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. XLV. (Book 45)

To find what Scale a Plat or Draft is drawn by, the Content of the Ground being given. (Book 45)

SUppose a piece of Ground or Field to be 30 Acres, and I measure this Plot by a Scale of 10 in the Inch, and by that Scale it makes but 17 Acres and 3 Roods, or 17 Acres and •• now the Question is, What Scale was it drawn by? The work is somewhat difficult by na∣tural Arithmetick, but by Artificial, and the Line of Numbers more easie.

Example, by that excellent Table the Table of Logarithmes.

First find out the mean proportional Number between the true quan∣tity of Acres, (viz. 30.) and the Quantity of Acres found by the sup∣posed Scale (viz. 17 ¾) which you may do thus: Adde the Logarithmes of these two Numbers together, the half of that summe is the Log. of the mean Proportional required; as thus:

• The Log. of 30 is 1.47712
• The Log. of 17 ¾ is 1.24919
• The 2 summes added together 2.72631
• The half of the Log: 1.36315
• The Number answering to—
• this Logarithme is 23. 08/100:

This Number is the mean proportional Number of 30 and 17 ¾:

Having thus found the mean proportional Number to be 23.08, the Rule in the Second place is thus:

As the Log. of this 17 Acres ¾, found by the supposed Scale, is to the Log of the mean Proportional (23.08) of the true Quantity of Acres, and the supposed Quantity, so is 10 (the supposed Scale) to the true Scale; as thus:

• The Logarithme of 17 ¾ is 1.24919
• The Log. of 23. 08/100 is 1.36315
• The Log. of 10 is 1.00000
• 1.11396

The Number answering this Logarithme is 13; which tells me that the true Scale that this Plat was drawn by, is a Scale of one Inch put or divided into 13 parts.

The way to work the Golden Rule, or Rule of Three by Log. is, to adde the Log. of the third Number and second Number together, and to substract the Log. of the first Number, and then the Number answering the Log. that remains after Substraction, is the fourth Num∣ber.

But here I have made no Addition, but substracted the first Number ont of the second and third, which is all one in Operation.

How to work the same Question on the Line of Numbers.

Having found the Poynt on your Line, which is 30, and the Poynt representing 17 ¾, find out the poynt which is the midst between these two; and that very poynt is the Mean Proportional between these two Numbers; which here is 23 and better; for if you set one poynt of your Compasses on 23 08/100, that Extent from thence to 30 will also reach from that to 17 ¾.

Then (as before is shewed) say, as 17 ¾: is to 23 08/100, So is 10 to 13 (the true Scale sought:)

Therefore extend your Compasses from 17.75: to 23 08/100, that Extent will reach from 10 to 13.

Thus you may see how readily this Question is wrought by Log. without Multiplication or Division, and also by the Line of Numbers, with two turns of your Compasses.

Example Second.

If a piece of Ground, or the side of a House be 100 Acres, Rods, or the like, and you measure it by a Scale of 12, and find it to be but 56 22/100, if you would know what Scale, in proportion to this, the Draft or Plot was drawn by, then work by the aforesaid Rules thus:

Find the Mean proportional Number of 100 and 56.22 thus, by ta∣king the half of them 2 Log. and the Number answering that, is the Mean proportional Number, as thus:

• Logarithme of 100. is—2.00000
• Log. of 56. 22. is—1.74989
• 74 98/100 the mean Proportional 1.87494
• Log. of 12 is—1.07918
• Added—2.95412
• The Number answering this Log. 1.20423

Now the nearest in whole Numbers is 16, which is the Scale by which the Draft or Plot was drawn.

This also you may work by Gunters Line, as is before shewed; for if you take the middle poynt between 100 and 56.22, you will find it to be neer 75; for if you set one poynt on or near 75, and extend the other to 100, that Extent will reach from 75 also to 56.22; so that 75 is the mean Proportional Number between them two.

Then if you extend your Compasses from 56.22, to 75, that Ex∣tent will reach from 12 (the supposed Scale) to 16, the true Scale: thus having shewed you several hints of this Line of Numbers, I will here shew a few more.