Waters Superficies. Square then, halfe the Subtendent of that watry
Superficies (which Subtendent must haue the equall partes of his
measure, all one, with those of the Semidiameter of the earth to your
watry Superficies): Subtracte this square, from the first: Of the
residue, take the Rote Square. That Rote, Subtracte from your first
Semidiameter of the earth to your watry Superficies: that, which
remaineth, is the heith of the water, in the middle, aboue the leuell.”
Which, you will finde, to be a thing insensible. And though it were
greatly sensible, *
[* Note.]
yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides
twelfth booke, noted: you may reduce all, to a true Leuell. But, farther
diligence, of you is to be vsed, against accidentall causes of the
waters swelling: as by hauing (somwhat) with a moyst Sponge, before,
made moyst your hollow Pyramis or Cone, will preuent an accidentall
cause of Swelling, &c. Experience will teach you abundantly: with great
ease, pleasure, and cõmoditie.
Thus, may you Double the Cube Mechanically, Treble it, and so forth, in
any proportion.
[Note this Abridgement of Dubbling the Cube. &c.]
Now will I Abridge your paine, cost, and Care herein. Without all
preparing of your Fundamentall Cubes: you may (alike) worke this
Conclusion. For, that, was rather a kinde of Experimentall demõstration,
then the shortest way: and all, vpon one Mathematicall Demonstration
depending. “Take water (as much as conueniently will serue your turne:
as I warned before of your Fundamentall Cubes bignes) Way it precisely.
Put that water, into your Pyramis or Cone. Of the same kinde of water,
then take againe, the same waight you had before: put that likewise into
the Pyramis or Cone. For, in eche time, your marking of the lines, how
the Water doth cut them, shall geue you the proportion betwen the
Radicall sides, of any two Cubes, wherof the one is Double to the other:
working as before I haue taught you:
[* ☞ Note.]
* sauing that for you Fundamentall Cube his Radicall side: here, you may
take a right line, at pleasure.”
Yet farther proceding with our droppe of Naturall truth:
[To giue Cubes one to the other in any proportion,
Rationall or Irrationall.]
+you may (now) geue Cubes, one to the other, in any proportiõ geuẽ:
Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedon
of Copper or Tinne: with one Base wãting, or open: as in our Cubike
Coffen. Frõ the bottome of that Parallelipipedon, raise vp, many
perpendiculars, in euery of his fower sides. Now if any proportion be
assigned you, in right lines: Cut one of your perpendiculars (or a line
equall to it, or lesse then it) likewise: by the 10. of the sixth of
Euclide. And those two partes, set in two sundry lines of those
perpendiculars (or you may set them both, in one line) making their
beginninges, to be, at the base: and so their lengthes to extend vpward.
Now, set your hollow Parallelipipedon, vpright, perpendicularly,
steadie. Poure in water, handsomly, to the heith of your shorter line.
Poure that water, into the hollow Pyramis or Cone. Marke the place of
the rising. Settle your hollow Parallelipipedon againe. Poure water into
it: vnto the heith of the second line, exactly.
[* Emptying the first.]
Poure that water * duely into the hollow Pyramis or Cone: Marke now
againe, where the water cutteth the same line which you marked before.
For, there, as the first marked line, is to the second: So shall the two
Radicall sides be, one to the other, of any two Cubes: which, in their
Soliditie, shall haue the same proportion, which was at the first
assigned: were it Rationall or Irrationall.
Thus, in sundry waies you may furnishe your selfe with such straunge and
profitable matter: which, long hath bene wished for. And though it be
Naturally done and Mechanically: yet hath it a good Demonstration
Mathematicall.
[=The demonstrations of this Dubbling of the Cube, and of the
rest.=]
Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones,
in the proportions assigned: and like Pyramids or Cones, are in
proportion, one to the other, in the proportion of their Homologall
sides (or lines) tripled. Wherefore, if to the first, and second lines,
found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in
continuall proportion: that fourth line, shall be to the first, as the
greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of
Euclide. If Pyramis to Pyramis, or Cone to Cone, be double,
[I. D.
= * Hereby, helpe your self to become a præcise practiser.
And so consider, how, nothing at all, you are hindred
(sensibly) by the Conuexitie of the water.=]
then shall * Line to Line, be also double, &c. But, as our first line,
is to the second, so is the Radicall side of our Fundamentall Cube, to
the Radicall side of the Cube to be made, or to be doubled: and
therefore, to those twaine also, a third and a fourth line, in
continuall proportion, ioyned: will geue the fourth line in that
proportion to the first, as our fourth Pyramidall, or Conike line, was
to his first: but that was double, or treble, &c. as the Pyramids or
Cones were, one to an other (as we haue proued) therfore, this fourth,
shalbe also double or treble to the first, as the Pyramids or Cones were
one to an other: But our made Cube, is described of the second in
proportion, of the fower proportionall lines:
[= * By the 33. of the eleuenth booke of Euclide.=]
therfore * as the fourth line, is to the first, so is that Cube, to the
first Cube: and we haue proued the fourth line, to be to the first, as
the Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to
the Cube, as Pyramis is to Pyramis, or Cone is to Cone.
[I. D.
= * And your diligence in practise, can so (in waight of
water) performe it: Therefore, now, you are able to geue
good reason of your whole doing.=]
But we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or
treble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was
to be demonstrated. And of the Parallelipipedõ, it is euidẽt, that the
water Solide Parallelipipedons, are one to the other, as their heithes
are, seing they haue one base. Wherfore the Pyramids or Cones, made of
those water Parallelipipedons, are one to the other, as the lines are
(one to the other) betwene which, our proportion was assigned. But the
Cubes made of lines, after the proportiõ of the Pyramidal or Conik
_homologall_ lines, are one to the other, as the Pyramides or Cones are,
one to the other (as we before did proue) therfore, the Cubes made,
shalbe one to the other, as the lines assigned, are one to the other:
Which was to be demonstrated. Note.
[* _Note this Corollary._]
* This, my Demonstratiõ is more generall, then onely in Square Pyramis
or Cone: Consider well. Thus, haue I, both Mathematically and
Mechanically, ben very long in wordes: yet (I trust) nothing tedious to
them, who, to these thinges, are well affected. And verily I am forced
(auoiding prolixitie) to omit sundry such things, easie to be practised:
which to the Mathematicien, would be a great Threasure: and to the
Mechanicien, no small gaine.
[* The great Commodities following of these new Inuentions.]
* Now may you, +Betwene two lines giuen, finde two middle proportionals,
in Continuall proportion: by the hollow Parallelipipedon, and the hollow
Pyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre
right lines may be found, proportionall in any proportion assigned, of
which, shal be produced a Parallelipipedon, æquall to the
Parallelipipedon giuen. Hereof, I noted somwhat, vpon the 36.
proposition, of the 11. boke of _Euclide_. Now, all those thinges, which
_Vitruuius_ in his Architecture, specified hable to be done, by dubbling
of the Cube: Or, by finding of two middle proportionall lines, betwene
two lines giuen, may easely be performed. Now, that Probleme, which I
noted vnto you, in the end of my Addition, vpon the 34. of the 11. boke
of _Euclide_, is proued possible. Now, may any regular body, be
Transformed into an other, &c. Now, any regular body: any Sphere, yea
any Mixt Solid: and (that more is) Irregular Solides, may be made (in
any proportiõ assigned) like vnto the body, first giuen. Thus, of a
_Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_,
may a Giant be made: and that, with any gesture, by the Manneken vsed:
and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make
one, of the same Mould (in any assigned proportion) bigger or lesser.
[* ☞]
Now, may you, of any * Gunne, or little peece of ordinaũce, make an
other, with the same _Symmetrie_ (in all pointes) as great, and as
little, as you will. Marke that: and thinke on it. Infinitely, +may you
apply this, so long sought for, and now so easily concluded: and
withall, so willingly and frankly communicated to such, as faithfully
deale with vertuous studies.+
[Such is the Fruite of the Mathematicall Sciences and Artes.]
Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte:
and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he
can, by order, Descend, to frame Naturall thinges, to wonderfull vses:
and when he list, retire home into his owne Centre: and there, prepare
more Meanes, to Ascend or Descend by: and, all, to the glory of God, and
our honest delectation in earth.
Although, the Printer, hath looked for this Præface, a day or two, yet
could I not bring my pen from the paper, before I had giuen you
comfortable warning, and brief instructions, of some of the Commodities,
by _Statike_, hable to be reaped: In the rest, I will therfore, be as
brief, as it is possible: and with all, describing them, somwhat
accordingly. And that, you shall perceiue, by this, which in order
commeth next. For, wheras, it is so ample and wonderfull, that, an whole
yeare long, one might finde fruitfull matter therin, to speake of: and
also in practise, is a Threasure endeles: yet will I glanse ouer it,
with wordes very few.
This do I call +‡Anthropographie‡+. Which is an Art restored, and of my
preferment to your Seruice. I pray you, thinke of it, as of one of the
chief pointes, of Humane knowledge. Although it be, but now, first
Cõfirmed, with this new name: yet the matter, hath from the beginning,
ben in consideration of all perfect Philosophers. +Anthropographie, is
the description of the Number, Measure, Waight, figure, Situation, and
colour of euery diuerse thing, conteyned in the perfect body of MAN:
with certain knowledge of the Symmetrie, figure, waight,
Characterization, and due locall motion, of any parcell of the sayd
body, assigned: and of Nũbers, to the sayd parcell appertainyng.+ This,
is the one part of the Definition, mete for this place: Sufficient to
notifie, the particularitie, and excellency of the Arte: and why it is,
here, ascribed to the Mathematicals. Yf the description of the heauenly
part of the world, had a peculier Art, called _Astronomie:_ If the
description of the earthly Globe, hath his peculier arte, called
_Geographie_. If the Matching of both, hath his peculier Arte, called
_Cosmographie:_ Which is the Descriptiõ of the whole, and vniuersall
frame of the world: Why should not the description of
[MAN is the Lesse World.]
him, who is the Lesse world: and, frõ the beginning, called
_Microcosmus_ (that is. _The Lesse World._) And for whose sake, and
seruice, all bodily creatures els, were created: Who, also,
participateth with Spirites, and Angels: and is made to the Image and
similitude of _God_: haue his peculier Art? and be called the _Arte of
Artes_: rather, then, either to want a name, or to haue to base and
impropre a name? You must of sundry professions, borow or challenge
home, peculier partes hereof: and farder procede: as, God, Nature,
Reason and Experience shall informe you. The Anatomistes will restore to
you, some part: The Physiognomistes, some: The Chyromantistes some. The
Metaposcopistes, some: The excellent, _Albert Durer_, a good part: the
Arte of Perspectiue, will somwhat, for the Eye, helpe forward:
_Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other
(in certaine thinges) will be Contributaries. And farder, the Heauen,
the Earth, and all other Creatures, will eche shew, and offer their
Harmonious seruice, to fill vp, that, which wanteth hereof: and with
your own Experience, concluding: you may Methodically register the
whole, for the posteritie: Whereby, good profe will be had, of our
Harmonious, and
[Micro Cosmus.]
Microcosmicall constitution.
[* ☞]
The outward Image, and vew hereof: to the Art of _Zographie_ and
Painting, to Sculpture, and Architecture: (for Church, House, Fort, or
Ship) is most necessary and profitable: for that, it is the chiefe base
and foundation of them.
[* Lib. 3. Cap. 1.]
Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, or
no. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke
in the 27. and 28. Chapters, of the second booke, _De occulta
Philosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther.
Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so
long agoe pronounced: of so many a Philosopher repeated: and of the
_Wisest_ attempted: And then, you will perceaue, how long agoe, you haue
bene called to the Schole, where this Arte might be learned. Well. I am
nothing affrayde, of the disdayne of some such, as thinke Sciences and
Artes, to be but Seuen. Perhaps, those Such, may, with ignorance, and
shame enough, come short of them Seuen also: and yet neuerthelesse they
can not prescribe a certaine number of Artes: and in eche, certaine
vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly
rise vp: and there was no such order taken, that,
[☞]
All Artes, should in one age, or in one land, or of one man, be made
knowen to the world. Let vs embrace the giftes of God, and wayes to
wisedome, in this time of grace, from aboue, continually bestowed on
them, who thankefully will receiue them: _Et bonis Omnia Cooperabuntur
in bonum._
+‡Trochilike,‡ is that Art Mathematicall, which demonstrateth the
properties of all Circular motions, Simple and Compounde.+ And bycause
the frute hereof, vulgarly receiued, is in Wheles, it hath the name of
_Trochilike:_ as a man would say, _Whele Art_. By this art, a Whele may
be geuen which shall moue ones about, in any tyme assigned. Two Wheles
may be giuen, whose turnynges about in one and the same tyme, (or equall
tymes), shall haue, one to the other, any proportion appointed. By
Wheles, may a straight line be described: Likewise, a Spirall line in
plaine, Conicall Section lines, and other Irregular lines, at pleasure,
may be drawen. These, and such like, are principall Conclusions of this
Arte: and helpe forward many pleasant and profitable Mechanicall workes:
[Saw Milles.]
As Milles, to Saw great and very long Deale bordes, no man being by.
Such haue I seene in Germany: and in the Citie of Prage: in the kingdome
of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all
maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight,
Spring, Man or Beast, moued. Take in your hand, _Agricola De re
Metallica:_ and then shall you (in all Mines) perceaue, how great nede
is, of Whele worke. By Wheles, straunge workes and incredible, are done:
as will, in other Artes hereafter, appeare. A wonderfull example of
farther possibilitie, and present commoditie, was sene in my time, in a
certaine Instrument: which by the Inuenter and Artificer (before) was
solde for xx. Talentes of Golde: and then had (by misfortune) receaued
some iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the
same, and presented it vnto the Emperour _Charles_ the fifth.
_Hieronymus Cardanus_, can be my witnesse, that therein, was one Whele,
which moued, and that, in such rate, that, in 7000. yeares onely, his
owne periode should be finished. A thing almost incredible: But how
farre, I keepe me within my boundes: very many men (yet aliue) can tell.
+‡Helicosophie‡+, is nere Sister to _Trochilike:_ and is, +An Arte
Mathematicall, which demonstrateth the designing of all Spirall lines in
Plaine, on Cylinder, Cone, Sphære, Conoid, and Sphæroid, and their
properties appertayning.+ The vse hereof, in _Architecture_, and diuerse
Instrumentes and Engines, is most necessary. For, in many thinges, the
Skrue worketh the feate, which, els, could not be performed. By helpe
hereof,
[* Atheneus Lib. 5. cap. 8.]
it is * recorded, that, where all the power of the Citie of Syracusa,
was not hable to moue a certaine Ship (being on ground) mightie
_Archimedes_, setting to, his Skruish Engine, caused _Hiero_ the king,
by him self, at ease, to remoue her, as he would.
[Proclus. Pag. 18.]
Wherat, the King wondring: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει
λέγοντι πιϛευτεόν. _From this day, forward_ (said the King) _Credit
ought to be giuen to Archimedes, what soeuer he sayth._
+‡Pneumatithmie‡ demonstrateth by close hollow Geometricall Figures,
(regular and irregular) the straunge properties (in motion or stay) of
the Water, Ayre, Smoke, and Fire, in theyr cõtinuitie, and as they are
ioyned to the Elementes next them.+ This Arte, to the Naturall
Philosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_
is not in the world. And that, all Nature, abhorreth it so much: that,
contrary to ordinary law, the Elementes will moue or stand. As, Water to
ascend: rather then betwene him and Ayre, Space or place should be left,
more then (naturally) that quãtitie of Ayre requireth, or can fill.
Againe, Water to hang, and not descend: rather then by descending, to
leaue Emptines at his backe. The like, is of Fire and Ayre: they will
descend: when, either, their Cõtinuitie should be dissolued: or their
next Element forced from them. And as they will not be extended, to
discontinuitie: So, will they not, nor yet of mans force, can be prest
or pent, in space, not sufficient and aunswerable to their bodily
substance. Great force and violence will they vse, to enioy their
naturall right and libertie.
[To go to the bottom of the Sea without daunger.]
Hereupon, two or three men together, by keping Ayre vnder a great
Cauldron, and forcyng the same downe, orderly, may without harme descend
to the Sea bottome: and continue there a tyme &c. Where, Note, how the
thicker Element (as the Water) giueth place to the thynner (as, is the
ayre:) and receiueth violence of the thinner, in maner. &c. Pumps and
all maner of Bellowes, haue their ground of this Art: and many other
straunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of this
Feat, (called commonly _Pneumatica_,) goodly workes are extant, both in
Greke, and Latin. With old and learned Schole men, it is called
_Scientia de pleno & vacuo._
+‡Menadrie‡, is an Arte Mathematicall, which demonstrateth, how, aboue
Natures vertue and power simple: Vertue and force may be multiplied: and
so, to direct, to lift, to pull to, and to put or cast fro, any
multiplied or simple, determined Vertue, Waight or Force: naturally,
not, so, directible or moueable.+ Very much is this Art furdred by other
Artes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: in
some, by _Trochilike_: and in other, by _Helicosophie_: and
_Pneumatithmie_. By this Art, all Cranes, Gybbettes, & Ingines to lift
vp, or to force any thing, any maner way, are ordred: and the certaine
cause of their force, is knowne: As, the force which one man hath with
the Duche waghen Racke: therwith, to set vp agayne, a mighty waghen
laden, being ouerthrowne. The force of the Crossebow Racke, is
certainly, here, demonstrated. The reason, why one mã, doth with a
leauer, lift that, which Sixe men, with their handes onely, could not,
so easily do. By this Arte, in our common Cranes in London, where powre
is to Crane vp, the waight of 2000. pound: by two Wheles more (by good
order added) Arte concludeth, that there may be Craned vp 200000. pound
waight &c. So well knew _Archimedes_ this Arte: that he alone, with his
deuises and engynes, (twise or thrise) spoyled and discomfited the whole
Army and Hoste of the Romaines, besieging _Syracusa_,
[=Plutarchus in Marco Marcello.=]
_Marcus Marcellus the Consul_, being their Generall Capitaine.
[=Synesius in Epistolis.=]
Such huge Stones, so many, with such force, and so farre, did he with
his engynes hayle among them, out of the Citie.
[=Polybius.=]
[=Plinius.=]
[=Quintilianus.=]
[=T. Liuius.=]
And by Sea likewise: though their Ships might come to the walls of
_Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with his
mighty Stones hurlyng:
[=* Athenæus.=]
what with Pikes of * 18 fote long, made like shaftes: which he forced
almost a quarter of a myle. What, with his catchyng hold of their Shyps,
and hoysing them vp aboue the water, and suddenly letting them fall into
the Sea againe:
[= * Galenus.=]
[=Anthemius.=]
what with his * Burning Glasses: by which he fired their other Shippes a
far-of: what, with his other pollicies, deuises, and engines, he so
manfully acquit him selfe: that all the Force, courage, and pollicie of
the Romaines (for a great season) could nothing preuaile, for the
winning of Syracusa. Wherupon, the Romanes named _Archimedes_,
_Briareus_, and _Centimanus_. _Zonaras_ maketh mention of one _Proclus_,
who so well had perceiued _Archimedes_ Arte of _Menadrie_, and had so
well inuented of his owne, that with his Burning Glasses,
[Burning Glasses.]
being placed vpon the walles of Bysance, he multiplied so the heate of
the Sunne, and directed the beames of the same against his enemies Nauie
with such force, and so sodeinly (like lightening) that he burned and
destroyed both man and ship. And _Dion_specifieth of _Priscus_,
a _Geometricien_ in Bysance, who inuented and vsed sondry Engins, of
Force multiplied: Which was cause, that the _Emperour Seuerus_ pardoned
him, his life, after he had wonne Bysance: Bycause he honored the Arte,
wytt, and rare industrie of _Priscus_. But nothing inferior to the
inuention of these engines of Force, was the inuention of Gunnes.
[Gunnes.]
Which, from an English man, had the occasion and order of first
inuenting: though in an other land, and by other men, it was first
executed. And they that should see the record, where the occasion and
order generall, of Gunning, is first discoursed of, would thinke: that,
“small thinges, slight, and cõmon: comming to wise mens consideration,
and industrious mens handling, may grow to be of force incredible.”