The second Booke ofGeometry.Of a Line.

Therefore

11.Congruall or agreeable Magnitudes are equall.8.ax. j.

A lesser right line may agree to a part of a greater, but to so much of it, it is equall, with how much it doth agree:Neither is that axiome reciprocall or to be converted: For neither in deede are Congruity and Equality reciprocall or convertible. For a Triangle may bee equall to a Parallelogramme, yet it cannot in all points agree to it: And so to a Circle there is sometimes sought an equall quadrate,although incongruallor not agreeing with it: Because those things which are of the like kinde doe onely agree.

12.Magnitudes are described betweene themselves, one with another, when the bounds of the one are bounded within the boundes of the other: That which is within, is called the inscript: and that which is without, the Circumscript.

Now followeth Adscription, whose kindes are Inscription and Circumscription; That is when one figure is written or made within another: This when it is written or made about another figure.

Homogenea, Homogenealls or figures of the same kinde onely betweene themselvesrectitermina, or right bounded, are properly adscribed betweene themselves, and with a round. Notwithstanding, at the 15. booke ofEuclides ElementsHeterogenea, Heterogenealls or figures of divers kindes are also adscribed, to witt the five ordinate plaine bodies betweene themselves: And a right line is inscribed within a periphery and a triangle.

But the use of adscription of a rectilineall and circle, shall hereafter manifest singular and notable mysteries by the reason and meanes of adscripts; which adscription shall be the key whereby a way is opened unto that most excellent doctrine taught by the subtenses or inscripts of a circle asPtolomeyspeakes, or Sines, as the latter writers call them.

1.A Magnitude is either a Line or a Lineate.

The Common affections of a magnitude are hitherto declared: TheSpeciesor kindes doe follow: for other then this division our authour could not then meete withall.

Lines.

2.A Line is a Magnitude onely long.

As areae.io.anduy.such a like Magnitude is conceived in the measuring of waies, or distance of one place from another: And by the difference of a lightsome place from a darke:Euclideat the 2d j.defineth a line to be a length void of breadth: And indeede length is the proper difference of a line, as breadth is of a face, and solidity of a body.

3.The bound of a line is a point.

Euclideat the 3.d j.saith that the extremities or ends of a line are points. Now seeing that a Periphery or an hoope line hath neither beginning nor ending, it seemeth not to bee bounded with points: But when it is described or made it beginneth at a point, and it endeth at a pointe. Wherefore a Point is the bound of a line, sometimeactu, in deed, as in a right line: sometimepotentiâ, in a possibility, as in a perfect periphery. Yea in very deede, as before was taught in the definition ofcontinuum, 4e.all lines, whether they bee right lines, or crooked, are contained or continued with points. But a line is made by themotion of a point. For every magnitude generally is made by a geometricall motion, as was even now taught, and it shall afterward by the severall kindes appeare, how by one motion whole figures are made: How by a conversion, a Circle, Spheare, Cone, and Cylinder: How by multiplication of the base and heighth, rightangled parallelogrammes are made.

4.A Line is either Right or Crooked.

This division is taken out of the 4 d j. ofEuclide, where rectitude or straightnes is attributed to a line, as if from it both surfaces and bodies were to have it. And even so the rectitude of a solid figure, here-after shall be understood by a right line perpendicular from the toppe unto the center of the base. Wherefore rectitude is propper unto a line: And therefore also obliquity or crookednesse, from whence a surface is judged to be right or oblique, and a body right or oblique.

5.A right line is that which lyeth equally betweene his owne bounds: A crooked line lieth contrariwise.4.d. j.

Now a line lyeth equally betweene his owne bounds, when it is not here lower, nor there higher: But is equall to the space comprehended betweene the two bounds or ends: As hereae.is, so hee that makethrectum iter, a journey in a straight line, commonly he is said to treade so much ground, as he needes must, and no more: He goethobliquum iter, a crooked way, which goeth more then he needeth, asProclussaith.

Straight Line.

6.A right line is the shortest betweene the same bounds.

Linea recta, a straight or right line is that, asPlatodefineth it, whose middle points do hinder us from seeing both the extremes at once; As in the eclipse of the Sunne, if a right line should be drawne from the Sunne, by the Moone, unto our eye, the body of the Moone beeing in the midst, would hinder our sight, and would take away the sight of the Sunne from us: which is taken from the Opticks, in which we are taught, that we see by straight beames or rayes. Therfore to lye equally betweene the boundes, that is by an equall distance: to bee the shortest betweene the same bounds; And that the middest doth hinder the sight of the extremes, is all one.

7.A crooked line is touch'd of a right or crooked line, when they both doe so meete, that being continued or drawne out farther they doe not cut one another.

Crooked Lines.

Tactus, Touching is propper to a crooked line, compared either with a right line or crooked, as is manifest out of the 2. and 3.d3. A right line is said to touch a circle, which touching the circle and drawne out farther, doth not cut the circle, 2d3. as hereae, the right line toucheth the peripheryiou. Andae. doth touch the helix or spirall.Circles are said to touch one another, when touching they doe not cutte one another, 3.d3. as here the periphery dothaej.doth touch the peripheryouy.

Therefore

8.Touching is but in one point onely. è 13. p3.

This Consectary is immediatly conceived out of the definition; for otherwise it were a cutting, not touching. SoAristotlein hisMechanickessaith; That a round is easiliest mou'd and most swift; Because it is least touch't of the plaine underneath it.

9.A crooked line is either a Periphery or an Helix.This also is such a division, as our Authour could then hitte on.

Periphery.

10.A Periphery is a crooked line, which is equally distant from the middest of the space comprehended.

Peripheria, a Periphery, or Circumference, aseio.doth stand equally distant froma, the middest of the space enclosed or conteined within it.

Therefore

11.A Periphery is made by the turning about of a line, the one end thereof standing still, and the other drawing the line.

Generation of Periphery.

As ineio.let the pointastand still: And let the lineao, be turned about, so that the pointodoe make a race, and it shall make the peripheryeoi. Out of this fabricke dothEuclide, at the 15. d. j. frame the definition of a Periphery: And so doth hee afterwarde define a Cone, a Spheare, and a Cylinder.

Now the line that is turned about, may in a plaine, bee either a right line or a crooked line: In a sphericall it is onely a crooked line; But in a conicall or Cylindraceall it may bee a right line, as is the side of a Cone and Cylinder. Therefore in the conversion or turning about of a line making a periphery, there is considered onely the distance; yea two points, one in the center, the other in the toppe, which therefore Aristotle namethRotundi principia, the principles or beginnings of a round.

12.An Helix is a crooked line which is unequally distant from the middest of the space, howsoever inclosed.

Examples of Helix.

Hæc tortuosa linea, This crankled line is ofProcluscalledHelicoides. But it may also be calledHelix, a twist or wreath: TheGreekesby this word do commonly either understand one of the kindes of Ivie which windeth it selfe about trees & other plants; or the strings of the vine, whereby it catcheth hold and twisteth it selfe about such things as are set for it to clime or run upon. Therfore it should properly signifie the spirall line. But as it is here taken it hath divers kindes; As is theArithmeticawhich is Archimede'es Helix, as theConchois, Cockleshell-like: as is theCittois, Iuylike: TheTetragonisousa, the Circle squaring line, to witt that by whose meanes a circle may be brought into a square: The Admirable line, found out byMenelaus: The ConicallEllipsis, theHyperbole, theParabole, such as these are, they attribute toMenechmus: All theseApolloniushath comprised in eight Bookes; but being mingled lines, and so not easie to bee all reckoned up and expressed,Euclidehath wholly omitted them, saithProclus, at the 9.p. j.

Mingled Lines.

Perpendicular Lines.

13.Lines are right one unto another, whereof the one falling upon the other, lyeth equally: Contrariwise they are oblique. è 10. d j.

Hitherto straightnesse and crookednesse have beene the affections of one sole line onely: The affections of two lines compared one with another arePerpendiculum, Perpendicularity andParallelismus, Parallell equality; Which affections are common both to right and crooked lines. Perpendicularity is first generally defined thus:

Lines are right betweene themselves, that is, perpendicular one unto another, when the one of them lighting upon the other, standeth upright and inclineth or leaneth neither way. So two right lines in a plaine may bee perpendicular; as areae.andio.so two peripheries upon a sphearicall may be perpendiculars, when the one of them falling upon the other, standeth indifferently betweene, and doth not incline or leane either way. So a right line may beperpendicular unto a periphery, if falling upon it, it doe reele neither way, but doe ly indifferently betweene either side. And in deede in all respects lines right betweene themselves, and perpendicular lines are one and the same. And from the perpendicularity of lines, the perpendicularity of surfaces is taken, as hereafter shall appeare. Of the perpendicularity of bodies,Euclidespeaketh not one word in hisElements, & yet a body is judged to be right, that is, plumme or perpendicular unto another body, by a perpendicular line.

Therefore,

Figure for demonstration 14.

14.If a right line be perpendicular unto a right line, it is from the same bound, and on the same side, one onely. ê 13. p. xj.

Or, there can no more fall from the same point, and on the same side but that one. This consectary followeth immediately upon the former: For if there should any more fall unto the same point and on the same side, one must needes reele, and would not ly indifferently betweene the parts cut: as here thou seest in the right lineae. io. eu.

15.Parallell lines they are, which are everywhere equally distant. è 35. d j.

Parallelismus, Parallell-equality doth now follow: And this also is common to crooked lines and right lines: Asheere thou seest in these examples following.

Parallel Lines.

Parallell-equality is derived from perpendicularity, and is of neere affinity to it. Therefore Posidonius did define it by a common perpendicle or plum-line: yea and in deed our definition intimateth asmuch. Parallell-equality of bodies is no where mentioned inEuclides Elements: and yet they may also bee parallells, and are often used in the Optickes, Mechanickes, Painting and Architecture.

Three Parallel Lines.

Therefore,

16.Lines which are parallell to one and the same line, are also parallell one to another.

This element is specially propounded and spoken of right lines onely, and is demonstrated at the 30.p. j.But by an addition of equall distances, an equall distance is knowne, as here.

1.A lineate is a Magnitude more then long.

A New forme of doctrine hath forced our Authour to use oft times new words, especially in dividing, that the logicall lawes and rules of more perfect division by a dichotomy, that is into two kindes, might bee held and observed. Therefore a Magnitude was divided into two kindes, to witt into a Line and a Lineate: And a Lineate is made thegenusof a surface and a Body. Hitherto a Line, which of all bignesses is the first and most simple, hath been described: Now followeth a Lineate, the other kinde of magnitude opposed as you see to a line, followeth next in order.Lineatumtherefore a Lineate, orLineamentum, a Lineament, (as by the authority of our Authour himselfe, the learnedBernhard Salignacus, who was his Scholler, hath corrected it) is that Magnitude in which there are lines: Or which is made of lines, or as our Authour here, which is more then long: Therefore lines may be drawne in a surface, which is the proper soile or plots of lines; They may also be drawne in a body, as the Diameter in a Prisma: the axis in a spheare; and generally all lines falling from aloft: And therforeProclusmaketh some plaine, other solid lines. So Conicall lines, as the Ellipsis, Hyperbole, and Parabole, are called solid lines because they do arise from the cutting of a body.

2.To a Lineate belongeth an Angle and a Figure.

The common affections of a Magnitude were to be bounded, cutt, jointly measured, and adscribed: Then of a line to be right, crooked, touch'd, turn'd about, andwreathed: All which are in a lineate by meanes of a line. Now the common affections of a Lineate are to bee Angled and Figured. And surely an Angle and a figure in all Geometricall businesses doe fill almost both sides of the leafe. And therefore both of them are diligently to be considered.

3.An Angle is a lineate in the common section of the bounds.

SoAngulus Superficiarius, a superficiall Angle, is a surface consisting in the common section of two lines: Soangulus solidus, a solid angle, in the common section of three surfaces at the least.

Angles.

[But the learned B.Salignacushath observed, that all angles doe not consist in the common section of the bounds, Because the touching of circles, either one another, or a rectilineal surface doth make an angle without any cutting of the bounds: And therefore he defineth it thus:Angulus est terminorum inter se invicem inclinantium concursus: An angle is the meeting of bounds, one leaning towards another.] So isaei.a superficiall angle: [And such also are the anglesouy.andbcd.] so is the angleo.a solid angle, to witt comprehended of the three surfacesaoi.ioe.andaoe.Neither may a surface, of 2. dimensions, be bounded withone right line: Nor a body, of three dimensions, bee bounded with two, at lest beeing plaine surfaces.

4.The shankes of an angle are the bounds compreding the angle.

ScèleorCrura, the Shankes, Legges, H. are the bounds insisting or standing upon the base of the angle, which in the Isosceles only or Equicrurall triangle are so named ofEuclide, otherwise he nameth themLatera, sides. So in the examples aforesaid,ea.andei.are the shankes of the superficiary anglee; And so are the three surfacesaoi.ieo.andaeo.the shankes of the said angleo. Therefore the shankes making the angle are either Lines or Surfaces: And the lineates formed or made into Angles, are either Surfaces or Bodies.

Lunular Angles, etc.

5.Angles homogeneall, are angles of the same kinde, both in respect of their shankes, as also in the maner of meeting of the same:[Heterogeneall, are those which differ one from another in one, or both these conditions.]

Therefore thisHomogenia, or similitude of angles is twofolde, the first is of shanks; the other is of the manner of meeting of the shankes: so rectilineall right angles, are angles homogeneall betweene themselves. But right-lined right angles, and oblique-lined right angles between themselves, are heterogenealls. So are neither all obtusangles compared to all obtusangles: Nor all acutangles, to all acutangles, homogenealls, except both these conditions doe concurre, to witt the similitude both of shanke and manner of meeting.Lunularis, a Lunular, or Moonlike corner angle is homogeneall to aSystroidesandPelecoides, Hatchet formelike, in shankes: For each of these are comprehended ofperipheries: The Lunular of one convexe; the other concave; asiue. The Systroides of both convex, asiao. The Pelecoides of both concave, aseau. And yet a lunular, in respect of the meeting of the shankes is both to the Systroides and Pelecoides heterogeneall: And therefore it is absolutely heterogeneall to it.

6.Angels congruall in shankes are equall.

This is drawne out of the10. e j. For if twice two shanks doe agree, they are not foure, but two shankes, neither are they two equall angles, but one angle. And this is that whichProclusspeaketh of, at the 4. p j. when hee saith, that a right lined angle is equall to a right lined angle, when one of the shankes of the one put upon one of the shankes of the other, the other two doe agree: when that other shanke fall without, the angle of the out-falling shanke is the greater: when it falleth within, it is lesser: For there is comprehendeth; here it is comprehended.

Notwithstanding although congruall or agreeable angles be equall: yet are not congruity and equality reciprocall or convertible: For a Lunular may bee equall to a rightlined right angle, as here thou seest: For the angles of equall semicirclesieo.andaeu.are equall, as application doth shew. The angleaeo.is common both to the right angleaei.and to the lunaraueo.Let therefore the equall angleaeo.bee added to both: the right angleaei.shall be equall to the Lunularaueo.

Lunulars equal to right lined angles.

The same Lunular also may bee equall to an obtusangle and Acutangle, as the same argument will demonstrate.

Therefore,

7.If an angle being equicrurall to an other angle, be also equall to it in base, it is equall: And if an angle having equall shankes with another, bee equall to it in the angle, it is also equall to it in the base. è8. & 4.p j.

For such angles shall be congruall or agreeable in shanks, and also congruall in bases.Angulus isosceles, orAngulus æquicrurus, is a triangle having equall shankes unto another.

Angles.

8.And if an angle equall in base to another, be also equall to it in shankes, it is equall to it.

For the congruency is the same: And yet if equall angles bee equall in base, they are not by and by equicrurall, as in the angles of the same section will appeare, as here. And so of two equalities, the first is reciprocall: The second is not. [And therefore is this Consectary, by the learned B.Salignacus, justly, according to the judgement of the worthy Rud.Snellius, here cancelled; or quite put out: For angles may be equall, although they bee unequall in shankes or in bases, as here, the anglea.is not greater then the angleo, although the angleohave both greater shankes and greater base then the anglea.]

And

9.If an angle equicrurall to another angle, be greater then it in base, it is greater: And if it be greater, it is greater in base: è52 & 24.p j.

As here thou seest; [The angleseai.anduoy.are equicrurall, that is their shankes are equall one to another; But the baseeiis greater then the baseuy: Therefore the angleeai, is greater then the angleuoy. And contrary wise, they being equicrurall, and the angleeai.being greater then the angleuoy.The baseei.must needes be greater then the baseuy.]

Equicrurall Angles.

And

Inscribed Angle.

10.If an angle equall in base, be lesse in the inner shankes, it is greater.

Or as the learned MasterT. Hooddoth paraphrastically translate it.If being equall in the base, it bee lesser in the feete (the feete being conteined within the feete of the other angle) it is the greater angle.[That is, if one angle enscribed within another angle, be equall in base, the angle of the inscribed shall be greater then the angle of the circumscribed.]

As here the angleaoi.within the angleaei.And the bases are equall, to witt one and the same; Thereforeaoi.the inner angle is greater thenaei.the outter angle.Inneris added of necessity: For otherwise there will, in the section or cutting one of another, appeare a manifest errour. All these consectaries are drawne out of that same axiome of congruity, to witt out of the10. e j. asProclusdoth plainely affirme and teach: It seemeth saith hee, that the equalities of shankes and bases, doth cause the equality of the verticall angles. For neither, if the bases be equall, doth the equality of the shankes leave the same or equall angles: But if the base bee lesser, the angle decreaseth: If greater, it increaseth. Neither if the bases bee equall, and the shankes unequall, doth the angle remaine the same: But when they are made lesse, it is increased: when they are made greater, it is diminished: For the contrary falleth out to the angles and shankes of the angles. For if thou shalt imagine the shankes to be in the same base thrust downeward, thou makest them lesse, but their angle greater: but if thou do againe conceive them to be pul'd up higher, thou makest them greater, but their angle lesser. For looke how much more neere they come one to another, so much farther off is the toppe removed from the base: wherefore you may boldly affirme, that the samebase and equall shankes, doe define the equality of Angels. ThisPoclus,

Therefore,

11.If unto the shankes of an angle given, homogeneall shankes, from a point assigned, bee made equall upon an equall base, they shall comprehend an angle equall to the angle given. è 23. p j.& 26.p xj.

[This consectary teacheth how unto a point given, to make an angle equall to an Angle given. To the effecting and doing of each three things are required; First, that the shankes be homogeneall, that is in each place, either straight or crooked: Secondly, that the shankes bee made equall, that is of like or equall bignesse: Thirdly, that the bases be equall: which three conditions if they doe meete, it must needes be that both the angles shall bee equall: but if one of them be wanting, of necessity againe they must be unequall.]

This shall hereafter be declared and made plaine by many and sundry practises: and therefore here we bring no example of it.

12.An angle is either right or oblique.

Thus much of the Affections of an angle; the division into his kindes followeth. An angle is either Right or Oblique: as afore, at the 4e ij.a line was right or straight, and oblique or crooked.

Right Angle.

13.A right angle is an angle whose shankes are right (that is perpendicular) one unto another: An Oblique angle is contrary to this.

As here the angleaio.is a right angle, as is alsooie.because the shankeoi.is right, that is, perpendicular toae.[The instrument wherby they doe make triall which is a right angle, and which is oblique, that is greater or lesser then a right angle, is the square which carpenters and joyners do ordinarily use: For lengthes are tried, saithVitruvius, by the Rular and Line: Heighths, by the Perpendicular or Plumbe: And Angles, by thesquare.] Contrariwise, an Oblique angle it is, when the one shanke standeth so upon another, that it inclineth, or leaneth more to one side, then it doth to the other: And one angle on the one side, is greater then that on the other.

Therefore,

14.All straight-shanked right angles are equall.

[That is, they are alike, and agreeable, or they doe fill the same place; as here areaio.andeio.And yet againe on the contrary: All straight shanked equall angles, are not right-angles.]

The axiomes of the equality of angles were three, as even now wee heard, one generall, and two Consectaries: Here moreover is there one speciall one of the equality of Right angles.

Semicircular Right Angles.

Angles therfore homogeneall and recticrurall, that is whose shankes are right, as are right lines, as plaine surfaces (For let us so take the word) are equall rightangles. So are the above written rectilineall right angles equall: so are plaine solid right angles, as in a cube, equall. The axiome may therefore generally be spoken of solid angles, so they be recticruralls: Because all semicircular right angles are not equall to all semicircular right angles: As here, when the diameter is continued it is perpendicular, and maketh twice two angles, within and without, the outter equall betweene themselves, and inner equall betweene themselves: But the outer unequall to the inner: And the angle of a greater semicircle is greater, then the angle of a lesser. Neither is this affection any way reciprocall, That all equall angles should bee right angles. For oblique angles may bee equall betweene themselves: And an oblique angle may bee made equall to a right angle, as a Lunular to a rectilineall right angle, as was manifest, at the6 e.

The definition of an oblique is understood by the obliquity of the shankes: whereupon also it appeareth; That an oblique angle is unequall to an homogeneall right angle: Neither indeed may oblique angles be made equall by any lawe or rule: Because obliquity may infinitly bee both increased and diminished.

15.An oblique angle is either Obtuse or Acute.

One difference of Obliquity wee had before at the9 e ij. in a line, to witt of a periphery and an helix; Here there is another dichotomy of it into obtuse and acute: which difference is proper to angles, from whence it is translated or conferred upon other things and metaphorically used, asIngenium obtusum, acutum; A dull, and quicke witte, and such like.

Acute Angle.

Obtuse Angle.

16.An obtuse angle is an oblique angle greater then a right angle. 11. d j.

Obtusus, Blunt or Dull; As hereaei.In the definition thegenusof bothSpeciesor kinds is to bee understood: For a right lined right angle is greater then a sphearicall right angle, and yet it is not an obtuse or blunt angle: And this greater inequality may infinitely be increased.

17.An acutangle is an oblique angle lesser then a right angle. 12. d j.

Acutus, Sharpe, Keene, as hereaei.is. Here againe the samegenusis to bee understood: because every angle which is lesse then any right angle is not an acute or sharp angle. For a semicircle and sphericall right angle, is lesse then a rectilineall right angle, and yet it is not an acute angle.

1.A figure is a lineate bounded on all parts.

So the triangleaei.is a figure; Because it is a plaine bounded on all parts with three sides. So a circle is a figure: Because it is a plaine every way bounded with one periphery.

Figures.

2.The center is the middle point in a figure.

In some part of a figure the Center, Perimeter, Radius, Diameter and Altitude are to be considered. The Center therefore is a point in the midst of the figure; so in the triangle, quadrate, and circle, the center is,aei.

Centers of Triangle, Quadrate, And Circle.

Centrum gravitatis, the center of weight, in every plaine magnitude is said to bee that, by the which it is handled or held up parallell to the horizon: Or it is that point whereby the weight being suspended doth rest, when it is caried. Therefore if any plate should in all places be alike heavie, the center of magnitude and weight would be one and the same.

3.The perimeter is the compasse of the figure.

Or, the perimeter is that which incloseth the figure. This definition is nothing else but the interpretation of the Greeke word. Therefore the perimeter of a Triangle is one line made or compounded of three lines. So the perimeter of the trianglea, iseio.So the perimeter of the circleais a periphery, as ineio.So the perimeter of a Cube is a surface, compounded of sixe surfaces: And the perimeter of a spheare is one whole sphæricall surface, as hereafter shall appeare.

Perimeters.

4.The Radius is a right line drawne from the center to the perimeter.

Radius, the Ray, Beame, or Spoake, as of the sunne, andcart wheele: As in the figures under written areae,ai,ao. It is here taken for any distance from the center, whether they be equall or unequall.

Examples of Radius.

5.The Diameter is a right line inscribed within the figure by his center.

As in the figure underwritten areae,ai,ao. It is called theDiagonius, when it passeth from corner to corner. In solids it is called theAxis, as hereafter we shall heare.

Examples of Diameter.

Therefore,

6.The diameters in the same figure are infinite.

Although of an infinite number of unequall lines that be only the diameter, which passeth by or through the centernotwithstanding by the center there may be divers and sundry. In a circle the thing is most apparent: as in the Astrolabe the index may be put up and downe by all the points of the periphery. So in a speare and all rounds the thing is more easie to be conceived, where the diameters are equall: yet notwithstanding in other figures the thing is the same. Because the diameter is a right line inscribed by the center, whether from corner to corner, or side to side, the matter skilleth not. Therefore that there are in the same figure infinite diameters, it issueth out of the difinition of a diameter.

And

7.The center of the figure is in the diameter.

As here thou seesta, e, ithis ariseth out of the definition of the diameter. For because the diameter is inscribed into the figure by the center: Therefore the Center of the figure must needes be in the diameter thereof: This is byArchimedesassumed especially at the 9, 10, 11, and 13Theoremeof hisIsorropicks, orÆquiponderants.

Centers in Diameters.

This consectary, saith the learned Rod. Snellius, is as it were a kinde of invention of the center. For where the diameters doe meete and cutt one another, there must the center needes bee. The cause of this is for that in every figurethere is but one center only: And all the diameters, as before was said, must needes passe by that center.

And

8.It is in the meeting of the diameters.

As in the examples following. This also followeth out of the same definition of the diameter. For seeing that every diameter passeth by the center: The center must needes be common to all the diameters: and therefore it must also needs be in the meeting of them: Otherwise there should be divers centers of one and the same figure. This also doth the sameArchimedespropound in the same words in the 8. and 12 theoremes of the same booke, speaking of Parallelogrammes and Triangles.

Meeting of Diameters.

9.The Altitude is a perpendicular line falling from the toppe of the figure to the base.

Altitudo, the altitude, or heigth, or the depth: [For that, as hereafter shall bee taught, is butAltitudo versa, an heighthwith the heeles upward.] As in the figures following areae,io,uy, orsr. Neither is it any matter whether the base be the same with the figure, or be continued or drawne out longer, as in a blunt angled triangle, when the base is at the blunt corner, as here in the triangle,aei, isao.

Altitudes.

10.An ordinate figure, is a figure whose bounds are equall and angles equall.

In plaines the Equilater triangle is onely an ordinate figure, the rest are all inordinate: In quadrangles, the Quadrate is ordinate, all other of that sort are inordinate: In every sort of Multangles, or many cornered figures one may be an ordinate. In crooked lined figures the Circle is ordinate, because it is conteined with equall bounds, (one bound alwaies equall to it selfe being taken for infinite many,) because it is equiangled, seeing (although in deede there be in it no angle) the inclination notwithstanding is every where alike and equall, and as it were the angle of the perphery be alwaies alike unto it selfe: whereupon of Plato and Plutarch a circle is said to bePolygonia, a multangle; and of AristotleHolegonia, a totangle, nothing else but one whole angle. In mingled-lined figures there is nothing that is ordinate: Insolid bodies, and pyramids the Tetrahedrum is ordinate: Of Prismas, the Cube: of Polyhedrum's, three onely are ordinate, the octahedrum, the Dodecahedrum, and the Icosahedrum. In oblique-lined bodies, the spheare is concluded to be ordinate, by the same argument that a circle was made to bee ordinate.

11.A prime or first figure, is a figure which cannot be divided into any other figures more simple then it selfe.

So in plaines the triangle is a prime figure, because it cannot be divided into any other more simple figure although it may be cut many waies: And in solids, the Pyramis is a first figure: Because it cannot be divided into a more simple solid figure, although it may be divided into an infinite sort of other figures: Of the Triangle all plaines are made; as of a Pyramis all bodies or solids are compounded; such areaei.andaeio.

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