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 vnifoorme solides Geometricall , of his owne inuention , hitherto not mentioned of by any Geometricians .

ALthough Geometrie for the certayntie thereof haue suche a priuiledge as fewe other Sciences , béeing so fortified with Demonstration that no Precepte or Rule thereof for the veritie can be reproued , yet least some , either of ignoraunce or malice shoulde affirme it vnprofitable , not seruing to any necessarie vse in a publike Weale , I thought good somewhat to say thereof , and firste the sentence of Plato written at the entrie into his Schole commeth to my remembraunce , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . therby excluding al such as were ignoraunt of Geometrie as vnable or vnméete to attayn higher secrets or misteries of Philosophie . Aristotle also intreating of Morall Philosophie , in the fifth Booke of his Ethikes , with geometricall Figures most beautifully paynteth out Iustice , discerning and seuerally comparing sundrie partes thereof with Geometricall and Arithmeticall proportions . But his Interpreters , whose works are yet extante , for want of skill in these matters , haue so blemished , darkned and defaced his meaning , that scarcely any resemblance thereof shall to the Reader appeare . In sundrie other his works also of naturall Philosophie , as the Physikes , Meteores , de Coelo & Mundo . &c. ye shal finde sundrye Demonstrations , that without Geometrie may not possibly be vnderstanded . And to leue Philosophie , how necessarie it is to attayne exacte knowledge in Astronomie , Musike , Perspectiue , Cosmographie and Nauigation , with many other Sciences and faculties , who so meanely trauayleth therein shall soone finde . But to omitte talking of Geometrie in generall , and to speake priuately of this Treatize , I thinke good first to open the order and effect thereof , and then somewhat to saye of the applyances . Like as euery body geometricall is inuironed with Superficies , and euery Superficies inclosed with lines , so that it consisteth of these thrée , Longitude , Latitude , and Profunditie , and without consideration of these thrée no solide may be measured : so is this Treatise also diuided into thrée Bookes . The first entreating of lines sheweth sundry meanes to measure all maner lengthes , heightes , distances , and profundities . The seconde termed Planimetra settes foorth diuerse meanes and rules to measure manyfolde superficies , playne Conuex and Concaue , whether they be compounde with straighte or circulare lines , or mixte of bothe . In the thirde named Stereometria , is sette out the exacte mensuration of sundry solides , replenished with a number of rules and preceptes , gathered out of Euclide , Archimedes and Appolonius Pergeus his Conykes , wherein the Reader shall not a little delighte himselfe with the finenesse and subtiltie of their inuentions , especially if he indeuor himselfe to search out the reason , cause and demonstration of them . And nowe somewhat to speake of the commoditie of these conclusions , as the skilfull in Architecture can applye the Stereometria to serue his turne in preordinance and forecasting both of the charges , quantities and proportion of all parcels necessarily appertayning to any kinde of buyldings : so Planimetra maye serue for disposing all manner grounde plattes of Cities , Townes , Fortes , Castles , Pallaces or other edifices . The Marshall of the fielde shall also moste spéedely thereby appoynt place conuenient for his Campe , distributing euery parte thereof according to the number of his men , horse , cariage . &c. Also in suruaying , parting and diuiding of Lands and woods , it is most requisite aswell for exact as spéedye dispatch therin , and therof we haue notable recorde in Histories how much this science auayled the Egiptians when as by the inundation of Nylus their whole countrey was so drouned that with slime of the water all their Bounds and marks were defaced : Yet certayne wise men ayded with knowledge in this science , founde out and distributed to euery man his owne . The other parte named Longimetra the ingeniouse practizioner wil apply to Topographie , fortification , conducting of mines vnder the earth , and shooting of greate ordinance . So that as there is no kinde of man , of what vocation or degrée so euer he be , but shall finde matter both to exercise his witte and diuersly to pleasure him selfe , so surely for a gentleman especially that professech the warres , aswell for discoueries made by sea , as fortification , placing of Campes , & conducting of Armies on the lande , how necessarie it is to be able exactly to describe the true plattes , symetrie and porportion of Fortes , campes , townes and countryes , coastes & harboroughes , I thinke there are none so vnskilfull , but will confesse these Geometricall mensurations most requisite . But if any there be so rude , ignoraunte and vnlearned , or so much blinded with selfe liking , that can not be content to acknowledge any thing requisite in a perfite souldier that is wanting in them selues ( for suche surely they are that moste arrogantly will maintayne this fonde opinion ) let them but regard and marke the renoumed Captaine Alexander the great , who had this knowledge in such high estimation , that he seldome or neuer in his manifoulde conquests wold attempt any thing , whether it wer for t , toune or countrey , but first he would haue the exacte Topographie therof , and thervpon inuente , deuise and after execute his warlyke Polycies . On the contrarie side , for negligence of this knowledge we may reade how Aniball the most worthie and famouse Souldier that euer hath béen ( all circumstances duely wayed ) was by that paynefull , graue and right valiaunt Captayne Quintus Fabius Maximus ledde and trayned into a fielde inclosed with stepe hilles and déepe riuers , where this noble Consull had so inuironed his Hoste by fortifying of two mountaynes , that he with all his Armie were in maruaylouse ieopardie to haue perished for famine , had not the craftie witte of Aniball inuented a present pollicie to escape suche imminente calamitie . Cyrus also that greate Kinge and mightie Monarche , was he not through ignoraunce of Topographie , euen in pursuing of victorie intrapped and discomfited with al his power by the Scithian quéene Thomiris at the riuer of Oaxis ? I passe ouer innumerable examples of all ages , manyfestly declaring the greate aduauntage or disaduauntage a Captayne may receyue , aswell inuading as defending by the strong or weake situation of places , and by the foreknowledge or ignoraunce of them : neither is there any liberall or frée minde , whether he be of profession Warlike or Ciuile , that will not take great delight and pleasure to sée how by Arte a man may measure the distances of places remote and farre a sunder , approching nighe none of them , and that aswell , yea and farre more exactly than if with Corde or pole he should paynfully passe them ouer . Herevpon did the Poetes fayne Atlas of suche huge and mightie personage , sustayning and vpholding the earth and mightie Masse of heauenly spheres , for that this man notwithstanding he were imprisoned in a mortall carkasse , and thereby detayned in this moste inferior and vilest portion of the vniuersall world , fardest distante from that passing pleasant and beautifull frame of celestiall Orbes , yet his diuine minde ayded with this science of Geometricall mensurations , founde out the quantities , distances , courses , and straunge intricate miraculouse motions of these resplendent heauenly Globes of Sunne , Moone , Planets and Starres fixed , leauing the rules and preceptes thereof to his posteritie . Archimedes also ( as some suppose ) with a glasse framed by reuolution of a section Parabolicall , fired the Romane nauie in the sea comming to the siege of Syracusa . But to leaue these celestiall causes and things doone of antiquitie long ago , my father by his continual paynfull practises , assisted with demonstrations Mathematicall , was able , and sundrie times hath by proportional Glasses duely situate in conuenient angles , not onely discouered things farre off , read letters , numbred péeces of money with the very coyne and superscription thereof , cast by some of his freends of purpose vppon Downes in open fieldes , but also seuen myles of declared what hath béen doon at that instante in priuate places : He hath also sundrie times by the Sunne beames fired Powder , and dischargde Ordinaunce halfe a myle and more distante , whiche things I am the boulder to reporte , for that there are yet liuing diuerse ( of these his dooings ) Oculati testes , and many other matters farre more straunge and rare which I omitte as impertinente to this place . But for inuention of these conclusions , I haue hearde him say , nothing euer helped him so muche as the exquisite knowledge he had by continuall practise attained in Geometrical mensurations . And for science in great Ordinance especially to shoote exactly at Randons ( a qualitie not vnméete for a Gentleman ) without rules Geometrical , and perfect skill in these mensurations , he shall neuer know any thing : thus haue I partely declared the pleasure and commoditie that any well disposed minde may reape by these thrée bookes of my fathers . But somewhat to say concerning the last Treatize of the fiue regulare bodies and their Metamorphosis adioyned by my selfe , for that I knowe some , yea many shall there be that will not spare to say , as they haue done by others , it is a fonde toy , a meare curiouse trifle , seruing to no vse or commoditie : surely I meane not greatly to labour to attayne the good opinion of suche ▪ hauing learned howe the disprayse of the ignoraunt and vitiouse is no lesse commendation than the prayse of the wise and vertuouse , béeing a precedent infallible from the beginning and alway contynuing that ignoraunce enuieth knowledge , and vice vertue . Euery man disliketh or extolleth things accordingly as they are agreable or repugnant to his appetite , and rather would I enioyning the company of Euclide , Archimedes , Appollonius Pergeus , and other geometers writing of the like toyes & curiosities be condemned , than in publishing matter agreable to suche mens phantasie and conceypt , be of them highly commended , and of the learned laught at or contemned , but howsoeuer Epicurus , Midas , or their like , giuen onely to lucre and worldlye pleasure liste to thinke thereof , I nothing mistruste of suche as couet the vnderstanding of matters harde and difficile , desiring the knowledge of things somewhat passing the reache and capacitie of the common sorte , wherein onely the nature of man surmounteth beastly kinde , or by proofe and assay in cases of like difficultie able to iudge them selues , this Treatize shall not be mislyked but thankfully receyued , and for the reste perswasions are but vayne , for as no words can adde stomacke or make the cowarde valiaunt , so surely suche two footed Moules and Todes whom destinie and nature hath ordayned to craule within the earth , and suck vpon the muck , maye not possibly by any vehement exhortation be reduced or moued to taste or sauour any whitte of vertue , science , or any suche celestiall influence , my hope is if any fault escape , as in suche long and intricate tediouse calculations of irrationall numbers maye happen to be some , the discrete and modest Reader will rather of curtesie amende it , than with enuiouse cauillation , vngratefullye requite my paynefull trauayles , whereby I shall be prouoked not onely to publishe the demonstrations of these and many moe straunge and rare Mathematicall Theoremes , hytherto hidden and not knowen to the worlde , but also to imprinte sundry other volumes of my fathers , which he long sithence compiled in the Englishe tong , desiring rather with playne and profitable conclusions to store his natiue language and benefite his Countrey men , than by publishing in the Latin rare and curiouse demonstrations , to purchase fame among straungers .