l e m e . l i b r a r y . u t o r o n t o . c a s t c 6 8 5 8 v e r . 1 . 0 ( 2 0 1 9 ) A GEOMETRICAL Practise, named PANTOMETRIA, diuided into three Bookes, Longimetra, Planimetra and Stereometriea, containing Rules manifolde for mensuration of all lines, Superfices and Solides. With sundry straunge conclusions both by instrument and without, and also by Perspectiue glasses, to set forth the true description or exact plat of an whole Region: framed by Leonard Digges Gentleman, lately finished by Thomas Digges his sonne. Elementes of Geometrie, or Diffinitions. A Poynt I call whiche cannot be diuided, whose parte is nothing. A Lyne is a length without breadth or thicknesse, whose ex­ tremities are two poyntes. THe shortest drawen between two Poyntes is a streight line, the contrary are crooked lines. A poincte A right line Crooked lines A Superficies is that hath length and breadth onely, beeing bounded or determined with lines. A Playne Superficies is that whiche lieth equally and euen­ ly betweene his lines or boundes. A superficies. A Playne Angle is the inclination of two lines lying in one playne Superficies, concurring or meeting in a poynt. IF those lines that containe the Angle be straight, it is called a right lined Angle, and those two lines his containing sides, but if a third straight line be drawne crosse the former from one to the other, that shall be called the subtending side. ABC the contayned angle AB BC the contayninge sides AC the syde subtendent. OF straight lined angles there are three kindes, the Ortho­ gonall, the Obtuse and the Acute Angle. WHen any right line falleth Perpendicularly vpon an o­ ther, that is to say, making the Angles on either side e­ quall, eche of those Angles is an Orthogonall or right Angle, and that falling line a Perpendicular. AB is the Perpendicular DC the ground line. A Perpendicular A right angle BAC the right angle contained of the Perpendicular, and one part of the ground line equall to BAD the right angle contained of the Perpendicular, and the other portion of the grounde line, and therfore both Orthogonall. Longimetra The Brode or Obtuse Angle is greater than the Orthogonal. Abroode The Acute or sharpe, is lesser than the right angle. a sharpe A Figure is comprehended within limites and bounds, whe­ ther it be one or many. A Circle is a plaine figure, determined with one line, which is called a Circumference, in whose mids there is a point named his Centre. From the which all right lines drawne to the circumference are equall. A circle. A circumferens. A Semicircle or halfe Circle, doth conteine both the Dime­ tient and Centre of his circle, with the precise halfe of his circumference. A Right line drawne through the Centre vnto the Circum­ ference of both sides, is named his Diameter or Dimeti­ ent, the halfe of it is called his Demidiameter. "Demidiameter" not found in OED. An halfe circle Dimetient ALl Straight lines besides the Diameter in any Circle pulled from one part of the Circumference to the other, be called cordes. THe portion of the Circumference from that corde compre­ hended, is named an Arcke. A Touch line is that toucheth a circle in a Pointe. an arck A corde Diametre A corde A touche lyne A triangle EMong Right-lined figures, suche as haue onely three si­ des are Triangles, whereof there be sundrie sortes bearing seuerall names, according to the diuersitie of their sides and Angles. IF the Triangles three sides be euery of them of like length, it is called an Equilater Triangle. ISoscheles is such a Triangle as hath onely two sides like, the thirde being vnequall, and that is the Base. Schalenum hath three vnequall sides. "Schalenum" not found in OED. A Rightangled Triangle is suche a one as hath one Righte Angle. AN Obtuseangle Triangle hath one obtuse angle, and is called Ambligonium. Oxigonium hath all acute or sharpe Angles. THere be also foure sided Figures called Quadrangles, whose Opposite sides and angles are equall, suche are na­ med Paralelogrammes, whereof there are but foure sortes. IF all the sides be equall, and al the angles right, than is that Paralelogramme called a square. If one side containing the right Angle, be longer than the o­ ther containing side, then is that figure called a Rectangle. If all the sides be equall, and no angle aright, then is it called Rhombus. BUt if it haue only the Opposite sides equall, and the other that containe an Angle vnequall, it shall be named Rhom­ boides. All other quadrangles are Trapezia. "trapezia" antedates earliest OED quotation (1631). POlygona, are such figures as haue moe than foure sides, whose angles if they be all like and equall, they are termed Equiangle Polygona. A figure of manie sides AL other plain Superficies, whether they be enuironed with straight or crooked lines, shalbe named irregulare figures. Longimetra WHen two right lines drawne in one plaine Superficies, are so equedistantly placed, that though they were infi­ nitely extended on either side, yet would neuer meete nor con­ curre, they shall be called Paralleles. Paralleles. A Quadrant is the fourth part of a Circle, included with two Semidiameters commonly diuided in 90. portions, which partes are named grades or degrees. The second kynde of Geome­ trie called Planimetra. HAVING accomplished the first part called Longimetra, con­ tayning sundrie rules to measure lengthes, breadthes, heigths, depthes, and distances: I thinke it meete now to proceede to the seconde kynde named Planimetra, wherein ye shall haue rules for the mensuration of all manner playne Figures, to know their contente superficiall. And forasmuch as there is no Superficies, but is enuironed with lines either streight or curue. And all Figures com­ prehended with streight lines may be resolued into Triangles: It seemeth most meete, first to teache the measuring of Triangles as followeth. The fyrst Chapter. Of Triangles. AS there are three kind of Angles, so is there also three kind of Triangles: the first is called Orthogonium, hauing one of his angles a right: the other Ambligonium contayning one obtuse angle: the third kinde is called Oxigonium, whose three angles are all acute. Of right angled Triangles also there are two kindes, for eyther it hath two equall sides, and then is it called Isoscheles, or three vnequall, and that is Scalenum. "Ambligonium" not a headword in the OED (but see quotation of 1570 under "obtuse-angled." The thirde kynde of Geome­ trie named Stereometria. IN THIS thirde booke ye shall receyue sundrie rules to mea­ sure the Superficies and Crassitude of solide bodies, whereof, although an infinite sorte of differente kyndes might be ima­ gined, yet shall I only entreate of such as are both vsually re­ quisite to be moten, and also may sufficiently induce the inge­ nious to the mensuration of all other solides what forme or figure soeuer they beare. And forasmuch as in setting foorth their seuerall kyndes, I haue chosen to vse the accustomable and auncient names well knowen to any trauelled in Geometrie, rather than to forge newe Englishe wordes which can neither so breefly nor so aptly expresse the like effecte, least to the com­ mon sorte any obscuritie might growe, I thinke good to adioyne euery of their diffinitions, DIFFINITIONS. 1 A Solide body is that hath lengthe, breadth and thicknesse boun­ ded or limited with Superficies. 2 Lyke solides are such as are encompassed with superfices that are lyke and of equall number. 3 A Prisma is a solide Figure comprehended of playne Superficies, whereof two are equall, like, and Parallele, the reste Parallelogrammes. "Prisma" not a headword in the OED. 4 A Pyramis is a solide Figure enclosed with many playne Superfi­ cies rysing from one, and concurring or meeting in a pointe. 5. A Sphere is a grosse or solide body comprehended of one coruex Superficies. In the middes whereof there is a pointe from whence all right lines to the same superficies extended, are equall. 6. That poincte is called his Center, and a streight line by that Cen­ tre passing throroughe this solide bounded on eyther side with the conuex superficies is called the Diameter of that Sphere. 7 Also intellectually ye may thus conceyue a Sphere to be made. Suppose a semicircle (his diameter remayning immouably fixed) to be turned round about til it returne to the place whence it firste beganne to moue, the figure so described, is a Sphere. 8 Likewise, if a right angled triangle (one of the contayning sides re­ mayning fixed) be turned circularly about the Figure so described, it is called a Cone. 9 The right line that remayneth fixed is the Axis. 10 The circle described by the other contayning side is the Base. 11 The third line or Hypothenusa, is the side of the Cone. 12 If a right angled parallelogramme (the one of the sides conteyning a right angle remayning immouable) be circularly turned, the Figure so described, is a Cylinder, and the immouable side is his Axis. 13 The Bases are the Circles by reuolution of the two opposite Pa­ rallele sides described. 14 The altitude or heigth of any solide body, is a line perpendicular­ ly falling from the toppe or highest parte of the solide vppon the playne whereon the body lyeth. 15 This perpendicular or line of altitude in directe solides falleth within the body, and vppon the base, but in declyning solides, it falleth without the bodies and bases. 16 As the concourse of lines maketh a playne angle, so the concurring or meeting of many superficies in a pointe, maketh a solide angle. 17 In euery solide body a right line drawen from one solide angle to one other is called a line Diagonall. But if it passe betweene opposite an­ gles, it is named the Diameter. A Mathematical Discourse of Geometrical solides. The Preface. ALthough the Rules and Preceptes already geuen abun­ dantly may suffice for the exact mensuration of any man­ ner Solides or vessels that are vsually occupied, or may be proponed to be moten, yet for the satisfaction also of such as deliting in matters only new, rare and difficile, seeke to reache aboue the common sorte, I haue thought good to adioyne this Treatise of the 5 Platonicall bodies, meaning not to discourse of their secrete or mysticall appliances to the Elementall regions and frame of Celestiall Spheres, as things remote and farre distant from the Methode, nature and certaintie of Geometrical demonstration, only heere I intend Mathematically to conferre the Super­ ficiall and Solide capacities of these Regulare bodies with their Circum­ scribing or inscribed spheres or Solides, & Geometrically by Algebraycall Calculations to search out the sides, Diameters, Axes, Altitudes, and lines Diagonal, together with the Semidimetients of their Equiangle Bases, containing or contained Circles, and as in the Problemes ye shal perceiue by Preceptes and Examples, the quantities and proportions of all these lines, Superficies and Crassitudes, with numbers Rationall and Radicall expressed, so haue I by Theoremes sowne the seede of Rules innumerable, wherin the studious may delite him selfe with infinite varietie: Finally I shall in the last Chapiter by Diffinitions and Theoremes partly set forthe the forme, nature and proportion of other fiue vniforme Geometricall So­ lides, created by the transformation of the fiue bodyes Regulare or Plato­ nicall: but not in so ample manner, as the noueltie or difficultie of the mat­ ter requireth, meaning as I see these the first frutes of my studies liked and accepted, to bestowe time and trauaile vppon an other peculiare Volume, which shall conteyne, not onely the demonstrations of these Theorems of spherall solides, but also of Conoydall, Parabollical, Hyperbollical and Ellepseycal circumscribed & inscribed bodies, & many mo lynes & solides produced by the sections of these, and reuolution of their sections: but to re­ turne to this present Treatise, let no man muse the writing in the English toung, I haue retayned the Latin or Greeke names of sundry lines and figures, as cordes Pentagonall, lines Diagenall, Icosaedron, Dedocae­ dron, or such like, for as the Romanes and other Latin writers, not with­ standing the copiouse and abundante eloquence of their toung, haue not shamed to borrow of the Gretians these and many other termes of arte: so surely do I thinke it no reproche, either to the English toung, or any English writer, where fitte words fayle to borrow of them both, but rather should we seeme therby to do them great iniurie, these beeing in deede certayne testimonies and memorials where such sciences firste tooke their originall, and in what languages and countreys they chieflye florished, which names or words how straunge soeuer they seeme at the firste acquayntaunce, by vse will grow as familiar as these, a triangle, a circle, or suche like, which by custome and continuance seeme meere En­ glish, yet to auoyde all obscuritie that may grow by the noueltie of them, I haue adioyned euery of their diffinitions, and so proceded to Problems and Theoremes with suche methode, as howe obscure or harde soeuer they appeare at the firste, through the rarenesse of the matter: I doubt not but by orderly reading, the ingeniouse Student, hauing any meane taste of cossicall numbers, shall finde them playne and easie. Diffynitions. PRoportion is a mutuall or enterchangeable relation of two magnitudes, being of one kind, compared togither in respecte of their quantities. The second diffinition. When the proportion of two magnitudes is such as may be expressed with numbers, then is it certaine & apparant and here is called rational: But when the proportion is such as cannot be expressed with numbers, but with their rootes onely, then is that proportion certayne also, but not apparante, and therfore here I name it surde or irrationall. The thirde diffinition. When there be three suche magnitudes or quantities that the first to the second retayne the same proportion that the second doth to the third, those quantities are saide to be proportionall, and the first to the thirde retayneth double the proportion of the first to the second, and the seconde is named meane proportionall betweene the first and the last. The fourth Diffinition When foure magnitudes are likewise in continual proportion, the first & fourth are the extremes, and the second and thirde the meanes, and the extreames are sayd to haue triple the proportion of the meanes. The fifth diffinition. Any lyne or number is sayde to be diuided by extreame & meane pro­ portion, when the diuision or section is suche or so placed, that the whole line or numer retayne the same proportion to the greater parte, that the greater doth to the lesser. The sixth diffinition. A lyne is sayde to be equall in power with two or moe lynes, when his square is equall to all their squares. The seuenth diffinition. A lyne is sayd to matche a superficies in power, when the square of that line is equall to the superficies. The eyght diffinition. When any equiangle triangle, square, or Pentagonum is in suche sorte described within a circle, that euery of their angles touche the cir­ cumference, their sides are called the trigonal, tetragonall and pentago­ nall Cordes of that circle. The ninth diffinition. About euery equilater triangle, square, or Pentagonum, a circle may be described, precisely touching euery of those figures angles, and that circle shall be called the circumscribing or contayning circle. The tenth diffinition. Also within euery of these equiangle figures a circle may be drawen, not cutting but only touching euery of their sides, this is called the in­ scribed circle. The eleuenth diffinition. Any right line drawen from angle to angle in those equiangle figures passing through the superficies, I name the line diagonall. The twelfth diffinition. A right line falling from any angle of these superficies perpendicu­ larly to the opposite side shalbe named that playnes perpendiculare. The. 13. Diffinition. TETRAEDRON is a body Geometricall encompassed with fowre equall equiangle tri­ angles. Tetraedron. The fourtenth Diffinition. HEXAEDRON or CVBVS is a solide fi­ gure, enclosed with sixe equall squares. Cubus. The fiftenth Diffinition. OCTAEDRON is a body comprehended of eight equall equiangle triangles. Octahedron The 16. Diffinition. ICOSAEDRON is a solide Figure, vnder twentye equall equiangle triangles conteyned. Icosaedrum The. 17. Diffinition. DODECAEDRON is a solide com­ prehended of twelue equall equiangle pentago­ nall Superficies. The. 18. Diffinition. These fiue bodies are called regular, and about euery of them a sphere may be described, that shall with his concaue peripherie exactly touche e­ uery of their solide angles, and it shall be called their comprehending or circumscribing sphere or globe, and these solides shalbe called the inscri­ bed or conteyned bodies of that sphere. The. 19. Diffinition. Also within euery of these regulare bodies a sphere may be described that shall with his conuex superficies precisely touche all the centres of those equiangle figuers wherewith these bodies are enuironed, and such a one I terme their inscribed or conteyned sphere, and those bodies shall be termed the circumscribing solides of that sphere. The. 20. Diffinition. The semidiameter of this inscribed sphere, forasmuche as it is the very Axis or Kathetus of euery Pyramis, hauing his base one of the equi­ angle playnes, and concurring in the centre, of which Pyramides (intel­ lectually conceyued) these bodies seeme to be compounded, it shal be na­ med the Axis or Kathetus of that body. The 21 diffinition. Euery of these bodies side, I call any one of those equall righte lines wherewith these equiangle Figures are enuironed that comprehende and include these bodyes. The 22 diffinition. Any one of the Figures wherewith these solides be enuironed is cal­ led the base of that solide. The 23 diffinition. A line falling from any solide angle of these bodyes perpendicularlye on the opposite playne or base, shall be named that solides Perpendi­ culare. The 24 diffinition. The power of any line gyuen is sayde to be diuided into lines re­ tayning extreame and meane proportion, when two suche lines are found as both make their squares ioyned togither, equall to the square of the line giuen, and also holde such proportion one to another as the two partes of a line diuided by extreame and meane proportion. The 25 diffinition. One of these regulare solides is saide to be described within an other, when all the angles of the internall or inscribed body at once touche the superficies of the comprehending or circumscribing regulare solide. The. 25. Probleme. A Metamorphosis or transformation of the fiue regulare bodies. HItherto haue I onely intreated of the fiue regulare bodies. Theoretically and practically opening sundrie meanes to search out the proportion and quantities of their sides, diameters, axes, perpendiculares, altitudes, and contentes both superficiall and solide, and that not onely in these so­ lides considered by them selues, but also conferred with other, aswell by inscription as circumscription, both of themselues and their spheres: so as I suppose hardely any question may be proponed concerning these bo­ dies, whiche by the formal problemes or theoremes may not be resol­ ued. Yet before I finishe this Treatise I thought good to adioyne one Chapter of the transformation of these Solides into suche bodies (as though they may not be tearmed regulare, for that Euclide hath demon­ strate onely fiue, and no mo possibly to be founde or imagined) yet haue they such vniforme composition and conuenience with these solides, that they are not onely enuironed with equilater and equiangle superficies as they be, but also haue all their sides equall, and one comprehending sphere exactly and at once touching all their solide angles, they are also capable of the regulare bodies, and onely herein different, that whereas the regulare solides be inuironed with one kinde of playnenes and re­ ceiue one internal sphere, and so consequently one axis. These Transfor­ med bodies are incompassed with seuerall kinde of playnesse, and haue se­ uerall axes and contayned spheres, of which their diuersitie aryseth ma­ nyfolde mo straunge, rare, and differente kinds of proportions, than may in a very large volume (I will not saye be demonstrate, but onely by Theoremes) be declared. Transformed regulare solydes I call them, both to auoyde the forging of newe names, which I should be inforced to vse for distinction sake, and also bicause they seeme to be created by the vnifourme section of the regulare bodies, and may by addition of certaine proportionall equall Pyramides, be reduced to the regulare solides, and are in sundry proportions and proprieties so agreable and resemblante to those regulare solides, whose names they beare, that they seeme onely to lose the fourme, and yet still to retayne the nature of them. I meane not here in so ample maner as the noueltie of the matter requireth to in­ treate of them, but onely by diffinitions and Theoremes open so muche as may be sufficient to explane the composition, fourme, nature, and pro­ portion of these, and also giue light to the ingeniouse infinitely to pro­ ceede for inuention of the like, whose vse and appliance may be manyfold to conclusions no lesse straunge than necessarie: but thereof in due place for the nature of these transfourmed solides peruse their seuerall diffini­ tions and Theoremes immediately ensuing. Of Tetraedron transformed. The firste Diffinition. TEtraedron transformed is a solide incompassed with foure equall equilater triangles, and foure equall equiangle playnes Heragonall, hauing equall sides with the triangles. Of the transfigured Cube. The seconde Diffinition. A Transfourmed Cube is a figure geometrical enuironed with 6 equi­ angle Octogonall and 8 equilater triangular playnes or bases, whose sides are all equall. Of Octaedron transformed. The 3 Diffinition. A Transfigured Octaedron is a Geometricall Figure incompassed with 14 bases, whereof 8 are equall equiangle Hexagonall playnes, and the other 6 are equall squares. Of the transfigured Icosaedron. The fourth Diffinition. ICosaedron transfigured, is a solide bodye incompassed with 32 Equi­ angle and equilater bases, whereof 20 are Hexagonall, and the other 12 Pentagonall playnes. Of Dodecaedron transformed. The fifthe Diffinition. A Transformed Dodecaedron is a massie or solide figure, comprehended of 12 equiangle dexagonal, and 20 equilater triangular bases. "dexagonal" not found in OED.