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197

MEN

Sure measuring squared timber, unskilful mea-

f or dually take one-fourth of the circumference, or girt,mu] .. ® Sl( le of a mean square; which quarter-girt thereforet}) e 'P le< ^ by itself, and the product multiplied by the length,thi k CCOUnt the solidity, or content: when the breadth andpr ett . ness are nea rly equal, this method will give the solidity^ near the truth ; but if the breadth and thicknessby r cons >derably, the error will be so great, that it oughtK ° jneans to be neglected.

8q Uar “us, suppose we take a balk, 24 feet long, and a footif t * , e throughout; and consequently its solidity 24 cubic feet:each 1* * be slit exactly in two, from end to end, makin;

Piece

aoli ,.«"~~ 6 inches broad, and 12 inches thick, the true^eth d eacb w ’b be 12 feet; but, by the quarter-girtquart wou W amount to much more ; for the false

tbiek 6r being equal to half the sum of the breadth and,i 8 §i es S) in this case will be 9 inches, the square of whichpli e g’, w bich being divided by 144, and the quotient multi-e aeh ^ tbe length, we obtain 13£ feet for the solidity of27 ’ and consequently the two solidities together make

,; e et. instead of 24.

of n ° a .' n ’ suppose the balk to be so cut, that the breadth

Ij ei , e e P lece may be 4 inches, and that of the other 8 inches,that p true content of the less piece will be 8 feet, andnnethod f 6 Skater 16 feet. But, proceeding by the otherSq Uare ’ quarter-girt of the less piece will be 8, whose1<J4 > 64, multiplied by 24, and the product divided byicethof? 1 feet, instead of 8. And by the sameinstead’ p oontent of the greater piece will be 16f feet,of 2d e °‘ ^6* And the sum of both is 274 feet, instead

the r, r F ber > the less piece be cut only 2 inches broad, andV° U [j ater 10 inches; the true content of the less pieceother jj, e *^ ^ eet > and that of the greater 20. But, by the? inc]) P „ et , 0 ^’ the quarter-girt of the less piece would beinstead ’^ose square, 49, being divided by 6, gives 8£ feet,c ont ent ° f ^ or t be content. And, by the same method, theSo that tbe greater piece would be 20^, instead of 20 feet.

“ jj en ei f s nm would be 28-4, instead of 24 feet,j^een jr Ce , ^ ' s ev ident, that the greater the proportion be-hy nsi no . ® breadth and depth, the greater will the error be,hie sat^g 116 ^ se method ; that the sum of the two parts, byhnrt s ; s Method, is greater, as the difference of the same twot^o Part 6ater ’ an< ^ consequently the sum is least when thej^hally are e qnal to each other, or when the balk is cut• er not h W ° ’ an< ^ lastly, that when the sides of a balkS*rt nieth t° Ve an ' nob or two f rom each other, the quarter-* a at , 0( may then be used, without inducing an error(l ^ e °f any material consequence.

( * r/l b er IV .—To find where a piece of round tapering

accor dino * cut ’ so t hat f he tw0 parts, measured separately ,Haters 7-j^ e common method of measuring, shall produceh' J n t/ le than when cut in any other part, and greater

Oie i er) '~~,p ut it through exactly in the middle, or at half_e h g v ! and the two parts will measure to the most6 common method.a a, 2 f ee j e '' 'Supposing a tree to girt 14 feet at the greatertl( l th at at Oie less, and consequently 8 feet in the middle ;

«n,y?> b y the

*2 128 W

H 0 S g r Pan :

is 32 feet.

common method, the whole tree measuresbut, when cut through at the middle, the

ah , -- sij ttl • ® easu res to 121, and the less part to 25 feet;it ■ ’ 8 the nj 8 ^et ’ wb ich exceeds the whole by 18 feet,y 1 * 0 t\vo tbat it can be made to measure to by cutting

3e

“ Demonstration. —Put g = the greatest girt, g = theleast, and x = the girt at the section; also i. = the wholelength, and z — the length to be cut off the less end.

“ Then, by similar figures, l : z :: G —g : x — g. Hence

x = GZ ^ gZ + g. But (g + xf. z + (g -f x)\ (l — z)

— a maximum ; whose fluxion being put equal to nothing,and the value of x substituted instead of it, there resultsZ— |t. Q. E. D.

“ Corollary. —By thus bisecting the length of a tree, andthen each of the parts, and so on, continually bisecting thelengths of the several parts, the measure of the whole will becontinually increased.

“Problem V. — To find where a tree should le cut, sothat the part next the greater end may measure to the mostpossible.

“ Rule. —From the greater girt take 3 times the less ;then, as the difference of the girts is to the remainder, so isone-third of the whole length, to the length from the less endto be cut off.

“ Or, cut it where the girt is one-third of the greatestgirt.

“ Note. —If the greatest girt do not exceed three times theleast, the tree cannot be cut as is required by this problem.For, when the least girt is exactly equal to one-third of thegreater, the tree already measures to the greatest possible ;that is, none can be cut off, nor indeed added to it, continuingthe same taper, that the remainder or sum may measure to somuch as the whole : and when the least girt exceeds one-third of the greater, the result by the rule shows how muchin length must be added, that the result may measure to themost possible.

“ Example. —Taking here the same example as before, weshall have, as 12 : 8 :: ^ : 7^ = the length to be cut off.;and consequently the length of the remaining part is 24 -|;also J s 4 = 4| is the girt at the section. Hence the contentof the remaining part is 135|4 feet; whereas the whole tree,by the same method, measures only to 128 feet.

“ Demonstration _Using the same notation as in the last

demonstration; we have here also x — G - Z + g, and

(g + xf. (l — x) = a maximum ; which, treated as before,gives z = X a L - - Ajl<1 x = ~ L 9 - z + 9 i G > by

substituting the above value of z. Q. e. d.

“Problem YI.— To cut a tree so as that the part next thegreater end may measure, by the common method, to exactlythe same quantity as the whole measures to.

Rule —Call the sum of the girts of the two ends vandtheir difference d, then multiply by the sum of d and 4 s,thus :

d X (d + 4 s).

From the root of the product take the difference betweend and 2 s, thus :

s/ (d? + 4 ds) — 2 s + d.

Then, As twice d —is to the remainder,

So is the whole length—to the length to be cut off;.