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.

The .28. Chapter.

To searche the beste proportion or simetrie of many places vvith the true distance approchyng neere none of them by the instrument named Theodelitus.

THis instrumēt vpon his staffe or otherwise in the field pla∣ced the index béeing in his diameter, let it direct your sight to some one place ••hich ye wil mesure. Truly in my fanta¦sy it wer more cōmodious if his dimetiēt or diameter wer first laid in a straight line, bringing the sight to y^e vttermost place toward the left hand, so conueying your index to euery 〈◊〉〈◊〉 or marke on the right syde, noting diligently the angle or angles of 〈◊〉〈◊〉 vpon some state stone or table prepared, which angles here I call grades or degrées from the dimetient apparantly cut by the lyne 〈◊〉〈◊〉 whilest he is broughte to euery marke. This performed, resorte to backe side of your instrument where necessitie requireth a circle or a ••••••icircle to be made, deuided exactly in 360, or if it be an halfe circle in degrées or portions, euen as your Theodelitus here is, from whose 〈◊〉〈◊〉 must finely drawe those angles of position noted before in your 〈◊〉〈◊〉 taken by youre instrument, so that after youre purpose hadde, they 〈◊〉〈◊〉 be cleane put out, then pull the index the instrumente vnmoued to∣••••••d the right hand, at pleasure obseruing through the fights some marke 〈◊〉〈◊〉 yardes from you or lesse as yée lyste. There shall bée youre seconde 〈◊〉〈◊〉, notyng vppon youre slate the angle of position from the dimetient 〈◊〉〈◊〉 lyne fiduciall, directing to the seconde place or marke, whyche ••word muste drawe in the back side from the Centre at large euen as you 〈◊〉〈◊〉 it in your instrument, then conuey your Theodelitus from thence to second marke or standyng place, causyng the diameter iustely to note 〈◊〉〈◊〉 first abiding. And here euen as tofore ye must search Angles of 〈◊〉〈◊〉 agayne, and marke them in the table or slate, which doon, resorte to the 〈◊〉〈◊〉, and vpon the last angle being the line directing to the seconde stan∣••••••g place, draw a circle as far or néere to the other as ye lyst: Or a 〈◊〉〈◊〉 duided in 180 degrées. Whose diuisyons muste take theyr begyn∣••ge at the lyne whyche is Diameter of the semicircle. Nowe drawe 〈◊〉〈◊〉 visuall lynes or angles of position laste taken by your instrument ••arge, and see where the lines méete, or a lyke toucheth his like. So 〈◊〉〈◊〉 you the due proportions, for the distaunce yée shall woorke thus, 〈◊〉〈◊〉 the lyne that goeth from the centre of the one circle or semicircle to

the other, in as many portions as ye thinke mete, or rather in so many 〈◊〉〈◊〉 fynd certaine measure, and by those parts diuide the lines betwixt eue••ry place of which ye require the length. Then multiply the portions that are betwéene any two sections or places in the distance of your two stati••ons, which I imagined here .300 pace, and make partition by those part•• that are betwixt the two centres, so haue ye the true length or distance 〈◊〉〈◊〉 two suche places. In like maner ye must do of the reste.

[illustration]

Example.

WHerby all thing may better appeare, ABC are the markes in the fielde to be measured, D the first abyding or standing place, wher ye shall set the cen∣tre of your instrument, his diameter lying directly agaynste A the firste marke EFGH the foure visuall lines running by the angles of position of the instrumēt vnto all the markes: the first noting no degre or portion: the second 20, the third 40, the fourth 90 degrees of the instrument, which directeth to the seconde station M, where ye shal now set the centre of your instrument, the diameter lying right agaynst your first abyding. here the lines visuall IKL running to the markes: out new angles of position agayne. The first noteth 55 degrees, the seconde 74, the third 85 grades. Now, if ye marke diligently where these lines crosse the other, ther is the true proportion of suche places, from those sections or crossings draw right lines as appeareth by the Figure. Now to get the distāce, ye shall seeke out the space betweene the two stations DM: being 300 pase or yardes, notwithstanding it is diuided but into 18 partes: also betwene the markes or places A and B, are cōtay∣ned 11 such partes. Seeing that I am ignorant what number of yardes be contay∣ned in those 11 portions, I am compelled to worke by the rule of proportion thus, 18 bringeth 300 yardes, what shall 11 bring mee? your quotient sheweth 183 and 2/6 that is ⅓, which maketh a foote: so betweene A and B are contayned 183 pases and a foote. Thus of all other aswell of DA, DB, DC, MC, MB, MA, as of CB, CA, the Reader must not bee ignorant that euen as I haue supposed by this figure, the instrument in the field placed and all things perfor∣med by it with all diligence marked vppon a slate or such like, so ought it to be ex∣actly drawen on the backeside of your instrument, eauen as the Figure afore de∣clareth.

Certes most excellent and farre passing all other is this kinde of mea∣suring, requiring great exercise, the distāce of one place had I am brought into the knowledge of many with the beste proportion.