may be made at pleafure; and we cannot fuppofe that Pores can change their Pofition, as a Surface may be altered, at diferetion ; or how a' Ray of Light, in paffing through a triangular Prifm, can be nearly perpendicu- lar to both Planes, 1 am at a lofs to devife. For my part, I look on it as ridiculous and prefuming, to tell us that Glafs and all tranfparent Bodies are full of Pores and minute Interftices, through which the Rays of Light have a free paflage j becaufe, thefe Pores muft: be in every part of it, and in all Direflions ; neither can there be any folid part, between one Pore and the contiguous or adjoining Pores ; for, if there be, it muft neceflarily impede, or entirely flop the progrefsof fome Rays of Light in their paflage to the Eye? and, confequently, will prevent the Vifion of the Objefl from being perfect. - Where- fore, lince the Pores are fo very numerous, have no folid part between them, and that, in all Direflions; for it is plain, that they are not confined to one pofi- tion, but muft lie in all diredtions, which is evident ; for, < turn the Glafs or Cryf- tal (C D) as you pleafe, the perception of the Objedtis as diftindl as before ; from which it is clear, that the Pores muft be the fame in all Diredlions, and then, I would afk, what becomes of the folid Body ? for, in reality, it muft be all Pores, It is an infult on our Underftandings, on our Reafon, and on common Senfe, to fuppofe, or for any perfon to attempt to perfuade us, that Glafs, or Adamant, the moft compadt of Stones, is more pervious, or that the Pores are more diredt than in foft Wood, or in any Wood; which every one, who has confidered It, H knows Book I. 30 OBJECTIONS TOTHE Plate II. knows to be full of Pores (like Veins in the animal Body) through which the Juices pafs for its nutriment; and, in many kinds of Wood, they are perceivable by the naked Eye, and lie in dire£l Lines. Can any Perfon perfuade himfelf, that the Pores are more numerous, or more dired, or that they are more capacious, fo as to admit a freer paflage, through Adamant, for the moft fubtle Fluid whatever (which I will fuppofe Light to be) than through a thin piece of Wood ; in which the Pores are obvious, and clearly vifible? Air, which is much denfer than Light, or Water, ftill denfer than Air, has a free paflage through ; but not through Glafs, I prefume. Yet, if you oppofe it to a Candle (being cut very thin acrofs the Pores) you may perceive, indeed, a few fcattered Rays of Light pafs through, but very far from having diftincl Vifion of the Candle, or even the out Line of the Flame, only. The reafon is very obvious; becaufe, the parts, between the vifible Pores, being more condenfed and compaft, receive the Light which falls on them, and either abforb or refledt it; which, therefore, does not pafs through to the Eye; confequently, the 'parts of the Objeft, which are in a dired line to the Eye, cannot be feen ; therefore, the Vifion of the Objedt is imperfed. As this is evidently the cafe when it is oppofed to a luminous Body, which is Teen by its own Light ; what will be the confequence when it is oppofed to an opake Body, which (as we are told) is feen only by means of refleded Light ? Why, that there is not the leafl: appearance of it to be feen at all ; not even through fine oiled Paper, which, I muft needs fuppofe, is infinitely more porous than any Glafs or Stone whatever. Whereas, through perfedly pellucid Subflances, every vifible Point, in any Ob- jed whether luminous or opake, is diflindly feen, without the leafl: impediment ; confequently, if Light be material, and a Ray is tranfmited from every Point, in a vifible Surface, to the Eye (for every phyfical point may be feen) and if thofe Rays are conveyed through tranfparent folid Bodies, they mufl neceflarily pafs through Pores, dired in all pofitions. Now I am fully convinced, that the Pores in Glafs, &c. are of thofe Ingenious Gentlemen’s own creating; who, when they are at a lofs for proof of certain Hypothefes (for want of better) they imagine Bodies to poflefs fuch and fuch qua- lities as may befl anfwer their purpofe. But, are thofe Chimeras, of their own fertile imaginations, to pafs on the World for real exiftencies? are the Conclufions drawn from fuch Premifes candid? by no means, they are very difingenuous, in- fomuch that, I deny it to be in the power of any Man, to give ocular or other de- monflrative proof that there are Pores in Glafs or tranfparent Stones ; and, I do believe that the mofl pellucid fubflances are the freefl from Pores ; for all porous Bodies are compreflible into lefs compafs, which neither Glafs nor Stones can pof- fibly be ; nor Water, which is perfedly tranfparent. Pores are, in my opinion, rather the caufe of Opacity than of Tranfparency in Bodies, feeing that, they abforb the Light in their recefles. Yet, I do not fuppofe that all Bodies, as Wood or Metals, which are the freefl of the kind from Pores, have any degeee of tranfparency, but when they are exceeding thin ; as leaf Gold, &c. but, they approach nearer to tranfparency than the more grofs and porous kinds. The real caufe of Tranfparency, and how Vifion is conveyed through tranfparent Bodies, are (I am firmly perfuaded) among the hidden myfteries of Nature, which is not given Man to explore. 6. It is, I believe, a Paradox, not eafily accounted for ; that, if two triangular Prifms, of Glafs or other diaphanous Subftance, having equal Angles, are fo placed Fig. 12. together (as ABC, BCD) that the outfide Faces (A B and CD) arc parallel, we have dired Vifion through them both (from the Eye, at E, to an Objed at F); whereas, if either of them (as BCD) be taken away, the Objed is lofl to fight from that Point of View, by reafon of the fuppofed refradion of the Rays of Light, in palling through the Prifm ABC. Now, 6 Sea. IV. MATERIALITY OF LIGHT. Now, if the Pores, through which the Rays E a, E b, &c. pafs, go direai^ through both Prifms (from a to c, and from b to d) how can they be varied by the removal of one of them ? the taking away of one Prifm can Certainly have no ef- fea on the Pores of the other, fo as to alter their Direaioii ; yet (whatever be the C^ufe) it is certain, if either Prifm (as BCD) be removed, that the Vifion, of the Objea F, is turned afide out of a Right Line, and totally loft to fight, from that Station ; for, it will be feen by an Eye at £, and not at E, and appear to be in the direaion E F. Or, the Objea, at F, being removed to Gj will appear to be at F, in the direaion of E F J and F will be its apparent place. Again; if tlie Pores go direa through Glars,'&c. the Rays of Light do not ftop at the Surface, and confequently, they cannot fufter either Refleaion or Re- fraaion by it. For, I prefilfhe, the true definition of a Pore is a final 1 Cavity or Interftice, which admits a free paflage for Fluids. If, therefore, they do enter, and pafs freely through, how can the furface affea them ? or how can the Rays of Light, if Light is a Body, be turned afide, within the Glafs, in any other direc- tion than that of the Pores ? It is affirmed and allowed, that the Angle of Refraaiori is always to that of Incidence in a certain Ratio or Proportion ; and fince the Angle of Incidence may be any Angle at pleafure, it necefl’arily follows, that Light palfes freely through in all Direaions ; which (according to the eftabliffied Hypothefis, that it is corporeal) implies Pores in all Direftions, according to one of the Doaor’s Definitions; and, if it were poffible for Pores to go right or direa through, in all Direaions (which is repugnant to reafon and abfurd to fuppole) the whole Prifm muft be all Pores ; which omniporous quality, being attributed to any kind of bodily Subftance, I am perfuaded, no Man, in his lenfes, will acquiefee in. 7 . I (hall juft give one more objeaion to the materiality of Light, and conclude this Seaion, and Subjea. Suppofe the Eye, at E, viewing an Objea, A B. There is fuppofed to be Rays Fig. T ofi Light, A E, BE, &c. tranfmited from every point to the Eye, forming an Image of the Objea on tlie Retina, which is generally allowed to be inverted ; although a late writer has given fome reafons to the contrary. Be that as it may, it is certain, that if thofe Rays of Light enter the Eye, they muft pafs through the Aperture or Pupil, and converge to a Point (E) within, before they can diverge againto form an Image of the Objea ; at a c b. Now if the Eye be fo fituated (in refpeft of Diftance) to the Objea, that the extreme Rays, AE and BE, incline to each other in an Angle (AEB) not exceed- ing 40 Degrees ; it is certain, that the Eye is capable of taking in the whole of that Objea at one View ; although, every part cannot be diftinaiy feen at once. Every phyfical Point, in the furface of the Objea, as C, D, F, &c. is fuppofed to trarfmit a Ray of Light to the Eye, as well as in all other Direaions, at the fame moment of Time. Confequently, the whole fyftem of Rays generate a Cone or Pyramid of Rays, dole wedged together in every part ; which, all enter the Eye at the fame inftant, paffing through a Point of the fame dimenfion as one fingle Ray, at E, the Vertex of the Pryamid or Cone ; ’ which Circumftance is fo egregioufly abfurd, that it is fufficient, in my opinion, to refure all that can be alledged con- cerning the Rays of Light being material. For, can a great quantity of Air (the rareft Fluid that we are acquainted with) pafs through a Pin-hole in an inftant? Will it not (like Water) require more or lefs time, according to the dimenfion of the Hole, to pafs through ? can any force drive the fame quantity of Air through a Hole of half an inch in Diameter, in the fame Time as through one of an Inch ? No ; the ratio of the Time required, with the fame force, will ever be in proportion to the area of the Aperture; that is, the ratio of different Apertures, to each other, is equal to the ratio of the Time required, with equal force. And OBJECTIONS, &c. Book I. And yet, a Pyramid of Light, whofe Bafe Is equal to St. Paul’s Cathedral, nay, millions ot times greater (for the Eve can take in, not only the Sun, but, a great part of the Hemifphere, at once) and its Altitude of any dimenfion, as far as the fixed Stars; yet, I fay, this prodigious Pyramid of Light, a material Body, a Fluid, can pafs (without any known impulfe) through ten thoufand imaginary Pores in the Cornea, in their pafiage to an imaginary Point within the Eye; through which, the whole is conveyed in an inflant, to the feat of Vifion ; and what becomes of it after? the Eye muft be very capacious to contain it all. Nay, more; the very fame Bafe, or Gbje£t, can fend forth millions of Pyramids of Light, to other Eyes all around, at the fame Time, and in all Directions. Impoffible! being Matter. Can any Perfon form an Idea, to comprehend how the lame Body, or Surface, fimply, can emit, or reflect what it receives, this inftant, from any other Body, in innumerable directions at the fame Inltant ? Let thole, who can, find belief and give credit for it. Much more might be fald in fupportof this Argument; but, as It is not diredtiy to the purpofe of Perlpedtivc, I fhall not trefpafs any longer on the Readers time and patience, Jn this Digrefhon from the Subject. I lhall only beg leave to draw from it this Conclufion ; that Light is not an exifting Medium compofed of Par-r tides; which, being reflefled from Objeds, in all Diredions, and ftriking on the Organs of Sight, conveys the Vifion of them to the Mind, and occafions the fenfe of Seeing ; intimating, by means of the different qualities of its heterogeneous Rays, not merely the exiftence of the furrounding Objeds, but, they are alfo luppoled to excite, in the Mind, the idea of Colours on their Surfaces ; which, otherwife, have no exiftence: ftrange Dodrine ! But, with fubmiflion, I think, that the perpetual exiftence of fuch a Medium is repugnant to the notion of luminous Bodies emiting Light, inceffantly ; which, proceeding from them, progrefiively, in Right Lines, excites Vifion ; not inftan- taneoufly, but propag; te 1 in Time : which, opinion, is more confiftent with the notion of its being refedtid from other Bodies. For, how a ftagnating Medium, a Fluid, can be lo aduated, as to be refleded, from Objeds, in all Diredions withiil itfelf, and confequently, in dired Oppofition to its firft, or incident motion, and with fuch amazing velocity, is beyond the reach of human reafon to conceive ; much lefs to comprehend and explain. It is the dlftinguiflilng Property of lucid or luminous Bodies to difpenfe Light all around them ; but how, or in what manner, it is not my intention to enquire into ; being well affured, that the attempt would be as fruitlefs as prefumptuous. Such Phsenomena are, and ever will be, to Man, impenetrable and infcrutable ; Myfteries npt to be unfolded but by infinite VVifdom itfelf. 7 o us, there is a large field of knowledge, open and in view, whereon to exercife our reafbning faculties, and which lies within our reach ; let us not, then, ftep afide into intricate Mazes and Labyrinths, out of which it is impoffible to extricate ourfelves ; in which, the far- ther we wander the more we are bewildered ; till, wearied with the vain purfuit, we are, at laft, obliged to own, that all our boafted knowledge is but to know how little can be known. SECT. Sea. V. 33 OF REFRACTED VISION. SECTION V. Of Refracted Vision. T he laft Seaion touched, though very llightly, on that part of Optics called Catoptrics, or reflefted Vifion. In this I fhall briefly touch on Dioptrics or refraded Vifion; both which, I look on, in many refpeds, as Deceptions in Vifion. For it is evident, that, in the cafe of Refledion, on the furface of Water, &c. or polilhed Mirrours, we do not fee the Objed, but only ib Image or Appearance. So likewife, in the cafe of Refradion through tranfparent Mediums, folid or fluid, wc do not fee the real Objed, but its Image or Reprefentation only; which is likewife a Deception ; feeing that, the Objed appears, in fome cafes, larger, in others, fmaller than it really is, and always appears to be where it ' really is not ; although, we imagine that we are looking diredly at the Objed. Perfpedive Reprefentations, are alfo manifefl; Deceptions in Vifion. 'The bufi- nefs of Perfpedive is to reprelent Objeds on a Plane Surface, by the rules of Optics ; which, in the true Point of View, will give the Idea of a real folid Objed (being properly and judicioufly (haded) having the appearance of projedures and rece- dings, of one part before or behind another ; and having alfo the fame hue or teint of Colour, the Deception becomes ftronger ; infomuch, that it is poflible fo to deceive the Eye, as to imagine the Reprefentation to be real and fubftantial. In looking at Objeds through a Telefcope, either refleding or refrading; al- though we level it exadly to the Objed, in a Right Line, and imagine we look diredly at the Objed, through the Tube; yet, it is plain that we do not fee the Objed. For, in a Refledor, we look diredly at a concave Mirrour, which is placed dired between the Objed and the Eye ; confequently, in that cafe, it is impolfible to fee the Objed, but only its refleded Image on the Mirrour, which is firfl; received on another Mirrour, at the other end through which we look, and refleded again to the former, oppofite to the Eye, (a moft curious and ingenious in- vention ; which was firfl; conftruded by Dr. Gregory, and thence called the Gregorian Telelcope.) The Image, of the Objed received on the Mirrour, is magnified, by means of convex glafles in the fmall Tube ; fo that, we do not fee even the refleded Image on the Speculum, but the Image of that, only, magnified, to a great degree. In relped of the rafrading Telefcope, the Objed itl'elf is magnified, in the fame manner as the Image on the Mirrour, in the Refledor ; fo that, we do not, in either cafe, fee the real Objed, but only its magnified Image, between the Objed GlaflTes and the Eye. I fiiall illuflrate it by a Angle magnifying Lens, I. Suppofe an Objed, as A B, and a convex Lens, CD, placed between the Objed and the Eye, at E. It is manifefl:, fince Vifion is conveyed in Right Lines from the Objed to the Eye, that, if we faw the real Objed, by the Vifual Rays A E and B E &c. it would appear, in proportion to the Lens G D, of the magnitude only; but we find, that it appears larger than its real magnitude, in propor- tion to the Lens. For, if the Rays A a and Bb fall perpendicularly on the convex Surface, CHD towards tiie Objed, they will pafs diredly to the other Surface, which is concave towards the Objed, cuting it at a and b, and are thence refraded, to the Eye at E, as a E, b E ; then, if and B 5 be drawn parallel to the Axe of the Lens (E I) and Ea, Eb are produced, cuting A and BB in ^ and B, then, is yf B the ap- parent place of AB, or its Image; which, being feen under a larger Angle B than the real Objed A B, if the Lens was removed, or Plane, it will confequently appear larger; by Theorem iff. 1 Hence 34 6 f refracted vision* Book I. Plate IL Nence it is manlfeft, that the real Obje£l, A B, is not feen ; for it is not poflible Fig 14 Point, A, can be feen under the refraded Lines A a, aE ; and if Ea, E b ’ be produced, it is evident that the Objcd A B, in that place, muft be larger, to appear equal to a b, on the Lens. This way of determining the Image or the apparent place of the Objed, is ac-^ cording to Smith, P. 51. Art. 139. which, in fome refpeds, feems right (He does not indeed determine the Refradion by drawing a right Line, from A or B, to the Center of either Surface ; nor does he give any certain Rule to determine the Refrac- tion. It is impofiible to afcertain the Point where any incident Ray, from an Objed, cuts the Lens ; feeing that, the inclination of its Surfaces is continually varying, . (from the Center to its Extremes) but I find there are Cafes in which it is very exceptionable. E'or, according to this method of determining the apparent mag- nitude, or place of the Objed, it can never appear larger than at the diftance of the Lens. But it is certain, that the Eye and the Objed may be fo fituated, in refped of the Lens, that notwithftanding, the Objed, A B, is confiderably lefs than the Lens, it will appear larger ; and confequently, its apparent place is on this Side (in the focus of the incident Rays) between the E'.ye and the Lens, as at « E the Eye being removed to E; for ab, the Image of A B, appears, to the Eye at E, under a greater Angle (^a E b') than the Lens C D ; and therefore, A B, or its Image ab, appears, to the Eye at E, larger than the Lens CD, which it never could do, if its apparent place was on the other fide of the Lens. In this Cafe, the Objed A B appears ered ; but, if another Objed, F G, be placed beyond the Focus of the Lens, and Right Lines F G^, be drawn, through the Center, I, of the oppofite Surface, confequently perpendicular to it ; and if F G, GF be drawn parallel to the Axe cuting ¥ f and G^ in G and F; then Is EG the Image, or apparent place of F G, which, in this Cafe, is inverted ; and confequent- ly, the real Objed F G is not feen ; for the extreme F is feen at f, and G at g, on the furface of the Lens, in a contrary diredion to their true places. The contrary effed is produced through a concave Lens, in which, the Objed always appears Icfs than its real magnitude, according to its Diftance; which, be- ing but the reverfe of the other, *tis needlefs to illu Urate it. It is evident, that Vifion is conveyed through any tranfparent Medium whatever, obliquely fituated, in refraded or broken Lines ; confequently, in fuch Cafe, the Objed never is where it appears to be. For, the Point A, which is fuppofed to be feen by the Eye at E, appears to be at A or a, and not at A where it really is ; and the Point S, which is leen diredly through the Center of the Lens, although it be feen in the Right Line E S (for there is no refradion perpendicularly thro’ parallel Surfaces of any kind) yet, its apparent place is at S; and, if either the Eye or the Lens be removed, the apparent place of the Objed is .varied. . • 2. Lenfes of all kinds, excepting fuch as are plane on both Sides, or fuch Me- iiifcus, (hollow on one Side and convex on the other) as being portions of concen- tric Spheres, confequently parallel Surfaces, partake of the nature and properties of Prifms, whofe Surfaces are inclined to each other. Wherefore, the Rays, in paffing through them, are more or lefs refraded, according as the Surfaces are more or lefs inclined j that is, the lefs the radius of the Sphere, to which the Lens is formed, the greater is the magnifying power ; becaufe, the refrading Angle is greater ; and, confequently, projeds the Rays to a greater diftance from the Center. Hence it becomes a Paradox, how a large Objed (I mean one, which, accord- ing to its Diftance, occupies, or appears equal to the whole furface of the Lens) can be regularly magnified ; for fince it is manifeft, that through parallel Surfaces there is no fenfible Refradion (except the Surfaces are confiderably diftant from each other and feen through obliquely, as Fbce, Fig. 12.) and the furfaces of every Lens are parallel at the Vertex, and nearly fo at a fmall Diftance from it ; and, the greater the angle of Inclination, the Refradion is greater; wherefore (the furfaces of a convex Lens, being leaft inclined at its extremes, i. e. the refrading Angle is Se£l. Vt OF REFRACTED ViSlON* the greateft ; and, feeing it is continually varying, from the Vertex to the Ex- tremes) it feems reafonable, to conclude, that the Objedl would be more diftorted about the edges of the Lens than near the middle ; which I do not find to be the Cafe. Query; by what means is the Objedt magnified equally? 3. There is another circumflance, which I do not remember to have feen no- ticed, by thofe who treat of the properties of Prifms, in refpe6l of Refradiort ; and which, 1 do not fee a reafon for. Right Lines, feen through a triangular Prifm, or any two inclined plane Surfaces, being parallel to the interfering Line of the two Planes, that is, to the refrading Angle of the Prifm, do not appear Right Lines, but, curved ; and they are more or lefs curved, as the Angle of Refrarion is greater or lefs ; the Curve being concave towards the refra£ling Angle. Now it is reafonable to fuppofe, that the fyftem of Rays from any Right Line to the furface of the Prifm, which is a Plane, would alfo form a Plane; which, by its interfedion with the furface of the Prifm, muft neceffarily generate a Right Line, in any pofition of the Prifm ; which, would be the Reprefentation of the original Line ; and, when the Angle of Refradiion is parallel to the original Line, they will alfo cut the other Surface, towards the Eye, in a Right Line, parallel to the oppofite ; how is it then that it appears curved ? The vertex of the Curve, whatever it be, is where it would be cut by a Plane, palling through the Eye, perpendicular to the Original Line and the Planes of the Prifm. I have already deferibed a Prifm, in the laft Sedion, and the nature of Re- fradion by it, as much as is my defign, in refped: of the deception of Villon through a Prifm. Thole who would be more acquainted with the properties of Prtfms and Lenfes, in general, 1 refer to Sir Ifaac Newton’s, or, where they are treated more at large, to Smith’s Optics; where, if he has patience to go through it, he will find enough to exercife his patience on. I do not mean to difparage the Work, for I believe it to be the bell of the kind, in many refpeds ; yet, I think it deficient, in fome Cafes ; but, in refped: of Colour, &c. it is of a piece with the reft, and, being dwelt too long on, becomes tedious, if not trifling. 4. As to the Phaenomenon of Colours feen through a Prifm it is really furprifing ; but, it is only the Edges of Objeds that are tinged or produce the Colours. A perfedly plane Surface is not at all varied in its Colour, when feen through a Prifm ; except at the Edges, when it is joined or oppofed to a darker Body ; and thofe Edges are moft coloured when they are parallel to the refrading Angle; for, when they are perpendicular to it, and feen direct through, they have no Colour at all, but the natural Colour of the Body, whatever it be. Alfo, the Colours feen through the Prifm, do not always follow in the fame order as when a Beam of the jSun’s Rays pafl'es through ; if a fmall Objed be oppofed to the Light (as, a Bar iti a Safti- VVindow) being parallel to the Prifm, the Objed is wholly loft in the Colours, between the Red and Blue ; the Red being towards the refrading Angle ; an in- tenfe Red always follows after any Line or Edge of an Objed which is oppofed to, or which obftruds the Light in one pofition of the Prifm, the brighter Colours fucceeding; but, being reverfed, there is a deep Purple, and Blue following; the other Colours are very faint. The Colours are more intenfe and vivid the brighter the Objed, or the greater the oppofition and obftrudion of the Light by an opake Objed. If a flip of White Paper be laid on any dark Body, being feen through a Prifm, its Edges are bordered, one with Red, the other Blue, according to the refracting Angle ; but being laid on another, of the fame Colour, none is perceived, except it makes a Shade on the other, not lying clofe, which Shade occafions Colour.’ Alfo, if a ftrong, black Line be drawn thereon, the Red and Blue pro- ceed contraryways from the Line, which is quite loft between them ; the Colours being produced, on any Surface, only where there is an oppofition with a darker Body. How, then, it is poffible to deduce or to draw a conclufion, from fuch Experiments, that all Objeds, which we perceive, are by Nature of the fame, or have no Colour at all, fave what is effeded by the Light refleded from their Surfaces ? Book I. 36 OF REFRACTED VISION. Plate IL 5. As ill Nature, there is necelTarlly Rcfra^lion at the common Surface of any two different Mediums, whether folid or fluid, except when we look perpendicu- larly through the Surface, or Surfaces, at the Objeft, although the true reafon of it be to us unkno*wn ; fo it would be fuperfluous and foreign to my Defign to multi- ply Cafes wherein it happens. I (hall, therefore, only take notice of one common Cafe, which is fo very common, though but little attended to, as to be obvious to every Perfon w'ho is blefled with the faculty of Seeing. Let any Perfon put a ftreight Stick into Water, flanting in any diredlion, and it immediately appears to be broken, at the furface of the Water; the part within, ap- pearing to take a different diredion to that which is out of the Water; and, the more it deviates from a Perpendicular, the more it is refraded or broken, till the Stick makes an Angle, with the furface of the Water, of about 45 Degrees ; and then, the Refradion is continually lefs, which is evident ; for, when the inclination of the Stick, to the Surface, is fuch, that the Angle it makes with the Surface is very fmall, confequently, the Refradion cannot be great ; and being immerfed perpen- dicularly, there is no Refradion ; which is the fame thing as looking diredly into the Medium, at an Objed in the Water, or through a Body of Glafs, very thick; in which Cafe, the Objed appears to be confiderably nearer, but, it is in the fame Diredion as we fee it. Alfo, when we look at the Heavens of Stars, &c. thofe which are in the Zenith are feen in a Right Line; and, the farther they are from that Point, the more the line of diredion is refraded or broken, fo chat, they are fuppofed not to be in that place in the Heavens where they appear to be; which Refradion, is faid to be owing to the Atmofphere of Air and Vapours, which fur- rounds the Earth, being denfer than the upper Regions. 6. It is a common, though curious and entertaining. Experiment, to immerfe a piece of Money or other Objed in a Bafon full of Water; which will appear to be railed confiderably higher than when the Vefl'el is empty. Fig. 15. Let ABC D reprefent a Veffel, fuppofe of Glafs ; and fuppofe E an Eye looking at an Objed, at F, at the bottom of the Veffel, being empty. W'hilff the Eye remains fixed, at E, fo that the Objed at F, can juft be feen over the edge, at C, let there be Water poured gently into the Veffel; as the Veffel fills, the Objed will appear, to rife, gradually, to G, fo as to be feen quite clear of the edge of the Veffel, in the diredion EG. Now it is certain, that the Objed is at F, which appears to be at G ; confe- quently, if the real Objed be feen at all, it is feen in the refraded Lines FI, IE, where the Right Line EG, to its apparent place, cuts the furface of the Water; but, more probably, the Image of the Objed, only, is feen at G. Confequently, a ftreight Stick or Wire being put into the Water, from E to F, will appear broken at the Surface (at C) and appear to go in the Diredion C G. Again. Let the Eye be removed to E, the Water remaining in the Veffel, fb that the Objed is apparently feen over the edge, at C ; and, whilft the Eye re- mains fixed, let the Water be drawn gently off, by means of a Cock, or other wife, at the bottom ; the Objed, apparently at G, will gradually fink lower in the Vef- lel, and totally difappear from that Station. Hence it is manifeft, that when the Water is in the Veffel, the real Objed, at F, is not feen by an Eye at E ; for, if it was poflible, it muft be feen through the Side of the Veffel, at H, which it is plain is not the Cafe; and, hence it is plain, that the Objed being feen, apparently at G, is a manifeft Deception in Vifion. The Cafe is the very fame, in Objeds feen oblique through Glafs, or any other pellucid Subftance; excepting fome fmall variations in the degree of Refradion. 7. Refradion in Water, according to all writers on Optics, is fubjed to one In- variable Law, without any fenfible error, but it is not to be demonftrated mathe- matically. 1 have made the Experiment myfelf, as accurately as it will admit of, and find it to be nearly as follows, 2 If 37 Sea. V. OBJECTIONS TO PORES IN WATER. If AB be fuppofed the fiirface of Water, and CE, DE or FE an incident Fio-. i6, “Bay of Liglit falling on it, at E ; on which Point, fuppofe a Circle be dcicribed, and the Perpendicular, EG, drawn. Then, if CE be produced to c, and ac be drawn, parallel to AB; take ab equal to three fourths of a c, and draw bd perpendicular to ac, cuting the Circumference in d; the Ray CE is fuppofed to be refraded, at E, and go in the diredlion PAl, into the Water. By the fame -Rule, DE goes in the dir-dion Ee, and FE in Eg. But, the lame Rays CE, DE, &c. will alfo be reflected in equal Angles, GEH ■ equal DEG, &c. Can the fame Ray be both reflected and refracted? impolfible; yet, an Objed (0) may be feen at H, by the refledted Rav E H, as well as at e, by ‘the refracted Ray Ee. It is however certain, that if a Circle be clefcribed on the Plane AGBg, and Lines are drawn thereon, as in this Figure; being immerfed perpendiculai ly in Water, to the Diameter AB; CEd, DEe, and FEg will appear Right Lines; which is very furprifing. But yet, I do not fee which way this Experiment proves, that Rays of Light go in thole directions into the Water; the whole Mafs being illumined in every part, if the Eye, being immerfed in Water, at e, fees the Object 0 over the edge of the Veflel, or any other obflacle, at E, on the Surface (which, i prefume is the Cafe) it indicates, that the Objedt is not leen in a Right Line ; or, rather, that its Image is leen at E, on the-furface of the Water. 8. Before I conclude this Sedion and SubjeO:, I cannot help taking notice of an extraordinary paffage ; which is in the Conclulion drawn from Prop. 8, Part 3, of tjie fecond Book of Newton’s Optics, Page 69; concerning the extraordinary poroflty of Water ; which is 19 times lighter, and confequently, he fays, 19 times rarer than Gold ; and Gold is fo rare, that Water may be forced through its Pores. tir, as he was informed, by an Eye witnefs, a Globe of Gold being filled with Water and fodered up (but of what thickiiefs he does not tell us ; I have beard fay above a quarter of an Inch) and, being prefled wdth great force, the Water fqueezed through its Pores; and flood, all over its Surface, in multitudes of fmall drops, like Dew. From which, he concludes that Gold has more Pores than folid Parts; but how fuch a conclulion can polfibly be drawn I cannot conceive. I Ihould fup- pofe, that, in fuch Cafe, the Water would fpout out In flreams, rather than fland on the Surface like Dew. Yet Gold will not admit either Air or Light through ...its Pores, though much rather Fluids than Water; which, he alfo concludes, from the fame Experiment, has above forty times more Pores than Parts ; and con- fequently, Gold (the compadeft Metal we know of) according to that Ratio, con- tains above twice as much empty Space as Matter. It is no wonder that Water is pellucid, being fo extremely porous, and admits Light fo_ freely thro.ugh it^ Pores; but, it is fomewhat furprizing that it is not compreflibi^* which all porous fubflances muft be. And 1 think it alfo furprizing, that a Man of Sir Jfaac’s fagacity Ihould advance fo P. 13. i. El. wherefore, the Angle b O B is greater than ObP. — — 12. j. But, the Angle OBP, being external, in refpefl of the Triangle b O B, is equal to bO B-pObP; wherefore, it is greater than double ObP. 10. i. But, if the Angle abc be equal half ABC, ac will not be half AC. 3. 6. El. Confequently, fince the Angle A B C is more than double abc (and AB is equal to a b) AC is ftill more than double a c. Therefore, the apparent magnitude of AC to a c is more than duplicate; feeing it is more than half ac, and is alfo feen at lefs than half the diftance. Now I am far from fuppofing, that this Author was fo deficient in Geometry as thefe Examples feem to indicate ; nay, I am well convinced he was not, as is evi- dent in the next Book ; and his Provifos are fome extenuation of the errors ; he ought, then, to have told us that they are nearly fo, by approximation, and not that they are lo, in exprefs Terms. But, in any Cafe whatever, there is not an equal ratio, or nearly, between the increafe and decreafe of the Perpendicular AC, and the Angle A O C. As this and the preceding Article are the only paffages which any way tend to advance the Theory of Perfpedlive (having been quoted, by Air. Kirby, for that, though but little, purpofe) 1 have, therefore, been more particular in my remarks on them. And, as what I have advanced in the two laft Seflions, and part of the firft, is not direflly to the purpofe of Perfpeflive, it will, I know, by fome, be deemed Impertinent and foreign to the Defign of this Treatlfe ; let them, if they pleafe, pafs it over, and proceed immediately to the Subjefl, which is not vitiated by it. As I do not intend to publilh a Treatife on Optics, and as Perfpedive has a near affinity to that Science, what exceptions I have always had, to fundry pafla- ges in the works of optical writers, I thought proper to give here, where I was treating on an eflential part of the Theory of Perfpedlive, and a branch of the Science of Optics. 1 hope I have not trefpaffed, too much; on the time and patience of the candid and unprejudiced Reader. I ffiall now proceed to Book the fecond, which treats on the Theory of Perfpedive ; where, I ffiall endeavour to make fome amends for his time fpent, 1 hope not loft, in perufing this Digreflion. BOOK B K [ 40 O O Of the Theory of Perfpe8:ive, redlilinear and curvilinear. SECTION I, Containing a general INTRODUCTION to PERSPECTIVE. T O define the Terms of Art peculiar to any Science, with brevity and perfpi- cuity, I have always looked on as a particular excellence in the Work; but I have frequently found, that an atfeftation of brevity has left the Term, intended' to be explained, rather obfcure, at lead: doubtful ; whilft others, endeavouring to render it clear by a multiplicity of words, have, at lad:, involved it in perplpity. Although my defign is to be as brief as the nature of the Subjeft will admit of, yet I am afraid I daall rather be thought prolix, than otherwife, in fome of the fol- lowing Definitions or explanation of the Terms; but certain 1 am, that, the Time fpent in acquiring a perfedl knowledge of all the Terms I have defined, will not be loft, as they contain many ufeful Lelfons to a young Student. 1 have endea- voured to explain every Term in the mod familiar manner ; not faying more than vi'as necedary, -yet, enough to be clearly underdood. To be too brief is worfe than prolixity, fo it be not tedious and trifling; the one leaves us doubtful, the other, probably, makes it clear at lad. i may, perhaps, be particular and fingular in my opinions ; but, I think it better to define each Term feparately, than to include feveral in one Definition, as is wery frequent ; and I always choofeto name the Term, I mean to define, fird, rather than end the Definition with it, or name it promifeuoufly. My reafon for which^ is, that ’tis much eafier to find when refered to j or, when there is occafion to look for any particular Definition, without reference, the Number not being known. The Order in which the Terms are defined is, with me, a material circumdance; begining at the Foundation, and going gradually on in a regular fucceflion ; never ufmg any Term, if poflible to avoid it, in the explanation of another, which has not already been defined. In general, I find them promifeuoufly jumbled together, without Order ; going, as it were, from one end of the Science to the other, to and again; too hadily introducing, perhaps, fome new or favourite Term, before others which feem neceifary to be fird known, in order to prepare the way, by removing fome impediment. How I have fucceeded, mud be left to the decifion of the candid and impartial Reader. I am no fwourer or admirer of new and uncommon Terms ; nor, indeed, do I think that every writer, on any Science, has a right to impofe new Terms ; unlefs. he has found out fome new Principles, on which it 'was not poflible for him to ex- patiate, by the Terms already known and in ufe ; as Dr. Brook Taylor has done in Perfpedlive ; on which, I have fpoke more largely in another place. It was im- poffible for him to convey the Ideas he intended to inculcate by the old Terms, aiid therefore, he was under the neceflity of inventing and enforcing new ones ; which arc extremely expreflive of the thing meant. But, if every writer, on that or any other Subjed , (who, becaufe he knows fomething of it, imagines he knows more -3 r tliaii .Ik Plate III. Scfi:. I. A GENERAL INTRODUCTION, &c. than any who have wrote before him) was to take the liberty to Impofe new and \inmeaning Terms, of his own, fuited only to his own trifling Ideas of the SubjefV, the Science would, by that means, become perplexed and intricate; each Perfon,. w'ho happened to receive his knowledge of it from different Books, would, confe- quently, underftand and call the fame Thing by different Names ; than which, efpecially when they are abfurdly or falfly named, nothing tends more to perplex, and involve the Science in obfeurity. I have already, in the Preface, given my reafons for omiting, in this Treatile, 'geometrical Definitions and Problems ; becaufe, I fuppofe the Reader already toler- ably verfed in Geometry; if not, I advife him firfl: to ftudy it, at lead; Pradlical Geometry ; without which, it is ufelefs to attempt, and impoflible to fucceed in the Study of Perfpedlive : the better he is acquainted with Euclid, the greater progrefs will he make in Perfpeflive ; of which, Geomety is the foundation. To affift him in it, I have compiled and compofed a Volume (mentioned in the Preface) which may be called an abridgment ; it, neverthelefs, contains all that is eflential. My chief aim, in that work, was to make it ufeful, the Study of it pleafant, and attainable by any tolerable Capacity, and applicable to various ufes in Life ; par- ticularly fubfervient to this Treatife of Perfpeflive, as I always refer to it for De- monftration : in which cafe, it may be deemed a part of this Work, and ought always go together. But, although I have, there, fully defined a Plane (Def. 6th.) yet, as it is fo very eflential in PerfpeiStive, it was by no means proper to omit it here ; feeing that, on it the whole Theory of Perfpeflive is built. A PLANE is a perfeflly even, {freight, and regular Surface, which is neither convex nor concave in any part ; but agrees, in every part, with a Right Line or {freight Ruler, applied, any how, to the Surface. 2. By the motion of a Right Line, a Plane may be conceived to be generated ; either by a diredf, lateral motion, or by fuppofing it whirled around (fo as not to generate a Cone) on any Point in it, 3. It is eafy to conceive, that, if the Eye of a Perfon be in a Plane, or in a con- tinuation of it, the neareff extremes or limits, towards the Eye, hide all the relf of the Plane ; for, the whole Plane vanifhes, and is loff to an Eye in the Plane j which, appears but a Right Line, extended in length to the apparent dimenfions of the Plane, confidered as having only length and breadth : thicknefs or fubffance is the property of Solids, a Plane has none; ’tis the Surface, only, that is confidered. N. B. The Piffure (in PerfpeRive) is always underlfood to be a Plane. There- fore, the Board, Canvas, or Paper, on which we draw, is the Plane of the Piflure. N. B. 2. A Plane may be of any Figure; and, in Perfpeffive, it Is frequently confidered as being infinitely extended, without regard to Figure, or to its limits. A Cube Is bounded by fix Planes, which are all Squares ; as A, B, X. Fig. i.Platelll. Every other Parallelopipid has alfb fix Planes, which are all Parallelograms ; cither right angled, as Fig. 2, or acute angled, as Fig. 3. A Prism is a Solid, whofe Bafe, A, and Top, B, are either fimilar Triangles*, Quadrangles t» or Poligons:|; of any Number of Sides. The other Planes or Sur- faces are all Parallelograms. (As X, X.) Fig. t Fig. I. 2, & 3. + Fig. 5 Pyramids have their Bafes (A) of any Number of Sides. The other Planes, of Fig. 6. the Sides, are all Triangles. (As Y, Y.) 4 “N. B. No Solid can be formed of Planes, having lefs than four; and that mull be a Pyramid, L ~ Of 42 A GENERAL INTRODUCTION Book II. PlatellL Of PLANES and their POSITIONS, in GENERAL. T. Horizontal is the fird: and mod; natural pofition of Planes ; fuch, are all Planes which are parallel to the Horizon; confequently, all horizontal Planes are parallel amongil ihemfelves. As Z, H, H. 2. Vertical. All Planes which are perpendicular to, or which cut the Horizon at light Angles, are called vertical Planes. As V. Fig. 7, 8, and 9. Vertical Planes may be in all pofitions, in refpedl of each other, viz. parallel, per- pendicular or inclined ; as may be conceived, by revolving a vertical Plane, on a RightLine, AB, perpendicular to the Plane of the Horizon; confequently, they will, if produced, all pals through the Zenith and Nadir * of our Horizon. 3. Inclined. All Planes whatever, which are neither parallel nor perpendicular to the Horizon, are Inclined Planes. F or, if a Plane cut the Horizon, or would if produced, in an acute Angle, it Is not vertical or perpendicular ; confequently, it inclines to the Horizon on one fide more than the other ; which, Inclination, is always meafured on that fide making the acute Angle ; as, for Example. Tig. 9. The Plane X inclines to the Horizon, in the Angle BC A ; and it is alfo faid to incline to a vertical Plane, in the Angle BCE; which Angle, if they have the fame Interfedtion, CD, or parallel Interfeftions with the Horizon, is the Comple- ment of it sinclination to the Horizon. See N. B. Def. 13th. Geo. 4. It may not be improper, here, to obferve (for It Is neceffary to know and un- derftand well) that the Angle of inednation, of one Plane to another, can be mea- fured, only in a Plane to which both the other are perpendicular; or, which is the fame thing, if a Line be drawn in each Plane, from the fame Point in their common Interledlion, and perpendicular to it, an acute Angle maile by rhefe Lines is the In- clination of the Planes ; tor it is evident, that a Plane p fling through thole two t 2. 7. El. Lines will be perpendicular to the common Inteifedlion -f' and conlequentiy to both Planes. To illuftrate it. Suppofe the horizontal Plane Z raifed up into the Pofrion of X, inclined to the Horizon ; the line C D, on which it was luppofed to turn, may be confidered as the common Interfedlion of the two Planes, X and Z It is evident, that the point A will, in that motion, have deferibed the Ark AB; and, if the Angle ACD be aRightone '^as it is fuppofed) BCD is flill a Right Angle; § Def. II. wherefore, A C'and C B are both perpendicular to C D ; § and the Plane ABC, de- Cieom. feribed by the motion of C B, Is alio perpendicular to C D, f and confequently, to jg'L ’ the two Planes X and Z.|: Tlierefore, the Angle ACB, in the Plane ABC, is the Angle of Inclination of thofe two Planes. Draw A D, at pleafure, cuting the common Interfedlion, C D, in D j and, from the fame point D, draw DB, in the inclined Plane X. Dem. Now, if ACD be a Right Angle, A D C is acute — Cor. 3. 10. i. El. wherefore, A D is longer than AC ; and alfo, B D than BC — P. 1 2. i . El. But, the Chord, or Subtenfe, A B, fubtends both the Angles, ACB and ADB; confequently, the angle A DB is lets than ACB. — — Cor. to 14. i. El. Or, fuppofe BF perpendicular to the Plane Z ; and let FD be drawn. Then, a Plane DBF, palTing through BF, is perpendicular to the Plane Z, but not to X. And, becaufe F D is longer than F C, and B Dthan B C, it is manifeft, that the Angle B C F is greater than B D F. But, the Plane CBF is perpendicular to both -the Planes, X and Z (as before) and, confequently, to their Interfedion C D. * Imaginary Points in the Heavens, diametrically oppofite to each other ; the one perpendicular over our Heads, the other under our Feet, in the lower Hemifphere. 4 From .Fig. 7. -Fig. 8. Sedl. I. TO PERSPECTIVE. 43 From which It Is clear, that the Angle made by a Plane ciitlng two other Planes, perpendicular to their common Interfedlion, is the Angle of Inclination of thofe two Planes ; feeing that, the Angle made by any other Plane, paffing through AB or B F, will necefl'arily be lefs, the greater the Inclination of AD, or F D, to C D. To fet this matter In the cleareft Light poffible, it being fo very elTential In Pcr- fpedlive, as well as in other Sciences and Arts, I have added the following Figure. Let ABC and C B D be two redlangular Planes, cuting each other In B C, their Fig. lo. common Interfedtion. Let EF and FH be two Right Lines, one^in each Plane, perpendicular to their Interfedlion B C, at the fame Point, F. Wherefore, the Plane E I F’ G, paffing through thofe Lines, Is perpendicular to both Planes, AC and C D ; § and, the Angle EFG, made by that Sedlilon, Is the § 2. 7. LI. largeft that can poffibly be made by a Plane which is perpendicular to either of them. For, fuppofe the Line E FI perpendicular to the Plane AC B, only ; and, a Plane EIKL to pafs through that Line, it will be perpendicular to the Plane AC;f and be- f 9. 7. El. caufe the Plane E F'G Is perpendicular to both the Planes AG and C D, and palfes through the fame Point E, it will, alfo neceffiarily, pafs thro’ the Line EH; wherefore, EH is the common Interfedlion of thole two Planes,^ EIFG and EIKL produced. I fay, that the Angle E F H, made by the Plane EIFG, is greater than E K H. Dem. Now, EH, the common Interledlion of the two Planes IFG and IKL, together with the Interfeftions EK and K H, of the Plane I KL with the two. Planes A C and C D, form a Triangle ; and fo does the fame line EH with the two Interfedllons E F and F H, made by the Plane IFG, with the fame Planes A C and C D. But HF K is a Triangle, and, the Angle FI F K is prefumed to be a Right one ; wherefore, FI K is longer than H F and, for the fame reafon, E K is 1 12. i.El. longer than E F ; and conlequently, the Triangle E F H, having one Side (EFI) common with the other Triangle E K H, and, having the other tw^o Sides, EF^ and F Ff, Icfs than the two Sides EK and K H, refpettively, they, therefore, contain a huger Angle, Wz. EFH than E K H. — — Cor. to 14. i. El. » But, the Plane IFG is perpendicular to both the Planes, AG and CD ; and the Plane I K L is perpendicular to one of them (AC) only. Therefore, the Angle EFG, made by the fedion of a Plane which Is per- Fig. 10. pendiculai to both the other, is larger than any Angle made by any other Sedlion, of a Plane perpendicular to one of them only ; and confequently, the Sedflon £ F' G meafures the true Angle of Inclination of the Planes ABC and C B D. N.B. The Angle made by the fedlion of a Plane inclined to both Planes, A C and C D, may be either greater or lefs than EFG. 5. Although It Is not poffible to conceive an Idea of a Plane abfl:ra£led from one or other of the three Politions I have explained, yet, in the application of Planes, in Perfpedive, as in Geometry, no particular regard Is had to them ; for one Plane is faid to be perpendicular to another, if it makes right Angles with the other Plane, as H to VA each of which is faid to be perpendicular to the other, notwithdanding Fig. 1 1. one of them (FI) is horizontal. The Planes X and Y are alfo faid to be perpendicular to each other, although both are inclined to the Horizon ; and, whatever their Inclination to the Florizon may be, it matters not, if they make right Angles with each other, as at C. For, if the Plane Y was turned up, on AB, its interfedlion with the Horizon, into the vertical Politlon W, and, along with it, the Plane X, into the horizontal Pofi- tion Z ; their Politlon, in refpedt of each other, is not altered, if the Angle, at D, be ilill a Right one, as before, at C. * E I is the true Interfe^^ion, perpendicular to the Plane A'C ; and confequently, E H would be I E produced. But, as it would run into the Figure below, I thought it beft to difpenfe with ir, as the Di-monllrarion is the lame. ! 6 . . I I Vi I! 44. A GENERAL INTRODUCTION Book II. PlateIII.6. So llkewlfe, one Plane Is fald to incline to another, if they do not interfe£l at right Angles, as H and X, or would not if produced, as X and V. The Plane X being inclined to both H and V, (Art. 3. of Planes) they are, for the lame reafon, both inclined to X ; yet, one is horizontal and the other vertical ; for, the inclination of two Planes is mutual. So that, when it is fald that one Plane is perpendicular or inclined to another, it means nothing more, than, that they are at right Angles, or otherwife with each other ; no regard being had to the hori- zontal or vertical Pofition of either; except the Pofition of one is previoufly known or determined, to which, the other is faid to be perpendicular or inclined. 7. In Perfpedive, It is alfo frequently fald, that. Lines are perpendicular to certain Planes ; whereas, if the Plane be vertical, it is eafy to conceive, from what has been faid, that all Lines, which are perpendicular to a vertical Plane, are horizontal, Fig. 12. and parallel amongd themfelves; as A B, E F, and CD, to the Plane GIK. Yet, the Planes, in which thefe Lines are, may be either horizontal, as ABFE; vertical, as EFCD ; or inclined to the Horizon, as A B C D. If the Plane be horizontal, the Lines perpendicular to it are really perpendicular i.e. to the Horizon; as A J, ED and FC. But, If the Plane be inclined to the Horizon, then, the Lines, which arc perpendicular to it, are alfo inclined to the Horizon, yet parallel amongft themfelves. Suppofe the Plane ILMN vertical, and perpendicular to the inclined Plane GH; the Lines LI and MN, which are at right Angles with its Interfeflion, IN, are per- pendicular to the Plane G H. But the line IP, which is perpendicular to the Hori- zon, is inclined to the Plane G H, in the Angle LI O. And OI, at right Angles with I P, Is horizontal ; but, it is alfo inclined to the Plane GH, in the Angle OIN equal LIP. Dem. Let N I be produced, towards H. LI is perpendicular to N H, Then, becaufe LIN is a Right Angle, LIH is, alfo, a Right one - C. 2. i. El. Conlequently, LIO, equal LIN — OIN, is equal to PIH, or LIH— LIP. Every Right Line, therefore, which is neither parallel nor perpendicular to a Plane, is confequently, inclined to that Plane ; and its Inclination may be known, by drawing a Perpendicular from the extreme, or any other Point, as M, in the Line 'I M, to the Plane GH; the Complement of the Angle IMN, i.e. I ML 4. I. El. equal MIN*, is the Inclination of I M to the Plane G H. Or, If a Plane (ILMN) be drawn, through the Line IM, perpendicular to the Plane GH ; the Angle MIN, which the line IM makes with IN, the interfefllon of the perpendicular Plane with G H, is the Angle of its inclination to the Plane G H. N. B. IN is the Seat of the Line I M or 10 , and alfo of ML, or, of the Plane N L, on G H ; produced by the Interfeflion of a perpendicular Plane paffing through the Line, as above. Therefore, the Angle which any Line makes with its Seat, on a Plane, is the Angle of its inclination to that Plane. 8. PQRS is a horizontal Plane, cuting the inclined Plane KPQ in the Line PQ. But, PQ is not the Seat of the Plane PR, on KQ, it being inclined to KPQ, in the Angle SPT ; for, if ST and RU be drawn, perpendicular to the Plane KPQ, the Lines P T and Q U (joining the Points, T and U, where the perpen- diculars cut the Plane, with P and Q) are the Seats of the Lines PS and QR; T U is the Scat of the Line R S, or of the Plane T U R S ; and confequently, PQUT is tlie Seat of the Plane PQRS, on KPQ. Alfo, turs is the Seat of that part which is over the Ground Plane. I would advife the young Student, who is not well verfed in thefe things, to make them familiar to him; for which, the reading over a fecond time, with due attention to the Figures, will be fufficient. 3 I have Se£l. I. TO PERSPECTIVE. 45 I have been more particular on this Subjeft, becaufe I have frequently known Pupils to be miflead, by calling Lines and Planes perpendicular, Imagining them to be really fo, i. e. to the Horizon^ from the common acceptation of the Term, Perpendicular, (to hang down, as a plumbLine) not confidering the Pofitlon of that Plane or Line, to wiiich, the other Planes or Lines are fald to be perpendicular; and to which, due attention mull firll be given. 9. Suppofe the Objefl AIKC to be, in the lower part, a right angled Paral- Fig. 13. lelopiped (the moll general form for Buildings, or the feveral Parts of a Building) the Planes B E D C, and A I B, of the Front and. End, and their oppohtes, are ver- tical; for they are perpendicular to the Horizon, or Ground (confidered as a Plane) on which it Hands. Now, the Lines AB and FE, in the Plane AI B, are perpendicular to the Plane B E D C ; and the Lines B C, ED, and I K, are all perpendicular to the Plane A I B (i. e. the originals of thofe Lines are fo, in the real Objedl) yet, they are all parallel to the Horizon, and the three lall, parallel between themlelves, though in dilferent Planes ;§ for, a Plane may pafs through any tw'o Lines which are parallel. (Ax. 5.) ^ El* 10. In the Practice of Perfpeclive, it .is often necellary to fuppofe the Obje£b, we are delineating, tranfparent, as if the whole Objedl was Glafs.; and, the Planes or parts of the Building, which are adjacent, are luppofed to be feen through the hither Planes ; as ABGH, the ground Plane, and FEDG parallel to it ; AFGH parallel (or otherwife) to B E D C, and H G D G oppofite to A F E B. Thefe fix Planes, forming a right angled Parallelopiped, compofe the Body of the Building ; every Angle of which, A, B, E, D, &c. is a folid Right Angle ; each being compofed of three plane Right Angles, as ABC, ABE, andEBC, of the Angle B; confequently, AB, BC, and B E are each perpendicular to the other. By means of this fuppofed tranfparency, the.connee, by the interfe6lion of the Rays, is the Orthographic Projeiftion. ’ Fig. 14. The Lines A a, B b, &c. are parallel to the Llori'zon and to each other; which. No. 2. falling perpendicular on tlie vertical Plane V, ddefibe the Orthography of the ' Obje6; A B C, on that Plane. Of this kind of Projection there may be infinite variety ; for, if either the Plane or the Objedl be turned, though ever fo little, the P'igure will be varied on it. The Figure, aefb, thus projedled, may likewife be called the Seat of the Objedt A B C on that Plane ; as a e c b on the Ground Plane. a is the Seat of the Angle A, and b of the Angle B, &c ; a b is the Seat of the Side A B, a e of A E ; and, the Diagonal a f of the Side A F. a e may alfo be confidered as the Projedtion, or Seat, of the Plane A E C, and a b of A B C ; for, if they were produced, they would cut the Plane V in thofe Lines. N. B. The Seat of a Point, Line, or Plane, may be had on any Plane in whatever Pofition; by draw- ing Right Lines from each extreme of the Line, &c. perpendicular to the Plane. b f ,011, the Plane Z, and hf on the Plane Y, are Llie Seats of the fame Line B F, in the Obje^ ABC; and fo of the reft. V. Ortho- Sea. II. IN GENERAL. 47 Orthographic Projeaion Is ufually called the Elevation. But, when it ex- hibits the End of a Hnikiing, or part of the End only, ihewing the Projeaures of the feveral parts, from the main Body of the Building, the Contour of the curves of Moulding, &c. it is called a Profile. And, when the Building, or other Objea, is luppofed to be cut by a vertical Plane, in any direaion through the Building, the hither part being fuppofed to be removed, and the infide expofed to view, Ihewing the thicknefs, &c. of tlie Walls and Floors, the ftruaure of the Roof and proportion of the Timbers, &c. it is called, a Section. All thefe different kiiuE of Projeaion are entirely geometrical, and fuppofes the Eye of the Speaator at an infinite Diftance ; the projeaing Rays being parallel. SCENOGRAPHY is the Projeaion made by a Cone or Pyramid of Rays, Right Lines, from every Angle of the Objea, converging to a Point. As O A, OB, &c. Thole Rays being cut by a Plane (X) palling, m any direaion, between the Ob- ^ jea and the Vertex (O) the Figure projeded, by the Rays, on that Plane, is called the Scenographic Projeaion of the Objea ABC. If the Vertex, (O) of this Pyramid of P^ays, be confidered as. int, there will be true, linear perfpedive, Reprefentation of all the Objeds, on that Plane ; which, is confidered as the Pidure. Now, the perform- ance of this, by geometrical Rules, is what is properly called Perspective. * Every Object, having length, breadth, and thicknefs or depth, comes tinder jhe Denontination, geomotnc.illy, of a Solid; the concavity, (as of a Box, or Houle, Sic.J cot being confidered ; but only the external honn, connit’.n^ uf Planes or pnher Surfaces, varioufly diijwfed. Book II. PlateV. I’g* 15* OF P R O J;E C TI O N N.B. Every Reference, to Figure 15, refers likewife to the Apparatus. Let BF IK L be an Object havlug tho three Dimenfions, length, breadth, and thickncfs ; which may be luppofed a Building, or what you pleafe, to be rspre- fented on the Plane, or Piifture, MN O P, ftanding perpendicular on the Ground Plane, S RZ ; whofe InterfeClion with it, (MP) is at right Angles .with the line of Station, (SL.) The Pi6ture, M N O P, is, therefore, direct, between the Object and the Spectator, ES. EA, EB, ’EF, &c may be confidered as Vifual Rays^ ki which the vihon of the Objeft, B P" I L, us conveyed to the Eye, at E. 2. It is obvious, that if this Plane, or Piflure, be tranfparent, the Reprefentatlon, a if c, on that Plane, of the Objeft, B F I L, on the other fide, would, by means of the Right Lines E A, El, EF, &c. from each Angle of the Object to the Eye, exactly coincide, and agree in every part, with the original Objeft. For, as it is not polfible for Vifion to be conveyed but in RightrLines to the Eye (except through denfe,''refra£ting Mediums) the Angle A, of the Objefl, muft neceflarily appear at a, on the Picture, B will appear at b, and F at.f, &c. where the' Plight Lines, from the Angles of the Objedt to the'Eye, pafs through the Pic- ture ; and is the fuppofed reafon for the genefis of a Point on the Pidlure, in The- ory. The Right Line a b, or b g, on the Pidure, joining the reprefentatlons of the Angles, or Points, A and B, or B and G, in the Objed, will alfo coincide with, and hide the Original Tine, A B, orBG, from the Eye, -at E ; and is, therefore, its Reprefentation ; and fo of all the other Lines on the Pidure. This, I think, needs no other Demonftration, for it is ocular, and ' evident, that the Angle, made by the Pyramid of Rays forming a folid Angle (A E F, F Pi C) at E, is, not only equal, but, theiame, under whicli, both the Objed and its Reprefentation are feen. 'SoTlkewife, the reprefentation, fi or f d, onthe Pidure, of any 'Line, FI or FD, in the Objed, is feen under the fame Plane Angle, lEF or FED. Confe- quently, fi coincides with FI, and fd with..FD; the reprefentations, abgh of the Plane A B G H, f g h i of F G H I, ..and b f c of the Plane B F C, coincide with each other, refpedively ; and coniequently, the whole Reprefentation, aifc, or Projcdion of the Objed, "A I P^C, . perfedly coincides with the Original, in the Point of View' E, in that Pofition^-and Situation of the.-Pidure and Objed. MTerefore, fince the Eye' is affeded, in the>fame manner, by the Lines and Angles on the Pidure, as by the correfponding Lines and Angles of the Objed, it is evident, that, if the Pidure had the fame degree of Light and Shade, and alfo the true teint of Colour, as the Objed, it would be impoffible for the Eye, at E, to diPinguifli whether it was a Pidure, delineated on the Plane MN QP, or the real Objed, on the others fide, that was perceived. 2. Hence it is raanifed, that there is, and may be, great deception in^ Vifion ; and alfo, that there may be various Reprefentations, of the fame Objed, from. the fame Station or Point ot View; which, notwithftanding their difference in figure and dimenfions, will have the fame Appearance in the true Point of View. For, if any other Plane, as 'MNOP, be placed betw^een the-Eye and the Ob- jed, not parallel to MNOP, the Reprefentation, of the lame Objed, proieded on that Plane, by its interfedion with the Vifual Rays, will not only be Imaller, but, alfo, very different in- Figure and Proportion. P'or, having drawn the Right Lines S A, SB, SC/' from the Foot of the 5 pedator,.at S, to each angle of tlie Objed, towards the Pidure, on the Ground Plane, they w.ill determine* the extreme width of the reprefentation of that Objed on each Pidure; and alfo the proportion, of the reprelentation of the Plane A G to that of G C, which is, as k 1 to 1 m, on the Pidure MNOP, where the lines S A, SB, and SC cut the bottom edge of tJie Pidure; and, on MNOP, the Proportion is as no to op; but, MPisnotpa- jralleltoMP; confequently, the proportion, of no to op, is. not as kl to Im. 7 Tf S^. IL- i: N: a E N. E K a Ei If the Pi^lure was placed on MQ, it is obvious that the ReprefentatioH'Wouid' be ftill lefs, and alfo different in its Figure and Proportion; as the parts Q r, rq, intercepted between the lines S A, SB, and SC, fufRciently evince. Thus, may there be as many different Reprefentations, of the fame Objeil, as^ you pleafe, and from the fame Point of View ; from the different Pofition and Diflance of thePidure; all. which, will affedl the Eye. alike,. at the Point E, in the Vertex of the Optic Cone or Pyramid of Rays, PROJECTED PERSPECTIVE, or PROJECTION'. If the VifualRays EB, EC, E D, &c. arc fuppofed to be produced, or proje^led Fig. I4'«, beyond the Objed, ABC D (a quadrangular Pyramid) and there fall on, or are cut No. 4.. by a Plane (V) the Reprefentation (abdc) of the Objedj projefted on that Plane,., is the Proje(^ion of the Objedl ABC. Projeded Perfpedive is the very fame, except in the operation,., as common Per- fpedive, i, e. when the Rays are cut by a Plane pafling between the Objed and the Eye, with only this difference, that, in common Perfpedive, the Reprefentation is always lefs than the Objed; becaufe the fedion of the Rays, by the Plane X, is jq-g on this fide, towards the Eye ; in projeded Perfpedive, the Reprefentation mull: ^ neceflarily be larger than the Objed, becaufe the Plane of the Sedion (V) is be- ^ yond the Objed ; but, if the Planes are parallel between themfelves, whether the Rays are cut on this or on the other fide of the Objed, or both, the Reprefenta- tions will be perfedly fimilar. a, b, c, and d are the projedive Reprefentations of the feveral Angles of the Py- ramid, A,B,C, D ; which, joined by Right Lines, is the Projedion of that Objed- on the Plane V. F G H I is the Plan or Seat of the Objed A B C D, on.the Ground Plane, to which, the Bale, A C d D, is parallel. By means of the Seat and the' Station; Point, S, the Reprefentation, abed, on. the Plane V, may be projededi The difference between Perfpedive and Projedion is very obvious. In Perfpedive, the Objed, to be reprefented, is always fuppofed to be beyond^ the Pidure; in Projedion, the Pidure is beyond the Objed ; which is projeded,. or fuppofed to be thrown forward to the Pidure, and which is full as rational to fuppofe. Nor is there occalion, in this Cafe, to fuppofe the Pidure tranfparent, as in the former, when the Pidure is fuppofed to be on this fide of the Objed,. and feen through ; which indicates the meaning of the term Perfpedive. The difference in the operation is very little, and is illuftrated by frequent ufe in the pradical part of this Work. TRANSPROJECTION. If the Vifuaf Rays, from the Objed (ABCD) are Fig. 14.. fuppofed to pafs through the Eye (at E) forming an oppofite Pyramid of Rays No. 6., (a.E b) and there fall on, or are cut by a Plane (YJthe Figure (a cbd) projeded on that plane, by its interfedion with the Rays, is the Tranfprejedion of- the Objed (ABC D). It is evident, that, in this kind of Projedion, the Reprefentation may be either larger or fmaller, or equal to the pbjed, as the Plane, Y, is removed farther from or nearer to the point E ; or, as the point E is removed nearer to,orfurther from the Objed. For, if the point E be in the middle, between the Objed and the Plane of the Sedion, the Reprefentation will be equal to the Objed ; and, whether it be nearer to, or further from the Objed, the Reprefentation will have that Proportion, to the Original, as their Diflances from the point E. It is alfo evident, that Projedions of this kind mud necelfarily be inverted; as the Rays ail pafs through one common Point (a is the tranfprojeded place of the Angle A, on the Plane Y, c of C, and b of the Vertex, B, of the Pyramid) in the lame manner as optical Philofophers endeavour to account for V’ilion ; by fup- pofing an Image, of ‘every Objed perceived, formed in the back part of the Eye,, N on. INTRODUCTION Book II. 50 PlatelV. the Pxetina, in the fame inverted Pofition. The Eye being in this refpedl like a Camera Obfcura, which is a kind of artificial Eye; in which, the Pifture is al- ways inverted. (See Page I o„ ift and 2nd Par. Alfo, fee Fig. 13, Plate II.) Notwithftanding, if the Plane Y, of the tranfprojetlive Piilure, be parallel to either the perfpedive, (X) or the projedive, (V) the Reprefentation thereon will be limilar to the other, or to both, if they are all parallel amongft themfelves. An ORIGINAL OBJECT is any Objed whatever, which is the Subjed of the Pidure we are delineating. Fig 15.. BFIL is an original Objed, of which, the Projedion, aifc, on the Plane MN O P, or aifc, on the Plane M NOP, are Reprefentations. Alfo, parts of an entire Objed, as a Column, a Chimney, &c. are the Originals of theiiTeparate Reprefentations. By ORIGINAL PLANE is meant, not only the Ground, or other Planes upon which Objeds are leated, but alfo, the plane Surfaces of original Objeds to be delineated. ABGH and BFC. &c. in the Original Objed BFIL, are Original Planes, as well as the Ground Plane (Z) on which it ftands. It may be neceffary to make a diftindion between Figures and Objeds, although every Plane Figure may be called an Objed ; but, I think, that Term is more properly, applicable to Solids than to Plane Figures. By ORIGINAL FIGURE, I fliall therefore mean, only. Figures in Original Planes, as Doors, Windows, &c. in Original Objeds; or the Figure of the Ori- ginal Plane itfelf. As F G H I, or BGFDC, &c. Any geometrical Plane f igure whatever, in Original Objeds, is, therefore, an Original Figure. ORIGINAL LINE is any I>ine in an Original Objed, whether Right Line or curved ; as the bounds or limits of all Original Planes, or Figures, are Lines. AB, BG, F G, F D, &c. are Original Lines, in the Original Planes ABGH and BFC, each, of which, is llkewife in another Plane ; for, each is the common Interfedion of two Planes ; confequently, each Line is in two Planes. By Th. 1.7. Eh the common Interfedion of two Planes is a Right Line, 3. i i.Euc. AB is the common Interfedion of the Ground Plane and the Plane ABGH, and is, therefore, in both Planes ; and, B G is, for the fame reafon, in both the Planes ABGH and BFC; and fo of all the reft. The Interfedion of two Lines is a Point ; and the extremes of Lines are alfo Points; w'herefore, the Angles F, G, PI, &c. of the Objed BFIL, are Points ; for they are the Interfedions, and alfo the extremes of Lines; and are called Original Points, All Objeds, whatever, that are formed by Art, as Buildings of all kinds, or other regular Objeds, (which are the fiteft fubjed for Perfpedive) are compofed either of Planes or of curved Surfaces, or both. Every Building and parts of Buildings come under the denomination of fome one or other geometrical Solid ; as Parallelopiped, Prifm, Pyramid, Cylinder, Cone, or Sphere ; or they are compounded of feveral, together. For, the Body of the Building is either one Parallelopiped, or it is compofed of feveral, varioufly difpofed, at the diferetion of the Archited. The Roofs are, generally, either triangular Prifms, or Pyramids. The Planes, of which Roofs are chiefly compofed, are either Triangles, Parallelograms, or Trapezia. The Planes of the Fronts and Ends of a Building are, for the moft part. Redangles; fometimes Pentagons (as BGFDC) or other Poligons ; which contain other geometrical Figures, varioufly difpofed, as Doors, Window?, &c. 4 which Se£l. II. TO PERSPECTIVE. 51 which are generally Re£langles, fometimes Circles, Ellipfes, or mixed Figures. Temples, in Gardens, Cupolas, &c. are either cylindrical, or poligonal Prifms ; the Roofs, of which, are either pyramidal, or fpherical, or mix’d curved Surfaces. Columns approach nearly to Cylinders ; in the lower part they are perfedly fo. Thus, may every part of a Building, or other regular Obje61:, be reduced to fome geometrical Solid or other. Solids are compofed of Planes or other Surfaces, all which, may again be reduced to their firft Principle, Lines; for, as the boun- daries of Solids are Planes, or other Surfaces, fo the bounds of Planes, &c. are Lines.t Alfo, the Figures in Planes, whether Doors, Windows, or other Fi- tBcf. 55 , gures, are formed of Lines ; all which, are Original Lines. So llkewife. Mouldings, Steps, &c. are reprefented by Lines ; which, in the Originals, are generated either by Planes, only, as ftreight Steps, or by Planes and curved Surfaces, as in Mouldings and circular Steps. Each Moulding, if it be right lined, is compofed of Planes and cylindrical Surfaces ; which, by their pa- rallel interfe(51:ions, generate Right Lines. Circular Mouldings, in Cornices, &c. are compofed of Planes, with cylindrical and other curved Surfaces; which, by their regular interfedlions, generate circular Lines. The Edges of Columns and other cylindrical Obje(5ls, which are reprefented by Lines, on the Pidure, have no real exiftence in the Originals, but are only appa- rent ; for there is no real Line ; as it is but one continued Surface, which returns again into itfelf, without cuting or interfeding, by which Lines are generated. The Lines which form the reprefentations of the Bafes, &c. of Columns are, in the Originals, fome of them real and fome only apparent; as the Contour of the curve of the Torus, &c. and have various forms, on the Pidure, according to the ftuatlon of the Eye, or Pidure. Having thus reduced compound Original Objeds into their frft Principles or Elements, viz. Planes and Lines; the next thing, to be confidered, is the Pofitlon and Situation of thofe Planes and Lines, in refped of the Pidure and of each other ; which being premifed and well confidered, the whole myflery of linear Perfpedive will be found comprifed in a fmall Compafs, both in Theory and Pradice. The Principles on which the Theory is built are few, but they are ge- neral, and applicable in all pofitions of the Pidure and fituation of the Objed, or j of the Eye. Wherefore, the Diftance and Situation of the Objed being deter- I mined j that is, the Station being fixed, from which an Objed is to be delineated, jl and the Pofition of the Pidure determined, the Reprefentation is alfo determin- able ; which, Reprefentation, is in proportion to the Diftance of the Pidure. Thus far I have proceeded, by way of Introdudion. I have called it an In- trodudion to Perfpedive, becaufe, it 'cannot be called a part or branch of that ' Science; feeing, all which it contains may be, and is, known to feveral, who are not acquainted with one Theorem, or any Rules for the pradice of Perfpedive. Neverthelefs, as this Work is intended to be a perfed Tutor for young Students, I am well convinced, that, the knowledge they will acquire by this Intro- dudion is by no means to be difpenfed with ; being as elfential to be previouUy known, as the Definitions of the Terms ufed in Perfpedive; wdilch are elemen- tary. It is certainly poffible, by Rules laid down, for a Perfon to pradice Per- fpedive without knowing what Perfpedive means ; but that is not compatible vvith my Defign, having entitled this Work a Compleat Treatife, which could not pol- fibly be, without accounting for tlie Rules given ; which I fliall do, as briefly as is j confiftent with the Subjed, not dwelling on any unneceflary part of the Science; 1 but certainly, every neceflary knowledge, previous thereto (which being, perhaps, ! no part of any other diftind Science) fliould firfl; be inculcated. SECTION 52 DEFINITION S* Book 11^ PlatelV. E O N III. Containing the ELEMENTS of PERSPECTIVE. I N order to invefligate the Theory of Perfpedive, with clearnefs and precifion,. it is necefi'ary to have recourfe to certain imaginary Planes, which may be con- ceived to pafs through the Eye of a Spedator, or Point of View, from which an Objeft is fuppofed to be delineated, in. all Politions as occafions require. The Piflure, as^it has already been obferved (under the Article Perfpeftive) is fuppofed to be between the Eye and the Obje6:. Three of thofe imaginary Planes, together with the Picture and any Original. Plane whatever, are fuppofed to be conftrufled as they are reprefented in Plate IV, Fig. 1 6, 17, 18, 19, 20 and 21. The Planes, thus conftrudled, I fhall firll: de- fine ; afterwards, the Lines generated by their Interfedtions ; and laflly, the Points produced by the interfedlions of the Lines; all, which, are fo elfentially necefl'ary,, that, in fhort, without the afiiftance of thefe five fundamental Planes, and the Lines and Points generated by their Interfedlions with each other, Perfpedlive would be a very intricate and perplexed Study j and, in Theory, a moll; imperfedt Science. I would particularly advertife young Students, not to pay the leafl regard to the general politions of the Planes I am about to define, but only their politions in refpedl of each other ; for which reafon I have given them in various Pofitions, and advife the Reader not to give particular attention to the firll:. For, in the Theory of Perfpedtive, the pofition they have to the Horizon is not conlidered, at all ; as the Theorems are general, and applicable in all pofitions and fituations of tlie Pic- ture, whatever ; fince (as Dr. Brook Taylor, in the Preface to his fecona Treatife, juftly obferves) all Planes, fimply as Planes, are alike imGeometry, and have ihe fame properties however fituated. DEFINITIONS. Fig. 16. Let ABGH be conlidered as a part, or a continuation of an Original 1 1 j^^Plane ; being fuppofed to be produced (if necellary) from any Original Objedt to 20 and 2 1. Pidture ; and, till it cuts a Plane, pafling through the Eye of a Spedlator, pa- rallel to the Pidlure. D E F I N I T I O N II. The PLANE of the PICTURE is conlidered as the Board, Paper, or Canvas, | on which is to be delineated the reprefentation of fome Original Objedl, or Plane Figure. As ABLM, or the Plane X, is the Pidlure. ; The Picture is, generally, and for the moll part, vertical, and diredl between ' the Eye and the Objedl to be delineated ; as in Fig. 16, 18 and. 19 ; but may be in any Pofition whatever; as in Fig. 17, 20 and 21. DEFINITION HI. Vx'\NISHING PLANE. If a Plane be imagined to pafs through the Eye, pa- rallel to any Original Plane, it is called the Vanishing Plane of that Original Plane ; or, limply, the Parallel of the Original Plane. As V, or I K L M, parallel to Z or ABGH, Fig. 16, 17, &c. E is the Eye. And, a Perpendicular (E C) from the Eye to its Interfedlion with the Pidlure, is the Radial of the Vanilhing Plane. D E F L- Sea. III. 53 DEFINITION iV. The DIRECTING PLANE is an imaginary Plane, fuppofed to pafs through the Eye of a Speaator, parallel to the Pidure. G HI K, or the Plane Y, parallel to the Pidure (X) is the Direding Plane ; the Eye of a Spedator being fuppofed at E, in the Direding Plane. N. B. The Diftance of thcDireifing Plane, from the Plfture, is always equal to the Diftance of the Eye (as E C) and being parallel to the Pidlure, it makes equal Angles with the Original Plane, as the Pidlure. (I HA equal MAN). DEFINITION V. ^ The VERTICAL PLANE (ECDF) is fuppofed to pafs through the Eye* perpendicular to the Original Plane and to the Pidure. Wherefore, it cuts all the four preceding Planes at right Angles, in all pofitions of the Pidure whatever. DEFINITION VI. RADIAL PLANE is an imaginary Plane, palling through the Eye and any original Right Line, whatever. As EV/D, or ECPD; which, being produced, would pafs through the original Line ON, or QP. Fig. 20. In Fig. 16. the Original Plane (AG) is fuppofed horizontal, and the Pidure (AL) vertical; confequently, the Varnilhing Plane (M K) and Direding Plane (KHj have the fame Pofitions, and cut each other at right Angles. In Fig. 1 7. fuppofe the Pidure (A B L M) and the Direding Plane (H I K G) in the fame vertical Pofition ; and fuppofe the four Planes, vh. the Original Plane, the Pidure, the Vanifhing and the Direding Planes, fo fixed together, as to be moveable on their Interfedions, A B, GH, IK, and LM, as on hinges ; and then, let us fuppofe, the Pidure and Direding Plane (and along with them the Vanilhing Plane) pulhed into an inclined pofition, on either fide of the original, vertical Po- fition; the Interfedions, AB and G H, remaining as they were, unmoved. It is evident, that the parallelifm of the Planes is not deftroyed by this motion ; for, the Vanilhing Plane (IK KM) is dill horizontal as before, though re- moved lower, to iklm; and, the Direding Plane, HikG or H/iiTG, is dill parallel to the Pidure, A ml B or A MB, both being inclined to the Original Plane and its Vanilhing Plane, in equal Angles. In Fig. 18. the Pidure (X) and the Direding Plane (Y) are dill vertical, but the Original Plane (Z) and its Vanilhing Plane (V) are inclined to them, and to the Horizon; making equal Angles with each other, as in the former Cafe. Nor is there underdood to be any difference, in their Pofition, between this and the preceding Figure ; for, either Plane (A L or H K) may be fuppoled the Pidure, and the other the Direding Plane ; by which means, it has both the Po- fitions of the former ; only, fuppofing the Original Plane horizontal, the Pidure and Direding Plane are confequently inclined. In Fig. 1 9. both the Original Plane and Pidure (N B H, and A B L M) are fup- pofed vertical; therefore, the Vanilhing and Direding Planes are alfo vertical; and the vertical Plane (CDFE) in this Cafe, is confequently horizontal. In Fig. 20. the Planes are all inclined to the Horizon, and alfo to each other ; excepting the Vertical Plane; which always cuts the other four at right Angles, and is, therefore, perpendicular to them all. This Figure 1 recommend, more particularly than any of the other four, to be contemplated by the young Student (although it is the fame in all) in order to di- vell him, entirely, of partiality to any particular Pofition, refpeding the Horizon. In Fig. 21. The four primary Planes are all moveable, and may be put into all the Pofitions of the former; either at right Angles, as Fig. 16; or inclined, on either fide, as in Fig. 17* in any Angle at pleafure. O As DEFINITIONS. Book II* '54 Plate IV. Planes are all marked with the fame Charadlers, as in the five preceding Figures, it would be fuperfluous to particularize them here; and, if a Plane be fuppofed to pafs through the four Points, E, C, D, and F, it will be vertical or per- pendicular to them all, in every Pofition, N. B. Any one of the four Planes may be the Original Plane, the oppofite one is, confequently, its Vanifning Plane ; either of the other may be the Pidture, and the oppofite to it is the Direfting Plane ; E C D F is ftill the Vertical Plane, Of LINES, generated by the Interfedlions of the five elementary Planes. DEFINITION VII. Fig. 1 6, 17, 18, 1 9, 20 and 21. DEFINITION IX. PARALLEL of the EYE is the Line 1 K, in which the Vaniflilng Plane (V) and the Diredling Plane (Y) Interfedl each other. As both thefe Planes are imagined to pafs through the Eye (at E) confequently, their Interfedion (I K) pafies through the Eye ; and, becaufe the Direding Plane is parallel to the Pidure, the Parallel of the Eye is parallel to the Pidiire. INTERSECTION of the PICTURE, with an Original Plane, is the Line in which any Original Plane cuts the Pidure ; or, in which an Original Plane, being produced, would cut the Pidure. AB ; is the Interfedion of the Pidure ABLM. with the Original Plane ABGH, DEFINITION VIII. VANISHING LINE is a Line produced by the Interfedion of an imaginary Plane, pafling through the Eye parallel to any original Plane, with the Pidure. LM, the Interfedion of the Vanilhing Plane, IKLM or V, with the Pidure ABLM or X, is the Vanilhing Line of the Original Plane, ABGH or Z. DEFINITION X. DIRECTING LINE is the Line GH, in which, an Original Plane (Z) cuts, or would, if produced, cut the Direding Plane (Y). DEFINITION XL Fie. 16 VERTICAL LINE is the Line CD, in which, the Vertical Plane ’ (E C D F) cuts the Pidure ; at fedion of the Original Plane. DEFINIT ION XII. The DIRECTOR of an Original Line. If an Original Line be produced till it cuts the Direding Plane, a Right Line pafling through the Eye and that Point is the Diredor of the Original Line. E D (Fig. 20 and 21) is the Diredor of the Line N O, being produced to D, right Angles with the Vanifliing Line and Inter- DEFINITION XIII. VISUAL RAY. With optical writers, this Term fignlfies an imaginary Ray of Light ; by which, V ifion is fuppofed to be conveyed from the Objed to the Eye ; therefore, in Perfpedive, it is a Right Line imagined to be drawn from any Point, in an Objed, to the Eye. EA, El, EF, &c. (Frig. 15.) and EN, EO, &c. (20 and 21.) are Vifual Rays. D E' F I N I T I O N XIV.- . RADIAL LINE is the parallel of any Original Line, producing its Vanifliing Point. (See Def. XXII.) As EV (Fig. 20, and 21) parallel to N O. D E F I- 8ea. Ill; 55 DEFINITION XV. DIRECT RADIAL is a Right Line, from the Eye or Point of View, perpen- dicular to the Pidlure. As EC. N. B. If the Original Plane be at right Angles with the Picture (as in Fig. i 6 .) the Direct Radial (EC) is the common Interfe£tion of the Vanilhiilg Plane, (IKLM) and the Vertical Plane (E C D F) ; and is, always, in the Vertical Plane. Of POINTS, and their Diftance from the Eye. definition XVI. The POINT of SIGHT is that Point where the Eye, of a Spectator, ought to be placed to look at *a Pi£lure ; for, in that Point, only, can a perfpedtive Pidure he feen perfedlly. E is the place of the Eye, or Point of View, to look on the Picture A B L M. It is the Point where the three imaginary Planes, viz. the Vanhhing, the Di- recting, and the Vertical Planes, interfeCt; confequently, the Eye is in all the three. Or, it is the Point of InterfeCtion between the Parallel of the Eye (I K) and the Prime Director (EP') N. B. It is the Vertex of the optic Pyramid of Rays (E A, E B, E F, &c.) the only Point in which Fig* 15* the Images, or Reprefentations (a i fc,) on the Planes or Piftiires, MN OP or OP, can exhibit a true Appearance of the Original Objedl (B F I L) on the other fide. DEFINITION XVII. The CENTER of the PICTURE is the Point C, in which a'perpendicular Line from the Eye, or Point of Sight, is cut by the Picture. T'he Direft Radial (E C) being perpendicular to the Pidture, is, therefore, in the Vertical Plane Fig* ^ (E C D F) confequently, the Center of the Picture, produced by the Perpendicular EC, is in the Ver- tical Line (C D) and, when the Original Plane is perpendicular to the Pidlure, it is the InterfsCtion of the Vertical Line (CD) and its Vanilhing Line (L M). DEFINITION XVIII. DISTANCE of the PICTURE, or principal Diftance, is the Direifl Radial, or perpendicular Line (E G) from the Eye to the Pi6lure, or to its Center. N. B. The Center and Diftance of the Pidlure are moil effential, and ought to be well underftood j for, except the Interfering and Dire£ting Points, all the reft are dependant on them. DEFINITION XIX. CENTER of a VANISHING LINE is the Poit;t where it is cut by a per- pendicular Line from the Eye or Point of Sight. EC (Fig. 15 and 16) or E C (Fig. 18, 19 and 20) being a Perpendicular from the Eye (E) tothe Vanifhing Line (L M) C or C is, therefore, its Center. DEFINITION XX. DISTANCE of a VANISHING LINE, is the Perpendicular, EC or EC, from theEye to its Center ; the Ihorteft Line that can be drawn to the Vanifhing Line* DEFINITION XXL POINT of INTERSECTION is that Point in which any Original Line (be- ing produced) cuts the Pidlure ; or the Plane of the Pidfure produced, if necefl'ary. FIsT. 2D, I is the Interfedting Point of the Line N O, produced to the PI6lure ; B and F are and 21. the Interfering Points, of the Lines A B and F I, on the Pidlures M N O P. Fig. 15. DEFINITION XXII. VANISHING POINT. If a Line be drawn from the Eye, parallel to any original Right Line, the Point, where It cuts the Pirure, is the Vanifliing Point of that Original Line. (See Theo. 2. Sedt. 3. Book i.) 5 NO ^6 DEFINITIONS and AXIOMS. Book IL Plate IV. N O is an Original Line, in the Original Plane HNBG; EVisa Line from the pjp. Eye, parallel to N O, cuting the Picture ; V is, therefore, the Vanifliing Point of and 2^ * NO, andEV is its Radial, or Parallel of the Original Line ; confequently it is in the Vanifhing Plane, or parallel of that Plane the Original Line is in. Fig. 15. EC or EV being parallel to the Original Lines AB, G H, FI, &c. C or V, where it cuts the Pi£lure, is, therefore, the Vanilhing Point of thofe Lines. N. B. The Radial EC, or E V, is the Diftance oftheVanifhing Point, C or V. DEFINITION XXIII. DIRECTING POINT is that Point in which any Original Line, being pro- duced, would cut the Dire- : .'t' b • % \ % i 1 ' ,• A.- ,\ \ r ■ •fc 1 ?^ ( .if ’ <> ,' 'f-‘< » ■ • 4 > . Y ft-- • % • #: ^ 1 - ■■?i' 0 ;3. •If •. j ,1 A-* TV ■:^f1 ti '. 'Si*, ' ;«' • T RECTILINEAR PERSPECTIVE. Sc£t. IV. EX. 2. The Planes A HGB and C DKL are perpendicular to MNO P , only ; ’ their Vanirtiing Plane (STOP) therefore, paU'es through its Radial (EG) and their Vanifliing Line (0 P) through the Center (C) of that Pidlure. 6i As thefe two Examples are particular Cafes; vjz. when the Original Plane is either horizontal or vertical, I fliall give another general Cafe, which will illu- ftrate the Theorem univerfally. Raife up the Planes of No. 2. AONB is the Pidure, G E H the Diredling Plane, and E the Eye. An Original Plane, IKLM, is moveable on the Line TU, its common Sedion Fig. 15. with the Ground, or other horizontal Plane, at right Angles with A B, the Inter- No. 2. feftion of the Picture with the Plane Z. It is evident that, as the Original Plane, IKLM, is turned (on T U) it is al- ways perpendicular to the Plane or Pidure (A ON B) ^ for, in every pofition of the t 9* 7* El. Plane, in that revolution, the Line TU is Hill perpendicular to the Piduie. Now, if the Plane IKLM, revolving, pafs through the Eye (at E) and, being all the while parallel to the Original Plane, it muft revolve on E C, the Direct Radial, which is parallel toTU; and, in every Pofition, its Interfedion with the Pidure muft necefiarily pafs through C, the Center of the Pidure; pjoduced by the Perpendicular E C. Let the Original Plane (JKLM) be raifed, making the Angle P QR. If the Plane IKLM, paffing through the Eye, makes the lame Angle with the Horizon, i. e. if it be parallel to the Plane IKLM, it will cut the Pidture in O N, the Vanilhing Line of that Plane, and pafs through C, the Center of the Pidure; making the Angle, N C M, with LM, the Vanilhing Line of horizontal Planes, equal to the Angle PQR, of the Original Plane with the Ground Plane. After the fame manner, the Vanilhing Lines of all Planes, which are perpendi- cular to the Pidure, may be afcertained and drawn on the Pidure ; knowing the Angle which the Original Plane makes with any horizontal Plane ; or with a ver- tical Plane, which is perpendicular to the Pidure. N. B. The Center of the Pidure is the Center of all Vanifliing Lines, of Planes perpendicular to the * Pidure; and they have all the fame Radial and Diftance (E C) by Def. 19. and 20. ■COR. Hence it Is evident^ that the Center of the Pidiure is the V anifiing Point of -all Lines which are perpendicular to the Pidiure, For, EC, a perpendicular Line from the Eye, producing the Center of the Ei£«*5* pidure, is the common Radial of all Vanilhing Planes, that are perpendicular to the Pidure ; and confequently, parallel to all Lines that are perpendicular to the Pidure (for, they are all parallel amongft themlelvcs, and to the Dired Radial, EC) asAB, GH, FI, &c. Therefore, lince E C is parallel to all fuch Lines, the Point (C) which it pro- duces on the Pidure, is their Vanlfhing Point ; by Def. 22. But, C is the Center of the Pidure; confequently, the Center of the Pidure is the Vanilhing Point of ail Lines that are perpendicular to the Pidure. in this fingle and particular Vanilhing Point, many Artifts feem to reft all their knowledge of Perfpedive (commonly called the Point of Sight) and, whenever a number ofLines tend to the fame Point, it is, by them, called a Point of Sight; not conlidering, or perhaps knowing, that the Center of the Pidure is a Vanilhing •Point, in common with all other, only by virtue of the general Definition (Def. 22.) And, becaufe the knowledge of it is more general, and, as they imagine, the pradice much eafier, we find it more ufed than any other; as moft regular Ob- jeds, particularly Buildings, are right-angled ; fo that, having one Side or Front parallel to the Pidure, (as is ufually the cafe) all the horizontal Lines in the ad- joining Sides are, confequently, perpendicular to the Pidure, and therefore vanilh in its Center ; which, trom the certainty of the Pofition of fuch Lines, is deter- ■tnined and fixed at pleafure, though often very injudicioully. Q .‘‘E- THE- 62 THE THEORY OF Book II. Plate VI. THEOREM V. Original Planes, whofe common Interfe6tion is parallel to the Pic- ture, have parallel Interfedlions with the Picture, and parallel Vanifning Lines. DEM. The common Interfeftion of the Original Planes being parallel to the Pic- ture, a Plane may be iuppofed to pafs tlirough that Interleftion which is alfo parallel to the Picture. Now, the Original Planes being produced, through their common SeSlion, to the Picture, each Plane will cut the Picture in a Line parallel to their com- mon Section (8.7. El.) Wherefore, fince the Interfeftion of each Original Plane, with the Pidure, is parallel to the common Sedion of both Planes, tliey are, confequently, parallel between themfelves. Q. E. D. 4. 7. EL 2. But the Vauilhing Line of each Original Plane is parallel to its Interfed:ioii with the Pidure. - _ _ _ _ Theorem 3. And the Interfedions are parallel between themfelves. Proved. Therefore the Vanilhing Lines are paiallel between themfelves. Fig. 22. EX. abed and edef are two Original Planes, whofe common Sedion (cd) is pa- rallel to the Pidure (A B L M) ; g h i is a Plane, fuppofed to pafs through their common Sedion, parallel to the Pidure. Now, the Plane abed being produced to the Pidure, cuts it in LM; and the Plane c d e f, being produced, cuts it in wherefore, fince the Pidure and the Plane g h i are parallel, and are both cut by the Planes Ad c B and La b M, ' ' . the Interfedions (^AB and L M) are, each, parallel to the common Sedion, c d ; and therefore they are parallel between themfelves. EX. 2. LMis the Vanifhing Line of the Plane edef, and LMofthe Planes V, U, & W; which are, refpedively, parallel to the Interfedions A B, J F, &c. (by Theo. 4.) and therefore they are parallel, alfo, between themfelves. 3. After what has been faid, it is manifeft that the Direding Lines of Planes, in fuch Cafe, are alfo parallel. For, fince their Interfedions with the Pidure are parallel, confequently, if they were produced till they cut the Direding Plane, their Interfediions with it would likewife be parallel ; the Direding Plane being parallel to the Pidure. COR. r. The common Interfe^l ion of a. Plane inclined to the Horizon^ 'with any hori- zontal Plane whatever^ being parallel to the Pidiure ; the Vanijhing Line, of that inclined Plane, is a Line parallel to the Horizon. , For, becaufe the common Interfedion is parallel to the Pidure, the Vanifh- ing Lines of both Planes are parallel to each other j by Theorem. But, one of thofe Planes is horizontal, therefore its Vanifhing Line is parallel to the Horizon | by Def. 8. Confequently, the other Vanifhing Line is, alfo, parallel to the Horizon. 'COR. 2. All Original Planes, that have parallel Vaniflnng Lines, have the fame Vertical Plane ; and, confequently, the fame Vertical Line j and, alfo, the fame Parallel of the Eye. IKLM is the parallel of .the Plane \V, cuting the Pidure in I>M, the Vaiiiihing Line of W ; and, IKLM, is the Parallel, or Vanifhing Plane of the Sea. IV. RECTILINEAR PERSPECTIVE. ^'3 the Original Plane X, paffing through the Eye (at E) and cuting the Piaure in L M, the Vanifhing Line of that Plane. Alfo, I K 1 m being parallel to the Horizon, its Seaion with the Piaure (1 m"' is the Vanifhing Line of hori- zontal Planes; to which, LM and LM arc parallel, by the Theorem. But, RST is the Vertical Plane of all thofe -Original Planes, and of all other parallel to them; i. e. it cuts them all at right Angles, being per- pendicular to their common Sedions (c d, f g, &c.) and it cuts the Piaure in CD the Vertical Line of them all; for the lame Plane cannot cut the Pic- ture in two Lines. I K is alfo the Parallel of the Eye of all thofe Planes ; for, it is the com- mon Seaion of all their Vanilhing Planes with the Direaing Plane. This Theorem, and tlie laft Corollary, may alfo be illuflrated by Figure 15. 1 . B G, the common Interfeaion of the two Planes A B G H and B F C, is Fig. 1 5. parallel to both Piaures (M N O P or O P) their Interfeaions(/yE and with MNOP, are therefore parallel ; alfo their Vanifhing Lines, R U and OP, on ’ that Piaure. The ether (MNOP) being parallel to the Plane B P' C, has but one Vanilhing Line (OP) which is parallel to the Interfeaions B G and MN. 2, The two Planes, A B G H and B F C, being vertical, a Plane vertical to them, is, confequenrly, horizontal ; for, no other Plane can cut both, or either of them and the Piaure, perpendicularly. (See Fig. 19.) Wherefore, tliey have the fame Vertical Plane ; which, in reality, Is hori- zontal ; and, confequently, they have alfo the fame Vertical Line ; which, in this Cafe, Is the. Vanilhing Line of horizontal Planes ; viz. C X or V Y, on either Pidure, refpeaively. N. B. It is the fame in all Pofuiens, whatever, of the Original Planes. THEOREM VI. 'The Planes, which produce the Vanifhing Lines of two Original Planes, are inclined to each other in the fame Angle as the Ori- ginals ; and have their common Interfedion, paffing through the Eye, parallel to the common Interfeclion of the Original Planes. ' DEM. For, firft, fince the Planes, producing the Vainfhing Lines of any two Ori- ginal 'Planes, are, refpeaively, parallel to the Originals, they have confe- quently the fame Inclination to each other, as the Origjnals. For, If all the Planes are produced (viz. the Original Planes and their Pa- rallels) they will cut each other in parallel Lines, and form a Parallelepiped, between their Interfeaions. - - - - 8. 7. El. Therefore, they have the fame Inclination to each other, refpeaively. 13. 7 * EX. U, or W, and X, are Original Planes, and KM, KM their Vanifhing Planes ; being parallel to them, refpeaively. The Angle, which the Original Plane (X) makes with W, is dPN (P d Fig. -2. and P N being both perpendicular to their- common Sedion P Q) and it Is equal to the Angle LKL which their Vanifhing Planes (KM and YiMj make with each other Becaufe KL is parallel to NP, and KZ, to d P, the Angle LKL is equal to dPN ^ j f r. Alfo, the Angle L K L is equal to L d P, which Is equal to d P N -f. t 4* k-- Therefore, the Planes, extended through thofe Lines, being refpedively parallel, are inclined to each other in the fame Angle; i, e. KM has the fame incli- nation to 'K M, as the Plane X has to U or W. 4 Secondlv. 6 ^ Plate V. 15 - No. 2. No. 3. No. 2. Fig- 23. Fig. 15- THE THEORY OF Book II. Secondly. Becaufe, in this Cafe, the common Seclions (c d, N O, &c.) of the Original Planes are parallel to the Picture, the Interfedlion (I K) of the .Va- nhhing Planes, palTing through the Eye (E) is alfo parallel to the Piflure; and, conTequently, to the common Seftionsof the Original Planes. For it is in the Diredtlng Plane, which is parallel to the Pidlure; by Def. 4. To illuftrate this Theorem more familiarly, by moveable Planes. 1. Ralfe up the Plane of the Piilure AONB, perpendicular, or otherwlfe, to the Horizon. The DireTing Plane (G E H) will be parallel to it; and I K L M to the Plane of the Horizon. Then, if the Plane ADFBizny Original Plane) whofe Interfedlion {AB) with the Horizon, or any horizontal Plane, is parallel to the Pidture, be ralfed up, on ABt making any Angle (P Q R) with the Horizon, and, a Plane, (/i^ON) pafs through the Eye (at E) parallel to that Original Plane; it is manifeh:, that this Vaniihing Plane (//i O N) makes the fame Angle with I K L M, as AD F B with the Horizon ; and IK (the parallel of the Eye) the Interfeclion of the Vaniihing Planes, is parallel to AB^ the common In- ter fedion of the Original Planes. 2. The Pidlure, &c. remaining in the fame Polition, perpendicular to the Ground Plane, and to the common Interfe61;ion (T U) of the Original Plane IKL M; making any Angle with the Horizon, at pleafure. Then, the Vaniihing Plane (IKLM) being placed parallel to the Original Plane IK L M, it is evident, that it will make the fame Anp^le with the Horizon, or with a Plane palling through the Eye, parallel to the Horizon, and cuting the Pidlure in L M (the Vaniihing Line of horizontal Planes, as the Original Plane (^I K L M) makes with the Horizon. Alfo, EC (the Diredl Radial) the Interfe I T E O R E M IX. ^Thef Reprefentation, on ' the Picture, of a Line parallel to the Pic- ture, is parallel to the Original; and, if has that proportion to the V Original, as the Diftance of the Picture to the Diftance of a Plane, ' paffing through the Original Line, .paralkl to the Picture. IDEM. The PerfpeCtive Projection of every Right Line on the Picture, is a Right Line ; and it is produced, in Theory, by the InterfeCtion of a Plane, with t the Picture, paffing through the Eye and the Original Line. - Theo. 8. Wherefore, fince the Original Line is, in this cafe, ffippofed parallel to the ' Picture, a Plane may pafs through that Line, aifo parallel to the PiClure. But, if two Planes being parallel, are both cut. by another .Plane, their In- terfeCtions are parallel. - - - - - 8. 7. El. But the Original Line is one oPthe BeCtions, ' arid its Projection, on the Fic- ' ture, is the other (the cuting Plane being fuppofed to pafs through the Ori- ginal I.»ine and the Eye). Therefore, the Reprefentation of a Line, which is parallel to the Picture, is parallel to the Original. Q. E. D. Fig, 22. EX. N O is an Original Line, parallel to the Picture (A L MB) E is the Eye. .Imagine a Plane, paffing , through ■ the Original Line NO, parallel to the Picture ; and, E N, E O, Vifual Rays, from the Eye, to each extreme of the Original Line ; which, are in the. lame Plane with NO. - Ax. 8. ' Wherefore, N £ O is. a Radial Plane,, paffing through the Original Line and the Eye, and cuting the PiCture in no, the Reprefentation of NO. - Th. 8, But, the Plane Y is fuppofed parallel to the Picture, and N O is in that' Plane ; confequently, n o, its Reprefentation, which is the interfeClion of the Radial PLine (N E O) with the PiCture, is parallel to the Original Line (N O). For, they are the common Sections of two parallel Planes,* by another Plane. (In the fame manner, pq, the Reprefentation of P.Q, may be proved parallel to PjQ. -4 Sea. IV. THE THEORY OF PERSPECTIVE. 69 EX. 2 . AH, BG, CD, &c. are Original Lines, parallel to both Piaures (M NOP or 0 P) EB and EG are Viiual Rays, to the Line BG ; which are all in the fame Radial Plane, curing the Piaures in bg, and which are, therefore, the Reprefentations of BG, on each Piaure, refpeaively (by Theo. 8th.) Bur, BG, tlie Original Line, is perpendicular to the Horizon or Ground Plane, (S M Z) and the Piaures are vertical Planes. And, the Radial Plane GEB is alfo vertical, feeing it pafles through G B. 9. El. Gonfequently, the common feaions of thofe Planes are Right Lines, perpendi- cular to the Ground Plane. . _ Cor. to 9. 7. El. aiul, confequently, they are parallel between themfelves. - 3. 7. Therefore, b g or dg, the Reprefentatlon of B G, is parallel to B G. Alfo, cd, orcd^ is parallel to C D ; and ah, or a h, to A H. t After the fame manner, the Reprefentatlon of any other Line (B C, G D, or G F, kc.) in the Plane B F C, which is parallel to the Pi£lure MNOP, only, may be proved parallel to their refpedlive Originals. DEM. 2. The Reprefentatlon, no (of N O) is proved parallel to N O, by the ift Part. Wherctore, the Triangles NEO, nEo, are Emilar ; and n o : N O : : E n : E N, or as E o : E O. - - Cor. 3. 2. 6. El. But, ECf is perpendicular, to thePidure AB, and confequently, to the PI aue O N S, which is parallel to the Pid:ure, cuting the Pidure in C, its Center, and the Plane O N S in f. Ef is therefore the fhorteft Line that can be drawn to thofe Planes, and confequently meafures their Didances. 12. i. El. For, EC is the diitauce of the Pidure, and E f of the Plane O N S. But, if two or more Right Lines are cut by parallel Planes, they wdl be cut proportionally (whether they proceed'from one Point or not) - 10. 7. El. \yherefbre, EC : E f : : E u : E N, or, as Eo Is to E O. But, no: NO:: E n : E N. Th. no: N O : : E C : E f, by equality of Ratios. That is, the Reprefentatlon, no, has that Proportion to N O, its Original, as the Diftance of the Pidure, E C, has to E f, the Diftance of a Plane paffing through the Original Line parallel to the Pidure. N. B. The fame Dernonftration holds good if be conlidered as- the Original Line, on this Side of the Pifture, and projefted to the Pi£lu: e. (See Projedted Perfpedlive, page 49 ) COR. I. From the former party it is evident, that the Projediions of any number of Lines, m hich are parallel among f themfelves and to the Pidlure, are alfo parallel amongf themfelves and to the Originals. A H, BG, and C D, are parallel amongft themfelves, and to both Pidures; their Reprefentations, on both, are therefore parallel. Alfo, BC and GD are parallel between themfelves and to thePidure MNOP only; their Reprefentations, on that Pidure, are therefore parallel between themfelves and to the Originals. COR. 2. From the fecond part of this Pheorem, may he clearly deduced ; that if an Original Line parallel to the Picture, be any how divided, the Reprefentations of the fever al P arts will have the fame Proportion to each other, and to the whole Re- prefenialion, as the Parts of theOrtginal Line have to each other, and to the whole Line. If from the divifions, i and 2, of the Line C D, the Vifual Rays E i, E 2 be drawn, they w ill cut the Reprefentations, c d, on both Pidures, to which the Original Line is parallel, in the lame Ratio, in the Points y and z. For, becaufe c d Is parallel to CD, the Triangles CED, cEd are fimilar. And, for rhe .ame reafon, CE i, C E 2, are fimilar to cEy, cEz, &c. C. 3. 2. o. El. Wherefore, cy : C 1 : : c z : C 2 : i. e. as c d : C D : - 4.. 6. El. And, confequently, cy:yz;zd::Ci:i2:2D;or, as cd to CD. S COR. Fig. 15. Fig. 22. No. 2. Fig. 15. 70 Plate VI. Fig; 22. No. 2. Fig. IS- Fig. 22, No. 2. I. 3. El. Fig. 15- THE THEORY OF BooklU COR. 3. Hence it is manifeji, that if the Eye be moved, fill keeping the fame Dfance from the PiSlure ; i. e. let the Eye be any where in the Directing 'Plane; the whole Reprefentation, and each Segment, will fill' have ' the fame proportion to the Original Pine, For, by the fecond part of the Theorem, the reprefentation of a Line parallel to the Figure has the fame Proportion to the Original, as the Diftanceof the Pic- ture to the Diftance of a Plane, pafling through that Line, parallel to the Pic- ture. And the Diftance of either is not varied, whilft the Eye is in the Di- re£ling Plane therefore, &c. GOR. 4. The Angle, which" the Reprefentations of any two Original Lines, that are parallel to the Piliure, make with each other, is equal to the Angle made by the Original Lines. For the Rfrefentations are refpellively parallel to their Originals. In Fig. 22. No. 2. Let the Original Lines, O N and S R, be produced till they interfc£l (in P) making the .Angle OPS, Becaufe n o is parallel to N O and r s to R S, being produced, they will make the Angle ops equal to O P S, made by the Original Line. 5. 7. El. In Fig. 15. the gf d, on the Pi£lure M N OP, to which the Plane B F C is parallel, is equal to. the Angle G F D ; 2a\Afgd to F G D, &c. For, fg is parallel to F G,.fd to F D, and to G D ; by the Theorem. COR. 5. Original Figures, in Planer- which are< parallel to the Picture, have' their Reprefentations fmilar‘to the Originals. • The Plane NOS is parallel to the Pidfure, A B wherefore, the Reprefenta- tion op is parallel to the Original Line, O P'; ps is allb parallel to P S ; and if o'^j O S be drawn, os will be parallel to O S, - by Theo. Part ift. By Cor. 4. the Angle ops is equal' to OPS; andi by the fecond Part of the Theorem, op : O P : : p s ; P S ; for, each is as E C to E f. Confequently, theTriangles pos, POS are equi-angular ; the Angle o is equal O, and s equal S ; ' and confequently, o s : O S : : o p : O P, or as p s : PS. Therefore, the Reprefentation, p o Sj is fimilar to the Original T riangle, POS. For, S E O P is a Pyramid, cut by a Plane (A B) parallel to its Bale (POS), The Plane B F C, in the Objet Vertical and DireSling Planes, will have jheir Reprefentations parallel to the Vertical Line, (C Dl) Becaufe, they all have the fame Diredor (EF) to which the Vertical Line (CD) is parallel; being produced by the fed ion of the Vertical Plane (EC DP) with the Pidure and Direding Plane. - , - - - 8, 7. El. As PY, parallel to R S, cuting E F, the Prime Diredor, in f. Ev is its Radial and P v its indefinite Reprefentation. COR. 4. All Lines which cut the Parallel of the Eye rf any Original Plane, have their Reprefentations parallel to the Vanifhing Line of t/jat Plane. Becaufe, the Parallel of the Eye is the Diredor of all fuch Lines, and it is parallel to the Vanifhing Line; by Theorem 2nd. TU is an Original Line, cuting IK, the Parallel of the Eve, of the Original Plane NBH, in k ; Ex is its Radial, x is therefore its V'’aiiifhing Point, and Ux its indefinite Reprefentation; which is parallel to LM, the Vanifhing Line of the Plane NBH. For, X Def. 11 , X 82 t Thco. 2. THE THEORY OF Book II. For, I K is parallel to L M •f*, and it is the common Se6lion of the Plane, I K L M and EkUx ; therefore, U x is parallel to LM, by Theorem 5th. COR. 5. T'he Reprefentaiion of any Original Lme makes equal Angles with the In- terfedlion and Vaniping Line^ of the Plane it is in^ as the Diredtor^ of that Line, makes with the Parallel of the Eye and the Diredling Line of that Plane. Becaufe the Reprefcntation is parallel to its Director, by the Theorem; and becaufe they cut parallel Lines in parallel Planes. COR. 6. If the Reprefentations of any two Lines are parallel, the Originals are either parallel between ibsmfelves and to the Pidiure, or they have the Jame Diredior. For, if the Original Lines are parallel between themlelves and to the Picture, their Reprefentations will be parallel (Theo. 9.); but, if they are not parallel to the Picture, they muft have the lame Director; feeing, there can be but one Line drawn in the Diredling Plane, parallel to both Reprefentations. N. B. Lines, which are parallel to the Pi£lure, have no Diredling Point, but the Direftor of everv fuch Line, is a Line drawn through the Eye parallel to the Original Line. For, the Reprefentations are parallel to the Original ; by Theorem gth. This, lad: Theorem (and the Corollaries deducible from it) contains the whole Theory of the Diredling Planer as, in it and the thirteen preceding Theorems is contained the whole knowledge of redlilinear Perfpedlive ; or, at lead:, all that I conceive to be really ufeful, in delineating. It has been my aim, not, merely, to amufe or to drew my knowledge in it, but to give ufeful Indrudlion ; and, I dare venture to affirm, that if the whole of this Theory be clearly underflood, the Student will feldom be at a lofs in Pradlice. It is a midaken notion which many entertain of Perfpedlive, that, the Theory is unneceffary to a Pradlitioner. It is certainly poffible to pradlice Perfpedlive, in all common Cafes, without being able to account, or give a reafon for any rule that is followed ; for, the Rules, being deduced from the Theory, will, undoubtedly, if ftridlly followed, produce certain effedls though we know not how to account for it : as there are many Perlons very acute in Menfu ration. Gauging, Surveying, &c. who know nothing of Geometry, the foundation of the whole. The caie is very different in Perfpedlive; for I am well convinced that it is of great ufe to under- fland the Theory well, in the fird place; and that, the Pradlice will, by that means, be fooner acquired, and more fecurely retained. For want of Theory, the Pupil is frequently bewildered, and knows not what he is about ; every different Example appearing difficult and drange, though, perhaps, founded on the lame invariable Principles. In fhort, the neared and mod certain road, to Perfpedlive, is to go through the Theory to Pradlice : and, I will venture to dake all my knowledge m it, that when acquired, the lofs of Time (if it be any) will never be regreted. I fhall give three more Theorems, on Circles and fpherical Bodies, and then pro- ceed to Pradlice. If any Perfon require further knowledge, or a more extenfive Theory, I refer him to the elaborate Work of Mr. Hamilton, which is deferving of the highed encomiums, if it was as ufeful as It is ingenious and learned in the Science ; for, he has certainly faid all that can be faid of It, in Theory : and, 1 am perfuaded, more than any other Perfon would ever have thought on, and much, more than is of real ufe j for, I think I have omited nothing that can be ufeful or neceffary to be known, by any Pradlitioner, whatever. SECTION m 6 CURVILINEAR PERSPECTIVE. SECTION V. Of the THEORY of CURVILINEAR PERSPECTIVE. J N this ‘^c£lion, I fhall chiefly confider the Theory of Perfpefllve relative to cir- cular Objeds ; which are the mofl: common, and mod ufeful of all curve lined Figures. Other Curves cannot be comprifed in any certain Theory, by which their perfpedive Rcprefentations can, with certainty, be afcertained ; or if they could, it would anlwer no purpofe to an Artiff, feeing that, irregular curved Figures or Objeds but ieldom occur, in Pradice. f would not be under flood to mean the curves of the apparent Contours of human or other Figures (endowed with Life or not) which occur in almofl every Pidure, but which, can never be reduced to Rules, for Pradice, from an eflablifhed Theory , but, irregular curved Figures, in Planes or other Surfaces ; winding or ferpeiitine Rivers, Rocks, Mountains, Trees, &c. which are bounded by irregular curved Lines and Surfaces, cannot be reduced to Pradice, in delineating them, by any Theory in the Science of Perfpedive. Not- wirhflanding they may be delineated with great accuracy, by any Perfon, who is a little accullomed to flietch by fight only, by means of an Apparatus, which I fhall dcfcribc in an Appendix to this Work. To treat, at large, of the various Curves which the Reprefentation of a Circle mav take, fuch a^ the Parabola or Hyperbola, is foreign to my Defign ; as it fo rarely afl'umes thole forms Nor is the knowledge thereof of any real ufe in deli"?* neating ; feeing tliat, the frnall part of the Rcprefentations of fuch Circles as are or can be reprefented, when they do affume either, could not readily be diflinguifhe4 from a portion of an Eliipfis ; which Curve, as it is the mofl general and ufeful, fo it is the eafiell to deferibe, and the only one of real ufe, in Perfpedive. In order to a clear underflanding of the nature of an Eliipfis and its Properties, it is neceffary to be acquainted with the Conic Sedions ; fince every Reprefentation of a Circle or Sphere, in Perfpedive, is either one or other of the Sedions of a Cone. But, as a thorough invefligation of it is not neceffary, here, I fhall refer the Reader, who defires to be perfedly acquainted with the Conic Sedions, to Mr. Steel’s, or to a later Work, by Mr. Emerfon. Neverthelefs, I find it impoffible to treat the Theory of the Circle, in Perfpec- tive, without having recourfe to them, in fome degree ; therefore I fhall, in the firfl place, define what is a Cone, and, the difference between a Right Cone and a fcalene or oblique Cone ; for, without that knowledge, all that can be faid of it would be to little or no purpofe. The methods of deferibing an Eliipfis, and a’l which appertains to it, are treated fully, yet briefly, in fix Problems in an Appendix to the pradical Part (Book ifl) of my Treatife on Geometry ; together with a concife Theory, of its mofl eflential Properties ; to which I refer the Reader ; I fhall confider it, here, only as being the perfpedive Reprefentation of a Circle or Sphere. DEFINITIONS. A CONE is a geometrical Solid, whofe Bafe is a Circle, which terminates In a Point, called its Vertex ; and, a Right Line, palling through the Vertex and the center of its Bafe, is called its Axe or Axis*. * The Axis of any thing is either a real or imaginary Right Line palling through its middle, in a certain and determined Fofition. If the Objed be a Plane Figure, its Axe may be either perpendicular to, or in the Plane of the Figure. If the Axe be in the Plane of the Figure, it is divided, by the Axe, into two equal and fimilar Figures. Any Diameter of a Circle may be its Axe, an Elliplis has but two Diameters w’hich are viz. the TcaufYCrfe and its Conjugate, at Right Angles with each other. It 84 THETHEORYOF Book U. Plate VII. I*^ confidered a-s a Pyramid whofe Bafe is a Polygon of an infinite num- ber of Sides ; every Sedlion of which, by a Plane parallel to its Bafe, will, confe- quently, be a fimilar Figure ; wherefore, the fediion of a Cone, parallel to its Bafe, is a Circle. 2. A Right CONE is that which is formed, or fuppofed to be generated* by the revolution of a right angled Triangle, on one of its Legs. \ ABC is a right angled Triangle; C is the Right Angle; B C is, therefore, perpendicular to A C, its Bafe. , If you fuppofe the Triangle ABC to be revolved quite around, on BC, as an Axis, its Bafe (A C) will deferibe the Circle A E D F, which is the Bafe of the Cone ABD; and the Hypothenufe (A B) will have deferibed ' the Surface of the Cone ; which is every where various, from the Vertex (B) to the Periphery of its Bafe ; its Sides are equal every where, as BA, BE, B D, &c. The Perpendicular (B Cj wliich is the Axe of the Cone, remains at red ; one extreme in B, the Vertex, the other in C, the Center of its Bafe. 3. An Oblique CONE has, alfo, a Circle for its Bafe, but its Axe is Inclined to its Bafe, as B C to AD. As if the Vertex (B) of a Right Cone, was drag’d on one fide, out of its perpendi- cular pofition ; confequently, the real Axe, of fuch a Cone, does not pafs through tlae center of its Bafe, yet B C is called its Axe. A Right Line bifeding any Angle of a Triangle is called its Axe; wherefore, j 5 E, brfedling the Angle A B D, is the Axe of the Triangle A 15 D, and confe- quently of the Cone ; for it pafles through the middle of fuch a Solid, whofe ro- tundity is elliptical, feeing, its Dimenfions, perpendicular to its Axe, are unequal; and, the more the Axe is inclined to the Bafe, the more its Dimenfions differ (its Bafe remaining the fame) and the real Axe is removed farther from the Center (C) towards A. ’ If a Sedion of an oblique Cone be made, by a Plane, perpendicular^to its real Axe (5 E) it is confidered as an oblique Sedion of it ; which Sedion is an Ellipfis, ,2nd B E would pafs through its Center ; confequently, the Solid would revolve re- gularly on BE, its real Axe. Whereas, on B C it would revolve very irregularly and unequal; but the Cone, ABD, would be equally poifed on B C, in a hori- zontal Pofition; wherefore, every Sedlon of a Cone, through the Axe BC, bi- feds the Cone; for the Triangles, ABC, CBD, are equal, in every Sedion, through B C ; by Prop, j 8. i. El. ‘ THEOREM L The Reprefentation, or perfpedl\^ Projedion, of every Circle, in a Plane to which the Pidure is parallel, is a Circle. The fcenographic Projedion, or perfpedive Reprefentation of an Objed, is the Sedion of the Optic Cone, or Pyramid of Rays, by a Plane, paffing between the Eye and the Objed. (See Sceuography, P. 47.) Fig. 28, ADGH be a Circle, in any Original Plane (Z) E is the Eye, and E A, E B, &c. Vifual Rays, from every Point in the Circumference, to the Eye, forming an oblique Cone (A E F), DEM, Now, If the Cone of Rays be cut by a Plane (X) parallel to the Plane Z, in which is the Original Circle; adgh, the:.Sedion of -the Rays, by that Plane, is a Circle. q For, Fig. 27. No. I. No. 2. tkx i K , t CURVILINEAR PERSPECTIVE. Sea. V. For, it is the fedtion of a Cone, by a Plane, parallel to its Bafe ; and, whether it be a Right or Oblique Cone, every fuch Sedion is a Circle ; feeing that, the Cone, a E f, cut off by the Plane X, is fimilar to the larger Cone (AEF). This isotherwife demonftrable, from Theorem 9, and Corollaries ; in which it is demonftrated, th^t the Reprefentation of every Plane Figure, parallel to the Piaure, is fimilar to the Original. Every Diameter in the Original Circle (A F, D H, &c.) being equal, the Re^ prefentations (a f, d h, &c.) have that proportion to their Originals, as the Dif- tanceof the Pidure to theDiftance of the Plane of the OriginalCircle. (Theo. 9.) confequently they are alfo equal therefore, the Reprefentation, adgh, is a Circle. Q. E. D. For, all Circles are fimilar Figures. COR. Hence It is manifejl^ that the Reprefentations, in Perfpedlive, of various Circles^ in the fame Plane, parallel to the Pi 5 lure, have all the fame Ratio amongd themfelves, and to each other, as the Original Circles. 85 T H E O R E M II. . The Reprefentation of a Circle, in a Plane not parallel to the Pidfure, is an Elliphs ; except in one certain Point of View, in which, its Reprefentation is, alfo, a. Circle. . Let A D G be an.Original Circle in the Plane Z ; E is the Eye, and E A, E B, &c. pig. 2S. Vifual Rays forming (as before) an Oblique Cone. C is the Center of the Circle. DEM. If the Cone of Rays be cut by a Plane (Y) which is confidered as the Pidure, palling through both lides of the Cone, not parallel to the Bafe, it is an oblique Sedion ; and, confequently, every fuch Sedion, of a Cone, is an Elliplis ; except,' as above. - ^ Prop. 90. P.'73. Em. Con. Sec. Wherefore, the Reprefentation, adg, on the Plane Y, (which is an oblique Sedion of the Cone of Rays E A, E B, ED, &c.) is an Elliplis. Q. E. D. Every Circle, which is vilible, appears an Elliplis, except when the Axe of the Eye is perpendicular to the Plane of the Circle, and palfes through its Center ; for, in that Cafe, only, the Vifual Rays, from the Eye to every Point in the Circum- ference, are equal, and confequently they generate a Right Cone, the Axe of the Eye being the Axe of the Cone. In every other polition, whatever, they mud: ne- cefl'arily form an Oblique Cone, feeing that, the Axe of the Eye (which is always the Axe ot the Cone) mull be inclined to the Plane of the Circle, if it be not perpendi* cular; and confequently, the Vifual Rays, forming the furface of the Cone, have various Inclinations to the Plane of the Circle, and therefore they are unequal. Wherefore, lince the fedion of an Oblique Cone, perpendicular to its central Axe, is an Elliplis, and the more the Axe is inclined to the Bafe, the more excentric is the Sedion ; feeing that, if the Eye be nearly in the Plane of the Circle, the Sedion approaches nearly to a Right Line ; and,* the Sedion, made by a Plane in that poli- tion, is the only true Appearance of the Circle ; or rather, by a fpherical Surface, per- pendicular to the'Axe, which truly meafures the Optic Angle (the Radius being equal to the Dillance) under which, the folid dimenlions of the Cone, every way, are feen. COR. jf the height of the Pye, or its di/lance from the DireBing Line, be taken a mean Proportional, between the difance of the nearejl convex part of the Circum- ference and the difance of the far thef concave part, from the DireBing Line, the Reprefentation will then be a Circle. Y Let S6 Plate VIL Fig. 29. + 5. 6. El. I! 4- »* ^ ^9 I « El. § C.5. 10. 1. S Cor. 3* Theo. 9. THE THEORY OF Book IL Let A B be a Diameter of the Circle, A D G, perpendicular to the Direding Line (K L) B S is the diftance of the Circle from it. Make S E : S A : : S B ; S E ; i. e. let S B : SE ; ; SE : S A, by Pr. 30. Geo. At E, as the Point of View, the Reprefentatioii of the Circle A DG, on any Plane (X) parallel to the Plane K E L, will be a Circle, and in no ether Point whatever, on that Plane, and at the diftance B S. DEM. Draw BI parallel to SE; and, fuppofe AEB afedion, by a Plane pafling through C E, the Axe of the Cone, and the Diameter A B, of its Bafe ; and alfo through S. . Then, becaufe B I is parallel to S E, the Triangles A IB, AES arefimllar; Wherefore, - - - - AB;BI::AS:SE. But, by Conftrudion, - - AS:SE::SE;SB. Wherefore, - - - - - SE:SB:;AB:BI; and therefore, the Triangles A I B, B ES arefimilart, for the Angle AB 1 is equal to E S Bj) ; E B S is equal A I B, and I A B equal B E S; feeing that, the Sides which fubtend thofe Angles are proportional. But, the Angle B E S is equal E B I for they are alternate ; wherefore, E B I is equal I A B (of the Cone AEB) and confequently, the leffer Cone (a E b) cut off by the Plane X (parallel to B I, and the Plane K E L) is fimilar to the Cone BE A ; for, the Angle E (at the Vertex) is common to both ; the Angle E b a (equal E B 1 ) is equal E A B, confequently, E a b is equal E B A§, ~ and therefore, the Cone (A E B) is cut fub-contrary, by the Plane X. But, if an Oblique Cone be cut fub-contrary, the Sedion is a Circle, Therefore, the Reprefentation, a h b g, on the Plane X, of the Original Circle A D B F, is a Circle. Q, E. D. Prop. 89. P. 73: Em. Con. Sec, N. B. The Center (s) of the Reprefentation, is not the Reprefentation of the Center (C) of the Ori- ginal Circle; for it is at c, where the Vifual Ray E C cuts ab, the Reprefentation of the Dia- meter A B. Wherefore, C E is not the real Axe ; feeing that, it cannot be the Axe of the Cone AEB, and alfo, of the leffer Cone (aE b) which is cut fub-contrary, and is therefore fimilar to AEB; confequently, the true Axe is common to both Cones. So likewife, df, the Reprefentation of the Diameter DF, in the Original Circle, is not a Diameter of the Reprefentation ; but, gh, the Reprefentation of the Chord GH, (which bifefts ab) is its Diameter perpendicular to ab. ' Wherefore, the Reprefentation of the Segment GBH is the lower Semicircle, gbh; and, gah, the upper Semicircle, reprefents the large Segment (GAH). It is evident, that, if the Eye be raifed, as at E*, the Diameter a b, in the Re- prefentation, will be lengthened ; feeing, that the Vifual Rays, E*A, E*B, cut the Picture more oblique ; whiift the other Diameter, g h, remains of the fame length §; and confequently, the Reprefentation of the Circle, from that Point of View, will be an Ellipfis, and ab its Tranfverfe Diameter, or Axe. But, if the Eye be lowered, to E^, the Diameter ab (now ce) will be fliorter than the Diameter g h (which, being parallel to the Interfedtion of the Pidlure, will have the fame length, feen from any Point in the Diredling Plane, as above) and it will then be the Tranfverfe Axe; and ce (the Conjugate to it) is the fliorteft: Diameter of the Ellipfis. If the Eye be removed on either Side of the Point E, the Reprefentation wilt be an Ellipfis ; for, the Sedlion will not, then, be fub-contrary (the Pidlure remain- ing as before). The Reprefentation (a b) of the Diameter A B, will ftill be a Diameter of the Ellipfis, becaufe it will pafs through its Center ; but it will be neither the tranfverfe nor the conjugate Axe, for they are always at right angles with each other. Hence, the Point of View, from which the Reprefentation of a given Circle, on any Picture, however fituated, will be a Circle, is eafily determined. 5 The Sea. V. CURVILINEAR PERSPECTIVE. 87 The Original Plane (K AL) in which the Circle is fituated, being produced if neceflary, cuts the Picture in MN, its Interfedioii. Having fixed on the Diltance, or Station Point, S (at pleafure) in the Diameter AB, produced, which is perpendicular to the Interfediion of the Pidture; let K L be drawn, through S, parallel to M N ; K L is the Diredling Line* - Def. ic. Draw S E perpendicular to K L, and parallel to the Pidliure. Make SE a Mean Proportional, between S A and BS ; - - Pr. 30. Geo. E is the place of the Eye, or the Point required ; from which, a Circle, agb h, (on the Plane X) whole Center is c, and Diameter a b, will truly reprefent the Original Circle A D B F, on the Plane A K L. iV. B. The diftance of the Pifture, in this Cafe, is not material ; for, the Point E being fixed (as above) every Seftion of the Cone of Rays EA, EF, EB, 5cc. parallel to the Plane X, or to the Direft- ing Plane, KEL, being cut fub-contrary, will, confequently, be a Circle. From what has been advanced, it is evident, that the Reprefentation of every Circle, in Perfpedlive, is fome one or other of the Conic Sedtionsj which, being a diftindt Science, would not be proper to enter on here ; but, from that confideration and the preceding Theorems, the two following Corollaries may be deduced. COR. I. If the Circumference of the Original Circle touches the DireBtng Line^ in a Point only^ and is not cut by it, the Reprefent ation of that Circle will he a Parabola. For, whether the Original Circle (AD B F) touches the Diredling Line, in S, (the Station Point) or in any other Point (B) it is the fame ; fince that Point, and alfo the Eye (E) which is the Vertex of the optic Cone of Rays, are both in the Diredling Plane (Y) it is evident, that the Diredling Plane touches the Cone in the Right Line EB, from its Vertex (E) to itsBafe (at B). Wherefore, fince the Pidlure (X) is always parallel to the Diredling Plane the Cone, AEB, is cut by a Plane parallel to its Side (EB) the Curve pro- duced by every fuch Sedlion is a Parabola. Prop. 76. P. 223. Em. Con. Sec* That part of the Original Circle (MG AN) which lies beyond the Pidlure, is rcprefented perfpedlively, and falls above the Interfedlion M N (as M a N) ; the remainder of the Circle, lying between the Pidlure and the Diredling Line, is pro- jedled below the Interfedlion ; as 1 is the reprefentation of the Original Point L* Every other Point (as K) is projedled further from the Interfedlion as it lies nearer to the Diredling Line (B S) and, except the Point B, in which it touches the Diredl- ing Line, may be fuppofed to have a Reprefentation, on the Pidlure, though at an immenfe diftance ; but, the Point B can have none ; becaufe, the Line (E B) which ftiould produce it, is parallel to the Pidlure, feeing, it lies in the Diredling Plane, and therefore can never cut the Pidlure. The Reprefentation of which Point, only* is wanting to compleat the Figure and form an Ellipfis ; but, for want of that Point, in the Reprefentation, it is kept open, and falls off in Right Lines, nearly, at a Diftance, to all fenfe, infinite. Hence it is plain, that, on account of the Diftance, but a fmall portion of that part of the Curve which falls below the Interfedlion, can be reprefented, in a Pic- tuie } and the part which can, together with the part which lies beyond the Pidlure, differs fo little, in its Reprefentation, from an elliptic Curve, that, in delineating, it would be needlefs to deviate from it ; and, I queftion that ever a diftindlion was made, in Pradlice. Nor can it ever be of ufe, but in delineating the infide of a large Rotunda, Circus, or circular Area; when the Spedlator may be fuppofed to ftand on the hither part of the Circumference, or in a Line which is a Tangent to it, and parallel to the Pidlure j in which cafe, only the farther, concave part of the Curve can, properly, be reprefented. Fig. 30, t Def. 4 . COR. 88 THETHEORYO F Book 11 . Plate VII, COR. 2. If the Diredilng Line (KL) cuts the Original Circle in two Farts ; that Firr, 2 1 . (K F A H L) which is on the fame fide with the PiUure, will^ in its Re- ^ * prefientation^ form an Hyperbola^ below the Vanijhing Line (O P) of the Plane of the Circle ; the other part (K B L) of the Circle {if it be reprefented on the fame Picture ) will be tranfiprojedled, and form an oppofite Hyperbola^ above the Vanifij- ing Line, which is equal and fimilar to the other. For, if the Directing Line paffes through the Center of the Original Circle, and the Eye, which is the Vertex of the Cone, is in the Dire£ling Plane ; in which Cafe, the Se£lion, made by the Pidture, is parallel to the Axe of the Cone, and the op- pofite Hyperbolas are equal, and equally diftant from the Vanifliing Line (O P). But, when the Circle is cut unequally, .by the Diredling Line (K L) the Cone (B E A F) is alfo unequally cut by the Diredling Plane (KEL) the Sedlion of which, with the Cone, is the Triangle KEL; thePidture (N OPM) being parallel to the Direfling Plane, cuts the larger Segment of the Circle, in MN, and the Cone, in the Curve f g aN, which is an Hyperbola. Prop. 104. P. 160. Em. G. Sec. .If the Sides (A E, F E, BE, &c.) of the Cone BE A, are produced through the Vertex (E) forming an oppofite Cone (aEbd) the Piflure being produced, will alfo cut that Cone, produced ; its Seflion, with it, is the oppofite Hyperbola, or tranfprojefled Reprefentation of that part of the Circumference of the Circle, (K BL) which lies on the other Side of the Direfling Line, by means of the Rays B E b, QEq, &c. produced to the Piflure ; which, notwithftanding it is the Re- prefentation of the lelfer Segment of the Circle, is generated by a fimilar Gone,, (a E b) and, although the Seflion is made at a greater diftance from their common Vertex, it is equal and fimilar to the perfpeflive and projeflive Reprefentation (fMga N d) of the larger Segment. This Curve, like the Parabola, can never be generated, or of ufe, but when the Area of the Circle is fo large, that the Speflator is fuppofed to ftand within it ; and that Segment (M G A H N) which lies beyond the Piflure, only, is required ; and which differs fo little from an elliptic Curve, in its Reprefentation, (MgahN) that the diflinflion is not very obvious, and feldom, if ever, regarded. Thefe are all the variety of Curves which a Circle in Perfpeflive can afiTume, however fituated in refpefl of the Piflure, or of the Eye; the chief of which is the Ellipfis. Every Circle, which the Eye is capable of taking, in at one View, it is manifefl, has the Appearance of an Ellipfis ; except, when the Axe of the Eye is perpendicular to the Plane of the Circle, and pafiTes through its Center; in which po- fition, only, it can appear, to the Eye, a true Circle ; although all Reprefentations of 4 Theo. I. Circles, in Planes parallel to the Pidure, are Circles'!'; but, being feen oblique, when the Eye is in the true Point of View, they have the Appearance of Ellipfes. THEOREM III. The Reprefentation, in Perfpefliv’e, of a Globe, or Sphere, is an Elliplis*; except when the Center of the Sphere coincides with the Center of the Piflure; in which Cafe, only, it is a Circle. The Cone of Rays from the Eye, as its Vertex, to the apparent Circumference of a Sphere, is always a Right Cone. For, a Sphere can have but one Pofition, either to a Plane, Line or Point ; and confequently, the Diameter, every way, always prefents itfelf to the Eye, and always forms a Right Cone. * I have found it very difficult, nay almoft impoffible, to convince fome Perfons of this plain and well known truth ; becaufe, as they truly obferve, the Diameters of a Sphere always appear equal ; which is as much as to fay, that the Vifual Rays, under which a Globe is fuppofed to be feen, always form a Right Cone, i. e. whofe Axe is perpendicular to its Bafe ; which is not fo with Circles, but in one po- fition only. But, fuch Perfons feem to forget, that the Reprefentation is on a Plane, and that Plane is confidered as the Plane of the Seilion ; which muft, confequently, cut every Cone oblique, but that which has its Axe perpendicular to the Picture. Now, Sea. V. CURVILINEAR PERSPECTIVE. 89 DEM. It Is evident, that, when a Sphere Is fo fituated. In refpea of the Piaure and the Eye, that, the Direft Radial coincides with the Axe of the Cone, (I. e. when it pafles through the Center of the Sphere) the Seaion of the Cone, made by the Piaure, is parallel to every apparent Diameter of the Sphere, which is the Bafe of the Cone ; and, being equal every way. It is confequently a Circle. But, the DIrea Radial is the Axe of the Cone, which pafles through the Center of its Bafe, and alfo through the Center of the Piaure. Confequently, the Center of the Piaure coincides with the Center of the Sphere, in this Seaion ; and therefore, the Reprefentation is a Circle. Theo. ifl. For it is the Seaion of a Cone, parallel to its Bafe. 2. But, if a Sphere be fo fituated that the Direa Radial does not pafs through its Center, the Axe of the Cone mufl: be inclined to the Piaure ; which, being the Plane of the Seaion, the Cone of Rays are cut obliquely by the Piaure. But the oblique Seaion of a Cone, through both its Sides, is an Elliphs. Therefore, the Reprefentation of a Sphere, w'hofe Center is not the Center of the Piaure, is an Elliplis. Q. E. D. For it is an oblique Seaion of a Cone. EX. Let A B be a Sphere and C its Center (fuppofed to be feen). Let E be fup- pofed the Eye of a Speaator, and EA, E B, Vifual Rays, from the Eye, to the apparent Diameter, AB*. AEB may, therefore, be fuppofed a Sec- tion of a Right Cone, through its Axe, EC, which is an llofceles Triangle ; the Sides, EA, EB, &c. of a Right Cone being equal Now, if FG be the Piaure (which is a Plane) parallel to A H B I (the Bafe of the Cone) the Line ab, in which the Plane AEB cuts that Plane, is a Diameter of that Seaion ; which Seaion, being every way equal, is, confe- xjuently, a Circle. But, E C, the Axe of the Cone, being perpendicular to its Bafe, Is perpen- dicular to the Piaure §, and to its Seaion, P'G, with the Triangle AEB; the Point, c, where the Piaure is cut by the Axe of the Cone, EC, is the Reprefentation of C, the Center of the Sphere. And, becaufe A B is a Diameter of the Bafe of the Cone, and C its Center, A B Is bifeaed in C ; cotflequently, a b is alfo bifeaed In c ; for, the Triangles CEA, C E B are congruous II ; and, becaufe ab is parallel to A B, the Tri- angles a E c, AEG, and c E b, CEB, are all fimilar But, E is fuppoled the Eye of a Speaator, and, E c, being perpendicular to FG (the Piaure) is the Direa Radial f ; and the Point c, where it cuts the Piaure, is the Center of the Piaure §, which is alfo the Center of the Repre- lentation of the Sphere, on that Piaure ; therefore it is a Circle. Theo. i. Now, if the Cone A EB be fuppofed to be cut by any other Plane, palling through the Diameter h i, and confequently through c, the Center of tlte Piaure and of the Reprefentation ; that Seaion wdll be an Elliplis. Let S D be a Seaion of another Plane or Piaure, with the Triangle AEB, cuting the Rays E A, EB, in e and d ; then is e d, in that Seaion, the repre- fentation of the apparent Diameter AB, of the Sphere, and c is the reprefen- tation of its Center, as before, in the Seaion ab; but, ed is not bifeaed in c. DEM. Let ace be an Ifofceles Triangle; ac=ec; and, ac=cb;th. ec—cb: Ax. 3. El. But, becaufe bcE, in the Triangle bEc, is a Right Angle, cbE is acute-j-, and, the Angle cbd, in the Triangle deb, is, confequently, obtufe. C. 2. i. i. Wherefore, c d, fubtending the obtufe Angle, is greater than cb; 12.1. El. And confequently, ec + cd, equal e d, is greater than ac-feb, equal a b. But, the Seaion LI El, which is vertical to A E B, cuts both Piaures in the fame Line, hi, equal to ab; as above. - * See the Note to Art. 4. Page 14, on Diredt Vilion, Z There Pr. go Eio. Fig. 32j t Def. 2, § 6. 7. El. II 7. I. El. X 2. 6. El. ^ Def. 15. § Def. 17. + C. 3cl io. j. El. T HE THEORY OF, &c. 90 Book II. plate VIL Therefore,,.. the Sedioii, ehdi, is an Ellipfis ; for, the Diameter e d is larger than any other, in that Section. but, E S, perpendicular to S D, is the Di reft Radial ; wherefore, S is' the 11 Dcf. 17. Center of that Pidurelj; and it does -not coincide with the Center of the Sphere (the Eye being at E) confequently, the Reprefentation of a Sphere cannot be a Circle, except when the Center of the Pidure coincides with the Center of the Sphere, in the Center of its Reprefentation. Fig. 33. EX. 2. To illuftrate this further. Let X, Y, and Z be three Globes, whofe Centers are all in the fame Right Line, parallel to ah, the Interfedion of the Pidure ; whofe Center is C, and EC its Diffance. Draw the Tangents E A, E B, ED, 6 ic. to the three Globes, the Chords, A'B, D F, and GH, of thofe Tangents, are the apparent Diameters of each; ' which, it may be obferved, are ftill turned towards the Eye ; and are con- fidered as the Diameters of the Bafes of the three Cones, AE B, DEF, and'- ; G E H, which are all Right Cones. Now, fince they are all cut by the fame Plane (of which a h is a Sedion) each Cone, except the firfl, A E B (whofe Axe, EX, coincides with the Dired • Radial, EC) is cut oblique; and, confequently, one Diameter, of each Sedion, is larger, as the Globe is farther from the perpendicular EG ,• as d f, and g h,* the Reprefentations of a Diameter of each Globe (Y, and Z) df being larger than a b, and g h ftill larger than d f. § Cor. 3.’ But, the other Diameters, perpendicular to thefe, are equal, in all§, for they ^ j h. 9. are fuppoied parallel to the Pidure, and equally diflant from it; confequently, . the Reprefentation of the Globe X, only, which is in the middle of the Pidure, or, the Center, C, of its Reprefentation, in the Center of thePidure^ is a Circle ; becaufe, the Diameters perpendicular to each other are equal :-* all others (as.of .Y and Z) are Eilipfes, becaufe they are oblique Sedions of~ Cones, whofe Diameters arc proved to be unequal. N. B. It is the fame however the Globes are fituated ; whether above, below, or-fidewaysof the Center of the Picture. If they are equally removed from the Perpendicular EC, and equally d if! ant from i the Pi< 3 ure, the Globes being equal, their Reprefentations are equal and pertedtly fimilar, though differently fituated ; and the farther they, arei remote, from. the Perpendicular, the more excentric is.the Ellipfis. It is obvious and demonftrable, that' a b, th'e reprefentation of the Diameter A B, is bifeded in C, the Center of the Pidure, and the reprefentation of the Center of the Globe ; for, the Triairgle a E b is Ifofceles. But, df, and g h are not bifeded; im i and k, the reprefentations of the Centers of the Globes Y and Z. For, becaufe the Right Lines EY and EZ (from the Eye to the Centers of the Globes) are perpendicular to the Chords, DF and GH, the Angles DEF, . GEH are bifeded by thofe Lines ; becaufe the Chords are bifeded. C. y. i . El. But, in the Triangles dEf and gEh, becaufe the Angles, at E, are biledcd, by the Right Lines E i, Ek; df and gh are. cut, by thofe Lines, in the Ratio of the other fides of the Triangles; - - 3' 6. EL. and confequently, di;if::Ed:Ef; and gk:kh: :Eg:Eh.- ■||ri2.i.El. But E g is lafs than Eht> becaufe, the Angle ghE is acute, and Egh is obtufe ; w'herefore, g k is lefs than k h ; and alio, d i than i f. Therefore, d f and g h are not bileded, by the Lines E Y and E Z. And confequently, fince they reprefent Diameters of the Spheres, Y and Z, their. Reprefentations are not Circles ; confequently they are Eilipfes. . SE CTI ONf A^ REEUT A'-TI QN. O.F- ERRORS/.&c, S E C T L G H VL Containing a full ' refutation of feveral Errors and 'abfurd Opinions, , which many Artifts entertain of Perfpedive; and, . therefore, look, on it as an imperfect and fallacious Science. I Shall, in this Se£lion, in the firfl: place, explain the realbn why the Reprefenta- tions of the Diameters of Columns, on a Pidfure which is parallel, or nearly - fo, to the Columns, are continually larger the farther they are removed from- the Center of the Pidure, and conlequently from .the Eye. As this is a particular circumftance, which many Perfons feem inclined todifpute,.. or, if it be admited, they look on it as an imperfedion in Perfpedive, I lhali en- deavour, and doubt not, to -make it appear conlbnant to reafon and Perfpedive, . to their entire fatisfadion. It has fo near affinity to what has been faid, in re- fpedof a Sphere, that the fame Diagram might have done for both ; but, in order -• to avoid miftakes, and to keep the Ideas diftind and feparate, I have given another. Let V, X, Y, and Z be theSedicMis of four Columns, by a horizontal Plane, in . which is the Eye, at E. Let L M be a Sedion of the Pidure, parallel to the Co- '34'^* lumns, and El, E K, EA, EB, &c. Vifual Rays, from the Eye to the apparent Diameters-of the Columns, which are ftill turned towards the Eye, as. Globes, It is evident, that the Vifual Rays cut the Pidure more oblique, the farther the Colu-mns are from the Perpendicular E C andy notwithhanding the optic; Angles DE F, GE.H are lefs, as the Columns are continued, their reprefentative Diameters, d f, and g h, intercepted between the Vifual Rays, continually increafe ; . and would, if the Columns were continued, till the Interval between them was -loft ;; the Reprefentations of their Diameters flill increaling till they touch and cut .each- other, For, the Space from Center to Center, of the Columns, are equal, and are, . confequently, reprelented fo on a parallel Pidure, ,df continued inhiiirely ; what,, then,, can become of the Space between the Columns, if it be not added to^their Diameters, in the Reprefentations on the Pidure .h I prefume, no Pcrfon will fay that there is any imperfedion in Perfpedive, in , th is Cafe ; I do affirm there is none in P-erfpedive ; the bulinefs, of which, is to re- prefent Objeds, truly, on a Plane; according to their Magnitudes, Diflances, and> Situations in refped of each other, oLthe Eye,.and of.the. Piclurej where, then,, is the imperfedion in this ? If.a Perfon, not knowing how to choofe a proper Difcance, take, into the Pidure,; more than the Eye is capable of taking in at one View; or if, through ignorance, the Pidure be ablurdly fituated, . in relped of the Objed, is the fault in Perfpedive,. or in his Judgment ? In Perfpedive there is not, nor can be, on the Principles here laid, down, any, the- lead error, if the Elements of Euclid. are to be dependejj^ on, upon- which the whole Fabric is ereded; if one falls, the other fiills with it. The Art of Perfpedive is to reprefent Objeds on a Plane, by geometrical Rulesi- according to their Situations, &c. (See Perfpedive, Page 47 .) It is-; well known (or ought to be) that no Perfpedive Reprefentation can ap- pear perfed, i. e. it cannot truly reprefent the Original Objed, though ever fa^ accurately delineated, but when the Eye is in, the true Point of View. Suppofe, then, E to be that Point; and A E B, D EF, ,&c. the Optic Angles, under which the Columns X, Y, Z (being equal) are feen. It is alfo, 1 prefume, allowed, that Objeds appear to have the fame proportion to each other, refpedively, as the Angles under which they are feen. Tli. i . S. 3 . D. Vifion, But,. A llEFUTATION OF ERRORS Book II. 92 Plate VIE But, the Angle DEF is lefs than the Angle AEB; becaufe their Subtenfes are equal, AB=:D F (ruppofing the full Diameter of a Column tobefeen) and the Vi- fual Rays EB, ED, EF, &c. the Sides ot thofe Angles, flill longer, the farther -tC.14. 1 El. they are from the Perpendicular, E C-f. How comes it then, that the Diameter of that Column (on the Pifture) is the largeft, which is feen under the leaft Angle? I'he realon is obvious ; becaufc, the Pidure, L M, cuts thofe Rays moft oblique, where the Angle is the leaRj in the Points g and h, &c. DEI\L If EF and EH are produced till they cut a Right Line drawn through the Centers of the Columns (which is parallel to the Pi 61 :ure) in F and iL ; E D and EG cuts that Line in D and G ; all which are beyond the Circumferences of the Circles; wherefore, DF and GH Are, each larger than a Diameter; and, if it was not fufficiently obvious, it would be ealy to prove that G H is larger than DF. But, A B is lefs than a Diameter ; confequently, ab is lefs than d f, and df than gh; for ah is parallel to AH. - - - 4. 6. El. Now, if the Eye (at E) be turned towards the Column V ; or, if the fituation of the Piclure be changed to NM, then is S, where ES cuts N M, the Center of thatPidure; on which, it is evident, that, ik, the reprefentative Diameter of the Column V, being neareft the Center, is the leaft ; which, on the PidureLM is equal to that of Y ; for, they are equally diftant from the Center of that Pidure ; and on the Pidure NM, the reprefentation of the Diameter of the Column Z, is contiderably larger than g h on the other. Yet, to the Eye, at E, both thefe Pidures truly reprelent the Diameters of the four Columns V, X, Y, andZ; V and Y appear equal on both, they being equally diftant from the Eye; X, the neareft, will appear the largeft, and Z, the fartheft, from the Eye, will appear the leaft. DEM. With any radius, as ES, on E as a Center, defcribe an Ark of a Circle, cuting all the Vifual Rays El, EK, &c. from the four Columns. The parts, ab, df, intercepted between the Rays, are the true pro- portions of the apparent Diameters of the Columns, and confequently, of ;their Reprefentations, on both Pidures. But the Ark ab is the greateft, ik is equal to df, and g his the leaft. Wherefore, the Angle aEb is greater than dEf (equal lEk) and gEh is the leaft; and confequently, the apparent Magnitudes of the Columns, X, Y, and Z, are in the fame Ratio. _ _ - Theo. i. Dired Vifion, It is unneceflary to enforce this, by dwelling longer on it ; as it is certain, if either of thefe Pidures be viewed from any other Station, they could not reprefent the four Columns V, X, Y, andZ, in the Pofition and Situation they are in. Suppofe the Eye removed to (the Point of View ought always to be oppoftte to the middle of the Pidure) and, from that Station, to view the Pidure NM. Draw the Vifual Rays E^’^r, W b, &c. and on E^ defcribe an Ark of a Circle, cuting them, in i, 2, 3, &c. Theflighteft glance of the difproportion of their Ap- pearance, from that Station, is fufficient convidion, that ab (which appears the largeft from the true Point of View) whofe apparent Magnitude is the Ark 3 4, does not appear half fo large as df', and^h appears larger than df (which ought to appear the leaft) as the Arks 34, 56, and 7 8, fufficiently evince. That the true, apparent magnitudes oif the Columns V, X, Y, and Z, from the Point of View, E, are the portions ab, df, &c. of the Ark idh, is manifeft, when we confider, that the Pidure, on which the Columns are reprefented, is a Plane ; but that, their true Appearances can only be reprefented on a fpherical Surface ; i. e. on the Surface of a Sphere, the Reprefentation and the Appearance are the fame, the Eye being in its Center. Does any one imagine that there is a real Arch in the Heavens, which has that Appearance ? in which, the Stars, &c. appear, equally diftant from the Eye in its Center, The Celeftial Globe, for inftanec, is a Pidure of the Heavens, Planets, 4 Stars, 93 Se£l. VI. AND ABSURD OPINIONS. i Stars, &c. cacli Rcprefentation of a Star, on its Surface, if they are truly de.* i pidlured,- would, to an Eye in the Center' of the Sphere, exactly coincide, and be j in the fame Right Line with its Original in the Heavens j and, their apparent Dif- tances, from each other, are meafured by an Ark of the Sphere ; whereas, their real Diftances from the Eye and from each other, refpedively, have not the leaft nity to their Diftances, as reprefented on the Surface of the Globe or from its Center. Wherefore, to an Eye at E, the true Point of View for either Picture [ (LM or MN) each Diameter, being feen under its true Angle, and the fame as its Original, will appear lefs and lefs, the farther they are diftant from the Eye, or from the Center of the Pidure ; although their Reprefentations are continually larger and larger, on thofe Pidures, as the original Columns recede. And this will ever be the cafe, on a Pidure parallel to the Columns, in fome degree, at any diflance of the Picture; but, at a proper Diftance, for taking in the whole, the difference Is fo little, and that ftill lefs as the Diftance is increafed, that, it is and ought to be defpenfed with, by making them equal : but, I muli obierve, it is not, then, true Perfpedive. 2. Mcthinks I hear fome carping Critic fay, I mull: allow, then, that Perfpedive Is fomewhat defective; by no means ; I have not yet given up the Point in debate. I fay, that, although, at any Diftance, there muft and wdll be a difference, though fcarce perceptible at a proper Diftance, yet I would never advife a Perfon, who would reprefent a row of Columns, in full Front, to make the leaft difference ia their Diameters; for, fince they fupport an Entablature, which is reprefented per- fectly horizontal, their Pedeftals orBafes the fame, confequently parallel, it would be very improper to make the Columns differ in width when they are equal in height, for this reafon ; becaufe it is impoflible to confine the Eye to the true Point of View, always; from which if you deviate, ever fo little, the whole Reprefen ta- tion is diftorted and imperfeCt. But, if it were poffible to confine the Eye, I would not flep the leaft afide from Perlpedive, on any account ; let the Rcprefentation be ever fo diftorted and prepof- terous, it will, and muft, if truly reprefented (by the Rules hereafter prefcribed) appear to the Eye, in the Point of View, as the Original. If we may be allowed to take liberties, in any Cafe, where (hall we draw the line between the perfeCf and imperfect Rcprefentation ? for the whole is more or lefs diftorted ; confequently, the Rules of PerfpeCtive are not to be depended on, at all. To what, then, muft we have recourle ? the Eye is not, in many Cafes, a competent judge ; we fhould, if we follow^ its ‘dictates. Implicitly, have as many Points of View, in a Picture, as Objects ; becaufe, if every ObjeCt be reprefented exaCtly as it appears to the Eye, on the fame Plane or Picture, there can be no point of View for the whole; and confequently, in a long, connected, and continued ObjeCt, compofed of Planes and Right Lines (as in Buildings of any kind, of regular Architefture) it would be beyond the power of Art to conneCt the feveral detached pieces or projeCtures fo, together, as to compofe one entire and uniform Picture of the whole. Notwithftanding an ingenious Author has treated it ludicroufly, in a fuppofed Dialogue between a Lady and an Artift, who was determined to abide by the Rules of PerfpeCtive ; his Argument has not the leaft weight, and muft be imputed to him, as not having a right notion of PerfpeCtive. The Eye is confidered as a Point ; -therefore, whether we fuppofeit confined to a Pin-hole or not ’tis the fame; for he muft allow (or he was very unfit to write on PerfpeCtive) that there can be but one Point of View for a Picture, in which it can be perfectly feen, as intended by the Artift ; confequently, when the Eye is not in that Point, the ObjeCts muft necefla- rily appear more or lefs diftorted, according as they are fituated nearer to or farther from the Center of the Pidure, or that Point which is oppoftte to the Eye, 3. I fliall, however, for the fake of the Argument, allow each ObjeCt to be truly repre- fented, as they appear to the Eye, on a Plane PiClure of a tolerable length, as ah: Globes are the fitteft SubjeCl to expatiate on, becaufe they are every w'ay the fame. A a Let Fig. 33. Plate VII. Let'C be the Center of the Pidure, tlie Eye being at E ; and therefore, the Pic- ture may be fuppo^ed to be extended equally towards a as to h. ' Now it is certain, that the three Globes appear round, let the Eye be fituated where it may; but they cannot, or ought not to be foreprefented,on the Plane ah, to be viewed by the Eye at E; for if they were, they would not appear rouwd, but gibbous, or Egg like, landing ereft, all but the Globe X, in the Center ; and the more fo, as they are farther removed from it ; the Eye muft be removed oppolite to each, and confe- quently, there would be as many Points of View as there are Objeds, which is an abfurd Hypothefis, in one Pidure. Suppofc, from the Point of View E, I would reprefent the three Globes, X, Y, and Z, as they appear ; that is, the outline of each to be a Circle ; the Eye, and its Axe, EC, muft be turned towards Y and Z, as EY, EZ; and confequently, the Pidure is turned with it, into the Pofirion bl, and Im. For, the Pidure muft be perpendicular to the Axe of the Eye, if the Objed be reprefented as it appears; in which Cafe, there are three diftind Pidures, viz. a b, b 1, and 1 m; each having a ■diftind Center, and the fame Diftance, EC or Ec; on which Pidures, each Globe is reprefented by a Circle, and the fartheft from the Eye (Z) is the leaft in its Re- prefentation, as they really appear. It muft be obvious, if the Globes were reprefented fo on one Plane (of which ahisfuppofed a Sedion, and C its Center) that they would be under a worle predicament than thofe reprefented ftridly perfpedively : fuch a Pidure could not appear true in any Point of View, whatever, every Objed having a diftind and fe- parate one. Suppofe them reprefented on the Plane ah, as they appear to the Eye, at E, i. e. round ; the Repreientation of the Globe Y leis than that of X, and, of Z, lefs than Y, as they are reprefented on bl and Im; would they appear in the proportion they are reprefented, atE, or in any other Point ? No, certainly; for, at E, the Reprefentations would appear lefs than the Original Objeds, and not round (except X, only) Y would appear more round than Z, and Z would alfo appear rounder than any other, more remote from C; but they would, all, except X, appear eliptical Splieroids, and not Globes (which is obvious, to any Pei foil tolerably acquainted with Optics or Dired Vifion) the fartheft, from X, dill more fo than the laft. Now, let the Eye be removed to E, or oppofite to Z ; Y, or Z, would then ap- pear round, but not of the fame proportion, as from the Point of ViewE; but, at E, neither X nor Z would appear round ; and, although equally diftant from the Eye, X would appear much the largeft ; where, then, in this Gafe, muft the Eye be placed, to fee thofe Reprelentations fuch as the real Objeds appear? There cannot be a Point determined, for each Reprefentation has a feparate Point of View. Can, then, the Pidure a h, in this Cale, be a true perfpedive Reprefentation of the three Globes, X, Y, and Z, as they appear to the Eye? certainly no, but each Reprefentation, on the Plane ah, is as much a diftind and feparate Pidure, as rhe three Pidures, ab, bl, and Im; the difference is only, that the three Pic- tures have but one Point of View, and the Pidure ah has three, equally diftant from it; by fuppoftng the three Pidures placed in a Right Line, ah. 4. If what I have here advanced be not fufficient, to dlveft thofe Artifts of their ab- furd notions of Perfpedive, I fhall give them one obfervation more, which they have not, perhaps, confidered with that attention it requires; and then leave them to purfue their own way, if it appears to them more eligible and reafonable, and will produce a better and more agreeable Reprefentation, of any Objed whatever. They quarrel and find fault with Perfpedive, but without reafon ; becaufe, it is an infallible and moft perfed Science. They would have all Objeds reprefented, in Perfpedive, exadly as they appear to the Eye; there is no fuch thing to be done ; ’tis notin the power either of Art or Science to reprefent, on a Plane, any ftngle Objed, except a Sphere, or a plane Circle, having the Eye oppofite to its Center, and rhe Pidure parallel to it, as it appears; and yet, Perfpedive will give a true ; nd juft Reprefentation of every regular Objed. The Sea. VI. AND ABSURD OPINIONS. 95 The real, and the only caule of all their errors, and falfe notions of Perfpeaive, is their not rightly dihinguifhing between the Reprefentation of an Objea, on a Plane, and the true Appearance of it ; two diflina things, which can never be united, on a Plane Surface or Pidure. The perfpeclive Reprefentation of any Objed is the fedion of the Cone or Pyramid of Rays, by a Plane ; and the Appearance of an Obied is the fedion of the Rays by the furface of a Sphere, only j to which, every Vilual Ray, from the Eye to the Objed, muft be Perpendicular; confcquentiy, the portions of the Arks, intercepted between the Vifual Rays, meafure the Optic Angles, under which, every part of the Reprefentation is leen (the Ark a, or dh^ Fig. 3:? and 34, may befuppofed a fedion of a Spiiere, cuting the Vilual Rays E A, E B, &c). It is therefore manifeft, that the true Reprefentation, and, at the fame time, the true Appearance, can only be reprefented on a fpherical Surface, the Eye ‘being in its Center; but that is not a perfpedive Reprefentation. 5. I exped it will, again, by fbme Perfons, be alledged, here, that Perfpedive, then, is not fufficient, to reprefent Objeds as they appear to the Eye. 1 affirm that it is. Let us, therefore, once more enquire, candidly, what is meant by Per- fpedive, and what efteds it is expeded to produce. Perfpedive is a reprefentation, on a Plane, of an Objed, or Objeds, in a fixed and determined Pofition and Point of View. (See Perfpedive, Page 4p.) I have already fhewn the bad efied of viewing a perfpedive Pidure, out of rhe true Point of View ; from which, if w'e deviate, we cannot exped that the leveral parts of a Pidure can vary their Bearings and Proportions, to each other, as the real Objeds ; no, certainly, that muft be a reality, not a Perfpedive ; which is but a Deception, a Reprefentation of a real Objed, on a Plane ; and, which, can never re- prefent the Objed, truly, from any other Point of View, but that for which it was delineated. . I have alfo fhewn, to ocular convldion (by the Apparatus) that there may be us many various Reprefentations of the fame Objed, from the fame Station, or Point of View, as there can be Pofitipiis of the Pidure ; all which, are true Per- fpedive, and wilfeaffed the Eye alike, in the true Point of Vievv ; which, is the Vertex of the Pyramid of Rays; feeing that, every correrponding Line, on each Pidure, is feen under the fame plane Angle as the Original, and every Surface, as well as the whole Objed, is feen under the fame folid Angle, or Pyramid of Rays, as the correlponding original Surfaces, or as the original Objed. ‘ . What, then, is it we are caviling about ? would ye have a real, folid Objed on a Plane, or a Reprefentation of it only, in a certain Pofition ? Certainly then, if the Eye be not in the true Point of View, the Pidure does nor truly exhibit an ap- pearance of the intended Objed, at the fixed Station; and, although it mav be a jufi:, perfpedive Reprefentation, it ma}-, neverthelels, be a very dilforted and dii- agreeable Pidure ; not owing to any fault, or imperfedion in Perfpedive, but to the choice of the Situation of the Pidure, or the D. fiance and Pofition of the Objed and the Pidure. Does not almoft every Objed, except a Sphere, appear different from every different Point of View? and can any Perlon be fo unreafonable as to exped, that a Repre- fentation of a folid Objed, on a Plane Surface, can appear truly to reprefent the Original in any otlier Point of View, hut in the Vertex of the Pyramid of V''ifual Rays, under which the Objed itfelf is llippofed to be feen ? the very fuppofirion is abfurd to the lafi degree ; becaufe, no two parts of the Reprefentation can, at the fame time, be leen under the true Optic Angle, in any other Point ; confe- quently, the Reprefentation, mufi appear erroneous. 6. Now, although the true Reprefentation, and alfo the true Appearance are de- pidured on a fpherical Surface, yet, I affirm that fuch a Pidure is fubjed to much greater imperfedion than a true Perfpedive, on a Plane Surface; becaufe, if the Eye be the leafi removed out of tiie Center, the whole Appearance and Effed is 3 deftroyed, Book n. j6 AREFUTATIONOFERRORS Plate VII. deftroyed, and exhibits a much worfe Image of the Obje6l, than a perfpe(fllve Re- prefentation, on a plane Pidure, can poflibly exhibit, in any Point of View; which is fo very obvious, that it is needlefs to point out the reafon. For, fuppofe the Eye, at Ey viewing the fphericalPidure, a^y6; the VifualRays.E a, E dy &c. fhew, atfiift fight, how much worle fuch a Pidure muft appear, than that on the Plane, a h, from the fame, though falfe Point of View ; not one of them will appear round, and the Appearance, of all, is prepollierous. From the circumftance 1 have mentioned, in refped of the true Reprefentatlon and Appearance being depidured, at once, on a fpherical Surface, fome Artifts ima- gine, that the Reprelentation on a Plane ought to be fo delineated ; it cannot be ; ’tis impoflible, in the nature of things. Suppofe a true Reprefentation of a long Building, In full front, delineated on a fpherical Surface, and it were poffible, after\yards, to reduce the fpherical Surface to a Plane ; is any Perfon fo weak as to fuppofe, that fuch a Reprefentation would appear like the Original, in any Point of View? he mull be weak, indeed, and have ftrange miftaken notions of Perfpec- tive, who can ; and yet 1 have heard this Point ftrenuoufly fupported, or rather argued for, fupported it could not be ; for, any thinking Perfon (who can think with propriety about it) mull be fenfible, that, whatlhould reprefent Right Lines will be curved, and, the whole, will give the Idea of a Rotunda, or externally round Building; feeing that, the extremes would fall off, not in Right Lines but curved, and they would appear lefs than the real Objedl ; to fay nothing of the almoll im- poflibility of producing or delineating luch a Pidure, at ail, or by any means; I Ihould be glad to be informed, how, or by what Rules. 7. There may be alledged another circumftance, which, I think, mull convince the moll obdinate, in Perfpedtive. I am perfuaded, that, if the Eye be fixed in a Point, at a proper Dldance from a tranfparent Plane, placed between the Eye and an Obje( 5 l, whilll the Hand traced, accurately, every Line of the Objefl, as it appeared on the tranfparent Plane ; fuch a delineation, all mull allow, would be a true one. Let thofe, who are not otherwife to be convinced, try the Experiment. I will dake all my knowledge In Perfpeftive, that every Reprefentation of a Right Line is a Right Line, on the Plane ; that Columns or Cylinders of equal magnitude, and parallel to the Plane, will be larger as they are more remote from that Point, on the Plane, to which the Eye is oppolite ; that the Reprefentation of a Ciicle or Sphere, feen oblique, is an Elliplis; that Obje< 5 ls of equal magnitude, and equally diflant from the Pi£lure parallel to them, however otherwife lituated or elevated will be reprefented equal ; with various other circumllances; all which may be fully proved to ocular convi£lIon, which will not admit of the lead doubt. Surely then, if by the rules of Perfpedllve the fame thing be elFedted, which cer- tainly will, in every refpeft, it mud exhibit true Reprefentations of Objefls. It has been obferved, that the bufinefs of Perfpe£live Is to produce the Figure of a fe£lion of the Cone or Pyramid of Rays, by a Plane, in any determined Po- lition ; which, if the Rules it preferibes be truly followed, it will mod certainly efFe< 5 l, without any fenfible error. For, wherever any Point, or Angle of an Obje8. I fhall now take notice of another great difficulty, which feems to be a Rum- bling Block to many Artifls ; who, one would imagine, would not hefitate one moment, to determine about it with propriety ; which is, to reprefent, on a ver- tical Piaure, the appearance of a direa Defcent ; which, Mr. Kirby has affirmed impoffible, in the nature of things, to be done ; that it is a ftrong inftance of the in- sufficiency of Perlpeaiye, and that, we muft have rccourfe to Experience, only, in luch Cafes ; intimating, that it is not poffible, by the rules of Perfpeaive, to give the Reprefentation of a defcending Plane ; which is fo ridiculous an affertion, that, any Perfon, who underftands Geometry tolerably, will eafily be -convinced of the contrary. For, firft, we are to confider whether the Difcent (which I ffiall fuppofe a Plane) is perceivable or not. If this defcending Plane can be feen, at all, from any fixed Station, it may, undoubtedly, be reprefcnted on the Piaure, from that Station, ' by the ftria Rules of Perfpeaive, or there is no truth in Perfpeaive ; either it is a perfea and infallible Rule, or it is no Rule at all. If the Plane can be feen, it is a fubjea of Perfpeaive; if it cannot be feen, it is no fubjea for a Piaure; which needs no Demonftration. To tell us, what defcends, and we aaually know to go .down-hill in Nature, will, if ever fo correaiy drawn, appear to rife, upwards, on the Piaure, is fiying nothing to the purpofe, the expreffion is vague and nuga- tory ; for, if the Plane defcended fo much as not to appear to rife on the Piaure, it could have no place or reprefentation thereon; but if it can be feen at all. It muft, necelfarily and unavoidably, appear to rife ; or rather, it muft, really, rife on the Piaure, for, the Appearance is to defcend. And yet, he fays we muft have recourfe to Experience. What kind of Expe- rience can teach us to do, what is impoffible (as he fays) in the nature of things, to be done? Experience will teach us all that can be done; Experience, in Per- fpeftive, will teach us to delineate regular Objeas which we never faw, yea, which never exifted ; but no Experience whatever, can make us perform impoffibilities ; to reprefent Objeas which are loft to fight In the Originals, from the determined Station. But, how can any Perfon be fo unreafonable, as to require that to be done, in a Piaure, which is fo liable to falfe conftruaions, or determinations, in Nature ; for, a level, or horizontal Plane will appear either to afcend or defcend, by the affinity and concurence of Lines and Objeas conneaed with them ; as for Example. When we are going down a regular defcent, towards a level Plane, of Land or Water, the greater the defcent, the more the level before us will appear to rife ; yea, fo ftrong is the Deception, that we are not eafily convinced of the con- trary ; except it be Water, before us, which, we are certain, cannot afcend. Likewife, when we are going up a gentle afcent, towards a level Plane, which is in fight, it will appear to defcend ; more efpecially when we fee the boundary Lines of both, as In regular Walks, in a Garden, &c. Nor is there any thing furprifing in it, If we confider, that, in the former Cafe, it is owing to the much greater breadth which we fee, of the level Plane, and having the fame appearance as an Afcent, before us, when we ftand or move on a level. In the other Cafe, the con- trary effed:, in appearance, is the caufe of the Deception. To draw two parallel and horizontal Lines acrofs the Piclure, and to give an Idea, that the fpace, between them, reprefents a defcending Plane (of a certain length) without Ihape, bounds or limits, fideways, or any Objed fituated on the inclination, to bias the judgment, is Indeed impoffible ; but that is giving too great latitude to the meaning of the Expreffion. 1 fay, it is abfolutely impoffible, in this Cafe, to give, by Art, an Idea of a defcending Plane, fimply as fuch. For, although a proper gradation of Light and Colour may make a fmall fpace, on the Pidure, .appear to reprefent a great length, yet, ’tis not poffible to fay whether it defcends, B b or r rl O >J O F £ n R O xR s Book n. ;Fiate VII or reprefents a level Surface; -bccaufe, the fame fpace, on the Picture, may repre- '■fent an equal length, on a declining Plane, to a Spectator Handing, as is requifite to repreient a level Ground, to a Perfon laid along, or firing ; confequently, it is not pofiible to fay, with abfolute certainty, whether it reprefents one or the other. And yet, I queftion if afkilful and ingenious Painter, in aerinl Perfpedtive (who had critically obferved and copied Nature) might not, by the effedl of Colour, fimply, even in this Cafe, deceive the Eye, and give the appearance of a Defcent. But if there are Objeds fituated on the inclined Plane, or, if the fhape or figure of t..e Plane., itfelf, is to be deferibed ; whatever can be feen of fuch Objeds (whether Tops or Bottoms it matters not) they may and can be reprefented, truly and exadly, by the infallible Rules of Perlpecfive ; and that, on the fame inva- u'iable Principles, as the moft common and ordinary Cafes, whatever, are fubjed to. To illufbrate what 1 have advanced, by a f.mple geometrical Scheme, will not be vei-y difficult ; which may be confidered as a vertical Sedion through the whole. Each Plane is, tlierefore, reprefented by a Right Line, making the true Angle of ■dhe Inclination of the Planes (as I fhall call them) with each other. •Fig. 35. Eet A F reprefent the Plane of the Horizon, and A B a Defcent, as a dope Bank, ,or any other Declivity ; making the Angle ABG with the Horizon. Let CD be ,a Sedion of the Pidure, E the place of the Eye, and EF its Altitude above the Horizon, at the Diftance EC from the Pidure, whofe Center is at C. Now, if the Eye (at E) be fo fituated, that a continuation of the inclined Plane would pafs through the Eye (as B AE) it is evident that the Plane (A B) from that Station, is totally loft to fight; its whole Reprefentation being the Line of its In- ■?- Cor. 2. terfedion with the Pidure-f. Confequently, if the Eye be moved further back (asat Theo. 2, £) its appearance, or place on the Pidure, would defeend ; which cannot be, unlefs .the Plane be confidered limply, a Plane, without lubftance, and feen on the under fide ; its apparent width from that Station is a D; but if the Eye be removed, nearer to the Pidure, in any diredion above B AE) the Reprefentation of the Plane rifeson the Pidure. At E% its width on the Pidure is ab, as the Rays, E^ A, E’^B, evince. If the Eye be moved back again, in the diredion BE% to E^, it is plain that, b, the apparent height of B, remains the fame, wherever the Eye is fituated in that Line; but the reprelentative width, a b, of A B, will be increafed, towards D, as it re- cedes, if the Pidure be at any diftance from the Defcent. Suppofe the Eye brought forward again to ; the width of A B from that Station is ab\ and, if the Plane was .continued infinitely, aV is its whole apparent breadth, E'‘V being parallel to AB. p. I prefume, this is intelligible and clear, hitherto. I will, next, fhew thatftis im- poftible for the Eye to judge, merely from the width of its Reprefentation on the; Pidure, whether the Plane be defeending or afeending, or perfedly on a level. From the fituation of the Eye at E% the reprefen tative width, on the Pidure, ,of the Slope, AB, \% ab \ whereas, if the Plane was more inclined to the Horizon, i. e. if the Defcent was more gradual, as A I, it would reprefent a lefs fpace ; if it was horizontal, the fame ab reprefents the length AH, only; but, if the Plane afeended from the Pidure, as A K, it ftill reprelents a fhorter length (AK) on that Plane (the Arks li, H h, &c. ftiew how much each is fhorter than the other) and AL, making right Angles, BLA, AL Eh with the Vifual Ray E'*B is the fhorteft length, from the Point A, that can be reprefented, by ab, from that Station, ■fttuatlon, and pofition of the Pidure. Now, if any Objed (as X) be fituated on the Inclination A'B; It is evident. If a continuation of the Plane of its Top (MN) paffes above or through the Eye, at 'E% it cannot be feen from that Point; whereas, if the fame, or any equal and fimilar Objed be fituated on the Horizontal Plane A H, the top, OP, maybe feen from that Station, not with (landing the Objed is more elevated ; becaufe, a continuation of the Plane of the Top, OPQ., falls below the Eye, at E% The figure of the inclined Plane, itfelf, or the figure of any Objed fituated upon it, is deferibed, perfpedivcly, in the fame manner, and on the fame Princi- ples as on a horizontal Plane ; which is exemplified in the pradical part of this .Work.' ' ScA. \ AND ABSURD OPINIONS. 99 10. Ill the laft place, I (hall demoiillrate, that the reprefentatlons of Ohje.6ts, wlilch are elevated perpendicularly, above the Horizon, have the fame proportion^ on a vertical Piiflure, as thole of the fame Magnitude, fituated on or near the Horizon ; the Objedf being parallel to the Pidlure. It is a mrftaken notion which feveral Perfons entertain, that the parts of a Building, which are elevated high above the Horizon, appear to diminidi in a greater ratio than thofe which are extended horizontally ; luch an opinion may be ealily refuted. If the fecond Part of the ^rh Theorem be well confidered. It is fuf- ficient refutation. L-et A G be fuppofed a high Obelilk, and A B, B D, &c. feveral equal Divihons thereon. Let E S be a Spedfator, E the Eye, andag, or. <7^, aSedlionofthe Pidlure ; which, being vertical, is parallel to the Objedl A G. Now, if the Vifual Rays E A, EB, &c. be drawn, they will cut the PIdlure 411 a, b, d, &c. then is f g, the reprefentation of F G, equal to a b, the re- ,prefentation of AB, or to bd, the reprefentation of BDf. p'or, fince the parts AB, BD, &c. of the Original Object, AG, are equal, the re- prefentations of them, on the Picture, being parallel to the Objedt, are alfo equal. But, AB is an Objedt, diredt before the Eye, at E, on the Horizon AS; and, F G is one of equal length, elevated greatly above it ; therefore, the Repre- fentations of equal Objedts at an equal Diftance from the Picture, and parallel to the Pidture, are equal. Q. E. D. The fame Objedt may be confidered as a Building, extended horizontally, and AEG a horizontal Plane, in which is the Eye, and alfo the Vifual Rays EA, EE;&c. ag, or agt is a Sedtion of the Pidture, as before. The interfedtions of the Vi- fual Rays with the Pidture, it is evident, are the fame. 1 1. There is yet another point of controverfy I have fome times been entertained with; which is, that, when we ftand, oppofite the middle, near a long Building, or range of Buildings in a Right Line, the horizontal Lines, in the Cornice, &c. appear to .decline towards either End, yet make no Angle; and therefore, they •imagine that the reprefentation of thofe Lines, in fuch Cale, will be curved. What a poor Idea muft aPerfon have of Perfpedlive, who advances and is really prepof- fefled of fuch an Opinion. 1 , (hall fay very little on this Point, becau-fe, the>eighth Theorem is full and perfedt ‘Demonftration, that th^ Reprefentation (on thePifture) of every Right Line, is a ■Right Line; and, it is fo if extended infinitely. Becaufe, a Plane may be fuppofed 'to pals through any Right Line and the Eye ; and, the Interfedlion of this imaginary Plane, with the Pidture, is the indefinite Reprefentation of every Line it paffes .through; for, the Eye being in a continuation of it, the whole Plane is loft to light t* Therefore, the Reprefentation of every Right Line is a Right Line. The truth, of which, any Perfon may foon be convinced of, by applying a per- •fedlly ftreight Ruler, before his Eye, parallel, or otherwife, to a Right Line, in an •Objedt, of any length, and imagine the edge of the Ruler to cut the Pidture, which is a Plane ; then certainly, if the Ruler coincides with the Original Line, lYom one end to the other‘(which it undoubtedly will) the Right Line, in which it is fuppofed to cut the Pidture, is the Reprefentation of the Original Line. It is the liime as if a tranfparent Plane was placed between the Eye and the Objedt, and any Right Line, in the Objedt, traced on it, exadtly, whilft the Eye is fixed in a Point j the •Line, fo deferibed, will be a Right Line. Thofe Perfons never confider (but ’tis plain they are not furnlfhed with the -means) that the Pidture, being placed in the true Point of View (confequently at its .proper Diftance) will appear the fame as the Original; for the ratio of the Parts is ;always in proportion to the Diftance. Confequently, every part of the Pidture, Teing feen under the fame Angle as the Original, will have the fame Appearance, fin every -refpedt ; and confequently, the Right Lines, on it, will appear to decline relther way, or both ways, the fame as in the Objedt, • .What I ivould fignify by the proportion of Obje^s, in this place, is, fimply, length and breadth. I Relpedting Fig. 36, f 4. 6. El. + Art. 3. of a Plane- lOO A REFUTATION OF ERRORS Book II. 1 2. Refpei 5 ling the appearance of parallel Right Lines direft before the Eye, there is | fomething paradoxical, which is not eafily reconciled to reafon ; for it is manifeft, ^ from the foregoing, and the eighth Theorem, that every Right Line appears a Right ' Line, when the Eye is diredled to it ; and yet it is certain, that parallel Right Lines, ^ however htuated, appear to converge, and confequently to approach each other. ; Suppofe two, parallel, horizontal Lines, direft before the Eye. It is manifefl-, - that their Reprefentations, on a Pidlure parallel to them, are alfo parallel Right ) Lines. Notwithftanding which, it is certain, that the neared: Diftance fubtends a greater Angle, at the Eye, than any other ; and that, the Angle fubtended, by their perpendicular Diftance, is continually lefs, the farther they are extended on either ; fide. How, then, can they appear Right Lines? feeing that, they appear wider ' afunder, direft before the Eye, than towards their extremes ; neverthelefs, fince a * Plane may pafs through each Line and the Eye, they muft: neceflarily appear Right Lines, and confequently parallel ; feeing they fubtend equal Angles, at equal Diftances from thofe parts which are neareft: to the Eye. This very extraordinary Phicnominon may be better conceived, by imagining the r Eye in the Center of a concave Sphere; and, imagine two great Circles, which are 1 Meridians, or longitudinal Circles, making an Angle of 15 or 20 Degrees, or more, ij Now, the Eye being in the common Center of both Circles, each appears a Right Ji Line, which way foever the Eye be turned, towards the Circumference. But, the I Meridians interfedf and crofs each other, in the Poles ; which, being diametrically , * oppofite, cannot both be feen at the fame time ; but, in looking towards either, ■ the Circumferences of the Circles will appear like two Right Lines, converging to L a Point ; the Pole, in which the Circumferences interfe£l:. Now, fuppofe the Eye diredfed to the Equator of thole Meridians ; and imagine two equal Chord Lines drawn, one in each Circle, at equal Diftances from the Eye, and parallel between themfelves. Thofe Chord Lines, it is manifeft, will appear to coincide with the Circumferences of the Circles, the Eye being in their .■ common Center ; and would, if they were continued beyond the Circles, infinitely. But, it is certain, that the Circumferences, being meridian Circles, crofs each other twice; wherefore, the parallel Lines, one in each Plane of the Circles, will appear i to converge to a Point towards each extreme, which are their Vanilhing Points; and, which, muft be infinite, before they can appear to coincide with the Poles of the Circles. Thus, it is evident, that Right Lines do always appear Right Lines, , and are reprefented by Right Lines ; notwithftanding which, parallel Right Lines appear, when diredf before the Eye, to converge to a Point, towards each extreme, I have now, I hope, fully refuted thofe truly ridiculous and abfurd Opinions, of Perlpefftive, which many have imbibed, and are not eafily divefted of; they, rather, j obftinately perfift in them, without being able to give a folid reafon for their Opi- nions, and are determined not to give up their Prejudices, at any rate, right or wrong. ' What ftrange infatuation muft pofl'efs that Perfon, who, having no argument of ! weight, to fupport his falfe notions of Things, has recourfe only to Sophiftry ; arid, <\ becaufehe cannot come at fterling Truth, himfelf, imagines there is no fuch thing to be found. In points of natural Philofophy, &c. where no certain Criterion can j be obtained, to fix our aflent, it is no wonder that we meet with fo many, widely different. Opinions; and, though few of the arguments advanced have the leaft foun- dation in Reafon, ‘‘tis amazing with what eagernels and warmth each Aflailant at- tacks his Opponent. But, in Sciences, purely mathematical, all muft agree ; when ; Truth appears, there is no refiftance can be made; we cannot with-hold our affent | ■ fo prevalent is her influence. To luch as are open to Convidion, and are defirous of coming at Truth, I think . I have laid enough for their convidlion, on the Points debated ; but if they are not, all that can be faid is to no purpofe, ’tis wafteof Time and Words. I lhall, there- ^ fore, leave them to enjoy their Opinions, and pleafe themfelves with their great |jj Sagacity; and proceed to lay down certain and infallible Rules for the Pradtice of« Perlpeclive, deduced from a perfect and well founded Theory ; which, if truly followed, and proper attention be given to the Leffbns, contained in the Introduc- tion, will moft certainly produce Harmony and true Effedl, of any regular Objed, fo far as comes within the province of linear Perfpedive. 5 IIL B O O K * (* Of: the Praftice of Perspective., S E C T I O N 1. An INTRODUCTORY PREFACE. I Come, novvT, to the pra6tical, and, in that, the ufeful part of Perfpeftive; to which, the foregoing Book is an Introdu6lion, only, but a very neceffary Introduction ; infomuch, that, without the knowledge inculcated by it, we Ihould proceed in ignorance and uncertainty. Neverthelefs, thofe Perlons who have not ftudied'' Geometry, and have not, now, perhaps, either leifure or inclination to fludy it (though, in my Treatifeof Geometry, they will find it neither fo abftrufe, tedious, or dry a Study, as many look on it to be, having treated that moft ufeful branch of • Science in a more familiar and intelligible manner, than has been done heretofore) r lay, that, without a fufficient fund of Geometry, to perceive and be clear in the Demonfirations, if they will but treafure up, in memory, the Theorems and Co- rollaries, and take all for granted (as they may depend on the Truths contained iii- them) they will find the great advantage ofiit, in PraClice. Every branch of Science is in two parts, theoretic and pradical. Theory teaches the knowledge of all that is • neceflary for Pradlice, in Speculation ; the* other a:'plies that Knowledge to real ufe. It is neceflary, firff, to know how,, be-^ fore we can perfo'-m any thing ; but notwithftanding, many Perfons may be faid to know PeripeClive (as a mathematical Science) yet know not- how to apply it to Practice, with fuccels. So, every Art, dependant on Science, may be^ acquired-^ wirhour tiie Tiieorv of it^ by cuftom or habit ; as in mechanic Trades, derived from Geometty and Mechanics ; all which^ require time and application, to -be- coQie fiuinliar to us. So likewife, PertpeClive may be practiced, without being laid, properly, that we underftand it ; feeing, we may not be able to give a fuffi- cient or latisfaCtory realon for the effects ofht, in any-inftance. And, being well verfed in the Theory, we lhall neverthelefs find, that it will require time and afli- duity to apply it, in all Cafes that may occur, in Practice, although founded on I the lame invariable Principles, without familiar Leflons, in every Cale, being given. I But we lhall, certainly, with the affi fiance of Theory, be able to comprehend, and to underltand the various methods of applying the Rules, better than without it ; as very little reflection, when we are at a lofs, will fet us right again, and eiw force the univerfality of its Principles. It is not, to be wondered at, that the heft of Treatifes on Perfpective has been the leafi underftood; viz. that, by Dr. Brook Taylor; becaufe it is very obfeure. I call It the beft, on account of his general Principles, not from its real utility, to Artifts, in refpeCt of practicing from its Rules. It can fcarce be faid, with propriety, that PerfpeClive exlfted before ; it certainly was not properly underftood, as a Science. How Infipid and imperfeCt are all the Books on the Subject before his, , except Gravefande’s and Ditton’s ; both which feem to touch on the fame Prin-> ciples, but are far fhort of a perfeCt, general Syftem. There are, in that fmall Treatifcj Rules fufficient, inalmoft all Cafes, for Plane- Objects; but, various Examples requiring various ways of applying them, he has. iV)t made itfo ufeful as it would have been, had he, inftead of refering to former.- Broblems, fhewii how to apply it there. As, in Example 3 rd, Page 24, Book the • K firfl, Fig..2j. Ws are told to make CB reprefeiit a Line, equal to. that which is. C q repre-. 102 AN INTRODUCTORY PREFACE. Book in. reprefented by GA (by Prop. 15 .) Now, 1 queftlon that one Perfon in fifty (who nnderftands the whole) ever law the leaft Affinity between the two Examples; the different fitiiatlons of the Lines makes it a very different Operation, though built on the lame Propofition of Euclid. Refpecling plane Obje0:s, only, Perfpedlive is foon acquired ; knowing how to delineate Figures in horizontal Planes, it is the fame in any other ; having found their Vanifhing Lines, with their Centers and Diftances ; which I have firft fhewn, in Pra 6 tice (as in Tlieory) how to find, ‘in all ufeful Cafes whatever. But, without the embellilhments of Mouldings and other architeilural Ornaments, in Build- ings, &c. all the reft would be to little purpofe. ThatTreatife comprehends only Perfpedlive in Plano ; but 1 have, in this Book, fhewn how to apply it in every ne- celfarv Cafe that can be deviled; and doubt not, to render it, by that means, the moft ufeful work, of the kind, yet publifhed in any Language, that we know of. Brook Taylor has, indeed, to his immortal Fame (in Perfpeflive) furnifhed us with new and extenfive Principles; but his Work, at beft, is imperfeift, and greatly deficient. His Theory is too concife, and is not regularly digefted. It may perhaps, by fome, be objedted, that I have made too much of the Theory. I could have faid much more, but know not where to curtail it; the Examples, given for llluftration rather than Demonftration, I am perfuaded, will not be found unnecefl'ary, to fome, or trifling to any. One Example, in each Cafe, to fome Perfons, v./ould be fufficient; toothers there can fcarce be too many, fo they are various, and not a repetition of the former. I (hall be guilty of the fame fault (if it be a fault) in Pradice ; I had rather fay too much than not fay enough, yet I would not be tedious ; becaufe, all that can be faid, to fome, will be too little, or rather too much, feeing it will be all to no purpofe. To fleer the middle courfe is a difficulty not eafily obviated ; but, it is my fixed defign to aim at it ; others muft determine how I have fucceeded. It is the opinion of many Artifts, that the whole of ufeful Perfpedlive may be comprifed in a little Compafs; that nothing is a greater difcouragement to the ftudy of it, than to fee a voluminous Work on the Subje(fl. ’Tis certain, that the Principles, on which the Theory of it is founded, are contained in a very fmall Compafs ; and, I would recommend Brook Taylor’s Epitome for that very reafon, which contains /Vz Yet, notwithftanding that valuable Treatife has been fo long publiflied, it is, at this time, but little known, and lefs underftcod ; which is a fufficient reafon, with me, tofuppofe that he has not faid enough on the Subject. The elaborate Vv^ork of Mr. Hamilton is fpun out to an immoderate length ; yet to as little ufe as the other. ’Tis my Defign, to comprife the whole of ufeful Perfpedtive in this Book. Neither of thefe Authors, I am perfuaded, had either taught or pradtifed the Art of delineating ; and confequently, they were not qualified for treating it in an eafy and familiar manner ; one great requifite in a Work of this kind. I have had experience in both, and am well convinced, that, to make it ufeful, it cannot be comprifed in fo little compafs as many imagine ; that it requires frequent repetitions of the fame Leflbn, fomewhat diverfified, to fami- liarize the Rules, in various Cafes; without which, not one in twenty will ever be a Proficient. Dr. Taylor truly fays, it is much better for the Student to devife Examples, himfelf, in particular Cafes, than to go through thofe of others ; but how tew are capable of doing fo ? nay, I find, many are not able to comprehend them at any rate, nor by any means; and therefore, to make fuch a Work really ufeful, variety of Cafes and Examples muft be devifed, for Praftice and Experience. In Practice, our Author has given fome Problems, containing the moft elegant and general Rules that can be, notwithftanding they are but feldom pradliced ; be- caufe he has not fhewn, properly, how to apply them : that fhall be my care to do, where they can be applied ufefully. His Diagrams are, in general, very imperfedt, and badly devifed ; ’tis evident he was no pradlitioner or delineator, himfelf, even i-ii Plane Objects (for he has given us no other). But, he has departed from his 3 own Sea. I. AN INTRODUCTORY PREFACE. 'OwnPi-inciples, in Example V. Fig. 19. Part II. having projeaed the Dodecahe- dron, in Perfpeaive, by means of the khnography and Orthography (as by the old Authors) inftcad of Vanilhing Lines and Vaninilng Points ; which is much more mafleily, elegant, and perfea ; and is what the difference chiefly confifts in. Such Subjedls are indeed of little ufe, except to familiarize us to find Vanifhing Lines and Points in all pofitions of Planes and Lines, to the Pidure, and fituations •of the Objed, or Pidure. Some Perfons arefo bigoted to the old Authors, that they cannot be reconciled to the new Terms, by Brook Taylor; nor indeed to his new Principles, till they find their Excellence by experience. It is not to be wondered at. ’Tis not eafy to divell: any Perfon of old habits and methods of pradice, though ever fo abfurd ; becaufe it is impoffible that they can fee the difference, at firft, and confequently cannot judge of it; but it is furprizing that they are not to be prevailed on to try ; and if they do, it is with feeming reludance, and with a fixed refolution to prefer, and perfevere in their old Prejudices. I am as much againft capricious innovations in Science as any Perfon ; but, if there be an appearance of any acquifition to it, we ought, candidly and unprejudiced, to make a fair trial of their merits; without which w-e cannot judge of their Excellence. In refped of the new Terms given us by Dr. Brook Taylor, fuch as Interfedion of the Pidure, Center of the Pidure, Vanifhing Lines and Points, &c. (together with Direding Plane and Line, which are moft effential, in Theory, though but of little ufe, lu Pradice) I am of opinion that no better Terms could poflibly be devlfed ; nor any other lb expreffive of what is meant by them. How narrow, how limited is the Bafe Line and Horizontal Line (the only Vanifhing Line known to the old writers on Perfpedive) when compared with them! What difference is there, either in Theory or Pradice, between the horizontal Vanifhing Line and any other, of Planes perpendicular to the Pidure? None at all, feeing, they have the fame Center and Difiance (Th. 4.); nor indeed in any other, except in finding them ; the Pradice, in all, is the fame, in every refped. It was impoffible for him to make the Principles of Perfpedive general, but by general Terms; which does not regard any Pofition either of. the Pidure or of the Original Plane, fince all Planes (Amply as Planes) are the fame. The Interfedion of the Pidure includes every Interfedion whatever, as well as the Bale Line, and they are all of the fame ufe. Vanifhing Line is not only general, but is, at the fame time, fo Ample and expreffive, that it conveys its utility at once to the Mind. In the fecond Theorem of the Arfl Book, it is proved, that parallel Riglit Lines, ho'vvever Atuated, appear to approach towards each other ; and, confeauently, if they are produced, inAnitely, they will appear to meet, and vanifh, in a Point at an inAnite Diflance. So likewife, parallel Planes appear to meet each other, and to vanifli in a Right Line, (fuppofed to be inAnite) or, properly, in a Point. Now, if this Theorem be well conAdered and underftood (together with the fore- going) it will be found to be the foundation of the new Principles of Perfpedive. For if a Plane be fuppofed to pafs through the Eye, parallel to any Plane, whatever, or any number of parallel Planes, and being produced, or continued till they are loft: to Aght, they will ail appear to unite, and to meet the Plane paffing through the Ey^, at an inAnite Diflance. But, if the Eye be in the continuation of a Plane, the whole of that Plane is lod to fight, and appears but a Right Line (Art. 3. of a Plane, P.41,) And, the In- terfedion of two Planes is a Right Line, (Ax. 3.) Wherefore, if a Plane (which may be conAdered as the Pidure) be Atuated any bow; the Line, in which this imaginary Plane would cut the Pidure, is that in which the parallel Planes unite, and vanifh; confequently it reprefents an inAnite Diflance; and confequently, the Line, fo produced, is their Vaniflaing Line ; for they cannot, if continued in- Anitely, appear to go beyond it. Hence, I 10^ i A'N^ IN^TROvD.UXT.OrR-Y. PREFACE. Book ni.^ 3D^4 Hence, tbofe Planes-are, very* aptly, faid to vanifh, they being loft to fight.. Therefore, all Parallel Planes have the fame Vaniftiing Line. (Theo. 3rd.) Alfo, if a Right Line be fuppofed to pafs through the Eye parallel to any num- ber of Lines, they will appear to converge towards that Line, and^to meet it in* one Point, at an infinite Diftance (Theo. II. Direft Viiion.) Wherefore, if this Right Line from the Eye cuts a Plane, anyhow fituated, it- will cut the Plane in a Point only ; which reprefents a Point at any Diftance what- ever, in that di region, and confequently, it reprefents the Point in which the Lines, parallel to it, converge ; which is infinite. Therefore, it is their Vaniftiing Point ; for, they are loft to fight before they appear to reach it, feeing it is infinite. And, iince the Line, producing that Point, pafi'es through the Eye, the whole Line is loft to fight, feeing that one Extreme is at the Eye ; and the Extremes of Lines are Points. Therefore, the Point, in which it cuts the Pidlure, is its whole Reprefentation ; and confequently, ail Lines, parallel to it, tend to that Point. Now I muft own, that I cannot conceive any Term fo fit to exprefs that Line, or. Point, in which parallel Planes, or Lines, meet each other,, as Vaniftiing Line, and Vvaniftiing Point; becaufe they are truly faid to vanifti in them. For the fame, reafon, perhaps, Mr. Noble, the laft writer on Perfpedlive, (except Ferguflbn) has, made ule of the Terms Entering Line, and Entering Point ; leeing that, the Plane, or Line, begins at the one and vanifties in the other. Had this Author been the in- veiator of thofe Principles, 1 ftiould not have found, fault with the Names he had given them.; but, fince there were Names already given, by the Author, which are more fignificant, I muft blame even an attempt to alter them ; becaufe, a multipli- - city of Names, for the fame Thing, occafions a confufion of Ideas, in the Mind, of their fignifieation and ule ; and cannot poflibly be of advantage to the Science. In the procefs of this Book, after fome neceflary obfervations, on the proportion of the.Pifture, the Height and Diftance of the Eye, &c. in the third Sedtion, I have, firft, ftiewn how to determine on the Pofition of thePidlure, in refpedlof the Objedf and the Eye, the Station being previoufly determined; then, how to pre- pare the Piclure, for Pradlice, according to the Principles contained in the foregoing Theory. The Pidlure being, prepared, the Situation and Diftance of the Objedl, and the Pofition of the Picture being determined, the following Problems, in that Sedtion, fhew how to find the Interledtions and Vanifhing Lines of Planes, in all Pofitions to the Pidture, if they are not parallel to it (for all fuch have no Vaniftiing Line. Theo. ift.) with their Centers and Drftances. Then, to fix the Vaniftiing Points of - certain Lines in thofe Planes, and determine their Diftances; by which, , the Lines* vaniftiing in them, are proportioned. The Pofition, Situation, and Diftance of the Pidture (in refpedt of the Objedland the Eye) being determined, the Original Lines, in the Objects, are. produced (if neceftary, and not parallel to the Pidture) to. their Interfering Points ; which are always found in the Iiuerfedtion of the Plane they are in (Theo. loth) or being much inclined to the Pidture they do not, perhaps, fall within the compafs of the Pidture ; then, other expedients are u fed, to find the reprefentation of fome prin- cipal Point in the Line, from which Point, the indefinite Reprefentation is drawn (Theo. i2th.) And laftly, the finite Parts, which reprefeut certain portions of Lines in the Original Objedt,-are determined, by Theorem 13 th; by whicii means, theObjedt is completed; proceeding from one Plane, or Face of the Objedt, to another ; drawing all the Figures, in each, by means of their refpedlive Vaniftiing Lines. Each two adjoining Faces, having one Line common to both, the Vanilh- ing Point, of that Line, is in both Vanilhing Lines (Theo. loth) confequently,. it is in their common Interfedlioii (Cor. 2nd. Theo. 7 th.) by the help of which, die: Vauifhiiig Lines of contiguous Faces are determined., z:: Ih; Sea. I. AN INTRODUCTORY, PREFACE. In Seaion 4th, I Ivave briefly illuftrated all the remaining, praaical. Problems in Brook Taylor’s Eflay, refpeaing the proportioning of Right Lines, peripedively, and Ihewn their great and extenfive utility ; each of which, founded on tlie moft folid and permanent Principles, is of immenfe value. For, without know- ing the whole of Perfpeaive, or praaicing by its Rules, rigidly, an Artifl, who is ac- 'cuftomed to Iketch, by fight, whatever lie fees before him, with feeming accuracy, may, by thefe Problems, reaify any. errors, in right lined, or circular. Plane Od- jeas, from the known proportion ofone part to another ,* the affinity oflhe Planes and Lines to each other, being known, and the ratio of one Part to another j which, may frequently be obtained, when the true meafures of thofe Parts cannot. Thefe Problems contain all the Rules neceffiary forPradice. They may l>e com- pared to the five fundamental Rules in Arithmetic, by which all others are worked ; and, a Perfon might, with as much propriety, imagine that he had given Arith- metic enough, for every Occafion, in thofe Rules, as Brook Taylor had of Per- fpeftive, in his firft Eifay ; whereas, the Rules, he has there given, are no more than the Elements of pradlical Perfpeflive, This Seflion contains, alfo, various Expedients; viz, for determining Vanifh- ing Points, when they fall beyond the limits of the Pidlure, geometrically and arithmetically, i. e* to determine their Diftance from the Center of the Vanifliing Line and from the Eye, both which are neceffiary to be known ; howto draw Lines to a Vaniffiing Point which is beyond the bounds of the Pidure, &c. In Sediou 5 th, thofc Rules are applied to real ufe, in delineating all kinds of Plane Figures; and, on Planes in various Pofitions. Firffi, by means of the Figure being geometrically drawm, in the Original Plane. Secondly, without it, by their known Proportions, their pofition to the Pidure, and the properties of the Figure, being regular. In the 6 th Sedion, from Plane Figures, I have proceeded to Solids, compofed of Planes, of various Figures ; and in various Pofitions. In the 7 th, I have applied them to ftreight Mouldings, compofed of Planes and cylindrical Surfaces, in Cornices, Entablatures, &c. Sedion the 8 th treats, of curve lined Objeds, ' in general. The 9 th (hews how to apply the wffiole to compound Objeds, in regular pieces of Architedure, and Buildings of various kinds. The I o th is for internal Views, and horizontal Pidures, in Ceiling Pieces, &c. The 1 1 th is adapted for the particular Profeffions of Cabinet-makers, Coach- :makers, &c. and for Machines, in general. The 1 2 th, and laft, is on inclined Pidures and Planes, in general 5 and, applied 'to Fortification, or military Architedure. N. B. A Scale of equal Parts is always adapted, or determined on, of the Propor- tion we intend* to delineate the Objeds, on the Pidure. L. ' The methods of making Scales, for various purpofes, are given in the Ap- pendix to my Treatife of Geometry. I i ' ^ D d I ? ^ V H .i 105 S e" C T I o N S E C T io6 Plate V. ION II. A preparatory and elementary INTRODUCTION, to the PRACTICE of PERSPECTIVE, A S this Book is intended fora compleat pradical Treatife, I have treated it in fuch a manner (in this Sedion) as if no Theory, or Elements, had been given, tor which reafon, I have defined a few more Terms, which are fuited to Pradicc only; as there are feveral Terms in the theoretic Lift which may be omited here; and, there are alfo, in Pradice, feveral which are not ufeful, in Theory. I have deduced from the Definitions fuch ufeful Leffons, which, if carefully attended to, will contain all the neceflary Theory for a Praditioner. PERSPECTIVE, is the Art of delineating the true Reprefentations of Objeds, on a Plane Surface, by geometrical Rules; according to thePofition, and Diftaiice of the Objeds, in refped of the Pidure and of the Eye. (See the Apparatus.) The Perspective Representation of an Object, is the Sedion of the Pyramid of Rays, A El, by a Plane, in any Pofition; which, is the Subjed of this Third Book. For, the Pidure of every Objed, truly delineated in Perfpec- tive, is fuppofed to be fo fituated, in refped of the Objed and the Eye, that if Vifual Rays, or Threads, i. e. if Right Lines were drawn from each Angle, or other Point in the Objed, to the Eye, they would pafs through the correfponding Points in the Reprefen ration, on the Pidure. As EA, EB, EF, &c. cut the Pidures in h^f\ a, b, f, &c. which are, there- fore, the Perfpedive Reprefentations, of the Points, A, B, F, &.c. on each Pidure. It is evident, that the Perfpedive Reprefentation of an Objed is nothing more than the Figure projeded on a Plane, by its Interfedion with the Vifual Rays from the Eye to the Objed ; wherefore, the whole bufinefs, of pradical Perfpedive, is to find the true Figure of the Sedion of the Rays, in all Pofitions, whatever, of the Pidure (which is confidered as the interfeding Plane) and in every fitualion of the Objed, or of the Eye. In order to which, it is neceflary to reconfider, well, the conftrudion of the elementary Planes; as in Plate IV ; or, to make them more familiar, I have given another, which is better fuited for Pradice ; as they are the foundation of the whole, and tlie origin of all the Lines and Points ufed in Pradice. Alfo the general In- trodudion to this Work fhould be attentively perufed, and tolerably well under- ftood, by every one who would be a Proficient in the Pradice of Perfpedive. Let B FI L be an Original Objed, reprefenting a Building, and, ES a Spedator, viewing that Objed, from the Point of View E (the Eye.) It is manifeft, that whilft the Eye remains fixed, in that Point, the Objed can- not vary in its Appearance. But, it is alfo obvious and demonftrable, that every different Sedion of the Pyramid of Rays, by a Plane in different Pofitions, will ex-' hibit a different Pidure; fb very different, indeed, they may be, that they can fcarcely be fuppofed to be Reprefentations of the fame Objed, much lefs from the fame Point of View and Pofition of the Objed, when viewed dired. Compare the Pidures, MN O P and OP; how different are the Images of the fame Objed, to each other. The one, on MNOP, has the Reprefentation, bgfdc, of the end of the Building, BFC, fimilar to the Original, in every re- fped; the Reprefentation, of the Front, AHGB, is diftorted, anddrag’d out to a prepofterous length, when the Eye is oppofite to it ; whilft the other Pidure exhibits a pleafing and natural Appearance, in almoft any Point of View* but moft fo in the true one. Yet, both Pidures affed the Eye alike, in the true Point of View, and appear the fame as the Original Objed, itfelf; each Line, Plane, 10 / Sea.II. AN ELEMENTARY I NT R O D U C T I O N, &c. or Figure (It is evident) being feen under the fame Optic Angle as in the ObjedV, Conlequently, the Pidlure being a true Perfpedtive Reprelentation of the Original, if it had llkewife the fame degree of Colour, Light and Shade, it would not be poflble for the Eye, at E, to diftinguifh whether the Image iaif c, or a i f c) delineated on a Plane, or tlie Objed itfelf (A 1 F C) was prefented to view. Having, thus clearly, explained what Perfpedive Is, I fliall next define all the praftical Terms, by means of which, the whole procefs is performed, and the Re- prefentations effeded. Let the Plane A I K B (which may be fuppofed the Pidure) be ralfed up into a vertical Pofition, i. e. perpendicular to the Plane it is fixed to ; which is the moft natural, moft general, and convenient Pofition. Alfo, raife up the Plane G K I H, parallel to the Pidure. This Plane is called, the Direding Plane (Def. 4.) and is very ufefiil. In Theory. In it, the Eye is always fuppofed to be, as at E; and if the Plane N/iiCL be turne^l down, parallel toABGH, the Line /iC meeting I K, in the Direding Plane; then, EC is the Diftance of the Eye from the Pidure (equal FD); and E F, (equal CD) is its height above thePlane ABGH, on which, the Spedator (EF) is fuppofed to ft and. Therefore, at whatever Diftance the Eye is from the Pidure (as E C) a Plane G K I H is fuppofed to pafs through the Eye, parallel to the Pidure. The Plane V being turned up, perpendicular to the Pidure, is confidered as a part of the prime or chief Vertical Plane; and, N/iiTL is called the Horizontal Plane; both which are fuppofed to pafs through the Eye, at E. N. B. This Conftrudion of the elementary Planes, fo eflential in Theory, is, alfo, very necelTary to be well confidered by every Pra£litioner, who may not have an inclination to go through the Theory; he will, moil certainly, hnd his account in it. DEFINITIONS. DEF. A. The GEOMETRICAL, or GROUND PLANE is that Plane on which an Original Objed, intended to be delineated, is feated. AsGHZZ; on which the Plans, ACDF' and X Y Z (fuppofed to be the Seats of fome Objeds) are geometrically drawn, which are to be reprefented on the Pidure, A I K B. On this Plane is always drawn, or fuppofed to be drawn, the Figures of the Seats or Plans of Ob- jefts fituated on it, in their true geometrical Proportion, according as they are iituated, in refpedl of each other and of the Pifture i and they are (in common Perfpe£livc) always underftood to be beyond the Pifture. In the Apparatus, the Plane ZSZ is the Ground Plane, reprefenting the real Ground, on which the; Objed, BFIL, (lands. The Redangle A B CL is its Plan, or Seat on the Ground Plane. Note. If the Objed Hands on a Floor or Table, &c, it is confidered as the Ground Plane. DEF. B. The GROUND LINE, or BASE LINE, is a Right Line on the Pidure, drawn parallel to its Top and Bottom, i. e. parallel to the Horizon. As AB, Fig. 37. No. i and 3. (See Interfedion. Def. 7. in the Theory.) ThePlane AIKB, being ereded, reprefents the Pidure, and ABZZ, being horizontal, is confidered as the Ground Plane. Therefore, the Line AB is the Interfedion of the Pidure with the Ground Plane. And, fince all Objeds which are to be reprefented, on the Pidure, are fuppofed to be beyond it, they muft neceiTarily appear above the Line of Interfedion; for which reafon, it is, very properly, called the ■Ground Line. Note. On the Ground Line is applied all the meafures, geometrically, of all Figures (as Plans, &c.) of Objeds on the Ground Plane, to be delineated ; and frequently, for other horizontal Planes, whofe Interfedion with the Pidure is not given or drawn. N. B. Of thefame ufe is the Interfedllon of any other Original Plane with the Pidure, viz, for propor- tioning Lines and Figures drawn on that Plane, DEF. C. The HORIZONTAL LINE, is a Right Line drawn parallel to and above the Ground Line ; as ML. (See Vanifhing Line Def. 8 j or G.) The Horizontal Line is called fo by way of eminence, or for diftindion from all other horizontal Lines, being the firft, or principal Vanilhing Line ; from its fixed and determined Pofition and Diftance, all nthcxVanifhing Lines, whatever, are determined ; the Vertical Line is determinable without it. PI. VIII. fig- 37 - No. I. n. viir. F'g- 37- f Thco. 3 j J Theo. 2. § Thco, I, t Dcf. 4. AN ELEMENTARY INTRODUCTION Book III. N. B.’Its Diftance from the Ground Line Is always equal to the height of the- Eye from the Ground, be- ing confidered as a Plane, on which the Objedf, to be delineated, is feated. So that, whether we be liting, ftrnding, or elevated, the height of the Eye, from the Ground Plane, being determined, and AE, or DC, being made equal to it (by the Scale of Proportion) NL drawn through the Point E, or C, parallel 10 A b, is the Horizontal Line; or Vanifhing Line of horizontal Planes, Note. It is fuppofed (in Theory) to be produced by a horizontal Plane (N/A'L) pafling through the Eye and curing the Pi£ture, in N L, their mutual Interfedlion. DEF. D. The VERTICAL LINE, is a Right Line drawn at right angles with the Horizontal and Ground Line ; cuting the Piclure into two equal Parts, perpendicularly. As E D. (See Def. ii.) DEF. E. CENTER of the PICTURE (or POINT of VIEW) is the Point C, in which the Horizontal and Vertical Lines cut each other. (See Def. i6 and 17.) N. B. The real Point of View is the place of the Eye, wher; it ought to be fixed when viewing a per- fpedUve Piiflure. The Point C, on the Pidu'e, oppofite to it, where a Perpendicular from the Eye would cut tlie Pi< 51 :uie, is, therefore, its Center; and is, generally, underftood to be the Point of Sight; i. e. to which the Eye muft be oppofite. Hence, Iftlie Center of the PiiSlure, C, be firft determined (as it frequently is) then a Right Line drawn through C, parallel to the Horizon, as L M, is the Horizontal Line ; and, another Line, drawn through C, perpendicular to it, as E D, is the Vertical Line. DEF. F. PARALLEL of the EYE. If EC be taken, in the Vertical Line, equal to the Diflance of the Pi£lure ; and, through the Point E (which is con- fidered as the Eye) if a Right Line be drawn, parallel to the Horizontal Line, as IK, it is called the Parallel of the Eye, of horizontal Planes. (See Def. 9.) DEF. G. VANISHING LINE, of any Original Plane, is a Right Line on the PiiRure, fuppofed to be produced by an imaginary Plane paffiiig through the Eye, or Point of View, parallel to the Original Plane ; the Line, in which fuch a Plane would cut the Picture, is the Vanilhing Line of that Original Plane. COR. Hence, the Horizontal Line is the Vanilhing Line of the Ground Plane, and all other horizontal Planes. For, it is produced by a horizontal Plane (N IKh) pafling through the Eye, and cuting thePifturc. COR. 2. And hence, all Planes, which are parallel betw^een themfelves, (but, not to the Pidure) have the fame Vanilhing Line-f-. For, there can be but one Plane pafling through the Eye parallel to them all; and fince It can produce but one Line on the Picture, by its interfeftion with it, that Line is, confequently, the. V anifhing Line of them all. * N. B. The Vanifhing Line, and the Parallel of the Eye, of any Original Plane, are parallel to theJnter- feflion of that'Plane with the Piclure L Therefore, LM, the Horizontal Line, and IK, the Parallel of the Eye, of horizontal Planes, are both parallel to A B, the Ground Line; which is the Interfedlion of the Groui;d Plane, with, the Pidlure. N. B. 2. Onginal Planes which are parallel to the PiiSlure have no Vanifhing Line, nor Interfeftion §. DEF.H. VISUAL RAY is a Right ^ Line .fuppofed to be drawn from the Eye to any Point in an Original Objc>ii.'ance ES (equal DF) is fufficient for that Pidure, i. e. for the Objed AC. ix th re are moreObjeds, as X and Z, extended to thefulMength of the Pidure, ■u uherude, it is little enough; they being feen under the Angle DEF, which '■e whole length of the Pidure (ubtends, viz. 55 Degrees, on the Ark df; and - f'.e leaft Diftance I would ever ufe, in fuch Cafe, when the Objeds extend the uil width of the Pidure; although it is more than Mr. Kirby’s ^reateft, •ES PRELIMINARY OBSERVATIONS. ”3 Sea. Ill, ES is therefore the DIftance of the Piaure D F, and alfo of the Objeas, or of a Plane palling through the neareft Planes of X and Z ; in whicli Cale, it is evi- dent that the Piaure mufl; be as large as the Objeds themfclvcs, being applied clofe to them. If they are fuppofed real Buildings, and the Scale of proportion be determined on, viz. a loth or 12 th part, or any otlier ; take Ef (one third part of E S) for the Diftance, and draw df parallel to DF ; then is El equal df; for Ef:df:: ES:DF; I. e. they are equal, and confequently, the Objeds x, y, andz, being reduced to the fime Proportion will all fubtend the lame Angles, refpeftively, as the Originals. By which means the Pidure may be delineated of any Pi'oportion. Now, if the Station (E) be determined, from which the Objed AC Is to be drawn, the Pofition of tlie Pidure is alfo determined. For, ES, the Station Line, bifeding the Optic Angle, A EC or DEF, ought always to be perpendicular to the Pidure; confequently, the Pofition of the Pidure, DF' or d f, is. determined ; i. e. it muft be perpendicular to ES. P'or, when the Station, or Point of View, is "lixed, from which we are de- termined to delineate the Objed, is it not moft rational to fuppofe the Pidure, on which the Objed is to be reprefented, placed dl red before the Eye? (as MNOP Fig. 154’ in the Apparatus) or is it more eligible to place it parallel to either Plane, AEG II or BFC (as MNOP) i. e. to A B or 13 C, as DC or BG, whofe Center, or pjg,, -3^ Point of View, is at C or G. The very fuppoiitlon of it Is abfurd to the lalt de- ^ gree ; and yet, this abfurdlty Is commited by every ArtlP; who places the Point of View at either Extreme, or perhaps entirely off the Pidure, as is frequent. The difference mUft be obvious to every Perfon who confiders it. For, the Pic- ture being-placed dired, according to D F, the Optic Angle is no more than A EC, •or ^Ec, about 23 or 4 Degrees; but, being placed in the Pofition DC, theOptic Angle is, aEg, 102 or 3 Degrees; for, EC, perpendicular to D'C, bifeds the Gptic Angle, on that Pidure; as E S bifeds the Angle A EC, on the Pidure DF ; notwithftanding, the Objed, ABC, does -not occupy one fourth part of •the Angle ^ E g, Jn the Apparatus, S P, on the Ground Plane, bifeds theOptic Angle, pSt, •under which the Objed is feen, according to the Pofition of the Pidure MNOP; for, the Angle, at the Station Point (S) is the fame as in the Horizontal Plane, at the Eye. How to prepare the PICTURE for I iX ^ X X Let A'lKB (when rai fed perpendicular to the Ground Plane) be fuppofed the Pidure; alfo, let the Direding Plane, G H I K, and Horizontal Plane, IKLM, be placed parallel to the Pidure, and the Ground Plane. Suppofe the Figures A D, and X Y Z, on the other fide, are to be reprefented on the Pidure; and the Points a, b, c, &c. where the Vifual Rays EA, EB, &c. would pafs through the Pidure, to be geometrically determined thereon. In order to which, fcveral preparatory Lines are weceffary to be drawn on the Pidure ; the Center and Diftance muff always be known or determined, together with the pofition or fituatlon of the Objed, in refped of the Pidure and of the Eye, and alfo its height, above the Ground Plane. Now, E is the place of the Eye, EC is the Diftance of the Pidure ; and EF is -the height of the Eye; the Point C, where the Perpendicular E C cuts the Pidure, is its Center. ML, the Line in which the Horizontal Plane cuts the Pidure, is the Horizontal Line (Def. C) or the Vanifhing Line of horizontal Planes ; and A B, in which the Pidure cuts the Ground Plane, is the Ground Line (Def. B.) confe- quently parallel to the Horizontal Line. Thefe two are the firft and moll ufeful Dines ; the Center of the Pidure (C) in this Cafe, is in the Horizontal Line, the Pidure being vertical. F f k The 1 14. PI. VIII. Fig, 37. No. 3. TO PREP/, RE THE PICTURE Book III, The next Line to be confidcred, a id of ute, is the Vertical Line, ED (Def. D.) If the Plane V be turned up, ptrpendl uljr to the Pidure, it will pafs through, the Eye, at E, and conlequent'y through EC, cuting the Piduie in EG, the Vertical Line; which, being produced will pafs through D. This Line always paffes through the Center of the Pidture, and is of the fame ufe for all vertical Planes, which are perpendicular to the Pidure, as the Horizon- tal Line for horizontal Planes; i.e. it is their ''’anilhmg Line, which has, confe- quently, the fame Center (C) and Diltance (E The conftrudion of thefe five Planes fhould be well attended to, and the Lines they generate, by tlieir Interfedlions ; three of wdiich, v'z. the Ground Line, the Horizontal Line, and the Vertical Line, are already produced on the Pidlure ; and the Parallel of the Eye, I K (whole real place is cut of the Pidlure) is tranfpoied to the Pidture, by turning upthe Horizontal Plane on M L, its Interfedlion, till it coincides with, or falls into the Pidlure ; and with it the Diredt Radial, or Diftance, (EC) together with the Eye (F) which falls into the Vertical Line, at E. When the perpendicular pofition of the Planes has been confidered attentively. Jet them be pufhed either from or towards you, keeping the Diredling and Hori- zontal Planes, joined in I K ; the Vertical Plane (V) may (fill be fuppofed to cut them all at Right Angles, and generates flill the fame Line on each. In all Por- tions, i. e. let the Angles they make with each other be what it may, the Planes are ftill parallel to each or her relpcdlively ; and their Interledlions are flill parallel amongfl therafelves. (Theo. 2nd.) Now, let the Pidlure be turned down, on AB, its Interfedlion with the Ground Plane, till they coincide; and let the Horizontal Plane, with the Eye (E) and the Parallel of rhe Eye (/A”) be allb turned down into the Pidure (as it is rcpre- fented on the Pidlure) ; alfo, let the Vertical Plane (V) be turned down on either fide, into the Pidure, and with it the Dircd Radial v^C) i. e. the Diftance of the Pidure, falling into the Horizontal Line, with the Eye at E. And, laftly, let that part of the Ground Plane, which lies beyond the Pidure (on which the Seats of Objeds are geometrically drawn out) be luppofed to be turned, on AB (its Interfedion with the Pidure) quite over to the wther fide ; and imagine it, ftmply, a Plane, without thicknefs ; lo that, all the Figures, defcribed on it, are feen on the other Side, inverted ; as and X Y Z; in which Cafe, it is obvious, that they have the fame Poiition, or Situation to the Pidure, as before; and the Lines ( which are not parallel to the Pidure) being produced, cut the Pidure, in the Interledion AB, in the fame Points, as before. Thus, is the Pidure prepared for Pradice, in common Perfpedive ; and all the elementary Planes are reduced to one Plane, viz. the Pidure^. All the fame Lines and Points, may be feen in No 3. which is divefted of that apparent intricacy of Planes on Planes, conlequently it is more fimple and intelligible; having only the preparatory Lines, anlwering to the InterfeiSions of the elementary Planes with the Pidure, on it. The Figures below A B, the Ground Line, are Plans of Objeds, on the Ground Plane, intended to be delineated. To prepare which, let the Ground Line (A B) be firft drawn, at fuch a conve- nient Diftance from the bottom of the Pidure, and parallel to it, as to allow room, below it, for drawing the Plans of Objeds (as X, Y, Z) intended to be delineated, in their true geometrical Proportion, Place, and determined Pofition to the Pic- ture, if neceftary. Tlie fpace, below AB (as AFBj is not confidered as a part of the Pidure, but of the Ground Plane, whofe real place is beyond the Pidure. * The Dire(5ting Pl.nre (G H I K) not being of ufe in common PerfpeiSive is fuppofed to be turned •down, or'rcGao\ed uul of the way. Through Sea. III. FOR PRACTICE. 115 Through the middle of the Piaure, draw ED, perpendicular to the Ground Line, div ding the Piaare into two equal Parts. This Line is the Prime vertical Van Ihing Line, in which, the Center of the Piaure, or Point of View, Is always (thougli not always in the Horizontal Vanilhmg Line, but when the Piaure is vertical, as it is now luppofed to he); make CD equal to the determined height of the Eye, G is the Center ; through which, draw ML, the Horizontal Line, parallel to A B, the Ground Line. Then, with the Radius EC (the determined Diftance) deferibe a Semicircle, cuting the Horizontal and Vertical Lines in £, E, and F \ i. e. make QE, &c. equal to the Diftance of the Piaure, which areconfidtred as the Eye, tranfpofed to the Piaure* (generally uiiclerltood by the Points of Diftancej and, through E, draw I K, the Rirallel of the Eye, parallel to the Horizontal Line ; then is the Piaure prepared, having all the fixed Lines and Points determined thereon. It only remains, now, to find other Interfeaions, Vanifhing Lines, and Vanifliing Points, necelfary for delineating the Objeds intended. The Ground Line, or Interfeaion of the Piaure with the Ground Plane, is the firft and principal Interfeaion ; and the only one, in general, made ufe of as fuch. But, the Interfeaions of other Planes are often wanted, and particularly vertical Jn- terleclions ; on which, the meafures of the heights of Objedts are applied ; although they are frequently made ufe of, yet very few conbder them as Interfeaions; and confequently, they do not fee the generality of the Principles and Rules by which the Work is performed ; as they look on the operation in Planes which are vertical or inclined, in a different light from fuch as are horizontal ; whereas, if it be well confidered, they will find it is the very fame ; for, wherever the geometrical mea- fures are applied, in proportioning any Line, it is confidered as an Interfeclion of fome Plane, in which that Line is or may be fituated. And, fince the Interfedllon of any Plane determines the Pofition of its Vanifhing Line, I have flaew'ii how to find the Interftdlions of Original Planes, in all necefl'ary and generally ufeful Cafes. Inclined Planes, in general, I have referved for the lad Sedlion ; but, as various Planes, which are perpendicular to the Piflure, are inclined to the Horizon, and vertical Planes are frequently inclined to the Picture, I (hall not confider them, in either Cafe, as inclined Planes, but fuch only as are inclined to both. As the Horizontal and Vertical Lines both pafs through the Center of the Pic- ture, they are, therefore, the Vanilhing Lines of Planes perpendicular to the Pidtuiet (being vertical) which, from their fixed and invariable Pofition, de- termine a 1 other Vanifh-ing Lines whatever. Therefore, in the fullowing Problems, I have, in the fird place, (hewn how, by them to determine the Vanidiing Lines of all Planes that are perpendTcular to the Pic- ture, either with or without the Interfedfiou ; which are fubjedt to one general Rule. Secondly, how to find the Vanifhing Lines of vertical Planes,' in all Pofitions to the Pidture (except parallel) aijd alfo, of certain Planes inclined to tiie Horizon, which are lubj^dt to the fame invariable Rule, as vertical Planes. The fifth Problem (hews how to find the Vanifhing Lines of Planes any how inclined, both to the Horizon and to the Pidlure (being vertical) from their known inclination to the Horizon, &c. as fpecified in the Problem. All which, are fre- quently neceffary in common Piadlice. f Thco. 4, * The Eye, or Point of Diftance, may be any where in the Circumference of a Circle, whofe Radius is EC, as occafion requires ; or, according tn the Pofition of original Planes, which are perpendicular to the Pidfure, and their Vanifhing Lines j all which, have the fame Center and Diftance (Theo. 4th ) 2 PROBLEM 1 16 Plate IX. Fig, 3g. f Cor. T. 9. 7'. El. t ^ JO* 11 4. I. Ei. § Theo. 4. Fi?:* ^5* No. 2. 1 9 * 7 - THE E L E M ENTS ;P R O B L E M I. Book III The Interfering Point of any Line, in a Plane which is perpendicular to the Picture, being given, together with the Angle of its Incli- nation to the Horizon ; to determine its Interfedtion-and Vanifhingr Line 3 the Center of the Pidlure being given. Let I be the Interfering Point of feme Line, given; C is the Center of the | Pirure ; and X is the Angle of Inclination of the Plane to the Horizon. f Through C, draw A B, the Horizontal Line, parallel to the Bottom of the Pidlure; and, through I, draw I D parallel to A B. i, Make the Angle UIF equal to the given Angle X; and, I F is the Interferion a required. f Through ,C, the Center, draw G H parallel to I F ; GH is the Vanldiing Line, j' Note. The Vaniflalng Line may be determined, by making the Angle ACG or B C H equal to the Inclination known ; without the Interledlion. ■ / DEM. Let I D be fuppofed the Ground Line, and 1 the InterfeeSling Point of the common Interfedion of the Ground Plane and the other I lane ; which, feeing both Planes are perpendicular to the Pidure, is ,.aifo perpendicular to the PiiSiure f. And, lince I is the Interfe ting Point, of a Line in the Plane, the lnterfe(Eiion of the Plane mufi: ne- ceflarily pafs through that Point J;; and confequentlv, it will make the fame Angle with the Ground Line and PJoiizoiUal Line, as the Planes make with each other (DI F, equal BLF|i) equal to the given Angle X. (Se-e Article 4th, of Planes and their Politions. Page 42.) Therefore, I F is the Intcifeciing Line of that Pianc. But, the Vaniflaing Lines of all Plane§, which are perpendicular to the Pi£lure,pafs through the Center of the Picture^. And, the Vanifhing Line of ever)- l^Iane is parallel to its Interfedlion. (Theo. 2.) Therefore, GH is the Vaniflimg Line of a Plane perpendicular to the Pifture, and .inclined to the Horizon in-the Angle H C B equal FID, equal X. N. B. If F be the Interfefting Point of any other Line in that Plane, and E F be drawn parallel to the Horizontal Line; the Angle EFI, equal FID (equal X) produces the fame Interfeftion. Alfo, if K be the Interfedling Point of any Line in another Plane, parallel to the former ; then, AK, j parallel to I F, is its InterftAion. For, all parallel Planes bave'parallel Interfedions. But they have the fame Vanifliing Line (GH). If EC, or EC, be equal to the Diftance of the Pifture; A K, or EF, may be coniidered as the parallel of the-Eye ; and £ C, or EC, (perpendicular to G H, or A B) is the Vertical Line. BCHOL. If the Angle of Inclination be greater or lefs, the procefs is the fame. But, if the Angle be increafed or enlarged to a Righf one, the Plane is no longer inclined, but vertical; and, if the Interfering Point, J, be in a Line, pafling through C, perpendicular to A B, the Interfection and Vanifhing Line are the j lame, the Vertical Line E J ; which is the Vanifhing Line of all vertical Planes that are perpen- dicular to tbc PlAure. See this Problem illuftrated by moveable Planes,; Fig. 15. No. 2. IKLM is the Grlglnal Plane, T U is its Interfeiflion with the Ground Plane, or other horizontal Plane; I is its Interfering Point, and IN is the Interlerion of \ that Plane with the Pidlure j making the Angle NIB equal to the Angle PQH, of the Original Plane with the Ground Plane. And, if IKLM (the Vanilhing Plane of all Planes which are perpendicular to the Pidlure) be placed parallel to the Original Plane, IKLM, it will cut the Pirurein ON, the Vanifhing Line of IKLM, in that Pofition. "For, it paffes through the Eye (at E)and the Diredl Radial (EC) ; co ,it is perpendicular to the Piruret, and paffes through C, its Center. Hence ; all Vanifhing Lines, which pafs through the Center of the Piflure, have the fame Center and Diitance, viz. the Center and Diftance of the Pidture. nfequently Note. If the Plane inclined on the other Side of its Interfeftion T U, the Angle of its Inclination is on the other fide of the Vertical Line ; to which, particular regard muft always be had. made PROBLEM 7 Sea. III. OF PRACTICAL PERSPECTIVE.* J'7 ROB E M ir. The VaniHiing Line of a Plane being given, with its Center and Diftance ; and the Angle of inclination which any Original Line,' in that Plane, makes with the InteiTeaion, of the Plane it is in, with the Piaure ; to find the Vanifliing Point of that Line, and to determine its Diftance. A B is the given Vanifliing Line, and C is its Center. Fig, 45 . Draw C E perpendicular to A B, and equal to the Diftance of the Vanifliing Line’ given or found. Through E draw G H (the Parallel of the Eye) parallel to the Vanifliing Line. Make the Agle GE A equal to the Angle which the Original Line makes with the Interfeflion, cuting the Vanifliing Line in A, the Vanifliing Point fought ; and E A is its Diftance, E being confidered as the Eye. Note. Regard muft always be had, on which flde the Original Line inclines, and the Angle GEA, or HEB muft be made accordingly. DEM. Imagine the Plane AGHB turned over, on AB, till E coincides with the Eye; then is GH in its true Place (ftill parallel to the Picture) ; and t A will be parallel to the Original Line, produ- cing its Vanifliing Pointf ; and making the fame Angle with the Vanifliing Line ( AB) and Parallel f Def. 22. of the Eye, as the Original Line makes with the Incerfedtion and Directing Line, of the Plane it is in | 1 . || Theo. 1 1. EX. The Pidlure, A I KB, being ereded, turn over the Vanifliing Plane, IKN L,' Fig. 37. parallel to the Original Plane. Then, the Radials E N, E O, E L, &:c. are refpedively parallel to their Ori- ginals, XY, YZ, andXZ, in the Triangle XYZ; and confequently, they make equal Angles, rcfpedively, with NL, and JA, as the Originals make with A B, the Iiiterfedlon of the Plane they are in. No. I. PROBLEM . III. The Interfering Point of any Line, in a vertical P’ane, being given, and the Angle of Inclination of the Plane to the Picture, to find its Interfedlion and Vanifhing Line ; the Center and Diftance of the Pirure being given, and the Pidure fuppofed vertical. Let I be the interfering Point given, and C the Center of the Pidure. Through C, draw AB, the Horizontal Line, and E D the Vertical Line at right angles ; and through I, the Interfeding Point given, draw I H, parallel to E D ; which is the Interfedioii required. Make C E equal to the Diftance of the Pidure; and make the Angle CE A equal to the Complement of the given Angle of Inclination ; cuting AB in A. Or (having drawn EG parallel to A B) make GEA equal to the given Angle ; and through A, draw F G parallel to 1 H, which is the Vanifliing Line required. DEM. Bectufe the Pidure and the Original Plane are both vertical, their Interfcdion is perpendicular to the Horizon and confequently, fincc I is the Interfeding Point of fome Line in that Plane, I H per- t Cor. to pcndicular to FI, the Ground Line, is the Interfedion ; by Theorem loth. 9 * ?• And, becaufe EAC (equal AEG) is equal to the Angle of Inclination, EA is the Radial, or Pa- rallel of the common Interfedion of the vertical Plane with horizontal Planes, producing its Vanifli- ing Point. (By Prob. 2.) Therefore, F G, drawn through A, parallel to I H, or ED (i. e. per- pendicular to F I) is the Vanifliing Line (Theo. 2nd). G g For, xi8 the elem ents BookllL Plate IX. For, imasine the Piflure FGHI turned up, vertical, and t'-c Triangle AEC vertical to it, i. e. horizontal; reprefenting a part of the Horizontal Plane ; b is the E e. I'hen, a vertical Plane, pafiing through tire Eye and the Line E A, will cut the Pifture in F G, the Vanifhing Line, which is parallel to IH, the Interfe 61 ion ; by Theorem 2nd. For, they are the Sedlions of parallel Planes with the Pifture (Def. 8). N. B. If the Interfedting Point of any other Line, in the Plane, had been given, as B or H, the Inter- fe£lion I H would be the fame. And confequently, if the Vanifhing Point of that Line, or any other, as F, or G, in any vertical Plane, be given or found, the Vanilhing Line is determined. For, it pafies through that Point (Th. lo.) perpendicular to the Ground Line, from its Pofition. Example, by the Apparatus. Eig. 15* f Theo. 5. I 4. 7. El. If the Planes A BG H and BFC were produced, they would cut the Picture MNOP, in BK and A’E, the Interledlions of thofe Planes; which are parallel be- tween themlelves, hecaufe their common Seflion, BG, is parallel to the Picture and, hecaufe BG is vertical, they are, alfo coiilequently, vertical, being parallel to B G relpectively Andy if a Plane (RSTU) be fuppofed to-pafs through E (theEyc.of the Specta- tor) parallel to BFC, curing the PiClure MNOP, produced, 111 R U, it is the Vanifhing Line of that Plane, parallel to its InterfeCtion, BE. Alio, a Plane (STOP) parallel to ABGH, cuts both PiCtures, in OP and OP; the Vanifhing Lines of that Plane, on both ; parallel to R U or W, t.e. to BK. No. 3 Fig. 40. When the Original Plane is inclined both to the Horizon and the PiCture, having its common Se6tion with horizontal Planes, parallel to the Pi^Iure; it is a fimilar Cafe to this ; and by turning the Pi(^lure fideways, on I H, (Fig. 40 ) ’lis the very'' fame in every refpedl ; as it is fully illuftrated in Fig. 15. No. 3. EX. ADPB is the Original Plane; A By its Interfedhon with the Ground Plane, is parallel to the Pidlure; i.e. to AB. Let it be railed up, making the Angle PQ^R, with the Horizon. AONR is the Pidiure, which being railed up vertical, 'the Plane XON /, paffing through the Eye, at E, will be parallel to the original Plane, ADFB -, and cut thePidtu-re in ON, its Vanifhing Line, parallel to the Horizontal Line, JAM. And, if the Original Plane was produced, it would alfo cut the Pidlure in a Line parallel to L M, or A B (i.e. toOt') below A B, the Ground Line, But, if the Original Plane inclined tov/ards the Pidlure, on this fide of AB-, then, itsVanifh- incr Line would fall below the Horizontal Line, and the Interfedlion above it, SCHOL. If the Interredioii of an Original Plane, in any pofition whatever, be given, and the Inclination of that Plane to 'the. Pidure known, its Vanifhing Line is determined as by this Problem; feeing that, the Vanifhing Line of a Plane is always parallel to its Interledion with the Pidure. Confequently, if the Interfedion, be parallel either to the Horizontal or Vertical Line, .the Vanifliing Line fought is alfo parallel to them. But, when the Inteifedion of the Original Plane is not given; and which, by reafon of the great Diftance of the Plane, from the Pidure, or Inclination to it, cannot be had, imr its Pofition afeertained ; then other Expedients are ufed, to find the Vanillung Lines of fuch Planes; viz. by finding the Vanifhing Points of two Lii>es in the Original Plane. (See Prob. 5.) N. B. The Center of every vertical Vanifhing Line is the Point In which it Is cut by the Horizontal Line; and the Center of every Vanifhing L'ne, which is horizontal, is the Point in which it is cut by the Vertical Line (by the 7 th Theorem) the Pidfure being vertical. The Diftance of 'Very Vanifning Line which does not pafs through the Center of the Pidlure, is the Hvpothenufe of a Right angled Triangle (as AEC) vvhofe Bafe and Perpendicular are the Diftance of the Picture, and the Diftance, AC or EC, betw en the Center of the Pidlure and the Vanifhing Fine. For, in refpedf of the. Vanifhing Line, F G, of vertical Planes, if CE be the Diftance of the Pic- ture, E A is the D fiance of the Vanifliing Line, and A is its Center. Bur, if G H b(i a V ariifhing Line of a Plane inclined to the Horizon, in the Angle G E A, and to the Pidlure, in A EC, having the lame Diftance, A E, E is its Center, andACis the Diftance of the Pidlure, 2 PRO- Sc^i, nr, OF PRACTICAL PERSPECTIVE, PROBLEM IV. The Vanifhing Line of a Plane, its Center and Diftance, being given,, and the' Vanlfhing Point of fome Line in that Plane; to find the Vanifhirig Point of other Lines, making a given Angle with that Line, whofe Vanifhing Point is given. AB is the Vanifliln’g Line, C is its Center, and A’the given VamQiing Point. Fig. 41. Draw CE perpendicular to A B, and equal to the Diftance of- the Vanifhing Line, given or found. Join AE ; and make AEB, or AED, equal to the Angle, which the Original Lines make with each other, cuting the Vanifliing Line in B or D, the Vanifhing Point fought. ; ‘ ‘ » DEM.i Itnaglne the Triaiigte AEB turned up, on AB, perpendicular to the Pidlure (if AB be confidered as die Vanifhing Line of a Plane perpendicular to the Pidlurej or, making the fame Angle with the PiAurc, IS the Plane, of which, AB is the Vanifhing Line. T hen is AEB the Parallel of whatever Plane the Original Lines are in, producing its Vanifhing Line ABji E coincides With the Eye, and EA, EB, &c. are rel'peftively parallel to the Original Lines ; feeina they pafs through the Eye and the Vanifhing Points of thofe Lines ( Def. 22 .) But, thcLtadials of twoLin^s producing their Vanifliing Points, make the fama Angle, at the Eye, as the Original Lines make with each other f. Therefore, B, or D is the Vanifhing Point required. N. B. If the Angle AEB be obtufe, it is not the Angle of Inclination of the Lines. In which Cafe, re- gard ought particularly to be had to the Pofition of the Original Lines, in refpeft of each other, and of the Pifture. For, AippofeFG to be the Reprefentation of a Line, whofe Vanifhing Point is A ; then, if FB be drawn, AFB reprefents an _Angle equ^l to AEB; confequently, the obtufe Angle is towards the Pifturt*. In which Cafe, let the Angle of Inclination be made on the other Side ; that is, produce AE to 1, and make I E B equal to the given Angle ; for if the Angle AE B (i. e. A E D) was made equal to the Angle of Inclination, the Point D (in that Cafe) would not be the Vanifhing Point required. PROBLEM V. The Angle of the Inclination of a Plane to the Horizon, together with the Angle which its Interfedtion, with any horizontal Plane, makes with the Picture, being given, to find its Vanifhing Line ; the In- terleftion of the Plane not being given, nor its Pofition known. The Center and Diftance of the Pidture are given. Let A B be the Horizontal Line, and C the Center of the Pifture. Fig. 42- Draw C E perpendicular to A B, and equal to the Diftance of the Pidlure ; and, through E, draw DE parallel to A B. Draw A, making the Angle DEA equal to the Angle which the Interfeflion of the inclined Plane (with horizontal Planes) makes with the Picture, cuting the Horizontal Line in A, the Vanilhing Point of the common Interfedtion. Draw E B, perpendicular to A E, cuting A B in B ; and, through B, draw B G perpendicular to A B, indefinite. MakeBFequal BE; and make the Angle BFG equal to the Inclination of the Original Plane to the Horizon, cuting BG in G, and draw AG, the Vanifli- ing Line fought. This Problem, for finding the Vanifhing Lines of Planes, cafually inclined to the Horizon and to the Pi£lure, is univerfal, and applicable in all Cafes, when the Angles are determinable ; as in the Roofs of Buildings, Pediments, &c. which, being frequently neceffary, in common Subjects, couW not be difpenfed with ♦ Def. 8. ■j* Theo. II., 120 THE ELEMENTS Book III. Plate IX, with here ; othe.rwlrc, I fliouM have omitted it till the laft Set^ion, which treats more fully bn fueh • Subje£ls. I {bail tHerefore referve the Demonftiation of it till then ; where, every Cafe and Circumftancc, rig. 42* tefpefting indiiied Planes; are fully deinonftrated, and exemplified by nloveable Planes. N.B. The Center of the Vanifliing Line, AG, is determined by drawing CH perpendicular to the Vanifli- + Th, ing Line, cutirig it in H, its Center f, nor does it differ, in that refped, from any other, exdept in Pofition ; for, C H is a part of the Vertical Line of the Original Plane. And, if Cl be drawn, parallel to the Vanllhlng Line, and equal to the Diftance of the Picture, the Line I H, to the Center, is the Diftance of the V anilhing Line AG; which, Diftance, is as applicable to that Vanifliing Line, as the t)iftance of the Pidare (I C) to all Vanifhing Lines which pafs through its Center, i. e. of Planes perpendicular to the Pidure. Fig. 15.' EX. In the Apparatus, V W, the Vanifliing Line of the Plane of the Roof, HIFG, is determined by this Problem; V being the Vanifhing Point of the common Se£lion, G H, with a Horizontal Plane, DGH K ; and E W, i. e. the Radial or Right Line from the Eye, parallel to GF determines W, in RU produced, the Vanifhing Point of GF and HI, on the Piidure M N O P. P'or, £, in the Horizontal Line, reprefents the Eye tranfpofed to the Pidure ; E Y being equal to the Diftance of the Eye from the Point Y ; and YEW is equal to the Angle DG F of the Inclination of the Roof to horizontal Planes. Wherefore, firice V is the Vanifhing Point of one Line (GH or IF) in the inclined Plane, and W is the Vanifhing Point of another Line (GF or HI) in the fame Plane ; confequently, V W (a Right Line drawn through thofe Points) is the Vanifhing Line of the Plane GF I H (Theo; loth. Cor i.) For, a Plane pafting through the Eye, and thofe Vanifhing Points, would b® parallel to GFI H, and would cut the Pidlure in the Line V W ; which is, there- fore, the Vanifhing Line of that Plane. Def. G* SECTION m containing The ELEMENTS of PRACTICAL PERSPECTIVE. H AVING, in the foregoing Seiftion, fhewn how to find and determine the In- terfering and Vanifhing Lines, of Planes, in all common Cafes ; and alfb tfie Vanifhing Points of Lines, in any Plane whofe Vanifhing Line is given or found; by means of which Vanifhing Points, all original Right Lines (not parallel to the Piefture) have their Indefinite Reprefentations, on the Piifture, truly and accurately deferibed. In this Section 1 have fhewn how to cut off certain portions, from the indefinite Reprefentation, which are the perfpeiftive Reprefentations of certain portions, or legments, of Lines in the Original Objeft. Having well coiifidered, that moft regular Objedls are bounded by Planes, and the bounds of Planes are Lines* ; it is evident, that to find the Reprefentations of Lines, in all Pofitions, is to find the Reprefentation of the Figure, or Objeift, bounded or circumfc'ribed by thofe Lines. And, fince the extremes of Lines arc Points, it follows, that, if the two Extremes, of a Right Line, be found, the whole Line is determined; and, by finding fundry Points in curved Lines, the Reprefen- tation of the Curve is determinable. Wherefore, the whole, of praiflical Perfpedlive, confifts in finding the Reprefention of a Point, any how fituated. But, fince Points are the interfedions of Lines, and, to find the Reprefentation of a Point, in Perfpedlive, it muft, neceflarily, be fuppofed in fome Line ; hence it follows, that, to determine the Reprefentations of Lines, in all Cafes, is the whole fum and fubftance of Practical Perfpedive. 6 * See Page 50 and 51. Book II* Section U. Now, Sea. IV. OF PRACTICAL PERSPECTIVE. 121 Now, Right Lines can have but three Pofitions, in refpea of themfelves and of the Piaure, viz. they mu ft be either parallel, perpendicular, or inclined; and, having learnt how to manage Lines In all thefe Cafes, by the following Problems, there remains little more to be done ; for, by conftruaing a number of Lines toge- ther, properly, an Objea is formed. By Theorem 12. the Indefinite Reprefentation of a Right Line, not parallel to the Piaure, is a Line drawn through its Interfeaing and Vanifhing Points. But, fince the Interfeaing Point is not always wanted (nor Is It always attainable) If any Point in the indefinite Reprefentation be determined, a Line drawn through that Point, to the Vanifhing Point, is the fame; for, it would, if produced, pafs through the Interfeaing Point of the Original Line. In this Seaion, which contains the whole Subftance of praaical, reailinear Per- fpeaive, I have fhewn how the indefinite Reprefentations of Lines, in all Pofitions, ace determined; and then, howto proportion them, in any given or known ratio to the Original ; and afterwards, how to manage them when the Vanifhing Point is not within the limits of the Piaure, by various Expedients. Let the Reader take particular notice, that I fhall, always (to favc repetition) in the following Problems, fuppole the Center of the Piaure to be given, and its DIftaiice known 3 except, In particular Cafes, when it is othei wife exprefted. The Diftance of the Piaure (being determined) is applied, in Praaice, by the fame Scale of Proportion to which the Piaure is delineated. Let it alfo be obferved, that 1 fhall always (in the Diagrams) make ufe of the initial Letters of the following Terms, viz. C for the Center of the Piaure, E for the Eye, in its firft or principal place on the Piaure, and E for its firft tranf- pofed place, in any Vanifhing Line, &c. and jE^ for the next tranfpolition, &c. and VL for any particular Vanifhing Line. But, feeing that the Ground Line, the Horizontal, and Vertical Lines never vary their Places, and are always ftronger drawn than the operative Lines, I think it needlefs to particularize them otherwife. PROBLEM VL How to find the Reprefentation of a Point whofe Seat on the Pidlure is given, and its Diftance from the Pi6l;ure known. Let C be the Center of the Pidlure, and S the Seat of the Original Point. t?- ^ big. 43. Draw a Right Line CS, through the Center of the Pi^fture, and the given Seat, indefinitely beyond S. Draw C A, at pleafure ; and SB parallel to GA. Make AC equal to the diftance of the Piflure, and SB equal to the diftance of the Original Point from its Seat. Draw A B, which will cut CS in b, the Point fought. Or, if the Original Point be between the Eye and the Picture, make SB equal to its diftance as before, and draw AB, which produce to the Pidure, cuting it in Then is b the projecled Reprefentation, of the Point B, on the Pi 61 :ure. Compleat the Parallelogram A C S D. DEM. AC is equal to tlie Diftance of the Pifture, SB to tlie Diftance of the Original Point from its Seat, and AC is parallel to SB (Con.) Confequently, the Triangles ACb, and bBS are ftmilar. Wherefore, Sb : bC ; ; SB : AC ; and, confequently, S b : Sb + bC (i. e. SC) : : SB ; S Bq- A C, equal SD (i. e. BD.) that is, Sb;SC : : SB: BD. Alfo Si* : SC : : S A ; AD. - Theo. 13. Now, becaufe SB is the Diftance of the Original Point, from the Piefture, and A C is equal to the Diftance of the Picture; draw CE and SF, both perpendicular to CS, confequently parallel ; make CE equal CA, and SF to SB, and draw EF ; which will cut SC in the fame Point, b. For, the Triangles CEb, bFS are fimllar. Wherefore, Sb : bC ; : SF : E C 3 i. e. as S B : A C. ' H h Hence T H E ELE^1ENTS Book UL Place IX iFig.43. Tlieo. 12. Def. 25. t Ax. '7. '§ Theo. 1 3. and Tb. ii. 8. of £1. Pig. 44. Hence it Is evident, that the projection of tlie Point b does not, in the leaft, de- ;pend on the fituation of the Lines C A and SB, in relpect or C S, but on their pa- rallelilm and .proportion to each other; wherefore, if tire true Diltances are not known, but only their Ratio, the Point b, will be projected the fame. e. g Take C I at pleafure ; and make SG to 'Cl, as SB to AC, and draw iG; the •Point b will be projected the fame, on the Picture. For, Fnce S G rCI ; : SB : C A, and Sb : bC ; : S B : C A ; confequently, Sb : bC : ; S G : C I, and confequentl)', the Point b is the fame. Hence may be feen the univerfality of the 13th Theorem. For, conceiving E to be the Eye, and EC the Direct Pvadial, i. e. rhe.'D'ftance of the Pi(£lure, and SF the Diftance of the Original Point from its Seat, i. e. from the Picture, imagine the Triangle EC b to be turned up, on bC, till EC is perpendi- cular to the Picture, and fuppofe b KS turned back, on S b, till S F is alfo perpendicular to the Pidlure, on the other Side ; then is E in the true place of the Eye, and F is in the Place of the Original I’oin.t, and confequently, £F is a Vifua! Ray from the Eye to the Point; which it is evident will pafs through the Picture, in the Point h; and, lince Vifion is conveyed in Right Lines to the Eye, the Point F will ap- -pear, on the Pidlure, at b; which is, therefore, the peifpedive Reprefentation of the Point F. (See App.) Again, becaufe EC is parallel to SF and cuts the Picture in C, its Center, C is the Vanifhing Point of S F ; (Cor. 2. Def. L ) for, the Line S F is perpendicular to the Picture ; and S is its interfering Point ; (Del. K.) wherefore, S C is its indefinite Reprtfentationf ; and EF is a Vifual Ray from the Eye to the 'Original Point; (Def. H.) And, becaufe EC is parallel to SF, and EF cuts them both, they are, therefore, all in the fame Plane confequently, the Vifual Ray, E F, will cut the Picture fomewhere in SC, the Interfeftion of that But, in this Cafe (the Lines having different Vanilhing Points) the Vanilhing Line, of the Plane they are in, mull be had ; and, it mull alfo be obferved, that both Lines are in the fame Plane, or the Operation cannot be performed by one Vanilhing Line. PROBLEM X. • i t ( The VanifLing Line of a Plane being given, and the Reprefentations of two Lines in that Plane, having different Vanilhing Points, to cut off, from any Point in one of them, a portion equal to that re- prefented by the other, or in any known Ratio. The different pofitions or fituatlons of the Lines, to each other, may make this Operation appear very different ; for which rcafon I (hall give it varioufly, of which, the firft is according to Brook Taylor. I i Let Figi 46* + Prob. 32. Geometry. 126 THE elements Plate X. Fig* 47 * t Dtf. 224 15. 1. El. Fig. 48* § Prob. 8* Book ill. Let A B be £l given Reprefentation of a Line, and F G the indefinite Reprefenta- tion of another Line. NM is the given Vanifhing Line, and C its Center. It is required to cut off (from the Point F) a portion, which (hall be to that repre- fented by A B in a known Ratio. Produce A B to its Vanifhing Point, D; and, through F, draw A H, to the Vanifhing Line, cuting it in H. Draw C E perpendicular to the Vanifhing Line and equal to its Diftance, and join E D and E G, by Right Lines. Make E K and E L in the Ratio required ; i, e make EK to E L as the Original Line, reprefented by AB, is to the other; and draw EM, or EN, parallel to K L. D raw B H and F D, interfering in I ; draw 1 M, or N I, curing F G, in O or P. Then, FO, or F P, reprefents a Line having that Proportion, to the Line which AB reprefents, as EL to E K. DEM. If N M be conlidered as the Vanifhing Line of a Plane perpendicular fo the Picture; imagine the 'friangle N E M turned up perpendicular to the Pifture; or, if the Plane in which the Original Lines are fituated, be inclined to the Pidture, let NEM be fuppofed parallel to it ; then, E is in the true place of the Eye. Now, ED and E G are the Radials of AD and F G, producing their Vaniflring Points, confequentljr they are parallel to them, refpedlively ; t and make the fame Angle (DEG) at the Eye, as the Original Lines make with each other. (Cor. i. Theo. 6.) And, becaufe E M is parallel to KL, D, G, and M are the Vanifhing Points of the three Sides of a ^'riangle (EKI ) the Radials, ED, KG, and £ M being ptrallel to them, 1 efpeftively. Bur, the three Lines, FI, FO, andlO, vanifh in thofe Points, refpe-Iively ; confequently, FIO reprefents a Triangle fimilar to EKL; the Angle DFG reprefents the Angle DEG (Prob. 4.) and confequently, I M reprefents a Line parallel to E M ; i. e. to KL. Therefore, the other two lides, FlandFO, or F P, have that Proportion to each other, perfpec- tively, as EK to EL. But, becaufe AB and FI, A F and B I have the fame Vaniflting Points, D and H, refpedlively, ABIF reprefents a Parallelogram (Cor. i. Theo. 3.) confequently, FI repreftnts a Line equal to the Original of AB;| and therefore, FO, or F P, reprefents a Line which has that i loportion to the Original of A B, as E K to EL. N. B. If the fituation of the Line AB was fuch, that a Right Line, joining A and F, was parallel to the Vanifliing Line N M, then B I mufi be drawn parallel alfo. But, if ab, the given Line, be fo fituated, that the Line a F is fo much inclined to the Vanifhing Line as not to reach it within the compafs of the Pifture, take any Point (J) in the Vanifhing Line, and draw Ja, Jb, indefinite; draw AD, at diferetion, cuting them in A and B, and proceedas before. For, *AB and ab having the fame Vanifhing Point (D) alfo, A a and Bb have the fame Vanifh* ing Point (J) ab BA reprefents a Parallelogram; and confequently, A B, ab, reprefent equal Lines. Cafe 2nd. Let the given Lines (AB and FP) be fo fituated as to crofs each other ; and, it is required to cut off, from the Point (I) of their Interfe£lion (or any other) a part (IF or IP) equal, or in any Ratio to the Original of A B. Draw Ac parallel to the Vanifhing Line (MD) indefinite; and, from the Point H, or any other, at diferetion, in the Vanifhing Line, draw HI and HB, cuting A c in c and b ; then, A b is to b c in the Ratio of the Original of A B to B I §. Make a c equal to A b and draw a H, cuting A B in d ; d I reprefents the fame meafure as AB, viz. ac equal to Ab. Produce A B to its Vanifting Point (D) and F I, to H; if it be not already done. C being the Center of the Vanifhing Line, draw C E perpendicular to H D, and equal to its Diftance ; and draw ED, EH. Make EK, EL, equal, or in the Ratio required, and join KL; to which, draw EQ parallel, giving the Vanifhing Point G. Draw G d, and produce it, cuting I F in F ; then, if E K be equal E L, I F re- prefents a Line equal to the Original of d I, that is, of A B, as it was required ^ or, whatever Ratio EK has to EL, FI reprefents the fame to AB, Again. Draw mn, through I, parallel to A c, cuting aH in m. Make In equal Im ; draw nH cuting AD in O ; and draw OG, cuting FH in P. I P reprefents a Line equal to I F, equal A B j i. e, to their Originals ; or in the Ratio of E K to E L, whether equal or otherwife. DEM. Se£l;. IV. OF PRACTICAL PERSPECTIVE. 127 DEM. Becaufc a m and F I have the fame Vanifliing Point (H) they rcprcfent parallel Lines ; and, be- caufe m n is parallel to ac and to ihe Vanilhing Line HD, a m I c reprefents a Parallelogram; confequently, the Originals of m I a', cl ac are equal. But, In was made equal I m, and nH reprefents a Line parallel to aHj confequently, lO and Id reprefent equal Lines, i See Problem 8th. ) And, fince OP and Fd have the fame Vanifhing Point (G) they alfo reprefent parallel Lines and confequently, lOP, I cl F reprelent equal Angles t; wherefore, they aie iimilar Trianglesj for the Angles P I 6, d I F, arc equal But, they ahb reprefent congruous I riangles, for IP;IF';:IO:ld::PO:dF; and I O reprefents a Line equal to I d ; confequently, I P reprefents a Line equal to I F, and PO to d F. If it bad been required to cut off, from the Point J, in F H, a portion, In a certain Ratio to A B ; draw B J. If B J be parallel to HD, draw DJ cuting Ab in a; then Ja reprefents an equal Line as A B. But, if B J be not parallel to the Vanifhing Line, produce it to its Vanifhing Point, and proceed as by Fig. 47, ufing J a for A B, as I d for A B, before. N. B. If K L was parallel to the Vaniihing Line, then FO and dF would alfo bfe parallel; for EG would be parallel, and confequently, could not produce a Vaniftiing Point. Suppofe it was required to cut off, from the Point a (Fig, 47.) in the Line aF, a part, equal, or In any known Ratio, to the Original of A B. Draw A a till it cuts the Vanifhing Line in J, and draw J B ; alfo draw aD to the Vanifhing Point of A B, Then a b reprefents an equal meafure as A B, and a F may be cut in any R atio to ab, as in the next Cafe; which is the jrd Example, Prop. 16, of Brook Taylor, mentioned in the Preface to this Book. Cafe ■^rd, Let AC or C B be the given Line. It is required to cut off from the Point C, a part, which reprefents an equal meafure as the other. D and G are the Vanifhing Points of the two Lines, and E is the Eye. Draw ED and EG, and bifeifl the Angle DEG, by the Line E PI. Draw A which will cut C G in the Point B, as required ; or G A in A. This Cafe, or this pofition of the Lines, is but another Example of this Problem, which Brook Taylor calls, finding the Reprefentatioa of a Circle from the reprefentation of one Radius given. It is certain that the whole Circle may be- comjdeted, from a Radius given, though not by this Cafe only, but by both, as I fhall exemplify. Let AD and FG be drawn indefinite ; C, their InterfefUng Point, is the repre- fentation of the Center, and C B is the Radius given, in the Pofition by Brook Taylor. He fays, “ make C A to reprefent a Line equal to that reprefented by C B (by the tSHiY’ riz. by .the firft Cafe of this ; “ i. e. bifefl the Angle E O D and, to make it uill lefs intelligible, in his Diagram, the Angle DEG 'i. e. E O D, in his Treatlfe) is not bifedled, nor trifefted, but is cut, very nearly, in the Ratio of 2 to 5 which negligence, in a Perfon of his fagacity, is to me furprizing. It is dempiiftrated in the 3rd of ,the 6th of El. that, if any Side of a Triangle be' divided in the Ratio of the other two Sides, a Right Line joining the oppofite Augle and the^pplnt of Seftion will bifedl that Angle. But, there may, veryjuftly, bean exception to this ; for, it is necefl’ary that the greater Segment be contiguous to the greater Side, and confequently the lefler Segment to the leaft Side. In the Triangle DEG, if DG be divided, in H, in the Ratio of DE to EG* theti, E H hifecls the Angle D E G ; becaufe G H, the greater Segment, is con- tiguous to the greater Side, E G ; which, it is obvious, could not be otherwife. The Conftru6lion, for proof of the Affertion, is to produce either Side, as G E* and make E J equal to E D, and join J D ; which is proved to be parallel to E H. For, becaufe E DrzE J the Angle E J D— JD E (9. i. El.) and, becaufe D E G is equal to them both (10. I.) and DEHrrHEG (Con.) J D E=D E H (Ax. 4.) therefore, EH is parallel to JD (4. i.) But, J E is the Radial of C B, produced, and E D of A C ; and E J is made equal to E D ; alfo, E H is a Right Line from the Eye, parallel to J D (i. e. to K L, as in the former Cafe) producing the Vanilh- ing Point of the Line AB (i, e. oflO, and F d, in the two former Cafes) ; by means of which Va- nilhing Point (H) C A and C B are cut, reprefenting equal portions of thofe Lines. Hence may be leen the affinity between the two Cafes, in bifeding the Angle DEG. For §C.I.T.3; J 2 . I. F.1. Fig. 49. 128 THE ELEMENTS Book III. Plate X. Fig. 49. 1 1. I. El. Fig. 50- •J; 2. 1. El. § lO. 3. El. ForEJ=:ED, i.e. EKtoEL, and E H is parallel to K L ; confequently, A C B reprcfents a Tri- angle fimilar to* K E L, or JED; and confequently, D, G, and H are the Vaniftiing Points of the three Sides A C, C B and AB; which is Ifofceles, by Conftruflion. And, fince D CG reprefents an Angle equal to D EG (Prob. 4*) confequently, ACG reprcfents an equal Angle to D E J f ; therefore, the Sides, containing the external Angle D E J, are, in this Cafe, made in the ratio (E K to E L) of the Originals, which AC and CB reprefent; and fince they are equal, confequently, AC and CB reprefent equal Lines. Now, C A and CB reprefent equal Lines, from the fame Point C; wherefore, if C be confidered as the Center of a Circle, in Perfpedive, CA and CB are Radii of the Circle ; and this is all that Brook Taylor has done towards finding the whole Reprefentation, which is far from being fufficient. In Be produced, make CF reprefent a part equal to what CB reprefents, and C I to C A, or C B ; then, B F, and A 1 reprefent Diameters of the Circle. Proceed as in Cafe ift. by making EjftT equal to EL, and draw EN parallel to draw AN and BN, cuting BF and AD in F and I, the Points fought. Or, having drawn AN, only, cuting BF in F ; draw PTI, cuting AD in 1. The four Points, A, B, I, and F, are all in the Circumference of a Circle, whofe Center is C; for, C A, C B, &c. reprefent equal Lines. But, thefe four Points are not fufficient, for compleating a Circle in Perfpedive. Draw HC and NC indefinite, and cut off, from the Center (C) C a, Cb, C i, and C f, reprefenting alfo equal meafures, to thefe reprefented by C A, &c. e. g. Make E M equal to E L ; join M L, and draw E O parallel to M L. Al.o, make EL equal E/f; and draw EP parallel to KL. Draw AO, O I, BP, and P F, cuting the indefinite Lines in a, i, b, and f. The Angle N E H being blfefled, by E Q, (hews the Affinity to the firfl: Cafe. Througli the eight Points, A, a, B, b, I, i, F, and f, an Ellipfis may be deferibed, which will be the reprelentation of a Circle in Perfpedfive ; from the given repre- fentation of one Radius, A G or C B. PROBLEM Xr. From three Points given, in the Circumference of a Circle, to find the Reprefentation- of the Circle ; having the Vanifliing Line of the Plane the Circle is in, and the place of the Eye. A, B, and C are the three given Points,' L M is the Vanifhing Line, and E is the place of the Eye. E S is the Diflance of the Vanifhing Line. Produce A B and C B, cuting the Vanifhing Line in I and K ; draw E I and E K. Make the Angles K E L and I E M each equal I E K, producing the Points L and M, in the Vanifhing Line. Draw A K and A M, C L and C I, interfedVmg in D and.F ; the Points DandF are in the Circumference, which pafles through A, B, and C. DEM. Becaufe the Anf^les K E L, KEI, and I EM are -equal (Con.) and E is the Eye, the Angles L D K, KB I, and IFM reprefent equal Angles (Prob. 4.) and confequently ADC, ABC, and A P' C alfo reprefent equal Angles Therefore, thofe Angles touch the Circumference § ; for they are in the fame Segment, or ftand on the fame Ark, A b C. ( Now, here are five Points, A, D, B, F, and C, in the Circumference ; but they are not fufficient for deferibing the true Curve of the Ellipfis. 'Therefore, draw A H at pleafure, cuting the Vanifhing Line, in H. Make the Angle H E J equal I E K, and draw C J, cuting AH in G, which is another Point in the Circumference of a Circle, 7 Thus Sea. IV. OF PRACTICAL PERSPECTIVE. 129 Thus may as many Points be found in the Circumterence as are iieceflacy to de- fcribe the whole Curve; which, palfing through all the Points, A, G, D, B, &c. will be an Ellipfis ; for, it is the Reprefentatiou of a Circle in Perl'peaive, feeing that thole Points are all in its Circumference. If there be too much Space between any two Points, to defcribe the Curve with gr.'ater accuracy, another Point may readily be found ; as a, or b, between A and C. Draw a B, or b B, at pleafure, cutingthe Vanilhing Line in S, or O. Join S E, or OE, and make the Angles SEH and SEN, or OKP, equal to lEK, cutingthe Vanilhing Line in H, N, or P, relpedlively. Draw HD, or N F, curing Sa in a; or P F, curing O b in b ; which Points, a and b, are alfo in the Circumference. ' For, becaufe A D and CB have the fame Vanilhing Point (K) alfo, AB and CF having the fame Va- nilhing Point I, the Originals of the Arks, DB and BF, are each equal A C f- ; for AD and CB' re- prefent parallel Lines it; confequently, BaD or F, and BAD or F, or BbF, reprefent equal Angles §, each being equal to the Original of /^BC; i. e. lEK. Othewife; draw MC, indefinite ; produce ME, and make the Angle QER equal to I E K ; ER, being produced, will cut the Vanilhing Line, fomewhere, if it be not parallel to it. If E R be parallel to the Vanilhing Line, draw A a alfo parallel ; or to the Point R, in which ER would cut the Vanilhing Line, curing MC in a. For, the Angle Q_E R -pR E M =: two Right Angles -}•. And, the Angle MaR (i. e. AaC) reprefents the obtufe Angle M £ R ; and, ABC reprefents an Angle equal to (^E R. But, the oppofite Angles of every Quadrilateral, infcribed in a Circle, are equal to two Right Angles Confequently, A a C reprefents an Angle in the oppolite Segment ; for, AaC and ABC are op - pofite Angles, in the Quadrilaierai ABCa. t Cor. to 10. 3. El. X Cor. I. Theo. 3« § Cor. 2. 9. 3. El. t I. I. El. t 3 ‘ There Is not, perhaps, in the whole Book, a more elegant Problem than this, which induced me to give it a place, and to perfedt it. Its utility is not fo great ; yet it may frequently be applied, by thofe who do not care to be confined Ifriftly to the Rules of Perfpeftive, in every refpedl. For, having, by any means, ob- tained the reprefentations of three or more Points, which are known to be in the Circumference of a Circle, the whole may, readily and accurately, be determined ; if they know the Diftance of the Pidure, and the Vanilhing Line of the Plane of the Circle. Brook Taylor has made a lame affair of this elegant Problem, notwithftanding his Principles are the fame ; by reafon of the fhort Difiance he has taken for the Eye, and the prodigious Dimenfions of the Circle, it is the mofidiftorted and pre- pofierous Diagram in the whole Book. He finds no more than one Point, on each fide, and leaves his Readers to find out the reft ; nor does he go about it in a proper manner, there being no occafion (if the Diftance was fufficlent) to draw another Line (dO) at pleafure, in order to make another Angle, equal to DOE; but, to make equal Angles withEO and DO, on either fide, continually. One thing I am much furprized at; he fays, “make the Angle dOe equal to DOE; or, having made an Infirument, containing the Angle DEO, turn it round the Center, O, till it comes into the Pofition dOe,” &c. which, is fo ungeometrical, that I could fcarce conceive it to be the expreffion of fo great a Man. Having had occafion, in this Problem, for a Vanilhing Point (J, or R) which was not within the compafs of the Piflure, the next Problem fliews how to deter- mine the Diftance of fuch Vanilhing Points from the Center of the Pidure, or Va- nilhing Line, and alfo from the Eye. PROBLEM XII. The Vanilhing Line of a Plane being given, and the place of the Eye, with a Line, from the Eye, inclining to the Vanilhing Line ; to find the Center of the Vanifhing Line, and to determine the Dif- tance of that Point, in which the inclined Line would cut the Va- nilhing Line, from the Center, and from the Eye. AB THE ELEMENTS Book III, * 3 ^ Plate X. Fig- 5^- f 2.>and 4. of 6. £1. r 60 El. § 9. 6. El. •{■ fame. Theo. II. §-4. 60-E]. t 4 Prob. 2. 7. 6. hi. 20. I. Prob. 2. 7. 6.-L1. A B is the Vanilhing Line given, and E D is the inclined Line from the Eye, atE. Db is confidered as the bounds, or limits of the Pidure, and B is the Point in which ED, produced, would cut the Vanifhing Line, AF, produced. Draw EC perpendicular to the Vanifhing Line, cuting it in C, its Center. Take C G any part of C E, a half, a third, or fourth, as E D is lefs or more in- clined to A B, and draw GF parallel to E D. Then, asCG:CE::CF:CBt; that is, if C G be a third part of C E then C F is a third part of C B, and G F of E B ; or, in whatever Ratio C G is of C E. Or, drawn E A perpendicular to ED. cuting the Vanilhing Line in A. Then, £nd a third Proportional to A C and C E (Prob. 31. Geo.) it will beCB. For, A E B is a Right Angle (by Conftrudlion) and EC is perpendicular to A B; therefore, as AC:CE:;CE:CB:|;; Alfoas AC: AE::CE;EB; i e. EB is a fourth Proportional to AC, CE,& AE, Jn Numbers, :they are thus determined, by a Scale of equal Parts. 'FirB. Take A C and C E by the Scale ; fquare CE, and divide the Produdf by A C, which, will give CB § ; for AC, C E, and C B are three Proportionals. Or, if the Triangle C F' G be ufed, multiply C F by C E, and divide the Produdl by C G ; the Quotient will be C B -f*. For, C G, C F, C E, and C B are four Proportionals. Secondly. Since AC:CE::AE:EB, confequently they are four Propor- tionals. Wherefore, multiply CE into A E, and divide the Produdt by AC, the Quotient will be EB, the Diflance of the Vanilhing Point, B, from the Eye. If the DiBance of the Pidlure be known (equal C E) and the Angle of the incli- nation to the Piflure, of one Side of a right angled Objedt, be determined; the yaniBiing Points, and their Diftances, are determinable. Let AB be the Horizontal vanilhing Line, and C the Center of the Pidlure. Draw C E perpendicular to A B, and equal to the DiBance given, or determined. Make the Angle C E A equal to the Angle given (or to its Complement, accord- ing to .which fide the Line inclines) cuting the VanlBiing Line in A, the Vanilh- ing Point of one fide [j; ; from which all the reft are determinable. Or, if the DiBance, C E, be fo great, that it cannot be laid down on the Pic- ture (as is frequently the Cafe) take CG half, a third, a fourth, or any portion of 'C E, and proceed as before j making the Angie C G H equal to C E A, the given Angle. Then will C H be half, a third, &c. of AC, or whatever portion CG was taken of C E § ; by which means, the diftance of the Vaniftiing Point A, from -the Center, C, is afcertained. Make H G F a Right Angle, i. e. draw GF perpendicular to G H, cuting the Vanifhing Line in F ; then wdll C F be alfo half, a third, &c. of CB, the diftance ■of the other Vaniftiing Point (B)from the Center (C.) Alfo, GF will be the fame portion of E B. Ijy' Thus may the real Diftances, be found, and the place of the Eye tranfpofed to the Pitfture, as yit will be exemplified in Practice, when their real places cannorbc had thereon ; all which may be found arithmetically, as follows. Let the Diftance of the Pidlure, C E, be given, equal 6, 5 (Feet, or what you pleafe) and let the Vanifhing Point A be determined as above, or at difcretion, on the Pidure; viz. AC equal 2, 6, from which all the reft may be determined. FIrft j to find the' Diftance of the Vaniftiing Point, A, from the Eye. Square A C and CE, which being added together, the fquare Root of that Pro- dud will be AEt ; for A E fquare is equal to A C added to C E fquare Secondly; to find the Diftance of the other Vaniftiing Point B, from the Center. Square CE, and divide that Produd by AC, the Quotient will be C B §. For, AC:CE::CE:CB. Alio, A C : A E,:.: A E : A B. Thirdly ; to find the Diftance of the Point B from the Eye. Having obtained AE, by the firft, multiply C E by A E, and divide that Pro- dud by A C, the Quotient wdll be E B ; for A C : C E : ; A E ; E B. Sea.III. OF PRACTICAL P E R S PEC T I'v E. The Dlftance of each Vanifhing Point, A and 'B, from the Eye (E) are as nc- ceflary to be had as their places on the Pi6lure,, the ufe of which I Ihall exemplify ^nd explain in its proper place, ' ' \t • i Fir/l. A C and C E being given, to find A E ; the Diftance of A. Square AC, viz, 2,6; alfo, fquare CE, viz. 6,5 AE—yi / multiplied by 2,6 multiplied by 6,5 7 156 52 325 390 fquared 49 6,76 Square of A C, 42,25- Square of AC added 6,76 ‘ X ' T ] r Sum of both )49,oi(7, fq. Root;=A E, * 1 Secondly. To find CB; AC and CE being given Thirdly. To find EB. Square CE; 6,5 Multiply CE=:6, 5 ■. j G,5 ■ by AE=zj divide by ACz=2, 6)42, 25(16, 25=CB divide by A C=:2,6)45,5(i 7,5 rrEB, Or ; having found C B (by the 2 nd) add the two Squares, of C E and C B, into one Sum ; the fquare Root, of which, will be E B. (By 20. i, El.) When the two Vanifhing Points, A and B, are known, to find the Diftanceof the Pifture, C E ; the Center being given. Multiply C B by A C, and extrad the fquare Root of that Product, it will, give CEthe Diftance of the Pi< 9 :ure; by Prob. 4. 7, 6. El. • To find the Diftance, C E, geometrically ; A and B being given, the vanifhing Points of Lines at right Angles with each other^ and C, the Center. Draw CE, perpendicular to AB; and, having bife6led AB, on F (the point of bifedlion) with the Radius A F, deferibe the Ark A E, cuting C E in E, C E is the Diftance required, of the Picture, or of the Vanifhing Line, A B. For, AC:CE::CE:CB; confequently, A EB is a Right Angle; by 7. 6. El. f PROBLEM XIII, 1 31 To draw a Line to a Vanlftiing Point which is not on the Picture,* its Diftance from the Center being given, or fome Line tending to the Point; the Vanifhing Line of the Plane it is in, being alfo given. Let A B be the given Vanifhing Line, and C its Center. ' Let I be the Interfefling Point of fome Line ; or the Reprefentatlon of any other pj Point in the Line, given or found. r It is required to draw a Right Line, from the Point I, tending to B, which is out of the Piflure ; by the known Diftance of the Vanifhing Point B, from C, the Center of the Vanifhing Line. D b is the limits of the Piflure. . * By the Vanifhing Point not being on the Pifture is meant, only, that it does not fall within its pre- feribed Bounds or Limits. But the PiAure may be imagined to be produced, as occafton require, fo that, the Vanifhing Points are always fuppofed to be on the Plane of the PiAwre. • ,r: ' Take ■ X bO 6 * 3 * Plate X. Fig- 51* No. I. No. 2, X 2 . 6. El. § Cor. to 6. 6. El. No. 3&4. tHE ELEMENTS Book III. Take C F equal half, a third, or any other equal part of C B. Draw C I, and F J parallel to C I. Make F J : C I : : F B : C B ; i. e. make J F equal half, two thirds, or the fame Complement of IC, as F B is of C B. Draw I J which will tend to the Point B. Or ; tiiake C It to C I, as C F is to C B ; i. e. if C F be a third part of C B, make C K one third of C 1 . Join K F, and draw I J parallel to K F. DEM. Pirlli In the T riangle C B I, becaufc J F is parallel to I C, C F : F B : : I J ; J B. - 2. 6. FI. Confequcntly, CF : CF + FB (equal CB) : : I J : I J +IB (equal IB) i.e. as JF is to 1C. 2. and 4.6. Therefore, IjB is a Right Line; for C B I is a right lined Triangle. and. In the Triangle C B 1 , becaufe CK:KI;;CF:FB (Con.) KF is parallel to I B. Q_E.D, 2ndly ; by a Right Line tending to the Point B ; the Diftance not being known. L'^t 1 J be a Line tending to the Point B, in the Vanifblng Line, AB. It is required to draw,, from the Point a, a Line tending to the fame Point. Draw aC and aT, at pleafure, making an Angle laC, cuting the Vanifhing Line in any Point (C) and the given Line at I. Draw IC. Take any Point ("K) in IC, and draw KF parallel to 1 J, cuting the Vanifhing Line in F. Draw K L parallel to la, and join LF. A Right Line, a b, drawn through a, parallel to L F, will tend to the Point B. Note. The Point Is given at a and <7, both without and between the given Lines ; and the proceft would be the fame if the Point was given on the other Side of the Vanithing Line ; or any other Line, inftead of the Vanifhing Line, which tended to the fame Vanifhing Point. DEM. In the Triangles I aC, I a C, becaufe KL is parallel to la, C L : L a : : C K : K I. 2. 6. El But, C K : K I : : C F : F B ; for K F' is parallel to I B, by Conftruftion. Confequcntly, CL:La::CF :FB; therefore, L F is parallel to a B. E. D. Or, the fame thing may be done in another manner. Let A B and C D be two given Lines ; E is a Point given, through which it is re- quired to draw a Line tending to that Point, in which A B and CD would interfedt. Draw A C, at pleafure (through E) and B D, parallel to A C, at difcretion. Draw either Diagonal, as A D ; draw E F parallel to C D, and F G to A B, cuting BD in G ; a Right Line, drawn through E and G, will tend to the fame Point with AB and C D, DEM. In the Triangle ADC, becaufe E F is parallel to CD, the Sides AC and AD, are cut propor- tionally, in E and Ff ; i. e. AE^EC : : AF : FD; and becaufe F G is par. to A H, AF : FD: ;BG : GD; wherefore, BG:GD::AE;EC; and conftquently, AB, EG, and C D will tend to the fame Point. 2ndly. When A falls without the two Lines. Becaufe £F is parallel to CD, ACrCA:: ADcD/*; and becaufe FG is parallel to AB, BD ;DG: ; AD iD/*, i.e. as AC is to CE-, therefore, &c. Hence it appears, that nothing more is required than to find a fourth Propor- tional. For, having drawn a Right Line (AC) at pleafure, through the given Point, E, cuting any two Lines which tend to the fame Vanifhing Point ; and, at a proper Diftance, another (BD) parallel to the former (the farther off the better) BD (or LM) being divided, in G, as A C is divided in E ; then, if a Right Line be drawn through the Points E and G (or g) it mufl: neceffarily tend to the fame Point ; for Right Lines, proceeding from the fame Point, cut parallel Lines pro- portionally §. This method is the moft eligible, becaufe it is the readieft. There are various ways of performing this ; of which, thofe by Brook Taylor are very ingenious, and do not require any parallel Lines to be drawn. Let AB and CD be two given Lines, tending to one Vanifhing Point (P.) It Is required to draw a Line, through E, which fhall reprefent a Line parallel to the Originals of A B and CD; and conlequently, it muft tend to the fame Va- nifhing Point ; feeing they are not parallel. 7 Firfl, :Sea. IV. PRACTICAL PERSPECTIVE. m Firfl, when the Point (E) is between the two Lines. Through the Point E draw two Lines, at pleafure, cutingthetwo given Llnes> No. 3. iin A, G. C, and F ; and, through A, C, and F, G, draw two Lines, meeting at a. Draw a H and a B, at pleafure, cuting the given Lines in I, H, B, and D ; join BI and DH, in ter feeing at K. A Right Line drawn through E and K will tend to thefame Point, with AB &CD. 2nd. When the Point E is fituated without the two given Lines, A B and C D. No. Draw EA and £Fat pleafure, cuting the two Lines, BF and Cl, in C, A, F, 'and G ; join AG and CF, interfeding at a. Draw any other Line, BA", indefinite ; and, at D, where it cuts C/, draw DH, through a ; alfo, draw B/, through a, Laflly, through H and I, draw HK, cuting BD produced, at K. A Right Line, drawn through E and K, will reprefent a Line parallel to the Originals of AFzi\d CF; and confeqiaently it will tend to the fame Van. Point, P^ Although Dr. Taylor has given Demonftration of all the Problems, previous to this, he has not favoured us with a Demonflration of it. But a little confideratioa will make it very obvious, on infpedion of the Figure, In No. 3, becaufe A B and C D have the fame Vanifhing Point, they reprefent parallel Lines ; and if a be confidercd as theVaniftiine Point of AC, F G, Sec. they, confequently, reprefent parallel Lines. Wherefore, AFGC and IHBD reprefent Parallelograms; and confequently, E and K, the Inter- ^ feftions of their Diagonals AG, GF, &c. reprefent the Centers of thofe Parallelograms f. f l6. .. And fincethe Parallelograms are between the fame Parallels, therr Centers are, confequently, equally diftant from each ; therefore, EK reprefents a Line parallel to the Originals of AB and CD.^ In No. 4 ; becaufe B Fand Cl reprefent parallel Lines, the Lines j^G, CF, Sec. which pafs through the Point a are all cut proportionally in that Point ; wherefore, the Originals of B H and / D, of A F ‘J-^.B.Gor. t., and CG are in the fame Ratio refpeftively, i. e. the Originals of E H:£> I: : AF’.CG\ cojifequently, 2. 6. M, £, A\ EC -. : EF\ EG, u e. as KH'. Kl\ for, KH:KI : : HQ -. ID-, i. e. as AF\ CG\ viz. as £// is to EC. Wherefore, fince EA\EC\ KH’.K I, confequently, EC\ C A :i.K I : IH; or, as an^y other Line drawn from AT, whofe < riginal is parallel to EA, would be cut, by BF and Cl. But, if KH be fuppofed parallel to EA, and AH to Cl, CyfFf/ reprefents a Parallelogram ; where- fore JH \s equal Cy/.; confequently, IK reprefents a Line equal to FC; and confequently, £K repre- fents a Line parallel to Cl. For, EC IK alfo reprefents a Parallelogram, 5. If it be required to draw feveral Lines to a Vanifhing Point which is out of the Pidlure, from various Points, in a given Line, Let A, B, and C, be three Points in the Line, AC; let DE and FG be two No. 5^. Lines which tend to the fame Vanifhing Point, P. At any Diftance, at difcretion, draw HL parallel to AC, and draw AH, DX, BK, and CL parallel to either Line (as FG) cuting HL, in the Points H, I, K, & L. Afllrme any Point (G) in the fame Line, FG, and draw GI, cuting DE in E; and, through E, draw ac parallel to AC and HL j laftly, draw G H, G K, and G L, cuting a c, in the Points a, b, and c. Then, if Right Lines A a, Bb, and Cc are drawn, they will tend to the Lame Point, P. For, confidering the Originals of AC and HL to be parallel to the PitSIurc, and FG as the Vanifh- ing Line of fome Plane; becaufe AH, DI, &c. are parallel fo FG, and Ha, IE, Sec. have the fame Vanifhing Point (G) they are all in parallel Planes, of which F G is the Vanifhing Line; and, becaufe ac is parallel to AC and HL, the Original of ac is parallel to their Originals, and coBfequentiy may be in the fame Plane with either (Ax. 5.) wherefore, A a, DE, Sec. reprefent parallel Lines, in the fame Plane, AacC, or in parallel Planes AHa, DIE, 6cc. of which F G is confidered as the Vanifli- ing Line; and confequently, they vanifh in the fame Point in that Line. There are various other Expedients might be given In this Setflion ; but, I (hall omit them, till they occafionally occur in the Work; when they will be more in- telligible and better underftood ; and, being immediately applicable, in Praftice, they will, at the fame time that we acquire them, (hew their ufe ; by which ;means, they will be deeper rooted in the Mind, and the application of them, in •future Examples, will be more familiar. L 1 SECTION SECT ION V, * 54 - Of the P’RACTICE of PERSPECTIVE, refpeaing PLANE FIGURES. I J 'N the lafl Seclion, I have exemplified and Illuftrated the Elements of Praflical Prrfpeftive, according to Brook Taylor 4 which, notwithllanding they are fo excellent in themfelves, and of great utility in Praaice, yet they do not lay a foundation whereon to begin; but teach, only, how to proportion one Part by another, either given or found, on the Picture; fo that, a Novice, in thefe matters, cannot poffibly apply them tojrealufe. Nor, in my Opinion, would any Perfbn ever be made a Pra6titioner, from that Treatife, unlels he was en- dowed with an extraordinary Talent, and a very comprehenfive Capacity; being quite converfant in Geometry, and particularly acquainted with the Doctrine of Proportion ; having a clear Idea of Planes and their Interfedlions with each other, and of Right Lines cut by Planes. Before 1 proceed to Figures, 1 have (hewn in three Problems, how to find the perfpedtive Reprefentations of Right Lines, in the three Pofitions, parallel, per- pendicular, and inclimd to the Pidlure ; having the Interfedtion and Vanifhing Line of the Plane they are in (their Diflance from the Picture being known and the Pofition of the inclined Line; that is, the Angle of its inclination to the Pic- ture, or to the Interfecllon, and its interfedling Point, or the leat of fome Point in it ; or the Reprefentation of fome Point in the Line.) How to proportion ^ , them, that is, to cut oft fuch Portions, or Parts, as are the true perfpedlive Repre- fentations .of certain Parts in the Originals, (in which the whole foundation of Pradlical Perfpedfive confifts) is contaiiK*d in the lafl Sedtion (Prob. 8th, 9th and roth) and are exemplified in this. Prob. 6lh (hews how to find the reprefentation of a Point on tlie Pidlure, any how fituated; its Seat on the Pidlure and its Difiance from its Seat being given; which, in reality, contains the whole; as it will be found hereafter. I (hall, in the next Problem, give a more familiar and introdudlory Leflbn, how to find the leprefentation of a Point, fituated on the Ground Plane; which is, undoubtedly, the firft Plane to be confidered. At the fame time, let it be obferved, that there is not the lead: difference, in the Operation, between the Ground Plane and any other, whole Inteiicdlion and Vanilhing Line is given, and its Diflance known ; as it wall be (hewn. Having, therefore, in the preceding, and in the next four. Problems, given the Elements of the whole, and fully demonftrated it ; I (hall not trouble the Reader •with the Demonflration of every Operation, in the following Work ; but only in particular Cafes, which may not readily be deduced from the Lregoing; as it would •only fw'ell the Work to an enormous bulk, and would be of no ufe to the Prabli- tioner. Therefore, where I fee occafion, I (hall refer to the elementary Problems, or Theorems, for Demonfiration, and to (Iiew how^ each particular Problem is ap- plicable in Pradice, in various Cafes, in the cour(e of the Work. Let it, here, again, be obferved, that whenever any Point, Line, or Figure, is given in the Original Plane (the Interfedion of the Plane being allb given) it is luppofed to be fo fituated on the other fide of the Interfedion, as it would be, if .the Original Plane was turned over, on the Interfedion, to the other Side of the Pidure, making the fame Angle with it; and (if it be not perpendicular) inclined -towards the lame Part. See Fig, 37 ; the Triangle X Y Z, on the other Side of the Pidure, is inverted, .on this Side, tor couvenicncy in Pradice. PROBLEM j Seft. V. PRACTICAL PERSPECTI VE, &c. C Oj PROBLEM XIV. To find the Reprefentation of a Point, fltuate on the Ground Plane, or other horizontal Plane^', its real Place being given thereon ; the Interfe, and Va- nhhing Line, ECG, is given, or found ; whether the Plane it is in be perpen- dicular to the Piflure, or inclined, having the Center and Diftance of the Vanifti- ing Line, or of the Picture, the Procefs is the fame. a and b are the Seats of G and H, and EH or Ec are Vifual Rays ; the reft: is obvious, on infpedtion of the Figure. But, if the Original Plane be inclined to the Pidure, a and b are not the Seats of G and H PROBLEM XVIL To find the Reprefentation of a Line, any how inclined to the Pic- ture, fituated in a Plane perpendicular to the Pidurej having the Interfedion and Vanilhing Line given. Let AB be an Original Line in the Ground Plane, whofe Interfedion is LI, and Vanilhing Line, VC; C is the Center of the Pidure. Draw C E, perpendicular to the Vanilhing Line, and equal to the Diftance of the Pidure. Produce B A, to its interfeding Point, I ; and draw E V, parallel to AB, cuting the Vanilhing Line in V, the Vanilhing Point of AB. Draw IV, the indefinite Reprefentation of A B ; and the Vifual Rays, A E, BE, being drawn, will cut it in a and b, the Reprefentations of the two Extremes, A andB. ab is the finite perfpedive Reprefentation of A B. Or, the Seats of the extreme Points being found, by drawing A d and B D, perpendicular to the Interfedion ; d C and D C, being drawn, projed the fame Points a and b, as before. Or, make V E equal to V E, 1 G equal I A, and I F equal to I B, Draw G E and F E, cuting I V in the fame Points, a and b. DEM. Sea. V. ■ APPLIED TO PLANE FIGURES. * 3 ? DEM. Becaiifc E V is a Right Line, from the Eye, parallel to the Original Line, and cuts the Pi^lute in V, V is the Vanifhing Point of the Line ABf ; and I being its interfering ^'oint, IV is its in- t Dcf. 22. definite Reprefentation;!; ; a and b are the Intcrferions of the Vifual Rays, with the Pidlure, as jThco. I2, proved in the former Problems. Therefore, a b is the Reprefentation of A B. G E and F E alfo projedl the fame Points, a and b. For V £' is parallel to FI, and equal to EV, the Dlftance of the Vanifliing Point; alfo, IG is equal to I A, and IF to 1 B ; by Conftru£lion. ' Wherefore, if ££■ and HF be drawn, they will be parallel ; for theTrianglc-s, EV E and BIF, are . fimilar irofceles; and lince IF, IB are refpeclively parallel ro E V and V^", conf. BF is paralL! to EE. Therefore, E is the Vanilhing Point of BF; and conf. F b ; bE: : Bb:bE, i.e. as Ib : b V, &c. The laft Method of proportioning inclined Lines will be found the moft conve- nient, in Pradice, of all other; and it may alfo be obferved, that if the Reprefen- tation, a, of any Point, A, in the Original Line, be given or found, and the In- terfeFB. For, BE is the indefinite Reprefentatlon of the Diagonal, BD, £ being its VanKhing Point; for E£ is parallel to BD, (C £ being equal to C E, and AD to AB) and DF being parallel to the Pic- ture, its Reprefentation, d f, is conftquently parallel to the Interfedion, (Prob. ib.) 2. If another fquare be required, draw fE, cuting AC in g, and draw gh parallel to df. By which Expedient any length of A C may be obtained. If the length AG had been required, equal twice AD ; AG on the Interfedion being made equal twice A B, and GE drawn, gives the fame Point g ; as before- Or, if CE be bifeded, at K, BK is the indefinite reprefentation of the Diagonal BG ; for EK is parallel to B G ; the Angle CEK is equal GBH, and, K is the Vanifhing Point of that Diagonal; draw BKj cuting A C in g, as before. 3. If the Square be at fomc Diflance from the Pidure, as ABED, produce the two perpendicular Sides to the Interfedion, cuting it in a and b ; alfo, produce the Diagonal FAlol, its interfeding Point ; or, make al equal 2. A, and proceed as in the former Cafe. The Figure explains the reft. COR. Hence, a Pavement of Squares, having their Sides parallel to the Pidure, may be delineated, with great facility. Let A B he the Ground Line, and E C D the Horizontal Line. Take the geometrical meafure of a Square, and apply it, on the Interfedion, A B, as often as it is required, from A to B; as a, b, &c. From each Divlfion, draw Right Lines to the Center, AC, aC, &c. and draw a Diagonal, from A or B, to the Eye, at E or D, cuting each indefinite Repre- fentation, in the Points h, c, &c. through which, draw the Lines F G, HI, &c. parallel to the Interfedion, or Vanifhing Line, and the feveral Figures X, Y, Z, &c. are the Reprefentations of Squares on the Pidure; which may be repeated, by drawing other Diagonals to any length or width- N n Now, PlateXII. 57 - No. I. *42 Plate XII Pi^- 5^- PRACTICAL PERSPECTIVE Book III. 9 Now, if one of thefe Squares (as Y or Z) be confidered fingly, it has fcarce the appearance of a Square, leaving no other, contiguous to it, to biafs the judgment ; but, by the affinity of the whole, the Eye (being accufiomed to fee Objeds as they appear, in all fituations) is not offended, and readily gives the affent ; al- though it is certainly capable of determining, that the feveral Reprefentations of Squares, are not Squares, but have the' appearance, only, of Squares, in certain Pofitions and Situations. PROBLEM XX. To find the Reprefentatlon of a Square, or other RetRangle, whofe Sides are all equally inclined to the Pidure. « Firft, by the Original Figure being geometrically drawn, in its determined Po- fition to the Picture j the Interfedion of the Plane it is in being given. Let A B C D be a Square, the Sides of which are equally inclined to the Pidure. C is the Center of the Vanilhing Line^jEE, or of the Pidure j £, £, are the tranfpofed places of the Eye, to the Vanifhing Line, viz. C E equal GE. As one Angle of the Square (A) touches the Interfedion, it is, confequently, the Interleding Point of the two Sides A B and A D. Let the other two Sides be produced to the Interfedion, at F and G. Draw AE, both ways; alfb, draw FEandGE, diagonal ways, cuting each other. Abed is the Reprefentation of A BCD. For the Point A being in the Pi£lure, is its own Reprefentation, AE, AE are Indefinite Reprefen- tations of AB and AD; and FA and G A, of FC and CG (Theo. 12) confequently, they cut each other in the reprefentations of the feveral Angles, B, C, and D. (Cor. 7. Th. 12.) If the Redangle H I K L be at fome Diftance from the Pidure, the Sides being produced to their interfeding Points, a, c, d, and the indefinite Reprefenta- tions, aE,-hR, &c. being drawn (as in the Figure) give the Reprefentation hikl of that Redangle ; its Sides being parallel to the Sides of the Square. E parallel to the Diagonal, I L, produces'its Vanifhing Point. Note. In this Cafe, it may be obferved, that there is no neceflity for the Eye, i. e. the Diftance of the Pidure or Vanifhing Line, being placed above it, but placed equally on either Side of its Center, C, in the Vani(bing‘Line, asCA. Squares, or other Rectangles (wherever they are fituated in the Original Plane) having the fame Pofition to the Pidure, have their Sides parallel, and confequently, they ^...have the fame V'’anifhing Points. N. B. The Diagonals of the Square are, in this Cafe, the one (B D) parallel, and the other (AC) per- pendicular to the Pidure ; confequently, their Reprefentations arc either parallel, as b d, or vani& in its Center, as A c. Let it be, here alfo, particularly noticed, that A, being the place of the Fye, in the Vanifhing Line (commonly called the Point of Diftance) is the Vanifhing Point of the Diagonal of a Square, whofe Sides are parallel and perpendicular to the Pidure. - Cdnfequently, i( the Diagonals are parallel and perpendicular to the Pidure, they are the Vanifhing Polntsof its Sides. .By which Points, all Lines perpendicular to the Pidure are proportioned, perfpedively ; as it may V be obferved in the preceding Problems, 2hd. How to find the Reprefentation of a Square, in this Pofition, not having the Figure drawn out geometrically, only its meafure and place known. , Let A be the InterfetSling Point of an Angle of the Square, fituate on the left ffde of the Station Line; at the Diftance A J. Make AF and AG each equal to the Diagonal, and proceed as before. , . Or, by the meafure of its Sides. Make ■Sea. V. APPLIED TO PLANE FIGUPES. ' Make £F equal to the Diagonal of a Square, whofe Side is CE ; i. e. make EF ♦equal to the Dilhince of the Vanilhing Point E, from the Eye, equal £E. Make Af equal to AB and draw fF, cuting the indefinite Reprefentation AE iin b, the Reprefentation of the Angle B. Draw bd parallel to FG, the Interfe^tioH, cuting AF in d; and, laftly, draw b E and d F, diagonal ways, cuting each other, which compleats the Figure. ‘►COR. Hence, a Pavement of Squares, diagonal- ways, may be delineated. Let AB be the Ground Line and ED the Horizontal vanilhing Line. Fig. 58. No. K Make Aa, ab, &c. equal to the Diagonal of the Square; make CE, and CD •each equal to the Diftance of the Piflure (C being the Center.) Draw AE, aE, he. and AD, aD, &c. cuting each other in the reprefentations •of Squares, placed diagonal-ways. Tluough d, where AD and BEinterfedl, draw ef parallel to AE; and where it cuts the feveral Lines, drawn from a, b, c, &c. viz. in e, a, &c. draw ^D, and Ed, &c. contrary ways; by which means they may be continued at pleafure. PROBLEM XXL To find the Reprefentation of any Re6:angle, obliquely fituated to the Picture. Firfl:, by having the Original drawn in the Geometrical Plane, in its true place and pofition, in refpedl of the Picture and of the Eye. - Fig, 55* . J ABCD is the Reflangle to be delineated; FT is theinterfedtion of the Plane it is in; KL is the Vanilhing Line, and C its Center. The Diftance is known. , Make CE, equal to the Diftance, and perpendicular to KL. ^ , ,• Draw EK and EL, parallel to the Sides of the Redtangle, AB and BC, re- fpedtively, producing their Vanilhing Points, K and L. The Original being at fome Diftance from thePidture, produce every Side, DA, iCB, &c. to the Interl'edlion, cuting it in F, G, H, and I, their Interfedting Points. Draw the indefinite Reprefentations FL, GL, H K, and IK.; which, by their rfnutual Interfedlions, give the Reprefentation, abed, of ABCD.; Method 2nd: By the Seats of every Angle, on the Pi(fture, and their Dif- :tances from their Seats, refpedUvely. If the Original be in a Plane perpendicular to the Pidlurc, the Seats of all its /Angles are in the Interfedlion of the Plane they are in. Let <7, b, c, and d, be the Seats of the Angles A, B, C, and D,' refpedcively. ’Draw 3 C, &c. to the Center; make ab equal to the Diftance of the Point A, from its Seat, be equal to the Diftance of B, cd of C, and dj ot the Angle D. Make CE and CF each equal to the Diftance of the Pidlure, .and draw bE, eF, dF, and fE, cuting aC, bC, &c. in tlie Points a, b, c, and d, refpedlively ; which -are the reprefentations of the feveral Angles A, B, C, and D. (Pr. 14. MetJi. j.) Join the Points a and b, b aridc, .&c. as in the Figure ; and the Quadrilateral, abed, is the reprefentation of the Redlangle, ABCl 3 , fituated as in the Figure. .Note.' This Method, is ufed by all the old writers on PerfpeAlve; particularly in the Jefuits, In which procefs it may be'obferved, that there is no occalion for the Vanlfhing Points, K and L; for if they made ufe of Vanifhing Points, at all, they were found by producing the Sides' ba, be, &c. to the Vanifliing Line ; i. e. to the Horizontal Line, for they knew ivo other Vaniflaing Line. Which Vanifhing Points, fo produced, they called accidental Points* jMlTIIOD PRACTICAL perspective 144 Book III. Plate XII. Method 3rd. By the known Proportion of the Original ; its Pofition, Situa- Plg* t-O. Diftance being determined. The Original Figure, ABCD, is drawn out in the Geometrical Plane, to (hew how the meafures, &c. are applied, but it is, otherwife, of no ufe in the operation ; as the careful and accurate obl'erver will perceive. IN, the Interfetflion ; KL, the Vanifhing Line; C, its Center; and CE its Diftance, are given, the fame as before. ' Through E, draw HI, parallel to the Vanilhing Line ; draw EK and EL, making the Angles, HEK, and I EL, equaL to the known Inclination of the Sides of the Redlangle (AB and BC) to the Interfedlion, refpedlively ; i. e. make HEK equal to ABF, and ILL equal to CBG; FG being parallel to the Inter- 44.1. El. fedtion, the Angles ABF and CBG are equal to their inclination to itf. Make DB equal to JB; I. e. to the diftance of the nearell Angle (B) to the Station Line, and draw BC; alfo, make Bd equal to its diftance from the Inter- fedlion (equal BP) and, C E being made equal to CE, draw dE, cuting BC in b, the Reprefentation of the Angle B. (Prob. i4th.) D raw the Indefinite Reprefentation s, bK and bL, from the Point b. It remains, now, to cut off (from b) b a and be, in the perfpedlive Reprefenta- tions of A B and BC, the Originals. Make KE' equal to KE, and L£^ equal to LE; which Points are ufed for proportioning all Lines which vanifh in K and L, refpedlively. (Prob. 17.) Draw E'b, cuting the Interfedlion in b; make equal A B, and draw aE*f cuting bKin a, the Reprefentation of A. Alfo draw E*b cuting the Interfedllon in b'' and make c equal BC (the other Side, whofe vanifhing Point is L) and draw cE" cuting bL in c, the Reprefen- tatron of C, another Angle of the Figure. * From the Points a and c, draw aL and cK, Interfedling at d, which com- pleats the Figure ; for, the Originals of thofe Sides, being parallel to the Originals of a b and b c, they have, confequently, the fame vanifhing Points, refpedively. SCHOL. Th is Method, which is perfe£fly confonant to the new Principles by Brook Taylor, may ap- pear to fome Perfons more difficult than either of the other, particularly the firfl-, which is performed, alfo, entirely on the fame Principles; the difference is very obvious, notwithffanding the effedt is the fame, as it may be feen, by comparing the Figures ; or by comparing both, with the 17 th Problem, which contains all the various Methods of proportioning inclined Lines, in general. > In the firft, there is a neceflity for having the Original Figure drawn geometrically, either on the Pidlure or fomewhere apart ; for which there is not, frequently, room to fpare on the Pidlure ; and we are liable to errors, in transfering the interfefting Points; but, without the Original being placed in the very Pofition, we cannot draw the Radials, or Parallels from the Eye, producing the Va- nifhing Points ; as by that Method. Whereas, by the laft Method, the Angles, being known, are made equal to the Originals; in which Cafe, the Radials would be parallel to the original Lines being in their true places. Or, if there be not room on the Pidfure, or interfere with the Objedls, the Vanifhing Points and their Diftances, from the Center and ffom the Eye, are all determinable, by Problem 12th; the Vanifhing Points, KandL, and their Diftances KA* and LA* being afeertained by it. There is, likewife, no need for the Original figure on the Pidlure, but being any where drawn out, geometrically, or its meafures being known, and the Pofition determined, they are applied to the Pidlure with the greatell facility, for which, a little Pradlice wdll render it quite ffuniliar ; regarding always, carefully, to make ufe of the proper Points for particular Lines, and to dif- tinguifii between the Vanifhing Points and thofe winch are ufed, for Vifual Rays, i.e. for cuting indefinite Lines in the Ratio required, PROBLEM XXII. To find the Reprefentation of a regular Pentagon, one Side being parallel to the Pidlure. PI. XIII. A B CDF be a regular Pentagon, having one Side (AB) parallel to the In- Fig. 61. terfedlion (IK) confequently parallel to the Pidure. LM is the Vanifhing Line, and E the Eye. 7 Produce PlateXm. Q. fy’ XJfaltt'H Jfay a ^ \ ' / fn \a Irf'r 76 / f ya N. “NVy***^ / t X. ■/ < i\ \ N. / / A ' / ^s. / Xs. \ / 1 ~T~v — \ ' t \ / X J / \ / • j ' / ‘ y^ K^/ j t \ 1 C ’‘l\ b V''-X>. XT' ^ ^ Se£t. APPLIED TO PLANE FIGURES.. HS Produce the inclined Sides, F D, CD, A F, and BC, to the Interfe6lIon, eat- ing it in G, H, I, and K» Draw EL, EM, &c. refpedlively parallel to them, producing the VaniHiing^ Points, L, M, N, and O, of each Side. Draw the indehnite Reprefentations GL, HM, IN, and KO ; producing, by their Interfeftions, the Figure, ijed, of the Original FJCD, (No. i.) and IJdcK of IF DCK {Ho. •/.) fo that, the reprelentation of one Side (AB) is wanting, in each ; which, on account of its parallelifm to the Pl£lure, has neither Interlefl- jng nor Vanidaing Point -j*; and muft be found by Problem l6; or, by drawing a Vifual Ray, EA, curing the indefinite Reprefentation, IN, of the Side A F, in a; or, makeNP equal NE, and I^/ equal I A, and draw a?. (Prob. 17.) Or, by pro- ducing either Diagonal, AD or BD; being parallel, refpedively, to the Sides BC, and AF, O and N are their Vanifhing Points. Having, by any of tliefe Methods, found a, the Reprefentation of A, draw a b, parallel to the Interleclion, cuting KO in b; which compleats the Figure. t + Cor, tat TT iico.- 1©. j Secondly. How to determine all the Vanifliing Points in this Pofition, without the Original Figure on the Pidure. Through E, draw RS parallel to the Vanilhing Line; and make the Angles REL, LEO, &c. each of 36 Degrees ; i. e. divide the Ark of the Semicircle, R23S, into five equal Parts, and produce the Lines E i, E 2, &c. to the Vanilh- ing Line, cuting it in L, O, &c. the V anifhing Points of the Sides and Diagonals of a Pentagon, having one Side parallel to the Pidure. SCHOL. The reafon of this is obvious ; for three cf thofe Angles, Wz. REN, is equal to the Angle of a Pentagon; and fince one Side is parallel to the Picture, R E is parallel to it; and confequently, EN will be parallel to AF, another Side; the Angle REN being equal B AF. Alfo, the Angle which any two Sides, not contiguous (as A F andCB) make with each other,j^ viz. FJC, is equal to one of thofe Angles; wherefore, EN being parallel to one of thofe. Sides, EO is confequently parallel to the other. (By 6. l. El.) Alfo to the Diagonals AD andBD.^ This Is fully explained, after Problem nth, in the 4th Book of Elements. To find the Reprefentation when every Side is inclined to the Piilure (the Original being In its truc- Place) has nothing particular ; for, if every Side be produced to its interfedling Point, and their Vanilh- ing Points are found by drawing parallels to every Side, from the Eye, the indefinite Reprefentations being drawn, produce the Reprefentation of the Original; as in the laft Problem, or Prob. 18. If the Vanifhing Point of any Side be found, all the reft are determined as before ;-by producing the Radial of that Side and deferibing a Semicircle ; and dividing it into five equal Parts, as in this Figure. PROBLEM XXIII. ‘ How to find the Reprefentation of a Pentagon, having only one Side given, in its true Place and Pofition, in the Geometrical Plane. Let A B be the given Side. It is required to find the Reprefentation of a regular Pentagon, whofe Side is equal to AB; and inclined to the Picture in the Angle BDG, or to the Interfec- tion GD. KM is the Vanifhing Line, C is its Center, and CE its Diftance. Through E, draw EH parallel to the Vanilhing Line, and make the Angle •HEN equal to BDG, the given Angle, producing the Vanifhing Point I. Produce BA to D ; draw the Indefinite Reprefentation DI ; and find the finite part a b, the reprefentation of A B (by Prob. 17)!. e. draw the Vifual Rays AE and BE; or, make IP equal IE ; and, DF, DG, equal to DA, and DB, refpec- tivelyj draw FP and GP, cuting DI in a and b. Produce IE, and defenbe a Semicircle N^zcO; divide the xArk inta five equal Parts, at a, c, and^; draw E^, \Lb^ &c. to the Vanilhing Line, producing; the Vanifhing Points, K, L, &c, of the remaining Sides of the Poligon. Having obtained a b, the Reprefentation of AB, as above; drawaMand bL- the indefinite Reprefentations of two Sides, from tliol'e Points; for,, laM repre- fents an Angle (lEM) equal to the Angle of a Pentagon ; and IbL reprefents an. Angle equal to lEL; which is equal to an external Angle of a Pentagon. O o Dra\^ 14^ FI. XIIL Fig. 62. Fig. 63. No. 2. PRACTICAL PERSPECTIVE. Book III. DravvaK, cuting bL in c; be is another Side of the Pentagon; for, ac re- prefents a Diagonal (which, in a Pentagon, is parallel to the oppolite Sidej whole Vanilhing Point is K. Draw 1 f, through c, cuting aM in f; cf reprefents a Diagonal parallel to the Side a b ; and confequently, a f reprefents another Side. Laftly; draw fK, andbM cuting it in d, and draw cd, w'hich compleats the Figure, abedf, required. The Vanilhing Point which Is out of the Pitlure) of the Side cd, and its parallel Diagonal b f, has not been wanted in this procefs. Nf and Ob are Diagonals, EN, Ec, orEO and Eii, being fuppofed Sides of a Pentagon; and EQ, EK are Parallels to them, from the Eye, producing their Vanifhing Points. CSeeProb. lO.) Note. If ab or be, or any Side in the Reprefentation had been given, the whole Figure may be com- pleted, as above, without the Interfedion. ' PROBLEM XXIV. To find the Reprefentation of a regular Hexagon, fituated any how to the Pidure. There is no Figure whatever, except a Square, eafier to deferibe than a Flexagon. To have the Original Figure drawn in the Geometrical Plane, it is evident, is the fame as has already been explained, in the foregoing ; I fliall, therefore, beg leave to pafs over that defeription. Let A B be a Side, given, of a regular Hexagon ; the Original of which, is pa- rallel to the Pidure. IK is the Fanifhing Line, Make lEK an equilateral Triangle, whofe Perpendicular is CE; as follows. Through E draw HL parallel to the Vanifhing Line, and, on E, delcribe a Se- micircle; which, divide into three equal Arks, at a and b. Draw E^ and Ei^, and produce them to the Vanilliing Line, cuting it in I and K j which, are all the Vanifhing Points required for this Pofition. Draw AI and BK; make Ab and Ba each equal to A B, in AB produced, both ways; and draw a I and bK. Draw BI and AK, or AC and BC, cuting them in F and G, and join FG, which is parallel to AB, and compleats the Figure. ADFGCB. SCHOL. A regular Hexagon having an even number of Sides, the oppofite ones are confequently pa- rallel; and the Diagonal, which pafTes through the Center, is alfo parallel to them ; andnecaufeAB is parallel to the Vanifliing Line, its Original is parallel to the Pifture; and confequently, the op- pofite Side F G is alfo parallel to AB ; and, becaufe they are equal, AF and BG are parallel, and confequently their Originals were perpendicular to the interfeefion; therefore they vanifli in the Center of the Vanifhing Line. (Cor. to Theorem 1 1.) The two Sides, A D and CG, and the Diagonal, B F, reprefent parallel Lines; confequently they have the fame Vanifhing Point, I; alfo BC, AG, andDF; all which may be feen in the Hexagon ABCGAZ), which may be fuppofed the Original Figure, one Side A B, being in the Picture. By drawing the Diagonals AG, BG and GD, the Hexagon is divided into fix equilateral Triangles, AOB, BOG, &c. and lEK has its three Sides refpeftively parallel to them; co'nfequently, lince one Side in each Triangle is parallel to [the Picture, and has no Vanifiring Point (Theo. i.) £I and EK arc parallel to the other two Sides of each Triangle (which are equallyinclined) and confequently, to the Sides AZ), CG, and BG, DFj of the Hexagon ; I and K are, therefore, their Vanifhing Points. COR. Hence, a Pavement of regular Hexagons, or equilateral Triangles, maybe eafily delineated. Let AB be the Interfedion of the Pi6lure, or the Ground Line of the Pavement; and let A a be the determined meafure of one Side of a Hexagon. Makeab, be, &c. equal to A a, as often as there isoccafion*. If C be the Center of the Picture (its Diftance being known) find the Vanifhing Points, D and E, as I and K above. Draw St^. V. A P P L 11' n TO PLANE FIGURES. M 7 Draw AD, aD, bD, &c. iihcl aE, bP', See. and, where they interfe'd, draw Lines parallel to the Interleflion, as fg, hi, &c. which divide the whole into the rcprelentatlons of equilateral Triangles j fix of which, around the fame Point o (as a^.■^^’/b) form a Hexagon; and tlius, as many may bs deferibed as are required; for which, the inlpedtion of the Figure is fufficieht. Cafe fecond. When the given Side is inclined to the Pi£lure, in any Angle. Let AB be the Side given, DP' is the Interfeiflion, and GL the Vanifhing Line Fig. 64. ©f the Plane it is in ; E is the Eye. Findab, the reprefentation ofAB, by any of the former Problems, its Vanifh- ing Point is G ; draw GE, and pioduce it ; on E delcribe a Semicircle, abed. Make ab and be each equal a E, and draw Eb, Ec, cuting the Vanifhing Line, in H and I the Vanifhing Points of the other Sides. Drawal and bH, the indefinite Reprefentations of two Sides. For, baf reprefents the Angle GEI, which is the Angle of a Hexagon ; equal aE^ + ^Ec, twice 60 Degrees ; and abc reprefents the Angle HE ? - f ■ ; (p. r' o B L E M',' XXVI. r 'I - - ' ■ ■ ' ^ r.' The Interfeciion of a vertical Plane being' given, and its Inclination to the PiOiure, to find the Reprefentation of any Figure in the Plane, • . . - - ’f Fig. 66. Let AB .be the given Interfedlion, and C the Center of the PitSlure, Through C, draw SD perpendicular to the Interfedion, aiid'CE perpendicular to SD, i. e. parallel to A B, and equal to the Dilfance of the Pifture. Make CES equal to the Complernent of the Angle of Inclination, cuting CS in S; and, through S, draw MN parallel to AB ; MN is the Vanilhing Line, Sis its Center, and ES is its Dilfance. (See Prob. 3 rd.) ' If it be required to find the Reprefentation of a Square, whofe Side is parallel to the Pi6lure, and equal to AB; at the diflance from the Interfedlion. Draw A Sand BS ; make SE‘ &’S£^ equal SE, and draw ^E‘, cuting AS in F. Draw F E\ cuting BS in H ; and draw F G and HI, parallel to AB. Then, FGHI reptefents a Square, in that Plane, at the diftance from the Interfedion of the Plane of the Original. ' 2 If Sea. V, APPLIED TO PLANE FIGURES, 149 If another Square be required, draw another Diagonal HE' or I Eh Or, if a greater length than a Square be required, make SO to SE (equal SE) as one Side of the Redangle, required, is to the other, and draw HO, cutting AS in K, and draw KL. For £ 0 , being drawn, is parallel to the Original of H K. See this illuftrated by moveable Planes. (Pig- 37, No. i.) Turn up the Plane AFDC, (on the other Side of the Picture) at right angles with the Ground Plane, RS is its interfedion with the Pidure, and A B, with the Ground Plane. AB and FD, being perpendicular to the Pidure, vanifh in its Center, and, if the Vertical Plane, V, be turned up, perpendicular to the Pidure, £Q_ will be parallel to the Diagonals, AEand BD, and produce their Vanllhing Point ; and, Q, on the other Side of the Center, C (in the Vertical Line) equally diftant from C, is the Vanifliing Point of the other Diagonals, FB and EC. The reft is obvious ,; if the Redangles AFEB and BEDC were Squares, their Diagonals would va- ^nifti in E ; CE being equal to the Diftance of the Pidure. Second. Let ab be the given Side of a regular Heptagon (in the Plane AIHB) parallel to the Horizon ; its Vanishing Point is S ; it is required to delcribe the Heptagon. SE is perpendicular to the Vanifhing Line MN. Having made SE equal SE, make the Angles SEM, SEN each equal to the external Angle of a Heptagon, and bifedt them, by the line EO and EP ; alfo, make the Angles MEQ, N E R, equal MEO, &c. i. e. having deferibed a I'emi- circle, on E, divide the Semi- circumference into feven equal parts, and make ab, be, and cd, each equal one feventh part. EQ and ER, being produced to the Vani(hing Line, MN, will generate two Vanilhing Points, equally diftant from S ; which, with M, O, S, P, and N are the Vanilhing Points of the Sides and Diagonals; none of them being, in this Cafe, parallel to the Pidlure. DrawaM; and Nc, through b. Indefinite ; draw Pc, through a, cuting N c in c, and draw cS, cuting aM at d. be and ad are Sides of the Heptagon, whofe Va- nifhing Points are M and N. Draw cQ, and Rd indefinite (by Prob. 13.) and, through a, draw Ne, cuting cQate; and draw eS, cuting the oppofite Side, at f ; and, laftly, draw eO and Pg, through f, cuting eO at g, which compleats the Figure afeeb, the reprefen- tation of a regular Heptagon, whofe Sides are all inclined to the Pidure. PROBLEM XXVIL To find the Reprefentation of any irregular Figure (entirely on the Principles of Brook Taylor) from the knowui dimenfions of the Figure, and the Angles which every Side makes with the adjoining Side or Diagonal ; the Place, Pofition, Situation, and Diftance of the Figure, in refpecd of the Pidlure and of the Station Line, being given ; in a Plane which is inclined to the Pidure, in a given Angle, the Center and Diftance of the Pidure being given, and the Inter- fedion of the Plane of the Figure. Note. The Original Figure {A.B D FG) being geometrically drawn, in the Original Plane, is of no other ufe (in the following Operation) than to determine its Situation and Polition ; otherwife it is not ' necelTary. '1‘he Place of the meareft Angle (A) and the Inclination of a contiguous Side (A B, or AG) in refpeft of the luterfedlion, being all that is wanted; the reft are determinable, in refpeft of themfelves ; as it will be exemplified. C is the Center of the Pidure, H/ is the luterfedion of the Plane of the Ori- ginal Figure, and X is the Angle of its Inclination to the Pidure. P ^ A is Fig. 67. 150 PRACTICAL PERSPECTIVE Book III. PL XIV. Fig. 67. f Prob. 2. Jj; Prob. 4. § Prob. 6 . I Dcf. 22. A is tlie place of the neareft Angle of the Figure, and JAB is the Angle of the inclination of the Side A B, to the Interfedion ; A B is the Diftance of the Angle A from the InterfetRion, and A e is its Diflance fiom the Station Line. Through C, draw EC perpendicular to the Interfe£lion, which is the Vertical Line of the Original Plane of the Figure. (Def. D.) Draw CE' perpendicular to EC, and equal to the Diflance of the Pidlnre. Make the Angle CE'C equal to Z (the Complement of X) cuting EC in C; and, through C, draw DG, the Vanifhing Line of the Plane of the Figure; C is its Center, and CE ‘ its Diflance. (Def. rg and 20.) CE, on the Vertical Line, being made equal to CE’ ; through E, drawERpa- rallel to the Vanifhing Line ; the Parallel of the Eye of that Plane. Draw ED.; making the Angle R ED, equal JAB ( 1 . e. to BKH, the Inclina- tion of the Side AB to the lnterfe6lion) cuting the Vanifhing Line in D ; the Va- nifhing Point of ABt> snd make the Angle DEG equal BAG, cuting the Vanifh- ing Line in G, the Vanifhing Point of AG Alfo, make the Angle DEH, equal to ABH, the Supplement of ABD; and H is the Vanifhing Point of BD; by the fame. Make the Angle GEI equal AG/, the Supplement of AGF; cuting the Va- nifliing Line in 1 , the Vanilhing Point of FG. Laftly, make the Angle lEF equal to DFG ; EF is the Radial of FD ; which, if produced, would cut the Vanifliing Line, in the Vanifhing Point ofFD. Thus, are the Vanifliing Points D, G, H, and I, produced, by the known In- clination of one Side to the Interfedlion, and the Angles which each Side makes wdth the adjoining Side, (by Prob. 4.^ The place of the Angle A being given in the Geometrical Plane (which is in- clined to the Pidlure, in the Angle E'CC; and to the Horizon, in the Angle CE'C) find its reprefentatioa on the Pidlure (Prob. 14.) Draw AB, perpendicular to HI, and draw BC. Make BL equal to BA, and CE equal to CE', and draw LE, cuting BCin a, the reprefentation of the original Point A. Or, draw BA parallel to CE', and make BA equal BA ; produce AB, and draw JlS parallel to HI, or perpendicular to BS; S is the Seat of the point A, on the Pjdlure; and SA its diflance from its Seat. Dra\<^ SC, and y/E' cuting it in a, the Reprefentation of A§, as before. Draw aD and uG the indefinite Reprefentations of AB and AG, from the Point a ; which, if produced, would pals through their luterfedling Points, iiTSc L. Make D £' equal DE ; and, through a, drawB'^ to the Interfedlion, cuting it in a. Make a b equal AB, and draw ^£‘, cuting aD in b ; ab reprefents a Line equal to AB (Prob. 7, or 17.) Draw bH. Produce DE, and make EK and EL refpedlively equal to AB and BD; or in the ratio of AB to BD ; and draw EM parallel to KL, cuting the Vanifhing Line, in M, the Vanifliing Point of the Diagonal AD. Draw a M. cuting b H in d. b d reprefents a Line in the ratio to that which a b re- prefents, as EK to EL, i. e. as AB to BD. (Prob. 10*.) For, EK is parallel to A B, and EL to B D ; wherefore KL, confequently EM, is parallel to the Dia- gonal AD; therefore, M is its vanilhing Point j] ; and abd reprefenis the Triangle A 5 1>. (Pr. 18.) Now, F, the Vanifhing Point of FD (being much inclined) is not in the PicSlure. Draw d /at pleafure, cuting the Vanifhing Line, and EF, in e and/; and draw gi parallel tod/ (at any diflance, at difcretion) and make hi to g h, as ed is toe/ * From the Pofition of AT? and BD, the meafure of A5, or its Ratio to BD, is taken beyond the Eye, in the Radial of AB, poduced j as in the third Cafe, of that Problem. C-g- 6 Se£l. V. APPLIED TO PLANE FIGURES. e. g. Draw the Diagonal fh 'y and draw d^ parallel to the Vanilhing Line, cutingy7», produced, in k \ draw /’/, parallel to FE, cuting gi in /; hih a fourth Proportional to / w \ t t' J 4 < • ..-1 K \ -r Jv Sea. VI. APPLIED TO PLANE SOLIDS. *55 If another, of equal dimenfions, be required, fituated on one fide of the Station Line, at the Diftance LJ (as ABE H) and difrant from the PidureL^, equal LA, IVlake LM equal Ali; and, on LM, defcrlbe the Square LMNO, as before. Draw OC, NC, &c. and draw aE, cuting LC in a, and MG in e. Draw ab parallel to LM ; and ad, bc^ toMN ; alio, draw cd parallel to ab. i. e. having drawn ab parallel to LM, deferibe a Square onab; and, ate, delcribe the Square e fgh, parallel to abed, which compleats the Figure. Or, having drawn LC, MC, and aE, compleat the Plan afeb. (Prob, jq.) Delcribe a Square on a b, and draw cG and dC ; draw ig parallel to ad, and gh parallel to cd, which compleats the F'igure. D EM. aEcc] bt-ino; parallel to the PicSture (by Hvpotbefis) is fimilar to the Original, Of ; and its Sides have that proportion to the Originals, as the Diftance of the Picture (CE) to th.- Diftanceof the Plane abed ; viz. as E C + L z/ ; i. e. a b (or a d) : LM : ; EC : E'J + LA. (Thco. g.) But, LMNO is a Square ; therefore, abed is a Square. And, fince it is parallel to the Piiffure, tlic Originals of the Sides a f , d g, &c. are perpendicular to thePifture ; confequently they vanifh in its Center, Ct. Aiih. afgd icprefents a Square in a vertical Plane, perpendicular to the PiiSlure ; for LO is the Inrerfeflion of that Plane and ACL' its Vanifhine Line j| ; wherefore, C£, being made equal to CE, £ is the Vauifhing Point of Lines (in that Plane) inclined to the Interfedlion in half a Right Angle; confequently, if one Side (ad) be parallel to the laterfeifion (L O) and O P being made equal La, P £ palfes through a Diagonal of that Side f, therefore, afgd reprefents a Squa.^e, ; and Ob being e^ual L a, E is a Diagonal of the Top, d g g h, which alfo reprefents a Square, parallel to afeb, and ON is the Interfeflioii of that Plane, with the Pi£lure. Therefore, age reprefents a Cube, having three Faces feen, each of which, is the reprefentation of a Square. Qg E. D. for the oppofite Faces are alfo Squares. PCL's Intcrfeclion of a Diagonal Plane cdfe, wbofe Vanifhing Line would pafs through £. The Cube AGHB being fituated on the Station Line (which, on the Pidture, coincides with the Vertical Line) has but two Faces feen ; the FrontAP'DB, and Top, F G H D, the Eye being above ir. The other Faces, AFGfC and BDHi, feeing they apparently incline towards the Vertical Line, -are confequently loll to fight in this Polidon ; as is obvious, fuppofing the Sol id tr.anfparenc. EXAMPLE I. To draw the Reprefentation of a high Wall, the End being parallel to the Picture, and ' at fame D fance from iL This Objecl, being a right angled Parallelopiped, is delineated as the Cube, Let PJ be equal to the Diftance of the Wall from the Station Line, on the Left. Make PQ equal to its thicknefs, and draw PC and QJ^ ; make Vd equal to the didance of the end of the Wall, from the Pidure ; draw dE cutjug P C in R, and draw R S parallel to P Q. Now, fince P is the interfeding Point of RF, the common Interfedion of Plane of the Wall (which is fuppofed vertical) with the Ground ; PT, perpendi- cular to theGround Line, orparallel to the Vertical, is the Interfedion of that Plane-l-. And, fince the Wall is perpendicular to the Horizon, confequently, itb Angles or Corners, R U and S V, are parallel to PT or Q X the Interfedions of the Pkmes thole Lines are in (for they are parallel to the Pidure, being vertical) therefore, draw RU and S V parallel toPT, i. e. to the Vertical Line. Make PT equal to the known height, and draw TC, cuting RU In U, and draw V U parallel to RS ; then RU V S, reprefents the End of the Wall, which is parallel to the Pidure (on the Ground Line). Laflly. Make PM (on the Ground Line) equal to the Diffance of the fartl.e:’ end of the Wall from the Pidure, or ^/M equal to its known length, D * u »v* + Cor. 5. '1 hco. g. f Cor. td Theo. 4. J Thco. 10. a Th. g& 10. t See N.Bi Prob* 20, Fig, 71. 4 156 ,PRACTICAL P'E R.SPECTl VE Book iir. Plate 'XV. Draw EM, cutlifg PC, the indefinite Reprefentation of its common Sedlon Fie; -o with the Ground Plane, in F ; and draw FY, parallel to R U, which compleats ‘ the Wall. ^ Or, if C F (on the Vertical Line) be made, to CE or E, as the height of the Wall to its Length, .F will be the Vanilhing Point of the Diagonal, HY, which cuts 1 C, in the Point Y, for its length j draw F Y parallel to RU. Let it be obferved, here, that if there be not room enough, on the InterfeSion, to fet off the whole ' length of the Wall, equal rfM, take half the Diftance, PJ (equal Ff, in the Plan below) make C e equal half C £ (the Dihanceof the Picture) and draw Jc, projecting the fame Poii.t F. 1 If the half meafure be too much, take a third, or fourth, or any equal portion, of PM (equal Pm) the .fame part being taken of C E will anfwer the fame purpofe (fee Prob. i j.) E X A M P L E II. How to reprefent feveral Parallelopipeds, as Bloch of Stone, ^c. for the Bafement of a fuilding, ranged in a Right Line, perpendicular to the Pidlure. Pig. 71. W, X, Y, and Z, are the Plans, or Seats of the Blocks on the Ground Plane. Let ABD, the Front of the firP, be clofe to the Pidure, i. e. in the fame Plane ; confcquently, in its geometrical Proportion, of height and width. Draw AC, BC, and DC. Make ha, on the Interfedion, equal to the length of the full, ab equal to the fpace between them, and be equal to the fecond. Draw <2 F, /^E, and c E, cuting AC in F, G, and H ; draw FI, GK, and HL perpendicular, cuting BC; andlM, KN, and L O, paralLl to B D, cuting DC, in M, N, and O, and compleat them, as in the Figure; having regard to the firlt hiding part of the next, and that of the third, &c.. Now, in order to continue them further, to any length, this Expedient maybe ufed, when the meafures exceed the limits of the Pidure, and, confequently, can- not be applied to the Inteiiedion, or Ground Line. D raw H g, from the fartheft corner of the fecond Parallelopiped parallel to the Interfedion, and draw <2 C and C, cuting it in e and g. If it be required to continue them of the fame dimenfions, and fpace, make Hf equal eg j or, hd being made equal to a b, draw ^C, cuting gH, at f, and draw fE and g £, cuting A C in a and b. Compleat the Parallelopiped ab cd as in the Figure. After the fame manner they may be repeated as often as you pleafe by drawing bg parallel to g H, cuting dL, and bO, \x\ f and g, and then proceed as before. This Expedient is, in many cafes, better than applying the half- meafures, or other portion, on the Ground Line; as they may be continued infinitely, within the compafs cf Kb (one Block and Space) by drawing G h (inftead of g H) which, being cut by a C, at i, needs not to be tranfpofed, as e to f. It muff be obvious, that Gh (or gH) is cut in the fame proportion to the fecond Block (at the hither end, cr the farther) as is to the firft ; and confequently, i E, hE, &c. will cut AC in the fame Points, as if the full meafures were applied on the Ground Line; and fo of ^b, to the next. EXAM- Sea. V7<. APPLIED TO PLA:NE- SOLIDS* 157 EXAMPLE III. To. draw fever al long Parallelopipeds^ parallel amongjl 'themfelves^ and perpendicular to the Pidlure ; reprefenting large Joijls, fupporting a rough Floor ( over HeaaJ, Let W, X, Y, and Z be the ends of the Joifts, of equal dimenfions, and equally Fig. 72. fpaced ; and, fi nee they are fuppofed to be parallel to the Pi6lure, therefore, they have their geonmetrieal proportion on the Picture. (Cor. 5, Th. 9.) Let them be drawn accordingly, by a Scale of equal Parts. Then, becaufe the Joifts are perpendicular to the Picture, in refpe^l of their length; (i. e. they are horizontal, and at right angles with the Piflure) draw aC and bC, &c. from every Angle,, neceffary, as in the Figure.; and determine their length, by Prob. 15, i. e. make AB, orA B, equal to the required length, and draw BE, curing A C in D ; or in D. Through D, draw F G parallel to the Vanifhing Line (C E) which determines the under Sides, P'e, &c. and the perpendicular Lines, de, fee. compleat them. The divifions of the Boards are determined by Prob. 8th ; i. e. take any mea- fure, A a, and repeat it, on AB, as often as there is occafion, if the Boards are fup- poled equal ; otherwife A a, ab, &c. muft be made in the lame ratio as the Boards, to each other ; and at the laft divilion, e, draw eP), curing the Vanilhing Line in H;‘ and draw <2 H, bW, &c. curing AD n\ i, 2, 3, &c; through which, draw Right Lines parallel to AB, i. e. to the Vanilhing Line. Or, divide AB into the fame number of Parts (as Ai, 12) and draw Lines to E,. which will produce the fame, as is obvious ; feeing that it does not depend on the dimenfions of the Parts, but on their number and ratio to each ether. (See Prob. 8.^ Note. A I, 12, on A B, being each equal to a fifth Part of JB, is the geometrical width of the Boards,,^ in proportion to the ejidsof the Joifts, and to Ac, the fpace between them., PROBLEM XXX, To draw the Reprefentation of a Cube, or other right angled Paral- lelopiped, any how inclined to the Pidlure. Firft. Let the Square ABCD be the Plan of a Cube, in the Geometrical T j* Plane, equally inclined to the Pidlure, and to the Interledlion, at the Angle A. Lie down the Diftance of the Pidure, CE from the Center, as ufual; E and E are the Vanilhing Points of all the Lines of its Sides. (Prob. 20.) Draw AE, both ways, A being the interfedling Point of two Sides, AB and AD. Find the perfpedlive PlanAj^CD (by Prob. lo) or of the two Sides, AB, AD, only. Draw AF perpendicular to the Ground Line, and equal to AB, a Side of the Cube, and draw F E, both ways. Draw B G and D H, parallel to A F ; and draw GE and H E diagonal-w’ays, cut- ing each other, at I, which compleats the Cube, AGH. Otherwife, without drawing the perfpedtive Plan, by means of the Vanilhing Lines of its Faces. Draw AF perpendicular to the Ground Line. Then, AF is the Interfeflion of twe) Faces of the Cube. Make A F equal to a Side, AB. Draw AE and FE, both ways; and through E, on either fide, draw the Va- nilhing Line of the contiguous Face of the Cube. (Prob. 3.) EE, or EF. Make E E or EE, on either Side, equal EE', and draw A E or A E, cuting the indefinite Reprefentation FE, in G or H, and draw EG, or D H, perpendicular. Draw B D parallel to the Interfeclion, and D H parallel to AF ; and laftly , draw G E and FI E, interfering at I, as before. R r AEGHD 15 ^ Plale XV. Pig- 7i* f Theo. 9. Cor. 1. Fig. 74. PRACTICAL PERSPECTIVE. Book IIL ABG HD is the Reprefentation of a Cube, in the pofition required ; fltuated on. a horizontal Plane, and its vertical Faces equally inclined to the Picture. DEM. For, ABCD the given Plan is a Square, wherefore ABCD reprefents a Square (Prob. 20) and FG I H alfo reprei'ents a Square, in a Plane parallel to A SCi). (Th. 3. and Cor. 1.) And, they are between the fame Parallels, AF, BG, Ac. therefore equal - - - (18. i. El.) But, AFisthe Interfeftion of the vertical Planes, A AG F andA_Z)HF; and E A is the Vanifhing Line of A A GF (Theo. 2 and 10) E is its Center (Def. 19) and EE ‘ its Diftance (Cor. 2. 7.) Wherefore, E is the Vanifhing Point of Lines inclined to the Interfeflion of that Plane, in half a-! Right Angle or 45 Degrees; confequentiy, it is the Vanifliing Point of the Diagonal of a Square, in that Plane, whofe Sides AF and AG are parallel to thePtdlure (Prob. 19 ;} and, AA and FG, perpen- dicular to them, vanifh in E, the Center of the Vanifhing Line, EA. - - - (Cor. to Th. ii.) I'herefore, AFGA reprefents a Square, at right Angles with ABCD and A F HZ), which (after the fame manner) may alfo be proved to be the reprefentation of a Square ; and their oppofites, D¥L\G and CIGA, being between the fame parallel Planes, are alfo reprefentations of Squares ; confequentiy,-. A A GHZ) reprefents a Cube. (See Defin. Page 41.) Ch E. D. ac fg reprefents another Cube of the fame dimenfions, and in the fame pofition to the Picture ; whole fituation in refpe£l of the former, is as the Plan, W, toBD;. which, on account of the Ihort Diftance (CE) nas a diftorted appearance,, being remote from the Station Line, though wholly within the Radius of the Diflance. By means of the Interfection {^A F) of the Face a d f g, or a f of the Face abed (y^F, or af^ being made equal to the Side of the Cube) the proportion of the pa- rallel Lines (ad andfg, or be) in thofe Planes are determined. For, AFy ad, and fg; alfo af, ad, and be are in the fame Plane, and are all parallel to the Pidture ; confequentiy, they are parallel between themfelves, and to their Originalsf. EXAMPLE IV. How to draw Parallelopipods ranged in a Right Line ^ and inclined to ihePiSiure obliquely^ Let X and Z be the Plans of right angled Parallclopipeds ; their inclination to the Pidlure is the Angle AlA, which they make with the Ground Line, CK. C being the Center of the Pidlure, and CF its Diflance, find the Vanifhing Points, G and H, of the Sides and Ends of the Parallelograms, X and Z, (Prob. 2, and 4.) and lie down their Difiances on tlie Horizontal Lme (Pr. 1 2.) at£ ’ andFL Transfer all the mealures, I A, IB, occ. of the Ohjecls, and the fpaces between them, to the Interfedlion, at A^ B, and C; and, having drawn the indefinite Repre- fentation, IG, of the whole length, draw AE \ BE \ &c. cuting it in a, b, and c. Becaufe the Angle I touches the Pidlure, IF, perpendicular to the Ground Line, is the Interfedlion of the Plane of the Fronts, and aho of the End, I K. Make ID equal to the height of the Objeds, and draw DG, DH, and IH. Make \K equal IK, and draw K E^ cuting IH ; Ik reprefents IK. (Pr. 17.) Draw ad, be, and c f, parallel to I D ; and, where they cut D G, In d, e, and f, draw d H, e H, and f H ; and from i, where k i cuts DH, draw iG, cuting them in j, h, and g, which compleats the upper faces. h 1 , parallel to ID, &c. cuting kG, compleats the End behL If another Objed be required, in the fame Line, feeing that the meafures on-- the Ground Line, would exceed the limits of that Pidure, take GF ' half GF h and draw F c till It cuts the Interfedion, at a. Make aB equal to half the fpace between the Objeds, and Be equal half the width of the next Objed, and draw B and c F % cuting I G in ^ and c. If it be required higher than the reft, make IF equal to its known height, and' draw F G. Draw perpendiculars from b and Cy cuting F G in ^ and F Draw Sea. VI. APPLIED TO PLANE SOLIDS. 159. Draw cuting kG at and draw j and dl parallel to bg^ which com- pleats that Object. Or, if c n be drawn parallel to the Ground Line, and BG cuting it in n ; di- vide cn, at in, as AC (equal AC) is divided in B ; and, from m and n,. draw to cuting IG in the fame Points, b and c, as before. By either of thefe Expedients they may be continued to any length. Note. G, the Vanifliing Point of the front Line IC, and the parallels to it (h-ing much inclined; may be fuppofed to be beyond the limits of the Pifture ; confequently, Ic, D f, 6cc. are tending to a Va- rifhing Point which is noton the Pifture, and muft, therefore, be drawn by means of the Expedients,, in the 13th Problem; which are very convenient when but few Lines are wanting. But, a better Expedient than them all is (when the Point is not very remote) to fix a Lath to the Drawing Board, or ftraining Frame of the Picture, and continue the Vanifliing Line as far as is re- quifite, on the Lath j fix a Pin in the Vanifning Point, and, with a long Ruler, draw the Lines 1 G, DG, &c. for no other Expedient, whatever, can be fo true, i. e. it cannot be performed fo accurately as to have the Vanifliing Point itfelf. For large Work (as in Scenery) a fine, flnooth Cord, fixed in the Vanifliing Point, is a good Expedient. Prifms of all kinds may be reprefented after the manner of the foregoing, which are right angled Prifms, having found the Plans of their Bafes, on the Ground Plane or other Plane, whether they be triangular, quadrangular, or multangular ; of which one or two Examples will be fufficient. EXAMPLE V. FI. XVi To reprejent a hexagonal Prifm^ perpendicular to the Ground Plane. The Situation, Diftance, and Pofitlon being determined, or AB being a Side, Fig- 75 *^ given or found, in Perfpedlive, whofe Radial is E V, find the Vanifhing Points, I and K, of the other Sides, and compleat the Plan, ABCDEF, of itsBafe (byPr. 24.) Draw A a, Bb, &c. perpendicular to the Ground Plane, i. e. to its Vanifhing Line (IK) or Interfedlion (GD.) Produce any Side, as AB, to its interfedting Point, (G) and draw G H, perpen- dicular to the Ground Line. G H is the Interfeclion of the Plane of the Face AabB, orghofBbcC (Pr. 3) for it is vertical ; and G, or g, is the Interfedfing Point of one Line in the Piane§, §Th.2.C.i. Make GH equal to the known height of the Prifm, and draw H V -f* cuting A a t Prob,.i3.. and B b in a and b ; the Original of a b being parallel to the Original of AB. Then, becaufe A a and B b are the common fedtions of the contiguous Faces with AabB, draw al and bK, cuting the other Perpendiculars F f and C c, in f and c ; which compleat as many Faces as can be feen. The otherFaces, which are all Parallelograms (SeeDef. Page4i) F e, Ed, and dC are fuppofed to be feen through, the Objeef being imagined tranfparent ; which, in fome Cafes, is a neceflaiy Expedi- ent ; by which means, the connedlion of the feveral Parts are more accurately determined. EXAMPLE VI. To reprefent a pentangular Prifm, laid along on the Ground Plane ; the Interfebiion and the Seat of the Objebl, on the Ground Plane, being given. AB is the given Side of the Pentagon, and BD the Seat of one Face of the Prifm. Fig. 76. C being the Center of the Pldfure, and CE its Diftance, find the Va.nifhing Points, 1 and K, of AB and AD (Prob. 2) by making the Angles JE I and O EK 2 equal, PI. XVI. Fig. 76. 7 T- PRACTICAL PERSPECTIVE' Book ni- equal, refpeclively, to and DAD ; or, having determined one, fikdt the. other, by Prob. 4. making the Angle lEK equal BAD, a Right Angle. Make a b to reprefent AB (Prob. 17.) and, on ab, conftrudl a Pentagon, in- a vertical' Plane, on the lame Principles as in Prob. 23. I being the Vaniihing Point of ab; FG, paffing through I, is the Vanifhing-. Line of the Plane of the end of the Prifm (being perpendicular to the Ground Plane) I is its Center, and El its Diflance. (Prob. 3.) As there is not room, on the Pidlure, beyond the Vanifhing Line, FG,. to find- the Vanifhing Points of the Sides of the Pentagon, take IE on this Side equal to- IE, through which draw MN (the parallel of the Eye, of that Plane) parallel to^ F G, and, on E delcribe a Semicircle ; which, divide into five equal parts. Make be, and^?^, de, each equal to a fifth part,, and draw E b^ Ec, &c. which, produce to the Vanifiiing Line, producing the Vanifhing Points, F,G,&H, by which, the Pentagon ab ede is completed, as in the Figure (by Prob. 23.) Draw aK indefinite, of the Side A D ; and make af to reprefent a length equal to AD (Prob. 17) alfo, draw eK and dK. Draw H g, through f, and g F, cuting dK in h, which compleats the Figure. For, afg e and eg h d reprefent Parallelograms, which are Faces of the Prifm afhdh; and abede reprefents an End which is a regular Pentagon ; to which the other Faces are perpendicular, lEK being a Right Angle (which laK repre- fents, Prob. 4) equal BAD. s The Expedient of turning over the Vanifhing Plane, F £ H, to find the Vanifhing Points of the Side of the Pentagon, is a very ufeful and neceffary Expedient; becaufe it often happens, that there is not room for turning it over on the other Side of the Vanifhing Line, if it be remote from the Center. And, notwithftanding it interferes with the Obje£t in this Diagram, yet being done firfi, a-nd the Vanifhing Points found (as in the Figure) all the operat.ve Lines (for finding th. m) are rubed out before we begin to draw the Objeft; or rather, they are never drawn, for, the Vanifliing Points are determined (asF.) by applying a ftreight Ruler to E and b, &c. In Fig. 37, the horizontal Vanifhing Plane, N/.^L, or the vertical Plane V, being turned over on their Interfedlons with the Pl£lure (which are the Vanifhing Lines they produce) it is obvious that the Angles, formed by the Radials, EN, EO, &c. are the fame on both Sides, and confequently, they pro- duce the fame Vanifhing Points, N, M, &c. whether the Vanifhing Plane be turned up or down, to the Right or to the Left. EXAMPLE VIL To reprefent a quadrangular or hexagonal Pyramid or any other\ Let W and Z be the Plans of the Bafes of two Pyramids, whofe Altitudes are known, the one a, Iquare Bafe, the other a Hexagon ; two Sides, of which, are pa- rallel to two Sides ot the Square. Find the perfpeSlive Plan ABCD, of the Bafe, W (by Prob. 21.) If the Original Figure be drawn in the Ground Plane, the fhorteR way is by pro- ducing its Sides to the Intcrfe 61 ;ion, AK ; and having found the Vanifhing Points, M and N, of the Sides, draw their indefinite Reprefentations (Method ift.) Draw the Diagonals, AC and B D, interfeding at H, the Center of the Bafe. D raw HI perpendicular j and where the Diagonal AC cuts the Interfedion, at A, d raw AF, allb perpendicular to the Ground Line. Make AF equal to the height of the Pyramid, and draw FG, cuting the Per- pendicular HI at I, and draw A I, BI, and DI, which compleat the Figure. If (as in this Example) on account of the Diagonal, AC, having very little in- .clination to the Ground Line, the Altitude cannot be determined with accuracy; and the other Diagonal, BD, is fo much inclined as not to cut the Interfedion within the Pidure ; it may be thus determined. From A • v:. . ',4 ■ -1^ * - ' ' i. «» • ✓ , j K* /' SeCt. VI^ A PPL I EXT TO PL.A'NE SOLIDS; From either Vanlfhing Point, M or N, draw a*Right Line througli the Center of the Bale, till it cuts the Ground Line, at* a ; draw a f parallel and equal to AP\ and draw fN, cuting HI in the fame Point, I, for the Vertex of the Pyramid, as before. Or, as in the following.. Second. Having found the perfpedlive Plan ABCDFG, of the hexagonal Bafe, Z' (Prob. 24)*M and O being two Vanifhing Points ; the other is out of the Pi£lure. From any Point, P, in the Vanifhing Line, draw P H, cnting the Ground Line, in K. Draw K L perpendicular to the Ground, Line, and HI parallel to KL. Make KL equal to the known Altitude of that Pyramid, and draw LP, cuting HI in I ; and draw A I, BI„ &,c. as before, which compleat that Obje6t. Prifms and Pyramids are very ufeful Subjefts for prafilcal LelTorrs. By the firfl-, we learn to form P.i- villions, Temples, Cupolas, Alcove-;, Sec. which are, generally, of fume prilmatic Form ; by the other we may form a SpirCj Obelifk, or Pyramid, of any Figure. As thofe Obj.efts admit of infinite variety (as nuny as there may be vario.us Poligons) it would be to little purpole to give more Examples of them, feeing that, if a Poligon of any number of Sides be drawn, in Perfpecfivc, for the Bafe (by the various Problems of the laft Section) the reft is the fame as in the Examples given. It may, ' I prefume, have been obferved, how inconvenient it would be to have the Ground Plans of all the Objects, to be delineated, geometrically drawn, in the Geometrical Plane; befides, the impeftibility of having room for them, when the Objects extend to any confiderable Diftance ; as the Plans, W, X, Y, and Z, of the Par.dielopipeds in the fecond Example (Plate XV.) evince. It may alfo be obferved,. that they are of no other ufe than to know how the Objects arefituated, in re- fpedt both of the Pi£ture and of each other; but-fince it is obvious, from what has been done, that they are not abfolutely ncceflary to be drawn, feeing that, if their meafures are applied to the InterferSlion of the Plane they are in, it will anfwer the purpofe ; therefore I fhal', in future Examples, where they are not abfolutely necelTary, do without them. Yet it mufl not be undeiftond, that we can do entirely with- out, which is impoftible ; but, in many Cafes, their Meafures and Diftances, only, are requifite. In others, the Situation ani Pofition of one Object to another, being irregular, require a true and correft Plan to be firft drawn (but noton the i ifture) f.om which all the Vanifhing Points may be afeertained j as every Line, ia Perfpective, is determined from its geometrical Proportion, and the pofition of the Ob*- je£t to the Pidure; confequently, they muft be known or imagined, otherwife we have no foundation to* build on. EXAMPLE VIII. 7 * 0 reprefent Steps with Kirhs at the ends ; as at the entrance into a Houfe, Gfev Firfl, when they are parallel to the PL£lure; and the Pidure clofe to them. Fig. 78 Is a Sedion of the Steps, in their geometrical proportion, with the In- elinatloa of the Kirbs to the Horizon, and to the Pidure. Take AB, on the Ground Line, equal to the width, or more properly, to the length of the Steps; and, AD and BJ to the thicknefs of the Kirbs. . 79 "- Make AF equal to their height, and deferibe the Redangles AFGD, and BHl J, geometrically ; which are fuppofed to be in the Pidure. C is the Center of the Pidure, and CE the Diflance. Draw** AC and BC. Make equal to the Diflance of the firfl Step from the Front of the Kirb ; and a b, be., and c d, each equal to the breadth of a Step, and draw ^ E, E, &c. cuting BC in a, b, c, &c. the perfpedive breadths of the Steps, on the Ground. Make Be, ef, &c. each eq'ual to the height of a Step, and draw eC, fC, &c. cuting the Perpendiculars, from a, b, c, &c. at I, m*, &.c. and transfer them, by the parallel Lines a f , bg, &c. on the Ground, to the other end, cuting AC at f, g, h, &c. from which Points, draw the Perpendiculars fr, gs, &c. and, from 1 , m, &c. draw Lines parallel to AB, cuting them at r, s, &c. draw r C, sC, &c. > cuting them again, at s, t. See. and the parallel Lines /r, ms, &c. at the internal Angles, being drawn, complca? the Stops. S3 The liSz FI. XVI. fig. 79. -f Thco..9. Fig. 80, PRACTICAL PERSPECTIVE Book III. The upper Step having a greater breadth than the other, make equal to its breadth, and draw DE cuting BC at e j the Perpendiculars d o and ep cut >6 C in o and p ; and o u, p v, being drawn parallel to the Ground Line, cut the corref- ponding Perpendiculars i u and k v, which determine it ; viz. o p v u. If the Kirbs are fquare, they are right angled Parallelopipeds, and are deline- ated, as defcribed in Problem the 29th. AFGDand BHIJ are the Fronts,. A F V k and BH p e are the Sides, which inclofe the Steps ; and, F vwG and H p q I are their upper Faces. If they incline with the Steps (as is common) make the Re6langles AK and' B L equal to the Ends ; fuppole, equal to the height of a Step. • Draw CC perpendicular to the Horizontal Line; and, CE being equal to the- diftance of the Pifture, make the Angle CEC equal to the Inclination of the Steps,, cuting CC at C, which is the vanidaing Point of their Inclination. Draw KC, LC, See. enting G G, I C, &c. in x y, &c. which compleat the in'* dined Faces of the Kirbs, as in the Figure ; by drawing x u and o y. In this Example, the front of each Step, being parallel to the Piiflure, are fimilar to each other ; an(J have that Proportion to each other, re;pectively, as their feveral Diftances, i. e. the fecond Step, is, in length and height, to the fit (t, as C E istoCE + aZi; and, the fecond istothefirft, asCE is toCE-f- a c. Sic. t V L, palling through C parallel to the Horizontal Line, is the Vanilhing Line of a Plane inclined to the Horizon, in the Angle CE C, equal ECL, and C is its Center (fee big. i^. No, 3, P. 72.) For C C is perpendicular to V L ; and E C producing the point C, is the Radial of fuch Lines in the in- tllnedPlane as are perpendicular t6its Interfeflion KL;. therefore,. C' is their Vanilhing Point.(Cor.Th. ii.f Case 2nd. E X A M P L E IX. PFhen the Steps are inclined to the Plciure ; the hcllnation being determined, alfo their Situation, in rejpedl of the Station dne^ In this Example I fhall fuppofe, that there is not room on the Pl£l.ure for afeertaining the Vanilhing Points and their Diftances, as ufual ; in order to apply the iith Problem. C being the Center of the Pidure, and KCL the Horizontal Line, drawCE perpendicular, aid take CE any equal part (fuppofe one third part) of the deter- mined Didance of the Pidure. Make the Angle CED equal to the Inclination known; and CD will be one third part of the didance of the Vanilhing Point L, of one Side ; which being de- termined, all the reft are determinable, arithmetically (by Prob. 12.) Or, make D EF a Right Angle ; C F will be a third part of CK and FE will be a third part' of its D. dance, by the fame. Sec. Having thus obtained the Vanidiing Points, K and- L, and their Didances E ^ and jE * being laid down (CE ‘ being equal to three times the difference between F G and F E ; and C E ^ to three times the difference between C D and D E) then, pro- ceed as ufual; AE being the Interaction of the inward Angle of the Kirb. Draw AK, the Indefinite Reprefentation of the Front, and A Lof the End. Let AJ be the length of the Steps (eq. A B, Fig 79) draw J£' cuting AK at .eS. ^ake A a equal to the didance of the fird Step ; ab, and b c. Sec. each equal to the breadth of a Step, and draw &c. cutting it in a, b, &c. and, feeing there is not room on the Interfedtion Ac, to apply the meafure of the top Step, which, fuppofe equal to Ac, draw djf parallel to Ab, andi'L cuting it in f; and draw / E cuting A L at B. Draw the Perpendiculars a j, bf, Sec. indefinite. OnAE, takeAE, B C, A:c. each equal to the height of a Step, and draw EL, CL, Sec. cuting the Perpendiculars from a, b, &c, as in the Figure. Draw Sea. vr. APPLIED TO PLANE SOLID 3.' 1.^3 D raw aK, and cuting it In k; draw kl perpendicular, andjIC cnting- it jn 1 ; draw IL, and cK curing it in m, and fK cnting the Perpendicular, ra a at n, and proceed, after the lame manner, thioughout. Or, having drawn all the Lines eK, fK, &cc. indefinite, draw IL; and m iv. perpendicuLir to the Ground Line; then n L, and op, &c. by proceeding after that manner the Steps are completed. The Kirb being equal to B T (as in the former Figure) draw BE', cuting AK at D i j^D reprefeiits the thicknels of the Kirb, at that end. The Original of the Point, G, being fuppofed on this fide of the Pidfure, and being in the Ground Plane, the Reppefentation (G) i< conrequently below the Ground Line, AB. (SeeN.B. Pron S Fig. 45. The ineafurc being applied on this fide, i,e. below AB, as A H, fuppofes AB proportioned perlp^dti 'tly ; but, being at the other Extreme, as aB, it is projtdled ; and is, conlequently, laiger than the Original.) Make Ad eq;uar to BJ, and draw E^d, cuting KA produced, in G; AG Is the projedlive reprefentatioii of the end of the Kirb, the Original, of which, is equal to A d, equal B J. Draw the Perpendiculars GH, y^F, iiud DG, indefinite; and, AE being made equal to the height of the Kirb, draw EK, cuting them in E, and G; and, being produced, it cuts G H at H.. EL cuts a Perpendicular from B, at I ; and, if EL, GL, and HL be drawn, KI, cuting them, compleats the upper Faces, EJ and EI/6H, Draw gb per- pendicular, cuting GL, at^, which compleats the End. If the Kirbs are inclined with the Steps, as in the former Cafe, draw LV, perpen- dicular to KL ; which is the Vanilhing Line of the Planes AEIB, ^EJ, &c. Make L£' equal to its Diftance, and draw E'V, making the Angle LE^V" equal to the inclination of the Steps, cuting L V in V, the Vanilhing Point of the inclined Sides*. KV is the Vanifhing Line of the Inclination. (Theo. lo. Cor. i.jb AB being madeequal to the height of the Front, draw KB, cuting DG, and GH, in K, H, B, and E; from which Points, draav KV , HV, &c. cuting the horizontal Lines GL, HL, &c. in s, r, t, and u *, through which, draw Ku, or join the Points rs, tu, only. Note. The Pi£lure might, with equal propriety, have been fuppofed on this fide of the Kirb, entirely;. Out, as it frequently happens, in PraAice (the Pifture being fixed inadvertently) that fome parts of the Object would project on tliisfide; I thought it necelTary to give an Example how to proceed in that Cafe. SCKOL. In this Example, it is obvious that, the Plane A £ 18 , in which, the Steps are perfpedively proportioned, being fartlier removed from the Vanifhing Line, V H, is fitter fqr the purpofe than GHL; and, if the Original of GH was in the Pklure, the meafures on AE would be lefs thaa the full meafure of the Steps, feeing it would be beyond the Pidlure. ABob is the Seellon made by the Piflure, with the inclined Kirb, and fiiews how much is fuppoted to projedl on this fide. Ab is the feftion of the Bottom, confequently, parallel to the Horizontal Line ; ab is the feftion of the End, which is vertical, therefore parallel to V L its Vanilhing Line and Bo is theSeftion with the inclined part, confequently, parallel to the Vanifliing Line, K V, oL the inclination of the Steps. (Theorem and.) '■* This will be clearly demonftrated and made manifefi, in the 12th and laft Sedlion of this Book. N. B. The Vanifhing Line (V W) of the Roof of the Objeft in the Apparatus, is deterrrrined in thii manner (as KV) and is in a fimilar Pofition, on the direft Pidlure, Ad NOP. V, the Vanifhing Point of HG, anfwcrs to K, in this Figure;, Y anfwers toL, and W to V. SECTION 164 Book III.- P R A G T I C A'L P E R S P E C T I V E Plate XV IL SECTION VIL- Gf the application of Perfpedive, to MOULDINGS, &c,. foregoing Se£tion contains many ufeful Leffons in plane Solids, which,’ f[ may be confidered as parts of Buildings, &c. In this, I intend to (hew how the more decorative parts are formed, fuch as PedeRals for Columns, Cornices and Entablatures, &c. of the various Orders ; without which embelliihments,. a Building feems naked and cleftitute of Ornament, I'hele Leilbns, of Steps, are very necelEiry to the art of delineating Mouldings ; which, when vve know how to manage well, particularly fuch Steps as return at the Ends, or on four Sides, forming mitre Angles, the greateft difficulty is fur- mounted. For Mouldings, breaking round-a Pedeftal, or the internal and external Angles of a Cornice, &c. properly conlidered, are but fo many Steps, one above or below another, of different dimenlions ; formed by the Fillets,, between the cylindrical parts, which are, properly, the Mouldings ; and are effedled only by Light and Shade. The Fillets, between them, are narrow Planes, cuting the curved Surfaces in parallel Lines ; which being defcribed, by the Rules given, and the mitre Angles of the Mouldings drawn, the bulinefs of the linear part is done, . and nothing remains but to give the appearance of folidity, convexity, and conca- vity, by a proper difpofition of Light and Shade. EXAM A L E XI. How to delineate fquare Steps, returning on every Side. Fig, 8iv Firfl, when they are parallel to the Pidlure. AB is their length, ABCD is half the geometrical Plan of the firfl Step j FGHI of the fecond, and KLM N of the third; and let OP be the meafureof a right angled Block, on the upper Step. C being the Center, draw AG and BC; and AE, which is a Diagonal, repre- fents AQ, in which are the Seats of G, L, and O, the Corners of the Steps. Compleat the Square ADC’B (Prob. 19) and draw the other Diagonal BD; which, it is evident,, tends to a Point on the left hand of the Center, equal to CE,. the Diftance of the Fldure. Produce P'G, KL, &c. to the Ground Fine, 6Uting it in g, /, &c. i. e. make- ' ^ fo breadth of the Sttps, and drawee, /C, &c. cuting the Diagonals at G, L, H, M, &g. the peripedive Seats, on the Ground* ' Plane, of the feveral corners of the Steps. Draw AF perpendicular to AB, and make A a, ai 5 , and ic, equal to the height of the Steps, as in the former Examples ; and, having drawn the Perpendiculars from G, L, &c. draw aE, bE, and cE, curing them at c and e, g and i. Draw ab, cd, ef, &c. parallel to A B, cuting Perpendiculars from B, H, &.C. at b, d, f, &c. which give the reprefentations of the Fronts, coirefpcnding with GH and LM. AabB, being in the Pidure, has 'its- full dimenlionSj in height and length ; the other tv.'oare in proportion to theirDiflances.- [laving obtained the F'ronts (by means of an imaginary Plane paffing through t Prob, ag. the Diagi nal AC, which AcFC reprefentb) finifh the Iquare of theTop, iklm-|-. '1 ben, liravv fC and bC, till they cut iikand df, which reprelent the return of rile Steps on that Sule, lo far as they can be ieen ; on the other Side they are leen to their full extent, the Point of view being on that Side,. 2 Draw Sea. VII. APPLIED TO SQUARE STEPS. Draw aC and -eC, &c. cntlng Perpendiculars from D, /, &c. at f-, and m ; and, having compleated the Face g m i, which is fomewhat feen, from j and draw Lines parallel to the Ground Line, cuting the adjacent Step, which termi- nate their appearance, and compleat the Steps. 1 6^ To reprefent the Block on the upper Step; draw the Perpendiculars Or, and Pq indefinite; make rr/ (on the vertical I nterfeaion, AF) equal to its height, and draw ^E, cuting Or, in r; onc Kutahlature, with an internal and external Angles^ On account of the variety of Parts, and the many breakings of the Fillets, around the Triglyphs, in the Doric Prize, it is nrcelfary to have it deferibed, geomethcally ; fo that, the Parts may be clearly un- derftood. Indeed every Moulding, delineated in Perfpedtivc, ought firft to be geometrically drawn^ to the Scale of the Drawing ; although the meafures may be applied from a Scale only. I have fhewn, in the 12 th Example, how a perfpedtive Plan may be drawn either above or below the Work ; by which the projedtures of the Mouldings, &c. may be determined at the Angles, as exemplified in the 13th. But, that procefs (though the moft to be depended on, for accuracy) is attended with ex- traordinary trouble, which may be leffened greatly, when we are tolerably acquainted with Mouldings. A Specimen of both will be fhewn, in the following Example, Let A B be the height of the Entablature, according to the Scale of the Drawing4 Let it be bifeded at C; and AC, or CB, is a Module, or Diameter of the Order. AC divided into 60 Minutes, is the Scale; by which the whole is proportioned, QM, the projedlure of the whole Cornice, is a Diameter; the reft is proportioned as by the Scale AB. D F is the Architrave, FG is the Prize, with its Triglyphs, and GMQ_is the Cornice. H I is the projedlurc of a Mutule or Modilion, and K of the whole Planceer ; Which, being feen in Perfpedlive (the Eye being below it) has a fine efteA, and adds g^rcatly t9 its auguft appearance. 1 No. g. • I .10/1 - i/o/ /o/(';in/J, /'.fi/yi/l’',/ IIIAX / ' Sea. VII, APPLIED TO ARCHITECTURE. No. 2 is a geometrical Plan of the Cornice, with an internal Angle at S, and an external, at T ; fhewing the Planceer of the Mutules, and panneling between them •, tor, uniefs we know the true geo- metrical form, It is not poflible to deferibe any, thing, perfpedlively. Each return of the Cornice is the fame, having two Mutuleseach way, except the laft, on the Right, which is I'uppofed unlimited; and, on the Left, it'is limited by the bounds of the Picture; which, on account of its diftance from the Center, would bediftorted, if it was continued much farther. B AFG is a perfpeftive Plan of the whole; which is the beft method of proceeding, if accuracy be re- quired in the feveral parts, and we do not gru*!ge the time fpent in doing it. It is formed as follows. At any Diftance from the Horizontal Line, £iCE, either above or below it, draw AB parallel to it ; one Side, in this Example, being parallel to the Piflure. Having determined on the Place of the Angle a, in the Defign, take A perpen- dicular over it, and draw AQ,^ indefinite ; C being the Center of the Pidure. Take AD equal ST, in the geometrical Plan ; and, having made CE in the Horizontal Line, equal to the Diftance of the Pi6lure (equal ES) draw D E, cur- ing AC in F; then, AF reprefents the length oi AD (equal ST, No. 2) and F is the internal Angle of the extreme Moulding in the Cornice. Draw FG parallel to AD, and DC curing it, in G, the external Angle ; for FG alfo reprefents an equal length as AF-f, each equal to ST. Make AC equal MQ (No. 1) the projeifture of the whole Cornice, and draw CC and AE curing it, at V. The Diagonal FX (i, e. DF) alfo cuts CC at X ; from which, draw X Y parallel to AB. The Angles K, L, and M, of the Corona (K, in the Profile) are determined by making Ad equal to MN, and drawing dC, cuting the Diagonals AW and FX ; by means of which, they are carried around, to L and M, and, by the fame means, the whole Plan is compleated, as in the Figure. To dTcribe every Step, by which the whole perfpedfive Plan is formed, would be as tedious as it would beufelefs; feeing that, the various Leflbns, already given, are fufficient for any right-lined Figure whatever. The Plans of the Mutules, Z, Z, are the reprefentations of Squares having one Side parallel to the Picture (found by Prob. 19) their places are determined as follows. Having made Ae equal MO (in the Profile) draw eC cuting the Diagonals AW and FX at S and JV, and draw JVM parallel to A B. Mdkt Aa equal to J M, in the Profile (equal T i in the geometrical Plan) make ab, be, &c. equal to the width of the Mutules and the fpaces between them (12, 23, &c. in the Plan) and from each Point, a, b, &c* draw Lines to C, the Center, cuting a parallel Line from S, in R, P, &c. For, the returning Side, AF, make Af, fg, &c. equal Ad, ah, &c. and drawy'E, gE, &c. cxximg AF, at j, a, 3, from which, draw Lines parallel to A B, cuting 6 ' C at F, V, &c. and HC at m, n, &c. For the other parallel Side, F G, make Dm, ml, Ik, alfo equal to u4a, ab, &c. from which draw Lines to the Center ; cuting JV M, at X, T, Z, For the apparent width of each Moulding, &c. in Front, draw MS perpendi* cular to the Horizontal Line; take SE equal to FC, on the Horizontal Line ; and, from every Angle, G, H, K, &c. in the Profile, draw EG, EH, &c. cuting M S in the feveral Points, g, h, k, &c. as in the laft. Example. Being thus prepared, we now proceed with the Reprefentation. a is the determined Angle of the extreme Moulding ; draw the Fillet a z of the tipper Moulding (which is in the Pidure) geometrical. From V, X, and Y (in the perfpedfive Plan) draw perpendicular Lines ^indefi- nite ; which are the Angles of the Ground Planes * of the whole. I Then, from the feveral Angles H, K, &c. in the perfpedtive Plan, draw Lines 1 perpendicular ; and, from the correfponding parts on the vertical Sedion, M S, I draw Lines parallel to the Horizon, cuting them. e. g. I * By Ground Planes, here, is not meant the real Ground or Floor, as defined, but the vertical I Planes, from which the whole Cornice, &c. projeck j called fo by Workmen. li 171 No. 2. No. 3. f Prob, 19. Fig. 85. Fig. 86. From 172 PRACTICAL PERSPECTIVE Book III. Plate From K draw a Perpendicular, Xc, and from k (where K E cut M S) draw ho- j jj rizontal Lines curing it, at c, the reprefentation of the external Angle of the Corona. Draw cC, cuting the Perpendicular from L at 1, the internal Angle of the Corona ; and from 1, draw parallel Lines, cuting the Perpendicular from M at m, the other external Angle of the fame. Then, from h (where HE cuts CM) draw horizontal Lines cuting the Per- pendicular from H (in the Plan) at h, the reprefentation of the Angle H. Draw hC cuting a Perpendicular from I, at j, and the horizontal Line j n gives the other external Angle, n. For the Mutules, draw a Perpendicular from S, In the Plan (where they would meet If they were continued to the Diagonal V) and from i in the Scale Line, MS, draw parallel Lines cuting it, at f; which mufl be returned at all the Angles, as if It was a continued Facia. From O, P, &c. draw perpendicular Lines cuting them, in the feveral Points, 0, p, &c. which are the front Faces of the Mutules. Draw rC, 7C, &c. cuting the horizontal Line hh ; where they fall againfl that Facia. In the returning Side, draw Perpendiculars from PP, &c. till they cut fC, the returning Facia, and fo of all the reft ; which are heft defcribed by the Fi- gure ; obferving that, if from the feveral parts, in the firft or neareft front of the perfpedive Plan (PyfV) perpendicular lines be drawm, and from thecprrefpond- ing parts in the Seclion, or Scale Line, MS, parallel Lines cuting them; and, by carefully remarking the mitre Angles of each Member, on A V, the true per- fpedfive proportions of them are carried round as many Breaks, in the Objedl, as are required; by means of the Vanifhing Point C (the Center) for one return, the other parallel; the Originals of them being, in this Example, parallel to the Pidure, and the Objed right-angled. The Mouldings around the Mutules are determined, at their projedlures, from the Plan, above (at O, P, &c.) and the width, in Front, from the Sedlion of the Pidlure, MS (at m) by means of which, and the Vanilhlng Point C, they are very eafily continued around the Mutules, from one Angle to the next, adjoining., Firft, drawing parallel Lines, from m, till they cut Perpendiculars from 0, P, &c. 1 at &c. and JC, /aC, &c, cuting Perpendiculars from J, at r andy ; through which they are drawn, parallel, to the Angle at and from ^ they are continued, after the fame manner, around the whole. Every thing being corred, and very particular In the Figure, makes it unneceflary to give a further defcription of it. To defcribe every Line in the whole Procefs, would take feveral Pages, be extremely tedious and apparently prolix. For if, as 1 have before obferved, the Student be not acquainted with Archl- tefture, nor underftands the feveral Parts of the Order, it will be impoflible for him to fucceed in this Example ; but if he is, it will be found lufRcicntly intelligible. The Architrave, of this Order, it is needlefs to fay any thing about ; feeing It is compofed of plane Facias, only; projedling, one over another, like Steps, of dif- ferent heights and projedtures, and is managed in the fame manner ; which, from inlpedlon of the Figure, may readily be defcribed. The Frize and TrIglyphs have nothing difficult in them,, efpeclally in this Pofi- tion. In the parallel Faces they are geometrically proportioned and fpaced (Theo. 9. Cor. 5) and in the returning Side, thifeir places are determined from the Plan above, or by the loth Problem; dividing the upper Line of the Frize, st, in the fame Ratio, perfj3e6lively, as the front Line, ns, is divided, at o, p, q (Prob. 8.) At No. 4, I have given two Triglyphs, more at large ; in order, that the Parts i may be more diftindlly made out ; in which, after dividing the front one, as in the Profile, geometrically, for the flutings (which are right angled) the indented Angles ; may be obtained, as in the Plan ABD, above; by drawing a Line from each Angle ; a and to the two Diftance Points, if they are within reach, cuting at c. Or, Sea. VII. APPLIED TO MOULDINGS. Or, if one Diflaiice (£) only ; bifea the widtii of tlie Flute, at and draw d c, to the Center, cuting a E at f, as'before ; ej\ &c. drawn, to /{, give the red. For the Triglyph in the returning Side ; having drawn the front Line, ad, to its vanifhing Point, c ; from each angle, a, b, &c. -in the front Triglyph, draw lines to E, cuting ad, at b, c, and d. Or, if it be convenient to have the Didance of the Eye on the other Side, it would be better; draw ah parallel to AB, and take the divifions, from a, at c f, g and h, equal to the geometrical meafures; and, from them, draw Lines to the other Diftance, as in the Figure (fuppohng it within reach) which give the Lme Points, more accurately defined. (See Prob. 8.) From the feveral divifions, on each Side, draw Lines perpendicular, as in the Figure .; by which means, the Triglyphs, X and Y, are compleated. The Perpen- diculars, from c and /&, (in the Plan, above) give the indented Angles in the front Triglyph ; in the other they are not feen. This operation is fuppofed either at the top or bottom of the flutings. The inclined Lines, at the Top, are determined, truly, by drawing a Pcrpendl- •-cular at the middle point, making de equal to half its width, and drawing a line to c, the Center, cuting the Perpendicular from Cy at /; join ^/and bf. It is unnecefiary to be more particular, or their vanilhing Points may be eafily determined ; by making tlie lame An^le at the Eye, as the inclined Lines make, either with the horizontal or verticarLinc (Prob. 4.) and where it cuts a Line, pafling through E, perpendicular (the Vanifhing Line of an inclined Face) is the Vanilhing Point oiaf. Either being fotrncl, determines the other ; and, the inclined Planes of Y being parallel to thofe of X, have the fame Vanifliing Lines, refpeclively, pafhng through tho Eye in the Points of Diftance. "I'he Vanifhing Points of the inclined Lines, in the front Triglyph, -vanifh below the Horizon, and thofe in the returning one (Y) which are feen (at f, g, h) vanifh at the fame Diftance above it, at ©• Although this method will feldom be practifed, yet I affirm it to be the beft, being by much the readieft: and the moft correft. Nor, is there any occafion to be at the trouble .of forming the Plan, to get the indented Angle; for, the F'lutes being proportioned ztab (as above) and the V anifhing Points determined, the Lines, af and bf, being drawn to their refpedtive vanifhing Points, determine at the fame time, the internal Angle. The Tenia (V and Z) both above and below, breaks regularly around; as it may be feen, and would be needlefs to explain ; fimilar Subjeds having fo frequently been repeated. Such minutias are beft deferibed by the Figure, only, being accu- rately drawn, and well defined. The Abacus (W) which may be fuppofed over a Column, is a Square ; and the entire Capitals, over the Pilafters, Y and Z, at the internal and external Angles, although they are detached Mouldings, are managed (from the Profile) after the fame manner as If they were continued ; their perlpeftive Plans being delineated, as above; from which, the doted Lines, correlponding with the Profile Section, (MS) determine their feveral Angles. Method 2nd. EXAMPLE XVIL How fo reprefent the fame thing 'without having the Profile drawn, or a perjpeciive Plan. Let AB be the given height, which divide into two Modules, and Into Minutes, as before (the reft being fuppofed not drawn,) The Scale (A M) of the projedure of the Mouldings, &c. at the top, is alfo neceflary. From a, the determined Angle, draw ak perpendicular, and transfer all the imeafures of the feveral Mouldings, from the Scale, to ak ; at b, c, d, &c. Make a^, ab, &c. equal to the feveral projedures MN, NO, &€. and proceed as in the Plan, above; draw'ing a£ for the diagonal Line of the top ; and, from the feveral divifions a, h, c, &c. draw Lines to the Center (C) cuting the Diago- nal, at e, j\ gy &:c. from which, draw perpendicular Lines; and, from the mea- lures on a k, draw diagonal Lines, to £; which give the fame Angles c, h, &c. as before j from which they are carried around the feveral Faces. X X The Book llli. Plate XIX. Fig. 87. PRACTICAL PERSPECTIVE The perfpe£live Plan, in this, is fuppofed to be formed on the Top of the Cor- nice, the fame as above; and fince ak is the Interfe£i:ion of' a Plane, paffing thro’ the Diagonal or mitre Angle of the Cornice (one Side, in this Cafe, being parallel to the Pidure) E, the Diftance Point, is the vanifhing Point of that Diagonal, confequently, all horizontal Lines, in that Plane (being parallel) vanilh in E. Wherefore, aE, bE, &c. are each, the indefinite Reprefentation of a Diago- nal of a Square, in different horizontal ‘Planes, whole Interfeding Points are a, b, &c. and, the diftance of each Angle from the Pidure, is a^, ab, &c. which be- ing transfered, perfpedively, to the Diagonal ag, each Angle, in its proper place, ■will be perpendicularly oppofite to the Points e, f, &c. as in the Example. And thus may the whole be compleated, when we are a little verfed in Mould- ings, by transfering the mealures, perfpedively, from one Diagonal to another, which will be further illuftrated in the following Example. EXAMPLE XVIII. How to reprefent a Cornice when it is inclined to the Picture, on both Sides. If the laft Example be tolerably well underftood, in the laft Procefs, this will be found eafy, being performed by the fame means. It muff be obfcrvcd, that the fiift Method, refpefting the Profile, cannot be applied here, nor in any Cafe; but when the Mouldings are parallel to the Picture. 7 he extra Plan may be ufed in all Pofitions; but I (hall in the following Lxampfes, do without; as it is only fuppofinrj- the 1 Ian formed at the Top, from which the Perpendiculars are drawn, and the feveral Angles of the Facias and Fillets determined. Let AEBbea Profile of the Cornice, to be drawn, and AC the Side of one Plane, on which it is to be projetfted. ML is the Horizontal Line, S is the Center of the Pitfture, and L the Vanifh- ing Point of AC\ which is fuppofed the upper edge of the Plane from which the Cornice is to proje-fl. Let the Top be fuppofed a Square. S being the Center of the Picture, and L the Vanifhing Point of one Side of a Right Angle (the Diftance of the Pidture being known) find M the Vanifhing Point of the other Side ; by Prob, 12. SM is a third Proportional to SL and the Diftance of the Pidure (found by Prob. 32, Geo.) Alfo, find N the Vanifhing Point of the Diagonal ; by making MN to N L as •one Side of the Triangle (whofe Perpendicular, at S, is the Diftance of the Pic- ture) to the other. As N, or O (Fig. 51.) GN, or EO bifedling the Angle FGH, or AEB. Draw y^M and CM, and cuting CM at E; draw LE, till it cuts .^M at D. ACKD is the Top, of a fquare Prifm, from which the Mouldings project. From each Angle, A, C, and D, draw perpendicular Lines, which reprefent the Corners of the Prifm; the Profile being proportioned to the neareft, AB. FromE {AB being equal to AB) draw EL and EM, cuting the Perpendicu- lars, from C and D, at G and H. ACGB and AD HB are the Grounds or Seats of the Cornice on the two Planes, which are feen. The ProjeCture of the Cornice, around the Prifm, is next to be determined. Draw the Diagonal DC, and produce it both ways ; alfo, produce N A indefi- nite, and draw A\i parallel to the Horizon ; on which, take the diagonal pro- jeclures of the Cornice, as on A K, in the Profile. ? EK being made equal and perpendicular to AE; confequently, AK, the Diagonal of the Square A JKE, of the projefture AE, is the mitre Angle of a Right Angle, for the Cornice AEB. From the feveral Projeftu res, C, D, &c. drawCG and DF, parallel to AB, and produce them to the Diagonal, cuting it in H and I. Make A\i, hi, and ik refpeftively equal to AH, HI and IK; and having made N O equal to the Diftance of the Vanifhing Point N, of the Diagonal, draw Oh, Oi, and Ok,__till they cut HA, produced, at h, /, and £ ; from all which, ' draw Sea. VII. >75 applied to Mouldings. draw Lines to both Vani(hing Points, L ^nd M, cutihg the other Diagonal, CD* produced both ways, at F, /, /, &c. which form the perfpcaive Plan of the principal Mouldings. £ F, FG, and HI, being drawn, are diagonal Lines of each Angle, from the lower edge of the Fillet, at the top, to the bottom of the Cornice. Let it be obferved, that N, being the Vanlfhing Point of the Diagonal of a Square, i. e. of a Line bifeding the Right Angle, is the fame whether the Top be a Square or other Reftangle; but, if it be not a Square, CD, produced, would not be the Mitre of the other Angles ; becaufe it would not bife^fl them ; neither would AN pafs through the oppofite Angle, or Mitre KL', yet N would be the Vaniftiing Point of both, becaufe they would ftill be parallel. The others are alfo parallel, and confequently, they have the fame Vanilliing Point ; which, being much inclined, is at a great diftance. Having Ihewn how the Mouldings are projected forward, on the Diagonal AE, p* and transfered to the other Diagonal ; fuppofe No. 2 the fame thing, prepared in the fame manner ; being diverted of feveral preparatory Lines, which, altogether, render the Work confufed, and are fuppofed to be rubed out. On AB, take a, b, and c, equal to the feveral heights of the Mouldings (as at a, b, and c, in the Profile) and, from N, projed them (as N a, &c.) till they cut Perpendiculars, from I and h, m d, e, f and g- ; and, from them, draw Lines to both Vanifhing Points, L and M, cuting Perpendiculars, from k, /, and m, n, &c. at o, p, q, &c. which compleat the Facias and fmall Fillets ; and there re- mains only, to join them by curved Lines, of the fame kind as in the Profile. If the Mouldings are very large, as many Points may be found, in the Curves, as are hecelTary to defcribe them with accuracy ; by Perpendiculars from AE, and meafures transfered to AB, from corre- fponding parts in the Profile. But, they may be as well performed by a careful Hand; for after all at- tempts at exailnefs, in fuch minutias, a judicious Perfon, in Mouldings, and Perfpe£live, would defcribe them as perfcflly, regarding the Pofition in which they arefeen. Let it be obferved, that the Curves are always flatter, in mitre Angles, than in the Profile, and more fo, the lefs the Angle ; likewife, in the diagonal Seftion, AEB, they are flatter than in CFG^ becaufe, the Plane of that Diagonal is nearer to a coincidence with the Eye ; for if it be in the Plane (however fituated) they are Right Lines. The Mouldings, in the diagonal Plane ^ A 5, ’'feeing that, the full meafures are applied, on AB, and projefled forward to Ad, are larger than the Originals; that is, than the given meafure; as Aabcd, than Aabc, on AB', equal Aab, &c. in the Profile. The Picture cuts the plane of the top inef. In which Cafe, if S be fuppofed in the Pidlure, AqcB on one Side, and y^fgiA on the other, are feftions of the Mouldings by the Pidlure; and, the whole, efgS, of that Sedtion, being on this fide of the Pidlure, is projedled, to the Pifture. AEeB, CApG, and HIc, are diagonal Sedlions through the mitre Angles; and, jklm is a Section perpendicular to the Plane ACGB. If what I have advanced be well underrtood, all that follows, refpetrting right-lined Mouldings, will be eafy and intelligible, almort from infpedlion of the Figures. EXAMPLE XIX. How to reprefent an Entablature with Modillons in the Cornice, obliquely Jituated to the PiElure, having internal and external Angles. ABC is the geometrical Profile of the Cornice j F is the front of a Modilion, 3g^ and FG is its Profile. C D is the Frize, and DE the Architrave ; with two Facias. In this Example, I fhall fuppofe the whole Cornice to be beyond the Pidlure, and touching it at the Angle A. AFG is the Interfertion of the plane of the top, HL is the horizontal Va- nirtiing Line, C is its Center, and GE the Dirtance of the Pidlure. H and L are the Vanilhing Points of the Sides ; i. e. of the horizontal Lines of the Mouldings ; M and N are the Dirtances of thofe Vanifhing Points, for proportioning Lines which vanilh in them, refpedtively ; and O is the Vanirtiing Point of the Diagonal* bifecrting the Angle, HEL, made by the Radials of the Sides. Draw the vertical Interfeftion, AE, of the diagonal Plane, parting through the mitre Angle ; and, transfer all the meafures of the heights of the Mouldings, from the Profile, to c, d, &c. equal Be, cd, &c, alfo, their projedtures, to A a, ab, bB. equal A a, a b, andbB. 5 Draw Book irL j ‘;6 PiateXi: Pig.'SS. PRACTICAL PERSPECTIVE Draw Indefinirc; and from all the Points ti, &c. draw Lines to N, Cluing XL at I, 2, 3; from which, draw Lines to H, curing a Diagonal, AO, at /’ &c. the perfpedlive Seats of the projeclures ; i. e. of the mitre Angles of the Corona, K, and the F'acia at F, in the Profile. From the feveral Angles, k, 1 , &c. obtained, as in the foregoing Examples, draw Lines to both Vanifhing Points, H and L, indefinite; and proportion them to their refpeiRive lengths, by means of the Diflances, M, and N, refpedlivelv, of thofe Vanifhing Points. The length being determined, according to the number of Mod lions, con- tained, orotherwile; make XD, equal to the flrfl: break, and draw DN, curing yfL at f, the internal Angle ; and draw fO, the Diagonal Line of that Angle, .in the plane of the Top ; indefinite, Draw bL, curing that Diagonal, at g; from which, draw a Perpendicular.; and c L curing it at e, and join e f ; the Diagonal of the Projedlure. Draw He, H f, and H g, indefinite; and, make f E to reprefent a Line in the proportion of the Original, offjFtoXf; (by Prob. 10) or, draw M f , and produce it to the Interfedfion, at B ; make BB to X D in that proportion, and draw F M, curing H f produced, at F. (Prob. 17.) DrawFO, curing Hg, produced, at h ; from which, draw a Perpendicular, curing H e, produced, at i ; and draw F L, h L, and i L, indefinite. Produce N F to the Interfedlion, and make FG equal to the length of the front Line of the Cornice ; draw GN, cuting FL at m ; and, G6 being made equal to the projedlure of the Cornice (A B, in the Profile) draw 6 N, cuting FL at ''n ; and n H, cuting h L, at p ; from which, draw a Perpendicular, cuting i L at q. Thus, are all the Angles at the top and bottom of the Cornice x)btained ; and, having let off the meature of the upper Fillet, at X, let it be continued around, by means of the Vanifhing Points H and L; then draw diagonal Lines, Fi, and - m q, by which the Mouldings are all determined (as at A C in the Profile) having, before, obtained their places on the diagonal Line, X c, at thefirft Angle. The perpendicular Lines of the Corona fall below it, from the Point where they are cut by the upper Line of the fame, at k. On the returning Side, on the left hand, the Cornice is fuppofed to fill againft, or is cut off by a Plane, parallel to the front Planes, and confequently has not a mitre Angle. Draw XH, indefinite; make Xa reprefent a length equal to what the whole projeefs from the Plane (by means of Xf, Prob. lo ; or, by its meafure, Pr. 17.) Draw aL and bH cuting at r ; from which, draw a Perpendicular. D raw c H cuting it at d, and draw ad, which will terminate all the Mould- ings ; as the diagonal Lines determine the mitre Angles ; by drawing Lines from every Ar.gle, on Xc, to the Vanifhing Point H. The Architrave, F, H, /, K, being compofed of plane Facias, with one Mould- ing, only, on each Face, needs no particular defeription, T 1 le Modilions are determined, in the fame manner, as the Mutules, in Ex. 16. Make Xg, on the Interfedlion, AF, equal to Ag, in the Profile, and draw gN, cuting X/J), at o, and draw oH cuting the Diagonal XO, at f; and, from draw a Perpendicular, giving the mitre Angle, S, of the front Planes, of the Modi- lions. Let it be continued around the feveral Breaks (as the Corona) in Pencil Lines. Draw /L, cuting the next Diagonal, at /. J"/ is the Seat, on the top, of the front of the Modilions on that Side; which may be continued around, from one Diagonal to the other, by means of the Vanifhing Points H and L. IMake X/; equal to A b, in the Profile, and draw cuting XD at 2; and draw' 2H, cutingy'/ at h.. From N, projedt the Point h to the Interfedfion, at i ; and make y L &c. refpedtively equal to the width of the Modilions, and to ■the fpace between them, alternately, as often as is requifite ; from all which, draw Lilies to the Point N, cuting the Scat, yV, in their perfpedlive widths, at ;/z, «, &c. from which, draw Perpendiculars, giving the feveral Fronts, x, x, as in the Figure. I The Sea. VII. APPLIED TO MOULDINGS, &c 177 The Modllions yy and z z, in the returning Side and Front, are determined •after the fame manner, viz. thole in the Side, y, y, and alfo v, v, by means of ►the Point M, and the front ones, z, z, by the Point N ; their meafures being fet 'offat 1, 2, 3, &c. on the Interfeaion of the Top, as thofe at x. Having thus obtained the Fronts, draw Lines to the refpedlve Vanlfhing Points LI and L, till they cut the Plane from which they projea ; and, having drawn Right Lines for the edge underneath, firft, the curved Lines may be drawn by hand, as in the P'igure, regarding the different appearance of each, as they recede. If the Parts are large, and require to be accurately projedled, the Moulding, around the Modilions, may be managed the fame as in the laff: Example, by a per- fpedive Plan, above the Cornice. But, a judicious Perfon will fave that unne- 'cefl'ary trouble j for, having obtained their mitre Angles, as at s, t, u, &c. the front Moulding being drawn, as any other continued Moulding, they may be re- turned at the tides, and meet the inner Moulding, fufficiently correct without the trouble of planing them. In this Example, there is all the variety that is requifite for fuch Cornices ; but, as the Corinthian Modilion may, to fome, appear more difficult than the Ionic, I ffiall give a fpecimen how it may be delineated, as briefly as poffible. In the following Leffbn, I ffiall not deferibe, over again, how the Mouldings arc ■to be drawn ; and, I have (in order to have the Modilions larger) omited the upper Mouldings of the Cornice, and the Corona ; having retained only the Planceer ; which I ffiall conffder as the Plane, on which the Modilions are feated, and AB the Interfedling Line of that Plane. The Vaniffiing Points, H and L, are allb the Vaniffiing Points of this Figure ; the Center or Point of view is at S, and the Diftance is ES. Draw the vertical Interfedion of the Diagonal, AP'; and, by a geometrical Scale of the Proportions, fet off the heights of the Modillions and Mouldings at a, b, &c. alfo, make ab, be, &c. equal to the width of the Modilions and fpaces between; and transfer them, by means of the Point P (the diffance of the Va- niffiing Point H) to A 5 (the firff: doted Line) which is the Seat of the fronts of the Modillions; and finiffi the fquare Blocks efgh, .&c. which enclofe the Mo- dillions, as in the former Example; by which means, they are truly proportioned, perfpe^Ilvely, as in the Figure, which would otherwife, be fomewhat difficult. In the fide, of each Block, mult be drawn, by hand, the Profile of the Mo- dillion, perfpedively, as it is reprefented at X (the geometrical Profile) according •as they are contraded, or obliquely fituated to the Eye. In the Front, of each, is alfo deferibed the end of the Scroll, and of the Leaf. In ffiort, having firff; obtained the Blocks or Cafes which contain them, the reft muft be delineated by a judicious hand and Eye ; for fuch Figures will baffle all Rules, nor is it poffible to fubjed them to any, by which they may with greater certainty be deferibed j except the Profile, in the fide of each, as the Trufs in the next Example (No. 2) which is a fimilar kind of Figure. The Moulding breaking around the Modillions, is firff: deferibed as other conti- nued Mouldings, at the front, on which their mitre Angles are obtained, from the Interfedion AB; the meafures, on which, not being equal, on both fides, is owing to the front Lines of the Fillets, being in another Plane, ftanding forwarder than the front of the Modillions. The Dentils have nothing particular in them ; the proportion being known, of their width and fpaces between; a Right Line drawn through either Corner, as fg, parallel to the Interfedion, being divided in the Ratio, and Lines drawn to any Point, G, in the Vaniffiing Line, according as the meafures are taken, greater or lefs, will give their meafures perfpedively (as by Prob. 8) but, their true meafures can only be applied, at the Interfedion of the Planethey are in, with the Pidure -(drawn through C) andprojeded, by means of Vifual Rays, to the Eye, at P. Fig, 88, No. .2. Yy EXAMPLE PlateXIX. Fig. 89. Prob. 3* t Prob. 1 7 EXAMPLE XX. Ho'W to reprefent a Door Head^ with a Pediment, fupported by Trujfes or Confoks, Let A be the determined Angle of the Carnice of the Objedli, in the Picture. At No. I. is the true geometrical proportion of the whole, in Front. Through A, draw yfB perpendicular, the vertical Sedlion of a diagonal Plane, pafling through the mitre Angle, whofe Vanifhing Point is G-f-. H and L are the Vanifhing Points of the front and fide Lines, which are horizontal; S is the Cen- ter of the Picture, and SE its Dihance; I and K are the Diftance Points of each, refpedively, for proportioning Lines which vanilh in them;|; ; EG bifeds the Angle HEL; conf. G is the Vaniihing Point of horizontal, diagonal Lines. (Pr. 2j.) Thefe tilings, here preniifed or given, are fuppofed to be found and determined, from the known pofi- tion and fituation of the Object, as in the former Examples. (See Prob. J2.) All the meafures, from the Profile, are produced, or applied to the Interfeclion, and, by means of the Diagonal the mitre Angle of each Moulding is obtained, as in the foregoing Examples. To determine the breakings of the Mouldings around the Truffes, draw aZ), the Interfedion of the Plane they are in, above the Cimma reverfa ; and, by meansof the Point I, projed the Point rz to that Interfedion, at a. Make a b, be, and c D, refpedively, equal to the Breaks, and to the opening between them, geometrical (as at v u. No. i.) and draw b I, cl, &c. cuting ^ L, at b, c, and d, their perfpedive Proportions. Draw c H, indefinite ; and by means of the Point K (the Diftance Point of H) projed the Point c to^the Interfedion, at d. Make de equal to the returning Side, and draw e K, cuting c H, at ^ ; and, through e, draw L cuting ^ which determine all the mitre Angles. The Mouldings are deferibed, or delineated, as in the laft Examples. To determine the proportion of the Trufs, or Conlole ; of which X is the Pro- file, and Z the Front (No. i.) Through D, draw a horizontal Line, DF, and produce H f (from H) to that Interfedion, cuting it, at h. Draw ho, perpendicular; and, from h, fetoff, geometrically, all the propor- tions of the Truls, at i, k, &c- in refped of its height (as in the Profile at X) from which, draw Lines to H, cutingyVz, at k, /, m, &c. from which they are transfered to the other fide, by the Vanifhing Point L, at 'o, u, x; and, by the Point H, projeded forward ; which proportions the heights in the other Trufs. The geometrical projedures are perfpedively proportioned, m the fame manner as the returning Moulding, at ce, by projeding the Pointy (from K) to the Inter- fedion DF, at I, and making i, 2, 3, equal to the projedures in the Profile ; from which, draw Lines to K, cuting h H at g, h, &c. and from them, draw perpendicular Lines ; by which means, finding as many Points as are neceflary, the true perfpedive form may be determined. At No. 2, is fhewm (more at large) how the Trufs is deferibed, having all the fame Letters of reference ; and, in proportion to the other, as 3 to 2. If the width be not already determined, by the Mouldings, draw I h, rill it cuts the Interfedion, D F, at f. H, I, K, and L (No, 2) are fuppofed to be the Vamfhing Points, in the Horizontal Line, anlwering to thofe below. Make fh equal to the width of the Trufs, and draw hi, cuting ^ L, at /. In the Piofilc (No. 1) take as many Points, in the Curve (i, 2, 3, 4) as are neceflary ; from which, draw lines parallel to the Horizon, cuting ho, at i, k, 1 , and m 3 and alfo perpendicular. Let thofe meafures be transfered to ho (No. 2) ' 2 from Sea. VII. APPLIED TO MOULDINGS, &C. 179 from which draw lines to the Vanilhin^ Point H; and, having proportioned perfpeaively, as fh (No. i) by the Interlbaion DP (where their geometrical pro- portions are transfered, at i, 2, 3) draw perpendicular lines from the divifions on cuting the Lines from i, k, &c. relpeaively, at 1, 2, 3, and 4, through which, a Curve being defcribed will reprefent the out- line cf the Conlole. From I, 2, &c. thus obtained, draw Lines to L ; by which means, the exterior Curve may be defcribed, as in the Figure; and, alfo, transfered to the other Conlole, which muft be defcribed after the fame manner. To determine the Pediment. Having obtained the Angle its perfpefllve length (Prob. 17) equal twice hA, geometrical ; at I (the dirtance of the Eye from the Vanilhing Point L) make the Angle, LIV, equal to the Angle ol the Pediment, cuting VL (the Vanilhing Line of the front Planes, Prob. at V, and let off an equal Diflance on the other fide L ; from which Points, Lines drawn through A and By determine the middle, at C; all the Mouldings vanilh in thole Points. The height and true pitch, or inclination, may he thus determined. Make BE equal AB ; i. e. make AE equal to twice the height of the Pedi- ment. Draw BF alfo perpendicular, and draw EL, cuting it at F. Draw.^^F and EE, interfedfing at C; ACB \s the true perfpebfive outline. Or, having bifedted AB, perfpedlively at G (by Prob. 8) draw a Perpendicular ; and, from B, draw BL, cuting it, at C as before ; and draw AC and BC. From all the Mouldings at B C, draw Lines to the Vanifh'ng Point L, cuting CG ; and the projedlures of the Mouldings being alfo perfpedfively propor- tioned, on CH, at r, s, t ; perpendicular Lines, from them, will cut others cor- refponding, from CC, drawn to H, in the true mitre Angles; and, from thofe Angles, draw Lines to both Vanilhing Points (V) of the Pediment, which will give all the Mouldings, with accuracy, beyond any other method whatever. The horizontal width of the Frize and Architrave, are proportioned to the Trufs and Mouldings, onfn-y by drawing Lines toL; the perpendicular widths of the Architrave (being parallel to the Pidture) and the opening of the Door, by the Vanilhing Points of the Diagonals of a Square (as at Y) in the Vanilhing Line V L of that Plane; regard being had to the projedlure of the Mouldings (See Pr. 26.) The recefs of the Door, pq, is determined (by Prob. 10) by its width, m p. Produce the Radial LE; make EM equal to the width, and EN to the recefs ; or, make EM and EN, in the ratio of one to the other. Join MN, and draw EG parallel to MN. O is the Vanifhing Point of a Dia- gonal of the Soffit, or head of the Door Cafe ; and mQ being drawn, cuts pH at q, as required. The reft is obvious, from the Figure. Th is Method, when it is applicable, is preferable to any other; for, having obtained the true proportion of one Line (mpj in any Plane, any other Line (pq) in that Plane is eafily determined, by the loth Problem. Indeed, it may always be applicable (provided, tha Diagonal Line mq (i. e, MN) be not very oblique to the Piftore ; and confequently, its Vanilhing Point (O) very remote ; for, if the whole Dif- tance (SE) cannot be ufed, half, or any other portion, may be taken, with equal propriety ; and the Point, O, afeertained the very fame (by Prob. 12.) Otherwifc; if no Line, in the fame Plane, be found or determined, there is but one general Method; which was applied, to proportion the return of the Moulding ce^ or the front Line at b, c, and d, viz. by the true meafure applied to the Interfedtion of the Plane it is in (Prob. 17) which being frequently ufed, would be needlefs to repeat ; particular regard being had to the true Iiiterfeftion of either Plane, mpq, of the Soffit (which is horizontal) or of the vertical Plane, W, of the Doorjamb. For, fince pq is the common feftion of both Planes, it is confequently in both ; and therefore, either Interfedlion will anfwer the famepurpofe. PQ_is the Interfe£tion of the horizontal Plane mpq; its true Diftance, from any other, being known, equal DQ_; a Line drawn through Q, parallel to the Horizon, is its Interfeiffion with the Pifture; for, all parallel Planes have parallel Interfedfions (8.7. El.) And, if Hp be produced, till it cuts PQj a Line drawn through P, perpendicular, is theinterfeclion of the vertical Plane W. •For, pq cuts the Pidlure at B ; therefore, P is its interfe£ling Point (Def. K) confequently, PR is the Interfeftion of the vertical Plane, W, that Line is in (Prob. 3.) and, P Qjof the horizontal Plane, mq. EXAMPLE PRACTICAL PERSPECTIVE Book III i8o PlateXIX. Fig. 90. EXAMPLE XXI. How to delineate a Block-Cornice, and to break the fame, or any other, around a Bow Window ; ’which is half a hexagonal Prifm. HL is the Horizontal vanilhing Line, C the Center of the PIdure, and CE is its Diftance ; H is the vanilhing Point of the Front of the Bow, and L of the Side of the Building, at right angles with it. Draw EH and EL making a Right Angle (HEL) and, with any Radius, on E, defcribe an Ark, x, y, %, cutingEH, at x. Make xy and jvz each equal to and draw Ey, cuting the Horizontal Line, at M ; and Ez, which, would cut it, if produced; M, and N (the fuppofed Point where Ez would cut HL, produced) are the vanilhing Points of the Sides and Diagonals of the Hexagon. Bifed the Angle v E;r, (i.e. HEL) at u, and draw Eu, to P; which is the Vanifhing Point of the Diagonal of the Right Angle. Let AD be the dnterfedion, of the neared: Angle of the Objed, with the Pic- ture; and A the determined height of that Corner j let A a be the height of a Plinth, above the Cornice; and ab the height of the Cornice, of which, X is the Profile. The Dimenfions of the Bow window, and other parts of the Building, being known, it would almod be fuperfluous to defcribe how it is to be determined in Perfpedive; fufficient inftrudions for that purpofe are contained in the 4th and 5th Examples. However, as the application of them to real Objeds, may not, to fome, be familiar, 1 (hall, briefly, defcribe it. At No. I. is a Plan of the Window, to a Scale of half the meafures applied in delineating ; and about one fixtieth part of the real Objed. ak is the pofition of the Pidure, applied clofe to the Angle a, and cuting the Building at the Angle c ; wherefore, great part of the Cornice is projeded to the Pidure, feeing it is on this fide, ak. Draw AF the Interledion of the Plane of the Top, parallel to the Horizontal . Line ; make HD equal to E H (the Diftance of the Vanilhing Point H) and draw A H and A L indefinite. Make AB equal to the (hort returning Plane (V) which is at right angles with W, the long fide of the Building (ah, continued) i.e. make A B equal twice ab in the Plan, and draw BD, cuting AH, at B. Draw N 5 indefinite (Prob. 13) and having found the point P, the diftance of N {by Pr. 12) draw P 6 to the Interfedion AF, cuting it at G. Make on the Interfedion AF, equal to twfice be (No. 1) and draw Be, cuting NB at C*. Draw CH, indefinite; and draw CD cuting AF at d; make ^/F equal to Ge (i.e. to the Front) and draw F D, cuting CH at D. From all the Angles, B, C, and D, draw Perpendiculars, which give all the Faces of the Prifin, that can be feen (Y, and Z) i.e. b c, and cd (No. i.) As there is not room, on the Interfedion, to fet off the whole meafure of the Plane W (equal four times ah. No. i) take AG a third part of it, and L£ one third of EL, and draw GE, cuting AL at /f; from which draw a Perpendicular. Thus, having determined the Building, we now proceed to the Cornice. From X, the Profile, transfer all the meafures, to AD, of the heights of the Mouldings, at a b. Draw the Diagonal Pa ; and determine the mitre Angle of ihe Mouldings (as in Example 17th) dl'Aabh. * Becaufe the point C (in SC) is beyond the Interfedion, A F, it is confidered as projeded to the Pidure, its place being on this fide, in the Original ^ as ak (No. i) the Pidure, cuts the Angle c. 3 Draw Sea. VII. APPLIED TO MOULDINGS, &c. i8i Draw aH and bH, cuting Bh, at c and h. Draw Nc and Nh, till they cut Cg, at d and g ; and draw dH and gH, cuting D f, at e and f. Alio, from i, draw iL, reprefenting a Facia, below the Cornice ; and i H, cuting Bh, and continue it around each Face of the Hexagon. The mitre Angles of the Hexagon are thus obtained. In every regular Poligon, Right Lines drawn through the Center, bifefl the Angles of the Poligon ; and in a Hexagon, they divide it into fix equilateral Triangles, each Diagonal being parallel to two oppofite Sides. (See No. i.) ArEy and yEz are two fuch Triangles ; conlequently, Ea-, Ey, and Esr, are parallel to all the Sides and Diagonals of the Hexagon -j-, producing the Vaiiilhing Points, H, M, and N, of the Diagonals or mitre Angles. But, the Angle, at b (No. i.) where the Hexagon joins with the Building is internal, and its Mitre is g b, produced; confequently parallel to fc; therefore, they have the fime Vanifhing Point (M ) a b is the Angle of the Building (which is a Right one) where the Mouldings are firh projected; ch Is the Internal, and dg, ef, two external Angles of the Hexagon. Produce Se, Sd, and Me, indefinite; and draw Ha, cuting Me, produced, at c. Draw N^-, cuting Sd at and eH cuting Se, produced, at y*; a, c, e, andyi are the extreme Angles of the Cornice ; which being obtained, the reft is ma- naged as in the former Examples ; by transfering all the meafures, of the heights of the Mouldings, from ab toch, and from ch to dg, &c. alfo, tjie projectures of the Mouldings, from the Diagonal to cc, from cc to de, and from de to e/*; by means of the vanifhing Points H and N ; from which Diagonals, perpendi- culars being drawn, give the Corona, at which is carried around in the fame manner ; and alfo the fmall Moulding at b, h, g, f. t Cor. I. Th. 6. The Blocks, are obtained after the fame manner as the Modillons, In Ex. 19th. The internal Angle at q being obtained, as above, and the hither face of the firft Block deferibed (by means of the Vanifhing Point L) through the outer Angle draw pq parallel to the Horizon, and fet off geometrically, the number of Blocks and fpaces between them, at i, 2, 3, &c. from which, draw Lines to fome point (P) in the Horizontal Line, cuting py, at b, &c. the perfpe£l:ive proportions of the fronts of the Blocks, on the Face Z. On the other Sides, they may be determined after the fame manner, or other- ways; as on the Face Y, by their Seat, BC, at the top, projefted to it from the lnterie£lion AF; how they are flnifhed is befl: explained by infpedion of the Figure. On the Side W, they are determined from their true meafures, by drawing pjg^ mn, through n, parallel to AF ; mn Is the Interfedllon of the Plane they are in with the Pidfure (which pq is not) therefore, having (from the point Q) projedfed the firft to mn, at r, fet off the true geometrical meafures, from r towards m, at r, s, &c. and, by means of the fame point Q, projedf them to rL; i. e. draw rQ, sQ, &c. cuting rL, which give their true places, on that Side. When the full meafures, on nm, exceed the bounds of the Pidfure, make ufe of the half meafures and diftance, &c. as in former Examples. The Blocks, in the returning Side, V, are determined, after the fame manner, from the true meafures; by means of the point D, the Diftance Point of the va- nifhing Point, H, of that Side. The Windows, and the Mouldings around them, are determined as the Door, See. in the foregoing Example ; by fetting off the true Ipace and meafure of the Moulding from 1 to k, and transfering them from one Plane to the other, as In the Figure. Ge and ^/F, on the Interfedtion A F, being the true geometrical meafure of each Plane; divide them in the true meafures of Piers, Mouldings, and Safli Squares, at /, k, /, and m ; from which draw Lines to the Points D and O, re- fpedlively, cuting BC and CD, at «, 0, p, y; and, from them. Perpendiculars being drawn, determine their perfpedlive proportions on thofe Planes, at r, s, &c. and the proportions of the Safh Squares are projedfed to their true places, from i and u, by means of the Vanifhing Points L and Q, refpedllvely. Z z I cannot i 82 PRACTICAL PERSPECTIVE Book III. 1 cannot help remarking-, liere, that the apparent Intricacy of reprefenting Mouldings, in PerfpeiSfive, with accuracy, has led fome Perfons to imagine that the whole Work is intricate, and the Diagramscom- plicated; imagining they mu ft neceflarily go through the whole, ta be acquainted with it. No fncli thing is abfolutely necellary. IF a Perfon be well acquainted with the Principles of the Theory, and the Elements of Pra^ftice, contained in the 3d, 4th, and 5 th Se^fions of this third Book, he will find no myftery in their application to any Objedt whatever ; the different Examples, here given, fhew how they may be applied, wdth fuccefs, in various cafes, and under various circumfiances, which might not readily occur, to a Perfon not thoroughly converfant in it. T hey muft not imagine, becaufe there are a multi- plicity of Lines in a Plate, in which there are fuch var cty of Parts in the Objedb, that the whole is in- oomprehenfible ; let them ft rft make themfelves Maftersof the fimple Leflbns and Examples, before they attempt a complex one, or it w ill be impoffible for them to fucceed in it. Can they imagine, that in fuch a Subject, as the 86 th: Figure, where every effential part is minutely defchbed, it can be done without Lines, by contemplating the Figure only ? Or, more particularly, the three laft Figures, in which, every horizontal Line is inclined to the Pidture ; every ©ne of which muft be perfpedtively pro- portioned, to its place and pofition. Although the defeription is full and explicit (the preceding Leffons being underftood) yet, to give a minute inveftigation of the 89th Figure, would fill, at leaft, half a dozen Pages, and would appear the moft prolix and tedious defeription imaginable. Notwithftanding the operative Lines in them neceflarily crofs each other, yet, if they are traced to their proper References, that intricacy (which fome objedl to) will no longer exift. It is obvious, that, to have avoided it, wholly, the Figures muft have been much Imaller, the Plates much larger, or their number greatly augmented, which would have enhanced the price, unnecelfarily. If they imagine that’ others have managed thofe matters more Amply, let them compare, and fee what they can make of Pozzo, the only Author, amongft the Italians, who has attempted it; nor has he attempted the defeription, but the delineation only, and that, in the moft eafy pofitior.s ; wherefore, his elegant Defigns, convey no better inftrudlions, than any other good Print, by infpeftion. Mr. Kirby, in his fiiftEffay, has run away from the Subjeft before he has^well begun it; and, in his pompous Work, what has he done ? nothine, to any purpofe. Mr. Highmore, in his 37 th Plate, has delineated a Cornice, at No. 3. tolerably, pa- rallel to the Pifture ; but No. 4, being inclined, is intolerable, and his defeription of it more fo. In fhort, a complicated Subjecl: cannot be deferibed without Words, nor can the operation be performed without Lines; although, as 1 have obferved, in the Preface, not half the Lines, which appear in thefe Diagrams, are neceffary to be drawn at all,, in the operation. s E C T I 0 N VIII. 0 F C U R v ] [LIN EAR 0 B ■ f E C T S T he PeiTpe< 3 :Iv€ of curve-lined Obje< 5 ts, the Subject of this eighth Section, iS' the moft difficult, of all other; feeing that, Curve Lines cannot be projefted as Right Lines, by means of interfeding and vanishing Points, indefinite; neither can any portion he taken, or cut off, perfpedively, otherwife than by drawing a Chord Line from one Point to another in the Original, and finding its Vaniftiing Point ; or, by any means, finding the reprefentation of the extreme Points, from- their correfponding Points in the Original. There are various ways of projeding the reprefentation of a Circle, in Perfpec- tive; all which, do no more than find the reprefentations of various Points in the Circumference. For, by the Theory of curvilinear Perfpedive, it is fuppofed, that the defeription or delineation of a Circle, or Sphere, in Perfpedive, is fome one or other of the Sedions of a Cone ; as in Fig. 28, Plate 7 ; it is obvious, that if Right Lines (EA, EB, &c.) from the Eye to the feveral Points A, B, C, &c. in the Curve (’which is fuppofed to be a Circle) in the Plane Z; be cut by another Plane (X or Y) the Points a, b, c, &c. being joined, carefully, by a fteady hand will generate a Curve (adg) which, to the Eye at E, will (as it is-obvious it muft) exadlv coincide with the original- Curve ; feeing that, it is in the furface of the fame Cons, of which, the Eye is its Vertex. Hence it is manlfeft, that, the more Points there are found in the Reprefenta- tion, the more exadly may the Curve be deferibed; but after all, it depends greatly- on the Hand and Eye; infomuch that, without great nicety in both, the repre- kntations Sea VIII. APPLIED* TO ROUN-D OBJECTS. 183 fentatiops of curve-lined’ Objefls will have a lame, and very difagreeable ap- pearance. It is, therefore, no wonder to fee fuch bad Reprefentations of round Objects, as are to be met with ; but it is a matter of lurprize, tliat any Perfon, who attempts it, Ihould have fo little judgement as to turn the Carves the con- trary way; or to make them flatter in thofe parts, where, it is obvious that, they would have a greater curviture. Without having the lead notion how to projecl Curves, perfpectively, is it pof- fible fora Perfon, who has been ufcd to fketch at all, to place a cylindrical Objefi:, _ of any kind, ora Vafe, &c. before bim, and not fee, immediately, how the curves are to be deferibed? or, what part of the Objeft appears more or lels curved than others? is is not obvious to a common Eye? yet may we frequently fee Vafes re- prefented, whole greateft fwell, at the top, is a Right Line, whllfl the Curves of the lower part take a contrary dire6lion, to each other ; indicating, thar the Evo is between them ; in which Cafe, the top would be the moft curved. Would not fuch attempts at Peidpeflive be better let alone ? and content themfelves witn a per- fedly geometrical Reprefentation ? in which they would all be Right Lines ; lavs only, the external Figure of a vertical Sedlion through the middle. But, what can be faid for the performances of thofe Artifts, whofe Works are an honour to their Country and to the Age they lived in, to fee them, often,, greatly deficient in thole paiticularSi I could with to fee the Works of the pre- lent Age more perfedt ; which, in other refpedls, feem to vie with the mofl; ce- lebrated amongft the Autients ; yet do not pay fufficient attention to thofe neceR fary Appendages, which are ellentially requifite, to a perfedl Pidture. As a Circle is the firft and principal of curve Lines, fb it is the only one that- can be reduced to any certain Rules, in delineating it perfpedlively. And, of alii the various ways to projedt the reprefentation of a Circle, in Perfpedlive, the beft, and moff pradlicable, is to fuppofe a regular Odlagon to be inferibed, or the Cir- cumference divided into eight equal Parts; or, if very large, into fixteen. Irregular curved Objedls are not Subjedls for Peifpedtive; all attempts at a Spi- ral, or twifted Column, &c. by Perfpedtive Rules, would be in vain. Various other Objedls^, as Rocks, Mountains, Rivers, Trees,. &c. are not fit'Subjedls for* Perfpedlive ; and confequently Landfeape Views cannot be taken or delineated by its Rules; becaufe it is impoffible to have the true geometrical Figures and Pro- portions of fuch Objedls, as before mentioned. I fliall however, in an Appendix, deferibe the ufe and application of an Apparatus for taking Views, with eafe and. great exadlnefs; the beft calculated, for the purpofe, of any I know or have ever heard of. For, notwithflanding what many imagine and affirm, of the poffibility of taking Landfeape Views with accuracy, by Sight only, I know it is impoffible to be done ; and cannot conceive it to be any way derogatory to the abilities of the moft eminent Artift, to make ufe of any expedient ; by means of which, he may be enabled to make a more corredt Portrait, and Pidlure. I do not mean that he fhould, rigidly, deferibe every minutia of the Objedls, as in Trees, &c. by it; but F muft affirm, that he would take the apparent Magnitudes of the feveral Objedls,, their Figures, and their Bearings in relpedl of each other, with much greater accu- racy than it is poffible to do by Sight. PROBLEM I. To find the Reprefentation of a Circle ; the Original being given, im any Plane, whofe Vanifhing Line, its Center and Diftance are- given ; according to Brook Taylor. Firft; by means of the Vanifhing Line, and one Vanifhing Point only. Let AFG be a Circle, in the Geometrical Plane; of which GK is the Inter- Fig. gi* fedlion, and V£ the Vanilhing Line; C is its Center, and CE its Diftance.. No. 1 2 Draw 184 PRACTICAL PERSPECTIVE Book III. Plate XX. Draw AB, atpleafure, cutlng theTnteiTedion at a *, and through C, D, G, &c. Fi,T .01. (Points aflbmed at plealure, in the Circumference) draw Lines parallel to A B, ^ eating the Inteiieclion, at a, c, d, &c. i Cor.Th.3. Draw EV parallel to A B, producing the Vanilhing Point, V, of .thofe Lines t. Draw aV, c V, &c. and, to E, draw Vifual Rays from every Point, A, D, G, &c. curing the indefinite Reprelentations of the Lines paffing through thole Points, relpedlively, at a, E, _§•, &c' Having thus obtained as many Points as are necellary, a Curve {adifB) deferihed caretully through thofe Points, will be an Eliipfis, and it is the true Reprefenta- tion of the Circle AFG. Or, having drawn the parallel Lines AB, &c. and their indefinite Reprefenta- tions, as before; make VE equal to VE; alfo, make ab equal to aB, ge equal to gH, &c. and draw b£, e£, «Sec. which will give the fame Points as before. This needs no Demonllration ; feeing that, the Points a, h, c, d, &c. are projefted the fame, as in Prob. 6th, where it is fully demonftrated. And, that the Curve is an Eliipfis, is demonftrated in Th. 2iid. SeA.^; of Curvilinear Pcrlpeefive ; for it is evident, and manifeft, that the Vifual Rays E A, ED, he. from the Kye to every Pc.int in the Circumference, would cut the Pidfure in the coi refponding Points a, r, dy Sic. therefore the Curve adf, deicribed through thofe Points, is the true fedfion of the Cone of Rays, and confequently it is an Eliipfis. Secondly; by the Diredling Line and one Direflor. No. 2. Fig. 91. Let No. 2. be a Circle, nearly in the fame Pofitiontothe Eye (at E) as before; 'No. 3. having the fame Iiiterleclion, GKI, and the lame dillance of the Eye (SE.) Draw. CD parallel to the Interfedion, and diflant from the Eye equal to the , diftance of the Vanifhing Line (VL) from the Interfedion (K.I.) CD is the Di- reding Line of the plane of the Circle. (Def. lo.) Draw AD, at plealure, cuting the Dirediug Line at D; and, from feveral Points, B, F, &c. in the Circumference, draw Right Lines to D, cuting the Interledioa ate, d, e, &c. the Interledlng Points of thofe Lines. Draw ED; and, from the Interleding Points, K, c, &c. draw Right Lines pa- -rallelto ED, which are the indefinite Reprefentations of thole Lines, refpedively. Draw the Vifual Rays EA, EB, &c. as before, cuting thofe Lines at b, J] &LC. and deferibe a Curve througli them, which is the true reprefentation of the Circle AF H. Draw V L, the Vanilliing Line of the plane of the Circle. Dem. Becaufe the Lines AD, BD, &c. have the fame Diredllng Point (D) their Reprefentations are parallel between themlelves (Cor. i. Theo. 14.) Confequently, feeing that K a, cb. See. pafs through the Interfering Points, K, c, d, &c. parallel to ED, they are the indefinite Reprefentations of thofe Lines; and confequently, the Vifual Rays EA, &c. will cut them, in the fame Points as before, which is manifeft; for, if Ea, Eb, &c. be .drawn, parallel to AD, BD, &c. refpedlively, their Vaniftiing Points, a, b, &c. are produced ; by which, tne affinity between the different Methods of producing the fame thing is accounted for. PROBLEM- 11 . To deferibe the Reprefentation of a Circle, having the Reprefenta- tion of one Diameter given. Let AB be the given Diameter; let ECE be the Vanilhing Line of the Plane it is in ; C is its Center, and CE its Diftance. Produce A B to its Vanillfmg Point, Vj and bifed AB, perfpedively (Prob. 8) e. g. Craw Abe parallel to the Vanilhing Line, and take two equal Divlfions Ab, be., at pleafure. Through B, draw cB, and produce it to the Vanilhing Line at G, and draw ^G cuting AB, at C, the Center of the Circle; through which, draw DF, parallel to the Vanilhing Line, 4 Make lAv Sea. viir. APPLIED TO ROUND OBJECTS. 185 Make VG equal to VE, and draw AG, curing DF at D; and, makeCF eqUc.i to C D; or, through B, draw GP'. DF Is a Diameter of the Circle. Make C£, on both Sides, equal to C E ; draw CC indefinite; and, through D or P", draw ED or jEF, cuting CC produced, at Hj through which, draw ab pa- rallel to DF; and, through D and F, draw CD, Ci% cuting it at a and b. Draw the Diagonals a E and b£; which will pafs througli C, the Center of the Circle, and cut aC, and b C, at d and c; draw c d, cuting CH at I ; 1 Pd is a Diameter, and abed is the reprefentation of a Square, circumferibing the Circle. Make ££% and EE\ each equal to £E; from both which Points, draw Lines through D and F, cuting the Diagonals at e, f, g, and h ; which Points are alfo in the Circumference. Through the Points, A, H, f, F, g, I, h, D, and e, if a Curve bedeferibed, it tvlll be the true Reprefentation of a Circle, whofe Diameter is equal to ab. Dem. For, fuppofe ab the Interfeftion of the Plane the Original Circle is in ; and ECE its Vanifhing Line, E, E are the Vamihing Points of the Diagonals of a Square f, and the two Diagonals, ac and f Prob. ig. bd, cut each other in its Center, C|; which is, confequcntly, the Center of a Circle, inferibed. :j; i6. i.EI. And, bccaufe HI pafles through C the Center of the Circle, HI reprefents a Diameter perpendicular to ab, feeing it vaniflies in C-, and DF, palling through C, is alio a Diameter, parallel to the Piftiire, Alfo, becaufe CD is equal to CF, and E E^ is equal to EE.; E^ and E^ are the Diftance Points of the Diagonals ; and confequently, feeing CD is equal to CF, the diagonal Diameters, eg and fh, reprefent Lines equal to DF, (See Prob. 8 th, and loth. Cafe 3rd, for a further Demonftration.) PROBLEM IIL How to defcrlbe the Reprefentation of a Circle, in any Plane whofe Vanilhing Line, Center and Diftance are given ; and Interfedion. Let AB be the Interfeflion of a vertical Plane, and ECE its Vanifhing Line; Fig. 93. its Diftance is CE. Let AB be the Diameter given, Bifed AB, at D; and draw CD perpendicular to AB. Make DC equal to AD, and deferibe the Ark DEF, a fourth part of the Cir- cumference. Draw AC, cuting the Circumference, at E ; from which, draw a Perpendicular, Ee, to the Interfedion. Make Be equal to Ae, and draw AC, BC, and DC; alfo cC and eC; and draw the Diagonals A£ and B£, cuting AC and BC, at I andK ; draw IK ; and JF/f, through S, parallel to A B. Through the Points, D, F and H, alfo through a, e, c, and rf', where the Diagonals, and IK, are cut by cC, eC, and DC, if a Curve be deferibed, accu- rately, it will be an Ellipfis, and the true reprefentation of a Circle, viewed oblique. If the Circle be large, and eight Points are not fufficient ; bifeft the Arks, DEandEF, at f and g; from which, draw Perpendiculars to AB; and make b and d, equally diftant, as a and f, from D ; or from A and B. Draw aC, fC, dC, and bC. From A and G, &c. where the Diagonals are cut, draw Af 3 .nd Gg, &c. parallel to AB, cuting fC and a C at and g ; and the others at h, /, 1 , /, m, and w, through which Points, the Reprefentation •will alfo pafs, and may be deferibed with more accuracy. The affinity between the Original and the Reprefentation, which the corre- fponding Charaders particularize, is fufficient Demonftration. S reprefents the Center of the Circle, C. This Method of proje£ling the Reprefentation of a Circle is, of all others, the beft and moft praAi- cable. It is nearly the fame thing as Prob. 25 th; for, the eight Points a, D, e, F, b, c, d, and H be- ing joined by Right Lines, will be the Reprefentation of an OAagon, which is clrcumfcribed by the Circle; as, in the other, the Circle is inferibed, i. e. touches every Side, as this pafles through its Angles 3 A Notwith-' i86 PRACTICAL perspective Bcok III. Plate XX. , Notvvithftanding, the Methods, in Prob, ift, by Brook Taylor, are facile and fimple, yet I believe,. * they are fcarce ever ufcd in Pra£lice. If the Circle be large, and Diftance adequate thereto, they are utterly imprafticable ; becaufe there is a neceffity for having the whole Circle and Diftance, at once, in their true places. Wherea;, by the laft, the Diftance is applied on either, or on both lides of the Center, as in all other Cafes whatever. Nor is AB neceflarily the interfeftion of the Plane of the Original Circle. For, if the place of the Circle be determined, on the Pirfture (either its Center, or the neareft part of the Circumference, at D) a Line drawn through S or D, parallel to the Interfecftion, or VanKhing Line, anfvvers the fame purpofe. AB or F// being made equal to the known Diameter, in that place, and a quarter of a Circle defcribed, as DEF, ot that proportion (which, it muft be ob- vious, is rs fufficient as the whole) the reft is as already defcribed. Other Points, if requifite, may be taken, as f and g, and more, if necefi'ary. in common Pra£f ice (in a Circle not very large) eight Points being fufficient, •^here is no real neceftify for defcribing an Ark, geometrically; fore (ore) the Point where a Perpendicular, from E, cuts the Diameter, is diftant from A or B fomewhat more than onefeventh part of the whole Diameter, AB; lo that, in all common Cafes, it may be afcertained near enough. The 2nd Problem is uleful in many Cafes. Having obtained a Line, AB, by any means, on the Pic- ture; which being known to be the Diameter of a Circle, the whole Circle may be projecled by the ineans there defcribed ; or by Prob. loth, Cafe 3rd, with the greateft exadnefs. Having obtained the parallel Diameter, DF, the affinity with this laft Problem is difcernable, EXAMPLE XXII. How to reprejent a plain, circular Arcade, cafually inclined to the Pidiure. Fig. 94. Let VM be the Vanifhlng Line of the Horizon, C the Center of the Pidure, CE its Diftance, and A the Interfering Point of the hither Angle of the Steps. Through A, draw A J parallel to the Horizon; which may be confidered as the Ground Line; alfo, draw AG perpendicular, the vertical Interfedlion of a Dia- gonal Plane ; on which, fet off, from A, the feveral heights and proportions of the Ohje6l, at D, G, &c. t Ex. 13. Angles of the Steps being firft obtained, as at A, a, bf; their length may be acquired by Prob. 7 th. If they exceed the hounds of the Pidure, take any equal portion of their length, as Aj, one third part; alfo, V being the Vanifhing’ Point of tliat Side (lee Prob. 21, Meth. 3rd) and VE its Diftance, make VD one third of VE, and draw J D, curing the indefinite Reprefentation A V, at N. AN repiefents a length equal to thrice Aj. (Prob. 17.) The Steps are fmifhed as in the 13th Example. Next, the proportion of the Plinth, ecf, is determined by the fame ; and the others, gh, ik, and 1 , by Example 4 ; which, with the Piers, are fo many Paral- lelopipeds, of equal magnitude and equally fpaced. (See Example 4.) Their meafures are fet oft, at i, 2, 3, on the Ground Line; and projedled to their Seats, on the Ground, at 3, 4, 5, 6, &c. by means of the Point £, on one Side; the Diftance, VE, of the other Vanifhing Point (V) falls out of the Pidure. Their height, and the upper Border at H, are obtained, by their geometrical height, A D (as the Pedeftal in Example 3 th.) Thofe above are perpendicularly over the Plinths below; the Band, or Gaping on the Top, is determined from Gj and, by drawing Lines to both Vanifhing Points, V and Y. The laft is not in the Pidure, and muft he drawn by Prob. 13, or the Point muft be afcertained ; where, E Y, produced, would cut the Vanifhing Line V M. E is the Diftance Point, of the Eye, for the Vanifhing Point (Y) of the Lines AY, &c. To reprefent the Arches; which, in this Example, are inclined to the Pidure (and conlequently, their Reprefentations are Ellipfes) the laft Figure is adapted. D, on the vertical Interledion AG, is the height of the Ipring of the Arch. Make DE and EF equal to the meafures on AB (Fig. 93) i. e. make DF equal to tlie height of the Arch (half the fpace between the Piers, equal CF) and DE equal De (cqu al LE) and, by means of the Vanifhing Point of the Diagonal (M) transfer them to the Angle of the Building, at H, I, K; or, if HK be the deter- minate height of the Arch, at that Angle, the reft was unneceflary. From Seel. VIII* 187 APPLIED TO ROUND OBJECTS, From H, draw Lines to both Vanifhing Points, V and Y, cuting the Piers, at a, /i m, n, &c. Bifedl m n, perfpeclivcly, at S the center of the Arch. Draw Sp perpendicular, cuting a Line, from K to the Vanilhing Point, Y, at p, the crown of the Arch; alfo, draw no and mq perpendicular; mqoti is half a Square, circumferibing the Arch. Draw the Diagonals So, and Sq, cuting a Line from I, at r and s. A Curve deferibed through m, r, p, s, and n, is the front of the Arch, the Reprelentation of a Semicircle. If more Points are requllite, fee the laid Problem. The inner Curve is deferibed as the outer ; nn being the thicknefs of the Pier, obtained below ; from C and d in the corner Pier. From all the Points in the outer Curve, draw Lines to the Vanifhing Point V; and, the inner Diameter, anfwering to m n, being drawn, SV cuts it at /' the Center; from which, the Diagonals/o, //, being drawn, in the Rectangle mqon^ give the Points r, and j; and a Perpendicular from /gives />; then, a Curve drawn through w, r, /), r, and «, determines as much of the Soffit of the Arch as can be feen. After the lame manner, the Arches adf, &c. arc deferibed, in the returning Side; which vvould be fuperfluous to repeat over again; as the Lines themfelves fhew their ufe and application the fame. If VL"' be made equal to V E, on the Vaniffiing Line of the front Plane, EMs the Vanifhing Point of one Diagonal (as ce) in each front Arch. EXAMPLE XXIII. T 3 reprefent an upright Cylinder y of any given Dimenfiom. Let A B be its Diameter. The height of the Cylinder is fuppofed to be known. Let V be the Center, and VC the Diflance of the Pidure. Defcribe a Square ABDg (Prob. 19) and inferibe a Circle aceg (by Pr. 2 or 3.) At the two Fxtremes of the Ellipfis, a and d, draw Perpendiculars, which re- prefent Sides of the Cylinder. Their height muft be determined by a Perpendi- cular either from A or b, or any Point in the Circumference. Make A^andb/ each equal to the determined height of the Cylinder ; the Point is in the Circumference at the Top ; and, if A C be drawn, it will cut Perpendiculars from a and e, at a and f, which are in the fame Circumference. bN will cut a Perpendicular from f at /; and thus, as many Points as you pleafe may be obtained, as <2, by c, / e^f g, and -6, correfponding with a, b, See. below; through which, the upper Curve may be deferibed. Or, having obtained any three Points, the Curve a, b, Cy dy may be deter- mined, by Prob. it, with the greatefl accuracy. The Plinth, ABD is parallel to the Picture; and the Abacus, EFG, is equally inclined on both fides, each of which reprefents a Square (fee Prob. 29 and 30) the Curves are ilill the fame ; and if they are equally diflant from the Vanifhing Line (VC) they are equal and fimilar Ellipfis. It is obvious that, as the Diameter of the Cylinder depends on the Ellipfis, at either Bafe, it is very liable to error; as the Elliplis is contracted or lengthened, it will be fmaller or larger in Diameter. To obtain the true Diameters of feveral Columns, with accurac}^ they muft be drawn in their true geometrical propor- tions, as in the next Figure. Let A B, D F, and GPI be the Sections of three Columns by a horizontal Plane, in which the Eye may be luppoled to be, at E; and, let AH be a SeCtion of the Picture, i e. fuppofe a Plane Handing upright on AH parallel to tlie Columns ; EC perpendicular lo AH giv^es its Center, and Diflance. J Prob, 8. Fig. 95 ^ Fig. 96. The Book III. iS8 PRACTICAL PERSPECTIVE PlateXXI. The Vlfual Rays EA, EB, &c. being drawn, are in the fame Plane ; and the' Points A, By Dy where they cut the Pidure, It is evident, are the places, where the apparent edges of the Columns will appear, and conlequently, AB is the apparent width of the Column AB, DF of DF, and G H of GH; which, on ac- count of the Rays, EG, EH, interfeding the Pidure more oblique than thofe fromDF, has a larger Reprefentation; notwithftanding it is really further from the Eye, and from the Center of the Pidure, at C. If the Eye be moved to E, the doted Lines fhew the difference in their Pro- portions, from the two Stations or Points of View ; where, the difference between the Reprefentations of DF and GH are much larger than at E. If the Pidure be at ah, parallel to AH (the Eye being at E) the Proportions of the Reprefentations are in the fame Ratio, as their Diftances, E c to E C. The true pofition of the Pidure from that Station, is a b, whofe Center is at c, Ec perpendicular to ah, bifedsthe Optic Angle, aEh; which Optic Angle, of the other Pidures, Is lEh; c i being equal to c h. And thus the Diameters may be obtained on any Pidure, fituated to the Columns in any Angle, atpleafure zs(HAK) the apparent Diameters on AK, are where the Vifual Rays cut It, at A, /, ky /, K. See this matter more fully treated, in the fixth Sedion, of the Theory, Fig. 34. EXAMPLE XXIV. How to reprefent fever al Cylindersy jn the faine Right Line j inclined to the Piciure, at pleafure. F‘S- 97 - f Prob. 3» Let X, Y, and Z, be the Seats of three Columns, on the Ground Plane, between the parallel Lines, AB and CD; inclined to the Pidure in the Angle BAF, at difcretion. AF is the Interfedion ; and H L, the Horizontal Vanifhing Line ; C is the Center, and CE the Diftance of the Pidure. Find the Vanifhing Point, H, of the inclination of the Line of the Columns, A B (Prob. 17.) and make H E L a Right Angle; which bifed by the Right Line E O. L is the Vanifhing Point of the other Sides of the Squares; in which the Co- lumns, i. e. their Plans are infcribed ; and O of the Diagonals. It would be fuperfluous, after fo many Examples of the fame kind given, to fhew how the Reprefentations of thofe Squares are obtained; either by applying their feveral Diftances from A to a, b, c, &c. (as in Ex. 4) or, as the Square ABFD is found (Prob. 19) feverally, at difcretion. The doted Lines fhew how the laft is managed ; and it is the beft, when there are Circles to be infcribed ; as they are reprefented at afy eg, and eh. Which being obtained, feverally, by the laft Example, draw AK, perpendicu- lar, and equal to the height of the Columns. Find the correfponding Squares of the tops of the Cylinders, in which inferibe Circles perfpedively -f*, and draw perpendicular Lines, aiy cly emy &c. repre- fenting the apparent edges of the Cylinders. If they were to reprefent Columns, dlminifhing at the upper ends, the Squares containing the Circles mu ft be reprefented lefs, in proportion to their refpedive Dia- meters ; and the Lines, which are, in this Cafe, perpendicular, will be gently curved ; having obtained the Diameters above and below, a curved Ruler is the beft expedient, for drawing the apparent Lines of the fides of the Columns. Now, although this method of obtaining the apparent Diameters, if it be ac- curately performed, is ftridlly true ; for, the reprefentations of their Bafes and Tops being true Elllpfes, there cannot poftibly be any error ; yet, the performance of it is liable to great error, becaufe it is impoftible to deferibe the Ellipfis accu- rately; and therefore, the Diameters cannot, by this means, be truly obtained ; efpecially at the bottom, being fo near the Vanifhing Line ; but, much more ac- curately, by the laft Example. ^ Then, Mi Then, 1 FhtcX'K •iff' • T if' •'^(^ '■'A ■-:n » ' 1 ' ',. (• ♦ \ > v" ■• J ■• ■ ' • '/ •f' ■ ' [ . V ..i’C, >'4aC i/- iaL t .Se£l. VIII. APPLIED TO PvOUND OBJECTS. ' .1% Then, fince EC is the Diftance cf the Picture, produce it to the Interfeclion, at r t Tnrl in ation A C' and RD t-lip l-Jnri'/nn J- • r‘n^^n, p", and fromy, where i V, 2V, cut the Diagonal AL, draw Lines to Y, cuting &c at fySyty through which Points, on both fides, draw the Curves of the Abacus. At the Corner A, defcribe an Equilateral Triangle (x) perfpedively, and the fame at each Corner, B and Cj giving the mitre Angles of the Moulding, as at A and B; (No. i.) which, in this cafe, are not bifeded, having a greater pro- jedure at the Corners, than in front. This operation done, draw a V, and aY indefinite; to which, draw Perpendi- culars from g, by /, ky &c. alfb, from «, o, p, &c. and draw g Y, h Y, &c. cuting them at n, o, p, &c. through which Points, defcribe the Curves bnopq, and arstu. In refped of thefe Curves, the operation above is not abfolutely neceffary. After the fame manner, the Abacus may be compleated ; viz. by projedlng the curves of the Fillet, in the middle, and at the bottom, in their refpedive Planes. The places of the Leaves may be obtained by deferibing an Ellipfis in the repre- fentation of a Square, inclined to the Pidure, as the Abacus. 3 D This PRACTICAL PERSPECTIVE Book III.- J98 Plate may alfo be done above, at v, u, x, y, z. The Diagonals and the two XXII Diameters, whole Originals are perpendicular to the Fronts, give the apparent vvidth of each Leaf, in the firft row, allowing a little fpace between them, and they alfo give the middle of the upper Leaves j between each, is a kind of Ballufter, from which the Caulicola and Scrolls Ipring. The true places and middle of the Heads may be obtained, by defcribing repre- fentations of larger Circles, concentric with the other. At Z is a Leaf, in front, Defcrtbe two Ellipfes (cef and ghi) giving the greateft projefture of the Leaves, in their true places ; their heights above the Aftragal (FG) are taken,- from the Profiles (at X) to the greateft projedlure of the Heads (at a and b) which projedfures (ac^ and bd) refpe£lively, are added to the diameter of the Column at the Aftragal, for the diameters of thofe Circles, reprefented by cf and gi. Thefe Ellipfes being obtained, in their true places, infcribed in Squares (as HI) inclined to the Pi6lure as the Abacus, for the upper one ; the Diagonals and parallel Diameters, give the true middle of each head of a Leaf, at c, d, e, &c. Thofe of the lower row fall between them ; as defcribed, in the Plan above. The Volutes, at the corners of the Abacus, are, except in dimenfions, the fame as the Ionic, and muft be obtained the fame, by perfpe£live Plans, at ;e,jv, z. Ho w they fall into the Caulicola, with the other decorative parts of the Capital, the Figure only can defcribe; for, I fairly own that it is not in words to delcribe it, fo as to be of any fervice in delineating. Being well acquainted with the le- veral parts of the Capital, the Rules I have given are fufficient ; the reft is beft defcribed by a careful and attentive perufal of the Figure. Thofe who are not verfed in it, would do well to draw from a real Capital, firft, in all pofitions ; after which, they will be able to delineate it by Rule. SECTION IX. Shewing the application of the whole to entire BUILDINGS, and regular pieces of Architecture. I N this Sedllon, I have applied the whole of what has been already done to corn- pleat Objedls, and particularly to Architefture ; as being, of all other, the fiteft Subjedl, and gives the greateft luftre to Perfpedlive ; on account of the regular dif« pofition, and arrangement of the feyeral parts of a Building, and being compofed, chiefly, of plane Surfaces, which generate Right, Lines-. * It is, to me, furprizing, that Artifts, in general, have no better notions of Per- fpeut little from the fedlion by the Plane, on VY ; becaufe the Optic Angle, ASC, is but fmall, and therefore the extreme Rays, AS and CS, cut the Plane not very oblique ; which, being increared,occafions Diftorfion. SV, parallel to AB, and SY, to BC, deterrhine the Vaniftiing Points of ho- rizontal Lines, in the Building, i, 2, 3, &c, may be confidered as Interfedling Points of feveral Lines in the Objedi. * BS perpendicular to the Figure does not necefiarily fall oh the Angle B (in the Apparatus it is fomewhat on the right hand) for, if the Station was ever fo little on either lide, it would fall differently, as muft be obvious, from the method given of determining it. Having 799* Plate xxin. Fig. 106. f 1. 8. El. 200 PRACTICAL PERSPECTIVE Book 111 Plate XXIII. Fig. 107. Fig. 108. Having thus determined the true pofitlon of the Pidure from the Station S, an- fwering to the Pidlure, MNOP, in the Apparatus; which is direftly between the Eye and the Obje£l; I fhall, next, (hew the great abfurdity of placing it otherwife. Let every thing remain in the fame Pofition ; S is the determined Station for the fameObjeft; and confequently it mu ft appear the fame, whatever Pofition the Pidure is fuppofed to have, in refped of the Objed. But let it be obferved, again, that the Reprefentation may differ very widely ; as will be (hewn. It is ufual with all (who know a little of common Perfpedive only) to make one Face of the Objed parallel to the Pidure ; in which cafe (the Objed being right angled) the Center of the Pidure is, necelfarily, the VanKhing Point of horizontal Lines in the other Side. Confequently, if two Faces of the Objed are feen, the Pidure muft neceflarily be obliquely fituated, in refped of the Objed and the Eye. Now, can any Perfon conceive it rational, or eligible, to place the Pidure in the Pofition ED, parallel to the End of the Building, to be feen from the Sta- tion S; and yet, this is the very fuppofition, in the Reprefentation exhibited by Fig. 107; whofe Center is C, and Diftance CE (equal twice SE, io6) which Pofition anfwers to the Pidure MNOP, in the Apparatus. How egregioufly abfurd is the Idea of viewing a Pidure in that Pofition; and yet, in any other Point of View it cannot appear like the Objed reprelented. The diftortion of the Front of the Building is obvious, but much more fo, is the Face X, of the Bow Window; whiift the other (Z) is geometrical, as well as the end of the Building. The Pofition being determined, and the Center (C) fixed; and alfo, the Interlec- tlon BG, of the front Plane (Its diftance from the Center, equal twice EF, 106) the Reprefentation is delineated by Prob. 1 9 (as by Ex. i and 2) in which cafe, C, the Center of the Pidure, is the only VanKhing Point. Now, thofe Artifts, who make a kind of neceffity of delineating their Objeds after this method, cannot, I conceive, give a reafbn, why, in a general View, fuch Objeds muft ncceffarily be fo fituated to the Pidure, as to have fome Face parallel to it; or, why the Pidure (hould not be placed parallel to the Front (AB) if it muft be fo to either; excepting that, they rather choofe the one to be parallel than the other; for there is as much propriety in one, as in- the other, but the Reprefentation will be much more diftorted; becaufe the Diftance (SG) of the Pidure (landing on AG, is lefs than SE in the former; and, the much greater length of the Front (AB) occafions a greater obliquity of the Rays, with that Pidure. For, SG being the Dired Radial of that Pidure, ASG is but half the Optic Angle under which it is feen; but it is manifeft, that the Eye cannot take in as much on the right-hand, in which cafe, the whole Angle will be very obtufe. C A SE THE Second. Fig. 108 exhibits a Reprefentation of the fame Objed and from the fame Sta- tion, in that pofition of the Pidure; in which, the Front is geometrical; but the projedures of the Cornice, Steps, &c. are dragged out prepofteroirily ; and the Face X, of the Bow Window, isworfe diftorted than before. C is the Center of the Pidure, and CE its Diftance, equal to twice SG (106.) The whole of the Objed ought by no means to exceed the Point E, from the Center ; whereas, it extends above twice CE, which is prepofteroUS. KL is the Vertical Line; and the Angles, CEK, CEL, being made each equal to the Inclination of the Roof, determine the VanKhing Points of the Lines F G, and FD In the Roof; K is alfo the VanKhing Point of HI parallel to FG; the reft is determined as the former, in which, thofe Lines are parallel to the Pidure. In both thefe Pidures, the horizontal Lines, in the Face, X, of the Bow Win- dow, vanKh in the Point of Diftance, in the Horizontal Line; equal CE, on the other Side of C, becaufe the Figure of it is a regular Odagon, (See Prob. 25.) c CASE ^Scaix. APPLIED TO BUILDINGS. 20i •CASE iT H E T Ha R D. Example XXXII. Fig. 109 exhibits a true and-natural Reprefentatlon of the fame Obje£t from the fame Station, in the rnofi judicious pofition of the .Pidure ; and, in order to (hew ‘the affinity between this and the Pidure MNOP, in the Apparatus, I hav^.niadc ufe of the fame Letters for Reference, where tliey can be properly applied. V Y is the Horizontal vanifhhig Line, and C the Center of the Pidure. Draw CE perpendicular to V Y, and equal to the Diftance (equal twice S B, and make the Angles CEV, CEY equal, refpedively, to the Angles BSV, BSY (106) or, make CV equal twice BV, and CY twice BY ; V and Y are the Vaniihing Points of horizontal Lines in the Front and End of the Building-j-. t ’I’*'" Then, becaufe the Angle E,iven for projefling all kinds of Figures of which it is compofed ; and finding the Va- nilhing Points, &c. at any given Diflance and pofition ot the Piflure. Figure 106 exhibits a general Method for contrading the Faces, &c. of an Ob- jecl, and for obtaining the true place of each parr, in refped of its bearing with others (being detached) from any determined Station, wh.ch, in many caies, is the befl: method of proceeding, and the leaft liable to error, EXAMPLE XXXIX. ' How to refrefent Doors and Window Shutters, open, in any given Angle or at pie a Jure, Let A V be the indefinite Reprefentation of the Side of a Building in any hori- zontal or vertical Plane j which, from the point A, is required to be peripeclively divided into certain finite parts, reprelenting Windows, Piers, &,c. , By the 8th Problem, Ac and VE are drawn parallel, between themfelves (V^E may be confidered as the V anifhing Line, and Ac, the Interledion, of whatever Plane the Original Line is in) make VE equal to the Diflance of the Vanifhing Point (V) of the Line A V ; and, on Ac, take A a, ab, &c. equal to the knowii proportions of the Piers and Windows, or Apertures of any kind. Draw aE, bE, &c. cutmg A V in the Points a, b, c, &c. which are the re- prefentations of the Original Points, a, b, &c. Wherefore, if ab be fuppofed the geometrical proportion of a Window, then, ah reprefents the fame perlpec- tively ; and fo of any other divifion, as be of a Pier, &c. N. B. Let it be obferved, that it does not depend on the real meafurcs being: applied on Ac, but, that they are in the fame ratio, refpedtively, EY is pf the diftarice of the Yanifhing Point; agreeable to Theorem 13 th, which is frequently exemplififd in the preceding Work; and, whether AV repre- sents a Line perpendicular or inclined to the Pidture, there is no difference in the procefs. Let AB reprefent the aperture of the opening of a Door, in the fide of a Room, &c. which is required to be feen open, in any pofition, at pleafure. Now, AB, the width of the Door, is the radius of a Circle, which it wmuld defcrlbe if-the Door revolved quite around; and confequently, the Door itlelf (being a Retlangle) would deferibe a right Cylinder, of which, BC is its Axis. By Prob. lotb. Seel. 4th. or, Pr. 2, Setl. 8th, deferibe the reprefentation of a Semicircle, Aa^F, whofe Radius (given or found) isAB. It is manifefl, that AB the perfpedlive \yidth of the Door, would, in its femi- revolution (from AB to BF) deferibe the femi-Ellipfis Aa^ 7 F; that is, the point A (B being fixed) would deferibe the femi- circumference of a Circle; and confe- quently, in whatever pofition the Door is open, the point A will be fomewhere in that Semicircumference, as at a or <7; wherefore, aB, a^, &c. are alfo repre- lentations of Radii, of the fame Circle. Take the Point a or a, in the Circumference, at pleafure (according as you re- .quire the Door more or lefs open) and draw aB or aB, which produce to its Vanifhing Point, H or H. Draw the Perpendicular ad, or ad, and, through C, draw HC, or HC till it cuts ad or ad, at d or d; then is aBCd, or aBCd, the reprefentation of the Door^ open in the pofition required, Or, Scd. IX. APPLIED TO DETACHED BUILDINGS. aiS - Or, if the Angle of the opening, on either fide, be known or determined ; produce A B to its Vanifhing Point (V) ; S being the Center of the Pidlure, draw SE perpendicular, equal to its Diftance, and join EV ; make the Angle VEH, or V E H", equal to the Angle determined ; which will give the Vanifhing Point H, or H, of the top and bottom Lines of the Door, in the pofition required. It the Door be required more open, as making an acute Angle, on the otlier fide ; then, the Vanirtiing Point will be on the other fide of V, The procefs is' the fame, in every Pofition, After the very fame manner. Window Shutters, being vertical, are determined ; Fig, If5. the Radius, of each, being half the width of the Window, AdordD. ABCD is the aperture of a Window, to which there is required the reprefen- tations of folding Shutters, open in any pofition or angle, at pleafure. Produce AxD and BC, both ways, and bifedl either, perfpedively (as BC) and, make B^ and C^ each reprefent half BC. On ad and dg defcribe reprefentations of Semicircles, abcd^ d.v\^defg', in which, according to the determined angle or opening of the Shutter, take b ore, e,f, pr^; from which, draw perpendicular, and draw ^B, or to its Vanifhing Point, at S or V ; from which points, draw SA, or VD, and produce them cuting the Perpendiculars, at b or e, which com- pleats the Shutter AB/^b, or C^’eD. ABcc reprefents another, lefs open ; and C/fD is one parallel to the Picture, confequently, it has the true figure and proportion of half the Window, AB^a, and C^gD reprefeiit others, quite open, againft the Walk The Shutters Y and Z defcribe Semicircles in vertical Planes, as abed, or Fig. ii6^ abed', or, more properly, they defcribe half horizontal Cylinders, . The firft, being parallel to the Picture, are Semicirclest ; the others are perpen- t Theo. 9, dicular to the Pi^ure ; or they may be inclined, in any Angle whatever; the Va- -5' jiifhing Points of the deferibing Lines are in the vertical Vanifhing Line (VL) of the Plane of the Circle ; as at V and L, &c. the front Lines are parallel. |nthe former they are parallel to the Pidlurej the front Lines vanifh in the Center. SECTION X. OF INSIDE VIEWS, IN GENERAL, T H E application of the Rules of Perfpeflive, whether to the interior or ex-? terior parts of Objects, is the fame, for. Planes and Lines, of which Ob?- jeds are compofed, are ftill the fame however fituated; whether they form inter? nal or external Angles, of Objeds feen externally ; or, forming Rooms or Conca? vities of any kind. Neverthelefs, at firft Sight, there appears a difference ; for I believe that, many who have pradifed Perfpedive with tolerable fuccefs, in ex- « terior Objeds, are fomewhat puzled, at thp firft attempting an Infide View ; not knowing, rightly, where to begin or where to leave oft; how to place their Pic- ture, or determine its Diftance. • I have feen an attempt to reprefent an Odagon Building, internally, in which Pidure were introduced feven Sides or Faces, out of the eight; it was a misfortune that the other could not be feen alfo, quite around, and then it would have been 'a mafler-piece, indeed. However, from wdiat was feen, and the tendency of the Lines, it was manifeft, that the Diftance of the Pidure was not more than one fourth part of its width, or length ; and confequently, the Optic Angle was above 120 Degrees, which ought not on any account (except when the Eye is confined to the true Point of Vievy, always) to exceed 60, I ^Tis 2j6 PRACTICAL PERSPECTIVE Book III. Plate XXIX. ‘Tis the fame thing, if, in order to fee the whole of a Dome and Cupola, inter- nally, we advance fo far into the Building, that the reprefentations of fuch Sub- jects are only fit for horizontal Pictures, or Cieling-Pieces ; for, when they are reprefented on a vertical Picture, they appear (at a proper Diftance to take in the whole) as if falling, or not upright. Yet are thefe things to be met with in Pic- tures and Prints, by Men of fome diftinCtion in the Arts. That the PerfpeCtive Reprefentation of the infide of a fine Building is more difficult to manage than the exterior is certain, and is manifefted in theatrical performances; which, being reprefented on feveral detached Planes, it is impof- lible, by the Rules of Art, to make them correfpond in every Point of View ; but, they are very rarely connected in any one Point. There are, undoubtedly, many fine Performances of the kind ; yet, without attempting to difparage their Authors, I am confident, that, were they better acquainted with PerfpeCtive, they would produce better and more natural Reprefentations ; even that famed Scene in Cymon is a jumble of inconfiftencies, both in point of Defign and Execution ; although it was a bold attempt out of the common mode of Reprefentations. Few Artifts have made PerfpeCtive fo much their Study, to know how to pro- portion one part to another, on detached Scenes, fo, as to make the whole unite in the proper Point of View, whether the Reprefentation be internal or external ; indeed, from the conftruCtion of the Theatres, it is fcarce poffible to be done ; nor are the Rules of PerfpeCtive fufficient, for the purpofe, without a tolerable knowledge of Lines, geometrically. It is the leafl qualification of a Scene Painter to be excellent in Landfcape, in which a fmall knowledge of PerfpeCtive is requifite ; but, in order to execute Defigns in Architecture with correCtnefs, and a juft proportion of the feveral Parts, requires a thorough knowledge of Per- fpeCtive. Is it not furprizing, that all who are concerned, or any way en- gaged in Scene Painting, do not make PerfpeCtive their immediate ftudy ; being the balls, the very foul and exiftence of their Profeffion? yet, to my certain knowledge, feveral Artifts, employed in it, are not only totally ignorant of it, in Theory, but they are, almoft, wholly unacquainted with its Rules, which, to me, is moft unaccountable. In Infide Views, the bounds of the Picture limit the whole, every way, which renders the operation, in fome cafes, more difficult than external Views. In order to (hew the infide of a Room, Temple, &c. properly, a SeCtion is fuppofed to be made; that is, the hither Wall (or inclofure of any kind) is fup- pofed to be removed, and the whole Infide laid open to View ; fo that, a proper Station and Diftance may be taken, from which, the whole, or as much as is re- quired, may be feen under an Angle not too large ; for, to fuppofe that we can be within a Room and exhibit the whole, or nearly, is truly abfurd ; yet it has been attempted, by thofe who have not a juft notion of Per^eClive. In refpeCl of determining the places and proportiona of Doors, Windows, &c, it muft be obvious, there can be no difference whether they are interior or exte- rior; the whole of which is contained in Prob. 8 th, and .has been unlverfally ap- plied throughout the Work ; particularly in Ex. 2nd and 4th, alfo in the 16 th, 19 th, and 21 ft, in finding the places of the Mutules, Modilions, Blocks, &c. EXAMPLE XL. How to reprefent the injide of a plain Building', the Sehiion, by the Pidiure, being parallel, to the Knds of the Buildings. In reprefenting the Infide of any Room, Church, &c. it is ufual to take the Sta- tion in a Line drawn through the middle of the Building, which indeed appears the moft rational; but in fuch cafe, in a regular Building (one fide being a du- plicate of the other) it is not fo piClurefque, as when the Station is towards either fide, as at S; from which, the infide of the Room is to be viewed; by which means,’ either Side may be fhewn to greater advantage, being more oppofed to the Eye than the other. Fig. 5 Sea. X. APPLIED to INTERNAL VIEWS. i No. I. is the Plan of a Prifon ; bn the left hand are Doors into feveral Cells, Plate XXX. Fig. I. Juftice. V, U, &c. are Piers fupporting the Roof. The Elevation, and Seaions, are fuppofed to be undcrftood. Let ABbe the place of the Piaure; by which, a feaion of the Building is fup- poled to be made, and the hither Wall removed ; fo that, the whole of the Infide, beyond it, being open to view, is intended to be reprefented on the Piaure AB, from fome determined Station-, as at S; asfeen through the Piaure, being fuppofed a tranfparent Plane. SPj SQ, &c. may be fuppofed Vifual Rays cuting the Pic- ture, which give the true place of each part of the Building thereon ; indicat- ing alfo, what Parts can be feen and which not, being obftruaed by the Piers. Let AB be produced to b, ab is the Ground Line of the Piaure; but for the conveniency of making a correa perfpeaive Plan,. take AB at a proper diftance, below; in which, let A B be taken equal AB (No. i.j or in any other proportion toAB; it is, here, taken double, or twice. Let the Horizontal Line (ECG) be drawn, at a proper diftahce from ab, equal to the height of the Eye above it; in which, take DC double AC (No. i. where a perpendicular, from S, cuts A B) C is the Center of the Piaure ; alfo, make CE double SC (No. i.) Eis the Eye, or Point of Diftance, by which the feveral parts of the Piaure are proportioned ; C D and C F are each half the Diftance. HI, drawn through C, is the Vertical Line of the Piaure ; which determines the iituation of the Eye, in refpea of the Building. Let the perfpeaive Plan be drawn, making IB, See. each double the meafures in the Plan, for the thicknefs of the Piers, and their diftance from the Station Line. How it is perfeded, it would be impertinent to explain ; the Point E being ufed, where the meafures Cb, Cc, Sec. are the full meafures (that is, double thofe at ab, be, &c. in tne Plan, No. i.) and, when it is more convenient to apply the half meafures, make ufeof the Point D, or F, on either fide, as occafioii requires ; as A D, for the Door at P ; the full meafure being applied from A to b, gives the fame Point p, as the half meafure, at D. A^is equal to AQ (No. i.) which, by drawing a Line to D, cuts AC at the place of that Corner, in the perfpedlive Plan ; making A ^ reprefent twice AQ (No. I.) feen at the Diftance CE; or equal AQ, at the Diftance CD. . AC, or CC, being proportioned at b, c, d, &c. perfpedlivcly, for the feveral Piers, &c. Lines parallel to A B, being drawn, determine thofe on the other fide, at g, k. Sec. and againft the Wall, at n, o, &c. by which means theTlan is compleated, in Perfpedtive, from which the Pidlurc is delineated, as follows. The meafure, on AB, being transfered to ab (that is, refpecting the fpace be- tween the Wall and the Piers, the thicknefs of the Piers, and diftance between them, in front) draw aC, cC, Sec. and, at draw perpendicular, the ver- tical Sedlion of the plane of the Piers, on the right hand, with the Pidlure; which, being fartheft from the Vertical Line, is beft, for proportioning the heights of the Piers, Arches, Sec. which arefetoff, from at a, b, c. Sec. according to known, or determined, meafures of their heights, in proportion to the Plan. Draw .^C, aC; bC, &c. and, from the Plan below, draw perpendiculars from g, i. Sec. cuting them, at g, /, Sec. Perpendiculars being drawn from all the other correfponding parts in the Plan, as 1, m, n, o, &c. parallel Lines drawn through g, f, Sec. determine the true places of all the Plinths, and Borders, at the tops of the Piers, from which the Arches fpring. The Arches, on each fide the Avenue, In front, are Semicircles ; wherefore, their Centers being obtained, at S, by bifediing the Diameter R T, they are readily deferibed; and, by drawing Lines, from S, on each fide, to C, the Centers of all the receding Arches are determined, in thole Lines. ■3 I In &c. and on the right are Doors and a Window, into the Area, or open Yard; at the farther end is a paffage, at R, leading to the Keeper’s Apartments, and, at the End is a Stair-cafe ; a few Steps of which lead, through a Door, to the Courts of 2lS Plate XXX. PRACTICAL PERSPECTIVE Book III. In the returning Fronts, the Arches are femi-EHipfes ; their heights are the fame as the other, determined on. A B, at d ; Cd being produced to the crown of the Arch, at D, obtained from below, at N, projeded. For, feeing that the Pidlure, on A B (No. i.) is nearer to the Piers than the Wall, confequently, it does not cut the Arches in the middle j wherefore, all the Wall, with the elip- tical Window, on this fide of projedls through the Pidlure and is projeded again to it. Cis the Center of that Arch ; the Curve, Def, is determined after the fame manner as if they were circular, it being obvious, that the fame lengths, from i to k, or from f to /, between the Piers, may reprefent any length of the Indefinite Reprefentation, AC, according to the diftance of the Eye (Cor. Th. 12.) AC remaining the fame. Hence it is undeterminable whether thofe Arches reprefent Circles or EHipfes, of any proportion, the Diftance of the Pidure being unknown ; for the two middle Arches are more excentric than the other two, feeing they have the fame height, and the Piers are wider afunder*. The eliptical Windows, over the Arches, are defcribed after the fame manner ; as the firft on the right hand, half a Window, infcribed in the Redangle abgh, indicates fufficiently. The groined Arches, over the fide Avenues, have an appearance of difficulty without the reality. If a Line be drawn through the crown of the Arch, at c, to C, and, diagonal Lines being drawn between the corners of oppofite Piers, in the perfpedive Plan, as rt and ms, interfeding at u, from which, a perpendicular, being drawn, cuts cC at c, the center of the Groin. Or thofe Diagonals may be drawn, at the tops of the Piers, and having drawn perpendicular Ordinates to the circular Arch, at A, B, they may be transfered, to the Diagonals ; where, perpendiculars being drawn, and, to C, Lines drawn through the points a, b, c, &c. in the front Arch, cuting the diagonal Ordinates at a, b, c, &c. through which the Curve may be accurately defcribed, by encreafing the number of the Ordinates at difcretion. The Arches in the middle Avenue are alfo femi-Elllpfes, geometrically defcribed, being parallel to the Pidure ; the tranfverfe Diameter of the firft is FG, and CD half the Conjugate. Having determined feveral Points, i, 2, 3, at equal diftances, or otherwife draw the.Ordinates 01, A2, B 7, parallel to CD ; from which Points, A, B, C, &c. draw Lines to the Center, giving the Diameters of the other Arches, HI and KL, cut in the feveral Points a, b, c ; a,b,c\ at which Points, Ordi- nates being drawn, and, from i, 2, 3, and D, Lines drawn to C cuting them at I, 2, 3, through which thofe Curves may alfo be defcribed. The Apertures through thefe Arches, being Circles parallel to the Pidure ; their Centers being determined, and their Diameters, are only neceflary. The large Beam, at O, is in the middle of the opening, at MN, and tends to the Center, on which the Joifts reft, which decline to the Wall on each fide, geo- metrically ; their meafures and diftance from each other being known, are readily determined, by Prob. 10, drawing a Line through B, parallel to MN. The Stairs, at the far end, are managed by Example 8, in front ; the returning flight being parallel to the Pidure are inclined, geometrically, to the Horizon. The front Steps may be proportioned to the opening between the Piers; as at Y and Z, in the geometrical Plan ; the Hand-rails, of which, tend to a Vaniffiing Point, in the Vertical Line, at H ; as the Kirbs in Example 8 th. The Door is geometrically proportioned, on the Steps. The middle Avenue ofthis Building, being loftier thantheSides, the Reprefentation is limited by a Sedion of the Walls, and Timbers of the Roof, over the Sides, geo- * At No. 2. is half a circular Arch, that is, a Quadrant, or fourth part of the circumference of a Circle, in a Square abed ; c is the Center, abed is a ReAangle circuml'cribing a fourth part of an El- lipfis, of the fame proportion as the middle Arches to the circular ones at each fide ; in which, it is ob- vious that, the Diagonals, ac and ar, cut the Curves at the fame height, at e and e ; and if the circular Curve be divided into more parts, at i and 2, thofe being transfered by parallel and perpendicular Ordi- nates, cut the Diagonals of both in the fame proportion ; and confequently they give correfponding Points in the Curve. metrically o Plate XXX pM .'\;xO>\\N \v;-\;"<\.j V- '¥ -- \ _i ^ Se£l. Xi APPLIED TO INTERNAL VIEWS. 219 metrically drawn ; the lower part Is bounded by the limitsof the Pidure in length which, on account of the Station not being in a central Line, has its Center to- wards the left hand, out of the middle ; by which means, it becomes diftorted, on the right, being extended too far from the Center; by which the optic Angle is enlarged, and the Vifual Rays cut the Piflure too oblique, as SB in the Plan (No. I.) in which cafe, no more fhouldj properly, be taken into the Pidure than the Piers, the optic Angle ASf, being about 50 Degrees. EXAMPLE XLI. Is the Infide of a Church, in the fame pofitlon, as the foregoing Example. At Figure 2, No. i, is a half Plan of the Building, as far as the Sedion by the Pidure, on A B. The Piers fupport the Roof as in the laft Example, having Co- lumns on them, of the Ionic Order, with an Entablature all around the middle part of the Building, and a coved deling. At the far End is a femi-odagonal recefs, in which is the Altar and communion Table. It would be wholly ^fuperfluous to form a perfpedive Plan, as in the preceding Example, or to fix the Station, with all the other Preliminaries to the Ground Plan ; fuffice it to fay, that the fituatlon of the Eye is nearly the fame in refped of the Building and Pidure. C is the Center, and CE half the Diftance by which the whole is proportioned ; the full Diftance cannot be contained within the Pidure, on either fide. The places of the Piers, the Windows, &c. it is obvious, are obtained here, as in the foregoing, by applying the half meafures (that is, the full meafures of the Plan) to the Ground Line, AB, fufficiently corred, infomuch that, an extra Plan would be unneceflary. The Piers are fquare, having a Sub-plinth with a Bafe Moulding, and Impoft at the top; on which is a regular Pedeftal, which forms the front of the Gallery, receding fomewhat from the Pedeftal ; all which need no other defcription. The Pidure is fuppofed to cut the Galleries, by which, the Seats, &c. are laid open to view ; the Sedion is drawn geometrically, from which, all the Seats and Backs tend to the Center of the Pidure. The Columns, are fuppofed to be planed at the bottom of each Pier ; but, if a central Line be drawn (as K L) from the middle point of each/ and- the height of the Column be there afcertained, all the reft are determined by Lines drawn to the' Center, giving each its proper Diameter, in proportion to its height. The Bafes of the Columns, being above the Eye, are curved contrary to the iftual order when below it j the Curves are ftill concave towards the Vanifhing Line. The Capitals are according to the-moderri Tonic, in Example 29. The recedes of the lower Windows, -on the Floor, being lefs than the window Frames, cannot be a difficulty ; the floping part. makes up the difference. The Windows above are perpendicular over the other ; the heights of which are deter- mined, on the vertical Interfedions of the Walls, AD and BG; alfo the Gallery, and other parts of the Building. The Nich at the far End, with the Doors and circular Windows over them, ate all parallel to the Pidure, confequently they are all geometrically proportioned, in front. The receding of the Nich, that is, the perfpedive widths of the Faces within, are beft determined from a perfpedive Plan, formed below, the figure be- ing determined ; or, if the whole be drawn geometrically, and the Station fixed, as in the foregoing, Vifual Rays may be drawn, cuting the Pidure and giving their proportions thereon. ' ’ The Mouldings in the Impoft, runing around it, have their Vaniffiing Points, for two Faces', in the Points of Diftance. The middle Face is parallel, and in one, feen on the right hand the Lines tend to the Center. The Compartments in the Sides and in the Head are beft defcribed by infpedion ; the Lines, forming the Head, are drawn after the fame manner as in an odagonal Dome, feen externally, as de- fcribed in the 35 th Example. The tib t^latc XXX. t’RACTiCAL perspective Booklil; The femicTrcular Heads of the upper Windows haVe been frequently defcribed ; thofe below have a flat Arch, which is but little feen, by reafon of the fpherical receffes in the Cieling of the Gallery, which are rejprefented by femi-Ellipfes, in the Plane of the Cieling, ‘ The Entablature* fupported by the Columns, is formed by means of geometrical Sections, at X and Z j the true place being acquired from a Plan below ; or from the Ground Line and vertical Interfedions, at H and I, The Profile of the Motildings, in the Entablature, on one fide, with the large Cove, and a fmaller on the other fide, being drawn, it would be impertinent to de- feribe the delineation of every Line from each Angle to the Center of the Pidure, and how to determine the mitre Angles at the End, where they return, over the Arch; for which, fee Example i6, or 19. For the large Cove, draw Lines from M and N, the internal Angles of the Build- ing to C* curing Perpendiculars from the Angles of the Frize, or the Pilafters, below the Capital, at O and P ; from which, draw Lines to the Eye, at the full Diftance on each fide, and produce them ; then. Lines drawn from Q and R, to C, will cut thofe Diagonals at S and T, and determine the returning Line or Mould- ing, which bounds the plane Cieling. The mitre Angles of the Cove may be truly projeded by means of Ordinates from various Points (as a and h) in the geometrical Curve ; but, as it comes fo fmall, at the place,' it may be done corredly enough, without that procefs ; taking care that it does not curve too much, but leads truly into the Perpendicular and Diagonal, both which are Tangents to the Curve, as RN and N Z, in the geometrical Sedion, and muft never cut it. The Compartments, in the Cieling, are but firnple Figures, as Redangles, with Ellipfes inferibed, (fee No. i.) which are managed the fame as Circles, as it has been Ihewn in the laft Example, in refped of the Arches. The recefles in the Cieling, over the Galleries, ’ are no more than a repetition of the fame thing by a lefs Scale, having only an Architrave below the Cove, which returns at every Beam from the Columns to the Wall ; whofe mitre Angles are de- termined by the fame means, from a Profile of the Cove, at G. Thus, without the real procefs, I have defcribed every neceflary ftep to be taken in the delineation of this Objed ; for, as the lart: is, in refped of its Plan, the fame kind of Figure, the fame kind of Procefs will anfwer for both, in the general parts; and, for particular ones, the Examples referedto give particular Inftrudions. EXAMPLE XLII. Is the reprefentation of an elegant Room, having a large, circular. Bow Window, in the Side, and a Cove-Cieling, inclined to the Pidiure, In the two former Examples, the center of View, not being in the middle of the Pidure, it may be imagined is wholly owing to the Station being towards one Side more than the other ; but I (hall (hew, here, that, it is not ; for, let the Sta- tion be taken where you pleafe, the Point of View, or rather the Center of the Pidure may ftill be in the middle, and yet, take into the Pidure as much of the Objed ; a circumftance extremely Ample in itlelf, though to fome it may appear a myftery* In Plate 29, Fig. 117, is a Plan of the Room intended to be delineated. Let E be the Station determined on ; from which, if Right Lines are drawn to F and B, the extreme on each Side of the Room, intended to be delineated, FEB is the Optic Angle under which it is feen ; and, if AB be the Interfedion of the Pidure, EC, perpendicular to A B, determines its Center, at C, and EC is its Diftance ; in which cafe, the Center of View is not in the middle of the Pidure. But, if the Optic Angle be bifeded by the Right Line ER, then, FG, perpendicular to ER, is the true Pofitlon of the Pidure, S is its Center, and ES its Diftance; and the farther End is confequently inclined to the Pidure; as it is reprefented in the 31ft Plate. See Figure the firft* 5 ■ Let "^Secl. X; APPLIED TO INTERNAL VIEWS. ■227 Let A B be the Ground Line, which is not neceflarily the bottom of the Plfture;^ ibut, whatever falls on this-fide is fuppofed projected to the Pi£lure. Draw AD and BK, perpendicular (the vertical Interfedtions of Jthe Sides) on 'which, fet up all the meafures of the heights (by the Scale of Proportion) for the Dado and Mouldings, the Windows, Chimney Piece, the Entablature^ Gove, &c. On AD, defcribe, geometrically, 'the true Profile- Sedlion of all the Mouldings; as DEF for the Cornice,. &c. the Lines of which (in this Cafe) not being.perpen- 'dicular tO'the Pidure, the Seflion of it, with. the Pidlure, is not the-true Profile; -the deviation, in this, is inconfiderable. But, when the Lines are more inclined to ■the Piflure, an Expedient for truly proportioning the Mouldings may be neceflary* Let ABC (Fig. ii8, PI. ag) be the Profile cf the Cornice to be reprefented in the Pi£lurc. The feftion with the Piflure, being vertical, makes no variation in the heights of the Mouldings ; but, according to the inclination of the Picture, to the Building, their projeftures are varied' confiderably. From each projefture a, b, c, &c. draw Lines perpendicular to BC, confequently parallel amongft * themfelves; make the Angle BCD equal to the complement of the inclination of the Cornice to the Pic- ture; CD will be the extreme projefture of the fe£lion of the Cornice, whofe Inclination to the Pifture is ECD; and, FDC is the true Seflion, in that pofitionof the Pifture ; tire projefture of each Mould- ingbeing taken from CD, where the parallel Lines cut it, at bj c, &c. ■On the other fide, the heights of the Windows, &c. are fet up from B to T. Having drawn the Horizontal Line (VL) and fixed the Center of ^the .Pi(£l:urc, the Vanifliing Point V being determined as ufual ; which, on account of the in- tcllnation of the Sides of the Room, is notin the Center, as iscuftomary in ^Infidc Views; ;the Room, or Building of any kind, being right angled. iln this-'Cafe, thefe Preliminaries are heft determined by the geometrical 'Plan. If the . Ground Plan of any Building which we intend to delineate bc'drawn, rtruly geometrical, the Station may be determined, fo, as to fee fuch parts of the -Objeis to SDt, (Prob. 12.-) And, if the Angle DEK be bife(^d, by the Line EN, 4 N -is the Vanifhing Point of a Diagonal, or mitre Angle. Then, if AB (Fig. i, PI. 31) be equal to FG, make VL equd to DS; or, in whatever ratio A B (Fig. 1) listo FG, fo-make'VC to DS, and V will be the Vanifhing Point ofthe Sides-of the Room; that .is, of all theLorizontal Lines in ithofe Sides, -and of all .other Lines parallel to them, whether in the deling, or (On the Floor, or elfewhere. (Cor. i. Theorem 3.) ■Being thus prepared, we now proceed'to the delineation. 3 k: Draw Practical Perspective Book ill. Plate Draw the indefinite Lines AV and BV, and proportion them, by feting off the vy VT * meafures of the feveral parts from A or B, and drawing Lines to the Diftance pj j ’ Point of the Eye, placed on either Side (at £, or £V being half, and V£* °* * one third part of the Diftance of the Vanifhing Point V, there not being room, on the Picture, for the full meafure. MakeAa, ab, &c. each one third of the meafures of the Originals, and draw a£% &c. cuting AV at a, &c. which give the Angle of the Chimney, &c. For its projedlure, make Aa equal to, or fomewhat more than the real mea- fure (becaufe, AB being inclined, it cannot be equal, but is the Diagonal of the Inclination; i. e. as CD to CB, Fig. ii8) and draw a V, cuting a Line drawm from a to the other Vanifhing Point, at Make in 'the ratio* of one third part of the front of the Chimney (as CD to CB, ii8) and draw JV, cuting a parallel Line from ^ at c. Divide be in the proportion of the Trufl'es, &c. and draw Lines to E\ cuting in their perfpedive Proportions. Draw perpendiculars from a, b, and e ; and, from all the Angles of the Cor- nice, &c. draw Lines to V ; and return the Mouldings, at the feveral Angles of the Chimney, internally, at gh, and externally, at ik and Im, as deferibed in Example 19; alfo, at no, the length being obtained by a Perpendicular from f. To deferibe every particular would be fuperfluous and unnecelfary, previous Lelfons having been frequently and particularly explained. Having obtained the mitre Angle at no, return the Mouldings of the Entabla- ture, at the End, by means of the Vanifhing Point of the End ; or by Prob. 13. The Angle of the Cove may be obtained, with accuracy, having made a geo- metrical Sedlion, as in the laft Example, feting up its height from D, and making its proje£ture as CD to CB (Fig. 118) in refpedl of the true projedlure; then, draw a Line from the internal Angle to V, cuting the Perpendicular fnatp, and draw the Diagonal {pq) of the Angle, indefinite; ^and, from the extreme projeb- ture of the Cove, draw a Line to V, cuting it at q. But, becaufe the Vaniihing Point of that Diagonal is not determined, and is out d ■ of the Pidlure ; from^, draw a Line tending to the Vanifhing Point of the End; and, where it is cut, at r, make rs equal pr, and draw a Line from the Va- nifhing Point of the other Diagonal ; which will give the Pointy nearly *. The other Angle, x, is determined by the Vanifhing Point of the Diagonal xy-\. ^ ' After the fame manner as in the foregoing, viz. by drawing feveral Ordinates to the Curve, at t, u, &c. the perfpedive Curve at the Angles may be deferibed, and transfered from one Angle to the other. On the other Side, V £ being half the diftance of the Eye, the meafures on the Ground Line, AB, are applied half the real meafures, for proportioning the Piers, and the opening of the Bow Window, &c. from B, to e, f, &c. BI is the height of the Windows. Draw Ih tending to the Vanifhing Point of the End ot the Room (as for the projedure of the Chimney) and Bd being made equal to the recefs of the Window, draw d h ; regarding the procedures of the Mouldings. Having obtained the opening of the Bow Window at i k, which is a Segment of a Circle (by drawing e£ and f£, cuting BV) find S the reprefentation of its Center (bifeding ik perfpedively, at 2, and drawing -5*211 to the Vanifliing Point of the End, and 3 V, cuting it at S; B3 being fomewhat more than its Dif- t.mce from the Wall, viz. as CD to CB, Fig. 1 18) and draw Si, and Sk, which reprelent Radii ; by means of which, other Radii may be obtained, as SI, Sm, &c. ' making certain Angles with Sk perfpedively (Prob. 10, Cafe 3rd.) and thus, not only the Curve (k Imn) but alfo, the true place of each Window, &c. is acquired. The Curve at the Top may be deferibed by the fame means. If the Lines, on the Floor, in the receding of the W indows, tend to the Center S, draw SR perpendicular, and divide it, geometrically, into the feveral di- * This is not ftriflly true, jljr j not beihg parallel to the Vanifhing Line ; but, being fo little inclined, the deviation is inconhderable. rs is lets than pr', they may be truly proportioned by Prob. 8, Cor. i. f This Van'fhing Point is alfo out of the Pidure, being diftant from the Center as SNto SE (F/g. 1 17.) the Diltance of the Pidture. . I vifions Plate IXXI dyT Mia/Zon j^farcA . Sea. X. APPLIED TO INTERNAL VIEWS. ’ Z 2 j vlfions for the Mouldings, &c. and producing SI, &c. to the Vanllhlng Lliie ; then. Lines drawn from each divifion, on RS, to the refpedive Vanifhing Point of each Jamb, will divide each Window into its feveral divifions of Mouldings, Safh Squares, &c. and, by the fame means, all the Curves, in the Mouldings, See. at the top, may be deferibed with the greateft accuracy. If the Jambs of each Window be parallel between themfelves, then, a Right Line, So, biieaing the Angle ISm, &c. perfpeftively, will produce the Vaniflaing Point of each relpedfively, and each Jamb muft, in that Cafe, be divided geome- trically, on the Angle, into the feveral divifions required. The Entablature may alfo be proportioned on BK, the fame as on AD, on the other Side ; which will be more accurate, than to depend entirely on the propor- tions being carried around, from the other Side ; which, as they diminifh fo much, at the far End, are liable to error. The places and proportion of the Columns, maybe obtained by Prob. 8 th, thu5. The Angley at the foot of the Pilafter being obtained, draw /j parallel to the Ground Line, which divide, geometrically, in the Proportion required ; that is, y I, equal j 4, is the diftance of the Column, at the Plinth, from the Wall ; and I 2, equal 3 4, is the width of the Plinth of the Column ; from all which, draw Lines to V, cuting the inclined Line, in which the Columns ftand, in the per- fpedive proportions of the Columns and Spaces ; which may be compleated from the known proportions of the Order, as in former Examples. If r s be taken, geometrically, the width of the Door, &c. in proportion to the Columns and fpaces. Lines drawn to V will give the place of the Door, at the farther End, which may be compleated, by Example 20. The Cieling has nothing of difficulty in it, fave the Ornament, being compofed of Right Lines, regularly difpofed ; which, from the geometrical Figure, may eafily be determined. G H, may be conlidered and ufed as the Interfeftion of the Pidure with the Cieling, on which, the divifions are geometrically difpofed ; from which, draw Lines to V ; and the feveral divifions, in the length, being found perfpedlively, on GV orHV, Lines drawn to the 'other Vanifhing Point will cut each Line, drawn to V, in the ratio required ; iby which, the perfpe£live Figure is formed on the Picture. : EXAMPLE XLIII; Is the Reprefentation of the infde of the Piazza, Covenf Garden, from the farther corner of the entrance into the Playhoufe. 4 . .J J 1 The Pofition of the Picture being determined, and confequently, the Inclina- tion of the Lines, in which the Piers ftand, is. known. ’ Let C be the Center of the Pidure, EV is the Horizontal Line, and V is the Vanifliing Point of one Face of the Piers, found, or determined at pleafure ; the other is out of the Pidture, on the Left, found as ufual ; the Diftance of the Pic- Fig. 2. ture is fix Inches and a half, nearly*. Let AB be the Interfedtion of the Pidlure, i. e. the Ground Line, and, let S be the determined Seat ([on the Pidlure) of the corner of the Pier, on the Ground, its Diftance from the Pidlure being known. , Becaufe there is not room, on the Picture, to fet off its whole Diftance, take CE half the Diftance; and make SD half the diftance of the corner of the Pier; draw DE cuting SC at a, the true place of that Corner, on the Pidlure. Draw aV, the indefinite Reprefentation of one Side of the Piers, and a Y, to the other Vanifhing Point, whofe Diftance, from C, is nine inches and three fourths. * Let it be obferved, that the Diftance, here ufed, Is too little ; but, being a true Portrait of fo pub- lic a Place, 1 thought proper to difpenfe with it; by reprefenting it as it appears, from the Station deter- iiune4 above. See Prduninary Obfervations, Pagein. Draw I 224 PRACTICAL PERSPECTIVE Book III, plate XXXL Draw zd parallel to AB; and having rransfered the full meafure of the Piers and Arches from AB, the Ground Line, to a^ (by means of the Point C, or any other in the Vanifhing Line) at hy c, and dy draw Lines to £, the Diftance Point, of V, giving their places, atab, and cd, &c. the reft may be obtained to any length, required, by Example 4th. Note, zd may be the Ground Line,by a lefs Scale, On the returnining Angle, they are obtained by the lame meafures, applied on the other Side, by the Point F the diftance Point of Y. Produce Va to the Ground Line, cuting it at A, and draw AH perpendicular, on which, fet up, from A, the meafures of the heights of the Piers, &c. at G and I ; from which, draw Lines to V. Perpendiculars, froma, b, c, &c. cuting them at e, f, g, &c. gl'^e the Piers, on that fide. From, ae, the common Interfedtion, they are returned on the other Side, totheLeft, The circular Arches are all conftrudled by Exam. 21, as the one on the Left; and the Curves over them, by means of Ordinates ahy cd, &c. their heights be* ing fet olF from e to i and k, &c. the true Curve being determined (fee Fig. 1 19) and as many Ordinates drawn as are requifite, they are returned on both Sides. The eliptic Arches and Borders, from the Piers to the Wall may be thus defcribed. Produce Y a and Ye, and having found the Point J (Prob. 8.) in the middle of the whole width ; from J draw a Perpendicular, cuting Ye and'Yk, produced, at 1 and m ; m is the middle Point in the Crown of the Arch, Draw IV and mV; and, from the Vanifhing Point Y, produce Lines through g, j, &c. from each Pier at the foot of the Arches, cuting 1 V at n and p, &c. from which, draw Perpendiculars, cuting mV at o and q, &c. the height of each Arch, refpedlively ; as Im, no, and pq. Then, by means of feveral Ordinates, as aby cd, &c. perfpe£lively found, on the Bafe Line of the Arch, at a, c, e, &c. (fee Fig. 11 9, PI. 29, for the geome- trical conftrudion) their feveral heights are fet off on ek, and projected, by the Vanifhing Point Y, cuting Perpendiculars from a, c, e; by which means, half the Curve of the firft Arch is defcribed, as ebdfm. The other half is determined by the fame means, and all the other Arches after the fame manner, by drawing Lines from the Seats, a, c, e, &c. of the Ordinates, to V, cuting the Bafe Line of each Arch, as gn, at i, 2,3, &c. ; from which, Ordinates being drawn, and Right Lines from b, d, f &c. to V, cuting them at 4, 5, 6, &c. then, through their Interfedlions, the Curve g50vu is defcribed. Having obtained all the Arches, or Borders, the Groins are next to be deter- mined. The middle Points, r, f, t, &c. are in the Line m V (allowing fomc- what for the thicknefs of the Border) and by means of horizontal Lines, whofe Vanifhing Point is V, drawn from b, d, f, &c. their lengths, from thofe Points, as bg, dh, ft. See. may be determined, by the Point E, as in all other fimilar Cafes whatever. Or, perhaps more readily, by perpendicular Lines from the Dia- gonals, fu and gx, (at the foot of the Arch) where they are cut by Lines drawn ^^om a, c, and e, toW, cuting the former at g, h, and /, refpedtively. The Plan of the Plinth of the Column, being determined, from its known meafure and place (by its Seat on the Ground Line, or otherwife) there is nothing fingular in its conftrudlion ; fave the Blocks or Ruftics (W) which are each equal to a Diameter, in height ; the fpace between is the fame ; they proje£l equal to the Plinth of the Bafe. Y and V are the Vanllhing Points of the horizontal Lines, in the Cornice, &c. The Pitlure is fuppofed to cut the Column, confequentiy its full meafures are applied, on BL or fomewhat more, feeing it is projeded. The Piazza, on the right-hand, is feen to the End ; and, through the Arch, at K, is feen the front of the next, on the other fide' of James ^Street.* The dlftant view of the Church and adjacent Buildings, being fo little feen, are befi: Iketched from the place; as it would be attended with unnecefiary trouble to find their places, &c. perfpe£lively, from their true geometrical proportions and ‘ pqfitions. The Rufiics, in the Piers, are a Scale for proportioning them. EXAMPLE 3 Sca 51 . X. APPLIED TO INTERNAL VIEWS. 225 EXAMPLE XLIV. Is the Rcprefentatlon of a Stciircafe, internal ; f jewing the lefcencUng Stairs, direJf. This Leflcn, is intended, not merely as an Infide View, hut'in ordvr to fhew that a clefcent may be re- prefeiitecl by the Rules of i erfj'euive, on tlie fame Principles as any other Subjeff, whatever ; in which .1 have reduced to pradlicc, what was heretofore treated on (Art. 8. Se£l. 6. B. i. P. 57) refpefling a clown-hill Reprefentation. In this Example the WIiulow Side is parallel to the Pi^tire, and confequently, the Sides of the Staircafe are perpendicular to it; therefore, the Center of the Pic- ture is the Variilhing Point of horizontal Line®, in the Sides. C is the Center, and ML the Horizontal Vanifliing Line ; and V L, the Ver- tical Line, isthe Vanifting Line of the Sides ofthe Staircafe. The Vanlfhin«T iJnes of the afeendinp, and dcfceiidincr Planes, i. e. of the Stairs, are determined by Prob. 2, making CE equal to the ]>illanceof the Piflure, and the Angie CEF equal to the inclination ofthe Stairs to the Horizon,. Make CG equal C F, and through F and (.d, draw Lines parallel to the Horizon; one is the Vanifning Line of the afeent, the other of the defeent.; airJ becaufe they are, in this Cafe, parallel, the Vertical Line, V L, is common to them all"|-, and confe- quenrly, F and G are the Centers of thefe Vanifhing Lines, relpedfiveiy, and their Didance is EF, -equal EG. -(Theo. y, and Def. 20.) As this is a circumdance which has occafioned fome difputes amongd Artids, I lhall clifplay it in the bed light I can, and doubt not I (hall do it fatisfadloriiy. In order to which, let Fig. 120 reprefent a feFtion of the Staircafe, geometri- cal, by half the Scale of the Piflure; AI> is the defeent, from the Landing, at A D, on which the Spcclator is fnppofed to dand, at ED; F. is the Eye or Point of View, and EC ii the Didaiice of the'Pidure*, of which AH is a Lffion. AK is the flight of Stair.s immediately afeending, as AB is d.tTcending, and Kl isthe 'V.,nder fide of the next. flight, over tliem, parallel to A B. ' Now, bccaulLthe Vanifhing Point of every original Right Line is where a pa- rallel from the Eye cuts the PidureX, and EC the DireCt Radial, produces the j Def, 22i Center, C isthe V^anifhing Point of the tides of the Half pace BL, and of the ex- tretnes of the Steps, as may be feen in the Pidturc ; alio, EF', and EG, being pa- rallel refpedively to the Alcent, AK, and Defeent, AB, confequently, F' and G are the Vanifhiiig Points of all Lines, oi'i the Pidure, parallel to them (as the Hand-Rails; the VVainfooting at the Sides, &c.) and confequently. Planes paffing through the Eye and thofe Ltnes, refpedively, parallel to the Stairs, mud cut the Fidure in the Lines IK and MN.paffitig througli 'F' and G, refpeflively, and therefore they are their Vaniflalng Line.. (Def. G.) This I prefume IS intelligible and clear ; and, if Vifual Rays are drawn to the feveral parts of the Staircafe, they will determine what can be feen and what can- not ; as E'B determines how much the Defeent rlfes on the Pidure, which it cuts at b ; Ab is, tiierefore, its whole apparent width. The afeending Flight, parallel to AK, it IS evident, sis feen on tlie under fide, as the Vifuai Ray EM evinces; and confequently, it defeends on the Pidure. Let AB be the Interfedlon of the Pidure with the landing of the Stairs, whicli, in this Cafe, is the Ground Li'ie, the Pidure being fuppoled clofe to the Stairs; all that lies on this fide is projeded to the Pidure ; -as the Door on the Landing, and Cielhig, as well as the Floor ; and it is obvious, that, whatever appears to •defeend, below the Landing, has Its place on the Pidure above the Ground Line; confequently, it rifes on the Pidure. Make BD equal to the known width of the Stairs, the width of the whole, being reprefented by AB; the Station is determined by the VeriicaFLine, VL. * The Diftance of the Picture, it muft be obfervecl, is too little, for t2k.in<5 in fo much as is here repre tented ; but it is maaifeft, that, if the Dillancc was half as much more, or one third lower, wefhould •uot lee the defeendingbtairs at all, as the Kye would be either in the Plane of the Stairs, or on the oth» bide. 4 • 3 E Draw Plate XXXII. f Theo. Cor. 2. Fig. 120^ 226 PRACTICAL PERSPECTIVE. Book III. Plate DrawBG and DG, to the center of the Vanifbing Line MN; the Originals y vv'TT thofe Lines being perpendicular to the Interfedion, AB; in which Lines are all ^ ‘ the upper edges, or nofeings of the Steps j whofe places are obtained by Prob. 8 th. Make GE equal EG,* the Diftance of the Vanilhing Line MN, or IK (Def. 20) Make Ba, ab, &c. double Aa, ab, &c. (Fig. 120) and draw aE, b£, &c. cut- ing BG at a, b, c, &c. giving their places on the Picture; and parallel Lines, ad, &c. being drawn, give the edges of the Steps; the Ends, being at right angles with the Pidures, vanifh in its Center; therefore, draw Ca, Qb, &c. and produce them to the next Step; and thus the defcending Stairs are compleated. The Half paces, below, at X, and above, at Y, being horizontal ; their Sides, confequently, vanilh in the Center; alfo, the ends of all the Steps, every where. The vvidth of the Half pace being equal to the Steps, draw eC and y*E, cuting eC at which gives the Angle gh, at the farther end of the Staircafe. The height may be determined on a Perpendicular from B, as BJ, and drawing JC, The afcending Flight -is managed by Example 9th in every refped. The Va- nifhing Point of the* Hand Rails, Wainfcoting, &c. of that Flight is at F, as well as of the Lines de and fg, of the Plane defg, being parallel to thofe below; the other, with the Lines in the Plane Z, is at G. The height of the Hand Rails, &c. are fet up on the Interfedion B J, as at h ; or from D, as Di, allowing for the ramping Curve, equal i k. The panneling of the Wainfcot is divided in the ratio required, as in any other Cafe, by Prob. 8th, making ufe of the Diftance EG (not EC) for the inclined part. The Windows may be proportioned on gb, at the farther end, geometrical. The Cornices, and other Mouldings, are managed by the Lefibns in Sed. 7. The Architrave around the Door, being projeded, is thus done. Make B I, Im, equal, refpedively, to its diltance from the Steps, and width of the Architrave, and draw El, Em, projeding thofe meafures to CB, produced, cuting it at i and b, and giving t k for the Architrave. The deling is projeded in the fame manner ; and all beyond BP. AOPB is the Sedion of the whole by the Pidure ; OP is the Interfedion with the Cieling, and AO and BP with the fides of the Staircafe ; on which Interfec- tions, all the meafures are applied, for each Plane refpedively. e OF HORIZONTAL PICTURES. Horizontal Pieces are confidered, by many, as performed by Rules different from vertical Pidures; whereas, they are the fame in every refped, being founded on an univerfal Theory, which regards no particular pofition of the Pic- ture; but only, the pofitions of the Planes and Lines in Objeds, to the Pidure, and to the Eye. For, however the Pidure be fituated, the Eye being confidered as a Point, it muft either be in the Plane or out of it, and cannot vary in its po- fition, otherwife ; confequently, its Center and Diftance, being determined by a perpendicular from the Eye to the Plane, are the fame in all pofitions of the Plane, refpeding the Horizon, to which we are naturally partial; but, being properly confidered, we fhall find it of no confequence in the Theory of Perfpedive. Let the Pidure be fituated as it may, all Original Lines perpendicular to the Pidure vanilh in its Center (Cor. to Th. 4.) Planes, or Lines, being parallel to the Pidure, have no Vanifhing Line, or Vanifhing Point; but, in fuch Cafe, the re- prefentations of all Figures, in Objeds fo fituated, are fimilar to the Originals (Theo. 9, Cor. 5) alfb, all Original Lines, however inclined to the Pidure, va- iiifh in that Point, where a parallel Line from the Eye would cut the Pidure ; and confequently, all Lines, which are parallel, amongft themfelves, have the fame Vanifhing Point (Th. 3, Cor. i.) with every other circumftance in the The- ory given, which is quite general, and cannot poffibly regard any particular pofi- tion of the Pidure, to the Horizon. ♦ Suppofe GN produced and made equal to GE, 3 The Plate XXXB Hi^li/hd by T.Ma/lon 3farch 1 '!'^ Jjy S . ,1 •S ' I ' f:k ' ' . 5 •* t I ■> T' ',Wv ' 'i' ■ \ I 4> t-c. < « • f . xt» I I- r [-■ ^ ‘I f.-v I -. i.< 1 ; ..t/:'. 'Sea. X. APPLIED TO INTERNAL VIEWS. 22 / The Reprefentations of moft Objeds (in common Cafes) on a vertical Piaiirc, ..are managed by means of the Horizontal Vanifhing Line, only, but, in many 'Cafes, other Vanilhing Lines are requifite. A horizontal Plane, palfing through the Eye, which, in other Cafes, produce.s -the horizontal Vanilhing Line, is, in this Cafe, parallel to the Pidture, and con- •fequently cannot cut it; for it is the Diredling Plane, of Pidures in fuch pofition. •(Def. 4 ) The theoretic conftruflion of the elementary Planes is the fame, however the Figure be fituated ; all the difficulty feems to be in fixing the Vani'hing Lines, nn this Cafe, having no particular bias in refpefl of the Horizon. As, in all common Cafes, the Horizontal Line and Vertical Line cut each other, -at right angles, in the Center ; fo, in this, two Lines cuting each ocher, perpen- dicularly, in the Center, anfwer the famepurpofe, either of which may be con- ffidered as the Horizontal Vanifhing Line; for, if the Picture be changed to a ver- tical pofition, and, the Objeifls being fuppofed changed with it, keeping their po- fition to the Pi£lure, it is then a common Cafe, in every circumftance ; the placing of thofe Lines, mofl conveniently, is therefore the principal bufinefs. Now, if the Piece be viewed centrally, it is immaterial how they are drawn, for reprefenting a Dome only ; but, if we would reprefent the Infide of any prif- matic Objedl, whatever, it is moft eligible to draw the Vanifhing Lines parallel to its Sides, as AB and DE; becaufe they are, then, the Vaniflaing Lines of thofe Fig. I2i, Planes, refpedively. But, if the View be not central, then, one of the Lines (DE) generally palfes through the center (C) of its Bafe and divides the Dome equally ; which may be confidered as the Vertical Line of the PiiRure ; the other (FG) as the Horizontal Line; S being the Center of the Pidure, whofe Diftance is known. The chief difficulty, in Cieling pieces, is the reprefenting Objeds in fuch un- common Pofitions, they being feen on the under fides, inftead of looking dired at them, as flanding before us, upright, in common Cafes ; and are reprefented as if lying horizontal, in refped of a vertical or upright Pidure; in which, the dif- ference confirts, wholly. The Cielings, Soffits of Doors and Windows, Plan- ceers of Cornices, &c. are, in this Cafe, parallel to the Pidure, as, in other Cafes, they are perpendicular to it j being thus reverfed, and feldom pradifed, makes the difficulty greater. There is one particular circumftance attending Cieling pieces ; the Center ofthe Pidure is the only Vanifhing Point, generally ; not owing to the pofition of the Pidure, but of the Objeds to be reprefented. For the Bafe of the Dome, or whatever elfe is reprefented, is always parallel to the Pidure ; and confequently, all Lines perpendicular to the Bafe (being alfo perpendicular to the Pidure) vanifh in its Center ; and, fince the Objeds to be reprefented are always upright, there is no occafion for any other Vanifhing Point; as, all Right Lines, in the Objeds reprefented, are either parallel or perpendicular to the Pidure. Nor can it be otherwife, unlefs reprefented on an inclined Cieling; in which Cafe, there may be as much variety of Vanifhing Lines and Points, as in any other. Thefe Preliminaries being well confidered and digefted, the difficulty will vanifh, .and leave the whole, fubjed to the fame common Rules already laid down ; for .there are no other ufed, or necefl'ary. EXAMPLE XLV. How to reprefent on a Cieling^ or horizontal PiSlure, the reprefentation of a Gallery of Cotiimunlcation i with a Dome and Cupola, as feen from below. A BCD is a geometrical half Plan ofthe Gallery, &c. to be reprefented, which Flo-. 122. is a Square, fuppofe of 30 feet; and BFHC is a Sedion of the fame. The height, BF, to the Cove, is 18 feet, the Cove 3 feet, and the Bafe of the Dome (which is a Hemifphere) is 24 feet. Through the Arches, in the middle of three “Sides, Galleries are fuppofed to communicate to various Apartments, above. The 228 PRACTICAL PERSPECTIVE Book III. \aXII. ■D' I'lic Eufe of tke G.illery being a Square, and the Picture horizontal, confe- tyae.ntlv j aialld to its Bale, its rcnreicnration Is therefore a Square. D( Icribe a Square, A H I B, orThe diinenfions you intend ilie Pidtire, which is lit re, dpuLle that or the Plan. d'ne Stailon being dcrermined, and the Diflance of the Ficiure, which, In tlhs Cale, cannot admit oi mucii .variation, trom the true, to any Five wiiich views it. FOr, iluppofc, every Cieling Piece is calculated to be leen from fome particular Station, iredovv; and every Perion who would lee tiie Piece, trulv, inuft (land in the lame place ; conlequently, the dillerence, in Diflance, is equal to the different heights ot tlie SjK'Ritors. tuppole the Mall, or Saloon, to be a Cube of 30 feet ; then, allowing 5 feet for the hcigi'.t ot the. -Eye, its Diliance fiomthc Cieling i-, 25 feet, which is the DiEance of the Pidlure ; upon wliich, is to be rcprefcnted, a ct.iitinuation of the Walls, &c. wirii the Entablature, fiom B to F', near 20 feet, with the Cove, Dome, and Cupola, covering the wlio-lc. Now, the Eillance of tlte Piclure is 25 feet,” v.hicli is not equal to its width, hut cannot, heie, he more, unlcls tlie Specliitor be luppoled to lit or lie down. Then, , a-', in common bales, tfie lieight of the Eye is next to be determined, lb, in this, .the place of the Ityc in rel[)ed: of the Pitiure ; which, I u ould never ad- vile to be -in the middle of the Piece, as it cannot produce an agreeable Ueprefen- ' tatioii ; leenig that, thedcKigitudniai L-ines in the Dome (which 111 tliat Cale are re- prefented by llight Lines) will be but lo many Radii of a Circle tending to its Center, as aC, bC, i&c. (!' ip. 124) ..ud, tire latitudinal Circles will be lb many concentiic Circles, asabc, n/'c. See. alfo, every Side of the Iloom will advance equally towards the Center. Therefore, let tlie Station be deterniined fomewhat towards one Side, where . the. -Cieling may be feen molt advantageouily ; nor is it necclfa'-y to be lituated • centrally, ,iii that Side; allhcugh it is a deviation from my general Maxim, of fix- ing the Center or View in the middle of the PiRurc ; bur, fince tiie pofirion oi the PicRire cannot, in this Caie, be varied, 1 had rather dilpenfe with it chan have du- . plicate Reprelentations. Let C, then, be the Centerof View; fomewhat to the right hand. Let A B be confidered as the Ground Line ; and if, through C, a Right Line, EF, be drawn, piarallel to AB, it may be confidered as the Florizontal Line of the Pidlure ; and DO, perpendicular to AB, alfo paffing through C, is the VertiCcal Line, .i. c. they are the Vanifning Lines of the Sides of the fquare Room relpectivclv. Tile Galleries, on three Sides, which are lu.pported by Truffes from the Wails, are firff to be reprclbiited ; which -I have duacle to prqjedt but three feet, as they would lude the Walls too much if they p'rojebted more. Now, fince the Pidture is parallel to the Trulies, and they are fpacecl equally, divide the three Sides A FI, HI, and BI, geometrically, into as many Ipaces and Trufits as are to be repre-lenced, atX, X, and deferibe their Seats on the Pifture, truly geometrical. Then, the Lines which meafure their thicknefs or height, being perpendicular to their Bales, and to the Pidlure, vanilh in its Center, as in all common Cafes whatever. ikiakeCE equal to the Diflance of the Pidlure, in the Vanifliing Line EF ; draw the indefinite Reprelentations,. aC, bC, &c. and proportion them in height, making a// equal to the height, and draw^E, cuting aC at <6 ; which, by drawing parallel Lines, to A PI, PIJ, 6cc. will determine them all around. The Cieling of the Gallery, between each Trufs, is conlequently determined. The under fide of the Trufs is like an Ionic Modilion, having, aPo, a Cimma re- verfa, around the top ; which, the Figure fufficiently deferibes. The Cornice of the Gallery is delcaibedas any other ; fave only, that the Fa- cias, cr lipright Fillets, arc contradled, in this Cafe, as the under Sides, or Plan- ceers, in common,; the difference is owing to the pofition of the Pidlure only, the appearance is the iaiiie ; as .it is illullrated in Phg. 125. Let i Sect. X. APPLIED TO CiELlNO VlEVVl Let A B be the Profile of a Cornice, feen from the Point E, by means of the Vifual Rays A E, BE, &c. Ift he Rays re cat by a vertical Plane (Ab) the reprefentations of the feveral Angles, at C, I), See. are where the Ra s a e cut by the Plane, at c, cl. Sic. and a hor.zontal Plane, Ba, cuts the fame Rays at a, b, cl. Sec. It is manifen-, that the proportions of the Members, on tho c Planes or Piftures, differ confiderably, yet it is obvious, that each appears the fame, in the Point of View, E; the difference, iii the Reprefentations, ariling from the different Sedfions of the Rays, as in any other Cafe, For the iron railing which is in the plane of the Facia of the Cornice, from each internal Angle, k, draw kC. Make A d equal to its width, and de equal to its known height, from the Ciellng (equal twice r/f, Fig. 122) and draw eE, cuting dC at f; draw f g parallel to AH, and g h parallel to H I ; alfo fi, parallel toAB, cuting cC, where it joins the Wall, on the other Side. The divilions and proportions of the Railing are determined, geometrically, on dk"*. See. at Cydy See. fromwdiich, Lines are drawn to C. The Gallery, with the Railing, being compleated, we next proceed with the decorations of the Walls, feen beyond or over it. Eiaving drawn, geometrically, the Plans, or Seats of the Pilafters, with all tlie projedtures of the mouldings of the Pedeftal, &c. at Y, Y ; from each exterior Angle, draw Lines to C, as jC, 1 C, &c, and from 1 , fet off, firft, the height to the Plinth ; then, the meafures of the Plinth, Mouldings, Dado, &c. of the PedeftaJ, at m, n, &c. and draw mE, nE, &c. cuting 1 C in their feveral per- fpediive Proportions, at Jl Sec. from each of which draw Lines parallel to their Seat, j 1 , cuting j G, and return them at right angles. The Diagonals, or mitre Angles are, in this Cafe, parallel to the Pidure, i. e. to A I and BH ; by which mean^ the Mouldings are determined, every where. The Bafe Moulding is hid by the projefture of the Plinth, and alfo the Bafesof the Piladers, on this Side, being greatly forefhortened ; which,, where they can be feen,^^ are managed after the fame manner. From o, p. Sec. draw Lines to C ; make pq equal to the height of the Capital, frpna .the Cieling, and draw qE, cuting pC, at n, and compleat the fquare of the Abacus ; which being parallel to the Pidlure, the curviture of it is geometrical. The Mouldings of the Capital are managed as the other (except in the Abacus, which are all circular curves, being parallel to the Picture) their meafures of height, being obtained, from pq, at r, &c. and their projedlures from the Plans below, drawing Lines from each, to C. The Volute may be determined, from its Plan, below, in refpefl of its projec- tures; but to deferibe it, particularly, would baffle all defeription. ’ The Windows, in this Side, and Niches, &c. in the others, are determined geometrically, on AB, Sec. the perpendicular Lines of which, vanifh in C, and the circular beads are femi-Eilipfes, deferibed, by Prob. 3, Sedt. 8. The other Sides are managed aftpr the fame manner, from Plans below. ' ' ‘ ' For the Entablature. Let M, N, O, and P, be the Seats of its pvojedlure, on the Prclure, in the Diagonals AI and BH. Draw MC, NC, Sec. Make MQ equal Co its height from the Pidlure, and draw QE, cuting MC at n; draw,«o, opy &X'. parallel, refpedtively, to the Sides, of the Piiftyre, nopq is the extreme projediiure of the Cornice, around. , . Make QR, &c. equal to the meafures of the Cornice, Frieze, aud^Architrave, with their feveral Mouldings; and draw R.E, Sec. giving them perfpe^fively, on M‘G.‘ Their projeflures are determined on the, Diagonal, AM, geometrical > rr, on Diagonals from riy o, &c. parallel to AM, Sec. mn. Sec. being divided in the^fame ratio, as A M, (Cor. 2. Th. 9.) * d k is the TnterfeAlon of the Plane df gk of the Railing, dlftant from, AH, equal A d, l.e. equal to Its known diftancefrom the Walls ; confequently, d is the Interfecting Point of that Angle. For, becaufe the Pifture is parallel to the Bafe of the wlvoje, the feflion of every Plane with it, being perpendicular, is geometrically determined; therefore, the Interfefting Point of every Perpendicular Line in the Objefts, as d or k of the Angles of the Railing, is readily found, its diilance from egvh Wall being known, or deiennined j the whole Bafe being a true geometrical Plan. * • ■ 3 M The 23 ^ Plate XXXIIL No. 2. PRACTICAL PERSPECTIVE Book III. The Square of the Cleling above the Cove is thus determined. From Tand U, the Seats of the proje6lure of the Cove (in the Diagonals At and BH) draw TC, and UC; and, T V being made equal to its height, draw VE, cuting TC at r; draw r s parallel to the Cornice, and compleat the Square rstu‘, in which, infcribe a Circle j and, concentric with it, another, leaving a border around the Dome ; and alfo, compleat the fame within the Square r stu. For, becaufe the Cieling is parallel to the Pi 61 ure, its perfpedive form is truly geometrical ; confequently, one Side being perfpedlively found, the rell: are allb determined, in the ratio of rs to the Original. The Angles of the Cove are determined by Ordinates, on the Diagonals (if it were neceflary) geometrically proportioned, alike on each; and others, on AC, &c. perfpeflively j (See the geometrical conftrudlon, Fig. 126, which determines three Points, a, and c, in the Curve, equally fpaced.) but, as they approach fo nearly to right Lines, here, it is an unneceffary procefs. To reprefent the circular Cornice, at the foot of the Dome. The Frieze or Facia, below the Cornice, is perfedly cylindrical, its height is determined on AC, which reprefents a Perpendicular from that Angle; on which, every meafure of height may be transfered from AB, geometrical, as s, t, u, &c, of the feveral divifions in the Dome. From S, the Center of the Bafe, draw SC, in which Line are the Centers of every Circle in the Dome, &c. for, the reprefentations of all Circles, in Planes which are parallel to the Pidlure, are Circles; and, becaufe SC reprefents a Line perpendicular to the Bafe, in its Center, SC is the Axis of the Dome, and pafl'es through the Center of every Circle in it; which being determined, on SC, or transfered from AC, by lines parallel to the Diagonal AI, as si, t2, &c. draw lines through i, 2, &c. parallel to A B, Now vx is the Diameter of the Bafe of the Dome, cuting the Axis at o; and I, 2, &c. are the reprefentations of the feveral Centers, of Circles in the Cornice. From V, fet off the projeflures of the Cornice, in proportion to the Radius vo,. at &c. from which draw Lines to C, cuting parallel Lines from i, 2, &c. which give the Radius of each Circle in the Cornice, as i 2 6cc. by which the Cornice may be compleated ; being compofed, wholly, of Circles. The parallel Circles In the Dome are determined after the fame manner ; which are necellary, for defcribing the longitudinal Borders. In Fig. 122, draw as many Ordinates parallel to its Bafe, G I, as are neceflary, from the feveral points a, b, c, equally fpaced in the Curve, to the Axis, or central line CH, cuting it at d, e, f. Then, make oa, qb, and oc in the fame ratio to ov, as ad, &,c. to the Radius GI; from which draw lines to C, as^C, bC, cC ; and, from the perfpedlive Centers, 2, 3, 4, and obtained as above, draw pa- rallel lines cuting them at e, f and which give the Radius of each Circle in its true place, ‘viz. e 2, f ^4, and h 5, which, lafl:. Is the Bafe of the Cupola. Divide the Bafe of the Dome into eight equal parts, at a, b, c, &c. fo that, ah and de are parallel to AB, &c. and fet off the widths of the Borders regu- larly from thole Points (half on each Side) draw o a, o b, &c. and, from the Centers 2, 3, 4, draw lines parallel to oa, ob, &c. cuting their refpedlivc Cir- cumferences at i, k, 1 , m, through which. Curves being traced, they are the central Lines of each Border^. Their widths may alfo be fet off at each Circle, in proportion to its radius, as at the Bafe, by which they are compleated. The Compartments are done after the fame manner, from thofe central Lines, refpecling the Margins and Mouldings which are longitudinal; the parallel Mould- ings are in Circles parallel to the Bale. ♦ The Eye being in the Plane of thofe palling through d and h, they are, therefore, reprefented by Right Lines ; thofe palling through b and f, being fartheft removed from the Center, are the moft curved; for, being viewed centrally, they all appear Right Lines (as in Fig. 124) feeing the Eye ii in the Plane of each Curve. 7 The riixxxiii I’liMiii'iirRffl JiiilUrScu/p. >,0-D- T 0 1. ~llr=— lllTil^~~~"_ IL=J St£l:. XL APPLIED TO CIELING PIECES. 231 The femlclrcular Windows, at the Bafe of the Dpme, are but little fcen, and may be truly determined by Ordinates, tending to the Center, geometrically divided at the Bafe, and parallel Circles curing them, as at y, i, k, /, m. The Margins about them may be done by t!ie fame means. The Bafe of the Cupola being obtained, as above, the Windows In It, are geo- metrically divided at the Bafe, and Lines drawn to the Center. Their apparent height may be obtained on SC or AC, as the other parts. Thus have I defcribed the whole procefs of this deling Piece, from the be- gining to the End ; to fay more concerning it w^ould be fuperfluous, as it muft be obvious, that the Rules made ufe of are the fame as have been applied in all other Cafes, and Subjects whatever. SECTION XL Is applied to Houfliold Furniture, Wheel Carriages, Machines, &c. T he foregoing Sections contain the Elements and Rules for the praftice of Perfpedlive, in all common Cafes ; I have alfo fhewn how to apply them, in familiar Lefl'ons and Subjefts, adapted to any capacity, in the 6th, 7th, 8 th, , and QthSedlions; and, in the laft, to particular Subjedls, and in a particular fituation of the Picture ; fo that, there remains nothing more to be done in refpedt of the application of it, to all ufeful SubJeftSj almoft whatever. In this Section, which relates to Objedts in particular ProfeffionS, yet ufeful to Artifls, I Ihall only make fome curfory remarks,- in refpedl of the reprefentations of fuch Objeds as are here treated on. The former Lefl'ons, being well underftood, * will be found applicable to whatever Subjed the Artlfl choofes, or has occafion for* 'I'o multiply Cafes and Subjeds, in which the application of the Rules of Perfpec- tive may have the appearance of fomething particular or Angular, would be endlefs, as, in almoft every different Objed, there is occafion for lome variety In the applica- tion of its Rules ; which, neverthelefs, it Is obvious, are all founded on the fame univerfal Principles, though the Rules deduced from them are varioufly applied. It is a neceffary requifite, in many Profeffions, to be able to give a Defign, to a Centleman, of whatever maybe wanted, out of the common run of things ; and although a perfed Drawing may not be required, yet to g>ve a flight fketch, with propriety, is certainly an Accomplifhment which cannot well be difpenfed with. In an’ Archited it is abfolutely neceffary, to give a corred Defign, but they are gene- rally contented with geometrical Reprefentationsj which does not give the true appearance and effed; that can only be done perfpedively. Ceometrical Drawings to a Workman are neceffary ; but, to give a true Idea how the Objed will appear, when executed, from any particular Station, requires fomevvhat more j fo givefuch an Idea, without knowing fomething of Perfpedive, requires a much greater fhare of genius than falls to the lot of many. Next to the Archited, the Cabinet-maker and Upholder will find his account in a knowledge of Perfpedive ; nay, 1 am of opinion, that it is full as neceflary to the Furnifher as to the Builder; having almoft as large a field, and as various. There is nothing influences a Gentleman more in the favour of his Workman, when he is pleafed to want fomething whimfical and out of the way, than to take his Pencil and Iketch out the Idea the Gentleman had conceived, and was big with, yet could not bring it forth without afliftance. He who can do that, and, at 2^2 PRACTICAL PERSPECTIVE Book III. Plate XXXIV. Fig. 129. at the fametimej difplay a little modern tafte, in Ornament, being known, is cer- tain of fuccefs, or of employ, at lead. 1 have, therefore, given feveral pieces of p'orniture, of various kinds, for his practice, which will alio, frequently, be found ufeful to the Artift, to furnilh his Family, or Converfation Piece. The Organ Builder, and Coach-maker, may fometimes find it neceflary to giv^e their Defigns in Perfpedlive; a Ipecimen is given for each Profeffion. And, laltly, the Mathematical Inftrument-maker, Mill-wright, or Engine-builder of any kind, by Perfpedlive, may be able to fhew to much greater advantage, the feveral move- ments, and mechanifm of his Inventions. There are but very few Perfons, who have not had much practice, in Perfpec- • tive, know any thing more of it, than the application to Objects, whofe Planes are parallel and perpendicular to the Pidure, and confequently, have feldom occa- fion for any other Vanilhing Point than the Center of the Pidure ; they always, therefore, adapt every thing to that pofition, without thinking about the pro- priety of it. Some cannot think a Drawing is Perfpedive, unlefs an End of the Objed; is reprelented ; for which reafon, we feldom fee any Objed delineated in Peripedive without having an End feen, which, in a Building, or other Objed of a tolerable length, is abfurd to reprefent, the Front being parallel. Notvvith handing 1 have endeavoured to explode the abfurdity of the Pidure, being necefl'arily parallel to the front, or fome other face of the Objed, as not being pidurefque ; yet, in the following Cafes, I would rather prefer it j luch as giving a Deiign of a piece of Furniture, &c. to a Gentleman ; becaufe, the Gen- tleman, not having judgment in Perfpedive, might be greatly deceived in his Ideas of the proportion of the feveral parts to each other; whereas, the front of the Objed being parallel to the Pidure, the true geometrical proportion is preferved, which is moll eflentially requilite, in order to be clearly underftood. ' Figure 129 exhibits a reprefentatlon of a Pedehal and Vafe dired in front; the Eye is diredly oppolite to the Ball at the top, being about the height of a Perfoif liting ; for, to (land, when we view fuch Objeds as are wholly below the Eye, although we fee no imperfedion or dillortion in the Objed, itfelf; yet, un- lefs the Diftance be greater than we ufually Hand at, to fee fuch Objeds, it will be greatly diftorted in the Reprelentation, particularly in the lower parts, feeing that the Diftance, in fuch cafes, is freqAiently lefs than the height of the Eye. C is the Center of view, for this Objed, the Diftance is CV, which is lefs than the height of the Eye ; but, as the far feet, are not feen, and as there are no receding parts feen, below, no diftortion is obvious ; the Objed Handing in little compafs, the infide of the feet are fcarce feen at all, owing likewife to their ta- pering downwards,; otherwife they would be much diftorted, on the returning fides. AE is a Scale of Proportion, on' which the heights of the feveral parts are fet up, from A, to B, D, and E, for the Pedeftal. EH is the geometrical width, by the fame Scale. From E and H, if Lines be drawn to the Center, and FH be made equal to its depth, in proportion to the front, F V being drawn, or a Ruler applied to F and V, cuts HC at I ; from which a parallel Line is draw’n, cuting EC, and determines, the width of thy top. ’ The Mouldings in the front of this Objed are the chief difficulty; but the geo- metrical figure being known, the,placeof the parallel ftreight part is eafily*deter- mined, on EH; and the curved parts, whether convex or concave, are flatter or q,uicker, as they are nearer to or farther from the Vanifhing Line, and may be done lufficieiitly corred by hand ; indeed all attempts at accuracy, by rule, in luch mi- ll uti, as, being fo fmali, would be fruitlefs and loft labour ; experience and judgment are the beft and only guides. The place of the toot of the Vafe being got, on the Pedeftal, at FG, let the height of the greateft fwell be fet up, from E to J ; where, draw a Line parallel VO the Horizon ; which, if the fwell be equal to the ftreight part of the Pedeftal, 2 may Sefi. XI, At^PLIED TO INTERNAL VIEWS. may be ufed for the fide of a Square, enclofmg the Circle, making KL equal to its diameter. If the Vafe fwell more, it will projed on this fide KL, if lefs it will fall within it ; the real liieafure, in either cafe, is Hill the fame, on KL, whe- ther it be projected larger, on this (ide, or be diminifhed on the other. This curve being determined, all the others will be more or lefs curved, as they are higher or lower ; each of which muft be done by the fame means, if accuracy be required ; finding the true place and proportion of each, by drawing a parallel Line, as NO, according to its height, above or below the other. The height of the top is belt determined by a central Line, and adjulting the fcale of proportion to its diliance by means of the Vanilhing Point (V) of Diagonalsj and a perpendicular Line drawn through K j but, as it is, here, on a level with the Eye, it is deter- mined by the Horizontal Line ; and thus, this Objedt may be compleated. Figure 130 exhibits the reprefentation of a round Pier Table according to the Fig. 130. prefent tafte, with tapering, term Feet. No. I, is half the Plan of the bottom, for the place of the Feet, and No. 2 is the figure of the Top, geometrical, in inlaid work ; both together form a Semicircle; but, if it has not a folding Top, which, when open, form, together, a complete Circle, I would rather it fhould be fomewhat broader, than half the length ; equal to the narrow margin, around ; becaufe, the figure within it Ihculd be an entire Semicircle, which is not fo here ; confequently the Figure is not fo regular. Let the Center of view be determined, at C, and take F G equal to the diameter of the Top; as much below C as the Eye is fuppofed above the Table. The Eye is, here, fituated fomewhat to the left hand, fo that this Objedl is not viewed di- redl in fciont, that is, not centrally, as the foregoing. BifedfFG, at E, and draw EC, FC, and GC, and from E or F draw a Line to the diftance Point (which is not here fixed) cuting GC or EG, at K or D ; through which, draw IK parallel to the front Line. In (hort, complete a Semi- cllipfis (lEK) reprefenting a Semicircle, within the Redlangle FIKG; and, wdthlii it others’, reprefenting the feveral Borders, &c. in the Top of the figure ; which may be obtained by drawing perpendiculars from a, b, and c to FG, and, from yl, Bf C, draw Lines toG, cuting the inner curve of the border at a, c, from which draw Lines to D, On account of the inner Curve, ahcy not being a com- plete Semicircle, bD docs not fall in the Diagonal GD. - The Top being completed, take FG parallel and equal to FG, at the diftance of the breadth of the Frame, and within it, take a lefs Diameter, allowing for the proje(Sure of the Top; on which, deferibe the curve at the bottom of the Frame, and the fmall Aftragal ; from the extremes of the inner curve, draw perpendicular, to the lower curve Line exprefling the thicknefs of the Top. For the place of the Feet, take fg below, at the diftance Ff, equal to the height of the Table, in proportion to its width (FG) and having deferibed the re- prdentation of a Semicircle, of the diameter of the Frame, the place of the Feet, at i and k, are in the Angles (but let it be noticed, that, the Top not being more than a Semicircle, the curve iA:;rk is lefs, becaufe the Top hangs over^ behind the Table; fo that, d reprefents the Center) From' X (No. i.) draw perpendiculars to HE, cuting it at Zj which meafures transfer to fg, and draw Lines to C, cuting the curve, at x, which give the place of the front Feet. The Plans, at a;, x, being completed by nieans of their proper Vanilhing Points, in the Horizontal Line, draw perpendiculars from each corner, to the Frame j and, within the fquare of the Foot, deferibe another, anfwering to the thicknefs at the bottom, from all the corners of which, draw Lines to the Frame, where their full dimenfions are obtained. The Plinth, at a little height from the bottom, is equal to the thicknefs of the Foot, at the Frame, the Lines of which tend to proper Vanilhing Points, as the Plans, below. The Compartments in the Frame, and the margins of the Feet, &c. it would be trifling to deferibe, being divided in the middle, and at each Foot ; or other- wife, at pleafure, from the Plan, below; the Figure muft fupply the reft. 3 N Figure 234 PRACTICAL PERSPECTIVE Book 111 Plate XXXIV. Fig. 13 1. Fig. 132. Figure 131 reprefents a Chair, dire£l in front ; C is the Center* of view, the Dirtance is CE. GH is the Ground Line, in which, take G and H for the places of the fore Feet; where, make their Plans, and defcribe the front, GABH, truely geometrical. Find V and X, the Vanifhing Points of the Sides, AF and BD, refpediively (Prob. 2.) and having fet, off, from V, the diftance of that Vanilhing Point, on the Horizontal Line, make Ba equal to the depth of the Seat, and draw a Line from a to the Eye, curing BV at D, and draw FD parallel to AB, cuting AX which gives the Figure of the Seat, in the naked Frame ; but being fluffed, they muft be gently curved at the fides and behind, as on the left fide. The place of the back Feet, on the Floor, muft be obtained by the fame means, obferving that they fplay, backward, from the Seat, and confequently, are farther diftant from the front ; alfo the line of diredlion is thereby fomewhat varied. To determine the height of the Back, and the Elbow, draw GB perpendicular, and equal to its height. Draw BC or BX, cuting a perpendicular Line from K or k, at a or c ; through which, a parallel Line, ab, determines its height. As to the Figure, ’tis at each Perfon’s option, and depends wholly on the Eye. The height of the Elbow is fet up from A to d, and by drawing a Line to X, its dire£lion, and where it is joined to the Back are alcertained ; alfo, its place oil the Rail may be determined, as the whole length, at D. The reft depends en- tirely on the Hand and Eye; for, no Rules can poftibly determine its apparent Figure, perfpeftively. Figure 132 is the reprefentation of a Lady’s dreffing Table. In this Pidure, the Center of view is taken fomewhat to the left hand, at C, the Diftance is CE\ The height of the'Eye fuppofes the Perfon fitting. The meafure of its length, in front, may be let off, on the fame Ground Line, GH, in proportion to the Chair; or on another, at MN, being fuppofed to ftand farther from the Pidure ; on which acconnt, its geometrical Figure, in front, is by a lefs Scale, as MABN. AB, the front Line of the Top, being determined, let it be bifeded, and, AD and BF' made equal to half the rneafure, in any pofition required ; from all which , Points, draw to C ; make A 6 equal to the breadth of the Top, and draw ^E*, cut- ing AC at b ; draw bf parallel to AD. From b it is transfered, by a doted Line, to the other fide. The hollowing and thicknefs of the Covers, in front, are deter- mined geometrically, by perpendiculars to AD, or BF, at f and D, or F. Make A a equal to the diftance of the neft of Drawers, which rife, by a Spring, out of the Frame, and draw cuting AC, at a; draw ac parallel to AB, cut- ing BC; obferving, in this operation, that there is a margin, of above half an Inch allowed on each fide, lefs than AB. Draw the perpendiculars ae and cd, and proportion them to the length ac ; or fet up the full meafure on a Perpendicular, the breadth of the margin within, at B ; on which, the feveral Drawers maybe proportioned, at i, 2, 3, geometrical. The front of the Drawers, being pa- rallel to the Pidure, they are geometrically divided into as many Drawers and in what order you pleafe. The drefting Drawer, being drawn out, it confequently projeds on’ this fide the Pidure. Having drawn the fronts of the two Drawers, over it, draw CG, and produce it, indefinite; produce HG, and make GI equal to the meafure you in- tend the Drawer to projed, and draw E'^I, till it cuts CG, produced, at K ; draw KL parallel to GH, cuting CH produced, at L; KGHL reprefents the upper face of the Drawer, projeded towards the Eye. HO (the ring) being made equal to its depth, draw CO, till it cuts a perpendicular. from L, at J, which determines the front. The divifions, for the Glafs, in the middle, the Boxes, &c. are determined, by their meafures on GH, or by a larger Scale, on KL, as Kc being made equal to the firft divifion, eR'' determines its apparent breadth, at d. After the fame manner the reft are determined. The Sea. XL APPLIED TO INTERNAL VIEWS. ^35 The recefs, below the Drawers, being an elliptic curve, is thus determined. The height being determined, draw ag, the tranfverle Diameter of the ElllpfeSj defcribe the geometrical curve a3 5g; ag being bifeaed at d, d4 perpendicular to ag, is half the Conjugate Dlariieter. Take as many Points, i, 2, 3, &c. on each fide, as are neceffary, from which, draw Ordinates to the Tranfverle at a, b, c, &c. obferving to make g, f, and e, refpedUvely, the fame diftances from that extreme, as a, b, and c from the other; from all which, draw Lines to C, and determine the length of each perfpedtively, and refpedively, by means of the Point of Dif- tance E^; making hb reprefent b 2, and, cc reprefent C3, &c. parallel Lines, bg, &c. will determine thofe on the other fide, giving the Points a, c, &;c* through which the perfpedlive curve may be defcribed, as in the Figure* What remains is fufticiently explicit by infpedtion. Figure 133 reprefents a Lady’s Secretary and Library ; of which there needs no Fig. 133* particular deicription. It is viewed diredtly central, fothat, the Vertical Line cuts it into two fimilar Figures. Being parallel to the Pidlure the whole frontj of Drawers, &c. is geometrically drawn ; the middle part, with the Delk Drawer^ may be projedled, as the lafl: Figure ; or the Pidture may be fuppofed to be wholly on this tide, fo that, only the Fall of the Delk is projected. The Shelves, with the Books, recede from the front, it would be impertinent to Ihew how it is managed here ; the Books are difpofed at difcretion, and, the Door which is open is determined by Example 39 ; V is the Vanifhing Point* For the reft, the Figure is more explicit than the moft elaborate defcription* Figure 134 exhibits the reprefentation of a Bed with a Canopy-Teafter, viewed Fig. 134. diredl in front ; one half Ihews the naked Frame, the other half is furnilhed. The Front being parallel to the Pidture, it is geometrically proportioned. C is the Center, and C V the Diftance of the Pidlure, by which the reft is proportioned. The Column, on the left hand, being feen entire, is reprefented as in former Examples; the returning Lines, in the Bafe and Capital tend to the Center; alfo the inlide of the Foot. The length of the fide (GH) being determined, and the height of the Poll:, at A, draw AC, cuting a perpendicular from H, at B, the apparent height of the head Poft. The geometrical curve of the Laths being determined, the pierfpedlivfe curve, of the fide Lath, is determined by means of Ordinates, at, hz. See. the feveral heights being fet up from the corner, at A, and Lines drawn to the Center. The Laths forming the Canopy, diagonal wife, are determined as grhined Arches, by Ordinates from the Diagonals, D E and EF, E being the Center. The hither one being determiiied from its known figure, dividing 'DE perfpecr tively, at a, b, c, &c. as the Bafe of the Original is divided by Ordinates, at dif- cretion; by which means, the feveral heights a*?, hb, &c. are determined, and the Curve defcribed through the Points a, h, c, &c. Then, Lines drawn to the Center, from the Points a, b, c, &c. cut the other Diagonal (FEj at k, i, h, &c. at which Points, the Ordinates k^, \i. See. being drawn, and from a, b, &c. Lines drawn to the Center, cuting them, at k, /, h, &c. through which, the curve of the othet diagonal Lath is drawn. In refpe£l of the Drapery,, the Furniture of the other half, I lhall only obferve^ that it is proportioned as the other; but as for the outline figure and ornament, I will not attempt to give any Rules ; fuch Subjeds are alw^ays beft drawn from a real Objed, or they can only be done by- a Perfon of experience and judgment. The figures of the Vallens are geometrical, in front, and no other are feeh. The Canopy is but-little feen, though it rifes confiderably ; a Right Line, from b, ftiews how much, fave the middle, which appears over it ; on which a Vafe, or other ornament, at difcretion, may be drawn, crowning the whole. Figure PRACTICAL PERSPECTIVE Book III; 236 Plate XXXV. Pig- 135' Figure 135 is a Defign for a large Library Bookcafe. . In order to give an Idea of the real proportion and figure of fuch an Object, to a Perfon who, perhaps, knows not what Perfpedive means, and who has no other Idea of its proportion and form, than what the Figure really exhibits, it is mofl proper to give it parallel, as it is here reprefented, but by no means to fhew an End. The Station is central, and confequently, the Point of View is in the middle. The receding parts, and breaking of the Mouldings around them, fuf- Fciently indicate that it is perfpe£lively delineated j and, by reafon of its regular pofition, I am of opinion that it has a much more natural appearance, than when viewed oblique, from either end, which always occafions diftortion; as may be feen in Plate 23, Fig. to8, efpecially when the Objed is long. In refpedf of the delineation, let it be obferved, that the Plinths of the middle part and the two Ends are in one Plane, which are geometrically proportioned, by the Scale, The parts which recede are determined as ufual. At either Corner, as A, of the Plinth, draw the perpendicular AG; on which, fet up all the meafures of its height, as AB for the Dado part, with its Bafe and Sirbafe ; BD the height of the upper Doors, &c. and DF, of the Cornice, &c. To the Center, C, draw Lines from each divihon; make Be equal to the re- ceding of the upper part; or, becaufe the w'hole Diftance cannot be on the' Pic- ture, take CE half its Diftance, and Ba half Be, and draw aE, cuting BC at b. Draw bd perpendicular, cuting the Lines drawn from B, D, &c. to C, which gives the height of the Doors, &c. perfpeftively, according to their diftance from the Front. For their widths, make Ba, equal to the proje * :■ Figure 143 reprefents a Phaeton,' more forefhorteried, and going from the Pidure; FJg. rby which the hind part is more feen, as the fore part in the former Figure. A defcription of the method of "delineating it, it is obvious, would be, in a . great meafure, -a repetition, confequently tedious. The Pofition to the Pidure being fixed and its Proportions known, the method of proceeding is as in the fore- ■going Figure, in refped of the Wheels ai^d Carriage. ' A B is the- Ground Line, on which the Pidure, Handing perpendicular, Is fup- pofed to touch the hither hind Wheel, of which, B D is the Interfedion. Bci, and "Bb, tending to their refpedive Vanilhing Points, are proportioned, at a, b, c, &:c. as ufual, for the places and diameters of the Wheels, &c, which are fleferibed 'as the foregoing ; and are transfered to the other fide, by the Vanilhing Point of ► the Axles, the true figures of which, being in vertical Planes, are determined as • other Figures, from their known geometrical form. On ef is fet off, perpendicularly, the height of the hinder Axle, and -the ho- ■rizontal movement of the fore Wheels, determined at /by a perpendicular from e\ -being cut by a Line drawn fromf, to the Vanilhing Point ofthe Line of diredioii - of the Carriage, on the Ground, eg^ on the Ground, is a diameter of that Circle, ’Which being deferibed, it may be determined in its proper place by Perpendicu- lars; after .which, the reft of the -Carriage may be drawn by a Perfon who has ■►knowledge of the feveral parts. 3 P The ^42 Plate XXXVI. Fig. 144- PRACTICAL PERSPECTIVE Book ith The corners of the Chair, tJii - Lines are the fame. (Theo. 3.) ^ 6 PROBLEM Se£t. XII* OF INCLINED PLANES. 249 P R O B L E M VII. f • The Inclination of a Line to the PiElure being given^ and the Angle of inclination of any Plane that Line is in, to the Pitture ; to deterfnine the inclination of the Line, to the Interfedtion of the Plane it is in, wilb the PiSiure, Let AB be the given Line (in the Plane AEB) inclined to the Pidure in a given Angle. Make A BD equal to the Angle known, and, from any Pdint in AB, draw AC perpendicular to BD; make the Angle CAD equal to the Complement of the in- clination of the Plane to the Pidure, and ADC, is the real Angle. On AB defcribe a Semicircle ; and, with the Radius AD, defcribe the Ark DE, cuting the other at E ; i. e. make AE equal AD; draw EB, and ABE is the Angle required. Draw AE and produce it j make EF equal to CD, and draw BF. ; t . Dem. Turn up the Triangle ADC, on AC, and ACB, on AB, fill AD coincides with AE ; then, turn up the Plane X, till B C coincides with B F. Now, X being cbnlidered as the Plane of the Pifture, arid ACB as a Plane palling through the Line AB perpendicular to the Pifttire, ABC is the angle of its inclination to the Pifture, given. But, ADC is the inclination of the Planfe it is in, to the Pidlure, and E B is the Interfedlion of that Plane with the Pifturej confequently, ABE is the inclination of the given Line, AB, to theinterfedlion, EB. From thefe Problems it muft be obvious, that the pofitlon of the PIdure, to the Horizon, is of no confequehce in the Theory of Perfpedive ; but is very much fo in common Pradice ; .becaufe, all perpendicular Lines, in Objeds, are parallel to thePidure, being vertical; and horizontal Lines are eafily determined, whether perpendicular or inclined, their inclination being known. To an inclined Pidure, their inclination is the Angle they make with their Seats, on the Pidure, N. B. The Vanilhing Line or Interfeftion of fome Plane in the Objeft, muft be given, to determine others, if neceflary ; for which end, the Center and Diftance of the PiSure are abfolutely necelTary. The Vanilhing Line of horizontal Planes, being perpendicular to the Pifture, is therefore firft deter- mined, and the Interfe£lion of the Ground Plane ; which of all other are fitteft. Without them, there would be great difficulty in proceeding. For want of that conlideration, the work of Dr. Brook Taylor is almoft ufelefs to a Pradtitioner ; he not giving, properly, one fpecimen, how to find the reprefen- * tation of a Line, from its known length, fituation, and place, in refpedl of the Picture ; but only, by means of the Interfefting Points of other Lines, and drawing Vifual Rays, from the Eye to the Ori- ginal Points, in their true places, which is not praftical in many cafes ; or, to determine the Figure from fome Line given in it, on the Pidfure ; fo that, the Student knows not how or where to begin the procefs, from the geometrical proportions of the Objeft, and its known or determined place and pofi- tion to the Pidture, and to the Horizon. EXAMPLE 1 . To find the reprefentation of a right angled Paralkhpiped, whofe Sides are hnown, Jituated at fome difiance jrom the Picture ; whofe Faces are all inclined to the Picture, the inclination of one being determined, and the inclination of one Side in that Face, to the PiSlure. The Center and Lifiance of the Pidlure, with the Seat, and difiance if the neareji Angle of the Obje^, to the Pidlure, being given. Let C be the Center of the Pidure, and S the given Seat of the hither Angle. Find a, the reprefentation of that Angle. (Prob. 6. Sed. 4.) Through C, draw DF, at plealure, and CE, perpendicular to DF; make CE equal to the Dif- tance of the Pidure, and the angle CED equal to the Complement of the given Angle of inclination of a Face to the Pidure. Through D, draw AB perpendicular to DF (the Vanilhing Line of that Face, Prob. I.) afid, perpendicular to DE draw EF; F is the Vanilhing Point of Lines perpendicular to that Plane. (Prob. 2.) Fi id the reprefentation of that Face whofe Vanilhing Line is AB, D is its Cen- ter, and DE its Diftance. (Prob. 21.) as follows. Produce FD ; make DG equal to DE, and draw GH parallel to AB. 3 R Fig. 150. Plate xxxviir. Fig. 151. Make 250 Book III. PRACTICAL PERSPECTIVE Make the Angle HGA equal to the inclination of the given Line to the Inter- feaionof that Face, whole Vanifhing Line is AB (Prob. 7.) and, make AGB a right Angle ; A and B are the Vanifhing Points of its Sides ; and, F being the Vaniflaing Point of Lines perpendicular to that Face, AF, and BF are the Va- nifhing Lines of the other Faces ; as it is manifeft by Problem 4. Draw a A, a B, and aF, the indefinite Reprefentatlons of three Sides, forming the hither Angle ; how to proportion them, I (hall (hew, as follows. The meafures of thofe Sides are known, and the diftance of the Angle aj from the Piclure; which was found by its given Seat, S, and its Diftance Sa. ^ Now, A, B, and F, are the Vanilhing Points of the Sides of the Objedl, any one of which may be in the fame Plane with SC (Ax. 6.) wherefore, ACS is the Vanilhing Line of fuch a Plane. (Theo. 10.) Make ab to reprefent a Line in proportion to that which Sa reprefents, as the Original of ab is to the diftance of the Angle a (Prob. 10.) thus. Make CE equal to CE, and perpendicular to AC; draw AE, and produce it j make EM, to £N, in the ratio of the Side, to the diftance of the Angle aj join MN, draw EJ parallel to MN-, and draw SJ, cuting aA at b; fo (hall ab repre- fent a Side of the Objedl, whole length and inclination to the Pidure was given. Draw bB and bF ; and, by means of the Radials AG and GB, make ad, or be, to reprefent a Line, in proportion to the Original of ab, as one Side of that Face is to the other (Prob. lo.j'u/s;. as GI toGK; and draw Ad (through c) and dF. The Face abed, whofe Inclination was .given, being ccsnoleated ; find e or g, fo, that de, or bg, (hall reprefent the proportion of the other given Side, by means of ab, or ad, as before; the Center and Diftance of either Vafiifhing Line being determined, (by Prob. i.)‘as/ 5 ', the Center of B F, by a PerpertdicuLr froth C (Th. 4.) and SO the Diftarlce, in AS produced; making BO equal BG, and joining OF, BO and OF are the Radials of the Sides in th'at Face, forrhing ^ right Angle BOF, at the iEye. - Make OP to OQ in the ratio of the Originals of ad to de; draw PQ and OR parallel to PQ. Draw aR which will cut dFat ej and, through e, draw Bf, cuting a F at f ; and laftly, draw f A, cuting b F at g, which compleats the Paral- lelepiped, a b d f, required. For, becaufe of the Vanifhirg Points A, B, and F, the Sides ab, ad, and af, form a folid Right Angle, at a; AGB, ATF, and BOF being Right Angles, whichbad, baf, and daf reprefent, refpec- tively; and the other Sides vanifh in thofe Points, refpeftively; asdeandde, &c. which reprefent Parallels to ab, af, &c. (Cor. to Th. 3.) Therefore bdf reprefents a right angled Parallelapi’ped, whofe proportion was known, and pofition to the Picture determined. ac, ae, and ag, reprefent Diagonals, in each Face, refpeftively, which are in proportion to the Sides, as IK, PQ_and TV, refpeftively, to GI and GK, OP and OQ> TX andTY, refpe£tively. The Parallelepiped, bdf, is truly determined, according to its pofition given, in refpeft of the Piflure, its place in refpeft of the Eye, and rts proportion in refpeft of its Diftance ;^no regard being had to its pofition refpedling the Horizon. Wherefore it is obvious, that, to determine its Pofition, in that refpeift, the horizontal Vanifhing Line is efientially neceffar.y, and the Vaniftiing Lines of its Faces are deter- mined in refpeft oftheir pofition to the Horizon, as well as to the Pidlure, asfhall be exemplified iti the next. EXAMPLE II. To reprefent an Odlaedron^^ per/pedilvely^ ftuated on a Plane Inclined to the Horizon, and to the PiSiure ; the angle oj Inclination to the Horizon, being given, and the Angle •which its InterfeBion •with horizontal Planes, )nakes •with the PiBure ; together •with the Seat of the OhjeB on the inclined Plane ; and, its ftuation in refpeB of the PiSlure, Let ABC (No. 2.) be the Seat of that Face oh which the Objeft reft?, on the Plane Z, which is in- clined to the Horizon in the AngleX; DF is its (nterfedlion with a horizqntal Plane, and DFG the Angle that Interfedion makes with the Pidure ; FG is the Interfedion of the Pidure with the hori- zontal Plane. The Station Point is S, the Diftance is S G, and SE the Height of ‘the Eye. * An Odaedron is a regular Solid, one of the five Platonic Bodies, having eight Faces, which arc equal, equilateral Triangles j about which Solid, if a Sphere was circumfcribed, every Angle of the Solid would be in the Surface of the Sphere. _ _ , Its geometrical Confti udion is neceflary to be underftood, before it be poffible to deferibe it perfpedively. Its geometrical Plan, on the Plane on which it refts, is a regular Hexagon, A EBFCD (No. i.) ABC is the Face, on which it refts. DEF is the upper Face, EAD, EBF, and FCD are inclined Faces, above; and AEB, ADC, and BFC, below, out of fight; IGHC is Its geometrical Elevation, Or Sedion through EC, (hewing the Inclination of its Faces, to each other; viz. the Angle GIK toual IGH. ' Fig. 152. No, 2*. No. I, I'btteXXXVm Sea. XII. OF INCLINED PLANES. 25' Thefe preliminaries being detet-mined, let AB be the Interfeaioil of the hori- zontal Plane, or Ground Line ; and, at the height of the Eye (SE, No. 2.) draw the Horizontal Line, D F, parallel to AB ; let C be the Centet of the Piaure. Draw CF perpendicular, equal to the Diftance of the Piaure, and find the Va- nifhing Point, D, of the inclination given. (Prob. 2. Sea. 3.) Make DEF a right Angle, and through F, draw FG perpendicular to D F. FG is the Vanifhing Line of a Plane, to which, Lines vanifhing In D are perpendicular. (Prob. 3.) Make F R equal EF, and the Angle F £G equal X, (No. 2.) and draw DG, the Vanifhing Line of tlie inclined Plane, (as by Prob. 5. Sea. 3.) indefinite. Produce EC, cuting AB at S; make SB equal GF (No. 2.) draw BD, the inde- finite Reprefentation of DF, (No. 2.) and find the Points a and b, rcprefenting a and b, in which, BA and C A (No. 2.) cut DF. (Prob. 8.) Find the Center (C) and Diflance of the Vanifhing Line DK (Prob. i.) make CE* (in QC produced) equal tb its Diftance^ and draw DEk Make the Angle DEH equal BaD (No. 2.) H is the Vanifhing Point of AB (No. 2.) Make the Angles HE'^l and lE^K each of 60 Degrees ; H, I, and K ore the Va- nifhing Points, of the Sides of the two Faces ABC and DEF (No. 1.) i e. of AB, AC, and BC, (No. 2.) and coiifdquently of the Sides of'the oppofite Face. Find the reprefentation of that Face, a be (Prob. 18) having obtained the Poirits, A and c (reprefenting a, b, and D, No. 2.) by means of the InteiTec- tion AB; making DE‘, equal DE ; or, by the Interfeftion BJ, of the Face, a be. Find the Vanifhing Line HM, of the 'contiguous Face, \VhOfe common IntferTefci tion, with that found is ab, and its Vanifhing Point H. (by Prob. 5.) Its liicli** nation is the acute Angle GIK (No. i.) determined by a 'petpendicular' from E, and IG being made equal IC; which Angle is 70 DegiVeS. - * '' Find the Center (O) and Diftance (OE^) of the Vanifliing Line HM^ '(Prob. i.) D raw E+H, and find the.Vanifhing Points P and Q 5 as I and K, above ; or make . . equilateral Triangles, X and Y, at the Eye (E'^) and produce their Sides to the Vaniftiin^ Line H M, cuting it at P and Q. / ,j Through a and b, draw Q a^ andPb, cuting at d; giving ab d for that Face. The Point d is in the upper Face, and the Sides of oppofite Faces are parallel,* refpeflively, two and two; therefore, draw d I, anddKj ,*and Pc, cuting dK at e; draw e H and join a e, which compleats the Figure. N. B. Th is Figure having eight Faces, two and two of which are parallel, there are confequently (if none are parallel to the Pidfure) four Vanifhing Lines ; but here are only fwo required, DL, and HM; a third will pafs through I and P, and the fourth through K and Q, which, being produced, Would meet in the Vanifhing Point of a e. It may be obferved, that the three Sides 6 f each Face, yanifh in its refjfeAire Vanifhing Line, (Theo. 10.) and the Vanifhing Point of the common InterfedliOn, of any two Faces, is the interfec- tlon of the Vanifhing Lines of thofe Faces, (Cor. '2. Theb. 6 .) as H, of ib and ef, the interfedtioh of the Vanifhing Lines DL and HM ; aftd K, of DK and QK, the Vanifhing Point of d e, and bt. # - » The next Figure exhibits two of the fame Objects, on a level Platie (or any pjg^ Plane perpendicular to the Picture) fituate alike to the Pidlure, but on, different fides of the Station Line ; the Sides, in both, have confequently the fame Vaiiifti- ing Points, and, their Faces have the fame Vanifhing Lines, refpetftively. AB Is the Vanifhing Line of the Faces which are perpendicular to the Pifture,* and C its Center; ag is the Interfeftion of one of thofe Faces. The Angle a, of one Objeft, touches the Pidure, the other is at fome diftance beyond it. FG being the Vanifhing Line of a Plane perpendicular to the common Interfec - tion ab, and FE (equal EF) its Diftance; the Angle FEG is made equal to that of a diagonal Plane with the Faces, (equal HlC, No. 1.) ‘u;>. 55 Degrees. A G is the Vanifhing Line of the diagonal Plane abed, which is a Square. The Vanifhing Points, A, H, and B, of the Sides a b, a f , and b f, or e d, being found (as above) and EK drawn, perpendicular to EG, cuting GF, produced, at K, the whole is determined. B K and G H interfe To give or prefcribe Rules, abfolutely, for perfeding a Pidure, in refped of Light, and Shade,- is as impoffible as in refped of Colour ; yet, by adhering to Rea- •Ibn, and carefully obferving Nature, we may arrive at a tolerable degree ot per- fedion. In the firft place, it is necefliiry to coiifider how the Objed is fuppoled to be fituated to the luminous Body, from which it is to receive the Effed, It has been almoft a general Rule, amongft Artifts, to fuppofe Light to flow from the left hand to the right; bufthat is entirely arbitrary, and can have no -foundation in the nature of things, but merely an habitual cuftom; it Is, however, proper to Imagine it to flow from one hand or the other ; for, to fuppofe the Luminary, or luminous Body, -diredly oppofed, either on this or on the other flde of the Objed, can never produce a pleafing effed ; feeing that, in the firft cafe, it ffs almoft wholly illumined, in the other, it is wholly deprived of Lights an agree- able mixture of both, judicioufly dlfpofed, is what contrafts one Objed, or part of an Objed from the other, and renders the' whole agreeable to the Eye, as a ..pleafing imitation of Nature. 7 Let OF LIGHT A N D SHADE. Book IV. FlateX'L' Tliere is, moreover, an inconveniency arifing from the cuflom of.thading Piftures, on. the left hand, generally, leeing they cannot fuit both fides of a Room ; dor, being placed, in that refpedl, properly, adds greatly to the effed of the Piece. Let it be carefully obferved, that, that part of an Objed, to which any lumi- nous Body . is .moil; diredly oppofed, .will be the brighteft; but, *When, from the iituation of the Eye, it is much contracted, or .foreihorted, it may appear as dark, or perhapsdarker than otherparts, on which, the Light is not fo direct;. For Example. Let Z be the Flan or Bafe of fome prifmatic Objeft, and AB, ■&c. Rays of Light, falling thereon. Becaufe.thofe Rays fall more perpendicularly on the Face EC, than on ED, that Face is, confequentlj^ more illumined than ED, and BD thaiiDF, whllft FG is wholly deprived of Light; aud G H, being farther removed from the Light, will be darker than F G, .from the effed of iLight fimply, >or dired. Now, if a Spedator be fo fituated, at E, that the Face BC, which is moil illumined, is much contraded, it may in fome htuations of the Objed'to the Light, appear darker than BD, which is more oppofiteto the ‘Eye ; whereas, if the Station be at £, where both appear equally contraded, the Face BC will appear the brigbteft; but, if the Light was in the diredion of £B •they would be, an^ alfo apipear equally bright, without diftindion. Any d iredion to the Rays of Light may be given, at difcretion, as AB; and, by drawing others, parallel to A B, dt is-eafy to know which Face fhoiild be the brightell:, and which the next, &c. but to determine in what degree, poiltively,_ is not 'poflible ; a careful obfervation of Nature is the beft criterion by which to judge of that ; and, even in that cafe, it is not eafy to determine, without ex- perience and found judgment. After the fame ^manner, the 'inclination of the Rays of Light to the Horizon, being taken into conhderation, may be determined, whether horizontal Faces of •Objeds, below the Eye, or inclined Faces, are more illumined than the vertical. As Light flows from above, the whole Hemifphere being illumined (when the Luminary -itfelf is not muCh elevated) the horizontal faces' of Objeds, or fuch as are much inclmed, are, generally, the brighteft ; provided they are oppofite to ithat quarter .from which the principal Light flows. Notwithfhandiiig it is ufual to fliade the Roofs of Buildings, Pediments, &c. more than the vertical Planes, it can only ‘be from a fuppofition of their being of darker materials, as Lead, Slate, &c. lor, if they were compofed of the fame, that is, if they are of one uniform Colour (and otherwife, no pofifive determination can be made, in refped of Light, from obfervation) inclined Planes, will be brighter than either vertical or, horizontal, being, generally, more oppofed to the Light, .In refped of curved Surfaces, from the effed of Light, limply or diredly, being convex, thofe parts which are moft towards the luminous Body, are the brighteft; and confequently, thofe which are fartheft removed from the Light will (without refledion) be the daikeft. On concave Surfaces, ^the effed is reverfe. On cylindrical Surfaces, as Columns, ilreight Mouldings, &c. the Light is always parallel to the fldes, or edges of the Mouldings ; the greateft Light being on that part which is moft perpendicularly .oppofed to the luminous Body (as AB) from which it gradates regularly, on each fide ; confequently, that fide which is fartheft from the Light will be the darkeft, fuppofing the Luminary to be fituated on this fide of the Objed, on either hand. But, if it be fuppofed fituate on the other fide, the Edge, AB, towards the luminous Body, will be the brighteft; and it will be gradated from that Edge, not to .the other, as in No. 2. (which has not the appearance of an entire Column or Cylinder, but of a Segment, cut off parallel to its Axe) but, it will be darkeft fomewhat from the other Edge (as No. 3.) not owing to refledion from other Objeds, but from the luminous Body, beii)g on the other Side; which, as it is more or lefs dired, .'will oecafion .the darkeft part.to.be more or lefs removed from the middle, I Convex Setfl. I. OF LIGHT AND SHADE. Convex fpherical Surfaces are brighteft on that part to which the luminous Fig. 3. feody is perpendicular, from which it gradates every way equally ; infenfibly varying, to the extremes, provided no other Objefl interfered, to reflect Light. If a Hemifphere, or lefler Segment, be properly (haded, it will reprefent equally as well a concave as a convex Surface, by fuppofing the Light on one fide or on the other; that is, when the Eye is fo fituated as to fee the whole circumference of its Bafe, or nearly ; as Fig. 3. Refpeding ftreight mouldings, which are compofed wholly of Planes and cy- lindrical Surfaces, their different fituations, towards the flow of Light, occafion variety of Shades. As for example ; the Cima-reda (compofed of a convex and a concave cylindrical Surface) in its proper pofition, in a Cornice, &c. has two ifrong Shades and one Light, in the middle; whereas the very fame Moulding, reverfed, as for a Bafe, has two Lights, and one Shade, in the middle, with a faint one at each Edge. The Cima-reverfa, in its proper pofition, hastwo Lights and one flrong Shade, in the middle ; the fame Moulding reverfed, has two faint Shades and one Light, in the middle. Figure 4 is the Profile of a Cornice ; fuppofe the Light to flow in the direc- Fig, 4^* tion AB^ at diferetion. It is eafy to determine, what parts of the Cornice will be light, and which (haded. For, drawing feveral Lines, touching the projec- tures and prominances, parallel to AB, it is obvious that the edge a, obftru6ting the Light, mud: neceflfarily occafion a (Irong Shade, below; which gradually dies away into the Light, at b, in the middle, where the Rays fall oblique on it, and where, the prominance or fwell of the convex part occafions another Shade, at c ; begining faint, and gradually Ifrengthening, reverfely. The fame reafoning accounts for the Shade in the middle of the Cima-reverfa. The great projedure of the Corona, (X) throws all below in Shade, which gives great expreflion and force to the wholef making the upper part appear to (land off, from the Canvas or Paper. The whole, below, being immerfed in Shade, would be totally loft to fight, was it not, in fome degree, illumined by refledion ; on which I (hall fpeak in its place ; let it fuffice, here, to obferve, that the effeds are almoft wholly reverfed, and but faintly exprelTed. Let No. 2. reprefent the upper Mouldings reverfed, for a Bafe Moulding. No-; 2-, Here, it is obvious, that the Rays, falling on it in the fame diredion, illumines the whole ; infomuch that, no part can be faid, with propriety, to be in Shade ; neverthelefs, the parts b, &c. which are moft diredly oppofed, will be brighter than the other, which are faintly (haded, as the Surface falls off. I fhall juft make an obfervation on the prevalent Cuftom of (hading Mould- ings, in architedural Defigns, as ftrongly illumined by the Sun ; which entirely deftroys their proper effed, Suppofe AB (No. 3.) to reprefent a Ray from the Sun. It is obviousj that all JsJq. the part, from B to will be in Shade ; and, being a Shadow, projeded by the upper edge (BC) ftrongly defined by a Right Line (be) has not a very agreeable effed. The Shade, below, is alfo more fudden and hard. Now, if by means of Light and Shade, it is intended to give an Idea of Mouldings^ I would a(k, ferioufly, which has the moft natural appearance and effed? Suppofe the Profile cut off, or covered; would any Perfon conceive what this Moulding is intended to reprefent ? But, where is the neeeflity to fuppofe the Sun to (hind oil them ? as there are ftrong Shades when it does not ; nor are they intended as Pidures, but Defigns, which ought to exhibit what is intended, in the moft ex- preflive manner poflible. If my opinion might be allowed to have any weight, I (liould fuppofe that a Sedion (hewing the infide of a Building, geometrical, would be more expreflive, by means of penumbral, inftead of ftrong edgy Shades, which are by no means natural, in fuch Cafes. Nay, fo fond are fome Architeds of forced Effeds, that we frequently fee a ftrong right line of Light introduced into a Sedion of a Dome, 3 T through 258 THEORY, OF THE PERSPECTIVE Book- IV. Plate XL. Fig- 5- Fig. 6. through an opening at the Top (AB) as ftrongly defined on the part at C, which is nearly horizontal (where it is abfolutely impoflible for Light to come) as elfe- where, atD; where, an elliptical figure of the opening would, only, bedeferibed; the concave Cylinder, below, being fhaded without any edge at all. Similar to this are the forced and unnatural EfFedls, which may frequehtly be feen, in Prints and Paintings, of a ftrong Light being introduced, ftrikih'g in a Riglat Line, from the upper edge of the Cornice, acrofs the Side of a RdOtti, a§ if it was open at the top. Alfo, on exterior Objedts, a ftrong diagonal Shade fre- quently croli'es a Building, without the leafl: apparent Caufe for fuch an Effedt. . There is (to me) another great impropriety ; which, though it does not come immediately within my cognizance, I (hall beg leave juft to mention ; iv-Hich is, the introducing Landfeape, and fome appeararice of Perfpedlive, in a geometrical Defign. I am well afl'ured that it need but be 'pointed out, to convince any fen- lible Perfon of its inconfiftency, as a Pi ; for, fince it 'is obvious, that every part of the Sun muft emit Light; confequently, a Ray of Light emited from a, and paffing through the- Point A, will proje(ft- its Shadow to a ; and..a Ray .from b will project the fame Point to &c. while the Center (S) only, being confidered, will proje£l it to. B, which is the center of the Shade, and its real Shadow. For (fuppofing the Point to have fubftance) every other part of the Shadow, is m( re languid, the farther it is from the Center, B; confequently, at its extremes, it cannot be diftingiriftied from "the furroundtng Light ; feeing tl>at, every Point, in the Sun’s fur/ace, emits light. Wherefore, fince the ‘Triangle.’, aAB, are limilar, it will confequently be, as, Aa is to Aa, fo iy ai to a'b, that'is,'as AB to AS. (6. 6. 'El.) -Hence, it is eafy toi account for the Penumbra of Shadows ; winch, at a diftance, appear dlftin^Bv de- fined, but on approaching near, we find it otherwife; infomneh that, except where the Lines, in any ObjeA occafioning'the Shade, cut the Surface on which the Shadow is projeiled, we cannot trace a line at all ; and, the farther the Shadow is from that Point, the more penumbra! it becomes ; that is, the Itfs diftihiftly defined; till, at a confiderable diftance, it becomes infenfibly mixed with the Light. Let 26o THEORY OF'SHADOWS. Book IV, plate XL Fig. 8. Fig. 9. Fig. II. Let AB be the Horizon of the part at D, to which, the Sun is diredlly oppofite. The Ray SC is perpendicular to the Horizon, AB; and confequently, a perpendi- cular Line, as CD, to the Horizon of that place, can have no Shadow, but will be projedled towards the Center. But, if any other Tangent, as EF, be drawn ; the dired Ray SC, to the Globe, is inclined to it, in the Angle SDE; and therefore, Lines perpendicular to EF will proje(fl; Shadows. As CE to D. Toilluftrate this more clearly ; fuppofe the Plane AB, horizontal, a portion of the Earth’s Surface, to which the Rays are inclined, and SE the diredlion of a Ray of Light. The Angle of Inclination, to the Plane A B, isSED; and, the length of the Shadow of the Perpendicular CD, is DE. Hence it is manifeft, that the Shadow of a Right Line, on a Plane, is always a Right Line (r. 7. El.) for it is projeded by a Plane of Shade, occafioned by the Line; as CED, which cuts another Plane, whofe common Sedlion is the Shadow. And, it is evident, that, the greater the angle of Elevation is, the fhorter is the Shadow; for, FC projedls the fame Point, C, to G; confequently, being ele- vated perpendicularly over C, the Line CD will proje£l no Shadow. In projc£ling Shadows by the Sun, there are three Cases to be confidered, or fituations of the Luminary ; viz. it muft be either on this fide, or on the other fide, or in the Plane of the Pidure. In order to determine the Shadows of Objeds, delineated on the Pidure ; hav- ing firfi: confidered and determined on the fituation and altitude of the Luminary, the next thing requifite, if the luminous. Body, or Point (for it is always con - fidered as fuch) be not in the Pidure, is to find the Vanifhing Point of the Rays of Light ; but, when it is in the Plane of the Pidure, the Rays are parallel on the Pidure, and confequently they have no Vanifhing Point; the Angle of Elevation is, then, only neceffary to be confidered. Let FGHI be the Pidure, and EG the Diflance of the Pidure, Alfo, let SE be fuppofed a Ray of Light. Imagine a Plane (ESB) to pafs through the Luminary, and through the Eye, at E, cuting the Pidure in FG; in which Line, whether the Sun be on this, or on the other fide of the Pidure, as at S, its apparent, or tranfprojeded place mufl neceflarily be ; as at F or G. For, being on the other fide, at S (fuppofed at an immenfe Diftance) and being in the Plane ESB, confequently ES is in that Plane, and will cut the Pidure in their common Interfedion (FG) at F. Wherefore, F reprefents the Sun on the Pidure, and is, confequently, the Vanifhing Point of the Rays of Light. When the Sun is on this fide, at S', or S% being dill in the fame Plane, produced, confequently S', or S^'E, being produced, will alfb cut the Pidure in FG, as at G; and, becaufe all the Rays are parallel amongfl themfelves, and all Lines which are parallel have the fame Vanifhing Point, confequently, G is their Vanifhing Point. (Def. 22.) for S'.G is a Ray of Light, to which all other Rays are parallel ; or, it is a Right Line, pafiing through the Eye, parallel to them. N. B. In the former Cafe (when the Sun is on the other fide of the Pl£lure) it is obvious, that its place, in the Pidture, muft neceflarily be above the Horizontal Line, as it cannot be feen till it is above the Horizon ; fo, in this Cafe (feeing; it is tranfprojedled) being above the Horizon, its tranfprojeded place muft neceflarily be below the Horizontal Line; from which (in either Cafe) it is farther removed, as the Luminary is more elevated above the Horizon. Alfo, whether its real place be on the right hand, or on the left, its apparent place is the fame, in the former Cafe, but reverfed in the other. (See Transprojection, in the Introdudion; Page 52.) Hitherto I have proceeded introdudorily, which, I have called the Theory of Shadows ; feeing that, all which has been faid is theoretic. I fliall now proceed to pradice, and lie down fuch Rules as are neceffary, for projeding the Shadows of all regular Objeds, particularly fuch as are right lined; which may bedeferibed on plane S urfaces, with facility and certainty, by adhering to the following Rules. 5 First. Sea. III. PROJECTION OF SHADOWS. 26s. .First. The indefinite projeftion of the Shadow of a Right Line, on any Surface whatever, is the Interfedion of that Surface by a Plane, paffing through the luminous Point and the Line; which, for diftinaion lake, I fhall call the Plane of Shade. This is obvious in itfelf. 'Second. The Vanifhing Line of the P/ane of Shade.^ projeaing the Shadow of any Right Line, is a Right Line drawn through the Vanifhing Point of the Rays of Light, and the Vanilhing Point of the Line whofe Shadow is required (Th. 10.) Becaule, the Vanifhing Point of the Line, whofe Shadow is projeaed, and the Vanilhing Point of the Rays; projeaing the Shadow, are in tlie Plane of Shade. Third. The Vanifhing Point of the Shadow of any Right Line, on a Plane, is the interfeaing Point of the Vanifhing Line of that Plane, and the Vanilhing Line of the Plane of Shade. (Cor, 2. Theo. 6.) Becaufe, the common Interfeaion of thofe Planes is the Shadow required. 'N. B. When the Sun i« in the Plane of the PIdure, there being no Vanifliing Point of its Rays, a Right Line, drawn through the Vanilhing Point of any Line whofe Shadow is required, parallel to the given Ray, cuts the Vanilhing Line of the Plane of projedion, in the Vanilhing Point of the Shadow. Alfo, if the Original Line be parallel to the Pi£lure, and confequently has no Vanifliing Point; (the Sun being on either fide of the Pitlure) then, a Right Line drawn through the Vanilhing Point of the Rays, parallel to the Line whofe Shadow is required, cuts the Vanilhing Line of the Plane of projedlion, in the Vaniflring Point of the Shadow. But, when the Luminary is in the Plane of the PitSlure, and the Line, whofe Shadow is required, parallel to it, the Shadow has no Vanifliing Point; for it is, in fuch Cafe, necellarily parallel to the Figure i therefore, parallel to the Vanilhing Line of the Plane of projection. S E C T I O N III. the P R o j E c T I O N of redil inear S h a d o w b. P R O B L E M 1 . y Having the Angle of the Sutis Rlevation (X) and the Inclination, to the PiSlure, df a vertical Plane pajjing through its Center {equal Z) with the Center and Dijiance ■ of the Pidiure given ; to find the Sun's place on the Picture, or the Vanijlnng Point of the Rays of Light. \ Let CE be the Horizontal Vaniflilqg Line, and C the Center of the Piflure. Make CB equal to the Diflance; perpendicular to CE, and draw AB parallel to CE. Draw BD, making the Angle ABD equal to Z, curing CE, at D ; and through • D, draw FG, perpendicular to CE. FG is the Iiiterfedlion of a vertical Plane .. paffing through the Sun, with the Pi£ture, whofe inclination to it is equal Z. Make D E equal to DB, and the Angle DEF, or DEG, equal to X, the Angle - of ths Sun’s Altitude ; F or G is the Vanifliing Point fought. Dem. Turn up the Triangle CBD, perpendicular, and turn over tlie Plane PEG, or FG, till DE coincides with DB. Then, if the Sun be on the other fide of the Pidlure, EF is parallelto the Sun’sRays; becaufe the Luminary is in the Plane EFG produced, and, the Angle DEF is equal to its Elevation; confe- quently, EF produced would pafs through the Center of the Luminary, wherefore, F is the place of the Sun on the Pifture ; i. e. F reprefents the Sun confequently all the P^ays center there, and therefore it is their Vanifliing Point. For Its Diftance is infinite, to all fenfe. CASE 2d. When the Sun is on this fide the Pi£fure, being behind the Spe^fator, it cannot appear in the Pidfure, but its place is tranfprojefted to G j for, DEG is the Angle of its Elevation, and it is in a continuation of the Plane PEG; wherefore, EG is parallel to the Rays of Light; confequently, G, the rranfprojedled place of the Luminary, is their Vanifhing Point. Or, as BroolcTaylor, very pertinently, calls it, the Shadow of the Spedfator’s Eye, on the PidtufC j for E is the Eye, wherefore GE, being produced, would pafs- through the Sun’s Center. 262 PROJECTION OF SHADOWS - Book IV. Plate XLL PROBLEM II. ^he Vanlfhlng Point of the Sun s Rays being given y and the reprefentation of a Line perpendicular to fame Plane^ whofe V anifhing Line is given ; together with the Center and Difiance of the Picture ; to find the reprefentation of the Shadow of that Line^ on the Planey and its Vamfhing Point. Fig. ‘Let S be the Vanifliing Point of the Rays, VL the Vanlfhing Line-of the Plane, . and AB the given reprefentation of a Line perpendicular to that Plane, C is , the Center of the Picture, whofeDiftance is known. Find F, the Vanilhlng Point of Lines perpendicular to the Plane, whofe V«a- nifliing Line is VL (Prob. 2. Seit. 12. B. 3.) confequently, AB vanifhes in F; . draw SF, cutingPL at V. V is the Vaniihing Point of the Shadow. Draw VB and S A, interfeding at D j B D is the Shadow of AB, required. Dem. Whether S be. confidered as the Image of the Sun, on the Picture, or S its tranfproje£ted place, • SF is the Vanifliing Line of the Plane of Shade, for all Lines perpendicular to that Plane; feeing ■that, F is the Vaniflung Point of all fuch Litves ; wherefore, V, its interfeftion with VL, is the Va- nifliing Point of the Shadow, of all Lines perpendicular to the Plane whofe Vaniihing Line is V L. For, S V and AB reprefent parallel Lines (Cor, to Theo. 3.) wherefore, a Plane may pafs through both Lines (Ax. 5.) and confequently, S A, VB, AF, and SF are all in that Plane. But, F is the Vaniihing Point of Lines perpendicular to the Plane ; and, becaufe S reprefents a Point at an infinite Dillance in that Plane, S F is the Vanifliing Line of a Plane palfing through A B. But, S reprefents the Sun, the Vanifliing Point of the Rays of Light, and V is its Seat on the Plane (whofe Diftance is fuppofed infinite) confequently, V is the Vaniihing Point of the Shadow, BD; for, the Plane of Shade, SAF, projedting its Shadow, is parallel to a Plane palling through the Eye and the Points S and E, feeing they are at an infinite difiance; and the Shadow, BD, is the interfedlion of that Plane with the Plane of Projedion ; therefore its Vaniihing Point is V, the inter- fedion of their Vaniihing Lines, VL, and SF. (Cor. 2. Theo. 6.) This Problem is univerfal ; l ihall, next,, apply it to .Planes which are perpen- dicular, to the Pidiure. No. 2. Secondly. Let AB be a Line perpendicular to the Horizon, whofe Shadow is required, and S the reprefentation of the Sun, on the Pidiure.; or S its tranf- projedled Image, the Sun being fuppofed on either Side. Draw SV perpendicular to the Vaniihing Line; then, V is the Seat of the Lu- minary, its Diftance being fuppofed nfinite. Draw' VB and SA, as before, .interfedling at D ; BD is the Shadow required. For, becaufe A B, whofe Shadow is tO' be projeded, is perpendicular to the Horizon, and the Pidure is fuppofed vertical; it is parallel to the Pidure, cenfequently SW, parallel to AB, is the Vaniihing Line of the Plane of Shade ; wherefore, V is the Scat of the Luminary, the Vaniihing Point of the Shadow ; and, SD is a Ray of Light, projeding the Shadow of the Point A, which determines.its length, BD, This is univerfally applicable to all Planes which are perpendicular to the Pic- ture, whether they be horizontal, vertical, or inclined to the Horizon. Thirdly. When the Luminary is in the Plane of the Pidiure. In this Case, the -Rays having no Vaniihing Point, the inclination of the Rays to the Plane of projedlion, being determined, they are all parallel on the Pidiure ; -feeing, the diftance of the Vaniihing Point is infinite. No. 3® Draw BD parallel tO'the Vaniihing Line, and SA, cufing BD at'D, making the Angle SDB equal to the Angle of the Sun’s Elevation ; BD is the Shadow required. For, becaufe the Luminary is fuppofed in the Plane of the Pidure, it is confequently in every Plane parallel to the Pidure ; wherefore, the Plane of Shade, SDB, is parallel to the Pidure, feeing the Line AB is parallel ; and confequently, BD, its Shadow, is parallel to the> Vaniihing Liiy of the Plane of •-rrojedion, feeing it is parallel to the Pidure. VWhen Sevfl.illl. BY THE SUN, ON PLANES. 2621 When the Plane of projedion is parallel to the Pi(5lure, there can be no Shallow proje6ted on it, but when the Luminary is on this fide. Let AB reprefent a Line perpendicular to the Plane X; and, let S be the tranf- prqjecled Image of the Sun, on the Piflure ; 'C is its Center. Join CS. ;DrawBD, parallel to CS, and AS, cuting BD, at Dj BD is the Shadow required 'Bc-caufe the Plane (X) is parallel to the Picture, it has no Vanilhing Line; wherefore, feeing the diftanceof the Luminary isfuppofed infinite, its Seat, on that Plane, is alfo at an infinite Diftance. But CS, produced, is the Seat of a Ray of Light, projefting its Vanifhing Point, S; for it pafies throut^h the Eye, which is perpendicularly oppofite to'C; wherefore, the Seat of the Luminary is at an infinite diftance, in the Line SC, produced. Confcquently, the Shadows BD, DD, &c. are all parallel toCS; and S is the Vanifhing Point of the Rays, which determine the length of the Shadow. P R O B L E M III. To proje5i the Shadows of Right Lines, on a Plane to which they are parallel and, in- all pof lions to the Pidiure. ♦ j First, when the Lines.are pamllel to the Pifture, and the Luminary in the Plane of the Picture. The Shadows of Lines, on a Plane to winch they are parallel, cannot be determined, conveniently, without having their Seats on that Plane; and is the fame as finding the Shadows of Lines perpendicular to the Plane; or, the Shadow of one extreme of' the Line being found, the other is eafily determined. Let AB be a Line parallel to the Pi£ture, and parallel to the Ground Plane, on which the Shadow is to be projedted. Its Seat is ab. ' If the Sun be in the Plane of the Pi£lure, and if it was in the Zenith of that place, the Shadow of AB is But, if the altitude of the Sun be the Angle SCD; then, drawing AD, parallel to SC, the Shadow of AB is CD; for, the Shadow of a Perpendicular, Ka, is aL>; and CD is equal A B, feeing, ABCD, the Plane of the Shadow, is a -Parallelogram. 'After the fame manner, the Shadow of the inclined Line, ’EF (being parallel to the Pidlure) is projected ; by producing the Line till it cuts the Plane, at G ; then, drawing GI, parallel to the Vanifhing Line, and SH, SI, through F- and 'E, parallel to SC, cuting GI, at H undl ; HI is its Shadow. Secondly. If the Line AB be either perpendicular or inclined to the Pifture, and parallel to the Horizon, its Seat, at leaft of one extreme (B) muff be deter- , mined ; and as the Point B may be in fome vertical Plane, (as X) its Seat on the ' Ground is in the Interfedlion of that Plane, at b; and fuppofing the Plane X ^parallel to the Pidlure, the Luminary is in that Plane.; confequently, the Sha- dow of the extreme B will be fomewhere in the Jnterfection of the Plane. Draw SD, through B, making the Angle SDb equal to the altitude of the Sun, , giving the Point D, in the Interfeflion of the Plane X, for the Shadow of B. 'Then, becaufe AB is perpendicular to the Picture, C, is its Vanifhing Point; and, becaufe the Shadow is proie6fed on a Plane, to which AB is parallel, the Shadow is necefifarily parallH to A B, and confequently it has the fame 'Vanifhing Point, C, Wherefore, draw CD and produce it; draw AE parallel to SD, (the given Ray ofLight) cuting CD, produced, at E, the Shadow of A. DE is the Shadow of A B on the Ground ; to wliich it is parallel and equal. For the Plane of Shade ABDE is a right angled Parallelogram in Perfpe-ftive. EG is its ’Shadow on. an inclined Plane, to which A B is parallel; proje£fed by the fame Rays of Light, or. Plane of Shade. - SCHOL. Becaufe the Sun is in the Plane X, it cannot be faid to be illumined; confequently, no Shadow can be projedted on it; otherwife, if the Luminary was ever fo little on this fide, the Shadow „ of .AB will be firft pjojefted on it, from B to D, and then projected to E; in which Cafe, BD would . not be a Ray of Light, but a Shadow, and confeqwently, not ^parallel to AE, .2 Fig. i4„ No. No. -Case 264 PlateXLI Fig. 15. No. 2. 'No. 3. PROJECTION OF SHADOWS Book IV, Case the Second and Third. When the Luminary is on this fide, or on the other fide of the Pidlure. First. Let AB be parallel to the Pi6lure, and to the Plane, on which its Sha- dow is to be prcye, and the lirft on the other fide, at y, on the low Roof. The Shadows of the Piers of the Arcade, againR the Wall, w'ithin, are projeCled by means of the Vanilhing Point S, till they meet the wall, where they are upright. Lines drawn to ©, determine their height. (See SeCl. IV. for the Arches.) 'I bus, the Shadow of the whole Building is compleated ; in which are all thp variety of Examples, neceflary for projeCling the Shadows of rightlined ObjeCls. ^73 SECTION IV. Of the Shadows of curved Lines and curve-lined Objeds, on plane and curved Surfaces. \ S the Perfpe^live of curve-lined Objeds is more difficult and liable to error jT^ than right-lined, on account of the continual bending, and varying in the direction of curve Lines ; fo, their Shadows are with more difficulty projedted. As there are no vanifliing Points of the Original Lines, fo neither can there be va- uiffiing Points of the Shadows; alfo, when the Shadows, either of right or curved lines, are projedted on a curved Surface, there being no Vaniffiing Line, the figure of the Shadow cannot, fo certainly, be deferibed thereon. Neverthelefs, Rules may be prefcribed, for projedling Shadows, in fuch cafes ; which, if followed, will produce the true projedtion of as many Points, in the Curve, as are neceffary, for delcnbing the contour of the Shadow^ with tolerable accuracy.- The principal Shadow being determined, the minutias, of Mouldings, &c. may, byaPerfon who has judgment, be done by Hand, and in many cafes they tnuft ; for, I fairly own, that it is more than human patience can bear, nor indeed is it poITible, to projedt Shadows, in all Cafes, with mathematical exadlnefs. The Shadow of a Circle, projedled upon any Plane, td which the Circle is parallel^ is a Circle, of equal dimenfions ; in all fituations of the Luminary, whatever. For, whether its fituation be on this, or on the other fide^ or in the Plane of the Pidlure; whether it be more or lefs elevated, *tisftillthe fame; the projedling Rays, being parallel amongft themfelves, generate a Cylinder, for the rhoft part oblique; coufequently, being cut by a Plane parallel to the Circle, its projedlioti muft alfo be a Circle, feeing it is the oppofite Bafe of the Cylinder ; but, being cut oblique it is an Ellipfis. So that, in Arches, &c. parallel to the Pidlure, the Lu- minary being on this fide, and projedling the Shadow on a vertical Plane, parallel to the Arches, the Center of each being determined, the reprelentation of the Shadow of each Arch, may be deferibed with Compafles; which will be in pro- portion to tiie Arches, as the diftance of the Plane of the Arches, to the diftance of the Plane of projedlion. But, being projedled on the Floor, or any other Hane^ not parallel to the Arches, their Reprefentations are Ellipfes; except when the Cylinder, projedling the Shadow, is cut fubcontrary*; in which Cafe, the Shadow- will be a Circle. In all other pofitions of the Arches, to the Pidlure, and in all fituations of the Luminary, being projedled on a parallel Plane, the Shadows being Circles, their Reprefentations are Ellipfes, deferibed as the Arches, themfelves ; and being projedled on any other Plane, the Shadows being Ellipfes, their reprefentations are^ alio Ellipfes, more excentric, generally, though not always fo. The Shadow of a Sphere or Globe (as well as its perfpedlive Projedlion) on a Plane, is an Ellipfis, in all Cafes, whatever; except when a Right Line paffing through the Center of the Luminary, and of the Sphere, is perpendicular to the Plane of Projedlion. • An oblique Cylinder or Cone is cut fubcontrary, when the Plane of fedlion is fo (ituated, that the Axe of the Cylinder or Cone has the fame inclination to it, contrarywife. As, if the Sun be elevated 45 Degrees, and be fituated fo, that a vertical Plane, palling through its Center, is perpendicular to the Plane of the Arches, L e. to a vertical Circle ; then, if thefliadow be projedled on the Floor, or any ho- rizpntal Pkns,. it will be a Circle; for the Rays cut the Floor, in the fame Angle, viz. 45 Degrees. 3 Z EXAMPLE 374 Plate XLV. PROJECTION OF SHADOWS. Book IV. EXAMPLE VIII. To projeSl the Shadows of Cylinders^ on a horhontal Plane^ lying along ; another up~ right, Jo ftuated, that its Shadow crojfes one of the former. Fig. 29. No. 2. Fig. 30. The Bafes of the Cylinder (X and Y) being Inclined to the Pidure, their repre- fentations are Ellipfes, whofe Shadows, may be thus projected. Let ABC D be the reprefentation of a Square, circumfcribing a Circle, whole fides are parallel and perpendicular to the Horizon. Draw the Diago lals AC and BD, alfo the two Diameters ab and cd, parallel to the fides 01 the bqiiare ; and, from the Points e and f, draw perpendiculars to the Bale, A D. Find the Shadow of the Square A BCD (by Prob. 2. and 3 ) and draw the Dia- gonals, AC and DB, draw dS, alfo gS and hS, cuting the Diagonals; and through their Interfedion,' draw ^<2, to the Vanifliing Point of AD, &c. Then, if a Curve be defcribed through the eight Points, a, e, d, f, b, g, c, and d, it will be the true reprefentation of the Shadow of the Circle, a c b d. h Or, becaufe there is but half its Shadow required, proceed thus. ' ‘ Draw ab perpendicular, through its Center ; alfo, cd and ef parallel to ab.' ■ Find the Shadows of a, c, e, &c. (Prob. 2. 'Cafe 3.) and delcribe the Curve aeb. Then, the Sides of the Cylinder being' parallel to the Plane of projedion, draw cV,'a Tangent to the Curve; and, the Shadow of the other End being de- fcribed by the fame means, compleats the Shadow of that Cylinder. ■' j . ' •»"' I- . Second. The Shadow of the Cylinder Z, on account of its fituatlon, is but little feen. Its Bale is turned from the Light, in the other it was illumined, lo that, ’tis the other edge of the prcle which projedsthe Shadow. Third. The Side^ of the uprigln Cylludci being perpendicular ro the Horizon, draw AS, a Tangent to its Bafe, at A, cuting a V, the feat of the Side he, on the Ground,, of the Cylinder Z, at ; from which Point, draw ^b perpendi- cular, cuting be (the greateft procedure of the Cylinder Z) at b. But, on account of the inclination of the Rays of Light, the Shadow of the Side AE will be continued ’6n the Ground, till it falls into that of the other, aC a ; and it is continued down the fide, from b towards a, till it is loft, inlenfioly, in the lhade of the other Cylinder. - Take as many Points, b, c^ d, in the curve of the Bafe, of the Cylinder Z, as are neceflary ; . and others, anfwering to them, that is, of equal height, pt rfpedively, at B, C, D, on the Side, AE, of the upright Cylinder j from all which, draw Lines to S, cuting others, drawn from the correfponding Points b, c, d, to the Vanilhing Point V, at b, c, d, &c. through which, a Curve being defcribed, will be the Shadow of the Side AE, on the Cylinder Z. For, the Side of the Cylinder (AE) projedlng the Shadow, is perpendicular to the Plane of projtftion ; wherefore, the Plane of Shade, occafioned by that Line, vaniflies in O S; and, all honzontal Lines (in that Plane) from the Points A, B, C, D, &c. vanifli at S ; and, confequeritly, they muft cut others, b^, &c. drawn in the Side of the Cylinder Z, being of equal height. I'herefore, abed is the feftiori of the Plane of Shade, with the furface of the Cylinder Z, made by the Side, AE, of the upright Cylinder; and confequently, abcdi^it^ Shadow on that Surface. The upper Bafe (EF) of the upright Cylinder, is above the Eye, and, being parallel to the Plane of projedlion, its Shadow is a Circle ; wherefore, having ob- tained the Shadow of any Diameter, asef, ofEF, by drawing Eo and Fo ; the former cuting AS, at e, and drawing ef to the Vanifhing Point of EF ; then, the Shadow being a Circle, and proje6led on a Plane, not parallel to the Pifture, its reprefentation is an Ellipfis (Theo. 2. Seift. 5. Book I.) ' Defcribe the reprefentation of a Circle, whofe Diameter found is e f (Prob. 2. Seift. 8.) and fromG, draw a Tangent, tending to S, the Shadow of the other Side of the upright Cylinder, which alfo cuts the other in the fame manner, and com- pleats the Shadow, as much as can be feen. Let it be obferved, that the Shadow, . on the Cylinder Z, is darkeft, where the Light is ftrongeft. Plate XLV 1 Sea. IV. BY THE SUN, ON PLANES, &:c. 275 EXAMPLE IX. 4 J’o projeSi the Shadows of Right Lines , on a convex cylindrical Surface, Let AK be a Right Line, any how fituated, in refpea of the Cylinder X; its Fig. 31, Vanilhing Point is V. Its Shadow on the Cylinder is required ; © being the tranfprojeaed Image of the Luminary. Draw.V 0, curing the Horizontal Line at J, the Vanifhing Point of the Shadow on the Ground. Draw AJ, cuting the Bafe of the Cylinder at a; and, thjough a, draw q cuting AK at A. - Aa, on the Ground, is the Shadow of AA, part of the Line AK; ’beyond which, its Shadow is projeaed on the Cylinder. . Having obtained S, the Seat of the Luminary, (Prob. 2.) ahb, R, the Vanifh- ing Point of the Seat of AK, on the Ground, draw AR; and, from various Points, B, C, &c. (in AK) at difcretion, beyond A^ drawB 5 ,-CC, &c.; perpendi- cular; from all which Points, j5, C, &c. fo obtained, draw Lines to S, cuting the Bale of the Cylinder, at b, c, &c. from which, draw perpendicular up the Cylinder^; and from the feveral Points, B, Ci &c. draw lines to 0, , cuting^ the cotrelponding Perpendiculars, at c, d^ &c. through 'all which,, a Curve,; a 3 cj^/t^, ,heing de- fcribed 13 the Shadow of AF, part of A K, on the cylindrical Surface,. .;; The remaining part of its Shadow, falls. on the. Ground, behind the Cylinder. ^ f ■ * ] ' ' j f * j ■,»]■** * Second. FH, on the top of the Cylinder, is a Square, whicH may repfefenf the Abacas of the Tufean Capital; to project its Shadow on, the 'Cylinder and Floor.^ •Take as many Points f, g, h, &c. as aie necelTary, in the. low<"r lanes, which call: their Shadows on the Cylinder ; from, all which, draw tp S ; alfo, from :the Angle G, cuting the upper Bafe, pr Curve, at 1, z, 3, &c.. . , ? Draw perpendiculars from each Point, down the Cylinder,,, which are cut, by Rays from each Point, f, g, &c. to o, at f See. the Angle- G is projeded to 'Gi The Ray k o , touches the Cylinder, at whence, it isprojeded.tp K, on the Floor. Curve Lines, drawn through Gfgh, and Gik, are the Shadows of the 'Lines Gk and G 1 on the Cylinder; kF is projected on the Floor. ^ ^ Draw Sa, a Tangent at the Point a, cuting ‘the Ray, from k at iC/ zK ls the Shadow of the fide of the Cylinder on the Floor. Draw KFto the Vaniihing Point of GF, and F o, cuting It at F, the Sha- dow of the Angle F ; draw FL, to the Vanifhing Point of GH, and FI, the upper Line, on the other fide, which cafts the Shadow ; I o cuts it at 7 ; through which, draw /M, tending to the vaniihing Point of HI, and compleats the Sha- dow on the Floor. ■ ■ ’ ‘ .V. Third. NO is a Line projefling from the Cylinder, at pleafute; let P be its Vaniihing Point ; to project its Shadow, on the Cylinder. Draw PQ, perpendicular, cuting the horizontal vaniihing Line, in the vaniihing Point of the Seat of N O, on any horizontal Plane, imagined at difcretion (above or below the Vaniihing Line) cuting the Cylinder. Let Nu be a fedlion of^fuch a Plane with the Cylinder. Draw Oo, perpendicular, cuting TVu at o, and, through o, draw Qn, inde- finite ; from the extreme N, and from various other Points, as r, s, in N O, draw perpendicular, cuting Qn, at n, t, and u, from which, draw to S, cuting iVu, at N, /, and u. Draw Nn, tr^ and u j, perpendicular, and N o , &c. cuting them at «, r, and j ; through which Points, the Shadow of NO is deferibed. The reft is obvious, from the Figure. EXAMPLE 5 »ROJECTlON OF SHADOWS Book IV. ^76 Plate XLV. Fig. 32. EXAMPLE X. To projeB the Shadow of a Tufcan Bafe on the Floor ^ cafually fiuaied to the PiSiure, O is the tranfprojefted Image, and S the Seat of the Luminary, Draw AS and Bo, cuting at and draw ; Co, cuting ^V, determines the Shadow of the edge of the Plinth, BC, provided nothing elfe interfered. But, the Torus, &c. and Shaft of the Column unite their Shadows with it. For the Column, draw SB and SF tangents to its Bafe, on the Ground; from the points pf contrail, draw BD and FE, perpendicular; which determine where the total Shade, on the Shaft, begins. B D, and EF, are the Sides of the Co- lumn which project the Shadow, on the Ground. Draw D o , and E o , cuting BS, and FS, at d and e, the Shadows of D and E. DE, is a Diameter of the Column; and de being drawn, tending to the fame Vanilhing Point, may be confidered as its Shadow ; on which, as a Diameter found, deferibe the reprefentation of a Circle (Prob 3. Se£t. 8.) or, of a Semicircle, dfe, only, which terminates the Shadow. The Column being fuppofed cut, by a horizontal Plane, through DE, its Shadow is, confequently, a Circle ; being projected on a Plane to which it is parallel. The Shadow of the Fillet g h, may be projedled after the fame manner ; but, as fo little of it is feen, it would be unneceffary trouble. Draw a Tangent to its Bafe, at b, from S, and draw bg perpendicular, and g o , cuting b S, at gy the Shadow of the Point g. By the fame means the Sha- dows of other Points, may be obtained. The Shadow of the Torus is not eafy to deferibe; feeing its Surface is conti- nually varying, it is fcarce poflible to determine what part of it proje ; to which, the Shadow of that Line, on the Leaf Y is parallel ; and, on the LeafZ it tends to X in the Vanifhing Line of the Leaf, as the Sha- dow of A B, on t lie fame Leaf, to K; the Points where the Vanifhing Line of the Leaf is cut, by the Vanifhing Lines of the Plane of Shade of each Line (AB and C D) relpedively. ^ The Shadow of the Supporter, CE, is p'^ojeded firfl on the Floor, to f. w .ere it cuts the Wall ; and, if the Shadow of any other Point, in the Line, be j^uo- leded on the V/all, as at g, then, f^ is its bliadow on the Wall. Sea. V. BY CANDLE-LIGHT, ON PLANES, &c. For the Steps; froin O, Lines arc drawn through the extremes By Cy See, on either Side, till they cut the Floor, or leaves of the Screen, at c, &c. and from, a, by &c. to the Vanilhing Point of the Step?, on the Floor, aiyd on the Leaves, being parallel to them. Otherwife, they are drawn to that Point in the Vanhhing Line, of each Leaf, in which, the Vanifliing Line of the Plane of Shade of the Step cuts it. Or, drawing through both extremes, of each Step, as jB and D, &c. curing the Shadows of both Sides, on the fame Plane, the Va- nilhing Point is unneceflary ; but when it fallswithin bounds, it is the moft corredl. In this Cafe (viz. of Lines inclined to the Plane of Proje£l:ion) the Vanifhtng Line of the Plane of Shade is necelTary, when the Shadow isprojc£led on various Planes, but it is not fo eafily determined, as for Shadows projected by the Sun ; nothing more l)€ing required than to draw a Right Line through the Vanifltmg Points, of the Line and of the Rays, hecaufe its diftance is fuppofed infinite; and confe- quently, the fame Vanifliing Line of any Plane of Shade, ferves for all Lines which arc parallel, Pur, here, the Light being at a Ihort diftance, ic cannot poflibly be in the Vanifhing Line, which is at an infinite dlltance ; except apparently fo, when it happens to be fo fituated, in refpeft of the Eye. That V L is the Vanifhing Line of the Plane of Shade of A B, ismanifeft; and is determined either by means of the horizontal Vanifliing Line (HL) or the vertical Vanifliing Line (M N) of the Wall, W,_ on which the Shadow would be projedled, the Screen being out of the way. For, AB, cuts the Wall, being prodcced, at P, as the Floor at A ; and V O reprefents a Line parallel to A B ; which, is a Ray of Light in the fame Plane with AB, cuting the Floor, at G, and the Wall, at Q. Confequently, G A, produced, is the Shadow of AB, on the Floor; and, PQ_produced, is its Sha- dow on the Wall ; the former cuts the Vanifliing Line of theF'loor, at L, the latter cuts the vertical Vanifhing Line at M ; both which, are. in the fame Right Line palling through V; and, fince L and M are Vanifliing Points of the Shadow, V L is, confequently, the Vanifhing Line of the Plane of Shade, occafioned by the Line AB (Theorem lo.) 2. The Shadow of the Screen, on the Wall and deling, may be thus projeded. Through, Ay B, Sec. at the bottom of the Screen, draw SAy SB, &c. and pro- duce them to the Interfedion of the Wall (W) with the Floor, cuting it at i, 2, Sec. from which Points, draw Perpendiculars up the Wall; and, through the Angles Fy Gy Sec. at the top, draw OF, OG, &c. cuting the Perpendiculars, cor- relponding with AFy See. at f, g. Sec. and draw fg, gb. Sec. But the Ray O/, cuts the Cieling ; wherefore, having found f, the Seat of the Light on the Cieling, draw fi, to that Point where the Perpendicular, from 4, cuts the Interfedion of the Wall with the Cieling; and O/, produced, limits the Shadow, at/; let it be produced, alfo, till it cuts the Perpendicular at k, and join )6k, cuting the Interfedion aty, and join Ji. This procefs would be indifpenhbly neceffary, provided the Lines, HI, &c. were not parallel to the Ceiling; but being lb, they are drawn either parallel to the Line (as for HI) or to the fame Vanifliing Point (as for IK). Aifghj ikl, are the extremes of the Shadow of the Screen, on the Floor, Wall, and Cieling. By the fame means, the Shadow of A B, may be projeded on the Screen, thus. Draw SAy SB, &c. cuting AF, the Seat of AB on the Floor, at n, o, p, andq; from which, draw Perpendiculars, cuting AB, at Ay By Sec. through which Points, draw OAzy Sec. cuting the correfponding angles of the Screen, at a, b, c, &c. which, being joined by Right Lines, give the fame Shadow as before. This procefs, though fliorter, is bv no means fo correA and maflerly ; bur, as it is performed in lefs room, it maybe applied when the inclination is fuch, that the Vanifliing Points are very remote. 3. The Shadow of the Table, on which the Candle Hands may be thus determined. Having obtained the Seat (s) of the Light on the Table, and (S) on the Floor, through s, draw a b, parallel to the Horizon, alfo c d, and e f at pleafure, cuting the Horizontal Line, at H, and, F' ; and through S, on the Floor, draw H c, and F^, indefinite; alfo parallel to a b. Then, Rays drawn, from O, through the extremes of thofe Lines, v/z. Oa, Oc, &c. (being produced) will cut the correfponding Lines in the Shadow of each,, refpedlively, by which means as many points may be obtained as are neceffary, for obtaining the true curve of the Shacloiv, which is a Circle, the Table being circular. 'i iie 28B PROJECTION OF S H A D O W S, &c. Book IV. Plate XLVIL The Candle fiandiog towards one edge of the Table, the Rays proceeding from it, around its Circum- ference, form a icalen.*, or oljbque Cone; which being projefted to the Floor (to which the 7 ' able is f.ip.iofed parallel) the Sefiion, thenon (which is its Shadow) is confeciuentiy a Circle; and, being feeu oblique, ICS Reprefencation is an Fllipfis (Theo. 2. Sedt. ^5.) No. 4. 4. The Shadow of the concave edge of the hollow Cylinder (X) on the interior Surface, may be projecled, in the following manner. Find s, the Fat of the Light on the Plane of its Safe, and draw ja, jb, &c. at pleaiure, curing the circuinlerence on both lides ; from the Points, d, e, &c. draw Kight Lines to the Vanjfhing Point of the fides of the Cylinder, and draw Oa, Ob, &c. curing them, refpedlively, at a, b, &c. through which, the contour of the Shadow may be detciibed. No. 5. 5. For the Shadow of the edge, of the conical Vefiel (W) on the Interior Surface. If Right Lines be drawn from S, the Seat of the Light, on the Plane of its upper Bafe, cuting it on both fides, and making vertical SeTions through them, down the Sides, within; then, draw O a, fitc. curing the oppofite, correfponding lines, at rz, &c. through which, the Shadow may be deferibed, on the Side. Where they fall on the Bottom, join cd. See. Ob, Oc, &c cuts thofe Lines, atj^ &c. through which, the curve of the Shadow is delcribed, thereon. By taking leveral Points (g, h, Sec.') in the exterior Curve, and finding their Seats on the Floor, its Shadow {gbi) may be delcribed on the Floor. No. 6. 6. The Shadows of Globes, it is noteafy to deferibe, with certainty. If a Tangent be diawn, from the Light (O) to any part of its Surface (as a) and, through that Point, a Sedlion, by a Plane, be delcribed, perpendicular to the Axis, (OC), (which, I freely own, is not ealy to do, being obliquely fituated^) then, take as many Points in its Circumference as are necefi'ary (a, b, d, &c.) and find their Seats, (J^, B, D, Sec ) on the Floor, or Cieling ; and, through S, or f, the Scat of the Light, draw Syf, SB. Sec. and Oa, Ob, Sec. cuting them, refpec- tivcly, at a, b, d. Sec. a Curve deferibed through a, b, Sec. will be an Ellipfis, the true Shadow of the Sphere ; being an oblique ledtion of the Cone of Rays. No. 7. The Shadow in the Nlch is deferibed as by Sunfhine, with little variation. ^ Find the Seat (/) of the Light on its Plane, (Prob. i.) frpm which, draw the Ordinates, ab, cd, &c. on which Ordinates deferibe Sedlions through the head of the Nich, perpendicular to the Plane, and draw Oa, Oc, &c. giving the Points a. c. Sec. a Curve delcribed through thofe Points is the true contour of the Sha- dow, in tlie Headp. Draw perpendicular, andjoiny^. The Shadow of the edge of the Vefiel, def, falls within the Nich ; the refl, with the Stool, falls on the Wall, the Legs being firfl: projeded on the Floor, through the Feet, by means of S, the Seat of the Light thereon. * The Shadow of a Globe or Sphere, it is obvious, is no more tlian the Shadow of a plane Circle, which is not the full diameter of the Globe, becaufe, 7'angents to a Sphere, from the fame Point can- not touch both extremes of the fame Diameter, feeing they would be parallel. The Circle projeding the Shadow is that, whofe Circumference is the Bafe of the Cone of Rays. + In this Example, the Light is fo fituated as to call: very little Shadow into the Nich; but, if the Candle was more oblique to it, the contour of the Shadow would be nearly the fame figure, the Light being below the head of the Nich ; as a tangent to the curve of the Head would always touch iton that fide of the Vertex towards the I..ight. Whereas, by Sunfliine, the Light being elevated above the Nich, a Tangent touches it on the other fide, and gives a different figure of the Shadow. And, if the Candle be elevated above it, the contour of the Shadow would be nearly the fame ; the difference arifing only from the feat of the Light, on the Plane of the Nich, which, in this is finite, in the other, at an infi- nite diftance; and alfo from the Rays of Light, which in this are diverging, in the other they are con- verging, the Light being on this fide of the Pidure. SECTION I Sea VI. OF REFLECTION, IN GENERAL. 289 SECTION VI. vOf refleQ:ed Light ; and of the refledled Images of Objeds, on Water, and polifhed, plane Surfaces j Keeping, &c. Plate XLVIII, I N treating on 'this Subjea it be necelTary to confider, In the firfl place, what is meant by Refledion, fimply, or confiderecl abdradcdly. 'Reflection, in a phylical fenfe, fignifies a rebounding of Matter, byPercufiion; i. e. when an elaflic Body, * in motion, ftrikes another Body, alfo elaflic ; it re- bounds, from that other Body, in a different diredion from that in which it was at firfl impelled ; making an Angle, with its firfl diredion, greater or lefs, accord- ing to the obliquity in which it flrikes the Surface of the other Body, If a Globe flrikes a Plane, or the Surface of another Globe (or any Surface whatever) perpendicularly, it will rebound from the other Surface, perpendicu- larly, in dired oppofition to its firfl, or incident motion; as if A falls, perpendi- cular, to the Plane EF, flriking it at B, it will rebound again, from B, towards A. But, it is found, from experiment, that if the incident motion be from C to B, oblique to the Plane, or other Surface, it will rebound, or be refleded from B towards D, alfo oblique; making the Angle A BD, with a perpendicular, at that Point (called the Angle of Refledion) equal to A BC' (called the Angle of Inci- dence) or, which is the fame thing, the Angle CBE, in which dt inclines to the .Plane, in its incident motion, is equal to DBF, in which it refleds from the Plane. And this is the fame, in all pofitions of the Plane, whether horizontal, vertical, or inclined. Hence, Light is faid to be refleded, from one Surface to .another j and, probably, from that confideration it is imagined to be material. Without taking Matter into confideration, it is -.certain, that any Surface (not wholly opake) being oppofed to a luminous Body, becomes illumined itfelf; and, in an inferior degree, illumines other Objeds, in vicinity with it. The firfl; and grand inflance, of wdiich, is the Moon and other Planets ; as tliey are more or lefs Illumined, by the Sun, or rather, the more their illumined Surfaces are towards the Earth, the more the Earth is illumined; even to fuch a degree (though but refleded) as to projed Shadows, flrongly defined. The Cafe is perfedly ■fimilar in refped of other Bodies', on the Earth. For, however any Surface be fituated to the Sun, being illumined by its Light, that Surface illumines others, which are near it, more or lefs, according as that Surface is fituated to the Sun, and as they are fituated in refped of each other. Without which. Bodies, or the Surfaces of Bodies, which are not illumined, diredly from the Sun or other luminous Body, would be fo totally immerfed in Shade as not to be vifible, ex- cepting their exterior Figure or Outline. Now, admitingRays of Light to be emited from any luminous Body, atS, and Fig, 48; falling, diredly, on another Body, or plane Surface, at A, they are faid to be r-refleded, diredly, again, towards S ; but, failing on it oblique, as at B, they are refleded towards D, making the Angle DBE equal SB A. Hence arifes an objedion to the Newtonian Syflem, refpeding the refledion of Light, from one Body to another, and from that other Body to the Eye ; by which means, only, it is conjedured, that Objeds, not illumined, diredly from the Sun, or other Light, become vifible. * By Elaftic, in this place, is meant;; {iiT>ply, hard Bodies only, as Stones, Metals, &c. of which, feme are more elaftic than others. Lead, or pur« Gold, yields to the ftroke, and therefore, does not rebound like Tin, or Tin like Copper, nor that like bon, or hard Steel. So Clay will not rebound like Free Stone, nor that like Marble; becaufe, after Percaflion, the more remote parts of Matter are ftill in motion, till the Body is comprelTed by the Stroke; which deprives it, either -wholly or in j)art, of Motion j-and confequently, it rebounds with lefs force, or not at all. /. D I.et Fig- 47- 29i^ Plate XLVIIL Fig. 48* OF REFLECTED LIGHT. Book IV. Let AB be a Plane Surface, directly oppofed to the Sun ; and, fuppofe AB the utmoft limits of the Plane. Let X be an Objeft, having one plane Face (CD)’ parallel to AB ; and fuppofe no other Objedl, or Body, near. Now, the diftance of the Sun being luppofed infinite, its Rays, confequently, fall perpendicularly on AB, in every part. Lets A, iS’B, &c. reprefent Rays of Light from the Sun. Then, according to the Maxim laid down, they are refleded, diredly back again, towards S (which, being material, is fomewhat repugnant to our Ideas of Matter; feeing that, it is continually flowing, with equal and unremited velocity, from the Sun to AB, how can it return back, in dire£t oppofition ?) confequently, the Objedl X being fituated fo, that none of the refledled Rays can poffibly fall on the Surface CD,- that Surface is wholly invifible, to any Eye at E, or E. ^ery; whether it would be vifible or not? 1 am of opinion that it would be clearly vifible, to any Eye, on this Side CD; not only in refpefl of its Figure, but that, the Surface, CD, would be illumined, by means of Refleftion, from A B ; when, according to the general Maxim, it could not, feeing no Rays, from AB, are reflected to CD. Yet, I am fully per- fuaded that it may be leen ; not only, diredlly, at E, but alfo oblique, in any diredion, as at £ ; although it is manifeft, that the Rays, from AB, rnufi: be re- fledled oblique, to CD, and again to E, or £, in various Angles and Diredlions ; and yet, the original Rays all fall perpendicularly on AB. Now, if CD be feen, at all, it is manifeft it mull; be illumined; and it is evi- dent, that it cannot be fo from the Sun, diredlly, confequently, it mull: be from Refledion; and fince no other Body is near, it rnufi be from the Surface AB. Hence, then, it is manifeft, that, Light is refleded in other diredions, from one Objed to another. How or by what means it is refleded, I will not attempt to enquire ; but ftiall only make a few Obfervations, refpeding the effeds it produces, on Objeds, fo efl'ential to the perfeding a Pidure, or true Portrait of Nature. It had, formerly, been cuftomary, witli many, to reprefent Objeds, immerfed in Shade, fo very obfeurely, as fcarce to be diftinguifhable ; whereas, it is not lb in Nature. In clear Day-light,' when the Sun does not fhine out, we fee Objeds in their true Colours, and every part is diftind; but, when th# Sun breaks out, and darts its Rays on thofe parts which are oppofite, the Colours are more in- tenfely vivid, occafioned by the refulgent luftre of the Sun-Beams. Neverthelefs, thofe parts which are prevented from receiving that additional Light, and appear to be in Shade, cannot poffibly be deprived of what Light they had before, but muft rather receive additional Light on them ; the difference, then, can only arife, from the fplendor of the furrounding Light, which dazzles the Eye, and renders thofe parts obfeure, which, before, were diftindly feen. But, when out of the full glow of Sun-fhine, we perceive every Objed or parts of Objeds, which are in Shade, as diftindly as before ; and in the fame Colours, though greatly dif- ferent from thofe on which the Sun fhines. Some, again, of late Years, run into the oppofite extreme, and make too little diftindion, between the fulleft Light and the ftrongeft Shade ; by which means, their Pieces look flat, and do not produce, on the Mind, a juft Idea of what is in- tended. For, although it is certain, that, in Sunfliine, every part receives additi- onal Light ; yet, the contraft is fo ftrong, from the brilliancy of the Colours, where the Rays fall, that, the other. Parts appear as if deprived of Light, in comparifon with thefe ; and, fince that vivid luftre cannot be given by Art, and Colour, the difference muft be made by keeping the other parts fomewhat under. Refpeding the degree of refleded Light on Objeds, it is not eafy to determine ; • that being more or lefs, according to the fituation of the Objed, in refped of others ; alfo, according as that other Objed is fituated to the Light. If one Objed be fo fituated to another, that the Surface of one, being diredly oppofed to the Light, is alfo much oppofed to the (haded fide of the other Objed ; the PluteXLVni T7wfjfal&)n S^>rOfxj7S‘ :>ii:(ouiv£ • \ I % \ I k Sedt. VI. OF REFLECTED LIGHT, &c. 291 the refledled Light, on the other, will be flronger than when they are more obliquely (ituated ; either the one, in refpedlof the Light, or the other, in refpedt of the illumined Surface. For example. The Cylinder, AB, being fituated near the Wall X, on which Fig. 49, the Light is diredl; the (haded Edge (AB) has a much flronger refledlion, than if the Plane (X) was more inclined to the Light ; or, being removed, and no other refledling Surface near, there would be no Refledlion on the Column, in this pofi- tion of the Light, fave what is received obliquely, from the Ground. Refpedling plane Surfaces. We frequently fee (in Prints and Drawings) the (haded face of a Pedehal, &c. very dark, at the hither edge, and gradated to- wards the other, fo, as to give it the appearance of concavity ; which cannot be, when no other Objedl is near. But, when one part of an Objedl projedls from the other, as the Plane X from Z, the Plane Z, being much oppofed to a (b ong Light, and confequently the Shadow of the Plane X, on Z, is very narrow; then, the Light, refle£led from Z to X, will be (Irongeft where it joins to Z, and gra- dually lefl'ened, towards the other edge ; but not in fo extravagant a degree, as may be feen in various Prints ; particularly in Mr. Kirby’s Perfpedlive of Architedlure. Now, the reafon for this is extremely obvious ; becaufe of the vicinity of the part which joins to Z, the Light refleded on it from Z is flronger than on thofe parts which are more remote ; and confequently, the refleded Light is more languid on the remote parts. But, if the Light was very oblique on Z, (b as to call a great breadth of Shadow, on the Plane; in that Cafe, the Light, refleded to the Plane X, will be more faint ; infomuch that, there is barely a diftindion between the Shadow, and the Objed occahoning that Shadow ; which there always (hould be, in fome degree, the Shadow being darked ; becaufe, the Shadow cannot receive any advantage from Refledion, which the Objed does. In refped of Mouldings; the edge of the Facia, AB, being in full Light, calls its Shadow on the Cavetto below ; confequently, the Mouldings cannot have their effed from the dired Light, but from Refledion, only. On the returning Mould* ing; the under Facia or Planceer (BC) by means of Refledion, from below, is brighter than the vertical Facia, over it j alfo the vertical Face of each Fillet, is darker than the horizontal ; which in other cafes, having the Light diredly on them, is confequently brighter. In Figure 4, Plate 40, the great projedure of the Corona (X) deprives the Mouldings, below, of Light, entirely, from above, which are diflinguifhable only by means of Refledion, from below ; confequently, the Ovolo is brightefl towards the lower edge, and the Cavetto, towards the upper ; and, they are more or lefs fo, according as they are oppofed to fome illumined Body, refleding Light on them. It may, therefore, be made a general Rule, with very little exception, that, the effed of refleded Light on Objeds is reverfe tq the dired Light ; particu- lar regard being had to the fltuation of the Objed which occafions the refledion. For, certainly, in whatever degree an Objed is illumined, or from whatever caufe, whether diredly or refleded, the effed of it, on the Objed, is the fame, in pro- portion to the degree of Light. But when (as is frequent) various Objeds_ refled Light on the fame Objed, no one being particularly predominant ; the Light by that means is fcattered and confufed, fo that, there is little or no Shade on the Objed j infomuch that, the parts can fcarce be diflinguifhed, one from another. The fame effed is produced from various luminous Bodies, varioufly (ituated to an Objed, or from various inlets of Light, on Objeds, each deftroying the effed occafioned by the other. In a Nich (fee Plate 43.) the Shadow is (Irongeft at the edge or outline, and gra- dually foftened towards that (ide of the Nich, which occa^ns the Shadow ; owing to the ftrong Refledion from the other Side, on which the JL.ight falls, dired. 7 Fig- 50* The OF REFLECTED IMAGES, ON WATER. Book IV. iiq2 fPlate XLVIII. The Shade (without Sun-fhIne)on a convex or concave fegment of a Cylinder, or Sphere, is the fame ; each may be the other, by fuppofing the Light on either Hand. More might be laid in refpedl of refleHence, we have an unerring Rule, for reprefenting the refleded Images of Ob- ■jeds on polhhed Surfaces. But it mufl beobferved, that, if the Objed: be at fomc diflance, beyond the Water, the meafure of the Objed mull be applied from its . Seat, on the Plane of the furface of the Water, not from the Water edge, as when it is in the Water or clofe to it. Alfo, that the Image reprefented on the Water, is not fimilar to the Reprefentation of the Objed, on the Pidure, which I lhallfhew,; except when a plane Figurc.only is reprefented, parallel to the Pidure. Is, E X A M P L E -s. ,52; ^ -Eet X be an Objed in the Water, having one Face (X) parallel to the Pid:ure. ’No. !• Produce the Sides, BA and CD, making AB and CD refpedively equal to them; alfo, make the perpendicular '£F .equal to EF, and join BJS and CE; i. e. make the Figure A£D fimilar to AED, inverted, from the Line AD where the Surface of the V/ater cuts the Objed:. The Image of the Door, &:c. is alfo fimilar, I, f‘ * For, the Angle BbA — EbF (the Anngle of incidence, tothe Angle of rcfiedlion) and, Ab 5 — EbF ! (2. I. Fl.) wlierefo'C EbArrAbA (Ax. 3. El.) then, bccaufe the Angles at A are right (AB being fup- ( |>o!d perpendicular 10 the Surface of the Water) and Ab is common to the two Triangles, ABb and •A Z) b, confequently, AS is equal. to AB; for the Triangles are congruous (tl. j.El.) I .For Sea. VI. OF REFLECTED IMAGES, ON WATER. 293 For the Face AG, being perpendicular to the Piaure, the horizontal Line, BG, vanifhcs in the Center; wlierefore, BG being parallel to the Water, its Image will be parallel to the Line, and conlequently, it has the fame Vanilhing Point. Draw BC, meeting GH, produced at G, which compleats the Image cf that Face, and which, it is obvious, is not fimilar to the Reprelentation ABGH ; feeing that, the Angles which are acute, in one, are obtufe in the other. Ifikewile, a fide of the Roof is rcprefented in the Objecl, which, it is manifell cannot be feen in the Water, bv Retledlion. The Piles, at U, which are perpendicular, have their Images alfo equal, each to tiie Reprefentation ; likewife, the Pile W is reflefted equal, and equally inclined, be- ing parallel to the Picture. But, being otherwife inclined, as Y and Z, one hang- ing towards, the other from the Picture, the Seats of their tops, on tiie furface of the Water, being determined, at a, produce the Perpendicular ba, making equal to ab, and draw from 6, to the Timbers, where they are cut by the Surface. The Vanifhing Point of the reflected Image is at the lame diftance on the other •fide of the horizontal Line, as that of tlie Objecl:, wdietherit be above or below it. The Image of the Bridge, on the Water, may be obtained after the fame man- ner, particularly the Piers. The refle<£led Images of all horizontal Lines tend to 3* the fame Point, in the horizontal Vanifliing Line, as the Reprefentations; for, Right Lines, being parallel to the Surface of the Water, their Images are either pa- rallel to them, or they have the fame Vanifhing Points, refpedlively. The horizontal Line, HI, of the Bridge, vanifhesat O ; wherefore, PQ, drawn perpendicular through O, is the Vanifhing Line of the Plaue of the face of the Bridge; IK vanifhes at P, and the inclined Line at the hither end at Q, equally diftaut from O, being equally inclined. Wherefore, k/ being made equal to Ik, the horizontal Line has the lame Vanifliing Point (O) as its original, and the in- clined Line IK, tends to Q, reprefenting a parallel to GH, at the hither end. For the Arches ; draw as many perpendicular Lines, ab, cd, &c. as are necef- fary, and produce them in the Water; making each equal to its correfponding one, ah equal ab, &c. by which means, as many Points, in the Curve, as you pleafe, may be obtained, and the image of the Arch fleferibed through them ; each being a Semi-Ellipfis, of which, ik isa Diameter. The Crane, &c. on the Wharf, hanging over the Vfater, has its Image reflefled, by means of perpendicular Lines from each part, and finding their Scats on the No. 4. furface of the Water (as w, of M) repeating the lame meafure downwards. For the Hogfhead ; draw as many Lines (reprefenting lines parallel to th cHo- rizon) as are necefl'ary, tending to the Vanifhing Point V of the horizontal Dia- meter ab; then, by means of perpendiculars from each extreme of thole Lines, their Images are acquired, and the curve of the Head delcrlbed through them. If other Lines are drawn, from one head to the other, to the Vanifliing Point X, of its Axe, the whole Image may be deferibed, as in the Figure. The Warehoufes, &c. ftaiiding atfome diftance from the Water, have their re- '• fleeted Images deferibed as the other Objefls ; by fuppofiiig the furface of the Water No. 5. produced, and the Seat of each Angle determined thereon; as of A, B, C, the Angles of the Warehoufe W ; which are made equal, downward, from their Seats b, r,) relpeflively, and the horizontal Lines have the fame Vanifliing Points. The Spire of the Church, at T, is reflected on the Water; becaufe, its Seat (S) on the level of the Water, being obtained, the height (ST) being applied ]^o. 6. downward reaches the Water, otherwife it would not. Some Perfons have exprefled furprize, that, Objefts, at a great diflance, fhoulJ have theirlmages refleiTcd on Water which is near. The inoft diflant ObjecTs, which can be feen, will be reflected ; provided, that the Eye be properly fituated, iiear the Surface, and there be no otlier Objedts intervening, i, e. when Right Lines 4 E from 294 OF REFLECTED IMAGES, ON MIRROURS. BookIV. Plate XLVIil. from the Ohjecl and from the Eye, making equal Angles with the furface of the Water, meet on the Water j otherwife, the ObjeiSts cannot be refleded. It feems Indeed rather furprizlng, that a Surface, very much inclined to the Horizon, and nearly approaching to perpendicular with thePidure, as a gentle Aflent can be re- fledcd ; but, the Eye being fo fituated, as above, and the furface of the Water beinsoii a level with the foot of the hill, its inclined Surface will be refleded thereon. It has been objeded; that the Refledioiis are too ftrong, in the Plate; I grant they are, as a Pidure ; but my defign is to (hew the figure of the Image on the Water, rather than the effed, confidering the Water not fimply as Water, but as a Miriour. Neverthelels, in clear Water, perfedly at reft, the Images of Objeds on its Surface will be found as diftind as the Objeds themfelves appear, according to their Diftance; fomewhat differing in Colour, rather darker, in the cleareft: Water. Thus much may fufiice, for Reflsdion on a horizontal Surface; which is the fame, whether it be a clear Fluid (as Water) or other plane Minour. Secondly. Of Reflection on Mirrours, vertical and inclined. Tig. 53. Let W be a Mirrour, parallel to the Pidure, whofe Center is S; and X, an Objed, whole refleded Image is required. In this Cafe, it muft be obferved (as in the former) that the Image, of any Point in an Objed, is refleded to the fame diftance from its Seat on the Mirrour, as the original Point from its Seat ; fo, that diftance is, here, reprefented, audits Seat obtained, perfpedively (in the former it is geometrical) and, becaufe the re- ceding Surface is. parallel to the Pidure, the Center is, conlequently, theVanifh- ing Point of Lines producing its Seat, being perpendicular thereto. Draw Lines, from the Feet, A, D, &c. of the Chair to S, the Center ; and, where they cut the Interfedion of the Mirrour with the Floor, at a, d, &c. make za reprefent a length equal Aa, &c. (Prob. 8.) and draw ah and dc perpendicular; then, BS and CS, curing thofe Perpendiculars, refpedively, give their refleded Images. For, as every Objed appears to be as far on the other fide, as it really is on this fide, from the Mirrour ; confequently, in this Cafe, it muft be fb perfpedively. The refleded Images of horizontal Lines, as the rails of the Chair, &c. (in this Cafe) tend to a Vanilhing Point in the Horizontal Line, at the fame diftance from the Center of the Pidure, on the contrary fide, as the Vanilhing Point of the Re- prefentation. If V be the Vanifliing Point of BC, the front Rail; then, in SG, a diftance being taken equal to SV, is the Vanifliing Point of be, the refleded Image of BC. I is the Vanifhing Point of the fide Rail CE; SK being made equal SI, K is the Vanifliing Point of its refleded Image. After the fame man- ner, the middle Point of the top Rail inay be obtained, by'a Perpendicular, ef, to the Floor, and as many other Points as are neceflary. On a vertical Alirrour, inclined to the Pidure, there is no difference in the pro- cefs, but only in the Vanilhing Point of Perpendiculars to the Mirrour; and, if the Vanifhing Point of Perpendiculars be determined, ’tis the fame in ail inclinations. Cl Tjhc. a Mirrour inclined to the Horizon, andcafually inclined to the Pidure. ^ It is required to find the refleded Image of the Chair, on the Mirrour Z. V being the Vanifhing Point of the bottom of the Alirrour, which is hori- zontal, find the Vanifhing Line (VL) of the Mirrour, its inclination to the Horizon being known (Prob. 5. Sed. 3.) and alfo, the Vanifhing Point (F) of Lines perpen- dicular to the Alirrour (Prob. 2. Sed. T2.) Then, having found the Interfedion (CD) of the Mirrou-r with the Floor, pro- duced (its Vanilhing Point is V) and G, the Vaiiifhing Point of Perpendiculars thereto, draw AG, DG, &c. cuting CD, at D, &c. and through them, draw uih and Dc, tending to the Vanifliing Point of the fides of the Mirrour*, indefinite. * This Vanifhing Point is the Interfecftion of the Vanilhing Line (VL) of the Mirrour, and the ycrtical Vanilhing Line RG, produced. Draw Sea. VI. KEEPING, AND AERIAL PERSPECTIVE. Draw AF, DF, &c. curing them, refpeaively, in the correfponding Points, a, d, &c. (their Seats on the Mirrour*) beyond which, their Images are rcprcfented equal, inPerfpeaive ; by making a^, reprefent an equal meafiire as A a, &c. (Prob. 8.) Find the reprcfentation of the Interfeaion of the plane of tiie Front, with tlic Mirrour (which will pafs through V) and the Interfcaing Points of AB and DC, therein, by producing them ; from which Points, draw Lines through rrandr/, in- definite, and BF, CF, &c. curing them in their refpeaive Images on the Mirrour, Then, having found the Interledion EB, of the Plane of the fide of the Chair, ■with the Mirrour, and the interfeding Points, E and B, of G t\ and FB, draw Ea ^nd Bb indefinite and, from F and G, draw lines to F, the Vhuiifliing Point of Lines perpendicular to the Mirrour, cuting them at /’and^. Thus, as many Points (as e, at the topj may be determined as occafion requires, from their Seats on the Floor; by which means, the rcfled:ed Image of any Objed, 'whatever, may be reprefented on the Mirrour. BC is parallel to the Mirrour, wherefore, the Image (be) of BC, has the fame Variifhing Point, in the Horizontal Line ; otherwife, it would not ; for, although all horizontal Lines, whatever, vanifli in the Horizon of the Pidure; in this Calc, ■the Mirrour is a Pidure, and the reprcfentation of the Chair is confidered as the original Objed, in relped of that Pidure, each Plane having its peculiar Vanifh- ing Line thereon. PQ is the Interfedion of the Plane of the Mirrour with the Cieling ; by means of which, the Cage, hanging from the Cieling, at N, is refleded. If its Seat, on the Cieling be determined, the Image of each Angle on the Mirrour is reprefented, indefinite, and their reprefeutativc lengths are determined, by drawing lines from ^ach extreme to F, the Vanifhing Point of perpendiculars to the Mirrour ; as no is the Seat of NO, on the Mirrour, and«o is its indefinite Image. After the fame manner, the corners of the Cage may be reprefented on the Mirrour. Of Keeping ; and the effed: of Distance. The Term, Keeping, in the Art of Painting, in general, is ufed tofighify a juft ■and proper fubordination of all the parts of a Pidure to the principal Objed; in refped of Magnitude, Colour, and diftindnefs of Parts, The magnitudes of Objeds, in refped of each other, perfpedively, are various, according to the Station from which they are viewed ; and confequently, the fub- ordination of Colour, &c. is not in proportion to the Objeds, but to their Diftance from the Eye. It is abfolntely impoffible to lie down Rules, by which the Artift may, with certainty, produce the defired effed ; feeing that, in hazy or foggy weather, or in a mifty morning, &c. Objeds are lefs diftind, at a fliort Diftance, than, in a clear Day, at a much greater; wherefore, no proportion or degree can be determined. If Objeds, of known magnitude, appear not to be far diftant, in ■the Pidure, and yet, their parts not dlftindly defined, in comparifon with others, in the Fore-ground ; it implies, that the Air is more grofs and hazy, than if the parts were more perfed. Keeping is, in a great meafure, fynonimous with Aerial Perfpedive ; which fignifies a diminution of Light, Colour, and diftindnefs of the parts of Objeds, in a regular gradation, as linear Perfpedive of Magnitude ; owing to the effedf ot Air, between the Eye and the Objed; which, being a Medium, obftruds the fight, in foine degree, at any Diftance; confequently, as there muft be a greater quan- tity of Air between the Eye and diftant Objeds than near ones, fo their parts arc lets difcernable ; the Lights and Shades are infenfibly mixed, and, at a great Dil- tance, it is all a confufed jumble, of Light, Shade, and Colour, without diftindion. It is cuftomary with many, in delineating, and may frequently be feen in Archi- .tedural Defigns, &c. to make a confiderable difference, in Teint, between one .plane Surface and another; when the one, in the Original Objed, recedes but a Ah and Dc are the Scats of AB and CD on the Plane of the Mirrev, produced. few 295 Book IV. 296 plate XLVIIL Fig. 50. No. 2,. KEEPING, AND AERIAL PERSPECTIVE. Book IV. few Inches, which is abTard; for, if that fodden gradation of efFeft was conti- nued to the diftarce ot a few Yards, It would become totally black, before it wa> pofiible for the Air to have any apparent effect on it. Now it is far from being fo in Nature, v/hich is obvious to any Eye; for, if the materials are clean, and of an even and uniform Colour (without which no judgment can be made) I will ven- ture to affirm, that two Plane Surfaces (V and Y) parallel between themfelves, at feveral Feetdiftance from each other, and having a full Light on them, cannot be diflinguilhed one from the other, nor fo much as the Line or Edge (AD) feen, at a proper diftance for d Jineating ; provided the Light be not (on this fide) fo very oblique (as at S, No. 3 ) as to Piade the other, beyond where the Line OC cuts it. 1 fay, the Eye being fb fituated (at O) that the Angle C, appears to cut the receding Plane, at B, they will appear as one continued Surface. Ne- verthelefs, it may be necelfary, in delineating, to make a diftinCtion, in propor- tion to the Diffance ; but, if the Plane V has Mouldings, &c. on it, being cur, apparently, by the edge of the other, the Line is fufficiently defined without it. As it mufi; be obvious, that no pefitive Rule can be prefcrlbed, I fhall juft make a few.obfervations, and conclude the Subje£l ; and, with it, the Book, In refpedl of Magnitude ; it is evident, that Objeifts mry appear, in Perfpefllvc, to have any proportion to each other, although not greatly diftant. For Example; in Plate 27, of Chelfea College, the fartheft Building, with a Pediment, is not above two eleventh parts of the hither one (to which it is equal) in height ; and yet it does not offend the Eye, nor appear at too great a Dlftance. But, if the gradation of Light was in proportion, it would be too great; becaufe, not merely the height is to be con- fidered, but the Iquare of the height, which would reduce it to the proportion of thirty to one; but where we flaall fix the ftandard for unity, I am at a lofs to devlfe. Now, if we were at twice the Dlftance from the hither Building (the diftance between the two Objedls being fomewhat more than twice that) the proportion of .one to the other, would not be much greater than one to three; the gradation of Light, in that cafe, w'ould be nearly as one to ten, but one third part of the former. Yet, I prefume, that in two Pidures fo delineated, it would not be advifable to make that difference in the effed of Aerial Perfpedive. In fhort, as it is abfolutely impoffible to fix any criterion to determine the Ratio of the gradation of Light and the effed of Diftance, but muft ever be at the dif- cretion of the Artift, I ftiall only obferve, that the Objeds in the Fore-ground, as the hither end of the Bridge (Plate 48) being fuppofed near the Pidure, muft have its parts diftind and perfed, with ftrong Lights and Shades, which gradate to the other End. The Buildings on the Wharf, &c. Ifiould be lefs perfed, as they recede from the Eye. The Church, at a I^iftance, the Trees and Hills, one be- yond another, muft confequently be lels and lefs diftind, according to their Dif- tance ; till at laft they are fcarce diftlnguifliable from the Sky. For all which, the Artift can have uoother Rule than, carefully and judicioufly,- to copy Nature (the Left Miftrefs) which, iq that Cafe, is not always uniform, but infinitely variable. I N S. ?• .j .1 '•>'# V % 4 f . I # • < SfW^ 5^-6 • fpL/o ^(:)(