Digitized by the Internet Archive in 2018 with funding from Getty Research Institute https://archive.org/details/practicalperspecOOmoxo / J ' ). . \ V. By the Optic\s , How to Delineate all Bodies, Buildings, or Landskips, &c. By the Catoj>tric\s, How to Delineate confufed Appearences, fo as when / feen in a Mirror or Pollifht Body of Teaching ( any intended fhape , the reflection 1 fhall fhewaDefigne. By the r Dioptirc{s , How to draw parts of many Figures into one, when feen through a Glafs or Chriflal cut into ^ many Faces* Ufefullfora’j ^Painters, Engravers Jr chit eels, &c- and all others that are any waies inclined to Spe- culatory Ingenuity. By Jofeph Moxon Hyirographer to the Kjngs mofl Excellent Majejiy. LONDON. Printed by Jo/efh Moxon, and fold at lus fhop in duffel Jlreet, at the Signe of Atlas* 1670, 1 I.' '* j* ’ t - To the Worfhipfull WILLI ASM XftTEL^ Efq. Sir, I Trefnt you with this peece of Practical Perfpedtive: 3fot that I thinly ou un - skilfuli in this Science , for I know your affections to Mathematical Arts .are great> and your perfections therein greater than to learn any thing from me . But as you are a general in~ courager of ingenuous Studies, and have exprejl a particular rcfpeCl to me by your many Civilities,1 aL waxes reckoned my felf bound to an humble ackjaow** lodgement ; which Sir, if you will accept this as a tcl{cn of it will yet more oblige me to be Sir ? Your mofl humblefervant* Jofeph Moxon. f - , '•*, ■* -■ V < i To tfie Reader. A Mong the Ingenuous Sciences pradifed in this Ela¬ tion PerfpeChye ought not in right to have the leaft efteem• For that it is full of moft ray if? mg Specula = tions 1 think none that knows what it is will deny . 'But that it is moft abfoluteiy neceffary for all (Painters, Engrayers, Architects and others that haye occaftton to draw Drafts 8 F I 3^1 T1 0 3^ VI. Ichnographie is the Bafe or Bottom whereon any Body or Building is eredted. And therefore the Office of Ichnographie is to defcribe the Bot¬ tom or Bale of any Body, or Building. There be two forts of Ichnographies named in this Book, viz. the Geometrick Ichnographie-i and the Scenographick Ichnographie. T> 8 FI 3^1 TIO 3^ VII By the (yeometrick Ichnographie is meant the true Geometrick Bottom or Bafe of any Body or Building 3 as a Circle is the Bafe of a Column; and a Square of four equal fides is the Bafe of a Pedeftal 6cc. This Geometrick Ichnographie is not feen in Se&ion, or through a Glafs, unlefs it be a Glafs that lies parallel to the Bafe, and therefore makes no Seftion with it. But VET WIT. i Tragical Terjpecfhe* 5 T> SFI !NJTIO t AC VIU The Scenograpbick. Ichnographie is the appearence of the fame Bafe in Section, or through a Glafs erected perpendicularly on the Plain thereon it Bands: As allowing the Circle a to be the Geo- metrick Ichnographie of a Column, the Sceno- graphick Circles bed fhall be the Shortned Ico¬ nographies of the fame Column. T>8F I 3^1 T I 0 IX. The Orthographic of any Body or Building, is the fore-right fide of any Plain. That is, The C fide 4 BPraBical TerJpeBivc. fide or Plain that lies parallel to a ftraight line that may be imagined to pafs through the out* ward convex points of your two Eys, continued to a convenient length. As the Plain of the Cube in Oper. XIX. marked L is the Orthogra- pluck Plain of that Cube. Therefore the Office of Orthographie is to Delineate the foresight Plain. T> 8 F I KJ T I 0 ^ X. »* • , , The Scenographick Appearencc of any Figure* Body,or Building,is that fide that declines from, or makes angles with that ftraight line imagined to pafs through the two outward convex points of your two Eys aforefaid. Workmen generally call it the Return of a fore-right fide. As the Plains M and N of the fame Cube in Oper. XIX. are the Scenographick Plains or fides of that Cube. 1 herefore the Office of Scenographie is to de- fenbe a Plain,or other Figure that declines from the Orthographick Plain. That is, makes angles with the Plain of the Orthographie. The knowledge of Scenographie comprehends fo much of the Art of Perfpedtive, that without fcruple many Authors call it TerfpeElive, when in their works they frequently fay the Perfpedtive 'appearence, though they only treat of a Sceno= graphick Figure : As if they concluded, that he who arrives to fo great a perfedtion in Perfpe- dtive, as Scenographie underftands or with diffi¬ culty may underhand the whole Art of Per= fpedtive. The Scenographick Appearence differs from the Orthographick in this 5 The Orthographick reprefents the fide of a Body, or Building, as it is feen when the Plain of the Glafs Hands parallel to Tracheal Terjpechve- 5 to that fide. But Scenographie reprefents the fide of a Body, or Building, as it feems through a Glafs eredted not parallel to that fide. Thefe foregoing Definitions ferve to fhew you who the Lady Perfpedtive is, and her fair Handmaids Ichnographie , Orthographic , and Sccno= graphic : who all of them in their feveral ca¬ pacities drefs her fo curioufiy, that he who is ever admitted her prefence, fhall fee a Perfon beautifull enough to commit a rape upon his Ey. But fhe has a Language by her felf, which is one reafon fhe is no better underftood ; yet it is very eafie to learn, and to make it appear fo, I have compofed this fmal Dictionary following, CHAPTER- II. Containing Definitions ufed in this Artt D 8 F I TI 0 3 \CL The 8FI3^JTI0 ^C IL The Height. Is the Perpendicular diftance be¬ tween theBafe and theEy,or which is alloneThe Height of the Vifual point above the Bafe, be* caufe the Ey lies alwaies in the Horizontal line, as well as the Vifual point. Thus the height from C 2 & / 6 . ^Practical TerJpeBive . a to & in the aforefaid Figure is called the Height of the Ey. T> E F I 3\(J TI 0 3 \C lit The Vifual ^oint. This point is not that point the Ey is placed in, (as mod young Studients in this Art are apt to think.) But is a point in the Horizontal line wherein all the Occular raies u- nite. For Example, If you look on a long ftraight Brick wall, you will perceive the feveral laies of Mortar between the feveral courfes of Bricks feem to incline towards each other, as if they ran into one common Center : Or as if you were in a long ftraight Gallerey where the two Sides, the Floor, and Ceiling, feem to incline and touch each other in a point or common Center, which point reflected on a Glafs erected perpendicularly on the Bafe, is called the Vifual point. T his you may experiment if you lay the Mufcovie Glafs in Definition VI. Bat upon the Ob¬ ject abed, for then fa much of your Scenogra- phick Figure as is made of Vifual raies,as the lines a b and c d , fhall run into a point oppoftte to the point e: which point elevated to the height of the Ey is the Vifual point, as a forefaid. This point is by fome Authors called alfo the Center : and by others the Principal Point . Note, that where an Object is reflected,it is al¬ io reverted, as you may fee in Looking Glaifes and cleer water. And for this reafon it is T that Orthographick Figures are inverted in their See- nographie. As you may fee in mo ft of the Fi¬ gures throughout this Book. T> E F I 3^1 TI 0 3^ IIIl The Ho^riontal Line . By the Horizontal line is not FraBical < TerJpeBive 7 not meant the Bafe or Line of Earth, though in Aftronomy Sec. it is generally fo taken- But it is a Line proceeding from the Ey, (which in Men of a common fize is placed about five foot above the Bafe or Ground they (land upon,) into the Vifual point., and is Parallel to the Bafe, as the Line be in Chap.i .Vefin.s. is the Horizontal Line. 0 8 FI TIO V The Dift ante. The Diftance is that fpace on the Bafe contained between the Glafs and the point in the Bafe that lies diredtly under the Ey. As from the point a in the Bafe, to o at the Glafs is called the Diftance. Note that in many Fi¬ gures two points of Diftance is required, as in 0 - peration 10, is, 1 6, 17> 18 , and feveral others throughout this Book. DBF I JAC I TI 0 C VI. The SeBion Is a Plain of tranfparent matter eredted Perpendicularly on the plain of the Bafe, and ftands directly before you, that is is Parallel to a ftraight line imagined to pafs through the convex points of your two Eys: As is this Plain of Mufcovie Glafs erected before the Fi s gure - ■ trnmm 8 TraBical TerfpeBive . gure e. This Plain is by fome Authors called the Se Elion ,by others the Table, and by others the Glafi. They that call it a Section call it molt properly, Becaufe it cuts or interferes the Plain whereon it Hands, as here it cuts the Plain or Bafe abed. And they that call it a Table, call it fo allufively as they do other Planis whereon any thing is ei= ther painted or written. But they that call it a Glafs have fome good real on for it too ^ Becaufe we know as yet no other matter fo apt to de- monftrate the changing of an Ortho*graphick Fb gure into a Scenographick as Glafs: And there¬ fore though I confefs the word Section to be more proper than the word Glafs, yet becaufe (as I faid before) my defign is Practice : I fhall all a- long in this Book(when I have occafion to name this Plain)call it Qlafs and not Section. This Glafs or Section is abfolutely neceffary to be underftood by all young Studients in the Art of Perfpedtive. Becaufefas by what hath ah ready been faid)without knowing what it is they cannot know what Perfpedtive means: Neither can they know the reafon why the Scenogra¬ phick Pigure alters from the Orthographick, or one Scenographick Pigure from another. But by rightly underftanding this Glafs they may ea- fily perceive the Various affedtions of Sceno¬ graphick appearences. For firft, If the Glafs be placed near the Vifual point and far from the Objedf, the Pigure appearing therein will be ve¬ ry fmall: Becaufe all the raies which inclofe* both Orthographick and Scenographick Fi¬ gures though wider afunder at the Objedt run into the Vifual point as into a common Center: As you may fee in the firll Pigure,where the Cir¬ cle a appears much fmaler in the Plain or Glafs d e f g, which Hands neer the Vifual point and farther Practical ‘Perspective- 9 farther from the Object than in the Glafs h i k 1 , which (lands farther from the Vifual point and neerer the Objedt Secondly If the Vilual point be placed higher, though at the fame diftance from the Objedt the greater the Scenographick form appears: As you may fee if you place the Vifual point at m, and delineate the fame Circle Scenographically as d, it becomes bigger in length and bredth • for the Vifual raies being e- levated the feveral Perpendiculars eredted on the Glafs interfedt them in wider diftances, Be- caufe futher from the Glafs. As you may fee in the Scenographick Circle d. Thirdly, If this Glafs were fet oblique to the Plain, the Figure would again alter ; For if it Incline to the Vifu¬ al point the Scenographick appearence of this Circle (for Example,)fhall on the Glafs lengthen between the Vifual point and the Objedt.Becaufe the Plain of the Glafs gathers in more of the Vi¬ fual raies : And if the Glafs Recline from the Vi¬ fual point the Scenographick appearence fhall grow rounder and more like the Orthographick Figure. But if the Glafs be placed parallel to the Plain the Objedt lies on, the Scenographick and Orthographick Figure fhall appear the fame,- Becaufe the Figure is not feen in Section. And all this you may eafily experiment by eredting the Mufcovy Glafs in befim VI. of this Chap, be¬ tween your Ey and the Figure : For if you keep your Ey Reddy in one point, and with a black- lead pencil trace about the edges of the fame Fi¬ gure as you find it on the Glafs, you will have the true Scenographie of that Figure: As you may try by comparing it afterwards with the fame Figure drawn by the Rules of Perfpedtive, as they are taught in this Book, D 2 VEFlKlTi IO FraBical FerfpeBive* V E F I Sfl T I 0 JAC VII. The Vifual %aies, or Diametrals , are thofe lines which proceed from the Vifual point through a Glafs to any point either above or below the Horizon. As in Oper. II. the line c f e is a Vifual llaie or a Diametral: So is the lines c kg, and c If in Oper. III. © e f i no ^ vni. Lines of Difiance, or Diagonals , are thofe Lines that are drawn from the point of Pittance, to any point either above or below the Hori¬ zon. As m Oper. III. the line b k t, is a line of Pi¬ ttance : So is the lines b dlh. T> E F I T I 0 The ObjeEt. The Object is that Figure, Body, or Building you intend to defcribe in Perfpe- < 5 tive. As in Oper. II. the point d is the Objed:: So are the Squares d ef g, and l m no, in Oper. IIII. CHAPTER III. " Several eSAdethodical denies and Obfervations > that may facilitate your FraBice in Pcr- fpedive. i. A LL Lines that in the Objed: or Geome- jf\ trick Figure are ftraight lines, are in Sce- nographie alfo ftraight lines. 2. All Lines that in a Geometrick Figure are Perpendiculars, mu ft in Scenographie alfo be Perpendiculars. 3. All Lines that in a Geometrick Figure are Paral- PraBictfl C P erjJjeVnve* II Parallel to its Bafe, muft in Scenographie be alfo Parallel to the Bafe. 4. All Lines that in Geometrick Figures re¬ turn at Light Angles from the Orthographick or foresight fide’muft in Scenographick Figures be drawn from the Vifual point. 5. All Right Lines that in a Geometrick Fi¬ gure return from the Orthographick or fore- right fide,in a Scenographick Figure run into the Horizontal Line. And 6 . All Lines that in the Geometrick Figure are Parallel to that return-line, fhall in a Sceno¬ graphick Figure be drawn from that point tound in the Horizon. 7. If a Right Angle having one of its Tides a- gainft the Bafe be to be cut otf,fo as the two fides of it fhall remain equal: Then a line drawn from the point of Diftance fhall fo cut it off. 8 . If the Figure you defigne to draw (lands on your right hand, you muft place it on the right hand of the Vifual point. If on your left hand, you muft place it on the left hand of the Vifual point. But if on neither hand, the Vifual point muft be placed in the middle of it. 9. If you fet off the Height of Columns, Polls, or other things, you muft meafure the defigned Height from the BafeTine upwards in the Front or Orthographic, and a Vifual raie drawn from that point in the Front fhall fhew the Height of your Columns, or Pods, all the way behind the Orthographie,even to the Vifual point. As in Oper. XXIX. dec. This is a Rule in all Figures, afwell wherein there is an Orthographick fide placed before it, as where there is none. 1 o. If you are to make Circles,Arches,Croffcs, Crofs-Arches, or any other Figure in the Roof of any Room, you muft firft project the fame on E the 12 f radical 'Perfective- the Ichnographie, and fo with Perpendiculars from the eminent points of the fame, carry it up into the Ceiling: as in Oper. 20, 21, 22, 23, 24, 25, 26, 27, dec. And then from thofe feveral points carry on your Circle, or Arch dee. As in Oper. 9, 10, ii, 12, 13, 25, 26, &c. 11. If you would find the middle point of any Scenographick Square, Crofs it in the Diagonal Angles with two lines, and in the point where they cut each ocher is the middle of the Square. As in Oper. 17, 20, 21,22, 2 3, 24, 25, 26, 27, dec. 12. A Ground Plain 01 Squares is the lame a- hove the Horizontal line it is below : Only if it be farther diftant above or below the Horizon the Squares will be fo much the wider. As in Oper. 20, 21, 22, ^3, 24, 25, 26, 27, dec. and all other wherein Roofs, or Ceilings are defcribed. 7 be/e brief Ofules will much help you in your ( Practice. A CawiSivn or two for your more profitable reading this r £ook c '7 Ou muft take fpecial notice of the feveral |[ fhapes of Letters wherewith the feveral points of each Figure is marked 5 for in fome Figures there come fo many Letters, that I was forc’d to reitterate them : But then the Let¬ ters are made in an other fhape^either inCapitals, as A, B, or final Horn ran, as a, b ; when as other* wife they are fet down in Italick Letters,thus,^, b. For if you do not take notice that the Letter you feek in the Figure be of the fame fort with that you read in the Operation ,you may eafily miftake one part of the Figure for another. Secondly, P ractical c PcrjJ)eBive- 13 Secondly, The different Letters arcfometimes madetofhew the corrofpondent points of the Geometnck and Scenographick Figures 3 as in Oper. VI the feveral points in the Geometnck Hexagon are marked with Italick Letters, thus, d efgh't, but in the Scenographick Hexagon the refpondent points are marked with Roma in Let - ters thus d e f g h 1. You are fometimes directed in this Book to draw a line from the Point abed, See. when the point intended is not 111 the Plate, but lies out of the Plate in the Horizontal line. As 111 Oper. XXXIII. you are directed to draw a line from the point c, but it is thereby meant you fhould draw a line from that point where c fhould Hand, viz. where the line d 13 c if conti¬ nued would cut the Horizontal line,which is the Vifual point; and is in that Figure marked c. Take this advice in Drawing a Perfpective Fi¬ gure, wherein there comes many Lines: For I have obferved one main thing that makes PerfpeStive be thought io diff cult, is the mix¬ ture of many Lines dircSIing to many defigned points: Therefore when you draw fame bufie peece in PerfpeStive, you may for the directing your Ey, Draw the Diagonals in Red, the Vo fual Raies in Black, Perpendiculars in Green, or any other different Goullers from That you in¬ tend your Perfpedtive Figure fhall be of. There may be fome more Rules worth your Notice : but as yet not comeing to my memory. Take thefe for the preient3 and as the reft offer themfelves to me, I fhall prefent them to you. In the mean time, If you meet with any words of Art you underhand not. Look over th e Lear,' where I have digefted them in an Alphabetical order , and to them annexed their explanation; E 2 An 14 Tragical c PerJj>eBive. An Explanation offuch lin-ufual words as you may find in this Book. Arch. A peece ota Circle. As in Oper. x is an Arch of a Circle. Axis, is that ftraight line whereon any Body moves circularly, as in the line a a in Oper at. xi. is the Axis of the Door. Bafe. See Chap, x Defin. i Broken beams. See Chap, i Dejin. 4. Capital. The top of a Column. Catuzes. efinjig. axareCatu7.es. Catop- trick. See Chap. 1 Dejin. 3. Center. The middle point of a Circle, or any other figure. Cilinder. A ftraight round Body, flat at either end. Fig. 53. Column. The upright Pillars markt xabyct in jig. 50. Curr.pcfite Order. an Order in Architedf. See Vignola. Cone. Fig y4- Corinthian Order. An Or¬ der in Architect..See Vignda. Cube. A Square Body of fix equal (ides, as 1 K L m jig. 19 are Cubes. Diagonals. See Chap. xDejin. 8. Diagonal Squares. The fquares in jig. 1 fi¬ at c all Diagonal Squares, becaufe all their tides are Diagonal lines. Diametrial. See Chap, x Dejin. 7. Dioptricif See Chap. 1. Dejin. 4. Direct beam. See - Chap. 1. Defin. x. Difiance. See Chap, z Defin. y. Dorkk Order. An Order in Arcni edt. See Vignola. . . Equilateral Triangle. A Triangle of equal fides, as are the Triangles d e f in fig 5- . , . ' ■ Geometrick. Figure or Body, is a Figure or Body that hath its true dcmenfions. Ground line. See Operat. 51. Ground-plain. A plain lull of Squares as fig. iy.16. Height. See Chap, z Defin. z. Hexagon. A figure of fix equal fides is a Hexa¬ gon; as the two figures in fig. 6. are Hexagons. Horizon. See Chap. x. Defin. 4. Ichnography. See Chap. 1 Defin. 6. Inter fell ion. T wo lines eroding one ano¬ ther are laid to mterfect each other, lonick Order. An Order in Architect. See Vignola. Lum'mous Body. Light body, as in fig. yi.ya.to 56. at a is the Luminous body. Newel- is the upright poft a pair of winding ftaii s winde about. Object. See Chap. Defin. 9. Occult. Occult lines or Arches are dark lines or Arches: which are only drawn for diredtion to make the Figure up by: thefe are throughout this book reprefented by prickt lines and arches. Oil agon, a Fi¬ gure of eight equal fides is an Oct agon. As are the two Figures in Fig. 8. Opa¬ eons body. A Dirk body As in Fig yi. yi. 73 74. yy. 56. the bodies obftruching beams of Light are Opacous bodies. Optick. See PerJfieBi've. Chap. 1 Defin. 1. Orthography . See Chap. 1 Defin. 9* Parallel. Parallel lines are ftraight lines that are all the way equally diftant from one another; as the Baft - and Horizon throughout this book, are all parallel to to one another. Parallellepippedon. is a lquare body that hath its fohr fides longer than its two ends, As in fig. yfi. the lquare body fig. Fedejtal. A lquare body whereon aColumn is fet, as d in fig ■ zS. Pentagon. A Figure of five equal fides, as are the two figures in fig. 7. Perpendicular. A line that falls plumb upon another line, without leaning to one fide or another is a Perpendicular line. Perffieclive. See Chap, x Defin. 1 Pilajlers. Square Pilla’ J that ufually ftand behind Columns to bear Arches &c. Quadrat. A fquare figure of four equal fides. Quadrant. A figure containing the fourth part of a Circle. Radius. Half the Diameter of a Circle. Re fielded beams. See Chap. I Defin. 3. Right Angle, is made by two lines exadtly Perpendicular to one another. Stenography. See Chap. I Defin 10. Section. See Chap x Defin 6. Semi¬ circle Haifa Circle. Station. The place you ftand on., Tetrahedron, is a Body whofe four Plains are four Triangles, as in fig. yy. Tube, a long hollow inftrument wherein is ufually a Glals fitted, to obferve ob¬ jects through. Tufcan Order. An Order in Architect. See Vignola. Vi final point. See Chap, x Defin. 3. Fdfiial rates. See Chap, x Defin. 7. O PER AT. 15 *P rattical T erffeBive. OTET^ATIO^ I. Height andldxikanccgiven, To find the fcitna¬ tion firjl of a Point 9 fecondly of a Line, and thirdly of a Square, in the flaff or SeBion. T He given Height from the groundto the Ey. or which is all one from the Bafe to the Horizon is a b, the Diftance from the Foot to the Glafle is a c. 1 he given point or O bjedt to be reprefented in the Glafs or Section is cl. The Glafs ero¬ ded perpendicularly on the Bafe is efgh. Draw a Vifual ray from the Ey at the point b to the Ob]eel at the point cl, as b d, and another line from the point of Station a to the fame point d, as a d; then creel a Perpeneicular on the point where this line a d cuts the line of Sedfion as at c , and where this Perpendicular cuts the' Vifual ray b d as here it does in i is the point that the Object d appears in the Glafs. To find the Scit nation ofa Line in the Sett ion or Glafs, You mull have the Height given, as ab, and the Diftance as a c, And as you were taught before to find the place of one point in the Glafs, fo now by the fame Rule the Operation muft be doubled to find the place of two points in the Glafs, and then a ftraight line drawn between thofe two points is the line required. Example. The given Line is de, ft hereforc from the Vifual point b I draw two ftraight lines or Vifual rays to the two points at the two ends of the line d e, and twd other ftraight lines from the point of Station a to the fame points d e. And from the points where thefe two lines interfedl the Glafs (as here they do in the points cf) I eredl Perpendi¬ culars into the two Vifual raies, and where thefe Perpendiculars inter* fedt the two Vifual raies (as here they do at gh) is the points de re¬ prefented in the Glafs, and a ftraight line drawn between the points g and h reprefents in the Glafs the ftraight line d e. To findthefeituation ofa Square figure in the Glafs. You have the given Height a b, the Diftance c. Draw Vifual rays from the Vifual point b to every angle of the Square d ef g, and draw alfo lines from the point of Station a to every angle of the Square : and where thefe lines of Station cut the line of Sedlion (as here they do in the points hi kl) credt Perpendiculars into each refpedlive Vifual ray, and where thefe Perpendiculars cut each refpedlive Vifual ray (as here in the points mno p) fhall be each refpedlive point of the Square in the Glafs, and ftraight lines drawn from point to point fhall include the Square. Thefe three Operations (though in effedl but one) are fifth acnc F to 16 Practical Eerfieffive*- to fhew the nature and quality of the Appearence in Sefbion or Glafs but this way is not much ufed among Artifts; nor is it fo ready for ufe as laying the Figure along on the Bafe-line drawn Parallel to the Horizon; becaufe this way you mull draw feveral Bafe-lines, even fo many as you have given points in your Figure; for in this Square you have four Bafe-lines,asthe lines d a,e a, fa,ga, are all Bafe-lines,which together with the line of Section and the feveral Perpendiculars ere¬ cted on than would fo pefter a large Defigne that it would be bodi tedious to draw and difficult to underhand when drawn : Therefore I fnall hereafter in the reft of this Book leave out the Glafs thus creeled, and fhew you how to perform the fame by laying the Figure along on the Bafe. OTET^ATIOt H- Height andDi&ancc given, to find the true See- nographicplace of a given Poin t. T He given Height is ab, the Biftance given is b c, The Point gi¬ ven is d. To let this Point in its true Scenographick place. Draw a line to reprefent the Bafe, and at the fdven Height a b above it draw a line parallel to the Bafe, to reprefent the Horizon, as c b-, Then draw a Perpendicular to the Bafe a a and Hori¬ zon as be into a as the line h a, and placing one foot of your Ccmpaftes on the point a open the other to the point d, and deferibe the prickt arch de$ then lay a ftraight Ruler to the point of Diftance/? and to efcribe a Circle in Perfpeitive. D Efcribe firfta Geometrick Circle, and divide it in to twelve e- qualparts as 152,3,4,5-, 6,7,8,9,10,11,12,through each one of which divifions let fall a line Perpendicularly on the Bafe, as through the point 1, draw the line which alfo runs through the point 7, through the point 2 which alfo paries through the point 6; and fo for the reft,as you may fee in the Figure. Then from the point 2 to 3 draw a Diagonal Line and continue it into the Bafe, and from the point 1 to 4 draw another Diagonal Line, which alfo continue into the Bafe. So is the Ichnography of your Circle fitted for your purpofe. For Tragical PerJ^eSive* 2 3 For defcribing the Scenographie of this Circle, let fall a Perpendi¬ cular upon die Bafe from die Vifiial point c, and from that Perpendi¬ cular fct off on the Bafe each way the diftance of each Perpendicular from the point i in the Geometrick Circle; and to thofe diftances draw Diametrals Lines from the Vifual point c: then fet off on die Balethe diftances of the feveral Diagonal Lines from the point i in the Geometrick Circle, and tb thofe diftances draw Diagonal Lines from the point of diftance b. And the interfedhons of thofe Dia go - nai Lines with the refpedtive Diametral Lines are the points through wliich the Sceno graphic of the Circle is to pafs. Note, That though this Geometrick Circle b'eplaced on the Bafe, it is all one in effeft as if it had been placed under the Bafe, as the other Geometrick Figures delineated in Perfpechve ate.* for the diftance of every point from the Bafe would be the fame. In the foregomg Figures you have all the points of the Figures drawn up into the Bafe by Perpendiculars ere£ted,and in this for want of room to place it under the Bafe the feveral points are transferred downwards to the Bale from the point 7, on either fide the Perpendicular 1 7. OTET^ATIO^ x. Another way to draw the Perfpeilive of a Circle at any AJJtgried Height and Diftance. D Raw your Bafe and Horizon, as hath been taught before.-and having alfigned your Vifual point,and points Diftance,Draw on a Center in the Bafe a Semi-circle; This Semi circle muft: be included in half a Quadrat, as in the Figure; and divided into as many equal parrs as you pleafe, (die more the better) tliis Figure is Divided into eight, as defgbiklm, and through every point ofdivifion muft; be a line drawn Perpendicular to the Bafe and continued to the outmoft ver ge of the half Quadrat .• Thefe feveral Perpendiculars muft be crofted at right Angles with other lines Paral¬ lel to the Bafe, and drawn through thofe points of divifion in the Se¬ mi-circle that are equidiftant to one another: So is your Geometrick Semi-circle finifht. I hen from the Vifual point draw Diametral lines to the feveral points where every Perpendicular touches the Bafe, and alfo two Diagonal lines from the points of diftance b b to the points where the Semi-circle touches the Bafe, fo /hall die interfeclions of thofe two lines with the Vifual lines be the points through which you muft draw lin es above the Bafe Parallel to the Bafe ( as in Oper. X V .) and the inrerfedhons of thofe fdpedtive Parallels with the refpedlive Diametral or Vifual lines are die points through which the Scenogra- phick Semi-circle is to run, as at def g h i kl m ,which is one half of die Scenographick Ground. The other Scenographick Semi circle is drawn through the interfedlions of the Diametrals with the Parallels in the under half of the Scenographick Ground. OPE R; 24 ‘Practical c PerJj>eUtive. OTE%_,ATIO^C XL Height and Diftance given, to ‘Draw the Per- fpetftive of a Flat Oval- I vide the Circumference of the Oval into fo many equal parts as you pleafe, the more the better; (In this exam¬ ple we have 16,) and mark them with 123 45 &c. from thefe points raife Perpendiculars into the Bale, and trans¬ fer the diftance of each Perpendicular from the Bafe to the Bale.- Then from the point of diftance c draw Biametrals 40 the feveral Perpendiculars in the Bafe, and Diagonals to the transferred di- ftances ofthofe Perpendiculars, and the mutual interfe&ions of theDiagonais with therefpeSliveDiametralsfhall give the feveral points through which theScenographick Oval mud pads.-as where the Diagonal drawn to the point x on the Bafe cuts the Diametral drawn to the point 1 is one point through which part of the Cir¬ cumference of the Stenograph ck Ovalmult be drawnand where the Diagonal drawn to the point 2 on the Bafe cuts the Diametral drawn to the point 2 is another point through which part of die Circumference of theScenographick Oval mud: pafs. 1 he like is to be underftood of all the other Diametral and Diagonal* Inter* feStions. OTE^JTIO^C XII. To Deferibe an irregular Figure in Perfpe£tive ? Height and Diflanc cgiven* T *t He Angles of this Figure are defghiklnmo , which ereSt Perpendicularly into the Bafe (as before,) and alfo trans¬ fer their Perpendicular didances from the Bafe into the Bafe, for then Diametralsdrawn ro die Perpendiculars and Dia¬ gonals to the transferred diftances cut each other re fpeShvely in the Scenographick. points where the Angles ofdie Figure mull: be placed, and lines drawn from Angle to Angle fhall include the Figure. As you may fee by the corresponding Romain and Itallick Letters. OTB 7^ 25 'Tragical Terjpe&ive- 0TE%JL?10S \C XIII. To begirt the former Figure with Parallel Lines about it ? in Perfpeddve: Height and Di- flan cc given- T PIefetwo Figures are inferred to Ihew that whether your Fi¬ gures are regular or irregular, Circumfcribed or not Cir- cumfcribed,.the working remains the fame in all refpedls .* For as in the foregoing Figures, fo in this, Perpendiculars eredted from the Angles, and the difance of the Angles from the Bale transferred to theBafe, and Diametrals draVvn to the Perpen¬ diculars in the Bafe,and Diagonals to the transferred diflances cut each other in thofe points the Scenographick Angles mull be placed, from whence lines drawn to each other relpedlively Ihall begirt the former Figure with a line Scenographically Parallel to it. I have been the more copious upon thefe Figures becaufe they come often in ufe in the defaming of Ground Plots, whereon to raife any Strudlure in Gardening, and in Fortifications. &c. OTSTRATI 0 XIIII. Shewing thatfrom one (feometrickJFigure you may make two or more Scenographick Figures by Elevating of Horizon. Y Ou were taught in Oper. VIII. to redu ce the Geometrick Odla - gon d ef gh ikl into Scenographie, which is here alfo noted with its cotrefponding Romain Letters. But you may make yet another appearance of this Figure,ifyou alter the Height though youobfervethe fame diftance, for here the Horizontal line is elevated into the upper be, and Diametrals drawn to the fame Per¬ pendiculars, and Diagonals to the fame transferred diflances, by their mutual interfedlions give the points FGHII^L which are the Angles of rhefecond Odtagon, and lines drawn from each Angle to its next include the Figure. This way of working may be of great ufe in drawing Covers for Boxes, Covers for Pulpits, and for feveral other defigncs,for according to your purpofe you may either elevate or deprefs the Horizontal line, or make as many elevations as you pleafe. Hi OVER - 2 6 Practical TerfpeBive* OTE%JTION^ XV, ToTefcribe the Scenographie Plain with Seven Quadrats or Squares in Lengthy and Seven in fBredth therein: Height and Di¬ ftance given * ^ 'He given Height i s ab, and die given Diftance cb, draw the |j Bafe a a and Parallel to it at the given Height the Horizon- - JiL nil line b h, then fet off the diftance with your Compares from the middle of the Horizontal line both waies, as from the Vi- fual point cloth: And keeping one foot of your Ccmpaftes ftiil in the Vifua) point, fet oftyour defigned length of the Bafe as at aa on either fide malting fnal occult arches or markes on the Bafe; then draw two DiagonalJines from the points of diftance^ to the points a a, and two Diametrals from the Vifual point c to the points a a, and where the Diagonals b a cut the Diametrals ca,a.s here in the points i/i/andtftzfhall be the four Angles of this Scenographick Plain.and a ftraight line drawn from d todfhall be one fide of this Plain.- The Bafe line a a another fide, and two lines drawn viz.one from a tod on die right hand, and another from a to don thelefthand fhall be the other two Tides that include this Square. Then divide the Bafe in¬ to Seven equal parts, as from atoefghi k c;and by laying a Ruler on the Vifual point rand each of thefeparts efgh i k draw by thefideof it Diametral lines through the Plain, and to the points where thofe Di¬ ametral lines interfeeft the two Diagonallines ha and b a (as here they do in 12. 3 4 $ 6 d. 11345-6 d)l ay a Ruler and draw lines by die fide of it Parallel to the Bafe,fo fhall your work be fimihed. OTET^ATIO^ XVI Height and Difiance given , To Describe the Sceno graphic of a Plain full of Squares with their Angles pointing againjl the Bafe. T He given Height is a b, the given diftance c ADraw the Bafe a a as before,and at the given Height Parallel to it the Horizon¬ tal line as bb-, then take the middle of the Horizontal line for the Vifual point as at c,and fet offyour diftance as at b on ei¬ ther fide c: Then let fall a Perpendicular through the Bafe as c d,2.nd from the point d divide your Bafe into as many equal parts as you pleafe bodi waies (as here into 6 each way marked with 12345 ;a;) then Traci teal TerJpeBive. 2 7 Then draw the rwo Diametrals to the points a a, and two Diagonals from the points of Diflance bb to act, and where thefe twoDiagonalsd^cut the Diametrals ca(as here in ee and. a a) /hall bethefonr angles of the Scenographick Plain ; and aftraight line drawnfrom etoe fliallbeone fide of thePIain, the.Bafe aa is ano- ther fide, and part of the Diametrals viz. from a to e on either fide /hall be the two other Tides of this Scenographick Plain. Then laying a ftraight Ruler to both the points of Pittance and all the devittons on the Bafe, Draw lines from thofe devifions to the Dia¬ gonals , fo /hall half your Plain be filled with Squares. To fill the other half, You mutt lay your ftraight Ruler to the feveral in- terfe&ions of the Diagonals with die two Diametrals on either fide the Plain, anddraw ftraight lines into the fide of the Plain e e, fo Ihall the other half of your Plain be filled alfo. .. r-. I he making of thefe two Scenographick Plains full of Squares is of great ule in Perfpedive, for the ready defigning and placing all forts of Buildings and moveables in their proper diftances one from another, and in their various appearences to the Ey: For in the de¬ figning of Stools, Tables, Crofies, and many other Figures your la¬ bour is as much facilitated as by coppying a Piclure when it and what you draw on are devided into fquares, and is indeed vertually the fame thing. For inftance, If you defigne each fide of one of thefe Squares to be an Inch, a Foot, or a Yard&c. long ; then what Objeft loever is placed one Square diftant from another is faid to be either an Inch a Foot or a Yard&c. diftant .* What Objeitt is pla¬ ced two Squares off is placed two Inches &c. diftant. And fo for any number of Squares. Again, For the various appearence to the Ey. All Buildings or Moveables whofe fore Tides are feituate in an Orthographick Plain are drawn upbn the Squares in Operat. X V. becaufe the fore and back fides of thofe Squares ly in an Orthographick line. As you may fee the Cubes in Operat. XIX. and the Buildings in O perat.XX. XXI. XXII. XXIII. X XIIII. XX V. XX VI. XXVIII. XXIX. XXX. XXXI. XXX VIII. and the Moveables in Operat. XXXIX. But thofe Buildings or Move¬ ables whofe corners ly in the Orthographick Plain, that is, whofe front declines from an Orthographick line are belt placed on this Scenographick Plain that hath its angles againft the Bafe. As you may fee in Operat. XL. X LI. XL11. and X LIX. . In thefe two Plains you may fee how the feveral diftances of the Objeft alters its appearence .* for all the feveral appearences of thofe Squares are but the feveral appearences of the Squares gh / m in the laftOperation,and in this, whofe points are transfer red into the Bafe, and made as in the foregoing Operations. I OP E R and one right before . f-W'”' His is performed upon a Ground-plain of nine Squares in i length and nine in bredih, (drawn as by Operat. X V.) I /hall not need fay any thing of the Tides of the WalL.it having been taught in the lafl Operation .• But the Doors Handing partly open and partly /hut require a little inflrudion to /hew the defigning of them in Per fpefli ve. You mufl underlland that a Door being hung upon hinges de- feribes in its opening and /hutting a Semi-circle. As for Example, If it Hand wide open with the back fide of it againfi the Wall, it fetches the fweep of a Semi circle before it can/hut upon the Door¬ frame that runs range with the Wall. Therefore having the width you intend your Door /hall be of, and pitcht upon what place in the fideofa Wall it/ha 11 move, (as on its Axis,) you need but deferibeon a loofe paper a Geometrick Semi-circle in the fame number of Squares thatyou intend your Door/hall deferibe its Semi-circle in, andobferve the points of the Parallels and Perpendiculars that this Semi-circle cuts,for the Scenographick Semi-circle mufi pafs through the fame points of the Diametral and Diagonal lines de¬ ferred on the Scenographick Ground-plain.- As you may fee the Door as. dd, containing fix Squares (as from the center a to e three, and from a to/three) deferibes on the Ground-plain the Semi circle elhidgf, which if you dr aw a Geometrick Semi -cir cle in fix Squares you will find that this Scenographick Semi circle pafies through the fame points of Diametrals and Diagonals as your Geometrick Se¬ mi circle does dirough Parallels and Perpendiculars. Then confi- der how neer open or /hut the Door is you intend to deferibe; that is, what angle it makes with the Wall; for the fame quantity you muH fet off on your Scenographick Semi-circle; as here it Hands open by the diflance jdg: Therefore draw a flraight line between a the cen¬ ter the Door moves on and d the edge of the Door, and prolong it into the Horizon,anddiat line dai hall be the under fide of the Door: Then deferibe thefame Semi circle of er the Door that there is on the Ground-plain, by ereefing Perpendiculars from as many points as you pleafe of the Semi circle on die Ground-plain, and by drawing firaight lines from each of thofe points into the center .a and pro¬ longing them into the Horizon; for where the Perpendiculars cut flraight lines drawn from thofe feveral points in the Horizon through the upper centers of the Door and fo prolonged Ihali be the points that the upper Scenographick Semi-circle elih dgf muff run through. If you would have the Door Hand wider open that you may fee more of die entrance as at ii. Draw (as before) a flraight line from i to the center a and prolongit into the Horizon,as ia k, fofhal/be K the 3 2 tPraBical Perfftl Uve. the under edge of the Door; and if from the point t you eredl the Per¬ pendicular/ i for the edge of the Door, a rfraight line drawn from the point k in the Horizon to the point a in the upper center of the Door and prolonged ihall give the .ineia for the upper edge of die Door. Thus mayyoudefcrihea Door handing open to what width you pleafe, as atnaee the Dodr hands wide open with its backfide a- gainlf the Wall; at#a li it /hews lefs open; at^ h it /hewsa Door whole edge lies almofl in the fame liiie its A xis does, and fo the Door /huts neerer and neerer,as atgg a a the Door appears quite /kit.-the Door on the other fide is made) uft after the fame manner, and there¬ fore needs no more Defcriptions. The Door in the middle hands alfo on three Squares, and alfo o- pens throu gh fix, and if you defcribe the Scenogtaphick Semi-circle m p q as you were taught juft now you may fet this Door alfo open to what width yen pleafe. Thus, Erect a Perpendicular on the point of the Semicircle you intend to open the Door to, and prolong it quite through the Door as r n s, and with your Compaftes m eafure on the Perpendicular, the dihance between that point in the Semi-circle and the Diameter of the Semi-circle,which is the line m n, where the Squares end; for that dihance fet offon the fame Perpendicular from the Bafe as from r, /hail give a point through which a firaight line drawn from the Center of the Door o to die Horizon /hall be the point in the Horizon from whence a hraight line drawn to the upper end of the Doors Axis lhall give the /hortning of the upper edge of the Door, and the points where the two hraight lines cut the Perpen¬ dicular /hall be the points that /hall/hew die ihortning of the fore edge of the Door. Above this Door is made a round Hole of the fame width the Door is, and of the thicknefs of die wall; which becaufe the lines by which it is made, are fo plain and the manner of working fo eafiel /hall for¬ bear farther Inftru&ions on it. Under this Arched Roof are drawn Parallel prickt lines to /hew , the devilionsofthe Roof: and by thefedevifions either widi Com¬ pares, or el fe a ft eddy hand, you may defcribe the Arches as you may fee them iii die Figure. . - By the making the forefaid Doors may be underfto od how to make Cafements Handing open at anywiddi; and feveral other Operations pleafant and neceffary for all that Study Perfpeftive. OTEP^JtlO^C XXII. To "Defcribe in Perfpeclive the infide of mo Chambers one above the other. T His is alfo performed by help of a Ground-plain, as in the for¬ mer Figure.- but the difference between tills and die for¬ mer is. i. This hath another Chamber over it. %. A Portal Traci teal TerfyeBive. 33 Portal and Hole on one fide of the Wail, and another right before you on the middle of the Ground-plain. 3. Steps placed juft within this Portal. 4. A fquare Hole in the middle of the Ground- plain. 5. Shelves on the other fide of the Chamber. 1. For what concerns barely the Chamber pver the lowermoft Room you may fee by the Figure t hat its Operation is the lame with die former;where you fee all the three fidcs of the Chamber wholeiy, but becaufe the CeeJing abed of the lower Room lies fo much above the Ey, you only feefomuch of the upper Chamber as is un-obfeured by the Ceeling. That is, all above the Catuzes a d refting on the two Ca tuzes e /the reft are only noted with prickt lines, as th ey wo uld ap - pearif the Ceeling were netinterpofed between the Ey and them.- an d the Joyfts over head are wrought as in die Jaft Qperat. . %. For die Portal on the fide of the Wall you mayfee that it ftands on three Squares on the Ground-plain, as doth the Portal right before you. Now to make the Arch over this Portal correfpuid with the Arch ofthe Portal right before you, viz. of the fame height as chat Portal is,do thus. Draw a line juft w here the Arching begins paral¬ lel to the Horizon asg^and prolong it to the corner of t ie fore-right Wall as at g: Then laying a Pvuler to the Vifuai point and the point gdderibe the line mg, then draw another Parallel line on the top of the Arch of the Portal and prolong it alfo to the corner of the fore¬ right Wall as to.% and laying a Ruler to that point i and the Vifuai point draw the Diametral line n oi, then ered two Perpendiculars nl and om from the fides ofthe Portal to interfed thele Diametrals,fo fhall you by croifing the Angles of interfedion have the Center/, and upper andlower bounds of the Arch; which a fteddy hand may eafily reduce into form. T his way of working is exad enough for Painters or o ther Artifi¬ cers. But if you will be yet more precife, See XXV. XXVI. The Hole m the fide of the Wall in theupper Chamber is made af¬ ter thefame way, viz- by ereding Perpendiculars to the deftgned width, as here it is three Squares on the Ground-plain, (the fame with the Portal) and by drawing Diametrals from the Vifuai point for the depth of the Hole; for where the Perpendiculars and Diame¬ trals interfedeach otherIhall be the Angles of the Square that in¬ cludes the Hole, and ftr aight lines drawnfrom Angle to Angle fhajl interfed: each other in the Center of the Hole. By die fame way of working you have the back fide of this round Holedeferibedto the thicknefs ofthe Wall, andalfo the back fide of the Portal (as you may fee in the Figure. „• In the other fide of the Wall on the Ground-plain is made ahother half round Hole;which being performed after thefame manner needs no further Explanation. 3. Juft within the Portal are placed four Steps going ftr aight up vvards;which are thus made. Draw the two Hope lines qq and rr (to what Hope you pleafe) parallel to each other for the width of the Steps: then affigne a depth for die hr ft Step,as to s, and Parallel to K 2 the 34 < PraUical < PerJfteffi f Vc' the Bafe draw aline forthe bredthof the Step, as ts, then draw Dia- metrals from the points s and /to the Hope line qq, fo Jhall the firft Step be made. The reft of the Steps are made after the fame manner; obferving thar all die feveraiSteps on their feverai Perpendiculars are equal in depth t o the firft. 4. in the middle of the Ground-plain is made a fquare Hole as v x^z, which may ferve for an Entrance down a pair of Stairs leading to fotne Vault or Cellar.Thefe Stairs are in length three Squares,and in depth the transferreddiftance of one Square, as you may fee the depth of the firft Step downwards is vx ,which is the transferred di¬ ftance o f vy, the bredth of the Step is a 1 a; fo that a Diametral drawn from the Vifual point to x and io proion ged di re£ts you to the fecond Step at b,and by placing one foot of your Compares at the point c, which is in the fame Diametral the point v is,and extending the odier to the point d (which alfo is in the fame Diametral the pointy is) you transfer the diftance of dc to eb which is the diftancey v trans¬ ferred to v x, and another Perpendicular erefted from the point h to e, (hews the depth of the fecond Step. *The reft of the Steps are made after the fame manner. 5. On the other fide of the Chamber againft the Wall isfet three Shelves,two above the Horizontal line,and one below,as ef g .• The further ends of them refts in the Wall right beforeyou,and the hither end is faftned to a Poll erected Perpendicularly on the Ground-plain; tlie bredth ofthefe Shelves are one Square on the Ground plain, as y ou may fee by the line h i, from which Diametrals prolonged to the Poftthe feverai Shelves are deferibed. Here you may note that by the feverai places of thefe three Shelves you have feverai appearances of them;for the Shelf g lying under die Horizon fhews its upper Plain. The Shelf f lying a little a- bove the Horizon fhews its under Plain, though much fore-ihortned. The Shelf e lying pretty high above the Horizon difeovers the under Plain more intirely. But on the other fide the Wall you have the Shelf k, which becaufe it ftands juft in the Horizontal line difeovers nei¬ ther the upper or under Plain,but only the very edge of the Shelf. 0Te%jTI03 \C XXIII. The manner of placing Columns behind one another in Perfp^dtive. Hough this may be performed by a Ground-plain of See- t nographick Squares,(as before,) yetbecaufein this Opera¬ tion no work comes between the two rows of Collumns we Ihall need deferibe no more of the Ground-plain than will ferve our turn. Therefore having drawn your Horizontal line to your defigned Height VraBkal Verfpe&ive. 3 5 ... ,* , Height above theBafe,and fee off your points ofDiftance at equal diftance on eitlier fide the VifuaJ point. Draw the Diamctrals ak,bb, cc,dd, Then from the points of Diftance draw the Diagonals e d, and fa, and at thepoints ofinterfedion of thefe Diagonals with the Dia¬ metrals draw ftraight lines parallel to theBafe, and thefe ftraight lines continued to the outmofl Diametrals fliall make four Sceno¬ graphick Squares, whereon fobr Columns muft be placed; as here the Squares al and cd and g and h&cc. , ^ , If you would have more Columns ftand behind thefe four. Draw Diagonal lines from theinnermoft Angle of the hindmoft Squares already made, as from g and and tnofe Diagonals Ihall interfed the Diametrals in c d .andba, from .which angles of interfedion ftraight lines drawn parallel to the Bafe and Horizon ftiall make the Scenographick Squares whereon two other Columns fhall ftand. , , By this manner of working you may fet more Columns behind thefe, till you aimolt fill the whole Ground plain. Having thus projeded thefe Scenographick Squares whereon the Columns muft ftand, you muft ered Perpendiculars from every angle of thefe Squares through the Horizontal line and fo upwards to the height your outmpft Column ought to be, as here to the line tk parallel to the Horizon; for the points where Diametrals in- terfed tlii$ line and the Perpendiculars ai and dk fhall be angles impendingperpendiculafly over the refpedive angles of each re¬ fpedive Square in die Ground-plain .• As here the angle i hangs di- redly over the angled, and the angle k hangs diredly over die angle d: Then from die points of Diftance draw Diagonal lines to die points/^ and where diofe Diagonals interfed the Perpendiculars ereded on the Diagonal angles of the Square in the Ground-plain muft another line be drawn parallel to the Horizon, and alfo o- ther Diagonal lines which /hall include the Squares that hang o ver the two firft Squares in the Ground-plain. The other Squares are found after thefame way, viz. by the mu¬ tual interfedions of die Diagonals widi the Perpendiculars. As riiay be feen in die Figure. , • \ The Arches over thefe Columns are very eafily made* for by drawing the Perpendicular dirough theVifual point in the Horizon, into /you have all die lines drawn parallel to the Horizon devid«4 into two equal parts, And by placing one foot of your Compaftes in the middle of the refpedive Parallel lines and opening the other to the angles of the Square that lies in die fame Parallel line you may deferibe thofe refpedive Arches. Asfor example. Ifyou placeone foot of your Compaftes in the middle point on the line i k and ex- tend the other to. the point lor k you may deferibe. the outmoft Se¬ mi-circle, andbyciofing the Compaftes to the other angle of the Square which lies in the fame line,you deferibe the feconq Semi-ck- eie; for the third Semi circle you may place one foot of yOurCom pafles in the middle of the next parallel line and extend the other to diepoint oh\ die fame Une^uiddefcribe the Semi circle o p. This work L i$ g6 TraBical TerfpeBive. is fo plain and Caifie to be under flood by the Figure, that I need di- icourfeno further uponit. In this Figure you may note, what correfpondencc the Squares a- bove the Horizon have with thofe below the Horizon. Yet as the Squares above the Horizon have a greater diflance from the Hori¬ zon than thofe below the Horizon have, it falls out that the upper .Squares are larger than the under Squares-- The reafon whereof youmay eafilyfee byeonfultingdie Figure. And as the Squares have a correfpondence, fo alfo have the Dia¬ gonals,- for the erodings of the Diagonals above the Horizon re¬ spond with the erodings of the Diagonals below the Hotizon. OTE^ATIO.?^ XXIIIL To defer ibe in Perfpe^ivc the appearance of four Piiafters, with four Beams upon them 5 And td ma\e an Arch over them . f ~|K His is alfo performed by a Ground-plain of Squares, which U confifls of feven in bredth and feven in depth-.- Defcribed as by Operat. X V. ThePilaflers a b and cd Handing each on one Square erefled perpendicularly on the angles of the Ground-piain,and divided upwards into three equal parts, asi i 3'. with Hraight lines parallel to the Horizon drawn through the fore, fide ofthem: But the lines that devide theinfide the Pilafters are Di- amctrals drawn from the Vifual point to the inmofl comer of the foredde of the Pilafler. As you may fee by the Figure. Upon thefe Pilaflers is laid a Frame offour Beams, ferving for bond to the Pilaflers, tis.efgb, two of which viz. f h are drawn from the Vifual point. And over thefe four Pilaflers is eredfced the Arches 1 kl and mnotohe fteen quite through; which Arches being to be made asthofein Operat. XXI. XXIII. needhereno furtherdifeourfe on them. Only you mull note, that what lines ofdevifionsyou make on the Orthographickor fore-fide of the Arch you mull drawfrom the center of each refpedlive Arch, and the devifionson the under fide of the Arch muft be dr awn by the fide of a Ruler laid to the Vi¬ fual point and to the devifion on the comer of the fore fide the Arch, as the devifions of the Pilaflers are. OTETijiriOi A(g XXV. The manner of making Side Arches raPerfpe&ive- H Aving drawn an Horizontal line, deferibe a Ground-plain of Squares, (as by Operat. XV.) and thereon draw a Scenogra- phick Circle (as you were taught by Operat. IX. X.)and mark it from the middle line with 1 % 3 4 5 on the Bafe; and on t*he *P raff teal ‘Perfpeffiveo the fide with a bedefg h: Then ered a PerpendicuJar on the corner of the Ground-plain of die height you intend your fide Wall fhall be as hi, and ered another Perpendicular on die corner of die Ground- plain as y 5-,and by drawing two Diametrals through the Vifual point the point 1 and the point y on the top, and from the Vifual point through ht o 5 on the Bafe, youfnall have the Scenographick appear¬ ance of the fide of the Wall wherein the Arch Ihail be made. Then from the point h (which you may fee by the Orthographick or fore, right Arch is the point where the Arching begins,) draw another Di¬ ametral into the Perpendicular line y y as h 1, and you will find the fame didance between 1 y on the PerpendicuJar as there is between 1 y on the Bafe .• But the reft of the diftances you muft transfer, as the Diftance between 1 2. on the Bafe you muft transfer from 1 to 2 on the P erpedicular, and the diftance 1 3 on the Bafe you muft trans¬ fer from 1 to 3 on the Perpenicular .* And fo for the reft. Then from the points a be defy on the Ground-plain ereft Per¬ pendiculars into the Ceding and where Diametrals drawn from the points 1 2 3 4 y interfed tliefe feveral Perpendiculars fhall be the points through which the Arch muft pafs. For Example. Where the Diametral drawn from the point 2 on the Perpendicularfnterfeds the Perpendicular ereded on the point a on the Bafe, as here it doth in a, fhall be one point through which the arching muft pals, and where the Diametrals drawn from the point 3 interfeds the Perpen-. dfcular erected on the point bon the Bafe, as here at b fhall be ano ther point through which the Arch muft pafs .• And fo alfo the in ter- fedionsfrom 4 and y with the Perperdiculars c d in the points c d, are points through which the Arch muft pafs. For finding the other points through which the Arch muft pafs,you muft go back again to 4 and fee where that Diametral interfeds the Perpendicular e for there fhall be another point, and where the Dia¬ metrals from 3 2 1 interfeds the Perpendiculars fgb fhall be the o- ther points through which the Arch is carried. Havyig finifht this Arch the other Arch is eafily made, for by draw¬ ing Parallels through the points 1 2 3 4 y in the top to the points kl wno , and alfo Parallels from the points a bcdefgh, Diametrals drawn to the points Itlmno will interfed the Parallels drawn from abedefghirithe pointsthrough which that Arch muft alfo pafs. On the Ceiling is made a Circle as it lies below on the Ground- plain of the Floor, as it appears in Perfpedive at that Height. On the right hand of the Ground-plain is ereded the fame two Arches that are on the left hand, whofe Operation is in all refpeds the fame.- Only on the right hand is drawn ftraight lines from point to point, and on the left arching lines which makes this on the right hand reprefent it felf(notasanArch,but)as part of a Fi¬ gure made up of eight Squares. r 1 ) OPE RAT- r&ftical TerfpeBke- OTET^ATIOH^ XXVI. The making of Crofs Arches in Perfpedtive- I N this Figure is defcribedfour Arches, Viz. one Orthographick Arch, which is the great Serai-circle marked 12 3 4 5 6 7 8 9, an Arch behind it Perfpe£Uvely deminiflied, marked 9 ab, and the two fide Arches,vVhofe Diametrals are 1 9 arid b 9. Thefc four Arches are required to be made before you can make Crofs* arching, As iri the Operation you will find. Therefore having de¬ ferred the two lide Arches (ds by the lad Operation) you mud de- vide the Orthographick Arch or Semicircle into eight equal parts, and frofti the Vifual point draw Diametrals to every one of them, as 12345-6789, Then from the points where the Perpendiculars cut the Diametrals of the fide Arch, as ini 12 3 4 5 6 7 8 9, Draw lines parallel to the Horizon to interfecl the refpedlive Diametrals of the Semi circle; for thofe refpeclive interfedtions fliall be the points through which the Crofs-arching mull be drawn. As for Example. Draw a line parallel to the Horizon from the point 1 in die fide Arch to theDiametral2 in thcOrdiographick Arch, and that fliall be one point through which the line of Crofs-arching mud run .• Then draw anodier parallel line from the point 3 in the fide Arch to the Diametral 3 of the Orthographick Arch, and there fliall be another pomt through which the Crofs arch mud run. Then draw another Parallel line from the point 4 in the fide Arch to die Diame¬ tral 4 of the Orthographick Arch, and there fliall be anodier point through which the Crofs -arch mud run. Do the like from the points 6 7 8 9, fo fliall you have all the points through which one of the , lines of the Crofs-arch mud run. , The other Crofs-arcliing viz. the Arch 1 8 9 on the other fide, is drawn by continuing the Parallels which you drew from the points 234 56789 of the fide Arch into the Diametrals 98765-4321, for at die interfe&ions of the Parallels widi thefe Diametrals fliall be the points through which the line of Crofs-arching mud pafs; which is indeed nodiing but the Arch on theothcrfide reverfed. This way of working is very eafie,and the truth of it very per. fpicuous. For as this line ofCrofs-Arcliing impends the Diagonals klmn on the Ground-plain, fo would you find that if Perpendiculars were eroded from the points 0 p &c. on die Ground-plain, they would run into the points where the Parallels and Diametrals cut each other above in die Crofling of the Ceeling.which is a proof diffident 10 verifie the truth of this Operation. 6PE7{%AT ■ 'FraBical TerJbeBive . OTET^ATIONf XXVII. To ere cl a StruBure on an 0<£iagonal Ichno- graphie. r-gp He Height is a l, the diftance h c. Having defcribed your I Groimd-pJain ScenographicaIly,as by Operat. VIII. Eredl on the two Orthographick Angles thereof dr! ce Perpendicu- larstoincludethefide, and through thefe two Perpendicu¬ lars draw a line parallel to the Bafe and Horizon, at the height you intend your upright work/hall break off, as d e.- And at the inter- fedion of this line with the Perpendiculars dr!ee, draw lines from the points of Diftance ^on your left hand to d, and from b on your right hand to e, and from the points where thofe lines of Diftance or Diagonals cut the Perpendiculars 1 1 and ffi draw Vifual lines to the Perpendiculars kfand g g; then from the point of Diftance b on your left hand draw a Diagonal from g to the Perpendicular h A And from the point ofDiftance b on yoUr right hand draw another Dia¬ gonal from k to the Perpendicular i/. And iaflly from i to h draw a line Parallel to the Bafe and Horizon, fo fhal) the Scenographick Ichnographie of this Building be transferred to the top and all the Jfidesofyour Perpendicular Building /hall have their Scenographick Heights. Y ou muft note that thofe Tides of the Ichnographies that ly behind theOrthographie are to be drawn with obfcure lines only, to dired you in the more true placing the Moldings, &c. in the reft of the work. The hollow of the Ichnographie is found in the Roof, by eroding Perpendiculars from every point of die Geometrick Circle,and every point of transferred diftance to the line of the Roof d e, for then by drawing Diagonals through every point of transferred diftance aforefaid, thofe Diagonals /hall cut the Perpendiculars in the feveral points through which the Circle in the Roof muft pafs. The Frame-work of the Spire (as you fee) is raifedon the upright Angles of the Figure in the Roof,and meets at the Perpendicular e- reded on the Center of the Scenographick Ichnogrophie, which you may find as by Operat. X X. if you lilt to ered more Columns behind thefe, you may chaw the Ground-plain of Squares necrer to the Vifual point, and by the Rules you have already placed thefe, fet as ma ny as you think good. OTST^JTIOl ^ XXIX To Defcri be in Perfpc where¬ on two perfons the one afeending the other defend¬ ing ]hall not come at one another . T SsHefe double Stairs are inferfed as a peece of Rarity, and de. feribed only in Orthographic, and not by the ftrickt Rules of Perfpeftive, left with many lines the work Ihould be obfeu- red. Yet fuch as lift to be curious therein may in thelaft(and feveral other) Operations find Rules whereby diey may effect their purpofe. Defcribe a Circle (or which is equivalent a Semi circle) for the Ich¬ nographie, as 123 &c. to 13, and in it deferibe a fmaller Circle, as a b c for die bignefs of the Newel: Then devide the great Semi circle into twelve equal parts, as 12 3 to 13, and draw lines from thofe de¬ vifions into the Semi-circle made for the bignefs of the Newel, fo will that Semi-circlealfo bedevidedinto twelve equal parts.* Then on every one of the points in the great Semi circle ere£t Perpendicu¬ lars, and thofe Perpendiculars Ihall lhe\v the ends of each refpeflive Step* As the Perpendicular at 1 1 bounds die outward end of the firft Tracticat Terjpettive. fir II Step; The Perpendicular at z abounds the fecond Step, &c. to 24. which makes good a whole Circle in the Ichnographie; and Per ¬ pendiculars erected from the inner Semi circle marks on the Newel the ends of the fame Steps. Work the fame way with the Steps on your right hand. (i .. . Youniay perceive that the winding of thefe Steps about the Newel may properly be compared to the winding of a thred about the Spindle ofa Screw,And that as thofe Spindles that have but one thred winde notfofuddenly about the Spindle as thofe that have twothreds; So thefe Stairs having two Afcents,each begun Dia¬ metrically oppofite to the other winde more fuddenly about than thofe that have but one Afcent; and therefore it is requifite the whole Circle bedevided into twenty fourparts for thefe fore of Stairs,when, as. odier Stairs have the Ichnographie devided blit into twelve or fix- teen equal parts. ,. The Newel is pierced through in feveral places only to let in Light. Who was the firfb Inventer ofthefe Stairs is not well known. But 2iSciamlurg\n France in the Kings PaLface is a pair ofdicm, made by Tiedro delBergo, and Jehan Cofin. They are alfo fet forth by Vignola, in his Book ofPerfpe£hve,froru whence this is taken. OTET^ATIO^ XXXVL To Delineate in Perfpe&ive a Platform wit!$ Columns? Pilaflers? Railes and Bannefters, as it isfeen from above ? where the By is placed di- reBly over it at fome reafonable height • Y Ou were fltewn in Ope rat. XXVI. that an Object placed far : bove the Ey deminilhes. It is the fame thing with an Objedi placed far below the Ey. For as you were told in Chap. If Defin. III. The Vifual raies do at laflmeet in a point; 1 here- fore in this Ope rat. the Ey being placed above the Edifice, the upper Railes feem larger and wider than the under Railes, and the Banne- (lets under the upper Railes in abc d appear greater than the Banne- flers in 1 X l.)and on that Scale you may meafure how much the angles is eroded above its Bafe, as here you will find a little more than one Square; dierefore in the Ground- plain at the dihance of a little more than one Square of that Parallel, under the line / k which is the bottom of the hind Feet, Draw an oc¬ cult Parallel inwards within the Bedhed, becaufe the Teher hopes inwards, as m /, and where that parallel cuts the Vifuai line dec fliall be the angularpoint, as at/, and a Perpendicular erected on diat point fhall cut the Vifuai line drawn to tho point / in the point g, which is the point that the hind part of the Frame of rhe Teher muh be drawn untofrom the two hind Pohs. See the Figure. Thefe Pohs are madefquare becaufe the hind Tides and corners of them may be represented by prickt lines, For in Frame-work it re quires that one half of the Rails &c.be drawn from the hind Tides and corners as well as from the fore Tides and corners, which by the help of thefe prickt lines you are directed to. OPE RJT? s ^ Tragical a Chair, ^Door, a Tho row-fare> and two Holes in a Wall* T Ffis Frame Hands widiits angles againH the Bafe upon feven Squares one way, and fix the other. It is Railed round the bottom,and hath one Shelf below the Horizon and two a- bove it. The manner of making it hath fufficiently been taught in the foregoing Operat. viz. ThePoHs are Perpendiculars c- refted on the Ground-plain;and the Rails, Shelves, &c.are all drawn from the points of Difiance a a on the right and left hand. The Shelves you mufi note do not projeifi over, but run range with the out fides of the upright Pofis- The TraBical the elevation of the Light above the T lain offhadovp being given- f ip' He Body of Light is a, the Ground-point b, the Cilinder c dep, fj ghik: The Cilindrick feSfion above the plain of Shadows -JL you muft bring iiito the Plain of Shadows, by letting fall Per¬ pendiculars from the four devifions above,viz. c de f, into the the plain of Sliadow, as at / m n o Then by drawing lines from the bo¬ dy of Light through the devifions iii the upper end of the Cilinder, as through c defa nd from the points g through /, /’through o,i through 71, and b through w; where the lines of Light cut the lines of Shadow, as here in the points^ r,fhall be the bounds of aScenographick arch, which joyned to the llraight lines h p and /rfliall include the Ihadow. You muff note, I hat theScenographick Figures edef made atthe top of this Cilinder, and Imno, made by the Perpendiculars let fall into the Ground are true ScenographickCircles; and that thecircle c dej appears fmaller than the Circle Imno, becaufe it is pofited neerer the Horizontal line, which alters the Scenographick appear¬ ance : as you may fee in the feveral Circles that arc made about the Newel of the WindingSrairs, in Operat.XX XIII. dhislmind you of, becaufe from the devifions in that Circle, and not from the Ground.point, the lines of Shadow are drawn to cut the lines ofLight. OTS^ATIOSVf LIIII. To find the fiadow of a Conefet on its pointy the ele¬ vation ofLight being given. t f| ^ He Body of Light is a. The Ground-point b, the Co needefg, j| Project the top of this Cone (which is a Circle) Scenographi- cally, as in Operat. IX. X. and by Perpendiculars let fall on the Ground, draw alfo the odier Circle hikl from the fame Vifual point, and point of Diftance.- Then from the Body of Light draw lines into the Ground, through the points defg of the upper Circle, and from the Ground poinr draw lines through die points h i /•/ ,in the Circle on the Ground; and where thefe lines interfeSf each other,as here in the points mn o,lhali be the points through which the Scenographickarch mnodrawnto meet the llraight lines/w,/#, and ko, Ihall be the track that fhall include the Ihadow. OP ERdT* being ajsigned. T He given Light is a, the Ground-point b, the angles of the Te- trahedron c def; Proje&the Triangle ede Scenographical- ly on the Ground, by Perpendiculars let fall from the an¬ gles, as into ghi, analet the length of the Perpendicular ef be the length of the Si dec,/; folhall the fidesg^, an dhi, be drawn from the fame point in the Horizontal line that the lines c e and ed were drawn from. Then from the body of Light draw lines in to the Ground, through die points ede-, and from the Ground-point draw lines through the points ghi; and where thefe lines interiedl each o- ther, as herein the points klm,d raw llraiglit lines from to k, from k to l, and from /to the fide df of the Tetrahedron, where it is cut by the prickt line m n. pofit ion of a Cilindcc The Elevation of Light T He Body of Light is a, the Ground-point b, the Staff c, by working as by Operat. LI. you find a flraight fhadow con- tinuedfrom/c,where thefhadow breaks againfl the Paral- lellepippedon,and and rifes Perpendicularly on the fide e f, and thence on the upper fide of the Parallellepippedon continues it felf Parallel to die Plain of Shadows as atg, and foloofes it felf behind the further fide of the Parallellepippedon. ' > ? . But the SafF hi makes another fafhionedShadow on the furface ofaCilinder;forby working as before, you find die fhadow of the Staf continue flraight from t to m, where the interpofition of theCilin- der breaks the lhadow,by receiving it on its own furface in the form of a Scenographick Semi-circlc, which is the Figure the Circumfe¬ rence of this Cilinder would make, if it were cut afunder in m n. tv OVER Jr: 62 fraBical TerfeBive. 0 TE%/T 10 ^C LVII. To draw a Figure which on the Elam fall he de~ formed : yet vphen feenfrom a defined point through a Holefallfew a Comely and proporti¬ onable f ape* |T Ake a Plain of fo many equal parts as you plcafe, as is the r |j/| firft Plain J B C V, which is devided into 8 in length, and v’..A 8 in depth, as you may fee markt down the fide and bot¬ tom with i z 3 to 8, and therein draw what you think good: as in the fecond Plain A BCD. Then make an other Plain of as many unequal parcsas is the third Plain cadb, (andm ark twofides of that alfo with i z 3 to 8.) 1 bus, Set off one ftde of your firft Plain A BCD at db of the third Plain, andin the middle of it as at e, ereft a Perpendicular as ate 4/, which pointy lhaii be the point ofDiftance: from this point ofDiftance draw two ftraight lines as fed, and fat: then fet oftyour intended height from the point ofDiftance,as/g,and draw the Diagonal linegcT to the point b, and where this Diagonal cuts the out line/c d, as here at c, draw the line c a Parallel Mod b, fo is die four Tides of your third Plain made. To divide this into fmaf fquares as your firft Plain is , do thus, Devide the line db into 8 e- qual parts,as 1 z 3 to 8 .* then lay a ftraidit Ruler to the point of Di- ftance /, and to every one of thefedeviftons fucceftively.and by the fide of it draw lines from the line ca to thefe devifions, andthefc lines Ihallreprefentthe Perpendiculars in the firft Plain: Then from the points where thefe Perpendiculars cut the Diagonal vcb draw lines Parallel to the lines ca and d b, and thefe lines fhaff reprefent the thwart lines that cut the Perpendiculars in the firft Plain. Having thus projected your Plain, drawn therein the figure that was drawn in your fecond Plaiigand becarefull to draw every ftroak thereof inits refponding Square and part of the Square, with the fame proportion to each of thefe Squares that you find in the fquares of the fecond Plain. And though the figure feem when drawn very di- ftortedandunproportionable.as in the fourth Plain abed, yet if y ou place one of you rEysdirefHy over the point fat the height of the line fg,and winck with the other Ey, you lhall fee (if your figure be exactly drawn) the perfefHhape of the figure in the fecond Plain. But that you may the better underftandme I have parted a thin Paft- board on the line h f g, which if you ereft perpendicularly, and look through the hole with one of your Eys clofe to that hole, you may fee the fame figure in this fourth Plain that is drawn in the fecond. OTE RJT. fraBical Terfpeftive. 63 OTEB^JTIO ^ L VIII. T'o draw? on a Quadrant aFigure feemingly confufed and without T)efgn,yet when of this Quadrant a Conefhall be made, a comely and proportionable Figure f ball appear . D Efcribe a Circle of the bignefs you intend the Bafe of your Cone fhail be, as the littie Circle 1x34 &c. to 8, Devidef this Circle into fo many equal parts as you wili(the more the better) this is devided into eight: and from the Center dr aw ftraight lines to every one of thefe devifions, as to 1 2 3. See. to 8. i hen devide one of thefe Semi-diameters alfo into eight e- qual parts, and through every one of thofe equal parts deferibeon the Center a circle concentrick to the outmoft, as the Circles 1 2 3 to 8. Then to make the appearence of thefe Lines and Circles on the outfide of a Cone,Defcribe a Quadrant whofe Radius fhall be twice the Diameter of rhis Circle, as the Quadrant on the right hand, and continue the under Radius to an extream length* Devide this Quadrant into two equal parts, and devide the lower part into eight equal parts; theneredf a Perpendicular on theBafe as at 8,andfrom the Center draw ftraight lines through the eight devifions aforefaid into the ere&ed Perpendicular. I hen placing one foot of your Com- pades in the point 8, open die other fuceflively to all the interfered points in the Perpendicular, and deferibe the feveral arches 17,26, 3 5,44, •>' 3 ,6 2, 71, 8 o. Then devide the outmoft arch into eight e- qual parts, as 1234767 8, and draw ftraight lines from the Center into each of thefe equal parts; So lh a 11 this Quadrant when cut clofe by the outmoft lines thereof,and rolled up in die form of a Cone,and placed diredlly before the Ey, fo as the light may fall round about it, fliew like the little Plain of Circles and ftraight lines, and be of the fame bignefs, and all the arches in this Quadrant fhall become con centrick circles equi-diftant from one another .-and the ftraight lines fhall proceed from the center both waies into the circumferance. Whence(byconfequence)itwillfollow, that what lines foever are drawn in any of tliefquares of the Plain of Circles,and the fame trnaf ferred to the Quadrant,in the fquare refponding to that fquare in the Circle, itmuft alfo appear in the Quadrant, when made in a Cone, in the fame lhape it doth in the Circle, As you may fee in the inlarged Circle and Quadrant, wherein a Figure is delineared. Of Catoptricky or TfefleBed Yearns* R Efle&ed Beams arefeen in a Glafs, or other matter curioufty poiliflit. If this Glafs or pollifht matter be exafHy flat the - Objed appearing in it, is exa£Hy like its Original. This you fee in common Looking Glaftes: But if the Glafs be not S flat 64 Tragical PerfpcBive* flatthe ObjeCtalters from the Original, and that moreorlefs accord¬ ing as the'Glals is more or lefs flat, Therefore in drawing Catop- trick Figures you mull confider the Shape of the furface of the Glafs which you intend to have the Reflection made upon, and for every furface you muftmake a particular Ichnographical Projection,which mult on the Glafs appear to be a Plain full of equal Squares.-on which ProjeCHonifyoutransferwhatfhailbe drawn on a Plain devided in¬ to the fame number of true Squares, though in the Projection it fnall {hew very diflorted and confufed; yet {hall the Reflection thereof on the Glafs fhew very regular,proportionable, and well compofed. I {hall give you one Example thereof in this next Operation. OTE%JTIO^ LXX. To T>raw an Appearance feemingly confufed and without dejigne whichJhall P^jfleU on a Cilinder Perpendicularly ereffed in a designed placeman intended Figure* O (— S Uppofe (For Example hike) it be the likencfs of the Figure A that is intended to be feen Reflected on a Cilinder. Firft de- vide every fide ofthe Figure into fo many equal parts as you pieafefhere we will take 12) and draw lines through thefe de- vifions to cut each other at right Angles.- Then if you make a Pro¬ jection whofe Reflection on a Cilinder Ihali fhew the fame Ihapc and number ofSquaresdt follows by confequencetharifyou transfer all the Lineaments in the Figure, A to this Projection, and place every Lineamentin its proper feituationfo as to correfpondwith the fame fpace from the top and Tides of this Plain, every one of thefe Linea¬ ments in the ProjeCtionfhall alfo appear in the Cilinder in the fame {hape and feituation they do in the Square Plain. Therefore to make this ProjeCtion,defcibe on the Center