Mathematical Instruments

Of the Construction and Use of Everard’s Sliding-Rule for Gauging.

This Instrument is commonly made of Box, exactly a Foot long, one Inch broad, and about six Tenths of an Inch thick.

It consists of three Parts, viz. A Rule, and two small Scales or Sliding-Pieces to slide in it; one on one Side, and the other on the other: So that when both the Sliding-Pieces are drawn out to their full Extent, the whole will be three Foot Long.

On the first broad Face of this Instrument are four Lines of Numbers; the first Line of Numbers consists of two Radius’s, and is numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 1. and then 2, 3, 4, 5, &c. to 10. On this Line are placed four Brass Center Pins, the first in the first Radius, at 2150.42, and the third likewise at the same Number taken in the second Radius, having MB set to them; signifying, that the aforesaid Number represents the Cubic Inches in a Malt Bushel: the second and fourth Center Pins are set at the Numbers 282 on each Radius; they have the Letter A set to them, signifying that the aforesaid Number 282 is the Cubic Inches in an Ale-Gallon. Note, The little long black Dots, over the Center Pins, are put directly over the proper Numbers. This Line of Numbers hath A placed at the End thereof, and is called A for Distinction-sake.

The second and third Lines of Numbers which are on the Sliding-Piece (and which may be called but one Line), are exactly the same with the first Line of Numbers: They are both, for Distinction, called B. The little black Dot, that is hard by the Division 7, on the first Radius, having Si set after it, is put directly over .707, which is the Side of a Square inscribed in a Circle, whose Diameter is Unity. The black Dot hard by 9, after which is writ Se is set directly over .886, which is the Side of a Square equal to the Area of a Circle, whose Diameter is Unity. The black Dot that is nigh W, is set directly over 231, which is the Number of Cubic Inches in a Wine-Gallon. Lastly, the black Dot by C, is set directly over 3.14, which is the Circumference of a Circle, whose Diameter is Unity.

The fourth Line, on the first Face, is a broken Line of Numbers of two Radius’s, numbered 2, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, the Number 1 is set against MB on the first Radius. This Line of Numbers hath MD let to it, signifying Malt Depth.

On the second broad Face of this Rule, are,

1. A Line of Numbers of but one Radius, which is numbered 1, 2, 3, &c. to 10, and hath D set at the End thereof for distinguishing it. There are upon it four Brass Center Pins: the first, to which is set WG, is the Gauge-Point for a Wine-Gallon; that is, the Diameter of a Cylinder, whose Height is an Inch, and Content 231 Cubic Inches, or a Wine-Gallon, which is 17.15 Inches. The second Center-Pin AG stands at the Gauge-Point for an Ale-Gallon, which is 18.95 Inches. The third Center-Pin MS stands at 46.3, which is the Side of a Square, whose Content is equal to the Inches in a solid Bushel. The fourth Center-Pin MR is the Gauge-Point for a Malt Bushel, which is 52.32 Inches.
2. Two Lines of Numbers on the Sliding-Piece, which are exactly the same as on the Sliding-Piece on the other Side the Rule, they are called C. The first black Dot something on this Side the Division of the Number 8, to which is set ☉c, is set to .795, which is the Area of a Circle whose Circumference is Unity; and the second, to which is set ☉d, stands at .785, the Area of a Circle, whose Diameter is Unity.
3. Two Lines of Segments, each numbered 1, 2, 3 to 100; the first is for finding the Usage of a Cask, taken as the middle Frustum of a Spheroid, lying with it’s Axis parallel to the Horizon, and the other for finding the Usage of a Cask standing.

Again, on one of the narrow Faces of this Rule, is,

1. A Line of Inches, numbered 1, 2, 3, 4, &c. to 12. each of which is subdivided into ten equal Parts.
2. A Line by means of which, and the Line of Inches, is found a mean Diameter for a Cask in the Figure of the middle Frustum of a Spheroid; it is figured 1, 2, 3, &c. to 7. at the End thereof is writ Spheroid.
3. A Line for finding the mean Diameter of a Cask in the Figure of the middle Frustum of a parabolic Spindle, which by Gaugers is called the second Variety of Casks; it is numbered 1, 2, 3, 4, 5, 6, and at it’s End is writ, 2 Variety.
4. A Line, by means of which may be found the mean Diameter of a Cask of the third Variety; that is, a cask in the Figure of two parabolic Conoids abutting upon a common Base: it is numbered 1, 2, 3, 4, 5, at the End thereof is writ 3 Variety.

And on the other narrow Face, is,

1. A Foot divided into 100 equal Parts, every ten of which are numbered; FM stands at the beginning of it, signifying Foot-Measure.
2. A Line of Inches, like that before spoken of, having IM set to the Beginning thereof, signifying Inch-Measure.
3. A Line for finding the mean Diameter for the fourth Variety of Casks, which is the middle Frustum of two Cones, abutting upon one common Base; it is numbered 1, 2, 3, 4, 5, 6. and at the Beginning thereof is writ FC, signifying Frustum of a Cone.

These are all the Lines on the four Faces of the Rule; but on the Backside of the two Sliding-Pieces are a Line of Inches from 13 to 36, when the two Sliding-Pieces are put Endways together, and against that the correspondent Gallons, or hundred Parts, that any small Tub, or such like open Vessel (from 13 to 36 Inches Diameter) will contain at one Inch deep: it’s Construction is the same as before delivered, in speaking of the Line of Ale Area on the Four-foot Gauging-Rod.

All the Lines of Numbers, before-described, may be put upon the Faces of this Sliding-Rule, as directed in the Construction of the Line of Numbers on Gunter’s Scale; only you must observe, that the first Radius of the broken Line of Numbers MD, begins directly under MB, and ends directly under the other MB; and that when either of the Lines of Numbers A or B are made, the Line MD from them may also be made. Example, The Distance from 1 to 2, on the Line A, laid off from 1 (towards the Left Hand) to 2, on the Line MD, will give the Division 2; the Distance from 1 to 3, on the Line A, will be equal to the Distance from 1 to 3; the contrary way on the Line MD: understand the same of other Divisions and Subdivisions. The reason of thus breaking this Line of Numbers, I shall shew in it’s Use.

The Line of Segments for the middle Frustum of a Spheroid lying, may be put upon the Sliding-Rule in the following manner: Take some Vessel lying, as a Butt, and fill it full of Water, then find it’s Content in Ale or Wine-Gallons (for it matters not which), take also it’s Bung-Diameter very exactly in Inches, or Tenths of Inches. Now to find against what Number, on the Line of Numbers of the Sliding-Piece, any Division of the Line of Segments must stand; suppose the Division 1, say, As Unity is to .01, So is the Content of the aforesaid Vessel in Gallons to a fourth Number (which will be the Gallons, or Gallons and Parts that are contained in such a Segment of the Vessel, as .01 is of a similar Vessel, whose Area is supposed Unity;) then let out of the Vessel as many Gallons of Water as that fourth Proportional directs, and having taken the Dry Inches, say, By the Rule of Three, As the Bung-Diameter; is to those Dry Inches found, So is 100 to a fourth Number; which will be the Number on the Line C, against which the Division 1 on the Segment-Line must stand.

Again, to find where the Division 2 must stand on the Line of Segments, say, As 1 is to .02, So is the Content of the aforesaid Vessel to the Gallons that must be taken out of it; then say, As the Bung-Diameter is to the Dry Inches, So is 100 to the Number on the Line C, against which the Division 2 must stand. Proceed in this manner for finding the Divisions 3, 4, 5, 6, 7, 8, 9, and when you come to find where the Division 10 must stand, you must say, As Unity is to the Vessel’s Content, So is .1 to the Number of Gallons to be taken out of the Vessel, and go on as before. Moreover, to find where the Division 20 must stand, say, As 1 is to the Content, So is .2 to the Number of Gallons to be taken out of the Vessel, &c. In this manner may the Divisions to 100 be found.

To find where the first Subdivision before 1 must stand, say, As 1 is to the Vessel’s Content, So is .002 to the Number of Gallons to be let out of the Vessel, and proceed as at first directed. And for the second Subdivision, make .003 the third Term of the Rule of Three, and proceed as before.

For the Subdivisions between 1 and 2, 2 and 3, &c. suppose 1 to be .0100, then the first Division from 1 will be .011, the second .012, the third .013, &c. which must be made the third Terms of the first Rule of Three, for finding where any of those Subdivisions must stand. And for the Subdivisions between 10 and 20, 20 and 30, you must suppose 10 to be .10, and 20 to be .20; then the first Subdivision from 10 will be .11, the second .12, the third .13, &c. which will be the third Terms in the first Rule of Three, for finding whereabouts these Divisions must stand.

The other Segment-Line, on the same Face of the Rule, may be made in the same manner as this, by setting the aforesaid Vessel upright, and making use of the Length instead of the Bung-Diameter.

The Construction of the four Lines on the narrow Faces of this Rule, is from the Rules that Everard hath laid down for finding the Contents of the four Varieties of Casks. For,

1. If there is a Cask in the Form of the middle Frustum of a Spheroid, half the Difference of the Squares of the Bung and Head-Diameter, added to the Sum and half Sum of the said Squares, divided by 3, will be the Square of the mean Diameter for a spheroidal Vessel; the Root of which will be the mean Diameter.
2. Three Tenths of the Differences of the Squares of the Bung and Head-Diameters, added to the Sum and half Sum of the said Squares, and the whole divided by 3, will be the Square of the mean Diameter of a Cask of the second Variety.
3. To the Sum and half Sum of the Squares of the Bung and Head-Diameters, add one Tenth of the Difference of the said Squares, which Sum, divided by 3, gives the Square of the mean Diameter of a Cask of the third Variety.
4. And Lastly, from the Sum and half Sum of the Squares of the Bung and Head-Diameters, substract half the Square of the Difference of Diameters, and the Remainder, divided by 3, will be the Square of the mean Diameter for the fourth Variety of Casks,

Use of Everard’s Sliding-Rule.

Use I.One Number being given to be multiplied by another, to find the Product.

Notation on the Lines of Numbers upon this Rule, is the same as before was shewn in the Use of the Carpenter’s Rule; therefore I shall not here repeat it, but proceed to solve this Use by the following Examples: Suppose 4 is to be multiplied by 6: Set 1 upon the Line of Numbers B, to 4 upon the Line A, and then against 6 upon B, is 24, the Product sought upon A. Again, to multiply 26 by 68, set 1 upon B to 26 upon A; then against 68 upon B, is 1768 on A.

Note, The Product of any two Numbers will have so many Places as there are in both the Numbers given, except when the lesser of them does not exceed so many of the first Figures of the Product, for then it will have one less.

Use II.One Number being given to be divided by another, to find the Quotient.

Suppose 24 is to be divided by 4, what is the Quotient? Set 4 upon B, to 1 upon A; then against 24 upon B, is 6 upon A, which will be the Quotient.

Again, let 952 be divided by 14: To find the Quotient, set 14 upon A, to 1 upon B, and against 952 upon A, you will have 68 the Quotient upon B.

Note, The Quotient will always consist of so many Figures as the Dividend hath more than the Divisor, except when the Divisor does not exceed so many of the first Figures of the Dividend; for then it will have one Place more.

Use III.Three Numbers being given, to find a fourth in a direct Proportion.

If 8 gives us 20, what will 22 give? Set 8 upon B, to 20 upon A; and then against 22 on B, stands 55 upon A, which is the fourth Number sought.

Use IV.To find a mean Proportional between two given Numbers.

Example. Let the two Numbers be 50 and 72; set 50 upon C, to 72 upon D; and then against 72 upon C, is 60 upon D, which is the Geometrical Mean between 50 and 72.

Use V.To find the square Root of any Number under 1000000.

The Extraction of the square Root, by help of this Instrument, is easier than any of the aforesaid Uses: for if the Lines C and D be applied one to another, so that 10 at the end of D, be even with 10 at the End of C; then those two Lines, thus applied, are like a Table of any Number upon C, you have the square Root thereof upon D.

Note, When the Number given consists of 1, 3, 5, or 7 Places of Integers, seek it in the first Radius on the Line C, and against it you have the Root required upon D. Example, Let the Number given be 144, I find this on the first Radius of the Line C, and against it is 12, the Root sought upon the Line D.

Use VI.The Diameter or Circumference of a Circle being given, to find either.

Set 1 on the Line A against 3.141, (where is writ C) on the Line B, and against any Diameter, on the Line A, you have the Circumference on the Line B, and contrariwise: As suppose the Diameter of a Circle be 20 Inches, the Circumference will be 62.831; and if the Circumference 94.247, the Diameter will be 30.

Use VII.The Diameter of any Circle being given; to find the Area, in Inches, or in Ale or Wine-Gallons.

Example. Let the Diameter be 20 Inches, what is the area? Set 1 upon D to .785, (where is set ☉d) and then against 20 upon D, is 314.159, the Area required. Now to find that Circle’s Area in Ale-Gallons, set 18.95 (marked AG) upon D to 1 upon C; then against the Diameter 20 upon D, is the Number of Ale-Gallons upon C, which is 1.11 Gallons. Understand the same for Wine-Gallons, by the proper Gauge-Point.

Use VIII.The transverse and conjugate Diameters of an Ellipsis being given, to find the Area in Ale-Gallons.

Example. Let the transverse Diameter be 72 Inches, and the Conjugate 50: Set 359.05, the Square of the Gauge-Point, upon B, to one of the Diameters (suppose 50 upon A); then against the other Diameter 72 upon B, you will have the Area upon A, which, in this Example, will be 10.02 Ale-Gallons, if instead of 359.05, you use 249.11, the Square of the Gauge-Point for Wine-Gallons.

Use IX.To find the Area or Content of a Triangular Superficies in Ale Gallons.

Let the Base of the Triangle be 260 Inches, and the Perpendicular, let fall from the opposite Angle, be 110 Inches; set 282 (marked A) upon B, to 130, half the Base upon A; then against 110 upon B, is 50.7 Gallons upon A.

Use X.To find the Content of an Oblong in Ale Gallons.

Suppose one of the Sides is 130 Inches, and the other 180; set 282 upon B, to 180 upon A; then against 130 upon B, is 82.97 Ale Gallons, the Area required.

Use XI.The Side of any regular Polygon being given, to find the Content thereof in Ale Gallons.

In any regular Polygon, the Perpendicular let fall from the Center to one of the Sides, being found and multiplied by half the Sum of the Sides, gives the Area. Example, in a Pentagon, suppose the Side is an Inch, then the Perpendicular let fall from the Center, will be found .837, in saying, As the Sine of half the Angle at the Center, which in this Polygon is 36 Degrees, is to half the given Side .5; So is the Sine of 36 Degrees taken from 90, which is 54 Degrees, to the Perpendicular aforesaid: whence the Area of a Pentagon Polygon, each of whose Sides is Unity, will be 1.72 Inches; which, divided by 282, gives .0061 the Ale Gallons in that Polygon. By the same Method you may find the Area of any other Polygon, whose Side is Unity in Ale Gallons. Now, suppose the Side of a Pentagon is 50 Inches, what is the Content thereof in Ale Gallons? Set 1 upon D, to .0061 upon C; then against 50 upon D, you have the area 15.252 Ale Gallons upon C.

Use XII.To find the Content of Cylinder in Ale Gallons.

Suppose the Diameter of the Base of a Cylinder is 120 Inches, and the perpendicular Height 36 Inches. Set the Gauge-Point (AG) to the Height 36 upon C; then against 120 the Diameter, upon D, is 1443.6 the Content in Ale Gallons.

Use XIII.The Bung and Head Diameters, together with the Length of any Cask, being given, to find it’s Content in Ale or Wine Gallons.

Suppose the Length of a Cask taken, as the middle Frustum of a Spheroid to be 40 Inches, it’s Head-Diameter 24 Inches, and Bung-Diameter 32 Inches. Substract the Head-Diameter from the Bung-Diameter, and the Difference is 8: then look for 8 Inches on the Line of Inches, upon the first narrow Face of the Rule; and against it on the Line Spheroid stands 5.6 Inches, which added to the Head-Diameter 24, gives 29.6 Inches for that Cask’s mean Diameter: then set the Gauge-Point for Ale (marked AG) upon D, to 40 upon C; and against 29.6 upon D, is 97.45 the Content of that Cask in Ale-Gallons. If the Gauge-Point for Wine (marked WG) is used instead of that for Ale, you will have the Vessel’s Content in Wine-Gallons.

If a Cask, suppose of the same Dimensions as the former, be taken as the middle Frustum of a parabolick Spindle, which is the second Variety, you must see what Inches and Parts on the Line marked Second Variety, stand against 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 Cask; then set the Rule as before, and against 29.1 Inches, you will have 94.12 Ale-Gallons for the Content of the Cask.

Again; if a Cask, suppose of the same Dimensions with either of the former ones, be taken as the middle Frustum of 2 parabolick Conoids, which is one of the third Variety, you will find against 8 Inches (the Difference of the Bung and Head-Diameters) on the Line of Inches, stands 4.57 Inches, on the Line called 3d Variety, which added to 24, the Head-Diameter, gives 28.57 Inches for the Cask’s mean Diameter: proceed as at first, and you will find the Content of this Cask to be 90.8 Ale-Gallons.

Lastly, If a Cask, suppose of the same Dimensions as before, is taken as the Frustums of 2 cones, which is the fourth Variety, look on another narrow Face of the Rule for 8 Inches, upon the Line of Inches; and against it, on the Line F. C, you will find 4.1 Inches, which added to 24, gives 28.1 for the mean Diameter of this Cask: proceeding as at the first, and you will find the Content of this Cask, in Ale-Gallons, to be 87.93.

Use XIV.There is a Cask posited with it’s Axis parallel to the Horizon, or Lying, in part empty; suppose it’s Content is 97.455 Ale-Gallons, the Bung-Diameter 32 Inches, and the dry Inches 8, to find the Quantity of Liquor in the Cask.

As the Bung-diameter upon C, is to 100 upon the Line of Segments L, So is the dry Inches on C, to a fourth Number on the Line of Segments: then As 100 upon B, is to the Cask’s whole Content upon A, So is that fourth Number to the Liquor wanting to fill up the Cask; which, substracted from the Liquor that the Cask holds, gives the Liquor in the Cask. Example; Set 32, the Bung-Diameter, on C, to 100 on the Segment Line L; then against 8, the Dry-Inches on C, stands 17.6 on the Segment Line. Now set 100 upon B, to the Cask’s whole Content upon A; and against 17.6 upon B, you have 16.5 Gallons upon A; and substracting the said gallons from 97.45, the Vessel’s whole Content, the Liquor in the Cask will be 80.95 Gallons.

Use XV.Suppose the aforesaid Cask’s Axis be perpendicular to the Horizon, or upright, and the Length of it be 40 Inches: to find how much Liquor there will be in the Cask, when 10 of those Inches are dry.

Set 40 Inches, the Length, on the Line C, to 100 on the Segment Line S; and against 10, the Dry-Inches, on the Line C, stands 24.2 on the Segment Line S. Not set 100 upon B, to 97.455, the Cask’s whole Content, upon A; and against 24.2 on B, you will have 23.5 Gallons, which are the Gallons wanting to fill up the Cask, and being substracted from the whole Content 97.455, gives 73.955 Gallons for the Quantity of Liquor remaining in the Cask.

Use XVI.To find the Content of any right-angled Parallelepipedon (which may represent a Cistern, or Using-Fat) in Malt-Bushels.

Suppose the Length of the Base is 80 Inches, the Breadth 50, and the Depth 9 Inches. Set the Breadth 50 on B, to the Depth 9 on C; then against the Length 80 on A, stands 16.8 Bushels on the Line B, which are the Number of Bushels of Malt contained in the aforesaid Cistern.

The broken Line of Numbers MD, is so set under the Lines A or B, that any Number on A or B multiplied by the Number directly under it on the Line MD, will always be equal to 2150.42, Number of Inches in a Malt-Bushel: from whence the Reason of the aforesaid Operation for finding the Number of Malt-Bushels, may be thus deduced. Let us call the Breadth a, the Length b, the Depth c, and the Number of Inches in the Malt-Bushel f; then the Malt-Bushels in any Utensil of the aforesaid Figure, will be expressed by $$\frac{abc}{f}$$. But by the Sliding-Rule the Operation is to set the Breadth a, to the Depth c; that is (from the aforementioned Property of the broken Line of Numbers MD), to $$\frac{f}{c}$$ on the Line A; and then against the Length b, on the Line A, will the Number of Malt-Bushels stand: therefore the Operation is but finding the fourth Term of this Analogy, by means of the Lines A and B, viz. $$\frac{f}{c}:a::b:\frac{abc}{f}$$.