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. XXXVII. (Book 37)

How to take the heighth of a Tree several wayes, the better to judge the worth of them, &c. (Book 37)

HAving shewed you how you may judge of Timber, whether it be sound or not, in the last Chapter, I will now shew you how to take the heighth, that you may the better know the worth of it; for where you have a Rule to go by, you may then the better ghess.

There be several wayes to take the Altitude of a Tree or Building that is perpendicular: as by a two-foot Rule, or two Sticks joyned in a right Angle, (that is square as the Figure A. B. C. having at A. a pin or hole to hang a Thred and Plummet on.

Suppose you were to take the height of X Y, first then hold that

end of your square marked with C. to your Eye, then goe backward or forward, till the Thred and Plummet hang just upon the middle of your Square, perpendicular, and your eye looking through two sights, or two Pins at A. and C. or over the ends of the Square, thus look to the very top of the Building at X.

(See Fig. 8, 9.)

Which found, with a Line and Plummet from your Eye at C, let fall to the Ground at D, measure the length of that Line, and adde it to the height, that Length to E; then measure the distance from E. to the foot of the Altitude, as at Y, and that (if your Ground be level) is the height of of X. Y.

Or take the Level from your Eye to the height, and adde that which is below the Level to the Height, &c. as the Line C. F. shew∣eth.

To find the height of a Tree, &c. by a straight Staffe, or by a Line and Plummet, the Sun shining, the Altitude perpendicular, and the Ground Level; if not, you must make the end of both the shadows level to each foot, which is soon done.

As if I should take the Level of B. at C. finding the very top of the shadow to End there, I measure the Distance from C. to B. and find it 60 foot; then at that very instant I set up a stick perpendicular, as E. D. 12 foot long, which I find to cast a shadow just 9 foot; and then the Rule orders it self thus: As 9 foot to 12, so 60 foot to 80, which you will find true, if you work it by Logarithmes, or by Rule and Compass thus; Set one point on 9, extend the other to 12, that Ex∣tent will reach from 60 to 80: Or if you work it by Natural Arith∣metick: as 9 is to 12, so 60 to 80.

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(See Fig. 10.)

The same may be done by Line and Plummet.

To take the Altitude or height by a Bole of Water, or by a Lookingglass placed parallel to the Horizon.

Place on the Ground a Bole of Water, or a Looking-glass, at a con∣venient distance from the Building or Tree, as far as you think the height is, then go back till you espie in the middle of the Water or Glass, the very top of the Altitude; which done, keep your standing, and let a Plum-line fall from your Eye till it touch the Ground, which gives the height of your Eye from the Ground. 2. Measure the distance from your Plummet to the Middle of the water. 3. The distance from the middle of the water to the foot of the Altitude.

Which Distances, if you have measured exactly straight and level, by Proportion you may find the Altitude required, thus:

As the distance from the Plummet level to the Center of the Water or Glass

Is to the height of your Eye from the Ground, which is the Length of your Plum-line,

So is the distance from the Center of the Water to the Base or foot of the Altitude exact perpendicular, to the very top of the height which gave the shadow to the Altitude; for if your Object be not upright, and you measure straight and level, and just under the top that gave the shadow: If you miss in any one of these, you are quite out in taking the height.

Example.

Suppose the Altitude A. B. the Glass or Bole of Water imagine to stand at the prick in the square C. you standing at D. your Eye at E. seeth the top A. in the middle of the square, your distance from D. to the middle of the square, is 7 foot and a half:

Your distance from your Eye to the Ground E. D. 5 foot.

The Distance from the middle of the square to the prick at the foot or base B. is 120 foot.

(See Fig. 11.)

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As 7.5: is to 5, so is 120 to 80 foot; or adde a Cypher to the 600, and a Cy∣pher to the 7 foot and ½, and divide as before.

Thus may you take the heighth exactly.

To take an Altitude accessible, at one station, by the Quadrant.

Suppose A. B. the Altitude as before, take your Quadrant, and look∣ing through the sights thereof go nearer or further from the Altitude, till you see the top at A. through your sights; and also that your thred at the same time fall just at the same distance, upon 45 degrees of the Limb of the Quadrant, then measure the distance, upon a level Line from your Eye, to the Altitude from the place where you stood, and if the Altitude be perpendicular that distance is the height.

But if it happen so that you cannot take sight at that distance, then goe nearer the Altitude, till the thred fall upon 63 deg. 26 min. in the Limb; this distance being doubled, and your height from your Eye to the Ground added, makes the height of the Altitude; if the Ground where you stand be level with the foot of the Altitude; if not, you must make it level.

Or if you find it most convenient to take your sight at a greater di∣stance than where the Line or Thred hangs or falls upon 45 degrees, then goe to the Complement of the last Examp. of 63 deg. 26, till the thred hang upon 26 deg. 34 min. in the Limb; the distance being mea∣sured, and the height of your Eye upon a level to the Altitude added, makes double the height of the Altitude.

These Rules be so plain, there needs no more Examp. but the larger your Quadrant, the better; and note that if the ground be not le∣vel, you must find the Level from your Eye to the foot of the Alti∣tude; and also measure the distance upon a level and straight Line, al∣wayes minding to adde what is below the level of your Eye, to the distance measured.

When you take an Altitude, make use of two of these Rules; the one will confirm the other; for the Rules are all true in themselves; therefore be you so in working them.

Thus having shewed you how to take an Altitude by the most use∣full Instrument, the Quadrant; I shall now shew you how to do it by the Doctrine of Triangles: And if you would be more satisfied in that most usefull and pleasant study, read these Learned mens Works: Mr. Bridges Trigonometria Britannica, Mr. Gellibrans Trigonometrie, Mr. Wings Astronomia Britannica, his Geodatus Practicus, Mr. Win∣gates Ʋse of the Rule of Proportion in Arithmetick and Geometry, or Mr. Newtons Trigonometria Britannica, pag. 51. whose Rules I shall observe, though the Examp. be my own, and as before, to take the height of a Tree.

The Angles and one Leg given to find the other in the Rectangular Triangle A. B. C. the Leg B. C. is inquired.

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The Terms of proportion are thus:

• As the Radius
• To the Leg given,
• So the Tangent of the Angle conterminate with the given Leg
• To the Leg inquired.

Illustration by Numbers.

• As the Radius:—10.0000000
• To the Leg A. B. 80.—1.9030893
• So is the Tangent of B. A. C. 45 d. 10.0001515
• To the Leg B. C. gives 80: 028/1000 1.9032408:

(See Fig. 12.)

You see the difference is not the 28^th part of 1000. and it is worth minding how it doth exactly agree with the first Examp. of the Qua∣drant, &c.

It may be wrought otherwise thus:

• As is the Sign of the Angle opposite to the given Leg
• To the Sign of the Angle opposite to the Leg inquired,
• So the Leg given
• To the Leg inquired.

Or thus:

• As the Tangent of the Angle opposite to the given Leg
• Is to the Radius,
• So is the Leg given
• To the Leg inquired.

Thus have I shew'd you how to take the height of a Tree, or any other Altitude, several wayes; now if you would judge the worth of a Tree standing, first take the height to the very top, or neer it; then

take the Height of the Length of Timber, so far as your Reason tells you you might measure it if it were down; substract the Length of the Timber from the Length of the height of the whole Tree, there then remains the Length of the head. Thus have you the Length of the Timber and Head; Next of all set a Ladder to your Tree, and girt it in such place as is most convenient, allowing for the Bark; then ac∣cording to the customary way of measuring, you may know the Quan∣tity of the Timber, and so consequently the worth of the Timber, ac∣cording to the price where the Timber is standing.

The Timber of the Tree may thus, easily, and neer to the Quanti∣ty be ghessed at: The head will be more difficult, because of the diffe∣rent Forms they grow in; and besides, some Timber-trees head much lower than others, so that for want of helping up, either by their not standing near others to draw or help one another up, or for want of pruning up while young, they head low, and run into great Arms of good lengths of Timber; with such Trees you must goe the higher in∣to the Arms, accompting them with the Timber as your Reason will best direct you.

Now then to estimate this head by Rule, I do judge, that if all the boughs of the head of most Trees were in an intire piece, from the place where they were cut off from the Timber, to the very top; the nearest (and I suppose exactest) Figure of any would be a Cone, or near to a Conical form that the head ends in: For we see that when a Tree is headed, it breaks out into a great many shoots, and as the Tree growes higher in the Lop, some of these shoots decay, still the more endeavouring to end in the figure of a Conical Body: and so the head of your Pollard-trees being greater than the Body, is occasion∣ed by the Sap swelling that place, endeavouring to break out nearest to where it was accustomed to go up the boughs, it searcheth for a passage, till it can contain it self no longer, and so swells the head.

This head commonly goeth with the boughs, and doth the better help them to be allowed this Form, whilest young; so that take a tree headed or never headed, it still ends in this Figure, nearer than any other, especially those that never were headed: this being then the nearest Figure part of the head can be reduced into, this being granted, it is as easily measured; for if you multiply the Basis by one third of the Altitude, the solid Content of the Figure is had, which you may value at such a price as Fire-wood beareth with you.

I will give you one Example, and it shall be of an Ash, which was felled in a place called the Old Orchard, by the Stables at Cashiobury: This tree I observed by several of the Rules before, and found it to

be 80 foot high from the ground to the top-shoot; I also observed the height of the Timber to be 56 foot long; by the same Rules then, setting a Ladder to this Tree about 25 foot high, I girthed it with a pack-thred (which place I took for the middle girth being the Tree did not taper) and it girthed 64 Inches upon the Bark; But most men that buy timber by the foot, have the Bark taken off at the girthingplace, or an Allowance for the Bark; but you may readily know the girth of the Tree under the Bark, though the Tree be standing or ly∣ing, without ever taking off the Bark, or making Allowance by ghess, as some doe; which to perform, find with your Penknife, or Prickers, the thickness of the Bark, or you may cut a hole thorow the Bark in the girthing-places, or two or three holes, and then observe the mean thickness: As on the foresaid Tree, the Bark was half an Inch thick, doubled makes one Inch, so then the tree is less by one Inch in the Diameter when the Bark is off; then by this general Rule, as 22 is to 7, so is the Circumference to the Diameter.

Examp. by the Line of Numbers.

Extend your Compasses from 22 to 7, the same extent will reach from 64 to 20 and near a half, for the Diameter of the Circumfe∣rence of 64.

Examp. by Arithmetick.

As 22 to 7, so is 64 to 20 8/22, the Diameter.

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But the Bark taking one Inch off from this Diameter, it is then 19 Inches and 8/22; then to find the true Circumference under the Bark, agreeing to this Diameter; say thus, As 7 to 22, so is 19 8/22, to 61, the Circumference, or near it.

For if you extend the Compasses from 7 to 22, the same will reach from 19 8/22 to 61, very near.

Or you may turn this 8 into a Decimal Fraction, for as 22 is to 100, so is 8 to 36 of 100 and near a ½.

Thus by four turns of your Compasses on the Line of Numbers you may in a moment find the Circumference under the Bark, which here we find to be 61, then according to the customary way of measuring, (though not the true way) take one fourth part of the Circumference, and say, As 12 is to this ¼ part of the Circumference in Inches, so

is the Length in feet twice repeated, to the content in feet or parts.

Examp.

The fourth part of 61 is 15 and ¼ Inches, extend your Compasses from the point 12. on your Line of Numbers, to 15 ¼: that extent will reach from 56 to 91 foot and neer a half, being twice turned to the Right hand, which must alwayes be if the ¼ of your Circumference be more than 12 Inches, if less, then to the Left hand.

Thus having found the Timber of this Tree to be 91 foot and a half (which must be valued according to the worth of Timber in the place where you are) here I will value it at 12 d. the foot, (though it was sold with courser Timber at an under-price) this at 12 d. the foot, comes to 4 l. 11 s. 6 d. Now to measure the head of this Tree accord∣ing to our supposed Rule aforesaid, if you take 56 (the Length of the Timber) from 80 (the height of the Tree) there remains 24 foot for the head: This 24 foot, which is the head, two parts of it I measure as a solid Cylinder, and one part as a Cone: for if you girt all the boughs a little above where they break out from the Timber, and adde the several girths together, they will girth more than the Timber where the head was cut off; two parts of the 24 foot is 16, which I measure by the same Rule I did the Timber, by the Line of Num∣bers, and the same Girt.

As 12 is to 15½ (the square in Inches,)

So is 16 (the Length in feet twice repeated) to 26 foot ¾ for the ⅔ of the Head:

Now for the ⅓ of the Head, which must be measured as a solid Cone, we must find the Basis, and multiply the Content thereof by one third of the Altitude.

The Circumference of the Basis was 61, but I will take it now for 60 Inches; then as before, As 22 is to 7, so is 60 (the Circumference) to the Diameter, which is 19 and a little more; that which is more I neglect, as not worth minding in such a business as this.

Half 60 is 30, half 19 is 9 and ½. Ex. to work it by the Line of Numbers:

Extend the Compasses from 1 to 9 and ½, the same will reach from 30 to 285, the Content in Inches of the Basis, then ⅓ of 8 foot is 32 Inches;

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Then say, As 1 to 32, so is 285 to 9120 〈 math 〉〈 math 〉 Inches, which divided by 1728 (the In. in one foot square) is 5 foot and ¼, and a little more; this added to 26 foot ¾, make 32 foot for the Head, which at a Groat per foot,

is 10 s. 8 d. But this must be valued according the Countrey you are in.

So according to these Rules, this Tree was worth 5 l. 2 s. 2 d. there was of the Head one Stack, and near a quarter, and 12 Faggots; it cost 3 s. per Stack to cut out, and was worth in the place where it stood, 10 s. You see how near the Rule agrees with this tree; but if the tree stands hanging down hill, it will then endanger the Timber in falling; or if you fear the Timber to be faulty, or some of the Arms blown off, you must judge-accordingly, &c.

It is oft found in your great Elms and Ashes, that they be hollow within and yet good Timber on the out-sides, especially some length of them toward or at the lower end: Now I will shew you how you may judge very near the Quantity of Timber that is in one of these trees, or a piece of them, according to the customary way.

Examp.

A piece of a Tree 12 foot long, and hollow and decayed wood at one end 12 Inches Diameter, at the other end hollow and decayed 6 Inches Diam. The piece was 26 Inches Diameter under the Bark. First I measure the piece as if all sound Timber, then the hollow and decayed by its Diameter; that being deducted from the piece, shew∣eth how much found Timber there is in foot and parts, measured the customary way.

As 7 to 22, so 26 to 81 5/7 the Circumference; the ¼ of it is 20 and ¼, and somewhat more: as 12 to 20 ¼ in Inches, so is 12 (the Length in feet twice Repeated) to 39 foot and ½, the Content as if it were all sound.

Then for the hollow, I take the mean Diameter to be 9 In. As 7 to 22, so 9 to 28 and 2/7, the Circumference of the hollow ¼ is 7 In. then as 12 to 7, so 12 foot to 4 and ¼ near; which taken from 39 foot and ½, leaves 35 foot and ¼ for the sound Timber of that piece.