The Way To Geometry电子书

The Way To Geometry

About Author:

The Authors Preface.

THE FIRST BOOKE OF Peter Ramus's Geometry, Which is of a Magnitude.

1. Geometry is the Art of measuring well.

2. The thing proposed to bee measured is a Magnitude.

3. A Magnitude is a continuall quantity.

4. That is continuum, continuall, whose parts are contained or held together by some common bound.

5. A bound is the outmost of a Magnitude.

6. A Magnitude is both infinitely made, and continued, and cut or divided by those things wherewith it is bounded.

7. A point is an undivisible signe in a magnitude.

8. Magnitudes commensurable, are those which one and the same measure doth measure: Contrariwise, Magnitudes incommensurable are those, which the same measure cannot measure. 1, 2. d. X.

9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given. Contrariwise they are irrationalls. 5. d. X.

10. Congruall or agreeable magnitudes are those, whose parts beeing applyed or laid one upon another doe fill an equall place.

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

The second Booke of Geometry. Of a Line.

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

2. A Line is a Magnitude onely long.

3. The bound of a line is a point.

4. A Line is either Right or Crooked.

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

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

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

8. Touching is but in one point onely. è 13. p 3.

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

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

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

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

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

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

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

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

The third Booke of Geometry. Of an Angle.

1. A lineate is a Magnitude more then long.

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

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

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

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

6. Angels congruall in shankes are equall.

7. If an angle being equicrurall to an other angle, be also equall to it in base, it is equall: And if an angle having equall shankes with another, bee equall to it in the angle, it is also equall to it in the base. è 8. & 4. p j.

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

9. If an angle equicrurall to another angle, be greater then it in base, it is greater: And if it be greater, it is greater in base: è 52 & 24. p j.

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

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

12. An angle is either right or oblique.

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

14. All straight-shanked right angles are equall.

15. An oblique angle is either Obtuse or Acute.

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

The fourth Booke, which is of a Figure.

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

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

3. The perimeter is the compasse of the figure.

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

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

6. The diameters in the same figure are infinite.

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

8. It is in the meeting of the diameters.

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

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

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

12. A rationall figure is that which is comprehended of a base and height rationall betweene themselves.

13. The number of a rationall figure, is called a Figurate number: And the numbers of which it is made, the Sides of the figurate.

14. Isoperimetrall figures, are figures of equall perimeter.

15. Of isoperimetralls homogenealls that which is most ordinate, is greatest: Of ordinate isoperimetralls heterogenealls, that is greatest, which hath most bounds.

16. If prime figures be of equall height, they are in reason one unto another, as their bases are: And contrariwise.

17. If prime figures of equall heighth have also equall bases, they are equall.

18. If prime figures be reciprocall in base and height, they are equall: And contrariwise.

19. Like figures are equiangled figures, and proportionall in the shankes of the equall angles.

20. Like figures have answerable bounds subtended against their equall angles: and equall if they themselves be equall.

21. Like figures are situate alike, when the proportionall bounds doe answer one another in like situation.

22. Those figures that are like unto the same, are like betweene themselves.

23. If unto the parts of a figure given, like parts and alike situate, be placed upon a bound given, a like figure and likely situate unto the figure given, shall bee made accordingly.

24. Like figures have a reason of their homologallor correspondent sides equally manifold unto their dimensions: and a meane proportionall lesse by one.

25. If right lines be continually proportionall, more by one then are the dimensions of like figures likelily situate unto the first and second, it shall be as the first right line is unto the last, so the first figure shall be unto the second: And contrariwise.

26. If foure right lines bee proportionall betweene themselves: Like figures likelily situate upon them, shall be also proportionall betweene themselves: And contrariwise, out of the 22. p vj. and 37. p xj.

27. Figures filling a place, are those which being any way set about the same point, doe leave no voide roome.

28. A round figure is that, all whose raies are equall.

29. The diameters of a roundle are cut in two by equall raies.

30. Rounds of equall diameters are equall. Out of the 1. d. iij.

The fifth Booke, of Ramus his Geometry, which is of Lines and Angles in a plaine Surface.

1. A lineate is either a Surface or a Body.

2. A Surface is a lineate only broade. 5. d j.

3. The bound of a surface is a line. 6. d j.

4. A Surface is either Plaine or Bowed.

5. A plaine surface is a surface, which lyeth equally betweene his bounds, out of the 7. d j.

6. From a point unto a point we may, in a plaine surface, draw a right line, 1 and 2. post. j.

7. To set at a point assigned a Right line equall to another right line given: And from a greater, to cut off a part equall to a lesser. 2. and 3. p j.

8. One right line, or two cutting one another, are in the same plaine, out of the 1. and 2. p xj.

10. The raies of the same, or of an equall periphery, are equall.

11. If two equall peripheries, from the ends of equall shankes of an assigned rectilineall angle, doe meete before it, a right line drawne from the meeting of them unto the toppe or point of the angle, shall cut it into two equall parts. 9. p j.

12. If two equall peripheries from the ends of a right line given, doe meete on each side of the same, a right line drawne from those meetings, shall divide the right line given into two equall parts. 10. p j.

13. If a right line doe stand perpendicular upon another right line, it maketh on each side right angles: And contrary wise.

14. If a right line do stand upon a right line, it maketh the angles on each side equall to two right angles: and contrariwise out of the 13. and 14. p j.

15. If two right lines doe cut one another, they doe make the angles at the top equall and all equall to foure right angles. 15. p j.

16. If two right lines cut with one right line, doe make the inner angles on the same side greater then two right angles, those on the other side against them shall be lesser then two right angles.

17. If from a point assigned of an infinite right line given, two equall parts be on each side cut off: and then from the points of those sections two equall circles doe meete, a right line drawne from their meeting unto the point assigned, shall bee perpendicular unto the line given. 11. p j.

18. If a part of an infinite right line, bee by a periphery for a point given without, cut off a right line from the said point, cutting in two the said part, shall bee perpendicular upon the line given. 12. p j.

19. If two right lines drawne at length in the same plaine doe never meete, they are parallells. è 35. d j.

20. If an infinite right line doe cut one of the infinite right parallell lines, it shall also cut the other.

21. If right lines cut with a right line be pararellells, they doe make the inner angles on the same side equall to two right angles: And also the alterne angles equall betweene themselves: And the outter, to the inner opposite to it: And contrariwise, 29, 28, 27. p 1.

22. If right lines knit together with a right line, doe make the inner angles on the same side lesser than two right Angles, they being on that side drawne out at length, will meete.

23. A right line knitting together parallell right lines, is in the same plaine with them. 7 p xj.

24. If a right line from a point given doe with a right line given make an angle, the other shanke of the angle equalled and alterne to the angle made, shall be parallell unto the assigned right line. 31 p j.

25. The angles of shanks alternly parallell, are equall. Or Thus, The angles whose alternate feete are parallells, are equall. H.

26. If parallels doe bound parallels, the opposite lines are equall è 34 p. j. Or thus: If parallels doe inclose parallels, the opposite parallels are equall. H.

27. If right lines doe joyntly bound on the same side equall and parallell lines, they are also equall and parallell.

28. If right lines be cut joyntly by many parallell right lines, the segments betweene those lines shall bee proportionall one to another, out of the 2 p vj and 17 p xj.

29. If a right line making an angle with another right line, be cut according to any reason [or proportion] assigned, parallels drawne from the ends of the segments, unto the end of the sayd right line given and unto some contingent point in the same, shall cut the line given according to the reason given.

30. If two right lines given, making an angle, be continued, the first equally to the second, the second infinitly, parallels drawne from the ends of the first continuation, unto the beginning of the second, and some contingent point in the same, shall intercept betweene them the third proportionall. 11. p vj.

31. If of three right lines given, the first and the third making an angle be continued, the first equally to the second, and the third infinitly; parallels drawne from the ends of the first continuation, unto the beginning of the second, and some contingent point, the same shall intercept betweene them the fourth proportionall. 12. p vj.

32 If two right lines cutting one another, be againe cut with many parallels, the parallels are proportionall unto their next segments.

33. If two right lines given be continued into one, a perpendicular from the point of continuation unto the angle of the squire, including the continued line with the continuation, is the meane proportionall betweene the two right lines given.

34 If two assigned right lines joyned together by their ends rightanglewise, be continued vertically; a square falling with one of his shankes, and another to it parallell and moveable upon the ends of the assigned, with the angles upon the continued lines, shall cut betweene them from the continued two meanes continually proportionall to the assigned.

35. If of foure right lines, two doe make an angle, the other reflected or turned backe upon themselves, from the ends of these, doe cut the former; the reason of the one unto his owne segment, or of the segments betweene themselves, is made of the reason of the so joyntly bounded, that the first of the makers be joyntly bounded with the beginning of the antecedent made; the second of this consequent joyntly bounded with the end; doe end in the end of the consequent made.

Of Geometry, the sixt Booke, of a Triangle.

1. Like plaines have a double reason of their homologall sides, and one proportionall meane, out of 20 p vj. and xj. and 18. p viij.

2. A plaine surface is either rectilineall or obliquelineall, [or rightlined, or crookedlined. H.]

3. A rectilineall surface, is that which is comprehended of right lines.

4. A rightilineall doth make all his angles equall to right angles; the inner ones generally to paires from two forward: the outter always to foure.

5. A rectilineall is either a Triangle or a Triangulate.

6. A triangle is a rectilineall figure comprehended of three rightlines. 21. d j.

7. A triangle is the prime figure of rectilineals.

8. If an infinite right line doe cut the angle of a triangle, it doth also cut the base of the same: Vitell. 29. t j.

9. Any two sides of a triangle are greater than the other.

10. If of three right lines given, any two of them be greater than the other, and peripheries described upon the ends of the one, at the distances of the other two, shall meete, the rayes from that meeting unto the said ends, shall make a triangle of the lines given.

11. If two equall peripheries, from the ends of a right line given, and at his distance, doe meete, lines drawne from the meeting, unto the said ends, shall make an equilater triangle upon the line given. 1 p. j.

12. If a right line in a triangle be parallell to the base, it doth cut the shankes proportionally: And contrariwise. 2 p vj.

13. The three angles of a triangle, are equall to two right angles. 32. p j.

14. Any two angles of a triangle are lesse than two right angles.

15. The one side of any triangle being continued or drawne out, the outter angle shall be equal to the two inner opposite angles.

16. The said outter angle is greater than either of the inner opposite angles. 16. p j.

17. If a triangle be equicrurall, the angles at the base are equall: and contrariwise, 5. and 6. p. j.

18. If the equall shankes of a triangle be continued or drawne out, the angles under the base shall be equall betweene themselves.

19. If a triangle be an equilater, it is also an equiangle: And contrariwise.

20. The angle of an equilater triangle doth countervaile two third parts of a right angle. Regio. 23. p j.

21. Sixe equilater triangles doe fill a place.

22. The greatest side of a triangle subtendeth the greatest angle; and the greatest angle is subtended of the greatest side. 19. and 18. p j.

23. If a right line in a triangle, doe cut the angle in two equall parts, it shall cut the base according to the reason of the shankes; and contrariwise. 3. p vj.

Of Geometry, the seventh Booke, Of the comparison of Triangles.

1. Equilater triangles are equiangles. 8. p. j.

2. If two triangles be equall in angles, either the two equicrurals, or two of equall either shanke, or base of two angles, they are equilaters, 4. and 26. p j.

3. Triangles are equall in their three angles.

4. If two angles of two triangles given be equall, the other also are equall.

5. If a right triangle equicrurall to a triangle be greater in base, it is greater in angle: And contrariwise. 25. and 24. p j.

6. If a triangle placed upon the same base, with another triangle, be lesser in the inner shankes, it is greater in the angle of the shankes.

7. Triangles of equall heighth, are one to another as their bases are one to another.

8. Upon an equall base, they are equall.

9. If a right line drawne from the toppe of a triangle, doe cut the base into two equall parts, it doth also cut the triangle into two equall parts: and it is the diameter of the triangle.

10. If a right line be drawne from the toppe of a triangle, unto a point given in the base (so it be not in the middest of it) and a parallell be drawne from the middest of the base unto the side, a right line drawne from the toppe of the sayd parallell unto the sayd point, shall cut the triangle into two equall parts.

11. If equiangled triangles be reciprocall in the shankes of the equall angle, they are equall: And contrariwise. 15. p. vj. Or thus, as the learned Mr. Brigges hath conceived it: If two triangles, having one angle, are reciprocall, &c.

12. If two triangles be equiangles, they are proportionall in shankes: And contrariwise: 4 and 5. p. vj.

13. If a right line in a triangle be parallell to the base, it doth cut off from it a triangle equiangle to the whole, but lesse in base.

14. If two trangles be proportionall in the shankes of the equall angle, they are equiangles: 6 p vj.

15. If triangles proportionall in shankes, and alternly parallell, doe make an angle betweene them, their bases are but one right line continued. 32 p. vj.

16. If two triangles have one angle equall, another proportionall in shankes, the third homogeneall, they are equiangles. 7. p. vj.

Of Geometry the eight Booke, of the diverse kindes of Triangles.

1. A triangle is either right angled, or obliquangled.

A right 2. A right angled triangle is that which hath one right angle: An obliquangled is that which hath none. 27. d j.

3. If two perpendicular lines be knit together, they shall make a right angled triangle.

4. If the angle of a triangle at the base, be a right angle, a perpendicular from the toppe shall be the other shanke: [and contrariwise Schon.]

5. If a right angled triangle be equicrurall, each of the angles at the base is the halfe of a right angle: And contrariwise.

6. If one angle of a triangle be equall to the other two, it is a right angle [And contrariwise Schon.]

7. If a right line from the toppe of a triangle cutting the base into two equall parts be equall to the bisegment, or halfe of the base, the angle at the toppe is a right angle: [And contrariwise Schon.]

8. A perpendicular in a triangle from the right angle to the base, doth cut it into two triangles, like unto the whole and betweene themselves, 8. p vj. [And contrariwise Schon.]

9. The perpendicular is the meane proportionall betweene the segments or portions of the base.

10. Either of the shankes is proportionall betweene the base, and the segment of the base next adjoyning.

11. If the base of a triangle doe subtend a right-angle, the rectilineall fitted to it, shall be equall to the like rectilinealls in like manner fitted to the shankes thereof: And contrariwise, out of the 31. p. vj.

12. An obliquangled triangle is either Obtusangled or Acutangled.

13. An obtusangle is that triangle which hath one blunt corner. 28.d i.

14. If the obtuse or blunt angle be at the base of the triangle given, a perpendicular drawne from the toppe of the triangle, shall fall without the figure: And contrarywise.

15. If one angle of a triangle be greater than both the other two, it is an obtuse angle: And contrariwise.

16. If a right line drawne from the toppe of the triangle cutting the base into two equall parts, be lesse than one of those halfes, the angle at the toppe is a blunt-angle. And contrariwise.

17. An acutangled triangle is that which hath all the angles acute. 29 d j.

18. A perpendicular drawne from the top falleth within the figure: And contrariwise.

19. If any one angle of triangle be lesse then the other two, it is acute: And contrariwise.

20. If a right line drawne from the toppe of the triangle; cutting the base into two equall parts, be greater than either of those portions, the angle at the toppe is an acute angle: And contrariwise.

The ninth Booke, of P. Ramus Geometry, which intreateth of the measuring of right lines by like right-angled triangles.

1. For the measuring of right lines; we will use the Iacobs staffe, which is a squire of unequall shankes.

2. The shankes of the staffe are the Index and the Transome.

3. The Index is the double and one tenth part of the transome.

4. The Transome is that which rideth upon the Index, and is to be slid higher or lower at pleasure.

5. If the sight doe passe from the beginning of one shanke, it passeth by the end of the other: And the one shanke is perpendicular unto the magnitude to be measured, the other parallell.

6. Length and Altitude have a threefold measure; The first and second kinde of measure require but one distance, and that by granting a dimension of one of them, for the third proportionall: The third two distances, and such onely is the dimension of Latitude.

7. If the sight be from the beginning of the Index right or plumbe unto the length, and unto the farther end of the same, as the segment of the Index is, unto the segment of the transome, so is the heighth of the measurer unto the length.

8. If the sight be from the beginning of the index parallell to the length to be measured, as the segment of the transome is, unto the segment of the index, so shall the heighth given be to the length.

9. If the sight be from the beginning of the transverie parallell to the length to be measured, as in the index the difference of the greater segment is unto the lesser; so is the difference of the second station unto the length.

10. If the sight be from the beginning of the transome perpendicular unto the height to be measured, as the segment of the transome, is unto the segment of the Index, so shall the length given be to the height.

11. If the sight be from the beginning of the Index parallell to the height, as the segment of the transome is, unto the segment of the index, so shall the length given be, unto the height sought.

12. If the sight be from the beginning of the Index perpendicular to the heighth to be measured, as the segment of the Index is unto the segment of the Transome, so shall the length given be to the heighth.

13. If the sight be from the beginning of the Index (perpendicular to the magnitude to be measured) by the names of the transome, unto the ends of some known part of the height, as the distance of the Names is, unto the rest of the transome above them, so shall the known part be unto the part sought.

14 If the sight be from the beginning of the Index perpendicular to the heighth, as in the Index the difference of the segment, is unto the difference of the distance or station; so is the segment of the transome unto the heighth.

15 Out of the Geodesy of heights, the difference of two heights is manifest.

16 If the sight be first from the toppe, then againe from the base or middle place of the greater, by the vanes of the transome unto the toppe of the lesser heighth; as the said parts of the yards are unto the part of the first yard; so the heighth betweene the stations shall be unto his excesse above the heighth desired.

17 If the sight be from the beginning of the yard being right or perpendicular, by the vanes of the transome, unto the ends of the breadth; as in the yard the difference of the segment is unto the differēce of the distance, so is the distance of the vanes unto the breadth.

The tenth Booke of Geometry, of a Triangulate and Parallelogramme.

1. A triangulate is a rectilineall figure compounded of triangles.

2. The sides of a triangulate are two more than are the triangles of which it is made.

3. Homgeneall Triangulates are cut into an equall number of triangles, è 20 p vj.

4. Like triangulates are cut into triangles alike one to another and homologall to the whole è 20 p vj.

5. A triangulate is a Quadrangle or a Multangle.

6. A Quadrangle is that which is comprehended of foure right lines. 22 d j.

7. A quadrangle is a Parallelogramme, or a Trapezium.

8. A Parallelogramme is a quadrangle whose opposite sides are parallell.

9. If right lines on one and the same side, doe joyntly bound equall and parallall lines, they shall make a parallelogramme.

10 A parallelogramme is equall both in his opposite sides, and angles, and segments cut by the diameter.

11. The Diameter of a parallelogramme is cut into two by equall raies.

12 A parallelogramme is the double of a triangle of a trinangle of equall base and heighth, 41. p j.

13 A parallelogramme is equall to a triangle of equall heighth and double base unto it: è 42. p j.

14 To a triangle given, in a rectilineall angle given, make an equall parallelogramme.

15 A parallelogramme doth consist both of two diagonals, and complements, and gnomons.

16 The Diagonall is a particular parallelogramme having both an angle and diagonall diameter common with the whole parallelogramme.

17 The Diagonall is like, and alike situate to the whole parallelogramme: è 24. p vj.

18. If the particular parallelogramme have one and the same angle with the whole, be like and alike situate unto it, it is the Diagonall. 26 p vj.

19. The Complement is a particular parallelogramme, comprehended of the conterminall sides of the diagonals.

20. The complements are equall. 43 p j.

21. If one of the Complements be made equall to a triangle given, in a right-lined angle given, the other made upon a right line given shall be in like manner equall to the same triangle. 44 p j.

22 If parallelogrammes be continually made equall to all the triangles of an assigned triangulate, in a right lined angle given, the whole parallelogramme shall in like manner be equall to the whole triangulate. 45 p j.

23. A Parallelogramme is equall to his diagonals and complements.

24. The Gnomon is any one of the Diagonall with the two complements.

25. Parallelogrames of equall height are one to another as their bases are. 1 p vj.

26 Parallelogrammes of equall height upon equall bases are equall. 35. 36 pj.

27 If equiangle parallelogrammes be reciprocall in the shankes of the equall angle, they are equall: And contrariwise. 15 p vj.

28 If foure right lines be proportionall, the parallelogramme made of the two middle ones, is equall to the equiangled parallelogramme made of the first and last: And contrariwise, e 16 p vj.

29 If three right lines be proportionall, the parallelogramme of the middle one is equall to the equiangled parallelogramme of the extremes: And contrariwise.

Of Geometry, the eleventh Booke, of a Right angle.

1. A Parallelogramme is a Right angle or an Obliquangle.

2. A Right angle is a parallelogramme that hath all his angles right angles.

3 A rightangle is comprehended of two right lines comprehending the right angle 1. d ij.

4 Foure right angles doe fill a place.

5 If the diameter doe cut the side of a right angle into two aquall parts, it doth cut it perpendicularly: And contrariwise.

6 If an inscribed right line doe perpendicularly cut the side of the right angle into two equall parts, it is the diameter.

7 A right angle is equall to the rightangles made of one of his sides and the segments of the other.

8 If foure right lines be proportionall, the rectangle of the two middle ones, is equall to the rectangle of the two extremes. 16. p vj.

9 The figurate of a rationall rectangle is called a rectinall plaine. 16. d vij.

Of Geometry the twelfth Booke, Of a Quadrate.

1 A Rectangle is a Quadrate or an Oblong.

2 A Quadrate is a rectangle equilater 30. d j.

3 The sides of equall quadrates, are equall.

4 The power of a right line is a quadrate.

5 If two conterminall perpendicular equall right lines be closed with parallells, they shall make a quadrate. 46. p. j.

6 The plaine of a quadrate is an equilater plaine.

7 A quadrate is made of a number multiplied by it self.

8 If three right lines be proportionall, the quadrate of the middle one, shall be equall to the rectangle of the extremes: And contrariwise: 17. p vj. and 20. p vij.

9 If the base of a triangle doe subtend a right angle, the powre of it is as much as of both the shankes: And contrariwise 47, 48. p j.

10 If the quadrate of an odde number, given for the first shanke, be made lesse by an unity; the halfe of the remainder shall be the other shanke; increased by an unity it shall be the base.

11 If the halfe of an even number given for the first shanke be squared, the square number diminished by an unity shall be the other shanke, and increased by an unitie it shall be the base.

12. The power of the diagony is twise asmuch, as is the power of the side, and it is unto it also incommensurable.

13. If the base of a right angled triangle be cut by a perpendicular from the right angle in a doubled reason, the power of it shall be halfe as much more, as is the power of the greater shanke: But thrise so much as is the power of the lesser. If in a quadrupled reason, it shall be foure times and one fourth so much as is the greater: But five times so much as is the lesser, At the 13, 15, 16 p xiij.

14 If a right line be cut into how many parts so ever, the power of it is manifold unto the power of segment, denominated of the square of the number of the section.

15. If a right line be cut into two segments, the quadrate of the whole is equall to the quadrats of the segments, and a double rectanguled figure, made of them both. 4 p ij.

17 If the side found be doubled, and to the double a unity be added, the whole shall be the gnomon of the next greater quadrate.

18 If from the halfe of the summe of the sides, the sides be severally subducted, the side of the quadrate continually made of the halfe, and the remaines shall be the content of the triangle.

19 If the base of a triangle doe subtend an obtuse angle, the power of it is more than the power of the shankes, by a double right angle of the one, and of the continuation from the said obtusangle unto the perpendicular of the toppe. 12. p ij.

Of Geometry, the thirteenth Booke, Of an Oblong.

1 An Oblong is a rectangle of inequall sides, 31. d j.

2 An oblong made of an whole line given, and of one segment of the same, is equall to a rectangle made of both the segments, and the square of the said segment. 3. p ij.

3 Oblongs made of the whole line given, and of the segments, are equall to the quadrate of the whole 2 p ij.

4 Two Oblongs made of the whole line given, and of the one segment, with the third quadrate of the other segment, are equall to the quadrates of the whole, and of the said segment. 7 p ij.

5. The base of an acute triangle is of lesse power than the shankes are, by a double oblong made of one of the shankes, and the one segment of the same, from the said angle, unto the perpendicular of the toppe. 13 p. ij.

6. If the square of the base of an acute angle be taken out of the squares of the shankes, the quotient of the halfe of the remaine, divided by the shanke, shall be the segment of the dividing shanke from the said angle unto the perpendicular of the toppe.

7. If a right line be cut into two equall parts, and otherwise; the oblong of the unequall segments, with the quadrate of the segment betweene them, is equall to the quadrate of the bisegment. 5 p ij.

8. If a right line be cut into equall parts; and continued; the oblong made of the continued and the continuation, with the quadrate of the bisegment or halfe, is equall to the quadrate of the line compounded of the bisegment and continuation. 6 p ij.

9. If the Mesographus, touching the angle opposite to the angle made of the two lines given, doe cut the said two lines given, comprehending a right angled parallelogramme, and infinitely continued, equally distant from the center, the intersegments shall be the meanes continually proportionally, betweene and two lines given.

The fourteenth Booke, of P. Ramus Geometry: Of a right line proportionally cut: And of other Quadrangles, and Multangels.

1. A right line is cut according to a meane and extreame rate, when as the whole shall be to the greater segment; so the greater shall be unto the lesser. 3 d vj.

2. If a right line cut proportionally be rationall unto the measure given, the segments are unto the same, and betweene themselves irrationall è 6 p xiij.

3. If a quadrate be made of a right line given, the difference of the right line from the middest of the conterminall side of the said quadrate made, above the same halfe, shall be the greater segment of the line given proportionally cut: 11 p ij.

4. If a right line cut proportionally, be continued with the greater segment, the whole shall be cut proportionally, and the greater segment shall be the line given. 5 p xiij.

5. The greater segment continued to the halfe of the whole, is of power quintuple unto the said halfe, that is, five times so great as it is: and if the power of a right line be quintuple to his segment, the remainder made the double of the former is cut proportionally, and the greater segment, is the same remainder. 1. and 2. p xiij.

6 The lesser segment continued to the halfe of the greater, is of power quintuple to the same halfe è 3 p xiij.

7 The whole line and the lesser segment are in power treble unto the greater. è 4 p xiij.

8 An obliquangled parallelogramme is either a Rhombus, or a Rhomboides.

9 A Rhombus is an obliquangled equilater parallelogramme 32 d j.

10 A Rhomboides is an obliquangled parallelogramme not equilater 33. d j.

11 A Trapezium is a quadrangle not parallelogramme. 34. d j.

12 A multangle is a figure that is comprehended of more than foure right lines. 23. d j.

13 Multangled triangulates doe take their measure also from their triangles.

14 If an equilater quinquangle have three sides equall, it is equiangled. 7 p 13.

The fifteenth Booke of Geometry, Of the Lines in a Circle.

1. A Circle is a round plaine. è 15 d j.

2 Circles are as the quadrates or squares made of their diameters 2 p. xij.

3. The Diameters are, as their peripheries Pappus, 5 l. xj, and 26th. 18.

4. Circular Geometry is either in Lines, or in the segments of a Circle.

5. If a right line be bounded by two points in the periphery, it shall fall within the Circle. 2 p iij.

6. If from the end of the diameter, and with a ray of it equal to the right line given, a periphery be described, a right line drawne from the said end, unto the meeting of the peripheries, shall be inscribed into the circle, equall to the right line given. 1 p iiij.

7. If an inscript do cut into two equall parts, another inscript perpendicularly, it is the diamiter of the Circle, and the middest of it is the center. 1 p iij.

8. If two right lines doe perpendicularly halfe two inscripts, the meeting of these two bisecants shall be the Center of the circle è 25 p iij.

9. Draw a periphery by three points, which doe not fall in a right line.

10. If a diameter doe halfe an inscript, that is, not a diameter, it doth cut it perpendicularly: And contrariwise: 3 p iij.

11. If inscripts which are not diameters doe cut one another, the segments shall be unequall. 4 p iij.

12 If two inscripts doe cut one another, the rectangle of the segments of the one is equall to the rectangle of the segments of the other. 35 p iij.

13 Inscripts are equall distant from the center, unto which the perpendiculars from the center are equall 4 d iij.

14. If inscripts be equall, they be equally distant from the center: And contrariwise. 13 p iij.

15 Of unequall inscripts the diameter is the greatest: And that which is next to the diameter, is greater than that which is farther off from it: That which is farthest off from it, is the least: And that which is next to the least, is lesser than that which is farther off: And those two onely which are on each side of the diameter are equall è 15 e iij.

16 Of right lines drawne from a point in the diameter which is not the center unto the periphery, that which passeth by the center is the greatest: And that which is nearer to the greatest, is greater than that which is farther off: The other part of the greatest is the left. And that which is nearest to the least, is lesser than that which is farther off: And two on each side of the greater or least are only equall. 7 p iij.

17 If a point in a circle be the bound of three equall right lines determined in the periphery, it is the center of the circle. 9 p iij.

18 Of right lines drawne from a point assigned without the periphery, unto the concavity or hollow of the same, that which is by the center is the greatest; And that next to the greatest, is greater than that which is farther off: But of those which fall upon the convexitie of the circumference, the segment of the greatest is least. And that which is next unto the least is lesser than that is farther off: And two on each side of the greatest or least are onely equall. 8 p iij.

19 If a right line be perpendicular unto the end of the diameter, it doth touch the periphery: And contrariwise è 16 p iij.

20 If a right line doe passe by the center and touch-point, it is perpendicular to the tangent or touch-line. 18 p iij.

21 If a right line be perpendicular unto the tangent, it doth passe by the center and touch-point. 19. p iij.

22 The touch-point is that, into which the perpendicular from the center doth fall upon the touch line.

23 A tangent on the same side is onely one.

24 A touch-angle is lesser than any rectilineall acute angle, è 16 p ij.

25 All touch-angles in equall peripheries are equall.

26 If from a ray out of the center of a periphery given, a periphery be described unto a point assigned without, and from the meeting of the assigned and the ray, a perpendicular falling upon the said ray unto the now described periphery, be tied by a right line with the said center, a right line drawne from the point given unto the meeting of the periphery given, and the knitting line shall touch the assigned periphery 17 p iij.

27 If of two right lines, from an assigned point without, the first doe cut a periphery unto the concave, the other do touch the same; the oblong of the secant, and of the outter segment of the secant, is equall to the quadrate of the tangent: and if such a like oblong be equall to the quadrate of the other, that same other doth touch the periphery: 36, and 37. p iij.

28. All tangents falling from the same point are equall.

29. The oblongs made of any secant from the same point, and of the outter segment of the secant are equall betweene themselves. Camp. 36 p iij.

30. To two right lines given one may so continue or joyne the third, that the oblong of the continued and the continuation may be equall to the quadrate remaining. Vitellio 127 p j.

31. If peripheries doe either cut or touch one another, they are eccentrickes: And they doe cut one another in two points onely, and these by the touch point doe continue their diameters, 5. 6. 10, 11, 12 p iij.

32. If inscripts be equall, they doe cut equall peripheries: And contrariwise, 28, 29 p iij.

The sixteenth Booke of Geometry, Of the Segments of a Circle.

1. A Segment of a Circle is that which is comprehended outterly of a periphery, and innerly of a right line.

2. A segment of a Circle is either a sectour, or a section.

3. A Sectour is a segment innerly comprehended of two right lines, making an angle in the center; which is called an angle in the center: As the periphery is, the base of the sectour, 9 d iij.

4. An angle in the Periphery is an angle comprehended of two right lines inscribed, and jointly bounded or meeting in the periphery. 8 d iij.

5. The angle in the center, is double to the angle of the periphery standing upon the same base, 20 p iij.

6. If the angle in the periphery be equall to the angle in the center, it is double to it in base. And contrariwise.

7. The angles in the center or periphery of equall circles, are as the Peripheries are upon which they doe insist: And contrariwise. è 33 p vj, and 26, 27 p iij.

8. As the sectour is unto the sectour, so is the angle unto the angle: And Contrariwise.

9. A section is a segment of a circle within cōprehended of one right line, which is termed the base of the section.

10. A section is made up by finding of the center.

11 The periphery of a section is divided into two equall parts by a perpendicular dividing the base into two equall parts. 20. p iij.

12 An angle in a section is an angle comprehended of two right lines joyntly bounded in the base and in the periphery joyntly bounded 7 d iij.

13 The angles in the same section are equall. 21. p iij.

14 The angles in opposite sections are equall to two right angles. 22. p iij.

15 If sections doe receive [or containe] equall angles, they are alike è 10. d iij.

16 If like sections be upon an equall base, they are equall: and contrariwise. 23, 24. p iij.

17 Angle of a section is that which is comprehended of the bounds of a section.

18 A section is either a semicircle: or that which is unequall to a semicircle.

19 A semicircle is the half section of a circle.

20 A semicircle is comprehended of a periphery and the diameter 18 d j.

21 The angle in a semicircle is a right angle: The angle of a semicircle is lesser than a rectilineall right angle: But greater than any acute angle: The angle in a greater section is lesser than a right angle: Of a greater, it is a greater. In a lesser it is greater: Of a lesser, it is lesser, è 31. and 16. p iij.

22 If two right lines jointly bounded with the diameter of a circle, be jointly bounded in the periphery, they doe make a right angle.

23 If an infinite right line be cut of a periphery of an externall center, in a point assigned and contingent, and the diameter be drawne from the contingent point, a right line from the point assigned knitting it with the diameter, shall be perpendicular unto the infinite line given.

24 If a right line from a point given, making an acute angle with an infinite line, be made the diameter of a periphery cutting the infinite, a right line from the point assigned knitting the segment, shall be perpendicular upon the infinite line.

25 If of two right lines, the greater be made the diameter of a circle, and the lesser jointly bounded with the greater and inscribed, be knit together, the power of the greater shall be more than the power of the lesser by the quadrate of that which knitteth them both together. ad 13 p. x.

26 If a right line continued or continually made of two right lines given, be made the diameter of a circle, the perpendicular from the point of their continuation unto the periphery, shall be the meane proportionall betweene the two lines given. 13 p vj.

27 The angles in opposite sections are equall in the alterne angles made of the secant and touch line. 32. p iij.

28 If at the end of a right line given a right lined angle be made equall to an angle given, and from the toppe of the angle now made, a perpendicular unto the other side do meete with a perpendicular drawn from the middest of the line given, the meeting shall be the center of the circle described by the equalled angle, in whose opposite section the angle upon the line given shall be made equall to the assigned è 33 p iij.

29 If the angle of the secant and touch line be equall to an assigned rectilineall angle, the angle in the opposite section shall likewise be equall to the same. 34. p iij.

Of Geometry the seventeenth Booke, Of the Adscription of a Circle and Triangle.

1. If rectilineall ascribed unto a circle be an equilater, it is equiangle.

2. It is equall to a triangle of equall base to the perimeter, but of heighth to the perpendicular from the center to the side.

3. Like rectilinealls inscribed into circles, are one to another as the quadrates of their diameters, 1 p. xij.

4. If it be as the diameter of the circle is unto the side of rectilineall inscribed, so the diameter of the second circle be unto the side of the second rectilineall inscribed, and the severall triangles of the inscripts be alike and likely situate, the rectilinealls inscribed shall be alike and likely situate.

5. If two right lines doe cut into two equall parts two angles of an assigned rectilineall, the circle of the ray from their meeting perpendicular unto the side, shall be inscribed unto the assigned rectilineall. 4 and 8. p. iiij.

6. If two right lines do right anglewise cut into two equall parts two sides of an assigned rectilineall, the circle of the ray from their meeting unto the angle, shall be circumscribed unto the assigned rectilineall. 5 p iiij.

7. If two inscripts, from the touch point of a right line and a periphery, doe make two angles on each side equall to two angles of the triangle assigned be knit together, they shall inscribe a triangle into the circle given, equiangular to the triangle given è 2 p iiij.

8 If two angles in the center of a circle given, be equall at a common ray to the outter angles of a triangle given, right lines touching a periphery in the shankes of the angles, shall circumscribe a triangle about the circle given like to the triangle given. 3 p iiij.

9. If a triangle be a rectangle, an obtusangle, an acute angle, the center of the circumscribed triangle is in the side, out of the sides, and within the sides: And contrariwise. 5 e iiij.

Of Geometry, the eighteenth Booke, Of the adscription of a Triangulate.

1. If right lines doe touch a periphery in the angles of the inscript ordinate triangulate, they shall onto a circle cirumscribe a triangulate homogeneall to the inscribed triangulate.

2. If the diameters doe cut one another right-angle-wise, a right line subtended or drawne against the right angle, shall be the side of the quadrate. è 6 p iiij.

3. A quadrate inscribed is the halfe of that which is circumscribed.

4. It is greater than the halfe of the circumscribed Circle.

5. If a right line be cut proportionally, the base of that triangle whose shankes shall be equall to the whole line cut, and the base to the greater segment of the same, shall have each of the angles at base double to the remainder: And the base shall be the side of the quinquangle inscribed with the triangle into a circle. 10, and 11. p iiij.

6 If two right lines doe subtend on each side two angles of an inscript quinquangle, they are cut proportionally, and the greater segments are the sides of the said inscript è 8, p xiij.

7 If a right line given, cut proportionall, be continued at each end with the greater segment, and sixe peripheries at the distance of the line given shall meete, two on each side from the ends of the line given and the continued, two others from their meetings, right lines drawne from their meetings, & the ends of the assigned shall make an ordinate quinquangle upon the assigned.

8 If the diameter of a circle circumscribed about a quinquangle be rationall, it is irrationall unto the side of the inscribed quinquangle, è 11. p xiij.

9 The ray of a circle is the side of the inscript sexangle. è 15 p iiij.

10 Three ordinate sexangles doe fill up a place.

11 If right lines from one angle of an inscript sexangle unto the third angle on each side be knit together, they shall inscribe an equilater triangle into the circle given.

12 The side of an inscribed equilater triangle hath a treble power, unto the ray of the circle 12. p xiij.

13 If the side of a sexangle be cut proportionally, the greater segment shall be the side of the decangle.

14 If a decangle and a sexangle be inscribed in the same circle, a right line continued and made of both sides, shall be cut proportionally, and the greater segment shall be the side of a sexangle; and if the greater segment of a right line cut proportionally be the side of an hexagon, the rest shall be the side of a decagon. 9. p xiij.

15 If a decangle, a sexangle, and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervaile the sides of the others. And if a right line inscribed do countervaile the sides of the sexangle and decangle, it is the side of the pentangle. 10. p xiiij.

16. If a triangle and a quinquangle be inscribed into the same Circle at the same point, the right line inscribed betweene the bases of the both opposite to the said point, shall be the side of the inscribed quindecangle. 16. p. iiij.

17. If a quinquangle and a sexangle be inscribed into the same circle at the same point, the periphery intercepted beweene both their sides, shall be the thirtieth part of the whole periphery.

Of Geometry the ninteenth Booke; Of the Measuring of ordinate Multangle and of a Circle.

1. A plaine made of the perpendicular from the center unto the side, and of halfe the perimeter, is the content of an ordinate multangle.

2 The periphery is the triple of the diameter and almost one seaventh part of it.

3. The plaine of the ray, and of halfe the periphery is the content of the circle.

4. As 14 is unto 11, so is the quadrate of the diameter unto the Circle.

5. The plaine of the ray and one quarter of the periphery, is the content of the semicircle.

6. The plaine made of the ray and halfe the base, is the content of the Sector.

7. If a triangle, made of two raies and the base of the greater section, be added unto the two sectors in it, the whole shall be the content of the greater section: If the same be taken from his owne sector, the remainder shall be the content of the lesser.

8. A circle of unequall isoperimetrall plaines is the greatest.

Of Geometry the twentieth Booke, Of a Bossed surface.

1. A bossed surface is a surface which lyeth unequally betweene his bounds.

2. A bossed surface is either a sphericall, or varium.

3. A sphericall surface is a bossed surface equally distant from the center of the space inclosed.

4. It is made by the turning about of an halfe circumference the diameter standeth still. è 14 d xj.

5. The greatest periphery in a sphericall surface is that which cutteth it into two equall parts.

6. That periphery that is neerer to the greatest, is greater than that which is farther off: And on each side those two which are equally distant from the greatest, are equall.

7 The plaine made of the greatest periphery and his diameter is the sphericall.

8 A plaine of the greatest circle and 4, is the sphericall.

9 As 7 is to 22. so is the quadrate of the diameter unto the sphericall.

10 The plaine of the greatest periphery and the ray, is the hemisphericall.

11 If looke what the part be of the ray perpendicular from the center unto the base of the greater section, so much the hemisphericall be increased, the whole shall be the greater section of the sphericall: But if it be so much decreased, the remainder shall be the lesser.

12 The varium is a bossed surface, whose base is a periphery, the side a right line from the bound of the toppe, unto the bound of the base.

13 A varium is a conicall or a cylinderlike forme.

14 A conicall surface is that which from the periphery beneath doth equally waxe lesse and lesse unto the very toppe.

15. It is made by turning about of the side about the periphery beneath.

16 The plaine of the side and halfe the base is the conicall surface.

17 A cylinderlike forme is that which from the periphery underneath unto the the upper one, equall and parallell unto it, is equally raised.

18 It is made by the turning of the side about two equall and parallell peripheries.

19 The plaine of his side and heighth is the cylinderlike surface.

Geometry, the one and twentieth Book, Of Lines and Surfaces in solids.

1 A body or solid is a lineate broad and high 1 d xj.

2 The bound of a solid is a surface 2 d xj.

3 If a right line be unto right lines cut in a plaine underneath, perpendicular in the common intersection, it is perpendicular to the plaine beneath: And if it be perpendicular, it is unto right lines, cut in the same plaine, perpendicular in the common intersection è 3 d and 4 p xj.

4 If three right lines cutting one another, be unto the same right line perpendicular in the common section, they are in the same plaine 5. p xj.

5 If two right lines be perpendicular to the under-plaine, they are parallells: And if the one two parallells be perpendicular to the under plaine, the other is also perpendicular to the same. 6. 8 p xj.

6 If right lines in diverse plaines be unto the same right line parallel, they are also parallell betweene themselves. 9 p xj.

7 If two right lines be perpendiculars, the first from a point above, unto a right line underneath, the second from the common section in the plaine underneath, a third, from the sayd point perpendicular to the second, shall be perpendicular to the plaine beneath. è 11 p xj.

8. If a right line from a point assigned of a plaine underneath, be parallell to a right line perpendicular to the same plaine, it shall also be perpendicular to the plaine underneath. ex 12 p xj.

9. If a right line in one of the plaines cut, perpendicular to the common section, be perpendicular to the other, the plaines are perpendicular: And if the plaines be perpendicular, a right line in the one perpendicular to the common section is perpendicular to the other è 4 d, and 38 p xj.

10. If a right line be perpendicular to a plaine, all plaines by it, are perpendicular to the same: And if two plaines be unto any other plaine perpendiculars, the common section is perpendicular to the same. e 15, and 19 p. xj.

11. Plaines are parallell which doe leane no way. 8 d xj.

12. Those which divided by a common perpendicle. 14 p xj.

13. If two paires of right in them be joyntly bounded, they are parallell. 15 p xj.

14. If two parallell plaines are cut with another plaine, the common sections are parallels, 16 p xj.

The twenty second Booke, of P. Ramus Geometry, Of a Pyramis.

1. The axis of a solid is the diameter about which it is turned, e 15, 19, 22 d xj.

2. A right solid is that whose axis is perpendicular to the center of the base.

3. If solids be comprehended of homogeneall surfaces, equall in multitude and magnitude, they are equall. 10 d xj.

4. If solids be comprehended of surfaces in multitude equall and like, they are equall, 9 d xj.

5 Like solids have a treble reason of their homologall sides, and two meane proportionalls. 33. p xj. 8 p xij.

6 A solid is plaine or embosed.

7 A plaine solid is that which is comprehended of plaine surfaces.

8 The plaine angles comprehending a solid angle, are lesse than foure right angles. 21. p xj.

9 If three plaine angles lesse than foure right angles, do comprehend a solid angle, any two of them are greater than the other: And if any two of them be greater than the other, then may comprehend a solid angle, 21. and 23. p xj.

10 A plaine solid is a Pyramis or a Pyramidate.

11 A Pyramis is a plaine solid from a rectilineall base equally decreasing.

12 The sides of a pyramis are one more than are the base.

13 A pyramis is the first figure of solids.

14 Pyramides of equall heighth, are as their bases are 5 e, and 6. p xij.

15 Those which are reciprocall in base and heighth are equall 9 p xij.

16 A tetraedrum is an ordinate pyramis comprehended of foure triangles 26. d xj.

17 The edges of a tetraedrum are sixe, the plaine angles twelve, the solide angles foure.

18 Twelve tetraedra's doe fill up a solid place.

19. If foure ordinate and equall triangles be joyned together in solid angles, they shall comprehend a tetraedrum.

20. If a right line whose power is sesquialter unto the side of an equilater triangle, be cut after a double reason, the double segment perpendicular to the center of the triangle, knit together with the angles thereof shall comprehend a tetraedrum. 13 p xiij.

The twenty third Booke of Geometry, of a Prisma.

Of Geometry the twentie fourth Book. Of a Cube.

Of Geometry the twenty fifth Booke; Of mingled ordinate Polyedra's.

Of Geometry the twenty sixth Booke; Of a Spheare.

Of Geometry the twenty seventh Book; Of the Cone and Cylinder.