Inftrumental arithmetic is the method of calculation by inftruments made for that purpose, for a quicker dispatch of business, and for the help of thofe who are deficient in common arithmetic. The most common inftruments are, the carpenter's rule, Coggeshall's fliding rule, Gunter's line, the gauging rule, and the diagonal rod.

1. The carpenter's rule confifts of two equal pieces, each a foot in length, connected together by a joint; one fide of this rule is divided into inches, and eighths of an inch; on the fame fide are also several plane fcales, divided into twelfth parts by diagonal lines, for planning dimenfions in feet and inches.

2. On the other fide of this rule, on one piece, are set all the principal lines which are on Cogg hall's fliding rule: namely, four lines marked ABC, and D refpectively; the two middle ones being on a flider, which runs in a groove.

Thefe

These four lines are logarithmic lines; the three marked A, B, C, are all equal; and are called double lines, because the numbers fet upon them run from rto 10, twice: the lowest line D is a fingle line, the numbers running from 4 to 40. It is called the girt line, from its use in cafting up the contents of trees, and timber. This line also may be used for gauging; as upon it, at 17.15, is marked W. G. and at 18.95, A. G. for the wine and ale gauge points.

Upon the other part of this fide is a table of pounds, fhillings, and pence, showing the value of a load, or 50 cubic feet of timber, at all prices from fixpence to two fhillings a foot.

In the use of thofe lines it must be ftrictly observed, that when i at the beginning of a line is accounted unity, then the 1 in the middle of the line will stand for 10, and the 10 at the end of the line will ftand for too: and when the 1 at the beginning of the line ftands for 10, the 10 in the middle of the line will ftand for 100, and the 10 at the end of the line 1000, &c. and all the intermediate divisions are altered in proportion.

The edge of the rule is divided decimally, each foot being divided into an hundred equal parts: by this measure dimenfions are taken in feet, and decimals of a foot, which is by much the best way.

3. Gunter's line is a line of figures, exactly the fame as the. three fingle lines on the carpenter's rule, and therefore needs no further description. Its ufe is to multiply, or divide numbers, to perform the rule of three direct, &c. It was formerly fet by itself on the carpenter's rule.

The figures 1, 2, 3, 4, &c. fometimes ftand fimply for themselves, at other times they fignify 10, 20, 30, &c. again at other times, 100, 200, &c. or 1000, 2000, &c.

4. The gauging rule ferves to compute the contents of cafks, or any other veffels, after the dimenfions have been taken. It is a fquare rule, with logarithmic lines on the fides, baving three fliders running in grooves in three of the fides.

Upon

Upon the first face of this rule are three lines: two, marked A and B, for multiplying and dividing; and one, marked MD, for malt depth, as it ferves to gauge malt. The middle line' B is upon a flider, and is a double line, marked at each edge of the flider, like that on the carpenter's rule: these three lines are all of the fame radius, or distance from 1 to 10, each line containing twice the length of the radius. A and B are numbered exactly alike, and the numbers may be increased or decreased at pleasure: thus the first number 1 may either stand for an unit, 10, or 100, &c. or 1, or o1, or coi, &c.: but whatever the first number is, the middle number will be ten times as much, and the laft number 100 times as much, as on the carpenter's rule.

The 1 on the line M D is oppofite 215 on the other lines, which are the cubic inches in a malt bufhel; and the divifions on this line are numbered retrograde to thofe of A and B.

Upon the lines A and B are feveral other marks and letters: thus, on the line A are M B, for malt bufhel, opposite the aforefaid number 215: and A, for ale, oppofite 282, the cubic inches in an ale gallon; and upon the line B, oppofite 231,is W, for wine, the cubic inches in a wine gallon; alfo S I, for fquare infcribed at 707, which is the fide of the fquare infcribed in a circle, whofe diameter is 1; and S E, for fquare equal, at .886, being the fide of a fquare equal to the fame circle: alfo C, for circumference, at 3.1416, which is the circumference of the fame circle.

Upon the fecond face, which is that oppofite the firft, are a flider, and four lines marked D, C, D, E, at one end, and at the other end marked root, fquare, root, cubic; the lines C and E, containing refpe&ively the fquares and cubes of the opposite numbers of the lines L, D: the radius of the line D is double to that of C, and triple to that of E; fo that whatever the first one on D denotes, the first on C is the fquare of it, and the first on E the cube of it; thus, if D begins with 1, C and E will begin with 1; but if D begins with 10, C will begin with 100, which is the fquare of 10, and E with 1000, the 3 E

VOL. I.

cube

cube of 10, and so on. Upon the line C, at .0796, is marked ac for the area of a circle, whofe circumference is 1; and ad Alfo at .7854, for the area of a circle, whofe diameter is 1. upon the line D are W G for wine gauge, at 17.15, and A G for ale gauge, at 18.95; and M R for malt round, at 52.32; these three being the gauge points of round and circular measure, and are found by dividing the fquare roots of 231,282, and 2150.4 by the fquare root of .7854; alfo MS for malt fquare is marked at 46.37, the malt gauge point for fquare measure, being the fquare root of 2150.4.

Upon the third face of this rule are three lines, one marked

N upon the flider; and two upon the stock, marked S S and S L, for fegment standing and segment lying, which serve to ullage standing and lying casks.

Upon the fourth face are a scale of inches, and three other fcales, marked first variety, fecond variety, and third variety; the scale for the fourth variety being on the infide of the slider on the third face: the use of thefe lines is to find the mean diameter of casks of each of the four different forms or shapes of the fides.

Befides thofe lines, there are two others on the infide of the two first fliders, being continued from one flider to the other one of these is the scale of inches, from 12 to 36, and the other a scale of ale gallons between the corresponding numbers 435 and 3.61, which forms a table to show, in ale gallons, the contents of all cylinders, whose diameters are from 12 to 36 inches, their common altitude being 1 inch.

5. The diagonal rod is a square rule, commonly four feet long, and folding together by joints; this inftrument is used both for gauging cafks, and computing their contents; and this is performed by taking one dimenfion only; namely, the diagonal of the cafk, that is, the length from the middle of the bung-hole to the meeting of the head of the cask, with the stave oppofite the bung-hole; and is the longest line within the cask from the middle of the bung. On one face of this rule is a scale of inches to take this diagonal by, oppofite

which are placed the areas, in ale gallons, of circles to the correfponding diameters, in like manner as the lines on the under fide of the three fliders in the fliding rule.

On the oppofite face are the fcales of ale and wine gallons, to show the contents of casks, from having the diagonal. All the other lines on this inftrument are the fame as those on the fliding rule, and to be used in the fame manner.

TO MULTIPLY NUMBERS TOGETHER.

Suppose the two numbers 24 and 12; draw the ilider out, till on the line B is oppofite to 12 on the line A; ther against 24 on the line B ftands 288 on the line A, which is the product of the two numbers.

Note. In any operations, when a number runs beyond the end of the line, seek it on the other radius, that is, the other' part of the line, which will be the tenth part of the required number.

PROBLEM II.

TO DIVIDE NUMBERS.

To divide 312 by 24; draw out the flider till the divifor 24 on B be opposite to the dividend 312 on A, then against Ion B stands 13, the quotient on A.

Suppose to square 12: fet 1 on the line B to 12 on A, then against 12 on B stands 144 on A.

PROBLEM IV.

TO EXTRACT THE SQUARE ROot.

Set 1 or 100, &c. on C to 1 or 10, &c. on D; then against every number found on C stands its square root on D.