A geometrical practise, named Pantometria diuided into three bookes, longimetra, planimetra, and stereometria, containing rules manifolde for mensuration of all lines, superficies 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. Who hathe also thereunto adioyned a mathematicall treatise of the fiue regulare Platonicall bodies, and their Metamorphosis or transformation into fiue other equilater vniforme solides Geometricall, of his owne inuention, hitherto not mentioned of by any geometricians.

Diffynitions.

PRoportion is a mutuall or enterchangeable relation of two magnitudes, being of one kind, compared togither in respecte of their quantities.

The second dffiinition.

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 thrée 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 betwéene the first and the last.

The fourth Diffinition

When foure magnitudes are likewise in continual proportion, the first & the 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 number 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 penpendiculare.

The .13. Diffinition.

TETRAEDRON is a body Geometricall

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encompassed with fowre equall equiangle tri∣angles.

The fourtenth Diffinition.

HEXAEDRON or CVBVS is a solide fi∣gure,

[illustration]

enclosed with sixe equall squares.

The fiftenth Diffinition.

OCTAEDRON is a body comprehended

[illustration]

of eight equall equiangle triangles.

The .16. Diffinition.

ICOSAEDRON is a solide Figure, vnder

[illustration]

twentye equall equiangle triangles conteyne••▪

The .17. Diffinition.

DODECAEDRON is a solide com∣prehended

[illustration]

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 figures 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 séeme 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.