:;S ^

1 !!i;.

r.ii

I l

m

I I

MEN

178

MEN

being done, the formula may be reduced to .8805a 2 j- b* 4 . c *

(ai + ac + ic)H-g-» which gives Theorem X.

Rule 2.

Notes on the Solidities of Bodies.

The method of the middle section has never been noticedby any writer in a practical way. There can be little doubtbut this method took its rise from that of equidistant ordi-nates, first given by Sir Isaac Newton , as we are informedby Shirtcliffe, in his Art of Gauging , in the followingwords :

“ I shall lay down a proposition for measuring planes orsolids by approximation, a thing of the greatest importanceto this part of science, of any that was ever brought for thatpurpose, since it may be said to contain the whole art ofgauging, and that of coppers, stills, tuns, as well as all kindsof casks, whether full or partly empty, either standing orlying.

“ Of measuring curvilinear planes and solids, byapproximation.

“ Proposition. — If mq = y', n r = y', p s = y'" repre-sent three equidistant perpendicular ordinates to the axis ofa curve, M N p, whose equation isyzra-J- bx + c x?, wherex stands for any abscissa, QT, and y its ordinate, to ; then,calling Q S the distance of the extreme ordinates, /, themeasure of the space, QM ps, will be thus expressed :

“ Theorem, q m p s = y' + y" + 4 y X i

o

“ For by the preceding principles, the quadrature of thecurve, whose abscissa is x, and ordinate a + b x + c x?, will

b OC C 2C*

be found a + — + — X %; and this, when x becomes l, is2 o

b i

i

a. +"■— + X l = 6 a + 3 bl+ 2 c* x 4 = Q 5 trs.

A 6 O

But from the equation of the curve we have these threeequations,

l( = a;

bl cP

y =a + — + — ;y “=a + bl + cP.

And by taking the difference of those above, we havebl cPy — a — ~2 + ~Y’

bl 3 cP

y -;/=—+ _.

And the difference of the last gives

i

y

a , . o l

2 y + a = —.

Whence 2 cl- — 1 x jT — 2 y + a ; and by the thirdequation above, b l = y"' — a — cl 2 ; therefore 3 bl =

3y“ — 3 a —6 X f — 2y" + a = —3y"' + 12/ —9a;put these for 2 c P and 3 b l, in the above expression of the

area, and then we have, QMPS = 6 a + 36/-t-2c/ 2l l

x-Q = y- i -4y' + y - 1 - g- q. e. d.

“Corollary 1 .—The same method of demonstr^ ^extends to any number of equidistant ordinates ; 80 0 fdenotes the sum of the extreme ordinates, B the 8those next to them, c the sum of the two next *° J®the last, and so on; then we shall have the ^tables of areas, for the several numbers of ordinates P reto them, viz., for

2. A

3. a + 4 b X

4. A + 3b X —

5. 7 a + 32 B + 12 c x ;

19 a -p 7 5 b -j- 50 c X

7. 41 a 4- 216 b 4-27 c 4- 272 d X

8. 751 a 4- 3577 b 4- 1323 c 4- 2989 d x

17280

8ec.

9,10

“ This method was invented by Sir Isaac New *- 0 b b/published by Mr. Jones in 1711 ; and since proseeu jj 0 )eMr. Coats, Mr. De Moivre , and by Mr. Stirling, in a .treatise entirely built thereon, where such as desire ainsight into this matter, may find it sufficiently exP 1and applied to some of the most intricate parts of 111matics.” oI)( jin?

This table errs in the last term of the areas corresp®to the ordinates 3, 5, 7, &c., viz., 4 b should only h e L af e12 c should only be 6 c, and so of the rest; that is < 41double what they ought to be. t

This error has also escaped the notice of that greamatician, Mr. Emerson, see p. 29 of his Differential ,,published with his Conic Sections, 1767, where he c0 rsame table as in Shortcliffe’s Gauging; but it has b e c 0 p/>rected by Dr. Hutton. See his Mensuration , lar£ eprinted in 1770. . . giin pl eS |

To show how this rule may be derived in the omanner, in order to approximate any curvilinear sur ^ 0 b 8which the ends are parallel, let aedba be bounde ^side by a parabolic curve, bcd; and let A b and E ? j e tdiameters, or straight lines, parallel to its axis ; anbe perpendicular to a b and e d ; also let a e be b lS ^ j e( p Pp ; and let r c be drawn parallel to a b or E D ; aDbe drawn cutting rcinG;

and let A b = q,fc = b,and EDzc;

also, A f, or fez m :

Cb + C

now f G will be equal to —-— >

a 4- c ; '

and the part G C will be b -g'"

but the space bcdb is equal to a parabola,

of * biC “

t )i«

base is A E,2 b — a-

or 2 m, and the height G C, or