QVI IN ME VIVIT PRO ME MORTUUS EST. 
ANNO SALUTIS NOSTRAE J616.   portrait of Aaron Rathborne   


ARTIFEX  
Arithmetica, Geometria  
THE SURVEYOR in Four books by AARON RATHBORNE  
Thesaurun & talentam ne abscondas in agro.  
Inertia stremia  
LONDON Printed by W: Stansby for, W: Burr. 1616.  W.H. fe.  engraved title page   

TO THE HIGH AND MIGHTY PRINCE CHARLES, Prince of WALES, Duke of CORNWALL, YORK, ALBANY and ROTHSAY: Marquis of ORMONT, Earl of ROSSE; and Baron of ARMANOCH: High Seneschal of SCOTLAND: Lord of the ISLES; and Knight of the most noble Order of the GARTER.  
IF in former ages (most noble Prince) the studies Mathematical were held meet for Princes; I doubt not but in these, they may implore (by your favourable admittance) the approbation and defence of your gracious patronage; which emboldens me the rather, to presume the dedication of these my labours to your highness protection. Wisdom is defined by CICERO to be divinarum atque humanarum rerum scientia, of the former part of which definition (being the most absolute) I will leave to speak unto those who can better writ; yet will thus much aver, that no man shall obtain the absolute perfection thereof, being absolutely ignorant of the rules, rudiments and principles of Mathematical discipline, as the due consideration of that sacred and mystical Unity and Trinity, may well approve. And how available and important they are, for the attaining to that humanarum rerum scientiam, in Peace or War, is worthily witnessed by PLATO, VEGETIUS, LIVIUS, and other Authors; who testify of LYCURGUS, and that famous Syracusean ARCHIMEDES; by the one, what excellent Laws and Ordinances were established and ministered, in the time of peace; and by the other, what more then wonderful devices and stratagems were wrought against the invincible forces of MARCELLUS, in the time of War, which they worthily impute to this their scientiae mathematicarum. But should my weakness here undertake to illustrate the excellency of that worth, which all worthy men admire; and that to your Highness, whose judgement is best able to discern; were but to deprave the one and the other, and rest in mine own reproach. Wherhfore, assuring myself, of your highness love and affection to these Arts, and your respect to the Professors thereof; with your power and ability of defence, against the malignant courses of malicious detractors, I presume in all humility to entreat your Patronage of these my labours, which in all duty I prostrate at your highness feet; with continual invocation to the Prince of Princes, ever to preserve your Princely dignity.  
Your highness most humble and devoted, AARON RATHBORNE.   


THE HIGH & MIGHTY PR. CHARLES PR. OF WALES. D. OF CORN: YOR: ALB: & ROTHS: MARQ: OF ORM: etc.   portrait of Charles I 

To whom great Prince can else this work be due.  
Than you, now placed where All is in you ● view  
And being the rule of what the bcoble do.  
Are both the Scale, & the Surveyor too.   
honey SOIT QVI MAL Y PENSE  royal blazon or coat of arms  Fra Delaram Sculp.   

The Preface.  
I Am not ignorant (friendly Reader) that hitherto, in writing, never any man pleased all; nor will I expect to be the first. To persuade the courteous, were causeless, for they are naturally kind; and to dissuade the captious, were bootless, for they will not be diverted: Let the first make true use of these my Labours, and they shall found much pleasure and profit therein; let the last (if they like not) leave it, and it shall not offend them. To make apology or exornation, in defence or commendation of the subject whereof I treat, were needless; it being already, in the world's opinion, of sufficient ability and reputation, both to defend and commend itself: Only of my manner and order in handling the same, I will say somewhat, as briefly as I may, for thy better instruction and understanding thereof.  
In general, I have disposed and digested the same into four Books; whereof, the two former tend specially to the principles and rules of Geometry, with performance thereby of many useful and necessary conclusions; and the two later, to matter of survey, with many instrumental conclusions, tending as well to that, as divers other purposes.  
Moore particularly, in my first Book, I begin with the Matters, Grounds, and Elements of GEOMETRY, as the definitions and terms of art belonging thereunto; most fitting first by the practitioner to be learned and well understood: then next have I placed therein divers Geometrical THEOREMS, as the foundations, grounds, and reasons whereon the practic part dependeth.  
In the second (having formerly laid the foundation) I show the means and practic operation of many necessary conclusions and Geometrical PROBLEMS, as the distinction, application, and division of Lines and Angles; and the description, reduction, addition, inscription, transmutation, division, and separation of all forms and kinds of superficial Figures, with their several dimensions. And considering, that as well in the THEOREMS of the first Book, as the PROBLEMS in the second, I wholly omit (for brevities sake, and avoiding confusion to the learner) their several demonstrations, using only explication and construction; I therefore express in the Margin against those THEOREMS and PROBLEMS, where, how, and in what place of EUCLID, RAMUS, and other Authors, to found their several demonstrations at large: and likewise, at the end of each construction, I have inserted the like numbers and notes of reference, from the THEOREMS in the first Book, to the PROBLEMS in the second; and the contrary: whereby most plainly and readily is found, and had, as well the reason and ground of any PROBLEM proposed, as the effect and operation of any THEOREM delivered.  
In the third Book I begin with the description of the several Instruments fit and usual for Survey; and of their several uses: wherein somewhat have I spoken (though too sparingly) concerning their abuse; being now grown shamefully general, by the multitude of simple and ignorant persons (using, or rather abusing, that good plain Instrument, called the Plain Table) who having but once observed a Surveyor, by looking over his shoulder, how and in what manner be directs his sights, and draws his lines thereon; they presently apprehended the business, provide them of some cast Plain Table, and within small time after, you shall hear them tell you wonders, and what rare feats they can perform; yea, and will undertake (or I will for them) that for ten groats a day, and their charges defrayed, they shall be able to undo any man they deal with; or at leastwise, to do him such wrong and prejudice, as perhaps he might, with more ease, and less loss, have given ten pounds a day to one that would have spoken less, and performed more. But what should I say more of them then thus, Monoculi inter caecos oculissimi sunt, and so will I leave the blind, with tumbling the blind into the mire.  
I further describe in this third Book, the composition and use of an Instrument of mine own, which I call the Peractor, and of a Chain, which I call the decimal Chain, with the divisions and parts thereof: which rightly understood and practised (I dare boldly say without ostentation) is far more useful and absolute for speed and exactness, than any yet ever used: And I will maintain by sufficient demonstration, that no man (using not the same, or the like) shall attain to the same or the like perfection, for precise exactness in any dimension, as I will thereby perform.  
And I further show therein, the best, speediest, and exactest means, for the survey and instrumental mensuration of a Manor, or any other superficial content whatsoever, by divers and several means; with many extraordinary observations and courses, therein to be had and taken, not usually known or practised; as by the argument of that Book more particularly appeareth.  
Wherein by the way I would advise the Reader, who desireth to make use thereof, and to profit himself thereby, in reading and practising; to take the Chapters before him as they lie in order; for that I have strived to place them in such an orderly and methodical form, as the one necessarily follow the other in use and practise; well knowing that disorder and irregularity in this kind, breedeth not a little trouble and confusion to the weak practitioner.  
The fourth and last Book, consisteth of the legal part of Survey; wherein I first show what a Manor is and the several parts thereof, with the appendants thereunto; how the same is created and maintained, and how and by what means destroyed and discontinued: also the several sorts and kinds of estates whereby any lands or tenements may be holden, and the several tenors, rents and services depending on those estates: I further show therein the order and manner of keeping Courts of Survey, with the entry of the tenants evidence and estates; and the orderly and artificial manner of engrossing the same, with many other necessary rules and observations tending to those purposes, as more at large also appeareth by the argument of the same Book.  
And here, as before, would I advise the practitioner, to observe the like course in reading and practising the rules and instructions of this Book, as I have formerly directed for the third, for that I have strictly observed the like decorum in placing the Chapters each after other, as of necessity they are to be used and practised.  
Now might I here much enlarge and protend this Preface, in explicating the wonderful use of the two former Books, in the performance of infinite conclusions Geometrical, so far passing this subject of Survey, as it in itself exceedeth the meanest matter of dispose, which (to avoid prolixity) I will here forbear, leaving the consideration thereof to thine own judgement, when thou shalt find therein by thy diligent practice the sweet and pleasing taste of such sense-beguiling fruit.  
And further might I amplify the same, not only in declaring the great and infinite pleasure, with no less profit, which the true knowledge, use, and understanding of the two later Books may bring (aswell to Surveyors, as all owners and occupiers of land in general) but also of the antiquity and necessity of Survey; howsoever slighted by many, who will not bestow a penny in points, or two pence in tape, or the like, but they will number the one, and measure the other, before they pay for either, and yet will disburse many thousands in a purchase, without the certain knowledge of either quantity, quality, or value thereof: (and these are those which are called penny-wise, etc.) whereby it often happens (as I have often known) that a valuable purchase being made, within few weeks after, the money hath been raised of the woods, and the lands perhaps immediately sold for much more than the money disbursed; and the same again vented at the third hand, hath yielded a double value: and all this unseen, and unsurueyed, with what disadvantage to the first vendor, I will leave to the consideration of my young Master, who hath thus offended in selling all, and resteth now in repentance, with full resolution not to offend in the like. And the like have I known of a purchase made, when a moiety of the charge could scarcely be raised.  
But to spend time to this purpose, were to little end, and therefore will I end this purpose; only entreating thee gentle Reader, that as I have thus employed mine idle hours, to found thee hours of employment; if thou reap either pleasure or profit by these my pains, to afford me thy good opinion (for Virtus laudata crescit, & honos alit arts) which is all I crave.  
AARON RATHBORNE. From my Lodging, at the house of M. ROGER BURGIS, against Salisburie-house-gate in the Strand, this sixth of November, 1616.   

THE SURVEYOR The first Book.  
THE ARGUMENT OF THIS BOOK.  
THis first Book consisteth of two parts; the former whereof entreateth only of the first Matters, Grounds, and Elements of GEOMETRY, as the distinction of Lines, Angles, Triangles, and other Figures, with their DEFINITIONS, showing what they are: The second part containeth diverse Geometrical THEOREMS, tending chiefly and most fitly to the subject and matter prosecuted in the subsequent Books, whereby the ingenious Practitioner may readily conceive and apprehended the ground and reason of the Precepts, Rules, and Problems therein delivered.  

THE FIRST PART.  

DEFINITION I A Point is that which is the lest of all Materials, having neither part nor quantity.  
Between Unity in ARITHMETIC, 
Euc. 1. Def. 1.  and this Point in GEOMETRY, there is a near resemblance: but that, more simple and pure; this, material; and (although the lest that can be imagined) requireth position and place, as this Point A.   

DEFINITION II A Line is length, without breadth-or thickness.  
THis is the first quantity in GEOMETRY, 
Euc. 1. Def. 2.  and may be divided into parts, in respect of his length, but admitteth no other division or dimension; and hath for his terms and limits that Geometrical point formerly spoken of. And of these lines are there two sorts; namely right, as the line A. and crooked, or spherical, as the line B.  
 diagram   

DEFINITION III A Superficies is that which hath only length and breadth.  
THis is the second quantity in GEOMETRY, 
Euc. 1. Def. 5.  having two several dimensions, namely, length and breadth, without depth or thickness (for that belongs to a solid or body, being the third quantity in GEOMETRY, and impertinent to this place, whereunto is attributed all three dimensions, as length, breadth, and thickness.) And as a line is limited with points, so is a Superficies with lines, and a Solid or Body with Superficies. As the figure A. being a Superficies, hath for his length B. C. or D. E. and for his breadth B. D. or C. E. which four lines are the bounds, limits, and terms of the same Superficies.  
 diagram   

DEFINITION FOUR An Angle is the congression or meeting of two lines in any sort, so as both make not one line.  
 diagram  diagram  diagram  diagram  
GEnerally of Angles (in respect of their lines) there are three sorts; 
Euc. 1. Def. 8.  namely, rightlined, spherical, and mixed. As the Angle A. is called a rightlined Angle, being composed of two right lines; the Angles B. and C. are spherical, or crooked Angles; and the Angle D. a mixed Angle, being caused of both.   

DEFINITION V If a right line fall on a right line, making the Angles on either side equal, each of those Angles are called right Angles: And the line erected is called a Perpendicular line unto the other.  
AS in general there are three sorts of Angles, in respect of their lines, 
Euc. 1. Def. 10.  as aforesaid; so in particular, of rightlined Angles there are three sorts, in respect of their quantities; namely, an Orthogonall, or right Angle; an Obtuse, or blunt Angle; and ●n Acute, or sharp Angle: whereof, according to this definition, the line A. B. ●●ling on the line C. D. and making the Angles on either side equal, namely, the Angle A. B. C. on the one side equal to the Angle A. B. D. on the other side; those two Angles are called right Angles: And the line A.B. erected on C.D. without inclination to either side, is called a Perpendicular, or plumb line. And here note further, that usually an Angle is described by three letters; of which, the second or middle letter representeth always the Angle intended. PROB. 4.5.6.9.  
 diagram   

DEFINITION VI An Angle which is greater than a right Angle, is an obtuse Angle.  
Every Angle in general (not being a right Angle) whether greater or lesser, is called an Obliqne Angle: 
Euc. 1. Def. 11.  but particularly, if greater than a right Angle, it is called an Obtuse Angle; if lesser, an Acute Angle: As the Angle B. D. C. (being greater than the right Angle A. D. C.) is an Obtuse Angle; for it containeth it, and also the Angle A. D. B.  
 diagram   

DEFINITION VII. An Acute Angle is that which is less than a right Angle.  
THis Definition is manifest by the former Diagram, wherein the Angle A. D. B. is an Acute Angle, 
Euc. 1. Def. 12.  being less than the right Angle A. D. E. for the same right Angle containeth it, and also the Angle B. D. E. being likewise an Acute Angle.   

DEFINITION VIII. A Figure is that which is contained under one o● many limits.  
AS these three figures, A. B. and C. whereof the first is contained under one limit; 
Euc. 1. Def. 14.  the second, under three; and the third, under four; and the like of others. Wherein is to be noted, that of two right lines, no Figure can be contained.  
 diagram  diagram  diagram   

DEFINITION IX. A Circle is a plain Figure, and contained under one line, which is called the Circumference thereof.  
A Circle, 
Euc. 1. Def. 15. R●m. 1.15.  of all other Figures, hath the priority, being of all most perfect and absolute; and therefore most fitting first to be defined: as the Figure A. in the last Diagram.   

DEFINITION X. The Centre of a Circle is that point which is in the midst thereof: from which point, all right lines, drawn to the Circumference, are equal.  
AS the point B. in this Figure is the Centre thereof, 
Euc. 1. Def. 16.  from whence the lines B. A. B. C. and B. D. being drawn to the Circumference (and as many others as are s● drawn) are all equal, and are semidiameters to the same Circle. PROB. 34.1.  
 diagram   

DEFINITION XI. The Diameter of a Circle is a right line, passing by the Centre through the whole Circle, and divideth the same into two equal parts: Either half of which Diameter, is called the Semidiameter of the same Circle.  
AS the line A. B. C. in this circle is the diameter thereof, 
Euc. 1. Def. 17.  for that it passeth by the centre B. through the whole circle, as from A. to C. and also divideth the circle into two equal parts, the one half towards D. the other towards E. Either half of which Diameter, as A. B. or B. C. is called the Semidiameter of the same Circle. PROB. 34.  
 diagram   

DEFIN. XII. A Semicircle is a Figure contained under the Diameter of a Circle, and the semicircumference of the same Circle.  
AS supposing the circle A. B. C. the Diagram of the last DEFINITION, 
Euc. 1. Def. 18.  to be divided into two equal parts by the diameter thereof A. C. whereby two Figures are projected, namely, A. D. C. and A. E. C. Now by this DEFINITION, each of these Figures are Semicircles, for that the one of them is contained under the diameter of the same circle A. C. and the semicircumference A. D. C. and the other under the same diameter, and the semicircumference A. E. C.   

DEFIN. XIII. A Segment, Section, or Portion of a Circle, is a Figure contained under a right line, and part of the circumference, either greater or less than the Semicircle.  
AS the diameter of a circle passing by the centre thereof, 
Euc. 1. Def. 19.3. Def. 5.  divideth the same into two equal parts; so any right line, drawn from any one part  diagram of the circumference to any other part thereof (and not passing by the centre) divideth the circle into two unequal parts, which are called Segments, Sections, or Portions of a circle: As in this circle, the figure A. C. B. because it is contained under the right line A. B. and the lesser part of the circumference A. C. B. is called a lesser Segment, Section, or Portion of a circle; and the figure A. D. B. because it is contained under the right line A. B. and the greater part of the circumference A. D. B. is called a greater Segment, Section, or Portion of a circle. Here note also, that these parts and such like of the circumference so divided, are commonly called Arches, or arch lines. And all lines (less than the diameter) drawn and applied as the line A. B. are called cords, or cord lines, of those Arches which they so subtend; or Subtenses, because they subtend both segments. THEOR. 73. PROB. 34.   

DEFIN. XIIII. Equal Circles are such as have equal Diameters, or whose lines, drawn from their Centres, are equal.  
AS these two circles A. and B. are equal, 
Euc. 3. Def. 1.  their diameters being equal, namely, C. D. and E. F. or their Semidiameters, (which, according unto this DEFINITION, are lines drawn from the centres unto their circumference) as A. C. or A. D. and B. E. or B. F.  
 diagram  diagram   

DEFIN. XU. A right line is said to touch a circle, which touching it, and being extended or produced, doth not cut the circumference thereof.  
AS the right line A. B. being drawn by the point C. doth there only touch the circle, 
Euc. 3. Def. 2.  and being produced unto B. cutteth not the circumference thereof. This line is commonly called a Tangent, or Contingent line: whereof there is great and infinite use, in many Conclusions, Geometrical and Astronomical, especially in the mensuration and resolution of Triangles, as well rightlined, as spherical, by the Canons and Tables of Synes, Tangents, and Secants, in that behalf calculated.  
 diagram   

DEFIN. XVI. An angle of a Section, or Segment, is that which is contained under a cord line, and the arch line of the same Section.  
AS the angles A. B. C. and B. A. C. in the lesser segment of this circle, 
Euc. ●. Def. 6.  are angles of a Section, because they are contained under the cord line A. B. and the arch line A. C. B. Also the angles D. B. A. and D. A. B. in the greater Segment are angles of the same Segment, by the like reason. And all angles of this kind are called mixed angles, because they are contained under a right line and a crooked. Of which two Segments, the lesser hath always the lesser angle; and the greater, the greater angle.  
 diagram   

DEFIN. XVII. An angle in a Section, or Segment, is when two right lines are drawn from any point in the arch line, to the ends or extremes of the cord line; the angle in that point of the arch line is called an angle in a Section or Segment.  
AS the angle A. B. C. in the lesser Segment is an angle in a Section, 
Euc. 3. Def. 7.  or Segment, by reason that the two right  diagram lines B. A. and B. C. are drawn from the point B. in the arch line to the ends or extremes of the cord line A. C. And also the angle A. D. C. in the greater Segment is an angle in a Section, or Segment, because the two right lines D. A. and D. C. are drawn from the point D. in the arch line to the ends or extremes of the cord line A. C. And here note, the greater Section hath in it the lesser angle, and the lesser Section the greater angle, contrary to the mixed angles in the precedent DEFINITION mentioned. And here also is to be noted, by the declaration of this and the former DEFINITION, the difference between an angle of a Segment, and an angle in a Segment; the first being called a mixed angle, and this a right lined angle.   

DEFIN. XVIII. If two right lines be drawn from any one point in the circumference of a Circle, and receive any part of the same circumference, the angle contained under those two lines is said to belong and to be correspondent to that part of the circumference so received.  
AS the angle B. A. C. contained under the right lines A. B. and A.C. drawn from the point A. and receiving the circumference B. D. C. by this DEFINITION is said to belong, 
Euc. 3. Def. 8.  subtend, and pertain unto the circumference B. D. C. And if right lines be drawn from the centre to the former points B. and C. then is that angle said to be in the centre of a circle, as the angle B. E. C. Which angle likewise subtendeth the same circumference B. D. C. and is always double in quantity to the former angle, drawn from the circumference. And the circumference B. D. C. is also the measure of the quantity or greatness of the angle B. E. C. in the centre.  
 diagram   

DEFIN. XIX. A Sector of a Circle is a figure contained under two right lines, drawn from the centre of a Circle, and under part of the circumference received of them.  
AS in the last DEFINITION, 
Euc. 3. Def. 9 Ram. 16.3.  the figure B. E. C. is the Sector of a circle, because it is contained under the two right lines E. B. and E. C. drawn from the centre E. and under part of the circumference, namely, B. D. C. received of them.   

DEFIN. XX. Right lined figures are such, as are contained under right lines, of what number soever, above two.  
AS those which follow, 
Euc. 1. Def. 20.  being contained under three, four, five, or more sides, who take their denominations, as well of the number of their angles, as of their sides; so a figure contained under three lines, in respect of his sides, is called a three sided figure; and in respect of his three angles is called a Triangle: and so of the rest. Where is to be noted, that every right lined figure hath as many angles as it hath sides.   

DEFIN. XXI. An Equilater Triangle is that, which hath three equal sides.  
AS a Triangle is the first of all right lined figures (for under less than three right lines can no figure be contained: 
Eus. 1. Def. 24. Ram. 8.8. ) So  diagram of all Triangles, the Equilateral Triangle is most simple and absolute, having equal lines and equal angles; containing every of them a Sextans of a Circle, which is 60 degrees. Triangles have their denominations, differences, and appellations, as well of their angles as their sides: As this Triangle A. in respect of his three equal sides, is called an Equilater Triangle; and in respect of his sharp angles, is termed an Oxigontum, or acute angled Triangle. But this appellation (in mine opinion) to this kind of Triangle, is needless; for that the name of an Equilateral Triangle doth imply the same, seeing it can contain no other but acute angles: for if in any Triangle there be either a right or an obtuse angle, the sides cannot be all equal, and therefore by this DEFINITION no Equilater.   

DEFIN. XXII. An Isosceles is a Triangle, which hath only two equal sides.  
 diagram  diagram  diagram  
THis is the second sort of Triangles, 
Eue. 1. Def. 25.  and hath two sides only of one length, the third being either longer or shorter: As these three Triangles A. B. and C. have every of them two equal sides, and are therefore called Isosceles. But in respect of their angles, the Triangle A. is called an Orthigonium, or right angled Isosceles, for that his angle at the point D. is a right angle. Also the Triangle B is called an Ambligonium, or an obtuse angled Isosceles, for that his angle at the point G. is a blunt or obtuse angle. And likewise the Triangle C. is called an Oxigonium, or acute angled Isosceles, because all his angles are acute or sharp. Also this Triangle is called an Equicrural Triangle, in respect of his two equal sides.   

DEFIN. XXIII. A Scalenum is a Triangle which hath all his sides unequal.  
 diagram  diagram  diagram  
THis is the third kind of Triangles, 
Euc. 1. Def. 26.  and hath all his sides of several lengths: As these three Triangles A. B. C. have every of them all their sides unequal, and therefore called Scalena. But in respect of their angles, the triangle A. is called an Orthigonium, or right angled Scalenon, for that his angle at the point D is a right angle. Also the triangle B. is called an Ambligonium, or an obtuse angled Scalenon, for that his angle, at the point I is a blunt or obtuse angle: And lastly, the triangle C. is called an Oxigonium, or an acute angled Scalenon, because all his angles are acute or sharp. It is to be noted generally in all Triangles, that in comparison of any two sides of a Triangle, the third side is called the Base; as of the Triangle A. in respect of the two lines E. D. and E. F. the line D. F. is the base: In regard of the two lines F. D. and F. E. the line E. D. is the base; and in respect of the two lines D. E. and D. F. the line E. F. is the base.   

DEFIN. XXIIII. A Square, or Quadrat, is a four sided figure, whose sides are all equal, and angel's all right angles.  
AS the figure A. B. C. D. is a Square, 
Euc. 1. Def. 30. Ram. 12.2. 2. Con. 1.  or Quadrat, because all the lines thereof are equal, and all the angles right angles. These four sided figures likewise, as well as Triangles, take their appellation partly of their sides, and partly of their angles; as by their several DEFINITIONS hereafter appeareth.  
 diagram   

DEFIN. XXV. A long Square is that whose angles are all right angles, and whose opposite sides only are equal.  
THis figure differeth little from the Square, 
Euc. 1. Def. 31. Ram. 1.13.  or Quadrat, last defined, having all equal angles like unto it; but the sides are unequal. As in this figure A. B. C.D. all the angles are right angles, and the opposite sides only are equal, as the length A. B. is equal to the length C. D. and the breadth A. C. to the breadth B. D. but compare them otherwise, and they are unequal.  
 diagram   

DEFIN. XXVI. A Rhombus (or Diamond) is a figure with four equal sides, but no right angle.  
AS this figure A. B. C. D. is a Rhombus, having all his sides equal, and likewise the opposite angles; 
Euc. 1. Def. 32. Ram. 8.14.  but the angles at A. and D. are acute angles, and those at B. and C. obtuse. Between a Square, or Quadrat, and this figure, is much resemblance, either kind having all  diagram sides equal; and likewise their angles in general quantity; but different in particular quality; that having four right angles, this two obtuse; and two acute angles; yet are they in general quantity equal to four right angles: for by how much the two acute angles are defective or wanting of two right angles, by so much are the obtuse angles abounding or exceeding. This figure is described by the connexion of two Equilater Triangles, by any two of their sides, as appeareth by the pricked diagonal line B. C. which being omitted and left out, this figure remaineth perfect, and hath his acute angles equal to those of an Equilater, namely, 60. degrees, and the obtuse angles double thereunto. PRO. 57   

DEFIN. XXVII. A Rhomboydes (or Diamond like) is a figure, whose opposite sides and opposite angles are only equal, and bathe no right angles.  
AS this figure A. B. C. D. is a Rhomboydes, 
Euc. 1. Def. 33. Ram. 9.14.  and hath his sides A. B. and C. D. opposite and equal, and likewise A. C. and B. D. but hath no right angle: For the angles at the points A. and D. are acute, opposite, & equal; and likewise the angles, at the points B. and C. are obtuse, opposite, and equal.   diagram 
Note here, that the four figures last before defined, namely, a Square, a figure of one side longer, a Rhombus, and a Rhomboydes, are commonly called Parallellograms; of which four, the two former are called right angled Parallellograms. PROB. 90.   

DEFIN. XXVIII. All other four lined figures, besides those formerly defined, are called TRAPEZIA, or Tables.  
AS all figures, of four sides, 
Euc. 1. Def. 34. Ram. 10.14.  which are made at adventure, without respect or regard of equality, or inequality, or observation of order, either in their lines or angles; which are therefore called irregular figures: as these figures A. and B. are.  
 diagram  diagram   

DEFIN. XXIX. Many sided figures are those which have more sides than four.  
 diagram  diagram  diagram  
OF these, 
Euc. 1. Def. 23. Ram. 11.14.  may infinite sorts be described, by addition of lines: but if they contain above four sides, they are generally called Polygona, but particularly according to the number of their sides: As the figure A. is called a Pentagon, because it is contained of five sides; the figure B. a Sexagon, being contained under six lines; and C. is called a Septagon, because it is contained under seven sides. And the like of others. PROB. 62.95.   

DEFIN. XXX. Either of those Parallellograms, which are about the diameter of a Parallelogram, together with the two supplements, is called a Gnomon.  
RIghtly to conceive this DEFINITION, 
Euc. 2. Def. 2.  it is requisite first to understand, what those Parallellograms are which are said to be about the diameter of a Parallelogram; and likewise, what supplements are. For the first, those are said to be parallellograms about the diameter, which  diagram have for their particular diameters part of that which the whole parallelogram hath: And supplements are such, as are without the whole diameter, the diameter passing between them, and cutting them not. As in the parallelogram A. B. E. D. the particular parallellograms H. K. E. F. and A. C. H. G. are said to be about the diameter, because they have for their particular diameters part of the whole diameter A. E. as A. H. and H. E. And the supplements are the two parallellograms C. B. K. H. and G. D. F. H. because they are without the whole diameter A. E. which passeth between them, and cutteth them not. Now take away either of those particular parallellograms, which soever it be, and the other remaining, together with the two supplements, is that which by this DEFINITION is called a Gnomon. PROB. 101. THEOR. 5.   

DEFIN. XXXI. That r●●ht lined figure is said to be inscribed in another right lined figure, which hath every angle touching every side of the figure wherein it is inscribed.  
AS in these two figures, the Triangle A. B. C. is  diagram  diagram said to be inscribed within the Triangle D. E. F. because every of his angles A. B. and C. doth touch every side of the Triangle D. E. F. Likewise, the square G. H. I K. is said to be inscribed within the greater square L. M. N. O. because every of his angles G. H. I K. toucheth every side of the same greater square. The like consideration is to be had of circumscribing one right lined figure about another.   

DEFIN. XXXII. A right lined figure is inscribed within a circle, when every angle of the inscribed figure toucheth some part of the circles circumference.  
ALl the angles of a regular right lined figure, inscribed in a circle, or the sides of the like figure circumscribed about a circle, 
Euc. 4. Def. 3.  may easily touch the circumference thereof, by reason of the perfection and uniformity of a circle. As the Triangle A. B. C. is inscribed in the circle A. B. C. and also the square D.E.F.G. is circumscribed about  diagram  diagram the circle A. for that every angle of the Triangle inscribed, and every side of the square circumscribed, toucheth some one point of the circumference of the circle. And the like consideration is to be had of circles inscribed or circumscribed within or about any right lined figure. PROB. 112.113.   

DEFIN. XXXIII. The altitude of a figure, is the parallel distance between the top of a figure and the Base.  
 diagram  diagram  diagram  
AS the height or altitude of this Triangle A.B.C. is the space or distance between the line D.E. drawn parallel to the Base B.C. by the highest point of the same Triangle, 
Euc. 6. Def. 4.  as by A. and the same Base B.C. which parallel distance is equal, and the same thing with the perpendicular A. F. And the like of the rest.   

DEFIN. XXXIIII. Parallel lines are such, as being drawn on any plain Superficies, and produced either way infinitely, do never meet or concur.  
AS these right lines A. and B. 
Euc. 1.35. Ram. 2.11.5.11.  which being produced and drawn forth infinitely, by reason of their equal  diagram  diagram and parallel distance the one from the other, will never meet or concur; and therefore are called parallel lines. PROB. 2.3.   

DEFIN. XXXV. A right line is said to be divided by extreme and mean proportion, when the lesser part, or segment thereof, is to the greater, as the greater is to the whole line.  
As the line A. B. being so divided in the point C. that the lesser part, or segment, C. B. hath the same proportion  diagram to the greater part, or segment, A. C. as the same greater part hath to the whole line A.B. then is the same line A. B. divided by an extreme and mean proportion. The means how to divide a line in such sort, is hereafter taught by the 20. PROB. of the next book. This line is of wonderful and infinite use in many Geometrical operations, as appeareth manifestly almost through the whole thirteenth book of Euclid. PROB. 20.   

DEFIN. XXXVI. The power of a line is the square of the same line, or any plain figure equal to the square thereof.  
AS the power of the line A. B. is the square of the same line, namely, the figure  diagram A. B. C. D. or any other plain figure equal thereunto. And so great power and ability is a line said to have, as the quantity of the square it makes: As this line A. B. containing 4. the power thereof is 16. In this kind is the diagonal or diameter of a square (as the line A. D.) said to be double in power to the side of the same square, for that a square made of the diagonal, is double in quantity to the square made of the side. And likewise the line which subtendeth the right angle in an Orthigonall Triangle, is said to be equal in power to both the containing sides: as the line A.D. which subtendeth the right angle A. C.D. in the Triangle A.C.D. is equal to both the squares made of the two containing sides, namely of A.C. and C.D. PROB. 23.   

DEFIN. XXXVII. To divide a given line in power, is to find two other lines, whose squares together shall be equal to the square of the given line, but the square of the one to the square of the other, to be in any proportion required.  
AS if A. B. were a line given, and it were required  diagram to divide the same line in power, according to the proportion of 2. to 3. It is hereby intended to find two other lines, as C. A. and C. B. whose squares together are equal to the square of the given line A.B. but the square of the one, namely C. A. is to the square of the other C. B. in such proportion, as 2. to 3. that is, the square of C. B. containeth the square of C. A. once and a half. The means how to perform the same, is hereafter taught in the 23. PROB. of the 2. Book. PROB. 23.   

DEFIN. XXXVIII. To enlarge a line in power, is to found another line, whose square shall have any proportion required (of the greater inequality) to the square of the given line.  
AS suppose the line 2 B in the former Diagram were given to be enlarged in power, as 3. to 5. It is hereby intended to found out another line, as the line C. B. whose square shall bear such proportion to the square of the given line 2 B. 
These lines are of infinite use in many Geometrical Conclusions.  as 5 to 3. which is a proportion of the greater inequality, and is called Superbipartiens tertias: that is, as 5 contains 3. so the square of C. B. the line sought for, containeth the square of 2 B. 1 ⅔·S The working whereof is taught in the 24. PROB. of the 2. book.   

DEFIN. THIRTY-NINE. A mean proportional line is that, whose square is equal to the right angled Parallelogram, or long Square, contained under his two extremes.  
A Mean proportional line  diagram is so termed, in respect of the relation it hath to two other lines, which are called his extremes, for of a mean without extremes, or extremes without a mean, there is no comparison. As in this Diagram, the perpendicular A.C. of the right angled Triangle A. B. D. is a mean proportional line between the two segments of the Base B.C. and C.D. his extremes, because the square of the same line A. C. namely, A. E. F. C. is equal to the long square, contained under the lines B. C. and C. D. (for the line C. G. is equal to the line B. C.) for as B. C. is to A. C. so is A. C. to C. D. Also the line A. B. is a mean proportional between B. C. the segment of the Base lying next unto it, and B.D. the whole Base; for as B. C. the lesser segment of the Base, is to B. A. so is B. A. to B. D. the whole Base. And lastly, the line A.D. is a mean proportional, between C. D. the segment of the Base, lying next unto it, and B.D. the whole Base; for as C. D. the greater segment of the Base, is to A.D. so is A. D. to B. D. the whole Base.   

DEFIN. XL. Like right lined figures are those, which have equal angles, and proportional sides about those equal angles.  
AS in these two right angled Parallellograms, 
Euc. 6. Def. 1.  the angle in the point  diagram A. of the greater, is equal to the angle E. of the lesser; likewise the angle B. to the angle F. and C. to G. and D. to H. And moreover, the side A. B. hath that proportion to the side A. C.  diagram as E. F. hath to E.G. and A.C. to C.D. as E. G. to G. H. and so of the rest: Wherhfore these two Parallellograms are called like right lined figures: and so of Triangles and all other figures, of what kind soever. PROB. 45.   

DEFIN. XLI. Reciprocal figures are such, as have the sides of either to other mutually proportional.  
As the Parallellograms A. and B. have  diagram their sides mutually proportional; 
Euc. 6. Def. 2.  that is, as the side C. D. of A. is to the side G. H. of B. so is the side G.I. of B. to the side C. E. of A. and therefore are they called Reciprocalls: for as 20. is to 15.  diagram an antecedent of A. to a consequent of B. so is 12. to 9 an antecedent of B. to a consequent of A.   

DEFIN. XLII. The quantity or measure of an angle, is the arch of a circle, described from the point of the same angle, and intercepted between the two sides of that angle.  
As in the Triangle A. B. C. the measure or quantity of the angle B. A. C. is the arch B. D. or E. F. for the  diagram circumference of every circle (whether greater, or lesser) is divided into 360 equal parts, which are called degrees, and every degree into 60 scruples or minutes, and every minute into so many seconds, etc. Which parts or degrees are greater or lesser, as the circles, whose parts they are, are greater or lesser; and those arches which contain the same number of parts or degrees in equal circles, are equal; and in unequal circles, they are called like arches; as the arches B. D. and H. G. are equal; but the arches B. D. and E. F. are like arches: for as B.D. is 50 degrees in the greater circle, so is E. F. 50 degrees in the lesser circle. And the like of others. PROB. 8.   

DEFIN. XLIII. The Quadrant of a circle is the fourth part thereof, or an arch containing 90 degrees.  
AS the arch K. B. D. in the former Semicircle is a Quadrant of that whole circle, or a fourth part thereof, and containeth 90 degrees.   

DEFIN. XLIIII. The Compliment of an arch, less than a Quadrant, is so much as that arch wanteth of 90 degrees.  
As the Compliment of the arch B. D. 50 degrees in the former Semicircle, is the arch K. B. 40 degrees.   

DEFIN. XLV. The Excess of an arch, greater than a Quadrant, is so much as the said arch is more than 90 degrees.  
AS the Excess of the arch H. K. B. D. 130 degrees, is the arch H. K. 40 degrees more than a Quadrant, that is more than K. D.   

DEFIN. XLVI. The Compliment of an arch, less than a Semicircle, is so much as that arch wanteth of a Semicircle, or of 180 degrees.  
AS the arch H.K. B.D. is an arch less than a Semicircle, and containeth 130 degrees, and the Compliment thereof to a Semicircle, is the arch H.G. 50 degrees, which is so much as the arch H. K. B. D. wanteth of a Semicircle, or of 180 degrees.   

DEFIN. XLVII. The Compliments of Angles are as the Compliments of Arches.  
AS the arch K. B. is the compliment of the arch B. D. to a Quandrant, and the arch B. D. of the arch K. B. So the angle K.A.B. 40 degrees is the compliment to a right angle of the Angle B.A.D. 50 degrees; and likewise the same angle B. A. D. of the same angle K.A.B. And in this sense is the third angle of any Triangle said to be the compliment of the other two, to two right angles, or a Semicircle: For the three angles of any Triangle are equal to two right angles; as is hereafter declared.    

❧ The second Part. Instructions concerning this Part.  
THis second Part consisteth of diverse Geometrical THEOREMS, or approved Truths; which are the Foundations, Grounds, and Reasons, whereon the Practic part dependeth. For as in the general course and tract of all designs, before the undertaking or execution of whatsoever action, the fittest means for an orderly performance, is judicially to consider; first, the Property, Passion, Nature, and kind of the intended enterprise; then, the best and most immediate means how to effect the same, and the Causes, Grounds, and Reasons, why, by those means such effects may be wrought; and afterwards, to put in execution: So before we enter into the Practic part, I will first here premise diverse THEOREMS concerning this subject, whereby the ingenious practitioner may most evidently conceive and understand the ground and reason of all the Rules and Problems in the following Books contained. Wherein I use only Explication and Construction, omitting (for brevity sake, and avoiding confusion to the Learner) their several Demonstrations; yet with such ample notes of direction in the Margin, as the Reader may readily found in EUCLID, RAMUS, and other Authors, their Demonstrations at large. And for their further ease and help, I have at the end of every Construction inserted the like notes of Reference from these THEOREMS to the following PROBLEMS, and the like from those to these; that having here the reason or cause, he shall there most readily found the effect; or seeing there the effect, he may as speedily understand the cause or reason thereof, Scire enim, proprie est, remper causam cognoscere.  

THEOREM I If any two right lines cut the one the other, the opposite or vertical angles are ever equal; and both the angles, on one and the same side of either line, are either of them right angles, or (being both taken together) are equal to two right angles.  
SVppose that B. E. and C.  diagram F. are two right lines, 
Euc. l. 1. p. 13.15. Ram. 5.8. Con 2. Ceul. 2.9.  which cut the one the other in the point A. Than I say, first, that the opposite or vertical angles are equal, namely, the angle B. A. C. to the angle F. A. E. and the angle B. A. F. to the angle C. A. E. for they are every of them right angles: 
Si ovotcunque recte in eodem puncto m●●●o seize in●ersecent omnes in communi sectione quatuor rectis aequabuntur.  and let the right line D. G. be likewise drawn, cutting the line B. E. in A. Than I further say, that both the angles, taken together on one and the same side of either line, is equal to two right angles, as the angles B. A. D. and D. A. E. on the upper side of the line B. E. and also the angles B. A. G. and G. A. E. on the neither side of the same line, are respectively equal to two right angles; for they consist of the right angles formerly mentioned. And the like of the angles on either side of the line D. G. PROB. 114, 117, 118.   

THEOREM II A right line, falling on two parallel right lines, maketh the outward angles on contrary sides of the falling line equal; and likewise the inward and opposite angles on the contrary sides of the same line; and also the outward angle, equal 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 equal to two right angles.  
LEt the right line E. F. fall on the  diagram two parallel right lines A. B. and C. D. Than saith this THEOREM, 
Euc. 1. p. 29. Ram. 7.9. Pit. 1.38.  first, 
Lineae eidem parallelae inter se sunt parallelae.  that it maketh the outward angles on contrary sides of the falling line, namely, the angles A. H. E. and F. G. D. to be equal; and likewise the inward and opposite angles on the contrary sides of the same line, as the angles A. H. F. and E. G. D. And also, that the outward angle, as A. H. E. is equal to the inward and opposite angle, on one and the same side of the falling line, namely, to E. G. C. And lastly, that the inward angles on one and the same side, as the angles E. G. C. and A. H. F. are equal to two right angles. PROB. 50.   

THEOREM III If a right line be divided into two equal parts, half the square of that whole line is double to the whole square of half the same line.  
SVppose A. B. to be a right line, and  diagram let the same be divided into two equal parts in the point C. Than I say, that half the square of that whole line, namely, the Parallelogram A. B. D. E. (for the whole square is A. B. F. G.) is double to the whole square of half the same line, namely, to the square A. C. D. H. as is manifest by the Diagram.   

THEOREM FOUR A right line being divided by chance, the square of the whole line is equal to both the squares made of the parts, and also to two rectangle figures, comprehended under the same parts.  
LEt the right line A. B. be divided  diagram by chance in the point C. Than I say, 
Euc. 2.4.  that the square of the whole line, namely, A. B. D. E. is equal to both the squares made of the parts, namely, to the squares A. C. G. H. and H. K. F. E. (for H. K. is equal to C. B.) and also to the two rectangle figures, comprehended under the same parts, namely, to the rectangle figures C. B. H. K. and G. H. D. F.   

THEOREM V The Supplements of those Parallellograms which are about the diameter in every Parallelogram, are always equal the one to the other.  
SVppose the figure A. B. D. E. in the former THEOREM, 
Complementa sunt aequalia. Euc. 1.43. Ceul. 2.81.  be a Parallelogram, whereof the diameter is A. E. and let the Parallellograms about the same diameter (according to the declaration of the 30. DEFINITION) be A. C. G. H. and H. K. F. E. Than I say, that the supplements of those Parallellograms, namely, the supplements C. B. H. K. and G. H. D. F. are equal the one to the other. PROB. 18.87.88.105.   

THEOREM VI In right angled Triangles, the square of the side subtending the right angle, is equal to both the squares of his containing sides.  

In triangulo rectangulo figura ad besin descripta aequatur fi●uris ad crura similibus similiterque sitis.  LEt the Triangle A. B. C. be a  diagram right angled Triangle, whose angle, at the point B. is a right angle; and let the line A. C. be the side subtending the same right angle, and B. A. and B. C. his containing sides. Than I say, 
Euc. 1.47. Pit. 1.50.  that the square of the side subtending the right angle, which is the square A. C. D. E. is equal to both the squares of his containing sides, namely, to the squares A. G. F. B. and B. H. K. C. PROB. 23.24.25.30.36.38.52.65.99.100.101.102.104.106.  
This former THEOREM, and the two next following, 
Nota.  are of infinite and wonderful use in most Geometrical Conclusions; especially in TRICONOMETRIE, or the supputation of Triangles, by the Canons thereof; as those excellent Tables of Logarithimes, or those of Synes, Tangents, and Secants, in that behalf calculated; and therefore especially to be regarded; and the most excellent properties and passions thereof to be well understood and practised.   

THEOREM VII. In obtuse angled Triangles, the square of the side subtending the obtuse angle, is greater than both the squares of the containing sides, by two rectangled figures, comprehended under one of the containing sides (being continued) and the line of continuation, from the obtuse angle to a perpendicular let fall thereon.  
LEt the Triangle A. B. C. be an obtuse angled Triangle, 
In triangulis obtusangulis basis plus potest cruribus duplici rectangulo ex allero crure & eius continuatione ad verticis perpendicularem E. 2. p. 12.  whose angle at the point C. is obtuse; and let the line A. B. be the side subtending the same obtuse angle, and A. C. and C. B. his containing sides; whereof, let A. C. be the side continued, and C. L. the line of continuation from the obtuse angle at the point C. to the perpendicular let fall thereon B. L. Now I say, that the square of the  diagram side, subtending the obtuse angle, namely, A. D. E. B. is greater than both the squares of the containing sides, namely, B. K. C. H. and A. C. F. G. by two rectangle figures, (which is all one, with one twice taken) comprehended under one of the containing sides (being continued) and the line of Continuation, namely, C. L. G. M. being twice taken. PROB. 44.   

THEOREM VIII. In acute angled Triangles, the square of the side subtending the acute angle, is less than both the squares of the containing sides by two rectangle figures, comprehended under one of the containing sides (whereon a perpendicular falleth) and that segment of the same side which is between the perpendicular and the acute angle.  

Theorema generale est ad investigationem perpendicularis intra triangulum cadentis data trium laterum quantitate.  LEt A. B. C. be an acute  diagram angled Triangle, having the angle at the point A. acute; let B. C. be the side subtending the same angle, and A. B. and A. C. the containing sides: also let B. L. be the perpendicular, 
Euc. 2.15. Ceul. 2.84.  A. C. the side whereon it falleth, and A. L. the segment thereof between the perpendicular and the acute angle A. Now I say, that the square of the side subtending the acute angle, namely, B. D. C. E. is less than both the squares of the containing sides, which are F. B. G. A. and A. C. H. K. by two rectangle figures (being all one, with one twice taken) comprehended under one of the containing sides A. C. (whereunto A. H. is equal) and the segment A. L. namely, A. L. H. I twice taken. PROB. 41.   

THEOREM IX. In rectangle Triangles, if from the right angle a perpendicular be let fall unto the Base, it shall divide the Triangle into two Triangles, like unto the whole, and also the one like unto the other.  
LEt A. B. C. be a recte angle  diagram Triangle, 
Euc. 6.8. Ram. 5.12.13.  whose angle at the point B. is a right angle; from whence, let the perpendicular B. D. be let fall to the Base A. C. Than I say, the perpendicular so falling, shall divide the Triangle into two Triangles, that is, A. B. D. and B. C. D. like unto the whole Triangle A. B. C. and also the one Triangle like unto the other; which is (according to the 40. DEFINITION) with equal angles, and proportional sides about those equal angles. PROB. 19.23.24.25.30.38.   

THEOREM X. An Isosceles, or a Triangle of two equal sides, hath his angles at the Base equal; and the equal sides being produced, the angles under the Base are also equal.  
LEt A. B. C. be an Isosceles, 
Euc. 1.5. Ceul. 2.3.4.  or a  diagram Triangle, whose two sides A. C. and A. B. are equal, and let A. C. be produced to D. and A. B. to E. I say then, that-his angles at the Base, namely, A. B. C. and A. C. B. are equal; and that the angles under the Base, as E. B. C. and D. C. B. are also equal, the one unto the other. PROB. 40.   

THEOREM XI. All equiangle Triangles have their sides, containing equal angles proportional, and their sides subtending equal angles, are of like proportion.  

Hoc Theorema praecipuum est totius Trigonometriae fundamentum.  SVppose  diagram  diagram A. and B. to be two equiangle triangles, that is, 
Euc. 6.4. Ram. 5.12.7, 9 Pit. 1.46. Ceul. 2.62.  having the angle D. equal to the angle G. and C. to F. and E. to H. Than I say, they have their sides, which contain those equal angles proportional; as D. C. and D. E. in the Triangle A. are proportional to G. F. and G. H. in the Triangle B. because they contain equal angles, namely, D. and G. for as D. C. is to D. E. so is G. F. to G. H. and the like of the rest: also their sides subtending equal angles, are of like proportion, as D. C. and G. F. subtending equal angles E. and H. and C. E. and F. H. subtending equal angles D. and G. are of like proportion: for as D. C. is to C. E. so is G. F. to F. H. And the like of the other sides and angles. PROB. 30.38.45.65.   

THEOREM XII. In any two Triangles compared, if two sides of the one be equal to two sides of the other, and the Base of the one to the Base of the other; they shall also have the angles contained under their answerable equal sides, the one equal to the other in either Triangle.  
LEt A. B. C. and D. E.  diagram  diagram F. be two Triangles compared, 
Euc. 1.8.  having two sides of the one equal to two sides of the other, as A. B. and A. C. of the one, equal to D. E. and D. F. of the other, and also the Base B. C. of the one, equal to the Base E. F. of the other. Now I say, they shall have their angles contained under answerable equal sides (as the angle A. contained under A. B. and A. C. equal to the angle D. contained under the answerable equal sides D. E. and D. F.) to be equal the one to the other. And the like of the rest.   

THEOREM XIII. If any side of a Triangle be continued, the outward angle made by that continuation, is equal to the two inward and opposite angles: And the three inward angles of any Triangle are equal to two right angles.  
LEt A. B. C. be a Triangle, 
Euc. 1.32. Ram. 6.9. Pit. 1.48.49. Ceul. 2.20.   diagram whereof let any of the sides be produced, as B. C. to D. Than I say, that the outward angle, made by that production or continuation, as the angle A. C. D. is equal to the two inward and opposite angles, namely, the angles C. A. B. and C. B. A. And also, that the three inward angles of any Triangle, as C. B. A. B. A. C. and A. C. B. are equal to two right angles. PROB. 111.114.117.118.   

THEOREM XIIII. In every Triangle, two of his angles, which too soever be taken, are less than two right angles.  
AS in the Diagram of the former THEOREM, take any two angles, 
Euc. 1.17.  as those at the points A. and C. or C. and B. or B. and A. and they are less than two right angles; for by the same former THEOREM all three of them are equal to two right angles.   

THEOREM XU. In every Triangle, two sides thereof (which too soever be taken) are greater (being joined together as one line) than the third side remaining.  
LEt A. B. C. (the Diagram of the 13. THEOREM) be a Triangle, whereof take any two of the sides, as A. B. and A. C. I say, 
Euc. 1.20.  those two sides being taken and joined together as one line, are greater than the third side remaining, namely, B. C. And the like of any other two, taken together. Whereby it is manifest, that under all three lines (without respect of quantity) a Triangle cannot be contained. PROB. 42.   

THEOREM XVI. In all Triangles, the greater side subtendeth the greater angle, and the lesser side subtendeth the lesser angle.  
Let A. B. C. be a Triangle, 
Trianguli maius latus subtendit maiorem angulu●s.  having  diagram the side A. C. greater than the side A. B. and less than the side B. C. Than I say, 
Euc. 1.18.19. Ram. 6.11. Euc. 1.47.48. Pit. 1.5. Ceul. 2.19.  that the angle A. B. C. being subtended by the greater side A. C. is greater than the angle A. C. B. being subtended by the lesser side A. B. And also, that the angle A. B. C. being subtended by the lesser side A. C. is lesser than the angle B. A. C. subtended by the greater side B. C.   

THEOREM XVII. If two sides of one Triangle be equal to two sides of another Triangle, and the angle contained under the equal sides of the one, be greater than the angle contained under the equal sides of the other; then the Base also of the one (namely, of that which hath the greater angle) shall be greater than the Base of the other.  
LEt there be two Triangles, A. B. C. and D. E. F. which have  diagram  diagram two sides of the one Triangle, as A. B. and A. C. equal to two sides of the other Triangle D. F. and D. E. and let the angle F. D. E. contained under the equal sides of the one be greater than the angle B. A. C. contained under the equal sides of the other. 
Euc. 1.24.  Than I say, that the Base F. E. of the one (namely, of that which hath the greater angle) is greater than B. C. the Base of the other.   

THEOREM XVIII. If a Triangle be equicrural, or having two equal sides; a perpendicular let fall from the angle contained under those equal sides to the Base, and continued, shall divide as well the same Base and angle, as also the measure of that angle, into two equal parts: Et contra.  
LEt A. B. C. be a Triangle, 
Pit. 1.23.  whose sides A. B. and A. C. are equal, and let fall a perpendicular from the angle, included by those equal sides, as A. E. to the Base B. C. and let the same be continued to D. Now I say, that a perpendicular foe let fall, shall divide as well the same Base B. C. and angle B. A. C. as also the measure thereof, namely, the arch line B. D. C. into two equal parts. PROB. 10.11.40.  
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THEOREM XIX. If a Triangle hath two equal sides, the power of one of those equal sides exceedeth the power of the perpendicular let fall on the Base from the angle it subtendeth, by the power of half the Base.  
LEt A. B. C. be a Triangle, having  diagram two equal sides, B. A. and B. C. and let B. F. be a perpendicular let fall to the Base A. C. from the angle it subtendeth A. B. C. Than I say, that the power of one of those equal sides, namely, the square A. B. D. E. exceedeth the power of the perpendicular, namely the square B. H. F. G. by the power of half the Base, namely, the square F. C. K. L. PROB. 36.40.41.64.   

THEOREM XX. If the power of one side of any Triangle be equal to both the powers of the other two sides, the angle contained under those two other sides, is a right angle.  
THis THEOREM is the converse of the 6 THEOREM, and therefore the explication and construction thereof serveth here. 
Euc. 1.48.    

THEOREM XXI. If a right line divide any angle of a Triangle into two equal parts, and if also the same line divide the Base, the segments of the Base shall have such proportion the one to the other, as the other sides of the Triangle have: Et contra.  
LEt A. B. C. be a Triangle, and let the right line A. D. divide the angle B. A. C. of the same Triangle into two equal parts; 
Euc. 6.3. Ceul. 2.61.  and also let the same line divide the Base B. C. Than I say, the segments of the Base, namely, B. D. and D. C. shall have such proportion the one to the other, as the other sides of the Triangle have, namely, A. B. and A. C. for such proportion as B. D. hath to D. C. the same hath A. B. to A. C.  
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THEOREM XXII. If a right line be drawn parallel to any side of a Triangle, the same line shall cut the sides of that Triangle proportionally.  
LEt A. B. C. be a Triangle, 
Euc. 6.2. Ram. l. 6. p. 9.5. p. 13. Con. 1.2. & 3 Pit 1.47.45. Ceul. 2.27.  unto one of the sides whereof A. B. is drawn a parallel line D. E. Wherhfore the same line doth cut the sides of that Triangle A. C. and B. C. proportionally: for first, as A. E. is to E. C. so is B. D. to D. C. also, as A. E. is to B.D. so is E. C. to D. C. and as A. C. is to A. E. so is B. C. to B. D. PROB. 12.13.14.15.16.22.98.  
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THEOREM XXIII. The superficial content of every right angled Triangle, is equal to half that right angled Parallelogram, which hath his length and breadth equal to the containing sides of the right angle; or whose length is equal to the subtending side, and breadth to the perpendicular, drawn from the right angle to the same side.  
LEt A. B. C. be a rectangle Triangle, 
Euc. 1. D. f. 27. Ram. 8.2.  whose angle at  diagram the point C. is a right angle, whereof the containing sides are A. C. and C. B. the subtending side A. B. and the perpendicular drawre from the right angle to the same side, is C. E. Now I say, the superficial content or Area of this right angled Triangle is equal unto half that right angled Parallelogram (namely, A. D. C. B.) which hath his length C. B. and breadth A. C. equal to the containing sides of the right angle; or whose length A. B. is equal to the subtending side, & breadth A. F. to the perpendicular line, drawn from the right angle to the same side, as the Parallelogram A. B. F. G. PROB. 39.52.92.102.106.   

THEOREM XXIIII. The Area or superficial content of every Equilater Triangle, is equal to half that long square, whose length and breadth is equal to one of the sides and the perpendicular.  
LEt A.B.C. be an Equilater Triangle, and A. E. the perpendicular thereof. Now I say, that the superficial content thereof is equal to half that long square D. F. B. C. whose length B.C. and breadth B. D. is equal to one of the sides, and the perpendicular. PROB. 37.  
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THEOREM XXV. All Triangles, of what kind soveur, are equal in their superficial content unto half that right angled Parallelogram, whose length and breadth is equal to the perpendicular, and the side whereon it falleth.  
Let A. B. C. be a Triangle, whose perpendicular is A. F. and the  diagram side whereon it falleth B. C. I say, that this Triangle A. B. C. is equal in his superficial content unto half the right angled Parallelogram D. E. B. C. whose length B. C. and breadth B. D. are equal to the perpendicular, and the side whereon it falleth. PROB. 39.41.44.72.77.99.   

THEOREM XXVI. Triangles which consist on one and the same Base, or on equal Bases, and in the same parallel lines are equal the one to the other.  
LEt A. B. C. D. B. C. and E. B. C. be three  diagram Triangles, 
Triangula in aequali besi & intra easem parallelas sunt aequalia.  consisting on one and the self-same Base B. C. (or on equal Bases, 
Euc. 1.38.  which is all one thing) and in the same parahell lines A. E. and C. B. Now I say, that all those three Triangles, A. B. C. D. B. C. and E. B. C. and as many more as may be drawn on the same Base, or a Base equal thereunto, and in the same parallel lines, are all equal the one to the other. PROB. 26.27.28.29.46.47.73.74.75.79.80.81.93.103.107.110.   

THEOREM XXVII. If Triangles and Parallellograms have one and the same Base, or equal Bases, and be in the same parallel lines, the Parallellograms shall be double to the Triangles.  
LEt B. C. D. and F. C. D. be two Triangles, 
Euc. 1.41. Ceul. 2.25.  and  diagram let A. B. D. C. and B. E. D. C. in this same Diagram be two Parallellograms, which Triangles and Parallellograms have one and the same Base C. D. and are in the same parallel lines A. F. and C. D. Now I say, that either of those two Parallellograms are double to either of those two Triangles. PROB. 76.77.92.110.   

THEOREM XXVIII. If a Triangle hath his Base double to the Base of a Parallelogram, and that they are both in the same parallel lines, then are they both equal the one to the other.  
LET A. B. C. D. be a Parallelogram, whose Base is C. D. and let A. E. D. and B. E. D. be two several Triangles, whose Bases E. D. are double to the Base of the Parallelogram (for E. C. and C. D. are equal) and who are within the same parallel lines with the Parallelogram A. B. C. D. Than I say, that either of those Triangles are equal to the same Parallelogram. PROB. 29.78.91.110.  
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THEOREM 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.  
LEt A. B. C. be an Equilater Triangle, 
Euc. 13.12.  whose perpendicular is A. E. let fall from the angle B. A. C. to the subtendent side B. C. Now I say, that the power of the side of the same Triangle, namely, B. C. G. H. which is the power or square of the side B. C. is to the power of the perpendicular thereof, namely, A. D. E. F. (which is the power or square of the perpendicular A. E.) in proportion Sesquitertia, or as 4. to 3. For of what parts the line B. C. or B. A. containeth in power 8. of such parts B. E. (which is the half of B. C.) containeth in power 2. Wherhfore the perpendicular A. E. being the residue, containeth in power of such parts 6. (for the squares of the lines A. E. and B. E. are by the 6. THEOREM equal unto the square of the line A. B. whereunto B. C. is equal.) Now 8. to 6. is Sesquitertia: wherefore the power of the line B. C. is to the power of the line A. E. in Sesquitertia proportion. So is the square A. D. E. F. ¾· of the square B. C. G. H. PROB. 36.  
 diagram   

THEOREM XXX. The diagonal line, or Diameter of any Square, is double in power to the side of the same Square.  
LEt A. B. C. D. be a Square, whose diagonal line, or Diameter, is the line A. D. Now I say, that the same line A. D. is double in power to the side of the same Square, that is, the Square A. D. E. F. is double to the Square A. B. C. D. PROB. 99.102.106.  
 diagram   

THEOREM XXXI. A Square, whose side is equal to the Diameter of any other Square, is double in content or superficial quantity to that other Square.  
THe explication hereof, is manifest by that of the last: For let A. D. E. F. in the last Diagram be a Square, whose side A. D. is equal to the Diameter of another Square, as the same line A. D. is the Diameter of another Square, namely, A. B. C. D. Wherhfore I say, that the Square A. D. E. F. is double in content, or superficial quantity, to that other Square. PROB. 102.106.   

THEOREM XXXII. All parallellograms have their opposite sides, and angles equal one to another; and their Diameters divide them into equal parts.  
AS these two parallellograms  diagram  diagram A. and B. have their opposite sides and angles, 
Eu. 1.34. Ram. 10.6. Ceul. 2.25.  equal one to another, as in the figure A. the sides C. D. and E. F. are opposite & equal, and likewise D. F. and C. E. Also the angles thereof at the points C. and F. are opposite and equal, and likewise those at D. and E. And moreover, their diameters divide them into equal parts, as the diameters C. F. and D. E. do either of them divide the parallelogram A. into two equal parts: And the like explication and construction is to be made of the figure B. PROB. 115.116.   

THEOREM XXXIII. Parallellograms which consist on one and the same base, or on equal bases, and in the same parallel lines are equal the one to the other.  
LEt A. B. C. D. and E. A. C. D. be two parallellograms, 
Eu. 1.35, 36.  which consist on one & the same base, namely C. D. (or on equal bases which is one and the same thing) and in the same parallel lines, namely E. B. and C. B. Now, I say, that those two parallellograms are both equal the one to the other. PROB. 89.90.  
 diagram   

THEOREM XXXIIII. Every Rhombus and Rhomboydes is equal to the long square, whose length is one of the sides, and breath equal to the parallel distance.  
LEt G. H. I K. be a Rhombus, and A. B. C. D. a Rhomboydes. I say, the Rhombus G. H. I K. is equal to the long square G. H. L. M. whose length is one of the sides G. H. and breadth the parallel distance  diagram  diagram H. M. And also that the Rhomboydes A. B. C. D. is equal to the long square E. F. C. D. whose length is one of the sides C. D. or A. B. and breadth, the parallel distance E. C. or F. D. PROB. 58, 59, 60, 61, 89, 90.   

THEOREM XXXV. Parallellograms and Triangles, within the same parallels, are in such proportion the one to the other as their bases are.  
LEt A. B. C. D. and B. E. D. F. be two parallellograms, 
Triangula vel parallelogramma aequealta sunt ut basis. Eu. 6.1. Ram. 10.13. Ceul. 2.26.   diagram within the same parallel lines A. E. and C. F. and let also B. C. D. and B. D. F. be two triagles within the same parallel lines. Than, I say, as the base C. D. is to the base D. F. so is the parallelogram A. B. C. D. to the parallelogram B. E. D. F. And so also is the triangle B. C. D. to the triangle B. D. F. PROB. 29, 110, 123, 124, 125, 126, 127, 128, 129.   

THEOREM XXXVI. A right line being first equally, and then un-equally divided; The square which is made of the part lying between those sections, together with the right angled parallelogram, contained under the unequal parts of the whole line; are equal to the square of half the whole line.  
LEt A. B. be a right line divided, 
Hereby is demonstrated that equation, of the greatest and lest karectes or numbers, and their equal●tie to the middle. Of great use is th●● 〈◊〉 in the rules of Algebar. Eu. 2. ●.  first equally  diagram in the point C and then un-equally as in the point D. The square, I say, which is made of the part lying between those sections C. and D. namely the square E. F. G. H. together with the right angled parallelogram A. D. F. I contained under the unequal parts of the whole line as A. D. and D. B. are equal to the square C. B. K. H. being the square of half the whole line, as of A. C. or C. B.   

THEOREM XXXVII. Two right lines being drawn in a circle, and the one intersecting the other, either equally or unequally howsoever; The rectangle figure contained under the parts of the one line, shall be equal to that, contained under the parts of the other.  
LEt A. C. and B. D. be two right lines drawn in the circle A. B. C. D. and let the one intersect the other unequally at all adventures in the point E. I say, 
The wonderful properties of a circle here by appeareth. And many strange conclusions geometrical from hence may be gathered. Eu. 3.35. Ceul. 2.50.  that the rectangle figure contained under the parts of the one line, namely under A. E. and E. C. being the parts of the line A. C. shall be equal to that contained under the parts of the otherline, namely under the parts B. E. and E. D. of the line B. D. And the like if those lines had intersected the one the other equally.  
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THEOREM XXXVIII. In all right angled parallellograms, the length thereof being enfolded in the breadth, produceth the Area or superficial content of the same.  
LEt A. B. C. D be a parallelogram right angled, I say, the length there of A. B. 40. being enfolded in the breadth A. D. 20. produceth the Area or superficial content of the same 800. PROB. 51.55.  
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THEOREM THIRTY-NINE. Every regular Poligon is equal to the long square, whose length and breadth is equal to half the perimeter, and a perpendicular drawn from the centre to the middle of any side of the same.  
LEt the Sexagon A. B. C. D. E. F.  diagram be a regular Polygon, whose three sides (being half the Perimeter) contain 18. and the perpendicular 6. G. 5⅕· This Polygon is equal to the long square H. 6. G. I whose length H. 6. or I G. is equal to half the Perimeter, and breadth H. I or 6. G. to the perpendicular 6. G. PROB. 64.95.   

THEOREM XXXX. If two or more right lines, are cut by divers parallel right lines; the intersegments of those lines so cut shall be proportional the one to the other.  
LEt A B. and A HUNDRED be two right lines, 
Pit. 1.39.  being cut by  diagram divers parallel right lines as QUEEN E. S H. T L. and the rest; I say, the Intersegments of those lines so cut, as A F. and A G. A I and A K. F M. and G N. and the rest are proportional the one to the other; that is to say, if A F. be ¼· of A B. then is A G. ¼· of A C. and if A M. be ¾· of A B. then shall A N. be ¾· of A C. The reason is, because the right line S H. cutteth off ¼· of the whole parallelogram Q E P R and the right line V O. ¾· thereof; and consequently the like parts, from all lines drawn overthwart those parallels. And the like consideration is to be had of all the other intersegments so by those lines cut out.   

THEOREM XLI. Three right Lines being proportional, a Square made of the Mean, is equal to the right angled figure, contained under the Extremes.  
LEt A. B. and C. be three right lines proportional in continual proportion, 
Euc. 6.17. & l. 7.20. Ram. 12.4. Pit. 1.43.  so that as A. is to B. so let B. be to C. Than I say, that the Square, namely, D. E. F. G. made of the Mean B. shall be equal to the right angled figure, namely, H. I K. L. contained under the two Extremes A. and C. as appeareth by the Diagram. PROB. 79.83.84.85.86.88.95.99.109.129.130.131.  
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THEOREM XLII. Four right lines being proportional, the right angled Parallelogram, contained under the two Means, is equal to the right angled Parallelogram, contained under the two Extremes.  
LEt A. B. C. and D. be four right lines proportional, 
Euc. 6.16. Pit. 1.42.  so that as A. is to B. so let C. be to D. Than I say, that the right angled Parallelogram, namely, E. F. G. H. contained under the two Means B. and C. shall be equal to the right angled Parallelogram, namely, I K. L. M. contained under the two Extremes A. and D. as appeareth by the Diagram. PROB. 49.56.87.88.93.  
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THEOREM XLIII. Of any three proportional right Lines, the Square which is made of the Mean, and that which is made of either of the Extremes, have such proportion the one to the other, as the two Extremes have.  
LEt A.B.C.D. and E. F. be three  diagram right lines, 
Eu. 12.2. D.P. 1 Ceul. 2.75.  in continual proportion: so that as A. B. is to C. D. so let C. D. be to E. F. Than I say, that the Square (namely, C. D.I.H.) which is made of the mean C.D. and that which is made of either of the extremes A. B. or E.F. have such proportion the one to the other (respectively) as those extremes have: For the same proportion as the greater extreme A. B. hath to the lesser extreme E. F. the same hath the square A. B. G. H. to the square C. D. H. I and that, to the square E.F. I K. which in this Diagram is Dupla sesquiquarta, as thereby appeareth. And the like consideration is to be had of the proportion of Circles, whose Diameters are so proportionable. PROB. 79.81.82.94.129.130.131.   

THEOREM XLIIII. If a rational right line be divided by an extreme and mean proportion, either of the segments is an irrational residual line.  
LEt the line A. B. be a rational right line, 
Euc. 13.6.  and let the same be divided by  diagram an extreme and mean proportion in the point C. Than I say, that either of the segments, namely, A.C. and C. B. is an irrational residual line. PROB. 20.   

THEOREM XLV. If a right line be divided by extreme and mean proportion, the whole line hath the same proportion to the greater segment, as the same greater segment hath to the lesser.  
LEt A. B. be a right line, divided by extreme and mean proportion,  diagram as in the point C. I say then, that the whole line A. B. hath the same proportion to the greater segment A. C. as the same greater segment hath to the lesser segment C. B. for as A. B. is to A. C. so is A. C. to C. B. PROB. 20.   

THEOREM XLVI. If a right line be divided by extreme and mean proportion, the Rectangle figure, comprehended under the whole line, and the lesser segment, shall be equal to the Square made of the greater segment.  
LEt A. B. be a right line, 
AB. 10.  
AC. 15— √ 125.  
CB. √ 125-5.  and let the same be divided by extreme & mean proportion in the point C. Than, I say, the Rectangle figure, namely, A. C. G. H. comprehended under the whole line A. G. (being equal to A. B.) and the lesser segment A. C. shall be equal to the Square, namely, C.B.E.F. made of the greater segment C.B. DEF. 35. PROB. 20.   

THEOREM XLVII. Two right lines being drawn 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 divide the one the other by an extreme and mean proportion: and the greater segments of either of them shall be equal to the side of the Pentagon.  
LEt A. B. and C. D. be two right lines, 
Eu. 13.8.  drawn in the Equilater Equiangles  diagram Pentagon, A. D. B. E. C. and let the line A.B. subtend the angle A. D. B. and the line C. D. the angle C. A. D. being two of the next immediate angles. I say then, that those two lines, by their intersection in the point F. shall divide the one the other by an extreme and mean proportion. And the greater segments of either of them, as the segments F. B. and F. C. shall be either of them equal to the side of the Pentagon A.D.B.E.C. PROB. 20.21.48.62.63.119.120.   

THEOREM XLVIII. Like Triangles are one to the other in double proportion that the sides of like proportion are.  
LEt A.B.C. and D.E.F. be  diagram two like Triangles, 
Eue. 6.19.  and let the angle A. of the one be equal to the angle D. of the other; the angle B. of the one, to the angle E. of the other; and the angle C. of the one to the angle F. of the other; and as the side A.B. is to the side B. C. so let the side D. E. be to the side E. F. so are B. C. and E. F. sides of like proportion. Now I say, that the proportion of the Triangle A. B. C. unto the Triangle D. E. F. is double the proportion of the side B. C. to the side E. F. PROB. 45.   

THEOREM XLIX. All like right lined figures whatsoever, are the one to the other in double proportion, that the sides of like proportion are.  
LEt A. and B. be two  diagram right lined figures like, 
Euc. 6.20.  having the angle at the point C. equal to the angle at the point H. and the angle at the point D. equal to the angle at the point I the angle E. to the angle K. and so of all the rest. And also as the side D. E. is to E. F. so let I K. be to K. L. etc. so are the sides E. F. and K. L. sides of like proportion. Than, I say, that the proportion of the figure A. unto the figure B. is double, the proportion of the side E. F. to the side K. L. PROB. 45.   

THEOREM L. All angles in equal circles, whether they are in the centres or circumferences, have the same proportion one to the other as the circumferences have wherein they consist: And so are the sectors, which are described on the centres.  
LEt A.B.C. and F. G. H. be two equal  diagram circles whereof let D. and E. be their centres; 
Euc. 6.33.  and let the angles which are in their centres be B. D. C. and H. E. G. and the angles which are in their circumferences B. A. C. and H. F. G. and let the sectors described on their centres be D. B. C. and E. H. G. Than, I say, that the angles B. D. C. and H. E. G. in the centres, and the angles B. A. C. and H. F. G. in the circumferences, have the same proportion one to the other, as the circumferences have wherein they consist, that is, as the circumference B.C. hath to the circumference H. G. And the same proportion also hath the sector D. B. C. to the sector E. H. G.   

THEOREM LIVELY If on the end of the Diameter of a circle, a perpendicular be raised, it shall fall without the circle, between which, and the circumference, another right line cannot be drawn to the Diameter, and the angle within the circle is greater, and that without the circle is lesser, than any acute angles made of right lines.  
LEt A. B. C. be a circle, 
Euc. 3.16.  whose diameter is A. C. and on the same diameter  diagram let the perpendicular D.C. be raised. Than, I say, first, that the same perpendicular D. C. shall fall without the circle; and that between the same perpendicular and the circumference B. C. another right line cannot be drawn to the diameter A.C. And also that the angle within the circle, namely A. C. B. is greater, and that without the circle, namely B.C.D. is lesser than any acute angles made of right lines.   

THEOREM LII. If a right line be a tangent or touch line to a circle, and another right line be drawn by the centre to the point of touch, it shall be a perpendicular to the tangent: And if a perpendicular Bee let fall from the centre to the tangent, it shall fall in the point of touch.  
LEt the right line A. B. be a tangent, 
Euc. 3.18.  or touch line to the circle D. E. C. let the  diagram point of touch be C. and the centre of the circle E. and let another right line, as D.C. be drawn by the centre E. to the point of touch C. Than, I say, that the same line so drawn by the centre to the point of touch, shall be a perpendicular to the tangent A. B. And that the perpendicular E. C. being let fall from the centre E. to the tangent A. B. shall fall in the point of touch C. PROB. 31.33, 114, 115, 116, 117, 118.   

THEOREM LIII. If a right line be drawn in a circle and not by the centre thereof, another right line bysecting the same by right angles shall pass by the centre of the same circle. And if from the centre a perpendicular be let fall on a right line drawn in the same circled not by the centre; the perpendicular shall divide the same line into two equal parts.  
LEt A. B. D. C. be a circle whose centre is E. and let B. C. be a right line drawn in the same circle, and not by the centre  diagram thereof, and let another right line as A. D. bysect the same by right angles in the point F. Than, I say, that the same line A. D. shall pass by the centre of the circle. Also from the centre E. let fall the perpendicular E. D. on the right line B. C. drawn in the same circled not by the centre; Than, I say, further that the perpendicular E D. shall divide the same line B.C. into two equal parts.   

THEOREM liv. If one angle be placed in the circumference of a circle, and another in the centre thereof, and are both subtended by one part of the circumference. That angle in the centre shall be double to that in the circumference.  
LEt A. B. C. be a circle, and let one angle be placed in the circumference thereof,  diagram 
Angulus in centro duplus est anguli in peripheria, in candem peripheriam insistentis. Euc. 3.20.  as the angle B. A. C. and another in the centre thereof, as the angle B. E. C. and let them both be subtended by one part of the circumference as B. D. C. Than, I say, that the angle B.E.C. in the centre, shall be double to the angle B. A. C. in the circumference.   

THEOREM LU. All angles consisting in one and the same segment of a circle are equal the one to the other; If in a semicircle, they are right angles; If in a lesser segment, they are greater than a right angle; If in a greater segment, they are lesser. And also the angle of a greater segment, is greater than a right angle, and the angle of a lesser segment is less than a right angle.  

Anguli in cadem sectione sunt aequales.  LEt A. B. C. D. E. F. be a circle, and let A. E. be the diameter thereof, which divideth  diagram the same into two semicircles or equal segments; Than, I say, that the angles A. C. E. and A. D. E. consisting in one and the same segment, 
Eu. 3.21.31  are equal the one to the other, and being in a semicircle they are both right angles: 
Ceul. 2.46.  Let also the line or cord A. C. divide the same circle into two unequal segments, as A. B. C. the lesser segment, and A. F. E. D. C. the greater, 
Note here the difference between an angle in a segment, and an angle of a segment, See Def. 16.17.  I say, the angle A. B. C. in the lesser segment is greater than a right angle, and the angle A. E. C. in the greater segment is less than a right angle. And also F. A. C. an angle of the greater segment is greater than a right angle, and the angle B. A. C. being an angle of the lesser segment, is less than a right angle. PROB. 19.23.24.25.30.38.52.65.   

THEOREM LVI. If a right line be a tangent to a cirle, and another right line be drawn from the touch (crossing the circle) to what point soever in the circumference; the angles caused by intersection or meeting of those two lines, are equal to the angles consisting in the alternate segments of the circle.  
LEt A. B. C. D. be a circle, and the right line E. F. a tangent to the same circle; 
Euc. 3.32.   diagram and let another right line as B. D. be drawn from the touch, namely, the point D. crossing the circle to what point soever in the circumference as the point B. Than, I say, the angles caused by intersection or meeting of those two lines E. F. and B. D. are equal to the angles consisting in the alternate segments of the circle; that is, the angle B. D. F. shall be equal to the angle B.A.D. and the angle B. D. E. to the angle B. C. D. in the alternate segments. PROB. 32.33.111.   

THEOREM LVII. If from a point without a circle, two right lines be so drawn, that the one be a tangent to the circle, and the other divide the same circle into two equal or unequal parts: The rectangle figure contained under the whole line which divideth the circle, and that part thereof lying between the utter circumference and the point, is equal to the square made of the tangent line.  
LEt B. C. F. be a circle, and without the same, take a  diagram point at all adventures, as the point A. from whence let two right lines be so drawn, that the one be a tangent to the circle, as A. C. and the other divide the same circle, as A. B. F. Than, I say, that the rectangle figure contained under the whole line A. F. and that part of the same line, lying between the utter circumference & the point, as B. A. namely, the rectangle figure A. H. G. F. is equal to the square made of the tangent line A. C. namely, to the square A. E. D. C.   

THEOREM LVIII. If from a point without a circle, two right lines be drawn to the concave circumference of the circle, they shall be reciprocally proportional with their parts taken without the circle. And another right line drawn from the point as a tangent to the circle, shall be a mean proportional between either whole line, and the utter segment thereof.  
LEt B. E. F. D. be a circle, and without the same circle take a point at all adventures, as at A. and from that point to the concave circumference of the circle, draw the two right lines A.E. and A. F. And let another right line be drawn from the same point as a tangent to the circle, as the line A.D. Than, I say, first, that these two lines A.E. and A.F. are reciprocally proportional with their parts taken without the circle, that is, as A. E. is to A. F. so is A. C. to A. B. And moreover, that between the lines A. F. and A. C. or between the lines A.E. and A.B. the tangent A. D. is a mean proportional.  
 diagram   

THEOREM LIX. Every circumference of a circle, is more than triple his Diameter, by such a proportion as is more than 10/71· and less than 1/7 of the same, the nearest rational proportion whereof is 22. to 7.  
IN the former Diagram, 
Corol: Euc. 12.1.  let B. C. D. F. E. be the circumference of a circle, and C. F. the Diameter thereof; I say, the same circumference is more than triple the Diameter C. F. by such a proportion as is more than 10/71· and less than 1/7· of the same; and that the nearest rational proportion thereof, is 22. to 7. Wherhfore to know the quantity of the circumference, multiply the Diameter by 22. and divide the Factus by 7. the Quotient resolveth the question. PROB. 34.67.   

THEOREM LX. Every circle is near equal to that right angled Triangle, of whose sides (containing the right angle) the one is equal to the semidiameter, and the other to the circumference of the same circle.  

Corol. Euc. 12.1.   diagram  

The precise squaring of a circle was never yet found out; and therefore in this and the 4. next Theorems following, this word (Near) is used. But all Conclusions hereby wrought, are without any apparent error.  LEt A. C. D. be a circle, and A. E. B. a right angled Triangle, whose angle at the centre B. is a right angle, and whose sides containing the right angle, namely, A. B. and E. B. the one is equal to the Semidiameter, as A. B. and the other to the circumference, as E.B. Than I say, that the Triangle A. E. B. is near equal unto the same circle. PROB. 68.68.   

THEOREM LXI. The Square made of the Diameter of a Circle, is in that proportion to the circle (very near) as 14. to 11. And therefore every circle is near 11/14· of the square about him described.  
LEt A. B. C. D. be a circle, 
Corol. Euc. 12.1.  and E.F.G.H. a square made of the diameter A. C. or B. D. Than, I say, that the square E. F. G. H. is very near in the same proportion to the circle A. B. C. D. as 14. to 11. And theree fore, the circle very near 11/14· of the same squar-about him described. PROB. 68  
 diagram   

THEOREM LXII. Every circle is near equal to the long square, whose length and breadth are equal to half the circumference, and half the Diameter; or to the whole Diameter, and 11/14· thereof.  
LEt A. C. D. be a circle; 
Corol. Euc. 12.1.  And let H. A. B. G. be a  diagram long square, whose length H. A. is equal to half the circumference, and breadth A. B. to half the diameter; And let also K. A. D. E. be another long square, whose length is the whole diameter A. D. and breadth 11/14· thereof, namely, K. A. Than, I say, that either of those two long squares are near equal to the circle A. C. D. PROB. 68   

THEOREM LXIII. Every semicircle is near equal to the long square, whose length and breadth is equal to half the arch line, and the semidiameter.  
LEt E. B. F. be a semicircle, 
Corol. Euc. 12.1.  whose semidiameter is the  diagram line B. C. and the half of whose arch line is equal to the line A. B. or D. C. Than, I say, that the long square, namely, A. B. C. D. (whose length A. B. is equal to half the arch, and whose breadth is B. C. the semidiameter) is near equal to the semicircle E. B. F. PROB. 69.   

THEOREM LXIIII Every sector of a circle, is near equal to that long square, whose length and breadth is equal to the semidiameter, and half the arch-line of the same sector; or the half semidiameter, and the whole arch line.  
LEt B. A. C. be the sector of a circle, 
Cor. Eu. 12.1.  and let D. B. A. E. be a long square,  diagram whose length B. A. or D. E. is the semidiameter or equal thereunto, and whose breadth D. B. or E. A. is equal to half the arch-line B. C. And let also H. F. A. G. be another long square, whose length H. F. or G. A. is equal to the whole arch-line B. C. and whose breadth F. A. or H. G. is half the semidiameter, or equal thereunto. Than, I say, that either of those two long squares is near equal to the sector B. A. C. PROB. 70.   

THEOREM LXV. All circumferences of circles, have the same proportion the one to the other, as their diameters have.  
LEt A. and B. be two circles, 
This Theor. is of excellent use in the forming of Mill-wheels, clocks, crane's, and other engines for waterworks, etc.  whereof let HUNDRED D. and  diagram E. F. be their several diameters, I say, that the same proportion that the diameter C. D. of the circle A. hath to the diameter E. F. of the circle B. the same proportion hath the circumference of A. to the circumference of B. PROB. 65.66. 
Cor. Eu. 12.1.    

THEOREM LXVI. All circles have the same proportion the one to the other, 
Omnes figurae fimiles circulis inscriptae sunt, ut quadrata à diametris circulorum quibus inscribuntur. Euc. 12.2.  as the squares of their Diameters have.  
LEt A. and K. be two circles, and let the squares circumscribed about them, be the several squares of their Diameters. Than, I say, that the circle A. hath the same proportion to the circle K. as the squares of their Diameters have, namely, as the square B. C. D. E. hath to the square F. G. H. I PROB. 65, 66, 104.108.  
 diagram   

THEOREM LXVII. If in a circle be described a quadrilaterall figure, the opposite angles thereof shall be equal to two right angles: and being intersected with two diagonalls, the right angled figure made of those diagonalls, is equal to the two right angled figures, comprehended under the opposite sides of the quadrilaterall figure.  
LEt A.B.C.D. be a circle, 
Exempla illustrissima babebis, Pit. lib. 2. p. 32.33.35.36.37.38 Anguli oppositis sect onibus aequantur duobus rect●s.  and let therein be described the quadrilaterall figure A. B C. D. let also the same figure  diagram be intersected with the two diagonals A. C. and B. D. Than, I say, first, that the opposite angles at the points A. and C. are equal to two right angles, 
Eu. 3.22. Pit. 1.54.  and likewise the opposite angles at the points B. and D. And also that the right angled figure made of the diagonalls A. C. and B. D. is equal to the two right angled figures (taken together) comprehended under the opposite sides A. B. and D. C. and under A. D. and B. C. 
This Prop. is of very great use in trigonometric.    

THEOREM 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.  
LEt A. B. E. C. be a circle, whereof F. B. is the semidiameter, and let A. B. C.  diagram be an equilater triangle inscribed in the same circle. Than, I say, that the power of the side of the equilater triangle A.B.C. namely, the square H. A C. G. hath to the power of the semidiameter F.B. namely, the square F.B.D.E. triple proportion, that is, as 3. to 1. For the square H. A. C. G. containeth the square F. B. D. E. three times.   

THEOREM LXIX. A triangle inscribed in a circle, hath every of his angles equal to half the arch, opposite to the same angle.  
LEt D. be a circle, 
Pit. 1.53.  and let A. B. C. be a triangle, inscribed at all adventures in the same circle. Than, I say, that the  diagram triangle A. B. C. hath every of his angles equal to half the arch, opposite to the same, as the angle at the point A. is equal to half the arch B. F. C. opposite thereunto, the angle at the point B. is equal to half the arch A. G. C. and the angle at the point C. is equal to half the arch A. E. B. For, the whole of every circle is 360. degrees, whereof the half is 180. and the three inward angles of every right lined triangle, is equal to two right angles, which is 180. degrees. THEOR. 13. PROB. 48.119.120.   

THEOREM LXX. If in a rectangle-triangle a perpendicular be drawn from the right angle to the base, the same perpendicular is a mean proportional between the sections of the base: And the side annexed to either section, shall be a mean between the same section and the whole base.  
LEt A. B. C. be a rectangle-triangle, 
Euc. Coral 6.8. Ceul. 2.63.  right angled at B. from whence let the perpendicular B. D. be drawn to the base A. C. Than, I say, that  diagram the same perpendicular B. D. is a mean proportional between the sections of the base, namely, between A. D. and D. C. And also that the side A. B. annexed to the section A. D. is a mean proportion between A. D. and the whole base A. C. and that the side B. C. annexed unto the section D. C. is a mean proportion between the same section D. C. and the whole base A. C. For, as A. D. is to A. B. so is A. B. to A. C. etc. PROB. 17, 19, 23, 24, 25, 30, 38, 43, 65, 66, 109.   

THEOREM LXXI. If in equal parallellograms, one angle of the one, be equal to one angle of the other, the sides which contain those equal angles, shall be reciprocal.  

Parallelogramma aequiangula aequalia, sunt lateribus reciproca: & contra.  LEt A. B. C. D. and E. F. G. C.  diagram be two parallellograms, equal the one to the other, 
Euc. 6.14.  and let the angle B.C.D. of the one be equal to the angle E. C. G. of the other. Than, I say, that the sides which contain those equal angles, are reciprocally proportional, that is, as D. C. to C. G. so is E. C. to B. C. PROB. 88.93.   

THEOREM LXXII. In rectangle-triangles, the figure which is made of the subtending side of the right angle, is equal unto both the figures made of those sides, which contain the right angle, so as those three figures are like, and in like sort described.  
LEt A. B. C. be a triangle, 
Euc. 6.31.  whose angle at the point C. is a right  diagram angle. Than, I say, 
Ceul. 2.79.  that the equilater triangle E. B. A. which is made of B. A. the subtending side of the right angle C. is equal unto both the equilater triangles, made of the containing sides B. C. and C. A. namely, to the triangles B. F. C. and A. C. G. taken together: And the like of squares, and all other like figures, in like sort described.   

THEOREM LXXIII. In all plain triangles, the sides are in proportion the one to the other, as the subtenses of the angles opposite thereunto; or as the sins of the angles opposite to those sides.  
LEt D. be a triangle, and let there be circumscribed about the same triangle the circle A. B. C. by means whereof the  diagram side A. B. is made the subtense of the angle A. C. B. that is, of the arch A. E. B. which is opposite to the same angle A. C. B. Also the side B. C. is made the subtense of the angle B. A. C. that is, of the arch B. F. C. which is opposite to the same angle B. A. C. and lastly, the side A. C. is made the subtense of the angle A. B. C. that is, of the arch A. G. C. which is opposite to the same angle A. B. C. Than, I say, that the side A. B. is in proportion to the side B. C. as the subtense of the angle A. C. B. to the subtense of the angle B.A.C. for the sides and subtenses, are one and the same. And likewise of the sins of those angles; which sins are the one half of their subtenses, and what proportion the whole hath to the whole, the same hath the half to the half. CHAP. 14.3.   

THEOREM LXXIIII. Every right lined figure, or plat, consisteth of more sides by two; then the number of triangles, whereof the same figure is composed.  
LEt A.B.C.D.E.F.G. be a right lined figure, 
Ram. 10.1.   diagram and let the triangles, whereof it is composed, be 1, 2, 3, 4, and 5. Than, I say, that the same right lined figure consisteth of more sides by two, than the number of triangles, whereof it is composed. For the sides thereof are seven, and the triangles five; as is apparent by the Diagram. CHAP. 37.3.    
The end of the first Book.   

THE USE AND OPERATION OF THE FORMER THEOREMS.  
The second Book.  
THE ARGUMENT OF THIS BOOK.  
THIS Book consisteth of divers Conclusions, or Geometrical PROBLEMS, here duly placed, by observation of natural course; the cause being formerly, in the first Book, amply expressed, and here the effect as fully made manifest, having either to other due relation. This Book is divided into four Parts, wherein most plainly, briefly, and methodically, is expressed the practic operation of the precedent THEOREMS; as the Distinction, Application, and Division of Lines and Angles, and the Description, Mensuration, Reduction, Addition, Inscription, Transmutation, Division, and Separation of all sorts and forms of superficial Figures, according to their several kinds.  

THE FIRST PART. Of the Properties, Passions, Dispositions, Applications, and Divisions of Lines and Angles.  

PROBLEM I Two right lines given, being unequal; to take from the greater a line equal to the lesser.  
LEt a. and B. be two unequal right lines  diagram given, 
Euc. 1.3.  whereof let A. be the greater; from which it is required to take a line equal to the lesser.  
First, join the two given lines together in such sort, as thereby they make any kind of angle, as C. D. E. and making the centre D. and the space D. E. (the length of the line B.) describe the arch line F. E. which shall cut off from the greater line, the line F. D. equal to the lesser line B. which was required to be done. DEF. 10.   

PROBLEM II To a right line given, to draw a parallel line at any distance required.  
SVppose the right line given, 
Euc. 1.31.  to be A. B. unto which line it is required to have a parallel drawn:  diagram Open your Compass to the distance required, and setting one foot in the end A. strike an arch line on that side the given line, whereon the parallel is to be drawn, and the like in the end B. as the arch lines C. and D. and by the convexity of those arch lines, draw the line C. D. which shall be parallel to the given line, as was required. DEF. 34.   

PROBLEM III To perform the former Proposition at a distance required, and by a point limited.  
LEt A. B. be a right line given, 
Euc. 1.31.  whereunto it is  diagram required to have a parallel line drawn at the distance, and by the point C. Place one foot in C. from  diagram whence take the shortest extension to the line A. B. as C. E. at which distance, place one foot in the end B. and with the other strike the arch line D. by the convexity of which arch line, and the limited point C. draw the line F. G. which shall be a parallel to the given line A. B. the thing required. DEF. 34.   

PROBLEM FOUR To erect a perpendicular on any part of a right line given.  
LEt A. B. be a right line given,  diagram and let C. be a point therein, 
Euc. 1.11. Ceul. 2.7.  whereon it is required to erect a perpendicular, Open the Compass to any convenient distance, and setting one foot in the point C with the other mark on either side thereof, the equal distances C. E. and C. F. Than opening the Compass to any convenient wider distance, with one foot placed in the points E. and F. strike two arch lines, crossing each other, as in D. from whence draw the line D. C. which shall be the perpendicular required. DEF. 5.   

PROBLEM V To raise or let fall a perpendicular to a line given, from a point either above or beneath the same line.  
LEt A. B. 
Euc. 1.11.  be a line given, and let D. be a point above the same line: It is required from  diagram the point D. to let fall a perpendicular line to the given line A. B. At any indifferent distance placing one foot in the point D. describe an arch line, intersecting the given line twice, as the arch line H. intersecteth the given line, in the points E. and F. Than either with the same, or some other convenient distance, by placing the one foot in those points E. and F. strike two arch lines, crossing each other, as in G. By which point, and the given point D. draw the line D. C. which shall be a perpendicular to the given line. And the like construction is to be used, if the point were beneath the given line. DEF. 5.   

PROBLEM VI Upon the end of a line given to raise a perpendicular.   diagram 
To perform the same another way.  
LEt A.B. in the former Diagram be a right line given, and B. the end thereof, whereon a perpendicular is to be raised. From the end B. prick out any five equal distances, and opening the Compass to 4. of them, with one foot in B. strike an arch line towards C. Than opening the Compass to all 5. Divisions, with one foot in the third Division, cross the same arch line in C. from whence draw the line C. B. which shall be the perpendicular required.   

PROBLEM VII. To divide a right line given, into two equal parts.  
LEt A. B. be a right line given, 
Euc. 1.10. Ceul. 2.6.  which is to be divided  diagram into two equal parts: Open the Compass to more than half of the given line, and placing one foot thereof in either of the ends A. or B. with the other strike an arch line towards D. and another towards C. then place one foot in the other end of the given line, and with the same distance cross the two former arches in D. and C. by which intersections, draw the line D. C. which shall divide the given line A. B. as was required.   

PROBLEM VIII. Upon a right line given, on a point therein limited, to make an angle equal to an angle given.  
LEt A. B. be a right line given, and C.  diagram a point therein limited, and let H. F. G. be an angle given: It is required on the right line A. B. and on the point therein C. to describe an angle equal to the angle given H. F. G. At any convenient distance setting one foot in F. the given angle, 
Euc. 1.23.  strike the arch line H. G. and at the same distance placing one foot in the limited point C. make the arch line D. E. Than take the distance from G. to H. and place that distance on the last drawn arch line from E. which endeth in D. by which point draw the line D. C. which shall include the angle D. C. E. upon the given line A. B. on the point therein limited HUNDRED being equal to the given angle H. F. G. the thing required. DEF. 42.   

PROBLEM IX. To make a right angle upon a line given, and on a point in the same line limited.  
LEt A. B. be a line given, and let B. be a point therein limited. It is required on the line A. B. and to the point in it limited B. to describe a right angle. By the sixth PROB. on the point B. raise the perpendicular C. B. which with the given line shall make the right angle A. B. C. on the line A. B. and to the point therein limited B. which was required. DEF. 5.  
 diagram   

PROBLEM X. To divide an angle given into two equal parts.  
LEt A. B. C. be an angle given,  diagram to be divided into two equal parts. 
Euc. 1.9. Ceul. 1.5.  Having opened the Compass to any convenient distance, place one foot in B. and with the other cross the two lines B. A. and B. C. in the points D. and E. upon which two points, strike two arch lines, at any equal distance, crossing one the other, as at F. From whence to the angle B. draw the right line B. F. which shall divide the given angle into two equal parts, as was required. THEOR. 18.   

PROBLEM XI. To divide a right angle given into three equal parts.  
LEt A. B. C. be a right angle given,  diagram to be divided into three equal parts. At any convenient distance, with one foot in B. cross the line B. C. as at E. and at the same distance on the points B. and E. strike two arch lines, crossing one the other in the point D. by which point, and the angle B. draw the line B. G. Than, by the tenth last before going, divide the angle G. B. C. into two equal parts, with the line F. B. So shall those two lines F. B. and G. B. divide the right angle given into three equal parts; which was the thing required. DEFINITION 21. THEOREM 18.   

PROBLEM XII. To divide a right line given into divers equal parts, as many as shall be required.  
LEt B. C. be a right line given,  diagram and let it be required to divide the same line into six equal parts. First from B. draw a line at all adventures, making an angle of any quantity with the given line, as the line B. A. making the angle A. B. C. Than by the 8. of this book make the angle D. C. B. equal to the angle A. B. C. and from B. towards A. and likewise, convenient distance, make 5. equal spaces (that is, one always less than the number of parts required) and from point to point respectively draw lines, intersecting the given line: So shall you divide the same into six equal parts required. THEOR. 22.   

PROBLEM XIII. To divide a right line given proportionally, according to any proportion required.  
LEt A. be a  diagram right line given, 
Euc. 6.10, 11, 12. Ceul. 2.67.  and let it be required to divide the same into two such parts, that the greater may be in proportion to the lesser, as the line B. is to the line C. First, make an angle of any quantity, as E. D. F. whereof make the side F. D. equal to the given line A. then place on the other side the line C. from D. to H. and the line B. from H. to E. from E. to F. draw the line E. F. and lastly, by the 3. of this Book, draw a parallel line to E. F. by the point H. as H. G. cutting F. D. in G. So shall you divide F. D. (being equal to the given line A.) in the point G. in such sort that the greater segment F. G. hath the same proportion to the lesser G. D. as the line B. hath to the line C. which was required. THEOR. 22.   

PROBLEM XIIII. From a right line, given to cut off any parts required.  
LEt A. B. be a right line given; 
Euc. 6.9. Ceul. 2.66.   diagram and let it be required to cut off from the same line 4/9 parts there of. Make any angle on the end B. by drawing the line E. B. on which line from the point B. make any 9 equal distances (or so many always as the Nomen importeth, which is here 9) From the ninth distance, draw the line E. A. and to that line by the fourth distance from B. draw the parallel line D. C. by the 3. of this Book, cutting the given line A. B. in C. So have you cut off the segment C. B. which is 4/9· parts of the given line A. B. the thing required. THEOR. 22.   

PROBLEM XU. To find a third line in continual proportion unto two lines given.  
LEt A. and B. be two lines given; and let it be required to find a third line, to be in such proportion to A. as A. is to B. Make an angle of any quantity, as H. E. C. then place the line A. from E. to D. and the line B. from E. to F. and  diagram draw the line D. F. place also the line A. from E. to G. and lastly, 
Euc. 6.11. Ceul. 2.68.  by the 3. of this Book, by the point G. draw the line G. C. parallel to F. D. So shall E. C. be a third proportional line to the two given lines, the thing required. THEOR. 22.   

PROBLEM XVI. To find a fourth proportional line to three lines given.  
LEt A. B. and C.  diagram be three lines given; and let it be required to found a fourth line, having such proportion to A. as B. hath to C. Make an angle of any quantity, as D. G. K. And seeing it is the greater extreme, which is sought, 
Euc. 6.12: Ceul. 2.69.  place first the lesser extreme C. from G. to H. and the lesser mean B. from G. to F. then draw the line F. H. and place the greater mean A. from G. to I by which point I draw the line E. I parallel to F. H. which cutteth D. G. in E. So have you E. G. the fourth proportional line required. THEOR. 22.   

PROBLEM XVII. To find a mean proportional line between any two lines given.  
LEt A. and B. be two lines given, 
Euc. 2.14. & 6.13. Ram. 16.19. Ceul. 2.64.   diagram between which it is required, to find a mean proportional line. First, join the two given lines together, so as they make both one right line, as C. F. D. meeting in the point F. then describe thereon the semicircle C. E. D. and on the point F. by the 5. of this Book erect a perpendicular to cut the circumference in E as F. E. which shall be the mean proportional required. DEF. 39 THEOR. 70.  
Note here, that if from the two points C. and D. to any one point in the limb be drawn two right lines (which by the 55. THEOR. make a right angle) as the lines C. E. and D. E. meeting in the point E. Than are those two lines mean proportionalls, that is, the line C. E. between C. F. and C. D. and the line D. E. between D. F. and D. C. THEOR. 70.   

PROBLEM XVIII. To find two mean proportional lines, between any two right lines given.  
LEt A. and B. be two right lines given, 
Euc. 1.43.  between  diagram which it is required to find two mean proportional lines. On the end of the line A. by the 6. of this book place the line B. perpendicularly, making a right angle, as the angle E. C. D. then draw the diagonal line E. D. whereon describe the semicircle E. G. D. making F. the centre, then enlarge the lines C. E. and C. D. towards L. and M and taking in your compass the given line B. the lesser extreme, place one foot in D. and with the other strike through the limb of the semicircle in G. and on the point G. lay your ruler, turning it up and down on that point till by drawing the line H. K. you may cut the two lines C. L. and C. M. equidistantly from the centre F. So shall E. H. and D. K. be two proportional lines, between the given lines A. and B. as was required. THEOR. 5.   

PROBLEM XIX. To find out in a line given, the two extremes of a mean proportional given: So as the same mean be not greater than half the given line.  
LEt the mean proportional given be A. and the line given B. C. It is required in the line B. C. to find two extremes, between which the line A. shall be a mean proportional. Upon the given line B. C. describe the semicircle B. F. C. then at the distance of the given mean, by the second of this book draw a line parallel to B. C. (which  diagram of necessity must either touch or cut the semicircle) as the line D. E. cutting the semicircle in F. From which point F. by the 5. of this book, let fall the perpendicular F. G. which shall so divide the given line B. C. in the point G. that the line given A. shall be a mean proportional between the two segments B. G. and G. C. the thing required. THEOR. 9.55.70. PROB. 83.   

PROBLEM XX. To divide a line given by an extreme and mean proportion.  
AB. 12.  
CB. 12.  
BD. 6.  
EBB. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 180.— 6.  
A 18— 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 180.  
To perform the former Prob. arithmetically.  
LEt the number given to be divided by extreme and mean proportion be 12. First, enfold the square thereof in 5. the factus is 720. divide that by 4. the quotus is 180. from the square root whereof deduct half the given number, the remainder is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 180-6. and this is the greater portion or section, which being deducted from the given number, there remaineth 17— 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 180. for the lesser portion; both which taken together makes 12. the given number.   

PROBLEM XXI. To found the lesser segment of a line, divided by extreme and mean proportion, when only the greater is given; and consequently, to found the whole line.  
LEt A. B. be the greater segment given. 
Euc. 12.2.  It is required to found the lesser segment, and consequently,  diagram the whole line. With the give line A. B. and half the same line, by the 9 PROB. make on the point B. the right angle A. B. C. Than continued the given line infinitely towards D. and at the distance A.C. with one foot in A. cut the continued line in D. from whence cut of D. E. equal to half the given line: So is E.A. the lesser segment, and consequently, E.B. the whole line; as was required.  
Or otherwise, thus.  
LEt A. D. be the greater segment given, it is required to find out the lesser, and consequently, the  diagram whole line. On the given line A. D. by the 63. PROB. hereafter expressed, describe the Pentagon A. D. B. E. C. and against any two, the next immediate angles, draw subtending lines, which shall intersect the one the other, as the lines A.B. and D. C. in the point F. So shall F. A. or F. D. be the lesser segment; F. B. or F. C. the greater; and consequently, A. B. or D. C. the whole line, which was required. THEOR. 47.   

PROBLEM XXII. Having the greater or lesser segment of a line divided by extreme and mean proportion given, to found the other segment, and so the whole line.  
LEt M. be the lesser segment of such a line given; 
Euc. 6.2.  It is required to found the other segment,  diagram and so the whole line. First, by the 20. PROB. divide any line by extreme and mean proportion, as the line A.B.C. Than make an angle of any quantity, as E. D. F. and place the lesser segment of the divided line A. B. from D. to K. and the greater B.C. from D. to I and draw the line I K. Than place the lesser segment given M. from K. to H. and so working forwards, as by the 16. PROB. is taught, you shall produce I G. the greater segment, which added to K. H. makes the whole line required. And the like course is to be held if the greater segment were given, and the rest sought. THEOR. 22.   

PROBLEM XXIII. To divide a right line given in power, according to any proportion given in two right lines.  
LEt the right line given be C. D. and the proportion given, 
Eu. 1.47.3.31.6.4.6.8.6.20. Cor. 2.  that between the two right  diagram lines A. and B. It is required to divide C. D. in power, according to the proportion of A. to B. First, by the 13. PROB. divide the given line C. D. in the point F. proportionally, as A. to B. Than on the same given line describe the semicircle C. E. D. and from the point F. raise a perpendicular, to cut the limb in E. from which point draw the lines E.C. and E.D. which two lines together, shall be equal in power to the given line, and the power of the line E. D. shall be in such proportion to the power of the line E.C. as A. to B. which was required. DEF. 36.37. THEOR. 6.9.55.70.  
To perform the same another way, according to any proportion required, between two numbers given.  
LEt A. B. be a right line given, and let it be required  diagram to divide the power thereof in proportion, as 2. to 3. First, add the two given terms together, which make 5. wherefore divide the given line A. B. into 5. equal parts, then describe thereon the semicircle A. C. B. and on the second part erect the perpendicular 2. C. to cut the limb in C. from which point draw the line C. A. which shall be 2/5· parts of the power of the given line; and the line C. B. which shall be 3/5· parts of the power of the same given line A. B. which was the thing required. DEFIN. 36, 37. THEOR. 6, 9, 55, 70.   

PROBLEM XXIIII. To enlarge a line in power, according to any proportion required.  
LEt C. D. be a line given, 
Eu. 1.47.3.31.6.4.6.8.6.20. Cor. 2.  to be enlarged in power, according to  diagram the proportion of A. to B. Seek first by the 16. PROB. a line, which shall bear the same proportion to the given line C. D. as B. to A. which will be found to be C. E. and thereon describe the semicircle C. F. E. and, on the point D. erect a perpendicular, to cut the limb in F. and lastly, draw the line C. F. which shall be in power to the given line C. D. as B. to A. as was required. DEFIN. 38. THEOR. 6, 9, 55, 70.  
To perform the same another way.  
LEt A. B. be a line given; 
Euc. 10. Def. 5.1.47.  and let it be required,  diagram to increase the same line A.B. in power, as 2. to 5. By the ninth PROB. with the given line make the right angle D.A. B. placing D. A. equal to A. B. Than draw the line D.B. and by the 23. PROB. before going, take half the power of the given line, which is A. E. and place it perpendicularly on the line D. B. as B. F. And lastly, draw the line D. F. which shall have such proportion in power to the given line A. B. as 5. hath to 2. as was required. DEFINITION 38. THEOREM 6, 9, 55, 70.   

PROBLEM XXV. To divide the circumference, or found out all the cord lines of a circle, not exceeding the tenth.  
LEt A.B. C.D. be a circle given; whereof it is required to found the cord lines. First, the Diameter A. B. being drawn, divideth the circle into two equal parts: The next is found by opening the Compass to the semidiameter, and with one foot in A. strike through the limb at F. and E. and draw the  diagram line F. E. which will divide the circumference into three equal parts. Than draw the line C. D. dividing the Diameter A. B. into two equal parts at right angles, 
Euc. 1.47.2.6.4.6.4.15.13.9, 10.  and draw the line C.A. which shall be the side of an inscribed square. Than setting one foot in I and at the distance I C. cross the Diameter in G. and draw the line C.G. which shall be the fift part, or the side of a Pentagon. The sixth part, or the cord of a Sextans, is the semidiameter. The seventh part, or the side of an inscribed Septagon, is half the third, as F.I. or I E. The eight being the side of an inscribed Octogon, is the line B.H. which is the cord of half the Arch or Quadrant B.H.D. The ninth is a cord of the third part of the arch F.L.A.E. as F.L. And the tenth is the line K.G. which is the greater segment of the semidiameter, divided by extreme and mean proportion. THEOR. 6, 36, 55. PROB. 118, 119, 121.   

PROBLEM XXVI. To draw a line from an angle in a Plot given, which shall take in as much as it cuts off.  
LEt A.B.C.D.E. be a Plot given. It is required by a line drawne-from an angle, 
Euc. 1.37.  in the  diagram same Plot, to take in as much as shall be cut off by the same line. Increase the line C. D. at pleasure towards F. Than from the point A. draw the line A.D. making the Triangle A.E. D. and by the point E. draw the line E.F. parallel to A.D. cutting the increased line in F. from whence draw the line F. A. which shall take in the Triangle H. and cut off the Triangle G. each equal to other, as was required. THEOR. 26.   

PROBLEM XXVII. To draw a line in such sort, as to retain the superficial quantity of a given Triangle, and yet altar the Base to any possible length required.  
LEt A.B.C. be a Triangle given, 
Euc. 1.37.  and let it be required to draw a line in such sort, as to retain the superficial quantity of the same Triangle, and yet altar the Base to the length of the line F. First, increase the Base B.C. to D. making B. D. equal to the given line F. Than draw the line A. D. and by the 3. PROB. by the point C. draw a parallel line to A. D. as E.C. cutting the side A.B. in E. And lastly, draw the line E.D. including the Triangle E. B. D. which retaineth the superficial quantity of the given Triangle A. B. C. and yet the Base altered, to the length required. THEOR. 26. PROB. 47.74.  
 diagram   

PROBLEM XXVIII. To draw a line in such sort, as to retain the superficial quantity of a given Triangle, and yet altar the altitude to any possible height required.  
LEt A.B.C. in the former Diagram be a Triangle given; 
Euc. 1.37.  and let it be required to draw a line in such sort, as to retain the superficial quantity of the same Triangle, and yet altar the altitude to the height of the line G. At the distance of G. draw the line E. H. parallel to the Base B.C. which shall cut the side A. B. in E. from which point draw the line E.C. Than continued the Base at pleasure towards D. and from the point A. draw a line parallel to the line E.C. as A.D. cutting the line of continuation in D. from which point draw the line E.D. which shall make the Triangle E.B.D. retaining the superficial quantity of the given Triangle A.B.C. and yet the altitude altered to the height required. THEOR. 26.   

PROBLEM XXIX. To found two right lines in such proportion as two figures given.  
LEt the Trapezium, 
Euc. 1.17.6.1.1.38.1.41.   diagram A.B.C.D. & the Parallelogram F.G.K. H. be two figures given. It is required to find 2. right lines in such proportion the one to the other, as those two given figures are. By the 26. PROB. reduce the Trapezium A.B.C.D. into the Triangle A. C. E. and let fall the perpendicular A.O. Than increase the Base of the Parallelogram at pleasure towards M. and place on that increased line K.L. equal to the Base K.H. and draw the line F. L. including the Triangle F. L. H. equal to the Parallelogram F.G. K.H. Than by the 27. PROB. reduce the Triangle F. L.H. to the Triangle N.M.H. making the Base thereof M.H. equal to the Base C.E. of the first reduced Triangle; and lastly, let fall the perpendicular N.P. So have you the two perpendiculars N. P. and A. O. in such proportion the one to the other, as the Parallelogram to the Trapezium given. THEOR. 35, 26, 28.   

PROBLEM XXX. To found two right lines in such proportion the one to the other, as two given Squares.  
LEt A.B.D.C. and E.F. G. D. be two Squares given. It is required, 
Euc. 6.4.8. Ac Cor. 19 Euc. 12.2. This Prob. is general for all figures reduced into Squares. Which may also be thus wrought. Seek a third proportional to the sides C. D. and D.G. which shall have such proportion to either of them, being the other extreme, as the squares have the one to the other. Euc. 12.2.   diagram to found two right lines in such proportion the one to the other, as the two given squares. Continued the side D. G. of the greater square given to H. making G. H. equal to the side of the lesser square given: Than draw the line F. H. subtending the right angle F.G.H. from which right angle let fall a perpendicular, cutting the line F.H. in K. So shall H.K. and K.F. be two lines in such proportion, as the two given squares, as was required. THEOR. 6, 9, 11, 55, 70.   

PROBLEM XXXI. To draw a touch line to a circle given, from a point assigned.  
LEt A. be the circle given, 
Euc. 3.17.  and B. the point assigned. It is required  diagram from the point B. to draw a touch line to the circle A. Draw the line A.B. from the centre to the point assigned, and divide the same into two equal parts in the point E. and at the distance E.A. or E. B. cross the limb in C. by which point and the point assigned B. draw the line B.D. which shall be a touch line to the circle A. as was required. THEOR. 52.   

PROBLEM XXXII. To apply a line unto a circle given, in such sort, as thereby to cut off a segment, wherein may be placed an angle, equal to an angle given.  
LEt A. be a circle given. 
Euc. 3.34. Ceul. 2.48.49.  It is required, to apply thereunto a line in such sort, as to cut  diagram off a segment from the same circle, wherein may be placed an angle equal to the angle B. By the PROB. last before going, draw the touch line C. E. D. and on the point E. by the eight PROB. make the angle F. E. D. equal to the given angle B. by drawing the line F.E. So shall the segment F.G.E. contain an angle, drawn from any point in the arch thereof, equal to the given angle B. as the angle E. G. F. the thing required. THEOREM 56.   

PROBLEM XXXIII. To describe, upon a line given, such a segment of a circle, as shall contain an angle, equal to an angle given.  
LEt the line given be A. B. and it is required to  diagram describe thereon such a segment of a circle, 
Euc. 3.33.  as shall contain an angle equal to the given angle F. To the end B. of the given line A.B. draw the line G. B. making by the 8. PROB. the angle A. B. G. equal to the given angle F. Than on the point B. erect a perpendicular to the line G.B. by drawing the line D.B. and on the end A. of the given line make the angle E.A.B. equal to the angle D. B. A. and draw the line A.E. which shall cut the perpendicular D. B. in C. which shall be the centre: Therefore on the point C. at the distance C.A. or C.B. describe the arch A. D. E. B. which with the given line shall make the segment of a circle, containing an angle equal to the given angle F. For if two lines be drawn from any one part of the circumference to the ends of the given line, they shall contain such an angle, as those at the points D. and E. which was required. THEOR. 52.56.   

PROBLEM XXXIIII. A segment of a circle being given; to found out the centre, and consequently the diameter and the whole circle.   diagram 
To found out Arithmetically the diameter of the whole circle, (the segment A. B. C. being given) and consequently the other parts; work thus: Suppose the cord line A. C. to be 12. and the perpendicular B. 2.4. Square half the cord line, which makes 36. which divided by the perpendicular 4. quoteth, 9 whereunto add the same perpendicular, which makes 13. the length of the whole diameter, whereby the rest of the parts are easily known.  
To find the extension of the arch line A. K. B. C. and to deliver the same in a right line, work thus; Divide the cord A. C. into four equal parts, and place one of those parts on the arch from A. to K. and from K. draw a line to the third part in the cord line, as K. 3. which taken double, shall be equal to the arch line, A. K. B. C. THEOR. 59    

The second Part.  
OF the making and description of all sorts of superficial figures, with their several and particular mensurations.  

PROBLEM XXXV. To make an equilater triangle, the side thereof being given.  
LEt A B. be a right line given, and it is required to  diagram make an equilater triangle, whose side shall be equal to the same line. 
Euc. 1.1.  At the distance A. B. setting one foot in A. strike an arch line towards D. and at the same distance with one foot in B. cross the same arch line in D. and from the intersection draw the lines D. A. and D. B. which with the given line A. B. shall make the equilater triangle A. B. D. as was required. DEF. 21. PROB. 36.   

PROBLEM XXXVI. To found the perpendicular of an equilater triangle Arithmetically, the side being given.  
IN the former Diagram let the side be given 8. It is required to found the perpendicular. 
By this Prob. the perpendicular of an Isoseeles is also found, the side and base being given.  Square the side 8. makes 64. then square the half base, 4. makes 16. which deduct from 64. rests 48. whose square root √. 48. near rational 6. 9/10 is the length of the perpendicular D C. required. THEOR. 6.19.29.   

PROBLEM XXXVII. The perpendicular and side of an equilater triangle being given, to find the Area or superficial content.  
IN the former Diagram the perpendicular √. 48. and the side 8. is given, and the Area is required, Multiply the whole of either by half, the other as √. 48. by 4. the product is √. 768. near rational, 27. 39/54· the superficial content required. THEOR. 24.  
Or thus, without the perpendicular.  
MVltiply 4. the half of one of the sides squarely, it makes 16. and the same product by the former half, 
Ram. 8.8.  makes 64. and that by half the perimeter which is 12. the product is 768. from whence extract the square root, which is near rational 27. 39/54· the superficial content as before.   

PROBLEM XXXVIII. To make a right angled triangle, the two containing sides being given.  
LEt A. and B. be two right lines given, for the containing sides of a right angled triangle required to be made. By the 9 PROB. of the two given sides A. and B. make a right angle, as the angle C. F. E. then draw the subtending line C. E. 
Euc. 3.31.  So have you included the right angled triangle C. F. E. with containing sides equal to the given lines A. and B. which was required. THEOR. 9.6.11.55.70.  
 diagram   

PROBLEM THIRTY-NINE. The perpendicular and base of a right angled triangle given to found the superficial content.  
IN the former Diagram the perpendicular F. G. 12. and the base C. E. 25. is given, and the content required. Multiplle the whole of either by half the other, as 12. by 12. ½·S or 25. by 6. the product is 150. the superficial content required. Or multiply the whole by the whole, the Product is 300. whereof take half being 150. as before. THEOREM. 23.25.  
Or thus, without the perpendicular.  
MVltiply the containing sides 15. by 20. the product is 300. whereof take half for your demand. Or multiply the half of the one in the whole of the other, the Product is 150 as before. THEOREM. 23.   

PROBLEM XL. To make an Jsoseeles triangle on a right line given.  
LEt the right line given be A. B. whereon it is required to describe an Isosceles triangle. Open the compass at pleasure, and placing one foot in A. with the other strike an arch towards C. and at the same distance placing one foot in B. cross, 
Euc. 1. Def. 25.  the former arch in C. and draw the lines C. A. and C. B. which shall include the Triangle required. THEOR. 10.18.19.  
 diagram   

PROBLEM XLI. The perpendicular and base of an Isosceles Triangle, given to found the area, or superficial content Arithmetically.  
IN the former Diagram, let C. E. the perpendicular √. 336. near rational 18. ⅓·S be given (or found by PROB. 36.) and let the base given be 16. It is required Arithmetically to find the content. Multiply the whole of either by half, the other as √. 336. by 8. the production is √. 21504. near rational, 146. 47/73· the superficial content required. Or multiply the whole by the whole, as √ 336. by 16. the Product. is √ 86016. whereof take the half which is √. 21504. and near rational 146. 47/73· as before. THEOR. 8.19.25.  
Or otherwise without the perpendicular, thus.  
IN the same former Diagram. Add all three sides together, which makes 56. whereof take half which is 28. then take the difference of each side from that half, as 8.8. and 12. And enfold those 3. each in other makes, 768. which multiply by 28. the former half, the Product is, 21504. whereof take the square root, which is near rational, 146. 47/73· for the superficial content as before, which rule is general for all right lined Triangles whatsoever.   

PROBLEM XLII. To make a Triangle of three unequal sides, the lines being given, so as the two shortest together be longer than the third line.  
LEt A. B. and C. be three lines given, whereof a Triangle is required to be made. 
Euc. 1.22.  Place the line A. from E. to F. then taking in the compass, the line B. with one foot in F. make an arch towards D. and at the distance of the third line, and with one foot in E. cross the arch line in D. from which intersection to E. and F. draw the lines D. E. and D. F. So shall you include the Triangle D. E. F. whose sides are equal to the given lines A. B. and C. as was required. THEOR. 15.  
 diagram   

PROBLEM XLIII. To find the perpendicular of any Triangle Arithmetically, the sides being given.  
IN the former Diagram let the sides given be E. D. 6. D. F. 8. and F. E. 10. It is required to find the perpendicular. Square the three sides severally, which make 36. 64. and 100 then add the square of the base, E. F. 100 to the square of one of the sides, as to 36. the square of the side F. D. which makes 136. from whence subtract the square of the other side D. F. 64. rests 72. whereof take the half 36. which divide by the base 10. producing 3. ⅗·S for the lesser segment of the base E. G. 
Euc. 1.47.  The square of which segment 12 24/25· being deducted from the square of E. D. 36. first added, the remainder is 23. 1/25 whose radix is 4. ⅘·S the length of the perpendicular D. G. required. THEOR. 70.   

PROBLEM XLIIII. The perpendicular and base of any Triangle being given, to find the area or superficial content thereof Arithmetically.  
IN the same former Diagram let the perpendicular D. G. 4. ⅘·S & the base E. F. 10. be given, and let it be required to found the Area or superficial content of the Triangle D. E. F. Multiply the whole of either by half the other, as the whole perpendicular 4. ⅘·S by the half base 5. the Product shall be 24. the superficial content required. Or multiply the whole by the whole, as 4. ⅘·S by 10. the Product is 48. whereof take half, which is 24. as before the superficial content. THEOR. 25.7.  
Or thus without the perpendicular.  
Add the three sides together 6. 8. and 10. making 24. whereof take 12. the half, and then the difference of each side from that half, as 6. 4. and 2. and enfold those differences each into other, which bringeth 48. that multiplied by the former half 12. produceth 576. The radix whereof is 24. the superficial content as before.  
This Proposition holdeth general in all Triangles, and is the fittest and most meet for the mensuration of all plots and irregular figures; 
Regula generalis.  whatsoever, being first reduced into Triangles, by drawing lines from angle to angle after the usual manner.   

PROBLEM XLV. To make a Triangle upon a line given, like unto another Triangle given.  
LEt the Triangle A. B C. and the line E. F. be given. It is required on the line E. F. to make a Triangle like unto the Triangle given. 
Euc. 1.37. Ceul. 2.78.  Upon the point E. by the 8. PROB. describe an Angle equal to the Angle A. B. C. and on the point F. describe another Angle equal to the Angle A. C. B. in the given Triangle, whereby shall be included the Triangle D. E. F. upon the line given E. F. like unto the Triangle given A. B. C. that is with equal angles and lines proportional, as was required. DEF. 40. THEOR. 11.49.  
 diagram   

PROBLEM XLVI. To make a Triangle equal to another Triangle given, upon the same Base, having an Angle equal to an Angle given.  
LEt A. B. C. be a Triangle given, whose base is B. C. and let the  diagram Angle given be D. It is required upon the base B. C. to describe a Triangle equal to the Triangle given A. B. C. having an Angle equal to the given Angle D. 
Euc. 1.28.1.37.  By the point A. by the 3. PROB. draw a parallel line to the base B. C. as E. A. and on the point C. describe an Angle equal to the Angle D. whereof the base to be one of the containing sides, and draw the line E. C. till it intersect, the parallel line in E. And lastly, from the point E. draw the line E. B. which shall include the Triangle E. B. C. equal to the given Triangle A. B. C. upon the same base, and having the Angle thereof E. C. B. equal to the given Angle D. as was required. THEOR. 26.   

PROBLEM XLVII. To make a Triangle: equal to another Triangle given, with a base or perpendicular limited.  
Work this Problem in all respects according to the doctrine taught in PROB. 27. and 28. THEOR. 26. 
Euc. 1.37,    

PROBLEM XLVIII. To make an Isosceles Triangle upon a line given, whose Angles at the base, shall be either of them double to the third Angle.  
LEt A. B. be a line given, whereupon it is required to describe a Triangle, whose Angles at the base shall be either of them double, to the third Angle. Suppose the given line A. B. to be the greater segment, 
Euc. 4.10.  of a line divided by extreme and mean proportion, and by the 21. PROB. find out the whole line, at the distance of which line so found, with one foot in A. strike an arch line towards C. and at the same distance with one foot in B. cross that arch line in C. from which intersection, draw the lines C. A. & C. B. which shall include the Triangle required. THEOR. 47.69.  
 diagram   

PROBLEM XLIX. To make a Triangle equal to a Parallelogram given, upon a line limited, and with an Angle equal to an Angle given.  
LEt the Paralelogram  diagram given be A. B. C. D. the line given E. C. & let the Angle given be M. It is required upon the line given to make a Triangle equal to the parallelogram given, 
Euc. 1.23.6.12.6.16.  having an Angle equal to the given Angle M. Take the given line E. C. for the base, and by the 16. PROB. reason thus reciprocally. If half the given line E.C. yield H. D. the breadth of the parallelogram: what gives A.B. the length thereof: The answer shall be F. G. the perpendicular of the Triangle to be made; at which distance by the 2. PROB. draw the line K. F. parallel to the base E. C. and on the end E. by the 8. PROB. describe an Angle equal to the given Angle M. and draw the line E. F. which shall cut the parallel line in F. from which point of intersection draw the line F. C. which shall include the Triangle E. F. C. equal to the parallelogram given, upon the line E. C. given and having the Angle F. E. C. thereof equal to the given Angle M. as was required. THEOR. 42.   

PROBLEM L. To make a Square upon a line given for the side thereof.  
LEt C. D. be a line given, whereon it is required to describe a Square. By the 6. PROB. on the end of the given line C. erect the perpendicular B. C. equal to the given line C.D. at which distance, 
Eu. 1.46.  with one foot in B. strike an arch line towards A. then with one foot in D. cross that arch line in A. And lastly, draw the lines A. B. and A. D. which shall include the square A. B. C. D. upon the line given for the side thereof C.D. as was required. THEOR. 2.  
 diagram   

PROBLEM LIVELY The side of a Square being given, to find the Area or superficial content Arithmetically.  
LEt A. B. 16. be the side of a square given, whereof the superficial content is required. Multiply the line given 16. in itself, the product is 256. the superficial capacity of the Square A. B. C. D. required. THEOR. 38.   

PROBLEM LII. To make two Squares which shall be equal the one to the other; and also to two unequal Squares given.  
LEt the right lines A. and B. be the two sides of two unequal squares given; and let it be required to make two other squares, which shall be equal unto them, and also the one of them equal to the other. By the 9 PROB. of the two given lines A. and B. make a right Angle, 
Euc. 1.6.1.32.  as the Angle C. E.D. and draw  diagram the subtending line C. D. on which line describe the semicircle C. E. F. D. and on the middle of C. D. erect the perpendicular F. G. to cut the limb in F. From which point draw the two lines F. C. and F. D. which shall be the two sides of two squares, equal to the two given squares, and also the one of them equal to the other; as was required. THEOR. 6.23, 55.   

PROBLEM LIII. To describe a Square in such sort as it shall pass by any three points given.  
LEt A. B. and C. be three points given, by which it is required to  diagram make a square to pass. First, by the two nearest points, 
Euc. 1.12.1.31.  as B. and C. draw a line at length as the line B. G. then by the third point A. by the 3. PROB. draw the line D. E. parallel to the line B. G. and on the point B. raise the perpendicular B. D. to cut the line D. E. in D. then at the distance D B. mark the points E. and F. between which draw the line E. F. which shall include the square D. E. F. B. passing by the three given points, as was required. PROB. 3.6.   

PROBLEM liv. To make a long Square or right angled Parallelogram, the length and breadth being given.  
LEt A. and B. be two right lines given for the length and breadth, it is required to make a right Angled parallelogram, whose length shall be A. and breadth B. By the 9 PROB. of the two given lines A. and B. make a right Angle, as the Angle C. F.E. and at the distance of the line A. with one foot in C. make an arch line towards D. and at the distance of the line B. with one foot in E. cross, the former arch line in D. from which intersection draw the lines D.E. and D. C. which shall include the right angled parallelogram C. D. E. F. having the length and breadth equal to the given lines; as was required. THROR. 55.  
 diagram   

PROBLEM LU. The length and breadth of a right angled Parallelogram or long Square being given, to find the Area or superficial content thereof Arithmetically.  
IN the former Diagram let the length given be A. 20. and the breadth B. 10. Multiply 20. by 10. the Product will be 200. for the Area or superficial capacity of the Parallelogram, C. D. E. F. which is the thing required. THEOR. 38.   

PROBLEM LVI. To make a Parallelogram, whose length is limited, equal to a Triangle given, with two opposite Angles each equal to an Angle given.  
LEt the Triangle given  diagram be F. E. C let the length limited be D. C. & let the Angle given be M. It is required upon the given line to make a Parallelogram equal to the Triangle given, 
Euc. 1.23.6.12.6.16. Ceul. 2.82.  having two opposite Angles each equal to the Angle given. By the 16. PROB. reason thus reciprocally; If the given line D. C. yield the perpendicular of the Triangle F. G. what gives half the base E. C. the answer shall be H. D. the breadth of the Parallelogram to be made; at which distance draw the line A. B. Parallel to the given line D. C. then on the point C. by the 8. PROB. describe an Angle as the Angle B. C. D. equal to the given Angle M. and draw the line B. C. which shall cut the Parallel line A. B. in B. from whence mark out the line A. B. equal to the given line D. C. and lastly draw the line A. D. which shall include the Parallelogram, A. B. C.D. equal to the Triangle given upon the line D. C. given, and having two opposite Angles, namely, A. and C. each equal to the Angle given M. as was required. THEOREM 42.   

PROBLEM LVII. To make a Rhombus, the side being given.  
LEt the line given be A. B. whereon it is required to describe a Rhombus. 
Euc. 1.1.  At the  diagram distance A. B. with one foot in B. describe the arch line D. C. & at the same distance setting one foot in A. Cross, the Arch line in D. on which point placing the compass at the former distance cross the arch line in C. And lastly, draw the lines D A. D C. and C B. which shall include the Rhombus, A. B. C. D. on the given line A. B. as was required. DEFINITIOM 26.   

PROBLEM LVIII. The side of a Rhombus being given to find out the Area or superficial content thereof Arithmetically.  
IN the former Diagram, let the side A B. or D C. 16. be given, and let it be required to found the Area or superficial content thereof. By the 36. PROB. find out the perpendicular D. E. √. 192. and near rational, 13.23/26· and multiply the same by the given side, 16. the Product shall be √. 49152. and near rational, 221.311/442 for the superficial content required. THEOR. 34.   

PROBLEM LIX. To make a Rhomboydes the length and breadth being given in two right lines.  
LEt the length given be the line D. C. and the breadth, the line G. of which length and breadth it is required to describe a Rhomboydes. At the distance of the given breadth G. and from one of the ends of the given length, as from C. choose a point as B. and  diagram at the same distance with one foot on the other end of the given length, 
Euc. 1.31.  as on D. strike an arch towards A. then at the distance of the given length with one foot in the point B. cross the former arch in A. And lastly, draw the lines A. D. A. B. and B. C. which shall include a Rhomboydes of the length and breadth given. PROB. 3.   

PROBLEM LX. To make a Rhomboydes with lines limited, having two opposite Angles, equal to an Angle given.  
LEt the limited lines for the length & breadth be D.C. & G. as in the former Diagram, and let the Angle given be F.C. B. and let it be required to make a Rhomboydes of such length and breadth, and with two opposite Angles equal to that given. 
Euc. 1.23.1.31.  On the end C. of the given length D.C. by the 8. PROB. protract an Angle equal to the given Angle, as D. C. B. making the line B. C. equal to the given breadth G. And so work forward in all respects as in the former Problem. PROB. 3.8.   

PROBLEM LXI. A Rhomboydes given to find the superficial content Arithmetically.  
LEt the Rhomboydes given be that in the former Diagram, It is required to found the superficial content thereof. Take the given length D. C. or A. B. 20. and seek out (as hath been formerly taught) the parallel distance or perpendicular B. E. 12. which multiply the one by the other, the Product is 140. the Area of the given Rhomboydes required. THEOR. 34.   

PROBLEM LXII. To describe a Pentagon, having sides and Angles equal.  
LEt any obscure line be drawn as the right line A. B. and at any convenient distance place thereon four equal parts or divisions, as from HUNDRED to 4. and at the distance of two of them, on the second part, as a centre, describe an obscure circle,  diagram on which centre raise the perpendicular D. 2. to cut the limb in D. then at the distance D. 1. with one foot in 1. cross the line in K. and with the distance D. K. mark the limb of the Circle in the points D. E.F. G. and H. And lastly, draw lines from point to point, which shall include the Pentagon required. DEF. 29. THEOR. 47. PROB. 20.21.25.   

PROBLEM LXIII. To describe a Pentagon upon a line given.  
LEt A. B. be a right line  diagram given, 
Euc. 13.8.  whereon it is required to describe a Pentagon. Suppose the line given, to be the greater segment of a right line divided by extreme and mean proportion. And by the 21. and 22. PROB. find the whole line; which let be A. C. and at the distance of A. C. with one foot in B. strike an arch towards H. and another towards K. and with the foot in A. strike one towards L. then take the distance of the given line, and on A. and B. cross the Arches at H. and L. and on H. or L. cross the Arch in K. And lastly, draw lines from each intersection to other, which shall enclose the Pentagon, as was required. THEOR. 47. PROB. 20.21, 22, 25.   

PROBLEM LXIIII The side of a Pentagon being given, to find the superficial content Arithmetically.  
IN the former Diagram, let the given side be 10. It is required to find the Area of that Pentagon. By the 36. PROB. (supposing an Isosceles described on any side of the Pentagon, whose top is the centre, as A. D B.) seek the perpendicular D. G. 6.22/25· which multiplied in half the perimetrie 25. produceth 172. the Area required. This rule is general in all kind of regular polligons, 
A general Rule.  of how many sides so ever; aswell for their superficial content, as finding their perpendicular. THEOR. 19.39. PROB. 36.   

PROBLEM LXV. To make two like Figures, bearing the one to the other, any proportion assigned in two right lines.  
LEt A. and B. be two right lines given, and let it be required to make two  diagram like Triangles, Squares, Circles or other like figures, having such proportion the one to the other, 
Eue. 6.4.6.8.  as A. to B. make of the two given lines one right line, as D. E. and describe thereon the Semicircle C.D.E. and on the point F. where the 2. given lines meet, erect the perpendicular C. F. to cut the limb in C. from whence draw the lines C. D. and C. E. which two lines shall be the sides of two equilater Triangles, or of two Squares, or other like figures or the diameters of two circles, bearing such proportion the one to the other, as the two given lines; which was required. THEOR. 6.11, 55, 66, 70, 65.   

PROBLEM LXVI. Two Circles being given, to make one Circle equal to them both.  
LEt A. and B. be the Diameters of two Circles given. It is required  diagram to make one Circle equal to them both. With the lines A. and B. by the 9 PROE. make a right Angle, as D. C. E. then draw the subtending line D. E. And lastly, on the line D. E. describe the Circle D. C. E. which shall be equal to the two given circles as was required. THEOR. 65.70, 66.   

PROBLEM LXVII. The Diameter of a Circle being given, to found the circumference thereof Arithmetically.  
IN the former Diagram, let the Diameter given be 14. It is required to find the circumference thereof. Multiply the Diameter given 14. by 22. the Product is 308. which divided by 7. bringeth 44. the circumference required. Or multiply 14. the Diameter by 3. 1/7· the Product is 44 as before. THEOREM. 59  
If the circumference be given, and the Diameter required, It appeareth by this rule, that the circumference 44. being multiplied by 7. and the product divided by 22. bringeth 14. the Diameter.   

PROBLEM LXVIII. The Diameter and Circumference of a Circle being given, to found the Area, or superficial content thereof Arithmetically, divers ways.  
IN the former Diagram let the Diameter of the Circle D. C. E. be 14. and the Circumference thereof 44. It is required to found the superficial content. Multiply the Semicircumference 22. by the Semidiameter 7. the Product will be 154. the superficial content required. THEOR. 62.  
Or multiply the whole Circumference 44. by the Semidiamenter 7. the Product will be 308. whereof take half, which is 154. as before. THEOREM. 60.  
Or multiply the square of the Diameter 196. by 11. the Product will be 2156. which divided by 14 bringeth 154. as before. THEOR. 61.   

PROBLEM LXIX. The Diameter and Arch-line of a Semicircle given, to found the Area thereof.  
LEt A. B. C. be a Semicircle given, 
A B. √. 98.  
A. E. √. 24. ½·  
E. D. √. 24. ½·  
F. E. 7.— √. 24. ½·S  whose Diameter is A. C. and the Arch line A.B.C. It is required to find the Area of the Semicircle. Multiply half the arch line 11. by the Semidiameter 7. The Product will be 77. for the Area required. THEOREM. 63.   

PROBLEM LXX. The Semidiameter and Arch line of a Sector of a Circle given to found the Area.  
IN the former Diagram, let B. C. D. be the Sector of a Circle, whose Semidiameter is D. C. or D. B. and the arch line B. C. and it is required to find the Area. Multiply the Semidiameter 7. by half the arch line B. C. 5. ½· the Product is 38. ½· for the Area required. THEOR. 64.   

PROBLEM LXXI. Any Segment or part of a Circle being given, to find the superficial content thereof.  
IN the former Diagram, let A. F. B. E. be the Segment of a Circle, the content whereof is required. By the 34 PROB. find out the Centre, and then draw the lines D. A. and D. B. and cast up the whole Figure A. F. B. D. as in the last Problem which will be 38. ½· then find the superficial content of the Triangle, A.B.D. by the 41. PROB. which is 24. ½ and deduct the same out of the whole content 38. ½ resteth 14. for the superficial content of the given Segment as was required.  
By this rule (observed with discretion) may all manner of Segments or parts of a Circle, whether greater or lesser than a Semicircle, be easily measured without further instruction.  
But here is to be noted, that the precedent rules concerning the mensuration of Circles, 
Note.  and their several parts, are not exactly true: for that the proportion between the Diameter and circumference is irrational; and the squaring of a Circle or the means thereof (other then mechanically) not yet discovered or found out; yet of such sufficient preciseness as no notable or apparent error can be made or found in the conclusions thereby wrought.   

PROBLEM LXXII. An irregular plot or Figure being given, to find the Area or superficial content thereof.   diagram 
This I hold the best manner of Mensuration of plots, aswell for expedition, as exactness in avoiding errors, often happening by multiplicity of numbers, and many multiplications.  
THus in this second part, have I taught the Description and Mensuration of all manner right lined superficial Figures, according to the strict and precise rules and precepts of absolute Art: yet seeing that dispatch and expedition in business of import, is much more requisite than needless niceness; I would not have my Surveyor ignorant or unfurnished of such other ready and perfect helps (though more mechanike) as may yield him ease, and save much labour in furthering his intents. To which end, for the speedy and exact mensuration of all superficial figures, I would have provided a Protractor in Brass, whose Scale should contain in length about 8. or 10. inches with equal Divisions on the edge of either side, of 12 in an inch on the one side, and 11. on the other, being numbered by tens after the usual manner of those kind of works (which for mine own part is the Scale, I never use in all my first drawn plots, whether the quantity be small or great, well knowing the inconveniency of smaller Scales) having placed thereon a Sextans of a Circle most excellent for many uses, as the speedy laying down of any Angle required, or the ready finding of any Angle given, etc. The order and making whereof is well known to Master Elias Allen, who for myself and friends hath made of them. By this Scale with help of the middle line thereof, Orthigonally drawn to the edge; you shall readily raise a perpendicular, and as instantly receive the length thereof, and most speedily obtain the base of any Triangle, or the side of any Figure given by applying the edge of the Scale thereunto: which is much more facile and speedy than the former Precepts; and to be preferred for exactness, speed and perfection, before the ordinary course with Scale and Compass. But doubting to exceed the Scale and Compass of my intended purpose, I will here conclude the second part of this my second Book.    

The third Part.  
OF the Reduction and Translation of all manner of superficial figures, from one form unto another, retaining still their first quantity.  

PROBLEM LXXIII. To reduce one triangle into another, on the same base, but having an angle equal to an angle given.  
LEt A. B. C. be a triangle given, 
Euc. 1.37.  and let  diagram the angle given be E. and the base of the given triangle B.C. on which base it is required to reduce the given triangle to another, having an angle equal to the angle E. From the point A. by the 3. PROB. draw a parallel line to the base B. C. as the line A. D. then on the point C. by the 8. PROB. make the angle D. C. B. equal to the given angle E. and draw the line D. C. cutting the parallel line in D. and lastly, draw the line D. B. which shall include the triangle D. B. C. equal to the given triangle A. B. C. upon the same base B. C. and having an angle equal to the given angle E. as was required. THEOR. 26.   

PROBLEM LXXIIII. To reduce one triangle into another, upon a base equal to a base given.  
LEt the triangle given be A. B. C. and F. the given base, 
Eu. 1.37.   diagram whereon it is required to reduce the given triangle. Take the given base F. and place it from B. to E. and from the point E. to the top of the given triangle A. draw the line A.E. then increase the side A. B. of the given triangle towards D. and from the point C. by the 3. PROB. draw a parallel line to A. E. as D. C. cutting the increased side in D. and lastly, draw the line D. E. which shall include the triangle D. B. E. equal to the given triangle A. B. C. upon the base B. E. equal to the given base, as was required. THEOR. 26. PROB. 27.  
If the given base be greater than the base of the given triangle, work in all respects, as is taught in PROB. 27.   

PROBLEM LXXV. To reduce one triangle to another, of any possible height required.  
LEt the triangle A. B. C. in the former Diagram be given, 
Ceul. 3.5.  and let the height required be the line G. of which height it is required to reduce the given triangle into another of the same quantity At the distance of the given height G. by the 2. PROB. draw the line H. K. parallel to the base B. C. then increase the side A. B. till it cut the parallel line H. K. in D. from which point draw the line D. C. then by the point A. draw the line A. E. parallel to D. C. cutting the base in E. and lastly, from the point E. draw the line E. D. which shall include the triangle D. B. E. equal to the given triangle A. B. C. and of the height G. as was required. THEOR. 26. PROB. 28.  
If the given height be less than the height of the given triangle, work in all respects, as is taught in PROB. 28.   

PROBLEM LXXVI. To reduce a triangle given into a square.  
LEt A. B. D. be a triangle given. 
Euc. 1.31.  It is required to reduce the same into a Geometrical square. By the 17. PROB. find out a mean proportional line between the base B. D. and half the perpendicular A. C. which shall be the line E. D. on which line by the 50. PROB. describe the square E. F. G. D. which shall be equal to the given triangle. THEOR. 25.27.  
 diagram   

PROBLEM LXXVII. To reduce a triangle given, into a right angled parallelogram.  
LEt A. B. C. be a triangle given. 
Euc. 1.42. Ceul. 2.28. & 2.80.  It is required to reduce the same into a right angled parallelogram. From the angle A. let fall to the base B. C. the perpendicular A. F. then take half thereof for the breadth, and the whole base B. C. for the length, with which breadth and length by the 54. PROB. describe the right angled parallelogram D. E. B. C. which shall be equal to the given triangle, as was required. THEOR. 25.  
 diagram   

PROBLEM LXXVIII. To reduce a tringle given, into a parallelogram, having an angle equal to an angle given.  
LEt A. B. C. be a triangle given, and let  diagram the angle given be D. It is required to reduce the same triangle into a parallelogram, having an angle equal to the angle D. By the 3. PROB. from the point A. draw the line E. A. parallel to the base B. C. then divide the base B. C. into two equal parts in the point G. on which point by the 8. PROB. describe the angle F. G. B. equal to the given angle D. and draw the line F. G. cutting the parallel line in F. from whence draw the line F. E. equal to G. B. and lastly, draw the line E. B. which shall include the parallelogram E. F. G. B. equal to the given triangle, having an angle F. G. B. equal to the given angle D. as was required. THEOR. 28.   

PROBLEM LXXIX. To reduce a triangle given into a Rhombus.  
LEt the triangle given be A. B. C. and it is required to reduce the  diagram same into a Rhombus. 
Euc. 1.37.  
Cor. 6.19.  By the 3. PROB. draw the line A. H. parallel to the base B. C. then divide the base B. C. into two equal parts in the point E. on which point at the distance E. B. or E. C. strike an arch towards D. and on the point C. at the same distance cross the former arch in D. by which intersection and the point C. draw the line H. C. to cut the parallel line in H. then by the 17. PROB. found out a mean proportional line between C. D. and C. H. which is C. K. upon which line C. K. by the 57 PROB. describe the Rhombus G. K. C. F. which shall be equal to the triangle given, as was required. THEOR. 26.41.43.   

PROBLEM LXXX. To reduce a square given into a triangle, having an angle equal to an angle given, and that on a line given.  
LEt the square given be A. B. C. D. the  diagram angle given E. and let the given line be F. 
Euc. 1.37.  On which line it is required to reduce the given square in to a triangle, having an angle equal to the angle E. First, continued the side of the given square D. C. to G. making G. D. equal to D. G. and draw the line G. B which shall include the triangle B. G. C. equal to the given square; then take the given line F. and lay it down from HUNDRED to O. and by the 74. PROB. make the triangle M. O. C. then by the third PROB. draw the line M. N. parallel to the base G. C. and on the point C. protract an angle equal to the given angle E. as N. C. O. and draw the line C. N. to cut the parallel in N. and lastly, from the point N. draw the line N. O. which shall include the triangle N. O. C. equal to the given square, having an angle as N. C. O. equal to the given angle, and that on the line O. C. equal to the given line F. as was required. THEOR. 26.   

PROBLEM LXXXI. To reduce a square given into a triangle, with angles equal, and lines proportional to a triangle given.  
LEt the given square  diagram  diagram be A. B. C. D. and let the given triangle be E. F. G. and let it be required to reduce the same square into a triangle, 
Euc. 1.37. Cor. 6.19.  with angles equal, and lines proportional to the given triangle. According to the first Part of the last PROB. make the triangle B. H. C. equal to the given square, then continued the side A. B. of the given square towards K. and on the point H. protract the angle M. H. D. equal to the angle F. in the given triangle, drawing the line M. H. at length to cut the continued side A. B. in K. then on the point C. protract the angle M. C. H. equal to the angle G. in the given triangle, and draw the line C. M. to cut the line M. H. in M. which shall include the triangle M. C. H. with equal angles to the given triangle, but of greater content than the given square; wherefore by the 17. PROB. find out a mean proportional line between H. K. and H. M. which is H. N. and from the point N. by the 3. PROB. draw the line N. O. parallel to M. C. which shall include the triangle N. O. H. equal to the given square, and having equal angles and lines proportional to the given triangle, as was required. THEOR. 26.43,   

PROBLEM LXXXII. To reduce a square into an equilater triangle.  
LEt the square given be A. B. C. D. and it is required to reduce  diagram the same into an equilater triangle. 
Euc. 1.37. Cor. 6.19.  Double the side D. C. by increasing the same to G. and draw the line G. B. to include the triangle B. G. C. equal to the given square, then at the distance of the side of the square, with one foot in C. strike an arch towards I and at the same distance with one foot in D. cross the same arch in I and by the intersection and the point C. draw the line K. C. at length to cut A. B. in H. and take C. H. and place the same from C. to E. then by the 17. PROB. find out the mean proportion between C. E. and C G. which is C. F. at which distance describe the equilater triangle K. F. C. which shall be equal to the given square, as was required. THEOR. 26. ●5. 43   

PROBLEM LXXXIII. To reduce a square given, into a right angled parallelogram or long square, the length and breadth being limited in a right line: So as the side of the square exceed not half the line given.  
LEt the square given be A. B. C. D. and let the right line given be F. C. 
Euc. 1.47.6.13.  It is required to reduce the same square into a long square, whose length and breadth together shall be equal to F. C. Upon the given line F. C. place the given square, as in the Diagram, then describe the semicircle F. E. C. to cut the side A. B. of the given square in E. from which point let fall the perpendicular E. G. to cut the given line in G. So shall F. G. be the length, and G. C. the breadth of the long square to be made, of which length and breadth by the 54. PROB. describe the parallelogram F. G. H. K. which is equal to the given square, and of the length and breadth required. THEOR. 41. PROB. 19  
 diagram   

PROBLEM LXXXIIII. To reduce a square given into a long square, whose breadth is limited in a right line given.  
LEt the square given be A. B. C. D. and let F. be the right line  diagram given. 
Euc. 1.47, 6, 13.  It is required to reduce the same square into a long square, whose breadth shall be equal to the given line F. Continued the side C. D. of the given square towards K. and place the given line F. from D. to G. and from the point G. draw the line G. A. which divide equally in N from which point draw the perpendicular N. H. to cut the line of continuance in H. on which point at the distance H. G. describe the semicircle K. A. G. to cut the line of continuance in K. so shall K. D. be the length sought for, with which length, and the given breadth F. or D. G. by the 54. PROB. describe the parallelogram L. M. D. K. which shall be equal to the given square, as was required. THEOR. 41. PROB. 19   

PROBLEM LXXXV. To reduce a square given into a long square, whose length is limited in a right line given.  
IN the former Diagram, 
Euc. 1.47, 6, 13.  let that square be the square given, and the length given the right line P. First, continued D. C. as before towards K. and make K. D. equal to the given line P. and draw the line K. A. upon the middle whereof S. raise the perpendicular SAINT H. to cut the line of continuance in H. on which point describe the semicircle, as before, to cut D. C. in G. so shall D. G. be the breadth sought, of which breadth and the given length, make the long square, as before. THEOR. 41. PROB. 19   

PROBLEM LXXXVI. To reduce a long square given into a geometrical square.  
LEt the long square given be A. B. C. D. and it is required to  diagram reduce the same into a geometrical square. 
Euc. 2.14. Ceul. 2.37.  Continued the side D. C. of the long square given towards H. and let the breadth B. C. of the long square be placed on the line of continuation from C. to H. then on D. H. describe the semicircle D. F. E. H. and increase the breadth of the long square C. B. till it intersect the limb in E. so shall E. C. (being the mean proportional between D. C. and C. H.) be the side of the square sought, wherefore, on the line E. C. by the 50. PROB. describe the geometrical square F. E. C. G. which shall be equal to the long square given, as was required. THEOR. 41. PROB. 19   

PROBLEM LXXXVII. To reduce one long square given into another, whose length or breadth is limited in a right line given.   diagram 
Or otherwise, thus.  
IN the former Diagram, let it be required to reduce the long square there given, into another long square, 
Euc. 6. 1●.  whose breadth shall be the given line E. By the 16. PROB. reason thus. If E. the given breadth give A. D. the breadth of the given square, what gives A. B. the length thereof, the answer shall be G. C. or K. H. the length sought for, with which length and the given breadth, make the long square K. H. C. G. as before. THEOR. 42.   

PROBLEM LXXXVIII. To reduce one long square given into another, whose length and breadth shall have proportion the one to the other, as two given lines.  
LEt the long square given be A. B. C. D. 
A B. 72.  
B C. 60.  
E F. 90.  
E G. 48.  and let the proportion given be such, 
Euc. 6.12.13.14.  as that between the two right lines M. and N. It is required to reduce the same long square into another long square, whose length and breadth shall have proportion the one to the other, as those two given lines. By the 17. PROB. seek out the mean proportion between the two given lines M. and N. which is Q. seek also the mean proportion between A. D. and A. B. the breadth and length of the long square given, which is the right line P. then by the 16. PRO. reason thus. If Q. gives P. what gives N. the answer shall be E.F. for the length sought; and again, if Q. gives P. what gives M. the answer shall be E. G. for the breadth sought; of which length and breadth E.F. and E.G. by the 54. PROB. make the long square E. F. C. G. which shall be equal to the long square given, and the breadth to the length, in such proportions as the line M. to the line N. as was required. THEOR. 5, 41, 42, 71.   

PROBLEM LXXXIX. To reduce a Rhombus into a geometrical square.  
LEt the Rhombus given be A. B. C. D. and let it  diagram be required to reduce the same into a geometrical square. 
Euc. 1.36.  First, by the 17. PROB. find out a mean proportional line between D. C. the side of the given Rhombus, and the parallel distance, or perpendicular line B. E. which shall be the line F. C. upon which line, by the 50. PROB. describe the geometrical square F. G. H. C. which shall be equal to the given Rhombus A. B. C. D. as was required. THEOR. 33, 34.   

PROBLEM XC. To reduce a Rhomboydes given, into a Geometrical Square.  
LEt the Rhomboydes given be A. B. C. D. and it is required to reduce the same into a Geometrical Square. 
Euc. 1.36.  Let fall the perpendicular B. E from the Angle B. to the Base D. C. between which Base and Perpendicular, by the 17. PROB. find out a mean proportional line, which shall be F.C. upon which line by the 50. PROB. describe the Square F. G. H. C. which shall be equal to the given Rhomboydes as was required. DEF. 27. THEOR. 33.34.   

PROBLEM. XCI. To reduce a Rhomboydes given into a Triangle, having an Angle equal to an Angle given.  
LEt the Rhomboydes given be A. B. C. D. and let the Angle given be E. F. G.  diagram It is required to reduce the same Rhomboydes into a Triangle, 
Eu. 1.41.  having an Angle equal to E. F. G. First, increase the line A. B. towards E. and also the base D. C. towards F. and make C. F. equal to D. C. then on D. protract the Angle E. D. F. by the 8. PROB. equal to the given Angle E. F. G. and draw the line D. E. to cut the increased line in E. And lastly, from E. draw the line E. F. which shall include the Triangle E. D. F. equal to the Rhomboydes given, and having an Angle equal to the given Angle as was required. THEOR. 28.   

PROBLEM XCII. To reduce a Trapezium given into a right angled Parallelogram, or into a right angled Triangle.  
LEt the Trapezium given be A. B. C. D. which is to be reduced into a right angled Parallelogram, or into a right angled Triangle. First, 
Euc. 1.41. Ceul. 2.29.  draw the diagonal line B. D. then by the 2. PROB. by the points A. and C. draw the lines E. F. and H. G. Parallel to B. D. and by the point B. and  diagram D. draw E. H. and F. G. to cut the two last lines Orthigonally, so shall you include the Parallelogram E. F. G. H. which is double to the Trapezium given, wherefore divide the same into two equal parts, by drawing the line K. L. So have you the right angled parallelogram E. F. L. K. or K. L. G. H. equal to the Trapezium given. And the diagonal line E. G. being drawn, shall include the right angled Triangle, E. H. G. or E. F. G. likewise equal to the same Trapezium as was required. THEOREM 23.27.   

PROBLEM XCIII. To reduce a Trapezium given into a Triangle, upon a line given, and having an Angle equal to an Angle given.  
LEt the Trapezium given be A. B. C. D. the given line E. and let the Angle given be F. It is required to reduce the same Trapezium into a Triangle, 
Euc. 1.23.1.13.1.37.1.44.6.12.  on the given line E. and having an Angle equal to the given Angle F. First, increase the base D. C. at  diagram length towards H. then draw the line A. C. and from the point B. by the 3. PROB. draw the line B. H. parallel to the line A. C. to cut the increased line in H. and then draw the line A. H. which shall include the Triangle A. D. H. equal to the Trapezium given, but not having the Angle nor Base required; wherefore take the given Base E. and place it from D. to Q. and by the 16. PROB. reason thus; if D. Q. the given Base gives D. H. what gives A. G. the perpendicular; the answer will be the line P. for the perpendicular of the Triangle sought, at which distance draw the line M. O. parallel to the line D. H. then upon the point D. protract an Angle equal to the given Angle F. as N. D. Q. and draw the line D. N. to cut the parallel line in N. And lastly, draw the line N. Q. which shall include the Triangle N. D. Q. equal to the Trapezium given, upon a line given, and having an Angle equal to an Angle given, as was required. THEOREM 26.42.71. PROBLEM 3.8.16.74.   

PROBLEM. XCIIII. To reduce a Trapezium into a Triangle, which shall be like unto another Triangle given.  

Euc. 1.37.1, 23. Cor. 6.19. Ceul. 3.1.  LEt the Trapezium given be A. B. C. D. and let the Triangle  diagram  diagram given be E. F. G. It is required to reduce the same Trapezium into a Triangle, which shall be like unto the given triangle E. F. G. First, by the 93. PROB. last before going; reduce the Trapezium given into the triangle A. D. H. which is equal thereunto, then by the 81. BROB. reduce the same triangle A. D. H. into the triangle N. D. K. which shall be equal to the Trapezium given, and like unto the given triangle E. F. G. as was required. THEOR. 26.43. PROB. 38.16.93.   

PROBLEM XCV. To reduce an equiangled Poligon given, into a Geometrical Square.  

Euc. 2.14. Ceul. 2.30.  LEt A. B. C. D. E. be a Pentagonal Poligon given, to be reduced into a Geometrical  diagram square. By the 17. PROB. found out the mean proportional line between half the perimetrie of the given Poligon, and the perpendicular thereof M. N. being let fall from the Centre to the middle of any side, 
This rule is general for the reducing of all rectangle poligons.  which mean proportional is the line G. H. whereon by the 50. PROB. describe the square F.G.H.K. which shall be equal to the given Poligon, as was required. DEFINITION. 29. THEOREM 39.41.   

PROBLEM XCVI. To reduce a plot given into a Triangle, with lines drawn from an Angle assigned.  
LEt A. B. C. D. E. be a plot given, and let the Angle assigned be A. It is required from the Angle A. to reduce the same plot into a Triangle. First, increase the side C. D. of the given plot, of convenient length both ways towards F. 
Caeterorum polygonorum rectangulorum reductio & demonstratio buic est similima: semper enim bac reductione sigillat in unum latus detrabitur. Euc. 1.37. Ceul. 3.1.2.  and G. then draw the line A. D. and by the 3.  diagram PROB. by the point E. draw the line E. G. parallel to the line A. D. to cut the continued line in G. from which point, draw the line G. A. then draw the line A. C. and by the point B. make B. F. parallel to A. C. to cut the line of continuance in F. from which point, draw the line F. A. which shall include the Triangle A. F. G. equal to the given plot, with the lines A. F. and A. G. drawn from the Angle A. assigned as was required. THEOR. 26.   

PROBLEM XCVII. To reduce a Figure given, into a Lunula or Figure of a Lunular form.  
LEt A. B. C. D. be a square given, and let it be required to reduce the  diagram same into a Lunula. Draw the diagonal A. C. and on C. the end thereof by the 6. PROB. erect the perpendicular E. C. equal to A. C. then continued the side A. B. to E. and on the point B. at the distance B. A. or B. E. describe the Semicircle A. F. E. And lastly, on the point C. at the distance C.A. or C.E. describe the Arch line A.G.E. which shall include the Lunula A.F.E.G. equal to the given square, as was required.   

PROBLEM XCVIII. To reduce an irregular Figure given, into a greater or lesser form, according to any given proportion.  
LEt A.B. C.D.E.F. be an irregular  diagram Figure given, and let the proportion given be that between M. and N. It is required to reduce the same Figure into a lesser, to be in such proportion to that given as M. to N. First reduce the given Figure into Triangles, by drawing right lines from any one angle, as from F. to all the opposite Angles, as B.C. and D. then by the 23. PROB. divide one of the sides as F.E. in power, as M. to N. so that the power of F. L. may be to the power of F. E. as M. to N. Than by the point L. draw the line L. K. parallel to E. D. to cut F. D. in K. and in like sort proceed with the rest, as K. I I H. and H. G. drawing them parallel to their answerable sides; so shall you include the Figure F. G.H. I.K.L. being like unto the Figure given, and in proportion to it, as the line M. to N. as was required. But suppose the lesser plot were given, and let it be required to reduce the same into a greater, according to the proportion of N. to M. then increase all the lines from F. towards A. B. C. D. and E. and by the 24. PROBLEM enlarge the line F. L. in power as N. to M. which set from F. to E. and by the point E. draw the line E. D. parallel to L. K. to cut F. D. in D. and in like sort proceed with the rest. So shall you include the irregular Figure A.B.C.D.E.F. like to that given, and of the proportion, as was required. THEOREM. 22.   

PROBLEM XCIX. To reduce an irregular Figure given, into a Geometrical Square.   diagram  diagram 
LEt A.B.C.D.E.F.G. be an irregular Figure given, to be reduced into a Geometrical Square. First, draw the lines B. F. and C. E. dividing the given Figure thereby into two Trapezias, and one Triangle, namely, A.B.F.G. B.F.E.C. and C.D.E. then cross those Trapezias with the diagonal lines B. G. and C. F. and let fall perpendiculars thereon from the Angles A. F. E. and B. and likewise from D. to the Base of the Triangle, then by the 17. PROBLEM find out the mean proportional, between half the diagonal B. G. and the two perpendiculars thereon falling, which shall be the line P. Also the mean proportional between half the diagonal C. F. and the two perpendiculars thereon falling, 
Euc. 2.14.6.13.1. 47.  which shall be the line Q. and likewise the mean proportional between half the Base C. E. and the perpendicular thereon falling from D. which shall be the line R. then by the 9 PROBLEM describe a right Angle at pleasure, as H. I K. and take the line P. and place the same, from I to L. and also the line Q. from I to M. and draw the line L. M. which line place from I to O. and also the line R. from I to N. And lastly, draw the line N. O. which shall be the side of a Square, equal to the given Figure as was required. THEOREM 6.25.41.30.  
Here might I fitly insert the Reduction and manner of translation of large and spacious plots, from one Scale to another, divers ways, with many other works of this nature, fit to be known, which for some special reasons, I will refer unto the later end of my next Book. And in the mean space will here conclude the third part of this Second Book.    

The fourth Part.  
HOw divers superficial figures, of several forms, are brought into one figure, and one form: Also to subtract one figure from another, and thereby to know how much the one exceedeth the other in quantity; and likewise, hereby is taught, the inscription and circumscription of one figure within and without another; and the division and separation of figures, into any parts required.  

PROBLEM C. Two Geometrical squares being given, to add them together into one square.  
LEt the two given squares  diagram be A. B. C. D. and D. E. F. G. and let it be required to add them together into one square. 
Euc. 1.47.  First take a side of either of the given squares, as A. D. and D. G. and by the 9 PROB. make thereof the right angle A. D. G. (as they are already placed in this Diagram) then draw the diagonal line A. G. and on that line by the 50. PROB. describe the square A. K. H. G. which shall be equal to the two given squares, as was required. THEOR. 6.   

PROBLEM CI. Two Geometrical squares being given, to add them together in such sort, as the one shall be a Gnomon unto the other.  
LEt the two given squares be A. B. C. D. and E. F. G. H. and let it be required to add them together, 
Euc. 1.47. Ceul. 2.38.  in such sort that the square E. F. G. H. shall be a Gnomon unto the other. Increase the side C. D. of the greater square to M. making D. M. equal to the side of the lesser square, then draw the subtending line A. M. which take and lay down from HUNDRED to N. and thereon by the 50. PROB. describe the square O. P. C. N. which shall perform what was required. DEF. 30. THEOR. 6.   

PROBLEM CII. To add divers squares together into one geometrical square.  
LEt A. B. C. D. and E. be the sides of five squares given, 
Euc. 1.47.6.31.  and  diagram let it be required to add them together into one geometrical square. 
divers figures of what form or kind soever, being by the ●ormer rules reduced into squares, may hereby instantly be added together into one.  First, by the 9 PROB. make a right angle at pleasure▪ as F. G. H. then (beginning with the lest sides first) take the line E. and place it from G. to N. and the line D. from G. to O. and draw the line N. O. whose square shall be equal to both the squares of E. and D. then take N. O. and place it from G. to P. and the side C. from G. to M. and draw the line M. P. which line place from G. to Q. and the side B. from G. to L. and draw the line L. Q. which line place from G. to R. and the line A. from G. to K. and lastly, draw the line K. R. and thereon by the 50. PROB. describe a square which shall be equal to all the five squares, whose sides were given, as was required. THEOR. 6.23.30.31.   

PROBLEM CIII. To add two given triangles together into one, which new composed triangle, shall have his perpendicular, equal to that of one of the given triangles.  
LEt the two given triangles be A. B. C.  diagram and E. F. G. and let it be required to add those 2. triangles into one, 
Euc. 1.37.1.38.  which shall have his perpendicular equal to that of the triangle A. B. C. By the 75. PROB. reduce the triangle E. F. G. to the triangle H. F. K. of equal height to the other given triangle, then increase the base B. C. of the triangle A. B. C. from C. to D. making C. D. equal to F. K. the base of the reduced triangle H. F. K. and lastly, draw the line A. D. which shall include the triangle A. B. D. equal to both the given triangles, and having the same perpendicular A. N. as the given triangle A. B. C. as was required. THEOR. 26. PROB. 28.   

PROBLEM CIIII divers Circles being given, to add them together into one Circle.  
LEt A. B. and C. D. be two Circles given, 
Euc. 1.47.12.2.  and  diagram let it be required to add them both into one Circle, or to make one Circle, which shall be equal to them both. Take the Diameter A. B. and by the 6. PROB. raise it perpendicularly on the end of the other Diameter C. D. as E. C. then draw the subtending line E. D. on which line as a Diameter describe the Circle E. D. which shall be equal to the two given Circles as was required. And in like sort, by help of the 102. PROB. may be added as many Circles as shall be required; For Circles are added by their Diameters, as Squares by their sides. THEOREM 6.66.   

PROBLEM CV. Two long Squares being given, to add them together into one long Square, whose breadth shall be equal to that of one of the long Squares given.  
LEt the two given long Squares be  diagram A. B. C. D. and E. F. G. H. and let it be required to add those two Squares together into one Square, 
Euc. 1.43.  whose breadth shall be equal to the given long Square A. B. C. D. First, increase the side G. F. of the greater long Square given towards K. making F. K. equal to the breadth of the lesser given Square, and so working on by the 87. PROB. reduce the given long square E. F. G. H. into the long square M. N. H. L. then on the Line M. L. by the 54. PROB. describe the long square O. M. L. P. equal to the given long square A. B. C. D. So shall the long Square O. N H. P. be equal to the two given long Squares, and the breadth thereof O. P. or N. H. equal to the breadth of the lesser long Square given, as was required. THEOREM. 5.   

PROBLEM CVI Two Geometrical Squares being given, to subtract the one out of the other, and to produce the remainder in a third Square.  
LET the two given Squares be A. B. C. D. and E. F. G. C. and let  diagram it be required to subtract the lesser out of the greater, 
Euc. 1.47.6.31.  and to produce the remainder in a third square. Continued out at length the side C. G. of the lesser given square towards M. and at the distance of the side of the greater given Square with one foot in the Angle F. of the lesser square strike an Arch line through the line of continuance in M. And lastly, by the 50. PROB. on the line G. M. describe the Square H. K. M. G. which shall be the remainder of the greater given Square, the lesser being subtracted from the same, as was required. THEOREM. 6.23.30.31.   

PROBLEM CVII. Two Triangles being given to subtract the one out of the other, and to leave the remainder in a Triangle of equal height to one of the given Triangles.  
LEt the two Triangles given be A. B. D. and E. F. G. 
Euc. 1.37.1.38.  And let it be  diagram required to subtract the Triangle E. F. G. out of the Triangle A. B. D. and to leave the remainder in a Triangle of equal height to the Triangle A. B. D. By the 75. PROB. reduce the Triangle E. F. G. to the triangle H. F. K. of equal height to the other given triangle, then take the Base F. K. of the same reduced triangle, and place it from D. to C. And lastly, draw the line A. C. which shall include and subtract the Triangle A. C. D. (equal to the lesser given triangle E. F. G) from the greater given triangle A. B. D. and leave the remainder in the triangle A. B. C. of the same height of the given triangle A. B. D. as was required, THEOR. 26. PROB. 28.   

PROBLEM CVIII. Two circles being given, to subtract the one out of the other, and to produce the remainder in a third circle.  
LEt the two circles given be A. B. and  diagram B. C. and let it be required to subtract the circle B. C. out of the circle A. B. and to produce the remainder in a third circle. 
Euc. 1.47.12.2.  Take the diameter B. C. and by the 6. PROB. raise it perpendicularly on the point B. as D. B. then at the distance of the diameter A. B. with one foot in D. strike an arch through the diameter B. C. in E. then at half the distance of B. E. describe the circle F. G. on the point B. So have you subtracted the circle B. C. out of the circle A. B. and produced the remainder in a third circle F. G. as was required. THEOR. 6, 66.   

PROBLEM CIX. A geometrical square and a triangle being given, to subtract the triangle from the square, and to produce the remainder in a square.  
LET the Geometrical square given be A. B. C. D. and let the triangle given be E. C. F. and let it be required to subtract the triangle from the square, 
Eu. 1.47.2.14.6.18.  and to produce the remainder in a square. By the 17. PROB. found out the mean proportional, between the perpendicular of the given triangle E. N. and half the base C. F. which shall be the line O. P. which place from C. to H. on which point H. at the distance of the side of the given square, strike an arch through the base C. F. as at M. and lastly, upon the line C. M. describe by the 50. PROB. the square G. K. M. C. So have you subtracted the given triangle E.C.F. from the given square A.B.C.D. and produced the remainder in the square G.K.M.C. as was required. THEOREM 6, 41, 70.   

PROBLEM CX. A triangle and a long square being given, to subtract the long square from the triangle, and to produce the remainder in a triangle of equal height, to the triangle given.  
LEt the given triangle be A.B.C. and  diagram the long square given D. E. F. C. and let it berequired to deduct the long square from the triangle, 
Euc. 1.38.1.41.6.1.  and to produce the remainder in a triangle, of equal height to the triangle given. First, increase the side F. E. of the long square towards K. and placing E.F. from E. to H. draw the line H.C. including the triangle H. C. F. equal to the long square given; which triangle by the 75. PROB. reduce into the triangle K.G.F. of equal height to the given triangle A.B.C. then take the base G.F. of the last found triangle, and place it from C. to M. on the base of the given triangle, and lastly, draw the line A.M. to include and subtract the triangle A.M.C. (equal to the given long square) from the given triangle A.B.C. & producing the remainder in the triangle A.B.M. of equal height to the triangle given, as was required. THEOR. 26, 27, 28, 35.   

PROBLEM CXI. Within a circle given to inscribe a triangle, with angles equal, and lines proportional, to a triangle given.  
LEt the circle given be A. B. C. and the triangle  diagram  diagram given D.E.F. and let it be required within the given circle to inscribe a triangle, 
Euc. 3.32.4.2. Ceul. 2.53.  with equal angles and lines proportional to the given triangle By the 31. PRO. draw the given circle, the right line G. H. touching the same in the point C. upon which point unto the line C.H. describe the angle A. C. H. equal to the angle D. E. F. and likewise on the same point, to the line G. C. describe the angle B. C. G. equal to the angle E. D. F. and lastly, draw the line B. A. which shall include the triangle A. B. C. inscribed within the given circle, with equal angles and lines, proportional to the triangle given, as was required. THEOR. 56, 13.   

PROBLEM CXII. To describe a Circle about a Triangle given.  
LEt the Triangle given be A.B.C. about which it is required to describe a Circle, 
Euc. 4.5.   diagram upon the middle point of any two sides of the Triangle by the 5. PROB. erect perpendiculars, which being produced will intersect each other, as E. D. and F. D. in the point D. which point of intersection shall be the Centre; whereupon at the distance from thence to any one of the Angles describe the Circle A.B.C. which shall circumscribe the Triangle as was required. DEF. 32.  
Or otherwise the Centre may be found as is taught in the 34. PROB.   

PROBLEM CXIII. To inscribe a Circle within a Triangle given.  
LEt the Triangle given be A. B.C. within which Triangle it is required  diagram to inscribe a Circle. By the 10. PROB. divide any two Angles of the given Triangle into two equal parts as A. B.C. and A. C.B. by drawing the lines B. D. and C. D. intersecting in D. which point of intersection shall be the Centre, 
Euc. 4.4: Ceul. 2.55.  whereon at the nearest distance from thence to any side of the Triangle describe the circle D. which shall be inscribed within the Triangle, as was required. DEF. 32. PROB. 10.   

PROBLEM CXIIII. To describe a Triangle about a Circle given, which shall be like unto a Triangle given.  
LEt the Circle given be H. K. M. and let the Triangle given be E. F. G. It is required to describe a triangle about the given Circle, 
Euc. 4.3. Ceul. 2.54.  like unto the given triangle. First, continued the Base F. G. of the given Triangle both ways towards  diagram  diagram N. and O. making the two outward Angles E. F. N. and E. G. O. then from the centre D. of the given circle draw to any part of the limb a semidiameter as D.M. to which line upon the centre D. describe the angle H. D. M. equal to the angle E.F.N. and also the Angle K.D.M. equal to the angle E.G.O. then by the points H. K. and M. draw right lines, (making right angles with the three semidiameters) which will intersect each other in the points A.B. and C. including the triangle A. B. C. like unto the given triangle, and circumscribed about the given circle, as was required. THEOR. 1.13.52.   

PROBLEM CXV. To describe a Square about a Circle given.  
LEt the circle given be A. B. C. D. and let it be required to describe a  diagram square about the same circle. 
Euc. 4.7. Ceul. 2.57.  Draw the two Diameters A. C. and B. D. cutting each other at right angles in the centre 1. then by the 3. PROB. by the points A. and C. draw parallel lines to the Diameter B. D. And likewise by the points B. and D. draw parallel lines to the Diameter A. C. which shall intersect each other in the points E.F.G.H. and include the square as was required. THEOR. 32.52.   

PROBLEM CXVI. Within a Square given to inscribe a Circle:  
LEt the square given be E.F.G.H. in the former Diagram, within which it is required to inscribe a circle. Draw the diagonal lines E. G. and F. H. interfecting each other in 1. the centre, on which point at the distance of the shortest extension to any side, describe the circle A B C D. within the square as was required. THEOR. 32.52.   

PROBLEM CXVII. About a Square given to circumscribe a Circle.  
LEt the Square given be A. B. C. D. in the former Diagram, about which it is required to circumscribe a Circle. Draw the diagonal lines A.C. and B. D. intersecting each other at right Angles in the point I the centre, 
Euc. 4.9.  on which point, at the distance from thence to any of the points A.B.C. or D. describe the Circle A.B.C.D. which shall circumscribe the Square as was required. THEOREM 1.13.52.   

PROBLEM CXVIII. To inscribe a Square within a Circle given.  
LEt the Circle given be A.B.C.D. in the former Diagram, and let it be required to inscribe a square within the same Circle. Draw the Diameters A. C. and B. D. crossing each other at right Angles in the centre I then between the points A.B.C. and D. draw the right lines A B. B C. CD. and D A. including and inscribing the square A.B.C.D. within the given Circle as was required. THEOREM. 
Euc. 4.6.  1.13.52. PROBLEM 25.   

PROBLEM CXIX. To inscribe a Pentagon, within a Circle given.  
LEt A. B. C. D. E. be a Circle given, 
Euc. 4.11. Ceul. 2.58.  within which it is required to inscribe  diagram a Pentagon. By the 25. PROBLEM Find out the fift cord line of a Circle, at which distance passing through the limb of the circle note five marks as at the points A. B. C. D. and E. And lastly, from point to point draw five right lines, which shall include and inscribe the Pentagon A. B. C. D. E. as was required. THEOR. 47.69. PROB. 48.   

PROBLEM CXX. About a Circle given to circumscribe a Pentagon.  
LEt the Circle given be A.B.C.D. in the former Diagram, about which, it is required to circumscribe a Pentagon. Let first a Pentagon be inscribed as before, and from the Centre N. draw right lines to every Angle of the inscribed Pentagon, 
Euc. 4.12.  as to A.B.C.D. & E. on which five points, draw lines Orthigonally to those lines issuing from the centre, which will intersect each other in the points F. G. H. I K. and circumscribe the Pentagon about the given circle as was required. THEOR. 47.69.   

PROBLEM CXXI. To inscribe a Sexagon within a circle given.  
LEt the circle given be A. B. C. D. E. F. 
Eu. 4.15. Ceul. 2.63.  within which it is required to inscribe  diagram a Sexagon. By the 25. PROB. find out the sixth cord of a Circle, which is always the semidiameter of the same circle; wherefore at the distance of the semidiamenter; Divide the limb of the circle into six equal parts, as in the points A. B C.D.E. and F. and then from point to point, draw right lines, which shall include and inscribe the Sexagon A.B.C. D.E.F. within the given Circle, as was required. PROB. 25.   

PROBLEM CXXII. To circumscribe a Sexagon about a Circle given.  
LEt the Circle given be A.B.C.D.E.F. in the former Diagram, about which it is required to circumscribe a Sexagon. First, divide the limb of the Circle into six parts, as was taught in the last PROB. in the the points A. B. C. D. E. and F. and draw Diameters from each opposite point to other, making in all three Diameters, then at every end of those Diameters draw lines Orthigonally unto them, which will intersect each other in the points G.H.I.K.L. and M. and include the Sexagon circumscribed about the given circle as was required. PROB. 25.   

PROBLEM. CXXIII. To divide a right lined Triangle given, into any number of equal parts required, from a point limited in any side of the same Triangle.  
LEt A. B. C. be a right lined Triangle given; let the limited point be D.  diagram in the Base B. C. and let it be required from the same limited point D. to divide the given triangle into five equal parts. First, divide the Base B. C. into five equal parts, as in the points E. F. G. and H. (or into as many as shall be required) then from the limited point to the opposite angle, draw a right line as A D. unto which line, by the points E.F.G. and H. draw Parallel lines, as E I F K. G L. and H M. And lastly from the points I K. L. and M. to the limited point D. draw the lines I D. EDWARD D. L D. and M D. which shall divide the given triangle into five equal parts from the limited point D. as was required. THEOREM. 26.35.   

PROBLEM CXXIIII. To divide a given Triangle by a line issuing from an angle assigned, in  diagram any proportion required.  
LEt the Triangle given be A.B.C. the angle assigned A. and the proportion that between the two right lines F. and G. 
Euc. 6.1. Ram. 10.13 Ceul. 3.8.  It is required to divide the same triangle into two parts by a line issuing out of A. the one part having proportion to the other, as F. to G. By the 13. PROB. divide the Base B C. as F. to G. the point of which division will fall in D. From which point draw the line D. A. So is the given Triangle divided into two parts, having proportion the one to the other, as F. to G. For as the line F is to the line G. so is the triangle A. C. D. to the Triangle A. B.D. as was required. THEOR. 35. PBOB. 13.  

A. E. 64.  
B. C. 80.    

PROBLEM. CXXV. A Triangle being given, and the Base thereof known, to divide the same into two parts by a line from an angle assigned, according to any proportion given in two numbers.  
LEt the triangle given be A. B. C. in the former Diagram, whose Base is B.C. 80. and the angle assigned A. and let it be required to divide the same into two parts, the one in proportion to the other Sesquidtera, that is, as 3. to 2. First, add together always the Termini rationis, which is 3/2· makes 5. then multiply the Base 80. by 3. the greater term, the factus will be 240. which divided by the former 5. quoteth 48. for the greater segment of the Base D C. 
Euc. 6.1. Ra. 10.13.  and that deducted from 80. the whole Base resteth 32. the lesser segment B D. from which point D. draw the line D A. which shall divide the given Triangle into two parts, as 3. to 2. that is, the greater A C D. shall be 3/5· parts, and the lesser A B D. 2/5· of the whole given Triangle, as was required. THEOR. 35.   

PROBLEM CXXVI. To divide a Triangle of known quantity given, into any two parts, from an Angle assigned, according to any number of Acres, roods and Perches required.  
LEt the Triangle be A B C. in the former Diagram, whose quantity is 16. Acres; let the Angle assigned be A. and let it be required to divide the same into two parts twixt H. and I viz. to H. 9 Acres. 2. roods, 16. Perches: and to I 6. Acres, 2. roods, 24. Perches. First, measure the Base B C. 80. Perches; then reduce the whole quantity of the Triangle into Perches, 
Euc. 6.1. Ra. 10.13.  by multiplying 160. (the number of Perches in an Acre) by 16. (the numbet of Acres in the Triangle given) the Product will be 2560. Reduce also one of the parts into Perches, as the greater part 9 Acres, 2. roods, 16. Perches; the reduction will be 1536. Perches. And then by the rule of proportion reason thus. If 2560. (the whole quantity) require 80. (the whole Base; whatpart of the Base will 1536. Perches (being the part of H.) require. Multiply 1536. by 80. and divide the Product by 2560. the quotus will be 48. the part of the Base sought, which placed from C. to D. and the line D A. drawn, will divide the given Triangle into the parts required. THEOR. 35.  
Or to lay out one of the parts given (whereby the other is known) work thus.  
LEt it be required to lay out the lesser part, which containeth 1024. perches, divide that number 1024. by half, the perpendicular A E. 32. the quotus will be 32. for the part of the Base to be cut off for that part; which placed from B. to D. and the line A. D. drawn, performeth the work. Or double the quantity given 1024. and divide that double by the whole Base 64. the work will be the same.   

PROBLEM CXXVII. To divide a Triangle given into two parts by a line drawn, from a point limited in one of the sides, in any proportion required.  
LEt A B C. be the Triangle given, the point limited E. in the Base B C. and let the proportion be that between I and K. It is required from the point E. to draw a line, which shall divide the same Triangle into two parts, having 
A G. 40. 
Euc. 6.1. Lam. 10.13 Coul. 3.9.   
B C. 42.  
B D. 12.  
D C. 30.  
B E. 30.  
F H. 16.  proportion the one to the other, as I to K. First, from the given point E. to the opposite angle A. draw a right line, as A. E. then by the 13. PROB. divide the Base B. C. as I to K. the point of which division will fall in D. from whence draw the line D F. parallel to A. E. cutting the side A B. in F. And lastly, from thence draw a right line to the limited point E. as FLETCHER E. which shall divide the given Triangle into two parts, and include and separate the Triangle F E B. from the Trapezium A F. E C. having such proportion one to the other, as the two given lines I and K. as was required. THEOR. 35.   

PROBLEM CXXVIII. To divide a Triangle given into two parts, according to any propertion given in two numbers from a point limited in any side thereof, Arithmetically.  
LEt the Triangle given be A B C. in the former Diagram. Let the point limited be E. in the Base B C. and let it be required to divide the same into two parts, in proportion one to the other, dupla sesquialtera, which is, as 5. to 2. First, let the Base B. C. be divided according to the proportion given thus; Add together the two given terms of the proportion 5. and 2. makes 7. then multiply the Base of the Triangle B. C. 42. by 2. the lesser term, the Product is 84. which divided by 7. the sum of the terms quoteth 12. for the lesser Segment of the Base B. D. which deducted from 42. the whole Base resteth 30. the greater segment D. C. then (considering the lesser part is to be laid towards B.) measure the distance from the given point E. to B. which admit 30. and by the rule of Proportion reason thus: If the distance E. B. 30. gives B. D. 12. 
Eu. 6.1. Ram. 10.13.  the lesser segment what gives A G. 40. the perpendicular of the given Triangle, and multiplying 40. by 12. and dividing the Product by 30. the answer will be 16. at which distance draw a parallel line to B E. cutting the side A B. in F. from which point draw the line F E. which shall divide the given Triangle in such sort that the Triangle F B E. shall be 2/7· parts, and the Trapezium A F. E C. 5/●· parts thereof, that is the Trapezium, containing the Triangle F B E. twice and a half, according to the proportion required. THEOR. 35.   

PROBLEM CXXIX. To divide a Triangle of any known quantity given, into two parts, from a point limited in any side thereof, according to any number of Acres, roods and Perches.  
LEt the Triangle given be A B C. in the last Diagram, whose quantity let be 5. Acres, 1. Rood, 0. Perch. Let the limited point be E. in the Base B C. and let it be required from the same point to divide the Triangle into two parts between M. and N. viz. to M. 3. Acres, 3. roods, 0. Perches thereof, and the residue to N. 1. Acre, 2. Roods, 0. Perch. First, reduce the quantity of N. being the lesser, 1. Acre, 2. Roods, 0. Perch, into Perches, 
Euc. 6.1. Ra. 10.13.  which makes 24. Perches, than (considering on which side of the limited point this part is to be laid, as towards B.) measure that part of the Base from E. to B. 30. Perches, whereof take half, which is 15. and thereby divide 240. the part of N. the quotus will be 16. the length of the perpendicular F. H. at which parallel distance from the Base B C. cut the side A B. in F. from whence draw the line F E. which shall include the Triangle, F. B. E. containing 1. Acre, 2. Roods, 0. Perch, the part of N. and so shall the Trapezium A F E C. contain the residene, namely, 3. Acres, 3. roods, 0. Perch, the part of M. as was required. THEOR. 35.   

PROBLEM CXXX. To divide a given triangle by a parallel line, to one of the sides, according to any proportion given.  
LEt A. B. C. 
A D. 48.  
B C. 54.  
B E. 24.  
B F. 36.  
E C. 30.  
H G. 32.  be a triangle given, and the proportion that between I and K. and let it be required to divide the same triangle into 2 parts, 
Euc. 6.10.6.13. Cor. 6.19. Ceul. 3.10.  by a line drawn parallel to the side A. C. the one to be in proportion to the other, as the line I to the line K. By the 13. PROB. divide the line B. C. in E. as I to K. then by the 17. PROB. find the mean proportion between B. E. and B. C. which let be B. F. from which point by the 3. PROB. draw the line F. H. parallel to the side A. C. cutting the side A. B. in H. which line shall divide the triangle given into two parts, the trapezium A. H. F. C. having such proportion to the triangle H. B. F. as the line I to the line K. which was required. THEOREM 43, 41. PROBLEM 13, 17.   

PROBLEM CXXXI. To divide a given triangle by a parallel line, to one of the sides, according to any proportion given in two numbers Arithmetically.  
LEt the triangle given be A. B. C. in the former Diagram, and let it be required to divide the same by a parallel line, 
Euc. 6.10.6.13. Cor. 6.19.  to one of the sides into two parts, to be in proportion the one to the other, Sesquiquarta, that is, as 5. to 4. First, let the base B C. 54. be divided, according to the proportion given, as is taught in the 127. PROB. so shall the lesser segment be B E. 24. and the greater segment E C. 30. then find out a mean proportional between B E. 24. and the whole base B C. 54. by multiplying 54. by 24. whose product will be 1296. the squa e root whereof is 36. the mean proportional sought, which in the former Diagram is B F. then by the rule of proportion, reason thus: If B F. 36. gives B E. 24. what A D. 48. the answer is H G. 32. at which distance draw a parallel line to the base, to cut the side A B. in H. from whence draw the line H F. parallel to A C. which shall divide the given triangle into two parts, in such sort, as the trapezium A. H. F. C. shall be 5/9· parts, and the triangle H. B. F. 4/9· thereof, that is the trapezium, containing the triangle H. B. F. once and a quarter, as 5. doth 4. which was required. THEOR. 43, 41. PROB. 13, 17.   

PROBLEM CXXXII. To divide a triangle of any known quantity, given into two parts, by a parallel line, to one of the sides, according to any number of Acres, roods, and Perches required.  
LEt the triangle given be A. B C. in the last Diagram, 
Euc. 6.10.6.13. Cor. 6.19.  whose quantity is 8. Acres, 0. Roods, 16. Perches, and let it be required to divide the same (by a parallel line to one of the sides, as the side A. C.) into two unequal parts, between M. and N. viz) to M. 4. Acres, 2. Roods, 0. Perches, thereof, and the residue to N. being 3. Acres, 2. Roods, 16. Perches. First, reduce both quantities into perches, which will be 720. and 576. then reduce both those numbers by abbreviation into their least proportional terms, which is 5/4· and according to that proportion divide the base B C. 54. of the given triangle, as is taught in the two last Problems, which will be in E. then seek the mean proportional between B E. and B C. as is taught in the last Problem, which is B F. 36. of which 36. take the half, and thereby divide 576. the lesser quantity of perches, the quotus will be H G. 32. at which parallel distance from the base, cut the line A B. in H. from whence draw the line H F. parallel, to the side A C. which shall divide the triangle given into two parts; the trapezium A. H. F. C. containing the part of M. and the triangle H. B. F. the part of N. as was required. THEOR. 41, 43.  
HEre might I now much enlarge and amplify this latter part, for the dividing of superficial figures of all forms and kinds: But seeing that all irregular figures and plots are most conveniently to be reduced into triangles, before the contents thereof can be had, or any division thereof made; and generally in matter of Survey (whereunto my purposes chief tend) all figures are found irregular; I will content myself (and entreat my Surveyor to be likewise satisfied) with these few former instructions, which being well understood, with due observation of the precedent rules, will serve his turn to whatsoever purpose can be required. But who so desireth further satisfaction, and more variety in this kind, I refer him to M. john Speidels Book, entitled his Geometrical Extraction, lately by him divulged; (wherein he hath taken more pains to excellent purpose, than this age, I fear, can afford him recompense) and to himself M. Henry Brigs, M. Thomas Bretnor, M. Io. johnson, and others, who are here amongst us Professors, and excellent Teachers of these Arts. And thus, I conclude, this second Book.    
The end of the second Book.   

THE EXACT OPERATION OF INSTRUMENTAL DIMENSIONS BY DIVERSE MEANS.  
The third Book.  
THE ARGUMENT OF THIS BOOK.  
THis Book tendeth chief to matter of Survey, wherein is first described and declared the several Jnstruments, fit for that purpose (with their use in practice) as the Theodelite, the Plain table, and Circumferentor, whereunto I have added an absolute Instrument, which I call the Peractor, together with the making and use of the Decimal Chain, used only by myself; then is taught the use of a necessary Field-booke; the taking, protracting, and laying down of angles divers ways; the reducing of customary measures into Statute measure, and the contrary, showing their difference; the use of the table of Synes, and the divided sights on the Circumferentor, withsupply of those defects in the ordinary use of the Plain table, by ignorant persons; the means to take Altitudes, Longitudes, Latitudes, and inaccessible Distances, aswell by synicall computation, as divers other means; the Dimension and plotting Instrumentally of all manner superficial figures, and irregular plots, by divers and sundry ways, with their several protractions accordingly: how with the Plain table exactly to take a plot of the largest Forest on one sheet of paper, without altering thereof: the means of Survaying and plotting of a Lordship or Manor, with the orderly handling of the same: the ways and means Instrumentally, to reduce hypothenusal, to horizontal lines, and the contrary: with the best and exactest courses to be held in the dimension, and plotting of Mountainous, and un-even grounds: the manner of enclosing, dividing, and laying out of Commons, wastes, or common Fields, into any parts required: the ordering of a plot after the protraction thereof: the reducing of plots from a greater to a lesser form, and the contrary: and lastly, the speedy reduction of perches into Acres, and those again into perches: Most exactly and artificially wrought, by the best and most immediate means for those purposes.  

CHAP. I Of the several Instruments in use, meet for Survey; which of them are most fit for use, and somewhat concerning their abuse.  
BEfore we enter the fields to survey, I hold it necessary we provide us of fitting furniture for the purpose, jest by our neglect therein, those by whom we have employment receive no less loss and prejudice, than ourselves shame and reproach: Wherhfore let us first consider what Instruments are most usual, and then of those, what most fit for our present purpose. The Instruments now most in use are the Theodelite, Plaine-table and Circumferentor, whereunto I will add one more, which I call the Peractor, used only by myself, and certain friends by my directions; Of all which I will hereafter make brief and particular descriptions, aswell concerning their several parts and composition, as of their use in practice. Nor will I exclude or wholly neglect the Familiar Staff of M. john Blagrave, and the Geodeticall staff and topographical glass of M. Arthur Hopton (though now together dead) or any other Instrument which are, or hereafter may be by the artor industry of any man, artificially invented or composed for his own, or others use. For, as I tie not myself to the use of any one Instrument, at all times, nor on all occasions, but for a large and spacious business use the Circumferentor or Peractor (as for many reasons most necessary and convenient) and for a lesser (where many small Enclosures and Town-ships are) the Plaine-Table; although either of the other will well serve for performance of either kind (and therefore if possibly you may, in one and the same business, use ever one and the same Instrument, for avoiding many inconveniences:) So will I not limit my Surveyor to the use of any one Instrument, but refer him to all or any at his pleasure, well knowing they are all composed and framed on one and the same Theorical ground: and although in performance and dispatch, one may be more speedy than another (wherein I find much difference, with the same exactness) yet all or any of them artificially handled, are to excellent use, aswell in survey of lands, as the performance of many other excellent conclusions Geometrical. Wherein I cannot by the way but condemn the folly of divers, who (deeming themselves more wise and skilful, than any other shall have just cause to conceive) having some small superficial use or insight into some one of these Instruments, are only wedded to that, and ignorant of the others use, will condemn them as unmeet and insufficient, the defect consisting in their own understanding. But I must needs acknowledge (which could I as easily reform, I should deserve well at the hands of many, who pay dearly for it) that by the ignorance of divers, using (or abusing) these instruments, infinite gross and palpable errors and abuses are daily committed, to the great loss and prejudice dices of many, who receive secret and insensible blows (found in their estates, though unfelt in themselves) who for my part shall pass unpitied (save only for their weakness) the fault being chief their own, when out of a penurious sparing, to yield better satisfaction to those which better can; they can be well contented blindly to swallow frogs for flies, and to pay triple damage (nay perhaps ten fold) another way, so they neither feel nor found it in their open view. And thus much concerning the several Instruments in general. Next of their Definitions, Parts, and Compositions particularly.   

CHAP. II Of the Theodelite and his several parts, with the Description and Composition thereof.  
THis Instrument consisteth of four several parts; As first, the Planisphere or Circle, whose limb is divided into 360. equal parts or divisions, called degrees, without which it is fitting equidistantly to draw and describe six concentricke lines or circles with cross Diagonals, by whose intersections are had the parts of a degree: upon which Planisphere, and within the divisions before specified, there is described the Geometrical square, being the second part thereof both which together, or either of them severally serve to excellent purpose, for the dimension of lengths, bredthes and distances. The third part is a Semicircle or Quadrant perpendicularly raised, and artificially placed on the former Planisphere (or more properly on the Index thereof) to be moved about circularly at pleasure on all occasions; which semicircle or quadrant hath the limb thereof equally divided into parts or degrees, with like concentricke Circles and Diagonals to those of the Planisphere: And the fourth and last part is a Scale, described within the same Semicircle or Quadrant, whose sides are divided into divers equal parts, the more the better, and to best purpose; which two later parts serve chief for the mensuration of altitudes and profundities; All which together composed, make an excellent Instrument meet for many purposes, especially for the description of Regions and Countries, or other spacious works. This Instrument would ask a far more exact and particular description, which for three principal reasons I will here omit; First, for that I hold it (although a general and fit Instrument for all mensurations, yet in particular) for this our business of Survey, not altogether so fitting and commodious as the rest before named, by reason of the multiplicity of Divisions therein contained, which will be so much the more troublesome in use and protraction: Secondly, for that M. Thomas Digges in his Pantometria, hath made a large and particular description thereof: And thirdly and lastly, for that M. Allen, who makes of these and all other Geometrical Instruments in metal, is so well acquainted with the parts and composition thereof, that whosoever purposeth to use the same, needeth only to bespeak it of him, without further instruction or direction to him. The use and employment whereof shall be hereafter touched. And thus much for this Instrument and the description thereof.   

CHAP. III The Plain table and his several parts, with the description and composition thereof.  
THis Instrument for the plainness and perspicuity thereof, and of his easy use in practice, receiveth aptly the name, and appellation of the Plain Table. A most excellent and absolute Instrument, for this our purpose in Survey; but with all, by reason of this his plainness (offering itself at first view, in some measure, to the weak understanding of meanest capacities, inciting them thereby to the practice thereof) is more subject to abuse, than all, or any of the rest: For notwithstanding that these, by the common and ordinary use and practice of this Instrument, may easily attain to a reasonable truth in Dimension and plotting of regular and even Plains; yet, if they come to irregular and un-even forms, as Hills and Dales, they are so far unfitting for the true mensuration thereof, that many using this Instrument, neglect the means, and those proper parts of the same Instrument (being the sights hereafter specified, whereof they know no use) which is their only help and aid in this kind. But, jest this relation should be held digression, let's return to the purpose. This Table itself is divided into five parts, or small boards, whereof three are in the breadth, and the other two employed as ledges in the length, to keep the rest together, whereunto a jointed frame is artificially applied, for the fastening and keeping plain thereon, an ordinary sheet of paper for use in the fields; of which length and breadth, or rather less, as 14½· inches in length, and 11. in breadth, the whole Table together should consist. But for my purpose, I would have made of these three boards in breadth, with help of ledges, to be thereon glued, to the backside thereof, and strong joints or gemmowes, fastening them together, an artificial box; which at any instant being opened, and the ledges fastened on, is fit for use in the Fields, and afterwards those ledges taken away, may be as instantly turned back wards, and enclosed as before, fit for the keeping of lose papers and small tools, till further occasion. Which if any desire to use, M. john Thomson in Hosier-Lane, will artificially fit him. To this Instrument doth also belong a Ruler with two fights thereon, which by divers men are used of divers kinds, but by the best Plain Table men (as occasion serves) those of M. JOHN GOODWYNS invention, that excellent and honest Artist, whose living Name (though himself be dead) I cannot remember without good respect. This Ruler is to contain in length about 16. inches, or as long or longer than the table, for drawing parallel lines on the paper fastened thereon, by the equal and opposite divisions on the frame; it is likewise to contain in breadth about two inches or upwards, and in thickness half an inch; the sights thereof are double in length the one to the other, the longer containing about 12. inches, and the shorter half as much; on the head or top of which shorter sight, must be placed a wire or brass pin; and to this sight there must be fastened a thread and plummet, to place the Instrument Horizontally; through the longer sight must be made a slit, extending almost the whole length of the sight. These two sights thus prepared, are to be placed perpendicularly on the Ruler, by square mortesses, made to that purpose through the same, in such sort as the brass pin on the shorter sight, and the slit through the longer sight, be precisely over the fiducial edge of the Ruler; either sight being equidistantly placed from each end of the Ruler, and the space between the sights to be exactly the just length of the longer sight, or to speak more properly of the divided part of the same sight; (which is always to be understood when I speak of the length thereof.) Upon this longer sight there is to be placed a vane of brass, to be removed up or down at pleasure; through which a small sight hole is to be made, answerable to the slit in the same sight, and the edge of the vane.  
By these sights thus placed on the Ruler, there is projected a Geometrical square, whose side is the divided part of the long sight, or the distance between the two sights. In the middle of which long sight, overthwart the same, there is drawn a line called the line of level, dividing the side of the projected square into two equal parts, also the same side is on this sight divided into 100 equal parts, which are numbered upwards and downwards, from the line of level, by five and ten to 50. on either side, which divisions are called the Scale.  
There is also on the same sight expressed another sort of division, representing the hypothenusal lines of the same square, as they increase by units, and are likewise numbered upwards and downwards from the line of level, from one to 12. as 1, 2, 3, etc. signifying 101, 102, 103, etc. which declare how much an hypothenusal or slope line, drawn over the same square (that is, from the pings head to any such division) exceedeth the direct horizontal line, being the side of the same square.  
There is moreover a third sort of divisions on this sight, representing the degrees of a quadrant (or as many as can well be expressed on the same sight being 25.) which are (as those before) numbered, from the line of level upwards and downwards, by five and ten to 25. which divisions are called the quadrant.  
Likewise there may be placed on the surface, or upper part of this Ruler, the table of Synes, mentioned in the description of the Circumferentor, next hereafter following; very necessary for use, as shall hereafter partly appear.  
Yet would I further advise concerning this Instrument; that in plain and level grounds, where is no use of reducing Hypothenusall lines; instead of those long sights formerly expressed, to have used such double sights as are hereafter mentioned in the description of the Peractor, which are far more ready, and less cumbersome for use, especially in rough and boisterous weather. And likewise, when occasion shall be offered, by irregular and un-even grounds, to reduce those hypothenusal lines, to use in stead of those long sights, the quadrant hereafter likewise specified in the same description, which when need requires may be easily fixed to the Ruler; and as soon laid by, having no further use, for of all other means in taking of heights, and especially for reducing of those hypothenusal lines, I hold that quadrant artificially handled to be most ready and exact, all conclusions being speedily wrought therewith without Arithmetic, or other collateral computation whatsoever.  
To this Instrument belongeth divers other ordinary parts, whereof I shall not need particularly to speak, seeing most men (if not too many) already know them, being of ordinary use; as the Socket, the Box and Needle with the staff or foot thereof, etc.   

CHAP. FOUR The Circumferentor with his parts, description, and composition.  
THis Instrument for expedition and portabilitie, exceedeth far the rest, and nothing inferior to any for exactness, if care and art be used; but not so vulgarly used (though much more general for use) as the plain Table is; the full and perfect use thereof, not lying so open and apparent to all men's understanding, as the other is.  
It is made and framed of well seasoned box, containing in length about eight inches, in breadth half as much, and in thickness about 4/3· of an inch; the left side whereof is besyled, and divided into divers equal parts, most fitly of 12. in an inch, to be used as the Scale of a Protractor, the Instrument itself serving well to protract the plat on paper, by help of the needle, and the degrees of angles and lengths of lines, taken in the field, and entered into your field Book, as shall be hereafter shown.  
About the middle in the surface thereof, a round hole is to be turned of the depth of half an inch, whereof the Diameter to be about 3½· inches, to place a card and needle therein, to be covered over with clear glass. The best card for this purpose is that, divided in the limb into 120. equal parts or degrees, with a Dial according to the Azimuths of the Sun, wherein the hours are numbered, and the months named, serving very aptly to show the time of the day.  
Besides, on the surface hereof is placed the table of Synes, calculated from the quadrant of a circle so divided, as the card before specified; the arch of this quadrant being 30. degrees, and the semidiameter or total sign thereof, is divided into 1000 equal parts, and numbers placed accordingly, answering to every degree and half degree, serving to express the length of every right sign for the Dimension and Supputation of Triangles, as shall be partly hereafter shown, and in the mean space the Table is here expressed.  

Tabula Sinuum.  ½ 26  334  609  824  959 1 52 7 358 13 629 19 839 25 969  78  383  649  853  972 2 104 8 407 14 669 20 866 26 978  130  430  688  879  983 3 156 9 454 15 707 21 891 27 987  182  477  725  902  991 4 208 10 500 16 743 22 913 28 994  233  522  760  923  997 5 259 11 544 17 777 23 933 29 998  284  566  793  943  999 6 309 12 588 18 809 24 951 30 1000  
There is also hereunto belonging two sights, double in length, the one unto the other, the longer containing about seven inches or upwards, being placed, and divided in all respects, as those formerly mentioned in the description of the plain Table; the shorter sight of these having one property and use, which that (as needless) hath not, being this, in the edge thereof, towards the upper part is placed a small wire, representing the centre of a supposed circle, the semidiameter whereof is that part of the same sight, between the wire, and a perpendicular point on the edge of the Instrument underneath the same; which part is imagined to be divided into 60. equal parts, and according to those divisions is the right edge of the Instrument, divided and numbered from the perpendicular point by five and ten, as 5.10.15. etc. And also from the same point on the surface and upper edge of the Instrument is perfected, the degrees of a Quadrant supplying the residue of those which could not be expressed on the long sight from 25, to 90. by ten, etc. For the use of these Divisions is there also belonging hereunto an Index or small Rule, at the one end whereof is a Centre hole to place on the Wire in the edge of the shorter sight, and at the other end a sight is placed, answerable to the fiducial edge of the same Index, which edge is likewise divided according to those divisions on the edge of the Instrument. This short sight is to have a plummet of lead hanging in a fine thread, serving to place the Instrument horizontally. Where note, that these sights, and their several divisions serve only for Altitudes, Profundities, and the reducing of hypothenusal lines to horizontal, which is to excellent purpose, and full of use. But when there is no occasion or use of these (as seldom there is in respect of other use) I would always have ready, such double sights as are hereafter expressed in the description of the Peractor, which doubtless are of excellent use, as I there explain; with this caution, that you always carefully keep one and the same part of your instrument forwards, and reccon ever your degrees in one and the same end of your Needle.  
The foot of this Instrument is that with three staves jointed in the head, and to be taken a sunder in the middle with brass sockets, according to the usual order, most fit for all Instruments, except in such cases, as in the description of the Peractor is excepted. And these are the several parts of this Instrument, and thus is it composed.   

CHAP. V The Peractor, with his several parts Description and Composition.  
AS I will not with the cunning Wine-taster forbear commendations, fearing too many partners; So will I refrain wooing any to use what I much affect, further than reason, and their own judgement shall rule them therein. But thus much will I boldly speak and maintain of this Instrument, that for general use, perspicuity, speed and perfection, it may well compare with any hitherto in use. It consisteth of a Planisphere in brass, much like unto that of the Theodelite, but where the limb of that is divided into 360. parts or degrees; this is only into 120. (so that each of these containeth three of those) and these subdivided into halves and quarters towards the limb thereof, without which divisions, there are drawn and described three concentricke Circles, being crossed with Diagonals, by whose intersections are exactly expressed the third part of every degree; whereby, and by tripling the former degrees cut (if occasion require) is had exactly the degrees of the Theodelite considering that ten of these contain thirty of those, wherewith in matter of Survey we shall little need to trouble ourselves. And here have we large and spacious degrees with their exact parts to ¼·S (which in others we want) by means whereof, and with help of such a chain, as I always use (which shall be next hereafter expressed) I will boldly approve and maintain, to work with much more facility and exactness, and come nearer the precise truth then any other can possibly do, not using the same or the like, which will appear most manifest in practice to all men's understanding.  
There is also an Index hereunto belonging, fixed on the Centre as that of the Theodelite, with two sights thereon placed, with sliding loops, either of them alike, and of like length, and either of them double sighted, the one having a slit beneath, and a thread above, the other a thread beneath, and a slit above, serving to look backwards and forwards at pleasure, without turning about or stirring the Instrument, when the Needle is at quiet; whereby I save near half the labour, and half the time that any man shall spend with other sights, for that hereby I need plant mine Instrument but at each other Angle, which is no small help for expedition, and such a means for exactness rightly handled, as few will imagine without due proof; and that without trouble of sending one before, and leaving another behind, as is usually accustomed. Neither use I these sights with this Instrument only, but with all others as the Theodelite, Plaine-Table, and Circumferentor.   diagram 
Than is there also hereunto belonging other usual parts, as a box of brass on the Centre to contain a Card and Needle therein, such as is formerly expressed for the Circumferentor, to be likewise covered over with clear glass, and close stopped with read wax about the edge thereof, to defend the Needle from Wind, Weather, and Air, the only enemies thereunto.  
Also a brass Socket to be screwed on with four screw pings on the back of the Instrument, which Socket aught to be precisely turned, and the head of the staff therewith, (I mean the brass part thereof) which will never otherwise turn evenly and nimbly about as it aught, the one within the other, without iercking and starting, which much troubleth the Needle in finding his natural point and place of rest. And if any doubt the truth of his Needle, let him take back sights for his better satisfaction therein.  
And lastly, for this as for the rest the like staff is to be provided as before is spoken of, which for all purposes is the absolute best, except only for water levels, and the works thereunto belonging, wherein it is necessary at all times, and at each several station to keep the instrument at one and the same horizontal distance, which otherwise may breed much cumber, and no less incertainety in those conclusions; wherefore for those and the like occasions, a foot with one staff, having three iron pikes therein, after the old order is best, and to best purpose.  
The making of this Instrument and the rest in brass are well known to M. ELIAS ALLEN in the Strand; and of those in wood to M. JOHN THOMPSON in Hosyer Lane.  
And thus have I briefly described these several Instruments with their particular parts, laying them before my Surveyor to take his choice as his fancy leads him; But in mine opinion all are better than any; so shall he best know what is best for his purpose.  
And now let us consider, what other necessaries are yet to be provided, before we begin our business, for fear we are to seek when occasion serves for use. As the Chain, Protractor, Field-booke, and the Scale and Compass whereof we will further speak.   

CHAP. VI The making and division of a Chain, called the decimal Chain.  
THis is the chain before spoken of in the Peractors description; which for conveniency in carriage, and avoiding casualties often happening to break it (though made of a full round wire) I would advise should contain in length but only two statute Poles or Perches, or three if you please at the most. In the dividing whereof it is to be considered, that the statute Perch or Pole, (which here we call an unite, or (Comencement) containeth in length 16 ½ feet. which is, 198. inches: This quantity is first to be divided into 10. equal parts called Primes, so shall every of these Primes contain in length 19 inches, and ⅘· of an inch: And then these Primes to be every of them subdivided into other 10. equal parts, which we will call Seconds: and so every of these Seconds shall contain in length one inch, and 49/50· parts of an inch. And thus is the whole Perch unite or comencement divided into 100 equal parts or links called Seconds.  
Which Chain so divided is thus to be distinguished and marked: First, at the end of every fifth Prime, or fiftieth second or link, which is the end of every half Pole, let a large curtain ring be fastened, so shall you have in the whole Chain (if but two Poles) three of those rings; the middlemost being the division of the two Poles, which in a Chain of this length is easily and readily discerned from those rings of the half Poles, though all of one greatness. Than at the end of every Prime, that is, at the end of every tenth second, or link, let a small curtain ring be placed, and not those rings of brass wire, as is usual in other chains, which, with every bush and twig are continually broken off, and lost.  
By those distinctions this Chain is now divided into these three terms, Unites, Primes and Seconds, whose Characters are these 0.1.2. So that if you would express 26. Unites, 4. Primes, and 5. Seconds, they are thus to be written 26 0· 4 1· 5 2· or together thus, 26 0 4 1 5 2· or more briefly thus, 26 4 ● 5 ●. making pricks or points only over the Fractions, whereby the rest may be conceived to be Unites, or Intigers, and the first point Primes, and the next seconds.  
But besides these divisions for mine own use, I always at the end of every 2 ½· Primes, which is the 1/4· of a Pole, sow on a small read cloth, or the like, (thrust through the ring of the chain, and at every 7 ½· Primes being the ¾· of a Pole; the like of yellow, or some other apparent colour, where with being once acquainted, and thereunto enured, you shall most speedily at the first view reckon the quantity of every ring, remembering that if it be the next ring, short of the read, it is two Primes, if the next over, three, if the next short of the yellow, it is 7. Primes, if the next over, 8. if the next short of a great half ring, it is 4. the next over, 6. And lastly, if the next short of the middle great ring, it is 9 and if the next over, 1. and so of the rest; Wherein is to be noted that your Chain thus marked, is always to be used with one and the same end forwards.  
This Chain thus divided and marked, you have every whole Pole equal to 10. Primes, or 100 seconds, every ¾· of a Pole equal to 7 ½· Primes, or 75. seconds, which is ¾· of 100 every half Pole equal to 5. Primes, or 50. seconds, which is ½· of 100 and lastly every ¼· of a Pole equal to 2 ½· Primes, or 25. seconds, which is ¼· of 100  
And here is to be noted, that in the ordinary use of this Chain for measuring and plaiting; I observe only Unites and Primes (but on necessity) which is much more exact than the ordinary use; but having occasion to make division or separation of lands, or for the dimension of common fields in their several parts by furlongs or wents and rigs, I use my Seconds; wherein, what exactness and most excellent use I found, I will refer to those who can discern the difference between a portion less than two inches, the length of my second, and that of 6. inches, and 3/●6· of an inch, the lest part of the best and exactest Chain now commonly used; but with those of the last of long twelves I will not meddle.  
But here me thinks I hear the Adversary question, to what purpose serves this niceness of inches in Instrumental observation, when coming to your protraction with a small Scale, you are not able to distinguish feet? I answer (and to purpose) thus: If by your ordinary Chain you take observation in your Field-bookes of ¼· ½· and ¾· and few or none otherwise, (or if they do, to small purpose, as they afterwards handle the matter) than I say, I taking my observation of 1. 2. 3. or 4. Primes, or of 6. 7. 8. or 9 Primes, can in my Protraction with a small Scale and Protractor (yet mine I must confess is none of the lest) easily distinguish and express how much less or more than ¼· ¾· or ¾· those quantities are, as may easily appear, with due observation of my former notes. Yet may it be further said, What is all this to purpose, if there be not as exact a means to obtain or get the true superficial content in casting up the plat, it being thus exactly laid down? I may answer again, Better one mischief then many; neither will I suffer this; for be well assured, I will not be so careful in that, and altogether careless in this; the means whereof in due time I may hereafter show, being unfit for this place; having already enlarged my Chain in length more by a Pole, than first I meant; and therefore purpose now to be no longer chained therein.  
The making of this Chain is well known to M. CHRISTOPHER JACKSON at the Sign of the Cock in Crooked-Lane, who by my directions hath made of them for me, and hath the scantling thereof.   

CHAP. VII. Of the Protractor and the Scale thereof.   diagram 
Provide for this Protractor a fine thin piece of brass well polished, in form of a long square, as the Figure E FLETCHER EDWARD I which (for conveniency in use) aught to contain in length from G. to H. about 5 ¼· inches, and in breadth from L. to B. somewhat better than 3 ½· inches, whereupon draw two lines as G H. and L B. cutting each other precisely at right Angles in the point D. dividing the Square into four equal parts, then on the point D. as a Centre, at the distance of D L. or D B. describe the Semicircle A B C. (for it is not material, or of necessity, that the Diameter thereof should agreed with the Diameter of the Card in the Instrument, as M. Hopton would have it in the 62. Chap. of his Top. glass) then divide the limb of the Semicircle A B C. into 60. equal parts or degrees, numbering them by five and ten in the outward space to 60. and in the inward space from 60. to 120. as in the figure, the first numbers serving for the East side, and the later (being the opposite degrees) for the West side of the whole Circle, so is a labour saved of dividing the other side which serves to no purpose; then let the edges of the Scale as E G. E F. and F H. be somewhat besiled, and made very smooth, and precisely parallel to the first drawn lines respectively, and above the rest, let care be had that the line L B. be made exactly perpendicular to the edge E F. of the Scale, or otherwise great errors may ensue in the use; then divide the Parallel degrees at either end of the Scale, between E G. and F H. And let the Scale of 12. be placed on the edge E F. and the sale of 11. on the edge of the back side, which are most necessary and meetest for use of any other; and lastly, cut out the square about the Centre D. and likewise that between the Semicircle & the pricked lines, having care that the line G H. be left perfect, and even with the Diameter A C. being the meridian line, and the guide of the rest. And so is this work finished.  
Yet would I have beside in some spare place of this Protractor, or on the back side thereof, the Sextans described, which is mentioned in the next Chap.  
Here is it to be noted, that this PROTRACTOR serveth without alteration or any difference, aswell for the PERACTOR as the CIRCUMFERENTOR. But if you would have it for the THEODELITE, then must the limb of the Semicircle be divided into 180. equal parts, and numbers placed accordingly, which is all the difference.  
And here also would I not have you forget to provide that long Protractor formerly mentioned in the conclusion of the second part of my second Book, which will stand you in good stead.  
Also to these there belongs a protracting pin made of a needle (according to the Centre hole of the Protractor) to be placed in the end of a small turned stick; or of ivory, as best likes you. And so are you thus far fitted: wherefore to the rest.   

CHAP. VIII. Of the ordinary Scale with the Sextans thereon described, very necessary for use.   diagram 
FOr this purpose let a Ruler of Brass or Box (but brass the better) be provided, as the Figure A B C D. which let contain in length about 7. or 8. inches, and in breadth about two inches, or somewhat less, whereupon on the one side let be placed two Scales, the one of 11. the other of 12. in an inch; and on the same side, let there be also described a Sextans or the sixth part of a Circle, whose chord E FLETCHER, (which is always equal to the Semidiameter of the same Circle) let contain in length about two inches or less, and let the limb be divided into 60. equal parts or degrees, and numbered by five and ten, as in the Figure. On the other side thereof there may be placed (after the order of these) divers other Scales, as of 16. 20. 24, etc. as you think fitting. So have you a necessary Instrument for many purposes. And this Sextans also would I have described on some spare place of your Protractor.  
To this must you have provided a neat pair of Compasses of brass, with fine steel points, which must always be ready serving for infinite occasions.  
Besides these ordinary Compasses, it is very fitting to be also provided of a pair of Callem Compasses, with screws to altar the one leg at pleasure, wherein to fasten a pen, black lead, a steel point, or the like, very fit for many purposes.   

CHAP. IX. Of a Ruler, for the reducing of Plaits.  
ALthough we are not as yet fit for the use of this Rule, yet seeing our business now in hand, is to provide us of necessaries: It is no ill rule to take our business before us: Wherhfore repair to Master JOHN THOMPSON in Hosier lane, who without further Instructions will furnish you, only this before you go: Let the Ruler be made of dry box, if you may, of a yard in length, and let the equal divisions thereon be of 12. in an inch, to be numbered with double numbers, as he useth for me. So will it serve you to good purpose, aswell for casting up of large plaits. etc.   

CHAP. X. The order of making of a necessary and fitting Field-booke, serving aswell for the Peractor and Circumferentor, as for the Theodelite, with the ordering and use thereof in the Fields.  
THis Book may consist of half a choir of paper, to be bound (most aptly for use) in a long Octavo: Let it be ruled towards the left margin of every side, with four lines, so shall you describe three Collums, the first serving for the degrees, the ●●cond (according to my Chain) for Vnits': and the third and last for Primes; or according to the accustomed use, for degrees, poles, and parts of a Pole.  
The order of using it is thus: Suppose you are to survey the Manor of Beauchampe, and are to begin with these five several parcels numbered in this plat or figure, with 1.2.3.4.5. being several grounds of several Tenants, and of several natures, whereof you are to make several observation, as appeareth. First, for the title of your Book begin thus: Manner. de Beauchampe in Com. Ebor. Pro Rege incip. 24. Junii, 1616.  
The Manor of Barnesey  
The North field.  
The east field  
The south field  
But seeing that practice is much more instructive (in works of this nature) than many words; I will refer the rest to your own travel; which by comparing those former notes of the Field book with the plat, and often protracting the same according to those degrees and lengths, the whole course is very easily understood. The manner and order of which protraction is hereafter taught in the 39 Chapter of this Book.  
And here note further that in practise you shall found many helps, which are too tedious here to express, as the taking in of divers severals together, when they lie in such sort divided with regular lines and hedges, that by only taking true notice of their several ends, as you pass by them, you shall most easily and speedily sever them on your plat: All which with many others (to avoid prolixity) I must refer to your own finding out by diligence and practice.  
And thus are we now reasonably well furnished of necessary implements for our purpose, and therefore fitting to prepare us to practise: but yet before we go into the Fields, we will consider of some necessary conclusions and observations, fit to be known and remembered.   

CHAP. XI. To lay down an Angle of any quantity required; or to found the quantity of any Angle given, by the Sextans and the Scale.   diagram 
Again, suppose C A B. be an angle given, & let it be required to know the quantity thereof. Extend the Compass to the chord of the Sextans as before, and at that distance with one foot in A. describe the Arch line C B. to cut both sides of the given Angle, as in C. and B. then opening the Compass to C B. and applying them at the same distance, to the degrees in the Sextans. It will appear that the quantity of the Angle is 40. degrees, the thing required.  
If the Angle given or required happen to be more than the whole Sextans, or above 60. degrees, yet take still the chord of the same Sextans, and describe the Arch line as before; and first place the whole Sextans, (which is the chord thereof) on the Arch line from B. so far as it will extend beyond C. and thereunto on the same Arch line, add so many degrees more as the Angle given or required, containeth degrees above 60. So shall you perform what was required.   

CHAP. XII. To lay down an Angle of any quantity required, or to find the quantity of any Angle given, by the Protractor.   diagram 
Again, suppose D C B. in the former Diagram be an Angle given, and let it be required to know the quantity thereof. Place the Centre of the Protractor as before in the Angle C. and the Meridian line thereof on the line C B. and having the semicircle upwards, note what degree on the edge of the Protractor is cut by the line D C. which you shall found to be 30. Showing that the given Angle D C B. containeth 30. degrees, the thing required.  
But here is to be noted, that these degrees thus taken by the Protractor belonging to the Peractor or Circumferentor, are not the true degrees of a Circle; for one degree of a Circle, is but the 360. part thereof, and one of these degrees thus taken are the 120. part, so that one of these containeth three of those; Wherhfore if you are to take the quantity of an Angle (according to the degrees of a Circle) by those Protractors, take always a third part upon the Semicircle, of the number given or required; as in the former example, where 30. is given, take 10. and so shall you find the Angle E C. B. in the last Diagram, to be an Angle of 30. degrees, and to be a third part of the Angle D C B. As may be proved, if you apply thereunto a Protractor belonging to the Theodelite; yet notwithstanding these Protractors and degrees in all our occasions in the use of the Circumferentor and Peractor are always to be used, which will tend to one and the same purpose.   

CHAP. XIII. The reducing of statute measure into Acres of any customary measure required, and the contrary, showing the difference between them.  
BY the Statute of 33. Ed. 1. It was ordained that an Acre of ground should contain 160. square Perches, to be measured by the Pole of 16½· feet, which is the measure now received, and generally allowed of, and is commonly called Statute-measure: yet notwithstanding in many places of this Kingdom, there are divers other sorts retained and claimed as customary, whereof some are greater, and some less than that by Statute. Wherhfore I hold it very fitting, and a main point belonging to a Surveyor, readily to reduce these quantities from the one to the other, whereby the difference may appear; whereof in practice he shall found often use; which to effect work thus. Suppose there are 5. Acres, 2. roods, 20. Perches, of 18. feet to the Pole given (called woodland measure; and let it be required to know the quantity thereof by statute measure, being of 16 ½· First, find out the lest proportional terms, between 18. and 16 ½·S which by their abbreviation, by 1 ½·S you shall find to be 12. and 11. then reduce your given quantity into the lowest denomination, which is Perches, so shall your 5. Acres, 2. roods, 20. Perches, be 900. Perches. And considering that the same proportion which the square of 12. bears to the square of 11. the like proportion bears the Acre of 18. foot Pole to that of 16 ½·S therefore square those two terms 12. and 11. which produceth 144. and 121. then multiply the given quantity 900. Perches by 144 the greater square (because the greater measure 18. is to be reduced into the lesser 16 ½·S the Factus is 129600. which divided by the lesser square 121. quoteth 1071 9/121· Perches; which reduced into Acres, is 6. Acres, 2. roods, 31. Perches, and 9/121· parts of a Perch; for the quantity required in statute measure, whose difference by deducting that from this, appeareth to be 1. Acre, 0. Rood, 11. Perches 9/121·  
But suppose the given quantity had been statute measure, and the same required to be reduced into woodland measure; then should you have multiplied the 900. Perches given by 121. the lesser square (because the lesser measure 16½· were to be reduced into the greater 18.) whereof the Product is 108900. which divided by 144. the greater square quoteth 756 ¼· Perches, which reduced into acres, is 4. Acres, 2. roods, 36 ¼· Perches, for the quantity by woodland measure; whose difference by deducting this from that appeareth to be 0. Acre, 3. Rood, 23 ¾· Perches.  
And the like course is to be held in all respects, with all other quantities of what proportion soever; as those of 12. 20.24½·S and 28. foot to the Pole, of all which several sorts I have found in divers places, whose difference; 〈◊〉 every Acre, from that of 16 ½·S appeareth by this Breviat following.  
An Acre measured by the Pole of these feet, 

12  
18  
20  
24  
24½  
28   Containeth of Statute measure 

0— 〈◊〉— 4 76/121·  
1— 0— 30 50/121·  
1— 1— 35 85/1089·  
2— 0— 18 62/121·  
2— 0— 32 133/174·  
2— 3— 20 4/●·    
And here is it not amiss to note the benefit and use of your two Scales of 11. and 12. in an inch formerly described in the 8. Chapter, which will serve you now to purpose. For if in your Surveys (as often happening) you meet with woodland grounds, whose quantities are required to be of the Acre of 18. foot Pole, and yet plaited with the rest: In such case you may measure those Wood-lands with the Pole of 16½· and likewise plate the same with the Scale of 12. as the rest, but to cast up the contents of those woodland grounds by the Scale of 11. which will produce the desired quantity; By reason that if 11. Perches be measured in a right line with the 18. foot pole, the same length containeth 12. Perches measured with the 16½ foot pole. But if you are constrained in the measuring of your Wood-lands, to use the Pole of 18. foot; then must you protract and lay down the same in your plat by the Scale of 11. which otherwise will not join with your other works; and the same likewise to be cast up by the same Scale of 11. as before. So shall you obtain the true quantity thereof in Acres, after the measure of the 18. foot pole required.   

CHAP. XIIII. Of the Table of Sines expressed on the Circumferentor.  
THis Table (as is specified in the description of the Circumferentor (Chap. 4.) serveth for the calculation, resolution and dimension of Triangles; not in respect of the Area or superficial content thereof; but for the finding out of the unknown sides and angles of the same; by means whereof, all manner of quantities in mensuration of altitudes, profundities, longitudes and latitudes are exactly known and discovered, considering that none of these can be had or obtained instrumentally, without description of Triangles.  
Wherhfore let it first be considered, that by the 73. THEOREM of the first book, the sides in all plain Triangles are in proportion the one to the other, as the Sins of the Angles opposite to those sides.   diagram 
Again, if A C. 866. the Sine of the Angle A B C. gives A B. 1000 the Sine of the Angle A C B. what gives A C. 40. and multiplying 1000 by 40. the Product is 40000. which divided by 866. quoteth 46 82/433· for the length of A B. the hypothenusal line, as was required.  
And thus much for a small taste only of this little Table, which may serve to induce and incite a willing mind, not only to the use and exercise thereof; but to the further consideration and practice of the infinite use of those most excellent Tables and works de BARTHOLOMAEO PITISCO GRUNBERG. now partly translated into English by M. RALPH HANDSON; and of those Tables, and more than admirable invention of logarithms, by that divine and noble Writer, the Lord MARCHISTON, whose name and honour will never out.   

CHAP. XU. Of the congruity in use between the Peractor and Circumferentor; and the means to found the quantity of an Angle by either of them.  
THese two Instruments in use differ little or nothing, considering the degrees of either are equally numbered, although those of the Circumferentor are placed on the Card, and these of the Peractor, on the limb of the Planisphere, whereby they are so much the larger, and thereby the fit for use: only herein they differ, the degrees observed and taken by the Peractor, are ever cut by the edge of the Index, moved about till through the sights thereon, the object be found, the Needle being always kept on one degree, and that most fitly on the Meridian line in the Card, the North end (being that with the Cross) lying ever over the Flower deluce, and the south end pointing to the beginning of the degrees: and the degrees observed and taken by the Circumferentor, are always cut in the Card by the South end of the Needle, playing about at pleasure. whilst the Instrument and the sights thereof are directed to the purposed mark. By means of which diversity, there is a divers means to be used, in taking the true quantity of an Angle, by these two Instruments, as followeth.  
By the 49. DEF. of the First. The quantity or measure of an Angle, is the Arch of a Circle, described from the point of the same Angle, and intercepted between the two sides of that Angle, which is found by the Circumferentor thus.   diagram 
Place your Instrument at B. in the former Diagram. (the Index standing on the Diameter where the degrees commence) then turn about the Instrument on the staff (the Index remaining) towards A. till your sight be parallel to the line B A. and there your Instrument fixed, remove the Index, directing your sight towards C. to be likewise parallel to B C. where observe what degree the edge of the Index cutteth, which will be 22. the quantity required. And here note an exquisite dispatch.   

CHAP. XVI. To take any horizontal distance at two stations, by Cynical computation.   diagram 
Than by the Rule of Proportion, reason thus. If A B. 707. the Sine of 15. degrees yield 20. the stationary distance, what A C. 839. the Sine of B. 19 degrees, and multiplying 839. by 20. and dividing the Product by 707. the quotus will be 23. 519/707· for the distance A C. And again, if A B. 707. the Sine of C. 15. degrees, yield 20. the stationary distance, what B C. 978. the Sine of A. 26 degrees, and multiplying 978. by 20. and dividing the Product by  d s A. 26 978. B. 19 839. C. 15 707. A B. 20. Perches. 707. the quotus will be 27 471/707· the distance B C. required.  
The Theorical ground and reason of this work dependeth on the 13. and 73. THEOREMS of the first book.  
Where note in all works of this nature, that if any of the three Angles be an obtuse Angle, containing above 30. degrees, than (seeing the Table of Sines exceedeth not 30.) deduct the excess of the obtuse Angle above 30. out of 30 (as if it were 44. the excess whereof above 30. is 14. which deduct out of 30. there remaineth 16. and of that remainder seek the Sine in the Table, which serves the turn. The reason hereof is, because the right Sine of the Arch in the greater or lesser Quadrant are all one and the same thing.  
Likewise note always in your working by the golden Rule, that the Sine of the Angle opposite to the Stacionary line (as in this example 707.) must be your first proportional number; and most fitly (though it may be otherwise, transpositis terminis medijs) the distance between the two stations the second; and the Sine of the Angle, opposite to the side, whose length you seek the third.  
And note also, that not only this, but all other the like Propositions are to be performed, aswell by the Peractor and Theodelite, as the Circumferentor, the Table of Sines being had in any void paper, or much rather those small Tables of logarithms, or of Pitiscus, which are imprinted by themselves in small volumes, being most excellent pocket-companions for infinite Conclusions, aswell Geometrical as Astronomical.  
And if any desire the performance of this Proposition, or the like by protraction; let him diligently observe the doctrine of the next.   

CHAP. XVII. To take the distance aswell between divers several places remote from your place of being, as between your being, and those several places, by the help of two stations.   diagram 
SVppose A.B. and C. be three places given remote from your place of being, which let be at D. and let it be required at D. to find the several distances, aswell between A B. and B C. as between D A. D B. and D C. First place your Instrument at D. and directing your sight to A. observe what degree is there cut, either by the Needle of the Circumferentor, or by the Index of the Peractor, which let be 32 ½· degrees, to be noted for your first observation, them turning your sight to B. make the like, where you find 21. degrees, and the like towards C. observing 15. degrees. Than your second Station, (not being limited) make choice thereof with such discretion (if the place will afford it, as at E.) that your Stationary distance be no less at the lest then ⅛· of the other distances from you, how much greater (with reason) makes no great matter; and as near as you may, let it make a right Angle with the first observation of your first Station, then for the last work of that station direct your sight to E. observing the degrees cut 120. then take up your Instrument, and leaving a mark at D. measure from D. to E. the stationary distance, which suppose 48. then at E. plant your Instrument precisely as at D. using the help both of your Needle and back sights herein, looking back to your mark at D. whereof special care is to be had, or main errors may ensue: which done, direct your sight first (as at the first station) to A. observing the degree there cut 45. the like to B. 31 ½· degrees, And lastly, to HUNDRED 15. degrees. So have you finished; if you omit not the collection of your several observations, which in your Field Book or otherwise are thus to be expressed.  
Than provide a clean sheet of paper, and according to these collections laid before you, protract the several angles or degrees here observed, as is taught in the next.  
Obs. 

1. St.  d 1 32 2 21. 3 9¼· 4 120.  
dist. p. 48.  
2. St. 1 45. 2 31 ½· 3 15.     

CHAP. XVIII. To protract any number of Angles or degrees taken by the Peractor, Theodelite or Circumferentor, at several observations.  
LEt the Angles or degrees taken be those expressed in the former Chapter, and let it be required to protract the same, whereby the quantity of each several distance there sought for, may appear. First, on your paper provided, draw a right line at pleasure, as FLETCHER G. in the former Diagram; then laying your Field-Booke before you, with the former observations, make choice of any point in the line F G. to represent your first Station, as at D. then applying the Scale of your Protractor to that line, lay down your stacionary distance 48. Perches from D. to E. representing the place of your second station; and placing your Protractor with the Centre on the point D. (the Semicircle upwards) turn it about thereon, till that degree on the Protractor which was taken from the first to the second station (which in this example is 120.) lie precisely on the line F.G. and then look in your Field-Booke for the degrees observed at your first station, which were 32 ½· 21. and 9 ¼·S (for the fourth 120. that is supposed always to fall on the first drawn line F G.) and against those several degrees on the limb of your Protractor, by the edge thereof with your protracting pin, make several pricks or points, as at H. I and K. then by the point D. and those three several pricks with the scale of your Protractor, and protracting pin, draw out at length the lines D H. D 1 and D K. so have you finished your first observations; then place your Protractor on the point E. in all respects as before, at D. and there mark the degrees of your second observations, as 45. 31½ and 15. as before, at the points L. M. and N. whereby, and by the point E. draw out at length the lines E L. E M. and E N. till they intersect the three former lines, drawn from D. as in the points A.B. and C. by which intersections from point to point, draw the lines A B. and B C. So have you finished your Protraction. And by applying the scale of your Protractor, (whereby the stacionary distance was laid down) to any line or distance, the several quantities will appear to be as they are expressed in the Diagram, on the several lines thereof, as was required.  
But here is to be noted, that if those former observations were made and taken by the Theodelite, than this Protraction is to be made and laid down by the Protractor belonging to the Theodelite, being divided into 360. degrees, as is before mentioned, which is to be performed in all respects, according to the rules and instructions before delivered.   

CHAP. XIX. To take any accessible altitude by the Circumferentor or Plain Table with the divided sights.   diagram 
But it happeneth oftentimes that the altitude required is of that height, that you cannot produce the vane low enough, to see the summitie of the height, as before. In which case you are to use the Index to be placed on the wyerpinne in the edge of the shorter sight, and turning it up and down close by the right edge of the Instrument, till through the sight thereof, and by the wire pin you espy the summytie of the given height, and then note the parts cut on the same edge of the Instrument, by the fiducial edge of the Index. For the same proportion which the parts cut, bear to 60. (the imagined parts on the edge of the shorter sight) the like hath the measured distance to the altitude required. Wherhfore, multiply the same measured distance by 60. and divide the Product by the parts cut, the quotus showeth your demand. And if you desire to know the Visual or Hypothenusall line, multiply the measured distance by the parts cut on the edge of the Index, and divide the Product by the parts cut on the edge of the Instrument; the quotus showeth what you desire. For what proportion the parts cut on the edge of the Instrument, bear to those cut on the edge of the Index, the like doth the measured distance to the Visual line.  
And here is to be noted, that of this later work the plain Table hath no use; and therefore of all other Instruments most unfit for these purposes of Altitudes and Profuncities, without help of the quadrant specified in the description thereof, CHAP. 3.  
Or with the Circumferentor, by Protraction, thus.  
PLace your Instrument at C. as before, and there observe the quantity of the Angle of altitude, which being gotten, protract and lay down the same as hath been taught, and on the base line from C. to B. lay down the measured distance 110. at the end whereof, as on the point B. either by the 6. PROBLEM of the second Book, or with help of your Protractor erect a perpendicular line as A B. to cut the other side of the protracted Angle, as in A. and with applying your Scale thereunto, the altitude appeareth, as was required.   

CHAP. XX. To take any accessible altitude divers ways, by the Peractor and the Quadrant thereof.  
SVppose M N. to be a perpendicular height given; and let it be required from O. to found the Altitude thereof. Place your Instrument precisely horizontal at O. as before is taught, then move your Quadrant up and down, till through the small round hole, in the end of the sight towards you at B. on the Quadrant (as it is described in the 5. Chapter) and by the pin in the great round hole of the other end at A. you espy the Summytie M. of the given height, where letting your quadrant rest, measure the distance O N. which suppose to be 48. then look on the side C D. of the quadrant, for the 48. line, reckoning from the Centre P. and passing down by that line to the edge of the handle or Index (which suppose to stand now on the line P F. drawn from the Centre of the quadrant) note what line (passing from the other side A.B. of the quadrant) the former line 48. meeteth and intersecteth on the edge of the Index, supposed as before the line P. F. and you shall found it somewhat more than 34. wherefore I conclude, that the altitude A B. is so much: and noting what part of the Index is there cut, you shall found it somewhat more than 58 4/5· the length of the visual line, or hipothenusall O. M. Where is to be noted, that in all works wrought with this quadrant, the side thereof A. B. representeth the perpendicular height, the side C. D. the horizontal distance, and the Index or handle, the hipothenusall or visual line.   diagram 
Or Sinically thus.  
Having placed your Instrument as before at O. By the degrees on the limb of the quadrant, observe the angle of altitude MON. 35. degrees 1′9. and measuring the distance ON. 48. as before (by the help of Pitiscus, or any other Canon) Reason thus: If ON. the radius 100000. yield 48. the measured distance; what M N. 70848. the tangent of the Angle MON. 35. degrees, 1′9. and multiplying the tangent 70848. by 48. the measured distance, you shall produce 3400704. which parted by the radius 100000. quoteth 34 704/100000 or in lesser terms, 34 22/3125· the altitude M N required.  
Again, if O N. the radius 100000. gives 48. the measured distance, what OH M. 122554. the secant of the Angle M O N. 35. degrees, 1′9. and multiplying the same secant 122554. by 48. and parting the Product by the radius 100000. you have 58 ●2592/100000· or in lesser terms, 58 2581/3125· being somewhat more than 4/5· the length of the visual or hipothenusall line O.M. as before.  
Or by Protraction, thus.  
PLacing your Instrument at O. as before, observe the Angle of altitude M O N. and measure the distance from O. to N. And then proceed to protracting thereof, as is taught in the later part of the last Chapter before going; and the work is finished.   

CHAP. XXI. To found out any inaccessible height by the Peractor, Theodelite, or Circumferentor.   diagram 
IT may oftentimes happen that inaccessible heights may be required, when by reason of waters, trenches, danger of shot, or many other impediments, a man cannot approach to the Base of the altitude required; yet of necessity to be had and known, which to perform, in a most absolute and exact manner, work thus.  
Suppose B C. to be a perpendicular height given, unto the Base, whereof HUNDRED (by some impediment) you may not approach nearer than D. yet the altitude is required; wherefore place your Instrument at D. precisely horizontal, and observe the Angle of altitude, as is before taught, which let be 53. degrees, 8′. then looking backwards, make choice in a right line from C. by D. of a second station, which let be A. and measure the distance from D. to A. which suppose 40. then at A. place your Instrument as before, and, likewise observe there the Angle of altitude, which suppose 36. degrees, 2′. so is your Instrumental work already finished. Than repairing to your Canon of Triangles, find the compliments of the Tangents of those two Angles taken, which of 53. degrees, 8′. the angle first observed, is 74991. and of 36. degrees, 2′. the Angle of your last observation, is 137470. between which two compliments take the difference, by deducting the lesser from the greater, which will be 62479. and then (for as much as the same proportion which the difference of the compliments 62479. beareth to the radius 100000. the like hath the measured distance between your two stations D A. 40. and the required altitude) multiply the radius 100000. by 40. the measured distance, the Product is 4000000. which parted by 62479. the difference of the compliments quoteth 64 1244/1000300· for the required altitude.  
And if it be required, to have the length of a scaling ladder to extend from D. to B. or the length of the visual line A B. or the inaccessible distance between D. and C. by respective observation of what was taught in the last Chapter, they are easily resolved.  
To perform the same by Protraction.  
Having taken both the Angles of altitude, and measured the stationary distance A D. 40. as before; protract the same thus; Draw a right line out at length, as E F. in the former Diagram, then on any point thereof, as at D. protract the first Angle taken, 53. degrees, 8′. (as before taught) then from D. towards E. lay down the stationary distance 40. to end in A. on which point A. protract your other Angle 36. degrees, 2 '. and continued forth the upper sides of those two Angles, till they interfect each other as in B. from which point B. by the 6. of the second, or by help of your Protractor, let fall a perpendicular, to cut the line E F. as B C. in C. which line B C. shall represent the given altitude, the height whereof, and the seveall lengths and distances in the whole work contained, is speedily had by applying thereunto the same scale, whereby your measured distance was laid down.  
And here is to be noted, that in all works of this kind, it is very requisite to take your stationary distance as large as conveniently you may, for that otherwise by reason of the acuitie of the Angle, as here of A B D. you shall hardly discern the true point of intersection by the lines B A. and B D. whereby, or from whence you may precisely let fall the perpendicular B C. as before in his due place: by neglect whereof main errors may ensue.  
Wherhfore a most excellent, absolute and exact course is that in the former part of this Chapter, for the performance of all manner of conclusions of this kind, and to be preferred before all others.  
And here now might I much enlarge this work, by inserting several Propositions for the taking and finding out of distances in heights, with the mensuration of profundities divers ways; all which and infinite other conclusions are fully included within the limits of these few former instructions, and with diligent observation and practice thereof may be well understood and performed; for whoso can take one height artificially, may perform another, and by deducting the one from the other, may decern the difference: and he that can skilfully take an altitude, by the same rule may perform a profundity, being the one a direct conversion of the other, without alteration or any difference, either in the Theorical ground, or practic operation thereof: wherefore to make great shows, or accumulation of needless varieties to one and the same purpose, were but expense of time unto myself, and cause of confusion to the learner, seeing fewest Precepts (so effectual) are fittest, aswell for apprehension as retention.  
But before I pass further, let this be remembered, that in all the former observations in taking of heights: the height of your Instrument is always to be added to the altitude taken.   

CHAP. XXII. To take the plot of a Field at one station; taken in any part thereof, from whence you may view all the Angles, and measuring from the station to every Angle.   diagram 
SVppose ABCDEFG. to be a Field, whereof it is required to take the Plot. First, cause papers or other marks to be placed directly in every Angle; and then make choice of some such convenient place within the same, as from whence you may best view the several Angles thereof, and there as at X. Plant your Instrument; if it be the Peractor, Theodelite, or Plaine-Table, fasten the Instrument to the staff with the scrue-pinne, that it stir not till your work be finished, the needle standing on the Meridian line of the Card, if the Circumferentor, that care is already taken; but admit the Peractor; then direct your sight, by turning the Index to any one Angle at your pleasure, as first to A. and observe the degree there cut by the edge of your Index, which let be 10. and with your Chain measure from your station to that Angle, which suppose 30. then direct your sight to B. and perform the like, and so to C.D. E.F. and G. till you have finished; still entering as you pass your severell observations, aswell of degrees as lines into your Field-booke, as was formerly taught in Chap. 10. which when you have finished shall appear to be thus.  
Which being laid before you, shall most speedily and exactly be protracted and laid down, as is taught in the next.  
d o 1 10 30 — 26 33 2 39 ½ 41 — 51 36 — 66 47 5 81 18 8 105 ½ 41 5   

CHAP. XXIII. To protract and lay down the observations made in the last Chapter, or any other taken in the like sort.  
TAke a clean sheet of paper, and draw a right line thereon at pleasure as PEA Q. in the Diagram of the last Chapter, representing the meridian Line; and laying your Field-booke before you with the former observations; First, place the Centre of your Protractor (the Semicircle upwards) on any point of the line as on X. with the Meridian line of the Protractor directly over that on the paper; where keeping your Protractor fixed, note all the degrees under 60. taken in your Field-booke, as 10.26.39 ½·S and 51. against which several degrees on the limb of your Protractor by the edge thereof, make several pricks or points with your protracting pin, as at H.I.K. and L. Than considering that those degrees are under 60. and therefore to lie on the East side of your plot or meridian line P Q. lay the edge of the scale of your Protractor on the Centre X. and every of those points H.I.K. and L. and draw from the Centre X. under the meridian line P Q. the several lines to A.B.C. and D. laying them down of the several lengths, belonging to them, according to your Field-booke, as X A. 30 0 X B. 33 0· 2 1· X C. 41 0· and X D. 36 0· making pricks or points at the end of every several length, as at A.B.C. and D. and then from point to point, draw the lines A B. B C. and C D. so have you finished the work on the East side of your Meridian. Than place again your Protractor in all respects as before, and note all the other degrees being above 60. as 66.81. and 105 ½ which by the edge of the Semicircle mark out as before, as at M. N. and O. and seeing they belong to the West side of your work, draw your lines from the Centre X. upwards towards those three several points, laying them down with their several lengths observed in your book, as X E. 47 0· 5 1· X F. 18 0· 8 1· and X G. 41 1· 5 1· making points as before at the end of every length; and lastly from point to point draw the lines A G. G F. FLETCHER E. and E D. So shall you enclose the Figure A B C D E F G. with equal Angles and proportional lines to the measured field as was required.   

CHAP. XXIIII. To take the plot of any Field at one station in any one Angle thereof, from whence may be seen all the other Angles of the same Field; and measuring from the station to every Angle.   diagram 
SVppose A B C D E F G. to be a field, the plat whereof is required to be taken. First, cause whites or marks to be placed directly in every Angle, then make choice of the most convenient Angle, from whence you may best view all the rest, as at A. where place your Instrument as before is taught, and directing your sight to one of the next Angles on either hand as to B. observe the degrees cut by the edge of the Index, which let be 24 ½· degrees and measure that line A B. 33 0· 4 1· then turning your Index to the next, as to C. make the like observation of 13 ¼· degrees. and measure the length A C. 55 0 ·2 1· and in like manner proceed to the rest, as to D.E.F. and G. still expressing in your Field-booke your several Angles and Lines as before is taught, which having finished will thus appear.  
And then protract, and lay down the same in all respects, according to the instructions of the last Chapter. The order whereof appeareth by the Diagram, d o 1 24 ½ 33 4 13 ¼ 55 2 6 47 — 116 69 2 104 65 6 95 ½ 41 6   

CHAP. XXV. To take the Plot of a Field at one station in any Angle, from whence the rest may be seen, and by measuring the sides of the Perimeter.   diagram 
SVppose A B C D E F G. be a Field to be plotted. First; set up marks as before, and choose an Angle fittest for your purpose, from whence you may see all the rest, and there plant your Instrument as at A. then direct your sight to one of the next Angles, on either hand, as to B. and note the degree there cut, 24 ½·S and measure the length of that line A B. 33 0· 4 1· then direct your sight to C. and note the degree there cut 13 ¼·S and measure the line from the Angle B. to the Angle C. 33 0· and in like manner work forwards to D.E. and F. and then note (having finished at F.) that you have yet remaining two lines to measure, namely, F G. and G A. and but one degree to be taken, as from A. to G. (The reason whereof dependeth on THEOR. 74.1.) wherefore measure the line F G. 33 0· 5 1· and express the same in your Book without any degree; and lastly, directing your sight to G. observe the degree cut 95 ½·S and measuring the line A G. place the length thereof in your Book 41 0· 6 1· against the last degree taken. So have you finished your Field-worke, and your observations stand thus.  
Which are to be protracted and laid down as is taught in the next.  
d o 1 24 ½ 33 4 13 ¼ 33 — 6 21 — 116 36 5 104 41 8 — 33 5 95 ½ 41 6   

CHAP. XXVI. To protract and lay down the observations had, according to the work in the last Chapter, or any other taken by the like means.  
THe several degrees and lengths so had and taken as before, and expressed in your Field book, being laid before you, work thus. First, according to your degrees taken, and as is taught in Chap. 23. draw out at length your several lines A B. A C. A D. A E. A F. and A G. as in the former Diagram, then opening your Compass on your Scale, take therein the first length 33 0· 4 1· at which distance with one foot in A. strike with the other an Arch through the line A B. cutting the same in B. then take your second length 33 0· at which distance with one foot in B. cross the line A C. in C. and draw the line B C. then take the third length 21 0· and with one foot in C. cross the line A D. in D. and draw the line C D. and in this manner work forwards, laying down every length, and drawing each line till you have enclosed the Figure A B C D E F G. which shall be a like Figure to the measured Field.   

CHAP. XXVII. To take the plot of a Field at two stations, where all the Angles cannot be seen at one, and measuring as in the 22. Chapter.   diagram 
SVppose this Figure be a Field to be plotted, which lieth in such sort, as from no one place all the Angles thereof can be seen. In such case make choice of a place for your first station, where may be viewed as many Angles thereof as possibly you may, which let be M. where you may see the several Angles at A.B.C.D.E. and F. then plant your Instrument in M. and there observe all those Angles, and measure the several lines, beginning from M. to A. and ending from M. to F. as is taught in the 22. Chapter, so have you finished the work of your first station. Than (before you remove your Instrument) make choice of some other convenient place for your second station, from whence you may see all the other Angles not formerly seen, as those at I.H. and G. which let be N. unto which place direct your sight, and observe the degree cut 65. then measure the stationary distance M. N. 40 0· and leaving a mark at M. remove now your Instrument to N. where place it precisely as it stood at M. with help of your needle and back sight, then observe your several degrees, and measure the several lengths from N. your second station to I.H. and G. as before, and your Field-worke is finished. So as you remember always to express your observations in your Field-booke, which shall thus appear.  
To be protracted and laid down as is taught in the next.  
d o 1 83 ¾· 29 — 109 ½· 19 2 1. Sta. 10 ¼· 26 8 28 ½· 30 8 41 ¼· 31 5 55 ½· 16 4 sta. dist. 65 40 — 74 ¼· 26 — 2. Sta. 46 ½· 29 — 33 ¼· 27 5   

CHAP. XXVIII. To protract and lay down any observations taken, according to the work of the last chapter.  
ON a fair sheet of paper draw a right line at length in any convenient place thereof, as K.L. in the Diagram of the last Chapter, which line is not to be taken as before for a meridian line, but supposed to be drawn according to the degree taken from the first to the second station; and therefore call it the stationary line, and thereon lay down your stationary distance 40 0· as from M. to N. which two points shall represent your two stations, then on M. the point of your first station, place the centre of your Protractor (the Semicircle upwards) turning it about till the degree 65. on the limb of your Protractor (being the degree taken from the first to the second station) cut precisely the stacionary line drawn, and there keeping it firm and immovable, mark out the several degrees of your first station; according to your Field-booke; and so work on in all respects as is taught in Chap. 23. which effected, remove your Protractor to N. the point of your second station, where placed precisely, as at the first, work forwards with the degrees and lines of your second station, as before; and so have you finished.   

CHAP. XXIX. To take the Plot of a Field at divers stations in divers Angles, where all cannot be seen from one, and to measure as in the 24. Chapter.   diagram 
SVppose this Figure be a Field to be plotted, whose angles cannot be seen from any one Angle thereof, wherefore imagine you are now standing in the Angle A. from whence you view and consider what Angles may there be conveniently seen and taken, which you found to be those at B. C. D. and E. then directly in those Angel's cause marks to be placed, and planting your Instrument as before is taught in A. direct your sight first to B. then to C. after to D. and lastly to E. noting the several degrees cut towards each several Angle, and measuring as in the 24. Chapter, from your station to every of those Angles severally, and your work of that station is finished. Now for that you ended your last work at E. remove your Instrument to that Angle, and there plant it precisely, as at the first station, using both your needle and backe-sight for your help therein; And here consider what Angles from hence may be perfectly seen and taken, which on view had, you find to be all the residue not formerly taken, as F.G.H.I.K.L. and M. wherefore having your marks placed, take your several observations and measure your several lines to every of these Angles, as to those of the first station, taking them in order as they lie, and you have finished. But suppose at this station, you could have seen only those Angles at F. G. H. and I then here must you have finished those, and removed your Instrument to I for a third station, and there to have performed the rest, or as many as there you might, and if any remaining, to take a fourth, and a fift station, etc. till you have finished, wherein many words are needless, the matter being apparent. Your observations of this work are these.  

A general Note.  And let it be noted, that where for brevity sake in mine instructions, aswell here as else where, I omit to express particularly, the several degrees and lengths observed and taken between each station and the several Angles, (which would be no less tedious, then troublesome in breeding confusion) I observe a due order in the placing of these observations of degrees and angles, according to the order of the letters about the plot, as the first degree and length 116 ½· 28 0· 4 1· belong to the first line A B. the second to the second A C. and the like of all the rest, which are to be protracted and laid down, as appeareth in the next.  
d o 1 116 ½ 28 4 101 41 8 85 ¼ 55 8 74 ½ 41 8 86 22 — 75 ½ 32 8 69 1/● 22 6 59 31 5 45 ¼ 42 5 36 ½ 44 4 23 30 4   

CHAP. XXX. To protract and lay down any observations taken, according to the work of the last Chapter.  
THe speediest and exactest course for protracting works of this nature, consisting of divers stations is thus. First, draw a right line at length on your paper, to represent the meridian line, as N O. in the last Diagram, whereon placing your Protractor, work in all respects as is taught in Chap. 23. for the observations of your first station; so shall you finish as much of this work as is included by the lines A B. B C. C D. and D E. the work of your first station; then by the point E. where you left, draw another meridian line as PEA Q. which (by PROB. 3.2.) make Parallel to the first line, N O. and then on the point E. place your Protractor in all respects, as at the first, & work with the rest as before, whereby you shall finish the work of your second station, and perfect the Figure A B C D E F G H I EDWARD L M. with equal Angles, and proportional lines to the measured field. And if there were more stations to be used in the Field-worke, then at the point where the work of each station endeth, you are to draw another meridian line parallel to the rest. Or before you begin your protraction, you may draw divers parallel lines on your paper, representing so many meridians, and by help of your parallel divisions, placed at either end of the Scale of your Protractor, you shall on any point falling either upon or beside those Meridian's place your Protractor parallel as you please. And this kind of protraction may be used in stead of that taught in the 28. Chapter, as the better, though either will serve, and both tending to one end.  
Hitherto have I taught after a perfect and exact manner the mensuration of severals (by divers means) where one field or close only is to be taken by itself: But if many severals (as a whole Lordship or Manor) were to be measured and plotted together: I hold not these former courses the fittest: but rather those which shall be hereafter taught. But first I will deliver some few directions and examples, for the dimension of severals after another order, by intersection of lines at several stations as followeth.   

CHAP. XXXI. To take the plot of any field at two stations, so as all the Angles may be seen from both stations, by measuring only the stationary distance.   diagram 
SVppose A B C D E F G. be a Field, the plot whereof is required to be taken. First, make choice of two such convenient places for your Stations, as from whence you may see all the Angles about the Field; with these further considerations, that the distance between your stations be of convenient length, the longer the better, that they lie towards the middle of the field; and that neither of them lie interposed in a right line between the other, and any Angle of the Field; but to be chosen with such discretion, as all lines drawn from either station to the several Angles, may intersect each other with as large angles as you may, which let be the two points H. and I and causing marks to be placed in every Angle, plant your Instrument at H. as is before taught, and directing your sight to A. observe the degree there cut; and the like to B. C. D. E. F. and G. and also to I the second station, then take up your Instrument, leaving a mark at H. from whence measure the stationary distance to I where placing your Instrument precisely as at H. observe likewise all the degrees cut by your Index, directed to each several Angle as before: Of all which several observations keep notice in your Field-booke as hath been often mentioned; wherewith on a clean sheet of paper by the directions of the 18. and 30. Chapters, the plot thereof is speedily protracted, and your business fully finished.  
And here note the accuitie of divers of those Angles in the Diagram, caused by the intersection of the pricked lines, notwithstanding all care had therein; and what inconveniences may hereby grow, without good regard; and yet are these Angles usually drawn by many, who make a poor shift therewith.   

CHAP. XXXII. To take the plot of any Field remote from you at two stations, when either by opposition you may not, or some other impediment you cannot come into the same.  
SVppose A B C D E  diagram F. be a Field, whereinto by no means you can be suffered to enter, yet of necessity must the plot thereof be had. In such case make choice of any two places, either near hand, or further off, all is one; so from thence you may well decern the several Angles of the same Field; and let your stationary distance be the full length of the Field, at the lest if possibly you may; which two places let be G. and H. First plant your Instrument at G. and by directing your sight in order one after other to A.B.C.D.E. and F. the several Angles of the Field, observe the several degrees there cut, as is before taught; then turn your sight to H. your second station, and note the degree there cut; which done, take up your Instrument, leaving a mark at G. and measure from thence to H. your stationary distance, and there plant your Instrument in all respects as before, and make the like observations to all the several Angles of the Field, as formerly at G. So have you finished your Field-worke, which is to be protracted and laid down according to your Instructions of the 18. Chapter.  
And here note, by reason of the length of the stationary distance, how excellently the lines issuing from thence, intersect each other, which of necessity makes the conclusion absolute.   

CHAP. XXXIII. To take the Plot of any Field by making observation at every Angle, and measuring only one line, but no part of the perimeter.   diagram 
d o 1 109 21 7 ½ 118 ½   10 ½   88 ½   106   74 ½   93   49   75 ½   35 ¼    
Which is to be protracted, as is taught in the next.   

CHAP. XXXIIII. To protract and lay down any observations taken, according to the work of the last Chapter.  
THis kind of protraction is somewhat different from all the rest formerly taught, wherefore observe it thus. First, draw divers parallel lines over all your paper, of convenient distance one from another (not exceeding the breadth of the Scale of your Protractor) which shall represent the Meridian's; then with your observations laid before you, take a point in any convenient place of your paper, whether upon or beside any parallel line, it is not material, as at A. in the last Diagram, then thereon place the Centre of your Protractor (the Semicircle upwards) turning it about on your protracting pin, till you found the match or opposite parallel divisions on either end of the scale, to lie either precisely upon any one line, or equally distant over or beneath the same; then look in your book what are the first two degrees, which is 109. and 118½· against which two degrees, by the edge of your Protractors Semicircle, make several pricks on your paper, whereby, and by the point A. draw the two right lines A F. and A B. out at length, and from A. to F. lay down your measured length 21 0· 7 1 ½·S then on the point F. place the Centre of your Protractor precisely as before, by help of your divisions on the end of the Scale, and found in your Field-booke what degrees you have (more than the first already done) marked with points or other marks in the margin (as was taught in the Field-worke) which are 10 ½· 106.93. & 75 ½·S against all which degrees on your Protractor make several pricks as before, whereby, and by the point F. draw out at length the several lines F B. F C. F D. and F E. remembering (as you are taught in Chap. 23.) always to draw those lines proceeding from all the degrees under 60. downward or towards you from the point F. and those above 60. upwards from the point F. Than note, that by drawing out the line F B. you have intersected the line A B. (formerly drawn) in the point B. on which point now place your Protractor as before, and find in your Book the second degree unmarked in the margin (for the first A B. is already done, and likewise all those which are marked) which is 88 ½·S against which, on your Protractor make a prick, and thereby, and by the point B. draw the line B C. till it intersect the line F C. in C. then place your Protractor on the point C. as before, and find in your book the next degree unmarked, which is 74 ½·S and against that degree on your Protractor make a prick, whereby, and by the point C. draw out at length the line C D. to cut the line F D. in D. and in like manner proceed with the rest; so shall you include the Figure A B C D E. like unto the measured Field.  
Where note, that now the points in the margin serve you to purpose, at an instant distinguishing those degrees taken at each angle towards the point F. from the others unmarked, representing those of the Perimeter.  
This kind of work well handled, is very exact and artificial.   

CHAP. XXXV. To take the Plot of any Field at divers stations, measuring only the stacionary distances.  
 diagram  
SVppose this irregular Figure to represent a Field, whose plot is to be taken. First, cause marks to be placed in every Angle thereof, and then (remembering those cautions and considerations in this behalf delivered in Chap. 31.) make choice of your first station, which let be at N. where planting your Instrument, direct your sight to as many Angles severally as lies there within your view, as to A.B.C. and D. and take observation of the several degrees there cut, and then (before you remove your Instrument) make choice of your second station, from whence you may not only see all those former angles, but as many more as possibly you may, which let be at O. then directing your sight to O. observe the degree there cut, and taking up your Instrument, leave there a mark, and measure the distance from thence to O. where again plant your Instrument as before at N. and then directing your sight to those former Angles A. B. C. and D. note severally what degrees are cut, and you have finished for those Angles. Now consider what other Angles you can here espy as those at E. M. L. and K. and thither likewise direct your sight severally, and make your several observations as before, which done (your Instrument remaining) make choice of your third station, (from whence you may not only see those Angles at E. M. L. and K. but as many other unfinished Angles as you may) which let be at P. then directing your sight to P. observe your degree thither, leave a mark at O. take up your Instrument, and measure from O. to P. where again plant your Instrument, and make your observations of those former Angles E. M. L. and K. so have you thus far wrought. And now again considering what other Angles you can here espy, you shall find within your view F. G. H. and I being all the residue unfinished; wherefore here also make your several observations of these last Angles, and choice of your last station, as at Q. whither directing your sight, and making observation, take up your Instrument, and with a mark at P. measure the distance to Q. where lastly plant your Instrument, and making your observations of those last several Angles at F. G. H. and I as before; your Field-worke is finished. And by the directions of the 18. and 30. Chapters, you may protract the same accordingly.   

CHAP. XXXVI. To take the Plot of a Forest, or any spacious Common or Waste, of whatsoever quantity, by the Plain Table, on one sheet of paper without altering thereof.   diagram 
SVppose A. B. C. D. E. F. G. H. I to be some large irregular ground, and it is required to plot the same. First, a sheet of paper being placed on your Plain Table after the usual manner, Take a point at all adventure about the middle thereof as at Q. in the Diagram, and thereon describe four concentricke Circles of convenient distance, the uttermost extending near the breadth of the Table, whereby you shall produce three several circular spaces, which are thus to be employed; In the uttermost, express the several lengths of all such lines, which in passing forward in your work shall be found declining or extending outwards, as the lines D E. G H. and I A. which lines always issue from an Angle greater than a Semicircle, the inclination whereof causeth the declination of the line: the second which is the middlemost space, let serve for the number of each line; and in the third and innermost space, express the several lengths of all such lines, as in passing forwards shall be found inclining or extending innards, as all the other lines in the figure not formerly mentioned.  
And these thus understood, proceed forwards in this manner. Plant your Instrument after the usual order in the Angle, where you purpose to begin, which let be A. and let the Centre Q. on your Table, always in your whole work represent your place of standing, then lay your Ruler on the Centre Q. and turning it about, thereon direct your sight to I and by the edge of the Ruler, draw the line Q. K. (which shall be your enclosing line) extending it to the uttermost Circle, and in the middle space, on that line (which is to bear no number) place the Ciphero. then keeping still the Centre point Q. turn about your Ruler, and direct your sight to B. and by the edge of your Ruler draw the line Q N. and in the middle space on that line express the Figure 1. the number of the first line you go on, so have you on the Table the Angle KING Q N. equal to the Angle I A B. in the field, then take up your Instrument, and measure the line A B. 29 0· 2 1 ½·S which length express on your first line Q N. and for as much as the line is inclining, place the same in the innermost space, as in the Diagram; then place your Instrument precisely at B. and turn it about on the staff (your Ruler lying on your first line Q N. till you find the same line to lie parallel with the line A B. of the Field, having Q. towards you, and N. towards the Angle A. (for the Centre Q. in every Angle must represent the point of the same Angle) than your Table there fixed, direct your sight to C. and draw your line which shall be the line Q R. and express thereon in the middle space the figure 2. for the second line, so have you also the Angle N Q R. on the Table, equal to the second Angle A B C. in the Field: then take up your Instrument, and measure the line B HUNDRED 12 0· 5 1· which place on the second line Q R. in the innermost space, and planting your Instrument at C. direct your last line Q R. towards B. as you did, the first towards A. and then take the quantity of this Angle, as before, and you shall have the Angle R Q L. on your Table, equal to B C D. in the Field. And thus proceed from angle to angle, making always the last side of the last Angle to be the first of the next, with all their points concurring in the Centre Q. and measuring and expressing the several lengths of each line, as before; you shall obtain the quantity of every Angle, and the length of every line throughout the whole work: whereby you may speedily protract and lay down the same, as is taught in the next.  
But in the mean space it is to be noted, that if in your former work it happen at any time, that one line fall directly on any other, formerly drawn on your Table (as in the former Diagram, the seventh line Q R. falleth on the second line formerly drawn) then in such case express the number of the same last line in the middle space, where the first is numbered, with a stroke between them; but place the numbers, expressing the length of the same last line, without the uttermost Circle, if it be declining (as in the former example 22 0· 2 1·) and within the innermost, if it be inclining.  
This kind of mensuration of spacious works with the Plain Table (which likewise may be wrought with any other Instrument according to Chapter 15.) is wonderful necessary both for speed and exactness, if artificially handled: but if it be required to have notice taken of the several lands and grounds abutting and confining hereon, you must then have provided a Field-booke for that purpose.   

CHAP. XXXVII. To protract and lay down any observations taken, according to the work of the last Chapter.  
Having before you the paper of your Field-worke as you wrought on the Table: If you imagine that one sheet of paper will not serve the turn, you may with mouth-glue, lay as many together as you please; and then (considering which way your work will extend) draw a right line accordingly on your paper, whereon, with your Scale and Compass, lay down the length of your first line Q N. 29 0· 2 1 ½·S as the line A B. in the Diagram of the former Chapter, and then on the end B. of that line, by PROB. 8.2. protract an Angle, equal to the Angle N Q R. as the Angle A B. C. and on the side B C. of that Angle, place by your Scale the length of your second line R Q. 12 0· 5 1· from B. to C. and on the point C. where your last length ended, protract another Angle, equal to the Angle R Q L. your third Angle taken, as the Angle B C D. and on the side C D. extended, place the length of your third line L Q. 14 0· 9 1· from C. to D. And so proceed from Angle to Angle, protracting your Angles equal (and in order by the number of lines) to those answerable in your concentricke Circles, and laying down duly by your Scale on each line, the length thereunto belonging, you shall produce the Figure A B C D E F G H I like unto the measured quantity.  
And having thus finished; if you doubt or make question, whether you have wrought exactly or not, and desire to be resolved therein, make approbation thus. Collect the quantity of all your Angles in your whole work, and adding them together, note the total thereof; which in this former work is 1260. then multiply 180. (the number of degrees in a Semicircle) by a number less by 2. then the number of Angles in your work, 
If your degrees be those of 120. as observed by the Peractor, or Circunferentor, then in stead of 180. you must take 60.  which here is 7. (for the number of Angles is 9 as appeareth by the line O Q. which showeth the number of the last line, and consequently of the Angles) and if the product of that multiplication agreed with the former total; then by THEOR. 74.1. you have done rightly, otherwise not; as the Product of 180. by 7. is 1260. agreeing with your first number; and therefore may you confidently affirm to have wrought exactly, which rule is general for all other plots, and superficial figures whatsoever.  
Thus hitherto have we dealt in the plotting and dimension of severals; and that by such several, exact and artificial ways and means, as may most sufficiently serve aswell for the absolute performance thereof, as (if well understood and practised) of many other excellent conclusions. But were you to survey and plot great quantities, and many severals together as a whole Lordship or Manor, or to deal in mensuration of impassable woodgrounds, wherein you are debarred from crossing over or working within the same; I cannot advise you therein to the use of these former Precepts, (though otherwise to excellent purpose) but rather to use and observe the means and courses prescribed and taught in the next following Chapter, which of all other is the most general, absolute and exact, for the mensuration of all manner superficial Figures, of what form, quantity, or number soever; and therefore to be observed with good respect.   

CHAP. XXXVIII. To take the plot of a Lordship or Manor, consisting of divers severals, of what nature or kind soever, whether Wood grounds or other.  
THe precepts and instructions taught, and delivered in the 10. Chapter of this book, concerning the description and use of a necessary Field-booke, might well serve with diligent observation for the performance of this work: But seeing that there (according to the proposed matter) my chief endeavours tended rather to the explanation thereof, then to the form and order of mensuration, we will here make use of the Figure there expressed; and by inserting certain necessary observations there omitted (as needless to that purpose) refer you for a full satisfaction, to the consideration and due observation of those, and these conjoined. Wherhfore suppose that the figure or Diagram there expressed, were a Manor, or part of a Manor to be measured and plotted. First, writ your Title, as there is mentioned then planting your Instrument in A. direct your sight to B. and having observed the degree there cut, take up your Instrument, and measure to B. entering your degree and length into your book, then plant not your Instrument at B. but only measure from thence to C. and there place your Instrument, & direct ypur sight backwards to B. observing your degree; but with this special regard, that in taking your back sights, you always reckon the degree cut by one and the same end of the Index, as you reckon on, when you direct your sight forwards; or otherwise you take the opposite degree to that you should, which will much trouble you in protraction: then here consider, that you are to leave the bounder, which you went against from A. where you began, to this place, & therefore draw a single line, and as you are there taught, writ the bounder past, as in the example of your book; then direct your sight to D. and observing your degree, measure thither, which having entered, make there another single line, for that here you leave that bounder also, which let be expressed; then measure from D. to E. and there plant your Instrument, and as before direct your sight backwards to D. and observe the degree with the former caution, which done, turn your sight to F. and having your degree, measure from E. to F. and likewise from F. to G. expressing in your Book those lengths severally, then plant your Instrument at G. and taking your degree from thence backwards to F. as before, here strike a single line, and writ your third bounder, then take your degree to H. and measure thither, where also (that being your last Angle) you must plant your Instrument, and work as before to A. where you began, and then strike a double line, signifying you have finished that Field. And in like manner proceed with all the rest from one Close to another, till you have finished the whole work, as you are most plainly directed in the 10. Chapter, remembering always to measure every line; and to place your Instrument at each other Angle, taking your backe-sights to that very point or mark, whereunto you directed your sight last before. So shall you most exactly, and with great expedition perform your desire. And your work is to be protracted as is taught in the next.   

CHAP. THIRTY-NINE. To protract and say down a Plot of many severals, of what quantity or number soever.  
ACcording to the quantity of your plot, or the largeness you suppose it will be of, glue papers together; but if very large, lay first together but 4. or 8. sheets only, and rule them all over with parallel lines, representing Meridian's of such convenient distance, as they exceed not the breadth of your Protractors Scale. Than laying your Field-booke before you, suppose you are to protract the observations mentioned in the tenth Chapter, and considering towards what point of the Compass your work will most incline or extend, begin your protraction accordingly; as in that example; it inclineth towards the East, and north-east from the place of beginning; wherefore begin your protraction towards the south-west part of your plot; and there make a point, whereon place the Centre of your Protractor (with the Semicircle either upwards or downwards, as you best fancy) and holding your protracting pin in that point, move about your Protractor thereon, till you find one and the same parallel division on either end of your Protractors Scale, to lie either directly upon any one parallel line, or equidistantly above or beneath the same; and there, with your left hand keep firm your Protractor, whilst you find in your Field-booke the first degree 65. against which, on the limb of your Protractor place the point of your protracting pin, and there keep it, bringing the edge of your Protractors Scale thereunto with the first division of the Scale on your chosen point, and then draw a line by the edge of your Scale of your first length in the book 20 0· 2 1· as the line A B. But with this respect, that (as before is taught) the lines belonging to every degree under 60. be drawn from the Centre point downwards, or inclining towards you, and the lines belonging to every degree above 60. (as the last line A B.) be drawn from the same point upwards, or reclining from you: Than place your Protractor on the point B. (being the end of your last line) in all respects as before at A. and finding your next degree and length in your book to be 68 16 0· 6 1· against 68 degrees on your Protractor, place the point of your protracting pin, and applying the Scale of your Protractor thereunto, with the beginning of the divisions thereof on the point B. draw your line of the length 16 0· 6 1· as B C. and here (considering you are to leave the bounder you went against) make a small stroke or mark at C. with your pen, and finding in your Book, at the end of your last length the figure (1) place that at C. with black lead (the use whereof shall partly appear) then place your Protractor on the point HUNDRED (the end of your last line) as before, and found your next degree and length 36 ½· 22 0· 5 1· and against that degree on your Protractor, place the point of your protracting pin, and bringing the edge of your Protractor thereunto as before, from the point C. draw the line C D. of the length 22 0· 5 1· which serving to a degree under 60. is to be drawn from the point C. downwards as before. And so proceed with your several degrees and lines, in order as you find them in your book, till you come to the point A. where you first began, and having wrought truly, you shall there justly enclose your first several. Than look in your Book where you are to begin your next enclosure; and you shall be thereby directed to begin from Nᵒ. (1) Wherhfore seek in your last protraction where you placed that number, which you shall find at C. and there you are to begin your second parcel; wherewith, and with the rest proceed in all respects as with the first. Small practice with good observation (whereunto I will leave you) is much more available than many words. And therefore will I cease to spend further time herein.   

CHAP. XL. The order and means of measuring and taking the several and particular quantities in common fields, with a brief instruction concerning the use of my Chain.  
THe whole plot and quantity of common fields are to be taken and plotted as they lie among other the adjacent grounds, according to the directions of the 38. Chapter, without regard of the several and particular quantities therein contained; which afterwards are to be had and obtained after this manner.  
Let a Book be purposely provided, which call your Common-Field-Booke to be ruled as in the example, containing eight Collums. The first towards the 
The West field art. Broad furlong.  Tenant's names. Breadth. Length. Quantity 0 1 2 0 1 2 a. r. p. (1) Io. Woods from the Church lane east wards free. 1 3 2 16 2 — 0-0-21 (〈◊〉) Wil Browne by Copy. 6 8 2 16 2 — 0-2-30- (3) Fra. Jacksons for 3. lives. 8 4 1 14 2 3 0-3-0 (4) Tho. Coakes for years. 7 5 2 15 4 2 0-2-36 (5) ●il. jones at william. 5 6 3 15 4 2 0 2-7 left hand, serving for the Tenants names and the tenure whereby they hold the same lands; the next three for the breadth of every parcel entitled with these Signs, or Characters, 0.1.2. signifying Unites, Primes and Seconds, as is taught in the 6. Chapter of this Book; the next three for the length with the like Title; and the eighth or last towards the right hand, for the content of each several parcel, the length and breadth being multiplied together. In this work there is no use of any other Instrument, than your chain only: And beginning with any one furlong or went, express first in your book the name of the field, & then of the same your first furlong, and so the rest of the Title, as in the example, then in the first Collum writ the Tenant's name, whose land you first measure, & withal from what place you begin, & on what point of the compass you pass from thence, whereby you shallbe able afterwards (observing the same course in the beginning of every furlong) to about and bound every parcel if need require, & likewise in the same first collum, express by what tenure it is held, them consider how the whole furlong lieth if all of one length, then need you take the length but once for all, although there were twenty Tenants lands in the same; but if irregularly, as in some places shorter, and others longer; then at every second or third breadth, (or oftener if occasion require) take the length thereof, and express the same under your title of length, as for the several bredthes, you may only cross over the whole furlong about the middle thereof, taking every man's breadth, and entering the same as you pass, unless you find extraordinary difference between the bredthes at either end; if so, then measure the breadth of both ends, adding those two bredthes together, whereof take half for your breadth, & enter it in your book, or you may enter both bredthes, and take half thereof when you cast up the contents. And thus proceed from one furlong to another, till you have finished the whole field. And when you have done (or at any time after at your pleasure) by multiplying those lengths and bredthes together (which is most speedily and exactly performed, as hereafter followeth according to the order of decimal multiplication) you have your several contents to be expressed in the last Collum. And lastly, number all the several parcels in the whole book by Figures in the margin, from 1. forwards as in the example, which will serve you to good purpose, in the collecting of every man's parcels together, as shall be hereafter declared.  
And here shall you found (and in all other works of this nature) most excellent use of my decimal chain described in the sixth chapter of this book.  
But jest you should be absolutely ignorant of the manner and order of casting up of the several contents, according to the lengths and bredths so taken and observed as before, and consequently the chain with the several parts and fractions thereof may stand you in little stead, I will here briefly touch the order thereof, in two examples thus.   diagram  diagram  diagram  diagram 
And after this manner (with due observation) may you most easily and aptly apply this chain, and the several parts and fractions thereof to all the ordinary rules of Arithmetic, as Addition, Subtraction, Multiplication, and Division, working any dimension thereby, as if they were Intigers, or whole numbers.  
And thus much for a taste only of the necessary and infinite use of this Chain thus divided.  
All this time hitherto in our former mensurations have we walked in plain and even levelles, wherein the Plain Table artificially handled, (whereof many using it, are to seek) is an excellent Instrument. But suppose we are now traveling into Wales, or any other place where are mountainous and uneven grounds; then must we of necessity, either leave that Instrument behind us, or use those means, or the like expressed in the description thereof, Chap. 3. Unless we have insight in that excellent art, which many plain plain Tablemen have (wanting those means) at an instant to convert the highest mountains to plain and level grounds, pressing them down, and enforcing them on a Plain sheet of paper to lie level with the rest; which they easily perform by only thrusting out their bordering grounds from their due and natural place, where, ab initio they have remained.  
Yet let us consider of some better means for the performance thereof, which shall be hereafter taught.   

CHAP. XLI. To reduce Hipothenusall to horizontal lines by the Peractor.  
SVppose  diagram A B C D. be a hill, or mountain to be protracted and laid down in your plot amongst your other grounds: It is apparent by the Figure, that the hipothenusall lines A B. and B C. cannot be laid down exactly in a right line between the other grounds which bounder on this hill at the points A. and C. Wherhfore we are to found the true level and horizontal distance between A. and C. which is a right line, extending overthwart the ground whereon the hill standeth. Which to perform, work thus. Plant your Instrument at A. the foot of the hill precisely horizontal, by help of your plummet (wherein great care must be had) then 'cause a mark to be placed on the top of the hill at B. to be of equal height from the ground, with the Centre of your Quadrant, whereunto direct your sight, moving the Quadrant up and down, till you perfectly decern the same, where letting it stand, measure the Hipothenusall line A B. which suppose to be 6/30· than seek 30. on the Index of your quadrant, and note what line issuing from the left side C D. of the Quadrant is cut by the same division or number of 30. on the edge of the Index, which you shall find to be 18. and that number is to be observed, and kept for the horizontal line A D. to be protracted and laid down in stead of the hipothenusall A B. And if the same hill from B. continued not plain and horizontal, but descendeth again on the other side, as this from B to C. then must that hipothenusall line B C. be likewise taken, by planting your Instrument at B. and causing a mark to be placed at C. as before, and then direct your sight to the mark, and measure the hipothenusall line B C. which suppose to be 40 0· then note as before what line cutteth 40 on the Index, as 32. and take that for your horizontal line C D. which added to the former 18. maketh 50 0· for the whole line A C. which is to be protracted and laid down for the two lines A B. and B C.  
And if at any time it happeneth (as often it may) that the measured distance of the hipothenusall line exceedeth the greatest number on the Index: In such case take half, or a third part of the measured distance, and finding that number on the Index, note what line from the left side of the quadrant intersecteth therewith on the edge of the Index, and the double or triple of the number of that line, is your horizontal line sought for.   

CHAP. XLII. To reduce hipothenusall to horizontal lines by the Circumferentor, or by the Plain Table, with use of those means expressed in the description thereof.  
PLant your Instrument as before, at the foot of the hill, and let a mark be placed in the top thereof, in manner as is directed in the last, then directing your sight to that mark, move the vane up and down on the longer sight, till through the small hole thereof, and by the pings head in the shorter sight, you espy the mark, then note among the hipothenusall divisions, what is then cut by the edge of the Vane, which suppose to be 7. signifying (as is expressed in the description, Chap. 3) 107. Than measuring the hipothenusall line, which suppose to be 40. by the rule of proportion, reason thus. If 107. the hipothenusall in the Instrument, yield 100 the side of the square thereby projected, what 40. the hipothenusall measured and by increasing 100 by 40. and parting the Product by 107. your answer will be 37. and very near ⅖· the length of your horizontal line sought for. And the same course in all respects which you have here held in this Angle of ascension, the like is to be observed in all works whatsoever for Angles of descension.  
Or otherwise it may be performed thus.  
BY any Instrument take the quantity of the Angle, either ascending or descending as before taught, and then by the 11. or 12. Chapter protract the same Angle, and on the one side, from the point thereof, lay down the measured length of your hipothenusall line, and from the point where those measures end, by PROB. 5.2. let fall a perpendicular on the other side of the Angle; and the Segment of that other side, between the intersection of the perpendicular, and the point of the Angle shall be your horizontal line required.  
And thus having showed the means for the reducing of these lines; let us now consider of the application and use thereof; which for our present purpose is to find and express the true content of irregular and uneven grounds, and withal notwithstanding their irregularity, in protraction to plot and lay them down in such sort, as neither in themselves they exceed the bounds of their own Perimeter (but may truly enclose though expressed in Plano) nor displace or thrust out of order the grounds adjacent. And seeing it is impossible, and against the rules of art and nature, precisely to express and limit a solid, or body within the bounds or terms of one visual superficies, which is comprised and composed of many: It is therefore not to be expected, we should truly express the irregular capacity of mountainous and uneven grounds in a plain sheet of paper: for if the plot be exact and answerable to the rest; the superficial content must needs be wanting, or if the true content of lines and angles be expressed, the plot of necessity must be erroneous: yet notwithstanding we are now to resolve of some direct and immediate course, aswell for the obtaining of the true superficial content, as for the orderly expressing and laying down of such disordered Figures, which shall be amply and plainly taught and delivered in the next following Chapter.   

CHAP. XLIII. The best and exactest means for the dimension & protraction of mountainous and uneven grounds, and the obtaining of their true Contents by the Plain Table.   diagram 
But yet are we further to consider, that notwithstanding we have observed the difference between the hypothenusal & horizontal lines happening in the Perimeter of this figure, whereby we are able to place the same in his due situation, yet are there within the compass of this Perimeter many hills & dales whereof we have hitherto taken no notice, saving only in compassing them about: And if we should, with these lines already had, cast up the superficial content after the usual manner, we should come far short of the true quantity thereof; which to redress, work thus. Before you take your work from the table, reduce the same into the largest Trapezium you may, by drawing the lines A M. M H. H K. & EDWARD A. as in the former Diagram, then cross the Trapezium with the diagonal line A H. and thereon let fall the Perpendiculars M N. and KING L. then by direction of your Instrument or otherwise let those lines be exactly measured with the chain over Hills & Dales in a right extension▪ which by reason of the unevenness of the ground you shall found to contain much more in length then your lines already laid down; as the line A▪ H. in the plot containing but 51 0· 4 1· by measure is found to be 60 0· 2 1· also M N in the plot but 22 0· 3 1· by measure 24 0· 4 1· and L KING in the plot but 24 0 ·8 1· by measure 28 0 ·2 1· which several lengths so found by measure express in figures severally on the lines in the plot whereunto they properly belong; and by those lengths so expressed cast up (after the usual manner) the superficial content of the Trapezium, whose true quantity you shall find to be 9 Acres, 3. roods and 23. Perches. But let the same be cast up according to those lines protracted it will contain thereby but 7. Acres, 2. roods, and 10. Perches. Whereby is manifest the usual errors committed by omission of these means. Than cast up the several contents of those small triangles about the Perimeter, adding those left out, and abating those taken in by the Trapezium, and your work is finished.  
Yet were it necessary to distinguish these from the rest in your plot, shadowing them off with Hills, etc. and likewise to express therein the true content, with some note of instruction concerning the same, for that otherwise a stranger not herewith acquainted, applying your Scale to the plot, may causelessly tax you of errors committed.  
And here note that notwithstanding I have directed this work to be wrought with the plain Table; (by reason that these former courses and observations are more usually omitted in the use of that Instrument than any other) yet I doubt not but he who can perform the same by that, (with due observation of what hath been formerly delivered) will be able to effect the same by any other Instrument. For your observations being had and taken in the field, and then laid down accordingly; you shall thereby afterwards measure your diagonal and perpendicular lines in the field; as before is directed.   

CHAP. XLIIII. To divide a Common of pasture, or a common field into any parts required.  
SVppose A B C D E F G H be a stinted pasture, or a common field in the use and occupation of three men, as A. B and C. and let it be required to divide the same between them proportionally, either according to their several stints (being a stinted pasture) or according to their several quantities, being an arable field in common: First measure and plot the whole quantity, and as is taught in the second part of the second Book, find the superficial content thereof which suppose to be 4268. Perches, or by reduction 26. acres 2. roods 28. perches, then seek the content of the high ways A E and H E passing through the same, which let be 341. perches to be deducted out of the whole quantity: and there resteth 3927. Perches, which to divide according to their several portions, reason thus with the rule of proportion. If all the whole number of beast-gates of A. B and C. together yield the whole quantity 3927. what the number of those belonging to A. and the answer will be his part: & working thus severally for their three several parts suppose them thus, to A. 924. Perches, to B. 1798. and to C. 1205. Than as  diagram is taught by the ten last Problems of the second Book, divide your plot accordingly, by the lines I M. and KING L. laying every man's part in such place as they shall mutually agreed: which being performed on your plot, you are then to effect the like on the ground, wherein you are only to lay out the lines I M. and KING L. in their due places, which is thus to be performed. Find out by your Scale the distance on the plot between the next angle, and the beginning of your first line as G M 3 0 ·6 1· then found also the angle G. in the field, and with your chain measure out that length from G. on the line G F. to end in M. where place your instrument, and finding by your Protractor on what degree the line M I passeth on the plot, on the same degree of your Instrument place the Index: and looking through your sights, cause dooles and marks to be made and placed in a rig●t line from M. to I and for the more exactness, and your better satisfactio, note in your plot also what angle is next unto the end I of the line M I as A. from whence take your distance by the Scale on your plot to I 18 0· 8 1· and finding the Angle A. in the field, from thence measure by your chain that length 18 0· 8 1· on the line A B. which if you find to end in I is an infailable assurance you have truly wrought; if not, reform it by reducing the end I of your line M I to that place. And by the like means, you are to lay out the line KING L. and more, if more required. And after this sort shall you lay out and divide any common or waste whatsoever; into what parts soever the same is required, to be divided, and laid out.   

CHAP. XLV. To know the hour of the day by the Peractor or Circumferentor with the Sun.  
PLace the Index of your Peractor on the Meridian line of the Planisphere, and direct the North part of the Instrument towards the Sun, turning it about on the Staff till the shadow of the third, or shining of the Sun fall directly through the sight, on the fiducial edge of the Index or Meridian line: then observe in the Card the next concentric circle above the present month, and note what houre-line (which are those spherical lines issuing from the centre) intersecteth that concentric circle, under the North end of the needle; which point of intersection, lying directly under that end of the needle, showeth directly the hour of the day, expressed at the upper end of that houre-line, or being short or over, it showeth how much the time of the day is short or over that hour.  
And the like is performed by the Circumferentor with turning the North end of the Instrument towards the Sun, till through the sight, the Sun shineth on the Meridian line; and making your observations as before.   

CHAP. XLVI. The ordering of a plot after the protraction thereof.  
THe order and course of protraction and laying down of plots according to the diversity of observations taken, is already in this Book sufficiently declared; and likewise the means of obtaining the true superficial content of all sorts of Figures by the Second part of the Second book; (the conclusion of which part, I would advise you to make use of for your further ease therein.) Now it resteth (the plot being thus protracted and cast up) next to consider whether it were fitting, to draw it fair of the same Scale and scantling wherein it is, or first to have it reduced into a lesser form. If the business be not so large, but that with conveniency it may be drawn in the first Scale wherewith it was plotted and cast up, then provide you a piece of ordinary new linen cloth of a reasonable fineness, and thereon passed you clean paper, according to the order of Maps (but those for the most part are too course) of the largeness of your plot; which being well washed with Allome water, and dried, fasten your rough plot, slenderly thereon with mouth glue at each corner thereof; and with a bodkin or pin of brass (which I hold the best for these purposes) trace out all your lines with a reasonable hand; for a light hand with discretion will make sufficient impression on your new plot, whereby you may well discern to draw your lines either with Pen or pencil as you please; and then express your houses, buildings, woods, rivers, waters, ways, and all other remarkable things in their due proportion perspectively; not placing your houses and trees every way, whereby here the tops and there the bottoms shall seem standing upwards, as is usually accustomed; and then garnish your plot about with some neat border, and within with Copartments for your Scale and Title; and in some convenient place thereof describe a Card, showing the situation according to the points of the Compass; and then let it be neatly coloured about the several lines, lightly washed off, and not daubed all over, as some painting Surveyors use. But above the rest, forbear much writing in your plots either of names or quantities (being absurd and gross) but only numbers of reference to your engrossed book concerning the same, if you make any; if not, than it may serve, as Chalk on a trencher.  
But if it be required to have your first plot reduced into a lesser or greater form; use the help of the next Chapter.   

CHAP. XLVII. To reduce any plot from a greater to a lesser quantity, and the contrary.  
IN the third part of my Second book I have at large declared the manner and means as well of reducing and translating of all superficial Figures from one form into another, retaining the same quantity; as one quantity into another, retaining the same form. But those being of particular figures, are not so meet or necessary for the Reduction or translation of a plot consisting of many (being often in use.) Wherhfore I will here deliver you a very speedy and exact means for the performance thereof. According to the proportion whereunto you would reduce your first plot, lay papers together with mouth-glue; as if you would reduce it into a fourth part, make your new plot to be in quantity a fourth part thereof. Than place your new made plot on your rough plot in such sort, and with such discretion, as the middle of the one may be about the middle of the other, whereby all the work of the one may be reduced the better into the other, and there fasten the one to the other slenderly at the corners with mouth-glue, in such sort, as when occasion serveth, any one of those glued places may be easily taken asunder, and the paper folded in; then take your reducing ruler mentioned in Chap. 9 and fasten the same with a needle or blue pin stricken into the table about the middle of the plots through the centre hole of the same rule, in such manner as both the plots together may be turned about at pleasure upon the table; wherein you must take great care of renting or tearing out your centre point in the plots; which to prevent, would require to be strengthened with a small piece of a Card or past-boord, to be glued thereon, underneath the first plot. And being thus prepared, you shall find good part of the work in your first plot, to lie without the utter edges of your clean paper, which let be first reduced; wherein (having resolved into what proportion you will make your Reduction, as into ¼· of the first) work thus; bring the edge of your Ruler to any Angle in your first plot, and note what number of division on the edge of your Ruler is there cut, which admit 40. then take half thereof which is 20. and against that division by the edge of your Ruler make a prick on your clean paper, then remove your Ruler to the next Angle in your first plot, and note the division there cut, whereof take likewise the half, and against that half by the edge of the Ruler make another prick; and between these two pricks draw a line, which shall represent the line between those two Angles of your first plot; and so proceed from Angle to Angle, and from Close to Close, till you have reduced all the work on your old plot, lying without the edges of the new, or so much as lies without the same towards any one side or corner thereof; and then ungluing one of your corners or sides, fold in backwards towards your old plot, that part of your new plot as is wrought; so shall you come to work that which formerly lay under the same; and thus by folding in, and working one side after another to the centre; whilst the other sides are fastened together with the old plot, you shall speedily reduce the whole plot into your desired proportion: For in taking the half of every line in your first plot your new by THEOR. 49.1. shall be ¼ of the old.  
And this kind of Reduction hold I the speediest and exactest of all other: Which by small practice, you shall much better find and understand, then with many words of Relation.  
And after this manner may you make your Reduction, into any other proportion, or reduce your plot from a lesser to a greater form in any proportion required; by increasing your second number, proportionally, as in this work you decreased the same.   

CHAP. XLVIII. To reduce any number of Perches given into Acres, and the contrary.  
SVppose 5496. Perches were given to be reduced into Acres. First after the usual manner (considering that a Statute Acre containeth 160. square Perches) divide the given number by 160. the quotus will be 34. and the remainder 56. which remainder divided by 40. (the number of Perches in a Rood) quoteth 1. and the remainder 16. So is the whole Reduction 34. Acres, 1. Rood, 16. Perches.  
Or more briefly thus.   diagram 
And if you would reduce these Acres, roods, and Perches, into their least Denomination, as into Perches. First multiply your number of Acres 34. by 160. the product is 5440. then multiply the number of roods 1. by 40. produceth 40. which together with the number of Perches 16. added to 5440. makes 5496. the first number. And the like of all others.  
Here might I now much enlarge this work by showing many other necessary conclusions fit for a Surveyor to know, as the mensuration of Timber, Board, Glass, Pavements, and the like; also the several ways and means of plotting of countries and large continents; of carrying of mines and trenches under ground; of waterworks, and the conveying of the water from any Fountain to appointed places of whatsoever possible distance; of the taking and making of the forms and models of Plotformes, Forts, Castles, Houses, and the like. But seeing that not only these, but infinite other Conclusions Geometrical, may be easily performed and wrought by the former rules and instructions well understood and practised; and for that they are without the scope and limits of Survey, whereunto I chief bend the subject of this Book; I will leave them to your own endeavours, and diligent practice. And having thus performed at large the Mathematical part of Survey in general; we will next consider of the Legal; and in the mean space here conclude this Third Book.   
The end of the third Book.   

THE LEGAL PART OF SURVEY.  
The fourth Book.  
THE ARGUMENT THEREOF.  
I Would not be mistaken, or have it understood; that I here undertake (as a Lawyer) to instruct or teach the rules or Jnstitutions of the Law (being out of mine element) but as a Surveyor, briefly and truly to express and deliver herein what I hold fit and meet for a Surveyor to know and understand. As first what a Manor is, and the several parts and members thereof, with the appendants thereunto: Next, the perquisites, casualties and profits of Court, and their several natures: Than the diversity of estates, whereby any Lands or Tenements may be holden, occupied or enjoyed; and the several tenors depending on those estates; with the Rents and services incident and belonging to those tenors: Also what reprises, payments, and deductions may be issuing out of a Manor, and the considerations thereof to be had: Likewise what courses are to be observed and taken, before the beginning of a Survey: The order and manner of keeping those Courts: The entry of the Tenant's evidence and estates; and the orderly manner of engrossing the same: with other brief and necessary Rules, and Observations tending to those purposes.  

CHAP. I Of a Manor with his several parts, and of the name and nature thereof: how made and maintained, and how discontinued and destroyed.  
AS in my works concerning the Mathematical part of Survey, comprised in the three former Books, I premise the definitions, principles and grounds thereof: so in this Legal part I hold it answerable to order, and a good decorum, (before we abruptly enter on the Survey of a Manor) first to consider what a Manor is, and the several parts thereof (jest being questioned of our present employment, we discover our own weakness in undertaking we know not what) and then to inform ourselves of the several natures, qualities and conditions, of the estates, tenors, and services of land; and of the several profits, rents, and commodities thereunto incident and appertaining; with such other meet and necessary observations, as are most fitting for a Survey or to know and understand, before he assume and take upon him the name, or at lest the office or function to a Surveyor belonging. Of all which in order; and first of a Manor what it is, and of the parts thereunto belonging.  

Manor whence derived.  Concerning the Derivation or Etymology of the word, I will not stand, whether it be of Maneo manner to remain in a settled place; or of Mano manare to proceed or spread abroad out of the bounty of those Prince's liberality who in the beginning bestowed them; or of Manuarius gotten by labour of the hand; which I hold the best; because there is more skill in getting then keeping; and with Manerium I will not meddle, seeing (as I take it) Manors were created before the word was made: But from whencesoever derived, A Manor is now that which hath thereunto belonging, 
What a Manor is. Parts of a Manor.  messages, Lands, Tenements, Rents, Services & Hereditaments; whereof part are demeans, being those which anciently and time out of mind, the Lord himself ever used, occupied and manured with the Manor house; the residue are Freeholds, Farms, and customary or copyhold tenements; and these have commonly divers services besides their rents properly belonging thereunto, whereof I will hereafter speak.  

Parts of a Manor.  There is moreover belonging to a Manor a Court Baron, and to divers a Court Leer; which is of more worth and efficacy, and is always granted from the King, or held by prescription. To these Courts, and consequently to the Manor is there usually belonging; Fines, issues, amerciaments, heriots, waives, strays, excheates, reliefs and other perquisites and profits of Court; whereof likewise I will further speak.  

Appendants to a Manor.  Besides, those there are often appendent and belonging to a Manor (which are not of necessity to be taken as the proper parts thereof) Wards, Marriages, advowsons, patronages, free-gifts or presentations of parsonages, Vicarages, Chapels, Prebends, etc. also Commons of Pasture, Moors, Marshes, free Warrens, Customs, Liberties, Franchises, and Privileges; likewise yearly Rents, suits of Court, tenths and services issuing and reprised out of other Manors. And of these, a Manor is neither made by them; nor destroyed or marred for want of them; wherefore they are termed rather appendants than parts of a Manor.  

Not present means to make a Manor.  Neither do those parts formerly named, properly of themselves make a Manor: For should any man at this day allot and appoint out any competent quantity of Land, and divide the same into demeans and tenement Lands; in feoffing Tenants in Fee of some part, and granting others by copy of Court-Roll, and perfecting the rest which before is said to belong unto a Manor; yet all this will not make a Manor; for that it is the office of time by long continuance to make and created the same. 
Continuance of time may perfect a Manor.   

How a manor may be destroyea and dismembered  But a Manor at this day may be dismembered, and utterly destroyed both in name and nature, by escheating the Freeholds, and Copieholds; for if of Freeholds or Copieholds there are not two at the lest, then are there no Suitors, and if no Suitors, no Court, and consequently no Manor, and then may it be termed a Signiory, which can keep no Court Baron at all.  

How one Manor may be divided, and made divers Manors.  Also it is to be understood, that one Manor may be divided into divers Manors; whereof we have divers examples at this day; as where a Manor descendeth to coheirs, and they make division and partition thereof; allotting to every of them demeans and services; whereby every of them hath a several Manor, and may keep several Courts Baron thereon, as if anciently entire.  

How divers Manors may be reduced into one Manor.  And in like manner two distinct and several Manors may be conjoined and made one entire Manor, if formerly the one held of the other; and that Manor so holding of the other do escheat; but otherwise not. And thus much concerning the name, nature, and parts of a Manor.   

CHAP. II Of Perquisites Casualties and profits of Court, and their several natures.  
IN the former Chapter I declare, that (among other things) there is belonging to a Manor a Court Baron at the lest, and to some a Leete or Law-day, commonly called the view of Frank pledge: Now herein will I show what perquisites, casualties, and profits are incident and belonging to those Courts; wherein I would have it understood, that it is not of necessity, that all these hereafter mentioned, must be in every Manor, but that they may be in any. And first of Fines.  

Fines of Land.  

Fines of post mortem.  FInes of lands are of divers kinds; As first, if a man holding to him and his heirs, or otherwise certain Lands and Tenements, by Copy of Court Roll, according to the custom of the Manor dye, his heir upon his admittance by the Lord, shall pay a Fine for such his admittance: And these Fines are of two sorts, either certain, or arbitrable; if certain, as one or two years rent, or the like; there is then no other question to be made, but the Lord by his Steward is to admit him, and he to pay such certain Fine accordingly; if uncertain or arbitrable, then is the Tenant to undergo what Fine the Lord shall in reason impose or require; and these are called Fines of post mortem.  

Fines of alienation.  Also a Tenant by Copy of Court Roll, hath not power to Alien or cell his estate or interest unto any other man, without he surrender the same into the Lords hands to the use of him unto whom he shall so cell the same; for which Alienation the Lord is also to receive a Fine, which in some Manors are likewise certain, and in others arbitrable, but being arbitrable, they are usually rated at a lower and more reasonable value then those after death, and these are commonly called Fines of Alienation.  
Likewise if a customary Tenant let or set his lands unto another for any term of years, not warranted by the custom, he is first to obtain licence of his Lord in this behalf; and is to pay a Fine in respect thereof.  
And moreover if the Lord of a Manor grant a Lease of any lands unto a Tenant for any number of years or for life or lives; and besides his annual Rent, make composition for a Fine to be in hand paid; this is also a Fine of Lands.  
Also in some places, the Custom is, that if a customary Tenant alien and make surrender of his whole estate he shall pay and yield unto the Lord the best beast he hath, 
Asarewell paid.  or a certain piece of money, in name of a Farewell. And in some places as well freeholders, as customary Tenants on every alienation shall pay a certain sum of money for a Fine in name of Offare, 
Offare & onfare  and onsare, and all these and the like are Fines of Land.   

Americiaments.  
A Merciaments are also perquisites of Court, whereof there are divers sorts; which in general are such Fines, penalties, and amerciaments as by the homage or afferers of the Court Leete, or otherwise are imposed on such Tenants as are found offenders within the Manor; As if the Free suitors, Copie-holders', or other Tenants, make default or be absent from the Lords Court, they are therefore amerced. Wherein is to be noted, that many Free-sutors make composition, 
Common Finet quid.  and are at their Fine certain in respect of their service of suit of Court; and these are called common Fines.   

Heriots.  
AN Heriot is properly the best beast which any heriotable Tenantis possessed of at the time of his death, whether it be Horse, Ox, Cow or the like; for which in many places a sum of money is paid by ancient composition, and in some places for default of live cattle (or the best beast not being to the Lords mind) it is in his choice to take the best of any other goods, implement, or commodity the Tenant hath at the time of his decease.  
Of these Heriots there are two sorts, Service and Custom: 
Heriot Service.  Heriot service is commonly mentioned and expressed in the Tenants grant: and therefore the Land answerable for satisfaction thereof; 
Heriot Custom.  and Heriot custom is that which time out of mind hath been ever paid upon and after the death of any Tenant dying seized of any such heriotable Lands: And these Heriots of either kind, are by the homage of every Court to be presented as they fall due; and seized by the Lords Bay life accordingly.  
And it is to be understood, 
One Tenant may be chargeable with divers Heriots.  that if a Tenant dieth seized of divers tenements or lands, which have been anciently charged with divers Heriots; the Lord at the time of the death of such Tenant shall receive so many several Heriots, as those lands at any time then-tofore were anciently charged or chargeable to yield.  

Heriotable lands divided, are severally chargeable.  And moreover, if any heriotable Tenement shall be severed & divided into divers parts, amongst several Tenants; the Lord shall have of every such Tenant particularly a several Heriot, for and in respect of those several heriotable parcels; which the Lord may seize and take, 
Lib. Ass. 27.24.  wheresoever he shall find the best for his best advantage.   

Reliefs.  
Reliefs are like wise accounted amongst perquisites of Court: but seeing it is a special service tied to the tenure of Lands; I will here omit to speak thereof; referring you for your satisfaction therein to the Title of Warde, Marriage and Relief, in the 4. Chap. following.   

escheats.  

Escbcates what they are.  THese are likewise perquisites of Court; and are such as if a Freeholder or Copieholder of inheritance, commit any manner of Felony, and be thereof attainted; his Lands are escheated and forfeited to the Lord of the Manor of whom they are holden; but the Lord shall not immediately enter thereinto; for the King is first to have annum diem & vastum; after which time expired, it than remaineth to the Lord and his heirs for ever.  
Also if any such Tenant dye without heir general or special, all his Lands and Tenements shall fall unto the Lord by escheat, to remain unto him and to his heirs for ever.   

Forfeitures.  

Forseituret of divers kinds.  FOrfeitures are of divers kinds; As if a Copieholder or customary Tenant deny, or will fully refuse the payment, doing or performance of his Rents, Services and Customs; or if he fell or cut Timber on his Copie-hold Lands contrary to the custom; or do or commit waste in the houses or otherwise; or if he grant or cell his Copie-hold estate by deed; or alien or let the same without licence of the Lord, beyond limitation of the Custom; In all or any of these, the customary Tenant shall forfeit his Copie-hold estate into the Lords hands: Which offences are to be found and presented by the homage at the next Court; and thereupon seizure made accordingly.  
Also Tenant for term of years, life, or lives, may forfeit his estate for making a larger estate of Freehold than he hath, or for not performance of such provisoes and conditions, as are expressed in his Lease or Deed, if any be.   

Waives.  
IF any man feloniously steal or take any goods or chattels of what nature or kind so ever; and by earnest prosecution he is enforced in flying to leave the same behind him; these goods are called Waives or waived goods; and in what place soever they are so left and waived, they shall be taken and seized for the use of the Lord of that Manor, if by his grant, charter or prescription, they belong unto him (or otherwise they are the Kings) and being so seized by the Bailiff or other Officer, they are to be presented and found at the next Court by the Homage there. 
Fresh suit. Waived goods restored.  But if the right owner make fresh suit after the thief, and attaint him at his suit for stealing thereof, he shall have his goods again, although they be waived. And the like in all respects is if any goods be taken by an officer, from any whom he suspecteth to have stolen the same, though there be no pursuit made or prosecuted.   

Estraies.  
EStraies are when a Horse, an Ox Sheep, or any other cattle of what kind so ever come into a Lordship or Manor, no man knowing from whence, nor the owner thereof; such are to be seized to the King's use, or to the use of the Lord of the Manor who hath the same by grant or prescription; and if the owner come and make claim within a year and a day, than he shall have the same again, paying for the charge thereof; or else after such time expired, the property thereof shall be to the King; or the Lord of the Manor having the same by grant or prescription; So that Proclamation be thereof made in the next markets and the Parish Church, according to the Laws in that behalf.   

Pleas and Process of Court.  
THese are where the Lord of a Manor in the Court Leete, or Court Baron, holdeth plea of his Tenants for actions of debt, of trespass or other causes, not exceeding the value of xlˢ. debt and damage.  
And under this title of perquisites is comprised all other casualties whatsoever, which may happen to grow or arise within any Manor; as profits arising by mines of Copper, Tin, Lead, Cole, and quarries of Stone; also by sale of Woods, Turbarie, and Pannage; likewise profits of Fairs and Markets, Fishing, 
Casualties may become certain.  Fowling and the like. All or any of which may become certain; by being let and disposed of for yearly rents.    

CHAP. III Of the diversity of estates, and their several natures.  
Having already showed in the two former Chapters what a Mannoris, with the several parts thereof, and the appendants thereunto: I hold it fitting here now to consider of Estates; as how and by what means a man may be estated either in a Manor or any other Lands or Tenements: wherein it is to be understood that all estates in general consist of two principal kinds as Freeholds and Chattels: which more particularly are sub divided into divers other parts or branches; as first Feesimple and Feetaile, which are termed Freeholds of inheritance: also estates After possibility of issue extinct; By Courtesy; In dower; and for term of life: Which four last mentioned, are called Freeholds, but not of Inheritance: Likewise estates by Copy of Court Roll, being claimed & held by custom, and are divided into the like parts, as Freeholds at the common Law: and lastly estates For term of years, and at will, which two last are Chattels. Of all which briefly in order as followeth.  

1. fee-simple.  
FEe simple of all other estates is the most large, ample, and absolute that we have in this Kingdom, 
defined.  or that can by our Laws be invested or made; and is that which is granted to any man and his heirs for ever, without any further or other limitation of use or uses; and therefore if such Tenant hath issue of his body, the land descendeth to him, if not, to the next of kin within the degrees of limitation hereafter specified.  
But if a man purchase in fee-simple to him and his assigns for ever, omitting this word heirs; here hath he but an estate for term of life; for heirs is the word which carrieth the inheritance.  
Yet it is otherwise if lands be so devised by Will; for the Law intendeth that learned counsel cannot always be present in such cases; and therefore is such devise construed for the best, according to the Testators meaning and intention, and not to the strict letter of the Will.  
Also if Lands be granted to any man with a woman in Franke-marriage, this word implieth an estate of inheritance without mention or addition of the word Heirs; Or to a man and to his blood the like.  
And here is to be considered, who are those which are said to be within the degrees of limitation before spoken of; that is, who are understood to be a man's heirs by the common Law. Suppose A. B. dieth seized of a state of inheritance without issue of his body: Neither his brother, or sifter of the half blood, 
The half blood. Abastard no beire. Lineal and collater all descent.  nor their issue shall be his heir; nor his bastard; nor his father, mother, grandfather, or grandmother; for inheritance may lineally or collaterally descend, but by no means lineally ascend by our Laws; but the brother or sister of the father of A.B. (which is called a collateral descent) shall be his heir; and then they dying seized without issue, the father of A.B. shall have the land as heir to his uncle or aunt, but not as heir unto him.  
Likewise it is to be understood, that by the laws of this Realm the eldest son is wholly to inherit; and he dying without issue, the second son, and so of the rest; 
Coparceners.  and if no sons but daughters, they shall jointly inherit as coparceners; but if no issue neither son nor daughter, then shall the eldest brother be heir, for want of such all the sisters; and in default of them, the uncle by the father's side, if the Land came by the father, or be of the purchase of him so deceased: But if there be no heir of the father's side, the purchased lands go to those of the mother's side; But if none such, than all those lands shall Escheat to the Lord of whom they are holden. 
Escheat.    

2. Fee Tail.  

Fee tail of two kinds. General tail.  THis estate of Fee-tail is divided into two kinds or sorts General and special. The first, being Fee-tail general, is when Lands or Tenements are granted unto any man, and the heirs of his body begotten without limitation, or express mention made by what woman; wherefore if such Tenant marrieth divers wives, and hath issue by them severally; they shall all be capable of the Inheritance of those Lands. But if it be mentioned and expressed in the grant by what woman, his Heirs shall proceed or be begotten, as if the gift be made to A. B, and to the heirs of his body lawfully begotten on Chis' wife, this is an especial Entail, for any of his issue begotten by another woman, 
Special tail.  shall not inherit by force or means of this grant or tail. And the like in all respects if Lands be granted to a woman in the like kind.  
Also if Lands be granted unto AB. and C. his wife, and to the heirs of their two bodies lawfully begotten; here are the man and his wife joynt-purchasers, and this is also a special tail both in him and her.  
Likewise if any man grant Lands or Tenements to another man with his Daughter in Franke-marriage, this is also a special tail; 
Frank marriage.  and both the man and woman shall be here Tenants in the special tail, for the word Franke-marriage implieth as much.  
Also if Lands be granted to a man and the heirs Males of his body; 
Descent by beires males.  this is an estate tail, and here the Female shall not inherit.   

3. After possibility of issue extinct.  
IF Lands or Tenements be granted to a man and to his wife, 
Free-bolds.  and to the Heirs of their two bodies lawfully begotten, and either of them dye without such issue between them; then is he or she surviving Tenant in tail of those Lands, but are without all hope and past possibility of having such Heir to inherit those Lands as was limited in the grant; & therefore is he or she so surviving and overliving the other, called Tenant in tail after possibility of issue extinct: and from and after the death of him or her so surviving; the estate tail so made and granted unto them, shall be utterly void, extinct and dead, as if the same were never granted; and the estate of inheritance of and to those lands, shall revert and turn unto the first Donor thereof and his heirs.   

4. By courtesy.  
IF a man marry a wife, being an Inheritrix, and hath issue by her, and she die; by our laws he shall hold, occupy, and enjoy such lands as his wife died seized of either in fee-simple, or Fee tail, during his natural life; and he is called Tenant by courtesy of England, because no other Nation admitteth the like estate. Wherein the Law requireth that such issue be vital, & brought forth into the world alive, although it immediately die, and also it is requisite that the husband be in actual and real possession of those lands, and seized of them in the right of his wife, at the time of her death, or otherwise he shall not be admitted Tenant by courtesy thereunto.  
But if any such Tenant by courtesy commit or suffer any stripe or waste, he is punishable in that behalf, by action of waste.  
Also it is to be understood, that no man can be Tenant by courtesy of a reversion: for if a woman solye seized in Fee, granteth a Lease to A. B. for term of his life, and afterwards marry and hath issue, and then dye; the Tenant or Leasee for life surviving; her husband in this case shall not be Tenant by courtesy.   

5. In Dower.  
BY the common Law of this Kingdom, if a man marry a wife, and at any time during the time of coverture he be directly and lawfully seized, either by purchase or descent of any lands or tenements, either in feesimple or fee-tail, and being so seized die, his wife shall be endowed of a full third part of all those lands and tenements, during her life; 
Dower at the common Law. Dower by custom.  and being thus endowed, she is called Tenant in Dower, and this is by the common Law.  
Besides this, there is Dower by custom, for in some places the woman shall have a moiety, and in some places more, and in others less during her life, of all the lands her husband was seized of at any time during the coverture, according to the custom of the place.  
But if the wife be not above the age of nine years, at the time of her husband's death; the common Law will not permit her endowment.  

A Woman shall have no dower.  And for divers causes a woman may be defeated of her dower; as if she or her husband, commit treason, murder, or felony, and be thereof attainted (yea, though they have their pardon:) also if she forsake her husband and live incontinently; and be not again reconciled without constraint of law; or if she detain and withhold the deeds and evidences from the heir, of those lands, whereof she claimeth dower, and the like.  
And some things there are, whereof a woman is not capable of endowment; as of Commons, Annuities, Estovers sans number, Homages, Services, and the like.  
There are beside these other kinds of dowers, as one called dowment, Ex assensu patris: another termed dowment ad ostium Ecclesiae: and a third, de la plus bell part: As appeareth at large by our common Laws; whereunto I refer you.   

6. For term of Life.  
A Tenant for term of life, is he, who holdeth lands or tenements, either for term of his own life, or for term of another man's life; but for distinctions sake, he who holdeth for his own life, is termed barely Tenant for life, and he that holdeth for another's life, is called tenant for term d'autervie, that is of another's life.  
And if either of these kind of tenants commit or suffer waste, 
Waste.  the leasor or he in reversion, shall bring and maintain against him an action of waste, and thereby recover triple damages.   

7. By Copy of Court roll.  
THese tenants are such, as in divers Manors hold lands and tenements to them and to their heirs, some in the nature of feesimple, others in fee-tail, or for term of life or lives, at the will of the Lord, according to the custom of the Manor; and in some Manors they hold by copy for term of years: And all these have no other evidence to show concerning the tenure of their lands, save only the copies of the rolls of their Lord's Court; and therefore are they called tenants by Copy of Court roll.  

Alienation of estate.  And if any of these tenants alien or cell his lands or estate by deed, he shall absolutely forfeit the same into the Lords hands: wherefore if he will alien his copyhold estate, he must come into the Lord's Court and there surrender the same into the Lords hands, to the use of him unto whom he alienateth the same.  
But in divers Manors the surrender may be made out of Court, 
Surrender out of Court.  unto any copyhold tenant, in presence of two of the homage (to the use as aforesaid) who are to present the same unto the Steward at the next Court, and admittance made accordingly.  
And these tenants can neither sue or be sued, in any of the King's Courts, by writ or otherwise, for these lands so holden. But they must implead and sue for the same, by way of plaint, in the Lords Court.  
And some are of opinion that these tenants are but in the nature of tenants at will of the Lord; who at his pleasure may displace them, 
Tenants at william.  and they without remedy, but by the Lord's favour. Yet others are of a contrary mind, who maintain, that if any such customary tenant (but those for years, their term being expired; 
Remedy for copyholder pat off.  paying and doing their services) shall without just cause be ejected and displaced by the Lord; he may bring and maintain his action of trespass against him, at the common Law.  
And if any of these cut timber, growing on his lands, 
Waste.  without licence of the Lord (but only for repair of his tenement) it is a waste, and an absolute forfeiture.  
And in most Manors if any such tenant shall farm or let out his land, for any longer time than a year, without the Lords licence, 
Forfeiture.  it is likewise a forfeiture unto the Lord But of these and many other the like, we are to be guided according to the custom of the Manor, where such tenants are.  
And generally these tenants, 
Base tenure.  for that they have no freehold at the common Law, but by custom, are termed tenants of base tenure.  
And thus much concerning freeholds and estates of inheritance; and next of Chattels.   

8. Term of years.  
A Tenant for term of years, is he, unto whom an estate is granted of lands, for any number of years agreed upon, between the Lord and Tenant; which term is always expressed in the lease so granted.  
On which lease there is usually reserved some annual rent, 
Rend reserved.  payable either half yearly or quarterly, according to their contract. For the recovery and obtaining of which rent, if it happen to be ariere and un-paid, 
Distress or action of debt.  the leasor is at his choice; whether he will enter and distrain, or bring his action at the common Law for the same.  
And in these leases for term of years, whether by writing or otherwise, 
No livery of seisin.  there need no livery of seison, but the tenant may immediately enter by virtue of his lease without further ceremony. But in leases for term of life or lives, it is otherwise.  
Also if this tenant commit or suffer waste; 
Waste.  the leasor may bring his action of waste against him; wherein he shall recover locum vastatum, and his triple damages.  
And if this tenant shall grant unto any other man a greater or larger estate in those lands he holdeth, than he hath therein himself; whereby he conveyeth the feesimple to himself, he shall forfeit his lease and the state, and term of years therein granted.   

9 Tenant at william.  
A Tenant at will, is he, unto whom lands or tenements are granted, to hold at the will of the Lord or leasor, by whom they are granted. And this tenant may be displaced or put out at any time, 
Displaced at pleasure.  without further notice, at the Lords pleasure; yea, although he hath tilled and sown his grounds. 
Liberty to take his corn and goods.  Yet in this case the law alloweth him free liberty of ingress, egress, and regress, aswell to take, cut, and carry away his corn when it is ripe; as to take and carry away his goods, and householdstuff, within convenient time; without punishment of committing trespass, or otherwise; for that be knew not his Lord's intention or time of entrance. But with tenant for term of years it is otherwise.  

Like remedy for rent.  And the Lord or leasor here hath the like remedy against this his tenant at will, for his rent, if it be behind and un-paid, as he hath against the tenant for term of years, last before mentioned.  
And it is to be noted, that this tenant at will is not by the law charged or chargeable with reparations, 
No action of Waste.  as is the tenant for years; and therefore no action of waste lieth against him; unless he wilfully committeth waste, by pulling down the buildings, or felling of timber, etc. In which case it is held, 
Trespass.  that the leasor may bring his action of trespass, and recover his loss sustained.  
And thus much briefly concerning estates in land; whereof the two later kinds are termed Chattels real; and all movable goods are called Chattels personal; as appeareth by this breviate following.  
An ANALYSIS or brief resolution of all estates in general.  
They consist of 

Freeholds, 

At the common Law. 

fee-simple, 

By purchase.  
By descent.    
Fee-tail, 

General.  
Special.    
Freehold, 

After possibility of issue extinct.  
Courtesy of Eng.  
Dower.  
Term of life.  
Term d'auter vie.      
By Custom. 

These are divided as freeholds at the common Law.      
Chattels, 

Real. 

Term of years.  
Wardship of lands.  
Tenure at william.    
Personal. 

Corn, cattle, Money, Plate,  
Householdstuff, etc.          

CHAP. FOUR Of the diversity of Tenors, and their several natures, with the services belonging.  
Having informed ourselves, as before, of the diversity of estates (for all lands whatsoever, consist of some of those, formerly mentioned) let us here, next, consider of the several tenors and services thereunto particularly belonging; which are main and principal observations to be had and used in survey of a Manor, and most meet and necessary for a Surveyor to know and understand. Whereof in order as followeth.  

1. Knight's service.  
THis tenure of Knight's service includeth Homage and Fealty, and commonly Escuage; and whoso holdeth any lands or tenements by this service, is bound by the Laws of this our Realm, to do unto his Lord Homage and Fealty; 
Homage.  being a service of the greatest humility and respect, that can be performed by a tenant unto his Lord: and for the most part, he is to pay Escuage, called Scutagium, that is, service of shield; which is to be assessed by authority of Parliament, as shall be hereafter declared.  

How the tenant shall do homage  When the tenant shall do homage to his Lord, the Lord shall sit, and the tenant kneeling down before him, on both his knees, and holding both his hands between his Lords hands, shall say thus: I become your man, from henceforth, of life and member, and of earthly honour, and to you will be faithful and true, and faith to you shall bear for the lands I hold of you (saving my faith which I own and bear unto our Sovereign Lord the King) and then the Lord so sitting, shall kiss him.  

How fealty is to be done.  Fealty is as much to say, as Fidelitas, or fidelity. In doing whereof, the tenant shall lay his right hand on a book, and say thus: I will be unto you my Lord, faithful and true, and faith to you shall bear, for the lands and tenements which I claim to hold of you, and duly shall do and perform unto you the customs and services which I aught to do, at the terms assigned, as help me God; and then he shall kiss the book. And it is to be noted, that Homage must be done unto the Lord himself personally; but Fealty may be made to the Steward of the Court, or to the Bailiff thereof. Also, tenant for term of life shall do Fealty, but not Homage.  

What duty belongeth to Escuage.  As concerning Escuage: he that holdeth his lands by a whole fee of Knight's service; when the King goes in person to the wars, he is bound to be with him by the space of forty days, sufficiently appointed for the wars: and he that holdeth by the moiety of the fee of Knight's service, is bound by his tenure to be with the King in such sort as before, by the space of twenty days; and so proportionally, according to the quantity and rate of his tenure.  
And it is to be understood, that after the King's return from the wars, a Parliament is usually called; 
Parliament.  by authority whereof, it is assessed, what and how much, in money, every man that holdeth by a whole fee of Knight's service (who was neither personally, nor any man for him, with the King) shall pay unto the Lord of whom he holdeth his land, by Escuage: and according to such assessment, every tenant shall pay to his immediate Lord, 
What Escuage to be paid.  after the rate and proportion of his tenure: and this money thus assessed, is called Scutage, or Escuage; for which, the Lord unto whom the same is due, may distrain for nonpayment thereof.  
Yet some tenants by custom, 
Custom.  are not otherwise bound, but to pay a moiety or third part of what shall be assessed, as aforesaid.  
And in some places the custom is, that whatsoever be assessed by Parliament, yet their Escuage is certain, and they pay neither more nor less, but such a sum of money, 
Escuage certain  as five shillings, or the like; and this is called Escuage certain: and this tenure is Socage tenure, and not Knights service; the Escuage whereof is always uncertain, and so called. And this Escuage uncertain (belonging always to Knight's service) draweth unto it Ward, Marriage, and Relief, as hereafter appeareth.   

2. Ward, Marriage, and Relief.  
AS formerly appeareth, Knight's service (the tenure last before mentioned) draweth unto it Ward, Marriage, and Relief; and therefore I hold it fitting here next to treat thereof. Wherhfore, first it is to be understood, that if a man hold any lands or tenements by this tenure, and dieth, his heir male being within the age of one and twenty years; the Lord of whom those lands are holden, shall have the Ward, that is, the custody and keeping of those lands so holden of him, 
Ward.  to his own use and behoof, without account, until the heir come to the full age of one and twenty years: for it is intended by the Law, that until he attain to that age, he is not fit or able to perform such service, as by this tenure is required.  
And if at the time of the death of such tenant the heir be unmarried, 
Marriage.  the Lord shall not only have the wardship of his body and lands, but the bestowing of him in marriage.  
And if a tenant by Knight's service die, and leave an heir female, being of the age of fourteen years, 
Heir female.  or upwards; then the Lord shall not have the ward of such heir, 
The full age of a woman.  neither of her body nor lands; because a woman of that age may have a husband able to perform the services required by this tenure.  
But if such an heir be under that age of fourteen years, and unmarried at the time of her ancestors death, then shall the Lord have the wardship of her lands so holden of him, till she attain to the age of sixteen years; by force of an act of Parliament, in the Statute of Westminster 1 Cap. 12.  
And note here the great difference between the ages of Males and Females: for the Female hath these several ages appointed unto her by the Law. First, 
Diversity of the ages of women.  at seven years of age the Lord her father may distrain his tenants for aid to marry her. Secondly, at nine years of age she is dowable. Thirdly, at twelve years of age she is held able to assent to matrimony: Fourthly, at fourteen she is able to have her land, and shall be out of ward, (if she be of this age at the death of her ancestor.) Fiftly, at sixteen she shall be out of Ward, though she were under fourteen years of age at the death of her ancestor: and sixtly, at one and twenty years she is able to make alienations of her lands or tenements. 
Man's ages.  But the Law limiteth to the Male, only two ages, that is, at fourteen years to have his lands holden in Socage: and at one and twenty to make alienations.  
As concerning Relief, 
Relief.  if a man hold his lands by Knight's service and dieth, his heir male being of the age of one and twenty years, or his heir female of the age of fourteen years, the Lord of whom such lands are holden, shall have relief of the heir.  
Also it is to be noted, that all Earls, Barons, or other the Kingstenants (holding of him in chief by Knight's service) if they die, their heir being of full age, as aforesaid, they aught to pay the old relief for their inheritance, 
What reliefs are to be paid, and how.  that is, the heir of an Earl for a whole Earledom an hundred pounds; the heir of a Baron for a whole Barony a hundred marks; the heir of a Knight five pounds, and he that hath less shall give less, according to the old custom of Fees. And the like is to be understood and observed of all others, that hold such lands immediately of any other Lord  
And also a man may hold lands of a Lord by two Knights fees, and then the heir being of full age at the death of his ancestor, shall pay to his Lord for relief ten pounds.   

3. Castle guard.  
IT is also to be understood, that a man may hold lands by Knight's service, 
Defined.  and not by escuage, nor pay escuage for the same: But he may hold by Castle guard, which is to keep a tower or some other place of his Lord's Castle, on a reasonable warning, when the Lord heareth of the approach of his Enemies.  
This is likewise Knights service, and draweth thereunto ward, marriage and relief, in all respects as the common Knight's service doth, before mentioned.   

4. Grand Sergeantie.  
ALso there is another kind of tenure in Knight's service, 
Defined.  which is called Grand Sergiantie, and this is where a man hold any lands and tenements of the King, by some such certain services, as he aught to perform in proper person, as to bear the King's Banner, or his Spear, or to conduct his Host, or to be Sewer, Carver, etc. and such service is called Grand Sergeantie, that is, a high or great service, because it is the most honourable and worthy service that is: for whoso holdeth by escuage, is not tied by his tenure to perform any other more special service, than another holding by escuage, but he that holdeth by grand Sergeantie, is tied to perform some special service to the King.  
And if a man hold land of the King by grand Sergeantie, and dye; his heir being of full age, then shall his heir pay unto the King, not only five pounds, as he that holdeth by escuage, but also the clear yearly value of such lands, as he shall hold by grand sergeanty.  
And also in the borders of Scotland, divers hold their lands of the King by cornage, 
Cornage.  which is to blow a horn, to give notice to the Country of the Enemy's approach, which service is also a kind of grand sergeanty.  

No tenure in grand sergeanty, but of the King.  And it is to be understood that none can hold by this tenure of grand sergeanty, of any other Lord save only of the King.   

5. Petty Sergeantie.  
Petty Sergeantie is, 
Defined.  where a man holdeth lands or tenements immediately of the King, and for his service in respect thereof, is bound to pay unto the King yearly, a Bow, a Spear, a Dagger, or some such other small thing belonging to the war. And this service is in effect no other than Socage, because the tenant is not tied to perform any personal service, but to pay somewhat yearly, as a rent is paid. Wherhfore this service of petty sergeanty, 
Not Knights service.  is no Knights service. Yet can it not be held of any other Lord save the King only, aswell as grand sergeanty.   

6. Homage ancestrel.  
IF a man and his Ancestors, whose heir he is, have holden lands or tenements of another, and his ancestors time out of mind, by homage, and have done unto him homage, this tenant thus holding, is called tenant by homage ancestrel, by reason of the long continuance which hath been by title of prescription, aswell concerning the tenancy in the blood of the tenant, as concerning the Signiory in the Lord  

This tenure extinct.  And it is to be noted, that if this tenant by homage ancestrel, shall at any time alien those lands unto another, although he immediately, or at any time after, purchase them again, he shall no longer hold by this tenure, because he hath discontinued, but shall from thenceforth hold it by the accustomed homage.   

7. Socage tenure.  
TEnure in Socage is, 
Defined.  where a man holdeth lands or tenements of a Lord by certain service, for all manner of services; so as the service be not Knight's service: As where a man holdeth of his Lord by fealty and certain rent, for all manner of services; or else, where a man holdeth by homage, fealty and certain rent, for all manner of services; for homage by itself maketh not Knight's service. Also a man may hold his lands by fealty only, which is likewise tenure in socage. For every tenure that is not tenure in Chivalry, is tenure in Socage.  
These tenants were tied in ancient time every of them with their ploughs by certain days in the year, to blow and sow their Lords demeans, for which cause this tenure was called Socagium, or seruicia socae, 
Why so called.  which is the same with Caruca, one Soak or one plough land. But now that service is by mutual consent, between the Lord and Tenant, in most places, 
Converted to rent.  converted to an annual rent, yet the name of Socage still remaineth.  

Escuage certain.  Also if a man holdeth by escuage certain, as is before spoken, he holdeth in effect but by socage.  
And further, it is to be understood, that when a tenant in socage dieth, the heir is to pay unto the Lord, of whom those lands are holden, a relief, 
Relief.  that is to say, the value of one years rend, besides the yearly rent, for the payment of which relief, the Lord may at his pleasure immediately distrain.   

8. Frank Almoigne.  
THis tenant in Frank Almoigne, or free alms, 
Defined.  is where an Ecclesiastical person holdeth lands of his Lord, in pure and perpetual Alms, which tenure began in ancient time, thus: If a man being seized of certain lands and tenements in his demesne, as of Fee, should thereof enfeoff an Abbot and his Covent, or a Prior and his Covent, or any other Ecclesiastical person, as a Dean of a College, or Master of an Hospital, or the like, to have and to hold the same lands to them and their successors for ever, in pure and perpetual alms, or in frank alms, in these cases the tenements should be holden in frank Almoigne. By force of which tenure, 
Service to be done.  those tenants which hold lands thereby, were bound to make Orisons and Prayers, and to do other divine services, for the souls of their Grantors and Feoffors, etc. and therefore discharged by the Law, to do or perform any other profane or corporal service, as fealty, or the like.  

Otherwise since the statute.  But it is now otherwise, since the Statute called Quia emptores terrarum, An. 18. ED. 1. So as now no man can hold in frank Almoigne, but by force of such grants as were made before that Statute.   

9 Burgages tenure.  
A Tenure in Burgages is, where an ancient Borough is, 
Defined.  whereof the King is Lord, and they which have tenements within the same Borough, hold of the King, by paying a certain yearly rent, which tenure in effect is but socage tenure. And the like is, where any other Lord spiritual or temporal is Lord of such Borough.  
And it is to be noted, that for the most part such ancient Boroughs and Towns, have divers and sundry customs and usages, 
divers customs.  which other Towns have not. For some Boroughs have a custom that the youngest son shall inherit before the eldest, which custom is commonly called Borough English. And in some places the woman by the custom of the Borough there, shall have all such lands and tenements in Dower, as her husband at any time during the coverture stood seized of.  

divers customs contrary to the course of common Law.  There are divers other customs in England, which are contrary to the course of the common law, which being probable and standing with reason, are good and effectual, notwithstanding they are against the common law.  

No custom without prescription.  But no customs are allowable, but those, as have been used by prescription, or time out of mind.   

10. Ancient Demesne.  
THere is likewise another tenure, 
Defined.  called ancient demesne, and the tenants who hold by this service, are freeholders by Charter, and not by copy of Court roll, or by the verge after the custom of the Manor, at the will of the Lord  
And these are such tenants as hold of those Manors, which were Saint EDWARD'S the King, or which were in the hands of King WILLIAM the Conqueror, which Manors are called the ancient Demesnes of the King, or the ancient demesnes of the Crown of England.  
And to such tenants as hold of those Manors, the Law granteth many large privileges and liberties, 
Quit of toll.  as to be quit of toll and passage, and such like impositions, usually demanded and paid of and by other men, for their goods and cattle, bought and sold in Fairs and Markets by them; also to be quit and free of tax and tallage granted by Parliament, 
Free of tax.  except it please the King to tax ancient demeans, when he thinketh fit, for great and urgent occasions. And divers other privileges are belonging to this tenure, wherein I refer you to our Laws.  
And if such tenant be at any time distrained, to do and perform unto their Lord any such other service or duty, which they or their ancestors have not been accustomed to do, 
A wit of Monstraverunt.  they shall sue out a Writ, called a Monstraverunt, directed to the Lord, commanding him that he distrayne them not to do other services or customs, than they have been accustomed to do.  
And it is further to be understood, that in the Exchequer there is a book remaining, 
Doomsday book.  called Doomes-day-booke, which book was made in the time of S. EDWARD the King, and all those lands which were in the seisin and in the hands of the said S. EDW. at the time of the making of the said book, are ancient demesnes.  
And thus much concerning the diversity of tenors and services. Now next let us consider of the rents thereon usually reserved, and the several kinds thereof.    

CHAP. V Of Rents, and their several natures.  
COnsidering that on every tenure there is usually some rent or other reserved: I hold it not unfitting to say somewhat here concerning the same. And first, it is to be understood, that as there are diversity of tenors, so likewise of rents; as one sort which is called a Rend service, 
Diversity of Rents.  another Rend charge, and a third Rent seck, or Redditus siccus, a dry rent.  

Rend service.  As concerning rend service, it fitly hath the name, for that it is usually tied and knit to the tenure; and is, as it were, a service, whereby a man holdeth his lands or tenements, or at lest, when the rents are unseparably coupled and knit with the service. As for example; where the tenant holdeth his land of the Lord by fealty and certain rent, or by homage, fealty, and by certain rent, or by any other kind of service and certain rent, this rent is called rend service.  

Distress of common right.  And here is to be noted, that if at any time this rend service be behind and unpaid, the Lord of whom the lands or tenements are holden, whether in fee simple, fee tail, for term of life, for years, or at will, may of common right enter and distrain for the rent, though there be no mention at all, nor clause of Distress put in the Deed or Lease.  
The nature of this rend service I say is to be coupled and knit to the tenure; and therefore, where no tenure is, there can be no rent service: wherefore, if at this day I be seized of lands or tenements in fee simple, and make a Deed of feoffment thereof unto another in fee simple, and reserve by the same Deed a rent, this can be called no rent service; for that there can be now no tenure between the Feoffor and Feoffee. But it is otherwise of feoffments made before the Statute of Quia emptores terrarum, 
Anno 18. Ed. 1.  formerly mentioned. For before the making of that Statute, if any man had made a feoffment in fee simple, and had reserved thereon unto himself a certain rent, although it had been without Deed, here had been created a new tenure between the Feoffor and Feoffee, and the Feoffee must have holden of the Feoffor, who by means thereof, might of common right have distrained for such rent: but since the time of that Statute, there can be no such holding or tenure created or begun; and consequently, 
No rent service can be now reserved on gift in fee.  no rent service can at this day be reserved upon any gift in fee simple; except in the King's case; who being chief Lord of all, may, and ever might, give lands to be holden of him.  
Thus it is apparent, that at this day no subject can reserve any rend service unto himself, unless the reversion of those lands, so by him granted, be still in himself: as where he granteth them in fee tail, or maketh but a lease for term of life, or for years, or else at will; for in all these cases, the reversion of the fee simple remaineth still in him: wherefore, if any rent be here reserved, it is to be called a rend service; and of common right is distrainable, although there be no clause of Distress comprised in the Deed or Lease.  
And if a man shall absolutely and wholly grant away in fee simple any lands or tenements by him so holden, leaving no reversion thereof in himself; and yet shall reserve unto himself in his Grant an annual rent; with a clause of Distress in his Deed indented, That it shall be lawful for him to distrain for the same, if need require; this rent (in regard, that the land is therewith charged) is called a rend charge: 
Rend charge.  But he cannot distrain for this rent of common right, as before for the other, but only by force and virtue of his Deed indented.  
And if there be no such clause of Distress contained in the Deed, then is this rent so reserved called a rent seck. 
Rend seck.   
Also, if a man standing seized of lands and tenements in fee simple, will grant either by Indenture, or poll Deed, an yearly rent unto another, issuing out of the same lands, whether it be in fee simple, fee tail, for term of life, for years, or at will, with clause of Distress; then this rent is called a Rent-charge, and he unto whom this rent is granted, may for default of payment enter and distrain.  
And it is further to be understood, that if a man make a lease unto another for term of life, and reserve thereon unto himself an yearly rent, and afterwards granteth that rent unto A.B. reserving the reversion of the lands unto himself; 
Rend seck.  this rent is but a Rent seck: for that A. B. who hath the rent, hath nothing in reversion of the land.  
And if a man giveth lands and tenements in tail, and reserve to him and his heirs a certain rent; or else make a lease for term of life, reserving certain rent; if he grant the reversion to another, and the tenant attorne accordingly, the whole rent and service shall pass by this word reversion, because the rent and service in such case be incident to the reversion, and pass by the grant of the reversion; 
Rend charge.  and here is the rend a Rent-charge. But if he had granted the rent only, it had been then a Rent seck.   

CHAP. VI Of Reprises and Deductions.  
AS we have formerly understood, what several rents, profits, and commodities may yearly arise or grow out of any Manor to the Lord thereof; so is it as fitting to consider, what Reprises, Deductions, Payments, Charges, and Duties, may be yearly issuing or going out of any Manor from the Lord thereof: For otherwise, in the conclusion of our Survey, or in making a perfect Constat, or Particular, (such duties not being reprised) the true value of the Manor may oftentimes seem greater than in truth it is; which would tend much to our shame and discredit.  
These Reprises and Deductions are never certain, or in all Manors alike; but in this more, and in that less: yet in one and the same Manor they are commonly the same, and usually such as these here following.  
Reprises are any manner of Rents, either in Money, Capons, Hens, Pepper, Cummin seed, or the like, issuing and paid out of one Manor to another: also, Suits of Court, or annual fines for the same; and the like may be issuing and payable to a Sheriffs Turn or Hundred; also Pensions or Portions to Ecclesiastical livings: likewise a rent may be issuing for way-leave, or some particular Passage; also for Watercourses, or placing of Pipes for conveyance of Water: likewise yearly Fees to Officers, as Stewards, Receivers, Bailiffs, Collectors, Keepers, etc. and also stypends, salaries, or annuities to chaplains, or the like: All which are ever to be deducted and reprised out of the total value of a Manor.  
And having thus furnished and informed ourselves, first of the Mathematical part of Survey, by the three former Books; and thus far of the Legal, as to know what a Manor is, and the several parts thereof; and likewise of all estates in general; and what Tenors, Services, and Rents are thereunto incident, appertaining, and belonging; let us proceed in an orderly and formal course; supposing we are now to undertake the survey of a Manor, which is to be performed as followeth.   

CHAP. VII. Observations and courses to be held and taken, before the beginning of a Survey.  
IT is first to be considered, for whom the business we undertake, is to be performed: if for the King, then are we to obtain Commission from his Majesty out of such Court or Courts as is requisite, according to the tenure of the lands to be surveyed, as the Exchequer, Duchy, etc. In declaring the form of which Commissions, I need not spend time, for that they are usual, and of ordinary course (in such cases) granted out of those Courts. Yet seeing, that for the most part, those Commissions give power to the Surveyor, by reference to certain articles annexed; and according to the efficacy and force thereof, the power and authority of the Surveyor is limited; it behoveth to have those articles as ample, full, & forcible, as you may devise; not knowing with what people you are to deal (who often prove obstinate) nor the nature, estate, or condition of tenancy (for the most part variable.) Which articles let be these here following, or the like in effect.  
Articles to be inquired of, and courses to be observed and held by A. R. in this present Commission named, for the better effecting and execution of his majesties service, in surveying of his highness Honours, Lordships, or Manors of A. and B. in the County of C. and of all Castles, Houses, Parks, Messages, Lands, Tenements, and Hereditaments thereunto belonging and appertaining.  
1. FIrst, the said A.R. is to enter into the said Honours, Lordships, and Manors, and all and singular other the premises, and every of them, and into every part and parcel of them, and every of them, and to make a survey of the quantity, quality, and yearly value thereof, and of every part, parcel, and member thereof respectively.  
2. Also the said A.R. is to call before him all such as now are or formerly have been Stewards, Bailiffs, Reeves, or Collectors of all or any his majesties issues, rents, revenues, and profits within the premises, and their deputies, and every or any of them, and to charge them on their oaths to deliver in unto him true and perfect rentals of all and every their several collections: and likewise to call before him all and every such person and persons, as have or are suspected to have any Evidences, Court Rolls, rentals, Books of Survey, Couchars, Terrars, Escripts, Write, or Mynuments whatsoever, touching or concerning the said Manors, Messages, Lands, Tenements, and Hereditaments, and every or any of them; and all and every such person and persons to examine upon their oaths concerning the same writings, and every of them: and also to demand, require, and receive of them, all and every such Books, rentals, and other Write, as he shall so found to be in their hands or custody: And if any shall make refusal of the delivery thereof, to certify his or their name and names, and the reason of such his or their refusal, to the Lord Treasurer of England, and Chancellor of the Exchequer, that speedy and due courses may be therein held and taken accordingly. But this is to be understood of Books and Write not being in the hands of the present Steward or Stewards of any of the premises, nor in the custody of any of his majesties Officers of his highness Courts of Record at Westminster; whereof he is only to take and extract notes, for his better instruction and information concerning the premises.  
3. Likewise he is to inquire, what are the several limits, butts, and bounds of all and singular the premises, and to express the same accordingly; and what Lord or Lords are conjoining or boundering thereon; & whether they or any of them have or do intrude or encroach upon or within the limits or bounds aforesaid, or the liberties or privileges comprised within the same.  
4. Also, whether the premises, or any part thereof, doth lie or extend into any other Manor; and whether any other Manors, Messages, Lands, or Tenements do lie within the limits or bounds of the premises; whose, and what they are; and to make perfect distinctions thereof particularly.  
5. What Castles, and other Manor or Mansion houses his Majesty hath within the same; in what estate of reparations the same now are and be; and if decayed or wasted, by whom the same hath been committed, & to what value; what demesne land's now are, or heretofore have been, belonging or appertaining to the said Houses, and in whose tenure and occupation the same now are; by what right or title they claim or challenge to hold; what several rents they pay in respect thereof; and what is the true quantity, quality, and yearly value of the premises.  
6. What Forests, Parks, and Chases his Majesty hath within the premises; what number and store of Game are in them; what Officers are thereunto belonging; what Fees they receive in respect thereof; in what estate of reparations, the Houses, Lodges, Walls, Pales, and Fences are; what is the quantity, quality, and yearly value thereof by the acre; what juistments, or what cattle, as Oxen, Cows, Horses, or the like, are usually depastured within the same; who hath the disposal thereof; and what is the value of a Beast-gate there.  
7. Also, what Moors, Marshes, Heaths, Wastes, or Sheep-walks, his Majesty hath of, in, upon, or belonging to the premises; what are the several quantities thereof; how many Sheep may be kept on those walks; and what is a Sheepe-gate worth.  
8. He is also to inquire, what freeholders there are within and belonging to the premises; what Manors, Messages, Lands, or Tenements they hold thereof, and what are their several quantities; and likewise, by what several tenors, rents, and services they hold the same.  
9 Also, what other estates there are; as tenants for term of life, or lives, years, or at will; what customary or copyhold tenants, or what other tenants there are within the premises; what lands they do severally hold, and the true quantity, quality, and yearly value thereof severally, and what yearly rents they pay for the same.  
10. Also, what are the several customs concerning the customary tenants; whether their fines upon death or alienation be certain, or incertain, and arbitrable; and if certain, what Fines they usually pay on every death or alienation of Lord or Tenant; and how, and in what manner, do those customary lands descend after the death of an ancestor.  
11. What Reliefs, Heriots, Fines, or other duties are paid, or answerable, upon or after the death or alienation of any Freeholder, Copyholder, or other tenant within the premises; how and by whom are they usually collected and disposed of; and what may be the value thereof in Communibus annis.  
12. Whether any customary tenants (whose lands are hariotable) have severed, aliened, divided, or dismembered the same, who hath the use and occupation thereof, and what are the several quantities, qualities, and yearly values of the same.  
13. Also, what are all and every the customs in general of, within, or belonging to the premises, and how, by what means or for what cause, may a copyholder or customary tenant, forfeit or loose his customary estate.  
14. What Commons there are, of, within, or belonging to the premises, whether stinted or unstinted; if stinted, then how, by what means, and according to what rate and proportion, how many beast-gates they contain, the value of each beast-gate, and the quantity, quality, and value of the whole.  
15. What arable fields and meadows there are, which lie in common, what are their several names, and of the several furlongs and wents therein contained; also how and in what manner they are kept and used; whether is it lawful for any tenant at his pleasure to enclose any part thereof, without leave of the Lord; how are they employed when the corn and grass is taken away, how stinted, and what is the eatage thereof worth by the beast-gate, or sheepe-gate, after the corn and grass is so taken off, as aforesaid.  
16. What woods or woodgrounds his Majesty hath within the premises, what grounds have been heretofore wood, and now converted to other uses, how long since & by whom, what wastes and spoils have been had or made of his majesties woods, how long since, by whom, and of what value; whether may any profit by pannage be made orraysed, by, or within the same woods, and what the profit or value thereof may yearly be.  
17. Also what tenants there are within the premises, who demise or let any part or parcel of their lands or tenements unto under-tenants, either for their whole term or any part thereof, and what fines and rents have or do they receive for the same.  
18. Likewise, what lands, tenements, rents, services, or other profits, are concealed or detained from his Majesty, how long since, by whom, and what the yearly value thereof is.  
19 What lands, tenements, leases, or other estates of, or in the premises, have been or are escheated, or forfeited to his Majesty, by whom, when, for what cause, and in whose occupation the same now are, and what is the value thereof.  
20. What fines, issues, amerciaments, perquisites of Court, heriots, waifs, strays, felons goods, and other casualties do yearly accrue and grow unto his Majesty out of the premises, by whom the same is, or hath been collected, gathered, and received, and what is and hath been the value thereof yearly in communibus annis.  
21. Also what enclosures and Incrochments have been heretofore made of, in, or upon any of his majesties commons, wastes, or other grounds, how long since▪ by whom, what rents are paid for the same, and what the yearly value thereof is.  
22. What Corne-mills, Fulling-mills, or other mills, his Majesty hath within the premises, who hold the same, what rends they pay, what is the yearly value thereof, what customs are thereunto belonging, and in what estate of reparations are all and every those Mills.  
23. What Markets and Fairs are there within the premises, on what days kept, what tolls are belonging to the same, by whom the same is collected and received, and what yearly profit ariseth thereby unto his Majesty.  
24. Also what Warrens, Fishings, Fowling, Hawking, Hunting, or other Royalties his Majesty hath within the premises, by whom the same is occupied or enjoyed, what rents are yearly paid for the same, and what is the yearly value thereof.  
25. What quarries of stone, mines of Tin, Led, Cole, or other mines his Majesty hath within the premises, who hath the use and occupation thereof, what rends they pay for the same, and what the yearly value is.  
26. What Mosses of peat or turf, what Broome, Heath, Furze or Flag, are within the premises, belonging to his Majesty, what are the rents and yearly values thereof.  
27. What Aduowsons', Patronages, Free-gifts or presentations of Parsonages, Vicarages, chapels or Prebends, or what Impropriations, are appendent or belonging to the premises, who is or are the present Incumbent or Incumbents, who hath the use of such Impropriation, what rent is paid for the same, and what is the yearly value thereof.  
28. Whether any Tenant or other person or persons whatsoever have ploughed up, cast down, removed or taken away any meere-stone, balk, hedge-row, or land-share, between the demeans of the premises, and any other messages, lands, or tenements, or between any the freeholds, and the tenement or customary lands, or between any of the premises, and the lands of other Lords, by whom such offence was committed, and where, and in what place and places those altered bounders aught to stand and remain.  
29. Also what Officers his Majesty hath within the premises, what fees do they yearly receive in respect thereof, what rends, deductions, reprises, or other payments or sums of money, are yearly paid, reprised, or issuing out of his majesties revenues of the premises, and to whom, for what cause, and to what end and purpose are the same so paid.  
30. And lastly, the said A. R. is to make all and every other such further and other inquiry and inquisition of, for, and concerning all and every such matters and things whatsoever, as in his discretion shall be held fit and requisite, for the better effecting and execution of his majesties service, in surveying of the premises.  
These Articles or the like, being drawn and fair written in Parchment by the Surveyor (the commission being to be taken out of the Exchequer) a brief warrant is to be directed to one of the Remembrancers, and written under the Articles to this effect.  
M. I O. These are to will and require you immediately to 'cause a Commission to be made, and directed to A. R. for the survey of his majesties Honours, Lordships, and Manors of A. and B. in the County of C. and of all Castles, Houses, Parks, Messages, Lands, Tenements, and Hereditaments, thereunto belonging or appertaining; whereunto is to be annexed the abovementioned Articles. Whereof fail you not: and these shall be your warrant in this behalf. From the Court, etc.  
Which Warrant is to be signed by the Lord Treasurer, or Chancellor of the Exchequer, and delivered to the Remembrancer accordingly.  
But if the business undertaken, be not for the King, but for a private man, then in regard that a Surveyor hath no power by any authority of Surueyorship, to be granted unto him by any such private man, to minister an oath, or perform such other duties as are requisite, it is fitting either that the Steward of the Manor, which is to be surveyed, join with him, in calling a Court Baron, and Court of Survey, to be there held (wherein the Steward is to give the charge and to deliver Articles, and minister oaths, aswell concerning the Court Baron as Court of Survey.) Or otherwise, the Surveyor is to have a commission, grant, or deputation from the Lord of the Manor under his hand and seal of the office of Steward and Surveyor of his Manors, Lands, and Tenements, for a certain term, or during pleasure: And then may the Surveyor, of himself execute all those offices and duties fit and requisite for a Steward and Surveyor, to do and perform. Which commission, grant, or deputation, let be thus, or to the like effect.  
OMnibus ad quos hoc praesens scriptum pervenerit A. B. de C. Comit. E. Armig. salutem. Sciatis me praefat. A. B. tam pro sincero amore & benevolentia qua iamdudum affectus sum, erga A. R. de cuius provida circumspectione, pia sedulitate, ac singulari in hac parie prudentia merito plurimum confido, quam pro diversis alijs causis & considerationibus, ex mera & spontanea voluntate mea dedisse & per praesentes concessisse eidem A. R. Officium Seneschal. sive Seneschalciam omnium & singulorum Domin. Manner. & haereditament. meorum quorumcunque in Comit. F. & custod. sive officium tenendi omnes & omnimodas Cur. Baron. Letar. Vis. franc. pleg. Dominior. & Manner. praedict. & eorum cuiuslibet, ac gubernationem & superuisionem eorundem. Ac ipsum A. R. generalem ac capital. Seneschal. ac Superuisorem meum omnium Curiarum, Dominiorum, Maneriorum, & haereditament. meorum praedict. facio, constituo, & ordino per praesentes. Habend. tenend. gaudend. exercend. & occupand. Officia praedict. cum pertinentijs, à dat. praesentium durant bene placito meo. Mando insuper universis & singulis Ballivis, Praeposit. Firmarijs, tenentibus & occupatoribus meis praemissor. & eorum cuilibet, quod praefat. A. R. de tempore in tempus, assistentes sint, obedientes, & auxiliantes in omnibus prout decet durant. termin. praed. In cuius ret testimonium huic praesenti scripto meo sigillum meum apposui. Dat. etc.  
Or to the same purpose in English.  
And being thus authorized, we may now proceed.   

CHAP. VIII. What courses are first to be held in the beginning of a Survey.  
COnsidering how precious time is, and withal, how chargeable these employments are to those whom it concern; it behoveth a Surveyor (respecting his credit and reputation) so to appoint and dispose of his business in an orderly course, as no time be idly lost, or vainly spent therein. Wherhfore, first let the Bailiff of the Manor be called, and a Warrant or Precept directed and delivered unto him, to summon as well a Court Baron (if need require) as a Court of Survey; to this, or the like effect.  
Branton. A. R. Seneschal. & Superuis. Manerij praed. Ballivo eiusdem, salutem: Tibi praecipio pariter & mando, quod diligenter praemonere facias omnes tenentes infra Manner. praedictum, tam residentes quam non residentes, atque omnes tenentes custumarios Manerij praedicti, quod sint coram me in hac part sufficienter deputato apud Branton pradictam, die Lunae secundo die Septembr. proximè futuro post datum huius, Non solum ad faciend. sectam suam ad Curiam Baron. & Superuis. sed etiam ad producend. & ostendend. omnes Literas, Chartas, Instrumenta, Indentur. copias Cur. Rotul. ac al. evidenc. unde tenere vendicant seperal. terr. & tenement. suas de Manerio praedicto; & omnia alia quae eye incumbent, & pertinebunt; & haec nullatenus omittas, & habeas ibi hoc praeceptum: Datum sub sigillo meo vicesimo quarto die Augusti, Anno Regni etc.  
Or to the like effect in English.  
Wherein let a convenient time be limited, as five or six days at the lest after notice given for the tenants appearance, that they may the better prepare themselves, and be the more inexcusable, if they happen to make default.  
Than are you to receive from the Bailiff all such rentals as he hath concerning his whole collection, as well such as are ancient, and of former times, as those of his last collection; which you are diligently to compare together, noting the difference: and if the later be lesser, than what decays of rent there are, and how occasioned; if greater, than what increment of rent, and whereon raised; which you are carefully to note and express, when you come to engross your rental.  
Next would I have you to reduce your rental to an Alphabetical form: wherein, use all the modern Tenants names; not omitting the ancient; which will be a great help for the speedy dispatch of your entries; and the ready finding of any Tenant's name, or rend, as you are to use them: which would be written thus, or in the like manner.  

Atkinson Thomas, late Brownes,— xx s.  
Armstrong William, late Tomlinsons',— v s.  
Bennet john, late Brights,— xv s.  
Branthwait Edward, late Finches.— xxij s.   
And in this sort proceed Alphabetically with all the whole rental; which is much available, where many Tenants are.  
And having thus prepared your rental in a readiness, against you have occasion to use it; you are to spend the residue of the time, until the day appointed for your Court, in conferring with the Steward concerning the present estate of the Manor; and in diligent view and search of the Lords Evidences and Court Rolls; taking them orderly before you, and from year to year briefly to express in a Book, for that purpose provided, the several Customs, Estates, Tenors, Rents, and Services, and all other remarkable things. So shall you be able fully to inform yourself of the nature, quality, estate, and condition thereof; and to understand, what articles are now most fit and apt to be presented unto your Homage, or jury, to be inquired of, when you have given them their charge; which you have now also fit opportunity to writ in a readiness for them against that time. And these articles would I have to be such as are expressed in the last Chapter, or so many thereof as you hold fitting and necessary for the purpose; and (if need require) to add and insert such others thereunto, as you shall found meet and requisite, for as much as by the last article annexed to your Commission, you have power and authority to make such further and other inquisition, as in your discretion shall seem fitting.  
Also, now have you convenient time, either to ride or walk abroad, and to take a respective view of the situation and extent of the Manor; whereby you shall be able to inform yourself, where, how, and in what sort you may with most conveniency begin, continued, dispose of, and perform your Instrumental mensuration, either by yourself or servants, whom you employ therein: wherein, for many respects, I would have nothing done or performed, before the first day of your Court be past; when as you have read and made known your Commission, and settled an orderly course with the tenants, for their attendance, aid, and assistance in that behalf.  
And thus, and in this like manner, may you spend the time to good purpose, till your Court day come.   

CHAP. IX. The order of keeping a Court of Survey.  
IF a Court Baron be kept with your Court of Survey, as is ever most fitting, then are you first to enter the style of the Court in this manner.  
Branton. CVria Baronis & Superuis. A. B. Armiger, ibidem tent. die Lunae, videlicet secundo die Septem. Anno Regni JACOBI, Dei gratia Angliae, Franciae, & Hyberniae Regis fidei defence. etc. xiv. & Scotiae 50. tent. per A. R. Seneschallum & Superuis.  
After the style of the Court thus entered, you shall 'cause the Bailiff, who serveth the Court, to make Proclamation by crying once Oyes, and then shall you will him to say, thus; All manner of persons, who were summoned to appear here this day, to serve the Lord of the Manor, for his Court now holden, draw near and give your attendance, and every one answer to his name, as he shall be called, upon the pain and peril that may fall thereon.  
Than by your rental, call them all severally by their names, marking those which are absent to be amerced. Which done, 'cause the Bailiff to make another Oyes, and willing them to draw near, and keep silence whilst the Commission be read; let the same be read unto them, and likewise the Articles thereunto annexed, if it be for the King.  
Than out of those tenants which are present, make choice of the most sufficient for your jury, wherein your best course is, formerly to inform yourself, and to take special notice, who are most fitting for your purpose, and to have their names ready written in a paper by themselves, which you may now thereby call accordingly. But being for the King, you have always a writ of assistance directed to the Sheriff of the shire, requiring him to return you a sufficient jury: yet may you without him by virtue of your Commission, impannell any jury at your own pleasure. Than direct the Foreman of the jury, to lay his hand on the Book, and swear him as followeth, or to the like purpose.  
YOu shall diligently inquire and make true presentment of all such matters, as on the Lord's behalf of this Manor, shall be given you in charge, you shall neither for favour, fear, affection, or other partial respect whatsoever, forbear to present what you aught to find, or find what you aught not to present, you shall herein keep the Lords counsel, your own, and your fellows, and in all things according to a sincere and upright conscience, you shall present the truth, the whole truth, and nothing but the truth, as by evidence and your own knowledge you shall be induced, to the best of your power, so help you God, and by the contents of the Book, which he is to kiss.  
And after the Foreman is thus sworn by himself, then 'cause three or four of the rest of the jury, to lay their right hands together on the Book, and give them their oath, as followeth.  
THe same oath which A. B. your Foreman before you, for his part, hath made and taken, you and every of you, for your parts shall truly keep and perform to the uttermost of your powers, so help you God. And 'cause them severally to kiss the Book.  
And in like manner swear all the rest. And all being sworn, 'cause the Bailiff to number them, as you read their names. Than 'cause him also to make the third Proclamation, and say thus: All you that be here sworn, draw near and hea●e your charge, and all the rest keep silence.  
Than make your exhortation, and deliver the charge of a Court Baron, after the usual manner. Which being finished, you are to address your speech unto them, concerning the present business of Survey, as occasion shall be offered, whereof to prescribe you any form or precedent, were to little purpose, seeing it is to be framed and directed to such ends and purposes, as the present cause requires, which you shall always find different and variable, and therefore I refer the same to your own discretion, deeming you now able and fitting sufficiently to perform the same in any kind.  
And then deliver unto them the Articles which you have ready drawn, according to the directions of the last Chapter, which is their charge concerning the business of Survey, relating unto them, that as they receive these Articles (whereof they are to inquire) in writing, so are they to answer the same in writing under their hands and seals particularly by a day, now to be limited and appointed, which for many reasons is most fitting to be, about the time of your concluding the business. Which day is to be expressed under their Articles, and your name subscribed thereunto.  
And now are you to take order, and give special directions unto all the tenants for their attendance, aid, and assistance, in your instrumental mensuration, appointing them by turns, how, when, and where you are to use their help and assistance, wherein you are to deal with such discretion, as you neither fail of their help, when occasion serveth, nor oppress them with grievance by their overmuch attendance.  
And having thus far proceeded, the rest of this day may be spent in entering their several deeds, evidences, and estates, in manner as shall be hereafter declared.  
But before you discharge the tenants, you are to consider (according to the number of them) in what time or how many days, you shall be able to enter their estates, and if they consist of divers Towneships, as large and spacious Lordships usually do; then your best course is to appoint them several days for their attendance, and bringing in of their evidence by several Towneships, for it would be no less troublesome to yourself, then distasteful to the tenants, to require their general and daily attendance until the business were wholly finished.  
And now may you aiourne the Court unto the next day, (or such other time as you think fitting) by causing the Bailiff to make proclamation to that purpose, and the like from time to time, till you have ended your business.  
The next day you may begin your mensuration in the fields, either by yourself or those whom you employ to that purpose, according to the instructions of the third Book. But it were fitting for your own part to be employed in entering of the tenants estates, until you have finished, or you may spend such time therein, as when the weather is not fitting to stir abroad, or in the mornings and evenings, as you shall find meetest for your purpose.   

CHAP. X. The order and manner of entering the Tenant's evidence, and their several estates.  
FIrst, it is to be considered, that most Manors (as if formerly spoken) consist of divers Towneships or particular parts, and the tenancy of those Towneships of divers estates, as Freeholds, copyholds, etc. Wherhfore, I hold it sitting and an orderly course, that not only every of those Towneships, but the several estates therein, be entered and taken severally and particularly by themselves, that is, all of one and the same Towneship and estate under one and the same title, for avoiding of confusion. As, suppose you are to survey the Manor of Branton, which consisteth of these several Towneships or parts, Branton, Bodley, and Sutton, and within those Towneships, are divers tenants, holding their lands by several estates, as Freehold, Copyhold, etc. Than would I have you make your several entries under those several titles whereunto they properly belong, as under the title of Branton towneship Freehold, enter all those which are of that Towneship, and of that nature: and under the title of Branton towneship Copyhold, enter all the copyholds of that towneship, and the like of all the rest.  
And these entries I would have made in lose sheets of paper at large, keeping them always sorted, according to the several Towneships and estates, till you have finished all your entries, and then to file them together orderly in a Book, each Towneship following other.  
In which several entries observe this course▪ having written your title as before, in the head or top of the sheet, then enter the tenants name, and the very words of grant, as they are in his Deed, Copy, or Leaf, which is to be written from the margin the whole breadth of the sheet, leaving only towards the right hand a space, wherein is to be expressed the rents, and services, and in the margin always express the tenure.  
And considering, that in few or no Deeds, Evidences, Copies, or Leases, the lands are particularly expressed by particular names, closes, and quantities, as the tenant now holdeth the same, and as you shall found them in your instrumental mensuration; having entered the effect of the Deed, Copy, or Lease, according to the purport thereof: it is fitting to question with the tenant, what several parcels he holdeth, for and under the services and rents contained in every Deed, Copy, or Lease particularly; as, what Meadow, what Arable, what Pasture, and their several names and quantities, as he esteemeth them; and if he know not what acres they contain (as most tenants will seem ignorant thereof) let him express of his Meadow how many days mowing, of his Arable how many days ploughing, and of his Pasture how many Beast-gates, and the like: for although it be not greatly material for these their given quantities, in respect you measure every particular; yet this kind of entry will serve you to good purpose, as shall hereafter appear. Also, you are to question him concerning his Pastures, as well those in several, as the Pastures in common, what every Beast-gate is worth by the year in every of them severally; whereof you may otherwise also inform yourself, jest you be deceived. The form of which entries let be in this manner.  
BRANTON Towneship Freehold.  
Socage,  
ANTHONY BORN holdeth freely to him and his heirs for ever, by deed indented, bearing date 14. die januarij, Anno Reg. Reginae Elizabeth. etc. 30. made and granted by and from WILLIAM BATEMAN, of etc. All that Mesuage or Tenement (expressing the very words of Grant) On which Grant is there a deed of feoffment of the same date, with livery of seisin thereon passed accordingly; by the yearly rents and services of Fealty, & iij. s.  

Particular.  
THe mansion-house, outhouses, and scite, consisting of two Orchards, three Gardens, and two Yards or Garths. containing together— 3. r.;  
A Close of Meadow, called Broad Meade, containing— 10. a.  
Another called Whitethorn close, containing— 8. a.  
Meadow in the common Meadow, called Long mead, three parcels containing— 12. d.  
A Close of Arable, called Bennets, containing— 7. a.  
Another of Arable, called the High Close,— 15. a.  
Arable in the South field, in six parcels, which contain all together— 4. a. 3. r.  
Arable in the North field, five parcels containing— 6. 2. 2. 1.  
One Close of Pasture, called the Ox Pasture, containing 30. beast. gates, at 13. s. 4. d. le gate,— 30. gates.  
Another called the Calf Close, containing 12. beast-gates, at 8. s. le gate,— 12. gates.  
In the great common Moor, at 2. s. 8. d. le gate,— 20. gates.  
In the common Cow-pastures, at 6. s. 8. d. le gate,— 10. gates.  
On the Downs depasturing, for 150. sheep, at 3.d. le gate,— 150. gates.  
Common sans stint on the Moors.   
And in this order, and under this Title, enter all the Freeholds within the Towneship of BRANTON: But withal, observe this course in all your entries; that these particulars, in the entering of them thus in your rough Book, exceed not, nor extend past half the breadth of every sheet or leaf, or little further, because directly after the several contents, before specified, there is particularly to be expressed afterwards, the true quantities of every parcel found by measure; and after that, the several and particular yearly values thereof, as shall be hereafter shown.  
In like manner, let your tenants for life or lives, in every Towneship, be entered under their due and several titles thus.  
BODLEY Towneship for lives.  
THOMAS HOCKLEY holdeth by Indenture, bearing date 23. die Novembr. Anno Reg. Reginae Elizabeth. 32. made and granted by and from A. B. All that Mesuage or Tenement (using the words of Grant) for the term of the natural lives of the said THOMAS HOCKLEY now aged 50. years; of JANE his wife now aged 42. years, and JOHN their son now aged 30. years, successively each after other: and payeth rend per annum— xx. s.  
Particular.  
ANd here writ the particulars in form as before.  
Than under the foot of those particulars make a brief Memorandum of such necessary observations as are to be noted, thus.  
There is due unto the Lord on the death of every of them dying tenant in possession, the best Beast Nomine Heriot. The tenant is to pay his rent quarterly, or within one and twenty days, on pain of xx. s. Nomine poenae; or within forty days, on pain of forfeiture by proviso. To do all manner of reparations (except great timber.) Not to let the whole, or any part, without the Lords licence. The Lord warranteth the premises against him and his heirs.  
And the like notes may be expressed under all your other entries.  
Where note, that special care is to be always had in the entrance of these grants for lives, whether they are all joint purchasers, as all named in the words of grant, or whether only granted to one during all their lives; wherein there is great difference; which is to be noted, and the very words of grant to be ever precisely expressed in your entries as before.  
As concerning your Copyhold Tenants let them be entered thus.  
SUTTON Town-ship Copyholds.  
W. B. holdeth the Copy of Court roll bearing date viii. die Junii Anno Regni Regis JACOBI Angliae iij. of the surrender of C. D. One messsage or tenement (according to the words of the Copy) late E. F. and before that G. H. To him and his heirs at the will of the Lord according to the custom of the Manor. For which he paid fine XLs. And payeth rend per annum. XXX ss  
Particular.  
Than writ your particular as before: and after that such brief Memorandums as you shall find fitting, according to the former directions in that behalf.  
And the like course are you to hold with Tenants for term of years, and at will; and all the rest: until you have fully finished all your entries. Than place all your leaves in order; and if you be assured that all are entered (which will easily appear by your rental, if you always cross the same, as you enter them severally) then writ a new rental according to this your rough Book, and join the same to the beginning of your Book, and file all together; But let your rental be alphabetical; or else if your book be very large, make an alphabetical Index or Table of all the Tenants names, to be placed in the beginning of the Book, before the Rental, with numbers of reference to the number of leaves; whereby instantly you may turn to any Tenants name therein as you shall have occasion to use the same.  
And thus have you finished with the Tenants concerning their several entries; but have not yet fully perfected your rough Book; whereby you may be able to engross the same. Wherhfore proceed therein as is taught in the next.   

CHAP. XI. The means and order of perfecting the Book of entries last mentioned, and the due placing therein of the several contents of every particular found by measure through the whole Manor, with the valuation thereof.  
WE are now to suppose before we begin this work, that not only all the Instrumental mensuration throughout the whole Manor is finished; but also the first plot drawn; and the several contents thereof cast up, and expressed therein; with the proper and particular numbers, severally belonging thereunto, according to the instructions of the Third Book.  
Which being effected, we are first to make an Index or alphabetical Table of all the Tenants names; whereunto is to be added the Lords and the Parsons, the one for Demesnes, the other for glebe; which Index is thus to be composed. If one sheet of paper will not serve, you are to take two, three or more, and with mouth-glue fasten them end to end, making thereof a long scroll or schedule of the breadth of the whole sheet. And with a small margin towards the left hand, let it from thence be ruled with black or read Ink overthwart the whole paper, of the distance of lines in ordinary writing; between which lines down by the same left margin writ all the Tenants names each under other alphabetically, leaving all the rest of the ruled Paper to be thus employed.  
First, take your Field-booke, and beginning where you first began your work in the fields, take all the numbers before you expressed in the margin thereof, as appeareth in Chap. 10.3. and noting to what Tenants name they belong in your Field-booke, against the same name place them in your Index between the ruled lines; whereby you shall speedily express and reduce the several numbers representing the several fields and closes throughout the whole Manor against every man's name in the Index to whom they properly belong.  
Than take your common Field-booke (mentioned in Chap. 40.3. and with it perform the like; but make a stroke or other mark for distinction between these and the former numbers; so have you also every man's particular parcels lying in the common fields expressed against his name. And thus is your Index perfect and fit to be employed as followeth.  
Take now your rough Book of entries, and turn to the first Tenants particular therein entered; and look in your Index what numbers are belonging to that Tenant; also take your Field-booke, and comparing those numbers in your Index with the same in the margin of your Field-booke, you shall found therein the several names of the fields and closes belonging to that Tenant; and the like names shall you found in your Book of entries, in that Tenants particular: wherefore express those several numbers against every particular parcel in the margin of your entries, whereunto they properly belong: and the like perform in all respects with your common field-booke, for those parcels lying in the common fields. Than lay your rough plot before you, and finding those several numbers in your plot, note the several content and quantity of every several parcel of ground, expressed in the plot represented by those numbers; and those quantities express and writ down in figures particularly in your book of entries, to every parcel whereunto they belong next after the quantities delivered by the tenant. As for the particular quantities in the common fields, you shall not found them in the plot, but in your common field-booke, according to the direction thereof in Chap. 40.3. before recited; which let thereby be expressed accordingly.  
And the like course in all respects is to be holden with all other the entries throughout your whole book.  
But it is to be considered, that one tenant may within this Manor hold lands of several estates, and by several rents and services, as Freehold, Copyhold, Tenement Lands, etc. yet are they all comprised and represented within and by those numbers expressed in the Index. In such case you are to compare these numbers with the particular names expressed in your field-booke, and those with the entries; and you shall most easily distinguish the one from the other, and express and assign to every of them their due and proper number and quantity, as before.  
Also, it is here to be considered, that we have not yet spoken of any means to express each man's particular quantity in the common stinted pastures, or sheep-walks, whereof only the general quantity is taken by measure, and expressed in the plot. Which to perform, work thus.  
Suppose there is in the Manor a stinted Cow-pasture, wherein every tenant hath a certain number of Beast-gates, some more, and some less, which are usually rated and stinted, either according to their rents, or after their quantities of known grounds, or their parts in the common arable fields: and imagine this pasture is found to contain by measure 212. Acres, 3. Roods, 20. Perches; first, collect out of your entries all the number of gates in the same pasture; which added together, let contain in the whole 116. Than reduce your measured quantity into the lowest denomination, as Perches (according to the directions of Chap. 48.3.) whereby you shall found the same to be 34060. Perches: and supposing the tenant, whose quantity you seek, hath in the same pasture 10. beast-gates; by the rule of proportion reason thus. If 116. the whole number of beast-gates, give 34060. Perches, the whole quantity, what gives 10. gates; and by increasing 34060. by 10. and parting the product by 116. your answer will be 2936. Perches, and a small quantity more, the proportional quantity belonging to 10. gates; which reduced into acres (by Chap. 48.3. before recited) is 18. Acres, 1. Rood, 16. Perches. And in like manner work with all the rest. And here is to be noted, that having thus gotten the proportional quantity belonging to a beast-gate, and truly understanding the value of a beast-gate, you shall be able at pleasure, and most certainly, to express by the acre the true yearly value thereof.  
And thus have you perfected your entries, for the true and certain quantities, according to measure; and now resteth the valuation.  
The best, speediest, and most certain means for your valuation, in mine opinion, is thus: Let it first be considered, that all grounds generally consist in quality of these three kinds, Meadow, Arable, and Pasture; and supposing every of these kinds likewise to consist of three sorts in value and goodness; as the first and best sort; the second and mean; and the third and worst sort: In your instrumental mensuration, when you writ in your field-booke the title of every field or close, consider with yourself, which of those three sorts the same field or close consisteth of; if of the best sort, express in some place of your title the figure 1. if of the second sort, the figure 2. and if of the third and worst sort, the figure 3. and having informed yourself by the best means you can (which I hold not fitting here to relate) of the general value, what the best sort of Meadow, Arable, and Pasture, is worth by the acre, and the like of the other sorts, throughout the whole Manor; then, according to those rates, pass over your whole book of entries, and value every particular parcel by itself; which, by finding in each title of your field-booke of what sort they are, is most speedily and exactly performed. And thus are your entries thus far perfected.  
But yet, before we proceed to the engrossing hereof, or rather before we finish with the tenants concerning their entries, it is to be inquired and considered, what other profits and commodities, besides these lands and tenements, are demised and granted by the Lord to any tenant within the Manor for yearly rent, or otherwise; which likewise are to be entered and expressed in your rough book, with the rents and yearly values thereof: as Ours of Tin, Led, Copper, Coal, etc. Quarries of Stone, Fishing, Fowling, Hawking, and Hunting, juistments, Herbage, and Pannage, free Warrens, Customary works, or Services, profits of Fairs and Markets, and mosses of Peat or Turf; all or any of which, and the like, may be within a Manor, and disposed and let for yearly rents, which by no means are to be omitted. All which premises, and the several quantities, rents, and values thereof, are here to be summed up, and their several totals expressed.  
Than are you to express the several reprises issuing out of this Manor, being such as are mentioned in the sixth Chapter of this Book. All which being likewise summed up, the total thereof is to be deducted from the former value, and the clear remainder expressed.  
Next are you to consider, if any of those profits and commodities, last before named, or the like, are within this Manor, and not let by lease, or otherwise, for any certain yearly rent; and if any such be, then are they to be here mentioned and expressed as casualties, and the yearly value thereof estimated, what they may or are likely to prove worth by the year.  
Also, the names and quantities of the common Fields, common Meadows, stinted Pastures, and all other unstinted Commons, are here to be expressed; and of those unstinted Commons, how they are accustomed, held, and occupied; whether peculiar to the Lord and tenants of this Manor; or whether any other Lord or Lords, and their tenants, have rake, escape, eatage, or other interest therein; and the butts, bounds, and limits thereof severally.  
Also, what woods and underwoods' are within the Manor, and their several values.  
Than would there be entered an abstract, in nature of a Custome-roll, showing briefly all the ancient customs of and belonging to the Manor. And also a Suit-roll of all the free suitors, etc.  
And lastly, a true and perfect description of all the out-bounds and limits of the whole Manor.  
And having thus perfected your rough Book, you may now call your jury, and receive their verdict, as in the next.   

CHAP. XII. The manner and order of receiving the Juries verdict, and the courses therein to be observed.  
WHen you are drawing towards a conclusion of perfecting your Book, according to the directions of the last Chapter; it were not amiss, that you hasten the jury, in perfecting their verdict, jest you be forced to spend time idly in attendance for the same: Which when they have effected, considering, that for the most part they are unacquainted with matter of form, though in effect and substance they may answer your desire and expectation, according to the articles delivered unto them; you are to call them before you, and reading, examining, and comparing the articles, with their several and particular answers thereunto, reduce the same into an orderly form of an Inquisition, observing still the substance of what they have found and presented: and then having read the same unto them, with their approbation and allowance thereof, cause your Clerk to engross the same accordingly in parchment, and then let them again consider thereof; and having set their hands and seals thereunto, demand of them, if they are mutually agreed on this their verdict; which when they have acknowledged, receive the same from the Foreman, and dismiss your Court.  
And here have you finished what here need to be performed: and now may you leave the Manor of BRANTON; and repairing homewards, may there perfect your plot, as you are formerly directed by the third Book; and engross your Survey, as is hereafter declared in the next.   

CHAP. XIII. The form and order of engrossing a Survey.  
TO prescribe and direct one certain and settled form and course herein for all in general, were impossible, in respect of the variety of occasions offered, according to the nature of the business, and the disposition of those for whom the same is performed: wherefore, the performance hereof must mainly depend upon the judgement, skill, and discretion of the Surveyor. For mine own part, I never yet for any two Lordships or Manors limited myself to one and the same form; but ever framed my course as the cause required: as in one Manor, where I found a commixture of other lands and tenements within the same, being holden of other Lords; here of necessity must I abutt and bound every several particular thereof: but in another, which I found entire, I hold it needless. Again, I found in one Manor divers and several Townships and parts, and those to consist of several estates and tenors; in such case, these are to be severally distinguished and divided, according to their several parts. Another shall you found sole and entire, which is to be ordered accordingly: and many other such like differences shall you found, which will minister occasion to altar any settled form.  
Besides, the will and disposition of him by whom you are employed, shall often 'cause you to altar your course: one perhaps approving of the form you use; another will have it in the nature of your ancient Terrars; a third, in order of a Particular, or Constat; and a fourth, it may be, in a fourth form; for Quot homines, tot sententiae. And again, one, for his own understanding, will have it in English; and another, of better understanding, will require it in Latin. And certainly it were very requisite, although your rough book be drawn in English, that always your engrossed book be written in Latin; unless the contrary be specially required. Yet in these mine examples and directions following, I hold it most fitting to deliver the same in English, for the better understanding of those who have most need; considering, that a reasonable Surveyor may be lame of that leg.  
But notwithstanding such varieties often happen; yet will we for a generality propose these rules and directions following; which I hold most meet and fitting to be observed and held in a formal and well ordered Survey.  
To which purpose, let us now suppose we are to engross a Survey of the Manor of BRANTON; according to the rough book thereof, specified in the tenth Chapter of this Book; wherein first begin with the title, which let be thus, or to the like effect.  
An exact and perfect Survey and view of the Manor of BRANTON, in the County of D. being parcel of the possessions of A. B. who holdeth the same of our Sovereign Lord the King, as of his Manor of G. in free and common Socage, and by the yearly rent of xiii. s. iiij. d. Had, made, and taken there, as well by Inquisition, and the oaths of a sufficient jury in that behalf, as by the view and particular mensuration of all and every the Messages, Lands, and Tenements, of, within, and belonging to the same. Anno Domini 1616. Annoque Regni Regis JACOBI, Angliae, etc. 14.  
By A. R. Superuis.  
NExt after this, or the like Title, in the following leaf, are you to writ and express and Index or Alphabetical Table of all the Tenants names (as hath been formerly taught) with numbers of reference against each name, in what leaf or leaves of the Book each Tenants particular is to be found: But notwithstanding, this Index is to be placed in your Book first and next after the Title; yet is it most convenient and fitting, to collect and writ the same (and also the rental next hereafter following) after the whole Book be engrossed; before which time, you shall not know how to place your numbers of reference therein, according to the number of the leaves.  
Than after this let next be placed a general rental of the whole Manor, but to be divided into such towneships and parts, as your Book is divided into; wherein first express your rents of such demesnes, as are let in lease, than the rents and services of your freeholds of inheritance. Thirdly, of your customary or copyhold tenants. Fourthly, of the tenants for life or lives. Fiftly, of those for term of years: and sixtly and lastly, your tenants at will, wherein let every of these bear their several titles, and under the foot of each kind, let the total thereof be collected and expressed, and in the end or foot of the whole rental, express first the total of every kind, and after that the general and total sum of all together. And if any rents or other reprises be issuing out of this Manor, you may here express them particularly under the title of reprises, which let be deducted out of the former total, and express the clear remainder.  
And thus is our rental finished; but to be collected and written (as I formerly noted) after the body and substance of the Book be engrossed.  
And your rental being thus finished, you may next place (if you please) the out-bounder of the whole Manor; and if any of the towneships or parts thereof lie dispersed and remote (as in many places you shall find them perhaps twenty miles distant, from the chief and principal part of the Manor; and sometimes in another County) it were very fitting and necessary, to express severally the several out-bounders of those towneships and parts. And it is to be noted, that in the expressing of these bounders, a main and principal care is to be had, that you use, observe, and keep the old and ancient names of such meres, marks, and bounds, as have been anciently used and accustomed; for that innovation in this kind is very dangerous for many causes; yet if you find the ancient meres, marks, and bounds, to very few and slender; or any of them decayed and worn out of knowledge, you may add as many more as in discretion you shall find cause; but by any means omit not, or leave out any of those which are ancient and noted bounds. If you think good, these bounders may be placed after, or in the end of the book; which being no matter of necessity whether (so it be had at all) I leave to your discretion.  
And now are you to begin with the body and substance of the book; and first of all with the Manor or mansion house, and the scite thereof; wherein you are to consider, whether the same be in the Lords own hands and occupation; or whether let by Lease, or otherwise unto any Tenant or Tenants, and to enter the same accordingly, as followeth under this Title.  
BRANTON DEMESNES.  
A B. Esquire is Lord of this Manor, and hath at this present in his own hands and occupation, the Manor or mansion house with the scite thereof; and so much of the demesnes thereunto belonging as are hereafter particularly expressed. Which Manor with all and singular the appurtenances, he holdeth of our Sovereign Lord the King, as of his Manor of G. in free and common Socage, and by the yearly rent of xiijs iiijᵈ  
Particulars.  
THe Manor or mansion house called Branton Hall, being fairly built with free stone, and all offices thereunto belonging, with two Stables, one Oxe-house, and a Dove-house; also the scite consisting of three fair gardens, two orchards, two courts, and three out-yards, lying all together between the high street of Branton south, and the Oxe-pasture hereafter mentioned North; abutting East, on Long meadow, and West on the scite of the Parsonage. And containeth together five Acres, two roods, and twenty Perches. a. r. p. 5-2-20.  
Valet per anum— xx livre  
Than next unto the house and scite, express the Parks (if any be) with the number of Deer therein; what number of Aunteller, and what of rascal Dear; also what number of beasts may be therein juisted without prejudice to the game; and also what pannage; and these may you particularly aburt and bound as before if need require; which is most easily and speedily done, having before you the rough plot of the whole Manor, and in the conclusion hereof express the quantity and yearly value as before.  
Than after these Parks, enter particularly all such several fields and closes of the demeans, as the Lord hath in his own use at the time of this Survey; which you may particularly abutt and bound as before, and express the several quantities, and values thereof: But herein for order's sake it were fitting first to enter all the meadow grounds particularly each after other, than the enclosed arable grounds, and next the pastures; and if any of the demesnes are lying in the common fields, then to express them particularly with their quantities and values; also you are to express what woods the Lord hath; and what right or custom the Tenants have or claim therein, either for depasturing or otherwise: and lastly, what wastes the Lord hath within the Manor. And at the foot of this particular, express the total quantity and value thereof.  
But it is to be noted, that all these particulars are to be collected (by help of the numbers in your Index) out of your field book for the names, and out of the rough plot for the several quantities, for that it is needless to enter these lands which are in the Lords hands into your rough book of entries; and the like course also is to be holden for your Glebe lands.  
And if any of the Demesne Lands are in Lease, let them also be entered under the former Title, in this manner.  
C. D. holdeth by Indenture of Lease, bearing date the twentieth day of january An. R. Regis jacobi &c. secundo, made & granted by and from A. B. unto the said C. D. All those lands, etc. (using the very words of grant) for the term of one and twenty years, commencing from and after the Feast of the Birth of our Lord God last passed, before the date of the same Lease, for and by the payment of the yearly rent of— thirty livre  
Particular.  
THen here enter the several and particular Closes, which you may abutt and bound, as before, expressing the particular quantity and value of every several Close, and at the foot of the particular express the total quantity and value, and if the yearly value exceed the rent reserved, deduct the rent from the total value, and express the clear remainder, thus.  
Valet ad demittend.— x●.  
And under this particular express a brief memorandum of the several covenants, clauses, conditions, and provisoes in the lease contained, after this manner.  
The Tenant is to pay his rent quarterly, or within one and twenty days after every Feast, on pain of forfeiture, by proviso to that purpose. He is to do all manner of reparations (except great timber) not to let or set without licence of the Lord The Lord maketh special warranties against his father, himself and his heirs, etc.  
And the like course is to be held for all other Leases, after the particulars expressed.  
And here also under this title of Demesnes, are you to enter all such Mills, Ours of Tin, Lead, Copper, Cole, etc. also quarries of Stone, Slate, and the like; also Fishing, Fowling, Hawking and Hunting; likewise juistments, Herbage, Pannage, free Warrens, customary Works or Services, profits of Fairs and Markets, and also mosses of peat or Turf, and the like, as are let and Markets, and also mosses of peat or Turf, and the like, as are let and demised by the Lord, to any Tenants within the Manor by lease for yearly rend or otherwise: all which (being thus let) are in the nature of demeans, and are to be particularly entered and expressed accordingly, with their several Rents and the yearly values thereof.  
But all of those last mentioned (excepting Mills) are to be severed and distinguished from the Demesne lands, because they are not matters of firm, stable, and certain perpetuity: For notwithstanding, that during the terms of the several leases thereof made and granted, the Tenants may be charged and bound to pay several yearly rents for the same, which for the time being are certain; yet perhaps at the end and expiration of those terms, they may be of little or no value at all; or on the other side of far greater worth and value then now they are, as often happeneth by those mines of Tin, Lead, Copper, Coals, and the like. Wherhfore notwithstanding they are entered under this general title of Demesnes; yet for destinctions sake, let them pass more particularly under this title of Casualties made certain.  
And after all these demesnes are thus entered and engrossed, make a brief conclusion thereof underneath the same, in this or the like manner.  

Conclusion of the Demesnes.  
The Demesnes of this Manor, consisteth of 

Lands, Tenements, & Mills in the use of 

The Lord 

Quantity— 1320 a.— oh r.— o. p.  
Value— 660 li.— oh s.— o. d.    
The Tenant's 

No. of Tenants— 5.  
Quantity— 163 a.— 0 r.— 0. p.  
Rend— 54 li.— 6 s.— 8 d.  
Value— 81-10-0.  
Ad demitt.— 27-3-4.      
Casualties made certain. 

No. of Tenants— 6.  
Rend— 25 li.— 8 s.— 6. d.  
Value— 94-18-0.  
Ad demitt.— 69-9-6.      
And after the Demesnes are thus entered and engrossed, than next unto it place the rectory or Parsonage, and then the Vicarage (if any be) under the proper title thereunto belonging, after this manner.  
The rectory of BRANTON.  
A. B. Clarke, being Parson there, holdeth the rectory of the gift of the Lord of this Manor (if it be so, and if otherwise, express it accordingly) who hath the gift, nomination and presentation thereof, as in the right of this Manor, as often as the same shall happen to be voided, which is valued in the King's books per annum. Lvi ●  
Particular.  
THe Parsonage or Mansion-house with the outhouses belonging thereunto, as a Barn, Stable, Oxe-house, and a Dovecoate, with the scite thereof, consisting of two Gardens, an Orchard, and three out-Yards, which lie together between the high street of Branton South, and a field called the Oxe-pasture North, abutting towards the East on the scite of this Manor, and West on a Lane there leading out of the high street into the Oxe-pasture aforesaid, and containeth together, one Acre and three roods. 1 a.— 3 r.— 0. p.  
Valet per annum— iij li.  
And in this sort let every particular parcel of glebe-Land be expressed with the butts and bounds thereof, which by help of the plot and field-Booke lying before you (being directed thereunto by the numbers in your Index) is instantly and exactly performed: For these glebe-Lands; and the Demesnes which are in the Lords hands, are never entered in your rough Book of entries. Wherein is always to be observed; that you express the true quantity and yearly value of every particular parcel; and in the foot of the particular, the total quantity and value as before. Yet is it not usual neither of these nor the 〈◊〉 holds of Inheritance to express any value at all; which I will refer to your own discretion, and the will and disposition of those by whom you are employed. And in like manner are you to express the Vicarage if any such be.  
And having thus finished your Parsonage, Vicarage and glebe-Lands, proceed next unto the Freeholds within this Towneshippe; which are to be entered and engrossed after this order, and under this title following.  
BRANTON Freeholds.  
Socage.  
A.B. holdeth freely to him and his heirs for ever by deed indented bearing date xxviij. die Marcijs Anno Regni Regis JACOBI Angliae, etc. Sexto made and granted by and from C. D. All that messsage or tenement (expressing the very words of grant) By the yearly rents and services of— Fealty & v. s  
Particular.  
THe Mansion house, outhouses and the scite thereof consisting of one Garden, two Orchards and three out yards, lying together between the high street of Branton North, and the common field called the South-field South; abutting Easton the Churchyard, and West on a lane leading into the South field. And containeth three roods and thirty perches. a. 1. p. 0. 3. 30.  
And thus proceed ●●h every parcel belonging to this freeholder; which being finished, at the foot of this particular express the quantity and value thereof: But as concerning the valuation of Freeholds, unless it be specially required, by reason of some purchase thereof to be made; or a possibility of escheat, or the like, you need not trouble yourself therewith.  
And having perfected your particular, express underneath the same, a brief Memorandum of such necessary observations as you shall found fitting, aswell concerning the Tenant's evidence, as what Heriots, Reliefs and other Duties and Services the Tenant aught to yield, do and perform unto the Lord on every death or alienation.  
And in like manner under the same Title ●nter all other the freeholders within this towneship; after which, collect and express together their several quantities in one total sum, and likewise their values (if it be required as before:) But in these and all others, as I have formerly noted, I would always have an orderly course holden in placing the particular lands of one and the same nature and quality together, as first (after the house and scite) all the meadow grounds, etc.  
And thus having entered and engrossed your freeholds of this towneship, let next be entered your copyholds or customary Tenants, after this manner, and under this Title.  
BRANTON copyholds.  
A. B. holdeth by copy of Court roll, bearing date 26 Februarij Anno Regni Regis jacobi Angliae etc. 4 to. of the surrender of C. D. All that messsage or tenement etc. (using the very words of the Copy) to him and his heirs at the will of the Lord, according to the custom of the Manor. For which he paid Fine on his admittance viˡⁱ per annum, —————. xlˢ.  
Particular.  
LEt your particulars here be entered in all respects as b●fore with the several butts and bounds thereof, expressing the quantity and value of every several parcel: and in the foot of the whole particular express the totals as before; then out of the total value (which admit to be x li.) deduct the Rent, and express the remainder thus.  
Valet in toto per annum x livre  
Viz. and dimittend viii two.  
Than under this particular thus perfected, make (as before) a brief Memorandum of such necessary observations as are sitting, as what heriot (if any be due on the death of the Tenant) what fines on death or aliena●●●; and what other services the Tenant oweth, etc. And after this order and under this title enter all the rest of the copyholds within this towneship.  
And the like course in all respects is to be holden in entry of the Tenants for life, or lives; for term of years; and those at the Will of the Lord; whereof to make several demonstrations, and to deliver several examples, were but great labour to small purpose, seeing they tend all to one and the same end.  
Wherhfore take this for a brief and general rule, that all lands whatsoever, and the tenancy thereof, consist of one of these seven kinds, which in every Manor where they are, are to be used as your several titles, and aught to be placed in the engrossing of your Book, each after other, as here they are expressed, viz. 

1. Demesnes.  
2. Gleabelands:  
3. Freeholds.  
4. Customary.  
5. For lives.  
6. For years.  
7. At william.    
And having after this form and order entered and engrossed the several Lands and Tenements, lying within this Towneship of BRANTON, under the several titles last before mentioned, collect your total of every kind, and in the end of this Towneship make your conclusion to this or the like purpose following.   

Conclusion of the Towneship of BRANTON.  
This Towneship consisteth of  

Demesnes in the use of 

The Lord 

Quantity— 1320 a.— 0 r.— 0. p.  
Value— 660. li.— 0 s.— 0. d.    
The Tenants 

Lands 

No. of Tenants— 5.  
Quantitie-163 a.— 0 r.— 0. p.  
Rend— 54 li.— 6 s.— 8. d.  
Value— 81— 10— 0.  
Ad demitt— 27— 3— 4.    
Casualties made certain, 

No. of Tenants— 6.  
Rend— 25 li.— 8 s.— 6. d.  
Value— 94— 18— 0.  
Ad demitt.— 69— 9— 6.        
Glebe lands Quantity— 56 a— 0-0.  
Freeholds of inheritance, 

No. Tenants.— 7.  
Quantity— 230 a.— 2 r.— 0. p.  
Rend— 13. s. 6. d.— 3. II pepper.    
Customary lands, 

No. Tenants.— 16.  
Quantity— 340 a.— 3 r.— 0. p.  
Rend— 17-16-5.  
Value— 152-13-4.  
Ad demitt.— 134-16-11.    
Tenement lands. 

For lives 

No. Tenants.— 12.  
Quantity— 432 a.— 2 r.— 0. p.  
Rend— 143 li.— 6. s.— 3. d.  
Value— 220-0-0.  
Ad demitt.— 76-13-9.    
For years 

No. Tenants.— 23.  
Quantity— 624 a.— 2 r.— 0. p.  
Rend— 156 li.— 13 s.— 9 d.  
Value— 310-18-0.  
Ad demitt.— 154-4-3.    
At will 

No. Tenants.— 8.  
Quantity— 120 a.— 0 r.— 0. p.  
Rend— 42 li.— 5 s.— 6. d.  
Value— 58-10-0.  
Ad demitt.— 16-4-6.       
And having thus finished this Towneship, proceed in the like form and order in all respects, with all other the towneships and several parts of and belonging to the whole Manor, observing still after every towneship and part, to make such or the like conclusion, as is last before specified; and in the end of all a full and general conclusion of the whole Manor; not forgetting first to enter all reprises, issuing, and going out of the same, which is to be deducted out of the whole value, as is before declared.  
And after this conclusion thus perfected, you are to remember and express all such necessary observations, as are fitting, according to the directions in that behalf delivered, in the latter end of the 11. Chapter of this Book, and your work is finished.  
Now might I here enlarge and amplify this work with many rules and examples, tending to these purposes, but presuming that what I have formerly delivered (being well understood and practised) may sufficiently serve a reasonable capacity; I will forbear to pester the practitioner in reading, or myself in writing of needless varieties; and therefore will here conclude my labours, and expose them to thy good liking.     
FINIS.  

 

  

  

  

   

Man's works have faults, since ADAM first offended,  
And those in these, are thus to be amended.  
ERRATA.  
PAge 35 line 31. for C B. read C D. p. 38. lin. last, for cut out, r. cut. p. 42. l. 9 for equiangles, r. equiangled. p. 77. l. 19 for F D. r. E D p. 110. l. 39 for draw the, r. draw to the. p. 118. l. 11. for 24. r. 240. p. 127. l. 17. for 4/3 r. ¾▪ p. 133. l. 10. for 3/2· or 5/4· r. ½ or ¾· p. 146. l. 13. for 15. r. 9 ¼· p. 162. l. 38. for O Q r. X. Q. p. 204 l. 16. for every other such, r. every such.  
In the Diagrams.  
Page 109. in Diagr. 109. place K. in the angle opposite to C. In Diagr. of Chap. 29.3. near unto K. place O. opposite to N.