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40. to find a mean Proportional between tWoNumbers; Sup- poie between 50 and 72 : Set 50 on C to 72 on D; then againft 72 on C you have 60 on D, which is the Mean required.
50. To extract the Square Root of a Number. Apply the Lines C and D to one another, fo as 10 at the End of D be even with 10 at the End of C j then are thefe two Lines a Table, fliewing the Square Root of any Number lefs than 1000000 by Infpe&ion : For againft any Number on C, the Number aniwering to it on D, is the Square Root thereof. Note, If the given Number confift of 1, 3,5, or
7 Places, feek it on the firft Radius, on the Line C, and againft it is the Root required at D.
6°. Either the diameter, or Circumference of a Circle, be- ing given; to find the other: Set 1 on the Line A againft 3, 141 (to which is writ C) on the Line B; and againft any Diameter on the Line A, you have the Circumference on B j and contrariwiie : thus, The Diameter being 20 Inches, the Circumference will be 62.831 Inches 5 and the Circumfe- rence being 94, 237, the Diameter will be 30.
7 . The Tiameter of a Circle given, to find the Area in In- ches, or in Ale or IVine-Galkns. Suppofe the Diameter 20 In- ches, what is the Area? Set 1 upon D, 10785 ( noted d) on C; then againft 20 on D is 314.159 the Area re- quired. Now to find that Circle's Area in Ale- Gallons, fet 18-25 (mark'd AG) upon D, to 1 on C; then againft the Diameter 20 upon D, is the Number of Ale-Gallons on C, viz. 1. II. The fame may ferve for Wine-Gallons, Re- gard only had to the proper Gauge-Point.
8°. "The two Diameters of an EUipfis being given, to find the Area in Ale-Gallons • Suppofe the Tranfverle Diameter 72 Inches, and the Conjugate 50: Set 359.05 the Square of the Gauge-point on B, to one of the Diameters (fuppofe 50) on A; then againft the other Diameter 72 on B, you will have the Area on A, viz. 10.02 Gallons, the Content of this Ellipfis at one Inch deep. The like may be done fur Wine-Gallons, if inftead of 359-05 you ufe ,,249.11 the Square of the Gauge-Point for Wine-Gallons.
9°. To find the Area of a Triangular Surface in Ale-Gal- lons : Suppofe the Bafe of the Triangle 260 Inches, and the Perpendicular let fall from the oppofite Angle no In- ches; Set 282 (mark'd A) upon B to 130, half the Bafe on A : Then againft no on B is 50.7 Gallons on A.
io Q . To find the Content of an Oblong in Ale-Gallons-. Suppofe one Side 130 Inches, and the other 180; let 282 on" B to 180 on A; then againft 130 upon B is 82.97 Ale- Gallons, the Area required.
Ii ff . To find the Content of a regular ^Polygon in Ale- Gallons, one of the Sides leing given: Find the Length of the perpendicular let fall irom the Centre to one of the Sides : This multiplied by half the Sum of the Sides, gives the Area. For an Inftance : Suppofe a Pen- tagon, whole Side is 1 Inch; here, the Perpendicular wilt be found 837^ by faying, As the Sine of half the Angle at the Centre, which in this Polygon is 36 , is to half the given Side 5, fo is the Sine of the Complement or" 36 viz, 54 to the Perpendicular aforefaid. Whence the Area of a Pentagon, whole Side is Unity, will be found 1.72 Inches, which divided by 282, give 1.0061, the Ale-Gallons in that Polygon.
12 . To find the Content of a Cylinder in Ale-Gallons : Suppofe the Diameter of the Bafe of the Cylinder 120 In- ches, the Perpendicular Height will be 36 Inches; Set there- fore the Gauge-Point (A G) to the Height, 36 on C j then aguinfl: 120, the Diameter on D, is found 1443.6, the Con- tent in Ale-Gallons.
1 3 . The Sung, and Head-Diameters, of any Cask, toge- ther with its Length, being given 5 to find its Content in Ale, or Wine Gallon?*
i°. Suppoie the Length of a Cask taken, (as the Middle Fruftum of a Spheroid which is the firft Cafe or Variety') be 40 Inches, its Head-diameter 24 Inches, and Bung-diameter 32 Inches; 1'ubtracl the Head-diameter from that of the Bung; the Difference is 8. Look, then, for
8 Inches on the Line of Inches, on the firft narrow Face of the Rule; and againft it, on the Line Spheroid ftands 5.6 Inches, which added to the Head-Diameter 24, gives 29.6 Inches for that Cask's Mean-Diameter : Set therefore the Gauge-Point for Ale (mark'd A G) on D, to 40 on C; and againft 29.6 on D, is 97.45, the Content of the Cask in Ale-Gallons. If the Gauge-Point for Wine ( mark'd "W G) be ufed inftead of that for Ale; you will have the VefTel's Content in Wine-Gallons.
2°. If a Cask, of the fame Dimenfions as the former be taken (as the Middle Fruftum of a parabolic Spindle, which is the Second Variety) fee what Inches, and Parts, on the Line mark'd Second Variety, ftand againft the Difference of the Bung and Head-Diameters, which in this Example is 8; and you will find 5.1 Inches, which added to 74, the Head-Diameter makes 29.1 Inches, the Mean-Dia- meter of the Cask; Set therefore the. Rule, as before, and
againft 29.I Inches, you will have 94,12 Ale-Gallons for the Content of the Cask. *
3 . If the Cask taken be the Middle- Fruftum of two parabolic Conoids, which is the Third Variety; againft 8 Inches, the Difference of the Head and Bung- Diameters on the Line of Inches, you will find 4.57 Inches on the Line called Third Variety, this added as before to 24, gives 28.57 f° r *he Cask's Mean-Diameter : Proceeding as be- fore, you will find the Content 90.8 Gallons.
4 . If the Cask taken be the Fruftums of two Cones which is the Fourth Variety, againft g Inches on the Line of Inches, you will find on the Line mark'd F C, 4.1 \ n _ ches to be added to 24 Inches : The reft, carried on as be- fore, gives the Content of the Cask 87.93 Ale-Gallons.
140. A Cask partly empty, lying with its Axis parallel tn the Horizon; to find the Quantity of Liquor therein : Find its whole Content, as above; which fuppofe 97.455 Gallons; and fuppofe the Inches, left dry, 8, and the Bung-Diameter 32 : Then, as the Bung-Diameter on C is to 100 on the Line of Segments L, fo are the dry Inches on C to a fourth Number on the Line of Segments: And as 100 upon B is to the Cask's whole Content on A, fo.is that fourth Number to the Liquor wanting to fill up the Cask 5 which, fub- tracled from the whole Contents of the Cask, gives the Li- quor remaining therein. JE. Gr. Set 32, the Bung-Diame- ter on C, to 100 on the Segment Line L; then againft 8, the dry Inches on C, ftands 17.6 on the Segment Line; Set therefore 100 on B, to the Casks whole Content on A; and againft 17.6 on B, you have 16.5 Gallons on A; fub- trafting therefore the faid Gallons from 97.45, the Veffel's whole Content 3 the Liquor in the Cask will be 80.95 Gallons.
1 5 . A Cask partly fianding upright, or with its Axis per- pendicular to the Horizon, to find the Liquor therein. Sup- pole the Length of the Cask 40 Inches, and 10 of them dry; let 40 Inches, on the Line C, to 100 on the Segment Line S; and againft 10, the dry Inches on the Line C, ftands 24-2 on H the Segment Line. Set, then, 140 on B, to 97.455, the Cask's whole Content on A; and againft 24.2 on B, you will have 23.5 Gallons, which is what is wanting to fill up the Cask : This, therefore, fubtra&ed from the whole Content 97.455, gives 73.955 Gallons, for the Quantity of Liquor remaining in the Cask.
16 . To find the Content of any right-angled 'Parallelepiped (E. gr. a Cifiem, Uting-Fat, or the like) in Malt-Sufiiels. Suppofe the Length of the Bafe 80 Inches, the Breadth 50, and Depth 9 Inches : Set the Breadth 50 on B, to the Depth 9 on C; then againft the Length 80 on A, ftands 16.8 Eufhels on B, the Number required.
Ccggffijal's Sliding-Rule, is principally ufed in mea- furing of the Superficies, the Solidity of Timber, &c. See Measuring, oc.
It confifts of two Rulers, each a Foot long, which are framed, or put together, various ways; fometimes they are made to flide by one another, like Glaziers Rules : Sometimes 3, Groove is made in the Side of a common two-foot Joint- Rule, and a thin Slidhg-'Piece put in, and Coggejlwl's "Lines added on that Side : But the moft ufual and commodious way, is to have one of the Rulers flide along a Groove made along the Middle of the other, as 'tis reprefented in Table Surveying, Fig. 18.
On the Sliding-§\&z of the Rule are four Lines of Num- bers, three whereof are double, that is, are Lines to two- Radius's, and one, a fingle broken Line of Numbers: The three firft, mark'd A, B, C, are figured 1, 2, 3, $£c. to 9; then 1, 2, 3» t§C t0?<>■ Their Conftm&ion, Ufe, £J?e. are the fame as thofe on Everard's Sliding-Rule. The fingle Line, called the Girt-Line, and noted D, whofe Radius is equal to the two Radius's of any of the other Lines, is broke for the eafier meafuring of Timber, and figured 4, 5, 6, 7, 8, 9, io, 20, 30, £5^. 4 to 5. It is divided into io Parts, and each 10th fubdivided into 4, and fb on to 10, &c.
On the back-fide of the Rule, are, i°. A Line of Inch- Meafure, from i to \i; each Inch being divided and fub- divided. 2 . A Line of Foot-Meafure; confifting of one Foot, divided into 100 equal Parts, and figured 10, 20» 30, Sfa The Back-fide of the Sliding-T'iece is divided into Inches, Halfs, &c>\ and figured from 12 to 24; Co that when Aid out, there may be a Meafure of two Foot.
Ufe of Coggefhal'i Sliding-Rule, m Meafuring 'Flam Superficies,
1. To meafure a Square. Suppofe, j& gr. the Sides be each 5 Feet; Set j on the Line B, to 5 on the Line A; then againft 5 on the Line B, is 25 Feet; the Content of the Square on the Line A.
2. To meafure a long Square. Suppofe the longeft Side 18 Foot, and the fhorteft 10 : Set 1 on the Line B, to 10 on the Line A; then againft 1 8 Foot, on the Line B, is 180 Feet, the Contents on the Line A, .
3, To
