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.

Example.

Lette ABCDE represente an Hexagonall imperfecte Pyramis encom∣passed with twoo equiangle Hexagonal Parallele playnes and 6 Isoscheles Tra∣pezia,

euery of their Bases being the sides of the

[illustration]

••reater Hexagonum 12, euery side of the lesser Hexagonum 6. Now, applying rules or lines ••nto the sides BF, GE, I finde them to con∣••urre at H. Thus haue you two Pyramides, ••he greater hauing his base an equiangle Hexa¦••onum, whose side is 12: the other is the Pyra∣mis HFG, whose base is the Hexagonū FG, ••uery of his sides being 6, the altitude of the les∣ser Pyramis is 15 the altitude of the greater ••0. Also by the rules giuen in Planimetra, yee ••hall finde the area of the lesser Hexagonum ••3 ⅗, the area of the greater 294 ⅖, and so con∣sequently by multiplication of them in the third ••arte of their altitudes as was taughte before in ••he mensuration of whole Pyramides, the solide contente of the lesser will be 468, the crassitude of the greater is 3744, the one deducted for the other, leueth 3276, the magnitude of that vnperfecte Pyramis FBCDE.

A note how by supputation to get the perpendiculares or altitudes of those Pyramides.

FOr that it were tedious & painfull in a solide of great quantitie, by rule or line to searche out these perpendiculares, ye shall by Arithmetike at∣tayne them thus: multiplie the greater Hexagonall side by the side of the vnperfect Pyramis, and the producte diuide by the difference or ouerplus of the sides Hexagonall, the quotient sheweth the side of the whole great Pyramis, from whose square (if ye deduct the square of the circles semidi∣ameter that encloseth the greater Hexagonum whiche ye were taught in Planimetra how to finde) the roote quadrate of the remainder is the alti∣tude of the greater Pyramis, which augmented agayn by the lesser Hexa∣gonall side, and the offcome diuided by the greater, bringeth in the quoti∣ent, the altitude of the lesser Pyramis.