by new deuise, but by good reason and authoritie: for, +Astrologie, is
an Arte Mathematicall, which reasonably demonstrateth the operations and
effectes, of the naturall beames, of light, and secrete influence: of
the Sterres and Planets: in euery element and elementall body: at all
times, in any Horizon assigned.+ This Arte is furnished with many other
great Artes and experiences: As with perfecte _Perspectiue_,
_Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4.
Elementes, the Arte of Graduation, and some good vnderstãding in
_Musike_: and yet moreouer, with an other great Arte, hereafter
following, though I, here, set this before, for some considerations me
mouing. Sufficient (you see) is the stuffe, to make this rare and
secrete Arte, of: and hard enough to frame to the Conclusion
Syllogisticall. Yet both the manifolde and continuall trauailes of the
most auncient and wise Philosophers, for the atteyning of this Arte: and
by examples of effectes, to confirme the same: hath left vnto vs
sufficient proufe and witnesse: and we, also, daily may perceaue, That
mans body, and all other Elementall bodies, are altered, disposed,
ordred, pleasured, and displeasured, by the Influentiall working of the
_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth
_Aristotle_, in the first of his _Meteorologicall_ bookes, in the second
Chapter: _Est autem necessariò Mundus iste, supernis lationibus ferè
continuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà prima
putanda omnibus est, vnde motus principium existit._ That is: +_This
=[Elementall]= World is of necessitie, almost, next adioyning, to the
heauenly motions: That, from thence, all his vertue or force may be
gouerned. For, that is to be thought the first Cause vnto all: from
which, the beginning of motion, is._+ And againe, in the tenth Chapter.
_Oportet igitur & horum principia sumamus, & causas omnium similiter.
Principium igitur vt mouens, præcipuum[que] & omnium primum, Circulus
ille est, in quo manifeste Solis latio, &c._ And so forth. His
_Meteorologicall_ bookes, are full of argumentes, and effectuall
demonstrations, of the vertue, operation, and power of the heauenly
bodies, in and vpon the fower Elementes, and other bodies, of them
(either perfectly, or vnperfectly) composed. And in his second booke,
_De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & prima
latio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea
nam[que] & continua est, & duobus motibus fit:_ In Englishe, thus.
+_Wherefore the vppermost motion, is not the cause of Generation and
Corruption, but the motion of the Zodiake: for, that, both, is
continuall, and is caused of two mouinges._+ And in his second booke,
and second Chapter of hys _Physikes_. _Homo nam[que] generat hominem,
at[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans
generation._+ Authorities may be brought, very many: both of 1000. 2000.
yea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_,
_Wise_, and godly men, for that Conclusion: which, daily and hourely, we
men, may discerne and perceaue by sense and reason: All beastes do
feele, and simply shew, by their actions and passions, outward and
inward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally,
the Elementes, and all thinges of the Elementes composed, do geue
Testimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions,
vertues, and naturall motions, depend of the Actiuitie of the heauenly
motions and Influences. Whereby, beside the specificall order and forme,
due to euery seede: and beside the Nature, propre to the Indiuiduall
Matrix, of the thing produced: What shall be the heauenly Impression,
the perfect and circumspecte Astrologien hath to Conclude._+ Not onely
(by _Apotelesmes_) τὸ ὁτὶ]. but by Naturall and Mathematicall
demonstration τὸ διότι. Whereunto, what Sciences are requisite (without
exception) I partly haue here warned: And in my _Propædeumes_ (besides
other matter there disclosed) I haue Mathematically furnished vp the
whole Method: To this our age, not so carefully handled by any, that
euer I saw, or heard of. I was,
[* Anno. 1548 and 1549. in Louayn.]
(for * 21. yeares ago) by certaine earnest disputations, of the Learned
_Gerardus Mercator_, and _Antonius Gogaua_, (and other,) therto so
prouoked: and (by my constant and inuincible zeale to the veritie) in
obseruations of Heauenly Influencies (to the Minute of time,) than, so
diligent: And chiefly by the Supernaturall influence, from the Starre of
Iacob, so directed: That any Modest and Sober Student, carefully and
diligently seking for the Truth, will both finde & cõfesse, therin, to
be the Veritie, of these my wordes: And also become a Reasonable
Reformer, of three Sortes of people: about these Influentiall
Operations, greatly erring from the truth.
[Note.]
Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and
the third +Light Practisers+. The first, & most cõmon Sort, thinke the
Heauen and Sterres, to be answerable to any their doutes or desires:
[1.]
which is not so: and, in dede, they, to much, ouer reache. The Second
sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare
any Sway in Generation
[2.]
and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and
_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so
farre in distance, so manifold in their motions, so constant in their
periodes. &c.) they assigne a sleight, simple office or two, and so
allow vnto thẽ (according to their capacities) as much vertue, and power
Influentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres,
hanged vp (for Signes) in London, for distinction of houses, & such
grosse helpes, in our worldly affaires: And they vnderstand not (or will
not vnderstand) of the other workinges, and vertues of the Heauenly
_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband
man: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as
the Porpentine doth: nor will allow these perfect, and incorruptible
mighty bodies, so much vertuall Radiation, & Force, as they see in a
litle peece of a _Magnes stone_: which, at great distance, sheweth his
operation. And perchaunce they thinke, the Sea & Riuers (as the Thames)
to be some quicke thing, and so to ebbe, and flow, run in and out, of
them selues, at their owne fantasies. God helpe, God helpe. Surely,
these men, come to short: and either are to dull: or willfully blind:
or, perhaps, to malicious. The third man, is the common and vulgare
_Astrologien_, or Practiser: who, being not duely, artificially, and
perfectly
[3.]
furnished: yet, either for vaine glory, or gayne: or like a simple dolt,
& blinde Bayard, both in matter and maner, erreth: to the discredit of
the _Wary_, and modest _Astrologien_: and to the robbing of those most
noble corporall Creatures, of their Naturall Vertue: being most mighty:
most beneficiall to all elementall Generation, Corruption and the
appartenances: and most Harmonious in their Monarchie: For which
thinges, being knowen, and modestly vsed: we might highly, and
continually glorifie God, with the princely Prophet, saying. +_The
Heauens declare the Glorie of God: who made the Heauẽs in his wisedome:
who made the Sonne, for to haue dominion of the day: the Mone and
Sterres to haue dominion of the nyght: whereby, Day to day vttereth
talke: and night, to night declareth knowledge. Prayse him, all ye
Sterres, and Light. Amen._+
In order, now foloweth, of +‡Statike‡+, somewhat to say, what we meane
by that name: and what commodity, doth, on such Art, depend. +Statike,
is an Arte Mathematicall, which demonstrateth the causes of heauynes,
and lightnes of all thynges: and of motions and properties, to heauynes
and lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance
(as the chief sensible Instrument,) Experience of these demonstrations
may be had: we call this Art, _Statike:_ that is, _the Experimentes of
the Balance_. Oh, that men wist, what proffit, (all maner of wayes) by
this Arte might grow, to the hable examiner, and diligent practiser.
“Thou onely, knowest all thinges precisely (O God) who hast made weight
and Balance, thy Iudgement: who hast created all thinges in _Number,
Waight, and Measure_: and hast wayed the mountaines and hils in a
Balance: who hast peysed in thy hand, both Heauen and earth. We therfore
warned by the Sacred word, to Consider thy Creatures: and by that
consideration, to wynne a glyms (as it were,) or shaddow of
perceiuerance, that thy wisedome, might, and goodnes is infinite, and
vnspeakable, in thy Creatures declared: And being farder aduertised, by
thy mercifull goodnes, that, three principall wayes, were, of the, vsed
in Creation of all thy Creatures, namely, _Number_, _Waight_ and
_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes
(auncient, famous, and to humaine vses most necessary,) are, all ready,
sufficiently knowen and extant: This third key, we beseche thee (through
thy accustomed goodnes,) that it may come to the nedefull and sufficient
knowledge, of such thy Seruauntes, as in thy workemanship, would gladly
finde, thy true occasions (purposely of the vsed) whereby we should
glorifie thy name, and shew forth (to the weaklinges in faith) thy
wondrous wisedome and Goodnes. Amen.”
Meruaile nothing at this pang (godly frend, you Gentle and zelous
Student.) An other day, perchaunce, you will perceiue, what occasion
moued me. Here, as now, I will giue you some ground, and withall some
shew, of certaine commodities, by this Arte arising. And bycause this
Arte is rare, my wordes and practises might be to darke: vnleast you had
some light, holden before the matter: and that, best will be, in giuing
you, out of _Archimedes_ demonstrations, a few principal Conclusions, as
foloweth.
+1.+
+The Superficies of euery Liquor, by it selfe consistyng, and in
quyet, is Sphæricall: the centre whereof, is the same, which is the
centre of the Earth.+
+2.+
+If Solide Magnitudes, being of the same bignes, or quãtitie, that
any Liquor is, and hauyng also the same Waight: be let downe into
the same Liquor, they will settle downeward, so, that no parte of
them, shall be aboue the Superficies of the Liquor: and yet
neuertheles, they will not sinke vtterly downe, or drowne.+
+3.+
+If any Solide Magnitude beyng Lighter then a Liquor, be let downe
into the same Liquor, it will settle downe, so farre into the same
Liquor, that so great a quantitie of that Liquor, as is the parte of
the Solid Magnitude, settled downe into the same Liquor: is in
Waight, æquall, to the waight of the whole Solid Magnitude.+
+4.+
+Any Solide Magnitude, Lighter then a Liquor, forced downe into the
same Liquor, will moue vpward, with so great a power, by how much,
the Liquor hauyng æquall quantitie to the whole Magnitude, is
heauyer then the same Magnitude.+
+5.+
+Any Solid Magnitude, heauyer then a Liquor, beyng let downe into
the same Liquor, will sinke downe vtterly: And wilbe in that Liquor,
Lighter by so much, as is the waight or heauynes of the Liquor,
hauing bygnes or quantitie, æquall to the Solid Magnitude.+
+6.+
[I. D.
The Cutting of a Sphære according to any proportion assigned
may by this proposition be done Mechanically by tempering
Liquor to a certayne waight in respect of the waight of the
Sphære therein Swymming.]
+If any Solide Magnitude, Lighter then a Liquor, be let downe into
the same Liquor, the waight of the same Magnitude, will be, to the
Waight of the Liquor. (Which is æquall in quantitie to the whole
Magnitude,) in that proportion, that the parte, of the Magnitude
settled downe, is to the whole Magnitude.+
By these verities, great Errors may be reformed, in Opinion of the
Naturall Motion of thinges, Light and Heauy. Which errors, are in
Naturall Philosophie (almost) of all mẽ allowed: to much trusting to
Authority: and false Suppositions. As, +Of any two bodyes, the heauyer,
to moue downward faster then the lighter.+
[A common error, noted.]
This error, is not first by me, Noted: but by one _Iohn Baptist de
Benedictis_. The chief of his propositions, is this: which seemeth a
Paradox.
+If there be two bodyes of one forme, and of one kynde, æquall in
quantitie or vnæquall,
[A paradox.]
they will moue by æquall space, in æquall tyme: So that both theyr
mouynges be in ayre, or both in water: or in any one Middle.+
Hereupon, in the feate of +Gunnyng+,
[N. T.]
certaine good discourses (otherwise) may receiue great amendement, and
furderance.
[The wonderfull vse of these Propositions.]
In the entended purpose, also, allowing somwhat to the imperfection of
Nature: not aunswerable to the precisenes of demonstration. Moreouer, by
the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth,
the Fire, may be nerely, knowen, how light or heauy they are (Naturally)
in their assigned partes: or in the whole. And then, to thinges
Elementall, turning your practise: you may deale for the proportion of
the Elementes, in the thinges Compounded. Then, to the proportions of
the Humours in Man: their waightes: and the waight of his bones, and
flesh. &c. Than, by waight, to haue consideration of the Force of man,
any maner of way: in whole or in part. Then, may you, of Ships water
drawing, diuersly, in the Sea and in fresh water, haue pleasant
consideration: and of waying vp of any thing, sonken in Sea or in fresh
water &c. And (to lift vp your head a loft:) by waight, you may, as
precisely, as by any instrument els, measure the Diameters of _Sonne_
and _Mone. &c._ Frende, I pray you, way these thinges, with the iust
Balance of Reason. And you will finde Meruailes vpon Meruailes: And
esteme one Drop of Truth (yea in Naturall Philosophie) more worth, then
whole Libraries of Opinions, vndemonstrated: or not aunswering to
Natures Law, and your experience. Leauing these thinges, thus: I will
giue you two or three, light practises, to great purpose: and so finish
my Annotation _Staticall_. In Mathematicall matters, by the Mechaniciens
ayde, we will behold, here, the Commodity of waight.
[The practise Staticall, to know the proportion, betwene
the Cube, and the Sphære.]
Make a Cube, of any one Vniforme: and through like heauy stuffe: of the
same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to
the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne,
Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and like
waight throughout.) And you may, by Say Balance, haue prepared a great
number of the smallest waightes: which, by those Balance can be
discerned or tryed: and so, haue proceded to make you a perfect Pyle,
company & Number of waightes: to the waight of six, eight, or twelue
pound waight: most diligently tryed, all. And of euery one, the Content
knowen, in your least waight, that is wayable. [They that can not haue
these waightes of precisenes: may, by Sand, Vniforme, and well dusted,
make them a number of waightes, somewhat nere precisenes: by halfing
euer the Sand: they shall, at length, come to a least common waight.
Therein, I leaue the farder matter, to their discretion, whom nede shall
pinche.] The _Venetians_ consideration of waight, may seme precise
enough: by eight descentes progressionall, * halfing, from a grayne.
[I. D.
* For, so, haue you .256. partes of a Graine.]
Your Cube, Sphære, apt Balance, and conuenient waightes, being ready:
fall to worke.❉. First, way your Cube. Note the Number of the waight.
Way, after that, your Sphære. Note likewise, the Nũber of the waight. If
you now find the waight of your Cube, to be to the waight of the Sphære,
as 21. is to 11: Then you see, how the Mechanicien and _Experimenter_,
without Geometrie and Demonstration, are (as nerely in effect) tought
the proportion of the Cube to the Sphere: as I haue demonstrated it, in
the end of the twelfth boke of _Euclide_. Often, try with the same Cube
and Sphære. Then, chaunge, your Sphære and Cube, to an other matter: or
to an other bignes: till you haue made a perfect vniuersall Experience
of it. Possible it is, that you shall wynne to nerer termes, in the
proportion.
When you haue found this one certaine Drop of Naturall veritie, procede
on, to Inferre, and duely to make assay, of matter depending. As,
bycause it is well demonstrated, that a Cylinder, whose heith, and
Diameter of his base, is æquall to the Diameter of the Sphære, is
Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of
the waight of the Sphære, adde halfe so much, as it is: and so haue you
the number of the waight of that Cylinder. Which is also Comprehended of
our former Cube: So, that the base of that Cylinder, is a Circle
described in the Square, which is the base of our Cube. But the Cube and
the Cylinder, being both of one heith, haue their Bases in the same
proportion, in the which, they are, one to an other, in their Massines
or Soliditie. But, before, we haue two numbers, expressing their
Massines, Solidities, and Quantities, by waight: wherfore,
[* =The proportion of the Square to the Circle inscribed.=]
we haue * the proportion of the Square, to the Circle, inscribed in the
same Square. And so are we fallen into the knowledge sensible, and
Experimentall of _Archimedes_ great Secret: of him, by great trauaile of
minde, sought and found. Wherfore, to any Circle giuen, you can giue a
Square æquall:
[* =The Squaring of the Circle, Mechanically.=]
* as I haue taught, in my Annotation, vpon the first proposition of the
twelfth boke, And likewise, to any Square giuen, you may giue a Circle
æquall:
[* =To any Square geuen, to geue a Circle, equall.=]
* If you describe a Circle, which shall be in that proportion, to your
Circle inscribed, as the Square is to the same Circle: This, you may do,
by my Annotations, vpon the second proposition of the twelfth boke of
_Euclide_, in my third Probleme there. Your diligence may come to a
proportion, of the Square to the Circle inscribed, nerer the truth, then
is the proportion of 14. to 11. And consider, that you may begyn at the
Circle and Square, and so come to conclude of the Sphære, & the Cube,
what their proportion is: as now, you came from the Sphære to the
Circle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough
one hole drawẽ, as the maner is) if you make a Square figure & way it:
and then, describing theron, the Circle inscribed: & cut of, & file
away, precisely (to the Circle) the ouerplus of the Square: you shall
then, waying your Circle, see, whether the waight of the Square, be to
your Circle, as 14. to 11. As I haue Noted, in the beginning of
_Euclides_ twelfth boke. &c. after this resort to my last proposition,
vpon the last of the twelfth. And there, helpe your selfe, to the end.
And, here, Note this, by the way.
[Note Squaring of the Circle without knowledge of the
proportion betwene Circumference and Diameter.]
That we may Square the Circle, without hauing knowledge of the
proportion, of the Circumference to the Diameter: as you haue here
perceiued. And otherwayes also, I can demonstrate it. So that, many haue
cumberd them selues superfluously, by trauailing in that point first,
which was not of necessitie, first: and also very intricate. And easily,
you may, (and that diuersly) come to the knowledge of the Circumference:
the Circles Quantitie, being first knowen. Which thing, I leaue to your
consideration: making hast to despatch an other Magistrall Probleme: and
to bring it, nerer to your knowledge, and readier dealing with, then the
world (before this day,) had it for you, that I can tell of. And that
is, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done:
[To Dubble the Cube redily: by Art Mechanicall: depending
vppon Demonstration Mathematicall.]
+Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or
a Cone: perfectly fashioned in the holow, within. Wherin, let great
diligence be vsed, to approche (as nere as may be) to the Mathematicall
perfection of those figures. At their bases, let them be all open: euery
where, els, most close, and iust to. From the vertex, to the
Circumference of the base of the Cone: & to the sides of the base of the
Pyramis:+
[=I. D.=
=The 4. sides of this Pyramis must be 4. Isosceles
Triangles alike and æquall.=]
+Let 4. straight lines be drawen, in the inside of the Cone and Pyramis:
makyng at their fall, on the perimeters of the bases, equall angles on
both sides them selues, with the sayd perimeters. These 4. lines (in the
Pyramis: and as many, in the Cone) diuide: one, in 12. æquall partes:
and an other, in 24. an other, in 60, and an other, in 100. (reckenyng
vp from the vertex.) Or vse other numbers of diuision, as experience
shall teach you.+
[=I. D.=
=* In all workinges with this Pyramis or Cone, Let their
Situations be in all Pointes and Conditions, alike,
or all one: while you are about one Worke. Els you
will erre.=]
+Then, * set your Cone or Pyramis, with the vertex downward,
perpendicularly, in respect of the Base. (Though it be otherwayes, it
hindreth nothyng.) So let thẽ most stedily be stayed.+ Now, if there be
a Cube, which you wold haue Dubbled. Make you a prety Cube of Copper,
Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or
Cubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so
proportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone,
will be hable to conteine the waight of them, in water, 3. or 4. times:
at the least: what stuff so euer they be made of. Let not your Solid
angle, at the vertex, be to sharpe: but that the water may come with
ease, to the very vertex, of your hollow Cone or Pyramis. Put one of
your Solid Cubes in a Balance apt: take the waight therof exactly in
water. Powre that water, (without losse) into the hollow Pyramis or
Cone, quietly. Marke in your lines, what numbers the water Cutteth: Take
the waight of the same Cube againe: in the same kinde of water, which
you had before:
[=I. D.=
=* Consider well whan you must put your waters togyther:
and whan, you must empty your first water, out of your
Pyramis or Cone. Els you will erre.=]
put that* also, into the Pyramis or Cone, where you did put the first.
Marke now againe, in what number or place of the lines, the water
Cutteth them. Two wayes you may conclude your purpose: it is to wete,
either by numbers or lines. By numbers: as, if you diuide the side of
your Fundamentall Cube into so many æquall partes, as it is capable of,
conueniently, with your ease, and precisenes of the diuision. For, as
the number of your first and lesse line (in your hollow Pyramis or
Cone,) is to the second or greater (both being counted from the vertex)
so shall the number of the side of your Fundamentall Cube, be to the
nũber belonging to the Radicall side, of the Cube, dubble to your
Fundamentall Cube: Which being multiplied Cubik wise, will sone shew it
selfe, whether it be dubble or no, to the Cubik number of your
Fundamentall Cube. By lines, thus: As your lesse and first line, (in
your hollow Pyramis or Cone,) is to the second or greater, so let the
Radical side of your Fundamẽtall Cube, be to a fourth proportionall
line, by the 12. proposition, of the sixth boke of _Euclide_. Which
fourth line, shall be the Rote Cubik, or Radicall side of the Cube,
dubble to your Fundamentall Cube: which is the thing we desired.
[☞ God be thanked for this Inuention, & the fruite ensuing.]
For this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking the
holy and glorious Trinity: hauing greater cause therto, then
[* Vitruuius. Lib. 9. Cap. 3.]
* _Archimedes_ had (for finding the fraude vsed in the Kinges Crowne, of
Gold): as all men may easily Iudge: by the diuersitie of the frute
following of the one, and the other. Where I spake before, of a hollow
Cubik Coffen: the like vse, is of it: and without waight. Thus. Fill it
with water, precisely full, and poure that water into your Pyramis or
Cone. And here note the lines cutting in your Pyramis or Cone. Againe,
fill your coffen, like as you did before. Put that Water, also, to the
first. Marke the second cutting of your lines. Now, as you proceded
before, so must you here procede.
[* Note.]
* And if the Cube, which you should Double, be neuer so great: you haue,
thus, the proportion (in small) betwene your two litle Cubes: And then,
the side, of that great Cube (to be doubled) being the third, will haue
the fourth, found, to it proportionall: by the 12. of the sixth of
Euclide.
[Note, as concerning the Sphæricall Superficies of the Water.]
Note, that all this while, I forget not my first Proposition Staticall,
here rehearsed: that, the Superficies of the water, is Sphæricall.
Wherein, vse your discretion: to the first line, adding a small heare
breadth, more: and to the second, halfe a heare breadth more, to his
length. For, you will easily perceaue, that the difference can be no
greater, in any Pyramis or Cone, of you to be handled. Which you shall
thus trye. _For finding the swelling of the water aboue leuell._
[☞]