SOCIUS MERCATORIS: OR THE Merchant's Companion: IN THREE PARTS.

The First, Being a plain and easie Introduction to Arithmetick, Vulgar and Decimal, the Extraction of the Square and Cube Roots, with a Table of 200 Square Roots, and their Use in the Resolution of Square Equations.

The Second, A Treatise of Simple and Compound Interest and Rebate, with Two Tables for the Calculation of the Value of Leases or Annuities, payable Quarterly, the one for Simple, the other Compound Interest, at 6 per Cent. per Annum, with Rules for making the like for any other Rate.

The Third, A new and exact way of Measuring Solids in the Form of a Prismoid and Cylindroid, with the Frustums of Pyramids and of a Cone: Whereunto is added, some Practical Rules and Examples for Cask-Gauging.

By JOHN MAYNE, Philo-Accomptant.

Nunquam nimis quod nunquam satis dicitur.
Sen.

LONDON, Printed by W G. for N. Crouch, in Exchange-Alley, over against the Royal-Exchange in Cornhill. 1674.

To his Honoured Friend THO. WILLIAMS, M. D. Physician in Ordinary to His S. Majesty.

SIR,

THough the happiness which I formerly enjoyed in your Converse, hath been, to my great loss, for some years discon­tinued; yet I easily perswade my self, that the Favour of a Great Prince, and the Best Master in the World, has not wrought such a change upon your even Virtue, but that you will still descend to remember him whom you were once pleased to honour with the Name of Friend. This Confidence has embol­d'ned me to present you with this small [Page] trifle; too mean indeed and trivial for your acceptance, but that I know you are wont to admit of any thing that proceeds from an honest undesigning Gra­titude. And though I am not at all inclin'd to vanity from the merits of the Work it self, yet I am proud that it affords me an opportunity to discover the lasting impressions, which your many Favours have made upon my Breast. Geometry, with Arithmetick her Wo­man, are Beauties, that having Truth written in their Foreheads, dare ap­pear in the Court of the greatest Mo­narch, and I doubt not but they will find very easie and courteous admit­tance into your Appartment; where if they shall afford you any divertisement when you return wearied from your ingenious Elaboratory, I shall then ac­compt that I have written to very good purpose. However, they certainly as­sure you, that it is impossible the teeth of Time should obliterate the honourable esteem conserved for you, in the heart of,

Sir, Your most humble Servant, IOHN MAYNE.

THE PREFACE.

I Shall not trouble thee, Reader, nor my self, with a long Apology for the publishing this Treatise. How demurely soever I should pretend to the contrary, I fear thou wilt still be apt to imagin, that I had a tolerable good opinion of it, before I ventur'd it to the Press; and truly I my self cann't well conceive, how any man should be over ambitious of being publickly laught at. If it be in any measure suited to the General Good, (for which I intended it,) I may at least expect thy pardon; but if upon the perusal thou shalt find it otherwise, I ingenu­ously acknowledge my self to have been mistaken. My Design in this Work is, to render the Rules of those excellent Arts, which the Title-page pretends to, so plain and obvious, as that they may be easily apprehended without the Assistance of a living Master. And if there were nothing new in the whole, but the perspicuity of the Principles, and easiness of the Method (which out of civility to my self I must deny) yet those alone are sufficient to vindicate me in this Publication; and I hope thou wilt not be angry, that I am a Well-wisher to thy Vnderstanding. For when I consider'd, that among the many good Books of this Nature, that are abroad in the [Page] World (though written by Persons of greater knowledge than I dare pretend to) some were so learned and obscure, as not to be understood, unless by those who have already made a considerable proficiency in these things; others, so voluminous and prolix that they fright the endeavours of such who cannot spend their whole time this way: I was willing, according to my abilities, to ob­viate both these inconveniences, and accordingly applied my self to the composure of something, which for its plainness and brevity might be ac­commodated to those of mean Capacities and small Leisure; and this Book is the result of those Contemplations. Whether I have accomplish'd my purpose or no, I make thee the Iudge; requiring only that thou censure impartially of the Author and his Endeavours, without being offended that he is desirous to do thee a courtesie. I shall not here expatiate in the praise of the Arguments I treat of, nor give thee one line of Encomium: though out of the great affection I bear to these Arts, I find a strange inclination in my self to be rhetorical, yet I am resolv'd not to affront thee; for truly their usefulness and excellency is so uni­versally known, that to tell thee of it as a new thing, were to suppose thee a Person of more than ordinary ignorance; only (as I said before) I must be so civil to my self too, as to inform thee, that (besides the Introduction to Arithmetick and the Treatise of Interest, of which I challenge no more than thou shalt find thy self very willing to give me) that Part which concerns the mea­suring [Page] of Solids, viz. the Prismoid, Cylindroid, &c. is wholly new, and never before made publick. The bulk is bigger than at first by me intended; but to gratifie the Book-seller, the Vulgar Arith­metick was an Appendix, though previous to the rest. But if one, or other, or all, prove either profitable or pleasant to thee, I am sufficiently oblig'd to subscribe,

Thy Friend, John Mayne.
From my House at the Golden-Ball in Shaws-Court, near St. Georges Church, Southwark; Iuly 29. 1673.

To the Ingenious Author, concerning his DECIMALS.

1
SIr, by your Art, and Pythagorean Pen,
I'd prove a Metempsychosis agen;
And were His Soul of Decimals but made,
2
As Plato's Soul o'th' world of Seven is said,
I'd swear 'twas slunk to you; but that you shew
3
More Skill than e're his rambling fancy knew.
Let roving Rabbies praise their Seven and Four;
4
We'l shew them Mysteries enough and more:
The Heav'nly Orbs are Ten, their Motions all
5
Conspire to make a perfect Decimal:
This is their Musick, and they shall be thus,
6
In spight of Tycho or Copernicus.
'Tis said the Muses are but Nine, but who
7
(Rather than fail) cann't add Apollo too?
Thus may we range the world, and quickly find,
8
We all to th' number of our Fingers bind.
Thus Logick all the wandring Species brings,
9
And places under tenfold Heads of things.
Thus I, to give the Author praise in all,
10
Reduce my Verses to a Decimal.

On his GAUGING.

YOur Circles, Sir, would make my folly ghess,
You were a Conjurer, though you wo'n't confess.
And Gauging is the rugged dev'lish Name
Of some Hobgobling Imp, the very same
That brought in Custome; but what e're he be,
He's a rare Fellow at the Rule of Three:
He doth just square the Circle; nay so true,
That the King's Right is given to a Cue.
There's none else such Impossibles can do:
You give the King's, I give this Right to you.
I. W.

On his worthy Friend Mr. J. Mayne, the Author of this BOOK.

Ingenious Artist, whither do'st aspire?
Or why t' outvye the Ancients do'st desire?
Have they not left enough to following Ages?
No: Thou their Master art, they but thy Pages.
My feeble Muse can never soar so high,
As thy Deserts herein extend, nor nigh.
Yet give me leave hereof to speak my mind:
No Man could better teach us in this kind,
Each Part so useful, and so plain I find.
T. W. [...].

To his ingenious Friend the Author Mr. John Mayne.

WHo reads thy Book with an im­partial Eye,
Will see how plain, and how ingeniously
Thy Rules are fram'd; here every Child may learn
Arithmetick, which doth the Truth discern.
The Iudges of our Realm could not dispence
To all Men Iustice, were't not fetch'd from hence:
Those Sons of Mars that furrow Neptune's Brow,
Unto this Science must their Labours bow:
The wealthy Merchant, and all Traders, hence
Must calculate their Gain, or their Expence:
The greedy Miser, here may plainly see
His Pelf's increase at Compound Vsurie:
The Purchaser of Farms, may also here,
Value his Lordships, whether cheap or deer.
Thy Squares and Cubes, methinks, so plain do seem,
That I old Euclid should thy Father deem.
All Humane Arts, Mechanical and Free,
For this Companion are oblig'd to Thee.
By Lines and Numbers, we our Buildings bring
In due proportion, framing every thing.
By these our Wooden Walls and Towers are fram'd,
Which guard our Island, and the Seas command:
[Page]These fill our Stores with rich and costly things,
Born from both Indies under Canvas Wings:
These fortifie our Towns with Forts, by Line;
By these we learn our Foes to undermine:
By these th' Excise and Customs we do scan,
Without Injustice to the Trading Man.
Thanks to our Author then, that hath set forth
These Arts so plain, and of abundant worth:
Which do to Sea and Land such Profit yield,
In Court, in City, Garrison, and Field.
Hugh Handy, Philomath.

A TABLE OF THE CONTENTS.

The First BOOK.
NOtation or Numeration.
pag. 1
Addition.
4
Subtraction.
8
Multiplication.
12
Division.
17
Reduction.
20
The Rule of Three.
22
The Rule of Practice.
28
Notation of Decimals.
34
Addition of Decimals.
37
Subduction of Decimals.
38
Multiplication of Decimals.
ib.
Division of Decimals.
41
Reduction of Decimals.
48
A Table of Reduction.
52
The Golden Rule.
54
The Double Golden Rule.
61
Of the Square Root.
66
A Table of Square Roots.
71
Of Quadratique Equations.
80
Of the Cube Root.
91
The Second BOOK.
OF Simple Interest.
pag. 99
Prop. 1. To find the Interest of any Sum, forborn any time, at any given rate.
100
Prop. 2. To find the present worth of any Sum, due at any time hereafter, at any given rate of Interest.
102
Prop. 3. Having the Principal, Amount, and Rate of In­terest, to find the Time of forbearance.
103
Prop. 4. Having the Principal, the Time, and the Amount, to find the Rate.
105
A Table of the Amounts of 1 l. from one to twelve months.
107
To find the Interest or Discompt of any Sum of Money by that Table.
107, 108
A Table for Equation of Time.
110
A more exact way of Equation.
111, 112, 113
A Decimal Table of the present worth of 1 l. per Quarter for 124 Quarters, at 6 per Cent. per Annum, Sim­ple Interest.
114
The Vse of the Table.
117
Of Compound Interest.
118
Prop. 1. To find the Increase of any given Sum, forborn any known Time, at a known Rate per Cent. per Annum.
118
Prop. 2. The Amount of a Sum of Money, the Rate of Interest per Cent. per Annum, and the Time being known, to find what was the Principal.
120
Prop. 3. The Principal, the Time, and the Amount of a Sum of Money being known, to find the Rate of In­terest per Cent. per Annum.
121
Prop. 4. The Principal, the Rate, and the Amount being known, to find the Time in which it hath so increa­sed.
122
Of Compound Interest Infinite.
123
[Page] Prop. 1. To find the present worth of an Estate in [...] Simple, at any Rate of Interest per C. per Ann.
123
Prop. 2. To find what Free-hold Estate any Sum of Money will buy, as any Rate of Interest per C. per Ann.
126
Prop. 3. An Estate being offered for a Sum of Money, the annual Rent being known, to find what Rate of In­terest the Purchaser shall have for his Money.
126
To find how many years purchase any Free-hold Estate is worth at any given Rate of Interest.
128
The number of years purchase being propos'd, to find the Rate of Interest it is offered at.
129
A Decimal Table for the Valuation of Leases or Annuities [...] payable Quarterly, at 6 per Cent. per Annum, Interest upon Interest for 31 years.
130
The way of Making the Table before-mentioned, for this, or any other Rate of Interest.
133
The Vse of the Table.
134
Six Questions performed by aid of the Canon of Loga­rithms.
137
The Third BOOK.
THe Definition of a Prismoid.
pag. 151
To find its Solidity
153
Prop. 1. An Example in Numbers.
ib.
Prop. 2. The Inversion of the former Solid upon its opposite Base, the Rule and Example.
157
Prop. 3. Another Example.
161
Prop. 4. The Inversion.
163
The Definition of a Pyramid.
171
To find its Solidity.
172
A Table of Divisors for Reduction of the Polygons, and the Cone, to Cubick Inches or Gallons.
173
An Example of a Trigon.
174
Of a Tetragon.
179
Of a Pentagon.
18 [...]
Of a Cone.
184
[Page]To find the Fall of a Conical Tunn.
186
Of Ca [...]k-Gauging.
191
To find a Casks Length.
192
To find the Head-diameter.
193
To find the Diagonal.
ib.
To find the Content as Spheroidal.
194
As Parabolical.
196
As Conical.
197
By a Table of Area's.
199, 200, &c.
To find the Vllage.
202, 205
ARITHMETICK IN Whole …

ARITHMETICK IN Whole Numbers.

NOTATION.

IT is necessary, that all Persons that would acquaint themselves with the Nature and Use of Numbers, do first learn to know the Characters by which any Quantity is expressed.

These Characters are in number nine, who with a Cypher are the Foundation of the whole Art of Arithmetick. Their form and denomi­nation as in this Example.

0. Cypher
1. One.
2. Two.
3. Three.
4. Four.
5. Five.
6. Six.
7. Seven.
8. Eight.
9. Nine.

[Page 2]These Characters standing alone express no more than their simple value, as 1 is but one, 2 standing by it self signifies but two, and so of the rest; but when you see more than one of those Figures stand together, they have then another signification, and are valued ac­cording to the place they stand in, being dig­nified above their simple quality, according to the Examples in this Table.

                Unites.
              Tens. 1
            Hundreds. 1 2
          Thousands. 1 2 3
        Ten Thousands. 1 2 3 4
      Hundred Thousands. 1 2 3 4 5
    Millions. 1 2 3 4 5 6
  Ten Millions. 1 2 3 4 5 6 7
Hundred Millions. 1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8 9

The denomination of Places according to this Table, must be well known, and are thus exprest; those standing in the place of Unites, signifie no more than their value before taught; but standing in the second place toward the left hand, they are increased to ten times the value they had before, 1 or One in the Unite place signifies but One; if it stand in the second place toward the left hand, and a Cypher be­fore [Page 3] it thus 10, it hath ten times its simple value, and is called Ten; if 2 stand in the place of the Cypher thus 12, it is then Twelve, being Ten and two Unites; 1, 2, or 3, standing in third place, with Figures or Cyphers toward the right hand of it, doth signifie Hundreds, as 100 is One hundred, 123 is One hundred twenty three, 321 is Three hundred twenty one, 213 is Two hundred thirteen; and so any three of the other Figures have like value, according to their Stations, the first to the right hand in the Unite place signifies so many Unites, the second, or that in the place of Tens, is increased to ten times its simple value, and in the third place, or place of Hundreds, any Figure there standing hath a hundred times the value it would have had were it in the Unite place.

The fourth place is the place of Thousands, any Figures standing there, with three Figures or Cyphers to the right hand of it, is so many Thousands as simply it contains Unites, so 3000 is Three thousand, 9825 is Nine thou­sand eight hundred twenty five, &c.

The fifth place is Ten thousands, and any five Figures placed together, are to be read after this manner: Example.

45326 Forty five thousand three hundred twenty six.

12345 Twelve thousand three hundred forty five.

The sixt place hath the denomination of [Page 4] Hundred thousands, and those six in the Table that stand in a rank are to be read, One hun­dred twenty three thousand four hundred fifty six.

The seventh is the place of Millions, and the seven in the Table are, One million two hundred thirty four thousand five hundred sixty seven.

And the eighth Rank of Figures are to be read, Twelve millions three hundred forty five thousand six hundred seventy eight.

The ninth rank is, One hundred twenty three millions four hundred fifty six thousand seven hundred eighty nine. And so any greater number of places, every figure one place more toward the left hand, is increased ten times in value more than in the place it stood before.

ADDITION.

ADdition, is a gathering or collecting of several Numbers or Quantities into one Sum, by placing all Numbers of like Deno­mination under one another, carrying all above ten to the next place, as in these Examples.

[...]

[Page 5]There is likewise another kind of Addition, that is not of whole Quantities, wherein is necessary to be known the number of Parts the Integer or whole Number is divided into, as Pounds and Shillings, every Pound is di­vided into 20 Shillings, and one Shilling is divided into twelve Pence, one Penny into four Farthings.

Now being to add a Number of Pounds and Shillings together, they are thus set down with a small Line or Point between them.

3-5

6:16

If these be added together, observe in casting up your Shillings, so many times as you have 20 in the Shillings, you must carry Unites to the Pounds, and set down the Remainder, being under 20, as in these Examples.

l. s.
3: 5
6: 16
10: 01

l. s.
4: 17
3: 15
5: 9
6: 12
19: 13

In the first Example, I find in adding the Shillings together, they make 21, so I set down 1 and carry 1 Pound to the Pounds: In the second Example, I find among the Shillings 53, which is 2 Pounds 13 Shillings, so I set down 13 under the Shillings, and 2 to the Pounds.

[Page 6]Any number of Shillings and Pence being to be added together, if your number of Pence amount to above 12, carry 1 to the Shillings, and set down the remainder under the Pence; if they make above 24, carry 2 Shillings, and set down the remainder, as before.

Examples.

s. d.
1: 6
2: 7
4: 1

s. d.
8: 9
2: 8
3: 10
15: 03

s. d.
1: 7
2: 6
3: 9
4: 8
5: 11
18: 05

In the first Example, you carry one Shilling; in the second, two; and in the third, three.

In Addition of Pence and Farthings, carry so many times four as you find in the number of Farthings to the Pence, setting down the remainder under the Farthings, as in these Examples.

[...]

[Page 7]When you would know the Sum of any number of Pounds, Shillings, Pence, and Far­thing, they are to be placed thus: [...]

Addition of Weight and Measure is perfor­med after the same manner.

  • 16 Ounces Averdupois, make a Pound.
  • 28 Pounds, make a Quarter.
  • 112 Pound, or 4 Quarters, make an Hun­dred gross.
  • 20 Hundred, make a Tun.

Examples.

[...]

Where observe, that so oft as I find 16 Oun­ces, I carry 1 to the Pounds; so often as I find 28 Pounds, I carry 1 to the Quarters; and as many times as I find 4 in the Quarters, so many times 1 do I carry to the Hundreds.

SUBTRACTION.

SVbtraction is the taking a lesser Number from a greater, and exhibits the Remainder.

In Subtraction the Numbers are placed one under another, as in Addition, thus: [...]

The first of these Numbers is called the Minorand, the second the Subducend, and the third Number, or the Number sought, is the Residuum.

8 The Minorand
6 The Subducend
2 The Residuum or Remainder

EXAMPLES of COINS.

[...]

But when the number of Pence or Shillings, are greater than the number that stands over it [Page 9] in the Minorand, you must borrow the next Denomination, as in this Example.

[...]

This Example I work after this manner, saying 9 d. out of 3 d. I cannot have, where­fore I borrow 1 s. from the Shillings, and sub­duct the 9 d. from that, and there will remain 3 d. which added to the other 3 d. maketh 6 d. I place therefore 6 d. in the Place of Pence, and proceed saying, 1 s. that I borrowed and 19 is 20 from 1 I cannot, wherefore I borrow 1 l. from the Pounds, and subduct from that the 20 s. and there remains nothing but the 1 s. which I place under the Shillings, and say, 1 that I borrowed and 6 is 7 from 7 and there remains nothing, then I place a Cypher under the 6, and say, 1 from 2 and there remains 1, which I set down, and 1 from 1 and there re­steth nothing. After this manner is performed Subduction of Weight and Measure.

Examples.

[...]

[Page 10] [...]

By which Examples, the Learner may per­ceive, that where the number to be subducted is greater than the number standing over it, I then borrow one from the next greater denomination, adding the remainder, if any be, to the lesser number before-mentioned, and setting them underneath those of like denomination with them.

The Proof of Subtraction is by adding the Subducend and Remainder together, and their Aggregate must always be equal to the Mino­rand, as you may see by the last Example.

I could here add many more Examples of Weight and Measure, but to the ingenious Practitioner I hope it will be enough, all other being wrought af [...]er the same manner, respect being had to the number of lesser denomina­tions contained in each greater. As

In Troy Weight,
  • 24 Grains make a Penny-weight.
  • 20 Penny-weight one Ounce.
  • 12 Ounces one Pound.

[Page 11]

Long Measure.
  • 4 Nails make a Quarter of a Yard.
  • 4 Quarters one Yard.
  • 5 Nails one Quarter of an Ell.
  • 4 Quarters one Ell.
  • 12 Inches a Foot.
  • 3 Feet a Yard.
  • 16½ a Perch.
  • 40 Perches a Furlong.
  • 8 Furlongs make an English Mile.

Liquid Measure.
  • 8 Pints make a Gallon.
  • 63 Gallons make a Graves Hogshead.
  • 4 Hogsheads make a Tun.
  • 36 Gallons make a Beer Barrel.
  • 32 Gallons make an Ale Barrel.

Dry Measure.
  • 8 Gallons of Corn make a Bushel.
  • 8 Bushels make a Quarter.

MULTIPLICATION.

MVltiplication is a kind of Addition, and resolveth Questions to be performed by Addition in a different manner: In order where­unto, it is necessary the Learner do well ac­quaint himself with this Table; the having this Table perfectly by heart, will make both this Rule and Division also very facile, other­wise they will be both troublesome and unplea­sant.

1 2 3 4 5 6 7 8 9 10 11 12
2 4 6 8 10 12 14 16 18 20 22 24
3 6 9 12 15 18 21 24 27 30 33 36
4 8 12 16 20 24 28 32 36 40 44 48
5 10 15 20 25 30 35 40 45 50 55 60
6 12 18 24 30 36 42 48 54 60 66 72
7 14 21 28 35 42 49 56 63 70 77 84
8 16 24 32 40 48 56 64 72 80 88 96
9 18 27 36 45 54 63 72 81 90 99 108
10 20 30 40 50 60 70 80 90 100 110 120
11 22 33 44 55 66 77 88 99 110 121 132
12 24 36 48 60 72 84 96 108 120 132 144

In the first Rank of this Table, you have an Arithmetical Progression from 1 to 12, [Page 13] and also in the first Column toward the left hand downwards. This Table doth at first sight exhibit the Sum of any number, so often repeated as you shall require, provided the numbers do neither of them exceed 12.

Multiplication hath three Members, thus called, a Multiplicand, a Multiplicator, and a Product: The Multiplicand, is the number to be repeated; the Multiplicator, is the number of times the first is to be repeated; and the Product, is the Sum of the Multiplicand so often repeated. As for Example.

A Countrey-man sold 6 Bushels of Wheat for 5 s. how many Shillings ought he to receive?

By Addition 6 must be 5 times set down thus: [...]

Or 5 six times repea­ted thus: [...]

But by Multiplication it is done thus:

6 The Multiplicand.

5 The Multiplicator

[Page 14]Now if you look in the Table precedent, in the first Column find 5, then look in the first Rank for 6, and cast your Eye down to their Angle of meeting, and you will find 30 standing under 6 and against 5, I then con­clude that 5 times 6 is 30; that is called the Product, and they will stand thus:

6 The Multiplicand.

5 The Multiplicator.

30 The Product.

But when you have a number to multiply, greater than any in the Table, as for Example:

A Gentleman having forborn his Rent of a Farm, at 157 l. per Quarter, for 3 Quarters, what ought he to receive?

The Multiplication will stand thus:

157 The Multiplicand.

3 The Multiplicator.

471 The Product.

I then say, 3 times 7 is 21, I set down 1 and carry 2; then, 3 times 5 is 15 and 2 is 17, I set down 7 next the 1, and carry 1; saying, 3 times 1 is 3 and 1 is 4, as in the Example before-going; and the Product is 471 l.

[Page 15]There is yet more variety, of which take these Examples following.

If 65 Ships do carry 536 Men in every Ship, how many Men will there be in all?

[...]

I say 5 times 6 is 30, set down 0 and carry 3; then 5 times 3 is 15 and 3 is 18, set down 8 and carry 1; then 5 times 5 is 25 and 1 is 26, which I set down: Then for the next Fi­gure, I say, 6 times 6 is 36, I set down 6 one place short of the former rank, and carry 3; then 6 times 3 is 18 and 3 is 21, set down 1 and carry 2; again, 6 times 5 is 30 and 2 is 32, these I set down: Then draw a line, and cast them up as they are placed, and the Sum is the Product and Answer to the Question, viz. 34840 Men.

In Multiplication, always make the lesser Number the Multiplicator, for it is all one whether I multiply 5 by 15, or 15 by 5, the Product is always the same.

[Page 16] If 128 Men of War have each made 746 Shot, how many Shot were made in all?

[...]

Begin as before with the Unites place, and say, 8 times 6 is 48, set down 8 and carry 4; 8 times 4 is 32 and 4 is 36, set down 6 and carry 3; then 8 times 7 is 56 and 3 is 59, which set down: Then go forward with the 2, (but remember to place your remainder one Figure short of the former) saying, 2 times 6 is 12, set down 2 under the 6 and carry 1; 3 times 4 is 8 and 1 is 9, which set down; twice 7 is 14, which set down: Also then, once 6 is 6, which place under the 9; once 4 is 4, which set under the 4; and once 7 is 7, which set under the 1: Then cast th [...] up, as in Addition, and the Sum is the Product, and answers the Question, viz. 95488 Shot.

If any number be to be multiplied by 1 with Cyphers, it is but adding so many Cyphers to the Multiplicand as there is in the Multipli­cator.

As for Example.

If 35678 be to be multiplied by 10, add one Cypher to the Multiplicand, thus, 356780; if by 100, add two Cyphers, thus, 3567800; &c.

[Page 17]And when any number is to be multiplied by any other number, that hath Cyphers an­nexed, always place the Cyphers immediately under the Line, as in these Examples.

[...]

DIVISION.

DIvision is also a kind of Subduction, and informs the Querent, how many times one number is contained in another.

There is in Division these three things to be observed, viz. the Dividend, the Divisor, and the Quotient. The Dividend is a number to be divided into parts, the Divisor is the quan­tity of one of those parts which the former is to be divided by, the Quotient is the number of such parts as the Dividend doth contain. There is also by accident a fourth number in this Rule necessary to be known, which is a Remainder, and that happens when the Divi­dend doth not contain an equal number of such quantities as it is divided by, as when 15 is to [Page 18] be divided by 4, the Dividend is 15, the Di­visor is 4, and there is a Remainder 3.

In Division you may place your numbers thus.

[...]

Multiplication is positive, but Division is performed by essays or tryals, after this manner: [...]

Here I first inquire how many times 3 I can have in 14, I find 4 times, I place 4 in the Quotient, and then mul­tiply the Divisor by that 4, placing the Product underneath the Dividend, as in the Example; say, 4 times 5 is 20, set down a Cypher under the 6 and carry 2, then 4 times 3 is 12 and 2 is 14, which I set down also, as in the Example; then subduct this Pro­duct from the Figures standing over them, and set down the Remainder.

[...]

Then for a new Divi­dend, I bring down the next figure, and postpone that to the Remainder, and inquire how many times 3 in 6, I cannot have twice, bacause I [Page 19] cannot have twice 5 from 5, I say then once, and place 1 in the Quotient, proceeding as be­fore saying, once 5 is 5, which I place under the first 6 toward the right hand, and once 3 is 3, which I set down under the other 6; subducting these as the former, I find the Re­mainder to be 31.

A [...]ter which I bring down the next figure in the Dividend, and postpone it to the Remain­der, as in this Example: [...] Then I inquire how many times 3 in 31, I sup­pose 9 times, placing 9 in the Quotient I multi­ply again, saying 9 times 5 is 45, 5 and carry 4; then 9 times 3 is 27, and 4 is 31; these being set down, as before directed, and subducted, there will remain nothing. I then conclude, that the Di­visor is so often contained in the Dividend as i [...] expressed in the Quotient, viz. 419 times.

For further Instructions, take these Exam­ples.

[...]

[Page 20] [...]

REDUCTION.

REduction is twofold, viz. bringing greater denominations into smaller, and that by Multiplication, as Pounds into Shillings, Shillings into Pence, &c. Also lesser deno­minations are reduced into greater, by Division as Pence into Shillings, Shillings into Pounds Minutes into Hours, Hours into Days, and Days into Years, &c.

Having any number of Pounds to reduc [...] into Pence, multiply them by 240.

Example.

In 869 Pounds how many Pence?

[...] Answ. 208560 Pence.

[Page 21] In 2486 Shillings how many Farthings?

[...] Answ. 119328 Farthings.

How many Minutes are there in 9476 Hours?

[...] The Answer 568560 minutes.

How many Pounds, Shillings, and Pence, are contained in 22929 Farthings?

[...]

[Page 22] In 544542 Cubique Inches, how many Beer Barrels, Firkins, and Gallons?

Inches in 1 B. Bar. 10152)544542(53:2:5 Bar. firk. g. [...] Inches in 1 Firkin 2538)6486 [...] Inches in 1 Gallon 282)1410 [...]

THE RULE OF THREE.

THis Rule is so called, because herein are three numbers given to find a fourth; of these three numbers, two are always to be mul­tiplied together, and their Product is to be divided by the third, and the Quotient exhibits the fourth number, or the number sought.

And here note, That of the three given numbers, if that number that asketh the Question be greater than that of like denomi­nation with it self, and require more, or if i [...] be less, and require less, then the number of like denomination is the Divisor.

[Page 23]Or, if the number that asketh the Question be less than that of like denomination, and require more; or if it be more, and require less, then the number that asketh the Question is the Divisor.

Example.

If 3 Yards of Sarcenet cost 15 s. what shall 32 Yards cost?

Which 3 numbers if you please may stand thus: [...]

Here you may see the term that asketh the Question is greater than that of like denomi­nation, being 3, and the other 32, and also requires more, viz. a greater number of Shil­lings; therefore, according to the Rule, the first term, or the term of like denomination to that which asketh the Question, is the Di­visor.

[Page 24]And the Answer is 160 Shillings, which being divided by 20 will be found 8 l.

Again,

If 32 Ells of Holland cost 160 s. what shall 3 Ells cost?

[...]

In this Question (being the Converse of the former) you may see the term that asketh the Question, here 3, is lesser than that of like de­nomination, being 32 Ells, and also requires less; therefore the first term here also is the Divisor.

And the Answer is 15 s.

If 36 Men dig a Trench in 12 Hours, in how many Hours will 144 Men dig the same?

[...]144)432(3 Hours, the fourth number. [...]

[Page 25]In this Question, the term that asketh the Question is greater than that of like denomi­nation, and requireth less; wherefore the term that asketh the Question is the Divisor.

If 144 Workmen build a Wall in 3 Days, in how many Days will 36 Workmen build the same?

[...]

This Question you may perceive to be the Converse of the former, here the term that asketh the Question is less than that of like de­nomination, and requires more, the term that asketh therefore is the Divisor.

If 125 lb. of Bisket be sufficient for the Ships Company for 5 Days, how much will Victual the Ship for the whole Voyage, being 153 Days?

This Question is of the same kind with the first Example; here the two terms of like de­nomination [Page 26] are 5 Days and 153 Days, the term that asketh the Question being more than the term of like denomination, and also requiring more; so, according to the general Rule, the term of like denomination to that which asketh the Question is the Divisor. It matters not therefore in what order they ar [...] placed, so you find your true Divisor; but if you will you may set them down thus: [...]

The Answer is 3825 lb. weight of Bisket.

[Page 27] A Ship having Provision for 96 Men during the Voyage, being accompted for 90 Days, but the Master taking on boord 12 Passengers, how many Days Provision more ought he to have?

Which is no more than this:

If 96 Men eat a certain quantity of Provision in 90 Days, in how many Days will 108 Men eat the same quantity?

[...]

The Answer is 80, so that for 108 Men he ought to have 10 Days Provision more.

If the Assize of Bread be 12 Ounces, Corn being at 8 s. the Bushel, what ought it to weigh when it is sold for 6 s. the Bushel?

[Page 28] [...]

In this Question, the term inquiring being less than the term of like de­nomination, and requiring more; therefore is the term so inquiring the Divisor.

The Answer is 16 Ounces.

THE RULE OF PRACTICE.

IT is necessary that the Learner get these two Tables perfectly by heart, which are only the aliquo [...] parts of a Pound and of a Shilling.

The Parts of a Shilling.
d. q.  
0 1 Forty eighth.
0 2 Twenty fourth.
0 3 Sixteenth.
1 0 Twelfth.
1 2 Eighth.
2 0 Sixth.
3 [...] Fourth.
4 0 Third.
6 0 Half.

[Page 29]

The Parts of a Pound.
s. d. q.  
0 00 1 The Nine hundred and sixtieth.
0 00 2 The Four hundred and eightieth.
0 00 3 The Three hundred & twentieth.
0 01 0 The Two hundred and Fortieth.
0 01 2 The Hundred and sixtieth.
0 02 0 The Hundred and twentieth.
0 03 0 The Eightieth.
0 04 0 The Sixtieth.
0 05 0 The Forty eighth.
0 06 0 The Fortieth.
0 08 0 The Thirtieth.
0 10 0 The Four and twentieth.
1 00 0 The Twentieth.
1 03 0 The Sixteenth.
1 04 0 The Fifteenth.
1 08 0 The Twelfth.
2 00 0 The Tenth.
2 06 0 The Eighth.
3 04 0 The Sixth.
4 00 0 The Fifth.
5 00 0 The Fourth.
6 08 0 The Third.
10 00 0 The Half.

Having these Tables perfectly in memory, any Question propounded will be readily re­solved, only by dividing the given number of [Page 30] Yards, Ells, Feet, Inches, Gallons, Quarts, Pounds, or Ounces.

Of which take some Examples.

145 Ells of Cloth at 3d. per Ell. 36 s. 3 d.

Three Pence being the fourth part of a Shilling, I divide the number by 4, and the quote is the number of Shillings it is worth.

728 at 4.d.

242s. 8d.

Four Pence being the third part of a Shilling, I divide by 3.

654 at 6d.

327 s.

Here take the half.

321 at 1d. 2q.

40s. 1d. 2q.

Here the eighth part.

[...]

Here take the sixth and the eighth part of the quote.

Having any number of Shillings to reduce into Pounds, cut off the last figure toward the [Page 31] right hand by a line, and the figures on the left hand of the line are so many Angels as they express Unites; draw a line under them, and take the half of them, and you have the num­ber of Pounds.

Examples.

[...]

Any Commodity, the value of 1 Yard being the aliquot part of a Pound, is thus cast up:

836 Yards of Broad Cloth at 6s. 8d. per Yard.

278l. 13 [...]. 4d.

Take the one third part, and that is the An­swer in Pounds: 3 in 8 twice, and carry 2; 3 in 23 seven times, and carry 2; 3 in 26 eight times, and carry 2; the third part of 2 l. is 13 s. 4 d. where always observe, that the Re­mainder is always of the same denomination with the Dividend.

[Page 32]654 lb. of Cloves at 5s. per lb.

163l. 10s.

Take the fourth part.

9464 Gall. of Brandy at 3s. 4d.

1577l. 6s. 8d.

The Sixth.

Where the Price is not aliquot.

625 at 3s. per Ounce. [...]

Here I take the tenth and the half of that tenth.

348 Dollers at 4s. 6d. [...]

The fifth and the eighth of that fifth.

245 lb. at 2s. 3d. [...]

The tenth and the eighth of that quote.

To cast up the amount of any Commodity, sold for any number of Farthings by the Pound, [Page 33] I borrow from the Dutch a Coin called a Guil­der, whose value is 2 s. English.

Then if a Question be proposed of the Amount of an Hundred weight of any Com­modity, by the Hundred Gross, viz. 112 lb. so many Hundred as there be, the Amount is so many Guilders so many Groats, as there are Farthings in the price of 1 lb.

As for Example.

A Hundred weight of Iron is sold for 5 Farthings the Pound, comes to 5 Guilders, that is 10 s. and 5 Groats, which together is 11 s. 8 d.

Again.

A Hundred weight of Lead is sold for 2 d. Farthing the Pound, that is 9 Guilders and 9 Groats, which is 21 Shillings.

But if it be the subtil Hundred, it is then but so many Guilders so many Pence: As if a Hundred weight of Tobacco be sold for 5 d. Farthing the Pound, the Hundred comes to twenty one Guilders and twenty one Pence, that is forty three Shillings and nine Pence.

ARITHMETICK IN DECIMALS.

NOTATION.

Integers. Decimals.
3 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9
Thousand Millions. Hundred Millions. Ten Millions. Millions. Hundred Thousands. Ten Thousands. Thousands. Hundreds. Tens. Unites. Tenths. Hundredths. Thousandths. Ten Thousandths. Hundred Thousandths. Millioneths. Ten Millioneths. Hundred Millioneths. Thousand Millioneths.

AS in Whole Numbers, the value or denomination of Places do increase by Tens, from the Unite place toward the left hand; so in Decimals, the value or denomination of Places do decrease [Page 35] by Tens, from the Unite-place toward the right hand, according to the precedent Table.

A Fraction or broken Number is always less than a Unite, as Pence are parts of a Shilling, and Shillings of a Pound, Inches of a Foot, and Minutes of an Hour, &c.

Fractions are of two kinds,

And are thus called

  • Vulgar, &
  • Decimal.

A vulgar Fraction is commonly expressed by two Numbers set over one another, with a small line between them, after this manner ⅔, the uppermost being called the Numerator, and the lower the Denominator.

The Denominator expresseth into how many parts the Integer or whole Number is divided, and the Numerator sheweth how many of those parts is contained in the Fraction.

Example.

If the Integer be a Shilling ⅔ is 8 d.

If it be 1 l. or 20 Shillings, it is 13 s. 4 d.

If a Foot, it is then 8 Inches.

Or if an Hour, it will be 40 Minutes.

A decimal Fraction hath always a common Number for a Numerator, and a decimal Number for its Denominator.

A decimal Number is known by Unity, wi [...]h [Page 36] one or more Cyphers standing before it, as 10, 100, 1000, &c.

A decimal Fraction is known from a whole Number by a point, or some other small mark of distinction, whether it stand alone, or be joyn'd with whole Numbers; as in these fol­lowing Examples.

[...]

Or else with a point over the head of Unity, or the Unite place; as in these Examples.

[...]

In decimal Fractions, the Numerators only are set down, the Denominator being known by the last Figure in the Numerator.

Example.

.2 is Two tenths.

.25 is Twenty five Hundredths.

.257 is Thousandths.

.2575 is Ten Thousandths, &c.

As Cyphers before a whole Number have no value, so Cyphers after a decimal Fraction are of no signification: But Cyphers before a decimal Fraction, are of special regard; for [Page 37] as Cyphers after a whole Number do increase that Number, so before a decimal Fraction they diminish the value of that Fraction.

Example.

.25 Twenty five hundredths.

.025 Twenty five thousandths.

.0025 Twenty five ten thousandths.

Each Cypher so added removing the Fraction further from Unity, making it ten times less than before.

ADDITION.

ADdition in Decimals, whether in pure De­cimals, or whole Numbers mixt with De­cimals, differs not from Addition in whole Numbers, only care must be had to the sepe­rating lines or points, that all places of like denomination stand one under another, both in the Addends and in the Sum; as in these Examples.

[...]

SUBTRACTION.

AS in Addition, so in Subtraction care must be had to the placing each Figure under that of like denomination with it self, then it is the same with Subtraction in whole Num­bers.

Examples.

[...]

MULTIPLICATION.

MVltiplication in whole Numbers serveth instead of many Additions, and teacheth of two Numbers given to increase the greater as often as there are Unites in the lesser.

It likewise consists of three Requisites, viz. a Multiplicand, a Multiplicator, and a Product.

[Page 39]The Multiplicand is the Number to be in­creased.

The Multiplicator is the Number by which it is to be increased.

And the Product is the Sum of the first Number so often repeated as there are U­nites in the second.

In decimal Fractions, or whole Numbers mixt with Fractions, the two first Numbers are called Factors, and the last is called the Fact.

Multiplication, whether in decimal Fracti­ons, or whole Numbers mixt with Fractions, differeth not (in the Operation) from Multi­plication in whole Numbers. The last Figures in both the Factors may be placed under one another, without respect to the distinction of places, or places of like denomination standing under one another, as in Addition and Sub­duction; yet from the Product must be cut off by a line or point so many places as there are Figures in decimal Fractions in both Factors of the last Figures standing toward the right hand.

Examples.

[...]

[Page 40]If it happen when the Multiplication is ended, that there be fewer Figures in the Pro­duct [...] than there are places in Decimals in both the Factors, then put Cyphers before the Pro­duct till the number of places be equal to those in both the Factors: As in these

Examples.

[...]

Whereby may be observed, That the Multi­plication of two Fractions doth not increase them as in whole Numbers, but they are here­by made less, and the Fact is removed further from Unity than either of the Factors.

If a whole Number be to be multiplied by a decimal Number, put so many Cyphers after the whole Number as there are in the decimal Number, and that Number will be the Product. If 48 be multiplied by 10, it will be 480; by 100, 4800; &c.

In multiplying decimal Fractions, or mixt Numbers, by a decimal Number, you need only remove the point or seperating line so many places toward the right hand as there be Cy­phers in the decimal Number. If you multi­ply .2845 by 10, the Fact will be 2.845; by 100, it will be 28.45; by 1000, 284.5; &c.

DIVISION.

DIvision, both in whole Numbers and Fra­ctions, is by young Practitioners found to be more difficult than any of the four Species; it will therefore require a little more industry in the Learner: But when once had, there will appear small difference between the Operation herein, as in any the precedent.

Division is also constituted by three Requi­sites, and a fourth by accident, viz. a Divi­dend, a Divisor, and a Quotient: The fourth is a Remainder, which doth not always happen to be.

The Dividend is the Number to be divided.

The Divisor is the Number by which the other is to be divided.

The Quotient is the Number found out by the Division.

And the Remainder is that which is left of the Dividend after the Division is ended, and is always less than the Divisor.

Example.

If 12 be to be divided by 4, then is 12 the Dividend, 4 the Divisor, and the Quotient will be 3.

If 13 be divided by 3, then 13 is the Divi­dend, 3 the Divisor, 4 the Quotient, and there will be a Remain, which is here 1.

[Page 42]Decimal Fractions, or mixt Numbers, are divided after the same manner as whole Num­bers are divided, only care must be had in gi­ving a true value to the Quotient. To perform which, observe well this General Rule.

The first Figure in the Quotient is always of the same denomination with that Figure which stands (or is supposed to stand) over the Unity place in the Divisor.

As to the manner of placing your Figures, and the way of dividing, there are many pub­lished by divers Writers of Arithmetick: The way of placing the Divisor under the Dividend, is the most apt for giving a value to the Quo­tient; but the rasing of Figures, and repeating the Divisor so often, is found an inconvenience; which to avoid, observe the following Ex­amples.

Being to divide 2487.04 by 53.6, I place them in this order: 53.6 ) 2487.048 (

Then I consider if the Divisor were placed under the Dividend, the Unity place in the Divisor, here 3 would stand under the 8 in the Dividend, I then set a mark over the head of the 8, and conclude the first Figure in the Quotient to be of the same denomination with it, which is Tens, in whole Numbers.

[Page 43] [...]

Having thus found the value of the first Fi­gure in the Quotient, I proceed to the division, and inquire, how many times 5 in 24? I find 4; I then set 4 in the Quotient, and go back, multiplying the whole Divisor by that Figure, and subduct the Product out of the Dividend, placing the Remainder underneath as part of a new Dividend: Thus 4 times 6 is 24, from 27, and there remains 3, which I place under the 7; again, 4 times 3 is 12, and 2 that I borrow­ed is 14, from 18, and there remains 4, which I place under the 8, as in the Example; then 4 times 5 is 20, and 1 I borrowed is 21, from 24, and there remains 3, which I place under the 4. For my new Dividend, I bring down the next Figure, here a Cypher, and post­pone it to the Remainder, and the Example will stand thus: [...]

Then proceeding in my Division, I ask, how many times 5 in 34? finding 6 times, I then place 6 in the Quotient, and as before say, 6 times 6 is 36, from 40, and there remains 4, which I set down under the Cypher; then 6 times 3 is 18, and 4 I borrowed is 22, from 23, [Page 44] and there remains 1, which I place under the 3; then 6 times 5 is 30, and 2 I borrowed is 32, from 34, and there will remain 2, which I place under the 4; then to this Remainder I bring down the next Figure in the Dividend, postponing it as I did the Cypher, and they will stand thus: [...]

I now inqui [...]e, how many times 5 in 21? and find 4 times, I then place 4 in the Quotient, and go on as before; there being yet a Remain­der, I add a Cypher, and proceed as before; and find, upon the adding one Cypher, my Di­visor greater than the Dividend, I place a Cy­pher in the Quotient: Example.

[...]

Having placed a Cypher in the Quotient, I add another to the Dividend, and make it 800; and then inquire, how many times 5 in 8? finding once, I put 1 in the Quotient, working as before: Where note, So long as there is a Remainder, if you add Cyphers and work after this manner, you may have as many Decimals as you please.

[Page 45]It doth often happen in Division, in decimal Fractions, or mixt Numbers, that the Unite place in the Divisor will stand beyond all the significant Figures in the Dividend, either to­ward the right hand or toward the left; in which case, that you may the better find out the value of the first Figure in your Quotient (according to the precedent General Rule) add Cyphers to the right or to the left hand of the Dividend, till you come over the Unity place in the Divisor, and what value or denomination that place is of, that is the denomination of the first Figure in the quote; as in these

Examples.

[...] [Page 46] [...]

If in Division in whole Numbers, there happen to be a Remainder, it is the Numerator of a Common Fraction, and the Divisor is the Denominator, and this Fraction is part of the quotient.

Example.

If you divide 66 by 8, the quotient will be 8 and [...]/8, according to the way of Vulgar [Page 47] Fractions, but in Decimal Fractions it will be 8.25.

Common way. [...] Decimal way. [...]

If you be to divide a whole by a decimal Number, cut off so many places by a mark, as there are Cyphers in the decimal Number: If 468 be divided by 10, the quote is 46.8; by 100, 4.68; and by 1000, quotes .468.

If a decimal Fraction, or a mixt Number, be to be divided by a decimal Number, remove your line or point so many places toward the left hand, as there are Cyphers in your decimal Number, supplying the vacant places with Cy­phers, if there be occasion: 69.5 divided by 10, is 6.95; by 100, it will be .695; by 1000, .0695; and by 10000, quotes .00695; &c.

Division being the Converse of Multipli­cation, as multiplying a mixt Number or deci­mal Fraction by a decimal Number, you remove your mark of distinction toward the right hand; so in dividing a decimal Fraction or mixt Number by a decimal Number, the mark is removed toward the left hand, as in the fore­going Examples.

REDUCTION.

TO reduce a vulgar Fraction into a decimal Fraction, your Rule is: Divide your Nu­merator by your Denominator, and the Quotient will be a decimal Fraction of the same value with the vulgar Fraction. So ¼, if reduced into a decimal Fraction, will be .25.

Example.

[...]

Here note, That only the even parts of an Integer will be exactly reduced into a decimal Fraction, as ½, 2/8, 2/16, &c. In all Surds, there will be some Remainder, but if you carry your decimal Fraction to four or five places, making the last one more than it is, if the sixth Figure be above 5, or else leave them out, and your Calculation will come near the truth; but if any desire to be more exact, he may take as many as he please.

[Page 49] Examples.

[...]

To reduce any decimal Fraction out of a greater denomination into a lesser, multiply the Fraction by those parts of the Integer into which you would have it reduced; as .65 being the parts of a Pound, you would know how many Shillings are contained in the Fraction, multiply it by 20: If you desire the Pence therein contained, multiply it by 240; or if Farthings, multiply by 960, the number of Farthings in a Pound or 20 Shillings.

[Page 50] [...]

The decimal parts of a Foot are reduced, by multiplying them by 12; if parts of a Foot Square, by 144; and the decimal parts of a Foot Solid, by 1728, the Cubick Inches in a Foot of Solid. The decimal parts of a Pound, are re­duced by 16, the Ounces in a Pound Aver­dupois; and 12, the Ounces in a Pound Troy. The decimal parts of a Beer Barrel by 36, and by 32 reduceth the parts of an Ale Barrel, into Gallons; and Gallons into Pints, by 8; Gallon [...] into Cubick Inches, by 282; and for Win [...] Gallons, by 231, the number of Cubick In­ches in such a Gallon, &c.

As greater denominations are reduced to lesser, by a multiplication of the several part [...] of the Integer; so lesser denominations ar [...] [Page 51] reduced to greater, by division. Any number of Shillings are reduced into Pounds, and the decimal parts of a Pound, if you divide them by 20; and Pence, if divided by 240.

Example.

[...]

Hours are reduced into the decimal parts of a Day, if you divide them by 24, the Hours in a Day Natural; and Minutes into the parts of an Hour, if divided by 60.

Perches are reduced into the decimal parts of an Acre, if you divide them by 160, the number of Square Poles or Perches in an Acre; and any number of Feet into Poles, and the decimal parts of a Pole, if you divide them by 16.5 the Feet in a Pole, or by 158.25 the number of Square Feet in a Square Pole; but if Wood-land Measure by 18, or if a Square Pole by 324, the Square Feet in a Pole or Perch of such Measure.

Any number of Inches are reduced into the parts of a Beer Barrel, if divided by 10152; and into Ale Barrels and parts, by 9024; &c.

For the ease of the Reader here is made a Table of English Coin reduced into the decimal parts of a Pound sterling.

[Page 52]

A Table of Reduction of English Coin, the Integer being one Pound.
Shil­lings. Deci­mals.   Pence. Decimals of a Pound.
19 .95   11 .0458333
18 .9   10 .0416667>
17 .85   9 .0375
16 .8   8 .0333333<
15 .75   7 .0291667>
14 .7   6 .025
13 .65   5 .0208333
12 .6   4 .0166667>
11 .55   3 .0125
10 .5   2 .0083333<
9 .45   1 .0441667>
8 .4      
7 .35      
6 .3      
5 .25   Far­things. Decimals of a Pound.
4 .2  
3 .15   3 .003125
2 .1   2 .0020833
1 .05   1 .0010417>

The Vse of the Table.

Having any Quest. wherein Pounds, Shillings & Pence, are required to be under one denomi­nation, viz. Pounds, and the parts of a Pound: First seek in the Column of Shillings for your Shillings, and set down the Fraction that stands against it; then in the Column of Pence, seek your Pence; in the Farthings, your Farthings; add all these together, and the Sum is the decimal Fraction desired.

Example.

What is the decimal Fraction for 17 s. 9 d. ¾?

First as the decimal parts of a Pound seek for 17 s. and the Fraction against it in the other Column is .85;

Which set down thus
.85
Then against 9 d. I find
.0375
And against 3 Farthings
.003125
Their Sum is
.890625

Which is the Number required, and is the decimal Fraction for 17 s. 9 d. ¾, as parts of a Pound.

Again, having a decimal Fraction in the parts of a Pound, and its desired to know the value thereof in lesser denominations: Let it be the Fraction before found, viz. .890625: I seek in the Table of Fractions for the neerest to it, [Page 54] and find .85, and against it 17 s. I then set .85 down, and subduct it from the other, and there remains .040625; I look over the Table again, and find the next neerest is .0375, against it 9 d. I subduct that; and find the Remainder .003125, stand against 3 Farthings.

[...]

So finding the value of any other decimal Fraction: If any thing remain after the last subduction, being less than a Farthing, I cast it away as of small regard.

THE GOLDEN RULE.

THis Rule is called the Rule of Three, be­cause herein are three Numbers given, to find a fourth. It is also called the Rule of Pro­portion, for as the first is in proportion to the second, so is the third to the fourth: And the Converse.

This Rule is called the Golden Rule for its excellent use in the Solution of Questions of various kinds, and great advantage is made of it in almost all kind of Calculations Arithmetical.

[Page 55]Two of the three Numbers given in every Rule of Proportion are of one denomination, and the third is of the same kind with the fourth sought; and one of the two Numbers that are of like species doth always ask the Question.

Arithmeticians distinguish this Rule by two denominations, one they call the Direct, and the other the Inverse or Backer Rule of Three.

One of the three given Numbers of like de­nomination in any Rule of Proportion is a Di­visor, the other remaining two are Multipliers. To find which of the forementioned Numbers is the Divisor, take these following Rules.

  • 1. If that Term to which the Question is an­nexed be more than that of like denomina­tion, and also requires more; or if it be less, and require less than the Term of like denomination; then that Term of like de­nomination to that which asketh the Que­stion is the Divisor, and the Question is in the Direct Rule of Three.
  • 2. If the Term which asketh the Question be more than that of like species, and requires less; or less, and requires more; then that Term which asketh the Question is the Di­visor, and the Question is in the Backer or Inverse Rule of Three.

Having by the precedent Rules discovered the Divisor, multiply the other two Numbers, and [Page 56] divide by the Divisor, your quote will be the Answer to the Question.

Note, If any of the Numbers given be in several denominations, they must be redu­ced into one, either greater or lesser, as before directed.

Example.

Quest. 1. If 12 ½ Yards of Taffaty cost 5 l. 7 s. 9 d. 3 q. what shall 5 ½ Yards cost?

In this Example, of the three Numbers gi­ven there are two of like denomination, and they are 12 ½ and 5 ½, the latter of which is the Term which asketh the Question, known always by the words what or how much. And this Term is less than that of like kind with it self, and also requires less, therefore according to the precedent Rule, this Question is in the Golden Rule Direct. These three Numbers may be placed in what order you please, pro­vided you mistake not your Divisor, but accor­ding to the general way, being reduced into De­cimals, and of one species, they will stand thus:

Yards. l. Yards.
12.5 5.390625 ∷ 5.5

Then, as before directed, multiply the second and third Numbers, and divide by the first, and the quotient exhibits the fourth Proportional or the Number sought.

[Page 57] [...]

The Answer is 2 l. 7 s. 5 d. 1 q.

Quest. 2. If 6 Yards of Broad Cloth cost 4 l. what shall 32 Yards cost?

Here the Term which asketh the Question is greater than the Term of like denomination, and requires more; therefore the Term of like denomination to the Term that asketh the Question is the Divisor.

[...]

[Page 58] [...]

The Answer is 21 l. 6 s. 8 d.

Quest. 3. If 320 Men raise a Breast-work in 6 Hours, in what time will 750 Men do the same?

Here the Term that asketh the Question is more than the Term of like denomination, and requires less; therefore the Term that asketh the Question is the Divisor, and this is the Backer Rule of Three.

[...]

The Answer is 2 Hours, 33 Minutes, and 36 Seconds.

[Page 59]Quest. 4. If 756 Men dig a Trench in 12 Hours, in how many Hours will 126 dig the same?

Here the Term that asketh the question is less than the Term of like denomination, and requires more; then according to the Rule the Term demanding is the Divisor, and this que­stion is also in the Inverse Rule of Three.

[...]

The Answer is 72 Hours.

There is sometimes four Numbers given in a question, yet is it but a Single Rule of Three, for one of the four Numbers is of no signifi­cation, and might as well have been left out.

Example.

Quest. 5. If 10 Workmen build a Wall 40 Foot long in 3 Days, in what time might 50 Men have done the same?

Here note, there is four numbers given, and yet there is but three to be used in working [Page 60] the question, you must therefore find which those 3 are that are necessarily to be used: Thus,

First, you must take the Term that asketh the question, here 50 Workmen; secondly, you must have the Term of like denomination with it, which is 10 Workmen; thirdly, the Term sought, being Days; you must take the Term of like denomination with that also, which is here 3 Days: The superfluous Term then in the question is 40, which might have been left out, and they will then stand thus: [...]

The Answer is Half a Day or 12 Hours.

This question is in the Rule of Three Inverse.

Quest. 6. If 100 l. gain 6 l. in 12 Months, what shall 32 l. gain in the same time?

In this question the 12 Months is the super­fluous Term, being of no use in the Calcula­tion, the Terms required being 100 l. 6 l. and 32 l.

Note, Though the Terms in this question be all Money, and so may seem to be of one species, yet they are not; 100 l. and 32 l. [Page 61] are of one kind, being both Principal, and the other Term is of the same deno­mination with the Term sought, viz. Gain or Interest.

[...]

The Answer is 1 l. 18 s. 4 d. 3 q. ferè.

And this question is in the Direct Rule of Three, the Term that asked the question being less than the Term of like denomination, and also requiring less, &c.

THE DOUBLE GOLDEN RULE.

THis Rule is called the Double Golden Rule, or Double Rule of Three, because it re­quires two distinct Calculations, before you can answer the question.

And in this Rule there are five Numbers given to find a sixth sought.

This differs not in the operation from the Single Rule, only the Calculation is twice re­peated.

Of the five Numbers given, the question is sometimes annexed to two, and sometimes but to one.

[Page 62]If the question be annexed to two of the five given Numbers, then are there two of the other three of the same species with those that ask the question, and the third is proportional to the Number sought.

For the due regulation of these two Calcula­tions, wh [...]n the question is annexed to two of the five Numbers, take these Directions.

First, take one of the Numbers demanding, and let that ask the question in the first opera­tion; secondly, take that of the same species, and also that of the like quality with the re­spondent, of these three constitute your first Rule of Proportion; then find which is your Divisor, according to your Rule pag. 55. and proceed to find the fourth in proportion.

Then for your second Rule of Three, take the other of the two Numbers to which the question is annexed, and let that ask the question; take also the Number of like kind, and the fourth Num­ber found in the first Calculation; judge which is your Divisor, and work accordingly; the last Quotient will be the sixth Number, or the Num­ber sought.

Example.

If a Trench be 20 Perches in length, and made by 12 Men in 18 Days; how long may that Trench be, that shall be wrought be 48 Men in 72 Days?

[Page 63]Here the question is annexed to two of the five Numbers, viz. 48 Men and 72 Days; now according to the foregoing direction, take one of the two Numbers inquiring, 48, and say, [...]

Then take the other of the two Numbers in­quiring, and say, [...]

If 6 Lighters bring 60 Tuns of Ballast in 5 Tides, how many Tun will 15 bring in 12?

[...]

[Page 64] If a Man travel 160 Miles in 4 Days, when the Days are 10 Hours long; in how many Days will he travel 195 Miles, when the Days are 14 Hours long?

[...]

When a Question is stated in the Double Rule of Three, so that there is but one Number in­quiring,

First, take that Number, and let it ask the question in the first Rule; take also the Number [Page 65] of like denomination, together with the Num­ber joyn'd to that of like denomination; and of these three Numbers constitute your first Rule of Proportion.

Secondly, let that Number which was found in the first Operation, ask the question in the second; then take the Number of like denomi­nation to it, and also the Number joyn'd with that like Number; of these three is your se­cond compounded; find your Divisor, and pro­ceed; the last quote exhibits the Answer.

Example.

If 4 Crowns at London make 2 Ducates at Venice, and 8 Ducates at Venice make 20 Pa­tacoons at Genoa; how many Patacoons at Genoa will make 120 Crowns at London?

[...]

Of the Square Root.

A Square is a plain Superficies bounded with four right Lines of equal length, the Angles also are equal, being all right Angles, as (a b c d) [...]

The measure of a Square is by a Square, that is, when it is known how many Square Inches, Feet or Perches, is contain'd in any Superficies, the Content or Area of the said Superficies is then said to be known. And in a Square, it is found by multiplying the length by the breadth, which being equal, it is called Squaring of a Number, and by the Learned Dr. Pell, Involution, and the Product or Area is the second Power; now the Side of [Page 67] such a Square is by Geometricians called a Root or the first Power.

Let the Side a b be 222 Inches, Feet, or Perches, &c.

[...]49284 the Area or (q) square. Number.

Now having the Area of a Square or Square Number given, and the Side or Root be requi­red.

This is called the Extraction of a Square Root, and also Evolution of the second Power.

Let the Number be as before 349284.

The first thing to be done in the Extraction of a Root is punctation, or pointing the Num­ber given; which is thus done, first set a point over the Unite-place, and omitting one point every other Figure thus, 349284; there be­ing three points in the Number, intimates three figures in the Root

To proceed then, enquire the greatest Square Number contained in those figures, under the first point on the left hand; the greatest Square Number in 34 is 25, whose Root is 5, which place in the quotient for the first figure in the Root, subduct its Square out of 34, and set the Remainder 9 underneath as in the Example.

[Page 68] Example.

[...]

The first figure in the Root thus found, the rest are found by Division; for a Divi­dend bring down the figures under the nex [...] point, and postpone them to the last Remain­der, and the Example will stand thus, your Divisor being double the Root found.

[...]

Then I proceed to Division, always suppo­sing the last Figure in my Divisor standing unde [...] the last save one in the Dividend; the Number to be subducted from the Dividend must always be the Square of the last Figure in the Root, and the Divisor multiplied by the last Figur [...] in the Root, so added together as in this Ex­ample, viz. so that the Unite-place in the last Number stand one place further to the righ [...] hand.

Nine times the Divisor
90
And 9 multiplied by it self
81
981

[Page 69]Which being subducted from the Dividend will remain 11, as part of a new Dividend, to them bring down the two next figures, and the Example will stand thus: [...]

The Divisor as before is double the whole Root found, and for the Number to be sub­ducted, after you have made enquiry how many times the Divisor will be found in the Dividend, if so placed as aforesaid, it will here be found once, then place [...]r in the quotient for the third figure in the Root, the Number to be subducted will be as before, and the Example will stand thus: [...]

The Divisor multiplied by the last Figure found, and the Square of that Figure placed as before directed.

[Page 70]Rectangle — 118

Square — 1

1181

1181 subducted from the Dividend 1184

There will remain but — 3

Which sheweth the Number was not a Square Number; but if you desire to have it further, add two Cyphers to the Remainder for a new Dividend, double your whole quotient for a new Divisor, and you may have as many De­cimals as you please.

[Page 71]

Tabula Laterum Quadra­torum ab Unitate ad 200.
Qua­drata. Latera.
1 1, 00000, 000000
2 1, 41421, 356237
3 1, 73205, 080757
4 2, 00000, 000000
5 2, 23606, 797750
6 2, 44948, 974278
7 2, 64575, 131106
8 2, 82842, 712474
9 3, 00000, 000000
10 3, 16227, 766017
11 3, 31662, 479036
12 3, 46410, 161514
13 3, 60555, 127546
14 3, 74165, 738677
15 3, 87298, 334621
16 4, 00000, 000000
17 4, 12310, 562562
18 4, 24264, 068712
19 4, 35889, 894354
20 4, 47213, 595500
21 4, 58257, 569496
22 4, 69041, 575982
23 4, 79583, 152331
24 4, 89897, 948556
25 5, 00000, 000000
26 5, 09901, 951359
27 5, 19615, 242271
28 5, 29150, 262213
29 5, 38516, 480713
30 5, 47722, 557505
31 5, 56776, 436283
32 5, 65685, 424948
33 5, 74456, 264654
34 5, 83095, 189485
35 5, 91607, 978310
36 6, 00000, 000000
37 6, 08276, 253030
38 6, 16441, 400297
39 6, 24499, 799840
40 6, 32455, 532034
41 6, 40312, 423743
42 6, 48074, 069841
43 6, 55743, 852430
44 6, 63324, 958071
45 6, 70820, 393250
46 6, 78232, 998313
47 6, 85565, 460040
48 6, 92820, 323028
49 7, 00000, 000000
50 7, 07106, 781185
51 7, 14142, 842854
52 7, 21110, 255093
53 7, 28010, 988928
54 7, 34846, 922835
55 7, 41619, 848710
56 7, 48331, 477355
57 7, 54983, 443527
58 7, 61577, 310586
59 7, 68114, 574787
60 7, 74596, 669241
61 7, 81024, 967591
62 7, 87400, 787401
63 7, 93725, 393319
64 8, 00000, 000000
65 8, 06225, 774830
66 8, 12403, 840464
67 8, 18535, 277187
68 8, 24621, 125124
69 8, 30662, 386292
70 8, 36660, 026534
71 8, 42614, 977318
72 8, 48528, 137424
73 8, 54400, 374532
74 8, 60232, 526704
75 8, 66025, 403784
76 8, 71779, 788708
77 8, 77496, 438739
78 8, 83176, 086633
79 8, 88819, 441732
80 8, 94427, 191000
81 9, 00000, 000000
82 9, 05538, 513814
83 9, 11043, 357914
84 9, 16515, 138991
85 9, 21954, 445729
86 9, 27361, 849550
87 9, 32737, 905309
88 9, 38083, 151965
89 9, 43398, 113206
90 9, 48683, 298050
91 9, 53939, 201417
92 9, 59166, 304663
93 9, 64365, 076099
94 9, 69535, 971483
95 9, 74679, 434481
96 9, 79795, 897113
97 9, 84885, 780180
98 9, 89949, 493661
99 9, 94987, 437107
100 10, 00000, 000000
101 10, 04987, 562112
102 10, 09950, 493836
103 10, 14889, 156509
104 10, 19803, 902719
105 10, 24695, 076596
106 10, 29563, 014099
107 10, 34408, 043279
108 10, 39230, 484541
109 10, 44030, 650891
110 10, 48808, 848170
111 10, 53565, 375285
112 10, 58300, 524426
113 10, 63014, 581273
114 10, 67707, 825203
115 10, 72380, 529476
116 10, 77032, 961427
117 10, 81665, 382639
118 10, 86278, 049120
119 10, 90871, 211464
120 10, 95445, 115010
121 11, 00000, 000000
122 11, 04536, 101719
123 11, 09053, 650641
124 11, 13552, 872566
125 11, 18033, 988750
126 11, 22497, 216032
127 11, 26942, 766958
128 11, 31370, 849898
129 11, 35781, 669160
130 11, 40175, 425099
131 11, 44552, 314226
132 11, 48912, 529308
133 11, 53256, 259467
134 11, 57583, 690279
135 11, 61895, 003862
136 11, 66190, 378969
137 11, 70469, 991072
138 11, 74734, 012447
139 11, 78982, 612255
140 11, 83215, 956620
141 11, 87434, 208704
142 11, 91637, 528781
143 11, 95826, 074310
144 12, 00000, 000000
145 12, 04159, 457879
146 12, 08304, 597359
147 12, 12435, 565298
148 12, 16552, 506060
149 12, 20655, 561573
150 12, 24744, 871392
151 12, 28820, 572744
152 12, 32882, 800594
153 12, 36931, 687685
154 12, 40967, 364599
155 12, 44989, 959799
156 12, 48999, 599680
157 12, 52996, 408614
158 12, 56980, 508998
159 12, 60952, 021292
160 12, 64911, 064067
161 12, 68857, 754045
162 12, 72792, 206136
163 12, 76714, 533480
164 12, 80624, 847487
165 12, 84523, 257867
166 12, 88409, 872673
167 12, 92284, 798332
168 12, 96148, 139682
169 13, 00000, 000000
170 13, 03840, 481041
171 13, 07669, 683062
172 13, 11487, 704860
173 13, 15294, 643797
174 13, 19090, 595827
175 13, 22875, 655532
176 13, 26649, 916142
177 13, 30413, 469565
178 13, 34166, 406413
179 13, 37908, 816026
180 13, 41640, 786500
181 13, 45362, 404707
182 13, 49073, 756323
183 13, 52774, 925847
184 13, 56465, 996625
185 13, 60147, 050874
186 13, 63818, 169699
187 13, 67479, 433118
188 13, 71130, 920080
189 13, 74772, 708488
190 13, 78404, 875209
191 13, 82027, 496109
192 13, 85640, 646056
193 13, 89244, 398945
194 13, 92838, 827718
195 13, 96424, 004377
196 14, 00000, 000000
197 14, 03566, 884762
198 14, 07124, 727947
199 14, 10673, 597967
200 14, 14213, 562373

The Use of the precedent Table is princi­pally for the ease of the industrious Artist; when he hath the Extraction of a Square Root in the Solution of any Question, it is but seeking the given Number in the Table, and just against it he shall find the Root. By the [Page 80] subsequent Examples will it plainly appear, how useful such a Table to 1000 Roots would be in quadratique Equations, and in the Cubes also, which (were there incouragement given to the Sons of Art) I doubt not but some in­genious Person would enrich the World there­with; these being long since Calculated by Mr. Henry Briggs of Oxford, and given me by my honoured Friend, Mr. Iohn Collins, his desire being to have them made more publick, and the conveniency of such a Table (before mentioned) shewn, by some Examples upon this.

Of Quadratique Equations.

Mr. Dary, in his Miscellanies, chap. 8. saith to this, or the like purpose:

1. When any Equation propos'd is incumbred with Vulgar Fractions, let it be reduced to its least Terms in whole Numbers, if possible; if not, let it be brought to its least Terms in Deci­mals.

2. It is evident from divers Authors, That if any Quant [...]ty shall be signed −, then the Square Root, or the Root of any even Power of such Quantity so sign'd, is inexplicable, for they cannot be generated from any Binomials that shall be equal.

As for Example.

− 9 being a Negative can be made of no­thing (if taken as a Square Number) but + 3 [Page 81] and − 3, which Roots are not equal, they being neither both Affirmatives nor both Nega­tives.

3. When you have cleared the Equation by the Second hereof, and that the Co-efficient in the highest Power is taken away, or be Unity, then will quadratique Equations resolve themselves into the four following Compen­diums.

4. Let your Equation be so reduced, that the highest Power stand on the left side alone, the sign + being always annexed, or supposed to be annexed.

Example, Quesita a.

First Equation.

[...]

Second Equation.

[...]

[Page 82]Third Equation.

[...]

Fourth Equation.

[...]

Illustration by Numbers, Quesita a.

First Equation.

[...]

[Page 83] Proof of the Affirmative.

The Square of 9 = 81

Six times the Root = 54

To which add 27

81

Which was to be proved.

Proof of the Negative.

[...]

Example 2.

[...]

[Page 84]Second Equation.

Example 1.

[...]

Example 2.

[...]

[Page 85]Third Equation.

Example 1.

[...]

Example 2.

[...]

[Page 86]Fourth Equation.

Example 1.

[...]

Example 2.

[...]

[Page 87]But if in a Square Equation there happen to be a Coefficient annexed to the highest Power, it is resolved by transferring the Coefficient with the Sign of Multiplication to the other side.

Admitting the Equation be [...] Then the Coefficient 2 being transferred (as before directed) they will stand as in this Example.

First Equation.

[...]

The Root of +25 being +5, then is +5+3 = 8, and a=+4, the Affirmative Answer. And +3−5 is =−2, and a=−1, the Ne­gative Answer.

The Proof is easie:

First, if a be = 4, 2 aa is = + 32, and 6 a is =+24, to which +8 being added, the Sum is +32 which was to be proved.

Again, a=−1, then 1 aa is =−2, whereto +8 being added, the Sum is =+6, which also was to be done.

[Page 88] Second Equation.

[...]

Now + 13 + the √ 121, viz. + 11 is =+24, the ¼ whereof is =+6= a, and aa = 36, and 4 aa=+144, +26 a=+156, to which if −12 be added, the Sum will be +144 also.

Again, If to +13 you add − the √ 121, viz. −11, 4 a will be =+2, and consequently + a=+½, 4 aa is then =+1, and +26 =+13, to which add −12, and the Sum is =+1, which was to be proved.

Third Equation.

[...]

[Page 89]−3+13 =+5 a, here a=+2, 5 aa = +20, −6 a=−12, to which add +32, the Sum is also +20.

Again, −3−13=−16=−5 a, and a= −3.2, 5 aa=+51.2: Also −6 a being = +19.2, to which add +32, the Sum is = +51.2.

Fourth Equation.

[...]

Which was to be done.

[Page 90] Note, Always where there is no Sign annexed to any Term in the Equation, the Sign + is supposed to be annexed.

I have been the larger in these Examples, that the young Analist may with the more ease ap­prehend the several kinds by this variety; in some o [...] the surd Roots I have on purpose omit­ted the large number of Places, four or five being sufficient for use in most cases; but if any desire to be more exact, he may take them as far as he pleaseth, or the Table doth exhibit.

Of the Cube Root.

THe Cube is a Solid, and hath three dimen­sions, length, breadth, and depth, and is inclosed by six plain square Superficies.

Example.

[...]

Let the Side a, b, or c, d, &c. be 125: To find the Content in Solid Feet or Inches, is the Involution of the Side or Root. Thus: [...]

The Solidity.

And this is called the Third Power.

[Page 92]The Evolution hereof, is also termed the Extraction of the Cube Root, wherein observe first your punctation, omitting two, point every third Figure.

Example.

[...]

The first Figure in the Root is found by ta­king the greatest Cube Number, contained in the Figure or Figures that stand under the first Point towards the left hand, here 71, whose Root is 4, therefore that 4 must be placed in the Quotient as the first Figure in the Root, and the Example will stand thus: [...]

Then the Cube of 4 is 64, which subduct out of the first Figures, and set down the re­mainder if any be. The first Figure found in this peculiar manner, the rest are found by Di­vision thus: The Dividend consists of the re­mainder, if any be, and the three Figures under the next Point postponed; the Divisor is always three times the Square of the Root, and three times the Root it self: These two Numbers being so to be added together, as that the Unites of the first stand over the Tens of the second.

Three times the (q) square. of the √ = 48

Three times the √ = 12

Divisor 492

[Page 93]Then will the Example stand thus: [...]

Then proceed to Division, always supposing the last Figure in the Divisor to stand under the last save one in the Dividend, and enquire, how many times 4 in 7? place 1 in the Quotient. Then for your Number to be subducted out of the Dividend, it always consists of three Num­bers, viz.

Three times the (q) square. of the first Figure = 48

Multiplied by the second 1

Product of the first by the second 48

3 times the (q) square. of the second by the first 12

And the Cube of the second 1

The Subducend 4921 [...]

The (q) square. of 41 = 1681

3

(q) square of 41 × 3 = 5043

Then for a new Dividend, bring down the three next Figures, postponing them as before.

The Divisor thrice the (q) square. of the √ 41 = 5043

And three time the √ 41 = 123

50553

[Page 94]Which being set on the left hand the Divi­dend, stands thus: [...]

Then enquire, how many times 5 in 30? you will find [...] times, which place in the Quotient. Your Subducend is as before,

3 times the Square of 41 × 5 = 25215

3 times the Square of 5 × 41 = 3075

And the Cube of 5 = 125

The Subducend 2552375

Which being subducted from 3032125

There will remain 479750

Which shews the Number was not a Cube Number; if you add three Cyphers, and work as before, you may have as many Decimals Fra­ctions as you please.

In this Extraction I have not taken the same Number the Cube first mentioned did produce, but by adding another Figure, made the Number greater, that it might take in all Cases; but in the following Extraction it is explicated.

[Page 95] [...]

Three times the Square of the Root = 3

Three times the Root is also = 3

The first Divisor = 33

Three times the Square of 1 = 3

2

Three times the Square of 1 × 2 = 6

Three times the Square of 2 × 1 = 12

The Cube of 2 = 8

The first Subducend = 728

[...]

Three times the Square of the Root = 432

Three times the Root = 36

The second Divisor = 4356

[Page 96]Three times the Square of the √ 12 = 432

5

Three time the Square of 12 × 5 = 2160

Three times the Square of 5 × 12 = 900

The Cube of 5 = 125

The second Subducend = 225125

A SHORT TREATISE OF …

A SHORT TREATISE OF SIMPLE & COMPOUND INTEREST: WITH TWO TABLES FOR THE CALCULATION OF The Value of Leases or Annuities by Quarterly Payments, at 6 per Cent. per Annum. By Iohn Mayne.

London, Printed by William Godbid, for Nath. Crowch, in Exchange-Alley. M. DC. LXXIII.

Of Simple Interest.

QUestions in Simple Interest are wrought by the Double Rule of Proportion, wherein five Numbers are given to find the sixt.

And if you put P=100 Principal, and T for Twelve Months, G=6 l. the Rate of Inte­rest, and p = any other Sum greater or lesser, t = any other Time (above or under Twelve Months) and also g = to the Gain thereof at that Rate.

Then if any one Term of these six be un­known, it is explicated by the other five (like Symbols having the same denomination) as in this Equation.

PTg = ptG

That is the Fact of 100 l. multiplied by one year, and that Product by 6 the Interest of 25 l. for 4 years, is equal to the Fact of 25 mul­tiplied by 4 years, and that Product by 6 the Interest of 100 l. for one year.

[Page 100] Example.

[...]

Which was to be proved.

Now forasmuch as the usual Questions of Simple Interest, are proposed from a Sum pre­sently due to the Gain thereof, & contra; it will be requisite you put A = the Amount of a Sum, forborn or due hereafter, and then you will have A= p+g, as in the former Equation.

Example.

P T P+G

As 100 in 1 to 106

So 25 in 4 to 31 = p+g

From the precedent Analogism will arise these four Propositions.

Prop. I.

A Sum presently due =p, being forborn a certain time = t, at a certain rate = G, per Cent. per Annum: Q. the Amount = A?

[Page 101]Equation A = [...]

That is, the given Sum multiplied by the given Time, and that Product again multiplied by the given Rate of Interest, the last Product divided by the Principal, viz. 100, in the Time, viz. 1, exhibits the Gain of that Sum in that Time.

Illustration.

Quest. 1. 25 l. being forborn 18 Months, at 6 per Cent. per Annum; what doth it a­mount to?

[...]

The Answer being 27 l. 5 s. the Amount in that time.

Quest. 2. If 175 l. be forborn for 7. Years, at 6 per Cent. per Annum, Simple Interest; what will it amonut to at the end of the said time?

[Page 102] [...]

The Answer = 248.5 That is 248 l. 10 s.

Prop. II.

A Sum of Money = A, due at a certain time hereafter = t, at a certain Rate of Interest = G, per Cent. per Annum. Q. The pre­sent worth = p?

Equation, p= [...]

That is, the Fact of the Amount multiplied by the Principal, 100, in the Time, viz. 1 Year, divided by the said Principal multiplied into the said Time, more the Rate of Interest multiplied into the Time of Forbearance, the Quotient is equal to the present worth.

Example.

Quest. 1. If 248 l. 10 s. be due at the end of 7 years, what is it worth in ready money, discompting Interest, at 6 per Cent. per Annum.

[Page 103] [...]

The Answer is 175 l.

Quest. 2. If 950 l. be due at the end of 12 years, what is it worth in ready money, at 9 per Cent. per Annum Simple Interest?

[...]

The Answer is 456 l. 14 s. 7 d. ¼ ferè.

Prop. III.

A Sum presently due = p, having been forborn [...] time unknown = t, did amount to a cer­tain Sum = A, at a Rate of Interest = G, per Cent. per Annum. Q. the Time of for­bearance = t?

[Page 104]Equation, [...]

That is to say, the Amount less the Principal, so increased, multiplied by 100, and that Product divided by the Fact of the before-mentioned Principal, and Rate of Interest, quotes the Time of forbearance.

Example.

Quest. 1. If 175 l. hath been forborn till with the Interest at 6 per Cent. per Annum it is increased to be 248 l. 10 s. Q. How long hath it been forborn?

[...]

The Answer is 7 years.

Quest. 2. If 25 l. hath been forborn till it is amounted to 27 l. 5 s. at 6 per Cent. per Annum, Simple Interest. Q. In what time is it so increased?

[...] [Page 105] [...]

The Answer 1 year and an half.

Prop. IV.

A Sum of Money = p, being forborn a certain time = t, and at the end of that Term did amount to a Sum = A. Q. At what Rate of Interest?

Equation, [...]

Or from the Amount subduct the Principal, and the Remainder multiply by 100, that Pro­duct divided by the Principal multiplied by the Time, the Quotient will be = G the Rate of Interest, per Cent. per Annum.

Illustration.

Quest. 1. If 250 l. forborn 3 year [...] and 6 months, did amount to 324 l. 7 s. 6 d. at what Rate of Interest did it so increase?

[Page 106] [...]

The Answer is 8 l. 10 s.

Quest. 2. If 175 l. being forborn 7 years, did amount to 248 l. 10 s. what Rate of Simple Interest per Cent. per Annum was it ac­compted at?

[...]

The Answer is 6 l.

If one month be taken for the 1/12 of a year, the business of Interest and Rebate is very easily performed by a small Table of the A­mounts of 1 l. for any number of months, not exceeding 12; which Table is made by this Analogy, 100. 106::1.1.06.

[Page 107]

A Table of the Increase of 1 l. at 6 per Cent per Ann.
Months. Value.
12 1.06
11 1.055
10 1.05
9 1.045
8 1.04
7 1.035
6 1.03
5 1.025
4 1.0 [...]
3 1.015
2 1.01
1 1.005

If the Question be of the Amount of any Sum forborn any number of months, at 6 per Cent. per Annum, multiply the given Sum by the Tabular Number for that time, and the Product answers the Question.

Example.

If 125 l. be forborn 10 months, what will it amount to?

The Principal = 125

The Tabular Number for 10 months = [...]

The Answer is 131 l. 5 s.

[Page 108]If the Question be only what is the Interest of any Sum for any time, then multiply the Sum for that time by the Tabular Number less an Unite.

Example.

What is the Interest of 125 l. for 10 months?

The Principal = 125

The Tabular Number less an Unite = .05

6.25

The Answer is 6 l. 5 s. prout suprà.

For Discompt or Rebate of any Sum to be forborn, the present worth is found by dividing the given Sum by the Tabular Number.

Example.

What is the present worth of 131 l. 5 s. due at the end of 10 months?

[...]

The Answer is 125 l.

But if any desire to be more exact, let him multiply the Interest of 1 l. for 1 day (which [Page 109] is .000164384) by the number of days, and that Product by the given Sum, and the last Product will be the Interest for that Sum for­born the time given.

Example.

What is the Interest of 125 l. forborn from the Tenth of March to the Tenth of Ianuary following, viz. 305 days?

[...]

The Answer is 6 l. 5 s. 4 d ferè.

Discompt is performed by Division, viz. get the Amount of 1 l. for the time required, by which divide the given Sum, and the Quote is the present worth.

Of Mean Time.

It hath been a custome amongst Merchants, in their Contracts upon Sale of Commodities, to agree upon divers times of payment, as two three-months, three six-months, &c. Now to find a time between these, wherein the whole Sum may be paid at one entire Payment with­out detriment to either Party, the subsequent Table doth shew upon the first inspection.

A Table for Equation of Time.
  1 is 1.5
  2 is 3
  3 is 4.5
  4 is 6
2 5 is 7.5
  6 is 9
  7 is 10.5
  8 is 12
  9 is 13.5
  1 is 2
  2 is 4
  3 is 6
  4 is 8
3 5 is 10
  6 is 12
  7 is 14
  8 is 16
  9 is 18
  1 is 2.5
  2 is 5
  3 is 7.5
  4 is 10
4 5 is 12.5
  6 is 15
  7 is 17.5
  8 is 20
  9 is 22.5
  1 is 3
  2 is 6
  3 is 9
  4 is 12
5 5 is 15
  6 is 18
  7 is 21
  8 is 24
  9 is 27
  1 is 3.5
  2 is 7
  3 is 10.5
  4 is 14
6 5 is 17.5
  6 is 21
  7 is 24.5
  8 is 28
  9 is 31.5
  1 is 4
  2 is 8
  3 is 12
  4 is 16
7 5 is 20
  6 is 24
  7 is 28
  8 is 32
  9 is 36
  1 is 4.5
  2 is 9
  3 is 13.5
  4 is 18
8 5 is 22.5
  6 is 27
  7 is 31.5
  8 is 36
  9 is 40.5
  1 is 5
  2 is 10
  3 is 15
  4 is 20
9 5 is 25
  6 is 30
  7 is 35
  8 is 40
  9 is 45

[Page 111]The manner of making this Table, is no more than adding one Term to the given num­ber of Terms, and take half the Sum.

Example.

Is three four-months given, add 4 to 12, the Sum will be 16, half that Sum, viz. 8 months, is the equated Time of Payment.

This indeed is but an approximation, though near enough the truth for practice. That excel­lent Accomptant Mr. Collins, in a Sheet Printed Anno 1665. hath taught a more exact way of Equation: Simple Interest, prop. 4. Compute (saith he) all the present worths, and then by proportion, if all those present worths amounted to the total of all those Payments, what did 1 l. amount to in the said time? From the result subtract an Vnite, the Remainder is the Interest of 1 l. for the time sought, which divide by the Interest of 1 l. for one Day, and the Quote is the Number of Days sought.

Example.

A Merchant sold Wines for 300 l. and hath given the Vintner three six-months for Pay­ment, viz. to pay 100 l. at the end of 6 months, another at 12, and the third 100 l. at 18 months end▪ the Question is, At what time may this Vintner pay 300 l. together, without detriment to himself or the Mer­chant.

[Page 112] [...]

100l. at 6 months is worth — 97.087

100l. at 12 months is worth — 94.34

100l. at 18 months is worth — 91.826

The present worths = 283.253

[Page 113] [...]

The Interest of 1 l. for the time is .059123786

The Interest of 1 l. for 1 day is = .000164384

[...]

The Answer is 359 days and a half, ferè.

By the Table, three six-months gives twelve months for the equated time, which you find above five days less than a year by this Calcu­lation.

[Page 114]

A Decimal Table of the present worth of One Pound, Quarterly Pay­ment, at 6 per Cent. per Annum, Simple Interest, for 124 Quarters.
1 .985222
2 1.956095
3 2.913033
4 3.856429
5 4.786662
6 5.704093
7 6.609071
8 7.501928
9 8.382985
10 9.252550
11 10.110919
12 10.958377
13 11.795197
14 12.621643
15 13.437970
16 14.244421
17 15.041234
18 15.828636
19 16.606846
20 17.376077
21 18.136533
22 18.888413
23 19.631907
24 20.367201
25 21.094474
26 21.813898
27 22.525642
28 23.229868
29 23.926732
30 24.616387
31 25.298981
32 25.974656
33 26.643553
34 27.305804
35 27.961542
36 28.610893
37 29.253979
38 29.890922
39 30.521837
40 31.146837
41 31.766032
42 32.379529
43 32.987432
44 33.589841
45 34.186856
46 34.778572
47 35.365082
48 35.946478
49 36.522847
50 37.094275
51 37.660848
52 38.222645
53 38.779748
54 39.332235
55 39.880180
56 40.423658
57 40.962742
58 41.497501
59 42.028005
60 42.554321
61 43.076514
62 43.594649
65 44.108787
64 44.618992
65 45.125321
66 45.627833
67 46.125686
68 46.621636
69 47.113036
70 47.600841
71 48.085103
72 48.565872
73 49.043199
74 49.517133
75 49.987721
76 50.455010
77 50.919048
78 51.379877
79 51.837543
80 52.292088
81 52.743556
82 53.191986
83 53.637421
84 54.079898
85 54.519459
86 54.956140
87 55.389980
88 55.821014
89 56.249280
90 56.674811
91 57.097644
92 57.517813
93 57.935349
94 58.350287
95 58.762658
96 59.172494
97 59.579826
98 59.984684
99 60.387099
100 60.787099
101 61.184713
102 61.579970
103 61.792897
104 62.363522
105 62.751872
106 63.137972
107 63.521849
108 63.903529
109 64.283035
110 64.660394
111 65.035628
112 65.408763
113 65.779820
114 66.148824
115 66.515796
116 66.880760
117 67.243736
118 67.604747
119 67.963814
120 68.320956
121 68.676196
122 69.029553
123 69.381047
124 69.730697

The Use of the precedent Table is principall [...] to shew the present worth of any Lease or A [...] ­nuity, payable Quarterly, for any term [...] years under 31, at 6 per Cent. per Annu [...] Simple Interest.

[Page 117] Example.

There is a Lease for 18 years to be sold, of the yearly value of 160 l. payable Quarterly, viz. 40 l. per Quarter, what is this Lease worth in ready money, allowing the Pur­chaser 6 per Cent. Simple Interest?

72 Quarters per Table = 48.565872 [...]

The Answer is 1942 l. 12 s. 8 d. ¼ ferè.

The Inversion of the Question, viz. What Quarterly Payment for 18 years will 1942 l. 12 s. 8 d. ¼ purchase?

As the former was done by Multiplication, where the Product exhibits the Answer; so if the Sum proposed be divided by the Tabular Number, the Quote gives your Answer.

Example.

[...]

The Answer is 40 l.

Of Compound Interest.

AS Simple Interest is performed by a Serie [...] of Musical, so is Compound Interes [...] wrought by a Rank of Geometrical continua [...] Proportionals. The operation whereof by th [...] Canon of Logarithms, take under these four Considerations.

Prop. I.

If you shall put p = the Logarithm of a Prin­cipal or Sum forborn, and t = the time o [...] forbearance in years, quarters, months, or day, r = the Logarithm of the Rate of Interest, per cent. per annum, per mensem, or per diem, a = the Logarithm of the Amount of the said Principal for the said time, at the Rate also aforesaid: Then Q. The Amount = a?

Equation, a = rt + p.

That is, Multiply the Logarithm of the Rate by the Number of Years, Quarters, &c. to which Product add the Logarithm of the Principal, and the Aggregate is equal to the Logarithm of the Amount.

[Page 119] Example.

Quest. 1. If 175 l. be forborn 7 years, what will it amount to at 6 per Cent. per Annum, Compound Interest?

Log. of the Rate = 0,02530586=r [...]

Log. of the Sum = 2,24303805 = 175 = p [...]

The Answer 263 l. 2 s. 8 d. ¼ ferè.

Quest. 2. If 1000 l. be forborn for 6 months, at 6 per Cent. per Annum, Compound In­terest, what will it amount to?

Log. of the former Rate divided by 12, the months in a year, is = 0,00210882 = r [...]

Add the Log. of 1000 viz. 3,00000000

1029.563 = Log. Amount 3,01265292 = a

The Answer 1029 l. 11 s. 3 d. ferè.

Prop. II.

A Sum of Money unknown, being forborn a cer­tain time = t, at a given Rate of Interest = r, is amounted to a given Sum = a; Q What was p?

Equation, p = a − rt.

From the Logarithm of the Amount, sub­duct the Logarithm of the Rate, multiplied by the time, and the Remainder is the Logarithm of the Principal.

Example.

Quest. 1. If 263 l. 2 s. 8 d. ¼ be the Amount of a Sum forborn 7 years, at 6 per Cent. per Annum, Compound Interest, what was the Principal?

Log. of the Rate = 0,02530586 = r [...]

Log. of the Amount 2,42017910

Log. of the Principal 2,24303805 = p = 175

The Answer 175 l.

[Page 121] Quest. 2. If 102 l. 11 s. 3 d. be the Principal and Interest of a Sum of Money forborn 6 months, at 6 per Cent. per Annum, Compound Interest, what was the Principal?

Log. of Rate for 1 m o. 0,00210882 [...]

Log. of 1029.563 = 3,01265292

Log. of the Principal 3,00000000 = 1000

The Answer 1000 l.

Prop. III.

A Sum of Money = p, being forborn for a time = t, did amount to a given Sum = a, at a Rate of Interest unknown: Q. The Rate per Cent. per Annum = r?

Equation, [...]

Divide the Logarithm of the Amount, less the Logarithm of the Principal, by the Time, and the Quote is the Logarithm of the Rate.

Example.

If 25 l. forborn 4 years, did amount to 31 l. 11 s. 2 d. ¼; at what Rate of Compound Interest did it so increase?

[Page 122]Logarithm of the Amount = 1,49808345

Logarithm of the Principal = 1,39794001

a − p divide by 4 ) 0,10014344

The Log. of the Rate = r = 0,02503586

Prop. IV.

A Sum of Money being forborn, at a given Rate, for a time unknown, but the Amount is known, how long was it so forborn?

Equation, [...]

Example.

If 1000 l. be increased to 10 [...]9 l. 11 s. 3 d. at 6 per Cent. per Annum, Compound Interest, in what time was it so increased?

[...]

The Answer 6 months.

It may here be expected that I should lay down the Construction of the Logarithms, having made use of them in these Calculations, but this being design'd a small Enchiridion, and there being large Volumns of that Subject in the World already, by several more learned Pens, I think it unnecessary to say any thing [Page 123] further thereof, for as they are of excellent use, so are they easie to be had.

COmpound Interest Infinite, may be so called as it relates to divers equal Payments at equal times, but the number of those equal times are infinite, (i. e.) when an Estate in Fee-Simple shall be sold for ever. Now there being usually an interval of time, between the Purchasers Payment and the reception of his first Rent, be it yearly, half yearly, or quar­terly;

Any Question of this Nature may be wrought by the following Analogism:

Putting V = the Rent (yearly or quarterly)and S = the Price paid for the Land, also R = the Common Factor of the Rate of Interest, per Cent. per Annum.

Hence then may arise these three Propositions.

Prop. I.

There is a Fee Simple to be sold, what is it worth in ready money, so that the Purchaser may have 6 per Cent. per Annum, Compound Interest, allowed for his money.

[Page 124] Quest. 1. There is a Manour to be sold of the clear yearly value of 969 l. 18 s. what Sum of ready money is this Estate worth, 6 per Cent. per Annum Compound Interest being allowed the Purchaser for his money?

Equation, [...]

The Annual (or Quarterly) Payment, divided by the Ratio, less Unity, exhibits the Sum in the Quotient.

[...]

The Answer is 16165.

Quest. 2. There is an Estate of 969 l. 18 s. per Annum ▪ payable Quarterly, what is it worth in ready money, allowing the Pur­chaser 6 per Cent. per Annum Compound Interest?

[Page 125] [...]

The Answer is 16524 l. 2 s. 6 d. ferè.

The difference between Yearly and Quarterly Payments in this Purchase raiseth the value 359 l. 2 s. 6 d.

☞ Having the increase of 1 l for a Year, at any Rate of Interest, the Biquadrate Root of that Increase, is the Increase of 1 l. for a Quarter at Compound Interest.

Prop. II.

A Sum of money lying ready for a Purchase, and it be desired to know what Free-hold Estate such a Sum will purchase, if laid out at a given Rate per C. per Ann. Compound Interest.

Theorem, V = S × R − 1.

Or, in other terms, the Sum of Money mul­tiplied by the Rate, less Unity, the Product shall be equal to the Annual half quarterly or quar­terly Payment.

Quest. A Gentleman upon Marriage of his Daughter promiseth to lay out 1600 l. for a Free-hold Estate, to be settled upon her and her Heirs, provided he meet with such a Pennyworth as shall bring 8 per Cent. per Annum, Compound Interest for his money: Q. What Annual Rent must it be?

[...]

The Answer 128 l. per Annum.

Prop. III.

An Estate being offered for a certain Sum of money, the annual Rent is also known: Q What [Page 127] Rate of Interest upon Interest shall the Pur­chaser have for his money?

Equation, V ÷ S = R − 1.

The annual Rent being divided by the Sum demanded, quotes the Rate less Unity.

Example.

Quest. 1. There is a Free-hold Estate to be sold for 1600 l. the yearly Rent being 128 l. what Rate of Interest shall the Purchaser have for his money?

[...]

Quest. 2. Admit there be a small Farm to be sold of the Value of 35 l. per Annum for 500 l. what Rate of Compound Interest shall the Purchaser have for his money at that price?

[...]

[Page 128]Furthermore, if it be inquired how many years Purchase any Annuity is worth, putting R = the Ratio as before, and Y the number of Years, the Rule is: [...]

That is, Divide Unity by the Ratio less 1, and the Quote informs the Number of Years.

Example.

There is a Free-hold Estate to be sold, Q. How many Years Purchase is it worth at 5 per Cent. per Annum?

[...]

The Answer is 20 Years Purchase.

What is it worth at 6 per Cent. p [...]r Annum?

[...]

The Answer is 16 Years, and ⅔ of a Year.

[Page 129]Again, if an Estate be offered at any num­ber of Years Purchase, and it be demanded what Rate of Interest the Purchaser shall have for his Money, the Rule is: [...]

That is, Divide Unity by the number of Years propos'd, and the Quote exhibits the Ratio, less Unity.

Example.

An Estate is offered at 20 Years Purchase, what Rate of Interest shall the Purchaser then have?

[...]

The Answer is 5 per Cent. per Annum.

There are many Tables of Compound In­terest Printed in sundry Books for the valuation of Leases and Annuities, but they are generally made for yearly Payments, when indeed by the common and most usual Covenants in Leases the Tenant is obliged to pay quarterly; and in Leases of great value, there will be found a con­siderable difference in the true worth, (so great, that 25 l. per Quarter is as good as 102 l. 5 s. per Annum.) I have therefore presented the Reader with a Table fitted to such Quarterly Payments, the Use of which Table I doubt not but will be very easily found by the Examples that follow.

[Page 130]

A Table of Interest, for the Valuation of Leases or Annuities for Quarterly Payments, at 6 per Cent. per Annum, Compound Interest, for 31 Years.
1 .985538
2 1.956824
3 2.914064
4 3.857459
5 4.787213
6 5.836372
7 6.609520
8 7.496573
9 8.373700
10 9.238139
11 10.090079
12 10.929724
13 11.757176
14 12.572685
15 13.376402
16 14.168496
17 14.949134
18 15.718484
19 16.476707
20 17.223982
21 17.960417
22 18.686219
23 19.401524
24 20.106484
25 20.801250
26 21.485968
27 22.160789
28 22.825841
29 23.481282
30 24.127242
31 24.763844
32 25.391256
33 26.009595
34 26.618988
35 27.219206
35 27.811474
37 28.394861
38 28.969764
39 29.536352
40 30.094714
41 30.645034
42 31.187396
43 31.721914
44 32.248000
45 32.767870
46 33.279531
47 33.783794
48 34.280753
49 34.843000
50 35.253244
51 35.728999
52 36.197819
53 36.659861
54 37.115237
55 37.564028
56 38.006330
57 38.442234
58 38.871893
59 39.295222
60 39.712487
61 40.123777
62 40.529043
65 40.928469
64 41.322133
65 41.710087
66 42.092531
67 42.469245
68 42.841611
69 43.206601
70 43.567307
71 43.922768
72 44.273138
73 44.618415
74 44.958701
75 45.194062
76 45.624577
77 45.950311
78 46.271331
79 46.587715
80 46.899521
81 47.206817
82 47.509668
83 47.808141
84 48.102298
85 48.392200
86 48.677877
87 48.959492
88 49.236993
89 49.510486
90 49.780023
91 50.045601
92 50.307460
93 50.665471
94 50.819753
95 51.070356
96 51.317335
97 51.560742
98 51.841984
99 52.037044
100 52.271047
101 52.499677
102 52.725986
103 52.949021
104 53.134301
105 53.385474
106 53.598963
107 53.809375
108 54.016743
109 54.221113
110 54.422527
111 54.620966
112 54.816601
113 55.009461
114 55.199474
115 55.386751
116 55.571297
117 55.753185
118 55.932443
119 56.109107
120 56.283219
121 56.454811
122 56.623921
123 56.790588
124 56.954843

[Page 133] The Calculation of a Number in the precedent Table, by aid of the Canon.

The Question being, What is the present worth of 1 l. per Quarter for 21 Years?

The Logarithm of the Increase of 1 l. at 6 per Cent. for three months = 0,0063264664

The Number of Quarters = 84 [...]

Discompt of 1 l. for 84 Qrs. 0,5314231776

Then by the Rule of Proportion:

If 1,467384617 com. arithm. 9,8334569382
have 100 l. for its Principal, 2,0000000000
what shall 1 l. have for its Pr. 0,0000000000

68.148601 = 1,8334569382

The Log. of the Discompt = 0,5314231776

[...]

The Answer is = 48 l. 2 s. 0 d. ½ ferè.

And after this manner may a Table be Calcu­lated, or the Value of a Lease for any Number of Years, may be found at any Rate of Inte­rest required.

The VSE of the TABLE.

This Table sheweth the Discompt of 1 l. per Quarter at 6 per Cent. per Annum, Com­pound Interest, and if the Tabular Number for so many Quarters as the Lease is to continue be multiplied by the Quarterly Payment, that Product is the present Value of that Lease in ready money.

Example.

A Lease of 40 l. per Annum ( viz. 10 l. per quarter) for 21 years, being to be sold, what is it worth in ready money?

84 quarters per Table = 48.102221

The quarterly Rent = 10 [...]

The Answer is 481 l. 0 s. 5 d. [...]/4; ferè.

But if the question be, What quarterly Rent for 21 years will a given Sum purchase? Then divide the given Sum by the Tabular Number for so many quarters.

Example.

[...]

[Page 135]A Gentleman having a Lease of certain Church Lands, worth 200 l. per Annum more than the reserved Rent, for 14 years to come, surrenders the same, upon condition the Chapter shall make him a new Lease for 31 years without a present Fine, but advan­cing the old Rent 10 l. per quarter during the whole term of 31 years; what doth he gain by the bargain, accompting Compound Interest on both sides?

56 quarters per Table = 38.006

The quarterly Rent = 50 [...]

124 quarters per Table = 56.955 [...]

The Answer is 377 l. 18 s. the new Lease being so much more worth than the old one.

240 l. is demanded for the Lease of a House for 7 years, the Tenant offers 100 l. and an advance of Rent equivalent to the rest of the Fine required, what ought this Rent to be?

[Page 136] [...]

The Advance of Rent ought to be 6 l. 2 s. 8 d. per quarter.

There is a Lease of 200 l. per Annum, viz. 50 l. per quarter, for 13 ¼ years, to be sold, what is it worth at 6 per Cent. Simple, and what at 6 per Cent. Compound Interest?

Simple.

53 quarters per Table = 38.779748 [...]

Compound.

53 quarters per Table = 36.659861 [...] Which subtracted from 1938.987440 Leaves 105.99439

Whereby it appears, that it is cheaper to the Purchaser at Compound Interest than at Simple Interest by 106 l.

[Page 137] Six Questions performed by the aid of the Canon of Logarithms.

Quest. 1. A Gentleman pays 350 l. for a Lease in Reversion, to commence at the end of 13 years and a quarter, and to continue for 21 years and 3 quarters, what quarterly Rent may he lett the Premises for, after he comes to be in possession thereof, so as to gain 8 per Cent. Compound Interest for his money?

The Logarithm of 350 l. = 2,544008

Worth of 1 l. forborn 53 quarters = 0,442865

Log. of the Increase of 350 l. = 2,986873

Worth of 1 l. for 87 quarters = 1,621420

Log. of 23.198 = 1,365453

The Answer = 23 l. 3 s. 11 d. ½ ferè.

Quest. 2. A Citizen having taken a Lease of a House and Shop for 21 years, at 370 l. Fine, and 100 l. per Annum, viz. 25 l. per quarter, Rent, at the end of two years is willing to leave it for 300 l. and the old Rent, or to have such an increase of Rent, during the whole term yet to come, as may reim­burse him his Fine paid, with Compound Interest at 6 per Cent. per Annum: What [Page 138] ought he to receive in advance of Rent, and what doth he offer to lose of his Fine paid in taking 300 l.

2,568202 = 370l.

1,682165 = The worth of 1 l. per quarter for 84 quarters. [...]

0,886037 The Advance of Rent ought to be 7 l. 13 s. 10 d. ¼ ferè.

1,652198 The worth of 1 l. per quarter for 76 quarters. [...]

2,545235 The present Fine ought to be 350 l. 18 s. 10 d.

Whereby it appears, there is 50 l. 18 s. 10 d. offered to be lost in putting off the House and Shop aforementioned.

Quest. 3. A sells a House to B for 800 l. to be paid with Interest upon Interest by 100 l. per Annum, viz. 25 l. per quarter, how many quarters Rent ought B to pay before A is satisfied for his 800 l. with Compound Interest at 6 per Cent. per Annum, and what ought the last Payment be?

[Page 139] [...]

The last Payment 13 l. 3 s. 4 d. ferè.

[Page 140] Quest. 4. A lends unto B a certain Sum of ready money, and accepts a Rent Charge of 40 l. quarterly for 7 years in satisfaction, finding it paid him his Principal with Interest upon Interest at 8 per Cent. within 13 l. 4 s. 6 d. what was the Money lent?

1,602060 The Logarithm of 40 l.

1,565196 The Logarithm of the worth of 1 l. quarterly for 48 quarters. [...]

3,167256 The Logarithm of the worth of 40 l. per quarter for 48 quarters. [...]

0,177140 The Logarithm of the increase of 1 l. forborn 28 quarters.

2,986188 The Logarithm of 968.7.

The Money lent was 968 l. 14 s.

Quest. 5. A Testator leaving one Son and two Daughters, bequeaths out of his Estate (be­ing 600 l. per Annum for 11 years) to his eldest Daughter 500 l. per Annum for 4 years next coming, at the end whereof, to his younger Daughter 300 l. per Annum for 7 years, and to his Son the Remainder of the [Page 141] Estate for the whole time: Q. Which had the greatest Portion, and by how much, calculating their several Annuities at 6 per Cent. Compound Interest?

0,539716 Logarithm of the worth of 1 l. per annum for 4 years.

2,698970 The Logarithm of 500 l. [...]

3,238686 = 1732.55. [...]

0,746820 The Logarithm of the worth of 1 l. per annum for 7 years.

2,477121 The Logarithm of 300 l. [...]

0,101232 The Logarithm of the worth of 1 l. forborn 4 years. [...]

3,122709 = 1326.47. [...]

0,896905 The Log. of the present worth of 1 l. per annum for11 years. [...]

2,896905 = 788.68

[Page 142]0,746820 The Log. of the present worth of 1 l. per annum for 7 years.

2,301029 The Logarithm of 200 l. [...]

0,101232 The Logarithm of the worth of 1 l. forborn 4 years. [...]

The Sons Portion —1673:00:05

The eldest Daughters Portion 1732:11:00

The youngest Daughters — 1326:09:04¾

[...]

Proof.

0,896905 The Logarithm of the worth of 1 l. per annum for 11 years.

2,778151 The Logarithm of 600 l. [...]

3,675056 The Logarithm of 4732 l.

Quest. 6. A Merchant sold 16 Kintals of Cy­prus Cottons for 320 l. to be paid at two six-months; the Buyer having Money by him, offers to pay the Money presently, [Page 143] provided the Merchant allow him Discompt at 6 per cent. Compound Interest. Q. What ought the Merchant to receive?

2,505149 The Logarithm of 320l.

0,018979 The Logarithm of 1 l. forborn 9 months, the equated time, acor­ding to the Table of Mean Time, Pag. 110. [...]

2,486170 The Logarithm of 306.316.

The Answer 306 l. 6 s. 4 d. ferè.

STEREOMETRY: OR, A New and the most Practical Way of GAUGING TUNNS, &c.

STEREOMETRY: OR, A New and the most Practical Way OF Gauging Tunns In the form of a PRISMOID & CYLINDROID: ALSO The Frustums of Pyramids and of a Cone. Together with The Art of CASK-GAUGING. By Iohn Mayne.

London, Printed by William Godbid, for Nath. Crowch, in Exchange-Alley. M. DC. LXXIII.

TO THE Young Geometrician.

I Hope by this time thou art so suffici­ently acquainted with the Nature and Use of a Decimal Fraction, that any Operation in the six Species, viz. Addi­tion, Subtraction, Multiplication, Di­vision, Involution and Evolution of the Second and Third Powers, will not ap­pear difficult to thee; and these being fa­miliar, any Calculation in Arithmetick, Geometry, Trigonometry, or other Ma­thematical Arts, will not seem strange: Amongst the many pleasant Walks in this Tempe, I have made it my present design to give thee some diversion in that part of Solid Geometry called Gauging, and herein passing by those Blossoms that kiss the hand of every Passenger, I have endea­voured (and I hope not altogether without success) to shew thee how to gather a Rose without danger of its Thorn: For the In­vention, [Page] the World is obliged to the Inge­nious Mr. Michael Dary, the Roots of these, and many other choice Mathematical Flowers, lying crowded together in a small Trea [...]ise called Dary's Miscellanies, Prin­ted 1669. Here, as in the former Part, thou hast both Precept and Example in the plainest method I could possibly express them. That they may by no means seem obscure to any ingenious Student, is the hearty desire of

Thy Friend, J. M.

The Explanation of the Signs or Characters.

  • + More.
  • − Less.
  • = Equal.
  • > Greater.
  • < Lesser.
  • × Multiplied.
  • √ Square Root.
  • (q) square Square.
  • ⊙ Circle.
  • ∷ Proportional.
  • ∽ Difference.

[Page] [Page 151]STEREOMETRY: OR, A New and the most Practical Way of GAUGING TUNNS, &c.

A plain and easie Method for finding the Solid Content of a Prismoid.

DEFINITION.

BY the word Prismoid is to be understood a Solid contained under six plain Sur­faces, whereof the two Bases ought to have these three qualifications:

  • 1. Rectangular Parallelograms.
  • 2. Parallel.
  • 3. Alike Situate.

i. e. So situate, that the Rectangular Conjugates in both Bases may be inserted by two and the same Planes, and a Right Line extended from the Center of one Base to the other may be called the Axis, and the other remaining four Planes are the Peri­patasma. [Page 152] Under this Definition is compre­hended the Frustums of Pyramids and Prisms.

Note also, If the Peripatasma be not made by the four flat Sides (spoken of before) but shall be constituted by Curveture from Circles or Elipse's, the Solid is then called a Cylindroid, and under this Definition is comprehended Frustums of Cones and Cylinders.

PROBL.

If in a Prismoid you put

  • C = the whole Content thereof.
  • A & B = the two Rectangular Conjugates above.
  • G & H = the two Rectangular Conjugates be­low.
  • A & G opposite = their two Correspondents one above the other below inserted by one Plane.
  • B & H opposite = their two Correspondents a­bove and below, and also inserted by one Plane.
  • P = the perpendicular height of the Prism or Prismoid.
  • K = the increment of any two Diameters, to be taken between A and G in the same Plane with them, at one Inch distance of the perpen­dicular.
  • L = the increment of any two Diameters, to be taken between B and H in the same Plane with them, at one Inch distance of the perpendi­l [...]r.

[Page 153]Then,

Analogism.

[...]

Or,

The Rectangle of the two Diameters at the Base multiplied into the Perpendicular, more the Semi-sum of G L and H K, ( viz. those two Diameters multiplied into their altern in­crements) multiplied into the Square of the Per­pendicular, to which add one third of the Rect­angle of K L (i. e.) the two increments multi­plied into the Cube of the Perpendicular is equal to the Content in Cubick Inches.

By which Theorem you find three fixed or stationary Numbers, which Mr. Dary calls reserved Coefficients, wherefore you shall find them hereafter called by that denomination: These three reserved Coefficients thus multiplied into the Perpendicular, the Product is equal to the whole Content, or by any part of the Perpendicular gives the Solidity of that part.

Prop. I.

Having a Tunn in the form of a Prismoid, the Dimensions being,

Inches.
  • above. A = 126 and B = 144
  • below.G = 102 and H = 108
  • P = 60

[Page 154] What is the Solidity of this Tunn in Cubick Inches?

First then to find the three reserved Co­efficients.

[...]

i. e. The difference between A and G (the two opposite Diameters above and below) di­vided by the Perpendicular quotes K, the in­crement of any two Diameters to be taken be­tween them, at one Inch distance in the Perpen­dicular, and in the same Plane.

[...]

That is, the difference between B and H (the two Diameters opposite the one above the other below) divided by the Perpendicular quotes L, the increment of any two Diameters to be taken between them, at one Inch distance in the Per­pendicular, and in the same Plane with them.

.6 = L

.4 = K [...]

.24 = The Rectangle of LK [...]

.08 = ⅓ of LK, the first Coefficient. [Page 155] [...]

Now having found these reserved Coeffi­cients, I proceed, and finding that ⅓ of K L must be multiplied by the Cube of the Perpen­dicular, I begin with it, and call that the first Coefficient; then ½ G L H K being to be multi­plied by the Square of the Perpendicular, I add that to the first Fact, and call it the second Co­efficient; lastly, G H being to be multiplied by the Perpendicular, I add that to the second Fact, and call it the third Coefficient; then will the Work stand thus:

[Page 156] Example.

[...]

The Answer 866160 Cubick Inches.

Now admitting this Tunn have but 33 wet Inches, what is the Content thereof?

[...] [Page 157] [...]

The Answer = 423248.76 Cubick Inches.

Prop. II.

Having a Tunn in the form of a Prismoid, the Dimensions being,

Inches.
  • above.G = 102 and H = 108
  • below.A = 126 and B = 144
  • P = 60

What is the Solidity in Cubick Inches?

To find this Tunns Solidity, the Rule is: [...] i. e. The Fact of A B (the rectangular Con­gates [Page 158] at the Base) multiplied by the Perpendi­cular, from whence subduct the Semi-sum of the two Facts (A and its altern decrement, B and its altern decrement, multiplied into the Square of the Perpendicular) more the one third of the Rectangle of K L, viz. the two decrements, multiplied into the Cube of the Perpendicular, and that Remainder is the Content in Cubick Inches.

To find the Coefficients.

[...]

This Rule being the Converse of the former, these Numbers K and L which before were Affir­matives are now become Negatives (then in­crement, now decrement;) the greater Con­jugates being subducted from the lesser makes the Dividends so much less than nothing, and consequently the Quotes, the Divisor being an Affirmative, yet these two Negatives being multiplied together, their Fact becomes Affir­mative, according to the Rule of Algebra, the Signs of the Factors being homogeneal (or alike) makes the Fact more, as in this

[Page 159] Example.

[...]

The Factors in these Rectangles being hete­rogeneal (or unlike) the Fact is made less.

[...]

These two Factors being both Affirmatives, the Fact is +.

[Page 160]With these three reserved Coefficients I pro­ceed to the Calculation, according to the pre­cedent Theorem.

[...] The Answer = 866160 Cubick Inches.

But if this Tunn have only 27 Inches of the Perpendicular wet, the Content then being required:

[...] [Page 161] [...]

Prop. III.

There is a Tun in the form of a Prismoid, the Dimensions are,

Inches.
  • above.A = 132 and B = 144
  • below.G = 108 and H = 108
  • P = 60

What is the Solidity in Cubick Inches?

[Page 162] [...] [Page 163] [...] The Answer = 911520 Cubick Inches.

Prop. IV.

There is a Tunn in the form of a Prismoid, the Dimensions being,

Inches.
  • above.G = 108 and H = 108
  • below.A = 132 and B = 144
  • P = 60

What is its Solidity in Cubick Inches?

The reserved Coefficients are found to be: [...] [Page 164] [...]

[Page 165] The Calculation.

[...] The Answer = 911520 Cubick Inches.

Here note, if any of the precedent Tunns be cloathed by Curveture, ( i. e. the Bases being Circular or Elliptical) the last Product ought to be divided by 1.27324, then will the Quotient exhibit the Cubick Inches in that Solid. But if the Question be Ale Gallons, let your Coefficients be divided by 282; if Beer Bar­rels be required, divide the Coefficients by 10152, the number of Inches in a Beer Bar­rel. In all flat sided Figures, and for those Solids, whose Peripatasma is constituted by Circles or Ellipsis, the Divisor for Beer Bar­rels is 12926>, for Ale Barrels 11490>, [Page 166] and for Ale Gallons 359; of which take these Examples.

What number of Beer Barrels and Gallons doth the last mentioned Tunn contain?

The three Coefficients for Beer Barrels divi­ded by 10152 are:

  • The first = +.000007880220646<
  • The second = −.0067375888>
  • The third = +1.8723404<

The three Coefficients for Gallons being divi­ded by 282 are:

  • The first = .0002837>
  • The second = .2425532>
  • The third = 67.4042553>

The Coefficients being thus fitted, the Calcu­lation is after this manner:

For Barrels.

[...] [Page 167] [...]

The Answer is 89 Barrels, 3 Firkins, and 1 Gallon, or 89 Barrels and 28 Gallons.

For Gallons.

[...]

[Page 168]The Answer is 3232 Gallons, which divided by 36 quotes 89 Barrels 28 Gallons, as before.

Example.

[...]

A New Way of GAUGING THE Frustum of a Pyramid, &c.

A New Way of GAUGING THE Frustum of a PYRAMID OR CONICAL TVNN.

[Page] [Page 171]A New Way of GAUGING THE Frustum of a Pyramid, &c.

DEFINITION.

A Pyramid is a Solid Figure, contained under many Superficies, whereof one is the Base, and the rest arise from the Base to the Vertex, and there meet in a Point.

The Frustum of a Pyramid is a Solid, cut with a Plane parallel to the Base, and the part cut off is also a Pyramid.

The Frustum of a Cone, may not improperly be termed the Frustum of a round Pyramid, (the Base being circular) nor do I think it an Heresie to call a Cylinder a round Prism.

The Frustum of a Pyramid, whose Bases are in the form of any ordinate Polygon, being alike, and alike situate, and also if a Right [Page 172] Line may be every where applied in the Peri­patasma from Base to Base, moreover a Right Line being extended from the Center of one Base to the other, may be called the Axis.

Then if you put

  • S = the whole Solidity.
  • B = a Side above.
  • A = a Side below.
  • P = the Perpendicular.
  • d = the common Addend at one Inch distance of the Perpendicular, and is thus made [...], that is, the difference between a Side above and a Side below, divided by the Perpendicular, quotes the increment, &c.
  • G = the Divisor.

The Rule is: G)AA p+A dpp+⅓ ddppp(=S

Or, in other terms:

To the Square of the Side multiplied by the Perpendicular, add the Fact of one Side in the In­crement multiplied by the Square of the Per­pendicular, more ⅓ of the q. of the Increment in the Cube of the Perpendicular, and the Ag­gregate divided by the Polygons respective Di­visor, the Quote will be the Solidity.

And further it is to be well observed, if your Frustum of a Pyramid stand upon its greater Base, the Rule then is thus varied: G):BB p−B dpp+⅓ ddppp(=S

[Page 173]That is to say:

From the Square of a Side at the Base mul­tiplied by the Perpendicular, subduct the Rect­angle of one of those Sides in the Decrement multiplied by the Square of the Perpendicular, more one third of the Square of the Decre­ment in the Cube of the Perpendicular, and that Remainder divided by the Divisor proper to the form of the Base, the Quote is equal to the Solidity.

Note also, that p may be put for a part of the Perpendicular, and the Answer will be the Content of that part required.

G) or the Divisors for these 8 Regular Polygons, and the Cone.
For Cubick Inches. For Ale Gallons.
Trigon 2.30940 Trigon 651.2000
Tetragon   Tetragon 282.0000
Pentagon .58123 Pentagon 157.2600
Hexagon .38497 Hexagon 108.5400
Heptagon .27513 Heptagon 77.5867
Octogon .20710 Octogon 58.4022
Nonagon .16176 Nonogon 45.6163
Decagon .12997 Decagon 36.6515
Cone 1.27324 Cone 359.0500

[Page 174]If your Tunn be the Frustum of a Cone:

Let A or B be the Diameter at the Base, and d the Increment or Decrement of any two Dia­meters between A and B, at one Inch distance of the Perpendicular, and the Divisor as per Table.

I shall only give you some Examples of the three first, and the Cones Frustum, which I think will be sufficient to inform any ingenious Practitioner how to perform the rest.

The Trigon.

Admit a Tunn be in the form of an equilateral Triangle, the Dimensions being,

  • A = 126 Inches, the length of a Side above,
  • B = 108 Inches, the length of a Side below,
  • P = 60 Inches, the Perpendicular,
  • Q. The Content in Ale Gallons?

The Coefficients are found, according to the precedent directions, thus: [...] [Page 175] [...]

These three divided by the Divisor for Ale Gallons, viz. 651.2, are as followeth: [...] The Answer = 1201.9020 Ale Gallons.

[Page 176]A Tunn of the same Dimensions standing upon its greater Base, the Coefficients are thus found: [...]

Being divided by the same Divisor with the former, they are: [...]

[Page 177]And are thus used: [...] The Answer as before = 1201.9080 Ale Gallons.

Now if each of these Tunns have 30 Inches of the Perpendicular wet, how much do they contain?

The First.

[...] [Page 178] [...] The Answer = 537.84750 Ale Gallons.

The Second.

[...] The Answer = 663.9630 Ale Gallons.

Proof.

[...]

The difference being less than a Pint.

The Tetragon or Square Pyramid.

There is a Tunn in the form of the Frustum of a Square Pyramid,

  • A = 144 Inches, the length of a Side above,
  • B = 108 Inches, the length of a Side below,
  • P = 60 Inches, the Perpendicular,
  • Q. The Content in Gallons?

The Coefficients being found by the former Rule and Example, viz.

P)A−B(= d.

⅓ of dd = the first Coefficient .12

dB = the second Coefficient 64.8

BB = the third Coefficient 11664

These three being divided by 282 the Cubick Inches in the Ale Gallon, are equal to [...] [Page 180] [...] The Answer = 3400.8960 Ale Gallons.

If the Tunn stand upon its greater Base, the Coefficients then are P) B − A (= − d, and the ⅓ d d = the first, A d = the second, and A A = the third, which divided by 282 the number of Cubick Inches in an Ale Gallon, they do appear to be =+.00042553 And if the Content be required [...] The Answer = 3400,855680

[Page 181]If 40 Inches of the Perpendicular be wet in the first Tunn, and 20 in the latter, and it be demanded what they contain in Ale Gallons.

The first.

[...]

The second.

[...]

Proof.

[...]

The Pentagonal Pyramid.

A Tunn in the form of the Frustum of a Pyramid, whose Bases are in the form of a Pentagon,

  • A = 144 Inches, the length of each Side above,
  • B = 108 Inches, the length of each Side below,
  • P = 60 Inches, the Perpendicular,
  • Q. The Content in Ale Gallons?

The three Coefficients found in the last Ex­ample, viz. [...]

For Ale Gall.

[...] The Answer = 6099.46920 Ale Gall.

[Page 183]Let another Frustum of a Pyramid of the same Bases and Altitude, stand upon its grea­ter Base, and the Content in Ale Gallons be demanded.

The Coefficients so found and divided as be­fore directed, are as followeth:

[...]The Answer = 6099.45120

If 50 Inches of the Perpendicular in the first Tunn be wet, and 10 Inches in the last, what is the Content in Ale Gallons?

[Page 184]The first.

[...]

The last.

[...]

Proof.

[...]

A Tunn in the form of a Frustum of a Cone, the Bases being alike and alike situate, as in the precedent Examples, the Dimensions being,

  • A = 144 Inches, the diameter above,
  • B = 108 Inches, the diameter below,
  • P = 60 Inches, the Perpendicular,
  • Q. The Content in Ale Gallons?

p)A−B(= d. [Page 185] [...]

These three divided severally by the Divisor proper to a Cone, as in the Table mentioned, viz. 359.05 they quote: [...] [Page 186] [...] The Answer = 2671.0440 Ale Gall.

And that the young Gauger may not be ob­liged to Dray-men to repleat the horizon with liquor, of such Tunns whose Bases are not posi­ted parallel thereto, (as indeed most are not be­ing made with a Drip or Fall) let him take this Example, a b c d e f g h m x, a Cone or Pyramid.

[figure]

[Page 187]Having the length of each Line in this Dia­gram, and the Content of the whole Cone or Pyramid in Cubick Inches, Gallons, or Bar­rels, &c. the Quantity of the Hoof h c b d is found by this Analogy:

As the Cube of the Line a c, to the whole Solidity:

So is the Cube of a Geometrical Mean between a c and a h, to the Content of the Cone or Pyramid cut off:

Which subducted from the whole, the re­mainder is the Content of the Hoof.

Some of the Lines being given, the rest are to be found.

Example.

There is a Tunn taken as the Frustum of a Cone,

  • c d = 144 Inches, the greater Base,
  • e f = 108 Inches, the lesser Base,
  • g b = 60 Inches, the depth.

Admitting the Base were raised 3 Inches, as the Line c h, it is then necessary to take another Diameter between c h and e f to find c d.

To find a b the Cones Axe.

As 18, the semi. diff. of Diam. co. ar. 8,744728

to 72, the semi-diam. at the Base: 1,857332

So is 60, the Tuns depth, 1,778151

to 240, the Cones Axe. [...]

Log. of the Sum of their (q) square. [...] [Page 188] The Logarithm of ac 250.567 2,398924

The Logarithm of ab 247.567 2,393692 [...]

A Geometrical Mean 2,396308

To find the Content of the whole Cone.

The Square of 144 4,316724

⅓ of ab = 80 1,903090

359 &c. complement arithm. 7,444842 [...]

Content in Ale Gallons 4620 ferè 3,664656

The Line a c, the Line a h, and the Content being known, to find the Content of the Fall c b d h.

As the Cube of ac, co.ar. 2,803228

to the whole Content: 3,664656

So is the Cube of the Mean, 7,188924 [...]

to the Content of the Cone cut off. 3,656808

[...] 82.6 the Content of the Fall.

Or thus:

As the Cube of ac, co. ar. 2,803228

to the whole Cones Solidity: 3,664656

So is the Cube of ah, 7,181076 [...]

to the lesser Cones Solidity. 3,648960 [...]

The whole Cone = 4620

The Frustum mxhcbd = 164

The being ½ the Fall or Hoof = 82

Some Practical RULES …

Some Practical RULES & EXAMPLES FOR CASK-GAVGING.

Some Practical RULES & EXAMPLES FOR CASK-GAVGING.

THe Corner-stone in the whole Fabrick of Cask-Gauging, as full, was long since laid by Mr. Oughtred, taking a Cask to be the Frustum of a Sphe­roid, under which capacity they are generally received, though indeed there have been, and daily are found some Cask differing in form, and really are more Parabolical than Spheroidal, I shall therefore lay down a plain Method for the performance of the Work ( viz. finding their Content) under these four Considerations:

  • As Spheroidal,
  • As The Frustum of a Parabolical Spindle,
  • As The Frustum of a Parabolical Conoid,
  • As The Frustums of two Cones abutting upon one common Base.

[Page 192] These severally, with and without a Table of Area's of Circles.

And forasmuch as the Dimensions must be the first thing known, before the Content can be found, I shall therefore shew the young Tyro, how by some of the Dimensions to find the rest, if any obstruction prohibit the taking of all.

The Boung-diameter, and Head-diameter, and Diagonal, to find the Casks length.

First subduct the semi-difference of Diame­ters from the Boung-diameter, and Square the Remainder, which Square subduct from the Square of the Diagonal, and the Remainder is the Square of the Casks semi-length.

Example.

Let BD be the Boung-diameter = 29 Inches,

HE be the Head-diameter = 23 Inches,

BE be the Diagonal = 35.3836>Inches,

SD the semi-difference = 3 Inches:

Q. the Length = LT?

The Square of BE = 1252

Square of BD − SD = 676

Sqaure of semi-length = 576 √ 24 [...]

48 = LT

This very Quest. was intended by Mr. Smith, p. 176. but through a Mistake it was left out.

[Page 193] The Boung-diameter, Diagonal, and Length, to find the Head-diameter.

The Rule.

From the quadrupled Square of the Diagonal subduct the Square of the Length, (which done) the Square Root of the Remainder is equal to the Sum of the Boung-diameter and one Head-diameter.

Example.

The Square of Diagonal = 5008

Square of the Length = 2304 [...]

Remainder = 2704 √ = 52

Subduct the Boung-diameter = 29 [...]

Head-diameter = 23

The Head-diameter, Boung diameter, and the Length, to find the Diagonal.

The Rule.

To the Square of the Semi-length add the Square of the Boung diameter, less the Semi-difference of Diameters, and the Square Root of their Sum is equal to the Diagonal.

[Page 194] Example.

Square of Semi-length = 576

Square of 26 = 676 [...]

Square of Diagonal = 1252 √ 35.3836>

A Cask taken as the Frustum of a Spheroid, cut with two Plane Parallels, each Plane bisecting the Axis at right Angles,

  • B the Boung-diameter = 29 Inches,
  • H E the Head-diameter = 23 Inches,
  • L T the Length = 48 Inches:

Q. The Content in Wine Gallons?

The Rule.

To the doubled Square of the Boung-diame­ter add the Square of the Head-diameter, their Aggregate multiply by the Length, and to the Product add the tenth part of it self, more one third of that tenth part, and from the Sum cut off as many places toward the right hand as were in the Multiplicand.

Example.

The doubled Square of 29 = 1682

The Square of 23 = 529 [...]

The Aggregate = 2211 [Page 195] [...] The Answer = 120.2784 Wine Gallons.

Another way.

The (q) square. of the Boung-diameter = 841

The (q) square. of the Head-diameter = 529 [...]

The Sum of their Squares = 1370

The Semi-sum = 685

The Semi-diff. of Squares = 156 [...]

Their Aggregate = 2211

The Length = 48 [...]

16788

8844 [...]

The Product = 106128

The tenth part = 106128

The [...] part or ⅓ of [...] = 35376 [...]

The Answer in Wine Gall. 120.2784

[Page 196] The same Cask being taken as the Frustum of a Parabolical Spindle, the Content may be thus found.

(q) square. of the Boung-diameter = 841

(q) square. of the Head-diameter = 529 [...]

Their Aggregate = 1370

The Semi-sum = 685

The tenth part of the Diff. = 312 [...]

The Length = 48 [...]

The Product = 1001376

Tenth of the Product = 1001376

⅓ of [...]

The Answer in Wine Gall. 113.48928

If taken as the Frustum of a Parabolical Conoid, cut as before mentioned, the Content may be found as in this

Example.

(q) square. of the Boung-diameter = 841

(q) square. of the Head-diameter = 529 [...]

The Sum = 1370 [Page 197] [...]

The Product = 98640

Tenth part of the Product = 98640

[...]

The Answer in Wine Gall. 111.7920

If a Cask of the same Dimensions be taken as the middle Frustum of two Cones abutting upon one common Base, cut with two Planes paral­lel, and each bisecting the Axis at Right Angles, the Content in Wine Gallons may be found as in this

Example.

(q) square. of the Boung-diameter = 841

(q) square. of the Head-diameter = 529 [...]

The Sum = 1370

The Semi-sum = 685 [...]

Semi (q) square. of diff. of Diam. = 18 [...] [Page 198] [...] The Answer in Wine Gall. 110.8128

For finding the Capacity of these, or any other Vessels, it is convenient to have always in readiness a Table of Area's of Circles in Wine and Ale Gallons: I think it unnecessary to swell this intended small Volume with them, there being two lately Printed, exactly Calculated to every tenth part and quarter of an Inch, and also a Table of Area's of Segments of a Circle, by my good Friend Mr. Iohn Smith, in his Book of Gauging, to whom in gratitude I am obliged to render my hearty acknowledgment for many favours and kind assistances in these Studies; yet that you may be able to find any Area of a Circle upon demand, in Wine or Ale Gallons, without a Table, take this

Rule.

Divide the q. of the Diameter by 294.1 for Wine, and by 359.05 for Ale Gallons, and the Quotient exhibits the Area. Or, saith Mr. Smith, Multiply the q. of the Diameter by .0034 for [Page 199] Wine, and by .0027851 for Ale Gallons, and the Product exhibits the Area in such Gallons.

As in these Examples.

The Diameter of a Circle = 21.7: Q. The Circles Area in Wine Gallons?

[...]

The Diameter of a Circle = 26.8: Q. The Area in Ale Gallons?

[...]

For finding the Capacity of a Cask, taken as Spheroidal, by a Table of Area's of Circles in Gallons.

[Page 200] Example.

A Casks Boung-diameter = 29 Inches, Head-diameter = 23, and the Length = 48 Inches: Q The Content in Wine Gallons?

⅓ of the Area of the Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

Semi-sum = .7763

Semi ∽ = .1768 [...]

Area of Mean Circle = 2.5058

The Length = 48 [...]

The Answer in Wine Gall. 120.2784

Another way.

⅓ of the Area of Boung ⊙ = 1.9062

⅓ of the Area of Head ⊙ = .5996 [...]

The Area of Mean ⊙ = 2.5058

The Length = 48 [...]

The Answer = 120.2784

That is, 120 Gallons, 1 Quart, and ¼ of a Pint, ferè.

[Page 201] To find the solid Content of a Cask, when taken as the middle Frustum of a Parabolical Spindle, &c.

The Dimensions as before.

⅓ of the Area of Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

Semi-sum = .7763

Tenth of the Difference = .03535 [...]

Area of Mean ⊙= 2.36435

The Length = 48 [...]

The Answer in Wine Gall. 113.48880

That is, 113 Gallons, and almost 2 Quarts.

And as the Frustum of a Parabolical Conoid, the Capacity is thus found:

⅓ of the Area of the Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

Semi-sum = .7763 [...]

Area of the Mean ⊙= 2.3290

The Length = 48 [...]

The Answer in Wine Gall. 111.792

[Page 202] If a Cask be taken as the middle Frustum of two Cones, abutting upon one common Base, &c.

The Dimensions as before.

⅓ of the Area of the Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

The Semi-sum = .7763 [...]

[...] of Area of 6 the ∽ of Diam. .0204 [...]

Area of Mean ⊙= 2.3086

The Length = 48 [...]

The Answer in Wine Gall. 110.8128

The Ullage, or Wants in a Cask, may be found under these two Considerations:

  • 1. A Cask standing on the Head, with the Diameters parallel to the Horizon.
  • 2. A Cask lying with the Axe parallel to the Horizon.

Prop. I.

In a Cask standing on the Head, with the Dia­meters parallel to the Horizon, some Liquor remaining, to find how many Wine Gallons it is.

[Page 203]Here are these five things necessary to be known:

  • 1. The Diameter at the Boung.
  • 2. The Diameter at the Head.
  • 3. The Length of the Cask.
  • 4. The Depth of the Liquor.
  • 5. The Diameter of the Liquors superficies.

Example.

[...]

The Diameter o p is thus found, first find the Axis of the whole Spheroid e f, thus; from the Square of half the Boung-diameter (n h) subduct the Square of half the Diameter at the Head, and extract the Square Root of the Re­mainder: Then by the Rule of Proportion, say, As that q √, is to n h, the Semi-boung-diameter: So is n i, the Casks Semi-length, to e n half the Axis sought.

[Page 204] [...]Diameter of the Liquors Superficies 27.4

Having found the Diameter of the Liquors Superficies:

Then, to ⅔ of the Area of that Circle = 1.70173

Add ⅓ of Area of the Head Circle = .59953 [...]

The Depth of Liquor = 11.6 [...]

The Answer in Wine Gallons = 26.694616

Which subducted from the whole Content, leaves the Ullage or Wants.

Prop. II.

A Cask lying with its Axe parallel to the Hori­zon, and having some Liquor remaining in it, to find the Content of the said Liquor in Gal­lons.

[...]

Let the Dimensions be as before.

In this Proposition there is five Requisites attending:

  • h g the Diameter at the Boung = 29.
  • a b the Diameter at the Head = 23.
  • i k the Length = 48.
  • s g the Depth of Liquor = 11.6.

And the Content of the whole Cask in Gal­lons.

Then by the help of a Table of Area's of Seg­ments of a Circle, whose Area is Unity, and the Radius divided in the Ratio of 1.0000 Parts, say by the Rule of Proportion: [...] versed Sine or Arrow of Segment.

[Page 206]Then seeking in the Table you will find .4000, and right against it under the Title Area you will find .37353. Then say: whole Content

1.0000. .37353 ∷ 120.2784. 44.9276

the Liquor remaining.

The Inversion of the Question, viz. To find the Liquor wanting.

As 29. 17.4 ∷ 1.0000. .6

Again,

the Ullage.

As 1.0000. .62647 ∷ 120.2784. 75.3509

The Liquor remaining = 44.9276 [...]

Which together make 120.2785 the Casks whole Capacity.

FINIS.

ERRATA.

PAg. 3. l. 4. r. in the third. P. 5. l. 22. r. 20: 13. P. 7. l. 17. r. 42: 3: 08: 15; l. 19. r. 33: 0: 05: 04. P. 9. l. 1. r. borrow of the. P. 13. l. 14. r. 5 s. the bushel. P 16. l. 13. r. 2 times. P. 19. l. 1. r. 5 from 6. P. 31. l. 19. dele always. P. 39. l. 22. r. in the deci­mal Fractions of both Factors. P. 50. l. 15. r. solid measure. P. 51. l. 15. r. 272.25. P. 52. l. 15. r. .0041667. P. 67. l. 14. dele as before. P. 82. l. 14. r. + a. P. 188. l. ult. r. ½ being.

THe Rules herein mentioned, and other Mathematical Arts, are taught by the Author, viz. Arithmetick, Vul­gar, Decimal, and Logarithmetical; the Doctrine of Triangles, Plain and Spherical; the Use of the Globes, Qua­drant, Sector, and other Mathematical Instruments; Fair Writing, and Mer­chants Accompts, by way of Debitor and Creditor; also the Art of Short Wri­ting.

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