Excvse me, gentle reder if oughte be amisse, straung paths ar
not troden al truly at the first: the way muste needes be
comberous, wher none hathe gone before. Where no man hathe geuen
light, lighte is it to offend, but when the light is shewed
ones, light is it to amende. If my light may so light some
other, to espie and marke my faultes, I wish it may so lighten
them, that they may voide offence. Of staggeringe and stomblinge,
and vnconstaunt turmoilinge: often offending, and seldome
amending, such vices to eschewe, and their fine wittes to shew
that they may winne the praise, and I to hold the candle,
whilest they their glorious works with eloquence sette forth, so
cunningly inuented, so finely indited, that my bokes maie seme
worthie to occupie no roome. For neither is mi wit so finelie
filed, nother mi learning so largly lettred, nother yet mi
laiser so quiet and vncombered, that I maie perform iustlie so
learned a laboure or accordinglie to accomplishe so haulte an
enforcement, yet maie I thinke thus: This candle did I light:
this light haue I kindeled: that learned men maie se, to
practise their pennes, their eloquence to aduaunce, to register
their names in the booke of memorie I drew the platte rudelie,
whereon thei maie builde, whom god hath indued with learning and
liuelihod. For liuing by laboure doth learning so hinder, that
learning serueth liuinge, whiche is a peruers trade. Yet as
carefull familie shall cease hir cruell callinge, and suffre
anie laiser to learninge to repaire, I will not cease from
trauaile the pathe so to trade, that finer wittes maie fashion
them selues with such glimsinge dull light, a more complete
woorke at laiser to finisshe, with inuencion agreable, and
aptnes of eloquence.
And this gentle reader I hartelie protest where erroure hathe
happened I wisshe it redrest.
TO THE MOST NO-
ble and puissaunt prince Edwarde the
sixte by the grace of God, of En-
gland Fraunce and Ireland kynge, de-
fendour of the faithe, and of the
Churche of England and Ire-
lande in earth the su-
preme head.
It is not vnknowen to youre maiestie, moste soueraigne lorde,
what great disceptacion hath been amongest the wyttie men of all
nacions, for the exacte knoweledge of true felicitie, bothe what
it is, and wherein it consisteth: touchynge whiche thyng, their
opinions almoste were as many in numbre, as were the persons of
them, that either disputed or wrote thereof. But and if the
diuersitie of opinions in the vulgar sort for placyng of their
felicitie shall be considered also, the varietie shall be found
so great, and the opinions so dissonant, yea plainly
monsterouse, that no honest witte would vouchesafe to lose time
in hearyng them, or rather (as I may saie) no witte is of so
exact remembrance, that can consider together the monsterouse
multitude of them all. And yet not withstandyng this repugnant
diuersitie, in two thynges do they all agree. First all do agre,
that felicitie is and ought to be the stop and end of all their
doynges, so that he that hath a full desire to any thyng how so
euer it be estemed of other men, yet he estemeth him self happie,
if he maie obtain it: and contrary waies vnhappie if he can not
attaine it. And therfore do all men put their whole studie to
gette that thyng, wherin they haue perswaded them self that
felicitie doth consist. Wherfore some whiche put their felicitie
in fedyng their bellies, thinke no pain to be hard, nor no dede
to be vnhonest, that may be a meanes to fill that foule panche.
Other which put their felicitie in play and ydle pastimes, iudge
no time euill spent, that is employed thereabout: nor no fraude
vnlawfull that may further their winning. If I should
particularly ouerrunne but the common sortes of men, which put
their felicitie in their desires, it wold make a great boke of
it self. Therfore wyl I let them al go, and conclude as I began,
That all men employ their whole endeuour to that thing, wherin
thei thinke felicitie to stand. whiche thyng who so listeth to
mark exactly, shall be able to espie and iudge the natures of al
men, whose conuersacion he doth know, though thei vse great
dissimulacion to colour their desires, especially when they
perceiue other men to mislyke that which thei so much desire:
For no man wold gladly haue his appetite improued. And herof
commeth that seconnde thing wherin al agree, that euery man would
most gladly win all other men to his sect, and to make them of
his opinion, and as far as he dare, will dispraise all other
mens iudgementes, and praise his own waies only, onles it be when
he dissimuleth, and that for the furtherance of his own purpose.
And this propertie also doth geue great light to the full
knowledge of mens natures, which as all men ought to obserue, so
princes aboue other haue most cause to mark for sundrie
occasions which may lye them on, wherof I shall not nede to
speke any farther, consideryng not only the greatnes of wit, and
exactnes of iudgement whiche god hath lent vnto your highnes
person, but also y^e most graue wisdom and profound knowledge of
your maiesties most honorable councel, by whom your highnes may so
sufficiently vnderstand all thinges conuenient, that lesse shal
it nede to vnderstand by priuate readying, but yet not vtterly
to refuse to read as often as occasion may serue, for bokes dare
speake, when men feare to displease. But to returne agayne to my
firste matter, if none other good thing maie be lerned at their
maners, which so wrongfully place their felicity, in so miserable
a condition (that while they thinke them selfes happy, their
felicitie must nedes seme vnluckie, to be by them so euill
placed) yet this may men learn at them, by those .ij. spectacles
to espye the secrete natures and dispositions of others, whiche
thyng vnto a wise man is muche auailable. And thus will I omit
this great tablement of vnhappie hap, and wil come to .iij.
other sortes of a better degre, wherof the one putteth felicitie
to consist in power and royaltie. The second sorte vnto power
annexeth worldly wisdome, thinkyng him full happie, that could
attayn those two, wherby he might not onely haue knowledge in
all thynges, but also power to bryng his desires to ende. The
thyrd sort estemeth true felicitie to consist in wysdom annexed
with vertuouse maners, thinkyng that they can take harme of
nothyng, if they can with their wysedome ouercome all vyces. Of
the firste of those three sortes there hath been a great numbre
in all ages, yea many mightie kinges and great gouernoures which
cared not greately howe they myght atchieue their pourpose, so
that they dyd preuayle: nor did not take any greatter care for
gouernance, then to kepe the people in onely feare of them,
Whose common sentence was alwaies this: _Oderint dum metuant_.
And what good successe suche menne had, all histories doe
report. Yet haue they not wanted excuses: yea Iulius Caesar
(whiche in dede was of the second sorte) maketh a kynde of
excuse by his common sentence, for them of that fyrste sorte,
for he was euer woonte to saie: #eiper gar adikeim chre:,
turannidos peri kallistom adikeim, t' alla d' eusebeim chreo:m.#
Whiche sentence I wysshe had neuer been learned out of Grecia.
But now to speake of the second sort, of whiche there hathe been
verye many also, yet for this present time amongest them all,
I wyll take the exaumples of kynge Phylippe of Macedonie, and of
Alexander his sonne, that valiaunt conquerour. First of kinge
Phylip it appeareth by his letter sente vnto Aristotle that
famous philosopher, that he more delited in the birthe of his
sonne, for the hope of learning and good education, that might
happen to him by the said Aristotle, then he didde reioyse in
the continuaunce of his succession, for these were his wordes
and his whole epistle, worthye to bee remembred and registred
euery where.
#Philippos Aristotelei chaireim.#
#esthe moi gegonota huom. polle:m oum tois theois charim echo:,
ouch houto:s epi te: genne:sei tou paidos, ho:s epi to: kata te:m se:m
he:likiam autom gegonenai elpizo: gar autom hupo sou graphenta kai
paideuthenta axiom esesthai kai heimo:m kai te:s to:m pragmato:m
diadoche:s.#
That is thus in sense,
Philip vnto Aristotle sendeth gretyng.
You shall vnderstande, that I haue a sonne borne, for whiche
cause I yelde vnto God moste hartie thankes, not so muche for
the byrthe of the childe, as that it was his chaunce to be borne
in your tyme. For my trust is, that he shall be so brought vp
and instructed by you, that he shall become worthie not only to
be named our sonne, but also to be the successour of our
affayres.
And his good desire was not all vayne, for it appered that
Alexander was neuer so busied with warres (yet was he neuer out
of moste terrible battaile) but that in the middes thereof he
had in remembraunce his studies, and caused in all countreies as
he went, all strange beastes, fowles and fisshes, to be taken
and kept for the ayd of that knowledg, which he learned of
Aristotle: And also to be had with him alwayes a greate numbre
of learned men. And in the moste busye tyme of all his warres
against Darius kinge of Persia, when he harde that Aristotle had
putte forthe certaine bookes of suche knowledge wherein he hadde
before studied, hee was offended with Aristotle, and wrote to
hym this letter.
#Alezandros Arisotelei eu pratteim.#
#Ouk ortho:s epoie:sas ekdous tous akroamatikous to:m logo:m, tini
gar dioisome:n he:meis to:m allo:m, ei kath' hous epaideuthe:men
logous, houtoi panto:m esontai koinoi, ego: de bouloi me:m am tais
peri ta arista empeiriais, e: tais dunamesi diapherim. erro:so.#
that is
Alexander vnto Aristotle sendeth greeting.
You haue not doone well, to put forthe those bookes of secrete
phylosophy intituled, #akroamatikoi#. For wherin shall we excell
other, yf that knowledge that wee haue studied, shall be made
commen to all other men, namely sithe our desire is to excelle
other men in experience and knowledge, rather then in power and
strength. Farewell.
By whyche lettre it appeareth that hee estemed learninge and
knowledge aboue power of men. And the like iudgement did he
vtter, when he beheld the state of Diogenes Linicus, adiudginge
it the beste state next to his owne, so that he said: If I were
not Alexander, I wolde wishe to be Diogenes. Whereby apeareth,
how he esteemed learning, and what felicity he putte therin,
reputing al the worlde saue him selfe to be inferiour to
Diogenes. And bi al coniecturs, Alexander did esteme Diogenes
one of them whiche contemned the vaine estimation of the
disceitfull world, and put his whole felicity in knowledg of
vertue, and practise of the same, though some reporte that he
knew more vertue then he folowed: But whatso euer he was, it
appeareth that Socrates and Plato and many other did forsake
their liuings and sel away their patrimony, to the intent to
seeke and trauaile for learning, which examples I shall not need
to repete to your Maiesty, partly for that your highnes doth
often reade them and other lyke, and partly sith your maiesty
hath at hand such learned schoolemaysters, which can much better
then I, declare them vnto your highnes, and that more largely
also then the shortenes of thys epistle will permit. But thys
may I yet adde, that King Salomon whose renoume spred so farre
abroad, was very greatlye estemed for his wonderfull power and
exceading treasure, but yet much more was he estemed for his
wisdom. And him selfe doth bear witnes, that wisedom is better
then pretious stones . yea all thinges that can be desired ar
not to be compared to it. But what needeth to alledge one
sentence of him, whose bookes altogither do none other thing,
then set forth the praise of wisedom & knowledg? And his father
king Dauid ioyneth uertuous conuersacion and knowledg togither,
as the summe of perfection and chief felicity. Wherfore I maye
iustelye conclude, that true felicity doth consist in wisdome
and vertu. Then if wisdome be as Cicero defineth it, _Diuinarum
atq; humanarum rerum scientia_, then ought all men to trauail
for knowledg in matters both of religion and humaine docrine, if
he shall be counted wyse, and able to attaine true felicitie:
But as the study of religious matters is most principall, so I
leue it for this time to them that better can write of it then I
can. And for humaine knowledge thys wil I boldly say, that who
soeuer wyll attain true iudgment therein, must not only trauail
in y^e knowledg of the tungs, but must also before al other
arts, taste of the mathematical sciences, specially Arithmetike
and Geometry, without which it is not possible to attayn full
knowledg in any art. Which may sufficiently by gathered by
Aristotle not only in his bookes of demonstration (whiche can not
be vnderstand without Geometry) but also in all his other
workes. And before him Plato his maister wrote this sentence on
his schole house dore. #Ageometre:tos oudeis eisito:#. Let no man
entre here (saith he) without knowledg in Geometry. Wherfore
moste mighty prince, as your most excellent Maiesty appeareth to
be borne vnto most perfect felicity, not only by reason that God
moued with the long prayers of this realme, did send your
highnes as moste comfortable inheritour to the same, but also in
that your Maiesty was borne in the time of such skilful
schoolmaisters & learned techers, as your highnes doth not a
little reioyse in, and profite by them in all kind of vertu &
knowledg. Amongst which is that heauenly knowledg most worthely to
be praised, wherbi the blindnes of errour & superstition is
exiled, & good hope conceiued that al the sedes & fruts therof,
with all kindes of vice & iniquite, wherby vertu is hindered, &
iustice defaced, shal be clean extrirped and rooted out of this
realm, which hope shal increase more and more, if it may appear
that learning be estemed & florish within this realm. And al be
it the chief learning be the diuine scriptures, which instruct
the mind principally, & nexte therto the lawes politike, which
most specially defend the right of goodes, yet is it not possible
that those two can long be wel vsed, if that ayde want that
gouerneth health and expelleth sicknes, which thing is done by
Physik, & these require the help of the vij. liberall sciences,
but of none more then of Arithmetik and Geometry, by which not
only great thinges ar wrought touching accomptes in al kinds, & in
suruaiyng & measuring of landes, but also al arts depend partly
of them, & building which is most necessary can not be w^{t}out
them, which thing considering, moued me to help to serue your
maiesty in this point as wel as other wais, & to do what mai be
in me, y^t not only thei which studi principalli for lerning, mai
haue furderance bi mi poore help, but also those whiche haue no
tyme to trauaile for exacter knowlege, may haue some helpe to
vnderstand in those Mathematicall artes, in whiche as I haue all
readye set forth sumwhat of Arithmetike, so god willing I intend
shortly to setforth a more exacter worke therof. And in the
meane ceason for a taste of Geometry, I haue sette forthe this
small introduction, desiring your grace not so muche to beholde
the simplenes of the woorke, in comparison to your Maiesties
excellencye, as to fauour the edition thereof, for the ayde of
your humble subiectes, which shal thinke them selues more and
more dayly bounden to your highnes, if when they shall perceaue
your graces desyre to haue theym profited in all knowledge and
vertue. And I for my poore ability considering your Maiesties
studye for the increase of learning generally through all your
highenes dominions, and namely in the vniuersities of Oxforde
and Camebridge, as I haue an earnest good will as far as my
simple seruice and small knowledg will suffice, to helpe toward
the satisfiyng of your graces desire, so if I shall perceaue
that my seruice may be to your maiesties contentacion, I wil not
only put forth the other two books, whiche shoulde haue beene
sette forth with these two, yf misfortune had not hindered it,
but also I wil set forth other bookes of more exacter arte,
bothe in the Latine tongue and also in the Englyshe, whereof
parte bee all readye written, and newe instrumentes to theym
deuised, and the residue shall bee eanded with all possible
speede. I was boldened to dedicate this booke of Geometrye vnto
your Maiestye, not so muche bycause it is the firste that euer
was sette forthe in Englishe, and therefore for the noueltye a
straunge presente, but for that I was perswaded, that suche a
wyse prince doothe desire to haue a wise sorte of subiectes. For
it is a kynges chiefe reioysinge and glory, if his subiectes be
riche in substaunce, and wytty in knowledge: and contrarye waies
nothyng can bee more greuouse to a noble kyng, then that his
realme should be other beggerly or full of ignoraunce: But as
god hath geuen your grace a realme bothe riche in commodities
and also full of wyttie men, so I truste by the readyng of
wyttie artes (whiche be as the whette stones of witte) they
muste needes increase more and more in wysedome, and
peraduenture fynde some thynge towarde the ayde of their
substaunce, whereby your grace shall haue newe occasion to
reioyce, seyng your subiectes to increase in substance or
wisdom, or in both. And thei again shal haue new and new causes
to pray for your maiestie, perceiuyng so graciouse a mind
towarde their benefite. And I truste (as I desire) that a great
numbre of gentlemen, especially about the courte, whiche
vnderstand not the latin tong, or els for the hardnesse of the
mater could not away with other mens writyng, will fall in trade
with this easie forme of teachyng in their vulgar tong, and so
employe some of their tyme in honest studie, whiche were wont to
bestowe most part of their time in triflyng pastime: For
vndoubtedly if they mean other your maiesties seruice, other
their own wisdome, they will be content to employ some tyme
aboute this honest and wittie exercise. For whose encouragement
to the intent they maie perceiue what shall be the vse of this
science, I haue not onely written somewhat of the vse of
Geometrie, but also I haue annexed to this boke the names and
brefe argumentes of those other bokes whiche I will set forth
hereafter, and that as shortly as it shall appeare vnto your
maiestie by coniecture of their diligent vsyng of this first
boke, that they wyll vse well the other bokes also. In the meane
ceason, and at all times I wil be a continuall petitioner, that
god may work in all english hartes an ernest mynde to all honest
exercises, wherby thei may serue the better your maiestie and
the realm. And for your highnes I besech the most mercifull god,
as he hath most fauourably sent you vnto vs, as our chefe
comforter in earthe, so that he will increase your maiestie
daiely in all vertue and honor with moste prosperouse successe,
and augment in vs your most humble subiectes, true loue to
godward, and iust obedience toward your highnes with all
reuerence and subiection.
At London the .xxviij. daie of Ianuarie. M. D. L I.
_Your maiesties moste humble seruant
and obedient subiect,
Robert Recorde._
+THE PREFACE,+
declaring briefely the commodi-
tes of Geometrye, and the
necessitye thereof.
Geometrye may thinke it selfe to sustaine great iniury, if it
shall be inforced other to show her manifold commodities, or els
not to prease into the sight of men, and therefore might this
wayes answere briefely: Other I am able to do you much good, or
els but litle. If I bee able to doo you much good, then be you
not your owne friendes, but greatlye your owne enemies to make
so little of me, which maye profite you so muche. For if I were
as vncurteous as you vnkind, I shuld vtterly refuse to do them
any good, which will so curiously put me to the trial and profe
of my commodities, or els to suffre exile, and namely sithe I
shal only yeld benefites to other, and receaue none againe. But
and if you could saye truely, that my benefites be nother many
nor yet greate, yet if they bee anye, I doo yelde more to you,
then I doo receaue againe of you, and therefore I oughte not to
bee repelled of them that loue them selfe, althoughe they loue
me not all for my selfe. But as I am in nature a liberall
science, so canne I not againste nature contende with your
inhumanitye, but muste shewe my selfe liberall euen to myne
enemies. Yet this is my comforte againe, that I haue none
enemies but them that knowe me not, and therefore may hurte
themselues, but can not noye me. Yf they dispraise the thinge
that they know not, all wise men will blame them and not credite
them, and yf they thinke they knowe me, lette theym shewe one
vntruthe and erroure in me, and I wyll geue the victorye.
Yet can no humayne science saie thus, but I onely, that there is
no sparke of vntruthe in me: but all my doctrine and workes are
without any blemishe of errour that mans reason can discerne.
And nexte vnto me in certaintie are my three systers,
Arithmetike, Musike, and Astronomie, whiche are also so nere
knitte in amitee, that he that loueth the one, can not despise
the other, and in especiall Geometrie, of whiche not only these
thre, but all other artes do borow great ayde, as partly
hereafter shall be shewed. But first will I beginne with the
vnlearned sorte, that you maie perceiue how that no arte can
stand without me. For if I should declare how many wayes my
helpe is vsed, in measuryng of ground, for medow, corne, and
wodde: in hedgyng, in dichyng, and in stackes makyng, I thinke
the poore Husband man would be more thankefull vnto me, then he
is nowe, whyles he thinketh that he hath small benefite by me.
Yet this maie he coniecture certainly, that if he kepe not the
rules of Geometrie, he can not measure any ground truely. And in
dichyng, if he kepe not a proportion of bredth in the mouthe, to
the bredthe of the bottome, and iuste slopenesse in the sides
agreable to them bothe, the diche shall be faultie many waies.
When he doth make stackes for corne, or for heye, he practiseth
good Geometrie, els would thei not long stand: So that in some
stakes, whiche stand on foure pillers, and yet made round, doe
increase greatter and greatter a good height, and then againe
turne smaller and smaller vnto the toppe: you maie see so good
Geometrie, that it were very difficult to counterfaite the lyke
in any kynde of buildyng. As for other infinite waies that he
vseth my benefite, I ouerpasse for shortnesse.
Carpenters, Karuers, Ioyners, and Masons, doe willingly
acknowledge that they can worke nothyng without reason of
Geometrie, in so muche that they chalenge me as a peculiare
science for them. But in that they should do wrong to all other
men, seyng euerie kynde of men haue som benefit by me, not only
in buildyng, whiche is but other mennes costes, and the arte of
Carpenters, Masons, and the other aforesayd, but in their owne
priuate profession, whereof to auoide tediousnes I make this
rehersall.
Sith Merchauntes by shippes great riches do winne,
I may with good righte at their seate beginne.
The Shippes on the sea with Saile and with Ore,
were firste founde, and styll made, by Geometries lore.
Their Compas, their Carde, their Pulleis, their Ankers,
were founde by the skill of witty Geometers.
To sette forth the Capstocke, and eche other parte,
wold make a greate showe of Geometries arte.
Carpenters, Caruers, Ioiners and Masons,
Painters and Limners with suche occupations,
Broderers, Goldesmithes, if they be cunning,
Must yelde to Geometrye thankes for their learning.
The Carte and the Plowe, who doth them well marke,
Are made by good Geometrye. And so in the warke
Of Tailers and Shoomakers, in all shapes and fashion,
The woorke is not praised, if it wante proportion.
So weauers by Geometrye hade their foundacion,
Their Loome is a frame of straunge imaginacion.
The wheele that doth spinne, the stone that doth grind,
The Myll that is driuen by water or winde,
Are workes of Geometrye straunge in their trade,
Fewe could them deuise, if they were vnmade.
And all that is wrought by waight or by measure,
without proofe of Geometry can neuer be sure.
Clockes that be made the times to deuide,
The wittiest inuencion that euer was spied,
Nowe that they are common they are not regarded,
The artes man contemned, the woorke vnrewarded.
But if they were scarse, and one for a shewe,
Made by Geometrye, then shoulde men know,
That neuer was arte so wonderfull witty,
So needefull to man, as is good Geometry.
The firste findinge out of euery good arte,
Seemed then vnto men so godly a parte,
That no recompence might satisfye the finder,
But to make him a god, and honoure him for euer.
So Ceres and Pallas, and Mercury also,
Eolus and Neptune, and many other mo,
Were honoured as goddes, bicause they did teache,
Firste tillage and weuinge and eloquent speache,
Or windes to obserue, the seas to saile ouer,
They were called goddes for their good indeuour.
Then were men more thankefull in that golden age:
This yron wolde nowe vngratefull in rage,
Wyll yelde the thy reward for trauaile and paine,
With sclaunderous reproch, and spitefull disdaine.
Yet thoughe other men vnthankfull will be,
Suruayers haue cause to make muche of me.
And so haue all Lordes, that landes do possesse:
But Tennaunted I feare will like me the lesse.
Yet do I not wrong but measure all truely,
All yelde the full right of euerye man iustely.
Proportion Geometricall hath no man opprest,
Yf anye bee wronged, I wishe it redrest.
But now to procede with learned professions, in Logike and
Rhetorike and all partes of phylosophy, there neadeth none other
proofe then Aristotle his testimony, whiche without Geometry
proueth almost nothinge. In Logike all his good syllogismes and
demonstrations, hee declareth by the principles of Geometrye. In
philosophye, nether motion, nor time, nor ayrye impressions
could hee aptely declare, but by the helpe of Geometrye as his
woorkes do witnes. Yea the faculties of the minde dothe hee
expresse by similitude to figures of Geometrye. And in morall
phylosophy he thought that iustice coulde not wel be taught, nor
yet well executed without proportion geometricall. And this
estimacion of Geometry he maye seeme to haue learned of his
maister Plato, which without Geometrye wolde teache nothinge,
nother wold admitte any to heare him, except he were experte in
Geometry. And what merualle if he so muche estemed geometrye,
seinge his opinion was, that Godde was alwaies workinge by
Geometrie? Whiche sentence Plutarche declareth at large. And
although Platto do vse the helpe of Geometrye in all the most
waighte matter of a common wealth, yet it is so generall in vse,
that no small thinges almost can be wel done without it. And
therfore saith he: that Geometrye is to be learned, if it were
for none other cause, but that all other artes are bothe soner
and more surely vnderstand by helpe of it.
What greate help it dothe in physike, Galene doth so often and
so copiousely declare, that no man whiche hath redde any booke
almoste of his, can be ignorant thereof, in so much that he
coulde neuer cure well a rounde vlcere, tyll reason geometricall
dydde teache it hym. Hippocrates is earnest in admonyshynge that
study of geometrie must prepare the way to physike, as well as
to all other artes.
I shoulde seeme somewhat to tedious, if I shoulde recken vp,
howe the diuines also in all their mysteries of scripture doo
vse healpe of geometrie: and also that lawyers can neuer
vnderstande the hole lawe, no nor yet the firste title therof
exactly without Geometrie. For if lawes can not well be
established, nor iustice duelie executed without geometricall
proportion, as bothe Plato in his Politike bokes, and Aristotle
in his Moralles doo largely declare. Yea sithe Lycurgus that
cheefe lawmaker amongest the Lacedemonians, is moste praised for
that he didde chaunge the state of their common wealthe frome
the proportion Arithmeticall to a proportion geometricall,
whiche without knowledg of bothe he coulde not dooe, than is it
easye to perceaue howe necessarie Geometrie is for the lawe and
studentes thereof. And if I shall saie preciselie and freelie as
I thinke, he is vtterlie destitute of all abilitee to iudge in
anie arte, that is not sommewhat experte in the Theoremes of
Geometrie.
And that caused Galene to say of hym selfe, that he coulde neuer
perceaue what a demonstration was, no not so muche, as whether
there were any or none, tyll he had by geometrie gotten abilitee
to vnderstande it, although he heard the beste teachers that
were in his tyme. It shuld be to longe and nedelesse also to
declare what helpe all other artes Mathematicall haue by
geometrie, sith it is the grounde of all theyr certeintie, and
no man studious in them is so doubtful therof, that he shall
nede any persuasion to procure credite thereto. For he can not
reade .ij. lines almoste in any mathematicall science, but he
shall espie the nedefulnes of geometrie. But to auoyde
tediousnesse I will make an ende hereof with that famous
sentence of auncient Pythagoras, That who so will trauayle by
learnyng to attayne wysedome, shall neuer approche to any
excellencie without the artes mathematicall, and especially
Arithmetike and Geometrie.
And yf I shall somewhat speake of noble men, and gouernours of
realmes, howe needefull Geometrye maye bee vnto them, then must
I repete all that I haue sayde before, sithe in them ought all
knowledge to abounde, namely that maye appertaine either to good
gouernaunce in time of peace, eyther wittye pollicies in time of
warre. For ministration of good lawes in time of peace Lycurgus
example with the testimonies of Plato and Aristotle may suffise.
And as for warres, I might thinke it sufficient that Vegetius
hath written, and after him Valturius in commendation of
Geometry, for vse of warres, but all their woordes seeme to saye
nothinge, in comparison to the example of Archimedes worthy
woorkes make by geometrie, for the defence of his countrey, to
reade the wonderfull praise of his wittie deuises, set foorthe
by the most famous hystories of Liuius, Plutarche, and Plinie,
and all other hystoriographiers, whyche wryte of the stronge
siege of _Syracusae_ made by that valiant capitayne, and noble
warriour _Marcellus_, whose power was so great, that all men
meruayled how that one citee coulde withstande his wonderfull
force so longe. But much more woulde they meruaile, if they
vnderstode that one man onely dyd withstand all Marcellus
strength, and with counter engines destroied his engines to the
vtter astonyshment of _Marcellus_, and all that were with hym.
He had inuented suche balastelas that dyd shoote out a hundred
dartes at one shotte, to the great destruction of _Marcellus_
souldiours, wherby a fonde tale was spredde abrode, how that in
Syracusae there was a wonderfull gyant, whiche had a hundred
handes, and coulde shoote a hundred dartes at ones. And as this
fable was spredde of Archimedes, so many other haue been fayned
to bee gyantes and monsters, bycause they dyd suche thynges,
whiche farre passed the witte of the common people. So dyd they
feyne Argus to haue a hundred eies, bicause they herde of his
wonderfull circumspection, and thoughte that as it was aboue
their capacitee, so it could not be, onlesse he had a hundred
eies. So imagined they Ianus to haue two faces, one lokyng
forwarde, and an other backwarde, bycause he coulde so wittily
compare thynges paste with thynges that were to come, and so
duely pondre them, as yf they were all present. Of like reason
did they feyn Lynceus to haue such sharp syght, that he could
see through walles and hylles, bycause peraduenture he dyd by
naturall iudgement declare what commoditees myght be digged out
of the grounde. And an infinite noumbre lyke fables are there,
whiche sprange all of lyke reason.
For what other thyng meaneth the fable of the great gyant Atlas,
whiche was ymagined to beare vp heauen on his shulders? but that
he was a man of so high a witte, that it reached vnto the skye,
and was so skylfull in Astronomie, and coulde tell before hande
of Eclipses, and other like thynges as truely as though he dyd
rule the sterres, and gouerne the planettes.
So was Eolus accompted god of the wyndes, and to haue theim all
in a caue at his pleasure, by reason that he was a wittie man in
naturall knowlege, and obserued well the change of wethers, aud
was the fyrst that taught the obseruation of the wyndes. And
lyke reson is to be geuen of al the old fables.
But to retourne agayne to Archimedes, he dyd also by arte
perspectiue (whiche is a parte of geometrie) deuise such glasses
within the towne of Syracusae, that dyd bourne their ennemies
shyppes a great way from the towne, whyche was a meruaylous
politike thynge. And if I shulde repete the varietees of suche
straunge inuentions, as Archimedes and others haue wrought by
geometrie, I should not onely excede the order of a preface, but
I should also speake of suche thynges as can not well be
vnderstande in talke, without somme knowledge in the principles
of geometrie.
But this will I promyse, that if I may perceaue my paynes to be
thankfully taken, I wyll not onely write of suche pleasant
inuentions, declaryng what they were, but also wil teache howe a
great numbre of them were wroughte, that they may be practised
in this tyme also. Wherby shallbe plainly perceaued, that many
thynges seme impossible to be done, whiche by arte may very well
be wrought. And whan they be wrought, and the reason therof not
vnderstande, than say the vulgare people, that those thynges are
done by negromancy. And hereof came it that fryer Bakon was
accompted so greate a negromancier, whiche neuer vsed that arte
(by any coniecture that I can fynde) but was in geometrie and
other mathematicall sciences so experte, that he coulde dooe by
theim suche thynges as were wonderfull in the syght of most
people.
Great talke there is of a glasse that he made in Oxforde, in
whiche men myght see thynges that were doon in other places, and
that was iudged to be done by power of euyll spirites. But I
knowe the reason of it to bee good and naturall, and to be
wrought by geometrie (sythe perspectiue is a parte of it) and to
stande as well with reason as to see your face in common glasse.
But this conclusion and other dyuers of lyke sorte, are more
mete for princes, for sundry causes, than for other men, and
ought not to bee taught commonly. Yet to repete it, I thought
good for this cause, that the worthynes of geometry myght the
better be knowen, & partly vnderstanding geuen, what wonderfull
thynges may be wrought by it, and so consequently how pleasant
it is, and how necessary also.
And thus for this tyme I make an end. The reason of som thynges
done in this boke, or omitted in the same, you shall fynde in
the preface before the Theoremes.
The definitions of the principles of
_GEOMETRY_.
Geometry teacheth the drawyng, Measuring and proporcion of
figures. but in as muche as no figure can bee drawen, but it
muste haue certayne boundes and inclosures of lines: and euery
lyne also is begon and ended at some certaine prycke, fyrst it
shal be meete to know these smaller partes of euery figure, that
therby the whole figures may the better bee iudged, and
distincte in sonder.
[Sidenote: A poincte.] _A Poynt or a Prycke_, is named of
Geometricians that small and vnsensible shape, whiche hath in it
no partes, that is to say: nother length, breadth nor depth. But
as their exactnes of definition is more meeter for onlye
Theorike speculacion, then for practise and outwarde worke
(consideringe that myne intent is to applye all these whole
principles to woorke) I thynke meeter for this purpose, to call
a _poynt or prycke_, that small printe of penne, pencyle, or
other instrumente, whiche is not moued, nor drawen from his
fyrst touche, and therfore hath no notable length nor bredthe:
as this example doeth declare.
[Illustration: {"therefore" symbol}]
Where I haue set .iij. prickes, eche of them hauyng both length
and bredth, thogh it be but smal, and thefore not notable.
Nowe of a great numbre of these prickes, is made a _Lyne_, as
you may perceiue by this forme ensuyng. ........................
where as I haue set a numbre of prickes, so if you with your pen
will set in more other prickes betweene euerye two of these,
[Sidenote: A lyne.] then wil it be a lyne, as here you may see
-------- and this _lyne_, is called of Geometricians, _Lengthe
withoute breadth_.
But as they in theyr theorikes (which ar only mind workes) do
precisely vnderstand these definitions, so it shal be sufficient
for those men, whiche seke the vse of the same thinges, as sense
may duely iudge them, and applye to handy workes if they
vnderstand them so to be true, that outwarde sense canne fynde
none erroure therein.
Of lynes there bee two principall kyndes. The one is called a
right or straight lyne, and the other a croked lyne.
[Sidenote: A streghte lyne.] _A Straight lyne_, is the shortest
that maye be drawenne between two prickes.
[Sidenote: A crokyd lyne.] And all other lines, that go not
right forth from prick to prick, but boweth any waye, such are
called _Croked lynes_ as in these examples folowyng ye may se,
where I haue set but one forme of a straight lyne, for more
formes there be not, but of crooked lynes there bee innumerable
diuersities, whereof for examples sum I haue sette here.
[Illustration: A right lyne.]
[Illustration: _Croked lynes._]
[Illustration: Croked lines.]
So now you must vnderstand, that _euery lyne is drawen betwene
twoo prickes_, wherof the one is at the beginning, and the other
at the ende.
[Illustration]
Therefore when soeuer you do see any formes of lynes to touche
at one notable pricke, as in this example, then shall you not
call it one croked lyne, but rather twoo lynes: [Sidenote: an
Angle.] in as muche as there is a notable and sensible angle by
.A. whiche euermore is made by the meetyng of two seuerall
lynes. And likewayes shall you iudge of this figure, whiche is
made of two lines, and not of one onely.
[Illustration]
So that whan so euer any suche meetyng of lines doth happen, the
place of their metyng is called an _Angle or corner_.
Of angles there be three generall kindes: a sharpe angle, a
square angle, and a blunte angle. [Sidenote: A righte angle.]
_The square angle_, whiche is commonly named _a right corner_,
is made of twoo lynes meetyng together in fourme of a squire,
whiche two lines, if they be drawen forth in length, will crosse
one an other: as in the examples folowyng you maie see.
[Sidenote: A sharpe corner.] _A sharpe angle_ is so called,
because it is lesser than is a square angle, and the lines that
make it, do not open so wide in their departynge as in a square
corner, and if thei be drawen crosse, all fower corners will not
be equall.
[Sidenote: A blunte angle.] _A blunt or brode corner_, is
greater then is a square angle, and his lines do parte more in
sonder then in a right angle, of whiche all take these examples.
[Illustration: Right angles.]
[Illustration: Sharpe angles.]
[Illustration: Blunte or brode angles.]
And these angles (as you see) are made partly of streght lynes,
partly of croken lines, and partly of both together. Howbeit in
right angles I haue put none example of croked lines, because it
would muche trouble a lerner to iudge them: for their true
iudgment doth appertaine to arte perspectiue, and as I may say,
rather to reason then to sense.
But now as of many prickes there is made one line, so _of
diuerse lines are there made sundry formes, figures, and
shapes_, whiche all yet be called by one propre name, [Sidenote:
A platte forme.] _Platte formes_, and thei haue bothe _length
and bredth, but yet no depenesse_.
And _the boundes_ of euerie platte forme are lines: as by the
examples you maie perceiue.
Of platte formes some be plain, and some be croked, and some
parly plaine, and partlie croked.
[Sidenote: A plaine platte.] _A plaine platte_ is that, whiche
is made al equall in height, so that the middle partes nother
bulke vp, nother shrink down more then the bothe endes.
[Sidenote: A crooked platte.] For whan the one parte is higher
then the other, then is it named a _Croked platte_.
And if it be partlie plaine, and partlie crooked, then is it
called a _Myxte platte_, of all whiche, these are exaumples.
[Illustration: A plaine platte.]
[Illustration: A croked platte.]
[Illustration: A myxte platte.]
And as of many prickes is made a line, and of diuerse lines one
platte forme, [Sidenote: A bodie.] so of manie plattes is made
_a bodie_, whiche conteigneth _Lengthe, bredth, and depenesse_.
[Sidenote: Depenesse.] By _Depenesse_ I vnderstand, not as the
common sort doth, the holownesse of any thing, as of a well,
a diche, a potte, and suche like, but I meane the massie
thicknesse of any bodie, as in exaumple of a potte: the
depenesse is after the common name, the space from his brimme to
his bottome. But as I take it here, the depenesse of his bodie
is his thicknesse in the sides, whiche is an other thyng cleane
different from the depenesse of his holownes, that the common
people meaneth.
Now all bodies haue platte formes for their boundes, [Sidenote:
Cubike.] so in a dye (whiche is called _a cubike bodie_) by
geomatricians, [Sidenote: Asheler.] and an _ashler_ of masons,
there are .vi. sides, whiche are .vi. platte formes, and are the
boundes of the dye.
[Sidenote: A globe.] But in a _Globe_, (whiche is a bodie rounde
as a bowle) there is but one platte forme, and one bounde, and
these are the exaumples of them bothe.
[Illustration: A dye or ashler.]
[Illustration: A globe.]
But because you shall not muse what I dooe call _a bound_,
[Sidenote: A bounde.] I mean therby a generall name, betokening
the beginning, end and side, of any forme.
[Sidenote: Forme, Fygure.] _A forme, figure, or shape_, is that
thyng that is inclosed within one bond or manie bondes, so that
you vnderstand that shape, that the eye doth discerne, and not
the substance of the bodie.
Of _figures_ there be manie sortes, for either thei be made of
prickes, lines, or platte formes. Not withstandyng to speake
properlie, _a figure_ is euer made by platte formes, and not of
bare lines vnclosed, neither yet of prickes.
Yet for the lighter forme of teachyng, it shall not be vnsemely
to call all suche shapes, formes and figures, whiche y^e eye
maie discerne distinctly.
And first to begin with prickes, there maie be made diuerse
formes of them, as partely here doeth folowe.
[Illustration:
A lynearic numbre.
Trianguler numbres
Longsquare numbre.
Iust square numbres
a threcornered spire.
A square spire.]
And so maie there be infinite formes more, whiche I omitte for
this time, considering that their knowledg appertaineth more to
Arithmetike figurall, than to Geometrie.
But yet one name of a pricke, whiche he taketh rather of his
place then of his fourme, maie I not ouerpasse. And that is,
when a pricke standeth in the middell of a circle (as no circle
can be made by compasse without it) then is it called _a centre_.
[Sidenote: A centre] And thereof doe masons, and other worke
menne call that patron, a _centre_, whereby thei drawe the
lines, for iust hewyng of stones for arches, vaultes, and
chimneies, because the chefe vse of that patron is wrought by
findyng that pricke or centre, from whiche all the lynes are
drawen, as in the thirde booke it doeth appere.
Lynes make diuerse figures also, though properly thei maie not
be called figures, as I said before (vnles the lines do close)
but onely for easie maner of teachyng, all shall be called
figures, that the eye can discerne, of whiche this is one, when
one line lyeth flatte (whiche is named [Sidenote: A ground
line.] the _ground line_) and an other commeth downe on it, and
is called [Sidenote: A perpendicular.] [Sidenote: A plume lyne.]
a _perpendiculer_ or _plumme lyne_, as in this example you may
see. where .A.B. is the grounde line, and C.D. the plumbe line.
[Illustration]
And like waies in this figure there are three lines, the grounde
lyne whiche is A.B. the plumme line that is A.C. and the _bias
line_, whiche goeth from the one of them to the other, and lieth
against the right corner in such a figure whiche is here .C.B.
[Illustration]
But consideryng that I shall haue occasion to declare sundry
figures anon, I will first shew some certaine varietees of lines
that close no figures, but are bare lynes, and of the other
lines will I make mencion in the description of the figures.
[Illustration: tortuouse paralleles.]
[Sidenote: Parallelys]
[Sidenote: Gemowe lynes.]
_Paralleles_, or _gemowe lynes_ be suche lines as be drawen
foorth still in one distaunce, and are no nerer in one place
then in an other, for and if they be nerer at one ende then at
the other, then are they no paralleles, but maie bee called
_bought lynes_, and loe here exaumples of them bothe.
[Illustration: parallelis.]
[Illustration: bought lines]
[Illustration: parallelis: circular. Concentrikes.]
I haue added also _paralleles tortuouse_, whiche bowe contrarie
waies with their two endes: and _paralleles circular_, whiche be
lyke vnperfecte compasses: for if they bee whole circles,
[Sidenote: Concentrikes] then are they called _concentrikes_,
that is to saie, circles drawen on one centre.
Here might I note the error of good _Albert Durer_, which
affirmeth that no perpendicular lines can be paralleles. which
errour doeth spring partlie of ouersight of the difference of a
streight line, and partlie of mistakyng certain principles
geometrical, which al I wil let passe vntil an other tyme, and
wil not blame him, which hath deserued worthyly infinite praise.
And to returne to my matter. [Sidenote: A twine line.] an other
fashioned line is there, which is named a twine or twist line,
and it goeth as a wreyth about some other bodie. [Sidenote:
A spirall line.] And an other sorte of lines is there, that is
called a _spirall line_, [Sidenote: A worme line.] or a _worm
line_, whiche representeth an apparant forme of many circles,
where there is not one in dede: of these .ii. kindes of lines,
these be examples.
[Illustration: A twiste lyne.]
[Illustration: A spirail lyne]
[Illustration: A touche lyne.]
[Sidenote: A tuch line.]
_A touche lyne_, is a line that runneth a long by the edge of a
circle, onely touching it, but doth not crosse the circumference
of it, as in this exaumple you maie see.
[Sidenote: A corde,]
And when that a line doth crosse the edg of the circle, then is
it called _a cord_, as you shall see anon in the speakynge of
circles.
[Sidenote: Matche corners]
In the meane season must I not omit to declare what angles bee
called _matche corners_, that is to saie, suche as stande
directly one against the other, when twoo lines be drawen a
crosse, as here appereth.
[Illustration: Matche corner. Matche corner.]
Where A. and B. are matche corners, so are C. and D. but not A.
and C. nother D. and A.
Nowe will I beginne to speak of figures, that be properly so
called, of whiche all be made of diuerse lines, except onely a
circle, an egge forme, and a tunne forme, which .iij. haue no
angle and haue but one line for their bounde, and an eye fourme
whiche is made of one lyne, and hath an angle onely.
[Sidenote: A circle.]
_A circle_ is a figure made and enclosed with one line, and hath
in the middell of it a pricke or centre, from whiche all the
lines that be drawen to the circumference are equall all in
length, as here you see.
[Illustration]
[Sidenote: Circumference.] And the line that encloseth the whole
compasse, is called the _circumference_.
[Sidenote: A diameter.] And all the lines that bee drawen crosse
the circle, and goe by the centre, are named _diameters_, whose
halfe, I meane from the center to the circumference any waie,
[Sidenote: Semidiameter.] is called the _semidiameter_, or
_halfe diameter_.
[Illustration]
But and if the line goe crosse the circle, and passe beside the
centre, [Sidenote: A cord, or a stringlyne.] then is it called
_a corde_, or _a stryng line_, as I said before, and as this
exaumple sheweth: where A. is the corde. And the compassed line
that aunswereth to it, [Sidenote: An archline] is called _an
arche lyne_, [Sidenote: A bowline.] or _a bowe lyne_, whiche
here marked with B. and the diameter with C.
[Illustration]
But and if that part be separate from the rest of the circle (as
in this example you see) then ar both partes called cantelles,
[Sidenote: A cantle] the one the _greatter cantle_ as E. and the
other the _lesser cantle_, as D. And if it be parted iuste by
the centre (as you see in F.) [Sidenote: A semyecircle] then is
it called a _semicircle_, or _halfe compasse_.
[Illustration]
Sometimes it happeneth that a cantle is cutte out with two lynes
drawen from the centre to the circumference (as G. is)
[Sidenote: A nooke cantle] and then maie it be called a _nooke
cantle_, and if it be not parted from the reste of the circle
(as you see in H.) [Sidenote: A nooke.] then is it called a
_nooke_ plainely without any addicion. And the compassed lyne in
it is called an _arche lyne_, as the exaumple here doeth shewe.
[Illustration: An arche.]
Nowe haue you heard as touchyng circles, meetely sufficient
instruction, so that it should seme nedeles to speake any more
of figures in that kynde, saue that there doeth yet remaine ij.
formes of an imperfecte circle, for it is lyke a circle that
were brused, and thereby did runne out endelong one waie, whiche
forme Geometricians dooe call an [Sidenote: An egge fourme.]
_egge forme_, because it doeth represent the figure and shape of
an egge duely proportioned (as this figure sheweth) hauyng the
one ende greate then the other.
[Illustration: An egge forme]
[Illustration: A tunne forme.]
[Sidenote: A tunne or barrel form] For if it be lyke the figure
of a circle pressed in length, and bothe endes lyke bygge, then
is it called a _tunne forme_, or _barrell forme_, the right
makyng of whiche figures, I wyll declare hereafter in the thirde
booke.
An other forme there is, whiche you maie call a nutte forme, and
is made of one lyne muche lyke an egge forme, saue that it hath
a sharpe angle.
And it chaunceth sometyme that there is a right line drawen
crosse these figures, [Sidenote: An axtre or axe lyne.] and that
is called an _axelyne_, or _axtre_. Howe be it properly that
line that is called an _axtre_, whiche gooeth throughe the
myddell of a Globe, for as a diameter is in a circle, so is an
axe lyne or axtre in a Globe, that lyne that goeth from side to
syde, and passeth by the middell of it. And the two poyntes that
suche a lyne maketh in the vtter bounde or platte of the globe,
are named _polis_, w^{ch} you may call aptly in englysh, _tourne
pointes_: of whiche I do more largely intreate, in the booke
that I haue written of the vse of the globe.
[Illustration]
But to returne to the diuersityes of figures that remayne
vndeclared, the most simple of them ar such ones as be made but
of two lynes, as are the _cantle of a circle_, and the _halfe
circle_, of which I haue spoken allready. Likewyse the _halfe of
an egge forme_, the _cantle of an egge forme_, the _halfe of a
tunne fourme_, and the _cantle of a tunne fourme_, and besyde
these a figure moche like to a tunne fourne, saue that it is
sharp couered at both the endes, and therfore doth consist of
twoo lynes, where a tunne forme is made of one lyne, [Sidenote:
An yey fourme] and that figure is named an _yey fourme_.
[Illustration]
[Sidenote: A triangle]
The nexte kynd of figures are those that be made of .iij. lynes
other be all right lynes, all crooked lynes, other some right
and some crooked. But what fourme so euer they be of, they are
named generally triangles. for _a triangle_ is nothinge els to
say, but a figure of three corners. And thys is a generall rule,
looke how many lynes any figure hath, so mannye corners it hath
also, yf it bee a platte forme, and not a bodye. For a bodye
hath dyuers lynes metyng sometime in one corner.
[Illustration: A]
Now to geue you example of triangles, there is one whiche is all
of croked lynes, and may be taken fur a portion of a globe as the
figur marked w^t A.
[Illustration: B]
An other hath two compassed lines and one right lyne, and is as
the portion of halfe a globe, example of B.
[Illustration: C]
An other hath but one compassed lyne, and is the quarter of a
circle, named a quadrate, and the ryght lynes make a right
corner, as you se in C. Otherlesse then it as you se D, whose
right lines make a sharpe corner, or greater then a quadrate, as
is F, and then the right lynes of it do make a blunt corner.
[Illustration: D]
Also some triangles haue all righte lynes and they be distincted
in sonder by their angles, or corners. for other their corners
bee all sharpe, as you see in the figure, E. other ij. sharpe
and one blunt, as is the figure G. other ij. sharp and one blunt
as in the figure H.
[Illustration: E]
[Illustration: F]
There is also an other distinction of the names of triangles,
according to their sides, whiche other be all equal as in the
figure E, and that the Greekes doo call _Isopleuron_, [Sidenote:
#isopleurom#.] and Latine men _aeequilaterum_: and in english it
may be called a _threlike triangle_, other els two sydes bee
equall and the thyrd vnequall, which the Greekes call
_Isosceles_, [Sidenote: #isoskeles#.] the Latine men _aequicurio_,
and in english _tweyleke_ may they be called, as in G, H, and K.
For, they may be of iij. kinds that is to say, with one square
angle, as is G, or with a blunte corner as H, or with all in
sharpe korners, as you see in K.
[Illustration: G]
[Illustration: H]
[Illustration: K]
Further more it may be y^t they haue neuer a one syde equall
to an other, and they be in iij kyndes also distinct lyke the
twilekes, as you maye perceaue by these examples .M. N, and O.
where M. hath a right angle, N, a blunte angle, and O, all
sharpe angles [Sidenote: #skalenom#.] these the Greekes and
latine men do cal _scalena_ and in englishe theye may be
called _nouelekes_, for thei haue no side equall, or like long,
to ani other in the same figur. Here it is to be noted, that in
a triangle al the angles bee called _innerangles_ except ani side
bee drawenne forth in lengthe, for then is that fourthe corner
caled an _vtter corner_, as in this example because A.B, is
drawen in length, therfore the angle C, is called an vtter angle.
[Illustration: M]
[Illustration: N]
[Illustration: O]
[Illustration]
[Illustration: Q]
[Sidenote: Quadrangle] And thus haue I done with trianguled
figures, and nowe foloweth _quadrangles_, which are figures of
iiij. corners and of iiij. lines also, of whiche there be diuers
kindes, but chiefely v. that is to say, [Sidenote: A square
quadrate.] a _square quadrate_, whose sides bee all equall, and
al the angles square, as you se here in this figure Q.
[Sidenote: A longe square.] The second kind is called a long
square, whose foure corners be all square, but the sides are not
equall eche to other, yet is euery side equall to that other
that is against it, as you maye perceaue in this figure. R.
[Illustration: R]
[Sidenote: A losenge] The thyrd kind is called _losenges_
[Sidenote: A diamond.] or _diamondes_, whose sides bee all
equall, but it hath neuer a square corner, for two of them be
sharpe, and the other two be blunt, as appeareth in .S.
[Illustration: S]
The iiij. sorte are like vnto losenges, saue that they are
longer one waye, and their sides be not equal, yet ther corners
are like the corners of a losing, and therfore ar they named
[Sidenote: A losenge lyke.] _losengelike_ or _diamondlike_, whose
figur is noted with T. Here shal you marke that al those squares
which haue their sides al equal, may be called also for easy
vnderstandinge, _likesides_, as Q. and S. and those that haue
only the contrary sydes equal, as R. and T. haue, those wyll I
call _likeiammys_, for a difference.
[Illustration: T]
[Illustration]
The fift sorte doth containe all other fashions of foure
cornered figurs, and ar called of the Grekes _trapezia_, of
Latin men _mensulae_ and of Arabitians, _helmuariphe_, they may be
called in englishe _borde formes_, [Sidenote: Borde formes.]
they haue no syde equall to an other as these examples shew,
neither keepe they any rate in their corners, and therfore are
they counted _vnruled formes_, and the other foure kindes onely
are counted _ruled formes_, in the kynde of quadrangles. Of
these vnruled formes ther is no numbre, they are so mannye and
so dyuers, yet by arte they may be changed into other kindes of
figures, and therby be brought to measure and proportion, as in
the thirtene conclusion is partly taught, but more plainly in my
booke of measuring you may see it.
And nowe to make an eande of the dyuers kyndes of figures, there
dothe folowe now figures of .v. sydes, other .v. corners, which
we may call _cink-angles_, whose sydes partlye are all equall as
in A, and those are counted _ruled cinkeangles_, and partlye
vnequall, as in B, and they are called _vnruled_.
[Illustration: A]
[Illustration: B]
Likewyse shall you iudge of _siseangles_, which haue sixe
corners, _septangles_, whiche haue seuen angles, and so forth,
for as mannye numbres as there maye be of sydes and angles, so
manye diuers kindes be there of figures, vnto which yow shall
geue names according to the numbre of their sides and angles, of
whiche for this tyme I wyll make an ende, [Sidenote: A squyre.]
and wyll sette forthe on example of a syseangle, whiche I had
almost forgotten, and that is it, whose vse commeth often in
Geometry, and is called a _squire_, is made of two long squares
ioyned togither, as this example sheweth.
[Illustration]
And thus I make an eand to speake of platte formes, and will
briefelye saye somwhat touching the figures of _bodeis_ which
partly haue one platte forme for their bound, and y^t iust round
as a _globe_ hath, or ended long as in an _egge_, and a _tunne
fourme_, whose pictures are these.
[Illustration: The globe as is before.]
Howe be it you must marke that I meane not the very figure of a
tunne, when I saye tunne form, but a figure like a tunne, for a
_tune fourme_, hath but one plat forme, and therfore must needs
be round at the endes, where as a _tunne_ hath thre platte
formes, and is flatte at eche end, as partly these pictures do
shewe.
_Bodies of two plattes_, are other cantles or halues of those
other bodies, that haue but one platte forme, or els they are
lyke in foorme to two such cantles ioyned togither as this A.
doth partly eppresse: or els it is called a _rounde spire_, or
_stiple fourme_, as in this figure is some what expressed.
[Sidenote: A rounde spier.]
Nowe of three plattes there are made certain figures of bodyes,
as the cantels and halues of all bodyes that haue but ij.
plattys, and also the halues of halfe globys and canteles of a
globe. Lykewyse a rounde piller, and a spyre made of a rounde
spyre, slytte in ij. partes long ways.
But as these formes be harde to be iudged by their pycturs,
so I doe entende to passe them ouer with a great number of other
formes of bodyes, which afterwarde shall be set forth in the
boke of Perspectiue, bicause that without perspectiue knowledge,
it is not easy to iudge truly the formes of them in flatte
protacture.
And thus I made an ende for this tyme, of the definitions
Geometricall, appertayning to this parte of practise, and the
rest wil I prosecute as cause shall serue.
THE PRACTIKE WORKINGE OF
+sondry conclusions geometrical.+