The manner of raising, ordering, and improving forrest-trees also, how to plant, make and keep woods, walks, avenues, lawns, hedges, &c. : with several figures proper for avenues and walks to end in, and convenient figures for lawns : also rules by M. Cook.

CHAP. XXXIX. (Book 39)

Of several Superficial Figures, and how they are to be measured. (Book 39)

TO speak of all sorts of Figures will be far beyond my intentions, there being so very many irregular Figures which have many un∣equal sides and angles; but they may all be brought into parts of some of the Figures following, and Measured like them; I shall shew you one Useful Prob. especially to make your Ovals by, whether they be made from two Centres, or four; and then I shall touch at some Su∣perficial Figures. (See Fig. 30.)

Suppose three pricks or points given (so they be not in a strait line) to find a Centre to bring them into a Circle. This may be done seve∣ral ways, viz. either by Circles, or by raising Perpendiculars; as if the points at A. B. C. were to be brought into a Circle: Draw a line from A. to B. and in the middle of that line raise a Perpendicular, as the line D. E. which you may soon do; for if you open your Com∣passes to any convenient distance, and set one point in B. draw the Arch 1. and 2. then setting one point in 4. draw 3. and 4. where these cross draw the line E. D. Do the same with the points B. C. and where the two Perpendicular lines meet is the Centre, as at F, &c. Superficial Figures that are irregular and right-lined, are such whose Sides or Angles are un-equal, of which some are triangles, or triangular Figures; and here Note, that there are five sorts of triangles, which are thus Named and known:

• 1. Isocheles hath two of the sides unequal.
• 2. Scalena hath the three sides unequal.
• 3. Orthygone hath one Right and two Acute Angles.
• 4. Ambligone hath one Obtuse and two Acute Angles.
• 5. Oxygone hath three Acute Angles, or Equilateral triangles.

(See Fig. 31.)

Every triangle is half of a square, whose Length and Breadth is equal to the Perpendicular, and Side cut by the Perpendicular; as is plain in the first Figure shewed by the pricked lines: therefore to Measure

any triangle, raise a Perpendicular from the Base, to the greatest An∣gle:

Then Multiply the whole Base by half the Perpendicular, or the whole Perpendicular by half the Base, and the Product is the Content. Or thus, take the whole Base and whole Perpendicular, and Multiply one by the other, the half of that Summe is the Content of the triangle, &c.

Square or Quadrangular Figures are these following.

• 1. A Geometrical square; this hath Right Angle, and sides equal.
• 2. An Oblong-square, which hath equal opposite sides and Rectang.
• 3. A Rhombus, hath equal Sides, and unequal Angles.
• 4. A Rhomboides, having unequal Sides, and Angles opposite, equal.
• 5. Trapezia, Are all other four-sided Figures.

(See Fig. 32.)

The first is Measured by Multiplying one of the Sides in its self. In the Second the length Multiplyed by the breadth, gives the Con∣tent:

The three last may be turned into two triangles each, and so Mea∣sured as is before said.

Polygones are these Figures following: as the end of a Tree hewed into five equal sides, this is called a Pentagone; of six sides, Hexagone; seven sides, Heptagone, eight sides, Octagone; nine sides, Enneagone; ten sides, Decagone; twelve sides, Dodecagone. To Measure any of these, take half the perimeter (that is, half the Compass about) and the perpendicular drawn from the Centre to the middle of any one of the sides, Multiply the one by the other, and it giveth the Content.

Circular Figures are these, which be thus Named:

• 1. The Circle, is near Equal to a square, made of ½ Diameter, and ½ Circumference.
• 2. The Semi-Circle, to a square made of half the Arch line, and ½ Semi-diameter.
• 3. The Quadrant, or fourth part of a Circle.
• 4. The Segment, Arch, or part of a Circle.

The first is Measured by Multiplying the Semi-circumference by the Semi-diameter. The second, by Multiplying the Radius or Semi-dia∣meter by ¼ of the Circumference of the whole Circle. The third, by Multiplying the Radius by ⅛ of the Circumference of the Circle that it was made of. The fourth by Multiplying the Radius by ½ the length of that Arch-line: thus have you the Content or Area of each.

To find the Diameter of any Circle, or the Circumference, by ha∣ving

one given; the lowest Number is, as 7 is to 22. so is the Dia∣meter to the Circumference; or as 22 is to 7. so is the Circumfe∣rence to the Diameter.

To find the Length of an Arch-line Geometrically.

This Problem is Useful to be known, for to Measure the Quadrand, Segment of a Circle, or Oval; for the Oval is made of parts of the Circle.

First, Divide the Chord-line of the part of the Circle into four equal parts; then set one of these parts from one End of the Chord-line, also set one of the four parts from the Angle in the Arch-line; then from one point to the other, draw a Line; the length of this Line is half the length of the Arch-line.

(See Fig. 33.)

Examp. A. B. the Chord-line, Divided into 4 parts, one of the 4 parts set from B. to C. and one part set from A. to D. then draw the Line C. D. which Line is half the length of the Arch-line A. D. B. which was to be found out.

Thus may you Measure this part of a Circle, or the like; but if the part of a Circle be greater than a Semi-circle, then Divide the Arch∣line into two Equal parts, and find the length of one of these, as is afore-said; which doubled giveth the length of the whole Arch-line: This Rule will assist you to Measure the Oval, whether it be made from two Centres or four, &c.

There is no regular Figure but may be Reduced into some of these Figures afore-said, therefore I shall shew you the Use of some Geome∣trical Figures, which are very Useful; not Questioning but that you Understand the first Rules in Geometry, as, to draw a parallel Line, to Raise a perpendicular-line from another, &c. for those things are out of my intended Discourse, therefore if you be to seek in them, consult with Euclid and others.

How to Raise a Perpendicular at the end of a Line, by which you make a Square, very Ʋseful also to set off a square∣line from a strait-line, in any Garden, Walk, House-end, or the like.

(See Fig. 34.)

Examp. If you be desired to set off a square-line at B. from the Line A. B. take six Foot, Yards, or Rod, and Measure from B. to C. in your strait-line, then take eight of the same Measure and set from B. to D.

and ten of the same, holding one end at C. bring the Line B. D. till it just touch the Line C. D. at D. so have you an Exact Square made by 6. 8. and 10. See Euclid first Book, Prob. 47. and p. 35. Math. Recreations, p. 93.

(See Fig. 35.)

This you may do in other Numbers that bear the like proportion; for Euc. tells you, that the square made of the side subtending the Right Angle, is Equal to the squares made of both the sides containing the Right Angle; for 10 times 10 is a 100. and 6 times 6 is 36. and 8 times 8 is 64. so 36 and 64. make 100. equal to the subtended square. There be several other ways to Raise a Perpendicular at the end of a Line, but this being so easie, and the most useful, I shall not name any other.