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.

Admitte ABCD the wyne vessell partly filled, ADCE the liquour, ABCE the defaulte or emptynesse, FGHI my smaller proportionall vessell pre∣pared as I haue tofore in the last Chapiter declared, LM the straight rodde di∣uided into 1000 parts, which being lette perpendicularly fall from B till it touch the opposite side at D, I fynde BD 800 partes, ED that is so much as is w••tte of the rodde 600 partes, lykewyse placing the rodde in my lesser vessel at G, ad∣mit I finde the diameter or profunditie GI 200 partes, this number multiplyed by 600, the partes of fulnesse yeldeth 120000, whiche diuided by 800, bringeth in the quotient 150, the partes of fulnesse for my smaller vessell: fillyng therefore the proportionall small vessell with liquour till it ryse vnto K, that is to the 150 parte in my diuided rodde LM, finally by measure I make triall what quanti∣tie of liquour is in my little vessell. Admit it 3 pottles: Nowe to lerne the quan∣titie of liquour conteyned in the great, first I multiply BD the greater diame∣ter

beyng 800 in his square, thereof aryseth 512000000, and that agayne by 3, so haue I 1536000000, which diuided 8000000 (the cube of GI the lesse dimetiēt) produceth 192, so many pottles is there in that wyne vessell. Nowe if ye desire to knowe how many pottles or gallons more will fill it full, ye may by the last Chap∣ter serche how many gallons the whole vessell will conteyne, and from that with∣draw 96 gallons, the quantitie of liquour already therin conteyned, the remayn is youre desire. I thynke it

[illustration]

not necessarie in this case to adioyne any farther exam∣ple, for that this was in the former Chapter plainly both by rule and example alredy declared. This kind of men∣suration serueth not only for wine vesselles, whereof there are sundry fourmes, but also for all manner bodies, what kinde of fashion soeuer they be of, and likewyse for their fragments or parts, and it is grounded vppon this Theo∣reme. All lyke solides retain among themselues triple proportion of theyr lyke or correspondente sides, that is to say, looke what proportion the cube of one side in the lesser solide, retayneth to the cube of his correspondent side in the greater solide, the same proportion doth the lesser body retayne to the greater: so that with the ayde of the golden rule the firste three beeing knowne, the fourthe is redyly founde: And as this rule serueth for the whole bodies, so is it also to be applyed to all fragments or partes: so that the Superficies wherwith the solides are diuided be lyke, and make equall angles with the sides and Superficies of eyther solides, for thereupon it must con∣sequētly ensue that those fragmēts correspondētly cōpared are also proportionall.

Thus muche I thought to adioyne touching the reason and demonstra∣tion of this rule, wherby the ingeniouse practizer might the better retaine in memorie the operation, and also bée able to applye the same to sun∣dry other vses not héere mentioned yf occasyon bee offered. And for as muche as thys kynde of measurynge dependeth wholly vppon the smalle proportionall Uesselle, whose quantitie is supposed knowen,

I thinke it not amisse to giue one rule generall for all small vessels how irregular soeuer they bée (their fashion not regarded) exactly to finde their solide quantities.