sci. rev. reader (‘02/03/07) 02-p5 dee - harvard...

Sci. Rev. Reader (‘02/03/07) 02-P5_Dee

1

*Preliminary draft for student use only. Not for citation or circulation without permission of edi-

tor.

There is (gentle Reader) nothing (the word of God only set apart) which so much beau-

tifieth and adorneth the soul and mind of man, as doth the knowledge of good arts and

sciences: as the knowledge of natural and moral Philosophy.

- Billingsley, ‘Translator to the Reader’

The Elements of Geometry of Euclid of Megara1

1 Billingsley and Dee were mistaken in calling Euclid of the Elements ‘Euclid of Megara’. Thelatter was a philosopher (properly called Eucleides of Megara) and a contemporary of Plato.


2

2-P5) John Dee (1527-1608), ‘Mathematical Preface’ (1570)2

Fig. 2-P5.1) Title page of Henry {??} Billingsley’s The Elements

2 Dee, John. The Mathematical Preface to the Elements of Geometrie of Euclid of Megara(1570), ed. & introduction Allen G. Debus. New York: Science History Publications, 1975. Pp.


3

All things which are, and have being, are found under a triple diversity general.

For either they are deemed Supernatural, Natural, or, of a third being. Things Super-

natural are immaterial, simple, indivisible, incorruptible, and unchangeable. Things

Natural are material, compounded, divisible, corruptible, and changeable. Things Su-

pernatural are of the mind only comprehended: Things Natural are able to be perceived

of the sense exterior. In things Natural, probability and conjecture hath place: But in

things Supernatural, chief demonstration and most sure Science is to be had. By which

properties and comparisons of these two, more easily may be described, the state, con-

dition, nature and property of those things which we before termed of a third being:

which, by a peculiar name also, are called Things Mathematical. For these, being (in a

manner) middle between things supernatural and natural, are not so absolute and ex-

cellent as things supernatural: Nor yet so base and gross as things natural: But [they]

are things immaterial: and nevertheless by material things able somewhat to be signi-

fied. And though their particular Images, by Art, are agreeable and divisible: yet the

general Forms, notwithstanding, are constant, unchangeable, untransformable, and in-

corruptible. Neither of the sense, can they, at any time, be perceived or judged. Nor

yet, for all that, in the royal mind of man, first conceived. But, surmounting the imper-

fection of conjecture, weaning and opinion: and coming short of high intellectual con-

ception are the Mercurial fruit of Dianoeticall discourse in perfect imagination subsist-

ing.3 A marvelous neutrality have these things Mathematical and also a strange partic i-

pation between things Supernatural, immortal, intellectual, simple and indivisible and

things natural, mortal, sensible, compounded and divisible. Probability and sensible

proof may well serve in things Natural and is commendable. In Mathematical reason-

ings a probable Argument is nothing regarded. Nor yet [is] the testimony of sense any

whit credited. But only a perfect demonstration—of truths certain, necessary, and invin-

1-55.3 ‘Dianoetical’ is a term from Aristotelian philosophy having to do with ways of thinking or variousforms of wisdom, or paths to wisdom; e.g., techné, epistemé, phronesis (practical), sophia (wis-dom), nous (intelligence).


4

cible, universally and necessarily concluded—is allowed as sufficient for an Argument

exactly and purely Mathematical. [ . . . ]

Neither Number nor Magnitude have any Materiality. First, we will consider of

Number and of the Science Mathematical to it appropriate, called Arithmetic: and after-

ward of Magnitude and his Science, called Geometry. But that name contenteth me

not, whereof a word or two hereafter shall be said. How Immaterial and free from all

matter Number is, who doth not perceive? yea, who doth not wonderfully wonder at it?

For neither pure Element nor Aristotle’s Quinta Essentia [quintessence or aether] is able

to serve for Number as his proper matter. Nor yet the purity and simpleness of Sub-

stance Spiritual or Angelical will be found proper enough thereto. And therefore the

great and godly Philosopher Anitius Boethius said [ . . . ]: All things (which from the very

first original being of things, have been framed and made) do appear to be Formed by

the reason of Numbers. For this was the principal example or pattern in the mind of the

Creator. O comfortable allurement, O ravishing persuasion, to deal with a Science,

whose Subject is so Ancient, so pure, so excellent, so surmounting all creatures, so

used of the mighty and incomprehensible wisdom of the Creator in the distinct creation

of all creatures: in all their distinct parts, properties, natures, and virtues, by order, and

most absolute number, brought from Nothing to the Formality of their being and state.

By Number’s property, therefore, of us by all possible means (to the perfection of the

Science) learned, we may both wind and draw our selves into the inward and deep

search and view, of all creature’s distinct virtues, natures, properties, and Forms: And

also further arise, climb, ascend, and mount up (with Speculative wings) in spirit, to be-

hold in the Glass of Creation the Form of Forms, the Exemplar Number of all things

Numerable: both visible and invisible: mortal and immortal Corporal and Spiritual, Part

of this profound and divine Science had Ioachim the Prophesier attained unto: by Num-

bers Formal, Natural, and Rational, foreseeing, concluding and foreshowing great par-

ticular events, long before their coming.4 [ . . . ]

4 Dee is claiming here that the art of numbers may be used in prophesying, prognostication, andperhaps even conjuring—all of which he saw as part of the philosopher’s quest. See (vol. 2)00.0 Harkness “Dees of Mortlake,” pp. 000.


5

Of my former words easy it is to be gathered that Number hath a treble state:

One, in the Creator; an other in every Creature (in respect of his complete constitution);

and the third in Spiritual and Angelical Minds and in the Soul of man. In the first and

third state, Number is termed Number Numbering. But in all Creatures otherwise,

Number is termed Number Numbered. And in our Soul Number beareth such a sway

and hath such an affinity therewith that some of the old Philosophers taught Man’s Soul

to be a Number moving itself.5 And indeed in us, though it be a very Accident, yet such

an Accident it is, that before all Creatures it had perfect being in the Creator Sempiter-

nally [eternally]. Number Numbering therefore, is the discretion discerning and dis-

tincting [sic] of things. But in God the Creator this discretion, in the beginning, produced

orderly and distinctly all things. For his Numbering, then, was his Creating of all things.

And his Continual Numbering of all things is the Conservation of them in being: And

where and when he will lack an Unit there and then, that particular thing shall be Dis-

created.6 [ . . . ]

Both Number and Magnitude have a certain Original seed (as it were) of an in-

credible property and of man never able fully to be declared. Of Number and Unit, and

of Magnitude [and] a Point, do seem to be much like Original causes. [ . . . ]

This Science of Magnitude, its properties, conditions, and appertenances, com-

monly now is, and from the beginning, hath of all Philosophers been called Geometry.

But, verily, with a name too base and scant for a Science of such dignity and ample-

ness. And, perchance, that name, by common and secret content of all wisemen hith-

erto hath been suffered to remain that it might carry with a perpetual memory of the first

and notablest benefit, by that Science, to common people showed: Which was, when

Bounds and [measures] of land and ground were lost and confounded (as in Egypt,

yearly, with the overflowing of Nilus, the greatest and longest river in the world); or that

ground bequeathed were to be assigned; or ground sold were to be laid out; or (when

disorder prevailed) that Commons were distributed into severalties. For where upon

5 Recall the epigram to this chapter.6 Cf. Augustine’s discussion of the connection between god’s mathematical thought andthe existence of things. See 8.2 Augustine, “Literal Meaning of Genesis”, pp. 000 below.


6

these and such like occasions, Some by ignorance, some by negligence, Some by

fraud, and some by violence, did wrongfully limit, measure, encroach, or challenge (by

pretence of just content and measure) those lands and grounds: great loss, disquiet-

ness, murder and war did (full oft) ensue: Till, by God’s mercy and man’s Industry, the

perfect Science of Lines, Planes, and Solids (like a divine Justicier) gave unto every

man his own. The people then by this art pleasured and greatly relied in their lands’ just

measuring; and other Philosophers, writing Rules for land measuring, between them

both thus confirmed the name Geometria, that is, (according to the very etymology of

the word) Land measuring. Wherein, the people knew no further of Magnitude’s use but

in Planes, and the Philosophers of these had no fit hearers or Scholars further to dis-

close unto than of flat [or] plane Geometry. And though these Philosophers knew of

further use and best understood the etymology of the word, yet this name Geometria

was of them applied generally to all sorts of Magnitudes, unleast otherwhile of [i.e., ex-

cept] Plato and Pythagoras, When they would precisely declare their own doctrine.

Then was Geometria with them. [ . . . ]

But well you may perceive by Euclid’s Elements that more ample is our Science

than to measure Planes, and nothing less therein is taught (of purpose) than how to

measure land. Another name, therefore, must needs be had for our Mathematical Sci-

ence of Magnitudes, which regardeth neither clod nor turf, neither hill nor dale, neither

earth nor heaven but is absolute Megethologia: not creeping on ground and dazzling the

eye with pole perch [measuring rod], rod, or line but lifting the heart above the heavens

by invisible lines and immortal beams meeteth with the reflections of the light incompre-

hensible, and so procureth Joy and perfection unspeakable. Of which true use of our

Megethica, Megethologia, Divine Plato seemed to have good taste and judgement, and

(by the name of Geometry) so noted it and warned his Scholars thereof, as in his sev-

enth Dialogue of the Commonwealth [i.e., Plato’s Republic] may evidently be seen. [ . . .

] Verily (saith Plato) whosoever have, (but even very little) tasted of Geometry, will not

deny unto us, this; but that this Science is of an other condition, quite contrary to that,

which they that are exercised in it, do speak of it. And there it followeth of our Geome-

try. [ . . . ] It must needs be confessed (saith Plato), That [Geometry] is learned for the


7

knowledge of that which is ever, and not of that which in time both is bred and is

brought to an end. Geometry is the knowledge of that which is everlasting. It will lift up

therefore (O Gentle Sir) our mind to the Verity, and by that means it will prepare the

Thought to the Philosophical love of wisdom that we may turn or convert toward heav-

enly things [both mind and thought] which now otherwise then becometh us, we cast

down on base or inferior things. Chiefly, therefore, Commandment must be given that

such as do inhabit this most honorable City by no means despise Geometry. For even

those things [done by it] which, in manner, seem to be beside the purpose of Geometry,

are of no small importance.7 And besides the manifold uses of Geometry, in matters

appertaining to war, he addeth more of second unpurposed fruit and commodity, arriv-

ing by Geometry: saying: [ . . . ] But also we know that for the more easy learning of all

Arts it importeth much whether one have any knowledge in Geometry or not. Let us

therefore make an ordinance or decree that this Science of young men shall be learned

in the second place.8 This was divine Plato his Judgement, both of the purposed, chief,

and perfect use of Geometry and of his second depending [upon] derivative commodi-

ties. And for us, Christian men, a thousand thousand more occasions are to have need

of the help of Megethological Contemplations whereby to train our Imagination and

Minds, by little and little, to forsake and abandon the gross and corruptible Objects of

our upward senses and to apprehend, by sure doctrine demonstrated, Things Mathe-

matical. And by them readily to be holpen [helped] and conducted to conceive, dis-

course, and conclude of things Intellectual, Spiritual, eternal, and such as concern our

Bliss everlasting, which otherwise (without Special Privilege of Illumination or Revelation

from heaven) No mortal man’s wit (Naturally) is able to reach unto or to Compass.9

And verily by my small Talent (from above) I am able to prove and testify that the literal

Text and order of our divine Law, Oracles, and Mysteries require more skill in Numbers

and Magnitudes than (commonly) the expositors have uttered, [who] rather only (at the

7 [Find passage in Plato’s Republic and give citation??]8 [Find passage in Plato’s Republic and give citation??]9 Here Dee echoes Nicholas of Cusa’s argument for mathematics as an aid to the contemplationof the divine, see 1.3 Cusa, pp. 000 [very end of Chap. XI]


8

most) so warned and showed their own want therein. (To name any is needless and to

note the places is here no place, But if I be duly asked my answer is ready.) And with-

out the literal, Grammatical, Mathematical, or Natural verities of such places [i.e., pas-

sages in the Bible] by good and certain Art perceived, no Spiritual sense (proper to

those places by Absolute Theology) will thereon depend. No man, therefore, can doubt

but toward the attaining of knowledge incomparable and Heavenly Wisdom: Mathemati-

cal Speculations, both of Numbers and Magnitudes, are means, aids, and guides ready,

certain, and necessary. [ . . . ]

I will orderly recite, describe, and declare a great Number of Arts from our two

Mathematical fountains, derived into the fields of Nature. Whereby such Seeds and

Roses as lie deep hid in the ground of Nature are refreshed, quickened, and provoked

to grow, shoot up, flower and grow fruit infinite and incredible. And these Arts (shall be

such as upon Magnitude’s properties do depend more than upon Number. And by good

reason we may call them Arts and Arts Mathematical Derivative, for (at this time) I de-

fine An Art to be a Methodical complete Doctrine having abundancy of sufficient and

peculiar matter to deal with, by the allowance of the Metaphysical Philosopher: the

knowledge whereof, to humane state [i.e., for the human state], is necessary. And that I

account An Art Mathematical derivative, which by Mathematical demonstrative Method,

in Numbers or Magnitudes, ordereth and confirmeth his doctrine as much and as per-

fectly as the matter subject will admit. And for that I intend to use the name and prop-

erty of a Mechanician otherwise than (hitherto) it hath been used; I think it good (for dis-

tinction sake) to give you also a brief description [of] what I mean thereby. A Mechani-

cian or a Mechanical workman is he whose skill is, without knowledge of Mathematical

demonstration, perfectly to work and finish any sensible work, by the Mathematician

principal or derivative, demonstrated or demonstrable. Full well I know that he which

inventeth or maketh these demonstrations is generally called A speculative Mechani-

cian, which differreth nothing from a Mechanical Mathematician. So, in respect of di-

verse actions, one man may have the name of sundry Arts, as some time of a Logician

some times (in the same matter otherwise handled) of a Rhetorician. Of these trifles I

make (as now in respect of my Preface) small account to file these for the fine handling


9

of subtle curious disputers. In other places they may command me to give good rea-

son, and yet here I will not be unreasonable.

First, then, from the purity, absoluteness, and Immateriality of Principal Geometry

is that kind of Geometry derived which vulgarly is counted Geometry, and is the Art of

Measuring sensible magnitudes, their just quantities, and contents. This teacheth to

measure either at hand, and the practicer to be by the thing Measured, and so by due

applying of Compass, Rule, Square, Yard, Ell, Perch, Pole, Line, Gauging rod, (or such

like instrument) to the Length, Plain, or Solid Measured. [ . . . ] Moreover, of the former

knowledge Geometrical are grown the Skills of Geography, Chorography, Hydrography,

and Stratarithmetry.

Geography teacheth ways, by which, in sundry forms, (as Spheric, Plain or

other), the Situation of Cities, Towns, Villages, Forts, Castles, Mountains, Woods, Ha-

vens, Rivers, Creeks, and such other things, upon the outface [i.e., surface] of the

earthly Globe (either in the whole or in some principal member and portion thereof con-

tained) may be described and designed in commensurations Analogical to Nature and

verity and most aptly to our view may be represented. [ . . . ]

To conclude, some for one purpose and some for an other liketh, loveth, getteth,

and useth Maps, Charts, and Geographical Globes. Of whosoever, to speak suffi-

ciently, would require a book peculiar.

Chorography seemeth to be an underlying and a twig of Geography and yet,

nevertheless, is in practice manifold and in use very ample. This teacheth Analogically

to describe a final [i.e., small] portion or circuit of ground with the contents not regarding

what commensuration [or measure] it hath to the whole or any parcel without it con-

tained [i.e., beyond its borders]. [ . . . ] Some also term this particular description of

places, Topography.

Hydrography delivereth to our knowledge, on Globe or in Plain, the perfect

Analogical description of the Ocean Sea coasts through the whole world, or in the chief

and principal parts thereof, with the isles and chief particular places of dangers con-

tained within the bounds and Sea coasts described; as of Quicksands, Banks, Pits,

Rocks, Races, Countertides, Whirlpools &c. This dealeth with the Element of the water


10

chiefly, as Geography did principally take the Element of the Earth’s description (with

his appertenances) to task. [ . . . ]

Thus have I lightly passed over the Artificial Feats chiefly depending upon vulgar

Geometry and commonly and generally reckoned under the name of Geometry. But

there are other (very many) Methodical Arts, which declining from the purity, simplicity,

and Immateriality of our Principal Science of Magnitudes do yet nevertheless use the

great aid, direction, and Method of the said Principal Science, and have proper names

and distinct; both from the Science of Geometry (from which they are derived) and one

from the other. [ . . . ]

Now to proceed, Astronomy is an Art Mathematical which demonstrateth the

distance, magnitudes, and all natural motions, appearances, and passions proper to the

Planets and fixed Stars for any time past, present, and to come in respect of a certain

Horizon or without respect of any Horizon. By this Art we are certified of the distance of

the Starry Sky and of each Planet from the Center of the Earth, and of the greatness of

any Fixed star seen or Planet in respect of the Earth’s greatness. [ . . . ]

Astronomy was to us from the beginning commended, and in manner com-

manded by God him self. In as much as he made the Sun, Moon, and Stars to be to us

for Signs and knowledge of Seasons and for Distinctions of Days and years.10 [ . . . ]

Music, or Motion, hath his Original cause:11 Therefore, after the motions most

swift and most Slow, which are in the Firmament, of Nature performed and under the

Astronomer’s consideration. [N]ow I will Speak of an other kind of Motion, producing

sound, audible, and of Man numerable. Music I call here that Science which of the

Greeks is called Harmonics. Not meddling with the controversy between the ancient

Harmonics and Canonists.12 Music is a Mathematical Science which teacheth by sense

and reason perfectly to judge and order the diversities of sounds, high and low. Astron-

omy and Music are Sisters, saith Plato. As for Astronomy the eyes, So for Harmonious

10 See 8.1 Genesis 1,14, pp. 000 below.11 According to Pythagorean doctrine, the motion of the planets produces the inaudible ‘music ofthe spheres’. See 1.1 Iamblichus, pp. 000 above.


11

Motion the ears were made. But as Astronomy has a more divine Contemplation and

commodity than mortal eye can perceive, So is Music to be considered that the Mind

may be preferred before the ear. And from audible sound we ought to ascend to the

examination [of] which numbers are Harmonious and which not. And why either the one

are, or the other are not. I could at large in the heavenly motions and distances de-

scribe a marvelous Harmony, of Pythagoras’s Harp with eight strings. [ . . . ]

Of Cosmography I appointed briefly in this place to give you some intelligence.

Cosmography is the whole and perfect description of the heavenly and also elemental

part of the world, their homologal [i.e., corresponding] application and mutual collation

necessary. This Art requireth Astronomy, Geography, Hydrography, and Music.

Therefore, it is no small Art nor so simple as in common practice it is (slightly) consid-

ered. This matcheth Heaven and the Earth in one frame and aptly applieth parts Corre-

spondent, So as the Heavenly Globe may (in practice) be duly described upon the Geo-

graphical and Hydrographical Globe. And there for us to consider an Equinoctial Circle

and Ecliptic line, Colors [i.e., colures13] , Poles, Stars in their true Longitudes, Latitudes,

Declinations, and Verticality; also Climbs and Parallels; and by an Horizon annexed

and revolution of the earthly Globe (as the Heaven is by the Primovant [i.e., primum

mobile] carried about in 24 equal Hours) to learn the Risings and Settings of Stars [ . . . ]

a thing necessary for due manuevering of the earth, for Navigation, for the Alteration of

man’s body being, whole, Sick, wounded, or bruised.14 [ . . . ]

Of Astrology here I make an Art several from Astronomy, not by new device but

by good reason and authority: for Astrology is an Art Mathematical, which reasonably

demonstrateth the operations and effects of the natural beams of light and secret influ-

ence of the Stars and Planets in every element and elemental body at all times in any

Horizon assigned. This Art is furnished with many other great Arts and experiences, As

12 Dee chooses to pass over without comment the principal debate among ancient music theo-rists. See Barker, Greek Musical Writings . . . , pp. 00013 Colures are the two great circles drawn on the terrestrial or celestial sphere that intersect atright angles to one another at the poles.14 Dee is referring here, somewhat obliquely, to the importance of geography in medical astrol-ogy. See 3.2.3 Servetus “In Favor of Astrology”, pp. 000 below.


12

with perfect Perspective, Astronomy, Cosmography, Natural Philosophy of the 4 Ele-

ments, the Art of Graduation, and some good understanding in Music. [ . . . ]

In order now followeth of Statics, somewhat to say what we mean by that name

and what commodity doth on such Art depend. Statics is an Art Mathematical which

demonstrateth the causes of heaviness and lightness of all things and of motions and

properties to heaviness and lightness belonging. And for as much as by the [ . . . ] Bal-

ance (as the chief sensible Instrument) Experience of these demonstrations may be

had, we call this Art Statics; that is, the Experiments of the Balance. Oh, that men with

what profit (all manner of ways) by this Art might grow to the able examiner and diligent

practicer [i.e., practitioner]. Thou only knowest all things precisely (O God) who hast

made weight and Balance thy Judgement; who hast created all things in Number,

Weight, and Measure;15 [who] hast weighed the Mountains and hills in a Balance; [and]

who hast poised in thy hand both Heaven and earth. [ . . . ]

— Although the Printer hath looked for this Preface a day or two, yet could I not

bring my pen from the paper before I had given you comfortable warning and brief in-

structions of some of the Commodities [which] by Statics [are] able to be reaped; In the

rest I will therefore be as brief as it is possible, and with all describing them somewhat

accordingly. And that you shall perceive by this which in order cometh next. For

whereas it is so ample and wonderful that an whole year long one might find fruitful

matter therein to speak of and also in practice, is a Treasure endless; yet I will glance

over it with words very few. —

This do I call Anthropography. Which is an Art restored and of my preferment to

your Service. I pray you think of it as one of the chief points of Human knowledge. Al-

though it be but now first Confirmed with this new name; yet the matter hath from the

beginning been in consideration of all perfect Philosophers. Anthropography is the de-

scription of the Number, Measure, Weight, Figure, Situation, and color of every diverse

15 The passage in the Book of Wisdom (11, 20) read, “But you have disposed all things bymeasure and number and weight.” Both 8.2 Augustine, “Literal Meaning of Genesis”, pp. 000and 8.4a Pascal, “Vacuum/Geometrical Mind”, pp. 000 elaborate upon this passage. It also fig-ures into Riccioli’s frontispiece (Fig. 3.1.5.?? Riccioli’s Almagestum Novum, 1650).


13

thing contained in the perfect body of Man; with certain knowledge of Symmetry, figure,

weight, Characterization, and due local motion of any parcel of the said body assigned;

and of Numbers to the said parcel appertaining. If the description of the heavenly part

of the world had a peculiar Art called Astronomy; If the description of the earthly globe

hath his peculiar art called Geography; If the Matching of both has his peculiar Art called

Cosmography, Which is the Description of the whole and universal frame of the world;

Why should not the Description of him who is the Lesser world and from the beginning

called Microcosmus (that is, the Lesser World). [ . . . ] {last sentence correct? Check

against photocopy?}

[Yet] I will herein crave Judgement of two most perfect Architects, the one being

Vitruvius, the Roman who did write ten books thereof to the Emperor Augustus (in

whose days our Heavenly Archemaster [i.e. Jesus Christ] was borne) and the other

[being] Baptista Albertus [Leon Battista Alberti], a Florentine who also published ten

books thereof.16 [ . . . ] An Architect ( saith Vitruvius) ought to understand Languages,

to be skillful of Painting, well instructed in Geometry, not ignorant of Perspective, fur-

nished with Arithmetic, have knowledge of many histories, and diligently have heard

Philosophers, have skill of Music, not ignorant of Physike, know the answers of Law-

yers, and have Astronomy and the courses Celestial in good knowledge. [ . . . ]

— Life is short, and uncertain. Times are perilous &c. And still the Printer

awaiting, for my pen staying. All these things, with further matter of ungratefulness, give

me occasion to pass away to the other Arts remaining with all speed possible. —

The Art of Navigation demonstrateth how by the shortest good way, by the aptest

direction, and in the shortest time a sufficient Ship between any two places (in passage

Navigable) assigned, may be conducted; and in all storms and natural disturbances

chancing how to use the best possible means whereby to recover the place first as-

signed [i.e. the ship’s home port]. What need the Master Pilot hath of other Arts here

before recited, it is easy to know; as of Hydrography, Astronomy, Astrology, and Horo-

metry. Presupposing continually the common Base and foundation of all, namely Ar-


14

ithemetik and Geometry. So that he be able to understand and judge his own neces-

sary Instruments and furniture Necessary, Whether they be perfectly made or no; and

also can (if need be) make them himself. As Quadrants; The Astronomer’s Ring; the

Astronomer’s Staff; the Astrolabe universal; An Hydrographical Globe; Charts Hydro-

graphical true (not with parallel Meridians); The Common Sea Compass; the Compass

of Variation; the Proportional; and Paradoxal Compassers (of me invented for our two

Muscovy Master Pilots, at the request of the Company); Clocks with spring, hour, half

hour, and three hour Sandglasses; and sundry other Instruments; And also be able on

Globe, or Plane to describe the Paradoxal Compass; and duly to use the same to all

manner of purposes whereto it was invented. And also be able to Calculate the Planets’

places for all times.

Moreover, with Sun, Moon, or Star (or without) to be able to define the Longitude

and Latitude of the place which he is in, So that the Longitude and Latitude of the place

from which he sailed be given, or by him be known whereto appertaineth expert means

to be certified ever of the Ships way. &c.17 [ . . . ]

Now end I, with Archemastry. Which name is not so new as this Art is rare. For

an other Art under this [name] a degree (for skill and power) hath been endued with this

English name before. And yet this may serve for our purpose sufficiently at this present.

This art teacheth to bring to actual experience sensible all worthy conclusions by all the

Arts Mathematical purposed, and by true Natural Philosophy concluded; and both ad-

deth to them further scope in the terms of the same Arts, and also by his proper Method

and in peculiar terms proceedeth with help of the foresaid arts to the performance of

complete Experiences, which of no particular Art are able (Formally) to be challenged.

If you remember how we considered Architecture in respect of all common handworks,

some light may have thereby to understand the Sovereignty and property of this Sci-

ence. Science I may call it rather than an Art, for the excellency and Mastership it hath

16 For Vitruvius, see Fig. 1.4.5 Leonardo’s ‘Virtuvian Man’ (c. 1490). For Alberti, see pp. 000[Introduction] above.17 On the importance of determining longitude at sea to all seafaring nations, see 3.4.5 Varen-ius, “General Geography”, pp. 000 below.


15

over so many and so mighty Arts and Sciences. And [since] by cause it proceedeth by

Experiences and searcheth [for] the causes of Conclusions by Experiences and also

putteth the Conclusions themselves in Experience, it is named of some, Scientia Ex-

perimentalis. The Experimental Science. Nicolaus Cusanus termeth it so, in his Ex-

periments Statikal.18 [ . . . ]

This Art, certifieth by Experience complete and absolute; [whereas] other arts,

with their Arguments and Demonstrations, persuade and in words prove very well their

Conclusions. But words and Arguments are no sensible certifying, nor the full and final

fruit of Sciences practicable. And though some Arts have in them Experiences, yet they

are not complete [nor] brought to the uttermost [that] they may be stretched unto and

applied sensibly. As for example, the Natural Philosopher disputeth and maketh goodly

show of reason and the Astronomer and the Optical Mechanician put some things in

Experience, But [not] all that they may; nor yet sufficiently and to the utmost, those

which they do [put into experience]. There, then, the Archemaster steppeth in and

leadeth forth [ . . . ] the Experiences, by order of his doctrine Experimental, to the chief

and final power of Natural and Mathematical Arts. [ . . . ]

— Here, I must end, thus abruptly (Gentle friend, and unfeigned lover of honest

and necessary verities.) For they who have (for your sake and virtuous cause) re-

quested me (an old forewarn Mathematician) to take pen in hand (through the confi-

dence they reposed in my long experience and tried sincerity) for the declaring and re-

porting somewhat of the fruit and commodity to be attained unto by the Art Mathemati-

cal; even they, sore against their will, are forced for sundry causes to satisfy the work-

men’s request in ending forthwith. He so feareth this so new an attempt and so costly,

And in matter so slenderly (hitherto) among the common Sort of Students, considered

or esteemed. —

[ . . . Yet] great Comfort and good hope may the Universities have, by reason of

this English Geometry and Mathematical Preface, that they (hereafter) shall be more

regarded, esteemed, and resorted unto. For when it shall be known and reported that

18 For a discussion of the main currents of ‘experimental philosophy’ in the seventeenth century,


16

of the Mathematical Sciences only, such great Commodities are ensuing (as I have

specified), and that indeed some of you unlatined students can be good witnesses of

such rare fruit by you enjoyed (thereby); as either before this was not heard of, or else

not so fully credited, Well may all men conjecture that far greater aid and better furniture

to win to the Perfection of all Philosophy may in the universities be had, [they] being the

storehouses and Treasury of all Sciences and all Arts necessary for the best and most

noble State of Common-Wealths.

Besides this, how many a Common Artificer is there in these Realms of England

and Ireland that dealeth with Numbers, Rule, and Compass; Who with their own Skill

and experience already had, will be able (by these good helps and informations) to find

out and devise new works, strange Engines, and Instruments for sundry purposes in the

Common Wealth? or for private pleasure? and for the better maintaining of their own

estate? I will not (therefore) fight against my own shadow. For no man (I am sure) will

open his mouth against this enterprise. No man (I say) who either hath Charity toward

his brother (and would be glad of his furtherance in virtuous knowledge) or that hath any

care and zeal for the bettering of the Common State of this Realm. Neither any that

make attempt what the wiser sort of men (Sage and Stayed) do think of them. To none

(therefore) will I make any Apology for a virtuous act doing and for commending, or set-

ting forth Profitable Arts to English men in the English tongue. But unto God our Crea-

tor let us all be thankful, for that As he of his Goodness, by his power, and in his wisdom

hath Created all things in Number, Weight, and Measure, So to us of his great Mercy he

hath revealed Means whereby to attain the sufficient and necessary knowledge of the

[a]foresaid his three principal Instruments, Which means I have abundantly proved unto

you to be the Sciences and Arts Mathematical.

And though I have been pinched with straightness of time, that no way I could so

pen down the matter (in my Mind) as I determined, hoping of convenient leisure. Yet, if

virtuous zeal and honest intent provoke and bring you to the readying and examining of

this Compendious treatise, I do not doubt but as the verity thereof (according to our

see the ‘Introduction” to chapter five, pp. 000


17

purpose) will be evident unto you. So the pith and force thereof will persuade you and

the wonderful fruit thereof highly pleasure you. And that you may the easier perceive

and better remember the principal points whereof of my Preface treateth, I will give you

the Groundplate of my whole discourse in a Table annexed, from the first to the last

somewhat Methodically contrived.

If Haste hath caused my poor pen anywhere to fumble, You will (I am sure) in

part recompense (for my earnest and sincere good will to pleasure you) Consider the

rockish huge mountains and the perilous unbeaten ways which (both night and day, for

the while) [my pen] hath toiled and labored through to bring you this good news and

comfortable proof of Virtue’s fruit.

So I commit you unto God’s merciful direction for the rest, heartily beseeching

him to prosper your studies and honest Intents; to his Glory and the Commodity of our

Country Amen.


18

Fig. 2-P5.2) Dee’s Mathematical Table


19

[The main headings of Dee’s ‘Mathematical Groundplat’ are as follows:

The Mathematical Sciences and Arts consist of,

I) The Principal Fields

A) Arithmetic

1) Simple—dealing with number only

2) Mixed—combined with Geometry to prove arithmetical properties

B) Geometry

1) Simple—dealing with magnitude only

2) Mixed—combined with Arithmetic to prove geometrical properties

II) Derivative Fields, which have

A) Which Have the Names of the Principal Fields (above)

1) Arithmetic (fractions, proportions, trigonometry, roots, algebra)

2) Geometry of Measurement

a) of Objects at Hand, measuring:

i) lengths (Mecometry)

ii) surfaces (Embadometry)

iii) solids (Stereometry)

b) of Objects removed from Measurer

i) distance (Apomecometry)

ii) depth (Hyposometry)

iii) breadth (Platometry)

(All of which relate to Geodesy, Geography,

Chorography, Hydrography, Stratarithmetry)

B) Or which Have Their Own Proper Names

1) Perspective—properties of all radiation (direct, broken, reflected)

2) Astronomy—distances, magnitudes, and motions of stars & planets

3) Music—to order by reason and sense sounds high and low

4) Cosmography—description of the celestial and terrestrial world

5) Astrology—influences of planets & stars on ‘elemental bodies’

6) Statics—investigates the causes of lightness & heaviness of things


20

7) Anthropography—measurement of figure of the human body

8) Trochilics—properties of circular motion (wheels, pulleys, etc.)

9) Heliosophy—spiral lines on planes, cylinders, cones, spheres, etc.

10) Pneumatithmy—motions of fluids like air, smoke, water, etc.

11) Menadry—the multiplication of forces or weights in machines

12) Hypogeiody—how to lay mine-shafts, perpendicular or horizontal

13) Hydragogy—direction of water along canals, dykes, channels, etc.

14) Horometry—instruments for the measurement of time

15) Zography—construction and intersection of visual pyramids

16) Architecture—construction of buildings

17) Navigation—find pathways across the seas, maintain bearing, etc.

18) Thaumaturgy—to make strange & wonderful effects (natural magic)

19) Archemastry—method based on experience (experimental science)

sci. rev. reader (‘02/03/07) 02-p5 dee - harvard...

Documents