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zo being divided into 1000 equal parts, the number 2 is placed PART III. at 301 of these parts, and the number 3 at 477 of them ; and so on; because these are the logarithms of 2 and 3, when the logarithm of 10 is 1000. The distance between 1 and 10, on the line marked D, is three times as great as that between 1 and 10 on the line E, and twice as great as that on any of the other lines A, B, C, N, or MD; and therefore the numbers on E are the cubes of those opposite to them on D, and the numbers on the other lines are the squares of those on D. Any proportion may be wrought on the sliding rule, by setting the first term on the rule opposite to the third on the slider; and then, opposite to the second term on the rule, will be found the fourth term, or answer on the slider : Or it may be wrought on the rod, by extending the compaffes from the first to the third term, and that extent, laid the same way from the second term, will reach to the fourth term. The lines, marked SS, and SL, are used for ullaging standing and lying calks ; set the length of a standing calk, or the bung-diameter of a lying caik on N, opposite to 100 on the other lines, and opposite to the wet inches on N, take off the number on SS, if it be a standing calk, or on SL, if it be a lying calk; then having placed the content of the cak on A, opposite to 100 on B, look on B, for the number taken off, and opposite to it, on A, is the quantity of ale- gallons in the calk. In like manner, on the rod, extend the compaffes on the line of numbers, froin the bung-diameter to the wet inches; and that extent will reach on SL, from 100 to a number, and the extent on the numbers from 1 to the content of the calk, will reach from that number to the quantity of liquor in the cask, in ale-gallons.

There are also lines on the rod, and on the inside of the sliders, for shewing the contents of cylinders, at one inch deep, in ale and wine gallons; and belides these, on the fliding rule there are lines marked spheroid or ist variety, 2d variety, 3d variety, and conical variety, on the inside of the slider N, for finding the mean diameters of casks; that is, the number belonging to the variety, and opposite to the difference between the bung and head diameters, added to the head-diameter, gives the mean diameter, or the diameter of a cylinder of the same length and content with the cask. These varieties are the casks mentioned in the sth, 7th, 3d, and 2d rules; but the inean diameters found by the sliding rule are not exact. The most remarkable lines on the red, are those called Diagonals; for if the rod be put into a calk from the bung to the fartheft edge of the head, the numbers on these lines, which are cut by the bung, will thew how many ale and wine gallons the call can hold ; and for common casks, this method seems to give the content more exadly

thaa

X X 2

I'ART III. than any rule that depends on their forms, and very nearly the

fame with the gth rule, which in every case gives the true con-
tent.
's. Since on these instruments every thing is performed by pro-
portions, the multipliers mentioned in the first rule must be
converted into divisors, by annexing cyphers to I, and then di-
viding it by these multipliers, and the quotients will be divisors,
answering the fame purpose; but in using the line D, the square
roots of these divisors are used in the first rule, or the square
roots of 3 times the divisors, in working any of the varieties,
cr of 6 times the same divisors, in using the gth rule; these are
called Guage-points on D, and for cylinders, they are 18.95
tor ale-gallons, and 17.15 for wine; and for any of the varieties,
except the third, which is the same as for the cylinder, they are
32.8 for ale, and 29.7 for wine: and for the gth rule, they are
46.4 for ale, and 42 for wine gallons.

For example, let the depth of a vessel be 56 inches, and the mean diameter 42 inches ; set 56 on C, oppolite to 18.95 on D, then opposite to 42 on D, is 275 on C, which is the content in ale-gallons; or if so on C be set to 17.15 on D, then opposite to 42 on D, is 336 wine-gallons on C.

Again, let the length of a cask of the conical variety be 40 inches, and the bung-diameter 32 inches, and the head 24 inches : first set 24 on C, to 24 on D, and opposite to 32 on C, is 27.7 on D, the square root of the product of 32 and 24; therefore, set 40 on C, to 32.8 on D, and then oppofite to 56, and 27.7 on D, are 116.5, and 28.5 on C, of which the diffe. rence is 88 ale-gallons, the content. Whenever the base of a vefsel has two diameters, the square root of their produét is to be found on D, and used instead of them, in the same manner as 27.7 in this example.

Again, let the length be 49 inches, the bung-diameter 32, the head 26, and the diameter in the middle between the head and bung 30.4 inches; and working as in the 9th rule, set 40 on C, opposite to 42 on D, and then opposite to 26, and 32, and 60.8 on D, there are found on C, 15.3,

and
23.2,

and 84.8, which, added together, give 122.3 wine-gallons for the content: and in the same manner, may any other of the above rules be wrought on the sliding rule.

Some of these numbers are marked on the line D, and are called Guage-points, such as W.G at 17.15, and A. G at 18.95, which are the guage-points for wine and ale galloris, in finding the contents of cylindrical veilels ; also M.R at 52.32, being the guage-point for malt-busheis in the same vessels, and M.S aç 46.37, the guage-point for malt-buíbels, in finding the contents of square or rectangular vefiels : and in the same manner, may

any

any of the other guage-points be marked on the line D. And Part III. . besides the guage-points, there are several other numbers marked on the rule, such as M.B at 2150.4, the inches in a malt-bufhel, A at 282, the inches in an ale-gallon; these are marked on the line A: and on the line B, there are marked W at 231, the inches in a wine-gallon, S.I at 707, and S.e at 886, which are the sides of squares inscribed, and equal to the circle of which the diameter is 1003, and C at 3141.6, the circumference of the same circle: and on the line C are marked OC at .0796, the area of a circle of which the circumference is 1, and Od at .7854, the area of a circle of which the diameter is 1. The use of these is obvious; for if the area of a circle be required of which the diameter is 8, place i on D, opposite to .7854, or Od on C, then, opposite to 8 on D, there will be found 50.3 on C, the area required.

The use of the line MD, is for guaging re&angular vessels, of floors of malt: set the length on B, opposite to the breadth on MD, and then opposite to the depth on A, there will be found the content in malt-bushels on B. Thus, if the length of a floor of malt be 270 inches, its breadth 56 inches, and its mean depth 5 inches ; fet 270 on B, oppofite to 56 on MD, counted towards the left hand, then opposite to 5 on A, there is found 35 on B, the number of bushels in the floor. By the line E, the contents of similar vessels

may

be found from one another, and the dimensions of similar vessels

may found from their contents. Suppose the depth of a vessel containing 160 gallons to be 40 inches ; what is the content of a similar vessel, of which the depth is 36 inches ? set 40 on D, against 100 on E, then, opposite to 36 on D, there will be found 72.9 on E, which is the content. Again, let the depth of a vefsel be 30 inches, and that it is required to find the depth of a fimilar vefsel that shall contain twice as much; set 30 on D, to I on E, and against 2 on E, there will be found 37.8 on D, which is the depth required. Suppose, again, that the diagonal, a cak containing a hogshead, is 31 inches, and that it is required to find the diagonal of a similar cask that shall hold a puncheon; fet 31 on D, to 63 gallons on E, and against 84 gallons on E, there will be found 34.1 inches on D, which is the diagonal required.

It is evident, that the same rules will give the contents of veffels in the measures of foreign countries, if proper multipliers or guage-points be found for these measures; and these multipliers and guage-points are found from the number of inches in

any measure exactly in the same manner with those for English measures. For example, the sextier of Paris contains 384 çubical Paris inches; therefore, if .785398 be divided by

3842

be

PART I'I. 384, the quotient .0020453 is a multiplier for giving the con

tents in fextiers; or if 384 be divided by :785398, the quotient
488.924, will be a divilor for the same purpose, and the square
root 22.1 of this divisor will give the guage-point for cylinders
to be used on the line D: These numbers suppose that the vessel
is guaged in Paris inches, but if the vessel be guaged in English
inches, the sextier contains 464.8 such inches ; and therefore the
multiplier in this case will be .00169, and the guage-point on
D will be 24.3. In like manner, there being 1040 cubical
English inches in a ftechan of Amsterdam, or in an almuda of
Portugal, numbers may be found for finding the contents in the
measures of these countries; and so on.

When a cajk, not full, is lying with its axis parallel to the
horizon, it was fhown in Prob. 9. of this part, how to find the
content of the full or empty part; but it required the area of
the segment of a circle to be found, which being tedious, is
usually done by means of a table of circular segments, of which
the following is a specimen.

A TABLE

Of the Areas of Segments.

V. S. Segm. V. S. Segm. V. S. Segm. IV. S. Segm. IV. S. Segm.
OL 00133 11 04701 21 11990 31 20738 41 30319
02 100375 12 05339 22 12811 32 21667 42 31304
03 00687 13 06000 23 13647 33 22603 43 3229
04 01054 14 06683 24 14494 34 23547| 44 33284
05 01468 15 73871 25 15355 35 24498 45 34278
06 01924 16 0811126 16226 36 25455 46 35274
07 102417 17 8854 27 17109 37 26418 47 36272
08 -2944 18 09613 28 1 8002 38 27386 48 37270
09 103501 19 10390 29 18905| 39 28359 49 38270
10 104088 20 1182 30 198171 40 293371 50 139270

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To ullage a lying calk by the table of segments, divide the wet inches by the bung-diameter, and find the quotient in the table, in one of the columns marked V. S. and the area of the fegment nearest to it on the right side, is to be taken out of the table, and multiplied by the whole content of the cak; divide the produet by 8, and from the quotient cut off four figures for a decis mal, the rest of the figures shew the quantity of ale-gallons in the cak

Suppose the content of a cask to be 92 gallons, the bung-diameter 32 inches, and the wet inches 8; divide 8.co by 32, the

quotient

quotient is .25, opposite to which in the table is found 15355, PART III. which, multiplied by 92, and four figures cut off from the product, gives 141.3660; and this, divided by 8, gives 18 ale-gallons nearly for the quantity of liquor in it.

If the cask be above half full, divide the dry inches instead of the wet inches, and thus find the content of the empty part, which, subtracted from the whole content, gives the quantity of liquor in it.

Τ Α Β Ι Ε
Of Foreign Measures expressed in English Inches.

aune

aune

cane

auiie

Inches, 46.680 47.604 77.220 27.1 201 27.228 27.276 27.260 22.860 22.896 25.008 23.466 22.836 21.9721

$

Inches. Paris

foot

12.792 Lyons

13.458 Leyden

12.396 Amsterdam

11.304 Antwerp

11.352 Bruffels

10.828 Hamburgh

11.376 Lubeck

11.448 Denmark

12.504. Sweden

11.733 Dantzick

11.328 Riga

10.986 Ruslia

arsheen

9.090

pal. merc. Rome

9.791

pal. arch. 8.779 Venice

foot

13.944 Leghorn palm Genoa

9.900 Naples palm 10.332 Madrid

S palm 9.012
foot

12.012 Toledo

foot

10.788 Gibraltar

11.039 Lisbon

cavido 20. 122 Constantinople sh. pike 25.576 Persia Old Greek foot 12.087 lold Roman foot

11.604 Scripture palm

3.648

9.182

palon

cane

brace mer. 34.270 brace arch. 30.739 cane.

81.900 brace 26.460 brace 22.95 cane 88.290 brace

25.200

82.5601 vara 35.0401 vara 32.220

33.1201 33.000

27.220 arish 38.364 cubit 18.132 cubit 17.490 cubit 21.8831

vara

vara

gr. pike

FINI S.

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