|Title:||The surueyor in foure bookes by Aaron Rathborne.|
|Publication Info:||Ann Arbor, Michigan: University of Michigan, Digital Library Production Service
2011 December (TCP phase 2)
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The surueyor in foure bookes by Aaron Rathborne.
Rathborne, Aaron., Hole, William, d. 1624,
London: Printed by W: Stansby for, W: Burre, 1616.
|Alternate titles:||Surveyor Surveyor.|
The title page is engraved and signed: W.H. fe., i.e. William Hole.
The first leaf is blank; the last bears an engraved portrait of the author.
Leaf O3 is a cancel; verso line 5 has "58 4/5". Variant: O3 is cancellandum; line 3 has "54 4/5".
Reproduction of the original in the British Library.
Surveying -- Early works to 1800.
TO THE HIGH AND MIGHTY PRINCE CHARLES, Prince of WALES, Duke of CORNWALL, YORKE, ALBANY and ROTHSAY: Marques of ORMONT, Earle of ROSSE; and Baron of AR∣MANOCH: High Seneschall of SCOTLAND: Lord of the ILES; and Knight of the most noble Order of the GARTER.
THE SURVEYOR The first Booke.
THE FIRST PART.
DEFINITION I. A Point is that which is the least of all Materialls, hauing neither part nor quantitie.
DEFINITION II. A Line is length, without breadth-or thicknesse.
DEFINITION III. A Superficies is that which hath onely length and breadth.
DEFINITION IIII. An Angle is the congression or meeting of two lines in any sort, so as both make not one line.
DEFINITION V. Jf a right line fall on a right line, making the Angles on either side equall, each of those Angles are called right Angles: And the line ere∣cted is called a Perpendicular line vnto the other.
DEFINITION VI. An Angle which is greater then a right Angle, is an obtuse Angle.
DEFINITION VII. An Acute Angle is that which is lesse then a right Angle.
DEFINITION VIII. A Figure is that which is contained vnder one o many limits.
DEFINITION IX. A Circle is a plaine Figure, and contained vnder one line, which is called the Circumference thereof.
DEFINITION X. The Centre of a Circle is that point which is in the middest thereof: from which point, all right lines, drawne to the Circumference, are equall.
DEFINITION XI. The Diameter of a Circle is a right line, passing by the Centre through the whole Circle, and diuideth the same into two equall parts: Either halfe of which Diameter, is called the Semidiameter of the same Circle.
DEFIN. XII. A Semicircle is a Figure contained vnder the Diameter of a Circle, and the semicircumference of the same Circle.
DEFIN. XIII. A Segment, Section, or Portion of a Circle, is a Figure contained vnder a right line, and part of the circumference, either greater or lesse then the Semicircle.
DEFIN. XIIII. Equall Circles are such as haue equall Diameters, or whose lines, drawne from their Centres, are equall.
DEFIN. XV. A right line is said to touch a circle, which touching it, and being ex∣tended or produced, doth not cut the circumference thereof.
DEFIN. XVI. An angle of a Section, or Segment, is that which is contained vnder a cord line, and the arch line of the same Section.
DEFIN. XVII. An angle in a Section, or Segment, is when two right lines are drawne from any point in the arch line, to the ends or extreames of the cord line; the angle in that point of the arch line is called an angle in a Section or Segment.
DEFIN. XVIII. If two right lines be drawne from any one point in the circumference of a Circle, and receiue any part of the same circumference, the angle contained vnder those two lines is said to belong and to be correspon∣dent to that part of the circumference so receiued.
DEFIN. XIX. A Sector of a Circle is a figure contained vnder two right lines, drawne from the centre of a Circle, and vnder part of the circumference receiued of them.
DEFIN. XX. Right lined figures are such, as are contained vnder right lines, of what number soeuer, aboue two.
DEFIN. XXI. An Equilater Triangle is that, which hath three equall sides.
DEFIN. XXII. An Isosceles is a Triangle, which hath onely two equall sides.
DEFIN. XXIII. A Scalenum is a Triangle which hath all his sides vnequall.
DEFIN. XXIIII. A Square, or Quadrat, is a foure sided figure, whose sides are all equall, and angles all right angles.
DEFIN. XXV. A long Square is that whose angles are all right angles, and whose op∣posite sides onely are equall.
DEFIN. XXVI. A Rhombus (or Diamond) is a figure with foure equall sides, but no right angle.
DEFIN. XXVII. A Rhomboydes (or Diamond like) is a figure, whose opposite sides and opposite angles are onely equall, and bath no right angles.
DEFIN. XXVIII. All other foure lined figures, besides those formerly defined, are called TRAPEZIA, or Tables.
DEFIN. XXIX. Manie sided figures are those which haue more sides then foure.
DEFIN. XXX. Either of those Parallelograms, which are about the diameter of a Pa∣rallelogram, together with the two supplements, is called a Gnomon.
DEFIN. XXXI. That rht lined figure is said to be inscribed in another right lined fi∣gure, which hath euerie angle touching euerie side of the figure wherein it is inscribed.
DEFIN. XXXII. A right lined figure is inscribed within a circle, when euerie angle of the inscribed figure toucheth some part of the circles circumference.
DEFIN. XXXIII. The altitude of a figure, is the parallell distance betweene the top of a figure and the Base.
DEFIN. XXXIIII. Parallell lines are such, as being drawne on any plaine Superficies, and produced either way infinitely, doe neuer meet or concurre.
DEFIN. XXXV. A right line is said to be diuided by extreame and meane proportion, when the lesser part, or segment thereof, is to the greater, as the greater is to the whole line.
DEFIN. XXXVI. The power of a line is the square of the same line, or any plaine figure equall to the square thereof.
DEFIN. XXXVII. To diuide a giuen line in power, is to finde two other lines, whose squares together shall be equall to the square of the giuen line, but the square of the one to the square of the other, to be in any pro∣portion required.
DEFIN. XXXVIII. To inlarge a line in power, is to find another line, whose square shall haue any proportion required (of the greater inequalitie) to the square of the giuen line.
DEFIN. XXXIX. A meane proportionall line is that, whose square is equall to the right an∣gled Parallelogram, or long Square, contained vnder his two ex∣treames.
DEFIN. XL. Like right lined figures are those, which haue equall angles, and pro∣portionall sides about those equall angles.
DEFIN. XLI. Reciprocall figures are such, as haue the sides of either to other mu∣tually proportionall.
DEFIN. XLII. The quantitie or measure of an angle, is the arch of a circle, described from the point of the same angle, and intercepted betweene the two sides of that angle.
DEFIN. XLIII. The Quadrant of a circle is the fourth part thereof, or an arch con∣taining 90 degrees.
DEFIN. XLIIII. The Complement of an arch, lesse then a Quadrant, is so much as that arch wanteth of 90 degrees.
DEFIN. XLV. The Excesse of an arch, greater then a Quadrant, is so much as the said arch is more then 90 degrees.
DEFIN. XLVI. The Complement of an arch, lesse then a Semicircle, is so much as that arch wanteth of a Semicircle, or of 180 degrees.
DEFIN. XLVII. The Complements of Angles are as the Complements of Arches.
❧ The second Part. Jnstructions concerning this Part.
THEOREME I. Jf any two right lines cut the one the other, the opposite or verticall angles are euer equall; and both the angles, on one and the same side of either line, are either of them right angles, or (being both ta∣ken together) are equall to two right angles.
THEOREME II. A right line, falling on two parallell right lines, maketh the outward an∣gles on contrarie sides of the falling line equall; and likewise the in∣ward and opposite angles on the contrarie sides of the same line; and also the outward angle, equall to the inward and opposite angle on one and the same side of the falling line; and the inward angles on one and the same side equall to two right angles.
THEOREME III. If a right line be diuided into two equall parts, halfe the square of that whole line is double to the whole square of halfe the same line.
THEOREME IIII. A right line being diuided by chance, the square of the whole line is equall to both the squares made of the parts, and also to two rectan∣gle figures, comprehended vnder the same parts.
THEOREME V. The Supplements of those Parallelograms which are about the diameter in euery Parallelogram, are alwaies equall the one to the other.
THEOREME VI. In right angled Triangles, the square of the side subtending the right angle, is equall to both the squares of his containing sides.
THEOREME VII. In obtuse angled Triangles, the square of the side subtending the ob∣tuse angle, is greater then both the squares of the containing sides, by two rectangled figures, comprehended vnder one of the contai∣ning sides (being continued) and the line of continuation, from the obtuse angle to a perpendicular let fall thereon.
THEOREME VIII. In acute angled Triangles, the square of the side subtending the acute angle, is lesse then both the squares of the containing sides by two rectangle figures, comprehended vnder one of the containing sides (whereon a perpendicular falleth) and that segment of the same side which is betweene the perpendicular and the acute angle.
THEOREME IX. In rectangle Triangles, if from the right angle a perpendicular be let fall vnto the Base, it shall diuide the Triangle into two Triangles, like vnto the whole, and also the one like vnto the other.
THEOREME X. An Isosceles, or a Triangle of two equall sides, hath his angles at the Base equall; and the equall sides being produced, the angles vnder the Base are also equall.
THEOREME XI. All equiangle Triangles haue their sides, containing equall angles pro∣portionall, and their sides subtending equall angles, are of like proportion.
THEOREME XII. In any two Triangles compared, if two sides of the one be equall to two sides of the other, and the Base of the one to the Base of the other; they shall also haue the angles contained vnder their answerable e∣quall sides, the one equall to the other in either Triangle.
THEOREME XIII. If any side of a Triangle be continued, the outward angle made by that continuation, is equall to the two inward and opposite an∣gles: And the three inward angles of any Triangle are equall to two right angles.
THEOREME XIIII. In euerie Triangle, two of his angles, which two soeuer be taken, are lesse then two right angles.
THEOREME XV. In euerie Triangle, two sides thereof (which two soeuer be taken) are greater (being ioyned together as one line) then the third side re∣maining.
THEOREME XVI. Jn all Triangles, the greater side subtendeth the greater angle, and the lesser side subtendeth the lesser angle.
THEOREME XVII. If two sides of one Triangle be equall to two sides of another Tri∣angle, and the angle contained vnder the equall sides of the one, be greater then the angle contained vnder the equall sides of the other; then the Base also of the one (namely, of that which hath the greater angle) shall be greater then the Base of the other.
THEOREME XVIII. If a Triangle be equicrurall, or hauing two equall sides; a perpendicu∣lar let fall from the angle contained vnder those equall sides to the Base, and continued, shall diuide as well the same Base and angle, as also the measure of that angle, into two equall parts: Et contra.
THEOREME XIX. If a Triangle hath two equall sides, the power of one of those equall sides exceedeth the power of the perpendicular let fall on the Base from the angle it subtendeth, by the power of halfe the Base.
THEOREME XX. If the power of one side of any Triangle be equall to both the powers of the other two sides, the angle contained vnder those two other sides, is a right angle.
THEOREME XXI. If a right line diuide any angle of a Triangle into two equall parts, and if also the same line diuide the Base, the segments of the Base shall haue such proportion the one to the other, as the other sides of the Triangle haue: Et contra.
THEOREME XXII. If a right line be drawne parallell to any side of a Triangle, the same line shall cut the sides of that Triangle proportionally.
THEOREME XXIII. The superficiall content of euery right angled Triangle, is equall to halfe that right angled Parallelogram, which hath his length and breadth equall to the containing sides of the right angle; or whose length is equall to the subtending side, and breadth to the perpendicular, drawne from the right angle to the same side.
THEOREME XXIIII. The Area or superficiall content of euery Equilater Triangle, is equall to halfe that long square, whose length and breadth is equall to one of the sides and the perpendicular.
THEOREME XXV. All Triangles, of what kind soueur, are equall in their superficiall con∣tent vnto halfe that right angled Parallelogram, whose length and breadth is equall to the perpendicular, and the side whereon it falleth.
THEOREME XXVI. Triangles which consist on one and the same Base, or on equall Bases, and in the same parallell lines are equall the one to the other.
THEOREME XXVII. If Triangles and Parallelograms haue one and the same Base, or equall Bases, and be in the same parallell lines, the Parallelograms shall be double to the Triangles.
THEOREME XXVIII. If a Triangle hath his Base double to the Base of a Parallelogram, and that they are both in the same parallell lines, then are they both equall the one to the other.
THEOREME XXIX. The power of the side of an Equilater Triangle, is to the power of the perpendicular thereof let fall from any angle to the subtendent side, in proportion Sesquitertia, or as 4. to 3.
THEOREME XXX. The Diagonall line, or Diameter of any Square, is double in power to the side of the same Square.
THEOREME XXXI. A Square, whose side is equall to the Diameter of any other Square, is double in content or superficiall quantitie to that other Square.
THEOREME XXXII. All parallelograms haue their opposite sides, and angles equall one to an∣other; and their Diameters diuide them into equall parts.
THEOREME XXXIII. Parallelograms which consist on one and the same base, or on equall bases, and in the same parallel lines are equall the one to the other.
THEOREME XXXIIII. Euery Rhombus and Rhomboydes is equall to the long square, whose length is one of the sides, and breath equall to the parallel distance.
THEOREME XXXV. Parallelograms and Triangles, within the same parallels, are in such proportion the one to the other as their bases are.
THEOREME XXXVI. A right line being first equally, and then vn-equally diuided; The square which is made of the part lying betweene those sections, together with the right angled parallelogram, contayned vnder the vn-equall parts of the whole line; are equall to the square of halfe the whole line.
THEOREME XXXVII. Two right lines being drawne in a circle, and the one intersecting the o∣ther, either equally or vnequally howsoeuer; The rectangle figure contayned vnder the parts of the one line, shall be equall to that, con∣tayned vnder the parts of the other.
THEOREME XXXVIII. In all right angled parallelograms, the length thereof being infolded in the breadth, produceth the Area or superficiall content of the same.
THEOREME XXXIX. Euery regular Poligon is equall to the long square, whose length and breadth is equall to halfe the perimeter, and a perpendicular drawne from the center to the middle of any side of the same.
THEOREME XXXX. If two or more right lines, are cut by diuers parallel right lines; the in∣tersegments of those lines so cut shall bee proportionall the one to the other.
THEOREME XLI. Three right Lines being proportionall, a Square made of the Meane, is equall to the right angled figure, contained vnder the Ex∣treames.
THEOREME XLII. Foure right lines being proportionall, the right angled Parallelogram, contained vnder the two Meanes, is equall to the right angled Pa∣rallelogram, contained vnder the two Extreames.
THEOREME XLIII. Of any three proportionall right Lines, the Square which is made of the Meane, and that which is made of either of the Ex∣treames, haue such proportion the one to the other, as the two Extreames haue.
THEOREME XLIIII. If a rationall right line be diuided by an extreame and meane propor∣tion, either of the segments is an irrationall residuall line.
THEOREME XLV. If a right line be diuided by extreame and meane proportion, the whole line hath the same proportion to the greater segment, as the same greater segment hath to the lesser.
THEOREME XLVI. If a right line be diuided by extreame and meane proportion, the Rectangle figure, comprehended vnder the whole line, and the lesser segment, shall be equall to the Square made of the grea∣ter segment.
THEOREME XLVII. Two right lines being drawne in an equilater Equiangle Pentagon, in such sort as they subtend any two of the next immediate angles, those two lines by their intersections shall diuide the one the other by an extreame and meane proportion: and the greater segments of either of them shall be equall to the side of the Pentagon.
THEOREME XLVIII. Like Triangles are one to the other in double proportion that the sides of like proportion are.
THEOREME XLIX. All like right lined figures whatsoeuer, are the one to the other in double proportion, that the sides of like proportion are.
THEOREME L. All angles in equall circles, whether they are in the centers or circumfe∣rences, haue the same proportion one to the other as the circumferen∣ces haue wherein they consist: And so are the sectors, which are de∣scribed on the centers.
THEOREME LI. If on the end of the Diameter of a circle, a perpendicular bee raysed, it shall fall without the circle, betweene which, and the circumference, another right line cannot be drawne to the Diameter, and the angle within the circle is greater, and that without the circle is lesser, then any acute angles made of right lines.
THEOREME LII. If a right line bee a tangent or touch line to a circle, and another right line bee drawne by the center to the point of touch, it shall bee a perpendicular to the tangent: And if a perpendicular bee let fall from the center to the tangent, it shall fall in the point of touch.
THEOREME LIII. If a right line be drawne in a circle and not by the center thereof, another right line bysecting the same by right angles shall passe by the center of the same circle. And if from the center a perpendicular be let fall on a right line drawne in the same circle not by the center; the per∣pendicular shall diuide the same line into two equall parts.
THEOREME LIIII. If one angle be placed in the circumference of a circle, and another in the center thereof, and are both subtended by one part of the circumfe∣rence. That angle in the center shall bee double to that in the cir∣cumference.
THEOREME LV. All angles consisting in one and the same segment of a circle are equall the one to the other; If in a semicircle, they are right angles; If in a lesser segment, they are greater then a right angle; If in a greater segment, they are lesser. And also the angle of a greater segment, is greater then a right angle, and the angle of a lesser segment is lesse then a right angle.
THEOREME LVI. If a right line be a tangent to a cirle, and another right line be drawne from the touch (crossing the circle) to what point soeuer in the cir∣cumference; the angles caused by intersection or meeting of those two lines, are equall to the angles consisting in the alternate segments of the circle.
THEOREME LVII. If from a point without a circle, two right lines be so drawne, that the one be a tangent to the circle, and the other diuide the same circle in∣to two equall or vnequall parts: The rectangle figure contayned vn∣der the whole line which diuideth the circle, and that part thereof ly∣ing betweene the vtter circumference and the point, is equall to the square made of the tangent line.
THEOREME LVIII. If from a point without a circle, two right lines be drawne to the concaue circumference of the circle, they shall be reciprocally proportionall with their parts taken without the circle. And another right line drawne from the point as a tangent to the circle, shall bee a meane proportio∣nall betweene either whole line, and the vtter segment thereof.
THEOREME LIX. Euery circumference of a circle, is more then triple his Diameter, by such a proportion as is more then 10/71· and lesse then 1/7 of the same, the neerest rationall proportion whereof is 22. to 7.
THEOREME LX. Euery circle is neere equall to that right angled Triangle, of whose sides (containing the right angle) the one is equall to the semidiameter, and the other to the circumference of the same circle.
THEOREME LXI. The Square made of the Diameter of a Circle, is in that proportion to the circle (very neere) as 14. to 11. And therefore euery circle is neere 11/14· of the square about him described.
THEOREME LXII. Euery circle is neere equall to the long square, whose length and breadth are equall to halfe the circumference, and halfe the Diameter; or to the whole Diameter, and 11/14· thereof.
THEOREME LXIII. Euery semicircle is neere equall to the long square, whose length and breadth is equall to halfe the arch line, and the semidiameter.
THEOREME LXIIII. Euery sector of a circle, is neere equall to that long square, whose length and breadth is equall to the semidiameter, and halfe the arch-line of the same sector; or the halfe semidiameter, and the whole arch line.
THEOREME LXV. All circumferences of circles, haue the same proportion the one to the o∣ther, as their diameters haue.
THEOREME LXVI. All circles haue the same proportion the one to the other, as the squares of their Diameters haue.
THEOREME LXVII. If in a circle be described a quadrilaterall figure, the opposite angles thereof shall bee equall to two right angles: and being intersected with two diagonalls, the right angled figure made of those diago∣nalls, is equall to the two right angled figures, comprehended vnder the opposite sides of the quadrilaterall figure.
THEOREME LXVIII. The power of the side of an equilater triangle inscribed in a circle, hath to the power of the semidiameter of the same circle triple proportion.
THEOREME LXIX. A triangle inscribed in a circle, hath euery of his angles equall to halfe the arch, opposite to the same angle.
THEOREME LXX. If in a rectangle-triangle a perpendicular bee drawne from the right angle to the base, the same perpendicular is a meane proportio∣nall betweene the sections of the base: And the side annext to either section, shall bee a meane betweene the same section and the whole base.
THEOREME LXXI. If in equall parallelograms, one angle of the one, be equall to one angle of the other, the sides which contayne those equall angles, shall bee reciprocall.
THEOREME LXXII. In rectangle-triangles, the figure which is made of the subtending side of the right angle, is equall vnto both the figures made of those sides, which contayne the right angle, so as those three figures are like, and in like sort described.
THEOREME LXXIII. In all plaine triangles, the sides are in proportion the one to the other, as the subtenses of the angles opposite thereunto; or as the sines of the angles opposite to those sides.
THEOREME LXXIIII. Euery right lined figure, or plat, consisteth of more sides by two; then the number of triangles, whereof the same figure is composed.
THE VSE AND OPERATI∣ON OF THE FORMER THEOREMES.
THE FIRST PART. Of the Properties, Passions, Dispositions, Applica∣tions, and Diuisions of Lines and Angles.
PROBLEME I. Two right lines giuen, being vnequall; to take from the greater a line equall to the lesser.
PROBLEME II. To a right line giuen, to draw a parallell line at any distance required.
PROBLEME III. To performe the former Proposition at a distance required, and by a point limitted.
PROBLEME IIII. To erect a perpendicular on any part of a right line giuen.
PROBLEME V. To raise or let fall a perpendicular to a line giuen, from a point ei∣ther aboue or beneath the same line.
PROBLEME VI. Ʋpon the end of a line giuen to raise a perpendicular.
PROBLEME VII. To diuide a right line giuen, into two equall parts.
PROBLEME VIII. Vpon a right line giuen, on a point therein limited, to make an angle equall to an angle giuen.
PROBLEME IX. To make a right angle vpon a line giuen, and on a point in the same line limitted.
PROBLEME X. To diuide an angle giuen into two equall parts.
PROBLEME XI. To diuide a right angle giuen into three equall parts.
PROBLEME XII. To diuide a right line giuen into diuers equall parts, as many as shall be required.
PROBLEME XIII. To diuide a right line giuen proportionally, according to any propor∣tion required.
PROBLEME XIIII. From a right line, giuen to cut off any parts required.
PROBLEME XV. To finde a third line in continuall proportion vnto two lines giuen.
PROBLEME XVI. To finde a fourth proportionall line to three lines giuen.
PROBLEME XVII. To finde a meane proportionall line betweene any two lines giuen.
PROBLEME XVIII. To finde two meane proportionall lines, betweene any two right lines giuen.
PROBLEME XIX. To finde out in a line giuen, the two extremes of a meane propor∣tionall giuen: So as the same meane bee not greater then halfe the giuen line.
PROBLEME XX. To diuide a line giuen by an extreme and meane proportion.
PROBLEME XXI. To find the lesser segment of a line, diuided by extreame and meane proportion, when onely the greater is giuen; and consequently, to find the whole line.
PROBLEME XXII. Hauing the greater or lesser segment of a line diuided by extreame and meane proportion giuen, to find the other segment, and so the whole line.
PROBLEME XXIII. To diuide a right line giuen in power, according to any proportion gi∣uen in two right lines.
PROBLEME XXIIII. To inlarge a line in power, according to any proportion required.
PROBLEME XXV. To diuide the circumference, or find out all the cord lines of a circle, not exceeding the tenth.
PROBLEME XXVI. To draw a line from an angle in a Plot giuen, which shall take in as much as it cuts off.
PROBLEME XXVII. To draw a line in such sort, as to retaine the superficiall quantitie of a giuen Triangle, and yet alter the Base to any possible length required.
PROBLEME XXVIII. To draw a line in such sort, as to retaine the superficiall quantitie of a giuen Triangle, and yet alter the altitude to any possible height re∣quired.
PROBLEME XXIX. To find two right lines in such proportion as two figures giuen.
PROBLEME XXX. To find two right lines in such proportion the one to the other, as two giuen Squares.
PROBLEME XXXI. To draw a touch line to a circle giuen, from a point assigned.
PROBLEME XXXII. To apply a line vnto a circle giuen, in such sort, as thereby to cut off a segment, wherein may be placed an angle, equall to an angle giuen.
PROBLEME XXXIII. To describe, vpon a line giuen, such a segment of a circle, as shall con∣taine an angle, equall to an angle giuen.
PROBLEME XXXIIII. A segment of a circle being giuen; to find out the centre, and consequent∣ly the diameter and the whole circle.
The second Part.
PROBLEME XXXV. To make an equilater triangle, the side thereof being giuen.
PROBLEME XXXVI. To find the perpendicular of an equilater triangle Arithmetically, the side being giuen.
PROBLEME XXXVII. The perpendicular and side of an equilater triangle being giuen, to finde the Area or superficiall content.
PROBLEME XXXVIII. To make a right angled triangle, the two containing sides being giuen.
PROBLEME XXXIX. The perpendicular and base of a right angled triangle giuen to find the superficiall content.
PROBLEME XL. To make an Jsoseeles triangle on a right line giuen.
PROBLEME XLI. The perpendicular and base of an Isosceles Triangle, giuen to find the area, or superficiall content Arithmetically.
PROBLEME XLII. To make a Triangle of three vnequall sides, the lines being giuen, so as the two shortest together bee longer then the third line.
PROBLEME XLIII. To finde the perpendicular of any Triangle Arithmetically, the sides be∣ing giuen.
PROBLEME XLIIII. The perpendicular and base of any Triangle being giuen, to finde the area or superficiall content thereof Arithmetically.
PROBLEME XLV. To make a Triangle vpon a line giuen, like vnto another Triangle giuen.
PROBLEME XLVI. To make a Triangle equall to another Triangle giuen, vpon the same Base, hauing an Angle equall to an Angle giuen.
PROBLEME XLVII. To make a Triangle: equall to another Triangle giuen, with a base or perpendicular limited.
PROBLEME XLVIII. To make an Isosceles Triangle vpon a line giuen, whose Angles at the base, shall be eyther of them double to the third Angle.
PROBLEME XLIX. To make a Triangle equall to a Parallelogram giuen, vpon a line limi∣ted, and with an Angle equall to an Angle giuen.
PROBLEME L. To make a Square vpon a line giuen for the side thereof.
PROBLEME LI. The side of a Square being giuen, to finde the Area or superficiall con∣tent Arithmetically.
PROBLEME LII. To make two Squares which shall be equall the one to the other; and al∣so to two vnequall Squares giuen.
PROBLEME LIII. To describe a Square in such sort as it shall passe by any three points giuen.
PROBLEME LIIII. To make a long Square or right angled Parallelogram, the length and bredth being giuen.
PROBLEME LV. The length and breadth of a right angled Parallelogram or long Square being giuen, to finde the Area or superficiall content there∣of Arithmetically.
PROBLEME LVI. To make a Parallelogram, whose length is limited, equall to a Tri∣angle giuen, with two opposite Angles each equall to an Angle giuen.
PROBLEME LVII. To make a Rhombus, the side being giuen.
PROBLEME LVIII. The side of a Rhombus being giuen to finde out the Area or superficiall content thereof Arithmetically.
PROBLEME LIX. To make a Rhomboydes the length and bredth being giuen in two right lines.
PROBLEME LX. To make a Rhomboydes with lines limited, hauing two opposite Angles, equall to an Angle giuen.
PROBLEME LXI. A Rhomboydes giuen to finde the superficiall content Arithme∣tically.
PROBLEME LXII. To describe a Pentagon, hauing sides and Angles equall.
PROBLEME LXIII. To describe a Pentagon vpon a line giuen.
PROBLEME LXIIII. The side of a Pentagon being giuen, to finde the superficiall content Arithmetically.
PROBLEME LXV. To make two like Figures, bearing the one to the other, any proportion assigned in two right lines.
PROBLEME LXVI. Two Circles being giuen, to make one Circle equall to them both.
PROBLEME LXVII. The Diameter of a Circle being giuen, to find the circumference there∣of Arithmetically.
PROBLEME LXVIII. The Diameter and Circumference of a Circle being giuen, to find the A∣rea, or superficiall content thereof Arithmetically, diuers wayes.
PROBLEME LXIX. The Diameter and Arch-line of a Semicircle giuen, to find the A∣rea thereof.
PROBLEME LXX. The Semidiameter and Arch line of a Sector of a Circle giuen to find the Area.
PROBLEME LXXI. Any Segment or part of a Circle being giuen, to finde the superficiall content thereof.
PROBLEME LXXII. An irregular plotte or Figure being giuen, to finde the Area or su∣perficiall content thereof.
The third Part.
PROBLEME LXXIII. To reduce one triangle into another, on the same base, but hauing an angle equall to an angle giuen.
PROBLEME LXXIIII. To reduce one triangle into another, vpon a base equall to a base giuen.
PROBLEME LXXV. To reduce one triangle to another, of any possible height required.
PROBLEME LXXVI. To reduce a triangle giuen into a square.
PROBLEME LXXVII. To reduce a triangle giuen, into a right angled parallelogram.
PROBLEME LXXVIII. To reduce a tringle giuen, into a parallelogram, hauing an angle equall to an angle giuen.
PROBLEME LXXIX. To reduce a triangle giuen into a Rhombus.
PROBLEME LXXX. To reduce a square giuen into a triangle, hauing an angle equall to an angle giuen, and that on a line giuen.
PROBLEME LXXXI. To reduce a square giuen into a triangle, with angles equall, and lines proportionall to a triangle giuen.
PROBLEME LXXXII. To reduce a square into an equilater triangle.
PROBLEME LXXXIII. To reduce a square giuen, into a right angled parallelogram or long square, the length and bredth being limited in a right line: So as the side of the square exceede not halfe the line giuen.
PROBLEME LXXXIIII. To reduce a square giuen into a long square, whose bredth is limited in a right line giuen.
PROBLEME LXXXV. To reduce a square giuen into a long square, whose length is limited in a right line giuen.
PROBLEME LXXXVI. To reduce a long square giuen into a geometricall square.
PROBLEME LXXXVII. To reduce one long square giuen into another, whose length or bredth is limited in a right line giuen.
PROBLEME LXXXVIII. To reduce one long square giuen into another, whose length and bredth shall haue proportion the one to the other, as two giuen lines.
PROBLEME LXXXIX. To reduce a Rhombus into a geometricall square.
PROBLEME XC. To reduce a Rhomboydes giuen, into a Geometricall Square.
PROBLEME. XCI. To reduce a Rhomboydes giuen into a Triangle, hauing an Angle equall to an Angle giuen.
PROBLEME XCII. To reduce a Trapezium giuen into a right angled Parallelogram, or into a right angled Triangle.
PROBLEME XCIII. To reduce a Trapezium giuen into a Triangle, vpon a line giuen, and hauing an Angle equall to an Angle giuen.
PROBLEME. XCIIII. To reduce a Trapezium into a Triangle, which shall be like vnto another Triangle giuen.
PROBLEME XCV. To reduce an equiangled Poligon giuen, into a Geometricall Square.
PROBLEME XCVI. To reduce a plotte giuen into a Triangle, with lines drawne from an Angle assigned.
PROBLEME XCVII. To reduce a Figure giuen, into a Lunula or Figure of a Lunular forme.
PROBLEME XCVIII. To reduce an irregular Figure giuen, into a greater or lesser forme, ac∣cording to any giuen proportion.
PROBLEME XCIX. To reduce an irregular Figure giuen, into a Geometricall Square.
The fourth Part.
PROBLEME C. Two Geometricall squares being giuen, to adde them together into one square.
PROBLEME CI. Two Geometricall squares being giuen, to adde them together in such sort, as the one shall be a Gnomon vnto the other.
PROBLEME CII. To adde diuers squares together into one geometricall square.
PROBLEME CIII. To adde two giuen triangles together into one, which new composed tri∣angle, shall haue his perpendicular, equall to that of one of the giuen triangles.
PROBLEME CIIII. Diuers Circles being giuen, to adde them together into one Circle.
PROBLEME CV. Two long Squares being giuen, to adde them together into one long Square, whose breadth shall be equall to that of one of the long Squares giuen.
PROBLEME CVI. Two Geometricall Squares being giuen, to subtract the one out of the other, and to produce the remainder in a third Square.
PROBLEME CVII. Two Triangles being giuen to subtract the one out of the other, and to leaue the remainder in a Triangle of equall height to one of the gi∣uen Triangles.
PROBLEME CVIII. Two circles being giuen, to subtract the one out of the other, and to pro∣duce the remainder in a third circle.
PROBLEME CIX. A geometricall square and a triangle being giuen, to subtract the tri∣angle from the square, and to produce the remainder in a square.
PROBLEME CX. A triangle and a long square being giuen, to subtract the long square from the triangle, and to produce the remainder in a triangle of e∣quall height, to the triangle giuen.
PROBLEME CXI. Within a circle giuen to inscribe a triangle, with angles equall, and lines proportionall, to a triangle giuen.
PROBLEME CXII. To describe a Circle about a Triangle giuen.
PROBLEME CXIII. To inscribe a Circle within a Triangle giuen.
PROBLEME CXIIII. To describe a Triangle about a Circle giuen, which shall be like vn∣to a Triangle giuen.
PROBLEME CXV. To describe a Square about a Circle giuen.
PROBLEME CXVI. Within a Square giuen to inscribe a Circle:
PROBLEME CXVII. About a Square giuen to circumscribe a Circle.
PROBLEME CXVIII. To inscribe a Square within a Circle giuen.
PROBLEME CXIX. To inscribe a Pentagon, within a Circle giuen.
PROBLEME CXX. About a Circle giuen to circumscribe a Pentagon.
PROBLEME CXXI. To inscribe a Sexagon within a circle giuen.
PROBLEME CXXII. To circumscribe a Sexagon about a Circle giuen.
PROBLEME. CXXIII. To diuide a right lined Triangle giuen, into any number of equall parts required, from a point limited in any side of the same Triangle.
PROBLEME CXXIIII. To diuide a giuen Triangle by a line issuing from an angle assigned, in any proportion required.
PROBLEME. CXXV. A Triangle being giuen, and the Base thereof known, to diuide the same into two parts by a line from an angle assigned, according to any pro∣portion giuen in two numbers.
PROBLEME CXXVI. To diuide a Triangle of knowne quantitie giuen, into any two parts, from an Angle assigned, according to any number of Acres, Roodes and Perches required.
PROBLEME CXXVII. To diuide a Triangle giuen into two parts by a line drawn, from a point limited in one of the sides, in any proportion required.
PROBLEME CXXVIII. To diuide a Triangle giuen into two parts, according to any proper∣tion giuen in two numbers from a point limited in any side thereof, A∣rithmetically.
PROBLEME CXXIX. To diuide a Triangle of any knowne quantitie giuen, into two parts, from a point limited in any side thereof, according to any number of Acres, Roodes and Perches.
PROBLEME CXXX. To diuide a giuen triangle by a parallel line, to one of the sides, accor∣ding to any proportion giuen.
PROBLEME CXXXI. To diuide a giuen triangle by a parallel line, to one of the sides, accor∣ding to any proportion giuen in two numbers Arithmetically.
PROBLEME CXXXII. To diuide a triangle of any knowne quantitie, giuen into two parts, by a parallel line, to one of the sides, according to any number of Acres, Roodes, and Perches required.
THE EXACT OPERATION OF INSTRVMENTALL DIMENSIONS BY DI∣VERS MEANES.
CHAP. I. Of the seuerall Instruments in vse, meete for Suruaie; which of them are most fit for vse, and somewhat concerning their abuse.
CHAP. II. Of the Theodelite and his seuerall parts, with the Description and Composition thereof.
CHAP. III. The Playne table and his seuerall parts, with the description and com∣position thereof.
CHAP. IIII. The Circumferentor with his parts, description, and composition.
CHAP. V. The Peractor, with his seuerall parts Description and Composition.
CHAP. VI. The making and diuision of a Chaine, called the Decimall Chaine.
CHAP. VII. Of the Protractor and the Scale thereof.
CHAP. VIII. Of the ordinarie Scale with the Sextans thereon described, very neces∣sarie for vse.
CHAP. IX. Of a Ruler, for the reducing of Plats.
CHAP. X. The order of making of a necessary and fitting Field-booke, seruing aswell for the Peractor and Circumferentor, as for the Theodelite, with the ordering and vse thereof in the Fields.
CHAP. XI. To lay downe an Angle of any quantitie required; or to find the quan∣titie of any Angle giuen, by the Sextans and the Scale.
CHAP. XII. To lay downe an Angle of any quantitie required, or to finde the quanti∣tie of any Angle giuen, by the Protractor.
CHAP. XIII. The reducing of statute measure into Acres of any customary measure required, and the contrary, shewing the difference betweene them.
CHAP. XIIII. Of the Table of Sines expressed on the Circumferentor.
CHAP. XV. Of the congruitie in vse betweene the Peractor and Circumferen∣tor; and the meanes to find the quantitie of an Angle by eyther of them.
CHAP. XVI. To take any horizontall distance at two stacions, by Sinicall com∣putation.
CHAP. XVII. To take the distance aswell betweene diuers seuerall places remote from your place of being, as betweene your being, and those seuerall places, by the helpe of two stacions.
CHAP. XVIII. To protract any number of Angles or degrees taken by the Peractor, Theodelite or Circumferentor, at seuerall obseruations.
CHAP. XIX. To take any accessible altitude by the Circumferentor or Plaine Table with the diuided sights.
CHAP. XX. To take any accessible altitude diuers wayes, by the Peractor and the Quadrant thereof.
CHAP. XXI. To find out any inaccessible height by the Peractor, Theodelite, or Cir∣cumferentor.
CHAP. XXII. To take the plot of a Field at one stacion; taken in any part thereof, from whence you may view all the Angles, and measuring from the stacion to euery Angle.
CHAP. XXIII. To protract and lay downe the obseruations made in the last Chapter, or any other taken in the like sort.
CHAP. XXIIII. To take the plot of any Field at one stacion in any one Angle thereof, from whence may bee seene all the other Angles of the same Field; and measuring from the station to euery Angle.
CHAP. XXV. To take the Plot of a Field at one station in any Angle, from whence the rest may be seene, and by measuring the sides of the Perimeter.
CHAP. XXVI. To protract and lay downe the obseruations had, according to the worke in the last Chapter, or any other taken by the like meanes.
CHAP. XXVII. To take the plot of a Field at two stations, where all the Angles cannot be seene at one, and measuring as in the 22. Chapter.
CHAP. XXVIII. To protract and lay down any obseruations taken, according to the work of the last chapter.
CHAP. XXIX. To take the Plot of a Field at diuers stacions in diuers Angles, where all cannot be seene from one, and to measure as in the 24. Chapter.
CHAP. XXX. To protract and lay downe any obseruations taken, according to the worke of the last Chapter.
CHAP. XXXI. To take the plot of any field at two stations, so as all the Angles may be seene from both stacions, by measuring onely the stationarie di∣stance.
CHAP. XXXII. To take the plot of any Field remote from you at two stations, when eyther by opposition you may not, or some other impediment you cannot come into the same.
CHAP. XXXIII. To take the Plot of any Field by making obseruation at euery Angle, and measuring onely one line, but no part of the perimeter.
CHAP. XXXIIII. To protract and lay downe any obseruations taken, according to the worke of the last Chapter.
CHAP. XXXV. To take the Plot of any Field at diuers stations, measuring onely the sta∣cionary distances.
CHAP. XXXVI. To take the Plot of a Forrest, or any spacious Common or Waste, of what∣soeuer quantitie, by the Plaine Table, on one sheete of paper with∣out altering thereof.
CHAP. XXXVII. To protract and lay downe any obseruations taken, according to the worke of the last Chapter.
CHAP. XXXVIII. To take the plot of a Lordship or Mannor, consisting of diuers seuerals, of what nature or kind soeuer, whether Wood groundes or o∣ther.
CHAP. XXXIX. To protract and laye downe a Plot of many seuerals, of what quantitie or number soeuer.
CHAP. XL. The order and meanes of measuring and taking the seuerall and parti∣cular quantities in common fields, with a briefe instruction concer∣ning the vse of my Chaine.
CHAP. XLI. To reduce Hipothenusall to Horizontall lines by the Peractor.
CHAP. XLII. To reduce hipothenusall to horizontall lines by the Circumferentor, or by the Plaine Table, with vse of those meanes expressed in the descrip∣tion thereof.
CHAP. XLIII. The best and exactest means for the dimension & protraction of moun∣tainous and vneuen grounds, and the obtaining of their true Con∣tents by the Plaine Table.
CHAP. XLIIII. To diuide a Common of pasture, or a common field into any parts required.
CHAP. XLV. To know the houre of the day by the Peractor or Circumferentor with the Sunne.
CHAP. XLVI. The ordering of a plot after the protraction thereof.
CHAP. XLVII. To reduce any plot from a greater to a lesser quantitie, and the contrarie.
CHAP. XLVIII. To reduce any number of Perches giuen into Acres, and the con∣trarie.
THE LEGALL PART OF SVRVEY.
CHAP. I. Of a Mannor with his seuerall parts, and of the name and nature therof: how made and maintained, and how discontinued and destroyed.
CHAP. II. Of Perquisites Casualties and profits of Court, and their seuerall natures.
Fines of Land.
Pleas and Proces of Court.
CHAP. III. Of the diuersitie of estates, and their seuerall natures.
2. Fee Tayle.
3. After possibilitie of issue extinct.
4. By courtesie.
5. In Dower.
6. For terme of Life.
7. By Copie of Court roll.
8. Terme of yeeres.
9. Tenant at will.
CHAP. IIII. Of the diuersitie of Tenures, and their seuerall natures, with the seruices belonging.
1. Knights seruice.
2. Ward, Marriage, and Reliefe.
3. Castle garde.
4. Grand Sergeantie.
5. Petie Sergeantie.
6. Homage Ancestrell.
7. Socage tenure.
8. Franke Almoigne.
9. Burgage tenure.
10. Ancient Demesne.
CHAP. V. Of Rents, and their seuerall natures.
CHAP. VI. Of Reprises and Deductions.
CHAP. VII. Obseruations and courses to be held and taken, before the begin∣ning of a Suruey.
CHAP. VIII. What courses are first to be held in the beginning of a Suruey.
CHAP. IX. The order of keeping a Court of Suruey.
CHAP. X. The order and manner of entering the Tenants euidence, and their se∣uerall estates.
CHAP. XI. The meanes and order of perfecting the Booke of entries last mentio∣ned, and the due placing therein of the seuerall contents of euerie particular found by measure through the whole Mannor, with the valuation thereof.
CHAP. XII. The manner and order of receiuing the Juries verdict, and the courses therein to be obserued.
CHAP. XIII. The forme and order of ingrossing a Suruey.
Conclusion of the Demesnes.
Conclusion of the Towne-ship of BRANTON.