CHAP. I. NUMERATION upon the Line.

BEfore I shew you how to number upon the Line, it will be ne­cessary to let you understand how the [Page 3]Line is divided and numbered, as al­so, what those divisions and numbers set to them upon the Ruler, do sig­difie.

Know therefore, that the Line of numbers begins at the Figure One, and so proceeds successively from I to 2.3.4.5.6.7.8.9. and then on farther, by 1.2.3.4.5.6.7.8.9.10. at the end of the line.

The first 1. which standeth at the beginning of the line, representeth the One tenth part of any Unite or Intiger, as One tenth part of a Foot, One tenth part of a Yard, Ell, Perch, Mile, &c. Or it may signifie One tenth of a Year, Moneth, Hour, &c. Or the one tenth of a Pound, Shilling, or Penny, &c. Or the one tenth part of any thing either in Number, Weight, Measure, Tine, or the like. The Figure 2. signifies two tenth parts of any thing. The figure 3. three tenth parts. The figure 4. four tenth [Page 4]parts, &c. till you come to the se­cond 1. which standeth in the middle of the line, which 1. signifieth One whole Unite or Intiger, as One whole Foot, Yard, Perch, &c.

Now the other intermediate divisi­ons, those which stand between the fi­gures 1 and 2 (which are in number ten) do represent (each of them) one hundred part of one Unite or Intiger; so the first division beyond the figure 1, represents 11 hundred parts of the Intiget, the second division, 12 hun­dred parts of the Intiger, and so on; the figure 2 representing 20 hundred parts of the Intiger, and the next di­vision beyond 2 is 21 hundred parts, and so on, till you come to the figure 1 in the middle of the line, which re­presenteth one whole Intiger. The fi­gure 2 signifieth two whole Intigers; the figure 3, three whole Intigers, and so on, till you come to 10 at the end of the line, which signifieth ten whole [Page 5]Intigers: and the intermediate divi­sions, which stand between 1 and 2 in the middle of the line are (every of them) tenth parts of the Intiger. So the Rule contains 10 whole Inti­gers, every of which is divided into 10 parts.

But if upon the line you would count numbers of more places then two, (which are all numbers above 10) then the 1 which is at the begin­ning of the line, must be accounted one Intiger; and the 1 in the middle of the line, ten Intigers; and the 10 at the end, will be 100 Intigers.

But yet farther, if upon the Line you would expresse numbers of more places than three (which are all num­bers above 100). Then the 1 at the beginning of the line, is to be accoun­ted ten Intigers, the 1 in the middle a hundred Intigers, and the 10 at the end of the line 1000 Intigers.

And if you proceed yet farther; then the 1 at the beginning must be accounted for a hundred Intigers, that in the middle a thousand, and the 10 at the end of the line for 10000, ten thousand Intigers.

In this manner you might proceed farther, by counting the first 1 for 1000, 10000, &c. Intigers, but to four places is sufficient, which by a Rule of a competent length (as of two Foot) any question concerning measuring, may be by one exactly e­nough performed.

The Divisions and Numbers on the line being thus explained, it rest­eth now to shew you how to find that point upon the line, which shall re­present any number proposed, and that I shall shew you in these Propo­sitions following, which may fitly be called.

CHAP. II. MULTIPLICATION by the Line.

IN Multiplication, the Proportion is this:

• As 1 upon the Line, Is to one of the numbers to be multiplyed:
• So is the other of the numbers to be multiplyed, To the Product of them. Which is the number sought.

Example 1. Let it be required to multiply 5 by 7. The proportion is;

As 1 ∶ to 5 ∷ so is 7 ∶ to 35.

Therefore

Set one Foot of your Compasses in 1, and extend the other Foot to 5; with that extent of the Compasses, [Page 15]place one Foot in 7, and the other Foot will fall upon 35, which is the Product.

Example 2. Let it be required to multiply 32 by 9. The Proportion is;

As 1 ∶ to 9 ∷ so 32 ∶ to 288.

Set one Foot in one, and extend the other Foot to 9, that same extent will reach from 32 to 288, which is the product or sum of 32, being mul­tiplyed by 9.

Otherwise

Set one Foot in 1, and extend the other to 32, the same extent will reach from 9, to 288, as before.

Example 3. Let it be required to multiply 8 75/100 by 5 45/100. The Analogy or Proportion is;

As 1 ∶ to 8.75 ∷ so 6.45 ∶ to 56.44. fere.

Set one Foot in 1, and extend the other to 8.75, the same extent ap­plyed forward upon the line, will reach from 6.45 to 56.44. fere.

Or if you set one Foot in 1, and extend the other to 6.45: The same extent will reach from 8.75 to 56.44, almost (namely to 43 ¾) as before.

CHAP. III. DIVISION by the Line.

IN Division three things are to be minded, viz.

• The Dividend, or number to be divided.
• The Divisor, the number by which the Dividend is divided.
• The Quotient, which is the num­ber sought.

And so often as the Divisor is con­tained in the Dividend, so often doth the Quotient contain Unity.

For the working of Division, this is the Analogy.

As the Divisor is to Unity, or 1.

So is the Dividend to the Quotient.

Example 1. Let it be required to divide 35 by 7. The Proportion is;

As 7 ∶ to 1 ∷ so 35 ∶ to 5.

Set one Foot of the Compasses in 7, and extend the other Foot down­wards to one; that same extent will reach from 35 downwards to 5, which is the Quotient, and so many times is 7 contained in 35.

Otherwise

Extend the Compasses upwards from 7 to 35, that same extent will reach upwards from 1 to 5, as be­fore.

Example 2. Let it be required to divide 288 by 32. The Proportion is;

As 32 ∶ to 1 ∷ so 288 ∶ to 9.

Extend t [...]e Compasses downwards from 32 to 1, that same extent will reach downwards from 288 to 9, which is the Quotient.

Or extend the Compasses upwards from 32 to 288; the same extent will reach upwards from 1 to 9. as be­fore.

Example 3. Let it be required to divide 56.44 by 8.75. The Propor­tion is;

As 8.75 ∶ to 1 ∷ so 56.44 ∶ to 6.45.

Extend the Compasses downwards from 8.75 to 1, the same extent will reach downwards from 56.44 to 6.45.

Or extend them upwards from 8.75, to 56.44, the same will reach upwards from 1 to 6.45. as be­fore.

Note this in Division, That so ma­ny times as the Divisor may be orderly set under the Dividend in Arithmetical Work, so many [Page 19]places of figures shall be in the Quotient of your Division: As if 34785 were to be divided by 75, the Quotient shall consist of three figures only, namely of 463, because 75 can be but three times set orderly under 34785 in Arithmetical operation.

CHAP. IV. The GOLDEN-RULE Direct by the Line.

THis Rule may well be termed the Golden-Rule, it being the most useful of all others; for having three numbers given you, may, by it, finde a fourth in proportion to them, as by divers Examples following shall be made plain. And this Rule is per­formed upon the Line, with the like ease and exactness, as any of those [Page 20]before mentioned: And for the working of it upon the Line, this is the general Analogy or Proportion.

• As the first number given, Is to the second number given;
• So is the third number given, To the fourth number required.

Or

• As the first number given, Is to the third number given;
• So is the second number given, To the fourth number sought.

Wherefore

Alwaies, Extend the Compasses from the first number to the se­cond, and that distance or extent applied, the same way, upon the Line, shall reach from the third, to the fourth number required.

Or otherwise, Extend the Com­passes from the first number to the third, and that extent applyed the same way, shall also reach from the second to the fourth.

Either of these waies will effect the same thing, as by Examples fol­lowing shall be made appear.

And it is necessary thus to vary the proportion, sometimes, to avoid the opening of the Compasses too wide; for when the Compasses are opened to a very large extent, you can nei­ther take off any distance exactly, nor give so a good estimate of any parts required, as you may do when they are opened to a lesser distance: But this you will find out best by practice; and therefore I will now proceed to Examples.

Example 1. If 45 yards of Cloth cost 30 pound, what will 84 yards cost at the same rate?

As 45 ∶ to 38 ∷ so 84 ∶ to 56.

Extend the Compasses downward from 45 to 30, that extent will reach downward from 84 to 56 l. the price of 84 yards.

Or extend the Compasses upwards from 45 to 84, the same will reach from 30 to 56, as before.

Example 2. If 26 Acres of Land be worth 64 l. a year; what is 36 Acres of the like Land worth by the year.

As 26 ∶ to 64 ∷ so 36 ∶ to 88.615.

Extend the Compasses from 26 to 64, the same extent will reach from 36 to 88 615/1000 parts (which is about 12 s. 3 d. 2 q.) and so much is 36 Acres of the like Land worth by the year.

Example 3. If 100 l. yield 6 l. Interest for one year, or 12 moneths, what shall 75 l. yield?

As 100 ∶ to 6 ∷ 75 ∶ to 4.50.

Extend the Compasses from 100 to 6, the same extent will reach from 75. to 4.50. (or 4 ½) which is 4 l. 10 s, and so much will 75 l. yield in­terest in the year.

Example 4. If 75 l. yield 4 l. 10 s. interest for one year, or 12 moneths, what will 100 l. yield?

As 75 ∶ to 4.50 ∷ so 160 ∶ to 67.

Extend the Compasses downwards from 75 to 4.50, the same extent will reach from 100 to 6, and such inte­rest will 100 l. yield.

Many other Questions might be added, but the Rule (and man­ner of working it) is so plain, that it needs them not, and so general, that he which can re­solve one, may as well resolve any other, and therefore I shall say no more of it in this place.

CHAP. V. The GOLDEN-RULE Reverse by the Line.

IN this reverse or backward Rule of Three, this note is specially to be observed, That if the third num­ber be greater then the first, then will [Page 24]the fourth number be less then the second. And on the contrary, if the third number be less than the first, then the fourth number will be greater than the second. As by Examples will appear,

Example 1. If 12 Workmen, do any piece of Work in 8 daies, how many Workmen shall do the same piece of Work in 2 daies?

It is here to be noted, That in this Question 12 is not the first num­ber (though it be first named) but 2; for the middlemost terme of the three must be of the same kind with the fourth num­ber which is to be sought; as in this Example it is Men, there­fore 12 (which are men) must stand in the middle, or second place, because the fourth num­ber which is to be sought is also Men: And therefore the num­bers will stand thus,

For if 8 daies, require 12 men, then 2 daies (which is but a fourth part of 8 daies) shall require four time 12 men, that is, 48 men.

For here, Less requires More; that is, less Time, more Hands, and hence the Work is contrary to the direct Rule: Wherefore to effect it,

Extend the Compasses, from 2 to 8, the same extent will reach from 12 (the contrary way on the Line) to 48, which is the number of men that will effect the same piece of work in two daies.

Example 2. If one Close will graze 21 Horses for 6 weeks, how many Hor­ses will the same Close graze for 7 weeks?

Extend the Compasses from 6 to 7, (for you must alwaies extend your Compasses to numbers of one kind, or denomination, as here 6 and 7 are [Page 26]both Horses) the same extent will reach from 21 backward to 18, and so many Horses will the same Close graze for 7 weeks.

CHAP. VI. Of DUPLICATE PRO­PORTION by the Line.

DUplicate proportion is such a proportion as is between Lines and Superficies, or between Super­ficies and Lines.

CHAP. VII. OF TRIPLICATE PRO­POTION by the Line.

TRiplicate proportion is such a proportion as is between Lines and Solids, or between Solids and Lines.

CHAP. VIII. The Extraction of the SQARE-ROOT By the Line.

TO Extract the Square-Root, is to find a mean proportional Number between 1 and the number given, and therefore is to be found by dividing the space between them into two equal parts.

Example. Let it be required to find the Square-Root of 36.

Extend the Compasses from 1 to 36, the middle way upon the Line between these two numbers is 6, which is the Square-Root of 36. In like manner may you find the Square-Root of 81 to be 9, of 144 to be 12, of 256 to be 16; and of other numbers as in this Table.

If you suppose the number to have pricks over every second Figure, as is usual in the Arithmetical Operation, then if the last prick towards the left hand fall over the last Figure (which will always be when the number of Figures are odde) then it will be best to place Unity at the 1 in the middle of the line, so that the Root and the Square may both fall forward towards 10 at the end of the Line.

But if the number of Figures be even, it will then be best to place Unity at 10 at the end of the Line; so the Root and the Square both will fall backwards towards the middle of the Line.

CHAP. IX. The Extraction of the CƲBE-ROOT By the Line.

TO Extract the Cube-Root, is to find the first of two mean pro­portionals between 1 and the number whose Cube-Root you require, and is therefore to be found upon the Line, by dividing the space between them into three equal parts.

Example. Let it be required to find the Cube-Root of 216.

Extend the Compasses from 1 to 216, one third part of that distance shall reach from one to 6, which is the Cube-Root of 216. In like manner [Page 35]may you find the Cube-Root of 729 to be 9, of 1728 to be 12, of 110592 to be 48, of 493039 to be 79, &c. as in this Table.

CHAP. X. The Use of the Line, applied to SUPERFICIAL-MEASURE, Such as Board, Glass, Wainscot, Pavement, Hangings, Paintings, &c. of what kind soever.

THe Measures by which Board, Glass, Timber, Stone, and such like are measured, is by the Foot, a Foot containing 12 Inches, and each Inch into eight parts called halves, quarters, and half-quarters; but this kind of division not being consenta­neous or agreeable to the divisions upon your Line of Proportion, where between 1 and 2 is divided (not into 8, but) into 10 parts, the like be­tween 2 and 3 into 10 parts, and so between 3 and 4,4 and 5, &c. There­fore, [Page 37]I hold it requisite both for ease and exactness, to have every Inch on your two-foot Rule divided not into 8, but into 10 equal parts, which here­after (throughout this Book) we will call Inch-Measure.

Again, Whereas your Foot is di­vided into 12 equal parts called In­ches, I would have your Foot divided into 10 equal parts, and each of those parts sub-divided into 10 other equal parts, so will you whole Foot contain 100 equal parts, which will be agree­able to the divisions of your Line, and facilitate the work, as by the Exam­ples in this kind given will be made to appear; and this we shall hereafter call Foot-Measure.

But if any person be so wedded to Inches halves and quarters, that he will not be beaten out of his opinion, but persist therein, and yet is desirous to have knowledge in the use of this Line, I say, such person may have ad­ded [Page 38]to the side of his Inches halves and quarters (by way of faceing, as I term it) a line of Foot-measure, and also his Inches into 10 as well as 8, so that he may measure by one, and work upon his Line by the other. And this indeed will be necessary to be done upon the Rules of those ingeni­ous Artificers who need them not, for that they many times meet with wilful persons that will have them to measure their way, how disconsenta­neous to Reason soever it be.

In this nature would I have the Rule divided, and in this manner have I caused them to be made both for my self and others, a figure whereof I have inserted at the beginning of the Book.

And here note, That what is here said concerning dividing the Inch and Foot into 10 parts, the like is to be understood of the Yard, Ell, Pole or Perch, [Page 39]or any other measure whatso­ever.

These things being premised, we will now proceed to Examples.

CHAP. XI. OF YARD-MEASƲRE By the Line.

MAny Artificers, as Joyners, Painters, Plaisterers, Pavi­ers, Upholsters, measure and sell their Work, not by the Foot but by the Yard, it will be necessary to give Examples in this kind of Measure al­so. And here also it is requisite that your Yard be divided into 100 parts, [Page 44] [...] [Page 45] [...] [Page 46]and not into Halves, Quarters, and Nails; which supposed, take these Examples following.

Example 1. A Joyner hath Wains­coted a Gallery containing 130 Yards 25 parts about, and in height 15 Yards 50 parts; how many square Yards is in that Gallery? The proportion is,

• As 1 yard, to 15.50 yards the height;
• So 130.25 the compass in yards, 10 2018.87 the content in yards,

Extend the Compasses from 1 to 15.50 the breadth, the same extent will reach from 130.25 the length, to 2018.87; and so many Square Yards of Wainscoting is in that Gallery.

Example 2. A Painter hath paint­ed Landskip, or other Work, over the Wainscot of a Room, which is 1 Yard 75 parts of a Yard deep; how much [Page 47]in length thereof will make a Yard Square?

• As the breadth 1.75, is to 1 yard or 100 parts;
• So is 1, or any other numb. of yards, to the length of a yard square.

Extend the Compasses from 1 to 1.75, the same extent will reach from 100 (or one yard) to 75.14; and so much in length of that painting will make a yard square.

Example 3. A Plaisterer hath laid and beautified a Cieling containing 13 yards broad, and 63 yards 30 parts long; how many square yards is there in that Cieling?

• As 1 yard, to the breadth 13 yards;
• So the length 63.30, to the content.

Extend the Compasses from 1 to [Page 48]13, the same extent will reach from 63.30, to 823 almost; and so many square yards are there in such a Ciel­ing.

Note, It may so fall out some­times, that it will be required to measure some piece of Work, and to give an estimate of the quantity of the yards therein contained, when you have not a yard thus divided by you, but only your two-foot Rule; for the supplying whereof I will add this following Problem.

CHAP. XII. OF LAND-MEASƲRE By the Line.

THe usual Measures for Land are Chains, of which there are di­vers sorts; but the denominations of the quantity of Land is given in in Acres and Perches.

The Chains now most in use are principally two;

• • One containing 1 Perch in length,
  • The other 4 Per­ches in length,
  each of them di­vided into 100 links.

For the practice of them take these Examples.

I. By the 1 Pole Chain.

Example 1. There is a plat of ground 30 Perches broad, and 183 Perches long; how many Perches doth it contain?

• As 1, to 30 the breadth;
• So 183 the length, to 5490 the content.

Extend the Compasses from 1 to 30, that shall reach from 183 to 5490 the content.

Example 2. But the length and breadth of the same piece of ground being given as before in Perches, if it were required to find the content in Acres, then,

• As 160, to 30 the breadth;
• [Page 52]So 183, to 34 Acres 31/100 parts of an Acre.

Extend the Compasses from 160 to 30, the same shall reach (being ex­tended the same way) from 183 to 34.31, that is, 34 Acres, 31 hundred parts of an Acre, which is something above a Rood.

II. By the Four Pole Chain.

Example 1. A piece of Land con­taining 16 Chains 25 Links in breadth, and 57 Chains 30 Links in length, how many Acres doth it con­tain? The Analogy is,

• As 10, to 16.25 the breadth in Chains and Links;
• So is 57.30 the length in Chains and Links, to 93 Acres 925/10060 parts of an Acre.

Extend the Compasses from 10 to 16.25, the same extent will reach from 57.30, to 93.0925.

Example 2. The Base and Perpen­dicular of a Triangle being given in Chains and Links, to finde the content in Acres.

This is a right useful and necessary Proposition; for by it all manner of irregular plats of Land are cast up: But my intent here is not to teach Surveying, but to shew the use of the Line of Proportion.

Wherefore, Let the Perpendicu­lar of the Triangle be 7 Chains 50 Links, and the Base 45 Chains 75 Links, the proportion will be,

• As 20, to 7.50 the Perpendicular;
• So is 45.75 the Base, to 17.15 the content.

Extend the Compasses from 20 to [Page 54]7.50, that extent shall reach from 45.75 to 17.15, which is 17. Acres, and 15/100 parts.

Example 3. Having the length of any Furlong given, to finde what breadth it must have to make an Acre.

Let the length of the Furlong be 12 Chains 50 Links, then to finde the breadth for one Acre this is the Analogy.

• As 15.20 the length in Chains, is unto 10;
• So is 1 Acre, to 80 Links, which must be the breadth of the Furlong.

Wherefore,

Extend the Compasses from 10 to 12.50, the same will reach from 1 A­cre to 80 Links the breadth of the Furlong.

CHAP. XIII. Of the Mensuration of divers Regular SUPERFICIAL-FIGƲRES by the Line.

HAving sufficiently shewn the manner of measuring of such Superficial-Figures as are measured by length and breadth, I will now shew you how by the Line to measure some other Regular Figures, as ahe Circle, &c.

CHAP. XIV. II. Of the Triangle.

A Triangle is a Figure consisting of three sides and three angles, the longest side whereof we call the base; and a line drawn from the Angle op­posite to the base we call the perpen­dicular.

To measure Triangles there are se­veral ways; I will only shew you one or two to be done by the Line.

Example 1. There is a Triangle whose base is 14 foot, and his perpen­dicular 6 foot, I would know how many square Foot is contained in this Tri­angle. The proportion is, [Page 62]

• As 2, is to 6 the perpendicular;
• So is 14 the base, to 42 the Area.

Or,

• As 1, is to 3, half the base;
• So is 14, to 42 the Area.

Or,

• As 2, is to 6 the perpendicular;
• So is 7 half the base, to 42 the Area.

Or,

• As 1, is to 6 the perpendicular;
• So is 14 the base, to 84 the double Area.

All these ways produce the same effect, but the first is the best:

Wherefore,

The base of your Triangle being 14, and the perpendicular 6; Extend the Compasses from 2 to 6, the same extent will reach from 14, to 42 the Area.

CHAP. XV. The Use of the Line applied to SOLID-MEASƲRE Such as Timber, Stone, &c.

TImber and Stone are usually mea­sured by the same Rule or Mea­sure as Board and Glass are, namely, by Feet and Inches: Therefore such a Rule as was mentioned in the begin­ning of the Tenth Chapter is fit for this business also.

Before we come to shew the way of measuring of Stone or Timber, it will be necessary to premise thus much, That the base or end of every piece of Timber or Stone is (or must be suppo­sed) [Page 67]either exactly square, that is, eve­ry side alike, or else one of the sides longer than the other: wherefore the first thing to be done is to finde the Area, or superficial content of the base or end of any piece of Timber or Stone to be measured, which may be done several ways, either in Inch-Measure, as by the first Example of the first part of the tenth Chapter; or in Foot-Measure, by the first Ex­ample in the second part of the same Chapter; or both in Foot-measure and Inch-measure, as in the first Ex­ample of the third part of the same tenth Chapter; and therefore need not be here repeated again: Where­fore, we will proceed to our intended purpose of Measuring, first, by Inch-measure only; secondly, by Foot-measure only; and thirdly, by both together; as we did before in the measuring of Board, &c.

CHAP. XVI. How to measure Stone or Timber by the Line, by having the Square of the Base, and the Length of the Piece given, both in Foot and Inch Mea­sure.

HOw to finde the length of a Side of a Geometrical Square, that shall be equal to any Parallelo­gram or Long Square, is taught at the latter end of the Tenth Chapter of this Book, by which Rule it may at any time be found: That being done there, I shall only here begin with Examples.

Example 1. There is a Squared piece of Timber whose length is 183 Inches, and the Side of the Square equal to the base or end thereof is 25 Inches 45 parts; how many Foot doth that piece contain?

• 1. As 41.57, to 25.45, the side of the Square;
• So is 183, the length in Inches, to a fourth Number.

• 2. And that fourth Number,
• to 68.62, the content in Feet.

Extend the Compasses from 41.57, to 25.45, the side of the Square; the same will reach from 183, the length, to some other part of the Line; from whence if you again extend the same distance, the point will rest upon 68 Foot 62/100 parts of a Foot; and so ma­ny Foot is in the piece.

Example 2. Let the side of a Square, equal to the Base of a piece of Stone or Timber, be 2 Foot 12 parts, and the length of the same piece 15 Foot 25 parts; how many solid Foot is there in that piece?

• 1. As 1. to 2 Foot 12 parts the side of the Square;
• So 15 Foot 25 parts, the length, to a fourth Number.

• 2. And that fourth Number,
• to 68.62, the content in Feet.

Extend the Compasses from 1, to 2.12, the side of the Square; that will reach from 15.25, the length, to some other number on the Line, from whence the Compasses being extend­ed, the movable point will fall upon 68.62, the content, as before.

Example 3. The side of a Square, equal to the Base of a Stone, being 25 Inches 45 parts, and the length of that Stone 15 Foot 25 parts, how ma­ny Foot doth it contain?

• 1. As 12, to 25.45, the Square in Inches;
• So is 15.25 Foot, the length, to a fourth Number.

• 2. And that fourth Number,
• to 68.62, the content.

Extend the Compasses from 12, to 25.45 the side of the Square; the same will reach from 15.25, to some other point upon the Line, from whence the Compasses being extend­ed, the movable point will fall upon 68 Foot 62 parts, the content of the Stone.

Example 4. There is a piece of Timber whose side of the Square of the Base is 25 Inches 45 parts; how much in length of that piece will make a Foot solid?

• 1. As 25.45, the side of the Square, is to 41.57;
• So is 1 Foot, to a fourth Number.

• 2. And that fourth Number,
• to 6 Inches 67 parts.

Wherefore, Extend the Compas­ses from 25.45 the side, to 41.57; the same will reach from 1, to some other point, from whence the Com­passes being extended, will reach to 6.67, the length of a Foot solid of that piece of Timber.

Example 5. The length of the side of a Square, equal to the Base of a [Page 83]piece of Timber being 2 Foot 12 parts, to finde how much in length of that piece will make a Foot solid in Foot-measure.

• As 2.120, the side of the Square, is to 1.000;
• So is 1.000, to a fourth Number:
• And that fourth Number,
• to 0.471 parts of a Foot, to make a Foot square.

Extend the Compasses from 2.120 the side of the Square, to 1000; the same extent will reach from 1000, downwards, to some other point upon the Line, and from thence downwards, to 222 parts of a Foot; and so much in length will make a Foot solid.

CHAP. XVII. Concerning Timber that is bigger at one end than at the other, either Round or Square; and how to measure it.

CHAP. XVIII. Concerning the measuring of Re­gular Solids, as Cylinders, Globes, Cones, and such like.

CHAP. XIX. Concerning the Gauging of Ʋessels By the Line.

BEfore you can measure your Ves­sel, to finde the content there­of in Gallons or parts, you must finde the content thereof in solid Inches; and to effect this, you must finde the content of two third parts of a Circle, agreeable to the diameter at the Bung; and one third part of another [Page 105]Circle, agreeable to that of the dia­meter at the Head; these two added together, and multiplied by the length of the Vessel, that product will be the content of that Vessel in Inches.

Example. Let there be a Vessel whose

• diameter at Head, 18
• diameter at Bung, 32
• length is 40

And let the content thereof, first in Inches, and then in Gallons, be re­quired.

CHAP. I. OF SUPERFICIAL MEASURE,

BY Superficial Measure is meant all kind of flat Measure, such as is Board, Glass, Pavement, Hang­ings, Plaistering, Tyling, Land-mea­sure, &c. And these several things are measured by distinct Measures, as some by the Foot, others by the Yard, others again by the Ell, some by the Rod, and some by the Square: of all which I shall give Examples: and,

CHAP. II. OF SOLID MEASURE.

BY Solid Measure is meant such Measure as hath Length, Breadth, and Thickness, such as Timber, Stone, or the like. But before Timber or Stone can be measured, you must finde the content of the square of the base thereof, which is taught by the Problem at the end of the Tenth Chapter: but that being performed by Compasses, I will here shew how it may be (by these Lines thus dispo­sed) performed without; and that shall be my first Proposition or Exam­ple.

Example 1. Let a piece of Timber or a Stone be 80 parts of a Foot [Page 147]deep, and 3 Foot 75 parts broad; how much in length of that piece will make a Foot square?

Here (by any of the former Rules of Superficial Measure) finde at 80 parts broad how much in length will make a Foot, which you shall finde to be 1 Foot 25 parts: For,

If you finde 80 parts, the depth of the piece, in one Line, against it in the other you shall finde 1 Foot 25 parts. Take 1 Foot 25 parts of your Foot Rule, and measure it along the breadth of the piece, which is 3 Foot 75 parts, and see how often it is con­tained therein, which you shall finde to be three times; wherefore, you may conclude, that the square of the base of that piece of Timber whose depth is 80 parts, and whose breadth is 3 Foot 75 parts, is just 3 Foot.

Now the square of the base of the [Page 148]piece being thus obtained, you may finde the length of a Foot square thereof in this manner.

Example 2. Let the Square of the base of a piece of Timber or Stone be 3 Foot, how much in length of that piece will make a Foot so­lid?

Look for 3 Foot in one of your Lines, and in the other right against it you shall finde 33 parts and ⅓ part of a part; and so much in length will make a Foot solid.

Example 3. Let a piece of Stone or Timber be 2 Foot 50 parts broad, and 50 parts deep; how much of that Stone in length shall make a solid Foot?

By any of the ways before prescri­bed you shall finde that the depth of [Page 149]your Stone being 50 parts, it will re­quire 2 Foot in length thereof to make a Foot square: Wherefore, measure how often you can finde 2 Foot in the breadth of your Solid, which you may finde only once, and 50 parts more, which is one quarter of two Foot: Wherefore, the Square of this Solid contains 1 Foot 25 parts. Wherefore, Look in one of your Lines for 1 Foot 25 parts, and right against it you shall finde 80 parts; and so much in length will make a Foot solid.

Example 4. The Square of the Base of any regular Solid being given, together with the length of the same Solid, to finde how many solid Feet are contained in the same.

Let the forementioned Solid serve for this Example also, whose length [Page 150]was 35 Foot: We found that the Square of the Base was 1 Foot 25 parts, and that 80 parts in length would make one solid Foot: Where­fore, take 80 parts of your Rule, and run it along the piece so often as you can, which you shall finde to be 43 and 60 parts, which is just three Quarters; so that in this piece of Timber there is 43 Foot and three Quarters.

I might add many more Examples of this kinde, and some to o­ther purposes; but these are suf­ficient for the purpose intended; and so I shall conclude this Treatise, leaving the farther pra­ctise thereof to your self: For, ‘Usus optimus Magister.’

CHAP. III. OF CIRCULAR MEASURE.
By having either the Circumfe­rence or Diameter of any Circle given, thereby to finde the Side of a Square Equal to the same Circle; or the Side of a Square that may be inscri­bed within the same Circle.

IN the Thirteenth Chapter of this Book you have six Examples by having the Circumference or Diame­ter of any Circle given, thereby to finde the Side of a Square equal, the Superficial Content, &c. But I have [Page 152]seen upon some Two-Foot-Rules Lines to effect this thing by only o­pening the Compasses to the distance given upon one Line, and applying the same to some of the other Scales. One of those Scales is noted at the end thereof with C, signifying the Circumference of any Circle; the other with D, signifying the Diame­ter; the other with S, E, signifying Square Equal to the Circie; the other with S. W, signifying Square within.

Example. So that if you should have given you the Diameter of a Circle being 15 Inches, Out of the Line noted with D take 15 Inches, apply that distance to the Line noted with C, it will reach to 47 Inches and 13/100 parts of an Inch; and so much is the Circumference of that Circle.

Again, The Diameter being 15 Inches, as before, take that distance out of the Line D, and it will reach upon the Line S E, to 13 Inches 29/100 [Page 153]parts; and that shall be the Side of a Square equal to the Circle whose Di­ameter is 15 Inches.

Again, The Diameter being 15 Inches, if you take that distance out of the line noted with D, it will reach upon the Line S W, to 10 Inches 60/1 parts of an Inch; and that is the length of the Side of the greatest Square that can be drawn within that Circle whose Diameter is 15 Inches.

The like may be done if the Cir­cumference were given, by taking the Circumference thereof out of the Line noted with C, and applying it to the other Scales.

This I thought convenient to adde here, because sometimes these Lines are put upon Two-foot Rules.