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.

AB the two markes whose distance I would mete C, my first staffe, I my triangle made of 3 staues placed thereat, as you may see in the figure directing with the one conteyning side to A, the first marke, and with the other to D and E my seconde and thirde staues, H is the fine notche or mark vpon the side sub∣tendent to my angle, where the line visual from C to the seconde marke B passeth, my triangle I situate now at D as it was before at C, the one conteyning side lying euen with the errered staues, the other directeth to my forth staffe F, placed in aright line with E, the thirde staffe, and A the firste marke. Agayne my line visuall proceeding from D to H the subtill notche in the subtendente side

of the angle, extendeth to my fifte staffe G, situate exactlye betweene E the thirde staffe, and B the other marke: This done, I measure the distaunce be∣tweene my seconde and thirde staffe, finding it 20 foote, likewise betweene the fourth and fifth staffe 72 foote, finally betweene the firste and thirde staffe 65. pase, so that according to the rule before giuen, multiplying 65 in 72, I haue 4680 whiche diuided by 20 yeldeth in the quotient 234, so many pase is there betwene A and B. I haue not here set out the figure in iust proportion, answering to these numbers, for that is not requisite, but in suche fourme as may best open and make manyfeste the situation of the staues and triangle, wherein consisteth all the difficultie of this practise.

[illustration]

A proofe or demonstration Geometrical of the former conclusion.

TO auoyde tediouse recitalles of the premisses, I suppose the figure in all re∣spects made as it was before: the proposition that I will here demonstrate shall be, that DE retayneth the same proportion to EC, that GF doth to AB,

whiche admitted or assumed as a Lemma, the rule before giuen of multiplying FG in EC, and diuiding by DE, to produce AB may two wayes be approued geometrically by the 16 proposition of the 6 boke of Euclides elements, and arith∣metically by the 19 proposition of the 7 boke of the same elements, wherin I mind

[illustration]

not to vse mo words, considering it is nothing els but the calculation by the rule of three, whose demonstration hath been handled by diuerse, and well knowen to any, meanely in these matters trauayled. But how these DE and EC become to be proportionall with GF and AB hath not ben by any hitherto proued, this Lemma therefore or proposition I minde to demonstrate. Firste it is apparante that DF is equidistante to AC by the 2•• propositon of the firste of Euclides elements, the words are these.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. And may thus be englished, if any right line falling or passing through two other right lines, making the outwarde angle equal to the inwarde opposite on the same side, or the two inwarde angles ioyned togither equall vnto two right angles, those two right lines are parallele: but here the line EC passing through the lines AC, DF maketh the outward angle FDE equal to the inward opposite on the same side ACD by supposition, bicause they were bothe made with one angle of the triangle, I may therfore conclude by this Theoreme, that DF is equidistante to CA, and farther inferre by the seconde Theoreme of the sixth booke, that AF to FF hath the same proportion that CD hath to DE, the proposition is this. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

Yf a parallele line be drawen to any side of a triangle it shal proportionallye cut the two other sides, and if two sides of a triangle be proportionally diuided the line that coupleth those diuisions shal be parallele to the other side. It is manyfest by the first part of this theoreme that DF being as it was before proued parallele to AC one side of the triangle ACE that it doth proportionally diuide the two other sides CE, EA, in the poynts FD the like shall be proued of DG, for seeing the right line CE falleth on the two right lines DG, CB making the outwarde angle HDE equal to the inwarde and opposite angle HCD on the same side of the line CE which in the construction of the figure was supposed, it must needes follow by the 28 propositiō of the first booke of Euclide tofore recited, that DG is parallele to CB, and forasmuch as in the triangle BCE, DG is drawen parallele to the one side CB, it shal by the seconde proposition of the 6 booke of Euclide (be∣fore also recited) deuide the two other sides CE, EB proportionally, so that BG shall retayne the same proportion to GE that CD dothe to DE, and so conse∣quently the same that AF doth to FE, as it is playne by the eleuenth theoreme of the fifth booke of Euclide: his words be these.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. The sence thereof is this, that any two proportions beeing equall or agreable to any one are also equal betwene themselues, as here first it was proued that AF to FE bare the same pro∣portion that CD to DE, and now that BG to GE retaineth the same proportion that CD to DE: therfore by the theoreme last recited AF to FE and BG to GE, shall be proportional: so haue you now two sides of the triangle AEB pro∣portionall with two sides of the triangle FEG, and the angles conteyned of those sides equal. Therfore by the 6 theoreme of the 6 booke those two triangles are e∣quiangle, the words of that theoreme are these.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

Yf two triangles haue an angle in the one equal to one angle in the other, and a∣bout those equal angles the sides proportional, those two triangles shall be equi∣angle, and those angles, equal whose subtendent sides are proportional, Euclide also in his 4 proposition of the same booke sayth thus.

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

In equiangle triangles aswell the contayning as the subtending sides of equall angles are proportionall, I may therfore affirme (seeing the triangles ABE, EFG are equiangle) that AB hath the same proportion to FG, that AE hath to FE, but before it was proued that AF and FE were proportionall to CD and DE, conioynedly therefore AE to FE, shall retayne the same proportion that CE doth to DE, by the 18 proposition in the fifte Booke of Euclide, saying thus: 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

If magnitudes disioynedly or seperatly be proportionall, conioynedly or com∣pounded, they shall also bee proportionall, wherevppon I may finally inferre by the 11 proposition of Euclides fifte Booke tofore recited, that AB to GF retayneth the same proportiō that CE doth to DE, bicause they both obserue the same pro∣portion that AE doth to EF. And thus to conclude, it is manifest that AB, FG, CE, DE, are 4 quantities proportionall: whereof three knowen, the fourth AB by the rule of proportion is to bee found. Very like vnto this is the Demon∣stration of the other, and by the selfesame propositions to be proued, and therefore superfluous to vse mo wordes.