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Author: Euclid.
Title: The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
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Print source: The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
Euclid., Dee, John, 1527-1608., Candale, François de Foix, comte de, 1502-1594., Billingsley, Henry, Sir, d. 1606.
Imprinted at London: By Iohn Daye, [1570 (3 Feb.]]
Alternate titles: Elements. English
Subject terms:
Geometry -- Early works to 1800.
URL: https://name.umdl.umich.edu/A00429.0001.001
How to cite: For suggestions on citing this text, please see Citing the TCP on the Text Creation Partnership website.

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title page
The Translator to the Reader.
TO THE VNFAINED LOVERS of truthe, and constant Studentes of Noble Sciences, IOHN DEE of London, hartily wisheth grace from heauen, and most prospe∣rous successe in all their honest attemptes and exercises.
Here haue you (according to my promisse) the Groundplat of my MATHEMATICALL Praeface: annexed to Euclide (now first) published in our Englishe tounge. An. 1570. Febr. 3.
¶The first booke of Eu∣clides Elementes.
Definitions.
1. A signe or point is that, which hath no part.
. A line is length ithout breadth.
3 The endes or limites of a lyne, are pointes.
4 A right lyne is that which lieth equally betwene his pointes.
5 A superficies is that, which hath onely length and breadth.
6 Extremes of a superficies, are lynes.
7 A plaine superficies is that, which lieth equally betwene his lines.
8 A plaine angle is an inclination or bowing of two lines the one to the other and the one touching the other, and not beyng directly ioyned together.
9 And if the lines which containe the angle be right lynes, then is it called a rightlyned angle.
10 VVhen a right line standing vpon a right line maketh the angles on either side equall then either of those angles is a right angle. And the right lyne which standeth erected, is called a perpendiculer line to that vpon which it standeth.
11 An obtuse angle is that which is greater then a right angle.
12 An acute angle is that, which is lesse then a right angle.
13 A limite or terme, is the ende of euery thing.
14 A figure is that, which is contayned vnder one limite or terme, or many.
15 A circle is a plaine figure, conteyned vnder one line, which is called a cir∣cumference, vnto which all lynes drawen from one poynt within the figure and falling vpon the circumference therof are equall the one to the other.
16 And that point is called the centre of the circle, as is the point A, which is set in the middes of the former circle.
17 A diameter of a circle, is a right line, which drawen by the centre thereof, and ending at the circumference on either side, deuideth the circle into two equal partes.
18 A semicircle, is a figure which is contayned vnder the diameter, and vn∣der that part of the circumference which is cut of by the diametre.
19 A section or portion of a circle, is a figure whiche is contayned vnder a right lyne, and a parte of the circumference, greater or lesse then the semicircle.
20 Rightlined figures are such which are contayned vnder right lynes.
21 Thre sided figures, or figures of thre sydes, are such which are contay∣ned vnder three right lines.
22. Foure sided figures or figures of foure sides are such, which are contained vnder foure right lines.
23. Many sided figures are such which haue mo sides then foure.
24. Of three sided figures or triangles, an equilatre triangle is that, which hath three equall sides.
25. Isosceles, is a triangle, which hath onely two sides equall.
26. Scalenum is a triangle, whose three sides are all vnequall.
27. Againe of triangles, an Orthigonium or a rightangled triangle, is a tri∣angle which hath a right angle.
28. An ambligonium or an obtuse angled triangle, is a triangle which hath an obtuse angle.
29. An oxigonium or an acute angled triangle, is a triangle which hath all his three angles acute.
30 Of foure syded figures, a quadrate or square is that, whose sydes are e∣quall, and his angles right.
31 A figure on the one syde longer, or squarelike, or as some call it, a long square, is that which hath right angles, but hath not equall sydes.
32 Rhombus (or a diamonde) is a figure hauing foure equall sydes, but it is not rightangled.
33 Rhombides (or a diamond like) is a figure, whose opposite sides are e∣quall, and whose opposite angles are also equall, but it hath neither e∣quall sides, nor right angles.
34 All other figures of foure sides besides these, are called trapezia, or tables.
35 Parallel or equidistant right lines are such, which being in one and the selfe same superficies, and pro∣duced infinitely on both sydes, do neuer in any part concurre.
Peticions or requestes.
1 From any point to any point, to draw a right line.
2 To produce a right line finite, straight forth continually.
3 Vpon any centre and at any distance, o describe a circle.
4 All right angles are equall the one to the other.
5 VVhen a right line falling vpon wo right lines, doth make n one & the selfe same syde, the two inwarde angles lesse then two right angles, then shal these two right lines beyng produced length concurre on that part, in which are the two angles lesse then two right angles.
6 That two right lines include not a superficies.
Common sentences.
1 Thinges equall to one and the selfe same thyng: are equall also the one to the other.
2 And if ye adde equall thinges to equall thinges: the whole shalbe equall.
3 And if from equall thinges, ye take away equall thinges: the thinges re∣mayning shall be equall.
4 And if from vnequall thinges ye take away equall thinges: the thynges which remayne shall be vnequall.
5 And if to vnequall thinges ye adde equall thinges: the whole shall be vn∣equall.
6 Thinges which are double to one and the selfe same thing: are equall the one to the other.
7 Thinges which are the halfe of one and the selfe same thing are equal the one to the other.
8 Thinges which agree together are equall the one to the other.
9 Euery whole is greater then his part.
The first Probleme. The first Proposition. Vpon a right line geuen not beyng infinite, to describe an e∣quilater triangle, or a triangle of three equall sides.
An addition of Campanus.
The second Probleme. The second Proposition. Frō a point geuen, to draw a right line equal to a rightline geuen.
The 3. Probleme. The 3. Proposition. Two vnequal right lines being geuen, to cut of from the grea∣ter, a right lyne equall to the lesse.
The first Theoreme. The 4. Proposition. If there be two triangles, of which two sides of th'one be equal to two sides of the other, eche side to his correspondent side, and hauing also on angle of the one equal to one angle of the other, namely, that angle which is contayned vnder the equall right lines: the base also of the one shall be equall to the base of the other, and the one triangle shall be equal to the other triangle, and the other angles remayning shal be equall to the other an∣gles remayning, the one to the other, vnder which are subten∣ded equall sides.
The 2. Theoreme. The 5. Proposition. An Isosceles, or triangle of two equal sides, hath his angles a the base equall the one to the other. And those equal sides be∣ing produced, the angles which are vnder the base are also e∣quall the one to the other.
The third Theoreme. The sixt Proposition. If a triangle haue two angles equall the one to the other: the sides also of the same, which subtend the equall angles, shalbe equall the one to the other.
The 4. Theoreme. The 7. Proposition. If from the endes of one line, be drawn two right lynes to any pointe: there can not frō the self same endes on the same side, be drawn two other lines equal to the two first lines, the one to the other, vnto any other point.
An other demonstration after Campanus.
The fift Theoreme. The 8. Proposition. If two triangles haue two sides of th'one equall to two sides of the other, eche to his correspondent side, & haue also the base of the one equall to the base of the other: they shall haue also the angle contained vnder the equall right lines of the one, e∣quall to the angle contayned vnder the equall right lynes of the other.
The 4. Probleme. The 9. Proposition. To deuide a rectiline angle geuen, into two equall partes.
The 5. Probleme. The 10. Proposition. To deuide a right line geuen being finite, into two equall partes.
The 6. Probleme. The 11. Proposition. Vpon a right line geuen, to rayse vp from a poynt geuen in the same line a perpendicular line.
The 7. Probleme. The 12. Proposition. Vnto a right line geuen being infinite, and from a point geuen not being in the same line, to draw a perpendicular line.
The 6. Theoreme. The 13. Proposition. When a right line standing vpon a right line maketh any an∣gles: those angles shall be either two right angles, or equall to two right angles.
An othe demonstration after Pelitarius.
The 7. Theoreme. The 14. Proposition. If vnto a right line, and to a point in the same line, be drawn two right lines, no both on one and the same side, making the side angles equall to two right angles: those two right lynes shall make directly one right line.
An other demonstration after Pelitarius.
The 8. Theoreme. The 15. Proposition. If two right lines cut the one the other: the hed angles shal be equal the one to the other.
The conuerse of this proposition after Pelitarius.
The same conuerse after Proclus.
The 9. Theoreme. The 16. Proposition. Whensoeuer in any triangle, the lyne of one syde is drawen forth in length: the outwarde angle shall be greater then any one of the two inwarde and opposite angles.
An other demonstration after Pelitarius.
The 10. Theoreme. The 17. Proposition. In euery triangle, two angles, which two soeuer be taken, are lesse then two right angles.
The 11. Theoreme. The 18. Proposition. In euery triangle, to the greater side is subtended the grea∣ter angle.
The 12. Theoreme. The 19. Proposition. In euery triangle, vnder the greater angle is subtended the greater side.
The 13. Theoreme. The 20. Proposition. In euery triangle two sides, which two sides soeuer be taken, are greater then the side remayning.
The 14. Theoreme. The 21. Proposition. If from the endes of one of the sides of a triangle, be drawen to any point within the sayde triangle two right lines. those right lines so drawen, shalbe lesse then the two other sides of the triangle, but shall containe the greater angle.
The 8. Probleme. The 22. Proposition. Of thre right lines, which are equall to thre right lines geuē, to make a triangle. But it behoueth two of those lines, which two soeuer be taken, to be greater then the third. For that in euery triangle two sides, which two sides soeuer be taken, are greater then the side remayning.
An other construction, and demonstration after Flussates.
The 9. Probleme. The 23. Proposition. Vpon a right line geuen, and to a point in it geuen: to make a rectiline angle equall to a rectiline angle geuen.
An other construction and demonsation after Proclus.
An other construction also, and demonstration after Pelitarus.
The 15. Theoreme The 24. Proposition. If two triangles haue two sides of the one equall to two sides of the other, ech to his correspondent side, and if the angle cōtai∣ned vnder the equall sides of the one, be greater then the an∣gle contayned vnder the equall sides of the other: the base also of the same, shalbe greater then the base of the other.
The 16. Theoreme. The 25. Proposition. If two triangles haue two sides of the one equall to two sydes of the other, eche to his correspondent syde, and if the base of the one be greater then the base of the other: the angle also of the same cōtayned vnder the equall right lines shall be grea∣ter then the angle of the other.
The 17. Theoreme. The 26. Proposition. If two triangles haue two angles of the one equall to two an∣gles of the other, ech to his correspondent angle, and haue also one side of the one equall to one side of the other, either that side which lieth betwene the equall angles, or that which is subtended vnder one of the equall angles: the other sides also of the one, shalbe equall to the other sides of the other, eche to his correspondent side, and the other angle of the one shalbe equall to the other angle of the other.
The 18. Theoreme. The 27. Proposition. If a right line falling vpon two right lines, do make the alter∣nate angles equall the one to the other: those two right lines are parallels the one to the other.
The 19. Theoreme. The 28. Proposition. If a right line falling vpon two right lines, make the outward angle equall to the inward and opposite angle on one and the same syde, or the inwarde angles on one and the same syde, e∣quall to two right angles: those two right lines shall be paral∣lels the one to the other.
The 20. Theoreme. The 29. Proposition. A right line line falling vppon two parallel right lines: ma∣keth the alternate angles equall the one to the other: and al∣so the outwarde angle equall to the inwarde and opposite an∣gle on one and the same side: and moreouer the inwarde an∣gles on one and the same side equall to two right angles.
Pelitarius after this proposition addeth this witty conclusion.
The 21. Theoreme The 30. Proposition. Right lines which are parallels to one and the selfe same right line: are also parrallel lines the one to the other.
The 10. Probleme. The 31. Proposition. By a point geuen, to draw vnto a right line geuen, a parallel line.
The 22. Theoreme. The 32, Proposition. If one of the sydes of any triangle be produced: the outwarde angle that it maketh, is equal to the two inward and opposite angles. And the three inwarde angles of a triangle are equall to two right angles.
The conuerse of this proposition is thus.
The 23. Theoreme. The 33. Proposition. Two right lines ioyning together on one and the same side, two equall parallel lines: are also them selues equall the one to the other, and also parallels.
The 24. Theoreme. The 34 Proposition. In parallelogrammes, the sides and angles which are opposite the one to the other, are equall the one to the other, and their diameter deuideth them into two equall partes.
A Corrollary taken out of Flussates.
An addition of Pelitarius.
The 25. Theoreme. The 35. Proposition. Parallelogrammes consisting vppon one and the same base, and in the selfe same parallel lines, are equall the one to the other.
The 26. Theoreme. The 36. Proposition. Parallelogrammes consisting vpon equall bases, and in the selfe same parallel lines, are equall the one to the other.
The 27. Theoreme. The 37. Proposition. Triangles consisting vpon one and the selfe same base, and in the selfe same paralles: are equall the one to the other.
The 28. Theoreme. The 38. Proposition. Triangles which consist vppon equall bases, and in the selfe same parallel lines, are equall the one to the other.
An addition of Pelitarius.
An other addition of Pelitarius.
The 29. Theoreme. The 39. Proposition. Equall triangles consisting vpon one and the same base, and on one and the same side: are also in the selfe same parallel lines.
An addition of Flussates.
An addition of Campanus.
The 30. Theoreme. The 40. Proposition. Equall triangles consisting vpon equall bases, and in one and the same side: are also in the selfe same parallel lines.
The 31. Theoreme. The 41. Proposition. If a parallelograme & a triangle haue one & the selfe same base, and be in the selfe same parallel lines: the parallelo∣grame shalbe double to the triangle.
The conuerse of this proposition is thus.
An other conuerse of the same proposition.
The 11. Probleme. The 42. proposition. Vnto a triangle geuen, to make a parallelograme equal, whose angle shall be equall to a rectiline angle geuen.
The conuerse of this proposition after Pelitarius.
The 32. Theoreme. The 43. Proposition. In euery parallelograme, the supplementes of those parallelo∣grammes which are about the diameter, are equall the one to the other.
The 12. Probleme. The 44. Proposition. Vppon a right line geuen, to applye a parallelograme equall to a triangle geuen, and contayning an angle equall to a rec∣tiline angle geuen.
The conuerse of this proposition after Politarius.
The 13. Probleme. The 45 Proposition. To describe a parallelograme equal to any rectiline figure ge∣uen, and contayning an angle equall to a rectiline angle geuē.
An other more redy way.
The 14. Probleme. The 46. Proposition. Vppon a right line geuen, to describe a square.
An addition of Procus.
The conuerse thereof is thus.
The 33. Theoreme. The 47. Proposition. In rectangle triangles, the square whiche is made of the side that subtendeth the right angle, is equal to the squares which are made of the sides contayning the right angle.
An addition of Pelitarius.
An other addition of Pelitarius.
An other addition of Pelitarius.
An other addition of Pelitarius. The diameter of a square being geuen, to geue the square thereof.
The 34. Theoreme. The 48. Proposition. If the square which is made of one of the sides of a triangle, be equall to the squares which are made of the two other sides of the same triangle: the angle comprehended vnder those two other sides is a right angle.
¶The second booke of Eu∣clides Elementes.
The definitions.
1. Euery rectangled parallelogramme, is sayde to be contayned vnder two right lines comprehending a right angle.
2. In euery parallelogramme, one of those parallelogrammes, which soeuer it be, which are about the diameter, together with the two supplementes, is called a Gnomon.
An other more redy way after Pelitarius.
The 1. Theoreme. The 1. Proposition. If there be two right lines, and if the one of them be deuided into partes howe many soeuer: the rectangle figure compre∣hended vnder the two right lines, is equall to the rectangle fi∣gures whiche are comprehended vnder the line vndeuided, and vnder euery one of the partes of the other line.
¶ Principles.
The first Proposition.
The 2. Theoreme. The 2. Proposition. If a right line be deuided by chaunce, the rectangles figures which are comprehended vnder the whole and euery one of the partes, are equall to the square whiche is made of the whole.
An other demonstration of Campane.
An example of this Proposition in numbers.
The second Proposition.
The 3. Theoreme. The 3. Proposition. If a right line be deuided by chaunce: the rectangle figure cō∣prehended vnder the whole and one of the partes, is equall to the rectangle figue comprehended vnder the partes, & vnto the square which is made of the foresaid part.
An example of this Proposition in numbers.
The third proposition.
The 4. Theoreme. The 4. Proposition. If a right line be deuided by chaunce, the square whiche is made of the whole line is equal to the squares which are made of the partes, & vnto that rectangle figure which is compre∣hended vnder the partes twise.
An other demonstration.
An example of this Proposition in numbers.
The fourth Proposition.
The 5. Theoreme. The 5. Proposition. If a right line be deuided into two equall partes, & into two vnequall partes: the rectangle figures comprehended vnder the vnequall part of the whole, together with the square of that which is betwene the sectiōs, is equal to the square which is made of the halfe.
An example of this proposition in numbers.
The fifth proposition.
The 6. Theoreme. The 6. Proposition. If a right line be deuided into two equal partes, and if vnto it be added an other right line directly, the rectangle figure con∣tayned vnder the whole line with that which is added, & the line which is addedt, ogether with the square which is made of the halfe, is equall to the square which is made of the halfe line and of that which is added as of one line.
An example of this proposition in numbers.
The sixt Proposition.
The 7. Theoreme. The 7. Proposition. If a right lyne be deuided by chaunce, the square whiche is made of the whole together with the square which is made of one of the partes, is equall to the rectangle figure which is cō∣tayned vnder the whole and the said parte twise, and to the square which is made of the other part.
Flussates addeth vnto this Proposition this Corollary.
An example of this proposition in numbers.
The seuenth proposition.
The 8. Theoreme. The 8. Proposition. If a right line be deuided by chaūce, the rectangle figure com∣prehended vnder the whole and one of the partes foure times, together with the square which is made of the other parte, is equall to the square which is made of the whole and the fore∣said part as of one line.
An example of this Proposition in numbers.
The eight proposition.
The 9. Theoreme. The 9. Proposition. If a right line be deuided into two equall partes, and into two vnequall partes, the squares which are made of the vne∣quall partes of the whole, are double to the squares, which are made of the halfe lyne, and of that lyne which is betwene the sections.
¶ An example of this proposition in numbers.
The ninth Proposition.
The 10. Theoreme. The 10. Proposition. If a right line be deuided into two equal partes, & vnto it be added an other right line directly: the square which is made of the whole & that which is added as of one line, together with the square whiche is made of the lyne whiche is added, these two squares (I say) are double to these squares, namely, to the square which is made of the halfe line, & to the square which is made of the other halfe lyne and that whiche is added, as of one lyne.
¶ An other demonstration after Pelitarius.
¶ An example of this Proposition in numbers.
The tenth Proposition.
The 1. Probleme. The 11. Proposition. To deuide a right line geuen in such sort, that the rectangle figure comprehended vnder the whole, and one of the partes, shall be equall vnto the square made of the other part.
The 11. Theoreme. The 12. Proposition. In obtuseangle triangles, the square which is made of the side subtending the obtuse angle, is greater then the squares which are made of the sides which comprehend the obtuse angle, by the rectangle figure, which is comprehended twise vnder one of those sides which are about the obtuse angle, vpon which being produced falleth a perpendicular line, and that which is outwardly taken betwene the perpendicular line and the obtuse angle.
The 12. Theoreme. The 13. Proposition. In acuteangle triangles, the square which is made of the side that subtendeth the acute angle, is lesse then the squares which are made of the sides which comprehend the acute an∣gle, by the rectangle figure which is cōprehended twise vnder one of those sides which are about the acuteangle, vpō which falleth a perpendiculer lyne, and that which is inwardly ta∣ken betwene the perpendiculer lyne and the acute angle.
¶ A Corollary added by Orontius.
The 2. Probleme. The 14. Proposition. Vnto a rectiline figure geuen, to make a square equall.
¶The third booke of Eu∣clides Elementes.
Definitions. Equall circles are such, whose diameters are equall, or whose lynes drawen from the centres are equall.
A right line is sayd to touch a circle, which touching the cir∣cle and being produced cutteth it not.
Circles are sayd to touch the one the other, which touching the one the other, cut not the one the other.
Right lines in a circle are sayd to be equally distant from the cen∣tre, when perpendicular lines drawen from the centre vnto those lines are equall. And that line is sayd to be more di∣stant, vpon whom falleth the greater perpendicular line.
A section or segment of a circle, is a figure cōprehended vnder a right line and a portion of the circumference of a circle.
An angle of a section or segment, is that angle which is con∣tayned vnder a right line and the circūference of the circle.
An angle is sayd to be in a section, whē in the circumference is taken any poynt, and from that poynt are drawen right lines to the endes of the right line which is the base of the segment, the angle which is contayned vnder the right lines drawen from the poynt, is (I say) sayd to be an angle in a section.
But when the right lines which comprehend the angle do re∣ceaue any circumference of a circle, then that angle is sayd to be correspondent, and to pertaine to that circumference.
A Sector of a circle is (an angle being set at the centre of a circle) a figure contayned vnder the right lines which make that angle, and the part of the circumference re∣ceaued of them.
Like segmentes or sections of a circle are those, which haue equall angles, or in whom are equall angles.
The 1. Probleme. The 1. Proposition. To finde out the centre of a circle geuen.
Correlary.
The 1. Theoreme. The 2. Proposition. If in the circūference of a circle be takē two poyntes at all ad∣uentures: a right line drawen from the one poynt to the other shall fall within the circle.
The 2. Theoreme. The 3. Proposition. If in a circle a right line passing by the centre do deuide an o∣ther right line not passing by the cētre into two equall partes: it shall deuide it by right angles. And if it deuide the line by right angles, it shall also deuide the same line into two equall partes.
The 3. Theoreme. The 4. Proposition. If in a circle two right lines not passing by the centre, deuide the one the other: they shall not deuide eche one the other into two equall partes.
The 4. Theoreme. The 5. Proposition. If two circles cut the one the other, they haue not one and the same centre.
The 5. Theoreme. The 6. Proposition. If two circles touch the one the other, they haue not one and the same centre.
The 6. Theoreme. The 7. Proposition. If in the diameter of a circle be taken any poynt, which is not the centre of the circle, and from that poynt be drawen vnto the circumference certaine right lines: the greatest of those lines shall be that line wherein is the centre, and the lest shall be the residue of the same line. And of all the other lines, that which is nigher to the line which passeth by the centre is greater then that which is more distant. And from that point can fall within the circle on ech side of the least line onely two equall right lines.
¶ A Corollary.
The 7. Theoreme. The 8. Proposition. If without a circle be taken any poynt, and from that poynt be drawen into the circle vnto the circumference certayne right lines, of which let one be drawen by the centre and let the rest be drawen at all aduentures: the greatest of those lines which fall in the concauitie or hollownes of the circumference of the circle, is that which passeth by the centre: and of all the other lines that line which is nigher to the line which passeth by the centre is greater then that which is more distant. But of those right lines which end in the conuexe part of the circumfe∣rence, that is the least which is drawen from the poynt to the diameter: and of the other lines that which is nigher to the least is alwaies lesse then that which is more distant. And from that poynt can be drawen vnto the circumference on ech side of the least onely two equall right lines.
¶ A Corollary.
The 8. Theoreme. The 9. Proposition. If within a circle be taken a poynt, and from that poynt be drawen vnto the circumference moe then two equall right lines, the poynt taken is the centre of the circle.
¶ An other demonstration.
The 9. Theoreme. The 10. Proposition. A circle cutteth not a circle in moe pointes then two.
An other demonstration to proue the same.
The 10. Theoreme. The 11. Proposition. If two circles touch the one the other inwardly, their centres being geuen: a right line ioyning together their centres and produced, will fall vpon the touch of the circles.
An other demonstration to proue the same.
The 11. Theoreme. The 12. Proposition. If two circles touch the one the other outwardly, a right line drawen by their centres shall passe by the touch.
¶ An other demonstration after Pelitarius.
The 12. Theoreme. The 13. Proposition. A circle can not touch an other circle in moe poyntes then one, whether they touch within or without.
¶ An other demonstration after Pelitarius and Flussates.
The 13. Theoreme. The 14. Proposition. In a circle, equall right lines, are equally distant from the cē∣tre. And lines equally distant from the centre, are equall the one to the other.
¶ An other demonstration for the first part after Campane.
The 14. Theoreme. The 15. Proposition. In a circle, the greatest line is the diameter, and of all other lines that line which is nigher to the centre is alwayes greater then that line which is more distant.
¶An other demonstration after Campane.
The 15. Theoreme. The 16. Proposition. If from the end of the diameter of a circle be drawen a right line making right angles: it shall fall without the circle: and betwene that right line and the circumference can not be drawen an other right line: and the angle of the semicircle is greater then any acute angle made of right lines, but the o∣ther angle is lesse then any acute angle made of right lines.
Correlary.
The 2. Probleme. The 17. Proposition. From a poynt geuen, to draw a right line which shall touch a circle geuen.
¶ An addition of Pelitarius.
The 16. Theoreme. The 18. Proposition. If a right lyne touch a circle, and from the centre to the touch be drawen a right line, that right line so drawen shalbe a per∣pendicular lyne to the touche lyne.
¶ An other demonstration after Orontius.
The 17. Theoreme. The 19. Proposition. If a right lyne doo touche a circle, and from the point of the touch be raysed vp vnto the touch lyne a perpendicular lyne, in that lyne so raysed vp is the centre of the circle.
The 18. Theoreme. The 20. Proposition. In a circle an angle set at the centre, is double to an angle set at the circumference, so that both the angles haue to their base one and the same circumference.
The 19. Theoreme. The 21. Proposition. In a circle the angles which consist in one and the selfe same section or segment, are equall the one to the other.
The 20. Theoreme. The 22. Proposition. If within a circle be described a figure of fower sides, the an∣gles therof which are opposite the one to the other, are equall to two right angles.
The 21. Theoreme. The 23. Proposition. Vpon one and the selfe same right line can not be described two like and vnequall segmentes of circles, falling both on one and the selfe same side of the line.
The 22. Theorme. The 24. Proposition. Like segmentes of circles described vppon equall right lines, are equall the one to the other.
The 3. Probleme. The 25. Proposition. A segment of a circle beyng geuen to describe the whole cir∣cle of the same segment.
¶ A Corollary. Hereby it is manifest, that in a semicircle the angle BAD is equall to the angle DBA: but in a section lesse then a semicir∣cle, it is lesse: in a section greater then a semicircle, it is greater.
The 23. Theoreme. The 26. Proposition. Equall angles in equall circles consist in equall circūferences, whether the angles be drawen from the centres, or from the circumferences.
The 24. Theoreme. The 27. Proposition. In equall circles the angles which consist in equall circumfe∣rences, are equall the one to the other, whether the angles be drawen from the centres, or from the circumferences.
The 25. Theoreme. The 28. Proposition. In equall circles, equall right lines do cut away equall cir∣cumferences, the greater equall to the greater, and the lesse e∣quall to the lesse.
The 26. Theoreme. The 29. Proposition. In equall circles, vnder equall circumferences are subtended equall right lines.
The 4. Probleme. The 30. Proposition. To deuide a circumference geuen into two equall partes.
The 27. Theoreme. The 31. Proposition. In a circle an angle made in the semicircle is a right angle: but an angle made in the segment greater then the semicircle is lesse then a right angle, and an angle made in the segment lesse then the semicircle, is greater then a right angle. And moreouer the angle of the greater segment is greater then a right angle: and the angle of the lesse segment is lesse then a right angle.
Correlary.
¶ An addition of Pelitarius.
¶ An addition of Campane.
The 28. Theoreme. The 32. Proposition. If a right line touch a circle, and from the touch be drawen a right line cutting the circle: the angles which that line and the touch line make, are equall to the angles which consist in the alternate segmentes of the circle.
The 5. Probleme. The 33. Proposition. Vppon a right lyne geuen to describe a segment of a circle, which shall contayne an angle equall to a rectiline angle geuē.
The 6. Probleme. The 34. Proposition. From a circle geuen to cut away a section which shal containe an angle equall to a rectiline angle geuen.
The 29. Theoreme. The 35. Proposition. If in a circle two right lines do cut the one the other, the rect∣angle parallelograme comprehended vnder the segmentes or parts of the one line is equall to the rectangle parallelograme comprehended vnder the segment or partes of the other line.
The 30. Theoreme. The 36. Proposition. If without a circle be taken a certaine point, and from that point be drawen to the circle two right lines, so that the one of them do cut the circle, and the other do touch the circle: the rectangle parallelogramme which is comprehended vnder the whole right line which cutteth the circle, and that portion of the same line that lieth betwene the point and the vtter cir∣cūference of the circle, is equall to the square made of the line that toucheth the circle.
¶Two Corollaries out of Campane.
¶ Hereunto also Pelitarius addeth this Corollary.
The 31. Theoreme. The 37. Proposition. If without a circle be taken a certaine point, and from that point be drawen to the circle two right lines, of which, the one doth cut the circle and the other falleth vpon the circle, and that in such sort, that the rectangle parallelogramme which is cōtayned vnder the whole right line which cutteth the circle, and that portion of the same line that lieth betwene the point and the vtter circumferēce of the circle, is equall to the square made of the line that falleth vpon the circle: then that line that so falleth vpon the circle shall touch the circle.
¶ An other demonstration after Pelitarius.
¶The fourth booke of Eu∣clides Elementes.
Definitions.
A rectiline figure is sayd to be inscribed in a rectiline figure, when euery one of the angles of the inscribed figure toucheth euery one of the sides of the figure wherin it is inscribed.
Likewise a rectiline figure is said to be circumscribed about a rectiline figure, when euery one of the sides of the figure cir∣cumscribed, toucheth euery one of the angles of the figure a∣bout which it is circumscribed.
A rectiline figure is sayd to be inscribed in a circle, when eue∣ry one of the angles of the inscribed figure toucheth the cir∣cumference of the circle.
A circle is sayd to be circumscribed about a rectiline figure, whē the cir∣cumference of the circle toucheth euery one of the angles of the figure about which it is circumscribed.
A circle is sayd to be inscribed in a rectiline figure, when the circumference of the circle toucheth euery one of the sides of the figure within which it is inscribed.
A rectilined figure is said to be circum∣scribed about a circle, when euery one of the sides of the fi∣gure circumscribed toucheth the circumference of the circle.
A right lyne is sayd to be coapted or applied in a circle, when the extremes or endes therof, fall vppon the circumference of the circle.
The 1. Probleme. The 1. Proposition. In a circle geuen, to apply a right line equall vnto a right line geuen, which excedeth not the diameter of a circle.
The 2. Probleme. The 2. Proposition. In a circle geuen, to describe a triangle equiangle vnto a tri∣angle geuen.
The 3. Probleme. The 3. Proposition. About a circle geuen, to describe a triangle equiangle vnto a triangle geuen.
¶ An other way after Pelitarius.
The 4. Probleme. The 4. Proposition. In a triangle geuen, to describe a circle.
The 5. Probleme. The 5. Proposition. About a triangle geuen, to describe a circle.
Correlary.
The 6. Probleme. The 6. Proposition. In a circle geuen, to describe a square.
The 7. Probleme. The 7. Proposition. About a circle geuen, to describe a square.
The 8. Probleme. The 8. Proposition. In a square geuen, to describe a circle.
The 9. Probleme. The 9. Proposition. About a square geuen, to describe a circle.
¶ A Proposition added by Pelitarius.
The 10. Probleme. The 10. Proposition. To make a triangle of two equall sides called Isosceles, which shall haue eyther of the angles at the base double to the o∣ther angle.
¶ A Proposition added by Pelitarius.
The 11. Probleme. The 11. Proposition. In a circle geuen to describe a pentagon figure aequilater and equiangle.
¶ An other way to do the same after Pelitarius.
The 12. Probleme. The 12. Proposition. About a circle geuen, to describe an equilater and aquian∣gle pentagon.
¶An other way to do the same after Pelitarius, by parallel lines.
The 13. Probleme. The 13. Proposition. An equilater and equiangle pentagon figure beyng geuen, to describe in it a circle.
The 14. Probleme. The 14. Proposition. About a pentagon or figure of fiue angles geuen beyng equi∣later and equiangle, to describe a circle.
The 15. Probleme. The 15. Proposition. In a circle geuen to describe an hexagon or figure of sixe an∣gles equilater and equiangle.
¶ An other way to do the same after Orontius.
¶ An other way to do the same after Pelitarius.
Correlary.
The 16. Probleme. The 16. Proposition. In a circle geuen to describe a quindecagon or figure of fiftene angles, equilater and equiangle.
¶ An addition of Flussates to finde out infinite figures of many angles.
¶The fifth booke of Eu∣clides Elementes.
Definitions.
A parte is a lesse magnitude in respect of a greater magni∣tude, when the lesse measureth the greater.
Multiplex is a greater magnitude in respect of the lesse, when the lesse measureth the greater.
Proportion is a certaine respecte of two magnitudes of one kinde, according to quantitie.
Proportionalitie, is a similitude of proportions.
Those magnitudes are sayd to haue proportion the one to the other, which being multiplied may exceede the one the other.
Magnitudes are sayd to be in one or the selfe same propor∣tion, the first to the second, and the third to the fourth, when the equimultiplices of the first and of the third beyng compa∣red with the equimultiplices of the second and of the fourth, according to any multiplication: either together exceede the one the other, or together are equall the one to the other, or together are lesse the one then other.
Magnitudes which are in one and the selfe same proportion, are called Proportionall.
When the equemultiplices being taken, the multiplex of the first excedeth the multiplex of the second, & the multiplex of the third, excedeth not the multiplex of the fourth: then hath the first to the second a greater proportion, then hath the third to the fourth.
¶An other example.
Proportionallitie consisteth at the lest in three termes.
When there are three magnitudes in proportion, the first shall be vnto the third in double proportion that it is to the second. But when there are foure magnitudes in proportion the first shall be vnto the fourth in treble proportion that it is to the se∣cond. And so alwaies in order one more, as the proportion shall be extended.
Magnitudes of like proportion, are sayd to be antecedents to antecedentes, and consequentes to consequentes.
Proportion alternate, or proportion by permutation is, when the antecedent is compared to the antecedent, and the conse∣quent to the consequent.
Conuerse proportion, or propotion by conuersion is, when the consequent is taken as the antecedent, and so is compared to the antecedent as to the consequent.
Proportion composed, or composition of proportion is, when the antecedent and the consequent are both as one compared vnto the consequent.
Proportion deuided, or diuision of proportiō is, when the ex∣cesse wherein the antecedent excedeth the consequent, is com∣pared to the consequent.
Conuersiō of proportion (which of the elders is commonly cal∣led euerse proportion, or euersiō of proportion) is, whē the an∣tecedent is compared to the excesse, wherein the antecedent excedeth the consequent.
Proportion of equalitie is, when there are taken a number of magnitudes in one order, and also as many other magnitudes in an other order, comparing two to two beyng in the same pro¦portion, it commeth to passe, that as in the first order of mag∣nitudes, the first is to the last, so in the second order of magni∣tudes is the first to the last. Or otherwise it is a compari∣son of extremes together, the middle magnitudes being taken away.
An ordinate proportionality is, when as the antecedent is to the consequent, so is the antecedent to the consequent, and as the consequent is to another, so is the consequent to an other.
An inordinate proportionality is, when as the antecedent is to the consequent, so is the antecedent to the consequent: and as the consequent is to an other, so is an other to the ante∣cedent.
An extended proportionality is, when as the antecedent is to the consequent, so is the antecedent to the consequent, and as the consequent is to an other, so is the consequent to an other. Apertubate proportionalitie is, when, thre magnitudes be∣ing compared to three other magnitudes, it cōmeth to passe, that as in the first magnitudes the antecedent is to the conse∣quent, so in the second is the antecedent to the consequent, & as in the first magnitudes the consequent is to an other mag∣nitude, so in the second magnitudes is an other magnitude to the antecedent.
The 1. Theoreme. The 1. Proposition. If there be a number of magnitudes how many soeuer eque∣multiplices to a like number of magnitudes ech to ech: how multiplex on magnitude is to one, so multiplices are all the magnitudes to all.
The 2. Theoreme. The 2. Proposition. If the first be equemultiplex to the second as the third is to the fourth, and if the fifth also be equemultiplex to the second as the sixt is to the fourth: then shall the first and the fifth compo∣sed together be equemultiplex to the second, as the third and the sixt composed together is to the fourth.
The 3. Theoreme. The 3. Proposition. If the first be equemultiplex to the second, as the third is to the fourth, and if there be taken equemultiplices to the first & to the third: they shall be equemultiplices to them which were first taken, the one to the second, the other to the fourth.
The 4. Theoreme. The 4. Proposition. If the first be vnto the second in the same proportion that the third is to the fourth: then also the equemultiplices of the first and of the third, vnto the equemultiplices of the second and of the fourth, accordyng to any mnltiplication, shall haue the same proportion beyng compared together.
A Corolly.
The 5. Theoreme. The 5. Proposition. If a magnitude be equemultiplex to a magnitude, as a parte taken away of the one, is to a part taken away from the other: the residue also of the one, to the residue of the other, shal be e∣quemultiplex, as the whole is to the whole.
The 6. Theoreme. The 6. Proposition. If two magnitudes be quemultiplices to two magnitudes, & any pares taken away of them also, be aequemultiplices to the same magnitudes: the residues also of them shal vnto the same magnitudes be either equall, or equemultiplices.
The 7. Theoreme. The 7. Proposition. Equall magnitudes haue to one & the selfe same magnitude, one and the same proportion. And one and the same magni∣tude hath to equall magnitudes one and the selfe same pro∣portion.
The 8. Theoreme. The 8. Proposition. Vnequall magnitudes beyng taken, the greater hath to one and the same magnitude a greater proportion then hath the lesse. And that one and the same magnitude hath to the lesse a greater proportion then it hath to the greater.
¶ For that Orontius seemeth to demonstrate this more plainly therefore I thought it not amisse here to set it.
The 9. Theoreme. The 9. Proposition. Magnitudes which haue to one and the same magnitude one and the same proportion: are equall the one to the other. And those magnitudes vnto whome one and the same mag∣nitude hath one and the same proportion: are also equall.
The 10. Theoreme. The 10. Proposition. Of magnitudes compared to one and the same magnitude, that which hath the greater proportion, is the greater. And that magnitude wherunto one and the same magnitude hath the greater proportion, is the lesse.
The 11. Theoreme. The 11. Proposition. Proportions which are one and the selfe same to any one pro∣portion, are also the selfe same the one to the other.
The 12. Theoreme. The 12. Proposition. If there be a number of magnitudes how many soer propor∣tionall: as one of the antecedentes is to one of the cōsequentes, so are all the antecedentes to all the consequentes.
The 13. Theoreme. The 13. Proposition. If the first haue vnto the second the self same proportion that the third hath to the fourth, and if the third haue vnto the fourth a greater proportiō thē the fifth hath to the sixth: thē shall the first also haue vnto the second a greater proportion then hath the fifth to the sixth.
¶ An addition of Campane.
The 14. Theoreme. The 14. Proposition. If the first haue vnto the second the selfe same proportion that the third hath vnto the fourth: and if the first be greater then the third, the second also is greater then the fourth: and if it be equall it is equall: and if it be lesse it is lesse.
The 15. Theoreme. The 15. Proposition. Like partes of multiplices, and also their multiplices compa∣red together, haue one and the same proportion.
The 16. Theoreme. The 16. Proposition. If foure magnitudes be proportionall: then alternately also they are proportionall.
The 17. Theoreme. The 17. Proposition. If magnitudes composed be proportionall, then also deuided they shall be proportionall.
The 18. Theoreme. The 18. Proposition. If magnitudes deuided be proportionall: then also composed they shall be proportionall.
The 19. Theoreme. The 19. Proposition. If the whole be to the whole, as the part taken away is to the part taken away: then shall the residue be vnto the residue, as the whole is to the whole.
¶ ALemma or Assumpt.
Corollary.
The 20. Theoreme. The 20. Proposition. If there be three magnitudes in one order, and as many other magnitudes in an other order, which being taken two and two in eche order, are in one and the same proportion, and if of e∣qualitie in the first order the first be greater then the third, then in the second order the first also shall be greater then the third: and if it be equall, it shall be equall: and if it be lesse, it shall be lesse.
The 21. Theoreme. The 21. Proposition. If there be three magnitudes in one order, and as many other magnitudes in an other order, which being taken two and two in eche order are in one and the same proportion, and their proportion is perturbate: if of equalitie in the first order the first be greater then the third, thē in the second order the first also shall be greater then the third: and if it be equall it shall be equall: and if it be lesse it shall be lesse.
The 22. Theoreme. The 22. Proposition. If there be a number of magnitudes, how many soeuer in one order, and as many other magnitudes in an other order, which being taken two and two in ech order are in one and the same proportion, they shall also of equalitie be in one and the same proportion.
The 23. Theoreme. The 23. Proposition. If there be three magnitudes in one order, and as many other magnitudes in an other order, which beyng taken two & two in eche order are in one and the same proportion, and if also their proportion be perturbate: then of equalitie they shall be in one and the same proportion.
The 24. Theoreme. The 24. Proposition. If the first haue vnto the second the same proportion that the third hath to the fourth, and if the fift haue vnto the second the same proportion that the sixt hath to the fourth: then al∣so the first and fift composed together shall haue vnto the se∣cond the same proportion that the third and sixt composed to∣gether haue vnto the fourth.
The 25. Theoreme. The 25. Proposition. If there be foure magnitudes proportionall: the greatest and the least of them, shall be greater then the other remayning.
¶ The first Proposition.
¶ The second Proposition.
¶ The third Proposition.
¶ The fourth Proposition.
¶ The fifth Proposition.
¶ The sixt Proposition.
¶ The seuenth Proposition.
¶ The eight Proposition.
¶ The ninth Proposition.
¶The sixth booke of Eu∣clides Elementes.
Definitions.
1. Like rectiline figures are such, whose angles are equall the one to the other, and whose sides about the equall angles are proportionall.
2. Reciprocall figures are those, when the terme of proporti∣on are both antecedentes and consequentes in either figure.
3. A right line is sayd to be deuided by an extreme and meane proportion, when the whole is to the greater part, as the grea∣ter part is to the lesse.
4. The alitude of a figure is a perpendicular line drawen from the toppe to the base.
5. A Proportion is said to be made of two proportions or more, when the quantities of the proportions multiplied the one into the other, produce an other quantitie.
6. A Parallelogramme applied to a right line, is sayd to want in forme by a parallelogramme like to one geuen: whē the pa∣rallelogrāme applied wanteth to the filling of the whole line, by a parallelogramme like to one geuen: and then is it sayd to exceede, when it exceedeth the line by a parallelogramme like to that which was geuen.
The 1. Theoreme. The 1. Proposition. Triangles & parallelogrammes which are vnder one & the self same altitude: are in proportion as the base of the one is to the base of the other.
Here Flussates addeth this Corollary.
The 2. Theoreme. The 2. Proposition. If to any one of the sides of a triangle be drawen a parallel right line, it shall cut the sides of the same triangle proportio∣nally. And if the sides of a triangle be cut proportionally, a right lyne drawn from section to section is a parallel to the o∣ther side of the triangle.
¶ Here also Flussates addeth a Corollary.
The 3. Theoreme. The 3. Proposition. If an angle of a triangle be deuided into two equall partes, and if the right line which deuideth the angle deuide also the base: the segmentes of the base shall be in the same proportion the one to the other, that the other sides of the triangle are. And if the segmētes of the base be in the same proportion that the other sides of the sayd triangle are: a right drawen from the toppe of the triangle vnto the section, shall deuide the an∣gle of the triangle into two equall partes.
The 4. Theoreme. The 4. Proposition. In equiangle triangles, the sides which cōtaine the equall an∣gles are proportionall, and the sides which are subtended vn∣der the equall angles are of like proportion.
The 5. Theoreme. The 5. Proposition. If two triangles haue their sides proportionall, the triangs are equiangle, and those angles in thē are equall, vnder which are subtended sides of like proportion.
The 6. Theoreme. The 6. Proposition. If there be two triangles wherof the one hath one angle equall to one angle of the other, & the sides including the equall an∣gles be proportionall: the triangles shall be equiangle, and those angles in them shall be equall, vnder which are subten∣ded sides of like proportion.
The 7. Theoreme. The 7. Proposition. If there be two triāgles, wherof the one hath one angle equal to one angle of the other, and the sides which include the other angles, be proportionall, and if either of the other angles re∣mayning be either lesse or not lesse then a right angle: thē shal the triangles be equiangle, and those angles in them shall be equall, which are contayned vnder the sides proportionall.
The 8. Theoreme. The 8. Proposition. If in a rectangle triangle be drawen from the right angle vn∣to the base a perpendicular line, the perpendicular line shall deuide the triangle into two triangles like vnto the whole, and also like the one to the other.
Corollary.
The 1. Probleme. The 9. Proposition. A right line being geuen, to cut of frō it any part appointed.
The 2. Probleme. The 10. Proposition. To deuide a right line geuē not deuided, like vnto a right line geuen beyng deuided.
¶ A Corollary out of Flussates.
The 3. Probleme. The 11. Proposition. Vnto two right lines geuen, to finde a third in proportion with them.
¶ An other way after Pelitarius.
¶ An other way also after Pelitarius.
The 4. Probleme. The 12. Proposition. Vnto three right lines geuen to finde a fourth in proportion with them.
¶ An other way after Campane.
The 5. Probleme. The 13. Proposition. Vnto two right lines geuen, to finde out a meane proportionall.
¶ A Proposition added by Pelitarius.
The 9. Theoreme. The 14. Proposition. In equall parallelogrammes which haue one angle of the one equall vnto one angle of the other, the sides shall be recipro∣kall, namely, those sides which containe the equall angles. And if parallelogrammes which hauing one angle of the one, equal vnto one angle of the other, haue also their sides reciprokal, namely, those which contayne the equall angles, they shall al∣so be equall.
The 10. Theoreme. The 15. Proposition. In equal triangles which haue one angle of the one equall vn∣to one angle of the other , those sides are reciprokal, which in∣clude the equall angles. And those triāgles which hauyng one angle of the one equall vnto one angle of the other, haue also their sides which include the equall angles reciprokal, are al∣so equall.
The 11. Theoreme. The 16. Proposition. If there be foure right lines in proportion, the rectangle figure comprehended vnder the extremes: is equall to the rectangle figure contayned vnder the meanes. And if the rectangle fi∣gure which is contained vnder the extremes, be equall vnto the rectangle figure which is contayned vnder the meanes: then are those foure lines in proportion.
The 12. Theoreme. The 17. Proposition. If there be three right lines in proportion, the rectangle figure comprehended vnder the extremes, is equall vnto the square that is made of the meane. And if the rectangle figure which is made of the extremes, be equal vnto the square made of the meane, then are those three right lines proportional.
¶ Corollary added by Flussates.
The 6. Probleme. The 18. Proposition. Vpon a right line geuen, to describe a rectiline figure like, and in like sort situate vnto a rectiline figure geuen.
The 13. Theoreme. The 19. Proposition. Like triangles are one to the other in double proportion that the sides of lyke proportion are.
Corollary.
The 14. Theoreme. The 20. Proposition. Like Poligonon figures, are deuided into like triangles and equall in number, and of like proportion to the whole. And the one Poligonon figre is to the other Poligonon figure in double proportion that one of the sides of like proportion is to one of the sides of like proportion.
The first Corollary.
The second Corollary.
The 15. Theoreme. The 21. Proposition. Rectiline figures which are like vnto one and the same recti∣line figure, are also like the one to the other.
The 16. Theoreme. The 22. Proposition. If there be foure right lines proportionall, the rectiline figures also described vpon them beyng lyke, and in like sorte situate, shall be proportional. And if the rectiline figures vppon them described be proportional, those right lynes also shall be pro∣portionall.
An Assumpt.
The 17. Theoreme. The 23. Proposition. Equiangle Parallelogrammes haue the one to the other that proportion which is composed of the sides.
The 18. Theoreme. The 24. Proposition. In euery parallelogramme, the parallelogrammes about the dimecient are lyke vnto the whole, and also lyke the one to the other.
¶ An other more briefe demonstration after Flussates.
¶ A Probleme added by Pelitarius.
¶ An other Probleme added by Pelitarius.
The 7. Probleme. The 25. Proposition. Vnto a rectiline figure geuen to describe an other figure lyke, which shal also be equall vnto an other rectiline figure geuen.
The 19. Theoreme. The 26. Proposition. If from a parallelogramme be taken away a parallelograme like vnto the whole and in like sorte set, hauing also an angle common with it, then is the parallelogramme about one and the selfe same dimecient with the whole.
¶ An other demonstration after Flussates, which proueth this proposition affirmatiuely.
The 20. Theoreme. The 27. Proposition. Of all parallelogrammes applied to a right line wanting in fi∣gure by parallelogrammes like and in like sort situate to that parallelograme which is described of the halfe line: the grea∣test parallelogramme is that which is described of the halfe line being like vnto the want.
The 8. Probleme. The 28. Proposition. Vpon a right line geuen, to apply a parallelogramme equall to a rectiline figure geuen, & wanting in figure by a paralle∣logramme like vnto a parallelogrāme geuen. Now it beho∣ueth that the rectiline figure geuen, whereunto the parallelo∣grāme applied must be equall, be not greater thē that paralle∣logramme, which so is applied vpon the halfe lyne, that the defectes shall be like, namely, the defect of the parallelogrāme applied vpon the halfe line, and the defect of the parallelo∣gramme to be applied (whose defect is required to be like vn∣to the parallelogramme geuen).
¶ A Corollary added by Flussates.
The 9. Probleme. The 29. Proposition. Vpon a right line geuen to apply a parallelogramme equall vnto a rectiline figure geuen, and exceeding in figure by a pa∣rallelogramme like vnto a parallelogramme geuen.
The 10. Probleme. The 30. Proposition. To deuide a right line geuen by an extreme and meane pro∣portion.
An other way.
The 21. Theoreme. The 31. Proposition. In rectangle triangles the figure made of the side subtending the right angle, is equal vnto the figures made of the sides cō∣prehending the right angle, so that the sayd thre figures b b like and in like sort described.
An other way.
The conuerse of this Proposition after Campane.
The 22. Theoreme. The 32. Proposition. If two triangles be set together at one angle, hauing two sides of the one proportionall to two sides of the other, so that their sides of like proportion be also parallels: then the other sides remayning of those triangles shall be in one right line.
The 23. Theoreme. The 33. Proposition. In equal circles, the angles haue one and the selfe same pro∣portion that the circumferēces haue, wherin they cōsist, whe∣ther the angles be set at the centres or at the circumferences. And in like sort are the sectors which are described vppon the centres.
Corollary. And hereby it is manifest, that as the sector is to the sector, so is angle to angle by the 11. of the fifth.
¶The first Proposition added by Flussates.
¶The first Corollary.
¶ The second Corollary.
¶ The third Corollary.
¶ The second Proposition.
¶ The third Proposition.
The fourth Proposition
¶The fift Proposition.
¶The seuenth booke of Eu∣clides Elementes.
¶Definitions.
1 Vnitie is that, whereby euery thing that is, is sayd to be on.
2 Number is a multitude composed of vnities.
3 A part is a lesse number in comparison to the greater when the lesse measureth the greater.
4 Partes are a lesse number in respect of the greater, when the lesse measu∣reth not the greater.
5 Multiplex is a greater number in comparison of the lesse, when the lesse measureth the greater.
6 An euen number is that, which may be deuided into two equal partes.
7 An odde number is that which cannot be deuided into two equal partes: or that which onely by an vnitie differeth from an euen number.
8 A number euenly euen (called in latine pariter par) is that number, which an euen number measureth by an euen number.
9 A number euenly odde (called in latine pariter impar) is that which an e∣uen number measureth by an odde number.
10 A number oddly euen (called im lattin in pariter par) is that which an odde number measureth by an euen number.
11 A number odly odde is that, which an odde number doth measure by an odde number.
12 A prime (or first) number is that, which onely vnitie doth measure.
13 Numbers prime the one to the other are they, which onely vnitie doth measure, being a common measure to them.
14 A number composed, is that which some one number measureth.
15 Numbers composed the one to the other, are they, which some one number, being a common measure to them both, doth measure.
16 A number is sayd to multiply a number, when the number multiplyed, is so oftentimes added to it selfe, as there are in the number multiplying vni∣ties: and an other number is produced.
17 When two numbers multiplying them selues the one the other, produce an other: the number produced is called a plaine or superficiall number. And the numbers which muliply them selues the one by the other, are the sides of that number.
18 When three numbers multiplyed together ye one into the other, produce any number, the number produced, is called a solide number: and the numbers multiplying them selues the one into ye other, are the sides therof.
19 A square number is that which is equally equall: or that which is contay∣ned vnder two equall numbers.
20 A cube number is that which is equally equall equally: or that which is con∣tayned vnder three equall numbers.
21 Numbers proportionall are, when the first is to the second equemultiplex, as the third is to the fourth, or the selfe same part, or the selfe same partes.
22 Like plaine numbers, and like solide numbers, are such, which haue their sides proportionall.
23 A perfect number is that, which is equall to all his partes.
1 The lesse part is that which hath the greater denomination: and the greater part is that, which hath the lesse denomination.
2 Whatsoeuer numbers are equemultiplices to one & the selfe same nūber, or to equall numbers, are also equall the one to the other.
3 Those numbers to whome one and the selfe same number is equmulti∣plex, or whose equemultiplices are equall: are also equall the on to the other.
4 If a number measure the whole, and a part taken away: it shall also mea∣sure the residue.
5 If a number measure any number: it also measureth euery number that the sayd number measureth.
6 If a number measure two numbers, it shall also measure ye number com∣posed of them.
7 If in numbers there be proportions how manysoeuer equall or the selfe same to one proportion: they shall alo be equall or the selfe same the one to the other.
¶The first Proposition. The first Theoreme. If there be geuen two vnequall numbers, and if in taking the lesse continu∣ally from the greater, the number remayning do not measure the number going before, vntill it shall come to vnitie: then are those numbers which were at the beginning geuen, prime the one to the other.
The conuerse of this proposition after Campane.
¶The 1. Probleme. The 2. Proposition. Two numbers being geuen not prime the one to the other, to finde out their greatest common measure.
Corrolary.
¶The 2. Probleme. Th 3. Proposition. Thre numbers being geuē, not prime the one to the other: to finde out their greatest common measure.
¶Corollary.
¶The 2. Theoreme. The 4. Proposition. Euery lesse number is of euery greater number, either a part, or partes.
¶ The 3. Theoreme. The 5. Proposition. If a number be a part of a number, and an other nūber the selfe same part of an other number, then both the numbers added together shall be the selfe same part of both the numbers added together, which one number was of one number.
¶ The 4. Theoreme. The 6. Proposition. If a number be partes of a number, and an other number the selfe same partes of an other number then both numbers added together shall be of both numbers added together the selfe same partes, that one number was of one number.
¶ The 5. Theoreme. The 7. Proposition. If a number be the selfe same part of a number, that a part taken away is of a part taken away: then shall the residue be the selfe same part of the resi∣due, that the whole was of the whole.
¶ The 6. Theoreme. The 8. Proposition. If a number be of a number the selfe same partes, that a part taken away is of a part taken away, the residue also shall be of the residue the selfe same partes that the whole is of the whole.
¶An other demonstration after Flussates.
¶ The 7. Theoreme. The 9. Proposition. If a number be a part of a number, and if an other number be the self same part of an other nūber: then alternately what part or partes the first is of the third, the self same part or partes shall the second be of the fourth.
¶The 8. Theoreme. The 10. Proposition. If a number be partes of a number, and an other nūber the self same partes of an other number, then alternately what partes or part the first is of the third, the selfe same partes or part is the second of the fourth.
¶The 9. Theoreme. The 11. Proposition. If the whole be to the whole, as a part taken away is to a part taken away: then shall the residue be vnto the residue, as the whole is to the whole.
¶The 10. Theoreme. The 12. Proposition. If there be a multitude of numbers how many soeuer proportionall: as one of the antecedentes is to one of the consequentes, so are all the antecedentes to all the consequentes.
¶The 11. Theoreme. The 13. Proposition. If there be foure numbers proportionall: then alternately also they shall be proportionall.
¶The 12. Theoreme. The 14. Proposition. If there be a multitude of numbers how many soeuer, and also other num∣bers equall vnto them in multitude, which being compared two and two are in one and the same proportion: they shall also of equalitie be in one and the same proportion.
Proportionalitie deuided, is thus demonstrated.
Proportionalitie composed, is thus demonstrated.
Euerse proportionalitie, is thus proued.
¶ A proportion here added by Campane.
A Corollary.
¶The 13. Theoreme. The 15. Proposition. If vnitie measure any number, and an other number do so many times mea∣sure an other number: vnitie also shall alternately so many times measure the third number, as the second doth the fourth.
¶The 14. Theoreme. The 16. Proposition. If two numbers multiplying them selues the one into the other, produce any numbers: the numbers produced shall be equall the one into the other.
The 15. Theoreme. The 17. Proposition. If one number multiply two numbers, and produce other numbers, the num∣bers produced of them, shall be in the selfe same proportion, that the num∣bers multiplied are.
Here Flutes addeh thi Coollary.
¶The 16. Theoreme. The 18. Proposition. If two numbers multiply any number, & produce other numbers: the num∣bers of them produced, shall be in the same proportion that the numbers multiplying are.
¶The 17. Theoreme. The 19. Proposition. If there be foure numbers in proportion: the number produced of the first and the fourth, is equall to that number which is produced of the second and the third. And if the number which is produced of the first and the fourth be equall to that which is produced of the second & the third: those foure numbers shall be in proportion.
¶The 18. Theoreme. The 20. Proposition. If there be three numbers in proportion, the number produced of the ex∣tremes, is equall to the square made of the middle number. And if that nū∣ber which is produced of the extremes, be equall to the square made of the middle number, those three numbers shall be in proportion.
¶The 19. Theoreme. The 21. Proposition. The left numbers in any proportion, measure any other nūbers hauing the same proportion equally, the greater the greater, & the lesse the lesse.
¶The 20. Theoreme. The 22. Proposition. If there be three numbers, and other numbers equall vnto thē in multitude, which being compared two and two are in the selfe same proportion, and if also the proportion of them be perturbate, then of equalitie they shall be in one and the same proportion.
¶The 21. Theoreme. The 23. Proposition. Numbers prime the one to the other: are ye least of any numbers, that haue one and the same proportion with them.
¶The 22. Theoreme. The 24. Proposition. The least numbers that haue one and the same proportion with them: are prine the one to the other.
¶The 23. Theoreme. The 25. Proposition. If two numbers be prime the on to the other: any number measuring one of them shalbe prime to the other number.
¶The 24. Theoreme. The 26. Proposition. If two numbers be prime to any one number, the number also produced of them shall be prime to the selfe same.
¶The 25. Theoreme. The 27. Proposition. If two numbers be prime the one to the other, that which is produced of the one into him selfe, is prime to the other.
¶The 26. Theoreme. The 28. Proposition. If two numbers be prime to two numbers, eche to either of both: the numbers produced of them shall be prime the one to the other.
¶The 27. Theoreme. The 29. Proposition If two numbers be prime the one to the other, and ech multiplying himselfe bring forth certaine numbers: the numbers of them produced shall be prime the one to the other. And if those numbers geuen at the beginning multi∣plying the sayd numbers produced, produce any numbers: they also shall be prime the one to the other: and so shall it be continuing infinitely.
¶The 28. Theoreme. The 30. Proposition. If two numbers be prime the one to the other: then both of them added to∣gether, shall be prime to either of them. And if both of them added toge∣ther be prime to any one of them, then also those numbers geuen at the be∣ginning, are prime the one to the other.
¶ The 29. Theoreme. The 31. Proposition. Euery prime number is to euery number which it measureth not, prime.
¶ The 30. Theoreme. The 32. Proposition. If two numbers multiplying the one the other produce any number, and if also some prime number measure that which is produced of them: then shall it also measure one of those numbers which were put at the beginning.
A Corollary.
The 31. Theoreme. The 3. Proposition. Euery composed number, is measured by some prime number.
¶ An other way.
The 32. Theoreme. The 34. Proposition. Euery number is either a prime number, or els some prime number measu∣reth it.
¶The 3. Probleme. The 35. Proposition. How many numbers soeuer being geuen, to find out the least numbers that haue one and the same proportion with them.
A Corollary.
¶The 4. Probleme. The 36. Proposition. Two numbers being geuen, to finde out the lest nūber which they measure.
The 33. Theoreme. The 37. Proposition. If two numbers measure any number, the least nūber also which they mea∣sure, measureth the selfe same number.
¶The 5. Probleme. The 38. Proposition. Three numbers being geuen, to finde out the least number which they measure.
Corollary.
¶The 34. Theoreme. The 39. Proposition. If a number measure any number: the number measured shall haue a part after the denomination of the number measuring.
¶The 35. Theoreme. The 40. Proposition. If a number haue any part: the number wherof the part taketh his deno∣mination shall measure it.
¶The 6. Probleme. Th 41. Proposition. To finde out the least number, that containeth the partes geuen.
Corrolary.
¶ The eighthe booke of Eu∣clides Elementes.
¶The first Theoreme. The first Proposition. If there be numbers in continuall proportion howmanysoeuer, and if their extremes be prime the one to the other: they are the least of all numbers that haue one and the same proportion with them.
¶ The 1. Probleme. The 2. Proposition. To finde out the least numbers in continuall proportion, as many as shall be required, in any proportion geuen.
¶Corollary.
The 2. Theoreme. The 3. Proposition. If there be numbers in continuall proportion how many soeuer, and if they be the lest of all numbers that haue one and the same proportion with thē: their extremes shall be prime the one to the other.
The 2. Probleme. The 4. Proposition. Proportions in the least numbers how many soeuer beyng geuen, to finde out the least numbers in continuall proportion in the said proportions geuē.
¶ The 3. Theoreme. The 5. Proposition. Playne or superficiall numbers are in that proportion the one to the other which is composed of the sides.
¶An other demonstration of the same after Campane.
¶The 4. Theoreme. The 6. Proposition. If there be numbers in continuall proportion how many soeuer, and if the first measure not the second, neither shall any one of the other measure any one of the other.
¶The 5. Theoreme. The 7. Proposition. If there be numbers in continuall proportion how many soeuer, and if the first measure the last, it shall also measure the second.
¶ The 6. Theoreme. The 8. Proposition. If betwene two numbers there fall numbers in continuall proportion: how many numbers fall betwene them, so many also shall fall in continuall pro∣portion betwene other numbers which haue the selfe same proportion.
A Corollary added by Flussates.
¶ The 7. Theoreme. The 9. Proposition. If two numbers be prime the one to the other, and if betwene them shall fall numbers in continuall proportion: how many numbers in continuall pro∣portion fall betwene them, so many also shall fall in continuall proportion betwene either of those numbers and vnitie.
¶ The 8. Theoreme. The 10. Proposition. If betwene two numbers and vnitie fall numbers in continuall proportion: how many numbers in continuall proportion fal betwene either of them & vnitie so many also shall there fall in continuall proportion betwene them.
¶ The 9. Theoreme. The 11. Proposition. Betwene two square numbers there is one meane proportional number. And a square number to a square, is in double proportion of that which the side of the one is to the side of the other.
¶ The 10. Theoreme. The 12. Proposition. Betwene two cube numbers there are two meane proportionall numbers. And the one cube is to the other cube in treble proportion of that which the side of the one is to the side of the other.
¶ The 11. Theoreme. The 13. Proposition. If there be numbers in continuall proportion how many so euer, and ech multiplying himselfe produce certayne numbers, the numbers of them pro∣duced shall be proportinall. And if those numbers geuen at the beginning multiplying the numbers produced, produce other numbers, they also shalbe proportionall: and so shall it be continuing infinitely.
¶ The 12. Theoreme. The 14. Proposition. If a square number measure a square number, the side also of the one shall measure the side of the other. And if the side of the one measure the side of the other, the square number also shall measure the square number.
¶ The 13. Theoreme. The 15. Proposition. If a cube number measure a cube number, the side also of the one shall mea∣sure the side of the other. And if the side of the one measure the side of the other, the cube number also shall measure the cube number.
¶ The 14. Theoreme. The 16. Proposition. If a square number measure not a square number, neither shall the side of the one measure the side of the other. And if the side of the one measure not the side of the other, neither shall the square number measure the square number.
¶ The 15. Theoreme. The 17. Proposition. If a cube number measure not a cube number, neither shall the side of the one measure the side of the other. And if the side of the one measure not the side of the other, neither shall the cube nūber measure the cube number.
¶ The 16. Theoreme. The 18. Proposition. Betwene two like plaine or superficiall numbers there is one meane propor∣tionall number. And the one like plaine number is to the other like plaine number in double proportion of that which the side of like proportion, is to the side of like proportion.
¶ The 17. Theoreme. Th 19. Proposition. Betwene two like solide numbers, there are two meane proportionall num∣bers. And the one like solide number, is to the other like solide number in treble proportion of that which side of like proportion is to side of lyke proportion.
¶ The 18. Theoreme. The 20. Proposition. If betwene two numbers there be one meane proportionall number: those numbers are like plaine numbers.
¶The 19. Theoreme. The 21. Proposition. If betwene two numbers, there be two meane proportionall numbers, those numbers are like solide numbers.
¶ The 20. Theoreme. The 22. Proposition. If three numbers be in continuall proportion, and if the first be a square number, the third also shall be a square number.
¶The 21. Theoreme. The 23. Proposition. If foure numbers be in continuall proportion, and if the first be a cube nū∣ber, the fourth also shall be a cube number.
¶The 22. Theoreme. The 24. Proposition. If two numbers be in the same proportiō that a square number is to a square number, and if the first be a square number, the second also shall be a square number.
¶ The 23. Theoreme. The 25. Proposition. If two numbers be in the same proportion the one to the other, that a cube number is to a cube number, and if the first be a cube number, the second al∣so shall be a cube number.
A Corollary added by Flussates.
¶The 24. Theoreme. The 26. Proposition. Like playne numbers, are in the same proportion the one to the other, that a square number is to a square number.
The 25. Theoreme. The 27. Proposition. Like solide numbers are in the same proportion the one to the other, that a cube number is to a cube number.
¶A Corollary added by Flussates.
An other Corollary added also by Flussates.
¶The ninth booke of Eu∣clides Elementes.
¶The 1. Theoreme. The 1. Proposition. If two like plaine numbers multiplying the one the other produce any num∣ber: the number of them produced shall be a square number.
¶The 2. Theoreme. The 2. Proposition. If two numbers multiplying the one the other produce a square number: those numbers are like plaine numbers.
A Corollary added by Campane.
The 3. Theoreme. The 3. Proposition. If a cube number multiplying himselfe produce a number, the number pro∣duced shall be a cube number.
¶The 4. Theoreme. The 4. Proposition. If a cube number multiplieng a cube number, produce any number, the number produced shall be a cube number.
¶The 5. Theoreme. The 5. Proposition. If a cube number multiplying any number produce a cube nūber: the num∣ber multiplyed is a cube number.
¶ A Corollary added by Campane.
¶ The 6. Theoreme. The 6. Proposition. If a number multiplieng himselfe produce a cube number: then is that num∣ber also a cube number.
¶ The 7. Theoreme. The 7. Proposition. If a composed number multiplieng any number, produce a number: the nū∣ber produced shall be a solide number.
¶ The 8. Theoreme. The 8. Proposition. If from vnitie there be numbers in continuall proportion how many soeuer: the third number from vnitie is a square number, and so are all forwarde leauing one betwene. And the fourth number is a cube number, and so are all forward leauing two betwene. And the seuenth is both a cube number and also a square number, and so are all forward leauing fiue betwene.
¶ The 9. Theoreme. The 9. Proposition. If from vnitie be numbers in continuall proportion how many soeuer: and if th number which followeth next after vnitie be a square number, then all the rest following also be square numbers. And if that number which followeth next after vnitie be a cube number, then all the rest following shall be cube numbers.
¶ The 10. Theoreme. The 10. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, and if that number which followeth next after vnitie be not a square num∣ber, then is none of the rest following a square number, excepting the third from vnitie, and so all forward leauing one betwene. And if that number which olloweth next after vnitie be not a cube number, neither is any of the rest following a cube number, excepting the fourth from vnitie, and so all forward leauing two betwene.
¶ The 11. Theoreme. The 11. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, the lesse measureth the greater by some one of them which are before in the said proportionall numbers.
¶ The 12. Theoreme. The 12. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, how many prime numbers measure the least so many also shal measure the num∣ber which followeth next after vnitie.
An other more briefe demonstration after Flussates.
¶The 13. Theoreme. The 13. Proposition. If from vnitie be numbers in continuall proportion how many soeuer, and if that which followeth next after vnitie be a prime number: then shall no other number measure the greatest number, but those onely which are be∣fore in the sayd proportionall numbers.
An other demonstration of the same after Campane.
¶The 14. Theoreme. The 14. Proposition. If there be geuen the least number, whom certayne prime numbers geuen, do measure: no other prime number shall measure that nūber, besides those prime numbers geuen.
A proposition added by Campane.
¶ The 15. Theoreme. The 15. Proposition. If three numbers in continuall proportion be the least of all numbers that haue one and the same proportion with them: euery two of them added to∣gether shall be prime to the third.
¶ The first Proposition added by Campane.
The second Proposition.
¶ The third Proposition.
¶The fourth Proposition.
¶The fift Proposition.
¶The sixt Proposition.
¶The seuenth Proposition.
¶The 8. Proposition.
¶The 9. Proposition.
¶The 10. proposition.
The 11. proposition.
The 12. proposition.
The 13. proposition.
¶The 16. Theoreme. The 16. Proposition. If two numbers be prime the one to the other, the second shall not be to any other number, as the first is to the second.
¶The 17. Theoreme. The 17. Proposition. If there be numbers in continuall proportion how many soeuer, and if theyr extremes be prime the one to the other, the lesse shall not be to any other number, as the first is to the second.
¶The 18. Theoreme. The 18. Proposition. Two numbers being geuen, to searche out if it be possible a third number in proportion with them.
¶ The 19. Theoreme. The 19. Proposition. Three numbers beyng geuen, to search out if it be possible the fourth num∣ber proportionall with them.
¶ The 20. Theoreme. The 20. Proposition. Prime numbers being geuen how many soeuer, there may be geuen more prime numbers.
A Corollary.
¶ The 21. Theoreme. The 21. Proposition. If euen nūbers how many soeuer be added together: the whole shall be euē.
¶ The 22. Theoreme. The 22. Proposition. If odde numbers how many soeuer be added together, & if their multitude be euen, the whole also shall be euen.
¶ The 23. Theoreme. The 23. Proposition. If odde numbers how many soeuer be added together, and if the multitude of them be odde, the whole also shall be odde.
¶ The 24. Theoreme. The 24. Proposition. If from an euen number be takē away an euen number, that which remai∣neth shall be an euen number.
¶ The 25. Theoreme. The 25. Proposition. If from an euen number be taken away an odde number, that which remai∣neth shall be an odde number.
¶ The 26. Theoreme. The 26. Proposition. If from an odde number be taken away an odde number, that which re∣mayneth shall be an euen number.
¶ The 27. Theoreme. The 27. Proposition. If from an odde number be taken a way an euen number, the residue shall be an odde number.
¶ The 28. Theoreme. The 28. Proposition. If an odde number multiplieng an euen number produce any number, the number produced shall be an euen number.
¶ The 29. Theoreme. The 29. Proposition. I an odde number multiplying an odde number produce any number, the number produced shalbe an odde number
A proposition added by Campane.
An other proposition added by him.
¶ The 30. Theoreme. The 30. Proposition. If an odde number measure an euen number, it shall also measure the halfe thereof.
¶ The 31. Theoreme. The 31. Proposition. If an odde number be prime to any number, it shal also be prime to the dou∣ble thereof.
¶ The 32. Theoreme. The 32. Proposition. Euery nūber produced by the doubling of two vpward, is euenly euen onely.
¶The 33. Theoreme. The 33. Proposition. A number whose halfe part is odde, is euenly odde onely.
An other demonstration to proue the same.
¶ The 34. Theoreme. The 34. Proposition. If a number be neither doubled from two, nor hath to his half part an odde number, it shall be a number both euenly euen, and euenly odde.
¶ The 35. Theoreme. The 35. Proposition. If there be numbers in continuall proportion how many soeuer, and if from the second and last be taken away numbers equall vnto the first, as the ex∣cesse of the second is to the first, so is the excesse of the last to all the nūbers going before the last.
¶ The 36. Theoreme. The 36. Proposition. If from vnitie be taken numbers how many soeuer in double proportion continually, vntill the whole added together be a prime number, and if the whole multiplying the last produce any number, that which is produced is a perfecte number.
¶The tenth booke of Eu∣clides Elementes.
Definitions.
1 Magnitudes commensurable are suchwhich one and the selfe same mea∣sure doth measure.
2 Incommensurable magnitudes are such, which no one common measure doth measure.
3 Right lines commensurable in power are such, whose squares one and the selfe same superficies, area, or plat doth measure.
4 Lines incommensurable are such, whose squares no one plat or superfi∣cies doth measure.
5 And that right line so set forth is called a rationall line.
6 Lines which are commensurable to this line, whether in length and power, or in power onely, are also called rationall.
7 Lines which are incommensurable to the rationall line, are called ir∣rationall.
8 The square which is described of the rationall right line supposed, is rationall.
9 Such which are commensurable vnto it, are rationall.
10 Such which are incommensurable vnto it, are irrationall.
11 And these lines whose poweres they are, are irrationall. If they be squares, then are their sides irrationall. If they be not squares, but some other rectiline figures, then shall the lines, whose squares are equall to these rectiline figures, be irrationall.
¶ The 1. Theoreme. The 1. Proposition. Two vnequall magnitudes being geuen, if from the greater be taken away more then the halfe, and from the residue be againe taken away more then the halfe, and so be done still continually, there shall at length be left a cer∣taine magnitude lesser then the lesse of the magnitudes first geuen.
A Corollary.
¶An other demonstration of the same.
¶The 2. Theoreme. The 2. Proposition. Two vnequall magnitudes being geuen, if the lesse be continually taken from the greater, & that which remayneth measureth at no time the mag∣nitude going before: then are the magnitudes geuen incommensurable.
¶A Corollary added by Montaureus.
¶ The 1. Probleme. The 3. Proposition. Two magnitudes commensurable being geuen, to finde out, their greatest common measure.
¶Corollary.
Monta•reus reduceth this Probleme into a Theoreme after this maner.
¶The 2. Probleme. The 4. Proposition. Three magnitudes commensurable beyng geuen, to finde out their greatest common measure.
¶ Corollary.
This Probleme also Montaureus reduceth into a Theoreme after this maner.
¶The 3. Theoreme. The 5. Proposition. Magnitudes commensurable, haue such proportion the one to the other, as number hath to number.
¶ The 4. Theoreme. The 6. Proposition. I two magnitudes haue such proportion the one to the other, as number hath to number: those magnitudes are commensurable.
Corollary.
¶ An Assumpt.
¶ An other demonstration of the 6. Proposition.
¶The 5. Theoreme. The 7. Proposition. Magnitudes incommensurable, haue not that proportion the one to the o∣ther, that number hath to number.
¶The 6. Theoreme. The 8. Proposition. If two magnitudes haue not that proportion the one to the other that num∣ber hath to number, those magnitudes are incommensurable.
¶The 7. Theoreme. The 9. Proposition. Squares described of right lines commensurable in length, haue that pro∣portion the one to the other, that a square number hath to a square number. And squares which haue that proportion the one to the other that a square number hath to a square nūber, shall also haue their sides cōmensurable in length. But squares described of right lines incommensurable in length, haue not that proportion the one to the other, that a square number hath to a square number. And squares which haue not that proportion the one to the other that a square nūber hath to a square number, haue not their sides commensurable in length.
An other demonstration to proue the same.
An other demonstration of the same first part after Montaureus.
An other demonstration to proue the same.
¶ Corrollary.
¶ An Assumpt.
¶ The 8. Theoreme. The 10. Proposition. If foure magnitudes be proportionall, and if the first be commensurable vnto the second, the third also shal be commensurable vnto the fourth. And if the first be incommensurable vnto the second, the third shall also be in∣commensurable vnto the fourth.
¶ A Corollary added by Montaureus.
¶ The 3. Probleme. The 11. Proposition. Vnto a right line first set and geuen (which is called a rationall line) to finde out two right lines incommensurable, the one in length onely, and the other in length and also in power.
¶ The 9. Theoreme. The 12. Proposition. Magnitudes commensurable to one and the selfe same magnitude: are also commensurable the one to the other.
¶ An Assumpt.
¶ The 10. Theoreme. The 13. Proposition. If there be two magnitudes commensurable, and if the one of them be in∣commensurable to any other magnitude: the other also shall be incommen∣surable vnto the same.
¶ A Corollary added by Montaureus.
¶ An Assumpt.
¶ The 11. Theoreme. The 14. Proposition. If there be sower right lines proportionall, and if the first be in power more then the second by the square of a right line commensurable in length vnto the first, the third also shalbe in power more then the fourth, by the square of a right line commensurable vnto the third. And if the first be in po∣wer more then the second by the square of a right line incommensu∣rable in length vnto the first, the third also shall be in power more then the fourth by the square of a right line incommensurable in length to the third.
¶ The 12. Theoreme. The 15. Proposition. If two magnitudes commensurable be composed, the whole magnitude com∣posed also shall be commensurable to either of the two partes. And if the whole magnitude composed be commensurable to any one of the two partes, those two partes shall also be commensurable.
¶ A Corollary added by Montaureus.
¶ The 13. Theoreme. The 16. Proposition. If two magnitudes incommensurable be composed, the whole magnitude also shall be incommensurable vnto either of the two partes cōponentes. And if the whole be incommensurable to one of the partes componentes, those first magnitudes also shall be incommensurable.
¶A Corollary added by Montaureus.
¶An Assumpt.
¶ The 14. Theoreme. The 17. Proposition. If there be two right lines vnequall, and if vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line, and wanting in figure by a square, if also the parallelogramme thus appli∣ed deuide the line where vpon it is applied into partes commensurable in length: then shall the greater line be in power more then the lesse, by the square of a line commensurable in length vnto the greater. And if the grea∣ter be in power more then the lesse by the square of a right line commensu∣rable in length vnto the greater, and if also vpon the greater be applied a parallelogrāme equall vnto the fourth part of the square of the lesse line, and wanting in figure by a square: then shall it deuide the greater line in∣to partes commensurable.
¶The 15. Theoreme. The 18. Proposition. If there be two right lines vnequall, and if vpon the greater be applied a parallelograme equall vnto the fourth part of the square of lesse, and wan∣ting in figure by a square, if also the parallelograme thus applied deuide the line whereupon it is applied into partes incommensurable in length: the greater line shalbe in power more then the lesse line by the square of a line incommensurable in length vnto the greater line. And if the grea∣ter line be in power more then the lesse line, by the square of a line incommē∣surable in length vnto the greater, and if also vpon the greater be applied a parallelograme equall vnto the fourth part of the square of the lesse and wanting in figure by a square: then shall it deuide the greater line into partes incommensurable in length.
¶An assumpt.
An annotacion of Proclus.
¶A Corollary.
¶The 16. Theoreme. The 19. Proposition. A rectangle figure comprehended vnder right lines commensurable in lengthe, being rationall according to one of the foresaide wayes: is ra∣tionall.
¶The 17. Theoreme. The 20. Proposition. If vpon a rationall line be applied a rationall rectangle parallelogramme: the other side that maketh the breadth thereof shall be a rationall line and commensurable in length vnto that line wherupon the rationall parallelo∣gramme is applied.
¶An Assumpt.
¶The 18. Theoreme. The 21. Proposition. A rectangle figure comprehended vnder two rationall right lines com∣mensurable in power onely, is irrationall. And the line which in power contayneth that rectangle figure is irrationall, & is called a mediall line.
¶A Corollary added by Flussates.
¶An Assumpt.
¶ The 19. Theoreme. The 22. Proposition. If vpon a rationall line be applied the square of a mediall line: the other side that maketh the breadth thereof shalbe rationall, and incommensura∣ble in length to the line wherupon the parallelograme is applied.
¶ The 20. Theoreme. The 23. Proposition. A right line commensurable to a mediall line, is also a mediall line.
¶ Corollary.
¶ The 21. Theoreme. The 24. Proposition. A rectangle parallelogramme comprehended vnder mediall lines cōmen∣surable in length, is a mediall rectangle parallelogramme.
¶ The 22 Theoreme. The 25. Proposition. A rectangle parallelogramme comprehended vnder mediall right lines commensurable in power onely, is either rationall, or mediall.
¶A Corollary.
¶The 23. Theoreme. The 26. Proposition. A mediall superficies excedeth not a mediall superficies, by a rationall su∣perficies.
¶The 4. Probleme. The 27. Proposition. To finde out mediall lines commensurable in power onely, contayning a rationall parallelogramme.
The 5. Probleme. The 28. Proposition. To finde out mediall right lynes commensurable in power onely, contayning a mediall parallelogramme.
An Assumpt.
A Corollary.
¶An Assumpt.
¶ The 6. Probleme. The 29. Proposition. To finde out two such rationall right lynes commensurable in power on∣ly, that the greater shall be in power more then the lesse, by the square of a right line commensurable in length vnto the greater.
¶The 7. Theoreme. The 30. Proposition. To finde out two such rationall lines commensurable in power onely, that the greater shalbe in power more then the lesse by the square of a right line incommensurable in length to the greater.
¶ An Assumpt.
¶ The 8. Probleme. The 31. Proposition. To finde out two mediall lines commensurable in power onely, comprehen∣ding a rationall superficies, so that the greater shall be in power more then the lesse by the square of a line commensurable in length vnto the greater.
¶An assumpt.
¶The 9. Probleme. The 32. Proposition. To finde out two mediall lines commensurable in power onely, comprehen∣ding a mediall supericies, so that the greater shall be in power more then the lesse, by the square of a line commensurable in length vnto the greater.
1. ¶An Assumpt.
2. ¶An Assumpt.
3. ¶An Assumpt.
¶ The 10. Probleme. The 33. Proposition. To inde out two right lines incommensurable in power, whose squares ad∣ded together make a rationall superficies, and the parallelogramme contai∣ned vnder them make a mediall superficies.
¶ The 11. Probleme. The 34. Proposition. To finde out two right lines incmensurable in power, whose squares ad∣ded together make a mediall superficies, and the parallelogramme contay∣ned vnder them, make a rationall superficies.
¶ The 12. Probleme. The 35. Proposition. To finde out two right lines incommensurable in power, whose squares ad∣ded together, make a mediall superficies, and the parallelogramme contai∣ned vnder them, make also a mediall superficies, which parallelogramme moreouer, shall be incommensurable to the superficies made of the squares of those lines added together.
The beginning of the Senaries by Composition. ¶ The 2. Theoreme. The 36. Proposition. If two rationall lines commensurable in power onely be added together: the whole line is irrationall, and is called a binomium, or a binomiall line.
¶The 25. Theoreme. The 37. Proposition. If two mediall lines commensurable in power onely containing a rationall superficies, be added together: the whole line is irrationall, and is called a first bimediall line.
¶The 26. Theoreme. The 38. Proposition. If two mediall lines commensurable in power onely contayning a mediall superficies, be added together: the whole line is irrationall, and is called a second bimediall line.
¶The 27. Theoreme. The 39. Proposition. If two right lines incōmensurable in power be added together, hauing that which is composed of the squares of them rationall, and the parallelogrāme contayned vnder them mediall: the whole right line is irrationall, and is called a greater line.
An Assumpt.
¶The 28. Theoreme. The 40. Proposition. If two right lines incōmensurable in power be added together, hauing that which is made of the squares of them added together mediall, and the pa∣rallelogramme contayned vnder them rationall: the whole right line is ir∣rationall, and is called a line contayning in power a rationall and a mediall superficies.
¶The 29. Theoreme. The 41. Proposition. If two right lines incommensurable in power be added together, hauyng that which is composed of the squares of them added together mediall, and the parallelogramme contayned vnder them mediall, and also incommen∣surable to that which is composed of the squares of them added together the whole right line is irrationall, and is called a line contayning in power two medials.
¶An Assumpt.
¶An Assumpt.
¶The 30. Theoreme. The 42. Proposition. A binomiall line is in one point onely deuided into his names.
ollary added by Flussates.
¶The 31. Probleme. The 43. Proposition. A first bimediall line is in one poynt onely deuided into his names.
¶The 32. Theoreme. The 44. Proposition. A second bimediall line is in one poynt onely deuided into his names.
¶The 33. Theoreme. The 45. Proposition. A greater line is in one poynt onely deuided into his names.
¶The 34. Theoreme. The 46. Proposition. A line contayning in power a rationall and a mediall, is in one point one∣ly deuided into his names.
¶The 35. Theoreme. The 47. Proposition. A line contayning in power two medials, is in one point onely deuided into his names.
¶Second Definitions.
A first binomiall line is, whose square of the greater part exceedeth the square of te lesse part y the square of a line commensurable in length to the greater part, and the greater part is also commensurable in length to te rationall line first set.
A second binomiall line is, when the square of the greater part exceedeth the square of the lesse part by the square of a line commensurable in length vnto it, and the lesse part is commensurable in length to the rationall line first set.
A third binomiall line is, when the square of the greater part excedeth the square of the lesse part, by the square of a line cōmensurable in length vnto it. And neither part is commensurable in length to the rationall line geuē.
A fourth binomiall line is, when the square of the greater part exceedeth the square of the lesse by the square of a line incommensurable in length vn∣to the greater part. And the greater is also commensurable in length to the rationall line.
A fift binomiall line is, when the square of the greater part exceedeth the square of the lesse part, by the square of a line incommensurable vnto it in length. And the lesse part also is commensurable in length to the rationall line geuen.
A sixt binomiall line is, when the square of the greater part exceedeth the square of the lesse, by the square of a line incommensurable in length vnto it. And neither part is commensurable in length to the rationall line geuen.
¶ The 13. Probleme. The 48. proposition. To finde out a first binomiall line.
The 14. Probleme. The 49. Proposition. To finde out a second binomiall line.
¶The 15. Probleme. The 50. Proposition. To finde out a third binomiall line.
¶The 16. Probleme. The 51. Proposition. To finde out a fourth binomiall line.
¶The 17. Probleme. The 52. Proposition. To finde out a fift binomiall lyne.
¶The 18. Probleme. The 53. Proposition. To finde out a sixt binomiall line.
¶A Corollary added out of Flussates.
¶An Assumpt.
¶An Assumpt.
¶ The 36. Theoreme. The 54. Proposition. If a superficies be contained vnder a rationall line & a first binomiall line: the line which containeth in power that superficies is an irrationall line, & a binomiall line.
¶ The 37. Theoreme. The 55. Proposition. If a superficies be comprehended vnder a rationall line and a second binomi∣all line: the line that contayneth in power that superficies is irrationall, and is a first bimediall line.
¶The 38. Theoreme. The 56. Proposition. If a superficies be contayned vnder a rationall line and a third binomiall line: the line that contayneth in power that superficies is irrationall, and is a second bimediall line.
¶The 39. Theoreme. The 57. Proposition. If a superficies be contained vnder a rationall line, and a fourth binomiall line: the line which contayneth in power that superficies is irrationall, and is a greater line.
¶The 40. Theoreme. The 58. Proposition. If a superficies be contained vnder a rationall line and a fift binomiall line: the line which contayneth in power that superficies is irrationall, and is a line contayning in power a rationall and a mediall superficies.
¶The 41. Theoreme. The 59. Proposition. If a superficies be contayned vnder a rationall line, and a sixt binomiall line, the lyne which contayneth in power that superficies, is irrational, & is called a line contayning in power two medials.
An Aumpt.
¶ The 42. Theoreme. The 60. Proposition. The square of a binomiall line applyed vnto a rationall line, maketh the breadth or other side a first binomiall line.
¶ The 43. Theoreme. The 61. Proposition. The square of a first bimediall line applied to a rationall line, maketh the breadth or other side a second binomiall line.
¶ The 44. Theoreme. The 62. Proposition. The square of a second bimediall line, applied vnto a rationall line: maketh the breadth or other side therof, a third binomiall lyne.
¶Here follow certaine annotations by M. Dee, made vpon three places in the demonstration, which were not very euident to yong beginners.
¶A Corollary.
¶ The 45. Theoreme. The 63. Proposition. The square of a greater line applied vnto a rationall line, maketh the breadth or other side a fourth binomiall line.
¶The 46. Theoreme. The 64. Proposition. The square of a line contayning in power a rationall and a mediall super∣ficies applied to a rationall line, maketh the breadth or other side a fift bi∣nomiall line.
¶The 47. Theoreme. The 65. Proposition. The square of a line contayning in power two medialls applyed vnto a ra∣tionall line, maketh the breadth or other side a sixt binomiall line.
¶The 48. Theoreme. The 66. Proposition. A line commensurable in length to a binomiall line, is also a binomiall line of the selfe same order.
¶The 49. Theoreme. The 67. Proposition. A line commensurable in length to a bimediall line, is also a bimediall lyne and of the selfe same order.
¶ A Corollary added by Flussates: but first noted by P. Monta•reus.
¶The 50. Theoreme. The 68. Proposition. A line commensurable to a greater line, is also a greater line.
An other demonstration of Peter Montaureus to proue the same.
An other more briefe demonstration of the same after Campane.
¶ The 51. Theoreme. The 69. Proposition. A line commensurable to a line contayning in power a rationall and a me∣diall: is also a line contayning in power a rationall and a mediall.
An other demonstration of the same after Campane.
¶The 52. Theoreme. The 70. Proposition. A line commensurable to a line contayning in power two medialls, is also a line contayning in power two medialls.
¶ An Assumpt added by Montaureus.
An other demonstration after Campane.
An Annotation.
¶ The 53. Theoreme. The 71. Proposition. If two superficieces, namely, a rationall and a mediall superficies be cōposed together, the line which contayneth in power the whole superficies, is one of these foure irrationall lines, either a binomial line, or a first bimediall lyne, or a greater lyne, or a lyne contayning in power a rationall and a mediall superficies.
¶ The 54. Theoreme. The 72. Proposition. If two mediall superficieces incommensurable the one to the other be com∣posed together: the line contayning in power the whole superficies is one of the two irrationall lines remayning, namely, either a second bimediall line, or a line contayning in power two medialls.
¶ A Corollary following of the former Propositions.
Here beginneth the Senaries by substraction. ¶The 55. Theoreme. The 73. Proposition. If from a rationall line be taken away a rationall line commensurable in power onely to the whole line: the residue is an irrationall line, and is called a residuall line.
An other demonstration after Campane.
An annotation of P. Monta•re•s.
¶The 56. Theoreme. The 74. Proposition. If from a mediall line be taken away a mediall line commensurable in power onely to the whole line, and comprehending together with the whole line a rationall superficies: the residue is an irrationall line, and is called a first mediall residuall line.
An other demonstration after Campane.
¶The 57. Theoreme. The 75. Proposition. If from a mediall lyne be taken away a mediall lyne commensurable in power only to the whole lyne, and comprehending together with the whole lyne a mediall superficies, the residue is an irrationall lyne, and is called a second mediall residuall lyne.
An other demonstrtion more briefe after Campane.
¶The 58. Theoreme. The 76. Proposition. Irom a right line be taken away a right line incommensurable in power to the whole, and if that which is made of the squares of the whole line and of the line taken away added together be rationall, and the parallelogrāme contained vnder the same lines mediall: the line remayning is irrationall, and is called a lesse line.
¶ The 19. Theoreme. The 77. Proposition. If from a right line be taken away a right line incommensurable in power to the whole line, and if that which is made of the squares of the whole line and of the line taken away added together be mediall, and the paral∣lelogramme contained vnder the same lines rationall: the line remaining is irrationall, and is called a line making with a rationall superficies the whole superficies mediall.
¶ The 60. Theoreme. The 78. Proposition. If from a right line be taken away a right line incommensurable in power to the whole line, and if that which is made of the squares of the whole line and of the line taken away added together be medial, and the parallelo¦gramme contayned vnder the same lines be also mediall, and incommen∣surable to that which is made of the squares of the sayd lines added toge∣ther: the line remayning is irrationall, and is called a line making with a mediall superficies the whole superficies mediall.
An assumpt of Campane.
¶ The 61. Theoreme. The 79. Proposition. Vnto a residual line can be ioyned one onely right lyne rational, and com∣mensurable in power onely to the whole lyne.
¶ The 62. Theoreme. The 80. Proposition. Vnto a first medial residuall line can be ioyned one onely mediall right lyne, commensurable in power onely to the whole lyne, and comprehendyng wyth the whole lyne a rationall superficies.
¶ The 63. Theoreme. The 81. Proposition. Vnto a second mediall residuall line can be ioyned onely one mediall right line, commensurable in power onely to the whole line, and comprehending with the whole line a mediall superficies.
¶ The 64. Theoreme. The 82. Proposition. Vnto a lesse line can be ioyned onely one right line incommensurable in power to the whole lyne, and making together with the whole lyne that which is made of their squares added together rationall, and that which is contayned vnder them mediall.
¶ The 65. Theoreme. The 83. Proposition. Vnto a line making with a rationall superficies the whole superficies me∣diall, can be ioyned onely one right lyne incommensurable in power to the whole lyne, and making together with the whole line that which is made of their squares added together mediall, and that which is contained vnder them rationall.
¶ The 66. Theoreme. The 84. Proposition. Vnto a line making with a mediall superficies the whole superficies medial, can be ioyned onely one right line incommensurable in power to the whole line, and making together with the whole line that which is made of their squares added together mediall, and that which is contained vnder them mediall, and moreouer making that which is made of the squares of them added together incommensurable to that which is contayned vn∣der them.
¶ Third Definitions.
A first residuall line is, when the square of the whole excedeth the square of the lyne adioyned, by the square of a lyne commensurable vnto it in lēgth, and also the whole is commensurable in length to the rationall line first set.
A second residual line is, when the square of the whole excedeth the square of the line adioyned, by the square of a line commensurable vnto it in lēgth, and also the line adioyned is commensurable in length to the rationall lyne.
A third residuall line is, when the square of the whole excedeth the square of the lyne adioyned, by the square of a line commensurable vnto it in lēgth and neither the whole line, nor the line adioyned is cōmensurable in length to the rationall lyne.
A fourth residuall line is, when the square of the whole lyne excedeth the square of the lyne adioyned, by the square of a lyne incommensurable vnto it in length, and the whole lyne is also commensurable in length to the ra∣tionall lyne.
A fiueth residuall line is, when the square of the whole lyne exceedeth the square of the lyne adioyned, by the square of a lyne incommensurable vnto it in length, and the lyne adioyned is commensurable in length to the ra∣tionall lyne.
A sixth residuall line is when the square of the whole line, exceedeth the square of the line adioyned, by the square of a line incommensurable vnto it in length, and neither the whole line nor the line adioyned is commensu∣rable in length to the rationall line.
¶The 19. Probleme. The 85. Proposition. To finde out a first residuall line.
¶ The 20. Probleme. The 86. Proposition. To finde out a second residuall line.
¶ The 21. Probleme. The 87. Proposition. To finde out a third residuall line.
¶ The 22. Probleme. The 88. Proposition. To finde out a fourth residuall line.
¶ The 23. Probleme. The 89. Proposition. To finde out a fift residuall lyne.
¶ The 24. Probleme. The 90. Proposition. To finde out a sixth residuall line.
¶The 67. Theoreme. The 91. Proposition. If a superficies be contayned vnder a rationall line & a first residuall line: the line which contayneth in power that superficies, is a residuall line.
¶The 68. Theoreme. The 92. Proposition. If a superficies be contained vnder a rationall line and a second residuall line: the line which containeth in power that superficies, is a first mediall residuall line.
¶ The 69. Theoreme. The 93. Proposition. If a superficies be contained vnder a rationall line and a third residuall line: the line that containeth in power that superficies is a second mediall residuall line.
The 70. Theoreme. The 94. Proposition. If a superficies be contayned vnder a rationall lyne, and a fourth residuall lyne: the lyne which contayneth in power that superficies, is a lesse lyne.
¶The 71. Theoreme. The 95. Proposition. If a superficies be contained vnder a rationall line and a fift residual line: the line that cōtayneth in power the same superficies, is a line making with a rationall superficies, the whole superficies mediall.
¶The 72. Theoreme. The 96. Proposition. If a superficies be contayned vnder a rationall line and a sixth residuall line, the line which contayneth in power the same superficies is a line ma∣king with a mediall superficies the whole superficies mediall.
¶ The 73. Theoreme. The 97. Proposition. The square of a residuall line applyed vnto a rationall line, maketh the breadth or other side a first reiduall line.
¶The 4. Theoreme. The 98. Proposition. The square of a first mediall residuall line applied to a rationall line, ma∣keth the breadth or other side a second residuall line.
¶The 75. Theoreme. The 99. Proposition. The square of a second mediall residuall line applied vnto a rationall line, maketh the breadth or other side a third residuall line.
¶The 76. Theoreme. The 100. Proposition. The square of a lesse line applied vnto a rationall line, maketh the breadth or other side a fourth residuall line.
¶The 77. Theoreme. The 101. Proposition. The square of a lyne making with a rationall superficies the whole super∣ficies mediall applied vnto a rational line, maketh the breadth or other side a fift residuall lyne.
¶The 78. Theoreme. The 102. Proposition. The square of a lyne making with a mediall superficies, the whole superfi∣cies mediall applied to a rationall line, maketh the breadth or other side, a sixt residuall line.
¶The 79. Theoreme. The 103. Proposition. A line commensurable in length to a residuall line: is it selfe also a residuall line of the selfe same order.
¶ The 80. Theoreme. The 104. Proposition. A line commensurable to a mediall residuall line, is it selfe also a medial re∣siduall line, and of the selfe same order.
An other demonstration after Campane.
¶ The 81. Theoreme. The 105. Proposition. A line commensurable to a lesse line: is it selfe also a lesse line.
An other demonstration.
¶The 82. Theoreme. The 106. Proposition. A line commensurable to a lyne making with a rationall superficies the whole superficies mediall, is it selfe also a lyne making with a rationall su∣perficies the whole superficies mediall.
An other demonstration.
¶The 83. Theoreme. The 107. Proposition. A line cōmensurable to a line, making with a mediall superficies, the whole superficies mediall, is it selfe also a line making with a mediall superficies the whole superficies mediall.
¶The 84. Theoreme. The 108. Proposition. If from a rationall superficies be taken away a medialt superficies, the line which containeth in power the superficies remayning, is one of these two ir∣rationall lines, namely, either a residuall line, or a lesse line.
¶The 85. Theoreme. The 109. Proposition. If from a mediall superficies be taken away a rationall superficies, the line which contayneth in power the superficies remayning is one of these two ir∣rationall lines, namely either a first mediall residuall line, or a line ma∣king with a rationall superficies the whole superficies mediall.
¶The 86. Theoreme. The 110. Proposition. If from a mediall superficies be taken away a mediall superficies incom∣mensurable to the whole superficies, the line which containeth in power the superficies which remaineth, is one of these two irrationall lines, namely, either a second mediall residuall line, or a line making with a mediall su∣perficies the whole superficies mediall.
¶The 87. Theoreme. The 111. Proposition. A residuall line, is ot one and the same with a binomiall lyne.
¶ A Corollary.
¶ The 88. Theoreme. The 112. Proposition. The square of a rationall line applyed vnto a binomiall line, maketh the breadth or other side a residuall line, whose names are commensurable to the names of the binomiall line, & in the selfe same proportiō: & moreouer that residuall line is in the selfe same order of residuall lines, that the bino∣miall line is of binomiall lines.
Here is the Assumpt (of the foregoing Proposition) confirmed.
M. Dee, of this Assumpt, maketh (, that is, Acquisiuely,) a Probleme vniuersall, thus:
¶ A Corollary also noted by I. Dee.
An other demonstration after Flussas.
¶ The 89. Theoreme. The 113. Proposition. The square of a rational line applied vnto a residuall, maketh the breadth or other side a binomial line, whose names are commensurable to the names of the residuall line, and in the selfe same proportion: and moreouer that binomiall line is in the selfe same order of binomiall lynes, that the residual line is of residuall lynes.
The Assumpt confirmed.
An other demonstration after Flussas.
¶The 90. Theoreme. The 114. Proposition. If a parallelogrāme be cōtained vnder a residuall line & a binomiall lyne, whose names are commensurable to the names of the residuall line, and in the sele same proportion: the lyne which contayneth in power that super∣ficies is rationall.
¶Corollary.
¶An otr Flussas.
¶The 91. Theoreme. The 115. Proposition. Of a mediall line are produced infinite irrationall lines, of which none is of the selfe same kinde with any of those that were before.
An other demonstratio
¶The 92. Theoreme. The 116. Proposition. Now let vs proue that in square figures, the diameter is incommensurable in length to the side.
An other demonstration.
An other demonstration after Flussas.
Here followeth an instruction by some studious and skilfull Grecian (perchance Theon) which teacheth vs of farther vse and fruite of these irrationall lines.
An aduertisement by Iohn Dee.
¶The eleuenth booke of Eu∣clides Elementes.
Definitions. A solide or body is that which hath length, breadth, and thicknes, and the terme or limite of a solide is a superficies.
2 A right line is then erected perpendicularly to a plerficies, whē the right line maketh right angles with all the lines it, and are drawen vpon the ground plaine superficies.
3 A plaine superficies is then vpright or erected perpendicularly to a plaine superficies, when all the right lines drawen in one of the plaine su∣perficieces vnto the common section of those two plaine superficieces, ma∣king therwith right angles, do also make right angles to the other plaine superficies. Inclination or leaning of a right line, to a plaine superficies, is an acute angle, contained vnder a right line falling from a point aboue to the plaine superficies, and vnder an other right line, from the lower end of the sayd line (let downe) drawen in the same plaine superficies, by a certaine point assigned, where a right line from the first point aboue, to the same plaine superficies falling perpendicularly, toucheth.
4 Inclination of a plaine superficies to a plaine superficies, is an acute an∣gle contayned vnder the right lines, which being drawen in either of the plaine superficieces to one & the self same point of the cōmon section, make with the section right angles.
5 Plaine superficieces are in like sort inclined the onher, when the sayd angles of inclination are equall the one to the o
6 Parallell plaine superficieces are those, which being produced or exten∣ded any way neuer touch or concurre together.
7 Like solide or bodily figures are such, which are contained vnder like plaine superficieces, and equall in multitude.
8 Equall and like solide (or bodely) figures are those which are contained vnder like superficieces, and equall both in multitude and in magnitude.
9 A solide or bodily angle, is an inclination of moe then two lines to all the lines which touch themselues mutually, and are not in one and the selfe same supericies.
10 A Pyramis is a solide figure contained vnder many playne superficieces set vpon one playne superficies, and gathered together to one point.
11 A prisme is a solide or a bodily figure contained vnder many plaine superficieces, of which the two superficieces which are opposite, are equall and like, and parallells, & all the other superficieces are parallelogrāmes.
12 A Sphere is a figure which is made, when the diameter of a semicircle abiding fixed, the semicircle is turned round about, vntill it returne vnto the selfe same place from whence it began to be moued.
13 The axe of a Sphere is that right line which abideth fixed, about which the semicircle was moued.
14 The centre of a Sphere is that poynt which is also the centre of the se∣micircle.
15 The diameter of a Sphere is a certayne right line drawen by the cētre, and one eche side ending at the superficies of the same Sphere.
16 A cone is a solide or bodely figure which is made, when one of the sides of a rectangle triangle, namely, one of the sides which contayne the right angle, abiding fixed, the triangle is moued about, vntill it returne vnto the selfe same place from whence it began first to be moued. Now if the right line which abideth fixed be equall to the other side which is moued about and containeth the right angle: then the cone is a rectangle cone. But if it be lesse, then is it an obtuse angle cone. And if it be greater, thē is it an a cute∣angle cone,
17 The axe of a Cone is that line, which abideth fixed, about which the triangle is moued. And the base of the Cone is the circle which is described by the right line which is moued about.
18 A cylinder is a solide or bodely figure which is made, when one of the sides of a rectangle parallelogramme, abiding fixed, the parallelogramme is moued about, vntill it returne to the selfe same place from whence it be∣gan to be moued.
19 The axe of a cilinder is that right line which abydeth fixed, about which the parallelogramme is moued. And the bases of the cilinder are the circles described of the two opposite sides which are moued about.
20 Like cones and cilinders are those, whose axes and diameters of their bases are proportionall
21 A Cube is a solide or bodely figure contayned vnder sixe e∣quall squares.
22 A Tetrahedron is a solide which is contained vnder fower triangles equall and equilater.
23 An Octohedron is a solide or bodily figure cōtained vnder eight equall and equilater triangles.
24 A Dodecahedron is a solide or bodily fi∣gure cōtained vnder twelue equall, equilater, and equiangle Pentagons.
25 An Icosahedron is a solide or bodily figure contained vnder twentie equall and equilater triangles.
A parallelipipedon is a solide figure comprehended vnder foure playne qua∣drangle figures, of which those which are opposite are parallels.
¶The 1. Theoreme. The 1. Proposition. That part of a right line should be in a ground playne superficies, & part eleuated vpward is impossible.
An other demonstration after Fl•s••s.
¶The 2. Theoreme. The 2. Proposition. If two right line cut the ou to the other, they are ne and the selfe same playne superficies: & euery triangle is in one & the selfe same superficie.
¶The 3. Theoreme. The 3. Proposition. If two playne superficieces cutte the one the other: their common section is a right line.
¶The 4. Theoreme. The 4. Proposition. If from two right lines, cutting the one the other, at their common section, a right line be perpendicularly erected: the same shall also be perpendicular∣ly erected from the playne superficies by the sayd two lines passing.
¶The 5. Theoreme. The 5. Proposition. If vnto three right lines which touch the one the other, be erected a per∣pendicular line from the common point where those three lines touch: those three right lines are in one and the selfe same plaine superficies.
The 6. Theoreme. The 6. Proposition. If two right lines be erected perpendicularly to one & the selfe same plaine superficies: those right lines are parallels the one to the other.
¶An other demonstration of the sixth proposition by M. Dee.
A Corollary added by M. Dee.
Io. Dee his aduise vpon the Assumpt of the 6.
¶ The 7. Theoreme. The 7. Proposition. If there be two parallel right lines, and in either of them be taken a point at all aduentures: a right line drawen by the said pointes is in the self same superficies with the parallel right lines.
The 8. Theoreme. The 8. Proposition. If there be two parallel right lines, of which one is erected perpendicularly to a round playne superficies: the other also is erected perpendicularly to the selfe same ground playne superficies.
¶ The 9. Theoreme. The 9. Pro Right lines which are parallels to one and the selfe same right line, and are not in the selfe same superficies that it is in: are also parallels the one to the other.
¶ The 10. Theoreme. The 1 If two right lines touching the one the otheher right lines touching the one the other, and nolfe same superficies with the two first: those right lines cōtaine equall angles.
¶ The 1. Probleme. The 11. Proposition. From a point geuen on high, to drawe vnto a ground plaine superficies a perpendicular right line.
¶The 2. Probleme. The 12. Proposition. Vnto a playne superficies geuen, and from a poynt in it geuen, to rayse vp a perpendicular line.
¶ The 11. Theoreme. The 13. Prposition. From one and the selfe poynt, and to one and the selfe same playne superfi∣cies, can not be erected two perpendicular right lines on one and the selfe same side.
M. Dee his annotation.
¶ The 12. Theoreme. The 14. Proposition. To whatsoeuer plaine superficieces one and the selfe same right line is e∣rected perpendicularly: those superficieces are parallels the one to the other.
A corollary added by Campane.
¶ The 13. Theoreme. The 15. Proposition. If two right lines touching the one the other be parallels to two other right lines touching also the one the other and not being in the selfe same plaine superficies with the two first: the plaine superficieces extended by those right lines, are also parallells the one to the other.
¶ A Corollary added by Flussas.
The 14. Theoreme. The 16. Proposition. If two parallel playne superficieces be cut by some one playne superficies: their common sections are parallel lines.
A Corollary added by Flussas.
The 15. Theoreme. The 17. Proposition. I two right lines be cut by playne superficieces being parallels: the partes o the lines deuided shall be proportionall.
¶ The 16. Theoreme. The 18. Proposition. If a right line be erected perpēdicularly to a plaine superficies: all the superficieces extended by that right line, are erected perpendicularly to the selfe same plaine superficies.
¶ The 17. Theoreme. The 19. Proposition. If two plaine superficieces cutting the one the other be erected perpendicu∣larly to any plaine superficies: their common section is also erected perpen∣dicularly to the selfe same plaine superficies.
The 18. Theoreme. The 20. roposition. If a solide angle be contayned vnder three playne superficiall angles: euery two of those three angles, which two so euer be taken, are greater then the third.
The 19. Theoreme. The 21. Proposition. Euery solide angle is comprehended vnder playne angles lesse then fower right angles.
¶ The 20. Theoreme. The 22. Proposition. If there be three superficiall plaine angles of which two how soeuer they be taken, be greater then the third, and if the right lines also which con∣tayne those angles be equall: then of the lines coupling those equall right lines together, it is possible to make a triangle.
An other demonstration.
¶ The 3. Probleme. The 23. Proposition. Of three plaine superficiall angles, two of which how soeuer they be taken, are greater then the third, to make a solide angle: Now it is necessary that those three superficiall angles be lesse then fower right angles.
¶ The 21. Theoreme. The 24. Proposition. If a solide or body be contayned vnder sixe parallel playne superficieces, the opposite plaine superficieces of the same bo∣dy are equall and parallelogrammes.
The 22. Theoreme. The 25. Proposition. If a Parallelipipedō be cutte of a playne superfi∣cies beyng a parallel to the two opposite playne superficieces of the same body: then, as the base is to the base, so is the one solide to the other solide.
First Corollary.
Second Corollary.
Third Corollary.
The 4. Probleme. The 26. Proposition. Vpon a right lyne geuen, and at a point in it geuen, to make a solide angle equall to a solide angle geuen.
The 5. Theoreme. The 27. Proposition. Vpon a right line geuen to describe a parallelipipedon like and in like sort si∣tuate to a parallelipipedon geuen.
The 23. Theoreme. The 28. Proposition. If a parallelipipedō be cutte by a plaine superficies drawne by the diagonall lines of those playne superficieces which are opposite: that solide is by this playne superficies cutte into two equall partes.
¶ The 24. Theoreme. The 29. Proposition. Parallelipipedons consisting vpon one and the selfe same base, and vn∣der one and the selfe same altitude, whose standing lines are in the selfe same right lines, are equall the one to the other.
¶The 25. Theoreme. The 30. Proposition. Parallelipipedons consisting vpon one and the selfe same base, and vnder the selfe same altitude, whose standing lines are not in the selfe same right lines, are equall the one to the other.
The 26. Theoreme. The 31. Proposit. Parallelipipedons consisting vp∣on equall bases, and being vnder one and the selfe same altitude, are equall the one to the other.
Flussas demonstrateth this proposition an otherway taking onely the bases of the solides, and that after this maner.
¶The 27. Theoreme. The 32. Proposition. Parallelipipedons being vnder one and the selfe same altitude, are in that proportion the one to the other that their bases are.
A Corollary added by Flussas.
The 28. Theoreme. The 33. Proposition. Like parallelipipedons are in treble proportion the one to the other of that in which their sides of like proportion are.
¶ Corellary.
¶ Certaine most profitable Corollaries, Annotations, Theo∣remes, and Problemes, with other practises, Logisticall, and Mechanicall, partly vpon this 33. and partly vpon the 34. 36. and other following, added by Master Iohn Dee. ¶ A Corollary. 1.
¶A Corollary.
¶A Corollary added by Flussas.
¶The 29. Theoreme. The 34. Proposition. In equall Parallelipipedons the bases are reciprokall to their altitudes. And Parallelipipedons whose bases are reciprokall to their altitudes, are equall the one to the other.
M. Iohn Dee, his sundry Inuentions and Annotacions, very necessary, here to be added and considered.
A Thereme.
A Corollary logisticall.
A Probleme. 1.
Or we may thus expresse the same thing.
A Probleme 2.
Note.
The 30. Theoreme. The 35. Proposition. If there be two superficiall angles equall, and from the pointes of those an∣gles be eleuated on high right lines, comprehending together with those right lines which containe the superficiall angles, equall angles, eche to his corespōdent angle, and if in eche of the eleuated lines be takē a point at all auentures, and from those pointes be drawen perpendicular lines to the ground playne superficieces in which are the angles geuen at the begin∣ning, and from the pointes which are by those perpendicular lines made in the two playne superficieces be ioyned to those angles which were put at the beginning right lines: those right lines together with the lines eleua∣ted on high shall contayne equall angles.
¶ Corollary.
¶The 31. Theoreme. The 36. Proposition. If there be three right lines proportionall: a Parallelipipedon described of those three right lines, is equall to the Parallelipipedon described of the middle line, so that it consiste of equall sides, and also be equiangle to the foresayd Parallelipipedon.
¶New inuentions (coincident) added by Master Iohn Dee.
A Corollary. 1.
¶A Probleme. 1.
A Probleme. 2.
¶ The 32. Theoreme. The 37. Proposition. If there be fower right lines proportionall: the Parallelipipedons descri∣bed of those lines, being like and in like sort described, shall be proportio∣nall. And i the Parallelipipedons described of them, being like and in like sort described, be proportionall: those right lines also shall be proportionall.
¶ The 33. Theoreme. The 38. Proposition. If a plaine superficies be erected perpendicularly to a plaine superficies, and from a point taken in one of the plaine superficieces be drawen to the other plaine superficies, a perpendicular line: that perpendicular line shall fall vpon the common section of those plaine superficieces.
¶ Note.
¶ The 34. Theoreme. The 39. Proposition. If the opposite sides of a Parallelipipedon be deuided into two equall partes, and by their common sections be extended plaine superficieces: the commō section of those plaine superficieces, and the diameter of the Paral∣lelipipedon shall deuide the one the other into two equall partes.
A Corollary added by Flussas.
¶ The 35. Theoreme. The 40. Proposition. If there be two Prismes vnder equall altitudes, & the one haue to his base a parallelogramme, and the other a triangle, and if the parallelogramme be double to the triangle: those Prismes are equall the one to the other.
A Corollary added by Flussas.
¶The twelueth booke of Eu∣clides Elementes.
¶The 1. Theoreme. The 1. Proposition. Like Poligonon figures described in circles: are in that proportion the one to the other, that the squares of their diameters are.
¶ Iohn Dee his fruitfull instructions, with certaine Corollaries, and their great vse.
¶A Corollary. 1.
¶ A Corollary. 2.
¶ A Theoreme.
A Corollary. 1.
A Corollary. 2.
¶ The 2. Theoreme. The 2. Proposition. Circles are in that proportion the one to the other, that the squares of their diameters are.
¶ An Assumpt.
¶ A Corollary added by Flussas.
¶ Very needefull Problemes and Corollaryes by Master Ihon Dee inuented: whose wonderfull vse also, be partely declareth.
A Probleme. 1.
A Probleme. 2.
Note.
A Probleme. 3.
A Probleme. 4.
A Corollary. 1.
A Corollary. 2.
¶ A Probleme 5.
A Probleme. 6.
Note.
A Corollary.
A Probleme 7.
¶A Corollary.
A Probleme. 8.
¶A Corollary.
A Probleme. 9.
¶A Corollary. 1.
Note.
¶The key of one of the chiefe treasure houses, belonging to the State Mathematicall.
¶The 3. Theoreme. The 3. Proposition. Euery Pyramis hauing a triangle to his base: may be deuided into two Py∣ramids equall and like the one to the other, and also like to the whole, ha∣uing also triangles to their bases, and into two equall prismes: and those two prismes are greater then the halfe of the whole Pyramis.
¶An other demonstration after Campane of the 3. proposition.
¶The 4. Theoreme. The 4. Proposition. If there be two Pyramids vnder equall altitudes, hauing triangles to their bases, and either of those Pyramids be deuided into two Pyramids equall the one to the other, and like vnto the whole, and into two quall Prismes, and againe if in either of the Pyramids made of the two first Pyramids be still obserued the same order and maner: then as the base of the one Pyra∣mis is to the base of the other Pyramis, so are all the Prismes which are in the one Pyramis to all the Prismes which are in the other, being equall in multitude with them.
¶ The 5. Theoreme. The 5. Proposition. Pyramids consisting vnder one and the selfe same altitude, hauing tri∣angles to their bases: are in that proportion the one to the other that their bases are.
¶ The 6. Theoreme. The 6. Proposition. Pyramids consisting vnder one and the selfe same altitude, and hauing Pligoon figures to their bases: are in that proportion the one to the other, that their bases are.
The 7. Theoreme. The 7. Proposition. Euery prisme hauing a triangle to his base, may be deuided into three py∣ramids equall the one to the other, hauing also triangles to their bases.
¶ Corollary.
First Corollary.
Second Corollary.
Third Corollary.
Fourth Corollary.
Fiueth Corollary.
The 8. Theoreme. The 8. Proposition. Pyramids being like & hauing triangles to their bases, are in treble propor¦tion the one to the other, of that in which their sides of like proportion are.
Corollary.
¶ The 9. Theoreme. The 9. Proposition. In equall pyramids hauing triangles to their bases, the bases are recipro∣kall to their altitudes. And pyramids hauing triangles to their bases, whose bases are reciprokall to their altitudes, are equall the one to the other.
A Corrollary added by Campane and Flussas.
The 10. Theoreme. The 10. Proposition. Euery cone is the third part of a cilinder, hauing one and the selfe same base and one and the selfe same altitude with it.
¶ Added by M. Iohn Dee.
¶ A Theoreme. 1.
¶ A Theoreme. 2.
¶ The 11. Theoreme. The 11. Proposition. Cones and Cylinders being vnder one and the selfe same altitude, are in that proportion, the one other that their bases are.
¶ The 12. Theoreme. The 12. Proposition. Like Cones and Cylinders, are in treble proportion of that in which the di∣ameters of their bases are.
The 13. Theoreme. The 13. Proposition. If a Cylinder be diuided by a playne superficies being a parallell to the two opposite playne super∣ficieces: then as the one Cylinder is to the other Cylinder, so is the axe of the one to the axe of the other.
¶ The 14. Theoreme. The 14. Proposition. Cones and Cylinders consisting vpon equall bases, are in proportion the one to the other as their altitudes.
¶The 15. Theoreme. The 15. Proposition. In equall Cones and Cylinders, the bases are reciprokall to their alti∣tudes. And cones and Cylinders whose bases are reciprokall to their alti∣tudes, are equall the one to the other.
A Corrollary added by Campane and Flussas.
The 1. Probleme. The 16. Proposition. Two circles hauing both one and the selfe same centre being geuen, to in∣scribe in the greater circle a poligonon figure, which shall consist of equall and euen sides, and shall not touch the superficies of the lesse circle.
¶ Corollary.
¶An Assumpt added by Flussas.
Iohn Dee.
¶A Corollary added by the same Flussas.
¶ The 2. Probleme. The 17. Proposition. Two spheres consisting both about one & the selfe same cētre, being geuē, to inscribe in the greater sphere a solide of many sides (which is called a Polyhedron) which shall not touch the superficies of the lesse sphere.
¶ Corollary.
I. De.
¶ Master Dee his aduise and demonstration, reforming a great errour in the designation of the former figure of Euclides second Probleme: with two Corollaries (by him inferred) vpon his said demonstration.
A Theoreme.
A Corollary. 1.
2.
¶ Note.
¶ The 16. Theoreme. The 18. Proposition. Spheres are in treble proportion the one to the other of that in which their diameters are.
A Corrollary added by Flussas.
A Corollary added by M Dee.
¶ Certaine Theoremes and Problemes (whose vse is manifolde, in Spheres, Cones, Cylinders, and other solides) added by Ioh. Dee.
A Theoreme. 1.
A Theoreme. 2.
A Probleme. 1.
¶ A Corollary.
¶ A Probleme. 2.
¶ A Probleme. 3.
¶ A Theoreme. 3.
The Lemma.
Note.
A Probleme. 4.
Or thus more briefely omitting all cutting of the Cylinder.
A Probleme. 5.
A Probleme. 6.
A Probleme. 7.
A Probleme. 8.
An assumpt.
¶ A Theoreme. 4.
Note.
A Theoreme. 5.
A Probleme. 9.
A Probleme. 10.
A Corollary.
A Probleme. 11.
¶ An aduise.
A Probleme. 12.
¶ A Corollary. 1.
¶ A Corollary. 2.
¶ A Corollary. 3.
A Theoreme. 6.
A Theoreme. 7.
¶ Logistically. ¶
A Corollary. 1.
A Corollary. 2.
A Theoreme. 8.
Note. 1.
Note. 2.
A proposition added by Flussas.
¶ The thirtenth booke of Euclides Elementes.
The 1. Theoreme. The 1. Proposition. If a right line be deuided by an extreme and meane proportion, and to the greater segment, be added the halfe of the whole line: the square made of those two lines added together shalbe quintuple to the square made of the halfe of the whole lyne.
The 2. Theoreme. The . Proposition. If a right line, be in power quintuple to a segment of the same line: the dou∣ble of the sayd segment is deuided by an extreame and meane proportion, and the greater segment thereof is the other part of the line geuen at the be¦ginning.
Two Theoremes, (in Euclides Method necessary) added by M. Dee.
A Theoreme. 1.
A Theoreme. 2.
Note.
The 3. Theoreme. The 3. Proposition. If a right line be deuided by an extreme and meane proportion, and to the lesse segment be added the halfe of the gerater segment: the square made of those two lines added together, is quintuple to the square made of the half line of the greater segment.
Here foloweth M. Dee, his additions.
¶A Theoreme. 1.
Or, thus it may be demonstrated.
A Theoreme. 2.
Two other briefe demonstrations of the same.
¶A Corollary. 1.
¶A Corollary. 2.
A Corollary. 3.
Iohn Dee.
A Probleme.
The second way to execute this probleme.
Note
¶ The 4. Theoreme. The 4. Proposition. If a right line be deuided by an extreame and meane proportion: the squares made of the whole line and of the lesse segmēt, are treble to the square made of the greater segment.
¶ The 5. Theoreme. The 5. Proposition. If a right line be deuided by an extreame and meane proportion, and vnto it be added a right ine, equall to the greater segment, the whole right line is deuided by an extreame and meane proportion, and the greater segment thereof, is the right line geuen at the beginning.
A Corollary added by Campane.
A Corollary. 1.
A Corollary. 2.
¶What Resolution is.
¶ What Composition is.
Resolution of the first Theoreme.
Composition of the first Theoreme.
Resolution of the 2. Theoreme.
Composition of the 2. Theoreme.
Resolution of the 3. Theoreme.
Composition of the 3. Theoreme.
Resolution of the 4. Theoreme.
Composition of the 4. Theoreme.
Resolution of the 5. Theoreme.
Composition of the 5. Theoreme.
An Aduise, by Iohn Dee, added.
The 6. Theoreme. The 6. Proposition If a rationall right line be diuided by an extreme and meane proportion: eyther of the segments, is an irrationall line of that kinde, which is called a residuall line.
¶A Corollary added by Campane.
The 7. Theoreme. The 7. Proposition. If an equilater Pētagon haue three of his angles, whether they follow in or∣der, or not in order, equall the one to the other: that Pentagon shalbe equi∣angle.
The 8. Probleme. The 8. Proposition. If in an equilater & equiangle Pētagon two right lines do subtend two of the angles following in order: those lines doo diuide the one the other by an extreme and meane proportion: and the greater segments of those lines are ech equall to the side of the Pentagon.
¶ The 9. Theoreme. The 9. Proposition. If the side of an equilater hexagon, and the side of an equilater decagon or ugled figure, which both are inscribed in one & the selfe same circle, be added together: the whole right line made of them is a line diuided by a extreame and meane proportion, and the greater segment of the same is the side of the hexagon.
A Corollary added by Flussas.
Campane putteth the conuerse of this proposition after this maner.
¶ The 10. Theoreme. The 10. Proposition. If in a circle be described an equilater Pentagon, the side of the Pentagon containeth in power both the side of an hexagon and the side of a decagon, being all described in one and the selfe same circle.
¶A Corollary added by Flussas.
¶The 11. Theoreme. The 11. Proposition. If in a circle hauing a rationall line to his diameter be inscribed an equila∣ter pentagon: the side of the pentagon is an irrationall line, and is of that kinde which is called a lesse line.
¶ The 12. Theoreme. The 12. Proposition. If in a circle be described an equilater triangle: the square made of the side of the triangle, is treble to the square made of the line, which is drawen from the centre of the circle to the circumference.
A Corollary added by Campane.
A Corollary added by Flussas.
The 1. Probleme. The 13. Proposition. To make a Pyramis, and to comprehend it in a sphere geuen: and to proue that the diameter of the sphere is in power sesquialtera to the side of the Pyramis.
¶An other demonstration to proue that as the line AB is to the line BC, so is the square of the line AD to the square of the line DC.
¶Two Assumptes added by Campane.
First Assumpt.
¶Second Assumpt.
¶ Certaine Corollaryes added by Flussas.
First Corollary.
¶ Second Corollary.
Third Corollary.
¶ The 2. Probleme. The 14. Proposition. To make an octohedron, and to cōprehend it in the sphere geuen, namely, that wherein the pyramis was comprehended: and to proue that the diameter of the sphere is in power double to the side of the octohedron.
Certayne Corollaries added by Flussas.
First Corollary.
Second Corollary.
Third Corollary.
¶The 3. Probleme. The 15. Proposition. To make a solide called a cube, and to comprehend it in the sphere geuen, namely, that Sphere wherein the former two solides were comprehendd and to proue that the diameter of the sphere, is in power treble to the side of the cube.
An other demonstration after Flussas.
¶ Corollaryes added by Flussas.
First Corollary.
¶ Second Corollary.
¶ The 4. Probleme. The 16. Proposition. To make an Icosahedron, and to comprehend it in the Sphere geuen, wher∣in were contained the former solides, and to proue that the side of the Ico∣sahedron is an irrationall line of that kinde which is called a lesse line.
A Corollary.
¶ A Corollary added by Flussas.
¶The 5. Probleme. The 17. Proposition. To make a Dodecahedron, and to comprehend it in the sphere geuen, wherin were comprehended the foresayd solides: and to proue that the side of the dodecahedron is an irrationall line of that kind which is called a re∣siduall line.
¶ Corollary.
A further construction of the dodecahedron after Flussas.
¶ Certayne Corollaryes added by Flussas.
First Corollary.
¶Second Corollary.
Third Corollary.
¶The 6. Probleme. The 18. Proposition. To finde out the sides of the foresayd fiue bodies, and to compare them together.
A Corollary.
An Assumpt.
¶ A Corollary added by Flussas.
¶The fourtenth booke of Euclides Elementes.
The Preface of Hypsicles before the fourtenth booke.
¶ The 1. Theoreme. The 1. Proposition. A perpendicular line drawen from the centre of a circle to the side of a Pentagon described in the same circle: is the halfe of these two lines, name∣ly, of the side of an hexagon figure, and of the side of a decagon figure be∣ing both described in the selfe same circle.
¶ The 2. Theoreme. The 2. Proposition. One and the selfe same circle comprehendeth both the Pentagon of a Do∣decahedron, and the triangle of an Icosahedron, described in one and the selfe same Sphere.
¶ The 3. Theoreme. The .3 Proposition. If there be an equilater and equiangle pētagon, aud about it be described a circle, and from the centre to one of the sides be drawne a perpendicular line, that which is contayned vnder one of the sides and the perpendicular line thirty times, is equall to the superficies of the dodecahedron.
¶ Corollary.
¶ The 4. Theoreme. The 4. Proposition. This being done, now is to be proued, that as the superficies of the Do∣decahedron is to the superficies of the Icosahedron, so is the side of the cube to the side of the Icosahedron.
An other demonstration to proue that as the superficies of the Dodecahe∣dron is to the superficies of the Icosahedron, so is the side of the cube to the side of the Icosahedron.
This being proued, now let there be drawne a Circle comprehending both the Pentagon of a Dodecahedron, and the triangle of an Icosahedron, being both described in one and the selfe same Sphere.
Nowe will we proue that a right line being deuided by an extreme and meane proportiō, what proportiō the line cōtaining in power the squares of the whole line and of the greater segment, hath to the line containing in power the squares of the whole line and of the lesse segment, the same proportion hath the side of the cube to the side of the Icosahedron, being both described in one and the selfe same sphere.
Now will we proue that as the side of the Cube is to the side of the Ico∣sahedron, so is the solide of the Dodecahedron to the solide of the Icosa∣hedron.
If two right lines be diuided by an extreame and meane proportion, they shall euery way be in like proportion: which thing is thus demonstrated.
¶The fourtenth booke of Euclides Elementes after Flussas.
¶The first Proposition. A perpendicular line drawen from the centre of a circle, to the side of a Pentagon inscribed in the same circle: is the halfe of these two lines taken together, namely, of the side of the hexagon, and of the side of the decagon inscribed in the same circle.
A Corollary.
¶The second Proposition. If two right lines be diuided by an extreme and meane proportion: they shall be diuided into the selfe same proportions.
¶The third Proposition. If in a circle be described an equilater Pentagon: the squares made of the side of the Pentagon and of the line which subtendeth two sides of the Pentagon, these two squares (I say) taken together, are quintuple to the square of the line drawen from the centre of the circle to the circūference.
¶A Corollary.
The 4. Proposition. One and the selfe same circle containeth both the Pentagon of a Dodeca∣hedron, and the triangle of an Icosahedron described in one and the selfe same sphere.
The 5. Proposition. If in a circle be inscribed the pentagon of a Dodecahedron, and the trian∣gle of an Icosahedron, and from the centre to one of theyr sides, be drawne a perpendicular line: That which is contained 30. times vnder the side, & the perpendicular line falling vpon it, is equal to the superficies of that so∣lide, vpon whose side the perpendicular line falleth.
A Corollary.
The 6. Proposition. The superficies of a Dodecahedron, is to the superficies of an Icosahe∣dron described in one and the selfe same sphere, in that proportion, that the side of the Cube is to the side of the Icosahedron contained in the self same sphere.
An Assumpt.
¶ The 7. Proposition. A right line diuided by an extreame and meane proportion: what propor∣tion the line contayning in power the whole line and the greater segment, hath to the line contayning in power the whole and the lesse segment: the same hath the side of the cube to the side of the Icosahedron contayned in one and the same sphere.
¶ The 8. Proposition. The solide of a Dodecahedron is to the solide of an Icosahedron: as the side of a Cube is to the side of an Icosahedron, all those solides being described in one and the selfe same Sphere.
A Corollary.
¶ The 9. Proposition. If the side of an equilater triangle be rationall: the superficies shall be irra∣tionall, of that kinde which is called Mediall.
A Corollary.
¶The 10. Proposition. If a Tetrahedron and an Octohedron be inscribed in one and the self same Sphere: the base of the Tetrahedron shall be sesquitertia to the base of the Octohedron, and the supersicieces of the Octohedron shall be sesquialtera to the superficieces of the Tetrahedron.
¶The 11. Proposition. A Tetrahedron is to an Octohedron inscribed in one and the selfe same Sphere, in proportion, as the rectan∣gle parallelogrāme contained vnder the line, which containeth in power 27. sixty fower partes of the side of the Tetrahedron, & vnder the line which is subsesquioctaa to the same side of the Tetrahedron, is to the square of the diameter of the sphere.
¶The 12. Proposition. If a cube be contayned in a sphere: the square of the diameter doubled, is e∣quall to all the superficieces of the cube taken together. And a perpendicu∣lar line drawne from the centre of the sphere to any base of the cube, is e∣quall to halfe the side of the cube.
¶ A Corollary.
The 13. Proposition. One and the self same circle containeth both the square of a cube, and the triangle of an Octohedron described in one and the selfe same sphere.
A Corollary.
The 14. Proposition. An Octohedron is to the triple of a Tetrahedron contained in one and the selfe same sphere, in that proportion that their sides are.
A Corollary.
The 15. Proposition. If a rational line containing in power two lines, make the whole and the greater segment, and again containing in power two lines, make the whole and the lesse segment: the greater segment shalbe the side of the Icosahe∣dron, and the lesse segment shalbe the side of the Dodecahedron contayned in one and the selfe same sphere.
The 16. Proposition. If the power of the side of an Octohedron be expressed by two right line ioyned together by an extreme and mene proportion: the side of the Ico∣sahedron contained in the same sphere, shalbe duple to the lesse segment.
The 17. Proposition. If the side of a dodecahedron, and the right line, of whome the said side is the lesse segment, be so set that they make a right angle: the right line which containeth in power halfe the line subtending the angle, is the side of an Octohedron contained in the selfe same sphere.
A Corollary.
¶The 18. Proposition. If the side of a Tetrahedron containe in power two right lines ioyned to∣gether by an extreme and meane proportion: the side of an Icosahedron described in the selfe same Sphere, is in power sesquialter to the lesse right line.
¶ The 19. Proposition. The superficies of a Cube is to the superficies of an Octohedron inscribed in one and the selfe same Sphere, in that proportion that the solides are.
¶ The 20. Proposition. If a Cube and an Octohedron be contained in one & the selfe same Sphere: they shall be in proportion the one to the other, as the side of the Cube is to the semidiameter of the Sphere.
A Corollary.
An aduertisment added by Flussas.
¶The fiftenth booke of Euclides Elementes.
Diffinition. 1.
Diffinition. 2.
¶ The 1. Proposition. The 1. Probleme. In a Cube geuen to describe a trilater equilater Pyramis.
¶ The 2. Proposition. The 2. Probleme. In a trilater equilater Pyramis geuen to describe an Octohedron.
A Corollary added by Flussas.
¶ The 3. Proposition. The 3. Probleme. In a cube geuen, to describe an Octohedron.
¶A Coollary aded by •luss••.
¶ The 4. Proposition. The 4. Probleme. In an Octohedron geuen, to describe a Cube.
The 5. Proposition. The 5. Probleme. In an Icosahedron geuen, to describe a Dodecahedron.
An annotation of Hypsi•les.
¶ The 6. Proposition. The 6. Probleme. In an Octohedron geuen, to inscribe a trilater equilater Pyramis.
A Corollary.
Second Corollary.
¶ The 7. Proposition. The 7. Probleme. In a dodecahedron geuen, to inscribe an Icosahedron.
¶ The 8. Proposition. The 8. Probleme. In a dodecahedron geuen, to include a cube.
¶ The 9. Proposition. The 9. Probleme. In a Dodecahedron geuen to include an Octohedron.
¶ The 10. Proposition. The 10. Probleme: In a Dodecahedron geuen, to inscribe an equilater trilater Pyramis.
¶ The 11. Proposition. The 11. Probleme. In an Icosahedron geuen, to inscribe a cube.
¶The 12. Proposition. The 12. Probleme. In an Icosahedron geuen, to inscribe a trilater equilater pyramis.
¶The 13. Probleme The 13. Proposition. In a Cube geuen, to inscribe a Dodecahedron.
First Corollary.
¶ Second Corollary.
Third Corollary.
The 14. Probleme. The 14. Proposition. In a cube geuen, to inscribe an Icosahedron.
First Corollary.
Second Corollary.
Third Corollary.
¶Fourth Corollary.
¶ The 15. Probleme. The 15. Proposition. In an Icosahedron geuen, to inscribe an Octohedron.
¶ The 16. Probleme. The 16. Proposition. In an Octohedron geuen, to inscribe an Icosahedron.
¶ First Corollary.
Second Corollary.
¶ The 17. Probleme. The 17. Proposition. In an Octohedron geuen, to inscribe a Dodecahedron.
¶ The 18. Probleme. The 18. Proposition. In a trilater and equilater Pyramis, to inscribe a Cube.
¶ A Corrollary.
The 19. Probleme The 19. Proposition. In a trilater equilater Pyramis geuen, to inscribe an Icosahedron.
¶ The 20. Proposition. The 20. Probleme. In a trilater equilater Pyramis geuen, to inscribe a dodecahedron.
The 21. Probleme. The 21. Proposition. In euery one of the regular solides to inscribe a Sphere.
A Corollary.
An aduesment of Flussas
¶The sixtenth booke of the Elementes of Geometrie added by Flussas.
¶ The 1. Proposition. A Dodecahedron, and a cube inscribed in it, and a Pyramis inscribed in the same cube, are contained in one and the selfe same sphere.
¶ The The proportion of a Dodecahedron circumscribed about a cube, to a Dodeca∣hedrō inscribed in the same cube, is triple to an extreme & meane propartiō.
The 3. Proposition. In euery equiangle, and equilater Pentagon, a perpendicular drawne from one of the angles to the base, is deuided by an extreme and meane proporti∣on by a right line subtending the same angle.
¶ A Corollary.
The 4. Proposition. If frō the angles of the base of a Pyramis, be drawne to the opposite sides, right lines cutting the sayd sides by an extreme and meane proportion: they shall containe the bise of the Icosahedron inscribed in the Pyramis, which base shalbe inscribed in an equilater triangle, whose angles cut the sides of the base of the Pyramis by an extreme and meane proportion.
¶ A Corollary.
¶ The 5. Proposition. The side of a Pyramis diuided by an extreme and meane proportion, ma∣keth the lesse segment in power double to the side of the Icosahedron in∣scribed in it.
¶ A Corollary.
¶The 6. Proposition. The side of a Cube containeth in power halfe the side of an equilater trian∣gular Pyramis inscribed in the said Cube.
¶ The 7. Proposition. The side of a Pyramis is duple to the side of an Octohedron inscri∣bed in it.
¶ The 8. Proposition. The side of a Cube is in power duple to the side of an Octohedron inscri∣bed in it.
¶ The 9. Proposition. The side of a Dodecahedron, is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the sayd Dodecahedron.
¶The 10. Proposition. The side of an Icosahedron, is the meane proportionall betwene the side of the Cube circumscribed about the Icosahedron, and the side of the Dode∣cahedron inscribed in the same Cube.
¶The 11. Proposition. The side of a Pyramis, is in power Octodecuple to the side of the cube in∣scribed in it.
¶The 12. Proposition. The side of a Pyramis, is in power Octodecuple to that right line, whose greater segment is the side of the Dodecahedron inscribed in the Pyramis.
¶ The 13. Proposition. The side of an Icosahedron inscribed in an Octohedron, is in power duple to the lesse segment of the side of the same Octohedron.
¶The 14. Proposition. The sides of the Octohedron, and of the Cube inscribed in it, are in power the one to the other in quadrupla sesquialter proportion.
¶The 1. Proposition. The side of the Octohedron, is in power quadruple sesquialter to that right line, whose greater segment is the side of the Dodecahedron inscribed in the same Octohedron.
¶ The 16. Proposition. The side of an Icosahedron, is the greater segment of that right line, which is in power duple to the side of the Octohedron inscribed in the same Icosahedron.
¶The 17. Proposition. The side of a Cube is to the side of a Dodecahedron inscribed in it, in duple proportion of an extreame and meane proportion.
¶ The 18. Proposition. The side of a Dodecahedron is, to the side of a Cube inscribed in it, in con∣uerse proportion of an extreame and meane proportion.
¶ The 19. Proposition. The side of an Octohedron, is sesquialter to the side of a Pyramis inscri∣bed in it.
¶ The 0. Proposition. If from the power of the diameter of an Icosahedron, be taken away the power tripled of the side of the cube inscribed in the Icosahedron: the power remayning shall be sesquitertia to the power of the side of the I∣cosahedron.
A Corollary.
¶ The 21. Proposition. The side of a Dodecaedron is the lesse segment of that right line, which is in power duple to the side of the Octohedron inscribed in the same Do∣decahedron.
¶ The 22. Proposition. The diameter of an Icosahedron is in power sesquitertia to the side of the same Icosahedron, and also is in power sesquialter to the side of the Pyra∣mis inscribed in the Icosahedron.
The 23. Proposition. The side of a Dodecahedron is to the side of an Icosahedron inscribed in it, as the lesse segment of the perpendicular of the Pentagō, is to that line which is drawne from the centre to the side of the same pentagon.
¶ The 24. Proposition. If halfe of the side of an Icosahedron be deuided by an extreme & meane proportion: and if the lesse segment thereof be taken away from the whole side, and againe from the residue be taken away the third part: that which remaineth shall be equal to the side of the Dodecahedron inscribed in the same Icosahedron.
The 25. Proposition. To proue that a cube geuen, is to a trilater equilater pyramis inscribed in it, triple.
¶The 26. Proposition. To proue that a trilater equilater Pyramis is duple to an Octohedron in∣scribed in it.
¶ The 27. Proposition. To proue that a Cube is sextuple to an Octohedron inscribed in it.
¶The 28. Proposition. To proue that an Octohedron is quadruple sesquialter to a Cube inscri∣bed in it.
¶A Corollary.
¶The 29. Proposition. To proue that an octohedrō geuē, is tre∣decuple sesquialter to a trilater equila∣ter pyramis inscribed in it.
¶ The 30. Proposition. To proue that a trilater equilater Pyramis, is noncuple to a cube inscribed in it.
¶ The 31. Proposition. An Octohedron hath to an Icosohedron inscribed in it, that proportion, which two bases of the Octohedron haue to fiue bases of the Icosahedron.
¶ The 32. Proposition. The proportiō of the solide of an Icosahedron to the solide of a Dodecahe∣dron inscribed in it, consisteth of the proportion of the side of the Icosahe∣dron to the side of the Cube contayned in the same sphere, and of the pro∣portion tripled of the diameter to the line which conpleth the centers of the opposite bases of the Icosahedron.
¶ The 33. Proposition. The solide of a Dodecahedron excedeth the solide of a Cube inscribed in it, by a parallelipipedon, whose base wanteth of the base of the Cube by a third part of the lesse segment, and whose altitude wanteth of the altitude of the Cube, by the lesse segment of the lesse segment, of halfe the side of the Cube.
¶A Corollary.
¶The 34. Proposition. The proportion of the solide of a Dodecahedron to the solide of an Icosa∣hedron inscribed in it, consisteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron, and of the proportion of the side of the Cube to the side of the Icosahedron inscri∣bed in one and the selfe same Sphere.
The 35. Proposition. The solide of a Dodecahedron containeth of a Pyramis circumscribed a∣bout it two ninth partes, taking away a third part of one ninth part of the lesse segment (of a line diuided by an extreme and meane proportion) and moreouer the lesse segment of the lesse segment of halfe the residue.
¶The 36. Proposition. An Octohedron exceedeth an Icosahedron inscribed in it, by a parallelipi∣pedon set vpon the square of the side of the Icosahedron, and hauing to his altitude the line which is the greater segment of halfe the semidiame∣ter of the Octohedron.
¶ A Corollary.
The 37. Proposition. If in a triangle hauing to his base a rational line set, the sides be commen∣surable in power to the base, and from the toppe be drawn to the base a per∣pendicular line cutting the base: The sections of the base shall be commen∣surable in length to the whole base, and the perpendicular shall be commen∣surable in power to the said whole base.
¶ A Corollary. 1.
A Corollary. 2.
A briefe treatise, added by Flussas, of mixt and composed regular solides.
¶ First Definition.
¶ Second Definition.
¶The first Probleme. To describe an equilater and equiangle exoctohedron, and to contayne it in a sphere geuen: and to proue that the diameter of the sphere is double to the side of the sayd exoctohedron.
¶The 2. Probleme. To describe an equilater & equiangle Icosidodecahedron, & to cōprehend it in a sphere geuen: and to proue that the diameter being diuided by an ex∣treame and meane proportion, maketh the greater segment double to the side of the Icosidodecahedron.
¶ An aduertisment of Flussas.
¶Of the inscriptions and circumscriptions of an Icosidodechedron.
Of the nature of a trilater and equilater Pyramis.
Of the nature of an Octohedron.
Of the nature of a Cube.
Of the nature of an Icosahedron.
Of the nature of a Dodecahedron.
Faultes escaped.
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