true content of the lefs piece would be 4 feet, and that of the greater 20. But, by the other method, the quarter girt of the lefs piece would be 7 inches, whofe fquare 49 being divided by 6, gives 8 feet, inftead of 4, for the content. And by the fame method the content of the greater piece would be 20%, instead of 20 feet. So that their fum would be 28, instead of 24 feet.

Hence it is evident, that the greater the proportion is between the breadth and depth, the greater will the error be, by ufing the falfe method; that the fum of the two parts, by the fame method, is greater, as the difference of the fame two parts is greater, and confequently the fum is leaft when the two parts are equal to each other, or when the balk is cut equally in two; and laftly, that when the fides of a balk differ not above an inch or two from each other, the quarter girt method may then be used, without inducing an error that will be of any material confequence.

From the preceding examples it appears that this new method, which is very near the truth, is full as eafy in practice as the common falfe one. But there are many other reafons for changing the method; and one in particular, is the preventing of the fellers from playing any tricks with their timber, by cutting trees into different lengths, fo as to make them measure to more than the whole did; for, by the falfe method, this may be done in many refpects, as will appear in the three following problems, which contain the chief cafes of this artifice, but which however I do not explain to teach them to ufe thefe means.

PROBLEM

To find where a Piece of Round Tapering Timber must be cut, fo that the Two Parts, measured feparately, according to the Common Method of measuring, fball produce a Greater Solidity than when cut in any other Part, and

Greater than the Whole.

*Cut it through exactly in the middle, or at of the length, and the two parts will meafure to the moft poffible by the common method.

EXAMPLE.

Suppofing a tree to girt 14 feet at the greater end, 2 feet at the lefs, and confequently 8 feet in the middle; and that the length is 32 feet.

Then, by the common method, the whole tree meafures to only 128 feet; but when cut through at the middle, the greater part measures to 121, and the lefs part to 25 feet; whose fum is 146 feet; which exceeds the whole by 18 feet, and is the most that it can be made to meafure to by cutting it into two parts.

Uu3

PROBLEM

* DEMONSTRATION.

Put G the greatest girt, g = the leaft, and

the girt at the fection; alfo L = the whole length, and z = the length to be cut off the lefs end.

Then, by fimilar figures, L:z :: G−g : x~g, hence ∞ = Gz-gz

L

+g. But (g+x)2. z+(G+x)2. (L—≈)= a maximum; whofe fluxion being put equal to nothing, and the value of x fubftituted instead of it, there refults z =

COROLLAR Y

= { L. 2. E. D.

By thus bifecting the length of a tree, and then each of the parts, and fo on, continually bifecting the lengths of the feveral parts, the measure of the whole will be continually increased.

PROBLEM V.

To find where a Tree should be Cut, so that the
Part next the Greater End may measure

to the Moft Poffible.

From the greater girt take 3 times the lefs; then, as the difference of the girts is to the remainder, fo is of the whole length, to the length from the less end to be cut off.

Or, cut it where the girt is of the greatest girt.

Note. If the greatest girt do not exceed 3 times the leaft, the tree cannot be cut as is required by this problem. For, when the leaft girt is exactly equal to of the greater, the tree already measures to the greateft poffible; that is, none can be cut off, nor indeed added to it, continuing the fame taper, that the remainder or fum may measure to fo much as the whole And when the leaft girt exceeds of the greater, the refult by the rule fhews how much in length must be added, that the refult may measure to the most poffible.

*

DEMONSTRATION.

EXAMPLE.

Ufing the fame notation as in the last demonstration; we fhall

have here alfo x =

Gz-gz
L

+g, and (G+x)2. (L−x) = 2

maximum; which, treated as before, gives z=

G-g

G=8x+8= } G, by substituting the above value of x.

And x =
G-g
L

2. E. D

EXAMPLE.

32

I

Taking here the fame example as before, we fhall have as 128::: 7 the length to be cut off; and confequently the length of the remaining part is 24; alfo = 4 is the girt at the fection. Hence the content of the remaining part is 1354 feet; whereas the whole tree, by the fame method, meafures only to 128 feet.

To Cut a Tree fo as that the Part next the Greater End may meafure, by the Common Method, to exactly the fame Quantity as the Whole measures to.

* Call the fum of the girts of the two ends s, and their difference d. Then multiply d by the fum of d and 45, and from the root of the product take the difference between d and 2s; then, as 2d is to the remainder, fo is the whole length, to the length to be cut off the finall end.

And if s be taken from the faid root, half the remainder will be the girt at the fection.

U u4

EXAMPLE.

DEMONSTRATION.

Ufing ftill the fame notation, we fhall have s2L = (L—z). (G+x); hence, inftead of x, fubftituting its value + g, we

L obtain x = 2d

dz
L

; × (√/ (4s + d). d−2s+d). And hence

x =

2.E. D.

EXAMPLE.

Ufing ftill the fame example, we have s 16, d12, and the length L = 32; hence

L

— × (√(4$ +d). d − 25 + d) = 3 2 × (√76 × 12.

ad

6

1

20) = 11⁄2 √57–263 = 13′599118 = the length to be cut off; and confequently 18 400882 is the length of the remaining part. Alfo✔(45+ d). d − s = 2√57 -87099669 is the girt at the fection. Hence the girt in the middle of the greater part is 14+7·099669=10*549834, whose 4th part is 2-637458;

2

and confequently the content of the fame part is 2:6374582 × 18′400882 128, the very fame as the whole tree measures to, notwithstanding above part is cut off the length.

A NEW