The fourth Booke, which is of a Figure.
1. A figure is a lineate bounded on all parts.
So the triangle aei. 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.
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.
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 triangle a, is eio. So the perimeter of the circle a is a periphery, as in eio. 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.
4. The Radius is a right line drawne from the center to the perimeter.
Radius, the Ray, Beame, or Spoake, as of the sunne, and
cart wheele: As in the figures under written are ae, ai, ao. It is here taken for any distance from the center, whether they be equall or unequall.
5. The Diameter is a right line inscribed within the figure by his center.
As in the figure underwritten are ae, ai, ao. It is called the Diagonius, when it passeth from corner to corner. In solids it is called the Axis, as hereafter we shall heare.
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 center
notwithstanding 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 seest a, e, i this 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 by Archimedes assumed especially at the 9, 10, 11, and 13 Theoreme of his Isorropicks, or Æquiponderants.
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 figure
there 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 same Archimedes propound in the same words in the 8. and 12 theoremes of the same booke, speaking of Parallelogrammes and Triangles.
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 but Altitudo versa, an heighth
with the heeles upward.] As in the figures following are ae, io, uy, or sr. 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, is ao.
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 be Polygonia, a multangle; and of Aristotle Holegonia, a totangle, nothing else but one whole angle. In mingled-lined figures there is nothing that is ordinate: In
solid 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 are aei. and aeio.
12. A rationall figure is that which is comprehended of a base and height rationall betweene themselves.
So Euclide, at the 1. d. ij. saith, that a rightangled parallelogramme is comprehended of two right lines perpendicular one to another, videlicet one multiplied by the other. For Geometricall comprehension is sometimes as it were in numbers a multiplication: Therefore if yee shall grant the base and height to bee rationalls betweene themselves,
that their reason I meane may be expressed by a number of the assigned measure, then the numbers of their sides being multiplyed one by another, the bignesse of the figure shall be expressed. Therefore a Rationall figure is made by the multiplying of two rationall sides betweene themselves.
Therefore,
13. The number of a rationall figure, is called a Figurate number: And the numbers of which it is made, the Sides of the figurate.
As if a Right angled parallelogramme be comprehended of the base foure, and the height three, the Rationall made shall be 12. which wee here call the figurate: and 4. and 3. of which it was made, we name sides.
14. Isoperimetrall figures, are figures of equall perimeter.
This is nothing else but an interpretation of the Greeke word; So a triangle of 16. foote about, is a isoperimeter to a triangle 16. foote about, to a quadrate 16. foote about, and to a circle 16. foote about.
15. Of isoperimetralls homogenealls that which is most ordinate, is greatest: Of ordinate isoperimetralls heterogenealls, that is greatest, which hath most bounds.
So an equilater triangle shall bee greater then an isoperimeter inequilater triangle; and an equicrurall, greater then an unequicrurall: so in quadrangles, the quadrate is greater then that which is not a quadrate: so an oblong more ordinate, is greater then an oblong lesse ordinate. So of those figures which are heterogeneall ordinates, the quadrate is greater then the Triangle: And the Circle, then the Quadrate.
16. If prime figures be of equall height, they are in reason one unto another, as their bases are: And contrariwise.
The proportion of first figures is here twofold; the first is direct in those which are of equall height. In Arithmeticke we learned; That if one number doe multiply many numbers, the products shall be proportionall unto the numbers, multiplyed. From hence in rationall figures the content of those which are of equall height is to bee expressed by a number. As in two right angled parallelogrammes, let 4. the same height, multiply 2. and 3. the bases: The products 8. and 12. the parallelogrammes made, are directly proportionall unto the bases 2. and 3. Therefore as 2. is unto 3. so is 8. unto 12. The same shall afterward appeare in right Prismes and Cylinders. In plaines, Parallelogramms are the doubles of triangles: In solids, Prismes are the triples of pyramides: Cylinders, the triples of Cones. The converse of this element is plaine out of the former also: First figures if they be in reason one to another as their bases are, then are they of equall height, to witt when their products are proportionall unto the multiplyed, the same number did multiply them.
Therefore,
17. If prime figures of equall heighth have also equall bases, they are equall.
[The reason is, because then those two figures compared, have equall sides, which doe make them equall betweene themselves; For the parts of the one applyed or laid unto the parts of the other, doe fill an equall place, as was taught at the [10. e. j]. Sn.] So Triangles, so Parallelogrammes, and so other figures proposed are equalled upon an equall base.
18. If prime figures be reciprocall in base and height, they are equall: And contrariwise.
The second kind of proportion of first figures is reciprocall. This kinde of proportion rationall and expressible by a number, is not to be had in first figures themselves: but in those that are equally manifold to them, as was taught even now in direct proportion: As for example, Let these two right angled parallelogrammes, unequall in bases and heighths 3, 8, 4, 6, be as heere thou seest: The proportion reciprocall is thus, As 3 the base of the one, is unto 4, the base of the other: so is 6. the height of the one is to 8. the height of the other: And the parallelogrammes are equall, viz. 24. and 24. Againe, let two solids of unequall bases & heights (for here also the base is taken for the length and heighth) be 12, 2, 3, 6, 3, 4. The solids themselves shall be 72. and 72, as here thou seest; and the proportion of the bases and heights likewise is reciprocall: For as 24, is unto 18, so is 4, unto 3. The cause is out of the golden rule of proportion in Arithmeticke: Because twice two sides are
proportionall: Therefore the plots made of them shall be equall. And againe, by the same rule, because the plots are equall: Therefore the bounds are proportionall; which is the converse of this present element.
19. Like figures are equiangled figures, and proportionall in the shankes of the equall angles.
First like figures are defined, then are they compared one with another, similitude of figures is not onely of prime figures, and of such as are compounded of prime figures, but generally of all other whatsoever. This similitude consisteth in two things, to witt in the equality of their angles, and proportion of their shankes.
Therefore,
20. Like figures have answerable bounds subtended against their equall angles: and equall if they themselves be equall.
Or thus, They have their termes subtended to the equall angles correspondently proportionall: And equall if the figures themselves be equall; H. This is a consectary out of the former definition.
And
21. Like figures are situate alike, when the proportionall bounds doe answer one another in like situation.
The second consectary is of situation and place. And this like situation is then said to be when the upper parts of the one figure doe agree with the upper parts of the other, the lower, with the lower, and so the other differences of places. Sn.
And
22. Those figures that are like unto the same, are like betweene themselves.
This third consectary is manifest out of the definition of like figures. For the similitude of two figures doth conclude both the same equality in angles and proportion of sides betweene themselves.
And
23. If unto the parts of a figure given, like parts and alike situate, be placed upon a bound given, a like figure and likely situate unto the figure given, shall bee made accordingly.
This fourth consectary teacheth out of the said definition, the fabricke and manner of making of a figure alike and likely situate unto a figure given. Sn.
24. Like figures have a reason of their homologallor correspondent sides equally manifold unto their dimensions: and a meane proportionall lesse by one.
Plaine figures have but two dimensions, to witt Length, and Breadth: And therefore they have but a doubled reason of their homologall sides. Solids have three dimensions, videl. Length, Breadth, & thicknesse: therefore they shall have a treabled reason of their homologall or correspondent sides. In 8. and 18. the two plaines given, first the angles are equall: secondly, their homolegall side 2. and 4. and 3. and 6. are proportionall. Therefore the reason of 8. the first figure, unto 18. the
second, is as the reason is of 2. unto 3. doubled. But the reason of 2. unto 3. doubled, by the 3. chap. ij. of Arithmeticke, is of 4. to 9. (for 2/3 2/3 is 4/9.) Therefore the reason of 8. unto 18, that is, of the first figure unto the second, is of 4. unto 9. In Triangles, which are the halfes of rightangled parallelogrammes, there is the same truth, and yet by it selfe not rationall and to be expressed by numbers.
Said numbers are alike in the trebled reason of their homologall sides; As for example, 60. and 480. are like solids; and the solids also comprehended in those numbers are like-solids, as here thou seest: Because their sides, 4. 3. 5. and 8. 6. 10. are proportionall betweene themselves. But the reason of 60. to 480. is the reason of 4. to 8. trebled, thus 4/8 4/8 4/8 = 64/512; that is of 1. unto 8. or octupla, which you shall finde in the dividing of 480. by 60.
Thus farre of the first part of this element: The second, that like figurs have a meane, proportional lesse by one, then are their dimensions, shall be declared by few words. For plaines having but two dimensions, have but one meane proportionall, solids having three dimensions, have two meane proportionalls. The cause is onely Arithmeticall, as afore. For where the bounds are but 4. as they are in two plaines, there can be found no more but one meane proportionall, as in the former example of 8. and 18. where the homologall or correspondent sides are 2. 3. and 4. 6.
Therefore,
| 2 | 3 | 4 | 6 |
| 3 | 4 | ||
| 8 | 12 | 18 | |
Againe by the same rule, where the bounds are 6. as they are in two solids, there may bee found no more but two meane proportionalls: as in the former solids 30. and 240. where the homologall or correspondent sides are 2. 4. 3. 6. 5. 10.
Therefore,
| 2 | 4 | 3 | 6 | 5 | 10 |
| 4 | 3 | ||||
| 6 | 12 | 24 | |||
| 24 | 5 | ||||
| 30 | 60 | 120 | 240 | ||
Therefore,
25. If right lines be continually proportionall, more by one then are the dimensions of like figures likelily situate unto the first and second, it shall be as the first right line is unto the last, so the first figure shall be unto the second: And contrariwise.
Out of the similitude of figures two consectaries doe arise, in part only, as is their axiome, rationall and expressable by numbers. If three right lines be continually proportionall, it shall be as the first is unto the third: So the rectilineall figure made upon the first, shall be unto the rectilineall figure made upon the second, alike and likelily situate. This may in some part be conceived and understood by numbers. As for example, Let the lines given, be 2. foot, 4. foote, and 8 foote. And upon the first and second, let there be made like figures, of 6. foote and 24. foote; So I meane, that 2. and 4. be the bases of them. Here as 2. the first line, is unto 8. the third line: So is 6. the first figure, unto 24. the second figure, as here thou seest.
Againe, let foure lines continually proportionall, be 1. 2. 4. 8. And let there bee two like solids made upon the first and second: upon the first, of the sides 1. 3. and 2. let it be 6. Upon the second, of the sides 2. 6. and 4. let it be 48. As the first right line 1. is unto the fourth 8. So is the figure 6. unto the second 48. as is manifest by division. The examples are thus.
Moreover by this Consectary a way is laid open leading unto the reason of doubling, treabling, or after any manner way whatsoever assigned increasing of a figure given. For as the first right line shall be unto the last: so shall the first figure be unto the second.
And
26. If foure right lines bee proportionall betweene themselves: Like figures likelily situate upon them, shall be also proportionall betweene themselves: And contrariwise, out of the 22. p vj. and 37. p xj.
The proportion may also here in part bee expressed by numbers: And yet a continuall is not required, as it was in the former.
In Plaines let the first example be, as followeth.
The cause of proportionall figures, for that twice two figures have the same reason doubled.
In Solids let this bee the second example. And yet here the figures are not proportionall unto the right lines, as before figures of equall heighth were unto their bases, but they themselves are proportionall one to another. And yet are they not proportionall in the same kinde of proportion.
The cause also is here the same, that was before: To witt, because twice two figures have the same reason trebled.
27. Figures filling a place, are those which being any way set about the same point, doe leave no voide roome.
This was the definition of the ancient Geometers, as appeareth out of Simplicius, in his commentaries upon the 8. chapter of Aristotle's iij. booke of Heaven: which kinde of figures Aristotle in the same place deemeth to bee onely ordinate, and yet not all of that kind. But only three among the Plaines, to witt a Triangle, a Quadrate, and a Sexangle: amongst Solids, two; the Pyramis, and the Cube. But if the filling of a place bee judged by right angles, 4. in a Plaine, and 8. in a Solid, the Oblong of plaines, and the
Octahedrum of Solids shall (as shall appeare in their places) fill a place; And yet is not this Geometrie of Aristotle accurate enough. But right angles doe determine this sentence, and so doth Euclide out of the angles demonstrate, That there are onely five ordinate solids; And so doth Potamon the Geometer, as Simplicus testifieth, demonstrate the question of figures filling a place. Lastly, if figures, by laying of their corners together, doe make in a Plaine 4. right angles, or in a Solid 8. they doe fill a place.
Of this probleme the ancient geometers have written, as we heard even now: And of the latter writers, Regiomontanus is said to have written accurately; And of this argument Maucolycus hath promised a treatise, neither of which as yet it hath beene our good hap to see.
Neither of these are figures of this nature, as in their due places shall be proved and demonstrated.
28. A round figure is that, all whose raies are equall.
Such in plaines shall the Circle be, in Solids the Globe or Spheare. Now this figure, the Round, I meane, of all Isoperimeters is the greatest, as appeared before at the [15. e]. For which cause Plato, in his Timæus or his Dialogue of the World said; That this figure is of all other the greatest. And therefore God, saith he, did make the world of a
sphearicall forme, that within his compasse it might the better containe all things: And Aristotle, in his Mechanicall problems, saith; That this figure is the beginning, principle, and cause of all miracles. But those miracles shall in their time God willing, be manifested and showne.
Rotundum, a Roundle, let it be here used for Rotunda figura, a round figure. And in deede Thomas Finkius or Finche, as we would call him, a learned Dane, sequestring this argument from the rest of the body of Geometry, hath intituled that his worke De Geometria rotundi, Of the Geometry of the Round or roundle.
29. The diameters of a roundle are cut in two by equall raies.
The reason is, because the halfes of the diameters, are the raies. Or because the diameter is nothing else but a doubled ray: Therefore if thou shalt cut off from the diameter so much, as is the radius or ray, it followeth that so much shall still remaine, as thou hast cutte of, to witt one ray, which is the other halfe of the diameter. Sn.
And here observe, That Bisecare, doth here, and in other places following, signifie to cutte a thing into two equall parts or portions; And so Bisegmentum, to be one such portion; And Bisectio, such a like cutting or division.
30. Rounds of equall diameters are equall. Out of the 1. d. iij.
Circles and Spheares are equall, which have equall diameters. For the raies, which doe measure the space betweene the Center and Perimeter, are equall, of which, being doubled, the Diameter doth consist. Sn.
The fifth Booke, of Ramus his Geometry, which is of Lines and Angles in a plaine Surface.
1. A lineate is either a Surface or a Body.
Lineatum, (or Lineamentum) a magnitude made of lines, as was defined at [1. e. iij]. is here divided into two kindes: which is easily conceived out of the said definition there, in which a line is excluded, and a Surface & a body are comprehended. And from hence arose the division of the arte Metriall into Geometry, of a surface, and Stereometry, of a body, after which maner Plato in his vij. booke of his Common-wealth, and Aristotle in the 7. chapter of the first booke of his Posteriorums, doe distinguish betweene Geometry and Stereometry: And yet the name of Geometry is used to signifie the whole arte of measuring in generall.
2. A Surface is a lineate only broade. 5. d j.
As here aeio. and uysr. The definition of a Surface doth comprehend the distance or dimension of a line, to
witt Length: But it addeth another distance, that is Breadth. Therefore a Surface is defined by some, as Proclus saith, to be a magnitude of two dimensions. But two doe not so specially and so properly define it. Therefore a Surface is better defined, to bee a magnitude onely long and broad. Such, saith Apollonius, are the shadowes upon the earth, which doe farre and wide cover the ground and champion fields, and doe not enter into the earth, nor have any manner of thicknesse at all.
Epiphania, the Greeke word, which importeth onely the outter appearance of a thing, is here more significant, because of a Magnitude there is nothing visible or to bee seene, but the surface.
3. The bound of a surface is a line. 6. d j.
The matter in Plaines is manifest. For a three cornered surface is bounded with 3. lines: A foure cornered surface, with foure lines, and so forth: A Circle is bounded with one line. But in a Sphearicall surface the matter is not so plaine: For it being whole, seemeth not to be bounded with a line. Yet if the manner of making of a Sphearicall surface, by the conversiō or turning about of a semiperiphery, the beginning of it, as also the end, shalbe a line, to wit a semiperiphery: And as a point doth not only actu, or indeede bound and end a line: But is potentia, or in power, the middest of it: So also a line boundeth a Surface actu, and an innumerable company of lines may be taken or supposed to be throughout the whole surface. A Surface therefore is made by the motion of a line, as a Line was made by the motion of a point.
4. A Surface is either Plaine or Bowed.
The difference of a Surface, doth answer to the difference of a Line, in straightnesse and obliquity or crookednesse.
Obliquum, oblique, there signified crooked; Not right or straight: Here, uneven or bowed, either upward or downeward. Sn.
5. A plaine surface is a surface, which lyeth equally betweene his bounds, out of the 7. d j.
As here thou seest in aeio. That therefore a Right line doth looke two contrary waies, a Plaine surface doth looke all about every way, that a plaine surface should, of all surfaces within the same bounds, be the shortest: And that the middest thereof should hinder the sight of the extreames. Lastly, it is equall to the dimension betweene the lines: It may also by one right line every way applyed be tryed, as Proclus at this place doth intimate.
Planum, a Plaine, is taken and used for a plaine surface: as before Rotundum, a Round, was used for a round figure.
Therefore,
6. From a point unto a point we may, in a plaine surface, draw a right line, 1 and 2. post. j.
Three things are from the former ground begg'd: The first is of a Right line. A right line and a periphery were in the ij. booke defined: But the fabricke or making of them both, is here said to bee properly in a plaine.
The fabricke or construction of a right line is the 1. petition. And justly is it required that it may bee done onely upon a plaine: For in any other surface it were in vaine to aske it. For neither may wee possibly in a sphericall betweene two points draw a right line: Neither may wee possibly in a Conicall and Cylindraceall betweene any two points assigned draw a right line. For from the toppe
unto the base that in these is only possible: And then is it the bounde of the plaine which cutteth the Cone and Cylinder. Therefore, as I said, of a right plaine it may onely justly bee demanded: That from any point assigned, unto any point assigned, a right line may be drawne, as here from a unto e.
Now the Geometricall instrument for the drawing of a right plaine is called Amussis, & by Petolemey, in the 2. chapter of his first booke of his Musicke, Regula, a Rular, such as heere thou seest.
And from a point unto a point is this justly demanded to be done, not unto points; For neither doe all points fall in a right line: But many doe fall out to be in a crooked line. And in a Spheare, a Cone & Cylinder, a Ruler may be applyed, but it must be a sphearicall, Conicall, or Cylindraceall. But by the example of a right line doth Vitellio, 2 p j. demaund that betweene two lines a surface may be extended: And so may it seeme in the Elements, of many figures both plaine and solids, by Euclide to be demanded; That a figure may be described, at the 7. and 8. e ij. Item that a figure may be made vp, at the 8. 14. 16. 23. 28. p. vj. which are of Plaines. Item at the 25. 31. 33. 34. 36. 49. p. xj. which are of Solids. Yet notwithstanding a plaine surface, and a plaine body doe measure their rectitude by a right line, so that jus postulandi, this right of begging to have a thing granted may seeme primarily to bee in a right plaine line.
Now the Continuation of a right line is nothing else, but the drawing out farther of a line now drawne, and that from a point unto a point, as we may continue the right line ae. unto i. wherefore the first and second Petitions of Euclide do agree in one.
And
7. To set at a point assigned a Right line equall to another right line given: And from a greater, to cut off a part equall to a lesser. 2. and 3. p j.
As let the Right line given be ae. And to i. a point assigned, grant that io. equall to the same ae. may bee set. Item, in the second example, let ae. bee greater then io. And let there be cut off from the same ae. by applying of a rular made equall to io. the lesser, portion au. as here. For if any man shall thinke that this ought only to be don in the minde, hee also, as it were, beares a ruler in his minde, that he may doe it by the helpe of the ruler. Neither is the fabricke in deede, or making of one right equal to another: And the cutting off from greater Right line, a portion equall to a lesser, any whit harder, then it was, having a point and a distance given, to describe a circle: Then having a Triangle, Parallelogramme, and semicircle given, to describe or make a Cone, Cylinder, and spheare, all which notwithstanding Euclide did account as principles.
Therefore,
8. One right line, or two cutting one another, are in the same plaine, out of the 1. and 2. p xj.
One Right line may bee the common section of two plaines: yet all or the whole in the same plaine is one: And all the whole is in the same other: And so the whole is the same plaine. Two Right lines cutting one another, may bee in two plaines cutting one of another; But then a plaine may be drawne by them: Therefore both
of them shall be in the same plaine. And this plaine is geometrically to be conceived: Because the same plaine is not alwaies made the ground whereupon one oblique line, or two cutting one another are drawne, when a periphery is in a sphearicall: Neither may all peripheries cutting one another be possibly in one plaine.
And
9. With a right line given to describe a peripherie.
This fabricke or construction is taken out of the 3. Petition which is thus. Having a center and a distance given to describe, make, or draw a circle. But here the terme or end of a circle is onely sought, which is better drawne out of the definition of a periphery, at the [10. e ij]. And in a plaine onely may that conversion or turning about of a right line bee made: Not in a sphearicall, not in a Conicall, not in a Cylindraceall, except it be in top, where notwithstanding a periphery may bee described. Therefore before (to witt at the said [10. e ij].) was taught the generall fabricke or making of a Periphery: Here we are informed how to discribe a Plaine periphery, as here.
Now as the Rular was the instrument invented and used for the drawing of a right line: so also may the same Rular, used after another manner, be the instrument to describe or draw a periphery withall. And indeed such is that instrument used by the Coopers (and other like artists) for the rounding of their bottomes of their tubs, heads of barrells and otherlike vessells: But the Compasses, whether straight shanked or bow-legg'd, such as here thou seest, it skilleth not, are for al purposes and practises, in this case the best and readiest. And in deed the Compasses, of all
geometricall instruments, are the most excellent, and by whose help famous Geometers have taught: That all the problems of geometry may bee wrought and performed: And there is a booke extant, set out by John Baptist, an Italian, teaching, How by one opening of the Compasses all the problems of Euclide may be resolved: And Jeronymus Cardanus, a famous Mathematician, in the 15. booke of his Subtilties, writeth, that there was by the helpe of the Compasses a demonstration of all things demonstrated by Euclide, found out by him and one Ferrarius.
Talus, the nephew of Dædalus by his sister, is said in the viij. booke of Ovids Metamorphosis, to have beene the inventour of this instrument: For there he thus writeth of him and this matter:—Et ex uno duo ferrea brachia nodo: Iunxit, ut æquali spatio distantibus ipsis: Altera pars staret, pars altera duceret orbem.
Therfore
10. The raies of the same, or of an equall periphery, are equall.
The reason is, because the same right line is every where converted or turned about. But here by the Ray of the periphery, must bee understood the Ray the figure contained within the periphery.
11. If two equall peripheries, from the ends of equall shankes of an assigned rectilineall angle, doe meete before it, a right line drawne from the meeting of them unto the toppe or point of the angle, shall cut it into two equall parts. 9. p j.
Hitherto we have spoken of plaine lines: Their affection followeth, and first in the Bisection or dividing of an Angle into two equall parts.
Let the right lined Angle to bee divided into two equall parts bee eai. whose equall shankes let them be ae. and ai. (or if they be unequall, let them be made equall, by the [7 e].) Then two equall peripheries from the ends e and i. meet before the Angle in o. Lastly, draw a line from o. unto a. I say the angle given is divided into two equall parts. For by drawing the right lines oe. and oi. the angles oae. and oai. equicrurall, by the grant, and by their common side ao. are equall in base eo. and io. by the [10 e] (Because they are the raies of equall peripheries.) Therefore by the [7. e iij]. the angles oae. and oai. are equall: And therefore the Angle eai. is equally divided into two parts.
12. If two equall peripheries from the ends of a right line given, doe meete on each side of the same, a right line drawne from those meetings, shall divide the right line given into two equall parts. 10. p j.
Let the right line given bee ae. And let two equall peripheries from the ends a. and e. meete in i. and o. Then from those meetings let the right line io. be drawne. I say, That ae. is divided into two equall parts, by the said line thus
drawne. For by drawing the raies of the equall peripheries ia. and ie. the said io. doth cut the angle aie. into two equall parts, by the [11. e]. Therefore the angles aiu. and uie. being equall and equicrurall (seeing the shankes are the raies of equall peripheries, by the grant.) have equall bases au. and ue. by the [7. e iij]. Wherefore seeing the parts au. and ue. are equall, ae. the assigned right line is divided into two equall portions.
13. If a right line doe stand perpendicular upon another right line, it maketh on each side right angles: And contrary wise.
A right line standeth upon a right line, which cutteth, and is not cut againe. And the Angles on each side, are they which the falling line maketh with that underneath it, as is manifest out of Proclus, at the 15. pj. of Euclide; As here ae. the line cut: and io. the insisting line, let them be perpendicular; The angles on each side, to witt aio. and eio. shall bee right angles, by the [13. e iij].
The Rular, for the making of straight lines on a plaine, was the first Geometricall instrument: The Compasses, for the describing of a Circle, was the second: The Norma or Square for the true erecting of a right line in the same plaine upon another right line, and then of a surface and body, upon a surface or body, is the third. The figure therefore is thus.
Now Perpendiculū, an instrument with a line & a plummet of leade appendant upon it, used of Architects, Carpenters, and Masons, is meerely physicall: because heavie things
naturally by their weight are in straight lines carried perpendicularly downeward. This instrument is of two sorts: The first, which they call a Plumbe-rule, is for the trying of an erect perpendicular, as whether a columne, pillar, or any other kinde of building bee right, that is plumbe unto the plaine of the horizont & doth not leane or reele any way. The second is for the trying or examining of a plaine or floore, whether it doe lye parallell to the horizont or not. Therefore when the line from the right angle, doth fall upon the middle of the base; it shall shew that the length is equally poysed. The Latines call it Libra, or Libella, a ballance: of the Italians Livello, and vel Archipendolo, Achildulo: of the French, Nivelle, or Niueau: of us a Levill.
Therefore
14. If a right line do stand upon a right line, it maketh the angles on each side equall to two right angles: and contrariwise out of the 13. and 14. p j.
For two such angles doe occupy or fill the same place that two right angles doe: Therefore
they are equall to them by the 11. e j. If the insisting line be perpendicular unto that underneath it, it then shall make 2. right angles, by the [13. e]. If it bee not perpendicular, & do make two oblique angles, as here aio. and oie. are yet shall they occupy the same place that two right angles doe: And therefore they are equall to two right angles, by the same.
The converse is forced by an argument ab impossibli, or ab absurdo, from the absurdity which otherwise would follow of it: For the part must otherwise needes bee equall to the whole. Let therefore the insisting or standing line which maketh two angles aeo. and aeu. on each side equall to two right angles, be ae. I say that oe. and ei. are but one right line. Otherwise let oe. bee continued unto u. by the [6. e]. Now by the [14. e.] or next former element, aeo. & aeu. are equall to two right angles; To which also oea. & aei. are equall by the grant: Let aeo. the common angle be taken away: then shall there be left aeu. equall to aei. the part to the whole, which is absurd and impossible. Herehence is it certaine that the two right lines oe, and ei, are in deede but one continuall right line.
And
15. If two right lines doe cut one another, they doe make the angles at the top equall and all equall to foure right angles. 15. p j.
Anguli ad verticem, Angles at the top or head, are called Verticall angles which have their toppes meeting in the same point. The Demonstration is: Because the lines cutting one another, are either perpendiculars, and then all
right angles are equall as heere: Or else they are oblique, and then also are the verticalls equall, as are aui, and oue: And againe, auo, and iue. Now aui, and oue, are equall, because by the [14. e.] with auo, the common angle, they are equall to two right angles: And therefore they are equall betweene themselves. Wherefore auo, the said common angle beeing taken away, they are equall one to another.
And
16. If two right lines cut with one right line, doe make the inner angles on the same side greater then two right angles, those on the other side against them shall be lesser then two right angles.
As here, if auy, and uyi, bee greater then two right angles euy, and uyo, shall bee lesser then two right angles.
17. If from a point assigned of an infinite right line given, two equall parts be on each side cut off: and then from the points of those sections two equall circles doe meete, a right line drawne from their meeting unto the point assigned, shall bee perpendicular unto the line given. 11. p j.
As let a, be the point assigned of the infinite line given: and from that on each side, by the [7. e.] cut off equall
portions ae, and ai, Then let two equall peripheries from the points e, and i, meete, as in o, I say that a right line drawne from o, the point of the meeting of the peripheries. unto a. the point given, shalbe perpendicular upon the line given. For drawing the right lines oe, & oi, the two angles eao, and iao, on each side, equicrurall by the construction of equall segments on each side, and oa, the common side, are equall in base by the [9. e]. And therefore the angles themselves shall be equall, by the [7. e iij]. and therefore againe, seeing that ao, doth lie equall betweene the parts ea, and ia, it is by the [13. e ij]. perpendicular upon it.
18. If a part of an infinite right line, bee by a periphery for a point given without, cut off a right line from the said point, cutting in two the said part, shall bee perpendicular upon the line given. 12. p j.
Of an infinite right line given, let the part cut off by a periphery of an externall center be ae: And then let io, cut the said part into two parts by the [12. e]. I say that io is perpendicular unto the said infinite right line. For it standeth upright, and maketh aoi, and eoi, equall angles, for the same cause, whereby the next former perpendicular was demonstrated.
19. If two right lines drawne at length in the same plaine doe never meete, they are parallells. è 35. d j.
Thus much of the Perpendicularity of plaine right lines: Parallelissmus, or their parallell equality doth follow.
Euclid did justly require these lines so drawne to be granted paralels: for then shall they be alwayes equally distant, as here ae. and io.
Therefore
20. If an infinite right line doe cut one of the infinite right parallell lines, it shall also cut the other.
As in the same example uy. cutting ae. it shall also cut io. Otherwise, if it should not cut it, it should be parallell unto it, by the [18 e]. And that against the grant.
21. If right lines cut with a right line be pararellells, they doe make the inner angles on the same side equall to two right angles: And also the alterne angles equall betweene themselves: And the outter, to the inner opposite to it: And contrariwise, 29, 28, 27. p 1.
The paralillesme, or parallell-equality of right lines cut with a right line, concludeth a threefold equality of angles: And the same is againe of each of them concluded. Therefore in this one element there are sixe things taught; all which are manifest if a perpendicular, doe fall
upon two parallell lines. The first sort of angles are in their owne words plainely enough expressed. But the word Alternum, alterne [or alternate, H.] here, as Proclus saith, signifieth situation, which in Arithmeticke signified proportion, when the antecedent was compared to the consequent; notwithstanding the metaphor answereth fitly. For as an acute angle is unto his successively following obtuse; So on the other part is the acute unto his successively following obtuse: Therefore alternly, As the acute unto the acute: so is the obtuse, unto the obtuse. But the outter and inner are opposite, of the which the one is without the parallels; the other is within on the same part not successively; but upon the same right line the third from the outer.
The cause of this threefold propriety is from the perpendicular or plumb-line, which falling upon the parallells breedeth and discovereth all this variety: As here they are right angles which are the inner on the same part or side: Item, the alterne angles: Item the inner and the outter: And therefore they are equall, both, I meane, the two inner to two right angles: and the alterne angles between themselvs: And the outter to the inner opposite to it.
If so be that the cutting line be oblique, that is, fall not upon them plumbe or perpendicularly, the same shall on the contrary befall the parallels. For by that same obliquation or slanting, the right lines remaining and the angles unaltered, in like manner both one of the inner, to wit, euy, is made obtuse, the other, to wit, uyo, is made acute: And the alterne angles are made acute and obtuse: As also the outter and inner opposite are likewise made acute and obtuse.
If any man shall notwithstanding say, That the inner angles are unequall to two right angles: By the same argument may he say (saith Ptolome in Proclus) That on each side they be both greater than two right angles, and also lesser: As in the parallel right lines ae and io, cut with
the right line uy, if thou shalt say that auy and iyu, are greater then two right angles, the angles on the other side, by the [16 e], shall be lesser then two right angles, which selfesame notwithstanding are also, by the gainesayers graunt, greater then two right angles, which is impossible.
The same impossibility shall be concluded, if they shall be sayd, to be lesser than two right angles.
The second and third parts may be concluded out of the first. The second is thus: Twise two angles are equall to two right angles oyu, and euy, by the former part: Item, auy, and euy, by the [14 e]. Therefore they are equall betweene themselves. Now from the equall, Take away euy, the common angle, And the remainders, the alterne angles, at u, and y shall be least equall.
The third is thus: The angles euy, and oys, are equall to the same uyi, by the second propriety, and by the [15 e]. Therefore they are equall betweene themselves.
The converse of the first is here also the more manifest by that light of the common perpendicular, And if any man shall thinke, That although the two inner angles be equall to two right angles, yet the right may meete, as if those equall angles were right angles, as here; it must needes be that two right lines divided by a common perpendicular, should both leane, the one this way, the other that way, or at least one of them, contrary to the [13 e ij].
If they be oblique angles, as here, the lines one slanting or
obliquely crossing one another, the angles on one side will grow lesse, on the other side greater. Therefore they would not be equall to two right angles, against the graunt.
From hence the second and third parts may be concluded. The second is thus: The alterne angles at u and y, are equall to the foresayd inner angles, by the 14 e: Because both of them are equall to the two right angles: And so by the first part the second is concluded.
The third is therefore by the second demonstrated, because the outter oys, is equall to the verticall or opposite angle at the top, by the [15 e]. Therefore seeing the outter and inner opposite are equall, the alterne also are equall.
Wherefore as Parallelismus, parallell-equality argueth a three-fold equality of angles: So the threefold equality of angles doth argue the same parallel-equality.
Therefore,
22. If right lines knit together with a right line, doe make the inner angles on the same side lesser than two right Angles, they being on that side drawne out at length, will meete.
As here ae, and io, knit together with eo, doe make two angles aeo, and ioe, lesser than two right angles: They shall therefore, I say, meete if they be continued out that wayward. The assumption and complexion is out of the [21 e], of right lines in the same plaine. If right lines cut with a right line be parallels, they doe make the inner angles on the same part equall to two right-angles. Therefore if they doe not make them equall, but lesser, they shall not be parallel, but shall meete.
And
23. A right line knitting together parallell right lines, is in the same plaine with them. 7 p xj.
As here uy, knitting or joyning together the two parallels ae, and io, is in the same plaine with them as is manifest by the [8 e].
And
24. If a right line from a point given doe with a right line given make an angle, the other shanke of the angle equalled and alterne to the angle made, shall be parallell unto the assigned right line. 31 p j.
As let the assigned right line be ae: And the point given, let it be i. From which the right line, making with the assigned ae, the angle, ioe, let it be io: To the which at i, let the alterne angle oiu, be made equall: The right line ui, which is the other shanke, is parallel to the assigned ae.
An angle, I confesse, may bee made equall by the first propriety: And so indeed commonly the Architects and Carpenters doe make it, by erecting of a perpendicular. It may also againe in like manner be made by the outter angle: Any man may at his pleasure use which hee shall thinke good: But that here taught we take to be the best.
And
25. The angles of shanks alternly parallell, are equall. Or Thus, The angles whose alternate feete are parallells, are equall. H.
This consectary is drawne out of the third property of
the [21 e]. The thing manifest in the example following, by drawing out, or continuing the other shanke of the inner angle. But Lazarus Schonerus it seemeth doth thinke the adverbe alterne, (alternely or alternately) to be more then needeth: And therefore he delivereth it thus: The angles of parallel shankes are equall.
And
26. If parallels doe bound parallels, the opposite lines are equall è 34 p. j. Or thus: If parallels doe inclose parallels, the opposite parallels are equall. H.
Otherwise they should not be parallell. This is understood by the perpendiculars, knitting them together, which by the definition are equall betweene two parallells: And if of perpendiculars they bee made oblique, they shall notwithstanding remaine equall, onely the corners will be changed.
And
27. If right lines doe joyntly bound on the same side equall and parallell lines, they are also equall and parallell.
This element might have beene concluded out of the next precedent: But it may also be learned out of those
which went before. As let ae, and io, equall parallels be bounded joyntly of ai, and eo: and let ei be drawn. Here because the right line ei falleth upon the parallels ae, and io, the alterne angles aei and eio, are equall, by the [21 e]. And they are equall in shankes ae, and io, by the grants, and ei, is the common shanke: Therefore they are also equall in base ai, and eo, by the [7 e iij]. This is the first: Then by [21 e], the alterne angles eia, and ieo, are equall betweene themselves: And those are made by ai and eo, cut by the right line ei: Therefore they are parallell; which was the second.
On the same part or side it is sayd, least any man might understand right lines knit together by opposite bounds as here.
28. If right lines be cut joyntly by many parallell right lines, the segments betweene those lines shall bee proportionall one to another, out of the 2 p vj and 17 p xj.
Thus much of the Perpendicle, and parallell equality of plaine right lines: Their Proportion is the last thing to be considered of them.
The truth of this element dependeth upon the nature of the parallells: And that throughout all kindes of equality and inequality, both greater and lesser. For if the lines thus cut be perpendiculars, the portions
intercepted betweene the two parallels shall be equall: for common perpendiculars doe make parallell equality, as before hath beene taught, and here thou seest.
If the lines cut be not parallels, but doe leane one toward another, the portions cut or intercepted betweene them will not be equall, yet shall they be proportionall one to another. And looke how much greater the line thus cut is: so much greater shall the intersegments or portions intercepted be. And contrariwise, Looke how much lesse: so much lesser shall they be.
The third parallell in the toppe is not expressed, yet must it be understood.
This element is very fruitfull: For from hence doe arise and issue, First the manner of cutting a line according to any rate or proportion assigned: And then the invention or way to finde out both the third and fourth proportionalls.
29. If a right line making an angle with another right line, be cut according to any reason [or proportion] assigned, parallels drawne from the ends of the segments, unto the end of the sayd right line given and unto some contingent point in the same, shall cut the line given according to the reason given.
Schoner hath altered this Consectary, and delivereth it
thus: If a right line making an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned. 9 and 10 p vj.
Punctum contingens, A contingent point, that is falling or lighting in some place at al adventurs, not given or assigned.
This is a marvelous generall consectary, serving indifferently for any manner of section of a right line, whether it be to be cut into two parts, or three parts, or into as many parts, as you shall thinke good, or generally after what manner of way soever thou shalt command or desire a line to be cut or divided.
Let the assigned Right line to be cut into two equall parts be ae. And the right line making an angle with it, let it be the infinite right line ai. Let ao, one portion thereof be cut off. And then by the [7 e], let oi, another part thereof be taken equall to it. And lastly, by the [24 e], draw parallels from the points i, and o, unto e, the end of the line given, and to u; a contingent point therein. Now the third parallell is understood by the point a, neither is it necessary that it should be expressed. Therefore the line ae, by the [28], is cut into two equall portions: And as ao, is to oi: So is au, to ue. But ao, and oi, are halfe parts. Therefore au, and ue, are also halfe parts.
And here also is the [12 e] comprehended, although not in the same kinde of argument, yet in effect the same. But that argument was indeed shorter, although this be more generall.
Now let ae be cut into three parts, of which the first let it bee
the halfe of the second: And the second, the halfe of the third: And the conterminall or right line making an angle with the sayd assigned line, let it be cut one part ao: Then double this in ou: Lastly let ui be taken double to ou, and let the whole diagramme be made up with three parallels ie, uy, and os, The fourth parallell in the toppe, as afore-sayd, shall be understood. Therefore that section which was made in the conterminall line, by the [28 e], shall be in the assigned line: Because the segments or portions intercepted are betweene the parallels.
And
30. If two right lines given, making an angle, be continued, the first equally to the second, the second infinitly, parallels drawne from the ends of the first continuation, unto the beginning of the second, and some contingent point in the same, shall intercept betweene them the third proportionall. 11. p vj.
Let the right lines given, making an angle, be ae, and ai: and ae, the first, let it be continued equally to the same ai, and the same ai, let it be drawne out infinitly: Then the parallels ei, and ou, drawne from the ends of the first continuation, unto i, the beginning of the second: and u, a contingent point in the second, doe cut off iu, the third proportionall sought. For by the [28 e], as ae, is unto eo, so is ai, unto iu.
And
31. If of three right lines given, the first and the third making an angle be continued, the first equally to the second, and the third infinitly; parallels drawne
from the ends of the first continuation, unto the beginning of the second, and some contingent point, the same shall intercept betweene them the fourth proportionall. 12. p vj.
Let the lines given be these: The first ae, the second ei, the third ao, and let the whole diagramme be made up according to the prescript of the consectary. Here by [28. e], as ae, is to ei so is ao, to ou. Thus farre Ramus.
Lazarus Schonerus, who, about some 25. yeares since, did revise and augment this worke of our Authour, hath not onely altered the forme of these two next precedent consectaries: but he hath also changed their order, and that which is here the second, is in his edition the third: and the third here, is in him the second. And to the former declaration of them, hee addeth these words: From hence, having three lines given, is the invention of the fourth proportionall; and out of that, having two lines given, ariseth the invention of the third proportionall.
2 Having three right lines given, if the first and the third making an angle, and knit together with a base, be continued, the first equally to the second; the third infinitly; a parallel from the end of the second, unto the continuation of the third, shall intercept the fourth proportionall. 12. p vj.
The Diagramme, and demonstration is the same with our [31. e] or 3 c of Ramus.
3 If two right lines given making an angle, and knit together with a base, be continued, the first equally to the second, the second infinitly; a parallell to the base from the end of the first continuation unto the second, shall intercept the third proportionall. 11. p vj.
The Diagramme here also, and demonstration is in all
respects the same with our [30 e], or 2 c of Ramus.
Thus farre Ramus: And here by the judgement of the learned Finkius, two elements of Ptolomey are to be adjoyned.
32 If two right lines cutting one another, be againe cut with many parallels, the parallels are proportionall unto their next segments.
It is a consectary out of the [28 e]. For let the right lines ae. and ai, cut one another at a, and let two parallell lines uo, and ei, cut them; I say, as au, is to uo, so ae, is to ei. For from the end i, let is, be erected parallell to ae, and let uo, be drawne out untill it doe meete with it. Then from the end s, let sy, be made parallell to ai: and lastly, let ea, be drawne out, untill it doe meete with it. Here now ay, shall be equall to the right line is, that is, by the [26. e], to ue: and at length, by the [28. e], as ua, is to uo; so is ay, that is, ue, to os. Therefore, by composition or addition of proportions, as ua, is unto uo, so ua, and ue, shall be unto uo, and os, that is, ei, by the [27. e].
The same demonstration shall serve, if the lines do crosse one another, or doe vertically cut one another, as in the same diagramme appeareth. For if the assigned ai, and us, doe cut one another vertically in o, let them be cut with the parallels au, and si: the precedent fabricke or figure being made up, it shall be by [28. e.] as au, is unto ao, the segment next unto it: so ay, that is, is, shall be unto oi, his next segment.
The [28. e] teacheth how to finde out the third and fourth proportionall: This affordeth us a meanes how to find out
the continually meane proportionall single or double.
Therefore
33. If two right lines given be continued into one, a perpendicular from the point of continuation unto the angle of the squire, including the continued line with the continuation, is the meane proportionall betweene the two right lines given.
A squire (Norma, Gnomon, or Canon) is an instrument consisting of two shankes, including a right angle. Of this we heard before at the [13. e]. By the meanes of this a meane proportionall unto two lines given is easily found: whereupon it may also be called a Mesolabium, or Mesographus simplex, or single meane finder.
Let the two right lines given, be ae, and ei. The meane proportional between these two is desired. For the finding of which, let it be granted that as ae, is to eo, so eo, is to ei: therefore let ae, be continued or drawne out unto i, so that ei, be equall to the other given. Then from e, the point of the continuation, let eo, an infinite perpendicular be erected. Now about this perpendicular, up and downe, this way and that way, let the squire ao, be moved, so that with his angle it may comprehend at eo, and with his shanks it may include the whole right line ai. I say that eo, the segment of the perpendicular, is the meane proportionall between
ae, and ei, the two lines given. For let ea, be continued or drawne out into u, so that the continuation au, be equall unto eo: and unto a, the point of the continuation, let the angle uas, be made equall, and equicrurall to the angle oei, that is, let the shanke as, be made equall to the shanke ei. Wherefore knitting u, and s, together, the right lines us, and oi, shall be equall; and the angles eoi, aus, by the [7. e iij]. And by the [21. e], the lines sa, and oe, are parallell: and the angle sao, is equall to the angle aoe. But the angles sae, and aoi, are right angles by the Fabricke and by the grant; and therefore they are equall, by the [14. e iij]. Wherefore the other angles oae, and eoi, that is, sua, are equall. And therefore by the [21. e.] us, and ao are parallell; and us, and eo, continued shall meete, as here in y: and by the [26. e.] oy, and as are equall. Now, by the [32. e.] as ue, is to ua, so is ey, to as. Therefore by subduction or subtraction of proportions, as ea, is to ua, so is eo, that is, ua, to oy, that is as.
And
34 If two assigned right lines joyned together by their ends rightanglewise, be continued vertically; a square falling with one of his shankes, and another to it parallell and moveable upon the ends of the assigned, with the angles upon the continued lines, shall cut betweene them from the continued two meanes continually proportionall to the assigned.
The former consectary was of a single mesolabium; this is of a double, whose use in making of solids, to this or that bignesse desired is notable.
Let the two lines assigned be ae, and ei; and let there be two meane right lines, continually proportionall betweene them sought, to wit, that may be as ae, is unto
one of the lines found; so the same may be unto the second line found. And as that is unto this, so this may be unto ei. Let therefore ae, and ei, be joyned rightanglewise by their ends at e; and let them be infinite continued, but vertically, that is, from that their meeting from the lines ward, from ei, towards u, but ae, towards o. Now for the rest, the construction; it was Plato's Mesographus; to wit, a squire with the opposits parallell. One of his sides au, moueable, or to be done up and downe, by an hollow riglet in the side adjoyning. Therefore thou shalt make thee a Mesographus, if unto the squire thou doe adde one moveable side, but so that how so ever it be moved, it be still parallell unto the opposite side [which is nothing else, but as it were a double squire, if this squire be applied unto it; and indeed what is done by this instrument, may also be done by two squires, as hereafter shall be shewed.] And so long and oft must the moveable side be moved up and downe, untill with the opposite side it containe or touch the ends of the assigned, but the angles must fall precisely upon the continued lines: The right lines from the point of the continuation, unto the corners of the squire, are the two meane proportionals sought.
As if of the Mesographus auoi, the moveable side be au;
thus thou shalt move up and downe, untill the angles u, and o, doe hit just upon the infinite lines; and joyntly at the same instant ua, and oi, may touch the ends of the assigned a, and i. By the former consectary it shall be as ei, is to eo, so eo, shall be unto eu: and as eo, is to eu, so shall eu, be unto ea.
And thus wee have the composition and use, both of the single and double Mesolabium.
35. If of foure right lines, two doe make an angle, the other reflected or turned backe upon themselves, from the ends of these, doe cut the former; the reason of the one unto his owne segment, or of the segments betweene themselves, is made of the reason of the so joyntly bounded, that the first of the makers be joyntly bounded with the beginning of the antecedent made; the second of this consequent joyntly bounded with the end; doe end in the end of the consequent made.
Ptolomey hath two speciall examples of this Theorem: to those Theon addeth other foure.
Let therefore the two right lines be ae, and ai: and from the ends of these other two reflected, be iu, and eo, cutting themselves in y; and the two former in u, and o. The reason of the particular right lines made shall be as
the draught following doth manifest. In which the antecedents of the makers are in the upper place: the consequents are set under neathe their owne antecedents.
| The I. is Ptolemeys and Theons I. | |||
| The makers: | The reason made. | ||
| iu, | ye. | ||
| uy, | eo, | ia, | ao. |
| The II. is Theons VI. | |||
| au, | ey. | ||
| ue, | yo, | ai, | io. |
| The III. is Theons III. | |||
| ea, | ui. | ||
| au, | iy, | eo, | oy. |
| The IIII. is Theons II. | |||
| oa, | iu. | ||
| ai, | uy, | oe, | ey. |
| The V. is Ptolemys, II. Theons IIII. | |||
| iy, | ue. | ||
| yu, | ea, | io, | ao. |
| The VI. is Theons V. | |||
| eu, | ai. | ||
| ua, | io, | ey, | yo. |
The businesse is the same in the two other, whether you doe crosse the bounds or invert them.
Here for demonstrations sake we crave no more, but that from the beginning of an antecedent made a parallell be drawne to the second consequent of the makers, unto one of the assigned infinitely continued: then the multiplied proportions shall be,
The Antecedent, the Consequent; the Antecedent, the
Consequent of the second of the makers; every way the reason or rate is of Equallity.
The Antecedent; the Consequent of the first of the makers; the Parallell; the Antecedent of the second of the makers, by the [32. e]. Therefore by multiplication of proportions, the reason of the Parallell, unto the Consequent of the second of the makers, that is, by the fabricke or construction, and the [32. e.] the reason of the Antecedent of the Product, unto the Consequent, is made of the reason, &c. after the manner above written.
For examples sake, let the first speciall example be demonstrated. I say therefore, that the reason of ia, unto ao, is made of the reason of iu, unto uy, multiplied by the reason of ye, unto eo. For from the beginning of the Antecedent of the product, to wit, from the point i, let a line be drawne parallell to the right line ey, which shall meete with ae, continued or drawne out infinitely in n. Therefore, by the [32. e], as ia, is to ao: so is the parallell drawne to eo, the Consequent of the second of the makers. Therefore now the multiplied proportions are thus iu, uy, in, ey, by the 32. e: ye, eo, ey, eo. Therefore as the product of iu, by ye, is unto the product of uy, by eo: So in, is to eo, that is, ia, to ao.
So let the second of Ptolemy to be taught, which in our
Table aforegoing is the fifth. I say therefore that the reason of io, unto oa; is made of the reason of iy, unto yu, and the reason of ue, unto ea. For now againe, from the beginning of the Antecedent of the Product i, let a line be drawne parallell unto ea, the Consequent of the second of the Makers, which shall meete with eo, drawne out at length, in n: therefore, by the [32. e.] as io, is to ao; so is en, unto ea. Therefore now again the multiplied proportions are thus:
| ue, | ea, | ue, | ea. |
| iy, | yu, | en, | ue; |
by the [32. e]. Therefore, by multiplication of proportions, the reason of en, unto ea, that is, of io, unto oa, is made of the reason of iy, unto yu, by the reason of ue, unto ea.
It shall not be amisse to teach the same in the examples of Theon. Let us take therefore the reason of the Reflex, unto the Segment; And of the segments betweene themselves; to wit, the 4. and 6. examples of our foresaid draught: I say therefore, that the reason of oe, unto ey, is made of the reason oa, unto ai, by the reason of iu, unto uy. For from the end o, to wit, from the beginning of the Antecedent of the product, let the right line no, be drawne parallell to uy. It shall be by the [32. e.] as oe, is to ey: so the parallell no, shall be to uy: but the reason of no, unto uy, is made of the reason of oa, unto ai, and of iu, unto uy: for the multiplied proportions are,
| iu, | uy, | iu, | uy. |
| oa, | ai, | on, | iu. |
by the [32. e.]
Againe, I say, that the reason of ey, unto yo, is compounded of the reason of eu, unto ua, and of ai, unto io.
Theon here draweth a parallell from o, unto ui. By the generall fabricke it may be drawne out of e, unto ui.
It shall be therefore as ey, is unto yo, so en, shall be unto oi. Now the proportions multiplied are,
| ai, | io, | ai, | io. |
| eu, | ua, | en, | ai. |
by the [32. e.]
Therefore the reason of en, unto io , that is of ey, unto
yo, shall be made of the foresaid reasons.
Of the segments of divers right lines, the Arabians have much under the name of The rule of sixe quantities. And the Theoremes of Althindus, concerning this matter, are in many mens hands. And Regiomontanus in his Algorithmus: and Maurolycus upon the 1 p iij. of Menelaus, doe make mention of them; but they containe nothing, which may not, by any man skillfull in Arithmeticke, be performed by the multiplication of proportions. For all those wayes of theirs are no more but speciall examples of that kinde of multiplication.
Of Geometry, the sixt Booke, of a Triangle.
1. Like plaines have a double reason of their homologall sides, and one proportionall meane, out of 20 p vj. and xj. and 18. p viij.
Or thus; Like plaines have the proportion of their correspondent proportionall sides doubled, & one meane proportionall: Hitherto wee have spoken of plaine lines and their affections: Plaine figures and their kindes doe follow in the next place. And first, there is premised a common corollary drawne out of the [24. e. iiij]. because in plaines there are but two dimensions.
2. A plaine surface is either rectilineall or obliquelineall, [or rightlined, or crookedlined. H.]
Straightnesse, and crookednesse, was the difference of lines at the [4. e. ij]. From thence is it here repeated and attributed to a surface, which is geometrically made of lines. That made of right lines, is rectileniall: that which is made of crooked lines, is Obliquilineall.
3. A rectilineall surface, is that which is comprehended of right lines.
A plaine rightlined surface is that which is on all sides inclosed and comprehended with right lines. And yet they are not alwayes right betweene themselves, but such lines as doe lie equally betweene their owne bounds, and without comparison are all and every one of them right lines.
4. A rightilineall doth make all his angles equall to right angles; the inner ones generally to paires from two forward: the outter always to foure.
Or thus: A right lined plaine maketh his angles equall unto right angles: Namely the inward angles generally, are equall unto the even numbers from two forward, but the outward angles are equall but to 4. right angles. H.
The first kinde I meane of rectilineals, that is a triangle doth make all his inner angles equall to two right angles, that is, to a binary, the first even number of right angles: the second, that is a quadrangle, to the second even number, that is, to a quaternary or foure: The third, that is, a Pentangle, of quinqueangle to the third, that is a senary of right angles, or 6. and so farre forth as thou seest in this Arithmeticall progression of even numbers,
| 2. | 4. | 6. | 8. | 10. | 12. |
| 3. | 4. | 5. | 6. | 7. | 8. |
Notwithstanding the outter angles, every side continued and drawne out, are alwayes equall to a quaternary of right angles, that is to foure. The former part being granted (for that is not yet demonstrated) the latter is from thence concluded: For of the inner angles, that of the outter, is easily proved. For the three angles of a triangle are equall to two right angles. The foure of a quadrangle to foure: of a quinquangle, to sixe: of a sexe angle, to eight: Of septangle, to tenne, and so forth, form a binarie by even numbers: Whereupon, by the [14. e. V]. a perpetuall quaternary of the outer angles is concluded.
5. A rectilineall is either a Triangle or a Triangulate.
As before of a line was made a lineate: so here in like manner of a triangle is made a triangulate.
6. A triangle is a rectilineall figure comprehended of three rightlines. 21. d j.
As here aei. A triangular figure is of Euclide defined from the three sides; whereupon also it might be called Trilaterum, that is three sided, of the cause: rather than Trianglum, three cornered, of the effect; especially seeing that three angles, and three sides
are not reciprocall or to be converted. For a triangle may have foure sides, as is Acidoides, or Cuspidatum, the barbed forme, which Zonodorus called Cœlogonion, or Cavangulum, an hollow cornered figure. It may also have both five, and sixe sides, as here thou seest. The name therefore of Trilaterum would more fully and fitly expresse the thing named: But use hath received and entertained the name of a triangle for a trilater: And therefore let it be still retained, but in that same sense:
7. A triangle is the prime figure of rectilineals.
A triangle or threesided figure is the prime or most simple figure of all rectilineals. For amongst rectilineall figures there is none of two sides: For two right lines cannot inclose a figure. What is meant by a prime figure, was taught at the [11. e. iiij].
And
8. If an infinite right line doe cut the angle of a triangle, it doth also cut the base of the same: Vitell. 29. t j.
9. Any two sides of a triangle are greater than the other.
Thus much of the difinition of a triangle; the reason or
rate in the sides and angles of a triangle doth follow. The reason of the sides is first.
Let the triangle be aei; I say, the side ai, is shorter, than the two sides ae, and ei, because by the [6. e ij], a right line is betweene the same bounds the shortest.
Therefore
10. If of three right lines given, any two of them be greater than the other, and peripheries described upon the ends of the one, at the distances of the other two, shall meete, the rayes from that meeting unto the said ends, shall make a triangle of the lines given.
Let it be desired that a triangle be made of these three lines, aei, given, any two of them being greater than the other: First let there be drawne an infinite right; From this let there be cut off continually three portions, to wit, ou, uy, and ys, equall to ae, and i, the three lines given. Then upon the ends y, and u, at the distances ou, and ys; let two peripheries meet in the point r. The rayes from that meeting unto the said ends, u, and y, shall make the triangle ury: for those rayes shall be equall to the right lines given, by the [10. e v].
And
11. If two equall peripheries, from the ends of a right line given, and at his distance, doe meete, lines
drawne from the meeting, unto the said ends, shall make an equilater triangle upon the line given. 1 p. j.
As here upon ae, there is made the equilater triangle, aei; And in like manner may be framed the construction of an equicrurall triangle, by a common ray, unequall unto the line given; and of a scalen or various triangle, by three diverse raies; all which are set out here in this one figure. But these specialls are contained in the generall probleme: neither doe they declare or manifest unto us any new point of Geometry.
12. If a right line in a triangle be parallell to the base, it doth cut the shankes proportionally: And contrariwise. 2 p vj.
Such therefore was the reason or rate of the sides in one triangle; the proportion of the sides followeth.
As here in the triangle aei, let ou, be parallell to the base; and let a third parallel be understood to be in the toppe a; therefore, by the [28. e. v]. the intersegments are proportionall.
The converse is forced out of
the antecedent: because otherwise the whole should be lesse than the part. For if ou, be not parallell to the base ei, then yu, is: Here by the grant, and by the antecedent, seeing ao, oe, ay, ye, are proportionall: and the first ao, is lesser than ay, the third: oe, the second must be lesser than ye, the fourth, that is the whole then the part.
13. The three angles of a triangle, are equall to two right angles. 32. p j.
Hitherto therefore is declared the comparison in the sides of a triangle. Now is declared the reason or rate in the angles, which joyntly taken are equall to two right angles.
The truth of this proposition, saith Proclus, according to common notions, appeareth by two perpendiculars erected upon the ends of the base: for looke how much by the leaning of the inclination, is taken from two right angles at the base, so much is assumed or taken in at the toppe, and so by that requitall the equality of two right angles is made; as in the triangle aei, let, by the [24. e v], ou, be parallell against ie. Here three particular angles, iao, iae, eau, are equall to two right lines; by the [14. e v]. But the inner angles are equall to the same three: For first, eai, is equall to it selfe: Then the other two are equall to their alterne angles, by the [24. e v].
Therefore
14. Any two angles of a triangle are lesse than two right angles.
For if three angles be equall to two right angles, then
are two lesser than two right angles.
And
15. The one side of any triangle being continued or drawne out, the outter angle shall be equal to the two inner opposite angles.
This is the rate of the inner angles in one and the same triangle: The rate of the outter with the inner opposite angles doth followe. As in the triangle aei, let the side ei, be continued or drawne out unto o; the two angles on each side aio and aie, are by the [14 e v]. equall to two right angles: and the three inner angles, are by the [13. e.] equall also to two right angles; take away aie, the common angle, and the outter angle aio, shall be left equall to the other two inner and opposite angles.
Therefore
16. The said outter angle is greater than either of the inner opposite angles. 16. p j.
This is a consectary following necessarily upon the next former consectary.
17. If a triangle be equicrurall, the angles at the base are equall: and contrariwise, 5. and 6. p. j.
The antecedent is apparent by the [7. e iij]. The converse is apparent by an impossibilitie, which otherwise must needs follow. For if any one shanke be greater than the other, as ae: Then by the [7. e v]. let oe, be cut off equall to it: and let oi, be drawne: then by [7. e iij]. the base oi, must
be equal to the base ae; but the base oi, is lesser than ae. For by the [9. e], ia; and ao, (to which ae, is equall, seeing that oe, is supposed to be equall to the same ai: and ao, is common to both) are greater than the said oi; therefore the same, oi, must be equall to the same ae, and lesser than the same, which is impossible. This was first found out by Thales Milesius.
Therefore
18. If the equall shankes of a triangle be continued or drawne out, the angles under the base shall be equall betweene themselves.
For the angles aei, and ieo: Item aie, and eiu, are equall to two right angles, by the [14. e v]. Therefore they are equall betweene themselves: wherefore if you shall take away the inner angles, equall betweene themselves, you shall leave the outter equall one to another.
And
19. If a triangle be an equilater, it is also an equiangle: And contrariwise.
It is a consectary out of the condition of an equicrurall triangle of two, both shankes and angles, as in the example aei, shall be demonstrated.
And
20. The angle of an equilater triangle doth
countervaile two third parts of a right angle. Regio. 23. p j.
For seeing that 3. angles are equall to 2. 1. must needs be equall to 2/3.
And
21. Sixe equilater triangles doe fill a place.
As here. For 2/3. of a right angle sixe lines added together doe make 12/3. that is foure right angles; but foure right angles doe fill a place by the [27. e. iiij].
22. The greatest side of a triangle subtendeth the greatest angle; and the greatest angle is subtended of the greatest side. 19. and 18. p j.
Subtendere, to draw or straine out something under another; and in this place it signifieth nothing else but to make a line or such like, the base of an angle, arch, or such like. And subtendi, is to become or made the base of an angle, arch, of a circle, or such like: As here, let ai, be a greater side than ae, I say the angle at e, shall be greater than that at i. For let there be cut off from ai, a portion equall to ae,; and let that be io: then the angle aei, equicrurall to the angle oie, shall be greater in base, by the grant. Therefore the angle shall be greater, by the [9 e iij].
The converse is manifest by the same figure: As let the angle aei, be greater than the angle aie. Therefore by the same, [9 e iij]. it is greater in base. For what is there spoken
of angles in generall, are here assumed specially of the angles in a triangle.
23. If a right line in a triangle, doe cut the angle in two equall parts, it shall cut the base according to the reason of the shankes; and contrariwise. 3. p vj.
The mingled proportion of the sides and angles doth now remaine to be handled in the last place.
Let the triangle be aei; and let the angle eai, be cut into two equall parts, by the right line ao: I say, as ea, is unto ai, so eo, is unto oi. For at the angle i, let the parallell iu, by the [24. e v]. be erected against ao; and continue or draw out ea, infinitly; and it shall by the [20. e v]. cut the same iu, in some place or other. Let it therefore cut it in u. Here, by the [28. e v]. as ea, is to au, so is eo, to oi. But au, is equall to ai, by the [17. e]. For the angle uia, is equall to the alterne angle oai, by the [21. e v]. And by the grant it is equall to oae, his equall: And by the [21. e v]. it is equall to the inner angle aui; and by that which is concluded it is equall to uia, his equall. Therefore by the [17. e], au, and ai, are equall. Therefore as ea, is unto ai, so is eo, unto oi.
The Converse likewise is demonstrated in the same figure. For as ea, is to ai; so is eo, to oi: And so is ea, to au, by the 12 e: therefore ai, and au, are equall, Item the angles eao, and oai, are equall to the angles at u, and i, by the [21. e v]. which are equall betweene themselves by the [17. e].