SLIDING-RULE. 



mean diameter of a caflc of the third variety, !. e. of a 

 cafk in the figure of two parabolic conoids, abutting on a 

 common bafe ; it is numbered I, 2, 3, &c. and noted thitd 

 variety. 



On the other narrow face, marked f, are, i. A foot 

 divided into 100 equal parts, marked F M. 2. A line 

 of inches, 'ike that before-mentioned, noted I M. 3. A 

 line for findmg the mean diameter of a fourth variety of 

 caflis, which is the middle fruftum of two cones, abutting 

 on a common bafe. It is numbered i, 2, 3, &c. and noted 

 FC, fignifying frullum of a cone. 



On the back fide of the two fliding-pieces is a line of 

 inches, from 13 to 36, when the two pieces are put endwife ; 

 and againll that, the correfpondent gallons, or hundred 

 parts, that any fmall tub, or the like open vellel (from i j 

 to ^6 inches diameter), will contain at one inch deep. 



Mr. Overley, and other writers on this fubjeft, have 

 fuggefted fome improvements in the conllruftion of this 

 inilrument. See Overlcy's Young Gauger's Inftruftor, 

 p. 108. 



SLiDlNG-/?a/i?, Ufe'of Everard^s. — I. To multiply one 

 number by another. Suppofe 4 required to be multiplied 

 by 6 : fet I on the line of numbers B, to 4 on the line A ; 

 then, againll 6 upon B is 24, the produdl fouglit upon A. 

 Again, to multiply 26 by 68, fet i on B to 26 on A ; 

 then, againll 68 on B is 1768 on A, the produdl fought. 



2. To divide one number by another. Suppofe 24 to 

 be divided by 4 : fet 4 on B to i on A ; then againll 24 

 on B is 6 on A, which is the quotient. Again, to divide 

 952 by 14 ; fet 14 on A to i on B, and againll 952 on A, 

 you have, on B, 68, which is the quotient. 



3. To work the rule of three. If 8 gives 20, what will 

 22 give ? Set 8 on B to 20 on A, then againft 22 on B 

 (lands 55 on A, the number fought. 



4. To find a mean proportional between two numbers, 

 fuppofe between 50 and 72: fet 50 on C to 72 on D ; 

 then againll 72 on C you have 60 on D, which is the mean 

 required. 



5. To extraft the fquare root of a number. Apply 

 the lines C and D to one another, fo that 10 at the end of 

 D be even with 10 at the end of C; then are thefe two 

 lines a table, fhewing the fquare root of any number lefs 

 than looooooby inlpeftion : for againft any number on C, 

 the number anfwering to it on D is the fquare root of it. 



Note, if the given number confills of i, 3, 5, or 7 places 

 of integers, feek it on the firft radiue, on the line C, and 

 againll it is the root required at D. 



6. Either the diameter or circumference of a circle, 

 being given, to find the other. Set i on the line A againll 

 3.141 (to which is written C) on the line B ; and againll 

 any diameter ou the line A, you have the circumference 

 on B, and contrariwife : thus, the diameter being 20 inches, 

 the circumference will be 62.831 inches; and the circum- 

 ference beiDjT 94.247, the diameter will be 30. 



7. The diameter of a circle given, to find the area in 

 inches, or in ale or wine-gallons. Suppofe the diameter 

 20 inches, what is the area ? Set i upon D, to .785 

 ( noted d) on C ; then againll 20 on D is 3 1 4. i J9, the area 

 required. Now, to find that circle's area in ale-g.il!ons, 

 fet 18.95 (marked A G) upon D, to i on C ; then againll 

 the diami.tcr 20 upon D, is the number of ale-gallons on C, 

 viz. I.I I. The fame may ferve for wine-gallons, regard 

 being only had to the proper gauge-point. 



8. The two diameters of an ellipfis being given, to find 

 the area in ale-gallons. Suppofe the tranfvcrfe diameter 

 72 inches, and the conjugate 50 : fet 35905, the fquare of 

 the gauge-point, ou B, to one of the diameters (luppofo 50) 



Vol. XXXIII. 



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 rnay be done for wine- 

 gallons, if, inftead of 359.0J, you ufe 249.1 1, the fquare of 

 the gauge-point for wine-gallons. 



9. To find the area of a triangular furface in ale-gallons. 

 Suppofe the bafe of the triangle 260 inches, and the per- 

 pendicular let fall from the oppofite angle no inches: fet 

 282 (marked A) upon B, to 130, half the bafe, on A ; then 

 againll 1 10 on B is 50.7 gallons on A. 



10. To find the cont-nt of an oblong in ale-gallons. 

 Suppofe one fide i3oiiches, and the other 180: fet 282 

 on B, to 180 on A ; then againft 130 upon Bis 82.97 ale- 

 gallons, the area required. 



11. To find the content of a regular polygon in ale- 

 gallons, one of the fides being given. Find the length 

 of the perpendicular let fall from the centre to one of 

 the fides : this, multiplied by half the fum of the fides, 

 gives the area. For an inllance ; fuppofe a pentagon, 

 whofe fide is i inch, here the perpendicular will be found 

 .837, by faying, as the fine of half the angle at the centre, 

 which, in this polygon, is 36°, is to half the given fide 

 .5, fo is the fine of the complement of 36^ wz. 54°, to the 

 perpendicular aforefaid. Whence the area of a pentagon, 

 whofe fide is unity, will be found 1.72 inches, which, 

 divided by 282, gives .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 inches, 

 the perpendicular height 36 inches : fet therefore the 

 gauge-point (A G) to the height 36 on C ; then againll 

 120, the diameter on D, is found 1443.6, the content in 

 ale-gallons. 



13. The bung and head-diameters of any cadt, together 

 with its length, being given ; to find its content in ale or 

 wine-gallons, i. Suppofe the length of a caflc taken (as 

 the middle fruftum of a fpheroid, which is the firft cafe or 

 variety) be 40 inches, its head-diameter 24 inches, and 

 bung-diameter 32 inches : fubtradl the head-diameter from 

 that of the bung, the diflerence is 8. Look, then, for 3 

 inches on the line of inches, on the firll narrow face of the 

 rule ; and againll it, on the line fpheroid, ftands 5.6 inches, 

 which, added to the head-diameter 24, gives 29.6 inches 

 for that caflc's mean diameter : fet, therefore the gauge- 

 point for ale (marked AG) on D, to 40 on C ; and 

 againft 29.6 on D, is 97.45, the content of the call': in ale- 

 gallons. If the gauge-point for wine (marked WG) be 

 ufed inllead of that for ale, you will have the veflel's con- 

 tent in wine-gallons. 



2. If a calk, of the fame dimenfions as the former, be 

 taken (as the middle fruftum of a parabolic fpindle, which 

 is the fecond variety), fee what inches, and parts, on the 

 line marked fecond 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 24, the 

 head-diameter, makes 29.1 inches, the mean diameter of 

 the caflc : fet, therefore, the rule, as before, and againft 

 29.1 inches, you will have 94.12 ale-gallons for the con- 

 tent of the caflc. 



3. If the cade taken be the middle fruftum of two pa- 

 rabolic conoids, which is the third variety ; againll 8 

 inches, the diflerence 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 for the calk's mean diameter ; proceeding, as before, 

 you will find the content 90.8 g.illons. 



4. If the calk taken be the frullums of two cones, which 

 is the fourth variety, againft 8 inches on the line of inches, 



T you 



