Mathematical Instruments
Book I. Additions. Ch. I.

# Of the Construction and Uses of the Carpenter’s Joint-Rule, together with the Line of Numbers commonly placed thereon.

This Rule is usually made of Box, 24 Inches long, an Inch and a half, or an Inch and a quarter broad, and of a Thickness at pleasure; one Side of it is divided into 24 equal Inches, according to the Standard at Guildhall, London, and every one of these 24 Inches are divided into 8 equal Parts; that is, into halfs, quarters, and half-quarters: The half-inches are distinguished from the quarters, and the quarters from the half-quarters, by Strokes of different Lengths, and at every whole Inch are set Figures, proceeding from 1 to 24.

On the same Side of this Rule, is commonly placed Gunter’s Line of Numbers, of which more hereafter.

The other Side of the Rule hath upon it the Lines of Timber and Board-Measure, the Construction of which is as follows:

The Line of Timber-Measure begins at 8 and a half; that is, when the Figures of the Timber-Line stand upright to you, it begins at the left End at 8 and a half, and proceeds to 36, within an Inch, and $$\frac{3}{8}$$ of an Inch of the other End. It is made from a Consideration, that 1728 Inches make a solid Foot: for any Division, suppose 9, which signifies the Side of a Square is so placed against some one of the Divisions of Inches or Parts on the other Side, beginning from the right Hand, that it's Square, which is 81 Inches, multiplied by that Number of Inches and Parts, must make 1728 Inches, or a solid Foot; which in dividing 1728 by 81, must be placed against 21$$\frac{1}{3}$$ Inches from the right Hand. In like manner the Division for the Number 10, on the Line of Timber-Measure, must be placed against 17$$\frac{28}{100}$$ Inches on the other Side; because 1728, divided by the Square of 10, which is 100, gives 17$$\frac{28}{100}$$ and in like manner for all the other Divisions. But because a Square, whole Side is either 1, 2, &c. to 8 Inches, requires more than 24 Inches in Length to multiply it by, in order to make a solid Foot, or 1728 Inches; and since 24 Inches is the whole Length of the Rule, therefore there is a Table put upon the left end of the Rule, supplying a greater Length.

The upper Row of Figures, numbered 1, 2, 3, 4, 5, 6, 7, 8, are Inches, or the Lengths of the Sides of Squares; and the second and third Rows are the correspondent Feet and Inches to make up a solid Foot. It is made by dividing 144 Inches by the Squares of 1, 2, 3, 4, 5, 6, 7, 8; as the Square of 1 Inch is 1, by which dividing 144, the Quotient will be 144 Feet for the first Number of the second Row of Figures, and in like manner for the rest.

On, or next the other Edge of the Rule, you have the Line of Board-Measure; and when the Figures stand upright, you see it numbered 7, 8, 9, &c. to 36. which is just 4 Inches from the right Hand, it is thus divided; suppose the Division 7 is to be marked divide 144, which is the Number of Inches in a square Foot, by 7, and the Quotient will be 20$$\frac{4}{7}$$ Inches; whence the Division 7 must be against 20$$\frac{4}{7}$$ Inches on the other Side of the Rule. Again, to mark the Division 8, divide 144 by 8, and the Quotient, which is 18 Inches, must be placed on the Line of Board-Measure against 18 Inches on the other Side: proceed thus for the other Divisions of the said Line. But because the Side of a long Square, that is either 1, 2, 3, 4, 5 Inches, requires the other Side to be more than 24 Inches, which is the whole. Length of the Rule; therefore there is a Table placed at the other end of the Rule, made in dividing 144 Inches by each of the Numbers in the upper Row, and then each of the Quotients by 12, to bring them into Feet.

## Use.Use of the Carpenter’s Joint-Rule.

The Inches on this Rule are to measure the Length or Breadth of any given Superficies of Solid, and the manner of doing it is superfluous to mention, it being not only easy, but even natural to any Man; for holding the Rule in the left Hand, and applying it to the Board, or any thing to be measured, you have your Desire. But now for the Use of the other Side, I shall shew in two or three Examples in each Measure, that is, Superficial and Solid.

### Example I. The Breadth of any Superficies; as Board, Glass, or the like, being given: to find how much in Length makes a Square Foot.

To do which, look for the Number of Inches your Superficies is broad, in the Line of Board Measure, and keep your Finger there; and right against it, on the Inches Side, you have the Number of Inches that makes up a Foot of Board, Glass, or any other Superficies Suppose you have a Piece 8 Inches broad, how many Inches make a Foot? Look for 8 on the Board Measure, and just against your Finger (being set to 8) on the Inch-Side, you will find 18, and 10 many Inches long, at that Breadth, goes to make a superficial Foot.

Again, suppose a Superficies is 18 Inches broad, then you will find that 8 Inches in Length will make a superficial Foot; and if a Superficies is 36 Inches broad, then 4 Inches in Length makes a Foot.

Or you may do it more easy thus: Take your Rule, holding it in your left Hand, and apply it to the Breadth of the Board or Glass, making the End, which is next 36, even with one Edge of the Board or Glass, and the other Edge of the Board will shew how many Inches, or Quarters of an Inch, go to make a square Foot of Board or Glass. This is but the Converse of the former, and needs no Example; for laying the Rule to it, and looking on the Board-Measure, you have your Desire.

Or else you may do it thus, in all narrow Pieces under 6 Inches broad: As suppose 3$$\frac{1}{4}$$ Inches, double 3$$\frac{1}{4}$$, it makes 6 $$\frac{1}{2}$$ then twice the Length from 6$$\frac{1}{2}$$ to the End of the Rule, will make a superficial Foot, or so much in Length makes a Foot.

### Example II. A Superficies of any Length or Breadth being given, to find the Content.

Having found the Breadth, and how much makes one Foot, turn that over as many times as you can upon the Length of the Superficies, for so many Feet are in that Superficies: But it it is a great Breadth, you may turn it over two or three times, and then take that together; and so say 2, 4, 6, 8, 10, &c. or 3, 6, 9, 12, 15, 18, 21, ’till you come to the End of the Superficies.

## Use.The Use of the Table at the End of the Board-Measure

If a Superficies is 1 Inch broad, how many Inches in Length must there go to make a superficial Foot? Look in the upper Row of Figures for 1 Inch, and under it, in the second Row, you will find 12 Feet; which shews that 12 Feet in Length, and 1 Inch in Breadth, will make a superficial Foot.

Again a Superficies 5 Inches broad, will be found, in the said Table, to have 2 Feet and about 5 Inches in Length to make a superficial Foot; and a Piece 8 Inches broad, will have a Length of 1 Foot 6 Inches to make a superficial Foot.

## Use.Use of the Line of Timber-Measure

The Use of this Line is much like the former: for first you must learn how much your Piece is square, and then look for the same Number on the Line of Timber-Measure, and the Space from thence to the End of the Rule, is the true Length at that Squareness to make a Foot of Timber.

Example. There is a Piece that is 9 Inches square, look for 9 on the Line of Timber-Measure, and then the Space from 9, to the End of the Rule, is the true Length to make a solid Foot of Timber, and it is 21$$\frac{1}{3}$$ Inches.

Again, supposed a Piece of Timber is 24 Inches square, then 3 Inches in Length will make a Foot, for you will find three Inches on the other Side against 24: but if it is small Timber, as under 9 Inches square, you must seek the Square in the upper Rank in the Table, and right under you have the Feet and Inches that go to make a fold Foot, as was in the Table of Board Measure: As suppose a Piece of Timber is 7 Inches square, look in the Table for 7, in the upper Row of Numbers, and you will find directly under 2 Feet, 11 Inches, which is the Length of the Piece of Timber that goes to make a solid Foot: But if a Piece be not exactly square, viz. is broader at one Side than the other, then the usual way is to add them both together, and take half the Sum for the Side of the Square; but if they differ much, this way is very erroneous: for that half is always too great, which from hence will easily be manifest.

Let AC be the longest Side, CD the shortest, and BD, or AB, half their Sum, which is taken for the Side of the Square, that is, for the Side of a Square whose Area is equal to the Product of the two Sides AC, and CD, into one another, or the Rectangle under them: Now with the Distance BD, and on the Center B, describe a Semicircle; draw the Diameter EB, at right Angles, to AD, and from the Point C raise the Perpendicular FC; then it is manifest, per Prop. 13. lib. 6. Eucl. that FC is a mean Proportional between the Sides AC, CD; that is, FC is the true Side of the Square, which, per Prop. 15. lib. 3. Eucl. is much less than EB, or it's Equal AB, or BD.

The usual way likewise for round Timber, is to take a String, and girt it about, and the fourth part of it is commonly allowed for the Side of the Square, that is, for the Side of a Square equal to the circular Base, and then you deal with it as if it was just Square. But this way is also erroneous; for by this Method you lose above $$\frac{1}{5}$$ of the true Solidity. But for maintaining this ill Custom, they plead, The Overplus Measure may well be allowed, because the Chips cut off are of little Value, and will not near countervail the Labour of bringing the Timber to a Square, to which Form it must be brought before it be fit to use.

### The Description of Gunter’s Line or the Line of Numbers

The Line of Numbers is only the Logarithms transferred on a Ruler from the Tables, by means of a Scale divided into a great Number of equal Parts, and whereas in the Logarithms, by adding or substracting them from one another, the Quæsita is produced; so here, by turning a Pair of Compasses forwards or backwards, according to due Order on this Line, the Quæsita will in like manner be produced. The Construction of this Line I shall give in speaking of Gunter’s Scale.

As to the Length of the Line of Numbers, the longer it is, the better it is; whence it hath been contrived several ways: As first upon a Rule of two Foot, and a Rule of three Foot long, by Gunter, which (as I suppose) is the Reason why it is called Gunter’s Line; then that Line was doubled, or laid so together, that you might work either right on, or cross from one to another, by Mr Windgate; afterwards projected in a Circle, by Mr Oughtred, and also to slide one by another, by the same Author; and last of all, projected into a kind of Spiral, of 5, 10, or 20 Turns, more or less, by Mr Brown, the Uses being in all of them in a manner the same, only some with Compasses, as Mr Gunter’s and Mr Windgate’s; and some with flat Compasses, or an opening Index, as Mr Oughtred’s and Mr Brown’s; and some without either, as the Sliding-Rules.

The Order of the Divisions on this Line of Numbers, and commonly on most others, is thus; it begins with 1, and so proceeds with 2, 3, 4, 5, 6, 7, 8, 9; and then 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, whose Order of Numeration is thus: The first 1 signifies one Tenth of any whole Number or Integer, and consequently the next 2 is two Tenths; 3, three Tenths; and all the small intermediate Divisions are 100 Parts of an Integer, or a Tenth of one of the former Tenths; so that 1 in the middle is one whole Integer, the next 2, two Integers; and 10 at the end, 10 Integers: Thus the Line is in it’s most proper Acceptation, or natural Division.

But if you are to deal with a Number greater than 10, then 1 at the beginning must signify 1 Integer, and 1 in the middle 10 Integers, and 10 at the end 100 Integers. But if you would have it to a Figure more, then the first 1 is 10, the second 100, and the last 10 a 1000. If you proceed further, then the first 1 is 100, the middle 1 a 1000, and the 10 at the end 10000, which is as great a Number as can well be discovered, on this or most ordinary Lines of Numbers; and so far, with convenient Care, you may resolve a Question tolerably exact.

## Numeration on the Line of Numbers

### Any whole Number being given under four Figures, to find the Point on the Line of Numbers that represents the same.

First look for the first Figure of your Number amongst the long Divisions that are figured, and that leads you to the first Figure of your Number; then for the second Figure, count so many Tenths from that long Division forwards, as that second Figure amounts to; then for the third Figure, count from the last Tenth so many Centesmes as the third Figure contains; and so for the fourth Figure, count, from the last Centesme, so many Millions as that fourth Figure has Units, or is in Value, and that will be the Point where the Number propounded is on the Line of Numbers. Two or three Examples will make this manifest.

First, to find the Point upon the Line of Numbers representing the Number 12. Now because the first Figure of this Number is 1, you must take the 1 in the middle for the first Figure; then the next Figure being 2, count two Tenths from that 1, and there will be the Point representing 12.

Secondly, To find the Point representing 144, First, as before, take for 1 the first; Figure of the Number 144, the middle Figure 1; then for the second (viz. 4.) count four Tenths forwards, lastly, for the other 4, count four Centesms further, and that is the Point for 144.

Thirdly, To find the Point representing 1728. First, as before, for 1000 take the middle I on the Line. Secondly, for 7 reckon seven Tenths forwards, and that is 700. Thirdly, for 2, reckon two Centesms, from that 7th Tenth, for 20. And, Lastly, for 8 you must reasonably estimate that following Centesm to be divided into 10 Parts (if it be not expressed, which in Lines of ordinary Length cannot be done), and 8 of that supposed 10 Parts is the precise Point for 1728, the Number propounded to be found; and the like of any other Number.

But if you was to find a Fraction, you must consider, that properly, or absolutely, the Line only expresses Decimal Fractions; as thus, $$\frac{1}{10}$$, or $$\frac{1}{100}$$, or $$\frac{1}{1000}$$, and more near the Rule in common Acceptation cannot express; as one Inch, one Tenth, one Hundredth, or one Thousandth Part of an Inch, it being capable to be applied to any thing in a decimal way: (But if you would use other Fractions, as Quarters, Half-Quarters, &c. you must reasonably read them, or else reduce them into Decimals.)

## The fundamental Uses of the Line of Numbers

### Use I.Two Numbers being given, to find a third Geometrically proportional to them, and to three a fourth, and to four a fifth, &c.

Extend your Compasses upon the Line of Numbers, from one Number to another; which done, if that Extent is applied (upwards or downwards, as you would either increase or diminish the Number), from either of the Numbers, the moveable Point will fall upon the third proportional Number required. Also the same Extent, applied the same way from the third, will give you a fourth, and from the fourth a fifth, &c. For Example, let the Numbers 2 and 4 be proposed, to find a third Proportional, &c. to them: Extend the Compasses upon the first Part of the Line of Numbers, from 2 to 4; which done, if the same Extent is applied upwards from 4, the moveable Point will fall upon 8, the third Proportional required; and then from 8 it will reach to 16, the fourth Proportional; and from 16 to 32 the fifth, &c. Contrariwise, if you would diminish, as from 4 to 2, the moveable Point will fall on 1, and from 1 to $$\frac{5}{10}$$, or .5, and from .5 to .25, &c. as is manifest from the Nature of the Logarithms, and Prop. 20. lib. 7. Eucl.

But generally in this, and most other Work, make use of the small Divisions in the middle of the Line, that you may the better estimate the Fractions of the Numbers you make use of; for how much you miss in setting the Compasses to the first and second Term, so much the more you will err in the fourth; therefore the middle Part will be most useful: As for Example, as 8 to 11, so is 12 to 16.50, if you imagine one Integer to be divided but into 10 Parts, as they are on the Line on a two-foot Rule.

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

Extend your Compasses from 1 to the Multiplicator, and the same Extent, applied the same way from the Multiplicand, will cause the moveable Point to fall upon the Product; as is manifest from the Nature of the Logarithms, and Defin. 15. lib. 7. Eucl.

Example. Let 6 be given to be multiplied by 5; extend your Compasses from 1 to 5, and the same Extent will reach from 6 to 30, the Product sought. Again, suppose 125 is to be multiplied by 144; extend you Compasses from 1 to 125, and the moveable Point will fall from 144 on 18000 the Product.

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

Extend your Compasses from the Divisor to 1, and the same Extent will reach from the Dividend to the Quotient; or, extend the Compasses from the Divisor to the Dividend, the same Extent will reach the same way from 1 to the Quotient, as is manifest from the Nature of the Logarithms, and this Property, that as the Divisor is to Unity, so is the Dividend to the Quotient.

Example. Let 750 be a Number given, to be divided by 25 (the Divisor), extend your Compasses downwards from 25 to 1; then applying that Extent the same way from 750, and the other Point of the Compasses will fall upon 30, the Quotient sought. Again, let 1728 be given to be divided by 12; extend your Compasses from 12 to 1, and the same Extent will reach the same way from 1728 to 144.

If the Number is a Decimal Fraction, then you must work as if it was an absolute whole Number; but if it is a whole Number joined to a decimal Fraction, it is worked here as properly as a whole Number: As suppose 111.4 is to be divided by 1.728, extend your Compasses from 1.728 to 1, the same Extent, applied from 111.4, will reach to 64.5. So again, 56.4 being to be divided by 8.75, and the Quotient will be found to be 6.45.

Now to know of how many Figures any Quotient ought to consist, it is necessary to write down the Dividend, and the Divisor under it, and see how often it may be written under it; for so many Figures must there be in the Quotient: As in dividing this Number 12231 by 27, according to the Rules of Division, 27 may be written 3 times under the Dividend; therefore there must be 3 Figures in the Quotient: for if you extend the Compasses from 27 to 1, it will reach from 12231 to 453, the Quotient sought.

Note, That in this Use, or any other, it is best to order it so, that your Compasses may be at the closest Extent; for you may take a close Extent more easy and exact than a large Extent, as by Experience you will find.

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

Extend your Compasses from the first Number to the second; that done, the same Extent applied the same way from the third, will reach to the fourth Proportional sought, as is manifest from the Nature of the Logarithms, and Prop. 19. lib. 7. Eucl. from whence it may be gathered, that the third Number multiply’d by the second, divided by the first, will give the fourth sought.

Example. If 7 give e 22, what will 14 give? Extend your Compasses upwards from 7 to 14, and that Extent applied the same way, will reach from 22 to 44, the fourth Proportional required. Again, if 38 gives 76, what will 96 give? Extend your Compasses from 38 to 96, and the same Extent will reach from 76 to 192, the fourth Proportional sought.

### Use V.Three Numbers being given, to find a fourth in an Inverse Proportion.

Extend your Compasses from the first of the given Numbers to the second of the same Denomination; if that Distance be applied from the third Number backwards, it will reach to the fourth Number sought.

Example. If 60 give e 5, what will 30 give? Extend your Compasses from 60 to 30, and that Extent applied the contrary way from 5, will give 2.5 the Answer. Again, If 60 gives 48, what will 40 give? Extend your Compasses from 60 to 40; that Extent applied the contrary way from 48, will reach to 32, the fourth Number sought.

### Use VI.Three Numbers being given, to find a fourth in a duplicate Proportion.

This Use concerns Questions of Proportions between Lines and between Superficies; now if the Denominations of the first and second Terms are Lines, then extend your Compasses from the first Term to the second (of the same kind of Denomination): this done, that Extent applied twice the same way from the third Term, and the moveable Point will fall upon the fourth Term required, which is manifest from the nature of the Logarithms, and from hence, viz. Because the fourth Number to be found is only a fourth Proportional to the Square of the first, the Square of the second, and the third, it is plain that the third, multiplied by the Square of the second, divided by the third, will be the fourth Number sought.

Example. If the Area of a Circle, whole Diameter is 14, be 154, what will the Content of a Circle be, whose Diameter is 28? Here 14 and 28 having the same Denomination, viz. both Lines, extend the Compasses from 14 to 28, then applying that Extent the same way from 154 twice, the moveable Point will fall upon 616, the fourth Proportional or Area sought: Because Circles are to each other as the Squares of their Diameters, per Prop. 2. lib. 12. Eucl.

### Use VII.Three Numbers being given, to find a fourth in a triplicate Proportion.

This Use is to find the Proportion between the Powers of Lines and Solids; that is, two Lines being given and a Solid, to find a fourth Solid, that has the same Proportion to the given Solid, as the given Lines have to one another. Therefore extend the Compasses from the first Line to the second, and that Extent, applied three times from the given Solid or third Number, will give the fourth sought: Because the third multiplied by the Cube of the second, divided by the Cube of the first, will give the fourth.

Example. If an Iron Bullet, whose Diameter is 4 Inches, weighs 9 Pounds, what will another Iron Bullet weigh, whose Diameter is 8 Inches? Extend your Compasses from 4 to 8, that Extent applied the same way three times from 9, will give 72, the Weight of the Bullet sought. Because the Weight of homogeneal Bodies are as their Magnitudes, and Spheres are to one another as the Cubes of their Diameters, per Prop. 16. lib. 12. Eucl.

### Use VIII.To find a mean Proportional between two given Numbers.

Bisect the Distance between the given Numbers, which Point of Bisection will fall on the mean Proportional sought: Because the square Root of the Quotient of the two Extremes divided by one another, multiplied by the lesser, is equal to the Mean.

Example. The Extremes being 8 and 32, the middle Point between them will be found to be 16.

### Use IX.To find two mean Proportionals between two given Lines.

Trisect the Space between the two given Extremes, and the two Points of Trisection will give the two Means. Because the Cube Root of the Quotient of the Extremes divided by one another, multiplied by the lesser Extreme, will give the first of the Mean Proportionals sought, and that first Mean multiplied by the aforesaid Cube Root, will give the second.

Example. Let 8 and 27 be the two given Extremes, the two Means will be found to be 12 and 18, which are the two Means sought.

### Use X.To find the Square Root of any Number under 1000000

The Square Root of any Number is always a mean Proportional between 1, and the Number whole Root you would find; but yet with this general Caution, viz. If the Figures of the Number are even, that is, 2, 4, 6, 8, 10, &c. then you must look for the Unit at the Beginning of the Line, and the Number in the second Part or Radius, and the Root in the first Part; or gather reckon 10 at the end to be Unity, and then both Root and Square will fall backwards towards the middle in the second Length or Part of the Line: But if they be odd, then the middle 1 will be most convenient to be counted Unity, and both Root and Square will be found from thence forwards towards 10; so that according to this Rule the Square Root of 9 will be found to be 3, the Square Root of 64 will be found to be 8 the Square Root of 144 to be 12, &c.

### Use XI.To find the Cube Root of any Number under 1000000000

The Cube Root is always the first of two mean Proportionals between 1 and the Number given, and therefore to be found by trisecting the Space between them; whence the Cube Root of 1728 will be found 12, the Root of 17280 is near 26, the Root of 172800 is almost 56. Although the Point on the Line representing all the square Numbers is in one place, yet by altering the Unit, it produceth various Points and Numbers for their respective proper Roots. The Rule to find this, is in this manner: You must use Dots (or suppose them to be set) over the first Figure of the Left-hand, the fourth Figure, the seventh, and the tenth; now if by this means the last Dot to the Left-hand falls on the last Figure, as it doth in 1728, then the Unit must be placed at 1 in the middle of the Line, and the Root, the Square, and Cube, will all fall forwards towards the end of the Line.

But if it falls on the last but 1, as it doth in 17280, then the Unit may be placed at 1 in the Beginning of the Line, and the Cube in the second Length; or else the Unit may be placed at 10 in the End of the Line, and the Cube in the first Part of the Line. But if the last Dot falls under the last Figure but two, as in 172800, the Unit must always be placed at 10 in the End of the Line, and then the Root, the Square, and Cube, will all fall backwards, and be found in the second Part, between the Middle 1, and the End of the Line. By these Rules it appears that the Cube Root of 8 is 2, the Cube Root of 27 is 3, the Cube root of 64 is 4, of 125 is 5, of 216 is 6, of 345 is 7, of 512 is 8, of 729 is 9, of 1000 is 10, &c.