to require testimony: how nedefull, how frutefull, how skillfull a thing
_Arithmetike_ is? I meane, the Lawyers of all sortes. Vndoubtedly, the
Ciuilians, can meruaylously declare: how, neither the Auncient Romaine
lawes, without good knowledge of _Numbers art_, can be perceiued: Nor
(Iustice in infinite Cases) without due proportion, (narrowly
considered,) is hable to be executed. How Iustly, & with great knowledge
of Arte, did _Papinianus_ institute a law of partition, and allowance,
betwene man and wife after a diuorce? But how _Accursius_, _Baldus_,
_Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being
otherwise, notably well learned) do iumble, gesse, and erre, from the
æquity, art and Intent of the lawmaker: _Arithmetike_ can detect, and
conuince: and clerely, make the truth to shine. Good _Bartolus_, tyred
in the examining & proportioning of the matter: and with _Accursius_
Glosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hac
glossa difficilior: Cuius computationem nec Scholastici nec Doctores
intelligunt. &c._ That is: +_In the whole booke, there is no Glosse
harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor
the Doctours vnderstand. &c._+ What can they say of _Iulianus_ law, _Si
ita Scriptum. &c._ Of the Testators will iustly performing, betwene the
wife, Sonne and daughter? How can they perceiue the æquitie of
_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex
Falcidia_? How can they deliuer him, from his Reprouers: and their
maintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How
Iustly and artificially, was _Africanus_ reckening made? Proportionating
to the Sommes bequeathed, the Contributions of eche part? Namely, for
the hundred presently receiued, 17-1/7. And for the hundred, receiued
after ten monethes, 12-6/7: which make the 30: which were to be
cõtributed by the legataries to the heire. For, what proportion, 100
hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that
is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law,
require an expert _Arithmeticien_, to vnderstand the deepe Iudgemẽt, &
Iust determinatiõ of the Auncient Romaine Lawmakers. But much more
expert ought he to be, who should be hable, to decide with æquitie, the
infinite varietie of Cases, which do, or may happen, vnder euery one of
those lawes and ordinances Ciuile. Hereby, easely, ye may now
coniecture: that in the Canon law: and in the lawes of the Realme (which
with vs, beare the chief Authoritie), Iustice and equity might be
greately preferred, and skilfully executed, through due skill of
Arithmetike, and proportions appertainyng. The worthy Philosophers, and
prudent lawmakers (who haue written many bookes _De Republica:_ How the
best state of Common wealthes might be procured and mainteined,) haue
very well determined of Iustice: (which, not onely, is the Base and
foundacion of Common weales: but also the totall perfection of all our
workes, words, and thoughtes:) defining it,
[Iustice.]
“to be that vertue, by which, to euery one, is rendred, that to him
appertaineth.” God challengeth this at our handes, to be honored as God:
to be loued, as a father: to be feared as a Lord & master. Our
neighbours proportiõ, is also prescribed of the Almighty lawmaker: which
is, to do to other, euen as we would be done vnto. These proportions,
are in Iustice necessary: in duety, commendable: and of Common wealthes,
the life, strength, stay and florishing. _Aristotle_ in his _Ethikes_
(to fatch the sede of Iustice, and light of direction, to vse and
execute the same) was fayne to fly to the perfection, and power of
Numbers: for proportions Arithmeticall and Geometricall. _Plato_ in his
booke called _Epinomis_ (which boke, is the Threasury of all his
doctrine) where, his purpose is, to seke a Science, which, when a man
had it, perfectly: he might seme, and so be, in dede, _Wise_. He,
briefly, of other Sciences discoursing, findeth them, not hable to bring
it to passe: But of the Science of Numbers, he sayth. _Illa, quæ numerum
mortalium generi dedit, id profecto efficiet. Deum autem aliquem, magis
quam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse.
&c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudentiæ
dico, causam arbitramur? +That Science, verely, which hath taught
mankynde number, shall be able to bryng it to passe. And, I thinke,
a certaine God, rather then fortune, to haue giuen vs this gift, for our
blisse. For, why should we not Iudge him, who is the Author of all good
things, to be also the cause of the greatest good thyng, namely,
Wisedome?+_ There, at length, he proueth _Wisedome_ to be atteyned, by
good Skill of _Numbers_. With which great Testimony, and the manifold
profes, and reasons, before expressed, you may be sufficiently and fully
persuaded: of the perfect Science of _Arithmetike_, to make this
accounte: That
[☞]
of all Sciences, next to _Theologie_, it is most diuine, most pure, most
ample and generall, most profounde, most subtile, most commodious and
most necessary. Whose next Sister, is the Absolute Science of
_Magnitudes_: of which (by the Direction and aide of him, whose
_Magnitude_ is Infinite, and of vs Incomprehensible) I now entend, so to
write, that both with the _Multitude_, and also with the _Magnitude_ of
Meruaylous and frutefull verities, you (my frendes and Countreymen) may
be stird vp, and awaked, to behold what certaine Artes and Sciences, (to
our vnspeakable behofe) our heauenly father, hath for vs prepared, and
reuealed, by sundry _Philosophers_ and _Mathematiciens_.
Both, _Number_ and _Magnitude_, haue a certaine Originall sede, (as it
were) of an incredible property: and of man, neuer hable, Fully, to be
declared. Of _Number_, an Vnit, and of _Magnitude_, a Poynte, doo seeme
to be much like Originall causes: But the diuersitie neuerthelesse, is
great. We defined an _Vnit_, to be a thing Mathematicall Indiuisible:
A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And
farder, that a Point may haue a certaine determined Situation: that is,
that we may assigne, and prescribe a Point, to be here, there, yonder.
&c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to
be tyed to any place, or seat: diuisible or indiuisible. Agayne, by
reason, a Point may haue a Situation limited to him: a certaine motion,
therfore (to a place, and from a place) is to a Point incident and
appertainyng. But an _Vnit_, can not be imagined to haue any motion.
A Point, by his motion, produceth, Mathematically, a line: (as we sayd
before) which is the first kinde of Magnitudes, and most simple: An
_Vnit_, can not produce any number. A Line, though it be produced of a
Point moued, yet, it doth not consist of pointes: Number, though it be
not produced of an _Vnit_, yet doth it Consist of vnits, as a materiall
cause. But formally,
[Number.]
Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting,
is the workemanship of our minde: which, of distinct and discrete Vnits,
maketh a Number: by vniformitie, resulting of a certaine multitude of
Vnits. And so, euery number, may haue his least part, giuen: namely, an
Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be
giuẽ: by cause, infinitly, diuision therof, may be conceiued. All
Magnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine,
or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way)
represented: nor of Nature produced: But, as (by degrees) Number did
come to our perceiuerance: So, by visible formes, we are holpen to
imagine, what our Line Mathematicall, is. What our Point, is. So
precise, are our Magnitudes, that one Line is no broader then an other:
for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our
Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we
can haue Smaller, then either Arte or Nature can produce any: and
Greater also, then all the world can comprehend. Our least Magnitudes,
can be diuided into so many partes, as the greatest. As, a Line of an
inch long, (with vs) may be diuided into as many partes, as may the
diameter of the whole world, from East to West: or any way extended:
What priuiledges, aboue all manual Arte, and Natures might, haue our two
Sciences Mathematicall? to exhibite, and to deale with thinges of such
power, liberty, simplicity, puritie, and perfection? And in them, so
certainly, so orderly, so precisely to procede: as, excellent is that
workemã Mechanicall Iudged, who nerest can approche to the representing
of workes, Mathematically demonstrated?
[☞]
And our two Sciences, remaining pure, and absolute, in their proper
termes, and in their owne Matter: to haue, and allowe, onely such
Demonstrations, as are plaine, certaine, vniuersall, and of an æternall
veritye?
[Geometrie.]
This Science of _Magnitude_, his properties, conditions, and
appertenances: commonly, now is, and from the beginnyng, hath of all
Philosophers, ben called _Geometrie_. But, veryly, with a name to base
and scant, for a Science of such dignitie and amplenes. And, perchaunce,
that name, by cõmon and secret consent, of all wisemen, hitherto hath
ben suffred to remayne: that it might carry with it a perpetuall
memorye, of the first and notablest benefite, by that Science, to common
people shewed: Which was, when Boundes and meres of land and ground were
lost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of
_Nilus_, the greatest and longest riuer in the world) or, that ground
bequeathed, were to be assigned: or, ground sold, were to be layd out:
or (when disorder preuailed) that Commõs were distributed into
seueralties. For, where, vpon these & such like occasiõs, Some by
ignorãce, some by negligẽce, Some by fraude, and some by violence, did
wrongfully limite, measure, encroach, or challenge (by pretence of iust
content, and measure) those landes and groundes: great losse,
disquietnes, murder, and warre did (full oft) ensue: Till, by Gods
mercy, and mans Industrie, The perfect Science of Lines, Plaines, and
Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. The
people then, by this art pleasured, and greatly relieued, in their
landes iust measuring: & other Philosophers, writing Rules for land
measuring: betwene them both, thus, confirmed the name of _Geometria_,
that is, (according to the very etimologie of the word) Land measuring.
Wherin, the people knew no farder, of Magnitudes vse, but in Plaines:
and the Philosophers, of thẽ, had no feet hearers, or Scholers: farder
to disclose vnto, then of flat, plaine _Geometrie_. And though, these
Philosophers, knew of farder vse, and best vnderstode the etymologye of
the worde, yet this name _Geometria_, was of them applyed generally to
all sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and
_Pythagoras_: When they would precisely declare their owne doctrine.
Then, was
[* Plato. 7. de Rep.]
* _Geometria_, with them, _Studium quod circa planum versatur_. But,
well you may perceiue by _Euclides Elementes_, that more ample is our
Science, then to measure Plaines: and nothyng lesse therin is tought (of
purpose) then how to measure Land. An other name, therfore, must nedes
be had, for our Mathematicall Science of Magnitudes: which regardeth
neither clod, nor turff: neither hill, nor dale: neither earth nor
heauen: but is absolute _Megethologia_: not creping on ground, and
dasseling the eye, with pole perche, rod or lyne: but “liftyng the hart
aboue the heauens, by inuisible lines, and
[☞]
immortall beames meteth with the reflexions, of the light
incomprehensible: and so procureth Ioye, and perfection vnspeakable.” Of
which true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_
seemed to haue good taste, and iudgement: and (by the name of
_Geometrie_) so noted it: and warned his Scholers therof: as, in hys
seuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in
Latin) thus it is: right well translated: _Profecto, nobis hoc non
negabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæc
Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa
versantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer
haue, (but euen very litle) tasted of Geometrie, will not denye vnto vs,
this: but that this Science, is of an other condicion, quite contrary to
that, which they that are exercised in it, do speake of it._+ And there
it followeth, of our _Geometrie_, _Quòd quæritur cognoscendi illius
gratia, quod semper est, non & eius quod oritur quando[que] & interit.
Geometria, eius quod est semper, Cognitio est. Attollet igitur
(ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum
preparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contra
quàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendum
est, vt qui præclarissimam hanc habitãt Civitatem, nullo modo,
Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modo
esse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith
_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which
is euer: and not of that, which, in tyme, both is bred and is brought to
an ende. &c. Geometrie is the knowledge of that which is euerlastyng. It
will lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by
that meanes, it will prepare the Thought, to the Philosophicall loue of
wisdome: that we may turne or conuert, toward heauenly thinges =[both
mynde and thought]= which now, otherwise then becommeth vs, we cast down
on base or inferior things. &c. Chiefly, therfore, Commaundement must be
giuen, that such as do inhabit this most honorable Citie, by no meanes,
despise Geometrie. For euen those thinges =[done by it]= which, in
manner, seame to be, beside the purpose of Geometrie: are of no small
importance. &c._+ And besides the manifold vses of _Geometrie_, in
matters appertainyng to warre, he addeth more, of second vnpurposed
frute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin
etiam, ad Disciplinas omnes facilius per discendas, interesse omnino,
attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam,
secundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we
know, that for the more easy learnyng of all Artes, it importeth much,
whether one haue any knowledge in Geometrie, or no. &c. Let vs therfore
make an ordinance or decree, that this Science, of young men shall be
learned in the second place._+ This was _Diuine Plato_ his Iudgement,
both of the purposed, chief, and perfect vse of _Geometrie_: and of his
second, dependyng, deriuatiue commodities. And for vs, Christen men,
a thousand thousand mo occasions are, to haue nede of the helpe of *
[I. D.
* Herein, I would gladly shake of, the earthly name,
of Geometrie.]
_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and
Myndes, by litle and litle, to forsake and abandon, the grosse and
corruptible Obiectes, of our vtward senses: and to apprehend, by sure
doctrine demonstratiue, Things Mathematicall. And by them, readily to be
holpen and conducted to conceiue, discourse, and conclude of things
Intellectual, Spirituall, æternall, and such as concerne our Blisse
euerlasting: which, otherwise (without Speciall priuiledge of
Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally)
is hable to reach vnto, or to Compasse. And, veryly, by my small Talent
(from aboue) I am hable to proue and testifie, that the litterall Text,
and order of our diuine Law, Oracles, and Mysteries, require more skill
in Numbers, and Magnitudes: then (commonly) the expositors haue vttered:
but rather onely (at the most) so warned: & shewed their own want
therin. (To name any, is nedeles: and to note the places, is, here, no
place: But if I be duely asked, my answere is ready.) And without the
litterall, Grammaticall, Mathematicall or Naturall verities of such
places, by good and certaine Arte, perceiued, no Spirituall sense
(propre to those places, by Absolute _Theologie_) will thereon depend.
[☞]
“No man, therfore, can doute, but toward the atteyning of knowledge
incomparable, and Heauenly Wisedome: Mathematicall Speculations, both of
Numbers and Magnitudes: are meanes, aydes, and guides: ready, certaine,
and necessary.” From henceforth, in this my Preface, will I frame my
talke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well
can, (and also wil,) vse their vtward senses, to the glory of God, the
benefite of their Countrey, and their owne secret contentation, or
honest preferment, on this earthly Scaffold. To them, I will orderly
recite, describe & declare a great Number of Artes, from our two
Mathematicall fountaines, deriued into the fieldes of _Nature_. Wherby,
such Sedes, and Rotes, as lye depe hyd in the groũd of _Nature_, are
refreshed, quickened, and prouoked to grow, shote vp, floure, and giue
frute, infinite, and incredible. And these Artes, shalbe such, as vpon
Magnitudes properties do depende, more, then vpon Number. And by good
reason we may call them Artes, and Artes Mathematicall Deriuatiue: for
(at this tyme) I Define
[An Arte.]
+An Arte, to be a Methodicall cõplete Doctrine, hauing abundancy of
sufficient, and peculier matter to deale with, by the allowance of the
Metaphisicall Philosopher: the knowledge whereof, to humaine state is
necessarye.+ And that I account,
[Art Mathematicall Deriuatiue.]
+An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue
Method, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine,
as much & as perfectly, as the matter subiect will admit.+ And for that,
I entend to vse the name and propertie of a
[A Mechanitien.]
_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it
good, (for distinction sake) to giue you also a brief description, what
I meane therby. +A Mechanicien, or a Mechanicall workman is he, whose
skill is, without knowledge of Mathematicall demonstration, perfectly to
worke and finishe any sensible worke, by the Mathematicien principall or
deriuatiue, demonstrated or demonstrable.+ Full well I know, that he
which inuenteth, or maketh these demonstrations, is generally called _A
speculatiue Mechanicien_: which differreth nothyng from a _Mechanicall
Mathematicien_. So, in respect of diuerse actions, one man may haue the
name of sundry artes: as, some tyme, of a Logicien, some tymes (in the
same matter otherwise handled) of a Rethoricien. Of these trifles,
I make, (as now, in respect of my Preface,) small account: to fyle thẽ
for the fine handlyng of subtile curious disputers. In other places,
they may commaunde me, to giue good reason: and yet, here, I will not be
vnreasonable.
[+1.+]
First, then, from the puritie, absolutenes, and Immaterialitie of
Principall _Geometrie_, is that kinde of _Geometrie_ deriued, which
vulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible
magnitudes, their iust quãtities and contentes.+
[Geometrie vulgar.]
This, teacheth to measure, either at hand: and the practiser, to be by
the thing Measured: and so, by due applying of Cumpase, Rule, Squire,
Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to
the Length, Plaine, or Solide measured,
[1.]
* to be certified, either of the length, perimetry, or distance lineall:
and this is called, _Mecometrie_. Or
[2.]
* to be certified of the content of any plaine Superficies: whether it
be in ground Surueyed, Borde, or Glasse measured, or such like thing:
which measuring, is named _Embadometrie_.
[3.]
* Or els to vnderstand the Soliditie, and content of any bodily thing:
as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels,
small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale
vessells, &c, the Measuring, commonly, hath a peculier name: and is
called _Gaging_. And the generall name of these Solide measures, is
_Stereometrie_.
[+2.+]
Or els, this _vulgar Geometrie_, hath consideration to teach the
practiser, how to measure things, with good distance betwene him and the
thing measured: and to vnderstand thereby, either
[1.]
* how Farre, a thing seene (on land or water) is from the measurer: and
this may be called _Apomecometrie_:
[2.]
Or, how High or depe, aboue or vnder the leuel of the measurers stãding,
any thing is, which is sene on land or water, called _Hypsometrie_.
[3.]
* Or, it informeth the measurer, how Broad any thing is, which is in the
measurers vew: so it be on Land or Water, situated: and may be called
_Platometrie_. Though I vse here to condition, the thing measured, to be
on Land, or Water Situated:
[Note.]
yet, know for certaine, that the sundry heigthe of Cloudes, blasing
Starres, and of the Mone, may (by these meanes) haue their distances
from the earth: and, of the blasing Starres and Mone, the Soliditie
(aswell as distances) to be measured: But because, neither these things
are vulgarly taught: nor of a common practiser so ready to be executed:
I, rather, let such measures be reckened incident to some of our other
Artes, dealing with thinges on high, more purposely, then this vulgar
Land measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._
Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, or
Land Measuring: more cunningly to measure & Suruey Land, Woods, and
Waters, a farre of. More cunningly, I say: But God knoweth (hitherto) in
these Realmes of England and Ireland (whether through ignorance or
fraude, I can not tell, in euery particular)
[Note.]
how great wrong and iniurie hath (in my time) bene committed by vntrue
measuring and surueying of Land or Woods, any way. And, this I am sure:
that the Value of the difference, betwene the truth and such Surueyes,
would haue bene hable to haue foũd (for euer) in eche of our two
Vniuersities, an excellent Mathematicall Reader: to eche, allowing
(yearly) a hundred Markes of lawfull money of this realme: which, in
dede, would seme requisit, here, to be had (though by other wayes
prouided for) as well, as, the famous Vniuersitie of Paris, hath two
Mathematicall Readers: and eche, two hundreth French Crownes yearly, of
the French Kinges magnificent liberalitie onely. Now, againe, to our
purpose returning: Moreouer, of the former knowledge Geometricall, are
growen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and
_Stratarithmetrie_.
“+‡Geographie‡+ teacheth wayes, by which, in sũdry formes, (as
_Sphærike_, _Plaine_ or other), the Situation of Cities, Townes,
Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, &
such other things, vpõ the outface of the earthly Globe (either in the
whole, or in some principall mẽber and portion therof cõtayned) may be
described and designed, in cõmensurations Analogicall to Nature and
veritie: and most aptly to our vew, may be represented.” Of this Arte
how great pleasure, and how manifolde commodities do come vnto vs, daily
and hourely: of most men, is perceaued. While, some, to beautifie their
Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other
some, for thinges past, as battels fought, earthquakes, heauenly
fyringes, & such occurentes, in histories mentioned: therby liuely, as
it were, to vewe the place, the region adioyning, the distance from vs:
and such other circumstances. Some other, presently to vewe the large
dominion of the Turke: the wide Empire of the Moschouite: and the litle
morsell of ground, where Christendome (by profession) is certainly
knowen. Litle, I say, in respecte of the rest. &c. Some, either for
their owne iorneyes directing into farre landes: or to vnderstand of
other mens trauailes. To conclude, some, for one purpose: and some, for
an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &
Geographicall Globes. Of whose vse, to speake sufficiently, would
require a booke peculier.
+‡Chorographie‡+ seemeth to be an vnderling, and a twig, of
_Geographie_: and yet neuerthelesse, is in practise manifolde, and in
vse very ample. “This teacheth Analogically to describe a small portion
or circuite of ground, with the contentes: not regarding what
commensuration it hath to the whole, or any parcell, without it,
contained. But in the territory or parcell of ground which it taketh in
hand to make description of, it leaueth out (or vndescribed) no notable,
or odde thing, aboue the ground visible. Yea and sometimes, of thinges
vnder ground, geueth some peculier marke: or warning: as of Mettall
mines, Cole pittes, Stone quarries. &c.” Thus, a Dukedome, a Shiere,
a Lordship, or lesse, may be described distinctly. But marueilous
pleasant, and profitable it is, in the exhibiting to our eye, and
commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true
Symmetry: not approching to any of them: and out of Gunne shot. &c.
Hereby, the _Architect_ may furnishe him selfe, with store of what
patterns he liketh: to his great instruction: euen in those thinges
which outwardly are proportioned: either simply in them selues: or
respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also,
terme this particular description of places, _Topographie_.