The compleat surveyor containing the whole art of surveying of land by the plain table, theodolite, circumferentor, and peractor ... : together with the taking of all manner of heights and distances, either by William Leybourn.

GEOMETRICALL DEFINITIONS.

1. A Point is that which cannot be divided.

[illustration]

A Point or Signe is that which is void of all Magni∣tude, and is the least thing that by minde and understanding can be imagined and conceived, than which there can be nothing lesse, as the Point or Prick noted with the letter A, which is neither quantity nor part of quantity, but only the terms or ends of quantity, and herein a Point in Geometry differeth from Unity in Number.

2. A Line is a length without breadth or thicknesse.

[illustration]

A Line is created or made by the moving or drawing out of a Point from one place to another, so the Line AB, is made by mo∣ving of a Point from A to B, and according as this motion is, so is the Line thereby created, whe∣ther streight or crooked. And of the three kindes of Magnitudes in Ge∣ometry, viz. Length, Breadth, and Thicknesse, a Line is the first, consist∣ing of Length only, and therefore the Line AB, is capable of divisi∣on in length only, and may be divided equally in the point C, or un∣equally in D, and the like, but will admit of no other dimension.

3. The ends or bounds of a Line are Points.

[illustration]

This is to be understood of a finite Line only, as is the line AB, the ends or bounds whereof are the points A and B: But in a Circular Line it is other∣wise,

for there, the Point in its motion returneth again to the place where it first began, and so maketh the Line infinite, and the ends or bounds thereof undeterminate.

4. A Right line is that which lieth equally between his points.

[illustration]

As the Right line AB lyeth streight and equall between the points A and B (which are the bounds thereof) without bowing, and is the shortest of all other lines that can be drawn between those two points.

5. A Superficies is that which hath only length and breadth.

[illustration]

As the motion of a point produceth a Line, the first kinde of Magnitude, so the motion of a Line produceth a Superficies, which is the se∣cond kinde of Magnitude, and is capable of two dimensions, namely, length and breadth, and so the Superficies ABCD may be divided in length from A to B, and also in breadth from A to C.

6. The extreams of a Superficies are Lines.

As the extreams or ends of a Line are points, so the extreams or bounds of a Superficies are Lines, and so the extreams or ends of the Superficies ABCD, are the lines AB, BD, DC, and CA, which are the terms or limits thereof.

7. A plain Superficies is that which lieth equally between his lines.

So the Superficies ABCD lieth direct and equally between his lines: and whatsoever is said of a right line, the same is also to be understood of a plain Superficies.

8. A plain Angle is the inclination or bowing of two lines the one to the other, the one touching the other, & not being directly joyned together.

As the two lines AB and BC incline the one to the other, and touch one another in the point B, in which point, by reason of the

inclination of the said lines, is made the Angle ABC. But if the two lines which touch each other be without in∣clination, and be drawn directly one to the other, then they make no angle at all, as the lines CD and DE, touch each other in the point D, and yet they make no angle, but one continued right line.

[illustration]

¶ And here note, that an Angle commonly is signed by three Letters, the middlemost whereof sheweth the angular point: As in this figure, when we say the angle ABC, you are to understand the very point at B: And note also, that the length of the sides con∣taining any angle, as the sides AB and BC, do not make the angle ABC either greater or lesser, but the angle still retaineth the same quantity be the containing sides thereof either longer or shorter.

9. And if the lines which contain the angle be right lines, then is it called a right lined angle.

So the angle ABC is a right lined angle, because the lines AB and BC, which contain the said angle, are right lines. And of right lined Angles there are three sorts, whose Definitions follow.

10. When a right line standing upon a right line maketh the angles on either side equall, then either of those angles is a right angle: and the right line which standeth erected, is called a perpendicular line to that whereon it standeth.

[illustration]

As upon the right line CD, suppose there do stand another right line AB, in such sort that it maketh the angles on either side thereof equall, namely, the angle ABD on the one side, equall to the angle ABC on the other side: then are either of the two angles ABC, and ABD right angles, and the right line AB, which standeth erected upon the right line CD, without inclining to either part thereof, is a perpendicular to the line CD.

11. An Obtuse angle is that which is greater than a right angle.

[illustration]

So the angle CBE is an obtuse angle, because it is greater than the angle ABC, which is a right angle; for it doth not only con∣tain that right angle, but the angle ABE also, and therefore is obtuse.

12. An Acute angle is lesse than a right angle.

So the angle EBD is an acute angle, for it is lesse than the right angle ABD (in which it is con∣tained) by the other acute angle ABE.

13. A limit or term is the end of every thing.

As a point is the limit or term of a Line, because it is the end there∣of, so a Line likewise is the limit and term of a Superficies; and a Superficies is the limit and term of a Body.

14. A Figure is that which is contained under one limit or term or many.

[illustration]

As the Figure A is contained under one limit or term, which is the round line. Al∣so the Figure B is con∣tained under three right lines, which are the limits or terms thereof. Likewise, the Figure C is contained under four right lines, the Figure E under five right lines, and so of all other figures.

¶ And here note, that in the following work we call any plain Superficies whose sides are unequall, (as the Figure E) a Plot, as of a Field, Wood, Park, Forrest, and the like.

15. A Circle is a plain Figure contained under one line, which is called a Circumference, unto which all lines drawn from one point within the Figure, and falling upon the Cir∣cumference thereof are equall one to the other.

[illustration]

As the Figure ABCDE is a Circle, contained under the crook∣ed line BCDE, which line is called the Circumference: In the middle of this Figure is a point A, from which point all lines drawn to the Circumference thereof are equall, as the lines AB, AC, AF, AD: and this point A is called the center of the Circle.

16. A Diameter of a Circle is a right line drawn by the Center thereof, and ending at the Cir∣cumference, on either side dividing the Circle into two equall parts.

So the line BAD (in the former Figure) is the Diameter there∣of, because it passeth from the point B on the one side of the Cir∣cumference, to the point D on the other side of the Circumference, and passeth also by the point A, which is the center of the Circle. And moreover it divideth the Circle into two equall parts, namely, BCD being on one side of the Diameter, equall to BED on the other side of the Diameter. And this observation was first made by Thales Miletius, for, saith he. If a line drawn by the center of any Circle do not divide it equally, all the lines drawn from the center of that Circle to the Circumference cannot be equall.

17. A Semicircle is a figure contained under the Diameter, and that part of the Circumference cut off by the Diameter.

As in the former Circle, the figure BED is a Semicircle, because it is contained of the right line BAD, which is the Diameter, and of the crooked line BED, being that part of the circumference which is cut off by the Diameter: also the part BCD is a Semicircle.

18. A Section or portion of a Circle, is a Figure contained under a right line, and a part of the circumference, greater or lesse then a semicircle.

[illustration]

So the Figure ABC, which consisteth of the part of the Cir∣cumference ABC, and the right line AC is a Section or portion of a Circle greater than a Semi∣circle.

Also the other figure ACD, which is contained under the right line AC, and the part of the cir∣cumference ADC, is a Section of a Circle lesse than a Semicircle.

¶ And here note, that by a Section, Segment, Portion, or Part of a Circle, is meant the same thing, and signifieth such a part as is either greater or lesser then a Semicircle, so that a Semicircle cannot properly be called a Section, Segment, or part of a Circle.

19. Right lined figures are such as are contain∣ed under right lines.

 

20. Three sided figures are such as are contained under three right lines.

 

21. Four sided figures are such as are contained under four right lines.

 

22. Many sided figures are such as have more sides than four.

 

23. All three sided figures are called Triangles.

[illustration]

And such are the Triangles BCD.

24. Of four sided Fi∣gures, a Quadrat or Square is that whose sides are equal and his angles right. [As the Figure A.]

[illustration]

25. A Long square is that which hath right an∣gles but unequal sides. [As the Figure B]

[illustration]

26. A Rhombus is a Figure having four equall sides but not right angles. [As the Figure C.]

[illustration]

27. A Rhomboides is a Figure whose opposite sides are equall, and whose opposite angles are also e∣quall, but it hath neither e∣quall sides nor equal angles. [As the Figure D.]

[illustration]

28. All other Figures of four sides (besides these) are called Trapezias.

[illustration]

Such are all Figures of four sides in which is observed no equality of sides or angles, as the figures A and B, which have neither equall sides nor equall angles, but are described by all adventures without the ob∣servation of any order.

29. Parallel, or equidistant right lines are such which being in one and the same Superficies and produced infinitely on both sides, do ne∣ver in any part concur.

[illustration]

As the right lines AB and CD are parallel one to the other, and if they were infinitely extended on either side would never meet or concur toge∣ther, but still retain the same distance.