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. XL. (Book 40)

To Divide a Right Line given, according to any Proportion Required; and how to Divide Land or Woods, with some Ʋses of the four-pole Chain. (Book 40)

THE Line A is Required to be Divided into two such parts, which shall have proportion the one to the other, as the Line B. hath unto C.

(See Fig. 36.)

Make an Angle of any Quantity, as A. B. C. and let the side A. B. be Equal to the given Line A. then set the Line C. from A. to E. and the Line B. from E. to C. and from the point E. draw the Line E. D. parallel to C. B. cutting the Line A. B. in D. so shall B. D. bear the same proportion to D. A. as the Line B. hath to the Line C. this is the Golden Rule by Lines.

To Divide a Right Line in Power, according to any Proportion given.

(See Fig. 37.)

Draw the Line C. D. Equal to the two Lines A. & B. then Divide the said Line in the point F. in proportion as A. to B. then in the middle of the Line C. D. Describe the Semi-circle C. E. D. and on the point at F. Raise the Perpendicular F. E. cutting the Semi-circle in E. from that 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. C. shall be in such proportion to the power of the line E. D. as A. to B.

Many Men when their Woods are felled, sell their Wood by the Acre, or half-Acre, or sometimes two or three Acres; and sometimes Men Let their Land to Plow by the Acre, and sometimes Men purchase part of a Field by the Acre: I will here shew you how you may cut off parts of an Acre, or several Acres from a Field; or how far you must go in a VVood or Field to make an Acre, more or less, of several Fi∣gures; which will be Useful to be known of most Men, for several other Occasions.

Suppose a VVood, or part of it, should be in a Triangle, as the Fi∣gure following, which should contain 745 pole 42/100. that is 4 Acres 105 Pole, and near a ½. Of this VVood there is Sold 2 Acres, which is to be taken off from the Angle C. and to cut the line A. B. having Measured your Triangle, and found it as above-said, and also the Base∣line to be 84 pole, then by the Rule of Three work it thus:

(See Fig. 38.)

If 745' 42. (the Content of the whole Triangle) have for its Base 84 Pole or Rods; what shall 320 pole have for its Base? (that is 2 Acres) See it wrought by Logarith.

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Here you see that this Log. gives 36 pole and 6/100. so that you must go 36 pole and little more on the Base-line A. B. from A. to D. for

your 2 Acres; then the Angle A. C. D. is the two Acres, or 320 Pole, and the Angle C. D. B. is 2 Acres 105 pole, and near ½ a pole.

By the same Rule may you cut off what Number of poles you please, from the Angle D. to fall upon the Line C. B. or to fall upon the Line D. B. having but the length of the Lines given you.

But if it be Required to take off a part from a Triangle according to any proportion given, by a Line drawn parallel to any of the sides assigned.

As let A. B. C. be a Triangle containing 5 acres, and it is desired to cut off 2 acres, by a Line drawn parallel to A. B. first, on the Line A. C. draw the Semi-circle A. E. C. and the Diameter C. A. Divide into 5 equal parts, and from the point of 3 of these parts of that Line draw the perpendicular D. E. to cut the arch-line in E. then set the length of C. E. from C. on the Diameter-line, and it will reach to the point F. then from that point at F. take the nearest distance to the Line A. B. and set that distance off from B. to G. then draw the Line F. G. exactly parallel to A. B. so will the Triangle C. G. F. be 3 acres, and G. B. A. F. 2 acres, the thing propounded.

(See Fig. 39.)

This Rule in it self is exact, but in a large Field or Wood it is diffi∣cult to be done, because the Semi-circles and other Lines are very hard to be drawn exactly.

But if your Field, or part of it, be a square, and you are to take off some parts of it, you may do it to any Number of Rods desired, easily and exactly, thus:

Let the Field be never so great, Measure you onely that side of the Square whence you are to take off your part, exactly.

(See Fig. 40.)

Examp. It is Required to cut off 2 acres, or 320 Pole from a Field, or part of one, that is in form of the Square A. B. D. C. with a Line drawn parallel to the side A. B. Now, finding the side of the Square to be 32 Pole; Divide 320 (the parts you are to cut off) by 32, the side of the Square, and the Quotient will be 10; then set off 10 Pole from A. to E. and from B. to F. and the Square A. B. F. E. is 2 acres, as was Required.

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This is very Useful for several Men, and readily to be performed; but if these sides A. C. or B. D. do not go Square from the end A. B. then must you find the mid-line of the Square you are to take, and Divide the Summe of Poles you are

to take off by that; the first Example will assist you to find this mid∣line, and somewhat help you in the working.

This being such an usefull Prob. I shall shew you how to perform it another way, as in the last Examp. the side of the square 32 Pole, and you know 160 Pole make one Acre; then divide 160 by 32 (the side of your Square) the Quotient is 5; which tells you, that you must mea∣sure 5 Rod, or Poles, from the side of your Square on each End, to make one Acre, 10 Pole for two Acres, 15 Pole for three Acres, &c. which you see agrees with the former Rule.

But if it be required to take off the parts of a Square, and to have those parts in a Triangle, then the first and second Figures will assist you how to perform that.

To divide an Irregular Figure into any parts required; that is, to take what number of Rods you please from such a Figure.

As, if A B C D E. be the Figure of a Field or Wood, and it is de∣sired to take off the half of it from the Angle at A. the whole Figure is 705 Pole, then the half is 352.50, and the Triangle A D E. is but 290 Pole, which wants 62 Pole and a half of the half of the Field; therefore take 62. 50 from the Triangle A C D. by the Rules deli∣vered in the 38th. Fig. and there will be added the Triangle A D H, which being added to the Triangle A D E, will divide the Figure into two equal parts, the thing desired.

(See Fig. 41.)

Thus may you take half from any irregular Figure, or more, or less than half, and from what Angle desired, which will assist you well how to fell your Woods by the Acre, or to know how far you shall go into a Field, to take off any parts; the fourth Rule (which I found by my Practice) I commend for very good.

One Example I shall give you more, which shall be according to my fourth Rule: I try'd it in a Field near to Cashiobury-Park; this I was ordered to doe by my Lords Steward Mr. Sydenham, to take off three Acres from a small Field as exactly as I could, at one End ap∣pointed by him.

First I measured that End, and found it to be 37 pole and ¾, but observing the Hedges, I found them to splay off a little, so that about 6 Rod and a half, or little more, would be the middle, which 〈 math 〉〈 math 〉 I set off at each End, and found that Line to be 38 Pole long; then I divided 160 (the Poles in one Acre) by 38 (the Poles of the End of the Field) the Quotient was 4 and 8/38, which

8 of 38 I must turn into parts of my Chain, that is, into Decimal parts, thus: As 38 is to 8, so is 100 to 21 and 2 of 38, which 2 is not considerable; So that if the Chain be divided into 100 〈 math 〉〈 math 〉 Links, you must then goe 4 Pole and 21 Links at each End for to make one Acre of Ground; but if your Chain be a four pole Chain, divided into 100 Links, then with such a Chain you must goe 4 pole 5 links and ¼ and a little more, to make one Acre at 38 Pole long: Then for three Acres I must goe in breadth 12 Pole, 63/100 to make three Acres and a little more, see it proved.

Here you may see that 12 Pole 63/100 multiplyed by 38 〈 math 〉〈 math 〉 Pole, gives 479 Pole and 94/100, which being divided by 160 (the Poles in one Acre) gives in the Quotient 2 and 159; so then if you adde but 6 of 100, to the 94, it is just three Acres; for whereas I take, in the Decimal parts, but 21/100, I should take the 21 Links, and the 22th part of one of these Links, which niceness may be dispensed with.

From what hath been said, you may measure any stand∣ing Wood, or part thereof, especially if these parts be near to a Square or Triangle, if not, you may Reduce them to one of these.

Thus having spoke something how superficial Figures are to be measured, I shall give an Example or two of the Chain, and it shall be of the Four-pole Chain, divided into 100 parts; as suppose the Fi∣gure A B C D.

(See Fig. 42.)

This Figure may be measured several wayes; as first it may be put into two Triangles, and so measured, or else you may measure both the Ends, and half them, and so measure the Length in. the middle; you may measure also both the sides and half them, and then measure the breadth in the middle. But for Example: First I measure the side A B. and find it to be 15 Chains and 80 Links of the Four-pole Chain, the End B C. is 6 Chains 74 Links, the other side C D. is 12 Chains 50 Links, and the other End D A. is 6 Chains. Then adde the two sides together, of which take 〈 math 〉〈 math 〉 the half, that half is the mean Length: both sides added together, make 28 Chains 30 links; half of which is 14 Chains 15 links; then adde the Ends together, viz. 6 Chains, and 6 Chains 74 links, the total of both is 12 chains 74 links, then half of the

Ends added together, is 6 chains 37 links: Then multiply the mean Length by the mean Breadth, and cut off 5 Figures to the Right hand, and whatsoever Figures Remain to the Left hand, are Acres; and those 5 Figures cut off are parts of an Acre.

Thus may you know the Content of a Field without 〈 math 〉〈 math 〉 Division; as in the last Examp. 14. 15, multiplyed by 6. 37, gives 901355, then if you take off five figures, as the fractional parts, there remains 9, which is nine Acres two Pole and above ⅛ of a Pole. But you may easily know the fractional part of any Decimal fracti∣on thus: This belongs to 100000: for if the Decimal fraction have 5 Figures, the Integer is 6, the fraction 4, then the Integer 5. &c.

Then work it by the Rule of Three, or by your Line of Numbers, thus: As 100000 is to 1355, so is 160 (the square Poles in one Acre) to 2 Poles and neer ¼, but that you may be the better satisfied in this most useful Rule, if 100000 be Equal to one Acre, or 160 Pole, 〈 math 〉〈 math 〉

So that when any Fraction is, repair but to these Rules, and you may see what Number of poles is equal to it: you may proportion it to half-poles, &c. for, 〈 math 〉〈 math 〉

Not onely to prove this, but also to shew you how much readier this way is, than the 100 Links, to bring it into Rods or Poles, then divide it by 160, to bring the aforesaid Measure to the one Pole Chain and 100; multiply 14. 15 by 4, it gives 56. 60; and 6. 37 multi∣plyed by 4 gives 25. 48, which being multiplyed one by the other, gives 1442 1680/10000: I will neglect the Fraction as being not ¼ of a Pole, and divide 1442 (the Poles in that Measure) by the sq. Poles in one Acre (160 Pole) and the Quotient is 9 and 2 over, that is, 9 Acres 2 Pole

and a little more, as before. But how much the other way is readier than this, I leave the Reader to judge.

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Example the Second.

How to measure a Triangle with the Four-pole Chain, and never use Division.

As in the Triangle A B C. the Base A C. is 40 Pole, and the pricked Perpendicular Line is 20, the half is 10 Pole; Now when you have even Poles (as in this Examp.) you must adde two Cyphers to the length, and two to the breadth; or else you cannot take off the 5 Fi∣gures, or 5 Cyphers, as is before shewed; then taking the 5 Cyphers off after Multiplication, there remains 40: 〈 math 〉〈 math 〉 which sheweth you that such a Triangle, that hath such a Base, and such a Perpendicular, containeth four Acres of Ground: And if you work it the common way, you will find it to be true.

(See Fig. 43.)

But to assist you yet further to turn

	M.	C.	X.
	ac. ro. p.	ac. ro. p.	ro. po.
1	6. 1. 0	0. 2. 20	0. 10
2	12. 2. 0	0. 1. 0	0. 20
3	18. 3. 0	1. 3. 20	0. 30
4	25. 0. 0	2. 2. 0	1. 0
5	31. 1. 0	3. 0. 20	1. 10
6	37. 2. 0	3. 3. 0	1. 20
7	43. 3. 0	4. 1. 20	1. 30
8	50. 0. 0	5. 0. 0	2. 0
9	56. 1. 0	5. 2. 20	2. 10

Poles into Acres, observe this Table: The Denominations of the several Numbers are known by the Marks under which they are set, as all under Ac. are Acres, under Ro. are Roods, under Po. are so many Pole; and so the first Column under M. answereth to Thousands, that under C. to Hun∣dreds, that under X. to Tens, and the odde Pole (if any be) are set down under Pole: As e. g. 1442 Pole: To

know how many Acres by this Table: first for the One thousand in the Table under M. is 6 Acres one Rood, set that down as you see in the preceding Page; then four Hundred under C. and against 4 is 2 Acres 2 Roods, set that down; then in the Ta∣ble 〈 math 〉〈 math 〉 under X. and against 4, is one Rood, set that down; then the odde Poles set down al∣wayes under the Poles, as 2 under Poles; then summe them up, and you shall find it is 9 Acres 2 Pole, as before: This Table being so plain, there needs no more Examples.

A Table of superficial long Measure, from an Inch to a Mile, ac∣cording to the Standard of England.

Inch.
12	A foot.
36	3	A yard.
45	3 ¾	1 ¼	Ell.
198	16 ½	5 ½	4 ⅖	Pole.
7920	660	220	176	40	Furlong
63360	5280	1760	1408	320	8	Mile.

A Table of square Measure.

Acres.	4	160	4840	43560
	Rood.	40	1210	10890
		Pole.	30 ¼	272 ¼
			Yards.	9
				Feet.

An Example of the Table of long Measure.

Suppose you were to find out how many Inches were in a Pole long; look under Inches, and against Pole, there is 198; and so many Inches are in a Pole long, and 16 ½ Foot, 5 ½ Yards:

And in the Table of Square Measure, to know how many square Yards is in a Pole, look against Pole, and above Yards there is 30 ¼, the square yards in a Pole.

There be several other sorts of superficial Measures, as Pavings, Plaisterings, Wainscotings and Painting; which are to be measured by the Yard square, and may be measured by some of the Rules before shewed; your readiest way is by the Yard divided into ten parts, so will you odde Measure come into Decimal Fractions, which are as easily cast up as whole Numbers: Or if you measure by the Foot Rule, have it divided into 10 parts, and when you have found the Content in feet, divide it by 9, the Quotient will shew you how many yards; and if any remain, they be feet.

Some sorts of Work are measured by the square of 10 foot the side, so that such a Square is 100 foot; for ten times Ten is a Hundred: By this Measure is your Carpenters Work measured, as Floors, Partiti∣ons, Roofs of Houses; So also is Tiling and Slatting measured; this is very ready to measure and to cast up: for if you multiply the Breadth by the Length, so many hundreds as you find, so many Squares are there; and what remains are parts of a Square.

Board and Glass, &c. are measured by the foot, which may be divi∣ded into ten parts, which will be much easier to count up.

But if you would be more fully satisfied in the Rules of Surveying, see the work of Mr. Leyborn, Mr. Wing, Mr. Rathborn, &c.

Having the Length of a Field, to know what Breadth will make one Acre of Ground, by the Four-pole Chain and Line of Numbers.

Ex. The Length is 12 Chains 50 Links, to find the Breadth to make that Length just one Acre, do thus: Extend your Compasses from 12.50 (the Length) to 10, that Extent will reach from one to 80, which is the Breadth in Links to make one Acre; for 〈 math 〉〈 math 〉 if you multiply 12.50 by 80, it yields 100000; from which if you take off five Cyphers, there remains one, which is one Acre, &c.